Generic dijet soft functions at two-loop order: correlated emissions
We present a systematic algorithm for the perturbative computation of soft functions that are defined in terms of two light-like Wilson lines. Our method is based on a universal parametrisation of the phase-space integrals, which we use to isolate the singularities in Laplace space. The observable-dependent integrations can then be performed numerically, and they are implemented in the new, publicly available package SoftSERVE that we use to derive all of our numerical results. Our algorithm applies to both SCET-1 and SCET-2 soft functions, and in the current version it can be used to compute two out of three NNLO colour structures associated with the so-called correlated-emission contribution. We confirm existing two-loop results for about a dozen and hadron-collider soft functions, and we obtain new predictions for the C-parameter as well as thrust-axis and broadening-axis angularities.
Keywords:QCD, Soft-Collinear Effective Theory, NNLO Computations
Soft functions are an essential ingredient of QCD factorisation theorems. They describe the low-energy contribution to a scattering process, which is usually easier to compute than the analogous hard process in full QCD. Due to the eikonal form of the soft interactions, the soft functions can be represented by a vacuum matrix element of Wilson lines that point along the directions of the energetic, coloured particles in the scattering process.
As long as the underlying scale of the soft interactions is large enough, the soft functions can be calculated order-by-order in perturbation theory. At next-to-leading order (NLO) the calculation involves one-loop virtual and single real-emission contributions, which can be computed with standard techniques. Starting at NNLO and beyond, the singularity structure of the individual contributions becomes intricate and the divergences in the phase-space integrals overlap. The calculation of NNLO soft functions is needed for high-precision resummations and has attracted considerable attention in the past years Belitsky:1998tc (); Becher:2005pd (); Kelley:2011ng (); Monni:2011gb (); Hornig:2011iu (); Li:2011zp (); Kelley:2011aa (); Becher:2012za (); Ferroglia:2012uy (); Becher:2012qc (); vonManteuffel:2013vja (); Ferroglia:2013awa (); Czakon:2013hxa (); vonManteuffel:2014mva (); Boughezal:2015eha (); Echevarria:2015byo (); Luebbert:2016itl (); Gangal:2016kuo (); Li:2016tvb (); Campbell:2017hsw (); Wang:2018vgu (); Li:2018tsq (); Dulat:2018vuy (); Angeles-Martinez:2018mqh (). Very recently, first results for NLO soft functions have been presented Li:2016ctv (); Moult:2018jzp ().
Whereas most of these calculations were performed analytically on a case-by-case basis, a systematic approach that exploits the universal structure of the soft functions is currently missing. The purpose of our work is to fill this gap, and as a first step we focus on soft functions that arise in processes with two massless, coloured, hard partons. The soft functions for these processes can be written in the form
where and are soft Wilson lines, and and denote the directions of the hard partons with . For concreteness, we assume that the hard partons are in a back-to-back configuration (), and that the Wilson lines are in the fundamental colour representation. The definition in (1) contains a trace over colour indices as well as a function , which specifies what is measured on the soft radiation with parton momenta for the observable under consideration. We will also see later that it is irrelevant whether and are incoming or outgoing directions up to the order we consider, NNLO. Our method therefore equally applies to dijet observables in annihilation, single-jet observables in deep-inelastic scattering, and zero-jet observables at hadron colliders. For convenience, we will refer to all of these cases with two massless, coloured, hard partons as dijet soft functions in the following.
The key observation of our analysis is that the soft matrix element in the definition (1) is universal, i.e. independent of the considered dijet observable. It is therefore possible to isolate the implicit divergences in the phase-space integrals with a universal parametrisation, and to compute the observable-dependent coefficients in an expansion in the dimensional regulator numerically. The dependence of the soft function on the observable is thus entirely confined to the measurement function , which acts as a weight factor for the numerical integrations. We will discuss the specific form we assume for the measurement function in the following section, where we will also learn that it is crucial to understand its properties in the singular limits of the matrix element.
The goal of our analysis thus consists in devising an algorithm that allows for an automated calculation of dijet soft functions to NNLO in the perturbative expansion. At NNLO the double real-emission contribution consists of three colour structures, which are often referred to as correlated (, ) and uncorrelated () emissions. As the phase-space parametrisations that are needed to factorise the divergences are different in both cases, we will concentrate in this work on correlated emissions, leaving the uncorrelated emissions for a future study BRT (). For observables that obey the non-Abelian exponentiation (NAE) theorem Gatheral:1983cz (); Frenkel:1984pz (), a dedicated calculation of the uncorrelated-emission contribution is in fact not needed, and we can therefore present complete NNLO results for a number of and hadron-collider soft functions already in this work. This is, however, not true for observables that violate the NAE theorem, like jet-veto or grooming observables, which we will address in BRT () (preliminary results can be found in Bell:2018jvf ()).
As explained earlier, we aim at a numerical evaluation of bare dijet soft functions in an expansion in the dimensional regulator . It is, however, well known that the phase-space integrals for certain soft functions suffer from rapidity divergences that are not regularised in dimensional regularisation (DR). This typically arises whenever the soft radiation is constrained to have small transverse momenta. Several prescriptions for the regularisation of the rapidity divergences have been proposed in the literature (see e.g. Chiu:2009yx (); Becher:2011dz (); Chiu:2012ir (); Echevarria:2015byo (); Li:2016axz ()), and in this work we will use a variant of the analytic regulator introduced in Becher:2011dz (). Specifically, this results in a modification of the generic -dimensional phase-space measure of the form,
where is the rapidity regulator. The rapidity scale is introduced on dimensional grounds, similar to the renormalisation scale in conventional DR. The rapidity divergences then show up as poles in , and the renormalised soft function depends on two scales and . In the context of Soft-Collinear Effective Theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001yt (); Beneke:2002ph (), these soft functions are classified as SCET-2 observables, whereas those functions that are not sensitive to the rapidity scale (and are well-defined in DR) refer to SCET-1 observables.
With the dimensional and the rapidity regulator in place, the bare soft functions can be evaluated in a double expansion in and . The main result of our analysis is an integral representation of a generic dijet soft function, in which all divergences are factorised. After introducing standard plus-distributions, the expansion in the various regulators can be performed and the coefficients of this expansion can be evaluated numerically. For the numerical integrations, we developed a new stand-alone program called SoftSERVE, which uses the Divonne integrator of the Cuba library Hahn:2004fe (). The code contains a number of refinements to improve the convergence of the numerical integrations, which we will not discuss in detail in this work, but which are explained in the user manual of SoftSERVE. The SoftSERVE package is publicly available at https://softserve.hepforge.org/.
Although the main objective of our work is the computation of bare dijet soft functions, we go one step further and extract the ingredients that are needed in practical applications of resummation within SCET. To this end, we assume that the renormalised soft function obeys a multiplicative renormalisation group equation (RGE) in Laplace space, which allows us to define the (non-cusp) soft anomalous dimension for SCET-1 soft functions and the collinear anomaly exponent for SCET-2 soft functions. In a previous study Bell:2018vaa (), we derived integral representations for these quantities at the two-loop level for the same class of dijet soft functions we consider in the present work. Using SoftSERVE, which provides a script for the automated extraction of the resummation ingredients, it is thus possible to cross check the results of Bell:2018vaa (), and to in addition obtain the finite (non-logarithmic) term of the renormalised soft function for SCET-1 observables and the bare soft remainder function for SCET-2 observables (both in Laplace space). According to the standard counting of logarithms for Sudakov problems (see Table 1), the two-loop expressions of and are needed at next-to-next-to-leading logarithmic (NNLL) accuracy, while the two-loop constants and enter at the NNLL level. SoftSERVE thus allows for increased logarithmic accuracy of SCET resummations, as was shown for the event-shape angularities in Bell:2018gce (), where the improvement was from NLL to NNLL (using preliminary results for the angularity soft function that were published in Bell:2015lsf ()).
The outline of this paper is as follows: in Section 2 we define more precisely which dijet soft functions are amenable to our algorithm and we define the general properties as well as the specific form we assume for their measurement functions. In Section 3 we outline the technical aspects of the bare soft function calculation, and in Section 4 we specify the form we assume for the RGEs of both SCET-1 and SCET-2 soft functions. In Section 5 we examine several extensions of our formalism which are relevant, e.g., for multi-differential observables and soft functions that are defined in Fourier space. In Section 6 we briefly discuss the numerical implementation of our algorithm in SoftSERVE, and in Section 7 we present sample results for and hadron-collider soft functions. All of our numerical results were generated using SoftSERVE, and the explicit examples we consider in Section 7 illustrate both the versatility and the usage of our code (template files for all soft functions considered in this work are provided in the SoftSERVE package). Most of these results are in fact already available in the literature at NNLO accuracy, and they hence provide strong cross-checks of our code, while also allowing us to study its numerical performance. Moreover, we obtain new predictions for the C-parameter, as well as thrust-axis and broadening-axis angularities. We finally conclude in Section 8 and provide some technical details of our analysis in the Appendix.
2 Measurement function
2.1 General considerations
We are concerned with soft functions that arise in processes with two massless, coloured, hard partons. A typical factorisation theorem of a dijet observable takes the form
where the symbol denotes a convolution in some kinematic variables, and is a hard function that contains the virtual corrections to the Born process at the large scale of the scattering process. The jet functions and encode the effects from collinear emissions into the directions and of the hard partons, and the soft function describes the low-energetic interactions between the two jets (for incoming partons the collinear functions are often called beam functions). As the soft, long-wavelength partons cannot resolve the inner structure of the jets, the soft function only sees the directions of the hard partons as well as their colour charges. This is reflected in the definition (1) of the soft function, where the Wilson lines depend on the direction and the colour representation of the associated hard partons. For concreteness, we adopt the notion for dijet observables in the following, and assume that the Wilson lines are given in the fundamental colour representation (the notation can be generalised to other processes by means of the colour-space formalism Catani:1996vz (), and we will discuss some examples for hadron-initiated process below). We further assume that the hard partons are in a back-to-back configuration (, along with ), which is appropriate for both and hadron-collider kinematics.
The soft function in the factorisation theorem (3) has a double-logarithmic evolution in the renormalisation scale and, possibly, also the rapidity scale . In order to make the associated divergences explicit, we find it convenient to consider an integral transformation, which turns the convolution in (3) into a product. Apart from avoiding distribution-valued quantities, this considerably simplifies the solution of the associated RGEs. For many observables this is achieved by a Laplace transformation. Denoting the corresponding Laplace variable by , we write the generic measurement function in the definition (1) of the soft function in the form
where is a function of the final-state momenta that is specific to the observable. We thus assume that the distributions can be resolved by a single Laplace transformation, which implies that the soft function is differential in a single kinematic variable. We further assume that the Laplace variable has dimension 1/mass and that the measurement cannot distinguish between the two jets, i.e. the function is supposed to be symmetric under exchange. In addition, we allow for a non-trivial azimuthal dependence of the observable around the jet (or beam) axis. In other words, there may exist an external reference vector that singles out a direction in the plane transverse to the jet (or beam) direction. We finally impose two technical restrictions on the function , namely its real part must be positive and it must be independent of the dimensional and rapidity regulators and .
Before we turn to some examples, let us recapitulate the assumptions that underlie our approach:
Dijet factorisation theorem: We assume that the soft function is embedded in a factorisation theorem of the form (3), which refers to a process with two massless, colour-charged, hard partons. The hard partons can be in the initial or final state, and they are supposed to be in a back-to-back configuration (, ). The soft function for such dijet observables has a double-logarithmic evolution in the renormalisation scale and, possibly, also the rapidity scale .
Measurement function: We assume that the measurement function can be written in the form (4). Typically, this is achieved by taking a Laplace (or Fourier) transform of a momentum-space soft function, with being the associated Laplace (or Fourier) variable. In order to ensure that the phase-space integrals converge, we require that . More specifically, the function is allowed to vanish only for configurations with zero weight in the phase-space integrations, and it is furthermore assumed to be independent of the regulators and .
Mass dimension: We assume that the variable has dimension 1/mass, and the function , which only depends on the final-state momenta , must therefore have the dimension of mass. This requirement could easily be relaxed to any positive mass dimension in the future, although we have not encountered any example that requires such a generalisation so far.
- symmetry: We assume that the measurement cannot distinguish between the two jets, and the function is therefore symmetric under the exchange of and . This requirement could again easily be relaxed in the future, at the expense of doubling the number of input functions that need to be provided by the SoftSERVE user.
Single-differential observables: We assume that the soft function only depends on one variable apart from the renormalisation and rapidity scales and . Physically, this implies that the observable is differential in one kinematic variable. This requirement is in fact not strictly imposed in our approach (we will discuss multi-differential observables in Section 5), but we find it instructive to develop our formalism for this simplified class of observables first.
Azimuthal dependence: Although we allow for a general azimuthal dependence of the observable around the jet/beam axis, we point out that the function is allowed to depend only on one angle per emitted particle in the -dimensional transverse plane. This implies that the measurement is performed with respect to an external reference vector , and the angle is then introduced as the angle between and in the plane transverse to the jet/beam direction.222For general -jet soft functions with non-back-to-back kinematics, it was shown that two angles per emitted particle are required in the general case Bell:2018mkk (). In addition, the function may depend on relative angles between two emissions, which are defined as the angles between their respective transverse momenta and .
Conditions (A1) and (A2) can be viewed as the strongest assumptions of our approach, although the generalisation to jet directions with non-back-to-back kinematics is already in progress Bell:2018mkk () (which also requires an extension of (A6)). Whereas the generalisation of (A5) will be discussed in Section 5, we already mentioned that assumptions (A3) and (A4) could easily be relaxed in the future. We further point out that our formalism is not limited to observables that obey the NAE theorem. For observables that violate NAE, the uncorrelated-emission contribution becomes non-trivial BRT (); Bell:2018jvf (), but our method still allows for the calculation of the correlated-emission contribution, and hence it yields two out of three NNLO colour structures for NAE-violating observables.
Let us now see which type of observables fall into the considered class of soft functions. First, there are event-shape variables that obey a hard-jet-soft factorisation theorem of the form (3) in the two-jet limit. As an example we consider the C-parameter distribution, which was studied within SCET in Hoang:2014wka (); Hoang:2015hka (). In an appropriate normalisation, its Laplace-space soft function can be written in the form (4) with
where the plus- and minus-components represent the projections onto the and directions with and . This function indeed has the dimension of mass, it is symmetric under exchange, and it does not depend on the regulators and . It is furthermore strictly positive, except for the trivial configuration with all , which has zero weight in the phase-space integrations. The C-parameter is a single-differential observable with a trivial azimuthal dependence since the measurement is performed with respect to the jet axis itself.
As a second class of observables, we consider threshold resummation at hadron colliders. The classic example is Drell-Yan production, which was factorised in the form (3) using methods from SCET in Becher:2007ty () (the collinear functions are the standard parton distribution functions in this case). In position space, the corresponding soft function can be written in the form (1) with a weight factor , where is the total momentum of the soft emissions. The vector thus plays the role of the reference vector in this case, and in the threshold kinematics it can be expanded as in the centre-of-mass frame of the collision. As its spatial components vanish, the observable again has a trivial azimuthal dependence around the beam axis. In terms of , the position-space soft function can then be expressed in the form (4) with
and one easily verifies that assumptions (A1)-(A6) are again satisfied for this observable.
We finally consider another class of hadron-collider soft functions, which arise in the context of transverse-momentum resummation. Taking again the Drell-Yan process as an example, the corresponding SCET analysis, which now involves beam functions that describe the effects from energetic initial-state radiation, can be found in Becher:2010tm (). The respective position-space soft function can then be written in a similar form as the one for threshold resummation, except that the reference vector is now purely transverse to the beam direction. It therefore induces a non-trivial azimuthal dependence, which is precisely of the form we anticipated in (A6). Writing , one finds that this soft function can again be written in the form (4) with
With the usual exponential damping factor of a Fourier transform in mind, we can then argue that its real part is positive as required by assumption (A2). The function itself, however, now vanishes for non-trivial kinematic configurations, which single out a complicated hypersurface in the phase-space integrations. These configurations still have zero weight in the phase-space measure – as required by (A2) – but we will see later that our numerical results are less accurate for this observable in comparison with other examples that do not suffer from this problem. As the function is purely imaginary, we will also explain later in Section 5 that the numerical implementation of the transverse-momentum-dependent soft function in SoftSERVE requires special attention. One easily verifies that the remaining assumptions (A3)-(A5) are fulfilled for this observable.
The above examples should not be understood as an exhaustive list of observables that can be treated in our formalism; they should rather help to illustrate what kind of restrictions are imposed by assumptions (A1)-(A6). Other observables relevant e.g. for jet-veto resummation and jet-grooming observables also fall in the considered class of dijet soft functions. We will discuss further examples in Section 7.
2.2 Specific parametrisations
After these general considerations, we now present the specific form we assume for the measurement function in our calculation. At NNLO the measurement is performed on either zero, one, or two emitted partons.
According to (A3), the function is supposed to have the dimension of mass, and since it only depends on the final-state momenta , it must vanish if there is no emission. We can therefore write the zero-emission measurement function in the form
for all observables we consider.
For one emission, we have to find a phase-space parametrisation that allows us to control the implicit soft and collinear divergences in the phase-space integrations. We choose the variables
where is a measure of the rapidity, is the magnitude of the transverse momentum, and parametrises the azimuthal dependence around the jet axis (with as described in (A6)). The inverse of this transformation is then given by , and .
In terms of these variables, we will see in the following section that the soft divergence arises in the limit . The variable is in fact the only dimensionful quantity in this parametrisation, and according to (A3) the function must therefore be linear in . The collinear divergences, on the other hand, emerge in the limits and . The - symmetry from (A4) then allows us to focus on one of the collinear limits, of which we choose the former. It turns out that the function may vanish or diverge as , and that we have to control its scaling in this limit to properly extract the collinear divergence. Taken together, this motivates the following ansatz for the one-emission measurement function:
where the power is fixed by the requirement that the function is finite and non-zero in the limit . For one emission, the observable is thus characterised by a parameter and a function that encodes the angular and rapidity dependence.333It follows from (A2) that and that is assumed to be independent of the regulators and .
Finding a suitable phase-space parametrisation for the double-emission contribution is much more involved. On the one hand the divergence structure of the matrix elements is more complicated, and on the other hand the measurement function must be controlled in various singular limits (whereas we only had to consider the limit in (10)). Moreover, we find that different parametrisations are needed for correlated and uncorrelated emissions. In this work we focus on the former, for which we introduce the variables
|along with the angular variables|
The variables and are thus functions of the sum of the light-cone momenta, the quantity is a measure of the rapidity difference of the emitted partons, and is the ratio of their transverse momenta.444The variable should not be confused with the total transverse momentum of the emitted partons. In general the measurement function now depends on three angles since the emitted partons may not only see the reference vector , but they will also see each other. The angles in (11b) are then introduced as , , and , and the inverse transformation is now given by , , , , and for .
After using the symmetries under and exchange, we will see in the following section that the implicit phase-space divergences now arise in four limits:
, which corresponds to the situation where both partons become soft;
, which reflects the fact that one of the partons becomes collinear to the jet direction ;
, which implies that the parton with momentum becomes soft;
and , which means that the emitted partons become collinear to each other.
The first limit can in fact be treated in analogy to the limit in the one-emission case; since is the only dimensionful variable in the parametrisation (11), we know that the function must be linear in . Yet, we still have to control the measurement function in the remaining three limits to make sure that we can properly extract the associated divergences.
What helps us in this situation is the underlying assumption in the factorisation theorem (3) that the observable is infrared safe. The one- and two-emission measurement functions are therefore not independent from each other, and we will derive explicit relations between them in the limit where one of the partons becomes soft () or both partons merge into a single parton ( and ) below. The last two limits from the above list are thus, as we say, protected by infrared safety, which means that we are guaranteed that the measurement function does not vanish in these limits since it must fall back to the one-emission case (which does not vanish for a generic configuration of the remaining parton). We therefore only have to consider the limit explicitly, which can be treated similarly to the limit in the one-emission case. Our ansatz for the correlated double-emission measurement function therefore reads
where the function is supposed to be finite and non-zero in the limit . Interestingly, this is achieved by factorising the same power of the variable as in the one-emission case – see (10). We explain in Appendix A why this is so, and we address the physical meaning of the parameter in the next section.555We again demand that and that is independent of any regulators as required by assumption (A2).
In order to extract the divergences of the bare soft function, we find it convenient to map the phase-space integrations onto a unit hypercube in the variables (9) and (11). While this can easily be achieved by exploiting the - symmetry for one emission, we will see in the following section that this leads to two independent regions in the two-emission case, which we label by A and B. Our formulae therefore depend on two different versions of the two-emission measurement function, which are defined as
Further explanations about the origin of these regions and the different representations of the measurement function in region B can be found in Section 3.3.
Before we come back to the explicit examples that we discussed towards the end of the last section, we derive the constraints from infrared safety that we mentioned earlier. To this end, we write the variables and from (11) in terms of and from (9) and the analogous variables and for the second emitted parton,
In the limit in which the parton with momentum becomes soft, i.e. , we thus see that and . Infrared safety then tells us that the value of the observable should not change under infinitesimally soft emissions,
which leads to the relation
We can derive a similar relation between the one- and two-emission measurement functions in the limit in which the two partons with momenta and become collinear to each other. In this situation, which implies and , we see that and . As the value of the observable should again not change under collinear emissions,
Relations (16) and (18) follow from the fundamental assumption that the observable that was factorised in (3) must be infrared safe. These relations are thus expected to hold for all observables we consider, and – as argued before – they guarantee that the measurement function does not vanish in two of the critical limits that we discussed above.
Starting from the observable definitions in (5) – (7), we can now easily derive the measurement functions for the C-parameter, threshold and transverse-momentum resummation in the phase-space parametrisations that we use in our calculation. The result is shown in Table 2, which illustrates that some observables have a non-trivial rapidity dependence, while others are sensitive to the reference vector and therefore depend on the angular variables and . From these expressions, we can verify that the functions and are finite in the limits and , respectively, as this was the basis for extracting the corresponding values of the parameter . We indeed see that these values can differ among the observables, and we will learn later that the case always corresponds to a SCET-2 observable. While we have so far introduced this parameter on purely technical grounds, we will see in the following section that it is related to the power counting of the momentum modes in the effective theory. Moreover, we can also easily verify that the constraints from infrared safety, (16) and (18), are satisfied for the considered class of observables.
2.3 Interpretation of the parameter
We saw in the previous section that the parameter is related to the scaling of the observable in the soft-collinear limit, and we will indeed see later that it controls the double logarithmic contributions to the renormalised soft function. We also mentioned that the parameter allows us to distinguish between SCET-1 and SCET-2 observables, and we would like to understand why the values for the C-parameter () and threshold resummation () are different, given that both observables are defined within SCET-1.
To this end, we go back to the factorisation theorem (3), which emerges in an effective theory with hard, collinear, anti-collinear, and soft momentum modes. Denoting the small expansion parameter in the theory by , we associate the following power counting to the momenta with
where is the large scale in the process, and where we have allowed for a generic scaling of the collinear momenta that is controlled by a parameter .
The factorisation theorem (3) then tells us that collinear, anti-collinear, and soft modes contribute to the observable at the same power, i.e. the observable must have the same scaling in in the three regions.666One is often left with additive observables of the form , which leads to a multiplicative factorisation theorem in Laplace space. We know, however, that the observable scales as in the soft region, since it has mass dimension one – see (A3) – and the power counting in the soft region is directly tied to the mass dimension. This can easily be verified for the examples in (5) – (7).
Our goal consists in establishing a relation between the parameter and the power-counting variable , and for this purpose it is sufficient to focus on a single emission. In this case, the parameter controls the scaling of the observable in the collinear limit (in the soft region). In the collinear region, on the other hand, we can exploit the hierarchy between the light-cone components, , to express the observable in the form
where we have used that the observable has mass dimension one, and that we can always express the transverse momentum in terms of and through the on-shell condition. The parameter then varies among the observables, and we find for the C-parameter, for threshold and for transverse-momentum resummation.
We argued before that the observable must scale as in the collinear region as well, and since and for collinear momenta, we obtain . We can extract further information if we express (19) in terms of the variables from (9),
from which we learn that the parameter controls the scaling of the observable with the rapidity-like variable . This scaling must, however, match the one we obtain when we take the collinear limit in the soft region, which brings us to the desired relation,
We thus see that the parameter is directly related to the power counting of the modes in the effective theory via , when the soft scaling is fixed to . For the C-parameter, for instance, the collinear modes scale as Hoang:2014wka (), which implies that and hence , which is in line with what we have found in Table 2. For transverse-momentum resummation, on the other hand, the relevant soft and collinear modes have the same virtuality, and so which translates into for a SCET-2 observable. However, the third example from our list appears to be peculiar, since requires that , and the relevant collinear modes should therefore scale as . We recall, though, that the collinear functions for threshold resummation are the standard parton distribution functions, and the power counting of the collinear modes is therefore not related to the threshold parameter , but rather to the non-perturbative scale in this case. In other words, the relevant collinear modes scale as with for this observable, and since this is indeed consistent with .
3 Calculation of the bare soft function
The definition (1) of what we call a generic dijet soft function depends on a measurement function , whose explicit form we discussed extensively in the previous section, as well as a matrix element of soft Wilson lines. The Wilson line associated (e.g.) with an incoming quark that travels in the direction is defined as
where is the path-ordering symbol, is the soft gluon field and are the generators of SU(3) in the fundamental representation. The Wilson line associated with an outgoing quark has a similar representation, except that the integration now runs from to . This subtle difference leads to the opposite sign in the -prescription of the associated eikonal propagators, which – to the considered order in the perturbative expansion – is only relevant for the NNLO real-virtual interference. However, as we will see later, it turns out that the corresponding squared matrix element does not depend on the sign of this prescription, and our formulae therefore equally apply to soft functions with incoming and outgoing light-like directions. The Wilson lines associated with anti-quarks, moreover, are anti-path-ordered and their definition can be found in e.g. Chay:2004zn ().
At leading order in the perturbative expansion, the soft matrix element in the definition (1) is trivial. Together with the form (8) of the zero-emission measurement function, this implies that the soft function is normalised to one at leading order. At higher orders, the bare soft function is subject to various divergences, which we control by a dimensional regulator and a rapidity regulator that is needed only for SCET-2 observables. The latter is introduced on the level of the phase-space integrals as
which is in the spirit of Becher:2011dz (), except that our version respects the - symmetry that we assume on the level of the observable (see (A4)). In this regularisation, the purely virtual corrections are scaleless and vanish at every order in perturbation theory. We are thus left with a single real-emission contribution at NLO, and with mixed real-virtual and double real-emission corrections at NNLO. The bare soft function can hence be written in the generic form
where is the renormalised strong coupling constant in the scheme, which is related to the bare coupling via with and . We furthermore introduced the rescaled variable for convenience.
In the following we in turn address the computation of the single real-emission correction , the mixed real-virtual interference , and the double real-emission contribution for a generic dijet soft function.
3.1 Single real emission
In the normalisation (24), the single real-emission correction takes the form
where denotes the corresponding soft matrix element and is the one-emission measurement function. At NLO the matrix element receives contributions from the four cut diagrams in Figure 1, where the double lines represent eikonal propagators associated with the and Wilson lines. We find that the first two diagrams yield equal contributions, while the latter two vanish because they are proportional to or . The NLO squared matrix element is then given by
where we suppressed the -prescription of the eikonal propagators since it is irrelevant at this order.
We next decompose the gluon momentum in terms of light-cone coordinates
with , , and , along with . As the one-emission measurement function only depends on one angle in the -dimensional transverse space (see (A6)), the phase-space measure can be simplified as
with . As the -dependence is universal among the considered class of observables, this integration can also be performed explicitly. We further use the - symmetry of the observable, which implies in the given parametrisation, to map the -integration over the interval to an integral over . We then arrive at the following master formula for the computation of the single real-emission correction:
which is valid for arbitrary dijet soft functions that fall into the considered class of observables and which are characterised by the parameter and the function .
Upon expanding in the regulators and , the result exposes divergences, whose origin can be more clearly identified in (29). First, there is a soft singularity that arises in the limit and gives rise to the factor . Second, the -integral in (29) diverges in the collinear limits and , i.e. when the gluon is emitted into the or directions. Due to the - symmetry, we can focus on one of these limits, and from (30) we finally read off that the collinear divergence is not regularised in dimensional regularisation for , which is precisely the SCET-2 case we identified in Section 2.3.
For , on the other hand, the rapidity regulator can be set to zero, and the expansion of the SCET-1 soft function starts with a pole, whose coefficient is controlled by the parameter . This can be seen explicitly if we rewrite the divergent rapidity factor in terms of distributions according to
As the function is by construction finite and non-zero in the limit , the remaining integrations in the expansion of (30) are well-defined and suited for a numerical integration. We in fact already presented the SCET-1 NLO master formula in Bell:2015lsf (), and an earlier derivation along similar lines – although less general – was given in Hoang:2014wka (). A similar NLO formula, valid also for non-back-to-back configurations, was presented in Kasemets:2015uus ().
For SCET-2 soft functions with , it is evident from (30) that the -integration produces a pole. It is in this case important that the -expansion is performed before the -expansion, since the -regulator is supposed to regularise rapidity divergences only. The expansion of the factor therefore generates and terms, which yield and poles on the level of the bare SCET-2 soft function, whose coefficients are related since they descend from the same Gamma function.
3.2 Real-virtual interference
The mixed real-virtual contribution is structurally identical to the single real-emission term. We now start from
where the only difference is the soft matrix element , which can be calculated from the first diagram in Figure 2. Interestingly, the one-loop correction now depends on the -prescription of the eikonal propagators on the amplitude level, but this dependence drops out in the interference with the Born diagram (see also Catani:2000pi (); Kang:2015moa ()). One finds
which again resembles the NLO matrix element (26), except that its expansion now starts with a pole, which is to be multiplied with the and poles of the subsequent phase-space integrations. The very fact that the matrix element (33) does not depend on the rapidity regulator – which is implemented only on the level of the phase-space integrals in (32) – is a key advantage of the regularisation prescription from Becher:2011dz ().
The subsequent calculation then follows along exactly the same lines outlined in the previous section, and the master formula for the computation of the real-virtual interference becomes
which we again already presented in the SCET-1 case in Bell:2015lsf ().
3.3 Double real emissions
For the double real-emission contribution, we start from
where is the two-emission measurement function. The respective soft matrix element now follows from the two-particle cut diagrams in Figure 2, which give rise to three colour structures – , and – of which the latter two are covered in this paper. The corresponding squared matrix elements are given by