Gauge Freedom in Path Integrals in Abelian Gauge Theory

We extend gauge symmetry of Abelian gauge field to incorporate quantum gauge degrees of freedom. We twice apply the Harada--Tsutsui gauge recovery procedure to gauge-fixed theories. First, starting from the Faddeev--Popov path integral in the Landau gauge, we recover the gauge symmetry by introducing an additional field as an extended gauge degree of freedom. Fixing the extended gauge symmetry by the usual Faddeev--Popov procedure, we obtain the theory of Type I gaugeon formalism. Next, applying the same procedure to the resulting gauge-fixed theory, we obtain a theory equivalent to the extended Type I gaugeon formalism.


I. INTRODUCTION
The standard formalism of canonically quantized gauge theories [1][2][3][4][5] does not consider quantum-level gauge transformations. There is no quantum gauge freedom, since the quantum theory is defined only after the gauge fixing. Within the broader framework of Yokoyama's gaugeon formalism [6][7][8][9][10][11][12][13][14][15], we can consider quantum gauge transformations as q-number gauge transformations among a family of Lorentz covariant linear gauges. In this formalism, quantum gauge freedom is provided by a set of extra fields, called gaugeon fields. The gaugeon formalism has been studied not only in Abelian fields [6,7,10,[16][17][18] and Yang-Mills fields [11][12][13][14][15][19][20][21] but also in the Higgs model [8,21,22], chiral gauge theory [9], Schwinger's model [23], Rarita-Schwinger field [24], string theory [25,26], and gravity [27,28]. Yokoyama and Kubo [7] proposed two types of gaugeon theories for Abelian gauge fields, which they referred to as Type I and Type II theories. The Lagrangian of each theory has a gauge fixing parameter α that can be shifted from α to α + τ by a q-number gauge transformation. The tree level photon propagator can be expressed as where the parameter a is defined as a = εα 2 (ε = ±1) for Type I, (1.2a) a = α for Type II. (1.2b) In Type I theory, the q-number gauge transformation can change the absolute value, but not the sign, of the parameter a; in Type II theory the parameter a can be arbitrarily altered.
The Lagrangian of the Abelian gauge field A µ in Type I theory [6,17] is given by where F µν = ∂ µ A ν −∂ ν A µ , B is the Nakanishi-Lautrup field, c * and c are the usual Faddeev-Popov (FP) ghosts, α is the gauge fixing parameter, Y and Y * are gaugeon fields, and K and K * are FP ghosts for the gaugeon fields, which are introduced to ensure the BRST symmetry [17]. This Lagrangian permits the q-number gauge transformation where we vary the gauge fixing parameter α. The transformation is defined by where φ A collectively represents all fields andα is defined bŷ The Lagrangian and q-number gauge transformation in Type II theory are described in [7]. BRST symmetric Type II theory is given by [16].
The Lagrangian of the extended Type I theory, investigated by Endo [18], is given by where Y i and Y i * (i = 1, 2) are two sets of gaugeon fields, K i and K i * are two sets of FP ghosts for the gaugeon fields, and the constant α i are the gauge fixing parameters. The corresponding parameter a of the tree level photon propagator (1.1) is given by a = 2α 1 α 2 .
Thus, this theory extends the Type I gaugeon formalism by setting a as quadratic in the gauge fixing parameters (cf. (1.2a)). Because the parameter a (and its sign) can be changed into arbitrary values by the q-number gauge transformation ( this theory possesses some characteristics of Type II theory. The Lagrangian can also be written as where Y ± are defined by and Y ± * , K ± , K ± * , and α ± are defined similarly. Gaugeon theories for the Yang-Mills fields have been proposed by Yokoyama [11] and Yokoyama, Takeda and Monda [15]. The BRST symmetric theories have been obtained by Abe [19] and Koseki, Sato and Endo [20]. Although these theories are easily shown to be equivalent to the standard formalism in the Landau gauge (a = 0), their equivalence to the standard formalism in non-Landau gauges (a = 0) cannot be demonstrated. Therefore, these theories should be compared with the Abelian gaugeon theory, which is equivalent to non-Landau gauge theory (a = 0) as well as to the Landau gauge (a = 0) [17].
Sakoda [29] extended the gauge freedom of Yang-Mills fields using the gauge recovery procedure for gauge non-invariant functionals proposed by Babelon, Schaposnik and Viallet [30] and Harada and Tsutsui [31,32]. Sakoda's theory includes the two gauges of the standard formalism: the Landau gauge and a non-Landau a-gauge. Sakoda's theory considers the total Fock space, which embeds the Fock spaces of the both gauges of the standard formalism. In this theory, the q-number gauge transformation connects the Landau gauge and non-Landau a-gauge. Different from the gaugeon formalism, the q-number transformation of Sakoda's theory cannot arbitrarily change the gauge parameter, but allows only α = 0 and α = a.
In this paper, we further extend the gauge freedom to allow more flexibility in the gauge parameter than in Sakoda's theory. As a first step, we consider the Abelian gauge field.
Starting with the Faddeev-Popov path integral in the Landau gauge, we extend the gauge freedom by twice applying the Harada-Tsutsui gauge recovery procedure [32]. In contrast, Sakoda [29] applied this procedure once to the Yang-Mills field.
The remainder of this paper is organized as follows. Section 2 reviews the Harada-Tsutsui gauge recovery procedure for gauge non-invariant functionals [32] and Sakoda's path integral [29] of Yang-Mills fields. In Section 3, we extend the gauge symmetry of Abelian gauge fields twice using the Harada-Tsutsui gauge recovery procedure. In Section 4, we relate our theory to the gaugeon formalism and show that our theory is equivalent to the extended Type I gaugeon formalism.

II. PATH INTEGRAL OF THE GAUGE NON-INVARIANT FUNCTIONAL
A. Harada-Tsutsui gauge recovery procedure Harada and Tsutsui's procedure extends the gauge degrees-of-freedom of the gauge noninvariant functional [32]. We illustrate their procedure on a system of gauge non-invariant Yang-Mills fields A µ . Such a system might comprise massive Yang-Mills fields.
The action S 0 [A] of the system is not invariant under the gauge transformation, where g is a group-valued function. The usual path integral of the system is given by which leads to non-renormalizable propagators in the massive Yang-Mills case. Now, we promote the group-valued function g(x) to a dynamical variable, and define an extended action by which is now invariant under the extended gauge transformation, To factor out the divergent gauge volume, we require gauge fixing (if g = 1, Z div reduces to Z 0 ). Expressing the gauge fixing condition as the corresponding FP determinant ∆ FP [A, g] is given by Inserting (2.8) into (2.6) and factoring out the gauge volume, we obtain We can also consider 't Hooft averaging. Instead of the gauge fixing condition (2.7), we where C(x) is an arbitrary c-number function. Averaging the path integral over C(x) with the Gaussian weight The first line expresses the delta functional as a Fourier integral with respect to a field Φ.
Equation (2.12) yields renormalizable propagators in the massive Yang-Mills case.

B. Sakoda's method
Sakoda [29] extended the gauge freedom of the gauge-fixed Yang-Mills fields in the Landau gauge by applying the Harada-Tsutsui procedure. This method is briefly explained below.
The Landau-gauge Lagrangian of a Yang-Mills field A µ is given by where F µν is the field strength, B is the Nakanishi-Lautrup field, and c andc are the FP ghosts. We express the path integral as where (2.16) Since we consider a gauge fixed system, the functional I 0 [A, B] is not gauge invariant under the gauge transformation (2.17) Now, we promote the group-valued function g(x) to a dynamical variable and definẽ is divergent sinceĨ 0 [A, B, g] is now gauge invariant. To factor out the divergent gauge volume, we require gauge fixing. For this purpose, we consider the following gauge fixing where C is an arbitrary c-number function. The corresponding FP determinant ∆ FP [A, g, C] is then given by averaging with a Gaussian weight, we obtain where a is the gauge fixing parameter. The corresponding Lagrangian is given by Here, the determinant ∆[A g ] inĨ 0 [A, B, g] has been expressed in terms of the FP ghosts η andη: The term in the second line isQ-exact, and thus ignorable in the subspace kerQ; the remaining term is nothing but the a-gauge Lagrangian. To show that the subspace ker Q corresponds to the Fock space of the standard formalism of the Landau gauge, we denote the Landau-gauge fields A ′ µ by A ′ µ = A g µ and express the Lagrangian as where F ′ µν is the field strength of A ′ µ , D ′ µ is the covariant derivative corresponding to A ′ µ , and η ′ = η g . The term in the second line is ignorable in the subspace ker Q; the remaining term is the Landau-gauge Lagrangian. Thus, in Sakoda's theory, the subspaces of the total Fock space identify the Fock spaces of the standard theory of the a-gauge and Landau gauge.
The a-gauge field A µ and Landau-gauge field A ′ µ are connected through the q-number gauge transformation g(x). The q-number transformation of Sakoda's theory limits the gauge parameter to only two values, α = 0 and α = a. Considering this, Sakoda's theory differs from the gaugeon formulation of the Yang-Mills field [15,19,20]. (Strictly speaking, using Sakoda's q-number transformation g(x) we can define another q-number gauge transformation {g(x)} τ with an arbitrary real number τ . This transformation changes the gauge parameter a into a(1 − τ ) 2 in the tree level propagator of A µ . We do not, however, consider this transformation at present, since the transformed Lagrangian would have complicated terms and it would not be easy to analyze the theory in this gauge. 2 ) 1 We comment here that another expression would be helpful to analyze the subspace kerQ by using the BRST charge Q. The field Φ (rather than B) plays the role of Nakanishi-Lautrup field in this a-gauge theory. 2 In the Abelian limit, the situation becomes simple. The Abelian limit of Sakoda's theory is equivalent to the Abelian gaugeon formalism (see sections III A and IV A); the transformation {g(x)} τ becomes a usual q-number gauge transformation of the Abelian gaugeon formalism.

A. Sakoda's extension
For the selfcontainedness of this section, we repeat here Sakoda's arguments in the Abelian case.
We start with the Landau-gauge Lagrangian of the Abelian gauge field given by The path integral is expressed as where Since we consider a gauge fixed system, the functional I 0 [A, B] is not gauge invariant under the gauge transformation where θ is an arbitrary scalar function. Now, we promote the function θ to a dynamical variable and definẽ The functionalĨ 0 [A, B, θ] is invariant under the extended gauge transformation, where λ is an arbitrary scalar function. The formal path integral forĨ 0 [A, B, θ], The corresponding FP determinant is then given by Inserting (3.10) into (3.8) and factoring out the gauge volume, we obtain Expressing the delta functional as a Fourier integral with respect to Φ and applying 't Hooft averaging with a Gaussian weight, we obtain with where a is the gauge fixing parameter. The two FP determinants can be expressed in terms of two pairs of ghost fields Summarizing these results, we obtain the Lagrangian of the first extension of the gauge freedom as Because we extended the gauge freedom by the method of Sakoda [29], the above Lagrangian is the Abelian limit of Sakoda's Yang-Mills Lagrangian (2.26). As implied by Sakoda [29], This transformation satisfies the nilpotency δ 2 B = 0. We can also find δ andδ transformations satisfying δ B = δ +δ, as in the Yang-Mills case (2.29) and (2.30).

B. The successive extension
Starting from the path integral (3.13), we again extend the gauge freedom of the Lagrangian L 1st . The functional I 1 [A, B, θ, Φ] is not gauge invariant under the gauge transformation, (3.18) where χ is an arbitrary scalar function. Now, we promote the function χ to a dynamical variable and definẽ where we have used A χ µ + ∂ µ θ χ = A µ + ∂ µ θ. The functionalĨ 1 [A, B, θ, Φ, χ] is gauge invariant under the following extended gauge transformation: (3.20) where λ is an arbitrary scalar function. The formal path integral forĨ 1 [A, B, θ, Φ, χ], To factor out the divergent gauge volume, we require gauge fixing. We consider the following gauge fixing condition where C is an arbitrary c-number function. The corresponding FP determinant ∆ FP [A, χ, C] is then defined as Inserting (3.23) into (3.21) and factoring out the gauge volume, we obtain Expressing the delta functional as a Fourier integral with respect to φ and applying 't Hooft averaging with a Gaussian weight, we obtain with where a ′ is another gauge fixing parameter. The Lagrangian of the successive extension of the gauge degree of freedom is expressed by where the third ∆ has been expressed in terms of the ghost fields ξ andξ: This Lagrangian is invariant under the following BRST transformation, where α is a numerical parameter satisfying a = εα 2 (ε = a/|a|), we can rewrite the Lagrangian (3.16) as which is exactly (1.3). It should be noted that the field θ introduced as an extended gauge freedom plays the role of a gaugeon field.
Next, we show that the Lagrangian (3.28) of the successive extension is equivalent to the extended Type I Lagrangian (1.8). Redefining the fields as where α + and α − are numerical parameters satisfying we can rewrite the Lagrangian (3.28) as Lagrangian resulting from the first extension agrees with the Abelian limit of Sakoda's Yang-Mills Lagrangian, and is equivalent to that of Type I gaugeon theory. The scalar field θ introduced as an extended gauge degree of freedom plays the role of a gaugeon field Y .
The theory obtained by the second extension is equivalent to extended Type I theory if the signs (ε and ε ′ ) differ. The scalar fields θ and χ introduced as extended gauge degrees of freedom play the roles of gaugeon fields Y + and Y − .
One might think what happens when we further repeat Sakoda's extensions of gauge freedom. In the Abelian case, extending the gauge freedom three times or more would not yield any new features, at least from the viewpoint of the photon propagator (1.1).
For example, when we apply Sakoda's extension to the Lagrangian L 2nd (4.5), we obtain the third pair of gaugeon fields (Y 3 * , Y 3 ), corresponding FP ghosts (K 3 * , K 3 ), and the third extended Lagrangian, where ε 3 is a sign factor and α 3 a numerical parameter. The photon propagator following from (5.1) is again expressed as (1.1) where the parameter a is now given by Thus the third extension does not expand the region of the values of the parameter a; the second extension is enough to give the parameter a an arbitrary value. In the non-Abelian case, non-trivial features may appear when we apply Sakoda's extensions multiple times.
Exploring this possibility is our next task.