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ELSEVIER Nuclear Physics A663&664 (2000) 707o-710cwww.elsevier.nl/locate/npe

Chiral symmetry for baryons

A. Hosaka", D. Jido\ Y. Nemoto" and ~L Okab

"Numazu College of Technology, Ooka, Numazu 410-8501 Japan

bDepartment of Physics, Tokyo Institute of Technology, Meguro, Tokyo 1.52-8.5.51 Japan

Roles of chiral symmetry for the baryon sector are examined. It is shown that twodifferent assignments are allowed for linear representations of positive and negative paritybaryons. Using linear sigma models of SU(2)R x 8U(2)£, physical implications of the twoassignments are discussed.

1. INTRODUCTION

Chiral symmetry and its spontaneous breaking are very important for low energy hadronphysics. When positive and negative parity particles are grouped into a chiral multiplet,some of their properties are determined solely by the pattern of realization of chiralsymmetry [1]. For instance, (a,IT) are considered to belong to the (~,~) representation ofSU(2)R x 8U(2)£ group and their mass difference can be explained by the non-zero chiralcondensate (qq). Another example is (p,ad which belong to (1,0) EEl (0, 1) representation.

As compared to the meson sector, however, we know less about the role of chiralsymmetry for baryons. This is the issue we would like to address in this paper. Anovel feature for the baryon case is that there are two distinctive ways to assign chiralrepresentations to baryon (fermion) fields when there are more than two baryons. Herewe consider the ground state nucleon N(939) == IV and its would be parity partner.Presumably, the latter may be identified with the first negative parity nucleon N(l53,5) ==»:

2. CHIRAL ASSIGNMENTS AND LINEAR SIGMA MODELS

We assume that the nucleons belong to linear representations of the SU(2)R x SU(2)Lchiral group. Then the chiral transformation for a nucleon field IV is defined by

(1)

where R (L) is an element of SU(2)R (,5[7(2)£), and iVR UVL ) are the right (left) handedcomponents of the baryon field, NR.I, = (1 ± ''(5)/2 N, Eq. (1) is no more than thedefinition; the transformation for the "right" ("left") handed baryon is just. called the"right" ("left") transformation.

When there are two nucleons !VI and ;\'2, the situation changes; two chiral transforrna­t ions are possible. In the "naive" assignment, the transformation rule for both nucleons

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708c A. Hosaka et al./Nuclear Physics A663&664 (2000) 707c- 710c

are the sa me as (1 ):

NIR ----+ RNIR ,

NZR ----+ RNzR ,

NIL ----+ LNIL ,

Nn ----+ LNn . (2)

while in th e "mirror" ass ignment . th e right and left transformations are interchanged forthe secon d nucleon:

~' I R ----+ INI R .

~'ZR ----+ L~'ZR ,

V-' IL -----t L~II L .

I/JzL ----+ RI/Jn .

(4)

Such two assignments are allowed, since the chirality of the Loren t.z group is ind epend entof the nomenclature of R and L of th e internal chiral group S U(2) R x SU(2)L. Th e mirrorassignme nt was adopted by DeTar an d Kunihiro about a decade ago [2] in ord er to explaina lattice observat ion of non-zero baryon masses when chiral symmetry is restored [3] .

In ord er to see dynamical im plications of the two assignment s, we const ruct the linearsigma models. In the naive assignment , the chiral invarian t lagr angi an up to dimensionfour is given by

Cnaive = 01(jtPI + -¢;Z(j¢;2+ a7}l (fI + iTs T - if)'I/-'1+bltz(o +hsT ' if)~'2

+ c{0k rsfI + i T ' if)tPl - 4~1(-y50 + iT ' i)¢'d +CM •

where a , band c are unknown constants. T he last term CM is for the chiral mesons which,however. is not relevan t for the present discussion. 'When chiral sym met ry is spontaneou slybrok en with (0) = 00, th e non-diagonal mass term of (4) can be diagona lized by ,V+ an dN_ which ar e identified to the obse rved nucleons: N+ -+ Nand N_ -+ N *

(.5 )

with eigen values

m ± = ~o ( v (a+ bF+4c2=f(a - b)) (6)

Here the mixing an gle 0 is defined by sinh 0 = - (a + &)/2c. We see that the masses of thenucleons are generated by 00 and t hey te nd to become massless when chiral symmetry isrestored .

It turns out. that in th e physi cal basis of N+ an d N_, the lagrangian (4) reduces to thesum of two independen t terms for IV+ and iV_. T his implies that th e off-diagonal Yukawacoupling Dr.Nt N _ vani shes, which was considered to be th e origin of t he small coupling ofthe decay N (l .53.5) -+ ITN(9:~9 ) [4]. Th e decoupling of N+ and N_ ind icates th at they donot form a chiral mul t iplet , where chiral symmet ry is irrelevant to pro pert ies of the twonucleons.

Next we consider the mirror case. T he chiral invariant lagrangian for 4'1an d 4'2 aregiven by [2]

Lmirr or = 0 1if)~' 1 + 4~2 i f)~'2 + mO (4~215 ~'1 - Dn 5¢'2)

+a¢l(fI + h 5T · ifh\ +b02(0 - hST - if )d'2+ CM , (7)

A. Hosaka et al./Nuclear Physics A663&664 (2000) 707c-710c 709c

Table 1Comparison between the naive and mirror assignments.

definition

mass in the Wigner phasen N'N" couplingchiral partner

relative sign of g1]Nand gI]',yo9"NN'(p)

naive assignmentN1R --+ RN1R , NIL --+ LNILN2R --+ RN2R , N2L --+ LN2L

oo

N+ +-+ 15N+ , N_ +-+ 151'1/_positivedecrease

mirror assignment!/JIR --+ R1)IR , !/JIL --+ L!/JIL1'2R --+ L!/J2R , !/J2L --+ R!/J2L

nlo (finite)(a + b)/ cosh J

1'+ +-+l/J­negativeincrease

A novel feature here is the presence of the chiral invariant mass term, from which severalinteresting consequences follow.

As in the naive case we can diagonalize the mass matrix by 1P+ and1P_:

(1/)+ ) 1 (eS

/2

/5e-S/ 2)( 1Pl )

1/,- =,)2 cosh 8 /5e-S/2 _eS/ 2 1P2

with sinh c = -(a + b)a-o/2mo, whose eigen values are given by

m± = ~(J(a+b)2a-6+4m6=f(a-b)a-o).2

(8)

(9)

(11 )

(10)

Because of the presence of the mass term, the nucleons are no longer massless whenchiral symmetry is restored; rather, they have a finite mass mo. The origin of the massparameter rna is an interesting question as it should be determined by dynamics which isoutside of the chiral symmetry.

One can discuss further differences between the mirror and naive models. First, mesoncouplings between 1/'+ and 1/J- can be finite in the mirror model. However, the observedsmall coupling constant can be reproduced by tuning parameters appropriately [2J. Sec­ond, the commutation relations between the axial charges Q~ and the nucleon fields inthe mirror case

TO 1--:-(tanh8/5 v-'++-.h ,~'-)2 cos 0

T a 1--:-(- tanh 8 /51/'- +--4'+)2 cosh 8

imply that the axial charges of the positive and negative parity nucleons have oppositesigns. Furthermore, one can extend the present argument to finite density [5J. We sum­marize various properties of the naive and mirror models in Table 1.

3. DISCUSSIONS AND SUMMARY

It would be interesting whether the actual nucleons, N(939) and N(1535), belong tothe naive or mirror assignments by testing different implications as shown in Table 1. Onecandidate is to study density dependence of the properties of N (15:35) [.5], through, forinstance, eta productions from nuclei.

710c A. Hosaka et al./Nuclear Physics A663&664 (2000) 707c-710c

/rt /11 /11 /" " '1

/ // / / / / /

/ / / / / /

N* / N* / N N* / N / N /N* /

N

grr.N-N* 8,",,,(a) (b) (e)

Figure 1. Dominant diagrams for the IN ---+ 7iT/N, (a), (b) for the Born terms,and (c) for the Kroll-Ruderman type term. The 7i N* N* coupling is in (a), andthe 7i N N coupling is in (b).

Another candidate is to look at the 7i and "I productions. Let us consider a photopro­duction, ~(N ---+ 7i"lN. Main contributions to this process are depicted in Fig. 1, where thefirst two diagrams are the resonance pole terms and the third diagram is analogous to theKroll-Ruderman term. It is now sufficient to consider only the three diagrams in Fig. 1,if we remember the fact that among various time ordered diagrams leading contributionsare those in which intermediate particles are close to mass shell [6]. To extract the signof the If N* N* coupling, we can look at an interference effect between the two resonancecontributions. If the signs of the two couplings are the same, the two resonance termsadd constructively, while otherwise, they add destructively.

In summary, we have investigated the role of chiral symmetry for baryons. It was shownthat the two chiral assignments lead to baryon properties which are very different to eachother. In particular, the mirror assignment predicts non-trivial behavior for the massesof baryons. It would be an interesting question how these assignments are realized inphysical baryons.

REFERENCES

1. T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994).2. C. DeTar and T. Kunihiro, Phys. Rev. D 39, 2805 (1989).3. C. E. DeTar and .l. B. Kogut, Phys. Rec. Lett. 59, 3:39 (1987); Phys. Rev. D 36,2828

(1987); S. Gottlieb, W. Liu, D. Toussaint, R. L. Renkin, and R. L. Sugar, Phys. Rev.Lett. 59, 1881 (1987).

4. D. Jido, M. Oka and A. Hosaka, Phys. Rev. Lett. 80, 448 (1998).5. H. Kim, D. .lido and M. Oka, Nucl. Phys. A640 (1998) 77.6. 1\. Ochi, M. Hirata and T. Takaki, Phys. Rev. C 56, 1472 (1997), and private com­

munication from T. Takaki.