季向东 (Xiangdong Ji) Shanghai Jiao Tong University University of Maryland

Transcription

1 季向东 (Xiangdong Ji) Shanghai Jiao Tong University University of Maryland

2 1. Gauge symmetry and Feynman parton distributions 2. TMDs in light of gauge symmetry 3. Wigner distributions and angular momentum sum rule 4. Comments on the New gauge symmetry by X. S. Chen, F. Wang et al.

3 One of the hall-marks of gauge theories is that partial derivative,, acting on a charged field (operator) is not gauge symmetric, one has to use covariant derivative, D = + iea, where A is the gauge potential. In QM, the partial derivative generates particle s linear momentum (also the so-called canonical momentum), therefore, the linear momentum of a charged particle is not generally a gauge-invariant concept

4 In Feynman distribution, we are interested in parton s longitudinal momentum k +, which conjugates to a light-cone coordinate direction ξ -. Therefore the non-local operator is separated along the light-cone ξ- direction. The gauge symmetry is ensured with a gauge link connecting the two fields.

5 In momentum space, gauge symmetry is ensured by a quark with momentum l +, plus a quark with momentum (l-k 1 ) +, a gluon k 1, plus a quark with momentum (l-k 1 -k 2 ) +, a gluon with k 1+, and a gluon with k 2+.and so on. l-k 1 l-k 1 -k 2 - k n

6 In other words, one can define a gauge-invariant quark field by This quark field has a gauge-link all the way going to infinity along the ξ - direction. Under gauge transformation, this is a gauge-invariant quark field. In A + = 0 gauge, the gauge link disappears, all momentum is carried by the quark. Only in this gauge, the parton distribution becomes the quark distribution alone.

7 Form factors are defined through off-forward matrix elements of gauge-invariant local operators. Since hadrons are color-neutral, their momenta are gaugeinvariant. The elastic form factors describe the spatial distribution in the transverse plan b: ρ(b) The impact parameter space b is conjugate to the momentum transfer Δ=P -P

8 More complete information on quarks comes from GPDs describing the joint distribution in longitudinal momentum x and transverse coordinate b spaces GPD(x, b) Again, the gauge symmetry in the momentum x is ensured through longitudinally polarized gluons, or gluons in the + direction. GPD makes the probability sense because the transverse coordinates and longitudinal momentum commutes.

9 When considering longitudinal as well as transverse parton momenta, gauge symmetry becomes an important issue. The simplest way is to define a gauge-invariant quark field, with gauge link linking the field point to the infinity along a certain path. Ψ(x)

10 There are infinity number of possible gauge links, therefore, there are infinite number of gaugeinvariant TMDs. Most of them cannot be reduced to local operators in any known way, therefore cannot be calculated in usual lattice QCD approach. Most of them cannot be probed in any experiment because they do not emerge from any factorization theorems.

11 We consider only these TMDs when integrating over kt, the ordinary Feynman distribution is recovered q x, k T d 2 k T = q(x) This requires that in the integration process, the two gauge links combine in such a way that a straight-line along the light-cone direction is recovered. This can be done more formally than in realistic, because the integration is UV divergent.

12 Factorization in DIS generates a gauge link. The gauge link go along the light-cone direction ξ -, with a straight-line. A staple type of gauge link existing in infinity so that in singular gauges, it provides the necessary mechanism for gauge symmetry Belitsky, Ji and Yuan, Nucl.Phys.B656: ,2003. Factorization in SIDIS also generates a gauge link along the light-cone direction Ji, Ma, Yuan, Phys.Rev.D71:034005,2005. Clearly if one integrates over kt,one formally recovered the usual parton distribution.

13 Problem with such a gauge link is that there is no way one can recover local operators. In fact, one can try to take moments of transverse momentum of the distribution. The integral with the moment converts into a transverse derivative with respect to the quark fields. It is not easy to take such a derivative with a nonlocal gauge-link operator, therefore, one cannot reduce the moments into local operators.

14 To recover local operators, one can consider gauge links directly connect the two fields with a straighline. (Ji, Yuan, Xiong, to be published) Only with such a gauge link, local operators results Hence one may calculate the moments on lattice. The integration over kt also results the usual Feynman parton distributions.

15 With GPD (x bt) and TMD (x, kt), it might be possible to consider distributions joint in x, bt and kt, such a distribution must be phase-space quantum Wigner type of distribution. Belitsky, Ji and Yuan Phys.Rev.D69:074014,2004. Gauge link problem is the same as the TMD case. Only with proper gauge link, the Wigner distribution can recover the angular momentum such rule. To be published (Ji, F. Yuan, X. Xiong)

16 In the last few years, new gauge symmetry has been proposed by 王凡, 陈向松,etc.

17 X. Ji, Phys. Rev. Lett. 104, (2010) [1 pages] X. Ji, Phys. Rev. Lett. 106, (2011) [1 pages] No response has been published by PRL

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19 All the gauge-dependent part is in A pure. A phys contains purely physical d.o.f s. Thus one can construct all the physical observables using A phys Covariant derivative can simply be

20 And momentum The text example of nucleon momentum fractions carried by quarks and gluons is toppled.

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22 It is known that the covariant derivative P = i - ea/c represents the kinetic momentum of a particle. Feynman: Consider a particle near a solenoid. When the current is turned on suddenly, the A is Produce and electron gets a momentum kick, this cannot be i because the w.f. is continuous in time!

23 There is no known way to quantize the theory with both A pure, and Aphys as quantum mechanical degrees of freedom. Quantization only makes sense in Coulomb/radiation gauge where one can choose A pure =0 However, in any other gauge, no one knows how to make the quantization either in canonical way or in path integral formulation.

24 physical quantities become non-local Lorentz transformation of Aphys becomes a mess. All physical theory and observables must obey Lorentz symmetry, even when constructing with only physical degrees of freedoms. In fact, in any effective field theory with correct physical degrees of freedoms, one must go through constructions will all symmetry taken into account properly!

25 It is quite clear that no physical processes can ever be invented to measure the observables described in these papers. In atomic theory, the natural choice is Coulomb gauge where the vector potential is of high order in power counting. Therefore, the electron s three momentum becomes physical. This can be seen clearly in NRQED construction. This is similar to A+ =0 gauge in high-energy scattering.

26 Gauge symmetry requires that when discussing parton s momentum, one has to be clear what is gauge invariant. This is particularly relevant for TDMs The new proposal for gauge symmetry by Chen et al is neither theoretically viable nor experimentally relevant.