Asymptotic conservation law with Feynman boundary condition
aa r X i v : . [ h e p - t h ] J a n || Sri Sainath || Asymptotic conservation law with Feynman boundarycondition
Sayali Atul Bhatkar,
Indian Institute of Science Education and Research,Homi Bhabha Rd, Pashan, Pune 411 008, India.
E-mail: [email protected].
Abstract
Recently it was shown that classical electromagnetism admits new asymptotic conservationlaws [21]. In this paper we derive the analogue of the first of these asymptotic conservation lawsupon imposing Feynman boundary condition on the radiative field. We also show that the Feyn-man solution at O ( e ) contains purely imaginary modes falling off as { log uu n r , n ≥ } which areabsent in the classical radiative solution. The log u mode has also appeared in [22, 23] and violatesthe Ashtekar-Struebel fall offs for the radiative field [24]. We expect that new (log u ) m u m r -modes wouldappear in the Feynman solution at order O ( e m +1 ). Thus, all the other modes are expected topreserve the Ashtekar-Struebel fall offs. 1 ontents log ur mode in A σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 The ur mode in A σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 O ( e ) with Feynman propagator 17 log uur mode in A σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 Expectation at higher orders in e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Soft theorems in gravitational and gauge theories [1–6] are related to asymptotic conservation laws.In the classical theory these conservation laws take following form : Q + (ˆ x ) | I + − = Q − ( − ˆ x ) | I − + . (1)Here, the future charge Q + is defined at I + − i.e. the u → −∞ sphere of the future null infinitydenoted by I + and u = t − r is its null generator. Similarly, the past charge Q − is defined at I − + which is the v → ∞ sphere of the past null infinity I − and v = t + r is the null generator of I − .In [7–11], the authors discussed the symmetry underlying the leading soft photon theorem. They2lso showed that the corresponding charges obey an asymptotic conservation law as given in (1).This line of investigation was further extended in [12]; it was shown that classical radiative fieldat O ( e ) admits an infinite number of conservation laws. They also provided evidence that theseconservation laws are related to the infinite number of tree level soft theorems proved in [13, 14] .Thus, tree level soft theorems in QED can be related to asymptotic conservation laws.In four spacetime dimensions, soft theorems admit non trivial loop corrections beyond theleading order term [16–18]. In [19, 20], the authors derived the subleading soft theorems for loopamplitudes; these subleading terms involve logarithms of the soft energy and are absent in thetree level analysis. These terms are closely tied to the long range forces present in four spacetimedimensions. It is natural to ask if the loop level soft theorems are related to new asymptoticconservation laws. Accordingly in [21], we incorporated the effect of long range electromagneticforce on the scattering particles and showed that there exists a new family of conservation lawsfor classical electromagnetism. It is expected that these conservations laws are related to the looplevel soft theorems.The first of these conservation laws relates the coefficient of the log rr mode at the past to thecoefficient of the log ur mode at the future ( u is the retarded time). This asymmetry is due to theuse of retarded boundary condition. The radiation travels to the future and we have a radiative log ur mode at the future while there exists a coulombic log rr mode at the past. This is expected tochange in the quantum theory due to the use of Feynman boundary condition.In this paper, our aim is to derive the analogue of this conservation law after imposing Feynmanboundary condition on the radiative field. We show that the logarithmic modes obey the asymp-totic conservation equation given in (91) by calculating the radiative field generated by scatteringof n charged point particles. An interesting aspect is that the Feynman solution contains newmodes (purely quantum modes) that are absent in the classical solution obeying retarded bound-ary condition. In this paper, we also aim to study the structure of these quantum modes. Thoughwe obtain the explicit expression for these modes in this toy example of scattering of n chargedparticles we will argue that many features of these modes are universal. Outline of this paper
In this paper our plan is to calculate the asymptotic radiative field generated by a general scatter-ing event involving n number of charged point particles using Feynman propagator. Although thisis not a physical problem, the so obtained Feynman radiative solution is useful to illustrate manyinteresting properties of the quantum gauge field. We will use this toy example to study certainuniversal aspects of the quantum gauge field. We work perturbatively in the coupling ’ e ’ as wellas in asymptotic parameters 1 /r (or 1 /t ).In a scattering process some n number of charged particles come in to interact and eventuallymove away from each other. We can divide the entire spacetime into two parts : a bulk regionwhich is a sphere of radius R around the origin r = 0 such that the non trivial interaction betweenthe particles takes place within this sphere. In this region, the particles in general will move oncomplicated trajectories depending on short range forces present between them. The second regionis the asymptotic region r > R in which we can completely ignore the short range forces. In theasymptotic region, we need to include the effect of the long range electromagnetic interaction thatstarts at O ( r ). This is done perturbatively in e and 1 /r .In section 2, we start by reviewing the asymptotic expansion of radiative field generated by The subleading terms admit corrections in presence of non-minimal couplings [15]. ω → O ( e ) using Feynman propagator and discuss the new modes in the asymptoticfield that arise as a result of the long range interactions between the scattering particles. We alsodiscuss new modes that would possibly appear in the Feynman solution beyond O ( e ). In section7, we discuss asymptotic conservation equation obeyed by the O ( e ) logarithmic modes of theFeynman radiative solution. Finally we summarise our results in section 8. In this section we will obtain the asymptotic expansion of the radiative field near future null infinityusing retarded propagator. The flat metric takes following form in retarded co-ordinate system( u = t − r ) : ds = − du − dudr + r γ z ¯ z dzd ¯ z ; γ z ¯ z = 2(1 + z ¯ z ) . I + corresponds to the limit r → ∞ with u finite. We use ˆ x or ( z, ¯ z ) interchangeably to describepoints on S . We will often use following parametrisation of a 4 dimensional spacetime point(Greek indices will be used to denote 4d cartesian components) : x µ = rq µ + ut µ , q µ = (1 , ˆ x ) , t µ = (1 , ~ . (2)Hence q µ is a null vector.In Lorenz gauge, the radiation can be obtained from the equation (cid:3) A µ = − j µ . Using theretarded propagator, we get : A σ ( x ) = 12 π Z d x ′ δ + ([ x − x ′ ] ) j σ ( x ′ ) . (3)The subscript ’+’ indicates that we have choose the retarded root of the δ -function constraint i.e. t > t ′ . The retarded root is given by t ′ = t − | ~x − ~x ′ | . (4)The form of t ′ at large r is t ′ = u + O ( r ). Thus the field A σ ( r, u, ˆ x ) at large r gets contributionfrom t ′ ∼ u . The bulk region corresponds to | r ′ | < R or | t ′ | < R (as c = 1) and it contributes to A σ at | u | < R . It is a characteristic of the retarded propagator that the asymptotic field at large u does not get contribution from the bulk region | t ′ | < R . Thus we can focus only on the asymptotic( t ′ > R ) trajectories.Let us write down the form of the source current that describes our scattering event. Denotingthe respective incoming velocities by V µi , charges by e i and masses by m i (for i = 1 · · · n ). Restrict-ing ourselves to the leading order in coupling e , we ignore the effect of long range electromageticinteractions on the asymptotic trajectories. Thus an incoming particle has the trajectory : x µi = [ V µi τ + d i ]Θ( − T − τ ) . is an affine parameter, here T denotes the value of τ such that r ( − T ) = R hence the short rangeforces can be ignored for τ < − T . Similarly let us denote the asymptotic outgoing velocities ofthe particles by V µj , charges e j and masses m j (for j = n + 1 · · · n ), an outgoing particle has thetrajectory : x µj = [ V µj τ + d j ]Θ( τ − T ) . For outgoing particles T denotes the value of τ such that r ( T ) = R hence the short range forcescan be ignored for τ > T . The current is given by summing over all particles that participate inthe scattering. The asymptotic part of this current at O ( e ) can be written down as : j asym σ ( x ′ ) = Z dτ h n X i = n +1 e i V iσ δ ( x ′ − x i ) Θ( τ − T ) + n X i =1 e i V iσ δ ( x ′ − x i ) Θ( − T − τ ) i . (5)The asymptotic radiative field generated by the scattering process can be obtained as follows A σ ( x ) = 12 π Z d x ′ δ + ([ x − x ′ ] ) j asym σ ( x ′ ) , = n X i = n +1 π e i V iσ Θ( τ +0 − T ) p ( V i .x − V i .d i ) + ( x − d i ) + n X i =1 π e i V iσ Θ( − τ +0 − T ) p ( V i .x − V i .d i ) + ( x − d i ) . (6)Here, τ +0 = − ( V i .x − V i .d i ) − p ( V i .x − V i .d i ) + ( x − d i ) is the retarded root. To expand theradiative field around I + let us take the limit r → ∞ with u finite in eq.(6). Using τ +0 = u + q.d i | q.V i | + O ( r ) in (6) : A σ ( x ) | I + = − πr h n X i = n +1 e i V iσ V i .q Θ( u + q.d i − T ) + n X i =1 e i V iσ V i .q Θ( − u − q.d i − T ) + ... i + O ( 1 r ) . (7)At large values of u , we see that the r -term goes like u . This mode is resposible for the so calledmemory effect [25–27]. We use ’ ... ’ to denote u -fall offs that are faster than any (negative) powerlaw behaviour. Let us rewrite (7) a bit succinitly : A µ ( x ) = 14 πr A [1 , µ (ˆ x ) u + O ( 1 r ) , | u | → ∞ . Next we study the O ( r ) term. We need to expand the √ ( V i .x − V i .d i ) +( x − d i ) factor in (6) to O ( r ). We get A [2] σ ( u, ˆ x ) = − π n X i = n +1 e i V iσ ( V i .q ) h u [ V i + 1 V i .q ] + V i .d i + d i .qV i .q i Θ( u + q.d i − T ) + in .A [2] σ denotes the coefficient of the O ( r ) term in the radiative field. In above expression we have notwritten down the contribution of incoming particles to avoid clutter. It takes the same form as thecontribution of the outgoing particles but comes with a factor of Θ( − u ). From above expressionwe see that the leading term at | u | → ∞ is O ( u ), it is followed by a O ( u ) term and then thereare highly suppressed terms like δ ( u ) which we are not keeping track of. Thus, analogous to (7)we get A µ ( x ) = 14 πr h A [1 , µ (ˆ x ) u + 1 r (cid:2) A [2 , − µ (ˆ x ) u + A [2 , µ (ˆ x ) u (cid:3) i + O ( 1 r ) , | u | → ∞ . r and we see that the asymptotic expansion of theradiative field around I + takes following form A ret µ ( x ) | I + = 14 π ∞ X m =0 ,n =1 m We will use (98) to substitute for τ − − τ +0 and also rewrite above expression in a succinit form A ∗ asym σ ( x ) = i π n X j =1 η j e j V jσ p ( V j .x − V j .d j ) + ( x − d j ) log τ − − η j Tτ +0 − η j T . (17)Here η j = 1( − 1) for outgoing (incoming) particles. Next we will find the asymptotic expansion ofabove expression. Using (99) we have τ +0 | I + = u + q.d i | q.V i | + O ( 1 r ) , τ − | I + = 2 r | q.V i | + O ( r ) . Thus we get h log τ − − Tτ +0 − T i I + = log ru + O (1) . We find that there are logarithmic modes in the radiative field. This is an interesting result aswe will discuss at the end of this calculation. Let us write down the full asymptotic expansion of A ∗ asym σ . Using (98), it is seen thatlog τ − | I + ∼ log r + ∞ X m,n =0 ,m ≤ n. u m r n . Similarly log τ +0 | I + ∼ log u + ∞ X n =0 ,m = −∞ ,m ≤ n. u m r n . 8e will write down the expansion for A ∗ asym σ by substituting above expressions in (17). A ∗ asym σ ( x ) ∼ log ur ∞ X m =0 ,n =1 m 1) for outgoing (incoming) particles. We do not have an explicit form for A ∗ bulk σ .It will depend on the details of the scattering process and short ranges forces present between theparticles. We are not interested in these non-universal terms. Nonetheless we will see in the nextsection that A ∗ bulk σ has some universal modes and we will use a trick to calculate them.We have already studied the asymptotic expansion of above solution (including A ∗ bulk σ ) around I + . Let us comment on some important differences between the Feynman solution and the retardedsolution. The leading order term of (21) is O ( log rr ). If we study (8), we see that such kind of modesare completely absent in the retarded solution! The retarded solution discussed in (8) starts at O ( r ). The r - component of the Feynman solution given in (21) takes following form A σ ( x ) | I + ∼ πr [ log u + u + ∞ X n =1 u n + ... ] . (22)Above expression should be contrasted with (7). The log u mode is absent in the classical field. Itis very important to note that this mode violates the Ashtekar-Struebel fall offs for the radiativefield [24]. We expect that the presence of a log u mode is a general feature of the quantum radiativefield. One way to prove this statement is to relate the log u mode to the universal leading softmode as done in [22, 23]. We will discuss this derivation in the next section.9et us turn to the coefficient of the u r mode. A [1 , σ (ˆ x ) = − π n X i = n +1 e i V iσ V i .q Θ( u + q.d i − T ) − π n X i =1 e i V iσ V i .q Θ( − u − q.d i − T ) − π n X i = n +1 e i V iσ V i .q − i π n X j =1 [ η j e j log | q.V i | + η j e j log √ V jσ q.V j . (23)It is well known that the u r mode in the gauge field is related to the memory effect and is expectedto be controlled by the leading soft factor [25–27]. Apriori it might seem that (23) is in conflictwith the previous statement. But let us recall that the memory effect is proportional to the changeof gauge field from u = −∞ to u = ∞ . Using (23), we get∆ A [1 , σ = Z I + du ∂ u A [1 , σ = − π n X i =1 η i e i V iσ V i .q . Thus we see that ∆ A σ is in fact proportional to the leading soft factor. Though there are correctionsto the u r mode as seen in (23), these corrections do not contribute to the memory effect. As seen in(23), this mode also has an imaginary piece which is absent in the classical mode given in (7). This’quantum’ mode has appeared in the analysis of [23]. Performing a co-oordinate transformation of(23), we calculate the radial component of the radiative field. We have A r ( x ) | imag = − π n X j =1 η j e j log | q.V i | . (24)Above expression matches with the quantum A r mode discussed in eqn (69) of [23] in the contextof massless QED coupled to gravity. This hints that the coefficent of above mode is universal.We will obtain the explicit for of coefficients of the log u and u modes in the next section. Herewe wish to emphasise that these modes are not expected to give rise to memory effects as they donot contribute to ∆ A σ . In particular the ’quantum’ u -mode present in (22) does not contributeto the tail memory effect discussed in [29, 30]. It should also be noted that this u -mode appearsat O ( e ) and is not related to the long range electromagnetic interaction. The tail memory termappears at O ( e ) and is a direct consequence of the long range interactions.We will conclude this section after writing down the asymptotic expansion of the full solutionin (21). It is given by A σ ( x ) | I + = ∞ X m =0 ,n =1 m In the last section, we derived the asymptotic expansion of Feynman solution generated by scat-tering of n charged particles. As seen in (25), it contains log u mode and u n modes that are absentin the retarded solution given in (8). In this section, we will discuss the relation of such modes todiscontinuity of the quantum field in the soft limit and show that the expectation values of thesemodes (in the quantum gauge field) match with the corresponding coefficients in (25). In [22, 23],the log u mode in the quantum field was derived from the discontinuity of the leading soft mode.Extending this idea, we show that the u -mode is related to the discontinuity of the tree levelsubleading soft mode. log ur mode in A σ Let us start with the coefficient of the log ur mode. Using the radiative field calculated in (21), wesee that this mode arises from the second line of (21). The contribution from the first term in thesecond line of (21) is given by A [1 , log] σ ( x ) = i π n X j =1 η j e j V jσ V j .q . (26)Here A [1 , log] σ has been used to denote the coefficient of the log ur mode.In [22, 23], the log ur mode was derived in a very different manner; the authors argued that thequantum gauge field has a discontinuity in ω → u mode in the position space. Let us study the quantum gauge field ˆ A µ . In momentum space theexpansion of ˆ A µ is given byˆ A σ ( x ) = 1(2 π ) Z d p | ~p | [ a σ ( p ) e ip.x + a † σ ( p ) e − ip.x ] . (27)Here, a σ ( ω, ˆ x ) = P r =+ , − ǫ ∗ rσ (ˆ x ) a r ( ω, ˆ x ) such that a r ( ω, ˆ x ) is identified as the annihilation operatorfor the respective helicity photons and ǫ rσ (ˆ x ) is the polarisation vector. We will find the leadingorder term in above expression at I + by taking the limit r → ∞ with u finite. Using stationaryphase approximation it can be shown that the leading order term is O ( r ) and its coefficientˆ A [1] µ ( u, ˆ x ) is given by [11]ˆ A [1] σ ( u, ˆ x ) = − i π Z ∞ dω [ a σ ( ω, ˆ x ) e − iωu + a † σ ( ω, ˆ x ) e iωu ] . (28)Above expression can be rewritten asˆ A [1] σ ( u, ˆ x ) = − i π Z ∞−∞ dω ˜ A σ e − iωu , here ˜ A σ = [ a σ ( ω, ˆ x ) Θ( ω ) − a † σ ( − ω, ˆ x )Θ( − ω ) ] . (29)We define a function ˆ A [1]+ µ ( u, ˆ x ) that has contribution from only positive frequencies i.e.ˆ A [1]+ µ ( u, ˆ x ) = − i π Z ∞ dω ˜ A µ ( ω, ˆ x ) e − iωu . 11n above expression let us add an imaginary part to u to make the integral well defined at ω → ∞ .ˆ A [1]+ µ ( u, ˆ x ) = − i π Z ∞ dω ˜ A µ ( ω, ˆ x ) e − iω ( u − iǫ ) . We will find the behaviour of the field at large u using a common trick. ∂ u ˆ A [1]+ µ ( u, ˆ x ) = − π Z ∞ dω [ ω ˜ A µ ( ω, ˆ x )] e − iω ( u − iǫ ) , = − i π u − iǫ ) Z ∞ dω [ ω ˜ A µ ( ω, ˆ x )] ∂ ω e − iω ( u − iǫ ) , = − i π u − iǫ ) h [ ω ˜ A µ ( ω, ˆ x )] e − iω ( u − iǫ ) i ∞ + i π u − iǫ ) Z ∞ dω ∂ ω [ ω ˜ A µ ( ω, ˆ x )] e − iω ( u − iǫ ) . (30)For now we will focus on the first term (The second term will be studied in the next subsectionand is subleading at large u ). We have ∂ u ˆ A [1]+ µ ( u, ˆ x ) = i π u − iǫ ) lim ω → [ ω ˜ A µ ( ω, ˆ x )] + ... Hence ˆ A [1]+ µ ( u, ˆ x ) = i π [log u + iπ Θ( u )] lim ω → [ ω ˜ A µ ( ω, ˆ x )] + ... . Repeating similar steps for the negative frequency contribution we get (for ω < 0, we need to shift u to u + iǫ .) ∂ u ˆ A [1] − µ ( u, ˆ x ) = − i π u + iǫ ) lim ω → − [ ω ˜ A µ ( ω, ˆ x )] + ... Hence ˆ A [1] − µ ( u, ˆ x ) = − i π [log u − iπ Θ( u )] lim ω → − [ ω ˜ A µ ( ω, ˆ x )] + ... . Let us first study the u term. Collecting the positve and negative frequency terms, its coeffi-cient turns out to be ˆ A [1 , µ (ˆ x ) = − Θ( u )8 π (cid:2) lim ω → + ω ˜ A µ ( ω, ˆ x ) + lim ω → − ω ˜ A µ ( ω, ˆ x ) (cid:3) = − π (cid:2) lim ω → + ωa µ ( ω, ˆ x ) + lim ω → + ωa † µ ( ω, ˆ x ) (cid:3) . Above mode is responsible for the so called memory effect [26] and as expected its coefficient isproportional to the leading soft factor. Then we turn to the log u mode. Collecting the positveand negative frequency terms we have :ˆ A [1 , log] µ (ˆ x ) = i π (cid:2) lim ω → + ω ˜ A µ ( ω, ˆ x ) − lim ω → − ω ˜ A µ ( ω, ˆ x ) (cid:3) . u term is governed by the discontinuity in ω ˜ A µ as ω → A [1 , log] µ (ˆ x ) = i π (cid:2) lim ω → + ωa µ ( ω, ˆ x ) − lim ω → + ωa † µ ( ω, ˆ x ) (cid:3) . (31)Thus, both the modes log u and Θ( u ) are related to the leading soft mode. We can evaluate theexpectation value of above operator using leading soft theorem. < out | ˆ A [1 , log] µ (ˆ x ) S | in > = i π (cid:2) ǫ + µ ǫ − ν + ǫ − µ ǫ + ν (cid:3) n X j =1 η j e j V νj V j .q < out | S | in >, = i π n X j =1 η j e j V jµ V j .q < out | S | in > . (32)Here, | in > = | , , ..., n ′ > and < out | = < n ′ + 1 , ..., n | . We see that the expectation valuematches with our expression obtained from Feynman radiative solution in (26). It is clear fromthis derivation that the existence of the log u mode is tied to the ω -mode. Since the ω -mode isuniversal we expect that the log u mode is also universal. ur mode in A σ Next we turn to the ru mode of the Feynman solution A σ . This mode arises from the second lineof (21). It is interesting to note that this mode gets contribution from both bulk and asymptoticsources. Let us first write down the contribution from the first term in the second line of (21). Inthis term, we substitute the expression of τ +0 from (98) and use log( τ +0 − η i T ) = log u | q.V i | + q.d i u + η i q.V i Tu + ... , to get A [1 , σ ( x ) | asym = i π n X j =1 η j e j V jσ [ q.d j V j .q + η j T ] . (33) A [1 , σ denotes the coefficient of the ru mode of A σ . Let us turn to the contribution from the bulk. A ∗ bulk σ ( x ) = i π Z r ′ Hence we get A + σ ( x ) + A − σ ( x ) = 18 π n X i = n +1 Θ( u − T ) e i X h V iσ (cid:2) x − d i ) .C i X log τ +0 + ( x − d i ) .C i Xτ +0 (cid:3) + C iσ τ +0 i + 18 π n X i =1 Θ( − u − T ) e i X h V iσ (cid:2) x − d i ) .C i X log τ +0 + ( x − d i ) .C i Xτ +0 (cid:3) + C iσ τ +0 i + 18 π n X i = n +1 e i X h V iσ (cid:2) x − d i ) .C i X log τ − − ( x − d i ) .C i Xτ − (cid:3) + C iσ τ − i , (47)where X = [ ( V i .x − V i .d i ) + ( x − d i ) ] / . We can study the expansion of above expression around I + . Using (98) and (108) we have (cid:2) A + σ ( x ) + A − σ ( x ) (cid:3) | I + = ∞ X n =1 ,m = −∞ ,m We discussed that new modes arise in the asymptotic field at O ( e ) as a result of the long rangeinteractions between scattering particles. In this section we will obtain an asymptotic conservationlaw obeyed by such modes.Let us first discuss the Q conservation law derived in [21] for the retarded solution. In [21]it was shown that F rA (which denotes the ’ rA ’-component of the field strength) has followingexpansion near u → −∞ . F [ret] rA | I + − = 1 r [ u F ( u ) rA (ˆ x ) + log u F (log u ) rA (ˆ x ) + ... ] + O ( 1 r ) . (58)Similarly around the past null infinity near v → ∞ F [ret] rA | I − + = log rr [ v F (log r ) rA (ˆ x ) + ... ] + O ( 1 r ) . (59)The Q conservation law derived in [21] is given as follows F (log u ) rA (ˆ x ) | I + − = F (log r ) rA ( − ˆ x ) | I − + . (60)The future charge is defined as Q +1 = R d z Y A (ˆ x ) F [log u ] rA (ˆ x ) | I + − and the past charge is Q − = R d z Y A ( − ˆ x ) F [log r ] rA ( − ˆ x ) | I − + . These charges are O ( e ). In [22, 23], the authors started with these’classical’ charges and showed that upon quantisation these charges reproduce the full log ω -softtheorem [19] including the purely quantum modes.We aim to find the analogue of (60) that is obeyed by the Feynman solution. F rA calculatedusing (51) takes following form F rA | I + − = log rr F [log r ] rA (ˆ x ) + 1 r [ u log u F [ u log u ] rA (ˆ x ) + (log u ) F [(log u ) ] rA (ˆ x ) + u F [ u ] rA (ˆ x ) + log u F [log u ] rA (ˆ x ) + ... ] . (61)Similarly around the past null infinity we have : F rA | I − + = log rr F [log r ] rA (ˆ x ) + 1 r [ v log v F [ v log v ] rA (ˆ x ) + (log v ) F [(log v ) ] rA (ˆ x ) + v F [ v ] rA (ˆ x ) + log v F [log v ] rA (ˆ x ) + ... ] . (62)In this section we will derive the conservation equation obeyed by these modes.[ F [log u ] rA (ˆ x ) − F [log r ] rA (ˆ x )] | I + − = [ − F [log v ] rA ( − ˆ x ) + F [log r ] rA ( − ˆ x )] | I − + . (63)So that the future charge can be defined as ˜ Q +1 = R d z Y A (ˆ x ) [ F [log u ] rA (ˆ x ) − F [log r ] rA (ˆ x )] | I + − and thepast charge by ˜ Q − = R d z Y A ( − ˆ x ) [ − F [log v ] rA ( − ˆ x ) + F [log r ] rA ( − ˆ x )] | I − + . Let us find the full log ur -mode of A σ at the future null infinity. We need to expand all the termsin the Feynman solution given in (51) around I + . First we turn to the seventh and eighth linesof (51). Using (102), we have log τ | I + ∼ log u + O ( u ). Using (2), we get X = − rq.V i + O ( r ).23ubstituting the limiting value of X , we can read off the coefficient of the O ( log ur ) term in theseventh and eighth lines of (51) : − π n X i = n +1 Θ( u − T ) e i V iσ q.C i ( q.V i ) − π n X i =1 Θ( − u − T ) e i V iσ q.C i ( q.V i ) (64)It can be shown that above term contains the full soft factor correspoding to the log ω term [19].There are many more terms that contribute to the log ur -mode. These remaining terms areactually not related for the log ω soft theorem [19]. We will list them. From the first line of (51),using (113) we get i π n X j =1 η j e j V jσ ( V j .q ) [ q.d j ( V j .q ) + V j .d j ] . (65)From the second line of (51), we get using (111) and (113) − i π n X j =1 η j e j C jσ [ 1( V j .q ) − . (66)The third line of (51) does not have a log u term. From the fourth line of (51), we get using (111)and (113) − i π n X j =1 η j e j V jσ V j .q h q.C j [ − V j .q ) − V j ( V j .q ) + 12 ] + C j i . (67)Using (113), the fifth line of (51) gives − i π n X j =1 η j e j V jσ q.C j ( q.V j ) ln | q.V j | . (68)As shown in Appendix C, the sixth line of (51) and A ∗ bulk σ do not have any logarithmic modes. Wehave the full coefficient of the log ur term. A [log u/r ] σ | I + − = − π n X i =1 e i V iσ q.C i ( q.V i ) + i π n X j =1 η j e j V jσ ( V j .q ) [ q.d j ( V j .q ) + V j .d j ] − i π n X j =1 η j e j C jσ [ 1( V j .q ) − − i π n X j =1 η j e j V jσ q.C j ( q.V j ) ln | q.V j |− i π n X j =1 η j e j V jσ V j .q h q.C j [ − V j .q ) − V j ( V j .q ) + 12 ] + C j i . (69)Next we need to write down the coefficient of the log rr -term in A σ . From the first line of (51),using (113) we get − i π n X j =1 η j e j V jσ ( V j .q ) [ q.d j ( V j .q ) + V j .d j ] . (70)24rom the second line of (51), using (112) and (113) we get i π n X j =1 η j e j C jσ ( q.V j ) . (71)The third line of (51) does not have a log r term. From the fourth line of (51), using (112) and(113) we get i π n X j =1 η j e j V jσ ( V j .q ) q.C j . (72)The fifth line of (51) contributes as follows − i π n X j =1 η j e j V jσ q.C j ( q.V j ) ln(2 | q.V j | ) . (73)Substituting X = − rq.V i + O ( r ) in the eighth line of (51), we get − π n X i = n +1 e i V iσ q.C i ( q.V i ) . (74)We have the full coefficient of the log rr term. A [log r/r ] σ ( x ) | I + = − π n X i = n +1 e i V iσ q.C i ( q.V i ) − i π n X j =1 η j e j V jσ ( V j .q ) [ q.d i ( V i .q ) + V j .d j ]+ i π n X j =1 η j e j C jσ ( q.V j ) + i π n X j =1 η j e j V jσ ( V j .q ) q.C j − i π n X j =1 η j e j V jσ q.C j ( q.V j ) ln(2 | q.V j | ) . (75)25 .2 Modes at Past null infinity Next we need to derive the field configuration at past null infinity and then compare the twoexpressions. Analogous to (51), around I − we have A σ ( x ) = i π n X j =1 η j e j V jσ τ − − τ +0 h log 1 τ +0 − T − log 1 τ − − T i + i π n X j =1 η j e j C jσ ( τ − − τ +0 ) (cid:2) τ +0 log Tτ +0 − T − τ − log Tτ − − T (cid:3) + i π n X j =1 η j e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) log T (cid:2) T − τ +0 + 1 T − τ − (cid:3) + i π n X j =1 η j e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) (cid:2) τ − log Tτ − − T + 1 τ +0 log Tτ +0 − T (cid:3) + i π n X j =1 η j e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) (cid:2) − ln τ +0 ln( τ +0 − T ) + ln τ − ln( τ − − T ) + 12 [ln τ +0 − ln τ − (cid:3) − i π n X j =1 η j e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) h Li ( − T − τ − τ − ) − Li ( − T − τ +0 τ +0 ) i + 18 π n X i = n +1 Θ( v − T ) e i X h V iσ (cid:2) x − d i ) .C i X log τ − − ( x − d i ) .C i Xτ − (cid:3) + C iσ τ − i + 18 π n X i =1 Θ( − v − T ) e i X h V iσ (cid:2) x − d i ) .C i X log τ − − ( x − d i ) .C i Xτ − (cid:3) + C iσ τ − i + 18 π n X i =1 e i X h V iσ (cid:2) x − d i ) .C i X log τ +0 + ( x − d i ) .C i Xτ +0 (cid:3) + C iσ τ +0 i . (76)Here, X = [ ( V i .x − V i .d i ) + ( x − d i ) ] / . Let us write down the full coefficient of the log vr -mode. We need to take the limit r → ∞ with v = t + r finite. In this co-ordinate system, 4 dimensional spacetime point can be parametrised as: x µ = r ¯ q µ + vt µ , ¯ q µ = ( − , ˆ x ) , t µ = (1 , ~ . (77)¯ q µ is a null vector. From the first line of (76), we get using (117) i π n X j =1 η j e j V jσ ( V j . ¯ q ) [ ¯ q.d j ( V j . ¯ q ) + V j .d j ] . (78)From the second line of (76), using (116) and (117) we get − i π n X j =1 η j e j C jσ [ 1( V j . ¯ q ) − . (79)26he third line of (76) does not have a log v term. From the fourth line of (76), using (116) and(117) we get − i π n X j =1 η j e j V jσ V j . ¯ q h ¯ q.C j [ V j ( V j . ¯ q ) − q.V i ) + 12 ] − C j i . (80)The fifth line of (76) gives − i π n X j =1 η j e j V jσ ¯ q.C j (¯ q.V j ) ln | ¯ q.V j | . (81)From seventh line of (76) we get18 π n X i = n +1 Θ( v − T ) e i V iσ ¯ q.C i (¯ q.V i ) + 18 π n X i =1 Θ( − v − T ) e i V iσ ¯ q.C i (¯ q.V i ) . (82)Hence we have A [log v/r ] σ ( x ) = + 18 π n X i = n +1 e i V iσ ¯ q.C i (¯ q.V i ) + i π n X j =1 η j e j V jσ ( V j . ¯ q ) [ ¯ q.d j ( V j . ¯ q ) + V j .d j ] − i π n X j =1 η j e j C jσ [ 1( V j . ¯ q ) − − i π n X j =1 η j e j V jσ ¯ q.C j (¯ q.V j ) ln | ¯ q.V j |− i π n X j =1 η j e j V jσ V j . ¯ q h ¯ q.C j [ − V j . ¯ q ) + V j ( V j . ¯ q ) + 12 ] − C j i . (83)Next we turn to the log rr -mode. From the first line of (76), we get − i π n X j =1 η j e j V jσ ( V j . ¯ q ) [ ¯ q.d j ( V j . ¯ q ) + V j .d j ] . (84)From the second line of (76), we get i π n X j =1 η j e j C jσ V j . ¯ q ) . (85)The third line of (76) does not have a log r term. From the fourth line of (76), we get i π n X j =1 η j e j V jσ ( V j . ¯ q ) ¯ q.C j . (86)The fifth line of (76) gives − i π n X j =1 η j e j V jσ ¯ q.C j (¯ q.V j ) ln | q.V j | . (87)27e get following contribution from the last line of (76)18 π n X i =1 e i V iσ ¯ q.C i (¯ q.V i ) . (88)The total coefficient is A [log r/r ] σ ( x ) | I − = 18 π n X i =1 e i V iσ ¯ q.C i (¯ q.V i ) − i π n X j =1 η j e j V jσ ( V j . ¯ q ) [ ¯ q.d j ( V j . ¯ q ) + V j .d j ]+ i π n X j =1 η j e j C jσ V j . ¯ q ) + i π n X j =1 η j e j V jσ ( V j . ¯ q ) ¯ q.C j − i π n X j =1 η j e j V jσ ¯ q.C j (¯ q.V j ) ln | q.V j | . (89)Thus, from (69), (75), (83) and (89) we can indeed check that following modes are equal underantipodal idenfication.[ A [log u/r ] σ (ˆ x ) − A [log r/r ] σ (ˆ x )] | I + − = [ A [log v/r ] σ ( − ˆ x ) − A [log r/r ] σ ( − ˆ x )] | I − + . (90)Using co-ordinate transformation, it can be shown that the quantum gauge field obeys followingconservation equation :[ F [log u/r ] rA (ˆ x ) − F [log r/r ] rA (ˆ x )] | I + − = [ − F [log v/r ] rA ( − ˆ x ) + F [log r/r ] rA ( − ˆ x )] | I − + . (91)Compared to (90), the RHS of above expression has extra minus sign as it has an extra factorof ∂ A ¯ q µ due to co-ordinate transformation. Finally we have derived the ˜ Q -conservation equationsuch that the future charge is defined by ˜ Q +1 = R d z Y A (ˆ x ) [ F [log u/r ] rA (ˆ x ) − F [log r/r ] rA (ˆ x )] | I + − andthe past charge by ˜ Q − = R d z Y A ( − ˆ x ) [ − F [log v/r ] rA ( − ˆ x ) + F [log r/r ] rA ( − ˆ x )] | I − + .Let us state some important observations. ˜ Q +1 gets contribution from (69) and (75) and it isseen that it contains terms that are not related to the log ω mode. We suspect that such irrelevantterms would cancel from the Ward identity and the ˜ Q -charge will reproduce the full log ω softtheorem [19]. Nonetheless using ˜ Q +1 -conservation law given in (91) is not a satisfactory way ofidentifying the asymptotic charge. [22, 23] have a different prescription to define the asymptoticcharges. The classical law given in (60) is used to define asymptotic charges, these charges arethen quantised and the corresponding Ward identity is shown to be equivalent to the full log ω softphoton theorem. But it should be noted that (60) itself is violated in the quantum theory. Hence,using either of the conservation equations given in (60) and (91) to define asymptotic charges isnot completely satisfactory. It would be useful to have a first principles-based construction of theseasymptotic charges via asymptotic phase space techniques. In this paper we have obtained the radiative field produced by scattering of n charged point particlesusing Feynman propagator. This problem is unphysical but the Feynman radiative solution soderived is useful to illustrate interesting features of the quantum gauge field.28e showed in (22) that the r -term in the Feynman solution at O ( e ) has following behaviour A µ ( x ) | I + = 1 r [ log u + u + ∞ X n =1 u n + ... ] . Here, ’...’ denote terms that fall off faster than any power law in u . The log u and the u n -modesare purely quantum modes; they are absent in the retarded solution in (7). It should be noted thatthe log u mode violates the Ashtekar-Struebel fall offs for the radiative field [24] that ensure theexistence of a well defined symplectic form. We leave the investigation of this issue to the future.The log u mode is controlled by the discontinuity of the leading soft mode [22, 23]. Extendingthis idea, we showed that the u -mode is related to the discontinuity of the tree level subleading softmode in the quantum gauge field. It is also shown that the expectation value of the the u -modein the quantum gauge field matches with the corresponding coefficient in the Feynman solution in(22). This is an interesting result and it hints that the u n modes for m > ω n − soft modes. Hence we expect that the presence oflog u and u n modes is a general feature of the quantum gauge field.New modes are expected to appear in the radiative field as we go to higher orders in thecoupling. Including the effect of long range electromagnetic force on the scattering particles, weobtained the Feynman solution upto O ( e ) in (51). The r -term takes following form A σ ( x ) | I + ∼ r [ log u + u + ∞ X m =1 log uu m + ∞ X n =1 u n + ... ] . The log uu n -modes in the Feynman solution appear at O ( e ). These modes go to 0 as u → ±∞ anddo not violate the Ashtekar-Struebel fall offs [24]. It should also be noted that the log uu -mode isabsent in the retarded solution [21]. We studied the coefficient of the log uu quantum mode andshowed that this mode is related to the discontinuity of the loop level soft log ω -mode. As thelog ω -mode derived in [19] is universal, the log uu -mode should also be universally present in thequantum gauge field.We expect that new (log u ) m u m -modes would appear in the Feynman solution at O ( e m +1 ) respec-tively such that they are related to the respective discontinuities of the ω m − (log ω ) m universalsoft modes The r -term of the radiative field speculatively takes following form A σ ( x ) | I + ∼ r [ e log u + e u + ∞ X m =0 ,n =1 ,n ≥ m. (log u ) m u n ] , such that the m th term in the summation appears at O ( e m +1 ). Hence it is expected that all themodes in the Feynman solution except the log u mode should preserve the Ashtekar-Struebel falloffs.In section 7, we turned to the asymptotic conservation equation obeyed by certain O ( e )logarithmic modes in the Feynman solution. This equation has been derived in (91) and relatesthe difference in the coefficients of the log ur and log rr modes in F rA at I + − to the difference inthe coefficients of the log rr and log vr in F rA at I − + . It should be possible to prove (91) in generalby following analysis of [14] albeit with Feynman boundary condition. It is expected that thecorresponding charges ˜ Q should reproduce the log ω soft photon theorem derived in [19]. Thesequestions need to be pursued in the future. 29 Acknowledgements I am extremely thankful to Nabamita Banerjee and Alok Laddha for numerous discussions. I amdeeply grateful to my family for their constant support. Finally I thank the people of India fortheir enduring help to basic sciences. A Feynman propagator We are mostly familiar with the momentum space form of the Feynman propagator. With nor-malisation such that (cid:3) G = − δ ( x − x ′ ), it takes following form G ( x, x ′ ) = Z d p (2 π ) e ip. ( x − x ′ ) p − iǫ . (92)We can perform the p intgral according to the given presciption and get G ( x, x ′ ) = i Z d p (2 π ) ω [ e ip. ( x − x ′ ) Θ( t − t ′ ) + e − ip. ( x − x ′ ) Θ( t ′ − t )] (93)Here ω = | ~p | . Let us work in a frame s.t. ˆ x − ˆ x ′ || p z -axis. We have d p = ω dω dφ d cos θ and theintegral over dφ gives 2 π . G ( x, x ′ ) = i Z − d cos θ ωdω π ) [ e − iω ( t − t ′ −| x − x ′ | cos θ ) Θ( t − t ′ ) + e iω ( t − t ′ −| x − x ′ | cos θ ) Θ( t ′ − t )] . (94)Performing the cos θ integral G ( x, x ′ ) = 1 | x − x ′ | Z ∞ dω π ) h e − iω ( t − t ′ ) [ e iω | x − x ′ | − e − iω | x − x ′ | ]Θ( t − t ′ ) − e iω ( t − t ′ ) [ e − iω | x − x ′ | − e iω | x − x ′ | ]Θ( t ′ − t ) i . (95)In the last line, we recall that ω > 0, the integral is the standard Fourier transform integral Z dω e − iωu Θ( ω ) = − iu + πδ ( u ) . (96)Hence we get G ( x ; x ′ ) = 18 π Θ( t − t ′ ) | x − x ′ | h − it − t ′ − | x − x ′ | + πδ ( t − t ′ − | x − x ′ | ) + it − t ′ + | x − x ′ | i + 18 π Θ( t ′ − t ) | x − x ′ | h − it − t ′ − | x − x ′ | + πδ ( t − t ′ + | x − x ′ | ) + it − t ′ + | x − x ′ | i . We can rewrite above expression as G ( x ; x ′ ) = 14 π h i ( x − x ′ ) + πδ + ( ( x − x ′ ) ) + πδ − ( ( x − x ′ ) ) i (97)Here, the subscript ’+’ denotes the retarded root t − t ′ − | x − x ′ | = 0 while the subscript ’-’ denotesthe advanced root t − t ′ + | x − x ′ | = 0. 30 Perturbative solution The Green function for d’Alembertian operator is δ ([ x − x ′ ] ). We will find the solution of thisdelta function perturbatively in coupling e . Here, x ′ µ ( τ ) is the equation of trajectory that getscorrected as we go to higher orders in e . We will write down the perturbative solution for τ .At zeroth order, we have free particles : x ′ µi = V µi τ + d i . Hence, the root of delta function δ ([ x − x ′ ] ) is given by : τ ± = − V i . ( x − d i ) ∓ (cid:2) ( V i .x − V i .d i ) + ( x − d i ) (cid:3) / . (98) τ +0 satisifies retarded boundary condition while τ − satisifies advanced boundary condition. Let usstudy above expression in the limit r → ∞ with u finite. Thus, around I + , using (2) we get : τ +0 | I + = u + q.d i | q.V i | + O ( 1 r ) , τ − | I + = 2 r | q.V i | + O ( r ) . (99)Now we take r → ∞ limit of (98) keeping v finite, using (77), we get : τ +0 | I − = − r V i . ¯ q + O ( r ) , τ − | I − = v − ¯ q.d i ¯ q.V i + O ( 1 r ) . Next we include the leading order effect of long range electromagnetic force. We know that thefirst order correction to the trajectory is given by (43) : x ′ µi = V µi τ + C µi log τ + d i . Using the corrected trajectory, the solution of delta function δ ( | x − x ′ | ) is given by : τ + 2 τ V i . ( x − d i ) − ( x − d i ) = − x − d i ) .C i log τ + C i (log τ ) . Here we have used the fact that V i .C i = 0. Noting that C µi is O ( e ), the RHS of above equationcan be treated as a perturbation. Hence we substitute the zeroth order solution (98) in RHS ofabove equation; it leads to following equation for τ : τ + 2 τ V i . ( x − d i ) − ( x − d i ) = − x − d i ) .C i log τ ± . We ignored the C i term as it is O ( e ). Now, above equation is just a quadratic equation in τ andthe solution is given by : τ ± = − V i . ( x − d i ) ∓ (cid:2) ( V i .x − V i .d i ) + ( x − d i ) − x − d i ) .C i log τ ± (cid:3) / . (100)We have used a subscript 1 to denote that it includes the first order perturbative effects. We canexpand the squareroot to O ( e ) : τ ± = − V i . ( x − d i ) ∓ (cid:2) ( V i .x − V i .d i ) + ( x − d i ) (cid:3) / ± ( x − d i ) .C i X log τ ± . (101)31ere, we have defined X = [ ( V i .x − V i .d i ) + ( x − d i ) ] / and τ ± are given in (98). Expandingaround I + , we get : τ +1 | I + = − u + q.d i q.V i − q.C i q.V i log ( − u ) q.V i + O ( 1 u ) ,τ − | I + = − rq.V i + q.C i q.V i log r + O ( r ) . (102)Expanding (101) around I − , we get : τ +1 | I − = − r V i . ¯ q − ¯ q.C i V i . ¯ q log r + O ( r ) ,τ − | I − = v − ¯ q.d i ¯ q.V i + ¯ q.C i ¯ q.V i log vq.V i + O ( 1 v ) . (103) C Integral in section 6 Let us first write down the indefinite integral given in (50). Z dx log x ( x − τ − ) ( x − τ +0 ) = 2( τ − − τ +0 ) (cid:2) ln τ +0 ln( x − τ +0 ) − ln τ − ln( x − τ − ) (cid:3) + 2( τ − − τ +0 ) h Li (cid:18) − x − τ − τ − (cid:19) − Li (cid:18) − x − τ +0 τ +0 (cid:19) i − ln x ( τ +0 − τ − ) (cid:2) (cid:0) x − τ +0 (cid:1) + 1 (cid:0) x − τ − (cid:1) (cid:3) − τ +0 − τ − ) (cid:2) τ − log x ( x − τ − ) + 1 τ +0 log x ( x − τ +0 ) (cid:3) (104)Above integral is to be integrated from T to R for outgoing particles. Let us consider the upperlimit and show that the divergent terms (in the R → ∞ limit) indeed cancel and also find if thereis any finite contribution.2( τ − − τ +0 ) h ln τ +0 ln R − ln τ − ln R + Li (cid:18) − R − τ − τ − (cid:19) − Li (cid:18) − R − τ +0 τ +0 (cid:19) i + O ( ln RR ) . (105)Let us use following property of the dilogarithm function [34].Li ( x ) = − π − 12 log(1 − x ) [2 log( − x ) − log(1 − x )] + Li ( 11 − x ) . Thus we haveLi (cid:18) − R − τ +0 τ +0 (cid:19) = − π − 12 log( Rτ +0 ) [2 log( Rτ +0 − − log( Rτ +0 )] + Li ( τ +0 R )= − π − 12 log ( Rτ +0 ) + O ( 1 R ) . Hence (105) is equal to 1( τ − − τ +0 ) [ln τ +0 − ln τ − ] + O ( ln RR ) . Z RT dx log x ( x − τ − ) ( x − τ +0 ) = 2( τ − − τ +0 ) (cid:2) ln τ +0 ln 1 τ +0 − T − ln τ − ln 1 τ − − T + 12 [ln τ +0 − ln τ − ] (cid:3) − τ − − τ +0 ) h Li (cid:18) − T − τ − τ − (cid:19) − Li (cid:18) − T − τ +0 τ +0 (cid:19) i + ln T ( τ +0 − τ − ) (cid:2) (cid:0) T − τ +0 (cid:1) + 1 (cid:0) T − τ − (cid:1) (cid:3) + 1( τ +0 − τ − ) (cid:2) τ − log Tτ − − T + 1 τ +0 log Tτ +0 − T (cid:3) (106)Hence we can write down the result of the both integraks in (50). A ∗ asym σ ( x )= i π n X j = n +1 e j V jσ τ − − τ +0 h log 1 τ +0 − T − log 1 τ − − T i + i π n X j = n +1 e j C jσ ( τ − − τ +0 ) (cid:2) τ +0 log Tτ +0 − T − τ − log Tτ − − T (cid:3) + i π n X j = n +1 e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) log T (cid:2) T − τ +0 + 1 T − τ − (cid:3) + i π n X j = n +1 e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) (cid:2) τ − log Tτ − − T + 1 τ +0 log Tτ +0 − T (cid:3) + i π n X j = n +1 e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) (cid:2) − ln τ +0 ln( τ +0 − T ) + ln τ − ln( τ − − T ) + 12 [ln τ +0 − ln τ − ] (cid:3) − i π n X j = n +1 e j V jσ ( x − d j ) .C j ( τ − − τ +0 ) h Li ( − T − τ − τ − ) − Li ( − T − τ +0 τ +0 ) i + in . (107)Let us study the expansion of various terms in above expression. Using (98), it is seen that τ +0 | I + ∼ u [ 1 + 1 u + ∞ X ≤ m ≤ n,n =1 u m r n ] .τ − | I + ∼ r + u + r u + ∞ X ≤ m ≤ n +1 ,n =1 u m r n . log τ − | I + ∼ log r + ∞ X m,n =0 ,m ≤ n. u m r n . log τ +0 | I + ∼ log u + ∞ X n =0 ,m = −∞ ,m ≤ n. u m r n . (108)33( τ − − τ +0 ) 1 τ +0 log T ( T − τ +0 ) ∼ r [ O (1) + log u ] [ 1 u + 1 u + ... + ∞ X m = −∞ ,m ≤ n,n =1 u m r n ] . τ − − τ +0 ) 1 τ − log T ( T − τ − ) (cid:3) ∼ r [ O (1) + log r ] [1 + ∞ X To find the coefficients of log ur and log rr modes in A σ , we need to calculate some lower order termsin the asymptotic expansion of (51) explicitly. Here we list the asymptotic expansions of variousquantities that appear in (51). Around I + Let us start with the retarded root τ +0 = − V i . ( x − d i ) − (cid:2) ( V i .x − V i .d i ) + ( x − d i ) (cid:3) / . τ +0 | I + = − V i .x + V i .d i + rV i .q (cid:2) − uV i + V i .d i rV i .q + ( uV i + V i .d i ) r ( V i .q ) + ( x − d i ) r ( V i .q ) (cid:3) = − u + q.d i ( V i .q ) − u r ( V i .q ) − u V i r ( V i .q ) − u r ( V i .q ) + O ( ur ) . (110)Hence we have1 τ +0 = − ( V i .q ) u h − q.d i u + O ( 1 u ) − ur [ 12 + V i ( V i .q ) + 12( V i .q ) ] + O ( u r ) i . (111)Next we turn to the advanced root. τ − = − V i . ( x − d i ) + (cid:2) ( V i .x − V i .d i ) + ( x − d i ) (cid:3) / . Around future null infinity, we get using (2) τ − = − V i .x + V i .d i − rV i .q (cid:2) − uV i + V i .d i rV i .q + ( uV i + V i .d i ) r ( V i .q ) + ( x − d i ) r ( V i .q ) (cid:3) = − rV i .q + 2 uV i + 2 V i .d i + u + q.d i ( V i .q ) + O ( 1 r ) . τ − = − V i .q ) r h uV i + V i .d i r ( V i .q ) + u + q.d i r ( V i .q ) + O ( 1 r ) i . (112)Also we can write down the asymptotic expansion of following term.2 τ − − τ +0 = 1 r | V i .q | (cid:2) − uV i + V i .d i rV i .q + ( uV i + V i .d i ) r ( V i .q ) + ( x − d i ) r ( V i .q ) (cid:3) − , = − r V i .q (cid:2) r uV i V i .q + 1 r V i .d i V i .q + 1 r u + d i .q ( V i .q ) + O ( 1 r ) (cid:3) . (113). Around I − We start with the advanced root . τ − = − V i . ( x − d i ) + (cid:2) ( V i .x − V i .d i ) + ( x − d i ) (cid:3) / . Around past null infinity, we get using (77) τ − = − V i .x + V i .d i + rV i . ¯ q (cid:2) − vV i + V i .d i rV i . ¯ q + ( vV i + V i .d i ) r ( V i . ¯ q ) + ( x − d i ) r ( V i . ¯ q ) (cid:3) = v − ¯ q.d i ( V i . ¯ q ) − v r ( V i . ¯ q ) + v V i r ( V i . ¯ q ) − v r ( V i . ¯ q ) + O ( vr ) . (114)Hence we have1 τ − = ( V i . ¯ q ) v h q.d i v + O ( 1 v ) + vr [ 12 − V i ( V i . ¯ q ) + 12( V i . ¯ q ) ] + O ( v r ) i . 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