On path-dependence of the QCD correlation functions
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n PathDependence of the QCD Correlation Functions1 I. O. Cherednikov Universiteit Antwerpen, Antwerp, Belgium email: [email protected] Abstract—Gaugeinvariant hadronic and vacuum correlation functions in QCD contain the systems of Wil son lines and loops having complicated geometrical structure. Pathdependence propagates, therefore, into such important properties of the quantum correlators as the renormalizationgroup behaviour, lightcone peculiarities, evolution, etc. In the present paper, I briefly overview several instructive examples of the man ifestations of the structure of paths in the hadronic and vacuum correlation functions with explicit transverse momentum/distance dependence. In particular, the transversemomentum dependent (TMD) parton den sities and the skewed jet quenching parameter in Euclidean and Minkowski spacetime are addressed. DOI: 10.1134/S1063779614040042 1
1. INTRODUCTION: PATH AND SHAPE VARIATIONS IN THE WILSON LOOP SPACE
perturbative Makeenko–Migdal (MM) equations [3, 4]: ν 2 μ (4) δ ∂ x W 1 [ Γ ] = N c g dz δ ( x – z ) δσ μν ( x )
Loop space consists of colorless gaugeinvariant field functionals—expectation values of the products of n Wilson loops [1, 2], defined, in general, in the manifold of arbitrary integration paths {Γi}:
ᐃ n [ Γ 1, …Γ n ] 1 1 = 〈 0 ᐀ TrΦ ( Γ 1 ) … TrΦ ( Γ n ) 0〉 , Nc Nc
∫
∫ Γ
(2)
× W 2 [ Γ xz Γ zx ], supplied with the Mandelstam constraints
∑a ᐃ i
(1)
μ
Φ [ Γ i ] = ᏼ exp ig dz Ꮽ μ ( z ) . Γi
The gauge fields Ꮽµ belong to the fundamental repre sentation of nonAbelian gauge group SU(Nc). Although the Wilson loops (2) are gauge invariant, their definition gives rise to the functional dependence on path and to the additional singularities due to non trivial behavior in vicinity of obstructions, cusps or selfintersections. Moreover, the renormalization and conformal properties of the Wilson loops possessing lightlike segments (or lying completely on the light cone) are known to be more intricate than those of the Wilson loops defined on offlightcone integration contours. Therefore, study of the geometrical and dynamical properties of the loop space which can include, in general case, cusped lightlike Wilson exponentials, will provide us with fundamental infor mation on the renormalization group behavior and evolution of the various gaugeinvariant quantum cor relation functions. Finding an appropriate and complete set of equa tions of motion in the loop space is not straightfor ward. It is known that the Wilson loops obey the non
i i n i [ Γ 1 …Γ n i ]
= 0.
(3)
The MM set of equations follows from the Schwinger–Dyson equations being applied to the Wil son functionals i δ 〈 α' α''〉 = 〈 α' δS α''〉 , ћ
(4)
where the quantum action operator S defines varia tions of arbitrary matrix elements. The differential operations in the loop space are the area and path 2
derivatives [3] : δ Φ [ Γ ] ≡ δσ μν ( x )
lim
δσ μν ( x ) → 0
Φ ( ΓδΓ ) – Φ ( Γ )
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