Integral equation for gauge invariant
quark two-point Green’s function in QCD
IPN, Univ. Paris-Sud 11, CNRS/IN2P3,
F-91405 Orsay, France
Gauge invariant quark two-point Green’s functions defined with path-ordered gluon field phase factors along skew-polygonal lines joining the quark to the antiquark are considered. Functional relations between Green’s functions with different numbers of path segments are established. An integral equation is obtained for the Green’s function defined with a phase factor along a single straight line. The equation implicates an infinite series of two-point Green’s functions, having an increasing number of path segments; the related kernels involve Wilson loops with contours corresponding to the skew-polygonal lines of the accompanying Green’s function and with functional derivatives along the sides of the contours. The series can be viewed as an expansion in terms of the global number of the functional derivatives of the Wilson loops. The lowest-order kernel, which involves a Wilson loop with two functional derivatives, provides the framework for an approximate resolution of the equation.
PACS numbers: 12.38.Aw, 12.38.Lg.
Keywords: QCD, quark, gluon, Wilson loop, gauge invariant Green’s function.
Gauge invariant objects are expected to provide a more precise description of observable quantities than gauge variant ones. Generally, gauge invariance of multilocal operators is ensured with the use of path-ordered phase factors [1, 2]. In this respect, the closed loop operator, the so-called Wilson loop, showed itself a powerful tool for the investigation of the confinement properties of QCD [3, 4, 5]. The properties of the Wilson loop were studied in detail in a long series of papers [6, 7, 8, 9, 10, 11, 12, 13, 14, 15].
On the other hand, the usual machinery of quantum field theory, based on the Dyson–Schwinger integral equations [16, 17], does not apply in a straightforward way to Green’s functions of operators involving path-ordered phase factors. The main reason is related to the difficulty of obtaining the functional inverses of the nonlocal gauge invariant Green’s functions and thus of being able to define analogues of proper vertices, which play a crucial role in the formulation of integral equations. Expressions of gauge invariant quark-antiquark Green’s functions in terms of Wilson loops were obtained in the past [18, 19, 20] with the use of the Feynman–Schwinger representation of the quark propagator [21, 22, 23, 24]; these, however, could not be transformed into equivalent integral equations without the recourse to approximations related to the quark motion.
The purpose of the present paper is to investigate the possibilities of deriving integral or integro-differential equations for gauge invariant Green’s functions which might allow for a systematic study of their various properties. We concentrate in this work on the quark gauge invariant two-point function, in which the quark and the antiquark fields are joined by a path-ordered phase factor, but the methods which we shall develop are readily applicable to more general cases.
Our starting point is a particular representation of the quark propagator in the presence of an external gluon field, where it is expressed as a series of terms involving path-ordered phase factors along successive straight lines forming generally skew-polygonal lines. That representation is a relativistic generalization of the one introduced by Eichten and Feinberg in the nonrelativistic case ; it was already used in a previous work for deriving a bound state equation for quark-antiquark systems ; however, in the latter work, the bound state equation was derived by circumventing the explicit writing of an integral equation for the related Green’s function and of the neglected higher-order terms of the interaction kernel. One of the main properties of the above representation is that in gauge invariant quantities, at each order of the expansion, the paths of the phase factors close up to form a Wilson loop. Thus, the corresponding Green’s function becomes expressed, through a series expansion, in terms, among others, of Wilson loops having skew-polygonal contours with an increasing number of sides.
Several differences occur with respect to the formulation of the Dyson–Schwinger equations. First, for the reasons mentioned above, proper vertices are not introduced; instead, we work directly with Green’s functions; the various kernels that appear are written explicitly in terms of functional derivatives of the logarithm of the Wilson loop average and of the quark Green’s function. Second, starting from the simplest gauge invariant two-point function, constructed with a phase factor along a single straight line joining the quark to the antiquark, one generates, through the equations of motion, a chain of new gauge invariant two-point functions with phase factors along -sided skew-polygonal lines between the quark and the antiquark (). On the other hand, every such Green’s function (with skew-polygonal sides) can be related with the aid of functional relations to the lowest-order Green’s function () and thus, in principle, an equation involving only the latter Green’s function is possible to construct. The third difference arises at the level of the presence of nested kernels, which do not occur in the Dyson–Schwinger equations and which persist here due to background effects induced by the Wilson loops: each nested kernel is modified by its new background when inserted inside a higher-order Wilson loop as compared to its original expression. The remaining terms in the kernels have the property of conventional irreducibility.
The integral equation that we obtain is constructed as an expansion in terms of the global number of derivatives of the logarithm of the Wilson loop average. Although it involves an infinite series of kernels and Green’s functions, at each order of the expansion the explicit expressions of the kernels and of the relations between high-order Green’s functions with the lowest-order one can be obtained from definite formulas.
On practical grounds, an increasing number of derivatives of a Wilson loop, each derivative occurring on a different region of the contour, is generally expected to give a relatively decreasing contribution at short- and at large-distances. Therefore, the series expansion of the kernels in terms of functional derivatives of Wilson loops can also be considered as a perturbative expansion, the most important contribution coming from the lowest-order non-vanishing term and involving the smallest number of derivatives. That property allows us to consider solving the integral equation with appropriate approximations.
The plan of the paper is the following. In Sec. 2, we introduce the definitions and conventions that will be used throughout this work. Section 3 deals with the representation of the quark propagator in external field in terms of path-ordered phase factors. In Sec. 4, functional relations are established between various Green’s functions. In Sec. 5, the integral equation for the gauge invariant quark two-point function with a straight line path is established and the structure of the kernel terms is displayed. Section 6 deals with the question of analyticity properties of the Green’s function. A summary and comments follow in Sec 7. Two appendices are devoted to the presentation of the summation method with free propagators and the study of the self-energy function.
2 Definitions and conventions
We introduce in this section the main definitions and conventions that we shall use throughout this work.
We consider a path-ordered phase factor along a line joining a point to a point , with an orientation defined from to :
where , () being the gluon fields and the generators of the gauge group in the fundamental representation, with the normalization tr. A more detailed definition of is given by the series expansion in the coupling constant ; all equations involving can be obtained from the latter expression. Parametrizing the line with a parameter , , , such that and , a variation of induces the following variation of [, ]:
where and is the field strength, . The variations inside the integral lead to functional differentiation of , while the variations at the end points (marked points) are defined as leading to ordinary differentiation. The functional derivative of with respect to () is then :
For paths defined along rigid lines, the variations inside the integral in Eq. (2.2) are related, with appropriate weight factors, to those of the end points; a displacement of one end point generates a displacement of the whole line with the other end point left fixed. Considering now a rigid straight line between and , an ordinary derivation at the end points yields:
When considering path variations of gauge invariant quantities, with paths made of segments, the end point contributions involving the explicit terms disappear, being cancelled by similar contributions coming from neighboring segments or from variations of neighboring fields. The general contributions that remain at the end are those coming from the internal part of the segments represented by the integrals in Eqs. (2.4)-(2). We adopt the following conventions to represent such contributions:
The first equation above corresponds to a displacement of the end point of the segment (taking into account the orientation on the path), while the second equation corresponds to a displacement of the starting point of the segment. Equations (2.4) and (2) can be written as
If two phase factors and along segments are joined at the point (a marked point), then, with the aid of the previous notations, we have:
The Wilson loop, denoted , is defined as the trace in color space of the path-ordered phase factor (2.1) along a closed contour :
where the factor has been put for normalization. It is a gauge invariant quantity. Its vacuum expectation value is denoted :
the averaging being defined in the path integral formalism.
In perturbation theory, is given by the sum of all connected diagrams, the connection being defined with respect to the contour , after subtraction of reducible parts . Variations of due to local deformations of the contour can then be expressed in terms of variations of :
This property is also generalized to the case of rigid variations of paths (segments). If the contour is a skew-polygon with sides and successive marked points , , , at the cusps, then we write:
the orientation of the contour going from to through , , etc. (i.e., towards s with indices increasing by one unit). Then, according to the definitions (2.6)-(2.10), the notation means that the derivation acts on the internal part of the segment with held fixed (), while means that the derivation acts on the internal part of the segment with held fixed ().
The gauge invariant two-point quark Green’s function is defined as
and being the Dirac spinor indices, while the color indices are implicitly summed. In the present work we shall mainly deal with paths along skew-polygonal lines. For such lines with sides and junction points , , , between the segments, we define:
The simplest such function corresponds to , for which the points and are joined by a single straight line:
(We shall generally omit the index 1 from that function.)
The free propagator will be designated by (without color group content):
3 The quark propagator in external field
We shall use a two-step quantization method, by first integrating the quark fields and then, at a second stage, integrating the gluon fields through Wilson loops. The first operation yields among various quantities the quark propagator in the presence of an arbitrary external gluon field. The latter, designated by , or by for short, satisfies the usual equation
To exhibit a Wilson loop structure in gauge invariant quantities, it is necessary to describe the quark propagator in external field by means of path-ordered phase factors. To this end, we shall first introduce a representation, already used in Ref. , which combines path-ordered phase factors along straight lines and free quark propagators. At a later stage, to sum directly self-energy effects, we shall replace the free quark propagator by the full gauge invariant Green’s fuction (2.18).
The starting point of the representation is the gauge covariant composite object, denoted , made of a free fermion propagator (without color group content) multiplied by the path-ordered phase factor [Eq. (2.1)] taken along the straight line :
[: color indices.] The advantage of the straight line over other types of line is that under Lorentz transformations it remains form invariant and in the limit tends to unity in an unambiguous way. satisfies the following equation with respect to :
A similar equation also holds with respect to , with held fixed, with the Dirac and color group matrices acting from the right.
The quantity is the inverse of the quark propagator in the presence of the external gluon field . Reversing Eq. (3.3) with respect to , one obtains an equation for in terms of :
Using the equation with , or making in Eq. (3.4) an integration by parts, one obtains another equivalent equation:
Equations (3.4) or (3.5) allow us to obtain the propagator as an iteration series with respect to , which contains the free fermion propagator, by maintaining at each order of the iteration its gauge covariance property. For instance, the expansion of Eq. (3.4) takes the form:
In order to sum, for later purposes, self-energy effects, one can use for the expansion of the propagator , instead of the free propagator , the full gauge invariant Green’s function (2.18). To this end, we define a generalized version of the gauge covariant object [Eq. (3.2)], by replacing in it with [Eq. (2.18)]:
The Green’s function satisfies the following equations of motion:
[Notice that the orientation of the path in is from to .] Then satisfies the equation
from which one deduces the expansion of around :
[The integrations on intermediate points are implicit.] Using the equations of and relative to , or making in Eq. (3) an integration by parts, one obtains another equivalent equation:
4 Functional relations for Green’s functions
Let us consider the Green’s function [Eq. (2.17)]. Integrating with respect to the quark fields, one obtains:
The simplest case of this equation, corresponding to , is:
The quark field integration yields also a corresponding determinant, which is a functional of the quark propagator in the external gluon field . That determinant will not, however, play an active role in the subsequent calculations and hence will not explicitly appear in the various formulas that we shall meet; it will rather contribute as a background effect; in particular, it contributes to the evaluation of the Wilson loop averages, unless the quenched approximation is adopted. Therefore, the averaging formulas that we shall encounter should be understood with the presence of the quark field determinant. The expansions that will be used for the quark propagator in the external gluon field can also be repeated inside the quark field determinant if Wilson loop averages are to be evaluated.
Using now for Eq. (3.12), one obtains:
A graphical representation of this equation for is shown in Fig. 2.
The Green’s function satisfies the following equation of motion with :
A graphical representation of this equation for and is shown in Fig. 3.
5 Integral equation
The equations of motion of the gauge invariant Green’s functions [Eqs. (3.8) and (4)] involve in their right-hand sides as unknowns the rigid path derivative of the Green’s functions. The core of the problem amounts therefore to the evaluation of the rigid path derivative of Green’s functions. That task, however, is facilitated by the functional relations (4), which relate two successive Green’s functions with increasing index. They allow the evaluation of the rigid path derivative of a Green’s function in terms of a similar derivative of a Wilson loop average and the derivative of a Green’s function with a higher index. Systematic repetition of this procedure allows us therefore to express the rigid path derivative of a Green’s function in terms of a series of Green’s functions whose coefficients are functional derivatives of Wilson loop averages. One thus obtains chains of coupled integral (or integro-differential) equations between the various Green’s functions. At the end, each Green’s function can be expressed, at leading order of an expansion, by means of the functional relation (4), in terms of the lowest-order Green’s function , and thus an equation where solely the Green’s function would appear becomes reachable.
In the present work we are mainly interested by the simplest Green’s function and therefore we shall concentrate our considerations on the equation of motion of that quantity.
The rigid path derivative of along the segment is obtained from Eq. (4):
Eliminating in the right-hand side of the latter equation the product through Eq. (4), one obtains the equation
[Integrations on new variables in the right-hand sides are implicit.] For , one has:
We next evaluate, in Eq. (5), the action of the path derivation operators on . The operator acts here on the segment and therefore it can be brought without harm to the utmost right, where an equation similar to Eq. (5) (with a relabelling of some arguments, the point being now a junction point on the path of ) is used with and then is brought back to the left; during the last operation it is also submitted to the action of the operator . The resulting terms that involve are:
Next, one observes that is part of the equation of motion of [Eq. (4)], being now one of the fermionic ends of . Using the latter equation and making an integration by parts with respect to , one arrives at a simplified expression in which in the second derivative of the derivation , which is along the segment , is replaced by the derivation , which is along . [The delta functions of the equations of motion do not contribute here because of the existence of the difference term .]
The net result is, including also the resulting terms:
In the term containing , the factors with the derivatives with respect to and are treated in the same way as were those with and with ; they yield at the end the factor plus a term with having a similar structure than the one with above. Repeated use of the procedure described with , yields a series expansion in () where all terms have similar structures. One obtains: