CC BY
37
UDC 539.12; 537.8; 530.1:51-72
T. V Snegirev
DIVERGENCE OF THE EFFECTIVE ACTION OF FREE MASSLESS SPIN-3 THEORY
IN ANTI-DE SITTER SPACE
In this work we investigate the structure of the effective action for a simplest model of the higher spin fields, free massless spin-3 field theory in anti-de Sitter space. We calculate the divergence of the effective action in the one-loop approximation, which in 4-dimensional space can be reduced to finding De Witt’s coefficients.
Key words: quantization, effective action, one-loop approximation, higher spins, De Witt’s coefficients.
1. Introduction
One of open problems of modem theoretical high-er-energy physics, which attracts a wide attention, is the construction of higher-spin field theory and investigation its physical application. Till present time La-grangian formulation is only well developed for free higher-spin field theory in flat space and space of constant negative curvature - anti-de Sitter space (AdS) [1-2]. The importance of the Lagrangian formulation for higher-spin fields is primarily the fact that the La-grangian formulation is the basis for constructing a quantum theory and studies its quantum aspects.
It is well known that in quantum field theory the central object is the effective action that determines the quantum corrections to the classical action, quantum corrections to classical equations of motion and structure of Green’s functions. In terms of the effective action the procedure of the renormalization in quantum field theory and the structure of the Ward identities in gauge theories is naturally described (review [3-5]).
Calculation of the effective action in the higher-spin field theory has not been previously considered. It is therefore interesting to explore some aspects of such effective action for a simple field model of higher-spin, the free massless spin-3 field theory in AdS space.
In particular, the standard quantum field theory analysis shows that the effective action contains ultraviolet divergences, which in 4-dimensional space can be reduced to finding the DeWitt’s coefficients at coincident points. This article presents a general scheme for calculating the coefficients, as well as the final result for the model under consideration.
2. Quantization of massless spin-3 field theory in the AdS space
The free massless spin-3 field theory in the AdS space is described in terms of completely symmetric tensor fields 9^° . The classical action for such theory has the form
Sd = J ddxjg\ ji p^V2 9^ + |(Wv (V9V --3(V9rV^v -29^V2cp^ + 3(V(pr(Vqj), +
+2 r (d - 3)P">(lvo- 3r (d - 1)ip ^cp ^
and is invariant under the following gauge transformation §9^ = V^va + +Va^v ,
with the parameter being traceless ^ = 0.
Above used the following notation (V9)^v = V“9a^v, m = ma , v is covariant derivative, r is the radius of
t ^ t a^ ’ ~
the AdS space, d is dimensionality of the space.
Since in the case under consideration the algebra of gauge transformations is closed and the generators are independent we can use Fadeev-Popov’s method [3] for the quantization imposing the gauge and the introducing of the respective ghost fields. The gauge fixing function is chosen in the form
X,v = (V9),v - 2 V^cpv - 2 Vvcp^.
Then the gauge fixing action Sgf and ghost action Sgh are introduced in the standard way according to the general quantization scheme
Sg =-2 i i*"X,
s„ =i d-xjFic -’I* v
0tVv
Thus the quantum massless spin-3 field theory in the AdS space is described by a total action Stotal = Scl+Sgf+Sgh and represents the sum of the classical action Scl, the gauge fixing action Sgf and the ghost action Sgh. For the theory under consideration the explicit expression for the total action Stotal has the following form
Stotal = -2 f ddxj\g\^ (G^ + rX ^va,apy )9“Py +
+f ddxj\g\c ap M^ ^, where
3
G^va,a|3y (gv|3 gay }sym gpa gvag|3y )sym,
(1)
9
= (d-3)(g]1a gvP gay ~ ^ ( ~ ^(gia gvsgpy )Sym > (2)
MaP^v = 1 2 g»agvP + ^ g^P - d g^Vg“P P2 + 2rd(3)
BecmHUK Trny (TSPUBulletin). 2011. 8 (110)
3. One-loop effective action and the proper time
method
Effective action is the classical action plus quantum corrections. Since the model is described by a quadratic action, the effective action contains only one-loop quantum correction
r(1)[ÿ] = 2Trln(( + Sf^ÿ]) - iTrlnM^ap[ÿ]. (4)
Here we introduced notation S.. [m] = 8 S[m] .
. 89! 897
Expression (4) is formal, and to study its structure, we use a manifestly covariant proper time method also known as the method of Fock-Schwinger-DeWitt [6]. Within this method, we will search the effective action in the form
' A
r(1)[9] = -i f—eism2 f ddxj\g\ TrUB (s | x, x') (5)
2 o s
where function UA (s | x, x') is called Schwinger kernel. One satisfies the Schrödinger type equation (6), as well as the initial condition (7)
;duB (s)
-i-
ds
= (V28A + XC )UCB (s),
lim UB(5 | x,x') = 8j(x,x').
(6)
(7)
Further in order to find this function we choose it in such a form that could be convenient to go to a series expansion
U(s | x,x') A1/2 expl -^ |X<(x,x')(is)n, (8)
(4ns) ^ 2s)
where a(x, X) is chosen as a half square of the distance along the geodesic between x and x'. A(x, x') is convenient to choose in the form of Van Vleck-Morett’s determinant [5]
- det GIIV A( x, x') = 1V .
jg(x)g(x)
Then the structure of the effective action (5) taking into account (8) becomes
(i)d/2
2(4n)d/2 „=o
X j dd x-\l\~g\Tran j ids(is)
(n-d/2)-1 g-ism
. (9)
r(1) _ _
1 div ~
-^-TraQ _ m Trax + Tra2
(10)
2(4n)2 „_o-
where coefficients a0, a1, a2 are the DeWitt’s coeffi-
cients which determine the structure of one-loop divergences in the 4-dimensional space. Calculation of these coefficients for each particular theory is an independent task. Note that it is suffices to find the coefficient a2 for the massless theory.
4. Calculation of DeWitt’s coefficients and divergences of the effective action
The basic relations allowing to calculate the DeWitt’s coefficients are representation of the UB (s | x, x' ) (8), and two conditions imposed on this function: equation (6) and the initial condition (7). Eq. (6) gives us the recurrence relations for coefficients ai
aMV. aA
+ (n + 1)a^
= -A-1/2V2
(V1/2aBn ) + X^
H B( n+1) V / B( n+1) V Bn
and the initial condition fixes the coefficient a
c
C aBn
0
aBo( x>x') = 8B 8( x,x').
Finding the explicit form of the coefficients for each particular theory is independent and task is to calculate the geometric objects of the form limV„ ...VN c(x,x'), lim V„ ...VN IB (x,x') and
Vi Vk V 5 ^ X V x Vi Vk B V 5 7
x ^ x
lim V
...V A1
Vl Vk
2( x, x').
The direct calculation of the coefficients aj, a2 in the coincident points, which determine the divergent part of effective action, gives the following expressions 1
aB1| = --SBR + XB,
B1I x’=x 6
2<|, = -V IB ) -1 RXf
'x =x 3 3
B + XACXCB +
+—d2 (d -1)2--------d (d - l)(d - 3).
36 90
Forms of IB and XBA are defined by (l)-(3) for the our model. Note that the divergent part of the effective action does not include coefficients themselves, but their trace. Finally taking into account that our theory is massless we can write the divergent part of effective action, defined by (10)
p(i) _.
div
2(4n)2
J ddXyJ\g~\ f -245r2 + -1i
(11)
We see that the integral over s in expression (9) obviously diverges at the lower limit. With the help of some regularization the divergence can be identified,
4 - d
namely proceed as follows. Represent s = —-—, then
the divergent part of effective action is singular terms at and has the form
Thus we have found the first DeWitt’s coefficients at coincident points and explicitly calculated the divergent part of the effective action for the free massless spin-3 field theory in anti-de Sitter space in one-loop approximation. To eliminate the divergence it is necessary to renormalize the theory. Since we consider the free theory in an external gravitational field the renormalization has a simple form
r?[ g,v ] = lim ( [ g(K ]-r« [ g^ ]).
Expression T®[g ] is automatically finite.
2
o
4
5. Conclusion
In this work we investigated the structure of the effective action for the free massless spin-3 field theory in the anti-de Sitter space. It was found that the model contains only one-loop quantum correction (4) by virtue of the quadratic theory. Also analysing of the quantum effective action by the proper time method it was shown that the effective action contains an ultraviolet divergence which in 4-dimension is determined by DeWitt’s coefficents. The main purpose of this article was to find these coefficients and to determine the di-
vergent part of the effective action. The result is given by (11). In order to eliminate the divergent part must hold a renormalization procedure, which for our model has a simple way, one needs from the regularized effective action to subtract the divergent part.
Acknowledgment
Author is grateful to I. L. Buchbinder and V. A. Krykhtin for many useful discussions. The work was partially supported by the RFBR grant, project No. 09-02-00078.
References
1. Fronsdal C. Massless fields with integer spin // Phys. Rev. D. 1978. Vol. 18. P. 3624.
2. Fang J., Fronsdal C. Massless fields with half integral spin // Phys. Rev. D. 1978. Vol. 18. P. 3630.
3. Peskin M. E., Shroeder D. V. An introduction to quantum field theory / M.I. «R&C Dynamics», 2001. 784 p.
4. Noguchi A., Sugamoto A. Dynamical origin of duality between gauge theory and gravity // Tomsk State Pedagogical University Bulletin. 2004. Iss. 7. P. 59-61.
5. Buchbinder I. L., Odintsov S. M., Shapiro I. L. Effective action and quantum gravity / IOP Publishing, Bristol and Philadelphia, 1992. 656 p.
6. De Witt B. S. Dynamical theory of groups and fields / Gordon and Breach, New York, 1987. 288 p.
Tomsk State Pedagogical University.
Ul. Kievskaya, 60, Tomsk, Russia, 634061.
E-mail: snegirev@tspu.edu.ru
Received 14.03.2011.
Т. В. Снегирёв
РАСХОДИМОСТИ ЭФФЕКТИВНОГО ДЕЙСТВИЯ В ТЕОРИИ СВОБОДНОГО БЕЗМАССОВОГО ПОЛЯ СО СПИНОМ 3 В ПРОСТРАНСТВЕ АНТИ-ДЕ СИТТЕРА
Исследуется структура эффективного действия на примере простейшей полевой модели высших спинов, свободного безмассового поля со спином 3 в пространстве анти-де Ситтера. Вычисляются расходимости эффективного действия в однопетлевом приближении, которые в 4-мерном пространстве сводятся к нахождению коэффициентов де Витта.
Ключевые слова: квантование теории, эффективное действие, однопетлевое приближение, высшие спины, коэффициенты де Витта.
Снегирёв Т. В., аспирант.
Томский государственный педагогический университет.
Ул. Киевская, 60, Томск, Россия, 634061.
E-mail: snegirev@tspu.edu.ru
