Differential Dyson-Schwinger equations for quantum chromodynamics

Using a technique devised by Bender, Milton and Savage, we derive the Dyson-Schwinger equations for quantum chromodynamics in differential form. We stop our analysis to the two-point functions. The 't Hooft limit of color number going to infinity is derived showing how these equations can be cast into a treatable even if approximate form. This approach exploits a background field technique in quantum field theory.


I. INTRODUCTION
The main difficulty of quantun chromodynamics (QCD) is that, at low energies, the theory is not amenable to treatment using perturbation techniques. This implies that some non-perturbative methods should be devised to solve them. The most widespread approach is solving the equations of the theory on a large lattice using computer facilities. This permitted to obtain, with a precision of a few percent [1,2], some relevant observables of the theory. This method improves as the computer resources improve making even more precise the comparison with experiment. Use of numerical techniques is a signal that we miss some sound theoretical approach to compute observables.
A similar situation is seen for the correlation functions of the theory. Studies on the lattice of the gluon and ghost propagators, mostly in the Landau gauge, [3][4][5] and the spectrum [6,7] proved that a mass gap appears in a non-Abelian gauge theory without fermions.
Theoretical support for these results was presented in [8][9][10][11][12][13] providing closed form formulas for the gluon propagator. Quite recently, the set of Dyson-Schwinger equations for this case was solved, for the 1-and 2-point functions, and the spectrum very-well accurately computed both in 3 and 4 dimensions [13][14][15]. Confinement was also proved to be a property of the theory [14,16].
Indeed, the Dyson-Schwinger equations were considered, since the start, the most sensible approach to treat a non-perturbative theory like QCD at low-energies [18,19] and, more recently, [20]. In any case, the standard technique is to reduce the set of equations, that normally are partial differential equations, to their integral form in momentum space. Some years ago, Bender, Milton and Savage [21] proposed to derive the Dyson-Schwinger equations and treat them into differential form. This way to manage these equation was the one used to find the exact solution [13]. This technique appears more general as it permits to work out a solution to a quantum field theory also when a background field is present. This is a rather general situation when a non-trivial solution of the 1-point equation is considered.
Such a possibility opens up the opportunity of a complete solution to theories that normally are considered treatable only through perturbation methods. The idea is that, knowing all the correlation functions, a quantum field theory is completely solved.
The aim of this paper is to derive the Dyson-Schwinger equations for QCD in differential form. We obtain them for the 1-and 2-point functions. We show that, in the 't Hooft limit [22], the equation can be cast into a treatable form. It appears that a non-local Nambu-Jona-Lasinio model is the proper low-energy limit of the theory [23].
The paper is so structured. In Sec. II, we give the main equations and notations. In Sec. III, we derive the set of equations for the 1-and 2-point correlation functions. In Sec. IV, we discuss the 't Hooft limit. Finally, in Sec. V, we present the conclusions.

II. BASIC EQUATIONS
In QCD, one has the Lagrangian (to fix the notation we assume (1, −1, −1, −1) for the metric signature) where L inv denotes the classical gauge-invariant part, L gf the gauge-fixing terms and L F P the Faddeev-Popov (FP) ghost term characteristic of non-Abelian gauge theories: in the usual notation. We set α for the gauge parameter and D µ represents the covariant derivative whose explicit forms are given by

III. CORRELATION FUNCTIONS IN QCD
We use the approach devised in [21]. Indeed, to get the Schwinger-Dyson equations one has to start from the quantum equations of motion that have the form Letters a, b, c, . . . = 1, . . . , 8 are for gluons and i, j, k, l, . . . = 1, 2, 3 are for colors. The scalar product runs on the former. Flavors are identified with the letter q = u, d, s, c, t, b. We fix the gauge to the Landau gauge, α → 0, and c,c are the ghost fields. Averaging on the vacuum state and dividing by the partition function Z QCD [j,ε, ε,η q , η q ] one has The one-point functions are given by These represent the 1-point function for the gluon, ghost and quark fields respectively.
Deriving once with respect to the currents, at the same point because of the averages on the vacuum (see [21]), one has being W (η)ai Deriving twice one has This give us the first set of Schwinger-Dyson equations as Setting the currents to zero and noticing that, by translation invariance, is The Schwinger-Dyson equation for the two-point functions can be obtained by further deriving eq.(9). One has We have assumed qq = 0, q ′ q = 0 for q ′ = q, Y n+1µ (x, y, . . .)/δη j q (z). Note that, for n = 0, we omit the index, e.g. W 0 = W . This yields, setting currents to zero and using translation invariance,

IV. 'T HOOFT LIMIT
We will give here an approximate solution to the set of equations we obtained in the preceding section. We will follow the track already pursued in [13]. We start from the set (10) and take For symmetry properties of P 2 under exchange of indexes the set reduces to We have introduced the quark mass matrix We note that is degenerate with respect the color index. Now, we write the expected solution in the form in the Landau gauge being η a ν some numerical coefficients, φ(x) a scalar field and ∆(x − y) the propagator. We will get for the 1-point functions Now we notice that, for the η−symbols in SU(2) we can write η a µ = ((0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)), that yields η 1 µ = (0, 1, 0, 0), η 2 µ = (0, 0, 1, 0), η 3 µ = (0, 0, 0, 1), (19) so that Similarly, Finally, being g µν the Minkowski metric and δ µν the identity tensor. We note that the number of components of η a µ are identical to those of γ µ T a as expected and the equation is consistent. This permits to write for the 1-point functions The 1-point function acquires a mass term given by µ 2 0 = Ng 2 ∆(0). Now, we apply eq. (16) to the 2-point function set to obtain This set of equations can be solved in the 't Hooft limit N ≫ 1, keeping Ng 2 constant, and Ng 2 ≫ 1. Then, at the leading order, the equations simplify to where we used the exact solution [15] φ 0 (x) = 2µ 4 being µ and χ arbitrary integration constants and κ = . We have set m 2 = 2Ng 2 ∆(0) and taken the momenta p so that We just notice that it goes like 1/ Ng 2 and, for this reason, we write the second set of equations as This set of equations can be solved exactly and yields the self-consistency equations of the theory at the leading order. In order to present the solutions, we point out that the equation has the solution Here we have set .
The last equation is an integral equation to be solved iteratively. This will yield the corrections to the gluon propagator.
Similarly, for the quark mass we have to solve the self-consistent system of equations given the mass matrix that couples both. This set will yield the quark propagator. These equations can be written as (i / ∂ − m i q )q i 1 (x) = g 2 d 4 y ′ ∆ 0 (x − y ′ )T a γ ν jq j 1 (y ′ )γ ν T aŜji q (y ′ − x)q i 1 (x) (i / ∂ − m i q )Ŝ ij q (x − y) = δ ij δ 4 (x − y) + g 2 d 4 y ′ ∆ 0 (x − y ′ )T a γ ν kq k 1 (y ′ )γ ν T aŜki q (y ′ − x)Ŝ ij q (x − y) This can be solved iteratively. The first approximation forq i 1 (x) = q i 0 (x) is the free particle solution. The second equation will yield the approximation Σ i q (x, x) = g 2 d 4 y ′ ∆ 0 (x − y ′ )T a γ ν jq j 0 (y ′ )γ ν T aŜji 0q (y ′ − x) that is a non-local Nambu-Jona-Lasinio approximation provided we identifyŜ ji 0q (y ′ − x) with the free Dirac propagator.

V. CONCLUSIONS
We have derived the set of Dyson-Schwinger equations for QCD for 1-and 2-point correlation functions. We have seen that they can be cast in a treatable form in the 't Hooft limit. The main aim of this effort is to present a proof of confinement for the theory. We hope to show this in future works.