Medium-induced multi-photon radiation
aa r X i v : . [ h e p - ph ] M a y Medium-induced multi-photon radiation
Hao Ma, Carlos A. Salgado
Departamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela, E-15782Santiago de Compostela, Spain
Konrad Tywoniuk
Department of Astronomy and Theoretical Physics, Lund University, S¨olvegatan 14A, S-22362 Lund, Sweden
Abstract.
We study the spectrum of multi-photon radiation off a fast quark in medium in theBDMPS/ASW approach. We reproduce the medium-induced one-photon radiation spectrum indipole approximation, and go on to calculate the two-photon radiation in the Moli`ere limit. Wefind that in this limit the LPM effect holds for medium-induced two-photon ladder emission.
1. Introduction
To date, the medium-modification of the jet fragmentation function has only been calculatedfor the single-particle emission spectrum. In QED, the one-photon radiation off an energeticquark/electron propagating a dense, finite-sized medium was calculated as a multiple scatteringprocess in [1]. In the context of QCD, the corresponding gluon radiation process, includingmultiple gluon interactions, was calculated in [2, 3, 4, 5]. In general, since it does not interact,the photon spectrum is hardly suppressed by formation time effects, in contrast to the gluoncase. Thus, while the suppression of high- p ⊥ hadrons serves as a probe [6] of the hot and densemedium created in the wake of a heavy-ion collision [7, 8, 9, 10], photon production is used asa calibration due to its lack of interaction with the surrounding QCD matter [11].On the other hand, medium-modifications of more exclusive observables, e.g., two-particleemission, are hitherto unknown. Particularly in vacuum, the latter process comprise key featuressuch as the energy and angular ordering of QCD radiation which arise due to interference betweenemitters. In QED, the lack of ordering of the radiation can be traced back to the fact that thephoton does not carry charge. On the contrary, intensive medium-interactions of the quark couldinduce unprecedented interferences between subsequent emissions which, in principle, could bemeasured by experiment. Furthermore, the two-photon spectrum is expected to be less involvedthan the corresponding QCD process due to the sterile nature of the photons, and thus servesalso as a first attempt at computing multi-particle radiation in medium.Following the BDMPS/ASW approach [1, 2, 3, 4, 5], we focus on a highly energetic quarktraversing the medium with energy E q , which radiates a soft photon with energy ω and transversemomentum k ⊥ ( E q ≫ ω ≫ k ⊥ ). The medium is modeled as a collection of static scatteringcenters, given by the gauge field A a ( x ) t a described by a Yukawa potential with the Debyemass serving as the inverse screening length, where t a is the color matrix in the fundamentalrepresentation. Due to the interaction with the medium, the hard quark acquires an eikonalhase which is described by the Wilson line. Additionally, due to the Bloch-Nordsieck structureof the QED radiation vertex, we need to take into account corrections from the Brownian motionof the quark in the transverse plane.
2. Medium-induced one-photon radiation
Here we are interested in the leading contribution to the spectrum in the high-energyapproximation. In QED, one has to keep both order O ((1 /p + ) ) and order O (1 /p + ) termsin the radiation vertex, where p + ≃ √ E q is the quark light cone momentum. Accordingly, wederive the spin non-flip medium-induced one-photon radiation spectrum for a finite medium oflength L, as in [1].For the purpose of illustration, let us consider spectrum for a fixed number of mediumscatterings in the Bethe-Heitler and factorization limits. In the first case, the N scatteringcenters are well-separated, thus the radiation off these scattering centers is the sum of N Bethe-Heitler spectra, given by d I BH d p f ⊥ d k ⊥ d(ln x ) ∝ α em π X i x q i ⊥ k ⊥ ( k ⊥ − x q i ⊥ ) , (1)where p f ⊥ = P i q i ⊥ − k ⊥ and k ⊥ are the transverse momentum of the outgoing quark andemitted photon, respectively, while x is given by ω = xE q , q i ⊥ is the momentum transfer fromthe i th scattering center, and finally α em is the electromagnetic coupling. In the factorizationlimit, on the other hand, the scattering centers are not resolvable, thus the resulting spectrumis given by d I fact d p f ⊥ d k ⊥ d(ln x ) ∝ α em π x ( P i q i ⊥ ) k ⊥ ( k ⊥ − x P i q i ⊥ ) . (2)When i = 1, we get the expression of the opacity expansion at first order from both of thesetwo limits. The full medium-induced one-photon radiation spectrum interpolates between theabove two limits. This is how one incorporates the Landau-Pomeranchuk-Migdal (LPM) effect[12, 13], which accounts for the fact that the scattering centers in medium interfere destructivelyduring the formation time of an on-shell photon.The coherent nature of the spectrum becomes most pronounced in the so-called Moli`ere limit,i.e. i) for soft and small angle photon emission keeping the transverse momentum of the photonmuch larger than the total transverse momentum transfer from the medium, ii) in case of largelength and iii) small density of the medium at hand. In this approximation, we can perform thesum over all multiple soft scatterings and the spectrum then readsd I Mol d p f ⊥ d k ⊥ d(ln x ) ≈ α em π x k ⊥ exp (cid:26) − ( P i q i ⊥ ) n C L + (cid:27) = α em π x k ⊥ exp (cid:26) − ( p f ⊥ + k ⊥ ) n C L + (cid:27) , (3)where n is the density of the medium, C is the probability of the hard quark scattering and L + = √ L is the size of the medium. In eq. (3), we still find the characteristic x rapiditydependence, accompanied by the 1 / k ⊥ drop, as for the Bethe-Heitler and factorization spectrain the limit k ⊥ ≫ q ⊥ . Yet, where as the two latter vanish in the limit q ⊥ →
0, the Moli`erespectrum peaks at zero momentum transfer. This illustrates the fact that the radiation spectrumfor multiple soft scattering retains features which cannot be obtained from a calculation at fixedorder in opacity.
3. Medium-induced multi-photon radiation
We go on to calculate the radiation of two photons inside the medium in both the amplitude andthe conjugate amplitude for the situation of identical sequence of radiations in the amplitude and igure 1: The radiations are inside the medium in both amplitude and conjugate amplitude. Here p = p + , p = (1 − x ) p + and p = (1 − y ) (1 − x ) p + . conjugate amplitude, respectively. We denote this particular contribution as a ladder emission.The corresponding diagram is depicted in Figure 1. In particular, first a photon is radiatedwith transverse momentum k ⊥ and carrying a momentum fraction x of the parent quark, andsubsequently another photon with transverse momentum k ⊥ and carrying a momentum fraction y of the quark. In this case, the expression readsd I d p f ⊥ d k ⊥ d k ⊥ d(ln x )d(ln y ) = ( α em ) (2 π ) Re 2 x y (1 − x ) ( p + ) × Z b + x d y + Z b + y + d a + Z x + y + d b + Z x + b + d z + Z d ρ ⊥ d ρ ⊥ d ρ ⊥ × exp (cid:26) − Z y + x d ξ Σ( ξ , x ρ ⊥ ) − Z b + a + d ξ Σ( ξ , x ρ ⊥ ) − Z x + z + d ξ Σ( ξ , y ρ ⊥ ) (cid:27) × exp (cid:26) − i y (cid:18) p f ⊥ − − xx k ⊥ − − yy k ⊥ (cid:19) · ρ ⊥ − i x (cid:18) k ⊥ x + k ⊥ y (cid:19) · ρ ⊥ (cid:27) × exp (cid:26) i (cid:18) ¯ q − k ⊥ · k ⊥ y (1 − x ) p + (cid:19) ( y + − a + ) (cid:27) exp (cid:26) i (cid:18) ¯ q − k ⊥ · k ⊥ x p + (cid:19) ( b + − z + ) (cid:27) × (cid:18) ∂∂ρ ⊥ − i xy k ⊥ (cid:19) · (cid:18) ∂∂ρ ⊥ + i xy k ⊥ (cid:19) K ( ρ ⊥ , y + ; ρ ⊥ , a + | µ ) × (cid:18) ∂∂ρ ⊥ − i (1 − x ) k ⊥ (cid:19) · (cid:18) ∂∂ρ ⊥ + i (1 − x ) yx k ⊥ (cid:19) K (cid:18) xy ρ ⊥ , b + ; ρ ⊥ , z + | µ (cid:19) , (4)here ¯ q = x m q − x ) p + and ¯ q = y m q − y ) (1 − x ) p + are the reciprocals of the photon formation lengths, m q is the mass of the quark and Σ( ξ , x ρ ⊥ ) = n ( ξ ) σ ( x ρ ⊥ ) / µ = (1 − x ) x p + and µ = (1 − y ) y (1 − x ) p + . The path integrals are given by K ( ρ ⊥ , y + ; ρ ⊥ , x + | µ ) = Z D r ⊥ exp (cid:26)Z x + y + d ξ (cid:20) i µ r ⊥ − Σ( ξ , x r ⊥ ) (cid:21)(cid:27) , (5)where r ⊥ ( y + ) = ρ ⊥ and r ⊥ ( x + ) = ρ ⊥ are the boundary conditions. In the dipoleapproximation, where σ ( x ρ ⊥ ) ≈ Cx ρ ⊥ , the path integral is given as a solution to the harmonicoscillator with imaginary frequency Ω = (1 − i ) p n C x / (2 µ ) [1], namely K osc ( ρ ⊥ , y + ; ρ ⊥ , a + | µ ) = µ Ω π i sin(Ω ( a + − y + )) × exp (cid:26) i µ Ω [( ρ ⊥ + ρ ⊥ ) cos(Ω ( a + − y + )) − ρ ⊥ · ρ ⊥ ]2 sin(Ω ( a + − y + )) (cid:27) . (6)In this case, and furthermore in the Moli`ere limit, one can perform all integrations in eq. (4)analytically. Thus, the spectrum readsd I Mol d p f ⊥ d k ⊥ d k ⊥ d(ln x )d(ln y ) ∝ exp (cid:26) − ( k ⊥ + k ⊥ + p f ⊥ ) n C L + (cid:27) , (7)where | k ⊥ + k ⊥ + p f ⊥ | = P i q i ⊥ is the total momentum transfer from the medium. One candraw several important conclusions from this, seemingly simple, final result. The appearance ofthe Moli`ere factor in eq. (7) signals that the LPM effect still holds for the medium induced two-photon ladder emission. In particular, eq. (7) is not simply a superposition of two one-photonspectra, as one naively would have expected from independent radiation, e.g., in vacuum. Thus,the medium induces some coherence on the multi-emission spectrum. The details of thesecoherence effects are still under investigation.
4. Summary and outlook
We present the study of the spectrum of one- and two-photon radiation off a quark in medium.In the Moli`ere limit, we find similar features for both, namely the appearance of Gaussiandistribution of the total momentum transfer from the medium. In the latter case, this isquite surprising as the corresponding vacuum spectrum is a superposition of two independentemissions.The above conclusions hold, strictly speaking, only for the ladder emission diagram, depictedin Figure 1. For the two-photon case, there appears Feynman diagrams with much morecomplicated emission sequences, in addition to the diagrams including interference with vacuumemissions. We are still working on a complete, analytical solution of all diagrams.
References [1] U. A. Wiedemann and M. Gyulassy, Nucl. Phys. B (1999) 345-382.[2] R. Baier, Yu. L. Dokshitzer, S. Peign´e and D. Schiff, Phys. Lett. (1995) 277.[3] U. A. Wiedemann, Nucl. Phys. B , 409 (2000).[4] C. A. Salgado and U. A. Wiedemann, Phys. Rev. D , 014008 (2003).[5] N. Armesto, C. A. Salgado and U. A. Wiedemann, Phys. Rev. D , 064910 (2005).[6] C. A. Salgado, Prog. Theor. Phys. Suppl. : 355-363, 2007.[7] K. Adcox et al . [PHENIX Collaboration], Nucl. Phys. A , 184 (2005).[8] B. B. Back et al . [PHOBOS Collaboration], Nucl. Phys. A , 28 (2005).9] I. Arsene et al . [BRAHMS Collaboration], Nucl. Phys. A , 1 (2005).[10] J. Adams et al . [STAR Collaboration], Nucl. Phys. A , 102 (2005).[11] S. S. Adler et al . [PHENIX Collaboration], Phys. Rev. Lett. , 232301 (2005).[12] L. D. Landau and I. Ya. Pomeranchuk, Dokl. Akad. Nauk. SSSR (1953) 535, 735.[13] A. B. Migdal, Phys. Rev.103