- Email: [email protected]
Renormalization group and decoupling in curved space
PROCEEDIN& SUPPLEMENTS
ELSEVIER
Nuclear
Renormalization
Physics
B (Proc.
Suppl.)
127 (2004)
162-165
www.elsevierphysics.com
group and decoupling in curved space
E. V. Gorbar”*t aDepartamento de Fisica, ICE, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil I outline a series of results obtained in collaboration with I.L. Shapiro. The renormalization group equations and decoupling of massive fields in curved spacetime are studied and the cases of massive scalar, fermion, and vector fields are considered. In the higher derivative sector we arrive at the standard form of decoupling and obtain the renormalization group &functions in both UV and IR regimes as the limits of general expressions. For the cosmological and Newton constants the corresponding P-functions turned out to be unaccessible in the perturbative regime and, in particular, the form of decoupling remains unclear.
1. INTRODUCTION
2. NON-COVARIANT CALCULATION
The standard approach to the renormalization group in an external gravitational field has been formulated in [1,2] on the basis of the Minimal Subtraction (MS) renormalization scheme or the modified Minimal Subtraction scheme MS. The renormalization group and, in particular, the decoupling of massive quantum matter fields in an external gravitational field are especially relevant for such phenomena as the graceful exit from the anomaly-induced inflation [3,4] and the possible low-energy quantum dynamics of the cosmological constant [5]. It is well known [6] that in order to renormalize quantum matter in curved spacetime in addition to the standard Einstein-Hilbert action with cosmological constant one has to introduce also higher derivative terms (we use the Euclidean space notation)
s,,, =
s
The renormalization group for the parameters of the vacuum action can be established through the perturbative calculation of the quantum corrections to the propagator of gravitational perturbations h,, on the flat background gll,, = qpv + hpv, i.e. we have to calculate the graviton polarization operator. Since we are considering the polarization operator, the terms in the vacuum action (1) (which is the most general effective gravitational action of the second order in the curvature tensor) must be expanded up to the second order in h,, in order to determine to which effective gravitational action corresponds the polarization operator. As far as the GaussBonnet term does not contribute to the propagator, we can replace the Ci,,p-term by the expression 2W, where W = REV - g R2. Then the relevant bilinear expressions have the form
dzg1/2[-eGR - eA+
c pva4e1 c puao + R&R + &E +
&V2R],
(1) where CPvap is the Weyl tensor and E is the integrand of the Gauss-Bonnet topological invariant. In the present paper we shall focus our attention on the renormalization of f3i and 82 terms, the Newton and cosmological constants.
*On leave of absence from Theoretical Physics. tThis work was supported FAPEMIG (Brazil).
the
0920-5632/% - see front matter doi:10.1016/j.nuc1physbps.2003.12.031
0 2004 Elsevier
by
Bogolyubov the
PERTURBATIVE
Institute
research
grant
B.V
-_; ‘%v.a~) ha4+ ... ,
for from
All rights
reserved.
(2)
E. I? Gorbar/Nuclear
Physics
B (Pmt.
Suppl.)
127 (2004)
3. COVARIANT EFFECTIVE
-; q,,a,ao,l hap+ ... , J
d%g1/2R2 =
--(q&La,
J
d4x hp”” [ d,d,d,dp+rl,,r&‘2
+ o,pa,W2]
J
cPx~~/~w
(3)
hap + . . . ,
= Jd4x
(4)
h“” [ &,v,0rpa4
-&1~~~~~a4+ fa,a,a,a, - fhdpa + 4 hd,a,
+ 77aaa,av) a21haD+ ... ,
(5)
where the dots stand for the lower and higher order terms. In this section we consider only the case of massive scalar field nonminimally interacting with gravity (the /Rqi2/2 interaction term). The cases of massive spinor and vector fields will be considered in Sect. 4. After lengthy calculations in the dimensional regularization, subtracting divergences (together with the ln[4rp2/m2] terms), neglecting the O(n - 4) terms, where n is the spacetime dimension, and, finally, comparing the calculated polarization tensor with the corresponding expansions (2) - (5) of the most general effective gravitational action of the second order in the curvature tensor, we arrive at the following eXpressiOns for 81, 1’32,&, 0~ :
‘A=&?
15a4
[-2(E-;)2+;
-
l
2(4~)~
Jt~xp{$
ln(z)+i]
[i+
+im’R[i+
h($$)+l]
+ fCp,,p[&+
&jln ($)
+ k&a)]
(i+h(s))
&(a)=-
(E-3(1--3
1 216 (47r)s a2
a
f(l) =
cfivcup +
+~n(a)]~},
(8)
where $ = A, i = < - i, ~1is the dimensional regularization parameter, and
02=
v2 T-
8A
V2 m2 ’
(7)
2
15a4 + 45a2
kR(a) = A(< - i)” ’
where A=l-lln(E),
OF THE
The calculation of the polarization operator, presented in section 2, is rather complicated, therefore, it is useful to verify it by using a covariant method. To this end, we need to obtain the covariant effective action up to the second order in curvature. The one-loop effective action can be presented as the functional trace of the proper-time integral of the heat kernel K(s). The corresponding heat kernel expansion in powers of the curvature tensor, which takes into account all derivatives at each order of the expansion, has been obtained by Avramidi using summation of the Schwinger-Dewitt series [7] and by Barvinsky and Vilkovisky using generalized SchwingerDewitt technique [8]. The corresponding results were presented for the case of massless fields. Performing a rather straightforward generalization to the case of massive scalar field, we obtain that the effective action can be cast into the form
R[f
4 (4n)2
-$4+&-A]-
DERIVATION ACTION
@G=&(t-i),
e,=-L(X+-&), 42 = &
163
162-165
- $ (< - f)+
E. P Gorbar/Nuclear
164
Physics B (Pmt. Suppl.) 127 (2004) 162-165
It is easy to see that the above expression for the effective action perfectly corresponds to that derived in the previous section.
4. MASSIVE FIELDS
FERMION
AND
VECTOR
$1)
f ermion
C’p”a’ + R [&a)]
R}
,
where l&(a)
=
300Aa2 - 480 A - 40a2 + 19 a4 225 a4
a4 - 120A - 10a2 + 30Aa2 k;(a) = 270 a4 For the massive vector field, we find
7
= lili* pg.
(14)
It is easy to see that when this procedure is applied, e.g. in the case of massive scalar field, to the expressions (8), the $MS)-functions for A, l/G, 81,~ are exactly the usual ones. The disadvantage of the %iS scheme is that one can not control the decoupling of the massive quantum fields in an external gravitational field, hence one should use a mass-dependent scheme. The derivation of the p-functions in the massdependent scheme has been described, e.g. in [9]. In the massive scalar case, we obtain the following P-functions for the C2 and R2 terms (note that the renormalization group in curved space is defined in x space [1,2], therefore, the B-functions are, in fact, operators in x space; also we find convenient to use the notation p2 = -V2 in the P-functions presented below since then the formulas take the familiar form):
- A(a2 - 4)
-
(11)
GROUP
In the MS scheme the P-function of the effective charge C is defined as
/3&is)
By using the Barvinsky-Vilkovisky method described in the previous section, we arrive at the following effective gravitational action for the massive spinor field:
$(a)]
5. THE RENORMALIZATION EQUATIONS
a: = -(4:)2
ca4
(
+
&
-
-l;o)7
(15)
P28= -&{A($-;)[a2(E-;)2+& +~(E-~)(-$-~)+~+~(E-;)-~] +(J&)
+ Gvas 2
[
2
+ gin
($)
1 --&+&}. + 36a4
+ /+(a)]CPyaB
In the UV limit p2 > m2, the ,L?i- and @sfunctions agree with the MS-scheme result. In the IR limit p” << m2, we find that these P-functions are supressed by $
+Rbk where
kb(a)
[d+g(<-f)]+;(E-;)2
= -&+&-$+A+$,
s,IR
81
A
-18a2 - &.
(13)
p;JR
= - 1680;4a)2 =
-
1
12 (47r)2
. f [ ({-
+ @J i!>”
-
. (17) &(,-
i!)
E. F! Gorbar/Nuclear
Physics
In the massive fermion case, we find
+
10 4 + !)A]. ( is-a2 8
(20)
Again, the UV limit p2 >> m2 agrees with the m-scheme result. The IR limit p” < m2 is qualitatively similar to the scalar case in the sense that it shows the decoupling f,IR
-
p1
- - 4,,42
*5
For the massive vector, equal to
+ ‘(2)
’
the P-functions
(21)
are
1 ‘,” = 2(4~)~
+(--$+$-;+;)A],
+
-- 5 --7 a4 + 4a2
(23)
z+ 16
<)A]. 64
(24
Exactly as in the scalar and fermion cases the UV regime is consistent with the m-scheme result and the IR regime p2 << m2 demonstrates the decoupling of the loop contribution 4
v,IR
(25)
B (Proc.
Suppl.)
127 (2004)
162-165
165
The expressions (15)-(26) are the main results of our work and they explicitly exhibits the decoupling of massive particles in the vacuum quantum effects in curved spacetime. At the same time, taking the derivative with respect to momenta of those terms in the effective action, which correspond to the cosmological constant and Einstein-Hilbert terms, brings zero result. Therefore, we conclude that these ,&functions can not be calculated in the massdependent scheme through the perturbative expansion of the metric on flat background. Only more complicated calculations on the non-flat background can provide a reliable information about the renormalization group for the cosmological and Newton constants. REFERENCES 1. B.L. Nelson and P. Panangaden, Phys. Rev. D25 (1984) 1019; Gen. Rel. Grav. 16 (1984) 625. 2. I.L. Buchbinder, Theor. Math. Phys. 61 (1984) 393. 3. I.L. Shapiro, Int. J. Mod. Phys. Dll (2002) 1159. 4. I.L. Shapiro, J. Sola, Phys. Lett. B530 (2002) 10. 5. I.L. Shapiro, J.Solb, Phys. Lett. B475 (2000) 236; JHEP 02 (2002) 006. 6. I.L. Buchbinder, S.D. Odintsov, and I.L. Shapiro, Effective Action in Quantum Gravity, IOP Publishing, Bristol, 1992. 7. I.G. Avramidi, Yad. Fiz. (Sov. Journ. Nucl. Phys.) 49 (1989) 1185. 8. A.O. Barvinsky and G.A. Vilkovisky, Nucl. Phys. B282 (1987) 163. 9. A.V. Manohar, Effective Field Theories, Lectures at the Schladming Winter School, UCSD/PTH 96-04 [hep-ph/9606222].
ELSEVIER
Nuclear
Renormalization
Physics
B (Proc.
Suppl.)
127 (2004)
162-165
www.elsevierphysics.com
group and decoupling in curved space
E. V. Gorbar”*t aDepartamento de Fisica, ICE, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil I outline a series of results obtained in collaboration with I.L. Shapiro. The renormalization group equations and decoupling of massive fields in curved spacetime are studied and the cases of massive scalar, fermion, and vector fields are considered. In the higher derivative sector we arrive at the standard form of decoupling and obtain the renormalization group &functions in both UV and IR regimes as the limits of general expressions. For the cosmological and Newton constants the corresponding P-functions turned out to be unaccessible in the perturbative regime and, in particular, the form of decoupling remains unclear.
1. INTRODUCTION
2. NON-COVARIANT CALCULATION
The standard approach to the renormalization group in an external gravitational field has been formulated in [1,2] on the basis of the Minimal Subtraction (MS) renormalization scheme or the modified Minimal Subtraction scheme MS. The renormalization group and, in particular, the decoupling of massive quantum matter fields in an external gravitational field are especially relevant for such phenomena as the graceful exit from the anomaly-induced inflation [3,4] and the possible low-energy quantum dynamics of the cosmological constant [5]. It is well known [6] that in order to renormalize quantum matter in curved spacetime in addition to the standard Einstein-Hilbert action with cosmological constant one has to introduce also higher derivative terms (we use the Euclidean space notation)
s,,, =
s
The renormalization group for the parameters of the vacuum action can be established through the perturbative calculation of the quantum corrections to the propagator of gravitational perturbations h,, on the flat background gll,, = qpv + hpv, i.e. we have to calculate the graviton polarization operator. Since we are considering the polarization operator, the terms in the vacuum action (1) (which is the most general effective gravitational action of the second order in the curvature tensor) must be expanded up to the second order in h,, in order to determine to which effective gravitational action corresponds the polarization operator. As far as the GaussBonnet term does not contribute to the propagator, we can replace the Ci,,p-term by the expression 2W, where W = REV - g R2. Then the relevant bilinear expressions have the form
dzg1/2[-eGR - eA+
c pva4e1 c puao + R&R + &E +
&V2R],
(1) where CPvap is the Weyl tensor and E is the integrand of the Gauss-Bonnet topological invariant. In the present paper we shall focus our attention on the renormalization of f3i and 82 terms, the Newton and cosmological constants.
*On leave of absence from Theoretical Physics. tThis work was supported FAPEMIG (Brazil).
the
0920-5632/% - see front matter doi:10.1016/j.nuc1physbps.2003.12.031
0 2004 Elsevier
by
Bogolyubov the
PERTURBATIVE
Institute
research
grant
B.V
-_; ‘%v.a~) ha4+ ... ,
for from
All rights
reserved.
(2)
E. I? Gorbar/Nuclear
Physics
B (Pmt.
Suppl.)
127 (2004)
3. COVARIANT EFFECTIVE
-; q,,a,ao,l hap+ ... , J
d%g1/2R2 =
--(q&La,
J
d4x hp”” [ d,d,d,dp+rl,,r&‘2
+ o,pa,W2]
J
cPx~~/~w
(3)
hap + . . . ,
= Jd4x
(4)
h“” [ &,v,0rpa4
-&1~~~~~a4+ fa,a,a,a, - fhdpa + 4 hd,a,
+ 77aaa,av) a21haD+ ... ,
(5)
where the dots stand for the lower and higher order terms. In this section we consider only the case of massive scalar field nonminimally interacting with gravity (the /Rqi2/2 interaction term). The cases of massive spinor and vector fields will be considered in Sect. 4. After lengthy calculations in the dimensional regularization, subtracting divergences (together with the ln[4rp2/m2] terms), neglecting the O(n - 4) terms, where n is the spacetime dimension, and, finally, comparing the calculated polarization tensor with the corresponding expansions (2) - (5) of the most general effective gravitational action of the second order in the curvature tensor, we arrive at the following eXpressiOns for 81, 1’32,&, 0~ :
‘A=&?
15a4
[-2(E-;)2+;
-
l
2(4~)~
Jt~xp{$
ln(z)+i]
[i+
+im’R[i+
h($$)+l]
+ fCp,,p[&+
&jln ($)
+ k&a)]
(i+h(s))
&(a)=-
(E-3(1--3
1 216 (47r)s a2
a
f(l) =
cfivcup +
+~n(a)]~},
(8)
where $ = A, i = < - i, ~1is the dimensional regularization parameter, and
02=
v2 T-
8A
V2 m2 ’
(7)
2
15a4 + 45a2
kR(a) = A(< - i)” ’
where A=l-lln(E),
OF THE
The calculation of the polarization operator, presented in section 2, is rather complicated, therefore, it is useful to verify it by using a covariant method. To this end, we need to obtain the covariant effective action up to the second order in curvature. The one-loop effective action can be presented as the functional trace of the proper-time integral of the heat kernel K(s). The corresponding heat kernel expansion in powers of the curvature tensor, which takes into account all derivatives at each order of the expansion, has been obtained by Avramidi using summation of the Schwinger-Dewitt series [7] and by Barvinsky and Vilkovisky using generalized SchwingerDewitt technique [8]. The corresponding results were presented for the case of massless fields. Performing a rather straightforward generalization to the case of massive scalar field, we obtain that the effective action can be cast into the form
R[f
4 (4n)2
-$4+&-A]-
DERIVATION ACTION
@G=&(t-i),
e,=-L(X+-&), 42 = &
163
162-165
- $ (< - f)+
E. P Gorbar/Nuclear
164
Physics B (Pmt. Suppl.) 127 (2004) 162-165
It is easy to see that the above expression for the effective action perfectly corresponds to that derived in the previous section.
4. MASSIVE FIELDS
FERMION
AND
VECTOR
$1)
f ermion
C’p”a’ + R [&a)]
R}
,
where l&(a)
=
300Aa2 - 480 A - 40a2 + 19 a4 225 a4
a4 - 120A - 10a2 + 30Aa2 k;(a) = 270 a4 For the massive vector field, we find
7
= lili* pg.
(14)
It is easy to see that when this procedure is applied, e.g. in the case of massive scalar field, to the expressions (8), the $MS)-functions for A, l/G, 81,~ are exactly the usual ones. The disadvantage of the %iS scheme is that one can not control the decoupling of the massive quantum fields in an external gravitational field, hence one should use a mass-dependent scheme. The derivation of the p-functions in the massdependent scheme has been described, e.g. in [9]. In the massive scalar case, we obtain the following P-functions for the C2 and R2 terms (note that the renormalization group in curved space is defined in x space [1,2], therefore, the B-functions are, in fact, operators in x space; also we find convenient to use the notation p2 = -V2 in the P-functions presented below since then the formulas take the familiar form):
- A(a2 - 4)
-
(11)
GROUP
In the MS scheme the P-function of the effective charge C is defined as
/3&is)
By using the Barvinsky-Vilkovisky method described in the previous section, we arrive at the following effective gravitational action for the massive spinor field:
$(a)]
5. THE RENORMALIZATION EQUATIONS
a: = -(4:)2
ca4
(
+
&
-
-l;o)7
(15)
P28= -&{A($-;)[a2(E-;)2+& +~(E-~)(-$-~)+~+~(E-;)-~] +(J&)
+ Gvas 2
[
2
+ gin
($)
1 --&+&}. + 36a4
+ /+(a)]CPyaB
In the UV limit p2 > m2, the ,L?i- and @sfunctions agree with the MS-scheme result. In the IR limit p” << m2, we find that these P-functions are supressed by $
+Rbk where
kb(a)
[d+g(<-f)]+;(E-;)2
= -&+&-$+A+$,
s,IR
81
A
-18a2 - &.
(13)
p;JR
= - 1680;4a)2 =
-
1
12 (47r)2
. f [ ({-
+ @J i!>”
-
. (17) &(,-
i!)
E. F! Gorbar/Nuclear
Physics
In the massive fermion case, we find
+
10 4 + !)A]. ( is-a2 8
(20)
Again, the UV limit p2 >> m2 agrees with the m-scheme result. The IR limit p” < m2 is qualitatively similar to the scalar case in the sense that it shows the decoupling f,IR
-
p1
- - 4,,42
*5
For the massive vector, equal to
+ ‘(2)
’
the P-functions
(21)
are
1 ‘,” = 2(4~)~
+(--$+$-;+;)A],
+
-- 5 --7 a4 + 4a2
(23)
z+ 16
<)A]. 64
(24
Exactly as in the scalar and fermion cases the UV regime is consistent with the m-scheme result and the IR regime p2 << m2 demonstrates the decoupling of the loop contribution 4
v,IR
(25)
B (Proc.
Suppl.)
127 (2004)
162-165
165
The expressions (15)-(26) are the main results of our work and they explicitly exhibits the decoupling of massive particles in the vacuum quantum effects in curved spacetime. At the same time, taking the derivative with respect to momenta of those terms in the effective action, which correspond to the cosmological constant and Einstein-Hilbert terms, brings zero result. Therefore, we conclude that these ,&functions can not be calculated in the massdependent scheme through the perturbative expansion of the metric on flat background. Only more complicated calculations on the non-flat background can provide a reliable information about the renormalization group for the cosmological and Newton constants. REFERENCES 1. B.L. Nelson and P. Panangaden, Phys. Rev. D25 (1984) 1019; Gen. Rel. Grav. 16 (1984) 625. 2. I.L. Buchbinder, Theor. Math. Phys. 61 (1984) 393. 3. I.L. Shapiro, Int. J. Mod. Phys. Dll (2002) 1159. 4. I.L. Shapiro, J. Sola, Phys. Lett. B530 (2002) 10. 5. I.L. Shapiro, J.Solb, Phys. Lett. B475 (2000) 236; JHEP 02 (2002) 006. 6. I.L. Buchbinder, S.D. Odintsov, and I.L. Shapiro, Effective Action in Quantum Gravity, IOP Publishing, Bristol, 1992. 7. I.G. Avramidi, Yad. Fiz. (Sov. Journ. Nucl. Phys.) 49 (1989) 1185. 8. A.O. Barvinsky and G.A. Vilkovisky, Nucl. Phys. B282 (1987) 163. 9. A.V. Manohar, Effective Field Theories, Lectures at the Schladming Winter School, UCSD/PTH 96-04 [hep-ph/9606222].