On covariant cubic vertices for irreducible HS fields with integer and half-integer spins on flat backgrounds

Possibilities to formulate consistent quantum gravity make an interest to the higher spin (HS) field theory more and more intense. Simultaneously, it may provide to discover both the physics beyond the Standard Model, with help of new elementary particles of matter and carriers of higher spin interactions and the massive HS fields appear by natural candidates to describe the problem of Dark Matter [1], [2], within. e.g. modified gravity concept [3], [4], [5] (alternatively to models of vector and sterile neutrinos [6]). These expectations are based on connection of HS fields with (Super(string Field Theory on constant curvature spaces [7], operating with infinite sets of massive and massless fields of (half)integer spins (for references and review see, e.g. [8], [9], [10]).

All these points are closely related with general problems in modern theoretical high-energy physics consisting in the construction of interactions with HS fields within gauge-invariant Lagrangian formulations as the most appropriate objects for quantization and calculating S-matrix elements. Whereas the cubic and quartic vertices for various HS fields with integer spins have been studied by many authors in the framework of different approaches (see, e.g., the papers for cubic [11], [12], [13], [14], [15], [16], [17], [18], [19] and for quartic vertices [20]), this problem for interacting irreducible HS fields with half-integer spins was considered only in terms of physical degrees of freedom in the light cone formalism in [21].

The aim of the note is to find the general covariant Lagrangian cubic vertices for massless totally-symmetric HS fields with half-integer (fermionic) and integer (bosonic) helicities in R ^{1 ,d -1} within BRST approach with incomplete (due to absence in it of the algebraic constraints) BRST operator elaborated for bosonic in [22] and for fermionic HS fields in [23]. BRST approach presents the realization of the AKSZ model [24] and was born due to BRST symmetry discovered in [25], [26] for the aims of Lagrangian [27] and then for the canonical quantization [28], [29] of the dynamical systems sub ject to constraints. The definitions and notations from [15], [16] are used for a metric tensor n ^v = diag (+ , — , … ， — ) ,gammamatrices ү ^м : { ү Н ,ү ^v } = 2 n *^v and e ( F ) , дһ н ( F ) , ( s ) 3 , for the values of Grassmann parity and ghost number of a quantity F , for the triple ( n i + 1 / 2 , П 2 + 1 / 2 , S 3 ) .

1. Cubic vertices for constrained bosonic and fermionic higher-spin fields

(dvdv, dfn"1"2)фм(s) = 0  0   (lo, 11, lii,g0 — d/2 — s)|ф) = (0,0,0,s)|ф),⑴

(үvdv, ү"1 )%(n)= 0  ^^  (to, ti,gf — d/2)|g = (0, 0,n))|办.

The vector | ф ) , ( |办 for half-integer spin) and the operators { 1 ₀ , 1 _{1 ;} 1₁bg ₀ } ; ( { t ₀ , 厶， g ^f } ) here are defined in the Fock space H ( H ^f ) with the Grassmann-even oscillators a _M , a + , ( [ a _M , a + ] = — n ^v ):

Is                                                                                     11

|ф) = £ s! ф"()Ha+i|0〉，(І0, 11, 111, g0)= (d dv, —ia dv, ^ a"a“, — ^ {a+, a"}), s>0 s

|办=Xn*n)na+i|0〉，(t0, t1,gf)=(行"a”, a*v，―2{a+,a"}).

n>0 '

In (4) it is introduced Grassmann-odd gamma-matrices [30], [23]: Yv = 7Yv,力2 = —1,行,Yv} = 0 (to provide fermionic character for t°, t1). The difference among two sets of operators above and vectors |ф〉 (scalar), |^) (Dirac spinor) is in the matrix-valued structure of the latter. The interacting Lagrangian formulation (LF) within BRST approach with total: QCot = (P2=1 QF|c + QC3)l2[d/2]), incomplete BRST operator in cubic approximation is given with accuracy up to linear in deformation parameter g, for 3-copies of massless HS fields: two fermionic 3"(П), i =1,2, and one bosonic ф"2^), including in respective vectors 彼⑴〉e Hfi), 0⑶〉e H⑶ of helicities (s)3, with own set of oscillators a俨，a+(k, k = 1, 2, 3 (with conventions П3 三 S3 and [i + 3 t i])

S [靠[( х ) з ]

2                                                “3

X S 02 Cn i [ x ^{( i )} ]+ S S 3 c [ x ^{(3 )} ]++ g / П dn^ ^ n e ( X ^{( e )} | V ⑶〉 ( s ) 3

i=1                                      k = 1

+ h.c.

S [1] | x ^{( i )} 〉 n i = Q F|c | Л ^{( і )} 〉 П і - g / ^П dn ^{( i + e )} Qi+1 ( л ^{( і +1)} | n i +2 ( x ^{( i +2)} | + (1 丿 e =1

分 2)) 1 ^V ( %) 3 ,

6 [1] | х В3%

Q C ³⁾ | A ⑶〉 S 3

-д/П

J e=1

dn ^{( e )} (ni { A ⑴ ¹ n 2 〈 x ^{(2) |} + ⁽¹

分 2)) 1 ^V ( ³⁾% ) 3

(for ( e,gh H ) [ Q C ³⁾ ,Q F|_c ] = (1 , 1) ； ( e,gh H ) x ^{( k )} = (1 — 6 * 3 , 0) ; ( е,дһ н )Л ^{( к )} = (“ , -1) ). These relations determine the irreducible gauge theory with non-Abelian gauge transformations (for SU (2) gauge group, 3 = 2 ² — 1 ) for the field vectors | x ^{( k )} 〉 s k (including the sets of auxiliary fields into joint configuration space), with gauge parameters |Л ^{( к )}%卜 from total Hilbert space H cot = ㊉ i H tOt|f ㊉ H t 3 t for H tot|f f ^{( i )} , related to i -th copy of fermionic and H _t ⁽ _o ³ _t ⁾ for bosonic fields. They, in turn, depend on pairs of ghost oscillators, Grassmann-even: q 0 i ) , p 0 i ) ; and Grassmann-odd: n ^{( k )} , P 0 * ) ; П ， ' , P ⁽ k ⁾⁺ ; n ⁽ k ⁾⁺ , P 1 k ) with nontrivial commutation relations:

[ q 0 i ⁾ ,p O i ⁾ ] = { n ^{( k )} , P 0 ^{k )} } = i,    { n ^{( k )} , P ^{( k )+} } = 1 , for , дһ н ( q o ,n o ,n ⁺ ) = - д һ н ( P o , Pq , P + ) = 1

according to BRST approaches with incomplete nilpotent BRST operators Q ， ³⁾ , q F )_c [23]

Q C ³⁾ = ( n o l o + n + l 1 + ЩІ ⁺ 1 + 1П + П 1 Р o ) ⑶； Q F|_c = ( q o t o + n o l o + n + l 1 + nil ⁺ 1 + 1 [ п + П 1 — q 2 ] P 0 ) ^{( i )} . (8)

Poincare irreducibility of the initial fields 3 ； ( n ) , i = 1 , 2 , and O^ Ss) is provided by imposing off-shell BRST-extended ү -traceless T 1 ^{( i )} , traceless L 1? constraints with respective spin constraints o Ck both on the field vectors | x ^{( k )} 〉， gauge parameters | Л ^{( к )} 〉 and on the three-vectors of | V ⁽³⁾ 沁％ , | V ⑶〉 (s% , | V ⁽³⁾ 〉 ( s ) 3 . The actions for free HS fields with s * included in | x ^{( k )} 〉 s k = | O ^{( k )} 〉 S k + O ( n ) in (5) and theirs equations of motions, gauge transformations (6), (7) for g = 0 , off-shell constraints have the form (with respective inner products 3l x 〉= R d d x^ * ( x ) x ( x ) [23]), first, for 成%) :

S Of/ x B ³ ) ] = У dn 0 ³⁾ S 3 ^{{ X}B\^Q c ³ 霜兀 ， 3叫 x B^s s = ⁰ ； gx BP^ = 3^{3) |} A c 〉 S 3 , L 11 ⁾ ( l x B 3 ⁾ 〉 , \ Л ⁽³⁾ 〉 ) = ( 1 / 2 a ⁽³F³⁾ + п ( ³⁾ Р 1 ³⁾ )( \ х В 3 ) 〉 , \ Л ⁽³⁾ 〉 ) = 0,

( \ х В° 〉 , \Л ⁽³⁾% ₃ = ( I O ⑶〉 S 3 —P ( ³⁾⁺ { n 0 ³⁾ \ o 1 ³⁾ 〉 S 3 - 1 + п""% - ? } , 。”*」)

^(for R dn ⁽³⁾ 三方 _n ( 3 ) );

then, in the second order approximation for spin-tensor fields &?_n )

s 02 cn i [ x ^{( i )} ] = /dn ( i ) n i { x ^{( i )} | Q F|c | X ^{( i )} 〉 S i , Q F|c | x ^{( i )} 〉 n i = 0 ； 6 o \ x ^{( i )} 〉 n i = Q Fj_c | Л ^{( і )} 〉 П і , (12) T 1 ^{( i )} ( \ x ^{( i )} 〉 , \ Л ^{( і )}» = ( ү “a« — in ^{( i )} p O i ⁾ — 2 q 0 i ) pf ) )( \ x ^{( i )} 〉 , \ Л ^{( і )}» = 0, (13) \ x ^{( i )} 〉 n i = \" ) 〉 n i + P ^{( i )+} ( q O ^{i )} Y \ x 1 i ⁾ ) n i - 1 — n ^{( i )} +處'〃— ) ， \ Л ^(і) 〉 П і = pf ⁾⁺ \ g ^{( i )} 〉 n i - 1 ,. (14)

Each copy of fields and gauge parameters is the proper eigen-vectors for the spin constraints with proper spin numbers n 〃， S 3 with incomplete spin operators o ^{( k )} , k = 1 , 2 , 3 :

o C^{k )} -—

(| x ^{( k )} 〉 n k , | Л ^{( к )} 〉 пк) =0 ,o C^{k )} = g ^{( k )} + n ^{( k )+} P 1 ^{k )} — n ^{( k )} P ^{( k )+} .         (15)

The three sets of operators { q F| _c , T ^{( i )} , L 1? , o ^{( i )} } for i = 1 , 2 and { Q C ³⁾ , L 1 ³⁾ ,o C ³⁾ } compose closed superalgebras. Note, the triplet LFs for free HS fields (9), (12) are equivalent for LFs with complete

BRST operators for the same HS fields [23] and reduces after resolution of the off-shell constraints to Fronsdal LF [31] for single field Ф^₃ ) and for Fang-Fronsdal LF [32] for ^^_n. ) -

The Noether identities for the cubic deformation of the fields (3(1)3(2)0(3)) of LF for the particles (0,Sk), k = 1, 2, 3 in the first degree in deformation (coupling) constant g1 ：     SoSi|(s)3 + S1(P2=1 S02K [x(i)] + SS3C [x(3)]) =0, transforms to the system of equations, which for x-local coinciding unknowns V(3) = V(3) = V(3)

IV ⑶〉(s)3(x) = П 脚)(x-Xk)V ⑶ П n0j) |0〉，|0〉三戏=1|0〉e, k=1

has universal form

(QCot, T(i), L11), Mk) - d-2) |V⑶〉(s)3 =位nk), QCot = XqFJc + QC3) • i2[d/2]

Thus, the BRST-closed, traceless and gamma-traceless vertex should be composed from ^a M ^{)+ ,}n i k ^{)+ ,} P 1 ^{k )+ ,} P 0 ^{k )} , q ^{( i ) ,}p 0 i ^{) ,} i = ^{1 , 2} of spin ^{( n} 1 ^{+ 1} / 2 ,n 2 ^{+ 1} / 2 ,^s 3 ) .

The solution of the equations (18) which covariantizes the light-cone cubic vertex [21] has two types of the vertices ( K -, and F -vertices) and are constructed from the Q ^t _c ^ot BRST-closed polynomials of the first order ( L ^{( k )} ) in oscillators а т? ,П ⁽ 2 ⁾⁺ ) and in momenta p ^kU = - id ^kU and ( Z ) of the third order in oscillators and the first order in p ⁽ m^{i )} u . Z a nd L ^{( k )} provide respectively Yang-Mills type (according to correspondence: L ⑴ “Щ ) 3 V 2 n 2) ° P 3 s 3) — " (Зит) * d ， V 2 n 2) ° P 3 s 3) ) and gravitational type interactions ((according to: Z^d n ) 或： 2 )遼 3 ) 1 現 ( n i - 1)/ "队 2 - 1) "噌 3 - 1) 。+ "照 ¹ ， ² , ^{3) ) amo}ng ^{the fields} and determined as

L(k) = (p戶。-p戶2))a(k)"+ - i(p0k+1) -P0k+2))n(k)+,

Z = l 11²⁾⁺ L ⁽³⁾ + cycle (1 , 2 , 3) , 厶 1, ⁺¹⁾⁺ = ； ( a ( ^{k )} “ ⁺ a ⁽ ₁ ^{k +1)+} - p ( ^{k )+} n ( k ⁺¹⁾⁺ ) + ( k ^— 1 k + 1 2 。)

Peculiarity of the vertices with half-integer spin is in the presence of the BRST-closed extended Dirac operator K ⁽¹²⁾ and operator F

K(12) = Y"b3) - iq(1) [p03) - p02) - 3P01)] + iq(2)球)- P^ - 3P02)],

F =(L⑶一2 [K(12), T(3)+] J , where

T 1 ⑶ + = ( 1 - i [ q 0 ²⁾ p 0 ¹⁾ + 婚喝 ¹ )》％。⑶ + i ( p 0 ¹⁾ + P 0 ²⁾ ) n ( ³⁾⁺ - 2P ( ³⁾⁺ (炉 + q 0 ²⁾ ) .

Thus, the parity-invariant solution (given by Figure 1) takes the form for cubic K -, and F -vertices

< ( 0 ,n i + 1 / 2 )

+ ............

( 0 ,s 3 )

Fig. 1. Interaction vertex | V(3)〉(s)3 for massless bosonic 0，|з)and two fermionic fields。；，几.)of helicities S3, ni, i = 1, 2. The terms in ” ... ” correspond to the auxiliary fields from |x(k)〉sk before constraints resolution s+1                    32

|VM⑶％)3 = K(12)    X    Z2(s-k+1) Y(L(i))ni-2(s-k+1)|0), s = X Пі + S3； k = S —2Smin + 1             i = 11

s 3

|VM⑶〉⑶3 = F           X           Z2(s-k) Y(L(i))ni—6i3 — 2(s-k)|0).

k=s —2(min(ni ,П2 ,S3 —1)) — 1

The vertices above are appropriate for cubic deformations of reducible Poincare group representations with integer and half-integer helicities, whereas for irreducible representations one should to impose additional traceless and gamma-traceless constraints (18) on the vertices [33]. Thus, the each vertex contains k -parameter family of vertices with fixed value of derivatives. For the case of d = 4 the number of different vertices is reduced. For S 3 = S min there are two terms with k = s - 2 s min + 1; s + 1 and one with k = s + 1 for S 3 > S min in the K -vertex without any parity-invariant F -vertex. Whereas for S 3 > S min there is only one F -vertex with k = s — 2 s min . Obtaining the interacting LF (5)-(7) in terms of only initial (spin)-tensors is simply reduced to calculation of the oscillators pairings.

Conclusion

The procedure for BRST approach with incomplete BRST operator to consistently construct covariant local cubic vertices for 3 copies of massless interacting HS fields: one with integer and two fields with half-integer spins on d dimensional flat backgrounds with preservation the Poincare group irreps for deformed gauge theory is developed. The parity invariant solution for the generating equations for cubic vertices are firstly found in the covariant form. They contains two types of three-vectors as the generating functions for the cubic vertex in the second order approximation for the LFs for fermionic fields. Of course, not all the vertices among K - and F -vertices will survive under requirement of gauge invariance of the cubically deformed LF in all orders in deformation constant g . For the spin S 3 = 2 the obtained solutions open the door for study the interactions of HS matter particles with gravity in covariant form. The problems to get covariant cubic vertices both for these kind of fields in the first-order LFs for the fermionic fields and for massive interacting HS fields also compose the current task’s list.

Acknowledgments: The author wish to acknowledge the Organizing Committee of the 18th Russian Gravitational Conference – International Conference on Gravitation, Astrophysics and Cosmology Conference, S.V. Sushkov for hospitality and wonderful scientific spirit.