IC/IR/96/25 INTERNAL REPORT XA9743267 (Limited Distribution)
United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
SU(3) SYMMETRY BREAKING CHIRAL QUARK MODEL AND THE SPIN STRUCTURE OF THE NUCLEON
X. Song1 International Centre for Theoretical Physics, Trieste, Italy.
ABSTRACT
The chiral quark model, suggested by Eichten, Hinchliffe and Quigg, and extended by Cheng and Li, can explain many unexpected proton flavor and spin structures rea sonably well. The SU(3) values of /3//8 and A3/A8 given in Cheng and Li’s description are, however, inconsistent with the experimental data. Taking SU(3)-symmetry-breaking effect into account and extending the SU(3) calculation, we found that not only the above inconsistency can be removed but also better agreement obtained between many other theoretical predictions and experimental results. A consequence of this breaking is that the strange sea would be slightly negatively polarized, As —(0.05 — 0.06), and the total quark spin would contribute about one half of the proton spin, All = 0.45 — 0.47.
MIRAMARE - TRIESTE August 1996
‘Permanent address: Department of Physics, Hangzhou University, Hangzhou 310028, People’s Republic of China. I. INTRODUCTION
One of the important goals in high energy physics is to reveal the internal structure of the nucleon. This includes to study the flavor and spin contents of the quark and gluon constituents in the nucleon and how these contents are related to the nucleon properties: spin, magnetic moment, elastic form factors and deep inelastic structure functions. In the late 1980’s, the polarized deep inelastic lepton nucleon scattering experiments [1] surpris ingly indicated that only a small portion of the proton spin is carried by the quark and antiquarks, and a significant negative strange quark polarization in the proton sea. Since then, a tremendous effort has been made for solving this puzzle both theoretically and ex perimentally (recent review see [2-4]). According to the most recent result [5,6], the quarks contribute about one third of proton’s spin, which is ony one half of the spin expected from the hyperon decay data (AE ~ 0.6) and the strange quark polarization is about —0.10, which deviates significantly from the naive quark model expectation. On the other hand, the baryon magnetic moments can be reasonably well described by the spin-flavor structure in the nonrelativistic constituent quark model. Most recently, the New Muon Collaboration (NMC) experiments [7] have shown that the Gottfried sum rule [8] is violated, which indicates that the d density is larger than the u density in the nucleon sea. This asymmetry has. been confirmed by the NA51 Collaboration experiment [9], which shows that u/d ~ 0.51 at x = 0.15. From the quark model of the nucleon, the density of u would be almost the same as that of d if the sea quark pairs are produced by the flavor-independent gluons (s could be different because of m, >> mUij). Many theoretical works, trying to solve these puzzles, have been published. Among these, the application of chiral quark model, suggested by Eichten, Hinchliffe and Quigg [10], and then extended by Cheng and Li [11], seems to be more promising. The chiral quark model was first originated by Weinberg [12] and then developed by Manohar and
Georgi [13]. In this model, they introduced an effective Lagrangian for quarks, gluons and
Goldstone bosons in the region between the chiral symmetry breaking scale (Axsb — 1 GeV)
2 and the confinement scale (Aqcd — 0.1 — 0.3 GeV). The great success of the constituent quark model in low energy hadron physics can be well understood in this framework. In the chiral quark model the effective strong coupling constant a, could be as small as 0.2-0.3, which implies that the hadrons can be treated as weakly bound states of effective constituent
quarks. The model gave a correct value for (G/t/Gv)n-P = (5/3)= 1.25, with g* ~ 0.75, and a fairly good prediction for baryon magnetic moments. The extended description given by Cheng and Li [11] can solve many puzzles related to the proton flavor and spin structures: a significant strange quark presence in the nucleon indicated in the low energy pion nucleon sigma term II. SU(3) SYMMETRY BREAKING EFFECT. A. Cheng-Li’s description In the chiral quark model, in the scale range between Axsb and AqcDi the relevant de grees of freedom are the quasiparticles of quarks, gluons and the Goldstone bosons associated with the spontaneous breaking of the SU(3) x SU(3) chiral symmetry. In this quasiparticle description, the effective gluon coupling is small and the important interaction is the cou pling among quarks and Goldstone bosons, which may be treated as an excitation of qq pair produced in the interaction between the constituent quark and the quark condensate. Note that these Goldstone bosons can be identified, in quantum numbers, with usual pseudoscalar 3 mesons, but they propagated inside the nucleon. The sea quark-antiquark pairs could also be created by the gluons. Since the gluon is flavor independent, a valence quark cannot change its flavor by emitting a gluon. Also the spin cannot be changed due to the vector coupling nature between the quarks and gluons. On the, other hand, emitting a Goldstone boson a valence quark could change its flavor (for charge meson, but not for neutral meson) and certainly change its spin because of pseudoscalar coupling between the quarks and Goldstone bosons. The presence of Goldstone bosons in the nucleon cause quite a different sea quark flavor-spin content from that given by emitting gluons from the quarks. Hence the chiral quark model may provide a better understanding to the above puzzles. According to Ref.ll, the effective Lagrangian describing interaction between quarks and Goldstone bosons can be written ( u \ 2 _ , Li = ffs#.? + vfpoqv ? (2.1) where A+ \ 7T~ K° (2.2) K~ K° 'TV? I and A, (i=l,2,...8) are the Gell-Mann matrices. If the singlet Yukawa coupling is equal to zero g0 = 0, the quark sea created by emitting 0" meson octet would contain more d quarks than u quarks (and less a quarks). The resulting u-d asymmetry seems to be consistent with u-d < 0 indicated by the NMC data which show a significant violation of the Gottfried sum rule. However, according to the U(3) symmetry, the singlet is as important as the octet in the quark meson interactions and its Yukawa coupling is equal to the octet coupling go = g8. In this case, one surprisingly finds that the flavor asymmetry in the sea disappears and the numbers of uu, dd and as are equal. Cheng and Li suggested a reason to introduce an unequal singlet and octet coupling go/gs = ( ^ 1. This unequality may arise from the subleading contributions in the 1 /Nc expansion [14]. Having the parameter £ and another 4 parameter a ~ |g8|2, a straightforward calculation yields the following results: u = 2 + u; d = 1 + d; s = 0 + 5 (2.3) where 2,1, and 0 denote the numbers of valence u, d and s quarks in the proton, and « = |(C2 +2C + 6). d=^(C + S) (2.4) ■5 = ^(C2 -2( + 10) (2.5) For quark spin contents in the proton, similar calculation gives Au = ^ - ^(8(2 + 37 )o (2.6) Ad = -- + -(C2 - l)a (2.7) As = —a (2.8) One can see that the emission of the Goldstone boson leads to a reduced polarization to each flavor of up, down and strange quarks: less positive for Au, less negative for Ad, and a negative polarization for the strange quarks. This result can be very well understood in t he chiral quark model. Taking a = 0.1 and ( = —1.2, they obtained a/d = 0.53 (expt. : 0.51 ± 0.04 ± 0.05) (2.9) Ig = J dx 2V = 0.236 (expt. : 0.235 ± 0.026) (2.10) f- s iTWTJTT) = 019 :0.18±0.03) (2.11) Using the same parameters, the quark spin polarizations are also consistent with the data. However, the SU(3) symmetry description yields h/h = 1/3 (expt. : 0.23), A3/A„ = 5/3 (expt. : 2.1) (2.12) In Ref. 11, Cheng and Li guessed that this inconsistency between theoretical prediction and data could be attributed by some SU(3) breaking effects. In the next subsection, we in troduce an SU(3) breaking effect arising from the quark mass difference and extend their description. We find that the inconsistency can indeed be removed. B. SU(3) symmetry breaking description We assume that the breaking of SU(3)-flavor symmetry arises from a mass difference between strange and nonstrange quarks. Since the strange quark is heavier, this breaking would cause a smaller probability of strange meson emission from a u (or d) quarks to various quark-meson states. Defining $(u —» 7r +d) as the probability amplitude of a ir+ meson emission from a u quark etc., we have |tf(u —» 7r +d)|2 = |$(d —» 7T~u)| 2 = \g6\2 = a, etc. (2.13) for nonstrange meson emission, and |vf(u -> Jt+s)|2 = |vl>{d Jfc°s)|2 = ea (2.14) for strange meson emission. The new parameter t denotes the ratio between the probability of emitting strange meson and that of nonstrange meson from the quarks, and 0 < e < 1. We assume that e is proportional to the ratio mu,d/ms, where mUid and m, are the constituent quark masses of nonstrange and strange quarks. In the following, we will show that a reasonable value e ~ 0.5 — 0.6 gives a good fit to the data. It is easy to see that the nonstrange quark numbers would not be affected by the SU(3) symmetry breaking effect, but the strange quark number in eq.(2.5) would be reduced. The results are: * = 5K2 + 2(4-6) (2.15) ^ = |(C2 + 8) (2.16) 5 = ^((2-2( + 10)-3o(l-e) (2.17) For the spin contents of the proton, instead of eqs.(2.6) —(2.8), we obtain Au = - — -(8(2 + 37) + —(1 — c) (2.18) Ad = —- + — K2 - 1) - -(1 - c) (2.19) 6 As = —a + a( 1 — e) (2.20) One can see that with the SU(3)-breaking effect, Au would be more positive, Ad more negative and As less negative. Defining /a = /„ — /j, /8 = /„ + fd — 2/,, Aa = Au - Ad, and A8 = Au + Ad - 2As, from (2.15)—(2.17) and (2.18) —(2.20), we have f^l h .,( 4o(l-<)i-0 ^ (2.21) J 1 + ttWti and ^ si_](2<^7)+^ ) (2.22) 3 1 - §(2£2 + 7) — a(l — e)' It is obvious that the correction factors, appearing in eqs (2.21) and (2.22), due to the SU(3)- breaking, are in the right direction, i.e. /3//8 decreases (provided that 1 + y(£ — 1) > 0) and A3/A8 increases. The SU(3) results can be recovered by taking e —* 1. III. NUMERICAL RESULTS AND DISCUSSION To maintain the agreement obtained in the SU(3) symmetry description, we choose a — 0.1 and £ = -1.2 used in Cheng and Li’s work. For the SU(3) breaking parameter e, we choose t = 0.5 - 0.6 as mentioned in the previous section. Having three parameters o, £ and ( in the SV(3)-breaking description, one obtains h/fs = 0.26 (expt. : 0.23), A3/A8 = 1.94 (expt. : 2.10) (3.1) the theoretical predictions are now much closer to the data than those from SU(3) description and the inconsistency shown in Ref. 11 is removed. The results for the quark flavor and spin contents in the proton are listed in Tables I and II respectively. Comparison with the SU(3) symmetry prediction and data are shown. For comparison, the results from the analysis given by Ellis and Karliner [15] are also shown. Several remarks are in order. (1) Assuming that the hyperon beta decay matrix elements are not sensitive to the S V(3) breaking introduced above (see Ref. [16]). we can write A3 = F+ D = Au — Ad and i Ag = 3F — D — Au + Ad — 2As. From the spin contents shown in Table II, we obtain F/D and other weak axial couplings. The results are listed in Table III It shows that our description gives a better agreement with the hyperon (3 decay data [17,18] as well. (2) The integrals of g\ and g" including QCD corrections can be written as P = ^Au + 1(2CS - Cyvs)A£ (3.2) /' = 7,1-^A, + 1(4CS + C„S)A£] (3.3) where g = 0.4565 and Cs(Q2), Chs (Q 2) are the QCD radiative correction factors given in Ref. 19. Taking a, = 0.35 at Q2 = 3.0 GeV3 and using the spin contents in Table II, the integrals P and Id are evaluated and listed in Table III. One can see that the SU(3)-breaking results are also better than those without SU(3)- breaking except for the integral of gd. Our prediction of Id is higher than both SMC and E143 data. In addition, our P value is closer to the SMC data, while the SU(3) symmetry prediction is closer to the E143 data. (3) For comparison, we also evaluated a quantity defined as < A? >= 2 < z > (3.4) which is a crude approximation of the asymmetry A\ measured in the deep inelastic lepton proton scattering, where < x > is the average value of the Bjorken variable x and can be taken as 0.5 — 0.7. Taking the numbers given in (2.15) to (2.20), we have < A\ >= 0.20 - 0.30, (e = 1.0) 0.24 -0.34 (e = 0.5) (3.5) the data from E143 jf1 A\{x)dx = 0.40 ±0.10 (3.6) seems to favor the symmetry-breaking description. (4) If we decompose the valence and sea contributions for the flavor-spin contents in the proton, from (2.15) —(2.17) we have u,eo = u = ^((2 + 2( + 6) (3.7) 8 daea = d=^(C+8) (3-8) s,ea = s - -(( - l)2 + 3ea (3.9) here the equality of quark sea and the antiquark sea is because of the sea must be flavorless. It is obvious that the chiral quark model predicts that the sea not only violates SU(3) flavor symmetry but also violates SU(2) symmetry, and s < u < d (3.10) except a special case: ( = 1 and e = 1. It yields a complete SU(3) symmetric sea: u = d = s, which was discussed in Ref. 10. (5) For sea quark spin contents, we have Au‘" = -?(8<2+37) +j(l-t) (3.11) Ai”“=+y(C!-l)-j(l-e) (3.12) As,ra = -ae (3.13) Fnlike the equality q,ea = q, required by the flavorless of the sea, we do not have similar equality for sea quark and sea antiquark polarizations, Ag,ea ^ Aq (q = u, d, s) (3.14) because all antiquark polarizations are zero Au = A d = As = 0 (3.15) in the chiral quark model (with or without SU(3) breaking described in this work). The smallness of antiquark polarization has been discussed by Cheng and Li [20] and could be tested by the future experiments. (6) In the SU(3) symmetry description, one has (C = 1, e = 1) Au,eo < 0, Adtea > 0. As,eo < 0 (3.16) 9 It implies that the sea quark of each flavor is polarized in the direction opposite to the valence quark of the same flavor. However, in the SU(3) breaking description, Ad* ea could be negative if 1 — e > |(C2 — 1) — 0.3. or e < 0.7. In this case, all sea quarks are spinning in the opposite direction with respect to the proton spin. This includes the cases (c = 0.5 or 0.6) discussed in this work. IV. SUMMARY In this note we introduced an SU(3) symmetry breaking effect into the chiral quark model. A breaking parameter c = |#(u —» fc+s)|2/|'P(u —> 7r +d)|3 < 1, which denotes a smaller probability of strange meson emission from a nonstrange quarks, is suggested. Taking e ~ 0.5 — 0.6, the /3//8 and A3/Ag values are much closer to the data. With the breaking effect, many other theoretical predictions are also in better agreement with the data. We note, however, that although the simple calculation carried out in this paper can account for many basic features of the spin and flavor contents of the nucleon, the model does not have power to predict the flavor and spin distributions in the nucleon. Furthermore, no (^-dependence can be discussed in this simple model. Since the predictions given by this model are quite close to the data, which implies that the basic picture given by the chiral quark model might be not far from the reality. Therefore it would be worthwhile to develop this model and reformulate it into a field theoretical form. Acknowledgements I would like to thank the International Center for Theoretical Physics for the award of Senior Associate Membership, which allows me to complete this paper during my visit at the Center from June to August 1996. I also thank P. K. Kabir, J. S. McCarthy and H. J. Weber for their helpful comments and suggestions. The author thanks D. Crabb and O. Rondon for providing the new E142 and E143 data. 10 REFERENCES [1] J. Ashman et ah, Phys. Lett. B206, 364 (1988); Nucl. Phys. B328, 1 (1989) [2] R. h. .laffe, Physics Today, September, 24 (1995). [3] M. Ansel mi no. A. Efremov and E. Leader, Phys. Rep. 261, 1 (1995) [4] F. E. Close, hep-ph/9509251, September 8 (1995) [.*>) B. Adeva et d. Phys.Lett. B302, 553 (1993); B320, 400 (1994) D. Adams et d., Phys.Lett. B329, 399 (1994); B336, 125 (1994). [6] P. L. Anthony et d. Phys.Rev.Lett. 71, 959 (1993); K. Abe et ol. Phys.Rev.Lett. 74, 346 (1995); 75, 25 (1995). [7] New Muon Collaboration, P. Amaudruz et d., Phys. Rev. Lett. 66, 2712 (1991; M. Arneodo et d.. Phys. Rev. D50, Rl, (1994). [8] K. Gottfried, Phys. Rev. lett. 18, 1174 (1967). [9] NA51 Collaboration, A. Baldit et d., Phys. Lett. B332, 244 (1994). [10] E. J. Eichten, I. Hinchliffe and C. Quigg, Phys. Rev. D45 2269 (1992); see also J. D. Bjorken, Report No, SLAC-PUB-5608 191 (1991) (unpublished) III] I . P. Cheng and Ling-Fong Li, Phys. Rev. Lett. 74 2872 (1995). [12] S. Weinberg. Phvsica 96 A, 327 (1979). Il l] A. Manohar and H. Cleorgi, Nucl. Phys. B234, 189 (1984). [I I] T. P. Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Claren don, Oxford, 1984), p.110. [15] J. Ellis and M. Karliner, Phys. Lett B34 1 397 (1995). [16] X. Song, P. K. Kabir and J. S. McCarthy, Phys. Rev. D54 2108 (1996) [17] Particle Data Group, L. Montanet et al.. Phys. Rev. D50 1173 (1994). [IS] F. E. Close and R. G. Roberts, Phys. Lett. b316 165 (1993). [19] S. A. Larin, Phys. Lett. B334, 192 (1994) [20] T. P. Cheng and Ling-Fong Li, hep-ph/9510235 11 Table I: Quark flavor contents for the proton in the chiral quark model (a = 0.1, £ = — 1.2) with (< = 0.5 and 0.6) and without (e = 1.0) SU(3) symmetry breaking. o to su II II Oi Quantity Data o b i e=1.0 u/d 0.51 ± 0.09 0.53 0.53 0.53 Ig 0.235 ± 0.026 0.236 0.236 0.236 /« — 0.51 0.50 0.48 U — 0.35 0.35 0.33 f. 0.18 ±0.03 0.15 0.15 0.19 h!h 0.23 ± 0.05 0.26 0.27 0.33 12 Table II: Quark spin contents for the proton in the chiral quark model (a = 0.1, £ = —1.2) with (c = 0.5) and without (e = 1.0) SU(3) symmetry breaking. Quantity Data (=0.5 e=0.6 (=1.0 An 0.8-1 ±0.05 (E143) 0.86 0.85 0.79 0.83± 0.05 (SMC) 0.85± 0.03 (E-K) A d -0.43± 0.05 (E143) -0.34 -0.34 -0.32 —0.44± 0.05 (SMC) —0.41± 0.03 (E-K) A.s -0.08± 0.05 (E143) -0.05 -0.06 -0.10 -0.09± 0.05 (SMC) -0.06± 0.04 (E142) —0.08± 0.03 (E-K) AT' 0.30± 0.06 (El 13) 0.47 0.45 0.37 0.20± 0.11 (SMC) 0.39± 0.10 (E142) 0.37± 0.07 (E-K) Ai/A» 2.09± 0.13 1.94 1.88 1.67 13 Table III: Hyperon beta-decay constants and the integrals of spin structure functions in the chiral quark model (o = 0.1, ( = —1.2) with (e = 0.5 and 0.6) and without (e = 1.0) SU(3) symmetry breaking. Quantity Data 6=0.5 (=0.6 6=1.0 F-D 1.2573± 0.0028 1.20 1.19 1.11 F+D/3 0.718± 0.015 0.70 0.70 0.67 F-D -0.340=1= 0.017 -0.29 -0.28 -0.22 F-D/3 0.25=1= 0.05 0.21 0.21 0.22 F/D 0.575± 0.016 0.61 0.62 0.67 Ip 0.136± 0.016 (SMC) 0.137 0.136 0.128 0.127± 0.011 (E143) In -0.031± 0.011 (E142) -0.021 -0.022 -0.021 -0.037 j= 0.014 (E143) ld 0.034± 0.011 (SMC) 0.053 0.052 0.049 0.042=1= 0.005 (E143)