Vector boson fusion at NNLO in QCD: SM Higgs and beyond
aa r X i v : . [ h e p - ph ] F e b DESY 11-153CP3-11-28LPN 11-51SFB / CPP-11-50September 2011
Vector boson fusion at NNLO in QCD: SM Higgs and beyond
Paolo Bolzoni a , Fabio Maltoni b , Sven-Olaf Moch c and Marco Zaro b a II. Institut für Theoretische Physik, Universität HamburgLuruper Chaussee 149, D–22761 Hamburg, Germany b Center for Cosmology, Particle Physics and Phenomenology (CP3),Université Catholique de Louvain,Chemin du Cyclotron 2, B–1348 Louvain-la-Neuve, Belgium c Deutsches Elektronensynchrotron DESYPlatanenallee 6, D–15738 Zeuthen, Germany
Abstract
Weak vector boson fusion provides a unique channel to directly probe the mechanism of elec-troweak symmetry breaking at hadron colliders. We present a method that allows to calculatetotal cross sections to next-to-next-to-leading order (NNLO) in QCD for an arbitrary V ∗ V ∗ → X process, the so-called structure function approach. By discussing the case of Higgs production indetail, we estimate several classes of previously neglected contributions and we argue that suchmethod is accurate at a precision level well above the typical residual scale and PDF uncertaintiesat NNLO. Predictions for cross sections at the Tevatron and the LHC are presented for a varietyof cases: the Standard Model Higgs (including anomalous couplings), neutral and charged scalarsin extended Higgs sectors and (fermiophobic) vector resonance production. Further results can beeasily obtained through the public use of the VBF@NNLO code. Introduction
The central theme of the physics program at the Large Hadron Collider (LHC) is the search forthe Standard Model (SM) Higgs boson and more in general the elucidation of the mechanism ofelectroweak symmetry breaking. The three dominant production mechanisms for the SM Higgsboson are, (in order of importance) gluon-gluon fusion via a top-quark loop, vector-boson fusion(VBF) and Higgs-Strahlung, i.e. associated production with W and Z -bosons (see e.g. [1, 2]).In extensions of the SM with a richer Higgs sector, such as in supersymmetry, or in stronglyinteracting light Higgs scenarios [3], the relative importance of the various channels might dependon the details or parameters of the model. In any case, VBF remains of primary importance,being the channel where longitudinal vector boson scattering gives rise to violation of unitarity ataround 1 TeV, if no other particle or interaction is present than what it is currently known fromexperiments.Precise knowledge of the expected rates for the Higgs boson production processes is an essen-tial prerequisite for any experimental search and, after discovery, will be a crucial input to proceedto accurate measurements. At a hadron collider such as the LHC, precision predictions need toinclude higher-order radiative corrections which usually implies the next-to-next-to-leading or-der (NNLO) in Quantum Chromodynamics (QCD) and the next-to-leading order (NLO) as far aselectroweak corrections are concerned. If accounted for, these higher-order quantum correctionsgenerally stabilize the theoretical predictions through an apparent convergence of the perturbativeexpansion and a substantially reduced dependence on the choice of the factorization and renormal-ization scales. For inclusive Higgs boson production in gluon-gluon fusion and the W H and ZH mode, the exact NNLO corrections in QCD are available [4–7] and have been shown to be veryimportant (see e.g. [8]).Higgs boson production via VBF is the mechanism with the second largest rate in the SM ando ff ers a clean experimental signature with the presence of at least two jets in the forward / backwardrapidity region [9–11] and a variety of Higgs boson decay modes to be searched for [12–15] . Inextensions of the SM which feature scalar or vector state(s) with reduced couplings to fermions,VBF can become the leading production mechanism. VBF is a pure electroweak process at leadingorder (LO) and it acquires corrections at NLO in QCD [16, 17] as well as in the electroweak sec-tor [18, 19] leading to a typical accuracy for the total cross-section in the 5 −
10% range. Recently,the NNLO QCD corrections for the VBF process have been computed [20] in the so-called struc-ture function approach [16], which builds upon the approximate, although very accurate, factor-ization of the QCD corrections between the two parton lines associated with the colliding hadrons.At NNLO the results are very stable with respect to variations of the renormalization and factor-ization scales, µ r and µ f , and can take full advantage of modern sets of precise parton distributionfunctions (PDFs) at the same accuracy. The small theoretical uncertainty for the inclusive rate dueto missing higher order QCD corrections as well as the PDF uncertainties are estimated to be atthe 2% level each for a wide range of Higgs boson masses.The purpose of the present article is two-fold. First, it provides an extensive documentationof the NNLO computation of Refs. [20, 21] and a broad phenomenological study for the LHC atvarious center-of-mass energies and for the Tevatron. Second, it exploits the universality of thestructure function approach to describe the production of any color-neutral final state from thefusion of vector bosons and applies it to a number of new physics models, e.g. with an extendedHiggs sector, or to vector resonance production (see e.g., [22, 23]).The outline is as follows: In Sec. 2 we define the VBF process as a signal, discuss carefullyits contributions and give arguments for the ultimate theoretical precision one could possibly aim1t in predictions of its rates. Sec. 3 is devoted to a discussion of the structure function approachat NNLO in QCD along with detailed computations of neglected contributions as a means of esti-mating the accuracy. In particular, we study the non-factorizable diagrams and the ones involvingheavy quarks, all of which are not accounted for in the structure function approach. Phenomeno-logical results for inclusive rates at the LHC and Tevatron are presented in Sec. 4. Extensions ofthe structure function approach to Higgs boson production via VBF in specific classes of modelsbeyond the SM (see e.g., [24] for charged Higgs bosons) as well as modifications due to anoma-lous couplings ( WW H , ZZH , etc.) and vector resonances are discussed in Sec. 5. Finally, weconclude in Sec. 6 and document some technical aspects and tables with VBF cross sections in theAppendices.
Processes in collider physics are always defined on simple conventions typically based on leading-order Feynman diagrams. While this normally poses no problems, it might lead to ambiguitieswhen decays of resonant states and / or higher order e ff ects are included. Consider, as a simpleexample, the class of processes which involve only the electroweak coupling α EW at the leadingorder, such as Drell-Yan (with the decay into two jets), single-top production and Higgs bosonproduction via coupling with a vector boson. Most of such processes can be easily defined at lead-ing order considering the corresponding resonant intermediate or final states, while some, such asVBF are (quantum-mechanically) ambiguous already at the leading order: pp → H j j with vectorbosons in the t -channel can interfere with pp → HV ( ∗ ) → H j j , i.e. , with Higgs associated produc-tion with a vector boson then decaying into two jets. Such interference, however, is quite smalleverywhere in the phase space and it can formally be reduced to zero by just taking the narrowwidth limit. This suggests that considering the two processes distinct is a handy approximation,at least at the leading order. For all of the processes in this class, i.e. , single-top and Higgs elec-troweak production, aiming at a better precision by including higher-order QCD e ff ects createsfurther ambiguities as it opens up more possibilities for interferences (one notable example is tW at NLO overlapping with t ¯ t production) and the reliability of the approximations made has to becarefully assessed.In general two complementary approaches can be followed. The first is to consider all in-terferences exactly, and introduce a gauge-invariant scheme to properly handle the width e ff ects.This can be consistently done at NLO order in QCD and electroweak corrections, following well-known and established techniques, such as the use of the complex-mass scheme [25]. This is thepath followed for instance in Ref. [19] for the calculation of QCD and electroweak e ff ects in theorder α EW process pp → H j j . The advantage of this approach is that all interference e ff ects arecorrectly taken into account in any region of phase space, including where tight cuts might createsignificant enhancements. This is the only way to proceed when interference e ff ects are similar toor larger than the corrections from higher orders. However, when such e ff ects are small, it o ff ersseveral drawbacks. The first is the unnecessary complexity of the calculation itself. The second isthat the operational separation between the two processes, which can be quite useful at the prac-tical level, for example in experimental analyses, is lost. In this context, even the definition ofsignal and background might not be meaningful and the distinction possibly leads to confusion.In this case another approach can be followed. Use a simple process definition and systematicallycheck the impact of higher-order corrections as well as those from interferences to set the ultimatepractical precision that can be achieved. In this section we argue that for vector-boson fusion this2rocedure is sound and can lead to a definition which is unambiguous to better than 1%, i.e. morethan su ffi cient for all practical applications at hadron colliders. HV ∗ V ∗ P P X X q q Figure 2.1:
Higgs production via the VBF process.
In short, we define VBF as the Higgs production for vanishing quark masses, through directcoupling to vector bosons in the t -channel, and with no color exchange between the two collidinghadrons, i.e. , all the processes that can be represented by the diagram in Fig. 2.1, where no heavy-quark loop is to be included in the blobs while additional vector bosons in a color singlet statemight appear at one or more loops (not shown).The definition above, while academic in nature, fits extremely well with what is looked for andassociated to in the current experimental searches. For example, it excludes the interfering e ff ectswith s -channel associated production, but includes the additional exchange of two gluons in a colorsinglet state in the t -channel. These two e ff ects, as we will argue in the following, are tiny and canbe neglected in the numerical evaluation of the total cross section. However, it is useful to keep inmind that our definition put them on a di ff erent ground. The main reason / motivation is that amongthe most important characteristics exploited in the experimental analyses is the presence of twohigh-invariant mass forward-backward jets and the absence of radiation in the central region. Boththese e ff ects are present in the case of the color-singlet component of double-gluon exchange inthe t -channel, while they are not typical of the s -channel contribution.The definition given above allows us to systematically classify processes for Higgs boson pro-duction as VBF and non-VBF. In the latter we include also possible interferences between ampli-tudes belonging to the VBF class and those in the non-VBF one.“VBF” processes: • Factorizable contributions in QCD, see Fig. 2.1. This class is evaluated exactly in this workfor massless quarks. It provides the bulk of all QCD corrections up to order α s to a precisionbetter than 1%. • Non-factorizable contributions in QCD. This class starts at order α s / N c . It is estimated inSec. 3.2 to contribute less than 1% to the total VBF cross section. • Electroweak corrections to diagrams in Fig. 2.1. These are relevant corrections which havebeen calculated in Ref. [19]. A combination with the NNLO QCD ones calculated in thiswork has been reported in Ref. [8]. 3igure 3.1:
Higgs production via the VBF process at LO in QCD. “Non-VBF” processes: • Single-quark line contributions, as calculated in Ref. [26]. These e ff ects are smaller than 1%in di ff erential cross sections. • Interferences in VBF itself known at NLO in QCD and electroweak, as calculated in Ref. [19].These e ff ects can be calculated at LO and are found to be very small. • Interferences between VBF and associated
W H and WZ production at NLO in QCD andelectroweak, as calculated in Ref. [19]. These e ff ects can be easily calculated at LO and arefound to be very small. • Interferences with the top-loop mediated Higgs production, Refs. [27, 28] and contributionscalculated in Sec. 3.3. These e ff ects are found to contribute less than 1% to the total crosssection. • t -channel vector boson production in presence of heavy-quark loops (triangles and boxes),see Sec. 3.3. These e ff ects are estimated to contribute less than 1% to the total cross section.We conclude this section by mentioning that the same kind of di ffi culties / ambiguities arisewhen the Higgs width becomes non-negligible, i.e. for large Higgs masses, let us say m H > ff ects with “background” processes canbecome important and need to be accounted for. Such e ff ects have been considered in full gener-ality at LO in studies of vector boson scattering [29, 30] and, for leptonic final states of the vectorbosons, in EW production of V V j j at NLO in QCD [31–34]. Let us discuss the radiative corrections to the VBF process with emphasis on the QCD contribu-tions. As mentioned above, at LO the Higgs production in VBF proceeds purely through elec-troweak interactions (see Fig. 3.1), with the cross section for pp → H j j being of order α EW .In the QCD improved parton model, higher order corrections arise. All NLO QCD contri-butions to the VBF cross section (order α EW α s ) can be treated exactly in the structure functionapproach, which we discuss in detail and extend to NNLO below in Sec. 3.1. At NNLO, i.e. atorder α EW α s , certain other contributions arise, which we estimate. These are diagrams involvinggluon exchange between the two quark lines (cf. Sec. 3.2), and diagrams involving closed heavy-quark loops (cf. Sec. 3.3). We also comment briefly on electroweak corrections at one-loop inSec. 3.4. 4igure 3.2: Higgs production via the VBF process at NLO in QCD.
The structure function approach is based on the observation that to a very good approximation theVBF process can be described as a double deep-inelastic scattering process (DIS), see Fig. 2.1,where two (virtual) vector-bosons V i (independently) emitted from the hadronic initial states fuseinto a Higgs boson. This approximation builds on the absence (or smallness) of the QCD interfer-ence between the two inclusive final states X and X . In this case the total cross section is givenas a product of the matrix element M µρ for VBF, i.e. , V µ V ρ → H , which in the SM reads M µν = (cid:16) √ G F (cid:17) / M V i g µν , (3.1)and of the DIS hadronic tensor W µν : d σ = S G F M V M V (cid:16) Q + M V (cid:17) (cid:16) Q + M V (cid:17) W µν (cid:16) x , Q (cid:17) M µρ M ∗ νσ W ρσ (cid:16) x , Q (cid:17) ×× d P X (2 π ) E X d P X (2 π ) E X ds ds d P H (2 π ) E H (2 π ) δ (cid:0) P + P − P X − P X − P H (cid:1) . (3.2)Here G F is Fermi’s constant and √ S is the center-of-mass energy of the collider. Q i = − q i , x i = Q i / (2 P i · q i ) are the usual DIS variables, s i = ( P i + q i ) are the invariant masses of the i -thproton remnant, and M V i denote the vector-boson masses, see Fig. 2.1. The three-particle phasespace dPS of the VBF process is given in the second line of Eq. (3.2). It is discussed in detail inSec. A.1.Higgs production in VBF requires the hadronic tensor W µν for DIS neutral and charged currentreactions, i.e. , the scattering o ff a Z as well as o ff a W ± -boson. It is commonly expressed interms of the standard DIS structure functions F i ( x , Q ) with i = , , i.e. F Vi with i = , , V ∈ { Z , W ± } and we employ the particle data group (PDG) conventions [35]. Thus, W µν (cid:16) x i , Q i (cid:17) = − g µν + q i ,µ q i ,ν q i F ( x i , Q i ) + ˆ P i ,µ ˆ P i ,ν P i · q i F ( x i , Q i ) + i ǫ µναβ P α i q β i P i · q i F ( x i , Q i ) , (3.3)where ǫ µναβ is the completely antisymmetric tensor and the momentum ˆ P i readsˆ P i ,µ = P i ,µ − P i · q i q i q i ,µ . (3.4)The factorization underlying Eq. (3.2) does not hold exactly already at LO, because interferencecan occur either between identical final state quarks ( i.e. , uu → Huu ) or between processes where5ither a W or a Z can be exchanged ( i.e. , ud → Hud ). However, at LO, these contributions can beeasily computed and they have been included in our results. On the other hand, simple argumentsof kinematics (based on the behavior of the propagators in the matrix element [36]) show thatsuch contributions are heavily suppressed already at LO and contribute to the total cross sectionwell below the 1% level, a fact that has been confirmed by a complete calculation even throughNLO [19]. Apart from these interference e ff ects, the factorization of Eq. (3.2) is still exact at NLO.This is due to color conservation: QCD corrections to the upper quark line are independent fromthose of the lower line. Fig. 3.2 (left) shows a sample diagram accounted for by the structurefunction approach, while the diagram Fig. 3.2 (right) vanishes at NLO due to color conservation, i.e. , Tr( t a ) = t a of the color SU( N c ) gauge group.The evaluation of Eq. (3.1) and Eq. (3.3) leads to the explicit result for the squared hadronictensor in Eq. (3.2) in terms of the DIS structure functions [16] (see also the review [1]) : W µν (cid:16) x , Q (cid:17) M µρ M ∗ νσ W ρσ (cid:16) x , Q (cid:17) = √ G F M V i ×× F (cid:16) x , Q (cid:17) F (cid:16) x , Q (cid:17) + ( q · q ) q q ++ F (cid:16) x , Q (cid:17) F (cid:16) x , Q (cid:17) P · q ( P · q ) q + q P · q − P · q q q · q + F (cid:16) x , Q (cid:17) F (cid:16) x , Q (cid:17) P · q ( P · q ) q + q P · q − P · q q q · q + F (cid:16) x , Q (cid:17) F (cid:16) x , Q (cid:17) ( P · q )( P · q ) ×× P · P − ( P · q )( P · q ) q − ( P · q )( P · q ) q + ( P · q )( P · q )( q · q ) q q + F (cid:16) x , Q (cid:17) F (cid:16) x , Q (cid:17) P · q )( P · q ) (cid:18) ( P · P )( q · q ) − ( P · q )( P · q ) (cid:19) . (3.5)At this stage it remains to insert the DIS structure functions F Vi with i = , , V ∈ { Z , W ± } .At NLO in QCD, explicit expression have been given in Ref. [16] using the results of Ref. [37].For the necessary generalization beyond NLO, let us briefly review the basic formulae. QCDfactorization allows to express the structure functions as convolutions of the PDFs in the protonand the short-distance Wilson coe ffi cient functions C i . The gluon PDF at the factorization scale µ f is denoted by g ( x , µ f ) and the quark (or anti-quark) PDF by q i ( x , µ f ) (or ¯ q i ( x , µ f )) for a specificquark flavor i . The latter PDFs appear in the following combinations, q s = n f P i = (cid:18) q i + ¯ q i (cid:19) , q vns = n f X i = (cid:18) q i − ¯ q i (cid:19) , (3.6) q + ns , i = (cid:18) q i + ¯ q i (cid:19) − q s , q − ns , i = (cid:18) q i − ¯ q i (cid:19) − q vns , (3.7)as the singlet distribution q s , the (non-singlet) valence distribution q vns as well as flavor asymmetriesof q ± ns , i . All of them are subject to well-defined transformation properties under the flavor isospin,see e.g. [38, 39]. 6or the neutral current Z -boson exchange the DIS structure functions F Zi can be written asfollows: F Zi ( x , Q ) = f i ( x ) Z dz Z dy δ ( x − yz ) n f X j = (cid:16) v j + a j (cid:17) × (3.8) × (cid:26) q + ns , j ( y , µ f ) C + i , ns ( z , Q , µ r , µ f ) + q s ( y , µ f ) C i , q ( z , Q , µ r , µ f ) + g ( y , µ f ) C i , g ( z , Q , µ r , µ f ) (cid:27) , F Z ( x , Q ) = Z dz Z dy δ ( x − yz ) n f X i = v i a i × (3.9) × (cid:26) q − ns , i ( y , µ f ) C − , ns ( z , Q , µ r , µ f ) + q vns ( y , µ f ) C v3 , ns ( z , Q , µ r , µ f ) (cid:27) , where i = , f ( x ) = / f ( x ) = x . The vector- and axial-vectorcoupling constants v i and a i in Eq. (3.8) are given by v i + a i = + (cid:16) − sin θ w (cid:17) u -type quarks , + (cid:16) − sin θ w (cid:17) d -type quarks , (3.10)and, likewise, in Eq. (3.9), 2 v i a i = − sin θ w u -type quarks , − sin θ w d -type quarks . (3.11)The coe ffi cient functions C i in Eqs. (3.8)–(3.9) parameterize the hard partonic scattering pro-cess. They depend only on the scaling variable x , and on dimensionless ratios of Q , µ f and therenormalization scale µ r . The perturbative expansion of C i in the strong coupling α s up to twoloops reads in the non-singlet sector, C + i , ns ( x ) = δ (1 − x ) + a s (cid:26) c (1) i , q + L M P (0)qq (cid:27) (3.12) + a s (cid:26) c (2) , + i , ns + L M (cid:18) P (1) , + ns + c (1) i , q ( P (0)qq −
0¯ ) (cid:19) + L M (cid:18) P (0)qq ( P (0)qq − (cid:19) + L R c (1) i , q + L R L M P (0)qq (cid:27) , C − , ns ( x ) = δ (1 − x ) + a s (cid:26) c (1)3 , q + L M P (0)qq (cid:27) (3.13) + a s (cid:26) c (2) , − , ns + L M (cid:18) P (1) , − ns + c (1)3 , q ( P (0)qq −
0¯ ) (cid:19) + L M (cid:18) P (0)qq ( P (0)qq −
0¯ ) (cid:19) + L R c (1)3 , q + L R L M P (0)qq (cid:27) , where a s = α s ( µ r ) / (4 π ) and i = , i.e. the towersof logarithms in L M = ln( Q /µ f ) and L R = ln( µ r /µ f ) (keeping µ r , µ f ), has been derived by renor-malization group methods (see, e.g. [40]) in terms of splitting functions P ( l ) i j and the coe ffi cients7f the QCD beta function, β l . In our normalization of the expansion parameter, a s = α s / (4 π ), theconventions for the running coupling are dd ln µ α s π ≡ d a s d ln µ = − β a s − . . . , = C A − n f , (3.14)with β the usual expansion coe ffi cient of the QCD beta function, C A = n f the number oflight flavors.Note, that the valence coe ffi cient function C v3 , ns in Eq. (3.9) is defined as C v3 , ns = C − , ns + C s3 , ns .However, we have C s3 , ns , ffi ces with C v3 , ns = C − , ns up to NNLO. In the singlet sector we have C i , q ( x ) = δ (1 − x ) + a s (cid:26) c (1) i , q + L M P (0)qq (cid:27) (3.15) + a s (cid:26) c (2) i , q + L M (cid:18) P (1)qq + c (1) i , q ( P (0)qq − + c (1) i , g P (0)gq (cid:19) + L M (cid:18) P (0)qq ( P (0)qq −
0¯ ) + P (0)qg P (0)gq (cid:19) + L R c (1) i , q + L R L M P (0)qq (cid:27) , C i , g ( x ) = a s (cid:26) c (1) i , g + L M P (0)qg (cid:27) (3.16) + a s (cid:26) c (2) i , g + L M (cid:18) P (1)qg + c (1) i , q P (0)qg + c (1) i , g ( P (0)gg − (cid:19) + L M (cid:18) P (0)qq P (0)qg + P (0)qg ( P (0)gg − (cid:19) + L R c (1) i , g + L R L M P (0)qg (cid:27) , where again i = , C i , q = C + i , ns + C i , ps , i.e. P (1)qq = P (1) , + ns + P (1)ps and c (2) i , q = c (2) , + i , ns + c (2) i , ps in Eq. (3.15). Note, that start-ing at two-loop order we have C i , ps ,
0. The DIS coe ffi cient functions c ( l ) i , k are known to NNLOfrom Refs. [41–44], likewise, NNLO evolution of the PDFs has been determined in Refs. [38, 45]and even the hard corrections at order α s are available [46, 47]. Accurate parametrizations of allcoe ffi cient functions in Eqs. (3.12)–(3.16) can be taken e.g. from Refs. [46, 47] and the splittingfunctions P ( l ) i j are given e.g. in Refs. [38, 45] . All products in Eqs. (3.12)–(3.16) are under-stood as Mellin convolutions. They can be easily evaluated in terms of harmonic polylogarithms H ~ m ( x ) / (1 ± x ) up to weight 4, see [48], and for their numerical evaluation we have used the F ortran package [49].For the charged current case with W ± -boson exchange the DIS structure functions F W ± i aregiven by, F W − i ( x , Q ) = f i ( x ) Z dz Z dy δ ( x − yz ) 1 n f n f X j = (cid:16) v j + a j (cid:17) × (3.17) × n δ q − ns ( y , µ f ) C − i , ns ( z , Q , µ r , µ f ) + q s ( y , µ f ) C i , q ( z , Q , µ r , µ f ) + g ( y , µ f ) C i , g ( z , Q , µ r , µ f ) o , F W − ( x , Q ) = Z dz Z dy δ ( x − yz ) 1 n f n f X i = v i a i × (3.18) Note, that with the conventions of Refs. [38, 45–47] both the pure-singlet and the gluon coe ffi cient functions aswell as the the splitting functions P (0)qg and P (1)qg in Eqs. (3.12)–(3.16) need to be divided by a factor 2 n f to account forthe contribution of one individual quark flavor (not the anti-quark). × Figure 3.3:
Examples of squared matrix elements contributing at NNLO to VBF involving a double gluonexchange between the two quark lines. × n δ q + ns ( y , µ f ) C + , ns ( z , Q , µ r , µ f ) + q vns ( y , µ f ) C v3 , ns ( z , Q , µ r , µ f ) o , where, as above, C i , q = C + i , ns + C i , ps and, also, C v3 , ns = C − i , ns up to two-loop order. The asymmetry δ q ± ns parametrizes the iso-triplet component of the proton, i.e. u , d and so on. It is defined as δ q ± ns = X i ∈ u − type X j ∈ d − type (cid:26)(cid:18) q i ± ¯ q i (cid:19) − (cid:18) q j ± ¯ q j (cid:19)(cid:27) . (3.19)Its numerical impact is expected to be small though. The respective results for F W + i are obtainedfrom Eqs. (3.17)–(3.18) with the simple replacement δ q ± ns → − δ q ± ns .The vector- and axial-vector coupling constants v i and a i are given by v i = a i = √ . (3.20)The coe ffi cient functions in Eqs. (3.17)–(3.18) including their dependence on the factorization andthe renormalization scales can be obtained from Eqs. (3.12)–(3.16) with the help of the followingsimple substitutions c (2) , + i , ns ↔ c (2) , − i , ns , c (2) , − , ns ↔ c (2) , + , ns and P (1) , + ns ↔ P (1) , − ns and so on. Again, all expres-sions for the coe ffi cient and splitting functions are given in Refs. [46, 47] and [38, 45], respectively.Eq. (3.2) with the explicit expressions for the DIS structure functions inserted provides thebackbone of our NNLO QCD predictions for Higgs production in VBF. However, as emphasizedabove, the underlying factorization is not exact beyond NLO and therefore, the non-factorizablecorrections need to be estimated. This will be done in the following. In order to assess the quality of the factorization approach, we now estimate the size of the non-factorizable contributions, i.e. those coming from diagrams involving the exchange of gluonsbetween the two quark lines and not included in the structure function approach. Neglecting in-terferences between t - and u -channel diagrams, which are kinematically suppressed, this class ofdiagrams vanishes at NLO because of color conservation, but contributes at NNLO for the firsttime. Here, the notion “class of diagrams” refers to a gauge invariant subset of the diagrams thatcontributes to a certain process. Examples of non-factorizable diagrams are shown in Fig. 3.3.Before we present a detailed numerical estimate of the size of the non-factorizable contributions,we briefly recall two general arguments from Refs. [20, 21] justifying their omission.The first argument is based on the study of the associated color factors. The possible colorconfigurations for factorizable and non-factorizable corrections are shown in Figs. 3.4 and 3.5, re-spectively, together with the associated color factors. The leading color factor of the factorizable9 ( N c − Figure 3.4:
Color configurations associated to non-factorizable double-gluon exchange corrections to VBFat NNLO. = (cid:16) N c − (cid:17) × = (cid:16) N c − (cid:17) × = − (cid:16) N c − (cid:17) × Figure 3.5:
Color configurations associated to factorizable corrections to VBF at NNLO. corrections is ( N c − / =
16, while we have ( N c − / = N c =
3) for the double gluon ex-change consisting of a double color-traces. Hence the non-factorizable corrections are suppressedby a factor O (1 / N c ) with respect to the leading factorizable ones.The second argument is based on the kinematical dependence of diagrams like those shown inFig. 3.3. Such contributions, see e.g., Fig. 3.3 (right), come from the interference of diagrams withone or two gluons radiated by the upper quark line with diagrams where gluons are radiated bythe lower line. Angular ordering in gluon emission, however, leads to radiation close to the quarkfrom which it is emitted, and since these quarks tend to be very forward (or backward) due to theexchange of a spin-1 particle in the t -channel, there is generally very little overlap in phase space.Those arguments have already been used in [50] to justify neglecting real-virtual double gluonexchange diagrams in the computation of NLO QCD corrections for Higgs in VBF in associationwith three jets.We now corroborate these considerations with a quantitative analysis. Let us express the totalcross section at order α s as σ NNLO = σ (cid:16) + α s ∆ + α s ∆ (cid:17) , (3.21)with ∆ = ∆ f act + ∆ non − f act . As already said, ∆ receives contributions mainly from factorizablecorrections. The non-factorizable ones that are not kinematically suppressed are exactly zero dueto color while interference e ff ects between amplitudes with identical quark lines in the final stateare highly suppressed (and not included beyond LO in our approach).The exact calculation of ∆ non − f act being out of reach, we estimate the ratio of the non-factorizablecontributions, ∆ non − f act , vs. the factorizable ones, ∆ f act , as follows R ≡ ∆ non − f act ∆ f act ≃ N c − R ≡ N c − ∆ non − f act , U (1)1 ∆ f act , U (1)1 , (3.22)where ∆ f act , U (1)1 = ∆ f act / C F and ∆ non − f act , U (1)1 denotes the “would-be” impact of the correctionscoming from non-factorizable diagrams at NLO, i.e. , the class of diagrams involving the exchange10igure 3.6: The four virtual topologies with one gluon exchange between the quark lines that would con-tribute at NLO, barring the vanishing color factor. Those diagrams represent the virtual part included in ∆ non − f actNLO . of one gluon between the two quark lines, computed as if the color factor were non-vanishing. Inother words, ∆ non − f act , U (1)1 can be thought of as the correction due to the gauge invariant class ofdiagrams where an extra U (1) gauge boson is exchanged between the two quark lines includingreal and virtual diagrams. The R ≃ R approximation assumes, of course, that the ratios willnot dramatically change in going from NLO to NNLO. As there is no substantial di ff erence inthe kinematics and no non-Abelian vertices enter at NNLO in diagrams where two gluons areexchanged in a color singlet in the t -channel, we can conclude that Eq. (3.22) should provide areasonable estimate.To compute ∆ non − f act we need to account for the four diagrams shown in Fig. 3.6 along withthe analogous real emission terms. As our argument only needs to provide an estimate of the cor-rections and it is based on the kinematics we can slightly simplify the calculation of the tensorintegrals by considering only vector couplings of the vector boson to the quarks, which eliminatesall but the scalar five-point functions. The latter can be reduced in terms of scalar four-point func-tions [51–53] and then evaluated, together with the scalar four- and three-point functions comingfrom the Passarino-Veltmann reduction, with the help of the Q cd L oop package [54]. For complete-ness, we stress that the results for the virtual diagrams have been checked against the amplitudeautomatically generated by M ad L oop [55], where machine precision agreement has been foundpoint by point in the phase space.The combination of the virtual and the real emission part, with the subtraction of the soft diver-gences (no collinear divergences occur in this class of diagrams), has been done via M ad FKS [56]that generates all the needed counterterms automatically and performs also the integration overphase space. In practice, the computation of the O ( α s ) part of the cross section that enters in ∆ non − f act , has been obtained as the di ff erence of the complete NLO and the corresponding Borncross sections, and special attention has been paid to controlling the uncertainty of the numericalintegration of the real emission contributions.In Tabs. 3.1–3.3 we present the results for the non-factorizable corrections, ∆ non − f act , at theTevatron and at the LHC, compared with the quark-initiated factorizable ones, ∆ f act , and we haveused the same color factor of the Born term. For the sake of simplicity, we have focused on the ud → udH channel, with Z -boson exchange and, as mentioned above, we have considered only thevector coupling of the Z -boson to the quarks. Interestingly, the numbers in Tabs. 3.1–3.3 displaya sudden change of sign for ∆ non − f act at around m H =
180 GeV = M Z , i.e. , the threshold of the h → ZZ process, which is due to the use of the zero-width approximation for the Z -bosons in theloop propagators, cf. also Refs. [58, 59]. A consistent inclusion of Z -boson width e ff ects, which isbeyond the scope of our analysis, would regularize this behavior.The results of Tabs. 3.1–3.3 show an R always well below unity. Once the O (1 / N c ) color11 H [ GeV] σ [pb] ∆ non − f act , U (1)1 ∆ f act , U (1)1 R
100 3 . · − . · − . · − . . · − . · − . · − . . · − . · − . · − . . · − − . · − . · − − . . · − · − . · − . . · − . · − . · − . Non-diagonal NLO QCD corrections to VBF at the Tevatron, √ S = .
96 TeV. Numbers havebeen computed ignoring the vanishing color factors of the diagrams in Fig. 3.6. The MRST2002 [57] NLOPDF set has been used. Renormalization and factorization scales have been set to M W . Integration errors, ifrelevant, are shown in parenthesis. m H [ GeV] σ [pb] ∆ non − f act , U (1)1 ∆ f act , U (1)1 R
100 2 . · − . · − . · − . . · − . · − . · − . . · − . · − . · − . . · − − . · − . · − − . . · − − . · − . · − − . . · − − · − . · − − . . · − · − . · − . . · − · − . · − . . · − · − . · − . . · − · − . · − . . · − · − . · − . Non-diagonal NLO QCD corrections to VBF at the LHC, √ S = M W . Integration errors, ifrelevant, are shown in parenthesis. suppression in Eq. (3.22) is taken into account, one finds an upper bound on R < ∆ f act on the total cross-section is at the 1% level, we estimate the contribution of theomitted non-factorizable corrections at most at the per-mil level, hence negligible in our scheme.Finally, note that ∆ non − f act decreases with increasing Higgs boson masses m H , so that, as expectedfrom the fact that the Higgs boson acts as a “kinematical de-correlator” between the two jets, thesize of the non-factorizable corrections becomes totally negligible for m H >
300 GeV.
Diagrams with heavy-quark loops can also provide contributions at NNLO in QCD that are notincluded in the structure function approach. Following our definition of VBF given in Sec. 2these contributions are classified in the strict sense as “non-VBF” processes. However, giventhat such e ff ects are genuinely new at NNLO and, moreover, that they have not been subject toextensive consideration in the context of VBF in the literature before, cf. the previous discussions12 H [ GeV] σ [pb] ∆ non − f act , U (1)1 ∆ f act , U (1)1 R
100 5 . · − . · − . · − . . · − . · − . · − . . · − . · − . · − . . · − . · − . · − . . · − . · − . · − . . · − . · − . · − . . · − · − . · − . . · − · − . · − . . · − − · − . · − − . . · − − . · − . · − − . . · − − . · − . · − − . . · − − . · − . · − − . . · − − . · − . · − − . . · − − . · − . · − − . . · − − . · − . · − − . . · − − · − . · − − . . · − − · − . · − − . . · − − · − . · − − . . · − · − . · − . . · − · − . · − . . · − · − . · − . . · − · − . · − . . · − · − . · − . Non-diagonal NLO QCD corrections to VBF at the LHC, √ S =
14 TeV. Numbers have beencomputed ignoring the vanishing color factors of the diagrams in Fig. 3.6. The MRST2002 [57] NLO PDFset has been used. Renormalization and factorization scales have been set to M W . Integration errors, ifrelevant, are shown in parenthesis. in Refs. [21, 60], they deserve detailed study here.We distinguish three di ff erent classes of contributions: the modulo squared of one-loop dia-grams with no extra radiation requiring a quark-gluon initial state, Fig. 3.7; the interference ofone-loop diagrams with an extra parton in the final state, Fig. 3.8, with the VBF real tree-leveldiagrams; and the interference of two-loop diagrams, Fig. 3.9, with VBF diagrams at the Bornlevel. Each of the three classes of loop diagrams has no soft / collinear divergencies and is gaugeinvariant, thus can be treated independently.Such contributions appear only for neutral weak currents. Moreover, in both boxes and trian-gles only the axial coupling of the Z -boson to the quarks survives, so that a mass-degenerate quarkdoublet gives zero contribution. Therefore only the top and bottom quarks need to be considered. Let us start with the diagrams shown in Fig. 3.7. A simple estimate [20] obtained in the limit m b → , m t → ∞ , where only the contribution from the triangle is parametrically relevant, associates to13 /b t/b Figure 3.7:
Next-to-next-to-leading order QCD corrections due to heavy-quarks ( t / b ) loops: pure one-loopdiagrams contributing through their modulo squared. Figure 3.8:
Next-to-next-to-leading order QCD corrections due to heavy-quarks ( t / b ) loops: one-loop plusextra parton diagrams interfering with VBF NLO real corrections. Figure 3.9:
Next-to-next-to-leading order QCD corrections due to heavy-quarks ( t / b ) loops two-loop dia-grams interfering with VBF LO diagrams. this class an e ff ect of less than one per-mil of the total cross section [21, 60]. By itself this resultis non-trivial given that the contributions of the diagrams in Fig. 3.7 are proportional to the quark-gluon parton luminosity, which is potentially large, especially at LHC. Thus, based on consideringthe triangle alone, it can be argued that contributions from the heavy-quark loops in Fig. 3.7 canbe safely neglected.We now investigate to which extent the conclusion above is confirmed by a complete calcula-tion of these contributions. Note that the corresponding one-loop diagrams are long known [61,62](see also [7]), although in a di ff erent kinematic regime ( i.e. , time-like) for the Z -boson. Our resultsconfirm previous findings and extend them for the first time to the t -channel regime. The contri-bution of these diagrams to Higgs boson production in VBF are shown in Fig. 3.10, where the sumof the triangle and the box, the triangle and the box alone and the limit m t → ∞ at √ S = √ S = .
96 TeV for the Tevatron have been plotted. The pole mass values m b = .
62 GeV, m t = . m t → ∞ , m b → -5 -4
100 200 300 400 500 600 700 800 900 1000 √ s = 7 TeV @ LHC tri+boxtriboxtri+box ¥ m H (GeV) s (pb) -4 -3
100 200 300 400 500 600 700 800 900 1000 √ s = 14 TeV @ LHC tri+boxtriboxtri+box ¥ m H (GeV) s (pb) -7 -6 -5
100 120 140 160 180 200 220 240 260 280 300 √ s = 1.96 TeV @ Tevatron tri+boxtriboxtri+box ¥ m H (GeV) s (pb) Figure 3.10:
Contribution of the heavy-quarks (t / b) loops to the total NNLO VBF cross section, at theLHC with √ S = √ S =
14 TeV (top-right) and at the Tevatron, √ S = .
96 TeV. Numbersare computed with the MSTW 2008 [63] NNLO PDF set. The renormalization and factorization scales havebeen set to M W . In this case we verified that, as reported also in [61], the contribution of the box goes to zero.Moreover, specially for large m H , the triangle alone is also an upper bound, while for small m H it approximates the total cross-section within a factor of 2. Therefore, given our aim to asses theimportance of these contributions relative to the VBF cross section, it gives a reasonable estimateof the exact value.Our results have been obtained by two independent computations of the squared amplitude,and also agree point-by-point in phase space with the results of a code automatically generated byM ad L oop [55]. The upshot is that the exact computation corroborates the findings of [20]. Theimpact of the diagrams in Fig. 3.7 on the VBF Higgs production cross section is always below theper-mil level of the Born contribution, and therefore can be safely neglected.The exact result for the VBF cross section in the quark-gluon channel at NNLO allows for acomparison with the Higgs-Strahlung case. In this case a sizable destructive interference betweenthe triangle and the box diagrams takes place [7, 62], while this does not happen for VBF (see alsothe plots in Fig. 3.10). It is instructive to find the origin of such a di ff erent behavior of the triangleand the box diagrams contributions in an s -channel process [62] compared to a t -channel one (cf.Fig. 3.10). To this aim it is useful to define the relative phase angle φ between the triangle and the15 H [ GeV] σ LO σ tri + box σ tri σ box σ ∞ tri + box
100 1 .
49 4 . · − . · − . · − . · −
120 1 .
22 3 . · − . · − . · − . · −
150 9 . · − . · − . · − . · − . · −
200 6 . · − . · − . · − . · − . · −
250 4 . · − . · − . · − . · − . · −
300 2 . · − . · − . · − . · − . · −
400 1 . · − . · − . · − . · − . · −
500 8 . · − . · − . · − . · − . · −
650 3 . · − . · − . · − . · − . · −
800 1 . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − Table 3.4:
Values of the contributions to the total NNLO VBF cross-section due to the heavy-quark loopdiagrams shown in Fig. 3.7 at the LHC, √ S = M W . Integration errors are below the 1% level. Cross-sections are inpb. m H [ GeV] σ LO σ tri + box σ tri σ box σ ∞ tri + box
100 5 .
08 2 . · − . · − . · − . · −
120 4 .
29 1 . · − . · − . · − . · −
150 3 .
40 1 . · − . · − . · − . · −
200 2 .
40 1 . · − . · − . · − . · −
250 1 .
76 9 . · − . · − . · − . · −
300 1 .
33 6 . · − . · − . · − . · −
400 8 . · − . · − . · − . · − . · −
500 5 . · − . · − . · − . · − . · −
650 2 . · − . · − . · − . · − . · −
800 1 . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − Table 3.5:
Values of the contributions to the total NNLO VBF cross-section due to the heavy-quark loopdiagrams shown in Fig. 3.7 at the LHC, √ S =
14 TeV. The MSTW2008 [63] NNLO PDF set has beenused. The LO cross-section, computed with LO PDFs, is also shown for comparison. Renormalization andfactorization scales have been set to M W . Integration errors are below the 1% level. Cross-sections are inpb. box amplitudes M tri and M box by |M tri + M box | = |M tri | + |M box | + |M tri ||M box | cos φ . (3.23)Recall that in the s -channel case in the large- m t limit M tri and M box are real and opposite insign thus implying φ ∼ π [62]. This remains true for the crossed amplitudes in the t -channel caseof Fig. 3.10 because, due to unitarity, the asymptotic behavior in the large- m t limit remains thesame. However, for finite top-quark masses and away from large- m t limit, both M tri and M boxacquire an imaginary part above the threshold √ S ≃ m t in the s -channel process. This is due16o the gluon-gluon scattering process opening to real top-quark pair-production. Thus, M tri and M box become complex numbers pointing in opposite directions in the complex plane within atolerance of less than 50 ◦ resulting in a strong destructive interference between them, see [62].The di ff erent findings in the VBF case can now be easily understood: in the t -channel processof Fig. 3.7 only M box can develop an imaginary part due to an intermediate real top-quark pairwhile M tri remains real. As a net result M tri and M box are likely to be almost orthogonal in thecomplex plane thus suppressing the interference term in Eq. (3.23).Finally, compared to the Higgs-Strahlung process, i.e. , the associated production of Higgs anda Z -boson in the gluon-fusion channel, the NNLO corrections in VBF from the heavy quarks areby far less important [7]. This di ff erence has also a simple physical explanation, because in theHiggs-Strahlung this contribution proceeds through a gg initiated s -channel, whereas VBF hasthe vector boson in the t -channel. The s -channel propagator enhances the small x -region wherethe gluon luminosity steeply rises. In the crossed case, i.e. , in VBF, the dominant contribution tothe cross section comes from e ff ective parton momentum fractions h x i ∼ − , where the gluonluminosity is not large yet. Let us now discuss the contributions coming from diagrams in Fig. 3.8 in the qq -channel and thecrossed qg one. These diagrams feature one initial (final) state on-shell gluon attached to the heavy-quark loop, together with the Z -boson and a time-like (space-like) o ff -shell gluon. We estimate theoverall contribution of this class of diagrams by computing only the triangles, yet keeping the m t and m b finite. This choice is justified by the fact that triangles are parametrically leading,especially for large values of m H and, as shown in Sec. 3.3.1, provide an approximate estimatefor the total contribution, although su ffi ciently accurate for our purposes. We also verified thatthe limit m t → ∞ , which we expect to be close to the upper bound for the cross-section, is indeedquite close ( ≃ Z -boson decay into hadrons [64] or hadro-production [65] in a di ff erent kinematic regime. As an additional simplification, we will consideronly the parton channel with an extra gluon in the final state, i.e. , the reaction qq ′ → qq ′ Hg , theother parton channels having similar or smaller impact. This latter assumption is supported bythe previous studies of Z -boson hadro-production [65], where it was shown that the entire class ofdiagrams in Fig. 3.8 is heavily suppressed with respect to the two-loop ones of Fig. 3.9 consideredbelow in Sec. 3.3.3.We have computed the necessary expressions for the interference of the one-loop diagramswith the tree-level ones analytically using M athematica and F eyn C alc [66] and have performedthe phase-space integration using M ad G raph [67, 68]. The respective numbers for the contributionto the VBF total cross-section of the process qq ′ → Hqq ′ g due to the heavy-quark triangle inFig. 3.8 at the LHC, at √ S = Finally, we consider the two-loop contributions of Fig. 3.9 where the heavy-quark loop is attachedvia two gluons to the light-quark line originating from one of the protons. These diagrams interferewith the Born VBF amplitudes. The e ff ective coupling to the light-quark line singles out the iso-17 H [ GeV] σ LO σ + glutri
100 1 .
49 5 . · −
120 1 .
22 4 . · −
150 9 . · − . · −
200 6 . · − . · −
250 4 . · − . · −
300 2 . · − . · −
400 1 . · − . · −
500 8 . · − . · −
650 3 . · − . · −
800 1 . · − . · − . · − . · − m H [ GeV] σ LO σ + glutri
100 5 .
08 2 . · −
120 4 .
29 2 . · −
150 3 .
40 1 . · −
200 2 .
40 1 . · −
250 1 .
76 7 . · −
300 1 .
33 5 . · −
400 8 . · − . · −
500 5 . · − . · −
650 2 . · − . · −
800 1 . · − . · − . · − . · − Table 3.6:
Values of the contributions to the total NNLO VBF cross-section due to the heavy-quark triangleplus gluon emission diagrams shown in Fig. 3.8 at the LHC, at √ S = √ S =
14 TeV (right).The MSTW2008 [63] NNLO PDF set has been used. The LO cross-section, computed with LO PDFs, isalso shown for comparison. Renormalization and factorization scales have been set to M W . Integrationerrors are below the 1% level. Cross-sections are in pb. triplet component of the proton in the squared matrix elements, since only the axial part of the Z -boson coupling contributes for non-degenerate heavy-quarks in the loop. That is to say, this classof diagrams is proportional to the (non-singlet) distribution δ q − ns of Eq. (3.19), which is generallysmall.In analogy to the previous Sec. 3.3.2, we aim at an estimate of the size of the contributionsof Fig. 3.9 rather than the exact result. To that end, we will restrict ourselves to the two-looptriangle diagrams. This choice is, of course, also driven by the fact that the two-loop double box inFig. 3.9 is currently unknown for the required VBF kinematical configuration and its computationwould be a tremendous task in itself.The two-loop triangles for the q ¯ qZ vertex have been computed in [64] for the case m b = m t ,
0, see also [65]. Rather compact results in terms of harmonic polylogarithms [48] (see [49]for numerical routines) for all kinematic configurations have been obtained in [69] and we usethe latter expressions to compute the numbers shown in Tab. 3.7 for the LHC at √ S = qq ′ → Hqq ′ at NNLO in QCD mediated by the two-loop triangle with the LO cross-section shows that suchcontributions are below the per-mil level.In ending this discussion of heavy-quark loop contributions we briefly remark that, of course,the diagrams shown in Figs. 3.7–3.9, also contribute in heavy-quark DIS, if the full neutral-currentreactions are considered, i.e. , both γ and Z -boson exchange at high Q . Currently available DISdata on heavy-quark production, however, is usually taken at Q values where these contributionsare not relevant. Their existence is an issue, though, to be recalled in the definition of variable-flavor number schemes, see e.g., [70]. We briefly discuss the electroweak corrections at one-loop. The combined strong and electroweakNLO corrections to Higgs production in VBF have been computed in [18, 19] and can be obtained18 H [ GeV] σ LO σ
100 1 .
49 8 . · −
120 1 .
22 7 . · −
150 9 . · − . · −
200 6 . · − . · −
250 4 . · − . · −
300 2 . · − . · −
400 1 . · − . · −
500 8 . · − . · −
650 3 . · − . · −
800 1 . · − . · − . · − . · − m H [ GeV] σ LO σ
100 5 .
08 2 . · −
120 4 .
29 2 . · −
150 3 .
40 1 . · −
200 2 .
40 1 . · −
250 1 .
76 9 . · −
300 1 .
33 7 . · −
400 8 . · − . · −
500 5 . · − . · −
650 2 . · − . · −
800 1 . · − . · − . · − . · − Table 3.7:
Values of the contributions to the total NNLO VBF cross-section due to the two-loop trinaglediagram shown in Fig. 3.9 at the LHC, at √ S = √ S =
14 TeV (right). The MSTW2008 [63]NNLO PDF set has been used.The LO cross-section, computed with LO PDFs, is also shown for compar-ison. Renormalization and factorization scales have been set to M W . Integration errors are below the 1%level. Cross-sections are in pb. via the program HAWK [71].In absence of a full calculation up to corrections of order α s α EW with respect to the Born am-plitude, which is currently beyond capabilities, a combination of EW and NNLO QCD correctionsis possible, yet formally subject to ambiguities. A pragmatic way to proceed is to follow two dif-ferent approaches, multiplicative or additive, and the corresponding di ff erences used to assess theimpact of neglected terms. Using a compact notation, we define σ QCDNNLO = σ + α s σ QCD + α s σ QCD (3.24)the total cross section at NNLO in QCD, σ EW = σ + α EW σ EW (3.25)the total cross sections including NLO EW corrections, and σ NLO + EW the result of the full calcu-lation at NLO in QCD and EW of Refs. [18, 19].The additive scheme amounts to simply define σ NNLO + EW ≡ σ NLO + EW + α s σ QCD , (3.26) i . e . , to add the missing α s terms to the full NLO + EW calculation. In this scheme no assumption onthe factorization of QCD and EW corrections is made and only terms that are known are included.It demands, however, the two terms to be evaluated with exactly the same EW ( G F , M Z , M W , . . . )and QCD (scales, α s , and PDFs) parameters, something possibly error-prone when di ff erent codesare used.The multiplicative scheme amounts to assuming that the QCD and EW corrections factorize toa very good approximation at NLO as well as at NNLO. In this case one can define σ NNLO + EW ≡ σ QCDNNLO + σ EW σ . (3.27)19his is a very handy approximation: it implies that the EW corrections can be evaluated indepen-dently as e ff ects such as scale and PDF choices are mostly canceled in the ratio σ EW /σ . Note,however, that it implies the inclusion of unknown higher order terms in the results. This is the ap-proach followed in Ref. [8] and the results relevant for Higgs production at the LHC can be foundthere. We have explicitly checked for a few values of the Higgs mass, that the di ff erences betweenthe two schemes are very small and totally negligible at the LHC. We are now in a position to present an extensive phenomenological analysis for the VBF produc-tion mechanism at the LHC at the center of mass energies of √ S = √ S =
14 TeV andthe Tevatron, √ S = .
96 TeV, employing the structure function approach up to the NNLO in QCD.This is a significant extension of our previous studies in Refs. [20, 21]. For the numerical resultswe use the following values for the electroweak parameters (see also [8]): The masses of the vec-tor bosons are M W = .
398 GeV and M Z = . G F = . · − and sin θ w = . Γ W and Γ Z , have been set to zero for simplicity, their e ff ect being of order 10 − or less on the totalrate. Moreover, we provide numbers for all PDFs currently available at NNLO accuracy in QCD,ABKM [70, 72], HERAPDF1.5 [73, 74], JR09 [75, 76], MSTW2008 [63] and NNPDF2.1 [77] al-ways using the default value of the strong coupling α s required by the respective set. A cut of1 GeV has been used to regulate the phase space integration over Q which extends to vanishing Q (see Sec. A.1). In the numerical evaluation we have checked explicitly that the results remainunchanged upon variations of this cut on Q , a fact readily understood by realizing that the e ff ective h Q i in VBF is much larger, typically h Q i ≃
20 GeV, see the discussion in [20].All results presented here can also be obtained through our publicly usable VBF@NNLOcode [78].
Let us start o ff with the Tevatron, where the cross section is roughly 0 . m H =
100 GeV and steeply falling as a function of m H . An exhaustive list of cross-sections,for di ff erent values of the Higgs boson mass in the range m H ∈ [90 , µ r = µ f = Q , where Q is the virtuality of the vector bosons which fuse into the Higgs, cf.Eq. (3.2). Our results in Tabs. B.1–B.5 show that the NLO corrections are always positive and nottoo large of the order of 1-2%, while the NNLO corrections are typically small of the order of ± µ r , µ f ∈ [ Q / , Q ]. Here we find markedly a clear improvement dueto the NNLO corrections computed. While we observe, e.g. for a Higgs mass of m H =
100 GeV,variations of order ± σ LO , this reduces to ± σ NLO , and to ±
2% for σ NNLO .The scale uncertainty increases slightly for larger Higgs masses, e.g. for m H =
250 GeV, we findorder ± ± ±
3% at LO, NLO and NNLO, respectively, see also Fig. 4.1. Ofcourse, other scale choices are possible relating µ r , µ f to the Higgs boson mass m H or the W -bosonmass M W , see Fig. 4.2 for a comparison. As discussed in [20] the present choice, i.e. , relating20 r and µ f to Q turns out to be the most natural one with the point of minimal sensitivity being µ r , µ f ≃ Q , as it exhibits the best pattern of apparent convergence of the perturbative expansion.Considering the size of the neglected contributions discussed at length above in Sec. 3 the residualtheoretical uncertainty due to perturbative QCD corrections at higher orders at the Tevatron is ofthe order 2-3%.The PDF dependence is clearly the dominating source of uncertainty at the Tevatron, the in-dividual PDFs report an uncertainity in the PDFs at NNLO (sometimes combined with the one inthe strong coupling α s ) of the order ± ff er by roughly 5% at low Higgs masses, e.g. m H =
100 GeV, which is increasingtowards larger Higgs masses, e.g., to 10% at m H =
250 GeV, see also Fig. 4.3. This is due to thelarger values of e ff ective h x i at which the parton luminosities are probed. In summary, thus, theuncertainty due to the non-perturbative parameters (PDFs, α s ) can be estimated to be of the order5% for a light Higgs boson at the Tevatron. s (pb) at Tevatron √ s = 1.96 TeVscale choice:Q/4 ≤ m R , m F ≤
4Q LONLONNLO -2 -1 -2 -1 s ( m R , m F )/ s NNLO (Q)0.850.90.9511.051.11.15 100 125 150 175 200 225 250 275 300 m H (GeV) Figure 4.1:
The total VBF cross sections at the Tevatron, √ S = .
96 TeV, at LO, NLO and NNLO in QCDwith the scale uncertainity from the variation µ r , µ f ∈ [ Q / , Q ]. The MSTW2008 [63] PDF set (68% CL)has been used. Numbers are in pb. Let us now present the numbers for the current and foreseen center-of-mass energies of the LHC.For a Higgs boson with a mass m H =
100 GeV the cross section is roughly 1 . .
5) pb at √ S = m H , e.g. to 0 . .
5) pb at m H =
500 GeV. Thisis illustrated in Figs. 4.4 and 4.7. The complete listings of cross-sections at LO, NLO and NNLOin QCD for Higgs boson masses in the range m H ∈ [90 , µ r = µ f = Q are given inTabs. B.6–B.10 ( √ S = √ S =
14 TeV), respectively.In analogy to the preceeding discussion in Sec. 4.1 our results in Tabs. B.6–B.10 demonstratea very good apparent convergence of the perturbative expansion, i.e. , the NLO corrections of theorder of 5%, while the NNLO corrections are of the order of ± (pb) at Tevatron, √ s = 1.96 TeVm H =120 GeV m R = m F = k Q LONLONNLO k = m R /Q -1 s (pb) at Tevatron, √ s = 1.96 TeVm H =120 GeV m R = m F = k m H LONLONNLO k = m R /m H -1 Figure 4.2:
Scale dependence of the VBF cross sections at the Tevatron, √ S = .
96 TeV at LO, NLOand NNLO in QCD for m H =
120 GeV and the choice µ r = µ f = κ Q (left) and µ r = µ f = κ m H (right). TheABKM [70] PDF set has been used. Numbers are in pb. s (pb) at Tevatron √ s = 1.96 TeV m R = m F = Q MSTW08ABKMJR09m H (GeV) s (pb) at Tevatron √ s = 1.96 TeV m R = m F = Q MSTW08HERAPDFNNPDF2.1m H (GeV) Figure 4.3:
The PDF uncertainity of the VBF cross sections at the Tevatron, √ S = .
96 TeV, at NNLO inQCD for PDF sets of ABKM [70], HERAPDF1.5 [73, 74], JR09 [75, 76], MSTW2008 [63] (68 % CL), andNNPDF [77]. All results have been normalized to the best fit of MSTW2008. display also an impressive perturbative stability with respect to scale variations, see Figs. 4.5 and4.8 also for a comparison of the scale choice relating µ r and µ f to Q and to m H , respectively. Thepreference to low scales is obvious from those plots, again with the point of minimal sensitivitybeing of the order µ r , µ f ≃ Q . Given that all the perturbative QCD corrections not accounted forin the structure function approach are very small and negligible, as discussed in Sec. 3, we canestimate the residual theoretical uncertainty due to uncalculated higher orders in QCD at the LHCto be of order 2% for the entire Higgs mass range up to m H = α s as provided by the respective PDF set). All PDF sets under consideration are displayed inFigs. 4.6 and 4.9. In comparison to the Tevatron, we see that over a large range of Higgs masses, m H ≃ . . .
300 GeV, there are rather small di ff erences between these sets only, because quarkPDFs are well constrained in the relevant x -region. Generally, the PDF uncertainties are larger forthe lower running energy of the LHC. The plots in Figs. 4.6 and 4.9 show that an almost constant2% PDF uncertainty can be associated to the cross section for the LHC for a light Higgs boson.22 (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤
4Q LONLONNLO -2 -1 -2 -1 s ( m R , m F )/ s NNLO (Q)0.850.90.9511.051.11.15 100 200 300 400 500 600 700 800 900 1000 m H (GeV) Figure 4.4:
The total VBF cross sections at the LHC, √ S = µ r , µ f ∈ [ Q / , Q ]. The MSTW2008 [63] PDF set (68% CL) hasbeen used. Numbers are in pb. s (pb) at LHC, √ s = 7 TeVm H =120 GeV m R = m F = k Q LONLONNLO k = m R /Q -1 s (pb) at LHC, √ s = 7 TeVm H =120 GeV m R = m F = k m H LONLONNLO k = m R /m H -1 Figure 4.5:
Scale dependence of the VBF cross sections at the LHC, √ S = m H =
120 GeV and the choice µ r = µ f = κ Q (left) and µ r = µ f = κ m H (right). The ABKM [70]PDF set has been used. Numbers are in pb. This deteriorates towards larger Higgs masses, e.g., the uncertainty being of order ± m H = α s ) needed for VBFprecision predictions are well under control. For larger Higgs masses, the PDF uncertainty is thedominating. 23 (pb) at LHC √ s = 7 TeV m R = m F = Q MSTW08ABKMJR09m H (GeV) s (pb) at LHC √ s = 7 TeV m R = m F = Q MSTW08HERAPDFNNPDF2.1m H (GeV) Figure 4.6:
The PDF uncertainity of the VBF cross sections at the LHC, √ S = s (pb) at LHC √ s = 14 TeVscale choice:Q/4 ≤ m R , m F ≤
4Q LONLONNLO -1 -1 s ( m R , m F )/ s NNLO (Q)0.850.90.9511.051.11.15 100 200 300 400 500 600 700 800 900 1000 m H (GeV) Figure 4.7:
The total VBF cross sections at the LHC, √ S =
14 TeV, at LO, NLO and NNLO in QCD withthe scale uncertainity from the variation µ r , µ f ∈ [ Q / , Q ]. The MSTW2008 [63] PDF set (68% CL) hasbeen used. Numbers are in pb. One of the most important features of the structure function approach resides in its universality:as long as the final state X in V ∗ V ∗ → X fusion process does not interact (strongly) with the quarklines, the cross section can be factorized using Eq. (3.2) and QCD corrections decouple from thenature of the fusion process. This property allows the computation of production cross sections atNNLO in QCD for a variety of processes relevant for new physics searches in a straightforwardway. In this section we present results for a few examples: Higgs production through anomalouscouplings which are relevant when new physics states are heavy and the e ff ects enter only at loop24 (pb) at LHC, √ s = 14 TeVm H =120 GeV m R = m F = k Q LONLONNLO k = m R /Q -1 s (pb) at LHC, √ s = 14 TeVm H =120 GeV m R = m F = k m H LONLONNLO k = m R /m H -1 Figure 4.8:
Scale dependence of the VBF cross sections at the LHC, √ S =
14 TeV, at LO, NLO and NNLOin QCD for m H =
120 GeV and the choice µ r = µ f = κ Q (left) and µ r = µ f = κ m H (right). The ABKM [70]PDF set has been used. Numbers are in pb. s (pb) at LHC √ s = 14 TeV m R = m F = Q MSTW08ABKMJR09m H (GeV) s (pb) at LHC √ s = 14 TeV m R = m F = Q MSTW08HERAPDFNNPDF2.1m H (GeV) Figure 4.9:
The PDF uncertainity of the VBF cross sections at the LHC, √ S =
14 TeV, at NNLO in QCDfor PDF sets of ABKM [70], HERAPDF1.5 [73, 74], JR09 [75, 76], MSTW2008 [63] (68 % CL), andNNPDF [77]. All results have been normalized to the best fit of MSTW2008. level; extra neutral and charged scalar production in extended Higgs sectors, such as in two Higgsdoublet and triplet Higgs models; neutral and charged vector fermiophobic resonance production.
V V H couplings
Due to its very peculiar signature with two forward tagging jets, whose angular correlations canalso be studied and related to the Higgs CP properties independently of the Higgs decay channel,VBF can be a powerful means to discover new physics residing at a scale Λ and responsible foranomalous VV H vertices [79]. The most generic structure for the
VV H vertex which generalizesEq. (3.1), has the form [80]: Γ µν ( q , q ) = M V (cid:18)(cid:16) √ G F (cid:17) / + a Λ (cid:19) g µν + a Λ (cid:16) q · q g µν − q µ q ν (cid:17) + a Λ ε µνρσ q ρ q σ , (5.1)where the e ff ective coe ffi cients a i are dimensionless and vanish in the SM. The VBF cross-sectionincluding anomalous vertices is known at NLO in QCD [81], and has been included in the program25BFNLO [82].As discussed in [79], at high energies one expects the e ff ective coe ffi cients a i to become formfactors, i.e. to display a dependence on the scattering energy such that unitarity would be preserved.However, since at the LHC the typical scales in a VBF process can be still considered small withrespect to a new physics scale of the order of 1 TeV, we will present numbers for constant valuesof a i and Λ =
500 GeV. With the vertex in Eq. (5.1), the expression corresponding to Eq. (3.5)becomes a bit more involved than its SM counterpart and it is given in the Appendix, Sec. A.2.Since a only changes the cross-section by an overall factor, we set it to 0. We therefore plotin Fig. 5.1 the total cross-sections at the LHC for di ff erent values of a and a . Uncertainties atNNLO are again found to be of the order of a few percent. -1 -1 -1 -1 H s (pb) at LHC(anomalous VVH coupling) √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0.02, 0)(a ,a ) = (0.05, 0) -1 -1 -1 -1 -1 -1 -1 H s (pb) at LHC(anomalous VVH coupling) √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0.02, 0)(a ,a ) = (0.05, 0) -1 -1 -1 m H (GeV) -1 -1 -1 -1 H s (pb) at LHC(anomalous VVH coupling) √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0.02, 0)(a ,a ) = (0.05, 0) -1 -1 -1 m H (GeV) -1 -1 -1 -1 H s (pb) at LHC(anomalous VVH coupling) √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0, 0.02)(a ,a ) = (0, 0.05) -1 -1 -1 -1 -1 -1 -1 H s (pb) at LHC(anomalous VVH coupling) √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0, 0.02)(a ,a ) = (0, 0.05) -1 -1 -1 m H (GeV) -1 -1 -1 -1 H s (pb) at LHC(anomalous VVH coupling) √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0, 0.02)(a ,a ) = (0, 0.05) -1 -1 -1 m H (GeV) H s (pb) at LHC(anomalous VVH coupling) √ s = 14 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0.02, 0)(a ,a ) = (0.05, 0) H s (pb) at LHC(anomalous VVH coupling) √ s = 14 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0.02, 0)(a ,a ) = (0.05, 0) m H (GeV) H s (pb) at LHC(anomalous VVH coupling) √ s = 14 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0.02, 0)(a ,a ) = (0.05, 0) m H (GeV) H s (pb) at LHC(anomalous VVH coupling) √ s = 14 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0, 0.02)(a ,a ) = (0, 0.05) H s (pb) at LHC(anomalous VVH coupling) √ s = 14 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0, 0.02)(a ,a ) = (0, 0.05) m H (GeV) H s (pb) at LHC(anomalous VVH coupling) √ s = 14 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO (a ,a ) = (0, 0)(a ,a ) = (0, 0.02)(a ,a ) = (0, 0.05) m H (GeV) Figure 5.1:
The total cross-section for Higgs production via VBF at the LHC, with √ S = √ S =
14 TeV (bottom). The values used for the parameters a , a are written on the plots. We have assumed Λ =
500 GeV. The MSTW 2008 [63] PDF set has been used. The uncertainty bands are obtained from thevariation of the renormalization and factorization scale in the interval Q / < µ f , µ r < Q , where Q is thevirtuality of the vector boson. Many extension of the SM feature an Higgs sector with an extra SU(2) doublet and / or triplet, thatlead to a particle spectrum including several neutral Higgs bosons, as well as single- and, possibly,26ouble-charged scalars. Because of the particular signature with forward jets, VBF can be a reallypowerful search channel also for these states.In a generic two Higgs doublet model (2HDM), such as that in the Minimal SupersymmetricStandard Model (MSSM) [83], the production rate of a CP-even light or heavy Higgs scalar viaVBF is equal to the SM one times overall factors [84–86]. Therefore, to obtain the total cross-section including QCD NNLO corrections for the h or H production, one just needs to rescalethe SM results given in Sec. 4.Pseudo-scalar states do not couple to vector bosons at the tree level, therefore VBF is notreally relevant in this case. On the other hand, it is of great phenomenological interest to study theproduction of charged Higgs bosons. As we will briefly see, there exist a class of models in whichcharged Higgs boson production via VBF can be observed. H + s (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤
4Q LONLONNLO -1 -1 m H (GeV) H ++ s (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤
4Q LONLONNLO -1 -1 m H (GeV) H - s (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤
4Q LONLONNLO -2 -1 -2 -1 m H (GeV) H -- s (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤
4Q LONLONNLO -2 -1 -2 -1 m H (GeV) Figure 5.2:
The total cross-section for charged Higgs production via VBF at the LHC, with √ S = Q / < µ f , µ r < Q , where Q is the virtuality of thevector boson. Plots refer to: H + production (top-left), H ++ production (top-right), H − production (bottom-left), H −− production (bottom-right). In models where the extra Higgs bosons are included in SU(2) doublets, like the MSSM orthe 2HDM, isospin conservation forbids (at the tree level) the appearance of W ± H ∓ Z and W ± H ∓ γ vertices, while they can be loop-induced [87]. However, in this case the corresponding total cross-section for VBF is totally negligible [88], leaving the associated production of a charged Higgs27 + K-factors:
LONLONNLO H (GeV) H ++ K-factors:
LONLONNLO H (GeV) H - K-factors:
LONLONNLO H (GeV) H -- K-factors:
LONLONNLO H (GeV) Figure 5.3:
The (N)NLO / LO (upper inlay) and NNLO / NLO (lower inlay) K -factors for charged Higgsproduction via VBF at the LHC, with √ S = Q / < µ f , µ r < Q , where Q is the virtuality of the vector boson. Plots refer to: H + production (top-left), H ++ production (top-right), H − production (bottom-left), H −− production (bottom-right). boson and a top-quark from a bottom-gluon initial state [89, 90] as the only phenomenologicallyrelevant production channel.This situation changes when models with isospin triplets in the Higgs-sector are considered.Many versions of such models exist in literature, with a very interesting particle content includingdouble-charged Higgs bosons. In many of such models the coupling of the charged Higgs bosons togauge bosons is allowed with a strength proportional to the ratio v ′ / v , where v ′ is the triplet vacuumexpectation value (vev) and v the SM one. Even though v ′ is constrained by electroweak data, inparticular by the ρ parameter of the SM, to be at most a few GeV, this bound can be loosenedby including more triplets [91]. In this case, the production of single- and double-charged Higgsbosons via VBF can be relevant at colliders [88, 92, 93]. The Lorentz structure of the VV H vertex28 + PDF errors:
JR09-VFABKM09MSTW08-68% CL m H (GeV) H ++ PDF errors: m H (GeV) H - PDF errors: m H (GeV) H -- PDF errors: m H (GeV) Figure 5.4:
The PDF uncertainties to the NNLO total cross section for charged Higgs boson production inVBF at the LHC, with √ S = H + production (top-left), H ++ production (top-right), H − production (bottom-left), H −− production (bottom-right) is, like the SM one in Eq. (3.1), proportional to the metric tensor (see, e.g., the Feynman rulesgiven in Ref. [94]), and it can be cast in the generic form Γ µν V i V j H = (cid:16) √ G F (cid:17) / m i m j F i j (cid:0) − ig µν (cid:1) , (5.2)where the dimensionless constant F i j depends on the particular model (in the SM the non-vanishingones are F ZZ = F W ± W ∓ = (cid:12)(cid:12)(cid:12) F i j (cid:12)(cid:12)(cid:12) = H ± , H ±± production via VBF at the LHC with √ S = K -factors are shown. In particularwe find that the NNLO K -factor for the nominal scale choice ( µ r = µ f = Q ) di ff ers from one forless than 1% in almost the whole mass range shown. From Fig. 5.4 it is clear that the uncertaintycoming from the PDFs is sizable, in particular for large values of the charged Higgs masses, wherethe PDFs are probed at larger values of x , i.e. , in a region, where the PDF uncertainties generallyincrease. 29 .3 Production of fermiophobic vector boson resonances In many extensions of the SM, additional vector bosons can appear in the particle spectrum, to-gether with or as an alternative to the Higgs boson, see e.g. [22, 23, 95, 96]. The existence ofheavy vector bosons, however, is severely constrained by electroweak precision data and directsearches and masses are pushed to the multi-TeV scale. This holds unless their couplings to (light)fermions are for some reason suppressed and only couplings to the SM weak vector bosons takeplace. In such models, where the new heavy vector resonances are dubbed “fermiophobic”, VBFcan become the dominant production channel.While a detailed phenomenological analysis is beyond our scope, we just limit ourselves topresenting a few results motivated by warped scenarios [97, 98] for the production of a neutral anda single-charged vector resonance, which couples to SM vector bosons via a trilinear vertex of theform Γ µ µ µ ( p , p , p ) = g h g µ µ (cid:16) p µ − p µ (cid:17) − g µ µ (cid:16) p µ − p µ (cid:17) − g µ µ (cid:16) p µ − p µ (cid:17)i . (5.3)This particular choice assumes that the new vector bosons are explicit or “hidden” gauge vectors ofa new sector. Di ff erent vertices are also possible (see e.g., [99,100]) and can be easily implementedin our calculation.The expression for the total cross-section in terms of structure functions and particle momentais given in Sec. A.3. The results we present are for neutral and charged vector boson production,which we call W ′± or Z ′ and which we assume not to be coupled to photons. (This is in factnot a real limitation as in practice forward jets are required entailing de facto a minimum Q forthe exchanged particle in the t -channel and such a cut could also be e ff ectively included in thestructure function approach). For the sake of simplicity, we also assume that no ZZZ ′ vertex existsin our model and that in the WZW ′ and WWZ ′ vertices the coupling constants are equal to thoseappearing in the respective WWZ
SM vertex: g WWZ ′ = g WZW ′ = g WWZ = g w cos θ w . (5.4)The corresponding results are shown in Fig. 5.5.30 W’/Z’ s (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO ZW > W’ + ZW > W’ - WW > Z’
W’/Z’ s (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO ZW > W’ + ZW > W’ - WW > Z’ m V’ (GeV) W’/Z’ s (pb) at LHC √ s = 7 TeVscale choice:Q/4 ≤ m R , m F ≤ LONLONNLO ZW > W’ + ZW > W’ - WW > Z’ m V’ (GeV)
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300 400 500 600 700 800 900 1000 m V’ (GeV) Figure 5.5:
The total cross-section for a vector resonance ( Z ′ or W ′ ) via VBF at the LHC, √ S = √ S =
14 TeV (bottom). The trilinear coupling between vector bosons is assumed to be equal to theone for
ZWW in the SM. The MSTW 2008 [63] PDF set has been used. The uncertainty bands are obtainedfrom the variation of the renormalization and factorization scale in the interval Q / < µ f , µ r < Q , where Q is the virtuality of the vector boson. The computation of NNLO in QCD predictions for production cross sections at the LHC is ingeneral a formidable, but nevertheless strongly motivated task. To date only a handful of verysimple processes (with 2 → V ∗ → V H . In this work we have argued that VBFprocesses provide a notable exception, being the only class of processes beyond one-body finalstates at the Born level (actually starting from a genuine three-body one) whose total cross sectioncan be calculated at NNLO through the use of currently available techniques and results. Wehave shown that the key reason for this unique possibility is that the structure function approachand the factorization approximation on which it is based, work extremely well up to order α s ,corrections being kinematically and parametrically (by α s / N c ) suppressed and de facto negligible.Considering also contributions coming from processes involving virtual heavy quarks, neglectedin the past, we find that the residual theoretical uncertainty from higher order QCD correctionsis at the 2% level over a wide range of Higgs boson masses. With the PDF uncertainties beingalso of the same order, one concludes that cross sections for this class of processes are amongthe most precise rate predictions available in LHC phenomenology. In this respect, our resultsstrongly encourage the e ff orts towards the calculation of the di ff erential NNLO rates in DIS, whichis now in sight [101]. A fully exclusive DIS computation at NNLO would put VBF on par withgluon-gluon fusion, and V ∗ → V H (see e.g., [102] for recent progress). It would allow to mimicexperimental selection cuts on the forward jets and to estimate non-trivial e ff ects such as centraljet-veto e ffi ciencies, for the first time at NNLO.Another very pleasant feature of the structure function approach is its universality: any weakboson fusion process to an arbitrary n -body final state X n (of particles not or very weakly in-teracting with the quark lines), i.e. , V ∗ V ∗ → X n , can be easily computed at the NNLO. As firstsimple applications we have considered fusion of V ∗ V ∗ to one-particle BSM final states: neutralscalar production with the full set of anomalous vector-vector-scalar couplings, single-chargedand double-charged scalars, neutral and charged heavy vector resonances. Other models and scat-31ering processes, such as two-body (scalar-scalar, scalar-vector, vector-vector, and fermion-anti-fermion) final states can be easily implemented. In fact, thanks to its modularity, the structurefunction approach to VBF cross sections via VBF@NNLO could be automatized using tools likeF eyn R ules [103] and M ad G raph a MC@NLO [55, 56, 105].Work in this direction is in progress. In the meantime, all results presented in this article can alsobe easily obtained through the public use of our VBF@NNLO code [78].
Acknowledgments
We thank R. Frederix, S. Frixione and V. Hirschi for their help in using M ad FKS [56] and M ad -L oop [55], and A. Akeroyd, G. Bozzi, C. Grojean, B. Kniehl and A. Vicini for discussions. TheFeynman diagrams in this article have been prepared using J axodraw [106] and A xodraw [107].This work is partially supported by the Belgian Federal O ffi ce for Scientific, Technical and Cul-tural A ff airs through Interuniversity Attraction Pole No. P6 /
11, by the Deutsche Forschungsge-meinschaft in Sonderforschungsbereich / Transregio 9 and by the European Commission throughcontract PITN-GA-2010-264564 (
LHCPhenoNet ). A Useful formulae
A.1 The VBF phase space
In this appendix we briefly document the parameterization for the phase space of the VBF process.We will first take the most general case, which is the production of an n -particles final state viaVBF with momenta K , . . . , K n , then we will specialize to the case of one particle.In the structure function approach we can consider the proton remnants as massive particles, andintegrate over their masses, which we label s , s .We will call P , P the momenta of the incoming protons, and P X , P X the momenta of the protonremnants. The choice of kinematical variables is guided by the requirement to resemble the DISkinematics as much as possible. We work in the hadronic center-of-mass reference frame, with P = (cid:16) √ S , , , √ S (cid:17) , P = (cid:16) √ S , , , − √ S (cid:17) . (A.1)Then, the Lorentz invariant phase space for this process is (cf. Eq. (3.2)), dPS = Y i = , ds i d P X i (2 π ) πδ (cid:16) P X i − s i (cid:17) dPS n ( K , . . . , K n ) (2 π ) δ P + P − P X − P X − X j = , n K j . (A.2)In Eq. (A.2) we have separated the phase space of the proton remnants and the one of the particlesproduced via VBF. In order to solve the kinematics, let q i be the momentum exchanged by each ofthe two protons, with the convention, as for DIS, that the direction of q i is incoming with respectto the proton vertex (see Fig. 2.1), q i = P X i − P i . (A.3)32e parameterize q i in terms of the two light-like momenta P i and the transverse components q ⊥ i : q = P · q P · P P + P · q P · P P + q ⊥ , (A.4) q = P · q P · P P + P · q P · P P + q ⊥ . (A.5)The Bjorken scaling variables x i for the DIS process are given by the scalar products P i · q i P i · q i = − q i x i = Q i x i , (A.6)where Q i = − q i . In analogy one can also define variables y i via the relation2 P · q = Q y , P · q = Q y , (A.7)so that y i = − Q i x i S ( Q i − q ⊥ i ) , (A.8)with q ⊥ i = q(cid:12)(cid:12)(cid:12)(cid:12) q ⊥ i (cid:12)(cid:12)(cid:12)(cid:12) .Then, the integration measure can be expressed as d q i = dQ i d q ⊥ i dx i x i , (A.9)and P X i = Q i x i − ! . (A.10)The phase space Eq. (A.2) reduces now to the following form, dPS = π ) Y i = , dQ i d q ⊥ i dx i x i dPS n ( K , . . . , K n ) (2 π ) δ q + q + X j = , n K j , (A.11)and the integrations on the transverse components can be cast in polar coordinates, d q ⊥ d q ⊥ = q ⊥ dq ⊥ d ϕ q ⊥ dq ⊥ d ϕ = π q ⊥ dq ⊥ q ⊥ dq ⊥ d ϕ . (A.12)The momentum-conservation condition imposed by the Dirac delta-function in Eq. (A.11) allowsto write a relation between the proton remnants variables and the total energy of the VBF products: S V BF = ( K + . . . + K n ) = ( q + q ) == Q Q S x x (cid:16) Q − q ⊥ (cid:17) (cid:16) Q − q ⊥ (cid:17) + Q Q S x x − Q − Q − q ⊥ q ⊥ cos ϕ . (A.13)Using this equation one can generate the phase space of the VBF-produced particles once the pro-ton remnants variables are fixed, simply in the same way as for the decay of a particle.33e will finally go through the details of the case in which only one particle with mass m andmomentum K is produced via VBF. In this case we simply have dPS ( K ) = d K (2 π ) πδ (cid:16) K − m (cid:17) , (A.14)and the d K integration can be eliminated using the momentum-conservation Dirac delta-function,so that we just need to solve the mass-shell condition imposed by the delta-function in Eq. (A.14)(which is nothing but Eq. (A.13) with S V BF = m ): m = Q Q S x x (cid:16) Q − q ⊥ (cid:17) (cid:16) Q − q ⊥ (cid:17) + Q Q S x x − Q − Q − q ⊥ q ⊥ cos ϕ . (A.15)This equation can, of course, be easily solved for cos ϕ , but in this case it would lead to an(integrable) singularity in the Jacobian, and therefore to numerical problems. Therefore, our choicehas been to write Eq. (A.15) as a second degree equation for q ⊥ : Aq ⊥ + Bq ⊥ + C = , (A.16)with A = S x x (cid:16) q ⊥ − Q (cid:17) Q Q , (A.17) B = − q ⊥ cos ϕ , (A.18) C = Q Q S x x − S x x q ⊥ Q + S x x − m − Q − Q . (A.19)Since two solutions of q ⊥ exist for each set of parameters x , x , Q , Q , q ⊥ , ϕ , two phase spacepoints are evaluated. We require “physical” phase space points to have positive q ⊥ and the numberof rejected points can be greatly reduced if one asks0 < Q < S x x (cid:16) Q − q ⊥ (cid:17) h Q − Q (cid:16) S x x − m (cid:17) + q ⊥ S x x i Q h Q − Q (cid:16) q ⊥ sin ϕ + S x x (cid:17) + q ⊥ S x x i , (A.20)which corresponds to the positivity of the discriminant of Eq. (A.16). The other parameters aregenerated in the following ranges: x · x ∈ " m S , , log x ∈ " log m S , , Q ∈ (cid:20) Q , (cid:16) √ S − m (cid:17) (cid:21) , q ⊥ ∈ [0 , Q ] , ϕ ∈ [0 , π ] , (A.21)where Q is a technical cut set to 1 GeV which prevents sampling PDFs and α s at too low scales.It also sets the lower bound for Q and we have checked the independence of the cross sectionresults on value of Q . The di ff erential cross-section from Eq. (3.2), multiplied by the Jacobiancorresponding to Eqs. (A.20)–(A.21), is integrated using VEGAS [108].34 .2 The VBF cross-section with anomalous V V H couplings
We report here the formula corresponding to Eq. (3.5) for the vertex in Eq. (5.1). We rewrite it as: Γ µν ( q , q ) = A g µν + A (cid:16) q · q g µν − q µ q ν (cid:17) + A ε µνρσ q ρ q σ , (A.22)where we have set A = M V (cid:18)(cid:16) √ G F (cid:17) / + a Λ (cid:19) , A = a Λ , A = a Λ . (A.23)Eq. (3.5), with the replacement M µν = Γ µν ( q , q ) has now the form: W µν (cid:16) x , Q (cid:17) M µρ M ∗ νσ W ρσ (cid:16) x , Q (cid:17) = X i , j = C i j F i (cid:16) x , Q (cid:17) F j (cid:16) x , Q (cid:17) . (A.24)Defining q · q = q , P a · q b = P ab , (A.25)the non-vanishing C i j read C = + q q q ! | A | + q Re (cid:16) A A ∗ (cid:17) + (cid:16) q q + q (cid:17) | A | + (cid:16) q − q q (cid:17) | A | , (A.26) C = P P q + q P − P q q ! | A | + P q q Re (cid:16) A A ∗ (cid:17) + P q − P q P ! | A | + P q − P q P − P q ! | A | , (A.27) C = P P q + q P − P q q ! | A | + P q q Re (cid:16) A A ∗ (cid:17) + P q − P q P ! | A | + P q − P q P − P q ! | A | , (A.28) C = P P s − P P q − P P q + P P q q q ! | A | − (cid:18) P P − q s (cid:19) q q q + s P P − P P q − P P q ! Re (cid:16) A A ∗ (cid:17) + q q q + s P P − P P q − P P q ! ε µνρσ P µ P ν q ρ q σ Re (cid:16) A A ∗ (cid:17) + ( q s − P P ) P P | A | − P P − q sP P ε µνρσ P µ P ν q ρ q σ Re (cid:16) A A ∗ (cid:17) + ( s q − P q P − P q P + P P q P P ! P P " s (cid:16) q q − q (cid:17) − P P − P P + P P ) | A | , (A.29) C = q s − P P P P | A | + " s P P (cid:16) q q + q (cid:17) + q − P q P − P q P − P P q P P Re (cid:16) A A ∗ (cid:17) + q P P ε µνρσ P µ P ν q ρ q σ Re (cid:16) A A ∗ (cid:17) + q q − P q P − P q P + s q q P P ! | A | + q q P P ε µνρσ P µ P ν q ρ q σ Re (cid:16) A A ∗ (cid:17) + q q − P q P − P q P + P P q q P P q ! | A | . (A.30) A.3 The VBF cross-section for a vector resonance
Finally, we give the formula for the total cross-section for the production of a vector resonance inthe structure function approach.We assume the usual gauge and Lorentz invariant tri-linear vertex: Γ µ µ µ ( p , p , p ) = g h g µ µ (cid:16) p µ − p µ (cid:17) + g µ µ (cid:16) p µ − p µ (cid:17) + g µ µ (cid:16) p µ − p µ (cid:17)i , (A.31)where all the momenta are taken to flow outside the vertex. Since in the processes that we haveconsidered the coupling g can always be factorized from the total cross-section, we will set it tounity.As in Sec. A.2, with the extra care to sum over the polarizations of the produced vector, we have W µν (cid:16) x , Q (cid:17) X λ (cid:16) M µρλ M ∗ νσλ (cid:17) W ρσ (cid:16) x , Q (cid:17) = X i , j = C i j F i (cid:16) x , Q (cid:17) F j (cid:16) x , Q (cid:17) , (A.32)where we have set M µνλ = Γ µνρ ( q , q , − q − q ) ǫ ρλ ( − q − q ) , (A.33)and the non-vanishing coe ffi cients C i j are C = q q q q q − q q − q q + q q + q q + q + m V ′ (cid:16) q − q (cid:17) (cid:16) q q + q (cid:17) , (A.34) C = − P q q − P q q − P q q + P q + P P q q q + P P q q − P P q q + P q q − P q q − P q q q + P q q − P q − P q q − m V ′ (cid:16) q − q (cid:17) (cid:18) P q q − P P q q + P q q + P q (cid:19) , C = C (1 ↔ , (A.36) C = P P q q n P P q q + P P q q − P P q − P P P q q q − P P P q q + P P P q q + P P q q + P P q + P P q q − P P P P q q + P P P P q q q − P P P P q q + P P P q q − P P P q q − P P P q q q − P P q + P P q q + P P q q + s q q ( q − q ) (cid:16) P P q − P P q + P P q (cid:17) − s q q ( q − q ) + m V ′ (cid:16) q − q (cid:17) (cid:18) P P q − P P q + P P q − s q q (cid:17) (cid:27) , (A.37) C = P P − P P q + P P q + P P q + P P q + P P q − P P q − s (cid:16) q q − q q + q + q q (cid:17) + m V ′ (cid:16) q − q (cid:17) (cid:18) − P P + s q (cid:19) . (A.38) B VBF cross sections
We present the VBF cross sections for the Tevatron in Tabs. B.1–B.5, and for the LHC in Tabs. B.6–B.10 and in Tabs. B.11–B.15, respectively. The PDF sets used and all other parameters are givenin the table captions. 37 H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
95 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
100 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
105 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
110 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
115 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
120 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
125 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
130 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
135 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
140 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
145 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
150 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
155 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.1:
Total VBF cross sections at the Tevatron, √ S = .
96 TeV at LO, NLO and NNLO in QCD.Errors shown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The ABKM [70] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
95 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
100 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
105 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
110 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
115 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
120 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
125 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
130 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
135 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
140 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
145 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
150 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
155 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.2:
Total VBF cross sections at the Tevatron, √ S = .
96 TeV at LO, NLO and NNLO in QCD.Errors shown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The HERAPDF1.5 [73, 74] PDF set has been used. Numbers arein pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
95 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
100 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
105 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
110 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
115 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
120 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
125 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
130 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
135 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
140 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
145 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
150 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
155 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.3:
Total VBF cross sections at the Tevatron, √ S = .
96 TeV at LO, NLO and NNLO in QCD.Errors shown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The JR09 [75, 76] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
95 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
100 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
105 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
110 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
115 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
120 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
125 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
130 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
135 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
140 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
145 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
150 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
155 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.4:
Total VBF cross sections at the Tevatron, √ S = .
96 TeV at LO, NLO and NNLO in QCD.Errors shown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The MSTW2008 [63] PDF set (68% CL) has been used. Numbersare in pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
95 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
100 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
105 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
110 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
115 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
120 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
125 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
130 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
135 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
140 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
145 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
150 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
155 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.5:
Total VBF cross sections at the Tevatron, √ S = .
96 TeV at LO, NLO and NNLO in QCD.Errors shown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The NNPDF2.1 [77] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
320 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
340 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
360 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
380 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.6:
Total VBF cross sections at the LHC, √ S = µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The ABKM [70] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
320 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
340 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
360 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
380 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.7:
Total VBF cross sections at the LHC, √ S = µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The HERAPDF1.5 [73, 74] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
320 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
340 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
360 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
380 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.8:
Total VBF cross sections at the LHC, √ S = µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The JR09 [75, 76] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 9 . + . − . + . − . · − . + . − . + . − . . + . − . + . − .
150 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − .
155 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
320 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
340 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
360 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
380 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.9:
Total VBF cross sections at the LHC, √ S = µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The MSTW2008 [63] PDF set (68% CL) has been used. Numbers arein pb. H [ GeV] σ LO σ NLO σ NNLO
90 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 1 . + . − . + . − . . + . − . + . − . · − . + . − . + . − . · −
155 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
160 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
165 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
170 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
175 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
180 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
185 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
190 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
195 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
200 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
210 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
220 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
230 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
240 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
250 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
260 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
270 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
280 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
290 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
300 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
320 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
340 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
360 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
380 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.10: Total VBF cross sections at the LHC, √ S = µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The NNPDF2.1 [77] PDF set has been used.Numbers are in pb. 47 H [ GeV] σ LO σ NLO σ NNLO
90 6 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
160 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
165 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
170 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
175 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
180 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
185 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
190 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
195 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
200 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
210 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
220 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
230 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
240 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
250 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
260 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
270 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
280 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
290 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
300 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
320 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
340 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
360 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
380 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.11:
Total VBF cross sections at the LHC, √ S =
14 TeV at LO, NLO and NNLO in QCD. Errorsshown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The ABKM [70] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 6 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
160 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
165 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
170 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
175 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
180 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
185 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
190 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
195 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
200 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
210 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
220 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
230 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
240 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
250 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
260 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
270 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
280 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
290 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
300 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
320 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
340 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
360 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
380 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.12:
Total VBF cross sections at the LHC, √ S =
14 TeV at LO, NLO and NNLO in QCD. Errorsshown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The HERAPDF1.5 [73, 74] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
160 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
165 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
170 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
175 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
180 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
185 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
190 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
195 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
200 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
210 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
220 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
230 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
240 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
250 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
260 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
270 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
280 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
290 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
300 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
320 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
340 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
360 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
380 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.13:
Total VBF cross sections at the LHC, √ S =
14 TeV at LO, NLO and NNLO in QCD. Errorsshown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The JR09 [75, 76] PDF set has been used. Numbers are in pb. H [ GeV] σ LO σ NLO σ NNLO
90 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
160 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
165 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
170 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
175 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
180 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
185 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
190 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
195 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
200 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
210 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
220 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
230 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
240 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
250 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
260 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
270 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
280 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
290 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
300 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
320 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
340 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
360 9 . + . − . + . − . · − . + . − . + . − . . + . − . + . − .
380 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 6 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.14:
Total VBF cross sections at the LHC, √ S =
14 TeV at LO, NLO and NNLO in QCD. Errorsshown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The MSTW2008 [63] PDF set (68% CL) has been used. Numbers arein pb. H [ GeV] σ LO σ NLO σ NNLO
90 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
95 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
100 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
105 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
110 5 . + . − . + . − . . + . − . + . − . . + . − . + . − .
115 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
120 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
125 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
130 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
135 4 . + . − . + . − . . + . − . + . − . . + . − . + . − .
140 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
145 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
150 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
155 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
160 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
165 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
170 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
175 3 . + . − . + . − . . + . − . + . − . . + . − . + . − .
180 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
185 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
190 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
195 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
200 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
210 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
220 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
230 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
240 2 . + . − . + . − . . + . − . + . − . . + . − . + . − .
250 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
260 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
270 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
280 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
290 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
300 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
320 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
340 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
360 1 . + . − . + . − . . + . − . + . − . . + . − . + . − .
380 9 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
400 8 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
450 7 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
500 5 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
550 4 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
600 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
650 3 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
700 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
750 2 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
800 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
850 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
900 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · −
950 1 . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − . + . − . + . − . · − Table B.15:
Total VBF cross sections at the LHC, √ S =
14 TeV at LO, NLO and NNLO in QCD. Errorsshown are respectively scale and PDF uncertainities. Scale uncertainities are evaluated by varying µ r and µ f in the interval µ r , µ f ∈ [ Q / , Q ]. The NNPDF2.1 [77] PDF set has been used. Numbers are in pb. eferences [1] A. Djouadi, Phys.Rept. , 1 (2008), arXiv:hep-ph / G35 , 033001 (2008).[3] G. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, JHEP , 045 (2007), arXiv:hep-ph / , 201801 (2002), arXiv:hep-ph / B646 , 220 (2002), arXiv:hep-ph / B665 , 325 (2003), arXiv:hep-ph / B579 , 149 (2004), arXiv:hep-ph / et al. , (2011), arXiv:1101.0593.[9] R. N. Cahn and S. Dawson, Phys. Lett. B136 , 196 (1984).[10] G. L. Kane, W. W. Repko, and W. B. Rolnick, Phys. Lett.
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