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Parity violation and the effective gravitational action in three dimensions
Volume 175, number 2
PHYSICS LETTERS B
31 July 1986
PARITY VIOLATION AND THE EFFECTIVE GRAVITATIONAL ACTION IN THREE DIMENSIONS ~ I. V U O R I O Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 5 May 1986
A diagrammatic computation of the parity-breaking gravitational effective action induced by massless fermions in three dimensions is presented. The lowest-order approximation to the Chern-Simons action is obtained, in agreement with a general index-theoretic result.
1. Introduction. Redlich [ 1 ] has shown that the Chern-Simons action is induced by fermions coupled to an external gauge field in three dimensions. The result has been extended by general and mathematical methods to any odd dimension 2k + 1 (k > 0) and also to the case of a gravitational background field [2]. An explicit calculation has thus far been lacking in the gravitational case. It is the purpose of this letter to report on such a computation in three space-time dimensions. It is not possible to obtain the complete C h e r n Simons gravitational action from perturbation theory, since the action is not a polynomial in the field variables. This is in contrast with the gauge field case. However, at least to lowest order in the metric perturbation h~v =-g~v - r/uv, our expectation is confirmed, as we shall see * 1 2. The effective action. The functional integral we are interested in is
z--fd~
dO exp i f d 3 x det
1. #t--
a '+
e~le a ~(3' D
<-
)
- Dv3'a)$ ,
(1)
where the forward and backward covariant derivatives are defined by Du ~k = (au + c°u)~'
~
= ~(~-~. - c°u)'
(2)
and cou is the spin connection: ~0
= 1 bc.. --~[Tb - c l e ~, e v 50 OJbct~ ,7 J bvV# c"
(3)
Our metric decomposition guy = *Tuv + huv induces an expansion for the dreibein eau = *Tau + ~hau + higher order corrections. To compute Z to second order in huv , there are three Feynman diagrams we must consider; they are displayed in fig. 1. The Feynman rules are the same as in ref. [4] (section 6), except that the chiral projection factor is omitted. Only the vacuum polarization graph (lc) contributes to the parity breaking term. In fact, when we evalThis work is supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC02-76ER03069. ,1 During the course of this investigation is was brought to my attention that a similar result has now been obtained independently [31. 176
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 175, number 2
PHYSICS LETTERS B
1o1
(b)
Fig. 1. The lowest order Feynman graphs.
(c)
uate graphs (la) or (lb), the external momenta
31 July 1986
q(i) factor out
of the loop integral and consequently
[a,b(h v,q (i), m, A) = F(huv,q(i))p(m, A), where P(m, A) is a polynomial in the fermion
mass m and the cutoff momentum A. Such a contribution will be cancelled in the Pauli-Villars subtraction scheme, which we employ to regulate our calculation. The Feynman integral for graph (1 c) is f _ _ d 3 p ...
Tr['¼ihUV(q)"/u(2P +q)u i(Ik÷m)(-¼i)ha~(-q)Tc~(-2P
- q)~ i(/b + ~ + m ) ]
(4)
II(q) =./(2703
(p2 _ m 2 + ie)[(p +q)2 _ m 2 + ie]
The Dirac algebra is realized by Pauli matrices 7 °=03,
3,1=io1,
7 2=io2,
(5)
and
Tr 7u7 v7 x = _2ieUVX.
(6)
The desired term (i.e. the term containing a trace of three 7's) can now be extracted from eq. (4). Denoting the parity-conserving and parity-breaking terms by PC and PB, respectively, we have Ieff =/e~fC + IPB = --i In Z, where
[PeB(h v,m)= "[" d3q (_@hUV(q)h~(__q)(_2ime .,q~t)f u
-1/~-~
\
'-"
d3p (2p+q)u(2p+q)# J (27r)" (p2 _ m 2 +ie)[(p +q)2 _ m 2 +ie]
) "
(7) The integration over the loop m o m e n t u m can be easily performed using a Feynman parametrization and it yields
3p A_IB_ 1 _
i 16~Tlrn
J'
~ d3p
j ( ~ ) 3 PvA-1B-1 -
-i 32~-ml
Jqv,
f ~)~pupy4_lB_ d3p _ +~ilml {r/ut~[1 +(1 -q2/am2)j] +(quqJq2)[1 +(3q2/4m 2 1 _ &r2-1 r/zCl~
1)J] },
(8)
where A = p 2 _ m 2 + ie, B = (p + q)2 _ m 2 + ie, A is a m o m e n t u m cutoff and 1 J=
f
dy[l -
(q2/4m2)(1
_ y 2 ) ] - 1 / 2 = 2(4m2/__q2)l/2tan-l((_._q2/4m2)l/2).
-1 We notice that J-+ constant X m/q for m ~ 0 and J - + 2 for m ~ _+oo. Because of the factor m on the right-hand side of eq. (7), IPeB(huv,m)would seem to vanish in the limit m -+ 0. 177
Volume 175, number 2
PHYSICS LETTERS B
31 July 1986
However, this is not all o f l eIB because/eft is divergent and we need counterterms. We use the Pauli-Villars method and introduce two regulator masses ,2 Mi (i = 1,2) and coefficients c i (c o = 1,M 0 = m) with the sum rules 2
i~__oCilMiIk = 0
(k = 0,1,2).
Then, in the limit m --> 0 and M i -+ ~ or M i ~ - ~ for i = 1,2 2 i=0 ~ czjPB(h v'M') = --ld(2-~-~ " f d3q ( 1 Mi hU~'(q)ha~(-q)eu~'vqT(qvq¢ - rlutfl2)) u l 128rr [Mil
=
1
M
1297r IM[
fd3x hU~(X)e rO.eIa,a~ho,a(x)_Elh~(x)l.
(9)
One may not immediately recognized this expression, but a functional derivation with respect to hUV(x) and symmetrization of the indices yields the linearized form of the topological mass term (cf. ref. [5] ), with a mass parameter (-16rrGu) -1 =M(647rlMI) -1 or 1//a = -~GM/IMI. Ie~f is therefore equal to the (lowest-order approximation of the) Chern-Simons action plus a possible surface term. This is consistent with the result in ref. [2]. It is our intention to discuss the relationship between the induced Chern-Simons action and the zero modes o f the Dirac equation in a separate publication [6]. The author wishes to thank L. Alvarez-Gaum6, E. Buturovic, D. Dtisedau and K. Lee for helpful conversations and R. Jackiw for suggesting this problem and for continued assistance. TPB hUt PC which is cubically ,2 Actually, one regulator mass would be enough for regulating ~eff, o , we need another one to regulate/eft, divergent.
References [1] [2] [3] [4] [5] [6]
178
A.N. Redlich, Phys. Rev. D29 (1984) 2366. L. Alvarez-Gaum~, S. Della Pietra and G. Moore, Ann. Phys. (NY) 163 (1985) 288. M.A. Goni and M.A. Valle, Universidad del Pais Vasco (Bilbao, Spain) preprint, Phys. Rev. D, to be published. L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1983) 269. S. Deser, R. Jackiw and S. Templeton, Ann. Phys. (NY) 140 (1982) 372. I. Vuorio, in preparation.
PHYSICS LETTERS B
31 July 1986
PARITY VIOLATION AND THE EFFECTIVE GRAVITATIONAL ACTION IN THREE DIMENSIONS ~ I. V U O R I O Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 5 May 1986
A diagrammatic computation of the parity-breaking gravitational effective action induced by massless fermions in three dimensions is presented. The lowest-order approximation to the Chern-Simons action is obtained, in agreement with a general index-theoretic result.
1. Introduction. Redlich [ 1 ] has shown that the Chern-Simons action is induced by fermions coupled to an external gauge field in three dimensions. The result has been extended by general and mathematical methods to any odd dimension 2k + 1 (k > 0) and also to the case of a gravitational background field [2]. An explicit calculation has thus far been lacking in the gravitational case. It is the purpose of this letter to report on such a computation in three space-time dimensions. It is not possible to obtain the complete C h e r n Simons gravitational action from perturbation theory, since the action is not a polynomial in the field variables. This is in contrast with the gauge field case. However, at least to lowest order in the metric perturbation h~v =-g~v - r/uv, our expectation is confirmed, as we shall see * 1 2. The effective action. The functional integral we are interested in is
z--fd~
dO exp i f d 3 x det
1. #t--
a '+
e~le a ~(3' D
<-
)
- Dv3'a)$ ,
(1)
where the forward and backward covariant derivatives are defined by Du ~k = (au + c°u)~'
~
= ~(~-~. - c°u)'
(2)
and cou is the spin connection: ~0
= 1 bc.. --~[Tb - c l e ~, e v 50 OJbct~ ,7 J bvV# c"
(3)
Our metric decomposition guy = *Tuv + huv induces an expansion for the dreibein eau = *Tau + ~hau + higher order corrections. To compute Z to second order in huv , there are three Feynman diagrams we must consider; they are displayed in fig. 1. The Feynman rules are the same as in ref. [4] (section 6), except that the chiral projection factor is omitted. Only the vacuum polarization graph (lc) contributes to the parity breaking term. In fact, when we evalThis work is supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC02-76ER03069. ,1 During the course of this investigation is was brought to my attention that a similar result has now been obtained independently [31. 176
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 175, number 2
PHYSICS LETTERS B
1o1
(b)
Fig. 1. The lowest order Feynman graphs.
(c)
uate graphs (la) or (lb), the external momenta
31 July 1986
q(i) factor out
of the loop integral and consequently
[a,b(h v,q (i), m, A) = F(huv,q(i))p(m, A), where P(m, A) is a polynomial in the fermion
mass m and the cutoff momentum A. Such a contribution will be cancelled in the Pauli-Villars subtraction scheme, which we employ to regulate our calculation. The Feynman integral for graph (1 c) is f _ _ d 3 p ...
Tr['¼ihUV(q)"/u(2P +q)u i(Ik÷m)(-¼i)ha~(-q)Tc~(-2P
- q)~ i(/b + ~ + m ) ]
(4)
II(q) =./(2703
(p2 _ m 2 + ie)[(p +q)2 _ m 2 + ie]
The Dirac algebra is realized by Pauli matrices 7 °=03,
3,1=io1,
7 2=io2,
(5)
and
Tr 7u7 v7 x = _2ieUVX.
(6)
The desired term (i.e. the term containing a trace of three 7's) can now be extracted from eq. (4). Denoting the parity-conserving and parity-breaking terms by PC and PB, respectively, we have Ieff =/e~fC + IPB = --i In Z, where
[PeB(h v,m)= "[" d3q (_@hUV(q)h~(__q)(_2ime .,q~t)f u
-1/~-~
\
'-"
d3p (2p+q)u(2p+q)# J (27r)" (p2 _ m 2 +ie)[(p +q)2 _ m 2 +ie]
) "
(7) The integration over the loop m o m e n t u m can be easily performed using a Feynman parametrization and it yields
3p A_IB_ 1 _
i 16~Tlrn
J'
~ d3p
j ( ~ ) 3 PvA-1B-1 -
-i 32~-ml
Jqv,
f ~)~pupy4_lB_ d3p _ +~ilml {r/ut~[1 +(1 -q2/am2)j] +(quqJq2)[1 +(3q2/4m 2 1 _ &r2-1 r/zCl~
1)J] },
(8)
where A = p 2 _ m 2 + ie, B = (p + q)2 _ m 2 + ie, A is a m o m e n t u m cutoff and 1 J=
f
dy[l -
(q2/4m2)(1
_ y 2 ) ] - 1 / 2 = 2(4m2/__q2)l/2tan-l((_._q2/4m2)l/2).
-1 We notice that J-+ constant X m/q for m ~ 0 and J - + 2 for m ~ _+oo. Because of the factor m on the right-hand side of eq. (7), IPeB(huv,m)would seem to vanish in the limit m -+ 0. 177
Volume 175, number 2
PHYSICS LETTERS B
31 July 1986
However, this is not all o f l eIB because/eft is divergent and we need counterterms. We use the Pauli-Villars method and introduce two regulator masses ,2 Mi (i = 1,2) and coefficients c i (c o = 1,M 0 = m) with the sum rules 2
i~__oCilMiIk = 0
(k = 0,1,2).
Then, in the limit m --> 0 and M i -+ ~ or M i ~ - ~ for i = 1,2 2 i=0 ~ czjPB(h v'M') = --ld(2-~-~ " f d3q ( 1 Mi hU~'(q)ha~(-q)eu~'vqT(qvq¢ - rlutfl2)) u l 128rr [Mil
=
1
M
1297r IM[
fd3x hU~(X)e rO.eIa,a~ho,a(x)_Elh~(x)l.
(9)
One may not immediately recognized this expression, but a functional derivation with respect to hUV(x) and symmetrization of the indices yields the linearized form of the topological mass term (cf. ref. [5] ), with a mass parameter (-16rrGu) -1 =M(647rlMI) -1 or 1//a = -~GM/IMI. Ie~f is therefore equal to the (lowest-order approximation of the) Chern-Simons action plus a possible surface term. This is consistent with the result in ref. [2]. It is our intention to discuss the relationship between the induced Chern-Simons action and the zero modes o f the Dirac equation in a separate publication [6]. The author wishes to thank L. Alvarez-Gaum6, E. Buturovic, D. Dtisedau and K. Lee for helpful conversations and R. Jackiw for suggesting this problem and for continued assistance. TPB hUt PC which is cubically ,2 Actually, one regulator mass would be enough for regulating ~eff, o , we need another one to regulate/eft, divergent.
References [1] [2] [3] [4] [5] [6]
178
A.N. Redlich, Phys. Rev. D29 (1984) 2366. L. Alvarez-Gaum~, S. Della Pietra and G. Moore, Ann. Phys. (NY) 163 (1985) 288. M.A. Goni and M.A. Valle, Universidad del Pais Vasco (Bilbao, Spain) preprint, Phys. Rev. D, to be published. L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1983) 269. S. Deser, R. Jackiw and S. Templeton, Ann. Phys. (NY) 140 (1982) 372. I. Vuorio, in preparation.