Numerical integration of subtraction terms
aa r X i v : . [ h e p - ph ] J un MITP/16-048
Numerical integration of subtraction terms
Satyajit Seth and Stefan Weinzierl
PRISMA Cluster of Excellence, Institut für Physik,Johannes Gutenberg-Universität Mainz,D - 55099 Mainz, Germany
Abstract
Numerical approaches to higher-order calculations often employ subtraction terms, bothfor the real emission and the virtual corrections. These subtraction terms have to be addedback. In this paper we show that at NLO the real subtraction terms, the virtual subtractionterms, the integral representations of the field renormalisation constants and – in the case ofinitial-state partons – the integral representation for the collinear counterterm can be groupedtogether to give finite integrals, which can be evaluated numerically. This is useful for anextension towards NNLO.
Introduction
Numerical methods are a promising path to higher-order corrections. The higher-order correc-tions are required for precision calculations in high energy physics. Within the numerical ap-proach one subtracts suitable approximation terms from the real emission contribution and thevirtual contribution. The subtraction terms for the real emission contribution at next-to-leadingorder (NLO) are well established [1–20]. More recently, it has become possible to use the sub-traction method for the virtual part at NLO as well [21–30]. The subtracted real emission contri-bution and the subtracted virtual contribution can be evaluated separately by numerical methods.The subtracted approximation terms have to be added back and give a finite contribution to thefinal result. Due to the universality of the singular limits, the approximation terms can be cho-sen as a sum of process-independent building blocks. When adding the approximation termsback, the integrations over the virtual loop momentum (for the virtual approximation terms) andthe unresolved phase space (for the real approximation terms) are independent of the process-dependent kinematics. At NLO the corresponding integrals are rather simple and the integrationover the loop momentum/unresolved phase space can be performed analytically once and for all.The situation changes at NNLO: An analytic integration of local subtraction terms is highlynon-trivial [31–37]. It is therefore a natural question to ask, if the integration over the loopmomentum/unresolved phase space can be done numerically. The computational costs for thenumerical integration of the subtraction terms will be small against the costs for the numericalintegration of the subtracted real emission contribution or the subtracted virtual contribution. Inthis paper we will study the issue at NLO. Let us stress that our motivation is to lay the foun-dations for an extension towards NNLO. If analytically integrated results for the approximationterms are available (as they are for NLO) it is more efficient to use these in actual NLO compu-tations. Our focus is therefore more on the principles and practicalities of the cancellations ofsingularities. We will soon see that due to some subtleties it is worth the effort to study theseissues at NLO.Taken separately, the integrations over the virtual approximation terms and the real approx-imations terms are divergent in four-dimensional space-time. When manipulating divergent in-tegrals, we will always use dimensional regularisation with D = − e space-time dimensions.Our final expressions will be finite and the limit e → D -dimensional loopmomentum space and the ( D − ) -dimensional unresolved phase space. The loop-tree dualitymethod [38–43] provides a technique to handle this situation.In the past there have been attempts to combine directly the virtual corrections with the realcorrections [44–47]. This has the disadvantage that one deals at all stages with kinematics ofan 2 → n process. Our approach first subtracts one set of approximation terms from the virtualcorrections and a different set from the real emission. We only combine the virtual approxima-tion terms with the real approximation terms. The approximation terms have a much simplerkinematic structure. At NLO this limits us to one-loop three-point functions (in the virtual case)and three external momenta (in the real emission case), independently of the number of hardparticles in the scattering process.Let us now discuss the subtleties of combining the virtual approximation terms with the real2pproximation terms. Our main interest is higher-order corrections in QCD. Therefore we dealwith massless gauge bosons and massless or massive fermions. Now let us consider a collinearsingularity from the real emission contribution. The two collinear particles will have transversepolarisations. On the other hand, for a collinear singularity in the virtual part one of the involvedparticles will have a longitudinal polarisation. These two pieces will not match. A second man-ifestation of the same problem is obtained by considering the g → q ¯ q splitting. In the collinearlimit this gives a singular contribution in the real emission part, however the corresponding limitin the virtual part is finite. The solution to both problems is to take the field renormalisationconstants into account in the form of un-integrated expressions. For massless fields, the a s -contributions to the field renormalisation constants are zero, however this zero comes from acancellation between ultraviolet and infrared regions. Effectively, the field renormalisation con-stants reshuffle ultraviolet with infrared transverse/longitudinal singularities and are needed fora local cancellation of singularities at the integrand level. We will explain these mechanisms indetail.If initial-state partons are present a further subtlety arises: The region for the collinear singu-larity from the virtual part does not match with the region for the collinear singularity from thereal part. The solution comes in the form of the collinear counterterm, which has to be included.In integrated form this counterterm has to parts: An x -dependent piece, leading to a convolutionin x , and an end-point contribution, proportional to d ( − x ) . We derive an integral represen-tation for both parts, such that on the one hand the integrand corresponding to the convolutionpart combines with the real part and on the other hand the integrand corresponding to the end-point contribution combines with the virtual part. In this way we achieve a local cancellation ofsingularities.This paper is organised as follows: In section 2 we introduce the setup and the notationand review known results. Sections 3-6 give the integral representations of all required ingredi-ents: We start in section 3 with the real approximation terms, followed by the virtual subtractionterms in section 4. Section 5 is devoted to the integral representation of the renormalisation con-stants. Section 6 discusses the collinear counterterm for initial-state partons. Having defined allingredients, we show in section 7 that the ingredients can be grouped together to give locallyintegrable expressions. However, local integrability does not mean that all contributions can beintegrated along the real axes. In the virtual approximation terms there can be thresholds, whichare avoided by a deformation into the complex plane. Section 8 discusses therefore contour de-formation. Finally, our conclusions are given in section 9. Various technical details are collectedin the appendix. 3 Notation and review of known results
Let us consider a 2 → n process. The contributions at leading and next-to-leading order arewritten in a condensed notation as h O i LO = Z n O n d s B , h O i NLO = Z n + O n + d s R + Z n + loop O n d s V + Z n O n d s C . (1)Here, d s B denotes the Born contribution, whose matrix elements are given by the square ofthe Born amplitudes with ( n + ) partons | A ( ) n + | , summed over spins and colours. Similarly, d s R denotes the real emission contribution, whose matrix elements are given by the square ofthe Born amplitudes with ( n + ) partons | A ( ) n + | . The term d s V gives the virtual contribution,whose matrix elements are given by the interference term of the renormalised one-loop amplitude A ( ) n + , with ( n + ) partons, with the corresponding Born amplitude A ( ) n + . The renormalised one-loop amplitude is given as the sum of the bare one-loop amplitude and the ultraviolet counterterm.We write d s V = d s Vbare + d s VCT . (2)Finally, d s C denotes a collinear counterterm, which subtracts the initial state collinear singular-ities. Taken separately, the individual contributions at next-to-leading order are divergent andonly their sum is finite. Within the numerical approach, one adds and subtracts suitably chosenpieces to be able to perform the phase space integrations and the loop integration by Monte Carlomethods: h O i NLO = Z n + (cid:0) O n + d s R − O n d s AR (cid:1) + Z n + loop (cid:0) O n d s Vbare − O n d s AV (cid:1) + Z n O n d s C + O n Z d s AR + O n Z loop d s AV + O n d s VCT . (3)The approximation term for the real emission part is denoted by d s AR , the approximation termfor the virtual part by d s AV . By construction, the expressions Z n + (cid:0) O n + d s R − O n d s AR (cid:1) and Z n + loop (cid:0) O n d s Vbare − O n d s AV (cid:1) (4)are numerically integrable. In this paper we are interested in the third term h O i NLO I + L = Z n O n d s C + O n Z d s AR + O n Z loop d s AV + O n d s VCT . (5)4n particular we show that this term can be integrated numerically as well. We will separatethis term into an ultraviolet part and an infrared part. The numerical integration of the formerpart is un-problematic and our focus lies on the numerical integration of the latter part. Asalready indicated by the notation, the integration over the phase space of n hard particles will becommon to all terms in eq. (5). However, d s AV involves an integration over the D -dimensionalloop momentum space, whereas d s AR involves an extra integration over the ( D − ) -dimensionalunresolved phase space. As these two terms are individually divergent, this requires a mappingbetween the loop momentum space and the unresolved phase space, such that non-integrablesingularities cancel locally in the combination.Let us now go into more details: We denote the phase space measure for n final-state particlesby d f n ( p a + p b → p , ..., p n ) = ( p ) D d D p a + p b − n (cid:229) i = p i ! n (cid:213) i = d D p i ( p ) D − q ( p i ) d ( p i − m i ) . (6)We have d s B = (cid:12)(cid:12)(cid:12) A ( ) n + (cid:12)(cid:12)(cid:12) d f n . (7)In order to keep the notation simple, we use the convention that the integral symbol includes theflux factor, the averaging factors for the spin and colour degrees of freedom of the initial-stateparticles, the symmetry factor for final-state particles and (in hadronic collisions) the integrationover the parton distribution functions. With this convention we have for example for hadroniccollisions Z n O n d s B = (cid:229) a , b Z dx f a ( x ) Z dx f b ( x )
12 ˆ sn s ( ) n s ( ) n c ( ) n c ( ) S Z d f n O n (cid:12)(cid:12)(cid:12) A ( ) n + (cid:12)(cid:12)(cid:12) . (8)The symmetry factor S is given by a product of factors ( n j ! ) , where n j denotes the number ofidentical particles of type j in the final state. The number of colour degrees of freedom of aparticle a is denoted by n c ( a ) . We have n c ( q ) = n c ( ¯ q ) = , n c ( g ) = . (9)The number of spin degrees of freedom of a particle a is denoted by n s ( a ) . In D = − e space-time dimensions we have within conventional dimensional regularisation n s ( q ) = n s ( ¯ q ) = , n s ( g ) = D − . (10)As long as we are dealing with finite quantities we may take the limit D →
4, yielding two spindegrees of freedom for a gluon in four space-time dimensions. We may write the phase spacemeasure for the real emission part as d f n + = d f n d f unresolved . (11)5here is some freedom in defining the real approximation terms. In this paper we consider forconcreteness dipole subtraction terms [3–7, 18–20], although our results can easily be translatedto all other local real subtraction schemes. In this case, d s AR is given as a sum over dipoles: d s AR = (12) (cid:229) ( i ′ , j ′ ) (cid:229) k ′ = i ′ , j ′ D i ′ j ′ , k ′ + (cid:229) ( i ′ , j ′ ) (cid:229) a ′ D a ′ i ′ j ′ + (cid:229) ( a ′ , j ′ ) (cid:229) k ′ = j ′ D a ′ j ′ k ′ + (cid:229) ( a ′ , j ′ ) (cid:229) b ′ = a ′ D a ′ j ′ , b ′ ! d f n d f unresolved . In this paper we use the convention that particles corresponding to a real emission event aredenoted with primes. The requirement of local subtraction terms implies that in general thedipole subtraction terms are matrices in spin and colour space. This is due to the fact that inthe factorisation of the matrix elements squared spin correlations survive in the collinear limit,while colour correlations survive in the soft limit. At NLO, the integration over the unresolvedone-particle phase space is easily performed analytically in ( D − ) dimensions. In a compactnotation the result of this integration is often written as d s C + Z d s AR = I ⊗ d s B + K ⊗ d s B + P ⊗ d s B . (13)After integration all spin-correlations average out, but colour correlations still remain, indicatedby the notation ⊗ . The terms with the insertion operators K and P do not have any poles inthe dimensional regularisation parameter e . All explicit poles in the dimensional regularisationparameter are contained in the term I ⊗ d s B .Let us now turn our attention to the virtual part. d s V is given by d s V = (cid:16) A ( ) ∗ A ( ) (cid:17) d f n . (14) A ( ) denotes the renormalised one-loop amplitude. It is related to the bare amplitude by A ( ) = A ( ) bare + A ( ) CT . (15) A ( ) CT denotes the ultraviolet counterterm from renormalisation. The bare one-loop amplitudeinvolves the loop integration A ( ) bare = Z d D k ( p ) D G ( ) bare , (16)where G ( ) bare denotes the integrand of the bare one-loop amplitude. Within the numerical approachalso the one-loop amplitude A ( ) can be calculated numerically. In order to avoid singularities inthe integrand, the subtraction method is used again: A ( ) bare + A ( ) CT = (cid:16) A ( ) bare − A ( ) soft − A ( ) coll − A ( ) UV (cid:17) + (cid:16) A ( ) CT + A ( ) soft + A ( ) coll + A ( ) UV (cid:17) . (17)6he subtraction terms A ( ) soft , A ( ) coll and A ( ) UV are chosen such that they match locally the singu-lar behaviour of the integrand of A ( ) bare in D dimensions. The term A ( ) soft approximates the softsingularities, A ( ) coll approximates the collinear singularities and the term A ( ) UV approximates theultraviolet singularities. These subtraction terms have a local form similar to eq. (16): A ( ) soft = Z d D k ( p ) D G ( ) soft , A ( ) coll = Z d D k ( p ) D G ( ) coll , A ( ) UV = Z d D k ( p ) D G ( ) UV . (18)Again, there is some freedom in defining these approximation terms. We use the approximationterms given in [23–27]. The approximation term d s AV is given by d s AV = d D k ( p ) D h A ( ) ∗ (cid:16) G ( ) soft + G ( ) coll + G ( ) UV (cid:17)i d f n . (19)At NLO the loop integration for the approximation term d s AV is easily performed analytically in D dimensions. One obtains d s VCT + Z loop d s AV = L ⊗ d s B . (20)The operator L contains, as does the operator I , colour correlations due to soft gluons. In addi-tion, the insertion operator L contains explicit poles in the dimensional regularisation parameter e related to the infrared singularities of the one-loop amplitude. These poles cancel when com-bined with the insertion operator I : ( I + L ) ⊗ d s B = finite . (21)Eq. (21) is a statement on the cancellation of singularities after the integration over the unresolvedphase space and the loop momentum space, respectively. In this paper we would like to achievea cancellation of singularities before these integrations. The amplitudes are vectors in colour space. It is convenient to define colour charge operatorsacting on the colour indices of the amplitudes as follows: The colour charge operators T i for theemission of a gluon from a quark, gluon or antiquark in the final state are defined byquark : T q → qg A (cid:0) ... q j ... (cid:1) = (cid:0) T ai j (cid:1) A (cid:0) ... q j ... (cid:1) , gluon : T g → gg A (cid:16) ... g b ... (cid:17) = (cid:16) i f cab (cid:17) A (cid:16) ... g b ... (cid:17) , antiquark : T ¯ q → ¯ qg A (cid:0) ... ¯ q j ... (cid:1) = (cid:0) − T aji (cid:1) A (cid:0) ... ¯ q j ... (cid:1) . (22)The minus sign for the antiquark has its origin in the fact that for an outgoing antiquark the(outgoing) momentum flow is opposite to the flow of the fermion line. The corresponding colour7harge operators for the emission of a gluon from a quark, gluon or antiquark in the initial stateare quark : T ¯ q → ¯ qg A (cid:0) ... ¯ q j ... (cid:1) = (cid:0) − T aji (cid:1) A (cid:0) ... ¯ q j ... (cid:1) , gluon : T g → gg A (cid:16) ... g b ... (cid:17) = (cid:16) i f cab (cid:17) A (cid:16) ... g b ... (cid:17) , antiquark : T q → qg A (cid:0) ... q j ... (cid:1) = (cid:0) T ai j (cid:1) A (cid:0) ... q j ... (cid:1) . (23)In the amplitude an incoming quark is denoted as an outgoing antiquark and vice versa. For thesquares of the colour charge operators one has T q → qg = C F , T g → gg = C A . (24)We also define the colour charge operator for the emission of a quark-antiquark pair from a gluonby T g → q ¯ q A (cid:16) ... g b ... (cid:17) = (cid:16) T bi j (cid:17) A (cid:16) ... g b ... (cid:17) (25)and T g → q ¯ q = T R . (26) C A , C F and T R are the usual SU ( N c ) colour factors, given by C A = N c , C F = N c − N c , T R = . (27)In squaring an amplitude we obtain terms proportional to T i · T k (with k = i ) and terms propor-tional to T i . We may re-express T i as a combination of terms involving only T i · T k with k = i .This can be done using colour conservation. We write for i ∈ { q , g , ¯ q } T i = − (cid:229) k = i T i · T k , (28)where the sum runs over all external coloured partons k excluding parton i . For the splitting g → q ¯ q we write T g → q ¯ q = − (cid:229) k = i T g → q ¯ q T i T i · T k . (29)We further denote by b the first coefficient of the QCD b -function, b = C A − T R N f , (30)and introduce for later convenience the constants g q = g ¯ q = C F , g g = b . (31)8n the real emission part there can be approximation terms corresponding to initial-state singu-larities with a flavour transition q → g or g → q . The averaging factor for the number of colourdegrees of freedom for the initial-state particle a ′ is determined from the real emission matrixelement with ( n + ) particles. When adding the real approximation terms back, it is withinthe dipole formalism common practice to take as averaging factor the number of colour degreesof freedom for the particle a in the Born amplitude with ( n + ) particles. This introduces acompensation factor in the integrated approximation terms. We have n c ( q ) n c ( g ) C F = T R , n c ( g ) n c ( q ) T R = C F . (32)In this paper we will not use this convention. We are interested in the local cancellation ofsingularities at the integrand level. It is therefore natural to work in the phase space of ( n + ) -final state particles and we simply keep the averaging factor corresponding to a ′ . The amplitudes are vectors in spin space as well. It is advantageous to set-up the subtractionmethod locally in spin space. This allows the use of optimisation techniques like helicity sam-pling [18, 19]. In QCD, both quarks and gluons have two independent spin states, which we canlabel by “ + ” and “ − ”. The polarisations of an external gluon are described by two polarisationvectors e ± µ , the polarisations of an outgoing quark are described by the two spinors ¯ u ± a , the onesof an incoming quark by u ± a . The polarisations of an outgoing antiquark are described by v ± a , theones of an incoming antiquark by ¯ v ± a . For the convenience of the reader we have listed explicitexpressions for all polarisation vectors and polarisation spinors in appendix A.Let us further denote by A x ( ..., i , ... ) the amplitude, where the polarisation vector of particle i has been removed. If particle i is a gluon, x is a Lorentz index, while in the case where particle i is a quark x corresponds to a Dirac index. Let us consider a one-loop integral with n external momenta { p , ..., p n } . In this sub-section itwill be convenient to take all particles as outgoing. Then, the momenta of the incoming particleswill have negative energy components. We further assume without loss of generality that thecyclic order of the external momenta is p , p , ..., p n . If this is not the case, a simple re-labellingof the momenta will achieve this. With the notation as in fig. (1) we define k j = k − q j , q j = j (cid:229) l = p l . (33)A generic one-loop integral can be written as I n = Z d D k ( p ) D P ( k ) n (cid:213) j = (cid:16) k j − m j + i d (cid:17) . (34)9 p ... p n − p n k n k k k n − Figure 1: The labelling of the momenta for a generic one-loop integral. The arrows denote themomentum flow. P ( k ) is a polynomial in the loop momentum k . The + i d -prescription in the propagators indicatesinto which direction the poles of the propagators should be avoided. The loop-tree duality tech-nique allows us to replace the integration over the D -dimensional loop momentum space by n integrations over the ( D − ) -dimensional forward hyperboloids [38]: I n = − i n (cid:229) i = Z d D − k ( p ) D − k i , P ( k ) n (cid:213) j = j = i h k j − m j − i dh (cid:0) k j − k i (cid:1)i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i , = q ~ k i + m i , (35)where h is a vector with h > h ≥
0. Alternatively, we may integrate over the backwardhyperboloids: I n = i n (cid:229) i = Z d D − k ( p ) D − k i , P ( k ) n (cid:213) j = j = i h k j − m j + i dh (cid:0) k j − k i (cid:1)i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k i , = − q ~ k i + m i . (36)Note the sign change in the i dh ( k j − k i ) -term.Typical ultraviolet subtraction terms are of the form I UV r = Z d D k ( p ) D P (cid:0) ¯ k (cid:1)(cid:0) ¯ k − µ + i d (cid:1) r , (37)with ¯ k = k − Q and µ UV an arbitrary mass. Q is an arbitrary four-vector independent of theloop momentum k . The quantity P ( ¯ k ) is again a polynomial in ¯ k . In eq. (37) there is only asingle propagator, but this propagator may be raised to the power r . Again, we may use the10esidue theorem to replace the integration over the D -dimensional loop momentum space by anintegration over the ( D − ) -dimensional forward hyperboloid [39]: I UV r = − i Z d D − k ( p ) D − ( r − ) ! (cid:18) dd ¯ k (cid:19) r − P (cid:0) ¯ k (cid:1)(cid:18) ¯ k + q ~ ¯ k + µ (cid:19) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ k = q ~ ¯ k + µ . (38)There are only a finite number of ultraviolet subtraction terms. The differentiations with respectto ¯ k in eq. (38) may be carried analytically once and for all. Note that we may take in eq. (38) theparameter µ to be complex. Alternatively, we may integrate over the backward hyperboloid: I UV r = i Z d D − k ( p ) D − ( r − ) ! (cid:18) dd ¯ k (cid:19) r − P (cid:0) ¯ k (cid:1)(cid:18) ¯ k − q ~ ¯ k + µ (cid:19) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ k = − q ~ ¯ k + µ . (39) We recapitulate some basic facts about phase space generation. Let us start from an n -partonconfiguration. In hadron collisions we have an integral of the form Z dz Z dz Z d f n ( z P a + z P b → p + ... p n ) f a ( z ) z f b ( z ) z M n ( { p } ) , (40)where we suppressed all factors not relevant to the discussion here. We denote by P a and P b the momenta of the incoming hadrons, the set { p } is given by { z P a , z P b , p , ..., p n } . Given P a and P b , we first generate the momentum fractions z and z and then the final state momenta { p , ..., p n } .Now let us look at an ( n + ) -parton configuration: Z dz Z dz Z d f n + (cid:0) z P a + z P b → p ′ + ... p ′ n + (cid:1) f a ( z ) z f b ( z ) z M n + (cid:0)(cid:8) p ′ (cid:9)(cid:1) , (41)with { p ′ } = { z P a , z P b , p ′ , ..., p ′ n + } . We would like to re-write this integral as an n -partonphase space integral plus some additional integrations. Using the phase space factorisation forfinal-state particles this can be done: Z dz Z dz Z d f n ( z P a + z P b → p + ... p n ) Z d f unres f a ( z ) z f b ( z ) z M n + (cid:0)(cid:8) p ′ (cid:9)(cid:1) . (42)Thus we first generate the momentum fractions z and z , then n final-state momenta { p , ..., p n } .Finally, using ( D − ) additional variables, we construct from the set { p , ..., p n } and the addi-tional variables the final-state momenta { p ′ , ..., p ′ n + } .11ow let us consider the case, where we use phase space factorisation with initial-state parti-cles. In this case we obtain a convolution in one variable, which we denote by x . We write d f unres = dx d f redunres , (43)where d f redunres is the measure for the remaining ( D − ) variables. We now have Z dz Z dz Z dx Z d f redunres Z d f n ( xz P a + z P b → p + ... p n ) f a ( z ) z f b ( z ) z M n + (cid:0)(cid:8) p ′ (cid:9)(cid:1) . (44)According to this expression, we would first generate the momentum fractions z and z , then thevariables of d f unres (including x ), then the intermediate momenta { p } = { xz P a , z P b , p , ..., p n } and finally the momenta { p ′ } = { z P a , z P b , p ′ , ..., p ′ n + } . We would like to switch the order andgenerate d f n before d f unres . We make the change of variables z = z ′ / x and obtain Z dz ′ Z dz Z d f n (cid:0) z ′ P a + z P b → p + ... p n (cid:1) (45) × Z dx Z d f redunres q (cid:0) x − z ′ (cid:1) f a (cid:16) z ′ x (cid:17) z ′ f b ( z ) z M n + (cid:0)(cid:8) p ′ (cid:9)(cid:1) . This allows us to generate d f n before d f unres . Consider now the case, where M n + factorises as M n + (cid:0)(cid:8) p ′ (cid:9)(cid:1) = Sing (cid:0)(cid:8) p ′ (cid:9)(cid:1) M n ( { p } ) , (46)where M n depends only on { p } . We are in particular interested in the case, where the singularfunction Sing ( { p ′ } ) is of the formSing (cid:0)(cid:8) p ′ (cid:9)(cid:1) = A ( x ) − d ( − x ) Z dy B ( y ) . (47)Plugging this in gives Z dz ′ z ′ Z dz z Z d f n (cid:0) z ′ P a + z P b → p + ... p n (cid:1) f b ( z ) M n ( { p } ) (48) × Z dx Z d f redunres (cid:20) q (cid:0) x − z ′ (cid:1) f a (cid:18) z ′ x (cid:19) A ( x ) − f a (cid:0) z ′ (cid:1) B ( x ) (cid:21) . Eq. (48) defines how to implement functions of the form of eq. (47). In particular this applies tothe cases, where A ( x ) and B ( x ) contain the same singular terms 1 / ( − x ) : A ( x ) = c − x + finite terms , B ( x ) = c − x + other finite terms . (49)12 The real approximation terms
In this section we define the real subtraction terms d s AR = (50) (cid:229) ( i ′ , j ′ ) (cid:229) k ′ = i ′ , j ′ D i ′ j ′ , k ′ + (cid:229) ( i ′ , j ′ ) (cid:229) a ′ D a ′ i ′ j ′ + (cid:229) ( a ′ , j ′ ) (cid:229) k ′ = j ′ D a ′ j ′ k ′ + (cid:229) ( a ′ , j ′ ) (cid:229) b ′ = a ′ D a ′ j ′ , b ′ ! d f n d f unresolved . The definition given here differs – when summed over the spins of the unobserved particles –from the original dipole subtraction terms [3] by finite terms. This is un-problematic as longas we add and subtract exactly the same quantity. The essential property of the subtractionterms is that they have the same singular behaviour as the matrix elements squared which theyapproximate. If an analytic integration of the subtraction terms is envisaged one may in a secondstep modify the approximation terms by finite terms in order to simplify the analytic integration.However, within the approach based on numerical integration discussed in this paper the secondstep is not necessary. The real approximation terms defined below have the additional pedagocialadvantage that they show manifestly, that all unresolved particles in the real approximation termshave transverse polarisations. This will be important for the cancellation of singularities.The real approximation terms are obtained from the singular limits of the real emission matrixelement squared. We have to consider soft and collinear limits. Let us start with the collinearlimit. We consider a splitting i → i ′ + j ′ . The collinear limit occurs only in massless case.However, if the masses of the particles are small against other invariants of the process, it isadvantageous to include approximation terms for the quasi-collinear limit [7, 18]. In the quasi-collinear limit we parametrise the momenta of the two quasi-collinear final-state partons i ′ and j ′ as p ′ i = zp + k ⊥ − k ⊥ + z m i − m ′ i z n p · n , p ′ j = ( − z ) p − k ⊥ − k ⊥ + ( − z ) m i − m ′ j − z n p · n . (51)Here n is a massless four-vector and the transverse component k ⊥ satisfies 2 pk ⊥ = nk ⊥ = p , p ′ i and p ′ j are on-shell: p = m i , p ′ i = m ′ i , p ′ j = m ′ j . (52)In the quasi-collinear limit we take terms of the order O ( k ⊥ ) , O ( m i ) , O ( m ′ i ) and O ( m ′ j ) to beof the same order. The collinear limit is a special case of the quasi-collinear limit, obtained bysetting m i = m ′ i = m ′ j =
0. If the emitting particle is in the initial state, the collinear limit isdefined as p ′ a = p , p ′ j = ( − x ) p + k ⊥ − k ⊥ − x n p · n , p a = xp − k ⊥ − k ⊥ x n p · n . (53)13ere, all particles are massless. In this paper we restrict ourselves to massless incoming partons,therefore we do not have to consider the generalisation to the massive quasi-collinear case forinitial-state partons.In the quasi-collinear limit we have to consider terms of order O ( k − ⊥ ) . In this limit the Bornamplitude factorises according tolim p ′ i || p ′ j A ( ) n + (cid:0) ..., p ′ i , ..., p ′ j , ... (cid:1) = gµ e S − e (cid:229) l i Split l i i → i ′ + j ′ ( p i , p ′ i , p ′ j , l ′ i , l ′ j ) T i → i ′ + j ′ A ( ) n ( ..., p i , l i , ... ) . (54)where the sum is over all polarisations of the intermediate particle. The quantity S e = ( p ) e e − eg E (55)is the typical phase space volume factor in D = − e dimensions and g E is Euler’s constant. Thevariables l ′ i and l ′ j denote the polarisations of the particles i ′ and j ′ , respectively. The splittingfunctions Split are given bySplit l i q → qg (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1) = ( p ′ i + p ′ j ) − m i ¯ u l ′ i ( p ′ i ) e / l ′ j ( p ′ j ) u l i ( p i ) , Split l i g → gg (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1) = p ′ i · p ′ j h e l ′ i ( p ′ i ) · e l ′ j ( p ′ j ) p ′ i · e l i ( p i ) ∗ + e l ′ j ( p ′ j ) · e l i ( p i ) ∗ p ′ j · e l ′ i ( p ′ i ) − e l ′ i ( p ′ i ) · e l i ( p i ) ∗ p ′ i · e l ′ j ( p ′ j ) i , Split l i g → q ¯ q (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1) = p ′ i · p ′ j ¯ u l ′ i ( p ′ i ) e / l i ( p i ) ∗ v l ′ j ( p ′ j ) . (56)Here we used the notation e / l ( p ) ∗ = e l µ ( p ) ∗ g µ , i.e. complex conjugation is only with respect tothe polarisation vector. We define the squares of the splitting amplitudes by (cid:2) P i → i ′ + j ′ (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1)(cid:3) ab = (cid:229) l , l ′ u la ( p i ) Split l ∗ Split l ′ ¯ u l ′ b ( p i ) for quarks, (cid:2) P i → i ′ + j ′ (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1)(cid:3) µ n = (cid:229) l , l ′ e l µ ( p i ) ∗ Split l ∗ Split l ′ e l ′ n ( p i ) for gluons. (57)The squared amplitude factorises in the (quasi-) collinear limit aslim p ′ i || p ′ j (cid:12)(cid:12)(cid:12) A ( ) n + (cid:12)(cid:12)(cid:12) = pa s S − e µ e A x ( ) n ∗ T i → i ′ + j ′ (cid:2) P i → i ′ + j ′ (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1)(cid:3) xx ′ A x ′ ( ) n . (58)Let us now consider the soft limit. We consider the case where particle j ′ becomes soft. In thesoft limit we parametrise the momentum of the soft parton p ′ j as p ′ j = l q (59)14nd consider contributions to | A ( ) n + | of the order l − . Contributions to | A ( ) n + | which are lesssingular than l − are integrable in the soft limit. In the soft limit a Born amplitude A ( ) n + with ( n + ) partons behaves as lim p ′ j → A ( ) n + = gS − e µ e e µ ( p ′ j ) J µ A ( ) n . (60)The eikonal current is given by J µ = (cid:229) i = j T i p ′ iµ p ′ i · p ′ j . (61)The sum is over the remaining n hard momenta p ′ i . The quasi-collinear splittings q → qg and g → gg have non-vanishing soft limits and a part of the soft limit is already approximated bythese terms. In addition we will need the terms which are singular in the soft limit, but not in the(quasi)-collinear limit. To this aim we set (cid:2) S q → qg (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1)(cid:3) ab = − (cid:16) p ′ i · e ′ j ∗ (cid:17) (cid:16) p ′ k · e ′ j (cid:17) + (cid:16) p ′ k · e ′ j ∗ (cid:17) (cid:16) p ′ i · e ′ j (cid:17)(cid:16) p ′ i · p ′ j (cid:17) (cid:16) p ′ i · p ′ j + p ′ j · p ′ k (cid:17) u l ′ i a ( p ′ i ) ¯ u l ′ i b ( p ′ i ) , (cid:2) S g → gg (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1)(cid:3) µ n = − (cid:16) p ′ i · e ′ j ∗ (cid:17) (cid:16) p ′ k · e ′ j (cid:17) + (cid:16) p ′ k · e ′ j ∗ (cid:17) (cid:16) p ′ i · e ′ j (cid:17)(cid:16) p ′ i · p ′ j (cid:17) (cid:16) p ′ i · p ′ j + p ′ j · p ′ k (cid:17) e l ′ i µ ( p ′ i ) ∗ e l ′ i n ( p ′ i ) − (cid:16) p ′ j · e ′ i ∗ (cid:17) (cid:0) p ′ k · e ′ i (cid:1) + (cid:0) p ′ k · e ′ i ∗ (cid:1) (cid:16) p ′ j · e ′ i (cid:17)(cid:16) p ′ i · p ′ j (cid:17) (cid:16) p ′ i · p ′ j + p ′ i · p ′ k (cid:17) e l ′ j µ ( p ′ j ) ∗ e l ′ j n ( p ′ j ) , (62)where we used the abbreviation e ′ l = e l ′ l ( p ′ l ) for l = i , j . In connection with crossing symmetryit is useful to define the following operation C i : e i ↔ e ∗ i , ¯ u i ↔ ¯ v i , u i ↔ v i , (63)which adjusts the polarisation vector or spinor of the i -th particle from the final to the initial stateand vice versa. We may now list the dipole subtraction terms. If both the emitter and the spectator are in the final state, the dipole approximation terms aregiven by D i ′ j ′ , k ′ = − pa s S − e µ e (64) A x ( ) ( ..., p i , ..., p k , ... ) ∗ T i · T k T i (cid:2) V i ′ j ′ , k ′ (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1)(cid:3) xx ′ A x ′ ( ) ( ..., p i , ..., p k , ... ) . V i ′ j ′ , k ′ are given for the various splittings by V i ′ q j ′ g , k ′ (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1) = C F (cid:2) P q → qg (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1) + S q → qg (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1)(cid:3) , V i ′ g j ′ g , k ′ (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1) = C A (cid:2) P g → gg (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1) + S g → gg (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1)(cid:3) , V i ′ q j ′ ¯ q , k ′ (cid:0) p i , p ′ i , p ′ j , p ′ k , l ′ i , l ′ j (cid:1) = T R (cid:2) P g → q ¯ q (cid:0) p i , p ′ i , p ′ j , l ′ i , l ′ j (cid:1)(cid:3) . (65)The mapped momenta p i and p k are defined in the massless case by p i = p ′ i + p ′ j − y − y p ′ k , p k = − y p ′ k , y = p ′ i · p ′ j p ′ i · p ′ j + p ′ i · p ′ k + p ′ j · p ′ k . (66)In the massive case we use p k = q l ( Q , m i , m k ) q l ( Q , ( p ′ i + p ′ j ) , m k ) (cid:18) p ′ k − Q · p ′ k Q Q (cid:19) + Q + m k − m i Q Q , p i = Q − p k , (67)where Q = p ′ i + p ′ j + p ′ k and l is the Källen function l ( x , y , z ) = x + y + z − xy − yz − zx . (68)Note that the particle type of the spectator is not changed and therefore m ′ k = m k . Eq. (67) reducesin the massless limit to eq. (66). If the emitter is in the final state and the spectator in the initial state, the dipole approximationterms are given by D a ′ i ′ j ′ = − pa s S − e µ e (69) A x ( ) ( ..., p i , ..., p a , ... ) ∗ T i · T a T i h V a ′ i ′ j ′ (cid:0) p i , p ′ i , p ′ j , p ′ a , l ′ i , l ′ j (cid:1)i xx ′ A x ′ ( ) ( ..., p i , ..., p a , ... ) . The dipole splitting function is related by crossing to the final-final case: V a ′ i ′ j ′ (cid:0) p i , p ′ i , p ′ j , p ′ a , l ′ i , l ′ j (cid:1) = V i ′ j ′ , a ′ (cid:0) p i , p ′ i , p ′ j , − p ′ a , l ′ i , l ′ j (cid:1) . (70)The mapped momenta p i and p a are defined by p i = p ′ i + p ′ j − ( − x ) p ′ a , p a = xp ′ a . (71)The variable x is given by x = p ′ i · p ′ a + p ′ j · p ′ a − p ′ i · p ′ j + (cid:16) m i − m ′ i − m ′ j (cid:17) p ′ i · p ′ a + p ′ j · p ′ a . (72)16n the massless case and in the case where m ′ i = m i and m ′ j = x = p ′ i · p ′ a + p ′ j · p ′ a − p ′ i · p ′ j p ′ i · p ′ a + p ′ j · p ′ a . (73) If the emitter is in the initial state and the spectator in the final state, the dipole approximationterms are given by D a ′ j ′ k ′ = − pa s S − e µ e (74) A x ( ) ( ..., p a , ..., p k , ... ) ∗ T a · T k T a h V a ′ j ′ k ′ (cid:0) p a , p ′ a , p ′ j , p ′ k , l ′ a , l ′ j (cid:1)i xx ′ A x ′ ( ) ( ..., p a , ..., p k , ... ) . The dipole splitting function is related by crossing to the final-final case: V a ′ j ′ k ′ (cid:0) p a , p ′ a , p ′ j , p ′ k , l ′ a , l ′ j (cid:1) = C ( a ′ , a ) V a ′ j ′ , k ′ (cid:0) − p a , − p ′ a , p ′ j , p ′ k , l ′ a , l ′ j (cid:1) . (75)The operation C is defined in eq. (63). The mapped momenta p a and p k are defined by p a = xp ′ a , p k = p ′ k + p ′ j − ( − x ) p ′ a , x = p ′ k · p ′ a + p ′ j · p ′ a − p ′ j · p ′ k p ′ k · p ′ a + p ′ j · p ′ a . (76)Note that we restrict ourselves to massless initial-state particles. This implies that the masses ofthe particles a , a ′ and j ′ are zero. If both the emitter and the spectator are in the initial state, the dipole approximation terms aregiven by D a ′ j ′ , b ′ = − pa s S − e µ e (77) A x ( ) ( ..., p a , ..., p b , ... ) ∗ T a · T b T a h V a ′ j ′ , b ′ (cid:0) p a , p ′ a , p ′ j , p ′ b , l ′ a , l ′ j (cid:1)i xx ′ A x ′ ( ) ( ..., p a , ..., p b , ... ) . The dipole splitting function is related by crossing to the final-final case: V a ′ j ′ , b ′ (cid:0) p a , p ′ a , p ′ j , p ′ b , l ′ a , l ′ j (cid:1) = C ( a ′ , a ) V a ′ j ′ , b ′ (cid:0) − p a , − p ′ a , p ′ j , − p ′ b , l ′ a , l ′ j (cid:1) . (78)In this case the mapped momenta are defined as follows: p a = xp ′ a , p b = p ′ b , x = p ′ a · p ′ b − p ′ j · p ′ a − p ′ j · p ′ b p ′ a · p ′ b , (79)17nd all final state momenta are transformed as p l = L p ′ l , (80)where L is a Lorentz transformation defined by L µ n = g µ n − (cid:0) K µ + ˜ K µ (cid:1) (cid:0) K n + ˜ K n (cid:1)(cid:0) K + ˜ K (cid:1) + K µ K n K , K = p ′ a + p ′ b − p ′ j , ˜ K = p a + p b . (81)Again we consider only the case of massless initial-state particles. Therefore the masses of theparticles a , a ′ , b ′ and j ′ are zero. In this section we give the virtual subtraction terms, which we split into an infrared part and anultraviolet part: d s AV = d s AV , IR + d s AV , UV , (82)with d s AV , IR = d D k ( p ) D h A ( ) ∗ (cid:16) G ( ) soft + G ( ) coll (cid:17)i d f n , d s AV , UV = d D k ( p ) D h A ( ) ∗ G ( ) UV i d f n . (83)The approximation terms are not unique and may be modified by adding finite terms. Thisfreedom is advantageous and can be used to improve the numerical stability when integratingover the subtracted virtual part [27]. In this paper our focus lies on the basic principles of thecancellation of singularities. In order to keep all formulae to a minimal length we quote theoriginal approximation terms from [25]. In this section we use the convention to take all particlesas outgoing. We may write the virtual infrared approximation terms as d s AV , IR = (cid:229) i (cid:229) k = i E i , k ! d D k ( p ) D d f n , (84)with E i , k = − pa s S − e µ e A ( ) ( ..., p i , ..., p k , ... ) ∗ T i · T k T i W i ( p i , p k , k i ) A ( ) ( ..., p i , ..., p k , ... ) (85)18 mitter p i spectator p k k i − k i k i +1 Figure 2: The momentum flow for the virtual infrared approximation terms. The kinematics isspecified by the three momenta p i , p k and k i .and W i ( p i , p k , k i ) = (86)2 i T i p i · p k h ( k i + p i ) − m i i k i h ( k i − p k ) − m k i − S i h ( k i + p i ) − m i i k i + S i (cid:0) ¯ k − µ (cid:1) . m i and m k are the masses of the external particles i and k , respectively. Furthermore, S i = i corresponds to a quark and S i = / k i − = k i + p i and k i + = k i − p k .Dimensionally regulated scalar loop integrals are invariant under Lorentz transformations anda shift of the loop momentum. This applies to the integral over the virtual infrared approximationterms. For the subtracted one-loop amplitude the loop momenta in the approximation terms hasto match the appropriate loop momenta in the one-loop amplitude. This is best achieved bydecomposing the one-loop amplitude into primitive one-loop amplitudes with a definite cyclicordering of the external legs and by matching the loop momenta in the approximation terms foreach primitive amplitude [48–53]. In adding the approximation terms back, we are in principlefree to shift the loop momentum or to do a Lorentz transformation. Thus, we may choose therelation between k i and k to be k µi = L µ n k n + a µ . (87)We may use this freedom for a cancellation of the divergences with the real emission part. We briefly comment on the virtual ultraviolet approximation terms: d s AV , UV = d D k ( p ) D h A ( ) ∗ G ( ) UV i d f n . (88)19he function G ( ) UV can be obtained from the Feynman diagrams for A ( ) by replacing in a Feyn-man diagram exactly one vertex or one propagator by the corresponding one-loop ultravioletapproximation term, summing over all replacement possibilities and over all Feynman diagrams.The basic approximation terms for vertices and propagators can be found in [25, 27]. In prac-tice, it is advantageous to compute G ( ) UV not from Feynman diagrams, but to use recurrencerelations [25, 27, 54]. The virtual ultraviolet approximation terms are of the form I UV r = Z d D k ( p ) D P (cid:0) ¯ k (cid:1)(cid:0) ¯ k − µ + i d (cid:1) r , (89)with ¯ k = k − Q , Q an arbitrary vector and µ UV an arbitrary mass. The quantity P ( ¯ k ) is a polyno-mial in ¯ k . Note that the integration in eq. (89) corresponds to a simple tadpole integral. In the MS-scheme the relation between the bare coupling g bare and the renormalised coupling g is given by g bare = Z g S − e µ e g . (90)The renormalisation constant Z g is given by Z g = + a s p (cid:18) − b (cid:19) e + O ( a s ) , (91)where a s = g / ( p ) . The scattering amplitudes are calculated from amputated Green functions.Let us first consider in massless QCD an amplitude with n q external quarks, n ¯ q external anti-quarks and n g external gluons. We set n = n q + n ¯ q + n g . Amplitudes with massive quarks will bediscussed later. The relation between the renormalised and the bare amplitude is given by A ( p , ..., p n , g ) = (cid:16) Z / (cid:17) n q + n ¯ q (cid:16) Z / (cid:17) n g A bare ( p , ..., p n , g bare ) . (92) Z is the quark field renormalisation constant and Z is the gluon field renormalisation constant.The Lehmann-Symanzik-Zimmermann (LSZ) reduction formula instructs us to take for the fieldrenormalisation constants the residue of the propagators at the pole. In dimensional regularisa-tion and for massless particles this residue is 1 and in an analytic calculation it is sufficient torenormalise the coupling: A ( p , ..., p n , g ) = A bare (cid:18) p , ..., p n , Z g S − e µ e g (cid:19) . (93)However Z = Z = / e -poles one finds in Feynman gauge Z = + a s p C F (cid:18) e IR − e UV (cid:19) + O ( a s ) , Z = + a s p ( C A − b ) (cid:18) e IR − e UV (cid:19) + O ( a s ) . (94)20n order to unify the notation we will write in the following Z i for the field renormalisationconstants, with the convention that Z i = Z if particle i is a massless quark and Z i = Z ifparticle i is a gluon. We further write Z ( ) i for the O ( a s ) -term: Z i = + Z ( ) i + O ( a s ) . (95)Thus Z ( ) q = a s p C F (cid:18) e IR − e UV (cid:19) , Z ( ) g = a s p ( C A − b ) (cid:18) e IR − e UV (cid:19) . (96)In massless QCD we may write the ultraviolet counterterm as A ( ) CT = " − a s p ( n − ) b e UV − (cid:229) i (cid:229) k = i T i · T k T i Z ( ) i A ( ) , (97)where we used colour conservation in the terms involving Z ( ) i .Let us now turn to the massive case. It is sufficient to consider the case of QCD amplitudeswith one heavy flavour, the generalisation to several heavy flavours is straightforward. There area few modifications. We have to take into account the heavy quark field renormalisation constant,which is given in conventional dimensional regularisation by Z , Q = + a s p C F (cid:18) − e UV − e IR − + m µ (cid:19) + O (cid:0) a s (cid:1) . (98)We write Z i = Z , Q if particle i is a massive quark. In this case we also set Z ( ) i = a s p C F (cid:18) − e UV − e IR − + m µ (cid:19) . (99)Secondly, the mass of the heavy quark is renormalised. For the heavy quark mass we have tochoose a renormalisation scheme. In the on-shell scheme the mass renormalisation constant isgiven in conventional dimensional regularisation by Z m , on − shell = + a s p C F (cid:18) − e UV − + m µ (cid:19) + O (cid:0) a s (cid:1) . (100)In the MS-scheme the mass renormalisation constant is simply given by Z m , MS = + a s p C F (cid:18) − e UV (cid:19) + O (cid:0) a s (cid:1) . (101)Again, we write Z ( ) m , scheme for the O ( a s ) -term of the mass renormalisation constant. In order topresent the generalisation of eq. (97) to the massive case it is convenient to define the quantity B ( ) ( p , ..., p n , g , m ) through A ( ) ( p , ..., p n , g , m + d m ) = A ( ) ( p , ..., p n , g , m ) + d mm B ( ) ( p , ..., p n , g , m ) + O (cid:0) ( d m ) (cid:1) . A ( ) CT = " − a s p ( n − ) b e UV − (cid:229) i (cid:229) k = i T i · T k T i Z ( ) i A ( ) + Z ( ) m , scheme B ( ) . (102)We may group the renormalisation constants into two groups, depending on whether or not theycontain in addition to ultraviolet divergences also infrared divergences. The field renormalisationconstants belong to the first group, these contain infrared divergences. The mass renormalisa-tion constants and coupling renormalisation constants belong to the second group, these do notcontain infrared divergences.We now introduce an integral representation for the counterterm from renormalisation. It isconvenient to separate d s VCT into two parts: d s VCT = Z loop (cid:0) d s VCT , IR + d s VCT , UV (cid:1) . (103)This separation is done as follows: d s VCT , UV contains for all renormalisation constants (fieldrenormalisation, coupling renormalisation and mass renormalisation) the terms, which lead ex-actly to the 1 / e UV divergences. In addition, d s VCT , UV contains finite terms from coupling renor-malisation and mass renormalisation, if for these parameters a renormalisation scheme differentfrom the MS-scheme is used. On the other hand, d s VCT , IR contains for the field renormalisationconstants the terms, which lead to the 1 / e IR divergences or finite terms. The splitting of the finiteterms is of course arbitrary, but a convenient choice. We may re-write d s VCT , IR as d s VCT , IR = (cid:229) i (cid:229) k = i F i , k ! d D − k ( p ) D − d f n , (104)with F i , k = − pa s S − e µ e (105)Re A x ( ) ( ..., p i , ..., p k , ... ) ∗ T i · T k T i [ X i ( p i , k i )] xx ′ A x ′ ( ) ( ..., p i , ..., p k , ... ) . The quantities [ X i ( p i , k i )] xx ′ are derived from the self-energy corrections on the external legs.However, there is a technical complication: The self-energy on an external leg is attached througha propagator with momentum p i to the Born amplitude. This propagator is exactly on-shell,leading to an 1 / d s VCT , UV contains all terms which lead to ultraviolet divergences. An integral rep-resentation for these terms can be found in ref. [25]. In addition, d s VCT , UV contains by definitionfinite terms from coupling renormalisation and mass renormalisation, if for these parameters arenormalisation scheme different from the MS-scheme is used. The most relevant applicationwould be the case of a massive quark, where the mass is renormalised in the on-shell scheme.We discuss the implementation of the finite terms in more detail in section (7.2).22 Factorisation
In the MS-scheme the collinear subtraction term is given by d s C = a s p e eg E G ( − e ) (cid:229) a ′ initial (cid:229) a ∈{ q , g , ¯ q } Z dx a e (cid:18) µ F µ (cid:19) − e P a ′ a ( x a ) d s B (cid:0) ..., x a p ′ a , ... (cid:1) . (106)The splitting functions P a ′ a ( x ) are given by P gq = n c ( q ) n c ( g ) C F h x + ( − x ) i , P qg = n c ( g ) n c ( q ) T R " + ( − x ) x , P qq = C F (cid:20) − x (cid:12)(cid:12)(cid:12)(cid:12) + − ( + x ) (cid:21) + C F d ( − x ) , P gg = C A (cid:20) − x (cid:12)(cid:12)(cid:12)(cid:12) + + − xx − + x ( − x ) (cid:21) + b d ( − x ) . (107)The splitting functions for anti-quarks are identical to the ones for quarks. The splitting functionsin eq. (107) are the spin-averaged splitting functions. We now look for an integral representationof the collinear subtraction term. The sought after integral representation has to fulfill two con-ditions: Firstly, it should match locally the singularities of the other contributions. Secondly, itshould integrate to produce exactly the same finite parts implied by eq. (106): d s C = a s p (cid:229) a ′ initial (cid:229) a ∈{ q , g , ¯ q } Z dx a (cid:20) e − ln (cid:18) µ F µ (cid:19)(cid:21) P a ′ a ( x a ) d s B (cid:0) ..., x a p ′ a , ... (cid:1) + O ( e ) . (108)Let us discuss the first point in more detail: The singularities have to match the correspondingsingularities of the real approximation term and the counterterm from field renormalisation. Thespin-averaged case in eq. (107) gives us some guidance: The x -dependent terms in the squarebrackets will match with the real approximation terms, while the end-point contributions pro-portional to g q = C F / P qq and g g = b / P gg will match with the counterterm from fieldrenormalisation. Thus we write d s C = d s CR + d s CCT , (109)where d s CR matches with the real approximation term and d s CCT matches with the countertermfrom field renormalisation. Between d s CR and d s AR the collinear singularities cancel, the softsingularity in d s AR cancels with the virtual part d s AV , the soft 1 / ( − x ) -singularities in d s CR aresoftened by the plus-distribution. Between d s CCT and d s VCT there is a cancellation of collinearsingularities, where both collinear particles have transverse polarisations. The self-energies con-tributing to d s VCT lead also to collinear singularities, where one particle has a longitudinal polar-isation. These singularities cancel with the virtual approximation term.23or d s CR we make the ansatz d s C = (cid:229) ( a ′ , j ′ ) (cid:229) k ′ = j ′ H a ′ j ′ k ′ + (cid:229) ( a ′ , j ′ ) (cid:229) b ′ = a ′ H a ′ j ′ , b ′ ! d f n d f unresolved . (110)As we would like to match locally the singularities we have to work with the spin-dependentsplitting functions (as opposed to the spin-averaged splitting functions appearing in eq. (106)).We may however sum over the polarisations of the unobserved particles a ′ and j ′ . In the fol-lowing we drop the adjustment factors n c ( g ) / n c ( q ) and n c ( q ) / n c ( g ) appearing in eq. (107) andadhere to the convention that the averaging for the colour degrees of freedom is performed withrespect to a ′ . The same applies to the averaging with respect to the number of spin degrees offreedom for initial-state particles. When integrating eq. (110), a factor 1 / x from the unresolvedmeasure is absorbed by the flux factor to produce the correct flux factor for the event with n final-state particles. The integral representation for H a ′ j ′ k ′ is given in section (6.1), the one for H a ′ j ′ , b ′ is given in section (6.2).For d s CCT we write d s CCT = (cid:229) i initial (cid:229) k = i K i , k ! d D − k i ( p ) D − k i d f n , (111)with K i , k = − pa s S − e µ e (112)Re A x ( ) ( ..., p i , ..., p k , ... ) ∗ T i · T k T i [ Z i ( p i , k i , p k )] xx ′ A x ′ ( ) ( ..., p i , ..., p k , ... ) . The integral representation for Z i ( p i , k i , p k ) is given in section (6.3). We first consider the case of an initial-state emitter and a final-state spectator. The spectator maybe massive ( m k = m ′ k ), all other particles are massless. We use the variables P = p ′ k + p ′ j − p ′ a and x = p ′ a p ′ j + p ′ a p ′ k − p ′ j p ′ k p ′ a p ′ j + p ′ a p ′ k , u = p ′ a p ′ j p ′ a p ′ j + p ′ a p ′ k , w = p ′ a p ′ j (cid:16) p ′ j p ′ k + m k (cid:17) p ′ j p ′ k (cid:16) p ′ a p ′ j + p ′ a p ′ k (cid:17) . (113)If we further set x = P P − m k , (114)24hen the variables u and w are related by u = − x − x x w . (115)We write H a ′ j ′ k ′ = − pa s S − e µ e (116) (cid:26) A x ( ) ( p a , ..., p k , ... ) ∗ T a · T k T a h Y a ′ j ′ k ′ ( x , w ) i xx ′ A x ′ ( ) ( p a , ..., p k , ... ) − d ( − x ) Z dy A x ( ) ( p a , ..., p k , ... ) ∗ T a · T k T a h Y a ′ j ′ k ′ , end ( y , w ) i xx ′ A x ′ ( ) ( p a , ..., p k , ... ) . The relation between the set of momenta { p ′ a , p ′ j , p ′ k } and the set { p a , p k } is as in section (3.3),in particular we have p a = xp ′ a . The expression in eq. (116) is of the form as in eq. (47) and canbe implemented as in eq. (48). In order to present the functions Y a ′ j ′ k ′ we factor out some commonprefactors and we write Y a ′ j ′ k ′ = − T a → a ′ j ′ ( − P ) x ( − x x )( − x ) w ˜ Y a ′ j ′ k ′ . (117)Then˜ Y a ′ g j ′ ¯ q k ′ = p / a ( [ − e − x ( − x )] " − w ln (cid:0) − P (cid:1) ( − x ) µ F xx ( − x x ) ! − w ) , ˜ Y a ′ q j ′ q k ′ = " − g µ n x + ( − x ) x u ( − u ) p ′ j p ′ k ♭ S µ n − w ln (cid:0) − P (cid:1) ( − x ) µ F xx ( − x x ) ! − g µ n − xx w , ˜ Y a ′ q j ′ g k ′ = p / a ((cid:20) − x − ( + x ) − e ( − x ) (cid:21) " − w ln (cid:0) − P (cid:1) ( − x ) µ F xx ( − x x ) ! − ( − x ) w ) , ˜ Y a ′ g j ′ g k ′ = " − g µ n (cid:18) − x − + x ( − x ) (cid:19) + ( − e ) ( − x ) x u ( − u ) p ′ j p ′ k ♭ S µ n × " − w ln (cid:0) − P (cid:1) ( − x ) µ F xx ( − x x ) ! . (118)Here p ′ k ♭ is a light-like vector defined by p ′ k ♭ = p ′ k − m k p ′ a p ′ k p ′ a . (119)The spin correlation tensor is given by S µ n = (cid:18) u p ′ jµ − − u p ′ k ♭ µ (cid:19) (cid:18) u p ′ j n − − u p ′ k ♭ n (cid:19) . (120)25he terms proportional to w ensure that the finite part is exactly as in eq. (108). Factorisationschemes different from the MS-scheme can be implemented by a suitable modification of thefinite terms.The end-point contributions Y a ′ j ′ k ′ , end are rather simple. They are zero for flavour off-diagonalsplittings: Y a ′ g j ′ ¯ q k ′ , end = , Y a ′ q j ′ q k ′ , end = . (121)For flavour conserving splittings we write in analogy with eq. (117) Y a ′ j ′ k ′ , end = − T a → a ′ j ′ ( − P ) x ( − x x )( − x ) w ˜ Y a ′ j ′ k ′ , end . (122)Then we have ˜ Y a ′ q j ′ g k ′ , end = p / a − x " − w ln (cid:0) − P (cid:1) ( − x ) µ F xx ( − x x ) ! , ˜ Y a ′ g j ′ g k ′ , end = ( − g µ n ) − x " − w ln (cid:0) − P (cid:1) ( − x ) µ F xx ( − x x ) ! . (123) We now consider the case of an initial-state emitter and an initial-state spectator. We use thevariables x = p ′ a p ′ b − p ′ a p ′ j − p ′ b p ′ j p ′ a p ′ b , v = p ′ a p ′ j p ′ a p ′ b , w = p ′ a p ′ j p ′ a p ′ j + p ′ b p ′ j . (124)The variables v and w are related by v = ( − x ) w . (125)We write H a ′ j ′ , b ′ = − pa s S − e µ e (cid:26) A x ( ) ( p a , p b , ... ) ∗ T a · T k T a h Y a ′ j ′ , b ′ ( x , w ) i xx ′ A x ′ ( ) ( p a , p b , ... ) − d ( − x ) Z dy A x ( ) ( p a , p b , ... ) ∗ T a · T k T a h Y a ′ j ′ , b ′ end ( x , y ) i xx ′ A x ′ ( ) ( p a , p b , ... ) . The relation between the set of momenta { p ′ a , p ′ b , p ′ j } and the set { p a , p b } is as in section (3.4),in particular we have p a = xp ′ a and p b = p ′ b . The expression in eq. (126) is of the form as in26q. (47) and can be implemented as in eq. (48). In order to present the functions Y a ′ j ′ , b ′ we factorout some common prefactors and we write Y a ′ j ′ , b ′ = − T a → a ′ j ′ p a p b ( − x ) w ˜ Y a ′ j ′ , b ′ . (126)Then˜ Y a ′ g j ′ ¯ q , b ′ = p / a ( [ − e − x ( − x )] " − w ln p a p b ( − x ) µ F x ! − w ) , ˜ Y a ′ q j ′ q , b ′ = " − g µ n x + ( − x ) x p ′ a p ′ b p ′ j p ′ a p ′ j p ′ b S µ n − w ln p a p b ( − x ) µ F x ! − g µ n − xx w , ˜ Y a ′ q j ′ g , b ′ = p / a ((cid:20) − x − ( + x ) − e ( − x ) (cid:21) " − w ln p a p b ( − x ) µ F x ! − ( − x ) w ) , ˜ Y a ′ g j ′ g , b ′ = " − g µ n (cid:18) − x − + x ( − x ) (cid:19) + ( − e ) ( − x ) x p ′ a p ′ b p ′ j p ′ a p ′ j p ′ b S µ n × " − w ln p a p b ( − x ) µ F x ! . (127)The spin correlation tensor is given by S µ n = p ′ jµ − p ′ j p ′ a p ′ a p ′ b p ′ bµ ! p ′ j n − p ′ j p ′ a p ′ a p ′ b p ′ b n ! . (128)The terms proportional to w ensure that the finite part is exactly as in eq. (108). Factorisationschemes different from the MS-scheme can be implemented by a suitable modification of thefinite terms.The end-point contributions Y a ′ j ′ , b ′ end are again rather simple. Y a ′ g j ′ ¯ q , b ′ end = , Y a ′ q j ′ q , b ′ end = . (129)For flavour conserving splittings we write in analogy with eq. (126) Y a ′ j ′ , b ′ end = − T a → a ′ j ′ p a p b ( − x ) w ˜ Y a ′ j ′ , b ′ end . (130)Then ˜ Y a ′ q j ′ g , b ′ end = p / a − x " − w ln p a p b ( − x ) µ F x ! , ˜ Y a ′ g j ′ g , b ′ end = ( − g µ n ) − x " − w ln p a p b ( − x ) µ F x ! . (131)27 k p a = − p i l ′ j = k i l ′ i = − k i − Figure 3: The kinematics for self-energy corrections for initial-state particles. In the collinearlimit the momenta l ′ i and l ′ j are on-shell. The momenta l ′ i , l ′ j and l k have positive energy. We now consider d s CCT , which we write as d s CCT = (cid:229) i initial (cid:229) k = i K i , k ! d D − k i ( p ) D − k i d f n , (132)with K i , k = − pa s S − e µ e (133)Re A x ( ) ( ..., p i , ..., p k , ... ) ∗ T i · T k T i [ Z i ( p i , k i , p k )] xx ′ A x ′ ( ) ( ..., p i , ..., p k , ... ) . The particle i is an initial-state particle and we write p a = − p i such that p a has positive energy.Particle k is the spectator. The spectator can either be in the final-state (in which case it can bemassive or massless) or in the initial-state (in which case it is assumed to be always massless).We will treat all cases simultaneously. To this aim we first set p ♭ k = p k − p k p k p a p a . (134) p ♭ k is always a massless momentum. We further define l k = p ♭ k , if particle k is in the final state,and l k = − p ♭ k = − p k = p b if particle k is in the inital-state. The definition is such that l k is alwaysa massless momentum with positiv energy. d s CCT has to match the collinear singularities of theself-energy corrections. These occur when the two propagators in the self-energy loop are on-shell. We define l ′ i and l ′ j as the on-shell momenta in the self-energy loop flowing in the directionof the hard-scattering process. In the singular collinear limit both l ′ i and l ′ j have positive energies.The kinematical situation is shown in fig. (3). Given p a , l k and k i we define l ′ i , l ′ j and l ′ k by l ′ i = p a − k i + yl k , l ′ j = k i , l ′ k = ( − y ) l k , (135)28ith y = − ( p a − k i ) l k ( p a − k i ) . (136)We will encounter the mapping in eq. (135) again in section (7.1.1), where it will be used to relatein the final-final case the virtual approximation terms to the real approximation terms. With thedefinition l i = p a , the inverse mapping { l ′ i , l ′ j , l ′ k } → { l i , l k , k i } is just – when restricted to { l i , l k } – the standard Catani-Seymour projection of eq. (66). The reason why this mapping is usefulfor the self-energies related to initial-state particles is as follows: For a collinear singularity theenergy flow across the cut of the self-energy diagrams has to be in the same direction for bothcut propagators. The momentum l ′ k will only be used to define the way the collinear singularityis approached. Given l ′ i , l ′ j and l ′ k we set y = l ′ i l ′ j l ′ i l ′ j + l ′ i l ′ k + l ′ j l ′ k , z = l ′ i l ′ k l ′ i l ′ k + l ′ j l ′ k . (137)It is easily checked that the two expressions for the variable y in eq. (136) and eq. (137) arecompatible. We further set P = l ′ i + l ′ j + l ′ k = p a + l k . If the initial-state particle is a quark wehave Z i ( p i , k i , p k ) ab = (138) = C F p / a yzP (cid:26) [ − ( + z ) − e ( − z )] (cid:20) − y ln (cid:18) P z ( − z ) µ F (cid:19)(cid:21) − ( − z ) y (cid:27) q (cid:0) E ′ i (cid:1) q (cid:0) E ′ k (cid:1) , in the case where the initial-state particle is a gluon we have Z i ( p i , k i , p k ) µ n = (139) = C A yzP " g µ n + ( − e ) S µ n l ′ i l ′ j − y ln (cid:18) P z ( − z ) µ F (cid:19)(cid:21) q (cid:0) E ′ i (cid:1) q (cid:0) E ′ k (cid:1) + T R N f yzP (" − g µ n − S µ n l ′ i l ′ j − y ln (cid:18) P z ( − z ) µ F (cid:19)(cid:21) + g µ n z ( − z ) y ) q (cid:0) E ′ i (cid:1) q (cid:0) E ′ k (cid:1) , where the spin correlation tensor is given by S µ n = (cid:0) zl ′ iµ − ( − z ) l ′ jµ (cid:1) (cid:0) zl ′ i n − ( − z ) l ′ j n (cid:1) . It is easily checked that the integrated expression gives Z d D − k i ( p ) D − k i K i , k = (140) − a s p e (cid:18) µ F µ (cid:19) − e g i A x ( ) ( ..., p i , ..., p k , ... ) ∗ T i · T k T i A x ′ ( ) ( ..., p i , ..., p k , ... ) + O ( e ) . Locally integrable combinations
Our aim is to combine the approximation terms such that they are locally integrable. In orderto achieve this, it is essential to take the field renormalisation constants into account. The lo-cal cancellation of singularities occurs separately for infrared and ultraviolet divergences. Formassless particles the a s -contribution of the field renormalisation constants is zero after the loopintegration. This does not imply that the integrand is identical to zero, it only implies that theintegrand is a function with possibly ultraviolet and infrared singularities, which integrates tozero within dimensional regularisation.Other manifestations, that the contribution from the field renormalisation constants are neededare:- The real approximation terms contain a divergent contribution from the splitting g → q ¯ q ofa gluon into massless quarks. The virtual approximation terms have no such contribution. Thedivergent part from the real approximation terms cancels with the contribution from the fieldrenormalisation constants.- In the collinear part of the real approximation terms all unresolved particles have transversepolarisations. In the collinear part of the virtual approximation terms one of the two collinearparticles has a longitudinal polarisation. These two contributions do not match. Again, thecancellation occurs through the contribution from the field renormalisation constants: The longi-tudinal part from the virtual approximation terms cancels with the longitudinal part from the fieldrenormalisation constants, the transverse part from the real approximation terms cancels with thetransverse part from the field renormalisation constants.- It is instructive to look at the explicit poles in e of infrared origin in massless QCD. Afterintegration one has for the various contributions d s AR = ( A ( ) ∗ a s p (cid:229) i (cid:229) k = i T i T k " − e (cid:18) | p i p k | µ (cid:19) − e − g i T i e IR A ( ) ) d f n + ..., d s AV , IR = ( A ( ) ∗ a s p (cid:229) i (cid:229) k = i T i T k " e (cid:18) − p i p k µ (cid:19) − e + S i e IR A ( ) ) d f n + ..., d s VCT , IR = ( A ( ) ∗ a s p (cid:229) i (cid:229) k = i T i T k (cid:20)(cid:18) − S i + g i T i (cid:19) e IR (cid:21) A ( ) ) d f n + ..., (141)where the dots denote ultraviolet poles and terms of order O ( e ) . The infrared poles cancel inthe sum of the three contributions. However, there is not a complete cancellation between d s AR and d s AV , IR alone.We would like to evaluate numerically the expression of eq. (5): h O i NLO I + L = Z n O n d s C + O n Z d s AR + O n Z loop d s AV + O n d s VCT . (142)We split this expression into two parts h O i NLO I + L = h O i NLO I + L , IR + h O i NLO I + L , UV , (143)30ith h O i NLO I + L , IR = Z n O n d s C + Z d s AR + Z loop d s AV , IR + Z loop d s VCT , IR , h O i NLO I + L , UV = Z n O n Z loop (cid:0) d s AV , UV + d s VCT , UV (cid:1) . (144)The two contributions in eq. (144) are separately numerically integrable. We may break up theterm h O i NLO I + L , IR into even smaller pieces, where an individual piece corresponds to an antennaand is separately numerically integrable. This is discussed in section (7.1). The term h O i NLO I + L , UV is discussed in section (7.2). In this sub-section we consider h O i NLO I + L , IR . (145)All terms contributing to eq. (145) can be written as colour dipoles, i.e. they are of the form − (cid:229) i (cid:229) k = i T i T k ..., (146)where i denotes the emitter and k denotes the spectator. We combine the colour dipole withemitter i and spectator k with the colour dipole with emitter k and spectator i . This forms acolour antenna with the hard particles i and k [11–14] and we may write eq. (145) as h O i NLO I + L , IR = − (cid:229) i < k T i T k h O i NLO , i , k I + L , IR . (147)Each antenna contribution is separately numerically integrable. We have to consider three typesof antenna structures. The two hard particles can either be both in the final-state, of mixed type(one in the final-state and the other in the initial-state) or both in the initial-state.Let us consider the contribution of d s AV , IR to a given antenna, i.e. a contribution of the form E i , k + E k , i = − pa s S − e µ e (148)2 Re A ( ) ( ..., p i , ..., p k , ... ) ∗ T i · T k (cid:20) W i ( p i , p k , k i ) T i + W k ( p k , p i , k k ) T k (cid:21) A ( ) ( ..., p i , ..., p k , ... ) . The loop integrals are three-point functions (where lower-point functions are considered as three-point functions with appropriate inverse propagators in the numerator). The momenta flowingthrough the loop propagators are given for E i , k by k i − = k i + p i , k i + = k i − p k . (149)31 mitter p k spectator p i k k − k k k k +1 = spectator p i emitter p k − k k +1 − k k − k k − Figure 4: The momentum flow for the virtual infrared approximation terms with emitter p k andspectator p i . A comparison with fig. (2) shows k i − = − k k + , k i = − k k and k i + = − k k − .and shown in fig. (2). For E k , i the momenta are given by k k − = k k + p k , k k + = k k − p i (150)and shown in fig. (4). We may use the freedom of Poincaré-invariance of the loop integrals ofeq. (87) and set k i = − k k . (151)This implies k i − = − k k + , k i + = − k k − . (152)Eq. (151) defines how E i , k and E k , i are integrated together.Our general strategy is as follows: We will write all integrals as integrals over the spatialcomponents of a momentum: Z d D − k ( p ) D − .... (153)For the virtual integrals this can be done using the loop-tree duality method. The loop-tree du-ality method will put one of the loop propagators on-shell. The task is to find suitable mappingsbetween the various contributions, such that all singularities cancel locally in ~ k -space and thelimit D → { p } a set of ( n + ) external momenta (including the two initial-state momenta), by { k } the single-element set of the on-shell loop momentum and by { p ′ } a set of ( n + ) externalmomenta. In the next sub-section we will define an invertible mapping f : { p } × { k } → (cid:8) p ′ (cid:9) , (154)such that the inverse mapping, when restricted to pf − : (cid:8) p ′ (cid:9) → { p } (155)32grees with the Catani-Seymour projections given in section (3), relating the ( n + ) -particlephase space to the n -particle phase space. We will use this mapping on the pre-image f − ( { p ′ } ) to associate a real configuration { p ′ } to a virtual configuration specified by { p } and k . Notethat there are points in { p } × { k } , which do not map to physical points { p ′ } . A typical examplewould be a loop momentum k in the ultraviolet region, leading to a configuration { ˜ p ′ } with final-state particles of negative energy. This explains the restriction on the pre-image f − ( { p ′ } ) . Inpractice, the correct physical region will be implemented by theta-functions.We will discuss the mappings for the three cases corresponding to a final-final antenna, afinal-initial antenna and an initial-initial antenny separately in the next three sub-sections. We consider the case, where p i and p k are final-state momenta, e.g. have positive energy com-ponents. Let us first consider the dipole with emitter i and spectator k . With the kinematics as infig. (2) we would like to have that in the collinear limit the momentum ( − k i ) has positive energyas well. Turned around this means that the momentum k i has negative energy in the collinearlimit. Thus, we use the loop-tree duality formula for the backward hyperboloids of eq. (36) toconvert the loop integrals into phase space integrals. This gives three backward hyperboloidswith origins at q i − = q i − p i , q i and q i + = q i + p k , plus an extra backward hyperboloid withorigin at Q corresponding to ultraviolet subtraction terms. The latter is free of infrared singular-ities. In order to combine the real approximation terms with the virtual approximation terms wedefine a mapping between the set of momenta { p i , p k , k i } and { p ′ i , p ′ j , p ′ k } . In the massless casewe use p ′ i = p i + k i + yp k , p ′ j = − k i , p ′ k = ( − y ) p k , (156)with y = − ( k i + p i ) p k ( k i + p i ) = − k i − p k k i − . (157)Note that the inverse mapping { p ′ i , p ′ j , p ′ k } → { p i , p k , k i } coincides with the mapping in eq. (66)when restricted to { p ′ i , p ′ j , p ′ k } → { p i , p k } .In the massive case we use p ′ k = a (cid:18) p k − Q · p k Q Q (cid:19) + b Q , p ′ j = − k i , p ′ i = Q + k i − p ′ k , (158)with Q = p i + p k . The constants a and b are given in appendix C. Again, this mapping canbe considered to be the inverse of eq. (67) together with supplementary information p ′ j = − k i .33 kk q i − q i q i +1 ~q i p i p k ~q th ~kk q i − q i q i +1 ~q i − p i − p k ~q th Figure 5: The integration regions for a final-final antenna. The upper picture corresponds to thedipole with emitter i and spectator k , the lower picture corresponds to the dipole with emitter k and spectator i . 34q. (158), which includes in the massless limit eq. (156), defines how the contributions E i , k and D i ′ j ′ , k ′ are integrated together.Writing the measure for the unresolved phase space as an integration over ~ k i (or the forwardmass hyperboloid for particle j ′ ) introduces a Jacobian factor: d f unres = d D − k i ( p ) D − ( − k i , ) J , (159)with J = (cid:2) l (cid:0) Q , m i , m k (cid:1)(cid:3) − D (cid:20) l (cid:18) Q , (cid:16) p ′ i + p ′ j (cid:17) , m ′ k (cid:19)(cid:21) D − p ′ i p ′ k (cid:16) p ′ i p ′ k + p ′ j p ′ k (cid:17) − m ′ k (cid:16) p ′ i p ′ j + m ′ i (cid:17) q (cid:0) E ′ i (cid:1) q (cid:0) E ′ k (cid:1) . (160)Let us now consider the dipole with emitter k and spectator i . With the kinematics as in fig. (4)we would like to have that in the collinear limit the momentum ( − k k ) has positive energy. Since k i = − k k this implies that the momentum k i has positive energy in the collinear limit. Thus, weuse the loop-tree duality formula for the forward hyperboloids of eq. (35) to convert the loopintegrals into phase space integrals. In the next step we have to relate the real emission integralsto the virtual integrals. This is straightforward. We may use the same mappings as in eq. (158)and eq. (156) with the roles of i and k exchanged. Thus, the integrations for E k , i and D k ′ j ′ , i ′ arerelated in the same way as the integrations for E i , k and D i ′ j ′ , k ′ . Taking into account the relationbetween E i , k and E k , i in eq. (151), we may relate the integration for D k ′ j ′ , i ′ to E i , k and obtain inthe massless case p ′ i = ( − y ) p i , p ′ j = k i , p ′ k = p k − k i + yp i , (161)with y = k i + p i k i + . (162)The geometric situation for the integration over the on-shell hyperboloids is sketched in fig. (5).The upper picture shows the contribution from the virtual approximation terms with emitter i andspectator k in the massless case. The integration is over three backward light-cones with origins at q i − , q i and q i + . The soft singularity resides in the integration over the backward light-cone withorigin at q i at the origin q i and is indicated by a red dot. The collinear singularities occur on thelines between q i − and q i (collinear singularity of i ) and between q i and q i + (collinear singularityof k ). The collinear regions are indicated in blue. There is a cancellation of singularities withinthe virtual dual contributions in the regions where two propagators are on-shell and have thesame sign in the energy component. These regions are indicated in green. There is a threshold35ingularity (indicated by an orange dot) at ~ q th . The threshold singularity is avoided by contourdeformation. The integration region for the real approximation term is the backward light-conewith origin at q i . The collinear singular region for the real approximation term with emitter i isthe line segment between q i − and q i .The lower picture shows the corresponding integration regions, where the roles of emitterand spectator are exchanged, i.e. emitter k and spectator i . Note that the soft and collinearsingularities occur in the same regions of D -dimensional loop momentum space. The integrationregion for the real approximation term is the forward light-cone with origin at q i . The collinearsingular region for the real approximation term with emitter k is the line segment between q i and q i + . Let us now consider a final-initial antenna. Without loss of generality we assume that p i is afinal-state momentum (i.e. a outgoing momentum with positive energy) and that p k correspondsto an initial-state momentum. With our conventions p k is an outgoing momentum with negativeenergy. In order to match the notation of section 3 we set p a = − p k . Thus p a is an incomingmomentum with positive energy. Let us start with the virtual dipole with emitter i and spectator k . As before we would like that in the collinear limit the momentum ( − k i ) has positive energy.Therefore we use the loop-tree duality formula for the backward hyperboloids of eq. (36) toconvert the loop integrals into phase space integrals. Next, we relate the integration for the realdipole D a ′ i ′ j ′ to the integration for the virtual dipole E i , k . We recall that we use the notation p k = − p a and p ′ k = − p ′ a . We define the mapping between the set of momenta { p i , p k , k i } and { p ′ i , p ′ j , p ′ a } by p ′ i = p i + k i − − xx p k , p ′ j = − k i , p ′ a = − x p k , (163)with x = p i p k + p k k i p i p k + p i k i + p k k i + m i − m ′ i + m ′ j . (164)Again, the inverse mapping { p ′ i , p ′ j , p ′ a } → { p i , p k , k i } coincides with the mapping defined ineq.( 71), when restricted to { p ′ i , p ′ j , p ′ a } → { p i , p k } .Expressing the measure for the unresolved phase space as an integration over ~ k i we find d f unres = d D − k i ( p ) D − ( − k i , ) J , (165)with J = p a p i p ′ a p ′ i q (cid:0) E ′ i (cid:1) q ( x ) q ( − x ) . (166)36 kk q i − q i q i +1 ~q i p i p k q c ~q c ~kk q i − q i q i +1 ~q i − p i − p k Figure 6: The integration regions for a final-initial antenna for the case where particle i is in thefinal state and particle k is in the initial state. The upper picture corresponds to the dipole withemitter i and spectator k , the lower picture corresponds to the dipole with emitter k and spectator i . 37et us now turn to the dipole E k , i with emitter k and spectator i . In the collinear limit we requirethat the momentum k i has positive energy. Thus, we use the loop-tree duality formula for theforward hyperboloids of eq. (35) to convert the loop integrals into phase space integrals. We thenrelate the real emission integrals to the virtual integrals. Following the same procedure as in thefinal-final case we find p ′ i = p i − k i − − xx p k , p ′ j = k i , p ′ a = − x p k , (167)with x = p i p k − p k k i p i p k − p i k i − p k k i + m ′ j . (168)Note that in this case particle i is the spectator and we have m ′ i = m i .The geometric situation for the integration over the on-shell hyperboloids for a final-initialantenna for the case where particle i is in the final state and particle k is in the initial state issketched in fig. (6). The upper picture shows the contribution from the virtual approximationterms with emitter i and spectator k in the massless case. The integration is over three backwardlight-cones with origins at q i − , q i and q i + . The soft singularity resides in the integration over thebackward light-cone with origin at q i at the origin q i and is indicated by a red dot. The collinearsingularities in the virtual terms occur on the lines between q i − and q i (collinear singularity of i )and between q i and q i + (collinear singularity of k ). The virtual collinear regions are indicated inblue. There is a cancellation of singularities within the virtual dual contributions in the regionswhere two propagators are on-shell and have the same sign in the energy component. Theseregions are indicated in green. There is also a cancellation of singularities within the virtualdual contributions at the point q c . The integration region for the real approximation term is thebackward light-cone with origin at q i . The collinear singular region for the real approximationterm with emitter i is the line segment between q i − and q i and matches with the correspondingline segment from the virtual term.The lower picture shows the corresponding integration regions, where the roles of emitterand spectator are exchanged, i.e. emitter k and spectator i . Note that the soft and the virtualcollinear singularities occur in the same regions of D -dimensional loop momentum space. Theintegration region for the real approximation term is the forward light-cone with origin at q i . Thecollinear singular region for the real approximation term with emitter k is now the line segmentindicated in purple. Note that the real collinear singular region (purple line) does not match withthe virtual collinear singular region (blue line segment between q i and q i + ). This mismatch iscompensated by the collinear counterterm for initial-state partons. We now consider an initial-initial antenna. The momenta p i and p k are outgoing momenta withnegative energies. In order to match the notation of section 3 we set p a = − p i and p b = − p k .38hus p a and p b have positive energies. Let us look at the virtual dipole E i , k . In the collinearlimit we require that the momentum ( − k i ) has positive energy. Therefore we use the loop-treeduality formula for the backward hyperboloids of eq. (36) to convert the loop integrals into phasespace integrals. Next, we relate the integration for the real dipole D a ′ j ′ , b ′ to the integration forthe virtual dipole E i , k . We recall that we use the notation p i = − p a , p ′ i = − p ′ a , p k = − p b and p ′ k = − p ′ b . We set p ′ a = − x p i , p ′ j = − k i , p ′ b = − p k , (169)with x = p i p k − p i k i p i p k + p k k i . (170)All final state momenta are transformed as p ′ l = L − p l , (171)where L − is the inverse Lorentz transformation to eq. (81). Explicitly we have (cid:0) L − (cid:1) µ n = g µ n + a (cid:0) K µ + ˜ K µ (cid:1) (cid:0) K n + ˜ K n (cid:1) + a (cid:0) K µ + ˜ K µ (cid:1) K n + a K µ (cid:0) K n + ˜ K n (cid:1) + a K µ K n . (172)The momenta K and ˜ K are given by K = p ′ a + p ′ b − p ′ j , ˜ K = p a + p b . (173)The coefficients are a = K (cid:0) ˜ K − K (cid:1) − ˜ K (cid:0) K + ˜ K (cid:1) , a = K − K (cid:0) ˜ K − K (cid:1) − ˜ K (cid:0) K + ˜ K (cid:1) , a = K − K − (cid:0) K + ˜ K (cid:1) (cid:0) ˜ K − K (cid:1) − ˜ K (cid:0) K + ˜ K (cid:1) , a = (cid:0) K + ˜ K (cid:1) (cid:0) ˜ K − K (cid:1) − ˜ K (cid:0) K + ˜ K (cid:1) . (174)Expressing the measure for the unresolved phase space as an integration over ~ k i we have d f unres = d D − k i ( p ) D − ( − k i , ) J , (175)with J = q ( x ) q ( − x ) . (176)39 kk q i +1 q i q i − ~q i p k p i ~q th ~kk q i +1 q i q i − ~q i − p k − p i ~q th Figure 7: The integration regions for an initial-initial antenna. The upper picture corresponds tothe dipole with emitter i and spectator k , the lower picture corresponds to the dipole with emitter k and spectator i . 40et us now turn to the dipole E k , i with emitter k and spectator i . In the collinear limit we requirethat the momentum k i has positive energy. Thus, we use the loop-tree duality formula for theforward hyperboloids of eq. (35) to convert the loop integrals into phase space integrals. Inrelating the real emission integrals to the virtual integrals we set now p ′ a = − p i , p ′ j = k i , p ′ b = − x p k , (177)with x = p i p k + p k k i p i p k − p i k i . (178)All final state momenta are transformed as p ′ l = L − p l . (179)The geometric situation for the integration over the on-shell hyperboloids for an initial-initialantenna is sketched in fig. (7). The upper picture shows the contribution from the virtual ap-proximation terms with emitter i and spectator k in the massless case. The integration is overthree backward light-cones with origins at q i − , q i and q i + . The soft singularity resides in theintegration over the backward light-cone with origin at q i at the origin q i and is indicated bya red dot. The collinear singularities in the virtual terms occur on the lines between q i − and q i (collinear singularity of i ) and between q i and q i + (collinear singularity of k ). The virtualcollinear regions are indicated in blue. There is a cancellation of singularities within the virtualdual contributions in the regions where two propagators are on-shell and have the same sign inthe energy component. These regions are indicated in green and purple. There is a thresholdsingularity (indicated by an orange dot) at ~ q th . The threshold singularity is avoided by contourdeformation. The integration region for the real approximation term is the backward light-conewith origin at q i . The collinear singular region for the real approximation term with emitter i is the line segment indicated in purple. Note that the real collinear singular region (purple line)does not match with the virtual collinear singular region (blue line segment between q i − and q i ).This mismatch is compensated by the collinear counterterm for initial-state partons.The lower picture shows the corresponding integration regions, where the roles of emitter andspectator are exchanged, i.e. emitter k and spectator i . Note that the soft and the virtual collinearsingularities occur in the same regions of D -dimensional loop momentum space. The integrationregion for the real approximation term is the forward light-cone with origin at q i . The collinearsingular region for the real approximation term with emitter k is the line segment indicated inpurple. Note that the real collinear singular region (purple line) does not match with the virtualcollinear singular region (blue line segment between q i and q i + ). This mismatch is compensatedby the collinear counterterm for initial-state partons.41 σ AV , IR dσ AR dσ VCT , IR softcollinear, longitudinal collinear, transversal Figure 8: The cancellation of infrared singularities within an antenna involving only final-stateparticles.
It is worth to summarise how infrared singularities cancel within an antenna. Let us first con-sider an antenna involving only final-state particles. In this case we have contributions fromthree terms: The virtual approximation term d s AV , IR , the real approximation term d s AR and theinfrared part from the field renormalisation constants, given by d s VCT , IR . The soft singularity ofthe antenna cancels between the virtual part d s AV , IR and the real part d s AR . The contribution fromthe field renormalisation constants does not contain any soft singularity. The real part has inaddition collinear singularities, where the two particles in the collinear splitting have transversepolarisations. These collinear singularities cancel with corresponding singularities from the fieldrenormalisation constants. On the other hand, the virtual part has collinear singularities, whereone of the two particles in the collinear splitting has a longitudinal polarisation. These singu-larities cancel as well with corresponding singularities from the field renormalisation constants.This mechanism is summarised in fig. (8).Let us now consider an antenna with initial-state particles. We have contributions from fourterms: As before, there are contributions from the virtual approximation term d s AV , IR , the realapproximation term d s AR and the infrared part from the field renormalisation constants, givenby d s VCT , IR . In addition we have a contribution from the collinear subtraction term d s C , whichsplits into an x -dependent convolution part d s CR and an end-point part d s CCT . As before, the softsingularity of the antenna cancels between the virtual part d s AV , IR and the real part d s AR . Howeverthe mechanism for initial-state collinear singularities is different: The collinear singularity withtwo transverse polarisations from the real part d s AR cancels with the corresponding singularityfrom the x -dependent convolution part d s CR . The collinear singularity with two transverse polar-isations from the end-point contribution d s CCT cancels with the corresponding singularity fromthe field renormalisation constants in d s VCT , IR , and finally the collinear singularity, where one ofthe two particles in the collinear splitting has a longitudinal polarisation cancels between d s VCT , IR σ AV , IR dσ AR dσ VCT , IR dσ C softcollinear, longitudinal collinear, transversal, x -dependentcollinear, transversal,end-point Figure 9: The cancellation of infrared singularities within an antenna involving initial-state par-ticles.and d s AV , IR . This is summarised in fig. (9). In this sub-section we consider the pure ultraviolet contribution h O i NLO I + L , UV . = Z n O n Z loop (cid:0) d s AV , UV + d s VCT , UV (cid:1) . (180)The term d s AV , UV contains the ultraviolet approximation terms for the vertices and the propaga-tors. To give an example, let us consider the ultraviolet approximation term for the quark-gluonvertex. This vertex has a leading colour contribution and a subleading colour contribution. Thesubtraction term for the leading colour contribution is given by [25] V V , UV qqg , lc (cid:0) µ , µ (cid:1) = iS − e µ e Z d D k ( p ) D i " g µ (cid:0) ¯ k − µ (cid:1) + ( − e ) ¯ k / ¯ k µ − µ g µ (cid:0) ¯ k − µ (cid:1) . (181)The quantity µ UV is an arbitrary mass. For the subleading colour contribution we have V V , UV qqg , sc (cid:0) µ , µ (cid:1) = iS − e µ e Z d D k ( p ) D i " ( − e ) ¯ k / g µ ¯ k / + µ g µ (cid:0) ¯ k − µ (cid:1) . (182)A complete set of ultraviolet approximation terms can be found in ref. [25]. The terms propor-tional to µ in the numerator are not divergent, but ensure that the finite part of the integratedexpression is proportional to the pole part. For the quark-gluon vertex approximation term this43mplies V V , UV qqg , lc = i ( p ) g µ ( ) (cid:18) e − ln µ µ (cid:19) + O ( e ) , V V , UV qqg , sc = i ( p ) g µ ( − ) (cid:18) e − ln µ µ (cid:19) + O ( e ) . (183)Now let us turn to d s VCT , UV . By construction, d s VCT , UV contains for all renormalisation constants(field renormalisation, coupling renormalisation and mass renormalisation) the terms, which leadexactly to the 1 / e UV divergences. In addition, d s VCT , UV contains finite terms from couplingrenormalisation and mass renormalisation. Let us first focus on the ultraviolet divergent terms.Note that we may obtain these terms from the renormalisation counterterms for all vertices andpropagators. The set of vertices and propagators for d s VCT , UV will correspond exactly to the setof vertices and propagators for d s AV , UV . This makes it easy to find an integral representationfor d s VCT , UV . We may use the integral representation for the ultraviolet approximation terms,substitute µ UV → µ and add a minus sign to obtain the integral representation for the contributionsto d s VCT , UV . For example V CT , UV qqg , lc (cid:0) µ (cid:1) = − V V , UV qqg , lc (cid:0) µ , µ (cid:1) , V CT , UV qqg , sc (cid:0) µ (cid:1) = − V V , UV qqg , sc (cid:0) µ , µ (cid:1) . (184)Choosing µ UV = µ ensures, that the counterterms just subtract out the 1 / e -pole. This can easilybe seen from eq. (183), yielding V CT , UV qqg , lc = i ( p ) g µ ( − ) e + O ( e ) , V CT , UV qqg , sc = i ( p ) g µ e + O ( e ) . (185)The terms d s AV , UV and d s VCT , UV contain as far as divergent contributions are concerned onlyultraviolet divergences. Moreover, the singular behaviour in the ultraviolet is up to a sign exactlyequal. Therefore, the sum of d s AV , UV and d s VCT , UV is integrable in the ultraviolet region andhence integrable everywhere.Let us now turn our attention to finite terms from renormalisation of the masses or couplings.In the MS-scheme these finite terms are absent. Therefore, if we take the strong coupling g andthe quark masses m in the MS-scheme nothing needs to be done. For the strong coupling the MS-scheme is the conventional choice. However, for the quark masses the use of the on-shell schemeis an alternative to the MS-scheme. We now discuss how to implement the on-shell scheme forquark masses in our framework. This will require only minor modifications. We start with the44ltraviolet approximation term for a massive quark propagator: − i S V , UV (cid:0) µ , µ (cid:1) = − iS − e µ e Z d D k ( p ) D i " − ( − e ) (cid:0) Q / + ¯ k / (cid:1) + (cid:0) − e (cid:1) m (cid:0) ¯ k − µ (cid:1) − ( − e ) ¯ k · ( p − Q ) ¯ k / (cid:0) ¯ k − µ (cid:1) + µ ( p / − m ) (cid:0) ¯ k − µ (cid:1) . (186)This approximation term integrates to − i S V , UV = − i ( p ) ( − p / + m ) (cid:18) e − ln µ µ (cid:19) + O ( e ) . (187)In order to find S CT , UV corresponding to a mass definition in the on-shell scheme one proceedsas before (i.e. adding an extra minus sign and substituting µ UV → µ ) and one adds an additionalfinite term, using the fact that S − e µ e Z d D k ( p ) D i − µ (cid:0) ¯ k − µ (cid:1) = ( p ) e eg E G ( + e ) (cid:18) µ µ (cid:19) − e = ( p ) [ + O ( e )] . (188)The required finite term is easily found by recalling that the counter-term leads to the Feynmanrule = i h ( p / − m ) Z ( ) − mZ ( ) m , on − shell i , (189)with Z ( ) m , on − shell given in eq. (100). Thus − i S CT , UV (cid:0) µ (cid:1) = − iS − e µ e Z d D k ( p ) D i " ( − e ) (cid:0) Q / + ¯ k / (cid:1) − (cid:0) − e (cid:1) m (cid:0) ¯ k − µ (cid:1) + ( − e ) ¯ k · ( p − Q ) ¯ k / (cid:0) ¯ k − µ (cid:1) − µ ( p / − m ) (cid:0) ¯ k − µ (cid:1) − µ m (cid:0) ¯ k − µ (cid:1) (cid:18) − + m µ (cid:19) (190)integrates to − i S CT , UV = − i ( p ) (cid:20) ( p / − m ) e + m (cid:18) − + m µ (cid:19)(cid:21) + O ( e ) (191)and implements the on-shell scheme for the quark mass.45 Contour deformation
Up to now we have defined integral representations for all contributions and maps between dif-ferent contributions such that the combination can be written as a single integral over Z d D − k ( p ) D − .... (192)We have achieved that all singularities, which would produce poles in the dimensional regulari-sation parameter e cancel locally at the integrand level. Therefore we may take the limit D → Z d k ( p ) .... (193)However, this does not yet imply that we can simply or safely integrate each of the three com-ponents of the loop momentum ~ k from minus infinity to plus infinity along the real axis. In thevirtual part there is still the possibility that some of the loop propagators go on-shell for real val-ues of the loop momentum. We have seen examples of these threshold singularities in the case ofa final-final antenna in fig. (5) or in the case of an initial-initial antenna in fig. (7). In the case ofa final-initial antenna we have cancellation between the various dual integrands. The thresholdsingularities are avoided by a deformation of the integration contour into the complex plane. Forthe loop three-momentum we write ~ k = ~ ˜ k + i ~ k (cid:16) ~ ˜ k (cid:17) , (194)where ~ ˜ k is real and ~ k ( ~ ˜ k ) defines the deformation. This introduces a Jacobian in integral over thevirtual approximation terms: Z d k ( p ) f (cid:16) ~ k (cid:17) = Z d ˜ k ( p ) (cid:12)(cid:12)(cid:12)(cid:12) ¶ k i ¶ ˜ k j (cid:12)(cid:12)(cid:12)(cid:12) f (cid:16) ~ k (cid:16) ~ ˜ k (cid:17)(cid:17) . (195)The deformation has to satisfy three requirements:1. The deformation has to match the i d -prescription of the dual propagators in eq. (35) andeq. (36).2. The deformation has to respect the ultraviolet power counting.3. The deformation has to vanish for soft and collinear singularities in order not to spoil thelocal cancellation of these singularities.Algorithms for the contour deformation can be found in the literature [22, 27–29, 42].46 Conclusions
In this paper we considered NLO calculations within a numerical approach. The numerical ap-proach employs subtraction terms both for the real emission contribution and the virtual contri-bution, such that the subtracted real emission contribution and the subtracted virtual contributioncan be integrated numerically. The subtraction terms have to be added back. In this paper weshowed that the various subtraction terms can be combined to give an integrable function, whichagain can be integrated numerically. Our motivation is not to improve NLO calculations. AtNLO, all subtraction terms are easily integrated analytically and in practical calculations it ismore efficient to use those. However, the situation is different at NNLO and beyond: There thetask of finding local subtraction terms is manageable, while the analytic integration of the localsubtraction terms is highly non-trivial. It is therefore desirable to have at NNLO and beyond amethod, which integrates the subtraction terms numerically. In order to achieve this, the subtrac-tion terms have to be combined in the right way with appropriate mappings between them. Thereare some subtleties related to field renormalisation and initial-state collinear singularities. In thispaper we studied these subtleties at NLO and obtained a clear picture how all singularities cancelat the integrand level.At a more technical level the new results of this paper include a mapping between virtualconfigurations and real configurations for all relevent cases, including initial-state particles andfinal-state massive particles. In addition we derived an integral representation for the collinearsubtraction term for initial-state particles, which matches locally with the singularities of theother contributions. Furthermore we presented a method on how to implement a mass definitionin the on-shell scheme within the numerical approach.With the results of this paper we can now split a NLO calculation into three parts, the sub-tracted virtual part, the subtracted real part and the combined subtraction terms. All three partscan now be evaluated numerically. Does this eliminate the need of any analytic calculation ofan integral? Not quite. While it is true that infrared singularities cancel between the real andthe virtual contributions at the integrand level and no integral needs to be computed analyti-cally for this to happen, there are singularities, which are absorbed into a redefinition of theparameters. These are the ultraviolet divergences, treated by renormalisation, and initial-statecollinear singularities, treated by a redefinition of the parton distribution functions. This intro-duces a scheme-dependence and each scheme has a well-defined prescription which finite termsare absorbed in a redefinition of the parameters and which not. The numerical approach has toreproduce the correct finite terms. This requires to add certain finite terms to the integral rep-resentations of some quantities in order to reproduce the correct finite terms of a given scheme.In order to find the correct finite terms for the integral representations we have to perform somesimple integrals analytically. The required integrals are tadpole integrals Z d D k ( p ) D ( k − m ) n , (196)47or the virtual case and Euler beta-function type integrals Z dx x n − e ( − x ) − e (197)in the real case. These two integrals are significantly simpler than the integrals required tointegrate all subtraction terms analytically and we might expect that this remains true at NNLOand beyond. Acknowledgements
This research was supported in part by the National Science Foundation under Grant No. NSFPHY11-25915.
A Polarisation vectors and polarisation spinors
We define the light-cone coordinates of a four-vector p µ as p + = p + p , p − = p − p , p ⊥ = p + ip , p ⊥ ∗ = p − ip . (198)In terms of the light-cone components of a light-like four-vector, the corresponding masslessspinors h p ± | and | p ±i can be chosen as | p + i = e − i f p | p + | (cid:18) − p ⊥ ∗ p + (cid:19) , | p −i = e − i f p | p + | (cid:18) p + p ⊥ (cid:19) , h p + | = e − i f p | p + | ( − p ⊥ , p + ) , h p −| = e − i f p | p + | ( p + , p ⊥ ∗ ) , (199)where the phase f is given by p + = | p + | e i f , ≤ f < p . (200)If the Cartesian coordinates p , p , p and p are real numbers, we have | p ±i † = e i f h p ±| , h p ±| † = e i f | p ±i , e i f = ± . (201)Spinor products are denoted as h pq i = h p − | q + i , [ qp ] = h q + | p −i . (202)Let q be a light-like four-vector. We define polarisation vectors for the gluons by e + µ = h q − | s µ | p −i√ h qp i , e − µ = h q + | ¯ s µ | p + i√ [ pq ] , (203)48ith s µ = ( ,~ s ) and ¯ s µ = ( , − ~ s ) , where ~ s = ( s , s , s ) are the Pauli matrices. The depen-dence on the reference four-vector q drops out in gauge invariant quantities. Under complexconjugation we have (cid:0) e + µ (cid:1) ∗ = e − µ , (cid:0) e − µ (cid:1) ∗ = e + µ . (204)For the spin sum we have (cid:229) l (cid:16) e l µ (cid:17) ∗ e ln = − g µ n + p µ q n + q µ p n p · q . (205)The reference four-vector q can be used to project any not necessarily light-like four-vector P ona light-like four-vector P ♭ : P ♭ = P − P P · q q . (206)The four-vector P ♭ satisfies ( P ♭ ) =
0. Let P be a four-vector satisfying P = m . We define thespinors associated to massive fermions by u ± = h P ♭ ± | q ∓i ( P / + m ) | q ∓i , ¯ u ± = h q ∓ | P ♭ ±i h q ∓ | ( P / + m ) , v ∓ = h P ♭ ± | q ∓i ( P / − m ) | q ∓i , ¯ v ∓ = h q ∓ | P ♭ ±i h q ∓ | ( P / − m ) . (207)These spinors satisfy the Dirac equations ( p / − m ) u l = , ¯ u l ( p / − m ) = , ( p / + m ) v l = , ¯ v l ( p / + m ) = , (208)the orthogonality relations ¯ u ¯ l u l = m d ¯ ll , ¯ v ¯ l v l = − m d ¯ ll , (209)and the completeness relation (cid:229) l u l ¯ u l = p / + m , (cid:229) l v l ¯ v l = p / − m . (210)We further have ¯ u ¯ l g µ u l = p µ d ¯ ll , ¯ v ¯ l g µ v l = p µ d ¯ ll . (211)In the massless limit the definition reduces to¯ u ± = ¯ v ∓ = h p ± | , u ± = v ∓ = | p ±i . (212)49 Self-energies and field renormalisation
B.1 The gluon self-energy
We first consider the gluon self-energy. With the notation k = k + p / k = k − p / − i P µ n (cid:0) p , µ (cid:1) = − ig S − e µ − D Z d D k ( p ) D i k k (cid:8) C A (cid:2) − p g µ n + p µ p n − ( − e ) k µ k n + ( − e ) g µ n (cid:0) k + k (cid:1)(cid:21) + T R N f (cid:2) p g µ n − p µ p n + k µ k n − g µ n (cid:0) k + k (cid:1)(cid:3)(cid:27) . (213)The self-energy may be written as − i P µ n = − i (cid:0) − p g µ n + p µ p n (cid:1) P (cid:0) p (cid:1) . (214)An analytic calculation of P ( p ) gives P (cid:0) p (cid:1) = g " b − C A + (cid:18) C A + T R N f (cid:19) e − e B (cid:0) p , , (cid:1) , (215)where B ( p , , ) is the scalar two-point function with masses m = m =
0, given for p = B (cid:0) p , , (cid:1) = ( p ) e eg E G ( e ) G ( − e ) G ( − e ) (cid:18) − p µ (cid:19) − e . (216)For p = B ( , , ) = . (217)The one-loop contribution to the field renormalisation constant Z g is given by Z ( ) g = P ( ) . (218)In dimensional regularisation this contribution is zero, due to a cancellation between ultravioletand infrared parts. Keeping track of the divergent ultraviolet and infrared parts we may write thiszero as Z ( ) g = a s p ( C A − b ) (cid:18) e IR − e UV (cid:19) . (219)Let us denote by P µ n UV ( p , µ , µ ) an ultraviolet approximation term to the one-loop self-energy. P µ n UV has the integral representation − i P µ n UV (cid:0) p , µ , µ (cid:1) = − ig S − e µ − D Z d D k ( p ) D i P µ n UV (cid:0) ¯ k , p , µ (cid:1)(cid:0) ¯ k − µ (cid:1) , (220)50here P µ n UV is a polynomial in ¯ k and p . The explicit expression can be found in ref. [25]. Here,we will only need the fact that with the choice µ UV = µ the ultraviolet subtraction term integratesto − i P µ n UV (cid:0) p , µ , µ (cid:1) = − i (cid:0) − p g µ n + p µ p n (cid:1) Z ( ) g , UV + O ( e ) , (221)with Z ( ) g , UV = a s p ( C A − b ) (cid:18) − e UV (cid:19) . (222)Let us now define the quantity ˆ P µ n ( p ,~ p ) ˆ P µ n (cid:0) p ,~ p (cid:1) = − g µ r + p µ q r + q µ p r p · q − p ( pq ) q µ q r ! (cid:0) − i P rs + i P rs UV (cid:1) (cid:18) − ig sn p (cid:19) . (223)Here, q is a light-like reference vector ( q = p is light-like. The loop integration inherent in ( P rs − P rs UV ) is by definition of P rs UV ultravioletfinite. Thus ˆ P µ n is finite for e <
0. It is also finite for e =
0, provided p =
0. For e = p = / p -pole from the definition in eq. (223). The former infrared singularity we would like to combinewith corresponding singularities from other terms in d s NLO I + L , IR , the latter pole requires specialtreatment.If we sandwich the analytic expression for ˆ P µ n ( p ,~ p ) between two amplitudes, where thepolarisation vector of the gluon has been amputated, we find for e < p → (cid:16) A µ ( ) ∗ ˆ P µ n (cid:0) p ,~ p (cid:1) A n ( ) (cid:17) = (cid:16) Z ( ) g − Z ( ) g , UV (cid:17) (cid:12)(cid:12)(cid:12) A ( ) (cid:12)(cid:12)(cid:12) + O ( e ) . (224)Thus, this expression contains exactly the terms from the field renormalisation constants, whichlead to the 1 / e IR divergences or finite terms. This is the contribution which we would like toinclude in d s VCT , IR . Note that the 1 / p -pole cancels after the (analytic) loop integration. How-ever, we would like to have an expression, where we can take the limit p → / p -singularity from the propagator. In order to arrive at anexpression suitable for numerical evaluation we will use a dispersion relation in the variable p for ˆ P µ n [44, 55]. Two properties of ˆ P µ n are relevant: First, for | p | < | ~ p | the function ˆ P µ n ( p ,~ p ) is analytic in p . Secondly, for large | p | the quantity ˆ P µ n behaves like a constant up to loga-rithmic corrections. Therefore we will use a dispersion relation with a subtraction. The startingpoint is Cauchy’s theorem:ˆ P µ n (cid:0) p ,~ p (cid:1) p − µ = p i I d ˜ p ˆ P µ n (cid:0) ˜ p ,~ p (cid:1) ( ˜ p − p ) (cid:0) ˜ p − µ (cid:1) , (225)with ˜ p = ( ˜ p ,~ p ) and where the contour is a small counter-clockwise circle around p . The factor1 / ( p − µ ) improves the large | p | -behaviour. µ is an arbitrary parameter, which may be51 e˜ p Im˜ p | ~p |−| ~p | Figure 10: The integration contour for the dispersion relation. The two small circles enclose thepoles of 1 / ( ˜ p − µ ) .complex. Ignoring the 1 / ( p · q ) -terms, which will vanish when contracted into the amplitude, wemay deform the contour as in fig. 10 and obtainˆ P µ n (cid:0) p ,~ p (cid:1) = p − µ p i ¥ Z | ~ p | d ˜ p " Disc ˆ P µ n (cid:0) ˜ p ,~ p (cid:1) ( ˜ p − p ) (cid:0) ˜ p − µ (cid:1) + Disc ˆ P µ n (cid:0) − ˜ p ,~ p (cid:1) ( − ˜ p − p ) (cid:0) ˜ p − µ (cid:1) (226) − (cid:0) p − µ (cid:1) ˆ P µ n (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = √ | ~ p | + µ − (cid:0) p − µ (cid:1) ˆ P µ n (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = − √ | ~ p | + µ . The last two terms subtract the residues at ˜ p = ± q | ~ p | + µ . The factor 1 / ( p − µ ) ensuresthat the half-circles at infinity give a vanishing contribution. Let us now consider the disconti-nuity Disc ˆ P µ n . We first note, that the ultraviolet approximation term P rs UV does not contributeto Disc ˆ P µ n . The ultraviolet approximation term P rs UV contains only tadpole integrals, which areindependent of p . Let us further denote by P µ n ( k , p , µ ) the numerator of the integrand of thegluon self-energy, i.e. − i P µ n (cid:0) p , µ (cid:1) = Z d D k ( p ) D i P µ n (cid:0) k , p , µ (cid:1) k k , (227) P µ n (cid:0) k , p , µ (cid:1) = − ig S − e µ − D (cid:8) C A (cid:2) − p g µ n + p µ p n − ( − e ) k µ k n + ( − e ) g µ n (cid:0) k + k (cid:1)(cid:21) + T R N f (cid:2) p g µ n − p µ p n + k µ k n − g µ n (cid:0) k + k (cid:1)(cid:3)(cid:27) , and define ˆ N µ n ( p , k , q ) in analogy with eq. (223):ˆ N µ n ( p , k , q ) = − g µ r + p µ q r + q µ p r p · q − p ( pq ) q µ q r ! P rs (cid:0) k , p , µ (cid:1) (cid:18) − ig sn p (cid:19) . (228)52orking out the discontinuity gives us then an expression which we can evaluate at p = P µ n (cid:0) p ,~ p (cid:1)(cid:12)(cid:12) p = = (229)1 ( p ) D − Z d D k Z d D k d + (cid:0) k (cid:1) d − (cid:0) k (cid:1) d D − (cid:16) ~ p − ~ k + ~ k (cid:17) ˆ N µ n ( ˜ p , k , q )( ˜ p − p ) (cid:16) − ˜ p µ (cid:17) + ( p ) D − Z d D k Z d D k d − (cid:0) k (cid:1) d + (cid:0) k (cid:1) d D − (cid:16) ~ p − ~ k + ~ k (cid:17) ˆ N µ n ( ˜ p , k , q )( ˜ p − p ) (cid:16) − ˜ p µ (cid:17) + µ ˆ P µ n (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = √ | ~ p | + µ + µ ˆ P µ n (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = − √ | ~ p | + µ , with ˜ p = ( k − k ,~ p ) and k = / ( k + k ) . Note that the ultraviolet approximation term P rs UV enters in ˆ P µ n , but not ˆ N µ n . Further note that in the last two terms in eq. (229) the UV-subtractedself-energy is evaluated at ˜ p = µ . These two terms give a finite contribution. The infraredsingularity is contained in the first two terms of eq. (229), for p > p < d D − k (or alternatively d D − k ):ˆ P µ n (cid:0) p ,~ p (cid:1)(cid:12)(cid:12) p = = g S − e µ e Z d D − k ( p ) D − [ X g ( p , k )] µ n . (230)This defines [ X g ( p , k )] µ n . The explicit expression is rather long and not reproduced here. How-ever, it can be extracted in a straightforward way from eq. (229). B.2 The massless quark self-energy
The self-energy for a massless quark is given by − i S ab = − ig C F S − e µ − D Z d D k ( p ) D i h − ( − e ) k / ab i k k . (231)We may write − i S ab = − ip / ab S ′ (cid:0) p (cid:1) . (232)An analytic calculation of S ′ ( p ) gives S ′ (cid:0) p (cid:1) = − a s p C F ( − e ) B (cid:0) p , , (cid:1) . (233)The one-loop contribution to the field renormalisation constant Z q for massless quarks is givenby Z ( ) q = S ′ ( ) . (234)53gain, this contribution is zero in dimensional regularisation due to a cancellation between ul-traviolet and infrared parts. Keeping track only of divergent ultraviolet and infrared parts onefinds Z ( ) q = a s p C F (cid:18) e IR − e UV (cid:19) . (235)Let us denote by S ab UV ( p , µ , µ ) an ultraviolet approximation term to the one-loop self-energy. S ab UV has the integral representation − i S ab UV (cid:0) p , µ , µ (cid:1) = − ig S − e µ − D Z d D k ( p ) D i P ab UV (cid:0) ¯ k , p , µ (cid:1)(cid:0) ¯ k − µ (cid:1) , (236)where P ab UV is a polynomial in ¯ k and p . The explicit expression can be found in ref. [25]. Withthe choice µ UV = µ the ultraviolet subtraction term integrates to − i S ab UV (cid:0) p , µ , µ (cid:1) = − ip / ab Z ( ) q , UV + O ( e ) , (237)with Z ( ) q , UV = a s p C F (cid:18) − e UV (cid:19) . (238)In analogy with the gluon self-energy let us consider the quantityˆ S ab (cid:0) p ,~ p (cid:1) = p / ag (cid:16) − i S gd + i S gd UV (cid:17) ip / db p . (239)For p = (cid:16) A a ( ) ∗ ˆ S ab (cid:0) p ,~ p (cid:1) A b ( ) (cid:17) = (cid:16) Z ( ) q − Z ( ) q , UV (cid:17) (cid:12)(cid:12)(cid:12) A ( ) (cid:12)(cid:12)(cid:12) + O ( e ) . (240)For large | p | the quantity ˆ S ab grows (up to logarithminc corrections) linearly with | p | . As inthe gluon case we will therefore use a dispersion relation with a subtraction. Let us further denoteby P ab ( k , p , µ ) the numerator of the integrand of the quark self-energy, i.e. − i S ab = Z d D k ( p ) D i P ab (cid:0) k , p , µ (cid:1) k k , P ab (cid:0) k , p , µ (cid:1) = − ig C F S − e µ − D h − ( − e ) k / ab i . (241)and define ˆ N ab ( p , k ) in analogy with eq. (239):ˆ N ab ( p , k ) = p / ag P gd (cid:0) k , p , µ (cid:1) ip / db p . (242)54ith the help of a dispersion relation we may re-write ˆ S ab as an expression which we mayevaluate at p = S ab (cid:0) p ,~ p (cid:1)(cid:12)(cid:12) p = = (243)1 ( p ) D − Z d D k Z d D k d + (cid:0) k (cid:1) d − (cid:0) k (cid:1) d D − (cid:16) ~ p − ~ k + ~ k (cid:17) ˆ N ab ( ˜ p , k )( ˜ p − p ) (cid:16) − ˜ p µ (cid:17) + ( p ) D − Z d D k Z d D k d − (cid:0) k (cid:1) d + (cid:0) k (cid:1) d D − (cid:16) ~ p − ~ k + ~ k (cid:17) ˆ N ab ( ˜ p , k )( ˜ p − p ) (cid:16) − ˜ p µ (cid:17) + µ ˆ S ab (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = √ | ~ p | + µ + µ ˆ S ab (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = − √ | ~ p | + µ , with ˜ p = ( k − k ,~ p ) and k = / ( k + k ) .Finally, using the loop-tree duality for the loop integrals in the last two terms, we may re-write all terms as an integration over d D − k :ˆ S ab (cid:0) p ,~ p (cid:1)(cid:12)(cid:12) p = = g S − e µ e Z d D − k ( p ) D − (cid:2) X q ( p , k ) (cid:3) ab . (244)This defines [ X q ( p , k )] ab . B.3 The massive quark self-energy
The self-energy for a massive quark is given by − i S ab = − ig C F S − e µ − D Z d D k ( p ) D i h − ( − e ) k / ab + (cid:0) − e (cid:1) m d ab i(cid:0) k − m (cid:1) k . (245)One expands the self-energy around p = m : − i S ab = − i h d ab S ( m ) + (cid:16) p / ab − m d ab (cid:17) S ′ ( m ) + ... i . (246)Then Z ( ) m , on − shell = − m S ( m ) , Z ( ) Q = S ′ ( m ) , (247)and one finds Z ( ) m , on − shell = a s p C F (cid:18) − e UV − + m µ (cid:19) , Z ( ) Q = a s p C F (cid:18) − e UV − e IR − + m µ (cid:19) . (248)55et us denote by S ab UV ( p , µ , µ ) an ultraviolet approximation term to the one-loop self-energy. S ab UV has the integral representation − i S ab UV (cid:0) p , µ , µ (cid:1) = − ig S − e µ − D Z d D k ( p ) D i P ab UV (cid:0) ¯ k , p , µ (cid:1)(cid:0) ¯ k − µ (cid:1) , (249)where P ab UV is a polynomial in ¯ k and p . With the choice µ UV = µ and by adding a suitable chosenfinite term we can ensure that S ab UV takes into account the ultraviolet divergence and the finite partdue to the on-shell mass renormalisation. The explicit expression is − i S ab UV (cid:0) p , µ , µ (cid:1) = ig C F S CT , UV , ab (cid:0) µ (cid:1) , (250)where S CT , UV is given in eq. (190). The ultraviolet subtraction term integrates to − i S ab UV (cid:0) p , µ , µ (cid:1) = − i h Z ( ) Q , UV p / − (cid:16) Z ( ) Q , UV + Z ( ) m , on − shell (cid:17) m i , (251)with Z ( ) Q , UV = a s p C F (cid:18) − e UV (cid:19) , Z ( ) m , on − shell = a s p C F (cid:18) − e UV − + m µ (cid:19) . (252)Let us consider the quantityˆ S ab (cid:0) p ,~ p (cid:1) = (cid:0) p / ag + m d ag (cid:1) (cid:16) − i S gd + i S gd UV (cid:17) i (cid:0) p / db + m d db (cid:1) p − m . (253)For p = m we haveRe (cid:16) A a ( ) ∗ ˆ S ab (cid:0) p ,~ p (cid:1) A b ( ) (cid:17) = (cid:16) Z ( ) Q − Z ( ) Q , UV (cid:17) (cid:12)(cid:12)(cid:12) A ( ) (cid:12)(cid:12)(cid:12) + O ( e ) . (254)Let us further denote by P ab ( k , p , µ ) the numerator of the integrand of the quark self-energy, i.e. − i S ab = Z d D k ( p ) D i P ab (cid:0) k , p , µ (cid:1) k k , P ab (cid:0) k , p , µ (cid:1) = − ig C F S − e µ − D (cid:20) − ( − e ) k / ab + (cid:18) − e (cid:19) m d ab (cid:21) . (255)and define ˆ N ab ( p , k ) in analogy with eq. (253):ˆ N ab ( p , k ) = (cid:0) p / ag + m d ag (cid:1) P gd (cid:0) k , p , µ (cid:1) i (cid:0) p / db + m d db (cid:1) p − m . (256)56ith the help of a dispersion relation we may re-write ˆ S ab as an expression which we mayevaluate at p = m :ˆ S ab (cid:0) p ,~ p (cid:1)(cid:12)(cid:12) p = m = (257)1 ( p ) D − Z d D k Z d D k d + (cid:0) k − m (cid:1) d − (cid:0) k (cid:1) d D − (cid:16) ~ p − ~ k + ~ k (cid:17) ˆ N ab ( ˜ p , k )( ˜ p − p ) (cid:16) − ˜ p µ (cid:17) + ( p ) D − Z d D k Z d D k d − (cid:0) k − m (cid:1) d + (cid:0) k (cid:1) d D − (cid:16) ~ p − ~ k + ~ k (cid:17) ˆ N ab ( ˜ p , k )( ˜ p − p ) (cid:16) − ˜ p µ (cid:17) + µ ˆ S ab (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = √ | ~ p | + µ + µ ˆ S ab (cid:0) ˜ p ,~ p (cid:1) p ( ˜ p − p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ p = − √ | ~ p | + µ , with ˜ p = ( k − k ,~ p ) and k = / ( k + k ) .Finally, using the loop-tree duality for the loop integrals in the last two terms, we may re-write all terms as an integration over d D − k :ˆ S ab (cid:0) p ,~ p (cid:1)(cid:12)(cid:12) p = m = g S − e µ e Z d D − k ( p ) D − [ X Q ( p , k )] ab . (258)This defines [ X Q ( p , k )] ab . C The momenta mapping from the virtual to the real space
In this appendix we determine the constants a and b in the mapping of eq. (158): p ′ k = a (cid:18) p k − Q · p k Q Q (cid:19) + b Q , p ′ j = − k i , p ′ i = Q + k i − p ′ k , (259)with Q = p i + p k . We require that p ′ i = m ′ i , p ′ k = m ′ k = m k . (260)We may use the first equation to eliminate a , doing so gives us a quadratic equation for b : a b + b b + c = , (261)57ith a = Q n m k Q + h m k ( k i Q ) + ( k i p k ) − ( p k Q ) i Q + k i Q h m k ( k i Q ) − ( p k Q ) − ( p k Q ) ( k i p k ) io , b = Q (cid:2) Q + k i Q (cid:1) h ( p k Q ) − m k Q i (cid:2) Q + k i Q − m ′ i + m ′ j + m k (cid:3) , c = m k Q h(cid:0) Q + k i Q − m ′ i + m ′ j + m k (cid:1) − ( k i p k ) i + m k Q ( k i p k ) ( k i Q ) ( p k Q ) − n Q + (cid:2) k i Q − m ′ i + m ′ j + m k (cid:3) Q − (cid:2) k i Q − m ′ i + m ′ j + m k (cid:3) Q + m k ( k i Q ) o × ( p k Q ) . (262)We thus have b = − b + √ b − ac a , (263)where the sign of the root is fixed by matching on the massless limit. The constant a is thengiven by a = Q h Q + k i Q − m ′ i + m ′ j + m k − (cid:0) Q + k i Q (cid:1) b i Q ( k i p k ) − ( p k Q ) ( k i Q ) . (264) References [1] Z. Kunszt, A. Signer, and Z. Trocsanyi, Nucl. Phys.
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