φ and K mesons in hot dense matter

Physics Letters B 271 ( 1991 ) 32-36 North-Holland PHYSICS LETTERS B

and K mesons in hot dense matter

J.-P. Bla izot a n d R. M 6 n d e z G a l a i n J

Service de Physique Th~orique 2. CEN-SA CLA Y, F- 9119 l G~f-sur- Yvette Cedex, France

Received 19 April 1991 : revised manuscript received 29 August 1991

A calculation of the K and ~) masses at finite temperature in the Nambu-Jona-Lasinio model suggests that the phase space for the ~ K K decay channel disappears for moderate values of the temperature. We discuss the relevance of this effect for the interpretation ofdilepton production in high-energy nuclear collisions.

The properties of mesons are expected to be affected by the presence of hot dense matter. This has been discussed by several authors, and is the subject of considerable current investigations. A particularly interesting issue is whether such modifications of meson properties could be used to diagnose the state of matter produced in high energy nuclear collisions. The purpose of this letter is to investigate the role of a possible modification of both ~) and K meson masses, and its implication concerning dilepton yields observed in nuclear collisions [ 1 ].

The total width F o f t h e 0 meson can be written as

r = r~ + rK + r, , (1)

where F ~ 4.4 MeV, F~ and FK are the partial decay

widths into muons and kaons respectively, and F, ac- counts for the residual channels. The kaon and muon branching ratios are respectively B K ~ - I ~ / F ~ 0 . 8 4 and B ~ - F ~ / F ~ 2 . 5 × 10 -4. The threshold of the two K channel is close to the 0 mass: A M = m o - 2 m k is about 28 MeV (averaged value for neutral and charged kaons). This relatively small value suggests that any modification of A~///will have a strong influence on the decay pattern of the ~) [ 2-4 ]. Let us sup- pose, for example, that the nuclear envi ronment induces a reduction of the phase space for the KK decay, i.e. A~/= too- 2ink decreases (we shall see below that

On leave of absence from lnstituto de Fisica, Facultad de lngenieria, Montevideo. Uruguay.

2 Laboratoire du Direction des Sciences de la Mati6re du Com- missariat/~ I'Energie Atomique.

there are models which lead to such a behavior). Then Fbecomes

F ' = F , + F k +F, , (2)

where we have assumed that Fr and F , are unaffected by the nuclear medium. Note, however, that F , could decrease slightly with temperature; indeed the size of the mesons is expected to grow with temperature: ac- cordingly, the square of the wave function at the ori- gin, to which the leptonic decay width is directly pro- portional, should decrease with temperature [ 5 ], As for F , it is dominated by the 9-~ channel and we have little to say about it; it is, however, much smaller than FK and a small variation of Fr should not affect the conclusions of the present discussion. The reduction of F k implies a reduction of the total width of the 0 ( F ' < F ) and a corresponding increase of the branching ratio B'~ = F , / F ' > B , . Thus, a decrease of AM causes the ~ to live longer and to produce, in its decay, less K's and more ~a's. In the extreme case where A M ~ 0, one expects an increase of the branching ra- t i o B ' ~ b y a f a c t o r o f o r d e r ~ ( 1 - 0 . 8 4 ) ~ 6 .

To assess the relevance of this effect to the understanding of d imuon production in nuclear collisions, we need an estimate of the modifications of meson masses in hot dense matter, as well as a comparison of the lifetime of the ~ with the time scale over which the effect can take place in an actual collision. We now discuss these two aspects in turn.

We have calculated the change in AM as a function of the temperature (T) , in the Nambu-Jona-Las in io model [6]. This model has been used successfully to

32 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

Volume 271, number 1.2 PHYSICS LETTERS B 14 November 1991

calculate meson masses and coupling constants [7 - 10] and it provides a simple description of the restoration of chiral symmetry at finite temperature [11,12] and density [10-14] . It has been the object of numerous investigations. Therefore we shall be very. brief in our presentation, emphasizing only those aspects which are relevant to the present discussion.

The lagrangian of the model, including the vector sector has the form [ 7,9 ]

L = q ( i ~ - too) q

l 8 + 2a ~ ;~ [(q2'q)2+ (qi75)o/q) ~ ]

1 8 + ~ ;;~ [ (@,,).,q)2+ ((17,,752N) 2 ] , (3)

where mo=diag(ml, nh, m~) is the current quark mass matrix for the light (1) and strange (s) flavors; 2 i are the Ge]l-Mann matrices with tr2,2j=2(Szj and 20= x/-~, a2 and h -~ are independent coupling constants for the scalar and the vector sectors, respectively. For simplicity, we do not include the six-fermion interaction induced by the U( 1 )~, anomaly [8]. This term is essential for understanding the rl-r I' mass splitting but has a minor role in the present calculation, as we discuss later.

A simple mean field treatment at finite tempera- lure yields the following "gap equation" for the constituent quark masses Mr ( f = 1, s ):

( ,n,-~..., f d3k 1 a n 1-~4~.. j=12 (27r)3 ~[, ( l - 2 n } ~ ) , (4)

A

where

, , ~= 1 ~ =,fl/r+ M~ (5) exp(fle~)+ 1 '

The presence of A by the integral sign in eq. (4) indicates that the diverging integral over k is cut-offat some value of A of the three-momenlum. A is to be considered as a parameter of the model, and all physical results depend on its value. In the approximation that we are using, the contributions of the various flavors to the constituent mass decouple so that one has one gap equation for each flavor.

The zero temperature limit o feq . (4) is obtained for fl-~ ov (then n ~ -+ 0). The corresponding constituent quark mass is denoted Mm. The first effect of

increasing the temperature is to reduce the domain of integration over the quark momenta in eq. (4). Since a 2 and m~-are kept constant when T increases, this effect is compensated by a diminution of the constituent quark mass (M~,< M~). When mr= 0, the constituent mass Mr eventually vanishes for some value T,.~xf3A/zr2-a "- of the temperature, corresponding to the restoration of chiral symmetry. As mr/Mm increases, the variation of Mr/Mfo gets smoother, with Mr going over to rnr at large values of the temperature. All this is illustrated in fig. 1 below. Similar effects occur when the baryonic density increases [ 9,10,12,14 ]. It is worth noticing that the integral in eq. (4) is equal to - ( ( l q ) / M f . Thus, the diminution of the constituent mass Mr with increasing temperature reflects that of the quark condensate, an effect which is expected on other grounds than the present model. In particular it obtains in chiral perturbation theory [ 15 ].

Mesons are described as qq excitations of the vacuum and their energies oo(q) are obtained as solu- tions of the equation

g=H(oz q)

where g is equal to a 2 o r b2c~ ~''" for scalar or vector excitations, respectively, and H(o~, q) is the "polari- zation operator" in the relevant (1Fq channel (Fbeing i;'5~ for n, i7~(24_+ i25) /x/2 for K + or 7 ; , ( 1 / x / 3 - 2 8 ) for 0). In order to obtain meson masses we simply solve eq. (6) for q=0 . A straightforward calculation gives [ 16]

f ( l - 2 n ~ , ) (7) d3k

H=({o) =48 (27r)3 4e~, 2 _co2 .1

I1K ( ~o l

f d3k 1 ( - e k ) - ( n , _ m s ) z] = 6 ( 2 ~ ) 3 ~ , [(E~ s _, ; 1

s 1

X ( n ~ , - n ~ ) e~-~k (~, - ~ ) ~ - c o 2

+ [ ( ~ + e~,)~- (m~-m~)-~]

- " - (~Td~--o/-)" (8)

33

V o l u m e 271, n u m b e r 1,2 P H Y S I C S L E T T E R S B 14 N o v e m b e r 1991

f d3k 1 1 H~¢(CO)= 12 (27~)3~] 4~s2 6,j2 CI '~ (1 -2n~) ' k

A

(9)

where C""= diag(0 , ,4, A, A ) with ,4 = 2 k 2 / 3 - 2e~,. 2 .

The polar izat ion opera tor H(co) has a non-vanish- ing imaginary part when (a) co>2M~ for ~, co>2M~ for 0 and co> (M~+M~) for K; or (b) when co< M~-M~ in the kaon operator . The first case corresponds to the decay of the meson into qq pairs. The second case, which is specific to non-zero temperature (or baryon dens i ty) , corresponds to scattering of mesons with quarks or ant iquarks present in the system. In the model, both processes are clearly un- physical since they involve explicit ly unconfined real quarks or ant iquarks. However, in the range of tem- peratures considered here, these imaginary parts do not play any significant role since meson masses will be found always in the co domain where H(co) is strictly real.

A simple analysis of eqs. ( 6 ) - ( 9 ) allows us to draw general conclusions about the behavior of meson masses with temperature. Let us consider first the case of the pion for which H(co) is given by eq. (7) . It is readily seen that, for mr=0 , the equat ion H~(co=0) = a 2 is equivalent to eq. (4) , implying that the pion mass vanishes. This result is of course a di- rect consequence of chiral symmetry breaking and holds at finite tempera ture as long as M f g 0. It may also be interpreted in terms of a strong at t ract ion re- sponsible for the large "b ind ing energy" of the pion. When the quark mass is different from zero, a slight imbalance appears, when one varies the temperature , between the var ia t ions in this binding energy and those of the consti tuent mass. However, since the pion binding is p redominan t ly a collective phenomenon, it is little sensit ive to a change in the const i tuent mass Mr, and this remains so as long as mr is small in comparison with M~. Thus, when one raises the temperature, one first sees a d iminut ion of the b inding energy which may be t raced back to the reduction of the k-integrat ion domain in eq. (7) . This induces an increase in the pion mass. A similar behavior is expected for the other pseudo-Golds tone bosons such as the kaon. In fact, we have per formed a numerical analysis which indicates that this behavior , namely the increase of the meson mass with increasing tern-

perature, obtains for mr/M~o<0.25, and tempera- tures not too close to T~. The case of vector mesons, such as the 9, is different. Firstly, the coupling constant b 2 does not affect at all the quark masses. Sec- ondly, for realistic values of the coupling, i.e. those needed to account for the exper imental masses, the b inding energy is much smaller; thus the variat ions of the vector meson masses will more strongly reflect the var ia t ions of the const i tuent masses; they will be found to decrease with increasing temperature , in agreement moreover with what is found in other ap- proaches [ 17 ].

Let us now turn to the numerical results. We consider a 2, b 2, m~, m~ and A as tempera ture independent parameters . (This is certainly reasonable for the current quark masses, but the possible variat ions of the coupling constants, and the cut-off, with the temperature introduce some uncertainty in the model predic t ions . ) Since we fix the meson masses to their empir ical values at zero temperature , the model has only two independent parameters which we choose to be the const i tuent light quark mass at zero temperature M~o and the cut-offA. This choice is mot iva ted in part by the fact that several physical quanti t ies ap- proximate ly scale as a function of T/M~o. Given M~o and A, we fix the other parameters as follows:

( 1 ) We use the gap equation for the light quark (eq, (4) for f = 1 at T = 0) and the pion mass equation (eqs, (6) , (7) at 7"=0, with c o = m ~ = 1 3 8 MeV) to f ind the light current quark mass m~ and the scalar coupling constant a 2.

(2) We use the gap equat ion for the strange quark (eq. (4) for f = s at T = 0 ) and the kaon mass equation (eqs. ( 6 ) - ( 8 ) at T = 0 , with co= inK=497 MeV) to find the strange current quark mass m~ and the strange const i tuent quark mass at zero tempera ture

(3) We use the 0 mass equat ion (eqs. ( 6 ) - ( 9 ) at T = 0 , with co=m,= 1019 MeV) to find the vector coupling constant b z.

We then calculate the tempera ture dependence of the observables at finite tempera ture as functions of the free parameters of the model (M~o and A ). Note, however, that one can fix A as a function of MLo by fit t ingf~ to its exper imental value 93 MeV [9]. For this value of A,fK is obta ined within a 20% accuracy [ 8 ]. In the present study, we explored a wider range in the cut-off, including that value which givesf~ = 93

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Volume 271, number 1.2 PHYSICS LETTERS B 14 November 1991

MeV. We chosed A between 700 and 1400 MeV and we used values ofM, o ranging from 300 to 600 MeV. In our calculations m, turns to be of the order of 3-6 MeV and m~ of the order of 80-120 MeV depending on the values of the free parameters M~o and A [ 16 ].

The consti tuent quark masses Ml and M~, obtained by solving eq. (4) for f = l and s, are displayed in fig. 1, as a function of the temperature. In fact Mr/M~ scales approximately with T/MIo and a small dependence in A is observed. As may be guessed from this figure, and confirmed by an explicit calculation with

m.=0 , chiral symmetry is restored when T/M~o~ 0.6-0.7.

Our detailed numerical calculations confirm the above discussion concerning the temperature dependence of meson masses. One observes a competit ive effect between the decrease of the consti tuent quark masses which tends to push the meson masses down, and the increase of the occupation factors which has the opposite effect, in the case of the pions and the kaons, the second effect wins, as expected for collective modes, and the result is a net increase in the masses (about 2% at T/MIo~0.5 for the K). On the contrary, the mass of the ~ decreases slightly with increasing temperature (about 0.5% at T/M~o~0.5). Our main result is summarized in fig. 2 which shows

A M = m o - 2ink as a function of T/3/llo . At a temperature of the order of 0.45M~o, i.e. well below the restoration of chiral symmetry, AM vanishes. We have

Mf/Mfo[ MS/Mso 1.B . . . . .

0.6-_

0.2-

0.2

M{/ H

O J, '~ O.fi 08 l.O " T/MI o

Fig. I. The constituent masses of the light (M~) and strange (3Is) quarks relative to their zero-temperature values M]o and M~o as a function of the ratio of the temperature T to the constituent light quark mass M,o. The free parameters vary as much as M]o~ 300-600 MeV and A ~ 700-1400 MeV and these variations are reflected in the shaded areas. Quark masses approximately scale with T/Mto and there is a small dependence in A. The small arrow indicates the temperature at which A,~I becomes negative (see fig. 2 ).

checked numerically that including the six-fermion interaction does not change qualitatively this result: with this term present, ZL44 vanishes for a temperature slightly lower than 0.45M, o.

Before we can draw any conclusion about the possible relevance of this effect for the understanding of d imuon production in nuclear collisions, we need to estimate for how long, in an actual collision, the medium can affect the 0. In order to do so, let us write the two coupled equations which govern the rate of d imuon production:

dN, dN, dt - - F N ° ' dt =F,N,, (10)

where N, and A~ are respectively the number o f t and d imuons at a given time t. Let us assume, for the pur-

pose of getting a simple estimate, that the 0 is sur- rounded by a medium hot enough to totally close the KK channel for a time r~, and that beyond that time

the 0 decays normally. Then it is not difficult to solve the previous equations to obtain the total number of produced dimuons in the form

N, =A~)B, [BB-~ ", - ( B 2 - l ) e x p ( - F ' r , ) ] , (11)

where No is the number of initially produced ~. The

factor in square brackets represents the correction to tile d imuon yield coming from medium effects. One sees that there is a competi t ion between the increase in the apparent branching ratio (B'~/B~ ~ 6) and the

' MeV

2o

l0

I I oi 0.2 0.3 0.4 ~o5

T/Mr o Fig. 2. The phase space for the decay of the 0 in two K's, ~k/_---- m,--2mK as a function of the ratio of the temperature 7 to the constituent light quark mass Mjo. The free parameters vary as much as in fig. 1. One observes an almost perfect scaling and a very small dependence in the cut-off. The KK channel is closed at a temperature T~ 0.45M]o.

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Volume 271, number 1,2 PHYSICS LETTERS B 14 November 1991

increase in the 0 l ifetime ( F ' ~ F / 6 ) . Because of this compet i t ion, the effect is actually much smaller than one could have expected. For example, taking a value Of Tl of 30 fro~c, one gets a 50% increase in the yield. Note that, in analogy with what is expected for the J / ~ meson in a s imilar context [ 18], ~ could be actually much smaller for O's with large Pv because of finite size effects and relat ivist ic t ime dilat ion. Nevertheless, the effect is not a priori negligible and might have to be taken into account in a quant i ta t ive calculation of di lepton yields.

In any case, this short s tudy suggests that med ium effects may have a strong influence on the proper t ies of strange mesons and this deserves further investigations. This is specially true since there are now con- flicting results in the l i terature concerning the way meson propert ies, and in par t icular their masses, are affected by tempera ture [15,17,3,4]. In chiral per- turbat ion theory [ 15 ], one takes into account s-wave n -n and K - n interactions, which provides an increase in the n and K masses propor t iona l to T 2 for low values of T. This predic t ion is qual i ta t ively similar to that of the N a m b u - J o n a - L a s i n i o model, but for different reasons. In the N a m b u - J o n a - L a s i n i o model, m e s o n - m e s o n interact ions are ignored, at least in the mean field calculation presented here, and the var ia t ion of meson masses arises from modif ica- t ions of the meson structure and the vacuum properties, such as the quark condensate or the consti tuent quark mass. In a recent paper, kissauer and Shuryak [3] argue that the dominan t K - n interaction is the resonant p wave (K*) . This leads to a strong modif icat ion of the optical potent ial describ- ing the propagat ion of the K in a n gas, which results in opposi te effects to those we have obtained, namely an increase in the 0 decay rate by a factor of 2-3. Whereas the effects of p-wave m e s o n - m e s o n interaction certainly dominates over those of the s wave,

it is not yet clear whether they domina te over those effects which we have considered in this letter, and clarif ication of this point is worth further study.

One of us (R .M.G. ) would like to acknowledge interesting discussions with G. Ripka.

References

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φ and K mesons in hot dense matter