arXiv:1204.4389v2 [hep-ph] 23 Apr 2012 ubri h rbl nrae ihtetmeaueof temperature the the with increases that fireball fact the the the reach in it account to number lose into region not Taking collision will the detectors. and through plasma pass QGP the they the when of in information elec- produced the through carry only the channel, state, with formation. tromagnetic final QGP interact not dilute the leptons identify can and to Since QGP cold signatures needs the the one in collisions, and directly the of measured in expansion be formed rapid decreasing the fireball the phase during the Considering density a relativis- and [12]. such through temperature collisions of be ion only - heavy realization can tic a The laboratories to in gas deconfinementtransition [11]. the a (QGP) to from plasma dense moving related in and is process hot which in matter, transition nuclear phase (QCD) modynamics exper- been yet not discovered. investi- has imentally pro- dimuonium is However, dimuonium colliders [10]. of - gated the possibility modern from at the emitted Recently, duction coherent [9]. of nuclei fusion the is nism ue nhayincliin hog ueelectromag- pure a process through collisions netic ion heavy in duced A γA uhsotrta h ietm fmo (2 of time life the than shorter much a to cay hstemo ekdcycnb goe ntepro- direct production like the dimuonium in proposed, Many ignored been have be mechanisms dimuonium. can decay of weak duction muon the thus ooy“unu”hsbe indt h on state bound the to signed been ( has “” nology iin[2], lision ytm ihBh ais52f,tebnigenergy binding the (1 fm, state QED ground 512 compact the radius most of the Bohr For of with QED. one of systems is calculations theoretically dimuonium electrody- standard are instance, with quantum properties 6] its of [2, of predicted system many ideal (QED), namics an is dimuonium usinta hte hr r tu unu”[]or [2] ( muonium” states “true 4] are natural [3, a there “dimuonium” to whether lead that of question nature leptonic the of ification o h states the for µ + ti eeal eivdta hr saqatmchro- quantum a is there that believed generally is It h bevto fpstoim( of observation The tnsfrances iunuscnas epro- be also can Dimuoniums nucleus. a for stands e → − ,wihhsbe icvrdi 90[] Since [5]. 1960 in discovered been has which ), γγ ( µ n 1 and ) + e µ + o olsos h rdcini otoldb h process the by controlled is production The collisions. ion ASnmes 61.e 57.j 12.38.Mh 25.75.Cj, 36.10.Ee, numbers: di PACS energies. energy LHC Wh transverse and the equation. 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Since s = (q + q )2 and the transition probability → 1 2 strange are much heavier than light quarks, the − 2 − (µ+µ )g 16πs (µ+µ )g amount of strange quarks in QGP is less than 10% of the Wqq¯ (s)= σqq¯ (s). 2 2 2 2 2 amount of light quarks at RHIC and LHC energies, and (s md mg) 4mdmg q − − − we consider only light quarks in the calculation (Nf = 2), (3) and their current mass is set to be zero. The scattering The medium created in high-energy nuclear collisions amplitude (q q p p k) for the process qq¯ µ+µ−g M 1 2 → 1 2 → evolves dynamically. In order to extract information can be written as about the medium by analyzing the dimuonium distri- butions, both the hot and dense medium and the dimuo- 2 µ 1 ν ν 1 µ = gQe v(q2) tγ γ + γ tγ nium production process must be treated dynamically. M /q k/ /q k/ h 2 − 1 − i In this paper, we treat continuous dimuonium production gνρ ρ ∗ in QGP self-consistently, including hydrodynamic evolu- u(q1) 2 u(p1)γ v(p2)ǫµ(k), (1) × (p1 + p2) tion of the QGP. Since dimuonium is a pure electromag- netic system, it can not be thermalized with the medium where q1, q2,p1,p2 and k are respectively the momenta which is governed by strong interactions. Thus its phase + − of q,q ¯, µ , µ and g, Q is the quark charge number, t space distribution should be controlled by a transport is the color matrix, g is related to the effective coupling equation. The transport equation should then be solved 2 constant via the definition αs = g /(4π), u, v, u and v together with the hydrodynamic equation which charac- are Dirac spinors, and ǫµ is the gluon polarization vector terizes the space-time evolution of the QGP. Considering satisfying ǫ∗ (k)ǫ (k) = g + k k /m2 with thermal µ ν − µν µ ν g that dimuonium is a heavy , we use a classical gluon mass mg =2gT/3 in the deconfinement phase [14]. Boltzmann-type transport equation to describe its evolu- For simplicity, we have dropped in (1) the flavor, color tion. The distribution f(pt, y, xt,η,τ b) as a function of and indices of quarks and . | transverse momentum pt and coordinate xt, longitudinal rapidity y and space-time rapidity η and proper time τ + + µ µ− µ µ− at fixed impact parameter b in a heavy ion collision is characterized by the equation ∂ 1 ∂ cosh(y η) + sinh(y η) + v f = β  − ∂τ τ − ∂η t · ∇t (4) with the dimuonium transverse velocity vt = pt/Et and g g 2 2 transverse energy Et = md + pt . Because the life time of dimuonium is muchp longer than the life time of QGP, the decay of dimuonium is ignored. Moreover, q q¯ q q¯ since dimuonium does not participate in strong inter- actions, the generated dimuonium interacts with QGP FIG. 1: The Feynman diagrams at tree level for the main only electromagnetically, the dissociation can then be ne- − dimuonium production process qq¯ (µ+µ )g in QGP. glected too. Therefore, we consider only the gain term → β(pt, y, xt,η,τ b) on the right hand side of the transport equation, | Introducing the total and relative momenta p = p1 +p2 ′ + − 3 3 3 − and p = (p1 p2)/2 of µ and µ , the amplitude 1 d k d q1 d q2 (µ+µ )g ′ − β = 3 3 3 Wqq¯ (s) (q1q2 p1p2k) can be expressed as (q1q2 pp k). 2Et Z (2π) 2Eg (2π) 2Eq (2π) 2Eq¯ M → + − M → Since the bound state (µ µ ) is non-relativistic, we can 4 (4) fqfq¯(1 + fg)(2π) δ (p + k q1 q2), (5) compute its production amplitude (q1q2 pk) by in- × − − M ′ → tegrating out the relative momentum p in the center-of- 2 2 2 2 where Eq = mq + q1, Eq¯ = mq + q2 and Eg = mass frame of the dimuonium [15], q q 2 2 mg + k are respectively quark, anti-quark and gluon 3 ′ 2 d p ′ ∗ ′ qenergies, and f , f and f are the thermal distribu- = (q1q2 pp k)ψ (p ), q q¯ g 3 µ M rmd Z (2π) M → q uµ/T tions for quarks and gluons, fq = 1/ e 1 +1 , 3/2   ′ 8√πa0 µ µ p q2 uµ/T k uµ/T ψ( ) = 2 ′2 2 , (2) fq¯ = 1/ e +1 and fg = 1/ e 1 . By (1 + a0p )   − using Bjorken’s hydrodynamics [16], the fluid velocity uµ ′ where md is the dimuonium mass, and ψ(p ) is the rel- appeared in the distributions and the temperature T in ative wave function for the ground state in momentum the distributions and gluon mass are functions of coor- x b space with a0 being the Bohr radius. From the known dinates ( t, η) at fixed and determined by the ideal scattering amplitude, we can calculate the dimuonium hydrodynamic equations [17] + − production cross section σ(µ µ )g(s) as a function of ∂ E + M = (E + p)/τ, qq¯ τ ∇ · − 3

∂τ Mx + (Mxv) = Mx/τ ∂xp, We can define the transverse energy distribution ∇ · − − f(E b) = dN(b)/(2πE dE ). It is shown in Fig.2 ∂τ My + (Myv) = My/τ ∂yp, t t t ∇ · − − for central| (b=0) Au+Au collisions at RHIC energy ∂τ R + (Rv) = R/τ (6) ∇ · − √sNN = 200 GeV and Pb+Pb collisions at LHC energy with the Lorentz factor γ = 1/√1 v2 and definitions √sNN = 5.5 TeV. We have taken the effective coupling E = (ǫ+p)γ2 p, M = (ǫ+p)γ2v and−R = γn as functions constant αs =0.3 (corresponding to T/Tc 1.5 2 [13]) − and the dimuonium mass m 2m = 211≃ MeV− (ne- of energy density ǫ, pressure p and density n of the d ≃ µ medium. To close the hydrodynamic equations, we need glecting the binding energy 1.4 KeV) in the numerical a equation of state to describe the nature of the QGP. We calculations. While the dimuoniums distribute in a wider take the result from the lattice QCD simulation with a region at LHC, they behave similarly at two energies. The result in the low Et region of Et < 1 GeV can be pa- phase transition temperature of deconfinement Tc = 190 − rameterized as a thermal distribution f(E ) e Et/Teff MeV [18]. Since the colliding energy is so high in heavy t ∼ ion collisions at RHIC and LHC, the net baryon density with a slope parameter Teff = 195MeV at RHIC and 240 in the fireball is rather small [12], we can simply set n =0 MeV at LHC. This indicates that the thermodynamic in- in the numerical calculations. The initial condition of formation of the medium carried by quarks and gluons is the hydrodynamic equations is controlled by the nuclear partly inherited by the produced dimuoniums and can be geometry which determines the impact parameter and used to signal the QGP formation in high energy nuclear the colliding energy which governs the ratio of soft to collisions. hard contributions [19]. Since we did not consider the initial dimuonium pro- 10-8 duction before the QGP formation and have neglected ) -2 the loss term in the transport equation (4), its analytic 10-9 solution becomes simple, 10-10 LHC ) (GeV

τ ′ ′ ′ t ′ β (pt, y, Xt(τ ),H(τ ), τ b) -11 f(pt, y, xt,η,τ b) = dτ | 10

′ dE | Zτ0 ∆(τ ) t ′ E -12

π 10 Θ(T (Xt,H,τ b) Tc) (7) RHIC × | − -13 with the definitions 10 dN/(2 -14 ′ ′ ′ 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Xt(τ ) = xt vt [τ cosh(y η) τ ∆(τ )] , ′ − ′ − − H(τ ) = y arcsinh (τ/τ sinh(y η)) , Et (GeV) − − ′ ∆(τ ) = 1 + (τ/τ ′)2sinh2(y η), (8) q − FIG. 2: The dimuonium transverse energy distribution dN/(2πEtdEt) in central Au+Au collisions at RHIC energy where the local temperature T (X ,H,τ ′ b) as a func- t | √sNN = 200 GeV and Pb+Pb collisions at LHC energy tion of time and coordinates at fixed impact parameter √sNN = 5.5 TeV. is determined by the evolution of the medium (6), the step function Θ indicates that the mechanism of gen- erating dimuonium discussed here can only take place in deconfined region, and the coordinate shifts x X and t → t η H in the solution (7) reflect the leakage effect in the 4.0 LHC transverse→ and longitudinal directions. The time integra- tion is from the initial time τ0 to τ. By integrating the 3.0 distribution over the phase space, we obtain the dimuo- N nium transverse momentum distribution at fixed impact 9 2.0 parameter b. For the finally observed dimuonium distri- 10 bution, τ should be so chosen that it is not earlier than 1.0 the end time τf of the QGP. RHIC 0.0 dN(b) f(pt b) = 0 100 200 300 400 | 2πptdpt Np τc 2 = d xtdydηEt cosh(y η) (2π)3 Z − FIG. 3: The rescaled dimuonium number 109N as a function f(pt, y, xt,η,τc b), (9) × | of participant number Np in Au+Au collisions at RHIC en- ergy √sNN = 200 GeV and Pb+Pb collisions at LHC energy where τc is an arbitrary time after τf , since the momen- √sNN = 5.5 TeV. tum distribution is unchanged for all τ > τf . 4

Fig.3 shows the momentum integrated dimuonium valid for the electromagnetic channel. For b 2R , the ≪ A yield N(b)=2π f(pt b)ptdpt as a function of the num- QGP is formed with a high temperature, long life time | + − ber of participantR Np in heavy ion collisions at and large size, the process qq¯ (µ µ )g becomes dom- → RHIC and LHC energies. The relation between Np and inant. Since the pure electromagnetic channel is not re- the impact parameter b can be easily determined by the lated to the fireball, the transverse energy distribution in nuclear geometry. With increasing centrality, the par- this channel can not show the thermodynamic behavior. ticipant number increases, and the temperature, the life In summary, we investigated the dimuonium produc- time and the space region of the formed QGP increase. tion in the QGP formed in relativistic heavy ion colli- As a result, the dimuonium yield goes up with centrality, sions. The dimuonium motion in the QGP is described due to the enhancement of the quark and gluon numbers. by a transport equation with the gain term character- + − For central collisions with maximum Np, the dimuonium ized by the production process qq¯ (µ µ )g, and the number is 1.3 10−9 at RHIC energy and becomes about space-time evolution of the plasma→ is controlled by ideal 3 times larger× at LHC energy, as shown in Fig.3. hydrodynamic equations. By solving the coupled trans- Let’s now compare the two dimuonium production port and hydrodynamic equations for high energy nuclear mechanisms in heavy ion collisions. One is through the collisions at RHIC and LHC energies, we found that while + − pure electromagnetic channel, A1A2 A1A2(µ µ ), the electrodynamics dominated dimuonium yield is not proposed by Ginzburg et al. [9], and the→ other is inside high enough, the transverse energy distribution inherits the formed QGP, qq¯ (µ+µ−)g, discussed here. For the thermodynamic behavior of the hot medium and can → impact parameter b > 2RA, where RA is the colliding be considered as an electromagnetic probe of the QGP. nuclear radius, there is no QGP formed, the production Acknowledgement: The work is supported by is only through the electromagnetic channel. However, the NSFC (Grant Nos. 10975084 and 11079024) and for b < 2RA, the colliding nuclei are broken, and the RFDP (Grant No.20100002110080 ). PZ thanks Prof. assumption of the replacement of the perturbation pa- Huanzhong Huang for the stimulating discussions in the rameter α by Zα with each exchange is no longer beginning of the work.

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