Resummation of the transverseenergy distribution in Higgs boson production at the Large Hadron Collider
Abstract:
We compute the resummed hadronic transverseenergy () distribution due to initialstate 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 nexttoleading 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.
Edinburgh 2014/07
IPMU 140051
ZUTH 09/14
MCnet1406
LPN14052
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 initialstate 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 initialstate 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 nexttoleading order (NLO) in QCD perturbation theory and also resummed to all orders in the QCD coupling . The resummation applies to leading, nexttoleading and some important nexttonexttoleading 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 fixedorder 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 vectorboson production [4, 5, 6] and closely related to the resummation of transverse momentum in vectorboson [7, 8, 9, 10] and Higgs production [11, 12, 13, 14].^{2}^{2}2The resummation of the jetveto 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 centreofmass energies and TeV;

Fixedorder 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 fixedorder 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 higherorder terms;

Monte Carlo studies of the effects of hadronlevel cuts on pseudorapidity and transverse momentum, with fixedorder 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 nexttoleading order in order to match our results to the fixedorder 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 transverseenergy distribution in the process at scale has the form
(1)  
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 transverseenergy distribution (1) is thus obtained by performing the inverse Fourier transformation with respect to the “transverse time”, . The factor is the perturbative and processdependent partonic cross section that embodies the allorder resummation of the large logarithms . Since is conjugate to , the limit corresponds to .
As in the case of transversemomentum resummation [20], the resummed partonic cross section can be written in the following form:
(2)  
Here is the cross section for the partonic subprocess of gluon fusion, , through a massivequark loop:
(3) 
where in the limit of infinite quark mass
(4) 
is the appropriate gluon form factor, which in the case of resummation takes the form [5, 6]
(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:
(6)  
(7)  
(8) 
Thus a calculation to NLO in involves the coefficients , , , and . The coefficients , , and read [21, 22]
(9) 
The coefficient for the Higgs transversemomentum spectrum is [23, 24]
(10) 
However, the value of the coefficient for the transverse energy in Higgs production could be different^{3}^{3}3We are grateful to Jon Walsh for a useful discussion on this point.. In Sec. 3, we will perform a fit to the fixedorder 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
(11)  
where and are the real and imaginary parts of
(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 EulerMascheroni constant. The resulting expressions for are
(13)  
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 transversemomentum resummation. In the notation of that paper, we have, for a renormalization scale ,
(14) 
where , , and is the lowestorder coefficients of the beta function:
(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
(16) 
where now
(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
(18) 
which is easily seen by expanding the exponential. This allows one to calculate the functions explicitly. They are given by [25, 26, 11]:
Now, the actual integral we require is
(20) 
and so we must introduce in the integrand. The analogue of Eq. (18) is
(21) 
where the generating function
(22)  
being the incomplete gamma function,
(23) 
The series represents power corrections, which we do not wish to include in the resummation, so we write instead
(24)  
where now
(25) 
i.e. we have chosen in (17), and
(26) 
Now
(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
(28) 
where was defined in Eq. (14). Now
(29) 
where , so
(30) 
where
(31)  
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:
(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
(33) 
instead of , and one must also replace in the coefficient of by
(34) 
We show in Appendix B that the vanishing of the transverseenergy 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
(35)  
and implies that
(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
(37) 
so that to NLO
(38) 
where, following the shift according to Eq. (32)
(39)  
Similarly, evaluating all PDFs at scale , we can write to NLO
(40) 
so that
(41)  
where
(42)  
Performing the Fourier transformation (1), we find terms involving the integrals
(43) 
with , which may be evaluated from
(44) 
with generating function
(45) 
Writing
(46) 
we have
(47) 
We can safely deform the integration contour around the branch cut along the negative real axis to obtain
(48)  
This gives