On the Cohomological Derivation of Yang-Mills Theory in the Antifield Formalism

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.


II. BRST FORMALISM
The BRST differential s is split into the Koszul-Tate resolution δ and the exterior derivative γ along the gauge orbits [22,23]: The Koszul-Tate differential maintains the equation of motion (Euler-Lagrange equation).
For any X and Y with Grassmann parity ε X and ε Y , we have: The BRST differential s is a nilpotent derivation: γδ + δγ = 0.

A. Antifield formalism
Let us consider the Lagrangian action It reads the equations of motion δS L 0 /δφ α0 = 0. This has gauge symmetries as δ ε φ α0 = Z α0 α1 ε α1 , where Z α0 α1 is a structure of the gauge group.
The field φ α0 with ghost number zero may imply ghost C α1 with ghost number one, as well as the ghosts of ghost C α2 with ghost number two, etc: We introduce antifield φ * α0 and antighosts C * A = C * α1 , . . . , C * α k of opposite Grassmann parity: Therefore, we can define the gauge variables as where Φ A is a set of fields including the original field, the ghost, and the ghosts of ghosts, Φ * A provides the antifields definition.
The action of the BRST differentials admits an antifield formalisms· = (·, S), where S stands for its generator and (, ) is the antibracket defined in the space of fields Φ A and antifields Φ * A by [28,29] The nilpotent expressions 2 = 0 becomes equivalent to the master equation (S, S) = 0.
We may construct a consistent interaction from S[Φ A , Φ * A ] in a deformed solution in powers of the coupling constant g [30]: of the master equation for the interacting theory S ,S = 0.
On substituting Eq. (15) into the master equation (16), we obtain the deformations of the master equations: where is a free action. We define the BRST differential s of the field theory by s· = (·, S 0 ). Using the last definition, Eq. (17)- (20) are rewritten as: We get all deformations of the master equation in the field theory.

III. DEFORMATIONS OF MASTER EQUATION
We consider the Yang-Mills Lagrangian action involving a set of massless fields A a µ as: where F a µν is the abelian field strengths defined by and η µν is the SO(1, 3) invariant flat metric in Minkowski space, and δ ab is a given symmetric invertible matrix with the following properties The equation of motion of (26) reads The action (26) is close according to an abelian algebra, and invariant under the gauge transformations (30). The BRST transformation provides ghosts C a , antifields A * µ a and antighosts C * a [20,21]: The BRST differential s consisting of δ and γ acts on A a µ , A * µ a , C a , and C * a : The classical master equation of the action (26) has the minimal solution by We will consider deformed solutions to the master equation (16) of the action (26). This comes to the minimal solution (34), when the coupling constant g vanishes.

A. First-order deformation
We recognized that the first-order deformation satisfies (23). Here S 1 is bosonic function with ghost number zero. We assume where a is a local function. The first-order deformation takes the local form where local current j µ shows the nonintegrated density of the first-order deformation related to the local cohomology of s at ghost number zero. To evaluate (36), we assume where (k) j µ are some local currents.
On substituting (37) and (38) into (36), we get It is decomposed into a number of antighost: Although we strictly impose the first expression in (40) with positive antighost numbers vanishes: where H I (γ) is the local cohomology of γ with pureghost number I. Term a I can exclusively reduce to γ-exact terms a I = γb I related to a trivial definition, that states a I = 0. This is plainly given by the second-order nilpotency of γ, which implies the unique solution for (41) up to γ-exact contributions, i. e. agh So, the nontriviality of the first-order deformation a I purposes the cohomology of the longitudinal differential γ at pureghost number equal to I, i. e. a I ∈ H I (γ).
To solve (40), we need to provide H I (γ) and H I (δ|d): where H I (δ|d) is the local homology of the Koszul-Tate differential δ with antighost number I.
For an irreducible situation, where gauge generators are field independent, we assume We then obtain This affords the first-order deformation as follows Let us consider (32) and (33). The local cohomology of γ at pureghost number one has a ghost C a , while pureghost number two shows two ghosts C a C b , i. e. {C a } ∈ H 1 (γ) and {C a C b } ∈ H 2 (γ). From here, we solve (46): where f a bc are the structure constants and antisymmetric on indices bc: This provides We also solve (48) by taking δ from a 1 : The last term in (53) vanishes due to antisymmetric property (51c). We obtain This shows Therefore, we get the first-order deformation up to antighost number two: Here the gauge generators are field independent, and are reduced to a sum of terms with antighost numbers from zero to two.

B. Higher-order deformations
We now solve the second-order deformation of the master equation, (24). We shall assume which takes the local form We shall use (56) to compute This provides the following results: namely, we get a set of equations Eqs. (62) and (64) lead to This implies the Jacobi identity: We also derive Equation (66) gives We can solve it by substituting γ of vector fields A a µ into ∂ µ C a : It provides We accordingly obtain the second-order deformation The Jacobi identity (68) shows (S 1 , S 2 ) = 0 → S 3 = 0.
We then find out that all orders higher than second shall vanish: We solve the Yang-Mills action by the first-and second-order deformations: This includes the gauge structures decomposed into terms with antighost number from zero to two. We can see the part with antighost number zero in the Lagrangian forms. The antighost number one corresponds to the gauge generators. Higher antighost numbers shows the reducibility functions appearing in the ghosts of ghosts. All functions with order higher than second will vanish.

C. Lagrangian and gauge structure
Setting Φ * A = 0 in (72), we read the entire Lagrangian action S L 0 : where F a µν is the field strengths defined by and f a bc are the structure constants of the Lie algebra. The gauge symmetries readδ which holds the following commutator: The gauge transformations remain abelian after consistent deformation. The antighost number one identifies the gauge transformations (76) by substituting C a with gauge parameter ε a . The antighost number two reads the commutator (77). The resulting model is a non-abelian Yang-Mills model constructed by abelian vector fields A a µ .

IV. CONCLUSION
In this paper, we studied a consistently deforming Lagrangian action of the Yang-Mills model in the framework of the antifield formalism. We used the BRST differential to rewrite the deformations of the master equation. The analysis showed that all orders higher than two are trivial. The deformations stopped at second-order provide the consistent interactions being abelian to order g. Upon dismissing antifields, the entire gauge structures of the interacting theory is being realized. Acknowledgment I have been partially supported by a grant from the Marie Curie European Community Programme during my stay at the University of Craiova.
Assuming X = Y , one can find For bosonic (commutative) and fermionic (anticommutative) variables, we have For any X, we have ((X, X), X) = 0, ∀X.