Gluodissociation and Screening of Upsilon States in PbPb Collisions at sqrt (s_NN) = 2.76 TeV

We suggest that gluon-induced dissociation and screening of the Y(nS) states explain the suppression of the Y(2S+3S) states relative to the Y(1S) ground state that has been observed by CMS in PbPb collisions at sqrt(s_NN)= 2.76 TeV at the CERN LHC. The minimum-bias gluodissociation cross sections of the 1S-3S states are calculated using a screened Cornell potential and a thermal gluon distribution. The 3S state dissolves due to screening before sizeable gluodissociation occurs, but for the 2S and 1S states there is an interplay between screening, gluodissociation, and feed-down from the chi_b(2P) and chi_b(1P) states. Based on a schematic approach, we find that the calculated suppression of the Y(2S) and Y(3S) states relative to Y(1S) is consistent with the CMS result, but allows for additional suppression mechanisms. The Y(1S) suppression through gluodissociation is, however, in good agreement with the CMS data.

The suppression of quarkonium states is one of the most promising probes for the properties of the quark-gluon plasma (QGP) that is generated in heavy-ion collisions at high relativistic energies. In the QGP the confining potential of heavy quarkonium states is screened due to the interaction of the heavy quark and the antiquark with medium partons and hence, charmonium and bottomium states successively melt [1] at sufficiently high temperatures T diss beyond the critical value T c ≃ 170 MeV.
Charmonium suppression has been studied since 1986 in great detail both theoretically, and experimentally at energies reached at the CERN Super Proton Synchrotron SPS, BNL Relativistic Heavy-Ion Collider (RHIC) [2,3,4,5], and CERN LHC [6,7]. The precise origin is still under investigation, in particular at LHC energies where regeneration due to statistical recombination of c andc in the quark-gluon plasma could be relevant, counteracting the J/Ψ dissociation in central collisions and contributing to the measured nearly flat suppression factor as function of centrality for p T > 0 [6].
Bottomium suppression is expected to be a cleaner probe. The Υ(1S) ground state with invariant mass 9.46 GeV is strongly bound, the threshold to BB decay is at 1.098 GeV. Its lifetime of 1. Υ suppression in heavy-ion collisions has recently been observed for the first time both by the STAR experiment at RHIC [9], and by the Compact Muon Solenoid (CMS) experiment at LHC [10]. The latter includes an observation of the enhanced suppression of the 2S + 3S relative to the 1S ground state, whereas the 1S suppression itself is considered in [11] by CMS.
This result is most likely not due to differences in the direct bottomium production mechanism in pp vs. PbPb collisions since nuclear modification of the parton distribution functions (shadowing) should affect all three states in a similar fashion [11].
In this Letter we investigate the suppression of Υ(1S), (2S), (3S) states at LHC energies due to screening and gluon-induced dissociation, including feed-down from the χ b (1P ) and χ b (2P ) states. Whereas gluodissociation below T c is not possible due to confinement, it does occur above T c where the color-octet state of a free quark and antiquark can propagate in the medium. The process is relevant below the dissociation temperature T diss that is due to Debye screening, and its significance increases substantially with the rising gluon density at LHC energies.
In the midrapidity range |y| < 2.4 where the CMS measurement [10] has been performed, the temperature and hence, the thermal gluon density is high, and causes a rapid dissociation in particular of the 2S and 3S states, but also of the 1S ground state. At larger rapidities up to the beam value of y beam = 7.99 and correspondingly small scattering angles where the valencequark density is high [12], nonthermal processes would be more important than in the midrapidity region that we are investigating here. Thermal gluons will also dissociate the χ b (1P ) and χ b (2P ) states which partially feed the Υ(1S) ground state in elementary collisions [13].
Due to the small velocity v ≪ c of the quarks in the bound state, the proper equation of motion for single-particle quarkonium states is the Schrödinger equation, with the color-singlet QQ quarkonium potential V QQ .
Reasonable parametrizations of the potential exist that have been tested in detailed calculations of the excited states.
In particular, the Cornell potential [14] has string and Coulomb part [15] is the string tension, and α ef f = 0.471 an effective Coulomb-like coupling constant that accounts for the short-range gluon exchange, respectively.
Although the string contribution to the potential vanishes for light quarkonia in the QGP above T c , it has to be considered at T > T c for heavy quarkonia that remain initially confined and are therefore not in thermal equilibrium with the plasma. Hence we maintain the string contribution in an approximate solution of the gluodissociation problem.
The string tension of quarkonium decreases with increasing temperature T in the quark-gluon medium. The screened potential can be written as [15,16,17] with r D (T ) the Debye radius, The number of colors is N c = 3, the number of flavors in the QGP taken as N f = 3, and the strong-coupling constant at the Υ(1S) mass α s ≃ 0.2. Because of the inverse proportionality of the minimum screening radius that permits a bound state to the heavy-quark mass [1], it is much more difficult to dissolve the Υ(1S) in the quark-gluon plasma through screening than the J/ψ(1S).   imaginary-valued contribution to the potential [18,17,19,20] due to Landau damping of the exchanged gluon as performed in [21] for the Υ(1S) and the The leading-order dissociation cross section of the QQ states through dipole interactions with hard gluons (E1 absorption of a single gluon) had been derived by Bhanot and Peskin (BP) [22]. In an operator product expansion, they calculate the gluodissociation cross section σ diss with pure Coulomb-like momentum eigenstates. This expansion is valid for sufficiently small bound-state radii. For an initial gluon of energy E (momentum p) the cross section is obtained from the Born amplitude A B using the optical Modifying the BP approach to approximately account for the confining string contribution, we use the singlet wave functions computed with Eq.(1).
Inserting a complete set of eigenstates of the adjoint (octet) Hamiltonian [15] the bottom quark mass) to calculate the dissociation cross sections of the Υ(1S, 2S, 3S) and the χ b (1P, 2P ) states [23], we obtain with the wave function overlap integral for the singlet radial wave functions g s n0 (r) of the b quark, and the adjoint octet wave functions g a k1 (r). The binding energy of the nS state is ǫ n , and the δ function accounts for energy conservation, For vanishing string tension σ → 0 and the corresponding values of the binding energy ǫ n , a pure Coulomb 1S wave function, and a simplification in the octet wave function, this expression reduces to the result in [22]. We can, however, evaluate it with the full octet wave function to obtain the Υ(1S) dissociation cross section σ 1S in terms of the BP expression σ BP with z = 1/(4q), and q = E/ǫ 1 − 1. The rhs approaches 1 for z → 0, recovering the BP formula. It approaches 0 for z → ∞, and agrees with the result obtained independently by Brambilla et al. in an effective field theory approach in the corresponding limit [24,25]. Their work also considers the thermal width of heavy quarkonia due to Landau damping, in addition to the break up of a color-singlet bound state into a quark-antiquark pair that is investigated here. In contrast to other assumptions, these authors find that breakup is the leading term as compared to Landau damping [25].
We obtain new results for the 2S and 3S states from eqs. states.
One should be prepared to expect modifications in the cross section values of the five states from next-to-leading order (NLO) contributions [26], where a gluon appears in the final state in addition to the b andb quarks, and hence, the phase space is larger than in leading order (LO). However, in [27] it was shown that the quasi-free process that corresponds to NLO is less important than LO for temperatures T >270 MeV.
Whereas the heavy quarkonium is not in thermal equilibrium with the QGP, it is reasonable to assume that the medium itself is thermalized due to with E(p) = (p 2 +m 2 g ) 1/2 , the gluon degeneracy g d =16, and the gluon density as the integral over the distribution function, n g = g d T 3 ζ(3)/π 2 for m g = 0.
Values for the thermal gluon density at temperatures 170, 200, 300 and 400 MeV and m g = 0 are n g = 1.25, 2.03, 6.85 and 16.23 fm −3 , respectively. The distribution function is shown in Fig. 2 (rhs scale).
The on-shell gluon energy (p 2 + m 2 g ) 1/2 is usually calculated assuming vanishing gluon mass m g = 0, but we shall also investigate the effect of a finite effective gluon mass, as has been suggested in quasi-particle models [28] based on lattice QCD results [29,30], with m g ≃ 0.5 − 1 GeV. It is argued in [31] that the effective mass of the gluons may initially be of the order of the gluon saturation scale, m g ≃ Q s , which is about 1 GeV at x Bjorken = 0.01.  Table 1.
The dissociation widths Γ(nS) of the nS states are then obtained by multiplying the average cross sections with the gluon density, Γ(nS) = n g · < σ nS diss >, and similarly for the χ b states. To compare with minimum-bias data, it is essential to consider the impact-parameter dependence. We assume a monotonic relation of the initial temperature T i (b) = T 0 (1 − b 2 /b 2 cr ) on impact parameter up to a critical value b cr where T i (b cr ) = T cr = 170 MeV, with T 0 500 MeV, and no gluodissociation beyond b cr , to obtain the temperature-dependent suppression factorR(nS, b) (and analogously for the χ b (nP ) states) prior to feed-down at impact parameter b aŝ Here With decay rates for the nS states from the particle data group -including the effect of different branching ratios into the µ + µ − detection channel for n =1, 2, 3 -, we calculate a decay cascade that matches the final populations measured by CMS for pp at 2.76 TeV [10], and thus provides initial populations which we use for the PbPb in-medium calculation at the same energy. Following the consideration of screening and gluodissociation of the five states, we calculate the radiative feed-down cascade in the medium for those states which have survived the strong-interaction processes at a given impact parameter b, to obtain the final yields in the presence of the QGP.
Our results for the suppression of the Υ(1S) state in PbPb relative to pp at 2.76 TeV are shown in Table 2 for several initial QGP temperatures T 0 ,  Table 2 with m g = 0 and 1 GeV, respectively.
As an example, we obtain for T 0 ≃ 800 MeV and finite effective gluon  Although screening of the strongly bound 1S ground state is negligible, we find that its gluodissociation is sizeable due to the strong overlap of the 1S gluodissociation cross section with the thermal gluon distribution. Its observed suppression factor R AA (1S) ≃ 0.62 in minimum-bias PbPb collisions [7] is mainly due to both direct gluodissociation of the 1S state, and to the melting and gluodissociation of the χ b (1P ) and χ b (2P ) states which partially feed the 1S state in pp, pp and e + e − collisions.
For a detailed comparison, one needs data with better statistics that is expected to become available from the 2011 PbPb run at the LHC. If it turned out to be possible to measure the populations of the 2S and 3S states very precisely, one could use this as a fairly accurate thermometer for the initial temperature T 0 of the quark-gluon plasma. On the other hand, substantial deviations from the experimental values might indicate that further mechanisms contribute to the suppression. It may, however, also turn out that the gluon distribution is not fully thermalized, in particular, in the longitudinal direction.