16 July 1998
.Physics Letters B 431 1998 254262
Modification of the f-meson spectrum in nuclear matter 1
F. Klingl, T. Waas, W. WeisePhysik-Department, Technische Uniersitat Munchen, D-85747 Garching, Germany
Received 4 September 1997; revised 19 March 1998Editor: J.-P. Blaizot
The vacuum spectrum of the f-meson is characterized by its decay into KK. Modifications of the KK-loops in baryonicmatter change this spectrum. We calculate these in-medium modifications taking both s- and p-wave kaon-nucleoninteractions into account. We use results of the in-medium K and K spectra determined previously from a coupled channelapproach based on a chiral effective Lagrangian. Altogether we find a very small shift of the f meson mass, by less than 10MeV at normal nuclear matter density r . The in-medium decay width of the f meson increases such that its life time at0rsr is reduced to less than 5 fmrc. It should therefore be possible to observe medium effects in reactions such as0pypfn in heavy nuclei, where the f meson can be produced with small momentum. q 1998 Elsevier Science B.V. Allrights reserved.
The study of in-medium properties of hadrons is a topic of continuing interest. Experiments planned at GSI . .HADES and running at CERN e.g. CERES detect dilepton pairs in high energy collisions of nuclei. Thisopens the possibility of investigating vector mesons in hot and dense hadronic matter. In order to explore suchmedium effects it is necessary that the vector mesons decay inside the hot and dense region of the collisionzone. The f meson with its small width of 4.4 MeV has a lifetime of about 45 fmrc which is obviously toolarge to observe any medium effects. On the other hand, we demonstrate in this note that medium modificationsare expected to increase the f width and shorten its lifetime to less than 5 fmrc at the density of normalnuclear matter. One then enters the range in which medium effects of slowly moving f-mesons could becomevisible. A reaction of particular interest is pyp fn in heavy nuclei. This process violates the OZI rule butsubstantial vf mixing makes the f production rate large enough to be well detectable. Such an experiment
w xcould be performed at GSI where a pion beam will be prepared for use in combination with HADES 1 .This paper presents a systematic calculation of the in-medium f meson self energy. We first review briefly
the vacuum properties of the f meson and discuss its self-energy. We then extend this self-energy to finitedensities by including the in-medium interactions of the decay kaons, taking into account both s- and p-waveinteractions of K and K with nucleons in nuclear matter. In the final part a brief summary and discussion of theresults will be given.
1 Work supported in part by GSI and BMBF.
0370-2693r98r$ see frontmatter q 1998 Elsevier Science B.V. All rights reserved. .PII: S0370-2693 98 00491-2
( )F. Klingl et al.rPhysics Letters B 431 1998 254262 255
2. The f meson in vacuum
The f meson is observed as a pronounced resonance in the strange quark sector of the electromagneticw xcurrent-current correlation function 2 . The Fourier transform of the correlation function is
4 i qP x < < :P q s i d x e 0 TT j x j 0 0 , 1 . . . .Hmn m nwhere in the present context j represents the strange quark current,m
1j sy sg s . 2 . .m m3Current conservation leads to a transverse tensor structure
q qm n 2P q s g y P q , 3 . . .mn mn 2 /q12 m . .which defines the scalar function P q sy P q . The low energy spectrum of the correlation function ism3 . w xwell described by Vector Meson Dominance VMD . We use our improved VMD approach of Ref. 4 which
ovac 2 2 2Im P q 1ya q ym . .f f f2Im P q s . 4 . . o2 2 2 vac 2g q ym yP q .f f fo Here we have introduced the bare mass m of the f meson, g sy3gr 2 ,y14 is its strong couplingf f
q yconstant, and a a constant which describes deviations from universality of the fe e and fKKfcouplings. This constant is close to unity, i.e. deviations from universality are small. For our present purposeextreme fine-tuning is not necessary and we can set a s1.f
The vacuum self-energy P vac of the f meson consists of three parts:fP vac sP q yvac qP 0 0vac qP vac , 5 .f f K K f K K f 3pL S
describing the coupling of the f to the KK and three-pion channels. The last term violates the OZI rule, butdespite the small vf mixing angle it contributes about 15 percent to the total f meson decay width. We
w xinclude its imaginary part as given in Ref. 4 but focus here on the more important parts of the self-energy .coming from the decay into KK channels. They are related by SU 3 to the rypp self energy and can be
w xwritten in the form of a one-loop integral 4,5 , up to subtraction constants:22 4yig d l 2 lyq 8 .
vac 2q yP q s y . 6 . . Hf K K 4 2 222 2 26 l ym q ie2p l ym q ie lyq ym q ie . . K . .K K
q y .The first term of the integrand involves a propagating K K pair; the second tadpole term ensures gaugeinvariance at the level of the hadronic effective theory. Here we have introduced the strong meson couplinggs6.5 and the charged kaon mass m s493 MeV. Evaluating this integral and applying regularization using aK
w xsubtracted dispersion relation 4 we get2g
vac 2 2 2 2 2 2q yRe P q sc q y q GG q ,m y4m , 7 . . .f K K 0 K K248p
32 2g 4m 2Kvac 2 2 2 2
q yIm P q sy q 1y Q q y4m , 8 . . .f K K K2 /96p q
( )F. Klingl et al.rPhysics Letters B 431 1998 254262256
w x 2where the subtraction constant c s0.11 has been fixed to give a best fit to data as explained in Ref. 4 . This0o . . w xleads to a bare mass m s910 MeV in Eq. 4 . In Eq. 7 we insert 4f
3 2 2(4m q2 2 2y1 arcsin 0-q -4m2 / 2 mq24m2 2 ~GG q ,m s . 9 .3 . 1q 1y( 22 q4m 21 2 2 2y 1y ln 4m -q or q -02 2 2 /q 4m
1y 1y( 2 qvac . . 0For P the same expressions as in Eqs. 7 , 8 hold, with the charged kaon mass m replaced by m .f K K K KS L
. 2 .Using these self-energies as input in Eq. 4 we plot the spectrum of the vacuum correlation function P q ino2 2 2 vac 2 y1 . . w .xFig. 4a dashed line . We also show the real part of the f meson propagator D q s q ym yP qf f f
.in Fig. 4b dashed line . The zero of Re D determines the physical mass of the free f meson.f
3. The f meson in medium
We choose a Lorentz frame with nuclear matter at rest. In the following we consider the case with the f ..meson at rest qs v,qs0 , so as to determine its in-medium mass. The tensor structure of the correlation
function then reduces to a term proportional to the spacelike Kronecker symbol d . All time components musti j1 ivanish, and one can single out a scalar function by taking the trace Ps P . The spectral function has a formi3
w x 2 2analogous to that in the vacuum 6,7 . One only needs to replace q by v and the vacuum self-energy by the 2 .in-medium self energy P v , r of the f meson, withf
2o2 2 2Im P v ,r 1ya v ym . .f f f2Im P v ,r s , 10 . . o2 2 2 2g v ym yP v ,r .f f f
where we use a s1 again as a good approximation. The difference between the vacuum and the in-mediumfw xself-energy defines the density dependent effective f-nucleon amplitude TT , as follows 7 :fN
rTT v ,r sP vac v 2 yP v 2 ,r . 11 . . . .fN f f .To leading order in density r this quantity reduces to the free forward f-nucleon scattering amplitude T vfN
. w xat qs0 , as implied by a general low-density theorem 15 , and we write in this approximation:P v ,qs0;r sP vac v 2 yr T v q . . . , 12 . . . .f f f N
where the dots represent terms of higher order in density.The primary modification of the self-energy P comes from the interactions of the intermediate K and Kf
.mesons with nucleons in the nuclear medium. The kaon propagators in Eq. 6 are then to be replaced by thein-medium propagators,
1 1"D l , l ; r s , 13 . .K 02 2 2 2 2
" " "l ym q ie l y l ym yS l , l ; r .K 0 K K 0
2 w xWe note in passing that a systematic chiral approach would have to include explicit axial vector meson degrees of freedom 16 .generalized to SU 3 . Their contributions to the imaginary part of the f meson self-energy is not expected to be significant, however, and
we prefer to absorb these features into the subtraction constant which is chosen to reproduce data.
( )F. Klingl et al.rPhysics Letters B 431 1998 254262 257
q y "where S are the K and K self-energies in nuclear matter or, correspondingly, those of K and K whereK 0 0.applicable . The kaon propagators have the following spectral representations at fixed kaon three-momentum l
" .A u , l ; r A u , l ; r . .0 0
"D l , l ; r s du y , 14 . .HK 0 0 /l yu q ie l qu y ie0 0 0 0 0with
yIm S " l , l ; r rp .K 0"A l , l ; r s . 15 . .0 2 22 2 2 " "l y l ym yRe S l , l ; r q Im S l , l ; r . .0 K K 0 K 0 .Eq. 14 includes the crossing symmetry relation
D " l , l ; r sD . yl , l ; r . 16 . . .K 0 K 02 2 .For fixed l we now substitute u s u q l in the integral 14 and get0
2 " 2 2 . 2 2 du A u q l , l ; r A u q l , l ; r . ."D l , l ; r s y . 17 . .HK 0 2 2 2 2 2 22 /yl 2 u q l l y u q l q ie l q u q l y ie0 0
The fKqKy in-medium self-energy for a f at rest is given by2 i g 2 d4 l
2q y q yP v , qs0; r s l D l , l ;r D vy l ,y l ;r q tadpole, 18 . . . .Hf K K K 0 K 043 2p .
where the tadpole part, not written explicitly, contributes only to the real part of P . Using the representationf . q17 for the kaon propagators the self-energy consists of four terms. For example, the first term for the K andKy becomes
2 2 2ig d l du du ly q1.q yP v , qs0; r s . H H Hf K K 4 2 2 2 22 26 yl yl2p . ( (u q l u q ly q
q 2 2 y 2 2( (A u q l , l ; r A u q l , l ; r / /q y. 19 .
2 2 2 2( (l y u q l q ie vy l y u q l q ie / /0 q 0 y .The remaining terms have similar forms. For positive v, only the term 19 contributes to the imaginary part
which is now easily evaluated 3:Im P q y v , qs0; r .f K K
q 2 2 y 2 22 3 2 2 2 ( (A u q l , l ; r A u q l , l ; r g d l du du l / /q yy qs ImH H H3 2 2 2 2 2 2 2 22 26 yl yl2p . ( ( ( (u q l u q l y u q l y u q l qvq iey q q y322 2 2 2
yg l v ,u ,u .q y2 2 q 2 2 y 2 2( (s du du A u qk , k ; r A u qk ,k ; r , 20 .H H / /q y q y 42 296p vyk yk1 2 2 2 2 2 22< < . .with k sl v ,u ,u r2v, where l a,b,c sa qb qc y2 aby2 acy2bc is the Kallen function. Forq y
3 Small vertex corrections, not treated explicitly, are taken care of indirectly by considering only the space components of Im P inaccordance with current conservation.
( )F. Klingl et al.rPhysics Letters B 431 1998 254262258
Fig. 1. Diagrams contributing to the leading s- and p-wave interactions of kaons in nuclear matter. The hyperon intermediate states Yinclude those of the baryon octet and decuplet.
.negative v the second term of Eq. 14 contributes to Im P . Crossing symmetry implies that the resulting .expression is the same as Eq. 20 .
Let us now first discuss contributions to Im P from s-wave KN and KN interactions for which the leadingfprocess is illustrated in Fig. 1a. The corresponding in-medium kaon spectral functions are determined using the
. w xcoupled channels approach based on the chiral SU 3 effective Lagrangian as described in Refs. 8,9 . Thisapproach successfully reproduces all available low-energy data of the KN as well as the coupled KN, pS, pL
w x .and hN, K L, K S systems. For alternative approaches see Refs. 1114 .For the Kq and K 0 modes in matter one finds that the spectral functions can be well approximated by a
w . w .q 0d-function with the free kaon mass replaced by m r sm r :K Kq 2 2 2 w 2 qA u q l , l ; r sd u ym r . 21 . . . . K
At rsr s0.17 fmy3 we find m qw ,535 MeV, nearly independent of l.0 K0yFor the K and K modes the spectrum generated by s-wave KN interactions has more interesting features.We give examples in Fig. 2 at rsr for two energies v of the external f meson. The KN system decays into0
0ypL and pS. The spectral functions A therefore have finite widths. The second interesting point is theK , K
y y3 yFig. 2. Spectral function A of K modes in nuclear matter at density r s r s0.17 fm for two energies v s1.0 GeV and v s1.2K 02 2 2. .GeV of the primary f meson taken at rest , as a function of the squared invariant mass u s k y k . The calculation includes s-wave KN0
. w xinteractions derived from chiral SU 3 dynamics with coupled channels 8,9 .
( )F. Klingl et al.rPhysics Letters B 431 1998 254262 259
( )F. Klingl et al.rPhysics Letters B 431 1998 254262260
.Fig. 3. Real and imaginary parts of the effective fN scattering amplitude T v,qs0;r in nuclear matter at density r s r . All s- andfN 0p-wave KN and KN interactions are included.
. . Fig. 4. a Spectrum of the strange quark current-current correlation function in vacuum dashed and in nuclear matter at r s r solid0. q y q y . q ycurve . The normalization is chosen such that the vacuum result can be compared directly to the ratio s e e K K rs e e
q y . w x . .m m for which data are taken from Ref. 10 . b Real part of the f meson propagator in vacuum dashed and in nuclear matter at r s r0 .solid curve .
( )F. Klingl et al.rPhysics Letters B 431 1998 254262 261
where the subtraction constant c s0 is fixed by the Thomson limit of the Compton amplitude involving the1 .strange quark current 2 .
Our results for the real and imaginary parts of T at rsr are shown in Fig. 3. Note that the lower plateaufN 0in Im T comes mainly from the fNK L channel, with an intermediate p-wave KNL coupling. ThefNsteep rise of Im T just above 0.9 GeV has two major contributions. About half of it comes from fNKKNfNwith s-wave interactions of the K and K in the nuclear medium. Roughly the other half originates from thefNK S ) process. The complexity of the low energy fN dynamics translates visibly into a highly structuredpattern for Re T below and around the f resonance.fN
We mention that s-wave interactions of the K in the medium must be treated to all orders in the density, evenat small r. In contrast, the iteration of p-wave interactions to higher orders in r has a significant effect only atenergies above the f resonance, at least for densities rQr . We estimate the uncertainties in the high energy0
.part of Im T at v)m to be at the 20-30 % level. However, this uncertainty has very little influence on thefN fin-medium spectral function of the f meson shown in Fig. 4a. In any case, the resulting in-medium mass shiftof the f stays within 1 % of its free mass.
.The predicted spectrum of the strange current-current correlation function, Eq. 10 , is shown at rsr in0Fig. 4 together with the real part of the in-medium f meson propagator,
1D v , qs0; r s . 27 . .f o2 2v ym yP v , qs0; r .f f
One observes a very small and insignificant shift of the resonance position by about 1 % of its free mass at.rsr . The primary in-medium effect is the broadening of the resonance due to inelastic fN reactions. Its0
width at v,m ,fIm Pf
G sy , 28 .fmf
.reaches G ,45 MeV at rsr and exceeds the free vacuum width of 4 MeV by about an order off 0magnitude. The f meson life time is reduced to
t sGy1 ,4.4 fmrc at rsr . 29 .f f 0Conseque...