Transverse spectrum of bremsstrahlung in finite condensed media

Transverse spectrum of bremsstrahlung in finite condensed media X. Feal*and R. A. Vazquez† Departamento de Física de Partículas & Instituto Galego de Física de Altas Enerxías Universidade de Santiago de Compostela, 15782 Santiago, Spain (Received 8 October 2018; published 2 January 2019) A formalism is presented in which the radiation of photons off high energy electrons during a multiple scattering process with finite condensed media can be evaluated for a general interaction. We show that the arising Landau-Pomeranchuk-Migdal suppression for finite size targets saturates at some characteristic photon energy. Medium coherence effects in the photon dispersion relation can be also considered leading to a dielectric suppression or transition radiation effects in the soft part of the spectrum. The main results of our formulation are presented for a Debye screened interaction and its well-known Fokker- Planck approximation, showing that for finite size targets or for the angular distributions of the final particles the differences between both scenarios cannot be reconciled into a single redefinition of the medium transport parameter (ˆ q). Our predictions are in very good agreement with the experimental data collected at SLAC. DOI: 10.1103/PhysRevD.99.016002 I. INTRODUCTION The Landau-Pomeranchuk-Migdal (LPM) suppression is a well-known effect that has been extensively studied. Interference phenomena in a multiple scattering scenario was initially considered by Ter-Mikaelian as the mechanism regulating the amount of scattering centers which can coherently emit as a single bremsstrahlung source [1].A classical evaluation of this effect for a semi-infinite medium was soon introduced by Landau and Pomeranchuk [2,3] and later completed by Migdal [4] for the quantum case by means of a Boltzmann transport equation for the electron. This calculation has shown [5] that except for the spin corrections for hard photons, the LPM suppression for an averaged target still agrees with the expected classical behavior of the infrared divergence. Further and more recent developments in various approaches have been introduced since then by Blankenbecler and Drell [6–8], Zakharov [9–15], the Baier-Dokshitzer-Mueller-Schiff- Peign´e group (BDMPS) [16], Baier and Katkov [17–19], and Wiedemann and Gyulassy [20], and extensive reviews were presented in [21,22]. We note, however, that all the existing calculations were done in the Fokker-Planck approximation, which both in the Boltzmann transport approach [4,16] and in the path integral formulation [9,20] lead to a Gaussian distribution of momenta. In this approximation, then, the transport properties of the medium have to be adequate to take into account the neglected large momentum tails of the original Debye screened or Coulomb interactions. Few works, on the other hand, considered the finite target case, which has always been problematic and sometimes misunderstood lacking a general formulation. Also, the angular distribution of the final particles has not been studied in general [20]. Taking into account these remarks, no result has ever been given beyond the Fokker- Planck approximation, that also accounts for the transverse photon and electron spectrum, and which includes in a natural way the finiteness of the target. In this paper, we have developed a formalism and a Monte Carlo code which allows for the computation of the bremsstrahlung spectrum of finite targets, arbitrary interactions and with a full control of the kinematics of the process, so that specific cuts on momenta of the final electron and photon can be applied. In Sec. II A we will briefly explain the LPM effect in a classical scenario and then we will review the formalism of the quantum case by considering finite lengths, general interactions and the angular distribution of the final particles. In Sec. II B we will make connection with the already existing works by showing that for semi-infinite mediums within the Fokker- Planck approximation the result of Migdal [4] and the equivalent rederivations by Bell [5] and Zakharov [10] are exactly reproduced in our formalism. In Sec. III, we will present and compare our results with the experimental data of SLAC [23]. Finally, we end in Sec. IV with some conclusions. *xabier[email protected] †[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 99, 016002 (2019) 2470-0010=2019=99(1)=016002(9) 016002-1 Published by the American Physical Society II. FORMALISM AND CALCULATION A. Finite length and general interactions In this section we will introduce the general formalism applicable to finite targets and general interactions. It has been predicted by Ter-Mikaelian [1] and Landau and Pomeranchuk [2] that at high energies the Bethe- Heitler cross section [24] stops being applicable to extended media. In order to understand this phenomenon we start with the emission amplitude for a process consisting in a collision with (n) sources MðnÞ em ¼−ie Zd4y¯ ΨðnÞ fðyÞγμAλ μðyÞΨðnÞ iðyÞþOðe2Þ;ð1Þ where in the Coulomb gauge Aλ μðyÞ¼NðkÞϵλ μeik·yis a free photon of momentum k, polarization ϵλ¼ð0;ϵλÞand normalization NðkÞ¼ ffiffiffiffiffiffiffiffiffiffiffi 2π=ω p,ΨðnÞ i;f ðxÞthe incoming and outgoing electron wave functions under the external field of the medium and e¼ffiffiffi α pthe electron charge. Since in the ω→0limit the number of photons diverges, in virtue of the soft photon theorem [25], the classical approximation holds [26] and we can replace JkðxÞ¼¯ ΨðnÞ fðxÞγkΨðnÞ iðxÞ→JkðxÞ≡vkðtÞδ3ðx−xðtÞÞ; ð2Þ where vðtÞ≡_ xðtÞis the electron velocity. Using the completeness relation X λ ϵλ iϵλ k¼δik −kikk ω2ð3Þ we can find a correspondence with a classical amplitude [5,27] MðnÞ em ¼−ieNðkÞZþ∞ −∞ dtk ω×vðtÞeiωt−ik·xðtÞ;ð4Þ One can consider the integration over time as the point in which the photon is emitted. This observation becomes manifest by letting the electron describe a discretized trajectory, with velocities vjfor j¼1;…;n cþ1and piecewise path xj¼xj−1þvj−1ðtj−tj−1Þ, where ncis the number of collisions. Equation (4) then produces MðnÞ em ¼eNðkÞ1 ωX nc j¼1 δjeiφj;ð5Þ where we find a superposition of ncsingle Bethe-Heitler like amplitudes [24] of the form δj≡k×vjþ1 ω−k·vjþ1 −vj ω−k·vj;ð6Þ interfering with a phase φj≡ωtj−k·xj. The evaluation of the square of (5) leads to a total emission intensity between the photon solid angle Ωkand ΩkþdΩkgiven by ωdI dωdΩk¼e2 ð2πÞ2X nc j¼1 δ2 jþ2ReX nc j¼1X j−1 i¼1 δj·δieiφj i;ð7Þ where we have split the sum in a diagonal and a nondiagonal contribution. The interfering behavior of the above sum is governed by the phase change between two arbitrary collisions or emission elements φj i≡φj−φi¼kμðxμ j−xμ iÞ¼Zzj zi dz kμpμðzÞ p0 ≃ωð1−βÞZzj zi dz þωZzj zi dz δp2ðzÞ 2βp2 0 ;ð8Þ where p0is the initial electron energy, δpðzÞis the accumulated momentum change of the electron at zwith respect to the photon direction and β¼jvj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m2 e=p2 0 p the electron velocity. This phase can be made maximal for large emission angles and/or photon frequencies, provided that φiþ1 i≫1for any two consecutive collisions. In that case the nondiagonal sum in (7) cancels and we are left with a totally incoherent superposition of (nc) single Bethe- Heitler intensities, with a maximal intensity of ωdIsup dωdΩk¼e2 ð2πÞ2X nc j¼1 δ2 j:ð9Þ In this regime emission decouples and all the scatterings can be considered to be independently emitting. In the opposite case, when the emission angle and/or photon energy are small so that the phase vanishes, the internal structure of the scattering is irrelevant. This observation becomes manifest by setting φj¼0in (5), so we are left with the first and last terms only and intensity acquires the minimum value ωdIinf dωdΩk¼e2 ð2πÞ2X nc j¼1 δj 2 ;ð10Þ which can be interpreted as a Bethe-Heitler intensity with a final velocity vncþ1due to the coherent deflection with all the medium centers. In this regime the entire medium acts as a single independent emission element. This behavior is a consequence of the well known soft photon theorem [25,28], although in the LPM literature it is known as the Ternovskii-Shul’ga-Fomin emission [29,30]. The suppression from the superior (incoherent) plateau of radiation (9) to the inferior (coherent) plateau (10) is known as the LPM effect for mediums of arbitrary size. Notice that in the X. FEAL and R. A. VAZQUEZ PHYS. REV. D 99, 016002 (2019) 016002-2 infinite medium limit (nc≫1) the coherent plateau can be neglected, since the soft photon theorem is not observed and then the suppression is infinite. The above classical arguments can be made quantitative and hold also for a quantum evaluation of the amplitude. By Fourier transforming electron states Ψi;fðxÞto the momentum space, we can write for the quantum amplitude (1) MðnÞ em ¼eNðkÞZd3pðzÞ ð2πÞ3Zl 0 dz exp ikμpμðzÞ p0d dz ×Sel snsðpðlÞ;pðzÞ;l; zÞfλ ss0ðzÞ kμpμðzÞ ×Sel s0s0ðpðzÞþk; pð0Þ;z; 0Þ;ð11Þ where we used the shorthand notation fλ ss0ðzÞ≡ϵλ μðkÞp0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi me p0−ω r¯ usðpÞγμus0ðpþkÞffiffiffiffiffiffi me p0 r:ð12Þ Here Sel s2s1ðp2;p 1;l2;l 1Þstands for the beyond eikonal evaluation of the elastic amplitudes for an electron to go from momentum p1to p2and from spin s1to s2due the amount of matter between l1and l2, thus given by SðnÞ sns0ðpn;p 0;zn;z 0Þ≡Y n−1 k¼1Zd3pk ð2πÞ3Y n i¼1 SnðziÞ sisi−1ðqiÞ; ð13Þ where we discretized the medium and thus qi≡pi−pi−1 is the 4-momentum transfer at the layer (i)ofnðziÞ scattering centers, and sum over intermediate spins is assumed. At high energies the amplitude for a change of momentum qat the ith layer verifies [31] SnðziÞ sisi−1ðqÞ¼2πδðq0Þδsi si−1βe−iqkzi ×Zd2yte−iqt·yexp−ig βX nðziÞ k¼1 χðyt−rk tÞ;ð14Þ where ytis an impact parameter, and rk¼ðrk t;z kÞis the position of the scattering center at the layer k. The external field characterizing the medium is given by (n) single Debye static sources with screening μd≃αmeZ1=3, coupled with strength g¼Zαto the electron, of the form χð1Þ 0ðxÞ≡Zþ∞ −∞ dsAð1Þ 0ðxÞ;A ð1Þ 0ðxÞ¼Zα jxje−μdjxj: ð15Þ The amplitude (11), which corresponds to a sum of the single emission elements shown in Fig. 1, can be squared and averaged over medium configurations of infinite transverse size R→∞in a finite length l, summed over final states, and averaged over initial states, leading to an intensity of emission in the photon solid angle Ωkand ΩkþdΩkand per unit of medium transverse size and unit time of ωdI dωdΩk¼e 2π2Y n i¼1Zd3pi ð2πÞ3Y n k¼1 ϕðδpk;δzÞ ×hnðyÞX n j¼1 δn jeiφj 0 2 þhsðyÞX n j¼1 δs jeiφj 0 2; ð16Þ where the spin nonflip currents δn jare given by (6) and the spin flip currents are given by δs j≡1 1−βk ˆ k·vjþ1 −1 1−βk ˆ k·vj :ð17Þ Here we have introduced explicitly the possibility of a medium with a refractive index, by using βk, the photon velocity. The functions hnðyÞand hsðyÞare the diagonal and nondiagonal sum in spins and helicities of the squared emission vertex (12), given by hnðyÞ¼1 2ð1þð1−yÞ2Þ;h sðyÞ¼1 2y2;ð18Þ and y¼ω=p0is the fraction of energy carried by the photon. They produce two contributions of the same order, the last one, however, only noticeable when y≈1due to hsðyÞ. In what follows we will neglect this contribution by assuming that y≪1. Within the same approximation we will assume also that the electron 4-momentum change in the emission vertex is negligible and β¼1unless otherwise required. The local elastic weights arising in the averaging of the square of (13) are given by ϕðδp;δzÞ¼expð−n0ðzÞδzσð1Þ tÞð2πÞ3δ3ðδpÞ þ2πδðδp0ÞβΣ2ðδp;δzÞ;ð19Þ where we can define the no collision probability expð−nðzÞδzσð1Þ tÞin the layer of length δzand density FIG. 1. Diagrammatic representation of the single emission elements appearing in a discretization in the variable z, of the amplitude MðnÞ em given at Eq. (11). TRANSVERSE SPECTRUM OF BREMSSTRAHLUNG IN …PHYS. REV. D 99, 016002 (2019) 016002-3 n0ðzÞtimes the forward distribution δ3ðδpÞ, and the collisional distribution Σ2ðq;δzÞafter an incoherent scattering with the centers in δz Σ2ðq;δzÞ≡Zd2xe−iq·xexpð−n0ðzÞδzσð1Þ tÞ ×ðexpðn0ðzÞδzσðxÞÞ−1Þ;ð20Þ which satisfies a Moliere’s equation with boundary condition Σ2ðq;0Þ¼0. The required single elastic cross sections at (19) and (20) can be shown to satisfy σð1Þ t≡ σð0Þwhere, at leading order in Zαusing (15) σðxÞ≡4πðZαÞ2 β2μ2 d μdjxjK1ðμdjxjÞþOðZαÞ3:ð21Þ The expression (16) is our main result. It can be directly used for numerical evaluation for arbitrary medium lengths or medium properties. In addition, it can be shown that in the continuum limit it can be cast as a combination of six path integrals [32]. These path integrals reduce to the well known Migdal’s or Zakharov results in the appropriate limits. In addition to a numerical evaluation of (16), we will derive also an heuristic formula for finite size targets to qualitatively understand the interference phenomena. The coherence length defined by the phase modulates the amount of scatterers which can be considered a single and independent unit of emission in the squared amplitude. We then define the length δl¼zj−ziin which the phase becomes larger than unity, which using (8) becomes φj i≃ω 2p2 0ðm2 eδlþˆ qðδlÞ2Þ≡1;ð22Þ then we get δlðωÞ≡m2 e 2ˆ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ8ˆ qp2 0 m4 eω s−1!:ð23Þ We also define the frequency ωcat which the coherence length becomes equal to l, the medium total length, thus ωc≃p2 0=ðm2 elþˆ ql2Þ, and the frequency ωsin which the coherence length becomes equal to a mean free path λ, ωs≃p2 0=ðm2 eλþˆ qlλ=2Þ. Since we assume the medium to be finite we further impose to (23) δlðωÞ¼lfor ω>ωc.In the coherence length δlðωÞthe internal scattering structure is irrelevant since the phase can be neglected, and the centers in δlðωÞact coherently like a single scattering source with the total equivalent charge in δlðωÞ. Since in the entire medium lthere are l=δlðωÞcoherence lengths, then we write the incoherent sum ωdI dωðlÞ¼ l δlðωÞe2ZdΩk ð2πÞ2Zd3δp ð2πÞ3 ×ðhnðyÞjδn 1j2þhsðyÞjδs 1j2Þϕðδp;δlðωÞÞ: ð24Þ By integrating in the photon solid angle Ωkand using (19), we find ωdIðlÞ dω¼l δlðωÞ e2 π2Zπ 0 dθsinðθÞFðθÞΣ2ðδp;δlðωÞÞ;ð25Þ where the electron momentum change is jδpj¼ 2p0βsinðθ=2Þand FðθÞ¼"1−β2cos θ 2βsinðθ=2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−β2cos2ðθ=2Þ p × log"ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−β2cos2ðθ=2Þ pþβsinðθ=2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−β2cos2ðθ=2Þ p−βsinðθ=2Þ#−1#: ð26Þ This last integral (25) can be numerically evaluated and the resulting intensity is exact for ω≫ωsand ω≪ωc.A simple interpolation formula in the Fokker-Planck approximation can be obtained from (25) by integrating its two asymptotic values, i.e., δlðωÞ≳land δlðωÞ≲λ, and then interpolating both regions. One finds ωdIðlÞ dω¼l δlðωÞ 2e2 π 1þnmðωÞ 3AþnmðωÞlogð1þAnmðωÞÞ;ð27Þ where nmðωÞ≃2ˆ qδlðωÞ=m2 eis a measure of the accumulated transverse momentum in a coherence length and A¼e−ð1þγÞ, with γEuler’s constant. The general behavior of these results can be summarized in Fig. 2where the photon intensity is pictured as a function of the photon frequency. Above the saturation frequency ωs the photon resolves all the internal structure of the scattering, the medium emits as a total incoherent sum of nc¼l=λ Bethe-Heitler intensities, where ncis the average number of collisions. In this regime, the photon intensity scales with l. Notice that total suppression can occur provided ωs becomes larger than p0, which causes that electrons with energies larger than plpm 0¼m2 e=n0σð1Þ texperiment the bremsstrahlung suppression in all their spectrum. For smaller frequencies the number of independent emitters, using (23), decreases with ffiffiffiffi ω pwhereas the charge of each element logarithmically grows with logð1=ffiffiffiffi ω pÞ. This suppression stops at ωc, where the coherence length (23) acquires the maximum value l, the medium emits as a single entity and intensity saturates to Weinberg’s soft photon theorem [25]. The presence of a medium modifies X. FEAL and R. A. VAZQUEZ PHYS. REV. D 99, 016002 (2019) 016002-4 the photon dispersion relation and substantially changes this picture in the soft limit. For the energies considered here the photon has velocity given by β2ðkÞ¼1−ω2 p ω2;ð28Þ where ω2 p≃4πZαn0=me≡m2 γis the plasma frequency, which can be interpreted as a photon mass mγ. This scenario induces an additional source of suppression due to the fact that the wavelength of a photon of frequency ωis now larger than in the vacuum case and thus kμpμðωpÞ≃kμpμð0Þþm2 γ 2ω:ð29Þ This extra term further suppresses the coherent plateau at ω<ωde, where kμpμð0Þ≡m2 γ=2ωde, i.e., ω2 de ¼ω2 plωc, since the denominators of (6) and (17), defined by (29), grow for smaller frequencies. This suppression is called the dielectric effect and holds for a totally homogeneous space or infinite medium. However, if the electron passes through vacuum to a medium and then again to vacuum, or in general through structured targets where density cannot be considered constant, then the definition (29) becomes local for each photon emission point. The photon emitted at the last leg (the vacuum) then satisfies mγ¼0, whereas the first leg photon satisfies mγ≠0, thus it can be shown that an interference destroys the dielectric suppression in the coherence plateau, dramatically enhancing the intensity for ω<ωde. This is called transition radiation [27,33]. Both of these effects have been implemented in our formalism and Monte Carlo. In Fig. 2we show qualitatively the dielectric effect and the transition radiation together with their characteristic frequency ωde. B. Semi-infinite length and Fokker-Planck approximation In Migdal’s calculation of the intensity [4] the angular distribution of photons is not considered, the medium is assumed to be semi-infinite l→∞and the Moliere’s equation satisfied by the elastic distributions is replaced by the Fokker-Planck equation. As a result, the electron distributions of momenta, which have a Yukawa form for a Debye screened interaction (15), are replaced by Gaussian distributions and the large pttails are neglected. We will briefly explain here how within these approximations Eq. (16) leads to Migdal’s result [4] and also the equivalent rederivations by Bell [5] and Zakharov [10]. We assume from here onwards a constant density n0ðzÞ≡n0so from (19) the electron mean free path is constant and reads λ≡1=n0σð1Þ t. In this elastic propagation the electron acquires a squared momentum transfer additive with the traveled length l. Indeed, from (19) we find ∂ ∂lhδp2ðlÞi ¼ n0σð1Þ thδp2ðδlÞi≡2ˆ q; ð30Þ where we defined the transport parameter ˆ q. Using (30) we can write then ˆ q¼ðη=2Þ×n0σð1Þ tμ2 d, where ηis a parameter to be determined a posteriori. Then the Fokker-Planck approximation for (20) is obtained as n0lσðxÞ≃n0lσð0Þ−1 2 ˆ qlx2ð31Þ so that (20) acquires a Gaussian form ΣG 2ðq;δzÞ≡2π ˆ qδzexp −q2 2ˆ qδz:ð32Þ Inserting (32) in (19) the integration in the internal momenta at (16) is now straightforward. It can be shown [32] that after taking the continuous limit δz→0, integrating in the final photon momentum and taking the l→∞ limit, the intensity of photons for y≪1is given by ωdIðlÞ dω¼l2e2 πffiffiffiffiffiffiffiffi ˆ qω 2p2 0 sZ∞ 0 dz exp −z ffiffiffi 2 ps ×sinz ffiffiffi 2 psþcosz ffiffiffi 2 ps ×1 z2−1 sinh2ðzÞ;ð33Þ which agrees with Migdal/Bell/Zakharov predictions [4,5,10]. In Migdal’s calculations, the parameter sis defined recursively, in terms of an implicit expression FIG. 2. Schematic representation of the bremsstrahlung regimes for several scenarios. Totally incoherent Bethe-Heitler superposition (BH), Landau-Pomeranchuk-Migdal suppression (LPM), totally coherent Bethe-Heitler superposition (mγ¼0), dielectric suppression (DE) and transition radiation (TR). See text for definition of the characteristics frequencies ωc,ωs,ωde. TRANSVERSE SPECTRUM OF BREMSSTRAHLUNG IN …PHYS. REV. D 99, 016002 (2019) 016002-5 which is related to the definition of ˆ q. Here, we define s≡ð2p0=m2 eÞffiffiffiffiffiffiffiffiffi ˆ q=ω pand introduce the parameter ηto account for the freedom in the definition of ˆ q. An estimation of the value of ηcan be given by considering the momentum transfer in a single collision δl≲λwhich is given, using (19),by hδp2ðδlÞi ¼ μ2 d2log2p0 μd−1¼μ2 dη;ð34Þ where the correction ηto μ2 dtakes into account the long tail of the Debye interaction (15) and a maximum momentum transfer of jδpj¼2p0is allowed in a single collision. High momentum changes are suppressed at high energies, however, due to the functions (6) and (17) in (16).We have checked that a maximum momentum transfer of jδpj≃2.5mematches the single emission and then we write for ηin (34) instead η¼2log2.5me μd−1¼2log2.5 αZ1=3−1:ð35Þ This effective momentum transfer under bremsstrahlung agrees with Bethe’s[24] estimation η¼2logð183=Z1=3Þ within less than 3% deviation in the range Z¼ð1;100Þ. The above relations hold, however, for the single scattering regime δz≤δl, so they can be used only to fix ηand thus ˆ q in the incoherent plateau. For the coherent plateau a (a) (b) (c) FIG. 3. Differential spectrum of photons emitted by an electron of p0¼8GeV traversing a sheet of gold of l¼0.023 mm for photon angles θk¼0.01=γ(a), θk¼2=γ(b) and θk¼10=γ(c), where γ≡p0=me, in the Monte Carlo (MC) evaluations of (16) in the Debye interaction (yellow), in the Fokker-Planck approximation with η¼8(purple), η¼4(dark grey), and η¼2(light grey), together with the corresponding path integral (PI) limits of (16) in the Fokker-Planck approximation (dot-dashed lines). Bands show the statistical uncertainty of the Monte Carlo. FIG. 4. Intensity of photons emitted by an electron of p0¼ 8GeV (top) and p0¼25 GeV (bottom) after traversing a sheet of gold of length l¼0.0038 mm in the Monte Carlo (MC) evaluation of (16) in the Debye interaction (squares), the Fokker- Planck approximation with η¼8(circles), η¼4(diamonds), and η¼2(triangles). Also shown our heuristic formula (25) and the Migdal prediction (33). X. FEAL and R. A. VAZQUEZ PHYS. REV. D 99, 016002 (2019) 016002-6 medium-length dependent fit for ηhas to be employed. Correspondingly, a single Fokker-Planck approximation can not fit both the upper and lower ends of the bremsstrahlung spectrum unless the medium length is very large, in which case the lower plateau occurs at very low frequencies and can be neglected. In our numerical calculations, we will choose the value of ηto match the Debye calculations in the incoherent plateau. III. RESULTS Expression (16) can be numerically evaluated for arbitrary interaction models, for finite size and arbitrarily structured targets with dielectric suppression and transition radiation effects included. We have developed a Monte Carlo code to evaluate this intensity by means of discretized paths with a typical step of δz¼0.1λ.Ina typical run we computed 104paths for 50 frequencies and 100 photon angles, spanning from ∼103steps for the shortest medium to ∼105steps for the largest. In order to check that our results are correct we implemented also the Fokker-Planck approximation (32) for (19) in this discretized approach and compared with the δz→0limit of (16), which within this approximation produces six integrable Gaussian path integrals extending the Boltzmann transport approach [4] to finite mediums [20]. We present our result for the Debye and Fokker-Planck cases for targets of lengths l¼0.0038, 0.023 and 0.2 mm, corresponding to an average of nc¼142, 862, and 7502 collisions, respectively, for electrons of p0¼8and 25 GeV, in order to compare to the SLAC data presented in [23,34,35].A systematic study and comparison with other experimental results will be presented elsewhere [32]. For gold we obtain an estimate for the Debye mass of μd¼16 keV, a transport parameter of ˆ q¼ðη=2Þ×1.89 keV3, and fix the effective parameter η¼8. The plasma frequency is ωp¼0.080 keV (see also [36]). In Fig. 3we show the differential photon intensity as a function of the photon energy for various fixed emission angles for an electron of p0¼8GeV traversing a gold sheet of l¼0.023 mm. The path integral limit is also shown, and an excellent agreement with the Fokker-Planck Monte Carlo evaluation is found. At low angles, the Fokker-Planck approximation overestimates the intensity by ∼20%. However, at larger angles the FIG. 5. Intensity of photons emitted by an electron of p0¼ 8GeV (top) and p0¼25 GeV (bottom) after traversing a sheet of gold of length l¼0.2mm in the Monte Carlo (MC) evaluation of (16) in the Debye interaction (squares), the Fokker-Planck approximation with η¼8(circles), η¼4(diamonds) and η¼2(triangles). Also shown our formula (25) and the Migdal prediction (33). FIG. 6. Intensity of photons emitted by an electron of p0¼ 8GeV (top) and p0¼25 GeV (bottom) after traversing a sheet of gold of length l¼0.023 mm in the Debye interaction (open squares) and the Fokker-Planck approximation with η¼8 (circles) with the dielectric and transition radiation effect included, compared to SLAC experimental data [23] (solid circles) rescaled. TRANSVERSE SPECTRUM OF BREMSSTRAHLUNG IN …PHYS. REV. D 99, 016002 (2019) 016002-7 Fokker-Planck approximation underestimates the intensity, in particular only half of the real emission is obtained for θ¼10γ−1. In Fig. 4we show the angular integrated spectrum for a sheet of gold of l¼0.0038 mm for electron energies of p0¼8and 25 GeV. We see that the parameter η¼8in the Fokker-Planck approximation matches the incoherent plateau but mismatches the coherent plateau. Also shown are the expression (25) and the Migdal prediction (33) both in the Fokker-Planck approximation. The predicted characteristic frequencies are ωc¼8MeV and ωs¼1.1GeV for p0¼8GeV, and ωc¼80 MeV and ωs¼11 GeV for p0¼25 GeV, being in good agreement with the obtained Monte Carlo results. In Fig. 5we show the same results for a sheet of gold of l¼0.2mm. We see that Migdal prediction becomes a good approximation for nc≥104, i.e., when the coherent plateau can be neglected. For this length we predicted ωc¼8keV and ωs¼60 MeV for p0¼8GeV, and ωc¼80 keV and ωs¼588 MeV for p0¼25 GeV. All these values are in well agreement with the Monte Carlo evaluation. In Fig. 6we show the dielectric and transition radiation effect implementation both in the Debye interaction and the Fokker-Planck approximation, and compare with SLAC data [23], for a sheet of gold of l¼0.023 mm and electron energies of p0¼8and 25 GeV. The characteristic frequency predictions ωde ¼0.6MeV for p0¼8GeV and ωde ¼1.9MeV for p0¼25 GeV and the comparison with experimental data are in very good agreement. The LPM characteristic frequencies are given in this case by ωc¼0.48 MeV and ωs¼418 MeV for p0¼8GeV, and ωc¼4.7MeV and ωs¼4GeV for p0¼25 GeV. In Fig. 7we show the intensity of bremsstrahlung for electron with a final transverse momentum jδpj<4MeV or jδpj>4MeV. We see that the Fokker-Planck results do not reproduce well the Debye calculation. Although large differences could be expected for the case of jδpj>4MeV, as the Fokker-Planck approximation underestimate the long tails of the transverse momentum distribution, it is perhaps more surprising to find that also cutting at low ptproduces different results: the Fokker- Planck result overestimates the emission in this case at large frequencies. IV. CONCLUSIONS A formalism implemented with a Monte Carlo method has been presented which is able to evaluate the bremsstrahlung intensity in a multiple scattering scenario under a general interaction. We have also found an heuristic formula which describes the LPM effect for finite size targets. Dielectric and transition radiation effects related to effective photon masses in the medium dispersion relation are included in this formalism if needed. Our results reproduce the experimental data of SLAC. We have shown that the Fokker-Planck approximation does not fit well the differential angular spectrum, especially if kinematical cuts are applied in the final particles. In the integrated spectrum, the Fokker-Planck approximation fails to reproduce the spectrum. If the ˆ qis fixed using the incoherent plateau, then the coherent plateau is not well reproduced, unless a length dependent definition of the transport properties of the medium is used. ACKNOWLEDGMENTS We thank J. Alvarez-Muñiz, N. Armesto, C. Salgado, and J. Sanchez-Guillen for reading the manuscript and for discussions. We thank the grant María de Maeztu Unit of Excellence of Spain and the support of Xunta de Galicia under the Project No. ED431C2017. This paper has been partially done under the Project No. FPA2017-83814-P of Ministerio de Ciencia, Innovación y Universidades (Spain). FIG. 7. Intensity of photons emitted by an electron of p0¼ 8GeV after traversing a sheet of gold of length l¼0.023 mm from electrons with final transverse momentum pt>4MeV in the Debye interaction (triangles) and the Fokker-Planck approximation with η¼8(diamonds), and with pt<4MeV in the Debye interaction (squares) and the Fokker-Planck approximation with η¼8(circles). X. FEAL and R. A. VAZQUEZ PHYS. REV. D 99, 016002 (2019) 016002-8 [1] M. L. Ter-Mikaelian, The interference emission of highenergy electrons, Zh. Eksp. Teor. Fiz. 25, 296 (1953). [2] L. Landau and I. Pomeranchuk, The limits of applicability of the theory of bremsstrahlung by electrons and of the creation of pairs at large energies, Dokl. Akad. Nauk Ser. 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