Low energy constraints on strange matrix-elements of the proton

Volume 223, number 3,4

PHYSICSLETTERSB

15 June 1989

LOW ENERGY CONSTRAINTS ON STRANGE MATRIX-ELEMENTS OF THE PROTON ~" V6ronique BERNARD Groupe de Physique Nucl~aire Thborique, Centrede RecherchesNuclbaires et UniversitbLouis Pasteurde Strasbourg, B.P. 20 Cr, F-67037Strasbourg Cedex, France

and UIf-G. MEIBNER Centerfor TheoreticalPhysics, Laboratoryfor Nuclear Science and Department of Physics, MassachusettsInstitute of Technology, Cambridge, MA 02139, USA

Received 20 March 1989

In the frameworkof a generalizedthree-flavorNambu-Jona-Lasiniomodelwe investigate proton axial-vectormatrix elements at low energies. We find a small value for (Plg?u7sslp), i.e. As~ -0.07. We summarize the constraints that chiral symmetry violation imposeson low-energystrangeness admixtures into the proton wavefunction.We also calculate the baryomagneticmoment of the proton.

1. Introduction Recently, much interest has been focused on the questions surrounding possible admixtures of strange operators into the proton's wavefunction. In particular, the standard analysis of the pion nucleon Z term (27~N) seems to indicate a large amount ofsYpairs in the proton [ 1-3 ]. This would lead to the surprising result that 1/3 of the nucleon mass comes from the strange sea, 8M~___350 MeV. Furthermore, recent EMC data [4] ~t suggest that strange quarks eat up most of the valence quark contribution to the spin of the proton. This has triggered a flurry of investigations [6,7 ], and various chiral models of the nucleon have been used to bridge the gap from the high to the low-energy domain [ 8,9 ]. In this paper, we want to continue our systematic study of strange-matrix elements within the framework of a generalized N a m b u This work is supported in part by funds provided by the US Department of Energy (DOE) under contract # DE-AC0276ER03069. ~l For reviews sec rcf. [5]. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

Jona-Lasinio model [ 10 ]. We will calculate the axialvector matrix elements ( p I~'u?sqjl P ) c (J = u, d, s) and argue that our results can be considered to be consistent with Jaffe's proposal [ 11 ] of a strong Q2 dependence of the isoscalar axial current due to the triangle anomaly. The generalized Nambu-Jona-Lasinio (NJL) model provides a natural laboratory to study the effects of dynamical and explicit chiral symmetry breaking when proper care is taken of the violation of the axial U ( 1 ) symmetry [ 10 ]. Quark-quark and quark-gluon dynamics is represented by effective multi-quark interactions with some strength parameters G and K. Beyond some critical value of G and/ or K, constituent (effective) quarks are formed nontrivially and chiral symmetry breaks to its diagonal subgroup. Pseudoscalar mesons appear as q~ bound states by virtue of Goldstone's theorem. The dynamical breakdown of chiral symmetry induces mixing of quark flavors and leads naturally to non-vanishing matrix elements like ( p I~s IP ), (P IYi~ssl p ) [ 11 ], , or
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the S t e r m , the isospin-violating part o f the p i o n - n u cleon coupling, the spin content o f the proton or the strange anomalous magnetic m o m e n t o f the proton, F ~2,p ) (0) [11]. In ref. [ 10 ] we demonstrated that there exist two generically different regimes o f the chirally asymmetric phase. These regimes are characterized by their very different behavior versus the introduction o f current quark masses, in particular o f the rather large strange quark mass [ m~/AQcD= O ( 1 ) ]. In the "barely broken regime" (BBR) with G>~ Gc~it ~2, non-linearities in ms are large and a considerable quenching of the strange scalar density in the proton takes place. This mechanism allows for a large Z term ( ~ 50 MeV) and a small s-quark content of the proton at the same time. In the "firmly broken regime" ( F B R ) characterized by G >> Gmt, non-linearities in m, are small and so is the 27 term. Both regimes also give a rather different prediction for the strange anomalous magnetic m o m e n t of the proton,

(N(p' )lgT, slN(p) )c ia~,,q" ---2,p,~ ,a(P' ) -~-£-N U(P) _ _ F (s) ( ,.t2 ~

(1)

with q a= ( p ' - - p ) : . We find F~S~ ( 0 ) = - 0 . 0 4 n.m. in the BBR as compared to - 0 . 1 4 n.m. in the FBR [ 11,12 ] ~3. In contrast, as we will demonstrate in what follows, the strange axial-vector matrix elements ( p Ig ~ s s l p ) is small and of similar value in both regimes. Furthermore, we will also study the quark mass dependence of the axial-vector operator which can be done without any complication in our model. We should point out here that the naive use o f the chiral limit can produce rather misleading results, something well known since the advent o f chiral perturbation theory [ 13 ]. Finally, we will summarize the predictions of the N J L model for the various strange matrix elements are related observables. In particular, we will also give a prediction of the "baryomagnetic" m o m e n t o f the proton which can be measured in vp scattering (cf. Kaplan and Manohar [7 ] ).

~2 For the moment, let us ignore the U( 1)Abreaking. Of course, a similar argument can be written down in the presence of two (or more) multi-quark coupling constants. ~3 For orientation, let us note that Kaplan and Manohar [ 7 ] estimate F ~ ~ 0.2.O( 1) from the chiral quark model. 440

15 June 1989

2. Calculation of axial-vector densities in the NJL model Our starting point is the generalized three-flavor N a m b u - J o n a - L a s i n i o lagrangian discussed in great detail in ref. [ 10 ], 8

o.~NjL = ~ ( i ~ - r h ) q + G

~

[ (~2"q) 2 - (~ys~.aq) 2]

a=0

-K{det[(t(l+85)q]+det[(l(l-Ts)q]},

(2)

with qT= (U, d, s) and rh =diag(rn,, md, rns) the bare quark mass matrix ~4. The effective multi-quark interaction consists of two terms, a conventional four-fermion coupling [ 14 ] and a six-fermion term to break the unwanted axial U (1) symmetry. G and K are coupling constants of dimension [ m a s s ] - 2 and [mass] -s, respectively. The quark masses are taken from experiment, it is worthwhile to notice that their values are determined at a renormalization scale/.t ~ 1 GeV [ 15 ] not far from the typical scale of dynamical chiral symmetry breaking, A x_~ 4nf~. The parameters G, K, and A can be fixed from an overall fit to the properties of the pseudoscalar Goldstone bosons (r~, K, q), the qq' mass splitting and the vacuum expectation values of the quark condensates (q~qi) o ( i = u, d, s). The cut-off A renders the divergent loop integrals finite, and the regularization procedure has to be chosen in h a r m o n y with the underlying symmetries of the lagrangian (2),

f#=SU(Nf)L®SU(Nf)R®SU(3)v®U(1 ) v ® ~ r (3) for Nf flavors. Here, color is only a global symmetry and special care has to be taken to avoid the appearance of colored Goldstone bosons. The four-fermion interaction in eq. (2) is indeed constructed so as to break the global chiral color symmetry. The six-fermion term in eq. (2) is of utmost importance for the problem of flavor mixing since in the Hartree-Fock approximation we chose to work with is entirely responsible for flavor mixing. Before proceeding with the calculation of axialvector matrix elements, we have to assemble some pertinent definitions of some quantities to be used. We will consider the axial currents g4 In what follows, we will work within the isospin limit m,=ma ~Dv/.

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PHYSICS LETTERSB

A~,=qTu?5"~2 q a

-

1

a

(4)

with a = 0 .... , 8 and Tr(2"2b)=2O "b. The following SU (3) flavor combinations are of interest:

G(3)gu=_½(N, sla~,su-dyu~,sdlN, s ) , G(S)g,

1

- 2x/~ (N,

(5a)

(Sb) G(°)g~,= ~( N, sl a?u~'su+d~'u~sd+:~u~sslN, s)

(5c)

with IN, s ) a nucleon of spin s and g~,the spin vector. From SU (3) symmetry one can deduce

G(3)=½(F+D)=0.625 , G~8)= ~

1

( 3 F - D ) =0.159

serious model of low-energy dynamics can say about the value of G co). Remember that in the canonical quark model with no strange quarks in the proton wavefunction, G c0) ~ (2x/~) G (8) ~ 0.18. An equivalent statement would be to say that the strange matrix element

G~S)~, =

slayu~,su+deuysd-2eyurss[N, s) ,

(6)

for the central values of F and D given by Kaplan and Manohar [7]. Then one can use the Ellis-Jaffe sum rule [ 16 ] to deduce the value for G co) via 1

f dxgef(x, Q2)=(PT i~1½Q2q~15~)3qp,) 0

1fi c(~)+ ~G~O)) =(-~a(~)+ g-,fi,

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(p, slg~75sl p, s) -= As~,

(9)

is very small. We will proceed to calculate the axialvector strengths G (°), G (3) and G (8) at low energies. In ref. [ 12 ] we pointed out that the six-fermion term does not contribute to GI s) because it has no ?uYs piece after Fierz transformation. There is, however, a subtlety connected with the four-fermion interaction due to the generalized Ward identities. It is not possible to attach the operator 28~u~5to the octet piece of the axial-vector interaction by resummation of fundamental bubbles because of the Ward identity. For the isoscalar axial-vector operator (2°?uTs) such an argument is not valid, and therefore it is possible to generate G~s) # 0 ~5. To be more specific, let us calculate the admixture of quark operators of a given flavor into constituent quarks of a different flavor. This is related to the fact that the NJL model does not confine quarks. Therefore, we will consider the equivalent axial-vector densities (Qiltb~5~,qjIQi), with Qi a constituent quark of flavor i. We find (Ul~28yu75qlU)

×(1

a~(Q2)n + O ( a ~ ) )

(7)

using the standard SU (3) decomposition of the quark charge operator Q and a~(Q 2) is the running coupling constant. As emphasized by Jaffe [ 11 ] the octet axial charges G (~) ( a = 1..... 8) are constants independent of Q2 because conserved currents have vanishing anomalous dimensions. The situation is, however, different for the isosinglet current A °u which is not conserved due to the triangle anomaly,

3g 2

0uA u° = q /2g 1--~2 ex,~p Tr (GX*G~a) .

u

1

u

1

= a2 8,u,su + a2 °,uT, [~-)loo VooOo[(128,~,,,q ] , (Wl#2°y,y, q l U )

=~2°,u,,u+ a2°,u?'5 [--£-~looVooOo[q)t°,.Y,q] , (lO) with

J= VooJoo= GJoo,

( 1 la)

(8)

Consequently, G co) may depend on Q2. To lowest order, the Q2 dependence of G (°) is weak [ 17 ] but in the non-perturbative regime it may show a strong Q2 dependence [ 11 ]. Here, we are interested in what a

~5 Actually, in ref. [ 12] we did not consider this possibility, but it remains true that the determinantal term does not contribute to G~S).

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PHYSICS LETTERSB

Joo= f ~d4 k Tr[~u~52oS(k)~uy5S(k) ] A

= 2N¢(Ru+Ra+R~), Oo[Ct28yuysq] I d4k = ~Tr[?~uY52oS(k)Y~,Y528S(k)]

(1 lb)

less coupling strength of the effective four-quark interaction. We have now assembled all tools to study the vector matrix elements (Ulayuysu[U), < U Ig~'~,sslU > .... ; these will be discussed in the following section.

3. Results and discussion

A

= lx~ N¢(Ru +Rd-- 2 R , ) , 0o [ ~]2°~'uy5q] =Joo,

( 1 lc) ( 1 ld)

First, we must fix parameters. We will use the parameters G, K and A determined in ref. [ 10 ]. These are A=I.0GeV,

where Si is the regularized quark propagator,

Si(k)=i f

d4k Mi (2n) 4 kZ-M2i

(i=u,d,s)

rh=6 MeV (12)

A

for constituent quarks of mass Ms. The latter is nonperturbatively generated in the spontaneously broken phase via the gap equation. We choose a covariant four-momentum cut-off, fA d 4 k = f ~ d4kO(A 2 - k 2 ) . The integrals Ri in ( I lb), ( 1 lc) can be written as

quq~'~q2 ]"

(13)

At q2=0, we have J , ( 0 ) = 0 and R~=M2L(0). For convenience, we define L(O)=-~ ~

16n 2

(14) •

(Ul~287uysqlW) -~

1 [ 2 2-~a(x,,l,,-x,L) ~ 1+ 1 -~a(2xulu 1 2-

2--

~ 2- ] , +xsls)

(15)

and (UI42°~'.75qlU)

N/~ ( 2

where 442

1+

G(O) G(3---5 = 0 . 3 4 ,

G(O) G(3-----5 =0.31 , As= - 0 . 0 7

Straightforward algebra leads to

la(2x2Tu+x~) ~ l-~a(2x,,I,,+X~Is)]' 1

x~=M~/A and

2~

a=

(16)

2 ~

3GA2/nz is the dimension-

GA2=3.43,

KA5=46.3,

(BBR), GA2=4.35, (FBR).

KA5=89.1, (17)

One notices that G and K are large, i.e. dynamical chiral symmetry breaking involves strong coupling. Therefore, our results should be considered illustrative since we work in the Hartree-Fock approximation not including higher-order corrections in G and/ or K. Keeping in mind that going from the constituent quarks to the proton G (°) and G (8) remain unchanged whereas G (3) is multiplied by (F+D), we find

As= - 0 . 0 6

MF-~A2.

In 1 + ~

1 =_

A=0.75GeV, r~=9 MeV

Ri=8M2ii(q2)t~u,+j~(q2)(t~uv

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G(8) G(3~ = 0 . 3 6 , (BBR) G(8) G(3~ = 0 . 3 5 , (FBR).

(18)

It is interesting to note that the results are very similar for the two regimes. In both cases, we find G (o)/ G(8)< 1 because of the strangeness admixture (remember that for As=0 one would have G(°)/ Gt8)= 1.15 ). These results can be immediately compared to the chiral bag model calculation of Hogaasen and Myhrer [8]. They find G(°)/G~3)=0.27, G(8)/ G (3) = 0.75, As = - 0 . 0 8 and argue that the spin of the proton comes to a large part from the valence quarks and partly from intermediate [qqqg) states and pionic corrections. Our model does indeed confirm their results for the valence quarks. In table 1, we summarize the strange matrix ele-

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Table 1 Strangeness admixtures into the proton wavefunction for various operators in the regime of "barely" and "firmly" broken chiral symmetry (BBR and FBR, respectively). The results for the physical strange quark mass as well as the chiral limit are given. BBR

y Y5

Ysu

FBR

rn~= 175 MeV

m,=0

m~= 175 MeV

ms=0

0.07 -0.55 -0.13

0.47 -1.8 -0.09

-0.05 -0.82 -0.17

0.14 -1.94 -0.17

ments for various operators calculated within the framework of the NJL model. To normalize with respect to the non-strange contributions, we utilize a generalization of Gasser's definition [ 1 ] for the scalar density via 2 ( p l g F , slp )

Y"=
(ar u+dF.d)IP> =

F~°) - ( 2 / x /r:3 )"F (8),~ F~ °) + (l/x//3)F(, 8) (19)

with ?~=~, ~5, ?~5 and F~ ~) (i=0, 3, 8) is the pertinent form factor (strength) of the respective proton matrix elements in the SU (3) flavor basis. From table 1 it becomes obvious that the quark mass dependence of the y,'s is indeed important (i.e. one should not take the naive chiral limit) and that these results are compatible with the quark model and OZI phenomenology. In table 2 we show some observables connected to these strangeness admixtures together with their chiral limit values. The most striking difference between the BBR and the FBR are the values

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for the pion-nucleon and kaon-nucleon 2~ terms as well as the strange anomalous magnetic moment of the proton. Current empirical values for 22~Nand the prejudice of F ~ (0) = 0 [ 18 ] seem to favor the regime of barely broken chiral symmetry. It will be of utmost importance to get a handle on both 22KNand F(~) 2,p to settle this question. Although the extrapolation from the threshold down to the unphysical Cheng-Dashen point of yet to come accurate KN scattering data will be difficult, the large discrepancy i xx . ~"~ "~'BBR/$-'FBR KN /z"~KN ~ 2 might help in getting a quick idea on bow the breakdown of chiral symmetry is indeed realized ~6. As emphasized by Kaplan and Manohar [7 ], elastic vp scattering can be used to extract the value of F~o)(0). These authors also point out that because the Zo couples to the operator a~u~sudTu75d-gTu75s, elastic vp scattering can give information about the "baryomagnetic" moments, /t°= ~#rX?q,

(20)

since the ordinary magnetic moments are related to /t3 and/ts. In our model, assuming additivity in the quark magnetic moments, we can also predict ( p l / t ° l p ) . We find ( p l / t ° l p ) =0.96 n.m.

BBR,

=0.90n.m.

FBR,

(21)

which is comparable to the isoscalar and isovector ~6 Here, we optimistically assume that all the difficulties with extracting the low-energy KN scattering amplitude might eventually be overcome.

Table 2 Observables related to strange matrix elements. The L"terms probe the scalar density, whereas the strange anomalous magnetic moment, the isospin-violating nN coupling and As are related to the operators ~,~,s, ~)'ss and ~A'ss, respectively, po is the baryomagnetic moment [4-7 ]. Notations as in the table 1. BBR

S,N [MeV] --rKN [MeV] F~Sp)(0) [n.m.] ( p l / ~ ° l p ) [n.m.]

~)gnNN/gnNN As

FBR

ms= 175 MeV

ms=0

m s = 175 MeV

ms=0

45.0 338.0 -0.05 0.96 0.008 -0.06

62.0 35.0 -0.04 0.97 --0.04

33.0 186.0 -0.14 0.90

34.0 13.0 -0.19 0.90 -0.07

0.006 -0.07

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m a g n e t i c m o m e n t s (/Zs = 0.44 n.m., #v = 2.35 n . m . ) o f the n u c l e o n . T h e b a r y o m a g n e t i c m o m e n t s o f o t h e r b a r y o n s can be calculated f r o m a d d i n g the p e r t i n e n t c o m b i n a t i o n s o f /~°v=0.96 ( 0 . 9 0 ) a n d ~ s ° = 0 . 9 4 ( 0 . 8 9 ) ( i n u n i t s o f the c o n s t i t u e n t s m a s s e s M ~ ~ a n d M f ~, respectively) for B B R ( F B R ) u s i n g S U ( 3 ) f wavefunctions. In s u m m a r y , we have s h o w n that at low energies there are small a d m i x t u r e s o f strange operators i n t o the p r o t o n w a v e f u n c t i o n , c o m p a t i b l e with the stand a r d q u a r k model. We have also given a n e s t i m a t e o f the " b a r y o m a g n e t i c " m o m e n t o f the p r o t o n w h i c h s h o u l d soon be extracted f r o m vp scattering data. We w o u l d like to t h a n k R.L. Jaffe a n d A. M a n o h a r for s o m e useful discussions. We also wish to t h a n k the m e m b e r s o f the I n s t i t u t e for T h e o r e t i c a l Physics o f the U n i v e r s i t y o f Bern for their k i n d hospitality, where part o f this w o r k was carried out. We are grateful to Jiirg G a s s e r for s o m e helpful remarks.

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J. Gasser, H. Leutwyler, M. Sainio and A. ~varc, Phys. Lett. B213 (1988) 85. [4] J. Ashman et al., Phys. Lett. B 206 (1988) 354. [5] R.L. Jaffe, in: IAP Proc. # 176 (1989); R.L. Jaffe and A. Manohar, MIT preprint CTP # 1706 (1989). [6] S.J. Brodsky, J. Ellis and M. Karliner, Phys. Lett. B 206 (1988) 309; C. Altarelli and G.G. Ross, Phys. Len. B 121 (1988) 391; F.E. Close and R.G. Roberts, Phys. Rev. Len. 60 (1988) 1471. [7] D.B. Kaplan and A. Manohar, Nucl. Phys. B 310 (1988) 527. [8] F. Myhrer and H. Hogaasen, MIT preprint CTP # 1635 (1988). [9] Z. Ryzak, Phys. Lett. B 217 (1989) 325; T. Hatsuda and I. Zahed, SUNY preprint (1989), unpublished; B.Y. Park and M. Rho, Z. Phys. A 331 (1988) 157; J. Ellis and M. Karliner, Phys. Len. B 213 (1988) 73; G.E. Brown, M. Rho and B.Y. Park, SUNY preprint ( 1989), unpublished. [10] V. Bernard, R.L. Jaffe and U.-G. MeiBner, Nucl. Phys. B 308 (1988) 753. [ 11 ] R.L. Jaffe, Phys. Lett B 193 (1987) 101. [12] V. Bernard and U.-G. MeiBner, Phys. Len. B 216 (1989) 392. [13] H. Pagels, Phys. Rep. 16 (1975) 219. [ 14] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 ( 1961 ) 345. [ 15 ] J. Gasser and H. Leutwyler, Phys. Rep. 87 ( 1982 ) 77. [ 16 ] J. Ellis and R.L. Jaffe, Phys. Rev. D 9 (1974) 1447. [ 17 ] J. Kodaira, Nucl. Phys. B 165 (1979) 129. [ 18 ] L.A. Ahrens et al., Phys. Rev. D35 ( 1987 ) 785.