Automatic predictions in the Georgi-Machacek model at next-to-leading order accuracy
Celine Degrande, Katy Hartling, Heather E. Logan, Andrea D. Peterson, Marco Zaro
IIPPP/15/72, DCPT/15/144, MCnet-15-34
Automatic predictions in the Georgi-Machacek modelat next-to-leading order accuracy
C´eline Degrande, ∗ Katy Hartling, † Heather E. Logan, ‡ Andrea D. Peterson, § and Marco Zaro
3, 4, ¶ Institute for Particle Physics Phenomenology, Department of Physics,Durham University, Durham DH1 3LE, United Kingdom Ottawa-Carleton Institute for Physics, Carleton University,1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada Sorbonne Universit´es, UPMC Univ. Paris 06, UMR 7589, LPTHE, F-75005, Paris, France CNRS, UMR 7589, LPTHE, F-75005, Paris, France (Dated: December 3, 2015)We study the phenomenology of the Georgi-Machacek model at next-to-leading order (NLO)in QCD matched to parton shower, using a fully-automated tool chain based on
Mad-Graph5 aMC@NLO and
FeynRules . We focus on the production of the fermiophobic custodialfiveplet scalars H , H ± , and H ±± through vector boson fusion (VBF), associated production witha vector boson ( V H ), and scalar pair production ( H H ). For these production mechanisms wecompute NLO corrections to production rates as well as to differential distributions. Our resultsdemonstrate that the Standard Model (SM) overall K -factors for such processes cannot in generalbe directly applied to beyond-the-SM distributions, due both to differences in the scalar electroweakcharges and to variation of the K -factors over the differential distributions. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] a r X i v : . [ h e p - ph ] D ec
1. INTRODUCTION
A deeper understanding of the scalar sector is a primary objective of the CERN Large Hadron Collider (LHC). Inaddition to precisely measuring the 125 GeV Higgs boson, Run II of the LHC will dedicate its efforts to searching forsigns of additional Higgs particles, which arise in a number of beyond-the-Standard-Model (BSM) scenarios. One suchscenario is the Georgi-Machacek (GM) model [1, 2], which extends the Standard Model (SM) with two scalar isospintriplets in a way that preserves the SM value of ρ = M W /M Z cos θ W = 1 at tree level. The phenomenology of theGM model has previously been studied in Refs. [3–28], including the application of a variety of constraints upon themodel parameter space. It has been shown to possess a decoupling limit, and can thus accommodate an SM-like 125GeV boson [21]. Furthermore, the tree-level couplings of this SM-like Higgs to fermions and vector bosons may beenhanced in comparison to the SM [25], a feature that cannot be accommodated in models that contain only scalarsin SU (2) singlet or doublet representations. The GM model can also be embedded in more elaborate theoreticalscenarios, such as little Higgs [29, 30] and supersymmetric [31–33] models, or generalized to larger SU (2) multiplets[34].The Georgi-Machacek model provides a useful benchmark framework for BSM Higgs searches. In addition to anSM-like scalar singlet h , the GM model also contains an extra scalar singlet H , a triplet H , and a fiveplet H underthe custodial symmetry. The structure of the model with respect to the custodial singlet and triplet states is similar tothat of the two Higgs doublet model (2HDM); as a result, the experimental searches and extensive analysis for 2HDMstates can often be recast in terms of the GM singlet and triplet scalars [25]. It is therefore particularly interesting tofocus on the custodial fiveplet states, H , H ± and H ±± . These scalars are fermiophobic and couple preferentially tovector bosons. As a result, the GM fiveplet contains two features that are absent from both the SM and the 2HDM:a doubly charged scalar H ±± and charged scalar states that couple to vector bosons. Consequently, the fermiophobicfiveplet states are produced primarily through the vector boson fusion (VBF) and associated production ( V H ) modes.This is in contrast to the 2HDM, where the heavy scalars are dominantly produced through associated productionwith a top quark or in top decays. These features lead to unique phenomenology and can be used to parametrizeeffects not captured by other common benchmark models.For the Georgi-Machacek model to be truly useful as an LHC benchmark, efficient and accurate calculationsmust be accessible to both phenomenologists and experimentalists. Great strides have been made in reducing boththeoretical and experimental uncertainties, making next-to-leading order (NLO) or higher order calculations standardpractice. Therefore, we describe the use of a fully-automated tool chain (which combines the FeynRules [35]and
MadGraph5 aMC@NLO [36] frameworks with the calculator
GMCALC [37]) to produce NLO differentialdistributions in the GM model, focusing on the examples of VBF,
V H , and H H production of the fiveplet states.In particular, we illustrate the insufficiency of extending the SM overall K -factors to BSM distributions, due totwo factors. First, differential K -factors can vary substantially for certain distributions (particularly in the case ofVBF). Second, the overall K -factors for differently-charged states can be somewhat different. These considerationsare important for accurately determining the effects of typical selection cuts, which is essential for measuring newstates in the event of a discovery.This paper is organized as follows. In the following section, we describe in more detail the scalar potential,spectrum, and couplings of the Georgi-Machacek model. In Sec. 3, we then outline the tools used for our fully-automated NLO calculations. Finally, in Secs. 4, 5 and 6, we present cross sections, K -factors, and differentialdistributions for VBF, V H , and pair production ( H H ), respectively, of the fiveplet states. We conclude in Sec. 7.For completeness, some details of the scalar potential of the GM model are collected in an appendix. The model filesfor the automated tool chain used to produce these results are publicly available on http://feynrules.irmp.ucl.ac.be/wiki/GeorgiMachacekModel .
2. THE MODEL
The scalar sector of the GM model [1, 2] consists of the usual complex isospin doublet ( φ + , φ ) with hypercharge Y = 1, a real triplet ( ξ + , ξ , ξ − ) with Y = 0, and a complex triplet ( χ ++ , χ + , χ ) with Y = 2. The doublet isresponsible for the fermion masses as in the SM.The scalar potential is chosen by hand to preserve a global SU (2) L × SU (2) R symmetry. This ensures that ρ = M W /M Z cos θ W = 1 at tree level, as required by precise experimental measurements [38]. In order to makethe global SU (2) L × SU (2) R symmetry explicit, we write the doublet in the form of a bidoublet Φ and combine the We normalize the hypercharge operator such that Q = T + Y/ triplets to form a bitriplet X : Φ = (cid:32) φ ∗ φ + − φ + ∗ φ (cid:33) , X = χ ∗ ξ + χ ++ − χ + ∗ ξ χ + χ ++ ∗ − ξ + ∗ χ . (1)The vacuum expectation values (vevs) are defined by (cid:104) Φ (cid:105) = v φ √ I × and (cid:104) X (cid:105) = v χ I × , where I is the unit matrixand the Fermi constant G F fixes the combination of vevs, v φ + 8 v χ ≡ v = 1 √ G F ≈ (246 GeV) . (2)These vevs are parametrized in terms of a mixing angle θ H according to c H ≡ cos θ H = v φ v , s H ≡ sin θ H = 2 √ v χ v . (3)The quantity s H represents the fraction of the squared gauge boson masses M W and M Z that is generated by thevev of the triplets, while c H represents the fraction generated by the usual Higgs doublet. The most general scalarpotential that preserves the custodial SU (2) symmetry may be found in Appendix A.After symmetry breaking, the physical fields can be organized by their transformation properties under the custodial SU (2) symmetry into a fiveplet, a triplet, and two singlets. The fiveplet and triplet states are given by H ++5 = χ ++ , H +5 = ( χ + − ξ + ) √ , H = (cid:114) ξ − (cid:114) χ ,r ,H +3 = − s H φ + + c H ( χ + + ξ + ) √ , H = − s H φ ,i + c H χ ,i , (4)where we have decomposed the neutral fields into real and imaginary parts according to φ → v φ √ φ ,r + iφ ,i √ , χ → v χ + χ ,r + iχ ,i √ , ξ → v χ + ξ . (5)The states of the custodial fiveplet ( H ±± , H ± , H ) have a common mass m and the states of the custodial triplet( H ± , H ) have a common mass m . Because the states in the custodial fiveplet contain no doublet field content, theydo not couple to fermions (i.e. they are fermiophobic).The two custodial singlets mix by an angle α , and the resulting mass eigenstates are given by h = cos α φ ,r − sin α H (cid:48) , H = sin α φ ,r + cos α H (cid:48) , (6)where H (cid:48) = (cid:114) ξ + (cid:114) χ ,r . (7)We denote their masses by m h and m H . The singlet h is normally identified as the 125 GeV SM-like Higgs bosondiscovered at the LHC [39–41]. Formulae for the masses m h , m H , m , and m , as well as the mixing angle α , may befound in Appendix A.The fiveplet states couple to vector bosons according to the following Feynman rules [8, 21, 42]: H W + µ W − ν : (cid:114) ig v χ g µν = 2( √ G F ) / M W (cid:18) − √ s H (cid:19) ( − ig µν ) , (8) H Z µ Z ν : − (cid:114) i g c W v χ g µν = 2( √ G F ) / M Z (cid:18) √ s H (cid:19) ( − ig µν ) , (9) H +5 W − µ Z ν : −√ i g c W v χ g µν = 2( √ G F ) / M W M Z ( s H )( − ig µν ) , (10) H ++5 W − µ W − ν : 2 ig v χ g µν = 2( √ G F ) / M W (cid:16) −√ s H (cid:17) ( − ig µν ) , (11)where we write the coupling in multiple forms to make contact with the notation of Refs. [8, 43]. The triplet vev v χ is called v (cid:48) in Ref. [8], and the factors F V V in Eq. (5.2) of Ref. [43] correspond in this model to F W + W − = − √ s H ( H production) , (12) F ZZ = 2 √ s H ( H production) , (13) F W ± Z = s H ( H ± production) , (14) F W ± W ± = −√ s H ( H ±± production) . (15)Note in particular that, for H , one cannot simply rescale the vector boson fusion cross section of the SM Higgs bosonbecause the ratio of the W W and ZZ couplings is different than in the SM.Additionally, two fiveplet scalars may also couple to a single vector boson through the following interactions: γ µ H +5 H + ∗ : ie ( p + − p + ∗ ) µ , (16) γ µ H ++5 H ++ ∗ : 2 ie ( p ++ − p ++ ∗ ) µ , (17) Z µ H +5 H + ∗ : ie s W c W (1 − s W )( p + − p + ∗ ) µ , (18) Z µ H ++5 H ++ ∗ : ies W c W (1 − s W )( p ++ − p ++ ∗ ) µ , (19) W + µ H + ∗ H : √ ie s W ( p + ∗ − p ) µ , (20) W + µ H +5 H ++ ∗ : ie √ s W ( p + − p ++ ∗ ) µ , (21)where all fields are incoming and in each case p Q is the incoming momentum of the scalar with charge Q . Note thatthese are independent of the mixing angle s H .There are theoretical constraints on the Georgi-Machacek model from considerations of perturbativity and vacuumstability [7, 13, 21], as well as indirect experimental constraints from the measurements of oblique parameters ( S , T , U ), Z -pole observables ( R b ), and B -meson observables [6, 13–15, 19, 25]. Currently the strongest of the indirectexperimental bounds arises from measurements of b → sγ , which constrain the triplet vev v χ ≤
65 GeV ( s H ≤ .
75) [25]. Additionally, the ATLAS like-sign
W W jj cross-section measurement, reinterpreted in the context of theGM model in Ref. [23], excludes a doubly-charged Higgs H ±± with masses in the range 140 ≤ m ≤
400 GeV at s H =0 .
5, and 100 ≤ m ≤
700 GeV at s H = 1, under the assumption of a 100% branching fraction for H ++5 → W + W + .An ATLAS search for singly charged scalars in the VBF production channel similarly excludes 240 ≤ m ≤
700 GeVfor s H = 1 under the assumption of a 100% branching fraction for H +5 → W + Z [44]. Additional constraints on v χ asa function of the BSM Higgs masses have been obtained in Ref. [26] using ATLAS data from several search channels.For the simulations that follow, we consider a single benchmark point in the GM model, generated using thecalculator GMCALC [37]. This point is allowed by all the constraints discussed above. We use the following valuesfor the scalar masses, mixing angles, and additional parameters M , as inputs: m h = 125 GeV , sin α = − . ,m H = 288 GeV , sin θ H = 0 . ,m = 304 GeV , M = 100 GeV ,m = 340 GeV , M = 100 GeV . (22)The parameters M , are dimensionful parameters in the scalar potential [see Eq. (A1)] that affect the values of thecouplings between scalars. The corresponding values for the underlying parameters of the scalar potential are givenin Appendix A. While we specify the complete parameter set, note that all the H V V couplings are proportional to s H . Therefore, both the VBF and V H production cross sections of the H states depend only on two parameters, s H and m , and the H H production cross sections depend only on m . At this parameter point the total widths of the As we consider only the fiveplet states in this work, we quote only the relevant interactions involving H scalar states and gauge bosons.A full set of Feynman rules for the GM scalar couplings may be found in Ref. [21]. Our benchmark point corresponds to the default point in
GMCALC . The choice of masses, mixing angles, and M , as input parameterscorresponds to the GMCALC input set 3. H states are about 0.3 GeV; therefore in our simulations we will take the final-state H particle(s) to be producedon shell.Finally, we choose the following set of SM inputs: M W = 80 .
399 GeV , M Z = 91 .
188 GeV , Γ W = 2 .
085 GeV , Γ Z = 2 .
495 GeV ,G F = 1 . × − GeV − . (23) α EM = 1 / .
35 is computed at tree level from M W , M Z , and G F .
3. COMPUTATIONAL FRAMEWORK
In this work we take advantage of a fully automated framework developed to study the phenomenology of BSMprocesses at NLO accuracy in QCD, including the matching to parton shower (PS). The framework is based on
MadGraph5 aMC@NLO [36]. In order to generate a code capable of computing NLO corrections to a BSMprocess, some extra information has to be provided besides the usual tree-level Feynman rules. This extra informationinvolves the ultraviolet (UV) renormalization counterterms and a subset of the rational terms that are needed in thenumerical reduction of virtual matrix elements (which are normally referred to as the R terms) [45]. The calculationof the UV and R terms starting from the model Lagrangian has been automatized via the NLOCT package [46],based on
FeynRules [35] and
FeynArts [47]. Once the UV and R Feynman rules have been generated, they areexported together with the tree-level Feynman rules as a
Python module in the Universal FeynRules Output (
UFO )format [48]. The
Python module can be loaded by any matrix-element generator, such as
MadGraph5 aMC@NLO .When the code for the process is written, the
UFO information is translated into helicity routines [49] by
ALOHA [50].
MadGraph5 aMC@NLO is a meta-code that automatically generates the code to perform the simulation of anyprocess up to NLO accuracy in QCD. The simulation can be performed either at fixed order or by generating eventsamples which can be passed to PS. The automation of the NLO QCD corrections has been achieved by exploiting the
FKS [51, 52] subtraction scheme to subtract the infrared singularities of real-emission matrix elements, as automated in
MadFKS [53]. Loops are computed by
MadLoop [54], which exploits the OPP [55] method as well as Tensor IntegralReduction [56, 57]; these are implemented in
CutTools [58] and
IREGI [59] respectively, which are supplemented byan in-house implementation of
OpenLoops [60]. Finally, the event generation and matching to PS is done followingthe
MC@NLO procedure [61]. Matching to
Herwig6 [62],
Pythia6 [63], Herwig++ [64], and
Pythia8 [65] isavailable.As a consequence, the only input needed to simulate processes in the GM model is the implementation of the modelin
FeynRules . We have validated our framework by comparing total cross sections at NLO for VBF with the resultsof the
VBF@NNLO code [43, 66, 67] and found agreement within the integration uncertainties.
4. VBF PRODUCTION
In the SM, VBF production has been calculated to a rather high level of accuracy: QCD corrections are knownup to next-to-next-to-leading order (NNLO) for the total cross section [43, 66, 68] and for differential observables atthe parton level [69–71]. The QCD corrections to the fully inclusive cross sections are fairly moderate, at the levelof a few percent. However, the corrections to differential observables are more significant, with NNLO correctionsreaching 5–10% relative to the NLO rate. At both the inclusive and differential levels, the computation of NNLOcorrections relies on the so-called structure-function approach [68], which neglects color- and kinematically-suppressedcontributions [43, 72–75] arising, for example, from the exchange of gluons between the two quark lines. Results atNLO in QCD including parton shower matching have been computed in Refs. [76, 77], where it has been found thatthe typical effect of the shower is to improve the description of jet-related observables by including the effect of extraradiation. NLO electroweak (EW) corrections are also known [78, 79] and are found to be comparable in size to theNLO QCD ones.The situation is less satisfactory for BSM scenarios like the Georgi-Machacek model. Although the total crosssection can be computed up to NNLO accuracy in QCD [66, 67, 80], no fully differential prediction exists beyondleading order (LO). As seen in the SM case, corrections to the inclusive total cross sections do not fully capture thebehavior at the differential level. In this section we aim to improve this situation, by presenting for the first time fullydifferential results at NLO in QCD including matching to the parton shower. Ordered in virtuality or in transverse momentum, with the latter only for processes with no light partons in the final state.
The code for VBF production of a fiveplet state in the GM model can be generated and executed in
Mad-Graph5 aMC@NLO with the commands > import model GM_UFO> generate p p > H5p j j $$ w+ w- z [QCD]> output VBF_h5p_NLO> launch
Note that we veto W and Z bosons in the s -channel with the $$ syntax. The example above generates the code for H +5 production. For the other states, H −− , H − , H , H ++5 , the code can be generated by replacing the H5p labelwith
H5pp~ , H5p~ , H5z , H5pp respectively.We present results for VBF in the GM model at the LHC Run II energy ( √ s = 13 TeV) at LO and NLO accuracy,in both cases matched to Pythia8 . We use the NNPDF 2.3 LO1 and NLO parton density function (PDF) sets [81]consistently with the order of the computation. We keep the renormalization and factorization scales fixed to the W boson mass, as the typical transverse momentum of the tagging jets is of the same order of magnitude. To obtain theuncertainty due to scale variations, we vary the renormalization and factorization scales independently in the range M W / ≤ µ R , µ F ≤ M W . (24)We recall that the computation of scale and PDF uncertainties in MadGraph5 aMC@NLO can be performedwithout the need of extra runs using the reweighting technique presented in Ref. [82]. We employ
FastJet [83, 84]to cluster hadrons into jets, using the anti- k T algorithm [85] with a radius parameter ∆ R = 0 .
4. A minimum jet p T of 30 GeV is required.In addition, we consider the effect of typical selection cuts used in VBF analyses. These VBF cuts require thatthere are at least two jets, and that the two hardest jets satisfy the conditions y j < . , | y j − y j | > . ,m ( j , j ) >
600 GeV, (25)where y j is the jet rapidity and m ( j , j ) is the invariant mass of the two jets. In Tables I and II we present the cross sections at the inclusive level and with the VBF cuts of Eq. (25), respectively,for the production via VBF of each of the fiveplet states. Results are shown at LO+PS and NLO+PS, together withthe fractional uncertainties obtained from scale variations. First, we note that the K factors without and with cutsare rather similar to each other. Furthermore, the K factors for the different fiveplet states are also rather similar,and lie around 1.1. The production of more negatively-charged Higgs bosons receives slightly larger QCD corrections;this effect, related to the cross section’s sensitivity to valence versus sea quarks, becomes slightly more pronouncedwhen VBF cuts are applied. The inclusion of NLO corrections also has the effect of reducing the scale uncertaintiesto the 1–2% level. The different dependence on the initial state quarks of the various processes is also reflected inthe efficiency of the VBF cuts. The fraction of events that survives the VBF cuts (tabulated under “cut efficiency”in Table II) varies from 44% in the case of H −− production to 47% in the case of H ++5 production, and is essentiallyunaffected by inclusion of the NLO corrections.We turn now to study the effect of NLO corrections on differential observables, focusing on the representative caseof H +5 production in VBF. In Figure 1, we show the LO+PS and NLO+PS distributions for a number of observables.In particular we consider the transverse momentum p T and pseudorapidity η of the Higgs boson ( H ) and of thehardest jet ( j ), as well as the invariant mass m ( j , j ) and azimuthal separation ∆ φ ( j , j ) of the two hardest jets.The shaded bands show the scale uncertainties at both LO and NLO. The VBF cuts of Eq. (25) have been applied.For each observable, we also show in the inset the differential K -factor: that is, the bin-by-bin ratio of the NLOprediction over the LO central value, with the shaded band reflecting the NLO scale uncertainty. As in the case of SMVBF Higgs boson production, the K -factor is in general not constant over the differential distributions. This effectis most visible for the hardest-jet observables. Therefore, a fully-differential computation at NLO+PS is stronglypreferable to ensure realistic signal simulations. Σ (cid:72) pp (cid:174) H (cid:43) jj (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS NLO (cid:144) LO P TH (cid:72) GeV (cid:76) Σ (cid:72) pp (cid:174) H (cid:43) jj (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS (cid:45) (cid:45) NLO (cid:144) LO (cid:45) (cid:45) Η H Σ (cid:72) pp (cid:174) H (cid:43) jj (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS NLO (cid:144) LO P Tj (cid:72) GeV (cid:76) Σ (cid:72) pp (cid:174) H (cid:43) jj (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS (cid:45) (cid:45) NLO (cid:144) LO (cid:45) (cid:45) Η j Σ (cid:72) pp (cid:174) H (cid:43) jj (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS
500 1000 1500 2000 25000.00.20.40.60.81.0
NLO (cid:144) LO
500 1000 1500 2000 2500 m (cid:72) j , j (cid:76) (cid:72) GeV (cid:76) Σ (cid:72) pp (cid:174) H (cid:43) jj (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS NLO (cid:144) LO (cid:68)Φ (cid:72) j , j (cid:76) FIG. 1: Differential distributions for VBF production of the H +5 boson, with the VBF cuts of Eq. (25) (see text fordetails). The distributions for other H states are very similar, differing primarily in overall normalization. Process LO (fb) NLO (fb)
Kpp → H −− jj . +5 . − . . +1 . − . pp → H − jj . +5 . − . . +1 . − . pp → H jj . +5 . − . . +1 . − . pp → H +5 jj . +5 . − . . +1 . − . pp → H ++5 jj . +6 . − . . +1 . − . TABLE I: Cross sections and K -factors for H ± VBF production, with scale uncertainties.
Process LO (fb) NLO (fb) K Cut efficiency pp → H −− jj . +7 . − . . +1 . − . pp → H − jj . +7 . − . . +1 . − . pp → H jj . +7 . − . . +1 . − . pp → H +5 jj . +7 . − . . +1 . − . pp → H ++5 jj . +7 . − . . +1 . − . TABLE II: Cross sections and K -factors for H ± VBF production, with scale uncertainties, after applying the VBFcuts given in Eq. (25). Also shown is the fraction of NLO events that survive the VBF cuts (“cut efficiency”). V H PRODUCTION
We now consider the associated production of a GM fiveplet state together with a W ± or Z boson. In the SM,the associated production of a Higgs boson with a vector boson is known to NNLO in QCD for the total crosssection [86–93]; the two-loop corrections increase the inclusive cross section by less than 5% at the LHC [87]. TheQCD corrections to the differential observables are also known to NNLO [94, 95], leading to increases of 5–20% incomparison with the NLO results. N LO threshold corrections of about 0.1% have also been calculated in Ref. [96].These results have been implemented along with the electroweak corrections [97, 98] in the vh @ nnlo code [99].In following sections, we present rates and distributions for V H production at NLO for the Georgi-Machacekmodel. The code for associated production of a fiveplet state in the GM model (in this example H +5 ) and a SM vectorboson ( W − or Z , decaying leptonically with l = e or µ ) can be generated and executed in MadGraph5 aMC@NLO with the commands > import model GM_UFO> add process p p > H5p l- vl~ [QCD]> add process p p > H5p l+ l- [QCD]> output VH_h5p_NLO> launch
In this case, we include the leptonic decay of the gauge bosons at the matrix-element level, so that spin correlationsand off-shell effects are automatically taken into account. As in the VBF case, the extension to the other states inthe Higgs fiveplet is straightforward. We set the renormalisation and factorization scales to the invariant mass of the(reconstructed)
V H system, µ R = µ F = M V H .We consider two sets of cuts. In the first case, we require only basic cuts on leptons and missing transverse energy.Leptons are required to satisfy the transverse momentum and pseudorapidity cuts p lT >
30 GeV and η l < . . (26)For W H associated production, we also cut on the transverse missing energy, reconstructed from neutrinos in theevent record: E miss T >
30 GeV . (27) Process LO (fb) NLO (fb)
Kpp → H − l + l − . +7 . − . . +1 . − . pp → H l + l − . +7 . − . . +1 . − . pp → H +5 l + l − . +7 . − . . +1 . − . pp → H −− l + ν l . +7 . − . . +1 . − . pp → H − l + ν l . +7 . − . . +1 . − . pp → H l ± ( − ) ν l . +7 . − . . +1 . − . pp → H +5 l − ¯ ν l . +7 . − . . +1 . − . pp → H ++5 l − ¯ ν l . +7 . − . . +1 . − . TABLE III: Cross sections and K -factors for V H production after the basic lepton identification cuts given inEqs. (26) and (27), with scale uncertainties. For the first three processes the Higgs is produced in association with a Z boson, and for the remainder with a W boson. Process LO (fb) NLO (fb) K Cut efficiency pp → H − l + l − . +7 . − . . +1 . − . pp → H l + l − . +7 . − . . +1 . − . pp → H +5 l + l − . +7 . − . . +1 . − . pp → H −− l + ν l . +7 . − . . +1 . − . pp → H − l + ν l . +7 . − . . +1 . − . pp → H l ± ( − ) ν l . +7 . − . . +1 . − . pp → H +5 l − ¯ ν l . +7 . − . . +1 . − . pp → H ++5 l − ¯ ν l . +7 . − . . +1 . − . TABLE IV: Cross sections and K -factors for V H production after applying the additional boosted-regime cutsgiven in Eq. (28). Also shown is the fraction of NLO events that survive the boosted-regime cuts (“cut efficiency”).In the second case, we consider a boosted regime, which is often used to enhance the signal-to-background ratioin SM V H searches [100, 101], by requiring the following additional cuts on the Higgs and the reconstructed gaugebosons’ transverse momenta: p HT >
200 GeV and p VT >
190 GeV, (28)as suggested in [102].
In Tables III and IV, we show the cross sections for
V H production of H states at LO+PS and NLO+PSwith basic cuts and with the additional boosted-regime cuts, respectively. The cross sections include the leptonicbranching fractions of the gauge bosons. Note that ZH ±± production of the doubly charged states is forbidden bycharge conservation. We find that the K -factors are larger than for VBF and, similar to the SM case, lie around 1 . K -factors without and with the boosted-regime cuts of Eq. (28) are essentially identical.We notice that processes with a more negatively-charged final state (which are therefore more sensitive to sea quarks)have slightly larger K -factors. As in the case of VBF, processes with a more positively-charged final state have alarger fraction of events which survive the cuts.In Figure 2, we present the LO+PS and NLO+PS distributions and K -factors for W − H +5 production under theboosted-regime V H cuts given in Eq. (28); the distributions for ZH +5 production are similar in shape. We show thetransverse momentum p T and pseudorapidity η of the Higgs, the transverse momentum of the reconstructed vectorboson (using monte carlo truth information) and the azimuthal separation ∆ φ between the lepton and the neutrino.In this case we find that the differential K -factors are generally constant over the distributions considered, with theexception of the Higgs pseudorapidity; in this case the K -factor has a maximum of around 1.4 in the central region,which reduces to a minimum of around 1.2 for a Higgs produced in the forward or backward regions.0 Σ (cid:72) pp (cid:174) WH (cid:43) (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS
200 300 400 500 6000.0000.0010.0020.0030.0040.0050.006
NLO (cid:144) LO
200 300 400 500 600 P TH (cid:72) GeV (cid:76) Σ (cid:72) pp (cid:174) WH (cid:43) (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS (cid:45) (cid:45) (cid:45) NLO (cid:144) LO (cid:45) (cid:45) (cid:45) Η H (cid:72) GeV (cid:76) Σ (cid:72) pp (cid:174) WH (cid:43) (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS
200 300 400 500 6000.0000.0010.0020.0030.0040.0050.006
NLO (cid:144) LO
200 300 400 500 600 P TW (cid:72) GeV (cid:76) Σ (cid:72) pp (cid:174) WH (cid:43) (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS NLO (cid:144) LO (cid:68)Φ (cid:72) l, Ν (cid:76) FIG. 2: Differential distributions for
W H +5 associated production, with the cuts in Eq. 28. The distributions forother H states or for associated production with a Z boson are very similar, differing primarily in overallnormalization. H H PRODUCTION
Finally we consider double Higgs production of two H states in the GM model. In contrast to the SM, pairproduction of the fiveplet scalars is generally dominated by Drell-Yan-like processes. The exception is H H pairproduction (which we therefore do not consider below), as there is no ZH H vertex due to the same symmetryconsiderations that forbid the ZHH coupling in the SM. H H pairs could be produced through VBF, and the H H , H +5 H − , and H ++5 H −− final states could also be produced via gluon fusion through an off-shell h or H . Theseprocesses have very small cross sections and are not considered here. In the SM, Higgs pair production is dominated by gluon fusion at the LHC. The rate of this production mode is known to be quitesmall, and receives important QCD corrections at NLO [103–106]. Corrections to the inclusive cross section have been obtained atNNLO [107–109], while corrections to differential observables are known at NLO [110]. The NNLO corrections to the inclusive crosssection are quite large, on the order of 20% [108] in comparison to the NLO result at 14 TeV. The effect of dimension-6 operators arisingfrom new physics has also been considered at NLO in Ref. [111], which found that the new couplings could alter the K -factors relevantto SM-like Higgs pair production by a few percent. Process LO (fb) NLO (fb)
Kpp → H −− H +5 . +4 . − . . +2 . − . pp → H − H . +4 . − . . +2 . − . pp → H −− H ++5 . +4 . − . . +2 . − . pp → H − H +5 . +4 . − . . +2 . − . pp → H H +5 . +4 . − . . +2 . − . pp → H − H ++5 . +4 . − . . +2 . − . TABLE V: Cross sections and K -factors for H H production, with scale uncertainties. The first two processesproceed through an s -channel W − , the next two through a Z and the last two through a W + . The code for the pair production of two fiveplet states in the GM model, for example H −− H +5 , can be generatedand executed in MadGraph5 aMC@NLO with the commands > import model GM_UFO> generate p p > H5pp~ H5p [QCD]> output H5mm_H5p_NLO> launch
Once again, the extension to the other combinations of states in the Higgs fiveplet is straightforward. We set therenormalisation and factorization scales to the invariant mass of the Higgs pair, µ R = µ F = M HH . We do not consideradditional cuts on these processes. In Table V we show the cross sections for H H production at LO+PS and NLO+PS, without cuts. In Figure3, we show the LO+PS and NLO+PS distributions and K -factors for H −− H +5 production. We show the transversemomentum p T and pseudorapidity η of the scalar H −− , and the invariant mass of the two scalars. The p T and η distributions of the other scalar H +5 are similar.As in the case of V H production, the differential K -factors are generally constant over the distributions considered,with the exception of the Higgs pseudorapidities; in this case the K -factor has a maximum slightly above 1.4 in thecentral region, which reduces to roughly 1.3 for a Higgs produced in the forward or backward regions.
7. CONCLUSIONS
We have presented cross sections, differential distributions, and K -factors for the production of fermiophobic fivepletscalars in the Georgi-Machacek model at NLO accuracy in QCD, including the matching to parton showers. Weconsidered production through VBF, V H , and H H associated production at the benchmark point of Eq. (22). Ourresults demonstrate the importance of a fully differential simulation at NLO+PS in order to accurately simulate thesignal at the LHC. Automated tools make such a simulation possible with a very limited effort. For what concerns V H and H H production, the description of the final state can be further improved by including the effect of theradiation of extra jets at NLO accuracy, for example using the Fx-Fx [112] or UNLOPS [113] merging technique,which are both automatized within MadGraph5 aMC@NLO . The model files for the automated tool chain used toproduce these results are publicly available on http://feynrules.irmp.ucl.ac.be/wiki/GeorgiMachacekModel . ACKNOWLEDGMENTS
We thank K. Kumar for helpful discussions about the Georgi-Machacek model and
FeynRules and we are gratefulto F. Maltoni and L. Barak for having encouraged us to pursue the present project. C.D. is a Durham InternationalJunior Research Fellow and has been supported in part by the Research Executive Agency of the European Union2 Σ (cid:72) pp (cid:174) H (cid:45)(cid:45) H (cid:43) (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS NLO (cid:144) LO P TH (cid:72) GeV (cid:76) Σ (cid:72) pp (cid:174) H (cid:45)(cid:45) H (cid:43) (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS (cid:45) (cid:45) (cid:45) NLO (cid:144) LO (cid:45) (cid:45) (cid:45) Η H Σ (cid:72) pp (cid:174) H (cid:45)(cid:45) H (cid:43) (cid:76) (cid:72) fb (cid:76) LHC 13 TeV LO (cid:43)
PSNLO (cid:43) PS
600 800 1000 1200 1400 1600 1800 20000.00.10.20.30.4
NLO (cid:144) LO
600 800 1000 1200 1400 1600 1800 2000 m (cid:72) H , H (cid:76) (cid:72) GeV (cid:76)
FIG. 3: Differential distributions for H −− H +5 associated production. The distributions for other H states are verysimilar, differing primarily in overall normalization.under Grant Agreement PITN-GA-2012-315877 (MC-Net). K.H., H.E.L., and A.P. were supported by the NaturalSciences and Engineering Research Council of Canada. K.H. was also supported by the Government of Ontariothrough an Ontario Graduate Scholarship. M.Z. is supported by the European Union’s Horizon 2020 research andinnovation programme under the Marie Sklodovska-Curie grant agreement No 660171 and in part by the ERC grantHiggs@LHC and by the ILP LABEX (ANR-10-LABX-63), in turn supported by French state funds managed by theANR within the “Investissements d’Avenir” programme under reference ANR-11-IDEX-0004-02.3 Appendix A: The scalar potential and masses of the Georgi-Machacek model
The most general gauge-invariant scalar potential involving these fields that conserves custodial SU (2) can bewritten as [21] V (Φ , X ) = µ † Φ) + µ X † X ) + λ [Tr(Φ † Φ)] + λ Tr(Φ † Φ)Tr( X † X )+ λ Tr( X † XX † X ) + λ [Tr( X † X )] − λ Tr(Φ † τ a Φ τ b )Tr( X † t a Xt b ) − M Tr(Φ † τ a Φ τ b )( U XU † ) ab − M Tr( X † t a Xt b )( U XU † ) ab . (A1)Here the SU (2) generators for the doublet representation are τ a = σ a / σ a being the Pauli matrices, thegenerators for the triplet representation are t = 1 √ , t = 1 √ − i i − i i , t = − , (A2)and the matrix U , which rotates X into the Cartesian basis, is given by [7] U = − √ √ − i √ − i √ . (A3)In the notation of the parameters of the scalar potential, our chosen benchmark point corresponds to values of µ = − (92 . , λ = λ = λ = λ = 0 . ,µ = (300 GeV) , M = M = 100 GeV. λ = 0 . , (A4)Here µ and λ have respectively been set using G F and M h = 125 GeV [see Eq. (A11)].The vevs are obtained by solving the minimization conditions, v φ (cid:20) µ + 4 λ v φ + 3 (2 λ − λ ) v χ − M v χ (cid:21) = 0 , µ v χ + 3 (2 λ − λ ) v φ v χ + 12 ( λ + 3 λ ) v χ − M v φ − M v χ = 0 . (A5)After electroweak symmetry breaking, the masses of the custodial fiveplet and triplet scalars are respectively givenby m = M v χ v φ + 12 M v χ + 32 λ v φ + 8 λ v χ ,m = M v χ ( v φ + 8 v χ ) + λ v φ + 8 v χ ) = (cid:18) M v χ + λ (cid:19) v . (A6)The mixing of the custodial singlets is controlled by the 2 × M = (cid:32) M M M M (cid:33) , (A7)where M = 8 λ v φ , M = √ v φ [ − M + 4 (2 λ − λ ) v χ ] , M = M v φ v χ − M v χ + 8 ( λ + 3 λ ) v χ . (A8) A translation table to other parametrizations in the literature has been given in an Appendix of Ref. [21]. Note that Refs. 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