EExtreme field physics and QED
Thomas Heinzl ∗ and Anton Ilderton † School of Mathematics and Statistics, University of PlymouthDrake Circus, Plymouth PL4 8AA, UK (Dated: October 26, 2018)We give a brief overview of the most important QED effects that can be studied in the presenceof extreme fields such as those expected at the Vulcan laser upgraded to a power of 10 Petawatts.
I. INTRODUCTION
Since the technological breakthrough of chirped pulse amplification [1] the power of optical lasers was increasedcontinuously, presently achievable intensities being around 10 W/cm . These are typical for lasers in the 1 Petawattclass such as the current Vulcan laser at the Rutherford-Appleton laboratory and amount to photon numbers ofabout 10 in a cubic laser wave length. By the correspondence principle (large quantum numbers) one expects theselaser beams to be very well described by classical electromagnetic fields. The associated (electric) field strength is ofthe order of 10 V/m. In such a strong laser field electrons are shaken so violently that their velocity approachesthe speed of light. One therefore concludes that the interactions between lasers and matter, in particular electrons,become relativistic.One may then go one step further and ask what would happen if, in the presence of an ultra-intense laser, onewould probe distances of the order of the electron Compton wave length, λ C = (cid:126) /mc . For length scales of thisorder one expects quantum effects to be all important, and one has to unify relativity and quantum mechanics in thepresence of a strong classical field. There is a theory at hand which does exactly this, namely strong-field QuantumElectrodynamics (QED). This is a generalisation of the standard quantum field theory of light and matter, QED, whichis at present the most accurately tested theory in physics. The tests in question, however, have almost exclusivelybeen done for weak fields, i.e. standard QED. Within a particle physics context the strong field sector has only beenaddressed in a single experiment, namely E-144 at SLAC (see [2] for an overview, and below). As this was utilising the50 GeV SLAC electron beam to produce back-scattered photons of 30 GeV it clearly was a high-energy experiment.Thus, the strong-field, low-energy regime of QED remains untested to date. As will become clear below, ultra-highintensity lasers are a unique tool to explore this uncharted region of the standard model. In the following we willshow that an upgrade of Vulcan to a power of 10 Petawatt will be a crucial step in this direction.
II. STRONG-FIELD QED
The fundamental QED interaction where a photon γ couples to an electron-positron pair ( e + e − ) is depicted inFig. 1. The coupling strength is determined by the elementary charge, e . γ e − e + e FIG. 1: The elementary interaction in QED.
Viewed as a process involving three real particles on their mass shell the Feynman diagram of Fig. 1 is forbidden byenergy-momentum conservation. A massless photon cannot ‘decay’ into an e + e − pair of total mass 2 mc . Accordingly, ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] S e p if one wants to create matter pairs from light one has to add some additional ingredient, such as extra particles or anexternal electromagnetic field. For our purposes we imagine that this field is provided by an ultra-high intensity laser.The effect of such a classical background field is to ‘dress’ the fermions which become effective (or quasi) particles withan effective mass m ∗ > m . Theoretically, they may be thought of as being solutions of the Dirac equation in a planewave (Volkov electrons [3]). In Feynman diagrams the electron lines become ‘fat’, as shown in Fig. 2, correspondingto a bare electron absorbing and emitting an arbitrary number of laser photons. FIG. 2: A dressed electron line describing the propagation of a Volkov electron.
A classical picture of this dressing process is the quiver motion of the electron in the laser field. At high intensitiesthis is so violent that the electron becomes relativistic acquiring a nontrivial γ -factor that increases its energy andmass. The intensity of the laser beam is measured by a universal ‘dimensionless laser amplitude’, a ≡ eEλ L mc . (1)In atomic physics a − is referred to as the Keldysh parameter, as reviewed recently in [4]. Obviously, a is a purelyclassical (no (cid:126) present) ratio of two energies; the energy gain of an electron moving across a laser wave length λ L inthe field E divided by its rest energy. The numerator does not appear to be Lorentz invariant, but there is indeed anequivalent covariant expression in terms of the energy-momentum tensor of the laser field [5]. One may say that thequivering electron becomes relativistic when its kinetic energy becomes comparable to its rest energy, i.e. when a (cid:39) a values achieved or expected at current and future high-powerlaser facilities. We have used the rule of thumb, a (cid:39) × P/ PW, relating a to laser power P in Petawatts [6]. TABLE I: Overview of current and future laser facilities: intensities I (in W/cm ) and a values (XFEL: X-ray free electronlaser at DESY, FZD: Forschungszentrum Dresden-Rossendorf, ELI: Extreme Light Infrastructure project, HiPER: High Powerlaser Energy Research facility).XFEL (‘goal’) FZD (150 TW) Vulcan/POLARIS (1PW) Vulcan (10PW) ELI/HiPER I a
10 20 70 200 5 × We mention in passing that for Vulcan (10 PW upgrade) intensities, a (cid:39) a (cid:29) optical lasers .In the remainder of this presentation we look at various strong-field QED processes paying particular attention tointensity effects signalled by the appearance of a . III. NONLINEAR COMPTON SCATTERING
The process in question is the collision of an electron and a high intensity laser beam such that a photon γ isscattered out of the beam. In terms of dressed electrons this is depicted on the left-hand side of Fig. 3 which, whenexpanded in the number of laser photons involved, becomes a sum of diagrams of the type shown on the right-handside representing the processes e + nγ L → e (cid:48) + γ . (2)Here, the electron absorbs an arbitrary number n of laser photons (energy (cid:126) ω L (cid:39) (cid:126) ω (cid:48) . We may pass from a quantum perspective to a classical electromagnetic wave picture as long as mc is The XFEL is ‘handicapped’ by its small wavelength, λ L (cid:39) . e − e − γ = ... + ... n γ L e − e − γ + ... FIG. 3: Feynman diagrams for nonlinear Compton scattering. the dominant energy scale in the rest frame of the electron. This classical limit is referred to as Thomson scattering.In terms of lab quantities, for the latter to be valid, one requires γ (cid:28) mc / (cid:126) ω L (cid:39) . . . where γ ≡ E p /mc isthe γ -factor of the electrons . It is important to emphasise that the processes (2) are not suppressed by any thresholdeffect. Thus, one can study intensity effects at arbitrarily low centre-of-mass energies both for photons and electrons.This is quite a unique feature of nonlinear Thomson or Compton scattering and singles out this process from a particlephysics point of view. FIG. 4: Shift of the (linear) Compton edge as a function of the Lorentz invariant x , for electron energy E p = 40 MeV anddifferent intensities a . In Fig. 4 we show the photon emission rates as a function of a suitably chosen Lorentz invariant, x , which basicallymeasures the energy of the scattered photons in any chosen reference frame. The peak at the very right correspondsto standard ( n = 1) low intensity ( a (cid:39)
0) Compton back scattering of laser photons colliding with a 40 MeV electronbeam. Using a polarised laser beam this may be used to produce polarised high energy photons of an energy given bythe linear Compton edge value, (cid:126) ω (cid:48) max (cid:39) γ (cid:126) ω L [7, 8]. Note that, in the lab frame, substantial energy is transferredfrom electrons to the scattered photons (blue-shift). In an astrophysical context such a process is referred to as ‘inverseCompton scattering’. This is to be contrasted with ‘normal’ Compton scattering (Compton’s original experiment)where the electrons are at rest in the lab and one observes a red-shift of the photon frequency ω L . The precise invariant statement is p · k (cid:28) m c where p and k are the 4-momenta of incoming electrons and photons, respectively. Notethat k is of order (cid:126) , k = (cid:126) ( ω L /c, k ). To study intensity effects one uses the (quantum) theory for high-intensity Compton scattering, developed in the1960’s. This is based on Volkov electrons as asymptotic scattering states [9, 10, 11] and may be found in the textbook[12]. The most striking experimental signal is a red-shift of the linear Compton edge, from 4 γ (cid:126) ω L to 4 γ ∗ (cid:126) ω L with γ ∗ ≡ γ / (1 + a ). This may be understood in terms of the electron mass shift [13] mentioned earlier, m ∗ = m (cid:113) a . (3)As the electron ‘gains weight’ ( m → m ∗ ) it will recoil less, reducing the energy transfer to the final state photon, hencethe red-shift in the maximum photon energy. This effect is illustrated in the photon spectrum of Fig. 4 for a = 20and a = 200, the latter value expected for the Vulcan 10 PW upgrade. It is interesting to consider what one wouldobserve in the lab frame. We have seen that back-scattering off high-energy electrons ( γ (cid:29)
1) produces a blue-shift(‘inverse’ Compton). On the other hand, high intensity ( a (cid:29)
1) produces a red-shift, hence works in the oppositedirection. It turns out that there is exact balance in the centre-of-mass frame of the Volkov electrons and the n laserphotons, that is when 4 γ ∗ (cid:39)
1. This can obviously be achieved by fine-tuning γ and a : for 40 MeV electrons theassociated a turns out just to be 200. Hence, for a of this order or larger one expects an overall red-shift, ω (cid:48) < ω L ,as the Volkov electron has become so heavy that it appears almost ‘static’ from the photons’ point of view.The dominant spikes in Fig. 4 correspond to single-photon absorption, n = 1 in (2). However, these spectra alsoshow further peaks corresponding to absorption of n = 2 , , . . . laser photons. In the laser community this effect iscalled higher harmonic generation. Their identification will depend crucially on the size of the background which maywash out the signals of higher harmonics. It will hence be important to simulate the scattering process numericallyusing realistic beam shapes.Both the red-shift and the relative width δ w ≡ ∆ ω (cid:48) /ω (cid:48) of the n = 1 spike depend sensitively on a , for a (cid:29) ω (cid:48) max ( a ) /ω (cid:48) max (0) (cid:39) /a and δ w ∼ a . Hence by tuning the initial electron energy E p and a onecan try to design radiation of a particular frequency ω (cid:48) and width. For example, using 40 MeV electrons and a laserwith a = 20 one obtains ω (cid:48) (cid:39) . δ w (cid:39)
20 %. It would be interesting to studyhow this radiation can be made more mono-energetic.Nonlinear Compton scattering (2) has been observed and analysed in the SLAC E-144 experiment [2, 14] using 47GeV electrons from the SLAC beam and a Terawatt laser with a (cid:39) .
4. This was a high energy ( γ (cid:39) ) and lowintensity ( a <
1) experiment (hence deep in the ‘inverse’ Compton regime). Photon spectra were not recorded andhence no red-shift was observed [15]. We reemphasize that this easily accessible process should be studied to a highprecision with high-power optical lasers, thus exploring the uncharted region of the standard model of low energiesand high intensities – see also Section VI.
IV. PAIR PRODUCTIONA. Stimulated pair production
The next process we will consider is pair production in the presence of an external (laser) field [16]. It is relatedto nonlinear Compton scattering (2) via crossing symmetry. In the language of Feynman diagrams this amounts toexchanging, in Fig. 3, the incoming electron with the outgoing photon line as illustrated in Fig. 5. e − e − γ −→ γ e − e + FIG. 5: Laser assisted pair production, formally obtained from nonlinear Compton scattering via crossing.
We are thus summing over all processes γ + nγ L → e + e − where a photon γ interacts with n laser photons andstimulates the production of an electron positron pair (see Fig. 6), sometimes called multi-photon Breit-Wheeler pairproduction [17]. Unlike (2) this process has a threshold as the energy 2 m ∗ c (cid:38) m ∗ appears here as the pairs are produced within the laser beam. Thisimplies that high intensity actually makes pair creation harder as the threshold is increased by a factor a [2]. Thus,it is preferable to use high energy photons for pair creation in collisions with lasers of moderate intensity. Again,this was precisely the setup of SLAC E-144 where photons of about 30 GeV were produced via nonlinear Compton(back)scattering (2) and subsequently used to create pairs according to Fig. 6 [2, 18]. ... n γ L γ e − e + FIG. 6: Stimulated multi-photon Breit-Wheeler pair production.
In Feynman perturbation theory, upon counting the number of photons in Fig. 6, one expects the associatedproduction rate (amplitude squared) to go like W n ∼ α n +1 a n for a (cid:28) α = 1 /
137 being the fine-structureconstant. For the n th subprocess to happen there is a threshold in photon number given by n = m ∗ c / (cid:126) ωω L (assuming backscattering). This was indeed confirmed by the SLAC experiment where the kinematics implied n = 5.On the other hand, within an all-optical setup one would require an astronomical number of photons of order n (cid:39) a . Hence, in perturbation theory, the lowest order rate contributing is the corresponding W n ∼ α n +1 whichis basically zero by power counting in α . However, for large a (cid:29) B. Spontaneous pair production
The most spectacular effect in strong-field QED is probably spontaneous vacuum pair production as first predictedby Sauter [19] and worked out in detail by Schwinger [20]. To see what is involved let us equate the rest energy ofan electron with its electromagnetic energy gain upon traversing a Compton wave length λ C in an electric field E , mc = eEλ C . This defines the critical field strength E c ≡ m c e (cid:126) = 1 . × V/m , (4)which translates into a critical intensity of I c = 4 × W/cm or a ,c = λ L /λ C (cid:39) . In a field of this magnitude itbecomes energetically favourable for the vacuum to break down by producing electron positron pairs in order to shieldand reduce the externally applied field. Note that (4) contains both the velocity of light, c , and Planck’s constant, (cid:126) ,signalling this is both a relativistic and quantum effect necessitating a QED treatment.Energy-momentum conservation rules out spontaneous pair creation by a single laser, hence one needs two (say,counter-propagating) beams to ‘boil the vacuum’ [21]. In Fig. 7 we have depicted one of infinitely many processesthat contribute and (in principle) need to be resummed to obtain the total answer . A crude model for the processis the tunneling of a positron from the Dirac sea through a potential of order mc /e . The resulting pair productionprobability is a typical tunneling factor proportional to exp( − πE c /E ) [19, 20] and implies an enormous suppressionfor fields below the critical value. On the other hand, there will be a pre-exponential factor which may be large [22].As the calculations can reliably only be done for idealised field configurations there is room for debate and differentpredictions. Some authors even find no exponential suppression at all [23, 24]. Technically, one proceeds in a more elegant fashion by calculating the effective action [20]. ... γ L γ L e − e + FIG. 7: Spontaneous vacuum pair production from two counter-propagating laser beams.
V. VACUUM POLARISATION
We have seen above that pairs will be produced once a critical field strength is reached (if not before). Intuitively onemay view this process as a break-up of the virtual e + e − ‘dipoles’ that are omnipresent as fluctuations of the vacuum,see Fig. 8. Collectively they produce what is known as vacuum polarisation which is a shielding effect making thephysical charge appear reduced as resolution decreases, i.e. for large distances. This has observable consequences suchas the Lamb shift and the electron and muon anomalous magnetic moments. −→ I > I c e + e − FIG. 8: Vacuum polarisation with virtual pairs in the loop (left-hand side) breaking up into real pairs above threshold(right-hand side).
In terms of Feynman diagrams vacuum polarisation is depicted as a fermion loop such as on the left-hand side ofFig. 8. Both for bare and dressed fermion lines one can associate a mathematical expression with this graph which,above threshold, develops an imaginary part signaling creation of real pairs as discussed in the previous section. Asphotons ‘disappear’ in this case pair creation is called an absorptive process. The real part, on the other hand, describeshow virtual pairs polarising the vacuum affect the propagation of probe photons and thus governs all dispersive effects.Real and imaginary parts of the vacuum polarisation diagram are related by the optical theorem (or Kramers-Kronigrelation) which connects the two sides of Fig. 8 in a precise quantitative manner.The most important dispersive effect probably is vacuum birefringence first discussed in Toll’s thesis [25]. Thepolarised vacuum hence acts as a medium with preferred directions dictated by the external fields. Accordingly, thereare two different refractive indices for electromagnetic probe beams of different polarisation states. These are n ± = 1 + α π (11 ± (cid:15) + O ( (cid:15) ν ) , (5)to lowest order in (dimensionless) intensity (cid:15) ≡ I/I c (cid:39) (10 − a ) and probe frequency ν ≡ ω/m . For an X-ray probeof ω = 5 keV and Exawatt class lasers one may achieve values of (cid:15) (cid:39) ν (cid:39) − .The idea [26] is to send a linearly polarised probe beam of sufficiently large frequency ω into a hot spot of extension d generated by one (or two counter-propagating) laser beams and measure the ellipticity signal, δ ∼ ω ( n + − n − ) caused by a phase retardation of one of the polarisation directions, see Fig. 9.The leading-order expression for δ is [26] δ = 3 . × (cid:18) dµ m (cid:15) ν (cid:19) , (6) d e BE
45 linear pol. elliptical pol. z l high I, w e + e FIG. 9: Schematic experimental setup to measure vacuum birefringence via an ellipticity signal. and grows quadratically with probe frequency ν , intensity (cid:15) and spot size d (taken to be the Rayleigh length). Evenif all these are maximised the effect is still extremely small for present and near-future facilities as shown in Table II. TABLE II: Expected ellipticity signals for some high-power laser facilities.facility Vulcan/POLARIS (1PW) Vulcan (10PW) ELI/HiPERellipticity δ × − × − − . . . − X-ray polarimetry is currently sensitive to ellipticities of about 10 − , the theoretical limit being 10 − . Thisrequires Exawatt facilities such as ELI or HiPER. However, if one could produce polarised γ beams of MeV energiesthe signal would go up by several orders of magnitude (with an expansion in ν = O (1) no longer possible). A possibleoption is Compton backscattering off the 3 GeV Diamond beam which would yield polarised photons in the 100 MeVrange. In this case one could even address questions such as the frequency dependence of the refractive indices whereanother Kramers-Kronig relation is expected between real and imaginary parts, the presence of the latter being tiedto anomalous dispersion, ∂n/∂ν < n e Heisenberg-Euler regimestandard QED e ( = 0)strong-fieldQEDpresently attainable (all optical)SLAC exp. FIG. 10: Parameter regime for strong-field QED: energy ν vs. field strength (cid:15) . High intensities explore the regime to the rightof the vertical axis with all-optical experiments staying close to the horizontal one (the Heisenberg-Euler regime). VI. CONCLUSIONS
As illustrated in Fig. 10 high-precision experiments using ultra-intense lasers would be testing an unknown regimeof the standard model characterised by extreme field strengths and low energies (at least in an all-optical setup ).It is not impossible that one could encounter surprises in doing so. One may even discover new weakly interactingsub-eV particles (WISPs) predicted by some theories beyond the standard model [28].In this contribution we have adopted a somewhat more conservative point of view by staying within QED, ‘only’assuming high intensities. Three areas of immediate interest have been identified, namely (i) nonlinear Compton orThomson scattering where intensity effects are not suppressed by any threshold, (ii) pair production and (iii) vacuumbirefringence. In the latter two cases, there are threshold and/or suppression factors to be overcome. For the Vulcan10 PW upgrade these are typically of the order of 10 − the inverse of which governs the Schwinger exponent. Anyideas on how to reduce the suppression would certainly be highly welcome. Acknowledgments
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A41 , 164039 (2008), 0711.1337. Using laser acceleration techniques for charged particles it is possible to reach energies E p (cid:39) (cid:126) ω > ∼ mc , i.e. ν > ∼∼