aa r X i v : . [ qu a n t - ph ] F e b Elastic electron-proton scattering in the presence of a circularlypolarized laser field
I Dahiri , M Jakha , S Mouslih , , B Manaut , , S Taj ∗ Sultan Moulay Slimane University, Polydisciplinary Faculty,Research Team in Theoretical Physics and Materials (RTTPM), Beni Mellal, 23000, Morocco. Faculty of Sciences and Techniques, Laboratory of Materials Physics (LMP), Beni Mellal, 23000, Morocco.
February 2, 2021
Abstract
Owing to recent advances in laser technology, it has become important to investigate fun-damental laser-assisted processes in very powerful laser fields. In the present work and withinthe framework of laser-assisted quantum electrodynamics (QED), electron-proton scattering wasconsidered in the presence of a strong electromagnetic field of circular polarization. First, wepresent a study of the process where we only take into account the relativistic dressing of theelectron without the proton. Then, in order to explore the effect of the proton dressing, we fullyconsider the relativistic dressing of the electron and the proton together and describe them byusing Dirac-Volkov functions. The analytical expression for the differential cross section (DCS)in both cases is derived at lowest-order of perturbation theory. As a result, the DCS is notablyreduced by the laser field. It is found that the effect of proton dressing begins to appear at laserfield strengths greater than or equal to 10 V/cm and it therefore must be taken into account.The influence of the laser field strength and frequency on the DCS is reported. A comparisonwith the Mott scattering and the laser-free results is also included.
Keywords: QED processes, Laser-assisted, Elastic scattering.
1. Introduction
The electromagnetic force is one of the four basic forces that govern our universe, along with gravity,strong and weak nuclear forces, and it is responsible for the attraction or repulsion of charged bodies.If we can understand how the particles that make up our universe interact with the electromagneticfield and how their properties change during its presence, we will be able to understand very wellthe radiation-matter interaction and we will also be able to discover new properties of the particles.This will enable us to gather as much information as possible about this universe and exactly howthe matter that composes it behaves. Since its invention in the 1960s, the laser, which is a new lightsource with excellent monochromaticity, high brightness, strong direction and coherence, has becomewidely used in various daily applications due to the special properties of its radiation. It is currentlyconsidered an indispensable tool for investigating physical processes in various fields covering atomicand plasma physics as well as nuclear and high-energy physics. Charged particle collisions in thepresence of laser have attracted a lot of interest in recent decades because of their broad applicationsand their contribution to the fundamental understanding of the atomic structure in detail. The vast ∗ Corresponding author, E-mail: [email protected] W/cm [1], and as a result, renewed theoretical attentionhas been given to the study of the basic processes in the presence of a laser field. In particular,electron scattering processes have played a fundamental role in the development of science, bothfrom a theoretical and experimental point of view. Electron scattering is a basic physical processwhose importance has long been recognized in atomic and molecular physics. The development ofhigh-power femtosecond lasers has made it possible to experimentally probe these scattering processesand to observe the multiple photon processes. The physics of laser-assisted electron scattering hasbeen extensively reported in the scientific literature. In order to find an overview of this area ofresearch, we refer the reader to several textbooks [2–4] as well as to some review papers [5–8] andreferences therein. The majority of these studies focused on the non-relativistic aspects of suchprocesses and at low or moderate field intensities. However, with the use of ultra-powerful lasers,the non-relativistic approach is no longer valid and therefore the relativistic treatment is necessaryeven in the case of non-relativistic electrons. In some early works on the scattering of electronsby a potential in a strong electromagnetic plane wave [9–11], the effects of the electron spin wereneglected, and the electron was considered as a laser-dressed Klein-Gordon particle instead of a Diracparticle. With regard to the relativistic treatment of electron scattering processes, the spin effectsof electron are investigated in Mott scattering in the presence of a strong circularly polarized laserfield by Szymanowski et al. [12, 13]. However, as they also pointed out, the scattering process in thisconfiguration is not so efficient and rich in detail as in the case of linear or elliptical polarization.The same process is treated by Li et al. in the presence of a linearly polarized laser field [14] aswell as in the Coulomb approximation [15]. The case of elliptical polarization has been consideredby Attaourti et al. [16]. The effects of the spin polarization in Mott scattering are investigatedby Manaut et al. [17, 18]. The resonances in laser-assisted Møller scattering were analyzed in thepresence of a powerful laser field in [19]. The scattering of an electron by a positron in the lightwave field was studied in the works [20, 21]. The scattering of an electron by a muon has beenstudied in the first Born approximation in the presence of a laser field with linear [22] or ellipticalpolarization [23–25]. Furthermore, several authors have studied other interesting processes, whetherin weak interactions such as particle decays [26–30] and the elastic scattering of a muon neutrino byan electron in the presence of laser field [31], or in quantum electrodynamics such as laser-assistedbremsstrahlung [32, 33] and pair production [34, 35]. The present work is devoted to the electron-proton scattering in the presence of a circularly polarized electromagnetic field. Electron-protonscattering is a basic example that is part of lepton-hadron scattering, which is another type that isno less important than the other types of electron scattering. The study of lepton-hadron scatteringat high energies and large momentum transfers allows us to extract more detailed information onthe internal structure of hadrons. The electron-proton scattering differs from that of Mott by thefact that the target (proton) is freely movable and therefore the recoil effect must be taken intoaccount. In this paper, we assumed the proton to be a Dirac particle which has no internal structure(point-like particle) and that the energy of the incoming electron was not high enough so that theinternal structure of the proton could be detected. At an early stage, Rosenbluth discussed in detailthe theory of elastic electron-proton scattering at very high energies (several 100 MeV) and foundthat the formula of the differential cross section (DCS) has to be modified by introducing electricand magnetic form factors representing the internal structure of the proton [36]. After that, manycalculations were performed to apply the radiative corrections [37–39]. To the best of our knowledge,only two articles have addressed the electron-proton scattering, where the proton is not consideredto be fixed, in the presence of an electromagnetic field. In 2014 and in a brief report, the recoil effectin relativistic scattering of an electron from a freely movable proton in the presence of a linearlypolarized laser field has been investigated in the first Born approximation by Liu and Li [40]. Inanother recent paper, Wang et al. studied the electron scattering from freely movable proton andpositive muon µ + in the presence of a radiation field and examined the dependencies of the DCSon the laser field intensity and the electron-impact energy [41]. In these two studies, it was foundthat the laser significantly contributed to the enhancement of the DCS. Apart from this, some earlystudies dealt with the same subject and considered the proton to be static as an atom or a positiveion H + [42, 43]. In this paper, we will address the circular polarization case of the laser field andshow how could the change of this polarization affect the process during particle scattering. Ourmain contribution through this is that we have considered the dressing of the proton and exploredits effects on the DCS. The remainder of the paper is structured as follows: we begin, in section2., with the most basic results of the electron-proton scattering in the absence of the laser field inorder to map the main steps to be followed during our treatment and to avoid distracting the readerby referring to other references for laser-free results. In section 3., we consider the electron-protonprocess in the presence of circularly polarized monochromatic laser field without dressing of proton,we developed the detailed theoretical formalism required to obtain the DCS. In section 4., in additionto the dressing of the incident and scattered electrons, we take also into account the dressing of theproton which introduces new modifications on the DCS. The results obtained in all cases are presentedand discussed in section 5.. Finally, our conclusions are summarized in section 6.. We note here thatwe have used, throughout this work, the metric tensor g µν = diag(1 , − , − , −
1) and atomic units inwhich one has ~ = m e = e = 1 where m e is the rest mass of the electron. For all k , the bold style k is reserved for vectors.
2. Electron-proton scattering in the absence of a laser field
This e - P scattering can be expressed as e − ( p ) + P ( p ) −→ e − ( p ) + P ( p ) , (1)the labels p i are the associated momenta, where the odd indexes ( i = 1 ,
3) stand for electron whilethe even ones ( i = 2 ,
4) stand for proton. We follow the usual steps of calculations and give theS-matrix element corresponding to this process in lowest order: S fi = − i Z d x ψ p ( x ) /A ( x ) ψ p ( x ) , (2)where ψ p ( x ) and ψ p ( x ) are the plane Dirac waves that describe, respectively, the initial and finalelectrons and can be written, when normalized to the volume V , as follows: ψ p ( x ) = 1 √ E V u ( p , s ) e − ip .x ,ψ p ( x ) = 1 √ E V u ( p , s ) e − ip .x . (3) /A ( x ) = A µ ( x ) γ µ with A µ ( x ) is the four-potential produced by the proton and it has the form A µ ( x ) = − Z d y D F ( x − y ) ψ p ( y ) γ µ ψ p ( y ) , (4)where D F ( x − y ) is the Feynman propagator for the electromagnetic radiation given by [44]: D F ( x − y ) = Z d q (2 π ) e − iq ( x − y ) (cid:18) − q + iε (cid:19) . (5)Then A µ ( x ) = Z d y d q (2 π ) e − iq ( x − y ) q + iε (cid:2) ψ p ( y ) γ µ ψ p ( y ) (cid:3) , (6)where ψ p ( y ) and ψ p ( y ) are the free initial and final states of the proton and they have the sameform as the electron waves (3): ψ p ( y ) = 1 √ E V u ( p , s ) e − ip .y ,ψ p ( y ) = 1 √ E V u ( p , s ) e − ip .y . (7)In equations (3) and (7), s i and E i ( i = 1 , , ,
4) denote the spin and energy, respectively. Substi-tuting equation (6) and the wave functions into the S-matrix element (2), this reads S fi = − i √ E E E E V Z d x d y d q (2 π ) e i ( p − p − q ) .x e i ( p − p + q ) .y q + iε M fi , (8)where M fi = (cid:2) u ( p , s ) γ µ u ( p , s ) (cid:3)(cid:2) u ( p , s ) γ µ u ( p , s ) (cid:3) . (9)The integration over four-dimensional spatial coordinates d x and d y can be carried out at once,resulting in Z d x e i ( p − p − q ) .x = (2 π ) δ ( p − p − q ) , Z d y e i ( p − p + q ) .y = (2 π ) δ ( p − p + q ) , (10)and the reminder integration over d q can be performed simply as follows: Z d q (2 π ) (2 π ) δ ( p − p − q )(2 π ) δ ( p − p + q ) q + iε = (2 π ) δ ( p − p + p − p )( p − p ) + iε , (11)and the total S-matrix element (8) reads S fi = − i √ E E E E V (2 π ) δ ( p − p + p − p ) q + iε M fi , (12)where q = p − p is the relativistic four-momentum transfer in the absence of the laser field. Tocalculate the scattering cross-section, we multiply the squared S-matrix element | S fi | by the densityof final states, and divide by the observation time interval T and the flux of the incoming particles | J inc. | and finally one has to average over the initial spins and to sum over the final ones. We obtain dσ = V d p (2 π ) V d p (2 π ) | J inc. | T (2 π ) V T δ ( p − p + p − p )16 E E E E V q |M fi | , (13)where |M fi | = 14 X s i |M fi | . (14)With the help of the following formula [44] d p E = 2 Z + ∞−∞ d p δ ( p − M c )Θ( p ) , (15)with Θ( p ) = (cid:26) p >
00 for p < M = 1836 .
152 672 45 × m e is the rest mass of the proton, the cross-section becomes dσ = 18(2 π ) E E E | J inc. | V q Z δ ( p − M c ) d p |M fi | , (16)with p + p − p − p = 0. By considering the flux incident of electron in the rest frame of proton | J inc. | = | p | c /E V and using d p = | p | d | p | d Ω f and E dE = c | p | d | p | , we get dσd Ω f = 18(2 π ) c M q | p || p | Z δ ( p − M c ) dE |M fi | . (17)The integration over dE can be performed using the following familiar formula [44] Z dxf ( x ) δ ( g ( x )) = f ( x ) | g ′ ( x ) | (cid:12)(cid:12)(cid:12)(cid:12) g ( x )=0 . (18)Thus we get (cid:18) dσd Ω f (cid:19) laser-free = 116(2 π ) M c q | p || p | |M fi | | g ′ ( E ) | , (19)where g ′ ( E ) = M + E c − E | p | c | p | F ( θ i , θ f , ϕ i , ϕ f ) , (20)with F ( θ i , θ f , ϕ i , ϕ f ) = cos( ϕ i ) sin( θ i ) cos( ϕ f ) sin( θ f ) + sin( θ i ) sin( ϕ i ) sin( θ f ) sin( ϕ f )+ cos( θ i ) cos( θ f ) , (21)and |M fi | = 14 T r [( c/p + c ) γ µ ( c/p + c ) γ ν ] T r [( c/p + M c ) γ µ ( c/p + M c ) γ ν ] . (22)
3. Electron-proton scattering in the presence of a laser field withcircular polarization
Let us now consider the e - P process (1) and assume that this scattering occurs in the presence of acircularly polarized monochromatic laser field, which is described by its classical four-potential A ( φ )that satisfies the Lorentz gauge condition ∂A ( φ ) = 0. It is expressed as follows: A µ laser ( φ ) = a µ cos( φ ) + a µ sin( φ ) , φ = ( k.x ) , (23)where k = ( ω/c, k ) is the wave 4-vector ( k = 0), φ is the phase of the laser field and ω its frequency.The polarization 4-vectors a µ and a µ are equal in magnitude and orthogonal such that: a µ = (0 , | a | , , ,a µ = (0 , , | a | , , (24)which implies ( a .a ) = 0 and a = a = a = −| a | = − ( c E /ω ) where E is the laser field strengthand c ≃
137 a.u is the velocity of light in the vacuum. The Lorentz gauge condition applied to thefour-potential yields k µ A µ = 0 , (25)which implies ( k.a ) = ( k.a ) = 0, meaning that the wave vector k is chosen to be along the z -axis. In such conditions, the electron state in the laser field is described by the following relativisticDirac-Volkov functions [45], where i = 1 , ψ p i ( x ) = (cid:18) /k /A c ( k.p i ) (cid:19) u ( p i , s i ) √ Q i V e iS ( q i ,x ) , (26)where S ( q i , x ) = − q i .x − a .p i c ( k.p i ) sin( φ ) + a .p i c ( k.p i ) cos( φ ) . (27) u ( p i , s i ) is the free electron bispinor normalized as ¯ u ( p i , s i ) u ( p i , s i ) = 2 c , and q i = ( Q i /c, q i ) is thedressed four-momentum acquired by the electron in the presence of the laser field q i = p i − a c ( k.p i ) k. (28)The square of this momentum is q i = m ∗ c , (29)where m ∗ is an effective mass of the electron inside the electromagnetic field m ∗ = r − a c . (30)The proton, due to its large mass, is much less influenced by the laser than the electron (at least,for the laser field strengths considered here). It will be treated, in the first Born approximation, as astructureless Dirac (spin- ) particle. Therefore, we will describe it in the first stage using the sameprevious unaffected plane waves given in equation (7). Then, the S-matrix element (2) becomes S fi = − i p Q Q E E V Z d x d y d q (2 π ) e i ( S ( q ,x ) − S ( q ,x ) − q.x ) e i ( p − p + q ) .y q + iε × (cid:20) u ( p , s ) (cid:18) /A/k c ( k.p ) (cid:19) γ µ (cid:18) /k /A c ( k.p ) (cid:19) u ( p , s ) (cid:21)(cid:20) u ( p , s ) γ µ u ( p , s ) (cid:21) . (31)The exponential term e i ( S ( q ,x ) − S ( q ,x )) can be recasted in a suitable form by introducing the followingparameters z = q α + α and φ = arctan( α /α ) , (32)where α = a .p c ( k.p ) − a .p c ( k.p ) ,α = a .p c ( k.p ) − a .p c ( k.p ) , (33)then we can write e i ( S ( q ,x ) − S ( q ,x )) = e i ( q − q ) .x e − iz sin( φ − φ ) . (34)Therefore, the S-matrix element becomes S fi = − i p Q Q E E V Z d x d y d q (2 π ) e i ( q − q − q ) .x e i ( p − p + q ) .y e − iz sin( φ − φ ) q + iε × (cid:2) u ( p , s ) (cid:0) C µ + C µ cos( φ ) + C µ sin( φ ) (cid:1) u ( p , s ) (cid:3)(cid:2) u ( p , s ) γ µ u ( p , s ) (cid:3) , (35)where C µ = γ µ − k µ a /kC ( p ) C ( p ) ,C µ = C ( p ) γ µ /k/a + C ( p ) /a /kγ µ ,C µ = C ( p ) γ µ /k/a + C ( p ) /a /kγ µ , (36)with C ( p i ) = 1 / c ( k.p i ). The three different quantities in equation (35) can be transformed by thewell-known identities involving ordinary Bessel functions J s ( z ), where z is their argument definedin equation (32) and s , their ordre, will be interpreted in the following as the number of exchangedphotons. Using such transformations [46] in equation (35), the matrix element S fi becomes S fi = − i p Q Q E E V ∞ X s = −∞ Z d x d y d q (2 π ) e i ( q − q − sk − q ) .x e i ( p − p + q ) .y q + iε × (cid:2) u ( p , s ) (cid:0) C µ B s ( z ) + C µ B s ( z ) + C µ B s ( z ) (cid:1) u ( p , s ) (cid:3)(cid:2) u ( p , s ) γ µ u ( p , s ) (cid:3) . (37)where B s ( z ) B s ( z ) B s ( z ) = J s ( z ) e isφ (cid:0) J s +1 ( z ) e i ( s +1) φ + J s − ( z ) e i ( s − φ (cid:1) / (cid:0) J s +1 ( z ) e i ( s +1) φ − J s − ( z ) e i ( s − φ (cid:1) / i . (38)Integrating over d x , d y and d q , we get S fi = − i p Q Q E E V ∞ X s = −∞ (2 π ) δ ( p − p + q − q − sk ) q + iε × (cid:2) u ( p , s ) (cid:0) C µ B s ( z ) + C µ B s ( z ) + C µ B s ( z ) (cid:1) u ( p , s ) (cid:3)(cid:2) u ( p , s ) γ µ u ( p , s ) (cid:3) , (39)with q = q − q − sk is the relativistic four-momentum transfer in the presence of the laser field.Because electron-proton scattering experiments are often carried out with a fixed target in the labo-ratory frame, we will evaluate the differential cross section (DCS) in the rest frame of the proton byfollowing the same steps as in the absence of the laser field in the previous section. This yields dσ = V d q (2 π ) V d p (2 π ) | J inc. | T (2 π ) V T δ ( p − p + q − q − sk )16 Q Q E E V q |M sfi | . (40)With the use of the formula (15) and the flux | J inc. | = | q | c /Q V , we get (cid:18) dσd Ω f (cid:19) with laser = + ∞ X s = −∞ dσ s d Ω f = 116(2 π ) M c q ∞ X s = −∞ | q || q | |M sfi | | g ′ ( Q ) | , (41)with p − p + q − q − sk = 0, and where g ′ ( Q ) = (cid:20) Q c + M + sωc (cid:21) − Q c | q | (cid:20) | q | F ( θ i , θ f , ϕ i , ϕ f ) + sωc cos( θ f ) (cid:21) , (42)and |M sfi | = 14 T r [( c/p + c )Γ sµ ( c/p + c )Γ sν ] T r [( c/p + M c ) γ µ ( c/p + M c ) γ ν ] , (43)where Γ sµ = C µ B s ( z ) + C µ B s ( z ) + C µ B s ( z ) , (44)and ¯Γ sν = γ Γ s † ν γ , = C ν B ∗ s ( z ) + C ν B ∗ s ( z ) + C ν B ∗ s ( z ) , (45)where C ν = γ C † ν γ = γ ν − k ν a /kC ( p ) C ( p ) ,C ν = γ C † ν γ = C ( p ) /a /kγ ν + C ( p ) γ ν /k/a ,C ν = γ C † ν γ = C ( p ) /a /kγ ν + C ( p ) γ ν /k/a . (46)By using the FeynCalc Mathematica package designed for the trace calculations [47–49], we cancomfortably program and calculate numerically the two traces appearing above in equation (43).The result we have obtained is as follows |M sfi | = M | B s | + M | B s | + M | B s | + M B s B ∗ s + M B s B ∗ s + M B s B ∗ s + M B s B ∗ s + M B s B ∗ s + M B s B ∗ s , (47)where the argument z of the various ordinary Bessel functions has been dropped for convenience andthe nine coefficients M , M , M , M , M , M , M , M and M are explicitly expressed, in terms ofdifferent scalar products, by M = 4( k.p )( k.p ) (cid:2) a ( k.p )( k.p ) − a c (2 c (( k.p )( k.p ) + ( k.p )( k.p ) M ) + ( k.p ) × ( k.p )( p .p ) − k.p )( k.p )( p .p ) + ( k.p )( k.p )( p .p ) + ( k.p )( k.p )( p .p ) − k.p )( k.p )( p .p ) + ( k.p )( k.p )( p .p )) + 2 c ( k.p )( k.p )(2 c M + ( p .p ) × ( p .p ) − c ( M ( p .p ) + ( p .p )) + ( p .p )( p .p )) (cid:3) , (48) M = 4 c ( k.p )( k.p ) (cid:2) − a .p )( a .p )( k.p )( k.p ) + ( a .p )( − a .p )( k.p )( k.p )+ 4( a .p )( k.p )( k.p )) + a ( c (2( k.p )( k.p ) + (( k.p ) + ( k.p ) ) M ) + ( k.p ) × ( k.p )( p .p ) − k.p )( k.p )( p .p ) + ( k.p )( k.p )( p .p ) − ( k.p )( k.p )( p .p ) − ( k.p )( k.p )( p .p ) + ( k.p )( − ( k.p )( p .p ) − ( k.p )( p .p ) + ( k.p )( p .p ) − k.p )( p .p ) + ( k.p )( p .p ))) (cid:3) , (49) M = 4 c ( k.p )( k.p ) (cid:2) − a .p )( a .p )( k.p )( k.p ) + ( a .p )( − a .p )( k.p )( k.p )+ 4( a .p )( k.p )( k.p )) + a ( c (2( k.p )( k.p ) + (( k.p ) + ( k.p ) ) M ) + ( k.p ) × ( k.p )( p .p ) − k.p )( k.p )( p .p ) + ( k.p )( k.p )( p .p ) − ( k.p )( k.p )( p .p ) − ( k.p )( k.p )( p .p ) + ( k.p )( − ( k.p )( p .p ) − ( k.p )( p .p ) + ( k.p )( p .p ) − k.p )( p .p ) + ( k.p )( p .p ))) (cid:3) , (50) M = M = 2 c ( k.p )( k.p ) (cid:2) a ( k.p )(( a .p )(( k.p ) + ( k.p )) − a .p ) + ( a .p ))( k.p ))+ 2 c (( a .p )( k.p )( c ( − ( k.p ) + ( k.p )) M + ( k.p )( p .p ) + ( k.p )( p .p ))+ ( k.p )( − ( a .p )( k.p )(( p .p ) + ( p .p )) + ( a .p )( c (( k.p ) − ( k.p )) M + ( k.p )( p .p ) + ( k.p )( p .p )))) (cid:3) , (51) M = M = 2 c ( k.p )( k.p ) (cid:2) a ( k.p )(( a .p )(( k.p ) + ( k.p )) − a .p ) + ( a .p ))( k.p ))+ 2 c (( a .p )( k.p )( c ( − ( k.p ) + ( k.p )) M + ( k.p )( p .p ) + ( k.p )( p .p ))+ ( k.p )( − ( a .p )( k.p )(( p .p ) + ( p .p )) + ( a .p )( c (( k.p ) − ( k.p )) M + ( k.p )( p .p ) + ( k.p )( p .p )))) (cid:3) , (52) M = M = − k.p )( k.p ) (cid:2) c ( k.p )(( a .p )( a .p )( k.p ) + ( a .p )( a .p )( k.p ) + ( a .p ) × ( a .p )( k.p ) + ( a .p )( a .p )( k.p ) − a .p )( a .p ) + ( a .p ) × ( a .p ))( k.p )) (cid:3) . (53)
4. Proton-dressing effect
So far, we only take into account the dressing of the incident and scattered electrons. In this sectionand in order to realize the laser-dressing effect on the proton, we will take into account the relativisticdressing of both particles involved in this process, the electron and the proton, and thus it will bedescribed together by the Dirac-Volkov plane waves. This will give rise to a new trace to be computed,but it will become clear that taking into account the relativistic dressing of the proton will simplyintroduce a new sum on the number of photons l that will be exchanged between the proton and thelaser field. In this case, the four-potential A µ ( x ) produced by the proton becomes A µ ( x ) = Z d y d q (2 π ) e − iq ( x − y ) q + iε (cid:2) ψ q ( y ) γ µ ψ q ( y ) (cid:3) , (54)where ψ q ( y ) and ψ q ( y ) are the Dirac-Volkov plane waves of the initial and final states of the dressedproton. They can be written, where i = 2 ,
4, as follows: ψ p i ( y ) = (cid:18) e P /k /A ′ c ( k.p i ) (cid:19) u ( p i , s i ) √ Q i V e iS ( q i ,y ) , (55)where e P = − e > A ′ laser ( φ ′ ) is the four potential of the laser field feltby the proton A ′ µ laser ( φ ′ ) = a µ cos( φ ′ ) + a µ sin( φ ′ ) , (56)0where φ ′ = ( k.y ) is the phase of the laser field. Q i is the total energy acquired by the proton in thepresence of a laser field, such that Q i = E i − a ω c ( k.p i ) . (57)The quantity S ( q i , y ) is given by S ( q i , y ) = − q i .y + a .p i c ( k.p i ) sin( φ ′ ) − a .p i c ( k.p i ) cos( φ ′ ) . (58)We do not have to repeat the steps leading to the DCS since they have already been presented inthe previous sections. Proceeding in the same way as before, we obtain for the unpolarized DCS (cid:18) dσd Ω f (cid:19) dressed proton = + ∞ X s,l = −∞ dσ ( s,l ) d Ω f = 116(2 π ) M ∗ c q ∞ X s,l = −∞ | q || q | |M ( s,l ) fi | | h ′ ( Q ) | , (59)where M ∗ is the effective mass of the proton acquired inside the laser field M ∗ = r M − a c , (60)and q = q − q − sk . In this case, the energy-momentum conservation q + q − q − q − ( s + l ) k = 0must be satisfied. The term |M ( s,l ) fi | is expressed by |M ( s,l ) fi | = 14 T r [( c/p + c )Γ sµ ( c/p + c )Γ sν ] T r [( c/p + M c )Λ µl ( c/p + M c )Λ νl ] , (61)with Λ µl = C ′ µ B l ( z P ) + C ′ µ B l ( z P ) + C ′ µ B l ( z P ) , (62)and C ′ µ = γ µ − k µ a /kC ( p ) C ( p ) ,C ′ µ = − C ( p ) γ µ /k/a − C ( p ) /a /kγ µ ,C ′ µ = − C ( p ) γ µ /k/a − C ( p ) /a /kγ µ . (63) z P = [1 /c ( k.p )] p ( a .p ) + ( a .p ) is the argument of the ordinary Bessel functions expressed inthe rest frame of the proton, and the phase φ P is defined by φ P = arctan (cid:18) a .p a .p (cid:19) . (64)The quantity h ′ ( Q ) is the derivative of the function h ( Q ) with respect to the energy Q given by h ′ ( Q ) = (cid:20) Q c + M ∗ + ( l + s ) ωc (cid:21) − Q c | q | (cid:20) | q | F ( θ i , θ f , ϕ i , ϕ f ) + ( l + s ) ωc cos( θ f ) (cid:21) . (65)1
5. Results and discussion
In this paper, we present a relativistic theoretical calculation, in the first Born approximation, tostudy the electron-proton scattering in the absence and the presence of a circularly polarized elec-tromagnetic field. Before presenting the results and their physical interpretation, we would like tonote that the various figures of the present work are plotted with the help of the programming lan-guage Mathematica. In this section, we will present all the numerical results obtained in both withand without laser; during that, we will follow the same arrangement that we adopted through theprevious sections. We will begin with the results obtained in the absence of the laser field and thenthose obtained in its presence. All the DCSs are given in atomic units.
We will start our discussion in this section with a comparison between the two DCSs of electron-proton scattering and the well-known Mott scattering. The latter can be seen as a particular case ofthe former, in which the target is fixed and heavy so that it creates a Coulomb field around whichall charged particles are affected. It has been found that the DCS expression of the electron-protonscattering becomes that of Mott scattering when the energy of the incoming electron is very smallcompared to the rest mass of the proton [44]. At this limit, the proton does not recoil and can thereforebe considered as a source of a constant external field. Conversely, when the electrons have high energy(the ultrarelativistic electrons) the recoil of the proton will inevitably change the expression of theDCS significantly. In the following, we will try to add some insights regarding the atomic number Z . In figure 1, we plot the DCS of the electron-proton scattering in the absence of the laser given inequation (19) and compare it with that of Mott scattering at different atomic numbers. From thisfigure, we see that the DCS of Mott scattering increases with increasing atomic number Z . The shapeof its curve is similar to that of a Gaussian function defined in the theory of statistic and probabilityas a normal distribution, with a peak at the final scattering angle ( θ f = 0 ◦ ). It is well-known thatthe DCS of Mott scattering, as expressed in theory, is quadratically dependent on the atomic number Z . When the atomic number Z = 11, we notice that the two DCSs of Mott and electron-protonscattering are equal. But, with the increase of the atomic number ( Z >
11 and therefore the targetbecomes very heavy), the Mott scattering becomes the most dominant and the valid method to studythe process and vice-versa, which means that whenever the atomic number is lower than 11 (
Z < Z = 11 can be considered as a limit separating the two scatterings and determining the validity ofone against the other. Figure 2 shows the DCS changes of the electron-proton scattering withoutthe presence of the laser in terms of scattering angle but at different kinetic energies. As expected,due to the unitarity of the S-matrix element, the DCS must be inversely proportional to the energy ofthe incoming electron, i.e. it decreases with increasing energy or vice-versa. The effects of relativityand spin have greatly contributed to these discrepancies that appear between both relativistic andnonrelativistic regimes. Just to get a complete and comprehensive overview of the DCS dependenceon the incoming electron energy and its final scattering angle, we have inserted in figure 3 a three-dimensional drawing (contour-plot) in which we represent the changes of the DCS in terms of twovariables at the same time, the energy of the incoming electron and its final scattering angle θ f . Fromthis figure, it can be seen that the peak of the curve is located around the value θ f = 0 and thatthe order of magnitude of the DCS decreases with increasing energy of the incoming electron. We2 Mott ( Z = ) Mott ( Z = ) Mott ( Z = ) Mott ( Z = ) e - P without laser - - -
20 0 20 40 600.00000.00010.00020.00030.0004 θ f ( degree ) DC S ( a . u ) Figure 1: The DCS of e - P scattering without laser given in equation (19) as a function of thescattering angle θ f compared to that of Mott scattering for different atomic numbers Z . The electronkinetic energy is T = 0 .
511 MeV. The geometry parameters are θ i = φ i = 15 ◦ , φ f = 105 ◦ .will end our discussion in this section by representing, in figure 4, the changes in momentum transfer | q | = | p − p | in the absence of a laser in terms of the scattering angle and for different kineticenergies. Momentum transfer is the basic physical quantity that characterizes all scattering theoriesand experiments, and it depends on the momentum of the projectile and the scattering angle. Wenote that the momentum transfer is a symmetric function with respect to the axis θ f = 0 ◦ and itscurve takes the lowest value at the same angle. It also increases almost linearly with the rise inkinetic energy and scattering angle. This behavior was clearly reflected in the DCS shown in theprevious figures since the DCS, according to its expression in (19), is inversely proportional to q .As we have seen previously, the DCS exhibits a strong peak at θ f = 0 ◦ and falls off very rapidly atlarge momentum transfers. In this section, we will try to show what happens when we embed the electron-proton scatteringin a circularly polarized electromagnetic field and what changes can occur in the DCS when theelectromagnetic field is present. The latter is characterized by its parameters which are field strengthand frequency. We will present here the results obtained in the case where we only dress the electronwithout the proton. The first thing we are accustomed to check in such processes occurring in theelectromagnetic field is to make sure that the DCS in the presence of the laser field is exactly equalto the corresponding one in the absence of the laser when we take the zero field strength ( E = 0 a.u)and without photon exchange ( s = 0). We give here two values of the DCS at ( E = 0 a.u and s = 0),DCS= 2 . T = 2 . . × − a.u at T = 0 .
511 MeV, for thegeometry parameters: θ i = φ i = 15 ◦ , θ f = 0 ◦ and φ f = 105 ◦ . Table 1 contains the values of the DCS( dσ s /d Ω f ) as a function of the number of photons exchanged s at different field strengths. We havechosen the laser frequency at ~ ω = 1 .
17 eV. According to these values, it can be said that the numberof exchanged photons enhances when the strength of the electromagnetic field E is increased. This isevident by the cutoff number which equals ±
10 ( s = −
10 photons for the negative part and s = +10photons for the positive part) in the case of E = 10 V/cm = 0 .
002 a.u, while it is equal to ±
100 in3 T = T = T = T = T = - - -
20 0 20 40 600.000000.000050.000100.000150.00020 θ f ( degree ) DC S ( a . u ) Figure 2: The DCS of e - P scattering without laser as a function of the scattering angle θ f for differentelectron kinetic energies with the same geometry parameters as in figure 1.the case of E = 5 . × V/cm = 0 .
01 a.u. Regarding the order of magnitude of DCS, we cansee that it decreases with increasing field strength. We can also conclude from the table that the DCSshows a symmetry of values with respect to s = 0, indicating that the photon absorption processes( s >
0) are exactly the same as the photon emission processes ( s < T of the incoming electron from T = 2 . T = 0 .
511 MeV and maintain the fieldstrength at E = 10 V/cm, we notice that the multiphoton process becomes more important and thecutoff number rises to the value s = ± θ f for different numbers of photons exchanged. The numbers of exchanged photonsthat we have summed over it are as follows: s = ± , ± , ± , ±
10, where the notation s = ± N meansthat we have summed over − N ≤ s ≤ + N . This figure shows that at s = ±
10, the DCS (withlaser) is exactly close to the DCS (without laser). But for example, at s = ±
2, we have severalorders of magnitude due to the difference between the two approaches and the case without laseris always the highest. Returning to the first case s = ±
10, the convergence achieved here provesthe validation of the sum rule demonstrated by Kroll and Watson [50]. Based on this figure, wenotice that the DCS with laser tends to approach that of the laser-free with increasing the numberof exchanged photons until it is exactly equal to it at a specific number of photons called the cutoffnumber in which the sum rule is fulfilled. The cutoff number is defined as a determined number ofphotons from which the DCS ( dσ s /d Ω f ) falls abruptly ( dσ s /d Ω f = 0 a.u), which can be attributedto the well-known properties of the Bessel function, which diminishes strongly when its argument z (equation (32)) is approximately equal to its order s . The cutoff number, according to the table 1,is s = ±
10 at E = 10 V/cm and ~ ω = 1 .
17 eV. In the non-relativistic regime, there is a rapidconvergence towards the DCS without laser. The situation is difficult and becomes more complicatedin the relativistic regime due to the lack of high-speed computers, and therefore we cannot sum overa very large number of exchanged photons. Hence, as we are limited in our computing capabilities,the convergence cannot be achieved numerically in this case. For instance, at T = 0 .
511 MeV it isnecessary to sum over number of photons − ≤ s ≤ +300 to recover the laser-free result. The sumrule implies that under particular conditions the DCS summed over all exchanged photons, in the4 - - -
20 0 20 40 601.01.21.41.61.82.02.2 θ f ( degree ) I n c i den t e l e c t r onene r g y ( M e V ) DCS ( a.u ) Figure 3: The variations of the DCS of e - P scattering without laser given in (19) as a function ofboth incident electron energy E and scattering angle θ f . The geometry parameters are the same asthose in figure 1. T = T = T = T = -
50 0 500200400600800100012001400 θ f ( degree ) M o m en t u m t r an s f e r ( a . u ) Figure 4: The momentum transfer | q | = | p − p | in the absence of the laser as a function of thescattering angle θ f for different electron kinetic energies. The geometry parameters are θ i = 0 ◦ , φ i = φ f = 45 ◦ and − ≤ θ f ≤ dσ s /d Ω f as a function of the number of photons exchanged s at different fieldstrengths. The laser frequency chosen is ~ ω = 1 .
17 eV. The geometry parameters are θ i = φ i = 15 ◦ , θ f = 0 ◦ and φ f = 105 ◦ . s DCS (a.u) E = 10 V/cm T = 2 . s DCS (a.u) E = 5 . × V/cm T = 2 . s DCS (a.u) E = 10 V/cm T = 0 .
511 MeV-10 0 -100 0 -300 0-8 9.53968 × − -50 1 . × − -200 3.12079 × − -6 1.08317 × − -20 0.150387 -100 4.29275 × − -4 2.60332 × − -10 0.00463691 -50 1.25848 × − -2 2.35595 × − -5 0.0132268 -20 4.11216 × − × − × − × − × − × −
10 0.00462091 50 1.25861 × − × −
20 0.149351 100 4.29368 × − × −
50 1 . × −
200 3.12214 × −
10 0 100 0 300 0presence of a laser field, is equal to the field-free DCS for the same scattering process, taken at thesame scattering angle and initial energy [50]. The same applies to figure 6, where the sum rule isfulfilled at the photon number ± E = 5 . × V/cm and the frequencyis ~ ω = 1 .
17 eV. The sum rule can be, mathematically, represented by the following formula: +cutoff X s = − cutoff dσ s d Ω f = (cid:18) dσd Ω f (cid:19) laser-free . (66)Figure 7 shows the summed DCS dependence on the laser field strength. The inset shows moredetails of the same figure at field strengths between 10 and 10 V/cm. The field strength E isinvolved to determine the DCS behavior via the argument z of Bessel functions in equation (47).Although the Bessel functions in the partial DCS of each multiphoton process ( dσ s /d Ω f ) oscillatewith the field strength, the summed DCS decreases gradually by orders of magnitude as the fieldstrength increases from 10 to 10 V/cm. At weak filed strengths, i.e. between 10 and 10 V/cm,and with regard to the frequency ~ ω = 1 .
17 eV, we notice that the laser has no effect on the DCS.However, the stronger the field strength and the more distortion of the electron state, the greater thereduction of the DCS. This means that the probability of scattering between the electron and protonis diminished in the presence of a circularly polarized laser field. This behavior has been verified ondifferent geometries. The first simple physical explanation to which we can attribute this behavioris as follows: the electron, when embedded in a circularly polarized laser field, of course moves inperpendicular electric and magnetic fields, but at the same time rotates on an orbit whose radiuscorresponds to the value of the laser field strength. In contrast, the results obtained in the case of thelinearly polarized laser field show that the laser makes a significant contribution to the enhancementof the DCS [40,41]. In this case, the electron moves only in perpendicular electric and magnetic fieldswithout any other movement, so that the probability of the electron being scattered with the protonwill be greater with respect to the linearly polarized laser field than that corresponding to a circularlypolarized laser field. Since this probability is directly related to the concept of the differential cross6
Without lasers =± =± =± =± - -
20 0 20 400.00.51.01.52.02.5 θ f ( degree ) DC S ( a . u ) Figure 5: The variations of the summed DCS of e - P scattering with laser given in (41) as a functionof the scattering angle θ f for different numbers of photons exchanged. The laser field strength andfrequency are E = 10 V/cm and ~ ω = 1 .
17 eV. The elecron kinetic energy is T i = 2 . θ i = φ i = 15 ◦ and φ f = 105 ◦ . The notation s = ± N means that we havesummed over − N ≤ s ≤ + N .section in scattering theory, the results obtained in both cases are consistent with this interpretation.In figure 8, we seek to illustrate the effect of laser frequency on the DCS. To this end, we haveplotted its changes in terms of laser field strength for two different frequencies, ~ ω = 0 .
117 eV and ~ ω = 1 .
17 eV. As can be seen from figure 8, the low laser-frequency affects the DCS when the fieldstrength exceeds the threshold of 10 V/cm, while the effect of a high laser-frequency on the DCSdoes not appear until the field strength is above 10 V/cm. Therefore, the first laser, which has alower frequency, can affect the DCS more than the second one even at lower field strengths. Hence,the effect of the laser on the DCS becomes weak at high frequencies; and in order to make a highlaser-frequency affecting the DCS, a very high field strengths are needed. For the sake of furtherillustration, like the 2-dimensional graphs, the contour graph in figure (9) shows more informationon the global variation and aspect of the DCS with respect to both laser field strength and frequencyfor the exchanged photons of about ±
10. In this type of 3D-graphics, it is the colors and contoursthat represent the changes in DCS on a two-dimensional plane. The DCS values are shown on theright of the graphic on the bar legend. According to the evolution of the laser frequencies, the effectof the laser field on the DCS decreases at high frequencies, while it becomes relevant as the laserfield strength increases at each fixed frequency. This is perfectly consistent with all that we havediscussed so far.
This section is reserved for the presentation and discussion of the results obtained in the case of thedressed proton. When we consider the interaction of both electron and proton with the electromag-netic field in the initial and final states, the theoretical calculation becomes difficult and somewhatcumbersome. The extraction of numerical and graphical results related to this situation requireshigh-speed and high-resolution computing facilities. Summing over two numbers of exchanged pho-tons, one for the electron ( s ) and the other for the proton ( l ), takes time and effort twice. Therefore,7 Without lasers =± =± =± =± - -
20 0 20 400.00.51.01.52.02.5 θ f ( degree ) DC S ( a . u ) Figure 6: The same as in figure 5, but for the laser field strength E = 5 . × V/cm. Thenotation s = ± N means that we have summed over − N ≤ s ≤ + N . ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■● Without laser ■ With laser ● ● ● ● ● ●■ ■ ■ ■ ■ ■ × - × - × - × - × - Laser field strength ( V.cm - )
10 100 1000 10 × - × - × - × - × - Laser field strength ( V.cm - ) DC S ( a . u ) Figure 7: The summed DCS of the e - P scattering versus the laser field strength for the electron kineticenergy T i = 0 . ± θ f = 90 ◦ and θ i = φ i = φ f = 0 ◦ .The laser frequency is ~ ω = 1 .
17 eV. The inset gives a clear zoom of the DCS variations for the fieldstrengths between 10 and 10 V/cm.8 ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■● ℏω = ■ ℏω =
10 100 1000 10 × - × - × - × - Laser field strength ( V.cm - ) DC S ( a . u ) Figure 8: The variations of the DCS of e - P scattering with laser given in (41) as a function of laserfield strength for two different frequencies ~ ω = 0 .
117 eV and ~ ω = 1 .
17 eV. The numbers of photonsexchanged is s = ±
5. The elecron kinetic energy is T i = 0 .
511 MeV. The geometry parameters are θ i = φ i = φ f = 0 ◦ and θ f = 90 ◦ . × × × × × Laser field strength ( V.cm - ) La s e r f r equen cy ( e V ) DCS ( a.u ) Figure 9: The variations of the DCS of e - P scattering with laser given in (41) as a function of bothlaser field strength and frequency for an exchange of ±
10 photons. The elecron kinetic energy is T i = 2 . θ i = 60 ◦ , θ f = φ i = 0 ◦ and φ f = 90 ◦ .9Table 2: Values of the two DCSs with and without the dressing effects of proton which are given,respectively, in equations (59) and (41) for different field strengths. The laser frequency is ~ ω =1 .
17 eV. The kinetic energy of the incoming electron is T = 2 . θ i = 60 ◦ , θ f = 1 ◦ , φ i = 0 ◦ , φ f = 90 ◦ . The two numbers of photons s and l over which we havesummed are − ≤ s ≤ +5 and − ≤ l ≤ +5. The DCS of Mott scattering is also included forcomparison.Field strength E (V/cm) (cid:0) dσd Ω f (cid:1) e − -dressing( ) (a.u) (cid:0) dσd Ω f (cid:1) (e − ,P)-dressing( ) (a.u) Mott DCS (a.u) withlaser ( Z = 1)10 . . . . . . × − . . . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − we will limit ourselves here to include, in table 2, a few relevant values for the two DCSs in case wedress only electron or electron and proton together. Looking at table 2, it is easy to see that the DCSsof the two cases are equal at the first three field strengths (10 , and 10 V/cm), which meansthat the proton dressing has no effect yet. Once we are above field strength 10 V/cm, we noticethat the two DCSs begin to differ significantly. These values justified the fact that the interactionof the proton with the laser field at field strengths between 0 and 10 V/cm was neglected in thiswork (section 3.) as well as in previous works related to this subject [40, 41]. Within this interval ofstrengths, no matter whether the proton dressing is considered or not, it is the same and no changeoccurs. But when higher field strengths are reached outside this interval ( E ≥ V/cm), theproton dressing must be taken into account. This can reasonably be explained by the heavier massof the proton compared to that of the electron. From table 2, it can be seen that the inclusion of theproton dressing has contributed to a significant reduction in DCS more than before ; and the natureof the electromagnetic field polarization plays an important role in this behavior. Comparing thesetwo DCSs with that of Mott scattering at Z = 1, we can see that, due to the recoil effect, there is adeviation between them, and that of Mott is always smaller.
6. Conclusion
In this work, we have studied and revealed the effects of the proton dressing in the relativistic electron-proton scattering in the presence of a circularly polarized monochromatic laser field. We started withchecking the results already known for electron-proton scattering in the absence of the laser field.After that, we moved to the case in which we apply the laser field, either to only the electron orto the electron and proton together. Our theoretical results show that the DCS of electron-protonscattering is greatly minimized by the presence of the laser field. In the case of the proton dressing,we have found that it has no effect at all as long as the field strength is below 10 V/cm and cantherefore be neglected. Despite all of this, we emphasize that the approach presented here may notprovide a realistic description of electron-proton scattering at very high energies. In this case, themodels considering proton as a structureless particle are no longer accurate, and it has therefore0become necessary to introduce the form factors responsible for the substructure of the proton (deepinelastic scattering). In any case, it seems that this issue is of particular interest and deserves afuture investigation.
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