Classical resummation and breakdown of strong-field QED
CClassical resummation and breakdown of strong-field QED
T. Heinzl, ∗ A. Ilderton, † and B. King ‡ Centre for Mathematical Sciences, University of Plymouth, Plymouth, PL4 8AA, UK
QED perturbation theory has been conjectured to break down in sufficiently strong backgrounds,obstructing the analysis of strong-field physics. We show that the breakdown occurs even in classicalelectrodynamics, at lower field strengths than previously considered, and that it may be curedby resummation. As a consequence, an analogous resummation is required in QED. A detailedinvestigation shows, for a range of observables, that unitarity removes diagrams previously believedto be responsible for the breakdown of QED perturbation theory.
Examining the transition to classical physics can helpus understand quantum theories, with topical examplesbeing the classical post-Minkowskian expansion of gen-eral relativistic dynamics [1, 2], classical double copy [3,4] and decoherence [5]. Whether classical or quantum,theories containing strong background fields are typi-cally analysed using background field perturbation the-ory [6, 7], where the strong background is treated ex-actly (as a classical field), while particle scattering onthe background is treated perturbatively. For quantumelectrodynamics (QED) in strong fields, this amounts toemploying the usual perturbative expansion in the finestructure constant, α (cid:28) , while fermion propagators are‘dressed’ exactly by the background: this is also calledthe Furry expansion [8]. It is an essential tool for the-ory and experimental modelling, and underlies numeri-cal particle-in-cell schemes used in astrophysics [9] andplasma physics [10].However, the Ritus-Narozhny (RN) conjecture sug-gests that the Furry expansion breaks down for suffi-ciently strong fields [11, 12]; for constant fields, as orig-inally formulated, one finds that the effective expansionparameter is not α , but αχ / [13], where χ is propor-tional to the background field strength. The conjecturehas been interpreted to hold for any background that canbe approximated as “locally constant”, i.e. constant overtypical “formation scales” [13, 14]. Due to the widespreaduse of the Furry, and related, expansions, it is crucial tounderstand their regime of applicability [15–19].The RN conjecture applies at high field strength, andlow energy [20, 21], suggesting it suits a classical analysis.Using this approach, we will show here explicitly that thebreakdown of perturbation theory occurs already in theclassical theory—and not just for constant fields. Cru-cially, this allows further progress than is possible in thequantum theory, as we can effectively resum the classicalperturbative series to all orders. We will show that thisresummation cures the unphysical behaviour associatedwith the breakdown of perturbation theory. As everyterm in the classical limit corresponds to some term inQED, our results have direct implication for the quantumtheory: in particular, we find that perturbation theorybreaks down for far lower intensities than predicted bythe RN conjecture. Classical.
A strong background, f µν = eF µν ext /m , for m and e the electron mass and charge respectively, ischaracterised by a dimensionless coupling ξ ∼ f /ω (cid:29) ,for ω a typical frequency scale of the background. Theclassical equations of motions in such a background are ¨ x µ = ( f µν + eF µν /m ) ˙ x ν , ∂ µ F µν = j ν , (1)in which F is the generated radiation field, x µ is the par-ticle orbit and j ν its current. (Note that c = 1 through-out.) The classical limit of the Furry expansion corre-sponds to treating f , therefore ξ , exactly, and e (madeappropriately dimensionless) perturbatively. The zeroth order equations describe the Lorentz orbit in the back-ground f , with no radiation. At higher orders, radiationand radiation back-reaction (‘RR’) appear [14, 22–27].The assumption behind the Furry expansion is simplythat these RR corrections, corresponding to higher pow-ers of α in QED, are subleading. We now give two exam-ples which show this is not the case, leading to a break-down of the perturbative expansion.First, an electron in a rotating electric field E ( t ) = E (0 , cos ωt, sin ωt ) can have a closed orbit, with energy mγ determined by ξ = eE / ( mω ) and ω [28]. TheLorentz force prediction for the energy is γ − ξ .Using Furry picture perturbation theory to calculate cor-rections to this, one finds it is not an expansion in somesmall parameter, but rather in powers of (cid:15) rad ξ , where (cid:15) rad := (2 / e / π )( ω/m ) , with leading behaviour γ − ∼ ξ (cid:0) − (cid:15) rad ξ + . . . ) . (2)Hence, for sufficiently strong fields, the corrections be-come larger than the supposedly dominant Lorentzforce contribution and the perturbative expansion breaksdown, signalled in (2) by the unphysical result γ − < .For our second example, consider an electron in anarbitrary plane wave (direction n µ , typical frequency ω , k µ := ωn µ , phase φ = k · x ) with transverse electric field a (cid:48) ( φ ) . According to the Lorentz force, i.e. zeroth orderin perturbation theory, the lightfront energy component n · p of the electron is conserved. The first perturbativecorrection to the final electron momentum p out is n · p out n · p = 1 − e π k · pm (cid:90) d φ | a (cid:48) ( φ ) | ≡ − ∆ (3) a r X i v : . [ h e p - ph ] J a n Exact ( LL ) LCFA ( LL ) Perturbative O ( Δ ) - ξ r ( ξ ) Δ = FIG. 1. Proportion, r ( ξ ) , of initial electron energy radi-ated in a -cycle circularly-polarised plane wave pulse a ( φ ) = mξ sin ( φ/ φ, sin φ ] for φ ∈ [0 , π ] and a ( φ ) = other-wise. The energy radiated is bounded by the initial electronenergy in the LL result, but is unbounded for the Lorentz re-sult. The LCFA discussed in the text is also shown: it char-acteristically over-predicts the radiated energy for ξ ∼ O (1) . The effective expansion parameter is ∆ ∝ ξ , which againmay not be small; the expansion breaks down for ξ (cid:29) ,signalled here by the unphysical behaviour n · p out < .In some cases it is possible to explicitly resum pertur-bative solutions to (1) [29]. A more general approach isto effectively resum the perturbative series by eliminatingthe electromagnetic variables from (1) to obtain the exactLorentz-Abraham-Dirac (LAD) equation for the electronorbit [30]. For our first example, LAD implies that γ satisfies the equation ξ = ( γ − (cid:15) rad γ ) [28]. Thisrecovers (2) if (cid:15) rad is treated perturbatively, but behavesas γ − ∼ ξ ( (cid:15) rad ξ ) − / for (cid:15) rad ξ (cid:29) , i.e. resumma-tion corrects the unphysical behaviour of perturbationtheory. For plane waves, the solution to the LAD equa-tion is not known, so we use the Landau-Lifshitz (LL)equation instead [31], which agrees exactly with LAD tolow orders and is adequate classically [32, 33]. (What isimportant is that both LAD and LL equations provideall-order results.) The exact solution to LL yields [34] n · p out n · p = 11 + ∆ > , (4)so that, comparing to (3) [35], resummation again fixesthe unphysical behaviour of perturbation theory.Note that the coefficient of the integral in (3) is (2 / αη , with the QED energy invariant η := (cid:126) k · p/m .This underlines that both perturbative and resummedresults have their origin in QED [36]. To begin mak-ing connections to QED we first need to understand inmore depth what changes when we go from low ordersof perturbation theory to all-orders results. To do so weconsider the energy-momentum K µ radiated by an elec-tron in a plane wave. This is calculated by inserting theLL solution for the electron orbit into the fully relativis-tic Larmor formula [33]. We focus for simplicity on thelightfront momentum fraction r := n · K/n · p . In Fig. 1, we show how the ratio r depends on the in-tensity of a pulsed plane wave. The leading, O (∆) , resultscales without bound as r ∼ ∆ ∼ ξ ; to this order in ∆ one has that r = 1 − n · p out /n · p , hence reaching r > reflects the unphysical behaviour in (3). The behaviourof the all-orders, or resummed, result, is completely dif-ferent: the total energy radiated is bounded by r ≤ ,as demonstrated by the plateau in Fig. 1. Thus the ef-fect of resummation is very clear, and physically sensible,but to help understand it we analyse the formation of theemitted radiation as a function of phase φ . Following theestablished procedure of expanding double phase inte-grals in the difference of two phases [37, 38], we developa locally constant field approximation (LCFA) for ourLL-corrected observables. Let ∆( φ ) be defined as in (3)but with the integral extending only up to φ , and define R := 1 + ∆( φ ) . Then we find the classically resummed,LCFA result [39] d r d φ = − e π m k · p (cid:90) ∞ d¯ s ¯ s R (cid:20) Ai ( z ) + 2 z Ai (cid:48) ( z ) (cid:21) , (5)where z = (¯ s R / ¯ χ ) / , ¯ χ = | a (cid:48) | k · p/m , and ¯ s = n · k out /n · p for k out the radiation wavevector. Notethe simple relation ¯ χ = χ/ (cid:126) relating the classical ¯ χ tothe quantum parameter χ . The LCFA is benchmarkedagainst the exact result and found to agree excellently inthe high-field limit in Fig. 1. If ∆ → ( R → ), (5) tendsto the classical limit of the O ( α ) quantum result [14].The ¯ s integral in (5) can be written as an integral overa low-frequency region, ¯ s ≤ , plus a high-frequency re-gion, ¯ s > . The integral over the low-frequency regionis exactly equal to the O ( α ) quantum result with recoiland spin set to zero. Hence, just like the QED result,it scales as ¯ χ / ∼ ξ / in the high-field limit typical ofthe RN conjecture. However, the high-frequency ( ¯ s > )contribution grows with a larger power, ∼ ξ , and thusdominates the scaling of the classical rate (5) in agree-ment with previous expectations [22, 40].In Fig. 2 we plot the local rate (5) for various ξ , andcompare to the perturbative (Lorentzian) result withoutRR. Fig. 2a shows that the higher the pulse intensity,the earlier the majority of radiation is emitted and hencethe quicker the electron is decelerated. Without RR, onthe other hand, the rate of radiation is symmetric withthe shape of the pulse: as much is emitted in the tail asin the rise. This is emphasised in Fig. 2b, in which wepick two phase points early and late in the pulse ( φ = 2 π and φ = 6 π as also indicated in Fig. 2a), and illustratehow the rate of emission at those points changes as ξ is increased. The perturbative scaling, ∼ ξ at small ξ , is corrected at large ξ to a scaling ∼ ξ − . Clearly,resummation in ∆ has changed the large- ξ behaviour.The origin of these different behaviours can be tracedback to the impact of RR corrections on the Airy argu-ment z in (5): note that it is the behaviour of the anal-ogous argument in QED results which determines the ξ = ξ = ξ = ξ = π π π π - - π π π π ϕ dr ( ϕ ) d ϕ ϕ = π ϕ = π a10 100 1000 1000010 - ξ dr ( ξ ) d ϕ ϕ = πϕ = π b FIG. 2. a) Classically resummed LCFA rate of radiation (5)(coloured lines), compared to the O (∆) result (faint graylines). b) Intensity scaling of (5) at two fixed points in thepulse (on the rising and falling edge respectively). The samefield and parameters were used as in Fig. 1. large- ξ asymptotic behaviour. Here we have,if ∆ (cid:28) : z ∼ (cid:18) ¯ sξ (cid:19) / ; if ∆ (cid:29) : z ∼ (¯ sξ ) / . If RR corrections are neglected, large ξ yields small Airyarguments ∼ (¯ s/ξ ) / ≡ ( s/ξ ) / , which leads to thepower-law ξ -dependence of the total emitted radiationassociated with the breakdown of perturbation theory( r > , recall Fig. 1). With RR, though, large ξ yieldsa large Airy argument which suppresses dr/dφ (leadingto r ≤ ). Hence, crucially, resummation reverses theasymptotic limit of the Airy functions compared to thatexpected from perturbation theory. We saw the physicalconsequence of this reversal above: the rate of radiationin the high- ξ region is suppressed as ξ − ; hence even inhigh- ξ pulses the radiation is mainly generated in the small - ξ regions in the rising edge of the pulse, where therate scales as ξ . The plateau in r at ever higher intensi-ties is a consequence of a balance between ever-strongerdecelerations over ever-shorter durations.The above examples advance our understanding of theRN conjecture significantly: we have seen that the break-down of perturbation theory in strong fields appears evenin non-constant backgrounds, that it occurs classically,and that it can be resolved by classical resummation. Quantum.
When intensity increases, quantum effectscan become relevant before large classical RR effects setin [32]. As we saw below (5), this can change the powerof ξ or χ in perturbative results, reducing the energy ra-diated compared to classical predictions [40], but it does not prevent perturbative breakdown, so that resumma-tion is still required. Comparing scales reveals that re-summation of classical effects becomes necessary, when αηξ ∼ , at far lower intensities ξ than required by theRN conjecture, α ( ηξ ) / ∼ (neglecting pulse length ef-fects in both cases). This implies that the contributionswhich fix unphysical behaviour in the QED Furry expan-sion must include at least those which fix its classicallimit. In this light we reconsider some of the observablesabove, but now in QED.The final momentum of an electron scattering off aplane wave is given in QED by the expectation value ofthe momentum operator ˆ P µ [41], (cid:104) ˆ P µ (cid:105) := (cid:88) f (cid:90) d P f (cid:16) π µ − σ µ + λn µ (cid:17) , (6)in which d P f is the differential probability to obtain afinal state f , π µ is the Lorentz force momentum of theelectron after traversing the wave [42], σ µ denotes thesum of momenta of any produced particles, and the coef-ficient λ is fixed by momentum conservation [43]. In theFurry expansion of (6), the O ( α ) contribution comesfrom elastic scattering ( σ = λ = 0) at tree level, yielding (cid:104) ˆ P µ (cid:105) = π µ as expected. Proceeding to higher orders in α , the RN literature has focussed on self-energy correc-tions to elastic scattering, known to contain terms scalinglike ( αχ / ) n at n -loops in a constant crossed field [13].These terms appear in (6) through the order α n coeffi-cients of π µ . The breakdown of the Furry expansion isclear already to first order in α . Including the one-loopelectron self-energy, one finds [43] (cid:104) ˆ P µ (cid:105) = π µ (cid:0) − mn · p αχ / + . . . (cid:1) , (7)with the correction dominating the leading term at large χ . However, as with any inclusive observable, there arefurther contributions to (6). At O ( α ) , one-photon emis-sion also contributes: ⟨ ˆ P µ ⟩ = π µ + 2 Im + ˆ P µ ˆ P µ Strikingly, this contribution removes exactly the αχ / -dependent self-energy terms from (cid:104) ˆ P µ (cid:105) . What remainshas a perturbative expansion in (cid:126) , and recovers classicalresults as (cid:126) → , including e.g. (4). (As shown in [43],this cancellation is required for the classical limit to existat all, as otherwise (cid:104) ˆ P µ (cid:105) would contain terms of order / (cid:126) .) Notably, we can argue that the cancellation of theself-energy terms must hold to all orders in α . Observethat the total coefficient of π µ in (7) is (cid:88) f (cid:90) d P f = P ( e − → anything ) = 1 . (8)Unitarity therefore removes, to all orders in perturbationtheory, the known loop contributions scaling with powersof αχ / . The same argument holds for other variablessuch as the quantum variance in the momentum, (cid:104) ˆ P (cid:105) −(cid:104) ˆ P (cid:105) , and the total outgoing momentum, obtained byremoving σ µ from (6).This points to a previously unexplored mechanism bywhich parts of the Furry expansion are brought back un-der control. It also reinforces our findings for the classicaltheory: some of the QED terms previously identified asleading to perturbative breakdown actually drop out, atleast for the observables considered above. Their RNscaling behaviour thus becomes irrelevant. The remain-ing terms, which contain the classical result (3), still needto be resummed. It is currently not known how to per-form this resummation in QED, as even going beyondknown O ( α ) results remains challenging. To see whatis involved, we sketch the calculation of O ( α ) contribu-tions to n · (cid:104) ˆ P (cid:105) from two-photon emission. We take theLCFA expression of [44, Eq. (59)], and insert the pho-ton lightfront momentum under the integral ( q + q inthe notation of [44]), which turns the probability intothe expectation value n · (cid:104) σ (cid:105) , see (6). We then examinethe (cid:126) → limit in order to compare with (4) expandedto second order in ∆ —note that in perturbation theory,both LAD and LL agree that the result is − ∆ + ∆ .Two-photon emission yields the expected classical term,albeit with a wrong coefficient, n · (cid:104) ˆ P (cid:105) n · p = 1 − ∆ + 729512 ∆ + . . . (9)as well as other classical terms, and terms of order / (cid:126) (plus a tower of (cid:126) corrections.) By the same logic asemployed at O ( α ) , the O ( α ) corrections to single photonemission [45], which we have not calculated here, andpossibly to the LCFA, will be responsible for removingunwanted terms and transforming the ∆ coefficient in(9) from about . to unity. Similarly subtle cancellationsalso occur in [46, 47]. This also confirms that photonloops contribute to the classical limit [36, 48]. Conclusions.
We have shown that the conjecturedbreakdown of perturbation theory in strong field QEDshould be expected to be a generic feature of backgroundfield perturbation theory. Furthermore, there is nothingintrinsically quantum about this result: the same prob-lems arise classically. We have also shown that resum-mation (achieved through the use of exact, or all-orders,results), can fix the unphysical behaviour of perturbationtheory, and that classical resummation becomes neces-sary at lower intensities than suggested by the RN con-jecture. The implication for strong field QED is that in order to obtain physically sensible results at high χ (reached through high ξ at fixed η [20, 21]) one shouldresum at least all classical contributions at each loop or-der. The relevant diagrams include loops and photonemissions, cf. [49]. We have also seen, for several natu-ral observables, that unitarity removes many previouslyconsidered diagrams scaling with powers of αχ / . In thecontext of the RN conjecture, it may thus be misleadingto look at only subsets of diagrams.Let us finally comment on the impact of resummationon photonic observables [50]. Starting with a probe pho-ton, a strong background field can cause helicity flip atone loop. In a constant crossed field this effect growslike αχ / at one loop, and higher loop corrections arebelieved to scale with higher powers of the same [13].It is a purely quantum effect, as is also seen by cuttingthe one-loop diagram to obtain the probability of pairproduction from a photon in a strong background. Ourclassical results are nevertheless relevant because higherorder corrections contain photon loops, which (as above)include classical contributions—their imaginary parts de-scribe real radiation emitted from the created pair [51],and we now know that such classical effects need to beresummed in the high-field limit. 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