Gravitational Bremsstrahlung in the Post-Minkowskian Effective Field Theory
Stavros Mougiakakos, Massimiliano Maria Riva, Filippo Vernizzi
aa r X i v : . [ g r- q c ] F e b Gravitational Bremsstrahlungin the Post-Minkowskian Effective Field Theory
Stavros Mougiakakos, Massimiliano Maria Riva, and Filippo Vernizzi Institut de physique th´eorique, Universit´e Paris Saclay CEA, CNRS, 91191 Gif-sur-Yvette, France (Dated: February 17, 2021)We study the gravitational radiation emitted during the scattering of two spinless bodies in thepost-Minkowskian Effective Field Theory approach. We derive the conserved stress-energy tensorlinearly coupled to gravity and the classical probability amplitude of graviton emission at leadingand next-to-leading order in the Newton’s constant G . The amplitude can be expressed in compactform as one-dimensional integrals over a Feynman parameter involving Bessel functions. We useit to recover the leading-order radiated angular momentum expression of [1]. Upon expanding itin the relative velocity between the two bodies v , we compute the total four-momentum radiatedinto gravitational waves at leading-order in G and up to order v , finding agreement with [2]. Ourresults also allow to investigate the zero frequency limit of the emitted energy spectrum. I. INTRODUCTION
The understanding of the dynamics of binary systemsand their gravitational wave emission has been crucial forthe extraordinary discovery of LIGO/Virgo [3, 4]. Thisfield has recently received a renewed attention, partic-ularly in application of the so-called post-Minkowskian(PM) framework [5–14], which consists in expanding thegravitational dynamics in the Newton’s constant G whilekeeping the velocities fully relativistic. This is comple-mentary to the post-Newtonian approach (see [15, 16]and references therein), where one expands in both ve-locity and G , since in a bound state these two are relatedby the virial theorem.Recently, many progresses have been made within thePM framework thanks to the application of several com-plementary approaches: in particular the effective one-body method [12, 13, 17, 18], the use of scattering am-plitude technics, such as the double copy [19–21], gener-alized unitarity [22–24] and effective field theory (EFT)[25–32] (see [33–41] for the quantum field theoretic de-scription of gravity), and worldline EFT approaches [42–46]. These developments concern the scattering of un-bound states but results can be extended to bound statesby applying an analytic continuation between hyperbolicand elliptic motion [47, 48]. Progresses have addressedthe conservative binary dynamics up to 3PM order [49–52], as well as tidal [53–59], spin [60–64] and radiationeffects [1, 65–71], and have spurred other new interestingresults (see e.g. [72–74] for an incomplete list).The culminating product of the scattering amplitudeprogram is the recent derivation of the 4PM two-bodyHamiltonian [75]. At this order, a tail effect is present[76–78] and manifests an infrared divergence propor-tional to the leading-order ( G ) energy of the radiatedBremsstrahlung, the gravitational waves emitted dur-ing the scattering of two masses approaching each otherfrom infinity. Studies on the leading-order gravitationalBremsstrahlung include [11, 79–84]. The full leading-order energy spectrum found in [75] was independentlyobtained in [2] using the formalism of [29], which derives classical observables from scattering amplitudes and theirunitarity cuts.In this paper we study the gravitationalBremsstrahlung using a worldline approach inspired byNon-Relativistic-General-Relativity (NRGR) [85] (see[86–90] for reviews) and recently applied to the PMexpansion [42–44, 52, 98]. In particular, we first definethe Feynman rules that allow us to derive the leadingand next-to-leading order stress-energy tensor linearlycoupled to gravity. From this we compute the classicalprobability amplitude of graviton emission, which isdirectly related to the waveform in Fourier space. Theamplitude is the basic ingredient for the computationof observables such as the radiated four-momentum andangular momentum, which we discuss in various limitand compare to the literature.Another article [99], whose content overlaps with ours,appeared while finalizing this work. II. POST-MINKOWSKIAN EFFECTIVE FIELDTHEORY
We consider the scattering of two gravitationally inter-acting spinless bodies with mass m and m approachingeach other from infinity. The gravitational dynamics isdescribed by the usual Einstein-Hilbert action. Neglect-ing finite size effects, which would contribute at higherorder in G (see e.g. [44, 53]), the bodies are treated asexternal sources described by point-particle actions. Weuse the Polyakov parametrization of the action and fixthe vielbein to unity. This has the advantage of sim-plifying the gravitational coupling to the matter sources[44, 100, 101]. Therefore, using the mostly minus met-ric signature, setting ~ = c = 1 and defining the Planckmass as m Pl ≡ / √ πG , we have S = − m Z d x √− gR − X a =1 , m a Z dτ a (cid:2) g µν ( x a ) U µa ( τ a ) U νa ( τ a ) + 1 (cid:3) , (1)where, for each body a , τ a is its proper time and U µa ≡ dx µa /dτ a is its four-velocity.To compute the waveform we need the (pseudo) stress-energy tensor T µν , defined as the linear term sourcing thegravitational field in the effective action [35, 85, 102], i.e.,Γ[ x a , h µν ] = − m Pl Z d xT µν ( x ) h µν ( x ) , (2)where h µν ≡ m Pl ( g µν − η µν ). From the Fourier trans-form of T µν , defined by ˜ T µν ( k ) = R d x T µν ( x ) e ik · x , onecan compute the (classical) probability amplitude of onegraviton emission with momentum k and helicity λ = ± i A λ ( k ) = − i m Pl ǫ ∗ λµν ( k ) ˜ T µν ( k ) , (3)where ǫ λµν ( k ) is the transverse-traceless helicity-2 polar-ization tensor, with normalization ǫ ∗ λµν ( k ) ǫ µνλ ′ ( k ) = δ λλ ′ (see definition in App. A). At distances r much largerthan the interaction region, the waveform is given interms of the amplitude as (see e.g. [103]) h µν ( x ) = − πr X λ = ± Z dk π e − ik u ǫ λµν ( k ) A λ ( k ) | k µ = k n µ , (4)where u ≡ t − r . The amplitude is evaluated on-shell,i.e. k µ = k n µ , with n µ ≡ (1 , n ) and n the unitary vectorpointing along the graviton trajectory.We can obtain the stress-energy tensor defined aboveby matching eq. (2) to the effective action computed or-der by order in G using Feynman diagrams. Let us nowintroduce the Feynman rules. Adding the usual de Don-der gauge-fixing term to eq. (1), S gf = Z d x (cid:20) ∂ ρ h µν ∂ ρ h µν − ∂ ρ h∂ ρ h (cid:21) , (5)where h ≡ η µν h µν , from the quadratic part of the gravi-tational action one can extract the graviton propagator, µν ρσk = i ( k + iǫ ) − | k | P µν ; ρσ , (6)where P µν ; ρσ ≡ ( η µρ η νσ + η µσ η νρ − η µν η ρσ ). We haveimposed retarded boundary conditions to account onlyfor outgoing gravitons. The cubic interaction vertex ofthe gravitational action can be found, for instance, in[35, 104].Thanks to the Polyakov form, the point-particle actioncontains only a linear interaction vertex. However, in or-der to isolate the powers of G , we parametrize the world-line by expanding around straight trajectories [44, 52],i.e., x µa ( τ a ) = b µa + u µa τ a + δ (1) x µa ( τ a ) + . . . , (7a) U µa ( τ a ) = u µa + δ (1) u µa ( τ a ) + . . . . (7b) Here u a is the (constant) asymptotic incoming velocityand b a is the body displacement orthogonal to it, b a · u a =0, while δ (1) x µa and δ (1) u µa are respectively the deviationfrom the straight trajectory and constant velocity of body a at order G , induced by the gravitational interaction.Moreover, we define the impact parameter as b µ ≡ b µ − b µ and the relative Lorentz factor as γ ≡ u · u = 1 √ − v , (8)where v is the relativistic relative velocity between thetwo bodies.The expansion of the worldline action in the secondline of eq. (1) generates two Feynman interaction rulesthat differ by their order in G . At zeroth order, we have(with R q ≡ R d q (2 π ) ) τ a = − im a m Pl u µa u νa Z dτ a Z q e − iq · ( b a + u a τ a ) , (9)where a filled dot denotes the point particle evaluatedusing the straight worldline. At first order in G we have1 τ a = − im a m Pl Z dτ a Z q e − iq · ( b a + u a τ a ) × (cid:16) δ (1) u ( µa ( τ a ) u ν ) a − i ( q · δ (1) x a ( τ a )) u µa u νa (cid:17) , (10)where the correction O ( G n ) to the trajectory is denotedby the order n inside the circle. The O ( G ) correctionto the velocity and the trajectory can be computed bysolving the geodesic equation obtained from the effectiveLagrangian at order G and for particle 1 it reads [44] δ (1) u µ ( τ ) = m m Z q δ − ( q · u ) e − iq · b − iq · u τ q B µ , (11) δ (1) x µ ( τ ) = im m Z q δ − ( q · u ) e − iq · b − iq · u τ q ( q · u + iε ) B µ , (12)where B µ ≡ γ − q µ q · u + iε − γu µ + u µ and we keep im-plicit the retarded boundary condition on the gravitonpropagator, 1 /q = 1 / [( q + iǫ ) − | q | ]. III. STRESS-ENERGY TENSOR
The radiated field can be computed in powers of G in terms of the diagrams shown in Fig. 1. The leadingstress-energy tensor is obtained from Fig. 1a and corre-sponds to the one of free point-particles, i.e.,˜ T µν Fig. a ( k ) = X a m a u µa u νa e ik · b a δ − ( ω a ) , (13)where we use the notation δ − ( n ) ( x ) ≡ (2 π ) n δ ( n ) ( x ) and forconvenience we define ω a ≡ k · u a , a = 1 , . (14) τ k (a) τ k (b) τ τ k (c) FIG. 1. The three Feynman diagrams needed for the compu-tation of the stress-energy tensor up to NLO order in G . Tocompute the symmetric one, it is enough to exchange 1 ↔ This generates a static and non-radiating contributionto the amplitude, proportional to δ − ( ω a ). While this con-tribution can be neglected when computing the radiatedmomentum, it must be crucially included for the compu-tation of the angular momentum, as shown below.At the next order we find˜ T µν Fig. 1b ( k ) = m m m Z q ,q µ , ( k ) 1 q (cid:20) γ − ω + iǫ q ( µ u ν )1 − γu ( µ u ν )1 − (cid:18) γ − k · q ( ω + iǫ ) − γω ω + iǫ − (cid:19) u µ u ν (cid:21) , (15)˜ T µν Fig. 1c ( k ) = m m m Z q ,q µ , ( k ) 1 q q (cid:20) γ − q µ q ν + (cid:0) ω − q (cid:1) u µ u ν + 4 γω q ( µ u ν )1 − η µν (cid:18) γω ω + 2 γ − q (cid:19) + 2 (cid:0) γq − ω ω (cid:1) u ( µ u ν )2 (cid:21) , (16)where µ , ( k ) ≡ e i ( q · b + q · b ) δ − (4) ( k − q − q ) δ − ( q · u ) δ − ( q · u ) , (17)and we have used momentum conservation, on-shell andharmonic-gauge conditions to simplify the final expres-sion. Of course, we must also include the analogous dia-grams with bodies 1 and 2 exchanged. The contributionin Fig. 1b comes from evaluating the worldline along de-flected trajectories while the one in Fig. 1c comes fromthe gravitational cubic interaction. We have checked thatthe sum of these two contributions is transverse for on-shell momenta, i.e. k µ ˜ T µν = 0 for k = 0, as expectedfor radiated gravitons. We have also verified that thefinite part of the stress-energy tensor agrees with that computed in [42] once the contribution from the dilatonis removed. IV. AMPLITUDES AND WAVEFORMS
We expand the amplitude defined in eq. (3) in powersof G , A λ = A (1) λ + A (2) λ + . . . . Given the definition (3)and the stress-energy tensor (13), the leading order reads A (1) λ ( k ) = − m Pl X a m a ǫ ∗ λµν ( n ) u µa u νa e ik · b a δ − ( ω a ) . (18)The NLO can be obtained by summing eqs. (15) and(16) and inserting the result in eq. (3). Integrating overone of the internal momenta, A (2) λ ( k ) = − m m m ǫ ∗ λµν ( n ) (cid:26) e ik · b (cid:20) (cid:18) − γ − k · I (1) ( ω + iǫ ) + 2 γω ω + iǫ I (0) + 2 ω J (0) (cid:19) u µ u ν + (cid:18) γ − ω + iǫ I µ (1) + 4 γω J µ (1) (cid:19) u ν − (cid:0) γI (0) + ω ω J (0) (cid:1) u µ u ν + 2 γ − J µν (2) (cid:21)(cid:27) + (1 ↔ , (19)where we have defined the following integrals, I µ ...µ n ( n ) ≡ Z q δ − ( q · u − ω ) δ − ( q · u ) e − iq · b q q µ . . . q µ n , (20) J µ ...µ n ( n ) ≡ Z q δ − ( q · u − ω ) δ − ( q · u ) e − iq · b q ( k − q ) q µ . . . q µ n . (21)(The indices inside these integrals must be changed whenevaluating the symmetric contribution (1 ↔ b µ = 0 and b µ = b µ and define the unit spatial vectors in the direction of v and of the impact parameter b , respectively e v ≡ v /v and e b = b / | b | , with e v · e b = 0. We also define v µ ≡ (1 , v e v ) so that u µ = δ µ , u µ = γv µ = γ (1 , v e v ) . (22)The energies of the radiated gravitons measured by thetwo bodies become, respectively, ω = k ≡ ω and ω = γω n · v . The amplitude simplifies to the follow-ing compact forms A (1) λ ( k ) = − m m Pl γv n · v ǫ ∗ λij e iv e jv δ − ( ω ) e ik · b , (23) A (2) λ ( k ) = − Gm m m Pl γv ǫ ∗ λij e iI e jJ A IJ ( k ) e ik · b . (24)After solving the integrals (20) and (21), for the functions A IJ we find A vv = c K (cid:0) z ( n · v ) (cid:1) + c h K (cid:0) z ( n · v ) (cid:1) − iπδ (cid:0) z ( n · v ) (cid:1)i + Z dy e iyzv n · e b h d ( y ) zK (cid:0) zf ( y ) (cid:1) + c K (cid:0) zf ( y ) (cid:1)i , (25) A vb = ic h K (cid:0) z ( n · v ) (cid:1) − iπδ (cid:0) z ( n · v ) (cid:1)i + i Z dy e iyzv n · e b d ( y ) zK (cid:0) zf ( y ) (cid:1) , (26) A bb = Z dy e iyzv n · e b d ( y ) zK (cid:0) zf ( y ) (cid:1) , (27)where K and K are modified Bessel functions of thesecond kind and we have introduced z ≡ | b | ωv , (28)and f ( y ) ≡ p (1 − y ) ( n · v ) + 2 y (1 − y )( n · v ) + y /γ . (29)The coefficients c , c and c depend on v and on therelative angles between the graviton direction and thebasis ( e v , e b ). Moreover, d , d and d depend also onthe integration parameter y . Their explicit form is givenin App. C. In eqs. (25) and (26) we have also includedthe non-radiating contribution proportional to a delta function, which may become relevant, for instance, whencomputing the radiated angular momentum at NLO.The waveform can be computed by replacing the am-plitude in eq. (4). We have not verified the full expres-sions with the waveform given in direct space in [83] butwe have checked that we recover their forward limit inFourier space. Moreover, we also find agreement with[83] for small-velocities . In this limit the exponential inthe above expressions can be expanded and the integralin y performed. We will use this limit to verify the totalradiated four-momentum below. V. RADIATED FOUR-MOMENTUM
In terms of the asymptotic waveform, the radiatedfour-momentum at infinity ( r → ∞ ) is given by [1, 83] P µ rad = Z d Ω du r n µ ˙ h ij ˙ h ij , (31)where a dot denotes the derivative with respect to theretarded time u and d Ω is the integration surface element.Using eq. (4) for the waveform, this can be expressedin a manifestly Lorentz-invariant way in terms of the am-plitude (3) as [42] P µ rad = X λ Z k δ − ( k ) θ ( k ) k µ |A λ ( k ) finite | , (32)where θ is the Heaviside step function and on the right-hand side we take only the finite part of the ampli-tude, excluding the terms proportional to a delta func-tion that do not contribute to ˙ h ij . Thus, at leading order |A λ ( k ) finite | = |A (2) λ ( k ) finite | + . . . and hence the radi-ated four-momentum starts at order G Since the modulo squared of the amplitude is symmet-ric under k → − k the four-momentum cannot depend onthe spatial direction b µ . Moreover, the energy measuredin the frame of one body is the same as the one measuredin the frame of the other one, hence the final result mustbe proportional to u µ + u µ . Using eq. (24), we can writeit as P µ rad = G m m | b | u µ + u µ γ + 1 E ( γ ) + O ( G ) , (33) To compute this contribution we have used this integral: Z q δ ( q · u ) δ ( q · u ) e − iq · b q µ q = b µ πγv | b | . (30) The signs in front of K and K of the last term of eqs. (2.9b)and (2.9c) of [83] are incorrect. We are using a different normalization of h µν with respect tothese references, which explains the absence of the prefactor(32 πG ) − . where E ( γ ) = 2 | b | π ( γ − X λ Z d Ω Z ∞ ω dω (cid:12)(cid:12) ǫ ∗ λij e iI e jJ A IJ ( k ) (cid:12)(cid:12) . (34)We confirm that at this order the result has homogeneousmass dependence and is thus fixed by the probe limit[2, 77, 83].Due to the involved structure of the y integrals ineq. (24), we were unable to compute E explicitly. Nev-ertheless, we can first compute the integrals in y in the v ≪ O ( v ), obtaining E π = 3715 v + 2393840 v + 6170310080 v + 3131839354816 v + O ( v ) . (35)The radiated energy in center-of-mass frame, P rad · u CoM ,where u µ CoM = m u µ + m u µ p m + m + 2 m m γ , (36)agrees with the 2PN results [77, 83, 105] while eq. (35)matches the expansion of the fully relativistic result re-cently found in [2]. This is a non-trivial check of our NLOamplitude (24).As an extra check, we can compute the leading-orderenergy spectrum in the soft limit, which is obtained byconsidering only wavelengths of the emitted gravitonsmuch larger than the interaction region, i.e. | b | ω/v ≪ E rad ≡ P this is given by dE rad dω (cid:12)(cid:12)(cid:12) ω → = 12(2 π ) X λ Z d Ω | ω A λ ( k ) ω → | . (37)In this limit the amplitude at order G receives contri-butions exclusively from the diagram in Fig. 1b, so it isnot affected by the gravitational self-interactions. Fromeqs. (24)–(27), it reads A (2) λ ( k ) ω → = − Gm m m Pl | b | γωn · v ǫ ∗ λij ( c e iv e jv + 2 ic e iv e ib ) . (38)Integrating eq. (37) over the angles by fixing some an-gular coordinate system and introducing the function I ( v ) ≡ − + v + v − v arctanh( v ) [1], we obtain dE rad dω (cid:12)(cid:12)(cid:12) ω → = 4 π (2 γ − γ v G m m | b | I ( v ) + O ( G ) , (39)which agrees with [71, 106]. We will come back to thisresult below. VI. RADIATED ANGULAR MOMENTUM
The angular momentum lost by the system is anotherinteresting observable as it can be related to the cor-rection to the scattering angle due to radiation reaction [1]. In terms of the asymptotic waveform this is given by[1, 107] J i rad = ǫ ijk Z d Ω du r (cid:16) h jl ˙ h lk − x j ∂ k h lm ˙ h lm (cid:17) . (40)As pointed out in [1], the waveform at order G is staticand can be pulled out of the time integration leavingwith the computation of the gravitational wave memory∆ h ij ≡ R + ∞−∞ du ˙ h ij . This can be related to the classicalamplitude by eq. (4),∆ h ij = i πr X λ Z dω π ǫ λij δ − ( ω ) ω A λ ( k ) ω → , (41)where from the right-hand side it is clear that only thesoft limit contributes to the gravitational wave memory.Moreover, since at this order the soft limit is uniquelydetermined by the diagram in Fig. 1b, the radiated an-gular momentum does not depend on the gravitationalself-interaction, confirming [1].To compute the radiated angular momentum, itis convenient to introduce a system of polar co-ordinates where n = (sin θ cos φ, sin θ sin φ, cos θ )and an orthonormal frame tangent to the sphere,with e θ = (cos θ cos φ, cos θ sin φ, − sin θ ) and e φ =( − sin φ, cos φ, ε ijk ǫ λjl ǫ ∗ λ ′ lk = − iλn i δ λλ ′ . The second term canbe rewritten by noticing that ǫ ijk x j ∂ k = i ˆ L i , where ˆ L i isthe usual orbital angular momentum operator, expressedin terms of the angles and their derivatives (see App. A).Using ǫ ∗ λ ′ lm ˆ L ǫ λlm = λ cot θ e θ δ λλ ′ , we obtain J rad = X λ Z d Ω(4 π ) ω A (2) λ ∗ ( k ) ω → ˆ J a (1) λ + O ( G ) , (42)where ˆ J ≡ λ ( n +cot θ e θ )+ ˆ L and we have introduced a (1) λ as the leading-order amplitude striped off of the deltafunction, i.e. defined by A (1) λ ( k ) = a (1) λ δ − ( ω ) e ik · b . (43)One can perform the angular integral in eq. (42) by align-ing e v and e b along any (mutually orthogonal) directionsand eventually obtains J rad = 2(2 γ − γv G m m J | b | I ( v )( e b × e v ) , (44)where J = m γv | b | is the angular momentum at infinity.This result agrees with [1].As noticed in [71], from eqs. (39) and (42) we observean intriguing proportionality between the energy spec-trum in the soft limit and the total emitted angular mo-mentum. We leave a more thorough exploration of thisresult for the future. VII. CONCLUSION
We have studied the gravitational Bremsstrahlung us-ing a worldline approach. In particular, we have com-puted through the use of Feynman diagrams, expand-ing perturbatively in G , the leading and next-to-leadingorder classical probability amplitude of graviton emis-sion and consequently the waveform in Fourier space.The next-to-leading order amplitude receives two con-tributions: one from the deviation from straight orbits,which can be expressed in terms of modified Bessel func-tions of the second kind; another from the cubic gravi-tational self-interaction, which we could rewrite as one-dimensional integrals over a Feynman parameter of modi-fied Bessel functions. When comparison was possible, wefound agreement with earlier calculations of the wave-forms [79, 82] in different limits.We have used the amplitude to compute the leading-order radiated angular momentum, recovering the resultof [1]. Moreover, we have computed the total emittedfour-momentum expanded in small velocities up to or-der v and we found agreement with the recent resultsof [2, 75]. Unfortunately we were not able to reproducetheir fully relativistic result, which we leave for the fu-ture. Nevertheless, we have built the foundations for analternative derivation of the recent results obtained withamplitude techniques.Another interesting limit is for small gravitationalwave frequencies, where the amplitude does not receivecontributions from the gravitational interaction. We havecomputed the soft energy spectrum recovering an intrigu-ing relation with the emitted angular momentum [71].Future directions include the study of spin and finite-sizeeffects and a more thorough investigation of the relationsbetween differential observables. ACKNOWLEDGEMENTS
We thank Laura Bernard, Luc Blanchet, Alberto Nico-lis, Federico Piazza, Khim Leong Wong, Pierre Vanhoveand Gabriele Veneziano for insightful discussions and cor-respondence. This work was partially supported by theCNES and by the Munich Institute for Astro- and Par-ticle Physics (MIAPP) which is funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) under Germany’s Excellence Strategy – EXC-2094 – 390783311.
Appendix A: Angular dependence
We can introduce the transverse-tracelesshelicity-2 tensors, normalized to unity, interms of the orthonormal frame tangent to thesphere, e θ = (cos θ cos φ, cos θ sin φ, − sin θ ) and e φ = ( − sin φ, cos φ, ǫ ± i ≡ √ ± e iθ + i e iφ ) , ǫ ± ij = ǫ ± i ǫ ± j . (A1)We can relate these tensors to the (real) plus and crossparametrization often used in the literature by ǫ plus ij = ǫ + ij + ǫ − ij , ǫ cross ij = − i (cid:0) ǫ + ij − ǫ − ij (cid:1) . (A2)For convenience, here we also explicitly report the ex-pression of the (orbital) angular momentum operator interms of the same polar coordinates,ˆ L x = i (sin φ∂ θ + cot θ cos φ∂ φ ) , (A3)ˆ L y = − i (cos φ∂ θ − cot θ sin φ∂ φ ) , (A4)ˆ L z = − i∂ φ . (A5) Appendix B: Integrals
To compute the integrals in eq. (20) we first need themaster integral I (0) , which can be solved by going tothe frame of body 2 as in eq. (22) and by removing thedelta functions by integrating in q and in the spatialmomentum along v . This leaves us with I (0) = − γv Z d q ⊥ (2 π ) e i q ⊥ · b | q ⊥ | + ω γ v = − πγv K (cid:18) | b | ω γv (cid:19) , (B1)where we can write | b | = √− b in a Lorentz-invariantfashion.We use this result to compute the descendant integrals I µ ...µ n ( n ) (see analogous examples in [29]). For instance,by the presence of δ ( q · u ) in the integrand, I µ (1) canonly be a sum of two pieces, one proportional to b µ andanother proportional to u µ − γu µ . The piece proportionalto b µ can be computed by taking the derivative of I (0) with respect to b µ and projecting it along b µ with propernormalization. It is easy to see that the other piece isproportional to I (0) upon projecting I µ (1) along u µ andtaking into account the first delta function.To compute the integrals in eq. (21), we can proceedanalogously. Although we were not able to solve the mas-ter integral J (0) in close form, we can express it in termsof an integral over a Feynman parameter as J (0) = Z dy e − iyk · b Z q δ − ( q · u + ( y − ω ) × δ − ( q · u + yω ) e − iq · b /q , (B2)where the integral in q can be solved similarly to I (0) . Appendix C: Coefficients
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