Chiral Symmetry and Axial Anomaly in Hadron and Nuclear Physics --- a review --
aa r X i v : . [ nu c l - t h ] J un Chiral Symmetry and Axial Anomaly in Hadronand Nuclear Physics — a review —
Teiji Kunihiro
Department of Physics, Kyoto University, Kitashirakawa, Sakyoku, Kyoto 606-8502, Japan
Abstract.
The important role played by chiral symmetry and axial anomaly in QCD in nuclearphysics is reviewed. Some recent topics on possible chiral restoration in hot and/or dense matter arepicked up. We also discuss so called effective restoration of chiral anomaly hot and/or dense matter,as may be seen in a character change of h ′ meson. Keywords: chiral symmetry, axial anomaly, QCD, hot and dense matter
PACS:
INTRODUCTION
Nuclear physics was primarily quantum many-body physics with the nuclear forcegiven, which is responsible for binding the nucleons against the repulsive Coulombforce between protons. Yukawa’s meson theory[1] was the first intended application ofquantum-field theory to the problem of the nuclear force. The salient ingredients of thenuclear force[2] are the tensor force[3] in the long range and the short-range repulsivecore[4]. The tensor force is generated by one-pion exchange between the nucleons. Theone-pion-exchange potential (OPEP)[3] reads V OPEP ( , ) = f m p t · t h ( s · s ) Y ( m p r ) + S Z ( m pi r ) i , (1)where Y ( x ) = exp ( − x ) / x ) , Z ( x ) = ( + / x + / x ) Y ( x ) and S = ( s · ˆ r ) s · ˆ r ) − s · s = √ (cid:2) s ⊗ s ] ( ) ⊗ [ ˆ r ⊗ ˆ r ] ( ) (cid:3) ( ) being the tensor operator which is constructedfrom the two second-rank tensors. The appearance of such an operator with a badsymmetry is due to the fact that the pion is a pseudo-scalar particle.Owing to the transformation properties of the tensor force, it only acts to the spin-triplet state but not to the singlet state, which is the reason why deuteron exists as aproton-neutron bound system although there are no di-neutron bound system: The sec-ond order contribution of the tensor force gives rise to an additional attraction betweenthe triplet state. This second-order effect of the tensor force is also an essential ingredientfor realizing the saturation property of the nuclear matter[5].Then why does pion is isovevtor and pseudo-scalar particle with the lightest massin the hadron world? These are all because the pion is the Nambu-Goldstone bosonassociated with dynamical breaking of chiral symmetry of QCD[6]. How important rolesdoes the chiral symmetry play in nuclear physics? Some answers may be found in [7, 8].Before answering this problem, we clarify the chiral symmetry and its spontaneousbreaking in QCD. Chiral Symmetry and Axial Anomaly in Hadron and Nuclear Physics — a review — June 28, 2018 1
HIRAL INVARIANCE OF CLASSICAL QCD LAGRANGIAN
The classical QCD Lagrangian reads L = ¯ q ( i g m D m − m ) q − F a mn F mn a . (2)The classical QCD Lagrangian with vanishing quark mass ( m →
0) is invariant underthe chiral transformation. The chiral transformation for N F -flavor quark field q f ( f = , , , , , N F ) is defined as a direct product of two unitary transformations U L and U R ; q L f ≡ − g q f → ( U L ) f f ′ q L f ′ , q R f ≡ + g q f → ( U R ) f f ′ q R f ′ . (3)Notice that the vector current ¯ q g m q = ¯ q L g m q L + ¯ q R g m q R is invariant under the chiraltransformation, although the Dirac mass term ¯ qq = ¯ q R q L + ¯ q L q R is not. If we neglect thecurrent quark mass term, the quark filed enters QCD only as a combination ¯ q g m D m q , andhence it become invariant under the chiral transformation. A warning is in order here;the axial U ( ) symmetry is explicitly broken by a quantum effect, which is known as U ( ) A anomaly[9].A quark bilinear operator F i j defined by F i j = ¯ q j ( − g ) q i = q j R q iL is transformedas follows, F i j → ( U L ) ik F lk ( U † R ) l j . (4)In the two-flavor case, the generators of the chiral transformation are given by theisospin charges Q a and the axial charges Q a ; Q a = Z dx ¯ q g t a q / , Q a = Z dx ¯ q g g t a q / . (5)We note the commutation relation, [ iQ a , ¯ qi g t b q ] = − d ab ¯ qq . (6)Then taking the vacuum expectation value of (6), we have h | ¯ qq | i = h | [ Q a , ¯ q g t a q ] | i , (7)which implies that if h | ¯ qq | i 6 =
0, then Q a | i can not be zero for some a . That is, chiralsymmetry is spontaneously broken! Indeed there is a following celebrated relation dueto Gell-Man, Oakes and Renner[10], f p m p = − m u + m d h | ¯ uu + ¯ dd | i , (8)which does indicate that the chiral symmetry is spontaneously broken in the QCDvacuum, because the pion decay constant f p ≃
93 MeV is finite.
Chiral Symmetry and Axial Anomaly in Hadron and Nuclear Physics — a review — June 28, 2018 2
OSSIBLE CHIRAL RESTORATION IN FINITE NUCLEI
One of the interesting nature of QCD is that the QCD vacuum can change along with aninclusion of external hard scale, which may be induced by baryon chemical potential,i.e.,the baryon density, temperature, strong magnetic field and so on. An interesting observa-tion is that a nucleus can provide a hard scale by its baryon density, which might cause achange of the QCD vacuum, and hence the chiral symmetry may be partially restored in afinite nucleus. Thus exploring possible evidence of partial restoration of chiral symmetryin the nuclear medium has become one of the most important and challenging problemsin nuclear physics[8, 11]. Relevant experimental studies include the spectroscopy ofdeeply bound pionic atoms [12], low energy pion-nucleus scatterings [13], and the pro-duction of di-pions in hadron-nucleus and photon-nucleus reactions [14, 15, 16]. Theseexperiments revealed the following anomalous properties of the pion dynamics in the nu-clear medium; (i) an enhancement of the repulsion p − -nucleon interaction[12, 13], (ii)an enhanced attraction of the p - p interaction in the scalar-isoscalar channel[14, 15, 16].In the theoretical side, possible relevance of the p - p interaction in a nuclear mediumwas first suggested in [17]. Weise and his collaborators showed that that the reductionof the temporal part of the pion decay constant in the nuclear medium F t p is intimatelyrelated to the anomalous repulsion (i) [18, 19]. It was also argued that the reduction of F t p is responsible for the phenomenon (ii) [20].Recently, Jido, Hatsuda and the present author[21] derived a novel sum rule for thequark condensate valid for all density, which sum rule is reduced to h ¯ qq i ∗ / h ¯ qq i = ( F t p / F p ) Z ∗ / p (9)in the low-density limit. Here h ¯ qq i ∗ is the quark condensate, F t p the (temporal) piondecay constant and the pion wave-function renormalization constant Z ∗ p all in the nuclearmedium. It is noteworthy that the Z ∗ p can be estimated with the use of the the iso-singletpion-nucleon scattering amplitude at low energy, and they found that Z ∗ p / = (cid:18) G ∗ p G p (cid:19) / = − g rr , (10)where G ( ∗ ) p is the (in-medium) pion coupling constant. Here the coefficient g = br / = .
184 with b = . ± . . (Parametrically, b is expressed as b = s p N F p m p + (cid:18) + m p m N (cid:19) p a p N m p . (11)Here s p N and a p N are the p -N sigma term and the iso-singlet scattering length, respec-tively.) Then the in-medium quark condensate is nicely expressed by the temporal piondecay constant and the pion coupling; h ¯ qq i ∗ = − F t p G ∗ p . (12)Now the s -wave p − -nucleus optical potential U s is parametrized as2 m p U s = − p [ + m p m N ]( b ∗ r + − b ∗ r − ) , Chiral Symmetry and Axial Anomaly in Hadron and Nuclear Physics — a review — June 28, 2018 3 − T (+) ∗ ( w = m p : m p ) r + − T ( − ) ∗ ( w = m p : m p ) r − , (13)where r ± = r p ± r n with r p ( n ) being the proton (neutron) density. Then the parameter b ∗ which can be extracted from the experimental data is expressed as b b ∗ = (cid:18) F t p F p (cid:19) . (14)Combining these relations, Jido et al[21] derived the following relation h ¯ qq i ∗ h ¯ qq i ≃ (cid:18) b b ∗ (cid:19) / (cid:18) − g rr (cid:19) . (15)Now one sees that the experimental evidence of the repulsive enhancement as given by b ∗ implies that the absolute value of the quark condensate in the nuclei is smaller thanthat in the vacuum, and hence the chiral symmetry is partially restored in the nuclei. EFFECTIVE
RESTORATION OF AXIAL SYMMETRY AT FINITETEMPERATURE AND DENSITY
How about other signals of the chiral restoration at finite density and/or temperature?The bottom line is that some hadrons are intimately related to the chiral symmetry andits dynamical breaking, and hence their properties may change along with the chiraltransition at finite density/temperature[7]. Such hadrons include the sigma meson[22].The chiral symmetry implies that the degeneracy of the vector and axial vector correla-tors as well as that in the scalar and pseudoscalar channels, which are parity partners.Thus exploring the possible tendency of the degeneracy in these opsite parity channelsshould be interesting in an extreme environment. The parity doubling in the baryon sec-tor may be affected by the underlying chiral symmetry[23]. Examining the propertiesof the negative-parity baryons such as N ∗ ( ) at finite density and/or temperatureshould be also interesting[24]. An interesting ingredient in this subject lies in the factthat N ∗ ( ) is strongly coupled with h meson. Thus the study of N ∗ ( ) is auto-matically to explore the properties of h meson in nuclei. One should also note that h meson is a mixing partner of h ′ ( ) , the nature of which is intimately related with theaxial anomaly of QCD[25].One of the fundamental properties of QCD is U ( ) A anomaly or axial anomaly[9].The ninth pseudoscalar meson h ′ which is almost flavor singlet with a mass as largeas 958 MeV is a reflection of the U ( ) A anomaly and the q vacuum owing to the in-stanton configuration. The mass of h and h ′ and their mixing property are realized withcombined effects of the anomaly, explicit and dynamical breaking of chiral symmetry.The instanton density as well as the quark condensates is expected to decrease at finitetemperature and density[26, 27]. Thus one should also explore the properties of the h - h ′ meson sector at finite temperature and/or density[28], which may show an effective restoration of U ( ) A symmetry[29, 28, 27] as seen in the mixing properties of h and h ′ mesons and their masses. Chiral Symmetry and Axial Anomaly in Hadron and Nuclear Physics — a review — June 28, 2018 4 he U ( ) A anomaly implies that the U ( ) A symmetry is explicitly broken by a quan-tum effect. In the context of the effective Lagrangian, there should exist a vertex whichviolates this symmetry. One of such an interaction is the six-quark interaction with adeterminantal form as introduced by Kobayshi and Maskawa[30] in 1970, L KMT = g D det i , j ¯ q i ( − g ) q j + h . c ., (16)where h.c. stands for Hermite conjugate. This vertex is contained in instanton-inducedquark interaction derived by ’t Hooft in 1976[31]; see [7] for a review. It was first shownby the present author [28] using a generaized Nambu-Jona-Lasinio model incorporatingthe Kobayashi-Maskawa-’t Hooft term (16) that the h and h ′ mesons change their natureowing to both the temperature dependence of the quark condensates and the possibledecrease in the KMT coupling constant g D with T . The coupling constant g D of theKMT term may be dependent on temperature and baryon chemical potential because theinstanton density is dependent on them[26, 27]. Such a possible temperature dependencecauses a temperature dependence of the mixing angle q h so that q h increases in theabsolute value and the mixing between the h and h ′ approaches the ideal one. Althoughthe h component in the physical h ′ decreases as T is increased, the h ′ mass decreasesgradually with increasing T , because the h tends to acquire the nature of the ninthNambu-Goldstone boson of the SU ( ) L ⊗ SU ( ) R ⊗ U ( ) A symmetry and decreases itsmass rapidly. This is an effective restoration U ( ) A symmetry first discussed by Pisarskiand Wilczek[29]. Such an anomalous decrease in the h ′ mass might have been observedin the relativistic heavy ion collisions at RHIC[32]. Recent studies on this problem arereviewed in [33]. BRIEF SUMMARY
The following is a summary what I wanted to say in this report: (1) The saturationproperty of the nuclear matter can be attributed eventually to chiral symmetry and itsdynamical breaking in QCD. (2) Hadrons are sort of elementary excitations on top ofthe nonperturbative QCD vacuum, and hence may change their properties along withthat of the QCD vacuum. (3) The QCD vacuum can change and even show a phasetransition(s) with an increase of temperature and/or baryon density, which in turn givesrise to a change of particle pictures of the hadrons in the system.
ACKNOWLEDGMENTS
I thank the organizers , in particular, Professor Ozawa to invite me to this interestingworkshop. This work was partially supported by a Grant-in-Aid for Scientific Researchby the Ministry of Education, Culture, Sports, Science and Technology (MEXT) ofJapan (No. 20540265) and by the Grant-in-Aid for the global COE program “ The NextGeneration of Physics, Spun from Universality and Emergence ” from MEXT.
Chiral Symmetry and Axial Anomaly in Hadron and Nuclear Physics — a review — June 28, 2018 5
EFERENCES
1. H. Yukawa, Proc. Phys. Math. Soc. Jap. , 48 (1935) .2. R. Tamagaki, Prog. Theor. Phys. , 91 (1968);R. V. Reid, Ann. of Phys. , 411 (1968) ;M. Taketani, R. Tamagaki, W. Watari, S. Machida, S. Ogawa, T. Ueda, W. Watari, M. Yonezawa, S.Furuichi and K. Nisimura, Prog. Theor. Phys. Suppl. (1967).N. Hoshizaki and S. Otsuki, Prog. Theor. Phys. Suppl. (1968).3. M. Taketani, J. Iwadare, S. Otsuki, R. Tamagaki, S. Machida, T. Toyoda, W. Watari and K. Nishijima,Prog. Theor. Phys. Suppl. (1956).4. R. Jastrow, Phys. Rev. , 165 (1950).5. As review articles, see, for example, H. A. Bethe, R Annu. Rev. Nucl. Sci. , 93 (1971) ;B. Day, Rev. Mod. Phys. , 495 (1978).6. Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345 (1961).7. T. Hatsuda and T. Kunihiro, Phys. Rept. , 221 (1994)8. T. Hatsuda and T. Kunihiro, arXiv:nucl-th/0112027.9. S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications (Cambridge University Press,UK, (1996);K. Fujikawa and H. Suzuki,
Path integrals and quantum anomalies
Oxford, UK: Clarendon, (2004).10. M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. , 2195 (1968).11. W. Weise, Nucl. Phys. A , 115 (2008) [arXiv:0801.1619 [nucl-th]].12. K. Suzuki et al. , Phys. Rev. Lett. , 072302 (2004) ; P. Kienle and T. Yamazaki, Prog. Part. Nucl.Phys. , 85 (2004).13. E. Friedman et al. , Phys. Rev. Lett. , 122302 (2004) Phys. Rev. C , 034609 (2005) .14. F. Bonutti et al. [CHAOS collaboration], Phys. Rev. Lett. , 603 (1996); Nucl. Phys. A , 213(2000); P. Camerini et al. [CHAOS collaboration], Nucl. Phys. A , 89 (2004) .15. A. Starostin et al. [Crystal Ball Coll.], Phys. Rev. Lett. , 5539 (2000) ; Phys. Rev. C , 055205(2002) .16. J. G. Messchendorp et al. , Phys. Rev. Lett. , 222302 (2002).17. T. Hatsuda, T. Kunihiro and H. Shimizu, Phys. Rev. Lett. , 2840 (1999).18. E. E. Kolomeitsev, N. Kaiser, and W. Weise, Phys. Rev. Lett. , 092501 (2003) .19. W. Weise, arXiv:nucl-th/0507058.20. D. Jido, T. Hatsuda, and T. Kunihiro, Phys. Rev. D63 ,011901 (2001) .21. D. Jido, T. Hatsuda and T. Kunihiro, Phys. Lett. B , 109 (2008). See also D. Jido, T. Hatsuda andT. Kunihiro, Prog. Theor. Phys. Suppl. , 478 (2007) [arXiv:0706.0258 [nucl-th]].22. T. Hatsuda and T. Kunihiro, Prog. Theor. Phys. , 765 (1985);Phys. Lett. B , 304 (1987).23. C. E. Detar and T. Kunihiro, Phys. Rev. D , 2805 (1989).24. H. Nagahiro, D. Jido and S. Hirenzaki, Phys. Rev. C , 035205 (2003).25. H. Nagahiro and S. Hirenzaki, Phys. Rev. Lett. , 232503 (2005).26. D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod. Phys. , 43 (1981).27. T. Schafer and E. V. Shuryak, Rev. Mod. Phys. , 323 (1998).28. T. Kunihiro, Phys. Lett. B , 363 (1989);Nucl. Phys. B , 593 (1991).29. R. D. Pisarski and F. Wilczek, Phys. Rev. D , 338 (1984).30. M. Kobayashi and T. Maskawa, Prog. Theor. Phys. , 1422 (1970);M. Kobayashi, H. Kondo and T. Maskawa, Prog. Theor. Phys. , 1955 (1971).31. G. ’t Hooft, Phys. Rev. D , 3432 (1976) [Errata; , 2199 (1978)]; Phys. Rep. , 357 (1986).32. R. Vértesi, T. Csörg˝o and J. Sziklai, Nucl. Phys. A , 631C (2009): arXiv:0905.2803.33. Roles of the U ( ) A anomaly at finite temperature and/or density is reviewed inT. Kunihiro, Prog. Theor. Phys.122