###### Abstract

We examine the contributions of soft gluons to the Higgs production cross section at the LHC in the Standard Model and its minimal supersymmetric extension. The soft gluon radiation effects of this reaction share many features with the Drell-Yan process, but arise at lowest order from a purely gluonic initial state. We provide an extension of the conventional soft gluon resummation formalism to include a new class of contributions which we argue to be universal, and resum these and the usual Sudakov effects to all orders. The effect of these new terms is striking: only if they are included, does the expansion of the resummed cross section to next-to-leading order reproduce the exact result to within a few percent for the full range of Higgs boson masses. We use our resummed cross section to derive next-to-next-to-leading order results, and their scale dependence. Moreover, we demonstrate the importance of including the novel contributions in the resummed Drell-Yan process.

CERN-TH/96-231

DESY 96-170

hep-ph/9611272

Soft Gluon Radiation in Higgs Boson
Production at the LHC
Michael Krämer^{1}^{1}1Present address:
Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, England,
Eric Laenen and Michael Spira

Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, FRG CERN TH-Division, CH-1211 Geneve 23, Switzerland

CERN-TH/96-231

DESY 96-170

hep-ph/9611272

November 1996

## 1. Introduction

The search for Higgs particles [1] is one of the most important endeavors for future high energy and hadron collider experiments. The Higgs boson is the only particle of the Standard Model (SM) which has not been discovered so far. The direct search at the LEP1 experiments via the process yields a lower bound on the Higgs mass of 65.2 GeV [2]. Theoretical consistency restricts the Higgs mass to be smaller than GeV [3]. The dominant Higgs production mechanism at the LHC, a collider with a c.m. energy of 14 TeV, is the gluon fusion process which is mediated by a heavy quark triangle loop at lowest order [4]. As an important step to increase the theoretical precision the two-loop QCD corrections have been calculated, resulting in a significant increase of the predicted total cross section by about 50 – 100% [5, 6]. The dependence on the unphysical renormalization and factorization scales decreased considerably by including these next-to-leading-order (NLO) corrections, resulting in an estimate of about 15% for the remaining scale sensitivity [5]. It is important to note, and we will demonstrate, that the NLO corrections are dominated by soft gluon radiation effects.

The minimal supersymmetric extension of the Standard Model (MSSM) is among the most attractive extensions of the SM. It requires the introduction of two Higgs doublets leading to the existence of five scalar Higgs particles, two scalar CP-even , one pseudoscalar CP-odd and two charged bosons . This Higgs sector can be described by fixing two parameters, which are usually chosen to be tg, the ratio of the two vacuum expectation values, and the pseudoscalar Higgs mass . Including higher order corrections to the Higgs masses and couplings up to the two-loop level, the mass of the lightest scalar Higgs particle is restricted to be smaller than GeV [7]. The direct search at LEP1 sets lower bounds of about 45 GeV on the masses of the MSSM Higgs bosons [2]. The dominant neutral Higgs production mechanisms at the LHC are the gluon fusion processes and the associated production with a pair which becomes important only for large tg [8]. The coupling of the neutral Higgs particles to gluons is again mediated by top and bottom loops, with the latter providing the dominant contribution for large tg, and squark loops, if their masses are smaller than about 400 GeV [9]. (In this paper we shall assume the squark masses to be 1 TeV, so that squark loops can safely be neglected.) The two-loop (NLO) QCD corrections to the gluon fusion mechanism have also been calculated [5, 9] and conclusions completely analogous to the SM case emerge. Soft gluon radiation effects again provide the dominant contribution to these corrections, for small tg. For large tg bottom mass effects will be of similar size due to the dominance of the bottom quark loops and we will not consider this regime in this paper.

In previous analyses the resummation of soft gluon radiation in the transverse momentum distribution of the Higgs bosons has been performed [10], which are significant at small . We consider universal soft gluon effects on the total production cross section and demonstrate that these dominate the NLO corrections both in the SM and the MSSM for small tg. The study of these effects in higher orders, and their resummation to all orders is our purpose in this paper.

Although our main focus is Higgs production, we will consider soft gluon effects in the Drell-Yan process for comparison. Higgs production shares many features with this reaction, apart from the species of leading initial state partons, e.g. it also proceeds at lowest order via a color singlet hard scattering process, and is a process at lowest order. The Drell-Yan process has been studied by performing exact perturbative QCD calculations up to next-to-next-to-leading order (NNLO) in Ref. [11] and in the context of soft gluon resummation in [12]. In this paper we present an extension of the conventional soft gluon resummation formalism, in which we use the Drell-Yan reaction to gauge its quality and importance. We then apply the extension to Higgs production to derive the first estimates of NNLO effects. These estimates are important in view of the size of the NLO corrections.

The paper is organized as follows: In section 2 we describe the construction of the resummed cross section and the extension of the soft gluon resummation formalism. In section 3 we present NLO and NNLO results from the expansion of the resummed cross sections for Higgs production and the Drell-Yan process. We conclude and present an outlook in section 4.

## 2. The Resummed Exponent

In this section we derive the resummed partonic cross section for Higgs boson production via gluon fusion. In order to set the stage, we must first discuss some preliminary approximations to the exact NLO calculation in Ref. [5].

In the Standard Model, the leading order (LO) process consists of gluon fusion into a Higgs boson via a heavy quark triangle loop, see Fig. 1.

Because the Higgs coupling to fermions is proportional to the fermion mass, the top quark strongly dominates this coupling, constituting about 90% of the total coupling. Our first approximation is to neglect the quark-antiquark and quark-gluon channels as these contribute at next-to-leading order (NLO) to the full cross section with less than 10% [5].

The exact NLO calculation in [5] was performed for the general
massive case, i.e. all masses were taken into account explicitly.
However, the following very useful approximation was identified,
involving the heavy top mass limit of the calculation. Let us define
the (NLO) K-factor^{2}^{2}2It should be noted that we use here NLO
parton densities and strong coupling in the NLO cross
sections and LO quantities in the LO cross sections for this
“hadronic” K-factor. This leads to K-factors smaller than two in
contrast with the “partonic” K-factor, which is defined with NLO
parton densities and strong coupling also in the LO cross section.
by

(1) |

where denotes the hadronic gluon-fusion cross section for the general massive case, calculated exactly in LO/NLO, and the scaling variables are defined by .

In Figs. 2 and 3 we compare with the approximation

(2) |

for scalar and pseudoscalar Higgs boson production, where the K-factor takes into account the top quark contribution to the relative QCD corrections only, in the limit of a heavy top quark. We observe that the approximation (2) is accurate to within 10% for the full Higgs mass range GeV of the SM Higgs boson as well as the pseudoscalar Higgs particle of the MSSM for small tg [At the threshold the pseudoscalar cross section develops a Coulomb singularity so that perturbation theory is not valid in a small margin around this value for the pseudoscalar mass [5].]. The same accuracy of the approximation emerges for the two scalar Higgs particles of the MSSM for small tg. Our second approximation then consists of assuming that is a valid approximation to for all orders, i.e. we will assume that the higher-order K-factor, when computed in the infinite top mass limit and combined with the massive LO cross section, will give a good approximation to the higher order cross section in the general massive case. In fact, we will see that at NLO the bulk of the K-factor is due to soft and collinear gluons, which do not resolve the effective coupling. The assumption that this persists to higher orders is supported by the validity of the infinite top mass approximation at NLO.

In the MSSM, the validity of these approximations depends strongly on the parameter tg. For the top quark contribution to the cross sections amounts to more than 70%. The heavy top quark limit is thus a reasonable approximation in this regime. For large values of tg the bottom loop contribution becomes significant so that the approximations are no longer valid. Still, the infinite top mass approximation deviates from the full NLO result, including bottom contributions, by less than 25% for as can be inferred from Fig.3.

We are now ready for the construction of the resummed partonic cross section, for which we will employ the methods of Ref. [14]. In order to retain similarity to the Drell-Yan case, we will denote the Higgs mass squared with throughout the text of this paper.

In the approximations outlined above, the regularized total Higgs production cross sections may be written in dimensions as []

(3) |

or, in terms of moments,

(4) |

where , , is the hadronic c.m. energy squared, is the dimensional regularization scale, and is the bare gluon distribution function. Note that the definition is such that the dependence on the moment variable in the parton densities is shifted by one unit compared to the dependence of the partonic cross section. In this way we remove an overall factor, emphasizing the soft gluon contribution to the partonic cross section.

In the approximations discussed in the beginning of this section, the -dimensional partonic cross sections can be cast into the form

(5) |

with the coefficients

(6) | |||||

(7) |

where is the bare strong coupling constant (with
dimension ) and denote the
modified top Yukawa couplings normalized to the SM coupling, which are
given in [5]. The factor in eq. (5)
stems from the effective coupling of the Higgs boson to gluons in the
heavy top quark limit, which can be obtained by means of low energy
theorems [5, 15]. They are given^{3}^{3}3The last two factors
in the large bracket originate from the anomalous dimension of the gluon
operators [16]. The top mass denotes the scale
invariant
mass .
We would like to
thank the authors of Ref. [17] for pointing out two errors in our
treatment of the strong coupling in eq.(8) and the anomalous mass
dimension of eq.(12) in an earlier version of this paper. The
numerical size of these errors is about 0.03% and thus negligible.
by [5, 16]

(8) | |||||

(9) |

where is the strong coupling constant in the scheme including flavors in the evolution; the couplings for 5 and 6 flavors are related by [18]

(10) |

denotes the QCD function and
its top quark contribution^{4}^{4}4The factors
include the
top quark contribution at vanishing momentum transfer, which
differs from the top quark contribution to the function by a finite amount at
[19]., which is given by [18, 20] [
is the number of light flavors]

(11) |

is the anomalous mass dimension including 6 flavors, which can be expressed as [21]

(12) |

Using these expansions the effective scalar couplings
are given by^{5}^{5}5This result agrees with Ref. [17].

(13) | |||||

where denotes the pole mass of the top quark. The scale dependence of the strong coupling in the lowest order cross section eq. (7) and the factor eq. (5) includes 5 light flavors, i.e. the top quark is decoupled. In the rest of the paper we identify

(14) |

Note further that the expression (5) is not yet finite for ; mass factorization and renormalization of the bare coupling in the Born cross section will be carried out after resummation. In eq. (5) we denote the correction factors by , which are defined in the infinite mass limit without the factorizing corrections to the effective coupling. They may be expanded as

(15) |

where we define, for the sake of convenience,

(16) |

Here is the renormalized strong coupling and we have chosen the renormalization scale equal to for the moment. From Ref. [5, 6] we can derive the following expression for the first two coefficients of the SM Higgs correction factor

(17) | |||||

(18) | |||||

(19) |

Note that we have implicitly redefined the scale by to eliminate factors and . The plus distribution in eq. (18) is as usual defined by

(20) |

We will now construct a resummed expression for by means of the methods described in Ref. [14]. Near the elastic edge of phase space the Higgs cross section in the infinite mass limit may be factorized into hard, soft and jet functions, in completely analogy with the Drell-Yan cross section. Following the arguments of Ref. [14] this leads to the Sudakov evolution equation

(21) |

In order to solve eq. (21) we must impose a boundary condition. We will shortly argue [14] that we may use the boundary condition for the moments, or in -space

(22) |

The solution to eq. (21) is then, in Mellin space,

(23) |

where as above . The formal Mellin inversion reads

(24) |

The solution (23) may be expressed as

(25) | |||||

with an arbitrary function. Note that is to be treated as part of any plus distributions in , because it arises from the originally -independent scale ratio. In deriving eq. (25) we have used the renormalization group invariance of the evolution kernel . Note further that we have expanded the full evolution kernel in the -dimensional running coupling constant, with . The defining equation of the -dimensional running coupling is

(26) |

with the boundary condition . Here and . The solution, linearized in , is

(27) |

with

The functions can be determined by choosing in eq. (25) and expanding the Sudakov equation to second order in the -dimensional running coupling

(28) | |||||

in terms of , using (see eq. (27))

(29) |

The resulting one- and two-loop coefficients of the evolution kernel can then be derived from low order calculations of the correction factors via

(30) |

and

(31) | |||||

Of course, unlike for Drell-Yan [11], the second order corrections to Higgs production have not yet been calculated. The above results can easily be inverted to derive , if the functions are known. The full functions may then be constructed by reexpanding the running coupling in , using eq. (27) with ,

(32) | |||||

In this way the one-loop coefficients can be straightforwardly determined from eqs. (18,19). They may be written as

(33) |

where the coefficient functions are regular functions of their arguments at . In the present case they are given by

(34) | |||||

(35) | |||||

(36) | |||||

(37) |

Given the limitations of the factorization theorem near the edge of phase space [22], from which the evolution equation (21) is derived [14], we might be tempted to immediately discard all terms of order . However, we wish to be careful with these terms, and we will examine them and their relevance later in this section.

As was shown in [14], the term and the plus-distributions in eq. (33) are separately renormalization group invariant. We are therefore free to choose different functions for these two terms in the general resummed expression eq. (25). The natural choice for the term is . Changes in generate terms at higher orders. The integral may then be carried out explicitly for this term in eq. (25). For the plus-distribution term however, the natural choice is , just as for Drell-Yan. Then, using in eq. (32), we absorb the factor in (33) into the boundary of the integral in eq. (25), involving only the running coupling. The term is of and will be neglected.

We now turn our attention to the function in (36). We have some freedom in its treatment, as different choices will only differ by constants or in . However, we will argue that among all such terms the ones generated by are universal and can legitimately be included in the resummed expression. To exhibit the importance of the different treatments of the residue function we choose three schemes which probe the full range of possibilities. After rescaling to incorporate the factor and combining the plus distribution with the Mellin transform in eq. (25), we see that the relevant function to approximate is [We are neglecting the coefficient , because it is contributing at .]. The three schemes are defined by

(38) | |||||

The minimal scheme involves replacing simply by , scheme includes all terms of in the exponent, whereas scheme includes in addition some terms in the exponent. Using the one-loop evolution kernel we can now construct the resummed expressions for the Higgs production correction factor in the three schemes. However, these expressions are still divergent for . The divergences are cancelled by mass factorization and renormalization for which we choose the scheme [23] throughout this paper. For this purpose we need the resummed gluon distribution [24]. To one loop accuracy it may be written as [14, 24]

(39) | |||||

where

(40) | |||||

and

(41) |

where is the mass factorization scale. As is well-known, the one-loop anomalous dimension is derived from the residue of the collinear singularity in the gluon operator matrix element. The function is related to that component of the residue which is proportional to the one loop coefficient of the QCD -function. The strong coupling in the LO cross section (7) has been left unrenormalized so far. The renormalization of this bare coupling is now performed in the scheme, via the replacement

where we explicitly show the renormalization scale . Thus the renormalization factorizes into several pieces according to the perturbative expansion of the QCD function,

(43) |

where

(44) |

Note that the above form of the -factor for the strong coupling constant, which is completely factorized from the correction factor, is very similar in form to the piece of the density, and we may thus combine the overall coupling constant renormalization with the mass factorization procedure. Restricting ourselves to next-to-leading order (i.e. putting ) and choosing one finds that simultaneous mass factorization and renormalization of the bare coupling in eq. (7) leads to

(45) |

Note that the dependence on the dimensional regularization scale drops out on the l.h.s. Because is infrared safe, we can return to four dimensions, and we find

(46) | |||||

(49) |

A few remarks are in order. First, the above expressions are formally not well defined because the integration paths in the exponent traverse a singularity, related to an infrared renormalon. Our purpose in this paper is however not the numerical evaluation of the resummed cross section; we consider the resummed formulae to be a generating functional for an approximation to the QCD perturbation series, rather than its approximate sum.

The second remark is that one may try to incorporate (a part of) the two loop evolution kernel. Some of the NNLO terms in the exact calculation can be inferred using renormalization group methods, such as used for Drell-Yan in Ref. [25]. A straightforward calculation along these lines leads to

(50) |

where is the two-loop component of the Altarelli-Parisi gluon-to-gluon splitting function [26]. Moreover, the mass factorization and renormalization of the overall coupling in the Born cross section can be carried out in a similar fashion as at one loop, leading to the modified distribution (for )