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The structure of the ϱ-meson in a Nambu-Jona-Lasinio approach beyond the mean-field approximation
Physics Letters B 272 (1991) 190-195 North-Holland
PHYSICS LETTERS B
The structure of the p-meson in a Nambu-Jona-Lasinio approach beyond the mean-field approximation S. Krewald, K. N a k a y a m a ~ and J. Speth 2 Institute ~'ir Kernphysik, Forschungszentrum Jiilich, H'-5170 Jiilich, FRG Received 3 April 1991; revised manuscript received 24 September 1991
The interpretation of mesons as pure q(t-states suggested by the Nambu-Jona-Lasinio model at the Hartree level is criticised. An extension of the model beyond the mean-field approximation is suggested for the vector-isovector channel. In the vicinity of the p-meson pole, the quark-antiquark contribution to the total polarization of the vacuum is found to be less than 50%.
Within the last few years, the lagrangian of N a m b u and Jona-Lasinio ( N J L ) [ 1 ] has been interpreted as an effective quark dynamic lagrangian and has been generalized to include interactions up to three flavours. Observables both in the mesonic [ 2-9 ] and in the nucleonic [ 10,11 ] sectors have been successfully described. The model shares one important feature of quantum chromodynamics, chiral symmetry, but does not produce asymptotic freedom, shows no confinement and has to be regularized. It has been pointed out, however, that despite the absence of confinement, the mass pattern for pseudoscalar, vector, and axial vector mesons agrees within 15% error with the experimental findings [3]. The masses of mesons above the quark-antiquark emission threshold have been identified with the positions of the resonances in the underlying quark-antiquark continuum [7,9,12,13]. The model has even been used to generate the form factors related to the quark currents in both space-like and time-like regions. In the case of the electromagnetic form factors of the pion, the important qualitative features of the experimental data have been reproduced [ 14,9,15 ]. A displeasing feature of NJL-type calculations performed in the mean-field approximation is that both the structure of the p-meson and the electromagnetic form factor
Also at: Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA.
190
of the pion are determined by the emission of quarks into the continuum. In order to remedy this situation, one has to go beyond the Hartree level. In other words, mesons should not be considered as pure quark-antiquark @1 states, but in principle must contain admixtures of q2cl2 states and even more complicated configurations, which allow the decay into colour-neutral objects. This is demanded by unitarity [ 16 ]. The dispersion relations can now be satisfied with physical colour singlet intermediate states generated by the non-qCl components of the meson wavefunctions. Here, we want to investigate whether a many-body approach beyond the Hartree level is compatible with the quark-antiquark interpretation of mesons, or if qualitative changes of the structure of mesons result. We concentrate on the vector-isovector channel which can be described with only two flavours. Thus, the number of free parameters is minimized in the present exploratory study. So far, all experimentally confirmed mesons are interpreted as members of qcl multiplets [ 17 ]. There are only very few non-qCl candidates [17]. The fo (975) meson might be a two-quark-two-antiquark state [ 18 ], or even more likely a kaon-antikaon molecule [ 19,20 ]. If one is exclusively interested in the widths of mesons, an explicit inclusion of non-q~t components of the meson wavefunctions can be avoided by bosonizing the NJL model via a heat-kernel expansion and
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 272, number 3,4
PHYSICS LETTERSB
thus generating effective lagrangians, which incorporate non-linear pseudoscalar and vector meson couplings, like the gauged non-linear sigma model. The width determined by the KSRF relation [21 ] is FKsRv= 140 MeV. The disadvantage of treating bosons as elementary fields is that any information about the intrinsic quark structure of the mesons is lost. We therefore keep the explicit quark dynamics at least in the vector-isovector channel. As long as one works with NJL-type lagrangians, the deconfinement of quarks cannot be avoided. The influence of quark emission processes can be minimized, however, if one realizes that the constituent quark mass is not well-determined at all. By choosing a sufficiently large constituent quark mass, the p-meson can be stabilized against the emission of quarks, as has been pointed out in refs. [7,8]. The relatively large constituent quark masses employed in refs. [7,8] have been criticised recently as a disturbing feature [22]. We would like to point out, however, that an even larger constituent quark mass has to be expected, as soon as one goes beyond the Hartree approximation. From a many-body theoretic point of view, the structure of mesons as described in the NJL model shows many analogies to the structure of nuclear excitations, if one replaces the vacuum by the ground state of the nucleus and the quarks (or antiquarks) by the particle (or hole) states of the nuclear shell model. The constituent quark mass, in this analogy, corresponds to the single particle energy in the shell model. As long as one works in a pure meanfield approximation, the single particle energies of the nuclear shell model may be identified with the experimental separation energies. If one goes beyond the mean-field approximation, the bare single particle energies must be increased, however, because the single particle states are dressed by the coupling to collective nuclear excitations which cause a considerable compression of the particle-hole energy gap [2325 ]. In the quark dynamical many-body problem, a similar effect is observed, because quarks are dressed by pions. The enhanced constituent quark mass helps to reduce the influence of deconfinement. Quark masses larger than a third of the mass of the nucleon have emerged even at the mean field level after introducing different cut-offs for the logarithmic and for the quadratic divergences [26]. Values up to
5 December 1991
mq = 1115 MeV have been used in the pseudoscalar nonet [27]. One basic problem of many-body theory is how to define suitable approximation schemes. An expansion in terms of the coupling constant obviously does not work for strong interactions. Within the framework of effective theories, such as the NJL model, a more adequate approach is to consider explicitly only those processes which show a rapid dependence on the total energy within the energy range under investigation and to include all other processes implicitly by a suitable choice of the parameters of the effective lagrangian. In this way, one is able to correlate classes of experimental observables, such as collective excitations of many-body systems. In a study of the vector-isovector channel below 1 GeV, a truncation at the qZq2 level seems reasonable, since the p-meson decays nearly exclusively into two pions. The explicit consideration of medium polarization effects beyond the mean field approximation is quite an involved problem. In the nuclear many-body problem, a successful approximation scheme is given by the well-known Bohr-Mottelson model which couples single particle states to collective degrees of freedom, i.e. the phonons [28]. In the quark dynamic many-body problem, an analogous approximation is to treat the pion as an elementary boson which couples to the quarks. Since we investigate the vector-isovector channel, a bosonization of the quark degrees of freedom in the scalar-pseudoscalar channel does not lead to double counting, as is known from nuclear field theory [29]. In NJL-type models at the Hartree level, an axial vector coupling term in the lagrangian induces a mixing between the pion and the a~-meson. This leads to a reduction of the quark axial-vector coupling constant ga and to an increase of the radius of the pion by approximately 10% [ 15 ]. Since we are primarily interested in the structure of the o-meson, we want to keep the structure of the pion as simple as possible. We therefore assume that the effect of the axial-vector coupling is already included in the bosonized pion. The model lagrangian employed in the present investigation is a hybrid one composed of the Gell-Mann-Levy sigma model [ 30 ] for the pseudoscalar interaction and a point-like fourfermion coupling for the vector interaction: 191
Volume 272, number 3,4
PHYSICS LETTERS B
~ - 9 ~/.)i~';'O/,~/"l- ~'~PGelI-M..... Levy'( 0", II')
(~,y 75r~u) ] , Y6~I~-M~.n L~y(~, ~) = --g~7(a+iTsr'~)~'
+ ½[ (a,,a)-' + (a,,n)=]
f ;~
"
m o -
m ;~ ]
(1) m~, -- m~
Now the question arises how to treat processes beyond the one-loop diagrams of the mean-field approximation. In a renormalizable theory, the procedure is clear. In a first step, one has to regularize all diagrams considered; in a second step, the divergent contributions can be absorbed by the renormalization. Form factors, defined by the vertex functions, emerge from the finite contributions only after the renormalization has been performed. We apply this procedure to the lagrangian eq. ( 1 ). This implies that a cut-off has to be attached also to those diagrams which contain pions as intermediate states. The lagrangian eq. (1) is not renormalizable, and therefore, the results depend explicitly on the cut off parameters, as is the case in all the NJL-calculations of vector mesons. The bosonization approximation leading to the effective lagrangian eq. ( 1 ) implies that a pion-quark-quark form factor like the one given by the NJL model at the Hartree level is no longer generated. Now one has to realize that in a theory with confined quarks, the meson-quark-antiquark vertices must receive important contributions from tchannel interactions, because form factors generated exclusively at the Hartree level by the iteration of quark-antiquark s-channel polarizations do not satisfy the dispersion relations. For this reason, we do not try to include a pion-quark-quark form factor derived at the mean-field level. The effect of replacing the cut-off employed in the present calculation by a pion-quark-antiquark form factor in diagrams involving pions is to enhance the contribution of the non-q~l polarization of the vacuum. Our model for the polarization of the vacuum in the vector-isovector channel is summarized in fig. 1 (left panel). In addition to the quark-antiquark polarization (fig. lb), a coupling to a 7t+~ - pair via 192
5 December 1991
quark triangles (fig. lc) is considered. The diagram (c) by itself is not gauge invariant. In the limit of infinitely heavy quarks, the diagram (d) reduces to a tadpole which is required for gauge invariance in a theory with only mesonic degrees of freedom. In the present calculation, we do not work out (d) explicitly, but ensure the transversality of the vector polarization by a subtraction. Meson exchanges in the tchannel are neglected in order to simplify the present approach. The electromagnetic form factor of the pion has the following contributions (fig. 1b ): (i) a bare photontwo-pion coupling via a quark triangle. This term does not depend on the strength g2 of the vector interaction in the lagrangian eq. (1). (ii) The photon couples via a quark-antiquark doorway to the p-meson, (iii) the photon couples via a x * ~ - doorway to the p-meson via two-quark triangles. The coupling of the photon to the analogue of (d) is effectively included in (iii) by the subtraction. The vector-isovector polarizability H"" due to two intermediate pions is given by f d4k , -iH,"(q=) = j ~ E, (k, q)D~(k+ ½q)
×D~(k-½q)E"(k, - q ) .
(2)
Here, E;'(k, q) denotes the coupling of the vectorisovector current to a rc+~ - pair via a quark triangle (see fig. lb, right panel (i)), while D~(k) denotes the pion propagator. As regularization, euclidean sharp cut-offs have been used both in three and four dimensions. The present model has three parameters, the constituent quark mass m q , the euclidean sharp cut-off R, and the vector coupling constant g2. The cut-offR is determined from the charge normalization as in ref. [ 14 ], while the vector coupling constant g~ gives the mass of the 9-meson. This leaves the constituent quark mass as a free parameter. In fig. 2, the electromagnetic form factor of the pion in the time-like region is compared with the data of Barkov et al. [ 31 ]. Employing a covariant cut-off of R = 666 MeV and a coupling constant g22= 5.0 GeV -z, a quark mass of mq = 417 MeV is required in order to reproduce the height of the cross section at the mass of the p-meson. The radius of the pion is found to be ( r ~ ) J / 2 = 0 . 5 4 fro, which is 18% smaller than the experimental value
Volume 272. number 3,4
PHYSICS LETTERS B
5 December 1991
/ //
p \\
(~)
(b)
7T \
(i)
"/T
+ (.)
(c)
, i
+
y
J
(d)
I
_y__
_~
(i.)
Fig. 1. (left panel ) The effective four-quark interaction in the vector-~sovector channel decomposed into an elementary point-like fourquark interaction (a), a quark-antiquark polarization (b), a two-pion polarization (c), and a selfenergy term (d). (right panel) The photon-two-pion coupling decomposed into an elementary coupling via a quark triangle (i), a quark-antiquark doorwa~ (ii), and a two-pion doorway ( iii ).
50[ £]ectromcgnetic c~ 40 form factor , of the pion
~
I soo 46o s6o 6oo 7oo 80o 900
2 [ (MeV)
Fig. 2. The electromagnetic form factor of the pion in the timelike region is shown as a function of the center of momentum energy 2E. Results are displayed for a constituent quark mass mq= 417 MeV (dashed), employing a four-dimensional euclidean sharp cut-off; and for mq=689 MeV, using a three-dimensional cut-off (solid). o f 0.66 fro. Since the threshold for the emission o f quarks comes at 834 MeV in the present calculation, the width o f the p-meson Fp===63 MeV is entirely d e t e r m i n e d by the decay into pions. Still one might
worry whether the results are sensitive to the presence of the unphysical decay channel in the near vicinity o f the v e c t o r - i s o v e c t o r resonance. In order to check this point, we performed another calculation with a larger constituent quark mass of mq = 689 MeV, employing a three-dimensional euclidean sharp cutoff R = 6 8 9 MeV and a coupling constant g~ = 7 . 3 G eV -2. N o w the quark emission threshold is energetically well separated from the p-meson. The results obtained, however, are qualitatively very similar to the ones obtained with the smaller constituent quark mass. The pion radius remains unchanged, while the width is F o ~ = 81 MeV. The model discussed here underestimates the experimental width F = 150 MeV [31 ]. In the present calculation, the t-channel interactions have been neglected. In the meson-exchange model for p i o n - p i o n scattering recently developped in ref. [20], the addition o f a t-channel p-meson exchange between the two pions has enhanced the width o f the p-meson by approximately 50%. If one increases the constituent 193
Volume 272, number 3,4
PHYSICS LETTERS B
quark mass, also the p i o n - q u a r k coupling constant via the G o l d b e r g e r - T r e i m a n relation, and therefore also the effect of the t-channel increases. One has to realize that the inclusion of tchannel meson exchanges is technically by far more complicated than the present model. A degeneracy of both the p-meson and co-meson masses is obtained in the NJL model, if one identifies the vector-isovector and the vector-isoscalar coupling strengths. In the present extension of the N a m b u model, such a degeneracy appears to be lost, because after turning offthe ~-~ interaction, the p-meson mass moves to 834 MeV for the covariant cut-off. One must keep in mind, however, that the m-meson may have non-q¢t c o m p o n e n t s as well. Because of G-parity conservation, the c0-meson mainly decays into three pions. Therefore one has to expect some q3ct3 contributions to the o>meson which may be estimated by considering p-~ interactions and which can induce a shift of the bare ~0-mass. In the NJL model at the Hartree level, the polarizability of the v a c u u m is entirely d e t e r m i n e d by the q u a r k - a n t i q u a r k degree of freedom. The present model treats both quark and colour singlet degrees of freedom on an equal footing. The importance of the quark degrees of freedom in polarizing the m e d i u m is given by the ratio P of the polarizability of the vacu u m due to quarks to the total polarizability:
g~qqincreases
P-
/Tqq (q2)
Hqq(q2)+H~+~ (q2)"
(3)
Here, H is defined via Hu, = (q~,q,- qZg,u ~ ) H. For both quark masses employed in fig. 2, the ratio P is displayed in fig. 3. The qualitative result obtained in the present calculation is that at least in the vector-isovector channel, it is not the q u a r k - a n t i q u a r k configuration which dominates the structure of the p-meson, but rather, colour singlet degrees of freedom contribute more than 50% to the polarizability of the vacuum. A major ~ + ~ - c o m p o n e n t in the p-meson has important consequences for nuclear physics. Inside a nuclear m e d i u m , the pion is polarized by the excitation of the A33resonance, which causes a density-dep e n d e n t modification of the effective mass of the pion. The mass of the p-meson, in turn, is lowered with increasing density via the two-pion c o m p o n e n t of the p-meson. This finding may have significant ef194
5 December 1991
5040-
Quark c o n t r i b u t i o n to the p o l a r i z a b i l i t y
..................... . . _ _
o~..?50-
~'2o100 5OO
460
6do 7do soo 9o0
2 £ ( MeV )
Fig. 3. The quark contribution to the polarizability of the vacuum in the vector-isovector channel is shown for the two calculations of fig. 2. fects on the equation of state of nuclear matter [ 32 ]. We thank Jifi Ho~ek and H. Reinhardt for useful discussions.
References [ 1] Y. Nambu and G. Jona-Lasinio,Phys. Rev. ! 22 ( 1961 ) 345; 124 (1961) 246. [2]A. Dhar, R. Shankar and S.R. Wadia, Phys. Rev. D 31 (1985) 3256. [3] D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188. [4] T. Kunihiro and T. Hatsuda, Phys. Lett. B 206 (1988) 385. [ 5] H. Reinhardt and R. Alkofer, Phys. Lett. B 207 (1988) 482. [6] V. Bernard, R.L. Jaffe and U.G. Meissner, Nucl. Phys. B 308 (1988) 753. [7] V. Bernard and U.G. Meissner, Nucl. Phys. A 489 (1988) 647. [8 ] M. Takizawa, K. Tsushima, Y. Kohyama and K. Kubodera, Nucl. Phys. A 507 (1990) 611. [9] S. Klimt, M. Lulz, U. Vogel and W. Weise, Nucl. Phys. A 516 (1990) 429; U. Vogl, M. Lutz, S. Klimt and W. Weise,Nucl. Phys. A 516 (1990) 469. [10] D. Dyakonov and Y.V. Petrov, Nucl. Phys. B 227 (1986) 457. [ll]Th. Meissner, E. Ruiz-Arriola, F. Gruemmer, H. Mavromatis and K. Goeke, Phys. Len. B 214 (1988) 312. [ 12 ] A.H. Blin, B. Hiller and J. da Providencia, Phys. Lett. B 241 (1990) 1. [13] E.M. Henley and H. Muether, Nucl. Phys. A 513 (1990) 667. [ 141 V. Bernard and U.G. Meissner, Phys. Rev. Lett. 61 (1988) 2296. [ 15] M. Lutz and W. Weise, Nucl. Phys. A 518 (1990) 156. [ 16] D. Toernqvist, Ann. Phys. (NY) 123 (1979) 1.
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[ 17 ] Particle Data Group, J.J. Hern~indezet al., Review of particle properties, Phys. Lett. B 239 (1990) 1. [ 18] R. Jaffe, Phys. Rev. D 15 (1977) 267. [ 19] J. Weinstein and N. Isgur, Phys. Rev. D 41 (1990) 2236. [20] D. Lohse, J.W. Durso, K. Holinde and J. Speth, Nucl. Phys. A516 (1990) 513. [21] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16 (1966) 255; Riazuddin and Fayazuddin, Phys. Rev. 147 (1966) 1071. [22] M. Takizawa, K. Kubodera and F. Myhrer, Phys. Lett. B 26l (1991) 221. [23] P. Ring and E. Werner, Nucl. Phys. A 21 l (1973) 198. [24] I. Hamamoto, Phys. Rep. 10 (1974) 63.
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[25] J.P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rep. 25 (1976) 83. [26] M. Jaminon, G. Ripka and P. Stassart, Phys. Lett. B 227 (1989) 191. [27] R. Alkofer and I. Zahed, Phys. Lett. B 238 (1990) 149. [28]A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II (Benjamin, New York, 1965 ). [29] H. Reinhardt, Nucl. Phys. A298 (1978) 77. [30] M. GeU-Mann and M. Levy, Nuovo Cimento 16 (1960) 705. [31 ] L.M. Barkov et al., Nucl. Phys. B 256 (1985) 365. [32] G.E. Brown, Nucl. Phys. A 522 ( 1991 ) 397.
195
PHYSICS LETTERS B
The structure of the p-meson in a Nambu-Jona-Lasinio approach beyond the mean-field approximation S. Krewald, K. N a k a y a m a ~ and J. Speth 2 Institute ~'ir Kernphysik, Forschungszentrum Jiilich, H'-5170 Jiilich, FRG Received 3 April 1991; revised manuscript received 24 September 1991
The interpretation of mesons as pure q(t-states suggested by the Nambu-Jona-Lasinio model at the Hartree level is criticised. An extension of the model beyond the mean-field approximation is suggested for the vector-isovector channel. In the vicinity of the p-meson pole, the quark-antiquark contribution to the total polarization of the vacuum is found to be less than 50%.
Within the last few years, the lagrangian of N a m b u and Jona-Lasinio ( N J L ) [ 1 ] has been interpreted as an effective quark dynamic lagrangian and has been generalized to include interactions up to three flavours. Observables both in the mesonic [ 2-9 ] and in the nucleonic [ 10,11 ] sectors have been successfully described. The model shares one important feature of quantum chromodynamics, chiral symmetry, but does not produce asymptotic freedom, shows no confinement and has to be regularized. It has been pointed out, however, that despite the absence of confinement, the mass pattern for pseudoscalar, vector, and axial vector mesons agrees within 15% error with the experimental findings [3]. The masses of mesons above the quark-antiquark emission threshold have been identified with the positions of the resonances in the underlying quark-antiquark continuum [7,9,12,13]. The model has even been used to generate the form factors related to the quark currents in both space-like and time-like regions. In the case of the electromagnetic form factors of the pion, the important qualitative features of the experimental data have been reproduced [ 14,9,15 ]. A displeasing feature of NJL-type calculations performed in the mean-field approximation is that both the structure of the p-meson and the electromagnetic form factor
Also at: Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA.
190
of the pion are determined by the emission of quarks into the continuum. In order to remedy this situation, one has to go beyond the Hartree level. In other words, mesons should not be considered as pure quark-antiquark @1 states, but in principle must contain admixtures of q2cl2 states and even more complicated configurations, which allow the decay into colour-neutral objects. This is demanded by unitarity [ 16 ]. The dispersion relations can now be satisfied with physical colour singlet intermediate states generated by the non-qCl components of the meson wavefunctions. Here, we want to investigate whether a many-body approach beyond the Hartree level is compatible with the quark-antiquark interpretation of mesons, or if qualitative changes of the structure of mesons result. We concentrate on the vector-isovector channel which can be described with only two flavours. Thus, the number of free parameters is minimized in the present exploratory study. So far, all experimentally confirmed mesons are interpreted as members of qcl multiplets [ 17 ]. There are only very few non-qCl candidates [17]. The fo (975) meson might be a two-quark-two-antiquark state [ 18 ], or even more likely a kaon-antikaon molecule [ 19,20 ]. If one is exclusively interested in the widths of mesons, an explicit inclusion of non-q~t components of the meson wavefunctions can be avoided by bosonizing the NJL model via a heat-kernel expansion and
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 272, number 3,4
PHYSICS LETTERSB
thus generating effective lagrangians, which incorporate non-linear pseudoscalar and vector meson couplings, like the gauged non-linear sigma model. The width determined by the KSRF relation [21 ] is FKsRv= 140 MeV. The disadvantage of treating bosons as elementary fields is that any information about the intrinsic quark structure of the mesons is lost. We therefore keep the explicit quark dynamics at least in the vector-isovector channel. As long as one works with NJL-type lagrangians, the deconfinement of quarks cannot be avoided. The influence of quark emission processes can be minimized, however, if one realizes that the constituent quark mass is not well-determined at all. By choosing a sufficiently large constituent quark mass, the p-meson can be stabilized against the emission of quarks, as has been pointed out in refs. [7,8]. The relatively large constituent quark masses employed in refs. [7,8] have been criticised recently as a disturbing feature [22]. We would like to point out, however, that an even larger constituent quark mass has to be expected, as soon as one goes beyond the Hartree approximation. From a many-body theoretic point of view, the structure of mesons as described in the NJL model shows many analogies to the structure of nuclear excitations, if one replaces the vacuum by the ground state of the nucleus and the quarks (or antiquarks) by the particle (or hole) states of the nuclear shell model. The constituent quark mass, in this analogy, corresponds to the single particle energy in the shell model. As long as one works in a pure meanfield approximation, the single particle energies of the nuclear shell model may be identified with the experimental separation energies. If one goes beyond the mean-field approximation, the bare single particle energies must be increased, however, because the single particle states are dressed by the coupling to collective nuclear excitations which cause a considerable compression of the particle-hole energy gap [2325 ]. In the quark dynamical many-body problem, a similar effect is observed, because quarks are dressed by pions. The enhanced constituent quark mass helps to reduce the influence of deconfinement. Quark masses larger than a third of the mass of the nucleon have emerged even at the mean field level after introducing different cut-offs for the logarithmic and for the quadratic divergences [26]. Values up to
5 December 1991
mq = 1115 MeV have been used in the pseudoscalar nonet [27]. One basic problem of many-body theory is how to define suitable approximation schemes. An expansion in terms of the coupling constant obviously does not work for strong interactions. Within the framework of effective theories, such as the NJL model, a more adequate approach is to consider explicitly only those processes which show a rapid dependence on the total energy within the energy range under investigation and to include all other processes implicitly by a suitable choice of the parameters of the effective lagrangian. In this way, one is able to correlate classes of experimental observables, such as collective excitations of many-body systems. In a study of the vector-isovector channel below 1 GeV, a truncation at the qZq2 level seems reasonable, since the p-meson decays nearly exclusively into two pions. The explicit consideration of medium polarization effects beyond the mean field approximation is quite an involved problem. In the nuclear many-body problem, a successful approximation scheme is given by the well-known Bohr-Mottelson model which couples single particle states to collective degrees of freedom, i.e. the phonons [28]. In the quark dynamic many-body problem, an analogous approximation is to treat the pion as an elementary boson which couples to the quarks. Since we investigate the vector-isovector channel, a bosonization of the quark degrees of freedom in the scalar-pseudoscalar channel does not lead to double counting, as is known from nuclear field theory [29]. In NJL-type models at the Hartree level, an axial vector coupling term in the lagrangian induces a mixing between the pion and the a~-meson. This leads to a reduction of the quark axial-vector coupling constant ga and to an increase of the radius of the pion by approximately 10% [ 15 ]. Since we are primarily interested in the structure of the o-meson, we want to keep the structure of the pion as simple as possible. We therefore assume that the effect of the axial-vector coupling is already included in the bosonized pion. The model lagrangian employed in the present investigation is a hybrid one composed of the Gell-Mann-Levy sigma model [ 30 ] for the pseudoscalar interaction and a point-like fourfermion coupling for the vector interaction: 191
Volume 272, number 3,4
PHYSICS LETTERS B
~ - 9 ~/.)i~';'O/,~/"l- ~'~PGelI-M..... Levy'( 0", II')
(~,y 75r~u) ] , Y6~I~-M~.n L~y(~, ~) = --g~7(a+iTsr'~)~'
+ ½[ (a,,a)-' + (a,,n)=]
f ;~
"
m o -
m ;~ ]
(1) m~, -- m~
Now the question arises how to treat processes beyond the one-loop diagrams of the mean-field approximation. In a renormalizable theory, the procedure is clear. In a first step, one has to regularize all diagrams considered; in a second step, the divergent contributions can be absorbed by the renormalization. Form factors, defined by the vertex functions, emerge from the finite contributions only after the renormalization has been performed. We apply this procedure to the lagrangian eq. ( 1 ). This implies that a cut-off has to be attached also to those diagrams which contain pions as intermediate states. The lagrangian eq. (1) is not renormalizable, and therefore, the results depend explicitly on the cut off parameters, as is the case in all the NJL-calculations of vector mesons. The bosonization approximation leading to the effective lagrangian eq. ( 1 ) implies that a pion-quark-quark form factor like the one given by the NJL model at the Hartree level is no longer generated. Now one has to realize that in a theory with confined quarks, the meson-quark-antiquark vertices must receive important contributions from tchannel interactions, because form factors generated exclusively at the Hartree level by the iteration of quark-antiquark s-channel polarizations do not satisfy the dispersion relations. For this reason, we do not try to include a pion-quark-quark form factor derived at the mean-field level. The effect of replacing the cut-off employed in the present calculation by a pion-quark-antiquark form factor in diagrams involving pions is to enhance the contribution of the non-q~l polarization of the vacuum. Our model for the polarization of the vacuum in the vector-isovector channel is summarized in fig. 1 (left panel). In addition to the quark-antiquark polarization (fig. lb), a coupling to a 7t+~ - pair via 192
5 December 1991
quark triangles (fig. lc) is considered. The diagram (c) by itself is not gauge invariant. In the limit of infinitely heavy quarks, the diagram (d) reduces to a tadpole which is required for gauge invariance in a theory with only mesonic degrees of freedom. In the present calculation, we do not work out (d) explicitly, but ensure the transversality of the vector polarization by a subtraction. Meson exchanges in the tchannel are neglected in order to simplify the present approach. The electromagnetic form factor of the pion has the following contributions (fig. 1b ): (i) a bare photontwo-pion coupling via a quark triangle. This term does not depend on the strength g2 of the vector interaction in the lagrangian eq. (1). (ii) The photon couples via a quark-antiquark doorway to the p-meson, (iii) the photon couples via a x * ~ - doorway to the p-meson via two-quark triangles. The coupling of the photon to the analogue of (d) is effectively included in (iii) by the subtraction. The vector-isovector polarizability H"" due to two intermediate pions is given by f d4k , -iH,"(q=) = j ~ E, (k, q)D~(k+ ½q)
×D~(k-½q)E"(k, - q ) .
(2)
Here, E;'(k, q) denotes the coupling of the vectorisovector current to a rc+~ - pair via a quark triangle (see fig. lb, right panel (i)), while D~(k) denotes the pion propagator. As regularization, euclidean sharp cut-offs have been used both in three and four dimensions. The present model has three parameters, the constituent quark mass m q , the euclidean sharp cut-off R, and the vector coupling constant g2. The cut-offR is determined from the charge normalization as in ref. [ 14 ], while the vector coupling constant g~ gives the mass of the 9-meson. This leaves the constituent quark mass as a free parameter. In fig. 2, the electromagnetic form factor of the pion in the time-like region is compared with the data of Barkov et al. [ 31 ]. Employing a covariant cut-off of R = 666 MeV and a coupling constant g22= 5.0 GeV -z, a quark mass of mq = 417 MeV is required in order to reproduce the height of the cross section at the mass of the p-meson. The radius of the pion is found to be ( r ~ ) J / 2 = 0 . 5 4 fro, which is 18% smaller than the experimental value
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/ //
p \\
(~)
(b)
7T \
(i)
"/T
+ (.)
(c)
, i
+
y
J
(d)
I
_y__
_~
(i.)
Fig. 1. (left panel ) The effective four-quark interaction in the vector-~sovector channel decomposed into an elementary point-like fourquark interaction (a), a quark-antiquark polarization (b), a two-pion polarization (c), and a selfenergy term (d). (right panel) The photon-two-pion coupling decomposed into an elementary coupling via a quark triangle (i), a quark-antiquark doorwa~ (ii), and a two-pion doorway ( iii ).
50[ £]ectromcgnetic c~ 40 form factor , of the pion
~
I soo 46o s6o 6oo 7oo 80o 900
2 [ (MeV)
Fig. 2. The electromagnetic form factor of the pion in the timelike region is shown as a function of the center of momentum energy 2E. Results are displayed for a constituent quark mass mq= 417 MeV (dashed), employing a four-dimensional euclidean sharp cut-off; and for mq=689 MeV, using a three-dimensional cut-off (solid). o f 0.66 fro. Since the threshold for the emission o f quarks comes at 834 MeV in the present calculation, the width o f the p-meson Fp===63 MeV is entirely d e t e r m i n e d by the decay into pions. Still one might
worry whether the results are sensitive to the presence of the unphysical decay channel in the near vicinity o f the v e c t o r - i s o v e c t o r resonance. In order to check this point, we performed another calculation with a larger constituent quark mass of mq = 689 MeV, employing a three-dimensional euclidean sharp cutoff R = 6 8 9 MeV and a coupling constant g~ = 7 . 3 G eV -2. N o w the quark emission threshold is energetically well separated from the p-meson. The results obtained, however, are qualitatively very similar to the ones obtained with the smaller constituent quark mass. The pion radius remains unchanged, while the width is F o ~ = 81 MeV. The model discussed here underestimates the experimental width F = 150 MeV [31 ]. In the present calculation, the t-channel interactions have been neglected. In the meson-exchange model for p i o n - p i o n scattering recently developped in ref. [20], the addition o f a t-channel p-meson exchange between the two pions has enhanced the width o f the p-meson by approximately 50%. If one increases the constituent 193
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quark mass, also the p i o n - q u a r k coupling constant via the G o l d b e r g e r - T r e i m a n relation, and therefore also the effect of the t-channel increases. One has to realize that the inclusion of tchannel meson exchanges is technically by far more complicated than the present model. A degeneracy of both the p-meson and co-meson masses is obtained in the NJL model, if one identifies the vector-isovector and the vector-isoscalar coupling strengths. In the present extension of the N a m b u model, such a degeneracy appears to be lost, because after turning offthe ~-~ interaction, the p-meson mass moves to 834 MeV for the covariant cut-off. One must keep in mind, however, that the m-meson may have non-q¢t c o m p o n e n t s as well. Because of G-parity conservation, the c0-meson mainly decays into three pions. Therefore one has to expect some q3ct3 contributions to the o>meson which may be estimated by considering p-~ interactions and which can induce a shift of the bare ~0-mass. In the NJL model at the Hartree level, the polarizability of the v a c u u m is entirely d e t e r m i n e d by the q u a r k - a n t i q u a r k degree of freedom. The present model treats both quark and colour singlet degrees of freedom on an equal footing. The importance of the quark degrees of freedom in polarizing the m e d i u m is given by the ratio P of the polarizability of the vacu u m due to quarks to the total polarizability:
g~qqincreases
P-
/Tqq (q2)
Hqq(q2)+H~+~ (q2)"
(3)
Here, H is defined via Hu, = (q~,q,- qZg,u ~ ) H. For both quark masses employed in fig. 2, the ratio P is displayed in fig. 3. The qualitative result obtained in the present calculation is that at least in the vector-isovector channel, it is not the q u a r k - a n t i q u a r k configuration which dominates the structure of the p-meson, but rather, colour singlet degrees of freedom contribute more than 50% to the polarizability of the vacuum. A major ~ + ~ - c o m p o n e n t in the p-meson has important consequences for nuclear physics. Inside a nuclear m e d i u m , the pion is polarized by the excitation of the A33resonance, which causes a density-dep e n d e n t modification of the effective mass of the pion. The mass of the p-meson, in turn, is lowered with increasing density via the two-pion c o m p o n e n t of the p-meson. This finding may have significant ef194
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5040-
Quark c o n t r i b u t i o n to the p o l a r i z a b i l i t y
..................... . . _ _
o~..?50-
~'2o100 5OO
460
6do 7do soo 9o0
2 £ ( MeV )
Fig. 3. The quark contribution to the polarizability of the vacuum in the vector-isovector channel is shown for the two calculations of fig. 2. fects on the equation of state of nuclear matter [ 32 ]. We thank Jifi Ho~ek and H. Reinhardt for useful discussions.
References [ 1] Y. Nambu and G. Jona-Lasinio,Phys. Rev. ! 22 ( 1961 ) 345; 124 (1961) 246. [2]A. Dhar, R. Shankar and S.R. Wadia, Phys. Rev. D 31 (1985) 3256. [3] D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188. [4] T. Kunihiro and T. Hatsuda, Phys. Lett. B 206 (1988) 385. [ 5] H. Reinhardt and R. Alkofer, Phys. Lett. B 207 (1988) 482. [6] V. Bernard, R.L. Jaffe and U.G. Meissner, Nucl. Phys. B 308 (1988) 753. [7] V. Bernard and U.G. Meissner, Nucl. Phys. A 489 (1988) 647. [8 ] M. Takizawa, K. Tsushima, Y. Kohyama and K. Kubodera, Nucl. Phys. A 507 (1990) 611. [9] S. Klimt, M. Lulz, U. Vogel and W. Weise, Nucl. Phys. A 516 (1990) 429; U. Vogl, M. Lutz, S. Klimt and W. Weise,Nucl. Phys. A 516 (1990) 469. [10] D. Dyakonov and Y.V. Petrov, Nucl. Phys. B 227 (1986) 457. [ll]Th. Meissner, E. Ruiz-Arriola, F. Gruemmer, H. Mavromatis and K. Goeke, Phys. Len. B 214 (1988) 312. [ 12 ] A.H. Blin, B. Hiller and J. da Providencia, Phys. Lett. B 241 (1990) 1. [13] E.M. Henley and H. Muether, Nucl. Phys. A 513 (1990) 667. [ 141 V. Bernard and U.G. Meissner, Phys. Rev. Lett. 61 (1988) 2296. [ 15] M. Lutz and W. Weise, Nucl. Phys. A 518 (1990) 156. [ 16] D. Toernqvist, Ann. Phys. (NY) 123 (1979) 1.
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[ 17 ] Particle Data Group, J.J. Hern~indezet al., Review of particle properties, Phys. Lett. B 239 (1990) 1. [ 18] R. Jaffe, Phys. Rev. D 15 (1977) 267. [ 19] J. Weinstein and N. Isgur, Phys. Rev. D 41 (1990) 2236. [20] D. Lohse, J.W. Durso, K. Holinde and J. Speth, Nucl. Phys. A516 (1990) 513. [21] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16 (1966) 255; Riazuddin and Fayazuddin, Phys. Rev. 147 (1966) 1071. [22] M. Takizawa, K. Kubodera and F. Myhrer, Phys. Lett. B 26l (1991) 221. [23] P. Ring and E. Werner, Nucl. Phys. A 21 l (1973) 198. [24] I. Hamamoto, Phys. Rep. 10 (1974) 63.
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[25] J.P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rep. 25 (1976) 83. [26] M. Jaminon, G. Ripka and P. Stassart, Phys. Lett. B 227 (1989) 191. [27] R. Alkofer and I. Zahed, Phys. Lett. B 238 (1990) 149. [28]A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II (Benjamin, New York, 1965 ). [29] H. Reinhardt, Nucl. Phys. A298 (1978) 77. [30] M. GeU-Mann and M. Levy, Nuovo Cimento 16 (1960) 705. [31 ] L.M. Barkov et al., Nucl. Phys. B 256 (1985) 365. [32] G.E. Brown, Nucl. Phys. A 522 ( 1991 ) 397.
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