# Resummation of soft and Coulomb corrections for production at the LHC

###### Abstract

In this paper, a combined resummation of soft and Coulomb corrections is performed for the associated production of the Higgs boson with a top quark pair at the LHC. We illustrate the similarities and critical differences between this process and the production process. We show that up to the next-to-leading power, the total cross section for production admits a similar factorization formula in the threshold limit as that for production. This fact, however, is not expected to hold at higher powers. Based on the factorization formula, we perform the resummation at the improved next-to-leading logarithmic accuracy, and match to the next-to-leading order result. This allows us to give NLL+NLO predictions for the total cross sections at the LHC. We find that the resummation effects enhance the NLO cross sections by about 6%, and significantly reduce the scale dependence of the theoretical predictions.

## 1 Introduction

After the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 Aad:2012tfa (); Chatrchyan:2012xdj (), a main task of particle physics is to investigate its properties. One particularly important property is the Yukawa coupling between the top quark and the Higgs boson, which is crucial to understand the origin of the large top quark mass. Precise knowledge of the top quark Yukawa coupling will also help us to constrain new physics effects in other couplings such as the Higgs boson self-coupling Goertz:2014qta (); Azatov:2015oxa (); Cao:2015oaa (); Shen:2015pha (). At the LHC, the top-Higgs Yukawa coupling can be probed by measuring the cross section for Higgs boson production in association with a top quark pair ( production). The observation of this process has been established very recently by the ATLAS Aaboud:2018urx () and CMS Sirunyan:2018hoz () collaborations. The measured cross sections are in agreement with theoretical predictions, although there are still large experimental uncertainties due to limited statistics. In the future, more precise measurements will be carried out at the upgraded LHC and the High Luminosity phase of the LHC (HL-LHC) Apollinari:2015bam (). For this reason, it is necessary to improve the theoretical understanding of this process.

At the leading order (LO), production can be initialized by or from the colliding protons, which has been studied many years ago Ng:1983jm (); Kunszt:1984ri (); Dicus:1988cx (). The next-to-leading order (NLO) quantum chromodynamics (QCD) corrections were calculated in Beenakker:2001rj (); Reina:2001bc (); Reina:2001sf (); Beenakker:2002nc (); Dawson:2002tg (); Dawson:2003zu (), while the NLO electroweak contributions were calculated in Yu:2014cka (); Frixione:2014qaa (); Frixione:2015zaa (). Given the complicated structure of this scattering process, it will be very challenging to perform an exact calculation of the next-to-next-to-leading order corrections. Therefore, a lot of efforts have been devoted to approximations of the higher order QCD corrections in various kinematic limits Kulesza:2015vda (); Broggio:2015lya (); Broggio:2016lfj (); Kulesza:2017ukk (). The derivation of an approximation often involves a factorization formula, which can be used to resum a class of large corrections to all orders in perturbation theory. For example, Refs. Broggio:2015lya (); Broggio:2016lfj (); Kulesza:2017ukk () have investigated the limit , where is the partonic center-of-mass energy and is the invariant mass of the system. In this limit, large logarithmic corrections are present at each order in perturbation theory. These corrections are resummed to all orders in the strong coupling up to the next-to-next-to-leading logarithmic (NNLL) accuracy Broggio:2016lfj (); Kulesza:2017ukk (). In this paper, we consider a different kinematic limit , where and are the masses of the top quark and the Higgs boson, respectively.

The limit under consideration is a bit different from the limit taken in Broggio:2016lfj (); Kulesza:2017ukk (). There are power-like corrections arising from exchanges of Coulomb gluons, besides logarithmic corrections coming from soft gluon emissions. This limit has been studied in Kulesza:2015vda (), where the soft gluon contributions are resummed, but the Coulomb gluon contributions are only incorporated at fixed-order. In this paper, we will derive a factorization formula which can resum simultaneously both kinds of higher-order corrections. The framework presented in this paper closely resembles that in production Beneke:2009ye (); Beneke:2010da (); Beneke:2011mq (). However, it should be emphasized that production is more complicated than production. In particular, in the threshold limit, the pair has a non-vanishing transverse momentum given by the recoil against the extra Higgs boson. This will lead to more involved interplay between the soft and Coulomb gluons, as will be clear later. The derivation of the factorization formula therefore requires new analyses other than those in Beneke:2009ye (); Beneke:2010da (). It is also much more difficult to calculate the hard function describing the contributions from hard gluons with typical momentum scale around .

The paper is organized as follows. In Sec. 2, we use the analytic form of the LO partonic cross sections to analyze their behavior in the threshold limit. In Sec. 3, we present the derivation of the factorization and resummation formulas using effective field theory methods. In Sec. 4, we show the numeric results based on our resummation formula. We conclude in Sec. 5.

## 2 Analyses of leading order results

The total cross section for inclusive production at hadron colliders can be expressed as Collins:1989gx ()

(1) |

where the sums are over all the partons within the colliding hadrons, i.e, ; is the center-of-mass energy of the collider; and are the top quark mass and the Higgs mass, respectively; is the factorization scale; and

(2) |

The above factorization formula involves the partonic cross section and the effective parton luminosity function . The definition of the latter is

(3) |

where is the non-perturbative parton distribution function (PDF) of the parton in the hadron . It is universal and can be extracted from experimental data. The partonic cross section can be calculated in perturbative QCD. In this work, we are interested in its behavior near the threshold limit , or . Here is the partonic center-of-mass energy defined by . To analyze this region, we define . The parameter goes to zero in the threshold limit, and represents the typical momenta of final state particles. To see this, consider the energy conservation condition in the partonic center-of-mass frame

(4) |

where and are the 3-momenta of the (anti-)top quark and the Higgs boson, respectively, and is the total energy of other emitted particles in the final state. In the limit , the 3-momenta of the (anti-)top quark and the Higgs boson becomes much smaller than their rest mass. The right side of the above equation can then be expanded in the small momenta and we obtain the following relation:

(5) |

We therefore have the power-counting

(6) |

This will be important for establishing the effective field theory description later in the next section. For the moment, we are going to investigate the behavior of the partonic cross sections in the limit .

The Born-level partonic cross section can be written as

(7) |

To the first order in , one can approximate the -functions in the above formula as

(8) |

and the squared-amplitudes can be expanded according to the counting in eq. (6). The integrals over the 3-momenta can then be carried out, and we arrive at approximate expressions of the partonic cross sections in the following form

(9) |

where is the vacuum expectation value of the Higgs field and is the strong coupling constant.

A crucial feature of eq. (9) is that the leading order partonic cross sections are proportional to in the threshold limit. This should be contrasted to the case of a similar process, production, where the threshold behavior is given by Czakon:2008cx (),

(10) |

That is, the Born partonic cross sections are linear in in the threshold limit. The different behaviors imply that the threshold region is less important in the case of production than for production. In order to study these different behaviors more precisely, we will numerically compare the approximate results to the exact ones in the following. For our numerical computations, we set , and Tanabashi:2018oca (). The strong coupling constant is evolved from the initial condition to the renormalization scale , where is the mass of the boson.

In Fig. 1, we numerically compare the approximate Born partonic cross sections for production with the exact ones, i.e, . The approximate results are obtained from Eq. (9). To calculate , we first employ the programs FeynArts Hahn:2000kx () and FeynCalc Shtabovenko:2016sxi (); Mertig:1990an () to generate the transition amplitudes and then use the Cuba library Hahn:2004fe (); Hahn:2014fua () to perform the phase space integration. Our numerical results are checked against the automatic program MadGraph5_aMC@NLO (MG5) Alwall:2014hca (). As shown in Fig. 1, the approximate and exact partonic cross sections approach each other as becomes small. They are both highly suppressed in the threshold limit, as can be expected. When grows larger, the approximate results start to overestimate the exact ones, indicating that there are important negative power corrections to the approximate formula.

In Fig. 2, we show a similar comparison for the production process. The exact results are obtained from Moch:2008qy (), while the approximate results are computed using Eq. (10). The first impression is that the small- region is less suppressed compared to the case, which is clear from the vs. behaviors. This means that the reliability of small- resummation can be quite different in these two processes, a fact not often mentioned in the literatures. Beside this, here we also observe that the approximate result in the channel overestimates the exact one as grows. Interestingly, for the subprocess, the approximate result underestimates the exact one for most values of , contrary to the case shown in Fig. 1. It is therefore possible that, for the sum of the two channels, the approximate result stays closer to the exact one than in the case of individual channels. This should be investigated at the hadron level as we are going to do below.

The partonic cross sections need to be convoluted with the parton distribution functions (PDFs) to arrive at the hadronic cross sections. It is therefore interesting to see how the approximate and exact results compare at the hadron level. We show in Fig. 3 the hadronic differential cross sections for production. From the left plot, we see again that the approximate results in both the and the channels overshoot a lot over the exact ones for large . It is also clear that the small- region is highly suppressed due to the behavior. In the right plot, we show the ratio between the approximate results and the exact ones as a function of . One can see that in the small- region, the approximate results are in good agreements with the exact ones. As goes above , the approximation quickly fails. On the other hand, we show in Fig. 4 the results for production. Again, we find that the small- region is more important in this case compared to production due to the behavior. Besides, one can also see the accidental cancellation between the power corrections in the and channel mentioned in the last paragraph. The above two facts lead to the observation that the small- expansion provides a reasonable approximation to the hadronic total cross section for production Bonciani:1998vc (); Moch:2008qy (); Beneke:2009ye (); Beneke:2010da (); Beneke:2011mq (); Piclum:2018ndt (). This is clearly not the case for production.

While the above analyses show that the small- region for production is not as important as that for production, it is still interesting to study the small- behavior of the cross section at higher orders in QCD. First of all, theoretically, production is similar but slightly different from production. The pair is recoiled against the Higgs boson and therefore has a non-vanishing transverse momentum already at the lowest order in QCD. The interplay between the soft gluons and the Coulomb gluons can therefore be a bit different from the case of production. This poses a question of how to properly factorize these contributions to all orders in perturbation theory, which was not addressed in the literature. Secondly, this process is closely related to the process at future electron-positron colliders, which receives important QED corrections, especially in the threshold region Denner:2003zp (). The investigation of soft and Coulomb gluons in the process can therefore be applied straightforwardly to the soft and Coulomb photons in the process. Finally, while the small- limit is not significant for the total cross section, it could be important if one specifically wants to study certain kinematic configurations sensitive to the threshold region by, e.g., vetoing additional jets.

In this work, we are going to study the threshold region by applying a cut on the variable. While this is not a physical cut (since cannot be measured), it simplifies the theoretical considerations. We define the following quantity Moch:2008qy ()

(11) |

Apparently, for sufficiently small , should be well approximated by the leading power expression of the threshold expansion. It is also obvious that as , approaches the total cross section defined in Eq. (1). The constrained cross section is the main object we are going to study in the rest of the paper. Before entering the technical details, we first investigate its leading order behavior against the variation of .

We again choose to work with the LHC. We take the exact results and the approximate results for the leading order partonic cross sections and , and plug them into Eq. (11). We denote the results as and , and plot their ratio in Fig. 5 as a function of . From the figure, we find that up to , provides a rather good approximation to the exact result. This fact should be kept in mind when we later combine the small- resummation and the fixed-order calculation.

At higher orders in perturbation theory, exchanges of Coulomb gluons and soft gluons lead to threshold-enhanced terms such as and . For small , these terms represent the dominant contributions to the constrained cross section . The rest of the paper will be devoted to deriving a factorization formula for the partonic cross section in the threshold limit, and resumming these enhanced terms to all orders in perturbation theory.

## 3 Factorization and resummation in the threshold limit

### 3.1 Higher order QCD corrections in the threshold limit

Beyond the Born level, the cross sections receive contributions from exchange of virtual gluons and emission of real gluons. We will investigate these contributions as a power expansion in in the threshold limit . In this limit, there will be -enhanced terms and -enhanced terms at higher orders in . Schematically, we are going to consider corrections of the form

(12) |

The collection of these terms are referred to as the improved next-to-leading logarithmic (NLL) corrections. Note that due to the presence of two kinds of terms, one needs to insist on a consistent logarithmic counting for both of them, which we take as . Using this counting, it can be seen that the NLL corrections include terms up to order . It is also clear from this counting that one needs to include formally next-to-leading power (NLP) terms besides the leading power (LP) ones in the power expansion. This greatly complicates the analysis of factorization, as will be clear below.

The behavior of higher order corrections in the threshold limit can be studied using the method of regions Beneke:1997zp (); Jantzen:2011nz (). We work in the partonic center-of-mass frame where the momenta of the two incoming partons are given by

(13) |

where and are two light-like vectors satisfying and . For a given momentum , we perform the light-cone decomposition as

(14) |

with and . We identify the following momentum regions relevant to our problem:

(15) |

These serve as the basis for constructing the effective field theoretic description of the process, and for deriving the factorization formula for the cross sections. At this point, it should be noted that there is a subtle difference between production here and production discussed in Beneke:2009ye (); Beneke:2009rj (); Beneke:2010da (); Beneke:2011mq (); Piclum:2018ndt (). In production, the 3-momentum of the pair is of the ultrasoft scale . This means that the rest frame is formally equivalent to the partonic center-of-mass frame. On the other hand, in production, the pair is recoiled by the Higgs boson and has a 3-momentum of the potential scale . Therefore, an ultrasoft mode in the partonic center-of-mass frame will become a potential mode in the rest frame. The impact of this difference on the factorization and resummation will be discussed in this section.

### 3.2 Effective field theories

In order to derive the factorization and resummation formulas in the threshold limit, it is useful to employ the language of effective field theories (EFTs). According to the momentum regions in Eq. (15), the relevant EFTs are the soft-collinear effective theory (SCET) and the non-relativistic QCD (NRQCD).

SCET Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt (); Beneke:2002ph (); Beneke:2002ni () describes the interactions among collinear, anticollinear and ultrasoft modes. At leading power and next-to-leading power in , the effective Lagrangians are given by

(16) | ||||

(17) | ||||

(18) | ||||

(19) |

where and denote the collinear and ultrasoft quark fields; and represent the collinear (ultrasoft) gluon fields, with their field strength tensors; is the collinear Wilson line.

To describe the interactions among the potential, soft and ultrasoft modes, we employ the potential non-relativistic QCD (pNRQCD) Pineda:1997bj (); Brambilla:1999xf (); Beneke:1999zr (); Beneke:1999qg (). The leading power and next-to-leading power effective Lagrangians can be written as Beneke:1999zr (); Beneke:1999qg (); Kniehl:2002br ()

(20) | ||||

(21) | ||||

(22) |

where and are Pauli spinor fields annihilating the top quark and creating the anti-top quark, respectively; are the chromoelectric components of the ultrasoft field strength tensor. The coefficient was calculated in Fischler:1977yf (); Billoire:1979ih () and is given by . The one-loop coefficient of the QCD -function is given in the Appendix. Note that in the pNRQCD power counting, and in Eq. (20) and (22) are considered as order . Worthy of particular attention is that in Refs. Beneke:1999zr (); Beneke:1999qg (); Kniehl:2002br (), the pNRQCD Lagrangian is derived in the rest frame of the quarkonium, where the heavy quark pair is recoiled by ultrasoft momenta. This is different from our case of production, where the top quark pair is recoiled by the Higgs boson. However, since the LO and NLO potentials in Eqs. (20)-(22) only involve the relative momentum between the heavy quark and anti-quark, in our case the LP and NLP Lagrangians take the same form. It should be stressed that this fact is not expected to hold beyond NLP. For example, as shown in Appendix B, new structures depending on the recoil momentum appear in the NNLP Lagrangian.

To derive the factorization formula, we first match the QCD amplitudes onto an effective Hamiltonian constructed out of the SCET and pNRQCD fields. Generically, we write

(23) |

where labels different color structures and for Lorentz structures, and are leading power and next-to-leading power effective operators describing the scattering process under consideration, while and are their Wilson coefficients arising from the hard region contributions. Note that the NLP effective Hamiltonian actually does not contribute to the cross section at next-to-leading power. The reason is that such a contribution would be given by the interferences between and , which vanish due to angular momentum conservation.

To obtain the Wilson coefficients , we need to calculate on-shell scattering amplitudes in the limit using both QCD and the effective Hamiltonian. In this limit, the loop integrals in the effective theories are scaleless and vanish in dimensional regularization. Therefore, we only need to calculate the QCD amplitudes up to NLO. For this calculation we employ the program packages FeynArts Hahn:2000kx (), FeynCalc Mertig:1990an (); Shtabovenko:2016sxi () and FIRE5 Smirnov:2014hma () to generate the amplitudes and perform the reduction to master integrals. The resulting master integrals can be evaluated to analytic expressions using Package-X Patel:2015tea (); Patel:2016fam (). The interface connecting Package-X, FIRE5 and FeynCalc is provided by FeynHelpers Shtabovenko:2016whf (). We renormalize the top quark mass in the on-shell scheme, and the strong coupling in the scheme. The Wilson coefficients can then be extracted after renormalizing the effective operators (which is equivalent to subtracting the infrared poles from the QCD amplitudes). For the purpose of this paper, we don’t need the explicit forms of individual Wilson coefficients and effective operators, but the combinations of them entering the cross section for production. These combinations will be given in the next subsection as the “hard functions”.

### 3.3 Factorization at leading and next-to-leading power

The discussion in the last subsection tells us that for the cross section up to NLP, we only need to consider the amplitudes of . At leading power, we use together with the LP Lagrangians and to calculate the cross section

(24) |

where labels the initial state, and represents phase-space integration over the momentum of the final state particle . In the LP Lagrangians (16) and (20), the interactions of the ultrasoft gluon with the collinear fields and heavy quark fields are encoded in the covariant derivative . Such interactions can be removed by the decoupling transformations Bauer:2001yt (); Beneke:2010da ()

(25) |

where and are ultrasoft Wilson lines in the fundamental representation along the directions implied by the subscripts, while are ultrasoft Wilson lines in the adjoint representation.

After the decoupling, the factorization of the LP cross section follows along the same line of arguments as the production Beneke:2010da (). The only difference comes from the appearance of the Higgs momentum . The factorization formula therefore reads

(26) |

where

(27) |

The potential function is given by

(28) |

where and are the projectors to the singlet-octet color states of the system, which are given by

(29) |

Note that since the interactions in the LP Lagrangian are spin-independent, the two fields in the definition of the potential function share the same polarization index , and similar for the two fields. It should be stressed that the potential function here is different from that in production, due to the presence of the recoil momentum .

The soft function in Eq. (26) is defined as

(30) |

where the color basis for the channel is given by

(31) |

and that for the channel is

(32) |

where and . This soft function is the same as that for production in Beneke:2010da (), since it does not feel the presence of the recoil momentum. It is diagonal in the color basis we have chosen. In practice, it is more convenient to quote its Laplace transform

(33) |

where

(34) |

Up to the NLO, the result reads Beneke:2010da ()

(35) |

where for , for , for (singlet), and for (octet). Note that the soft function actually does not depend on the index .

Finally, the hard function is defined as the product of LP Wilson coefficients projected onto the -wave spin structure determined by the potential function. In general, the hard function is not diagonal. However, since the soft function is diagonal, only the diagonal entries of the hard function contribute to the cross section. Furthermore, since the soft function does not depend on the index , we can project the diagonal entries of the hard function onto the singlet-octet basis as

(36) |

We have calculated these entries at NLO explicitly using the method described in the last subsection, and write the result as

(37) |

where , and the LO coefficients are given by

(38) |

The one-loop hard anomalous dimensions are given by

(39) |

The analytic expressions for the NLO coefficients are too tedious to be shown. To get some impression of their sizes, we quote the numeric values here with and :

(40) |

To achieve NLL accuracy for the resummation, we need to further consider next-to-leading power corrections to the cross section. These amount to contributions from the NLP interactions in and to the squared-amplitude in Eq. (24). In analogy to the arguments in Beneke:2009ye (); Beneke:2010da () for production, it can be shown that single insertions of give vanishing results due to angular momentum conservation, while does not contribute due to baryon number conservation. The terms in involve subleading potentials between the top and anti-top quarks. These contributions can be incorporated by upgrading the potential function to the NLO, which we will discuss in the next subsection. Finally, we need to consider the corrections induced by .

The terms in involve extra interactions between ultrasoft gluons and potential modes, which are not removed by the decoupling transform (25). In Beneke:2010da (), it was proved that these interactions do not contribute to the cross section at NLP. However, the arguments there rely on the fact that the partonic center-of-mass frame and the rest frame are the same, and therefore the potential function does not depend on an external 3-momentum. However, for production, the recoil momentum from the extra Higgs boson spoils the proof, and we need to reinvestigate the contributions from here.

We begin with an explicit diagram depicted in Fig. 6. Its contribution to the partonic cross section in the threshold limit can be written as (up to overall factors due to coupling constants, color factors, Wilson coefficients, etc.)

(41) |

where is given by

(42) |

where is defined in Eq. (27),

(43) |

and

(44) |

Note that we have suppressed the imaginary part in the propagators. We now observe that the last factor in the above expression does not depend on , , and , while the other factors do not depend on . Together with the fact that , we can conclude that after integrating over , , and , the function must be proportional to (multiplied by a function of and other scalar quantities). As a result, after performing the integration over as in Eq. (41), the contribution of this diagram to the partonic total cross section must vanish.

The argument above can be generalized to all contributions from a single insertion of in a more formal way. The cross section induced by can be written as

(45) |

where denotes time-ordered product. We can perform the usual decoupling transforms (25) to remove the leading power interaction between ultrasoft and potential modes. The remaining interaction is of the form from . As a result, we can write the cross section as