CC BY
30
UDC 530.1; 539.1
ON MASSIVE HIGHER SPIN INTERACTIONS
Yu. M. Zinoviev
institute for High Energy Physics, 142280, Protvino, Russia. E-mail: Yurii.Zinoviev@ihep.ru
We discuss a possibility to extend a Fradkin-Vasiliev formalism of constructing consistent cubic interaction vertices to the cases where vertex contains massive and/or massless higher spin fields. As an illustration we provide application of this formalism to the gravitational interactions of massless and partially massless spin-5/2 fields.
Keywords: higher spins, frame-like description, Fradkin-Vasiliev formalism.
1 Fradkin-Vasiliev formalism
Let us first of all briefly remind what is Fradkin-Vasiliev formalism [1,2] developed for the construction of consistent cubic interaction vertices among the massless higher spin fields. The main ingredient of this formalism is the so-called frame-like description of massless higher spin fields [3-5]. Its main features can be described as follows.
• Each massless higher spin field is described by a set of one-forms $ (physical, auxiliary and extra ones).
•
ô$ = DÇ + ...
where dots stand for the terms without derivatives.
can be constructed R = D A ...
where again dots stand for the terms without derivatives.
rewritten in a explicitly gauge invariant form L0 ~ ^RAR.
There exist three types of possible cubic vertices:
• trivial: L ~ RARAR, i.e constructed using gauge invariant two-forms only and thus trivially gauge invariant (hence the name);
• abelian: L ~ R A R A which contain one-form and whose gauge invariance (up to total derivative) follows from the Bianci identities;
• non-abelian: £~RA $ A $, which look similar to the ones in the Yang-Mills theories and whose gauge invariance requires introduction of nontrivial corrections to the gauge transformations.
All the non-abelian vertices can be constructed by the following deformation procedure:
curvatures in the form: AR ~ $ A $; transformations 6$ ~ $£; covariantly, i.e. 6R ~ R£;
•
Lagrangian L ~ ^ R Alt, which is just the free Lagrangian but with the initial curvatures replaced by the deformed ones.
Vasiliev has shown [6] that any non-trivial cubic vertex for massless completely symmetric fields with spins si, s2 and s3 having up to
N = s i + S2 + s3 - 2
derivatives can be obtained as a linear combination of abelian and non-abelian vertices.
The Fradkin-Vasiliev formalism was initially-developed and effectively applied for the construction of cubic vertices for the massless higher spin fields (see e.g. [7-12]). As we have seen the two main ingredients of this formalism are frame-like description and gauge invariance. But frame-like gauge invariant description exists for the massive higher spin fields as well [13,14]. Thus it seem natural to extend this formalism to the case of cubic vertices containing massive and/or massless fields. Some examples for electromagnetic and gravitational interactions of massive fields already exist [15-18]. In what follows we apply such formalism for gravitational interaction of partially massless spin-5/2 field
cUS cl
simplest non-trivial fermionic case.
Note that in the frame-like description most of the auxiliary and extra fields are mixed symmetry (spin-) tensors (7-traceless in fermionic cases)
ai...as-i,bi...bk
and this make all calculations rather involved (especially in the fermionic cases). One of the possible ways to simplify investigations is to restrict ourselves with particular space-time dimension d = 4 and use a multispinor frame-like formalism where all fields are still one forms but with all local indices replaced by spinor ones a ^ (aa). For the spin | case we are interested in here it means for example:
, (7^) = 0 ^ , ,
Q[ab], Y^ab = 0 ^ Qa3Y, Qa Pi.
Thus in what follows we will work in (A)dS4 space with background frame eaa and covariant derivative D normalized so that (A = —A2)
D A D^a = 2A2Ea3, Ea3 = 1 eaa A e3a.
2 Massless spin —
In this section we begin with the massless case that will be useful for the comparison. The free Lagrangian in AdS4 space can be written as follows [13]:
3A
Lo = 4aßa ea ß D',p3a 3 + —^«3* Ea 7 ^
- 2 ^aßa E' 3
4aßß + h.c.
öo^aßY = DiaßY + ej naßY + Xe(aè Ç3)Y è.
5oüaßY = DVaßY + X2e(aè .
physical frame field, while in spin-5/2 case it also put physical field on shell:
r Q
T = 0 ^ ü = ü(p) 0 — =0.
Op
Using these gauge invariant objects the free Lagrangian can be rewritten in the form:
Lo = a-tUaßY RaßY + avTcßY TaßY + h.c.
(5)
where coefficients ai,2 must be adjusted so that auxiliary field Q do not enter.
Now let us turn to the gravitational interactions. It turns out that deformations for the spin-| correspond just to the minimal substitution rules: D ^ D + w, e ^ e + h:
ARa3Y = co[w(as Q3y)s + X2h(aa ],
ATa3à = c0[w(aY 't^3)Yà + 3 i>a33 (6)
+Xh(a3 ^3 + hnà Qa3Y ].
At the same time deformations for the gravitational curvature and torsion have the form:
ARaß = b0[Q(aYs ü3)yS + 2X2^(aia p3^6
+X2^aa 3 43)a 3 ]
(7)
ATaa = 2b0 [üa3l i>ßYa + 2X4aß3 3 + h.c].
(1)
In this, non-trivial (on-shell) part of gauge transformations looks like:
This Lagrangian is invariant under the following local gauge transformations:
5HaßY = R(as nßY)S,
STaßa = R(aY £3)y" + Ra 3 iaßß,
(8)
(2) sRaß = 2boR(ajs V3)7ä.
We will need also an auxiliary field üaßY (though it does not enter the free Lagrangian) with the corresponding gauge transformations:
(3)
Now we consider the following interacting Lagrangian, which is just the sum free spin-5/2 and spin-2 Lagrangians with the initial curvatures replaced by the deformed ones:
Following the general procedure we construct two gauge invariant objects (similar to the curvature and torsion in the spin-2 case):
Ra3Y = DQa3Y + A2e(as ,
Ta3Y = D'^a3Y + Ae(at + esY Vta3S. (4)
Note that there is an essential difference between spin-2 and spin-5/2 cases as far as the zero torsion condition is concerned. In the spin-2 case this condition simply allows one to express Lorentz connection in terms of
L = a{R aß y RaßY + a^TaßY taßY +aoRaß Raß + h.c.
(9)
For the Lagrangian (9) to be invariant under the transformations (8) we have to put:
3a\co = 4aobo.
Note that the cubic vertex extracted from this Lagrangian contains terms with up to 2 derivatives in agreement with [19-21].
3 Partially massless spin —
Now let us turn to the partially massless spin-2 (recall that in four dimensions such field has four physical degrees of freedom corresponding to helicities ±2, ±2). Correspondingly, gauge invariant description requires two fields (main and Stueckelberg ones) and the free Lagrangian has the form:
iea 3 D3 - i0
Lo = Vaß
DV>a
VßY& - VaßaEaßVaßß]
+3a2[^aßaEaß- VaaßEaßr-]
-?a1^aE aß Vß + h.c.
ai
+ Y [?Vaßä E
\2
D^aßax + ale(a, f)à ß
tß)
(H)
-JlaßY 'j-aßä
DQaßi - E(a61
5
D^a/3à + eY'
*QaßY
+a.1eSaß Vß)di ß + a2e£l(a^ß),
T = 0 ^ ü = Ü(v) © ^ = o ^ v = v (v) ©
SS
SV=0
SS
SV = 0
aß y V aßY + RaßY DV aßY
+ ^ VaßYEYÖ VaßS .
-6a2naßinaßY ■ 12a2
Now let us turn to the gravitational interactions. Deformations for partially massless spin 2 again correspond to the minimal substitution rules D ^ D + w, e ^ e + h, while deformations for the gravitational curvature and torsion are defined up to the possible field redefinitions:
h** ^ hark + Kie3"VaY&VMs + K2ea"V3Y&VMs + ...
Non-trivial (on-shell) part of gauge transformations looks like:
(10)
Here ai2 = a22 = — 52-. This Lagrangian is invariant under the following gauge transformations:
SRaß = SR aßY
S j aßa
2b1R(a7s nß)Y& + b2 RaßY = coR(c
+
(aÄ nßY)&
(15)
coR(aY tß)Y* + coRé
aßß
+e7a na3Y + a2ea(ae
6o^a = DC + 3a2e3„ C3" + 3aieaA C.
Correspondingly, we will need two auxiliary fields (note thatVa3Y is a zero form):
6oQa3Y = Dna3Y, 6Va3Y = 6a2na3Y. (12)
Each of these four fields has its own gauge invariant object:
, V 3Y)s
The interacting Lagrangian is just the sum of free Lagrangians with deformed curvatures plus one possible abelian vertex:
o=
aR aßY R aßY + a2Taßä T'aßa + a,3#a^a + a4 VaßYEY6 VaßS +a5Taßä e~,à V aßY + iaoRaß Raß +a6Raß VaßY Vy + h.c.
(16)
(13)
= Di[>a + 3a2epà i"3" +3a1eaà- E3yVa3Y, Va3Y = DVa3Y - 6a2Qa3Y.
Similarly to the massless case, the zero torsion conditions not only allows one to express auxiliary-fields in terms of physical ones, but simultaneously put physical fields on shell:
At last, the free Lagrangian in terms of these gauge invariant objects looks as follows:
Lo = aiRa3Y Ra3Y + a2Ta3à Ta3&
+a,3Ma + aAVa3Y EYS Va3S
+a5Ta3à eY" Va3Y + h.c. (14)
Note that there is an ambiguity in the choice of coefficients due to identity:
0 « D(Ra3YVa3Y )
=D
Gauge invariance fixes all the coefficients in deformations as well as coefficient ae of abelian vertex in terms of gravitational coupling constant c0 hence we obtained one non-trivial vertex only. As in the massless case cubic vertex contains terms with up to two derivatives in agreement with [19].
Conclusion
Thus the Fradkin-Vasiliev formalism allows one systematically investigate cubic vertices for massless, partially massless and massive fields (in any combination). Here we use the multispinor framelike formalism. Thus means a restriction to d = 4 case, but allows us to describe bosonic and fermionic fields on equal footing with possible generalization to arbitrary spins. Let us stress some points where good
understanding is still lacking:
•
terms of gauge invariant objects;
Acknowledgement
This research has been supported by RFBR grant No. 14-02-01172.
References
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Received, 04.11.2012
Ю. M. Зиновьев
О ВЗАИМОДЕЙСТВИЯХ МАССИВНЫХ ПОЛЕЙ С ВЫСШИМИ СПИНАМИ
Мы обсуждаем возможность расширить формализм Фрадкина-Васильева построения непротиворечивых кубических вершин взаимодействия на случаи, когда вершина содержит массивные и/или безмассовые поля высших спинов. В качестве иллюстрации мы даем применение этого формализма к гравитационному взаимодействию безмассовых и частично безмассовых полей со спином 5/2.
Ключевые слова: высшие спины, реперный формализм, формализм Фрадкина-Васильева.
Зиновьев Ю. М., доктор физико-математических наук, ведущий научный сотрудник. Институт физики высоких энергий. 142280 Протвино, Московская область, Россия. E-mail: yurii.zinoviev@ihep.ru
