Polarized QED splittings of massive fermions and dipole subtraction for non-collinear-safe observables

Building on earlier work, the dipole subtraction formalism for photonic corrections is extended to various photon--fermion splittings where the resulting collinear singularities lead to corrections that are enhanced by logarithms of small fermion masses. The difference to the earlier treatment of photon radiation is that now no cancellation of final-state singularities is assumed, i.e. we allow for non-collinear-safe final-state radiation. Moreover, we consider collinear fermion production from incoming photons, forward-scattering of incoming fermions, and collinearly produced fermion-antifermion pairs. For all cases we also provide the corresponding formulas for the phase-space slicing approach, and particle polarization is supported for all relevant situations. A comparison of numerical results obtained with the proposed subtraction procedure and the slicing method is explicitly performed for the sample process e- gamma ->e- mu- mu+.


Introduction
Present and future collider experiments require precise predictions for particle reactions, i.e. for most of the relevant processes radiative corrections have to be calculated. This task becomes arbitrarily complicated if either the order in perturbation theory (loop level) or the number of external particles is increased, or both. In recent years the needed techniques and concepts have received an enormous boost from various directions; for a brief overview we refer to some recent review articles [1,2].
In this paper we focus on real emission corrections involving photons at next-to-leading order (NLO). Apart from the integration over a many-particle phase space, here the main complication is the proper isolation of the singular parts which originate from soft or collinear regions in phase space. To solve this problem at NLO, two different types of methods have been developed in the past: phase-space slicing (see, e.g., Ref. [3]) and subtraction [4][5][6][7][8] techniques. In the slicing approach the singular regions are cut off from phase space in the numerical integration and treated separately. Employing general factorization properties of squared amplitudes in the soft or collinear regions, the singular integrations can be carried out analytically. In the limit of small cutoff parameters the sum of the two contributions reproduces the full phase-space integral. There is a trade-off between residual cut dependences and numerical integration errors which increase with decreasing slicing cuts; in practice, one is forced to search for a plateau in the integrated result within some errors by varying the slicing cut parameters.
This cumbersome procedure is not necessary within subtraction formalisms which are based on the idea of subtracting a simple auxiliary function from the singular integrand and adding this contribution again. This auxiliary function has to be chosen in such a way that it cancels all singularities of the original integrand so that the phase-space integration of the difference can be performed numerically, even over the singular regions of the original integrand. In this difference the original matrix element can be evaluated without regulators for soft or collinear singularities. The auxiliary function has to be simple enough so that it can be integrated over the singular regions analytically with the help of regulators, when the subtracted contribution is added again. This singular analytical integration can be done once and for all in a process-independent way because of the general factorization properties of squared amplitudes in the singular regions. At NLO several subtraction variants have been proposed in the literature [4][5][6][7][8], some of which are quite general; at next-to-next-to-leading order subtraction formalisms are still under construction [9].
The dipole subtraction formalism certainly represents the most frequently used subtraction technique in NLO calculations. It was first proposed within massless QCD by Catani and Seymour [5] and subsequently generalized to photon emission off massive fermions [6] 1 and to QCD with massive quarks [7,8]. Among the numerous applications of dipole subtraction, we merely mention the treatment of the electroweak corrections to e + e − → 4 fermions [11], which was the first complete treatment of a 2 → 4 particle process at NLO. The formulation [5,7,8] of dipole subtraction for NLO QCD corrections assumes so-called infrared safety of observables, i.e. that all soft or collinear singularities cancel against their counterparts from the virtual corrections, either after parton-density redefi- In Section 6 we demonstrate the use and the performance of the methods presented in Sections 3,4,and 5 in the example e − γ → e − µ − µ + . A summary is given in Section 7, and the appendices provide more details on and generalizations of the formulas presented in the main text. In particular, the derivation of the factorization formulas for processes with incoming polarized photons splitting into light fermions and for the forward scattering of incoming polarized light fermions is described there.
2 Non-collinear-safe photon radiation off final-state fermions

Dipole subtraction and non-collinear-safe observables
For any subtraction formalism the schematic form of the subtraction procedure to integrate the squared matrix element λγ |M 1 | 2 (summed over photon polarizations λ γ ) for real photon radiation over the (N + 1)-particle phase space dΦ 1 reads where dΦ 0 is a phase-space element of the corresponding non-radiative process and [dk] includes the photonic phase space that leads to the soft and collinear singularities. The two contributions involving the subtraction function |M sub | 2 have to cancel each other, however, they will be evaluated separately. The subtraction function is constructed in such a way that the difference λγ |M 1 | 2 − |M sub | 2 can be safely integrated over dΦ 1 numerically and that the singular integration of |M sub | 2 over [dk] can be carried out analytically, followed by a safe numerical integration over dΦ 0 . In the dipole subtraction formalism for photon radiation, the subtraction function is given by [6] where the sum runs over all emitter-spectator pairs f f ′ , which are called dipoles. For a final-state emitter (final-state radiation), the two possible dipoles are illustrated in Fig. 1. The relative charges are denoted Q f , Q f ′ , and the sign factors σ f , σ f ′ = ±1 correspond to the charge flow (σ f = +1 for incoming fermions and outgoing antifermions, σ f = −1 for outgoing fermions and incoming antifermions). The implicitly assumed summation over τ = ± accounts for a possible flip in the helicity of the emitter f , where κ f = ± is the sign of the helicity of f both in |M 1 | 2 and |M sub | 2 . The singular behaviour of the subtraction function is contained in the radiator functions g (sub) f f ′ ,τ (p f , p f ′ , k), which depend on the emitter, spectator, and photon momenta p f , p f ′ , and k, respectively. The squared lowest-order matrix element |M 0 | 2 of the corresponding non-radiative process enters the subtraction function with modified emitter and spectator momentap where ± refers to a spectator f ′ in the final or initial state, and the same set {k n } of remaining particle momenta enters |M 1 | 2 and |M 0 | 2 . The modified momenta are constructed in such a way thatp Note that no collinear singularity exists for truly massive radiating particles f , because the invariant p f k does not tend to zero if the photon emission angle becomes small (for fixed photon energy k 0 ). In such cases the corresponding masses are kept non-zero in all amplitudes, in the subtraction functions, and in the kinematics, and the subtraction procedure works without problems. Collinear (or mass) singularities result if the mass m f of a radiating particle is much smaller than the typical scale in the process under consideration. In such cases it is desirable to set m f to zero whenever possible. In a subtraction technique this means that m f = 0 can be consistently used in the integral dΦ 1 λγ |M 1 | 2 − |M sub | 2 , but that the readded contribution [dk] |M sub | 2 contains mass-singular terms of the form α ln m f . If such mass singularities from collinear photon radiation do not completely cancel against their counterparts in the virtual corrections, the corresponding observable is not collinear safe. The dipole subtraction formalism as described in Ref. [6] is formulated to cover possible mass singularities from initial-state radiation, but assumes collinear safety w.r.t. final-state radiation.
In collinear-safe observables (w.r.t. final-state radiation), and only those are considered for light fermions in Ref. [6], a collinear fermion-photon system is treated as one quasiparticle, i.e., in the limit where f and γ become collinear only the sum p f + k enters the procedures of implementing phase-space selection cuts or of sorting an event into a histogram bin of a differential distribution. Technically this level of inclusiveness is reached by photon recombination, a procedure that assigns the photon to the nearest charged particle if it is close enough to it. Of course, different variants for such an algorithm are possible, similar to jet algorithms in QCD. The recombination guarantees that for each photon radiation cone around a charged particle f the energy fraction is fully integrated over. According to the KLN theorem, no mass singularity connected with final-state radiation remains. Collinear safety facilitates the actual application of the subtraction procedure as indicated in Eq. (2.1). In this case the events resulting from the contributions of |M sub | 2 can be consistently regarded as N-particle final states of the non-radiative process with particle momenta as going into M 0 Φ 0,f f ′ 2 , i.e. the emitter and spectator momenta are given byp → p f +k in the collinear limits, the difference λγ |M 1 | 2 − |M sub | 2 can be integrated over all collinear regions, because all events that differ only in the value of z f enter cuts or histograms in the same way. The implicit full integration over all z f in the collinear cones, on the other hand, implies that in the analytical integration of |M sub | 2 over [dk] the z f integrations can be carried out over the whole z f range.
In non-collinear-safe observables (w.r.t. final-state radiation), not all photons within arbitrarily narrow collinear cones around outgoing charged particles are treated inclusively. For a fixed cone axis the integration over the corresponding variable z f is constrained by a phase-space cut or by the boundary of a histogram bin. Consequently, mass-singular contributions of the form α ln m f remain in the integral. Technically this means that the information on the variables z f has to be exploited in the subtraction procedure of Eq. (2.1). The variables that take over the role of z f in the individual dipole contributions in |M sub | 2 are called z ij and z ia in Ref. [6], where f = i is a final-state emitter and j/a a final-/initial-state spectator. In the collinear limit they behave as z ij → z i and z ia → z i . Thus, the integral dΦ 1 λγ |M 1 | 2 − |M sub | 2 can be performed over the whole phase space if the events associated with |M sub | 2 are treated as (N + 1)-particle event with momenta . This can be formalized by introducing a step function Θ cut (p f , k, p f ′ , {k n }) on the (N + 1)-particle phase space which is 1 if the event passes the cuts and 0 otherwise. The set {k n } simply contains the momenta of the remaining particles in the process. Making the dependence on Θ cut explicit, the first term on the r.h.s. of Eq. (2.1) reads where we have decomposed the subtraction function |M sub | 2 into its subcontributions |M sub,f f ′ | 2 of specific emitter-spectator pairs f f ′ . Apart from this refinement of the cut prescription in the subtraction part for non-collinear-safe observables, no modification in |M sub | 2 is needed. Since its construction exactly proceeds as described in Sections 3 and 4 of Ref. [6], we do not repeat the individual steps in this paper. However, the modification of the cut procedure requires a generalization of the evaluation of the second subtraction term on the r.h.s. of Eq. (2.1), because now the integral over z f f ′ implicitly contained in [dk] depends on the cuts that define the observable. In the following two sections we work out the form of the necessary modifications, where we set up the formalism in such a way that it reduces to the procedure described in Ref. [6] for a collinear-safe situation, while the non-collinear-safe case is covered upon including extra contributions.

Final-state emitter and final-state spectator
For a final-state emitter i and a final-state spectator j with masses m i and m j the integral of g ij,τ (p i , p j , k), (2.5) where the definitions of Sections 3.1 and 4.1 of Ref. [6] are used. There the results for G (sub) ij,τ (P 2 ij ) with generic or light masses are given in Eqs. (4.10) and (3.7), respectively. In order to leave the integration over z ij open, the order of the two integrations has to be interchanged, and the integral solely taken over y ij is needed. Therefore, we definē ij,τ (p i , p j , k). (2.6) Note that no finite photon mass m γ is needed in the functionḠ (sub) ij,τ (P 2 ij , z) in practice, because the soft singularity appearing at z → 1 can be split off by employing a [. . .] + prescription in the variable z, This procedure shifts the soft singularity into the quantity G (sub) ij,τ (P 2 ij ), which is already known from Ref. [6]. Moreover, the generalization to non-collinear-safe integrals simply reduces to the extra term Ḡ (sub) ij,τ (P 2 ij , z) + , which cancels out for collinear-safe integrals where the full z-integration is carried out.
For arbitrary values of m i and m j a compact analytical result ofḠ (sub) ij,τ (P 2 ij , z) cannot be achieved because of the complicated structure of the integration boundary. Note, however, that only the limit m i → 0 of a light emitter is relevant, since for truly massive emitters no mass singularity results. The case of a massive spectator j is presented in App. A; here we restrict ourselves to the simpler but important special case m j = 0.
In the limit m i → 0 and m j = m γ = 0 the boundary of the y ij integration is asymptotically given by , y 2 (z) = 1, (2.8) and the functions and quantities relevant in the integrand g (sub) ij,τ behave as The evaluation of Eq. (2.6) becomes very simple and yields where P f f (z) is the splitting function, ij,τ (P 2 ij ) in the case of light masses, 12) with the auxiliary function L(P 2 , m 2 ) = ln m 2 P 2 ln 13) which are taken from Eqs. (3.7) and (3.8) of Ref. [6]. 2 Finally, we give the explicit form of the ij contribution |M sub,ij (Φ 1 )| 2 to the phasespace integral of the subtraction function, generalizing Eq. (3.6) of Ref. [6]. Whilep i ,p j , {k n } are the momenta corresponding to the generated phase-space point inΦ 0,ij , the momenta p i and k result fromp i via a simple rescaling with the independently generated variable z. The invariant P 2 ij is calculated via P 2 ij = (p i +p j ) 2 independently of z. The arguments of the step function Θ cut (p i , k,p j , {k n }) indicate on which momenta phase-space cuts are imposed.
For unpolarized fermions the results of this section have already been described in Ref. [13], where electroweak radiative corrections to the processes γγ → WW → 4 fermions were calculated. In this calculation the results for non-collinear-safe differential cross sections were also cross-checked against results obtained with phase-space slicing. Another comparison between the described subtraction procedure and phase-space slicing has been performed in the calculation of electroweak corrections to the Higgs decay processes H → WW/ZZ → 4 fermions [14].

Final-state emitter and initial-state spectator
For the treatment of a final-state emitter i and an initial-state spectator a, we consistently make use of the definitions of Sections 3.2 and 4.2 of Ref. [6]. In this paper we only consider light particles in the initial state, because the masses of incoming particles are much smaller than the scattering energies at almost all present and future colliders. Therefore, the spectator mass m a can be set to zero from the beginning, which simplifies the formulas considerably.
Before we consider the non-collinear-safe situation, we briefly repeat the concept of the collinear-safe case described in Ref. [6]. Following Eqs. (4.24) and (4.27) from there, the inclusive integral of g ia,τ (p i , p a , k), (2.16) where we could set the lower limit x 0 of the x ia -integration to zero because of m a = 0. Since, however, the squared lowest-order matrix element |M 0 | 2 multiplying g (sub) ia,τ in Eq. (2.2) depends on the variable x ia , the integration of |M sub | 2 over x = x ia is performed employing a [. . .] + prescription, (2.17) This integration, where the ellipses stand for x-dependent functions such as the squared lowest-order matrix elements and flux factors, is usually done numerically. Since the soft and collinear singularities occur at x → x 1 = 1 − O(m γ ), the singular parts are entirely contained in G (sub) ia,τ (P 2 ia ) in Eq. (2.17), and the upper limit x 1 could be replaced by 1 in the actual x-integration. For completeness we give the explicit form of the functions G (sub) ia,τ and G (sub) ia,τ in the limit m i → 0, 18) which are taken from Eqs. (3.19) and (3.20) of Ref. [6].
In a non-collinear-safe situation, the ellipses on the l.h.s. of Eq. (2.17) also involve z ia -dependent functions, as e.g. θ-functions for cuts or event selection. Thus, also the integration over z ia has to be performed numerically in this case, and we have to generalize Eq. (2.17) in an appropriate way. To this end, we generalize the usual [. . .] + prescription in the following way. Writing d n r g(r) for the [. . .] + prescription in the r i -integration in a multiple integral over n variables r k (k = 1, . . . , n), we can iterate this definition to two-dimensional integrals according to In the notation [g(r)] (r i ) +,(a) we omit the superscript (r i ) if g(r) depends only on the integration variable r i , and we omit the subscripts (a) or (a, b) if a = 1 or a = b = 1. This obviously recovers the usual notation for the one-dimensional prescription used above. Introducing a double [. . .] + prescription in x = x ia and z = z ia , we generalize Eq. (2.17) to If the functions hidden in the ellipses do not depend on z, the last two terms within the curly brackets do not contribute and the formula reduces to Eq. (2.17). We derive Eq. (2.21) and the explicit form of the two extra terms in two steps. In the derivation we quantify the previous ellipses by the regular test function f (x, z). The first step introduces a [. . .] + prescription in the x-integration of the l.h.s. of Eq. (2.21) after interchanging the order of the integrations, The upper limit x 1 (z) of the x-integration follows upon solving the explicit form of the limits z 1,2 (x) (given in Eq. (4.22) of Ref. [6]) for x. The full form of x 1 (z) is rather complicated for finite m γ , but in the following it is only needed for m γ = 0, where it simplifies to .

(2.23)
Note that soft or collinear singularities result from the region of highest x values, x → x 1 = max{x 1 (z)}, so that the first term in curly brackets in Eq. (2.22) is free of such singularities owing to the [. . .] + regularization. Thus, we can set m i → 0 in this part, i.e. in particular x 1 (z) → 1, yielding In the second step we introduce a [. . .] + prescription for the z-integration in both terms, ia,τ . (2.25) In the second equality we just reordered some factors and integrations. Since all integrals over the test function f are now free of singularities, i.e. the singularities are contained in the integrals multiplying f , we can set the regulator masses m γ and m i to zero in the arguments of f . Thus, we can write ia,τ .
( 2.27) Equation (2.26) is equivalent to the anticipated result (2.21), which was to be shown. The explicit results for G ia,τ (P 2 ia ) have already been given above in Eq. (2.18), the two remaining functions are easily evaluated tō The collinear singularity ∝ ln m i that appears in non-collinear-safe observables is contained in the functionḠ (sub) ia,+ (P 2 ia , z). The resulting ia contribution |M sub,ia (Φ 1 )| 2 to the phase-space integral of the subtraction function reads which generalizes Eq. (3.18) of Ref. [6]. Again, the arguments of the step function Θ cut (p i , k, {k n }) indicate on which momenta phase-space cuts are imposed. We recall thatΦ 0,ia is the phase space of momentap i (x) and {k n (x)} (without final-state radiation) with rescaled incoming momentump a (x) = xp a instead of the original incoming momentum p a . In the actual evaluation of Eq. (2.29), thus, the two phase-space points Φ 0,ia (P 2 ia , x) andΦ 0,ia (P 2 ia , x = 1) have to be generated for each value of x owing to the plus prescription in x. The relevant value of the invariant P 2 ia is then calculated separately via P 2 ia = (p i −p a ) 2 for each of the two points, so that P 2 ia results from the momenta entering the matrix element M 0 in both cases. 3 The variable z, however, is generated independently of the phase-space points and does not influence the kinematics in the matrix element.
The combination of the subtraction procedures described in this and the previous section has been successfully applied and compared to results obtained with phase-space slicing in the calculations of electroweak corrections to Drell-Yan-like W-boson production, pp → W → ν l l + X, and to deep-inelastic neutrino scattering, ν µ N → ν µ /µ + X, building on the calculations discussed in Refs. [15,16] and [17], respectively.

Phase-space slicing
In the phase-space slicing approach the soft and collinear phase-space regions are excluded in the (numerical) integration of the squared amplitude of the real-emission process. In the so-called two-cutoff slicing method the soft region is cut off by demanding that the photon energy k 0 should be larger than a lower cut ∆E which is much smaller than any relevant energy scale of the process. The collinear regions are excluded by demanding that each angle of the photon with any other direction of a light charged particle should be larger than the cut value ∆θ ≪ 1. Note that this phase-space splitting is not Lorentz invariant. In the soft and collinear regions the photon phase space can be integrated out analytically by employing the general factorization properties of the squared amplitudes, which are, e.g., discussed in Section 2.2 of Ref. [6] (including polarization effects). General results for the integral over the soft region can, e.g., be found in Refs. [18,19]. The integrals over the collinear regions for final-state radiation can be easily obtained from intermediate results of the two previous sections as follows.
The cuts defining the collinear region for the photon-emitter system of Section 2.2 translate into new limits for the integration variables y ij and z ij , The integrals of these functions over z = z ij are given by As it should be, in these results the dependence on the spectator particle j completely disappears, because it was only needed in the phase-space parametrization. We also note that the same results can be obtained from Section 2.3, where the limits on x ia and z ia are changed to Using the functionsḠ (sli) τ and G (sli) τ , the integral over the collinear photon emission cone around particle i reads where the momentap i and {k n } belong to the phase-space pointΦ 0 . Of course, apart from the polarization issue this is a well-known result which can be found in various papers [3]. 4 3 Collinear singularities from γ → ff * splittings

Asymptotics in the collinear limit
We consider a generic scattering process where the momenta of the particles are indicated in parentheses and λ γ = ± is the photon helicity. Here a is any massless incoming particle and f is an outgoing light fermion or antifermion. The remainder X may contain additional light fermions which can be treated in the same way as f . For later use, we define the squared centre-of-mass energy s, The collinear singularity in the squared matrix element |M γa→f X | 2 occurs if the angle θ f between f and the incoming γ becomes small; in this limit the scalar product , where m f is the small mass of f . Neglecting terms that are irrelevant in the limit m f → 0 the squared matrix element |M γa→f X (k, p a , p f ; λ γ )| 2 for a definite photon helicity λ γ = ± (but summed over the polarizations of f ) asymptotically behaves like where x = 1 − p 0 f /k 0 and Q f e is the electric charge of f . The matrix element Mf a→X corresponds to the related processfa → X that results from γa(→ ff * a) → f X upon cutting thef * line in all diagrams involving the splitting γ → ff * (see also Fig. 2). The incoming momenta relevant in the different matrix elements are given in parentheses. Figure 2: Generic diagrams for the splittings γ → ff * with an initial-state spectator a or a final-state spectator j.
Moreover, in Eq. (3.3) we assume a summation over τ = ±, where τ = ± refers to the two cases where the sign κf of thef helicity is equal or opposite to the photon helicity λ γ . The functions h γf τ (k, p f ), which rule the structure of the collinear singularity, are given by The derivation of this result is given in App. B.1. Note that the collinear singularity for kp f → 0 can be attributed to a single external leg (namelyf ) of the related hard processf a → X. Thus, there is no need to construct the subtraction function |M sub | 2 from several dipole contributions ∝ Q f Q f ′ . Instead we can construct |M sub | 2 as a single term ∝ Q 2 f . Nevertheless we select a spectator f ′ to the emitter f for the phase-space construction, which proceeds in complete analogy to the photon radiation case. We have the freedom to choose any particle in the initial or final state as spectator. In the following we describe the "dipole" formalism in two variants: one with a spectator from the initial state, another with a spectator from the final state. The two situations are illustrated in Fig. 2.

Initial-state spectator
The function that is subtracted from the integrand |M γa→f X (k, p a , p f ; λ γ )| 2 is defined as follows, with the radiator functions and the auxiliary quantity Here we kept the dependence on a finite m f , because it is needed in the analytical treatment of the singular phase-space integration below. The modified momentapf and {k n } entering the squared matrix element on the r.h.s. of Eq. (3.6) will only be needed for m f = 0 in applications with small values of m f . In this limit they can be chosen as 3.9) with the Lorentz transformation matrix Λ µ ν given by It is straightforward to check that |M sub | 2 possesses the same asymptotic behaviour as |M γa→f X | 2 in Eq. (3.3) in the collinear limit with m f → 0. Thus, the difference |M γa→f X | 2 − |M sub | 2 can be integrated numerically for m f = 0. The correct dependence of |M sub | 2 (and the related kinematics) on a finite m f is, however, needed when this function is integrated over θ f leading to the collinear singularity for θ f → 0. The actual analytical integration can be done as described in Ref. [6] (even for finite m a and m f ). Here we only sketch the individual steps and give the final result. The (N + 1)-particle phase space is first split into the corresponding N-particle phase space and the integral over the remaining degrees of freedom that contain the singularity, dφ(p f , P ; k + p a ) = x 1 0 dx dφ P (x);pf (x) + p a [dp f (s, x, y f,γa )], (3.12) with the explicit parametrization [dp f (s, x, y f,γa )] = s 4(2π) 3 (3.13) The upper kinematical limit of the parameter x = x f,γa is given by (3.14) but in the limit m f → 0 we can set x 1 = 1. While the integration of the azimuthal angle φ f of f simply yields a factor 2π, the integration over the auxiliary parameter 3.15) with the boundary (k, p f , p a ), (3.17) the result of this straightforward integration (for m f → 0) is For clarity we finally give the contribution σ sub γa→f X that has to be added to the result for the cross section obtained from the integral of the difference |M γa→f X | 2 − |M sub | 2 , Although formulated for integrated cross sections, the previous formula can be used to calculate any differential cross section after obvious modifications.
For the case of unpolarized photons this subtraction variant has already been briefly described in Ref. [17], where it was applied to the contributions to deep-inelastic neutrino scattering, ν µ N → ν µ /µ + X, that are induced by a photon distribution function of the nucleon N. Moreover, the method presented here was successfully used in the calculation of photon-induced real corrections to Drell-Yan-like W production (see Section 10 of Ref. [1] and Ref. [16]) and of photon-and gluon-induced real corrections to Higgs production via vector-boson fusion at the LHC [21]. All these results were also cross-checked against phase-space slicing.

Final-state spectator
As an alternative to the case of an initial-state spectator described in the previous section, we here present the treatment with a possibly massive final-state spectator j with mass m j , i.e. we consider the process (3.20) The initial-state particle a is assumed massless in the following, but all formulas can be generalized to m a = 0 following closely the treatment of phase space described in Section 4.2 of Ref. [6]. The subtraction function now is constructed as follows, with the radiator functions 3.22) and the auxiliary parameter The momentapf andp j are given bỹ 24) while the momenta of the other particles are unaffected. Note that this construction of momenta is based on the restriction m f = 0, which is used in the integration of the difference |M γa→f jX | 2 − |M sub | 2 .
In the integration of |M sub | 2 over the collinear-singular phase space, of course, the correct dependence on a finite m f is required. Owing to the finite spectator mass m j , this procedure is quite involved; we sketch it in App. B.2. Here we only present the results needed in practice. The cross-section contribution σ sub γa→f jX that has to be added to the integrated difference |M γa→f X | 2 − |M sub | 2 is given by (3.25) where the collinear singularity is again contained in the kernels (3.26) Of course, the singular contributions ∝ ln m f have the same form as in the case of an initial-state spectator discussed in the previous section.

Phase-space slicing
From the results of the two previous sections, the corresponding formulas for the phasespace slicing approach can be easily obtained. The collinear region, which is omitted in the phase-space integration, is defined by the restriction θ f < ∆θ on the fermion emission angle θ f in some given reference frame.
In Section 3.2 this constraint translates into new limits on the variable y f,γa , which modifies the result of the integral analogously defined to Eq. (3.17) to The cross-section contribution of the collinear region of f then reads The same result is obtained from Section 3.3 with App. B.2, where the new limits on Collinear singularities from γ * → ff splittings

Asymptotics in the collinear limit
We consider a generic scattering process where the momenta of the particles are indicated in parentheses. Depending on the particle content of the remainder X, there may be additional, independent collinear-singular configurations, but we are interested in the region where the invariant mass (p f + pf ) 2 = 2m 2 f + 2p f pf of the produced fermion-antifermion pair ff becomes of the order O(m 2 f ), where m f is small compared to typical scales in the process. The singular behaviour of the full squared matrix element |M ab→ff X (p f , pf )| 2 entirely originates from diagrams containing a γ * → ff splitting, i.e. the singularity is related to the subprocess ab → γX. For the matrix element of this subprocess we write M ab→γX = T µ ab→γX (k)ε λγ ,µ (k) * , where T µ ab→γX (k) is the amplitude without the photon polarization vector ε λγ,µ (k) * . In the collinear limit p f pf → 0 the light-like momentumk is equal to k = p f + pf up to masssuppressed terms. Neglecting terms that are irrelevant in the limit m f → 0 the squared matrix element |M ab→ff X (p f , pf )| 2 asymptotically behaves like where and N c,f is the colour multiplicity of f (N c,lepton = 1, N c,quark = 3). The momentum k ⊥ is the component of p f that is orthogonal to the collinear axis defined by k, i.e. kk ⊥ = 0, and becomes of O(m f ) in the collinear limit. An explicit prescription for the construction of k ⊥ can, e.g., be found in Ref. [8], where the analogous case of the gluonic splitting into massive quarks Q, g * → QQ, is worked out. It is important to realize that h ff ,µν in Eq. (4.2) is not proportional to the polarization sum E µν = λγ ε λγ,µ (k) * ε λγ,ν (k) of the photon, so that the r.h.s. is not proportional to the polarization-summed squared amplitude |M ab→γX | 2 of the subprocess. This spin correlation has to be taken care of in the construction of an appropriate subtraction function in order to guarantee a pointwise cancellation of the singular behaviour in the collinear phase-space region. The spin correlation encoded in h ff ,µν drops out if the average over the azimuthal angle φ f of the γ * → ff splitting plane around the collinear axis is taken. 5 Indicating this averaging by up to terms that are further suppressed by factors of m f . The averaged squared matrix element behaves as Since the collinear singularity for p f pf → 0 can be attributed to a single external leg (the photon) of the related hard process ab → γX, also in this case there is no need to construct the subtraction function |M sub | 2 from several dipole contributions. The function |M sub | 2 can be chosen as a single term ∝ Q 2 f . Nevertheless a spectator is selected for the phase-space construction, as in the previous section. In the following we describe the "dipole" construction in two variants: one with a spectator from the initial state, another with a spectator from the final state. The two situations are illustrated in Fig. 3.

Initial-state spectator
We define the subtraction function as and the auxiliary parameters The auxiliary momenta entering the amplitude for the related process ab → γX are given byp while the momenta of the other particles remain unchanged. In these equations we kept the dependence on m f , but of course in the numerical integration of |M ab→ff X | 2 −|M sub | 2 we can set m f to zero, because we are only interested in the limit m f → 0. For the integration of |M sub | 2 over the collinear-singular region, we need the m f -dependence of the spin average of h a,µν ff , (4.10) and an appropriate phase-space splitting, 11) where we have used the shorthands x = x ff ,a and z = z ff ,a . The explicit form of [dp f ] reads 4.12) with the integration limits for the variables x and z . (4.13) Separating the singular contributions as described in Section 2.3, we rewrite the integral of h a ff for m f → 0 as (4.14) The new functionsh a ff , etc., defined here are obtained from obvious substitutions and straightforward integrations, . (4.15) Using these functions the phase-space integral of the subtraction function reads where we have made explicit which momenta enter the cut function Θ cut (p f , pf , {k n }).
Concerning the phase-space integration over dΦ γ (P 2 , x) and its integration over the boost parameter x the same comments as made after Eq. (2.29) apply. There are actually two phase-space points for each x value to be generated (one for x < 1 and another for x = 1), each determining momentap a (x),k(x), {k n (x)} for the evaluation of P 2 and the matrix elements. The generation of the parameter z proceeds independently, and the squared amplitude |M ab→γX | 2 in Eq. (4.16) does not depend on z. Thus, if the full range in z is integrated over, i.e. if the collinear ff pair is treated as a single quasiparticle in the cut procedure, the last two terms in curly brackets do not contribute. In this case the fermion-mass logarithm is entirely contained in the H a ff contribution. According to the KLN theorem this contribution will be completely compensated by virtual O(α) corrections to the process ab → γX if collinear ff pairs are not distinguished from emitted photons.

Final-state spectator
Since the case with a massive final-state spectator j is quite involved, we here present the formalism for m j = 0 and give the details for the massive case in App. C.
For m f = m j = 0, the subtraction function can be defined as 4.18) and the auxiliary parameters The new momenta entering the amplitude for the related process ab → γjX are given bỹ (4.20) whereas all remaining momenta k n of particles in X remain unchanged. Equation (4.17) can be used to integrate the difference |M ab→ff jX | 2 − |M sub | 2 for massless fermions f . In order to integrate |M sub | 2 over the collinear-singular region, the dependence on m f has to be taken into account. Details of this procedure can be found in App. C. The result can be written in the form The momentak,p j , {k n } directly correspond to the generated phase-space point inΦ γ , while the parameter z is generated independently. The comments on the z-integration made at the end of the previous subsection apply also here. The squared amplitude |M ab→γjX | 2 in Eq. (4.21) does not depend on z, and thus, if the event selection for f and f is inclusive in the collinear region of the γ * → ff splitting, the integral over z trivially reduces to the factor H ff ,j (P 2 ).

Phase-space slicing
Here we again deduce the integral over the collinear phase-space region which is needed in the slicing approach. This region can, e.g., be defined by restricting the angle θ ff between the f andf directions to small values, θ ff < ∆θ ≪ 1.
In Section 4.2 this restriction leads to new limits in x ff ,a and z ff ,a , 1, (4.23) where k 0 = p 0 f + p 0 f is the energy in the ff system. This modifies the integrated results toH 4.24) whereH ff and H ff are defined analogously to Eq. (4.14). The integral of the squared matrix element over the collinear regions then reads The same results can be obtained from Section 4.3 with App. C, where the new limits on the integration variables are given by Collinear singularities from f → f γ * splittings

Asymptotics in the collinear limit
We consider a generic scattering process with the momenta of the particles and the (sign of the) helicity κ f = ± of the incoming fermion f indicated in parentheses. We are interested in the region where the squared momentum transfer ( , where m f is small compared to typical scales in the process. The singular behaviour of the full squared matrix element |M f a→f X (p f , p ′ f ; κ f )| 2 entirely originates from diagrams containing an f → f γ * splitting, i.e. the singularity is related to the subprocess γa → X. For the matrix element of this subprocess we write M γa→X (k, p a , λ γ ) = T µ γa→X (k)ε λγ,µ (k), where T µ γa→X (k) is the amplitude without the photon polarization vector ε λγ,µ (k). In the collinear limit p f p ′ f → 0 the momentumk is given by k = p f − p ′ f up to mass-suppressed terms. Neglecting terms that are irrelevant in the limit m f → 0 the squared matrix element |M f a→f X (p f , p ′ f ; κ f )| 2 asymptotically behaves like Figure 4: Generic diagrams for the splittings f → f γ * with an initial-state spectator a or a final-state spectator j, where f is a light fermion or antifermion.
The momentum k ⊥ is the component of k that is orthogonal to the collinear axis defined by p f , i.e. k ⊥ p f = 0, and becomes of O(m f ) in the collinear limit. A derivation of this factorization is described in App. D.1.
Note that h f f κ f ,µν in Eq. (5.2) is not proportional to the polarization sum E µν = λγ ε λγ ,µ (k) * ε λγ ,ν (k) of the photon, so that the r.h.s. is not proportional to the polarization-summed squared amplitude |M γa→X | 2 of the subprocess. This spin correlation has to be taken into account in the construction of an appropriate subtraction function in order to guarantee a point-wise cancellation of the singular behaviour in the collinear phase-space region. The spin correlation encoded in h f f κ f ,µν drops out if the average over the azimuthal angle φ ′ f of the f → f γ splitting plane around the collinear axis is taken. Details of this averaging process, which is indicated by . . . φ ′ f , are given in App. D.1. The result is (5.6) which is valid in four space-time dimensions up to terms that are further suppressed by factors of m f . Here P γf (x) is the splitting function Since the collinear singularity for p f p ′ f → 0 can be attributed to a single leg (the photon) of the related hard process γa → X, also in this case there is no need to construct the subtraction function |M sub | 2 from several dipole contributions. The function |M sub | 2 can be chosen as a single term ∝ Q 2 f , and a spectator is only used in the phase-space construction as previously. In the following we again describe the "dipole" construction in two variants: one with a spectator from the initial state, another with a spectator from the final state. The two situations are illustrated in Fig. 4.

Initial-state spectator
We define the subtraction function as and the auxiliary parameters Assuming again the incoming particle a to be massless and defining the needed auxiliary momenta for the related process γa → X are given bỹ where the Lorentz transformation matrix Λ µ ν is constructed from the momenta P µ and P µ as in Eq. (3.10). In these equations we kept the dependence on m f , but of course in the numerical integration of |M f a→f X | 2 − |M sub | 2 we can set m f to zero if we are only interested in the limit m f → 0. For the integration of |M sub | 2 over the collinear-singular region, we need the m f -dependence of its azimuthal average, with summation over τ = ± and 14) where we have used the shorthands x = x f,f a and y = y f,f a . An appropriate phase-space splitting is given by with the explicit form of [dp ′ f ] [dp ′ f (s, x, y)] =s 4(2π) 3 5.16) and the integration limits for the variables x and y (5.17) In the limit m f → 0 the integral (5.18) can be easily evaluated to 19) and the part to be added to the cross section reads

Final-state spectator
As an alternative to the case of an initial-state spectator, we now present the treatment with a final-state spectator j, i.e. we consider the process Figure 5: QED diagrams contributing to e − γ → e − µ − µ + at tree level.

Phase-space slicing
Finally, we derive the integral over the collinear phase-space region for the slicing approach. This region is defined by restricting the angle θ ′ f between the outgoing and incoming f to small values, θ ′ f < ∆θ ≪ 1. In Section 5.2 this restriction leads to new limits in y f,f a , which modify the integrated result to where the integral is defined analogously to Eq. (5.18). The cross-section contribution for the collinear scattering of f is given by The same results can be obtained from Section 5.3 with App. D.2, where the new limits on the integration variables are given by (5.31) 6 Application to the process e − γ → e − µ − µ + In this section we illustrate the application of the methods described in Sections 3,4,and 5 to the process e − γ → e − µ − µ + at a centre-of-mass energy √ s much larger than the involved particle masses, √ s ≫ m e , m µ . Of course, this process is not of particular importance in particle phenomenology, but it involves the three issues of (i) incoming photons splitting into light ff pairs, (ii) the collinear production of light ff pairs, and (iii) forward-scattered fermions and, thus, provides a good test process for these cases. As already mentioned in Section 2, our treatment of non-collinear-safe final-state radiation has already been tested in other processes.
To illustrate the formalism, it is sufficient to consider the process e − γ → e − µ − µ + in QED, where only the four diagrams shown in Fig. 5 contribute. The corresponding helicity amplitudes, including the full dependence on the masses m e and m µ , can be obtained from the treatment of e − γ → e − e − e + presented in Ref. [22] after some obvious substitutions. In the following we compare the result with the full mass dependence to results obtained with the described subtraction and slicing methods in various kinematical situations. Denoting the polar angle of an outgoing particle i by θ i and the angle between the two outgoing muons by α µµ , we distinguish the following cases: a) No collinear splittings Angular cuts: θ cut < θ e − < 180 • − θ cut and θ µ ± < 180 • − θ cut and θ cut < α µµ .
No collinear singularities are included, and the integrated cross section is well defined for vanishing fermion masses, i.e. none of the subtraction methods has to be applied. The difference between massive and massless calculations indicates the size of the fermion mass effects.
b) Collinear splitting γ → e − e + * Angular cuts: θ cut < θ e − and θ µ ± < 180 • − θ cut and θ cut < α µµ . The collinear splitting γ → e − e + * of the incoming photon is integrated over, so that the third diagram of Fig. 5 develops a collinear singularity for backward-scattered electrons. The methods of Section 3 are applied to the calculation with massless fermions.
For the numerical evaluation we set the fermion masses to m e = 0.51099907 MeV and m µ = 0.10565839 GeV, the fine-structure constant to α = e 2 /(4π) = 1/137.0359895, the beam energies to E = E e = E γ = 250 GeV, and the angular cut to θ cut = 10 • . In the subtraction and slicing methods the masses m e and m µ are neglected everywhere except for the mass-singular logarithms, i.e. the laboratory frame defined by the above beam energies coincides with the centre-of-mass system. For the fully massive calculation the two frames are connected by a (numerically irrelevant) boost along the beam axis with a tiny boost velocity of O(m 2 e /E 2 ). Our numerical results for the different kinematical situations and the various methods are collected in Table 1. In addition in Table 2 we show the analogous results for the situation where the energy of each final-state lepton l = e − , µ ± is restricted by E l > 10 GeV. All results are obtained with an integration by Vegas [24], using 25 × 10 6 events. While a simple phase-space parametrization is sufficient in the subtraction formalism, dedicated phase-space mappings are required to flatten the corresponding collinear poles in the slicing approach and when employing the full mass dependence of the matrix elements. The fully massive results have been checked with the program Whizard [25], where agreement within the integration errors has been found.
The results obtained with the different subtraction variants, where a spectator is chosen from the initial state (IS) or from the final state (FS), are in mutual agreement within the integration error, which is indicated in parentheses. Subtraction and slicing results are also consistent within the statistical errors as long as the angular slicing cut ∆θ is not chosen too large. For example, some of the slicing results for ∆θ = 10 −1 still show a significant residual dependence on ∆θ. In the chosen example, the integration errors of the subtraction and slicing results are of the same order of magnitude. However, we would like to mention that the subtraction approach is often more efficient, as e.g. observed in the applications of Refs. [14][15][16][17]21] mentioned above. This superiority of the subtraction formalism typically deteriorates if complicated phase-space cuts are applied, as in the chosen example, because the cuts act differently in the various auxiliary phase spaces and thus introduce new peak structures in the integrand.
Finally, we remark that the impact of mass-suppressed terms is significantly reduced if the cut on the lepton energies E l is applied. This cut guarantees that E l ≫ m l overall in phase space, so that mass-suppressed terms are proportional to m 2 l /Q 2 with Q ≫ m l . Without any restriction on E l , there are at least small regions of phase space where Q is not much smaller than m l , leading to larger mass effects. This feature is clearly visible in Tables 1 and 2 when comparing results based on the full mass dependence in the matrix elements with the subtraction and slicing results that are based on the asymptotic limit m l → 0.

Summary
The dipole subtraction formalism for photonic corrections is extended to various photon-fermion splittings where the resulting collinear singularities lead to corrections that are enhanced by logarithms of small fermion masses m f . Specifically, we have considered non-collinear-safe final-state radiation, collinear fermion production from incoming photons, forward-scattered incoming fermions, and collinearly produced fermionantifermion pairs. All formulas needed in applications are provided, only the scattering matrix elements for the underlying process and for relevant subprocesses have to be supplemented in the simple approximation of a massless fermion f . Particle polarization is taken care of in all relevant cases, e.g., for incoming fermions and photons. For the purpose of cross-checking results in applications, we also provide the formulas needed in the phase-space slicing method.
As an example illustrating the use and performance of the proposed methods we have explicitly applied the subtraction procedures to the process e − γ → e − µ − µ + and compared the results to those obtained with phase-space slicing. The presented subtraction variants will certainly be used in several precision calculations needed for present and future collider experiments such as the LHC or ILC.

Acknowledgement
We are grateful to Markus Roth for collaborating on the issue of Section 2 in an early stage of this work.

Appendix
A More details on non-collinear-safe final-state radiation Here we generalize the results of Section 2.1, where non-collinear-safe photon radiation off fermions is treated, to the situation where massive spectators in the final state exist. To this end, we only have to consider the case of final-state emitter and final-state spectator.
For m i → 0, m γ = 0, but m j = 0, the boundary of the y ij integration [given for the massless case in Eq. (2.8)] is given by , (A.1) and the functions relevant for the integrand g (sub) ij,τ behave as The evaluation of Eq. (2.6) now becomes non-trivial and yields For m j → 0, the results forḠ (sub) ij,τ (P 2 ij , z ij ) reduce to Eq. (2.10), as can be easily seen after realizing that η(z) = O(m j ) and σ(z) = 1 + O(m 2 j ) in this limit.
B More details on the subtraction for γ → ff * splittings B.1 Factorization in the collinear limit In this section we derive the asymptotic behaviour (3.3) of the squared amplitude |M γa→f X | 2 for the case where the outgoing light fermion flies along the direction of the incoming photon. We consider polarized incoming photons with momentum k µ and polarization vector ε µ λγ , where λ γ = ± is the sign of its helicity. We further introduce a light-like gauge vector n µ (n 2 = 0, nk = 0), i.e. ε µ λγ is characterized by ε µ −λγ = (ε µ λγ ) * , kε λγ = nε λγ = 0. (B.1) In the following we make use of the identity 6 where E µν (k) = ε µ + ε ν − + ε µ − ε ν + = −g µν + k µ n ν + n µ k ν kn (B.3) is the polarization sum of the photon in four space-time dimensions and ǫ µνρσ the Levi-Civita tensor with ǫ 0123 = +1. In a gauge for the photon where nk = O(k 0 ), it is easily shown by power counting that the logarithmic singularity arising from the phase-space region kp f = O(m 2 f ) (m f ≪ k 0 ) originates from diagrams in which the incoming photon collinearly splits into a light ff * pair. The generic form of such graphs is shown in Fig. 6. Assuming summation over the polarization of the outgoing fermion f , the squared matrix element, thus, behaves like |M γa→f X (k, p a , p f ; λ γ )| 2 Tf a→X (pf , p a ), (B.4) 6 This identity is easily proven using a representation of the polarization vectors by Weyl spinors. Employing the conventions of Ref. [23], we have εȦ B + = ε µ + σȦ B µ = √ 2nȦk B / kn and εȦ B − = ε µ − σȦ B µ = √ 2kȦn B / kn * for the polarization bispinors, so that ǫ µνρσ k ρ n σ = i 4 (ǫȦĖǫĊĠǫ BD ǫ F H − ǫȦĊ ǫĖĠǫ BF ǫ DH )σ μ AB σ νĊ D σ ρĖ F σ σ GH k ρ n σ = i 4 (kȦnĊǫ BD k X n X − ǫȦĊk B n D kẊ nẊ )σ μ AB σ νĊ D = i 4 (−kȦn B nĊ k D + nȦk B kĊ n D )σ μ AB σ νĊ D = i 2 kn kn * (ε µ + ε ν − − ε µ − ε ν + ) = i(kn)(ε µ + ε ν − − ε µ − ε ν + ). The only non-trivial step is the third equality which follows from a twofold application of Schouten's identity. and k X denoting the outgoing total momentum of X. The upper kinematical limit of the parameter x = x f j,γ is given by (B.14) The parameter κ is arbitrary, because the singular behaviour does not depend on it; in practice the independence of the final result on κ can be used as check. The auxiliary momenta entering the hard scattering matrix element for the subprocess ab → γjX also become more complicated, In order to integrate the subtraction function we need the azimuthal-averaged version of h µν ff ,j , 4) and an appropriate splitting of the phase space of the momenta p f , pf , p j , dφ(p f , pf , p j ; P ) = dφ(k,p j ; P ) [dp f (P 2 , y ff j , z ff j )], (C.5) [dp f (P 2 , y ffj , z ff j )] = 1 4(2π) 3P 4 P 2 − m 2 j y 2 y 1 dy ff j (1 − y ff j ) z 2 (y ff j ) and z 1,2 (y ffj ) are the z 1,2 of Eq. (C.2), evaluated as functions of y ff j . Up to this point, the full dependence on m f and m j is kept.
Since we want to keep the momentum flow in the collinear limit open, i.e. the z ff j integration should be done numerically, we have to interchange the order of y ff j and z ff j integrations in the singular phase-space integration over [dp f ]. For arbitrary masses m f and m j , this seems hardly possible analytically, so that we focus on the limit m f → 0 in the following, because this is the interesting case. We define H ff ,j (P 2 , z) =P where we were allowed to use m f = 0 in the prefactors and in the integration limits of z = z ff j . The relevant asymptotics of y 1,2 (z) for m f → 0 is The actual integration over y ff j yields H ff ,j (P 2 , z) = P f γ (z) 2 ln (C.10) The case m j = 0 given in Eq. (4.22) can be easily read off after realizing that η(z) = O(m j ). For the evaluation of H ff ,j (P 2 ) it is easier to integrate first over z and then over y ff j . The result is (C.11) which could also be derived from Eq. (5.36) of Ref. [8]. For m j = 0 this obviously leads to the form given in Eq. (4.22).