Virtual corrections to gg→ZH in the high-energy and large- m t limits
TTTP20-041, P3H-20-074
Virtual corrections to gg → Z H in the high-energyand large- m t limits Joshua Davies a , Go Mishima b , Matthias Steinhauser b a Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK b Department of Physics, Tohoku University, Sendai, 980-8578 Japan c Institut f¨ur Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT)Wolfgang-Gaede Straße 1, 76128 Karlsruhe, Germany
Abstract
We compute the next-to-leading order virtual corrections to the partonic cross-section of the process gg → ZH , in the high-energy and large- m t limits. We use Pad´eapproximants to increase the radius of convergence of the high-energy expansionin m t /s , m t /t and m t /u and show that precise results can be obtained down toenergies which are fairly close to the top quark pair threshold. We present resultsboth for the form factors and the next-to-leading order virtual cross-section. a r X i v : . [ h e p - ph ] N ov Introduction
At the CERN Large Hadron Collider (LHC), gluon fusion processes play an important roledue to the large gluon luminosities at high collision energies. As a consequence one oftenobserves that gluon fusion–induced processes provide a numerically large contributionto the theory predictions of production cross-sections. This is true even for processesfor which the leading-order (LO) contribution consists of one-loop diagrams. A primeexample of such a process is inclusive Higgs boson production, where the gluon-fusionchannel is about an order of magnitude larger than all other contributions.In this paper we consider the associated production of a Z and a Higgs boson, pp → ZH ,often called “Higgs Strahlung”. This process was of particular importance in the obser-vation of the Higgs boson’s decay into bottom quarks at ATLAS [1] and CMS [2]. At LOit is mediated by a tree-level process in which a quark and anti-quark annihilate. For thischannel, corrections up to next-to-next-to-next-to-leading (N LO) order are available [3](see also Ref. [4] and references therein) and are included in the program vh@nnlo [5, 6].Electroweak and QCD corrections are also included in the program
HAWK [7].Associated ZH production can also occur via the loop-induced gluon fusion process.Although formally of next-to-next-to-leading order (NNLO) with respect to pp → ZH , itprovides a sizeable contribution, in particular in the boosted Higgs boson regime in whichthe transverse momentum of the Higgs boson is large [8, 9]. Furthermore, the process gg → ZH provides sizeable contributions to the uncertainties of ZH production with asubsequent decay of the Higgs boson in a pair of bottom quarks, e.g., Ref. [10]. Beingloop-induced, gg → ZH is sensitive to physics beyond the Standard Model. In Ref [11]it was suggested that the gluon-initiated component of pp → ZH can be extracted bycomparing to W H production, which only has a Drell-Yan–like component. It is thusimportant to consider NLO QCD corrections to gg → ZH , requiring the computation oftwo-loop box-type Feynman diagrams with two different final-state masses ( m Z and m H )and the massive top quark propagating in the loops.An exact LO (one-loop) calculation was performed in [12]. At NLO only approximationsin the large m t limit are known [13,14]. In this work we consider the two-loop NLO virtualcorrections in the high-energy limit, expanding the two-loop master integrals for m t (cid:28) s, t ,where s and t are the Mandelstam variables. Furthermore, we also provide analytic resultsfor the form factors in the large- m t limit. A similar approach has been applied to therelated process of Higgs boson pair production, gg → HH , where a comparison to exactnumerical calculations [15] could be performed and good agreement was found, even forrelatively small values for the Higgs boson transverse momentum [16]. In Ref. [17] thehigh-energy expansion was successfully applied to the top quark contribution of the two-loop diagrams contributing to gg → ZZ .The remainder of the paper is organized as follows: In Section 2 we introduce our no-tation, and give our definitions for the form factors and the virtual finite cross-section. In [14] only the squared amplitude has been computed.
2n Section 3 we consider the quality of our approximations by comparing with the exactLO expressions. In Section 4, we briefly discuss the two-loop form factors in the large- m t limit and in Section 5 we discuss the form factors in the high-energy limit, and investi-gate the behaviour of Pad´e approximants constructed using this expansion. In Section 6we study the NLO virtual finite cross-section and apply our Pad´e scheme to extend theapproxmation to a larger kinematic region. Finally in Section 7 we conclude our findings.Auxiliary material can be found in the Appendix. In Appendix A we present analyticresults for the one-particle reducible double-triangle contribution and in Appendix B webriefly discuss out treatment of γ and the application of projectors to obtain the formfactors. In Appendix C we present the relations between the form factors and helicityamplitudes for the process gg → ZH . The expansions of the form factors are provided inan analytic, computer readable form in the ancillary files of this paper [18]. We consider the amplitude for the process g ( p ) g ( p ) → Z ( p ) H ( p ) where all momentaare assumed to be incoming. This leads to the following definitions for the Mandelstamvariables, s = ( p + p ) ,t = ( p + p ) ,u = ( p + p ) , (1)with s + t + u = m Z + m H . (2)Additionally, p = p = 0, p = m Z and ( p + p + p ) = m H . We also introduce thetransverse momentum ( p T ) and the scattering angle ( θ ) of the final-state bosons, whichare related to the Mandelstam variables as follows, t = − s − β cos θ ) + m H + m Z ,u = − s β cos θ ) + m H + m Z ,p T = ut − m H m Z s = s β sin θ , (3)where β = (cid:115) − m Z + m H s + ( m Z − m H ) s . (4)3 t b t Figure 1: LO and NLO Feynman diagrams contributiong to gg → ZH . Curly, wavy anddashed lines represent gluons, Z bosons and scalar particles (Higgs or Goldstone bosons),respectively. Solid lines stand for top or bottom quarks. Note that the latter are onlypresent in the triangle diagrams.We denote the polarization vectors of the gluons and the Z boson by ε λ ,µ ( p ), ε λ ,ν ( p )and ε λ ,ρ ( p ), in terms of which the amplitude can be written as M λ ,λ ,λ = A µνρ ε λ ,µ ( p ) ε λ ,ν ( p ) ε λ ,ρ ( p ) . (5)Due to charge-conjugation invariance, the vector coupling of the Z boson to the quarksin the loop does not contribute. The axial-vector coupling is proportional to the thirdcomponent of the isospin and thus mass-degenerate quark doublets also do not contribute.Since we consider the lightest five quarks to be massless, only contributions from the top-bottom doublet remain. Furthermore, each individual term of A µνρ is proportional to thetotally anti-symmetric ε tensor from the axial-vector coupling.At LO and NLO both triangle- and box-type diagrams have to be considered. Examplesof these diagram classes are shown in Fig. 1. In the box-type diagrams the Higgs bosoncouples directly to the quark loop, so diagrams involving the bottom quark are suppressedby their Yukawa coupling with respect to diagrams involving the top quark. This isnot the case for the triangle-type diagrams; here contributions from both the top andbottom quark loops must be considered. At NLO there is also the contribution fromthe one-particle reducible double-triangle contribution, shown as first diagram in thesecond row of Fig. 1. Note that in the numerical results discussed in the main part ofthis paper these contributions are excluded, however, we present exact analytical resultsin Appendix A. The contribution from the reducible double-triangle diagrams to the(differential) partonic cross- section is implemented in the computer program which comestogether with Ref. [14].In total one can form 60 tensor structures from the indices µ, ν, ρ , the independent mo-menta p , p and p and an ε tensor. Details of the computation of these 60 structuresand our treatment of γ in d = 4 − (cid:15) dimensions are given in Appendix B. After4pplying transversality conditions ( p i · ε λ i ( p i ) = 0), gauge invariance w.r.t. the gluons( p µ A µνρ = p ν A µνρ = 0) and Bose symmetry ( A µνρ ( p , p , p ) = A νµρ ( p , p , p )), 14 ten-sor structures remain which can be grouped such that one has to introduce 7 form factors.We follow the decomposition of Ref. [12] and write A µνρab ( p , p , p ) = iδ ab √ G F M Z s α s ( µ ) π ˜ A µνρ ( p , p , p ) , ˜ A µνρ ( p , p , p ) = (cid:40) (cid:16) s ε µνρα p α − p µ ε νραβ p α p β (cid:17) F ( t, u ) − (cid:16) s ε µνρα p α − p ν ε µραβ p α p β (cid:17) F ( u, t )+ (cid:18) p µ + m Z − ts p µ (cid:19) ε νραβ p α (cid:104) p β F ( t, u ) + p β F ( t, u ) (cid:105) + (cid:18) p ν + m Z − us p ν (cid:19) ε µραβ p α (cid:104) p β F ( u, t ) + p β F ( u, t ) (cid:105) + (cid:16) s ε µνρα p α − p µ ε νραβ p α p β + p ν ε µραβ p α p β + g µν ε ραβγ p α p β p γ (cid:17) F ( t, u ) (cid:41) , (6)where a, b are colour indices. Note that while the decomposition is the same, the formfactors in Ref. [12] have dimension 1 / GeV whereas we pull out an overall factor of 1 /s such that our form factors are dimensionless. Only F receives contributions from thetriangle-type diagrams discussed above. We represent the expansion coefficients of theform factors in the strong coupling constants with the following notation, F = F (0) + α s ( µ ) π F (1) + · · · . (7)At this point it is convenient to define new form factors which are linear combinations ofthose of Eq. (6), F ( t, u ) = F ( t, u ) − t − m Z s F ( t, u ) ,F ( u, t ) = F ( u, t ) − u − m Z s F ( u, t ) ,F − ( t, u ) = F ( t, u ) − F ( u, t ) ,F +2 ( t, u ) = F ( t, u ) + F ( u, t ) . (8)It is easy to see that F +2 ( t, u ) drops out in the squared amplitude and thus does notcontribute to physical quantities. It is furthermore convenient to introduce F − ( t, u ) = F ( t, u ) − F ( u, t ) ,F +12 ( t, u ) = F ( t, u ) + F ( u, t ) ,F − ( t, u ) = F ( t, u ) − F ( u, t ) , +3 ( t, u ) = F ( t, u ) + F ( u, t ) , (9)leaving six physically relevant functions: F +12 ( t, u ) , F − ( t, u ) , F − ( t, u ) , F +3 ( t, u ) , F − ( t, u ) , F ( t, u ) , (10)where only F +12 ( t, u ) has contributions from triangle-type diagrams. F − k ( t, u ) (with k =12 , ,
3) and F ( t, u ) are anti-symmetric w.r.t. the exchange of the arguments t , u , and F + k ( t, u ) (with k = 12 ,
3) are symmetric. At leading order F − ( t, u ) = 0, however it hasnon-zero contributions starting at NLO.For the computation of the one- and two-loop Feynman diagrams (some examples areshown in Fig. 1) in the high-energy limit, we proceed as follows: After producing theamplitude we perform a Taylor expansion in the Z and Higgs boson masses (since m Z , m H (cid:28) m t ), leaving one- and two-loop integrals which depend only on s , t and m t . Using integration-by-parts reduction techniques with the programs FIRE 6 [19] and
LiteRed [20], these integrals can be reduced to the same basis of 161 two-loop master inte-grals as in Refs. [21, 22], in which they were computed as an expansion in the high-energylimit to order m t . Inserting these expansions into the amplitude yields its high-energyapproximation. We use FORM 4.2 [23] for most stages of the computation. The calcula-tion is performed in the covariant R ξ gauge and we allow for a general electroweak gaugeparameter ξ Z which appears in the propagators of the Z boson and Goldstone boson χ .Thus only the triangle-type diagrams depend on ξ Z , and this dependence cancels uponsumming the Z and χ contributions.We also investigate the large- m t limit of the form factors. This expansion is straightfor-ward and proceeds in analogy to [17, 22]. The programs q2e and exp [24, 25] are used toproduce an asymptotic expansion for m t (cid:29) q , q , q , again performed using FORM , yieldingproducts of massive vacuum integrals and massless three-point integrals. Results for the gg → ZH amplitude, expanded to order 1 /m t , have been previously published in [14];here we provide one additional term in this expansion, to order 1 /m t , and provide resultsat the level of the form factors.The renormalization of the ultra-violet (UV) divergences proceeds in a similar way as forthe processes gg → HH [22] and gg → ZZ [17]. In particular, we work in the six-flavourtheory and renormalize the top quark mass on shell and the strong coupling α s in the MSscheme. In addition, our treatment of γ (see Appendix B for more details) requires thatwe apply additional finite renormalization to the axial and pseudo-scalar currents [26], Z A = 1 − α s π C F + O ( (cid:15) ) ,Z P = 1 − α s π C F + O ( (cid:15) ) . The subtraction of the infra-red poles proceeds according to Ref. [27], using the conven-tions of Refs. [17, 22]. The subtraction has the form [27] F (1) = F (1) , IR − K (1) g F (0) = ˜ F (1) + β log (cid:18) µ − s − iδ (cid:19) F (0) , (11)6ith β = 11 C A / − T n f / F (1) , IR is UV renormalized but still infra-red (IR)divergent. F (1) is finite. An explicit expression for K g is given in Eq. (2.3) of Ref. [22].After the second equality sign in Eq. (11) we make the µ dependence explicit. Note thatbelow we only need ˜ F (1) to construct the NLO cross-section.In analogy to the process gg → HH [28] we define the NLO virtual finite cross-sectionfor gg → ZH as˜ V fin = G F m Z s (cid:16) α s π (cid:17) (cid:88) λ ,λ ,λ (cid:40) (cid:104) ˜ A µνρ sub ˜ A (cid:63),µ (cid:48) ν (cid:48) ρ (cid:48) sub (cid:105) (1) + C A (cid:18) π − log µ s (cid:19) (cid:104) ˜ A µνρ sub ˜ A (cid:63),µ (cid:48) ν (cid:48) ρ (cid:48) sub (cid:105) (0) (cid:41) × ε λ ,µ ( p ) ε (cid:63)λ ,µ (cid:48) ( p ) ε λ ,ν ( p ) ε (cid:63)λ ,ν (cid:48) ( p ) ε λ ,ρ ( p ) ε (cid:63)λ ,ρ (cid:48) ( p ) , (12)where the form factors entering the amplitude ˜ A µνρ sub are the IR-subtracted finite formfactors ˜ F (1) of Eq. (11). The superscripts “(0)” and “(1)” after the square brackets inEq. (12) indicate that we take the coefficients of ( α s /π ) and ( α s /π ) , respectively, ofthe squared amplitude, in accordance with Eq. (7). For the discussion in Section 6 it isconvenient to introduce the α s -independent quantity V fin = ˜ V fin α s . (13) In this section we compare our high-energy expansion with the exact LO result. Weimplement this by using
FeynArts [29] to generate the amplitude and
FormCalc [30] tocompute it, performing a Passarino-Veltman reduction to three- and four-point one-loopscalar integrals. Schouten identities allow us to write the result in terms of the tensorstructures and form factors of Eqs. (6) and (8). We use
Package-X [31] to evaluate thePassarino-Veltman functions with high precision and to produce an analytic expansion inthe limit of a large top quark mass. We have verified that our implementation of the exactLO result reproduces the cross-sections provided in the literature [13,14,32]. Additionallywe have compared the large- m t expansion derived from this result with a direct expansionof the amplitude as described above.In Fig. 2 we show the squared amplitude at LO, as a function of √ s , for a fixed scatteringangle θ = π/
2. The solid black lines correspond to the exact result. The colouredlines correspond to different expansion depths in m Z and m H , as detailed in the plotlegend. All of the latter include high-energy expansion terms up to m t . The red curvedemonstrates that after including quadratic terms both in m H and m Z , the deviationfrom the exact result is below 1% (for √ s ∼ > m H corrections appear to bemore important than the m Z corrections. The inclusion of the quartic terms (shown asgreen and pink curves, best visible in Fig. 2(c)) improves the accuracy of the expansion,leading to an almost negligible deviation from the exact expression. These terms are much7
50 500 750 1000 1250 1500 1750 2000 s ( GeV )0.00000.00050.00100.00150.00200.00250.00300.00350.0040 | A | L O / s m Z , m H m Z , m H m Z , m H m Z , m H Pade (a)
800 1000 1200 1400 1600 1800 2000 s ( GeV )1.001.051.101.151.201.251.301.351.40 | A | L O /| A | L O , e x a c t m Z , m H m Z , m H m Z , m H m Z , m H m Z , m H m Z , m H m Z , m H , m Z m H (b)
800 1000 1200 1400 1600 1800 2000 s ( GeV )0.9981.0001.0021.0041.0061.0081.010 | A | L O /| A | L O , e x a c t m Z , m H m Z , m H m Z , m H m Z , m H m Z , m H m Z , m H m Z , m H , m Z m H (c) Figure 2: (a) LO squared amplitude and (b) ratio of expansions to the exact result.Plot (c) is a zoomed-in version of (b). Note that for better readability in (a) no quarticcorrections are shown; the black curve in (a) refers to the exact result and the blue curve(and associated uncertainty band) is the result obtained from Pad´e approximation.harder to compute, however, so in the NLO results we will restrict the approximation tothe quadratic corrections only. The solid blue curve and associated uncertainty bandin Fig. 2(a) shows the result of a procedure to improve the expansions based on Pad´eapproximants, which is discussed in more detail in Section 5. Here it is based on theexpansion to quadratic order in m H and m Z , confirming that our computation of theNLO amplitude only to this order is sufficient.In Fig. 2 and in the following sections, we use the following parameter values: m Z = 91 . , m t = 172 . , m H = 125 . , G F = 1 . × − GeV − . (14)8 NLO form factors: large- m t limit In this section we discuss the large- m t expansion of the form factors at NLO. While theexpansion of ˜ F +12 starts at m t , we find that the other form factors of Eq. (10) exhibit somecancellation in the leading contributions. In particular, ˜ F − , ˜ F − , ˜ F +3 and ˜ F start only at1 /m t , and ˜ F − starts at 1 /m t .We now present the leading terms of the large- m t expansion to establish our notation.For brevity, we restrict ourselves to ˜ F +12 . Expressions for the expansion of all form factorsto 1 /m t are provided in the ancillary files of this paper [18]. Our result reads˜ F + , (1)12 = − C A + sm t (cid:18) C A − C F (cid:19) + 1 m t (cid:18) C F s (cid:20) m H + 162 m Z − s (cid:21) + C A (cid:18) (cid:20) − (cid:26) m H + m Z − m H m Z (cid:27) + 4011 m H s + 6713 m Z s − s + 60 (cid:8) m H + m Z − s − t (cid:9) t (cid:21) − s log (cid:18) − sm t (cid:19) (cid:20) m H + 26 m Z − s (cid:21)(cid:19) , (15)where C A = 3 and C F = 4 / SU (3).We have verified that after constructing ˜ V fin in Eq. (12) we find agreement with the resultsof Ref. [14] up to order 1 /m t . We now turn to the high-energy expansion of the NLO form factors. The analytic expres-sions are large, so we refrain from showing them here but we provide analytic expressionsfor all form factors in the ancillary files of this paper [18]. For illustration we discuss inthe following the results for ˜ F +3 .In Fig. 3(a) we plot ˜ F +3 as a function of √ s . We can see that the high-energy expansionsof both the real (blue solid lines) and imaginary parts (green dashed lines) converge wellfor √ s ∼ >
800 GeV, which is in analogy to gg → HH [22] and gg → ZZ [17]. Note thatthe lighter-coloured curves include fewer m t expansion terms; the darkest lines show theexpansion up to m t .As in previous publications on gg → HH and gg → ZZ , we make use of Pad´e approx-imants to improve the description of the high-energy expansion. The methodology usedfollows that of Section 4 of Ref. [17] and so is not described in detail, but is only sum-marized briefly here. The expansion is used to produce 28 different Pad´e approximants,which are combined to produce a central value and error estimate for the approximation.In Fig. 3(a) the Pad´e results are shown as red bands. The width of the bands denotethe uncertainty estimates. For regions in which the high-energy expansion converges, the9
50 500 750 1000 1250 1500 1750 2000 s ( GeV )2520151050510 R e ( F + , ( ) )
250 500 750 1000 1250 1500 1750 2000 s ( GeV )10.07.55.02.50.02.55.07.510.0 R e ( F + , ( ) ) p T =200GeV p T =250GeV p T =300GeV p T =400GeV p T =450GeV p T =500GeV p T =600GeV p T =700GeV p T =800GeV m t m t (a) (b)Figure 3: Results for ˜ F +3 as a function of √ s for fixed θ = π/ p T (b).In (a) dashed and solid lines correspond to the imaginary and real parts, respectively.The red curves in (a) represent the Pad´e results. In (b) only the real part is shown,and the Pad´e results are shown as green dashed lines. The high-energy expansions up toorder m t and m t are shown in blue. The widely-separated pair of curves correspond to p T = 400 GeV.Pad´e-based approximation reproduces the expansion. However, it also produces reliableresults for smaller values of √ s , as can be expected from the comparison with the LOresult shown in Fig. 2(a).In Fig. 3(b) we show ˜ F +3 for the fixed values of p T = 200 , . . . ,
800 GeV. The blue (dashedand dotted) curves correspond to the high-energy expansions and the green (dashed)curves to the Pad´e results. For all Pad´e curves we also show the corresponding uncertaintyband, which for p T = 200 GeV is relatively large but for p T = 250 GeV the uncertaintyband is already quite small; it is completely negligible for higher values of p T . Note thatthe high-energy expansions are only shown for p T ≥
400 GeV; for lower p T values thecurves lie far outside of the plot range. For p T ∼ >
450 GeV the expansions converge andare very close to the Pad´e results. For p T = 400 GeV, while the expansions initially agreewith each other and the Pad´e close to √ s = 800 GeV, they diverge for larger values of √ s . We recall here that the high-energy expansion is an expansion in m t /s , m t /t and m t /u . For a fixed value of p T , increasing √ s can lead to values of t or u which are notlarge enough for convergence.In Section 6 we will apply the Pad´e procedure to the virtual finite cross-section, in orderto compare our results with a state-of-the-art numerical evaluation at NLO [33].10
50 500 750 1000 1250 1500 1750 2000 s ( GeV )0.000.010.020.030.040.050.060.07 f i n p T = 150GeV p T = 200GeV p T = 250GeV p T = 300GeV p T = 400GeV p T = 500GeV p T = 600GeV m t m t Figure 4: V fin as a function of √ s for eight values of p T , at the scale µ = s . Theuncertainty estimate of the Pad´e procedure is displayed as light-coloured bands. For p T ≥
350 GeV, we also show two high-energy expansion curves including terms up toorder m t and m t . Our starting point is ˜ V fin in Eq. (12). For the LO form factors we use the exact resultsand for the two-loop form factors we use the high-energy expansion. This allows us towrite V fin in Eq. (13) as an expansion in m t up to order m t . At this point we can applythe Pad´e approximation procedure as described in Section 5. In Fig. 4 we show V fin forseveral fixed values of p T as a function of √ s . For p T = 400 GeV and larger we also showtwo high-energy expansion curves, which include terms up to m t and m t . These curvesagree well with each other and with the Pad´e approximation which they produce. Forlower values of p T , these curves do not agree with each other, and are not visible within Note the double-triangle contribution, which is known analytically (see Appendix A and Ref. [14]),is not included in our numerical results for V fin . p T = 150 GeV Pad´e procedure produces stable results with an uncertainty of about10%. For p T = 200 GeV the uncertainties are notably smaller and for higher p T valuesthey are negligible. We advocate to use results based on our approach for p T ∼ >
200 GeV.For p T ≈
150 GeV the Pad´e approach provides important results for cross-checks. Forlower values of p T other methods have to be used for the calculation of gg → ZH , seeRef. [33].We have compared our results with those of Ref. [33], which are obtained numerically,but without making any expansions. In the high-energy region, for 104 kinematic pointswith p T ≥
200 GeV we agree to within 2-3%, and for 42 points with 150 ≤ p T <
200 GeVwe agree to within 10% and our values are consistent to within the uncertainty of thePad´e procedure. We also construct V fin using the large- m t expanded NLO form factorsof Section 4 and find that for 120 points with √ s ≤
284 GeV, we agree to within 1%.For larger values of √ s , as expected, the large- m t expansion starts to diverge significantlyfrom the numerical results as one approaches the top quark threshold at √ s ≈
346 GeV.
In this paper we consider two-loop NLO corrections to the Higgs Strahlung process gg → ZH . The corresponding Feynman integrals involve five dimensionful parameters ( s , t , m t , m Z and m H ) which makes an analytic calculation impossible with the current state-of-the-art techniques. Numerical computations are also very challenging, however they haverecently been achieved in Ref. [33]. In Section 6 we have discussed the cross-check of ourresults against these.In our approach we use the hierarchy between the top quark, Higgs and Z boson massesand perform an expansion for m t (cid:29) m Z , m H , effectively eliminating the dependence on m Z and m H from the integrals. We show at one-loop order that the expansion convergesquickly, which allows us to truncate the expansion at NLO after the quadratic terms. Asfar as the remaining scales are concerned, we concentrate on the high-energy region where s, t (cid:29) m t . We expand our master integrals in this limit such that we obtain results for theform factors including expansion terms up to m t . This allows us to construct, for eachphase-space point (e.g., a particular pair of √ s , p T values) a set of Pad´e approximants,which considerably extend the region of convergence of our expansion. Our approachprovides both central values and uncertainty estimates.We provide results for all form factors involved in the gg → ZH amplitude, both atone- and two-loop order. Furthermore, we provide relations between the form factorsand helicity amplitudes in Appendix C. The main emphasis of the paper is the IR sub-tracted virtual corrections to the partonic cross-section, which can be combined with other(e.g. Drell-Yan–like) corrections to gg → ZH . Our method provides precise results for p T ∼ >
200 GeV with almost negligible uncertainties. For lower values of p T our uncer-12ainties increase, in particular for smaller values of √ s . The analytic results for the formfactors obtained in this paper can be obtained in electronic form from Ref. [18]. Acknowledgements.
We thank the authors of Ref. [33] for making their results avail-able for comparison, prior to publication. The work of J.D. was in part supported bythe Science and Technology Research Council (STFC) under the Consolidated GrantST/T00102X/1. The work of G.M. was in part supported by JSPS KAKENHI (No.JP20J00328). This research was supported by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under grant 396021762 — TRR 257 “ParticlePhysics Phenomenology after the Higgs Discovery”.
A Double-triangle contribution
In this appendix we present results for the one-particle reducible double-triangle contri-bution, as shown in the first diagram in the second row of Fig. 1. We have computed thesix form factors in Eq. (10). The results can be expressed in terms of the functions (seealso Ref. [34] where the corresponding contributions for gg → HH were computed) B ( x, ,
0) = 2 − log − xµ ,B ( x, m , m ) = 2 + β x log β x − β x + 1 − log m µ ,C ( x, y, , m , m , m ) = 12( x − y ) (cid:18) log β x + 1 β x − − log β y + 1 β y − (cid:19) , (16)with β x = (cid:112) − m /x and β y = (cid:112) − m /y . Our explicit calculation shows that F +3 ( t, u ) = F − ( t, u ) = F ( t, u ) = 0. We furthermore find that F ( t, u ) is equal to F ( t, u )with F ( t, u ) = 8 m t m Z s ( m H − u ) ( m Z − u ) (cid:26) u (cid:2) B ( m H , m t , m t ) − B ( u, m t , m t ) (cid:3) + (cid:0) m H − u (cid:1) (cid:2)(cid:0) − m H + 4 m t + u (cid:1) C ( u, m H , , m t , m t , m t ) + 2 (cid:3) (cid:27) × (cid:20) B ( u, m t , m t ) − B ( m Z , m t , m t ) + B ( m Z , , − B ( u, , m t u − m Z m Z C ( u, m Z , , m t , m t , m t ) (cid:21) , (17)which can be used to construct F +12 ( t, u ), F − ( t, u ) and F − ( t, u ) (cf. Section 2). We notethat it is straightforward to expand the results in Eq. (17) in the large- and small- m t limit. 13he contribution of the double-triangle diagrams to the squared matrix element havebeen computed in Ref. [14] and implemented in the corresponding computer program, seeAppendix of Ref. [14]. B Pro jectors and γ gg → ZH amplitude contains a quark witheither an axial-vector coupling to a Z boson, or a pseudo-scalar coupling to a Goldstoneboson. The γ matrix present in these couplings is not defined in the d = 4 − (cid:15) space-time dimensions of dimensional regularization. The couplings are re-written in terms ofanti-symmetric tensors, according to [26], as γ µ γ = i ε µνρσ ( γ ν γ ρ γ σ − γ σ γ ρ γ ν ) ,γ = i ε µνρσ ( γ µ γ ν γ ρ γ σ + γ σ γ ρ γ ν γ µ − γ ν γ ρ γ σ γ µ − γ µ γ σ γ ρ γ ν ) , (18)which allow one to compute the loop integrals in two ways. The first is to ensure the ε tensors are not contracted until one can safely work in four dimensions, by solvingtensor loop integrals, performing UV renormalization and IR subtraction (as detailed inEq. (11)) and then finally contracting with the ε tensors.An alternative approach, which avoids the need to compute tensor integrals, is to projectthe amplitude onto the 60 possible rank-three Lorentz structures which can be formedfrom the three independent momenta and the ε tensor. Since each term of the projectorsonto these structures contains an ε tensor, the result of their contraction with the r.h.s. ofEq. (18) can be treated correctly in d dimensions. The 60 projectors act on the amplitudeas P µ µ µ i A µ µ µ = F i (19)to produce 60 form factors F i , which can be reduced to a minimal set (see the discussionaround Eq. (6)). Each projector P µ µ µ i can be written generically as C i, ε µ µ µ ν q ν + C i, ε µ µ µ ν q ν + · · · + C i, g µ µ ε µ ν ν ν q ν q ν q ν , (20)and we contract this general form with our amplitude. In the final result, we specify eachof the 60 sets of coefficients { C i,j } in order to obtain the 60 form factors F i . C Helicity amplitudes
In this appendix we describe how one can obtain the helicity amplitudes for the process gg → ZH from the tensor decomposition which we have introduced in Section 2. In14nalogy to Eq. (5) we introduce˜ M λ ,λ ,λ = ˜ A µνρ ε λ ,µ ( p ) ε λ ,ν ( p ) ε λ ,ρ ( p ) . (21)We furthermore specify the (contravariant) external momenta and polarization vectors asfollows: p = √ s , p = √ s − , p = √ s − y − y sin θy cos θ , p = √ s − y y sin θ − y cos θ ,ε + ( p ) = [ ε − ( p )] (cid:63) = 1 √ i , ε + ( p ) = [ ε − ( p )] (cid:63) = 1 √ − i ,ε + ( p ) = [ ε − ( p )] (cid:63) = 1 √ i cos θ sin θ , ε ( p ) = √ s m Z y y sin θ − y cos θ ,y = 1 + m Z − m H s , y = β = (cid:115) − m Z + m H s + ( m Z − m H ) s , (22)where ε denotes the longitudinal components of polarization vectors. Recall that allexternal momenta are defined as incoming, see Eq. (1). We have chosen the conventionfor the polarisation vector of p , following Ref. [35], such that ε + ( p ) → ε − ( p ) in thecenter-of-momentum frame. Furthermore, the polarization vectors satisfy (cid:88) λ ε λ ,µ ( p ) ε (cid:63)λ ,µ (cid:48) ( p ) = − g µµ (cid:48) + p ,µ p ,µ (cid:48) + p ,µ p ,µ (cid:48) p · p , (cid:88) λ ε λ ,ν ( p ) ε (cid:63)λ ,ν (cid:48) ( p ) = − g νν (cid:48) + p ,ν p ,ν (cid:48) + p ,ν p ,ν (cid:48) p · p , (cid:88) λ ε λ ,ρ ( p ) ε (cid:63)λ ,ρ (cid:48) ( p ) = − g ρρ (cid:48) + p ,ρ p ,ρ (cid:48) m Z , (23)which means that we have fixed the gauge for the external gauge bosons. With the abovechoice, some of the tensor structures in A µνρab ( p , p , p ), which are proportional to either p ν or p µ , are irrelevant because ε λ ( p ) · p = ε λ ( p ) · p = 0 . (24)This reduces the number of Lorentz structures in Eq. (6) from 14 to 8. We note that, aswe will see in Eq. (28), the dependence on F +2 drops out in the helicity amplitude.15n total there are 2 × × M − λ , − λ , − λ = − ˜ M λ ,λ ,λ , ˜ M ++ − = − ˜ M +++ (cid:12)(cid:12)(cid:12) θ → θ + π,y →− y , ˜ M + −− = − ˜ M + − + (cid:12)(cid:12)(cid:12) θ → θ + π,y →− y , (25)yielding the 4 independent helicity amplitudes. Note that the Mandelstam variables areinvariant under the simultaneous replacements θ → θ + π, y → − y and thus the formfactors do not change.For the evaluation of the ε tensor, we use the convention that ε = +1 , (26)and so some of the relevant contractions are as follows: ε µραβ ε µ + ( p ) ε ρ + ( p ) p α p β = − i s − cos θ ) ,ε µραβ ε µ + ( p ) ε ρ + ( p ) p α p β = i s − cos θ )( y − y ) ,ε ραβγ ε ρ + ( p ) p α p β p γ = − i s √ s √ y sin θ ,ε µραβ ε µ + ( p ) ε ρ ( p ) p α p β = i s √ s √ m Z y sin θ ,ε µραβ ε µ + ( p ) ε ρ ( p ) p α p β = − i m Z √ s √ θ ,ε µνρα ε µ + ( p ) ε ν + ( p ) ε ρ + ( p ) p α = i √ s √ θ ,ε µνρα ε µ + ( p ) ε ν + ( p ) ε ρ ( p ) p α = − i s m Z ( y cos θ + y ) . (27)In terms of the form factors, the 4 independent helicity amplitudes read˜ M + − = − i s m Z y sin θ ( sy F − − m Z F − )˜ M + − + = i s √ s √ y sin θ (1 − cos θ )(2 F − − y F − − y F +3 )˜ M +++ = i s √ s √ θ (cid:2) F − − ( y − y )(2 F − + 4 F + y F − + y cos θF +3 ) (cid:3) ˜ M ++0 = − i s m Z (cid:2) ( sy F − − m Z F − − m Z F ) cos θ − sy F +12 + m Z y sin θF +3 (cid:3) , (28)where we have omitted the arguments of the form factors, F ( t, u ).16hese helicity amplitudes have some general properties which are valid at any loop order.It is useful to introduce the partial wave decomposition of the amplitude [35] M λ a ,λ b ,λ c = 14 π (cid:88) J (2 J + 1) (cid:104) λ c | S J | λ a λ b (cid:105) d Jλ a − λ b ,λ c ( θ ) , (29)where λ a , λ b (= ± λ c (= ± , Z boson. J is the total angular momentum of the system, S J is the J -component of the S-matrix, and d JM,M (cid:48) ( θ ) is the Wigner small- d function (see, e.g. [36]).In the case of M ++0 , d Jλ a − λ b ,λ c ( θ ) = d J , ( θ ) = P J (cos θ ) , (30)where P J ( x ) are the Legendre polynomials, which are even functions for even J , and oddfunctions for odd J . Taking into account that F − , F − , F − and F are anti-symmetric in t ↔ u exchange and thus odd functions in cos θ , whereas F +12 and F +3 are symmetric in t ↔ u exchange and thus even functions in cos θ , we find that M ++0 is an even functionof cos θ . Using Eq. (29) and the property of the Legendre polynomials mentioned above,we conclude that only J -even components components contribute to M ++0 and thus thisamplitude is a J -even channel. In a similar discussion, it is also straightforward to showthat M + − , expanded in terms of d J , ( θ ) = P J (cos θ ) where P nJ is the associated Legendrepolynomial, is a J -odd channel, and that M +++ , expanded in terms of d J , ( θ ) = P J (cos θ ),is a J -even channel. On the other hand, M + − + does not have such a feature. In total,the squared amplitude should be symmetric in t ↔ u exchange due to the fact that thetwo initial state gluons are indistinguishable, and this symmetry is made apparent whensumming all of the squared helicity amplitudes.The contribution from the triangle diagrams is present only in M ++0 via F +12 , and this canbe understood in the following way. Due to the Landau-Yang theorem, the on-shell modeof the mediating Z -boson is forbidden; only the off-shell mode and the Goldstone bosonpropagate. Since the off-shell mode and the Goldstone boson behave as scalars underrotation, they do not appear in M + − ( J -odd) or M + − + ( J -indefinite). Furthermore,using the transversal condition of the Z -boson, ε ± ( p ) · p = 0 one can also derive that ε ± ( p ) · p = 0 and thus the transverse components of the final state Z -boson do notcouple to the scalar-scalar-vector interaction (see, e.g., the explicit form of the Feynmanrule for the Goldstone-Higgs- Z boson vertex). Because of this property, the contributionfrom triangle diagrams is absent in M +++ ; only M ++0 (and M −− , due to Eq. (25))contains such contributions. References [1] M. Aaboud et al. [ATLAS], Phys. Lett. B (2018), 59-86 [arXiv:1808.08238 [hep-ex]]. 172] A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. (2018) no.12, 121801doi:10.1103/PhysRevLett.121.121801 [arXiv:1808.08242 [hep-ex]].[3] M. C. Kumar, M. K. Mandal and V. Ravindran, JHEP (2015), 037[arXiv:1412.3357 [hep-ph]].[4] G. Heinrich, [arXiv:2009.00516 [hep-ph]].[5] O. Brein, R. V. Harlander and T. J. E. Zirke, Comput. Phys. Commun. (2013),998-1003 [arXiv:1210.5347 [hep-ph]].[6] R. V. Harlander, J. Klappert, S. Liebler and L. Simon, JHEP (2018), 089[arXiv:1802.04817 [hep-ph]].[7] A. Denner, S. Dittmaier, S. Kallweit and A. M¨uck, Comput. Phys. Commun. (2015), 161-171 [arXiv:1412.5390 [hep-ph]].[8] C. Englert, M. McCullough and M. Spannowsky, Phys. Rev. D (2014) no.1, 013013[arXiv:1310.4828 [hep-ph]].[9] R. V. Harlander, S. Liebler and T. Zirke, JHEP (2014), 023 [arXiv:1307.8122[hep-ph]].[10] G. Aad et al. [ATLAS], [arXiv:2007.02873 [hep-ex]].[11] R. V. Harlander, J. Klappert, C. Pandini and A. Papaefstathiou, Eur. Phys. J. C (2018) no.9, 760 [arXiv:1804.02299 [hep-ph]].[12] B. A. Kniehl, Phys. Rev. D (1990) 2253.[13] L. Altenkamp, S. Dittmaier, R. V. Harlander, H. Rzehak and T. J. Zirke, JHEP (2013), 078 doi:10.1007/JHEP02(2013)078 [arXiv:1211.5015 [hep-ph]].[14] A. Hasselhuhn, T. Luthe and M. Steinhauser, JHEP (2017) 073[arXiv:1611.05881 [hep-ph]].[15] S. Borowka, N. Greiner, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk and T. Zirke,JHEP (2016) 107 [arXiv:1608.04798 [hep-ph]].[16] J. Davies, G. Heinrich, S. P. Jones, M. Kerner, G. Mishima, M. Steinhauser andD. Wellmann, JHEP (2019), 024 [arXiv:1907.06408 [hep-ph]].[17] J. Davies, G. Mishima, M. Steinhauser and D. Wellmann, JHEP (2020), 024[arXiv:2002.05558 [hep-ph]].[18] .[19] A. V. Smirnov and F. S. Chuharev, [arXiv:1901.07808 [hep-ph]].1820] R. N. Lee, J. Phys. Conf. Ser. (2014), 012059 [arXiv:1310.1145 [hep-ph]].[21] J. Davies, G. Mishima, M. Steinhauser and D. Wellmann, JHEP (2018), 048[arXiv:1801.09696 [hep-ph]].[22] J. Davies, G. Mishima, M. Steinhauser and D. Wellmann, JHEP (2019), 176[arXiv:1811.05489 [hep-ph]].[23] B. Ruijl, T. Ueda and J. Vermaseren, [arXiv:1707.06453 [hep-ph]].[24] R. Harlander, T. Seidensticker and M. Steinhauser, Phys. Lett. B (1998) 125[hep-ph/9712228].[25] T. Seidensticker, hep-ph/9905298.[26] S. Larin, Phys. Lett. B (1993), 113-118 [arXiv:hep-ph/9302240 [hep-ph]].[27] S. Catani, Phys. Lett. B (1998) 161 [hep-ph/9802439].[28] G. Heinrich, S. P. Jones, M. Kerner, G. Luisoni and E. Vryonidou, JHEP (2017),088 [arXiv:1703.09252 [hep-ph]].[29] T. Hahn, Comput. Phys. Commun. (2001), 418-431 [arXiv:hep-ph/0012260 [hep-ph]].[30] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. (1999), 153-165[arXiv:hep-ph/9807565 [hep-ph]].[31] H. H. Patel, Comput. Phys. Commun. (2017), 66-70 [arXiv:1612.00009 [hep-ph]].[32] O. Brein, M. Ciccolini, S. Dittmaier, A. Djouadi, R. Harlander and M. Kramer,[arXiv:hep-ph/0402003 [hep-ph]].[33] L. Chen, G. Heinrich, S.P. Jones M. Kerner, J. Klappert and J. Schlenk, ZU-TH45/20, CERN-TH-2020-199, IPPP/20/57, P3H-20-076, KA-TP-21-2020, TTK-20-42,PSI-PR-20-21.[34] G. Degrassi, P. P. Giardino and R. Gr¨ober, Eur. Phys. J. C (2016) no.7, 411doi:10.1140/epjc/s10052-016-4256-9 [arXiv:1603.00385 [hep-ph]].[35] M. Jacob and G. C. Wick, Annals Phys.7