Resummation of the transverse-energy distribution in Higgs boson production at the Large Hadron Collider

# Resummation of the transverse-energy distribution in Higgs boson production at the Large Hadron Collider

Massimiliano Grazzini111On leave of absence from INFN, Sezione di Firenze, Sesto Fiorentino, Florence, Italy., Andreas Papaefstathiou, Jennifer M. Smillie and Bryan R. Webber
Physik Institut, Universität Zürich, Switzerland
Higgs Centre for Theoretical Physics, University of Edinburgh, UK
Cavendish Laboratory, J.J. Thomson Avenue, Cambridge, UK
E-mail: ,, ,
July 26, 2019July 26, 2019
July 26, 2019July 26, 2019
###### Abstract:

We compute the resummed hadronic transverse-energy () distribution due to initial-state QCD radiation in the production of a Standard Model Higgs boson of mass 126 GeV by gluon fusion at the Large Hadron Collider, with matching to next-to-leading order calculations at large . Effects of hadronization, underlying event and limited detector acceptance are estimated using aMC@NLO with the Herwig++ and Pythia8 event generators.

Higgs boson, Hadronic Colliders, QCD Phenomenology
preprint: Cavendish-HEP-14/01
Edinburgh 2014/07
IPMU 14-0051
ZU-TH 09/14
MCnet-14-06
LPN14-052

## 1 Introduction

The new particle discovered recently by the ATLAS [1] and CMS [2] Collaborations at the LHC looks very much like the Higgs boson of the Standard Model, although its properties remain to be fully explored. For this exploration, detailed predictions of the expected characteristics of Higgs production within the Standard Model will be essential, in order to optimize signal to background ratios and to search for any signs of new physics. One such characteristic is the amount and distribution of initial-state QCD radiation, which is predicted to be exceptionally high in production by gluon fusion and exceptionally low in vector boson fusion. A thorough understanding of initial-state radiation is therefore essential for the separation of these production mechanisms.

In the present paper we study the distribution of the total amount of transverse energy () emitted in Standard Model Higgs boson production by gluon fusion at the LHC. Results are presented at next-to-leading order (NLO) in QCD perturbation theory and also resummed to all orders in the QCD coupling . The resummation applies to leading, next-to-leading and some important next-to-next-to-leading logarithms of ((N)NLL) where is the hard process scale, taken to be the Higgs mass . Thus it improves the treatment of the small- region, where the fixed-order prediction diverges whereas the actual distribution must tend to zero as . By matching the resummed prediction to the NLO result valid at large , we provide a uniform description from the low- to the high- region.

Our approach follows on from Ref. [3], based in turn on the early work on resummation in vector-boson production [4, 5, 6] and closely related to the resummation of transverse momentum in vector-boson [7, 8, 9, 10] and Higgs production [11, 12, 13, 14].222The resummation of the jet-veto distribution has been considered in Refs. [15, 16, 17, 18, 19]. We make a number of improvements relative to Ref. [3], including:

• Predictions for the experimentally relevant Higgs mass of 126 GeV, at centre-of-mass energies and TeV;

• Fixed-order predictions to NLO, i.e. .

• Expansion of the resummation formula to NLO, and demonstration that to this order the structure of the logarithmic terms is consistent with the fixed-order prediction;

• Matching of the resummed and NLO predictions across the whole range of ;

• A constraint on the perturbative unitarity of the prediction, using the method of Ref. [11], which reduces the impact of logarithmic terms in the large- region;

• Studies of the effects of renormalization scale variation and unknown higher-order terms;

• Monte Carlo studies of the effects of hadron-level cuts on pseudorapidity and transverse momentum, with fixed-order matching to parton showers using aMC@NLO interfaced to Herwig++ and Pythia8.

The paper is organized as follows. In Sec. 2, we review the resummation procedure and then describe the necessary modifications to implement the unitarity condition mentioned above. This involves some changes in the formalism and the evaluation of new integrals in this prescription. In Sec. 3, we expand our resummed result to next-to-leading order in order to match our results to the fixed-order prediction at this accuracy. This renders our predictions positive throughout the -range. In Sec. 4, we investigate the distribution further through Monte Carlo studies. We first reweight Monte Carlo results to our analytic distribution and then investigate the impact of hadronisation and underlying event. We end the main text in Sec. 5 with conclusions and discussion. A number of appendices then contain supplementary results.

## 2 Resummation of logarithmically enhanced terms

Here we summarize the results of Ref. [3] as applied to Higgs boson production. The resummed component of the transverse-energy distribution in the process at scale has the form

 [dσHdQ2dET]res. = 12π∑a,b∫10dx1∫10dx2∫+∞−∞dτe−iτETfa/h1(x1,μ)fb/h2(x2,μ) (1) ⋅ WHab(x1x2s;Q,τ,μ)

where is the parton distribution function (PDF) of parton in hadron at factorization scale , taken to be the same as the renormalization scale here (we illustrate the impact of varying this scale in Sec. 3). In what follows we use the  renormalization scheme. To take into account the constraint that the transverse energies of the emitted partons should sum to , the resummation procedure is carried out in the domain that is Fourier conjugate to . The transverse-energy distribution (1) is thus obtained by performing the inverse Fourier transformation with respect to the “transverse time”, . The factor is the perturbative and process-dependent partonic cross section that embodies the all-order resummation of the large logarithms . Since is conjugate to , the limit corresponds to .

As in the case of transverse-momentum resummation [20], the resummed partonic cross section can be written in the following form:

 WHab(s;Q,τ,μ) = ∫10dz1∫10dz2Cga(αS(μ),z1;τ,μ)Cgb(αS(μ),z2;τ,μ)δ(Q2−z1z2s) (2) ⋅ σHgg(Q,αS(Q))Sg(Q,τ).

Here is the cross section for the partonic subprocess of gluon fusion, , through a massive-quark loop:

 σHgg(Q,αS(Q))=δ(Q2−m2H)σH0, (3)

where in the limit of infinite quark mass

 σH0=α2S(mH)GFm2H288π√2. (4)

is the appropriate gluon form factor, which in the case of resummation takes the form [5, 6]

 Sg(Q,τ)=exp{−2∫Q0dqq[2Ag(αS(q))lnQq+Bg(αS(q))](1−eiqτ)}. (5)

The functions , as well as the coefficient functions in Eq. (2), contain no terms and are perturbatively computable as power expansions with constant coefficients:

 Ag(αS) = ∞∑n=1(αSπ)nA(n)g, (6) Bg(αS) = ∞∑n=1(αSπ)nB(n)g, (7) Cga(αS,z) = δgaδ(1−z)+∞∑n=1(αSπ)nC(n)ga(z). (8)

Thus a calculation to NLO in involves the coefficients , , , and . The coefficients , , and read [21, 22]

 A(1)g=CA,A(2)g=16CA[CA(676−π22)−53nf],B(1)g=−16(11CA−2nf), C(1)gg(z)=14[CA(2−π23)+5+4π2]δ(1−z)≡c(1)gδ(1−z), C(1)gq(z)=C(1)g¯q(z)=12CFz. (9)

The coefficient for the Higgs transverse-momentum spectrum is [23, 24]

 ¯¯¯¯B(2)g=C2A(2324+1118π2−32ζ3)+12CFnf−CAnf(112+π29)−118CFCA. (10)

However, the value of the coefficient for the transverse energy in Higgs production could be different333We are grateful to Jon Walsh for a useful discussion on this point.. In Sec. 3, we will perform a fit to the fixed-order NLO result at small transverse energy, with this coefficient as a free parameter.

Returning to Eq. (1), we may recast it in a form with a real integrand as

 [dσHdQ2dET]res. = 1πs∫∞0dτe−F(R)g(Q,τ)[R(R)g(s;Q,τ)cos{F(I)g(Q,τ)+τET} (11)

where and are the real and imaginary parts of

 Fg(Q,τ)=2∫Q0dqq[2Ag(αS(q))lnQq+Bg(αS(q))](1−eiqτ). (12)

As explained in [3], the coefficient functions in Eq. (2) contain logarithms of , which are eliminated by the choice of factorization scale , where , being the Euler-Mascheroni constant. The resulting expressions for are

 R(R)g(ξ,τ) =∫1ξdx1x1{fg/h1(x1)fg/h2(ξx1)(1+αSπ2c(1)g) (13) +αSπ∫1ξ/x1dzz[fg/h1(x1)fs/h2(ξzx1)+fs/h1(x1)fg/h2(ξzx1)]12CFz}, R(I)g(ξ,τ) =αS2∫1ξdx1x1∫10dzz{2fg/h1(x1)fg/h2(ξzx1)Pgg(z) +[fg/h1(x1)fs/h2(ξzx1)+fs/h1(x1)fg/h2(ξzx1)]Pgq(z)} = αS2∫1ξdx1x1{2fg/h1(x1)fg/h2(ξx1)[2CAln(1−ξx1)+16(11CA−2nf)] +∫1ξ/x1dzz[4CAfg/h1(x1){fg/h2(ξzx1)[z1−z+1−zz+z(1−z)]−fg/h2(ξx1)z1−z} +{fg/h1(x1)fs/h2(ξzx1)+fs/h1(x1)fg/h2(ξzx1)}CF1+(1−z)2z]},

where and all PDFs and coefficient functions are understood to be evaluated at scale . We have defined for convenience.

### 2.1 Evaluation of the exponent

We now seek to evaluate the exponent of the form factor, (12), analytically. We will use the method of Ref. [11] where the analogous calculation was performed for transverse-momentum resummation. In the notation of that paper, we have, for a renormalization scale ,

 Gg(aR,L)≡−2∫Qb0/bdqq[2Ag(αS(q))lnQq+Bg(αS(q))]=Lg1(Y)+g2(Y)+aRg3(Y)+… (14)

where , , and is the lowest-order coefficients of the beta function:

 dlnaRdlnμ2R=β(aR)=−∞∑n=0βnan+1R. (15)

The term collects the LL contributions , the function resums the NLL contributions , the function controls the NNLL terms , and so forth. We will give the explicit form of the functions below. We can therefore deduce that in general

 −2∫QQ0dqq[2Ag(αS(q))lnQq+Bg(αS(q))]=2λg1(y)+g2(y)+aRg3(y)+… (16)

where now

 y≡2β0aRλandλ≡ln(Q/Q0). (17)

Now by expressing in terms of and relating this in turn to , we can write the integrand in (16) as a function of , and then use the result

 ∫QQ0dqqf(lnQq)=f(ddu)1u(eλu−1)∣∣u=0, (18)

which is easily seen by expanding the exponential. This allows one to calculate the functions explicitly. They are given by [25, 26, 11]:

 g1(y) = A(1)gβ0y(y+ln(1−y)), g2(y) = B(1)gβ0ln(1−y)−A(2)gβ20(y1−y+ln(1−y)) +A(1)gβ1β30(12ln2(1−y)+y+ln(1−y)1−y)+A(1)gβ0(y1−y+ln(1−y))ln(Q2μ2R), g3(y) = −A(3)g2β20(y1−y)2−B(2)gβ0y1−y+A(2)gβ1β30y(3y−2)−2(1−2y)ln(1−y)2(1−y)2 +A(1)gβ40(β21(1−2y)ln2(1−y)2(1−y)2+ln(1−y)(β0β2−β21+β211−y) +y(β0β2(2−3y)+β21y)2(1−y)2)+B(1)gβ1β20y+ln(1−y)1−y−A(1)g2y2(1−y)2ln2(Q2μ2R) +⎛⎝B(1)gy1−y+A(2)gβ0y2(1−y)2+A(1)gβ1β20(y1−y+(1−2y)ln(1−y)(1−y)2)⎞⎠ln(Q2μ2R).

Now, the actual integral we require is

 Fg(αS,λ)≡2∫Q0dqq[2Ag(αS(q))lnQq+Bg(αS(q))](1−eiqτ) (20)

and so we must introduce in the integrand. The analogue of Eq. (18) is

 ∫Q0dqqf(lnQq)(1−eiqτ)=f(ddu)J(Qτ;−u)|u=0 (21)

where the generating function

 J(Qτ;u) = ∫Q0dqq(qQ)u(1−eiqτ) (22) = 1u−(−iQτ)−uγ(u,−iQτ),

being the incomplete gamma function,

 γ(u,z)=Γ(u)−zu−1e−z∞∑k=0Γ(u)Γ(u−k)z−k. (23)

The series represents power corrections, which we do not wish to include in the resummation, so we write instead

 J(Qτ;−u) = 1u[(−iQτ)uΓ(1−u)−1] (24) = 1u[exp(λu+∑k=2ζkkuk)−1]≡1u[eλuZ(u)−1]

where now

 λ=ln(Qτiτ0), (25)

i.e. we have chosen in (17), and

 Z(u)≡exp(∑k=2ζkkuk)=τu0Γ(1−u). (26)

Now

 1u[eλuZ(u)−1]=Z(u)u[eλu−1]+1u[Z(u)−1] (27)

and the second term involves no logarithms, so again we drop it from the resummation. We show in Appendix A that the first term implies that

 Fg(αS,λ)≡2∫Q0dqq[2Ag(αS(q))lnQq+Bg(αS(q))](1−eiqτ)=−Z(ddλ)Gg(αS,2λ), (28)

where was defined in Eq. (14). Now

 Z(ddλ)=1+ζ22d2dλ2+ζ33d3dλ3+… (29)

where , so

 Fg(αS,λ)=−2λg1(y)−g2(y)−aR˜g3(y)+… (30)

where

 ˜g3(y) = g3(y)+π212aRd2dλ2[2λg1(y)] (31) = g3(y)−π23A(1)g(1−y)2.

The other terms from (29) contribute logarithms only at the level of and beyond, so we do not consider them.

Following Ref. [11], we can now enforce the ‘unitarity’ condition, as , by a shift of argument of the logarithm:

 λ→˜λ=ln(1+Qτiτ0)=12ln(1+Q2τ2τ20)−iarctan(Qττ0), (32)

so that now . We must apply a corresponding shift in the factorization scale of the parton distributions and coefficient functions (given explicitly below in Eqs. (3.1)). They are now evaluated at a scale of

 ˜μF=Q√1+Q2τ2/τ20 (33)

instead of , and one must also replace in the coefficient of by

 αS(˜μF)πarctan(Qττ0). (34)

We show in Appendix B that the vanishing of the transverse-energy distribution for implies a dispersion relation between the real and imaginary parts of its Fourier transform. This allows (11) to be written in the simpler equivalent form

 [dσHdQ2dET]res. = 2πs∫∞0dτe−F(R)g(Q,τ)cos(τET)[R(R)g(s;Q,τ)cos{F(I)g(Q,τ)} (35) −R(I)g(s;Q,τ)sin{F(I)g(Q,τ)}]σHgg(Q,αS(Q))

and implies that

 ∫∞0dET[dσHdQ2dET]res.=1sR(R)g(s;Q,0)σHgg(Q,αS(Q)), (36)

where, on account of (33), the parton distributions in are evaluated at scale .

## 3 Matching to fixed order

We now match the resummed expression derived above to the NLO perturbative expansion of the transverse energy distribution, taking care to avoid double counting of the terms already contained in the resummation.

### 3.1 Expansion of the resummed prediction

Performing the expansion of Eq. (30) in powers of , we find

 −2λg1 = 2A(1)gλ2aR+83β0A(1)gλ3a2R+O(a3R) −g2 = 2B(1)gλaR+2[A(2)g+β0B(1)g−β0A(1)gln(Q2μ2R)]λ2a2R+O(a3R) −aR˜g3 = π23A(1)gaR+2[B(2)g+23π2β0A(1)g−β0B(1)gln(Q2μ2R)]λa2R+O(a3R), (37)

so that to NLO

 Fg(αS,λ)=aRF1+a2RF2 (38)

where, following the shift according to Eq. (32)

 F1 = 2A(1)g(˜λ2+π26)+2B(1)g˜λ F2 = 83β0A(1)g˜λ3+2[A(2)g+β0B(1)g−β0A(1)gln(Q2μ2R)]˜λ2 (39) +2[B(2)g+23π2β0A(1)g−β0B(1)gln(Q2μ2R)]˜λ.

Similarly, evaluating all PDFs at scale , we can write to NLO

 Rg(τ)=R0+aR(R1+˜λR′1)+a2R(R2+˜λR′2+˜λ2R′′2) (40)

so that

 SgRg = R0+aRR1+a2RR2+aR(˜λR′1−F1R0) (41) + a2R[˜λR′2+˜λ2R′′2−(F2−12F21)R0−F1(R1+˜λR′1)]

where

 R0 = ∫1ξdx1x1fg(x1)fg(ξx1) R1 = ∫1ξdx1x1{2c(1)gfg(x1)fg(ξx1)+CF∫1ξ/x1dzfs(x1)fg(ξzx1)} R′1 = −∫1ξdx1x1{2fg(x1)fg(ξx1)[2CAln(1−ξx1)+16(11CA−2nf)] (42) +2CFfs(x1)fg(ξzx1)1+(1−z)2z]}.

Performing the Fourier transformation (1), we find terms involving the integrals

 Ip(ET,Q)=12π∫+∞−∞dτe−iτETlnp(1+Qτiτ0) (43)

with , which may be evaluated from

 Ip(ET,Q)=dpdupI(ET,Q;u)|u=0 (44)

with generating function

 I(ET,Q;u)=12π∫+∞−∞dτe−iτET(1+Qτiτ0)u. (45)

Writing

 1+Qτiτ0=zQτ0ET (46)

we have

 I(ET,Q;u)=−i2πET(QETτ0)u∫+i∞−i∞dzzuez−τ0ET/Q. (47)

We can safely deform the integration contour around the branch cut along the negative real axis to obtain

 I(ET,Q;u) = −uET(QET)uexp(uγE−τ0ET/Q)Γ(1−u) (48) = −uET(QET)uexp(−τ0ETQ−∞∑k=2ζkkuk).

This gives

 I1(ET,Q) = −1ETe−τ0ET/Q I2(ET,Q) = −2ETln(QET)e−τ0ET/Q I3(ET,Q) = −3ET[ln2(QET)−π26