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30
UDC 530.1; 539.1
One-loop divergences for the field of spin 3 on a de Sitter background in
nonminimal gauge
H. Popova, K. Stepanyantz
Physical Faculty, Moscow State University, Moscow, 119991, Russia.
E-mail: [email protected]
We consider the field of spin 3 on the de Sitter background in two dimensions and calculate the the divergent part of the one-loop effective action in a nonminimal gauge. For this purpose we construct a simple formula for the 62 coefficient of an arbitrary second order differential operator with small nonminimal terms. Using this formula we find that the divergent part of one-loop effective action for the considered theory is gauge independent.
Keywords: higher spins, Minakshisundaram-De Witt-Seeley coefficients, (anti)-de Sitter space.
1 Introduction
Higher spins are described by the fields ^MlM2...Ms which are totally symmetric and satisfy the condition
1 a fi
Pa fi ^5...^s
0.
(1)
The action for these fields on the flat background was obtained in fl] for bosons and in [2] for fermions. It is also possible to describe the higher spin fields on the (anti)-de Sitter background:
R
1
Uvafi
D(D — 1)
(9^a9vfi 9^fi 9 va
)R,
S
(-1)s
dDx\J—9 (Vap
aPMi ...Us )2 — ^ s(s — 1)
x(V,
apfifi U3...^s'' ¡afiU3...Us
)2 — s(Vap,
a^2...^s
)2 + s(s — 1)
xVaPaP^s Vfi
— 4 S(S — 1)(S — 2)
:(VaPafi
fiM4...Ms 2
)2
+ CiR(pMl...Ms ) + C2R(p.
77M3...M,
)2
action is gauge independent on shell. However, explicit calculations show that off shell the effective action depends on the gauge choice, see e.g. [4]. The choice of the minimal gauge considerably simplifies the calculations, especially on the curved background.
We will investigate the gauge dependence of the effective action for higher spin fields on the (A)dS background. For this purpose it is convenient to choose the A, ft gauge
S,
(2)
gf
( —1)s
J" d x 9 s(1 + A)[Vapa^2...^
where R = const. Then the action for bosons has the form:
— X(s — 1)(1 + ^)V(M2 paa^3...^s )]2.
(5)
(3)
In this paper we calculate the divergent part of the one-loop effective action in this gauge and show that the result does not depend on the parameters A mid ft.
2 Algorithm for calculating the one-loop divergences
In the one-loop approximation the effective action is given by
where the coefficients c^d c2 are defined by the invariance of the action under the transformations
Sp^i...^s = s(VMiaM2...Ms + VM2aMiM3...Ms + ...). (4)
The parameter aMlM2...Ms-1 is totally symmetric and traceless «fifiM3...Ms-1 = 0.
In [3] it was shown that under certain assumptions the consistent Lagrangian formulation for free bosonic totally symmetric higher spin fields is possible only in constant curvature Riemann space.
In order to calculate quantum corrections for the considered theory (as for all gauge theories), it is necessary to fix a gauge. It is well-known that the effective
r[P] = S [P] + - ft Tr ln D + O(ft2),
(6)
where Tr includes f dDx and the differential operator D is defined by
D
S2S
5<pi5<pj
(7)
Gauge (5) is called nonminimal, because the corresponding second variation of the classical action (with the gauge fixing terms) is given by a nonminimal operator
D ~ V + kUvVuVv + s»vu + w
(8)
in which terms with the largest number of derivatives differ from the Laplace operator.
From the mathematical point of view, calculation of the divergent part of the one-loop effective action corresponds to obtaining the Minakshisundaram-De Witt-Seeley coefficients [5]- [10]. For the second order operator D these coefficients are defined by
tr(x| exp(isD)|x) = -—^ \Di2 ( b0 + b2(is)
+&4(is)2 + O(s3) j.
(4nis)D/2
Using the dimensional regularization one-loop divergences can be related with certain Minakshisundaram-De Witt-Seeley coefficients. Really, taking into account that
ds
ln D = —i —exp(is(D + i0)),
o
it is easy to see, that, for example, for D « 2 the divergent part of the one-loop effective action can be written as
1
i-ioop 4n(D — 2)
J dDx^—gb2
D = VI + eKiv V^Vv + SlVu + W,
(12)
assuming e ^ 0.
The divergent diagrams are constructed using the
expansion
ln(D )
= ln(di + eKgvB^Bv) + ln (1 +
Bl + eK^v B^Bv
+ 2
+ 12
ln(di + eKivB^Bv), ln (1 +
1
Bl + e Kqv B^Bv
V
-V
ln(di + e KivB^Bv), ln(di + e KgvB^Bv),
(9)
ln 1 +
dl + e Klv didv
V
+..
(13)
After constructing the divergent diagrams we extract the logarithmically divergent terms and replace them according to the prescription
1
(10)
dD k
(2n)2k2 2n(D — 2)'
->•
(14)
In the curved space we use the expansion with respect to
hiv = 9iv — Viv
(15)
and after the calculation of the divergent part restore the covariant result, using the equations:
Similar formulas can be also written for other dimensions. For various differential operators Minakshisundaram-De Witt-Seeley coefficients can be found using the Schwinger^De Witt technique and its generalizations [11]- [14]. For a minimal operator heat kernel coefficients on the (A)dS background can be found using harmonic analysis on homogeneous spaces [15]. Using this method the one-loop effective potential for fields of arbitrary spin on the (A)dS background in four dimensions was calculated in [16] in the minimal gauge.
Here we use the generalization of the method proposed by G.t’Hooft and M.Veltman [17]. Using this technique b4 coefficient (without terms which are integrals of total derivatives) has been calculated for an arbitrary differential operator [18]. However, on the (A)dS background this algorithm does not work, because terms containing total derivatives are essential. In order to take into account terms with total
b2
which gives the divergences in two dimensions. Moreover, the nonminimal terms are considered to be small. Namely, we consider the operator
Riv 2 ldahva + dv dahia B^Bv h B h^v^j
+O(h
2\.
R = d^Bv hiv — d2 h + O(h2).
(16)
The result is
b2 = tr (w — 1 ViSl — 1 SiSl + 1R 11
1
e K aaW — —e VUSl + -e(K aaVlSl 1
+2Kiv ViSv ) + 8e(KaaSl + 2Kiv Si Sv )
—112 e Kaa R +1e Kiv Riv ). (17)
For e = 0 this formula gives the result for the minimal operator.
3 One-loop divergences for the spin 3 field in nonminimal gauge
In order to understand if the one-loop divergences for higher spin fields are gauge dependent, we investigate the simplest case D = 2, s = 3, considering the parameters ft Mid A to be small. Then the classical action with the gauge fixing terms in the lowest order
1
in ß and A is written as
S + Sgf = -^ J d2x^-g(^(VaP,
aYMiM2M3 )
-3
i— + A +2ß) (vc
+ 3A( Va^a^i M2 ) 6(A + ß)V a$aßM3 Vß ^Y1M3
3 R
+ Ö(A + 2ß)(VaPaßß)2 + "2(^MiM2M3 )2
+
3 — 4(1 + A + 2ß)
R($aaM3)
b2(main) — 3 (20 + 36ß — 0(A ß) j R.
b2(ghost) — g (13 + 18ß + o(ß)) R.
Combining the results for b2(ghost) and b2(main) one obtain the one-loop divergences
r
(œ)
1 —loop
1
J d2x^~g{b2(main) 2b2(ghost))
4n(D - 2) J d'2x^~g( — 2R + o(A,ß)). (22)
4n(D — 2) 1
(18)
The second variation of this expression with respect to the field paia2a3 is the differential operator, corresponding to one-loop diagrams with a loop of the spin 3 field. From this operator we construct matrixes eK S^, and W and substitute them into Eq. (17). Then we obtain the main part of the result
(19)
However, it is also necessary to take into account diagrams with a ghost loop. For the considered gauge the ghost Lagrangian is
8
Lgh = (S0oC^v — 2(SMSv + S oS p)Cov
8
— 2(S Sa + S oS v )can + (2 + 8)Rc^v ). (20)
Substituting the corresponding matrixes in formula (17) for the b2 coefficient we obtain the result for the ghost contribution:
(21)
This expression does not contain terms of the first order in A Mid ft- Therefore, in the considered approximation the result is gauge invariant.
4 Conclusion
We present a simple formula for the b2 coefficient of an arbitrary second order differential operator with small nonminimal terms. In particular, this formula allows to calculate terms with total derivatives. By the same method it is possible to find a coefficient b4. Now this work is in progress. Using the constructed formula in two dimensions we calculated a divergent part of the one-loop effective action for the field of spin 3 on the (anti)- de Sitter background in a nonminimal gauge. The result appeared to be gauge independent in the considered approximation (first order in the small parameters A Mid ft).
Acknowledgement
This work was supported by RFBR grant No 11-01-00296a, 12-02-00170 and 12-02-31128.
2
References
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Received 01.10.2012
E. Попова, К. Степаньянц
ОДНОПЕТЛЕВЫЕ РАСХОДИМОСТИ ДЛЯ ПОЛЯ СО СПИНОМ 3 В ПРОСТРАНСТВЕ ДЕ-СИТТЕРА В НЕМИНИМАЛЬНОЙ КАЛИБРОВКЕ
Мы рассматриваем поле со спином 3 в 2-мериом пространстве де-Ситтера и вычисляем расходящуюся часть однопетлевого эффективного действия. С этой целью мы конструируем простую формулу для коэффициента 62 при произвольном дифференциальном операторе второго порядка. Используя эту формулу, мы показали, что расходящаяся часть однопетлевого эффективного действия для рассматриваемой теории не зависит от калибровки.
Ключевые слова: высшие спины, эффективное действие, пространство (анти)-де-Ситтера.
Попова Е., кандидат физико-математических наук.
Московский государственный университет.
Физический факультет, кафедра теоретической физики, 119991 Москва.
E-mail: [email protected]
Степаньянц К., кандидат физико-математических наук.
Московский государственный университет.
Физический факультет, кафедра теоретической физики, 119991 Москва.
E-mail: [email protected]
