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2011
Quarkonium dissociation in quark-gluon plasma via ionization in a magnetic field
Kevin Marasinghe Iowa State University, [email protected]
Kirill Tuchin Iowa State University, [email protected]
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Abstract We study the impact of a magnetic field, generated in collisions of relativistic heavy ions, on the decay probability of a quarkonium produced in the central rapidity region. The quark and antiquark components are subject to mutually orthogonal electric and magnetic fields in the quarkonium comoving frame. In the presence of an electric field, the quarkonium has a finite dissociation obabilitypr .We use theWKB approximation to derive the dissociation probability. We find that the quarkonium dissociation energy, i.e., the binding energy at which the dissociation probability is of order unity, increases with the magnetic field strength. It also increases with quarkonium momentum in the laboratory frame owing to the Lorentz boost of the electric field in the comoving frame. We argue that J/ψ’s produced in heavy-ion collisions at the Large Hadron Collider with P⊥ > 9 GeV would dissociate even in vacuum. In plasma, the J/ψ dissociation in a magnetic field is much stronger because of the decrease of its binding energy with temperature.We discuss phenomenological implications of our results.
Disciplines Astrophysics and Astronomy | Physics
Comments This article is from Physical Review C 84 (2011): 044908, doi: 10.1103/PhysRevC.84.044908. Posted with permission.
This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/physastro_pubs/146 PHYSICAL REVIEW C 84, 044908 (2011)
Quarkonium dissociation in quark-gluon plasma via ionization in a magnetic field
Kevin Marasinghe and Kirill Tuchin Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Received 11 April 2011; revised manuscript received 2 September 2011; published 20 October 2011)
We study the impact of a magnetic field, generated in collisions of relativistic heavy ions, on the decay probability of a quarkonium produced in the central rapidity region. The quark and antiquark components are subject to mutually orthogonal electric and magnetic fields in the quarkonium comoving frame. In the presence of an electric field, the quarkonium has a finite dissociation probability. We use the WKB approximation to derive the dissociation probability. We find that the quarkonium dissociation energy, i.e., the binding energy at which the dissociation probability is of order unity, increases with the magnetic field strength. It also increases with quarkonium momentum in the laboratory frame owing to the Lorentz boost of the electric field in the comoving frame. We argue that J/ψ’s produced in heavy-ion collisions at the Large Hadron Collider with P⊥ > 9 GeV would dissociate even in vacuum. In plasma, the J/ψ dissociation in a magnetic field is much stronger because of the decrease of its binding energy with temperature. We discuss phenomenological implications of our results.
DOI: 10.1103/PhysRevC.84.044908 PACS number(s): 25.75.Cj
I. INTRODUCTION static if the distance over which it significantly varies is much larger than the quarkonium radius. We will refer to this approx- In their seminal 1986 paper [1], Matsui and Satz suggested imation as “quasistatic.” For a quarkonium with binding energy that quarkonium production in high-energy nucleus-nucleus ε and radius α /ε , the quasistatic approximation applies collisions can be used as an indicator for production of quark- b s b when ε τ/α 1. For τ = 2fmwegetε τ/α ≈ 23, which is gluon plasma (QGP). They argued that a quarkonium of radius b s b s comfortably large enough to justify the quasistatic approxima- r will dissociate in a QGP when the Debye screening radius tion, where we assumed that ε is given by its vacuum value. r (T ) becomes smaller then r. The observed effect would be b D As T increases, ε drops, while τ increases. The temperature an “anomalous” suppression of quarkonium yield. In practice, b dependence of ε is model dependent, but it is certain that even- determination of the screening radius r (T ) turned out to be b D tually ε vanishes at some finite temperature T . Therefore, a remarkably difficult problem even in a static medium. The b 0 only in the close vicinity of T , i.e., at very small binding ener- existing approaches to solve this problem include lattice QCD 0 gies, is the quasistatic approximation not applicable. We thus calculations of quarkonium correlators [2–7], construction of rely on the quasistatic approximation throughout the paper. potential models of quarkonium spectral functions [8–14], and use of effective field theory [15–17]. It is remarkable that in spite of much progress there still exists substantial uncertainty in the value of the J/ψ dissociation temperature and in the B0R/τ circulating around the direction of B0;hereR is the nuclear functional form of rD(T ); see, e.g., [18,19]. Another complica- radius and τ the relaxation time. This electric field generates circular tion arises due to “cold nuclear matter effects,” i.e., nuclear ef- currents that, by Lenz’s law, support the original magnetic field. fects that are independent of temperature and that would occur There are two types of generated current: (i) Foucault currents in the even if no plasma were formed. There is ongoing controversy QGP; (ii) a current of charged fermions produced via the Schwinger as to the nature of these effects, although there is agreement mechanism [35]. Note that lepton Schwinger pairs are certainly not that they lead to suppression of quarkonium yield [20–29]. in equilibrium with the QGP and therefore do not contribute to the It has been recently realized [30] that colliding heavy ions plasma electrical conductivity. In the first case, for a medium with electrical conductivity σ we estimate B ∼ jR ∼ σER ∼ σB R2/τ, produce a very strong magnetic field in the direction perpendic- 0 0 which implies ular to the reaction plane (defined as a plane containing the mo- menta of the ions and the impact parameter). This has a number τ ∼ σR2. (1) of interesting phenomenological consequences [30–33]. The Using the values of the electrical conductivities obtained in Ref. [36] strength of the magnetic field at the Relativistic Heavy Ion ∼ ∼ Collider (RHIC) is estimated as eB ≈ m2 , and at the Large and in Refs. [37,38]wegetτ 8fmandτ 0.4 fm, respectively. 0 π In the second case, recalling that the density of the Schwinger pairs Hadron Collider (LHC) as eB ≈ 15 m2 [30,34]. The relax- 0 π is n ∼ (eE)2τ,wehaveB ∼ envR ∼ e3B2R3/τ, which implies ation time of the magnetic field, neglecting the back reaction, is 0 0 2 3 of the order of the width of the two Lorentz-contracted nuclei τ ∼ e (eB0)R . (2) divided by c, which is ∼0.1 fm at the RHIC and ∼0.01 fm at We took into account that the Schwinger pairs are relativistic, v ∼ 1, the LHC. However, we argued recently [31] that the relaxation ∼ time of this field is actually much larger if the back reaction is since their longitudinal momentum is of order eEτ eB0R me for R = 5 fm. We estimate that at the RHIC τ ∼ 0.5 fm, while taken into account.1 In fact, the magnetic field can be treated as at the LHC τ ∼ 8 fm. A more accurate estimate of τ requires not only better knowledge of electrical conductivity in expanding 1 This can be seen as follows. Upon collision, the magnetic field B0 plasma, but also numerical solution of the magnetohydrodynamic rapidly decreases with time, inducing an electric field of strength E ∼ equations.
0556-2813/2011/84(4)/044908(8)044908-1 ©2011 American Physical Society KEVIN MARASINGHE AND KIRILL TUCHIN PHYSICAL REVIEW C 84, 044908 (2011)
A magnetic field has a threefold effect on a adequate for determining the quarkonium ionization energy quarkonium: due to the rapid variation of f with binding energy. Our result (1) Lorentz ionization. Suppose the quarkonium travels with for w is in agreement with previous calculations [41]. constant velocity in a magnetic field in the laboratory frame. In Sec. III we study the nonrelativistic approximation of Boosting to the quarkonium comoving frame, we find mutually the ionization probability w. We argue that it provides a orthogonal electric and magnetic fields given by Eqs. (3) and remarkably good estimate of relativistic formulas; see Fig. 2. (4). In the presence of an electric field the quark and antiquark This is an important observation as it allows us to include the have a finite probability to tunnel through the potential barrier, contribution of the quark spin interaction with the magnetic thereby causing quarkonium dissociation. In atomic physics field. A fully relativistic calculation that accounts for the spin such a process is referred to as Lorentz ionization. In the contribution is not yet available. This is discussed in Sec. III D. nonrelativistic approximation, the tunneling probability is of In Sec. III we also show that the relativistic formulas for w order unity when the electric field E in the comoving frame derived in Sec. II reduce to nonrelativistic formulas found in 1/2 3/2 the literature. satisfies eE m εb (for weakly bound states), where εb is the binding energy and m is the quark mass; see Eq. (26). In Sec. IV we calculate the dissociation energy of the J/ψ This effect causes a significant increase in the quarkonium in a magnetic field and claim that it strongly depends on the magnetic field and the J/ψ’s velocity V in the laboratory dissociation energy εd . The corresponding results for J/ψ are exhibited in Fig. 4. They suggest that a J/ψ in plasma frame. Our results are summarized in Fig. 4. In a strong dissociates at a much lower temperature then it would in the magnetic field such as the one expected to be produced at absence of a magnetic field. the LHC, a J/ψ moving with P⊥ > 9 GeV in the reaction (2) Zeeman effect. The energy of a quarkonium state plane is expected to dissociate because of the magnetic field depends on spin S, orbital angular momentum L, and total even in vacuum. At finite temperature, when the binding energy angular momentum J . In a magnetic field these states split; decreases, dissociation becomes prominent at lower transverse eB0 momenta. The magnetic field has no influence on the J/ψ the splitting energy in a weak field is M = gJz, where 2m dissociation when it moves perpendicularly to the reaction Jz =−J,−J + 1,...,J is the projection of the total angular momentum on the direction of the magnetic field, m is plane (i.e., parallel to the field) because the corresponding the quark mass, and g is the Lande´ factor depending on electric field vanishes in the comoving frame. The nontrivial J , L, and S in a well-known way; see, e.g., [39]. For azimuthal angle dependence of w may be an important source example, a J/ψ with S = 1, L = 0, and J = 1(g ≈ 2) splits of azimuthal anisotropy of J/ψ production in heavy-ion collisions. into three states with Jz =±1, 0 and with mass difference = = 2 In Sec. V we discuss the phenomenological significance of M 0.15 GeV, where we used eB0 15mπ (the value of magnetic field expected at the LHC [30,34]). Thus, the Zeeman our results. effect leads to the emergence of new quarkonium states in Before continuing with the main part of the paper, we plasma. list here some of our notational definitions in order to avoid (3) Distortion of the quarkonium potential in a magnetic confusion: V and P are the velocity and momentum of the field. This effect arises in higher-order perturbation theory and quarkonium in the laboratory frame; M is its mass; p is becomes important at field strengths of order B ∼ 3πm2/e3 the momentum of the quark or antiquark in the comoving [40]. This is 3π/α times stronger than the critical Schwinger frame; m is its mass; B0 is the magnetic field in the laboratory field. Therefore, this effect can be neglected at the present frame; E and B are the electric and magnetic fields in the collider energies. comoving frame; γL is the quarkonium Lorentz factor; and The purpose of this paper is the analysis of the Lorentz γ is a parameter defined in Eq. (21). We use Gauss units ionization of quarkonia. throughout the paper; note that the expressions eB, eE, and Our paper is organized as follows. In Sec. II we give a fully eB0 are the same in Gauss and Lorentz-Heaviside units. relativistic derivation of the Lorentz ionization probability w using the WKB approximation. We treat the force binding II. QUARKONIUM IONIZATION RATE q and q¯ into a quarkonium as a short-range one, i.e., 1/2 A. Comoving frame (Mεb) R 1, where εb and M are the binding energy and mass of the quarkonium, respectively, and R is the radius Consider a quarkonium traveling with velocity V in a 1/2 of the nuclear force given by R ≈ (αs /σ ) , where σ = 1 constant magnetic field B0.LetB and E be magnetic and GeV/fm is the string tension. For example, the binding energy electric fields in the comoving frame, and let the subscripts 2 of c and c¯ in a J/ψ in vacuum is εb = 0.64 GeV M/R = and ⊥ denote field components parallel and perpendicular to Mσ/αs ≈ 3 GeV. This approximation is even better at finite V, respectively. Then, ε J/ψ temperature on account of the decrease in b. Regarding = = × as being bound by a short-range force enables us to calculate E 0, E⊥ γLV B0, (3a) the dissociation probability w with exponential accuracy, B0 · V (V × B0) × V − B = , B⊥ = γ , (3b) w ≈ e f , independently of the precise form of the quarkonium V L V 2 wave function. This is especially important since solutions 2 −1/2 of the relativistic two-body problem for quarkonium are not where γL = (1 − V ) . Clearly, in the comoving frame B · readily available. We will argue that exponential accuracy is E = 0. If the quarkonium travels at angle φ with respect to the
044908-2 QUARKONIUM DISSOCIATION IN QUARK-GLUON PLASMA ... PHYSICAL REVIEW C 84, 044908 (2011) magnetic field in the laboratory frame, then with constant momentum along the direction of the magnetic field. = 2 − 2 + 2 = 2 2 B B0 cos φ 1 γL γL,E B0γLV sin φ. (4) The effective potential U(y) = ε0(y) − m + p corre- sponding to Eq. (6) is plotted in Fig. 1. We can see that the We choose the z and y axes of the comoving frame such that tunneling probability is finite only if E>0. It is largest when B = Bzˆ and E = E ˆy. A convenient gauge choice is A = B = 0. It has already been noted in Refs. [41–43] that the −Byxˆ and ϕ =−Ey. For future reference we also define a effect of the magnetic field is to stabilize the bound state. In useful dimensionless parameter ρ [42] spite of the linearly rising potential (at B>E), the tunneling E γLV sin φ probability is finite as the result of rearrangement of the QED ρ = = . (5) vacuum in the electric field. B cos2 φ 1 − γ 2 + γ 2 L L The ionization probability of quarkonium equals its tunnel- 2 − 2 = 2 ing probability through the potential barrier. The latter is given Note that (i) 0 ρ 1 because B E B0 0 and (ii) when the quarkonium moves perpendicularly to the by the transmission coefficient √ y magnetic field B , ρ = V . −2 1 −p2dy − 0 w = e 0 y ≡ e f . (7) In the nonrelativistic approximation one can also calculate B. WKB method the preexponential factor, which appears due to the deviation It is natural to study quarkonium ionization in the comoving of the quark wave function from the quasiclassical approx- frame. As explained in the Introduction, ionization is quantum imation. Such a calculation requires matching quark wave tunneling through the potential barrier caused by the electric functions inside and outside the barrier [44]. To determine field E. In this section we employ the quasiclassical, or WKB, the preexponential factor in the relativistic case, one needs to approximation to calculate the quarkonium decay probability solve the relativistic two-body problem, which is analytically w. For the gauge choice specified in Sec. II A the quark energy challenging [45]. Fortunately, as we argue later, one does not need to know the prefactor to make reliable estimates of the ε0 (ε0
− ∓ − 2 − 2 − 2 − 2 + 2 + 2 ε0E px B (ε0E px B) (B E ) ε0 m px y = . (9) 1,2 e(B2 − E2)
Define dimensionless variables 0 = ε0/m and q = px /m. equation [41] Integration in Eq. (7)gives ∂f (q ) m = 0. (11) ∂qm 2 2 − + 1 + q ( 0E − qB) f = 0 Using (10) and the parameter ρ defined in Eq. (5), we find 2 2 − 2 m e(B E ) ⎧ ⎫ 2 2 2 2 2 ⎨ − + − 2 − 2 + + 2 ⎬ ( 0E − qB) − (B − E ) − + 1 + q ρ( − ρq ) 0ρ qm 1 ρ 0 1 qm − 0 0 m ln e(B2 − E2)3/2 1 − ρ2 ⎩ ( − ρq )2 − 1 + ρ2 ⎭ ⎧ ⎫ 0 m ⎨ 2 2 2 2 ⎬ 0E − qB + (B − E ) − + 1 + q × 0 − 2 + 1 + q2 ln ⎩ ⎭ . = 0 m − 2 − 2 − 2 2 + + 2 . (12) ( 0E qB) (B E ) 0 1 q 1 − ρ2 (10) This is an implicit equation for the extremal momentum qm = qm( 0,ρ). Substituting qm into Eq. (10), one obtains f For different q’s, w = e gives the corresponding ionization f = f ( 0,ρ), which by means of Eq. (7) yields the ioniza- probabilities. The largest probability corresponds to the small- tion probability. The quasiclassical approximation that we est f , which occurs at momentum qm determined by the employed in this section is valid inasmuch as f (qm) 1.
044908-3 KEVIN MARASINGHE AND KIRILL TUCHIN PHYSICAL REVIEW C 84, 044908 (2011)
U m 20 0.2 E 0 15 0.1 0.2 2 m E B 2 10
y eE
0.2 0.4 0.6 0.8 1.0 m ∋ 1 f 0 0.5 0.1 5
0.9 0.2 0 B 0 E B 0.0 0.5 1.0 1.5 2.0 ∋ 0.3 b
= 2 FIG. 1. (Color online) Effective potential U(y) FIG. 2. (Color online) Dimensionless function fmeE/m versus 2 + + 2 + 2 − − 2 + 2 + 2 = b for different values of ρ. The solid line is the full relativistic m (px eBy) py eEy m px py for py 0, calculation; the dashed line is the nonrelativistic approximation. The p = m/6, B = m [except for the lowest (blue) line where B = 0]. x J/ψ = . The width of the potential barrier decreases with increasing E and binding energy in vacuum corresponds to b 0 68. increases with increasing B.1− corresponds to the binding 0 III. NONRELATIVISTIC APPROXIMATION energy in units of m. A very useful approximation of the relativistic formulas In order to compare with the results obtained in Ref. [41] derived in the previous section is the nonrelativistic limit using the imaginary time method, we can rewrite Eq. (12)in because (i) it provides a very good numerical estimate (see terms of an auxiliary parameter τ0 as Fig. 2), (ii) it allows us to eliminate the parametric dependence in Eqs. (10) and (12) and write f (q ) explicitly in terms of ρ 1 − ρ2 − 2 + 1 + q2 m 0 m and 0, and (iii) spin effects can be taken into account. τ0 = , (13a) ρ( 0 − ρqm) − tanh τ0 = 0 ρqm ρ . (13b) A. Arbitrary binding τ0 0ρ − qm The motion of a particle can be treated nonrelativistically if Taking advantage of these equations, Eq. (10) can be cast its momentum is much less than its mass. In such a case ε0 ≈ m into a more compact form: or εb = m − ε0 m. Additionally, the motion of a charged 2 particle in an electromagnetic field is nonrelativistic if E B. m τ0ρ fm = [1 − 0( 0 − qmρ)], (14) Indeed, the average velocity of a nonrelativistic particle is of eE 1 − ρ2 order v ∼ E/B = ρ. Thus, the nonrelativistic limit is obtained = where we denoted fm = f (qm). This agrees with the results of by taking the limits b εb/m 1 and ρ 1. In these limits Ref. [41]. Numerical solution of Eqs. (13a), (13b), and (14)is the extremum conditions (13a),(13b) reduce to shown in Fig. 2. 2 + q2 τ = b m , (18a) 0 ρ tanh τ ρ C. Special case: Crossed fields 0 = . − (18b) An important limiting case is that of crossed fields E = B. τ0 ρ qm Since also E ⊥ B (see Sec. II A), both field invariants vanish. Of two solutions to Eq. (18a) we pick the following one: Nevertheless, the quarkonium ionization probability is finite [41]. This limit is obtained by taking ρ → 1 in the equations =− 2 2 − from the previous section. Employing (13a) and (13b), we get qm τ0 ρ 2 b. (19) the following condition for the extremum: The sign of qm is fixed using (18b) by noticing that 2 − + 2 − = tanh τ /τ < 1. Elimination of q gives 0 1 2qm 3 0qm 0, (15) 0 0 m 2 − − 2 = 2 with the solution τ0 (τ0 coth τ0 1) γ , (20) = 1 − 2 + where qm 4 3 0 0 8 . (16) √ 2 Substitution into (14) produces γ = b . (21) ρ 3/2 2 m2 − 2 + 1 + q2 f = 0 m . γ is analogous to the adiabaticity parameter of Keldysh [46]. m − (17) 3 eE 0 qm Taking the nonrelativistic limit of Eq. (14) and using (19)we
044908-4 QUARKONIUM DISSOCIATION IN QUARK-GLUON PLASMA ... PHYSICAL REVIEW C 84, 044908 (2011) obtain This formula is valid when 2 3/2 2 2m (2 b) ρ m, eEρ ε 1/R. (31) f = g(γ ), (22) b m 3eE where g(γ ) is the Keldysh function [46] D. Contribution of quark spin 3τ 1 τ 2 1/2 g(γ ) = 0 1 − 0 − 1 . (23) So far we have neglected the contribution of quark spin. In 2γ γ γ 2 order to take into account the effect of spin interaction with
2 the external field, we can use the squared Dirac equation for a In Fig. 2 we show the dimensionless ratio fmeE/m as a bispinor ψ: function of the binding energy b (in units of m)forseveral values of ρ. The vacuum binding energy of J/ψ corresponds to [(ε − eϕ)2 − ( p − e A)2 − m2 + e · B − ieα · E]ψ = 0, b = 0.68. We observe an excellent agreement between the full (32) relativistic calculation and the nonrelativistic approximation. = = where At ρ 0.9 and b 0.68, the difference between the two lines is 10% and it can be further improved by considering σ 0 σ 0 = , α = . (33) higher-order corrections to fm [43]. 0 σ 0 −σ
The operators z and αy do not commute. Therefore, in order B. Weak binding to apply the WKB method for calculation of the ionization probability one actually needs to square (32), which leads to √ Of special interest is the limit of weak binding, γ 1, i.e., a differential equation of the fourth order in derivatives. The 2 b ρ. Expansion of Eq. (20)atsmallγ and τ0 yields problem becomes more tractable in the nonrelativistic case and for crossed fields. Spin effects in crossed fields were discussed 1 2 τ0 = γ 1 + γ (24) in Ref. [47]. 18 With quark spin taken into account, the nonrelativistic and substitution into Eq. (23) and subsequently into Eq. (22) version of Eq. (6) becomes yields 1 μ + 2 + 2 − − · =− 2 (px eBy) py eEy s B εb, (34) 2 m 3/2 2m s fm = (2 b) . (25) 3 eE and hence Hence, the quarkonium dissociation probability reads [44] μ 2 = − + · + − + 2 py 2m εb s B eEy (px eBy) , (35) 2 (2ε m)3/2 s w = exp − b . (26) 3 meE where μ is the quark magnetic moment and s is the projection of the spin in the direction of the magnetic field. For a Since the quasiclassical approximation employed in this paper point quark, μ = μ = eh¯ . The effect of quark spin on the B 2mc is valid if f (qm) 1, it follows that the binding energy must quarkonium dissociation probability can be taken into account satisfy → = − μ · by replacing εb εb εb s s B in formulas for fm.With 2/3 this replacement, all results of this section apply to a particle (eE) 2 εb ρ m. (27) with spin. Note that the effective binding energy ε decreases m1/3 b if the spin is parallel to the magnetic field and increases if it is Note also that we work in the approximation√ of the short-range antiparallel. In particular, in the case of weak binding, binding potential, meaning that 2 b 1/(mR) (see Sec. I). 2 (2ε m + 2seB)3/2 w = exp − b . (36) 3 meE s=±1/2 C. Strong binding Since the nonrelativistic limit provides a good approximation In the limit γ 1, (20) and (23) imply that of the full relativistic formulas, we will implement the quark γ 2 3γ spin dependence using the nonrelativistic prescription. τ = ,g(γ ) = . (28) 0 2 8 Substituting (28)into(22), we derive IV. DISSOCIATION ENERGY OF J/ψ ε2 B f = b . (29) Now, we would like to use the results of the preceding m eE E sections to analyze the dissociation probability w of the Thus, the quarkonium dissociation probability in the case of quarkonium as a function of its binding energy εb. We suppose strong binding is that the quarkonium moves with velocity V at angle φ with respect to the external magnetic field B0. Different binding ε2 B energies may correspond to either different quarkonium states w = exp − b . (30) eE E or the same quarkonium state at different temperatures.
044908-5 KEVIN MARASINGHE AND KIRILL TUCHIN PHYSICAL REVIEW C 84, 044908 (2011)
5 0.7 2 eB0 15m V 0.2 V 0.5 V 0.7 0.6 2 4 2 0.5 V 0.9 2 3 eB0 15mπ 0.4 ψ b J 2 m GeV f
d 0.3 2 eB 7m2 V 0.95 ψ 0 π 0.2 b J 4
1 0.1 2 eB0 mπ 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V ∋ b FIG. 4. (Color online) Dissociation energy of the J/ψ as a FIG. 3. (Color online) fm versus b for different values of the function of its velocity V in different magnetic fields B . We assumed = {− } 0 J/ψ velocity V . The dissociation probability is w exp fm .The that the J/ψ moves in the reaction plane. The vacuum dissociation = 2 magnetic field in the laboratory frame is taken to be eB0 15mπ . J/ψ energy is zero. The solid horizontal line indicates the vacuum binding moves perpendicularly to the field (i.e., in the reaction plane). Solid energy; the dashed horizontal lines correspond to the binding energy lines correspond to the full relativistic calculation; dashed lines to the at certain temperatures. nonrelativistic approximation. The J/ψ binding energy in vacuum corresponds to b = 0.68.
The dissociation probability is given by Eq. (7)asw = implies that most of the J/ψ’s moving in the reaction plane exp{−fm( b)}.Atfm( b) 1, the quarkonium is stable with with P⊥ > 9 GeV will dissociate in a magnetic field even in respect to Lorentz ionization because w 1. The dissociation vacuum. becomes probable only at values of binding energy b = d In a thermal medium, εb exhibits strong dependence on such that fm( d ) ∼ 1. In view of the steep dependence of temperature T .AtT = 0, εb equals its vacuum value. As fm( b)on b, the probability w( b) is in fact close to a step T increases εb drops until it vanishes at some T0 near Tc. function: it becomes 1 when b < d and is zero otherwise. The precise value of T0 as well as the functional form of Consequently, we introduce the quarkonium dissociation εb(T ) are model dependent; see, e.g., [18,19]. To illustrate energy in a magnetic field, εd = m d , as the solution to the the temperature effect, we indicated in Fig. 4 one-half and equation one-quarter of the J/ψ binding energy, εb/2 and εb/4. Suppose that the plasma temperature is such that the binding energy is = 2 eB one-half of that in vacuum. Then, at eB0 15mπ ,allJ/ψ’s f d − = 1, (37) 2m2 with V>0.71 (P⊥ > 3.1 GeV) will dissociate, while at eB0 = 2 7mπ this occurs at V>0.92 (P⊥ > 7.3GeV). where we took into account the contribution of quark spin So far in this section we have discussed the case of a (see Sec. III D). The negative sign in front of the spin term quarkonium moving in the reaction plane, i.e., at φ = π/2. in Eq. (37) is chosen to give the highest dissociation energy. Since the electric field in the comoving frame is proportional (Which of two particles c or c¯ has the higher probability to to sin φ [see (4)], the dissociation probability decreases at tunnel depends on the direction of the magnetic field.) The oblique angles. J/ψ’s moving parallel to the magnetic field solution of Eq. (37) depends on the electric and magnetic B0, i.e., at φ = 0, are not affected by electromagnetic field at fields in the comoving frame, which in turn depend on the all. This nontrivial azimuthal angle dependence translates into magnetic field in the laboratory frame, B0, and the quarkonium azimuthal anisotropy of the produced J/ψ’s, which we plan kinematics through the Lorentz transformation formulas (4). to address in a separate presentation. In Fig. 3 we show the function fm( b) for different velocities The size of the spin contribution is determined by the 2 V of J/ψ, assuming it moves perpendicularly to the magnetic dimensionless ratio ξ = eB/2 bm [see (36)]. Its magnitude field. In vacuum, M = 3.1 GeV and m = 1.87 GeV, so that for a J/ψ moving with a nonrelativistic velocity perpendic- = − = ≈ 2 εb 2m M 0.64 GeV. The corresponding dimensionless ular to the magnetic field B0 is ξ (eB0/240mπ )(0.68/ b). parameter b varies in the interval 0 b 0.68. Therefore, even in the strongest field expected at the LHC, = 2 = The numerical solution to Eq. (37) is plotted in Fig. 4. eB0 15mπ , the effect of spin is quite small at T 0. We see that the dissociation energy of the J/ψ increases ξ increases as the temperature of the plasma and/or the with its velocity and is larger in a strong magnetic field. At velocity of the J/ψ increases, owing to decrease of the = 2 = eB0 15mπ , which according to estimates may be reached binding energy from its vacuum value b 0.68 down to in heavy-ion collisions at the LHC [30,34], the dissociation zero. When ξ ∼ 1 nonrelativistic treatment of spin breaks energy equals the vacuum binding energy at V = 0.94. This down.
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V. DISCUSSION AND CONCLUSIONS dissociate with probability of order unity even in vacuum, where we rely on the magnetic field estimates of Ref. [30]. In this paper we chose the quarkonium proper frame This effect is more pronounced at finite temperature owing to to calculate the dissociation probability. In this frame the the decrease in binding energy and, as seen in Fig. 4, it occurs potential energy of, say, an antiquark (with e<0) is the in weaker fields. sum of its potential energy in the binding potential and its Although magnetic fields in pp and pA collisions are much energy in the electric field −eEy, where y is the electric field weaker than in AA collisions, they are still strong enough to direction. Since |e|Ey becomes large and negative at large cause J/ψ dissociation at sufficiently high momenta P⊥.A and negative y (far away from the bound state) and because truly spectacular feature of such a process would be J/ψ decay the quarkonium potential has finite radius, this region opens into two heavier D mesons. The validity of the quasistatic up for the motion of the antiquark. Thus there is a quantum approximation can perhaps be advocated for a prehadronic mechanical probability to tunnel through the potential barrier cc¯ state owing to its short production time. This intriguing formed on one side by the vanishing quarkonium potential and possibility will be addressed elsewhere. on the other by the increasing absolute value of the antiquark The effect of J/ψ dissociation in a magnetic field vanishes energy in an electric field. Of course the total energy of the in the direction parallel to the magnetic field, i.e., perpendicular antiquark (not counting its mass) is negative after tunneling. to the reaction plane. Therefore, J/ψ dissociation gives However, its kinetic energy grows proportionally to eEy as it a negative contribution to the total azimuthal asymmetry goes away. By picking up a light quark out of vacuum it can coefficient v . The size of this contribution is a subject of hadronize into a D meson. 2 our ongoing investigation. If we now go to the reference frame where E = 0 and Finally, we would like to mention the possibility of J/ψ there is only a magnetic field B (we can always do so since dissociation directly by electric fields induced in a plasma E
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