Subleading Power Rapidity Divergences andPower Corrections for q_{T}

Subleading Power Rapidity Divergences and
Power Corrections for

Markus A. Ebert, ebert@mit.edu Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA    Ian Moult, ianmoult@lbl.gov Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA    Iain W. Stewart, iains@mit.edu Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA    Frank J. Tackmann, frank.tackmann@desy.de Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA    Gherardo Vita, vita@mit.edu Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA    and Hua Xing Zhu zhuhx@zju.edu.cn Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Abstract

A number of important observables exhibit logarithms in their perturbative description that are induced by emissions at widely separated rapidities. These include transverse-momentum () logarithms, logarithms involving heavy-quark or electroweak gauge boson masses, and small- logarithms. In this paper, we initiate the study of rapidity logarithms, and the associated rapidity divergences, at subleading order in the power expansion. This is accomplished using the soft collinear effective theory (SCET). We discuss the structure of subleading-power rapidity divergences and how to consistently regulate them. We introduce a new pure rapidity regulator and a corresponding -like scheme, which handles rapidity divergences while maintaining the homogeneity of the power expansion. We find that power-law rapidity divergences appear at subleading power, which give rise to derivatives of parton distribution functions. As a concrete example, we consider the spectrum for color-singlet production, for which we compute the complete suppressed power corrections at , including both logarithmic and nonlogarithmic terms. Our results also represent an important first step towards carrying out a resummation of subleading-power rapidity logarithms.

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MIT-CTP 5087 DESY 18-207 December 19, 2018
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USACenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USATheory Group, Deutsches Elektronen-Synchrotron (DESY), D-22607 Hamburg, GermanyCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USADepartment of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China

1 Introduction

Observables in quantum field theory that are sensitive to soft and collinear emissions suffer from potentially large logarithms in their perturbative predictions. The structure of these logarithms depends on the observable in question. For a large class of phenomenologically relevant observables, these logarithms arise from emissions that are widely separated in rapidity, as opposed to, or in addition to, the more standard case of logarithms from a hierarchy of virtualities. At leading order in the associated power expansion, these rapidity logarithms can be resummed to all orders in using rapidity evolution equations. Historically these include the well-known massive Sudakov form factor Collins:1989bt (), Collins-Soper Collins:1981uk (); Collins:1981va (); Collins:1984kg (), BFKL Kuraev:1977fs (); Balitsky:1978ic (); Lipatov:1985uk (), and rapidity renormalization group Chiu:2011qc (); Chiu:2012ir () equations.

The resummation of such rapidity logarithms is necessary for a number of applications, including the spectrum for small in color-singlet processes (see e.g. refs. Ji:2004wu (); Bozzi:2005wk (); Becher:2010tm (); GarciaEchevarria:2011rb (); Chiu:2012ir (); Wang:2012xs (); Neill:2015roa (); Ebert:2016gcn (); Bizon:2017rah (); Chen:2018pzu (); Bizon:2018foh ()), double parton scattering (see e.g. refs. Diehl:2011yj (); Manohar:2012jr (); Buffing:2017mqm ()), jet-veto resummation (see e.g. refs. Banfi:2012jm (); Becher:2013xia (); Stewart:2013faa ()), recoil sensitive event-shape observables (see e.g. refs. Dokshitzer:1998kz (); Becher:2012qc (); Larkoski:2014uqa (); Moult:2018jzp ()), multi-differential observables (see e.g. refs. Laenen:2000ij (); Procura:2014cba (); Marzani:2015oyb (); Lustermans:2016nvk (); Muselli:2017bad (); Hornig:2017pud (); Kang:2018agv (); Michel:2018hui ()), processes involving massive quarks or gauge bosons (see e.g. refs. Ciafaloni:1998xg (); Fadin:1999bq (); Kuhn:1999nn (); Chiu:2007yn (); Chiu:2009mg (); Gritschacher:2013pha (); Hoang:2015vua (); Pietrulewicz:2017gxc ()), and small- resummations that go beyond the simplest applications of BFKL (see e.g. refs. Catani:1990eg (); Balitsky:1995ub (); Kovchegov:1999yj (); JalilianMarian:1996xn (); JalilianMarian:1997gr (); Iancu:2001ad ()). In all these cases, the resummation was performed at leading power (LP), and at present very little is known about the structure of rapidity logarithms and their associated evolution equations at subleading power.

There has been significant interest and progress in studying power corrections Manohar:2002fd (); Beneke:2002ph (); Pirjol:2002km (); Beneke:2002ni (); Bauer:2003mga (); Laenen:2008gt (); Laenen:2010uz (); Larkoski:2014bxa () both in the context of -physics (see e.g. refs. Mantry:2003uz (); Hill:2004if (); Mannel:2004as (); Lee:2004ja (); Bosch:2004cb (); Beneke:2004in (); Tackmann:2005ub (); Trott:2005vw (); Paz:2009ut (); Benzke:2010js ()) and for collider-physics cross sections (see e.g. refs. Dokshitzer:2005bf (); Laenen:2008ux (); Freedman:2013vya (); Freedman:2014uta (); Bonocore:2014wua (); Bonocore:2015esa (); Kolodrubetz:2016uim (); Bonocore:2016awd (); Moult:2016fqy (); Boughezal:2016zws (); DelDuca:2017twk (); Balitsky:2017flc (); Moult:2017jsg (); Goerke:2017lei (); Balitsky:2017gis (); Beneke:2017ztn (); Feige:2017zci (); Moult:2017rpl (); Chang:2017atu (); Boughezal:2018mvf (); Ebert:2018lzn (); Bahjat-Abbas:2018hpv ()). Recently, progress has been made also in understanding the behaviour of matrix elements in the subleading soft and collinear limit Bhattacharya:2018vph () in the presence of multiple collinear directions using spinor-helicity formalism. In ref. Moult:2018jjd () the first all-order resummation at subleading power for collider observables was achieved for a class of power-suppressed kinematic logarithms in thrust including both soft and collinear radiation. More recently in ref. Beneke:2018gvs () subleading power logarithms for a class of corrections in the threshold limit have also been resummed. In both cases the subleading power logarithms arise from widely separated virtuality scales, and their resummation make use of effective field theory techniques. Given the importance of observables involving nontrivial rapidity scales, it is essential to extend these recent subleading-power results to such observables, and more generally, to understand the structure of rapidity logarithms and their evolution equations at subleading power.

In this paper, we initiate the study of rapidity logarithms at subleading power, focusing on their structure in fixed-order perturbation theory. We show how to consistently regularize subleading-power rapidity divergences, and highlight several interesting features regarding their structure. In particular, power-law divergences appear at subleading power, which give nontrivial contributions and must be handled properly. We introduce a new “pure rapidity” regulator and an associated “pure rapidity” -like renormalization scheme. This procedure is homogeneous in the power expansion, meaning that it does not mix different orders in the power expansion, which significantly simplifies the analysis of subleading power corrections. We envision that it will benefit many applications.

As an application of our formalism, we compute the complete power-suppressed contributions for for color-singlet production, which provides a strong check on our regularization procedure. We find the interesting feature that the appearing power-law rapidity divergences yield derivatives of PDFs in the final cross section. Our results provide an important ingredient for improving the understanding of distributions at next-to-leading power (NLP). They also have immediate practical applications for understanding and improving the performance of fixed-order subtraction schemes based on the observable Catani:2007vq ().

To systematically organize the power expansion, we use the soft collinear effective theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001ct (); Bauer:2001yt (), which provides operator and Lagrangian based techniques for studying the power expansion in the soft and collinear limits. The appropriate effective field theory for observables with rapidity divergences is SCET Bauer:2002aj (). In this theory, rapidity logarithms can be systematically resummed using the rapidity renormalization group (RRG) Chiu:2011qc (); Chiu:2012ir () in a similar manner to virtuality logarithms. The results derived here extend the rapidity renormalization procedure to subleading power, and we anticipate that they will enable the resummation of rapidity logarithms at subleading power.

The outline of this paper is as follows. In sec. 2, we give a general discussion of the structure and regularization of rapidity divergences at subleading power. We highlight the issues appearing for rapidity regulators that are not homogeneous in the power-counting parameter, focusing on the regulator as an explicit example. We then introduce and discuss the pure rapidity regulator, which is homogeneous. In sec. 3, we derive a master formula for the power corrections to the color-singlet spectrum at , highlighting several interesting features of the calculation. We also give explicit results for Higgs and Drell-Yan production, and perform a numerical cross check to validate our results. We conclude in sec. 4.

2 Rapidity Divergences and Regularization at Subleading Power

Rapidity divergences naturally arise in the calculation of observables sensitive to the transverse momentum of soft emissions. In a situation where we have a hard interaction scale and the relevant transverse momentum of the fields is small compared to that scale, , the appropriate effective field theory (EFT) is SCET Bauer:2002aj (), which contains modes with the following momentum scalings

(1)

Here we have used lightcone coordinates , defined with respect to two lightlike reference vectors and . For concreteness, we take them to be and . Unlike SCET where the modes are separated in virtuality, in SCET the modes in the EFT have the same virtuality, but are distinguished by their longitudinal momentum ( or ), or equivalently, their rapidity . This separation into modes at hierarchical rapidities introduces divergences, which arise when or Collins:1992tv (); Manohar:2006nz (); Collins:2008ht (); Chiu:2012ir (); Vladimirov:2017ksc (). These so-called rapidity divergences are not regulated by dimensional regularization, which is boost invariant and therefore cannot distinguish modes that are only separated in rapidity.

Rapidity divergences can be regulated by introducing a rapidity regulator that breaks boost invariance, allowing the modes to be distinguished, and logarithms associated with the different rapidity scales to be resummed. The rapidity divergences cancel between the different sectors of the effective theory, since they are not present in the full theory. They should not be thought of as UV, or IR, but as arising from the factorization in the EFT. By demanding invariance with respect to the regulator, one can derive renormalization group evolution equations (RGEs) in rapidity. In SCET, a generic approach to rapidity evolution was introduced in refs. Chiu:2011qc (); Chiu:2012ir (). These rapidity RGEs allow for the resummation of large logarithms associated with hierarchical rapidity scales.

At leading power in the EFT expansion, the structure of rapidity divergences and the associated rapidity renormalization group are well understood by now, and they have been studied to high perturbative orders (see e.g. ref. Li:2016ctv () at three-loop order). Indeed, in certain specific physical situations involving two lightlike directions, rapidity divergences can be conformally mapped to UV divergences Hatta:2008st (); Caron-Huot:2015bja (); Caron-Huot:2016tzz (); Vladimirov:2016dll (); Vladimirov:2017ksc (), giving a relation between rapidity anomalous dimensions and standard UV anomalous dimensions. However, little is known about the structure of rapidity divergences or their renormalization beyond the leading power.111For some interesting recent progress for the particular case of the subleading power Regge behavior for massive scattering amplitudes in super Yang-Mills theory, see ref. Bruser:2018jnc ().

In this section, we discuss several interesting features of rapidity divergences at subleading power, focusing on the perturbative behavior at next-to-leading order (NLO). At subleading power there are no purely virtual corrections at NLO, and so we will focus on the case of the rapidity regularization of a single real emission, which allow us to identify and resolve a number of subtleties. After a brief review of the structure of rapidity-divergent integrals at leading power in sec. 2.1, we discuss additional issues that arise at subleading power in sec. 2.2. We discuss in detail the behavior of the regulator at subleading power, highlighting effects that are caused by the fact that it is not homogeneous in the power expansion. In sec. 2.3, we introduce the pure rapidity regularization, which regulates rapidity instead of longitudinal momentum and which we find to significantly simplify the calculation at subleading power. Finally, in sec. 2.4, we discuss the distributional treatment of power-law divergences, which arise at subleading power.

2.1 Review of Rapidity Divergences at Leading Power

We begin by reviewing the structure of rapidity divergent integrals at leading power. As mentioned above, we restrict ourselves to the case of a single on-shell real emission, which suffices at NLO. Defining , its contribution to a cross section sensitive to the transverse momentum of the emission is schematically given by

(2)

Here, we have extracted the overall behaviour, and is an observable and process dependent function, containing the remaining phase-space factors and amplitudes. The precise form of is unimportant, except for the fact that it includes kinematic constraints on the integration range of ,

(3)

For our discussion we take such that we can work in dimensions. In the full theory, eq. (2.1) is finite, with the apparent singularities for or being cut off by the kinematic constraints in eq. (3). In the effective theory, one expands eq. (2.1) in the soft and collinear limits specified in eq. (1). This expansion also removes the kinematic constraints,

(4)

such that individual soft and collinear contributions acquire explicit divergences as or . This is actually advantageous, since the associated logarithms can now be tracked by these divergences. To regulate them, we introduce a regulator , where is a parameter such that . By construction, inserting under the integral in eq. (2.1) does not affect the value of when taking in the full calculation. To describe the limit , we expand eq. (2.1) in the soft and collinear limits described by the modes in eq. (1). To be specific, the soft limit of eq. (2.1) is obtained by evaluating the integrand together with the regulator using the soft scaling of eq. (1), and expanding in ,

(5)

Since the leading-power result must scale like , the LP soft limit must be a pure constant, which implies that the kinematic constraints in eq. (3) are removed. This introduces the aforementioned divergences as or , which are now regulated by .

The analogous expansion in the collinear sectors is obtained by inserting the or scalings of eq. (1) into eq. (2.1), and expanding in ,

(6)

In this case, only the lower bound on is removed by the power expansion, while the upper limit is given by the relevant hard scale . The expansion of in the collinear limit can still depend on the momentum , as indicated by the functional form of , and likewise for the -collinear limit.

Without the rapidity regulator, the integrals in eqs. (2.1) and (2.1) exhibit a logarithmic divergence as or , which is not regulated by dimensional regularization or any other invariant-mass regulator. Since is fixed by the measurement, this corresponds to a divergence as the rapidity . The rapidity regulator regulates these divergence by distinguishing the soft and collinear modes. To ensure a cancellation of rapidity divergences in the effective theory, it should be defined as a function valid on a full-theory momentum , which can then be expanded in the soft or collinear limits. Since there are no divergences in the full theory, this guarantees the cancellation of divergences in the EFT expansion.

At leading power a variety of regulators have been proposed. Since the divergences are only logarithmic, and the focus has not been on higher orders in the power expansion, there are not many constraints from maintaining the power counting of the EFT. Therefore, a variety of regulators have been used, including hard cutoffs Balitsky:1995ub (); JalilianMarian:1997gr (); Kovchegov:1999yj (); Manohar:2006nz (), tilting Wilson lines off the lightcone Collins:1350496 (), the delta regulator Chiu:2009yx (), the regulator Chiu:2011qc (); Chiu:2012ir (), the analytic regulator Beneke:2003pa (); Chiu:2007yn (); Becher:2011dz (), and the exponential regulator Li:2016axz ().

At subleading power, we will discuss in more detail the application of the regulator, which can be formulated at the operator level by modifying the Wilson lines appearing in the SCET fields as Chiu:2011qc (); Chiu:2012ir ()

(7)
(8)

where and are soft and collinear Wilson lines. The operator picks out the large (label) momentum flowing into the Wilson line, is a rapidity regularization scale, a parameter exposing the rapidity divergences as poles, and a bookkeeping parameter obeying

(9)

Note that at leading power, one can replace in eq. (8), as employed in refs. Chiu:2011qc (); Chiu:2012ir (), while at subleading power we will show that this distinction is actually important. The regulator was extended in ref. Rothstein:2016bsq () to also regulate Glauber exchanges in forward scattering, where regulating Wilson lines alone does not suffice.

2.2 Rapidity Regularization at Subleading Power

We now extend our discussion to subleading power, where we will find several new features. First, while at leading power, rapidity divergences arise only from gluons, at subleading power rapidity divergences can arise also from soft quarks. Soft quarks have also been rapidity-regulated to derive the quark Regge trajectory Moult:2017xpp (). Here, since we consider only the case of a single real emission crossing the cut, this simply means that we must regulate both quarks and gluons. More generally, one would have to apply a rapidity regulator to all operators in the EFT, as has been done for the case of forward scattering in ref. Rothstein:2016bsq (). It would be interesting to understand if these subleading rapidity divergences can also be conformally mapped to UV divergences of matrix elements, as was done for the rapidity divergences in the leading power soft function in refs. Vladimirov:2016dll (); Vladimirov:2017ksc ().

Second, the structure of rapidity divergences becomes much richer at subleading power, placing additional constraints on the form of the rapidity regulator to maintain a simple power expansion. This more interesting divergence structure follows directly from power counting. For example, the subleading corrections to the soft limit can be obtained by expanding the integrand in eq. (2.1) to higher orders in . The power counting for soft modes in eq. (1) implies that the first power suppression can only be given by additional factors of or in eq. (2.1). At the next order, , one can encounter additional factors . The possible structure of rapidity-divergent integrals in the soft limit up to is thus given by222We can also have integrals with an additional factor of or , which however do not change the structure of the integrand and can thus be treated with the same techniques as at leading power.

(10)

where it is understood that . We can see that the limit only produces logarithmic divergences, while the power-suppressed corrections give rise to power-law divergences. The prototypical rapidity-divergent integral encountered in the soft limit is thus given by

(11)

where counts the additional powers of .

A similar situation occurs in the collinear sectors. In the -collinear limit, , the large momentum is not suppressed with respect to , such that the power suppression can only arise from explicit factors of . (Of course, can also give a suppression, but it can always be reduced back to .) Similarly, in the -collinear limit is unsuppressed, and power suppressions only arise from . However, the structure of the collinear expansion of is richer than in the soft case, because there is always a nontrivial dependence on the respective unsuppressed ratio . To understand this intuitively, consider the splitting of a -collinear particle into two on-shell -collinear particles with momenta

(12)

The associated Lorentz-invariant kinematic variable is given by

(13)

Expanding any function of in thus gives rise to additional factors of the large momentum . Thus, in general, expanding in the collinear limit can give rise to both positive and negative powers of that accompany the power-suppression in . These factors are of course not completely independent, as the sum of all soft and collinear contributions must be rapidity finite, i.e., any rapidity divergences induced by these additional powers of must in the end cancel against corresponding divergences in the soft and/or other collinear contributions. In summary, the generic form of integrals in the collinear expansion is given by

(14)
(15)

Here, and are regular functions as . At LP, only contributes, which gives rise to logarithmic divergences, while at subleading power for we again encounter power-law divergences. As we will see in sec. 2.4, these power-law divergences have a nontrivial effect, namely they lead to derivatives of PDFs in the perturbative expansion for hadron collider processes.

The presence of power-law divergences at subleading power also implies that more care must be taken to ensure that the regulator does not unnecessarily complicate the power counting of the EFT. For example, with the exponential regulator Li:2016axz (), or with a hard cutoff, power-law divergences lead to the appearance of powers of the regulator scale, and hence break the homogeneity of the power expansion of the theory.

Furthermore, at leading power one also has the freedom to introduce and then drop subleading terms to simplify any stage of the calculation. While this may seem a general feature and not appear very related to the regularization of rapidity divergences, we will see in a moment that this freedom, explicitly or not, is actually used in most of the rapidity regulators in the literature.

In summary, having a convenient-to-use regulator at subleading power imposes stronger constraints than at leading power. In particular, we find that the regulator

  • must be able to regulate not only Wilson lines, but all operators, including those generating soft quark emissions,

  • must be able to deal not only with logarithmic divergences, but also with power-law divergences without violating the power counting of the EFT by inducing power-law mixing,

  • and should be homogeneous in the power-counting parameter to minimize mixing between different powers.

The first requirement means one cannot use regulators acting only on Wilson lines, such as taking Wilson lines off the light-cone as in ref. Collins:1350496 (), the regulator as used in refs. Chiu:2009yx (); GarciaEchevarria:2011rb (), and the regulator as used in refs. Chiu:2011qc (); Chiu:2012ir (), while the regulator as modified and employed in refs. Rothstein:2016bsq (); Moult:2017xpp () and the analytic regulator of ref. Becher:2011dz () can be used. The second requirement is satisfied by all dimensional regularization type regulators, such as the regulator or analytic regulator, but not by those that are more like a hard cutoff, including the exponential regulator Li:2016axz (). To highlight the last point, in the following we discuss in more detail the properties of the regulator at subleading power.

2.2.1 The Regulator at Subleading Power

In the regulator, one regulates the momentum of emissions through the regulator function (see eq. (7))

(16)

For a single massless emission this corresponds to regulating its phase-space integral as

(17)

In the soft limit , the regulator is homogeneous in and therefore does not need to be expanded. The prototypical soft integral in eq. (11) evaluates to

(18)

Symmetry under implies that

(19)

This reflects the symmetry under exchanging , which is not broken by the regulator. One can easily deduce the behavior as from eq. (2.2.1). Since , a pole in can only arise if both functions have poles, which requires . A finite result is obtained if exactly one function yields a pole, which requires to be even. For odd , the expression vanishes at . Hence, the exact behavior for is given by

(20)

In particular, since the regulator behaves like dimensional regularization, it is well-behaved for power-law divergences and the soft integrals only give rise to poles from the logarithmic divergences.

In the collinear sector, the behavior is more complicated at subleading power, because the regulator factor is not homogeneous in . At leading power Chiu:2011qc (); Chiu:2012ir (); Rothstein:2016bsq (), one takes advantage of the fact that in the -collinear limit and in the -collinear limit, so that the expanded result correctly regulates the collinear cases, and makes it symmetric under the exchange . A fact that will be important for our analysis is that this power expansion induces higher order terms. These terms have never been considered in the literature since they are not important at leading power. However, at subleading power one can no longer neglect the subleading component of the regulator. Implementing the regulator at subleading power in the collinear limits thus requires to expand the regulator eq. (16) itself,

(21)

Applying this to the general LP integral in the -collinear sector, eq. (14) with , we obtain

(22)

and analogously for . Here, the first line is the standard LP integral, while the second line arises from expanding the regulator and is suppressed by . While it is also proportional to , the remaining integral can produce a rapidity divergence to yield an overall finite contribution.

In sec. 3, we will see explicitly that these terms from expanding the regulator are crucial to obtain the correct final result at subleading power. However, in practice they are cumbersome to track in the calculation and yield complicated structures. To establish an all-orders factorization theorem, the mixing of different orders in the power expansion due to the regulator becomes a serious complication. Hence, it is desirable to employ a rapidity regulator that is homogeneous in . We will present such a regulator in the following sec. 2.3.

2.3 Pure Rapidity Regularization

We wish to establish a rapidity regulator that is homogeneous at leading power such that it does not mix LP and NLP integrals, as observed in sec. 2.2.1 for the regulator. This can be achieved by implementing the regulator similar to the regulator of refs. Chiu:2011qc (); Chiu:2012ir (); Rothstein:2016bsq (), but instead of regulating the momentum with factors of , one regulates the rapidity of the momentum , where

(23)

To implement a regulator involving rapidity we use333 Note that we can implement the pure rapidity regulator in terms of label and residual momentum operators for example as (24) where the label momentum operator picks out the large momentum component of the operator it acts on, while picks out the or components. In this case, the operator (25) picks out the rapidity of the operator it acts on. factors of

(26)

Here we have defined a rapidity scale (\upsilon) which is the analog of the scale (\nu) in the regulator. Although is dimensionless, in contrast to the dimensionful , it still shares the same properties as pure dimensional regularization. In particular, it will give rise to poles in that can be absorbed in -like rapidity counterterms. To ensure independence of eq. (26), we introduced a bookkeeping parameter in analogy to the bookkeeping parameter in the regulator, see eq. (9) and ref. Chiu:2012ir ().

We call eq. (26) the pure rapidity regulator, and pure rapidity regularization the procedure of regulating rapidity divergences using eq. (26). When only the poles are subtracted we then refer to the renormalized result as being in the pure rapidity renormalization scheme.

If we want to make the rapidity scale into a true rapidity scale , then we can change variables as

(27)

With this definition eq. (26) becomes

(28)

and the factor regulating divergences depends on a rapidity difference between the scale parameter and .

It is interesting to consider the behavior of amplitudes regulated with eq. (28) under a reparameterization transformation known as RPI-III Manohar:2002fd (), which takes and for some, not necessarily infinitesimal, constant . For a single collinear sector, this can be interpreted as a boost transformation. Since RPI transformations can be applied independently for each set of collinear basis vectors they in general constitute a broader class of symmetry transformations in SCET. Prior to including a regulator for rapidity divergences all complete SCET amplitudes are invariant under such transformations. All previous rapidity regulators violate this symmetry. For the pure rapidity regulator in eq. (28) we have , so the transformation is quite simple.444 Any operators that are defined such that they transform under RPI-III, will do so by a factor , where is their RPI-III charge. The pure rapidity regulator therefore has an RPI-III charge of . This leads to rapidity-renormalized collinear and soft functions in SCET which carry this charge. When considering any observable like a cross section, the combined charge of the renormalized functions describing this observable is zero. It can be compensated by defining the rapidity scale to transform like a rapidity, . Therefore, the factor in the regulator does for RPI-III what the usual factor does for the mass-dimensionality in dimensional regularization.

As an example of the application of this new regulator, we consider again a real emission with momentum . The regulator function that follows from eq. (26) is given by

(29)

The real-emission phase space is then regulated as

(30)

A peculiar feature of the pure rapidity regulator is that it renders the prototypical soft integrals scaleless such that they vanish. That is, using eq. (29) in eq. (11), we obtain

(31)

The final integrals are scaleless and vanish for all integer values of , just like scaleless integrals vanish in dimensional regularization.555Technically one can find terms of the form , which can be set to zero via analytic continuation in the standard manner.

Considering the collinear sectors, the prototypical collinear integrals in eq. (14) with become

(32)

Although the regulator does not act symmetrically in the -collinear and -collinear sectors, the asymmetry is easy to track by taking and when swapping and . Since is homogeneous in , it does not generate any subleading power terms, in contrast to eq. (2.2.1) for the regulator. In particular, the LP integral becomes

(33)

where we used the standard distributional identity to extract the divergence. (See sec. 2.4 below for a more general discussion.) Taking , the analogous pole in the -collinear sector has the opposite sign, such that the poles cancel when adding the -collinear and -collinear contributions. This is a general feature in all cases where the soft contribution vanishes as in eq. (31).

Some comments about the features of the pure rapidity regulator are in order:

  • It involves the rapidity

    (34)

    and therefore breaks boost invariance as required to regulate rapidity divergences. The boost invariance is restored by the dimensionless rapidity scale, analogous to how the dimensionful mass scale in dimensional regularization restores the dimensionality.

  • Rapidity divergences appear as poles, allowing the definition of the pure rapidity renormalization scheme as a dimensional regularization-like scheme.

  • At each order in perturbation theory, the poles in and the -dependent pieces cancel when combining the results for the -collinear, -collinear, and soft sectors.

  • The pure rapidity regulator is homogeneous666In cases where it is possible to combine label and residual momenta in the phase space integral that needs to be rapidity regulated. in the SCET power counting parameter . Therefore it does not need to be power expanded, and hence does not mix contributions at different orders in the power expansion.

  • For the case of a single real emission considered here:

    • Soft integrals and zero-bin Manohar:2006nz () integrals are scaleless and vanish.

    • It follows that the poles and the dependent pieces cancel between the -collinear and -collinear sectors.

    • The results for the -collinear and -collinear sectors are not identical but are trivially related by taking and when swapping .

The introduction of this new pure rapidity regulator allows us to regulate rapidity divergences at any order in the EFT power expansion, while maintaining the power counting of the EFT independently at each order.

Although in this paper we will only use pure rapidity regularization for a single real emission at fixed order, we note that one can derive a rapidity renormalization group for the pure rapidity regulator by imposing that the cross section must be independent of . Similar to the regulator, this regulator is not analytical and can also be used to properly regulate virtual and massive loops. This will be discussed in detail elsewhere.

To conclude this section we note that the pure rapidity regulator can be seen as a particular case of a broader class of homogeneous rapidity regulators given by

(35)

where is an arbitrary parameter governing the antisymmetry between the -collinear and -collinear sectors. As for the pure rapidity regulator, this regulator is homogeneous in and renders the same class of soft integrals scaleless. However, it requires an explicit dimensionful scale to have the correct mass dimension. Note that for , eq. (35) only depends on the boost invariant product and therefore does not regulate rapidity divergences. For , it recovers the pure rapidity regulator and the dependence on cancels. Lastly, for and massless real emissions, eq. (35) essentially reduces to the regulator of ref. Becher:2011dz ().

2.4 Distributional Treatment of Power Law Divergences

To complete our treatment of rapidity divergences at subleading power, we show how their distributional structure can be consistently treated when expanded against a general test function. In particular, we will see that the power-law rapidity divergences lead to derivatives of PDFs.

In the collinear limit at NLP, we obtain divergent integrals of the form

(36)

which appear for both the regulator (with ) and the pure rapidity regulator (with ).

The function is defined to be regular for . If it is known analytically, we can in principle evaluate the integral in eq. (36) analytically and expand the result for to obtain the regularized expression. However, is typically not given in analytic form. In particular, for collisions it contains the parton distribution functions (PDFs) . Therefore, to extract the rapidity divergence, we need to expand in in a distributional sense. To do so, we first change the integration variable from to the dimensionless variable defined through , such that eq. (36) becomes

(37)

In eq. (37), the rapidity divergence arises as . For , it can be extracted using the standard distributional identity

(38)

where is the standard plus distribution and we remind the reader that its convolution against a test function is given by

(39)

For , these distributions need to be generalized to higher-order plus distributions subtracting higher derivatives as well. For example, for one obtains

(40)

where the second-order plus function regulates the quadratic divergence . Its action on a test function is given by a double subtraction,

(41)

In appendix B, we give more details on these distributions, generalizing to arbitrary . Note that the second-order plus function has also appeared for example in ref. Mateu:2012nk ().

Eq. (40) implies the appearance of derivatives of delta functions, , which will induce derivatives of the PDFs that are contained in . The appearance of such derivatives in subleading power calculations was first shown in ref. Moult:2016fqy () in the context of SCET-like observables. However, in such cases they arose simply from a Taylor expansion of the momentum being extracted from the PDF. Here, they also arise from power-law divergences, a new mechanism to induce derivatives of PDFs. Recently, power-law divergences inducing derivatives of PDFs have appeared also in the study of SCET-like observables involving multiple collinear directions at subleading power Bhattacharya:2018vph (). We believe they are a general feature of calculations beyond leading power.

In practice, the higher-order distributions can be cumbersome to work with. Instead, we find it more convenient to use integration-by-parts relations to reduce the divergence in eq. (37) to the linear divergence , which yields explicit derivatives of the test function. For the cases and we encounter in sec. 3, this gives

(42)
(43)

Equations (42) and (2.4) can be used to write the kernels fully in terms of a standard , but they must be applied within the integral to directly yield derivatives of the test function .

In our application in sec. 3, will always involve the PDF and vanish at . We can thus also write eqs. (42) and (2.4) as operator equations,

(44)
(45)

Note that the second relation is quite peculiar, as we have to add the boundary term proportional to , and thus cannot be interpreted as a distributional relation. In our calculation in sec. 3, this term will not contribute due to an overall suppression by , such that only the divergent term in eq. (45) needs to be kept.

3 Power Corrections for Color-Singlet Spectra

In this section we use our understanding of rapidity regularization at subleading power to compute the perturbative power corrections to the transverse momentum in color-singlet production at invariant mass , which is one of the most well studied observables in QCD. Schematically, the cross section differential in can be expanded as

(46)

where is the leading-power cross section and the NLP cross section. These terms scale like

(47)

and hence only the LP cross section is singular as . In particular, contains Sudakov double logarithms .

The factorization of in terms of transverse-momentum dependent PDFs (TMDPDFs) was first shown by Collins, Soper, and Sterman in refs. Collins:1981uk (); Collins:1981va (); Collins:1984kg () and later elaborated on by Collins in ref. Collins:1350496 (). Its structure was also studied in refs. Catani:2000vq (); deFlorian:2001zd (); Catani:2010pd (). The factorization was also studied in the framework of SCET by various groups, see e.g. refs. Becher:2010tm (); GarciaEchevarria:2011rb (); Chiu:2012ir (). Using the notation of ref. Chiu:2012ir (), the factorized LP cross section for the production of a color-singlet final state with invariant mass and total rapidity in a proton-proton collision can be written as777We suppress that for gluon-gluon fusion, and carry polarization indices.