Analytical spectrum of nonlinear Thomson scattering including radiation reaction
aa r X i v : . [ h e p - ph ] F e b Analytical Spectrum of Nonlinear Thomson Scattering IncludingRadiation Reaction
A. Di Piazza ∗ and G. Audagnotto Max Planck Institute for Nuclear Physics,Saupfercheckweg 1, D-69117 Heidelberg, Germany bstract Accelerated charges emit electromagnetic radiation and the consequent energy-momentum lossalters their trajectory. This phenomenon is known as radiation reaction and the Landau-Lifshitz(LL) equation is the classical equation of motion of the electron, which takes into account self-consistently radiation-reaction effects in the electron trajectory. By using the analytical solution ofthe LL equation in an arbitrary plane wave, we compute the analytical expression of the classicalemission spectrum via nonlinear Thomson scattering including radiation-reaction effects. Both theangularly-resolved and the angularly-integrated spectra are reported, which represent the exactclassical expressions of the spectra in the sense that neglected contributions are smaller thanquantum effects. Also, we have obtained a phase-dependent expression of the electron dressedmass, which includes radiation-reaction effects. Finally, the corresponding spectra within thelocally-constant field approximation have been derived.
PACS numbers: 12.20.Ds, 41.60.-m
I. INTRODUCTION
Maxwell’s and Lorentz equations allow in principle to describe self-consistently the clas-sical dynamics of electric charges and their electromagnetic field. However, even in the caseof a single elementary charge, an electron for definiteness, the solution of the self-consistentproblem of the electron dynamics and of that of its own electromagnetic field is plaguedby physical inconsistencies, which ultimately are related to the divergent self energy of apoint-like charge. In fact, the inclusion of the “reaction” of the self electromagnetic field onthe electron dynamics (known as radiation reaction) implies an unavoidable Coulomb-likedivergence when one evaluates the self field at the electron position [1–4]. However, thisdivergence can be reabsorbed via a redefinition of the electron mass, which ultimately leadsto one of the most controversial equations in physics, the Lorentz-Abraham-Dirac (LAD)equation [5–7]. In the case of interest here, where the external force is also electromagnetic,the LAD equation can be derived by eliminating from the Maxwell-Lorentz system of equa-tions the electromagnetic field generated by the electron. In this respect, solving the LAD ∗ [email protected] ~ = c = 4 πǫ = 1 are employed throughout and the metric tensor is η µν =diag(+1 , − , − , − II. ANALYTICAL SPECTRUM OF NONLINEAR THOMSON SCATTERING
Let us consider an electron (charge e < m , respectively), whose trajec-tory is characterized by the instantaneous position x ( t ) and the instantaneous velocity β ( t ) = d x ( t ) /dt . The electromagnetic energy E radiated by the electron per unit of an-gular frequency ω and along the direction n = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ) within a solidangle d Ω = sin ϑdϑdϕ is given by [see, e.g., Eq. (14.67) in Ref. [1]] d E dωd Ω = e ω π (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ dt n × ( n × β ( t )) e iω ( t − n · x ( t )) (cid:12)(cid:12)(cid:12)(cid:12) , (1)and we stress that this expression of the emitted energy is valid for an arbitrary trajectoryof the electron.Now, we assume that the electron moves in the presence of a plane-wave background field,described by the four-vector potential A µ ( φ ) = ( A ( φ ) , A ( φ )), where φ = ( n x ) = t − n · x ,with n µ = (1 , n ) and the unit vector n identifying the propagation direction of the planewave itself. We decide to work in the Lorenz gauge ∂ µ A µ ( φ ) = ( n A ′ ( φ )) = 0 with theadditional condition A ( φ ) = 0. Here and below the prime indicates the derivative withrespect to the argument of a function. By assuming that lim φ →±∞ A ( φ ) = , then theLorenz-gauge condition implies n · A ( φ ) = 0. Thus, the four-vector potential A µ ( φ ) can bewritten as A µ ( φ ) = P j =1 a µj ψ j ( φ ), where the four-vectors a µj have the form a µj = (0 , a j ) andfulfill the orthogonality conditions ( a j a j ′ ) = − δ jj ′ , with j, j ′ = 1 ,
2, and ( n a j ) = − n · a j =0, and where the functions ψ j ( φ ) are arbitrary (physically well-behaved) functions such thatlim φ →±∞ ψ j ( φ ) = 0.It is convenient first to express the emitted energy d E /dωd Ω as an integral over the laser4hase ϕ = ω φ , where ω is the central angular frequency of the plane wave (or, more ingeneral, an arbitrary frequency scale describing the time dependence of the plane wave).This is easily done because dφ ( t ) /dt = 1 − n · β ( t ) along the electron trajectory and oneobtains d E dωd Ω = e π ω ω (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ dϕ n × ( n × p ( ϕ )) p − ( ϕ ) e i ωω R ϕ −∞ dϕ ′ ε ( ϕ ′ ) − n · p ( φ ′ ) p − ( ϕ ′ ) (cid:12)(cid:12)(cid:12)(cid:12) , (2)where p µ ( ϕ ) = ( ε ( ϕ ) , p ( ϕ )) = ε ( ϕ )(1 , β ( ϕ )), with ε ( ϕ ) = m/ p − β ( ϕ ), is the electronfour-momentum and p − ( φ ) = ( n p ( φ )). In fact, it is in general convenient to introduce alsothe four-dimensional quantity ˜ n µ = (1 , − n ) / n µ , ˜ n µ , and a µj fulfill the complete-ness relation: η µν = n µ ˜ n ν + ˜ n µ n ν − a µ a ν − a µ a ν (note that ( n ˜ n ) = 1 and that, as we havealready seen, ( a a ) = ( a a ) = −
1, whereas all other possible scalar products among n µ ,˜ n µ , and a µj vanish). By using the quantities n µ , ˜ n µ , and a µj one can define the light-conecoordinates of an arbitrary four-vector v µ = ( v , v ) as v + = (˜ n v ), v ⊥ = − (( va ) , ( va )),and v − = ( n v ). Also, the four-dimensional scalar product between two four-vectors a µ and b µ can be written as ( ab ) = a + b − + a − b + − a ⊥ · b ⊥ .Now, we recall that the LL equation in an external electromagnetic field F µν = F µν ( x )reads [2] m du µ ds = eF µν u ν + 23 e h em ( ∂ α F µν ) u α u ν + e m F µν F να u α + e m ( F αν u ν )( F αλ u λ ) u µ (cid:21) , (3)where s is the electron proper time and u µ ( s ) = p µ ( s ) /m is the electron four-velocity. In thecase of the plane wave described above, we can introduce the central laser four-wave-vectoras k µ = ω n µ such that the laser phase reads ϕ = ( k x ). By indicating as p µ = ( ε , p ),with ε = p m + p , the initial four-momentum of the electron, i.e., lim ϕ →−∞ p µ ( ϕ ) = p µ ,the four-momentum p µ ( ϕ ) at the generic phase ϕ is given by [32] p µ ( ϕ ) = 1 h ( ϕ ) (cid:26) p µ + 12 η [ h ( ϕ ) − k µ + ω mη P µ ( ϕ ) − ω m η P ( ϕ ) k µ (cid:27) . (4)In this expression we have introduced the parameter η = ( k p ) /m and the functions h ( ϕ ) = 1 + 23 e η Z ϕ −∞ d ˜ ϕ ξ ⊥ ( ˜ ϕ ) , (5) P µ ( ϕ ) = F µν ( ϕ ) p ,ν , (6)5here F µν ( ϕ ) = Z ϕ −∞ d ˜ ϕ (cid:20) h ( ˜ ϕ ) ξ µν ( ˜ ϕ ) + 23 e η ξ ′ µν ( ˜ ϕ ) (cid:21) , (7)with ξ ⊥ ( ϕ ) = ( e/m ) A ′⊥ ( ϕ ) and ξ µν ( ϕ ) = ( e/m )[ n µ A ′ ν ( ϕ ) − n ν A ′ µ ( ϕ )]. Note that, assumingthat | ξ µν ( ϕ ) | ∼ | ξ ′ µν ( ϕ ) | as it is typically the case for standard laser fields, the term propor-tional to ξ ′ µν ( ϕ ) in F µν ( ϕ ) can be neglected according to Landau and Lifshitz reduction oforder [2] (see Ref. [20] for a situation where this term cannot be ignored). For this reasonwe write F µν ( ϕ ) = Z ϕ −∞ d ˜ ϕ h ( ˜ ϕ ) ξ µν ( ˜ ϕ ) (8)and we use this expression below. For the sake of later convenience, we also report here thelight-cone components of the four-momentum of the electron in the plane wave includingradiation reaction: p − ( ϕ ) = p , − h ( ϕ ) , (9) p ⊥ ( ϕ ) = 1 h ( ϕ ) [ p , ⊥ − m F ⊥ ( ϕ )] , (10) p + ( ϕ ) = m + p ⊥ ( ϕ )2 p − ( ϕ ) = 1 h ( ϕ ) m h ( ϕ ) + [ p , ⊥ − m F ⊥ ( ϕ )] p , − , (11)where F ⊥ ( ϕ ) = R ϕ −∞ d ˜ ϕ h ( ˜ ϕ ) ξ ⊥ ( ˜ ϕ ) [see Eq. (8)] as well as the corresponding longitudinalmomentum [ p k ( ϕ ) = n · p ( ϕ )] and the energy: p k ( ϕ ) = p + ( ϕ ) − p − ( ϕ )2 = p , − h ( ϕ ) (cid:26) m h ( ϕ ) + [ p , ⊥ − m F ⊥ ( ϕ )] p , − − (cid:27) , (12) ε ( ϕ ) = p + ( ϕ ) + p − ( ϕ )2 = p , − h ( ϕ ) (cid:26) m h ( ϕ ) + [ p , ⊥ − m F ⊥ ( ϕ )] p , − + 1 (cid:27) . (13)Before replacing Eq. (4) [or equivalently Eqs. (9)-(11)] in Eq. (2), it is convenient towrite the latter equation in the form d E d k = − e π Z dϕdϕ ′ ( p ( ϕ ) p ( ϕ ′ ))( k p ( ϕ ))( k p ( ϕ ′ )) e i R ϕϕ ′ d ˜ ϕ ( kp ( ˜ ϕ ))( k p ( ˜ ϕ )) , (14)where we have introduced the four-wave-vector of the emitted radiation k µ = ( ω, k ) = ω (1 , n ) and we have used the identity (see also Refs. [1, 34] on this) Z ∞−∞ dϕ ( kp ( ϕ ))( k p ( ϕ )) e i R ϕ −∞ d ˜ ϕ ( kp ( ˜ ϕ ))( k p ( ˜ ϕ )) = 0 . (15)Equation (14) is especially useful if one expresses the four-dimensional scalar products inlight-cone coordinates and exploits the fact that the electron four-momentum is on-shell,6.e., p ( ϕ ) = m . After a few straightforward manipulations, one can easily write Eq. (14)in the form d E d k = − e π m η Z dϕdϕ ′ e i k − p , − η R ϕϕ ′ d ˜ ϕ [ h ( ˜ ϕ )+ π ⊥ ( ˜ ϕ )] (cid:8) h ( ϕ ) + h ( ϕ ′ ) + [ F ⊥ ( ϕ ) − F ⊥ ( ϕ ′ )] (cid:9) , (16)where π ⊥ ( ϕ ) = 1 m (cid:20) p ⊥ ( ϕ ) − p − ( ϕ ) k − k ⊥ (cid:21) = 1 mh ( ϕ ) (cid:20) p , ⊥ − m F ⊥ ( ϕ ) − p , − k − k ⊥ (cid:21) (17)This expression shows that the effects of radiation reaction are all encoded in the function h ( ϕ ) [see Eq. (5)] and if radiation reaction is ignored, i.e., for h ( ϕ ) = 1, one obtains theclassical spectrum of Thomson scattering. This, in turn, can be obtained as the classicallimit of the spectrum of nonlinear Compton scattering as reported, e.g., in Ref. [36], whichis accomplished by neglecting the recoil of the emitted radiation (emitted photon in thequantum language) on the electron. More precisely, we recall here that Eq. (16) dividedby ω corresponds to the classical limit of the average number of photons emitted by theelectron per units of emitted photon momentum [37–39].As one can easily recognize, from Eq. (16) one can obtain the angularly-integrated energyemission spectrum d E /dk − by using the fact that d k = ( ω/k − ) dk − d k ⊥ and then d E dk − = − e π m η Z d k ⊥ ωk − Z dϕdϕ ′ e i k − p , − η R ϕϕ ′ d ˜ ϕ [ h ( ˜ ϕ )+ π ⊥ ( ˜ ϕ )] × (cid:8) h ( ϕ ) + h ( ϕ ′ ) + [ F ⊥ ( ϕ ) − F ⊥ ( ϕ ′ )] (cid:9) . (18)By noticing that ω = k + + k − / k ⊥ / k − + k − /
2, the integral in d k ⊥ is easily taken as it isGaussian. By passing for convenience to the average and the relative phases ϕ + = ( ϕ + ϕ ′ ) / ϕ − = ϕ − ϕ ′ , the resulting energy spectrum is given by d E dk − = − ie πη k − p , − Z dϕ + dϕ − ϕ − + i e i k − p , − η (cid:26)R ϕ − / − ϕ − / d ˜ ϕ [ h ( ϕ + + ˜ ϕ )+ F ⊥ ( ϕ + + ˜ ϕ )] − ϕ − hR ϕ − / − ϕ − / d ˜ ϕ F ⊥ ( ϕ + ˜ ϕ ) i (cid:27) × (cid:26) h (cid:16) ϕ + + ϕ − (cid:17) + h (cid:16) ϕ + − ϕ − (cid:17) + h F ⊥ (cid:16) ϕ + + ϕ − (cid:17) − F ⊥ (cid:16) ϕ + − ϕ − (cid:17)i (cid:27) × * m p , − ( ϕ − Z ϕ − / − ϕ − / d ˜ ϕ h p , ⊥ m − F ⊥ ( ϕ + + ˜ ϕ ) i) + 2 im η k − p , − ϕ − + i + , (19)where the shift of the pole at ϕ − = 0 towards the negative imaginary half-plane can beunderstood by imposing that the Gaussian integral converges [19, 34, 36].7e observe that the structure of the exponential function in the first line of this equationallows for introducing the concept of electron dressed inside a plane wave [40, 41] alsowhen radiation-reaction effects are important. Indeed, the phase-dependent electron squaredressed mass ˜ m ( ϕ + , ϕ − ) can be defined here as [see Eq. (19)]˜ m ( ϕ + , ϕ − ) = m ( ϕ − Z ϕ − / − ϕ − / d ˜ ϕh ( ϕ + + ˜ ϕ )+ 1 ϕ − Z ϕ − / − ϕ − / d ˜ ϕ F ⊥ ( ϕ + + ˜ ϕ ) − " ϕ − Z ϕ − / − ϕ − / d ˜ ϕ F ⊥ ( ϕ + + ˜ ϕ ) . (20)This expression generalizes the phase-dependent square electron dressed mass as reported,e.g., in Refs. [40–43], as it includes radiation-reaction effects.Equation (19) can be explicitly regularized. First, we integrate by parts the term pro-portional to [ h ( ϕ + + ϕ − /
2) + h ( ϕ + − ϕ − / /ϕ − and we obtain d E dk − = − ie πη k − p , − Z dϕ + dϕ − ϕ − + i e i k − p , − ˜ m ϕ + ,ϕ − ) m η ϕ − × (cid:28) ¯ h ( ϕ + , ϕ − ) (cid:26) m p , − [ p , ⊥ − h F ⊥ i ( ϕ + , ϕ − )] − m p , − ¯ h ( ϕ + , ϕ − ) (cid:27) + im η k − p , − h h (cid:16) ϕ + + ϕ − (cid:17) h ′ (cid:16) ϕ + + ϕ − (cid:17) − h (cid:16) ϕ + − ϕ − (cid:17) h ′ (cid:16) ϕ + − ϕ − (cid:17)i − m p , − ¯ h ( ϕ + , ϕ − ) (cid:2) ¯ F , ⊥ ( ϕ + , ϕ − ) + h F ⊥ i ( ϕ + , ϕ − ) − ¯ F ⊥ ( ϕ + , ϕ − ) · h F ⊥ i ( ϕ + , ϕ − ) (cid:3) + 12 h F ⊥ (cid:16) ϕ + + ϕ − (cid:17) − F ⊥ (cid:16) ϕ + − ϕ − (cid:17)i × (cid:26) m p , − [ p , ⊥ − h F ⊥ i ( ϕ + , ϕ − )] + 2 im η k − p , − ϕ − (cid:27)(cid:29) , (21)8here we have introduced the notation¯ h ( ϕ + , ϕ − ) = 12 h h (cid:16) ϕ + + ϕ − (cid:17) + h (cid:16) ϕ + − ϕ − (cid:17)i , (22) h h i ( ϕ + , ϕ − ) = 1 ϕ − Z ϕ − / − ϕ − / d ˜ ϕ h ( ϕ + + ˜ ϕ ) , (23)¯ F ⊥ ( ϕ + , ϕ − ) = 12 h F ⊥ (cid:16) ϕ + + ϕ − (cid:17) + F ⊥ (cid:16) ϕ + − ϕ − (cid:17)i , (24)¯ F , ⊥ ( ϕ + , ϕ − ) = 12 h F ⊥ (cid:16) ϕ + + ϕ − (cid:17) + F ⊥ (cid:16) ϕ + − ϕ − (cid:17)i , (25) h F ⊥ i ( ϕ + , ϕ − ) = 1 ϕ − Z ϕ − / − ϕ − / d ˜ ϕ F ⊥ ( ϕ + + ˜ ϕ ) , (26) h F ⊥ i ( ϕ + , ϕ − ) = 1 ϕ − Z ϕ − / − ϕ − / d ˜ ϕ F ⊥ ( ϕ + + ˜ ϕ ) . (27)Note that with these definitions, the square of the electron dressed mass can be simplywritten as ˜ m ( ϕ + , ϕ − ) = m (cid:2) h h i ( ϕ + , ϕ − ) + h F ⊥ i ( ϕ + , ϕ − ) − h F ⊥ i ( ϕ + , ϕ − ) (cid:3) . (28)At this point only the terms in the second line of Eq. (21) need an explicit regularization. Inthe absence of radiation reaction, this is achieved by imposing that the emission spectrumhas to vanish in the absence of the external field [19, 34, 36]. Here, due to the effect ofradiation reaction, we need a slightly more complicated regularization procedure. To thisend, we introduce the function H ( ϕ + , ϕ − ) = ϕ − h h i ( ϕ + , ϕ − ) = Z ϕ − / − ϕ − / d ˜ ϕ h ( ϕ + + ˜ ϕ ) (29)and notice that ∂H ( ϕ + , ϕ − ) ∂ϕ − = ¯ h ( ϕ + , ϕ − ) > ϕ + . Now, for any positive real number a , it is Z ∞−∞ dH H + i e iaH = 0 . (31)We have indicated the integration variable as H here because, by exploiting the result inEq. (30), we change variable to ϕ − and we obtain Z ∞−∞ dϕ − H ( ϕ + , ϕ − ) + i ∂H ( ϕ + , ϕ − ) ∂ϕ − e iaH ( ϕ + ,ϕ − ) = Z ∞−∞ dϕ − H ( ϕ + , ϕ − ) + i h ( ϕ + , ϕ − ) e iaH ( ϕ + ,ϕ − ) = 0 . (32)9his result shows that we can formally regularize the remaining terms of Eq. (21) bysubtracting the vanishing quantity h ( ϕ + ) (cid:26) m p , − [ p , ⊥ − F ⊥ ( ϕ + )] − m p , − h ( ϕ + ) (cid:27) × Z ∞−∞ dϕ − H ( ϕ + , ϕ − ) + i h ( ϕ + , ϕ − ) e i k − p , − η H ( ϕ + ,ϕ − ) (33)inside the integral in ϕ + . As it will be clear below, the additional front factor h ( ϕ + ) isincluded because for | ϕ − | ≪ H ( ϕ + , ϕ − ) ≈ h ( ϕ + ) ϕ − . The resulting regularizedexpression of the energy spectrum reads d E dk − = − ie πη k − p , − Z dϕ + dϕ − e i k − p , − ˜ m ϕ + ,ϕ − ) m η ϕ − × * ¯ h ( ϕ + , ϕ − ) m p , − [ p , ⊥ − h F ⊥ i ( ϕ + , ϕ − )] − m p , − ¯ h ( ϕ + , ϕ − ) ϕ − − m p , − [ p , ⊥ − F ⊥ ( ϕ + )] − m p , − h ( ϕ + ) H ( ϕ + , ϕ − ) /h ( ϕ + ) e − i k − p , − η ϕ − [ h F ⊥ i ( ϕ + ,ϕ − ) −h F ⊥ i ( ϕ + ,ϕ − ) ] + im η k − p , − ϕ − h h (cid:16) ϕ + + ϕ − (cid:17) h ′ (cid:16) ϕ + + ϕ − (cid:17) − h (cid:16) ϕ + − ϕ − (cid:17) h ′ (cid:16) ϕ + − ϕ − (cid:17)i − m p , − ¯ h ( ϕ + , ϕ − ) ϕ − (cid:2) ¯ F , ⊥ ( ϕ + , ϕ − ) + h F ⊥ i ( ϕ + , ϕ − ) − ¯ F ⊥ ( ϕ + , ϕ − ) · h F ⊥ i ( ϕ + , ϕ − ) (cid:3) + 12 ϕ − h F ⊥ (cid:16) ϕ + + ϕ − (cid:17) − F ⊥ (cid:16) ϕ + − ϕ − (cid:17)i × (cid:26) m p , − [ p , ⊥ − h F ⊥ i ( ϕ + , ϕ − )] + 2 im η k − p , − ϕ − (cid:27)(cid:29) , (34)where we have removed the now unnecessary shift + i d E /dk − vanishes if the external plane wavevanishes. III. THE EMISSION SPECTRUM WITHIN THE LCFA
In order to implement the LCFA, we use the same strategy as in Ref. [36] by expandingEqs. (16) and (19) for small values of | ϕ − | [recall that within the LCFA the problematic termproportional to 1 / ( ϕ − + i
0) can be integrated analytically, see, e.g., Refs. [34, 36], whereas10t is easier to perform the integration by parts of the terms proportional to 1 / ( ϕ − + i after the expansion for | ϕ − | ≪ ξ = | e | A /m = | e | E /mω [29, 33, 34], where A = E /ω and E are the amplitude of the vector potential A ⊥ ( ϕ ) andof the electric field E ⊥ ( ϕ ) = − ω A ′⊥ ( ϕ ) of the plane wave, such that ξ is the amplitude of ξ ⊥ ( ϕ ) = ( e/m ) A ′⊥ ( ϕ ) (see Refs. [19, 36, 44–55] for investigations about the limitations ofthe LCFA). Moreover, in the realm of classical electrodynamics one has to assume that thequantum nonlinearity parameter χ = η ξ is much smaller than unity [29, 33, 34]. Underthese conditions the LCFA is expected to be very accurate except possibly for extremelysmall emitted radiation frequencies, which we do not consider here [36, 51, 52]. This indeedwell overlaps with the regime where classical radiation-reaction effects are typically largebecause, apart from long laser pulses, radiation-reaction effects become large for ξ ≫ χ ≪
1, to be able to neglect quantum corrections.Under the above assumptions, as it is known, one has to expand the phases in Eqs. (16)and (19) up to the third order in ϕ − , whereas the leading-order expansion is sufficient forthe pre-exponential functions. The resulting angularly-resolved and angularly-integratedenergy spectra within the LCFA can be written as [see the appendix for details on the moreinvolved derivation of Eq. (36)] d E LCFA d k = e √ π m η Z dϕ + h ( ϕ + ) χ ( ϕ + ) q π ⊥ ( ϕ + ) × (cid:2) π ⊥ ( ϕ + ) (cid:3) K / (cid:18) k − p , − h ( ϕ + ) χ ( ϕ + ) (cid:2) π ⊥ ( ϕ + ) (cid:3) / (cid:19) , (35) d E LCFA dk − = 2 e √ π k − p , − Z dϕ + ε ( ϕ + ) p − ( ϕ + ) h ( ϕ + ) η × (cid:20) K / (cid:18) k − p , − h ( ϕ + ) χ ( ϕ + ) (cid:19) −
12 IK / (cid:18) k − p , − h ( ϕ + ) χ ( ϕ + ) (cid:19)(cid:21) . (36)Here, we have introduced the local quantum nonlinearity parameter χ ( ϕ ) = η | ξ ( ϕ ) | (thisequality holds in our units where ~ = 1 and it is easily checked that the above formulas donot explicitly contain ~ ), the modified Bessel function K ν ( z ) of order ν [56] and the functionIK ν ( z ) = Z ∞ z dz ′ K ν ( z ′ ) . (37)11s expected from the very meaning of the LCFA, the above Eqs. (35)-(36) can be obtainedfrom the corresponding expressions in the absence of radiation reaction by replacing thecomponents of the electron four-momentum obtained from solving the Lorentz equation inthe plane wave with the corresponding expressions obtained from solving the LL equation[see Eqs. (9)-(13)]. In particular, one can find that in the absence of radiation reactionEq. (36) has exactly the same form as the classical limit of the quantum energy emittedspectrum as computed in Ref. [33]. However, we point out that the quantities d E LCFA /d k and d E LCFA /dk − are not local in ϕ + because both the function h ( ϕ ) [see Eq. (5)] and thefunction F ⊥ ( ϕ ) [see the definitions below Eqs. (7) and (11)] are not local in the laser phase.This is also expected from the physical meaning of radiation reaction, with one of the mainphysical consequences being the accumulation effects of energy-momentum loss.As an additional remark, we notice that by taking the integral of Eq. (36) in dk − oneobtains that the total energy radiated is given by E LCFA = 23 e η Z dϕ + ε ( ϕ + ) h ( ϕ + ) ξ ⊥ ( ϕ + ) . (38)This result coincides with the total energy radiated also beyond the LCFA, a curious cir-cumstance, which also occurs in the absence of radiation reaction [33].Interestingly, the total minus component d K − dϕ + = Z ∞ dk − Z d k ⊥ k − ω d E dk − d k ⊥ dϕ + = Z ∞ dk − Z d k ⊥ d E d k dϕ + (39)of the four-momentum radiated classically per unit of laser phase by an electron in a planewave including radiation reaction has been recently computed within the LCFA in Ref. [57]in the different context of the so-called Ritus-Narozhny conjecture on strong-field QED [58–63]. According to Eq. (39), by defining d E LCFA /d k dϕ + as the integrand in Eq. (35), and byperforming the integral of this quantity over d k ⊥ one can easily show that d K − , LCFA dϕ + = 2 e √ π Z ∞ dk − k − p , − h ( ϕ + ) η (cid:20) K / (cid:18) k − p , − h ( ϕ + ) χ ( ϕ + ) (cid:19) −
12 IK / (cid:18) k − p , − h ( ϕ + ) χ ( ϕ + ) (cid:19)(cid:21) (40)in agreement with the result in Ref. [57]. 12 V. CONCLUSIONS
In conclusion, we have derived analytically the angularly-resolved and the angularly-integrated energy emission spectra of nonlinear Thomson scattering by including radiation-reaction effects. This has been accomplished by starting from the analytical solution of theLL in an arbitrary plane wave and by using the classical formulas of radiation by acceleratedcharges.The spectra are obtained as double integrals over the plane-wave phase. A particular,new regularization technique has to be used in order to regularize the angularly-integratedspectrum. We point out that the resulting spectra include higher-order classical radiativecorrections and can be considered as “classically exact” in the sense of the Landau andLifshitz reduction of order, meaning that neglected classical corrections are much smallerthan quantum corrections, which have been of course ignored from the beginning.Moreover, we have obtained a phase-dependent expression of the electron dressed mass,which includes radiation-reaction effects.Finally, the expressions of the angularly-resolved and the angularly integrated spectrawithin the locally-constant field approximations have been derived as well. These expres-sions have the property that are expressed as single integrals over the laser phase of thecorresponding expressions without radiation reaction with the electron four-momentum re-placed with its expression including radiation reaction. Thus, they turn out to be non-localexactly for the nature itself of radiation reaction giving rise to cumulative energy-momentumloss effects.
Appendix: Derivation of Eq. (36)
The staring point is Eq. (19) and in order to implement the LCFA there, we expand eachterm of the pre-exponent up to the leading order for | ϕ − | ≪
1, whereas we keep terms upto ϕ − in the phase (see, e.g., [36]): d E LCFA dk − = − ie πη k − p , − Z dϕ + h ( ϕ + ) Z dϕ − ϕ − + i e i k − p , − η h ( ϕ + ) ϕ − [ ξ ⊥ ( ϕ + ) ϕ − ] × (cid:20) ξ ⊥ ( ϕ + ) ϕ − (cid:21) (cid:26) m p , − h p , ⊥ m − F ⊥ ( ϕ + ) i + 2 im η k − p , − ϕ − + i (cid:27) . (41)13ow, we integrate by parts the only term containing 1 / ( ϕ − + i in the pre-exponent andwe obtain d E LCFA dk − = − ie πη k − p , − Z dϕ + h ( ϕ + ) Z dϕ − ϕ − + i e i k − p , − η h ( ϕ + ) ϕ − [ ξ ⊥ ( ϕ + ) ϕ − ] × (cid:28)(cid:26) m p , − h p , ⊥ m − F ⊥ ( ϕ + ) i (cid:27) (cid:20) ξ ⊥ ( ϕ + ) ϕ − (cid:21) − m p , − h ( ϕ + ) (cid:20) ξ ⊥ ( ϕ + ) ϕ − (cid:21) + im η k − p , − ξ ⊥ ( ϕ + ) ϕ − (cid:29) . (42)This equation is already regular and can been expressed in terms of modified Bessel functionsbut, for the sake of convenience, we integrate by parts the last term and have d E LCFA dk − = − ie πη k − p , − Z dϕ + h ( ϕ + ) Z dϕ − ϕ − + i e i k − p , − η h ( ϕ + ) ϕ − [ ξ ⊥ ( ϕ + ) ϕ − ] × (cid:28)(cid:26) m p , − h p , ⊥ m − F ⊥ ( ϕ + ) i (cid:27) (cid:20) ξ ⊥ ( ϕ + ) ϕ − (cid:21) − m p , − h ( ϕ + ) (cid:20) − ξ ⊥ ( ϕ + ) ϕ − (cid:21) (cid:20) ξ ⊥ ( ϕ + ) ϕ − (cid:21)(cid:29) = − ie πη k − p , − Z dϕ + h ( ϕ + ) Z dyy + i e i k − p , − h ϕ +) χ ( ϕ +) y (cid:18) y (cid:19) × (1 + 2 y ) (cid:26) m p , − h p , ⊥ m − F ⊥ ( ϕ + ) i − m p , − h ( ϕ + ) 1 − y − y y (cid:27) , (43)where we have introduced local quantum nonlinearity parameter χ ( ϕ ) = η | ξ ( ϕ ) | (see alsothe main text). At this point, we observe that the main contribution to the integral in y = ϕ − | ξ ⊥ ( ϕ ) | / | y | .
1. Moreover, we recall that within the LCFAwe are assuming that ξ ≫ ϕ + comes from the regions where F ⊥ ( ϕ + ) isat the largest. From the definitions below Eqs. (7) and (11), we obtain that F ⊥ ( ϕ ) = Z ϕ −∞ d ˜ ϕ h ( ˜ ϕ ) ξ ⊥ ( ˜ ϕ ) = em Z ϕ −∞ d ˜ ϕ h ( ˜ ϕ ) A ′⊥ ( ϕ )= em (cid:20) h ( ϕ ) A ⊥ ( ϕ ) − e η Z ϕ −∞ d ˜ ϕ ξ ⊥ ( ˜ ϕ ) A ⊥ ( ˜ ϕ ) (cid:21) , (44)which shows that | F ⊥ ( ϕ ) | . h ( ϕ ) ξ . In conclusion, we can consistently neglect the lastterm in Eq. (43) as compared to the second-last one within the LCFA (note that we do notmake any assumptions about the values of | p ⊥ | /m and p , − /m as compared with ξ ) and14e finally obtain the expression in the main text: d E LCFA dk − = 2 e √ π k − p , − Z dϕ + ε ( ϕ + ) p − ( ϕ + ) h ( ϕ + ) η × (cid:20) K / (cid:18) k − p , − h ( ϕ + ) χ ( ϕ + ) (cid:19) −
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