Научная статья на тему 'О взаимодействии массивных высших спинов в подходе Фрадкина-Васильева'

О взаимодействии массивных высших спинов в подходе Фрадкина-Васильева Текст научной статьи по специальности «Физика»

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Ключевые слова
ВЫСШИЕ СПИНЫ / КАЛИБРОВОЧНАЯ ИНВАРИАНТНОСТЬ / ВЗАИМОДЕЙСТВИЯ / HIGHER SPINS / GAUGE INVARIANCE / INTERACTIONS

Аннотация научной статьи по физике, автор научной работы — Зиновьев Ю. М.

Мы обсуждаем подход Фрадкина-Васильва к исследованию взаимодействий полей с высшими спинами. Изначально этот подход был развит для исследования взаимодействий безмассовых полей. Однако, используя реперный калибровочно инвариантный формализм для масивных полей, его можно применять для любой комбинации массивных и/или безмассовых полей. После краткого описания такого подхода мы рассмтриваем простейший возможный пример — самодействие и гравитационное взаимодействие для частично безмассового спина 2.

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ON MASSIVE HIGHER SPIN INTERACTIONS IN FRADKIN-VASILIEV APPROACH

Here we discuss Fradkin-Vasiliev approach for investigation higher spin elds interactions. Initially this approach was developed for investigation of massless elds interactions, but using frame-like gauge invariant formalism for massive higher spin elds it can be straightforwardly applied to any combination of massive and/or massless elds. After brief description of such approach we consider the simplest possible examples — self-interaction and gravitational interaction for partially massless spin 2 eld.

Текст научной работы на тему «О взаимодействии массивных высших спинов в подходе Фрадкина-Васильева»

UDC 530.1; 539.1

On massive higher spin interactions in Fradkin-Vasiliev approach

Yu. M. Zinoviev

institute for High Energy Physics, Protvino, Moscow Region, 142280, Russia E-mail: Yurii.Zinoviev@ihep.ru

Here we discuss Fradkin-Vasiliev approach for investigation higher spin fields interactions. Initially this approach was developed for investigation of massless fields interactions, but using frame-like gauge invariant formalism for massive higher spin fields it can be straightforwardly applied to any combination of massive and/or massless fields. After brief description of such approach we consider the simplest possible examples — self-interaction and gravitational interaction for partially massless spin 2 field.

Keywords: higher spins, gauge invariance, interactions.

1 Fradkin-Vasilev approach for massive fields

First of all, let us briefly recall what frame-like formalism for higher spins is [1-3]. Really it is just a natural and straightforward generalization of well-known frame-like formalism for gravity

eM“,WM“6 ^^ai...as_i,6i ...bk, o < k < S - 1

where now instead of frame eMa and Lorentz connection wMab one introduces a whole bunch of fields ^ai.„as_i,bi...^ o < k < s — 1. It is very important that all these fields are gauge ones so that each field has its own gauge transformation:

ai ...as_ i ,bi ...bk — d £ai...as_i,bi...bk +

where dots stand for the terms without derivatives. Moreover, each one has its own gauge invariant field strength (which we generally will call curvatures):

'J~f ai ...as_ i ,bi ...bk 7~) /T) ai ...as_ i ,bi ...bk I

— D[^$vJ + • • •

where again dots stand for the terms without derivatives. Remarkably that the free Lagrangian can be rewritten in terms of these gauge invariant curvatures as

Lo = RR

very similar to the usual Yang-Mills theories.

Now let us turn to the so-called constructive approach to investigation of possible interactions. Schematically it can be described as follows.

• Construct cubic vertex such that its free variations vanish on-shell:

Li — $$$ ^ ¿oA ~ 0

mations such that now all variations in the linear approximation vanish off-shell:

¿1$ — ^ SoLi + SiLo = 0

Note that these two steps are completely general and common to any constructive approach based on metriclike, frame-like or any other formalism one uses. But in a frame-like formalism there are two more steps.

vatures such that deformed curvatures transform covariantly:

AR — $$ =^ SR — R£

in terms of these deformed curvatures as

L - VRR

Such approach is straightforward and do allows one investigate possible interactions. But its first two (and the most hard) steps do not take into account the existence of gauge invariant curvatures though it is clear that in any gauge invariant theory the most simple and elegant formulation is the one in terms of gauge invariant objects. Roughly speaking, the Fradkin-Vasiliev approach [4,5] modifies the order of calculations to take advantages of these curvatures existence. Schematically, it can be described as follows.

sponding corrections to gauge transformations

A R — $$ © ¿i$ — =^ SR — R£

so that they transform covariantly. In this, there is still some ambiguity because this guarantees that the equations are gauge invariant, but not necessarily Lagrangean.

Yu. M. Zinoviev. On massive higher spin interactions in Fradkin-Vasiliev approach

Put them into the Lagrangian and require it to be gauge invariant

ties here taking into account the invariance of the Lagrangian):

C

1

1

Lo = 2 {- 2 {/ac+

1

1

+ 4Bab2 — 2 { MV} BabDMBv +

+m[{ £6V} ^MabBv + e^aBab//]

which is invariant under the following gauge transformations:

/ = + n/ + me/£, ¿o“/b = D,n

ab

„ab

SoBM = + m£M, SoBa = —2mn

For all four fields there exist gauge invariant curvatures:

:

ab

7 a

ab

D[M^]a6 - y e[M[aBv]6]

= D[:fv]a — “[:,v]a + me[,a

= D: Bab + 2m“/b

= D[:Bv] — B,v — m/[:,v]

AFMvab = 1[B[:[aBv]b]- e[:[aBb]cBv]c] +

+Q[Mc[aQv]b]c ATMva = 2Q[Mab/v]b - B[:aBv]

AB/b = “Mc[aBb]c, ABMv = -B[:a/v]

thus fixing all remaining ambiguities.

As we have seen the main ingredients of such approach are frame-like formalism and gauge invariance. But gauge invariant frame-like formalism exists for massive higher spin particles as well [6,7]. Thus such approach can be used for investigations of interactions for any combination of massless and/or massless fields. In the next section as an illustration we consider the most simple example — self-interaction and gravitational interaction for so-called partially massless spin 2 field.

2 Example: partially massless spin-2

First of all let us remind what is partially massless spin 2 [8-10]. It is an exotic representation that exists in de Sitter space only and has four physical degrees of freedom — helicities (±2 ±1)- In a frame-like gauge invariant formalism it can be described by the following free Lagrangian:

aa v]

In this, corresponding corrections to gauge transformations look like:

¿i“/b = -2nc[aOMb]c

¿i/:a = -2nab/:b + 2“/b£b - B/£

¿iBob

c[a Db]c

¿iB, = -B/C

It is not hard to check that under such corrected gauge transformations all curvatures transform covariantly:

ab

¿T:va

ab

SB,

= 2nc[aF,vb]c

= 2nabT,vb + 2FMvabeb -B[:,v]ae = nc[aB,b]c, SV = -B[:,v]aC

Gravitational interaction It turns out that in this case deformations for partially massless curvatures are minimal, i.e. correspond to standard minimal gravitational interaction:

ab

^[h[M[aBv]b] - e[,[aBb]ch„]c] --^[:c[a^]b]c

AT,va

AB/b

AS

-w[: /v]b - ^[/bV - mh[,aBv] -^:c[aBb]c

= B[,ahv]a + m/[,ahv]a

where hMa and wMa are usual gravitational frame field and Lorentz connection. But to find deformation for Riemann tensor and torsion requires much more work with the result:

AR

ab

= - 1Q[„c[aQv]b]C - 1 B[,[aBv]b] +

1

= -2“[: “v] - 4B[: Bv]

+ 1 e[:[aBb]cBv]c - -B[:[/b] +

In this, free Lagrangian can be written in terms of these curvatures as follows:

4

m [a nb]cr c m /- [a c b]

+ “2 e[:[ B ] /v] - — /■[:[ /v] ]

AT a AT

-“[:ab/v]b + B[,aBv]+ m/[,aBv]

r 1 / -r ab 77 cd , o f a^ bc ,

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Lo — 4 j abcd / Fa,S + 2 { ab } +

+m { ir}Va£abc

Self-interaction. By straightforward calculations it is not hard to find quadratic deformations for curvatures (note that we have already fixed all ambigui-

Here we again take into account invariance of the complete Lagrangian i.e. the sum of Lagrangian for massless gravitational spin 2 and partially massless spin 2 fields. Similarly to the self-interaction case, form the formulas given it straightforward to find appropriate corrections to gauge transformations and check that all deformed curvatures do transform covariantly.

Conclusion requires more work.

Thus Fradkin-Vasiliev approach provides effective Acknowledgment framework for investigation of cubic vertices for mass-

less and massive fields. It also allows investigate pos- The work was supported in parts by RFBR grant

sibilities to go beyond linear approximation, though it No. 11-02-00814.

References

[1] M. A. Vasiliev. Sov. J. Nucl. Phys. 32 (1980) 439.

[2] V. E. Lopatin, M. A. Vasiliev. Mod. Phys. Lett. A3 (1988) 257.

[3] M. A. Vasiliev. Nucl. Phys. B301 (1988) 26.

[4] E. S. Fradkin, M. A. Vasiliev. Phys. Lett. B189 (1987) 89.

[5] E. S. Fradkin, M. A. Vasiliev. Nucl. Phys. B291 (1987) 141.

[6] Yu. M. Zinoviev. Nucl. Phys. B808 (2009) 185, arXiv:0808.1778.

[7] D. S. Ponomarev, M. A. Vasiliev. Nucl. Phys. B839 (2010) 466, arXiv:1001.0062.

[8] S. Deser, A. Waldron. Nucl. Phys. B607 (2001) 577, arXiv:hep-th/0103198.

[9] S. Deser, A. Waldron. Phys. Lett. B513 (2001) 137, arXiv:hep-th/0105181.

[10] Yu. M. Zinoviev. arXiv:hep-th/0108192.

Received 01.10.2012

Ю. М. Зиновьев

О ВЗАИМОДЕЙСТВИИ МАССИВНЫХ ВЫСШИХ СПИНОВ В ПОДХОДЕ

ФРАДКИНА-ВАСИЛЬЕВА

Мы обсуждаем подход Фрадкина-Васильва к исследованию взаимодействий полей с высшими спинами. Изначально этот подход был развит для исследования взаимодействий безмассовых полей. Однако, используя реперный калибровочно инвариантный формализм для масивных полей, его можно применять для любой комбинации массивных и/или безмассовых полей. После краткого описания такого подхода мы рассмтриваем простейший возможный пример — самодействие и гравитационное взаимодействие для частично безмассового спина 2.

Ключевые слова: высшие спины, калибровочная инвариантность, взаимодействия.

Зиновьев Ю. М., доктор физико-математических наук, ведущий научный сотрудник.

Институт физики высоких энергий.

Ул. Победы, 1, Протвино, Моск. обл., Россия, 142280.

E-mail: yurii.zinoviev@ihep.ru

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