We propose a short summary of the present situation concerning the proton spin decomposition. We briefly discuss some of the main controversies about the issues of gauge invariance, uniqueness and measurability. As a conclusion, we argue that part of the controversies is actually undecidable

Lorcé, C.

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DOI 10.1393/ncc/i2013-11585-3Colloquia: QCDN12IL NUOVO CIMENTO Vol. 36 C, N. 5 Settembre-Ottobre 2013Canonical and kinetic decompositions of the proton spinC. Lorce´IPNO, Universite´ Paris-Sud, CNRS/IN2P3 - 91406 Orsay, France andLPT, Universite´ Paris-Sud, CNRS - 91405 Orsay, Francericevuto il 18 Aprile 2013Summary. — We propose a short summary of the present situation concerning theproton spin decomposition. We briefly discuss some of the main controversies aboutthe issues of gauge invariance, uniqueness and measurability. As a conclusion, weargue that part of the controversies is actually undecidable.PACS 11.15.-q – Gauge field theories.PACS 12.20.-m – Quantum electrodynamics.PACS 12.38.Aw – General properties of QCD (dynamics, confinement, etc.).PACS 13.88.+e – Polarization in interactions and scattering.1. – IntroductionThe last few years have seen numerous developments concerning the proper definitionof quark and gluon contributions to the proton spin. In particular, the status and thephysical relevance of the canonical angular momentum operators has been clarified thanksto the notion of gauge-invariant extensions. This allows one to render gauge invariantthe interpretation of Δg as the gluon spin contribution. Moreover, it has been shownthat one can access the canonical orbital angular momentum provided that one is able toextract experimentally either the Wigner distributions or particular twist-3 distributions.For a recent review of the discussions, see ref. [1].In this short letter, we summarize the recent developments about the proton spindecomposition and briefly discuss the issues of gauge invariance, uniqueness and mea-surability. In sect. 2, we present the suggestion made by Chen et al. to separate thegauge field into pure-gauge and physical terms. Although gauge invariant, this approachis not unique owing to the Stueckelberg symmetry which reflects the freedom in definingwhat is exactly meant by pure-gauge and physical contributions. In sect. 3, we recallthe kinetic and canonical gauge-invariant definitions of quark orbital angular momen-tum (OAM) and argue that there exist actually infinitely many inequivalent canonicalOAM operators, raising the question of deciding which is the physical one. In sect. 4,we show that the Wigner operator gives access to both the kinetic and canonical angularmomentum operators, provided that one uses the appropriate Wilson lines. Finally, weconclude this letter with sect. 5.c© Societa` Italiana di Fisica 1516 C. LORCE´2. – Chen et al. decomposition and Stueckelberg symmetryIn order to unambiguously define what is meant by gluon spin and orbital angularmomentum, Chen et al. proposed to separate explicitly the gauge degrees of freedomfrom the physical ones [2-5]Aμ(x) = Apureμ (x) + Aphysμ (x),(1)where the pure-gauge and physical parts satisfy specific gauge transformation lawsApureμ (x) → A˜pureμ (x) = U(x)[Apureμ (x) +ig∂μ]U−1(x),(2)Aphysμ (x) → A˜physμ (x) = U(x)Aphysμ (x)U−1(x).(3)Since Apureμ (x) is a pure gauge, it can be written asApureμ (x) =igUpure(x)∂μU−1pure(x),(4)where Upure(x) is some unitary gauge matrix with the gauge transformation lawUpure(x) → U˜pure(x) = U(x)Upure(x).(5)Clearly, in the gauge U(x) = U−1pure(x) the pure-gauge term vanishes.By construction, the decomposition (1) is gauge invariant A˜μ = A˜pureμ + A˜physμ . How-ever, it is not unique since we still have some freedom in defining exactly what we meanby “pure-gauge” and “physical”. The reason is that the pure-gauge and physical termsremain respectively pure-gauge and physical under the following transformation leavingAμ(x) invariantApureμ (x) → Apure,gμ (x) = Apureμ (x) +igUpure(x)U−10 (x) [∂μU0(x)]U−1pure(x),(6)Aphysμ (x) → Aphys,gμ (x) = Aphysμ (x)−igUpure(x)U−10 (x) [∂μU0(x)]U−1pure(x),(7)where U0(x) is a gauge-invariant unitary matrix. At the level of Upure(x), this transfor-mation readsUpure(x) → Ugpure(x) = Upure(x)U−10 (x).(8)While the ordinary gauge transformation acts on the left of Upure(x) as in eq. (5), this newtransformation acts on the right. It is therefore important to distinguish them. Notingthat the pure-gauge term Apureμ plays a role similar to the derivative of the Stueckel-berg field, we refer to this transformation as the Stueckelberg (gauge) transformation [1].Explicit realizations of the Chen et al. decomposition are usually non-local. Gauge in-variance is then assured by the use of Wilson lines whose path dependence is at the originof the Stueckelberg symmetry [6].CANONICAL AND KINETIC DECOMPOSITIONS OF THE PROTON SPIN 173. – Kinetic and canonical orbital angular momentumThere exist essentially two kinds of gauge-invariant quark orbital angular momentum.One is the kinetic OAM [7]Mμνρq,OAM(x) =i2ψ(x)γμx[ν↔Dρ](x)ψ(x)(9)and the other one is the canonical OAM [2,3]Mμνρq,OAM(x) =i2ψ(x)γμx[ν↔Dρ]pure(x)ψ(x),(10)where the covariant derivatives at the point x are defined as Dμ(x) = ∂μ − igAμ(x) andDμpure(x) = ∂μ− igAμpure(x). We used for convenience the notations a[μbν] = aμbν − aνbμand←→∂ =−→∂ −←−∂ . These two OAMs differ by a so-called potential term [4,5]Mμνρpot (x) = −g ψ(x)γμx[νAρ]phys(x)ψ(x),(11)which is usually non-vanishing. In the gauge U(x) = U−1pure(x), the canonical OAM simplyreduces to the same expression as in the definition of the Jaffe-Manohar OAM [8] andcan then be thought of as a gauge-invariant extension (GIE) of the latter [9-11].Contrary to the kinetic quark OAM, the canonical quark OAM is not Stueckelberginvariant, i.e. it depends on how one explicitly separates the gauge field into pure-gaugeand physical terms. There is consequently an infinite number of possible different def-initions of canonical OAM, all sharing the same formal structure (10). The reductionto the Jaffe-Manohar OAM occurs in different gauges, which implies that the differentcanonical OAMs are not equivalent. This raises the question of deciding which canonicaldecomposition is the physical one.4. – Wigner operator and its relation with orbital angular momentumThe gauge-invariant quark Wigner operator is defined as [12,13]W [γμ]q(x, k) ≡∫d4z(2π)4eik·z ψ(x− z2 )γμWC(x− z2 , x + z2 )ψ(x + z2 ).(12)It can be interpreted as a phase-space density operator. It then is natural to define thequark OAM density as [14,15]Mμνρq,OAM(x) =∫d4k x[νkρ] W [γμ]q(x, k).(13)In order to be gauge invariant, the definition of the Wigner operator involves a gaugelink. The consequence of this gauge link is that the Wigner distribution inherits a pathdependence. Using a straight gauge link in eq. (12) leads to the kinetic OAM Lz [6, 9].With the view of connecting the Wigner distributions to the Transverse-Momentumdependent parton Distributions (TMDs) [16, 17] appearing in the description of high-energy semi-inclusive processes like Semi-Inclusive DIS and Drell-Yan, it is more natural18 C. LORCE´to consider instead a staple-like gauge link consisting of two longitudinal straight linesconnected at x− = ±∞ by a transverse straight line. In this case, eq. (12) gives thecanonical OAM z appearing in the light-front GIE [6,9, 15,18].As emphasized in the previous section, there exist formally an infinity of gauge-invariant canonical quark OAM. Note however that the proton structure is usually probedin high-energy scattering experiments. Even though physics is invariant under rotations,actual high-energy experiments provide us with a specific direction and make thereforethe light-front GIE more natural, just like a Stern-Gerlach experiment provides us witha natural basis for describing the spin states.5. – ConclusionSeparating explicitly the gauge degrees of freedom from the physical ones led tothe notion of gauge-invariant extension, and allowed the definition of gauge-invariantcanonical angular momentum operators. This approach has been shown to be tightlyconnected with the use of non-local operators and Wilson lines. However, the gauge-invariant canonical operators are not unique owing to the Stueckelberg symmetry thatcan be thought of as the path dependence in the non-local approach. This is nicelyreflected in the definition of quark orbital angular momentum based on the Wigneroperators. Depending on the choice of the path for the Wilson line, one obtains differentgauge-invariant results. We stress that it is the experimental conditions that fix theWilson lines and therefore the gauge-invariant extension to use.∗ ∗ ∗I would like to thank E. Leader, A. Metz, B. Pasquini, L. Szymanowski, M. Waka-matsu and F. Wang for many helpful comments and discussions. This work was supportedby the P2I (“Physique des deux Infinis”) network.REFERENCES[1] Lorce´ C., Phys. Rev. D, 87 (2013) 034031.[2] Chen X.-S., Lu X.-F., Sun W.-M., Wang F. and Goldman T., Phys. Rev. Lett., 100(2008) 232002.[3] Chen X.-S., Sun W.-M., Wang F. and Goldman T., Phys. Rev. D, 83 (2011) 071901.[4] Wakamatsu M., Phys. Rev. D, 81 (2010) 114010.[5] Wakamatsu M., Phys. Rev. D, 83 (2011) 014012.[6] Lorce´ C., Phys. Lett. B, 719 (2013) 185.[7] Ji X.-D., Phys. Rev. Lett., 78 (1997) 610.[8] Jaffe R. L. and Manohar A., Nucl. Phys. B, 337 (1990) 509.[9] Ji X., Xiong X. and Yuan F., Phys. Rev. Lett., 109 (2012) 152005.[10] Ji X., Xu Y. and Zhao Y., JHEP, 08 (2012) 082.[11] Ji X., Xiong X. and Yuan F., arXiv:1207.5221 [hep-ph].[12] Ji X., Phys. Rev. Lett., 91 (2003) 062001.[13] Belitsky A. V., Ji X. and Yuan F., Phys. Rev. D, 69 (2004) 074014.[14] Lorce´ C. and Pasquini B., Phys. Rev. D, 84 (2011) 014015.[15] Lorce´ C., Pasquini B., Xiong X. and Yuan F., Phys. Rev. D, 85 (2012) 114006.[16] Meissner S., Metz A. and Schlegel M., JHEP, 08 (2009) 056.[17] Lorce´ C., Pasquini B. and Vanderhaeghen M., JHEP, 05 (2011) 041.[18] Hatta Y., Phys. Lett. B, 708 (2012) 186.

Canonical and kinetic decompositions of the proton spin

4 pages, contribution to the proceedings of the 3rd Workshop on the QCD Structure of the Nucleon (QCD-N'12), Oct 22-26, Bilbao, SpainInternational audienceWe propose a short summary of the present situation concerning the proton spin decomposition. We briefly discuss some of the main controversies about the issues of gauge invariance, uniqueness and measurability. As a conclusion, we argue that part of the controversies is actually undecidable

Canonical and kinetic decompositions of the proton spin

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