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Nuclear Physics B 887 (2014) 168-174

www. elsevier. com/locate/nuclphysb

Factorization of cubic vertices involving three different

higher spin fields

Y.S. Akshay, Sudarshan Ananth*

Indian Institute of Science Education and Research, Pune 411008, India Received 22 May 2014; received in revised form 5 July 2014; accepted 8 August 2014 Available online 12 August 2014 Editor: Stephan Stieberger

Abstract

We derive a class of cubic interaction vertices for three higher spin fields, with integer spins A1, A2, A3, by closing commutators of the Poincare algebra in four-dimensional flat spacetime. We find that these vertices exhibit an interesting factorization property which allows us to identify off-shell perturbative relations between them.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

Interactions of massless particles are, in general, very highly constrained. In flat spacetime backgrounds, there exist consistent cubic interaction vertices describing massless higher spin fields [1,2] and their couplings to gravity [3]. At the level of the equations of motion, there has been considerable progress in our understanding of higher spin theories in both flat and anti-de Sitter spacetimes. Fully interacting, non-linear equations of motion describing higher spin fields are known [4]. However, a consistent description of higher spin fields (A > 2), at the level of the action, remains elusive. Interesting attempts to address this problem using a Fock space approach and the Poincaré algebra include Refs. [2,3]. In this paper, we adopt a more direct method, conducive to our aim of identifying factorization properties and establishing perturbative ties in the space of higher spin theories.

* Corresponding author. http://dx.doi.Org/10.1016/j.nuclphysb.2014.08.002

0550-3213/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

In this paper, we work with the Poincare generators for (3 + 1)-dimensional flat spacetime in light-cone gauge. We write down a general ansatz for the cubic interaction vertex in a theory describing three different higher spin fields. Demanding closure of the Poincare algebra yields a class of generic higher spin cubic interaction vertices which we rewrite in momentum space, using spinor helicity notation.

In this spinor helicity language, interesting structures and relations are manifest. Specifically, the cubic vertex in higher spin theories may be obtained by simply multiplying the corresponding cubic vertices involving lower spin fields. Given two theories, one involving spins (Xi, X2, X3) and the other (Xi, X'2, X'3), one may obtain by the direct product of their cubic vertices, the cubic interaction term for a theory involving spins (Xi + X^, X2 + X'2, X3 + X'3). A converse of this property is the factorization of the higher spin vertex into the corresponding lower spin ones.

One obvious consequence of these properties is that the cubic vertex describing fields of spins (nX1, nX2, nX3) is the nth power of the (X1, X2, X3) cubic vertex. This is reminiscent of the KLT relations [7] and even more so of their off-shell extensions [8]. An interesting question is whether such relations extend to quartic and higher order vertices in higher spin theories. Further study along the lines described here requires the extension of the Poincare generators and the calculations presented here to order a2. There is, however, reason to believe that such an attempt might run into difficulties. The no-go theorems for higher spin theories [5] do not allow consistent interacting theories involving particles of spin greater than two. On the other hand, many of these results were derived assuming that locality and Lorentz invariance were manifest in the theory under consideration. Neither of these properties is manifest in light-cone gauge making it an ideal choice for the study of higher spin fields, a point we return to at the end of this paper. The cubic interaction vertices we derive here are not a comprehensive listing of all possible vertices1 since our aim here is not to be encyclopedic but to establish both factorization properties and perturbative links in higher spin theories.

2. Poincare generators

We define light-cone coordinates in (-, +, +, +) Minkowski spacetime by

x ± =

x0 ± x3

1 • 2 x — ix

The corresponding derivatives being d±, 9 and d. In four spacetime dimensions, all massless fields have two physical degrees of freedom for which we use the notation $ and $. We choose the field $ to have helicity X while the field $ has helicity -X. The generators of the Poincare algebra, in light-cone coordinates are

p~ = id— = —p+ j = l(x 9 — x d — X),

+ = —id + = —

p = —i9 p = —id,

j + = (x+d— x d+),

— 3+

= I x+-;--x d

j = x d — x —- + X —-j > d++ d+

and their complex conjugates. -1 is defined using the prescription in [9].

1 For such listings, see for example [6].

Using the free equations of motion 9+ = which is modified in the interacting theory. The Hamiltonian for the free field theory is

H = j d 3x H =-j d3x0dd0, (4)

where the second equality only holds for the free theory. This is rewritten as

H = j d3x H = j d3xd-0SH0, (5)

in terms of the time translation operator

H 0 = d+0 = {0, H}, (6)

where { ,} denotes the Poisson bracket. When interactions are switched on, Sh picks up corrections, order by order in the coupling constant a. The other generators that pick up corrections are

Sj+-0 = S0+- 0 - ix+H + °{a2), Sj-0 = S0-0 + ixSH0 + Sas0 + O (a2),

Sj-0 = S°-0 + ixSH0 + sa0 + O (a2). (7)

Here, Sa and Sa represent spin transformations. We assume these to be of the form 00 as this form agrees with the known transformations. At cubic order, these do not mix with any of the other terms and are therefore not relevant to the calculations in this paper.

3. Deriving cubic interaction vertices

We focus on the following three structures for cubic interaction vertices at order a.

SH01 ~ 0203; SH02 ~ 0103; SH03 ~ 0102, (8)

where the fields 0i, 02 and 03 have integer spins Ai, A2 and A3 respectively. The first of these structures, at the level of the action would correspond to terms of the form

S ~f d4x 010203 + c.c. (9)

We enhance this basic form with derivatives to arrive at the Ansatz

SH01 = aAd+v[dad +p02dbd+a03] + c.c., (10)

where \x, p, a, a, b are integers and A is a numerical factor that could depend on the variables and spins. Note that terms of the form 0203 in SH01 are independent of terms of the form 0203 and hence will not 'talk' to one another. The commutators

[Sj,SaH ]01 = 0,

[Sj+- ,SH]01 = -Sh 01, (11)

impose the following conditions on our Ansatz a + b = A 2 + A3 - A1,

H + p + a = -1. (12)

Since a, b > 0, the first of these conditions2 implies that the vertex cannot exist unless X2 + X3 > X1. Now, let X = X2 + X3 - X1 so the first equation of (12) reads a + b = X. There are precisely (X + 1) possible values for a pair (a, b). We now rewrite our Ansatz in (10) as a sum of these (X + 1) terms

= aJ2An d+^n [9nd+pnfo 9(x-n)d +o"03] + c.c. (13)

The next set of commutators are

[Sj- ,Sh = 0 [Sj + ,SH = 0, (14)

and yield the following conditions

J2An{(.Vn + 1 - Xi) d+(l-1)9( dnd +pn$29(x-n)d+°nfo)

+ (Pn + x2) d+"n( 9 (n+1) d+(pn-1)$2 9(x-n) d+°nfo)

+ (an + X3)d +^n (9nd+pnfc9(X-n+1)d+(on-1)fa)} = 0, (15)

J2An{n d+^n( 9(n-1) d+(pn+1)^2 9(x-n) d+°n^3)

+ (x -n)9nd+pn$29(x-n-1)d+^n+^fo)) = 0. (16)

These conditions are satisfied if the coefficients obey the following recursion relations. An+1 = An = (-1)(n+1)( X W

(n + 1)-n * - \n + 1 Pn+1 = Pn - 1; an+1 = an + 1; Mn+1 = in, (17)

with the last condition showing that in is independent of n. The following "boundary" conditions are also necessary.

Pn=x = -X2; an=0 = -X3. (18)

The solution of the recursion relations for p, a and i subject to (18) is

Pn = x - x2 - n; an = n - x3; in = x1 - 1. (19)

Thus (13) reads

SaHfo1 = a ¿(-1)n d+(x1-1) [9nd+(x-x2-n)fo2 9(x-n)d+(n-x3)fo] + c.c. (20)

H = j d3x SHfo1, (21)

2 Note that this reduces to the condition in [2] if we set À1 = À2 = À3.

the interaction Hamiltonian is

Ha = a j d?x ¿(-ir^^i d+Xi [dnd+(X-X2-n)<2 d(x-n)d+{n-x3)fo] + c.c. (22)

Notice that if we set Xi = X2 = X3 = X' in (22) with X' odd, Ha vanishes. Hence, a non-vanishing self-interaction Hamiltonian, for odd integer spins exists, if and only if we introduce a gauge group. However, a consistent non-trivial vertex, coupling three fields of different spins, exists irrespective of whether the spins are even or odd.

We note that if the action obtained from the above Hamiltonian is to describe a theory involving fields of odd integer spins with cubic self interaction terms, the existence of a gauge group is forced upon the theory. Interestingly, the three fields could, in principle, carry different gauge groups.

4. Factorization properties and perturbative relations

We now rewrite the above results in the language of spinor helicity [10] where a four-vector is expressed as a bispinor using paà = , with det(paà ) yielding -p^p^. The spinor products are

r(kl- - lk-) r(kl- - lk-) {kl) = ^2 '-Tkïn1 ; [kl] = ^ k/k-r2- (23)

Eq. (22) involves the sum of two kinds of terms: $$$ and In Fourier space, the coefficient of the second kind of term <1 (p)<2(k)<3(l)S4(p + k +1) up to a sign reads

p- (lk-- l-k)X2+X3-Xl, (24)

kX2 lX3

which may be rewritten as

1 {pk}(-Xl+X2-X3) {kl}(Xl+X2+X3) (lp}(-Xi-X2+X3). (25)

It is clear that (25) exhibits the nice factorization property described in the introduction. It follows from the expression that given vertices for spins (X1, X2, X3) and (X^, X'2, X'3), their product yields the vertex for (X1 + X2 + X'2, X3 + X'3). As a corollary, note that the coefficient for the coupling of three fields (nX1, nX2, nX3) is the n-th power of the coefficient for the coupling (Xi, X2, X3).

This factorization property is similar in spirit to the KLT factorization relations. However, it is important to note that while the KLT relations are on-shell relations concerning amplitudes, the above relations are off-shell and valid at the level of the action. The next logical step in this program of research is to attempt a derivation of consistent quartic interaction vertices for higher spin fields in four-dimensional flat spacetime. This step is likely to reveal whether this factorization property exists for higher orders in the interaction. In principle, this would involve the same procedure followed in this paper after incorporating correction terms at order a2. Such a derivation, if successful, would seem to suggest that many no-go results [5] apply primarily to quantum field theories in which both locality and Lorentz invariance are manifest. Light cone gauge formulations of higher spin theories [ii] are very interesting in this regard since locality and Lorentz invariance are no longer manifest and instead need to be checked. The factorization

property and the perturbative ties that follow are much broader than the results in [12,13]. They are also very similar in spirit to much of the work devoted to relating spin 1 and spin 2 theories [14,15] but we still have much to learn about higher spin theories perhaps through unitarity methods [16] and S-matrix studies [17].

Acknowledgements

We thank Hidehiko Shimada, Stefano Kovacs and Sunil Mukhi for discussions. We also thank the anonymous referee of [13] for suggesting that we generalize our results from cubic self-interactions to cubic interaction of fields with different spins. Y.S.A. is supported by an INSPIRE fellowship from the Department of Science and Technology, Government of India.

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