Classical Yang-Mills observables from amplitudes
Leonardo de la Cruz, Ben Maybee, Donal O'Connell, Alasdair Ross
aa r X i v : . [ h e p - t h ] S e p Prepared for submission to JHEP
Classical Yang-Mills observables from amplitudes
Leonardo de la Cruz, Ben Maybee, Donal O’Connell, Alasdair Ross
Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edin-burgh, EH9 3FD, Scotland
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The double copy suggests that the basis of the dynamics of general relativityis Yang-Mills theory. Motivated by the importance of the relativistic two-body problem,we study the classical dynamics of colour-charged particle scattering from the perspectiveof amplitudes, rather than equations of motion. We explain how to compute the change ofcolour, and the radiation of colour, during a classical collision. We apply our formalism atnext-to-leading order for the colour change and at leading order for colour radiation. ontents N particles 37 – 1 – Introduction
Future prospects in gravitational wave astronomy require perturbative calculations at veryhigh precision [1]. Remarkably, ideas and methods from quantum field theory offer apromising avenue of investigation [2–17]. Powerful techniques have been developed to com-pute scattering amplitudes, and we have learned how to extract classical physics efficientlyfrom amplitudes [18–51]. A crucial insight from quantum field theory is the double copy:the observation that scattering amplitudes in gravitational theories can be computed fromamplitudes in gauge theories . Since perturbation theory is far simpler in Yang-Mills (YM)theory than in standard approaches to gravity, the double copy has revolutionary potential.Yang-Mills theory, treated as a classical field theory, shares many of the importantphysical features of gravity, including non-linearity and a subtle gauge structure. In thisrespect the YM case has always served as an excellent toy model for gravitational dy-namics. But our developing understanding of the double copy has taught us that theconnection between Yang-Mills theory and gravity is deeper than this. Detailed aspectsof the perturbative dynamics of gravity, including gravitational radiation, can be deducedfrom Yang-Mills theory and the double copy [11, 21, 53, 54]. In fact the double copy extendsbeyond perturbation theory, leading to exact maps [8, 55–81] between classical solutionsof gauge theory and gravity, even when there is gravitational radiation present [9].So we are motivated to take another look at perturbative YM processes, particularlythose that connect to gravitational wave physics. However, the double copy is best un-derstood as a relationship between the scattering amplitudes of YM theory and gravity.Consequently it is useful to formulate classical YM dynamics in terms of amplitudes, ratherthan by solving the equations of motion. We explain how to do so in this article. Sinceour interest is in the classical theory, we systematically ignore non-perturbative quantumeffects. Nevertheless we identify observables which are well-defined in the classical theoryand which can be extracted unambiguously from perturbative scattering amplitudes toany order. Our approach is complementary to an effort to understand the double copyfrom a purely classical, worldline perspective [11, 54, 82–89]. Perhaps the simplest way ofimplementing the double copy for classical quantities will be to compute observables usingclassical worldline methods; extract the relevant amplitudes by comparing to our formulaefor observables in terms of amplitudes, and then double copy to gravity.Returning briefly to our underlying interest in gravitational dynamics, it is worthemphasising the importance of spin. The spin of the individual bodies in a compact binarycoalescence event influences the details of the outgoing gravitational radiation. This spinalso contains information on the poorly-understood formation channels of the binaries.Measurement of spin is therefore one of the primary physics outputs of gravitational waveobservations. It is fortunate, then, that spin can be incorporated rather naturally in theformalism of quantum field theory [41, 76, 90–104], even for very large (classical) spins.However the details of spin dynamics quickly becomes complex.In the Yang-Mills context we can also discuss the dynamics of spin. However Yang-Mills theories always involve colour, and colour is in many respects very analogous to spin. The double copy was thoroughly reviewed recently [52] – 2 –he dynamics of colour is actually a little simpler than spin, because the latter is tiedtogether with spacetime while colour is defined in its own vector space. We therefore payparticular attention to the dynamics of colour in Yang-Mills theory. We discuss in detailthe change of colour of a particle during a scattering event, and the radiation of colour tonull infinity. We also discuss more briefly the impulse (change of momentum) and radiationof momentum in YM theory. The methods we use build on ideas previously described byKosower and two of us in a previous paper [15] (KMOC).The classical equations of central interest in our paper describe the Yang-Mills fieldwith some gauge group coupled to several classical point-like particles. These particles eachcarry colour charges c a which are time-dependent vectors in the adjoint representation ofthe gauge group. Often known as Wong’s equations [105], the equations of motion for N of these particles following worldlines x α ( τ α ) and with velocities v α and momenta p α ared p µα d τ α = g c aα ( τ α ) F a µν ( x α ( τ α )) v α ν ( τ α ) , (1.1a)d c aα d τ α = gf abc v µα ( τ α ) A bµ ( x α ( τ α )) c cα ( τ α ) , (1.1b) D µ F aµν ( x ) = J aν ( x ) = g N X α =1 Z d τ α c aα ( τ α ) v µ ( τ α ) δ (4) ( x − x ( τ α )) , (1.1c)where the Yang-Mills field is A µ = A aµ T a and F aµν is the associated field strength tensor.Although our motivation lies in gravitational dynamics, classical YM theory is alsoimportant in other contexts. Its asymptotic symmetry group has recently received intensestudy both for its own intrinsic interest but also as a toy model for the asymptotic symmetrygroup of gravity [106–112]. The radiation of colour charge of interest to us is also importantin that context.A totally different application is to the quark-gluon plasma, a high-temperature phaseof matter which is described by the classical equations of Yang-Mills theory. Indeed itis in this context that the classical theory has its main application, as a tool to modeltransport phenomena in non-Abelian plasmas in the very high temperature regime [113–118]. The equations of motion also provide a successful approximation for calculating thegluon distribution functions from deeply inelastic, ultrarelativistic ion collisions [119–121],and radiation from classical particles has recently been calculated in this regime [122, 123].Our paper is organised as follows. In section 2 we review the theoretical treatment ofcolour in Yang-Mills theory, emphasising the importance of coherent states in the classicallimit. We will see that point-like particles emerging from an underlying matter field in arepresentation R of the gauge group have an adjoint-valued colour charge which naturallyhas dimensions of angular momentum. In section 3 we will set up expressions for theclassical and momentum impulse when scattering states have colour, and provide an explicitcomputation of these observables at NLO. In section 4 we turn to radiation, constructingthe total radiated colour charge in terms of amplitudes and explicitly recovering the classicalLO radiation current derived in [11]. In section 5 we reproduce our earlier results usingpurely classical methods. Finally, we discuss our results in section 6.– 3 – Review of the theory of colour
In this section we review the emergence of non-Abelian colour charges c aα — a vector inthe adjoint representation for each particle species α — from a quantum field theory withscalars ϕ α in any representation R α of the gauge group, coupled to the Yang-Mills field.We specialise to the case of two different scalar fields, so that α = 1 ,
2. Our action is S = Z d x X α =1 (cid:20) ( D µ ϕ † α ) D µ ϕ α − m α ~ ϕ † α ϕ α (cid:21) − F aµν F a µν ! , (2.1)where D µ = ∂ µ + igA aµ T aR . We have restored factors of ~ for convenience below. Thegenerator matrices (in a representation R ) are T aR = ( T aR ) ij and satisfy the Lie algebra[ T aR , T bR ] ij = if abc ( T cR ) ij . We take the metric signature to be mostly negative. For simplicity we begin with the case of just a single massive scalar. At the classical level,the colour charge can be obtained from the Noether current j aµ associated with the globalpart of the gauge symmetry. The colour charge is explicitly given by Z d x j a ( t, x ) = i Z d x (cid:16) ϕ † T aR ∂ ϕ − ( ∂ ϕ † ) T aR ϕ (cid:17) . (2.2)Notice that a direct application of the Noether procedure has led to a colour charge withdimensions of action, or equivalently, of angular momentum.It’s worth dwelling on dimensional analysis in the context of the Wong equations (1.1)since they motivate us to make certain choices which may, at first, seem surprising. TheYang-Mills field strength F aµν = ∂ µ A aν − ∂ ν A aµ − gf abc A bµ A cν (2.3)is obviously an important actor in these classical equations. Classical equations shouldcontain no factors of ~ , so we choose to maintain this precise expression for the fieldstrength when ~ = 1. By inspection it follows that [ gA aµ ] = L − . We can develop thisfurther; since the action of equation (2.1) has dimensions of angular momentum, the Yang-Mills field strength must have dimensions of p M/L . Thus, from equation (2.3),[ A aµ ] = r ML , [ g ] = 1 √ M L . (2.4)This conclusion about the dimensions of g is in contrast to the situation in electrodynamics,where [ e ] = √ M L . Put another way, in electrodynamics the dimensionless fine structureconstant is e / π ~ while in our conventions the analogue is ~ g / π ! It is possible to An equivalent point of view is that any factors of ~ appearing in an equation which has classical meaningshould be absorbed into parameters of the classical theory. We use the notation [ x ] for the dimensions of the quantity x . The symbols L and M stand for dimensionof length and mass, respectively. – 4 –rrange matters such that the YM and EM cases are more similar, but we find the presentconventions to be convenient in perturbative calculations.Continuing with our discussion of dimensions, note that the Yang-Mills version of theLorentz force, equation (1.1a), demonstrates that the quantity gc a must have the samedimension as the electric charge. This is consistent with our observation above that thecolour has dimensions of angular momentum.At first our assignment of dimensions of g may seem troubling from the perspective ofextracting classical physics from scattering amplitudes following the algorithm described byKosower and some of the authors (KMOC) [15]. The fact that g has dimensions of 1 / √ M L implies that the dimensionless coupling at each vertex is g √ ~ , so factors of ~ associatedwith the coupling appear with the opposite power to the case of electrodynamics (andgravity). However, because the colour charges are dimensionful the net power of ~ turnsout to be the same, and thus the KMOC approach is ultimately unaltered. We will also seebelow that the dimensionful nature of the colour clarifies the classical limit of this aspectof the theory. To see how this works we must quantise. Dimensional analysis demonstrates that ϕ has dimensions of p M/L , so its mode expansionis ϕ i ( x ) = 1 √ ~ Z dΦ( p ) (cid:16) a i ( p ) e − ip · x/ ~ + b † i ( p ) e ip · x/ ~ (cid:17) . (2.5)The index i labels the representation R . We have normalised the ladder operators byrequiring [ a i ( p ) , a † j ( q )] = 2 E p (2 π ) δ (3) ( p − q ) δ ij ≡ ˆ δ ( p − q ) δ ij , (2.6)and likewise for the antiparticle operators. Each ladder operator therefore has dimensionsof M − . We also write the Lorentz-invariant phase space measure asdΦ( k ) = d k (2 π ) π Θ( k ) δ ( k − m ) ≡ ˆd k ˆ δ (+) ( k − m ) , (2.7)and introduce the hat notation in the measure and the delta functionsˆd n p ≡ d n p (2 π ) n , ˆ δ ( n ) ( x ) ≡ (2 π ) n δ ( x ) , (2.8)in order to avoid a proliferation of factors of 2 π .After quantisation, the colour charge of equation (2.2) becomes a Hilbert space operatorwhich will be important below. To emphasise that this is an operator we write the quantisedform as C a : C a = i Z d x (cid:16) ϕ † T aR ∂ ϕ − ( ∂ ϕ † ) T aR ϕ (cid:17) = ~ Z dΦ( p ) (cid:16) a † ( p ) T aR a ( p ) + b † ( p ) T a ¯ R b ( p ) (cid:17) , (2.9)using the generators of the conjugate representation ¯ R , T a ¯ R = − T aR . The overall ~ factorguarantees that the colour has dimensions of angular momentum, as we require. It is– 5 –mportant to note that these global colour operators inherit the usual Lie algebra of thegenerators, modified by factors of ~ , so that[ C a , C b ] = i ~ f abc C c . (2.10)Now we turn to the action of the colour operator on the single-particle states | p i i = a † i ( p ) | i . (2.11)Acting with the colour charge operator of equation (2.9) we immediately see that C a | p i i = ~ ( T aR ) ji | p j i , h p i | C a = ~ h p j | ( T aR ) ij . (2.12)Thus inner products yield generators scaled by ~ : h p i | C a | p j i ≡ ( C a ) ij = ~ ( T aR ) ij . (2.13)The ( C a ) ij are simply rescalings of the usual generators T aR by a factor of ~ , and thussatisfy the rescaled Lie algebra in equation (2.10); since this rescaling is important for us,it is useful to make the distinction between the two.We may then write a generic single particle state as | ψ i = X i Z d Φ( p ) φ ( p ) χ i | p i i , (2.14)where the vector χ i labels a general colour state, and the normalisations are chosen suchthat Z dΦ( p ) | φ ( p ) | = 1 , X i χ i ∗ χ i = 1 . (2.15)The colour operator acts on these states as C a | ψ i = Z dΦ( p ) ( C a ) ij φ ( p ) χ j | p i i . (2.16)Furthermore, we define the colour charge of the particle as h ψ | C a | ψ i = χ i ∗ ( C a ) ij χ j . (2.17)Computing this charge and extracting its classical limit is the topic of the next section 2.3.As a final remark on these rescaled generators, let us write out the covariant derivativein the representation R . In terms of C a , the ~ scaling of interactions is precisely the sameas in QED (and in perturbative gravity): D µ = ∂ µ + i gA aµ T a = ∂ µ + ig ~ A aµ C a ; (2.18)for comparison, the covariant derivative used by KMOC in QED was ∂ µ + ieA µ / ~ [15].Thus we have arranged that factors of ~ appear in the same place in YM theory as inelectrodynamics, provided that the colour is measured by C a . This ensures that the basicrules for obtaining the classical limits of amplitudes will be the same as in KMOC [15].In practical calculations one can thus restore ~ ’s in colour factors and work using C a ’severywhere. However, it is worth emphasising that unlike classical colour charges, thefactors C a do not commute. – 6 – .3 Colour of point-like particles in the classical regime The classical point-particle picture emerges from sharply peaked quantum wavepackets. In[15], linear exponential generalisations of Gaussian wavepackets were chosen for relativisticmomentum space wavepackets. The essential property that must be satisfied was for theparticle to have a sharply-defined position and a sharply-defined momentum wheneverthe classical limit was taken. To understand colour, governed by the Yang-Mills-Wongequations in the classical arena, a similar picture should emerge for our quantum colouroperator in equation (2.9). Following the KMOC philosophy [15] we will first consider thefull quantum states and then the classical limit. We define the classical limit of the colourcharge to be c a ≡ h ψ | C a | ψ i . (2.19)Our focus in this section is on the colour structure of our particle. The full state is atensor product of colour and kinematics: | ψ i = X | ψ colour i ⊗ | ψ kin i , (2.20)but as we ignore kinematics for now, we simply write the colour part of the state | ψ colour i →| ψ i in the remainder of this section. Then, in the classical limit, the critical requirementson the colour part of the state are that h ψ | C a | ψ i = finite , h ψ | C a C b | ψ i = h ψ | C a | ψ ih ψ | C b | ψ i + negligible . (2.21)Since the colour operator explicitly involves a factor of ~ , another parameter must be largeso that the colour expectation h ψ | C a | ψ i is much bigger than ~ in the classical region. Thissituation is basically the same as for the usual classical limit of angular momentum: in thatcase we take the spin quantum number j large so that ~ j is a classical angular momentum.For colour, we similarly need the size of the representation R to be large. (We will see thisexplicitly in the case of SU (3) in a moment.) For the second requirement we must selectappropriate colour wavefunctions | ψ i which are analogous to the kinematic wavepacket inKMOC [15].Coherent states are the key to the classical limit very generally [124], including the caseof angular momentum, so we choose a coherent state to describe the colour of our particle.The states used in KMOC to describe momenta [15] can themselves be understood ascoherent states for a “first-quantised” particle — more specifically they are states for therestricted Poincar´e group [125–127]. However, not all definitions of coherent states areequivalent, so we need to specify in what sense our states are coherent. The definition weuse was introduced by Perelomov [128], which formalises the notion of coherent state forany Lie group and hence can be utilised for both the kinematic and the colour parts. Itis in this sense that the states used by KMOC are coherent for the Poincar´e group. Werefer the interested reader to [129] for details of the Perelomov formalism and to [130] forapplications.For the explicit construction of the appropriate colour states we will use the Schwingerboson formalism. For SU (2), constructing irreducible representations from Schwinger– 7 –osons is a standard textbook exercise [131]. One simply introduces the Schwinger bosons— that is, creation a † i and annihilation a i operators, transforming in the fundamentaltwo-dimensional representation so that i = 1 ,
2. The irreducible representations of SU (2)are all symmetrised tensor powers of the fundamental, so the state a † i a † i · · · a † i j | i , (2.22)which is automatically symmetric in all its indices, transforms in the spin j representation.For groups larger than SU (2), the situation is a little more complicated because theconstruction of a general irreducible representation requires both symmetrisation and an-tisymmetrisation over appropriate sets of indices. This leads to expressions which areinvolved already for SU (3) [132, 133]. We content ourselves with a brief discussion of the SU (3) case which captures all of the interesting features of the general case.One can construct all irreducible representations from tensor products only of funda-mentals [134, 135]; however, for our treatment of SU (3) it is helpful to instead make useof the fundamental and antifundamental, and tensor these together to generate represen-tations. Following [132], we introduce two sets of ladder operators a i and b i , i = 1 , , and ∗ respectively. The colour operator can then be written as C e = ~ (cid:18) a † λ e a − b † ¯ λ e b (cid:19) , e = 1 , . . . , , (2.23)where λ e are the Gell-Mann matrices and ¯ λ e are their conjugates. The operators a and b satisfy the usual commutation relations[ a i , a † j ] = δ ij , [ b i , b † j ] = δ ij , [ a i , b j ] = 0 , [ a † i , b † j ] = 0 . (2.24)By virtue of these commutators, the colour operator (2.23) obeys the commutation relation(2.10).There are two Casimir operators given by the number operators N ≡ a † · a , N ≡ b † · b , (2.25)with eigenvalues n and n respectively, so we label irreducible representations by [ n , n ].Na¨ıvely, the states we are looking for are constructed by acting on the vacuum state asfollows: (cid:16) a † i · · · a † i n (cid:17) (cid:16) b † j · · · b † j n (cid:17) | i . (2.26)However, these states are SU (3) reducible and thus cannot be used in our constructionof coherent states. We write the irreducible states schematically by acting with a Youngprojector P which appropriately (anti-) symmetrises upper and lower indices, thereby sub-tracting traces: | ψ i [ n ,n ] ≡ P (cid:16)(cid:16) a † i · · · a † i n (cid:17) (cid:16) b † j · · · b † j n (cid:17) | i (cid:17) . (2.27) Here we define a † · a ≡ P i =1 a † i a i and | ξ | ≡ P i =1 | ξ i | . – 8 – j . . . j n i i · · · i n Figure 1 : Young tableau of SU (3)In general these operations will lead to involved expressions for the states, but we canunderstand them from their associated Young tableaux (Fig. 1). Each double box columnrepresents an operator b † i and each single column box represents the operator a † i , and thusfor a mixed representation we have n double columns and n single columns.Having constructed the irreducible states, one can define a coherent state parametrisedby two triplets of complex numbers ξ i and ζ i , i = 1 , ,
3. These are normalised accordingto | ξ | = | ζ | = 1 , ξ · ζ = 0 . (2.28)We won’t require fully general coherent states, but instead their projections onto the[ n , n ] representation, which are | ξ ζ i [ n ,n ] ≡ p ( n ! n !) (cid:16) ζ · b † (cid:17) n (cid:16) ξ · a † (cid:17) n | i . (2.29)The square roots ensure that the states are normalised to unity . With this normalisationwe can write the identity operator as [ n ,n ] = Z d µ ( ξ, ζ ) (cid:16) | ξ ζ i h ξ ζ | (cid:17) [ n ,n ] , (2.30)where R d µ ( ξ, ζ ) is the SU (3) Haar measure, normalised such that R d µ ( ξ, ζ ) = 1. Itsprecise form is irrelevant for our purposes.With the states in hand, we can return to the expectation value of the colour operator C a in equation (2.9). The size of the representation, that is n and n , must be largecompared to ~ in the classical regime so that the final result is finite. To see this let uscompute this expectation value explicitly. By definition we have h ξ ζ | C e | ξ ζ i [ n ,n ] = ~ (cid:16) h ξ ζ | a † λ e a | ξ ζ i [ n ,n ] − h ξ ζ | b † ¯ λ e b | ξ ζ i [ n ,n ] (cid:17) . (2.31)After a little algebra we find that h ξ ζ | C e | ξ ζ i = ~ (cid:0) n ξ ∗ λ e ξ − n ζ ∗ ¯ λ e ζ (cid:1) . (2.32)We see that a finite charge requires a scaling limit in which we take n , n large as ~ → ~ n i fixed for at least one value of i . The classical charge is thereforethe finite c-number c a = h ξ ζ | C a | ξ ζ i [ n ,n ] = ~ (cid:0) n ξ ∗ λ a ξ − n ζ ∗ ¯ λ a ζ (cid:1) . (2.33) Note that the Young projector in equation (2.27) is no longer necessary since the constraint ξ · ζ = 0removes all the unwanted traces. – 9 –he other feature we must check is the expectation value of products. A similarcalculation for two pairs of charge operators in a large representation leads to the importantproperty h ξ ζ | C a C b | ξ ζ i [ n ,n ] = h ξ ζ | C a | ξ ζ i [ n ,n ] h ξ ζ | C b | ξ ζ i [ n ,n ] + O ( ~ )= c a c b + O ( ~ ) . (2.34)This is in fact a special case of a more general construction discussed in detail by Yaffe [124].In appendix A we prove equation (2.34), and show that the correction term is O ( ~ ). Thesame argument can be used to demonstrate an important property of the coherent statesin the classical limit, which is that the overlap h χ ′ | χ i is very strongly peaked about χ = χ ′ [124]. We have thus constructed explicit colour states which ensure the correct classicalbehaviour of the colour charges.In the calculation of the colour impulse and radiated colour in sections 3 and 4, wewill only need to make use of the finiteness and factorisation properties, so we will avoidfurther use of the explicit form of the states. Henceforth we write χ i for the parametersof a general colour state | χ i with these properties, and d µ ( χ ) for the Haar measure of thecolour group, whatever it may be. Now that we have reviewed the theory of colour for a single particle, it’s time to considerwhat happens when more than one particle is present. We will shortly discuss the dynamicsof colour in detail; here, we set up initial states describing more than one point-like particle.We take our particles to be distinguishable, so they are associated with distinct quan-tum fields ϕ α with α = 1 ,
2. We only consider two different particles explicitly, though it isno more difficult to consider the many-particle case. We also restrict to scalar fields, againfor simplicity. The action is therefore as given in equation (2.1). Both fields ϕ α must be inrepresentations R α which are large, so that a classical limit is available for the individualcolours.At some initial time in the far past, we assume that our two particles both have well-defined positions, momenta and colours. In other words, particle α has a wavepacket φ α ( p α )describing its momentum-space distribution, and a colour wavepacket χ α as described insection 2.3. In this initial state, both particles are separated by an impact parameter b (which must then be very large compared to the spatial spread of the momenta in thewavepackets [15]). We further assume that there is no incoming radiation (that is, novector boson) in the incoming state. Thus, the state is | Ψ i = Z dΦ( p )dΦ( p ) φ ( p ) φ ( p ) e ib · p / ~ | p χ ; p χ i = Z dΦ( p )dΦ( p ) φ ( p ) φ ( p ) e ib · p / ~ χ i χ j | p i ; p j i . (2.35)Notice that the state | Ψ i refers to a multi-particle state. We reserve the notation | ψ i forsingle particle states.We measure the colour of multi-particle states by acting with a colour operator whichis simply the sum of the individual colour operators (2.9) for each of the scalar fields. For– 10 –xample, acting on the state | p χ ; p χ i we have C a | p χ ; p χ i = | p i ′ p j ′ i (cid:0) ( C a ) i ′ i δ j ′ j + δ i ′ i ( C a ) j ′ j (cid:1) χ i χ j = Z dµ ( χ ′ ) dµ ( χ ′ ) (cid:12)(cid:12) p χ ′ ; p χ ′ (cid:11) h χ ′ χ ′ | C a ⊗ ⊗ C a | χ χ i = Z dµ ( χ ′ ) dµ ( χ ′ ) (cid:12)(cid:12) p χ ′ ; p χ ′ (cid:11) h χ ′ χ ′ | C a | χ χ i , (2.36)where C aα is the colour in representation R α and we have written C a for the colouroperator on the tensor product of representations R and R . In the classical regime, usingthe property that the overlap between states sets χ ′ i = χ i in the classical limit, it followsthat h p χ ; p χ | C a | p χ ; p χ i = c a + c a , (2.37)so the colours simply add. Now we move on to the dynamics of colour. Our focus in this section will be on the colourimpulse — that is, the total change in colour during a scattering event — leaving radiationof colour to the next section. We begin by setting up the colour impulse observable in thevein of [15, 95] before turning to explicit examples at LO and NLO.
A natural observable in Yang-Mills theory is the total change in the colour charge of oneof the massive scattering particles, h ∆ c a i = h Ψ | S † C a S | Ψ i − h Ψ | C a | Ψ i = i h Ψ | [ C a , T ] | Ψ i + h Ψ | T † [ C a , T ] | Ψ i , (3.1)where we have introduced the S and T matrices, related by S = 1 + iT , and utilised theoptical theorem. We call this observable the colour impulse , as it mirrors the structureof the momentum impulse ∆ p µ of [15] and angular impulse ∆ s µ of [95]. An immediatenovelty for this impulse is that it is a Lorentz scalar, instead transforming in the adjointrepresentation of the gauge group.Substituting the 2-particle wavepackets in equation (2.35) yields h ∆ c a i = Y i =1 , Z dΦ( p i )dΦ( p ′ i ) φ i ( p i ) φ ∗ i ( p ′ i ) e ib · ( p − p ′ ) / ~ × h p ′ χ ; p ′ χ | i [ C a , T ] + T † [ C a , T ] | p χ ; p χ i , (3.2)which we expand in terms of amplitudes by inserting complete sets of states, = X X Z dΦ( r )dΦ( r ) d µ ( ζ )d µ ( ζ ) | r ζ ; r ζ ; X ih r ζ ; r ζ ; X | . (3.3)– 11 –he set X could contain any number of extra gluon or scalar states, whose phase spacemeasures and sums over any other quantum numbers are left implicit in the summationover X .It is frequently convenient to write amplitudes in Yang-Mills theory in colour-orderedform; for example, see [136] for an application to amplitudes with multiple different externalparticles. The full amplitude A is decomposed onto a basis of colour factors times partialamplitudes A . The colour factors are associated with some set of Feynman topologies. Oncea basis of independent colour structures is chosen, the corresponding partial amplitudesmust be gauge invariant. Thus, h p ′ χ ; p ′ χ | T | p χ ; p χ i = h χ χ |A ( p , p → p ′ , p ′ ) | χ χ i ˆ δ (4) ( p + p − p ′ − p ′ )= X D hC ( D ) i A D ( p , p → p ′ , p ′ ) ˆ δ (4) ( p + p − p ′ − p ′ ) , (3.4)where C ( D ) is the colour factor of diagram D and A D is the associated partial amplitude.The remaining expectation value is over the colour states χ i . Using this notation and theaction of the colour operator in equation (2.36), we can write the colour impulse as h ∆ c a i = Y i =1 , X D Z dΦ( p i )dΦ( p ′ i ) φ i ( p i ) φ ∗ i ( p ′ i ) e ib · ( p − p ′ ) / ~ × (cid:20) i h [ C a , C ( D )] i A D ( p , p → p ′ , p ′ ) ˆ δ (4) ( p + p − p ′ − p ′ )+ X X Z dΦ( r i ) D C ( D ′ ) † [ C a , C ( D )] E A ∗ D ′ ( p ′ , p ′ → r , r ) × A D ( p , p → r ′ , r ′ , r X ) ˆ δ (4) ( p + p − p ′ − p ′ )ˆ δ (4) ( p + p − r − r − r X ) (cid:21) . (3.5)Finally, let us introduce the momentum mismatch q i = p ′ i − p i , and transfer w i = r i − p i .After integrating over the delta functions, we arrive at h ∆ c a i = i Z dΦ( p )dΦ( p )ˆd q ˆ δ (2 p · q + q )ˆ δ (2 p · q − q ) × Θ( p + q )Θ( p − q ) φ ( p ) φ ( p ) φ ∗ ( p + q ) φ ∗ ( p − q ) e − ib · q/ ~ × (cid:26) X D h [ C a , C ( D )] i A D ( p , p → p + q, p − q ) − i Y i =1 , X X Z ˆd w i ˆ δ (2 p · w i + w i )ˆ δ (4) ( w + w − r X )Θ( p i + w i ) × X D,D ′ D C ( D ′ ) † [ C a , C ( D )] E A D ( p , p → p + w , p + w , r X ) × A ∗ D ′ ( p + q, p − q → p + w , p + w , r X ) (cid:27) . (3.6)It is interesting to note that factors of expectation values of colour commutators, such as h [ C a , C ( D )] i , in the colour impulse play a similar role to that of the momentum mismatch q µ in the momentum impulse ∆ p µ [15]. – 12 –he momentum impulse in QED and gravity was discussed in detail in [15]. In Yang-Mills theory, the presence of colour leads to slight modifications of those KMOC expressions.The basic difference is the colour structure of the amplitude. The observable itself is builtfrom the (colour singlet) momentum operator P µ , so factors of C a appearing in the colourimpulse, equation (3.6), do not arise in the momentum case. We proceed by writing thefull amplitude as a sum over colour structures, finding h ∆ p µ i = i Z dΦ( p )dΦ( p )ˆd q ˆ δ (2 p · q + q )ˆ δ (2 p · q − q ) × Θ( p + q )Θ( p − q ) φ ( p ) φ ( p ) φ ∗ ( p + q ) φ ∗ ( p − q ) e − ib · q × (cid:26) X D q µ hC ( D ) i A D ( p , p → p + q, p − q ) − i Y i =1 , X X Z ˆd w i ˆ δ (2 p · w i + w i )ˆ δ (4) ( w + w − r X )Θ( p i + w i ) × X D,D ′ D C ( D ′ ) † C ( D ) E A D ( p , p → p + w , p + w , r X ) × A ∗ D ′ ( p + q, p − q → p + w , p + w , r X ) (cid:27) . (3.7)By construction, both impulse observables are well defined in the classical regime. Oncewavefunctions of the types described in section 2 are used, the details of the wavefunctionswill not be important. However, to extract expressions which are valid in the classicalapproximation, it is important to be aware that the commutators of the C a contain powersof ~ . In particular one must take care to expand all commutators of colour factors.All other powers of ~ appear as described by KMOC. In brief, the rescaled covariantderivative of equation (2.18) ensures that each factor of the coupling g is accompanied bya factor ~ − / ; all massless external and loop momenta are products of a factor of ~ and awavenumber; care must be taken with squares of massless momenta q in delta functions.Finally, small shifts of order ~ ¯ q to the dominant momenta of order m in wavefunctions canbe neglected; this is an example of a general property of coherent states in the classicallimit [124]. We therefore introduce the notation (cid:28)(cid:28) f ( p , p , · · · ) (cid:29)(cid:29) = Z dΦ( p )dΦ( p ) | φ ( p ) | | φ ( p ) | h χ χ | f ( p , p , · · · ) | χ χ i . (3.8)The nature of the wavepackets make evaluating these expectation values very easy in theclassical limit: the momentum phase space integrals are simply evaluated by replacingmassive momenta with 4-velocities, p i → m i u i [15], while the colour expectation valueis guaranteed by equation (2.34) to behave as a product of commuting classical colourcharges. We will still use single angle brackets to indicate expectation values which areonly over the colour states.Following this procedure, the colour impulse becomes h ∆ c a i → ∆ c a = i (cid:28)(cid:28)Z ˆd ¯ q ˆ δ (2 p · ¯ q )ˆ δ (2 p · ¯ q ) e − ib · ¯ q G a (cid:29)(cid:29) , (3.9)– 13 –here we define the colour kernel G a to be G a = ~ X D [ C a , C ( D )] A D ( p , p → p + q, p − q ) − i ~ X X Y i =1 , Z ˆd ¯ w i ˆ δ (2 p · ¯ w i + ~ ¯ w i ) ˆ δ (4) ( ¯ w + ¯ w − ¯ r X ) X D,D ′ C ( D ′ ) † [ C a , C ( D )] × A ∗ D ′ ( p + q, p − q → p + w , p + w , r X ) A D ( p , p → p + w , p − w , r X ) . (3.10)We designed this kernel so that it is of order ~ in the classical approximation. Clearly atLO only the first term, linear in the amplitude, contributes; the second integral contributesfrom NLO, where it is an integral over tree level diagrams, while the first term involvesone-loop amplitudes.In the same notation, the momentum impulse is h ∆ p µ i → ∆ p µ = i (cid:28)(cid:28)Z ˆd ¯ q ˆ δ (2 p · ¯ q )ˆ δ (2 p · ¯ q ) e − ib · ¯ q I µ (cid:29)(cid:29) , (3.11)with momentum kernel I µ = ~ ¯ q µ X D C ( D ) A D ( p , p → p + q, p − q ) − i ~ X X Y i =1 , Z ˆd ¯ w i ˆ δ (2 p i · ¯ w i + ~ ¯ w i ) ˆ δ (4) ( ¯ w + ¯ w − ¯ r X ) ¯ w µ X D,D ′ C ( D ′ ) † C ( D ) × A ∗ D ′ ( p + q, p − q → p + w , p + w , r X ) A D ( p , p → p + w , p + w , r X )(3.12)when the scattering particles carry colour. We may now compute the colour impulse explicitly. We begin at leading order (LO) forthe scalar YM theory defined by equation (2.1), moving to next-to-leading order (NLO)in the next subsection. We will strip coupling constants from amplitudes, writing ¯ A ( n ) D forthe charge stripped partial amplitudes at O ( g n +2 ). At LO the colour kernel is G a, (0) = ~ g X D [ C a , C ( D )] ¯ A (0) D ( p , p → p + ~ ¯ q, p − ~ ¯ q ) . (3.13)Here only the t -channel tree topology contributes, so the sum between colour and kinemat-ics is trivial; we simply have ¯ A = 4 p · p + ~ ¯ q ~ ¯ q , C (cid:16) (cid:17) = C · C , (3.14)and therefore the colour impulse factor is (cid:2) C a , C (cid:0) (cid:1) (cid:3) = [ C a , C b ] C b = i ~ f abc C c C b . (3.15) We adopt the convention that time runs vertically in Feynman diagrams. – 14 –nserting these expressions into the colour kernel, all factors of ~ cancel as expected for aclassical observable. The classical limit is∆ c a, (0)1 = − g (cid:28)(cid:28) f abc C c C b Z ˆd ¯ q ˆ δ ( p · ¯ q )ˆ δ ( p · ¯ q ) e − ib · ¯ q p · p ¯ q (cid:29)(cid:29) = g f abc c b c c u · u Z ˆd ¯ q ˆ δ ( u · ¯ q )ˆ δ ( u · ¯ q ) e − ib · ¯ q ¯ q . (3.16)Notice that while evaluating the large double angle brackets we obtained classical colourcharges as expectations values of the C α .The remaining integral is straightforward but divergent. While we use dimensionalregulation throughout the remainder of the paper to define divergent integrals, in this caseit is convenient to take a different approach.The logarithmic divergence in the colour may seem surprising at first. However, thespacetime position of the particle is also logarithmically divergent in four dimensions;this is simply the familiar divergence due to the long-range nature of 1 /r forces in fourdimensions. We therefore introduce a cutoff regulator L of dimensions length as follows.Consider the following quantity2 b ∂ ∆ c a, (0)1 ∂b = b µ ∂ ∆ c a, (0)1 ∂b µ = − ig f abc c b c c γb µ Z ˆd ¯ q ˆ δ ( u · ¯ q )ˆ δ ( u · ¯ q ) e − ib · ¯ q ¯ q µ ¯ q , (3.17)where γ = u · u . The integral on the RHS was evaluated in [15]. Using that result, it iseasy to show that the solution of the differential equation is∆ c a, (0)1 = γg f abc c b c c π p γ − (cid:18) b L (cid:19) , (3.18)where we have included the regulator explicitly. At NLO the classical colour kernel, with ~ ’s from couplings removed, is G a, (1) = g X Γ [ C a , C (Γ)] ¯ A (1)Γ ( p , p → p + ~ ¯ q, p − ~ ¯ q ) − ig ~ Z ˆd ¯ ℓ ˆ δ (2 p · ¯ ℓ + ~ ¯ ℓ )ˆ δ (2 p · ¯ ℓ − ~ ¯ ℓ ) C (cid:0) (cid:1) † [ C a , C (cid:0) (cid:1) ] × ¯ A ∗ ( p + ~ ¯ q, p − ~ ¯ q → p + ~ ¯ ℓ, p − ~ ¯ ℓ ) ¯ A ( p , p → p + ~ ¯ ℓ, p − ~ ¯ ℓ ) , (3.19)where Γ is a set of one-loop topologies which span the independent colour factors. By theanalysis of [15], the topologies relevant in the classical regime are . (3.20) See appendix B for the evaluation of these diagrams. – 15 –e will refer to these as the box B , cross box C , triangles T ij and non-Abelian diagrams Y ij respectively. The latter, involving the 3-gluon interaction vertex, are new to the Yang-Mills calculation. Of course these are not the only diagrams we must calculate; there isalso the product of trees, which we will view as a cut box | B , in the non-linear part of thecolour kernel. Note that we now have two distinct colour structures to calculate, one forthe loops and one for the cut box. We will investigate in detail how these structures affectthe cancellation of classically singular terms, but first let us work with the 1-loop, linearpiece. At NLO we need to calculate the 1-loop scalar amplitude A (1) = C (cid:16) (cid:17) B + C (cid:16) (cid:17) C + C (cid:16) (cid:17) T + C (cid:16) (cid:17) T + C (cid:16) (cid:17) Y + C (cid:16) (cid:17) Y . (3.21)A first task is to choose a basis of independent colour structures. The complete set ofcolour factors can easily be calculated: C (cid:16) (cid:17) = C a C a C b C b , C (cid:16) (cid:17) = C a C b C b C a , C (cid:16) (cid:17) = 12 C (cid:16) (cid:17) + 12 C (cid:16) (cid:17) = C (cid:16) (cid:17) , C (cid:16) (cid:17) = ~ C a f abc C b C c , C (cid:16) (cid:17) = ~ C a C b f abc C c . (3.22)At first sight, we appear to have a four independent colour factors: the box, cross box andthe two non-Abelian triangles. However, it is very simple to see that the latter are in factboth proportional to the tree colour factor of equation (3.14); for example, C (cid:16) (cid:17) = ~ C a f abc [ C b , C c ] = i ~ f abc f bcd C a C d = i ~ C (cid:16) (cid:17) , (3.23)where we have used equation (2.10). Moreover, similar manipulations demonstrate thatthe cross-box colour factor is not in fact linearly independent: C (cid:16) (cid:17) = C a C b (cid:16) C a C b − i ~ f abc C c (cid:17) = ( C · C )( C · C ) − i ~ C a , C b ] f abc C c = C (cid:16) (cid:17) + ~ C (cid:16) (cid:17) . (3.24)– 16 –hus at 1-loop the classically significant part of the amplitude has a basis of two colourstructures: the box and tree. Hence the decomposition of the 1-loop amplitude into partialamplitudes and colour structures is A (1) = C (cid:16) (cid:17) (cid:20) B + C + T + T (cid:21) + ~ C (cid:16) (cid:17) (cid:20) C + T T iY + iY (cid:21) . (3.25)This expression for the amplitude is particularly useful when taking the classical limit. Thesecond term is proportional to two powers of ~ while the only possible singularity in ~ at oneloop order is a factor 1 / ~ in the evaluation of the kinematic parts of the diagrams. Thus,it is clear that the second line of the expression must be a quantum correction, and canbe dropped in calculating the classical colour impulse. Perhaps surprisingly, these termsinclude the sole contribution from the non-Abelian triangles Y ij , and thus we will not needto calculate these diagrams. We learn that classically, the 1-loop scalar YM amplitudehas a basis of only one colour factor, and moreover depends on the same topologies as inelectrodynamics, so we have A (1) = C (cid:16) (cid:17) A (1 , QED) + O ( ~ ) , (3.26)in terms of the one-loop QED amplitude A (1 , QED) .The colour impulse factor in equation (3.19) therefore reduces to a single commutator.To calculate this we need to repeatedly apply the commutation relation in equation (2.10),which yields (cid:2) C a , C (cid:0) (cid:1) (cid:3) = [ C a , C b C c ] C b C c = i ~ f acd (cid:16) C d C b C c C b + C b C d C b C c (cid:17) = i ~ f acd (cid:16) C d C c ( C · C ) + (cid:16) C d C b + i ~ f bde C e (cid:17) (cid:16) C c C b + i ~ f bce C e (cid:17)(cid:17) = i ~ f acd (cid:16) C d C c ( C · C ) − i ~ f dbe (cid:16) C e C b C c − C e C b C c (cid:17) + O ( ~ ) (cid:17) . (3.27)The colour impulse factor is itself a series in ~ . The partial amplitude is also a Laurentseries in ~ , which is presented in appendix B. In brief, the leading term in this expansionis the apparently singular (enhanced by one inverse power of ~ ) part A (1 , QED) − ∼ O ( ~ − ),and the classical term A (1 , QED)0 ∼ O ( ~ − ). This has a very important consequence forthe impulse kernel — unlike in the QED case, the apparently singular term A (1 , QED) − inthe partial amplitude now contributes classically, because of the second term in the colourimpulse factor: G a, (1)1-loop = ~ g (cid:26) if acd C d C c ( C · C ) (cid:16) A (1 , QED) − + A (1 , QED)0 (cid:17) + ~ f acd f dbe (cid:16) C e C b C c − C e C b C c (cid:17) A (1 , QED) − (cid:27) . (3.28)However, there are still singular terms in the first line; their cancellation requires includingthe quadratic part of the colour kernel. – 17 – .3.2 Cut box Rather than viewing the second term in equation (3.19) as a product of trees, we will treatthe quadratic piece as a weighted cut of the box diagram, and define | B = − i ~ Z ˆd ¯ ℓ ˆ δ (2 p · ¯ ℓ + ~ ¯ ℓ )ˆ δ (2 p · ¯ ℓ − ~ ¯ ℓ ) × ¯ A ( p + ~ ¯ q, p − ~ ¯ q → p + ~ ¯ ℓ, p − ~ ¯ ℓ ) ¯ A ( p , p → p + ~ ¯ ℓ, p − ~ ¯ ℓ ) . (3.29)Using the tree in equation (3.14) we can Laurent expand this expression in ~ , as discussedin appendix B, which yields the leading terms | B − = − i p · p ) ~ Z ˆd ¯ ℓ ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ )¯ ℓ (¯ q − ¯ ℓ ) | B = − i p · p ) ~ Z ˆd ¯ ℓ ¯ ℓ · ¯ q ¯ ℓ (¯ q − ¯ ℓ ) n ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) o . (3.30)To determine the classical contributions we must calculate the associated colour impulsefactor, C (cid:0) (cid:1) † (cid:2) C a , C (cid:0) (cid:1) (cid:3) = i ~ ( C · C ) f abc C c C b = i ~ f abc (cid:16) C c C d + i ~ f dce C e (cid:17) (cid:16) C b C d + i ~ f dbe C e (cid:17) = i ~ f acd C d C c ( C · C ) + ~ f acd f dbe (cid:16) C e C b C c − C e C b C c (cid:17) + O ( ~ ) . (3.31)Clearly we have a similar situation to equation (3.28): the colour impulse factor is again anexpansion in ~ , and its leading term yields a classical contributions from | B . Meanwhile | B − also contributes classically from the correction to the colour structure — however,there is still a singular term: G a, (1)cut box = i ~ g f acd (cid:16) C d C c ( C · C ) (cid:16) | B − + | B (cid:17) − i ~ f dbe (cid:16) C e C b C c − C e C b C c (cid:17) | B − (cid:17) = 2 g f acd ( p · p ) Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ( C d C c ( C · C ) × (cid:20) δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) ~ + ¯ ℓ · ¯ q (cid:16) ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) (cid:17) (cid:21) − if dbe (cid:16) C e C b C c − C e C b C c (cid:17) ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) ) . (3.32) It now remains to calculate the full colour kernel, G a, (1) = G a, (1)1-loop + G a, (1)cut box . (3.33)The first priority is to study the classically singular terms which sit in both parts of thekernel. Recall that each part of equation (3.19) came with a different colour kernel. The– 18 –pshot of this fact is that, after explicit calculation, the singular terms now involve onecommon colour structure: G a, (1) − = i ~ g f acd C d C c ( C · C ) (cid:16) A (1 , QED) − + | B − (cid:17) + ~ g f acd f dbe (cid:16) C e C b C c − C e C b C c (cid:17) (cid:16) A (1 , QED) − + | B − (cid:17) . (3.34)As detailed in appendix B, the singular term in the expansion of the QED amplitudeoriginates entirely from box diagrams, and takes a neat form in terms of delta functions: A (1 , QED) − = i p · p ) ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) . (3.35)This expression is the same as the | B − term in equation (3.30). The coefficients are suchthat the terms in the first line of equation (3.34) cancel, ensuring the apparently singularpart of the colour kernel vanishes.However, an interesting new feature of the colour impulse is that the colour structurein the second line of equation (3.34) combines with the sum of the 1-loop singular termsto give a non-zero classical contribution: h G a, (1) − i O ( ~ ) = − ig f acd f dbe (cid:16) C e C b C c − C e C b C c (cid:17) ( p · p ) Z ˆd ¯ ℓ ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ )¯ ℓ (¯ q − ¯ ℓ ) . (3.36)With all possible singular terms safely dealt with, it remains to combine the O ( ~ )terms in equation (3.28) and equation (3.32). Conveniently, these all have the same colourfactor: G a, (1) = ig ~ f acd C d C c ( C · C ) (cid:16) A (1 , QED)0 + | B (cid:17) + h G a, (1) − i O ( ~ ) . (3.37)Now we can sum the diagrams in the partial amplitude, the explicit expressions for whichare given in appendix B. The result is G a, (1) = g Z ˆd ¯ ℓ ℓ (¯ ℓ − ¯ q ) ( if acd C c C d ( C · C ) × " ˆ δ ( p · ¯ ℓ ) (cid:20) m + ( p · p ) ¯ ℓ · (¯ ℓ − ¯ q ) (cid:18) p · ¯ ℓ − iǫ ) + i ˆ δ ′ ( p · ¯ ℓ ) (cid:19)(cid:21) + ˆ δ ( p · ¯ ℓ ) (cid:20) m + ( p · p ) ¯ ℓ · (¯ ℓ − ¯ q ) (cid:18) p · ¯ ℓ + iǫ ) − i ˆ δ ′ ( p · ¯ ℓ ) (cid:19)(cid:21) − if acd f dbe (cid:16) C e C b C c − C e C b C c (cid:17) ( p · p ) ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) ) . (3.38) Finally the observable, the colour impulse, is given by∆ c a, (1)1 = (cid:28)(cid:28) i Z ˆd ¯ q ˆ δ (2 p · ¯ q )ˆ δ (2 p · ¯ q ) e − ib · ¯ q G a, (1) (cid:29)(cid:29) . (3.39)– 19 –pon substituting the kernel in equation (3.38), we can average over the momentum andcolour wavefunctions implicit in the expectation value. For sharply peaked momentumwavefunctions and large SU ( N ) representations, this merely has the effect of sending p i m i u i , and replacing quantum colour factors with products of commuting classical charges.Hence we finally obtain the NLO colour impulse∆ c a, (1)1 = g Z ˆd ¯ q ˆd ¯ ℓ ˆ δ ( u · ¯ q )ˆ δ ( u · ¯ q ) e − i ¯ q · b ℓ (¯ ℓ − ¯ q ) × ( ˆ δ ( u · ¯ ℓ ) " f acd c c c d ( c · c ) m (cid:20) u · u ) ¯ ℓ · (¯ ℓ − ¯ q ) (cid:18) u · ¯ ℓ − iǫ ) + i ˆ δ ′ ( u · ¯ ℓ ) (cid:19)(cid:21) − f acd f dbe c b c c c e ( u · u ) δ ( u · ¯ ℓ ) + ˆ δ ( u · ¯ ℓ ) " f acd c c c d ( c · c ) m (cid:20) u · u ) ¯ ℓ · (¯ ℓ − ¯ q ) (cid:18) u · ¯ ℓ + iǫ ) − i ˆ δ ′ ( u · ¯ ℓ ) (cid:1)(cid:3) + f acd f dbe c e c b c c ( u · u ) δ ( u · ¯ ℓ ) . (3.40)This is found to agree with the result obtained by solving the classical equations of motion,which is discussed in section 5. It is evident that the usual (momentum) impulse in YM theory should be similar to theQED case discussed in [15]. But it is also natural to expect some new terms in the YMimpulse in view of the self-coupling of the YM field. Diagrams involving this self-couplingare present at NLO. In this subsection, we investigate the impulse in the YM case withthis thought in mind. We begin with equation (3.12) for the impulse kernel I µ , which nowonly involves colour factors of the partial amplitudes themselves.At leading order we can just reuse the expressions in equation (3.14), finding I µ, (0) = 4 g ( p · p )¯ q ¯ q µ C · C . (3.41)Then, substituting into equation (3.11) and taking the classical limit as before we have theLO momentum impulse∆ p µ, (0)1 = ig c · c Z ˆd ¯ q ˆ δ ( u · ¯ q )ˆ δ ( u · ¯ q ) e − ib · ¯ q ¯ q ( u · u )¯ q µ . (3.42)This expression is closely related to the NLO impulse in QED, which can be obtained fromthe YM case by replacing c · c with the product of the electric charges of the two particles.This relationship is natural, since at leading order the gluons do not self-interact.– 20 –ust as in the colour case, at NLO the momentum kernel has linear, 1-loop andquadratic, cut box components: I µ, (1) = ~ g ¯ q µ X Γ C (Γ) ¯ A (1)Γ ( p , p → p + ~ ¯ q, p − ~ ¯ q ) − ig ~ Z ˆd ¯ ℓ ˆ δ (2 p · ¯ ℓ + ~ ¯ ℓ )ˆ δ (2 p · ¯ ℓ − ~ ¯ ℓ ) ¯ ℓ µ C (cid:0) (cid:1) † C (cid:0) (cid:1) × ¯ A ( p + ~ ¯ q, p − ~ ¯ q → p + ~ ¯ ℓ, p − ~ ¯ ℓ ) ¯ A ( p , p → p + ~ ¯ ℓ, p − ~ ¯ ℓ ) . (3.43)The decomposition of the 1-loop amplitude onto the colour basis in equation (3.25)makes computing the first term (linear in the one-loop amplitude) in the impulse kerneltrivial; we have I µ, (1)1-loop = ~ g ¯ q µ C (cid:16) (cid:17) (cid:16) A (1 , QED) − + A (1 , QED)0 (cid:17) . (3.44)This means that the non-Abelian triangle Feynman diagrams do not contribute to theimpulse: a somewhat surprising result, since it is only in these diagrams that the self-interaction of the gluons appears.Meanwhile we will denote the kinematic terms in the quadratic piece of the momentumkernel | B µ , which has the same definition as equation (3.29) but dressed with an extra loopmomentum — explicit expressions are given in appendix B. Its colour factor is simply C (cid:16) (cid:17) † C (cid:16) (cid:17) = ( C · C )( C · C ) = C (cid:16) (cid:17) . (3.45)Thus there is only one relevant colour structure in the NLO momentum impulse, that ofthe box. The momentum kernel factorises accordingly: I µ, (1) = g ( C · C ) (cid:20) ~ ¯ q µ (cid:16) A (1 , QED) − + A (1 , QED)0 (cid:17) + | B µ − + | B µ (cid:21) . (3.46)This is just a colour dressing of the NLO momentum impulse in QED — in particular, thecancellation of singular terms between the cut box and 1-loop diagrams is guaranteed [15].Gathering all the terms from triangles, boxes and the cut box in appendix B and insertinginto equation (3.11), upon taking the classical limit in the now familiar way we find∆ p µ, (1)1 = g ( c · c ) Z ˆd ¯ ℓ ˆd ¯ q ˆ δ ( u · ¯ q )ˆ δ ( u · ¯ q )¯ ℓ (¯ ℓ − ¯ q ) " ¯ q µ ( ˆ δ ( u · ¯ ℓ ) m + ˆ δ ( u · ¯ ℓ ) m + ( u · u ) ¯ ℓ · (¯ ℓ − ¯ q ) ˆ δ ( u · ¯ ℓ ) m ( u · ¯ ℓ − iǫ ) + ˆ δ ( u · ¯ ℓ ) m ( u · ¯ ℓ + iǫ ) !) − i ¯ ℓ µ ¯ ℓ · (¯ ℓ − ¯ q ) δ ′ ( u · ¯ ℓ ) δ ( u · ¯ ℓ ) m − ˆ δ ( u · ¯ ℓ ) δ ′ ( u · ¯ ℓ ) m ! . (3.47)We have found that in the non-Abelian theory the final result for the impulse is identicalto QED [15] with the charge to colour replacement Q Q → c · c . In fact this resultfollows from the colour basis decomposition in equation (3.25) and in particular the factthat the non-Abelian triangle diagrams only contribute to the ~ suppressed second colourstructure. – 21 – Radiation
One of the strengths of studying impulse-like observables is that radiative phenomenaare naturally included, as explored in depth in [15]. Moreover, the double copy makesradiation in Yang-Mills theory a powerful tool for studying its gravitational counter-part [11, 21, 85]. The general discussion of radiation of momentum in KMOC [15] appliesdirectly to the Yang-Mills case, but weakly-coupled YM theory contains another interestingobservable: the total colour radiated to infinity. In this section, we study this radiation ofcolour in the quantum formalism. As an explicit example we compute the leading orderclassical colour current found in [11] from scattering amplitudes.
The construction and calculation of the colour impulse relied on the adjoint-valued colouroperator in equation (2.9) for a scalar field in representation R . To study the total radiatedcolour we need a similar operator for the gluon radiation field. This can easily be obtainedby restricting to the adjoint representation, namely by taking ( T a adj ) bc = if bac . Since thegluon field is real and has two helicity eigenstates, its colour operator is F a = i ~ f bac X σ = ± Z dΦ( k ) a b † σ ( k ) a cσ ( k ) , (4.1)where σ labels the helicity. Of course, it is also possible to derive this expression di-rectly from the Noether charge for vector fields in the adjoint representation, given inequation (5.15), in close analogy to our discussion in sections 2.1 and 2.2.This adjoint colour charge is of interest elsewhere in the literature since it plays a rolein the physics of YM theory at asymptotic infinity [106–112]. In this connection, the natureof the final state of the radiation is relevant [111]. Here, we do not compute this final stateexplicitly. Instead, we compute expectation values of operators on the final state.Following the KMOC route [15] to obtain an expression for the total colour chargeradiated from a scattering event leads to h R a col i = h Ψ | T † F a T | Ψ i , (4.2)where we made use of the fact that there are no gauge bosons in the incoming state ofequation (2.35). Before expanding in terms of on-shell scattering amplitudes, it is worthdemonstrating colour conservation in our formalism. At the operator level, assuming onlythe quantum fields corresponding to particles 1 and 2 are present in addition to the Yang-Mills field, the statement that colour is conserved is[ C a + C a + F a , T ] = 0 . (4.3)It then immediately follows that h ∆ c a i + h ∆ c a i = h Ψ | T † [ C a , T ] | Ψ i + h Ψ | T † [ C a , T ] | Ψ i = −h Ψ | T † [ F a , T ] | Ψ i = −h Ψ | T † F a T | Ψ i = −h R a col i , (4.4)– 22 –here the second line holds from the absence of gluon radiation in the incoming state. Totalcolour is therefore conserved in the quantum theory, as it must be given the associatedglobal symmetry.Let us proceed in expanding equation (4.2) by inserting complete sets of states; forleading order radiation we need to consider an extra explicit gluon with momentum k anda colour index, so we will take the set X in the resolution of identity in equation (3.3) tojust include the contribution X b,σ Z dΦ( k ) | k b , σ ih k b , σ | . (4.5)Note that higher order corrections could also be obtained by adding further states, but wewill just be interested in the lowest order case here. Using equation (4.1) and integratingover intermediate delta functions we find h R a col i = X b,c,σ Z dΦ( k )dΦ(˜ k )dΦ( r )dΦ( r ) d µ ( ζ )d µ ( ζ ) × h Ψ | T † | r r k b , σ ; ζ ζ ih k b , σ | F a | ˜ k c , σ ih r r ˜ k c , σ ; ζ ζ | T | Ψ i = − i ~ X b,c,σ Z dΦ( k )dΦ( r )dΦ( r )d µ ( ζ )d µ ( ζ ) f abc Υ ∗ b ( r , r ; k, σ ) Υ c ( r , r ; k, σ ) , (4.6)where Υ a ( r , r ; k, σ ) = Z dΦ( p )dΦ( p ) φ ( p ) φ ( p ) e ib · p / ~ ˆ δ (4) ( p + p − r − r − k ) × X D h ζ ζ |C a ( D ) | χ χ i A D ( p , p → r , r ; k, σ ) . (4.7)We have factorised the amplitude into colour structures C ( D ) and partial amplitudes A D ,as in equation (3.4). However, here the colour factor gains a free index from the externalgluon state.To take the classical limit of equation (4.6) and introduce radiation kernels we follow[15], finding R a col = − if abc X σ (cid:28)(cid:28) ~ − Z dΦ( k ) R ∗ b ( k, σ ) R c ( k, σ ) (cid:29)(cid:29) . (4.8)The large angle brackets, defined in equation (3.8), are the expectation value over theincoming scalar wavepackets, and thus include the colour states. The radiation kernelsinherit the colour index of the external gluon, and take the form R a ( k, σ ) = ~ Z ˆd q ˆd q ˆ δ (2 p · q + q )ˆ δ (2 p · q + q ) ˆ δ (4) ( k − q − q ) e ib · q / ~ × X D C a ( D ) A D ( p + q , p + q → p , p ; k, σ ) . (4.9)The powers of ~ are organised such that the radiation kernel will be O ( ~ ) and thereforeclassical in the limit. Note that because the colour charge has dimensions of angularmomentum, the ~ scaling here works out the same way as in the total radiated momentum.– 23 – .2 Leading order evaluation Let us explicitly compute the leading order radiation kernel for the scattering of two massivescalar particles, described by the action in equation (2.1). In the classical limit the LOkernel is given in terms of coupling constant stripped amplitudes by R a, (0) (¯ k ) = ~ g Z ˆd ¯ q ˆd ¯ q ˆ δ (2 p · ¯ q + ~ ¯ q )ˆ δ (2 p · ¯ q + ~ ¯ q ) × ˆ δ (4) (¯ k − ¯ q − ¯ q ) e ib · ¯ q X D C a ( D ) ¯ A (0) D ( p + q , p + q → p , p ; k, σ ) . (4.10)Clearly the terms in the amplitude which contribute to the classical radiation are those at O ( ~ − ). The shifts in the delta functions are important for obtaining this accurately. Therelevant amplitude is the non-Abelian extension of the 5-point tree studied in [15]; thiswas used in [21] to take the double copy and calculate radiation in Einstein gravity. Therelevant Feynman topologies for emission from particle 1 are (4.11)so we need to calculate the classical terms in the 5-point amplitude A (0) (¯ k a ) = X D C a ( D ) ¯ A (0) D ( p + q , p + q → p , p ; k, σ ) (4.12)= h C a (cid:16) (cid:17) A + C a (cid:16) (cid:17) A + C a (cid:16) (cid:17) A + (1 ↔ i + C a (cid:16) (cid:17) A .
Explicitly, the colour factors are given by C a (cid:16) (cid:17) = ( C a · C b ) C b , C a (cid:16) (cid:17) = ( C b · C a ) C b , C a (cid:16) (cid:17) = 12 C a (cid:16) (cid:17) + 12 C a (cid:16) (cid:17) , C a (cid:16) (cid:17) = ~ f abc C b C c , (4.13)with the replacement 1 ↔ C a (cid:16) (cid:17) = ( C a · C b ) C b + i ~ f bac C c C b = C a (cid:16) (cid:17) + i C a (cid:16) (cid:17) . (4.14)Hence the full basis of colour factors is only 3 dimensional, and the colour decompositionof the 5-point tree is A (0) (¯ k a ) = C a (cid:16) (cid:17) (cid:16) A + A + A (cid:17) + 12 C a (cid:16) (cid:17) (cid:16) A + 2 iA + iA (cid:17) + (1 ↔ . (4.15) The momentum routing is as indicated in equation (4.10). – 24 –iven that the second structure is O ( ~ ), it would appear that we could again neglectthe second term as a quantum correction. However, this intuition is not quite correct, ascalculating the associated partial amplitude shows: A + 2 iA + iA = − i ε hµ (¯ k ) ~ (cid:20) p · p ¯ q p · ¯ k p µ ~ + 1 ~ ¯ q ¯ q (cid:16) p · ¯ k p µ − p · p (¯ q µ − ¯ q µ ) − p · ¯ k p µ (cid:17) + O ( ~ ) (cid:21) , (4.16)where we have used p · ¯ q = p · ¯ k − ~ ¯ q / ~ downstairs. However, this will cancel against the extra power in the colourstructure, yielding a classical contribution. Meanwhile in the other partial amplitude thepotentially singular terms cancel trivially, as in QED, and the contribution is classical: A + A + A = 2 ~ ε hµ (¯ k )¯ q p · ¯ k (cid:20) p · p ¯ q µ + p · p p · ¯ k p µ (¯ q − ¯ q ) − p · ¯ k p µ + 2 p · ¯ k p µ + O ( ~ ) (cid:21) . (4.17)Summing all colour factors and partial amplitudes, the classically significant part of the5-point amplitude is¯ A (0) (¯ k a ) = X D C a ( D ) ¯ A (0) D (¯ k )= − ε hµ (¯ k ) ~ (cid:26) C a ( C · C )¯ q ¯ k · p (cid:20) − ( p · p ) (cid:18) ¯ q µ − ¯ k · ¯ q ¯ k · p p µ (cid:19) + ¯ k · p p µ − ¯ k · p p µ (cid:21) + if abc C b C c ¯ q ¯ q (cid:20) k · p p µ − p · p ¯ q µ + p · p ¯ q ¯ k · p p µ (cid:21) + (1 ↔ (cid:27) , (4.18)where we have used that ¯ q − ¯ q = − k · ¯ q since the outgoing radiation is on-shell. Finally,we can substitute into the radiation kernel in equation (4.10) and take the classical limit.Averaging over the wavepackets sets p i = m i u i and replaces quantum colour charges withtheir classical counterparts, yielding R a, (0) (¯ k ) = − g Z ˆd ¯ q ˆd ¯ q ˆ δ (4) (¯ k − ¯ q − ¯ q )ˆ δ ( u · ¯ q )ˆ δ ( u · ¯ q ) e ib · ¯ q ε hµ × (cid:26) c · c m c a ¯ q ¯ k · u (cid:20) − ( u · u ) (cid:18) ¯ q µ − ¯ k · ¯ q ¯ k · u u µ (cid:19) + ¯ k · u u µ − ¯ k · u u µ (cid:21) + if abc c b c c ¯ q ¯ q (cid:20) k · u u µ − u · u ¯ q µ + u · u ¯ q ¯ k · u u µ (cid:21) + (1 ↔ (cid:27) . (4.19)Our result is equal to the leading order current ˜ K a, (0) obtained in [11] by iteratively solvingthe Wong equations in equation (1.1a) and equation (1.1b) for timelike particle worldlines.We will show this explicitly in the next section.– 25 – Classical perspectives
In this section we compute the same classical observables, impulse and radiation, usingpurely classical techniques. These calculations are not too complex, and serve to verify theresults we obtained using scattering amplitudes. This gives confidence in applying thesequantum methods to gravity, for example, where the classical calculations can becomesignificantly more involved. We start with the colour and momentum impulses beforemoving to the total radiated colour charge, discussing its relation to asymptotic symmetries.
We start with the NLO colour impulse, initially in the more general case of a system of N interacting particles, and later restrict to the N = 2 case for comparison with earliersections of the paper.As discussed previously, the appropriate equations of motion for each particle’s world-line are the Yang-Mills-Wong equations in equation (1.1a) and equation (1.1b). We areseeking perturbative solutions, and therefore expand worldline quantities in the coupling: x µα ( τ α ) = b µα + u α τ α + ∆ (1) x µα ( τ α ) + ∆ (2) x µα ( τ α ) + · · · ,v µα ( τ α ) = u α + ∆ (1) v µα ( τ α ) + ∆ (2) v µα ( τ α ) + · · · ,c aα ( τ α ) = c aα + ∆ (1) c aα ( τ α ) + ∆ (2) c aα ( τ α ) + · · · . (5.1)Here ∆ ( i ) x µα indicates quantities entering at O ( g i ). Calculating higher order correctionsrequires solving for the gauge field A aµ ( x ), using equation (1.1c). Provided the particleworldlines remain well separated, we can find perturbative solutions using the field equationin the form [11] ∂ A aµ ( x ) = K aµ ( x ) ,K aµ ( x ) ≡ J aµ ( x ) + gf abc A b ν ( x ) (cid:0) ∂ ν A cµ ( x ) − F cµν ( x ) (cid:1) . (5.2)The current K aµ is conserved but gauge dependent; for simplicity we have chosen Lorenzgauge. Writing ∆ ( i ) A aµ (¯ ℓ ) for perturbative corrections to gauge-field quantities at order O ( g i − ), the LO and NLO gauge fields are solutions to the equations ∂ [∆ (1) A aµ ( x )] = ∆ (1) J aµ ( x ) ,∂ [∆ (2) A aµ ( x )] = ∆ (2) J aµ ( x ) + gf abc h ∆ (1) A b ν ( x ) (cid:16) ∂ ν ∆ (1) A cµ ( x ) − ∆ (1) F cµν ( x ) (cid:17)i , (5.3)respectively — note that the LO equation is the same as for Abelian electrodynamics. Wesolve by Fourier transforming , using tildes to represent the Fourier transformed quantity.Solving the LO equation, it is easy to show that the leading order field is∆ (1) ˜ A aµ (¯ ℓ ) = − g X α ˆ δ ( u α · ¯ ℓ ) e i ¯ ℓ · b α c aα u µα ¯ ℓ . (5.4) Our conventions for the Fourier transform are g ( x ) = R ˆd ¯ ℓ e − i ¯ ℓ · x ˜ g (¯ ℓ ) and ˜ g (¯ ℓ ) = R d x e i ¯ ℓ · x g ( x ) . – 26 –t will be useful to define the straight line trajectory y µα ≡ b µα + u µα τ α corresponding to theinitial unperturbed worldlines. With this definition the LO equations of motion become m α d ∆ (1) x µα d τ α = g c aα Z ˆd ¯ ℓ e − i ¯ ℓ · y α ∆ (1) ˜ F a µν (¯ ℓ ) u αν , d∆ (1) c aα d τ α = gf abc Z ˆd ¯ ℓ e − il · y α u µα ∆ (1) ˜ A bµ (¯ ℓ ) c cα , (5.5)while at NLO, O ( g ), we haved ∆ (2) x µα d τ α = gm α Z ˆd ¯ ℓ e − i ¯ ℓ · y α h ∆ (1) ˜ F a µν (¯ ℓ )∆ (1) v αν c aα + ∆ (1) ˜ F a µν (¯ ℓ ) × u αν ∆ (1) c aα + ∆ (2) ˜ F a µν (¯ ℓ ) u αν c aα − i ¯ ℓ · ∆ (1) x α ∆ (1) ˜ F aµν (¯ ℓ ) u αν c aα i , d∆ (2) c aα d τ α = gf abc Z ˆd ¯ ℓ e − i ¯ ℓ · y α h u α · ∆ (1) ˜ A b (¯ ℓ ) ∆ (1) c cα + ∆ (1) v α · ∆ (1) A b (¯ ℓ ) c cα + u α · ∆ (2) ˜ A b c cα − i ¯ ℓ · ∆ (1) x α u α · ∆ (1) ˜ A b (¯ ℓ ) c cα i . (5.6)These NLO equations involve the LO corrections to the fields and the particles’ colours,positions and velocities, so although they are not the main quantities of interest we willneed to integrate the expressions in equation (5.5); for example,∆ c aα ( τ α ) = Z τ α −∞ d τ ′ α d c aα d τ ′ α . (5.7)In performing these integrals one must include an iǫ convergence factor, so the definitionof y µα is modified such that ¯ ℓ · y α = ¯ ℓ · b α + ( u α · ¯ ℓ + iǫ ) τ α . This yields∆ (1) x µα ( τ α ) = ig X β = α c α · c β m α Z ˆd ¯ ℓ e i ( ¯ ℓ · b β − ¯ ℓ · y α ) ˆ δ (¯ ℓ · u β ) ¯ ℓ · u α u µβ − ¯ ℓ µ u α · u β ¯ ℓ (¯ ℓ · u α + iǫ ) , ∆ (1) c aα ( τ α ) = ig X β = α f abc c bα c cβ u α · u β Z ˆd ¯ ℓ e i (¯ ℓ · b β − ¯ ℓ · y α ) ˆ δ (¯ ℓ · u β )¯ ℓ ℓ · u α + iǫ ) . (5.8)We now have the information to determine the NLO field, which is∆ (2) ˜ A aµ (¯ ℓ ) = − g ℓ X β = α Z ˆd ¯ ℓ e i (¯ ℓ − ¯ ℓ ) · b α e i ¯ ℓ · b β ˆ δ ((¯ ℓ − ¯ ℓ ) · u α )ˆ δ (¯ ℓ · u β ) × ( c aα c α · c β m α ¯ ℓ − ¯ ℓ µ u α · u β ¯ ℓ · u α + ¯ ℓ · ¯ ℓ u α · u β u µα (cid:0) ¯ ℓ · u α (cid:1) − ¯ ℓ · u β u µα ¯ ℓ · u α + u µβ ! + if abc c bα c cβ ¯ ℓ ¯ ℓ µ u α · u β (¯ ℓ − ¯ ℓ ) − ℓ · u α u µβ (¯ ℓ − ¯ ℓ ) + u α · u β u µα ¯ ℓ · u α !) . (5.9)It is now very simple to use the Fourier transform of the Yang-Mills equation in equa-tion (5.2) to calculate the LO momentum space current. In this context it is useful to– 27 –ename the momentum of the field ¯ k , relabel ¯ ℓ = ¯ q , and introduce ¯ q = ¯ ℓ − ¯ ℓ . Then wefind that∆ (2) ˜ K a µ (¯ k ) = g X β = α Z ˆd ¯ q ˆd ¯ q ˆ δ (4) (¯ k − ¯ q − ¯ q )ˆ δ ( u α · ¯ q )ˆ δ ( u β · ¯ q ) e ib α · ¯ q e ib β · ¯ q × (cid:26) c α · c β ¯ q ¯ k · u α c aα m α (cid:20) − ( u α · u β ) (cid:18) ¯ q µ − ¯ k · ¯ q ¯ k · u α u µα (cid:19) + ¯ k · u α u µβ − ¯ k · u β u µα (cid:21) + if abc c bα c cβ ¯ q ¯ q (cid:20) k · u β u µα − u α · u β ¯ q µ + u α · u β ¯ q ¯ k · u α u µα (cid:21) (cid:27) . (5.10)This result was first obtained in Ref. [11]. Comparing against equation (4.19), we can seethat (up to an irrelevant overall sign) the two particle restriction of the current is equal tothe LO radiation kernel calculated using amplitudes. Returning to the impulse, we can skip over the current and substitute the NLO fieldinto equation (5.6). A straightforward but tedious calculation then yields the results forthe NLO corrections given in appendix C. The observable quantities, the impulses, aredefined by ∆ c aα ≡ Z ∞−∞ d τ α d c aα d τ α , ∆ p µα ≡ m α Z ∞−∞ d τ α d v µα d τ α . (5.11)Using the results for ∆ (2) c aα it is straightforward to show, after a redefinition of the inte-gration variables, that the NLO colour impulse takes the form∆ c a, (2) α = f abc Z ˆd ¯ q e ib · ¯ q e − ib · ¯ q ˆ δ (¯ q · u )ˆ δ (¯ q · u ) Z ˆd ¯ ℓ ¯ ℓ (¯ ℓ − ¯ q ) × ( ˆ δ (¯ ℓ · u ) " c · c c b c c m u · u ) (cid:0) ¯ ℓ − ¯ ℓ · ¯ q (cid:1) (¯ ℓ · u − iǫ ) ! + if bde c c c d c e ( u · u ) ¯ ℓ · u − iǫ + ˆ δ (¯ ℓ · u ) " c · c c b c c m u · u ) (cid:0) ¯ ℓ − ¯ ℓ · ¯ q (cid:1)(cid:0) ¯ ℓ · u + iǫ (cid:1) ! + if bde c c c d c e ( u · u ) ¯ ℓ · u + iǫ − if bde c c c d c e ˆ δ (¯ ℓ · u ) ¯ ℓ · u ¯ q ) . (5.12)Notice that the signs of the iǫ on the second and third lines in the above equation aredifferent. This is a simple consequence of a change of variables required to associate theloop momenta of the classical calculation to that derived from amplitudes (see appendixC). To see that this indeed is the same as our earlier result we must manipulate the iǫ factors in the denominators further. For the quadratic denominator we can replace1(¯ ℓ · u α + iǫ ) = i ˆ δ ′ (¯ ℓ · u α ) + 1(¯ ℓ · u α − iǫ ) , (5.13)and for the linear denominator we make the shift ¯ ℓ → ¯ q − ¯ ℓ , which simply has the effect ofchanging the sign of the iǫ . Then these terms can be averaged and combined to form a Notice that we set b = 0 in sections 3 and 4 using translation symmetry. The ¯ q · u α term can be ignored due to the delta function δ (¯ q · u α ). – 28 –elta function. These procedures are the same as for the momentum impulse [15] and arebriefly reviewed in appendix B. After these manipulations, the result from amplitudes inequation (3.47) matches the colour impulse computed classically in equation (5.12), up tothe term on the last line.This final term is spurious, and can be traced back to the non-Abelian correction tothe gauge field at NLO, shown in the bottom line of equation (5.9). We observe that theterm is proportional to the integral u · I = Z ˆd ¯ ℓ ˆ δ (¯ ℓ · u ) ¯ ℓ · u ¯ ℓ (¯ ℓ − ¯ q ) , (5.14)which vanishes on the support of ˆ δ (¯ q · u ) and ˆ δ (¯ q · u ). This can easily be seen by writing I µ = Au µ + B ¯ q µ ; then we have that A = 0, and hence u · I = 0.The momentum impulse follows through similarly, and for the case of two particleswe find that the result is the same as in Abelian electrodynamics, calculated by KMOC[15], but with the replacement Q Q → c · c . This is in agreement with the quantumcalculation of section 3.4. Our classical calculation of the NLO impulse relied on the order g gauge field in equa-tion (5.9). This gauge field in isolation is also of interest, because it is the leading radiationfield generated by the scattering of the two particles. It therefore describes the transportof momentum and colour by the classical YM field itself. This classical transport of colourdeserves more discussion.The Noether current for a vector field transforming in the adjoint of the colour groupis j aµ ( x ) = − f abc A bν ( x ) (cid:0) ∂ µ A cν ( x ) − ∂ ν A cµ ( x ) (cid:1) . (5.15)To measure the instantaneous rate of colour radiation at a time t during a scattering event,we surround the particles by a large sphere and measure the flux of j aµ across its surface,taking the limit that the radius of the sphere goes to infinity. More specifically, we areinterested in outgoing radiation from our scattering event, so we take this large radiuslimit at fixed retarded time u = t − r . We then integrate over all retarded times. Thus thesurface of integration — namely the null future boundary I + of Minkowski space — isthree-dimensional, parameterised by u and the coordinates on the two-sphere. The colourradiated to I + is R a col = Z I + ∗ j a = − Z ∞−∞ d u lim r →∞ Z dΩ r j ar , (5.16)where j a = j aµ d x µ and ∗ j a is its Hodge dual. Any additional vectorial dependence arising by regulating the divergent integral will have vanishing dotproduct with u . We use Bondi coordinates u , r , θ and φ . – 29 –o evaluate this integral, we need an expression for the asymptotic field. This satisfiesthe Yang-Mills equation, given in linearised form in equation (5.2). Since there is noincoming radiation in our situation, we impose retarded boundary conditions. Using thestandard large-distance expansion of the retarded Green’s function we readily find A aµ ( x ) = 14 πr Z d ω π e − iωu ˜ K aµ (¯ k ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ k ν =( ω,ω ˆ x ) + O (cid:18) r (cid:19) , (5.17)where r = | x | and t = x . It may also help the reader to record the derivative of the field,which is ∂ µ A aν ( x ) = − i πr Z d ω π ¯ k µ ˜ K aν (¯ k ) e − iωu (cid:12)(cid:12)(cid:12)(cid:12) ¯ k ρ =( ω,ω ˆ x ) + O (cid:18) r (cid:19) . (5.18)Hence upon integrating over delta functions the total radiated charge is R a col = i (4 π ) Z ∞−∞ d ω π Z dΩ f abc ˜ K b ν ( − ¯ k ) (cid:16) k ν ˜ K cr ( k ) − ¯ k r ˜ K cν (¯ k ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ¯ k =( ω,ω ˆ x ) . (5.19)This expression can be considerably simplified. Current conservation is k ν ˜ K aν ( k ) = 0,so the first term in parentheses vanishes. In the second term, we have k r = − ω . We canfurther exploit the symmetry of the integral, and reality of the current K aµ ( x ) to show R a col = i (2 π ) Z ∞ d ω ω Z dΩ ω f abc ˜ K bν ( − ¯ k ) ˜ K c ν (¯ k ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ k =( ω,ω ˆ x ) = i (2 π ) Z d ¯ k d¯ k δ (¯ k − | ¯ k | )2 | ¯ k | f abc ˜ K bν ( − ¯ k ) ˜ K c ν (¯ k )= Z dΦ(¯ k ) if abc ˜ K b ∗ ν (¯ k ) ˜ K c ν (¯ k ) . (5.20)Finally, we use completeness of the polarisation vectors to write the classical total radiatedcolour in precisely the same form as the quantum expression: R a col = − if abc X σ = ± Z dΦ(¯ k ) (cid:16) ε ∗ σ · ˜ K ∗ (¯ k ) (cid:17) b (cid:16) ε σ · ˜ K (¯ k ) (cid:17) c . (5.21)Comparing with equation (4.8) confirms that, in the classical limit, the radiation kernelcoincides (up to a possible sign) with ε σ · ˜ K a (¯ k ) at large distances.Let us finish with a few additional remarks on this radiated colour. In ordinary elec-trodynamics it is elementary that charge is connected with the current appearing in theequation of motion. Although we made use of the Noether current in our discussion above,it remains the case that the radiated charge is connected to the current K a in the lin-earised form of equation (5.2). It is easy to check, using the explicit asymptotic field ofequation (5.17), that the radiated charge is R a col = 1 g Z I + ∗ K a . (5.22)We may now make use of equation (5.2) in the formd ∗ F a = − ∗ K a (5.23)– 30 –o write R a col = − g Z I + d ∗ F a , (5.24)where F a is the linearised field strength. The radiated charge may therefore also be recon-structed by integration over the boundaries I + ± of I + as gR a col = Z I + − ∗ F a − Z I ++ ∗ F a = gc initial − gc final . (5.25)In other words, the radiated charge is the difference between initial and final charges, asmeasured by integrating the electric fields over large spheres in the far past and the farfuture: total colour charge is conserved, as we also saw using quantum mechanical methodsin equation (4.4).Although our focus was on radiation of global charge, some of the expressions aboveare also relevant in the discussion of the larger asymptotic symmetry group of Yang-Millstheory, see for example [106–110, 112, 137, 138]. It would be interesting to broaden ouranalysis to this context, particularly in the context of the infrared structure of loop ampli-tudes. In this article, we developed methods for computing classical observables in Yang-Millstheories from scattering amplitudes. This amounts to an extension of the scope of theKMOC formalism [15] to encompass perturbative Yang-Mills theory. In addition to theobservables familiar from electrodynamics and gravity, namely the momentum impulse andthe total radiated momentum, we constructed two new observables: the colour impulse andtotal radiated colour charge.Our underlying motivation is to understand the dynamics of classical general relativ-ity through the double copy. In particular, we are interested in the relativistic two-bodyproblem which is so central to the physics of the compact binary coalescence events ob-served by LIGO and Virgo. Consequently, we focused on observables in two-body events.Although we only considered unbound (scattering) events, it is possible to determine thephysics of bound states from our observables. This can be done concretely using effectivetheories [14]. We also hope that it may be possible to connect our observables more directlyto bound states using analytic continuation, in a manner similar to the work of K¨alin andPorto [32, 34].The emergence of the classical theory from an underlying perturbative quantum fieldtheory is surprisingly intricate. Coherent states play an important role in this story, asemphasised by Yaffe [124] in the context of large N theories. In section 2 we emphasisedthe role of coherent states in describing the colour structure of particles in the classicalapproximation. It is also important that the representation of the corresponding quantumfield is large. This is in exact analogy with the emergence of a classical spin from aquantum system, and indeed the states we used for colour can equally be used to describespin. Furthermore the physics of the colour impulse in YM theory is closely analogous to– 31 –he physics of angular momentum and the associated angular impulse [95, 96]. Since thestory for colour is a little simpler, we expect it to be a useful toy model for spin in gravity.We studied the impulse and its colourful counterpart at NLO in YM theory. One sur-prise in our work was that the part of the (four-point) amplitude which is relevant in theclassical theory is exactly proportional to the classical part of the QED four-point ampli-tude. Indeed the impulse at next-to-leading order in the YM case is basically equal to theQED case; the only difference is a charge-to-colour replacement. This is a little peculiarbecause it is natural to expect the non-linearity of the Yang-Mills field to enter at thisorder (and it does so in the quantum theory). Nevertheless the colour impulse, which isintrinsically non-Abelian by definition, is non-vanishing. Although it is constructed fromthe same one-loop amplitude, an interplay of colour commutators and classically singularterms in the amplitude results in an expression for the colour impulse which involves variousdifferent colour factors. We confirmed the results of our calculations by a direct classicalcomputation using the Yang-Mills-Wong worldline formalism. It is interesting to compareour methods to those of Shen [85], who implemented the double copy at NLO wholly withinthe classical worldline formalism following ground-breaking work of Goldberger and Ridg-way [11]. Shen found it necessary to include vanishing terms involving structure constantsin his work. Similarly, in our context, some colour factors are paired with kinematic nu-merators proportional to ~ . It would be interesting to use the tools developed in this paperto explore the double copy construction of Shen [85] from the perspective of amplitudes.Throughout our paper, we emphasised that scattering amplitudes can be used to de-termine classical YM observables. But so do Wong’s equations. We have not addressed thequestion of whether it is easier to find a particular observable from amplitudes or from theWong equations. This question isn’t really of interest to us since our goal is to understandgravity, where amplitudes are much easier to compute than any (known) classical proce-dure. But possibly our work offers a way to combine the advantages of classical equationsand the double copy. We provided explicit expressions for YM observables in terms ofamplitudes; given a determination of these observables from the Wong equations, then itis possible to solve for the (classical part of the) amplitude. If it is possible to compute acorresponding gravitational amplitude from the double copy unambiguously from the clas-sical parts of a Yang-Mills amplitude, then this method would allow for the computationof observables in gravity from the Wong equations. Compared to the worldline doublecopy of Goldberger, Ridgway [11] and Shen [85] this suggestion would implement the dou-ble copy in a more standard manner. Our methods may also shine light on the difficultyimplementing the double copy off-shell in the worldline theory discussed in [87], since onecould check proposals for implementing a worldline double copy against our formulae.Our expression for the colour impulse is in many ways similar to the KMOC expressionfor the ordinary impulse. In essence the impulse describes a transfer of a small amount ofmomentum ~ ¯ q , weighted by an amplitude of order 1 / ~ . Thus the momentum transferredby many gluons leads to a macroscopic impulse. In the colour case, the small momentumtransferred is replaced by a colour commutator. This is reminiscent of the transition fromfuzzy spaces to the continuum (see, for example [139, 140]): the momentum transfer inthe impulse is the Fourier transform of a derivative, corresponding to the commutator in– 32 – fuzzy space. Perhaps there is a clue here to how the double copy works.Turning to radiation of colour, a first comment is that the relevant amplitude is nolonger proportional to the QED case. This means that at NNLO the impulse will nolonger be proportional to the QED impulse, because the radiated momenta are genuinelydifferent in the two cases. One motivation for studying impulse and radiation togetheris that they are related by conservation of momentum, so the five-point radiation termscapture dissipative effects in the impulse. The physics of momentum conservation anddissipation is rich, so we look forward to further work in this area.Colour radiation is also interesting from the point of view of asymptotic symmetrygroups. Yang-Mills theory is an interesting toy model for gravity in this context, as pointedout in an early paper by L¨uscher [137]. It would be very interesting to study colour radiationat NLO, in particular to understand what becomes of the infrared divergences of the loopamplitudes, and their impact on soft theorems. The Yang-Mills case is particularly subtlein view of the presence of collinear divergences. In electromagnetism and gravity a first stepin these directions has recently been made [46], where the connection between quantumand classical soft theorems in electromagnetism and gravity was studied using radiationkernels. We look forward to future progress on these fronts. Acknowledgements
We thank Roger Horsley, David Kosower, Se´an Mee, Alexander Ochirov and SiddharthPandey for useful discussions, and Ingrid Holm for correcting some typos in our equations.BM and AR are supported by STFC studentships ST/R504737/1 and ST/T506060/1 re-spectively. LDLC and DOC are supported by the STFC grant ST/P0000630/1. Thisresearch was supported by the Munich Institute for Astro- and Particle Physics (MIAPP)which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda-tion) under Germany’s Excellence Strategy – EXC-2094 – 390783311. Some of our figureswere produced with the help of TikZ-Feynman [141].
A Charge factorisation in SU(3)
In this appendix we prove equation (2.34). Using the coherent states restricted to the SU (3) irreducible representation [ n , n ] in equation (2.29), we have h ξ ζ | C a C b | ξ ζ i [ n ,n ] = 1( n ! n !) h | ( ζ ∗ · b ) n ( ξ ∗ · a ) n ( a † λ a a − b † ¯ λ a b ) × ( a † λ b a − b † ¯ λ b b ) (cid:16) ζ · b † (cid:17) n (cid:16) ξ · a † (cid:17) n | i = 1( n ! n !) h | ( ζ ∗ · b ) n ( ξ ∗ · a ) n (cid:16) a † λ a a a † ¯ λ b a − b † ¯ λ a b a † λ b a − a † λ a a b † ¯ λ b b − b † ¯ λ a b b † ¯ λ b b (cid:17) (cid:16) ζ · b † (cid:17) n (cid:16) ξ · a † (cid:17) n | i . (A.1)– 33 –ote that we can consider the a and b terms separately since they commute. The termswith only two a operators (or b ’s) reduce to products of the form h ξ ζ | a † i a j | ξ ζ i [ n ,n ] = 1( n ! n !) h | ( ζ ∗ · b ) n ( ξ ∗ · a ) n a † i a j (cid:16) ζ · b † (cid:17) n (cid:16) ξ · a † (cid:17) n | i = n ξ ∗ i ξ j , (A.2)and similarly for expressions involving b i b † j . We made use of the fact that the states arenormalised. Next we have the term involving four a ’s (or b ’s), which yields h ξ ζ | a † i a j a † k a l | ξ ζ i [ n ,n ] = n ξ ∗ i ξ l δ j k + n ( n − ξ ∗ i ξ l ξ ∗ k ξ j . (A.3)In the limit where n is large we can replace the n ( n −
1) factor with n . Using this andreturning all λ and ~ factors we find ~ ( λ a ) ij ( λ b ) kl h ξ ζ | a † i a j a † k a l | ξ ζ i [ n ,n ] = ~ n ξ ∗ λ a ξ ξ ∗ λ b ξ + ~ n ξ ∗ λ a · λ b ξ . (A.4)Now, gathering all the terms with a pair of a ’s and a pair of b ’s, which are simplyproducts of the expressions in equation (A.2) contracted with Gell-Mann matrices, wehave that h ξ ζ | C a C b | ξ ζ i [ n ,n ] = ~ (cid:16) n ξ ∗ λ a ξ ξ ∗ λ b ξ + n ζ ∗ ¯ λ a ζ ζ ∗ ¯ λ b ζ − n n ξ ∗ λ a ξ ζ ∗ ¯ λ b ζ − n n ξ ∗ λ b ξ ζ ∗ ¯ λ a ζ (cid:17) + ~ (cid:16) ~ n ξ ∗ λ a · λ b ξ − ~ n ζ ∗ ¯ λ a · ¯ λ b ζ (cid:17) . (A.5)Recognising the charge expectation values h ξ ζ | C a C b | ξ ζ i [ n ,n ] from equation (2.33), thiscan be written as h ξ ζ | C a C b | ξ ζ i [ n ,n ] = h ξ ζ | C a | ξ ζ i [ n ,n ] h ξ ζ | C b | ξ ζ i [ n ,n ] + ~ (cid:16) ~ n ξ ∗ λ a · λ b ξ − ~ n ζ ∗ ¯ λ a · ¯ λ b ζ (cid:17) . (A.6)The finite quantity in the classical limit ~ → , n i → ∞ is the product ~ n i . The terminside the brackets on the second line is itself finite, but comes with a lone ~ coefficient,and thus vanishes in the classical limit. This then proves the factorisation property inequation (2.34). B Diagrams and amplitude expressions
In this appendix we gather the expressions necessary for calculating the 1-loop partialamplitudes introduced in section 3.3. We are only interested in the leading classical termsof the relevant topologies, and will not list quantum corrections.From equation (3.25) we know that the relevant 1-loop topologies for NLO observablesin YM theory are those which contributed to the analogous QED calculation in [15]. TheQED amplitude A (1 , QED) = B + C + T + T (B.1) The correction term vanishes in the classical ~ → – 34 –s constructed from the triangle T ij , box B , and cross box C diagrams. Beginning with thetriangles, the leading classical terms are iT = p − qp p + qp ℓ = i m ~ Z ˆd ¯ ℓ ˆ δ ( p · ¯ ℓ )¯ ℓ (¯ ℓ − ¯ q ) + O ( ~ ) ,iT = p + qp p − qp ℓ = i m ~ Z ˆd ¯ ℓ ˆ δ ( p · ¯ ℓ )¯ ℓ (¯ ℓ − ¯ q ) + O ( ~ ) . (B.2)We refer the curious reader to Ref. [15] for the detailed calculations: heuristically, the ~ expansion of the diagrams is conducted by rescaling ℓ → ~ ¯ ℓ and q → ~ ¯ q on the support ofthe delta functions in equation (3.9). Propagator denominators are expanded as a seriesin ~ . Noting that the loop integrals are symmetric under the replacement ¯ ℓ → ¯ q − ¯ ℓ , thischange of variables can be exploited to change the sign of the (Feynman) iǫ in massivepropagators. Then, averaging over the two expressions for the integral and applying theidentities iδ ( x ) = 1 x − iǫ − x + iǫ − i ˆ δ ′ ( x ) = 1( x − iǫ ) − x + iǫ ) (B.3)leads to the expressions in equation (B.2). This symmetrisation trick is unnecessary forcalculating the individual terms from box topologies, for which we choose the followingmomentum routing: iB = p + qp p − qp ℓ iC = p + qp p − qp ℓ i | B = p + qp p − qp ℓℓ − q (B.4)The cut box appears outside of A (1 , QED) , forming the quadratic part of the NLO colourkernel, and is defined in equation (3.30). However, these diagrams all have kinematiccoefficients of the form D = D − + D + O ( ~ ) , (B.5)where D − ∼ O ( ~ − ) and D ∼ O ( ~ − ). We choose to label the terms like this as the O ( ~ − ) terms are those which generally contribute classically, and we ignore the O ( ~ )– 35 –erms as they always act as quantum corrections. The O ( ~ − ) terms would give rise tocontributions to the impulse which are classically singular, and as shown in section 3.3 itis necessary to consider all three diagrams in order to see that they cancel. We have B − = i p · p ) ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ( p · ¯ ℓ − iǫ )( p · ¯ ℓ + iǫ ) ,C − = − i p · p ) ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) p · ¯ ℓ + iǫ )( p · ¯ ℓ + iǫ ) , | B − = − i p · p ) ~ Z ˆd ¯ ℓ ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ )¯ ℓ (¯ q − ¯ ℓ ) , (B.6)and B = i p · p ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ( p · ¯ ℓ + iǫ )( p · ¯ ℓ − iǫ ) × (cid:26) p − p ) · ¯ ℓ + ( p · p )¯ ℓ (cid:18) p · ¯ ℓ − iǫ ) − p · ¯ ℓ + iǫ ) (cid:19)(cid:27) ,C = − i p · p ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ( p · ¯ ℓ + iǫ )( p · ¯ ℓ + iǫ ) × (cid:26) p + p ) · ¯ ℓ − ( p · p ) (cid:18) ¯ ℓ ( p · ¯ ℓ + iǫ ) + ¯ ℓ − q · ¯ ℓ ( p · ¯ ℓ + iǫ ) (cid:19)(cid:27) , | B = i p · p ) ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ¯ ℓ n ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ˆ δ ′ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) o . (B.7)Here the delta functions in | B originate in its definition as the quadratic part of the colourkernel in equation (3.19). Applying the symmetrisation trick and equation (B.3) ensuresthat the sums of the box and cross box contributions can also be recast in terms of deltafunctions — hence, upon including the triangle contributions, the leading terms in theexpansion of the QED amplitude in equation (B.1) are A (1 , QED) − = i p · p ) ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) A (1 , QED)0 = 2 ~ Z ˆd ¯ ℓ ℓ (¯ q − ¯ ℓ ) ( i ( p · p ) (cid:20)
12 (¯ ℓ − ℓ · ¯ q ) (cid:16) ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ˆ δ ′ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) (cid:17) − (¯ ℓ − ¯ ℓ · ¯ q ) i ˆ δ ( p · ¯ ℓ )( p · ¯ ℓ − iǫ ) + i ˆ δ ( p · ¯ ℓ )( p · ¯ ℓ + iǫ ) ! (cid:21) + m ˆ δ ( p · ¯ ℓ ) + m ˆ δ ( p · ¯ ℓ ) ) . (B.8)A similar averaging procedure can also be applied to the cut box, yielding | B = − i p · p ) ~ Z ˆd ¯ ℓ ¯ ℓ · ¯ q ¯ ℓ (¯ q − ¯ ℓ ) n ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ˆ δ ′ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) o , (B.9)which is the result listed in equation (3.30).– 36 –inally, for the momentum impulse the cut box | B µ is dressed by a power of the loopmomentum, and thus | B µ − = − i p · p ) ~ Z ˆd ¯ ℓ ¯ ℓ µ ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ )¯ ℓ (¯ q − ¯ ℓ ) , | B µ = i p · p ) ~ Z ˆd ¯ ℓ ¯ ℓ µ ¯ ℓ (¯ q − ¯ ℓ ) n ¯ ℓ ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ¯ ℓ ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) o . (B.10)Shifting with ¯ ℓ → ¯ q − ¯ ℓ and averaging the two expressions, these can be written equivalentlyas | B µ − = − i g ( p · p ) ~ Z ˆd ¯ ℓ ¯ q µ ¯ ℓ (¯ q − ¯ ℓ ) ˆ δ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) , | B µ = i p · p ) ~ Z ˆd ¯ ℓ ¯ ℓ · (¯ ℓ − ¯ q )¯ ℓ (¯ q − ¯ ℓ ) ¯ ℓ µ (cid:16) ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ˆ δ ′ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) (cid:17) − i ( p · p ) ~ ¯ q µ Z ˆd ¯ ℓ (¯ ℓ − q · ¯ ℓ )¯ ℓ (¯ q − ¯ ℓ ) (cid:16) ˆ δ ( p · ¯ ℓ )ˆ δ ′ ( p · ¯ ℓ ) − ˆ δ ′ ( p · ¯ ℓ )ˆ δ ( p · ¯ ℓ ) (cid:17) . (B.11)Applying the analysis of [15], one would expect the non-Abelian triangles iY = p − qp p + qp qℓ iY = p + qp p − qp q ℓ (B.12)to contribute to the NLO observables. The series in ~ for these partial amplitudes containterms at O ( ~ − ); however, the decomposition of the full amplitude onto the colour basisin equation (3.25) shows that these always act as quantum corrections, and thus we neednot calculate these diagrams. C Colour deflection for N particles Here we give the full general N particle results for the colour deflection. The strategyto perform the calculation is iterative and follows Ref. [11]. However, here we have notintroduced an extra integration and performed a sum over integration labels as in [11]. Ascan be seen in equations (5.8), we are removing self-interactions. At NLO this leads toan important distinction between sums over particle species. Accordingly, in the followingexpressions we have separated the contributions according to the type of sum involved.– 37 –ur result for the NLO colour deflection is∆ (2) c aα ( τ α ) = N X β =1 ,β = αγ =1 ,γ = α Z ˆd ¯ q Z ˆd ¯ q ˆ δ (¯ q · u γ )ˆ δ (¯ q · u β ) e i ¯ q · b γ e i ¯ q · b β × e − i (¯ q +¯ q ) · ( b α + u α τ α ) H a, (2) A (¯ q , ¯ q ; u α , u γ , u β )+ N X β =1 ,β = αγ =1 ,γ = β Z ˆd ¯ q Z ˆd ¯ q ˆ δ (¯ q · u γ )ˆ δ (¯ q · u β ) e i ¯ q · b γ e i ¯ q · b β × e − i (¯ q +¯ q ) · ( b α + u α τ α ) H a, (2) B (¯ q , ¯ q ; u α , u γ , u β ) , (C.1)where ¯ q ij... = ¯ q i + ¯ q j + · · · andtr( A, B, C, D ) ≡
14 tr( /A /B /C /D ) . (C.2)Here, H a, (2) A (¯ q , ¯ q ; u α , u γ , u β ) = i f abc c bγ c cα c α · c β m α ¯ q ¯ q ¯ q · u α (¯ q · u α ) × ( u α · u γ tr(¯ q , ¯ q , u α , u β ) + ¯ q · u α tr(¯ q , u β , u α , u γ )) − f abc f cde c bγ c dβ c eα (cid:18) u α · u β u α · u γ ¯ q ¯ q ¯ q · u α ¯ q · u α (cid:19) , H a, (2) B (¯ q , ¯ q ; u α , u γ , u β ) = i f abc c bβ c cα c β · c γ m β ¯ q ¯ q ¯ q · u α (¯ q · u β ) (cid:18) u α · u β tr(¯ q , u β , ¯ q , u γ )+¯ q · u β tr(¯ q , u α , u β , u γ ) (cid:19) + f abc f bde c cα c dγ c eβ ¯ q ¯ q ¯ q · u α × (cid:18) − u β · u γ ¯ q · u α ¯ q + 2 u α · u γ ¯ q · u β ¯ q − u α · u β u β · u γ ¯ q · u β (cid:19) . (C.3)In order to reach the form of the colour impulse in section 5 we set N = 2 and perform thetime integration on the support of the on-shell conditions. To recover the final observableswe define the loop momentum as ¯ ℓ ≡ ¯ q and ¯ ℓ ≡ ¯ q + ¯ q in the first and second contributionsin (C.3) respectively. References [1] S. Babak, J. Gair, A. Sesana, E. Barausse, C. F. Sopuerta, C. P. Berry, E. Berti,P. Amaro-Seoane, A. Petiteau, and A. Klein,
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