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The decoupling theorem and minimal subtraction
Volume 100B, number 5
PHYSICS LETTERS
16 April 1981
THE DECOUPLING THEOREM AND MINIMAL SUBTRACTION* Burt A. OVRUT and Howard J. SCHNITZER Department of Physics, Brandeis University, Waltham, MA 02254, USA Received 21 October 1980 We show to the two-loop level that the decoupling theorem of Appelquist and Carazzone is valid, and a consistent light effective field theory exists, for quantum electrodynamics renormalized by minimal subtraction. It is also shown that irreducible, mixed light-heavy graphs must be included when integrating out the heavy fields to obtain the correct effective lagrangian for light particles.
Appelquist and Carazzone [1 ] have shown that, if there are two widely separated mass scales in a renormalizable field theory, then a consistent effective field theory for the light fields can exist. This "decoupling theorem" has since been extended to many important problems in theoretical physics. During the past year there have been several attempts to provide a systematic construction of effective field theories [2,3]. Among these attemps only those of ref. [3] are based on 't Hooft's minimal subtraction (MS) renormalization scheme [4] (or any of its modifications [5]). As an application of our formalism we have shown [6], that in a scalar field theory with spontaneous symmetry breaking, our methods and the decoupling theorem are valid, to at least the one-loop level. Although these new renormalization schemes may permit decoupling to occur, we claim that decoupling also occurs if minimul subtraction is used. In this paper we extend those results to show that the Appelquist-Carazzone decoupling theorem is valid, and a consistent light effective fieM theory exists in QED, at least to the two-loop level, when a minimal subtraction renormalization scheme is employed/ Consider the quantum electrodynamics of n h "heavy" fermions ffh, each with non-vanishing mass M, and n~ "light" fermions t ~ , each with zero mass. The lagrangian in covariant gauges, in terms of bare fields and parameters, is given by * Research supported in part by the Department of Energy under contract EY-76-5-O2-3230-A002.
"~ = -- ¼ (~u Av - Ov Au)2 - (1/2ct0) (0UA~u)2 + ~h i~¢h - M0t~h Ch +g0t~h'CCh + t~ i~¢~ +go ~ A ' ¢ ~ .
(1)
Summation over n h heavy fermions and n~ light fermions is understood. In all calculations we employ dimensional regularization [7] and minimal subtraction renormalization [4]. Extension of our results to modified MS schemes [5] is straightforward. To begin the construction of the effective field theory, we calculate the photon propagator dressed to the two-loop level. Gauge invariance guarantees that the polarization tensor, Iluv , is transverse, i.e., IIu~ (q) = II (q2) (quq v _ q2guv).
(2)
It follows that the photon propagator, Duv, is [8] - iDu,,(q) = [1 + II(q2)] -1 X ( - g u u + [1 - a ( l + I I ( q 2 ) ) ]
qlaqv/q 2} q - 2 .
(3)
Feynman graphs contributing to the polarization scalar, II(q2), to the two-loop level are shown in fig. 1. The one-loop graphs (A) and (D) are easily evaluated [9] *a, as are the coefficients of the poles associated with the subdivergences in graphs (B), (C), (E) and (F). The coefficient of the pole associated with the overall diver,1 The results of these papers must be modified to include massive fermions. 403
Volume 100B, number 5
PHYSICS LETTERS
16 April 1981
ii(q2)= "--',4
~
~
(A)
(c)
(e)
-
+ (E)
(o)
), __~.__
[ g2 4n
g4
~167r2 3 h
j
photon
_
heavy fermlon
_
light
fermion
Fig. 1. Graphs eontr~uting to the photon polarizationtensor, FII~v,to the two-loop level.
)+
g2 4 g4 4ne l-~n2 5 n~ + - ln-(16~r2) 2 /32
g2 20 161r2 9
n~2+O(q2/M2),
[ g2 4n
g4
fl(q2)=-- ~16~r2 3 la gences of the these graphs can be extracted from eartier work [10]. From these coefficients one can fired the beta function, photon anomalous dimension, and heavy-mass scaling function to the two-loop level. They are g3 4 g5 /3(g)- 16n 2 -~n T + (16rr2) 2 4nT, g2 4 nT + g4 4nT' 3'A(g)- 161r2 3 (167r2)2
'~M(g) =
6g2 16/r2'
(4)
(5)
(9)
(16zr2)2
4nh)lnM2 /22
+O(q2/M2).
(10)
Note that in the limit of vanishing q2, expression (10) is finite, which is not the case for the complete polarization scalar in (9), Let/)uvdenote the part of the photon propagator dressed by graphs involving at least one heavy-fermion line. It follows from eq. (3) that - i / ) v(q ) = [l+I](q2)] -1 X (-guy+ [1-a(l+l~I(q2))]
quqv/q2lq-2.
(11)
Note that in the momentum regime [q21 ~ M 2, dropping terms of O(q2/M2), (6)
respectively, where n T = n~ + n h. The scaling function associated with the gauge parameter o~is given by (g, ct) = --2trfA(g ).
4nh)lnM2 /3-~
where/32 = 4rr exp(--3,E)/a 2, 3'E is Euler's constant and p is 't Hooft's unit of mass. As discussed in refs. [3] and [6], the effective field theory for light particles is obtained by "integrating out" heavy fields from Green's functions describing light-particle processes. The heavy-fermioncontribution to the polarization scalar, which we denote by I], is
(v)
_
(
(167r2)2
(7)
Using the finite parts of graphs (A) and (D), the twoloop renormalization functions (4)-(7), and the fact that FI must satisfy the renormalization group equation
b~zv(q) Iq21~ M2-~[1+I~1(0)]-1Duv(q)e,
(12)
where Duv(q)e is an effective photon propagator, with an effective gauge parameter, given by
-iD,v(q)e = I-guy +(1-ae)quqv/q 2]q-2,
(13)
a e = a [1 +(I(0)l,
(14)
a
(P ~---~+~~g+TMM ~ + ~ ~--27A(g)) (l+II) =0, (8) we find that the renormalized polarization scalar, to the two-loop level for arbitrary space-like momenta q, is
404
167r2 ~ nh
(167r2) 2 4n h In
.
(15)
Given eqs. (12)-(15) we can now construct the effective light-field theory. The insertion of Duv between two light-fermion lines is equivalent, for [q2[ ~ M 2, to a tree graph of effective light fields with effective photon propagator
Volume 100B, number 5
PHYSICS LETTERS
Duv(q)e given by (13), and an effective coupling parameterg e defined by ,2 ge = [1 + fI(0)] -1/2g.
(16)
From this result it can be shown [11 ] that the effective fight-field lagrangian, valid for fight-particle processes up to the two-loop level, with external momenta q such that [q21 ,~M 2, is given by
16 April 1981
The total derivative o f g e with respect to Atis given by
dge - (At ~# + ~ 0g O-~-+ TM M - ~0 ) g e.
(21)
At-d--p---
Inserting ( 4 ) - ( 6 ) and (20) into (21"), we find that, to order gS,
dge=[ g3 4 n + At ~
g~5
4n~)
.~167r2 3 Q (167r2)2
.tge = _ 1 (~u Ave - Ov'4"ue)2 - (1/2t~e) (~UAue)2 g5
+ t~e i~0~e + g e ~ e ~ ( ~ e ,
(17)
where
Ave = [1 +[I(0)] 1/2Au,
(18)
~)~e = ~ ,
(19)
with Cte, 1](0) andg e given by (14), (15) and (16), respectively. Summation over n~ fight effective fermions in (17) is understood, and terms of O(q2/M 2) are ignored. The validity of effective lagrangian (17) can be explicitly checked by considering the scaling properties of its effective fields and parameters under the renormalization group. If (17) is correct, these quantities must scale according to "decoupled" renormalization group equations derived from the renormalization of purely effective light-field graphs. Using the renormalization group functions ( 4 ) - ( 7 ) , and the explicit expressions (14) and (16) for the effective parameters, we shall verify that the necessary "decoupling" takes place. First, consider the effective coupling parameter. Using (15) and (16) we find that this parameter is given, to order gS, by
ge =g + ~ g3 2 \167r2-3 nh 3 5(
g5
)
M2
(167r2)2 2nh h a ~ -
nh 41nM212
(22)
It might appear that the term proportional to nhn~, which is due to the non-cancellation of the two-loop (In) 2 term, destroys decoupling and invalidates the effective lagrangian. However, it is important to remem-
ber that the beta function associated with the effective coupling parameter must be expressed in terms of ge and not g.t Inverting eq. (20) we find that g = ge
ge3 n h In 342 + O(g5). 247r2 /i2
(23)
Substituting this expression for g into (22) we find that the unwanted term cancels, and that
At dge/dAt =/3e(ge) , /3e(ge)= ge3 4
+ - - gSe
(16rr2) 2 4n~,
(24)
which is the correct, two-loop, decoupled beta function for light fields. If one did not express (22) in terms ofge, one would erroneously believe that there is an anomalous two-loop logarithm, In M2]~ 2, which destroys the decoupling behavior for minimal subtraction. As demonstrated, this belief is absolutely false! Now consider the effective gauge parameter. Using (14) and (15) we find that, to order g 4,
(20)
,2 We note that ge is the "invariant charge" Green's function associated with the heavy-fermioncontribution to the f f ~ A u vertex, It has been rendered finite using pure minimal subtraction renormalization. This is the sense in which we use the term minimal subtraction. An alternative view is to demand that ge be the renormalized parameter of some suitable (clearly not MS) renormalization scheme. Such a scheme is probably equivalent to momentum subtraction at q2 = 0. These two views lead to identical physics.
M2
+ 9- -6 ~ nhn ~ In -t12 - .
5nh
(16 2)24nhlhaM2]'I i2J (25)
The total derivative of c% with respect to Atis given by d°e(~
Ate=
0
~
At~+fl~-gg+TMM~-ffI+6
~)
a e.
(26)
Inserting ( 4 ) - ( 7 ) and (25) into (26) we find that, to order g4, do¢e = _ + g4 2t~[ g2 4 n ~ At-d~-At \167r 2 3 (~67r2)2 4n~].
(27) 405
Volume 100B, number 5
PHYSICS LETTERS
Again, it is important to remember that the anomalous dimension associated with the effective photon fieM must be expressed in terms o f g e and a e and not g and a! Using (23) we can invert (25), and find g2 4 lnM " Z2~+-o9( g a )e nh 16rr2 3 /j-
O~ = O~e + Ote - -
(28)
Substituting (23) and (28) into (27), it follows that da e /a ~ - # = - 2 c t e ~'Ae(ge), g2
4
g4
~'Ae(ge) = 16rr2-~ n~ + (16n2) - ~
4n~,
(29)
which is the correct, two-loop, decoupled light anomalous dimension. We conclude from eqs. (24) and (29) that the Appelquist-Carazzone decoupling theorem is valid, and a consistent light effective field theory exists, in QED at least to the two-loop level when a minimal subtraction scheme is employed. Note that the irreducible, mixed light-heavy graphs (E) and (F) o f fig. (1) must be included when integrating out the heavy fieMs. Failure to include such terms in fI would change the definition ofge, and (24) would be altered to read ~t dge/d/a =/3(ge) - g5 nh/64rt2 '
(30)
which would invalidate the decoupling theorem. Similar results are valid in QCD *a ¢3 This follows from the results of refs. [3] and [6] above. A more detailed discussion for these issues and QCD will be presented in a subsequent publication.
406
16 April 1981
It is a pleasure to acknowledge Professor L. Abbott for many enlightening discussions.
References [1] T. Appelquist and J. Carazzone, Phys. Rev. Dll (1975) 2856. [2] C.K. Lee, Nucl. Phys. B161 (1979) 171; Y. Kazama and Y.-P. Yao, Phys. Rev. Lett. 43 (1979) 1562; N.P. Chang et al., City College preprints (1979). [3] B. Ovrut and H. Schnitzer, Phys. Rev. D21 (1980) 3369; S. Weinberg, Phys. Lett. 91B (1980) 51; P. Bin6truy and T. Schiicker, CERN preprints Ref. TH 2802, 2857. [4] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [5] J.C. Collins and A.J. Macfarlane, Phys. Rev. D10 (1974) 1201; A.J. Buras, Rev. Mod. Phys. 52 (1980) 218. [6] B. Ovrut and H. Schnitzer, Phys. Rev. D22 (1980) 2518. [7] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [8] See, for example, J.D. Bjorken and S.D. Drell, Relativistic quantum fields (McGraw-Hill, New York) pp. 302303. [9] P. Majumdar et al., Pl~ys. Rev. D21 (1980) 2203; W. Celmaster and R. Gonsalves, Phys. Rev. D20 (1979) 1420. [10] J. Rosner, Ann. Phys. 44 (1967) 11.
PHYSICS LETTERS
16 April 1981
THE DECOUPLING THEOREM AND MINIMAL SUBTRACTION* Burt A. OVRUT and Howard J. SCHNITZER Department of Physics, Brandeis University, Waltham, MA 02254, USA Received 21 October 1980 We show to the two-loop level that the decoupling theorem of Appelquist and Carazzone is valid, and a consistent light effective field theory exists, for quantum electrodynamics renormalized by minimal subtraction. It is also shown that irreducible, mixed light-heavy graphs must be included when integrating out the heavy fields to obtain the correct effective lagrangian for light particles.
Appelquist and Carazzone [1 ] have shown that, if there are two widely separated mass scales in a renormalizable field theory, then a consistent effective field theory for the light fields can exist. This "decoupling theorem" has since been extended to many important problems in theoretical physics. During the past year there have been several attempts to provide a systematic construction of effective field theories [2,3]. Among these attemps only those of ref. [3] are based on 't Hooft's minimal subtraction (MS) renormalization scheme [4] (or any of its modifications [5]). As an application of our formalism we have shown [6], that in a scalar field theory with spontaneous symmetry breaking, our methods and the decoupling theorem are valid, to at least the one-loop level. Although these new renormalization schemes may permit decoupling to occur, we claim that decoupling also occurs if minimul subtraction is used. In this paper we extend those results to show that the Appelquist-Carazzone decoupling theorem is valid, and a consistent light effective fieM theory exists in QED, at least to the two-loop level, when a minimal subtraction renormalization scheme is employed/ Consider the quantum electrodynamics of n h "heavy" fermions ffh, each with non-vanishing mass M, and n~ "light" fermions t ~ , each with zero mass. The lagrangian in covariant gauges, in terms of bare fields and parameters, is given by * Research supported in part by the Department of Energy under contract EY-76-5-O2-3230-A002.
"~ = -- ¼ (~u Av - Ov Au)2 - (1/2ct0) (0UA~u)2 + ~h i~¢h - M0t~h Ch +g0t~h'CCh + t~ i~¢~ +go ~ A ' ¢ ~ .
(1)
Summation over n h heavy fermions and n~ light fermions is understood. In all calculations we employ dimensional regularization [7] and minimal subtraction renormalization [4]. Extension of our results to modified MS schemes [5] is straightforward. To begin the construction of the effective field theory, we calculate the photon propagator dressed to the two-loop level. Gauge invariance guarantees that the polarization tensor, Iluv , is transverse, i.e., IIu~ (q) = II (q2) (quq v _ q2guv).
(2)
It follows that the photon propagator, Duv, is [8] - iDu,,(q) = [1 + II(q2)] -1 X ( - g u u + [1 - a ( l + I I ( q 2 ) ) ]
qlaqv/q 2} q - 2 .
(3)
Feynman graphs contributing to the polarization scalar, II(q2), to the two-loop level are shown in fig. 1. The one-loop graphs (A) and (D) are easily evaluated [9] *a, as are the coefficients of the poles associated with the subdivergences in graphs (B), (C), (E) and (F). The coefficient of the pole associated with the overall diver,1 The results of these papers must be modified to include massive fermions. 403
Volume 100B, number 5
PHYSICS LETTERS
16 April 1981
ii(q2)= "--',4
~
~
(A)
(c)
(e)
-
+ (E)
(o)
), __~.__
[ g2 4n
g4
~167r2 3 h
j
photon
_
heavy fermlon
_
light
fermion
Fig. 1. Graphs eontr~uting to the photon polarizationtensor, FII~v,to the two-loop level.
)+
g2 4 g4 4ne l-~n2 5 n~ + - ln-(16~r2) 2 /32
g2 20 161r2 9
n~2+O(q2/M2),
[ g2 4n
g4
fl(q2)=-- ~16~r2 3 la gences of the these graphs can be extracted from eartier work [10]. From these coefficients one can fired the beta function, photon anomalous dimension, and heavy-mass scaling function to the two-loop level. They are g3 4 g5 /3(g)- 16n 2 -~n T + (16rr2) 2 4nT, g2 4 nT + g4 4nT' 3'A(g)- 161r2 3 (167r2)2
'~M(g) =
6g2 16/r2'
(4)
(5)
(9)
(16zr2)2
4nh)lnM2 /22
+O(q2/M2).
(10)
Note that in the limit of vanishing q2, expression (10) is finite, which is not the case for the complete polarization scalar in (9), Let/)uvdenote the part of the photon propagator dressed by graphs involving at least one heavy-fermion line. It follows from eq. (3) that - i / ) v(q ) = [l+I](q2)] -1 X (-guy+ [1-a(l+l~I(q2))]
quqv/q2lq-2.
(11)
Note that in the momentum regime [q21 ~ M 2, dropping terms of O(q2/M2), (6)
respectively, where n T = n~ + n h. The scaling function associated with the gauge parameter o~is given by (g, ct) = --2trfA(g ).
4nh)lnM2 /3-~
where/32 = 4rr exp(--3,E)/a 2, 3'E is Euler's constant and p is 't Hooft's unit of mass. As discussed in refs. [3] and [6], the effective field theory for light particles is obtained by "integrating out" heavy fields from Green's functions describing light-particle processes. The heavy-fermioncontribution to the polarization scalar, which we denote by I], is
(v)
_
(
(167r2)2
(7)
Using the finite parts of graphs (A) and (D), the twoloop renormalization functions (4)-(7), and the fact that FI must satisfy the renormalization group equation
b~zv(q) Iq21~ M2-~[1+I~1(0)]-1Duv(q)e,
(12)
where Duv(q)e is an effective photon propagator, with an effective gauge parameter, given by
-iD,v(q)e = I-guy +(1-ae)quqv/q 2]q-2,
(13)
a e = a [1 +(I(0)l,
(14)
a
(P ~---~+~~g+TMM ~ + ~ ~--27A(g)) (l+II) =0, (8) we find that the renormalized polarization scalar, to the two-loop level for arbitrary space-like momenta q, is
404
167r2 ~ nh
(167r2) 2 4n h In
.
(15)
Given eqs. (12)-(15) we can now construct the effective light-field theory. The insertion of Duv between two light-fermion lines is equivalent, for [q2[ ~ M 2, to a tree graph of effective light fields with effective photon propagator
Volume 100B, number 5
PHYSICS LETTERS
Duv(q)e given by (13), and an effective coupling parameterg e defined by ,2 ge = [1 + fI(0)] -1/2g.
(16)
From this result it can be shown [11 ] that the effective fight-field lagrangian, valid for fight-particle processes up to the two-loop level, with external momenta q such that [q21 ,~M 2, is given by
16 April 1981
The total derivative o f g e with respect to Atis given by
dge - (At ~# + ~ 0g O-~-+ TM M - ~0 ) g e.
(21)
At-d--p---
Inserting ( 4 ) - ( 6 ) and (20) into (21"), we find that, to order gS,
dge=[ g3 4 n + At ~
g~5
4n~)
.~167r2 3 Q (167r2)2
.tge = _ 1 (~u Ave - Ov'4"ue)2 - (1/2t~e) (~UAue)2 g5
+ t~e i~0~e + g e ~ e ~ ( ~ e ,
(17)
where
Ave = [1 +[I(0)] 1/2Au,
(18)
~)~e = ~ ,
(19)
with Cte, 1](0) andg e given by (14), (15) and (16), respectively. Summation over n~ fight effective fermions in (17) is understood, and terms of O(q2/M 2) are ignored. The validity of effective lagrangian (17) can be explicitly checked by considering the scaling properties of its effective fields and parameters under the renormalization group. If (17) is correct, these quantities must scale according to "decoupled" renormalization group equations derived from the renormalization of purely effective light-field graphs. Using the renormalization group functions ( 4 ) - ( 7 ) , and the explicit expressions (14) and (16) for the effective parameters, we shall verify that the necessary "decoupling" takes place. First, consider the effective coupling parameter. Using (15) and (16) we find that this parameter is given, to order gS, by
ge =g + ~ g3 2 \167r2-3 nh 3 5(
g5
)
M2
(167r2)2 2nh h a ~ -
nh 41nM212
(22)
It might appear that the term proportional to nhn~, which is due to the non-cancellation of the two-loop (In) 2 term, destroys decoupling and invalidates the effective lagrangian. However, it is important to remem-
ber that the beta function associated with the effective coupling parameter must be expressed in terms of ge and not g.t Inverting eq. (20) we find that g = ge
ge3 n h In 342 + O(g5). 247r2 /i2
(23)
Substituting this expression for g into (22) we find that the unwanted term cancels, and that
At dge/dAt =/3e(ge) , /3e(ge)= ge3 4
+ - - gSe
(16rr2) 2 4n~,
(24)
which is the correct, two-loop, decoupled beta function for light fields. If one did not express (22) in terms ofge, one would erroneously believe that there is an anomalous two-loop logarithm, In M2]~ 2, which destroys the decoupling behavior for minimal subtraction. As demonstrated, this belief is absolutely false! Now consider the effective gauge parameter. Using (14) and (15) we find that, to order g 4,
(20)
,2 We note that ge is the "invariant charge" Green's function associated with the heavy-fermioncontribution to the f f ~ A u vertex, It has been rendered finite using pure minimal subtraction renormalization. This is the sense in which we use the term minimal subtraction. An alternative view is to demand that ge be the renormalized parameter of some suitable (clearly not MS) renormalization scheme. Such a scheme is probably equivalent to momentum subtraction at q2 = 0. These two views lead to identical physics.
M2
+ 9- -6 ~ nhn ~ In -t12 - .
5nh
(16 2)24nhlhaM2]'I i2J (25)
The total derivative of c% with respect to Atis given by d°e(~
Ate=
0
~
At~+fl~-gg+TMM~-ffI+6
~)
a e.
(26)
Inserting ( 4 ) - ( 7 ) and (25) into (26) we find that, to order g4, do¢e = _ + g4 2t~[ g2 4 n ~ At-d~-At \167r 2 3 (~67r2)2 4n~].
(27) 405
Volume 100B, number 5
PHYSICS LETTERS
Again, it is important to remember that the anomalous dimension associated with the effective photon fieM must be expressed in terms o f g e and a e and not g and a! Using (23) we can invert (25), and find g2 4 lnM " Z2~+-o9( g a )e nh 16rr2 3 /j-
O~ = O~e + Ote - -
(28)
Substituting (23) and (28) into (27), it follows that da e /a ~ - # = - 2 c t e ~'Ae(ge), g2
4
g4
~'Ae(ge) = 16rr2-~ n~ + (16n2) - ~
4n~,
(29)
which is the correct, two-loop, decoupled light anomalous dimension. We conclude from eqs. (24) and (29) that the Appelquist-Carazzone decoupling theorem is valid, and a consistent light effective field theory exists, in QED at least to the two-loop level when a minimal subtraction scheme is employed. Note that the irreducible, mixed light-heavy graphs (E) and (F) o f fig. (1) must be included when integrating out the heavy fieMs. Failure to include such terms in fI would change the definition ofge, and (24) would be altered to read ~t dge/d/a =/3(ge) - g5 nh/64rt2 '
(30)
which would invalidate the decoupling theorem. Similar results are valid in QCD *a ¢3 This follows from the results of refs. [3] and [6] above. A more detailed discussion for these issues and QCD will be presented in a subsequent publication.
406
16 April 1981
It is a pleasure to acknowledge Professor L. Abbott for many enlightening discussions.
References [1] T. Appelquist and J. Carazzone, Phys. Rev. Dll (1975) 2856. [2] C.K. Lee, Nucl. Phys. B161 (1979) 171; Y. Kazama and Y.-P. Yao, Phys. Rev. Lett. 43 (1979) 1562; N.P. Chang et al., City College preprints (1979). [3] B. Ovrut and H. Schnitzer, Phys. Rev. D21 (1980) 3369; S. Weinberg, Phys. Lett. 91B (1980) 51; P. Bin6truy and T. Schiicker, CERN preprints Ref. TH 2802, 2857. [4] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [5] J.C. Collins and A.J. Macfarlane, Phys. Rev. D10 (1974) 1201; A.J. Buras, Rev. Mod. Phys. 52 (1980) 218. [6] B. Ovrut and H. Schnitzer, Phys. Rev. D22 (1980) 2518. [7] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [8] See, for example, J.D. Bjorken and S.D. Drell, Relativistic quantum fields (McGraw-Hill, New York) pp. 302303. [9] P. Majumdar et al., Pl~ys. Rev. D21 (1980) 2203; W. Celmaster and R. Gonsalves, Phys. Rev. D20 (1979) 1420. [10] J. Rosner, Ann. Phys. 44 (1967) 11.