Contents
Abstract

We present an algorithm for the numerical calculation of one-loop QCD amplitudes. The algorithm consists of subtraction terms, approximating the soft, collinear and ultraviolet divergences of one-loop amplitudes and a method to deform the integration contour for the loop integration into the complex space. The algorithm is formulated at the amplitude level and does not rely on Feynman graphs. Therefore all required ingredients can be calculated efficiently using recurrence relations. The algorithm applies to massless partons as well as to massive partons.

MZ-TH/10-38

Numerical NLO QCD calculations

Sebastian Becker, Christian Reuschle and Stefan Weinzierl

Institut für Physik, Universität Mainz,

D - 55099 Mainz, Germany

1 Introduction

Multi-jet final states play an important role for the experiments at the LHC. An accurate description of jet physics is therefore desirable. Although jet observables can rather easily be modelled at leading order (LO) in perturbation theory, this description suffers several drawbacks. A leading order calculation depends strongly on the renormalisation scale and can therefore give only an order-of-magnitude estimate on absolute rates. Secondly, at leading order a jet is modelled by a single parton. This is a very crude approximation and oversimplifies inter- and intra-jet correlations. The situation is improved by including higher order corrections in perturbation theory.

At present, there are many next-to-leading order (NLO) calculations for processes at hadron colliders, but only a few for or more partons in the final state [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. It is desirable to have NLO calculations for processes in hadron-hadron collisions with in the range of . However, the complexity of the calculation increases with the number of final state particles. For any NLO calculation there are two parts to be calculated: the real and the virtual corrections. Almost without exceptions all examples cited above use the dipole formalism [50, 51, 52, 53] to subtract out the infrared divergences from the real corrections. The subtracted real correction term is then integrable in four dimensions and can be calculated numerically by Monte Carlo techniques. By now there are several implementations for the automated construction of subtraction terms [54, 55, 56, 57, 58, 59]. The required Born amplitudes can be calculated efficiently with the help of recurrence relations[60, 61, 62, 63, 64, 65, 66]. The calculation of the virtual corrections for QCD processes with many external legs has been considered to be a bottle neck for a long time. The past years have witnessed significant progress in this direction. The main lines of investigation focus on a perfection of the traditional Feynman graph approach [67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77] or are based on unitary methods [78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92].

In this paper we would like to discuss a third and purely numerical approach. To this aim we extend the subtraction method to the loop integration of the virtual corrections and we evaluate the subtracted virtual corrections numerically with the help of a suitable chosen contour deformation. The method is formulated in terms of amplitudes and does not rely on Feynman graphs. Therefore all ingredients can be calculated efficiently using recurrence relations. Purely numerical approaches have been discussed in the past [93, 94, 95, 96, 97, 98, 99, 100, 101, 102]. The literature focuses either on individual Feynman graphs and subtraction terms for individual graphs or on a contour deformation for infrared and ultraviolet finite amplitudes, where no subtraction terms are needed. Unfortunately the methods discussed in the literature for the subtraction terms on the one hand and for the contour deformation on the other hand are not compatible with each other and cannot be combined. Furthermore it is not clear if the methods discussed so far in the literature are sufficiently efficient to be applied to multi-parton processes. What is new in this paper is the development of compatible methods for the subtraction terms and the contour deformation and the combination of all relevant aspects into one formalism. Inspired by recent work on the structure of infrared singularities of multi-loop amplitudes [103, 104] we found that the soft and collinear subtraction terms can be formulated at the level of amplitudes, without referring to individual Feynman graphs [105]. This is a significant simplification and opens the door to an efficient implementation. In addition we need subtraction terms for the ultraviolet divergences. In this paper we present a set of ultraviolet subtraction terms which have the expected form of local counterterms and which are particularly well suited for the numerical contour integration in the sense that this set does not introduce additional singularities along the contour. The second important ingredient of our method is an algorithm for the contour deformation. The subtraction terms eliminate only singularities, where the contour is pinched but leave singularities where a deformation into the complex plane is possible. The contour deformation takes care of these remaining singularities. It is highly non-trivial to find a general algorithm which avoids these singularities and which leads to stable Monte Carlo results. We achieve this goal by first introducing Feynman parameters. After Feynman parametrisation the contour deformation of the loop momentum is straightforward. In order to avoid all singularities we have to deform the integration over the loop momentum and the integration over the Feynman parameters. For an amplitude with a large number of external particles we take additional measures to improve the efficiency of the numerical Monte Carlo integration. Our method works for massless and massive particles.

This paper is organised as follows: In section 2 we give an overview of the general ideas behind our approach. In a shortened form these ideas have been presented in [106]. In section 3 we provide a complete list of all subtraction terms. The infrared subtraction terms have been given for the first time in [105], we list them here again with a minor modification. The minor modification in the collinear subtraction terms adapts the subtraction terms to the chosen method of contour deformation. The ultraviolet subtraction terms are new. In section 4 we discuss in detail the contour deformation. Together with the subtraction terms this part constitutes the core of our method. In section 5 we discuss a few points which might help to understand our method or which might help to avoid possible pitfalls. Examples of a possible pitfall are diagrams like massless tadpoles, which give zero in an analytical calculation. These diagrams have to be included in the numerical calculation in order not to spoil the local cancellation of singularities. Section 6 discusses checks and simple examples. NLO results on more complicated processes will be published in a separate publication. Finally, section 7 contains a summary and our conclusions. In an appendix we have documented certain technical aspects of our method.

2 General setup

In this section we give an overview of our method and define our notation. In subsection 2.1 we start from the subtraction method for real corrections and present the extension to the virtual corrections. The pole structure in the dimensional regularisation parameter of one-loop QCD amplitudes is very well understood and recalled in subsection 2.2. Throughout this paper we work with colour ordered amplitudes. These are defined in subsection 2.3. Subsection 2.4 introduces the notation which we use for the kinematics. Our method works at the level of amplitudes. These can be calculated efficiently with the help of recurrence relations without relying on Feynman graphs. Recurrence relations are discussed in subsection 2.5. For the construction of the subtraction terms we need to know from which integration region divergences arise. This is reviewed in subsection 2.6.

2.1 The subtraction method

The starting point for the calculation of an infrared safe observable in hadron-hadron collisions is the following formula:

(1)

In this equation we have written explicitly the sum over the flavours and of the two partons in the initial state. In addition there is a sum over the flavours of all final state particles, which is not shown explicitly. The momenta of the two incoming particles are labelled and , while to denote the momenta of the final state particles. gives the probability of finding a parton with momentum fraction inside the parent hadron . is the flux factor, for massless partons it is given by . The quantity denotes the number of spin degrees of freedom of the parton and equals two for quarks and gluons. Correspondingly, denotes the number of colour degrees of freedom of the parton . For quarks, this number equals three, while for gluons we have eight colour degrees of freedom. The matrix element is summed over all colours and spins. Dividing by the appropriate number of degrees of freedom in the initial state corresponds to an averaging. is the phase space measure for final state particles, including (if appropriate) the identical particle factors. The matrix element is calculated perturbatively.

The contributions at leading and next-to-leading order are written as

(2)

Here a rather condensed notation is used. denotes the Born contribution, whose matrix elements are given by the square of the Born amplitudes with partons . Similar, denotes the real emission contribution, whose matrix elements are given by the square of the Born amplitudes with partons . gives the virtual contribution, whose matrix elements are given by the interference term of the one-loop amplitude , with partons, with the corresponding Born amplitude . denotes a collinear subtraction term, which subtracts the initial state collinear singularities. Taken separately, the individual contributions at next-to-leading order are divergent and only their sum is finite. In order to render the individual contributions finite, such that the phase space integrations can be performed by Monte Carlo methods, one adds and subtracts a suitable chosen piece [50, 51, 52, 53]:

(3)

The term in the first bracket is by construction integrable over the -particle phase space and can be evaluated numerically. The subtraction term can be integrated analytically over the unresolved one-particle phase space. Due to this integration all spin-correlations average out, but colour correlations still remain. In a compact notation the result of this integration is often written as

(4)

The notation indicates that colour correlations due to the colour charge operators still remain. The action of a colour charge operator for a quark, gluon and antiquark in the final state is given by

quark :
gluon :
antiquark : (5)

The corresponding formulae for colour charge operators for a quark, gluon or antiquark in the initial state are

quark :
gluon :
antiquark : (6)

In the amplitude an incoming quark is denoted as an outgoing antiquark and vice versa. The terms with the insertion operators and do not have any poles in the dimensional regularisation parameter and pose no problem for a numerical evaluation. The term lives on the phase space of the -parton configuration and has the appropriate singularity structure to cancel the infrared divergences coming from the one-loop amplitude. Therefore is infrared finite. We emphasise that this cancellation occurs after the loop integration has been performed analytically in dimensions. is given by

(7)

denotes the renormalised one-loop amplitude. It is related to the bare amplitude by

(8)

denotes the ultraviolet counterterm from renormalisation. The bare one-loop amplitude involves the loop integration

(9)

where denotes the integrand of the bare one-loop amplitude. In this paper we extend the subtraction method to the integration over the virtual particles circulating in the loop. To this aim we rewrite eq. (8) as

(10)

The subtraction terms , and are chosen such that they match locally the singular behaviour of the integrand of in dimensions. The first bracket in eq. (10) can therefore be integrated numerically in four dimensions. The term approximates the soft singularities, approximates the collinear singularities and the term approximates the ultraviolet singularities. These subtraction terms have a local form similar to eq. (9):

(11)

The contribution from the terms in the first bracket of eq. (10) can be written as

(12)

The integral on the right-hand side is finite. It is one of the key ingredients of the method proposed here, that this rather complicated and process-dependent integral can be performed numerically with Monte Carlo techniques. We recall that the error of a Monte Carlo integration depends on the variance of the integrand and scales with the number of integrand evaluations like . It is important to note that the error does not depend on the dimension of the integration region. Eq. (12) gives for an observable a contribution to the next-to-leading order prediction. The right-hand side corresponds to a -dimensional integral. (The phase-space integral is -dimensional, the loop integral -dimensional.) In practise this -dimensional integral is done with a single Monte Carlo integration. There is no need to evaluate for a given phase-space point the inner four-dimensional loop integral by a separate Monte Carlo integration. This is essential for the efficiency of the method.

The building blocks of the subtraction terms are process-independent. When adding them back, we integrate analytically over the loop momentum . The result can be written as

(13)

The insertion operator contains the explicit poles in the dimensional regularisation parameter related to the infrared singularities of the one-loop amplitude. These poles cancel when combined with the insertion operator :

(14)

The operator contains, as does the operator , colour correlations due to soft gluons.

2.2 Explicit poles in the dimensional regularisation parameter

The explicit poles in the dimensional regularisation parameter of the individual pieces are well known. These poles are either of ultraviolet or infrared origin. Let us consider an amplitude with external quarks, external anti-quarks and external gluons in massless QCD. We set . Obviously we have . After an analytical integration the poles in the dimensional regularisation parameter of a bare massless one-loop QCD amplitude are given by

is the first coefficient of the QCD -function and given by

(16)

The constants are given by

(17)

The colour factors are as usual

(18)

The poles of eq. (2.2) are cancelled by and the infrared poles obtained from integrating the real emission contribution over the unresolved phase space. The leading order QCD amplitude with partons is proportional to . For a massless QCD amplitude with partons the ultraviolet counterterm is given by

(19)

The insertion operator contains the infrared poles obtained from integrating the real emission contribution over the unresolved phase space. The insertion operator is given in massless QCD by

(20)

with

(21)

2.3 Colour decomposition

Amplitudes in QCD may be decomposed into group-theoretical factors (carrying the colour structures) multiplied by kinematic functions called partial amplitudes [107, 108, 109, 110, 111]. These partial amplitudes do not contain any colour information and are gauge invariant objects. The colour decomposition is obtained by replacing the structure constants by

(22)

which follows from . In this paper we use the normalisation

(23)

for the colour matrices. The resulting traces and strings of colour matrices can be further simplified with the help of the Fierz identity :

(24)

There are several possible choices for a basis in colour space. A convenient choice is the colour-flow basis [112, 113, 54]. This choice is obtained by attaching a factor

(25)

to each external gluon. As an example we consider the colour decomposition of the pure gluon amplitude with external gluons. The colour decomposition of the tree level amplitude may be written in the form

(26)

where the sum is over all non-cyclic permutations of the external gluon legs. The quantities , called the partial amplitudes, contain the kinematical information. They are colour ordered, e.g. only diagrams with a particular cyclic ordering of the gluons contribute. For the convenience of the reader we have listed the colour ordered Feynman rules in appendix A. Similar decompositions exist for all other Born QCD amplitudes. As a further example we give the colour decomposition for a tree amplitude with a pair of quarks:

(27)

where the sum is over all permutations of the external gluon legs. In squaring these amplitudes a colour projector

(28)

has to be applied to each gluon. At the one-loop level the colour decomposition is slightly more involved. Let us start with an example: The colour decomposition of the one-loop gluon amplitude into partial amplitudes reads [111]:

(29)

The one-loop partial amplitudes are again gauge invariant. The subleading partial amplitudes are related to the leading ones by

(30)

where the sum is over all shuffles of the set with the set . It is therefore sufficient to focus on the leading partial amplitudes . The leading partial amplitudes can be decomposed further into smaller objects called primitive amplitudes. The primitive amplitudes are separately gauge invariant. For the example of the one-loop gluon amplitude we have the decomposition of the leading partial amplitude into two primitive amplitudes, which are characterised by the particle content circulating in the loop:

(31)

For the primitive amplitude there is either a gluon or a ghost circulating in the loop, while for the primitive amplitude there is a quark circulating in the loop.

Similar decompositions exist for all other one-loop QCD amplitudes. If the external legs of a one-loop QCD amplitude involve a quark-antiquark pair we can distinguish the two cases where the loop lies to right or to the left of the fermion line if we follow the fermion line in the direction of the flow of the fermion number. We call a fermion line “left-moving” if, following the arrow of the fermion line, the loop is to the right. Analogously, we call a fermion line “right-moving” if, following the arrow of the fermion line, the loop is to the left.

Figure 1: Examples of diagrams regarding left-moving and right-moving primitive amplitudes: Diagrams (a) contributes to the left-moving primitive amplitude, while diagram (b) contributes to the right-moving amplitude.

Examples of diagrams regarding these types of primitive amplitudes are shown in fig. (1). It turns out that in the decomposition into primitive amplitudes a specific quark line is, in all diagrams which contribute to a specific primitive amplitude, either always left-moving or always right-moving [114]. Therefore in the presence of external fermions primitive amplitudes are in addition characterised by the routing of the fermion lines through the amplitude. In summary we can always write a full one-loop QCD amplitude as a linear combination of primitive amplitudes:

(32)

The colour structures are denoted by , while the primitive amplitudes are denoted by . In the colour-flow basis the colour structures are linear combinations of monomials in Kronecker ’s. In order to construct a primitive one-loop amplitude one starts to draw all possible planar one-loop diagrams with a fixed cyclic ordering of the external legs, subject to the constraint that each fermion line is either only left-moving or only right-moving. This set of diagrams is then further divided into the subset of diagrams with a closed fermion line and the ones without. The set of diagrams with a closed fermion line form a separate primitive amplitude.

In the following we will work exclusively with primitive amplitudes. In order to simplify the notation we will drop the subscripts and simply write

(33)

for a primitive one-loop amplitude. The full one-loop amplitude is just the sum of several primitive amplitudes multiplied by colour structures. We stress a few important properties of primitive one-loop amplitudes:

  1. Primitive amplitudes are gauge invariant.

  2. Primitive amplitudes have a fixed cyclic ordering of the external legs and a definite routing of the external fermion lines.

  3. The flavour of each propagator in the loop is unique: Either it is a quark propagator or a gluon/ghost propagator.

Our method exploits these facts. The first property of gauge invariance is crucial for the proof of the method. The second point ensures that there are at maximum different loop propagators in the problem, where is the number of external legs.

2.4 Kinematics

We introduce some notation which will be used throughout this paper. Let be a primitive amplitude with external legs. Since the cyclic ordering of the external partons is fixed, there are only different propagators occurring in the loop integral.

Figure 2: The labelling of the momenta for a primitive one-loop amplitude. The arrows denote the momentum flow.

With the notation as in fig. (2) we define

(34)

We can write the bare primitive one-loop amplitude as

(35)

is a polynomial in the loop momentum . The -prescription in the propagators indicates into which direction the poles of the propagators should be avoided. We denote by the set of indices , for which the propagator in the loop corresponds to a gluon. If we take the subset of diagrams which have the gluon loop propagator and if we remove from each diagram of this subset the loop propagator we obtain a set of tree diagrams. After removing multiple copies of identical diagrams this set forms a Born partial amplitude which we denote by .

We introduce two matrices, which depend on the external momenta (and the internal masses), but not on the loop momentum . The kinematical matrix is a -matrix and is defined by

(36)

The Gram matrix is a -matrix and is defined by

(37)

where the indices and take the values .

2.5 Recurrence relations

Although we use in the proof of our method the fact that a primitive amplitude can be written as a sum of Feynman diagrams, it is important to note that in the final formulae, which enter the numerical Monte Carlo program, only amplitudes occur – and not individual Feynman diagrams. There are several possibilities how these amplitudes can be calculated. A particular efficient method is based on recurrence relations. We first review the recursive method for tree-level partial amplitudes and discuss afterwards the necessary modifications for the computation of the integrand of a one-loop primitive amplitude.

We start with the computation of tree-level partial amplitudes. Berends-Giele type recurrence relations [60] build tree-level partial amplitudes from smaller building blocks, usually called colour ordered off-shell currents. Off-shell currents are objects with on-shell legs and one additional leg off-shell. Momentum conservation is satisfied. It should be noted that off-shell currents are not gauge invariant objects. Recurrence relations relate off-shell currents with legs to off-shell currents with fewer legs. As an example we discuss the pure gluon current, which can be used to calculate the pure gluon amplitude. The recursion starts with :

(38)

is the polarisation vector of the gluon corresponding to the polarisation . The recursive relation states that in the pure gluon off-shell current a gluon couples to other gluons only via the three- or four-gluon vertices :

(39)

where

(40)
Figure 3: The recurrence relation for the gluon current. In an off-shell current particle is kept off-shell. This allows to express an off-shell current with on-shell legs in terms of currents with fewer legs.

and and are the colour ordered three-gluon and four-gluon vertices

(41)

The recurrence relation is shown pictorially in fig. (3). The gluon current is conserved:

(42)

From an off-shell current one easily recovers the on-shell amplitude by removing the extra propagator, taking the leg on-shell and contracting with the appropriate polarisation vector. Similar recurrence relations can be written down for the quark and antiquark currents, as well as the gluon currents in full QCD. The guiding principle is to follow the off-shell leg into the “blob”, representing the sum of all diagrams, and to sum on the r.h.s of the recurrence relation over all vertices involving this off-shell leg and off-shell currents with less external legs.

There are only a few modifications needed to compute the integrand of a one-loop primitive amplitude. We can divide all vertices into two classes: either a vertex is directly connected to the loop or it is not. Again we can follow the off-shell leg into the “blob”. If the first vertex which we encounter belongs to the second class, then there is attached to this vertex a one-loop off-shell current with fewer legs. The other objects attached to this vertex are one or more tree-level off-shell currents. If on the other hand the first vertex belongs to the first class, then two edges of the vertex are connected to loop propagators. The object connected to these two edges is a tree-level off-shell current with two legs off-shell. It can be computed with methods similar to the computation for tree-level off-shell currents with one leg off-shell. We illustrate this principle in fig. (4) for the case of a toy theory with a single field and a single three-valent vertex.

Figure 4: The recurrence relation for the one-loop current in a toy model with a single field and a single three-valent vertex. The one loop currents are represented by an oval with a hole, tree-level currents are represented by an oval without a hole.
Figure 5: The recurrence relation for the ultraviolet subtraction terms in a toy model with a single field and a single three-valent vertex. Objects with a cross represent ultraviolet subtraction terms.

Within our method we also need an ultraviolet subtraction term. The complete ultraviolet subtraction term has a structure similar to ordinary ultraviolet counterterms: We can represent the ultraviolet subtraction term as a sum over diagrams, where each diagram has a tree-structure with exactly one propagator or vertex replaced by a basic ultraviolet subtraction term. The complete ultraviolet subtraction term can again be calculated recursively. As example we consider again the toy model with a single field and a single three-valent vertex. The recursion relation for the ultraviolet subtraction term is shown in fig. (5).

2.6 Singular regions of the integrand of one-loop amplitudes

In order to construct the subtraction terms we have to know in which regions of the integration domain for the loop momentum divergences arise. There are three types of singularities in one-loop amplitudes, which have to be approximated by suitable local subtraction terms. The singularity types are soft, collinear and ultraviolet. Let us briefly review under which conditions these singularities occur [115, 97, 67]. Soft singularities occur when a massless particle is exchanged between two on-shell particles. In the amplitude

(43)

this corresponds to the case

(44)

In that case the propagators , and are on-shell. The singularity comes from the integration region

(45)

A collinear singularity occurs if a massless external on-shell particle is attached to two massless propagators. In the amplitude of eq. (43) this corresponds to

(46)

In that case the propagators and are on-shell. The singularity comes from the integration region

(47)

where is a real variable.

Finally ultraviolet singularities arise when components of the loop momentum tend to infinity. We will work in Feynman gauge throughout this paper. Therefore any loop integral with propagators in the loop is maximally of rank . Power counting arguments show immediately that all diagrams with five or more propagators in the loop are ultraviolet finite. Therefore all ultraviolet divergent diagrams have four or less propagators in the loop. It can be shown that the ultraviolet divergent diagrams are only those which are propagator or vertex corrections. Of course this has to be the case for a renormalisable theory.

3 The subtraction terms

In this section we present all subtraction terms for the numerical calculation of one-loop QCD amplitudes. In subsection 3.1 we give the infrared subtraction terms for massless QCD. The generalisation to massive QCD is presented in subsection 3.2. The ultraviolet subtraction terms for massless and massive particles can be found in subsection 3.3. All subtraction terms are added back in integrated form. The sum of all integrated subtraction terms (plus the ultraviolet counterterms) defines the insertion operator . This is discussed in subsection 3.4.

3.1 The infrared subtraction terms for massless QCD

In this section we present the infrared subtraction terms for massless QCD. The extension of the infrared subtraction terms to massive particles is discussed in the next section. We start with the soft subtraction term, which is given by

(48)

We recall that denotes the set of indices , for which the propagator in the loop corresponds to a gluon.

The soft subtraction term is derived as follows: In the case where gluon is soft, the corresponding propagator goes on-shell and we may replace in all Feynman diagrams which have the propagator the metric tensor of this propagator by a polarisation sum and gauge terms:

(49)

Here denotes the on-shell limit of and denotes the sum over the physical polarisations:

(50)

is a light-like reference vector. We note that self-energy diagrams are not singular in the soft limit, therefore adding them to the loop diagrams will not change the soft limit. With the inclusion of the self-energy diagrams and a corresponding replacement as in eq. (49) the contribution from the polarisation sum in eq. (49) makes up a tree-level partial amplitude, where two gluons with momenta and have been inserted between the external legs and .

Figure 6: The configuration for the soft limit.

This is illustrated in fig. (6). In the soft limit this tree-level partial amplitude is given by two eikonal factors times the tree-level partial amplitude without these two additional gluons:

(51)

In the soft limit we may replace by and by . Eq. (51) then leads to eq. (48). The terms with and in eq. (49) vanish for the sum of all diagrams due to gauge invariance. Integrating the soft subtraction term we obtain

(52)

We have multiplied the left-hand side by , where is the typical volume factor of dimensional regularisation, is Euler’s constant and is the renormalisation scale.

Let us now consider collinear singularities. The collinear subtraction is given by

(53)

where the symmetry factors are given by

(54)

The function ensures that the integration over the loop momentum of the subtraction terms in eq. (53) is ultraviolet finite. The function has the properties

(55)

There are many possible choices for the function . We use here a choice which is compatible with the contour deformation discussed in sect. (4). We take the function as

(56)

is an arbitrary four-vector independent of the loop momentum and is an arbitrary scale. Since these two quantities are arbitrary, there are no restrictions on them, they even may have complex values. We will later choose purely imaginary with . This will ensure that the denominator of eq. (56) does not introduce additional singularities for the contour integration.

The collinear subtraction term is derived as follows: We have to consider configurations where two adjacent propagators go on-shell with a massless external leg in between.

Figure 7: Configurations for the collinear limit. Only diagrams (a) and (b) lead to a divergence after integration. Diagrams (c) and (d) are not singular enough to yield a divergence after integration.

These configurations are shown in fig. (7). The diagrams (c) and (d) of fig. (7), where an external gluon splits into a ghost-antighost pair or into a quark-antiquark pair are in the collinear limit not singular enough to yield a divergence after integration. Therefore we are left with diagrams (a) and (b). Let us first consider the splitting as in diagram (a). In Feynman gauge one can show that only the longitudinal polarisation of the gluon contributes to the collinear limit. The same holds true for the splitting of diagram (b). In this case the collinear limit receives contributions when one of the two gluons in the loop carries a longitudinal polarisation (but not both). The external gluon has of course physical transverse polarisation. It is well-known that the contraction of a longitudinal polarisation into a gauge invariant set of diagrams yields zero. The shaded “blobs” of picture (a) and (b) of fig. (7) consist almost of a gauge invariant set of diagrams. There is only one diagram missing, where the longitudinal polarised gluon couples directly to the other parton connected to the shaded blob. This is a self-energy insertion on an external line, which by definition is absent from the amputated one-loop amplitude. We can now turn the argument around and replace the sum of collinear singular diagrams by the negative of the self-energy insertion on the external line.

Figure 8: In the collinear limit the sum of the singular diagrams in Feynman gauge of the amputated one-loop amplitude is equal to the negative of the self-energy insertion on the external line where the gluon carries a longitudinal polarisation.

This is indicated in fig. (8) for the splitting and in fig (9) for the splitting.

Figure 9: In the collinear limit the sum of the singular diagrams in Feynman gauge of the amputated one-loop amplitude is equal to the negative of the self-energy insertion on the external line where one of the gluons carries a longitudinal polarisation.

The self-energy insertions on the external lines introduce a spurious -singularity. In order to calculate the singular part of the self-energies we regulate this spurious singularity by putting slightly off-shell, but keeping and on-shell and imposing momentum conservation. We can use the same parametrisation as in the real emission case:

(57)

with and . The singular part of the self-energies is proportional to

(58)

The terms with and correspond to soft singularities and have already been subtracted out with the soft subtraction term . In the collinear limit we therefore just have to subtract out the terms which are non-singular in the soft limit. These terms are independent of and lead to eq. (53). If we compare the soft and collinear subtraction terms for the integrand of a one-loop amplitude with the subtraction terms for the real emission, we observe that there are no spin correlations in the subtraction terms for the integrand of the one-loop amplitude. In the real emission case spin correlations occur in the collinear limit. This can be understood as follows: From the proof of eq. (53) one can see that in the collinear limit always one of the collinear gluons carries an unphysical longitudinal polarisation. Hence, there are no correlations between two transverse polarisations of a gluon.

Integrating the collinear subtraction term we obtain

(59)

3.2 The infrared subtraction terms for massive QCD

In this section we present the infrared subtraction terms for massive QCD. This is in particular relevant to top quark physics. There are only a few modificatio