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BNL-112229-2016-JA

Charm Degrees of Freedom in the Plasma

S. Mukherjee, P. Petreczky and S. Sharma

Brookhaven National Laboratory, Upton, NY 11973, USA

January 2016

Physics Department

U.S. Department of Energy USDOE Office of Science (SC), (NP) (SC-26)

Notice: This manuscript has been co-authored by employees of Brookhaven Science Associates, LLC under Contract No. DE- SC0012704 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

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This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Charm Degrees of Freedom in the Quark Gluon Plasma

Swagato Mukherjee, Peter Petreczky and Sayantan Sharma Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

Lattice QCD studies on fluctuations and correlations of charm have established that deconfinement of charm degrees of freedom sets in around the chiral crossover temperature, Tc, i.e. charm degrees of freedom carrying fractional baryonic start to appear. By reexamining those same lattice QCD data we show that, in addition to the contributions from quark-like exci- tations, the partial pressure of charm degrees of freedom may still contain significant contributions from open-charm and -like excitations associated with integral baryonic charges for temperatures up to 1.2 Tc. - become the dominant degrees of freedom for temperatures T > 1.2 Tc.

PACS numbers: 11.15.Ha, 12.38.Gc

Nuclear modification factor and elliptic flow of open- data with resummed perturbation theory results, which charm in heavy-ion collision experiments are im- are available only for zero quark masses, we normalize the portant observables that provide us with detailed knowl- off-diagonal flavor susceptibilities with the second order c 2 2 edge of the strongly coupled quark gluon plasma (QGP) charm quark susceptibility χ2 = (∂ p/∂µˆc ) calculated at [1]. Most of the theoretical models that try to describe µX = µc = 0. Such a normalization largely cancels the these quantities rely on the energy loss of heavy explicit charm quark mass dependence of the off-diagonal via Langevin dynamics [2–4]. However, the importance of susceptibilities and enables us to probe whether the u- possible heavy-light (strange) bound states inside QGP c flavor correlations can be described by the weak cou- uc uc have been pointed out in Refs. [5–8]. In particular, pres- pling calculations. In the weak coupling limit χ11, χ13 uc ence of such heavy-light bound states above the QCD and χ31 are expected to have leading order contributions 3 transition temperature seems to be necessary for the si- at O(αs) [21], where αs is the QCD strong coupling multaneous description of elliptic flow and nuclear mod- constant. This contribution is, strictly speaking, non- ification factor of Ds [5]. Presence of various perturbative but can be calculated on the lattice using hadronic bound states [9] as well as colored [10, 11] ones dimensionally reduced effective theory for high tempera- in strongly coupled QGP created in heavy-ion collisions ture QCD, the so-called electrostatic QCD (EQCD) [22]. have also been speculated in other other contexts. Similarly, in the weak coupling picture, the leading con- uc By utilizing various novel combinations of up to fourth tribution to χ22 arises from the so-called term 3/2 order cumulants of fluctuations of charm quantum num- and starts at O(αs ) [23]. Thus, it is generically ex- uc uc uc uc ber (C) and its correlations with (B), pected χ22  χ13 ∼ χ31 ∼ χ11 in the weak coupling and (S) lattice QCD stud- limit. As shown in Fig. 1 such an obvious hierarchy in ies [12] have established that charm degrees of freedom magnitude of the off-diagonal susceptibilities is clearly associated with fractional baryonic and electric charge absent in the lattice data for T < 200 MeV. However, start appearing at the chiral crossover temperature, Tc = for T & 200 MeV these lattice results are largely consis- 154±9 MeV [13–15]. Below Tc the charm degrees of free- tent with the weak coupling calculations, indicating that dom are well described by an uncorrelated gas of charm the weakly coupled quasi-quarks can be considered as the hadrons having vacuum masses [12], i.e. by the dominant charm degrees freedom only above this temper- resonance gas (HRG) model. Similar conclusions were ature. The fact that for Tc . T . 200 MeV the charm also obtained from lattice QCD studies involving the light degrees of freedom are far from weakly interacting quasi- up, down and strange quarks [16, 17]. quarks is also supported by lattice QCD studies of the On the other hand, lattice QCD calculations have also screening properties of the open-charm mesons. In this shown that weakly interacting quasi-quarks are good de- temperature range the screening masses of open-charm scriptions for the light quark degrees only for tempera- mesons also turn out to be quite different from the ex- tures T & 2 Tc [16, 18–20]. The situation for the heavier pectation based on an uncorrelated charm and a light charm quarks is also analogous. By re-expressing the lat- quark degrees of freedom [24]. tice QCD results for charm fluctuations and correlations From the preceding discussion it is clear that the up to fourth order from Ref. [12] in the charm (c) and up weakly interacting charm quasi-quarks cannot be the (u) quark flavor basis, we show the u-c flavor correlations, only carriers of charm quantum number for T ≤ 200 uc m+n m n defined as χmn = (∂ p/∂µˆu ∂µˆc ) at µu = µc = 0 in MeV. Such an observation naturally raises the question Fig. 1. Here, p denotes the total pressure in QCD, µu and whether charm excitations associated with baryon num- µc indicate the up and charm quark chemical potentials ber zero and one, exist in QGP for Tc . T . 200, withµ ˆX ≡ µX /T . In order to compare these lattice QCD along with the charm quasi-quark excitations carrying 2

C uc c HTLpt the open-charm sector, p , can be written as χmn/χ2 EQCD 0.5 C C m n: 22 p (T, µC , µB) = pq (T ) cosh (ˆµC +µ ˆB/3) + 13 0.4 C C 31 pB(T ) cosh (ˆµC +µ ˆB) + pM (T ) cosh (ˆµC ) , (2) 11 C C C 0.3 where pq , pB and pM denote the partial pressure of the quark-like, meson-like and baryon-like excitations, re- 0.2 spectively, and µB and µC = µc represents the baryon and charm chemical potentials. 0.1 Using combinations of up to fourth order baryon- charm susceptibilities it is easy to isolate the partial pres- 0.0 C C C sures of pq , pM and pB appearing in Eq. 2. For ex- ample, pC = 9(χBC − χBC )/2, pC = (3χBC − χBC )/2, -0.1 T [MeV] q 13 22 B 22 13 and pC = χC + 3χBC − 4χBC . The contributions of 160 180 200 220 240 260 280 300 320 340 M 2 22 13 these partial pressures compared to total charm pressure C C p (T, 0, 0) = χ2 is shown in Fig. 2 (top). For T . Tc uc the partial pressure of mesons, pC and the partial pres- FIG. 1. Off-diagonal quark number susceptibilities χnm nor- M c C malized by the second order diagonal charm susceptibility χ2 sure of , pB agree with the corresponding par- as a function of temperature [12]. The shaded band shows tial pressures from the HRG model including all the ex- the three loop hard thermal loop perturbation theory calcu- perimentally observed as well as additional lation for χ22/χ2; the width of the band corresponds to a predicted but yet unobserved open-charm hadrons with variation of the renormalization scale from πT to 4πT [23]. vacuum masses [12]. The contributions of pC and pC Also shown, as dashed lines, are the results of dimensionally M B reduced EQCD calculations for χ corresponding to temper- remain significant till T . 200 MeV. In fact, for T . 180 11 C C atures 1.32 Tc and 2.30 Tc from [22]. MeV the combined contributions of pM and pB exceeds C the contribution from pq . With increasing temperatures C C pM and pB deviate from the HRG model predictions. 1/3 baryonic charge. In the present work, we address This indicate that these charm meson and baryon-like this question by postulating that such open-charm me- excitations can no longer be considered as vacuum charm son and baryon-like excitations exist alongside the charm mesons and baryons. This is in line with the lattice QCD quasi-quarks in QGP and then investigate whether such studies on spatial correlation functions of open-charm an assumption is compatible with the exact lattice QCD mesons [24], which show significant in-medium modifi- results on charm fluctuations and their correlations. cations of open-charm mesons already in the vicinity of Charm fluctuations and their correlations with other Tc. The partial pressure of quark-like excitations is quite conserved quantum numbers can be measured on the lat- small for T ∼ Tc and becomes the dominant contribution tice through the generalized charm susceptibilities to pC only for T > 200 MeV. Since a charm-quark-like excitation does not carry a ∂i+j+kp(T, µ , µ , µ ) χXYC = Y C , (1) strangeness quantum number, the excitations carrying ijk ∂µˆi ∂µˆj ∂µˆk X Y C µX =µY =µC =0 both strangeness and charm quantum numbers are a much cleaner probe of the postulated existence of the whereµ ˆ = µ /T . For notational brevity we will sup- X X charm hadron-like excitations. In this sub-sector, the press the superscripts of χ whenever the corresponding pressure can be partitioned into partial pressures of |C| = subscript is zero. To check our postulates against the 1 meson-like excitations carrying strangeness |S| = 1 and lattice QCD results, throughout this study, we will use C = 1 baryon-like excitations with |S| = 1, 2, i.e. the lattice QCD data of Ref. [12] on up to fourth order C,S C,S=1 generalized charm susceptibilities, i.e. for i + j + k ≤ 4. p (T, µB, µS, µC ) = pM (T ) cosh (ˆµS +µ ˆC ) + To avoid introduction of unknown tunable parameters P2 pC,S=j(T ) cosh (µ − jµ + µ ) . (3) we simply postulate an uncorrelated, i.e. non-interacting j=1 B B S C gas of charm meson, baryon and quark-like excitations for Thus, the partial pressures of the strange-charm hadron- C,S=1 SC T & Tc. Owing to the large mass of the charm quark it- like excitations can be obtained as: pM = χ13 − BSC C,S=1 SC SC BSC C,S=2 self, compared to T ∼ 2 Tc, it is safe to treat all the quark, χ112 , pB = χ13 − χ22 − 3χ112 , and pB = BSC SC SC meson and baryon-like excitations as classical quasi par- (2χ112 +χ22 −χ13 )/2. In Fig. 2 (bottom) we show the ticles, i.e. within the Boltzmann approximations. Fur- fractional contributions of these partial pressures towards C C thermore, as discussed in Ref. [12], the doubly and triply the total charm partial pressure p (T ) = χ2 . Even in charmed baryons are too heavy to have any significant this sub-sector, contributions from the hadron-like exci- contributions to QCD thermodynamics in the tempera- tations are significant for T . 200 MeV. However, partial ture range of interest and we thus neglect their contribu- pressure for the S = 2 charm baryon-like excitations is tions. With these simplifications the partial pressure of negligible. 3

1.0 1.0

C C c1/p c2/p 0.8 C C C C 0.5 c3/p c4/p pq /p 0.6 C C pB/p 0.0 C C 0.4 pM/p

0.2 -0.5 T [MeV] 0.0 -1.0 150 170 190 210 230 250 270 290 310 330 0.3 C,S=2 C pB /p FIG. 3. Lattice QCD results for four constraints (ci) normal- C,S=1 C ized by the total charm pressure (see text). 0.2 pB /p C,S=1 C pM /p the strange-charm sub-sector, we have six generalized SC SC SC BSC BSC BSC 0.1 susceptibilities χ13 , χ22 , χ31 , χ112 , χ121 and χ211 . We can use three of these to estimate the partial pres- C,S=1 C,S=1 C,S=2 T [MeV] sures pM , pB and pB defined above, while the 0.0 remaining ones will provide three additional constraints 150 170 190 210 230 250 270 290 310 330 that can used to validate our proposed model. These constraints can be written as: FIG. 2. (Top) Fractional contributions of partial pressures of BSC BSC SC SC SC c2 ≡ 2χ121 + 4χ112 + χ22 − 2χ13 + χ31 = 0,(5a) charm quark-like (pC ), meson-like (pC ), and baryon-like (pC ) q M B c ≡ 3χBSC + 3χBSC − χSC + χSC = 0, (5b) excitations to the total charm partial pressure (pC ). (Bot- 3 112 121 13 31 BSC BSC tom) Fractional contributions of partial pressures of charm- c4 ≡ χ211 − χ112 = 0. (5c) C,S=1 strange meson-like (pM ), charm-singly-strange baryon-like C,S=1 C,S=2 Note that the above constraints hold trivially for a free (pB ) and charm-doubly-strange baryon-like (pB ) ex- citations to the total charm partial pressure (pC ). The charm-quark gas. It is assuring that our proposed model solid lines show the corresponding partial pressures obtained also smoothly connects to the HRG at Tc. In Fig. 3 from HRG model including additional quark model predicted we show the lattice QCD data for ci’s. Despite large er- charm hadrons (see text). rors on the presently available lattice data all the ci’s are, in fact, consistent with zero. Note that, since a possible strange-charm di-quark-like excitation will carry Having shown that there can be significant contribu- |C| = |S| = 1 but |B| = 2/3, the QCD data being con- tions from charm meson and baryon-like excitations to sistent with the constraint c4 = 0 actually tells us that the charm partial pressure in QGP, it is important to the thermodynamic contributions of possible di-quark- ask whether the addition of only these charm degrees like excitations are negligible in the deconfined phase of of freedom besides the charm quark-like excitations is QCD. sufficient to describe all available lattice QCD results One may speculate on the nature of these charm for up to fourth order charm susceptibilities. As dis- C C hadron-like excitations and, in particular, why their par- cussed previously in Ref. [12], the constraints χ4 = χ2 , BC BC SC SC tial pressures vanish gradually with increasing tempera- χ11 = χ13 , χ11 = χ13 are due to negligible contri- ture. A likely explanation may be that with increasing butions from |C| = 2, 3 hadron-like states and they do temperature the the spectral functions of these excita- not provide any independent constraint specific to our tions gradually broaden. A detailed treatment of ther- proposed model. The remaining four independent fourth C BC BC modynamics of quasi- with finite width was de- order generalized charm susceptibilities, χ2 , χ13 , χ22 BC veloped in Refs. [25–27]. It was shown that broad asym- and χ31 allow us the define the three partial pressures, C C C metric spectral functions lead to partial pressures that pq , pM and pB and one constraint are considerably smaller than those obtained with zero BC BC BC width quasi-particles of the same mass, and for suffi- c1 ≡ χ − 4χ + 3χ = 0, (4) 13 22 31 ciently large width the partial pressures can be made that has to hold if the model is correct. If we consider arbitrarily small. Thus, the smallness of the partial pres- 4 sure of charm quark-like excitations for T ∼ Tc may im- These findings may have important consequences for the ply that they have a large width for those temperatures, heavy quark phenomenology of heavy-ion collision exper- while the widths of the charm hadron-like excitations in- iments, especially in understanding the experimentally crease with the temperatures and these excitations be- observed elliptic flow and nuclear modification factor of come very broad for T & 200 MeV. Such a gradual melt- heavy flavors at small and moderate values of transverse ing picture is also consistent with the gradual changes of momenta [5–8, 28, 29]. the screening correlators of open-charm meson-like exci- Acknowledgments: This work was supported by U.S. tations with increasing temperature [24]. Department of Energy under Contract No. de-sc0012704. Finally, one may wonder whether the rich structure The authors are indebted to the members of the BNL- of the up to fourth order generalized charm susceptibil- Bielefeld-CCNU collaboration for many useful discus- ities can be described only in terms of the charm quasi- sions on this subject as well as for sharing of the lattice quarks without invoking presence of any other type of QCD data. charm degrees of freedom. In terms of charm quasi- quarks alone, the charm partial pressure will be pC /T 4 = 2 2 6/π mˆ c K2(m ˆ c) cosh(ˆµC +µ ˆB/3), wherem ˆ c = mc/T with mc being the mass of the charm quasi-. The lattice QCD results for the charm susceptibilities, [1] R. Rapp and H. van Hees, in R. C. Hwa, X.-N. Wang for example, the non-vanishing values of χSC , can only (Ed.) Quark Gluon Plasma 4, World Scientific, 111 mn (2010) (2009) arXiv:0903.1096 [hep-ph]. be described if the charm quasi-quark mass depends [2] G. D. Moore and D. Teaney, Phys. Rev. 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