The Higgs boson and the physics of WW scattering before and after Higgs discovery
aa r X i v : . [ h e p - ph ] M a r The Higgs boson and the physics of
W W scattering before and after Higgs discovery
Micha l Szleper
National Center for Nuclear Research,Ho ˙za 69, 00-681 Warszawa, Poland
Abstract
This work presents a comprehensive overview of the physics of vector bosonscattering (VBS) in the dawn of Run 2 of the Large Hadron Collider (LHC).Recalled here are some of its most basic physics principles, the historicalrelation between vector boson scattering and the Higgs boson, then discussedis the physics of VBS processes after Higgs discovery, and the prospects forfuture VBS measurements at the LHC and beyond. This monograph reviewsthe work of many people, including previously published theoretical work aswell as experimental results, but also contains a portion of original simula-tion-based studies that have not been published before. ontents
V V scattering . . . . . . . . . . 333.4 The
V V interaction and why it is still interesting . . . . . . . . . . . . . . 343.4.1 Higgs mass and couplings in
V V scattering . . . . . . . . . . . . . . 353.4.2 Gauge boson couplings in
V V scattering . . . . . . . . . . . . . . . 373.5 Beyond the Standard Model? . . . . . . . . . . . . . . . . . . . . . . . . . 49 V V scattering at the LHC 55 W Approximation and the Equivalence Theorem . . . . . 594.2.2 The “production × decay” approximation . . . . . . . . . . . . . . 604.3 Emission of a gauge boson off a quark . . . . . . . . . . . . . . . . . . . . . 614.4 Interaction of two gauge bosons . . . . . . . . . . . . . . . . . . . . . . . . 634.5 Gauge boson decay and possible final states . . . . . . . . . . . . . . . . . 644.6 The uniqueness of W ± W ± . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.7 Reducible backgrounds and selected experimental issues . . . . . . . . . . . 773 CONTENTS
V V scattering in LHC measurements at 8 TeV . . . . . . . . . . . . . . . . 95 t ¯ t production background . . . . . . . . . . . . . 1046.3 Modeling of the W +jets backgrounds . . . . . . . . . . . . . . . . . . . . . 1056.4 Modeling of the QCD multijet background . . . . . . . . . . . . . . . . . . 1066.5 W Z and ZZ as backgrounds to W W . . . . . . . . . . . . . . . . . . . . . 1076.6
W Z and ZZ as signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.7 Key uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.8 Higgs couplings in V V scattering . . . . . . . . . . . . . . . . . . . . . . . 1106.9 Anomalous triple gauge couplings . . . . . . . . . . . . . . . . . . . . . . . 1216.10 Anomalous quartic couplings . . . . . . . . . . . . . . . . . . . . . . . . . . 125
V V scattering at the FCC . . . . . . . . . . . . . . . . 1317.2 Higgs to gauge couplings at the FCC . . . . . . . . . . . . . . . . . . . . . 1337.3 Anomalous gauge couplings at the FCC . . . . . . . . . . . . . . . . . . . . 138 hapter 1Introduction “In the beginning there was symmetry.” - Werner Heisenberg
In the beginning there was symmetry and it was spontaneously broken. It is time for thisingeneous and revolutionary idea to find its way to the recognition and acceptance of thewide public. A recent google search for the phrase “in the beginning there was” producedseveral suggested endings for the query, including: the word, chaos, light, death, darkness,and even more improbable ideas. Only no symmetry. But unlike all these other ideas,this one has by now a very solid scientific basis.In his popular book “The Part and the Whole” (1969), Werner Heisenberg wrote: “Inthe beginning there was symmetry”. He expressed the opinion that basic symmetries ofthe world define the existing particle spectrum rather than the other way around, andcontrasted this view with “in the beginning there was the particle” for which he creditedDemocritus. While Heisenberg most probably did not mean electroweak gauge symmetry,a concept being back then in its early development and in which he admittedly did noteven show much interest, his point of view is even more actual today than it was back inthe 1960’s. Today we can tell his assertion was indeed correct. As of 2012, it even seemsthat we exactly know how this happens. And yes, it was that symmetry breaking act thatdefined what particles we have today. Not the other way around.
The concept of spontaneous symmetry breaking is a milestone in physics. The realiza-tion that certain key features of our physical world are not explicitly determined by anyfundamental laws of nature, but rather are the result of a spontaneous choice of a singlesolution that happened possibly just once, very early on in the history of the Universe,has a profound impact on our understanding of the world. Technically, spontaneous sym-metry breaking is a mode of realization of symmetry breaking in a physical system, wherethe underlying laws are invariant under a symmetry transformation, but the system as awhole changes under such transformations. It thus describes systems where the equationsof motion or the Lagrangian obey certain symmetries, but the ground state of the system,5
CHAPTER 1. INTRODUCTION i.e., the lowest energy solutions, do not exhibit that symmetry. Rather than being re-flected in the individual solutions, the symmetry of the equations is reflected in the rangeof possible a priori solutions, even if not observed in the physical world. According to theGoldstone theorem, spontaneous breakdown of a symmetry is necessarily associated withthe appearance of new spinless particles, the so called Goldstone bosons, one for eachgenerator of the symmetry that is broken. Unless the underlying symmetry is furtherbroken explicitly, the Goldstone bosons are massless. Conversely, if the symmetry is notexact, the bosons acquire mass, albeit are typically expected to be light. In the lattercase we talk of pseudo-Goldstone bosons. A well known example of this kind are thepions, which can be identified with pseudo-Goldstone bosons related to the spontaneousbreakdown of the chiral-flavor symmetries of QCD as a result of the strong interaction.The fact that pions are not entirely massless is related to the approximative character ofthe symmetry, which is due to the different quark masses. Still, it is well known that pionsare much lighter than all the rest of the hadron spectrum. Chiral symmetry breaking isan important example of spontaneous symmetry breaking affecting the chiral symmetryof strong interactions. It is responsible for over 99% of the mass of the nucleons, as itallows to comprise heavy baryons out of nearly massless quarks, and thus is in fact re-sponsible for the bulk of the Universe mass. Of course, by saying this we implicitly takefor granted the existence of that remaining 1%. Which is in fact far from trivial. Thewhole clue is that quarks, and leptons for that matter, are only nearly massless. Withoutit, no mass would be there whatever. It is perhaps not so paradoxical as it may seemat first glance that over 99% of the particle physicists’ effort in the last few decades, inparticular in the field of experiment, has been put to reveal the origin of that remaining1%. Finally, on July 4, 2012, the tiger broke free. By announcing the discovery of theHiggs boson [1], the ATLAS and CMS collaborations at CERN strongly suggested thatin fact this mass too originates from spontaneous symmetry breaking - this time affectingthe SU(2) × U(1) symmetry of electroweak interactions. The effect, as understood fromthe Standard Model perspective, is intrinsically connected with the so called Higgs mech-anism, which was originally proposed to explain how the weak gauge bosons W and Z acquire masses. Nevertheless, both historically and conceptually, the idea of the Higgsmechanism as a theoretical means to explain the weak boson masses, and that of theHiggs particle as its possible experimental consequence are two autonomous entities. Andthis distinction in a limited sense holds still today, i.e., after the Higgs boson discovery. Atechnical clarification is in place here. By the term ”Higgs mechanism” we will understanda mechanism of providing masses to vector bosons that is based on a particular methoddeveloped within the framework of Quantum Field Theory, namely by absorption of scalarfields that thus become the missing longitudinal degrees of freedom of the initially mass-less vectors. It works regardless of the rest of the model and what triggers electroweaksymmetry breaking in particular. As a matter of fact, the idea was initially introducedto particle physics in a somewhat different (and obsolete by now) context. Existence ofa physical scalar particle, the Higgs boson, can be regarded as a sufficient proof of cor-rectness of the Higgs mechanism to provide masses to W and Z bosons, and indeed is theonly proof of it available at hand and within reach of contemporary particle accelerators.Still, a Higgs boson is far from being a necessary condition for the Higgs mechanism betrue. Rather, it is a phenomenological consequence of a certain way of realization of thismechanism. Moreover, it does not definitely settle the question of whether this realization .2. SPONTANEOUS SYMMETRY BREAKING V V scattering processes, where V = W, Z , in the aftermath of Higgs discovery, particularlyin relation to existing scenarios of physics beyond the Standard Model (BSM). We alsobriefly overview other existing LHC results that directly or indirectly relate to the physicsof
V V scattering. In chapter 4 a detailed study of the
V V scattering process at theLHC from a phenomenological point of view is presented. Formal signal and backgrounddefinitions are given, and computational methods and problems in the evaluation of thesignal are discussed. Also discussed are detector-related backgrounds and the relevantdetector-specific capabilities and limiting factors that define the size of such backgroundsin a real experiment such as at the LHC. Our present knowledge of these effects, derivedfrom various available detector-specific analyses, is summarized. Finally, it is argued thatsame-sign
W W scattering in particular deserves special attention. Chapter 5 presents aselective review of existing literature on the subject, from the early pre-LHC works whichdiscussed
V V scattering mainly in the context of Higgsless models, up to the most recentpapers and post-Higgs discovery developments. Ultimately, all this knowledge is gatheredto sketch a tentative simulation-based analysis and present the up to date prospects forthe observarion of physics beyond the Standard Model via
V V scattering in the LHC at √ s = 13 TeV and beyond it.Any studies of detector-related effects affecting the evaluation of signal and back-grounds contained in this work are based solely on those results which have been officiallypresented by the relevant collaborations. This work reviews a lot of earlier work pub-lished by many people, including the work of theorists as well as results of experiments,but contains also an amount of original studies and results that have not been publishedbefore. For the latter, nothing else than publicly available simulation and analysis toolshave been used. Conceptually, this work is a continuation of the analysis presented inRef.[106], with substantial updates and improvements, and with a much extended scope. CHAPTER 1. INTRODUCTION hapter 2The Higgs boson in the StandardModel
This chapter sketches a vaguely historical derivation of the Standard Model and the Higgsboson as its key component.
Four-fermion contact interactions cannot exist in any complete theory of elementary par-ticles. This is because the corresponding coupling constant is forced to have the dimensionof cross section (i.e., inverse energy squared, in units where Planck’s constant is set to 1) , while the cross section in the lowest order of perturbative expansion must in turn beproportional to the square of the coupling constant. From simple dimensional analysisit immediately follows that asymptotically, at energies much larger than the masses ofthe particles involved, the total cross section is bound to be quadratically divergent withenergy. Unbound amplitude growth inevitably leads to unitarity violation at some energy.Unitarity is an imperative property of quantum systems which ensures the sum of prob-abilities of all possible final states evolving from a particular initial state is always equalto 1, and so it must hold in any acceptable theory. It is somewhat less straightforwardto show that four-fermion interactions also inevitably lead to non-renormalizable pertur-bation expansion, meaning that calculations of decay rates and cross sections suffer of anincreasing number of divergences arising from Feynman diagrams involving loops, and ul-timately making the theory lose predictive power. These facts were well known even whilethe Fermi theory of weak interactions, governed by a coupling constant G F expressed inGeV − , was the only existing one and indeed provided a good description of existing datain the low energy region. An improvement of Fermi’s model, inescapable from a purelytheoretical point of view, required the introduction of a force carrier, which necessarilyhad to be a vector boson, in analogy to the photon of QED. Note that in this case thelowest order weak interaction process is a second order process in the coupling constantwhich now governs the coupling of fermions to the intermediate vector bosons. It is easy In the framework of Quantum Field Theory, everything is measured in units of some power of en-ergy. The Lagrangian density has dimension 4, fermionic fields have dimension 3/2, bosonic fields havedimension 1. CHAPTER 2. THE HIGGS BOSON IN THE STANDARD MODEL to see that due to this the coupling constant itself now becomes dimensionless, prettymuch as the fine structure constant of QED, thus eliminating the theoretical shortcom-ings associated to the Fermi theory. Indeed, the similarity of weak and electromagneticinteractions suggested the possibility of a unified theory, where the photon and the W bosons were part of the same SU(2) multiplet. However, the prerequisite of correspon-dence of the intermediate vector boson model with the Fermi theory at low energy hasan important consequence: the weak force carriers must have a non-zero mass in orderto describe a short-range interaction. The success is only partial since non-zero mass isa source of two paramount problems and these provide the two in principle independentways to derive the full Standard Model as we know it today.For the sake of completeness one should mention here also another potential problemwith electroweak unification, which resided in the parity violating character of the weakinteractions. Parity violation effects were first observed experimentally in 1957. Thesolution consisted in enlarging the gauge group to SU(2) × U(1) to involve parity violatinginteractions. This had to be followed by the introduction of another neutral gauge boson,the Z . We know today that three weak force carriers, W + , W − and Z , are necessary todescribe accurately all the observed phenomenology of weak interactions. One problem with non-zero mass is related to the fact that SU(2) × U(1) gauge invarianceforbids explicit mass terms for gauge bosons. Gauge boson masses must be introduced tothe theory in a dodgy way and here is where spontaneous symmetry breaking comes in.The theory must be written so that the Lagrangian exhibits required gauge invariance,but its lowest energy solution cannot. This, however, poses another problem: how to avoidmassless spin-zero particles, excluded by experiment, which according to the Goldstonetheorem are bound to appear as a result of spontaneous symmetry breaking? In a paperof 1964, Higgs showed [2] that the Goldstone bosons need not physically appear in arelativistic theory when a local symmetry is spontaneously broken. Instead, they mayturn a massless vector field into a massive vector field. Regardless of how the relevantconcepts actually evolved from the historical point of view, it seems in principle clearthat the simplest implementation of this requires adding extra scalar fields to the theory.Namely, at least three scalar fields, playing the role of the would-be Goldstone bosons ofthe broken symmetry, are needed to endow the three gauge bosons, W + , W − and Z , withmass. Expressed the idea in simple words, each apparently massless gauge field and anapparently massless scalar field need to combine to form a massive vector field, while thetotal number of helicity states remains unchanged. This is the essence of the so calledHiggs mechanism, which more adequately is also referred to as the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism [3] [4], to properly honor the many contributors tothe idea in its presently known shape. There are no massless scalar fields left, as wouldbe predicted by the Goldstone theorem. In particle physicists’ jargon it is often saidthat those fields are “eaten up”. Exactly why longitudinal polarization is intrinsicallyconnected with the emergence of non-zero mass will be explained further below. For thetime being let us stick now to the most fundamental question: how does this happen? .2. THE HIGGS BOSON FROM THE PRINCIPLE OF GAUGE INVARIANCE Inclusion of three scalar fields is the minimum required to provide masses to three gaugebosons. This in itself carries no clues as to the origins of symmetry breaking and more-over, it inherently demands some additional terms in the Lagrangian to couple the threeGoldstone bosons to the gauge bosons and known fermions in a gauge invariant way. Onceagain now, real history put aside, the most general formalism that can be developed atthis point, derived from the basic principles of Effective Field Theory, is known as theElectroweak Chiral Lagrangian (EWChL) approach. Without getting here into too muchdetail of the EWChL (more information on the subject can be found, e.g., in Refs. [5]),let us only recall its main principles and basic features. The main idea is one of havinga low-energy effective parameterization of a full theory expressed in a model independentway. The general leading order (LO) Lagrangian in a practically useful form must beSU(2) L × U(1) Y -invariant and contain all the Lorentz-, C- and P-invariant operators up todimension 4 (in theorists’ jargon this means the dimension of the fourth power of energy).In such effective formulation the full Lagrangian can be symbolically written down in theform: L = L SM + L EW ChL = L SM + X i a i O i . (2.1)where L SM are the familiar pieces that emerge from the Standard Model Lagrangianin the infinite Higgs mass limit and L EW ChL is a collection of additional dimension-4operators expressed in terms of a 2 × UU = exp ( i~σ~πv ) . (2.2)In the above, ~π is a triplet of scalar fields, ~σ are Pauli matrices and v = ( √ G F ) − / ≈
246 GeV . Numerical coefficients a i play the role of effective new couplings. There isno explicit fundamental Higgs field included and, in the general case, the matrix is non-linearly parameterized with the three fields ~π .Each specific set of coefficients a i reproduces the full phenomenology associated toa given physical scenario. It can be shown that only 5 independent operators accountfor SU(2) L + R -conserving contributions. These coefficients may contribute to gauge bosonself-energies ( a ), triboson couplings ( a , a ) and effective four-boson couplings ( a , a ).If the a i ’s are understood as originating solely from new physics at a TeV scale, then theirtypical sizes are expected to be of the order 10 − − − . Precision electroweak data priorto the Higgs discovery defined more stringent experimental limits on their respective val-ues. Once data from LEP became available, it was noticed that there were actually onlytwo dimension-4 operators left in this Lagrangian that could modify the phenomenologyrelated to the mechanism of electroweak symmetry breaking at some higher energy with-out contradicting any of the existing low energy data from the electroweak sector. Thenumerical coefficients for these operators are the ones traditionally denoted as a and a .Thus, the relevant part of the Lagrangian was Note that this is exactly the quantity known as the Higgs vacuum expectation value, but it does nothave such interpretation in this framework CHAPTER 2. THE HIGGS BOSON IN THE STANDARD MODEL L EW ChL = a [ Tr ( V µ V ν )] + a [ Tr ( V µ V µ )] (2.3)where we have defined V µ = U ( D µ U ) † and D µ is the electroweak gauge covariant deriva-tive. For reasons that will become completely transparent in the next section, an effectivemodification of the four-boson couplings could be realized in terms of Higgs exchange(if we do not assume explicitly its existence in the model) or exchange of new, heavyparticles. Thus, our ignorance of the electroweak symmetry breaking mechanism couldbe effectively shown in terms of a two-dimensional ( a , a ) plane of which parts had beenalready excluded on theoretical grounds and other parts remained unexplored. The ef-fective Lagrangians by themselves could only describe accurately the electroweak physicsat low energy. They necessarily invoked some new physics to tackle the issues of renor-malizability and unitarity. Perturbative EWChL predictions can be extended to higherenergies using known techniques of unitarization, the two most commonly known classesof them are called Pad´e unitarization (a.k.a. Inverse Amplitude Method) and K-matrixunitarization (a.k.a. N/D Protocol). Typically this procedure leads to predictions of newresonances in the particle spectrum. The entire nature: masses, widths, couplings andspins of those resonances are in principle determined by the choice of ( a , a ), but inpractice some theoretical uncertainty related to the use of Chiral Perturbation Theory isbound to be present and manifest in that quantitative predictions depend on the unita-rization method that had been chosen. Here is where model-independence ends, becauseunitarization scheme is part of a model. This uncertainty becomes the larger the lighterthe predicted resonances, which certifies that the entire approach is for technical reasonsmostly suited for Higgsless scenarios (the term Higgsless here should be understood asanything not involving a physical scalar lighter than, say, 700 GeV). The entire formal-ism makes no a priori assumptions as to the nature or dynamics of the gauge symmetrybreaking mechanism. It is interesting to notice that for a specific choice of parameters, theStandard Model phenomenology could also be reproduced in principle . Correspondenceof the EWChL with the Standard Model has been in fact demonstrated, albeit only inthe heavy Higgs limit [6]. This correspondence is given by setting a = 0 and a being in-versely proportional to the Higgs mass squared. However, the resonance widths obtainedby applying e.g. the Inverse Amplitude Method are not exactly the same as the uniquelydetermined - for a given Higgs mass - Standard Model Higgs widths. Full correspondencebetween soft and hard electroweak symmetry breaking has not been demonstrated, at leastwithin known unitarization schemes. In any case, existence of a light scalar resonancemakes this kind of description of little practical use.The EWChL approach used to be an important theoretical framework to study theeffective phenomenologies in different scenarios of electroweak symmetry breaking. It al-lowed to confront their predictions with those of the Standard Model with a light Higgswithout running into strict model-dependence or into unphysicalness (like in the Higgs-less Standard Model). With the Higgs discovery, the minimum list of operators up todimension 4 has been completed. In principle the SM Higgs can be added to the EWChLby hand and the same formalism can still be applied with this modification. This is thesimplest possible upgrade of the EWChL formalism and indeed some studies of physicsbeyond the Standard Model have been carried in this language. Coefficients a and a can be reinterpreted as modifications to SM couplings which potentially induce simulta-neous existence of heavier resonances [7]. However, a somewhat different approach has .2. THE HIGGS BOSON FROM THE PRINCIPLE OF GAUGE INVARIANCE Extensions to the SM can be parameterized in terms of higher dimension operators. Onthis we will elaborate in a bit more detail in the next chapter. But for now let us stillback up to the Standard Model.
All the above being said, history went actually a different way. At this point in history,Higgs and independently Englert and Brout [3], had already come up with a somewhatarbitrary and yet elegant idea of the exact mechanism that triggers the gauge symmetrybreakdown, that could solve the problem in a remarkably economical way compared to thetechnical complicacy of the EWChL formulation. The idea was effectively incorporatedinto the theory of electroweak interactions by Weinberg [8] and ever since then it becamethe core of the Standard Model of elementary particles. The concept did not call for anynew phenomenology, with just one exception: the Higgs boson. And only one parametersuffices here for a complete quantitative description: the Higgs mass. To understand thewhole mechanism, let us first consider a toy model. The following explanation is modeledon the one from Ref. [9]. In relativistic field theory the simplest Lagrangian that canrealize spontaneous symmetry breaking is given by the addition of a complex scalar field φ such that L = ( ∂ µ φ ) ∗ · ∂ µ φ − V ( φ ) , (2.4)where V ( φ ) = µ φ ∗ φ + λ ( φ ∗ φ ) (2.5)Figure 2.1: The “Mexican hat” potential V ( φ ) - the case of λ > µ < Strictly speaking, as long as we do not include Majorana neutrinos. Those can be generated via adimension 5 operator. CHAPTER 2. THE HIGGS BOSON IN THE STANDARD MODEL and µ and λ are the mass and self-interaction coupling constant of the physical scalarparticles related to the field. The model is invariant under the global transformation φ ( x ) → φ ( x ) e iα . If µ >
0, this model describes just a self-interacting scalar field andnothing happens of special interest. If however µ <
0, then φ = 0 is a local maximum ofthe potential and therefore is bound to be an unstable state. The minimum of potential V has now the form of a circle defined by | φ | = − µ / λ . In other words, in the ground statethe value of φ is non-zero, its magnitude being actually v/ √ v = q − µ /λ , but witharbitrary phase. Thus, there will be a degenerate family of vacuum states, accordinglyto possible choices of the phase angle α . By choosing a particular minimum, say the onewhere φ is real and positive, one breaks the symmetry with respect to α . We can performa Taylor series expansion around this location. Defining two real shifted fields φ , suchthat φ = 1 √ v + φ + iφ ) , (2.6)the Lagrangian rewrites L = 12 [( ∂ µ φ ) + ∂ µ φ ) ] − V, (2.7)with V = − λv + λv φ + λvφ ( φ + φ ) + 14 λ ( φ + φ ) . (2.8)Because we have defined φ and φ so that the vacuum corresponds to a non-zerovalue of φ only, the model describes effectively two kinds of particles: φ of mass √ λv and the massless φ , along with their respective triple and quartic couplings. Particle φ is then the Goldstone boson related to breaking the initial symmetry of the system as aresult of the spontaneous choice of a vacuum state, while φ is an extra massive scalarparticle, prototype of the Higgs boson.Now comes the Higgs mechanism. By adding a massless gauge field into the picture,e.g., the electromagnetic field with potential A µ , the Lagrangian of the model can beexpressed as L = ( D µ φ ) ∗ D µ φ − F µν F µν − V ( φ ) , (2.9)where we can explicitly define the covariant derivative as D µ φ = ∂ µ φ − ieA µ φ , and F µν = ∂ µ A ν − ∂ ν A µ is the electromagentic field tensor. This Lagrangian is invariant under thelocal gauge transformations φ ( x ) → φ ( x ) e iα ( x ) , (2.10) A µ ( x ) → A µ ( x ) + 1 e ∂ µ α ( x ) . (2.11)Expansion, as before, around the chosen vacuum, yields a term of the form L = ... + 12 ( ∂ µ φ − evA µ ) + ..., (2.12) .2. THE HIGGS BOSON FROM THE PRINCIPLE OF GAUGE INVARIANCE B µ defined as B µ = A µ − ev ∂ µ φ , (2.13)with a mass equal to ev . There is no Goldstone boson left. Instead, the gauge fieldacquired mass by interaction with the scalar field φ .So what does it all have to do with providing masses to W and Z bosons while leavingthe photon massless in a way that does not violate gauge symmetry of the StandardModel? The key feature behind implementing this idea within the context of the Higgsmechanism resided in postulating a fourth scalar field, in addition to the three would-be Goldstone bosons discussed before, for the formation of an SU(2) isospin doublet ofcomplex scalar fields, usually denoted asΦ = 1 √ φ + φ ! = 1 √ w + iw h + iz ! . (2.14)Contrary to the general EWChL case, here a linear parameterization in the scalar fields isassumed. The vacuum is chosen so that h | Φ | i = √ (0 , v ) T , which means that it carriesa non-vanishing value of the neutral h field. Here v is exactly the same quantity we haveintroduced in the previous section. Note that consequently this vacuum carries a weakcharge, but no electromagnetic charge. In this case not the whole SU(2) × U(1) symmetryis broken. There is an unbroken subgroup related to the fact that Φ does not interactwith the photon. Like before, massless fields w , w and z are absorbed to form massterms for the apparenty massless weak bosons. The masses are effectively given by: M W = gv s πα √ G F sinθ W , (2.15) M Z = s πα √ G F sinθ W cosθ W . (2.16)Here we have introduced the “electroweak mixing (Weinberg) angle” defined via sinθ W = e/g, (2.17)the ratio of the original electromagnetic and the weak coupling constants. The photonremains massless. Since the value of sinθ W may be determined experimentally, e.g., froma measurement of fermion scattering processes, the above formulae represent in fact a prediction for the W and Z masses. The fourth field h is needed to trigger spontaneoussymmetry breaking by its non-vanishing vacuum expectation value. As a result, theshifted field H = h − v becomes a physical, massive, self-interacting scalar - the StandardModel Higgs boson. The mass of the Higgs boson is given by M H = √ λv (2.18)and hence it is not known a priori without knowledge of λ . However, everything else inthe theory is completely determined or at least calculable.Rewriting the Lagrangian in terms of physical particles, we see that the Higgs couplesto the gauge bosons:6 CHAPTER 2. THE HIGGS BOSON IN THE STANDARD MODEL L = ... + M W · W + µ W − µ · (1 + H/v ) + 12 M Z · Z µ Z µ · (1 + H/v ) + ... (2.19)with a coupling proportional to the mass squared. As a byproduct, it also couples tofermions generating their mass terms: L = ... + X f m f f ¯ f · (1 + H/v ) + ... (2.20)By construction, the coupling to fermions is proportional to the fermion masses.This completes the Standard Model of elementary particles and fundamental inter-actions, the most successful theory in modern physics, from the point of view of gaugeinvariance. The second reason why non-zero mass is a problem concerns polarization and the issue ofunitarity. As already mentioned, the unitarity condition is equivalent to the requirementof the sum of probabilities of all possible final states evolving from a particular initialstate be always equal to 1. This sum of probabilities must be in principle calculated toinfinite order in perturbative expansion, which is of course impossible to achieve. For thetechnical issues regarding the concept of unitarity and the connection between unitarityand renormalizability, the reader is referred to more topical literature, e.g., Ref. [10].In the following we will use the commonly accepted practical criterion of tree unitaritywhich demands for any 2 → B µ whose form is a wave-plane solution ofthe Klein-Gordon equation: B µ ( x ) = Cǫ µ ( p ) e − ipx (2.21)with the so called Lorenz condition , ∂ µ B µ = 0, imposed. Here p is the four-momentumof the particle, C is a normalization constant whose value is inessential at the presentmoment and ǫ µ is the polarization vector corresponding to the plane wave. In the bosonrest frame, ǫ µ can be decomposed into the individual Cartesian coordinates, where ǫ µx = (0 , , , , (2.22) ǫ µy = (0 , , , , (2.23) ǫ µz = (0 , , , , (2.24) Not Lorentz condition, as is often erroneously called. .3. THE HIGGS BOSON FROM THE PRINCIPLE OF UNITARITY ǫ µx and ǫ µy , ǫ µ + = 1 √ , , i, , (2.25) ǫ µ − = 1 √ , , − i, , (2.26)which correspond to two possible circular polarization states. Let us now suppose theboson moves along the z axis. The quantities ǫ µ + and ǫ µ − will now stand for the twodegrees of freedom of polarization transverse to the boson direction, while ǫ µz will becomethe longitudinal polarization and be further on denoted as ǫ µL . Translated into the languageof helicity, i.e., the projection of the boson’s spin onto its direction of motion, ǫ µ + , ǫ µL and ǫ µ − correspond to helicities +1, 0 and -1, respectively.The general expression for the three components of ǫ µ for a boson with mass M, energyE and 3-momentum p z directed along the z axis can be simply found by applying Lorentztransformation. It is however fully sufficient to consider that the transverse polarizationsneed not change, while ǫ µL , by definition directed along the momentum 3-vector, must beof the form ǫ µL = ( a | ~p | E , a ~p | ~p | ) = ( a p z E , , , a ) , (2.27)where a >
0. The normalization condition readily yields a = E/M , hence ǫ µ + = 1 √ , , i, , (2.28) ǫ µ − = 1 √ , , − i, , (2.29) ǫ µL = 1 M ( p z , , , E ) . (2.30)One can quickly verify that the above indeed satisfies the Lorenz condition expressed as p µ ǫ µ = 0.In this moment we have arrived at a very important conclusion. The requirement of ǫ L = 0 makes sense only if M = 0. For a massless boson there is no solution satisfyingthe Lorenz condition that would correspond to longitudinal polarization. And this iswhy, in the languauge of relativistic Quantum Field Theory, on-shell photons are purelytransverse. More generally, for a massless boson of spin J , Lorentz invariance forbidsother helicities than + J and − J .The form of ǫ L defines its key feature which lies in its energy dependence. It is clearthat at energies much larger than the boson mass, it grows indefinitely with energy, like ǫ L ∼ E , being a source of potentially fatal misbehavior of the gauge boson sector. Toelucidate the problem, let us consider a simple scattering process involving two on-shell,same-sign, longitudinally polarized W bosons:8 CHAPTER 2. THE HIGGS BOSON IN THE STANDARD MODEL
Figure 2.2: Feynman diagrams for the Standard Model process W + W + → W + W + : thefour- W contact interaction graph, the γ/Z -exchange graph and the Higgs exchange graph. W + L W + L → W + L W + L .In the lowest order, three subprocesses readily contribute to this process: the four- W contact interaction and t -channel (space-like) photon and Z exchange. The amplitude ofthe contact interaction part must be proportional to M ∼ ǫ L ǫ L ǫ L ǫ L ∼ s , (2.31)where s is the center of mass energy squared of the interacting bosons. Obviously itdiverges like the fourth power of energy and so, paradoxically, the electroweak theory leadsto similar difficulties as did before the Fermi theory. Even worse at first sight may lookthe diagram involving t -channel Z exchange, as one may expect in this case the leadingdivergence to be like ∼ E from an appropriate combination of all the longitudinal W and Z components. It can be shown, however, that the longitudinal part of the Z propagatorvanishes and the full contribution from the t -channel photon and Z exchange in fact alsogoes like s in the leading term. Moreover, by appropriate choice of the coupling constantfor the four- W contact interaction, which in practice is secured by the Standard Modelgauge invariance, the two leading terms can be made to cancel each other exactly. Itis worth to remember this point, since it will come back to us in further considerations.The triple gauge boson couplings, W W Z and
W W γ , are well constrained by experimentand we need not consider their variation at this point. The same can hardly be told ofthe quartic couplings which remain largely unconstrained from the experimental pointof view. Altogether, there are four quartic boson couplings allowed in the StandardModel:
W W W W , W W ZZ , W W Zγ and
W W γγ , and their values within the model arecompletely specified. The
W W W W coupling in itself can be probed experimentally atthe LHC in an independent way, via measurements of triboson production. Generally, it isexpected that new physics may manifest itself in changes of the effective quartic couplingsrelative to the Standard Model and therefore alter the Standard Model predictions fortriboson production, as well as the high energy behavior of
W W scattering amplitudes.For the sake of this chapter we will assume that quartic couplings correspond exactly totheir Standard Model values. Consequently, we are left with M Gauge = − g s M W + O ( s ) . (2.32) .3. THE HIGGS BOSON FROM THE PRINCIPLE OF UNITARITY H that canbe exchanged between the two W lines will result in an additional term M H = g HW W sM W + O ( s ) . (2.33)From dimensional analysis it follows that the coupling constant that governs the in-teraction of H with the W boson must have the dimension of energy. By looking at theexpressions for M Gauge and M H one easily notices that the leading divergences will cancelout exactly if and only if the condition g HW W = gM W is exactly fulfilled. Recalling that g in itself is related to the W mass, this in particular means that the scalar H must coupleto the W proportionally to M W . We already have such candidate: it is the StandardModel Higgs boson. Indeed, tedious calculations within the framework of the StandardModel yield the asymptotic result M Gauge + M H = g M H M W (2.34)at energies much larger than the Higgs mass.The same arguments apply to the opposite sign W boson scattering process W + L W − L → W + L W − L .In this case we have to take into account additional diagrams corresponding to s -channel(time-like) photon and Z exchange, as well as an s -channel Higgs exchange diagram.Without repeating the main points nor getting into detailed calculations we can immedi-ately write down the final results for the corresponding amplitudes: M Gauge = − g u M W + O ( s ) , (2.35) M H = g HW W uM W + O ( s ) . (2.36)where u is the familiar Mandelstam variable and we have used the high energy approxi-mation s + t + u = 0.Similarly, to the process of W ± Z scattering W ± L Z L → W ± L Z L ,the lowest order diagrams that contribute are the W W ZZ contact interaction, s - and t -channel W ± exchange and t -channel Higgs exchange. And likewise, the divergenceresulting from the sum of the former three is exactly canceled by the Higgs exchangediagram in the SM.0 CHAPTER 2. THE HIGGS BOSON IN THE STANDARD MODEL
With the ZZ scattering process the question is seemingly different, since in the SMit can only occur via Higgs exchange (both s - and t -channel). However, in any realhadron-hadron experiment this process cannot be separated from the dominant W + W − → ZZ process, where three additional graphs contribute in the lowest order, including the W W ZZ contact interaction, t -channel W ± exchange and s -channel Higgs exchange. Onceagain here, Higgs exchange provides cancelation of unwanted divergences.By introducing a Higgs boson with appropriate couplings to other particles, unitarityin the theory is established. This in turn completes the Standard Model from the pointof view of the unitarity principle. But there is more here. A Higgs boson is necessarybefore the energy scale of unitarity violation. A Higgs that is too heavy is useless in theSM. From these considerations an upper bound on the Higgs mass [11] was derived waybefore its actual observation. hapter 3Standard Model experimental statusand prospects for BSM The LHC has finished Run 1. Both ATLAS and CMS have produced their preliminary(now every day closer to being final) results based on combinations of the entire datasetsfrom 7 TeV and 8 TeV. Even if some of the results that have been published until noware not yet to be considered final, the most important findings are unlikely to changesignificantly until the LHC is restarted again with a higher energy (13 TeV) and collectsenough new data. To discuss physics of Run 2 of the LHC, it is important to realizewhat exactly has become known from Run 1 and within what uncertainty margins, thenhow these uncertainty margins translate into the potential of new discoveries in the forth-coming years. This is of course true not only for
V V scattering, but for the entire LHCphysics. But the relation between the Higgs boson and
V V scattering is special and sothis dependence is here even more strict. This chapter will review our current, most up todate, knowledge about the Higgs boson and summarize other measurements with director indirect impact on the physics of
V V scattering in the next years.
Four well known mechanisms of Higgs production at the LHC are: gluon-gluon fusion viaheavy quark loops, Vector Boson Fusion (VBF), Higgsstrahlung off a gauge boson andheavy quark fusion (also called t ¯ t - or b ¯ b -associated production). Their relative importancevaries with the Higgs mass and the kind of physics we want to study, to a lesser degree withthe actual proton beam energies. For a Higgs mass in the vicinity of 125 GeV, gluon-gluonfusion is by far the dominant production mode, with VBF contributing roughly an orderof magnitude less and the other modes less still. For Higgs-like resonance masses abovethe t ¯ t threshold, the relative amounts of gluon-gluon fusion and VBF become graduallycloser to unity, up to the point of the latter becoming over 1/3 of the total cross sectionat around 1 TeV.On the other end of the Higgs boson, the relative importance of different decay modesis driven by the respective mass thresholds for the decays into heavy particles. For M H < M W , as is indeed the case for the Standard Model Higgs, decays to fermions like the b or c quarks or to the τ leptons are strongly preferred as far as raw branching fractions212 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM are concerned. Background and event reconstruction efficiency issues define however ZZ and W W as being among the leading light Higgs decay modes to study, the only otherfully competitive channel being in fact γγ which occurs solely via loop corrections. In theHiggs-like resonance mass region above 150 GeV, decays into W W and ZZ become justabout the only relevant ones, and this assertion changes only marginally on the openingof the t ¯ t channel for masses larger than 350 GeV.Our process of interest is intrinsically connected with Vector Boson Fusion followedby decay into a pair of vector bosons. In the resonance region it is quite identical withit and so the W + W − and ZZ scattering modes are naturally the most widely studied todate. Of course, in an experiment we only know the bosons in the final state. The process ZZ → ZZ is in principle the most direct probe of the Higgs boson, as it only proceeds viaHiggs exchange in the Standard Model, but it cannot be separated from W + W − → ZZ .Specific VBF analyses have been performed in the low Higgs mass range and will be thebasis for future heavy resonance searches at 13 TeV. These studies have come up witha typical experimental VBF signature to search for. It consists of two energetic forwardjets and all the final Higgs decay products usually well isolated in the barrel region ofthe detector, the two direct decay products being typically reconstructed in oppositehemispheres. The purely electroweak character of the process means no QCD color flowoccuring in the event and reflects in a large rapidity gap between the two leading jets.The typical VBF signature used in Higgs searches does not explicitly discriminate betweenthe gauge boson polarizations. Indeed such discrimination is impractical in a kinematicregime where at least one of the gauge bosons must be off-shell. The spin and parity ofthe resonance can be nonetheless determined afterwards from the angular distributions ofthe decay products, where naturally the ZZ channel keeps the most complete informationavailable in the detector. Higgs signal has been independently observed with more than 5 σ significance, in bothATLAS [15] and CMS [14], in two decay modes: H → ZZ ∗ → l + l − l + l − (CMS: 6.5 σ ,ATLAS: 6.6 σ ) and H → γγ (CMS: 5.6 σ , ATLAS: 7.4 σ ). A third bosonic decay mode, H → W + W − , comes close (CMS: 4.7 σ , ATLAS: 4.1 σ ). Observed significances agree withSM expectations.The Higgs boson mass has now been precisely determined from a combination of datafrom the two most sensitive decay modes which not unexpectedly also provide the bestmass resolution: H → l and H → γγ . Its final values have been reported to be: M H = 125 . ± . stat ) ± . syst ) GeV (ATLAS) and M H = 125 . + 0 . − . ( stat ) + 0 . − . ( syst ) GeV (CMS).Higgs masses determined from the two channels separately are in satisfactory agreementat CMS, with the final mass difference being quoted as M γγH − M lH = − . + 0 . − . GeV.ATLAS observed a marginally larger difference whose statistical significance is likewiseweak, M lH = 124 . ± .
52 GeV vs. M γγH = 125 . ± .
50 GeV. It should be noted that,as far as the central values are concerned, M lH > M γγH for CMS, but M γγH > M lH forATLAS, which clearly favors statistical and systematic uncertainties rather than physics .1. HIGGS BOSON EXPERIMENTAL STATUS W W decay channel is assumed. A typical signature consists of two forward high energyjets (labeled (1)) with a large pseudorapidity gap, and two central leptons (labeled (2))with a large gap in the azimuthal angle. Quantities like ∆ η and ∆ φ are instrumental inisolating the process from the bulk of the background.4 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM as the most plausible interpretation of any possible mass shifts. There is no experimentalsupport to the idea of there actually being two nearly degenerate resonances, at leastwithin the present resolutions.A study of Higgs decays into τ + τ − revealed independent evidence of a Higgs signal atthe 4.5 σ level in ATLAS [16] and at the 3.2 σ level in CMS [17], both being compatiblewith the expectations for a ∼
125 GeV Standard Model Higgs boson and hence stronglysuggesting that the Higgs indeed does couple to fermions. On the other hand, no otherfermionic decay has been firmly and directly established on its own. The ones that havebeen directly searched for are the decays to b ¯ b , µ + µ − and more recently e + e − [18]. Analysisof the former does indeed reveal hints at a roughly 2 σ level, in consistency with SMexpectations. A CMS combination of data from the two most important fermionic Higgsdecays: τ + τ − and b ¯ b , does not yet reach the 5 σ significance level [19]. With the amountof data collected so far, lack of signal observation in decays to lighter fermions is fullyconsistent with the SM. Of course, decays H → γγ occurring at a rate roughly consistentwith the Standard Model indirectly suggest that Higgs couples to the top quark, too.Moreover, theory predicts the main Higgs production mechanism be gluon-gluon fusionvia top quark loops and so the total Higgs production rate is driven predominantly bythe Higgs coupling to the top. In other words, the simple observation of total Higgsproduction occurring at a rate roughly consistent with the SM is another (and actually,the strongest), albeit indirect, confirmation that the Higgs couples to fermions.Certain rare Higgs decays predicted by the Standard Model have been searched for aswell. Measurement of the rate of Higgs decaying into, e.g., Zγ would be a very interestingtest of the Standard Model, but so far data are of not enough statistical precision to doso [20].The width of the Higgs boson in the SM is fully determined by its mass. For a 125 GeVHiggs, the expected width is close to 5 MeV, which is unfortunately far beyond presentexperimental resolution. From an analysis of data in the resonance region of the 4-leptondecay channel, CMS found the observed resonance width in agreement with the detectorresolution width and placed a 95% CL upper bound on the intrinsic Higgs width at 3.4GeV. A novel method has been proposed to constrain the Higgs width by examinationof the 4-lepton mass spectrum away from the Higgs peak [21]. In the dominant gluonfusion process, Higgs off-shell production and decay into 4 leptons gets enhanced due tothe proximity of the Z pair production threshold. The ratio of cross sections for off-shell and on-shell Higgs production, σ ( gg → H ∗ → ZZ ) /σ ( gg → H → ZZ ∗ ) is directlyproportional to the Higgs width. Using this technique, CMS placed a much better upperbound on it at 22 MeV (95% CL) [22]. Crucial to the identification of the 125 GeV resonance with the SM Higgs is determinationof its spin and parity. The SM Higgs boson has spin-parity J P = 0 + . Different spin-parityhypotheses of the observed Higgs-like resonance have been severely constrained by thedata. The spin and parity of the Higgs resonance can be independently analyzed in eachdecay mode, based on angular distributions of the respective decay products. In CMS, thishas been achieved so far using the three leading decay modes [24]: H → ZZ ∗ → l + l − l + l − , .1. HIGGS BOSON EXPERIMENTAL STATUS H → W + W − → l + l − νν and H → γγ . By far the most sensitive of them is the 4-leptonchannel. Strictly speaking, J P is not measured , only the likelihood of different hypothesescan be determined relative to each other (one can of course argue that such procedurequalifies as being a measurement). Each J P hypothesis translates into specific predictionsof the angular distributions that are computed directly from the corresponding matrixelements. For every pair of hypotheses their relative likelihood of consistency with thedata can then be quantified. The procedure is more likely to end up in a conclusiveresult only as long as one of the two selected hypotheses is the correct one (and theother incorrect). Thus, in practice, each non-standard J P hypothesis is tested againstthe J P = 0 + hypothesis. A q value is then determined from data that is equal to therelative likelihood of the tested hypothesis against the reference J P = 0 + case. Statisticalsignificance of each result is determined by comparing the single q value obtained fromdata with its predicted probability distributions that are calculated under the assumptionsthat one or the other J P hypothesis is correct. The respective probability distributionsare obtained from a number of simulated “fake” experiments. The hypotheses that havebeen tested include J P = 0 + h (scalar with higher order couplings), 0 − (pseudoscalar),1 ± (vector and pseudovector), 2 ± m (tensor and pseudotensor with minimal couplings toSM particles - a graviton analogue) and 2 ± h (tensor and pseudotensor with higher ordercouplings). In addition, a maximum likelihood function can be defined in which a mixed J P state is allowed, e.g., 0 + with 0 − .Analysis of the 4-lepton channel is based on a technique described in detail in literature[23]. It exploits information on five angles that characterize the decay: two angles describethe orientation of the decay plane of one Z boson in the lab, a third angle the relativeazimuthal orientation of both Z decay planes and the last two angles describe the two Z decays in the respective Z rest frames. The 4-lepton channel alone allows to reject all of thetested hypotheses at a confidence level (CL) greater than 95%. In the H → γγ channel,Higgs spin correlates to the polar angle of the γγ pair in the Higgs rest frame. Accordingto the Landau-Yang theorem, decays of a massive vector into a couple of massless vectorsare forbidden, so J = 1 is here excluded. All spin-zero scenarios produce an identicalisotropic γγ distribution and therefore the J P = 0 − hypothesis cannot be studied usingthis channel. Results for the tensor hypotheses depend on the production mechanism,but currently none can be fully excluded at 95% CL. Finally, in the leptonic H → W W channel, analysis in a two-dimensional plane spanned by the event transverse mass andthe lepton-lepton mass was done. The exclusion of the pseudoscalar hypothesis from thischannel is marginal, but J P = 2 + m can be excluded at 95% CL or more in the cases wherethe preferred production mechanism is quark-antiquark fusion. From a combination of H → ZZ , W W and γγ results, the J P = 2 + m model is excluded at a 99.9% CL regardlessof the combination of the gluon-gluon and quark-antiquark production modes and otherspin-2 hypotheses are excluded at 99% CL or higher. Likewise spin-1 hypotheses areexcluded at more than 99.99% CL from the combination of decays H → ZZ and W W .The pseudoscalar hypothesis is excluded at 99.5% CL. This of course refers to pure J P hypotheses. The 95% CL limit on the fractional pseudoscalar cross section in the Higgsresonance is 0.43 and so a significantly mixed parity state is by all means allowed.A combined spin-parity analysis from the three main decay modes was also published The possibility of providing additional evidence based on the τ + τ − decay mode is being studied CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM by ATLAS [25]. This analysis excluded the graviton-inspired 2 + hypothesis at a morethan 99.9% CL, spin-1 hypotheses at 99.7% CL and the pure 0 − hypothesis at 97.8% CL.They do not quote numbers for the maximum allowed pseudoscalar admixture. Exclusionlimits have been also set on the hypothesis that the observed signal is shared between twonearly degenerate mass states. Finally and most importantly for the sake of this work, Higgs couplings have been probedvia measurements of branching fractions for the main decay modes: W + W − , ZZ , γγ and τ + τ − (and b ¯ b , in principle), and the respective production mechanisms. All Higgscouplings can in principle be inferred from fits to the observed rates in different combi-nations of Higgs production mechanisms and decay modes, where each full production × decay path can be parameterized as a function of the relevant couplings. However, dataare not precise enough to determine independently all the couplings with a reasonableaccuracy. For this reason, results are usually presented in one of two forms. In the firstapproach, events are categorized by final state, including the contributions from all pro-duction mechanisms and a single parameter µ for each final state is fit to the observedsignal yield. The quantity µ is the measured signal strength (cross section × branchingfraction) relative to the predicted SM signal strength. The most recent results of theoverall signal strength relative to the SM that is obtained from a simultaneous fit to allHiggs decay channels are [14] [15]: µ = 1 . ± . stat ) ± . syst ) + 0 . − . ( theo ) (CMS), and µ = 1 . ± . stat ) + 0 . − . ( syst ) (ATLAS).In the channels of most interest for us here, CMS results were: µ W W = 0 . + 0 . − . (from W + W − ) and µ ZZ = 1 . + 0 . − . (from 4 l ).ATLAS most recently published values are: µ W W = 1 . + 0 . − . ( stat ) + 0 . − . ( syst ) and µ ZZ = 1 . + 0 . − . .While consistent with the SM, these numbers still keep room for sizeable deviations.Fits of µ were also done in separate categories where events were tagged by productionmode, exploiting the distinct kinematic and topological signatures of VBF, Higgsstrahlungand t ¯ t -associated production (the largest “untagged” sample corresponds mostly to gluonfusion). They revealed consistency with the SM within rather large errors.The above are pure experimental results, with no model-dependence involved. How-ever, their relations to the genuine Higgs couplings are entangled. In the other approach,events were categorized according to their full production × decay chains and a simulta-neous theoretical fit of the corresponding cross sections × branching fractions was done tothe data in which only two parameters were allowed to vary freely: one to globally modifythe Higgs couplings to bosons, another to globally modify the Higgs couplings to fermions. .1. HIGGS BOSON EXPERIMENTAL STATUS SM σ / σ Best fit ± = 1.00 µ ZZ tagged → H ± = 0.83 µ WW tagged → H ± = 1.13 µ tagged γγ → H ± = 0.91 µ tagged ττ → H ± = 0.93 µ bb tagged → H ± = 1.00 µ Combined
CMS
Preliminary (7 TeV) -1 (8 TeV) + 5.1 fb -1 = 125 GeV H m Figure 3.2: Upper plot: Values of the best fit σ/σ SM for the combination (solid verti-cal line) and by predominant decay mode. The vertical band shows the overall σ/σ SM uncertainty. Lower plot: 68% CL contours for individual channels and for the overallcombination (thick curve) for the ( κ V , κ f ) parameters. The cross indicates the globalbest-fit values. The dashed contour bounds the 95% CL region for the combination. Theyellow diamond represents the SM expectation. Results from the CMS collaboration.The shown parameter space was here restricted to the first quadrant where the globalminimum of the fit was found. A second minimum was also obtained for κ f < CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM ) µ Signal strength ( -0.5 0 0.5 1 1.5 2
ATLAS
Prelim. -1 Ldt = 4.6-4.8 fb ∫ = 7 TeV s -1 Ldt = 20.3 fb ∫ = 8 TeV s = 125.5 GeV H m - + = 1.57 µγγ → H - + - + - + - + = 1.44 µ → ZZ* → H - + - + - + - + = 1.00 µ ν l ν l → WW* → H - + - + - + - + = 1.35 µ , ZZ*, WW* γγ→ HCombined - + - + - + - + = 0.2 µ b b → W,Z H <0.10.4 ± ± - + = 1.4 µ (8 TeV data only) ττ → H - + - + - + - + = 1.09 µ ττ , bb → HCombined - + - + - + - + = 1.30 µ Combined - + - + - + Total uncertainty µ on σ ± (stat.) σ ) theorysys inc. ( σ (theory) σ Figure 3.3: Upper plot: The measured signal strengths normalized to SM expectations forthe individual final states and various combinations. The best-fit values are shown by solidvertical lines. The total ± σ uncertainties are indicated by green shaded bands, with theindividual contributions from the statistical, systematic (including theory) and theoreticaluncertainties (from QCD scale, PDF, and branching ratios) are shown as superimposederror bars. Lower plot: Results of fits that probe different coupling strength scale factorsfor fermions and vector bosons, assuming only SM contributions to the total width: 68%CL contours from individual decay channels and their combination. Results from theATLAS collaboration. Images reproduced from Ref. [13]. .1. HIGGS BOSON EXPERIMENTAL STATUS κ V and κ f , consistent with unitywithin 1 σ ; the accuracy is roughly ∼
10% for κ V and ∼
20% for κ f (see Fig. 3.2). Fromone-dimensional parameter scans (in which the other coupling was set to its SM value),one gets the following 95% CL intervals: κ V ǫ [0.88, 1.15] and κ f ǫ [0.64, 1.16]. A similaranalysis was done by ATLAS [13] (see Fig. 3.3).It should be stressed here that this procedure is not completely model-independentbecause the content of the loops in gluon fusion and in H → γγ decays must be explic-itly assumed in order to relate production mechanisms with decay modes via the sameparameters: in the SM, the loops are dominated by respective contributions from the topquark and from the W boson. Agreement with the SM of the total Higgs signal strength,in particular in the dominant “untagged” category, as well as that of the H → γγ signalstrength, justifies the approach. More generally, the procedure can be regarded self-consistent for any model that does not involve significant contributions from unknownheavy particles within the present energy reach. This may in fact be the case in an inter-esting wide class of theories beyond the SM, known as SILH models, that we will discussfurther on. The procedure itself of scaling the Higgs couplings by only two independentfactors, κ V and κ f , is likewise consistent with the expected low-energy phenomenology ofthese models. Therefore, the above result is of special interest from this point of view. Adedicated test for the presence of BSM particles was carried based on γγ data. A fit to thedata where all tree-level couplings were assumed equal to their SM values, κ V = κ f = 1,and varied freely were the effective Higgs couplings to gluons and photons, κ g and κ γ ,revealed consistency with the SM within 1 σ .By reverting the procedure, the top coupling is probed by assigning a common signalstrength factor for the gluon fusion production mechanism, with addition of the little t ¯ tH production mode, because they both scale predominantly with the Yukawa coupling ofthe top quark in the SM. The assumption that the Higgs couples proportionally to thefermion mass has been indirectly supported by the data.Other tests included modified up-type to down-type fermion couplings, motivated bySUSY models, and modified independently top, bottom and τ couplings. No deviationsfrom the SM were observed. By contrast, all dedicated searches for a non-Standard Higgs to date gave negative results.Additional, heavy SM-like Higgses were excluded at the 95% CL or more up to themass of 710 GeV from a combination of data from ZZ and W W decays [27]. Likewise,no additional resonances have been observed in the γγ spectrum between 150-850 GeV[28].Dedicated searches were carried for neutral and charged Higgses within the frameworkof the Minimal Supersymmetric extension of the Standard Model (MSSM). The moststringent exclusion limits come from the search for the MSSM decay h, H, A → τ + τ − [29]for which the standard τ + τ − analysis was modified so as to maximize the sensitivity toBSM effects. An MSSM scalar Higgs differs from the SM Higgs in terms of the relativecontributions from different production mechanisms and decay branching fractions. Inparticular, b ¯ b -associated production followed by decay into a τ + τ − pair gets enhancedbecause Higgs couplings to down-type fermions and third generation fermions increase0 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM with tanβ . Additional exclusions were obtained from searches for the MSSM-specificeffects affecting the decays into b ¯ b and µ + µ − . Charged Higgses, predicted by the MSSM,were searched for in the decay channels H ± → τ ± ν , H ± → cs and H ± → tb [30]. Thecombination of all these results severely constrain the available MSSM parameter spacein the Higgs sector, although the hypothesis that the only discovered boson so far is infact the lighter of the two scalar Higgses of the MSSM cannot be ruled out completely.Other, non-minimal supersymmetric models have been constrained as well. This in-cludes in particular the Next-to-Minimal Supersymmetry (NMSSM), predicting a lightHiggs scalar decaying into a pair of light Higgs pseudoscalars, with a final state consist-ing of 4 muons, h → aa → µ . Such decay chain, once thought to be an alternativeto the SM/MSSM scenario that should attract physicists’ main attention in case theLHC fails to observe Higgs signal in one of its mainstream SM channels, is inconsistentwith the data [31]. An upper limit has been set on the cross section for standalonelight pseudoscalar Higgs production via gluon fusion followed by decay into a muon pair, σ ( pp → a ) Br ( a → µ + µ − ), which translates into further limits in the NMSSM parameterspace [32].Explicit searches for Higgs anomalous couplings have been carried. Higgs productionin association with a single top quark (and a light quark jet) is particularly sensitive tothe relative sign of the Higgs boson coupling to fermions and bosons and to the valueitself of the Higgs to top coupling. Such studies were carried independently based on the b ¯ b and γγ decay modes, but their results were inconclusive [26].Inconsistency of the Higgs boson with models assuming the existence of a fourthlepton generation, as well as fermiophobic Higgs models, was shown early on [33]. Otherdedicated searches include heavy scalar and pseudoscalar Higgses in a general two-doubletmodel (2HDM), doubly charged Higgses, invisible decays of the SM-like Higgs and leptonflavor violating decays and were translated into respective exclusion limits [34].To summarize, consistency of all the data with the Standard Model holds invariablyin what regards Higgs physics. Most key analyses have already been performed on thewhole 7+8 TeV dataset and so the main conclusions are unlikely to change significantlyanytime before late 2015. No hints of new physics have been observed, whether in theHiggs sector, or for that matter in the many non-Higgs related searches carried at bothATLAS and CMS (for a review of the latter the reader is referred elsewhere [35] [36]).On the other hand, plenty of room for new physics is still there to be unraveled at somehigher energy, or even possible to show up eventually at the currently available energy ifonly more LHC luminosity was available. There are no clear indications so far as to whatthis physics beyond the Standard Model might be. Contrary to SUSY, which may notprovide any measurable hints of new physics unless by an increase of energy, SILH modelsin general predict new physics in both ways. The second phase of the LHC, due to startin 2015, will increase both the energy and the luminosity and has chances to solve thepuzzle.There is one more important thing to learn from the spin-parity analyses in particular.Since the H → W + W − channel offers relatively little sensitivity to the Higgs spin-parity,the same weakness is bound to apply to measurements of W helicity in the final state.Even more difficult this will become in the most interesting high mass region wherethe W ’s are more boosted. Separation of W helicities in W W scattering requires othertechniques to be used. .2. ELECTROWEAK PHYSICS RESULTS Both ATLAS and CMS have produced a large number of results concerning gauge bosonproduction in general [37]. The most directly relevant for us are those concerning dibosonand triboson production. Their importance for the study of
V V scattering is twofold.Measurements of total cross sections for diboson production cross check our calculationsof irreducible background. More specific analyses of the respective kinematic distributionsallow to place limits on anomalous triple and quartic vector boson couplings.In what regards triboson production, 95% CL limits were set at CMS on anomalousquartic couplings for
W W γγ , W W Zγ [38]. These were based by searches for the
W W γ and
W Zγ final states, respectively. There is no directly equivalent limit so far on the
W W W W coupling, i.e., based on a measurement of triboson
W W W production, eitherfrom ATLAS or CMS.Results abound as far as diboson production is concerned. Let us review the mostimportant of them. The total inclusive W + W − cross section at 7 TeV as measured byCMS was found to be [39] σ ( pp → W + W − ) | T eV = 52 . ± . stat ) ± . syst ) ± . lumi ) pb,which is consistent within the errors with Standard Model predictions in the next-to-leading order, including the two main production mechanisms of quark-antiquark anni-hilation and gluon-gluon fusion. That the measured value is actually marginally higherthan the prediction can be ascribed to other production mechanisms such as: diffractiveproduction, double parton scattering, QED exclusive production, and Higgs boson pro-duction with decay to W + W − , expected to yield additional contributions up to about 5%altogether. The ATLAS Collaboration measured [40] σ ( pp → W + W − ) = 51 . ± . stat ) ± . syst ) ± . lumi ) pb.The total pp → ZZ cross section at √ s = 7 TeV was measured to be σ ( pp → ZZ ) | T eV = 6 . ± . . ( stat ) ± . . ( syst ) ± . lumi ) pb (CMS) [41], and σ ( pp → ZZ ) | T eV = 6 . ± . stat ) ± . . ( syst ) ± . lumi ) pb (ATLAS) [42].The inclusive W ± Z production cross section from CMS was [43] σ ( pp → W Z ) | T eV = 20 . ± . stat ) ± . syst ) ± . lumi ) pb,and from ATLAS it was [44] σ ( pp → W Z ) | T eV = 19 . ± . . ( stat ) ± . syst ) ± . lumi ) pb.All the ZZ and W Z cross sections are consistent with Standard Model NLO predictions.Sadly, there is no dedicated measurement of the much lower same-sign
W W produc-tion, although some bounds on it can be in principle indirectly inferred using a combinedmeasurement of
W W + W Z based on events with a W decaying leptonically and two jets.Unfortunately, these measurements [45] are of not enough precision to extract the tinysame-sign W W contribution. Errors labeled lumi are those related to the LHC luminosity measurement. CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM
Finally, the
W γ and Zγ cross sections are [46]: σ ( pp → W γ ) | T eV × Br ( W → lν ) = 37 . ± . stat ) ± . syst ) ± . lumi ) pb (CMS),and σ ( pp → Zγ ) | T eV × Br ( Z → ll ) = 5 . ± . stat ) ± . syst ) ± . lumi ) pb(CMS).The ATLAS collaboration does not quote their total cross section values, but restrictsthe measurements to a predefined fiducial region. In any case, no deviations from the SMwere observed [47].At 8 TeV, CMS measured [48]: σ ( pp → W + W − ) | T eV = 69 . ± . stat ) ± . syst ) ± . lumi ) pb, σ ( pp → ZZ ) | T eV = 8 . ± . stat ) ± . syst ) ± . lumi ) pb, and σ ( pp → W Z ) | T eV = 24 . ± . stat ) ± . syst ) ± . lumi ) pb [43].The W + W − value is slightly higher than the Standard Model NLO prediction of 57.3 ± . . pb, but again an extra 5% increase of this value is expected from the additionalcontributions calculated at the next-to-next-to-leading order, chiefly from Higgs bosonproduction. Explanations in terms of new physics have also been suggested, but are notquite convincing. The ZZ and W Z values agree with Standard Model NLO predictionswithin the errors. ATLAS showed: σ ( pp → W + W − ) | T eV = 71 . ± . stat ) + 5 . − . ( syst ) + 2 . − . ( lumi ) pb [49], σ ( pp → ZZ ) | T eV = 7 . ± . . ( stat ) ± . syst ) ± . lumi ) pb [50], and σ ( pp → W Z ) | T eV = 20 . ± . . ( stat ) ± . . ( syst ) ± . . ( lumi ) pb [51].Limits on anomalous triple gauge couplings, and in particular the ones of most directrelevance for us, namely W W Z and
W W γ , have been derived so far from the 7 TeV data.An up-to-date summary of these measurements, together with a set of references to theoriginal papers, is available in Refs. [56]. Also included in the summary are the respectiveresults from the TeVatron and LEP. In all these works, anomalous couplings were studiedwithin the formalism known as the effective Lagrangian approach, in which the mostgeneral form of the
W W Z/γ vertex is considered, including all terms that respect Lorentzinvariance and conserve C and P . The couplings are taken to be constant parameters ofthe Lagrangian and therefore independent of the boson momenta. The conceptual basisof this approach is explained in detail in Ref. [63]. Accordingly, experimental limits areset on five quantities: ∆ g Z , ∆ κ Z , λ Z , ∆ κ γ and λ γ . The first three of these modify the W W Z vertex, the following two modify the
W W γ vertex. In the SM, λ Z = λ γ =0 and g Z = κ Z = κ γ =1. Their actual values are determined from studies of diboson productionprocesses for which these vertices play a primary role, namely W W , W Z and
W γ . Ananomalous triple gauge coupling would be manifest in the rate of diboson productionat high boson p T and invariant mass. Typically, it is ascertained in CMS by a one-dimensional evaluation of the p T spectrum of the leading lepton or of the dijet (both from W/Z decay) or of the photon. For the theoretical calculation of the expected spectrum,either one or two anomalous parameters are varied at a time. Correlations betweencouplings that contribute to the same vertex are rather weak and so one-dimensional .3. OTHER RESULTS OF RELEVANCE FOR THE STUDY OF
V V
SCATTERING
W W Z vertex - compilation of results coming from LEP, TeVatron and LHC experiments. Imagereproduced from Ref. [56].While all these parameters may be probed independently in experiment, considerationsof gauge symmetry induce additional relations between them: λ Z = λ γ , (3.1)∆ κ Z = ∆ g Z − ∆ κ γ tan θ W . (3.2)From this it follows that, e.g., measurement of W W Z couplings could be translated into
W W γ couplings on theoretical grounds.
V V scattering
For the correct assessment of reducible backgrounds, several other measurements are asimportant. Let us only mention the ones we will directly refer to in this work.Top production has been measured in several final states, including different W decaychannels [52]. The most accurate inclusive t ¯ t production cross sections come from thedilepton final state. CMS reports [53] σ ( pp → t ¯ t ) | T eV = 162 ± stat ) ± syst ) ± lumi ) and4 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM
Figure 3.5: Current limits on the anomalous couplings that contribute to the
W W γ vertex - compilation of results coming from LEP, TeVatron and LHC experiments. Imagereproduced from Ref. [56]. σ ( pp → t ¯ t ) | T eV = 239 ± stat ) ± syst ) ± lumi ) (from 5.3/fb of data).ATLAS measured [54] σ ( pp → t ¯ t ) | T eV = 177 ± stat ) ± ( syst ) ± lumi ) and σ ( pp → t ¯ t ) | T eV = 238 ± stat ) ± syst ) ± lumi ).The numbers are in good agreement with NNLO+NNLL calculations by Czakon et al. [55],where the quoted uncertainty of the latter is roughly the size of experimental errors.A lot of other results indirectly relate to our subject. Important feedback is obtainedin particular from forward physics where jet multiplicity and kinematics obtained fromcommon event generators can be cross checked in detail against the data. These thingshowever play a rather secondary role for us and we need not go through them here. V V interaction and why it is still interesting
In the previous chapter we have sketched the derivation of the Higgs boson using twoindependent approaches: from the principle of SU(2) × U(1) gauge invariance and fromthe requirement of tree level unitarity of all Standard Model processes. The paramountphenomenological manifestation of the underlying model is the existence of a physicalscalar particle which couples to all known particles of non-zero mass in a completelydetermined way. Such particle manifests itself in a twofold way. At energies available inthe LHC to date it should be produced in proton-proton collisions via different physical .4. THE
V V
INTERACTION AND WHY IT IS STILL INTERESTING
V V scattering
The second phenomenological manifestation of the Standard Model Higgs boson residesin the high energy behavior of the
V V scattering amplitudes. In the Standard Modelthe
HW W coupling is chosen such that it fully cancels the quadratic divergencies thatappear after combining photon and Z exchange graphs with the four- W contact interac-tion. Thus, it is precisely the same particle which has been discovered at the LHC that issupposed to provide these exact cancelations. Violation of unitarity at some high enoughenergy will be the most extreme (and unrealistic) manifestation of the still existing prob-lem should this cancelation not be the case. But let us put questions of unitarity aside,as they in fact represent a technical issue. The entire high energy behavior of the V V scattering amplitude is a fully quantitative question and depends on many inputs. Totaland differential cross sections for the scattering of longitudinally polarized W and Z gaugebosons, at energies much larger than the masses of the latter, are a major experimentalfield where consistency of the Standard Model has to be tested. To this date we havepractically no experimental data to confirm that the Higgs boson indeed does its job,assigned to it by the Standard Model.A simple tree level calculation of the process W + L W + L → W + L W + L reveals two basicfacts. The total cross section as a function of the center of mass energy behaves differentlydepending on both the Higgs mass and Higgs couplings. As long as the HW W coupling isexactly 1, expressed in units of the value predicted by the Standard Model, the amplitudekeeps rising up to the energy equivalent to the Higgs mass (notice that for M H < M W it in practice never does so), then stays approximately flat. Phase space causes the crosssection fall for higher energies. In the absence of a Higgs boson the amplitude risesindefinitely and so does the cross section. It can be calculated that unitarity violationoccurs at about a 1.2 TeV energy and thus some new physics is bound to enter beforethis scale. Put in a more physical language, unless the scattering amplitude receives newcontributions that reduce the amplitude way before this point, the W W interaction beforethe scale of 1.2 TeV inevitably becomes strong. The term “strong” specifically means thatmultiple rescattering is likely to occur. This means a difference in the basic dynamics ofelectroweak symmetry breaking compared to the Standard Model case where it is supposedto be “soft” and a single Higgs boson exchange takes place instead. We would talk thenof a strongly interacting gauge sector. An even more interesting scenario occurs if the
HW W coupling is different from 1. As we already know, in such case the quadratic termsin the amplitude are not completely canceled and this incomplete cancelation must showup at a high enough energy. In general, the total cross section will rise up to the Higgsmass (not if M H < M W ), fall past the Higgs mass and rise up again at some energy. Thesituation again calls for new physics as the unitarity limit would be still inevitably hit atsome energy whose precise value depends on the value of the coupling. The unrealistic bynow, extreme case of no Higgs boson at all is technically equivalent to setting either an6 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM infinite Higgs mass or a zero coupling. Relative to the SM prediction, the cross sectionwill be enhanced at all energies as long as the
HW W coupling is lower than the SM oneand will reveal an energy pattern consisting of a depletion followed by a turning pointand an enhancement if the
HW W coupling is larger than the SM one. This is becausein the latter case there is an overcancelation of the quadratic divergence by the Higgsgraph which subtracts from the constant term in the total amplitude. At a certain energythe quadratic term becomes dominant anyway and asymptotically the cross sections fora given g HW W and for 1 − g HW W become the same.Figure 3.6: The total W + W + scattering cross sections as a function of the center of massenergy for different final (and initial) state polarizations and for different Higgs masses,including the limiting Higgsless case. Assumed are two on-shell, unpolarized, colliding W + beams. A cut on the scattering angle that corresponds to pseudorapidity of ± . W direction was applied. The individual W T W T + W T W L curves for each Higgs mass value coincide within the width of the blue line. Results ofMadGraph [125] calculations.Angular distributions of the scattered W ’s are also sensitive to the mass and couplingsof the Higgs boson. In the Standard Model with a light Higgs, the scattering occurspredominantly at small angles. A signature of any rise of the total cross section at somehigh energy is visible as the appearence of an additional component that tends to favorlarge scattering angles, with a local maximum at 90 o . Thus any deviation from theStandard Model in terms of the Higgs couplings would be, quite similarly like differentHiggs masses, observable as a correspondent excess in the rate of W L W L scattered atlarge angles. The excess is the more pronounced the higher the energy. In the abovedemonstration of the principles, we have arbitrarily chosen same-sign W W scattering (inthe next chapters we will see that this choice is in fact well motivated), but the same basicqualitative features are expected of the other scattering processes, involving W + W − , W Z .4. THE
V V
INTERACTION AND WHY IT IS STILL INTERESTING ZZ pairs.Figure 3.7: The total W + L W + L scattering cross sections as a function of the center of massenergy for different values of the HW W coupling, g HW W , Assumed here are two collidingon-shell, unpolarized W + beams and a 120 GeV Higgs boson. Coupling g HW W =1 (lowerblack curve) corresponds to the Standard Model. Blue curves represent g HW W <
1, thecurve for g HW W =0 is equivalent to the Higgsless case. Green curves represent g HW W > W + T W + X scattering (upper black curve, subscript X denotes any polarization, T or L ), its variations with the HW W coupling are containedwithin the line width. A cut on the scattering angle that corresponds to pseudorapidityof ± . W direction was applied. Results of MadGraphcalculations. V V scattering
As mentioned in the previous chapter, the high energy behavior of vector boson scatteringamplitudes is sensitive not only to the Higgs couplings to vector bosons (and Higgs mass),but also to the triple and quartic vector boson couplings. As much as the former are ZZ should be always understood as a sum of the amplitudes for the W + W − → ZZ and ZZ → ZZ scattering processes. CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM
Figure 3.8: Total W + L W + L scattering cross section as a function of the center of mass energyfor different values of the W W W W quartic coupling (labeled 4 W , blue curves) and the W W Z triple coupling (labeled
W W Z , green curves). The corresponding couplings arescaled by a constant factor relative to their respective Standard Model values. Assumedhere are two colliding on-shell, unpolarized W + beams and a 120 GeV Higgs boson. Acut on the scattering angle that corresponds to pseudorapidity of ± . W direction was applied. Results of MadGraph calculations. .4. THE V V
INTERACTION AND WHY IT IS STILL INTERESTING W + T W + X scattering cross section as a function of the center of massenergy for different values of the W W W W quartic coupling (labeled 4 W , blue curves) andthe W W Z triple coupling (labeled
W W Z , green curves). The corresponding couplings arescaled by a constant factor relative to their respective Standard Model values. Assumedhere are two colliding on-shell, unpolarized W + beams and a 120 GeV Higgs boson. Acut on the scattering angle that corresponds to pseudorapidity of ± . W direction was applied. Results of MadGraph calculations.0 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM
Figure 3.10: Examples of angular distributions of the scattered W + W + pairs (pseudo-rapidities with respect to the incoming W + W + direction) at different center of massenergies, depending on the value of the Higgs mass. SM-like couplings were assumed inall the cases. Top left: W + L W + L with a 120 GeV Higgs. Top right: W + T W + X (here sub-script X denotes any polarization, T or L ). Bottom left: W + L W + L with a 500 GeV Higgs. Bottom right: W + L W + L , Higgsless case. The blue curve in the last plot already involvesunitarity violation and therefore is unphysical. Results of MadGraph calculations. .4. THE V V
INTERACTION AND WHY IT IS STILL INTERESTING W + W + pairs (pseu-dorapidities with respect to the incoming W + W + pair direction) at different center ofmass energies, depending on the Higgs and gauge couplings. A 120 GeV Higgs bosonwas assumed in all the cases. Top left: W + L W + L with the HW W coupling equal to 0.8times its SM value.
Top right: W + L W + L with the HW W coupling equal to 1.2 times itsSM value.
Bottom left: W + L W + L with the SM W W W W coupling scaled by a factor of0.999 (the partially visible blue curve involves unitarity violation and therefore is unphys-ical).
Bottom right: W + T W + X with the SM W W W W coupling scaled by a factor of 0.9.Results of MadGraph calculations.2
CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM measured via Higgs partial width measurements, the latter can be probed independentlyvia measurements of diboson and triboson production. Consistency of the three types ofmeasurements: Higgs couplings, multiboson production and vector boson scattering athigh energy is an important closure test for any consistent physical theory and should berigorously tested.There is at least one fundamental difference between the phenomenology of scaledHiggs couplings and that of non-SM gauge couplings. The former manifests solely in W L W L pairs, ultimately as an enhancement with energy. In the latter, there is alwaysa combination of two effects. One is still the energy dependence of W L W L , which inthis case is even steeper because the leading divergence now goes like the fourth power ofenergy (to begin with, we are assuming a simple scaling of the SM couplings by a constantfactor), the other is the overall energy-independent normalization constant which affectsin principle all helicity combinations in the same way. This will be mainly observable in W T W T pairs, because they are the most abundant. Because however such normalizationshifts will be much better measurable in the total diboson production than in boson-bosonscattering, this effect is of lesser interest for us. Mixed W T W L pairs will be modified inboth ways: in the overall normalization and as a rise at high energy (remember that each W L intrinsically carries energy dependence!). Therefore, in the general case, both W T W X as well as W L W L may be of interest. Moreover, angular distributions in vector bosonscattering (VBS) processes exhibit similar qualitative features for W T W X and W L W L pairs in the scenario with a modified quartic coupling. The leading divergence in a VBSprocess is the same in case of an anomalous quartic coupling as for an anomalous triplegauge coupling. Put another way, for every anomalous quartic coupling, there is anequivalent value of the triple couplings that asymptotically produces the same effect. Aslong as we restrict ourselves to pure VBS processes and scaling individual SM couplings byconstant factors, energy dependence of W L W L pairs still carry the most information. Thisis because of their much steeper energy dependence which very quickly dwarfs any effectsin W T W X . But, as we will see in the next chapter, a clean VBS sample is impossible toisolate in a real experiment. And new physics is likely to modify different couplings in acorrelated way.New physics may mainfest itself in new interactions between gauge bosons. Theseinteractions should show up indirectly as certain combinations of modified effective gaugeboson couplings and Higgs to gauge couplings. We don’t know the underlying new physics,but we do have a theoretical machinery to parametrize it in a model independent way.This is where Effective Field Theory comes back. Once again, this general framework hasenough flexibility to describe the low energy phenomenology of new physics regardless ofwhat it really is.A modern effective quantum field theory for physics beyond the Standard Model canbe written down in terms of an extended Lagrangian [58] L = L SM + X i c i Λ O i + X j f j Λ O j + ... (3.3)where O i are dimension-six operators, O j are dimension-eight operators, the coefficients c i , f j are dimensionless and Λ is the energy scale of new physics. The Standard Modelis recovered in the limit Λ → ∞ and the entire model is bound to capture all the low-energy effects of physics beyond the Standard Model. By dimensional analysis one expects .4. THE V V
INTERACTION AND WHY IT IS STILL INTERESTING W + L W + L scattering cross section as a function of the center of massenergy for different values of the relevant dimension-6 operators in the W Effective FieldTheory approach. Varied are: C W / Λ (upper plot) and C B / Λ (lower plot). Assumedhere are two colliding on-shell, unpolarized W + beams and a 120 GeV Higgs boson. Acut on the scattering angle that corresponds to pseudorapidity of ± . W direction was applied. Results of MadGraph calculations.4 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM
Figure 3.13: The total W + T W + X scattering cross section as a function of the center of massenergy for different values of the relevant dimension-6 operators in the W Effective FieldTheory approach. Varied are: C W / Λ , C W W W / Λ (upper plot), C ˜ W / Λ and C ˜ W W W W / Λ (labeled CP W and CP W W W , lower plot). Assumed here are two colliding on-shell, un-polarized W + beams and a 120 GeV Higgs boson. The rises at high energy are due tothe W + T W + L combination, total normalization effects are predominantly due to W + T W + T .A cut on the scattering angle that corresponds to pseudorapidity of ± . W direction was applied. Results of MadGraph calculations. .4. THE V V
INTERACTION AND WHY IT IS STILL INTERESTING O i , O j are constructed from known fields,that is, particles of the Standard Model. Discovery of a new particle should result inrevision of the model and inclusion of additional operators.All related BSM phenomenology is described in a way which depends only on theratios c i / Λ or f j / Λ . However, practical usefulness of an effective quantum field theoryis restricted up to energies of order Λ. At energies higher than that, operators of arbitraryhigh dimension become important, because they are no longer suppressed. The quantityΛ characterizes a particular theory. While we don’t know the scale at which new physicssets in, parameters of the form c i / Λ and f j / Λ have calculable intrinsic validity boundsdefined by the unitarity condition. These bounds fix the maximum allowed value of thescale Λ that is relevant should new physics arise from any particular higher dimensionoperator. A prescription to apply K -matrix unitarization within the context of the effec-tive field theory has also been proposed [59]. In an alternative formulation, sometimescalled the Lagrangian approach [60], the anomalous couplings are taken to be constantLagrangian parameters. There is no explicit relation to the scale of new physics and thisformalism is applicable in the approximation in which these parameters do not dependon energy.The Effective Field Theory approach has been gaining wide recognition in recent time,with more and more studies of sensitivity to BSM being expressed in this language. Allpossible independent dimension-6 operators constructed from the known fields have beencatalogued [61]. There are just three dimension-6 operators that conserve both C and P and affect the interactions of gauge bosons. Following the notation used elsewhere inliterature, these can be written as: O W W W = Tr [ W µν W νβ W µβ ] , (3.4) O W = ( D µ Φ) † W µν ( D ν Φ) , (3.5) O B = ( D µ Φ) † B µν ( D ν Φ) . (3.6)In the above, Φ is the Higgs doublet field and W µν = ig σ a ( ∂ µ W aν − ∂ ν W aµ + gǫ abc W bµ W cν ) , (3.7) B µν = ig ′ ∂ µ B ν − ∂ ν B µ ); (3.8) σ a are Pauli matrices and g , g ′ are the SU(2) and U(1) gauge couplings, respectively. Twoadditional operators appear if we do not assume C and P conservation: O ˜ W W W = Tr [ ˜ W µν W νβ W µβ ] , (3.9)6 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM O ˜ W = ( D µ Φ) † ˜ W µν ( D ν Φ) . (3.10)Here the dual field strengths are defined as ˜ V µν = ǫ µνρσ V ρσ .Typically, one such operator modifies more than one interaction vertex and vice-versa,each interaction receives the contributions from more than one higher-dimension operator.There is some arbitariness in the way all these operators are defined. Recently it was noted[62] that the most useful formulation, at least from an experimental point of view, couldbe one obtained by choosing a basis of higher-dimension operators such that they matchclosely the measured processes, ideally in a one-to-one correspondence. Such approachwould allow to study one vertex at a time and spare from the additional work of combiningdata from different processes in order to study the potential effects of a single operator.This interesting approach is still in the lounge, waiting for being implemented in commonlyaccessible event generators and used in data analyses, and hence will not be applied inthis work.It is transparent that V V scattering processes are not the best channels to studyoperators involving modifications of triple gauge couplings. Much better statistical sig-nificance can be obtained by measuring the total diboson production, i.e., not necessarilyin the VBS mode. It is actually non-VBS diboson production that produces the moststringent limits on these parameters to present day, with currently existing data comingfrom LEP [63] and TeVatron experiments, as well as from Run 1 of the LHC. However,it is vital to know how these operators will affect the VBS measurements.All of the above operators modify triple vector boson couplings, in addition of some ofthem modifying the Higgs couplings and/or the quartic vector boson couplings. Namely, O W W W modifies also the quartic
W W W W coupling, while O W modifies both the quarticand the HW W coupling. By contrast, operator O B affects neither, but it may affect othercouplings, like HZZ or HZγ . The key point that we want to emphasize and elaboratefurther on in this work is that each of these operators affects W L W L and W T W X pairsdifferently. This fact will have a paramount importance in order to interpret correctlythe results of future measurements, which will - most probably - reveal a complicatedcombination of many effects (if anything!).It is not difficult to tell which operators can affect W L W L , W T W T or W T W L verticesstraight from their definition, even without deep knowledge of quantum field theory. A W field can be obtained either via a field strength W µν or via a Higgs field derivative. Everyappearance of the field strength in the operator corresponds to transverse helicity. Longi-tudinal helicities enter via covariant derivatives of the Higgs field ( D µ Φ). Consequently, O B can affect only the scattering of W L W L pairs, while O W affects all possible helicitycombinations. Operator O W W W has only field strengths in it, hence it affects directlyonly vertices involving W T W T . However, mixed pairs W T W L get also affected indirectly,via the t -channel scattering process with a Z T exchange, in which one vertex is bound tocomprise only transverse helicity states.Although a simple scaling of a triple or quartic gauge coupling by a constant factorproduces a divergence that goes like s , gauge invariance enforces cancelation of the ∼ s terms for all the dimension-6 operators [64]. Consequently, the leading divergences arealways proportional to s .In the language of higher dimensional operators, the Higgs to gauge couplings can be .4. THE V V
INTERACTION AND WHY IT IS STILL INTERESTING O Φ d = ∂ µ (Φ † Φ) ∂ µ (Φ † Φ) , (3.11) O Φ W = (Φ † Φ) Tr [ W µν W µν ] . (3.12)Both modify HW W and
HZZ vertices, but not pure gauge couplings. The first of themaffects only HW L W L vertices and will be further considered by means of a simple scalingof the HW W coupling by a constant for better transparency. The second one generatesanomalous HW T W X vertices and has no impact on W L W L . In addition, operator O Φ B = (Φ † Φ) B µν B µν (3.13)affects only HZZ , HZγ and
Hγγ of all the triple vertices and so it can be probed via ZZ scattering.Here and in the remainder of this work we are assuming that the Higgs boson is a purescalar. Possible admixtures from non-scalar components can be parameterized by meansof an additional set of higher dimension operators. They have been constrained by theLHC Higgs data at 7 and 8 TeV, using a combination of the most sensitive Higgs decaychannels. The limits were mainly driven by the ZZ and γγ channels, while standalonelimits from W W in the purely leptonic decay mode are actually the weakest because in thiscase crucial kinematic information escapes with the two undetected neutrinos. In fact, theentire visible
W W phenomenology very weakly depends on non-scalar admixture effectswithin the limits driven by the other decay modes. Although not necessarily negligibleon their own and although potentially important from the interpretative point of view,such effects cannot make any major impact on our considerations.It is also possible to reinterpret the anomalous couplings quoted in section 3.2 in thelanguage of the coefficients of dimension-6 operators. Based on Ref. [58], one gets thefollowing relations: c W W W / Λ = 2 λ γ g m W = 2 λ Z g m W , (3.14) c W / Λ = 2 ∆ g Z m Z , (3.15) c B / Λ = 2 " ∆ κ γ m W − ∆ g Z m Z = 2 ∆ κ γ − ∆ κ Z m Z . (3.16)It should be stressed that the above relations hold so long as we expect the dimension-6operators be dominant. Consideration of dimension-8 operators would generally renderthem not valid anymore.Precise determination of the corresponding limits on these coefficients from the mostup-to-date combination of all the existing LHC, TeVatron and LEP data is a complicated8 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM
Vertex, helicities O W W W O W O B O Φ d O Φ W O Φ B HW W , W L W L - v - v - - HW W , W T W X - v - - v - W W Z , W L W L - v v - - - W W Z , W T W X v v - - - - W W γ , W L W L - v v - - - W W γ , W T W X v (v) - - - - W W W W , W L W L - v - - - - W W W W , W T W X v v - - - -Table 3.1: Sensitivity to dimension-6 operators of the individual gauge and Higgs to gaugecouplings that contribute to W W scattering, decomposed into helicity combinations of theinteracting (initial and final)
W W pair. Note that these are not necessarily the helicitiesat a single vertex. Helicity-flip contributions ( W L W L → W T W X and W T W X → W L W L )have been ignored in this table. For the W ± W ± process these effects are only relevant atcenter of mass energies near the W W mass threshold and do not get enhanced by any ofthe dimension-6 operators. The same is not necessarily true for the W + W − process. Theentry marked as (v) stands for marginally sensitive, but not measurable.task that surpasses the scope of this work. It is also inessential for us in this moment. Infact, most of these limits so far have not changed dramatically since LEP times. Theirimprovement will be possible with an order of magnitude increase in integrated luminosityand doubled beam energy planned for LHC Runs 2 and 3. Without getting into too muchdetail and in accordance with the quoted relations, we can safely assume the alloweddimension-6 operator coefficients c W W W / Λ , c W / Λ and c B / Λ still be of order ± − .On the other hand, a clean study of quartic gauge boson couplings can be carriedwith interactions that do not have a triple vertex associated to it. These are describedusing dimension-8 effective operators. Dimension-8 operators are not necessarily just ahigher order correction to dimension-6 operators. Likely, they probe different physics.Anomalous triple couplings can result from averaging out unknown heavy particles inloops. Quartic couplings can be regarded as a window to electroweak symmetry breaking.They arise as a contact interaction manifestation of heavy particle exchange. It is quitepossible that quartic couplings deviate from the SM, but triple couplings do not. Theoperators of direct relevance for us are: O S, = [( D µ Φ) † D ν Φ] × [( D µ Φ) † D ν Φ] (3.17)and O S, = [( D µ Φ) † D µ Φ] × [( D ν Φ) † D ν Φ] , (3.18)because they only modify the W W W W and
W W ZZ vertices. A combination 8 c − v ( O S, − O S, ), where v is the Higgs vacuum expectation value and c is a dimensionless number,corresponds to a simple rescaling of the Standard Model quartic coupling by a factor c . .5. BEYOND THE STANDARD MODEL? W L W L pairs. Additional dimension-8 operators canbe constructed from field strength tensors and field derivatives or from field strengthtensors alone. These are: O M, = Tr [ W µν W µν ] × [( D β Φ) † D β Φ] , (3.19) O M, = Tr [ W µν W νβ ] × [( D β Φ) † D ν Φ] , (3.20) O M, = ( D µ Φ) † W βν W βν D µ Φ , (3.21) O M, = ( D µ Φ) † W βν W βµ D ν Φ , (3.22) O T, = Tr [ W µν W µν ] × Tr [ W αβ W αβ ] , (3.23) O T, = Tr [ W αν W µβ ] × Tr [ W µβ W αν ] , (3.24) O T, = Tr [ W αµ W µβ ] × Tr [ W βν W να ] . (3.25)We have only listed here the operators that affect the same-sign W ± W ± scatteringprocess. A full list of dimension-8 operators that can modify quartic gauge couplings,including those which can produce anomalous quartic vertices involving only Z ’s and γ ’s,that do not exist in the SM, can be found in Ref. [65]. Numerical coefficients behind theseoperators (usually denoted as f with the appropriate subscripts) are largely unconstrainedby experiment. The possibilities to study quartic couplings at LEP were very limited,while the TeVatron did not offer enough energy and luminosity. Vector Boson Scattering(VBS) at the LHC is the right place to probe them. Upon discovery of the Higgs boson, the Standard Model has been completed. Is thisreally the end of the story? Volumins of theoretical papers have been written to explainwhy the Standard Model cannot be the ultimate theory and we will not repeat thesearguments here. And yet, for improbable this may seem at first glance, the bare truth isthat hardly any experimental result in particle physics to the present date can be said tosupport the idea that the Standard Model needs any major change anywhere below thePlanck scale! Let us critically review what we currently have. The muon magnetic dipolemoment may be one such indication [66]. The magnetic dipole moment is a measure ofquantum effects that modify the effective strength of a charged particle interaction witha photon. These quantum corrections can be very precisely predicted in the frameworkof Quantum Electrodynamics (QED). Such calculations consistently reveal a lower valuethan the experimental world average, the discrepancy is currently at the level of 3.6 σ .This result, while very interesting, is still not significant enough, as well as too isolated0 CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM and indirect to be convincing on its own right. The last 15-20 years brought an explosionof neutrino physics projects, following the observation of neutrino oscillations by Super-Kamiokande. But neutrino masses, regardless of what they ultimately are, includingDirac or Majorana, can be in principle accomodated within the Standard Model if onlywe relax the massless neutrino prejudice which used to be sort of imposed by hand tothe theory before 1996. In a minimalistic scenario it would only require giving neutrinoswhat they always could have within the Standard Model framework and not make anyimpact on the rest of the theory. Finally, the much celebrated naturalness problem,i.e., keeping the Standard Model Higgs boson light despite its quadratically divergentradiative corrections from fermionic loops, is possible simply by invoking some kind ofanthropic principle (technically by assuming an enormous amount of fine tuning betweenthe “naked” mass and the radiative mass shift). Whether or not we find such solutionssatisfactory from the purely aesthetic point of view is, alas, a different question. Yetother claims for the necessity of physics beyond the Standard Model have been madeon purely theoretical grounds, like within the frameworks of Grand Unification Theories,Superstrings, etc., but they all lack any experimental evidence.More suggestive in this respect are in fact astrophysical observations. Evidence ofDark Matter in the Universe is firmly established and does call for new physics. It couldbe argued, though, that in principle nothing forbids adding extra particles to the StandardModel Lagrangian that would completely decouple from the known particles except viagravitation, without adding anything to our undestanding of the known part of the world.Overwhelming excess of matter over anti-matter in the Universe cannot be explained byStandard Model physics, either, at least in its presently known form. But it is still anopen question whether this asymmetry can be explained in terms of leptogenesis in thescenario of a strongly CP -violating neutrino sector. And that’s really all we have.Of the proposed extenstions of the Standard Model, Supersymmetry (SUSY) repre-sents the best known class of models. Originally proposed to tackle the technical problemof loop corrections to the Higgs mass, over thirty years later it still offers a wide rangeof valid models which to this date are neither confirmed nor excluded experimentally.Results of the LHC Run 1 have rendered the simplest SUSY models, such as the MSSMor the NMSSM, less popular, but more generalized models are still in the mainstream ofBSM searches. SUSY has been said to be the only known class of models that reduce exactly to the Standard Model at low energy, so as to possibly reveal no hints of itselfwhatsoever at the presently reachable energies. Depending on one’s point of view, this canbe found as much an advantage as a weakness. In fact, if SUSY is true, there is not muchto expect in the forseeable future from W W scattering, either, in terms of deviations fromthe Standard Model.A separate wide class of alternative candidates for physics Beyond the Standard Modelis known as the Strongly Interacting Light Higgs (SILH) models [67]. They are generallybased on the assumption that electroweak symmetry breaking is triggered by a light com-posite
Higgs, which emerges from a new strongly-interacting sector as a pseudo-Goldstoneboson. This implies the existence, at some higher energy scale, of an additional particlespectrum, characterized by a typical mass parameter
M >> M H and a coupling constant g , with g SM << g < π . The Higgs multiplet is assumed to belong to this “strong” sector.In the limit g SM = 0 the Higgs becomes an exact Goldstone boson. Ordinary StandardModel particles couple weakly to the strong sector. Models known as Little Higgs [68], .5. BEYOND THE STANDARD MODEL? L = L SM + L H + L V , (3.26)where L SM is our familiar Standard Model Lagrangian, L H describes additional interac-tions involving the Higgs boson and L V describes additional interactions involving gaugebosons only. These new interactions imply modifications of the cross sections and branch-ing fractions of the Higgs boson relative to the predictions of the Standard Model. Inparticular, Higgs couplings to known fermions and bosons are somewhat different than inthe Standard Model. In an effective formulation, the whole Higgs-related phenomenologyof SILH models can be described via the choice of a few numbers that parameterize ourignorance of the underlying physics. It was shown that general rules of SILH select justthree of them as the most important ones for LHC studies, which govern the leadingeffects expected in Higgs physics. In the following these are denoted as ξ = ( vg/M ) ( v =246 GeV is the Higgs vacuum expectation value), c y and c H . In terms of these parameters,the Higgs partial widths are modified with respect to the Standard Model as follows:Γ( h → f ¯ f ) SILH = Γ( h → f ¯ f ) SM [1 − ξ (2 c y + c H )] , (3.27)Γ( h → W + W − ) SILH = Γ( h → W + W − ) SM [1 − ξ ( c H − O ( g SM /g ))] , (3.28)Γ( h → ZZ ) SILH = Γ( h → ZZ ) SM [1 − ξ ( c H − O ( g SM /g ))] , (3.29)Γ( h → γγ ) SILH = Γ( h → γγ ) SM [1 − ξRe ( 2 c y + c H J/I + c H I/J + O ( g SM /g ))] . (3.30)Here I and J are loop functions describing Higgs radiative decays whose numerical valuesdepend mostly on the top quark mass. Their full definitions can be found in Ref. [67].The ξ parameter naturally ranges between 0 and 1, the two limiting cases correspondingto the Standard Model and technicolor theories, respectively. Note that to the lowestorder it is correct to say that SILH phemomenology in comparison with the StandardModel can be described as an overall modification of all the Higgs couplings to fermionsand another overall modification of all the Higgs couplings to gauge bosons. For example,the “fermiophobic Higgs” scenario is obtained by setting c H =0 and ξc y =1/2. Extractionof c y and c H is a main task for precision measurements of the Higgs production rate andbranching fractions. Given that c y and c H are typically numbers of the order of unity,the size of possible deviations from the Standard Model in terms of Higgs productioncross sections times branching fractions can amount even to ∼ CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM excluded in the light of present data and their existence will be subject to verification atthe LHC with √ s = 13 TeV.As we already know, an HW W coupling different from the Standard Model one reflectsin the predicted high energy behavior of
W W scattering amplitudes. Indeed, in SILHmodels the light Higgs unitarizes the amplitudes only partially or, better saying, it onlydefers the unitarity crisis to higher energies. We talk then of a partially strong
W W scattering. Relevant cross sections still grow above the Higgs mass, albeit slowlier, withan asymptotic behavior given in the lowest order of g /M s by A ( W ± L W ± L → W ± L W ± L ) = − c H g M s, (3.31) A ( W + L W − L → W + L W − L ) = c H g M ( s + t ) , (3.32) A ( W ± L Z L → W ± L Z L ) = c H g M t, (3.33) A ( W ± L W ± L → Z L Z L ) = c H g M s, (3.34) A ( Z L Z L → Z L Z L ) = 0 , (3.35)and up to scale of M , where new physical states are bound to appear and do the rest ofthe unitarization. Notice that the above amplitudes are proportional to the ones obtainedwithin the framework of a Higgsless Standard Model and in fact, up to the energy M : σ ( pp → jjW L W L ) SILH = ( c H ξ ) σ ( pp → jjW L W L ) Higgsless . (3.36)The immediate conclusion that can be drawn at this point is that all the previousstudies of W W scattering, which assumed a pure Higgsless Standard Model as a phe-nomenological laboratory, are not entirely obsolete even once the Higgs has been discov-ered. Their results remain completely valid as long as the predicted signal sizes are scaledby an appropriate factor dependent on the actual value of the
HW W coupling.The following toy model well illustrates the phenomenological complementarity be-tween SILH and SUSY. The most straightforward example of a general framework inwhich partially strong
W W scattering can take place is the two-doublet model (2HDM).In this framework, couplings of the light and heavy Higgs scalars to the W boson are givenby g SM sin ( β − α ) and g SM cos ( β − α ), respectively, where α is the Higgs mixing angle and tanβ is the usual ratio of vacuum expectation values. If the factor sin ( β − α ) is sufficientlysmall and the heavy Higgs sufficiently heavy, the relevant amplitudes can rise significantlyin between the energies corresponding to the masses of the light and heavy Higgses forpartially strong W W scattering to take place. The heavy Higgs will ultimately unitarizethis growth. This, however, is generally not the case in models involving SUSY, e.g., inthe MSSM the heavier the heavy Higgs is the closer to unity the factor sin ( β − α ) will beand vice-versa. Thus no appreciable W W scattering can be expected in the MSSM.Finally, it is always important to realize that different phenomenological features maybe directly linked to each other within certain classes of models, but need not be so in .5. BEYOND THE STANDARD MODEL?
W W scattering. The dynamics of electroweak symmetrybreaking ultimately still will remain an open question and can be possibly concluded onlyvia direct measurement of the
W W scattering cross section at high energy.For the sake of completeness one should mention also another predicted signature ofSILH models, namely enhanced production of Higgs pairs at high energy [71]. Measure-ment of double Higgs production can have important implications for spotting out ourposition on the electroweak phase diagram [72], but such effects can be hard to detect inpractice and do not belong to our main topic.The physics meaning of Higgs couplings larger than their Standard Model values hasbeen recently investigated as well. The case of
HW W >
CHAPTER 3. STANDARD MODEL EXPERIMENTAL STATUS AND PROSPECTS FOR BSM hapter 4
V V scattering at the LHC
This chapter provides a comprehensive overview of the
V V scattering process at the LHCfrom a phenomenological point of view.Longitudinal
V V scattering carries the most direct, quantitative information aboutthe details of the actual mechanism of electroweak symmetry breaking. The practicalchallenge lies in digging that information out. As we have no W beams, in any real ex-periment we have no control over the polarizations of the interacting pair. This means,assuming every helicity state is taken with equal probability, that a V L V L initial pair hap-pens in only 1/9 of all the V V cases. The majority of initial pairs are V T V T and V T V L states, of little sensititvity to the Higgs parameters and indeed to the Higgs existenceat all. Furthermore, because of the matrix element, interacting V L V L pairs make up nomore than 5% of the total interacting V V pairs, assuming the Standard Model is approx-imately correct and the Higgs boson light. Variations of the V T V T and V T V L scatteringcross sections as a function of the Higgs mass and HV V couplings amount to some ∼ V V interacting pairs is merely a potential background in our search. One way to proceedis to look for specific kinematic signatures associated to a hard 2 → assumption is then that any possible excess over the prediction is due to theadditional V L V L component. This indeed was the approach taken in many early phe-nomenological papers on the subject. It is clear that such approach requires very goodcontrol over the systematic errors related to the theoretical prediction. Measurement of V polarization based on the decay products is especially difficult for the W W processwhere crucial information escapes along with two neutrinos, although the methodologyis in principle known and applied in some analyses [74]. But in our kinematic regime ofinterest measurement of the final state polarizations will be a challenge. In what followswe will show that we can, nonetheless, to some extent measure the polarizations of theinitial state. More often than not polarizations are actually conserved in the scatteringprocess, at least in what regards W L W L versus all the rest. And this conservation holdsmost strictly in the W ± W ± process. This means, in particular, that W ± L W ± L pair in thefinal state can be produced almost exclusively from an initial W ± L W ± L pair (see Fig. 4.1).The only exception to this rule lies in the region of center of mass energies just above thedouble W mass threshold, where helicity-flip effects are more likely to occur, above all556 CHAPTER 4.
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SCATTERING AT THE LHC W T W X → W L W L . In the other direction, these effects are negligible altogether becauseof the relative smallness of W L W L . In any case, for center of mass energies above 400GeV, the admixture from helicity-flip effects is completely negligible. This is the primereason why it makes sense to separate W L W L pairs in the final state from the bulk of W W interactions and study their distinctive kinematic properties.It is much more complicated to separate a clean W L W L scattering process in W + W − ,for which still at an energy of a TeV about 20% of W L W L pairs come from the process W T W X → W L W L .Figure 4.1: Total W + W + → W + L W + L scattering cross sections in the SM as a function ofthe center of mass energy. Shown are the individual contributions of different initial polar-ization states to the final state consisting of purely longitudinal W + L W + L pairs. Subscript X denotes any polarization ( T or L ). Assumed are two on-shell, unpolarized, colliding W + beams. A cut on the scattering angle that corresponds to pseudorapidity of ± . W direction was applied. Results of MadGraph calculations. In a hadron collider,
W W scattering can occur via W emission off two colliding quarks.A lowest order diagram of the process is shown in Fig. 4.2 (left).The final state is characterized by the presence of two W bosons (more precisely: theirrespective decay products) and two jets. Regardless of how we technically define signaland background, it is clear that in practice we have no control over whether a specificpair of vector bosons has indeed interacted or not. A whole other class of events in whichtwo W bosons are produced and do not interact will inevitably be present and separablefrom the signal process on a statistical basis only, thus becoming the bulk of irreduciblebackground. There are two different approaches often adopted in literature regarding howthe signal can be formally defined. The kinematic approach defines the signal in termsof the expected kinematics of a hard 2 → .1. FORMAL SIGNAL DEFINITION W W scattering at the LHC (left) and examples ofgraphs contributing to the irreducible background (middle and right). The scatteringgraph contributes to the signal as long as W = W L , otherwise it contributes to theirreducible background as well.arbitrary and so no definition is really unique. Indeed, many different signal definitionshave been in use by experimentalists. Moreover, by the same token, it is assumed that thepart that does not fall into the signal window and hence formally defines the irreduciblebackground does not depend on the Higgs sector parameters, which in general need not beexactly true. A feature of this approach is that the Standard Model predicts some signal,too. Deviations from the Stanard Model will usually lead to different signal predictions;consistency of each prediction with real data in the predefined kinematic window can beassessed.A second, more generic approach defines the signal explicitly as the excess of W W pairsover the prediction of the Standard Model, apparently regardless of the actual physicalmechanism that leads to such excess. The Standard Model in itself, regardless of theactual physical process, is then the formal definition of the total irreducible background.As we saw, this background will be composed mainly of W T W T and W T W L pairs. Toreemphasize this point, the Standard Model signal is equal to zero by construction . Signalis BSM. The first approach is of course closer to what eventually will be done in a realexperiment. However, to study the problem from a conceptual point of view, the seconddefinition has at least two important advantages. First of all, it is unique as long as wefix the Standard Model Higgs mass that we use to define irreducible background. Second,it does not rely on any particular kind of interaction and there is no signal region defineda priori. The fact that we know what process is responsible for the possible appearenceof signal is a bonus we can make use of at a later stage, but not a prerequisite. Notethat some W L W L scattering is naturally predicted even in the Standard Model and sothe signal graph in itself is not fully equivalent to what we are for. The correspondencebetween the two approaches is clear and the translation of respective results into eachother is conceptually more or less obvious, although it has been sometimes the source ofsome confusion.For a better understanding of the full process from a theoretical point of view, onecan decompose its complete parton level description into three distinct, intrinsically con-nected parts involving W emission, interaction and decay. However, one should alwayskeep in mind that such factorization is approximative, its practical applicability is a sub-ject of study and any potential conclusions we would like to draw require independentconfirmation in the exact evaluation of the full process.8 CHAPTER 4.
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Factorization of the signal process into the subsequent steps of W emission, interactionand decay, as we would herewith like to do, is a useful means to study certain features,but not always a satisfactory way of quantitative description. Quantum-mechanically, itis clear that all paths leading from the initial pp state (which can be decomposed intothe many possible sub-states at the quark and gluon level) to any specific final state,say, jjµ + µ + νν , must be considered for the correct evaluation of the process. Clearly,the irreducible background must include not only graphs identical with the signal graph,with a dominant contribution from transverse W ’s, but likewise a large number graphsnot involving any W W interaction. Either of the two categories of events is not gauge-invariant on its own. Strong interference effects may occur, depending on the gauge,and so not only they cannot be treated separately, but neither can the signal. Correctcalculation of signal and irreducible background from first principles (i.e., Feynman rules)requires all these processes added at the level of amplitudes. Signal must be defined usingthe “subtraction method” which technically requires the computation of two total crosssections for (e.g.) pp → jjµ + µ + νν : one corresponding to the Standard Model, anotherone to the alternative scenario. The signal ultimately comes from subtracting the formerfrom the latter. Note that in general the signal can be positive inasmuch as it can benegative and that both make physical sense. And indeed, the signal is negative in certainregions of phase space. In addition to pure electroweak diagrams, ∼ α in the lowestorder (up to the level of W decay), background also includes mixed, electroweak-QCDprocesses, ∼ α α S . The minimal collection of those correspond to gluon exchange graphsbetween the two interacting quarks. Depending on the chosen final state, the numberof additional electroweak-QCD diagrams can vary widely and so does therefore the totalbackground cross section. All in all, the lowest order calculation of the pp → jjµ + µ + νν process, which is the simplest from the computational point of view, requires considerationof 5656 Feynman diagrams.The signal in the lowest order is a purely electroweak process. However, interferencebetween scattering and non-scattering diagrams applies in principle also for electroweak-QCD ones. The fact that signal (understood as BSM!) can indeed be calculated ignoringany QCD contributions, regardless of the relative amounts of the pure electroweak andelectroweak-QCD processes, is a present from nature rather than a rule of thumb. Inter-ference effects can be shown to cancel out to a good accuracy in the difference throughwhich we define the signal . This is because electroweak-QCD events populate mostlya different region of kinematic phase space than the purely electroweak signal events -the respective transverse momenta of scattered W ’s are nearly clean separated - and thisconclusion holds even for W + W − scattering where the total signal+background cross sec-tion is dominated by QCD contributions by an order magnitude. In our example process pp → jjµ + µ + νν , this allows to reduce the number of Feynman diagrams necessary for thecalculation of the signal to 5208 (however, another dedicated calculation is then neededto determine the term to subtract, which is not equal to the total irreducible backgroundanymore). Bear in mind of course that what we are for in this chapter is an estimate of the magnitudes of signaland background, not a precision measurement .2. COMPUTATIONAL ISSUES AND METHODS W Approximation and the Equivalence Theorem
Older literature made extensive use of the so called Effective W Approximation [75], withits nice acronym E W A (or more generally, Effective Vector Boson Approximation, EVBA).Its main advantage is that it renders the lowest order signal graph gauge invariant on itsown, under some approximative assumptions. The idea is similar to that of factorizationfor parton distribution functions. The total cross section is described in terms of densityfunctions for a polarized W being radiated by a fermion with a given momentum fraction,times the scattering amplitude for two bosons carrying these momentum fractions. Theboson is assumed to be radiated approximately collinearly at a high center of mass energy,so it is close to the mass shell and we can neglect the fermion masses. In the amplitudeof the scattering process it is then taken to be on shell. This means that the treatmentwill only necessarily be valid when √ s >> M W and so small virtualities of the gaugeboson may be neglected. The explicit expression for the density function can be derivedfrom the matrix element calculated for an on-shell boson being emitted off a fermion asa function of the fermion initial four-momentum and the momentum fraction carried bythe boson. The total process cross section to the level of the scattered gauge bosons isfinally given by integrating the scattering amplitude with the appropriate density functionover the full range of the momentum fraction. The E W A provides an effective way tocalculate the signal-like graphs standalone to a typical accuracy of 20-30%. A substantialliterature exists on the accuracy and conditions of applicability of the E W A. Although thevalidity of the approximations does not explicitly exclude any helicity states, the E W A byconstruction disregards non-scattering contributions and is therefore not able to predictthe total irreducible background levels. The latter still requires computation of the fullset of diagrams.The E W A technique is often coupled with the evaluation of gauge boson interactiondone using the Equivalence Theorem [76]. The Equivalence Theorem states that at an0
CHAPTER 4.
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SCATTERING AT THE LHC energy much larger than the vector boson mass, the amplitude for a process involvinginteraction of longitudinally polarized vector bosons on the mass shell is given by theamplitude in which these vector bosons are replaced by the corresponding unphysicalGoldstone bosons. Intuitively this is understandable as a consequence of the Higgs mech-anism. The vector bosons get their masses via absorbing the Goldstone bosons and sotheir longitudinal components retain the properties of the scalar interactions. Whetherthis naive intuition is really correct or not, the approximation is valid up to the leadingenergy term and it is applicable in every order of perturbation theory. The approximationis very useful because it is technically much easier to calculate amplitudes involving mass-less scalars than those involving massive vectors. The ratio of the actual vector bosonmass to the center of mass energy of the interaction defines its practical accuracy.However, several authors emphasized the importance of using full matrix elementcalculations in order to correctly reproduce the entire kinematics of the final state, whichlies at the basis of defining optimum selection criteria for the isolation of longitudinalsignal from transverse background. The advantages of the E W A and the EquivalenceTheorem naturally waned once full matrix element generation tools became available tothe public and fast computer clusters alike. Approximative techniques to evaluate the
W W interactions are rarely used in modern studies. × decay” approximation In quantum physics, full calculation of the process, say, pp → jjµ + µ + νν involves summingover all the possible paths leading to the final state. Note that in such, formally fullycorrect, treatment information on the individual W helicities is lost. We don’t even knowwhether there was a W + W + intermediate state at all at any time in the process. Asa matter of principle, helicity is well defined only for on-shell bosons. To what extentthe W ’s after interaction are on-shell and hence to what extent they can be sensiblyassigned longitudinal or transverse polarizations at all, is a crucial issue. Experimentally, W helicity manifests itself in angular distributions of the decay products - for examplethe charged lepton from W decay with respect to the mother W direction. It cannot bededuced on an event by event basis.The full process pp → jjµ + µ + νν can be reasonably expressed as a coherent sum of its W L W L and W T W T + W T W L contributions only so long as we can assign two W helicitiesto each event, even if only on paper. This is possible if and only if the scattered W ’s areproduced near enough the mass shell, or equivalently, off-shell effects, including graphs inwhich a W boson is exchanged in the t -channel, do not lead to significant changes of themeasurable kinematics of the final state. Only under this assumpton can the process beapproximately factorized into steps consisting of W W production and decay. It was shownthat in the kinematic region of interest for us, this approximation indeed holds to betterthan 10% both in shapes and normalizations, which is quite enough for our purposes.Because of this lucky fact, the characteristic features of final states associated to W L W L and W T W T + W T W L can be studied separately of each other. One can also hope for a moredetailed signal event selection that will be based not solely on the scattering kinematics,but also on the preferred W helicities. The on-shell approximation for the scattered bosonsis sometimes referred to as the “production × decay” approximation, as it technically .3. EMISSION OF A GAUGE BOSON OFF A QUARK × decay” approximation;cf. Fig. 4.3. Drawing by J. Kuczmarski.allows to reduce computational work to the reduced process pp → jjW + W + (where thetwo bosons are assumed to be exactly on-shell) in the first step and thus decrease thenumber of Feynman diagrams to consider from 5656 to 1428. Similar conclusions hold for W + W − and ZZ if only applied far enough from the Higgs resonance region where theagreement expectedly breaks down.Since in principle we can indeed sensibly assign specific polarizations to W W finalstates, it is legitimate to restrict the formal signal definition explicitly to W L W L pairs inall the computations, at least in studies concerning the source of electroweak symmetrybreaking. The practical advantage is one of avoiding large cancelations in the signaldefinition coming from the dominant and largely BSM-insensitive W T W T + W T W L states.In literature one also finds a similar approach under the name of Narrow Width Ap-proximation. Generally it has been shown to work for Standard Model processes with atypical accuracy of Γ /M , the ratio of the total width to the mass of the particle involved[77]. However, some implementations of the Narrow Width Approximation in event gen-erators consist merely of neglecting the non-resonant graphs, but with off-shell effects andspin correlations kept, and thus are not fully equivalent to our approach. The characteristic difference in the kinematics of the final states associated to the emissionof W L and W T off a quark are their different angular distributions. The W L W L and W T W T luminosity spectra calculated from pure emission probabilities from two quarks collidinghead-on at a fixed energy and without any further interaction assumed, already revealinteresting differences in their kinematic features - see Fig. 4.6.The longitudinally polarized W tends to be emitted at a smaller angle (hence smaller2 CHAPTER 4.
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SCATTERING AT THE LHC a) b)c) d)e) f)g) h)i) j)k) l)Figure 4.5: Kinematic distributions of final state muons from the pp → jjµ + µ + νν processat 14 TeV, obtained using the W on-shell approximation (labeled MadGraph+PYTHIA)and exact matrix element calculations (labeled PHANTOM). Shown are: pseudorapiditiesof the two muons (a-d), their transverse momenta (e-h), distances in the azimuthal angle(i,j) and invariant masses (k,l). VBF topological cuts were applied, including | η µ | < η jj >
4. In background calculations only electroweak processes were taken into accountand the Higgs mass was set 200 GeV. .4. INTERACTION OF TWO GAUGE BOSONS W L (left) and of a W T (right) in intervals of the W W invariant mass. Assumed isa pair of colliding quarks, each emitting a W boson, no W W interaction is taken intoaccount. Calculation done within the Effective W Approximation.transverse momentum) with respect to the incoming quark direction than the transverselypolarized W . As a consequence, the final quark accompanying a longitudinal W is moreforward than the one accompanying a transverse W . This effect is more pronounced thelarger the invariant mass of the W W pair, M W W . The transverse momentum distributionsof quarks associated with W L emission become narrower as M W W increases and the peakof the distribution gradually moves to lower values. No such trend is associated with W T emission, except for very large M W W , where in case of a fixed incoming quark energy theeffects of overall energy and momentum conservation become a limiting factor. Theseobservations suggest that our potential to separate the W L W L signal from the W T W T background increases with M W W already at the level of emission. Tagging two oppositeforward jets in a relatively narrow band of transverse momentum for a fixed value of M W W is the ideal technique to be used. The practical problem in implementing this conclusionin an experiment is that the absolute scale of transverse momentum of the emitted W is defined by the mother quark energy. Events can be efficiently discriminated based onthe transverse momenta of the outgoing jets so long as we have monochromatic quarkbeams . Total cross sections and angular distributions in the scattering process of two on-shell W bosons, depending on their energies and polarizations, were already discussed in theprevious chapter. Here we have just learned that in addition, since W L ’s tend to beemitted from a quark line in a more collinear way than W T ’s, the W L W L rest frame willbe approximately equivalent to the lab frame as long as we disregard highly asymmetricquark-quark collisions. Excess over the predictions of the Standard Model is thereforeexpected in the central region of the detector, as far as the scattered W directions are Obviously, we would be doing much better in a lepton collider, if only it had a similar energy reach! CHAPTER 4.
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SCATTERING AT THE LHC concerned. Let us now build a naive toy model of such process. For a
W W pair com-ing from nearly collinear emissions and scattered back-to-back at a large angle, we canapproximate M W W ≈ q M W + p (1) T p (2) T , (4.1)where p (1) T and p (2) T are the transverse momenta of the scattered W ’s understood as un-signed scalar quantities. The product p (1) T p (2) T (or its square root, to be more precise) is ameasure of M W W and this equivalence naturally works better for large M W W . Background Signal d σ / dp T W [ pb / G e V ] p TW [ GeV ] ( W + W + → W + W + ) M H = GeV M WW [ GeV ] d σ / dp T W [ pb / G e V ] p TW [ GeV ]( W + W + → W + W + ) higgsless − ( W + W + → W + W + ) M H = GeV M WW [ GeV ] Figure 4.7: The W + W + scattering cross sections for background and signal as a functionof the transverse momenta of the outgoing W , for different center of mass energies ( M W W ).Calculation done within the Effective W Approximation.A large value of p (1) T p (2) T is not only the kinematic region where deviations from theStandard Model are supposed to emerge (because of the s -divergence), but also inde-pendently where W L -associated jet kinematics is more easily distinguishable from the W T -associated jet kinematics. The branching fraction of W decay into any of the charged leptons with a correspondingneutrino is (10 . ± . Z boson decays into an oppositely charged lepton pairof a given flavor in (3 . ± . . ± . • Purely leptonic pp → jjW + W − → jjl + νl − νpp → jjW ± W ± → jjl ± νl ± ν .5. GAUGE BOSON DECAY AND POSSIBLE FINAL STATES pp → jjW ± Z → jjl ± νl + l − pp → jjZZ → jjl + l − ννpp → jjZZ → jjl + l − l + l − Leptonic W and Z decays are the preferred decay modes for a wide range of mea-surements involving gauge bosons, and among other things, provide some of themost sensitive means for Higgs studies. Their main limiting factor is low statisticsinduced by the small individual branching fractions. Practical viability of thesemodes crucially depends on our background rejection capability. This in general fa-vors the non-zero total charge states W ± W ± , W ± Z , and the four-lepton final state( ZZ ), which is the only one where the full kinematics of the process can be mesured.On the other hand, both the Z production cross section and its leptonic branchingfraction are lower than those of a W , hence production rates favor W + W − followedby W ± W ± . The purely leptonic channels are often regarded as the “gold-plated”modes in phenomenological literature because of their clean, distinctive signaturesand because the rough magnitudes of both the signal and the main backgroundscan usually be reasonably estimated without involving a full detector simulation. • Semi-leptonic pp → jjW W → jjjjlνpp → jjZW → jjjjlνpp → jjW Z → jjjjl + l − pp → jjZZ → jjjjl + l − These processes combine a hadronic decay of one gauge boson with a leptonic de-cay of the other. They are characterized by reasonable statistics and higher re-ducible backgrounds. Typically, control of the latter requires full detector simulationto handle, e.g., the dominant backgrounds from processes involving production of
W/Z +jets with a jet misidentified as a lepton. Additionally, at large
W/Z energies,the two jets originating from a hadronic decay tend to merge in the detector whichfurther reduces the signal isolation efficiency and adds extra backgrounds to be con-sidered. Early studies usually revealed these channels be somewhat less promising,overall, than purely leptonic. However, improvements in event reconstruction inLHC experiments and in particular the use of novel techniques of “jet pruning” [78]that allow to determine the mass of the original object producing the jet and there-fore distinguish QCD jets from W jets to a large accuracy, bring new interest tothe semi-leptonic channels again. These techniques have been demonstrated to beapplicable in the k T and Cambridge-Aachen jet reconstruction algorithms [79], butnot in the default anti- k T algorithm used in CMS and ATLAS. Since they have beenshown to offer great promise, reconsideration of the jet reconstruction algorithm tobe applied for VBS analyses is a potential possibility. Clearly a lot of rework needsto be done as dedicated reprocessing of all the past studies will be required, butin the end the semi-leptonic channels may prove very useful to icrease the totalsignificance of the signal. • Purely hadronic6
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SCATTERING AT THE LHC pp → jjW W → jjjjjjpp → jjW Z → jjjjjjpp → jjZZ → jjjjjj Despite their large branching fractions, these processes are completely overwhelmedby the multi-jet QCD background and in addition their study requires full detector-dependent modeling of event reconstruction effects. The purely hadronic modes aretherefore not considered for detailed studies at this time.In the above we assume l = e, µ . Decays into taus constitute yet a separate class ofspecific final states and signatures, but due to the relative complexity and lower identifi-cation efficiency they can be disregarded for the time being.In the search for the most promising channels to begin with, the crucial point isthe underlying physics and in particular existence or non-existence of heavy Higgs-likeresonances within the energy range of the LHC. Their existence of course favors the W + W − and ZZ channels and indeed these usually have been given the most attention,also as a byproduct of Higgs physics. Further on however we will mainly focus on analternative yet plausible scenario that such resonances, if any, are too heavy for directdetection at the LHC. In this case, the non-resonant modes W ± W ± (and W ± Z , in somesense) acquire not only equal importance, but as we will further see, their relative exoticitycan be well turned into an advantage. W ± W ± It is now time to explain that the apparently arbitrary choice of the pp → jjµ + µ + νν process as a particular example in many of our earlier considerations was in fact wellmotivated. The W ± W ± final state, with its ± W W scattering is the only process for which the cross-talk amplitudes, W T W X → W L W L and W L W L → W T W X , are completely negligible, mostly due to lack of any s -channel graphsthat contribute to the process. The latter also has other consequences. Contrary to otherdiboson states with two accompanying jets, production of the jjW ± W ± state in the lowestorder is dominated by only one physical mechanism at the quark level, namely a quark-quark interaction associated with a W ± emission from each colliding quark. Whether ornot these two W ± bosons do interact, information on their polarizations stays encodedin the kinematics of the two outgoing quarks (recall section 4.3), unless it is disturbedby a subsequent quark interaction. If only we knew the energies of the colliding quarks,appropriate cuts on the angles and transverse momenta of the two tagging jets wouldincrease the probability of choosing a W L W L state - regardless of their own final kinematicsand the rest of the process. By the same token, the only QCD contributions to theirreducible background are graphs ∼ α α S of the form of internal gluon exchange betweenthe two quarks. Not only they are negligible in the calculation of the BSM signal, asalready shown, but their contribution to the background can be reduced to below 10% Strictly speaking, W ± W ± is non-resonant as long as there are no bosonic isospin triplets, and hencedoubly charged bosons, in nature. .6. THE UNIQUENESS OF W ± W ± jjW + W − final state at the LHC.Figure 4.8: Examples of Feynman diagrams of purely electroweak processes that con-tribute to the process pp → jjW + W − , but have no equivalent for jjW + W + . Eventswhere both W ’s originate from a decay of a neutral particle contribute both to our def-inition of signal (left) and the irreducible background (right), but kinematicwise do notallow the distinction of W polarizations. The left graph is actually Higgs production viaHiggsstrahlung and is of little relevance once VBF selection criteria are imposed. How-ever, huge additional contributions to the irreducible background change significantly itsoverall kinematic distributions and mask the part of the background which is related tosingle W emissions from each colliding quark.A W + W − pair can come from virtual Z decay, as well as two consecutive emissionsoff a single quark. Even more importantly, the electroweak-QCD background receiveshuge additional contributions from graphs involving gluon-gluon and quark-gluon inter-actions. In fact, processes ∼ α α S dominate the total jjW + W − production by an orderof magnitude, prior to kinematic cuts. Usual non-VBF Higgs production graphs, e.g.,those involving Higgsstrahlung followed by Higgs decay into W + W − , also contribute tothe signal according to our working definition, but are of little use kinematicwise whenwe go to higher energies. All in all, signal in the W + W − mode can be expected lesswell kinematically separated from background, and background much larger. Assumingthe absence of new heavy Higgs-like resonances within the energy reach of the LHC at13/14 TeV, W + W − is a more difficult choice. For completeness one should notice that thechoice of same-sign W pair is also a powerful shield against the overwhelming reduciblebackground originating from t ¯ t production. Only second order effects, like leptonic B decays or lepton charge misidentification can lead to a non-zero t ¯ t background. Theseaspects will be elaborated further on.Assuming the pure Higgsless Standard Model scenario as the theoretical basis forthe definition and numerical computation of the signal, the total signal cross section for pp → jjW + W + at √ s = 13/14 TeV is roughly an order of magnitude smaller than the8 CHAPTER 4.
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Figure 4.9: Distributions of signal (left), electroweak background (middle) and QCDbackground (right) for the process pp → jjW + W + at 14 TeV as a function of jet pseu-dorapidities. Results of parton-level MadGraph calculations involving all processes ∼ α (left and middle) and all processes ∼ α α S (right). Interference between the two classeswas neglected for demonstration purposes.irreducible background. Basic kinematic signatures of the signal and irreducible back-grounds in terms of angular distributions of the two outgoing jets, confirm the usefulnessof the forward jet tagging technique (a basic VBF signature in the LHC) to isolate sig-nal from background. The requirement of two opposite-sided, large pseudorapidity jets,2 < | η j | < η >
4, suppresses the bulk of soft parton-parton collisions. It eliminatesmost of the electroweak background, and even more efficiently the electroweak-QCD back-ground - for an illustration of the basic topologies of signal and backgrounds, see Fig. 4.9.This, together with another basic topological requirement of two W bosons within theacceptance of the detector (which can be approximated quantitatively as | η W | <
2) hasan important effect at the quark level as it effectively selects a very specific configura-tion of the colliding quarks. Not only we have a single production mechanism - residualprocesses in which both outgoing W ’s originate from the same quark line, or not from aquark at all, are now completely suppressed - but also a common production kinematics.Energy distributions of the two quarks before interaction, usually preferring the lowestenergies, as dictated by proton PDF’s, now begin to peak quite strongly around roughly ∼ √ s ≈ uu → ddW + W + at a fixed energy.Comparison of the final state kinematics of the processes pp → jjW + W + at 14 TeVand uu → ddW + W + at 2 TeV, after no more than the basic VBF topological cuts definedabove, is very telling. The kinematics of the outgoing W bosons are indeed very similar, .6. THE UNIQUENESS OF W ± W ± pp → jjW + W + at14 TeV as a function of the energies of the two incident quarks after imposing basictopological cuts discussed in the text. Results of a parton-level MadGraph calculation.Figure 4.11: Schematic representations of the full set of processes which need be takeninto account for the evaluation of the signal in the “production × decay” approximation(left) and of the reduced set of processes which need be taken into account to learn thebasic kinematics of the signal process and of the irreducible background (right). Drawingsby J. Kuczmarski.0 CHAPTER 4.
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Figure 4.12: Distributions of transverse momenta of the jets associated to the emis-sion of W L W L signal (left) and W T W T background (right) in the quark level process uu → ddW + W + at an incident center of mass energy of 2 TeV. Results of MadGraphsimulations. Note similarity in gross features to the distributions resulting from the pureemission process shown in Fig. 4.6.Figure 4.13: Distributions of transverse momenta of the jets associated to the emission of W L W L signal (left) and W T W T background (right) in the full process pp → jjW + W − at14 TeV (bottom row). Results of MadGraph simulations. Note similarity in gross featuresto the distributions resulting from the quark process shown in Fig. 4.12 and from the pureemission process shown in Fig. 4.6. .6. THE UNIQUENESS OF W ± W ± uu → ddW + W + at 2 TeV. Left: the differential crosssections of background and signal in a two-dimensional space defined by the square rootsof the transverse momentum products of the two outgoing d quarks and the two outgoing W ’s, q p d T p d T versus q p W T p W T ; the color contours are equidistant and the scale rangesfrom zero (white) to 0.004 fb/GeV for the background and to 0.0002 fb/GeV for thesignal (purple). Upper right: the signal to background ratio from dividing the two leftplots; the vertical scale is logarithmic for better visualization and ranges from 0.03 (white)to 30 (purple).
Lower right: the distributions of the ratio p W T p W T / ( p d T p d T ) for signaland background. No kinematic cuts were applied. Results of a MadGraph calculation.Signal was calculated by considering longitudinal W + W + pairs only and subtracting theSM-based distributions from the Higgsless-based distributions.The fact that most features of the final state kinematics can be approximated with apicture of two colliding monochromatic quark “beams” has a very important phenomeno-logical consequence. Systematic differences in the kinematics of the tagging jets associatedto the emission of longitudinally and transversely polarized gauge bosons can indeed beobserved in an experiment. Partonic structure functions inside a proton in the first stepinevitably smear out the measured transverse momentum distributions and hide the infor-mation on W polarity. Remarkably, after basic VBF topological cuts this information can2 CHAPTER 4.
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SCATTERING AT THE LHC be unveiled again. Tree level calculations show that transverse momentum distributionsof jets in signal and background events indeed follow the same qualitative trends as out-lined before for the pure emission process of a longitudinal and transverse gauge boson,as well as in the signal and background in an ideal quark process, once VBF topologicalcuts are applied on the former - see in particular Figs. 4.12 and 4.13.To summarize our findings, signal is characterized by emissions of longitudinal W ’sfollowed by their hard interaction. Signatures of the first part are two opposite taggingjets at large pseudorapidities and with relatively low transverse momenta. The secondpart induces a large W W scattering angle which translates into small W pseudorapiditiesand large transverse momenta. Because their respective kinematics is severely constrainedby the sole physical mechanism and the energies of the colliding quarks, signal and back-ground events occupy rather restricted and largely separated regions in the phase spaceof the four final state particles, and in particular their transverse momenta. The signalcross section is nearly flat over a large range of the product q p W T · p W T and much morerapidly falling with p j T · p j T . By contrast, the background cross section is much steeper in q p W T · p W T than p j T · p j T . This is not unexpected, as we recall the expression q p W T · p W T directly correlates with the center of mass W W energy, M W W . The region of phase spacewhere all four transverse momenta are relatively low to moderate, typically p WT ∼ p jT ∼ p W T · p W T / ( p j T · p j T ) = const (4.2)to a fair accuracy corresponds to a line of constant signal to background ratio (S/B).As already mentioned when discussing hadronic decays, decays of energetic W ’s arehighly boosted in the lab and decay products are emitted in a nearly collinear way. Ourpractical measure of M W W in an experiment is then the product of the two transversemomenta of the visible charged leptons. Asymptotically for high energies, the latter isjust a numerically scaled down (by a factor of 4) version of the former. We have hencearrived in a heuristic way to the definition of an experimental dimensionless variable R p T = p l T · p l T p j T · p j T (4.3)whose fixed value indeed respresents a constant S/B to a good enough accuracy.Correspondence between R p T and the typical VBF signature is straightforward. Werecall that the latter includes two central back-to-back leptons (in case of leptonic decays)with high transverse momenta. Here however the specific cut value for these transversemomenta is now scaled with the values of transverse momenta of the jets, in a waythat is based on background rejection grounds. Physically this can be said equivalent toadding the requirement of high M W W and longitudinal polarizations. By contrast, theconventional VBF selection criteria are polarization-blind. This unique combination ofthe four transverse momenta is in fact more effective in separating signal from backgroundthan a combination of selection criteria imposed separately on the individual transversemomenta, because they scale with each other.All the above considerations are equally true for W − W − as for W + W + , althoughquantitative details differ due to the presence of two valence u quarks inside a proton. .6. THE UNIQUENESS OF W ± W ± pp → jjµ + µ + at 14 TeV after applying basic topological VBF cuts only. Left: thedifferential cross sections of background and signal in a two-dimensional space defined bythe square roots of the transverse momentum products of the two leading jets and the twooutgoing muons, q p j T p j T versus q p µ T p µ T ; the color contours are equidistant and the scaleranges from zero (white) to 0.48 · − fb/GeV for the background and to 0.7 · − fb/GeV for the signal (purple). Upper right: the signal to background ratio from dividing thetwo left plots; the vertical scale is logarithmic for better visualization and ranges from0.04 (white) to 40 (purple).
Lower right: the distributions of the ratio p µ T p µ T / ( p j T p j T )for signal and background. Results of a MadGraph simulation, processed by PYTHIA6 [126] for the effects of parton showering, hadronization and jet reconstruction, andfurther processed by PGS 4 for the effects of finite resolution in the measurement of jetand muon p T in a CMS-like detector. Signal was calculated by subtracting the SM-baseddistributions from the Higgsless-based distributions.4 CHAPTER 4.
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Figure 4.16: The kinematics of longitudinal signal and transverse background in the fullprocess pp → jjµ + µ − at 14 TeV after applying basic topological VBF cuts only. Left: the differential cross sections of background and signal in a two-dimensional space definedby the square roots of the transverse momentum products of the two leading jets andthe two outgoing muons, q p j T p j T versus q p µ + T p µ − T ; the color contours are equidistantand the scale ranges from zero (white) to 0.0011 fb/GeV for the background and to0.12 · − fb/GeV for the signal (purple). Upper right: the signal to background ratiofrom dividing the two left plots; the vertical scale is logarithmic for better visualizationand ranges from 0.0025 (white) to 2.5 (purple).
Lower right: the distributions of theratio p µ + T p µ − T / ( p j T p j T ) for signal and background. Results of a MadGraph simulation of pp → jjW + W − , processed by PYTHIA 6 for W decay into muons, the effects of partonshowering, hadronization and jet reconstruction, and further processed by PGS 4 for theeffects of finite resolution in the measurement of jet and muon p T in a CMS-like detector.The original PYTHIA 6 source code was modified to account for the correct, polarization-dependent, angular distributions for the decays W ± → µ ± ν . Signal was calculated byconsidering longitudinal W + W − pairs only and subtracting the SM-based distributionsfrom the Higgsless-based distributions. .6. THE UNIQUENESS OF W ± W ± pp → jjW + Z → jjµ + µ + µ − at 14 TeV after applying basic topological VBF cutsonly. Left: the differential cross sections of background and signal in a two-dimensionalspace defined by the square roots of the transverse momentum products, q p j T p j T versus q p µ + T p ZT where p ZT stands for the total transverse momentum of the two opposite-signmuons that reproduce the best Z mass. The color contours are equidistant and the scaleranges from zero (white) to 2 · − fb/GeV for the background and to 3 · − fb/GeV for the signal (purple). Upper right: the signal to background ratio from dividing thetwo left plots; the vertical scale is logarithmic for better visualization and ranges from0.004 (white) to 4 (purple).
Lower right: the distributions of the ratio p µ + T p ZT / ( p j T p j T )for signal and background. Results of a MadGraph simulation, processed by PYTHIA6 for parton showering, hadronization and jet reconstruction, and further processed byPGS 4 for the effects of finite resolution in the measurement of jet and muon p T in aCMS-like detector. Signal was calculated by subtracting the SM-based distributions fromthe Higgsless-based distributions.6 CHAPTER 4.
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Figure 4.18: The kinematics of longitudinal signal and transverse background in the fullprocess pp → jjZZ → jjµ + µ − µ + µ − at 14 TeV after applying basic topological VBF cutsonly. Left: the differential cross sections of background and signal in a two-dimensionalspace defined by the square roots of the transverse momentum products of the two leadingjets and the two Z bosons, q p j T p j T versus q p Z T p Z T where the transverse momenta of the Z bosons were reconstructed from pairs of opposite-sign muons reproducing the best Z masses. The color contours are equidistant and the scale ranges from zero (white)to 3 · − fb/GeV for the background and to 4 · − fb/GeV for the signal (purple). Upper right: the signal to background ratio from dividing the two left plots; the verticalscale is logarithmic for better visualization and ranges from 0.0008 (white) to 0.8 (purple).
Lower right: the distributions of the ratio p Z T p Z T / ( p j T p j T ) for signal and background.Results of a MadGraph simulation of pp → jjZZ , processed by PYTHIA 6 for Z decayinto muons, the effects of parton showering, hadronization and jet reconstruction, andfurther processed by PGS 4 for the effects of finite resolution in the measurement of jetand muon p T in a CMS-like detector. Signal was calculated by considering longitudinal ZZ pairs only. This study did not include the correct, polarization-dependent, angulardistributions for the decays Z → µ + µ − . Such effects cannot nonetheless change any ofour conclusions. .7. REDUCIBLE BACKGROUNDS AND SELECTED EXPERIMENTAL ISSUES W − W − is approximately one fourth of that of W + W + .By the same arguments it should be clear that R p T is a specific variable suited for thestudy of same-sign W W scattering, but not of other VBS processes, W + W − in particular.For a comparative study of R p T usefulness in different VBS processes, see Figs. 4.14 thru4.18. The meaning of R p T is not selection of a hard scattering process any more thanconventional VBF selections are. Rather, it is rejection of background of a specific type:the one related to gauge boson emissions off two colliding quarks in which at least one ofthe bosons is transversely polarized. The uniqueness of same-sign W W is that it is theonly process in which this type of background can be made its main component. Thisobservation is very important. For the purely leptonic decay modes, the whole signalsize, defined in terms of W L W L pairs and the unitarity limit is of order of 0.3 fb. Thismeans in any realistic scenario a low number of signal events to begin with. Feasibilityof signal detection, assuming luminosities measured in hundreds of inverse femtobarns, isthus mainly determined by the background rejection potential. By reducible background is meant all contributions that can mimick the signal in a realexperiment, but physically come from a different collection of particles in the final state.In other words, in an ideal detector the reducible background could be zero, but is notbecause of finite detector performance and event reconstruction capabilities. Followingthe standard background process classification used e.g. in CMS, the most importantpotentially dangerous reducible background sources in the study of
W W scattering athigh energies, for the purely leptonic decay modes, are: inclusive t ¯ t production, W +jetswith a jet misidentified as a lepton and QCD multijet events with two jets misidentified asleptons. For the purely leptonic decay modes, the key detector features that determine themagnitude of the reducible background are the purities of lepton reconstruction, includingthe charge, and to a lesser degree the efficiency of b quark tagging.Figure 4.19: Two lowest order graphs for the inclusive t ¯ t background.It must be stressed immediately at this point that we are considering the backgrounds8 CHAPTER 4.
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SCATTERING AT THE LHC that can be significant at 13/14 TeV after imposing all the discussed selecion criteria. Itis an obvious fact that use of looser criteria, e.g., as a preselection for a multivariate typeof analysis (MVA) will translate into additional and differently composed background toconsider.Top pair production at the LHC overwhelms the
W W scattering signal by severalorders of magnitude. We have already noted that the same-sign
W W scattering mode isadvantageous here in that a t ¯ t pair produces in principle always an opposite-sign gaugeboson pair, along with two b -jets. However, the initial t ¯ t production cross section is somuch larger than our potential signal that tiny effects associated to leptonic b decays ora charge mismeasurement of the lepton arising from W decay, can lead to measurableeffects which cannot be disregarded. As each top quark decays into a W and a bottomquark, it is clear that b -tagging efficiency plays an important role. A typical b -taggingalgorithm in a collider experiment is based on the most characteristic feature of B mesons,namely their short lifetimes, identifiable in the detector as subsequent decays occurringfrom a vertex which is displaced somewhat from the proton-proton interaction point.Many b -tagging algorithms have been developed and their performance studied in CMS,the most commonly accepted being the Combined Secondary Vertex (CSV) algorithm[80]. The algorithm relies on the reconstruction of secondary vertices together with thetrack-based lifetime information in a jet. For each track a 3-dimensional impact parameteris computed from its minimum distance of approach to the vertex, then tracks in a vertexare ranked based on a significance number equal to the value of the impact parameterexpressed in units of its uncertainty. Likelihood discriminants to identify the jet as a b -quark are based on the significance of usually the second-ranked (“High Efficiency”) orsometimes the third-ranked (“High Purity”) tracks. The threshold value is, as always,arbitary and allows to choose an optimum working point for each analysis based onthe general performance curve that correlates the tagging efficiency with tagging purity,the latter determined in terms of the efficiency for tagging a u − , d − , s − , c − or gluonjet. Since tightening the tagging criterion quickly leads to an avalanche increase of lightquark mistagging, the final b -tagging efficiency is determined mainly by the maximumacceptable tagging impurity. From the CSV performance curves we learn that signallosses can be kept up to or below 2% overall, while 50% of genuine b quarks get tagged.For a t ¯ t event, with two b quarks in it, this means a reduction factor of 0.25. Useful,but far insufficient to keep the t ¯ t background to manageable levels. Alternatively, a 0.10reduction factor can be obtained by allowing of a 10% loss of the signal. The fact that b -tagging efficiencies decrease in the forward/regions regions is not very disturbing becausetag jets usually do not originate from b quarks, as we will see further on. Because ofsteeply increasing impurity rates, further adjustments of these numbers leave rather littleroom for improvement.The bulk of the t ¯ t background must be in any case eliminated kinematicwise. Thetop quark mass defines a natural upper bound for the invariant mass of its visible decayproducts, in our case the jet and the lepton. There is of course an ambiguity here relatedto correlating the proper jet with the proper lepton, overall kinematic constraints howeverfavor a configuration in which relative ranks (defined by the respective p T values) of thejets and the lepton anticorrelate. Which is to say, more often than not, the largest- p T lepton with the second- p T jet and the largest- p T jet with the second- p T lepton reproduce .7. REDUCIBLE BACKGROUNDS AND SELECTED EXPERIMENTAL ISSUES t ¯ t background in the pp → jjW + W + process at 14 TeV, after applying basic topologicalVBF cuts. Results of a MadGraph calculation, processed by PYTHIA 6 for W decay intomuons, the effects of parton showering, hadronization and jet reconstruction. The originalPYTHIA 6 source code was modified to account for the correct, polarization-dependent,angular distributions for the decays W ± → µ ± ν .0 CHAPTER 4.
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SCATTERING AT THE LHC the top mass constraint. Furthermore, as in a typical t ¯ t production event, the two b -jetsdo not undergo any hard interactions, the two-jet invariant mass strongly prefers muchlower values than those typical of W W scattering. It was determined from simulationthat the combination of simple cuts: M l j >
200 GeV, M l j >
200 GeV, M jj >
500 GeV,together with b -tagging already reduces the t ¯ t background to manageable levels as long asit is only driven via effects like charge mismeasurement or leptonic B decays. Commonlyused for reduction of the t ¯ t background is the additional requirement of central jet veto.Its usual form is removal of events with any additional jets of p T larger than a predefinedthreshold and anywhere between the two tagging jets in pseudorapidity. Because W W scattering is a pure electroweak process, little jet activity in between the two tagging jetsis expected in signal. Note however that a certain form of the central jet veto is alreadyapplied via the requirement of two tagging jets. By construction we require them here tobe the two leading jets in the event, hence “central jet veto” proper in practice means anadditional cut only if at least one of the tagging jets has p T below the chosen threshold.Such cut provides another factor ∼ t ¯ t background rejection.Detector efficiencies in terms of lepton charge determination, especially at large trans-verse momenta ( p T ∼
300 GeV), are relatively poorly studied. This is because beamenergies of 7 or 8 TeV do not provide much data in this region and most mainstreamphysics analyses are very little sensitive to such effects anyway. Finally, because evalua-tion of potential backgrounds related to charge misidentification is in practice done usingvarious partly or wholy data-driven methods that do not require explicit knowledge ofthe misidentification probability per se. But the results from 7/8 TeV cannot be directlyapplied to 13/14 TeV because the relevant p T and η distributions differ. Correct chargereconstruction is generally easier in the central barrel region of the detector, which isadvantageous for us. A simulation-based study done in CMS, in which the rate of muonswith the reconstructed charge not equal to the generated charge was measured, revealedcharge misreconstruction be at the level of 10 − for muon p T up to 100 GeV and slowlyrising above [81]. An earlier study of cosmic muons passing through the whole detectorin which muon charges were reconstructed separately in the top and bottom halves anddisagreed, revealed this disagreement be already at the 0.5% level for p T of the order of300 GeV [82]. The two results do not disagree badly and taking into account the im-provements in muon reconstruction between the times of the two analyses, we can safelyassume a 99.7% muon sign matching probability as fully realistic in our kinematic rangeof main interest.The charge misreconstruction for electrons is known both from simulations and fromdata using a Tag-and-Probe technique to be well below 1% in the barrel region ( | η | < p T and gradually increasing with p T [84] [85]. The use of independentmethods to estimate the charge from a combination of various data from the centraltracker detector and the electromagnetic calorimeter was shown to significantly reduce theinefficiencies of the standard Gaussian-Sum Filter (GSF) track curvature method. Theefficiency still degrades somewhat with p T , but 99% gives the right order of magnitudeof what can be readily achieved as far as electron sign matching is concerned. ATLAS .7. REDUCIBLE BACKGROUNDS AND SELECTED EXPERIMENTAL ISSUES p T >
50 GeV.The jet → electron or photon → electron misidentification rates, commonly called“fake rates”, are usually determined using data driven methods. For an analysis of agiven final state, a control sample is defined from data which differs from the signal datain that the quality criteria used to formally define an object of a certain kind, say, anelectron or a muon, were loosened. The control sample is known to be composed mainlyof “fakes”. The fake rate in the signal region is calculated by scaling the measured back-ground distributions with the measured probabilities of each “fake” to pass the nominalquality criteria, usually as a function of its p T and | η | . Results of such methods are di-rectly applicable only in the context of their specific analyses. On the other hand, genericbut simulation-based studies exist in which misidentification probabilities were measuredrelative to any random jet of a given p T and | η | . The results naturally depend on thedetails of the electron selection criteria applied. In CMS, simulation work has shown [86]that a combination of stringent electron identification criteria based on: • track, electromagnetic and hadronic isolation, each defined as the p T / E T sum ofall tracks/clusters lying within a ∆ R =0.5 cone around the reconstructed electron,relative to the E T of the electron, • the geometrical matching of the track with the cluster in both pseudorapidity andthe azimuthal angle, • the ratio of the electromagnetic energy deposit to the electron momentum, E/p , oralternatively, | /E − /p | , • the electromagnetic cluster shape described in terms of the ratio of energy depositswithin a 3 × × . ± . · − on average and somewhat increasing with E T , as far as can be judgedfrom a rather low statistics. Moreover, due to the mechanisms of W +jets production atthe LHC, in these events only 27% of those “fake electrons” have the same sign as the W .This translates into W +jets background rates being nearly 3 times lower in W ± W ± thanin W + W − . The same study suggests that the electromagnetic cluster shape described interms of the pseudorapidity spread of the shower in 5 × E T >
50 GeV and larger than 90% for E T >
100 GeV, which is of main interestfor us. Given the large total W +jets and QCD multijet events cross sections, keepingthe fake rates low is imperative for the electron decay channels and so even allowing aslight decrease in the reconstruction efficiency is the better choice. Similarly, fake ratesof photons misreconstructed as electrons were determined from simulation to be (0.7 ± CHAPTER 4.
V V
SCATTERING AT THE LHC
It is worth noting that all the abovementioned variables, and a few additional ones, areused as discriminators in the standard electron-ID used in CMS analyses. Based on earlysimulation studies, ATLAS reported jet-to-electron fake rates of the order of 2 · − for jet p T >
100 GeV, with the Boosted Decision Tree techniques used for electron identificationand isolation, while keeping high electron-ID efficiency in this kinematic region [87]. Sincethis study was based on a simulated dijet sample, it is not possible to derive the chargecorrelation factor. Another study [88] reports on the possibility of a further reductiondown to the level of ∼ − at the expense of electron-ID efficiency decreasing to 67%.Unfortunately both numbers are p T -averaged.In view of everything above, we can tentatively assume for further considerations anaverage electron fake rate in our kinematic domain of ∼ − , times the appropriatesign factor, with a 90% electron-ID efficiency and a 99% charge reconstruction efficiency.However, one cannot completely trust Monte Carlo programs to study effects related, e.g.,to jet fragmentation at the LHC. Only real data in the appropriate kinematic domain willultimately determine the impurities. Further improvements in electron purity and signmatching, keeping a reasonably high overall reconstruction and identification efficiency,must be particularly encouraged and followed with a special attention, since they canbe the key for success in including the electron-electron and mixed muon-electron decaychannels to the W W scattering search at 13 TeV and can prove vital for the observationof signal.Fake rates of hadrons misreconstructed as muons in principle include two distincteffects. The first of them are punch-thru pions which reach the muon chambers. Theyare usually associated with hadronic activity around the fake muon track. The secondclass are real muons from pions or kaons decaying in the detector, often referred to as“non-prompt” muons. These are recognizable by a characteristic kink in the track, visibleat the point of the pion or kaon decay. By choice of appropriate isolation criteria botheffects can be suppressed to a negligible level. Measurements and simulations done withinATLAS [89] reveal a total fake rate for jets of less than ∼ − and dropping with jet p T , and about 10 − for single tracks. The latter however may be contaminated with real“prompt” muons from W and Z decays. In our further considerations we will disregardthese backgrounds.One last experimental issue that is definitely worth to mention at the present moment,particularly in the context of the planned future upgrades of LHC detectors, concernsjet reconstruction efficiency at large pseudorapidity. In Standard Model VBF processesat 14 TeV, pseudorapidity distributions for tagging jets peak between 2-3. This holdsapproximately equally true for V T V X as for V L V L pairs. This however does not implythat this region is of most interest for BSM search. Contrary, the best sensitivity toBSM effects is likely to be more forward. E.g., non-SM Higgs couplings will reflect ina wide pseudorapidity range for the tagging jets, going even all the way up to 5 for thesubleading jet. It is therefore important to have a good jet reconstruction in the entirepseudorapidity range and keep high performance in the most forward region for the wholeHigh Luminosity LHC program. .7. REDUCIBLE BACKGROUNDS AND SELECTED EXPERIMENTAL ISSUES pp → jjW + W + process at 14 TeV, with leptonic W + decay, afterapplying basic topological VBF cuts, namely ∆ η jj > | η l | < .
1. Shown are theSM spectra for W L W L and W T W X pairs and the W L W L spectra for the Higgsless signal.Results of a MadGraph calculation, processed by PYTHIA 6 for W decay into muons, theeffects of parton showering, hadronization and jet reconstruction. The original PYTHIA6 source code was modified to account for the correct, polarization-dependent, angulardistributions for the decays W ± → µ ± ν .4 CHAPTER 4.
V V
SCATTERING AT THE LHC hapter 5Simulation-based studiesvs. experimental results
The processes of
V V scattering have been lying in the interest of physicists for almost aslong as the Standard Model itself. Despite of there being many simulation-based analysesof
V V scattering at the LHC with 14 TeV, both at a phenomenological level or involvingelements of a full experiment-specific detector simulation, a vast majority of them needcritical revisiting in accordance to recent developments in our experimental knowledgeand in the available simulation tools.To begin with, in most older studies it was Higgs boson existence that was consideredthe biggest unknown of the model. Consequently, signal was calculated either in terms ofa pure Higgsless Standard Model, or a Higgsless Standard Model where only the unitarityof scattering amplitudes was enforced by hand, or a Standard Model with a very heavyHiggs, or finally within the framework of a particular alternative model of electroweaksymmetry breaking. Physicswise all these scenarios are now obsolete. This however doesnot imply that older studies should be sent to oblivion. It is rather straightforward toreinterpret the results of the former three classes of works in terms of a 125 GeV Higgswith different
HW W or HZZ couplings. For that the relevant signal figures should toa good approximation only be scaled down by calculable, coupling-dependent factors.In a similar manner it has been shown that in the regime of the LHC at 14 TeV, thesole unitarity bound produces a 20-25% reduction of the signal figures compared to apure Higgsless Standard Model. Moreover, in a non-resonant process like W ± W ± thisapproach is in fact approximately equivalent to assuming a heavy Higgs (with M H ≈ W ± W ± mass spectrum in a non-trivial way. More problematicis only reinterpretation of results that were obtained by assuming particular alternativemodels of electroweak symmetry breaking, but even there some of the analysis methodsthat have been worked out may remain useful today.Equally problematic are the old estimates of reducible backgrounds. In phenomeno-logical studies such backgrounds are, more often than not, treated either qualitatively,e.g., by suggesting certain cuts to suppress them, or considered only partially. Optimisti-cally, its supposedly largest component was studied in more detail (typically, inclusive t ¯ t production).Results based of full detector simulations for a specific experiment often differed widely856 CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS from results of purely phenomenological analyses, especially in the semileptonic decaychannels. The former have been evaluated using detector-specific simulation tools avail-able at the time of their publication, which is, in the early stages of software developmentfor ATLAS or CMS, to focus on these two. It is obvious that these tools have sinceimproved paramountly. But the improvements are usually difficult to quantify withoutredoing the whole simulation. Unfortunately this means that these older studies thatinvolved full detector simulation usually do not represent a valid reference to assess thebest current experimental sensitivity in the search for physics beyond the Standard Modelin
W W scattering.First, however, let us briefly recall and review the leading past works in the subject,focusing not that much on their numerical results, but rather with a special emphasis onwhat things of all those older studies remain completely valid today.
Physicists’ interest in
V V scattering clearly predates the LHC. Already in the early papersof Chanowitz et al. [92] it was noticed that scattering of same-sign longitudinally polar-ized W ’s is the most sensitive probe of effects related to the mechanism of electroweaksymmetry breaking. Several other authors, including Barger et al. [93] and independentlyDicus et al. [94] studied in detail the process of W + W + scattering in the context of theplanned Superconducting Super Collider (SSC). Their numerical results do not have adirect importance for us, but some of their qualitative observations are strikingly up todate. Among other things, they proposed kinematic cuts to keep the t ¯ t background undercontrol and stressed the importance of jet transverse momenta in the separation of thelongitudinal W W signal from the transverse
W W background. In particular, to tacklethe latter, cuts on the maximum allowed jet p T were discussed. Early studies exist alsofor electron-positron colliders [95].Systematic studies of the W W scattering phenomenology in the particular contextof the LHC started later in the 1990’s. Their main physical focus was observation ofsignal related to different scenarios of electroweak symmetry breaking on the assumedabsence of a Higgs boson. Despite their main underlying physics assumptions are nowimplausible, a large amount of knowledge is still contained in these studies and a lot ofthis knowledge remains valid in the context, e.g., of searches for new heavy resonancesor other experimental signatures in the absence of such resonances. To a large extent wecan still follow the general guidelines presented in those papers.The early paper of Barger et al. [93] provided the justification on theoretical groundsof some basic signal selection criteria in the context of heavy Higgs searches at the LHC.It is here that introduced and justified on theoretical grounds was the idea of a centraljet veto as a primary criterion to distinguish QCD-related backgrounds from the purelyelectroweak signal.Especially enlightening from the phenomenological point of view, although again fo-cused on various Higgsless scenarios, are the works of Bagger et al. [96]. They developedthe general methodology, introduced the “subtraction method” for the mathematical defi-nition of the signal and recapitulated on the basic experimental signatures. They proposedoriginal sets of kinematic cuts optimized for all the individual VBS processes separately .1. EARLY CALCULATIONS W and Z decay modes in their analyses andtermed them as “gold-plated”. Let us recall some of their main conclusions that are stillvalid today. From the different analyses by Bagger et al. it follows that depending on theactual physics scenario, any of the different scattering processes: W + W − , W ± W ± , W ± Z or ZZ , may turn out to be the most promising one, or even a combination of all of themcould be required. Models which predict heavy scalar resonances were found most easyto study in the ZZ and W + W − processes (in agreement with everything we have said sofar), heavy vector resonances should show up more efficiently in the W Z process, whilevery heavy resonances or scenarios with no such resonances at all would manifest only asan increase of the total event yield at large invariant mass and this increase is the mostpronounced in no else than W ± W ± . The significance of the ZZ channel is driven mainlyby the l + l − νν final state rather than the cleaner, but lower rate 4 l final state. The formeris nonetheless contaminated by a detector dependent background coming from Z + QCDjets events, which have not been explicitly taken into account in this analysis, exceptfrom assuming it be suppressable by applying a cut on missing transverse energy (MET).The significance of the W + W − process in its turn crucially depends on the efficiency ofsuppressing the overwhelming inclusive t ¯ t background using such techniques as b -taggingand central jet vetoing. In these works, the respective signals were calculated using theEffective W Approximation and the Equivalence Theorem, as well as assuming particu-lar scenarios of electroweak symmetry breaking, alternative to the Higgs model. All theanalyses were carried at a purely partonic level. Quantitative estimates of the requiredluminosity to observe a non-SM signal in the different channels vary from below 100 to250/fb for the LHC running at 14 TeV, but because of the approximative character ofthe relevant calculations they should be taken with care. All the signal scenarios involvestrong
W W scattering, in which they resemble a Higgsless Standard Model with rein-forced unitarity. Even though the authors perform an essentially counting experiment,with an analysis which is not optimized for resonance search (in most cases the resonanceis very broad anyway), differences of more than a factor 3 in the required luminositiesgive a rough idea of the degree of model dependence of the signal significances and henceof all their quoted results. The process ZZ → l is perhaps the most interesting both be-cause of its low background and because it offers the best event reconstruction and hencefull determination of the nature of the heavy resonance, but it also requires the largestluminosity for observation. It was estimated to be around 300/fb of LHC running at 14TeV to observe a 99% CL signal, which corresponds to roughly 4 σ . The sensitivity of the W ± Z process to non-SM physics was shown rather marginal. The authors suggest in factfocusing on the Drell-Yan process to enhance the significance of W ± Z in the search forheavy vector resonances. Meanwhile, the W ± W ± process fares poorly in scenarios withheavy scalars, but somewhat surprisingly turns to be the most efficient in scenarios withheavy vectors, in addition to non-resonant ones. Typical luminosities required to observesignal at a 99% CL oscillate roughly around 200/fb. Of other interesting observationsthat are worth recalling, the authors stress that signal is contained mainly at relativelylow p T of the tagging jets, typically p T ∼ M W /
2. Large jet p T thresholds usually appliedin various analyses of LHC data because of pile-up related background would thereforetranslate into low signal detection efficiency. As a possible alternative, single jet-tagging8 CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS was proposed, which of course would come at the expense of background rejection effi-ciency. However, good signal efficiency could be achieved with double-jet tagging if onlythe jet p T threshold could be lowered to 15 GeV.Unfortunately, the studies by Bagger et al. came too much ahead of their time andmany of their important conclusions, perhaps because of the obsolete by now computa-tional techniques they applied, got largely forgotten before the LHC started operation. Itis time now to rediscover the findings of this work and reevaluate them with modern andfully up to date simulation tools.Other classic works include the ones by Dobado et al. [97], focused specifically on ZZ and W Z production. Their studies were carried within the Electroweak Chiral Lagrangianapproach. They mainly elaborated on a unitarization technique based on the Inverse Am-plitude Method, in which new dynamic resonances appear in
V V scattering and enforceunitarization. They also applied the Equivalence Theorem and used the Effective W Ap-proximation in their calculations of the relevant VBS processes. A tentative analysis waspresented at the level of undecayed gauge bosons, by further assuming a 100% efficiencyin their reconstruction, and moreover only irreducible backgrounds were taken into ac-count. Since both their signal and background treatments are highly approximative, theirnumerical results cannot be considered but purely qualitative. In what’s important forus, however, they do confirm the importance of Drell-Yan production rather than VBSfor the
W Z process.Chanowitz et al. [98] in a series of follow-up papers focused on W + W + and W Z processes. The authors observed a complementarity of the W + W + and W Z processesas a function of the mass scale of the hypothetical new, heavy vector resonances. Thecombination of the two was shown to guarantee the “no-lose theorem”, meaning that signalwould be always observable one way or another, i.e., in at least one of the two processes.For the calculation of the
W Z signals they considered Drell-Yan as well as VBS. Here toocalculations were based on the Chiral Lagrangian Model and, as in the previous analyses,the Equivalence Theorem and the Effective W Approximation were used to evaluate VBSprocesses. Only irreducible backgrounds were explicitly considered. However, their workwas the first to mention the potential importance of detector dependent backgroundsrelated to lepton sign mismeasurement. They also followed up on the issue of separatingthe final state polarizations, but focused on purely leptonic cuts for this purpose. Thereason was simple: specific of their analysis was the treatment of VBS and Drell-Yantogether. They omit some typical VBS cuts, like forward jet tagging, which would killtheir Drell-Yan signal. Under these conditions, they finally found an LHC luminosity of140/fb guarantee the “no-lose” condition with a significance of at least 3 σ , which perphasdoes not have a direct meaning for us.In a ground breaking paper, Butterworth et al. [99] found that semi-leptonic decaymodes could be as promising as purely leptonic. They considered only the W + W − scat-tering process (note however than in semi-leptonic decays one of the W ’s has no measuredcharge, so in reality a sum of W + W − and W + W + is automatically implied) and calcu-lated the signal in several models within the Electroweak Chiral Lagrangian approach,that corresponded to the existence of heavy scalar or heavy vector resonances, as well asno resonances at all. The calculation was done using a modified version of the PYTHIAgenerator which indirectly involves the Effective W Approximation. Background eval-uation included t ¯ t production and radiative W +jets events, calculated likewise within .2. RECENT WORKS AND POST-HIGGS DISCOVERY DEVELOPMENTS W bosons decaying hadronically, the real performanceof this procedure may depend on additional effects, e.g., detector resolution, not studiedin this analysis. Using a newer PYTHIA version with improved parton showering and adedicated event reconstruction software used by CMS at the time, it was found [100] thatthese predictions were way too optimistic. It also indicated that a lot of work was stillrequired on the detector and reconstruction side.Some of the many other studies of the phenomenology of V V scattering before Higgsdiscovery are listed under Ref. [101].A lot of early simulation work, that in addition included simulated detector responseand event reconstruction, was done within the ATLAS collaboration [102]. In these stud-ies, signals were calculated using PYTHIA and the various backgrounds using such gen-erators as MadGraph and MC@NLO. The main focus was
W Z in different semi-leptonicand purely leptonic decay modes and
W W in the semi-leptonic decay mode in whichcase the signal and backgrounds were evaluated together with
W Z , where W → lν and Z → jj (in a real experiment, processes tend to be naturally grouped by final state asseen in the detector). They also produced a result for ZZ → l + l − νν , but only in thescenario of a Higgs-like resonance with a mass of 500 GeV. Unfortunately, no results frompurely leptonic decays of W W have been shown and no W ± W ± in particular. Their anal-yses included standard VBS selection criteria, not specifically optimized for gauge bosonpolarization. However, their p T threshold for the identification of tag jets vary between10 and 20 GeV, ensuring reasonably high acceptance for longitudinal bosons. The studyfocused mainly on Higgsless, resonant scenarios, many of the considered resonances wererelatively light and so the results must be regarded as out of date today. Interestingly,the only non-resonant scenario considered in this study (there is however no informationabout the exact parameter values used within PYTHIA to simulate this kind of signal)did not lead to promising results and was not even included in the table of results thatconcluded the study. A new generation of
V V scattering studies commenced with the introduction of universal,commonly accessible physics calculation tools, like MadGraph, CompHEP, ALPGEN,PHASE/PHANTOM or VBFNLO, which calculate the full matrix elements for a givenprocess. They replaced PYTHIA-based and other signal calculations done only in theEffective W Approximation. At the same time, since those generators often did nothave any alternative models of electroweak symmetry breaking explicitly implemented,signal calculations necessarily required a more generic, model-indpendent approach. Suchapproach was in pratice provided by considering a pure Higgsless Standard Model, or a0
CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS
Higgsless Standard Model with effective unitarization. This could be implemented eitheras a sharp cutoff or else assuming that the relevant scattering amplitudes saturate justbefore reaching the unitarity limit. Incidentally, the latter is phenomenologically similarto non-resonant Higgsless models. Therefore many of these newer studies are directlyrelevant for the case of a light Higgs boson with modidied couplings and no new resonanceswithin the mass range of the LHC. They require in principle only a scale factor for aneffective translation into a physically valid and up to date scenario.One of the earliest
W W scattering analyses that did not make use of approximativecomputation techniques was the work by Eboli et al. [112], in which purely leptonicdecays were studied in the
W W process and in all charge combinations. They find afull calculation of the scattering amplitudes necessary not only for a correct cross sectionevaluation, but also to describe accurately all correlations between final state particles. Anotable feauture of the presented analysis was the most complete available treatment ofinclusive t ¯ t background, it included contributions from processes with up to two associatedQCD jets computed at the matrix element level. The inadequacy of considering only pure t ¯ t production in the lowest order was shown. Conceptually the work was focused ona study of anomalous quartic vector boson couplings in which two exclusive workinghypotheses were considered in what regards Higgs existence. For a discussion of the mainconcept and of the obtained results, we will still come back to this work in the nextsection.Of the newer analyses at the phenomenological level, the works by Ballestrero etal. [103] clearly stand out and they also effectively triggered a lot of further, detectorlevel work within the CMS collaboration. In a series of papers they studied both thesemi-leptonic and purely leptonic decay channels. Calculations were done with the newlycreated PHANTOM program [127] which computes complete tree level matrix elementamplitudes for 2 → O ( α ), ( α α S ) and ( α α S ), wher-ever appropriate. All the analyses were carried in a manner which closely resemblesrealistic experimental analyses. Scattering processes were grouped by final state. Signalwas defined in terms of a VBF-like kinematics in the purely electroweak process wherethe final event yields were compared in the Higgsless and light Higgs cases. The fact thatthey typically assume M H = 200 GeV for the Standard Model case is a rather minor issue.Additionally, processes O ( α α S ) and ( α α S ) accounted for all the extra, non-scattering,background. In principle, the scope of background processes that can be taken into ac-count in this way includes the most basic t ¯ t production process without additional quarksor gluons. But as we already mentioned as will further see in the next chapter, such treat-ment is insufficient for an accurate account of inclusive t ¯ t background for VBF processes,since the bulk of events that can survive VBF cuts comes in fact from higher order dia-grams. The applied selection criteria selected general VBF events and were not optimizedfor longitudinal W/Z polarization. Moreover, a high p T threshold for tag jets was used,as had become already routine e.g. in Higgs searches. All the analyses were carried atthe partonic level and background treatment, as mentioned above, in practice includedonly irreducible backgrounds. The final sensitivity was evaluated from event counting orfrom a shape analysis of the invariant mass spectrum of the visibile gauge boson decayproducts. Not surprisingly, the most interesting results came from the purely leptonicstates. Again here the same-sign dilepton channel ( W ± W ± ) was shown to provide thebest discrimination between different scenarios, closely followed by ZZ → l + l − νν and .2. RECENT WORKS AND POST-HIGGS DISCOVERY DEVELOPMENTS W + W − . Obviously, quantitative comparisons are subject to further change once all re-ducible backgrounds are properly included. The analysis of semi-leptonic decays can onlybe treated as a demonstration because the jet merging issue which affects the decays ofhighly boosted gauge bosons in a real detector was not addressed in this study and becausefinal state radiation, leading to additional jet combinatorics, was not simulated. Perhapsthe most interesting result of Ballestrero et al. from our (biased) perspective resides inthat their works were among the first ones to explicitly consider the Strongly InteractingLight Higgs models as an alternative to either Higgsless or the Standard Model. Theirresults suggested a decrease of the signal size (redefined here for our purposes as theenhancement with respect to the Standard Model) by a factor 3-4 when compared to apure Higgsless scenario. Since effectively the only relevant feature of the considered SILHscenario was a modification of the HW W and
HZZ couplings by a factor numericallyclose to 0.7, compared to the SM, their result presented in this way can be treated asmodel independent. It is in fact a particular example of a scale factor that is necessaryto apply to all the former Higgsless-based studies to render them fully realistic.The CMS collaboration produced a full set of results, corresponding to the manydifferent final states, obtained using the general prescription of Ballestrero et al., withthe addition of the dedicated CMS event reconstruction software [104]. The final statesthat were considered corresponded both to semi-leptonic and purely leptonic decay modesof W + W − , W ± Z , ZZ and W ± W ± . The results were admittedly not very encouraging.However, e.g., the analysis of the same-sign channel was clearly suboptimal. Moreover, asalready stressed, the work was done at the time of rapidly changing CMS reconstructionand analysis tools and cannot be taken as the final word. Very derisable would be to havethese data reanalyzed with the most recent versions of the CMS software and using themost efficient selection criteria for a complete and up to date evaluation.In another analysis done at the phenomenological level, Zeppenfeld et al. [105] studiedleptonic decays of the W + W − , ZZ and W ± Z scattering pairs. They calculated signal andbackgrounds using the VBFNLO generator program, where signal was defined in terms ofa 1 TeV Standard Model Higgs or alternatively via a Warped Higgsless model with heavyvector resonances. Their most important conceptual innovation from today’s perspectivewas that their background treament included realistic modelling of t ¯ t +jets in additionto irreducible backgrounds. For the former they developed an original simulation-basedapproach which is suitable for VBF analyses. Initial and final state radiation processeswere simulated and double counting was avoided by defining mutually exclusive topologi-cal requirements for processes with 0, 1 and 2 QCD jets generated at the matrix elementlevel. Consequently, they found t ¯ t +jets the most important remaining background inthe W + W − channel, in contrast to what was assumed in many other studies. We willreview their method in further detail in the next chapter. A toy jet reconstruction byrecombination of the final state partons was also applied. Quite consistently with mostprevious studies, they found W ± Z the preferred channel for the vector resonance scenarioand W + W − closely followed by ZZ → l + l − νν for the heavy Higgs scenario. They finallyfound a very high signal significance after collecting 300/fb of data at 14 TeV withinthe considered scenarios. This analysis also lacks separate consideration of the W ± W ± process.The importance of the W ± W ± process as the one which guarantees the best realisticsignal to background ratios, and the possibility to further improve signal selection criteria2 CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS by careful study of specific signatures of W L W L and W T W X separately, including lowerthresholds on the p T of tag jets, was rediscovered in the paper by Doroba et al. [106].Many of the old observations of Bagger et al. were reconfirmed using a full tree levelmatrix element calculation of signal and backgrounds, including t ¯ t +jets, and includingrough estimates of some additional detector effects related to jet reconstruction and leptoncharge misidentification. Signal in this work was defined in terms of a Higgsless StandardModel with the unitarity condition implemented by applying appropriate weight factorsto generated events with a W W mass larger than 1.2 TeV. Translation of all the resultsinto the case of a light Higgs boson with modified couplings is straightforward.From 2012 onwards it has become clear that all we can realistically hope for in
V V scattering are the effects of non-SM Higgs (and gauge) couplings. Some general guidelinesfor a rough recomputation of all the predicted signal sizes were presented by Cheunget al. [107]. They considered all the different
V V scattering processes and calculatedscale factors to be applied to results of former Higgsless studies as a function of theactual
HW W and
HZZ couplings. Only purely leptonic W and Z decays were takeninto account. Computation of signal sizes was carried at the parton level and using theframework of the two Higgs doublet model (2HDM).Most recently, various studies have been concentrated on possible methods to enhancesignal significance via improvement of data analysis techniques as well as event recon-struction tools. Improvements in the analysis can be expected by means of applyingnovel techniques to explore the full shapes of signals and backgrounds in the multidimen-sional phase space spanned by the entire kinematics of visible particles in the final state.Measured multidimensional distributions can be compared to predictions arising fromparticular theoretical models calculated from the matrix elements. A likelihood functioncan then be defined to quantify the consistency of data with a predefined model. Suchapproach was used in a study by Freitas and Gainer [108]. The analysis they proposefalls into the category of Multivariate Analyses (MVA), which have become the standardin contemporary experiments like ATLAS or CMS, suplementing or in many cases com-pletely superseding the respective cut-based analyses. In fact, most Higgs related resultspublished by CMS or ATLAS to date have versions of MVA’s at their bases. The po-tential of discerning various theoretical models is quantified by a ∆ χ calculated for anytwo hypotheses. Focusing on W + W + scattering at √ s = 14 TeV, the authors found asignificant improvement in the LHC potential to discern SILH models from the StandardModel by using their own version of the Matrix Element Method (MEM). The reference inthis study was a one-dimensional analysis of the lepton-lepton invariant mass spectrum,as is routinely practiced in data analyses in HEP experiments. Expressed directly as afunction of the SILH parameter ξc H which governs the modification of Higgs couplingsto gauge boson in the lowest order, the expected ∆ χ rises approximately linearly from 0to 10 as the value of ξc H increases from 0 to 1. Note that effectively ξc H =0 is equivalentto the Standard Model, while ξc H =1 is equivalent to no Higgs. Meanwhile, a similar∆ χ obtained by considering solely the two-lepton invariant mass spectrum was foundlarger than 1 only for unrealistically large deviations from the Standard Model, beyond ξc H > χ in a counting experiment is bound to depend on the selec-tion criteria and ∆ χ in any analysis that does not exploit the full final state kinematicsis bound to depend on the event preselection used to measure the analyzed spectrum. .3. THE QUARTIC COUPLING PERSPECTIVE V V scattering, see next chapter.Also in the context of MVA’s, the subject of semi-leptonic decay modes was recentlyrevitalized by Cui and Han [109]. Their main theoretical focus was also SILH models ver-sus the Standard Model. They considered
W W scattering, with all charge combinationsincluded, and explored the jet substructure to separate the signal from various backgroundprocesses, including reducible backgrounds such as t ¯ t +jets and W +jets. A detailed studyof jet substructure provides an effective means not only to distinguish boosted W jetsfrom QCD jets to a large accuracy, but also to account for the different W polarizationsbetween signal and irreducible background. Wherever signal consists of longitudinallypolarized gauge bosons, the two partons from hadronic W decay tend to be emitted moreperpendicularly with respect to the W direction than in background events. Put anotherway, the p T ratio of the two partons from signal events tends to be larger than the cor-responding ratio from background events. For the purely electroweak processes of theStandard Model, the p T share of the two partons is usually highly asymmetric, while it istypically more balanced in the signal. In order to distinguish boosted W jets from QCDjets, the authors use the current state-of-the-art methods. They are based on the factthat a boosted W has two hard subjets (i.e., geometrical regions where hadronic energy isconcentrated), while a QCD jet has a single hard subjet. Subjets can be identified usingdedicated techniques known as filtering, pruning or trimming. Additionally, W decayproducts have no color altogether. On the other hand, QCD jets carry color charge andare color-connected to other partons in the event. This reflects in different transverse jetprofiles - QCD jets are typically much more diffuse. The most powerful way of separatingsignal from background is to combine different variables describing jet substructure: themasses and transverse momenta after jet pruning, planar flows, jet cone size dependencies,etc., to form an effective discriminator for the Boosted Decision Tree method. Overall,they found signal for the simplest Higgsless case possible to observe at more than 5 σ (from S/ √ S + B ) after 100/fb of data at 14 TeV, which is better than reported for the purelyleptonic decay modes. The corresponding result for SILH models scales like ( c H ξ ) . Thereis however considerable uncertainty related to their quantitative background evaluation.Although they considered both t ¯ t +jets and W +jets, as well as the irreducible jjW W background from processes ∼ α α S , to demonstrate the principles of operation of the W -jet tagging procedure, their final background numbers are likely to be underestimated.This is because these backgrounds were obtained solely via parton showering from thebasic t ¯ t , W + 1 jet and W W processes, respectively, generated with PYTHIA, when thisapproach is known to be insufficient. The efficiency of the W -jet tagging algorithm mayalso depend on detector resolution and pile-up. Nevertheless, this work clearly indicatesthe direction. It proves that with the current reconstruction and analysis tools, and en-visaging possible further refinements in the coming years, semi-leptonic decay channelswill offer additional discovery potential and should not be neglected. Three leading order graphs contribute to W ± W ± scattering in the Standard Model (fiveto W + W − ), including Z/ γ exchange, the W W W W (quartic) contact interaction and4
CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS
Higgs exchange. Before Higgs discovery, and even later until Higgs couplings were knownto enough precision, it was the Higgs exchange graph that represented the major puzzlein the entire picture and so a measurement of
W W scattering could be practically con-sidered equivalent to Higgs probing. As the by now discovered Higgs boson continues tofit Standard Model predictions with better and better precision, a shift of viewpoint isgradually taking place in what regards the physical motivation of studying
V V scatteringprocesses at the LHC. By assuming Higgs couplings known, e.g., exactly equal to theirvalues predicted in the SM or to the values measured in the LHC, and by moreover assum-ing that no new physics be directly observed within the energy range at consideration, wecan revert this reasoning and reformulate the problem in terms of the quartic
W W W W coupling. Indirect signs of new physics may include a modification of the effective four- W interaction term, leading to an anomalous coupling value. Such deviation would violatethe cancelation of the leading ∼ s terms in the W W scattering amplitudes between thecontact interaction graph and the Z/ γ exchange graph, producing a divergence propor-tional to the fourth power of energy. The observed energy dependence would therefore inprinciple be different than in the case of modified Higgs couplings. However, deviationsfrom the SM are likely to show up in more complicated forms than as a simple scalingfactor applied on the SM value. Generally, anomalous quartic couplings may be generatedas a contact interaction approximation of heavy particle exchange. The specific form ofthe operator that effectively contributes to the quartic vertex, plus the energy scale atwhich new physics sets in and places a natural cutoff for the relevant calculations, is akey question in order to assess the expected energy dependence. The actual value of the W W W W quartic coupling is currently very poorly constrained by experiment. As wenoticed before,
V V scattering, along with triboson production, provide the most directprobes of the quartic couplings.Interestingly, prior to Higgs discovery some authors studied Higgsless models in thelanguage of effective anomalous quartic couplings. The correspondence is straightforward.Deviations from a pure Higgsless SM, possibly arising from heavy particle exchange orsome kind of strong dynamics, were effectively parameterized as an anomaly in the quar-tic gauge vertex. As an example, based on such approach Godfrey [110] noticed early onthat for the lowest dimension operators that do not include photons the LHC will providethe most constraining measurements compared to e + e − , e − e − , γγ or e γ colliders. Fur-thermore, he found same-sign W ± W ± scattering be the best process to study the quarticcouplings. A similar approach was taken by Belyaev et al. [111], who basically redid thework of Bagger et al. in the language of anomalous quartic couplings. Today these studiesare however of historical interest only.By contrast, the work of Eboli et al. [112] retains its actuality because they haveconsidered the case of a 120 GeV Higgs as one of their two reference scenarios in thestudy of quartic couplings. It was also one of the earliest papers where the full analysiswas carried within the language of the Effective Field Theory with higher dimension(dimension-8 in this case) operators. As already mentioned, these authors consideredthe W ± W ± and W + W − processes and purely leptonic decays. The analogy to formerHiggsless studies is evident. They propose a full collection of selection criteria, which isin fact a variation of the familiar Higgsless selection criteria. They included: | η j | < η jj < η minj < η l < η maxj , MET >
30 GeV and p lT >
30 (100) GeV for same-sign (opposite-sign) leptons. Additional cuts: M jj > .4. V V
SCATTERING IN LHC MEASUREMENTS AT 8 TEV φ eµ > W + W − only. W helicities were not distinguished,but a moderate jet p T threshold for the tag jets (20 GeV) ensured good acceptance for W L W L pairs. Signals were calculated using the MadGraph generator with self-addedmodifications to include the anomalous terms. It must be stressed that in this analysisbackground evaluation included only irreducible background and inclusive t ¯ t productionfor W + W − and irreducible background only for W ± W ± . Moreover, all the analysis wasdone essentially at the parton level, with some experimental resolutions and estimatesof reconstruction efficiencies simulated on top. The results are therefore likely to be toooptimistic as far as overall background rejection is concerned. Unitarity constraints weresatisfied by imposing a sharp cutoff at M W W = 1.25 TeV. Final results were extracted bymerely counting the total event yield in the signal window. From a combination of bothprocesses and assuming an integrated LHC luminosity of 100 fb − , they came to predictthe following 99% CL limits: − < f S, Λ TeV − < − < f S, Λ TeV − < W ± W ± and W + W − is crucial, because the two coefficients studied from eachprocess separately show strong anticorrelation, especially in the same-sign process. Thiswas the first such detailed study that explicitly focused on the LHC sensitivity to anoma-lous quartic couplings and used the language of higher-dimension operators. Although itmay need minor updates in several places, it still remains the most complete phenomeno-logical analysis of its kind. V V scattering in LHC measurements at 8 TeV
Practically no simulation-based studies of
V V scattering on a detector-independent levelexist for pp collisions at 7 or 8 TeV. This is not just because these beam energies werenot really considered at the early stages of LHC planning, but rather decided later onas a compromise between current technical possibilities and physics needs. The mainreason is that it was known from rough order of magnitude estimates that 7/8 TeV wouldin fact not suffice to carry a truly conclusive measurement in terms of possible physicsbeyond the SM. After Higgs discovery, this became even more clear. This is why existingVBS-like measurements done at 8 TeV are still largely disconnected conceptually fromall the simulation work presented above and it is not always a trivial task to realize howthey in fact relate. Here we will examine first results from ATLAS and CMS concerning V V scattering that were obtained from an analysis of the 8 TeV data. In doing this ourmain point of interest will not be what the results tell us about physics, but rather whatwe can learn for future analyses at a higher beam energy.Preliminary results on vector boson scattering at 8 TeV have been produced both byATLAS [113] and CMS [114]. As the main ideas behind these two studies are closelyrelated and both analyses came out at a similar time (as usual, though, the ATLAS papercame first), we will discuss both of them simultaneously, making appropriate distinctionsonly when relevant. In both cases searches were carried for a loosely defined VBS-likesignature in the same-sign
W W scattering process and the purely leptonic decay channel.6
CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS
To partially tackle the inescapable problem of low statistics, applied VBS-like selectionwas in both cases minimal. The signature consisted of two reconstructed same-sign leptons(each of which could be either an electron or a muon) passing all the respective “high-quality” criteria and at least two jets within detector acceptance. The definitions ofdetector acceptance were marginally different for ATLAS and CMS, but could not play anymajor role in the final result. Minimum p T of 20 (25) GeV was required in ATLAS (CMS)for the leptons and of 30 GeV for the jets. In addition, a minimum missing transverseenergy of 40 GeV was required to account for the two neutrinos. The only additionalselection criteria applied on the data in order to separate pure electroweak jjW ± W ± production from processes involving gluon exchange was a cut on the jet-jet invariant mass, M jj >
500 GeV, and rapidity separation | ∆ y jj | > p T ) jets in the event. These criteria are alsoinstrumental in suppressing various sources of reducible background, chiefly inclusive t ¯ t production. At an average lepton p T that corresponds to the beam energy of 8 TeV,the charge of the muon is measured accurately to negligible levels. However, inefficiencyof electron charge determination can produce non-negligible detector background comingfrom e + e − pairs copiously produced at the Z boson peak. For this reason, the ee invariantmass was required to lie outside a band of 10 (ATLAS) or 15 (CMS) GeV around the Z mass. This cut affected only the electron-electron decay channel. Moreover, for anylepton pair its mass had to be larger than 20 GeV in ATLAS and 50 GeV in CMS. This isperhaps the most significant difference between the two analyses and was dictated by therespective detector capabilities in what concerns in particular the contamination from jetsmisreconstructed as leptons. Remaining inclusive t ¯ t background, entering via both leptonsign-flip effects and leptonic b quark decays, was effectively eliminated by standard b quarkvetoing. Details of the b -tagging techniques were developed independently by the twocollaborations, but both are based on combining the information from impact parametersignificance of the individual tracks with explicit secondary vertex reconstruction. Thebulk of background coming from W Z or ZZ +jets production was reduced by a veto on athird lepton. Here the cut depends on the efficiency and purity of lepton reconstructionand must be optimized in a detector-dependent manner. Consequently, it was somewhatstricter in ATLAS than in CMS: it involved any additional reconstructed leptons with p T > / jjW ± W ± production within a kinematic region consis-tent with vector boson scattering. There is no explicit distinction between W W scatteringand non-scattering
W W production at any stage of the two analyses. In fact, the ATLASpaper is conservatively entitled “Evidence of Electroweak Production...” and makes nomention of VBS anywhere in the paper abstract. Naturally, signal definition includes SMcontributions and so it contains what in all our previous considerations has been called ir-reducible background. There is a subtle way in which signal is not exactly the same thingin the two analyses. In the CMS analysis, there is no distinction between pure electroweak jjW ± W ± production and QCD mediated (gluon exchange) processes within the selectedkinematic window. Both are treated as an integral part of the signal. The ATLAS anal-ysis explicitly separates QCD production of jjW ± W ± as another class of background,as opposed to a purely electroweak process. How much of each we have in the sample .4. V V
SCATTERING IN LHC MEASUREMENTS AT 8 TEV jjW ± W ± production. Such differences are in fact within the statistical errorsof the signal sample collected in these studies. The future practical solution to the aboveproblems would be in applying selection criteria tight enough so that any QCD contri-bution, including the interference, would become negligible altogether. This is, however,not a viable option for the present energy and the accumulated statistics.For the background estimates both collaborations developed original methods whichdiffer rather widely. Whenever simulations are used, typically, leading order generatorswere applied (MadGraph, POWHEG, SHERPA or ALPGEN), and the results were nor-malized in terms of a constant factor to the next-to-leading order in QCD cross sectionsobtained, e.g., with VBFNLO. Uncertainties of the order of 10% were found within thesignal kinematic window for the main backgrounds. Differences in the respective LO gen-erators used by ATLAS and CMS are unlikely to play a major role, but the respective PDFand QCD scale choices are in fact one of the main components of the systematic errors.CMS background predictions are for the most part data-driven. The so-called “non-prompt” lepton backgrounds, originating from leptonic decays of heavy quarks, hadronsmisidentified as leptons ( W +jets), and electrons from photon conversions in the detectorwere deduced from a control sample defined by one lepton which passes the full leptonselection criteria and another lepton which fails these criteria, but passes a “loose lep-ton” selection. Fake rates for such loose leptons to pass the nominal lepton criteria werethen calculated and applied to the signal region. Similarly, the W Z background withtwo accompanying jets is predicted from a data control region requiring an additionallepton with p T >
10 GeV. Other background sources included triboson production, sign-flip effects and double parton scattering. They amounted to less than 10% of the totalbackground and were estimated from simulation. ATLAS background predictions for
W Z and ZZ +jets (labeled “prompt”), as well as photon conversion background, were drivenfrom full detector simulations and cross checked with the data in several same-sign dilep-ton control regions. Sign-flip backgrounds (part of which they include in the “conversion”category) and backgrounds involving leptons reconstructed from jets (collectively denotedas “other non-prompt”) were estimated directly from data.Interesting is the significant difference in the final background composition betweenATLAS and CMS. “Prompt” backgrounds, composed in 90% of W Z production with alost third lepton (either not reconstructed or falling outside detector acceptance), amountto as much as two thirds of the total background at ATLAS. By contrast, it is lessthan 20% in CMS. In absolute numbers, the remaining
W Z background is over 7 timeshigher in ATLAS than it is in CMS (7.5 ± ± e ± e ± , µ ± µ ± and e ± µ ± . Such difference cannot be explained solely by physics, i.e., details of8 CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS the applied selection criteria, although stronger cuts on | ∆ y jj | and M ll adopted by CMScontribute in the right direction. We recall that the third lepton veto used to suppress W Z was stricter in ATLAS than in CMS in terms of the p T threshold. However, differentefficiencies of lepton reconstruction and third lepton veto, connected to the respective“tight” and “loose” lepton identification criteria applied by the two experiments, mustbe doubtlessly causing the resulting discrepancy. By contrast, the amounts of “non-prompt” (including conversion) background predicted in the two experiments are roughlysimilar: 6.3 ± ± ± ± M ll and | ∆ y jj | ; to a lesser degree | M ee − M Z | and | η e | which affect only the electrondecay channels. A 5% difference exists in the total integrated luminosity recorded by thetwo detectors (20.3 fb − ATLAS vs. 19.4 fb − CMS). The rest of the difference is naileddown to be due to theoretical uncertainties, reconstruction efficiencies and resolutions.In particular, the most important systematic uncertainties in the signal predictions arethose related to the choice of PDF’s (7.7%) and the QCD scale (5%), jet energy scale andresolution (5%) and lepton efficiency (3%).To quantify the statistical significance of the signal, events yields were examined ineight separate intervals formed by 4 bins of M jj times two lepton charges. The observed(expected) signal significance in CMS is 2.0 σ (3.1 σ ). In ATLAS, the correspondingnumbers are 3.6 σ (2.8 σ ). The apparently large discrepancy between the actually observednumbers of events in the signal region: 12 events in CMS vs. 34 events in ATLAS isconsistent within the errors with all the earlier predictions.It is trivial to convince oneself that ATLAS and CMS 8 TeV results neither confirm nordisconfirm Higgs existence, let alone give any clue of the relevant Higgs coupling. Like-wise, they are hardly sensitive to anomalous triple gauge couplings within their presentexperimental bounds. Instead, they can be interpreted in terms of the first experimentalbounds on the quartic W W W W coupling. The fact that both analyses used a relativelyhigh jet p T threshold to protect from pile-up jets means that W L W L pairs were disfa-vored. This additionally reduces the sensivity to the HW W coupling and also to thosehigher dimension operators which modify only longitudinal gauge boson interactions. TheCMS collaboration obtained 95% CL limits on all nine dimension-8 operators that leadto anomalous contributions to the
W W W W coupling. Respective signal predictions werederived from MadGraph-generated samples in which one anomalous parameter was variedat a time. Limits were based on the measured lepton-lepton invariant mass spectrum of .4.
V V
SCATTERING IN LHC MEASUREMENTS AT 8 TEV p T , but did not reveal improvements insensitivity to the parameters in question. Here there was no specific optimization with therespect to the individual anomalous parameters, e.g., in what concerns the W W helicitycombinations they directly affect. The effect of these parameters on the background ismarginal and was neglected. This concerns also the
W Z background, since in our signalwindow it is dominated by QCD contributions rather than
W Z scattering. In addition tothe uncorrelated limits on individual parameters, limits were derived in the two-dimensionspace of f S, / Λ vs. f S, / Λ . Obtained contours show a strong anticorrelation betweenthese parameters. Because of this the limits on individual parameters obtained by one-dimensional projections of the correlated limits are weaker than their uncorrelated limitsby at least a factor of ∼
5. The reason is straightforward. Each scattering process probesin fact specific combinations of anomalous parameters. And the other way around, im-provement on the correlated limits can be only achieved by combining data from differentscattering processes: W + W − , W Z and ZZ , which probe different combinations of thesame parameters. Data at 8 TeV are however of not enough statistical power to studythe other processes.Figure 5.1: Left: observed and expected two-dimensional limits on the operator coeffi-cients a and a from an analysis of the process pp → jjW ± W ± at 8 TeV done by theATLAS collaboration - image reproduced from Ref. [113]. Right: observed and expectedtwo-dimensional limits on the operator coefficients f S, and f S, from an analysis of theprocess pp → jjW ± W ± at 8 TeV done by the CMS collaboration - image reproducedfrom Ref. [114].The ATLAS collaboration derived limits on the W W W W coupling expressed in termsof the a and a parameters of the Electroweak Chiral Lagrangian. The underlying model,as currently implemented in the WHIZARD generator, includes a 125 GeV Higgs bosonin addition to the traditional dimension-4 operators foreseen within the framework of theEWChL, and so non-zero values of a and a can be in principle reinterpreted as equivalent00 CHAPTER 5. SIMULATION-BASED STUDIES VS. EXPERIMENTAL RESULTS to anomalous quartic couplings. While formally data may still be interpreted in thislanguage, such treatment has not gained wide recognition in the physicists’ community.A more practical problem is that non-zero a and a still induce unitarity violation andprovide no built-in mechanism to restore unitarity. Results do depend on the arbitrarilychosen unitarization scheme, which is an intrinsic uncertainty of the model. ATLAS usedthe K-matrix unitarization procedure and did not quantify the theoretical uncertaintiesrelated to this particular choice. Effective equivalence of the phenomenological impactof non-zero ( a , a ) with that of dimension-8 operators from the Effective Field Theoryapproach was demonstrated. The relationships for the jjW ± W ± process are supposedlythe following [65]: a = v f S, , (5.1) a = v ( f S, − f S, )16Λ , (5.2)where v is the usual Higgs vacuum expectation value. With the above one can verify thatthe far endpoints of the 95% CL contours from ATLAS, approximately a = ± a = ∓ f S, / Λ ≈ ±
870 TeV − and f S, / Λ ≈ ∓ − , severaltimes weaker bounds than from the CMS analysis. Why such discrepancy? Partly becauseATLAS does see an excess of events with respect to SM predictions, while CMS sees adeficit. Another reason is that the CMS methodology does not assume any physicalcutoff Λ for the evaluation of the anomalous signals. The underlying assumption thatnew physics does not directly show up to the presently available energy is natural inthe light of no new physics having been actually observed in the LHC so far. However, f S, / Λ and f S, / Λ may lead to unitarity violation within the quoted limits. Put anotherway, even a BSM signal equivalent to hitting the unitarity limit at the highest availableenergy could not be observed with the present data. By saying earlier on that the resultsneither confirm nor disconfirm Higgs existence, we effectively meant exactly the samething. Data at 8 TeV do not provide enough sensitivity to establish really physicallymeaningful limits on the studied parameters. Formally calculable limits reflect the appliedunitarization procedure or lack of it and one should be extremely careful in drawingphysics conclusions. The exercises done by ATLAS and CMS serve as a demonstration ofprinciples and technical preparation for future measurements at higher energies. They setup and test the methodology. They reveal the main experimental and theoretical issuesto be addressed in such measurements. But their physics meaning on its own is for thetime being quite limited.The CMS collaboration derived also limits on the production cross section timesbranching fraction of a doubly charged Higgs decaying into W ± W ± . Doubly chargedHiggs bosons are expected in models that contain a Higgs triplet field. In such models,the W ± W ± scattering process would be a resonant one, in contrast to the SM and itsmost popular proposed extensions. hapter 6What can the LHC measure After delivering 5 fb − of proton-proton data at 7 GeV and 20 fb − at 8 TeV, the LHCentered the first long shutdown (LS1) phase from 2013 till the end of 2014 and has beendue to upgrades. LS1 included a large number of simultaneous activities concerning boththe injectors and LHC itself, aimed to ensure reliable operation at nominal parameteresfrom 2015. Most importantly the center of mass energy will now be nearly doubled andbecome 13 TeV. Early plans assumed a center of mass energy of 14 TeV and a lot ofearlier simulation work was in fact done under this assumption. As physics is unlikely tochange significantly between 13 and 14 TeV, these studies are mostly still valid and inthis work we will discuss 13/14 TeV simulation results in a complementary way, withoutmaking clear distinctions. The main priorities of LS1 are to repair and consolidate theinterconnects, bring all necessary equipment up to the level needed for 6.5 TeV per beam,repair leaks and other maintenance work required after 3 years of operation. Upgradeand maintenance activities in the machine are accompanied by concurrent upgrade andmaintenance activities on part of the individual detectors.The LS1 will be the first long shutdown of the LHC, part of a long term draft planwhich foresees operation until 2035, with several subsequent operation periods and longshutdowns. Run 2 of the LHC is due to start early in 2015 and last for the next 3 years withan intermediate luminosity of 10 cm − s − . Long shutdown 2 (LS2) is planned from mid-2018 until the end of 2019. After that, Run 3 of the LHC will proceed with nominal energyand nominal luminosity of 2 × cm − s − . The amount of proton-proton data collectedin Runs 2 and 3 is conservatively expected to be 300 fb − . Given the experience fromRun 1 and the excellent machine operation which surpassed conservative expectationsalready in the second year of running, it may possibly turn out even larger. After 2022,the machine will be due for another major upgrade for an order of magnitude increaseof luminosity, while keeping the same beam energy. This future phase is refered to asthe High Luminosity LHC (HL-LHC). Long shutdown 3 (LS3) is planned to last from2023 until late 2025 for the LHC and from 2024 until mid-2025 for the injectors. Thefollowing three machine operation periods, interspaced with long shutdowns 4 and 5, aimat delivering 3000 fb − of proton-proton collisions until 2035. This is the ultimate aim ofthe LHC.In this chapter we will try to answer the question of what can the LHC, operating at13/14 TeV, measure in the various V V scattering processes, having in mind everythingwe have learned so far both on the theory side and from existing measurements. Because10102
CHAPTER 6. WHAT CAN THE LHC MEASURE the physics motivation to study VBS processes has significantly changed only in the lastcouple of years and is still in the process of reformulation, up to date analyses are not soabundant and a comprehensive review of fully valid predictions for the LHC is rather hardto find. In an attempt to fill the hole, we will herewith sketch some analyses which aresupposed to be completely consistent with all our present knowledge, yet not involvinganything more than common simulation tools. For full transparency and in order to avoidusage of any experiment specific software, the analysis will be kept as simple as possiblefrom the point of view of the applied analysis tools. It will be a cut-based analysis. Whilewe do not assume that the future final analyses by the ATLAS and CMS collaborationswill indeed be done in this way, such simple analysis is accurate enough for our purpose,which is to evaluate the order of magnitude of possible signals from different BSM sources,shed light on the LHC potential to identify a physics scenario from the sole study of VBSprocesses, as well as to identify some of the main challenges and limiting factors in whatconcerns background rejection. Whenever possible and applicable, we will follow the ideasof earlier works by many authors, but all the results will be independently recalculatedwith modern simulation tools. This will include some detector resolution effects, so longas the latter do not involve full detector simulation.Our main focus here will be on the purely leptonic decays. This choice is mainlymotivated by pragmatism - these channels do not suffer from complicated, QCD-relatedsystematic uncertainties in event reconstruction and, as we have seen, the final signal tobackground ratio, with all major detector dependent effects included, is driven by merelya few experiment-specific factors: the purity of electron reconstruction, charge measure-ment efficiency for electrons and muons at high p T , and the efficiency of b -tagging. Allthese effects can be to a rough accuracy described in terms of simple numbers, with-out necessarily applying the entire methodology of event reconstruction used in a realexperiment. All other systematics can be either assumed known or play a lesser role.Our baseline to define the BSM signal and tune the necessary selection criteria will bethe Higgsless Standard Model, as we inherit from most of the classic studies. However,the ultimate goal is to find how this translates into realistic scenarios with modified Higgscouplings and anomalous triple and quartic gauge boson couplings, with all the relevantsimilarities and differences being taken into account. In previous chapters we have discussed in detail the formal definitions, the computationalmethods and issues for the complete calculation of the signal and irreducible background.A full set of signal selection criteria that are applicable to future data at √ s = 14 TeVfollows directly from our previous considerations: • at least two jets with 2 < | η j | < η j η j < • exactly two isolated same-sign/opposite-sign leptons with ∆ ϕ > . • M jj >
500 GeV, • M l j , M l j >
200 GeV, .1. MODELING OF THE SIGNAL AND IRREDUCIBLE BACKGROUND • b quark veto, • optional: central jet veto, • R p T > . p lT , small | η l | and large M ll .We recall that the first two criteria are basic topological VBF cuts, the next four arededicated t ¯ t suppression cuts (not exactly - requirement of large M jj also suppresses theirreducible background) and only the last item represents cuts, only one in the same-signcase, that separate the longitudinal W signal from the transverse W background.Figure 6.1: Actual signal cross section relative to the Higgsless cross section as a functionof the actual HW W coupling relative to its Standard Model value. Signal sizes weredetermined after applying all the signal selection criteria discussed in the text. Thepoints provide a good first approximation of how to scale the results of all the former
W W scattering studies which used the Higgsless hypothesis to evaluate signals in orderto reinterpret them in terms of a Higgs with modified couplings. Result of parton-levelMadGraph calculations for the process pp → jjW + W + with W + → µ + ν .From analyses available to date it can be inferred that the order of magnitude W + W + signal cross section, where signal is defined in terms of a pure Higgsless Standard Model,and by further assuming purely leptonic decays ( l = e, µ ), is close to 0.12 fb. This numberhas been recalculated using MadGraph 5. It includes effects associated to hadronization,final state radiation and jet reconstruction using an imitative simple jet cone algorithm.Sheer signal size is similar for W + W + and for W + W − . Physically realistic scale factorsrange from 0.8 for a pure Higgsless with unitarity or a heavy Higgs, to 0.31, 0.16 and0.06 for a light Higgs boson that couples to the W with a strength equal to, respectively,0.7, 0.8 and 0.9 of what is predicted in the Standard Model [115] (see Fig. 6.1). Becauseof the amplitude interference patterns, signals for HW W couplings larger than unity (in04
CHAPTER 6. WHAT CAN THE LHC MEASURE
SM units) are generally lower than for their mirror values. Finite detector resolutions inthe measurement of p T or η of the leptons and jets effectively play the role of a further ∼
10% reduction. Signal in the anomalous gauge coupling scenarios must be calculatedindependently and will be shown later on. Irreducible background levels for W + W + areof the order of 0.05 fb in a conventional cut analysis and can be shrinked at least to 0.02 fbby applying an R p T cut instead or even more sophisticated correlated variable techniques.The latter will be much closer to what eventually can be achieved using a MultivariateAnalysis in which the entire final state kinematics is exploited. For W + W − the irreduciblebackground amounts to about 0.11 fb and it is unlikely to improve in a significant way.Signal for W − W − is about a factor 4 lower than for W + W + , but backgrounds are atsimilar levels.Total cross sections for the signal (calculated within the Higgsless scenario) and irre-ducible background in the W + W + and W + W − processes, after each subsequent class ofselection criteria discussed in the text, are shown in Fig. 6.2. t ¯ t production background Reducible backgrounds in real experiments are typically determined using partly or wholydata-driven methods. This and the following sections discuss pure simulation-based re-sults and are not intended as a model for a future analysis of experimental data. Theirpurpose is merely to establish a suitable methodology to estimate all these backgrounds insimulation-based studies, before they can be cross checked against the data. The methodsdescribed here should be accurate enough to assess the orders of magnitude of the relevantbackgrounds and to study the main challenges related to background reduction.Calculations of the t ¯ t background are affected by large QCD-related uncertainties. Theinclusive t ¯ t production cross sections in proton-proton collisions at √ s = 7 and 8 TeVhas been measured by both ATLAS and CMS. These numbers provide the only currentlyavailable direct experimental bond to reduce the theory-based systematic uncertainties forthe predicted t ¯ t cross sections at 14 TeV. As the number of relevant Feynman diagramsgrows rapidly with the order in α S , the whole process cannot be accurately modelled inthe lowest order plus allowing initial and final state radiation. The leading parton levelsubprocesses that are complete missed in such approximative treatment are graphs leadingto an additional quark-jet in the final state, pp → t ¯ tq . These two classes of events donot involve any double-counting. Their coherent sum reasonably reproduces the totalcross sections at 7 or 8 TeV, as measured in the LHC. A more satisfactory description,developed especially for the study of inclusive t ¯ t production as a background to VBFprocesses, is based on explicitly considering three processes at the tree level: pp → t ¯ t , pp → t ¯ tj and pp → t ¯ tjj ( j denoting quarks and gluons alike) plus initial and final stateradiation. These processes represent the leading order contributing diagrams of inclusive t ¯ t production for the cases where 2, 1 or 0 tagging jets arise from b quarks, respectively.The three different topological configurations select three mutually exclusive subsamplesand thus double-counting is automatically avoided. It is actually the latter two classes Following the common convention, by pp we always mean the sum of all the corresponding interactionsat the parton level, i.e., quark-quark, quark-gluon or gluon-gluon, while the remnants of the protons areignored. Hence, e.g., pp → t ¯ t does not mean baryon number violation. .3. MODELING OF THE W +JETS BACKGROUNDS t ¯ t background. The contribution from pp → t ¯ t with both b quarks becoming tagging jets is minimal. That the two abovementioned methods produceconsistent results for 14 TeV has been verified.From completed simulation-based studies, that include also CMS-like detector resolu-tion effects, it can be inferred that the total top production background falling within thekinematic phase space defined by all the abovementioned signal selection criteria: basicVBF cuts, t ¯ t suppression cuts and the R p T cut, can be roughly parameterized as B t ¯ t = 12 f b · (1 − ǫ b − tag ) · (1 − ǫ sign ) · ǫ CJV . (6.1)Here the normalization factor includes the branching fractions of W decaying into elec-trons or muons and the proper selection efficiency, ǫ b − tag is the average efficiency of b -tagging, ǫ sign is the average efficiency of lepton charge reconstruction and ǫ CJV is the cen-tral jet veto factor, if applied. For example, setting ǫ b − tag =0.5, ǫ sign =0.995 and ǫ CJV =1,as expected for the same-sign mode, one gets B t ¯ t ∼ B decays are suppressed by a combination of kinematics andisolation criteria to much below this level. Another subclass of the inclusive t ¯ t produc-tion background that affects the same-sign mode is W + t ¯ t production, where one leptoncomes from W decay, another from top decay. This background was shown negligibleafter applying standard signal selection cuts. For ǫ sign ∼ ǫ CJV ∼ B t ¯ t ∼ √ s = 14 TeV, only W + W + carries the potential of signal lev-els above background fluctuations assuming luminosities measured in hundreds of inversefemtobarns.The total t ¯ t cross section after each of the analysis cuts discussed in the text is shownin Fig. 6.3 (top plot). W +jets backgrounds Jets misreconstructed as electrons are the primary source of these backgrounds. Thelowest order process of this kind that can mimick the signal is W + 3 jets, where inprinciple any of the three jets can be the fake electron. The kinematic regime we areprobing by applying the signal selection cuts strongly favors large- p T leptons. As a directconsequence, events in which the leading jet (where, as usual, we rank objects in a givenclass according to their p T ) is the one that gets misreconstructed make up over 90% ofall the cases of W +jets events falling kinematically within the signal phase space. Thesubleading jet as the fake electron accounts for just about the rest of it. For the samereasons, it is inessential to consider additional samples with more than three jets at thegeneration level. Given the large total cross section for W +jets at the LHC, the purityof electron reconstruction is a crucial number. The final amount of W + +jets eventsmimicking the signal can be predicted as being roughly B W + + jets = 5 pb · ǫ j − fake · f + / − , (6.2)06 CHAPTER 6. WHAT CAN THE LHC MEASURE in total, where ǫ fake is the overall probability of a jet being reconstructed as an electronsatisfying all the quality selection criteria and f + / − is the sign matching factor. For ǫ fake ∼ . · − and f + / − =0.27 ( W + W + ), this gives 0.08 fb. For f + / − =0.73 (opposite-sign), it is about 0.2 fb. It is not a negligible number and, not so unexpectedly, it is a moreimportant background source than top production for the same-sign mode. However, thisbackground is bound to affect different final states differently. Half of the total B W + jets is due in the jjee final state, the other half in the jjeµ final state (where signal is twicethe size of the jjee signal) and no contribution is possible to the jjµµ final state.The W − +jets background is typically a factor 2 lower due to the charge asymmetryin W production at the LHC. We assume then additional contributions of 0.1 fb for theopposite-sign and 0.04 fb for W − W − .Another class of background is related to a fake electron being reconstructed froma photon with an associated track. The leading order process than can generate suchevents is W jjγ . Its total cross sections is much lower than for W + 3 jets and additionalkinematic and combinatorial factors make it in fact negligible. This background is of theorder of B W jjγ = 0 . f b · ǫ γ − fake . (6.3)where ǫ γ − fake is the overall probability of a photon misreconstructed as an electron sat-isfying all the quality selection criteria and of the required charge. For ǫ γ − fake ∼ B W jjγ < W +jets cross section after each of the analysis cuts discussed in the text isshown in Fig. 6.3 (second plot). The leading order background process of this kind is jjjj with two of the four jets misre-constructed as electrons. Huge cross sections for QCD processes at the LHC compensatethe low probability of having two simultaneous fakes and so this background can proveoverwhelming. This result may seem surprising at first glance, but in fact it is dictatedby the very specific kinematic correlations we are looking for, significantly different fromthe ones typically observed in regular gauge boson physics analyses done on the 7 and 8TeV data. Again here, the kinematic regime we probe favors large p T and therefore fakesgenerated by the two leading jets account for 80-90% of all events satisfying the completeselection criteria, while the rest to a sub-percent level comes from the combination of thefirst with the third jets being reconstructed as fake electrons. For the same reasons itis also here inessential to consider higher order processes. In order to render the QCDmultijet background manageable, we further assume the following combination of cuts tobe applied in the jjee final state only: M ET >
60 GeV, M ee >
250 GeV, p j T >
30 GeV.The meaning of the first two cuts is straightforward. For the third cut, note that herein most cases j denotes really the third jet. Extra cuts bring a substantial reduction of .5. W Z
AND ZZ AS BACKGROUNDS TO
W W jjee signal, or equivalently well over 90% of thetotal signal. In total, B jjjj = 6 . nb · ǫ j − fake · f + / − , (6.4)where ǫ j − fake is the probability of a jet being reconstructed as an electron satisfying all thequality selection criteria and f + / − is a combinatorial factor equal to 0.25 for each same-signmode and 0.5 for opposite-sign. By assuming ǫ j − fake ∼ − we end up at B jjjj ∼ jjee final state, as does the W +jetsbackground concern the jjee and jjeµ final states in fixed proportions, comparison ofthe selected event yields will be an additional tool to disentangle the various backgroundsources and isolate the signal (provided enough statistical power). Yet another piece ofvaluable information will be provided by the study of the W − W − mode.The total QCD multijet cross section after each of the analysis cuts discussed in thetext is shown in Fig. 6.3 (third plot). W Z and
Z Z as backgrounds to
W W
Several previous analyses, in particular the ones by Chanowitz et al. [98], hinted on thepossibility that continuum
W Z production with one lepton which escaped detection, couldbe as well an additional significant background to W ± W ± . The subject was brought upagain in the recent analyses by ATLAS [113] and CMS [114]. The validity of this assertionstrongly depends on the applied selection criteria. In our case, the amount of remaining W + Z background with at least two associated jets after cuts gets reduced to about 0.04fb altogether, i.e., regardless of whether the negatively charged lepton from Z decay getsreconstructed or not. The geometrical condition of this third lepton falling outside ofthe accepted pseudorapidity range of | η | < W + Z background in the analysis of W + W + . For W − W − , the relativecontamination from W − Z is about a factor 2 larger from pure combinatorics. Similar isthe W Z contamination to W + W − , here however both W + Z and W − Z can contribute.Signal from W ± Z , if any, eventually adding up to signal from W W is of course a bonusrather than a problem.The total
W Z cross section after each of the analysis cuts discussed in the text isshown in Fig. 6.3 (bottom plot). Contaminations from ZZ are still smaller. W Z and
Z Z as signals
To estimate the amount of BSM signal for the
W Z and ZZ processes, we follow existingliterature on the subject, and the work of Bagger et al. [96] in particular. We can recalland confirm here some of their most elaborated and relevant signal selection criteria thatwere shown to exploit specific kinematic features of each of these processes in order toenhance S/B. In addition to requiring standard VBF topology and applying cuts againstinclusive t ¯ t background (for ZZ only a cut on M jj > GeV applies), the process specificcuts are the following. For
W Z :08
CHAPTER 6. WHAT CAN THE LHC MEASURE • M Z −
10 GeV < M l + l − < M Z + 10 GeV, • M T ( W Z ) >
500 GeV, • p ZT > M T ( W Z ), • M ET >
50 GeV, • p lT >
40 GeV.For ZZ → l : • M Z −
10 GeV < M l + l − < M Z + 10 GeV for both lepton pairs, • M l >
500 GeV, • p ZT > q M l − M Z for each Z , • p lT >
40 GeV.For ZZ → l + l − νν : • M Z −
10 GeV < M l + l − < M Z + 10 GeV, • M T ( ZZ ) >
500 GeV, • p T ( ll ) > M T ( ZZ ), • M ET >
250 GeV, • p lT >
40 GeV.In the above, M Z is the PDG Z mass, while all other symbols refer to reconstructedquantities. The transverse masses are defined as follows: M T ( W Z ) = [ q M ( lll ) + p T ( lll ) + M ET ] − [ ~p T ( lll ) + ~M ET ] , (6.5) M T ( ZZ ) = [ q M Z + p T ( ll ) + q M Z + M ET ] − [ ~p T ( ll ) + ~M ET ] . (6.6)Background is expected to be dominated by irreducible SM background for ZZ andadditionally Zt ¯ t +jets production for W Z . Under these assumptions and applying thecuts described above, background levels amount approximately to 0.027 fb for W ± Z ,0.003 fb for ZZ → l and 0.009 fb for ZZ → l ν . Higgsless signals would be of the orderof 0.009 fb, 0.005 fb and 0.012 fb, respectively. For W Z , background is relatively large andits kinematic separation from the signal, if by the latter we understand non-SM Higgscouplings, is marginal. This forces to use strict selection criteria which in turn wouldrequire very high luminosity to be successfully applied. The ZZ modes are relativelyclean, especially the 4 l , but clearly suffer of low statistics.The total cross sections of the signal and irreducible background for W Z and ZZ after each of the analysis cuts discussed in the text are shown in Fig. 6.4. In the eventof absence of new heavy resonances within reach, these processes are unlikely to improveour knowledge of the Higgs sector. .7. KEY UNCERTAINTIES A phenomenological analysis based on signal and background calculations done by matrixelement generators at the tree level is affected by specific uncertainties. These comepartly from theory itself and partly from imperfect knowledge of detector related effects.A detailed analysis of all the systematic erros is rather inessential at this point, butwe can outline the most important limitations to the accuracy of our predictions. Notaccidentally, some of them will translate into the limiting factors at the time of carryingthe real measurement.Total cross sections for proton-proton processes calculated in a given order in pertur-bative expansion are sensitive to the choice of such things as the set of parton distributionfunctions (PDF’s) and the QCD factorization and renormalization scales. Typically, thechoice of PDF’s by itself does not change numerical results by more than 5%. The fac-torization scale corresponds to the resolution at which the proton is being probed. Whencalculated to all orders in perturbative QCD, the hadronic cross section is independent ofthe scale. But at any finite order it must depend logarithmically on it [116]. Moreover,the dependence is usually significant at low orders in perturbation theory. The way toobtain a reliable prediction is to calculate higher-order corrections until the factorizationscale dependence is reduced. It was shown that calculations of diboson production in thevector boson scattering configuration carried at the next-to-leading order (NLO) in QCDare very weakly dependent on the scale. The residual uncertainty is of 2.5% in a typi-cal VBF kinematics (for W + W + ). Meanwhile, results of leading order (LO) calculationscan be made coincide with the former by a choice of the factorization scale equal to themomentum transfer of the t -channel electroweak boson [118]. This solution has recentlybeen implemented as an option in MadGraph 5. Deviations induced by setting the scaleto a fixed value, e.g., the Z mass are of order of 10%. Even more sophisticated recipes arecurrently devised [119]. These will allow further reduction of scale related uncertaintiesfor future studies and data analyses.Furthermore, a key problem in making precise perturbative QCD predictions is to setthe proper renormalization scale of the running coupling. A poor choice of the renormal-ization scale can manifest itself as a strong dependence on the ratio of the NLO crosssection to the LO cross section (the so called K-factor) [117]. As our process of interestis of purely electroweak nature, the signal predictions are affected only via uncertaintiesin the modeling of parton hadronization and final state radiation. In studies that do notinvolve detailed detector simulation, these are anyway dwarfed by imperfect modeling ofjet reconstruction procedures, calorimeter efficiencies and resolutions. In the predictionof QCD-related background, variations depending on the scale can easily amount to 30%without dedicated hard work. In our case we take clear advantage of the fact that thesebackgrounds are expected to be small after all cuts. For inclusive t ¯ t production we havealso an experimental ansatz since the total cross sections have been measured at 7 and 8TeV by both ATLAS and CMS. It is in any case the inclusion of t ¯ tj and t ¯ tjj processes inthe first place that ensures the total cross sections are consistent with the measurementswithin the errors of the latter. These errors are of the order of 5-10%.The overall smallness of background in the same-sign process is an advantage at thetime of the measurement, but a relative disadvantage for phenomenological studies. Thebackground is difficult to predict because it depends primarily on a combination of tiny10 CHAPTER 6. WHAT CAN THE LHC MEASURE detector effects rather than physics calculable from first principles. In case of t ¯ t this isnot only knowledge of exact b -tagging efficiencies as a function of jet p T and η , but aswe saw in the same-sign W W channel, the result is mainly driven by the efficiency oflepton charge identification. At the present moment this is only taken into account as anorder of magnitude estimate. Surely, a small change in efficiency can produce a significanteffect on the S/B ratio. Likewise, W +jets and QCD mulitjet backgrounds enter via thetiny effects of jets misidentified as leptons. These are strongly detector- and software-dependent and only an order of magnitude estimate can again be made at this point.And in any case they must be considered in simultaneous relation with the efficiency oflepton-ID for genuine leptons. It is of little use to evaluate such effects in more detail onpurely phenomenological grounds. However, we can at least define kinematic conditionsunder which the abovementioned backgrounds are small enough that their precise valuescan be measured using data-driven methods at the proper time, but will not jeopardizethe entire analysis. Ultimately, exact background levels and compositions will differ fromexperiment to experiment (in our case from ATLAS to CMS).In all the numerical predictions that are presented in this section, things like detectorefficiencies, if only different from unity by more than a few per cent, were taken into ac-count by simply scaling the final cross sections. Basic detector resolutions [83] [84] [91] canbe simulated with dedicated simulation tools, namely the PGS program [128]. In general,PGS-level results are over 10% lower than PYTHIA-level (generator+hadronization) re-sults. Meanwhile, results in the electron decay channels are consitent with those in muondecay channels, from which they differ only in the assumed resolution, to a few per centand hence this number can be used as a rather conservative upper limit of the resolution-related uncertainty. Differences between jet reconstruction algorithms and the effects ofchoosing a cone/cluster size parameters R are negligibly small. We recall that althoughthe current standard jet definition used, e.g., in CMS is set by the anti- k T alghorithm [90],which is not an available option in PYTHIA 6, but has been implemented in PGS 4, it islikely to be changed in the future for specific analyses in order to improve W tagging inhadronic decays. For the purpose of the studies presented here, either the k T or a simplecone algorithm [79] with R = 0 . p T threshold,important from the point of view of W L selection, requires a dedicated study within afully realistic pile-up simulation. Such study is currently under preparation. Triggeringefficiencies are not taken into account anywhere in this study, but purely leptonic decaymodes are obviously advantageous in this respect. They do not require any dedicatedVBF trigger based on hadronic signals. Instead, triggering on a single lepton should bequite enough and the fact that any of the two required leptons may fire the trigger makestrigger efficiency a minor issue. Since we want high- p T leptons, trigger thresholds shouldnot be a problem, either. V V scattering
The lepton-lepton invariant mass or transverse mass spectra (where applicable) of thesignal and backgrounds, after aplying all the selection criteria, are shown for all the
V V scattering processes in Figs. 6.5 thru 6.10. Respective signals were calculated within the .8. HIGGS COUPLINGS IN
V V
SCATTERING
HW W coupling be 0.7, 0.8 or 0.9 ofits SM value, which is still not ruled out by experiment, for W + W + in the purely leptonicdecay we can hope for a signal size of the order of 0.040, 0.020 or 0.008 fb, respectively,after all selection criteria; similar for W + W − , and about a quarter of that for W − W − .Total background levels may amount to 0.1 fb, 1.1 fb and 0.07 fb, respectively. Stickingto W + W + alone, this means roughly 12, 6 or 2 signal events after collecting 300 fb − ofdata over 30 Standard Model events. In terms of anomalous Higgs-to-gauge couplings andhaving in mind the present experimental bounds derived from Higgs measurements fromLHC Run 1, it already looks unlikely that W W scattering could provide a quantitativemeasurement on its own right. In order to observe frail hints of BSM anytime beforethe LHC enters in its High Luminosity regime (2025), it will be necessary to combinedifferent processes and different decay channels. Nonetheless, consistency cross checks to ∼
20% with precision measurements of Higgs production rates and decays will certainly beattainable and they should still be considered an important part of the physics programfor LHC Runs 2 and 3. Variations of the
HW W coupling of less than 20% will onlybe accessible with 3000 fb − of data. And of course, the more the Higgs boson appearsSM-like, the more confined gets W W scattering to the role of a consistency cross checkwith limited precision, as opposed to a true BSM search.In the ZZ channel, the primary focus will be direct search for new resonances. Onthe absence of such, BSM signal arising from a scaled HZZ coupling may consist of ahandful of events even after 3000 fb − . Not unexpectedly, the l + l − νν final state offers inprinciple more statistics than 4 l , but is harder to analyze. Here, however, special effortis required to include the semileptonic decays into the game. Under strict requirementsof two tagging jets in the endcaps, two additional jets in the barrel that reproduce the Z mass and no additional jet activity, QCD background levels may turn out controlable.Dedicated simulations are missing at the present moment.The W Z channel probes in principle both
HW W and
HZZ couplings in a combinedway, but its sensitivity is marginal. BSM signal levels are insufficient to be measurableeven with 3000 fb − .It is clear that the application of MVA’s enhances the possibilities to carry an optimalanalysis and isolate the signal. The final sensitivity depends on the kinematic separationof signal and background in the multidimensional phase space and on the overall signalstatistics, to a lesser degree on the amount of background. As long as the full kinematicinformation on each event is taken into account, quantitative results should in principlenot depend on the applied preselection of events, unless the latter suppresses too muchof the proper signal. It is therefore preferrable to reduce signal losses to minimum. Acut-based analysis, such as outlined in this work, represents in fact the current lowerlimit in the achievable sensitivity. In any case, what we have learned from the presentstudies, and will emphasize this point once again, is that a conventional analysis thatconsists of applying polarization-blind VBF selection criteria plus a shape analysis of thelepton-lepton invariant mass spectrum is suboptimal and should be replaced by a moresophisticated analysis that explores the full kinematics of the final state to deliver the bestfinal result. A lot of valuable information sits in particular in the jet spectra, far morethan whether the process was VBF or not. Correlated variables like R p T can be thoughtof as a first effective step towards fuller exploration of the entire kinematic phase of thefour particles in the final state. Given that BSM effects in W W scattering may sit at the12
CHAPTER 6. WHAT CAN THE LHC MEASURE
Figure 6.2:
Total cross sections for the signal and irreducible background after each subsequentclass of cuts proposed in the analysis. Red histograms are for pp → jjW + W + , blue histogramsfor pp → jjW + W − ; in both cases W decay into muons is assumed. The signal is calculatedunder the Higgsless hypothesis. Results reflect pure event kinematics, all detector efficienciesand inefficiencies (where appropriate) are assumed 100%. The meaning of the cut labels is thefollowing: • VBF: < | η j | < η j η j < | η µ | < . • ttbar: M jj >
500 GeV and M j µ , M j µ >
200 GeV, • ∆ ϕ : ∆ ϕ > . • R p T /leptonic: R p T > . W + W + or p µ T + p µ T > M µµ >
300 GeV for W + W − , • Mass/CJV: M µµ >
250 GeV for W + W + or centraljet veto with p T >
25 GeV for W + W − , • MET+ p j T : missing E T >
60 GeV and p j T > W decay into muons( jjW + W − only), the effects of parton showering, hadronization and jet reconstruction andfurther processed by PGS 4 for the effects of finite resolution in the measurement of jet andmuon p T in a CMS-like detector. The original PYTHIA 6 source code was modified to accountfor the correct, polarization-dependent, angular distributions for the decays W ± → µ ± ν . Thecorresponding results for the decays of W into electrons are typically consistent within a fewper cent and/or statistical fluctuations and are not shown here. .8. HIGGS COUPLINGS IN V V
SCATTERING
Total cross sections for the different main kinds of reducible background: inclusive t ¯ t , W +jets, QCD multijet and W Z , after each subsequent class of cuts proposed in the analysis.Red histograms are for pp → jjW + W + , blue histograms for pp → jjW + W − ; in both cases W decay into muons is assumed. Results reflect pure event kinematics, all detector efficiencies andinefficiencies (where appropriate) are assumed 100%. For details regarding this calculation andthe precise meaning of cut labels see caption of Fig. 6.2. CHAPTER 6. WHAT CAN THE LHC MEASURE
Figure 6.4: Total cross sections for the signal (red histograms) and irreducible background(blue histograms) after each subsequent class of cuts proposed in the analysis for theprocesses pp → jjW + Z → jjµ + µ + µ − (top), pp → jjZZ → jjµ + µ − µ + µ − (middle)and pp → jjZZ → jjµ + µ − νν (bottom) at 14 TeV. The signal is calculated under theHiggsless hypothesis. The meaning of the cut labels is the following: • VBF: < | η j | < η j η j < | η µ | < . • ttbar: M jj >
500 GeV and M j µ , M j µ >
200 GeV, • M jj : M jj >
500 GeV, • M Z : reconstructed Z mass(es) within 10 GeV, • M T : transverse mass(defined in detail in section 6.6) >
500 GeV, • M ZZ : M µ >
500 GeV, • p ZT : p ZT > M T for jjW + Z and jjZZ → jj l ν or > q M µ − M Z for jjZZ → jj l , • MET: missing E T >
50 GeV for jjW + Z or 250 GeV for jjZZ , • p lT : p lT >
40 GeV. Results of aMadGraph calculation, processed by PYTHIA 6 for W decay into muons ( jjZZ samplesonly), the effects of parton showering, hadronization and jet reconstruction and furtherprocessed by PGS 4 for the effects of finite resolution in the measurement of jet and muon p T in a CMS-like detector. .8. HIGGS COUPLINGS IN V V
SCATTERING pp → jjW + W + at 14 TeV, with W + → µ + ν (left) and with with W + → e + ν (right).Shown are the signal calculated under the Higgsless hypothesis and various contributionsto the background, normalized to 300/fb of data. Applied were respectively all the signalselection criteria foreseen for the same-sign muon channel (cuts 1-4 from Fig. 6.2, up toand including R p T ) and for the same-sign electron channel (all cuts 1-6 listed in Fig. 6.2).Signal was calculated by subtracting the SM jjW + W + sample (by definition also identicalwith irreducible background) from the Higgsless jjW + W + sample. The top productionbackground was simulated as described in section 6.2. The W Z background was obtainedfrom a dedicated jjW + Z sample with subsequent W and Z decays into muons. The b -tagging efficiency was assumed 50% for a single b quark, for the muon charge mis-ID probability a constant value of 0.3% was taken, all other efficiencies and purities wereassumed 100%. The W +jets background was deduced from dedicated jjjW + and jjW + γ samples, where either any of the jets or the photon was assumed to be misidentified asan electron. The QCD multijet background was deduced from a dedicated jjjj samplewhere any pair of jets was assumed to be simultaneously misidentified as electrons. Theprobability of a jet faking an electron was assumed 1 . · − with a further 27% probabilityof sign matching and that for a photon 0.7% with a 50% probability of sign matching.For a more detailed explanation of the procedure see sections 6.3 and 6.4. The electroncharge mis-ID probability for the evaluation of the top background was assumed to be 1%.Results of MadGraph simulations, processed by PYTHIA 6 for W decay into leptons (topproduction only), the effects of parton showering, hadronization and jet reconstructionand further processed by PGS 4 for the effects of finite resolution in the measurementof jet and lepton p T in a CMS-like detector. The original PYTHIA 6 source code wasmodified to account for the correct, polarization-dependent, angular distributions for theleptonic W decays.16 CHAPTER 6. WHAT CAN THE LHC MEASURE
Figure 6.6: Invariant mass distributions of the two leptons resulting from the process pp → jjW + W + at 14 TeV, with one W decaying into a muon and another decaying intoan electron (left) and with each W decaying either into a muon or an electron (right).Shown are the signal calculated under the Higgsless hypothesis and various contributionsto the background, normalized to 300/fb of data. For the left plot, applied were all thesignal selection criteria foreseen for the same-sign mixed muon+electron channel (cuts1-5 listed in Fig. 6.2). The W +jets background was deduced from dedicated jjjW + and jjW + γ samples, where either any of the jets or the photon was assumed to bemisidentified as an electron. The electron charge mis-ID probability for the evaluation ofthe top background was assumed to be 1%. The final µ + e + kinematics was deduced byaveraging out the distributions obtained in µ + µ + and the e + e + channels, which differedonly due to different detector resolution effects assumed during processing by PGS 4. Allthe remaining procedures and assumptions were identical as described in the caption ofFig. 6.5. The right plot was obtained by summing up the individual W decay channels. .8. HIGGS COUPLINGS IN V V
SCATTERING pp → jjW + W − at 14 TeV, with W ± → µ ± ν (left) and W ± → e ± ν (right). Shownare the signal calculated under the Higgsless hypothesis and various contributions tothe background, normalized to 300/fb of data. Applied were respectively all the signalselection criteria foreseen for the opposite-sign muon channel (cuts 1-5 from Fig. 6.2, upto and including CJV) and for the opposite-sign electron channel (all cuts 1-6 listed inFig. 6.2). Signal was calculated by subtracting a SM jjW + L W − L sample from a Higgsless jjW + L W − L sample. Irreducible background was calculated from a SM jjW + W − sample.The top production background was simulated as described in section 6.2. The W Z background was obtained from a dedicated jjW + Z sample with subsequent W and Z decays into muons and an additional factor 1.5 was assumed to account for jjW − Z (notsimulated). The b -tagging efficiency was assumed 50% for a single b quark, all otherefficiencies and purities were assumed 100%. The W +jets background was deduced froma dedicated jjjW + sample, where any of the jets was assumed to be misidentified as anelectron. An additional factor 1.5 was assumed to account for jjjW − (not simulated).The QCD multijet background was deduced from a dedicated jjjj sample where any pairof jets was assumed to be simultaneously misidentified as electrons. The probability ofa jet faking an electron was assumed 1 . · − with a further 73% probability of signmatching. For a more detailed explanation of the procedure see sections 6.3 and 6.4.Results of MadGraph simulations, processed by PYTHIA 6 for W decay into leptons(for signal, irreducible background and top production), the effects of parton showering,hadronization and jet reconstruction and further processed by PGS 4 for the effects offinite resolution in the measurement of jet and lepton p T in a CMS-like detector. Theoriginal PYTHIA 6 source code was modified to account for the correct, polarization-dependent, angular distributions for the leptonic W decays.18 CHAPTER 6. WHAT CAN THE LHC MEASURE
Figure 6.8: Invariant mass distributions of the two leptons resulting from the process pp → jjW + W − at 14 TeV, with one W decaying into a muon and another decaying intoan electron (left) and with each W decaying into either a muon or an electron (right).Shown are the signal calculated under the Higgsless hypothesis and various contributionsto the background, normalized to 300/fb of data. For the left plot, applied were all thesignal selection criteria foreseen for the opposite-sign mixed muon+electron channel, i.e.,cuts 1-5 from Fig. 6.2 (up to and including CJV). The final µe kinematics was deduced byaveraging out the distributions obtained in µ + µ − and the e + e − channels, which differedonly due to different detector resolution effects assumed during processing by PGS 4. Allthe remaining procedures and assumptions were identical as described in the caption ofFig. 6.7. The right plot was obtained by summing up the individual W decay channels. .8. HIGGS COUPLINGS IN V V
SCATTERING pp → jjW ± Z at 14 TeV, with subsequent leptonic decays of the gaugebosons. Shown are the signal calculated under the Higgsless hypothesis and the irreduciblebackground, normalized to 300/fb of data. Applied were all the signal selection criteriaforeseen for the W Z leptonic channel, no distinction was made between lepton flavors(e or µ ). Signal was calculated by subtracting the SM jjW + Z sample (by definitionalso identical with irreducible background) from the Higgsless jjW + Z sample. The crosssections were obtained by scaling the simulated jjW + Z sample by a factor of 1.5 toaccount for jjW − Z (not simulated). All the relevant efficiencies and purities were assumed100%. Results of MadGraph simulations, processed by PYTHIA 6 for parton showering,hadronization and jet reconstruction and further processed by PGS 4 for the effects offinite resolution in the measurement of jet and lepton p T in a CMS-like detector.20 CHAPTER 6. WHAT CAN THE LHC MEASURE
Figure 6.10: Invariant mass distrubution of the four leptons resulting from the process pp → jjZZ → jjl + l − l + l − (upper plot) and transverse mass distribution (defined in insection 6.6) from the process pp → jjZZ → jjl + l − νν (lower plot) at 14 TeV. Shownare the signal calculated under the Higgsless hypothesis and the irreducible background,normalized to 300/fb of data. Applied were all the signal selection criteria foreseen for the ZZ four-lepton channel and for the ZZ → l + l − νν channel, respectively; no distinction wasmade between lepton flavors (e or µ ). Signal was calculated by subtracting the SM jjZ L Z L sample from the Higgsless jjZ L Z L sample. All the relevant efficiencies and purities wereassumed 100%. Results of MadGraph simulations, processed by PYTHIA 6 for Z decayinto leptons, the effects of parton showering, hadronization and jet reconstruction andfurther processed by PGS 4 for the effects of finite resolution in the measurement of jetand lepton p T in a CMS-like detector. This study did not include the correct, polarization-dependent, angular distributions for the leptonic Z decays. Such effects are nonethelessunlikely to change any of our conclusions. .9. ANOMALOUS TRIPLE GAUGE COUPLINGS Whether or not we will be able to observe any
BSM signal is one question. Another one iswhether we will be able to correctly interpret the result in an independent and standaloneway, that is, not having to rely on other concurrent measurements and assume consistencywithin a given physics scenario.New physics may manifest itself e.g. in anomalous triple gauge couplings which maylikewise show up as an enhancement of
W W scattering at high invariant mass. Detailsof the shapes of kinematic distributions of the final state particles depend on the physicsscenario, but in general will also depend on specific values of the anomalous couplings,giving rise to annoying interpretative ambiguities. It is vital to study the dependencies ofthe individual kinematic variables and single out those of them that are mostly sensitiveto the scenario but not to numerical values and vice-versa.Updated 90% CL limits on the dimension-6 operators that lead to anomalous triplegauge couplings have been recently calculated [64]. We take the following values: c W W W / Λ ǫ [ − , .
9] TeV − , c W / Λ ǫ [ − . , .
6] TeV − , c B / Λ ǫ [ − , .
9] TeV − .The above limits come from a combination of LEP, the TeVatron and the LHC Run 1 data.They are asymmetric because in each case the central values of the relevant couplings weredetermined.It happens that anomalous couplings of roughly this size can produce BSM signalsin W W scattering of the same order of magnitude as would be produced by a
HW W coupling set to, e.g., 0.8 of its SM value. Assuming that the scale of new physics, Λ, isbeyond direct LHC reach and therefore that no cutoff is necessary in evaluating signalrates, the expected amount of signal for W + W + in the purely leptonic decay modes is closeto 0.050 fb for c W W W / Λ = − / TeV , 0.016 fb for c W / Λ = − / TeV (this includes0.011 fb of W L W L and 0.005 fb of W T W X signal) and 0.003 fb for c B / Λ = − / TeV . Inderiving these numbers we have applied exactly the same signal selection criteria as weused before, although W T W X signals may be possible to further improve with dedicatedoptimizations. As it was with HW W , the LHC sensitivity to triple gauge couplings in W + W + scattering is again at the very limit of present experimental bounds.The most important point is that signal in general, understood as a sum of all possibleBSM effects, may be as much manifest in W L W L pairs alone (if c B = 0), as in W T W X alone (if c W W W = 0), both W L W L and W T W X in roughly similar amounts (if c W = 0),or any combination of the above cases. Any of the above scenarios ultimately manifestsin an enhancement of W W scattering at high invariant mass, the latter being correlatedwith high lepton transverse momenta. Therefore, the R p T variable is still a good BSMprobe owing to its numerator. Because of the denominator, however, it always favors W L W L pairs over W T W X pairs. Without prior knowledge of the helicity composition22 CHAPTER 6. WHAT CAN THE LHC MEASURE
Figure 6.11: Distributions of log R p T in the process pp → jjW + W + at 14 TeV, withleptonic W + decay ( l = e, µ ) in different physics scenarios: Standard Model (black histo)and BSM signals for g HW W = 0 . c W / Λ = − / TeV (green histo) and c W W W Λ = − / TeV (blue histo). Applied were VBF topological cuts, including ∆ ϕ ll > W + decay, partonshowering, hadronization and jet reconstruction; no detector effects were included. Theoriginal PYTHIA 6 source code was modified to account for the correct polarization-dependent angular distributions for the W decays. Signals were calculated by subtractingthe SM sample from the corresponding BSM sample. In calculating the BSM distributionsit was assumed that the scale of new physics, Λ, is higher than the accessible energies andhence no cutoff was applied. .9. ANOMALOUS TRIPLE GAUGE COUPLINGS R p T distribution(bottom plot) in the process pp → jjW + W + at 14 TeV, with leptonic W + decay ( l = e, µ )in different physics scenarios. On the two upper plots, solid lines represent the leadingjet and lepton, respectively, and dashed lines represent the sub-leading jet and lepton,respectively. Vertical error bars represent the RMS. Each bin on the horizontal axisrepresents a physics scenario; from left to right: the Standard Model and BSM signals for g HW W = 0 . c W / Λ = − / TeV , c W W W / Λ = − / TeV and c W W W / Λ = − / TeV .Results of MadGraph 5 simulations, all conditions and assumptions as for Fig. 6.11.24 CHAPTER 6. WHAT CAN THE LHC MEASURE of the signal, an enhancement in the numerator may be just compensated by a largerdenominator. For a complete understanding of a future experimental result we need toexamine the individual transverse momenta of the four final state objects.What follows is a “quick and dirty” demonstration of principles how to extract themost physics information based on no more than four transverse momenta. Expressed ina logarithmic scale, individual p T distributions in any given physics scenario, as well asthe corresponding R p T distributions, can be to a first rough approximation described interms of two parameters: the mean value and the RMS, see Fig. 6.11. This allows firstsimple studies of kinematic separation between different scenarios, before more detailedanalyses become available. Since the expected signal statistics is anyway bound to besmall, any more detailed shape spectrum may not be even plausible in practice. Fromsuch comparisons (see Fig. 6.12 bottom) we infer that R p T by itself will be enough todistinguish the SM scenario from BSM with a handful of events, as long as the overallsignal size is significant enough. But it does not suffice to identify a BSM scenario.In particular, a pure c W W W scenario can produce an R p T spectrum indistinguishablefrom the one produced by an anomalous HW W scenario, despite the two signal samplesconsisting of different W polarizations. Moreover, outgoing lepton spectra are sensitiveboth to the physics scenario and to the numerical values of the anomalous coefficients andthis interplay is very sophisticated (see Fig. 6.12 middle). Consequently, e.g., numericalvariations within the c W W W scenario may be larger than differences between scenarios(cf., e.g., “
HW W ” with “ c W W W −
10” and “ c W W W −
5” in Fig. 6.12 bottom).Fortunately, jet transverse momenta are a direct measure of helicity composition of the
W W sample and nothing else. They do not depend on values of the anomalous coefficients(see Fig. 6.12 top). The trend is clear: the more W T the higher jet p T . A pure W L W L (“ HW W ”) signal gives log p j T = 1 . ± .
25 (RMS),log p j T = 1 . ± .
24 (RMS),while for a pure W T W X signal (e.g. “ c W W W − p j T = 2 . ± .
26 (RMS),log p j T = 1 . ± .
29 (RMS).Combined the information from the two jets, the two extreme cases can be distin-guished at a 3 σ level with as few as 7 isolated signal events and at 5 σ with 20 events.This will be possible with 3000 fb − , unfortunately not with 300 fb − . Approximatelyfour times this statistics is required to perform similar with a mixed helicity signal likein the case of c W = 0. From a back of the envelope calculation it follows that with 20isolated signal events (e.g., with g HW W = 0 . − ), the helicity compositionof the signal can be determined to better than 20%. With 60 events (e.g., g HW W = 0 . − ), it can be known to ∼ p T does increase with M W W for transverse polarization. This is why the leading jet p T of the c W signal is hardly different from the one in the SM, although the former has a relativelylarger W L W L component. But for any BSM scenario that manifests as a steady enhance-ment at high M W W up to the kinematic limit of available phase space, differences in thejet p T spectra for W T W X are effectively a second order effect. On the other hand, jet p T .10. ANOMALOUS QUARTIC COUPLINGS W L W L are very much independent of anything and are a unique signature oflongitudinal polarization.Once we have measured the helicity compisition and hence settled which BSM effectplays the dominant role (at least within the limited scope of scenarios considered here),the numerical value of the leading anomalous coefficient can be deduced by studying thelepton transverse momenta (or alternatively the lepton-lepton invariant mass, which isquite the same thing).In the above studies we have considered explicitly only negative values of the anoma-lous coefficients. Positive values of c W W W or c W happen to be more bound experimentallyand in addition the interference with SM diagrams is in this case destructive. The sig-nal will then consist of a slight depletion followed by very little enhancement within theallowed phase space, likely beyond LHC sensitivity.Variations of c B within the presently allowed range produce too little signal to bedetected at the LHC. As a matter of principle, c B affecting only W L W L pairs cannot bedistinguished from pure HW W scaling by means of the methodology proposed here.
Some of the dimension-6 operators discussed above modify also the gauge quartic cou-plings. It is conceivable that their values will be eventaully determined by non-VBSprocesses at the LHC and applied as background in more dedicated VBS analyses. Onthe other hand, quartic couplings can best de determined via VBS processes, along withtriboson production. Since sensitivity of VBS processes to anomalous triple couplings,including Higgs to gauge couplings, within their present bounds is rather slim, a largedeviation from SM predictions could in fact signal non-trivial contributions from physicsrelated to operators of yet higher dimension than 6, namely dimension 8. For this reasonit makes sense to go directly to the presently unbounded dimension-8 operators and studytheir potential consequences. These operators may affect VBS and triboson production,but not other processes. Such studies are currently in progress, some have already beenshown.In section 5.3 we have already discussed the LHC sensitivities to anomalous quarticcouplings based on an early study of W ± W ± carried at the phenomenological level. Mostrecently, sensitivity to quartic couplings in VBS processes has been studied by the Snow-mass 2013 study group [120]. Results were presented in the language of higher-dimensionoperators in Effective Field Theory. Studied were the respective sensitivities of the ZZ process to parameters f T, / Λ and f T, / Λ , of W Z to f T, / Λ and of W ± W ± to f T, / Λ .In Effective Field Theory, these coefficients scale dimension-8 operators L T, , L T, and L T, , respectively. In the same work, total cross sections for W ± W ± W Z and ZZ werecalculated varying many anomalous coefficients one at a time. Reportedly, f T, / Λ and f T, / Λ were the parameters that all the total cross sections were found most sensitiveto. The choice of f T, / Λ and f T, / Λ for ZZ was motivated by the fact that these pa-rameters are built uniquely from the neutral field strengths B µν and so they can only beprobed in this process. Note that all these operators are built from field strengths only( B µν or W µν ) rather than Higgs field derivatives, and consequently affect directly only thevertices involving transversely polarized states. The various anomalous coefficients were26 CHAPTER 6. WHAT CAN THE LHC MEASURE implemented in MadGraph matrix element calculations. Simulations included typicaldetector resolutions parameterized in the DELPHES program. Only irreducible back-grounds were considered, except for the W ± W ± process for which the analysis includedthe W Z background scaled by an additional factor 2 to account for other backgrounds.Applied were minimal selection criteria which consisted in principle of a jet-jet invariantmass cut, M jj > V V mass (for ZZ and W Z ) or of the 4-body invariant mass, M jjll (for W ± W ± ). Differ-ent pile-up conditions were simulated for the W ± W ± process, but results reportedly didnot vary much. The authors calculated also for each process the corresponding unitaritybounds as a function of the respective coefficient values. Results were presented withand without applying a sharp unitarity violation cutoff. Snowmass study results for 14TeV are summarized in Table 6.1. One notices in particular that an order of magnitudeincrease in integrated luminosity, between 300 and 3000 fb − , translates into an increaseof merely a factor ∼ σ
95% CL ZZ , f T, / Λ
300 fb − − − ZZ , f T, / Λ − − − ZZ , f T, / Λ
300 fb − − − ZZ , f T, / Λ − − − W Z , f T, / Λ
300 fb − − − W Z , f T, / Λ − − − W ± W ± , f T, / Λ
300 fb − − − W ± W ± , f T, / Λ − − − Table 6.1: The 5 σ significance discovery values and 95% CL limits for coefficienits ofdimension-8 operators with 300 and 3000 fb − of data at 14 TeV using different VBSprocess. Numbers in brackets correspond to imposing a unitarity violation cutoff. Resultsof the Snowmass13 study [120].Some other studies exist that include full detector simulation and improved backgroundevaluation. The ATLAS collaboration presented a new set of simulation-based studies for14 TeV, reformulated and updated after Higgs discovery [121]. Under the assumptionthat each scattering process will be mainly sensitive to new physics arising from just oneof those higher order operators, sensitivities to new physics have been evaluated for ZZ , W Z and W ± W ± in the purely leptonic decay modes. These studies were based on fullysimulated events, including detector effects related to jet clustering, pile-up, as well asparameterized reconstruction efficiencies and resolutions for the different physics objects.Early work was based on the EWChL approach and amplitude unitarization accordingto the model of Dobado et al. [97] to evaluate gauge boson scattering signals, and results .10. ANOMALOUS QUARTIC COUPLINGS a and a . More recently, a newer set ofanalyses was presented with all the results translated into the language of Effective FieldTheory.The quartic W W W W coupling was studied in terms of the f S, / Λ coefficient thatscales the effective operator O S, and is best probed via the same sign W ± W ± channel.Otherwise, the analysis was akin to the one carried by the Snowmass study. Various re-ducible backgrounds were estimated using a combination of simulation work with existingexperimental data. The most important of them were reportedly total jjW Z production, jjW ± W ± production via gluon exchange (QCD) graphs and several detector-dependentbackgrounds generally termed “mis-ID’s”. The latter class included photon conversion,jets faking leptons and lepton charge flips.Results were presented in terms of the 5 σ discovery reach and 95% confidence levelexclusion limits for expected luminosities of 300/fb and 3000/fb. With 300/fb, the 5 σ discovery limit was obtained at 10 TeV − , while the expected 95% CL exclusion limitis 6.8 TeV − . An order of magnitude increase in luminosity was found to translate intonearly an order of magnitude improvement of the exclusion limit, but only slightly morethan a factor 2 in the discovery reach. This is due to a non-trivial relation between f S, / Λ and the signal size. However, in the event of BSM observation with 300 fb − , theanomalous coefficient could be measured with a precision better than 5% with 3000 fb − ,which fully qualifies for the term precision measurement.Based on earlier studies at the phenomenological level, similar limits can be alsoexpected on f S, / Λ . It is important to realize that all these studies assume just one non-vanishing anomalous parameter at a time. It was also shown that f S, and f S, producesimilar signal and so are in fact anticorrelated. This anticorrelation is especially strong for W ± W ± . Combination of different scattering processes, in particular W ± W ± and W + W − ,is instrumental in restricting the allowed ranges of both parameters at a time so to becomparable with limits obtained from considering just one parameter at a time.The ATLAS study is the first complete detector-specific study of the unique physicscapabilities of W ± W ± scattering after Higgs discovery (unique in the sense that the samecannot be measured with possibly better precision in any other processes). Since a size-able fraction of the background is comprised by the various detector-specific “mis-ID’s”,ATLAS results cannot be directly transferred to CMS, but one can safely assume thatmore important at this stage is further detector-independent analysis optimization. Thepublished study most certainly keeps much room for improvements. First and foremost,it is polarization-blind. Moreover, a high p T threshold of 50 GeV for both tag jets wasused to protect against pile-up jets. This however at 14 TeV means automatical loss oftwo thirds of the W L W L sample. It should be noted that f S, produces BSM effects onlyin W L W L . Any lowering of the jet p T thresholds will reflect in improved sensitivities. Forexample, with the p T threshold applied on only one tag jet the W L W L scattering statisticsincreases by a factor 2. A one-dimensional evaluation of the 4-body invariant mass doesnot exploit the full relevant details of the final state kinematics. Lack of explicit consid-eration of ∆ ϕ ll allows a lot of unnecessary non-VBS contributions, in particular jjW Z ,without practically any gain in terms of signal. Finally, since f S, enhances high W W masses for W L W L pairs, it is plain to see that R p T will be as efficient a criterion to extractthe BSM signal as it was for the case of a scaled HW W coupling. The above remarksbecome even more important when one notices that the Snowmass study [120] revealed28
CHAPTER 6. WHAT CAN THE LHC MEASURE [TeV] jjll m1 2 3 4 5 E n t r i e s VBS ssWW (SM) -4 = 10 TeV S0 fSM VBS ssWW +SM ssWW QCDSM WZ + mis-ID Simulation Preliminary
ATLAS -1 L = 3000 fb ∫ ] -4 ) [TeV ν ± l ν ± l → ± W ± (VBS W Λ / S0 f0 2 4 6 8 10 ] σ S i gn i f i c an c e [ -1 -1
300 fb A TL A S SimulationPreliminary
Figure 6.13: Reconstructed 4-body mass spectrum in the SM and in the scenario with f S, / Λ = 10 TeV − (left) and BSM signal significance in standard deviations as a functionof f S, / Λ (right) from the pp → jjl ± l ± νν process at 14 TeV. Results of simulations doneby the ATLAS experiment, images reproduced from Ref. [121]. [TeV] ν m0.6 0.7 0.8 0.9 1 E n t r i e s -4 = 1.0 TeV T1 fSM VBS WZ +VBS WZ (SM)SM WZ QCD Simulation Preliminary
ATLAS -1 L = 3000 fb ∫ ] -4 ) [TeV - l + l ν ± l → Z ± (VBS W Λ / T1 f0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ] σ S i gn i f i c an c e [ -1 -1
300 fb
ATLAS
SimulationPreliminary
Figure 6.14: Reconstructed
W Z mass spectrum in the SM and in the scenario with f T, / Λ = 1 TeV − (left) and BSM signal significance in standard deviations as a functionof f T, / Λ (right) from the pp → jjl ± l + l − ν process at 14 TeV. Results of simulations doneby the ATLAS experiment, images reproduced from Ref. [121]. .10. ANOMALOUS QUARTIC COUPLINGS W ± W ± rate to f T, / Λ than to f S, / Λ or f S, / Λ .There may be no other way than to go to lower jet p T in order to conclude anything about f S, / Λ or f S, / Λ proper.Discovery reaches and expected exclusion limits for other dimension-8 operators werealso obtained from W ± Z → W ± Z and Zγγ production. From a study of
W Z scatteringexpected limits and discovery reaches were derived on f T, / Λ , for which this processis supposed to be most suited. It must be stressed however that the Snowmass studyrevealed W ± W ± be actually much more senisitive to this parameter than W Z . Againhere, selection criteria were reduced to a suitable combination of 3 leptons and a highjet-jet invariant mass. The expected 95% CL limits were found to be 0.7 TeV − and0.3 TeV − for 300 fb − −
1, respectively, while 5 σ discovery reaches were1.3 TeV − and 0.6 TeV − , respectively. The results are in fact in full agreement withthose of the Snowmass study. Because f T, directly affects only transversely polarizedpairs, high jet p T threshold may not be a problem here. However, drawing the resultfrom a one-dimensional analysis of the reconstructed W Z mass spectrum is beyond doubtsuboptimal.A similar study for the
W Z channel was recently presented by the CMS collaboration[122]. This analysis included fast simulation of detector response at high luminosity,with dedicated CMS-specific packages developed for the planned low luminosity and highluminosity phases of the LHC. More restrictive selection criteria than applied in theATLAS study included most of the typical VBF criteria for
W Z , namely ∆ η jj > Z mass. In addition,cuts were applied on the lepton-lepton and jet-lepton separation. The final result wasobtained from a shape analysis of the reconstructed one-dimensional W Z transverse massspectrum. This produced similar, if only slightly better, sensitivities than those reportedby ATLAS. A 5 σ discovery is expected for f T, / Λ down to 1 TeV − with 300 fb − andto 0.55 TeV − with 3000 fb − . This study too used a conservative jet p T threshold andmade no specific selection as to separate W/Z polarizations. Another interesting resultis that the Standard Model process of
W Z scattering after the proposed preselection canbe visible at 5 σ after collecting 185/fb of data. Hence, consistency with the SM can beprecisely tested in the event of absence of new physics.Once again it is important to realize the arbitrariness of such analyses in terms of theassumed choice of the appropriate parameter or parameters to be studied in relation toa particular scattering process. Strictly speaking, f S, and f S, modify also the W W ZZ vertex. Likewise, both vertices are sensitive to other dimension-8 operators, constructedeither from field strength tensors and Higgs field derivatives or from field strength tensorsalone. Most of these dependencies have not been explicitly studied so far. They maynot even add up coherently, but involve non-trivial interference effects. The task ofdisentangling the different contributions will be a long and complicated one. It willcertainly require a combination of all scattering processes. We are only at the beginningof the real work. The good news is that we can at least partly help this task witha technique to separate different polarization states via their respective jet transversemomentum spectra.Finally, even if we understand the VBS measurements as purely Standard Modelmeasurements, aimed solely at setting limits on various anomalous contributions (for theLHC such scenario cannot be disregarded), application of dedicated techniques to separate30
CHAPTER 6. WHAT CAN THE LHC MEASURE longitudinal from transverse gauge bosons will certainly result in better experimentallimits - at least for those operators which affect mainly V L V L pairs. hapter 7Beyond the LHC According to the European Strategy for Particle Physics, update of 2013,
Europe’s toppriority should be the exploitation of the full potential of the LHC, including the high-luminosity upgrade of the machine and detectors with a view to collecting ten times moredata than in the initial design, by around 2030 . The ultimate goal of the LHC is todeliver 3000 fb − of proton-proton data within the next 15-20 years, but until then beamenergy will stay at 13/14 TeV. For beyond the High Luminosity LHC timescale, the nextpriority outlined by the European Strategy for Particle Physics is pushing the energyfrontier. A “High Energy LHC” option has been studied as a possible next step after2035. The replacement of the NbTi dipole magnets with 20 T dipoles based on the novelHigh Temperature Superconductor (HTS) technology would allow reaching as high as33 TeV in the very same LHC ring. Meanwhile, a yet more ambitious project has beengaining momentum and attracting the attention of particle physicists. On February 12-15,2014, a kick-off meeting took place of the Future Circular Collider (FCC) Study groupin Geneva [123]. A total number of 341 physicists from all over the globe met to discussthe rationale and perspectives to build a new, more powerful collider that one day maypossibly become the LHC successor. The main aim of the project is building a new ringin the area of Geneva that will eventually collide protons at a center of mass energy of100 TeV. The process of setting up a new international collaboration was initiated. V V scattering at the FCC
The subject of Vector Boson Scattering is widely considered as one of the key physicstopics for the FCC (here and in what follows, we will refer as FCC to specifically theproton-proton option, more exactly known as FCC-hh - there exist also electron-electronand electron-hadron options that are being considered in parallel, possibly to be realizedsome day in the very same FCC tunnel).Producing any realistic simulation-based predictions for
W W scattering in proton-proton collisions at 100 TeV is connected to several theoretical issues. Event topologychanges as all the outgoing products of a collision get generally boosted more forward thanthey are in the LHC. This means in particular that the typical signature of a VBS event isnow modified so to extend to higher pseudorapidity of the two tagging jets. Early studiesindicate that the minimum jet pseudorapidity range to be covered extends at least up to | η | <
6. From the experimental point of view it is unlikely that useful jet reconstruction13132
CHAPTER 7. BEYOND THE LHC can be extended to yet higher | η | in a real detector, and so we will assume this minimumcoverage as a reasonable compromise between physics needs and technical possibilities.Leading order calculations of the process, say, pp → jjW + W + with the required kinematiccoverage may not be of enough accuracy for 100 TeV and use of NLO generators is officiallyencouraged and recommended by the FCC-hh group. However, because of relatively littleQCD contamination, the same-sign channel in the purely leptonic decay mode is the leastaffected by this uncertainty. Preliminary studies with the VBFNLO generator indicatethat LO versus NLO differences amount here to less than 10% [124]. The applicabilityof currently available PDFs constitute a source of additional uncertainty. All these issuesneed to be aggressively addressed. Furthermore, little is known at the present moment ofthe particle detectors and their performances for the FCC, although some first intelligentguesses as to what these potential detectors may (are bound to?) look like have alreadybeen presented. In any case, development of complete, realistic simulations for the FCC isa task for many years and many people. Nonetheless, it is already possible to get a roughglimpse of the possibilities and main advantages over the LHC. And once again here, wewill make the claim that a traditional analysis consisting of selecting VBF-topology eventsand studying the lepton-lepton invariant mass spectrum is by far a suboptimal strategy.The total cross section for pp → jjW + W + with two forward “tagging” jets is over 40times larger at 100 TeV than at 14 TeV. A hint of the FCC physics capabilities can begrasped by simply repeating quite the same analysis we have outlined in previous sectionsfor the LHC, with the slight alteration of the basic topological cuts which now will read:2.5 < | η j | < η j η j < | η l | < t ¯ t background for 100 TeV is currently missing; thereis however much to suppose that this background can be kept at a manageable levelhere too. The kinematic bounds from the top quark mass, which we have previouslyquantified as m j l <
200 GeV and m j l <
200 GeV are valid here as well, while thesignal region in 100 TeV collisions starts typically at significantly higher values. Stringentcuts like m jl >
400 or 500 GeV can be readily imposed if necessary to suppress the t ¯ t background to the desired level without being too costly to the signal. In all the followingconsiderations we will only discuss the signal and irreducible background, the latter beingdefined, as usual, as the Standard Model total W W production.It is plain to see that all the kinematic features that distinguish the BSM signal fromthe SM background are qualitatively still the same as we saw for 14 TeV. In particular, W L W L signal clearly populates a region of lower jet transverse momenta than W T W X background. The median of the leading jet p T distribution of the signal is found around100 GeV which poses no problems from the detector point of view. The sub-leadingjet distribution, however, has a median around 50 GeV. This means that requiring twotagging jets with p T >
50 GeV, as is assumed by default in some studies, automaticallymeans an unacceptable 50% signal loss in every study where we are interested in W L W L .Special effort must be dedicated in order to keep the machine pile-up under reasonablecontrol and make the low p T jets accessible to physics analysis. A machine operating ina 5 ns mode (option considered as a possible backup solution), and therefore having a5 times lower pile-up, would be clearly advantageous from this point of view. Ideally, .2. HIGGS TO GAUGE COUPLINGS AT THE FCC p T >
20 GeV level would be the goal that best corresponds to the physicsneeds. Alternatively, one should reconsider the concept of tagging only one forward jetand setting an algorithm to find the second jet off line. Such studies however have notbeen seriously started to the present moment. The leading lepton p T for the signal rangesvirtually from around 100 GeV above, hence triggering on a single lepton will not be aproblem . At the level of basic topological cuts, the two kinematic variables that offer the best sen-sitivity to the
HW W coupling are still ∆ ϕ ll and R p T . The former, as usual, selects hardscattering events, the latter separates BSM from SM contributions, the more effectivelyif BSM manifests in W L W L pairs. Since we are working here with energy-independentvariables, the respective signal and background regions can be taken to a first approxi-mation the same as we had before, before more detailed, dedicated optimization is done.Simple cuts like ∆ ϕ ll > . R p T > . HW W coupling being as close to the SM one as to afew per cent. The amount of irreducible background left will be close to 0.66 fb, whilesignal ranges from 2.54 fb to 0.69 fb, to 0.26 fb and to 0.06 fb for the scenario of the
HW W coupling being equal to 0.8, 0.9, 0.95 and 0.98 times its Standard Model value,respectively. For details, see Figs. 7.1 thru 7.4. Assuming an integrated luminosity of1000 fb − (the order of magnitude that is usually considered for the FCC-hh), a 3-4%deviation from the Standard Model coupling will be measurable with a 5 σ significance.Assuming 3000 fb − , we reach a 2% sensitivity. A combined shape analysis of the tworespective distributions will ultimately produce even more accurate results. As long aswe can restrict our analysis to the case of g HW W <
1, shape analysis in the lepton-leptoninvariant mass distribution gives little improvement in this measurement and is not muchmore efficient than a simple counting experiment as we have just done. Nonetheless, aninterpretative ambiguity may still exist between g HW W < g HW W > Z/γ exchange and the 4- W contact graphs on the other. To be more explicit, the spectrum montonously rises withrespect to the SM if g HW W <
1, while it initially falls and ultimately rises if g HW W > HW W coupling differs from unity enough to produce a statistically significant signal.Past it the g HW W > g HW W < If any triggering will still be used at all - some authors predict, based on the so called Moore’s law,that the need to have a first level trigger will disappear altogether by the time FCC-hh is starting. CHAPTER 7. BEYOND THE LHC
Figure 7.1: Distributions of pseudorapidities (two upper plots) and transverse momenta(two lower plots) of the leading and sub-leading jets from the pp → jjW + W + processat 100 GeV, with leptonic W + decay ( l = e, µ ). Shown are the distributions for theStandard Model irreducible background (black histos) and the BSM signal (red histos).BSM was defined in terms of the HW W coupling set to 0.9 of its SM value. Signalwas calculated by subtracting the SM sample from the BSM sample and multiplied by afactor 10 for better visibility. Only basic topological cuts were applied (see text). Resultof MadGraph simulations, processed by PYTHIA 6 for parton showering, hadronizationand jet reconstruction. No detector effects were taken into account. .2. HIGGS TO GAUGE COUPLINGS AT THE FCC
Distributions of transverse momenta of the leading and sub-leading (two upperplots) leptons, and invariant mass distributions of combinations of jets and leptons (two lowerplots) from the pp → jjW + W + process at 100 GeV, with leptonic W + decay ( l = e, µ ). Shownare the distributions for the Standard Model irreducible background (black histos) and the BSMsignal (red histos). BSM was defined in terms of the HW W coupling set to 0.9 of its SM value.Signal was calculated by subtracting the SM sample from the BSM sample and multiplied bya factor 10 for better visibility. Only basic topological cuts were applied. Result of MadGraphsimulations, processed by PYTHIA 6 for W decay into leptons, parton showering, hadronizationand jet reconstruction. The original PYTHIA 6 source code was modified to account for thecorrect, polarization-dependent, angular distributions for the W decays. No detector effectswere taken into account. CHAPTER 7. BEYOND THE LHC
Figure 7.3: Top: distribution of the lepton-lepton azimuthal separation, ∆ ϕ , and bottom:distribution of the ratio p l T p l T / ( p j T p j T ), from the pp → jjW + W + process at 100 GeV,with leptonic W + decay ( l = e, µ ). Shown are the distributions for the Standard Modelirreducible background (black histos) and the BSM signal (red histos). All assumptionsand conditions as in Fig. 7.2. For the lower plot, an additional cut on ∆ ϕ > . pp → jjW + W + process at 100 GeV,with leptonic W + decay ( l = e, µ ). Shown are the BSM signal (red histo) stacked on theStandard Model irreducible background (black histo). In addition to basic topologicalcuts, required was ∆ ϕ > . R p T > .2. HIGGS TO GAUGE COUPLINGS AT THE FCC S/ √ S + B ( S - BSMsignal, B - SM irreducible background) as a function of the actual value of the HW W coupling relative to its SM value, simulated in the pp → jjW + W + process at 100 TeV,with leptonic W + decay ( l = e, µ ), and assuming an integrated luminosity of 1000 fb − .All assumptions and conditions as in Fig. 7.4. In signal evaluation no unitarity cutoff wasapplied and so the leftmost points may be slightly overestimated.38 CHAPTER 7. BEYOND THE LHC
As mentioned before, the most sensitive probes of triple gauge couplings will come frommeasurements of total diboson production. Preliminary simulation work indicates thatthe relevant VBS modes will be able to independently cross check these results and pushthe sensitivity to well within the present limits in terms of anomalous operators O W W W , O W and even O B . Of course, these limits will be still improved by the LHC. More orless detailed quantitative estimates of such sensitivities are currently being worked outby many people, but perhaps are not the most urgent question at the present moment.In fact a far more important issue needs to be tackled. The analyses carried so far (likethe one we have just reported on in the previous section!) usually focus on a single BSMeffect or a single anomalous operator at a time. This is acceptable for LHC energieswhere the main question is whether we can observe any BSM effect given present boundson anomalous couplings, but our ability to identify a physics scenario wuthout relyingon other measurements is limited. The aim of the FCC is however to identify a physicsscenario. A single anomalous operator is unlikely what we will eventually observe in anexperiment. The key question to address is whether we can disentangle the different effectsfor a correct interpretation of the experimental result. This requires a careful comparativestudy of the phenomenology associated to the possible different scenarios.In the scenario with non-SM Higgs to gauge couplings, signal manifests solely in W L W L pairs rising with energy above SM prediction. It happens that a quartic W W W W cou-pling scaled by a constant factor will also be mostly observable in this way. The entirecontribution of the quartic vertex to the dominant W T W T scattering cross section is ratherminute, so the total rate varies very little with it. Mixed W T W L pairs get some energydependence in addition to an overall normalization shift, but this last effect is even lessappreciable than in W T W T because of lower absolute rates, and the first effect is dwarfedby a much steeper energy dependence coming from W L W L pairs. Consequently, it is thefunctional form of the W L W L energy dependence through which one must distinguish ascaled HW W coupling from a scaled
W W W W coupling. Various anomalous contribu-tions to the quartic coupling may however affect W T W X as well as W L W L .There is one result published from the Snowmass 2013 study [120] that is of directinterest for the FCC. The sensitivity of the W ± W ± scattering process to the coefficient f T, / Λ , measured in terms of the expected 5 σ discovery reach and 95% CL limit increasesby an impressive factor of 100 between 14 and 100 TeV, assuming the same integratedluminosity and excatly the same data analysis. By comparison, sensitivities to anomalouscoefficients studied in the W Z and ZZ processes were compared at beam energies of14 and 33 TeV and revealed improvement by merely a factor 1.2-1.8, depending on theanalysis. This already gives a glimpse of the superb physics capabilities of the FCC, butdoes not answer the question of being able to identify the scenario.A yet different story is the one with triple gauge couplings W W Z and
W W γ . Theyaffect W L W L , W T W T and W T W L pairs in different ways, as well as they affect both VBSand non-VBS processes. Since the VBS sample is a fraction of the non-VBS sample inabsolute counts and because the kinematics of their respective final states partly overlap,VBS signals can only be studied on their own right once stringent criteria are predefinedto suppress the unavoidable non-VBS contamination to a negligible level. For a correctevaluation of pure VBS signals, the non-VBS contribution must be negligible not only .3. ANOMALOUS GAUGE COUPLINGS AT THE FCC W ± W ± this can be effectively achieved bytightening the lepton back-to-back requirement to ∆ ϕ ll > .
8. This is because of twoclasses of non-scattering processes. One involves a u − u quark collision with one of thequarks interacting after W + emission, the other is u − ¯ d annihilation. They are negligiblein the SM, but become part of the non-VBS signal with anomalous W W Z and
W W γ couplings. Luckily, tightening ∆ ϕ ll does not significantly reduce the VBS signal at 100TeV. For the other V V scattering processes, the large number of diagrams potentiallycontributing to the non-VBS signal may prove this much more complicated.In any BSM scenario, new physics ultimately ends up enhancing the
V V scatteringcross section at a sufficiently high invariant mass. It remains true regardless of whetheror not this cross section gets depleted at some intermediate scale, depending on thesigns of the anomalous coefficients and therefore the pattern of interference between theindividual scattering diagrams. It is also true regardless of whether it is W L W L or W T W X the primary source of signal. It should be noted that even if W T W X contribute to the VBSsignal, it does not make W L W L any less important. Quite the contrary, the O W operatorproduces a similar amount of VBS signal for both helicity combinations, in clear contrastwith O W W W on one side and O B , O Φ d , O Φ W and the relevant dimension-8 operators onthe other. It makes the ability to separate the two samples experimentally a bonus ofspecial interest.As was the case for the LHC, use of the R p T variable, the way we did just above in thescenario with modified Higgs to gauge couplings, is effective for any BSM scenario thatenhances high W W invariant masses. This is because of the still holding strong correlationbetween M W W and the lepton transverse momenta. It is also always automatically moreeffective for W L W L signals than for W T W X signals because of the jet transverse momentain its denominator. However, if we allow both W L W L and W T W X signals, there is noway to separate these by looking at R p T alone without a priori knowledge of the physicsscenario. A study of respective signals associated to, e.g., the O W operator clearly showsthis interpretative ambiguity: the entire shape of the R p T distribution for a pure W L W L sample with, say, C W / Λ = -10/TeV almost exactly coincides with that of a pure W T W X sample with C W / Λ = -20/TeV (see Fig. 7.6). The ambiguity is solved by lookingat the individual jet transverse momenta, and chiefly the p T of the leading jet. Theseparation of the two helicity sub-samples is much better for 100 TeV than for 14 TeV.The maximum of the leading jet p T distribution is clearly shifted with respect to the SM tolower values for W L W L signals and to higher values for W T W X signals. A shape analysisof the leading jet p T distribution, measured from the events that pass the standard R p T cut, should suffice to determine the helicity composition to a satisfactory accuracy, enoughto resolve whether the signal is indeed W L W L -driven (via O B , pure HW W or W W W W )or W T W X -driven (via O W W W ) or mixed (via O W or any suitable combination). A peakat around 100 GeV of the measured excess over the SM (itself having a median around200 GeV) is an unequivocal sign of W L W L . A broader peak at around 300 GeV signals W T W X . The signal peak positions link directly to helicity and hardly vary with theactual physics scenario or specific values of the anomalous coefficients. The fact that40 CHAPTER 7. BEYOND THE LHC
Figure 7.6: The shapes of R p T distributions resulting from the pp → jjW + W + processat 100 TeV with leptonic W + decay ( l = e, µ ). Shown are: the Standard Model scenario(black histo, all helicity combinations summed up), the C W / Λ = -10/TeV additivesignals (blue solid histo - W T W X pairs, red solid histo - W L W L pairs) and the C W / Λ =-20/TeV additive signals (blue dashed histo - W T W X pairs, red dashed histo - W L W L pairs). For the sake of a convenient comparison, each distribution was individually scaledto the contents of its first bin, R p T <
2. VBF topological cuts (see text) and a cut on∆ ϕ ll > . .3. ANOMALOUS GAUGE COUPLINGS AT THE FCC pp → jjW + W + process at 100 TeV with leptonic W + decay ( l = e, µ ). Shownare the Standard Model scenario (black histo, all helicity combinations summed up),and the additive signals of C W / Λ = -10/TeV (blue histo - W T W X pairs, red histo - W L W L pairs). Applied were all signal selection criteria discussed in the text. Results ofMadGraph simulations, all assumptions and conditions as in Fig. 7.6.42 CHAPTER 7. BEYOND THE LHC
Figure 7.8: Mean values and RMS of the individual transverse momenta of the two jetsin the process pp → jjW + W + at 100 TeV, with leptonic W + decay ( l = e, µ ) in differentphysics scenarios. Solid lines represent the leading jet and dashed lines represent thesub-leading jet. Vertical error bars represent the RMS. Each bin on the horizontal axisrepresents a physics scenario; from left to right: the Standard Model and BSM signalsfor c B / Λ = − / TeV , c B / Λ = − / TeV , c B / Λ = 10 / TeV , c W / Λ = − / TeV , c W / Λ = − / TeV , c W / Λ = 10 / TeV , c W W W / Λ = − / TeV c W W W / Λ = − / TeV and c W W W / Λ = 2 / TeV . Applied were VBF selection criteria, including ∆ ϕ ll > . .3. ANOMALOUS GAUGE COUPLINGS AT THE FCC c W / Λ (see Fig. 7.8), is becausethe proportion of the selected W L W L and W T W X pairs changes likewise. The RMS ofthe log ( p j T ) distributions are close to 0.3 and 0.4 for W L W L and W T W X , respectively,making a clear distinction possible whenever signal itself becomes statistically significant.Larger widths are already a clear indication that signal is in fact a mixture of W L W L and W T W X , with two distinct sub-samples vaguely emerging from the spectrum. The sub-leading jet is a less powerful discriminator on its own because it receives a substantial W L contribution from W T W L pairs. Nonetheless it can be used as an additional consistencycross check. Once combined the information from the two jets, it turns out that as few as10 events suffice to distinguish a pure W L W L from a pure W T W X signal at the 5 σ level.In other words, with an isolated signal sample of N events, the helicity composition canbe deduced to a precision of ∼ / √ . N . That makes, e.g., for HW W = 0 .
95 and noother anomalous couplings (260 signal events in 1000 fb − ) a 4% measurement.In a given physics scenario, W L W L and W T W X signals do not differ significantly interms of the outgoing lepton kinematics. Somewhat different widths of the respective p T distributions (larger for W T W X than for W L W L ) are expected as a simple consequence ofthe angular distributions in W decay. These differences alone may however easily provenot significant enough or too entangled with other effects to be of much practical useunless we know beforehand the helicity composition of the selected sample from othersources. But once we have independently established the helicity composition, leptonsin the final state help resolve the remaining ambiguities concerning the physical scenarioand the sign of the anomalous coefficients. The W L W L -driven scenarios clearly differ inthe lepton transverse momenta and/or invariant mass distributions. For example, the me-dian of the leading lepton p T distribution is around 600 GeV for a purely HW W -drivensignal, but may become larger with new physics manifesting itself in modified
W W W W or W W Z couplings. These numbers are a direct consequence of the s -dependences of therelevant amplitudes plus a common phase space factor, and so they have also very littlesensitivity to the actual values of the parameters in question. It is the total signal ratethat determines the coefficient values. The lepton-lepton invariant mass also unambigu-ously fixes the sign of the relevant anomalous parameter in case an ambiguity exists onmeasuring the signal rate alone. Because of the sign of the interference terms betweenthe three basic graphs contributing to W + W + → W + W + , any of the following scenarios: g HW W < C W < C B <
0, produces a steady enhancement in the M ll spectrum,ultimately suppressed by phase space. By contrast, g HW W > C W > C B > C W W W , thesign may be more difficult to determine. In any case, there is a strong correspondencebetween invariant mass distrubutions and transverse momenta distrubutions of the out-going leptons and detailed simulation work will ultimately have to tell which approach ispreferrable.There are however also ambiguities that would take more effort to resolve. For exam-ple, the signal of C B < W W coupling by anappropriately chosen constant, although the former does not modify the
HW W couplingat all. Such ambiguities may be ultimately solvable only in the context of combining datafrom different processes.44
CHAPTER 7. BEYOND THE LHC
Figure 7.9: Transverse momentum distributions of the leading and subleading leptons(upper and middle plots) and lepton-lepton invariant mass distribution (lower plot) inthe pp → jjW + W + process at 100 TeV with leptonic W + decay ( l = e, µ ). Shown arethe Standard Model scenario (black histo, all helicity combinations summed up), and theadditive W L W L signals of C B / Λ = -20/TeV (red histo), C B / Λ = 20/TeV (green histo)and g HW W = 0 .
93 (blue histo). Applied were all signal selection criteria discussed in thetext. Results of MadGraph simulations, all assumptions and conditions as in Fig. 7.6. .3. ANOMALOUS GAUGE COUPLINGS AT THE FCC c/ Λ or c/ Λ , but one cannot separate the coefficient from the energy. However,if the data fit the theoretical curve in the entire kinematic phase space covered by theexperiment, then Λ must be at least equal to the W W invariant mass of the highest datapoint. Otherwise, data would indicate the appropriate cutoff value. On the other end,the value of Λ must be lower than a calculable upper limit defined by the unitarity condi-tion. Hence one could at least bound Λ from above and below. Nevertheless, for practicalpurposes the unitarity condition may be completely irrelevant in terms of evaluating theexpected sensitivity limits, because the FCC sensitivity reaches to anomalous coefficientvalues that do not lead to unitarity violation within the available energy scale [120].All in all, we have emphasized the primary importance of studying same-sign
W W interactions at the FCC. Moreover, it is the full event kinematics studied from a clean W + W + scattering sample with leptonic W decay, and most of all the transverse momentaof all four final state particles, that carry the bulk of the necessary information in orderto disentagle the underlying physics scenario and correctly interpret the results.46 CHAPTER 7. BEYOND THE LHC hapter 8Summary
The Higgs boson is an empirical fact. Moreover, based on all the data collected at Run 1of the LHC, it looks by all means consistent with the Standard Model one. In particular,Higgs couplings to vector bosons are consistent with SM ones to an accuracy of roughly ∼ V V scattering at high energies willultimately tell if this is indeed the case.The full phenomenology of
V V scattering at high energy depends on the Higgs mass,Higgs to gauge couplings, gauge boson triple couplings and gauge boson quartic couplings.With the present experimental bounds on these inputs, only the Higgs mass can be con-sidered definitely fixed for VBS studies. Effects from non-SM Higgs couplings and triplecouplings can still be observed at the LHC with √ s = 13 TeV with hard work and someluck and consistency checks can be done with new, more precise measurements of all therelevant quantities that will come directly from Higgs physics on one side and total di-boson production on the other. Agreement between these three classes of measurementscan be translated into the first real experimental limits on anomalous quartic gauge cou-plings. Alternatively, disagreement may signal existence of the latter. In the event ofabsence of direct observation of new resonances, the best process to study VBS-relatedphysics is same-sign W W scattering in the purely leptonic decay mode, but with furtherimprovements the semi-leptonic decay modes may prove equally important. However,with 300 fb − VBS processes on their own offer little possibilities to interpret the resultsin a standalone way, i.e., without relying on concurrent measurements. This is beacusethe BSM effects are bound to be tiny and statistics too low to carry more precise studies.The 300 fb − program is likely to end up as a Standard Model measurement, of a similarphilosophy as the ones already carried by ATLAS and CMS from 8 TeV data. However,the focus for the High Luminosity LHC program should be BSM and it is time now toplan an analysis strategy different from a Standard Model analysis. The High Luminos-ity program has chances to provide enough data for at least some physics scenarios bedistinguished from others based on studies of VBS processes alone. Among other things,this can be done by applying novel techniques to separate different helicity combinationsin the selected samples of V V pairs that we advocate in this work. In particular, shouldan excess over SM predictions be observed, it should be possible to tell whether this ex-14748
CHAPTER 8. SUMMARY cess is related to the mechanism of electroweak symmetry breaking or to other physics.Improvement in the sensitivity to new physics, and especially to those effects that affectmainly V L V L pairs, should be sought at low transverse momenta and large pseudorapidi-ties of the tagging jets. This should be taken into account in planning future machineand detector upgrade activities for the HL-LHC phase. Not least, even in the absence ofnew physics, application of analysis techniques that fully exploit vector boson helicitieswill result in better exclusion limits, at least for those scenarios that do not modify thedominant transverse polarizations.A qualitative improvement in sensitivity to BSM effects in VBS processes can only beachieved via further increase in beam energy. The FCC with its √ s = 100 TeV has allthe potential to observe many BSM effects in V V scattering and to identify the physicalsources of these effects. Consistency of VBS measurements with Higgs physics, dibosonproduction and triboson production measurements at the FCC will provide an ultimateclosure test of the Standard Model or the theory that will replace it. hapter 9Acknowledgments
The list of people I feel indebted to is long and, as always, subjective. But there are afew names I definitely must mention here.First and foremost, I thank Profs. Stefan Pokorski, Jan Kalinowski and S lawek Tkaczykfor their long collaboration, countless meetings and discussions throughout the years,always inspiring and enlightening. The complicated interplay between theory and ex-periment, the conceptual feedback I have received on the theory side (SP, JK) and thesupport on the practical, experimental side (ST), was a very stimulating experience andcrucial to complete this work. To Jan and S lawek I am also grateful for kindly agreeingto review parts of this work before publication and sending me their valuable comments.Any remaining mistakes, typos and other shortcomings that no doubt can still be found,are of course my fault.I thank my CMS colleagues from the NCBJ and the University of Warsaw for theirtolerance and patience to put up with me during all this time, as well as for all theirintellectual feedback. In particular, I am indebted to Prof. Krzysztof Doroba for hispioneering role in starting up the
W W related activities within the Warsaw CMS group,and to Prof. Jan Kr´olikowski, Micha l Bluj, Artur Kalinowski, Marcin Konecki, PiotrZalewski and many others for sharing their wisdom and experience in our group meetingsand informal discussions.And I thank Mayda Velasco, my former boss at Northwestern, for it is there that myadventure with CMS has started. 14950
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