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Mixed Lorentz anomaly in four-dimensional space
Volume 173, n u m b e r 2
PHYSICS LETTERS B
5 June 1986
M I X E D L O R E N T Z A N O M A L Y IN F O U R - D I M E N S I O N A L S P A C E S. Y A J I M A and T. K I M U R A Research Institute for Theoretical Physics, Hiroshima Universi(v, Takehara, Hiroshima 725, Japan Received 4 February 1986
By using the diagrammatical and the topological methods, the Lorentz anomaly is evaluated for chiral Weyl fermions interacting with combined gravitational and U(1) gauge fields in four-dimensional space. The results are consistent with that of Nieh in which signs of anomalous terms are taken correctly. The absence of the Lorentz anomaly, in the space-time with torsion, is also explained from the diagrammatical method.
In four-dimensional riemannian space, it is only expected that a gravitational anomaly appears when a U(1) gauge field couples with the gravitational field [ 1]. In fact Nieh has shown that the U(1) gauge field causes the breakdown of general covariance and local Lorentz invariance in gauge theories of chiral Weyl fermions [2]. The Lorentz anomaly is given by (Tab - Tba)= [i/12(4n') 2] [ e a b u v R f uv - eabuvRUVXofxp + 2 e a b u v f u v l x ~] ,
(1)
in which we have used the notation, R U v o o = FUvo,o - FUvo,p - F u o~pP°~vo + PUaoF'~vo ,
R = R uvuv '
f u r = a u , v - av, u .
(2)
The form (1) was presented in our previous paper [3] ,1, and differs from Nieh's result only in the sign of the first term of the RHS. Using the path integral and the expansion coefficient of the heat kernel, we have also found that the Lorentz anomaly vanishes when the torsion field couples with spln-g " 1 Weyl spinors, though the torsion behaves similar to the U(1) gauge field in the action [3]. To explain the origin of the different results of the case o f U(1) gauge field and that of torsion, it is desirable to evaluate the anomaly from another point of view, such as Feynman's diagram method. On the other hand, in this journal, Caneschi and Valtancoli [5] attempted to derive Nieh's result relying upon the topological method. They, however, concluded that they were unable to reproduce the result. We shall here evaluate critically the Lorentz anomaly with the aid of the diagrammatical and the topological methods and show that the results derived are consistent with (1) in which the relative sign o f the first and second terms is taken into account correctly. The peculiarity of the case of torsion is also explained from the diagrammatical method. Einstein anomalies in (4n + 2)-dimensional riemannian space are given by Alvarez-Gaum6 and Witten [ 1]. The anomalous vertex factor of the covariant Lorentz anomaly is obtained analogous to their discussion. We shall consider a massless Weyl spinor in four-dimensional riemannian space, S = f d 4 x ( d e t e b v ) ~1.l ( ~-L T a e a u Duff L - @LDUTaeaUt~L), - ~
(3)
+1 Thel RHS of eq. (1) is four times as large as the result in ref. [3] because the U(1) gauge field coupling was given through (0# + ~iau). It should be remarked that the signs ofRt~vO o and F~v are opposite to Nieh's in refs. [2,4]. 154
Volume 173, number 2
PHYSICS LETTERS B
5 June 1986
where Du = ~
1. ab ~ leo g °ab
-
~L = ~ P + '
,
~--
Du
41. a bta_ U a b , = ~.la + -516o
oab=~i[~/a,Tb ]
,
ffL=p_ff
,
P-+ = 1(1 +- ~'5 ) '
(4)
and 6oabla is the spin connection, the P+ are the projection operators of chirality, and we adopt the metric with signature (+, - , - , - ) . The classical conservation equation implied by local Lorentz invariance is Tab -
Tba= 0 ,
(5)
with Tab =---tab + 2Sab I~ '
in which 1. b u ( ~- 7 a D u P tab =-~le
~ - ~-+DuTaP
~b) ,
Sab t~ = leC u ~ ( O a b 7 c + 3'cOab)P_t~ .
(6)
Now we discuss the one-loop diagrams which contribute to the anomalies by the Pauli-Villars regularization. A massive fermion X is introduced so that the functional measure in the path integral formalism is invariant under the gauge transformations. Then the regularized e n e r g y - m o m e n t r u m tensor is given by (7)
(Tab)reg = Tab - T a b l ¢ ~ x •
Here the Lorentz anomaly appears, because for the massive regulator spinor the e n e r g y - m o m e n t u m tensor with insertion of P _ , Tab [ ¢ ~ x ' is not symmetric under the interchange of the indices a and b even formally. And it is only the regulator diagram that contributes to the anomaly. Note that the factor P_ is not inserted at vertices other than the anomalous vertex that this e n e r g y - m o m e n t u m tensor affects, in calculating the covariant anomalie due to one-loop diagrams. We postulate that the regulator field X obeys the equation of motion (iT'D - 3 4 ) x = 0. The antisymmetric component of the regularized e n e r g y - m o m e n t u m tensor is written as (Tab - Tba)reg =--(tab -- tba + Sab Ulu)i~_~x = _ (tab - t b a ) [ ~ x - l ( D u ~ ) ( Z ~ / t a a a b + [Oab, 7Ul ) P _ x 1
--~X(2OabT u + [ T " , O a b ] ) P _ D u x
1.
--
= ~iMXOab)'5X,
(8)
by using 07"D M)X = 0.We may,therefore, adopt ]i.iMtTab ~[5 as the anomalous vertex factor of the Lorentz anomaly. Anomalous terms are expected to appear as the finite terms in an infinite mass limit (114-+ ,,o) of the calculation of one-loop diagrams. Actually, the covariant Lorentz anomaly in the presence of a U(1) gauge field is derived by the collection of the contributions of diagrams in fig. 1. The lowest order interaction terms, -
Lint = _ ~ T # a # X -- ~n(OUu 1 1. -- l4r 1Uudpr %" f u p f"v O --4U~(~huu
--
12nUu~) r r ) [~TU(0U+ iaU)x _ ~ ( g u _ iaV)3,uX] ,
(9)
give vertices among spinors X, U(1) gauge fields a u and gravitons ~buu, in whichguv = r/u u + K~buu, so ea u = ~au lnOau. For simplicity's sake, we may impose the coordinate condition [6], -
au(~u,
1
Then the contribution from the convergent diagram fig. la, (la), is (1 a ) = --~ i~M(27r) -4
fd4p ( [ ( p
- q l )2 _ M 2 ] (p2 _ M 2) [(p + q2)2 _ 11,/2] }-1
× ( T r [ 3 ' 5 ° a b ( p - ql +M)3'U(P + M ) T ° ( P
+ q2 + M ) ] (2p - ql)udpuu(ql)ao(q2)
+ Tr[75 ° a b ( p -- ql + M)~/°(P + M ) T a ( P + q2 +M)] (2p + q2)UOuv(q2)ao(ql)) _1
1
-gX+i~Y,
(11) 155
Volume 173, number 2
PHYSICS LETTERS B
(a)
(b) -
5 June 1986
(d)
(c)
o.p
Fig. 1. Relevant diagrams contributing to the four-dimensional Lorentz anomaly in the presence of a u(1) gauge field in curved space• in which p = 7Upu, etc., the terms breaking either the gauge invariance or general covariance are neglected after the actual calculation, X and Y are covariant forms: X = [tc/(ar02 ] eabTa (q2rl~V - - q ~ ql)uCuv(ql)q2,raa(q2) ' y = [r/(4rr) 2 ] eab~ '~ [qluql,y~)av(ql) -- qluql,rCau(ql)]
q~aV(q2).
In the evaluation of the contributions from figs• 1 b - 1 d, there appears an integral fd4p(p2
l f ( 1 + A - M2) 2 = d4p (p2 _ )I,/2)2
2A ) (p2 _ M2)3 + "'" '
(12)
where the first term is a logarithmical divergent term and the second term a finite term. We, however, may consider only infite terms, since the logarithmical divergence disappears if we introduce more than two suitable massive regulator fields from the outset. In a similar way to the triangle diagram, the contributions from figs. 1b - 1 d are, respectively, expressed by ( l b ) = ~½ Y ,
(lc) = g Y,
(ld) = [1/3(4rO2]eab°°q2qoaa(q)
.
(13,
14, 15)
The combination of these contributions, (11) and (13), (14), can be collected in a covariant and gauge invariant form,
[K/6( 41r)2 ] [eab,,/6 (q2 ~?uu _ q~ q f ) ~ u v ( q l )q2,yaa (q2) -- eab'ra { ql uql ,rcP8u(ql ) -- qluql2t~)6tt(ql )} q~aV(q2)] , (16) and leads to the first and secondterms of the RHS in eq. (1), and the form (15) also corresponds to the third term. Therefore those contributions are consistently connected with our result (1), in which the Riemann-Chfistoffel curvature tensor in the linear approximation is
Ruvoa = ~t~(~v~)orpuo + ~u~)ogb~ --- ~u~od~u~ - ~u~orpz, o) .
(17)
Next, we shall confirm that the Lorentz anomaly vanishes in curved space with torsion (namely, In R i e m a n n Cartan space). The torsion field behaves as the axial U(1) gauge field,
V = 7UDu = ,yu [Ou - ll(¢°abu 1 ~,Xu] =TU(0u - ~ l"W ab uOab + ~' i 7 5 A u ) , " + K ab u)Oab+'~C
(18)
where K'~O u is the contorsion, Ka/3u = ½(C'~O u + COu '~ + Cu/3'~), Cu '~/3 is the torsion field, Cu ~/3 = -Cu/3'~, and A u • . . 1 is the dual torsion field,A u = ieuc~ovC ~ 13 ~. Therefore, the U(1) gauge field a u is replaced by 175Au, in the case of the torsion field. Then the contribution of the type of fig. 1a, (la'), becomes
(la') =-~iKM(2~r)-4fd4p{
[(p - q l ) 2 - M 2 ] (p2 _ M 2) [(p +q2)2 _ MZl }-1
X {Tr[T5oab(p - ql +M)TU(P +M)ToT5(P + q2 +M)] (2p - ql)vqbuv(ql)Ao(q2) + Tr[75°ab(p - ql +M)3'P75(P +M)TU(P +q2 + M ) ] (2p +q2)t'(p,u(q2)Ap(ql)} = O. 156
(19)
Volume 173, number 2
PHYSICS LETTERS B
5 June 1986
The vanishing of ( l a ' ) is verified by transposing the trace parts and changing the integration variables Pu into - P u ' analogous to the proof of Furry's theorem [7]. By a similar direct calculation, the contributions of the type of figs. 1b - l d also disappear. Consequently, there is no Lorentz anomaly in the space with torsion. On the other hand, the Lorentz anomalies with gauge fields in any even-dimensional riemannian space have been discussed by the topological method [5]. In general, if the Dirac operator is I •
V = 7 a e a U ( ~ u -- ~ l w
.ab
uUab
+Au),
(20)
where A u is either an abelian or a non-abelian gauge field, the index theorem for this operator is [8] index X7=
fA(n2) Tr exp(iF/21r).
(21)
By power expanding the integrand in (21), and keeping only the terms that have a (N + 2)-form for a given number N of dimensions, the anomaly with both gauge and gravitational fields is obtained as well as those which have a pure gauge or gravitational effect. In four dimensions, the sixth-order form obtained in the expansion of eq. (21) contains two terms, Tr F 3 and Tr F Tr R 2. The latter is the relevant term for anomalies which come into question, and it is obvious that only the U(1)contribution survives. If we want to transfer from the situation of having a gauge anomaly only to that of having a Lorentz anomaly alone, we may use the appropriate counterterm, IVloc = f c o l ( A , F ) c o 3 ( c o ,
R)
=fA Tr(¢o dw + 3 w 2
3
),
(22)
with the Chern- Simons form. Then the consistent Lorentz anomaly is derived by 80 Wloc = f F
Tr(0"d~).
(23)
Eq. (23) is recast in a covariant tensorial form, 0 abRab °aFU veuvoo .
(24)
At first sight, it seems that the form (24) has nothing to do with our result (1). But if we use the following formula; 1
euvoo6a~ =-~euvoaeab~xe
"),6Kh
1
I,$;,'yf,c;K~. ~,,t,b~t¢~.+pi~f~Kh +~'/6~t¢~. _ t ~ " / 6 ~ K h +
=-~eabKXV" u v " Oa
~UO~vo
~Uo~vO
-pa-Uv
-vo-Up
"/6 Kh
6vO 6Uo)
= Eabpa6~6v -- eabua6~6O + eabvo6~6o + eabuv67o~ -- 6abuo6~v~o + ea~'ouo6v6vo ,
(25)
where~ = 6~'I~ . _ ~6 = 6"Y66 ~.,
the form (24), to which the covariant Lorentz anomaly in the presence of gravity and the U(1) gauge field corresponds, is rewritten as
oab(eaboaRuv°°F uv + eabuvRFUV + 4 e a b v a R u ° F uu) •
(26)
Acting the Hodge-de Rham operator on the field strength 2-forms [9], we have AF = (d6 + 6 d ) F = ( A F ) u v ~1d X 1d A dx v = ( - F u v l o ° - R u o F ° v + R v o F P u + R u v o a F ° a ) l d x U A d x V =
(Fo u lpv)½ dxU
^dxV.
(27)
Hence, the form (26) again changes by this relation,
oab(eab..r6 R F v~ - eabv6 R v6 UVFu v + 2eabv6FV~ I° o + 2eab3, 6 ( A F ) 3"6) .
(28)
Here the eabuv(AF)UV in (28) can be removed by the counterterm eabuvcoab/aFOV I o" Finally, eq. (28) relates consistently to our result (1), contrary to Caneschi and Valtancoli's contention [5]. 157
Volume 173, number 2
PHYSICS LETTERS B
5 June 1986
We may conclude that the results of the covariant Lorentz anomaly by means of the heat kernel, diagrammatical and topological methods are consistent with one another. In four-dimensional riemannian space, the Lorentz anomaly in the presence of a U(1) gauge field is given by the form (1) and no Lorentz anomaly appears, even in the space with torsion.
References [1 ] [2] [3] [4] [5] [6 ] [7] [8] [9]
158
L. Alvarez-Gaum6and E. Witten, Nucl. Phys. B 234 (1983) 269. H.T. Nieh, Phys. Rev. Lett. 53 (1984) 2219. S. Yajima and T. Kimura, Prog. Theor. Phys. 74 (1985) 866. L.N. Chang and H.T. Nieh, Phys. Rev. Lett. 53 (1984) 21. L. Caneschi and P. Valtancoli, Phys. Lett. B 156 (1985) 93. T. Kimura, Prog. Theor. Phys. 42 (1969) 1191. C. Itzykson and J.B. Zuber, Quantum field theory (McGraw Hill, New York, 1980) p. 276. M.F. Atiyah and I.M. Singer, Ann. Math. 87 (1968) 485,546; 83 (1971) 1,119,139. A. Lichn6rowicz, G6om~trie des Groupes de Transformations (Dunod, Paris, 1958).
PHYSICS LETTERS B
5 June 1986
M I X E D L O R E N T Z A N O M A L Y IN F O U R - D I M E N S I O N A L S P A C E S. Y A J I M A and T. K I M U R A Research Institute for Theoretical Physics, Hiroshima Universi(v, Takehara, Hiroshima 725, Japan Received 4 February 1986
By using the diagrammatical and the topological methods, the Lorentz anomaly is evaluated for chiral Weyl fermions interacting with combined gravitational and U(1) gauge fields in four-dimensional space. The results are consistent with that of Nieh in which signs of anomalous terms are taken correctly. The absence of the Lorentz anomaly, in the space-time with torsion, is also explained from the diagrammatical method.
In four-dimensional riemannian space, it is only expected that a gravitational anomaly appears when a U(1) gauge field couples with the gravitational field [ 1]. In fact Nieh has shown that the U(1) gauge field causes the breakdown of general covariance and local Lorentz invariance in gauge theories of chiral Weyl fermions [2]. The Lorentz anomaly is given by (Tab - Tba)= [i/12(4n') 2] [ e a b u v R f uv - eabuvRUVXofxp + 2 e a b u v f u v l x ~] ,
(1)
in which we have used the notation, R U v o o = FUvo,o - FUvo,p - F u o~pP°~vo + PUaoF'~vo ,
R = R uvuv '
f u r = a u , v - av, u .
(2)
The form (1) was presented in our previous paper [3] ,1, and differs from Nieh's result only in the sign of the first term of the RHS. Using the path integral and the expansion coefficient of the heat kernel, we have also found that the Lorentz anomaly vanishes when the torsion field couples with spln-g " 1 Weyl spinors, though the torsion behaves similar to the U(1) gauge field in the action [3]. To explain the origin of the different results of the case o f U(1) gauge field and that of torsion, it is desirable to evaluate the anomaly from another point of view, such as Feynman's diagram method. On the other hand, in this journal, Caneschi and Valtancoli [5] attempted to derive Nieh's result relying upon the topological method. They, however, concluded that they were unable to reproduce the result. We shall here evaluate critically the Lorentz anomaly with the aid of the diagrammatical and the topological methods and show that the results derived are consistent with (1) in which the relative sign o f the first and second terms is taken into account correctly. The peculiarity of the case of torsion is also explained from the diagrammatical method. Einstein anomalies in (4n + 2)-dimensional riemannian space are given by Alvarez-Gaum6 and Witten [ 1]. The anomalous vertex factor of the covariant Lorentz anomaly is obtained analogous to their discussion. We shall consider a massless Weyl spinor in four-dimensional riemannian space, S = f d 4 x ( d e t e b v ) ~1.l ( ~-L T a e a u Duff L - @LDUTaeaUt~L), - ~
(3)
+1 Thel RHS of eq. (1) is four times as large as the result in ref. [3] because the U(1) gauge field coupling was given through (0# + ~iau). It should be remarked that the signs ofRt~vO o and F~v are opposite to Nieh's in refs. [2,4]. 154
Volume 173, number 2
PHYSICS LETTERS B
5 June 1986
where Du = ~
1. ab ~ leo g °ab
-
~L = ~ P + '
,
~--
Du
41. a bta_ U a b , = ~.la + -516o
oab=~i[~/a,Tb ]
,
ffL=p_ff
,
P-+ = 1(1 +- ~'5 ) '
(4)
and 6oabla is the spin connection, the P+ are the projection operators of chirality, and we adopt the metric with signature (+, - , - , - ) . The classical conservation equation implied by local Lorentz invariance is Tab -
Tba= 0 ,
(5)
with Tab =---tab + 2Sab I~ '
in which 1. b u ( ~- 7 a D u P tab =-~le
~ - ~-+DuTaP
~b) ,
Sab t~ = leC u ~ ( O a b 7 c + 3'cOab)P_t~ .
(6)
Now we discuss the one-loop diagrams which contribute to the anomalies by the Pauli-Villars regularization. A massive fermion X is introduced so that the functional measure in the path integral formalism is invariant under the gauge transformations. Then the regularized e n e r g y - m o m e n t r u m tensor is given by (7)
(Tab)reg = Tab - T a b l ¢ ~ x •
Here the Lorentz anomaly appears, because for the massive regulator spinor the e n e r g y - m o m e n t u m tensor with insertion of P _ , Tab [ ¢ ~ x ' is not symmetric under the interchange of the indices a and b even formally. And it is only the regulator diagram that contributes to the anomaly. Note that the factor P_ is not inserted at vertices other than the anomalous vertex that this e n e r g y - m o m e n t u m tensor affects, in calculating the covariant anomalie due to one-loop diagrams. We postulate that the regulator field X obeys the equation of motion (iT'D - 3 4 ) x = 0. The antisymmetric component of the regularized e n e r g y - m o m e n t u m tensor is written as (Tab - Tba)reg =--(tab -- tba + Sab Ulu)i~_~x = _ (tab - t b a ) [ ~ x - l ( D u ~ ) ( Z ~ / t a a a b + [Oab, 7Ul ) P _ x 1
--~X(2OabT u + [ T " , O a b ] ) P _ D u x
1.
--
= ~iMXOab)'5X,
(8)
by using 07"D M)X = 0.We may,therefore, adopt ]i.iMtTab ~[5 as the anomalous vertex factor of the Lorentz anomaly. Anomalous terms are expected to appear as the finite terms in an infinite mass limit (114-+ ,,o) of the calculation of one-loop diagrams. Actually, the covariant Lorentz anomaly in the presence of a U(1) gauge field is derived by the collection of the contributions of diagrams in fig. 1. The lowest order interaction terms, -
Lint = _ ~ T # a # X -- ~n(OUu 1 1. -- l4r 1Uudpr %" f u p f"v O --4U~(~huu
--
12nUu~) r r ) [~TU(0U+ iaU)x _ ~ ( g u _ iaV)3,uX] ,
(9)
give vertices among spinors X, U(1) gauge fields a u and gravitons ~buu, in whichguv = r/u u + K~buu, so ea u = ~au lnOau. For simplicity's sake, we may impose the coordinate condition [6], -
au(~u,
1
Then the contribution from the convergent diagram fig. la, (la), is (1 a ) = --~ i~M(27r) -4
fd4p ( [ ( p
- q l )2 _ M 2 ] (p2 _ M 2) [(p + q2)2 _ 11,/2] }-1
× ( T r [ 3 ' 5 ° a b ( p - ql +M)3'U(P + M ) T ° ( P
+ q2 + M ) ] (2p - ql)udpuu(ql)ao(q2)
+ Tr[75 ° a b ( p -- ql + M)~/°(P + M ) T a ( P + q2 +M)] (2p + q2)UOuv(q2)ao(ql)) _1
1
-gX+i~Y,
(11) 155
Volume 173, number 2
PHYSICS LETTERS B
(a)
(b) -
5 June 1986
(d)
(c)
o.p
Fig. 1. Relevant diagrams contributing to the four-dimensional Lorentz anomaly in the presence of a u(1) gauge field in curved space• in which p = 7Upu, etc., the terms breaking either the gauge invariance or general covariance are neglected after the actual calculation, X and Y are covariant forms: X = [tc/(ar02 ] eabTa (q2rl~V - - q ~ ql)uCuv(ql)q2,raa(q2) ' y = [r/(4rr) 2 ] eab~ '~ [qluql,y~)av(ql) -- qluql,rCau(ql)]
q~aV(q2).
In the evaluation of the contributions from figs• 1 b - 1 d, there appears an integral fd4p(p2
l f ( 1 + A - M2) 2 = d4p (p2 _ )I,/2)2
2A ) (p2 _ M2)3 + "'" '
(12)
where the first term is a logarithmical divergent term and the second term a finite term. We, however, may consider only infite terms, since the logarithmical divergence disappears if we introduce more than two suitable massive regulator fields from the outset. In a similar way to the triangle diagram, the contributions from figs. 1b - 1 d are, respectively, expressed by ( l b ) = ~½ Y ,
(lc) = g Y,
(ld) = [1/3(4rO2]eab°°q2qoaa(q)
.
(13,
14, 15)
The combination of these contributions, (11) and (13), (14), can be collected in a covariant and gauge invariant form,
[K/6( 41r)2 ] [eab,,/6 (q2 ~?uu _ q~ q f ) ~ u v ( q l )q2,yaa (q2) -- eab'ra { ql uql ,rcP8u(ql ) -- qluql2t~)6tt(ql )} q~aV(q2)] , (16) and leads to the first and secondterms of the RHS in eq. (1), and the form (15) also corresponds to the third term. Therefore those contributions are consistently connected with our result (1), in which the Riemann-Chfistoffel curvature tensor in the linear approximation is
Ruvoa = ~t~(~v~)orpuo + ~u~)ogb~ --- ~u~od~u~ - ~u~orpz, o) .
(17)
Next, we shall confirm that the Lorentz anomaly vanishes in curved space with torsion (namely, In R i e m a n n Cartan space). The torsion field behaves as the axial U(1) gauge field,
V = 7UDu = ,yu [Ou - ll(¢°abu 1 ~,Xu] =TU(0u - ~ l"W ab uOab + ~' i 7 5 A u ) , " + K ab u)Oab+'~C
(18)
where K'~O u is the contorsion, Ka/3u = ½(C'~O u + COu '~ + Cu/3'~), Cu '~/3 is the torsion field, Cu ~/3 = -Cu/3'~, and A u • . . 1 is the dual torsion field,A u = ieuc~ovC ~ 13 ~. Therefore, the U(1) gauge field a u is replaced by 175Au, in the case of the torsion field. Then the contribution of the type of fig. 1a, (la'), becomes
(la') =-~iKM(2~r)-4fd4p{
[(p - q l ) 2 - M 2 ] (p2 _ M 2) [(p +q2)2 _ MZl }-1
X {Tr[T5oab(p - ql +M)TU(P +M)ToT5(P + q2 +M)] (2p - ql)vqbuv(ql)Ao(q2) + Tr[75°ab(p - ql +M)3'P75(P +M)TU(P +q2 + M ) ] (2p +q2)t'(p,u(q2)Ap(ql)} = O. 156
(19)
Volume 173, number 2
PHYSICS LETTERS B
5 June 1986
The vanishing of ( l a ' ) is verified by transposing the trace parts and changing the integration variables Pu into - P u ' analogous to the proof of Furry's theorem [7]. By a similar direct calculation, the contributions of the type of figs. 1b - l d also disappear. Consequently, there is no Lorentz anomaly in the space with torsion. On the other hand, the Lorentz anomalies with gauge fields in any even-dimensional riemannian space have been discussed by the topological method [5]. In general, if the Dirac operator is I •
V = 7 a e a U ( ~ u -- ~ l w
.ab
uUab
+Au),
(20)
where A u is either an abelian or a non-abelian gauge field, the index theorem for this operator is [8] index X7=
fA(n2) Tr exp(iF/21r).
(21)
By power expanding the integrand in (21), and keeping only the terms that have a (N + 2)-form for a given number N of dimensions, the anomaly with both gauge and gravitational fields is obtained as well as those which have a pure gauge or gravitational effect. In four dimensions, the sixth-order form obtained in the expansion of eq. (21) contains two terms, Tr F 3 and Tr F Tr R 2. The latter is the relevant term for anomalies which come into question, and it is obvious that only the U(1)contribution survives. If we want to transfer from the situation of having a gauge anomaly only to that of having a Lorentz anomaly alone, we may use the appropriate counterterm, IVloc = f c o l ( A , F ) c o 3 ( c o ,
R)
=fA Tr(¢o dw + 3 w 2
3
),
(22)
with the Chern- Simons form. Then the consistent Lorentz anomaly is derived by 80 Wloc = f F
Tr(0"d~).
(23)
Eq. (23) is recast in a covariant tensorial form, 0 abRab °aFU veuvoo .
(24)
At first sight, it seems that the form (24) has nothing to do with our result (1). But if we use the following formula; 1
euvoo6a~ =-~euvoaeab~xe
"),6Kh
1
I,$;,'yf,c;K~. ~,,t,b~t¢~.+pi~f~Kh +~'/6~t¢~. _ t ~ " / 6 ~ K h +
=-~eabKXV" u v " Oa
~UO~vo
~Uo~vO
-pa-Uv
-vo-Up
"/6 Kh
6vO 6Uo)
= Eabpa6~6v -- eabua6~6O + eabvo6~6o + eabuv67o~ -- 6abuo6~v~o + ea~'ouo6v6vo ,
(25)
where~ = 6~'I~ . _ ~6 = 6"Y66 ~.,
the form (24), to which the covariant Lorentz anomaly in the presence of gravity and the U(1) gauge field corresponds, is rewritten as
oab(eaboaRuv°°F uv + eabuvRFUV + 4 e a b v a R u ° F uu) •
(26)
Acting the Hodge-de Rham operator on the field strength 2-forms [9], we have AF = (d6 + 6 d ) F = ( A F ) u v ~1d X 1d A dx v = ( - F u v l o ° - R u o F ° v + R v o F P u + R u v o a F ° a ) l d x U A d x V =
(Fo u lpv)½ dxU
^dxV.
(27)
Hence, the form (26) again changes by this relation,
oab(eab..r6 R F v~ - eabv6 R v6 UVFu v + 2eabv6FV~ I° o + 2eab3, 6 ( A F ) 3"6) .
(28)
Here the eabuv(AF)UV in (28) can be removed by the counterterm eabuvcoab/aFOV I o" Finally, eq. (28) relates consistently to our result (1), contrary to Caneschi and Valtancoli's contention [5]. 157
Volume 173, number 2
PHYSICS LETTERS B
5 June 1986
We may conclude that the results of the covariant Lorentz anomaly by means of the heat kernel, diagrammatical and topological methods are consistent with one another. In four-dimensional riemannian space, the Lorentz anomaly in the presence of a U(1) gauge field is given by the form (1) and no Lorentz anomaly appears, even in the space with torsion.
References [1 ] [2] [3] [4] [5] [6 ] [7] [8] [9]
158
L. Alvarez-Gaum6and E. Witten, Nucl. Phys. B 234 (1983) 269. H.T. Nieh, Phys. Rev. Lett. 53 (1984) 2219. S. Yajima and T. Kimura, Prog. Theor. Phys. 74 (1985) 866. L.N. Chang and H.T. Nieh, Phys. Rev. Lett. 53 (1984) 21. L. Caneschi and P. Valtancoli, Phys. Lett. B 156 (1985) 93. T. Kimura, Prog. Theor. Phys. 42 (1969) 1191. C. Itzykson and J.B. Zuber, Quantum field theory (McGraw Hill, New York, 1980) p. 276. M.F. Atiyah and I.M. Singer, Ann. Math. 87 (1968) 485,546; 83 (1971) 1,119,139. A. Lichn6rowicz, G6om~trie des Groupes de Transformations (Dunod, Paris, 1958).