Helicity Methods for High Multiplicity Subleading Soft and Collinear Limits
Abstract
The factorization of multileg gauge theory amplitudes in the soft and collinear limits provides strong constraints on the structure of amplitudes, and enables efficient calculations of multijet observables at the LHC. There is significant interest in extending this understanding to include subleading powers in the soft and collinear limits. While this has been achieved for low point amplitudes, for higher point functions there is a proliferation of variables and more complicated phase space, making the analysis more challenging. By combining the subleading power expansion of spinorhelicity variables in collinear limits with consistency relations derived from the soft collinear effective theory, we show how to efficiently extract the subleading power leading logarithms of jet event shape observables directly from known spinorhelicity amplitudes. At subleading power, we observe the presence of power law singularities arising solely from the expansion of the amplitudes, which for hadron collider event shapes lead to the presence of derivatives of parton distributions. The techniques introduced here can be used to efficiently compute the power corrections for jettiness subtractions for processes involving jets at the LHC.
MITCTP 5090 Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
1 Introduction
The factorization properties of multileg gauge theory amplitudes in the soft and collinear limits are essential for our theoretical understanding of these amplitudes, as well as for the calculation of multijet observables at hadron colliders. While the leading power soft and collinear limits have been extensively studied, very little is known about the subleading power factorization properties of multileg amplitudes, or multijet observables.
There has recently been significant progress in understanding the structure of power corrections in the soft and collinear limits Manohar:2002fd (); Beneke:2002ph (); Pirjol:2002km (); Beneke:2002ni (); Bauer:2003mga (); Hill:2004if (); Mannel:2004as (); Lee:2004ja (); Bosch:2004cb (); Beneke:2004in (); Tackmann:2005ub (); Trott:2005vw (); Dokshitzer:2005bf (); Laenen:2008ux (); Laenen:2008gt (); Paz:2009ut (); Benzke:2010js (); Laenen:2010uz (); Freedman:2013vya (); Freedman:2014uta (); Bonocore:2014wua (); Larkoski:2014bxa (); 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 (); BahjatAbbas:2018hpv (), including the first all order resummation of power suppressed logarithms for collider observables with soft and collinear radiation Moult:2018jjd () and more recently for the case of threshold Beneke:2018gvs (). However, complete calculations of the all orders structure of power suppressed terms have so far focused on the case of two backtoback jets, corresponding to color singlet production at the LHC, or dijet production in . Both for improving our theoretical understanding, as well as for practical applications for observables at the LHC, it is important to be able to extend these calculations to the multijet case.
Compact expressions for multipoint amplitudes are typically expressed using the spinorhelicity formalism DeCausmaecker:1981bg (); Berends:1981uq (); Gunion:1985vca (); Xu:1986xb (), and color ordering techniques Berends:1987me (); Mangano:1987xk (); Mangano:1988kk (); Bern:1990ux (). See e.g. Dixon:1996wi (); Dixon:2013uaa () for reviews. Due to the success of unitarity Bern:1994zx (); Bern:1994cg () and recursion Britto:2004ap (); Britto:2005fq () based techniques, a wealth of tree, one and twoloop multipoint amplitudes are known in QCD. However, for the most part, this wealth of data has not been exploited in the study of subleading power corrections to collider observables.
In this paper we provide a method to directly and efficiently compute subleading power logarithms for multijet event shape observables using known spinor amplitudes. First, we study the expansion of the twoparticle collinear limit to subleading powers in terms of spinor helicity variables, providing a convenient parametrization in terms of standard kinematic variables. Then, we exploit consistency relations derived in soft collinear effective theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001ct (); Bauer:2001yt (); Bauer:2002nz () to show that the leading logarithms at subleading power for a broad class of multijet event shape observables can be computed using only the twoparticle collinear limit, to any order in . The two particle collinear limit is particularly convenient from the perspective of multijet calculations, since it avoids the complicated phase space integrals that appear in soft limits. We use several simple examples to show explicitly how this can be done in an efficient manner. These techniques should enable a rapid extension of the availability of power corrections to multijet processes.
By extending to the multipoint case, we are also able to improve our theoretical understanding of subleading power corrections and factorization, since features that are specific to two backtoback jet directions no longer apply. In particular, we observe that at subleading powers, generic multipoint amplitudes exhibit power law, instead of logarithmic divergences. The proper treatment of these power law divergences in terms of distributions leads to derivatives of the parton distribution functions (PDFs) in hadron collider observables. An interesting feature about multipoint amplitudes is that these singularities arise already at the squared amplitude level, even if the corresponding phase space integrals are not themselves singular. This is a generic features whose treatment at fixed order provides the first step towards understanding their all orders structure for generic amplitudes.
An outline of this paper is as follows. In Sec. 2 we discuss the parametrization of the twoparticle collinear limit in spinorhelicity variables, showing how we can efficiently expand amplitudes to subleading powers in the collinear limit, and giving several concrete examples. In Sec. 3 we show how we can use consistency relations derived in SCET to extract subleading power logarithms for multijet event shape observables from the twoparticle collinear limit. In Sec. 4 we discuss the treatment of power law divergences which appear in the power expansion of amplitudes. We conclude in Sec. 5, and provide an outlook for a number of applications of the techniques discussed here.
2 Subleading Power Expansions of Spinors
In this section, we describe in detail the subleading power expansion of spinor helicity variables, focusing on the behavior and parametrization of the two particle collinear limit at subleading powers. While soft limits have been studied extensively (see e.g. Strominger:2017zoo () for a recent review), subleading power collinear limits are much less well studied, and therefore parametrizations of spinors in these limits are less widely known in the literature. A convenient parametrization of the two particle collinear limit at subleading powers was given in Stieberger:2015kia (); Nandan:2016ohb (). In this section, we review this parametrization, and make it explicit in terms of the standard momenta that are useful for calculations of observables in the collinear limit. In Sec. 3 we will apply this expansion to extract subleading power logarithms in event shape observables.
2.1 Subleading Power Collinear Limit
Here we will consider the subleading power expansion of the twoparticle collinear limit. We assume that we have two particles with momenta and that are collinear along a direction . It is convenient to work in lightcone coordinates, decomposing a given momentum as . Here is an auxiliary lightlike vector. As a concrete example we can take the vectors to be and . We then define particles collinear to the direction to have the momentum scaling
(1) 
where is some typical hard scale for the energy of the collinear radiation and is our power expansion parameter. Note that is a scaling parameter that determines the size of various contributions, and hence does not itself show up in expanded amplitudes. With this momentum scaling, it is straightforward to expand amplitudes expressed in terms of standard Mandelstam invariants. However, we would also like to be able to expand amplitudes expressed in terms of spinor helicity variables. We follow the notation of Dixon:1996wi ().
To expand particles and in the two particle collinear limit, we parametrize the full spinors as
(2)  
where and are momenta along and respectively, and the term dominates. The parameters and are such that , and and are complex parameters involving small combinations of momenta in which we will expand, with . Both and are needed in order to take generic collinear limits. The special case with corresponds to an additional kinematic restriction (discussed below), in which case Eq. (9) is identical to the decomposition in Refs. Stieberger:2015kia (); Nandan:2016ohb (). For square brackets we have the analogous decomposition
(3)  
where again the term dominates, and the indicates complex conjugation. Inverting Eqs. (9) and (3), one obtains
(4)  
Now, we solve for the quantities in terms of . Without loss of generality, we shall take and to be the four vectors and and use the following representations for the spinors Dixon:1996wi ()
(5) 
These correspond to using the Dirac basis of the gamma matrices, and by default we assume that the convention for spinor momentum labeling is always outgoing. Thus these momenta are positive for outgoing particles and negative for incoming particles.^{1}^{1}1For incoming particles , and conjugated spinors also get an extra minus sign, which ensures that spinor identities are valid for both positive and negative outgoing momenta. Solving Eq. (4) to obtain the we obtain
(6)  
where for , and are the expansion parameters.
Note that the scaling of collinear momenta makes it manifest that , , , and are quantities, and thus Eq. (9), allows us to safely expand amplitudes as a power series in and which are the only variables. One also observes the appearance of the energy fractions and of 1 and 2 respectively. The series thus obtained will be the limit of the amplitude when particles become collinear.
For only final state collinear particles (or only initial state collinear particles), we can exploit the freedom of choosing in order to make the total transverse momentum for all particles that are being taken in the collinear limit of Eq. (1), to be zero. For two final state particles this implies . Here , , and we can define the momentum fraction as
(7) 
where . With these assumptions,
(8)  
In this case the exact spinor decompositions become simpler:
(9)  
Since here , so the expansion has also been reduced to a single small parameter .
Another interesting case is the collinear limit between an emission with outgoing momentum and an initial state particle with outgoing momentum . Here and and we can define the momentum fraction for the emission relative to the initial particle, via
(10) 
where . In this case its natural to choose so that we have (rather than the sum of the two momenta). With these assumptions we have
(11) 
We also have , and , so that these spinors are already aligned with the collinear direction. In this situation there is still an expansion for and , and once again, the expansion is in the single parameter .
Employing these parametrizations of the spinors, we can efficiently expand in the two particle collinear limits. The usual leading power simplification that arises for an amplitude in the collinear limit can be illustrated with the MHV fourpoint gluon amplitude. In the limit where are collinear we have
(12) 
where the splitting function makes the displayed term . This result is valid for both outgoing and incoming particles. The terms in the ellipses in Eq. (12) are terms of higher power in the collinear limit. In the next few sections we illustrate results for subleading terms in the collinear limit through a couple of examples.
2.2 Example:
As an illustrative example, we shall derive the subleading collinear limits for the process of decay of a color singlet into 4 partons, which has all particles outgoing. For concreteness and simplicity, we shall take the singlet to be a Higgs, and the 4 partons to be two quarkantiquark pairs with differing flavors. At treelevel, only the following helicity confugurations contribute Kauffman:1996ix ():
(13)  
The conjugate helicity configurations can be obtained using parity. To illustrate the types of structures that the subleading power expansion yields we will consider two choices for the pairs of particles going collinear. In one case there will be no leading power collinear limit, and in the other case there is a leading power collinear, which gives a more complicated result.
We begin by analyzing the behavior of the amplitude when quark 1 and antiquark 4 become collinear. This particular collinear limit has no leading power, , term in the amplitude since there is no spinor product with in the denominator of Eq. (13). This makes extracting the nexttoleading behavior in the collinear limit straightforward, since one can just use the standard leading power expressions for the spinors, namely
(14)  
Substituting these into the amplitudes, we get the following expansion in
(15) 
Note that these results are expressed entirely in terms of the collinear spinors , , the momentum fraction , and the spinors for the other directions. These subleading power expressions t take a very simple form, due to the fact that there was no leading power term. In the first case where the quark and antiquark have opposite helicities, we see that the amplitude behaves like that for a scalar in the direction . In the second case when they have the same helicity, it behaves like an amplitude for a particle with spin 1 along . It would be interesting to understand this in more detail. Some work in this direction, involves representing subleading power collinear limits of gluon amplitudes in terms of mixed EinsteinYangMills amplitudes Stieberger:2015kia ().
For helicity configurations that do not have a leading power limit, it is also simple to get the power suppressed squared amplitude in the collinear limit to , since this comes only from the interference of the two suppressed amplitudes. Neglecting any color structures, we find that the amplitude squared have the following subleading terms:
(16) 
In this case, they involve only Mandelstam invariants with the direction , as well as the momentum fraction , but do not otherwise involve the substructure of the splitting. These can now be trivially integrated over the collinear phase space to obtain subleading power corrections for an event shape observable, as we will describe in Sec. 3.
The previous limit was particularly simple due to the fact that it did not have a leading power term. To illustrate a slightly more complicated example, we examine the behavior of the amplitudes in (13) when the quarks become collinear. This collinear limit has a leading power term, which is governed by the standard leading power collinear factorization. We must therefore keep all the subleading terms in the expansion of the spinors. In this case the required substitutions are
(17)  
Plugging these in to the amplitudes and expanding, we arrive at the following structure for the amplitudes
(18) 
where the leading power term , and each successive term acquires a power in , so .
The leading power amplitudes obey the well known factorization into a splitting function and a lower point amplitude
(19) 
where the tree level splitting amplitudes can be found summarized in Appendix II of Ref. Bern:1994zx (). For our example, the lower point amplitudes we require are
(20) 
and the relevant splitting functions are given by
(21) 
We therefore have
(22)  
and
(23)  
which implies
(24)  
as expected from Eq. (19).
More interesting are the subleading power terms. We find
(25)  
and
(26)  
These amplitudes have an interesting structure. First, note that they depend on both the and directions. These suppressed amplitudes have the interesting feature that they factorize as
(27) 
namely onto a lower point amplitude but involving the residual vector . It would be interesting to understand in general the factorization structure of these amplitudes, even at tree level. Some work in this direction at tree level has been done in Stieberger:2015kia (); Nandan:2016ohb (). It seems that this depends significantly on whether or not there exists a leading power collinear limit. However, for our purposes, it is sufficient to be able to expand the spinor amplitudes to subleading power.
To compute the cross section to , we must now consider the different interference terms, which gives a more complicated result. Noting that the color structure is identical for all helicity configurations, we simply sum over all possible configurations to get:
(28) 
where the order of the various terms here is given by . The leading term at is given by:
(29)  
where denotes the real part of . The factors turn out to appear frequently in the squared amplitudes, so it is worth getting some intuition by evaluating it explicitly. We have that
(30) 
Using the Dirac basis for the gamma matrices, we have that
(31)  
Thus, it follows that
(32)  
This enables us to obtain the following expression
(33) 
where are the components of the transverse momentum. Thus, we see that a particular simple expression follows for the following term
(34) 
where is the magnitude of the transverse momentum, and and are unit vectors in the plane transverse to . We thus gain some intuition for the appearance of the factor. Moreover, it becomes apparent that:

If the expression appears linearly, it will vanish upon integration to obtain the cross section, since it is odd in .

Secondly, it captures the effect of projecting the momentum from other sectors onto transverse components in the sector.
We now write the term :
(35) 
which as argued in the previous part vanishes upon integration. Finally the most interesting term is the subsubleading term :
(36)  
Since appears quadratically here, this amplitude does not vanish when integrated over the angle .
Using the parametrization of this section, one can efficiently expand any amplitude expressed in terms of spinors in the two particle collinear limit. As we will show below, this is in fact sufficient to derive the leading logarithms at subleading power for event shape observables at any order in .
3 Subleading Power Logarithms in Event Shape Observables
Having understood how to expand spinor amplitudes in the subleading power collinear limit, we would like to apply this to the calculation of subleading power logarithms for multijet observables. While our expansion techniques can be used quite generally, as an example of particular interest, we will consider the jet event shape jettiness, Stewart:2010tn (). The jettiness observable has received significant recent attention since it can be used to formulate a subtraction scheme for performing NNLO calculations with jets in the final state, known as jettiness subtractions Boughezal:2015aha (); Gaunt:2015pea (), which have been used to compute jet at NNLO Boughezal:2015dva (); Boughezal:2015aha (); Boughezal:2015ded (); Boughezal:2016dtm (); Campbell:2017dqk (), as well as inclusive photon production Campbell:2016lzl ().
The jettiness observable is defined as Stewart:2010tn ()
(37) 
where the minimum runs over . This observable can therefore be viewed as projecting the radiation in the event onto axes plus the two beam directions, as shown in Fig. 1. While more general measures are possible, the above choice is convenient for theoretical calculations, because it is linear in the momenta Jouttenus:2011wh (); Jouttenus:2013hs (). The are massless reference momenta corresponding to the momenta of the hard partons present at Born level,
(38) 
In particular, the reference momenta for the incoming partons are given by
(39) 
where
(40) 
Here, is the total momentum of any additional colorsinglet particles in the Born process, and and now correspond to the total invariant mass and rapidity of the Born system. A more detailed discussion of the construction of the in the context of fixedorder calculations and jettiness subtractions can be found in Ref. Gaunt:2015pea (). We will discuss this in detail below only for jettiness, which is the case of interest here.
For , with a typical hard scale, one is forced into the soft and collinear limits, and one can expand the cross section in powers of as
(41) 
The first term in this expression, contains the most singular terms, with the scaling
(42) 
These are referred to as the leading power terms, and a factorization formula Stewart:2009yx () describing these terms has been derived in SCET Bauer:2000ew (); Bauer:2000yr (); Bauer:2001ct (); Bauer:2001yt (); Bauer:2002nz (). It takes the schematic form
(43)  
Here are beam functions, are jet functions, and is the soft function, and denotes different partonic channels. The kinematic dependence on the jet directions is described by the hard function, , which is the infrared finite part of the squared matrix element for the jet process. We will compare this kinematic dependence to what we find later for the power corrections.
Beyond the terms described by this factorization formula, there are terms which scale as
(44) 
Since they are suppressed by powers of the observable, , we refer to them as power corrections. It has been shown that the calculation of these power suppressed terms can significantly improve the performance of jettiness subtractions. This has been explicitly illustrated in the case of color singlet production in Moult:2016fqy (); Boughezal:2016zws (); Moult:2017jsg (); Boughezal:2018mvf (); Ebert:2018lzn (). However, one would like to extend this to the case of multiple jets in the final state, where they are most needed.
The calculation of the power corrections for in the case of multiple jets is quite complicated. For applications, one would like to compute it to NNLO, namely with two additional emissions. Due to the presence of the multiple regions inherent in the jettiness definition, the multiparticle phase space becomes very complicated. Fortunately, in Moult:2016fqy () consistency relations were derived that show that the leading logarithms at subleading power can be computed to any order in by considering virtual corrections to the subleading power twoparticle collinear limits. While this was shown in the context of color singlet production, it also holds more generally. This implies that the parametrization of Sec. 2 for the two particle collinear phase space is in fact sufficient to obtain the full leading log result at subleading power. This is a remarkable simplification, as it enables the calculation to be performed at any order in as a sum over two particle collinear limits, instead of having to consider complicated multiparticle phase space integrals.
In this section we will show how to efficiently extract the leading logarithmic subleading power corrections for a process for which the helicity amplitudes are known by exploiting the consistency relations and applying the methods explained in Sec. 2.1. In Sec. 3.1 we review the consistency relations, which will allow us to derive all our results from the twoparticle collinear limit. In Sec. 3.2 we set up the phase space for integrating over the two particle collinear limit, and then in Sec. 3.3 we give an explicit example for . Since the goal of this paper is to illustrate the method, rather than perform a complete calculation for a particular process, we will illustrate it on simple tree level amplitudes. However, since we show that the leading logarithm can be extracted from the two particle collinear phase space, at higher orders one simply would consider the twoparticle collinear limit of a more complicated amplitude.
We also note that for the complete calculation of the power corrections for the jettiness observable, one must consider not only power corrections arising from i) the expansion of the matrix element, but also power corrections arising from ii) the expansion of the phase space, ii) the flux factors, and iv) the measurement definition. These sources of power corrections, and techniques for systematically organizing their expansion have been discussed in great detail in Ebert:2018lzn (). These later three types of corrections are primarily a bookkeeping exercise, and while they do have the same importance for the final result, they are not associated with the subleading power expansion of the amplitude. In this section, we focus purely on case i), illustrating how the leading logarithms can be extracted from expanding spinor amplitudes in the twoparticle collinear limits. The application to a full process of interest will be carried out in a future publication.
3.1 Consistency Relations
We begin by reviewing the consistency relations derived in Moult:2016fqy (), which will reduce the problem of computing the subleading power leading logarithms of multijet event shape observables to the calculation of phase space integrals over twoparticle collinear limits.
We consider the fixed order calculation of the cross section in SCET. In SCET, each particle is either soft, collinear, or hard, and each graph gives a result with a homogeneous scaling in , depending on the number of soft, collinear and hard particles. Explicitly, we can write the loop result for the cross section at subleading power, which we denote with the super script , as
(45) 
Here the ellipses include UV renormalization, and collinear PDF renormalization, which are not leading logarithmic effects, and so we will not discuss them further. In the first line, we have included a number of different PDF structures, including derivatives, which can exist in the final result. The origin of these terms will be discussed in Sec. 4. The consistency relations will hold separately for each different structure. Their arguments have been dropped, since they are not relevant for the current discussion. In this expression and label the scalings obtained from the contributing particles, i.e., hard, collinear, or soft, and is an integer. To be concrete, at one loop (), we have a single particle, which can be either soft, or collinear. We therefore have
soft:  
collinear:  (46) 
At two loops (), as relevant for NNLO we have the following possibility.
hardcollinear:  
hardsoft:  
collinearcollinear:  
collinearsoft:  