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A T-odd Momentum Correlation in Radiative \beta-Decay

We consider neutron radiative \beta-decay, n\to p e^- \bar{\nu}_e\gamma, and compute the T-odd momentum correlation in the decay rate characterized by the kinematical variable \xi = {\bf l}_\nu \cdot ({\bf l}_e\times{\bf k}) arising from electromagnetic final-state interactions in the Standard Model. Our expression for the corresponding T-odd asymmetry A_\xi^{SM} is exact in O(\alpha) up to terms of recoil order, and we evaluate it numerically under various kinematic conditions. Noting the universality of the V-A law in the absence of recoil order terms, we retain the parametric dependence on masses and coupling constants throughout, so that our results serve as a template for the computation of A_\xi^{SM} in allowed nuclear radiative \beta-decays and hyperon radiative \beta-decays as well. We consider the pattern of A_\xi^{SM} which appears in nuclear decays and show that A_\xi^{SM} can be suppressed in particular cases, facilitating searches for new sources of CP-violation in such processes.


I. INTRODUCTION
Radiative β-decay offers the opportunity of studying T-odd momentum correlations which do not appear in ordinary β-decay [1]. We consider a correlation characterized by the kinematical variable ξ = l ν · (l e × k), so that it is both parity Pand naively time-reversal T-odd but independent of the particle spin. Its spin independence renders it distinct from searches for permanent electric-dipole-moments (EDMs) of neutrons and nuclei. The inability of the Standard Model (SM) to explain the cosmic baryon asymmetry prompts the search for sources of CP-violation which do not appear within it and which are not constrained by other experiments. A triple momentum correlation in radiative β-decay is one such example, as we shall illustrate; under the CPT-theorem, T-violation is linked to CP-violation. A decay correlation, however, can be, by its very nature, only "naively" or "pseudo" T-odd, that is, only motion-reversal odd. As a result, although the appearance of a T-odd decay correlation can be engendered by sources of CP-violation beyond the Standard Model, it can also be generated without fundamental Tor CP-violation. In this paper we compute the size of the T-odd momentum correlation in radiative β-decay simulated by electromagnetic final-state interactions in the SM [2]. This is crucial to establishing a baseline in the search for new sources of CP-violation in such processes. Our work is motivated in large part by the determination that pseudo-Chern-Simons terms appear in SU(2) L ×U(1) gauge theories at low energies -and that they can impact low-energy weak radiative processes involving baryons [3][4][5]. In the SM such pseudo-Chern-Simons interactions are CP-conserving, but considered broadly they are not, so that searching for the Pand T-odd effects that CPviolating interactions of pseudo-Chern-Simons form would engender offers a new window on physics beyond the SM [6].
Searches for T-violating decay correlations in neutron and nuclear β-decay have a long history. The best experimental limits are on the so-called D term, which appears as the triple correlation DS · (l e × l ν ), where S is the polarization of the decaying particle [7,8]. These limits still greatly exceed the size of the D correlation expected from SM final-state interactions [9,10]. Radiative β-decay offers the possibility of forming a T-odd correlation from momenta alone; to our knowledge such a possibility was first considered in the context of K + l3γ decay [11]. The T-odd asymmetry computed in Ref. [11] from electromagnetic finalstate interactions has recently been recalculated and is in significant disagreement with the earlier result [12].
In this paper we evaluate the T-odd asymmetry in radiative β-decay from electromagnetic radiative corrections in the SM and focus on the neutron case: n(p n ) → p(p p )+e − (l e )+ ν e (l ν ) + γ(k). The motion-reversal-odd terms in the decay rate, which mimic the appearance of T-violation, are engendered by the interference of the tree-level amplitude with the imaginary part of the O(α) corrected amplitude, which is determined by the physical two-particle cuts and hence mediated by the scattering of particles on their mass shells [9,13,14]. In what follows, we detail the computation of the interference terms and their components, as well as the resulting numerical integration over the allowed phase space to yield the T-odd asymmetry A SM ξ . Our results are exact in O(α) up to corrections of recoil order, namely, up to terms of O(ε/M ), where ε is an energy scale which is small with respect to the nucleon mass M . This certitude is guaranteed by the small Q-value of the decay, so that ε M , and by Low's theorem [15]. The natural scale of hadron excitations is set by the pion mass m π ; consequently, in neutron radiative β-decay ε m π as well, and non-electromagnetic final-state interactions cannot contribute to the physical two-particle cuts. This is in con-FIG. 1: Contributions to n(p n ) → p(p p ) + e − (l e ) +ν e (l ν ) + γ(k) up to corrections of recoil order. The effective weak vertex is denoted by ⊗ and is controlled by the Fermi constant G F . The diagram enumeration is utilized in our calculation of the T-odd asymmetry.
tradistinction to K + l3γ decay for which such contributions are appreciable, albeit relatively small [16]. We relegate intermediate results essential for our final results but yet nonessential to the flow of our discussion to Appendices. Since we neglect all terms of recoil order, our results can be immediately translated to the computation of A SM ξ in nuclear and hyperon radiative β-decays as well. We consider, in particular, the pattern in A SM ξ predicated by variations in the Q-value of the decay as well as by the nuclear dependence of the axialvector coupling constant. We assess the size of undetermined corrections before offering a final summary of our results.

II. FORMALISM
We work in a simultaneous expansion in the electromagnetic coupling constant e and in ε/M , so that the leading contributions to neutron radiative β-decay are from the diagrams in Fig. 1. At this order the baryons are effectively structureless, and the contributions arise from bremsstrahlung off the charged particle legs of ordinary β-decay, yielding a gauge-invariant result [15]. Employing the notation and conventions of Ref. [17], the decay amplitude is: with [18,19] M 01 (l e , k, where µ is the photon polarization vector and λ ≡ g A /g V , noting that g V and g A are the vector and axial-vector weak coupling constants of the nucleon, respectively. We explicitly include the arguments in the momenta l e , k, and p p for later convenience. The branching ratio and photon energy spectrum for this process have been computed previously [18,19]. The expressions which follow from Eq. (2) are consistent with the experimental results [20,21]. The next-to-leading order terms in the small-scale expansion, i.e., those of O(ε/M ), have been computed in heavy-baryon chiral perturbation theory and are no larger than O(0.5%) of the leading-order result [19] -this is some twenty times smaller than the current experimental sensitivity [21]. In what follows we neglect all recoil-order terms and consider the O(α) corrections to the amplitude of Eq. (2). For future reference, employing lepton and hadron tensors, we note that [19] spins where M n , M p , and ω refer to the neutron mass, the proton mass, and the photon energy, respectively, and with m e the electron mass. In realizing the amplitudes from the Feynman rules we impose i (k) · k = 0 for each polarization state i of a real photon with momentum k, noting To effect the subsequent photon polarization sums, however, we employ QED gauge invariance and make the replacement i=1,2 µ i (k) ν * i (k) −→ −g µν throughout, without any supplemental conditions.
Denoting the O(α) correction to the amplitude by M loop the amended decay rate is determined by The T-odd triple momenta correlation ξ = l ν · (l e × k) in the decay rate can arise from the interference between the tree level amplitude M tree and the anti-Hermitian parts of the one-loop corrections to it, so that ultimately the interference term spins (2Re(M tree M * loop )) contains terms linear in ξ. Since we consider the decay and detection of unpolarized particles exclusively, spins |M| 2 T−odd is indeed characterized by terms linear in ξ. Evidently the induced asymmetry is suppressed by a factor of α ≡ e 2 /4π ∼ 1/137; explicit computation shows it to be much smaller still.
Before turning to the computation of |M| 2 T−odd let us consider its relation to a measurable quantity. Following Ref. [11], we define a T-odd asymmetry A ξ , namely, where N + is defined as the total number of decay events with positive ξ, and N − is defined as the number of events with negative ξ. Specifically, we compute where Γ ± contains an integral of |M| 2 over the region of phase space with ξ > < 0, respectively; the numerator is non-zero if and only if |M| 2 T−odd is non-zero. Working to corrections of O(ε/M ), the neutron radiative β-decay rate Γ in the neutron rest frame is where p p = M n − l e − l ν − k and E ν = M n − M p − E e − ω are fixed throughout. The precise form of Γ ± depends on the concrete choice of coordinate system. Choosing the direction of the electron momentum l e as the z direction and letting k and l e fix the x-z plane, then under this specific choice ξ > 0 corresponds to φ ν ∈ [0, π] and ξ < 0 corresponds to φ ν ∈ [π, 2π]. Thus we define and where E max e = M n − M p − ω, ω max = M n − M p − m e , and ω min is determined by the threshold energy of the detector. In our computation of |M| 2 we set M n = M p = M in terms which would yield corrections beyond leading order in the recoil expansion. We limit the integration over x k to the range [−c, c]; we discuss this as well as our choice for c in Sec. IV.

III. COMPUTATION OF spins |M| 2 T−odd IN LEADING ORDER
To compute the T-odd pieces, we need to obtain the anti-Hermitian parts of the oneloop diagrams Im(M loop ). We do this by performing "Cutkosky cuts" [13], which means we simultaneously put intermediate particles in the loops on their mass shells in all physically allowed ways and then perform the relevant intermediate phase space integrals and spin sums. Graphically speaking, after imposing the cuts, the anti-Hermitian part of a one-loop diagram can be viewed as the product of two physical tree-level processes. We have where n refers to the summation over all the possible cuts of the one-loop diagrams and dρ n and sn refer to the intermediate phase space integration and spin sums, respectively, for a cut which yields state n. The matrix elements M ni and M f n refer to the two treelevel diagrams after a physical cut. After excluding the physically unacceptable cuts, 14 FIG. 2: All two-particle cut contributions to n(p n ) → p(p p ) + e − (l e ) +ν e (l ν ) + γ(k) which appear in O(α) up to corrections of recoil order, using the syntax of Fig. 1. A "×" means that the intermediate particle has been put on its mass shell; two such symbols define the Cutkosky cut. The diagram enumeration is utilized in our calculation of the T-odd asymmetry. Note that the first number selects a particular Feynman diagram, and the second determines the particular twoparticle physical cut in that diagram.
FIG. 3: Compton scattering diagrams which appear in Im(M loop ) for γ − e cuts. We denote the two graphs by M d γe (l e , k , l e , k) and M c γe (l e , k , l e , k), respectively. The diagrams and amplitudes appropriate to γ − p scattering follow from replacing electron with proton variables. cut diagrams remain, and they are illustrated in Fig. 2. We evaluate them explicitly. The momenta labeled as k , l e , and p p refer to momenta of intermediate particles. In performing the Cutkosky cuts, each particle in a pair of particles is put on its own mass shell.
It is useful to categorize the cuts as per the sorts of processes involved. That is, M f n describes the manner in which select particles rescatter, so that we can have Compton scattering or electron-proton scattering, the latter with or without the emission of an additional photon. The family of diagrams given by (1), (2), (5.1), and (6.2) contain Compton scattering from the electron, as illustrated in Fig. 3, whereas the family comprised of (3), (4), (7.2), and (8.3) contain Compton scattering from the proton. In these families M f n is captured by one of the following expressions: or where ≡ (k ). Correspondingly, M ni is given by the tree-level neutron radiative β-decay amplitude, as per the form of M 01 and M 02 , with only some of the arguments changed. Technically we define a "family" to be those contributions to the T-odd correlation which cancel amongst themselves to yield zero when we replace or * by k or or * by k as per the Ward-Takahashi identities. Furthermore, there is an intermediate phase space integral over the kinematically allowed phase space. For γ − e scattering we have with P 0e ≡ l e + k, whereas for γ − p scattering we have with P 0p ≡ p p + k. Collecting the pieces, we have We denote the two graphs by M ef epγ (l e , p p , l e , k, p p ) and M ei epγ (l e , p p , l e , k, p p ), respectively. The diagrams and amplitudes appropriate to proton bremsstrahlung follow from exchanging electron and proton variables. for the "γ − e" cuts, and for the "γ − p" cuts. In addition to the families of Compton cuts, there are cuts in which M f n is determined by electron-proton scattering either with and without bremsstrahlung, and, correspondingly, M ni is determined by either nonradiative or radiative β-decay. Referring to Fig. 2, we see for cuts in which the electron and proton scatter with bremsstrahlung that diagrams (5.2) and (6.1) comprise the family associated with electron bremsstrahlung, as shown in Fig. 4, and (7.1) and (8.1) comprise the family associated with proton bremsstrahlung. In these families M f n is given by one of the following: or Moreover, M ni is given by neutron β-decay, as shown in Fig. 5, which, up to recoil-order corrections, reads: Collecting the pieces, we have and Im(M 7.1 ) = 1 8π 2 dρ epγ sep M pf epγ (l e , p p , l e , k, p p )M DK (l e , p p ) , for the "e − p − γ" cuts. The last family of cuts is given by (6.3) and (8.2) in Fig. 2. In this case M f n is given by e − p scattering, and we have M ep (l e , p p , l e , p p ) = −e 2ū e (l e )γ µ u e (l e ) g µν (l e − l e ) 2ū p (p p )γ ν u p (p p ) .
The corresponding M ni is given by M 01 (l e , k, p p ) for (6.3) and M 02 (l e , k, p p ) for (8.2). We for the "e − p" cuts. In these graphs the intermediate momenta satisfy l e + p p = l e + p p , so that the integral over the allowed phase space is slightly different from that in families with e − p scattering and bremsstrahlung. In particular, diagrams (6.3) and (8.2) are each infrared divergent when l e = l e ; this divergence cancels, however, as expected [22], once we construct A SM ξ . The expressions we have collected complete the building blocks of the computation of the T-odd correlation in O(α) up to recoil-order corrections. The spin-averaged T-odd correlation is and we report the contributions to it family by family as each family represents a QED gauge invariant group of contributions. We employ a subscript system to identify the contributions in a straightforward way. Since the T-odd correlations are given by the interference of treelevel diagrams, which are numbered in Fig. 1 as (01) and (02), with one-loop level diagrams, which are numbered in Fig. 2 as (1), (2) [23] and present the details, as well as all needed integrals, in Appendix A. In Appendix B we report concrete expressions for the final gauge-invariant combinations of the various contributions to |M| 2 T−odd which result after performing the trace calculations and employing the formulae of Appendix A for the intermediate phase-space integrals. We work to leading order in the recoil expansion throughout. Judging by the structure of the resulting expressions one can see that some families, namely, the γ −p family containing cuts (3)+(4)+(7.2)+(8.3), as well as the e−p−γ family containing cuts (7.1)+(8.1), do not have leading-recoil-order contributions, whereas others do and need to be considered carefully. The computations necessary to determine Im(M loop ) and the resulting T-odd interference term are involved, so that we employ the program "FORM" to compute analytic expressions for the traces [24]. We compute all of the diagrams with these methods as a check of our procedures -we verify that the expected cancellations do indeed occur.

IV. RESULTS
Before presenting our final results for the asymmetry, there are three important remarks to be made concerning our numerical evaluation of the integral of |M| 2 T−odd over the allowed phase space. First of all, we note that the contributions to the asymmetry from the e − p and e − p − γ cuts dominate the final numerical result. The γ − p contribution vanishes in leading order, whereas the γ − e contribution partially cancels -the latter observation comes from our detailed numerical evaluation of the asymmetry. Second, we note that the contributions from the diagrams of the e − p cuts each contain an infrared divergence; we regulate this by inserting a fictitious photon mass m γ . However, as we show in Appendix B, the infrared divergence cancels in the net contribution to the asymmetry from the e − p cuts. The remaining piece is thus finite and well-defined, and we can safely set m γ to zero. Finally, we note it is most convenient to choose a restricted range in the γ − e opening angle. As one can see from the formulae in the Appendix A, the solutions to the Passarino-Veltman equations become invalid if the opening angle θ eγ between the outgoing electron and the photon is exactly equal to 0 or to π. There is no physical divergence. Rather, the spatial components of the vector and tensor equations to determine the relevant coefficients become degenerate at such a boundary. Potentially one could remove this difficulty by solving the equations for infinitesimal values of θ eγ or (θ eγ − π) and then interpolating the solutions to the needed θ eγ = 0 and π points. In our present work, we simply choose a restricted range x k ≡ cos θ eγ ∈ [−0.9, 0.9], which spans the angular range over which the neutron radiative decay rate is largest [25].
We can now present our results for A SM ξ . Noting Eq. (8), we see that Eqs. (4), and (5) share a common factor of e 2 g 2 V G 2 F M 2 /2, making A SM ξ independent of the decaying particle's mass in leading order in the recoil expansion. As can be seen explicitly in Appendix B, all of the contributions to |M| 2 T−odd are found to be proportional to (1 − λ 2 ), so that the resulting asymmetry goes as (1 − λ 2 )/(1 + 3λ 2 ), up to small corrections, in this limit. The universal dependence on λ in |M| 2 T−odd stems from the special nature of the T-odd correlation. It is a real triple product in momenta arising from the interference of a tree-level diagram with an imaginary part of an one-loop diagram after summing over the particles' spins. To leading order in M , the only surviving contribution is obtained from the product of the symmetric part of the lepton tensor, which is determined by a trace containing γ 5 , namely, l ρ ν αβγδ + l δ ν αβγρ − g ρδ l ν µ αβγµ , where α, β, and γ refer to photon or lepton indices, with the symmetric part of the hadron tensor. The latter is proportional to (1 + λ 2 )p ρ p δ − λ 2 M 2 g ρδ , where p is a baryon momentum and p 2 = M 2 . As one can easily check, this special combination generates an overall (1 − λ 2 ) coefficient; the remaining (1 + λ 2 ) term cannot be of leading order once the photon spin sum is effected. We use λ = −1.2701 ± 0.0025 [26] in our numerical evaluation. For definiteness, the remaining input parameters we employ are m e = 0.510999 MeV, M n = 939.565 MeV, M p = 938.272 MeV, and α −1 = 137.0360these quantities can be regarded as exact for our current purpose [26]. We show our results for the T-odd asymmetry in neutron radiative β-decay in Table I and Fig. 6. We see that the asymmetry is rather smaller than α. We recall that the radiative β-decay rate grows as log ω min as ω min → 0, whereas the |M| 2 T−odd tends to zero in that limit. Consequently the small values of the asymmetry as ω min → 0 is reflective by the growth in the decay rate itself. Generally A SM ξ is determined by an interplay between λ and the energetics of the decay, along with the value of ω min . The universal (1 − λ 2 )/(1 + 3λ 2 ) behavior in λ makes for a rich pattern. If, for some nucleus, the associated value of λ were significantly different from unity, the T-odd effect could be considerably amplified, whereas for if λ ∼ 1, the T-odd effect could be substantially reduced, facilitating from this perspective at least the search for physics beyond the SM. Interestingly a "quenching" of the Gamow-Teller strength in nuclei in relation to shell-model predictions is experimentally established [27,28] -it derives from the presence of many-body correlations in the nucleus [29]. As an illustration, we consider the process 19 Ne → 19 F + e + + ν e + γ. We map our formalism to this new case by using the parameters M19 Ne = 17700.13744 MeV and M19 F = 17696.898613 MeV [30], as well as g eff A = 0.928, as determined by Refs. [31,32] with Ref. [33] for a translation from the conventions of those references to g eff A . The axial-vector coupling can be determined in β-decay from either the measured decay rate [31,32] or the β-asymmetry [34], and the determined values are in agreement, though the former method yields a more precise result. To determine [26] and ∆ V R = 0.0240 [35], though further refinements of the latter are known in principle [36,37]. We employ λ eff = 0.941 and present the asymmetries as a function of ω min in Table II. As one can see from the numerical results for fixed ω min , the asymmetry in 19 Ne radiative β-decay is in fact significantly smaller than that in the neutron case.
In our paper, we compute the O(α) contribution to the T-odd asymmetry, keeping only the leading terms in the recoil expansion. The accuracy of our calculation is limited by the uncertainties in the input parameters we employ, as well as by the numerical size of the neglected recoil-order contributions. Crudely we expect the latter to be reduced with respect to the leading-order contribution by a factor of O(E max e /M ) ∼ 1 · 10 −3 . The uncertainty in λ eff may well be the dominant error in some nuclei. Nevertheless, we can conveniently check the rough size of the recoil-order contributions in the neutron case by replacing the vertex γ µ (g V − g A γ 5 ) in the tree-level amplitude with the weak magnetism contribution, −iσ µν q ν F 2 (q 2 )/(2M ), where F 2 (0)/g V = κ v = 3.706, the isovector magnetic moment of  which is no larger than ∼ 4 · 10 −7 for ω min = 0.3 MeV.

V. SUMMARY
In this paper, we have computed the T-odd correlation in neutron radiative β-decay arising from SM physics. The T-odd correlation is characterised by the kinematical variable ξ = l ν · (l e × k); consequently, it is spin-independent -and thus fundamentally different from a permanent EDM. The mimicking T-odd correlation arises from the presence of electromagnetic final-state interactions when the intermediate particles are each put on its own mass shell. Taking the neutron case as a prototype, we have computed the leading-order result, which is of O(α, (ε/M ) 0 ), to the T-odd asymmetry exactly. In particular, our detailed analysis shows that the resulting T-odd asymmetry is controlled by (1 − λ 2 ), so that A SM ξ vanishes as λ → 1, suggesting that radiative β-decay studies in other systems could be employed to good effect. Indeed our procedure is completely general, so that the resulting formulae can be readily translated to hyperon and nuclear radiative β-decays as well. We have explicitly considered the example of 19 Ne radiative β-decay as a concrete illustration of the possibilities. Its lifetime is much shorter than the neutron's, making experiments more practical, and it should be possible to study such decays in a trapped atom experiment [38]. It also possesses a value of λ eff rather close to unity. The combined effect of these observations is a T-odd asymmetry roughly an order of magnitude smaller than that in neutron β-decay. In this case A SM ξ presents a much reduced background against which to search for new sources of CP-violation.
hospitality during the execution of this work. We acknowledge partial support from the U.S. Department of Energy under contract DE-FG02-96ER40989.

Appendix A: Intermediate Phase Space Integrals
The computation of the imaginary parts of the loop diagrams requires an integration over the allowed phase space of the intermediate momenta as fixed by the momenta of the final-state particles and energy-momentum conservation. In this Appendix we report the integrals which appear in the diagrams of Fig. 2 and label them as per the diagrams in that figure. For diagrams with cuts which yield Compton scattering from electrons our results can be compared to, and agree with, those of Refs. [11] and [12]. In what follows we report the integrals which arise from γ − e cuts: (1), (2), (5.1), and (6.2), and then the integrals which arise from the cutting of electron and proton lines to generate physical ep → epγ scattering, namely, (5.2) and (6.1), and ep → ep scattering, (6.3) and (8.2). The integrals associated with the rest of the cuts in Fig. 2 are not given explicitly because they do not contribute in leading order in the recoil expansion, as we note in the main body of the text. Nevertheless, we note the relationships between these integrals which appear in the large M p limit in order to make the cancellations associated with these terms transparent.
From diagram (1), defining P 0e ≡ l e + k, we have as well as From diagram (2) we have We apply the Passarino-Veltman reduction method to compute integrals which contain additional powers of the intermediate momenta [23]. That is, writing the values of a 2 and b 2 are fixed by the solution of the set of equations J 1 = a 2 m 2 e + b 2 l e · P 0e , l e · kJ 2 = a 2 l e · P 0e + b 2 P 2 0e . Moreover, where c 2 , d 2 , e 2 , and f 2 are given by the solution of the set of equations 0 = 4c 2 + d 2 m 2 e + e 2 P 2 0e + 2f 2 l e · P 0e , 0 = c 2 + d 2 m 2 e + f 2 l e · P 0e , a 1 = e 2 l e · P 0e + f 2 m 2 e , l e · kb 2 = c 2 + e 2 P 2 0e + f 2 l e · P 0e .
For integrals which depend on M p we report their form in the large M p limit for subsequent use. Note that M rather than M p appears in the limiting form because the n − p mass difference itself is of higher order in the recoil expansion. From diagram (5.1) we have with I 0e = (p p · P 0e ) 2 − M 2 p P 2 0e , noting as M p → ∞. In addition where a 5.1 and b 5.1 are given by the solution of the set of equations J 1 = a 5.1 M 2 p + b 5.1 p p · P 0e , l e · kJ 5.1 = a 5.1 p p · P 0e + b 5.1 P 2 0e .
In the large M p limit b 5.1 ∼ 1/M and a 5.1 ∼ 1/M 2 . We postpone discussion of the integrals from diagrams (5.2) and (6.1) to consider the integrals from the remaining diagrams with Compton cuts. From diagram (6.2) we have with I e = (p p · l e ) 2 − M 2 p m 2 e and J 6.2 ∼ π 2M |l e |k · l e log E e + |l e | E e − |l e | .
as M p → ∞. In addition = a 6.2 P µ 0e + b 6.2 l µ e + c 6.2 p µ p , where a 6.2 , b 6.2 , and c 6.2 are given by the solution to the set of equations J 2 = a 6.2 p p · P 0e + b 6.2 p p · l e + c 6.2 M 2 p , J 5.1 = a 6.2 l e · P 0e + b 6.2 m 2 e + c 6.2 p p · l e , l e · kJ 6.2 = a 6.2 P 2 0e + b 6.2 l e · P 0e + c 6.2 p p · P 0e . and in the large M p limit a 6.2 , b 6.2 ∼ 1/M and c 6.2 ∼ 1/M 2 . Finally L µν 6.2 = dρ γe k µ k ν (l e · k )(p p · k ) = d 6.2 g µν + e 6.2 p µ p p ν p + f 6.2 l µ e l ν e + g 6.2 P µ 0e P ν 0e + h 6.2 (p µ p l ν e + l µ e p ν p ) +i 6.2 (p µ p P ν 0e + P µ 0e p ν p ) + k 6.2 (l µ e P ν 0e where the coefficients which appear are given by the solution to set of the equations 4d 6.2 + e 6.2 M 2 p + f 6.2 m 2 e + g 6.2 P 2 0e + 2h 6.2 p p · l e + 2i 6.2 p p · P 0e + 2k 6.2 l e · P 0e = 0 , d 6.2 + f 6.2 m 2 e + h 6.2 p p · l e + k 6.2 l e · P 0e = 0 , e 6.2 p p · l e + h 6.2 m 2 e + i 6.2 l e · P 0e = a 5.1 , g 6.2 l e · P 0e + i 6.2 p p · l e + k 6.2 m 2 e = b 5.1 , d 6.2 + e 6.2 M 2 p + h 6.2 p p · l e + i 6.2 p p · P 0e = 0 , g 6.2 p p · P 0e + i 6.2 M 2 p + k 6.2 p p · l e = b 2 , d 6.2 + g 6.2 P 2 0e + i 6.2 p p · P 0e + k 6.2 l e · P 0e = l e · ka 6.2 . Note that the equations have been chosen to yield a self-consistent solution for the six coefficients.
The integrals associated with the γ − p cuts can be found if necessary by replacing the intermediate momentum l e by p p as well as l e by p p in the γ − e integrals we have provided. Specifically we note where P 0p ≡ p p + k, and as M p → ∞. Moreover, and whereas and so that The integrals in the remaining diagrams of Fig. 2 arise from cutting the electron and proton lines to generate physical ep → epγ or ep → ep scattering. The intermediate phase space integrals in these cases are more complicated than those associated with the Compton cuts; fortunately, closed-form expressions for the integrals in the large M p limit suffice to leading order in the recoil expansion. With P 0 ≡ p p + l e + k, we note for diagram (5.2) as M p → ∞. Moreover, and J 5.2 ∼ π 4M |l e + k| log m 2 e + l e · k − (E e + ω) 2 + (E e + ω) 2 − m 2 e |l e + k| m 2 e + l e · k − (E e + ω) 2 − (E e + ω) 2 − m 2 e |l e + k| (A22) as M p → ∞. In addition, where a 5.2 and c 5.2 are given by the solution to (m 2 e + l e · k)J 5.2 − so that in the large M p limit a 5.2 ∼ 1/M and c 5.2 ∼ 1/M 2 . Turning to the integrals from diagram (6.1) we have so that as M p → ∞ as well as where as M p → ∞ I 6.1 ∼ 2(m 2 e + l e · k)I 6.1 − 2I 5.2 − 2Ĩ 6.1 (A27) with I 6.1 = πk · l e M ω 2 (E e + ω) 2 − m 2 e + (E e + ω)k · l e 2k · l e log E e + ω + (E e + ω) 2 − m 2 e E e + ω − (E e + ω) 2 − m 2 e .