CC BY
39
UDC 530.1; 539.1
Extended BRST renormalization P. M. Lavrov
Tomsk State Pedagogical University, 634061, Kievskaya St. 60, Tomsk 634061, Russia
The renormalization of general gauge theories curved space-time backgrounds is considered within the Sp(2)-covariant quantization method. It is proven that gauge invariant and diffeomorphism invariant renormalizability to all orders in the loop expansion and the extended BRST symmetry after renormalization is preserved.
Keywords: Extended BRST symmetry, Sp(2) quantization, general covariance, renormalization.
1 Introduction
In Quantum Field Theory Green’s functions contain divergences [1,2]. Renormalization is one of important means to construct a suitable quantum version for formulating fundamental interactions exiting in the Nature.
Renormalization of general gauge theories within the Batalin-Vilkovisky formalism [3,4] has been proved in papers [5,6].
Renormalizations in curved space-time using the Dyson criterion [7] are under intense investigations beginning with paper [8] (see [9-13] and references therein). We are going to continue our investigation of gauge invariant renormalizability in curved space-time with the help of new concept of renormalizability [5]. In [14] it was done in the BV formalism [3,4]. We have extended these considerations to the case when a theory is defined in the presence of external backgrounds, in particular in curved space-time and proved that in this case the gauge invariant renormalizability is compatible with preserving general covariance.
In the present paper we consider the problem of gauge invariant renormalizability of general gauge theories in the Sp(2)-method [15-17] in the presence of a gravitational background field and to prove general covariance of renormalization.
The paper is organized as follows. In Section 2 the Sp(2) formalism the general gauge theories in the presence of an external gravitational field is considered. In Section 3 general covariance of renormalization in the Sp(2) method is proved. In Section 4 concluding remarks are given.
We use the condensed notations as given by DeWitt [18]. Derivatives with respect to sources and antifields are taken from the left, and those with respect to fields, from the right. Left derivatives with respect to fields are labeled by the subscript “I”. The Grassmann parity of any quantity A is denoted by e(A).
2 General gauge theories in curved space within Sp(2) formalism
Let us consider a theory of gauge fields A1 in an external gravitational field gMV. The classical theory is described by the action which depends on both dynamical fields and external metric,
So = So(A,g).
(1)
Here and below we use the condensed notation g = gMV for the metric, when it is an argument of some functional or function. The action (1) is assumed to be gauge invariant,
SoR =0, SAi = R“(A,g)Aa,
A“ = A“(x) (a = 1, 2,..., n), (2)
as well as covariant,
ÔS,
SSo
Jg So = JA sg Ai + ^ Sg gMv = 0
Sgu
(3)
where A“ are independent parameters of the gauge transformation, corresponding to the symmetry group of the theory. The diffeomorphism transformation of the metric in Eq. (3) has the form
5g= -gMadv- 3ag^
= -gM« Vv£“ - gvaVM£“
= -VM^v -VvCm . (4)
Here Ca are the parameters of the coordinates transformation,
r = T(x) (a = 1, 2,..., d). (5)
The generating functional Z(J, $*, $,g) of the Green functions can be constructed in the form of the functional integral
Z (J, $*, $ ,g) = i d$exp{ -i [sext ($, $*, $, g)
+Ja$/
.
Here $A represents the full set of fields of the complete configuration space of the theory under
consideration and $Aa, $a are antifields. Finally, where
Sext($, $*, $A,g) is the quantum action constructed ^
with the help of the solution S = S($, $*, $A,g) to w = I Ja
the master equations
l(5$
Ab
Aa
1(S, S)a + V aS = ¿HAaS,
S($, <, $,g)|*.=*=
^ = 0
So (A, g).
Here on the space of fields $A and antifields $Aa the extended antibrackets are defined
caW(J, $*, $) = 0,
and (7) 1
w -(r,r)a + Var = o.
-^, c {acb} =0, (13)
(14)
(15)
(F, G)a =
JF JG
(8)
Aa
— (F ^ G) ( —1)(e(F)+1)(e(G)+1).
In what follows we assume the general covariance of S = S($, $*, $, g),
ro JS A * JS JS
Jg S=Jg < +Jg <Aa + ¿g“Jg g;v = 0.(6)
Aa "WMv
Let us choose the gauge fixing functional F = In particular the extended antibrackets (8) obey the F($,g) in a covariant form
Jacobi identities
((F G){a H)b}( —1)(e(F)+1)(e(H)+1) +cycl.perm.(F, G, H) = 0,
+
where curly brackets denote symmetrization with respect to the indices a, b of the Sp(2) group:
A{aBb} = A“Bb + BbAa.
In addition the operators V“, A“ are introduced
St S
V
eab $
Ab
(5
A ,
Aa
(—1)£
J$A
Aa
ab ba
fc — — fc ,
-12 i _ _ab
fc =1 fcab = —fc .
One can find the algebra of the operators (10)
A{“Ab} = 0, A{“Vb} + V {“Ab} = 0, V {“Vb} = 0.
The action of the operators (10) on a product of functionals F and G gives
A“(F • G) = (A“F) • G + F • (A“G)(-1)e(F> +
+ (F, G)“(-1)e(F),
V{“(F,G)b} = (V{“F,G)b} - (-1)e(F}(F, V{“G)b}. Note that Sext satisfies the master equations
x(Seæi, Seæi)a + V aSeæi = iftAaSeæi.
caZ(J, $*, $) = 0,
Jg F = 0 ,
(17)
then the quantum action Sext = Sext($, $*, $, g) obeys ^ the general covariance too
0.
Jg Seæi
From the Eq. (18) and the assumption that the term with the sources JA in (6) is covariant
Jg (Ja$A) = (Jg Ja)$A + JA(Jg$A) = 0 ,
(19)
(10)
it follows the general covariance of Z = Z( J, $*, $, g). Indeed,
where e“b is the antisymmetric tensor for raising and lowering Sp(2)-indices
JgZ = H
Jg <Aa
JSe
J<Aa
JSext(
A
+
JSe
Jgg;v +
+ (Jg Ja)$A
Sext + JA<
} . (20)
Making change of integration variables in the functional integral, (20),
$A ^ $A + Jg$A , (21)
we arrive at the relation
rJSeXt Jg $A + Jg $A ^ +
L J$A
S$*
_ = JSext JSext
$ a—=-----1--------g«v +
+ g A a + Jguv g +
+ (Jg Ja)<A + Ja( Jg <A)
exp
(H)
Sext + JA<A
From gauge invariance of initial action (2) in usual manner one can derive the BRST symmetry and the Ward identities for generating functionals Z = Z($, $*, $, g), W = W($, $*, $, g^d , r = r($, $*, ^, g) in the following form
-
JgSext + Jg (Ja<A)
Sext + JA<A
0.
(22)
From (22) it follows that the generating functional of connected Green functions W(J, $*, $, g))
(12) W (J, <, $ ,g) = ^ In Z (J, <, $, g) — 99 —
(23)
x
x
obeys the property of the general covariance as well
JgW(J, <, $, g) = 0. (24)
Consider now the generating functional of vertex functions r = r($, $*, $, g)
r($, $*, $, g) = W (J, $*, $, g) — Ja$a ,
where A_ JW(J, <, $, g)
$
JJa
From definition of $A (26) and the general covariance of W (J, $*, $, g) we can conclude the general covariance of Ja$a. Therefore,
Jg r($, <, $, g) = Jg W (J, $*, $, g) = 0,
In its turn the one-loop renormalized action S1R is covariant
Jg S1R = 0 .
(32)
(25)
Ja = — . (»)
Constructing the generating functional of one-loop renormalized Green functions Z1(J, $*, $,g), with the action S1R = S1R($, $*, $, g), and repeating arguments given above, we arrive at the relation
Z1 = 0, W1 = 0, r 1 =0 .
(33)
(27)
the generating functional of vertex functions obyes the property of the general covariance too. So, in this Section it is proved that if an external gravitational background gMV does not destroy the gauge invariance of an initial action So = S0(A,g). then the generating functional of Green functions can be constructed with the help of solution to the Sp(2)-master equations in an usual way. Moreover, if we assume the general covariance of the initial action then we prove the general covariance of non-renormalized generating functional of Green functions as well as both the generating functional of connected Green functions and of vertex functions.
The generating functional of vertex functions r( = Ti($, $*, $,g) which is finite in one-loop approximation
ri = S+f +h2 [r12L+r(2)in] + O(h3), (34)
(2)
contains the divergent part r( and defines renormalS
S ^ S2R = S1R — H2r(12div.
Starting from (31), (32) and (33) we derive J r(2) =0 J r(2) = 0
r 1 — 0 , r 1 io« — 0 ,
(35)
(36)
g 1,div ’ g 1,fin
that means general covariance of the divergent and finite parts of r1 in two-loop approximation. Therefore the two-loop renormalized action S2R = S2R($, $*, $, g) is covariant
Jg S2R = 0.
(37)
3 Covariant renormalization in curved spacetime
Up to now we consider non-renormalized generating functionals of Green functions. We are going to prove the general covariance for renormalized generat- Sr = S - ^inr ing functionals. For this end, let us first consider the n=i
one-loop approximation for r = r($, $*, $, g),
Applying the induction method we can repeat the procedure to an arbitrary order of the loop expansion. In this way we prove that the full renormalized action, Sr = Sr ($, $*, $, g),
nF(n)
n— 1,div :
(38)
r = S+HtrdlV+rfiJ + °(h2):
(28)
where rd2 and r fin
S ^ S1R = S — Hr
(1)
div.
(29)
From (18) and (27) it follows that in one-loop approximation we have
Jg [rd2 +rfij =0
(1)
and therefore rdiV and rf1^ obey the general covariance independently
J r(1) = 0
Jgr div = 0,
J r(1) = 0
Jgr fin = 0 .
H
covariance
denote the divergent and finite parts of the one-loop approximation for r. The divergent local term r^iV gives the first counterpart in one-loop renormalized action S1R
Jg Sr = 0;
(39)
and the renormalized generating functional of vertex functions, rR = rR($, $*, $, g)),
rR = S + £ Hnrn’!)1,fin :
n=1
(40)
(30)
which is finite in each finite order in h, is covariant SgrR = 0 . (41)
Therefore, taking into account results of Section 4 we can state that in presence of an external gravitational field the gauge invariant renormalizability can be arrived with preserving general covariance of functional r (41).
4 Conclusions mations in the form
We have considered the general scheme of gauge - ¿в ФА = в“ФАМа
invariant and covariant renormalization of the quantum gauge theories of matter fields in flat and curved where form of the Sp(2) doublet of constant Grassspace-time. Using the Sp(2) formalism we have marm parameters we find invariance of renormalized
proved that in the theory which admits gauge invari- effective action Гд under these transformations on the
ant and diflteomorphism invariant regularization, these hypersurface ФАа = 0
two symmetries hold in the counterterms to all orders ^ г | = 0
of the loops expansion together with extended BRST B R 1ф*=о '
symmetry. Indeed let us define the renormalized ex- Note Qnce more th&t tQ obtain thege resultg we h&ve tended BRST operators s“ used the gauge invariant renormalizability of general
= (гд •)“ + V“ • gauge theories in the Sp(2) formalism without assum-
ing the use of regularization for which acting by Д“ on Then we find that s“ satisfy the extended BRST alge- a local functional gives zero [17].
bra
S{asb} = о Acknowledgments
due to the equations for ГR and identities for extended The work was supported by the LRSS grant
antibrackets and operators V“ existing in the Sp(2) for- 224.2012.2, the RFBR grant 12-02-00121 and the
malism [15-17]. Defining the extended BRST transfor- RFBR-Ukraine grant 11-02-90445.
References
[1] Bogoliubov N. N. and Shirkov D. V., The Theory of Quantized Fields (New York, Interscience, 1959).
[2] Collins J. C., Renormalization (Cambridge University Press, Cambridge, UK, 1984).
[3] Batalin I. A. and Vilkovisky G. A., Phys. Lett. 102B (1981) 27.
[4] Batalin I. A. and Vilkovisky G. A., Phys. Rev. D28 (1983) 2567.
[5] Voronov B. L„ and Tyutin I. V., Theor. Math. Phys. 52 (1982) 628.
[6] Voronov B. L., Lavrov P. M. and Tyutin I. V., Sov. J. Nucl. Phys. 36 (1982) 292.
[7] Dyson F. J., Phys. Rev. 75 (1949) 1736.
[8] Utiyama R. and DeWitt B. S., J. Math. Phys. 3 (1962) 608.
[9] Buchbinder I. L., Theor. Math. Phys. 61 (1984) 1215.
[10] Toms D. J., Phys. Rev. D26 (1982) 2713; Phys. Lett. B126 ( 1983) 37; Phys. Rev. D27 (1983) 1803..
[11] Panangaden P., Phys. Rev. D23 (1981) 1735; Nelson B. L., Panangaden P., Phys. Rev. D25 (1982) 1019.
[12] Buchbinder I. L., Odintsov S. D. and Shapiro I. L., Effective Action in Quantum Gravity (IOP Publishing, Bristol, 1992).
[13] Shapiro I. L., Class. Quant. Grav. 25 (2008) 103001.
[14] Lavrov P. M. and Shapiro I. L„ Phys. Rev. D81 (2010) 044026.
[15] Batalin I. A., Lavrov P. M. and Tyutin I. V., J. Math. Phys. 31 (1990) 1487.
[16] Batalin I. A., Lavrov P. M. and Tyutin I. V., J. Math. Phys. 32 (1991) 532.
[17] Batalin I. A., Lavrov P. M. and Tyutin I. V., J. Math. Phys. 32 (1991) 2513.
[18] DeWitt B. S., Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965).
Received 01.10.2012
П. М. Лавров
РАСШИРЕННАЯ БРСТ ПЕРЕПОРМИРУЕМОСТЬ
Перенормируемость калибровочных теорий общего вида на искривленном пространстве-времени изучается в рамках Эр(2)-ковариантного метода квантования. Доказывается, что калибровочная инвариантность, общая ковариантность и расширенная БРСТ симметрия сохраняются во всех порядках разложения по петлям.
Ключевые слова: Расширенная БРСТ симметрия, Зр(2)квантование, общая ковариантность, перенормировка.
Лавров П. М., доктор физико-математических наук, профессор.
Томский государственный педагогический университет.
Ул. Киевская, 60, Томск, Россия, 634061.
E-mail: [email protected]
