OZI violation in the nucleon, delta's and hyperons

Volume 240, number 1,2 PHYSICS LETTERS B 19 April 1990

OZI VIOLATION IN T H E NUCLEON, DELTA'S AND H Y P E R O N S

T. K U N I H I R O 1,2

Institute of Theoretical Physics, University of Regensburg, D-8400 Regensburg, FRG

and

T. HATSUDA 3

Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA

Received 6 December 1989; revised manuscript received 30 January 1990

The anomalous quark contents in the octet and decouplet baryons such as the ss content in the proton or the uu content in the f~- are examined on the basis of a chiral quark model, in which the constituent quark masses are given dynamically by the spontaneous breaking of the chiral symmetry. It is shown that the flavor mixing due to the UA( 1 )-anomaly gives rise to anomalous quark contents of all the flavors, while the short-range interaction between the constituent quarks (one-gluon exchange) tends to suppress (enhance) them for the decuplet (octet) baryons. The resulting nN sigma term is enhanced by about 20% due to one- gluon exchange.

Flavor mixing effects have been one of the main subjects of hadron physics. The smallness of the mixing seen systematically in mesons except for a few systems like the q-complex (q and q ' mesons) is summarized as the Okubo-Zweig- I izuka (OZI) rule [ 1 ]. Recently, much at tention has been paid to the flavor mixing effects or OZI-violating processes in baryons. The nN sigma term X~N is one of the quantities to which such a process may play a significant role. It is defined as the nucleon matrix element of the scalar operator

X~N = r h ( a u + d d ) N , (1)

with rh = (mu + m a ) / 2 being the average of the current quark masses without strangeness. The "experimental" value of it is reported as 51 _+ 5 MeV or 57 _+ 6 MeV dependent on the analysis [2 ].

The first order chiral perturbation, which can lead

Supported in part by BMFT grant 06 OR 762. 2 Permanent address: Faculty of Science and Technology, Ryu-

koku University, Seta, Otsu-city 520-21, Japan. 3 Supported by the US Department of Energy under Grant No.

DE-FG02-88ER40388.

tO the Ge l l -Mann-Okubo ( G M O ) mass relations for baryons, gives only half of the experimental value of Z~N if the OZI-rule in the proton is taken for granted, i.e.,

2(YS)N Y - ( a U ) N + ( d d ) N ~ 0 . (2)

Conversely, if one is eager to reproduce the experimental value relying on the first order chiral pertur-

bation, one must accept a huge s-quark content in the proton: y_~ 0.48.

In recent years, some authors, using SUr(3) ver- sions of the Nambu-Jona-Las in io (NJL) model [3,4], have shown that the full inclusion of the effects of the current quark masses give values as large as 40-50 MeV for the sigma term with a small strangeness content, i.e. y = 0 . 1 - 0 . 1 2 [5-7] : They started with the F e y n m a n - H e l l m a n theorem first noted by Gasser [ 8 ], by which the quark content in the proton is related to the proton mass Mp

0Mp (q iq i )P = Omi ' (3)

where rni ( i = u , d, s) is the current quark mass. The

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland ) 209


problem is to get a formula for Mp in terms of m, as precisely as possible. An addi t ive quark ansatz was adopted as a first approximat ion [9], i.e., M p = 2M~+M,~ where the M~ ( i = u , d, s) are the constituent quark masses and are identif ied with the dynam- ical masses generated by spontaneous chiral symmetry breaking (SzSB). The underlying picture for this procedure is, as is clearly stated in refs. [ 10,5], that the dynamics responsible for the SzSB is in the inter- mediate range of scale, and confinement hardly af- fects the observables related with the chiral symmetry and SzSB. It is worth ment ioning here that the success of the const i tuent quark model can be under- stood in terms of a nonl inear chiral quark model where SzSB is assumed [ 11 ]. The NJL model may be taken as a different version o fa chiral quark model, in which SzSB is given dynamically. In the addi t ive ansatz, the quark contents are given as

aa6 (gl~qi)e=2Q,,+Q,li with Qj,--- 0m, ' (4)

where @i is the ith current quark content in the j th const i tuent quark.

In this short report, we pursue the picture that the NJL model could be a field theoretical version of the consti tuent quark model, and extend the previous work to the octet and decouplet baryons by incorpo- rating the short-range interact ion due to one-gluon exchange ( O G E ) using some successful const i tuent quark model ~. We will show that the interact ion contr ibutes posit ively to the quark content in the proton for all flavors, and hence make the sigma term in closer agreement with the experimental value, while the ratio y remains as small as 0.12. As for the other baryons, OGE enhances or suppresses the anomalous content depending on the spin structure of the baryons. We clarify also that the appl icat ion of the F e y n m a n - H e l l m a n theorem is t an tamount to calculating the d iagram of a-meson exchange from the external fields to the constituent quarks. Needless to say, our model with OGE gives the octet and decuplet baryon masses precisely. This confirms the observa-

~ The effect of OGE on the sigma term has been examined by Nelson [ 12], who used a linear formula relating the current quark masses with the constituent ones. However, this is not always legitimate since the nonlinear dependence on the current quark masses is essential for the evaluation of the quark content in the proton in some models such as the NJL model.

t ion that the success of the G M O formula does not necessarily imply the val idi ty of the first order per- turbat ion with respect to ms [ 2 ].

Our effective lagrangian may be writ ten as follows:

S = £"~JL + SOGE + ~coNv • ( 5 )

Here 5eNJL is the NJL model to be specified later and is unders tood to be responsible for SzBS in the inter- media te range, and ~OOE and ScoNv denote the OGE and the force responsible for confinement, respec- tively. Our ansatz is, in accordance with refs. [ 13,11 ], that the dynamics could be treated in two steps; first SN~L is put on and thus SzBS is realized, then the short-range and the long-range interactions repre- sented by ~OOE and 5"c'oNv act. In the second process, we may utilize some const i tuent quark model which is successful in describing low-lying hadrons. The new aspect of our model in compar ison to the conven- t ional const i tuent quark model is that the collective nature of the vacuum and mesons can be taken into account naturally. Here we note that s imilar ideas are independent ly used to analyse the p ro ton -neu t ron mass difference in the med ium [14], and the effect o f the instanton on the nuclear forces [ 15 ].

Now let us write down ~NJL,

~'(~NJL =(t( iy'O- m )q

8

+ ~ lgs[((l)t~q)Z+(gli2aysq) 2] a = 0

+gD[de t #i( 1 --75)qj +h .c . ] , (6)

where the quark field qi has three colors (Arc = 3 ) and three flavors ( N r = 3 ) , 2a ( a = 0 . . . . , 8 ) are the Gell- Mann matrices with ;to = x /~ l . The second term is the explicit S U v ( 3 ) ( S U r ( 3 ) ) breaking part with m = diag(mu, ma, ms) being the current quark mass matrix. The last term is a reflection of the axial anomaly of QCD, which has SUn (3) ® SUR (3) invariance but breaks UA ( 1 ) symmetry [ 16 ]. This term is responsible for the mixing of the different flavors in the mean field approximat ion.

Our model contains several parameters to be fixed: The current quark masses m = diag ( mu, ma, ms), the coupling constants gs and gD, and the momen tum cutoffA characterizing the chiral symmetry breaking scale. We have assumed SUr(2)- invar iance , namely we set m~ =md =--~h and adopted the s tandard cur-

210


rent algebra value rh=5.5 MeV [17]. Another four parameters were determined to reproduce the four basic quantities m~= 138 MeV, f~=93 MeV, rnK= 495 MeV and mq, = 958 MeV. (gs, A ), ms and gD are fixed essentially by ( m ~ , f ~ ) , mK and mn,, respec- tively. The resulting parameter set becomes

A = 6 3 1 . 4 M e V , gsA2=3.666, gDAS=--9 .306,

m~= 135.3 MeV. (7)

Here we have used the three-momentum cutoff scheme for convenience only.

Once the parameters have been fixed, the model gives an extensive description of the observables related with the chiral symmetry and the UA ( 1 ) anomaly; see refs. [5,18 ] for details. Here we only quote the results for the constituent masses (M~, Md, Ms) for later convenience:

M , = M d = 3 3 5 M e V and M ~ = 5 2 7 M e V , (8)

which are quite consistent with the phenomenological values usually used in the non-relativistic quark model [ 19 ]. These are determined from the following self-consistency condition:

Mu =m~ - 2 g s ( ~ u ) --2gD (rid) ($s),

M d = m d --2gs (rid) --2gD (~S) ( a U ) ,

M~ =rn~ - 2 g s (2s) --2gD (aU) (o ld ) , (9)

where (~]sqs) ( i = u , d, s) are the quark condensate of the ith quark with mass Mi,

f d4p 1 (~]sq~) = - iN~ tr (27r) 4 ~ . p - M , (lO)

Now put ~OGE and L~CONV on, which is simply ac- complished by introducing the mass formula for the octet and the decuplet baryons in a constituent quark model, for instance, that of Isgur and Karl [20] ~2:

M B = M o + M i + + b ~_. ,~7, ~ • ( l l ) i i < j lVl iJV1 )

The meaning of the respective terms is as follows. 340 represents the contributions of the confinement po- tential and the short-range color-electric interaction, which are independent of the constituent masses. a / 2 M ~ denotes the kinetic energy of the constituent quarks, and the spin-spin term with the coefficient b is the color-magnetic interaction responsible for the octet-decuplet mass splitting. We give the explicit forms of the mass formula for some baryons in table 1.

For the constituent masses M~, we take our result in the NJL model, as promised. Other parameters, 340, a and b, in the mass formula can be determined by fitting the masses of the proton (938), the A(1232) and the f2(1672), for example. The resulting values are

~2 We note that, even if one takes a different kind of baryon mass formula, such as the one given by De Rtijula, Georgi and Glashow [ 19], the quantitative results given below hardly change [21,22].

Table 1 The explicit mass formulas for the octet and decuplet baryons in the Isgur-Karl model. Note that one can simply set M. = Md only when calculating the masses, but not when calculating the quark contents.

B MB

P A + A++ E+ ~*+ .-o ~*+

A o

Mo+ 2Mu+ Md+ ½a(2/Mu+ 1/Ma) +4b( 1/4M~ - 1/MuMa) Mo+ 2Mu + Md + ½a(2/Mu+ 1/Ma) +4b( I / 4M z + 1/2MuMd) Mo + 3Mu + 3a/Mu + 3b/M~ Mo + 2Mu + Ms + ½a( 2/Mu + l /M~) + 4b(1/4MZ~- l /MuMs) Mo + 2Mu + Ms + ½a( 2/Mu + l /Ms) + 4b( l /4M~ + l /2M,~Ms) Mo + M,, + 2Ms + ½a( l /Mu + 2/Ms) + 4 b ( 1 / 4 M ~ - l /MuMs) Mo + Mu + 2Ms + ½a(1/Mu + 2/Ms) + 4b(1/ 4M~ + 1/2MuMO Mo+Mu+Md+M~+ Oa(1/Mu+I/M,~+I/MD--3b/MuMd Mo + 3M~ + 3 a/M~ + 3b/M~

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a = (172.3 MeV) 2, M 0 = - 5 2 . 9 M e V ,

b= ( 176.5 MeV) 3 . (12)

Then one gets for other baryons the following masses:

A ( 1 1 1 5 ) = 1 1 1 4 ,

Z ( 1 1 9 3 ) = 1185, Z*(1385) = 1372,

E ( 1 3 2 0 ) = 1331, - - -*(1507)=1519, (13)

in MeV. The fits are excellent. Now applying the Feynman-Hel lman theorem with

this mass formula, we get the quark content in the proton:

( a ~ 3 ) (g ] /q , )p=2 1-- ~ + Q.i

( a 4b~ + l - ~ - ~ + ~ j Q a i. (14)

Here Qj~ can be obtained by differentiating eq. ( 11 ) with the use of (9) and (10). It turns out that Qj~ is essentially the propagator of the ~ meson #3 [ 1 8 ]:

Qj,=[I+Vo.H,~(O)]jT' , (15)

where V, is a symmetric matrix representing the ver- tices of the residual interaction in the G channel:

gs gD(SS) gD(dd)'~ V~=2 go(gS) gs gD ( Z / U ) / , (16)

go(dd) gD (~/U) gs t /

a n d / / o ( q 2 ) =diag( /~(q2) , /d(q2) , /~(q2) ) is polari- zation for the G-mode with ]i(q z) being the loop-in- tegral of the ith quark. The point of the derivation of eq. ( 15 ) is the simple identity

d ( qiqi ) - - = / , ( 0 ) . ( 1 7 )

dM,

Thus one sees that Qj~ represents the effect of the ~- meson exchange between the external operator and the constituent quarks.

One can see from eq. ( 14 ) that the gluon-exchange term has a positive contribution, though the kinetic term has a negative one to the sigma term. In general, if the nucleon matrix element of an operator is nega-

,3 The propagator for the c-mode in our model is given as D,,(q 2) = - [1 + V~.H,,(q2)] -l. Vo.

tive (positive) and is proportional to a negative power of Mg, it contributes positively (negatively) to the quark contents and hence to X~N. AS we will see later, this is also the case for other baryons.

A simple numerical calculation for the proton gives

( aU)p = 5.01 (4.73), ( d d ) p =4.01 (3.03),

(~S)p = 0 . 5 3 3 ( 0 . 4 5 9 ) , (18)

with

y = 0 . 1 1 8 ( 0 . 1 1 8 ) , (19)

where the numbers in parentheses are the results of the simple additive ansatz ( a = b = 0 ) . Thus one sees that the gluon-exchange effect enhances the matrix elements without changing the ratio y. ~'~N is evalu- ated to be

X~N=49.6(42.7) MeV. (20)

This is consistent with the empirical value within the error. A possible discrepancy could be attributed to clouds of pseudo-scalar (ps) mesons, which may give a further enhancement of about 10 MeV [23,8], though part of the effects of the ps-meson pairs might be substituted by that of sigma meson exchange.

Thus one sees that the rcN sigma term is essentially accounted for by G-meson exchange and the short- range interaction between the constituent quarks. We remark that both baryon masses and the large sigma term are reproduced with a small strangeness content in the proton. Note that the nonlinear effect of ms is fully included in our calculation via the Hartree equation ( 9 ), the nonlinearity of which is essentially important in giving the large sigma term, even at ms as large as 150 MeV, as first demonstrated in ref. [ 6 ] without OGE: If one takes the approximation in which the linear terms with respect to ms are kept, then one can only get 25 MeV at most for the sigma term ~4

AS we have obtained the mass formulas for various baryons in table 1 as well as Qsi and Qu(a),, we can calculate various quark contents in other baryons in- cluding hyperons by applying the Feynmann- Hellman theorem simply: The ith quark content in A +, for example, is given as

~4 The significance of taking the nonlinear dependence on ms into account has been also demonstrated in the SU(3) Skyrme model [ 24,25 ], contrary to the naive estimation [ 26 ].

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a 2b) (2Qui+ Qdi), (21) ( (liqi ),., += 1 2 M 2

which shows clearly that OGE together with the kinetic energy term act in reducing the quark contents as ment ioned before. (Recall that OGE contr ibutes posit ively to the A + mass. )

The numerical values o f quark contents in the octet and decuplet baryons are given in table 2, where the contents in the s imple addi t ive ansatz (a = b = 0 ) are also given in parentheses. One can see that there exist not only normal but also anomalous quark contents which cannot be expected in the naive quark model; the lat ter is due to the flavor mixing effect coming from the anomaly term, which is further enhanced by the OGE effect for the octet baryons ~5 General ly speaking, the strange-quark contents are smaller than the non-strange ones. Of course this is also the case for the anomalous quark contents: Com- pare the non-strange and strange quark contents in A + +, and also note that the non-strange contents in f~- is larger than the strangeness content in the proton, for instance. This is because the consti tuent quark mass of the s-quark is larger than that of the u- and d- quarks. It is remarkable that the quark contents are greatly suppressed in the decuplet baryons to whose masses the OGE contr ibutes positively. Recall that the large value of X~N is in t imate ly related with the fact that OGE contr ibutes negatively to the proton mass in our approach. Thus, it would be interesting to examine whether the rather small quark contents in the A's or "hA sigma te rm" S,,~zx=rh(~u+dd)~x, which is 19 MeV in our model, for example, could be obta ined in other approaches or hopefully in the exper iment : I f this is the case, the mechanism which gives the large Z~r~ shown in this repor t could be ver- ified. Phenomenological ly , the rather large non- strange quark contents seen in Z, E and f2 may be also interesting.

In summary, we have extended the previous esti- mat ion of the quark content in the pro ton with the use of the NJL model by incorporat ing the short-range interact ion due to one-gluon exchange: This is based

Table 2 The scalar contents of the u-, d- and s-quark in the baryons. The numbers in parentheses are those in the simple additive quark ansatz without OGE. The numbers with asterisks are anomalous quark contents in the baryons.

B (au)B (dd)~ (ss)B

P 5.01(4.73) 4.01(3.03) 0.533(0.459)* A + 2.13(4.73) 1.37(3.03) 0.207(0.459)* A ++ 2.89(6.42) 0.603(1.34)* 0.207(0.459)* Z + 3.68(4.56) 1.02(1.17)* 1.92(1.73) E *+ 2.38(4.56) 0.678(1.17)* 1.33(1.73) E ° 2.95(2.69) 1.06(1.00)* 3.09(3.00) E *° 1.66(2.69) 0.718(1.00)* 2.50(3.00) A ° 3.32(2.86) 3.32(2.86) 1.71(1.73) ~ - 0.721(0.828)* 0.721(0.828)* 3.72(4.27)

on the observat ion that the NJL model can be re- garded as a field theoretic version of the const i tuent quark model. We have shown that OGE contr ibutes posi t ively to the quark content in the proton, hence makes the nN sigma term closer to the exper imental value, while the strangeness content in the proton remains small (y ~ 0.12). We have also calculated quark contents in other baryons and shown that the anomalous quark contents are greatly suppressed in baryons to whose masses the OGE contributes positively.

It should be remarked here that the anomalous quark contents can be dependent on channels, i.e., even i f the strangeness content in the proton in the scalar channel is small, as is conf i rmed in this paper, it does not necessarily deny a large strangeness content in other channels like the axial vector one, as suggested by the recent EMC data [27]. The channel-dependent strangeness content may be related to the different mixing propert ies of the flavors in ~ and 11 mesons [ 18 ].

One of the present authors (T .K. ) wishes to ex- press his sincere thanks to the faculty members of the Facul ty of Science and Technology o f Ryukoku Uni- versity for allowing him to leave the universi ty for a year. He also acknowledges Wolfram Weise for his warm hospi tal i ty and interest in this work.

~5 Note that the effect of the kinetic term is always negative for the quark contents, which partly cancels the enhancement due to OGE in the octet baryons. The dd content ofZ + is a typical example of this cancellation.

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OZI violation in the nucleon, delta's and hyperons