Romanian Reports in Physics, Vol. 57, No. 2, P. 189–203, 2005
NUCLEAR PHYSICS. PARTICLE PHYSICS
CONSISTENT INTERACTIONS BETWEEN BF AND MASSIVE DIRAC FIELDS. A COHOMOLOGICAL APPROACH EUGEN-MIHÃIÞÃ CIOROIANU1, SILVIU CONSTANTIN SÃRARU2 Faculty of Physics, University of Craiova, 13 A. I. Cuza Str., Craiova 200585, Romania 1 E-mail:
[email protected] 2 E-mail:
[email protected] (Received September 19, 2003)
Abstract. Consistent interactions for a four-dimensional gauge theory, described in the free limit by an abelian BF-type model and one massive Dirac field, are approached in the framework of the deformation theory based on the local BRST cohomology. The deformation procedure leads to an interacting model with modified gauge transformations, whose algebra is open. Key words: consistent interactions, local BRST cohomology, BF theories.
1. INTRODUCTION A key point in the development of the BRST formalism was its cohomological understanding, which allowed, among others, a useful investigation of many interesting aspects related to the perturbative renormalization problem [1–4], the anomaly-tracking mechanism [4–8], the simultaneous study of the local and rigid invariances of a given theory [9], as well as to the reformulation of the construction of consistent interactions in gauge theories [10–13] in terms of the deformation theory [14–16], or, actually, in terms of the deformation of the solution to the master equation. The scope of this paper is to investigate the consistent interactions that can be added to a free, abelian, four-dimensional gauge theory, describing a single massive Dirac field and a BF-type model [17] involving one scalar field, two types of one-forms and one two-form. This work enhances the previous Lagrangian [18] and Hamiltonian [19–20] results on the study of self-interactions in certain classes of BF-type models. The resulting interactions are accurately described by a gauge theory with an open algebra of gauge transformations, which are second-order reducible, like in the case of the free model, but the new reducibility relations take place on-shell. It is also worth noting that the Dirac field gains non-trivial gauge transformations.
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Our strategy goes as follows. Initially, we determine in Section 2 the antifield-BRST symmetry of the free model, that splits as the sum between the Koszul-Tate differential and the exterior derivative along the gauge orbits, s = δ + γ. Next, in Section 3 we determine the consistent deformations of the solution to the master equation for the free model. The first-order deformation belongs to the local cohomology H 0 ( s | d ) , where d is the exterior space-time
derivative. The computation of the cohomological space H 0 ( s | d ) proceeds by expanding the co-cycles according to the antighost number, and by further using the cohomological groups H ( γ ) and H ( δ | d ) . We find that the first-order deformation is parametrized by three kinds of functions depending on the undifferentiated scalar field, which become restricted to fulfill certain equations in order to produce a consistent second-order deformation. We select only those solutions that ensure effective cross-couplings between the BF field spectrum and the Dirac field, and consequently infer that the remaining deformations, of order two and higher, can be taken to vanish. The identification of the interacting model is developed in Section 4, where it turns out that the cross-interactions between the Dirac field and the BF field spectrum are described by a generalized minimal coupling of the type current-vector field in an arbitrary ‘background’ of the scalar field. The role of the vector field is played by the one-form from the BF theory displaying an abelian U (1) gauge symmetry, while the above mentioned current is associated with the conservation law of the Dirac theory corresponding to a special class of one-parameter rigid symmetries that involve an arbitrary function of the scalar field. Meanwhile, both the gauge transformations corresponding to the coupled model and their algebra are deformed with respect to the initial abelian theory in such a way that the new gauge algebra becomes open and the reducibility relations take place on-shell. Section 5 closes the paper with the main conclusions. 2. FREE MODEL. BRST SYMMETRY
The starting point is represented by the free Lagrangian action
∫
(
S0 ⎡⎣ Aµ , H µ , ϕ, Bµν , ψ α , ψ α ⎤⎦ = d 4 x H µ ∂µ ϕ + 1 Bµν Fµν + 2
(
+ψ α i ( γ µ )
α
)
∂ − mδαβ ψβ ⎞⎟ , β µ ⎠
(1)
where we employed the notation Fαβ = ∂[ α Aβ] for the field strength of the oneform Aµ . We observe that (1) is written as a sum between the action of a fourdimensional abelian BF theory (involving two one-forms, one scalar field and one
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Interaction between BF and massive Dirac fields
191
two-form) and the action corresponding to a single massive Dirac field. The free action (1) is found invariant under the gauge transformations
δε Aµ = ∂ µ ε,
δε H µ = 2∂ ν εµν ,
δε ϕ = 0,
δε ψ α = 0,
δε B µν = −3∂ ρ εµνρ ,
(2)
δε ψ α = 0,
(3)
where the gauge parameters ε, εµν and εµνρ are bosonic, with εµν and εµνρ completely antisymmetric. The gauge transformations (2–3), are abelian and offshell second order reducibile. More precisely, we observe that if in (2) we make the transformations µνρ , εµν → εµν θ = −3∂ ρ θ
(4)
µνρλ , εµνρ → εµνρ ( θ ) = 4∂ λ θ
(5)
with θµνρ and θµνρλ arbitrary antisymmetric fields, the gauge transformations of the fields H µ and Bµν identically vanishes δε( θ ) H µ ≡ 0, δε( θ ) Bµν ≡ 0.
(6)
Moreover, if in (4) we perform the changes µνρλ , θµνρ → θµνρ ( ω) = 4∂ λ ω
(7)
with ωµνρλ arbitrary antisymmetric field, the transformed gauge parameters (4) identically vanish
εµν (θ
( ω)
)
≡ 0.
(8)
We remark that the BF theory alone is a usual linear gauge theory of Cauchy order equal to four, while the Dirac field is described by a linear, propagating theory, of Cauchy order equal to one, so the overall Cauchy order of the starting model is equal to four. In order to construct the BRST symmetry of this “free” theory, we introduce the field/ghost and antifield spectra
(
)
* , ψ* , ψ*α , Φ α 0 = ( Aµ , H µ , ϕ, Bµν , ψ α , ψ α ) , Φ*α 0 = Aµ* , H µ* , ϕ* , Bµν α
ηα1 = ( η, C µν , ηµνρ ) , ηα2 = ( C µνρ , ηµνρλ ) ,
(
)
(
)
* , η* η*α1 = η* , Cµν µνρ , * , η* η*α2 = Cµνρ µνρλ ,
* ηα3 = C µνρλ , η*α3 = Cµνρλ .
(9) (10) (11) (12)
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The fermionic ghosts ηα1 respectively correspond to the bosonic gauge parameters εα1 = (ε, εµν , εµνρλ ), the bosonic ghosts for ghosts ηα2 are due to the first-order reducibility relations (4–5), while the fermionic ghosts for ghosts for ghosts ηα3 are required by the second-order reducibility relations (7). The star variables represent the antifields of the corresponding fields/ghosts. Their Grassmann parities are obtained via the usual rule ε ( χ* ) = ( ε ( χ ) + 1) mod 2,
(13)
where we employed the notations
(
)
χ = ( Φ α 0 , ηα1 , ηα2 , ηα3 ) , χ* = Φ*α 0 , η*α1 , η*α2 , η*α3 .
(14)
Since both the gauge generators and the reducibility functions are fieldindependent, it follows that the BRST differential reduces to s=δ+γ ,
(15)
where δ is the Koszul-Tate differential, and γ means the exterior longitudinal derivative. The Koszul-Tate differential is graded in terms of the antighost number (agh,agh ( δ ) = −1, agh ( γ ) = 0) and enforces a resolution of the algebra of smooth functions defined on the stationary surface of field equations for the action (1), C ∞ ( Σ ) , Σ : δS0 / δΦ α 0 = 0. The exterior longitudinal derivative is graded in terms
of the pure ghost number (pgh, pgh ( γ ) = 1, pgh ( δ ) = 0) and is correlated with the gauge symmetry via its cohomology at pure ghost number zero computed in C ∞ ( Σ ) , which is isomorphic to the algebra of physical observables for the free model. The two degrees of the generators from the BRST complex are valued as pgh ( Φ α 0 ) = 0, pgh ( ηα1 ) = 1, pgh ( ηα2 ) = 2, pgh ( ηα3 ) = 3,
(
(
)
( )
( )
( )
(16)
pgh Φ*α 0 = pgh η*α1 = pgh η*α2 = pgh η*α3 = 0,
(17)
agh ( Φ α 0 ) = agh ( ηα1 ) = agh ( ηα2 ) = agh ( ηα3 ) = 0,
(18)
)
( )
( )
( )
agh Φ*α 0 = 1, agh η*α1 = 2, agh η*α2 = 3,agh η*α3 = 4,
(19)
while the actions of δ and γ on them read as δΦ α 0 = δηα1 = δηα2 = δηα3 = 0,
(20)
* =−1F , δA*µ = ∂ ν Bνµ , δH µ* = −∂ µ ϕ, δϕ* = ∂ µ H µ , δBµν 2 µν
(21)
5
Interaction between BF and massive Dirac fields
(
δψ*α = − i ( γ µ )
β α
)
(
∂ µ + mδβα ψβ , δψ*α = − i ( γ µ )
α β
193
)
− mδβα ψ α ,
* = ∂ H * , δη* * δη* = −∂ µ Aµ* , δCµν [µ ν ] µνρ = ∂[µ Bνρ ] , * * , δη* * * * δCµνρ = −∂[µCνρ ] µνρλ = −∂[µ ηνρλ ] , δCµνρλ = ∂[µ Cνρλ ] ,
(22) (23) (24)
γΦ*α 0 = γη*α1 = γη*α2 = γη*α3 = 0,
(25)
γAµ = ∂ µ η, γH µ = 2∂ νCµν , γBµν = −3∂ ρ ηµνρ , γϕ = 0,
(26)
γψ α = γψ α = 0, γη = 0, γC µν = −3∂ ρC µνρ , γηµνρ = 4∂ λ ηµνρλ ,
(27)
γC µνρ = 4∂ λC µνρλ , γηµνρλ = γC µνρλ = 0.
(28)
The overall degree of the BRST complex is named ghost number (gh) and is defined as the difference between the pure ghost number and the antighost number, such that gh ( s ) = 1 . The BRST symmetry admits a canonical action in a structure named antibracket, s⋅ = ( ⋅, S ) , where its canonical generator, S , is a bosonic functional of ghost number zero, that satisfies the classical master equation
( S, S ) = 0. The notation (,) signifies the antibracket, which is defined by decreeing
the fields/ghosts conjugated to the corresponding antifields. In the case of the free theory under discussion, the solution to the master equation takes the form
∫
(
* ∂ ηµνρ − 3C * ∂ C µνρ + S = S0 + d 4 x Aµ* ∂ µ η + 2 Hµ* ∂ νC µν − 3Bµν ρ µν ρ
)
* ∂ C µνρλ , +4η*µνρ ∂ λ ηµνρλ + 4Cµνρ λ
(29)
and we observe that it contains pieces with the antighost number ranging from zero to three.
3. BRST DEFORMATION PROCEDURE
3.1. GENERAL SETTING A consistent deformation of the free action (1) and of its gauge invariances (2–3) defines a deformation of the corresponding solution to the master equation that preserves both the master equation and the field/antifield spectra. So, if S0 ⎡⎣ Aµ , H µ , ϕ, Bµν , ψ α , ψ α ⎤⎦ + g d 4 xa0 + O ( g 2 ) ,
∫
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stands for a consistent deformation of the free action, with deformed gauge transformations δε Aµ = ∂ µ ε + gβµ + O ( g 2 ) , δε Hµ = 2∂ ν εµν + ρµ + O ( g2 ) , δε ϕ = = gβ + O ( g2 ) , δε B µν = −3∂ ρ εµνρ + gβµν + O ( g2 ) , δε ψ α = gσα + O ( g2 ) and δε ψ α =
= gλ α + O ( g2 ) , then the deformed solution
S = S + g d 4 xa + O ( g2 ),
∫
(30)
satisfies the master equation ( S , S ) = 0, where the non-integrated density of the first-order deformation starts like * β a = a0 + A*µ βµ + H *µ ρµ + ϕ* β + Bµν µν
µν
α
+ ψ*α σα + ψ *α λ + “more” .
α
The terms βµ , ρµ , β, β , σα and λ are obtained by replacing the gauge parameters ε, εµν and εµνρ respectively with the fermionic ghosts η, C µν and ηµνρ into the functions βµ , ρµ , β, βµν , σα and λ α . The master equation ( S , S ) = 0 holds to order g if and only if sa = ∂ µ j µ ,
(31)
for some local j µ . In order to solve this equation, we develop a according to the antighost number a = a0 + a1 + " + aI , agh ( aK ) = K , gh ( aK ) = ε ( aK ) = 0.
(32)
The number of the terms in the expansion (31) is finite and it can be shown that we can take the last term in a to be annihilated by γ
γaI = 0.
(33)
Consequently, we need to compute the cohomology of γ, H ( γ ) , in order to determine the component of the highest antighost number in a. From the definitions (25–28) it is simple to see that H ( γ ) is spanned by Fµν , ∂µ H µ , ϕ,
(
)
∂µ Bµν , ψ α , ψ α and the antifields χ* = Φ*α 0 , η*α1 , η*α2 , η*α3 , by their space-time
derivatives, as well as by the undifferentiated ghosts η A1 = ( η, ηµνρλ , C µνρλ ) . (The derivatives of the ghosts η A1 are removed from H ( γ ) since they are γ-exact, as can be seen from the first formula in (26), the last definition in (27), and respectively the first relation in (28).) If we denote by e M ( η A1 ) the elements with
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Interaction between BF and massive Dirac fields
195
pure ghost number M of a basis in the space of the polynomials in the ghosts η A1 , it follows that the general solution to the equation (33) takes the form
(
)
aI = µ I ⎡⎣ Fµν ⎤⎦ , ⎡⎣ ∂ µ H µ ⎤⎦ , [ ϕ] , ⎡⎣∂ µ Bµν ⎤⎦ , [ ψ α ] , ⎡⎣ψ α ⎤⎦ , ⎡⎣ χ* ⎤⎦ e I ( η A1 ) ,
(34)
where agh ( µ I ) = I and pgh ( e I ) = I . The notation f ([ q ]) means that f depends on q and its space-time derivatives up to a finite order. The equation (31) projected on antighost number ( I − 1) becomes ( I −1)
δaI + γaI −1 = ∂ µ m
µ.
(35)
Replacing (34) in (35), it follows that the last equation possesses solutions with respect to aI −1 if the coefficients µ I pertain to the homological space H I ( δ d ) , i.e., δµ I = ∂ µ lIµ−1 . In the meantime, since our free model is linear and of Cauchy order equal to four, according to the general results from [21–22] we get that H J ( δ d ) vanishes for J > 4, so we can assume the first-order deformation stops at the antighost number four ( I = 4 ) a = a0 + a1 + a2 + a3 + a4 ,
(36)
where a4 is of the form (34) with µ4 from H 4 ( δ d ) . 3.2. FIRST-ORDER DEFORMATION By direct computation, we infer that the most general representative of H 4 ( δ d ) can be taken of the type W C* + δ ( µ 4 )µνρλ = δδϕ µνρλ
2W
δϕ2
* * * ( H[*µCνρλ ] + C[µν Cρλ ] ) +
3 4 * + δ W H * H *H *H * , + δ W3 H[*µ H ν*Cρλ µ ν ρ λ ] δϕ δϕ4
(37)
with W ( ϕ ) an arbitrary function depending on the undifferentiated scalar field. On the other hand, the elements of pure ghost number equal to four of the basis in the space of polynomials in the ghosts η A1 are ηC µνρλ , ηαβγδ ηµνρλ .
(38)
In order to couple (37) to the second element in (38), we need some completely antisymmetric constants kαβγδ that, by covariance arguments, can only be
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proportional to the completely antisymmetric four-dimensional symbol εαβγδ . So, the most general form of the last representative from the expansion (36) is 2 ⎛ * * * * a4 = ⎜ δW Cµνρλ H[*µCνρλ +δ W ] + C[µν Cρλ ] + δϕ2 ⎝ δϕ 3 4 * + δ W H * H * H * H * ⎞ ηC µνρλ + + δ W3 H[*µ H ν*Cρλ µ ν ρ λ⎟ ] δϕ δϕ4 ⎠
(
)
(39)
2 ⎛ * * * * H[*µCνρλ + 1 ⎜ δM Cµνρλ +δ M ] + C[µν Cρλ ] + 2 ⎝ δϕ δϕ2
(
)
3 4 αβγδ , * + δ M H * H * H * H * ⎞ ηµνρλ ε H[*µ H ν*Cρλ +δ M µ ν ρ λ⎟ αβγδ η ] 3 δϕ δϕ4 ⎠
where the numerical factor ½ in the second term was taken for convenience, and the functions W and M are arbitrary functions of the undifferentiated scalar field. By computing the action of δ on a4 and by taking into account the relations (25–28) it follows that the solution to the equation (35) for I = 4 is precisely given by 2 3 ⎛ * + δ W H * C * + δ W H * H * H * ⎞ × 4 A C µνρλ + ηC νρλ + a3 = − ⎜ δW Cνρλ [ ν ρλ ] ν ρ λ⎟ µ 2 δϕ δϕ3 ⎝ δϕ ⎠ 2 ⎛ * + δ W H * H * ⎞ B* ⎞ C µνρλ − (40) +2 ⎛⎜ W η*µνρλ + 4 δW Hµ* η*νρλ + 6 ⎜ δW Cµν µ ν ⎟ ρλ ⎟ δϕ δϕ2 ⎝ ⎝ δϕ ⎠ ⎠
(
)
2 3 * + δ M H * C * + δ M H * H * H * ⎞ ηµνρ ε αβγδ . − ⎛⎜ δM Cµνρ [µ νρ ] µ ν ρ⎟ αβγδ η 2 δϕ δϕ3 ⎝ δϕ ⎠
By means of the equation (33) projected on the antighost number two (2)
δa3 + γa2 = ∂ µ m µ ,
(41)
the solution (40) and the definitions (25–28) lead to 2 ⎛ * + δ W H * H * ⎞ −3 A C µνρ + ηC µν − a2 = ⎜ δW Cµν µ ν⎟ ρ δϕ2 ⎝ δϕ ⎠ 2 * ⎞ C µνρ + ⎛ ⎛ δM C * + δ M H * H * ⎞ B µν + −2 ⎛⎜ W η*µνρ + δW H[*µ Bνρ ⎜ µν µ ν⎟ ]⎟ ⎜ δϕ δϕ2 ⎝ ⎠ ⎠ ⎝ ⎝ δϕ
(
⎞ +2 ⎛⎜ δM H µ* A*µ − M η* ⎞⎟ ⎟ εαβγδ ηαβγδ − ⎝ δϕ ⎠⎠ 2 ⎛ µαβ ηνγδ . * + δ M H *H* ⎞ ε − 9 ⎜ δM Cµν µ ν ⎟ αβγδ η 4 ⎝ δϕ δϕ2 ⎠
)
(42)
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Interaction between BF and massive Dirac fields
197
Next, we investigate the equation (33) projected on the antighost number one (1)
δa2 + γa1 = ∂ µ m µ ,
(43)
which, combined with (42), further yields
(
)
* C µν + ϕ*η + a1 = δW Hµ* ( 2 AνC µν − H µ η ) + W 2 Bµν δϕ
(44)
+2 ⎛⎜ δM Hρ* Bρα − MA*α ⎞⎟ εαβγδ ηβγδ + a1 , ⎝ δϕ ⎠ where a1 is the general solution to the “homogeneous” equation γa1 = 0.
(45)
Using the definitions (21–22), we easily find that a1 = iU ( ψ*α ψ α − ψ*α ψ α ) η − δU Hµ* ψ α ( γ µ ) δϕ
α β
ψβ η,
(46)
where U is an arbitrary function of the scalar field. The equation (33) projected on the antighost number zero reads as (0)
δa1 + γa0 = ∂ µ m µ .
(47)
Introducing (46) in (44), we obtain the solution for (47) to be of the type
(
a0 = − Aµ U ( ϕ ) ψ α ( γ µ )
α β
)
ψβ + W ( ϕ ) H µ + 1 εαβγδ M ( ϕ ) B αβ B γδ . 2
(48)
Combining the formulas (39), (40), (42), (44) and (48), the first-order deformation of the solution to the master equation for the model under study can be written in the form
)
(
α S1 = d 4 x ⎛⎜ − Aµ U ( ϕ ) ψ α ( γ µ ) ψβ + W ( ϕ ) H µ + 1 M ( ϕ ) εαβγδ B αβ B γδ + β 2 ⎝
∫
α + Hµ* ⎛⎜ δW ( 2 AνC µν − H µ η ) − δU ψ α ( γ µ ) ψβ η + 2 δM B µα εαβγδ ηβγδ ⎞⎟ + β δϕ δϕ δϕ ⎝ ⎠
+iU ( ϕ ) ( ψ*α ψ α − ψ*α ψ α ) η − 2εαβγδ M ( ϕ ) A*α ηβγδ +
2 * C µν + ϕ*η + ⎛ δW C * + δ W H * H * ⎞ −3 A C µνρ + ηC µν − +W ( ϕ ) 2 Bµν ρ ⎜ δϕ µν δϕ2 µ ν ⎟ ⎝ ⎠
(
)
(
2 * + W ϕ η* ⎞ C µνρ + ⎛ ⎛ δM C * + δ M H * H * ⎞ B µν + −2 ⎛⎜ δW H[*µ Bνρ ( ) µνρ ⎟ ⎜ ⎜ δϕ µν µ ν⎟ ] δϕ δϕ2 ⎝ ⎠ ⎠ ⎝⎝
)
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⎞ +2 ⎛⎜ δM Hµ* A*µ − M ( ϕ ) η* ⎞⎟ ⎟ εαβγδ ηαβγδ − ⎝ δϕ ⎠⎠ 2 2 ⎛ µαβ ηνγδ − ⎛ δW C * + δ W H * C * + * + δ M H*H * ⎞ ε − 9 ⎜ δM Cµν η µ ν αβγδ νρλ [ ν ρλ ] ⎟ ⎜ 4 ⎝ δϕ δϕ2 δϕ2 ⎠ ⎝ δϕ 3 ⎞ + δ W3 H ν* Hρ* H λ* ⎟ 4 AµC µνρλ + ηC νρλ + 2W ( ϕ ) η*µνρλC µνρλ + δϕ ⎠
(
)
2 ⎛ ⎛ * + δ W H * H * ⎞ B* ⎞ C µνρλ − +4 ⎜ 2 δW Hµ* η*νρλ + 3 ⎜ δW Cµν µ ν ⎟ ρλ ⎟ δϕ δϕ δϕ2 ⎝ ⎠ ⎝ ⎠ 2 3 ⎛ * + δ M H * C * + δ M H * H * H * ⎞ ηµνρ ε αβγδ + − ⎜ δM Cµνρ [ µ νρ ] µ ν ρ⎟ αβγδ η δϕ2 δϕ3 ⎝ δϕ ⎠ 2 ⎛ δ3W H * H *C * + * * C* * C* H C + ⎜ δW Cµνρλ +δ W + + µ νρλ µν ρλ [ ] [ ] [µ ν ρλ ] δϕ2 δϕ3 ⎝ δϕ
(
)
4 3 ⎞ * * + H[*µ H ν*Cρλ + δ W4 Hµ* H ν* Hρ* H λ* ⎟ ηC µνρλ + 1 ⎛⎜ δM Cµνρλ +δ M ] 3 2 δϕ δϕ δϕ ⎝ ⎠
(49)
2 δ4 M H * H * H * H * ⎞ ηµνρλ ε αβγδ ⎞ . * * * H[*µCνρλ +δ M ⎟ µ ν ρ λ⎟ αβγδ η ] + C[µν Cρλ ] + 2 4 δϕ δϕ ⎠ ⎠
(
)
It is by construction an s-co-cycle of the ghost number zero, such that S0 + gS1 is the solution to the master equation to order g. 3.3. HIGHER-ORDER DEFORMATIONS Next, we investigate the equations that control the higher-order deformations. The equation that controls the consistency of the first-order deformation reads as 1 S , S + sS = 0, ( ) 2 2 1 1
(50)
where S2 denotes the second-order deformation of the solution to the master equation for the deformed theory
( S, S ) = 0,
S = S + gS1 + g2 S2 + ….
Using (49), by direct computation, we obtain that ⎛ 4 ⎛ µνρλ δa X a ⎞ 1 S ,S =ε 4x d + U aµνρλ δ Ya ⎟ + ( ) ⎜⎜ ⎜ Ta µνρλ 1 1 a 2 δϕ δϕ ⎠ ⎝ a=0 ⎝
∫
∑
(51)
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Interaction between BF and massive Dirac fields
199
α ⎛ δ ( MU ) * ⎞ H σ ψ α ( γ σ ) ψβ − iMU ( ψ α ψ*α − ψ*α ψ α ) ⎟ ηµνρλ − +2 ⎜ β δϕ ⎝ ⎠
(52)
−2 M ( ϕ ) U ( ϕ ) ψ α (
)
α γµ
β
ψβ ηνρλ
),
where T0µνρλ = 4 A*µC νρλ + B µνC ρλ + H µ ηνρλ − 2η*C µνρλ − ϕ*ηµνρλ ,
(53)
( ) * B αβ − C * ηαβγ C µνρλ + + ( 2 H α* A*α + Cαβ ) αβγ * C αβµ − 2 H * C αµ ηνρλ + 3H * C αµν B ρλ , + ( 3Cαβ ) α α
(54)
αβγδ ηµνρλ + * C αβ + C * C αβγ + C * T1µνρλ = H α* H α + Cαβ αβγ αβγδC
T2µνρλ = H α*
(( Hβ*Bαβ − 3Cβγ* ηαβγ ) C µνρλ + 3Hβ*C αβµ ηνρλ ) +
( (
)
)
* C αβγ + H *C αβ + C * C αβγδ + 3C * C * C αβγδ ηµνρλ , + H α* 3Cβγ β βγδ αβ γδ
((
)
)
(55)
* C αβγδ ηµνρλ − H *ηαβγ C µνρλ , T3µνρλ = H α* Hβ* H γ*C αβγ + 3Cγδ γ
(56)
T4µνρλ = H α* Hβ* H γ* H δ*C αβγδ ηµνρλ ,
(57)
( (
)
* B αβ − 6 A* Aα ηµνρλ + U 0µνρλ = 1 η*αβγδ ηαβγδ + η*αβγ ηαβγ + Bαβ α 2 * ηαβµ − A B αµ ηνρλ + 1 B µν B ρλ η, + A*µ η + 3 Bαβ α 2 2
)
)
(
(58)
U1µνρλ = 1 ηC *µνρλ − AµC *νρλ + 3B*µν C *ρλ − 2η*µνρ H *λ ηαβγδ ηαβγδ 4 * η + 3 C * A − 3B* H * ηαβγ + 1 C * B αβ + A*α H * η ηµνρλ + (59) + 1 Cαβγ αβ γ α 2 2 αβ γ 2 αβ
((
) ( ) − ηB η ) + 34 ηC
( (
+ H α* Aβ B αβ ηµνρλ + 3 ηαβµ ηνρλ 2
(
(
αµ νρλ
)
))
* αβµ ηνρλ , αβ η
)
* η + H * A ηαβγ + H * B αβ η ηµνρλ + U2µνρλ = H α* 3 Cβγ β γ β 2 + 3 H α* Hβ*ηηαβµ ηνρλ + ( H *µ ( C *νρλ η + 3H *ν B*ρλ ) + 4 +3 1 C *µν η + H *µ A*ν C *ρλ ηαβγδ ηαβγδ , 4
(
) )
(60)
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Eugen-Mihãiþã Cioroianu, Silviu Constantin Sãraru
12
U3µνρλ = 1 H *µ H *ν ( 3C *ρλ η + 2 H *ρ Aλ ) ηαβγδ ηαβγδ + 2 + 1 H α* Hβ* H γ*ηηαβγ ηµνρλ , 2
(61)
U 4µνρλ = 1 ηH *µ H *ν H *ρ H *λ ηαβγδ ηαβγδ , 2
(62)
and
X ( ϕ) = W ( ϕ) M ( ϕ) , Y ( ϕ) = W ( ϕ)
δM ( ϕ ) . δϕ
(63)
Because none of the terms in (52) are BRST-exact, the consistency of the firstorder deformation (49) demands that W ( ϕ ) , M ( ϕ ) and U ( ϕ ) verify the equations W ( ϕ ) M ( ϕ ) = 0, W ( ϕ )
δM ( ϕ ) = 0, M ( ϕ ) U ( ϕ ) = 0, δϕ
(64)
such that we can conclude that the second-order deformation of the solution to the master equation can be chosen to vanish. So, the deformed solution to the master equation stops at order one in the parameter of the deformation, i.e., S = S + gS1 .
(65)
However, in order to obtain effective interactions among the BF and Dirac fields we must take M ( ϕ) ≡ 0 , (66) such that the equations (64) are satisfied for W ( ϕ ) and U ( ϕ ) arbitrary functions of the scalar field. 4. IDENTIFICATION OF THE INTERACTING THEORY
Replacing the solution (66) of the equations (64) into the first-order deformation (49), the fully deformed solution to the master equation (65), consistent to all orders in the coupling constant, becomes
(
(
)
)
(
)
α S = d 4 x ⎛⎜ ψ α ( i γ µ ) ∂ µ + igU ( ϕ ) Aµ − mδαβ ψβ + H µ ∂ µ ϕ − gW ( ϕ ) Aµ + β ⎝ ⎛ + 1 Bµν Fµν + Hµ* ⎜ 2 ⎛⎜ ∂ νC µν + g δW AνC µν ⎞⎟ − g δW H µ η − 2 δϕ δϕ ⎠ ⎝ ⎝
∫
− g δU ψ α ( γ µ ) δϕ
α β
* −3∂ ηµνρ + 2 gW ϕ C µν + ψβ η ⎞⎟ + Bµν ( ) ρ ⎠
(
)
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Interaction between BF and massive Dirac fields
201
+igU ( ϕ ) ( ψ*α ψ α − ψ*α ψ α ) η + gW ( ϕ ) ϕ*η + Aµ* ∂ µ η − * ⎛ ∂ C µνρ + g δW A C µνρ ⎞ + η* µνρλ − 2 gW ϕ C µνρ + −3Cµν ( ) ) ⎜ ρ ⎟ µνρ ( 4∂ λ η ρ δϕ ⎝ ⎠ 2 ⎛ * ηC µν + δ W H * H * −3 A C µνρ + ηC µν ⎞ − + g ⎜ δW Cµν µ ν ρ ⎟ δϕ2 ⎝ δϕ ⎠
(
)
* C µνρ − W ϕ η* −2 g ⎛⎜ δW H[*µ Bνρ ( ) µνρλC µνρλ ⎞⎟ + ] ⎝ δϕ ⎠ * ⎛ ∂ C µνρλ + g δW A C µνρλ ⎞ + +4Cµνρ ⎜ λ ⎟ δϕ λ ⎝ ⎠ 3 ⎛ 2 * + δ W H * H * H * ⎞ A C µνρλ − +4 g ⎜ δ W2 H[*µCνρ ] µ ν ρ⎟ λ δϕ3 ⎝ δϕ ⎠ 2 3 ⎛ * + δ W H * C * + δ W H * H * H * ⎞ ηC µνρ + − g ⎜ δW Cµνρ µ νρ µ ν ρ⎟ [ ] δϕ2 δϕ3 ⎝ δϕ ⎠
(67)
2 ⎛ ⎛ * + δ W H * H * ⎞ B* ⎞ C µνρλ + +4 g ⎜ 2 δW Hµ* η*νρλ + 3 ⎜ δW Cµν µ ν ⎟ ρλ ⎟ δϕ2 ⎝ δϕ ⎠ ⎝ δϕ ⎠ 2 ⎛ δ3W H * H *C * + * * * * + g ⎜ δW Cµνρλ + δ W2 H[*µCνρλ ] + C[µν Cρλ ] + [µ ν ρλ ] δϕ δϕ3 ⎝ δϕ
(
)
4 ⎞ + δ W4 Hµ* H ν* Hρ* H λ* ⎟ ηC µνρλ . δϕ ⎠
From the deformed solution to the master equation (67), we identify the entire gauge structure of the interacting gauge theory. The pieces of the antighost number zero of (67) represent the Lagrangian action of the deformed theory
∫
(
(
)
S L ⎡⎣ Aµ , H µ , ϕ, Bµν , ψ α , ψ α ⎤⎦ = d 4 x 1 Bµν Fµν + H µ ∂ µ ϕ − gW ( ϕ ) Aµ + 2 +ψ α
((
) β ( ∂µ + igU ( ϕ) Aµ )
α iγµ
− mδα
β
)
ψβ ⎞ .
(68)
⎟ ⎠
The terms of the antighost number one from (67) give us the deformed gauge transformations δε Aµ = ∂ µ ε, δε Bµν = −3∂ ρεµνρ + 2 gW ( ϕ ) εµν ,
(69)
δε ϕ = gW ( ϕ ) ε, δε ψ α = −iU ( ϕ ) ψ α ε, δε ψ α = iU ( ϕ ) ψ α ε,
(70)
α δε H µ = 2 ⎛⎜ ∂ ν εµν + g δW Aν εµν ⎞⎟ − gε ⎛⎜ δW H µ + δU ψ α ( γ µ ) ψβ ⎞⎟ . β δϕ δϕ ⎝ ⎠ ⎝ δϕ ⎠
(71)
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Eugen-Mihãiþã Cioroianu, Silviu Constantin Sãraru
14
We observe that the Dirac field becomes endowed with non-trivial gauge transformations, which can be regarded as the gauge version of a one-parameter rigid symmetry of the Dirac theory multiplied by an arbitrary function of the scalar field. The cross-interactions between the BF field spectrum and the Dirac field are expressed by a generalized minimal coupling with the one-form Aµ with abelianU (1) gauge transformations via the conserved current corresponding to the previously mentioned rigid symmetry in a ‘background’ of the scalar field. We notice that there appear two types of pieces with the antighost number two in (67). Ones are quadratic in the pure ghost number one fields, while the others are linear in the ghosts of ghosts. Analyzing the structure of the former kind of terms, we conclude that the deformed gauge algebra is open, while from the latter kind of terms we find that the first-order reducibility functions are modified with respect to the initial model and the first-order reducibility relations hold onshell. The presence of the terms with antighost numbers higher than two reveals, on the one hand, the higher-order gauge structure corresponding to the deformed gauge algebra and, on the one hand, the second-stage reducibility relations, which also take place on-shell. 5. CONCLUSION
To conclude, in this paper we have investigated the consistent Lagrangian interactions that can be introduced between an abelian BF-type theory and one massive Dirac field with the help of the deformation of the solution to the master equation combined with cohomological techniques. Starting with the BRST differential of the free theory, s = δ + γ , we fully compute the first-order deformation and show that we can take all the deformations of orders two and higher to vanish. The cross-interactions between the BF field spectrum and the Dirac field are described by a generalized minimal coupling. Our deformation procedure modifies the gauge transformations, as well as their algebra, which becomes open. Meanwhile, the reducibility relations take place on-shell, by contrast to the free model. Acknowledgments. This work was supported by the type AT grant 33547/2003, code 302/2003, from the Romanian Council for Academic Scientific Research (CNCSIS) and the Romanian Ministry of Education, Research and Youth (MECT).
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