Methods for evaluating physical processes in strong external fields at e+e- colliders: Furry picture and quasi-classical approach
MMethods for evaluating physical processes in strongexternal fields at e + e − colliders: Furry picture andquasi-classical approach Stefano Porto ∗ † II. Institut für Theoretische Physik, University of Hamburg, Luruper Chaussee 149, D-22761Hamburg, GermanyE-mail: [email protected]
Anthony Hartin
DESY, Deutsches Elektronen-Synchrotron, Notkestr. 85, D-22607 Hamburg, Germany
Gudrid Moortgat-Pick
II. Institut für Theoretische Physik, University of Hamburg, Luruper Chaussee 149, D-22761Hamburg, Germany;DESY, Deutsches Elektronen-Synchrotron, Notkestr. 85, D-22607 Hamburg, Germany
Future linear colliders designs, ILC and CLIC, are expected to be powerful machines for the dis-covery of Physics Beyond the Standard Model and subsequent precision studies. However, dueto the intense beams (high luminosity, high energy), strong electromagnetic fields occur in thebeam-beam interaction region. In the context of precision high energy physics, the presence ofsuch strong fields may yield sensitive corrections to the observed electron-positron processes. TheFurry picture of quantum states gives a conceptually simple tool to treat physics processes in anexternal field. A generalization of the quasi-classical operator method (QOM) as an approxima-tion is considered too.
Proceedings of the Corfu Summer Institute 2012 "School and Workshops on Elementary Particle Physicsand Gravity"September 8-27, 2012Corfu, Greece ∗ Speaker. † S. P. gratefully acknowledges support of the DFG throught the grant SFB 676 ”Particles, Strings, and the EarlyUniverse”. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] A p r ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto
1. Introduction
After the first successful data runs of the LHC a completely new energy region at the Terascaleenergy frontier has been touched and already one new boson, a Higgs boson, has been discovered.Thanks to the excellent luminosity and detector performance at the LHC, already a rather preciseand consistent mass measurement of the new particle has been achieved, m Higgs ∼
126 GeV [ , ] . (1.1)Such a mass for a Higgs boson is perfectly in agreement with mass predictions from high precisionanalyses in the electroweak sector. However, in order to clearly manifest whether the discoveredparticle is the Standard Model (SM) Higgs boson or whether the candidate points to an extendedmodel as, for instance, Supersymmetry, a precise measurement of all couplings, the total widthand branching ratios is necessary. Therefore corresponding measurements at an e + e − collider arehighly desirable.Designs for a planned future e + e − linear collider (LC) are set up to reach high √ s up to 1-1.5TeV for the ILC and about 3 TeV for CLIC, with a very high luminosity in the range of 10 − -10 − cm − s − [3, 4].The key arguments for a linear lepton collider are, for instance, the clean environment and theavailability of polarized beams in the initial state, as well as rather low background rates comparedwith corresponding rates at a hadron collider. These facts permit in principle a full reconstructionof the observed processes and enable unprecedented high precision measurements. Therefore theLC is expected to perform high sensitivity for precision physics in the Higgs boson, top quarkand electroweak gauge bosons sectors and in both direct, as well as indirect searches for newphysics Beyond the Standard Model (BSM). Thus, the planned linear collider physics potential canstrikingly contribute and even extend the high energy physics range of the LHC.In order to reach the high luminosity required by the comprehensive physics program, the LCapplies strongly squeezed and intense bunches of electrons and positrons at the Interaction Point(IP). At the ILC, for instance, about a factor 1000 more intense beams, i.e. more e − / e + per pulse,will be achieved than at the SLC. In order to optimize the outcome of such high precision physicsat a LC, one needs to know, however, in detail all processes occurring at the IP.Each charged bunch generates an intense collective electromagnetic field, that is felt by theinteracting particles from the oncoming beam. Due to its intensity, this bunch field can be consid-ered as one external electromagnetic field as a whole rather than to be composed by single photonsinteracting with the colliding leptons.We study the impact of such external fields on the actual physics processes in electroweak andBSM physics. To achieve this goal, a full comprehension of QED effects in an external field (nonlinear QED) is required.Processes in external electromagnetic fields have caused interest since the beginning of quan-tum electrodynamics, with the paradox of electron quantum tunneling in an arbitrary high potentialbarrier, observed by Klein in 1929 [5]. Sauter (1931) [6] showed that this effect depends expo-nentially on the intensity of the field in the barrier. Schwinger (1951) [7] interpreted this para-dox via the concept of a critical field: the Schwinger critical field ( E cr = m e / e (cid:39) · V/m, B cr = m e / e (cid:39) . · T, using natural units) corresponds to the intensity of an electromagnetic2 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto field at which the vacuum spontaneously creates electron-positron pairs. The vacuum that is dueto the external field full of virtual particles becomes unstable and transforms into a more stablecharged vacuum via producing real particles [8].The critical field can be reached in nature on the surface of pulsars and magnetars [9] andclose to superheavy nuclei [10]. Such conditions are very difficult to reproduce in a laboratory.In recent years, new experiments have been developed and designed where the Schwinger criticalfield condition could be achieved in the boosted rest frame of a test electron. In the context of lowenergy physics, this is for example the case for newly designed intense lasers like XFEL at DESY[11]. Regarding high energy physics (HEP), the first experiment to study the strong field regimeof nonlinear QED was the E-144 experiment at SLAC [12], in which 46.6 GeV electrons wereshot through an intense laser. A second example of sources for non linear QED effects in HEPexperiments are, as explained above, the interaction points (IPs) at future linear collider designs,ILC [3] and CLIC [4]. The Schwinger condition may be fulfilled, indeed, in the rest frame of theultrarelativistic particles that scatter at the IP.Quantum processes in intense electromagnetic fields have been studied in different researchareas like intense plasma and laser physics [13] (for a detailed review see also [14]), using the socalled Furry picture (FP) [15]. According to the FP, the external field is treated as a classical objectthat modifies the equations of motions. However, an analogous appropriate study is still lackingfor the case of linear colliders, and this is the aim of our studies.In these proceedings, we will address the question whether electroweak physics processes maybe affected by the presence of the very intense external electromagnetic fields at the IPs at collidersand whether these effects can be easily incorporated in the calculations.In section 2 we will introduce the parameters that need to be considered in general in thepresence of an external electromagnetic field, describing also the generated fields at the interactionpoints of a future linear collider; in section 3 we introduce the Furry picture formalism and explainhow to apply it to the discussed processes; in section 4 we sketch the Ba˘ıer-Katkov quasi-classicaloperator method as a possible alternative in the case of ultrarelativistic particles; we conclude insection 5.
2. Intense fields at linear colliders
In order to perform the high precision physics program planned for future linear colliders, veryhigh luminosity L is needed. Therefore extremely squeezed e + and e − bunches are required, L ∝ N e + N e − σ x σ y , where N e ± is the number of e ± per bunch and σ x , σ y are the transversal dimensions of the bunchpropagating along z .Each of the dense bunches ( N e ± ∼ and σ x , σ y ∼ nm) can be regarded as an electro-magnetic current generating a collective strong electromagnetic field at the IP.Correspondingly, the colliding particles at the IP will see a superposition of the collectivefields originating from the two beams. Due to the boosted particle frame, these fields can beapproximated by two almost anticollinear constant crossed fields; each particle will mainly see3 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto the constant crossed field from the oncoming bunch [16]. A constant crossed field is the limit forinfinite period of a plane wave field with momentum k = c ω , it has a trivial spatial dependence A µ ( k · x ) = a µ k · x and its electric ( E ) and magnetic ( B ) components are orthogonal and equal inmagnitude: E ⊥ B , | E | = | B | . (2.1)Due to momentum transfer, the presence of external electromagnetic field permits processesthat are kinematically not allowed, as, for instance, beamstrahlung (i.e. bremsstrahlung in theelectromagnetic field of a relativistic particle bunch) and coherent pair production. These externalfields can also affect the rate of the allowed processes like incoherent pair production.At previous accelerators LEP and SLC these effects have been considered and estimated withsome approximations. Beamstrahlung and coherent pair production have been treated via theBa˘ıer-Katkov quasi-classical operator method (QOM) [17] while incoherent pair production wasdescribed via the equivalent photon approximation (EPA) [18].However, effects from strong external fields do not only affect the previously mentioned “back-ground” processes at the IP, but they may also have direct consequences on the electroweak SM orBSM processes that are in the focus of future colliders. To our knowledge, this problem has neverbeen addressed at colliders. Only some topics have been discussed in the context of laser physics([19, 14] and references therein), astroparticle physics [9] and decays in extremely intense fields[20, 21].Therefore a correct knowledge of the QED effects within a strong external field environmentis required in order to optimize the physics program with the expected precision.For the description of the external electromagnetic field at the IP, it is useful to introduce fourLorentz and gauge invariants that one can compose with the external field strength tensor F µν andthe momenta of the propagating particle p µ . The considered propagating particle can also be aphoton. The probabilities of the processes with a single initial particle in a general external field(ex. beamstrahlung, coherent pair production) depend on the following quantities [22]: x = e (cid:112) ( A µ ) m = eEm ω (2.2) χ = em (cid:113) ( F µν p µ ) (2.3) F = F µν F µν = B − E (2.4) | G | = | F µν ˜ F µν | = | − E · B | (2.5)where m and e denote the mass and the charge of the propagating particle (for an initial photonone uses the mass and charge of the electron) and ˜ F µν = ε µνρσ F ρσ .The parameter x represents the work done by the external field in a Compton length λ C = eF ¯ h / mc in units of the energy ¯ h ω of the quanta of the external field. Therefore, for x (cid:28)
1, a lownumber of photons from the external field are expected to interfere with the interacting particles,while for x (cid:38)
1, multiphoton processes are favoured with a nonlinear dependence on the externalfield. Therefore x is called the classical (since it does not depend on ¯ h ) nonlinearity parameter.The χ parameter, often also called ϒ in the literature, represents in units of mc the workperformed by the field over the Compton length in the particle rest system. It parametrizes the4 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto
Machine LEP II SLC ILC-1TeV CLIC-3TeVEnergy (GeV) 94.5 46.6 500 1500 N (10 ) 334 4 2 0.37 σ x , σ y ( µ m) 190, 3 2.1, 0.9 0.49, 0.002 0.045, 0.001 σ z (mm) 20 1.1 0.15 0.044 χ average Table 1:
Lepton colliders parameters. N is the number of leptons per bunch, σ x , σ y are the transversaldimensions of the bunches, σ z presents the longitudinal dimension. E is the energy of the particles in thebunches. The parameters for ILC-1TeV are taken from a 2011 dataset. magnitude of the quantum nonlinear effects and is called the quantum nonlinearity parameter [14].In particular, in the case of ultrarelativistic particles, it describes the intensity of the external fieldin the particle frame in units of the Schwinger critical field: χ = γ L EE cr = γ L BB cr where γ L denotes the Lorentz factor and E , B the electric and magnetic components in the laboratoryframe. Highly energetic initial particles can see a critical regime also in less intense fields in thelaboratory frame, as long as χ ∼ O ( ) . This value would correspond to an external field of theorder of the Schwinger critical field, at which the vacuum is polarized. The parameters F and | G | instead describe respectively the relative magnitude and orientation between E and B .The probability W of a process with one initial particle in a constant background field dependsin general only on χ , F , | G | since correspondingly x (cid:29) F , | G | = p (cid:29) m e )in a relatively weak field compared to E cr , one has | F | , | G | (cid:28) min ( , χ ) [22], confirming that acrossed field is a good approximation for the field seen by the colliding particles at the IP of LCs.One can generalize the above picture also to processes with two initial particles in an external field.As a consequence, the probabilities of processes at the IPs simplify W ( χ , F , | G | ) (cid:39) W ( χ , , ) ,depending effectively only on the intensity of the external field. Moreover, constant crossed fieldsallow simpler analytical calculations and integrations and have been object of recent studies [23].The χ parameter varies during the collision since the bunches distort under the pinch effectand the disruption effects. According to [16], the average value for χ in a gaussian bunch is givenby: χ average ≈ Nr e γ L α em σ z ( σ x + σ y ) (2.6)where N is the number of leptons per oncoming bunch, α em the fine structure constant, r e theCompton radius, σ x , σ y the transversal dimensions of the bunches, σ z the longitudinal dimension.Using the IP beam-beam simulation program CAIN [24] one obtains the values presented in Tab. 1for ILC and CLIC, while for LEP II and SLC we used the approximations (2.6).It is clear that the electromagnetic fields at the IPs of future linear colliders would be much5 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto
ILC-1TeV CLIC-3TeV ∼ . · . · . · Table 2:
Pair production processes at the IP regions at ILC and CLIC. more intense than in the previous lepton accelerators , with values of χ up to order O ( ) . Such avalue for χ corresponds to the polarization of the vacuum. This unstable vacuum requires incor-poration within the calculations in a non-trivial way. A study of the effectiveness of the previouslymentioned quasi-classical and EPA approximations for processes in this physical case should beundertaken.The expected e + e − pair productions at linear colliders are listed in Tab. 2.
3. The Furry picture and its application
The intense external field experienced by each particle at the IP is characterized by a highphoton density and a corresponding wave function overlap, so that it can be seen as a classical external background field. For physics in intense fields, the effects of such a classical externalfield are taken into account exactly through the so called
Furry picture or representation (FP) ofquantum states [15]. The main idea of FP is to find the exact solutions of the equation of motion inthe external field, taking into account the latter non perturbatively. Then, one applies the solutionsin the Feynman-Schwinger-Tomonaga S-matrix theory [25] to calculate the probabilities of thephysical processes. We propose to use FP also to calculate the probabilities of processes at the IPof linear colliders. In the following, we briefly review the FP technique [26].In the usual Interaction (or Dirac) picture the time dependence is shared between the statevectors and the observables. In particular, the Hamiltonian is given by H = H + V (3.1)where H is the time-independent unperturbed Hamiltonian, describing the time evolution of ob-servables, and V is the interaction Hamiltonian, that regulates the time dependence of the states. InQED, V represents the gauge interactions between fermions and photons. The eigenstates of H are assumed to be the free states of the particle in the vacuum.In the FP the Hamiltonian is given by H = H + H ext + V = H B + V (3.2)where H ext represents the interaction of the fermions with the external classical field. In the FPthe considered state vectors are the bound states of the fermions in the external field, eigenstates of H B . The FP bound eigenstates are related by a canonical transformation to the free particle statesof the Interaction picture [15], they obey to different commutation relations but, in the limit of null At the LHC, the less dense bunches do not allow to reach the really high critical field corresponding to proton pairproduction (10 V/m). ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto external field, the usual commutation relations are recovered. The corresponding QED Lagrangiancan be written as: L = ¯ ψ ( i (cid:54) ∂ − e (cid:54) A ext − m ) ψ − FF − e ¯ ψ (cid:54) A ψ (3.3)where A µ ext is the classical external field and its interaction term is distinguished from the usualgauge interaction term. Note that there is no kinetic therm F ext F ext since A µ ext is a classical externalbackground field, not a dynamical field.From the Lagrangian (3.3) one derives the modified Dirac equation: ( i (cid:54) ∂ − e (cid:54) A ext − m ) ψ = Ψ Vp ( k · x ) = (cid:112) ( π ) ε p E p ( k · x ) u ( p ) (3.5)with E p ( k · x ) ≡ (cid:18) − e (cid:54) A ext (cid:54) k ( k · p ) (cid:19) exp (cid:20) − ip · x − i (cid:90) ( k · x ) (cid:20) e ( A ext ( φ ) · p )( k · p ) − e A ext ( φ ) ( k · p ) (cid:21) d φ (cid:21) , (3.6)where k is the momentum of the external field, p and ε p the canonical momentum and energy ofthe fermion while u ( p ) is the usual Dirac spinor solution.The solution (3.5) is valid for a vast class of classical external fields. However, the preciseexpression is known only for few configurations like the plane wave electromagnetic field, thecrossed electromagnetic field and the Coulomb field. This solution takes entirely into accountthe effects of the external electromagnetic field on the fermion. The Ψ V solutions constitute anorthogonal and complete system [28], for a review [29]. Solutions of analogue equations of motionfor charged scalars and vector bosons in an external field have been found, see [20] for a review. Ψ V and the analogue scalar and vector solutions can be used in perturbation theory to buildnew Feynman rules and diagrams in order to describe processes in an external field.The described solutions of the modified Dirac equation (3.4), of the analogue modified Klein-Gordon equation etc. can be used in perturbation theory to build new Feynman rules and diagramsin order to describe processes in an external field.In particular, here we give the QED FP-Feynman rules at the tree-level, built out of solution(3.5). The fermion two-point function in coordinate space is given by, see Fig. (1): G ( x , x (cid:48) ) = ( π ) (cid:90) + ∞ − ∞ d p E p ( k · x ) (cid:54) p + mp − m E p ( k · x (cid:48) ) e ip · ( x (cid:48) − x ) , (3.7)where the usual fermion propagator is sandwiched between E p factors ( E p = E † p γ ) coming fromthe Volkov solutions. It is interesting to note that there is a non trivial dependence on the coordinates x , x (cid:48) within which the fermion propagates, instead of their difference x (cid:48) − x [25].Often in laser physics [13], the FP fermion line is interpreted considering a bare fermion“dressed” by an arbitrary number of photons emitted or absorbed from the laser, Fig. 2.7 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto x x ′ Figure 1:
The FP fermion propagator. The double line represents the Volkov solution. = + + + . . .
Figure 2:
Interpretation of the electron propagator derived from the Volkov solution.
The QED vertex in Fig. (3), instead, is given by: − ie γ e µ = − ie ( π ) + ∞ ∑ r = − ∞ E p f ( r ) γ µ E p i ( r ) δ ( p f + k f − p i − r k ) . (3.8)Each term of the sum is given by the usual Dirac matrix γ µ , sandwiched between the factors E p , E p and multiplied by a δ -function with a momentum conservation law as argument. The lattercontains a term − rk that represents the momentum exchanged with the external field.In the case of a constant crossed field the sum in the vertex becomes an integral: − ie γ e µ = − ie ( π ) (cid:90) + ∞ − ∞ dr E p f ( r ) γ µ E p i ( r ) δ ( p f + k f − p i − r k ) . (3.9) p f p i k f Figure 3:
The FP QED vertex.
Initial and final (anti)fermions are described by the usual Dirac spinors u p , ¯ u p , v p , ¯ v p since the E p factors have been grouped from the Volkov spinors in Feynman rules (3.8) and (3.9). Thephoton propagator at tree level is unchanged, since the photon has null charge and it does notinteract directly to the external field.The above described Feynman rules are the tool to build every Feynman diagram that isneeded, at each order in perturbation expansion. For example, beamstrahlung and coherent pairproduction can be drawn as FP processes at the 1 st -order in perturbation theory, see Fig. (4).8 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto e − γe − (a) Beamstrahlung γ e + e − (b) Coherent pair production Figure 4: st -order FP processes. Writing down the amplitudes of these two processes, one obtains an expression with a sumcoming from the modified QED vertex (3.8). This leads to a naïve interpretation [30] for thenew Feynman diagram: it can be seen as sum over all the Feynman graphs characterized with theemission or absorption of r photons from the external see Fig. (5). Figure 5:
Naïve interpretation of FP beamstrahlung diagram.
This interpretation shows that even if the photons of the external field were not considered atthe beginning, they “appear” through the quantum interaction of the fermion with the external fieldencoded in the Volkov solution.Typically, performing probability calculations within the FP, one has to to handle integralsover Airy or Bessel functions coming from the E p factors, that can be simplified using the integral-representation properties of these special functions.As shortly described in Sec. 2, each colliding particle sees at the IP the superposition of twoexternal fields, but usually only the field from the oncoming bunch is considered. In order totake fully into account the effect of the intense fields of both colliding charge bunches at the IP of acollider, new particle wavefunctions are required. These are obtained by solving the Dirac equationminimally coupled to the fields of two, non-collinear, constant crossed fields. Such new solutionswill lead to different particle process transition probabilities. Technically, calculations using thenew solutions may prove simpler than those in one constant crossed field [31].For an expression of beamstrahlung process through the Furry picture in N collinear constantcrossed fields, see [32]. A new EM solver/generic event generator software,
IPStrong , is being In the case of a constant crossed field, and hence of a QED vertex with an integral, one can interpret saying thatthere has been the absorption/emission either of one photon with momentum rk or of r photons with momentum k . ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto developed to model the strong field processes at the IP at linear colliders with χ and the FP-crosssections as inputs [31].
4. Alternative: generalized quasi-classical operator method (QOM)
The conceptually simple FP recipe can lead to relatively complex analytical calculations al-ready at the first order, due to the –already mentioned– multidimensional integrations over specialfunctions and polynomials. Only a few tree-level two-vertices processes [33, 19] and one-loop 2-points amplitudes [28] are known, some of them are computed exploiting the optical theorem, seealso, [9] and [20].Asymptotic approximations of results in the FP involving highly energetic particles (‘ultra-relativistic’, i.e. Lorentz factor γ L (cid:29)
1) that are present, for instance, in the IP region of a linearcollider, are equivalent to calculations in a fully quasi-classical approximation [22].Therefore, Ba˘ıer and Katkov invented an effective method for the calculation of such processesin an external field using the a quasi-classical approach from the beginning. Their alternativeprocedure is well-known as quasi-classical operator method (QOM) [17], for a review [25] and[34]. The QOM is particularly powerful when considering ultrarelativistic initial state particles andits results are implemented in
CAIN and
GuineaPig [35] to estimate beamstrahlung and coherentpair production at linear colliders.The QOM has been originally applied to the case of radiation from a charged particle in anexternal magnetic field B (synchrotron radiation) [17]. The field is considered stationary and B = | B | (cid:28) B cr . From the classical theory the Larmor frequency ω and the peak frequency of thequasi-continuous spectrum ω are such that: ω ≈ | e | B ε , ω ∼ ω (cid:16) ε m (cid:17) (4.1)where ε is the energy of the electron.In this process one can identify two quantum effects: the quantum propagation of the electronand the quantum recoil on the electron due to the photon emission. The relevance of a quantumeffect is encoded by the commutation relations between the operators and by the correspondingdynamical variables in the uncertainty relations.Exploiting the non-commutativity between the velocity components of the electron in B , oneobtains the corresponding uncertainty relations ∆ v i ∆ v k ∼ e ¯ hB ε = BB cr γ = ¯ h ω ε , (4.2)where ¯ h ω is the unit energy interval between the possible electron levels in motion in the field B .Relations (4.1) and (4.2) show that the motion is increasingly classical for increasing energy ε .The non-commutativity between the electron and the emitted photon dynamical variables is oforder ¯ h ωε . (4.3)Considering however, χ ∼ ¯ h ωε (cid:38)
1, it is obvious that the classical theory cannot be applied for thequantum recoil of the emitted photon energy ¯ h ω [17].10 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto
Due to (4.2) and (4.3), the key idea of QOM is to consider the motion of the electron asbeing classical right from the beginning, whereas the quantum recoil from photon emission is notneglected in the calculation of the amplitude. Therefore this method is called quasi-classical .In order to implement the quasi-classical approach, the amplitudes are written in terms of operators instead of the corresponding dynamical variables. In particular, the electron solutionsof equation of motion in an external field are not written in terms of Volkov solutions, but in thesymbolic operator form [25] (operators are denoted by ˆ): ψ ( x ) = √ H u ( ˆ P ) e − i ¯ h ˆ Ht φ ( x ) (4.4)where φ ( x ) is the classical wavefunction of a spinless particle and u ( ˆ P ) = (cid:112) ˆ H + m (cid:32) ˆ H + m σ · ˆ P (cid:33) (4.5)is the usual Dirac spinor u ( p ) incorporating the kinematic momentum operator ˆ P = ˆ p − e A of theelectron within the vector potential A as well as the energy operator operator ˆ H = (cid:112) ˆ P + m . Note,ˆ P has not to be misinterpreted as the canonical (or generalized) momentum ˆ p that appears also inthe Volkov solutions. In (4.4) the information about the external field is implicitly encoded usingˆ P . The arbitrary two-component spinor w satisfies w ∗ w = U f i = e √ π √ ¯ h ω (cid:90) dt (cid:104) f | u † f ( ˆ P ) √ H ( α i · ε ∗ i ) u i ( ˆ P ) √ H | i (cid:105) e i ω t (4.6)where the bra (cid:104) f | and the ket | i (cid:105) are solutions of the Klein-Gordon equation in the given field, α i = γ γ i ( i = , , γ µ are the Dirac matrices and ε µ the photon polarization vector.In the probability calculations the operators are kept up to a later stage where further commu-tation relations may appear. According to the previous prescriptions, the commutation relations ofvelocity components of the electron moving in an external field are neglected, while the commuta-tion relations between electron and photon operators are left untouched.Eventually, one obtains an expression composed by a product of commuting operators, so thatthat they can be substituted by the corresponding classical values (c-numbers). The simplificationcomes in that the relatively complicated expressions derived from the exact solutions in an externalwave do not appear explicitly while the simple expression for the classical trajectory of the electronin the field is inserted.The results by Ba˘ıer and Katkov on synchrotron radiation, coherent pair production and anni-hilation of a pair into a photon [17] are consistent with the ones obtained by Nikishov and Ritus[30] and previously by Klepikov [36]. In the case of the beamstrahlung it has been shown that thetransition probabilities obtained using the FP and the QOM are asymptotically identical in the ultra-relativistic limit [37]. The QOM has been successfully applied to processes happening in media,in crystals with strong inter-lattice fields and also in super strong fields (common in astrophysics)[34], [38].The QOM method could in principle be generalized to processes other than only the photonradiation and cross-symmetric ones. Our present idea is to understand whether it is possible to11 ethods for evaluating physical processes in strong external fields at e + e − colliders Stefano Porto apply this kind of approach also to higher order processes, the main object of studies of futurelinear colliders, as it is possible with the Furry picture, and understand whether it is effectiveand practical. The process e − e + → µ − µ + is currently under study in the quasi-classical approach.This process has already been studied in Born approximation [39] and in the context of laser-drivenreaction from positronium using Volkov solutions [19].Another promising operator approach, based on the quasi-classical Green’s function of theDirac equation in an arbitrary is being developed lately in the context of laser and atomic physics[40, 41].
5. Conclusions and outlook
The planned linear collider will produce physics processes in the environment of very intenseelectromagnetic fields possibly exceeding the critical field introduced by Schwinger in the restframe of the colliding electrons and positrons.The unstable vacuum present at the interaction points might lead to a regime of nonlinearQuantum Electrodynamics, affecting the processes in the IP area. Such conditions therefore moti-vate to calculate all probabilities of the physics processes under fully consideration of the externalelectromagnetic fields affecting the vacuum.At previous lepton colliders, the much weaker external electromagnetic fields at the IPs did notneeded to be considered apart for background processes: the first order background processes asbeamstrahlung and coherent pair production, the second order incoherent pair production as well.At future linear colliders the external fields would be orders of magnitude higher so an estimateof the effects on all the processes is requested. As we have shown, indeed, the χ parameter, thatencodes the dependence of the probabilities on the intensity of the external field at the IP, is up to3 orders of magnitude higher at ILC and CLIC than at LEP. In particular at CLIC-3TeV, we wouldhave χ av ∼ .
34, describing a critical regime.A formally exact method to consider processes in a classical electromagnetic environment isgiven by the Furry picture of quantum states. The interaction with the external field is taken intoaccount not perturbatively and separately from the usual gauge interactions since fully incorporatedin modified equations of motion, the solutions of which are utilized in the usual S-matrix formalism.The quasi-classical operator method (QOM) offers an alternative to FP in the case of ultrarel-ativistic initial states; a generalization of this method to two vertex processes is now under study.
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