Generalized Loop Space and TMDs

The Standard Model describes the three (of four) basic interactions known in Nature in terms of the quantum fields which are constituted by representations of special unitary gauge groups of symmetry. However, the physical observables do not always coincide with the fundamental degrees of freedom of the Standard Model. Therefore it can be useful to switch to the loop space representation of the gauge theory, where the variables are inherently gauge invariant but the degrees of freedom are absorbed in the path/loop dependence. Over-completeness of this space requires the introduction of an equivalence relation which is provided by Wilson loop functionals operating on piecewise regular paths. It is well known that certain Wilson loops show the same singularity structure as some Transverse Momentum Dependent PDFs (TMDs), which are not renormalizable by the common methods due to exactly this singularity structure. By introducing geometrical operators, like the area-derivative, we were able to derive an evolution equation for these Wilson loops and we hope to apply this method in the future to find some renormalization schemes for TMDs.

Holonomy : Note that under gauge transformations : Holonomy : Note that under gauge transformations : Holonomy : Note that under gauge transformations :

Shuffle multiplication
Shuffle motivation Let ω 1 , · · · , ω k+l be in Ω then: where σ is running over all (k, l)-shuffles. This produces then a map A p × A p → A p , the algebra multiplication in the with Chen Iterated Integral associated Hopf algebra.

Properties Gel'fand topology
The Gel'fand topology on the spectrum of a unital, Abelian Banach algebra is Hausdorff.

Loop Group and Lie Algebra Topological Group
We can define a product on ∆ such that (∆p, ) has the structure of topological group that is Hausdorff and completely regular or Tychonov.

Group of Generalized Loops
We call the above mentioned topological group (∆p, ), the Group of Generalized Loops of M, based at p ∈ M,and we denote it by LMp.

Pointed Differentiation
A pointed differentiation is a pair (d, p) where d : U → Ω is a differentiation and p ∈ Alg(U, k).

Lie Algebra and Tangent Space
The pointed differentiations then form a tangent space T LMp and one is able to show that this space is isomorphic to the Lie Algebra of (∆p, ), i.e.

T LMp lMp
(1) We can define a product on ∆ such that (∆p, ) has the structure of topological group that is Hausdorff and completely regular or Tychonov.

Group of Generalized Loops
We call the above mentioned topological group (∆p, ), the Group of Generalized Loops of M, based at p ∈ M,and we denote it by LMp.

Pointed Differentiation
A pointed differentiation is a pair (d, p) where d : U → Ω is a differentiation and p ∈ Alg(U, k).

Lie Algebra and Tangent Space
The pointed differentiations then form a tangent space T LMp and one is able to show that this space is isomorphic to the Lie Algebra of (∆p, ), i.e.

Motivation
Path derivative

Terminal Endpoint Derivative
Let Ψ be a path functional on PM, with values in R (resp., C; gl(m)).We define the Terminal Endpoint Derivative of Ψ, at γ, in the direction of v ∈ T γ(1) M, as the limit: provided this limit exists independently of the choice of the vector field V ∈ X M, such that V (γ(1)) = v.

Motivation
Path derivative

Terminal Endpoint Derivative
Let Ψ be a path functional on PM, with values in R (resp., C; gl(m)).We define the Terminal Endpoint Derivative of Ψ, at γ, in the direction of v ∈ T γ(1) M, as the limit: provided this limit exists independently of the choice of the vector field V ∈ X M, such that V (γ(1)) = v.

Motivation
Path derivative

Terminal Endpoint Derivative
Let Ψ be a path functional on PM, with values in R (resp., C; gl(m)).We define the Terminal Endpoint Derivative of Ψ, at γ, in the direction of v ∈ T γ(1) M, as the limit: provided this limit exists independently of the choice of the vector field V ∈ X M, such that V (γ(1)) = v.

Terminal Endpoint Derivative
Given a loop functional Ψ on LMp, with values in R (resp., C; gl(m)), we define its Area Derivative, given by ∆ λ;(u,v) (q) · Ψ(γ), as the limit: t 2 provided this limit exists independently of the choice of the vector fields U, V ∈ X U, considered above.

Motivation
Area Derivative

Terminal Endpoint Derivative
Given a loop functional Ψ on LMp, with values in R (resp., C; gl(m)), we define its Area Derivative, given by ∆ λ;(u,v) (q) · Ψ(γ), as the limit: t 2 provided this limit exists independently of the choice of the vector fields U, V ∈ X U, considered above.

Motivation
Area Derivative

Terminal Endpoint Derivative
Given a loop functional Ψ on LMp, with values in R (resp., C; gl(m)), we define its Area Derivative, given by ∆ λ;(u,v) (q) · Ψ(γ), as the limit: t 2 provided this limit exists independently of the choice of the vector fields U, V ∈ X U, considered above.

Motivation
Wilson Quadrilateral (To be submitted for publication together with gravitational case)

Motivation
Defining a new derivative

Motivation
Defining a new derivative  One-loop result for Π One-loop result for Π