Exact final state integrals for strong field QED

Exact final state integrals for strong field QED

Victor Dinu Department of Physics, University of Bucharest, P. O. Box MG-11, Măgurele 077125, Romania

This paper introduces the exact, analytic integration of all final state variables for the process of nonlinear Compton scattering in an intense plane wave laser pulse, improving upon a previously slow and challenging numerical approach. Computationally simple and insightful formulae are derived for the total scattering probability and mean energy-momentum of the emitted radiation. The general form of the effective mass appears explicitely. We consider several limiting cases, and present a quantum correction to Larmor’s formula. Numerical results are plotted and analysed in detail.

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The recent and predicted progress in laser technology leading to very high peak intensities justify the need for a better understanding of so-called nonlinear QED, describing phenomena occurring in fields so strong that their effects cannot be treated perturbatively.

Unfortunately, the complexity of the processes inside these ultra-intense laser beams has meant that several simplifications have had to be used to make practical computations feasible. The laser beam is usually supposed to be in a coherent state, which can be well approximated by a classical field. Due to the relatively small frequencies, unless massive particles of very high Lorentz factor are involved, quantum effects are small, so a fully classical description may often be justified. For instance, in discussing the scattering of an electron in a laser beam, one may consider Thompson scattering instead of its quantum counterpart, nonlinear Compton scattering (NLCS) piata (). The classical approximation allows for a realistic description of the laser field and the inclusion of radiation reaction (RR) diprr (), but is unable to describe important quantum effects, such as nonperturbative pair creation from vacuum pere (), the trident process trident (), or vacuum birefringence biref ().

A treatment of these processes in the framework of nonlinear QED, even in a semiclassical approach, has not yet been performed without further approximations, such as replacing the laser field by an idealized plane wave, thus allowing for analytical (Volkov) solutions to the Dirac equation. This disregards the strong spatial focusing of the beam needed to attain high intensities. In addition, for a long time, results were restricted to infinite, monochromatic plane waves, or even, giving up periodicity, to a crossed field model niki1 (); niki2 (). Only recently, the more realistic short pulse plane waves came into use boca1 (); SHK (); SK ().

In boca1 (); MDP (); boca3 (), the photonic and electronic distributions resulting from NLCS were described in detail for some model pulses. In principle, by integrating these distributions, the total probability and expectation values, such as for the emitted radiation’s energy-momentum, can be obtained. However, previous papers did never plot these quantities, because of the great numerical challenge posed by this task. By a change in integration order and a different regularization, allowing for all final state integrals to be performed analytically, we obtain formulae that are not only very easy to compute, but offer a new understanding of the scattering mechanism, the effective mass’s role, the time coherence of the process and its relation to the classicality parameter. An extensive numerical exploration of the results, their dependence on the various parameters, and their limits, now becomes possible. The method we develop is quite general and shall be applied to other strong field QED processes in a future paper.


For our starting point, notations and conventions, we refer the reader to unip (). In short, and are the initial electron and final photon four-momenta. We opt for natural units, so . Define , where and is some characteristic frequency of the wave. Let be an invariant lightfront coordinate, used to describe the plane wave pulse by the four-potential . By transversality, . We choose , , and use lightfront notations such as . The final results for probability/momentum will prove to be manifestly Lorentz invariant/covariant. While working with , only the gauge changes keeping -only dependent are allowed. But the end results can be expressed in terms of the classical velocity, so they are gauge invariant. For a long pulse, one may choose the carrier frequency and the peak value of the envelope of equal to one. To compare very short pulses one may prefer to fix and so that the pulse’s range is of order and . Whatever our choice, should offer a reliable description of the peak intensity, so .

Classical motion:

Let be the kinetic momentum of a classical electron moving in this plane wave field, where


If we set , then is indeed the velocity of the particle before meeting the wave. As opposed to unipolar pulses that permanently accelerate the particle unip (), for the usual whole-cycle pulses, and , as long as RR is neglected.

Scattering probability:

In the Furry picture, the total NLCS probability, averaged over the initial electron’s spin and summed over all possible spin/momentum states of the final particles, is:


where , and we denoted the moving average of a function by . The formal expression (2) was obtained from formula C1 in unip (), as follows. Instead of the expressions C3, we used the unregularized integrals , with . Explicitation of all in the quadratic form C2 led to the inner double integral in (2), by writing as . Then, the and integrals were performed, eliminating the four dimensional delta function and imposing the lightfront conservation laws , . In the exponent the average of the classical momentum was identified. A change of variable from to led to the final result. See also boca1 (); SHK (); SK (). If we want to compute them first, all in the generic case, or at least , need to be regulated, damping the oscillations of the integrand with a convergence factor such as , , that can be discarded after a partial integration restricts to the length of the pulse boca1 (); unip ().

At first glance, in writing (2) we have added to the numerical complexity, constructing a double integral out of simple ones. But, in fact, by a change of quadrature order, the analytical integration over , and leaves us with only two easy integrals, instead of the initial four. In addition, we get rid of the rapid oscillations encountered when computing . Expressing in lightfront coordinates, we notice the integral over is Gaussian, if regulated by replacing in the exponent the factor by , then taking . The previous damping factor is now superfluous. Introducing the invariant number , the result is




We recognize the effective mass , first introduced by Kibble, that appears in the Volkov propagator kibble () and the Wigner function wig1 (); wig2 ():


A Lorentz and gauge invariant, depends on the averaging interval. For a whole-cycle finite pulse, the mass shift vanishes when . We now use the relation and the fact that is an even function of , so the result is indeed real. Changing variables from , to and , and noticing that , we get


A new analytical integration leads to:


where, in terms of trigonometric integrals,

Notice that the Lorentz invariant (7) depends on , and 4-products of the values of the function , but not on . Interestingly, we can rewrite the probability in terms of the classical velocity (1) as function of the proper time, eliminating all reference to the driving field and proving gauge invariance. Could the result be generalized to an arbitrary, not necessarily plane, wave? It is hard to answer, because of the many very different trajectories allowed inside a general field. The attractiveness of a plane wave derives from the simplicity of the law of motion it entails. The electron’s motion always looks the same, regardless of the initial position. A quantum computation for a general field would require the use of a wavepacket with some initial average position and momentum that only in the limit relates to a particular classical motion.

Radiated energy-momentum:

The same procedure can be applied to compute the expectation value of the emitted photon’s momentum,


Following the same method as for the probability, we are left with the manifestly covariant double integral




and the new functions used are

Notice that was computed with a distribution normalized not to unity, but to . Therefore it is not the average value of the momentum of one photon, assuming an emission has taken place, but the expected outcome per incident electron, counting both NLCS and ellastic scattering events. To compute the former, conditional probabilities are needed. This amounts to dividing the expectation value by . The standard deviation of or any of its higher moments can similarly be computed.

Periodic and adiabatic limits

Even though derived for a pulse, our formulae can provide the emission probability and radiated energy-momentum per cycle in the case of an infinite, periodic plane wave, described by . They can be obtained by truncating the pulse to a finite number of cycles , and considering the quickly reached limit of and . The results, that we shall denote by and , are given by (7) and (9) with the integral restricted to one period. A circularly polarized monochromatic wave provides a simple example, as is independent of , so . Consider now the modulated wave,


where is a periodic function, e.g. monochromatic, the envelope is a smooth function vanishing at infinity and controls the pulse length. Provided , (7) and (9) are practically proportional to , being well approximated by


Of particular interest are the limiting cases of small or . With today’s technology, is around the order of and only for the optical range a large is possible. If, for instance, , an electron energy of at least   is needed for to reach unity.

Perturbative limit

The results can be expressed as integrals containing the field’s Fourier transform and the Klein-Nishima probability of linear Compton unip (). However, to emphasize the role of , we prefer to work with a quadratic function related to the potential’s auto-correlation,


The weak field limits () of (7) and (9) are:


where , , and are the universal, positive functions,

Since is increasing for , 16 decreases with , having the classical upper bound

Classical limit:

This is expected when , since is proportional to , a fact obscured by the use of natural units. That is, the laser photon energies are much smaller than  in the electron’s rest frame. Now the major contribution to the integral (9) comes from small . By Taylor expansions, such as


one obtains the approximation




No classical equivalent is found for the second term in the integrand of (9), negligible in this limit. An equivalent form of (19) can be written by expressing everything in terms of a classical particle’s velocity as function of proper time,


When not only , but also is very small, equation (21) further reduces to:


The same result arises directly from classical electrodynamics, if one neglects radiation reaction. For , (22) is just the time integrated Larmor’s formula.

Figure 1: (color online) The function describing the strong field correction to Larmor’s formula

The function (20) is decreasing for . As shown in Fig. 1, it departs very quickly from its upper bound . The deviation is noticeable even for . It follows that a small doesn’t necessarily make classical Thompson scattering a good model. For strong fields, much less power is radiated than Larmor predicts. In CED, the spectral and angular distribution is also a double integral, but an interesting cancelling of interference terms leaves a single one after the frequency is integrated away. The transfer of energy and momentum to the field is well defined at each moment in time, there are no quantum uncertainties. Interestingly, this form of decoherence is also shown by the better approximation (21), that looks misleadingly classical, though the argument of the correction is in fact proportional to . Heuristically, is a coherence lighfront time scale. When small enough, it allows for a time-incoherent model of emission, but at high intensities the mass shift implied by (18) cannot be neglected in this coherence interval, hence the aforementioned correction. This happens when the peak electric field in the electron rest frame approaches the Schwinger critical value . Formula (21) could provide a general improvement to Larmor’s, valid in an arbitrary driving field, whose frequencies in the rest frame of the electron are similarly low compared to , so the emission can be viewed as incoherent in time, hence the product of the classical motion of a charged particle. This should not be confused to the second term in the radiated energy’s expansion in powers of , at fixed , that is non-local in time, as discussed in larm1 (); larm2 (). As for the total scattering probability, by an asymptotic expansion of the functions one gets the limit


In this case no decoherence is observed, as photon emission probability is not a classical concept.

Numerical results:

We start illustrating our results using a one-cycle, linearly polarized pulse, characterized by a Gaussian potential:

Figure 2: (color online) The NLCS probability (7) is plotted as function of , for the pulse (24) and , . The dashed line is the classical limit (23). The inset details the perturbative region of small .

The total NLCS probability is shown in Fig.2. The quadratic increase from the perturbative region, shown in detail in the upper left corner, quickly slows down as grows past unity. The higher the parameter , the lower is.  In general, (7) boundlessly grows with the length/intensity of the pulse. Even for one as short as (24) and the experimentally attainable , the result can easily surpass unity. In piazza (), this possibility was noticed, interpreted as a sign that multiphoton emission cannot then be neglected, and a re-normalization of the whole series of n-photon NLCS probabilities was suggested. Moreover, for a unipolar pulse, (7) shows the logarithmic IR divergence typical of Bremsstrahlung unip (); greger (). These problems can be dealt with by including a one loop self-energy diagram, that adds nothing to (9), but does contribute to the expectation value of the final electron’s momentum, even in the whole-cycle case rrqed (). For the general theory of the cancellation between real and virtual photon IR divergences, see FSY ().

Figure 3: (color online) The expectation value of , as a multiple of the electron rest energy, plotted as function of , for the pulse (24). The dotted line is the Thompson limit (22), while the dashed one is the better approximation (21).

To plot formula (9), we need more than just the invariants , . The frequency of any available laser that can reach the nonlinear regime is around the optical range. Let’s assume . It remains to know the incidence, that we choose head-on. While experimentally difficult, this gives the largest for a given pulse and electron beam. In Fig. 3 a comparison is drawn between the expectation value of the radiated energy and its two incoherent approximations. Both (22) and (21) overestimate (9), but the latter is a much closer match.

Let us now consider a modulated harmonic wave,


where describes the polarization state, is known as carrier-envelope phase (CEP) and controls the pulse length. We present computations for the symmetric, Gaussian envelope . For the carrier-envelope model to make sense, we assume is larger than, say, . Fig. 4, shows contour plots of the total probability (7) as function of pulse peak potential/length, for linear polarization and various values of the nonclassicality parameter . Again, we find that in experimentally realistic conditions, our result can easily surpass unity, signalling the need for taking into account higher order Feynman diagrams. We are interested in the region where stays well below unity, so the model can be given credit. This region grows with , once it gets close to unity. The influence of the polarization state and the convergence towards the adiabatic limit (13), as the pulse length increases, are shown in Fig. 5. The CEP has a very small impact on for this smooth potential.

Figure 4: (color online) Contour plot of the NLCS probability (7), in terms of the duration/strength of the pulse (25), for linear polarization and . The contours correspond to
Figure 5: (color online) The probability (7) divided by the length of the pulse (25), as function of polarization state, for , and . The curves correspond, in decreasing order, to .

In order to study (9), we now choose an optical frequency of , and specify the initial electron momentum, setting its Lorentz factor and polar angles and . We have already looked at the energy for a one-cycle pulse, so we consider a longer one. In Fig. 6 a plot of the radiated energy’s dependence on the initial electron energy is shown for the pulse (25) with , linear polarization and . In the classical regime, after becomes much greater than , the growth is quadratic, but at even higher energies quantum effects slow it down. For a while, it is well described by the local in time, incoherent approximation (21). Then, as goes past unity, we reach a highly nonlocal quantum radiation regime. It is interesting to notice that, at very high energies, the emission becomes stronger for lower incidence angles than for higher ones, because the corresponding smaller implies weaker quantum effects. Fig. 7 shows the increase of as the polarization is changed from linear towards circular. The influence of on (9) is due to the term in (1). For the example considered in Fig. 6, both this dependence and the one on the CEP are extremely weak. They can be noticeable for a very short pulse and small , as seen from the example in Fig. 8. As to the vector , for large , it practically has the direction of and the length , in the laboratory frame all emission being concentrated inside a very narrow cone. Significant differences arise only at low energies, where quantum effects are, however, small.

Figure 6: (color online) A plot of the expectation value versus the initial electron’s Lorentz factor for the pulse (25) with , and . The incidence angles are and . The head-on result is much larger at low energies, surpassed by the others at very high ones. The dotted/dashed lines are the approximations (22) and (21).
Figure 7: (color online) The relative increase in the average radiated energy when the polarization is changed from linear to circular (, solid line) / elliptic (, dashed line). The pulse is (25), , , , and , , (growing order at left)
Figure 8: (color online) The radiated energy’s dependence on the angle , for , and the pulse (25), with , and CEP (growing order at left). Solid/dotted stand for circular/linear polarization.


We have found a way to analytically integrate all final state variables out of the NLCS probability and expectation values. This not only allowed for the first time their detailed numerical exploration, by saving huge computational expense, but also revealed new insights into the structure of scattering processes in strong fields. We shed light on the role of the effective mass and the emission’s coherence in time. Simple results were found for the monochromatic, perturbative and classical limits. We derived a strong field correction to Larmor’s formula, arising from the mass shift. Computations performed under realistic conditions showed its usefulness, but also found large values for the probability, even surpassing unity. This suggests multiple scatterings and radiative corrections need to be considered. In a future paper, our method will be applied to these, as well as to other strong field processes.


The author thanks V. Florescu, A. Ilderton and G. Torgrimsson for useful discussions, and ESF-RNP-SILMI for support in attending conference FILMITh, Garching 2012


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