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Physics Letters B

www.elsevier.com/locate/physletb

BRST-BV approach to cubic interaction vertices for massive and massless higher-spin fields

R.R. Metsaev

Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospect 53, Moscow 119991, Russia

ARTICLE INFO

Article history:

Received 21 May 2012

Received in revised form 1 December 2012

Accepted 6 February 2013

Available online 8 February 2013

Editor: L. Alvarez-Gaume

ABSTRACT

Using BRST-BV formulation of relativistic dynamics, we study arbitrary spin massive and massless fields propagating in flat space. Generating functions of gauge invariant off-shell cubic interaction vertices for mixed-symmetry and totally symmetric fields are obtained. For the case of totally symmetric fields, we derive restrictions on the allowed values of spins and the number of derivatives which provide a classification of cubic interaction vertices for such fields. As by product, we present simple expressions for the Yang-Mills and gravitational interactions of massive totally symmetric arbitrary spin fields.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In view of the aesthetic features of higher-spin field theories [1] these theories have attracted considerable interest in recent time (for review, see, e.g., Refs. [2,3]). Further progress in higher-spin field theories requires, among other things, better understanding of interacting mixed-symmetry field theories. Although many interesting approaches to the interacting mixed-symmetry fields are known in the literature analysis of concrete dynamical aspects of such fields is still a challenging procedure. One of ways to simplify analysis of mixed-symmetry field dynamics is based on the use of light-cone gauge approach. In Refs. [4,5], using light-cone gauge approach, we found generating functions of parity invariant cubic interaction vertices for massive and massless fields of arbitrary symmetry. Also, we derived restrictions on the allowed values of spins and the number of derivatives, which provide the complete classification of cubic interaction vertices for totally symmetric massless and massive fields. We note however that light-cone gauge approach, while being powerful to study classical interacting field theories, becomes cumbersome when considering the quantization and renormalization of relativistic field theories. Therefore, from the perspective of quantum higher-spin field theories, it is desirable to obtain gauge invariant and manifestly Lorentz invariant off-shell counterparts of light-cone gauge cubic vertices in Refs. [4,5]. This is what we do in this Letter.

In order to obtain gauge invariant vertices of massive and massless fields we use the BRST-BV formulation of field dynamics. The BRST-BV approach turned out to be successful for the studying manifestly Lorentz invariant formulation of string theory [6]. In

this Letter, we demonstrate that it is the BRST-BV method that offers way for the straightforward Lorentz-covariantization of the light-cone gauge cubic vertices obtained in Ref. [4].

2. Review of BRST-BV approach to massive and massless field

We begin with reviewing the BRST-BV description of free fields propagating in flat space. In d-dimensional Minkowski space, an arbitrary spin field of the Poincare algebra is labeled by mass parameter m and by spin labels s1sv. For massless fields, m = 0, v = ], while for massive fields, m = 0, v = [d-1 ]. In order to discuss mixed-symmetry fields it is sufficient to set v > 1, while, for the discussion of totally symmetric, fields we can set v = 1. In what follows, a particular value of v does not matter.

Massless and massive mixed-symmetry fields. To streamline the BRST-BV description of mixed-symmetry bosonic fields we use a finite set of bosonic oscillators a^ and fermionic ghost oscillators bn, cn, n = 1,...,v, for the discussion of massless fields and a finite set of bosonic oscillators aA, Zn, and fermionic ghost oscillators bn, cn, n = 1,...,v, for the discussion of massive fields (for notation, see Appendix A). Using such oscillators and fermionic Grassmann coordinate 6, we introduce the following ket-vectors to discuss the mixed-symmetry massless and massive fields:

} = $(x,9, a, b, c)|0> massless field, } = $(x,9, a,Z, b, c)|0> massive field.

(2.1) (2.2)

E-mail address: metsaev@lpi.ru.

Ket-vectors (2.1), (2.2) are assumed to be Grassmann even. Infinite number of ordinary gauge fields depending of space-time coordinates xA is obtained by expanding ket-vectors (2.1), (2.2) into the Grassmann coordinate 6 and the oscillators a^, Zn, bn, cn. This is to say that ket-vectors (2.1), (2.2) involve fields of all spins. For these

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ket-vectors to describe fixed spin irreducible fields, some algebraic constraints must be imposed on the ket-vectors.1 But to avoid unnecessary complications, we do not impose any constraints on the ket-vectors. This implies that our ket-vectors actually describe reducible sets of massless and massive fields of all spins.

In the BRST-BV approach, gauge invariant action for free mass-less and massive fields (2.1), (2.2) and corresponding gauge transformations take the form [6]

V123 = V 010203,

(2.12)

S2 = 2 I ddxd0 Qb№>

<0 | — (|0 >)1 sm = q b\a>,

where the BRST operator Qb is defined by the relations

where the quantity V depends on i) derivative with respect to space coordinates p£; ii) derivatives with respect to Grassmann

(r) A (r) (r) (r)

coordinates p0r; iii) oscillators a() , Zn , b(), c(). It is the quantity V that we refer to as cubic interaction vertex. We now discuss our solution for the vertex V.

3. Cubic vertices for massive and massless fields

Up to this point we have considered massive and massless fields on an equal footing. Depending on the values of mass parameters entering cubic vertices, the cubic vertices can be separated into the following five groups:

Qb = 0 (□ - m2) + SApA + mS + Mpe, PA — d/dxA, pe — d/d0,

SA =J2 (cnU^ - a^Cn), S = (CnZn + ZnCn),

n=l n=l

M = £ CnCn, (2.6)

□ = pApA. The |A) (2.4) is considered to be Grassmann odd. For massless fields, the |A) depends on xA, 6, aA, bn, cn, while, for massive fields, the |A) depends on xA, 6, aA, Zn, bn, cn.

We now proceed to review the BRST-BV approach to cubic interaction vertices. To this end we consider the action

S = S2 + S 3,

where S2 is given in (2.3), while cubic interaction S3 can be presented as

S3 = Ud1 d2d3 №|<02|<03||Vi23>, dr = daXrd0r, (2.8)

|V 123 > = V123 j ddx S(d)(x - Xi)S(d)(X - X2)

S(d)(x -X3)|0>1|0>2|0>3. (2.9)

One can make sure that, under gauge transformation given by

s\&> = QB|A> - \0 *A> - |A*0 >,

— j d1 d2 <^1|<^2||V123>,

(2.10)

action (2.7) is invariant (to cubic order in fields) provided the vertex |V 123> satisfies the equations

QBot|V 123 > = 0, QBot —

Q B •

(2.11)

r=1,2,3

Eqs. (2.11) tell us that the vertex |V 123> should be BRST closed. These equations by themselves do not determine the vertex |V 123> uniquely. Vertices obtained via field redefinitions take the form QB°'|C123> and such vertices, which we refer to as BRST exact vertices, also satisfy Eqs. (2.11). Thus all that is required is to find solutions to Eqs. (2.11) which are not BRST exact. It is such solutions that we discuss in our Letter. Solutions for V123 (2.9) we find can be presented as

1 Detailed discussion of the constraints and mixed-symmetry massless fields via ket-vector (2.1) may be found in Ref. [7] (see also Ref. [8]). Discussion of other approaches to mixed-symmetry fields may be found in Ref. [9].

m1 = 0, m2 = 0, m3 = 0; m1 = m2 = 0, m3 = 0; m1 = m2 — m = 0, m3 = 0; m1 = 0, m2 = 0, m1 = m2, m1 = 0, m2 = 0, m3 = 0.

m3 = 0;

We study cubic vertices having mass parameters as in (3.1)-(3.5) in turn. In what follows we use the following notation for the operators constructed out of the oscillators and derivatives:

pr = pr+1 - pr+2,

P 0r =

[r ~ r + 3],

a(r)A _ <r)A

dn — an

-— Zn

(rs) (r)A (s)A

amn — ai

r, s = 1, 2,3, m, n, q = 3.1. Cubic vertices for three massless fields

We begin with discussing the parity invariant cubic interaction vertex for three massless mixed-symmetry fields (3.1). General solution of Eqs. (2.11) takes the form

v-v(Lm L(2) L(3) Z

v = v \Ln , Ln , Ln , Z

Q (11) Q (22) Q (33) \ mn , mn , mn

L(r) pA a(r) A p (r) Ln = pr an + p0rcn ,

7 — n (12) r (3) I n (23) r (*) _i_ n (31)r (' Zmnq = Qmn Lq + Qnq Lm + Qqm Ln

(23) (1)

,(31)r(2)

Q (rr) _ a(rr) , b mn — amn + u\

(r)c(r)

+ b(r)C + un Li

(r)„(r)

, (rr+1)

(rr+1)

- -b()c 2 bm ci

(rK(r+1)

—bn 2 n

(r+1) (r)

(3.10)

(3.11)

where V in (3.7) is arbitrary polynomial of quantities defined in (3.8)-(3.10). The quantities pA, p0r, ajnn are defined in (3.6). Quantities L(nr), Qi^1, and Zmnq are the respective degree 1, 2, and 3 homogeneous polynomials in the oscillators. Henceforth, degree 1, 2, and 3 homogeneous polynomials in the oscillators are referred to as linear, quadratic, and cubic forms respectively. All forms appearing in (3.7) are BRST closed but not BRST exact. This implies that vertex (3.7) cannot be simplified anymore by using field redefinitions. Comparing solution (3.7)-(3.11) with the one obtained in light-cone gauge (see expressions (5.2), (5.3) in Ref. [4]), we note that BRST-BV vertex (3.7) provides straightforward Lorentz-covariantization of the light-cone gauge vertex.2

2 For mixed-symmetry massless fields, BRST invariant linear forms were found in Ref. [10], while BRST invariant linear and cubic forms were studied in Ref. [11]. Interesting novelty of our representation for cubic forms Zmnq (3.9) is that the cubic forms can entirely be presented in terms of linear forms Ln(r) (3.8) and quadratic forms Q_mn+1> (3.11).

Vertices for totally symmetric fields. Because vertex (3.7) has the same structure as the light-cone gauge vertex in Ref. [4] we can use result in Ref. [4] to classify vertices of totally symmetric fields in a rather straightforward way. Namely, to consider the totally symmetric fields it is sufficient to use one sort of oscillators. Therefore we set v = 1 in (2.6) and ignore contribution of the oscillators with n > 1. Also we adopt the simplified notation for linear forms L(r) = L(p and cubic form Z = Z111.3 Now repeating analysis in Section 5.1 in Ref. [4], we find a vertex that describes interaction of massless spin s(1), s(2), s(3) fields,4

V (s(1), s(2), s(3); k) = Z1 (s -k) n (L(r))s<r)+2(k-s),

r=1,2,3

r=1 ,2,3

s — 2smjn < k < s, s — k even integer.

(3.12)

(3.13)

From (3.12), we see that there is 1-parameter family of vertices labeled by non-negative integer k which is the number of powers of the momenta pA, pgr entering the vertices. Detailed discussion of restrictions (3.13) may be found in Section 5.1 in Ref. [4].

Minimal derivative scheme:

Ln — P r an + i

r — 1, 2, 3,

0 (12)=a(n) V.mn — amn

, f(1) j (2) _ 1 b(1)c(2)

+ „ 2 Lm Ln -, um Ln

2m23 2

0 (23) _a(23) _ Zn

V.mn — amn

L(2) m-

_ 1 b(2)c(3) _ 1 b(3)c(2) 2Um Ln 2 n Lm ,

_1 bm)cm_1

2 m Ln 2 n m ,

- -b(2)c „ un L1

L(2)L(3) mn

z(3) 1

0 (31) _a(31) , L (1)__L L (3) L (1)

U-mn — amn + _ Ln „ 2 Lm Ln

(2) (1)

0 (rr) _ (rr) b(r) (r) b(r) (r) _ -, 2 Qmn — amn + bm cn + bn cm , r — 1, 2,

0 (33)_ a(33) _ t(3)t(3)+ b(3)c(3)+ b(3)c(3) V.mn — amn zm hn + um Ln + un Lm .

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

Vertex V (3.14) is arbitrary polynomial of linear and quadratic

forms appearing in (3.14). The quantities pA, p6r, a„ defined in (3.6). We make the following comments.

amJ are

3.2. Cubic vertices for two massless fields and one massive field

We now discuss the parity invariant cubic vertices for two massless and one massive mixed-symmetry fields with mass parameters as in (3.2). General solution of Eqs. (2.11) takes the form

V-V(L(3) 0 (12) 0 (23) 0 (31)l 0 (11) 0 (22) 0

— n , mn , mn , mn mn , mn , mn

(33) N

(3.14)

where we introduce two representations for linear and quadratic forms entering cubic vertex (3.14). These representations are referred to as massive field strength scheme and minimal derivative scheme. This is to say that, in the massive field strength scheme and the minimal derivative scheme, linear and quadratic forms appearing in (3.14) take the following form: Massive field strength scheme:

Lnr) — p A a() A +

r — 1, 2,

L(3) pAa(3)A, Ln — p3 an ,

0 (12)-a(U) + mn — amn +

' l(1)L(2) - -b(1)c(2) - -b _ 2 m Ln „um Ln „un

2m3 2 2

(1) (2)

(2) (1)

Q(23) _ (2) A (3) A . * r (2) A (3

mn — am an + 2 Lm p2 an

(2)A (3)A

(2) A (3)A

0 (31) - a mn —

(3)A (1)A

^ (3)Al(1) L

0 (rr) _ a (rr) b(r) c(r) + b (r) c(r) r- 1 2 mn — amn + um Ln + un Lm , ' — 1,

0 (33) -a(33) _ t(3)t(3) , b(3)c(3) 4- b(3)c( mn — amn zm zn + um Ln + un Lm

.(3),(3)

■¿3)„(3)

■¿3)„(3)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

i) Linear and quadratic forms appearing in (3.14) are BRST closed but not BRST exact. This implies that solution (3.14) cannot be simplified anymore by using field redefinitions.

ii) In massive field strength scheme, the forms L(3), Q(23), Q(31) are constructed by using the vector oscillators a(3)A which are BRST closed but not BRST exact. Such oscillators streamline the procedure for finding the vertex but increase number of derivatives entering the vertex.

iii) In the minimal derivative scheme, the linear and quadratic forms involve minimal number of derivatives. It is not possible to decrease the number of derivatives by adding BRST closed or BRST exact expressions to the linear and quadratic forms entering the minimal derivative scheme.

iv) Linear and quadratic forms in the minimal derivative scheme differ from the ones in massive field strength scheme by BRST exact quantities. This implies that vertex in the minimal derivative scheme is obtained from the one in massive field strength scheme by using field redefinitions.

v) Comparing minimal derivative solution (3.21)-(3.26) with the one obtained in light-cone gauge (see expressions (6.5)-(6.9) in Ref. [4]), we note that the BRST-BV vertex provides straightforward Lorentz-covariantization of the light-cone gauge vertex.

Vertices for totally symmetric fields. To consider totally symmetric fields we set v = 1 in (2.6) and ignore contribution of the oscillators with n > 1. Also we adopt the simplified notation for linear form L(3) = L13) and quadratic forms Q(rr+1) = Q 1(r1r+1). Using solution (3.21)-(3.26) and repeating analysis in Section 6.1 in Ref. [4], we find vertex that describes interaction of spin s(1), s(2), s(3) fields with mass parameters as in (3.2),

3 For totally symmetric massless fields, BRST invariant linear, quadratic, and cubic forms were discussed in Refs. [12,13] (see also Ref. [14]).

4 For arbitrary d, cubic vertices of massless arbitrary symmetry fields in flat space

were found for the first time in the light-cone gauge in Refs. [4,15,16] (for d = 4,

see Ref. [17]). Manifestly Lorentz invariant description of cubic vertices for totally

symmetric fields was obtained in Ref. [18], while the BRST description was given

in Ref. [13] (see also Ref. [12]). Manifestly Lorentz invariant on-shell vertices were

discussed in Ref. [19]. In the framework of BV approach, the discussion of some

particular cubic vertices may be found in Ref. [20]. Interesting use of the BRST tech-

nique for the studying interaction vertices may be found in Ref. [21].

V(s(1),s(2),s(3); t) — (L(3))r n (0(rr+1))'

r—1,2,3

s — E s(r),

r—1,2,3

a(r) — 1 (s — t) — s(r), r — 1, 2, 2( ; , , ,

ct(3) — 1 (s + t)— s(3), 2

r(r+2)

(3.27)

(3.28)

(0, s(3) - s(1) - s(2)) < r < s(3) - \s(1) -

max U, s

s - r even integer

(3.29)

(3.30)

From (3.27), we see that there is 1-parameter family of vertices labeled by non-negative integer t .

3.3. Cubic vertices for one massless and two massive fields with the same mass values

General solution of Eqs. (2.11) corresponding to the parity invariant cubic vertices for one massless and two massive mixed-symmetry fields with the same mass values (3.3) is given by

(1) ,(2) ,(3) n(12)

V = V(Lni), Ln , L

mnq mn

.(11) n (22) n(33))

Q(22), Q^, (3.31)

where linear, quadratic, and cubic forms appearing in (3.31) take the following form in the massive field strength scheme and the minimal derivative scheme. Massive field strength scheme:

L(r) - pAa(r)A

Ln = P 3 d n ,

r = 1, 2, Ln3) = pAan)A +

A (3)A

Q a2) - a' mn =

(1)Aa(2)A

Z L(1)a(2)Aa(3)A L(2)a(1)Aa(3)A, Zmnq = Lm an aq - Ln am aq ,

Q (rr) _ a(rr) _ Z(r)Z(r) b(r)c(r)+ b(r) (r)

^■¿mn — amn Zm hn + um Ln + un Lm ,

r = 1, 2,

(33) _a(33) b(3)c(3) b(3)c(3) mn — amn +um Ln +un Lm •

Minimal derivative scheme:

f(1) pAa(1)A mZ(1) , p c(1) Ln = p 1 an + mZn + p01 cn ,

L2 = pA a(2)A - mZn(2) + p

Ln3 = p A an3) A + p 03 43), Z(1)

Q (12) _a(12^ I Z^r(2) _

Qmn — amn + -, Ln

Z^L<rUZ(')Z(

„ um + Zm Zn

(1) (2)

_ Um Un _ Uli

2 bm cn 2 bn cm

Z _f(1) Q (23)

Zmnq = Lm Qnq

+ L® Q

+ L n ^-¿qm

(31) L(3) Q (12) + L q U-mn ,

Q (12) _a(12) (1)Z(2) _ 1 b(1)c(2) _ 1 b(2)c(1)

mn — amn + Zm hn 2 m Ln 2 n Lm ,

(rr+1)

Q (rr) -a(ll) _ Z(')Z(') , h(l)c(l) _i_ h(l)c

mn — amn Zm Zn + um Ln + un L;

Q (33)-a(33)+ b(3)c(3)+ b(3)c(3)

mn — amn + um Ln + un Lm •

(rr+1)

-1 b 2

(r) (r)

(r) (r+1)

(r+1L(r)

(r) (r)

(r) (r)

r = 2, 3; r = 1, 2,

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

(3.37)

(3.38)

(3.39)

(3.40)

(3.41)

(3.42)

(3.43)

(3.44)

(3.45)

Vertex V (3.31) is arbitrary polynomial of linear, quadratic, and

cubic forms appearing in (3.31). The quantities pA, p0r, a, are defined in (3.6). We make the following comments.

(r)A a(rs)

iv) Cubic form Zmnq (3.34) can be rewritten in terms of field strength for massless field,

Zmnq = Fq

(3)ABa(1)Aa(2)B, aa,

(3)AB A (3)B

— p 3 aq -

p3Baq(3)A.

(3.46)

v) Comparing minimal derivative solution (3.37)-(3.45) with the one obtained in light-cone gauge (see expressions (6.23)-(6.28) in Ref. [4]), we note that the BRST-BV vertex provides straightforward Lorentz-covariantization of the light-cone gauge vertex.

Vertices for totally symmetric fields. To consider totally symmetric fields we set v = 1 in (2.6) and ignore contribution of the oscillators with n > 1. Also we adopt the simplified notation for linear forms L(r) = L(p, quadratic form Q(12) = Q.jJ2'1, and cubic form Z = Z111. Using solution (3.37)-(3.45) and repeating analysis in Section 6.2 in Ref. [4], we find vertex that describes interaction of spin s(1), s(2), s(3) fields with mass parameters as in (3.3),

V (M) s(2) s(3). k . k )

V Is , s , s ; kmin, kmax)

= (Q(12))a Z> Y\ (L

r=1,2,3

-(1)_ k _k . _ s(2) r (2) — , , (1)

= kmax kmin s , r = kmax kmin s ,

r(3) — k .

1 —"min,

= S - 2s(3) - kmax + 2k

X = s(3) - kmin, S — J2

r=1,2,3

kmin + max(s(1), s(2)) < kmax < S - 2s(3) + 2kmin, 0 < kmin < s(3\

(3.47)

(3.48)

(3.49)

(3.50)

From (3.47), we see that there is 2-parameter family of vertices labeled by non-negative integers fcmjn and kmax which are the respective the minimal and maximal numbers of powers of momenta pA, p0r. Vertex V(0, 0,1; 1,1) = L(3) describes Yang-Mills interaction of massive spin-0 field, while vertex V(s, s, 1; 0, s) = (Q(12))s-1 Z is a candidate for Yang-Mills interaction of massive spin-1, s > 1, field. Vertices V(0, 0, 2; 2, 2) = (L(3))2 and V(1,1, 2; 1, 2) = L(3)Z describe gravitational interaction of the respective spin-0 and spin-1 massive fields, while vertex V(s, s, 2; 0, s) = (Q (12))s-2Z2 is a candidate for gravitational interaction of massive spin-s, s > 2, field. Recent discussion of electro-magnetic interaction of totally symmetric massive field may be found in Ref. [22]. Discussion of some particular cases of cubic vertices may be found in Ref. [23].

3.4. Cubic vertices for one massless and two massive fields with different mass values

i) Linear, quadratic, and cubic forms appearing in (3.31) are BRST closed but not BRST exact.

ii) In massive field strength scheme, the forms

LOO, l(2), q(12),

Z are constructed by using the vector oscillators a(1)^, a(2)^. These oscillators are BRST closed but not BRST exact.

iii) Linear, quadratic, and cubic forms in the minimal derivative scheme differ from the ones in massive field strength scheme by BRST closed and BRST exact quantities. This implies that vertex in the minimal derivative scheme is obtained from the one in massive field strength scheme by using field redefinitions and change of vertices basis.

General solution of Eqs. (2.11) corresponding to the parity invariant cubic vertices for one massless and two massive mixed-symmetry fields with different mass values (3.4) is given by

V-V(Lm L(2) q(12) q(23) q(31)\q(") q

v — v y n , u n , mn , mn , mn | mn , mn

(22) Q (33h Qmn I

(3.51)

where linear and quadratic forms appearing in (3.51) take the following form in the massive field strength scheme and the minimal derivative scheme.

Massive field strength scheme:

L(r) _ pAa(r)A

Ln — P 3 d n ,

r — 1, 2,

0 (12)- a

mn — m

0 (23) _

°mn — am an

(1)A_(2)A

22 m21 m22

L(3) PA a(3) A p c(3) Ln — P3 an + TP03 cn ,

L(2)L(3), Lm Ln ,

0 <31) - a

W.mn — a,

(3)A (1)A

_L(3) L(1) 2 m n ,

0 (rr) - a(rr) - t(r)t(n + b(r)c(r) + b(r)c

mn — amn Çm Çn + um Ln + un Li

0 m_am, b(3)c(3)+ b(3)c(3)

mn — amn + um Ln + un Lm .

Minimal derivative scheme:

J(1) PAa(1)A m2 Z(1) , P c(1) Ln — IP 1 an +--in + TP 01 cn ,

1 mj 1

(2) A (2)A m1 (2)

Ln) — TTA an)--1 in

L(3)_ PAa(3)a p c(3) Ln — P3 an + p03 cn ,

Z(1) Z (2 )

0 (12) _a(12) i L (2) _ L (1)

mn — amn + - Ln -,

2m1 2m2

m2 + m2 Z(1) Z(2) _ 1 b(1)c(2) _

+ -, sm Çn ~.um Ln

2m1m2 2

(r) (r)

(r) (r)

\(r)r(r) m,

r — 1,2 ,

(2) (1)

0 (23) — a(23) , z

mn — amn +

L(3) + ■ n+

(2) (3)

2(m2 — m2)

L( ) L

_1 b®c(3) _1 b(3)m

2 m Ln 2n Lm , Z(1)

0 (31) _ a(31) _ I^L(3) _

>imn — amn „ um

(3) (1)

-L(3) L

2(m2 — m2) _ 1 b(3)c(1^ 1

2 m Ln 2 n Lm ,

0 (rr) _ (rr) Z(r)Z(r) b(r) (r) b(r) (r) _ 1 2 °mn — amn Çm Çn + bm cn + bn cm , r — 1 2,

0 m_am, b(3)c(3)+ b(3)c(3)

mn — amn + m n + n m .

(3.52)

(3.53)

(3.54)

(3.55)

(3.56)

(3.57)

(3.58)

(3.59)

(3.60)

(3.61)

(3.62)

(3.63)

(3.64)

(3.65)

Vertex V (3.51) is arbitrary polynomial of linear and quadratic forms appearing in (3.51). The quantities pA, p6r, aj,r)A defined in (3.6). We make the following comments.

amn are

i) Linear and quadratic forms appearing in (3.51) are BRST closed but not BRST exact.

ii) In massive field strength scheme, the forms L(1), L(2), Q(12), Q(23), Q(31), are constructed by using the vector oscillators a(1)A, a(2)A. These oscillators are BRST closed but not BRST exact.

iii) Linear and quadratic forms in the minimal derivative scheme differ from the ones in massive field strength scheme by BRST exact quantities.

iv) Comparing minimal derivative solution (3.58)-(3.65) with the one obtained in light-cone gauge (see expressions (6.57)-(6.63) in Ref. [4]), we note that the BRST-BV vertex provides straightforward Lorentz-covariantization of the light-cone gauge vertex.

Vertices for totally symmetric fields. To consider totally symmetric fields we set v = 1 in (2.6) and ignore contribution of the oscillators with n > 1. Also we adopt the simplified notation for linear forms L(1) = L^, L(2) = L® and quadratic forms Q (rr+1) =

Q1('1r+1). Using solution (3.58)-(3.65) and repeating analysis in Section 6.3 in Ref. [4], we find a vertex that describes interaction of spin s(1), s(2), s(3) fields with mass parameters as in (3.4),

V(s(1),s(2),s(3); t(1),t(2))

— (L(1^r v(L(2^r^^ 0(rr+1))ff

r—1,2,3

a(1) — 1 (s(2) + s(3) — s(1) + t(1) — t(2))

a(2) — L (s(1) + s(3) — s(2) — r(1) + r(2))

a(3) — i. UV) + s(2) — s(3) — r(1) — t(2))

|s(1) —s(2) — r(1) + r(2)| < s(3) < s(1) + s(2) — r(1) — t(2),

s — r(1) — r(2) even integer,

s — s(1) + s(3) + s(3).

(3.66)

(3.67)

(3.68)

(3.69)

(3.70)

(3.71)

Thus, there is 2-parameter family of vertices (3.66) labeled by nonnegative integers t(1), t(2).

3.5. Cubic vertices for three massive fields

General solution of Eqs. (2.11) corresponding variant cubic vertices for three massive mixed (3.5) is given by

to the parity in-symmetry fields

V_V(Lœ L(2) L(3) 0(12) 0(23) 0(31)|0

— n , n , n , mn , mn , mn mn

0 (22) mn

0 (^ mn

(3.72)

where linear and quadratic forms appearing in (3 lowing form in the massive field strength scheme derivative scheme.

Massive field strength scheme:

L(r) - PA a(r)A

Ln — p r dn ,

0 (rr+1) _ a(r)A_(r+1)A

0 mn — am an ,

0 (rr) _ a(rr) _ Z(r)t(r) b(r)c(r)+ b(r) (r)

V.mn — amn Çm Çn + um Ln + un Lm .

.72) take the fol-and the minimal

(3.73)

(3.74)

Minimal derivative scheme:

L(nr) — p A anr) A +

cn( ) +

22 mr+1 — mr+2„ (r) Çn ,

(3.75)

I (rr+1)

(rr+1)

_l_ L(r+1) —_

2mr n 2mr+1

-(r+1)

Z(r)Z(r+1)

2mr mr+1

(mr + m2+1 — m2+2)

- -b(,)c _ um Li

(r) (r+1)

— -b

(r+1)c(r)

0 (rr) _ a(rr) _ Z(r)t(r) b(r)c(r)+ b(r) (r)

V.mn — amn Çm Çn + um Ln + un Lm .

(3.76)

(3.77)

Vertex V (3.72) is arbitrary polynomial of linear and quadratic forms appearing in (3.72). The quantities pA, p6r, a.

(r)A a(rs)

defined in (3.6). We make the following comments.

i) Linear and quadratic forms appearing in (3.72) are BRST closed but not BRST exact.

ii) In massive field strength scheme, the forms L(r), Q(rr+1) are constructed by using the vector oscillators a(r)A. These oscillators are BRST closed but not BRST exact.

iii) Linear and quadratic forms in the minimal derivative scheme differ from the ones in massive field strength scheme by BRST exact quantities.

iv) Comparing minimal derivative solution (3.75)-(3.77) with the one in light-cone gauge (see expressions (7.2)-(7.4) in Ref. [4]), we note that the BRST-BV vertex provides straightforward Lorentz-covariantization of the light-cone gauge vertex.

Vertices for totally symmetric fields. To consider totally symmetric fields we set v = 1 in (2.6) and ignore contribution of the oscillators with n > 1. Also we adopt the simplified notation for linear forms L(r) = L(p and quadratic forms Q(rr+1) = Q1('1r+1). Using solution (3.75)-(3.77) and repeating analysis in Section 7.1 in Ref. [4], we find a vertex that describes interaction of massive spin s(1), s(2), s(3) fields,

V(s(1), s(2), s(3); t(1),t(2),t(3))

= П (L(r)y(° (Q(rr+1))ff(r+2>,

r=1,2,3

a(r) = 1 (s + T(r) - T(r+1) - T(r+2)) - s(r),

s(3) - s(1) - s(2) + t(1) + t(2)

^ t(3) < s(3) - |s(1) - s(2) - t(1) + T(2)|,

s + ^ T(r) even integer;

r=1,2,3

(3.78)

(3.79)

(3.80)

(3.81)

r=1,2,3

Thus, there is 3-parameter family of vertices (3.78) labeled by nonnegative integers t(1), t(2), t(3) .

Finally, we make comment concerning the higher-order vertices and fully gauge invariant action. As is well known, the first step in the derivation of the full action is the finding of cubic interaction vertices. In this Letter, using the BRST-BV approach, we obtained Lorentz-covariant off-shell description of cubic vertices for massive and massless fields which were obtained in light-cone gauge in Ref. [4].5 Cubic vertices found in the present Letter provide nice starting point for the systematical study of higher-order vertices and fully gauge invariant action. In higher-spin gauge fields theory, derivation of fully gauge invariant action is very difficult technical problem. This is to say that fully gauge invariant action for higherspin (low-spin massless plus higher-spin massive) fields has been obtained so far only for the string theory [6]. By now, from application of BRST method to string theory, it is clear that BRST method provides systematical setup for the study of higher-order vertices in higher-spin gauge fields theory. But the derivation of those higher-order vertices requires, among other things, knowledge of cubic vertices found in the present Letter.6 We are planning to study BRST invariant higher-order vertices in future.

Acknowledgement

This work was supported by the RFBR Grant No. 11-02-00814. Appendix A. Notation

Our conventions are as follows. XA denotes coordinates in d-dimensional flat space-time, while pA denotes derivatives with respect to XA, pA = d/dXA. Vector indices of the Lorentz algebra so(d — 1,1) take the values A, B, C = 0,1,...,d — 1. To simplify

5 We think that the BRST-BV approach we discussed in this Letter might be useful for the studying interaction vertices of AdS fields. Recent discussion of interaction vertices of AdS fields may be found in Ref. [24]. Interesting use of BRST technique in AdS space may be found in Ref. [25].

6 Recent study of BRST invariant quartic vertices for higher-spin massless fields

may be found in Refs. [27,13].

our expressions we drop mostly positive flat metric tensor nAB in scalar products: XAYA = nABXAYB. We use fermionic Grassmann coordinate в, в2 = 0, and the corresponding derivative pg = 3/30, {pg,0} = 1. Integration in в is defined as /d00 = 1. We use a set of the creation bosonic operators aA, Zn and fermionic ghost operators bn, cn and the respective set of annihilation bosonic operators aA, Zn and fermionic ghost operators Cn, bn. These operators are referred to as oscillators in this Letter.7 (Anti)commutation relations, the vacuum, and Hermitian conjugation rules are defined as

[am ,aB] = n &mn, [Zm ,Zn] = &mn,

{bm, Cn} = $n

{Cm, bn} = &n

aA |0> = o, Zn |0) = o, bn|0} = 0, Cn|0} = 0,

A|_ - A

an — an ,

Zn = Zn ,

b\ = b n,

cn — cn.

(A.1) (A.2) (A.3)

For momenta and coordinates, we use the Hermitian conjugation rules pA1 = — pA, p0 = p0, XAt = XA, 01 = 0. Hermitian conjugation rule for the product of two arbitrary ghost parity operators A, B is defined as (AB)1 = B1 A1. The internal ghost number operator is defined as

NJP = в pg + Nc - Nb,

Nb = E bnCn, Nc = Cnbn.

The external ghost number operator N™' is defined on a space of ordinary x-depended gauge fields entering ket-vectors (2.1), (2.2) so that Nfp|$) = 0, where total ghost number operator is defined as Nfp = NFP + Nf. The |Л) appearing in (2.4), (2.10) satisfies the restriction (Nfp + 1)|Л) = 0.

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