On the light quark mass effects in Higgs boson production in gluon fusion

Production of Higgs bosons at the LHC is affected by the contribution of light quarks, that mediate the gg \to Hg transition. Although their impact is suppressed by small Yukawa couplings, it is enhanced by large logarithms of the ratio of the Higgs boson mass or its transverse momentum to light quark masses. We study the origin of this enhancement, focusing on the abelian corrections to gg \to Hg amplitudes of the form (C_F alphas L^{2})^n, where $L \in { ln(s/mb^2), ln(p_\perp^2/mb^2) }. We show how these non-Sudakov double logarithmic terms can be resummed to all orders in the strong coupling constant. Interestingly, we find that the transverse momentum dependence of these corrections is very weak due to a peculiar cancellation between different logarithmic terms. Although the abelian part of QCD corrections is not expected to be dominant, it can be used to estimate missing higher-order corrections to light quark contributions to Higgs boson production at the LHC.


Introduction
The Higgs boson discovered at the LHC by ATLAS and CMS collaborations almost four years ago [1,2] is a mysterious particle. Indeed, it seems to fit perfectly into the Standard Model (SM) of particle physics and its mass is numerically close to the weak scale v. However, the mechanism that would tie these two quantities together in a more general theory requires presence of other, relatively light, particles that couple to the Higgs boson. Such particles have not been observed so far and limits on their masses gradually become so tight that the "natural" relation m H ∼ v is endangered. Further exploration of Higgs boson properties, including its couplings and quantum numbers, will be essential for understanding to what extent the observed particle is indeed described by the Standard Model and, hopefully, for discovering clues as to what the mass scale of physics beyond the Standard Model can actually be.
An important observable in Higgs physics is the Higgs boson transverse momentum distribution. There are several reasons for that. On one hand, precise knowledge of the Higgs boson p ⊥ -distribution is important for understanding jet-vetoed cross sections and, more generally, observables subject to experimental constraints. The uncertainties in modeling the p ⊥ -distribution affect values of the Higgs coupling constants extracted from such fiducial quantities. Since the total inclusive Higgs boson production cross section is currently known through next-to-next-to-next-to-leading order in perturbative QCD [3], the uncertainty in the Higgs p ⊥ -distribution may become the dominant one when future experimental data is confronted with theoretical predictions for the Higgs boson production.
Further motivation for the precise description of the Higgs boson p ⊥ -spectrum comes from the observation that the p ⊥ -distribution is, potentially, a good observable for detecting relatively light (m ∼ m H ) colored particles that couple to the Higgs boson [4]. Indeed the contribution of a particle with the mass m ∼ m H to the Higgs boson production in gluon fusion is almost independent of p ⊥ for p ⊥ < m while for p ⊥ > m it rapidly decreases. Thus the p ⊥ -distribution of Higgs bosons or jets recoiling against it, may serve as a sensitive probe of this type of physics beyond the Standard Model.
High-precision theoretical predictions for Higgs boson p ⊥ -distribution within the Standard Model are necesary to pursue this program [5,6]. Unfortunately, despite significant progress in understanding the Higgs p ⊥ -spectrum in recent years, the overall situation is unsatisfactory. The main challenge is to describe the bottom quark contribution to the Higgs boson production in gluon fusion at moderate values of the transverse momentum. Indeed, the gg → H transition in the Standard Model is dominated by the top-quark loop, thanks to the large Higgs-top Yukawa coupling. Since the top quark mass is large compared to the Higgs mass, it is possible to integrate out the top quark and describe the Higgs production at sufficiently low transverse momentum in the effective field theory with a local ggH interaction. This reduces the number of loops in perturbative computations by one and allows us to push them to very high orders in QCD perturbation theory. Within this approximation, the Higgs p ⊥ -distribution has been evaluated through next-to-next-toleading order at high p ⊥ < m t [7,8] and to next-to-next-to-leading logarithmic accuracy at low p ⊥ < m H [9,10]. 1 At the same time understanding the bottom-quark contribution to gg → Hg turned out to be more involved. 2 Indeed, since m b ∼ 4.2 GeV, the point-like approximation for the bottom quark contribution to ggH vertex is only valid for tiny transverse momenta p ⊥ < m b . In a broader and more interesting momentum region p ⊥ > m b , the local vertex approximation for the bottom quark-mediated ggH interaction is invalid and we must deal with the computation of complicated box diagrams with internal masses. Calculation of such diagrams at two and more loops is beyond the reach of existing computational techniques. As the result, the gg → Hg amplitudes for p ⊥ > m b are only known in the leading (one-loop) approximation.
The bottom quark contribution to Higgs boson production is small, compared to the contribution of the top quark. However, it is still relevant phenomenologically because of the high precision of forthcoming experimental measurements of the Higgs-gluon coupling and because the bottom quark contribution is dynamically enhanced. Indeed, although the coupling of the bottom quark to the Higgs boson is small compared to the Higgs-top coupling, the n-loop bottom quark contribution to gg → Hg is enhanced by two powers of large logarithms per one power of For relevant values of the transverse momentum p ⊥ ∼ 30 GeV and the Higgs boson mass m H = 125 GeV, these logarithms can be numerically quite large L 2 ∼ 20 − 50. In fact, the magnitude of the double logarithmic corrections suggests that the all-order resummation may be necessary.
The origin of these logarithmically enhanced terms is currently not well understood. Although their double logarithmic nature suggests a mechanism similar to the Sudakov enhancement [12], as we explain below the mass suppression of the amplitude M gg→Hg ∼ m 2 b makes such an interpretation problematic. Contribution of bottom quarks to the Higgs boson production in gluon fusion was discussed in Refs. [13][14][15] in the context of p ⊥resummation. There it was pointed out that the standard technology of p ⊥ -resummation only applies for p ⊥ < m b , while for larger values of p ⊥ it is incomplete. The authors of Refs. [13][14][15] then used differences between various resummation prescriptions to estimate the uncertainty in the Higgs p ⊥ -distribution, caused by unknown higher-order QCD corrections to the bottom quark contribution.
The goal of this paper is to make a step towards a better understanding of the origin of double logarithmic corrections to the Higgs boson production, their computation in the two-loop approximation and to their resummation. Since these tasks are very challenging, we restrict our analysis to abelian QCD corrections, i.e. corrections associated with the abelian color factor C n F in the n-th order of QCD perturbation theory. Note that the abelian radiative corrections are generated by the coupling of virtual gluons to massive offshell quarks. As a consequence, these corrections are infra-red finite on their own, so that physical results can be obtained without the need to consider processes with additional soft and collinear radiation.
The paper is organized as follows. In the next section we introduce our notation. In Section 3 we describe evaluation of one-loop double logarithmic corrections to the bottom quark contribution to gg → Hg helicity amplitudes. In Section 4 we extend this analysis to two loops. In Section 5 we show how these logarithmic corrections can be resummed to all orders in the strong coupling constant. Numerical estimates of the corrections are given in Section 6. We conclude in Section 7.

Setup and notations
We consider the Higgs boson production in the process gg → Hg mediated by the bottom quark loop. The Higgs boson has a non-vanishing transverse momentum. The particle momenta are assigned in the following way Our goal is to find the double logarithmic contributions to helicity amplitudes in a kinematic situation where the energy of the final state gluon E 3 is much smaller than the energies of the colliding gluons E 1,2 and the Higgs boson mass. At the same time, we consider E 3 to be much larger than the mass of the quark that mediates the gg → H transition. When written in terms of kinematic invariants, these conditions imply where s = (p 1 + p 2 ) 2 , t = (p 1 − p 3 ) 2 , u = (p 2 − p 3 ) 2 and p 2 ⊥ = tu/s is the square of the transverse momentum of the Higgs boson or the gluon in the final state.
To illustrate this kinematic situation further, consider the production of a Higgs boson through a bottom quark loop accompanied by an emission of a soft gluon. We take m b = 4.2 GeV, √ s ≈ m H , p ⊥ ≈ 20 GeV and assume central production (small rapidity), so that with λ ∼ 0.25. We consider λ to be a small parameter and adopt the scaling rules Eq.(2.3).
In the limit λ → 0 the gg → Hg amplitude develops the 1/λ singularity, characteristic to the soft gluon emissions; this allows us to write the perturbative series for the amplitude in the following way In Eq.(2.4), we neglected all terms that are less singular than λ −1 α n s ln 2n λ in the λ → 0 limit. We are interested in the abelian part of the coefficients C n , which determine the double logarithmic approximation for the amplitude. The leading-order coefficient C 1 is well-known and can be extracted from the one-loop result for the gg → Hg amplitude [16]. In what follows, we explain how to obtain this coefficient without following the standard route of a one-loop computation. We then compute the two-loop coefficient C 2 and generalize the result to arbitrary n.
We begin by fixing the notation for helicity amplitudes. There are eight helicity amplitudes that are needed to describe g 1 g 2 → Hg 3 process. However, when the gluon g 3 is soft, the Higgs boson is effectively produced in the collision of two energetic gluons g 1 and g 2 . This can only happen when helicities of these gluons are equal. The constraint λ 1 = λ 2 leaves us with four helicity amplitudes which are pair-wise related by the parity conjugation. We take M +++ and M ++− as the two independent amplitudes that we need to compute.
It is convenient to write the amplitudes in such a way that their spin-helicity structure in the soft limit is factored out, and the remaining part only depends on the Mandelstam invariants of the process Two helicity-dependent form factors A ++± are given by the series in the strong coupling constant where the abelian and non-abelian parts are separated. Our goal is to compute abelian contributions at two loops and beyond.

One-loop helicity amplitudes in the double logarithmic approximation
In this section, we will study the double logarithmic contributions to the one-loop gg → Hg amplitude mediated by a quark of mass m. A well-known example of a double logarithmic enhancement is provided by the Sudakov logarithms [12]. However, the situation with gg → Hg is different. Indeed, as we will show, in contrast to the Sudakov logarithms [12] associated with the radiation of soft virtual gauge bosons by highly energetic on-shell charged particles, the double logarithmic enhancement of the gg → Hg amplitude is caused by a soft quark exchange. Such non-Sudakov double logarithms are typical for amplitudes that are mass-suppressed at high energy [17][18][19]. Since physics of these non-Sudakov double logarithmic corrections is not well-known, we begin by discussing the one-loop case in detail.
In total, there are ten one-loop Feynman diagrams contributing to the leading order gg → Hg amplitude, Fig.1. However, up to differences in color factors that ensure that the final result is proportional to structure constants f abc of the gauge group SU(3), diagrams that differ only by the direction of the quark flow in the loop give identical contributions. The number of relevant diagrams can be further reduced by a judicious choice of gluon polarization vectors. Indeed, each polarization vector can be chosen to satisfy two transversality conditions. It is convenient to require Explicit expressions for polarization vectors satisfying Eq.(3.1) in terms of spinor products are given in Appendix. Emission of a soft gluon g 3 off the gluon or quark line carrying large momentum p i can be described by an effective vertex proportional to p i · ǫ 3 . Thus the condition ǫ 3 · p 2 = 0 ensures that there are no soft gluon emissions off the gluon and quark lines carrying the external momentum p 2 . As the result, only diagrams shown in Fig.1 need to be considered.
To determine the double logarithmic asymptotic behavior of the amplitude we follow the original method of Ref. [12]. We start by evaluating the diagram Fig.1a together with the diagram with the opposite direction of the quark flow. By calculating the trace we find that the diagram is proportional to m b . The mass suppression is caused by the fact that the Higgs-quark interaction flips quark helicity. Since strong interactions conserve helicity in the massless limit, the mass term provides the required second helicity flip in the quark loop. This helicity flip is caused by a mass term of the soft quark propagator (3. 2) It follows from Eq.(3.2) that once the mass term is selected, the soft quark propagator becomes a propagator of a scalar particle, which is sufficiently singular at small momenta to develop a double logarithmic contribution. We note that if the mass term is taken from the quark propagator that carries large momentum, the double logarithmic contribution does not develop because the soft quark propagator 1/l is insufficiently singular. By virtue of a similar argument, the soft loop momentum l can often be neglected in the numerators of contributing diagrams, since we are only interested in the leading logarithmic enhancement. We note that the last feature is not generic (see e.g. the analysis of the diagram Fig.1c below). With all these simplifications it is straightforward to derive contributions of the diagram Fig.1a to the helicity-dependent form factors. They read is the three-point function with two of its legs off-shell. To compute C(s, t, m 2 b ) in the double logarithmic approximation, we follow Ref. [12] and introduce the Sudakov parametrization of the virtual momentum l = αp 1 + βp 2 + l ⊥ . We integrate over the transverse momentum components l ⊥ by taking the residue of the soft quark propagator pole This allows for a symmetric treatment of the soft and collinear parts of the double logarithmic contribution. The two remaining propagators in Eq.(3.4) become To obtain the double logarithmic contribution we require both α and β integrations to be logarithmic. This requirement is automatically satisfied for the integration over α. At the same time the integration over β is logarithmic only for β > |t|/s. Hence, in the double logarithmic approximation Eq.(3.4) reduces to where the intervals |t|/s < β < 1 and m 2 b /s < α < 1 are determined by the effective infrared and ultraviolet cutoffs of the logarithmic integral, and additional kinematic constraint αβ > m 2 b /s ensures that the pole of the soft quark propagator is in the integration domain. It is convenient to factor out the large logarithm L = ln (m 2 b /s) ≈ ln (m 2 b /m 2 H ) and introduce the normalized variables η = ln α/L and ξ = ln β/L. By using Eqs.(3.3,3.4,3.7) we find where τ t = ln m 2 b /|t| /L. Next, we consider the diagram Fig.1b. To compute this diagram in the double logarithmic approximation, we again pick up a mass term from the soft quark in the t-channel and neglect the momenta l and p 3 everywhere in the numerator. Then the contribution of the diagram Fig.1b reduces to . (3.10) We use the same Sudakov parametrization l = αp 1 + βp 2 + l ⊥ as before. Upon inspecting the infrared structure of Eq.(3.10) we find that the double logarithmic contribution can only be obtained when the propagator becomes independent of β. This leads to a constraint β < |t|/s. The logarithmic integration intervals become m 2 b /s < α < 1, m 2 b /s < β < |t|/s and we obtain (3.12) We will now discuss the diagram shown in Fig.1c where the gluon is emitted off the soft quark line. Similar to the previous case we deal here with the box diagram and need to "remove" one of its propagators to obtain the proper (logarithmic) scaling of the integrand. In fact, the underlying box diagram has two non-overlapping momenta regions that lead to a double logarithmic enhancement. These regions are characterized by the choice of the soft momentum in the diagram. Indeed, we can choose the soft momentum l in such a way that the momentum of the emitted gluon p 3 flows through the lower (upper) half of the quark loop Fig.1c in region I (II), respectively. Consider region I and choose the momentum decomposition l = αp 1 + βp 3 + l ⊥ . After omitting irrelevant terms, the quark propagators becomel 14) It follows from Eqs.(3.13,3.14,3.15,3.16) that the double logarithmic contribution can be obtained in two different ways: (i) for α < |u|/s only the p 3 term in the numerator of Eq.(3.15) should be kept (the "scalar" contribution) and (ii) for α > |u|/s the αp 1 term should be taken from the numerator in Eq. (3.15) to cancel an extra power of α that appears in the denominator of Eq.(3.16) in this limit (the "vector" contribution). 3 These two contributions are proportional to 17) and Note that the vector integral has the usual tensor structure of a color-dipole emission which gives A +++ = −A ++− , similar to all other diagrams. At the same time, the tensor structure of the scalar integral corresponds to the three-gluon configuration described by a local gauge invariant operator which does not contribute to the all-plus helicity amplitude. By crossing symmetry, the scalar contributions of the momentum regions I and II are equal. Therefore, the total scalar contribution of the diagram Fig.1c can be written in terms of the double logarithmic integral over the interval m 2 b /|t| < α < |u|/s, m 2 b /|t| < β < 1 that originates from region I. This gives whereas A (0),1c,s +++ = 0. At the same time, the vector contribution of region II vanishes due to our choice of the polarization vector for the gluon g 3 , p 2 · ǫ 3 = 0. As the result, the total vector contribution of the diagram Fig.1c is given by the double logarithmic integral over the interval |u|/s < α < 1, m 2 b /|t| < β < 1 from region I. It reads We are now in position to present the leading-order bottom-quark contribution to gg → Hg helicity amplitudes in the double logarithmic approximation. We sum the contributions of individual diagrams given in Eqs.(3.8,3.12,3.21,3.22) and obtain A (0) where we used τ = ln(m 2 b /p 2 ⊥ )/L. These results coincide with the double logarithmic limits of the one-loop amplitudes computed in Ref. [16] long time ago. 4 Our analysis identifies the origin of the double logarithmic enhancement of the gg → Hg amplitude mediated by a light quark. With this understanding, it is straightforward to extend the above calculation first to two loops and then to all orders in the strong coupling constant α s . We will describe how to do this in the next sections. The abelian part of the two-loop correction originates from a soft gluon exchange between virtual bottom quarks. The relevant two-loop Feynman diagrams are shown in Fig.2. We note that the one-loop correction to qqH vertex appears as a subdiagram in many two-loop diagrams in Fig.2. This correction develops a double logarithmic enhancement, so that the (properly normalized) qqH vertex in the one-loop approximation reads [12] V qqH = 1 + δV qqH , In Eq.(4.1), q 1 and q 2 are the momenta of the off-shell quark lines and we assume that m 2 b ≪ q 2 1 , q 2 2 ≪ q 1 ·q 2 . This expression and the one-loop analysis of the previous section can be used to easily compute the leading logarithmic part of the relevant two-loop diagrams.
We begin with the diagram Fig.2a. The external momenta of the vertex subgraph in this case are q 1 = p 1 − l and q 2 = p 2 + l, where l is the soft momentum of the quark loop. For l = αp 1 + βp 2 + l ⊥ we get q 2 1 = sβ, q 2 2 = sα, 2q 1 · q 2 ≈ s, so that where x = C F α s L 2 /2π and η, ξ and L are defined in the previous section. The double logarithmic integration over the soft quark momentum is the same as for the diagram Fig.1a and the correction to the helicity amplitudes is obtained by including δV 2a qqH factor into the integrand of the one-loop expression Eq. The diagram in Fig.2b is computed in a similar way. Virtualities of the quark lines become q 2 1 ≈ |t| and q 2 2 ∼ sα, and the one-loop vertex reads Substituting this result into Eq.(3.12), we obtain . (4.5) The double logarithmic contribution of the diagram Fig.2c is generated when the quark propagator between the emission vertex of the soft gluon g 3 and the qqH vertex becomes independent of the soft photon loop momenta. In this case the inner loop reduces to the one-loop vertex with an additional restriction on the integration region. We find The corresponding two-loop corrections to the amplitudes read (4.7) To compute the diagram Fig.2d in the double logarithmic approximation, we again insert the expression for the qqH vertex Eq.(4.1) into the one-loop diagram in Fig.1c. As was explained in the previous section this diagram receives the double logarithmic contributions from two independent momentum regions. In region I we parametrize the soft momentum as l = αp 1 + βp 3 + l ⊥ and find ln |t|β s ln α α > |u| s . (4.8) We note that the two integration regions, α < |u|/s and α > |u|/s, correspond to scalar and vector contributions, respectively. The expression for the qqH vertex in region II can be found in the same way. We insert these results into Eqs. (3.21,3.22) and obtain . (4.10) We note that the corresponding one-loop expression given in Eq.(3.21) includes equal scalar contributions from regions I and II. At two loops, contributions of regions I and II are not equal anymore; we separate them in Eq.(4.9) and indicate contribution of the region I by the corresponding integral and the contribution of region II by the t ↔ u symmetric term. The vector contribution Eq.(4.10) comes entirely from region I, as in the one-loop case. Diagrams shown in Fig.2e and Fig.2f are related by crossing symmetry and we only consider the evaluation of the former. This diagram receives the scalar contribution from region I; the double logarithmic term is generated when the propagator between the emission vertex of the gluon g 2 and the qqH annihilation vertex becomes independent on the soft momenta. Thus as in the case of the diagram Fig.2c the inner loop reduces to the one-loop vertex integral with an additional restriction on the integration region. The effective vertex in this case reads Since the scalar contribution to the all-plus helicity amplitude vanishes, we obtain A (1A),2e The result for the diagram in Fig.2f is also given by Eq.(4.12) since it is symmetric with respect to the replacement t ↔ u.
Taking the sum of all the individual contributions in Eqs.(4.3,4.5,4.7,4.9,4.10,4.12), we obtain the two-loop correction to the gg → Hg amplitude in the double logarithmic approximation where the new variable ζ = ln(u/t)/L parametrizes the dependence of the amplitudes on the soft gluon rapidity.

Resummation of the abelian double logarithmic contributions
The perturbative expansion parameter for the double logarithmic corrections x = C F αs 2π L 2 is not small numerically, x ∼ 1. For this reason, resummation of such corrections might be relevant. This problem is also quite interesting theoretically, since very little is known about the all-order structure of the power-suppressed non-Sudakov logarithms. Indeed, on the one hand, only few examples of the resummation of non-Sudakov double logarithmic corrections are known so far [17][18][19] and, on the other hand, systematic renormalization group analysis of the high-energy behavior of the on-shell amplitudes beyond the leadingpower approximation is still elusive for existing effective field theory methods.
The problem that we discuss in this paper is, however, simpler than the general case. As we pointed out already, to leading order in m b , higher-order double logarithmic corrections to the helicity amplitudes are caused by multiple soft virtual gluon exchanges and a single soft quark exchange. Thus we have to consider Feynman diagrams similar to Fig.2 but with multiple soft gluon exchanges between different quark lines. For the abelian part of the corrections we can use simple factorization properties of soft emissions in QED. It is well-known that in this case, upon summing over all relevant diagrams, integrations over soft gluon momenta factorize and the all-order result is given by the exponent of the single gluon contribution, given by the O(α s ) term in Eq.(4.1).
By using the expression Eq.(4.2) specific for the diagram Fig.2a we find the Sudakov exponent to be e −xξη . The all-order double logarithmic corrections to helicity amplitudes are then obtained by including this exponent into the integrand of Eq.(3.8). Upon integration over ξ, we obtain the resummed expression for helicity amplitudes in the form of the oneparameter integral The multiple gluon exchange diagrams related to Fig.2b and Fig.2c must be considered simultaneously. 5 After summing over all possible permutations of the soft gluon emission vertices, their contributions factorize and produce a product of the exponents of the oneloop contributions (4.4, 4.6). They combine into the exponential factor e −xξη identical to the previous case. By including it into the integrand of the one-loop expression Eq. (3.12) we get the all-order result For the diagram Fig.2d the Sudakov factor depends on whether a vector or a scalar contribution is considered, cf. Eq. (4.8). For the vector part, the Sudakov exponent is e −x(ξ+1−τt)η and the all-order result associated with the leading-order contribution Eq.(3.22) reads As in the case of the diagrams Fig.2b and Fig.2c, the scalar contributions from the momentum region I of the diagram Fig.2d combine with the diagram Fig.2e, exponentiate and produce a Sudakov factor e −x(1−τt−τu+τtτu+ηξ) . The Sudakov exponent of the scalar contribution from region II of the diagrams that combine Fig.2d and Fig.2e with additional soft exchanges is obtained by the replacement t ↔ u. The sum of these contributions is, therefore, given by 6 Double logarithmic corrections to the differential cross section We are now in position to estimate the effect of the corrections, computed in the previous section, on the differential cross section of the Higgs boson production in association with a jet. The total amplitude of this process is given by the sum of top and bottom contributions since contributions of lighter quarks are negligible. We therefore write Thanks to its large Yukawa coupling, the top quark provides the dominant contribution to the scattering amplitude. In the soft limit the real emission from inside the top-quark loop is power-suppressed i.e. the soft emission factorizes with respect to the gg → H amplitude. The result for this contribution is well known and in the limit of an infinitely heavy top quark reads There are O(α s ) corrections to this formula that, however, are not essential for us. The largest effect of the bottom quark on the differential cross section is caused by its interference with the top quark contribution. We find where dσ (0) gg→Hg is the top quark mediated cross section, and we neglect the finite top mass effects in the interference term. Note that since the leading bottom quark effect is due to the interference with the top quark mediated amplitude, to leading order in 1/m t , any additional real emission contribution involves the three-gluon interaction and does not contribute to the abelian part of the correction.
We can now use the result derived in the previous section for numerical estimates. It is convenient to express the correction to the cross section through the variables τ = ln(m 2 b /p 2 ⊥ )/L and ζ = ln(u/t)/L, which parameterize the dependence of the cross section on the transverse momentum and rapidity. We obtain where ellipsis stands for terms suppressed by higher powers of x.
We can use the result Eq.(6.6) to estimate the impact of the QCD corrections to bottom quark contributions to gg → Hg on the Higgs boson transverse momentum distribution. In principle, we should convolute the partonic cross section Eq.(6.4) with the parton distribution functions. However, we will now argue that, given the structure of the corrections shown in Eq.(6.6), this is not necessary. Indeed, within the accuracy of our approximation can be considered independent of the partonic center-of-mass energy. In addition, series in Eq.(6.6) shows very weak dependence on the rapidity of the soft gluon. Indeed, the function f in Eq.(6.6) does not depend on the gluon rapidity up to O(x). Moreover, at O(x 2 ) the rapidity-dependent part of the coefficient includes only high powers of τ . If the soft gluon is emitted at large rapidity, |ζ| ≈ 1 and τ ≪ 1. On the contrary, central emission with the large transverse momentum implies |ζ| ≪ 1 and τ ≫ 1. Therefore, the the rapidity-dependent term is small everywhere and can be neglected. After these modifications the function f depends only on the transverse momentum of the emitted gluon or the Higgs boson. As a result it remains unaffected by the integration over parton distribution functions if the transverse momentum of the Higgs boson is kept fixed. Therefore, we can write dσ pp→H+j dp 2 where L eff = ln(m 2 H /m 2 b ) and x eff = αsC F 2π L 2 eff . We emphasize that Eq.(6.7) only applies to the contribution of gg partonic channel to the production of the Higgs boson in proton collisions and that only abelian corrections are taken into account there.
We note that the series in Eq.(6.7) has peculiar structure. Indeed, the one-loop double logarithmic correction to dσ/dp ⊥ is independent on p ⊥ , thanks to a cancellation between p ⊥dependent contributions to individual helicity amplitudes Eq.(3.23), when the differential cross section is evaluated [15]. In principle, it could have been possible to interpret this result as an indication that the naive factorization of soft emissions extends to a region beyond p ⊥ > m b , at least inasmuch as the interference with the top quark loop is concerned. However, our result Eq.(6.7) shows that such an interpretation does not hold and that the cancellation of p ⊥ -dependent double logarithmic corrections does not persist beyond oneloop. In fact, starting from three loops, the double logarithmic corrections to the differential cross section start to depend on the rapidity of the emitted gluon as Eq.(6.6) shows.
To understand numerical impact of these corrections, we use m H = 125 GeV, m b = 4.2 GeV, α s = 0.12 and consider p ⊥ in the range m b < p ⊥ < 50 GeV. We note that the one-loop double logarithmic corrections reduce the cross section by about 16%. This is somewhat larger than the result of the full computation, but still in the right ballpark. The two-loop correction increases the result by about 1.5%. This is somewhat smaller than the next-to-leading order effect in gg → H cross section but, given the fact that we only consider the abelian contribution here, the two results are not inconsistent. 6 However, our main interest is in p ⊥ -dependent corrections and these corrections turn out to be quite small. In fact, the two-loop correction in Eq.(6.7) decreases by just about 0.2% when the transverse momentum varies from p ⊥ ∼ m b to p ⊥ ∼ 50 GeV. This tiny change is the result of a strong cancellation between τ 3 and τ 4 term in Eq.(6.7). When taken separately, these terms could have caused a change in the two-loop result that is closer to one percent. The three-loop correction in Eq.(6.7) changes the prediction by about −0.1% and its p ⊥ -dependent part is one order of magnitude smaller.

Conclusion
In this paper, we have studied the bottom-quark loop contribution to the production of the Higgs boson in association with a jet in gluon fusion in the double logarithmic approximation. This contribution is suppressed by the ratio of the bottom-quark mass to the Higgs boson mass but, at the same time, it is enhanced by two powers of large logarithms, ln(s/m 2 b ) or ln(p 2 ⊥ /m 2 b ), per one power of the strong coupling constant. As it is repeatedly emphasized in the literature, these terms may be important for phenomenoly, in particular for the description of the Higgs boson transverse momentum distribution in an interesting kinematic region m b < p ⊥ < m H . We have analyzed the abelian part of the double logarithmic corrections and computed the gg → Hg helicity amplitudes which incorporate these terms to all orders in α s .
Numerically, the abelian corrections appear to be moderate. For example, the two-loop corrections change the transverse momentum distribution by about two percent. However it is important to note that the p ⊥ -dependent part of these corrections is only about 0.2% due to the cancellation between different p ⊥ -dependent terms. Assuming that, up to an obvious change in the color factor C F → C A , the non-abelian corrections will be similar to the abelian ones, we estimate the yet unknown non-abelian corrections to be about three times larger. We conclude that the description of the Higgs boson transverse momentum distribution with a few percent precision requires a calculation of the O(α s ) logarithmically enhanced non-abelian corrections to bottom quark contribution while the all-order resummation is, probably, not important. Our analysis sets up a framework for such a calculation. A new element in the calculation of the non-abelian part is its infra-red sensitivity and a related need to account for the contribution of soft radiation.