QQuantum regime of laser-matter interactions atextreme intensities
Alexander Fedotov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute),115409 Kashirskoe sh. 31, Moscow, Russia
DOI: w ill be assigned A survey of physical parameters and of a ladder of various regimes of laser-matter inter-actions at extreme intensities is given. Special emphases is made on three selected topics:(i) qualitative derivation of the scalings for probability rates of the basic processes; (ii)self-sustained cascades (which may dominate at the intensity levels attainable with nextgeneration laser facilities); and (iii) possibility of breaking down the Intense Field QEDapproach for ultrarelativistic electrons and high-energy photons at certain intensity level.
One of the notable trends of the last decades was unprecedented growth of laser power and in-tensity accessible for experimental research. As of now, the most striking progress was achievedin the (near-)optical range, which corresponds to a typical wavelength λ (cid:39) µ m, frequency ν = c/λ (cid:39) Hz and photon energy (cid:126) ω (cid:39) W L (cid:39) . τ L (cid:39) . As a consequence, the peak laserpower is huge, P L = W L τ L (cid:39) J10 − s = 10 W ≡ , comparable to the net power produced by a large country. The average power, though, of courseremains low due to a small repetition rate ν R (cid:39) − ÷
10 Hz. Assuming that the pulses arefocused to diffraction limit, the expected peak intensity is estimated by (cf. [2]): I L = P L R (cid:39) P L λ (cid:39) W(10 − cm) (cid:39) W/cm . Furthermore, the ongoing construction or upgrade at such facilities as CLF (UK), Apollon(France), PEARL (Russia), ELI Beamlines (Czech Republic), ELI-NP (Romania), EP-OPAL(USA), QiangGuang (China) and alike is very promising in attaining laser intensities of theorder of 10 W/cm or even higher in the nearest future. Moreover, far-reaching exawatt-class facilities such as ELI [3] and XCELS [4] aiming at achieving the intensity (cid:38) W/cm are being also planned already. All this brings reasonable prospects on further advance ofexperimental studies of a variety of yet unexplored phenomena of laser-matter interactions atsuch extreme intensities. ≡ − s – such small durations became attainable after invention of Chirped Pulse Amplification [1]. SFHQ 2016 a r X i v : . [ phy s i c s . op ti c s ] D ec Basic parameters and physical regimes
Let us discuss the key parameters of the theory, and the characteristic intensity levels corre-sponding to the various non-perturbative regimes of high-intensity laser-matter interaction tobe compared to the experimentally accessed or expected values from the Introduction.Perhaps the main element of the strong field approach is a concept of external (classical)background field A µ . Obviously, this implies that the number of photons in relevant field modesis not changed essentially by absorption or scattering events, in particular is huge enough.Assuming ω is characteristic frequency and estimating laser pulse energy as W L (cid:39) E + H π V (cid:39) E π V , we arrive at ¯ N γ (cid:39) W L / (cid:126) ω (cid:29)
1, or E (cid:29) (cid:112) (cid:126) ω/V (cid:29)
1. This criterion is always satisfiedin cases of either ω = 0 (static field) or V = ∞ (infinitely extended field), but both areformally unphysical. For a tightly focused laser pulse we may assume V (cid:39) λ which results in E (cid:29) ω (cid:112) (cid:126) /c . Then the limitation on corresponding intensity reads I L = c π E (cid:38) W/cm and is almost ever satisfied with huge margins (it is perhaps enough to mention that typically¯ N γ (cid:39) cm − ), hence we adopt it wherever possible.Another important and instructive example is a strong field concept in atomic physics. Theatomic length (Bohr radius) and energy (Rydberg) are l at = (cid:126) /Zme = 5 . × − cm and E at (cid:39) Ze /l at = mZ e / (cid:126) (cid:39) Z = 1). In this context the field can be considered ‘strong’ if the work itproduces across the size of an atom eEl at (cid:38) E at , resulting in E (cid:38) E at ≡ Ze/l = m Z e / (cid:126) =5 × V/cm and I L (cid:38) c π E = 3 × W/cm . For such laser intensities material targets arerapidly ionized, thus turning into plasma. Laser-plasma interactions are usually simulated withParticle-In-Cell (PIC) codes.One of the most important parameters arises in a criterion of whether free electron motiondriven by laser field is relativistic or not. Consider the classical equation of electron motion d(cid:126)pdt = e (cid:18) (cid:126)E + (cid:126)vc × (cid:126)H (cid:19) . (1)From Eq. (1) the momentum of laser driven electron quiver oscillation can be estimated as p ⊥ (cid:39) eE/ω , hence the motion is relativistic if a ≡ p ⊥ /mc (cid:39) eE/mωc (cid:38) γ (cid:39) a (cid:38) E (cid:38) E rel ≡ mωc/e , or I L (cid:38) c π E (cid:39) × W/cm . Theresulting dimensionless parameter a admits a Lorentz- and gauge-invariant definition for aplane wave field, a = emc (cid:112) −A µ A µ , and a number of alternative interpretations. For example,one point is that for a (cid:38) a is often called the (dimensionless) classicalnonlinearity parameter. Due to its importance, it is also often used in laser physics communityto quantify the field strength and intensity (the latter via a ≈ × − λ [ µ m] (cid:113) I L [W/cm ]).In these terms the currently attained intensity level corresponds to a (cid:39) .Yet another interpretation arises in the framework of QED, where motion of an electronin a given external field A µ is described by a sum of diagrams shown in Fig. 1a. Here eachvertex corresponds to the factor − ieγ µ A µ , while each thin electron line (free propagator) to iS = γp − m (cid:39) /m . Hence, the expansion parameter of the QED perturbation theory is givenby e A /m , i.e. by the same parameter a as above. In this context, a (cid:38) A µ interaction regime. This can This is indeed invariant under gauge transformations δA µ ∝ k µ of a plane wave. SFHQ 2016 + + + ... (a) = + (b)
Figure 1: Exact propagator in external field (a) and a closed-form equation it obeys (b).be understood in a pictorial way by noticing that in presence of a large number of backgroundphotons in the relevant field mode the interaction vertex weight ( ∼ √ α in vacuum) shouldacquire an additional Bose factor (cid:112) ¯ N γ ( − ≈ (cid:112) ¯ N γ , √ α → √ α × (cid:113) ¯ N γ (cid:39) √ α × (cid:113) l C × λ × ¯ n γ (cid:39) e √ (cid:126) c × (cid:115)(cid:18) (cid:126) mc (cid:19) × πcω × E π (cid:126) ω (cid:39) a , (2)and hence is effectively replaced by a . Here we assumed that the electron probes the photonscontained in an effective tubular interaction volume of Compton width (cid:39) l C = (cid:126) /mc aroundthe classical electron trajectory of length λ . Note that (cid:126) totally cancels, thus assuring that a ,even though being an expansion parameter of QED perturbation theory, is nevertheless still ofpurely classical origin.Since the focus of the rest of the paper is on laser-matter interactions at the high-power laserfacilities, we always assume below that a (cid:29)
1. Under such condition all-orders summationwith respect to interaction with external field should be done (see Fig. 1a). This is naturallydone within the approach which I call the Intense Field QED (IFQED). Namely, it is easyto see that the ‘exact’ (with respect to interaction with external field), or ‘dressed’, electronpropagator obeys the equation shown schematically in Fig. 1b. The rules for computationof the QED amplitudes are then formulated as in ordinary QED, but with free fermion linesand propagators replaced with the dressed ones. This approach was tested experimentallymany times indirectly in atomic physics as well as directly in the late 90’s famous E144 SLACexperiment [5, 6].When laser intensity is further increasing, novel physical regimes should show up. Forexample, under the condition a i = ( Ze ) E/M ωc (cid:38)
1, or a = eE/mωc (cid:38) M/Zm (cid:39) M p /m ∼ × , the ions should also become relativistic. The corresponding intensity is I L (cid:38) × W/cm and is far beyond the currently attainable level. Another complication shouldshow up at even lower intensities. Assuming γ (cid:39) a and E ⊥ , H (cid:39) E , the radiation reactionforce (cid:126)F rad (cid:39) − e (cid:16) (cid:126)E + (cid:126)vc × (cid:126)H (cid:17) ⊥ γ m c (cid:126)v, (3)acting on electron, becomes (cid:38) eE for E (cid:38) (cid:0) m ω c /e (cid:1) / , or a (cid:38) (cid:0) mc /e ω (cid:1) / (cid:39) I L (cid:38) × W/cm . In this regime one should account for classical radiationreaction in simulations of laser-matter interactions. Accounting for both effects requires justadequate correction of the developed PIC codes, and this is now indeed a hot topic in the laserphysics community.Self-action of an electron was at the focus of classical theory for decades and, as well known,was one of original motivations for invention of quantum theory. Besides radiation reactionforce, it also implies existence of electromagnetic contribution into electron mass, classically of SFHQ 2016 E em ( r ) (cid:39) e /r , where r is the electron radius. In the limit of a pointlike electron( r →
0) this contribution diverges, but paradoxes appear in classical theory already when themass correction E em ( r ) (cid:38) mc , i.e. for r (cid:46) r e ≡ e /mc ( r e is called the ‘classical electronradius’). The radiation reaction force (3) may produce across the distance r e the work (cid:38) mc if the field strength in a proper reference frame exceeds E P (cid:38) E cr ≡ m c /e . These conditionsare considered as limits of applicability of Classical Electrodynamics.Regime a (cid:28) a (cid:38) χ (cid:28) χ (cid:38) (a) I L [W/cm ] PHYSICAL REGIME5 × (?) Sauter-Schwinger QED crit-ical field5 × Relativistic ions2 . × Massive self-sustained QEDcascades10 Quantum radiation reaction,pair photoproduction ( χ (cid:38) × Classical radiation reaction5 × Mid’2010s state-of-the-art × Relativistic electrons ( a (cid:38) × Strong field of atomicphysics (rapid ionizationand plasma formation)10 External (given classicalbackground) field concept < Week field quantum regime (b)
Table 1: Classification of physical regimes oflaser-matter interactions according to the val-ues of the key parameters a and χ (a) andthe ladder of various regimes successively at-tainable upon increasing laser intensity (b).Domain corresponding to quantum regime atextreme intensity is colored in gray. However, things are quite different in quan-tum theory (QED), which introduces a novelCompton scale l C = (cid:126) /mc ≈ r e . Eventhough electron still formally remains point-like (as required by relativity), it turns out itcannot be localized at size smaller that l C be-cause of uncontrollable disturbance of vacuum(pair creation). For example, the average of thecharge density operator in one-electron state isdelocalized to a size (cid:39) l C . As a consequence,the divergency of electron self energy is par-tially canceled by contribution of vacuum vir-tual pairs and becomes much weaker, E em ( r ) (cid:39) ( e /l C ) log ( l C /r ). In this sense, a pointlikecharge is effectively replaced by a cloud of vir-tual pairs of size (cid:39) l C = 1 /m (cid:39) × − cm.That is why we adopted the Compton lengthas the effective width of electron ‘trajectory’in our above estimation (2). Unlike in Classi-cal Electrodynamics, radiation reaction in QEDcan never outreach the Lorentz force (thus theknown paradoxes of the former are avoided),however the work produced by the field acrossthe distance l C can still exceed the rest en-ergy, eE S l C (cid:39) mc . The required field strength E S ≡ m c /e (cid:126) = 1 . × V/cm (note that E S (cid:39) E cr ) is called the Sauter-Schwinger,or QED critical field, and corresponds to laserintensity I L = c π E (cid:39) × W/cm , whichis far beyond the capabilities of the existingor prospective laser facilities. Besides laserphysics, the electric and magnetic fields (cid:38) E S may arise in heavy ion collisions, magneticfields H (cid:39) m c /e (cid:126) (cid:39) × G are also an-ticipated around compact astrophysical objects(magnetars).Since the electromagnetic field strengths arenot Lorentz invariant, it may seem not obvious for which reference frame the condition
E, H (cid:38) Since magnetic fields do not produce work, for them the criticality condition is formulated e.g. by demandingthe principle Landau level to be relativistic. SFHQ 2016 S should be formulated. In fact this criterion should be formulated in terms of Lorentzinvariants. For example, in absence of particles (i.e. in vacuum) the only field invariants are E − H = − F µν F µν and (cid:126)E · (cid:126)H = (cid:15) µνλ κ F µν F λ κ . In such a case the field strengths shouldactually exceed E S in a reference frame, for which either electric or magnetic fields vanishes orthey are parallel. However, in presence of a particle with 4-momentum p µ one extra Lorentzinvariant, usually called the dynamical quantum parameter, can be defined: χ = e (cid:126) m c (cid:113) − ( F µν p ν ) = γ (cid:114)(cid:16) (cid:126)E + (cid:126)v × (cid:126)Hc (cid:17) − ( (cid:126)v · (cid:126)E ) c E S . (4)It actually acquires several equivalent native physical meanings, e.g. in case of electron aratio of the electric field strength to E S in its proper (rest) frame, or proper acceleration inCompton units. Typically, in the lab frame E (cid:107) ∼ E ⊥ , where the components are denotedaccording to direction of particle momentum. Hence for ultrarelativistic particle E P (cid:107) ∼ E (cid:107) , E P ⊥ ∼ γE ⊥ (cid:29) E P (cid:107) and χ = E P /E S (cid:39) γE ⊥ /E S . Yet another interpretation stems from the factthat for radiating electron with χ (cid:46) χ (cid:38) χ (cid:38) a = eE/mωc (classical nonlinearity pa-rameter) and χ (cid:39) γE ⊥ /E S (dynamical quantum parameter) are independent and allow forclassification of various regimes of laser-matter interactions, see Table 1a. For instance, inSLAC experiment both a , χ ∼
1. However, if the electrons are not accelerated by exter-nal sources but driven by the field then, assuming E ⊥ ∼ E and γ ∼ a (cid:29)
1, we arrive at χ (cid:39) ( (cid:126) ω/mc ) a (cid:38) a (cid:38) (cid:112) mc / (cid:126) ω (cid:39) I L (cid:38) W/cm . The list of various laser-matter interaction regimes discussed above,along with the required laser intensities, given relative to the present day state-of-the-art level,is summarized in Table 1b. Further details can be found in the reviews [7–9]. Within the framework of IFQED approach, calculation of probabilities of a process is reducedto calculation of diagrams with ‘exact’, or ‘dressed’ electron external lines and propagators (seeFig. 1a), determined by equation in Fig. 1b. The latter equation can be solved exactly for justa few particular external field backgrounds (e.g. constant field, plane wave, Coulomb field),but even when it admits exact solution, application of the method usually results in extremelybulky intermediate calculations [10]. But as a rule, qualitative considerations may result ina deeper insight into the problem. Here I am going to demonstrate how at least some of theknown asymptotic expressions for probability rates of basic QED processes in strong externalfield could receive a simple-man derivation, based exclusively on kinematical and dimensionalarguments together with the uncertainty principle (see [11] for a more detailed presentation).In presence of external field the QED processes can be subdivided into field-modified (i.e.those which occur even in absence of the field) and field-induced. Let us focus below on the field- It should be stressed that the quantum regime of interaction with strong external classical background under discussion arises due to recoil in essentially multiphoton radiation processes (e.g. hard photon emission),as opposed to the (completely different!) week field quantum regime on bottom of Table 1b, where quantumeffects arise due to absorption or emission of individual laser photons.
SFHQ 2016 p k θϑ E (a) k p p θϑ E (b) Figure 2: Basic IFQED processes: photon emission (a) and pair photoproduction (b).induced processes only. The diagrams for the basic processes of this kind, single photon emissionand pair photoproduction, are shown in Fig. 2. For such sort of processes we can introducethe energy lack ∆ ε = (cid:80) ε f − (cid:80) ε i > t e ) is the time it takes for the field toprovide the amount of work required for a process to occur (symbolically, e (cid:82) t e (cid:126)E · d(cid:126)s (cid:39) ∆ ε ),and the second one ( t q (cid:39) / ∆ ε ) is the time for which according to the uncertainty principlethe final state can endure as a virtual one. As we will shortly observe, it turns out that in arelativistically strong laser field of our interest ( a (cid:29)
1) both these timescales are much smallerthan the laser period ∼ /ω , hence the laser field can be considered as locally constant . Inaddition, we also assume whenever possible that the particles are ultrarelativistic and movingtransversely with respect to the field, in such a case the field can be also considered crossed ( (cid:126)E ⊥ (cid:126)H and E = H ), and the total probabilities depend exclusively on the quantum dynamicalparameter(s) χ i . Furthermore, since any field is all the same equivalent to a constant crossedfield, we can replace it e.g. with the constant purely electric field directed orthogonally to themomentum of the initial particle. Then, by picking up the time gauge (cid:126)A ( t ) = − (cid:126)Et , the energiesof the charged particles can be written as ε (cid:126)p ( t ) = [( (cid:126)p − e (cid:126)A ( t )) + m ] / .Obviously, if t e (cid:29) t q then the process is (quasi-)classical, in particular its probability shouldbe exponentially suppressed. The probability of the process under such conditions can beestimated by the quasiclassical ‘imaginary time’ technique, W i → f ∝ (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − (cid:90) t ∗ ∆ ε ( it (cid:48) ) dt (cid:48) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) , (5)where t ∗ is determined by the conditions of energy-momentum conservation ∆ ε ( it ∗ ) = 0, ∆ (cid:126)p =0. It turns out that typically t ∗ (cid:39) t e , so that W i → f = O ( e − t e /t q ). If, on the contrary, t e (cid:28) t q ,then the process is essentially quantum and unsuppressed.Let us first demonstrate how the scheme works using a popular example of electron-positronpair creation from vacuum. For this case ∆ ε = 2 m , t e (cid:39) m/eE and t q (cid:39) /m . Note that for a (cid:29) t e = 1 /ωa (cid:28) /ω , thus the locally constant field approximation should work.For E (cid:28) E S = m /e we have t e (cid:29) t q , so that the process should be suppressed. The expectedsuppression factor is e − t e /t q (cid:39) e − E S /E . More precisely, assuming (for the sake of simplicityonly) (cid:126)p ⊥ = 0 we obtain: ∆ ε ( t ) = 2 √ m + e E t , ∆ ε ( it ∗ ) = 0, t ∗ = m/eE = t e (as announced Here (cid:126)p ⊥ denotes the component of the momentum which is orthogonal to the field. SFHQ 2016 bove), and W e − e + = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:40) − (cid:90) m/eE (cid:112) m − e E t (cid:48) dt (cid:48) (cid:41)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e − πm /eE , (6)in perfect agreement with the exact result of Sauter, Heisenberg-Euler, Schwinger and others.As a second illustration, consider pair photoproduction by a hard photon (see Fig. 2b). Forthis case, the energy lack can be written as∆ ε ( t ) = ε (cid:126)p (cid:48) ( t ) + ε (cid:126)p ( t ) − k ≈ (cid:112) ( k − p ) + e E t + m + (cid:112) p + e E t + m − k ≈≈ k − p + e E t + m k − p ) + p + e E t + m p − k = k (cid:0) e E t + m (cid:1) p ( k − p ) (cid:38) (cid:0) e E t + m (cid:1) k , (7)where the minimum is attained for p = p (cid:48) = k/
2. In the weak field limit the second term in thenumerator ( m ) should dominate over the first term ( e E t ) so that ∆ ε (cid:39) m /k . It is wellknown that for ultrarelativistic kinematics θ, ϑ (cid:39) m/k (cid:28)
1. Hence the characteristic scales t q (cid:39) / ∆ ε (cid:39) k/m (cid:28) t e (cid:39) ∆ ε/eEϑ (cid:39) m/eE, if the dynamical quantum parameter of the initial hard photon κ = eEkm (cid:46)
1. According to theabove in such a regime we expect that W e − e + = O ( e − / κ ). Indeed, looking for a stationarypoint we obtain ∆ ε ( it ∗ ) = 2 (cid:0) − e E t ∗ + m (cid:1) /k = 0, t ∗ = m/eE = t e , and W e − e + = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:32) − (cid:90) m/eE (cid:0) − e E t + m (cid:1) k dt (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e − m / eEk (cid:12)(cid:12)(cid:12) = e − / κ . (8)This is again in accordance with exact computations, but [unlike Eq. (6)] we are unaware ofother simple-man derivations of this result in previous literature.In the opposite (quantum, κ (cid:29)
1) strong field limit, according to Eq. (7), ∆ ε ( t ) (cid:39) e E t /k and angular spread arises mostly due to contortion of electron and positron trajectories by thefield, θ ( t ) , ϑ ( t ) (cid:39) eEt/k (cid:28)
1. Hence eEϑ ( t ) t (cid:39) ∆ ε ( t ) identically, i.e. t e is not fixed butarbitrary. At the same time, from its definition t q (cid:39) / ∆ ε ( t q ) (cid:39) k/e E t q we obtain t q (cid:39) (cid:18) ke E (cid:19) / ≡ meE κ / ≡ km κ / . (9)Hence, for κ (cid:29) t q , on dimensional grounds we have W e − e + ( κ (cid:29) (cid:39) e t q ∼ e m k κ / , W γ ( χ (cid:29) (cid:39) e m p χ / , (10)where the second formula gives the probability W γ of hard photon emission by electron (seeFig. 2a) for χ (cid:29)
1, as ultrarelativistic kinematics for both processes is exactly the same as soonas electron mass is neglected. These results are (up to numerical factor ∼
1) also in perfectagreement with exact calculations.
SFHQ 2016 ε of seedparticle external field (donates energy byparticles acceleration)Multiplicity N e − e + limited by ε exponentially increases ( ∝ e Γ t ),potentially up to macroscopic valueProceeds until. . . . . . secondary particleslose energy and χ • . . . particles escape (?); • . . . field depletion (?); • . . . thermolization (?);Similar to: Extensive Air Showers gas or dielectric dischargeTable 2: S(hower)-type vs. A(valanche)-type QED cascades in external field.
In case seed particle in a strong field has χ (cid:38)
1, it can emit photons with κ ∼
1, whichare in turn capable for pair photoproduction. In such a case these events may follow oneby one, thus forming a chain called a QED cascade. In principle, this is very similar to thefamiliar cascades generated by high energy particles in matter due to Bremsstrahlung and pairphotoproduction on nuclei (e.g., as in Extensive Air Showers). However, a novel distinctivefeature for cascades in a strong laser field is that such a field may not serve only a target, butin general is also capable for acceleration of charged particles, thus donating them energy andmaking the cascade self-sustained. The principle differences between these two types of cascades(which we abbreviate S- and A-type [12]) are summarized in Table 2. Here our point is thatproduction of A-type cascades starting from a certain intensity level may dominate in laser-matter interactions resulting in creation of macroscopic amount of pairs and hard photons,as shown in Table 1b, thus totally changing the landscape of laser-matter interactions [13].Moreover, since this process (at least in most probable scenarios) leads to depletion of externalfield, it can also prevent practical attainability of the Sauter-Schwinger critical field with opticallasers [14] (this point is indicated in Table 1b by a question mark).During the process of hard photon emission or pair photoproduction the values of energy ε and dynamical quantum parameter χ of a parental particle are partitioned among the secondaryones. As stressed above, the main distinctive feature of a self-sustained (A-type) cascade againstthe ordinary S-type cascade is the ‘acceleration stage’ where these ε and even more importantly χ are then restored back by the field before the next QED event takes place. In contrast toconsideration of a QED process itself, at this stage the difference between the actual field and aconstant crossed one should be necessarily taken into account (that is why we call our constantand crossed field approximation local ), as otherwise χ would be conserved exactly. At the sametime, motion of ultrarelativistic particles in between the QED events may be still consideredclassically . Concerning a problem of ε and χ evolution along a particle trajectory in a genericfield configuration, a non-trivial result is that (apart from a few artificial particular cases, e.g.linearly polarized purely electric field or a running plane wave field) for an initially slow particle It turns out that motion in a constant crossed field is classical exactly. In a general setting, for a subcritical( E (cid:28) E S ) electric field quasiclassical approximation breaks down only near the turning points, where particlesare slow. Ultrarelativistic motion in a subcritical magnetic field is also classical since particle occupies highLandau levels. SFHQ 2016 .0 0.5 1.0 1.5 2.0 t ( (cid:0)(cid:1) ) N e + e (cid:2) Run 1Run 2 Run 3 (a) t ( (cid:0)(cid:1) ) (cid:2) e without QED effects Run 1Run 2 Run 3 (b) t ( (cid:0)(cid:1) ) < (cid:2) e > Run 1Run 2Run 3 (c)
Figure 3: Simulation campaign of QED cascade generation: evolution of cascade multiplicity(a); parameter χ of the seed electron (b); and parameter χ averaged over the cascade (c) forthree independent Monte Carlo runs @ a = 2 × (from [16]).at time scales m/eE (cid:28) t (cid:28) /ω we always have ε ( t ) = mγ ( t ) (cid:39) eEt, E ⊥ ( t ) (cid:39) Eωt, χ ( t ) (cid:39) E ⊥ ( t ) E S γ ( t ) (cid:39) mc (cid:126) ω (cid:18) EE S (cid:19) ( ωt ) . (11)Hence it takes for such a particle t acc (cid:39) αE S /ωE to gain χ ∼
1, where for illustration purposewe have skipped the accidental dimensionless factor (cid:112) α mc / (cid:126) ω (cid:39) . t e (cid:39) /W γ ( χ ( t e ))for a free path time (i.e. typical time between the QED events), in the same manner we obtain t e (cid:39) ω (cid:18) αE S E (cid:19) / , χ ( t e ) (cid:39) (cid:18) EαE S (cid:19) / . (12)Obviously, for E (cid:38) αE S (or I (cid:38) α I S ∼ . × W/cm ) the time scales hierarchy t acc (cid:28) t e (cid:28) /ω is established, meaning that (i) photons are mostly emitted when χ ∼ /ω of the laser field. These are exactly the conditions singling out a QED cascade.Since due to acceleration the parameters of the particles in each generation remain the sameon average, the cascade multiplicity should grow exponentially N e − e + (cid:39) e Γ t . This qualitativepicture is fully supported by Monte Carlo simulations [15–17], see Fig. 3. In particular, onecan observe in Fig. 3c how the average parameter χ tends to a definitive value, which is in factin fairly good agreement with the estimate (12). The same agreement was observed for otherrelevant quantities, including the increment Γ (cid:39) /t e .The final stage of a self-sustained (A-type) cascade still remains poorly understood. Forlaser power 10PW (i.e. intensities ∼ W/cm and tight focusing) the duration of exponentialgrowth of cascade multiplicity is restricted either by driving laser pulse duration or by particlesescape from the focal region. However, for higher power (either higher intensity or weaker fo-cusing) it was demonstrated that cascade multiplicity may rapidly become macroscopic [17,18].When the density of created pairs exceeds the relativistic critical plasma density, the arisingelectron-positron plasma starts depleting the driving laser field. However, in an alternative sce-nario at high density the electron-positron-photon plasma may come to (quasi-)equilibrium dueto various relaxation processes. Such processes, however, as of now remain almost unexplored. SFHQ 2016 m = (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) αχ / [21] + (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) α χ log χ [21] + (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) α χ / log χ [22] + . . . + (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) α χ / log χ [23] ++ (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) α χ / [23] + (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) α χ log χ [24] + (cid:124) (cid:123)(cid:122) (cid:125) (cid:39) α χ / [24] + . . . Figure 4: Some 2nd and 4th order radiation corrections to electron mass operator M computedor estimated by the Ritus group in 1972-1980. The key results are enclosed in boxes. In ordinary QED, the running coupling constant α ( ε (cid:29) m ) (cid:39) E em ( ε ) /m (cid:39) α log( ε/m ) remainssmall within the whole reasonable region of energy (in particular, up to the Electroweak Theoryenergy scale), hence perturbation theory always works pretty well. However, as was noticedalready soon after the very birth of IFQED approach [19, 20], the leading order contributionsto the mass and polarization operators within IFQED are growing surprisingly fast with χ or κ (i.e. with both energy and field strength): M (2) ( χ (cid:29) (cid:39) αmχ / , P (2) ( κ (cid:29) (cid:39) αm κ / . (13)This can be easily traced back to Eqs. (10) via the optical theorem, and implies that for χ, κ (cid:38) α − / (cid:39) . × the radiation corrections cease to be small, M (2) (cid:39) m , P (2) (cid:39) m .In addition, in a proper reference frame t e ∼ W − γ (cid:39) t C , t γ ∼ W − e + e − (cid:39) t C , (14)meaning that the concept of radiation-free motion, obviously underlying the IFQED approach,could show up only at Compton scale, where localization is all the same impossible. If so, thisshould blow up the approach, thus making IFQED a truly non-perturbative theory like QCD.Systematic analysis of the actual expansion parameter of IFQED perturbation theory wasundertaken by the Ritus group [21, 22], and especially by Narozhny [23, 24], see Fig. 4. Bycomparing 2-nd and 4-th order contributions to the mass operator it was initially conjecturedthat the expansion parameter is M (4) /M (2) (cid:39) αχ / . However, further estimation of the 6-th order contributions identified M (6) /M (4) (cid:39) αχ / as the true expansion parameter. Thisconclusion is consistent with the also known data for polarization and vertex operators (notshown here for brevity). Origination of both parameters can be easily understood in terms ofthe qualitative approach of Sec. 3 [25]. Indeed, the estimations (9), (10) were based exclusivelyon the uncertainty principle and thus are valid for virtual processes as well. In a proper referenceframe the characteristic longitudinal and transverse sizes of a vacuum polarization loop areestimated by l (cid:107) , P (cid:39) ( m/k ) t q (cid:39) l C κ − / , l ⊥ , P (cid:39) ϑt q (cid:39) eEt q /k (cid:39) l C κ − / . Interestingly, for κ (cid:29) l C and moreover that l (cid:107) , P ∼ r e (classical electron Estimations for self energy are exactly the same because ultrarelativistic kinematics is similar for both cases. SFHQ 2016 adius!) for α κ / ∼
1. Now the aforementioned expansion parameters can be revealed exactlyas in ordinary QED as Coulomb to rest energy ratios ( e /l ) /m for transverse ( l = l ⊥ , P ) andlongitudinal ( l = l (cid:107) , P ) loop sizes, respectively. On the grounds of the optical theorem theremight be also a tight relation between the higher-order radiation corrections and cascades (sincecutting the former diagrams leads to the latter ones). However, for self-sustained cascades dueto the scaling (12) we have always αχ / (cid:39) E/E S (cid:46) αχ / (cid:38) Acknowledgments
I am grateful to the organizers of the Helmholtz International Summer School “QFT at theLimits: from Strong Fields to Heavy Quarks” (JINR, Dubna, 2016) for inviting me to givethis lecture, and to D. Blaschke, S.A. Smolyansky, A.I. Titov, A. Ilderton, T. Heinzl and A.A.Mironov for fruitful comments and inspiring discussions. The work was supported by RFBRgrant 16-02-00963a.
References [1] D. Strickland and G. Mourou, Opt. comm.
219 (1985).[2] V. Yanovsky et al , Optics Express .[4] .[5] D.L. Burke et al , Phys. Rev. Lett. et al , Phys. Rev. D60
309 (2006).[8] A. Di Piazza, C. M¨uller, K.Z. Hatsagortsyan, and C.H. Keitel, Rev. Mod. Phys.
249 (2015).[10] V.I. Ritus, Journ. Soviet Laser Research
497 (1985).[11] A.M. Fedotov, arXiv:1507.08512 (2015).[12] A.A. Mironov, N.B. Narozhny, A.M. Fedotov, Phys. Lett.
A378 , 1083 (2014).[14] A.M. Fedotov, N.B. Narozhny, G. Mourou, G. Korn, Phys. Rev. Lett. et al , Phys. Rev. STAB et al , Phys. Plasmas , 056706 (2016).[18] E.N. Nerush et al , Phys. Rev. Lett.
371 (1969).[20] V.I. Ritus, Sov. Phys. JETP
555 (1972).
SFHQ 2016
22] D.A. Morozov, V.I. Ritus, Nucl. Phys.
B 86
309 (1975).[23] N.B. Narozhny, Phys. Rev.
D20
D2112