Kaons and strange quarks in dense matter

Physics Letters B 278 (1992) 29-33 North-Holland PHYSICS LETTERS B

Kaons and strange quarks in dense matter -A-

M. Lutz , A. Ste iner a nd W. Weise Institute of Theoretical Physics, University of Regensburg, W-8400 Regensburg, FRG

Received 18 November 1991; revised manuscript received 2 January 1992

We study the change of the masses and decay constants ofK + and K- mesons in a dense nuclear medium, using the framework of the SU ( 3 ) Nambu and Jona-Lasinio model. We do not observe a tendency towards kaon condensation in this model.

Properties of mesons in dense and hot matter are subject of considerable current interest. In particular, mesons with strange quark components such as kaons and ~ mesons have been suggested as suitable indi- cators of the dense and hot environment in which they are produced in high energy nuclear collisions (see e.g. ref. [ 1 ] for a review). Recently Ko et al. and Brown et al. [2 ] have studied the effect that a de- creasing kaon mass with increasing nuclear density would have on the kaon yield in high energy heavy ion collisions. Their considerations are based on ear- lier suggestions by Kaplan and Nelson [ 3 ] that kaon condensation should occur at several times nuclear matter density Po = 0.17 f m - 3. The driving force for this is an attractive kaon-nucleon S-wave interaction, related to explicit chiral symmetry breaking by the large strange quark mass.

The primary aim of this paper is to investigate the density dependence of the kaon mass and decay constant. In this context we shall re-examine the kaon condensation scenario and also explore the behaviour of the strange constituent quark in dense matter.

Any modification of the K ÷ and K - masses will strongly influence the decay of the ~ meson in hot compressed matter, as pointed out in recent model calculations [4 -7 ] . We will comment on this issue later.

Our framework is the SU (3) Nambu and Jona- Lasinio (NJL) model including vector interactions

¢r Work supported in part by BMFT grant 06 OR 762.

and effects of the axial U (1)A anomaly. This model has been used previously [ 8,9 ] to evaluate the spec- troscopy as well as various other properties of the J ~ = 0 -+, 1 -+ meson nonets. Our calculation will be based on the extension of this model to finite density and temperature as presented in ref. [ 10 ].

The temperature dependence of the kaon mass at zero baryon density has already been investigated in the NJL model (though without vector interactions) [ 6 ]. It was found that the kaon mass experiences a slight increase with rising temperature T. In this model the kaon decay constant first decreases smoothly and then more rapidly at temperatures around T = 150-200 MeV where chiral symmetry is restored. An evaluation of the K + mass in dense matter (also excluding vector interactions) has been re- ported in ref. [ 11 ]. Here we will concentrate in particular on the density dependence of the K + - K - mass difference. We consider an isospin-symmetric system with equal densities of u- and d-quarks (P ,=Pd) . The corresponding nuclear matter density is

l + P = ~ (P, Pd) = ]P,,. The density of strange valence quarks will be kept equal to zero in this work (ps= 0).

We start from the effective lagrangian

LP=q( i~ 'uOU-mo)q+ ½Gs[ (q2~q)2+ (qiTs2~q) 2 ]

+ G D [ d e t q i ( l + 7 5 ) q j + d e t q i ( 1 - y s ) q j ] , (1)

where q = (q i )= (u, d, s) are the quark fields and mo = d i a g ( m °, m] , m °) is the current quark mass matrix. The local scalar-pseudoscalar interaction

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved. 29

Volume 278, number 1,2 PHYSICS LETTERS B 19 March 1992

(with coupling strength Gs) and the vector-axial- vector interaction (with coupling strength Gv) are invariant under chiral U(3 )L®U(3)R transforma- tions. Here the U(3)n . . . . generators 2, ( i=0 , 1, ..., 8) include the singlet 2o = x / ~ diag( 1, 1, 1 ) and the standard Gell-Mann matrices 2 ~ .... ,2s. The last term proportional to GD selectively breaks axial U(1)A symmetry but keeps chiral SU(3)L®SU(3)R.

As in ref. [ 10 ] we will use the lagrangian ( 1 ) in the thermal mean field approximation. The self-energies entering in the thermal quark propagators are 27= Am + 7oAE. The dynamical quark masses Am = m - mo and the energy shifts AE are determined self- consistently by the gap equations

Am. = - 2 G s (au) --2GD (rid) (Ys) ,

Amd = - 2 G s (rid) --2GD (aU) (.(S) ,

Am~ = - 2 G s (gs) --2Go (ftu) (rid) , (2)

and

AE. =2Gv ( u ' u ) ,

AEs =2Gv (s~s) .

AEd = 2Gv ( d f d ) ,

(3)

The expressions for the thermal quark condensates (~]gq~) and densities (q~q~) for each of the flavors i= u, d, s are

3 <0q)=- ~

A

f m x dpp2-~ -- [ 1 - n ( E p + & E ) - ~ ( E p - z ~ ) ] , 0

(4) A

( q t q ) = ~ dpp2[ n (Ep+AE)_~(Ep_AE)] , 0

(5)

with m~ = m ° + Arn~ and E~ ~) = ~ / p2+ m~. A charac- teristic cutoffA has been introduced to regularize the momentum space integrals. The quark and antiquark Fermi functions are

n(x) ={I + exp [ f l ( x - / t ) 1} - ' ,

~(x) = { 1 + exp [f l(x+ p) ] } -~ , (6)

with fl= (kaT)-~ and the chemical potential/2. We

are primarily interested in the low temperature limit f l- ~-~0 for which pi--- (q~,q~) = n-2(/ t~ - m 2 ) 3/2, or

~, =~/m~ + (~%)~/~. (7)

Meson modes of four momentum qU= (co, q) are described by the quark-antiquark T-matrix T(q)= K+KJ(q) T(q) where K is the two-body interaction kernel derived from the lagrangian ( 1 ) and J(q) are properly regularized quark-antiquark loop integrals [8-10] . The meson masses are determined by the singularity condition

det [ l - J (og , q = 0 ) K ] = 0 . (8)

We fix the parameters of the model by the require- ment that meson and vacuum properties are well re- produced at vanishing density and temperature. Us- ing GsA2= 3.7, Gv= 1.1Gs, GDA2(~s)=0.6, a three- momentum cutoff A = 0.75 GeV and current quark masses m ° = m ° = 4 MeV, m ° = 8 7 MeV, we find constituent quark masses m,=md=361 MeV, ms= 501 MeV, pseudoscalar meson masses m~= 139 MeV, mK=498 MeV, m , =519 MeV and m, ,=963 MeV, and decay constantsf~= 93 MeV, fK= 97 MeV. The chiral condensates are ( ~ u ) = ( d d ) = - ( 2 8 7 MeV) 3, ( g s ) = - (306 MeV)3; the resulting vector meson masses are mp=rno~=765 MeV, m x . = 8 6 4 MeV and ms= 997 MeV.

Vacuum and pion properties in hot compressed matter as they result from this model have been dis- cussed in detail in ref. [ 10 ]. In particular at temperature T = 0 the condensates (au) and (dd) are shown to decrease linearly with increasing density up to about p ~ 2po. The gap equations (2) imply a corresponding behaviour for the non-strange constituent quark masses. The variation of the condensate, d ( ~ u ) = - (3X~,p/m2f2~) (~u), is controlled by the pion-quark sigma term X,~u=m°(ulau+ddlu). This is the NJL analogue of the model independent result [ 12 ] that has 3X~, replaced by the pion-nucleon sigma term.S~u. Note that in a constituent quark model, we would indeed have ~,nN ~. 3X~,.

Let us now first discuss the strange constituent quark mass ms in dense matter. In absence of the fla- vor mixing, U(1)A breaking interaction ( G o = 0 ) , there is evidently no change in the strange quark condensate (gs), and correspondingly in ms (see eq. (2) ) , as long as ps=0. However, with the U( I )A breaking term turned on such that the q' mass is re-

30


produced, we see f rom eq. (2 ) that the var ia t ion o f rn~ becomes d i n s = - 2 G s d ( g s ) - 4 G D ( a U ) d ( a u ) in terms of the var ia t ions d ( ~ u ) and d ( ~ s ) o f the condensates (we have used ( t~u) = ( r i d ) ) . We find that the relat ive change of m~ with densi ty is

d m s _ 0 . 0 5 / ) . (9 ) ms Po

This l inear densi ty dependence of ms holds roughly up to twice the densi ty o f nuclear matter . It reflects the fact that the strange quark, through f lavor mixing dynamics , is sur rounded by a u- and d-quark cloud which experiences m e d i u m corrections. This effect is weak, however, since only about 10% of the strange const i tuent quark mass is due to the f lavor mixing interact ion in our model.

Next, we investigate the masses o f K + and K - mesons in matter . We look for poles (eq. ( 8 ) ) in the T- matr ix which describe the propagat ion o f u~-pairs (for K + ) or s~-pairs (for K - ). The matr ix e lement of the pseudoscalar in teract ion kernel in this channel is K = Gs + GD (t~U). Fo r Gv # 0 in eq. ( 1 ), mixing between pseudoscalar and axial-vector channels occurs. This mixing is proper ly taken into account as described in refs. [ 9,10 ].

Close to the kaon poles in the T-matrix, the inverse propagators of the K -+ modes have the form

D-+ (q) = (to_+ - T - A E ) e - q 2 - m 2 - H + - ( p , T,q) , (lO)

so that the energies of the K - modes are to+ = x/q2 + m ~ + H -+ _+ AE, including the energy shift due to the vector interaction. All other med ium effects are summar ized in the self-energies H -+ (p, T, q). We de- fine the kaon mass in the m e d i u m as

mK+_(p, T ) - x / m ~ + H +- (p, T, q = 0 ) . ( l l )

M e d i u m modi f ica t ions o f the K -+ masses enter through several compet ing mechanisms. First , the u- quarks in the K + experience the Pauli principle which tends to raise the K + mass. Note that this effect is suppressed in the K - with its sa valence structure. Second, the mass m, of the consti tuent u-quarks drops with increasing density. Together with the smooth re- duct ion o f the strange quark mass ms, this mechanism tends to decrease both kaon K + and K - masses. Final ly the vector in teract ion propor t iona l to Gv generates an energy shift AE=2Gvp,=3Gvp for the u-

quark (see eq. ( 3 ) ) . For the a-ant iquark this shift has opposi te sign. No such energy shift appears for the s-quark as long as ps= 0.

Let us now discuss the K + and K - masses as a funct ion of density, first in the absence of the vector coupling (i.e. with G v = 0 ) . In our case with vanishing strange quark density, but pu=Pd> 0, we then ex- pect a spli t t ing between mK+ and inK- due to the dif- ferent role played by the Pauli pr inciple in the K + mode as compared to the K - . It is instructive to examine this behav iour in the chiral U(3)L(~)U(3)R l imit with vanishing current quark masses and with the U ( 1 ) A breaking turned off (Go = 0). In this l imit the K+-K - mass difference to leading order in the densi ty p becomes

3p mK+--mK-= 4 f 2, (12)

with f=f ,=fK, the pseudoscalar meson decay constant .

When the energy shifts due to vector couplings are included together with non-zero current quark masses m ° and m °, the kaon energies to+ ( q = 0 ) result as shown in fig. 1. For compar ison the masses rn~+_, cal- culated with inclusion o f vector interact ions, are also shown. The energy to+ by the K + at rest now in- creases with densi ty (by about 10% at P=Po over its free space value mK~ 500 MeV) whereas the K - energy to_ decreases as roughly the same rate for p ~<Po. The influence of the vector interact ion is twofold. First, it generates an energy difference between u- and g-quarks. On the other hand it reduces the Pauli el'-

70o

600 ... " / 1 1 " ~ ' m K +

[ M e V ] . ~ . ~ 500 ~--- I

L, O0 w_(~=O) I I I I

0 1 2 density 9/9o (n.m.)

Fig. 1. Solid curves: The energies 09+ and co_ ofK + and K- mesons at rest as a function of density p (given in units of nuclear matter density Po = 0.17 fm- 3). Dashed curves: effective K ÷ and K- masses as a function of density.

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fect which generates the mass splitting (12). As a consequence the net effect of the vector interaction on the difference 09+-09_ of the K ÷ and the K - energies is small.

In fig. 1 we also show the effective masses m_+ (p) = o9_+ (q = 0 ) -T- AE. One observes that (o9 + + o9_ )q=o = inK+ + mK-, the minimal energy needed to produce a K+K - pair in matter, is a remarkably smooth function of density. This can be understood by studying the relative change of the sum o f K + and K - masses to leading order in the current quark masses mo, omitting the U ( 1 )A breaking interaction. In this limit

dmK+ + dmK- _ dm,~ + O ( m o ; GD) • ( 1 3 ) mK mK m,~

In ref. [ 10] it was shown that the relative change of the pion mass is extremely smooth at densities and temperatures below the chiral phase transition. Chiral symmetry protects the pion mass against rapid changes. A similar mechanism is at work in the kaon. The Gell-Mann-Oakes-Renner (GOR) relation [131,

2 1 m K f 2 K = - - ~ ( m ° + m ° ) ( a U + g s ) + O ( m g ) , (14)

can be shown to hold approximately in matter as well, with kaon mass and decay constant as well as the condensates properly replaced by density and temperature dependent quantities. The proof is based on the fact that the axial Ward identity from which the GOR relation is derived, remains valid in dense and hot matter.

Next, we study the decay constantsfK+ andfK- in dense matter. Their dependence on p reflects the change of the K + and K - intrinsic wave functions with increasing compression. The actual calculation is performed by evaluating the relevant in-medium matrix elements < 0 I Au I K(q) ) = iqufK of the axial current. Results are shown in fig. 2 together with the pion decay constant j% taken from ref. [ 10 ]. We observe that the density dependence offK- is far less pronounced than that off~. The changes offK+- with density combine with the variation of the masses in fig. 1 in just such a way that the GOR relation (14) is approximately satisfied even in dense matter. A detailed assessment of this result will be given elsewhere.

The splitting betweenfK- andfK+ seen in fig. 2 re-

f(9} 09 fK-

f(9=O) 0.8

05 f.\ \

\ I I I I I

0 1 2 density 9/9o (nm.)

Fig. 2. Decay constantsfx+ and fx- ofK + and K- mesons as a function ofdensityp. The corresponding result for the pion decay constant f~ (from ref. [10] ) is shown for comparison (dashed curve).

suits from the combined influence of Pauli effects and vector couplings. In the absence of vector and U ( 1 )A breaking interactions, the leading density dependence in the difference between the kaon decay constants is

3p +O(mo) (15) fl~--fK+ = 4mKft¢

with mK andfK in the denominator of the RHS, taken at p = 0 (inclusion of the vector interaction reduces the RHS by roughly a half). Furthermore we find that dfK+ + dfK- = df~ to leading order in the current quark masses (together with Go = 0). One should note that the asymmetry introduced by pu=pd#Ps implies an additional explicit breaking of chiral SU(3). The proportionality to my i in eq. ( 15 ) is thus not in con- flict with chiral perturbation theory.

From the behaviour of rnK, in fig. 1 there does not seem to be a tendency towards kaon condensation. This is apparently in contrast to the scenario suggested in refs. [2,3 ] and based on an attractive S-wave KNinteraction which causes a downward shift of both the K + and K - mass:

~KN dm~=m2K(p) --m2K(p=0) ~ -- ~--K p . (16)

Here XKN ~ 0 = ~ (mu + m °) (NI z2u+~slN) is the kaon- nucleon sigma term. It determines the variation of the condensate ( a u + g s ) as a function of density. In the NJL model the leading order contribution is given by d( ~u+gs) = ( ul~u+~slu)pu+ ( d[au+ gsfd)pa

32


= (N] au+gslN)p, where the last equal i ty follows in any consti tuent quark model of the nucleon. One finds for the relat ive change o f this condensate:

d(au+gs) ~KN (au+~s) ~ m~f2K p" (17)

Now, our result impl ies that the mass shifts din/c_+ are not jus t given by the change o f the condensate (au+gs) , as would be suggested by eq. (16) . Changes o f k a o n masses and intr insic wave functions cooperate such that the following relat ions hold:

dmK+_ df~+_ 1 d (au+gs ) ~'KN mx fK 2 (au+gs) 2m2xf2K p '

(18)

to leading order in densi ty and current quark masses. Let us also ment ion that the densi ty dependence of

the kaon mass impl ied by eq. (16) would not be compat ib le with the existing K + - n u c l e u s elastic scat- tering da ta which permi ts a change o f m~+ by not more than a few percent a t p ~ ½Po [ 14].

In conclusion, a consistent NJL model study of both the mass shifts and decay constants of the K + and K - in dense mat te r suggests a smooth densi ty dependence o f these quant i t ies at least up to twice the density o f nuclear matter .

We have also invest igated the tempera ture dependence of kaon masses and decay constants and found negligible effects for T~< 100 MeV. At this poin t our results essentially agree with those o f ref. [ 6 ].

A final r emark concerns the (9--,K+K - decay in dense mat te r from the poin t of view of the present investigation. In ref. [ 10 ] we found that the densi ty dependence o f the ~ mass follows that o f its s- and g- quark consti tuents: dm~-O.O5m~(p /po ) . With m~+ + mK- a lmost constant forp~<po, the (~--.K+K - decay channel will then be closed a l ready at densi t ies slightly above ½Po.

We would like to thank the Inst i tute for Nuclear

Theory, Univers i ty of Washington, Seattle, for hos- pi ta l i ty and support .

References

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Kaons and strange quarks in dense matter