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Gauge-invariant decomposition of nucleon spin

Gauge-invariant decomposition of nucleon spin. Masashi Wakamatsu , Osaka University : Urich, Spin2010. Plan of Talk. 1. Introduction 2. Gauge-invariant decomposition of covariant angular-momentum tensor 3. Observability of our nucleon spin decomposition

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Gauge-invariant decomposition of nucleon spin

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  1. Gauge-invariant decomposition of nucleon spin Masashi Wakamatsu , Osaka University : Urich, Spin2010 Plan of Talk 1. Introduction 2. Gauge-invariant decomposition of covariant angular-momentum tensor 3. Observability of our nucleon spin decomposition 4. Some phenomenological implications 5. Summary and conclusion

  2. gauge-invariantly decomposable into and ? Most people believe that the polarized gluon distribution is an observable quantity from polarized DIS measurements. On the other hand, it is often claimed that there is no gauge-invariantdecomposition of into and . 1. Introduction After 20 years of theoretical and experimental efforts, we now believe What carry the remaining 2/3 of the nucleon spin ? controversial question Because the gauge principle is one of the most important principle of physics, which demands that only gauge-invariants can be observed, how to reconcile these conflicting observations is a fundamentally important problem !

  3. No further decomposition of ! Two popular decompositions of the nucleon spin common Each term is not separately gauge-invariant !

  4. An especially important observation here is that, since one must conclude that Two popular decompositions of the nucleon spin common

  5. Each term is separately gauge-invariant ! It reduces to the gauge-variantJaffe-Manohar decomposition in a special gauge ! New gauge-invariant decomposition by Chen et al. X.-S. Chen et al., Phys. Rev. Lett. 103, 062001 (2009) ; 100, 232002 (2008). The basic idea with and Answer

  6. The quark part of our decomposition is common with the Ji decomposition. The quark and gluon intrinsic spin parts are common with the Chen decomp. A crucial difference with the Chen decomp. appears in the orbital parts However, in a recent paper (M.W., Phys. Rev. D81 (2010) 114010), we have shown that the way of gauge-invariant decomposition of nucleon spin isnot necessarily unique, and proposed another gauge-invariant decomposition : where “potential angular momentum” The QED correspondent of this term is the orbital angular momentum carried by electromagnetic field, appearing in the famous Feynman paradox in his textbook.

  7. Note that this potential angular momentum term is solelygauge-invariant ! since This means that one has a freedom to include the potential OAM term into the quark OAM part in our decomposition, which leads to the Chen decomposition.

  8. not the “canonical OAM” . Chen’s decomposition is not recommendable, because the knowledge of standard electrodynamics tells us that the momentum appearing in the Lorentz force equationof motion of a charged particle is the so-called “dynamical momentum” with the full gauge field, not the “canonical momentum” or its “nontrivial gauge-invariant extension” Similarly, the orbital angular momentum, accompanying the mass flow of a charged particle, is the “dynamical OAM”

  9. First extract and through GPD analyses. Next, extract the quark and gluon polarization and through polarized DIS measurements and identify them with and in our decomp . Then, the quark and gluon OAM can be known by subtracting and from and . This problem is of fundamental importance, especially because we are aware of the wide-spread statement that there is no gauge-invariant decomposition of , and that there is no gauge-invariant local operator corresponding to the 1st moment of ! Since the quark part of our decomposition coincides with the Ji decomposition, we may follow Ji’s well-known program toward full nucleon spin decomposition. What was lacking in this wishful argument is a rigorous proof of the identification

  10. Another important problem is as follows : Since our gauge-invariant decomposition (as well as Chen et al.’s one) was given in a particular or fixed Lorentz frame, we could not give a definite answer to the question whether our decomposition has a frame-independent meaning or not. The confirmation of frame-independence is very important. Otherwise, the decomposition cannot be thought to reflect an intrinsic property of the nucleon. We can show that these two questions can be answered simultaneously, by making full use of a gauge-invariant decomposition of covariant angular-momentum tensor of QCD in an arbitrary Lorentz frame

  11. 2. Gauge-invariant decomposition of covariant angular-momentum tensor Now, we attempts to generalize our gauge-invariant decomposition of the nucleon spin in a covariant form. Covariant generalization of the decomposition has twofold advantages. • It is essential to prove the frame-independence of the decomposition. • It generalizes and unifies the known nucleon spin decompositions in the market. • Jaffe-Manohar decomposition • Bashinsky-Jaffe decomposition • Chen et al. decomposition • Ji decomposition • Our decomposition To achieve this goal, we follow a similar idea as proposed by Chen et al. !

  12. However, the point of our argument is that we can postpone a concrete gauge-fixing until later stage, while accomplishing a gauge-invariant decomposition of based on the above conditionsonly. The starting point is the decomposition of gluon field Here, we impose the following conditions only. and As a matter of course, these conditions are not enough to fix gauge uniquely ! As expected, we again find the way of gauge-invariant decomposition is not unique. decomposition (I) & decomposition (II)

  13. Gauge-invariant decomposition (I) : with This decomposition reduces to any ones of Bashinsky-Jaffe, of Chen et al., and of Jaffe-Manohar one, after an appropriate gauge-fixing in a suitable Lorentz frame, which means that these three decomposition are all gauge-equivalent ! They are not recommendable decompositions, however, because the quark and gluon OAM parts in those do not correspond to any experimental observables !

  14. Gauge-invariant decomposition (II) : our recommendable decomposition with generalized potential OAM term ! The point is that we can show that the quark and gluon OAMs in this decomposition can be related to experimental observables !

  15. 3. Observability of our nucleon spin decomposition Inserting our decomposition (II) into the helicity normalization condition : where we obtain nucleon spin sum rule with The decomposition is not only gauge-invariant but also basically frame-independent.

  16. relation with actual high-energy observables ! We can prove that while with The quark OAM extracted from the combined analysis of GPD and polarized PDF is the “dynamical OAM”(or “mechanical OAM”) not the “canonical OAM” !

  17. We can also verify that with The gluon OAM extracted from the combined analysis of GPD and polarized PDF contains the “potential OAM”, inaddition to the “canonical OAM” ! The whole part may be called gluon “dynamical OAM” .

  18. The two decompositions yield the same fermion OAM in scalar diquark model, but not in QED (gauge theory). • The x-distributions of the fermion OAMs in the two decompositions are different even in scalar diquark model. 4. Some phenomenological implications (A) Toy model analysis • M. Burkardt and Hikmat BC, Phys. Rev. D79, 071501 (2009). Using two “toy models”, i.e. scalar diquark model and QED to order α, they compared the two kinds of fermion orbital angular momenta • in QED and QCD to order a

  19. (B) A phenomenological analysis of nucleon spin contents • “The role of orbital angular momentum in the proton spin”, M. Wakamatsu, Eur. Phys. J. A44 (2010) 297. based on Ji’s sum rule with puzzling observationin isovector channel ? An indication of Large ?

  20. Phenomenological estimate of isovector quark OAM based on Ji’s sum rule only unkown with neutron beta-decay PDF fits (MRST, CTEQ et al.) Lattice QCD estimate for : corresponding to isovector channel : no disconnected-diagram contributions !

  21. semi-empirical + lattice QCD predictions for The strong scale-dependence of quark-OAM would resolve above discrepancy. On the other hand, the prediction of refined cloudy bag model of Myhrer-Thomas typical from standard quark model with SU(6)-like structure ? How can we understand this significant discrepancy ? Thomas’ resolution scenario : Phys. Rev. Lett. 101 (2008) 102003. Their bag model prediction corresponds to low energy scale, must change sign between the two energy scales !

  22. The scale-dependence is extremely strong in the low energy scale, which throws doubts on the use of perturbative renormalization group equation at such scales. However, we have pointed out a danger of his resolution scenario. This choice of starting energy of evolution (or the matching energy scale with the low energy model) is rather arbitrary, although it gives phenomenologically successful description of parton distributions in the DIS region. To avoid these problems, we proposed a downward evolution from high energy side, without specifying the exact matching energy scale with low energy models. The point is, however, that the matching energy scale must at least be higher than the unitarity violating limit, where gluon momentum fraction becomes negative.

  23. Sensitivity to QCD coupling constant

  24. Why does the CQSM predict large and negative , then ? In our opinion, the problem may have deep connection with the existence of two kind of quark OAM ! Since the CQSM is an effective quark theory that contains no gauge field, one might naively think that there is no such distinction in the definition of quark OAM. However, it turns out that this is not necessarily the case. The point is that it is a highly nontrivial interaction theory of quark fields. To explain it, we recall the past analyses of GPD sum rules within the CQSM.

  25. CQSM analyses of GPD sum rules : • Isoscalar channel : J. Ossmann et al., Phys. Rev. D71,034001 (2005). • Isovector channel : M. W. and H. Tsujimoto, Phys. Rev. D71,074001 (2005). Isoscalar case : 2nd moment of where with

  26. Isovector case : 2nd moment of “canonical” with difference “dynamical”

  27. Inspired by the recent proposal by Chen et al., we find it possible to make a gauge-invariant decomposition of covariant angular-momentum tensor of QCD, including the gluon part, in an arbitrary Lorentz frame. Making use of this fact, we could show that our decomposition of nucleon spin is not only gauge-invariant but also practically frame-independent. We have also succeeded to convince that each piece of our nucleon spin decomposition precisely corresponds to the observables that can be extracted from combined analysis of the GPD measurements and the polarized DIS measurements, thereby giving a strong support to the standardly-accepted experimental project aiming at complete decomposition of the nucleon spin. 5. Summary and conclusion See, for the detail • M. Wakamatsu, arXiv : 1007.5355 [hep-ph], and references therein

  28. A practically very important lesson learned from our theoretical consideration is that the quark OAM extracted from the combined analysis of GPDs and polarized PDFs is the “dynamical OAM” not the “canonical OAM” or its non-trivial “gauge-invariant extension” At the moment, we do not know any means, with which we can extract the canonical OAMpurely experimentally, i.e. model independently. Still, one should keep in mind the existence of 2 kinds of quark OAM ! Naturally, this also means the existence of 2 kinds of gluon OAM ! “dynamical” and “canonical” gluon OAM

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