Coulomb corrections to photon and dilepton production in high energy pA collisions
CCoulomb corrections to photon and dilepton production in high energy pAcollisions
Kirill Tuchin Department of Physics and Astronomy, Iowa State University, Ames, IA, 50011 (Dated: November 4, 2018)We consider particle production in high energy pA collisions. In addition to the coherentinteractions with the nuclear color field, we take into account coherent interactions withthe nuclear electromagnetic Coulomb field. Employing the dipole model, we sum up theleading multiple color and electromagnetic interactions and derive inclusive cross sectionsfor photon and dilepton production. We found that the Coulomb corrections are up to 10%at √ s = 200 GeV per nucleon. I. INTRODUCTION
A pivotal feature of high energy pA and AA collisions at RHIC and LHC is large longitudinalcoherence length that by far exceeds radii of heavy nuclei. In QCD, color fields of nucleons ina heavy nucleus fuse to create an intense coherent color field, which has fundamental theoreticaland phenomenological importance. Since nuclear force is short-range, only nucleons along the sameimpact parameter add up to form a coherent field. Because the QCD contribution to the scatteringamplitude at high energy is imaginary, it is proportional to α s . Thus, the parameter that charac-terizes the color-coherent field is α s A / ∼
1, where A is atomic weight. The longitudinal coherencelength increases with the collision energy, but decreases with momentum transfer, so that at lowenergies color coherence is a non-perturbative phenomenon. At RHIC the longitudinal coherencelength is large even for semi-hard transverse momenta (a few GeV’s), authorizing application ofthe perturbation theory to color-coherent processes [1–3].Along with strong color field, heavy-ions also posses strong electromagnetic Coulomb field. Theelectromagnetic force is long range, so that all Z protons of an ion contribute to the field. Also, theQED contribution to the scattering amplitude is approximately real. As a result, the parameterthat characterizes the coherent electromagnetic field is αZ ∼
1. Since both parameters α s A / and αZ are of the same order of magnitude in heavy ions, electromagnetic force must be takeninto account along with the color one. This observation is a direct consequence of coherence whichenhances the electromagnetic contribution by a large factor Z . Not all particle production channelsin high energy pA and AA collisions are equally affected by the nuclear Coulomb field(s). Our a r X i v : . [ h e p - ph ] N ov main observation is that gluon emission off a fast quark is completely unaffected in the eikonalapproximation, whereas photon and dilepton production are moderately modified. The centralgoal of this article is to evaluate the magnitude of the Coulomb corrections to these processes.The article is structured as follows. In Sec. II we develop a formalism, inspired by the Glauber-Mueller model [4] that takes into account both color and electromagnetic coherence by means ofmultiple scattering resummation. This formalism is applied in Sec. III to calculate the scatter-ing amplitude of color-electric dipole of size r on heavy nucleus. The dipole-nucleus amplitude isemployed in Sec. IV and Sec. V to compute inclusive photon and dilepton cross sections corre-spondingly. We conclude in Sec. VI with a discussion of our results and their ramifications on pAand AA phenomenology.Strong electromagnetic interactions in pA and AA collisions were investigated before by manyauthors [5–11] who where concerned with pure QED contributions. In this paper we are moreinterested to study an interplay between the QCD and QED dynamics. II. GLAUBER MODEL
Let the nucleus quantum state be described by the wave function ψ A that depends on positions { b a , z a } Aa =1 of all A nucleons, where b a and z a are the transverse and the longitudinal positions ofa nucleon a correspondingly. (In our notation, transverse vectors are in bold face). If the proton–nucleus scattering amplitude i Γ pA is known for a certain distribution of nucleons, then the averagescattering amplitude is (cid:104) Γ pA ( b , s ) (cid:105) = (cid:90) Z (cid:89) a =1 d b a dz a | ψ A ( b , z , b , z , . . . ) | Γ pA ( b − b , z , b − b , z , . . . , s ) . (1)The scattering amplitude is simply related to the scattering matrix element S as Γ( b , s ) = 1 − S ( b , s ). The later can in turn be represented in terms of the phase shift χ so that in our caseΓ pA ( b − b , z , b − b , z , . . . , s ) = 1 − exp {− iχ pA ( b − b , z , b − b , z , . . . , s ) } . (2)At high energies, interaction of the projectile proton with different nucleons is independentinasmuch as the nucleons do not overlap in the longitudinal direction. This assumption is tan-tamount to taking into account only two-body interactions, while neglecting the many-body ones[12]. In this approximation the phase shift χ l ¯ lZ in the proton–nucleus interaction is just a sum ofthe phase shifts χ l ¯ lp in the proton–nucleon interactions and correlations between nucleons in theimpact parameter space are neglected. We have (cid:10) Γ pA ( b , s ) (cid:11) = (cid:68) − e − iχ pA (cid:69) = (cid:68) − e − i (cid:80) a χ pN (cid:69) = 1 − e − i (cid:80) a (cid:104) χ pN (cid:105) , (3)where in the last term (cid:104) . . . (cid:105) stands for an average over a single nucleon position in the nucleus,defined below in (7). To the leading order in coupling α s , the phase shift χ pN can be expanded as − iχ pN = ln(1 − Γ pN ) ≈ − Γ pN . Therefore, we can write (cid:10) Γ pA ( b , s ) (cid:11) = 1 − exp (cid:110) − (cid:88) a (cid:104) Γ pN ( b , s ) (cid:105) (cid:111) . (4)Strong and electromagnetic contributions decouple in the elastic scattering amplitude at theleading order in respective couplings: Γ pN = Γ pN s + Γ pN em . (5)Indeed, as we discuss below i Γ pN em is real, while i Γ pN s is imaginary, which is a consequence of thefact that SU (3) generators are traceless. Owing to (5) we can cast (4) in the form (cid:10) Γ pA ( b , s ) (cid:11) = 1 − exp (cid:110) − A (cid:10) Γ pN s ( b , s ) (cid:11) − Z (cid:10) Γ pN em ( b , s ) (cid:11) (cid:111) , (6)where Z is the number of protons. In the Glauber model we average over the nucleus using thenuclear density ρ as follows (cid:10) Γ pN s ( b , s ) (cid:11) = 1 A (cid:90) ∞−∞ dz a (cid:90) d b a ρ ( b a , z a )Γ pN s ( b − b a , s ) . (7)Neglecting the diffusion region, nuclear density is approximately constant ρ = A/ ( πR A ) for pointsinside the nucleus and zero otherwise. The range of the nuclear force is about a fm, which is muchsmaller than the radius R A of a heavy nucleus. Therefore, b ≈ b a and (cid:10) Γ pN s ( b , s ) (cid:11) = 1 A (cid:113) R A − b πR A ρ Γ pN s (0 , s ) . (8)In this approximation the total proton-nucleon cross section is σ pN ( s ) = 2 πR p Γ pN s (0 , s ), with R p being proton’s radius, so that (cid:10) Γ pN s ( b , s ) (cid:11) = 1 A ρT ( b ) 12 σ pN ( s ) , (9)where T ( b ) = 2 (cid:113) R A − b is the thickness function. It follows from (9) that A (cid:104) Γ pN s (cid:105) ∼ α s A / ,which implies that (6) sums up terms of order α s A / ∼ α s (cid:28)
1. Indeed, the leadingstrong-interaction contribution to the pN elastic scattering amplitude corresponds to double-gluonexchange. Note also, that the corresponding (cid:104) i Γ pN s (cid:105) is purely imaginary.Proton density in the nucleus is Zρ/A , hence (cid:10) Γ pN em ( b , s ) (cid:11) = 1 Z (cid:90) ∞−∞ dz a (cid:90) d b a ZA ρ ( b a , z a )Γ pN em ( b − b a , s ) (10)= 1 A ρ (cid:90) d b a T ( b a )Γ pN em ( b − b a , s ) . (11)Electromagnetic interaction is long-range, therefore all values of impact parameter b contributeto the total cross section. Moreover, the leading logarithmic contribution comes from impactparameters far away from the nucleus b (cid:29) b a ∼ R A . In this case, (cid:10) Γ pN em ( b , s ) (cid:11) = 1 A ρ Γ pN em ( b , s ) (cid:90) d b a (cid:113) R A − b a = Γ pN em ( b , s ) , b (cid:29) R A . (12)However, if we are interested in differential cross section at impact parameters b ∼ R A no suchapproximation is possible. The leading electromagnetic contribution to elastic pN scattering am-plitude arises from one photon exchange; the corresponding (cid:104) i Γ pN em (cid:105) is purely real. We note, that(7) sums up terms of order αZ ∼ α (cid:28) pA cross section can be computed using the optical theorem as follows σ pA tot ( s ) = 2 (cid:90) d b Im [ i Γ pA ( b , s )] = 2 (cid:90) d b (cid:8) − exp[ − A (cid:10) Γ pN s ( b , s ) (cid:11) ] cos[ Z (cid:10) i Γ pN em ( b , s ) (cid:11) ] (cid:9) . (13) III. DIPOLE-NUCLEUS SCATTERING
Similarly to the proton–nucleus scattering, one can consider scattering of color and electricsinglet q ¯ q pair (dipole) of size r off a heavy nucleus. Since a single gluon exchange is an inelasticprocess, the leading in α s contribution to the elastic scattering amplitude comes from the doublegluon exchange given by A (cid:10) Γ q ¯ qN s ( b , s ; r ) (cid:11) = 2 C F N c ρT ( b ) 12 πr α s ln 1 rµ , (14)where µ is an infrared scale and s the center-of-mass energy squared, while the leading in α termarises from a singe photon exchange given by Z (cid:10) i Γ q ¯ qN em ( b , s ; r ) (cid:11) = ZA ρ α (cid:90) d b a T ( b a ) ln | b − b a − r / || b − b a + r / | . (15)In this article we employ a simple but quite accurate “cylindrical nucleus” model (see e.g.[13, 14]). Namely, we set T ( b ) = 2 R A if b < R A and zero otherwise. The impact parameterintegrals in (13),(15) can now be taken exactly. In particular, integration over b a is described inAppendix. Since in QCD r (cid:28) R A we can neglect a very narrow region | b − r / | < R A < | b + r / | in which case (A3) yields for the electromagnetic term in the elastic dipole–nucleon scatteringamplitude (cid:10) i Γ q ¯ qN em ( b , s ; r ) (cid:11) = 2 α (cid:20) − b · r R A θ ( R A − b ) + ln | b − r / || b + r / | θ ( b − R A ) (cid:21) . (16)The total cross section for dipole-nucleus scattering has the same form as (13) and can be nowwritten as σ q ¯ qA tot ( s ; r ) =2 (cid:90) d b (cid:26) − exp (cid:2) − A (cid:10) Γ q ¯ qN s ( , s ; r ) (cid:11)(cid:3) cos (cid:18) αZ b · r R A (cid:19)(cid:27) θ ( R A − b )+ 2 (cid:90) d b (cid:26) − cos (cid:18) αZ ln | b − r / || b + r / | (cid:19)(cid:27) θ ( b − R A ) . (17)In the first line of (17) we can replace the cosine by one, because r (cid:28) R A , αZ ∼
1. The corre-sponding contribution to the cross section is σ q ¯ qA s ( s ; r ) = 2 πR A (cid:8) − exp[ − A (cid:10) Γ q ¯ qN s (0 , s ; r ) (cid:11) ] (cid:9) , (18)which is a purely QCD term, hence the subscript “s” for the “strong” interaction. Integral in thesecond line of (17) can be taken exactly and yields the QED contribution [15, 16] σ q ¯ qA em ( s ; r ) ≡ (cid:90) b max R A db b (cid:90) π dφ (cid:26) − cos (cid:18) αZ ln b + r / − br cos φb + r / br cos φ (cid:19)(cid:27) = 4 πr ( αZ ) ln b max R A = 4 πr ( αZ ) ln s m q m N R A , (19)where m q and m N are quark and nucleon masses correspondingly. Terms of order r /R A areneglected in (19). ∗ Energy dependence arises from the long distance cutoff b max = s/ (4 m N m q ) ofthe b -integral.The total cross section is thus simply a sum of the QCD and QED terms σ q ¯ qA tot ( s ; r ) = σ q ¯ qA s ( s ; r ) + σ q ¯ qA em ( s ; r ) . (20)From comparison of (18) and (19) it is clear that the QED contribution to the total cross section issuppressed relative to the QCD term by ( rαZ/R A ) log s , hence the Coulomb correction is largestfor soft processes with larger r . Since the largest dipole size is of order R p , the smallest suppressionfactor is of order ( αZ ) /A / log s , which for gold nucleus is about 0 . √ s = 200 GeV. Because, Z ∼ A , the relative contribution of the Coulomb correction increases with A . ∗ I would like to stress that approximation r (cid:28) R A holds only if the dipole size r is determined by a QCD scale. Forexample, in photon emission r ∼ /k , where k is photon’s momentum, in dilepton production r ∼ /M , where M is dilepton’s invariant mass. Therefore, only if k and M are at or above ∼ /R p ∼
200 MeV can this approximationbe used. The original calculation of [17, 18] was done in QED in the opposite limit of a point-like nucleus r (cid:29) R A . At high energies, dipole–nucleon scattering amplitude acquires energy dependence Γ q ¯ qN ∼ s ,where to the leading order in QCD ∆ s = 4 ln 2( α s N c /π ) [19, 20] and in QED ∆ em = (11 / πα [21, 22]. Since ∆ em (cid:28) ∆ s we can neglect the effect of QED evolution. A phenomenological wayto take QCD evolution into account is to parameterize the scattering amplitude in terms of quarksaturation momentum ˜ Q s and anomalous dimension γ as follows A (cid:10) Γ q ¯ qN s (0 , s ; r ) (cid:11) = 14 ( r ˜ Q s ) γ , (21)where ˜ Q s ≈ . A / GeV and γ ≈ .
63 [23]. Numerical value of the saturation momentum isknown from DIS and heavy-ion phenomenology (see e.g. [24]).
IV. PHOTON PRODUCTION
In this and the next section we discuss photon and dilepton production in high energy pA colli-sions. Photon production without electromagnetic corrections was calculated in [25]. If we assumethe validity of the collinear factorization on the proton side, the problem reduces to computing thephoton and dilepton production in qA collisions. † We adopt the following notations: four-momentaof incoming quark, photon and outgoing quark are q , k and k correspondingly; bold face denotestheir respective transverse components; z = k /q + . Transverse coordinates of incoming quark,photon and outgoing quark in the amplitude are u , x and x ; those in the complex conjugated am-plitude are distinguished by a prime; r = x − x , r (cid:48) = x (cid:48) − x (cid:48) , b = ( x + x ) / b (cid:48) = ( x (cid:48) + x (cid:48) ) / S ( b , r ) = 1 − Im [ i Γ q ¯ qA ( b , s ; r )] = exp[ − A (cid:10) Γ q ¯ qN s ( b , s ; r ) (cid:11) ] cos[ Z (cid:10) i Γ q ¯ qN em ( b , s ; r ) (cid:11) ] . (22)With these notations we can write down the double-inclusive cross section as follows [29–31] dσ qA → γqX d k d k dz = 12(2 π ) (cid:90) d u d u (cid:48) d x d x d x (cid:48) d x (cid:48) e − k · ( x − x (cid:48) ) − i k · ( x − x (cid:48) ) × φ q → qγ ( r , r (cid:48) , z ) (cid:2) − S (( x (cid:48) + u ) / , x (cid:48) − u ) − S (( x + u (cid:48) ) / , x − u (cid:48) )+ S (( x (cid:48) + x ) / , x (cid:48) − x ) + S (( u + u (cid:48) ) / , u − u (cid:48) ) (cid:3) , (23)where the square of the light-cone wave-function φ q → qγ ( r , r (cid:48) , z ) = 2 e f (2 π ) r · r (cid:48) r r (cid:48) − z ) z δ ( u − z x − (1 − z ) x ) δ ( u (cid:48) − z x (cid:48) − (1 − z ) x (cid:48) ) (24) † One should be cautious with the collinear factorization of dilute projectiles at high energies since it is not valid inexclusive processes, see e.g. [26], and is violated even in some inclusive processes [27, 28]. describes photon emission off quark in the chiral limit. According to (22),(9),(10) we have S ( b , r ) = exp (cid:26) − C F N c ρT ( b ) 12 πr α s ln 1 rµ (cid:27) cos (cid:26) ZA ρ α (cid:90) d b a T ( b a ) ln | b − r / − b a || b + r / − b a | (cid:27) . (25)Integration over the final quark transverse momentum k gives the single-inclusive cross section dσ qA → γqX d k dz = 2 αe f − z ) z (cid:90) d ˜ b (cid:90) d r (2 π ) (cid:90) d r (cid:48) (2 π ) e − i k · ( r − r (cid:48) ) r · r (cid:48) r r (cid:48) φ q → qγ ( r , r (cid:48) , z ) × (cid:104) −S (˜ b + z r / , z r ) − S (˜ b + z r (cid:48) / , z r (cid:48) ) + 1 + S (˜ b + z ( r + r (cid:48) ) / , z ( r − r (cid:48) )) (cid:105) . (26)where ˜ b = b − r / b a in (25) yields S ( b , r ) = e − ( ˜ Q s r ) γ θ ( R A − b ) + cos (cid:18) αZ ln | b − r / || b + r / | (cid:19) θ ( b − R A ) (27)up to terms of order r /R A . With the same accuracy, integration over b in (26) can now be doneexplicitly using (13),(22),(18),(19): dσ qA → γqX d k dz = αe f − z ) z (cid:90) d r (2 π ) (cid:90) d r (cid:48) (2 π ) e − i k · ( r − r (cid:48) ) r · r (cid:48) r r (cid:48) φ q → qγ ( r , r (cid:48) , z ) × (cid:104) σ q ¯ qA tot ( s ; z r ) + σ q ¯ qA tot ( s ; z r (cid:48) ) − σ q ¯ qA tot ( s ; z ( r − r (cid:48) )) (cid:105) . (28)Eq. (28) can be cast into a factorized form by employing the following identities [13, 32] (cid:90) d x e − i k · x x x = − πi k k , (29) (cid:90) d x (cid:48) x (cid:48) · ( x + x (cid:48) ) x (cid:48) ( x + x (cid:48) ) = π ln 1 x , (30)The result reads dσ qA → γqX d k dz = α (2 π ) e f k − z ) z (cid:90) d x e − i k · x ln 1 xµ ∇ x σ q ¯ qA tot ( s ; z x ) . (31)The electromagnetic contribution can be calculated exactly: dσ qA → γqX em d k dz = α (2 π ) e f πk − z ) z πz ( αZ ) ln s m q m N R A , (32)where we used (cid:90) d x ln 1 x e − i k · x = 2 πk . (33)To obtain a qualitative estimate of the QCD contribution to the inclusive cross section (31), notethat unless x < /k the exponent is rapidly oscillating. Furthermore, integrand is exponentiallysuppressed at zx > / ˜ Q s . We thus obtain dσ qA → γqX s d k dz ≈ α (2 π ) e f k − z ) z (cid:90) x dx ln 1 x ∂ x [ x∂ x ( σ q ¯ qA tot ( s ; zx ))] , (34)where x is the smallest of three scales 2 /k , 2 / ( z ˜ Q s ) and 1 /µ . Expanding (18) with (21) at small zx we find σ q ¯ qA s ( s ; zx ) ≈ πR A
14 ( ˜ Q s z x ) γ , (35)which upon substitution into (34) and combining with (32) produces dσ qA → γqX d k dz ≈ απ e f k − z ) z (cid:20) πz ( αZ ) ln s m q m N R A + 14 γk R A ˜ Q γs ( x z ) γ ln 1 x (cid:21) . (36)We see that the ratio of the QCD and the QED terms is of order ( R A ˜ Q s ) ( k / ˜ Q s ) η , with η = 1, if k (cid:28) ˜ Q s and η = 1 − γ , if k (cid:29) ˜ Q s . Thus, the QED interactions have the largest relative impactat small photon transverse momenta and in more peripheral events. Note also that the role ofQED interactions diminishes with energy because the saturation momentum increases as a powerof energy, whereas the QED contribution is only logarithmic. The distinct feature of z -dependenceof inclusive photon production cross section is that it vanishes in the eikonal limit z →
0, which isevident from (26). (cid:72) GeV (cid:76) (cid:82) Γ (cid:72) (cid:37) (cid:76) FIG. 1: Fraction of the QED contribution in the total differential photon production cross section. Solidline: Cu, dashed line: Au, √ s = 200 GeV, µ =1/fm, quark mass m q = 150 MeV. The relative magnitude of the Coulomb correction to the photon spectrum can be expressed interms of the ratio R γ = dσ qA → γqX em d k dz (cid:30) dσ qA → γqX d k dz , (37)which is plotted in Fig. 1. As expected, the Coulomb correction is largest at small k and forheavier nucleus. For p-Au collisions at √ s = 200 GeV it constitutes about 7% at small k . Atlarger energies it slowly increases as log s . V. DILEPTON PRODUCTION
Dilepton production by an incident quark is quite complicated because both the quark and theproduced dileptons interact with the electromagnetic field of the target nucleus, and quark alsointeracts with the nuclear color field. At large invariant mass M of produced dilepton pair, anintermediate process of photon splitting γ ∗ → (cid:96) + (cid:96) − can be factored out, which leads to significantsimplifications. This will be our assumption throughout this section. A detailed analysis of thisapproximation can be found in [27].Our notation scheme in this section follows the same pattern as in the previous one. Momentaof incident photon and outgoing leptons are q , k and k correspondingly; lepton’s light-conemomentum fraction is z = k /q + . Transverse coordinates of leptons are x and x ; r = x − x is dipole size, b = ( x + x ) / dσ γ ∗ A → (cid:96) + (cid:96) − d k d k = π (2 π ) (cid:90) dz (cid:90) d x d x d x (cid:48) d x (cid:48) e − i k · ( x − x (cid:48) ) e − i k · ( x − x (cid:48) ) φ γ ∗ → (cid:96) + (cid:96) − ( r , r (cid:48) , z ) × (cid:2) Q em ( x , x , x (cid:48) , x (cid:48) ) − S em ( b , r ) − S em ( b (cid:48) , r (cid:48) ) (cid:3) , (38)where the squared light-cone wave-function describing photon splitting into dilepton pair is givenby φ γ ∗ → (cid:96) + (cid:96) − ( r , r (cid:48) , z ) = 2 απ m (cid:26) r · r (cid:48) rr (cid:48) K ( rm (cid:96) ) K ( r (cid:48) m (cid:96) )[ z + (1 − z ) ] + K ( rm (cid:96) ) K ( r (cid:48) m (cid:96) ) (cid:27) . (39)The scattering matrix elements of electric dipole is (cp. (25)) S em ( b , r ) = cos (cid:8) Z (cid:10) i Γ q ¯ qN em ( b , s ; r ) (cid:11)(cid:9) = cos (cid:26) ZA ρ α (cid:90) d b a T ( b a ) ln | b − b a − r / || b − b a + r / | (cid:27) , (40)and that of electric quadrupole is Q em . The later is a complicated function of its coordinates.Explicit form of its QCD analogue can be found in [31]; it significantly simplifies in the large N c approximation [33]. If either x = x (cid:48) or x = x (cid:48) , the quadrupole reduce to a dipole, e.g. Q em ( x , x , x (cid:48) , x (cid:48) ) | x = x (cid:48) = S em (cid:0) ( x + x (cid:48) ) / , x − x (cid:48) (cid:1) . (41)0Upon integration over k and k , eq. (38) gives the total inclusive cross section that agrees withresults of [5].Since we are interested in invariant mass distribution, it is convenient to introduce another pairof independent momenta, photon transverse momentum q and the relative momentum of the pair (cid:96) , as follows q = k + k , (cid:96) = (1 − z ) k − z k . (42)Invariant mass of dilepton can be expressed as M = ( k + k ) = q + ( k − + k − ) − ( k + k ) = m + (cid:96) z (1 − z ) . (43)We took into account that in the light-cone perturbation theory q − (cid:54) = k − + k − because photonsplitting is only an intermediate step in dilepton production. Using these notations, the phase in(38) can be written as − i k · ( x − x (cid:48) ) − i k · ( x − x (cid:48) ) = − i (cid:96) · ( r − r (cid:48) ) − i q · ( b − b (cid:48) ) − i q · ( r − r (cid:48) )( z − / . (44)Factorization of photon decay assumes that (cid:96) ∼ /M and q < m (cid:96) [27]. Therefore, we can neglectthe last term in (44): − i k · ( x − x (cid:48) ) − i k · ( x − x (cid:48) ) ≈ − i (cid:96) · ( r − r (cid:48) ) − i q · ( b − b (cid:48) ) . (45)For an almost on-mass-shell photon, the transverse polarization is dominant. Expanding (39) atsmall m (cid:96) and keeping only the term dominant at small dipole sizes, we get φ γ ∗ → (cid:96) + (cid:96) − ( r , r (cid:48) , z ) ≈ απ r · r (cid:48) r r (cid:48) [ z + (1 − z ) ] . (46)Since (46), as well as the scattering factors are q -independent, we can integrate in (38) over q , whichin view of (45), yields (2 π ) δ ( b − b (cid:48) ). Moreover, since M is larger than the typical momentumtransfer ∆ ∼ √ αZ/b by a t -channel photon, i.e. ∆ (cid:28) M , we can expand the quadrupole amplitudeat small difference | r − r (cid:48) | (cid:28) | r + r (cid:48) | /
2, which yields Q em ≈ S em ( b , r − r (cid:48) ). (Other scatteringfactors in (38) do not depend on this difference). With these assumptions and approximations wederive at large Mdσ γ ∗ A → (cid:96) + (cid:96) − d (cid:96)d b = π (2 π ) (cid:90) dz [ z + (1 − z ) ] (cid:90) d rd r (cid:48) e − i (cid:96) · ( r − r (cid:48) ) απ r · r (cid:48) r r (cid:48) × (cid:2) S em ( b , r − r (cid:48) ) − S em ( b , r ) − S em ( b , r (cid:48) ) (cid:3) . (47)1We can take one of the two-dimensional integrals using (29),(30). This gives dσ γ ∗ A → (cid:96) + (cid:96) − d (cid:96)d b = π (2 π ) απ (cid:90) dz [ z + (1 − z ) ] 2 π(cid:96) (cid:90) d re − i (cid:96) · r ln 1 r ∇ r [1 − S em ( b , r )] . (48)To calculate the Laplacian appearing in the right-hand-side of (48) we use the expression for thescattering amplitude in the integrand of (17) (with Γ s = 0): ∇ r [1 − S em ( b , r )] =(2 αZ ) b R A cos (cid:18) αZ b · r R A (cid:19) θ ( R A − b )+ b ( b − r / ( b + r / (2 αZ ) cos (cid:18) αZ ln | b − r / || b + r / | (cid:19) θ ( R A − b ) (49)As mentioned before, at b < R A we can expand this expression in powers of r /R A , while at b > R A in powers r /b . We have ∇ r [1 − S em ( b , r )] ≈ (2 αZ ) (cid:20) b R A θ ( R A − b ) + 1 b θ ( R A − b ) (cid:21) . (50)Plugging this into (48) and employing (33) yields dσ γ ∗ A → (cid:96) + (cid:96) − d (cid:96)d b = 43 π α(cid:96) ( αZ ) (cid:20) b R A θ ( R A − b ) + 1 b θ ( R A − b ) (cid:21) . (51)Notice that the dilepton spectrum at a given impact parameter is energy-independent. This aconsequence of the quasi-classical approximation. Integration over impact parameter can be donedirectly in (48) using (33) and (19) if neglect a small contribution at b < R A . The result is dσ γ ∗ A → (cid:96) + (cid:96) − d (cid:96) = π (2 π ) απ (cid:90) dz [ z + (1 − z ) ] 2 π(cid:96) (cid:90) d re − i (cid:96) · r ln 1 r π ( αZ ) ln s m (cid:96) m N R A = 8 α π (cid:96) ( αZ ) ln s m (cid:96) m N R A . (52)The same formula is obtained by integration of an approximate formula (51) over b . This is because(19) assumes that b (cid:29) R A . Note that b -integrated cross section exhibits logarithmic dependenceon energy, which enters through the cutoff b max (see (19)).If there were no QED interactions of dilepton with the nucleus we would have instead of (47) dσ γ ∗ → (cid:96) + (cid:96) − d (cid:96)d b = π (2 π ) (cid:90) dz [ z + (1 − z ) ] (cid:90) d rd r (cid:48) e − i (cid:96) · ( r − r (cid:48) ) απ r · r (cid:48) r r (cid:48) = π (2 π ) (cid:90) dz [ z + (1 − z ) ] 2 απ (cid:96) = α π (cid:96) , (53)Changing the integration variable from (cid:96) to M we obtain the well-known QED result for virtualphoton decay probability dσ γ ∗ → (cid:96) + (cid:96) − d b = 2 α π dMM . (54)2The difference between the dilepton production cross section in the Coulomb field and in vacuumcan be expressed as the following ratio f ( (cid:96), b ) = dσ γ ∗ A → (cid:96) + (cid:96) − d bd (cid:96) (cid:30) dσ γ ∗ → (cid:96) + (cid:96) − d (cid:96)d b . (55)Using (51), (53) we derive that at large invariant masses f ( (cid:96), b ) = 4( αZ ) R A (cid:96) (cid:20) b R A θ ( R A − b ) + R A b θ ( R A − b ) (cid:21) . (56)As in the previous section, we express the relative magnitude of the Coulomb correction to thedilepton spectrum as a ratio R (cid:96) = f (cid:96) f (cid:96) , (57)which is plotted in Fig. 2 for electron-positron pair production by high energy virtual photonin a Coulomb field of gold nucleus. We observe that the relative contribution of the Coulombcorrections to dilepton production increases at smaller M ∼ (cid:96) and toward the nucleus boundaryand can reach 10% in semi-peripheral and peripheral collisions. (cid:123) (cid:72) GeV (cid:76) (cid:82) (cid:123) (cid:72) (cid:37) (cid:76)
FIG. 2: Fraction of the QED contribution in the e + e − dilepton production cross section in the Coulombfield of gold nucleus, A = 197, Z = 79. Solid line: b = 1 fm, dashed line: b = 3 fm, dashed-dotted line: b = 5 fm, dotted line: b = 7 fm. VI. DISCUSSION AND SUMMARY
In this article we investigated the role of electromagnetic Coulomb interactions in photon anddilepton production in high energy pA collisions. Among other important processes that receiveelectromagnetic corrections is gluon emission off a fast quark and q ¯ q production. Photon productionvanishes in the eikonal approximation, i.e. when valence quark moves strictly along the straight3line, corresponding to z →
0. In contrast, gluon production cross section diverges in this limit as1 /z giving the leading logarithmic term to the rapidity distribution. Therefore, QED contributionto gluon production appears only as a correction to a sub-leading order in α s and can be safelyneglected. In q ¯ q production via gluon splitting, Coulomb corrections come about already at theleading order because at least one fermion carries finite z .QED corrections to photon production are largest at small transverse momentum of photonand increase with energy and nuclear weight. In p-Au collisions at √ s = 200 GeV per nucleon,the Coulomb correction to photon production reaches 7%. Dilepton production receives QEDcontributions at two stages: at virtual photon emission, which is qualitatively similar to photonproduction, and at virtual photon splitting into a dilepton pair. The later can proceed even invacuum. We computed the Coulomb correction to this process and found that it is largest for smallinvariant masses M and increases with impact parameter. In p-Au collisions at √ s = 200 GeVper nucleon, the Coulomb correction is up to 10% at M ∼
200 MeV. An upshot of this is that theprompt photon yield extracted from the dilepton spectrum using the equation dN (cid:96) + (cid:96) − dM = α πM N γ ,is overestimated by about 10%.It is of a special interest to extend the analysis of this article to the initial stage of heavy-ioncollisions. At a qualitative level, we expect that the main features that we observed in pA scatteringare carried over to AA scattering. However, a quantitative estimate of the Coulomb corrections inheavy-ion collisions require further analytical investigation. Acknowledgments
This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-87ER40371.
Appendix A
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