A Determination of the Nucleon Tensor Charge

A Determination of the Nucleon Tensor Charge Leonard Gamberga , Gary R. Goldsteinb a b Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, U.S.A. Department of Physics and Astronomy, Tufts University, Medford, MA, USA arXiv:hep-ph/0111095v1 8 Nov 2001 (Received: November 9, 2018) Exploiting an approximate phenomenological symmetry of the J P C = 1+− light axial vector mesons and using pole dominance, we calculate the flavor contributions to the nucleon tensor charge. The result depends on the decay constants of the axial vector mesons and their couplings to the nucleons. 1 Introduction The spin composition of the nucleon has been intensely studied and has produced important and surprising insights, beginning with the revelation that the majority of its spin is carried by quark and gluonic orbital angular momenta and gluon spin rather than by quark helicity [1,2]. In addition, considerable effort has gone into understanding, predicting and measuring the transversity distribution, h1 (x), of the nucleon [3]. Transversity, as combinations of helicity states, |⊥/⊤ >∼ ( |+ > ±|− > ), for the moving nucleon is a variable introduced originally by Moravcsik and Goldstein [4] to reveal an underlying simplicity in nucleon–nucleon spin dependent scattering amplitudes. In their analysis of the chiral odd distributions, Jaffe and Ji [5] related the first moment of the transversity distriR1 a bution to the flavor contributions of the nucleon tensor charge: 0 (δq (x) − δq a (x)) dx = δq a (for flavor index a). This leading twist transversity distribution function, δq a (x), is as fundamental to understanding the spin structure of the nucleon as its helicity counterpart ∆q a (x). However, while the latter in principle can be measured in hard scattering processes, the transversity distribution (and thus the tensor charge) decouple at leading twist in deep inelastic scattering since it is chiral odd. Additionally, the non-conservation of the tensor charge makes it difficult to predict. While bounds placed on the leading twist quark distributions through positivity constraints suggest that they satisfy the inequality of Soffer [6], there are no definitive theoretical predictions for the tensor charge. In contrast to the axial vector isovector charge, no sum rule has been written that enables a clear relation between the tensor charge and a low energy measurable quantity. Among the various approaches, from the QCD sum rule to lattice calculations models [7], there appears to be a range of expectations and a disagreement concerning the sign of the down quark contribution. We present a new approach to calculate the tensor charge that exploits the approximate mass degeneracy of the light axial vector mesons (a1 (1260), b1 (1235) and h1 (1170)) and uses pole dominance to calculate the tensor charge [8, 9]. Our motivation stems in part from the observation that the tensor charge does not mix with gluons under QCD evolution and therefore behaves as a non-singlet matrix element. In conjunction with the fact that the tensor current is charge conjugation odd (it does not mix quark-antiquark excitations of the vacuum, since the latter is charge conjugation even) suggests that the tensor charge is more amenable to a valence quark model analysis. 2 The Tensor Charge and Pole Dominance The flavor components of the nucleon tensor charge are defined from the forward nucleon matrix element of the tensor current, hP, ST ψσ µν γ5 λa ψ P, ST i = 2δq a (µ2 )(P µ STν −P ν STµ ). 2 1 (1) We adopt the model that the nucleon matrix element of the tensor current is dominated by the lowest lying axial vector mesons a X h0 ψσ µν γ5 λ ψ MihM, P, ST |P, ST i λa 2 hP, ST ψσ γ5 ψ P, ST i = lim . 2 − k2 2 MM k 2 →0 µν (2) M The summation is over those mesons with quantum numbers, J P C = 1+− that couple to the nucleon via the tensor current; namely the charge conjugation odd axial vector mesons – the isoscalar h1 (1170) and the isovector b1 (1235). To analyze this expression in the limit k2 → 0 we require the vertex functions for the nucleon coupling to the h1 and b1 meson and the corresponding matrix elements of the meson decay amplitudes which are related to the meson to vacuum matrix element via the quark tensor current. The former yield the nucleon coupling constants gMNN and the latter yield the meson decay constant fM . Taking a hint from the valence interpretation of the tensor charge, we exploit the phenomenological mass symmetry among the lowest lying axial vector mesons that couple to the tensor charge; we adopt the spin-flavor symmetry characterized by an SU (6) ⊗ O(3) [10] multiplet structure. Thus, the 1+− h1 and b1 mesons fall into a (35 ⊗ L = 1) multiplet that contains J P C = 1+− , 0++ , 1++ , 2++ states. This analysis enables us to relate the a1 2 meson decay constant measured in τ − → a− 1 + ντ decay [11], fa1 = (0.19 ± 0.03)GeV , and the a1 N N coupling constant ga1 NN = 7.49 ± 1.0 (as determined from a1 axial vector dominance for longitudinal charge as derived in [12] but using gA /gV = 1.267 [13]) to the meson decay constants and coupling constants. We find √ 5 2 (3) fa1 , gb1 NN = √ ga1 NN , fb1 = Mb1 3 2 √ where the 5/3 appears from the SU (6) factor (1 + F/D) and the 2 arises from the L = 1 relation between the 1++ and 1+− states. Our resulting value of fb1 ≈ 0.21 ± 0.03 agrees well with a sum rule determination of 0.18 ± 0.03 [14]. The h1 couplings are related to the b1 couplings via SU (3) and the SU (6) F/D value, fb1 = √ 3fh1 , 5 gb1 NN = √ gh1 NN 3 (4) For transverse polarized Dirac particles, S µ = (0, ST ) these values, in turn, enable us to determine the isovector and isoscalar parts of the tensor charge, 2i fb gb NN hk⊥ δq v = √1 1 , 2MN Mb21 2i fh gh NN hk⊥ δq s = √1 1 , 2MN Mh21 (5) respectively (where, δq v =(δu − δd), and δq s =(δu + δd)). Transverse momentum appears in these expressions because the tensor couplings involve helicity flips that carry kinematic factors of 3momentum transfer, as required by rotational invariance. The squared 4-momentum transfer of the external hadrons goes to zero in Eq. (2), but the quark fields carry intrinsic transverse momentum. This intrinsic k⊥ of the quarks in the nucleon is determined from Drell-Yan processes and from heavy vector boson production. 3 Mixing In relating the b1 (1235) and h1 (1170) couplings in Eq. (4) we assumed that the latter isoscalar was a pure octet element, h1 (8). Experimentally, the higher mass h1 (1380) was seen in the K + K̄ + π ′ s 2 decay channel [13,15] while the h1 (1170) was detected in the multi-pion channel [13,16]. This decay pattern indicates that the higher mass state is strangeonium and decouples from the lighter quarks – the well known mixing pattern of the vector meson nonet elements ω and φ. If the h1 states are mixed states of the SU (3) octet h1 (8) and singlet h1 (1) analogously, then it follows that 3 (6) gh1 (1170)N N = gb1 NN , 5 √ with the h1 (1380) not coupling to the nucleon (for gh1 (1)NN = 2gh1 (8)NN ). These symmetry relations yield the results fh1 (1170) = fb1 , δu(µ2 ) = (0.58 to 1.01) ± 0.20, δd(µ2 ) = −(0.11 to 0.20) ± 0.20. (7) These values are similar to several other model calculations: from the lattice; to QCD sum rules; the bag model; and quark soliton models [7]. The calculation has been carried out at the scale µ ≈ 1 GeV, which is set by the nucleon mass as well as being the mean mass of the axial vector meson multiplet. The appropriate evolution to higher scales (wherein the Drell-Yan processes are studied) is determined by the anomalous dimensions of the tensor charge [17] which is straightforward but slowly varying. It is interesting to observe that the symmetry relations that connect the b1 couplings to the a1 couplings in Eq. (3) can be used to relate directly the isovector tensor charge to the axial vector coupling gA . This is accomplished through the a1 dominance expression for the isovector longitudinal charges derived in [12], √ 2fa1 ga1 NN gA . (8) = ∆u − ∆d = gV Ma21 Hence for δq v we have δu − δd = 2i 5 gA Ma21 hk⊥ , 6 gV Mb21 MN Mb1 (9) It is important to realize that this relation can hold at the scale wherein the couplings were specified, the meson masses, but will be altered at higher scales (logarithmically) by the different evolution equations for the ∆q and δq charges. To write an analogous expression for the isoscalar charges (∆u + ∆d) would involve the singlet mixing terms and gluon contributions, as Ref. [12] considers. However, given that the tensor charge does not involve gluon contributions (and anomalies), it is expected that the relation between the h1 and b1 couplings in the same SU (3) multiplet will lead to a more direct result δu + δd = 3 Mb21 v δq , 5 Mh21 (10) for the ideally mixed singlet-octet h1 (1170). These relations are quite distinct from other predictions. In conclusion, our axial vector dominance model with SU (6)W ⊗O(3) coupling relations provide simple formulae for the tensor charges. This simplicity obscures the considerable subtlety of the (non-perturbative) hadronic physics that is summarized in those formulae. We obtain the same order of magnitude as many other calculation schemes. These results support the view that the underlying hadronic physics, while quite difficult to formulate from first principles, is essentially a 1+− meson exchange process. Forthcoming experiments will begin to test this notion. Acknowledgments: This work is supported in part by Grant No. DE-FG02-92ER40702 from the U.S. Department of Energy. 3 References [1] J. Ashman et al. Nucl. Phys. B328, 1 (1989); B. Adeva et al., Phys. Lett. B 302, 533 (1993); P.L. Anthony et al., Phys. Rev. Lett. 71, 959 (1993) ; K. Abe et al., Phys. Rev. Lett. 72, 25 (1995). [2] X. Ji, Phys. Rev. Lett 78, 610 (1997). [3] V. Barone , A. Drago and P. G. Ratcliffe, arXive: hep-ph/0104283 and references contained therein . [4] G.R. Goldstein and M.J. Moravcsik, Ann. Phys. (NY) 98, 128 (1976); Ann. Phys. (NY) 142, 219 (1982); Ann. Phys. (NY)195, 213 (1989). [5] R. L. Jaffe and X. Ji, Phys. Rev. Lett. 67, 552 (1991); Nucl. Phys. B375, 527 (1992). [6] J. Soffer, Phys. Rev. Lett. 74, 1292 (1995); G. R. Goldstein, R. L. Jaffe and X. Ji, Phys. Rev. D 52, 5006 (1995). [7] H. He and X. Ji, Phys. Rev. D 52, 2960 (1995); Phys. Rev. D 54, 6897 (1996); S. Aoki, et al., Phys. Rev. D 56, 433 (1997); L. Gamberg, H. Reinhardt and H. Weigel, Phys. Rev. D 58, 054014 (1998); L. Gamberg, “Structure Functions and Chiral Odd Quark Distributions in the NJL Soliton Model of the Nucleon”, Proceedings of Future Transversity Measurements, RIKEN-BNL Workshop (2000). [8] L. Gamberg and G. R. Goldstein, arXive: hep-ph/0106178. [9] L. Gamberg and G. R. Goldstein, arXive: hep-ph/0107176, to appear in Phys. Rev. Lett. . [10] B. Sakita, Phys. Rev. 136, B1756 (1964); F. Gürsey and L.A. Radicati, Phys. Rev. Lett. 13, 173 (1964). [11] W.-S. Tsai, Phys. Rev. D 4, 2821 (1971). [12] M. Birkel and H. Fritzsch, Phys. Rev. D 53, 6195 (1996). [13] D.E. Groom, et al. (Particle Data Group), Eur. Phys. Jour. C15, 1 (2000). [14] V.M. Belyaev and A. Oganesian , Phys. Lett. B395, 307 (1997). [15] A. Abele, et al., Phys. Lett. B415, 280 (1997); D. Aston, et al., Phys. Lett. B201, 573 (1988). [16] A. Ando, et al., Phys. Lett. B291, 496 (1992); and references contained therein. [17] X. Artu and M. Mekhfi, Z. Phys. C45 (1990) 669. 4