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We present a brief review of the cohomological solutions of self-coupling interactions of the fields in the free Yang-Mills theory. All consistent interactions among the fields have been obtained using the antifield formalism through several order BRST deformations of the master equation. It is found that the coupling deformations halt exclusively at the second order, whereas higher order deformations are obstructed due to non-local interactions. The results demonstrate the BRST cohomological derivation of the interacting Yang-Mills theory.

Dirac’s pioneering approach [

To generalize constrained systems for canonical conditions and (anti-)com- mutative variables, Becchi, Rouet, Stora [

In this paper, we briefly review the construction of all consistent interactions of the free Yang-Mills theory determined from all coupling deformations of the master equation. We see that the resulting action presents deformed structures of the gauge transformation and yields a commutator for it. In Section 2, the BRST differential and the antifield formalism are introduced. Section 3 introduces the consistent interactions among the fields. We consider the BRST coupling deformations of the master equations in the antifield formalism in Section 4. In Section 5, we demonstrate its application to the massless Yang-Mills theory by calculating all several order deformation of the master equation. Section 6 presents a conclusion.

The gauge invariant in a phase space implies that the smooth phase space

The reduced space, by taking

whose cohomology is equal to the cohomology of the longitudinal differential

where

Any nilpotent derivation has a degree in a

The positive degree of the differential

with the following property

where the operators

such

The cohomology algebra of the differential

while the elements of its image subspace,

The cohomology algebra of

If the co-/homology

The zeroth cohomology group of the BRST differential

and also their anticommutation:

It means that the Koszul-Tate differential

The generator of the Koszul-Tate complex may be chosen in an equal number of freedom as the generator of the longitudinal exterior complex. It follows that they are canonically conjugate in the extended space of original and new generators of

which is called the Poisson bracket and defined as follows:

where

Equation (13) represents the BRST symmetry in the Hamiltonian formalism. The choice of

which is the master equation of the BRST generator in the Hamiltonian formalism.

To understand the consistent interactions among fields with a gauge freedom, we begin our study with a Lagrangian action:

where the action

The equations of motion then read

The equations of motion is then determined from the action principle:

Let consider the deformations of the action in such a way

that implies the deformation of gauge symmetries as

This provides the deformed gauge transformations:

Equation (18) and Equation (19) lead to the following expression:

Hence, the deformations by their orders are as follows:

which define the deformed gauge transformations that close on-shell for the interacting action, the so-called consistent interactions, while the original gauge transformations are reducible [

Assume that the gauge fields of consistent interactions are trivially defined to be the following sum:

we then obtain

which does not manifest an exact interacting theory. A theory is strict if the consistent deformations are merely proportional to its free theory action

where charges

It represents the unperturbed action by charges of the coupling constants.

Let us consider the gauge transformation defined by the Equation (17). The classical fields

which have the following ghost numbers,

It also implies antifields

The presentation of the gauge variables is therefore provided by

where a set of fields

The BRST symmetry is a canonical transformation, and defined by an antibracket structure:

where

The Grassmann parity and ghost number of the antibracket are, respectively:

The antifields are now considered as mathematical tool to construct the BRST formalism. The solution can be interpreted as source coefficient for BRST transformation, i.e., an effective action in the theory.

The fields and antifields establish the solution

Section 2 presented the master Equation (15) of the BRST generator in the Hamiltonian formalism. The gauge structure is now constructed through the solution

This shows the consistency of the gauge transformations. The master Equation (36) includes the closure of the gauge transformations, the higher-order gauge identities, and the Noether identities. The master equation maintains the consistent specifications on

Substituting the definition (35) into the master Equation (36) yields

We then derive

which are simplified as follows [

the so-called deformations of the master equation [

The Equation (40) implies that

The free gauge invariant action

by setting

It provides the solution

The BRST differential

Using the definitions (48), the deformations of the master equation are rewritten as follows:

which are the deformations of the master equation in terms of the BRST differential

Let us consider a set of

where

in such a way

where

The gauge transformation with the free equation of motion,

manifests an irreducible transformation by

while

The differential operator

The implementation of the BRST transformation in the minimal sector provides the field

which can schematically be illustrated:

We calculate the BRST-differential

The classical master Equation (47) of the action (50) holds the minimal solution (45) in such a way

We now consider the deformed solution of the master equation for the action (50) smoothly in the coupling constant

Let us assume

where

where

To evaluate Equation (60), we assume

where

obviously

They can be decomposed on the several orders of the antighost number:

The positive antighost number are strictly given as replacement for the first expression [

To proof it, let us consider

while

The objects

Moreover,

corresponding to a trivial definition, which states

Hence, the non-triviality of the first-order deformation

where

For an irreducible linear situation, where gauge generators are field independent, we assume that

where

The first-order deformation up to antighost number two are:

The

where

We now consider the Koszul-Tate differential

The local cohomology of the exterior longitudinal derivative

From (79), we then solve

by

where

The expression

We simply notice that

This indicates

To obtain

The last term in above relation vanishes, i.e.

since

while

Therefore, we derive

It shows

The results for the first-order deformation are summarized as follows:

Finally, we derive

The first-order deformations of the solution (

We now consider the higher-order deformations of the master equation for the action (50). The second-order deformation (

that takes the local form

Using the Equation (88) from Section 5.1, we calculate

while employing the following relations

and the definitions

They lead to the following expression

that is reduced to

We then decompose

namely,

We also define

From (91), it follows a set of equations

Equations (99) and (101) imply

and

The later expression is called the Jacobi identity. Similarly, we obtain

So, the Equation (103) remains to be solved:

We solve it by substituting the exterior longitudinal differential

Accordingly, we derive

Hence, the second-order deformations becomes

The Jacobi identity (105) obviously implies

Similarly, all deformations with orders higher than the second-order completely vanish:

As a result, the solution to the deformations becomes

We have determined the Yang-Mills theory from the first- and second-order deformations of the master equation. The solutions of the master equation, which entirely include the gauge structures, are decomposed into terms with the antighost numbers from zero to two. In other words, the part with the antighost number equal to zero represents the Lagrangian action, while the antighost number one is proportional to the gauge generators. The terms with higher antighost numbers provide the reducibility functions, where the on-shell relations become linear components in the ghosts for ghosts. It is shown that all functions with order higher than second vanish in this model.

Let us consider the Equation (109) and identify the entire gauge structure of the Lagrangian model that describes all consistent interactions in the

The antighost number zero of (109) shall provide the Lagrangian action of the interacting theory:

Accordingly, the Yang-Mills theory is characterized by the following non- abelian action:

where the non-abelian field strengths

and

So, the commutator among the deformed gauge transformations becomes:

The gauge symmetry remains abelian to order

The invariance of the action under the gauge transformations (113) is also obtained by the Noether identities

The antighost number one of the deformation of the master equation allows to identify the gauge transformations (113) of the action (110) by substituting the ghost

In this paper, we reviewed deformed gauge transformations in the framework of the BRST-antifield formalism characterized by the antibracket that acts similar to the Poisson bracket in the Hamiltonian formalism. We provided the BRST cohomology of the consistent interactions through several order deformations of the master equation. The BRST-antifield formalism in the cohomological space provides the generalized framework of consistent interactions among fields with a gauge freedom by any types of invariant action. We see that higher order deformations could be neglected due to non local interactions and their obstruction of consistent local couplings, which are associated with the anomalous gauge quantization. We demonstrated its functions by applying the BRST-antifield formalism to the

The author thanks the editor and the referee for their comments. Research of A. Danehkar is funded by the EU contract MRTN-CT-2004-005104. This support is greatly appreciated.

Danehkar, A. (2017) On the Cohomological Derivation of Yang-Mills Theory in the Antifield Formalism. Journal of High Energy Physics, Gravitation and Cosmology, 3, 368-387. https://doi.org/10.4236/jhepgc.2017.32031

For a function

The left derivative

For any

Considering Equation (32) and Equation (118), it follows that

Assuming

For bosonic (commutative) and fermionic (anticommutative) variables, we have

For any

Furthermore, the antibracket has the following properties: