Physics Letters B 286 ( 1992 ) 221-224 North-Holland P I-t Y $ I C S k E -l-I E R $ B

Thermodynamics of open and hidden charmed mesons within the NJL model

F.O. Got t f r i ed and S.P. Klevansky lnstitul f~r Theoretische Physik, Philosophenweg 19, W-6900 Heidelberg, FRG

Received 19 February 1992; revised manuscript received 4 May 1992

We consider effects of temperature on the masses of mesons containing charmed quarks, by considering the SUf( 3 ) Nambu- Jona-Lasinio model after formally replacing ms by rn o In the temperature range of interest, we find no Mott-like transition of the J~ T state into the D/) continuum, indicating that within this model, it is not the mechanism whereby J / T suppression occurs.

Over the last few years, serious attention has been given to the study of hadronic matter under condi- tions of extreme temperature and density, since this should have direct relevance for heavy-ion experi- ments. One aspect of the investigation relates to the modification of the hadronic spectra as the external parameters are applied. This has been studied exten- sively for the low-lying mesons in the (relativistic) field theoretical Nambu-Jona-Lasinio model (NJL) [ 1-4 ], and other chiral models, see for example ref. [5]. The temperature dependence of higher lying states, such as the charmonium spectrum, is usually investigated via the non-relativistic potential model (PM) [ 6-8 ]. In the potential model, the string ten- sion and coupling strengths are assigned a tempera- ture dependence, and it is this built-in screening mechanism that accounts for the variation (or lack thereof) of the hadron masses and the continuum background with temperature. On the other hand, other authors [ 9 ] have tried to combine a relativistic approach (via the MIT bag model) and the PM ap- proach to construct an adequate description for the temperature dependence of the D mesons, which have open charm. (Note that these approaches represent work done with effective models: for work done on extracting the temperature dependence of the had-

Supported in pan by the Deutsche Forschungsgemeindschafl under contract number Hu 233 / 4-1.

ronic spectra from QCD sum rules, the reader is re- ferred to refs. [ 10-12 ]. )

As is suggested in ref. [9], it would be useful to have a description of both light and heavy quarks in one model. We therefore investigate the physical consequences of including the "c" quark into the NJL model. At first sight this may seem inappropriate: the charmed quark should be regarded non-relativisti- cally, and it thus seems that invoking a fully SUf(4) symmetric relativistic field theory to include these particles is physically unreasonable. (A discussion of the bosonized Uf(4) NJL model without an explicit 't Hoofl terms is given in ref. [ 13 ].) We argue how- ever, that we are saved by the necessary presence of a regulator A for the model. Since A is of the order of 1 GeV, the quasiparticle energies accessible to the charmed quark are restricted to be less than Ec~ x / ~ + m c 2 -~ me, effectively rendering the charmed quark as non-relativistic within this model. The dy- namically generated part of the c quark mass in any event is found to lie below A. We remark further that our view of the regulator A follows that of ref. [ 14 ], i.e. we regard the cut-off as an approximate, if crude implementation of the property of asymptotic free- dom of quantum chromodynamics (QCD): by sup- pressing the interaction between quarks for large space-like momentum transfer, one simulates the be- havior of the running coupling constant of QCD. Since the SUf(4) symmetry in QCD is badly broken by the charm mass mo it is conceivable that the cou-

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved. 221

Volume 286, number 3,4 PHYSICS LETTERS B 30 July 1992

pling constants of the NJL lagrangian could depend on inc. We disregard any such possible effects here.

In what follows, we shall not examine the NJL la- grangian in a full SUf(4) symmetric form. Rather we restrict ourselves to a model study of the SUf(3 ) form, after replacing ms by m~ where appropriate, and iden- tifying the heavy mesons such as the D, J~ ~ a n d qc as those mesons with the appropriate quantum num- bers l ( Je ) . The three flavor NJL lagrangian is taken to be

8

~N~L = q ( i T " O u - M ) q + G , ~ [q2~q)2+ (t]i752~q) 2] a = 0

8 -G2 Y,

a = 0

[ ((tyu2aq) 2q- (qiT~,7~2~q) 2 ]

- K { d e t [ q ( l + 7 ~ ) q l + d e t [ q ( 1 - 7 ~ ) q ] } . (1)

Here M is the diagonal mass matrix (m o, md ,o mhO ) that breaks chiral symmetry explicitly, and GI, G2, and K are coupling strengths, the last of which con- trols the breaking of the UA( 1 ) symmetry of the three flavor interaction lagrangian, m ° refers to the cur- rent mass of the heavy quark. The 2 ~, a = l, ..., 8 are the Gell-Mann matrices and 2 ° = x ~ . It is useful to note that in the mean field approximation, the six- fermion interaction may be combined with the exist- ing four-fermion terms, which in the approximation of light quark isospin symmetry, mu=md, can be written as [ 15 ]

~NJL = q ( i ~ - M ) q

8

+ ~, [K~-)(£t2~q)2+K!+)((li752~q)2] t'=O

+ t (-) ~K m (q28q) (q2°q) + ! ~'(+) 2"~m (qi7528q) (qiy520q)

+ ' ( - ~Km )(q2°q)(qASq)+ 1I,'(+) ~'~m ((liT~20q)(qi~s28q)

8 - G 2 ~ [(@,2aq)Z+(@,7~A~q) 2] , (2)

a = 0

with the modified couplings constants

K6 +" ) =G1 +- ]K( (( t:ih )) + 2 (( au )) ) ,

KI +- ) =K~ + ) =K:] +" ) = G, ~ ½K<< h-h>> ,

K,~ + ) =K~ +" ) =K~, +') =K-~ -+ ) = G~ T- ~K<< tiu>>,

K~s+" ) =G, + ~K( << fih >> -4<< t~u>> ) , (3)

while the mixed flavor terms have the couplings K~m+ ~ = + (~x/5)K(((h-h)) - ( ( a u ) ) ) . The notation ((qq)) refers to the thermal averages of the condensate, taken in the grand canonical ensemble,

<<O>) Tr O exp [ - fl( H - Z, lz, N~) ]

Tr exp [ - f l ( H - Y , #,N,) ]

fl being the inverse temperature. Here/tg and N, are the chemical potential and number operators associ- ated with flavor i.

In the Hartree approximation, the self-consistency equations for the self-energy, or so-called gap equa- tions can be written for mu= rnd as

mu = m ° - 4 G , (( au)) + 2K(( au)) (( Eh )) ,

rnh =m ° --4Gl (( fih )) + 2K(( t~u)) (( t~u)) . (4)

It should be noted that this result differs from that quoted by ref. [4] by a rescaling of the coupling con- stant by a factor of two, since those authors use a Fierz-invariant form of eq. ( 1 ). The explicit form of the quark condensate is given as

A

3 f 2mi - 2rr2 dpp -~p (tanh ½flw;,+tanh ½flO)p~) • 0

(5)

(+-) =Ep,,. +fi,, and fi~ is the shifted chemical Here O~p,~ potential [ 3,4,15 ]

fli = f l i -4G2 (( q~, q, )~ (6)

that is given in terms of the quark densities

(( q~, qi )) = A

3 f 2 m i 2zr2 dp p ~ (tanh ½flmp+i-tanh ½flmpTi) . 0

(7)

A cut-offA restricting the three-momentum has been introduced. Eqs. ( 4 ) - (7) are solved self-consistently for the masses of the light and heavy quark. A nu- merical solution of these equations for rnh = ms has been undertaken in ref. [4]. Using their values for

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Volume 286, number 3,4 PHYSICS LETTERS B 30 July 1992

the cut-offand coupling strengths, i.e. A = 0.750 GeV, G~A2= 1.83, and KA5=8.54, we choose m , = m d = ° o 4.7 MeV and (for the moment) G2AZ=0. Setting first mO= o 1 3 1 M e V a n d t h e n m h ° = m c = ms = o 1.35 GeV, we obtain the temperature dependence of the dynam- ically generated strange and charmed masses in this model. This is depicted in fig. 1. At T= 0, the constit- uent quark masses are found to be mu=md=372 MeV, m~=571 MeV and me= 1.878 GeV. One notes from fig. 1 that the temperature dependence of the heavy quark mass is somewhat decoupled from that of the light quark masses: the expected SUf(2) be- havior of the light quarks remains intact. This can be understood directly on examining eq. (4). The light quark condensate contribution to mh is small in com- parison with m °. On the other hand, the mass m, is determined by the SUr(2)-like structure of the gap equation

m, = m ° +4K~ +) ( (au)) ,

with K~ + ) the effective coupling given in eq. (3). In the pseudoscalar meson sector, we are interested

in particular in determining the thermodynamic de- pendence of the D meson mass. This is obtained by solving the dispersion relation

1 -2K~+)HpS(k, 09) =0 (8)

for 09 when k = 0, and mh = m~. This state carriers the same quantum numbers I ( J e) = ½(0-) as do the kaons. The pseudoscalar polarization H p~, in imagi- nary time is given as

(1)

o

U3 6O (3

E

0 , , ~ = d

0 0.2 4

T (OeV)

Fig. 1. Temperature dependence of the u, s, a n d c quarks in the model. Parameters are given in the text.

HP~(k, 09,) = - N c 1 lira exp (i09, r/) fl,7~o ~ f dap (27[) 3

x tr V lS f (p+ k, to,, + v . )F2S f' (.p, co,,) , (9)

with F~ =F2 = its and f and f ' appropriate flavors. Here 09., v. are the Matsubara frequencies associated with bosons and fermions respectively, v. = + 2nz~/fl, 09.= + (2n+ 1 )rr/fl, n=0, 1, 2 ..... This expression can be evaluated explicitly: it is given in detail elsewhere [ 16 ]. The parameter set mentioned previously is used but with G2A2=l.38. One finds m . = 1 3 9 MeV, mK=527 MeV (496 MeV), mD=2.001 GeV (1.867 GeV) at T= 0. The values listed in parentheses are the experimentally known masses. The mass of the D meson can be adjusted to its empirical value by em- ploying a second cut-offA'= 1.08A to restrict the in- tegral in eq. (9). This does not affect the qualitative behavior of the mass of the mode with temperature. The results obtained are shown in fig. 2. Together with the meson masses, we have plotted the respective quark-antiquark thresholds. From the figure, one notes that all the bound states decay into free quarks at the same temperature, TD--~230 MeV. This un- physical process can be regarded as placing a bound on a validity of the model to temperatures lower than TD. It is also interesting to observe that the trends in the K and D masses are similar despite a change of one order of magnitude in mh in evaluating these.

The vector meson modes are obtained via the la- grangian ( 1 ) by solving the dispersion relation

4 b5 . . . . . . .

c~

o

co D U

E

G 0 O. 2 C,.-:

',sov) Fig. 2. Temperature dependence of the pseudoscalar n, K, and D mesons (solid curve). The dashed curves follow the respective quark-ant iquark continua. The DO cont inuum is given by the dotted curve. Parameters are given in the text.

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Volume 286, number 3,4 PHYSICS LETTERS B 30 July 1992

1 + 2G2HT(k, ~o) = 0 , (10)

where the transverse polarization is defined to be

liT= ~ ( kukp/k 2 -g~ , )H "p

and H up is the polarization calculated via (9) with

f'~ = ?~'75 and F 2 = ~PYs. This assumes that the state is bound or is an approximation to the mass if the state

is a resonance of narrow width. Strictly speaking, if the state is a resonance, the mass should be deter- mined via the spectral function [ 17 ]. This is unnec-

essary for our purposes. The parameter set used for the pseudoscalar mesons places the p meson at mp=744 MeV (770 MeV), while the J / ~ falls at 3.589 GeV (3.097 GeV) at T = 0 . Experimental val-

ues are again given in parentheses. Once again, a somewhat enlarged cut-off on eq. (10) can be used, A'= 1.27A, to bring msl~, to its experimental value at

T = 0 . The spectrum, together with the quark-an t i - quark threshold is also shown in fig. 2. The D/)

threshold is also indicated. Here we observe the fol- lowing : ( i) the J~ ~ i s a bound state and its mass falls slowly over the temperature range of interest. This is in accordance with the behavior predicted by QCD

sum rules [12 ], and is in approximate accord with the constant behavior predicted by the non-relativis- tic potential model in ref. [7]. (ii) Contrary to the effect reported in ref. [7] however, we find that the J~ ~ mass does not cross the DO threshold and thus no Mort-like temperature can be defined. Further, the trends of the J~ ~Umass and the D/) con t inuum in the temperature range in which this model should be valid ( T < 230 MeV), indicate that it is unlikely that such a crossing would be observed at higher values of T. This model (as with ref. [9] ) thus implies that no contr ibut ion to the J~ ~' suppression can be inferred via the Mott mechanism unless some ad-hoc explicit temperature dependence is incorporated into the in-

teraction. In the case presented here, where no Mort-

like t ransi t ion is observed, one would not expect to

see any kink in the ET dependence of the suppression ratio S, as defined in ref. [8]. As proposed by the

authors of refs. [7,8], the J/~P suppression would

probably require another physical origin, such as thermal activation. Nevertheless, high Ev measure-

ments could arbitrate between such a PM and the de- scription as given here.

We wish to thank J. Hfifner for providing the stim- ulation to this work and him and D. Blaschke (Ros-

rock) for many useful discussions.

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