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Flavor and Spin Contents of the Nucleon in the Quark Model with Chiral Symmetry



Flavor and Spin Contents of the Nucleon in the Quark Model with Chiral Symmetry
T. P. Cheng2 and Ling-Fong Li1
2

Department of Physics, Carnegie-Mellon University, Pittsburgh, PA 15213 Department of Physics and Astronomy, University of Missouri, St. Louis, MO 63121 (CMU-HEP94–30, hep-ph/9410345)

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arXiv:hep-ph/9410345v2 24 Oct 1994

A simple calculation in the framework of the chiral quark theory of Manohar and Georgi yields results that can account for many of the ”failures” of the naive quark model: signi?cant strange quark content in the nucleon as indicated by the value of σπN , the u-d asymmetry in the nucleon as measured by the deviation from Gottfried sum rule and by the DrellYan process, as well as the various quark contributions to the nucleon spin as measured by the deep inelastic polarized lepton-nucleon scatterings.

One of the outstanding problems in non-perturbative QCD is to understand, at a more fundamental level, the successes and failures of the non-relativistic quark model. An important step in that direction was taken years ago by Manohar and Georgi [1] when they presented their chiral quark model with an e?ective Lagrangian for quarks, gluons and Goldstone bosons in the region between the chiral symmetry breaking and the con?nement scales. They demonstrated how the successes of the simple quark model could be understood in this framework, which naturally suggests a chiral symmetry breaking scale ΛχSB ? 1 GeV , signi?cantly higher than the QCD con?nement scale. And it also allows for the possibility for a much reduced (e?ective) strong coupling αS , thus the result of hadrons as weakly bound states of constituent quarks. A meaningful calculation of the magnetic moments of the baryons (with great success) can also be carried out, etc. In this note we shall show that this framework, with a simple extension, can also remedy many of the ”experimental failures” of the simple quark model. It has been known for a long time that the measured value of the pion nucleon sigma term σπN indicates a signi?cant strange quark presence in the nucleon [2]. This failure of the simple quark model is further high-lighted when the deep inelastic polarized muon-proton scattering measurements made by EMC [3] indicated a signi?cant contribution to the proton spin by the strange quarks in the sea [4]Then came along the NMC [5] result showing that the Gottfried sum rule [6] is not satis?ed experimentally, indicating a nucleon sea that is quite asymmetric with respect to its u and d quark contents. This basic piece of physics is now con?rmed by a dedicated Drell-Yan experiment colliding protons on proton and on neutron targets [7]. An important feature of the chiral quark model is that the internal gluon e?ects can be small compared to the in-

ternal Goldstone bosons and quarks. As we shall demonstrate, this picture can account for, in terms of two parameters, the broad pattern of the observed ?avor and spin contents of the nucleon. In the chiral quark model, the dominant process is the ?uctuation of quark into quark plus a Nambu-Goldstone boson. This basic interaction causes a modi?cation of the spin content because a quark can change its helicity by emitting a spin zero meson, Fig. 1(a). It causes a modi?cation of the ?avor content because the Goldstone boson will in turn ?uctuate into a quark-antiquark pair, Fig. 1(b). Instead of using the parton evolution equation of the chiral quark theory, as has been carried out by Eichten, Hinchli?e, and Quigg [9], we will illustrate the basic physical mechanism with a schematic calculation, as was also considered by these authors. In the absence of interactions, proton is made up of two u quarks and one d quark. We now calculate proton’s ?avor content after any one of these quarks emits a quark-antiquark pair via Goldstone bosons, which have interaction vertices: LI = g8 q φq, ? ? πo η √ + √ π+ K+ 8 2 6 ? η πo π? ?√ +√ Ko ? λi φi = ? φ= ?, 2 6 2η ? o i=1 √ K K ? 6 q = (u, d, s), and the λ′ s are the Gell-Mann matrices. We have suppressed all the spacetime structure and have only displayed the ?avor content of the coupling. The probability amplitudes of meson emission from an u quark to various meson-quark states (as given by the coe?cients in front of the channel names) are 1 1 Ψ (u) = g8 π + d + K + s + √ π o u + √ ηu . 2 6 (1)

From this, we deduce that 1 ? 8 3 a is the probability of no meson emission with a being the probability of emitting 2 a π + or a K + : a ∝ |g8 | . By substituting the quark content of the mesons into the above equation, one obtains the proton’s ?avor composition (after one interaction): 8 2 2 1 ? a (2u + d) + 2 |Ψ (u)| + |Ψ (d)| 3 with |Ψ (u)|2 = 2 |Ψ (d)| = 1
2 9 2 9

14u + 2u + 5 d + d + s + s a, 14d + 2d + 5 (u + u + s + s) a .

(2)

In this picture the strange quarks in the sea is brought about by the ?uctuation of the valence u and d quarks into the s-quark-containing mesons: K ’s and η. An asymmetry develops between u and d distributions because there is an initial u-d asymmetry in the valence quarks and the u quark cannot ?uctuate into π ? (hence a ?nal state containing u) while the d quark cannot ?uctuate into π + , etc. Since the 1/Nc expansion, Nc being the number of colors, is thought to be a useful approximation scheme for QCD, it will be worthwhile to see what will it teach us here. The leading contribution comes from the planar diagrams. At this order, there are nine Goldstone bosons, including the unmixed diagonal channels: uu, dd and ss, all with the same couplings. If we express this in the language of SU(3), besides the octet there is also the singlet, with the singlet Yukawa coupling being equal to the octet coupling: g0 = g8 . When this ninth singlet η ′ meson (here it is not the physical η ′ , but the singlet meson in the planar approximation) is included in the computation, one ?nds a surprising result of a ?avor-independent sea [9]. Namely, the original u-d asymmetry due to the asymmetric π ± ?uctuations is just cancelled by the corresponding asymmetry due to the coherent neutral meson emissions. As a result, there is now an equal number of uu and d d, as well as ss pairs. Mathematically, this ?avor independence comes about as follows. Equating the coupling constants g8 = g0 in the extended vertex,
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nonplanar contributions, the equality between the octet and singlet couplings is broken: g0 /g8 ≡ ζ = 1 .We can now repeat the above calculation and express our result in terms of the two parameters— this coupling ratio ζ and a, the probability of π + emission: u = 2 + u, d = 1 + d, and s = s, with the antiquark contents being: u= a 2 a 2 ζ + 2ζ + 6 , d = ζ +8 , 3 3 a 2 s= ζ ? 2ζ + 10 . 3 (4) (5)

Let us review our phenomenological knowledge of the ?avor content of the nucleon: The deviation from the Gottfried sum rule for the deep inelastic lepton-nucleon scatterings is interpreted as showing an asymmetry between the u and d quarks in the nucleon sea:
1

dx
0

p n 2 F2 (x) ? F2 (x) 1 u?d . ? = x 3 3

(6)

The NMC measurements did indeed show the Gottfried integral being signi?cantly di?erent from one third: IG = 0.235 ± 0.026 . (7)

LI = g8

qλi φi q +
i=1

2 g0 qη ′ q 3

(3)

Here we have quoted the new New Muon Collaboration result published this year [5]. This u-d asymmetry has now been con?rmed by the NA51 Collaboration [7] at CERN in a Drell-Yan experiment of scattering protons on proton and on deuteron targets [8]. The measured ratio of muon pair production cross sections σpp and σpn can be expressed as the antiquark content ratio: u/d = 0.51 ± 0.04 (stat) ± 0.05 (syst). (8)

and squaring the amplitude, one obtains the probability distribution of
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This quantity is of particular interest in our model calculation, since it depends only on one parameter: u/d = ζ 2 + 2ζ + 6 . ζ2 + 8 (9)

|Ψ| ∝

2

( qλi q ) ( qλi q ) +
i=1

2 ( qq ) ( qq ) 3

which has the index structure as 2 ( λi )ab ( λi )cd + δab δcd = 2δad δbc . 3 i=1 The r.h.s., deduced from a well-known identity of the Gell-Mann matrices [10], clearly shows the ?avor independent nature of the result. While the mathematics of this ?avor asymmetry cancellation is fairly straightforward, the physics is more intriguing. It shows that the deviation from an SU(3) symmetric sea should, for the most part, spring from the nonplanar contributions. That nonplanar contributions are important for an adequate description of the nonperturbative QCD is to be expected. In fact the axial anomaly vanishes at the planar-diagram level. The resolution of the η ′ mass problem depends on the nonplanar physics [11]. However, with the admission of the 2
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Interestingly, with only the octet Goldstone contribution (thus ζ = 0) this ratio is ?xed to be 0.75, comparing rather poorly with the experimental result in Eq. (8). With the inclusion of the singlet contribution (ζ = 0), this expression still implies a lower bound for the u/d ratio of 1/2 at ζ = ?2, and an upper bound of 5/3 at ζ = 4. More relevantly, the experimental value in Eq. (8) implies that the coupling ratio ζ must be negative: ?4.3 ≤ ζ ≤ ?0.7 . Given the crudity of our calculation and the sensitivity of the quadratic relations, we shall merely illustrate our model calculation results in the subsequent discussion with the following simple parameter choice: a = 0.1 , and ζ = ?1.2 (10)

which corresponds to u/d = 0.53, and to ?xing a in Eq. (4) so that Eq. (6) will yield the central experimental value of the Gottfried sum:

IG =

1 2 1 4 u ? d = + (ζ ? 1) a = 0.235 . + 3 3 3 9
qa +qa ( q +q )

?u =

proton can also be calculated from Eq. (4) with the parameters of Eq. (10): fu ? 0.48, fd ? 0.33, fs ? 0.19. (11)

The fractions of quark ?avors fa ≡

in the

4 1 ? 8ζ 2 + 37 a, 3 9 1 2 2 ζ ? 1 a, ?d = ? + 3 9 ?s = ?a ?d = ?0.32, ?s = ?0.10.

(15) (16) (17)

which for the parameters of Eq. (10) yields the values: ?u = 0.79, (18)

Observationally, the strange quark fraction fs can be deduced from the phenomenological value [12] of the πN sigma term σπN = m N uu + dd N = 45 ± 10 M eV where m ≡ tion M8 ≡
1 2 1 3

(12)

(mu + md ) , and the SU(3) symmetry rela-

(m ? ms ) N uu + dd ? 2ss N . = MΛ ? MΞ ? ?200 M eV

(13)

Using the current algebra result of ms /m = 25 we obtain the ratio 1 ? 2y ≡ uu + dd ? 2ss uu + dd = 3mM8 5 ? . (m ? ms ) σπN 9

Since the 1988 announcement by EMC of their proton spin content result, there has been signi?cant new experimental developments in the measurement of the spindependent structure functions of the neutron, as well as that of the proton [13]. In the meantime further higher order perturbative QCD calculations have been carried out for such spin-dependent structure function sum rules [14]. Taking into account of these higher order results [hence the perturbative QCD predicted Q2 dependence through αs (Q2 )], Ellis and Karliner have recently shown [15] that all the diverse experimental measurements are consistent with each other, and that the fundamental Bjorken sum rule is veri?ed to about 12%. When the ?nal result, after using a ?avor SU(3) symmetry relation [similar to that of Eq. (13)], is expressed in terms of the individual quark contributions to the proton spin, ?uexp = 0.83, ?dexp = ?0.42, ?sexp = ?0.10, (19)

Keeping in mind that the scalar operator qq measures the sum of the quark and antiquark numbers (as opposed to q ? q, which measures the di?erence), we ?nd (fs )σπN = y ? 0.18 . 1+y (14)

The good agreement between this and the value in Eqs. (11) should be regarded as fortuitous in view of the fact the value in Eq. (14) involves the ?avor SU(3) symmetry relation (13). Thus an uncertainty of at least 20%, above and beyond that shown in Eq. (12), has to be included in this estimate. We now turn to the nucleon spin content. In the no interaction limit, the spin-up proton (p+ ) wave function 1 p+ = √ (2u+ u+ d? ? u+ u? d+ ? u? u+ d+ ) implies that 6 the probabilities of ?nding u+ (u quark in the spin-up 5 1 1 2 state), u? , d+ and d? in p+ are 3 , 3 , 3 and 3 , respectively. These values will be altered by the same meson emission processes of the chiral quark model as discussed above in connection with proton’s ?avor content. The 2 total probability of Goldstone emission being a 3 ζ +8 , the contributions of the various spin states, after one interaction, can then be read o? from
5 +3 1 1 2 5 2 1? a 3 ζ +8 3 u+ + 3 u? + 3 d+ + 3 d? 2 2 2 2 1 1 2 |Ψ (u+ )| + 3 |Ψ (u? )| + 3 |Ψ (d+ )| + 3 |Ψ (d? )| ,

where amplitudes Ψ (q± ) can be calculated in an entirely similar manner as that done for Eq. (2). The quark contributions to the proton spin ?q = q+ ? q? are:

we have a comparison with our result in Eq. (18). One of the principal outcome of the this recent round of experimentation and phenomenological analysis is the conclusion that the total quark spin contribution actually does not vanish: ?Σ ≡ ?u + ?d + ?s ? 0.31, which is to be compared to our model calculation result of 0.37 in Eq. (18). We should also note that ours is an SU(3) symmetric computation. The basic feature that the strange quark is heavier than the up and down quarks have not been taken into account. The meson emission corrections for each component of an SU(3) multiplet must, in such a calculation, be the same. Consequently, the naive quark model ratio of ?3 /?8 = 5/3 is unchanged (where ?3 = ?u ? ?d and ?8 = ?u +?d ? 2?s). It is about 25% lower than the phenomenological value [16] of (?3 /?8 )exp ? 2.1. And, the similarly de?ned ?avor-fraction ratio of f3 /f8 = 1/3 is greater than the experimental value of (f3 /f8 )exp ? 0.23. Inclusion of the SU(3) breakings will necessarily increase the number of parameters and complicate the model calculation. We postpone such an endeavor to a later stage, and choose to present our principal results without having them masked by this complication. Overall we ?nd it rather encouraging that this simple calculation in a theoretically well-motivated framework can account for many of the puzzling features discovered in recent years of the spin and ?avor contents of the nucleon. To us what is signi?cant is this broad pattern of agreement in one uni?ed calculation. Our e?ort 3

overlaps with many of the previous works [17], where these e?ects have been discussed as unrelated phenomena. What we wish to emphasize in this work is that the nonrelativistic quark model, when supplemented with the Goldstone structure, does yield an adequate approximation of the observed low energy physics [18]. A key ingredient in this implementation is the inclusion of the ninth Goldstone boson with a di?erently renormalized coupling g0 ? ?1.2g8. Presumably the success with such an inclusion shows that, above the con?nement scale, the nonplanar mechanism which endows the η ′ with a mass can still be treated as a perturbation as suggested by the 1/Nc expansion of QCD. With this ?rst encouraging result, it might be worthwhile to embark on a more elaborate ?eld theoretical calculation. This will involve more mass, cuto? parameters, and further phenomenological inputs, but it will also allow a more detailed comparison of the Q2 and x dependences of the structure functions between the chiral quark theory expectations and the experimental measurements. One of us (L.F.L.) would like thank Ernest Henley for useful conversation and for bringing Ref. [7] to our attention. This work is supported in at CMU by the Department of Energy (DE-FG02-91ER-40682), and at UM-St.Louis by an RIA award and by the National Science Foundation (PHY-9207026). ? E-mail addresses: stcheng@slvaxa.umsl.edu, and lfli@bethe.phys.cmu.edu ? Figure Caption: Fig. 1. Spin and ?avor corrections due to Goldstone boson ?uctuations. (a) A spin zero meson couples to quarks of opposite helicites. (b) Production of a (q ′ q ′ ) pair via Goldstone emission.

[1] A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984); S. Weinberg, Physica (Amsterdam) 96A, 327 (1979), Sec. 6; H. Georgi, Weak Interactions and Modern Particle Theory, (Benjamin/Cummings, Menlo Park, CA, 1984), Sec. 6.4 and 6.5. [2] T. P. Cheng, Phys. Rev. D 13, 2161 (1976); T. P. Cheng and R. F. Dashen, Phys. Rev. Lett. 26, 594 (1971). [3] European Muon Collaboration, J. Ashman, et al., Phys. Lett. B206, 364 (l988); Nucl. Phys. B328, 1 (1990). [4] J. Ellis and R. L. Ja?e, Phys. Rev. D 9, 1444 (1974). [5] New Muon Collaboration, P. Amaudruz et al. Phys. Rev. Lett. 66, 2712 (1991); M. Arneodo et al. Phys. Rev. D 50, R1 (1994). [6] K. Gottfried, Phys. Rev. Lett. 18, 1174 (1967). [7] NA 51 Collaboration, A. Baldit et al. Phys. Lett. B332, 244 (1994). [8] S. D. Ellis and W. J. Stirling, Phys. Lett. B256, 258 (1993).

[9] E. J. Eichten, I. Hinchli?e, and C. Quigg, Phys. Rev. D 45, 2269 (1992); see also J. D. Bjorken, Report No. SLAC-PUB-5608, 1991 (unpublished). [10] See, for example, T. P. Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics (Clarendon Press, Oxford, 1984), p.110. [11] G. Veneziano, Nucl. Phys. B159, 213 (1979). [12] J. Gasser, H. Leutwyler, and M. E. Sainio, Phys. Lett. B253, 252 (1991). [13] Spin Muon Collaboration, B. Adeva et al., Phys. Lett. B302, 533 (1993), D. Adams et al., Phys. Lett. B329, 399 (1994); E142 Collaboration, P. L. Anthony et al., Phys. Rev. Lett. 71, 959 (1993); E143 Collaboration, R. Arnold et al., preliminary results presented at St. Petersburg (Florida) Conference, June 1994. [14] J. Kodaira et al., Phys. Rev. D 20, 627 (1979); S. A. Larin, F. V. Tkachev, and J. A. M. Vermaseren, Phys. Rev. Lett. 66, 862 (1991); A. L. Kataev and V. Starshenko, CERN Th.7208-94, hep-ph/9403383. [15] J. Ellis and M. Karliner, CERN Th. 7324-94, hepph/9407287; also see F. Close and R.G. Roberts, Phys. Lett. B316, 165 (1993). [16] ?8 = 0.601 ± 0.038 from S. Y. Hsueh et al., Phys. Rev. D 38, 2056 (1988); ?3 = gA = 1.2573 ± 0.0028. [17] For the relevant prior investigations, besides of those already cited, see, for example, T. Hatsuda and T. Kunihiro, Phys. Reports (to be published); for σπN and the strange quark content, see J. F. Donoghue and C. R. Nappi, Phys. Lett. B168, 105 (1986); for the proton spin contents, see, e.g., S. J. Brodsky, J. Ellis, and M. Karliner, Phys. Lett. B206, 309 (1988); H. Dreiner, J. Ellis, and R. Flores, Phys. Lett. B221, 169 (1989); H. Fritzsch, Phys. Lett. B229, 122 (1989); S. Forte, Nucl. Phys. B331, 1(1990); for the Gottfried sum rule and Drell-Yan processes, see, e.g., E. M. Henley and G. A. Miller, Phys. Lett. B251, 453 (1990); S. Forte, Phys. Rev. D 47, 1842 (1993); E. Eichten, I. Hinchli?e, and C. Quigg, Ibid, R747 (1993); R. Ball and S. Forte, Nucl. Phys. B425, 516 (1994). [18] For a recent discussion of the quark model (without the Goldstone structure) with respect to the Q2 = 0 probes of the nucleon spin content, see F. Close, Talk at the 6th ICTP Workshop, Trieste (1993), Rutherford Appleton Lab Report RAL-93-034.

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This figure "fig1-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9410345v2

q+ (a)

q-

q q' q (b) q'



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