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2009 Multiphoton interactions in lepton photoproduction on nuclei at high energies Kirill Tuchin Iowa State University, [email protected]
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Abstract We study multiphoton effects in quantum electrodynamics in lepton photoproduction on heavy nuclei and nuclear medium at high energies. We are interested in energy, charge density, and nuclear geometry dependence of the cross sections. We use the impact parameter representation that allows us to reduce the problem of photoproduction to the problem of propagation of electric dipoles in the nuclear Coulomb field. In the framework of the Glauber model we resum an infinite series of multiphoton amplitudes corresponding to multiple rescattering of the electric dipole on the nucleus. We find that unitarity effects arising due to multiphoton interactions are small and energy independent for scattering on a single nucleus, whereas in the case of macroscopic nuclear medium they saturate the geometric limit of the total cross section. We discuss an analogy between nuclear medium and intense laser beams.
Disciplines Astrophysics and Astronomy | Physics
Comments This article is from Physical Review D 80 (2009): 093006, doi: 10.1103/PhysRevD.80.093006. Posted with permission.
This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/physastro_pubs/141 PHYSICAL REVIEW D 80, 093006 (2009) Multiphoton interactions in lepton photoproduction on nuclei at high energies
Kirill Tuchin Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA, and RIKEN BNL Research Center, Upton, New York 11973-5000, USA (Received 7 August 2009; published 23 November 2009) We study multiphoton effects in quantum electrodynamics in lepton photoproduction on heavy nuclei and nuclear medium at high energies. We are interested in energy, charge density, and nuclear geometry dependence of the cross sections. We use the impact parameter representation that allows us to reduce the problem of photoproduction to the problem of propagation of electric dipoles in the nuclear Coulomb field. In the framework of the Glauber model we resum an infinite series of multiphoton amplitudes corresponding to multiple rescattering of the electric dipole on the nucleus. We find that unitarity effects arising due to multiphoton interactions are small and energy independent for scattering on a single nucleus, whereas in the case of macroscopic nuclear medium they saturate the geometric limit of the total cross section. We discuss an analogy between nuclear medium and intense laser beams.
DOI: 10.1103/PhysRevD.80.093006 PACS numbers: 12.20. m, 25.20.Lj
The paper is structured as follows. In Sec. II we write the I. INTRODUCTION cross section in a mixed coordinate/momentum represen- In this article we offer a new perspective on the multi- tation by transforming the scattering amplitude to the photon interactions in lepton photoproduction on a heavy transverse coordinate space [11,12]. The cross section nucleus at high energies. Multiphoton effects arise due to then becomes a convolution of the photon wave function, multiple rescattering of the projectile photon on the pro- describing splitting of the photon into a lepton-antilepton tons of the nucleus. Each scattering contributes a factor of pair characterized by the transverse vector r, and the elastic Z to the cross section and must be taken into account for amplitude i llZ for the scattering of the lepton electric heavy nuclei. Traditionally, this is accomplished by solving dipole ll off the nucleus Z at some impact parameter B; see the Dirac equation in the external Coulomb field of the (1). The advantage of this representation becomes clear if nucleus. An explicit assumption of this approach is that the we recall that the trajectory of an ultrarelativistic particle in nucleus can be treated as a pointlike particle. The goal of an external field is a straight line. This implies that the this article is to solve the photoproduction problem in a dipole representation diagonalizes the scattering matrix. It general case, by explicitly taking the nuclear size into was shown in [10] that the lepton photoproduction ampli- account. This allows us to evaluate the finite nuclear size tudes take a rather simple form in the dipole representation. effects in lepton photoproduction. However, our main in- This representation was used in [13] for numerical study of terest is to use the developed formalism to investigate the the impact parameter dependence of the pair production. photoproduction in another extreme case—inside the nu- In Sec. III we use the Glauber theory for multiple clear medium (e.g. a thin film). We will argue that the scattering [2] to express the dipole-nucleus scattering am- nonlinear effects are very different in these two opposite plitude i llZ through the dipole-proton one i llp. This can cases. The formalism that we develop and use in this work be done if the nuclear protons can be considered as inde- can be also applied to the very interesting problem of pendent scattering centers. For large Z their distribution photoproduction on intense laser beams [1]. There the thus follows the Poisson law that leads to a particularly beam size can be tuned so that both limits can perhaps be simple expression for the amplitude (10). To avoid possible realized experimentally. confusion we emphasize that we consider ultrarelativistic Our approach to the multiphoton effects is based on the scattering on a ‘‘bare’’ nucleus, i.e. a nucleus stripped of all Glauber model [2], which allows a proper treatment of the electrons. Such nuclei are provided by the high energy nuclear geometry. In Sec. Vand the Appendix we show that heavy ion colliders such as RHIC and LHC. Therefore, the Glauber model calculation of the pair production cross the problem of screening of the nuclear Coulomb field by section agrees with the traditional approach pioneered by electrons—which is an important issue at high energy Bethe and Maximon [3], which is based on a solution of the photon-atom interactions—is not relevant here. Dirac equation in an external Coulomb field (see [4–10] for We calculate the scattering amplitude for dipole-proton recent developments). This is an important result of this scattering in Sec. IV. The momentum exchanged in the paper. It opens the possibility of a unified approach to the high energy collision is very small. The corresponding geometric effects in QED. In particular, in this paper we impact parameter is large. The maximal impact parameter 0 study the dependence of the lepton pair photoproduction bmax increases with energy. For a single nucleus, it always 0 on the density of charge sources and nuclear geometry. holds that bmax R. The region of large impact parame-
1550-7998=2009=80(9)=093006(9) 093006-1 Ó 2009 The American Physical Society KIRILL TUCHIN PHYSICAL REVIEW D 80, 093006 (2009) ters is where the dipole cross section picks up its logarith- matter was previously discussed by Lyuboshits and 0 mically enhanced energy dependence. Therefore, bmax is Podgoretsky in Ref. [15] who reached similar conclusions. the effective radius of electromagnetic (EM) interaction of A related problem has been recently addressed by the photon with the nucleus. This is very different from the Muller [16] who considered scattering of an intense laser strong interactions where the Yukawa potential drops ex- pulse on a heavy nucleus. He argues—basing on work done ponentially at distances 1=ð2m Þ much shorter than the with his collaborators [17]—that the resummation in the nucleus radius. density of the laser beam can be phenomenologically To make connection with the world of high photon significant if one studies the laser-nucleus interaction at densities, we consider photoproduction off a nuclear me- the LHC. 0 dium with transverse extent d bmax and the longitudinal L d extent . This can be thought of as a prototype of an II. PHOTOPRODUCTION IN THE DIPOLE MODEL intense laser pulse. Any photon can be converted into an electric dipole at the expense of an additional which, One of the Feynman diagrams for the photoproduction however, impacts only the total rate, but not the dynamics process is shown in Fig. 1. In the high energy limit s t, of the multiple scatterings. This motivated us to consider where s and t are the usual Mandelstam variables, the cross the photoproduction off the nuclear medium in more detail. section for the diagram in Fig. 1 can be written as [18] Also, as has been mentioned, this case is similar in many Z 2 Z 1 Z 1 d r llZ aspects to the well-studied dynamics of the strong tot ðsÞ¼ d ðr; Þ tot ðr;sÞ 2 2 0 interactions. Z Z Z 1 At first, in Sec. V we study the Born approximation with ¼ d2r d ðr; Þ d2B Imhi llZðr; B;sÞi: the leading logarithmic accuracy. Born approximation cor- 0 responds to scattering on only one proton in a nucleus. For (1) realistic nuclei, the photoproduction cross section reduces This formula is the Fourier transform of the momentum to the Bethe-Heitler formula (31)[14] with a characteristic r Z2 space expression for the scattering amplitude. Dipole size dependence on the proton number. In this case the is a separation of the l and l in the transverse configuration i llp elastic scattering amplitude is approximately real, see space (i.e. in the plane perpendicular to the collision axes); (25), corresponding to the dominance of elastic processes this is the variable that is Fourier conjugated to the relative over the inelastic ones. This is not difficult to understand: a transverse momentum k of the pair. We use the bold face to significant part of the cross section originates at distances distinguish the transverse two-vectors. Impact parameter B much larger than the nuclear radius; the nuclear density is a transverse vector from the nuclear center to the center there is small. On the contrary, in the case of the nuclear of mass of the ll pair; see Fig. 2.In(1) ðr; Þ is the square llp medium the amplitude i is approximately imaginary; of the photon ‘‘wave function’’ averaged over the photon see (32) and (33c). This implies the dominance of inelastic polarizations and summed over lepton helicities. It is given processes. Formally, this happens because of an approxi- by [12,18] mate cylindrical symmetry of the nuclear distribution 2 2 m 2 2 2 2 around a small dipole on one hand, and antisymmetry of ðr; Þ¼ fK1ðrmÞ½ þð1 Þ þK0ðrmÞg; llp the real part of the scattering amplitude Re½i with respect to inversion b ! b, see (12), on the other hand. (2) For the nuclear medium, the photoproduction cross section where m is a lepton mass. is the fraction of the photon is proportional to Z in the Born approximation. light-cone momentum taken away by the lepton. llZ Multiple rescatterings induce a correction to thepffiffiffi Born hi ðr; B;sÞi is the dipole-nucleus elastic scattering am- approximation that generally depend on energy s and plitude averaged over the proton positions (indicated by the nuclear charge Z. These effects are studied in Sec. VI. Expectedly, the multiphoton effects lead to rather different expressions for the resumed cross sections (40) and (43) for a single nucleus and for the nuclear medium. For a single nucleus the multiphoton processes resumed within the Glauber model do not depend on energy. In the Appendix, we demonstrate, using the method of [10], that the Glauber approach reproduces the result of Bethe and Maximon, i.e. unitarity corrections beyond the leading logarithmic approximation [3]. In the opposite case of the FIG. 1. Photoproduction of a lepton pair. Dashed lines denote nuclear medium, the cross section grows as a function of photons, solid lines denote fermions. The ellipsis indicates energy and Z until it saturates the black disk geometric summation over all proton numbers. Note that the t-channel limit. A problem of propagation of positronium through photons can hook up to either fermion or antifermion lines.
093006-2 MULTIPHOTON INTERACTIONS IN LEPTON ... PHYSICAL REVIEW D 80, 093006 (2009) b’ p the ll electric dipole with the nucleus. Indeed such an d approach proved very useful for the study of the strong b B interactions. O III. GLAUBER MODEL FOR MULTIPLE SCATTERING FIG. 2 (color online). Geometry of the dipole-nucleus colli- Let the nucleus quantum state be described by the wave sion in the transverse plane. p, d, and O denote proton, center of c b ;z Z Z function A that depends on positions f a aga¼1 of all mass of the dipole, and the nucleus center positions correspond- protons, where ba and za are the transverse and the longi- b0 B b b R ingly. ¼ , . tudinal positions of proton a correspondingly. If the dipole- nucleus scattering amplitude i llZ is known, then averag- h i symbol) and the dipole-nucleus cross section is, by ing over the proton positions is performed as virtue of the optical theorem, Z YZ llZ 2 2 Z h ðr; B;sÞi ¼ d badzajc Aðb1;z1; b2;z2; ...Þj llZ 2 llZ tot ðr;sÞ¼2 d B Imhi ðr; B;sÞi: (3) a¼1 llZ ðr; B b1;z1; B b2;z2; ...;sÞ: Note, that in the high energy limit, the dipole cross section does not depend on . Integrating over we get (6) 2 Z 2 The scattering amplitude is simply related to the scattering Z m d r 2 ðsÞ¼ K2ðrmÞþK2ðrmÞ llZðr;sÞ: matrix element S as ðs; BÞ¼1 Sðs; BÞ. The latter can tot 2 3 1 0 in turn be represented in terms of the phase shift , so that (4) in our case llZ Equations (1) and (2) admit the following interpretation. ðr; B b1;z1; B b2;z2; ...;sÞ At high energies, the processes of the Z scattering pro- llZ ceeds in two separated in time stages. The first stage is ¼ 1 expf i ðr; B b1;z1; B b2;z2; ...;sÞg: fluctuation of the photon into the lepton pair ! l þ l. (7) Let the photon, lepton, and antilepton 4-momenta be At high energies, interaction of the dipole with different q ¼ðqþ; 0; 0Þ, k ¼ð qþ;k ; kÞ, and k q ¼ðð1 protons in the nucleus is independent inasmuch as the Þqþ; ðk qÞ ; kÞ, with ¼ kþ=qþ. Here we use the light-cone momentum notation of the 4-vector p ¼ protons do not overlap in the longitudinal direction. This 0 3 assumption is tantamount to taking into account only two- ðpþ;p ; pÞ, where p ¼ p p . The four-scalar product 1 body interactions, while neglecting the many-body ones is p k ¼ 2 ðpþk þ p kþÞ p k. The light-cone time [2]. In this approximation the phase shift llZ in the dipole- span of this fluctuation is nucleus interaction is just a sum of the phase shifts llp in 1 1 x ¼ ¼ the dipole-proton interactions, i.e., þ k q k q m2þk2 m2þk2 þð Þ q þ 1 q llZ þ ð Þ þ ðr; B b1;z1; B b2;z2; ...;sÞ ð1 Þq X þ : ¼ 2 2 (5) 1 exp i llp r;s;B b : m þ k ¼ ð aÞ (8) a The largest contribution to the cross section arises, as we The problem of calculating the scattering amplitude of the will see later, from the longest possible x , referred to as þ dipole on a system of Z protons thus reduces to the problem the coherence length lc [19]. This corresponds to ¼ 1=2 2 of calculating of the scattering amplitude of the dipole on a and k ¼ 0, so that lc ¼ qþ=4m . Obviously, 1=lc ¼ llp i i 2 single proton . In this approximation, he i¼e h i, 4m =qþ is the minimum light-cone energy l transfer. The second stage of the Z process is interaction of the i.e. correlations between nucleons in the impact parameter lepton pair with the nucleus. In the nucleus rest frame its space are neglected. time span is of the order of the target size 2R, where R is We can rewrite (6) and (7)as Y the nucleus radius. We observe that lc 2R for suffi- llZ llp q m h ðr; B;sÞi ¼ 1 ½1 ðr;s;B baÞ : (9) ciently high þ for a lepton of any mass supporting a the physical interpretation of the two stage process. It now becomes clear why it is convenient to write the cross For a heavy nucleus with large Z we obtain section for the process Fig. 1 in a way in which the photon llZ llp h ðr; B;sÞi 1 expf Zh ðr;s;B baÞig: (10) wave function describing the splitting process ! l þ l is separated from the amplitude llZ describing interaction of This approximation is justified as long as 1 and
093006-3 KIRILL TUCHIN PHYSICAL REVIEW D 80, 093006 (2009) Z 1 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 . Indeed, expression in the exponent of (10) can 2 jb rj hRe½i llpðr; B;sÞ i ¼ d2b 2 R2 b2 ln 2 ; also be obtained by expanding Z 0 1 jb þ 2 rj i llZ ¼ Z lnð1 llpÞ (15) llp 1 llp 2 2 with b0 ¼ B b; see Fig. 2. Z½ 2ð Þ þ Z þ Oð ZÞ: The imaginary part of the dipole-proton elastic scatter- ing amplitude is depicted in Fig. 3(b). The corresponding analytical expression reads IV. DIPOLE-NUCLEUS SCATTERING AMPLITUDE 2 Z 2 Z 2 0 d l d l Now we turn to the calculation of the dipole-proton Im½i llpðr; b0;sÞ ¼ 2 2 l2 l02 amplitude; see Fig. 3. At high energies the t-channel i b0 1=2 r l i b0 1=2 r l Coulomb photons are almost real: l2 l2. This is the ðe ½ þð Þ e ½ ð Þ Þ well-known Weisza¨cker-Williams approximation that we ðe i½b0þð1=2Þr l0 e i½b0 ð1=2Þr l0 Þ: use throughout the paper. It allows us to determine the leading logarithmic contribution to the scattering ampli- (16) tude. In this approximation, the real part of the elastic dipole-proton scattering amplitude reads Averaging over the proton positions we derive Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Z 2 Z 2 llp 1 2 2 2 d l d l 0 0 hIm½i ðr;B;sÞ i ¼ d b 2 R b Re½i llpðr; b0;sÞ ¼ ½ei½b þð1=2Þr l ei½b ð1=2Þr l Z 2 2 l2 l2 Z 2 0 Z d l 0 0 1 dl 1 ei½b þð1=2Þr l ei½b ð1=2Þr l 0 2 ð Þ ¼ 2 J0 b þ r l l0 0 l 2 i b0 1=2 r l0 i b0 1=2 r l0 1 ðe ½ þð Þ e ½ ð Þ Þ: 0 J0 b r l : (11) 2 (17)
Taking the integral we arrive at The Weisza¨cker-Williams approximation breaks down for very small momenta of the t-channel photons. To 0 1 jb rj determine the cutoff transverse momentum lmin at which Re ½i llpðr; b0;sÞ ¼ 2 ln 2 : 0 1 (12) this approximation still holds, we use the Ward identity jb þ 2 rj applied to the lower vertex of Fig. 3(a). It gives p l ¼ p l p l 0 p M2=p ;p ; 0 M The nuclear density ðb;zÞ is normalized such that þ þ þ ¼ , where ¼ð Þ with being the proton mass. It follows that lþ ¼ pþl =p . Z 2 2 2 2 2 2 Thus, l ¼ lþl þ l l if l ðpþ=p Þl . Since d bdz ðb;zÞ¼Z: (13) 2 the minimum value of l is 4m =qþ [see (5) and the following text] we find Therefore, the average of the real part of the amplitude sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi over the proton position reads p ð2mÞ4 M2ð2mÞ4 Mð2mÞ2 l ¼ þ ¼ ¼ ; min p q2 p2 q2 s (18) þ þ Z Z z b0 1 r llp 1 2 0 j 2 j Re i r; B;s d b dz 2 ln ; 2 h ½ ð Þ i ¼ 1 s p q p q l Z z jb0 þ rj where ¼ð þ Þ þ. min corresponds to the 0 2 0 maximum possible impact parameter bmax 1=lmin.As (14) we will see in the next section, the energy dependence of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the dipole-nucleus amplitude arises from the logarithmic z 2 R2 b2 const where 0 ¼ . Since , we get dependence on these cutoff scales (this is why a precise 0 value of the proportionality coefficient between bmax and 1=lmin is not very important).
V. BETHE-HEITLER LIMIT Before we consider a general case it is instructive to discuss the scattering process in the Born approximation. We would like to (i) confirm that we reproduce the result of Bethe and Heitler [14] and (ii) determine the range of impact parameters B that gives the largest contribution to FIG. 3. The first two terms in the expansion of llp in . the cross section.
093006-4 MULTIPHOTON INTERACTIONS IN LEPTON ... PHYSICAL REVIEW D 80, 093006 (2009) The Born approximation corresponds to taking into Using b0 B b, see Fig. 2, we can write (15)as account only the leading terms in the coupling . jB 1 rj 1 Expanding the exponent in (10) we have hRe½i e eþpðr; B;sÞ i 2 ln 2 jB þ 1 rj Z llZ llp 1 2 llp 2 2 h ðr;B;sÞi ¼ Zh ðr;B;sÞi 2Z h ðr;B;sÞi þ : Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2b 2 R2 b2 (19) i llp jB 1 rj The real part of stems from a diagram with a single 2 ln 2 : ¼ 1 (21) photon exchange, while its imaginary part arises from the jB þ 2 rj double photon exchange (in the high energy limit these diagrams are purely real and imaginary correspondingly); To obtain the cross section we need to integrate over the see Fig. 3. Therefore, at the lowest order in , the imagi- impact parameter B. Expand the logarithm in (21)atB llp r=2 nary part of i corresponds to the two photon exchange as Fig. 3(b) that gives—according to (19)—contribution of jB 1 rj B r 2Z ln 2 O r3=B3 ; order to the cross section (3). On the other hand, 1 2 þ ð Þ (22) llp 2 jB þ rj B the real part of ði Þ corresponds (at the leading order in 2 ) to the product of the two diagrams Fig. 3(a) with single we derive photon exchanges; it is of the order 2Z2 1.1 Thus, in the B r Born approximation e eþp hRe½i ðr; B;sÞ i 2 : (23) Z B2 1 llZ ðr;sÞ¼2 d2B Z2hRe½i llpðr; B;sÞ i2 tot;B 2 Turning to the imaginary part we can write Eq. (17)as follows: þ ZhIm½i llpðr; B;sÞ i : (20) 2 Z d2l Z d2l0 Im i eþe p r; B;s h ½ ð Þ i ¼ 2 2 02 The reason for keeping a subleading term in (20) (the term 2 l l in the integrand proportional to Z) is that in the nuclear ðei½Bþð1=2Þr l medium the real part of the dipole-proton amplitude van- ei½B ð1=2Þr lÞðe i½Bþð1=2Þr l0 ishes, see Eq. (32), and then the main contribution stems i B 1=2 r l0 from the imaginary part of that amplitude. e ½ ð Þ Þ (24a) 1 jB 1 rj ¼ 4 2ln2 2 A. Single nucleus 2 1 jB þ 2 rj 0 Values of the dipole-proton impact parameter b can ðB rÞ2 b0 2 2 : range up to its maximal value max, which is determined B4 (24b) 0 as the inverse minimal momentum transfer, i.e. bmax 1=l l To the leading order in we can neglect the small absorp- min, where min is given by (p18ffiffiffi). For electron production at the collision energy of s ¼ 10 GeV we estimate tion effect given by (24b) and write for the scattering 0 bmax 20 nm, whereas the radius of the Gold nucleus is amplitude 0 7 fm. Thus for realistic nuclei bmax R. (In the case of B r photoproduction this approximation holds for energies h llpðr; B;sÞi i2 : pffiffiffi B2 (25) larger than about s ¼ 30 GeV.) We will demonstrate shortly that the large logarithmic Consequently, in the high energy and Born approximations contribution to the dipole cross section (20) comes from the dipole-nucleus cross section becomes the region b0 b and b0 r=2. Because by definition Z B r 2 b R, this is equivalent to B b0 maxfR; =2g, where llZ 2 2 2 ð Þ tot;Bðr;sÞ¼Z d B4 : (26) ¼ 1=m is the Compton wavelength of a lepton. Here we B4 used the fact that the characteristic value of the dipole size Denote R ¼ maxfR; =2g. Integration over B in the inter- is r 1=m since it corresponds to a sharp maximum of the val maxfR; r=2g 093006-5 KIRILL TUCHIN PHYSICAL REVIEW D 80, 093006 (2009) X Z 2 Z 1 Z 0 1 Z d r 1 2 jb rj ðsÞ¼ d ðr; Þ llZ ðr;sÞ hRe½i llpðr; B;sÞ i ¼ d2b0 ðB b0ÞL ln 2 tot;B 2 2 tot;B Z 0 1 l 0 jb þ 2 rj Z X 1 2 2 Z b0 1 r 4 3Z2m2 drr K2 rm K2 rm 2 j 2 j ¼ l 1ð lÞþ 0ð lÞ ðBÞL d b0 ln ¼ 0: 0 3 Z 0 1 l jb þ 2 rj bmax r2 ln ; (32) R (28) The last equation follows because the integrand is an odd b0 ! b0 where the sum runs over all lepton species. Taking inte- function under the reflection . The first nonvan- grals over the dipole sizes finally produces ishing contribution to the real part of the amplitude is proportional to the nuclear density gradient, which has X the largest value at the boundary. These small effects can Z 2 2 1 bmax ðsÞ¼ 4 3Z2m2 þ ln : be taken into account if the nuclear density distribution is tot;B l 3 3m4 3m4 R (29) l l l known. The leading contribution arises from the imaginary part In particular, for electrons of the amplitude that reads 1 Z 2 Z d2l Z d2l0 3 2 llp 2 0 Z eþe X 28 Z s hIm½i ðr; B;sÞ i L d b ! ðsÞ¼ ln : Z 2 2 l2 l02 tot;B 9 m2 2Mm (30) e e ðei½b0þð1=2Þr l ei½b0 ð1=2Þr lÞ i½b0þð1=2Þr l0 i½b0 ð1=2Þr l0 In a frame where nuclear proton moves with velocity v ¼ ðe e Þ p p = p p ð þ Þ ð þ þ Þ,p theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi minus component of its four- (33a) p M 1 v = 1 v s momentumpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is ¼ ð Þ ð þ Þ. Therefore, ¼ 1 Z d2l ¼ L4 2 ð1 eir lÞ 2M! ð1 vÞ=ð1 þ vÞ, where the incoming photon fre- Z l4 (33b) ! q =2 quency is ¼ þ . Equation (30) now reads 1 1 2 L4 2ð2 Þ r2 ln ; (33c) sffiffiffiffiffiffiffiffiffiffiffiffi Z 4 rlmin 3 2 Z!eþe X 28 Z ! 1 v tot;B ðsÞ¼ ln : (31) where in the last line we kept only the logarithmically 9 m2 m 1 þ v e e enhanced term. We see that in the case of the nuclear medium, the entire elastic dipole-proton amplitude is ap- This is the high energy limit of the formula derived by proximately imaginary and is given by Eq. (33c). Bethe and Heitler [14]. In the Appendix we discuss the Moreover, unlike (24b) it increases logarithmically with energy-independent correction to this equation. energy. Substituting (32) and (33c) into (20) we obtain in the Born approximation B. Nuclear medium 2 llZ ðr;sÞ¼4 Z 2r2 ln : Consider a nuclear matter having an arbitrary large tot;B rl (34) number of protons Z. We assume that the interproton min distance is much smaller than the Compton wavelength Note, that cross section (34)isZ times smaller then the one of electron, i.e. 1=3 . This allows treating the nu- given by (27). clear matter as the continuous medium of density . Let the 0 medium transverse size be d bmax and longitudinal size 0 2 VI. UNITARITY EFFECTS L lc. In the medium rest frame lc ’ bmax. Therefore, the nuclear medium has a form of a film with d L. What The unitarity relation applied to the elastic scattering does the lepton photoproduction look like in this case? The amplitude at a given impact parameter reads [we suppress key observation in this case is that the dipole-proton inter- the argument ðr; b;sÞ of all functions] action range is much smaller compared to the transverse llZ llZ 2 llZ size of the medium. Therefore, b0 B and then it follows 2Imði Þ¼j j þ G ; (35) from (15) that the real part of the elastic dipole-proton GllZ scattering amplitude vanishes. Indeed, changing the inte- where is the inelastic scattering amplitude. Using (10) gration variable b ! b0 in (15) and taking the limit of we can solve (35)as 0 small b we obtain llZ Z Im i llp G ¼ 1 e h ð Þi: (36) ffiffiffi 2 0 p The values of bmax at s ¼ 10 GeV for e, , and are It follows that the total, inelastic and elastic cross sections 20 nm, 0.4 pm, and 2 fm, correspondingly. are given by 093006-6 MULTIPHOTON INTERACTIONS IN LEPTON ... PHYSICAL REVIEW D 80, 093006 (2009) Z llZ 2 llp ZhImði llpÞi must not increase with energy. Therefore, in the case of tot ¼ 2 d Bf1 cos½ZhReði Þi e g; (37a) lepton pair photoproduction on a single nucleus the unitar- Z ity corrections do not grow with energy at the leading order llZ d2B 1 e ZhImði llpÞi ; in ¼ f g (37b) in 1 and Z 1. Actually, the Glauber model that we Z use in this paper also allows us to reproduce the subleading llZ 2 llp ZhImði llpÞi el ¼ d Bf1 2 cos½ZhReði Þi e (i.e. energy-independent) corrections to Eq. (30) that were first obtained by Bethe and Maximon [3]. This result can be Z Im i llp þ e h ð Þig: (37c) easily derived using the formalism developed by Ivanov and Melnikov in [10]. The derivation is outlined in the llp Appendix. In the framework of our approach it is straight- Zh i 1 2 When , deviation from the Born approxima- forward to calculate corrections of order Z to the Bethe- tion becomes large and the unitarity effects set in. We Maximon formula by taking into account the imaginary proceed to analyze the unitarity corrections in two extreme part of the amplitude (24b). This problem will be addressed cases. elsewhere. A. Single nucleus B. Nuclear medium In the case of photon-nucleus scattering, we derive using Conclusions of the previous section are dramatically (25) and integrating over the directions of B: reversed in the case of scattering off a nuclear medium. Z þ B r For the total dipole cross section we obtain after substitut- llZ e e Z ¼ 2 d2B 1 cos 2Z tot el B2 ing (32) and (33c) into (37) Z B max 2 ¼ 4 dBB½1 J0ð2Z r=BÞ : (38) llZ r;s 2 d2 1 exp L2 2r2 ln : 0 tot ð Þ¼ (42) rlmin Introducing the dimensionless variable x ¼ 2Z r=ð2BmaxÞ Similar results were obtained for an infinite medium in we can express this integral in terms of the generalized [15]. Substituting into (4) we derive hypergeometric function: Z 2 2 Z d r 2 ð2Z rÞ s 2 m2d2 K2 rm K2 rm llZðr;sÞ¼ fx2 F ½ð1; 1Þ; ð2; 3; 3Þ; x2 tot ð Þ¼ 1ð Þþ 0ð Þ tot 8 2 3 2 3 L2 2r2 lnð2=rl Þ þ 4ð2 2 lnx2Þg; (39) ½1 e min : (43) where is Euler’s constant. The total photoproduction At large Z the multiphoton effects start to play an impor- cross section of light leptons is obtained using (4). tant role. Actually, it is convenient to directly plug (38) into (4) It is instructive to study the unitarity corrections by and first integrate over r and then over B. The result can increasing the proton number Z while keeping the medium be expressed in terms of another dimensionless variable size given by L and d fixed. This corresponds to an increase y ¼ Z=ðBmaxmÞ as in the proton number density. We are interested to know at Z 3 2 what ’s the multiphoton effects become observable. In Z Z ðsÞ¼ ð2y 28y3 lnð2yÞ tot 9m2y3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1.0 þf 3 1 þ y2 þ 24y2 1 þ y2 e qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0.8 þ cosh½3lnðy þ 1 þ y2Þ g lnðy þ 1 þ y2ÞÞ: Z (40) 0.6 To find the high energy asymptotic we expand (40) at small 0.4 y and find 28 3Z2 1 41 12 0.2 Z s ln y2 : tot ð Þ¼ 2 þ þ þ (41) 9m 2y 42 35 0.0 109 1013 1017 1021 1025 In the leading logarithmic approximation that we use in Z this paper, only the leading logarithm in the square brack- ets of (41) can be guaranteed. Indeed, it reproduces the FIG. 4. Ratio of the lepton photoproduction cross section and Z Bethe-Heitler formula (30). However, Eq. (41) does allow itspffiffiffi Born approximation as a function of the ‘‘nucleus’’ charge . us to conclude that corrections to the leading logarithm s ¼ 10 GeV, L ¼ 1nm, and d ¼ 100 nm. 093006-7 KIRILL TUCHIN PHYSICAL REVIEW D 80, 093006 (2009) Fig. 4 we show the ratio of the total photoproductionpffiffiffi cross electromagnetic interactions in a high density system. A calculation to its Born approximation for s ¼ 10 GeV high density of charges can be realized also in ultrafast and L ¼ 1nm, d ¼ 100 nm. We observe that for electrons pulse lasers that emit pulses containing as many as N the deviation from the linear Born regime starts at about 1020 photons and having dimensions L 093006-8 MULTIPHOTON INTERACTIONS IN LEPTON ... PHYSICAL REVIEW D 80, 093006 (2009) 2 Z 2 Z Z m d r 2 one needs to take into account the fact that the smallest ¼ 2 K2ðmrÞþK2ðmrÞ d2R tot 2 3 1 0 longitudinal momentum transfer depends on and k; see (5). In the logarithmic approximation, we maximized (5) 1 jR rj 2i Z 1 jR rj 2i Z 1 with respect to both and k. Now it must be retained 2 R 2 R which means that (i) the regularization of the logarithmi- jR rj cally divergent integrals must be done in momentum space 4ð ZÞ2ln2 : R (A4) and (ii) the integral [see (1)] must be done after the R r regularization since the cutoff depends on it. We refer the Taking integrals over and yields interested reader to Ref. [10] for details. The result is Z 28 3 2 1 tot ¼ Z ½c ð1 i ZÞ 9m2 2 28 3Z2 s 109 c 1 i Z 2c 1 ; Z s ln : þ ð þ Þ ð Þ (A5) tot;Bð Þ¼ 2 (A6) 9 me Mme 42 where c ðzÞ is the logarithmic derivative of the function. As expected, the unitarity corrections are independent of energy in agreement with the results of Sec. VI A. 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