aa r X i v : . [ h e p - ph ] A ug In-medium QCD and Cherenkov gluonsI.M. Dremin Lebedev Physical Institute, Moscow, Russia
Abstract
The equations of in-medium gluodynamics are proposed. Their classicallowest order solution is explicitly shown for a color charge moving with con-stant speed. For nuclear permittivity larger than 1 it describes emission ofCherenkov gluons resembling results of classical electrodynamics. The choiceof nuclear permittivity and Lorentz-invariance of the problem are discussed.Effects induced by the transversely and longitudinally moving (relative to thecollision axis) partons at LHC energies are described.
The collective effects observed in ultrarelativistic heavy-ion collisions at SPSand RHIC [1, 2, 3, 4] have supported the conjecture of quark-gluon plasma(QGP) formed in these processes. The properties and evolution of this mediumare widely debated. At the simplest level it is assumed to consist of a setof current quarks and gluons. It happens however that their interaction isquite strong so that the notion of the strongly interacting quark-gluon plasma(sQGP) has been introduced. Moreover, this substance reminds an ideal liquidrather than a gas. Whether perturbative quantum chromodynamics (pQCD)is applicable to the description of the excitation modes of this matter is doubt-ful. Correspondingly, the popular theoretical approaches use either classicalsolutions of in-vacuum QCD equations at the initial stage or hydrodynamicsat the final stage of its evolution.The collective excitation modes of the medium may however play a crucialrole. One of the ways to gain more knowledge about the excitation modes isto consider the propagation of relativistic partons through this matter. Phe-nomenologically their impact would be described by the nuclear permittivityof the matter corresponding to its response to passing partons. Namely thisapproach is most successful for electrodynamical processes in matter. There-fore it is reasonable to modify QCD equations by taking into account collectiveproperties of the quark-gluon medium. For the sake of simplicity we considerhere the gluodynamics only. The generalization to quarks is straightforward. email: [email protected] he classical lowest order solution of these equations coincides with Abelianelectrodynamical results up to a trivial color factor. One of the most spec-tacular of them is Cherenkov radiation and its properties. Now, Cherenkovgluons take place of Cherenkov photons [5, 6]. Their emission in high energyhadronic collisions is described by the same formulae but with nuclear per-mittivity in place of the usual one. It should be properly defined. Actually,one considers them as quasiparticles, i.e. quanta of the medium excitationswith properties determined by the permittivity. The interplay of mediumproperties and velocity of the particle is crucial for the radiation field.Another important problem of this approach is related to the notion of therest system of the medium. The Lorentz invariance is lost if the permittivityis introduced . Therefore one has to choose the proper coordinate systemwhere its definition is at work. While it is simple for macroscopic media inelectrodynamics, one should consider partons moving in different directionswith different energies in case of heavy-ion collisions. It has direct impacton properties of emitted particles. The fast evolution of the medium and itsshort lifetime differ it from common electrodynamical examples.All these problems are discussed in what follows. At the beginning let us remind the classical in-vacuum Yang-Mills equations D µ F µν = J ν , (1) F µν = ∂ µ A ν − ∂ ν A µ − ig [ A µ , A ν ] , (2)where A µ = A µa T a ; A a ( A a ≡ Φ a , A a ) are the gauge field (scalar and vector)potentials, the color matrices T a satisfy the relation [ T a , T b ] = if abc T c , D µ = ∂ µ − ig [ A µ , · ] , J ν ( ρ, j ) is a classical source current, ~ = c = 1 and the metrictensor is g µν =diag(+,–,–,–).In the covariant gauge ∂ µ A µ = 0 they are written as (cid:3) A µ = J µ + ig [ A ν , ∂ ν A µ + F νµ ] , (3)where (cid:3) is the d’Alembertian operator. It was shown [8] (and is confirmedin what follows) that in this gauge the classical gluon field is given by thesolution of the corresponding Abelian problem. In principle, this deficiency is cured by the relativistic generalization of the notion of permit-tivity (e.g., see [7]). he chromoelectric and chromomagnetic fields are E µ = F µ , (4) B µ = − ǫ µij F ij , (5)or as functions of gauge potentials in vector notations E a = − gradΦ a − ∂ A a ∂t + gf abc A b Φ c , (6) B a = curl A a − gf abc [ A b A c ] . (7)The equations of motion (1) in vector form are written asdiv E a − gf abc A b E c = ρ a , (8)curl B a − ∂ E a ∂t − gf abc (Φ b E c + [ A b B c ]) = j a . (9)The Abelian equations of in-vacuum electrodynamics are obtained fromEq. (3) if the second term in its right-hand side is put equal to zero andcolor indices omitted. The medium is accounted if E is replaced by D = ǫ E in F µν , i.e. in Eq. (4) . Therefore the Eqs. (8), (9) in vector form aremost suitable for their generalization to in-medium case. The equations of in-medium electrodynamics differ from in-vacuum ones by dielectric permittivity ǫ = 1 entering there as △ A − ǫ ∂ A ∂t = − j , (10) ǫ ( △ Φ − ǫ ∂ Φ ∂t ) = − ρ. (11)The permittivity describes the matter response to the induced fields whichis assumed to be linear and constant in Eqs. (10), (11). It is determinedby the distribution of internal current sources in the medium. Then externalcurrents are only left in the right-hand sides of these equations.Now, the Lorentz gauge condition isdiv A + ǫ ∂ Φ ∂t = 0 . (12)The Lorentz invariance is broken if ǫ = 1 in front of the second terms inthe left-hand sides. Then one has to deal within the coordinate system where ǫ denotes the dielectric permittivity of the medium. The magnetic permittivity is put equalto 1 to simplify the formulae. substance is at rest. The values of ǫ are determined just there. To cancelthese requirements one must use Minkowski relations between D , E , B , H valid for a moving medium [7]. It leads to more complicated formulae, andwe do not use them in this paper.The most important property of solutions of these equations is that whilethe in-vacuum ( ǫ = 1) equations do not admit any radiation processes, ithappens for ǫ = 1 that there are solutions of these equations with non-zeroPoynting vector.Now we are ready to write down the equations of in-medium gluodynam-ics generalizing Eq. (3) in the same way as Eqs. (10), (11) are derived inelectrodynamics. We introduce the nuclear permittivity and denote it also by ǫ since it will not lead to any confusion. After that one should replace E a inEqs. (8), (9) by ǫ E a and get: ǫ (div E a − gf abc A b E c ) = ρ a , (13)curl B a − ǫ ∂ E a ∂t − gf abc ( ǫ Φ b E c + [ A b B c ]) = j a . (14)The space-time dispersion of ǫ is neglected here.In terms of potentials these equations are cast in the form: △ A a − ǫ ∂ A a ∂t = − j a − gf abc ( 12 (curl[ A b , A c ] + [ A b curl A c ]) + ∂∂t ( A b Φ c ) − ǫ Φ b ∂ A c ∂t − ǫ Φ b gradΦ c − gf cmn [ A b [ A m A n ]] + gǫf cmn Φ b A m Φ n ) , (15) △ Φ a − ǫ ∂ Φ a ∂t = − ρ a ǫ + gf abc (2 A b gradΦ c + A b ∂ A c ∂t − ǫ ∂ Φ b ∂t Φ c ) + g f amn f nlb A m A l Φ b . (16)If the terms with explicitly shown coupling constant g are omitted, one getsthe set of Abelian equations which differ from electrodynamical equations(10), (11) by the color index a only. Their solutions are shown in the nextsection. The external current is ascribed to a parton fast moving relative toother partons ”at rest”.The potentials are linear in g because the classical current J µ is linearalso. Therefore omitted terms are of the order of g and can be taken intoaccount as a perturbation. It was done in [9, 10] for in-vacuum gluodynamics.Here, the general procedure is the same. After getting explicit lowest ordersolution (see the next section) one exploits it together with the non-Abeliancurrent conservation condition to find the current component proportional to . Then with the help of Eqs. (15), (16) one finds the potentials up to theorder g . They can be represented as integrals convoluting the current withthe corresponding in-medium Green function. The higher order correctionsmay be obtained in the same way. We postpone their consideration for furtherpublications.The crucial distinction between Eq. (3) and Eqs. (15), (16) is that there isno radiation (the field strength is zero in the forward light-cone and no gluonsare produced) in the lowest order solution of Eq. (3) and it is admitted for Eqs.(15), (16) because ǫ takes into account the collective response (polarization)of the nuclear matter. We have assumed that no color indices are attachedto ǫ . It would correspond to the collective response of the color-neutral (onthe average) medium if color exchange between the external current J µ andmedium excitations is numerous and averages to zero. The lack of knowledgeabout the collective excitations of the nuclear medium prevents more detailedstudies. However it seems to be justified at least for Cherenkov effects. Cherenkov effects are especially suited for treating them by classical approachto Eqs. (15), (16). Their unique feature is independence of the coherence ofsubsequent emissions on the time interval between these processes.The problem of the coherence length for Cherenkov radiation was exten-sively studied [11, 12]. It was shown that the ω -component of the field of acurrent can be imitated by a set of oscillators with frequency ω situated alongthe trajectory. The waves from all oscillators add up in the direction given bythe Cherenkov angle θ independent on the length of the interval filled in bythese oscillators. The phase disbalance ∆ φ between emissions with frequency ω = k/ √ ǫ separated by the time interval ∆ t (or the length ∆ z = v ∆ t ) isgiven by ∆ φ = ω ∆ t − k ∆ z cos θ = k ∆ z ( 1 v √ ǫ − cos θ ) (17)up to terms which vanish for large distances between oscillating sources andthe detector. For Cherenkov effects the angle θ iscos θ = 1 v √ ǫ . (18)The coherence condition ∆ φ = 0 is valid independent of ∆ z . This is a crucial roperty specific for Cherenkov radiation only . Thus the change of color atemission vertices is not important if one considers a particular a -th compo-nent of color fields produced at Cherenkov angle. Therefore the fields (Φ a , A a )and the classical current for in-medium gluodynamics can be represented bythe product of their electrodynamical expressions (Φ , A ) and the color matrix T a . As a result, one can neglect the ”rotation” of color at emission verticesand use in the lowest order for Cherenkov gluons the well known formulaefor Cherenkov photons just replacing α by α S C A for gluon currents in prob-abilities of their emission. Surely, there is radiation at angles different fromthe Cherenkov angle (18). For such gluons one should take into account thecoherence length and color rotation considering corresponding Wilson lines[13].Let us remind the explicit Abelian solution for the current with velocity v along z -axis j ( r , t ) = v ρ ( r , t ) = 4 πg v δ ( r − v t ) . (19)In the lowest order the solutions for scalar and vector potentials are relatedso that A (1) ( r , t ) = ǫ v Φ (1) ( r , t ) , (20)where the superscript (1) indicates the solutions of order g .Therefore the explicit expressions for Φ suffice. Using the Fourier trans-form, the lowest order solution of Eq. (11) with account of (19) can be castin the form Φ (1) ( r , t ) = g π ǫ Z d k exp[ i k ( r − v t )] k − ǫ ( kv ) (21)The integration over the angle in cylindrical coordinates gives the Besselfunction J ( k ⊥ r ⊥ ). Integrating over the longitudinal component k z with ac-count of the poles due to the denominator and then over the transverse one k ⊥ , one gets the following expression for the scalar potential [14]Φ (1) ( r , t ) = 2 gǫ θ ( vt − z − r ⊥ √ ǫv − q ( vt − z ) − r ⊥ ( ǫv − , (22)Here r ⊥ = p x + y is the cylindrical coordinate, z is the symmetry axis.The cone z = vt − r ⊥ p ǫv − The requirement for ∆ φ to be a multiple of 2 π (or a weaker condition of being less or ofthe order of 1) in cases when Cherenkov condition is not satisfied imposes limits on the effectiveradiation length as it happens, e.g., for Landau-Pomeranchuk or Ter-Mikaelyan effects. These poles are at work only for Cherenkov radiation! etermines the position of the shock wave due to the θ -function in Eq. (22).The field is localized within this cone. The Descartes components of thePoynting vector are related according to Eqs. (22), (20) by the formulae S x = − S z ( z − vt ) xr ⊥ , S y = − S z ( z − vt ) yr ⊥ , (24)so that the direction of emitted gluons is perpendicular to the cone (23) anddefined by the Cherenkov angletan θ = S x + S y S z = ǫv − , (25)which coincides with (18).The higher order terms ( g ...) can be calculated using Eqs. (15), (16).The expression for the intensity of the radiation is given by the Tamm-Frank formula (up to Casimir operators) dEdl = 4 πα S Z ωdω (1 − v ǫ ) . (26)It is well known that it leads to infinity for constant ǫ . The ω -dependence of ǫ (its dispersion) usually solves the problem. For absorbing media ǫ acquiresthe imaginary part. The sharp front edge of the shock wave is smoothed. Theangular distribution of Cherenkov radiation widens. The δ -function at theangle (18), (26) is replaced by the Breit-Wigner shape [15] with maximumat the same angle (but | ǫ | in place of ǫ ) and the width proportional to theimaginary part. Without absorption, the potential (22) is infinite on thecone. With absorption, it is finite everywhere except the cone vertex andis inverse proportional to the distance from the vertex. For low absorption,the field on the cone increases as (Im ǫ ) − / (see [29]). Absorption inducesalso longitudinal excitations (chromoplasmons) which are proportional to theimaginary part of ǫ and usually small compared to transverse excitations.The magnetic permittivity is easily taken into account replacing ǫ by ǫµ inthe Breit-Wigner formula.In electrodynamics the permittivity of real substances depends on ω . More-over it has the imaginary part determining the absorption. E.g., Re ǫ for water(see [16]) is approximately constant in the visible light region ( √ ǫ ≈ . ω and becomes smaller than 1 at high energies tending to 1asymptotically. The absorption (Im ǫ ) is very small for visible light but dra-matically icreases in nearby regions both at low and high frequencies. Theo-retically this behavior is ascribed to various collective excitations in the waterrelevant to its response to radiation with different frequencies. Among them he resonance excitations are quite prominent (see, e.g., [17]). Even in elec-trodynamics, the quantitative theory of this behavior is still lacking, however.Then, what can we say about the nuclear permittivity? The partons constituting high energy hadrons or nuclei interact during thecollision for a very short time. Nevertheless, there are experimental indica-tions that an intermediate state of matter (CGC, QGP, nuclear fluid ...) isformed and evolves. Those are J/ ψ -suppression, jet quenching, collective flow( v ), Cherenkov rings of hadrons etc. They show that there is collective re-sponse of the nuclear matter to color currents moving in it. Unfortunately,our knowledge of its internal excitation modes is very scarce, much smallerthan in electrodynamics.The attempts to calculate the nuclear permittivity from first principles arenot very convincing. It can be obtained from the polarization operator. Thecorresponding dispersion branches have been computed in the lowest orderperturbation theory [18, 19, 20]. Then the properties of collective excita-tions have been studied in the framework of the thermal field theories (forreview see, e.g., [21]). Their results with additional phenomenological ad hocassumption about the role of resonances were used in a simplified model ofscalar fields [6] to show that the nuclear permittivity can be larger than 1 thatadmits Cherenkov gluons.Let us stress the difference between these approaches and our considera-tion. In Refs. [18, 19, 20, 21] the medium response to the induced current isanalyzed. Namely it determines the nuclear permittivity. The permittivity isthe internal property of a medium. Its quantitative description poses prob-lems even in QED. It becomes more difficult task in QCD where confinementis not understood. Therefore we did not yet attempt to compute the nuclearpermittivity and introduced it purely phenomenologically in analogy to in-medium electrodynamics. Our main goal is to study the medium response tothe external color current. Cherenkov effect is proportional to g accordingto Eq. (22) if ǫ is constant or chosen purely phenomenologically. However ǫ should tend to 1 at small g and Cherenkov effect disappears. Thus it isof the order of g at small g . Mach waves in hydrodynamics [22] are of thesame order. When the current J (3) is treated as external one in equations ofin-vacuum gluodynamics [8, 9, 10] the effect is proportional to g .We prefer to use the general formulae of the scattering theory [23] toestimate the nuclear permittivity. It is related to the refractive index n of the edium: ǫ = n (27)and the latter one is expressed [23] through the real part of the forward scat-tering amplitude of refracted quanta Re F (0 o , E ) asRe n ( E ) = 1 + ∆ n R = 1 + 6 m π νE Re F ( E ) = 1 + 3 m π ν πE σ ( E ) ρ ( E ) . (28)Here E denotes the energy, ν is the number of scatterers within a single nu-cleon, m π the pion mass, σ ( E ) the cross section and ρ ( E ) the ratio of real toimaginary parts of the forward scattering amplitude F ( E ). Thus the emis-sion of Cherenkov gluons is possible only for processes with positive Re F ( E )or ρ ( E ). Unfortunately, we are unable to calculate directly in QCD thesecharacteristics of gluons and have to rely on analogies and our knowledge ofproperties of hadrons. The only experimental facts we get about this mediumare brought by particles registered at the final stage. They have some fea-tures in common which (one can hope!) are also relevant for gluons as thecarriers of the strong forces. Those are the resonant behavior of amplitudesat rather low energies and positive real part of the forward scattering ampli-tudes at very high energies for hadron-hadron and photon-hadron processes asmeasured from the interference of the Coulomb and hadronic parts of the am-plitudes. Re F (0 o , E ) is always positive (i.e., n >
1) within the low-mass wingsof the Breit-Wigner resonances. This shows that the necessary condition forCherenkov effects n > ρ -meson shape at SPS [24] andazimuthal correlations of in-medium jets at RHIC [4, 25] were explained byemission of comparatively low-energy Cherenkov gluons [26, 27]. The partondensity and intensity of the radiation were estimated. In its turn, cosmic raydata [28] at energies corresponding to LHC ask for very high energy gluons tobe emitted by the ultrarelativistic partons moving along the collision axis [5].Let us note the important difference from electrodynamics where n < n > ω -dependence of n ) was taken into account. Otherwise theintensity of the radiation given by Eq. (26) diverges. It can be easily incor-porated in Eqs. (15), (16) (more precisely, in their Fourier components). Theformula (28) valid for n − ≪ In electrodynamics these quanta are photons. In QCD those are gluons. We can only say that Re F ( E ) ∝ g at small g that confirms above estimates. or larger n . The imaginary part of ǫ can be easily accounted. In principle, itmay be estimated from RHIC data (see [29]).Up to now we did not discuss one of the most important problems of thecoordinate system in which the permittivity is defined. The in-medium equations are not Lorentz-invariant. There is no problemin macroscopic electrodynamics because the rest system of the macroscopicmatter is well defined and its permittivity is considered there. For collisionsof two nuclei (or hadrons) it asks for special discussion.Let us consider a particular parton which radiates in the nuclear matter. Itwould ”feel” the surrounding medium at rest if momenta of all other partons(or constituents of the matter), with which this parton can interact, sum tozero. In RHIC experiments the triggers which registered the jets (created bypartons) were positioned at 90 o to the collision axis. Such partons should beproduced by two initial forward-backward moving partons scattered at 90 o .The total momentum of other partons (medium spectators) is balanced be-cause for such geometry the partons from both nuclei play a role of spectatorsforming the medium. Thus the center of mass system is the proper one to con-sider the nuclear matter at rest in this experiment. The permittivity must bedefined there. The Cherenkov rings consisting of hadrons have been registeredaround the away-side jet which traversed the nuclear medium. This geometryrequires however high statistics because the rare process of scattering at 90 o has been chosen.The forward (backward) moving partons are much more numerous andhave higher energies. However, one can not treat the radiation of such a pri-mary parton in c.m.s. in the similar way because the momentum of spectatorsis different from zero i.e. the matter is not at rest. Now the spectators (themedium) are formed from the partons of another nucleus only. Then the restsystem of the medium coincides with the rest system of that nucleus and thepermittivity should refer to this system. The Cherenkov radiation of suchhighly energetic partons must be considered there. That is what was donefor interpretation of the cosmic ray event in [5]. This discussion clearly showsthat one must carefully define the rest system for other geometries of theexperiment with triggers positioned at different angles.Thus our conclusion is that the definition of ǫ depends on the experimentgeometry. Its corollary is that partons moving in different directions withdifferent energies can ”feel” different states of matter in the same collisionof two nuclei because of the dispersive dependence of the permittivity. The ransversely scattered partons with comparatively low energies can analyze thematter with rather large permittivity corresponding to the resonance regionwhile the forward moving partons with high energies would ”observe” lowpermittivity in the same collision. This peculiar feature can help scan the(ln x, Q )-plane as it is discussed in [30]. It explains also the different valuesof ǫ needed for description of RHIC and cosmic ray data.These conclusions can be checked at LHC because both RHIC and cosmicray geometry will become available there. The energy of the forward movingpartons would exceed the thresholds above which n >
1. Then both typesof experiments can be done, i.e. the 90 o -trigger and non-trigger forward-backward partons experiments. The predicted results for 90 o -trigger geometryare similar to those at RHIC. The non-trigger Cherenkov gluons should beemitted within the rings at polar angles of tens degrees in c.m.s. at LHC bythe forward moving partons (and symmetrically by the backward ones). Thisidea is supported by some events observed in cosmic rays [28, 27]. The equations of in-medium gluodynamics (15), (16) are proposed. They re-mind the in-medium Maxwell equations with non-Abelian terms added. Theirlowest order classical solutions are similar (up to the trivial color factors)to those of electrodynamics (22), especially, for Cherenkov gluons. The nu-clear permittivity of the hadronic medium is related to the forward scatteringhadronic amplitudes and its possible generalization is discussed. This def-inition asks for the distinction between the different coordinate systems inwhich the Cherenkov radiation (and nuclear permittivity) should be treatedfor partons moving in different directions with different energies.This consideration has led to explanation of several effects observed atSPS, RHIC, cosmic ray energies and predicts new features at LHC [30]. Someestimates of properties of the nuclear matter formed in ultrarelativistic heavy-ion collisions have been done and are predicted.
Acknowledgements
I thank A.V. Leonidov for useful discussions. This work was supported inparts by the RFBR grants 06-02-17051-a, 06-02-16864-a, 08-02-91000-CERN-a. eferences [1] K.H. Ackermann et al (STAR), Phys. Rev. Lett. 86 (2001) 402.[2] F. Wang (STAR), J. Phys. G30 (2004) S1299.[3] A. Adare et al (PHENIX), arXiv:0705.3238; 0801.4545.[4] J.G. Ulery, arXiv:0709.1633.[5] I.M. Dremin, JETP Lett. 30 (1979) 140; Sov. J. Nucl Phys. 33 (1981)726.[6] V. Koch, A. Majumder, X.N. Wang, Phys. Rev. Lett. 96 (2006) 172302.[7] L.D. Landau, E.M. Lifshitz, Elektrodinamika sploshnyh sred , M., Nauka,1982, p. 362.[8] Yu.V. Kovchegov, Phys. Rev. D54 (1996) 5463.[9] Yu.V. Kovchegov, D.H. Rischke, Phys. Rev. C56 (1997) 1084;arXiv:hep-ph/9704201.[10] S.G. Matinyan, B. M¨uller, D.H. Rischke, Phys. Rev. C56 (1997) 2191;arXiv:nucl-th/9705024.[11] I.E. Tamm, I.M. Frank, Doklady AN SSSR 14 (1937) 107.[12] I.M. Frank,
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Note added in proofs.