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One-loop correction to vector boson masses in the Glashow-Weinberg-Salam model of electromagnetic and weak interactions
Volume 91B, number 1
PHYSICS LETTERS
24 March 1980
ONE-LOOP CORRECTION TO VECTOR BOSON MASSES IN THE GLASHOWWEINBERG-SALAM MODEL OF ELECTROMAGNETIC AND WEAK INTERACTIONS Francesco ANTONELLI a, Maurizio CONSOLI b and Guido CORBO a a Istituto di I.)'sica G. Marconi, Universit~ di Roma, Rome, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy b Istituto di Fisica, UniversiM di Catania, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Italy Received 11 January 1980
We present the computation of one-loop corrections to the vector boson masses in the Glashow-Weinberg-Salam model. We find a significant increase with respect to the values given by the tree approximation, for a reasonable range of the weak mixing angle.
In the G l a s h o w - W e i n b e r g - S a l a m model of the electromagnetic and weak interactions [1], all the physical quantities can be connected to three parameters of low-energy phenomenology: the Fermi coupling constant GF, the fine-structure constant a and the ~ parameter defined, at the tree level, as: o ( ~ e ~ u e ) / o ( v u e ~ rue ) = (~2 _ ~ + 1)/(~2 + ~ + 1).
(1)
The three independent parameters o f the model, namely the SU(2) coupling constant g, the charged vector boson mass M and the sine of the weak mixing angle s w, can be preliminarily fixed by the zeroth order relations: 2
S w
=
I( 1
__
~)
,
ot = gZs2/47r,
GF = g 2 / 8 M 2
(GF = G F / ~ / 2 ) ,
(2)
where ~ is the ratio a/b, with a and b given by the lagrangian coupling of the electron with the neutral vector boson, £int=(ig/4cw)WOg3,u(a + b T 5 ) e ,
a=4s 2-
1,
b=-m.
(3)
In terms of these quantities, the masses of the intermediate vector bosons are given (at the tree level) by: M 2=
4e2 8(1 - ~)GF
,
Mg=
4e2
(4)
8(1 - ~)GF [1 -- 1(1 -- ~)]
In this paper we report on the results o f a calculation Of one-loop corrections to relation (4). The fact that these corrections turn out to be substantial is interesting both from an experimental point of view (for the choice of the energy of the future e+e - machines) and from a physical standpoint, since a measurement of these masses would allow a test of the theory beyond the tree level. We have proceeded as follows: we considered three independent low-energy processes which can be used to define the three parameters of the theory, namely muon decay (to define the Fermi constant), Coulomb scattering of muons on electrons (to define a) and a typical neutral current process, i.e. rue scattering (to define the weak mixing angle 0w). The relations between the observed measured quantities and the basic parameters are themselves affected b~higher-order corrections; therefore, in order to obtain, from the measured quantities, a prediction for M 2 and M 0 , including corrections o f order g 2 , we have to compute corrections both to the chosen processes and to the vector boson propagators. 90
Volume 91B, number 1
PHYSICS LETTERS
24 March 1980
Let us first consider the corrections to muon decay. We can divide them into two parts. The first one is equivalent to a shift in the value of the Fermi coupling constant GF,
d F = (g2/8M2)(1 + 61),
(5)
and it does not depend on the kinematics of the process; the second one coincides with the standard (ref. [2]) radiative corrections to the process described by Fermi pointlike interaction. This separation requires a careful treatment of diagrams containing photon internal lines and can be done following, e.g., ref. [3]. What we compute here are the corrections of the first kind: the value of GF can be directly obtained from experimental data (muon lifetime) taking into account corrections of the second kind. A similar separation can be made for the neutral current process, rue scattering * 1 The corrections of the first kind consist in a shift of the effective Fermi constant and a shift of the value s 2 given by: S2w = 1(1 - ~)(1 + 53).
(6)
can be obtained by extrapolating the slope of the ratio o~ve/o rue at zero energy after subtraction of QED corrections, similarly to what is done for muon decay. Finally, the fine-structure constant a can be defined as the residue of the one-photon exchange pole in Coulomb scattering (e/a -+ e/a); the corresponding shift in a,
Ot= (g2s2 /47r)(1 + 62) ,
(7)
can he obtained isolating the pole terms in one-loop corrections to e/a -+ e/a. From the above discussion, it follows that the location of the pole, in the charged vector boson propagator, will be: 4e 2 3//2=-
( 1 + 5 1 6Z - 6 3 ) - H + ( M 2 ) , 8(1 - ~ ) d F and analogously for the neutral vector boson:
(8)
4e2 [ 1 + 5 1 - 5 2 - 6 3 + ( s w /2c w )26 3 ] Ho(-Mg), (9) 8(1 - ~)d F [ 1 - I(1 - ~)1 where "H = II/(27r)4i and the II's are the self-energy contributions. In the one-loop corrections (5's and self-energies), we use the lowest-order expressions (2) and (4) since we are working up to order g2 only. The cancellation between the ultraviolet divergences appearing in the 6's and in the self-energies will be shown elsewhere, together with the presentation of the details of the calculation. Here, we only report on the results:
Mg=
2142 =
= ~2
___ . [ 1 -- - -g2 t,~u xleptons 4e2 M + s;quarks+6~/+6]IWI)I vM 8(1 -- ~) GF 167r2
(10)
[ 1 -- g2 ,xleptonsa. 4e2 t~Mo " xquarks. ~Mo * 6Ho +6Wo) ] ' 8(1 - ~ ) G F F Z ~(1-- ~)] 167r 2
where ~leptons, 6quarks and 6 H are the contributions to the renormalization of the vector boson masses arising from leptons, quarks and the Higgs particle, respectively, while the 6W's are the remaining contributions (box diagrams, finite parts of vertex and self-energy corrections). The most important contributions come from fermions, as one could deduce by using renormalization group arguments. We find, for a typical lepton and quark doublet: +1 This is possible because, in both cases, electromagnetic radiative corrections to Fermi pointlike interactions are ultraviolet convergent. 91
Volume 91B, number 1
PHYSICS LETTERS
1eptons_ , 2 /m~) 2 - si] M - -~[ln(M
24 March 1980
xquarks "M = --~1 [ln(M2/m2) + 2 ln(M2/m 2 -- 51
,
(11)
•
A factor 3 has been introduced to take into account the color degrees of freedom of quarks. Finite corrections, proportional to the ratio m2/M 2 have been ignored everywhere. For the neutral vector boson, we find analogously: xleptons 2 -1 4 2 2 2 = --(3Cw) (4S w - 2Sw + 1) [ln(M0/m~) - s] , VMo 6quarks r~ 2 . - l r . 1 6 4 2 ' 2 2 4 4 2 Mo = --I.JCw) lU~Sw -- 4s w + 2) In(M6/m u) + (gSw - 2s w + l)
(12)
hl(M2/m2) -
1(710 4 + 10S2 - 5] -~sw
(13)
The contribution of light fermions is quite large and it increases the boson masses. From the Higgs particle we have:
1 m2 6H = ~ ~
3
m2
M2
1 m2
-+H(M), 4 m2H- --- M 2 l n -m~H
8~o-~-
3
~
m2
M2+7~H(Mo)
gm~TM21n mH 2
cw
'
(14)
where the first two terms come from the self-energy corrections to muon decay and the function H(M) comes from the vector boson self-energies on the mass shell, 31 1 m4 H(M) =-i-~ + 12 M 4
1 m2 + ( 1 m4 2 M2 12 M 4 (
1 m6+__ 1 m4 12 M 6 3 M 4
+ 0(4M2 - m2)
( + O(m2 -
l m6+ lm4
4M2)
12 M 6
3 M4
1 m2
,
m2
4M 2 )m~m2 1 [ l l n m2
M2 ]
-~+\
(4M2~m2) 1/2 m2 arctg
(~4M 2\\ m2m2Hll/2/.j
(15,
m2]{M2 ! -~ \ In ~m2H 1- - ~l(m2--4M211/2~HH ! In l l + [[(m ( m 2 - 4 M4M2)/m 2 ) / m 2 ]2l /]1/2 2}--
In the limiting case of very large and very small Higgs mass, we have: 65
1 m2
H(M) m~>>MZ72 H(M)
~-
31
m2
8M2
51n m2 ~:~ 6
1 M2
1M 2
6m2
3m2
1 m2 2M 2
3 m2 1 m2H 1 M \ mH 4M 2 n~-~---2~---arctg~)2~H H- .
In
m2 H M2 '
(16)
As one can see, the terms proportional to the ratio m2/M 2 exactly cancel, in the limit of very large Higgs mass, the corresponding ones in eq. (14). This is exactly the screening theorem of ref. [4],; No singularity arises in the limit of very small Higgs mass. The expressions for 6~/, 8W° can be decomposed as follows: = R M + IIW , R1 = - 6 - 6 c 2
RM=R1-R2-R3, M2
M20_M2
lnM2+c2wlnM2+l 7 M2 M2 4 1
+(4S4w- 20s 2 + 10) 4s--~wlnMM~,
92
(17)
8<
In--
2+
Jw
-
(18a)
Volume 91B, number 1
R2=--3
22 sw'
2 2 R3=-3Cw-3
1 +
4s2 -1F --L-4C2 4s 2
4C2 3 (4S2w 8S4)7 +-] ' 4s 2 - 1 4c 2
W
[56 2 llW = ~ 2 [-~-Sw - - - +
1 12c2
+1 c4
W
s2-s+
W
24c 2
W
1 [_4s24 c2\
25 3c 2
24 March 1980
PHYSICS LETTERS
2 29 Sw 3 c2
Analogously we find: 2 2
6w° =RM + (Sw/Cw)R3 + Hw
(18b,c)
W
ln-M2
W
1
)
( 4M2
12C2w \
M02 1/2 arctg ||-[[4M2--7 M 2 ) 1/2 )
1
M2
!
\\
-
(19)
M(~
'
(20)
2 w 2 +17 1 13 In M~22 + 22 - 58 2 + 8SwC 12c 2 9-3 -cw 36 c 2
ii w _
(4M2-M2) 1/2 + \
M2
((4M2-M2)l/2)I arctg \ \
M~
4+29 2 2 -- 1 ) . \ - ~- ~ - c2w - 4SwCw 12c2w
(21)
In table 1 we show the uncorrected and corrected values of the vector boson masses for a range of S2w. The main theoretical uncertainty in our O(g 2) results lies in the choice of the light quark (u, d, s) masses. In table 1 we have used the values suggested by current algebra [5]. The uncertainties coming from the choice of the presently unknown t and H masses are very small. We note that the leading O(g 4) corrections, proportional to ln2(Mw/mf) can be easily derived, using renormalization group techniques, starting from the results given here. We have been informed that similar calculations have been done independently by M. Veltman. The authors are deeply indebted to Profs. N. Cabibbo and L. Maiani for their encouragement and invaluable help.
Table 1 Uncorrected (M, Mo) and corrected (M, Mo) values of the vector boson masses. Masses are given in GeV. For quarks we have used the following values: m u = 0.01, m d = 0.01, m c = 1.75, m s = 0.25, m t = 20, m b = 15. We have also assumed m H = 10 [6]. For the Fermi coupling constant we assume the value [7]: GF = 1.1663 X 10 -5 GeV -2 . S2w
M
~r
Mo
-Mo
0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27
85.529 83.363 81.354 79.484 77.737 76.100 74.562 73.114 71.748
90.418 87.886 85.554 83.398 81.395 79.530 77.787 76.153 74.618
95.032 93.203 91.531 89.998 88.589 87.293 86.097 84.994 83.974
100.202 98.028 96.053 94.255 92.613 91.110 89.731 88.465 87.300
93
Volume 91B, number 1
PHYSICS LETTERS
References [1] S. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Proc. 8th Nobel Syrup. (Almqvist and Wiksell, Stockholm, 1968); S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [2] R.E. Behrends, R.J. Finkelstein and A. Sirlin, Phys. Rev. 101 (1956) 866; S.M. Berman, Phys. Rev. 112 (1958) 267; T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652. [3] A. Ross, Nucl. Phys. B51 (1973) 116; A. Sirlin, Nucl. Phys. B71 (1974) 29. [4] M. Veltman, Acta Phys. Pol. B8 (1977) 475; Phys. Lett. 70B (1977) 253. [5] H. Leutwyler, Phys. Lett. 48B (1974) 45; M. Testa, Phys. Lett. 56B (1975) 53; S. Weinberg, Rabi Festschrift (New York Academy of Science). [6] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; S. Weinberg, Phys. Rev. D7 (1973) 2887; J. Ellis, M.K. Galliard, D.V. Nanopoulos and C.T. Sachrajda, CERN preprint TH. 2634. [7] D.H. Wilkinson, Nature 257 (1975) 189; D.H. Wilkinson and D.E. Alburger, Phys. Rev. C13 (1976) 2517.
94
24 March 1980
PHYSICS LETTERS
24 March 1980
ONE-LOOP CORRECTION TO VECTOR BOSON MASSES IN THE GLASHOWWEINBERG-SALAM MODEL OF ELECTROMAGNETIC AND WEAK INTERACTIONS Francesco ANTONELLI a, Maurizio CONSOLI b and Guido CORBO a a Istituto di I.)'sica G. Marconi, Universit~ di Roma, Rome, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy b Istituto di Fisica, UniversiM di Catania, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Italy Received 11 January 1980
We present the computation of one-loop corrections to the vector boson masses in the Glashow-Weinberg-Salam model. We find a significant increase with respect to the values given by the tree approximation, for a reasonable range of the weak mixing angle.
In the G l a s h o w - W e i n b e r g - S a l a m model of the electromagnetic and weak interactions [1], all the physical quantities can be connected to three parameters of low-energy phenomenology: the Fermi coupling constant GF, the fine-structure constant a and the ~ parameter defined, at the tree level, as: o ( ~ e ~ u e ) / o ( v u e ~ rue ) = (~2 _ ~ + 1)/(~2 + ~ + 1).
(1)
The three independent parameters o f the model, namely the SU(2) coupling constant g, the charged vector boson mass M and the sine of the weak mixing angle s w, can be preliminarily fixed by the zeroth order relations: 2
S w
=
I( 1
__
~)
,
ot = gZs2/47r,
GF = g 2 / 8 M 2
(GF = G F / ~ / 2 ) ,
(2)
where ~ is the ratio a/b, with a and b given by the lagrangian coupling of the electron with the neutral vector boson, £int=(ig/4cw)WOg3,u(a + b T 5 ) e ,
a=4s 2-
1,
b=-m.
(3)
In terms of these quantities, the masses of the intermediate vector bosons are given (at the tree level) by: M 2=
4e2 8(1 - ~)GF
,
Mg=
4e2
(4)
8(1 - ~)GF [1 -- 1(1 -- ~)]
In this paper we report on the results o f a calculation Of one-loop corrections to relation (4). The fact that these corrections turn out to be substantial is interesting both from an experimental point of view (for the choice of the energy of the future e+e - machines) and from a physical standpoint, since a measurement of these masses would allow a test of the theory beyond the tree level. We have proceeded as follows: we considered three independent low-energy processes which can be used to define the three parameters of the theory, namely muon decay (to define the Fermi constant), Coulomb scattering of muons on electrons (to define a) and a typical neutral current process, i.e. rue scattering (to define the weak mixing angle 0w). The relations between the observed measured quantities and the basic parameters are themselves affected b~higher-order corrections; therefore, in order to obtain, from the measured quantities, a prediction for M 2 and M 0 , including corrections o f order g 2 , we have to compute corrections both to the chosen processes and to the vector boson propagators. 90
Volume 91B, number 1
PHYSICS LETTERS
24 March 1980
Let us first consider the corrections to muon decay. We can divide them into two parts. The first one is equivalent to a shift in the value of the Fermi coupling constant GF,
d F = (g2/8M2)(1 + 61),
(5)
and it does not depend on the kinematics of the process; the second one coincides with the standard (ref. [2]) radiative corrections to the process described by Fermi pointlike interaction. This separation requires a careful treatment of diagrams containing photon internal lines and can be done following, e.g., ref. [3]. What we compute here are the corrections of the first kind: the value of GF can be directly obtained from experimental data (muon lifetime) taking into account corrections of the second kind. A similar separation can be made for the neutral current process, rue scattering * 1 The corrections of the first kind consist in a shift of the effective Fermi constant and a shift of the value s 2 given by: S2w = 1(1 - ~)(1 + 53).
(6)
can be obtained by extrapolating the slope of the ratio o~ve/o rue at zero energy after subtraction of QED corrections, similarly to what is done for muon decay. Finally, the fine-structure constant a can be defined as the residue of the one-photon exchange pole in Coulomb scattering (e/a -+ e/a); the corresponding shift in a,
Ot= (g2s2 /47r)(1 + 62) ,
(7)
can he obtained isolating the pole terms in one-loop corrections to e/a -+ e/a. From the above discussion, it follows that the location of the pole, in the charged vector boson propagator, will be: 4e 2 3//2=-
( 1 + 5 1 6Z - 6 3 ) - H + ( M 2 ) , 8(1 - ~ ) d F and analogously for the neutral vector boson:
(8)
4e2 [ 1 + 5 1 - 5 2 - 6 3 + ( s w /2c w )26 3 ] Ho(-Mg), (9) 8(1 - ~)d F [ 1 - I(1 - ~)1 where "H = II/(27r)4i and the II's are the self-energy contributions. In the one-loop corrections (5's and self-energies), we use the lowest-order expressions (2) and (4) since we are working up to order g2 only. The cancellation between the ultraviolet divergences appearing in the 6's and in the self-energies will be shown elsewhere, together with the presentation of the details of the calculation. Here, we only report on the results:
Mg=
2142 =
= ~2
___ . [ 1 -- - -g2 t,~u xleptons 4e2 M + s;quarks+6~/+6]IWI)I vM 8(1 -- ~) GF 167r2
(10)
[ 1 -- g2 ,xleptonsa. 4e2 t~Mo " xquarks. ~Mo * 6Ho +6Wo) ] ' 8(1 - ~ ) G F F Z ~(1-- ~)] 167r 2
where ~leptons, 6quarks and 6 H are the contributions to the renormalization of the vector boson masses arising from leptons, quarks and the Higgs particle, respectively, while the 6W's are the remaining contributions (box diagrams, finite parts of vertex and self-energy corrections). The most important contributions come from fermions, as one could deduce by using renormalization group arguments. We find, for a typical lepton and quark doublet: +1 This is possible because, in both cases, electromagnetic radiative corrections to Fermi pointlike interactions are ultraviolet convergent. 91
Volume 91B, number 1
PHYSICS LETTERS
1eptons_ , 2 /m~) 2 - si] M - -~[ln(M
24 March 1980
xquarks "M = --~1 [ln(M2/m2) + 2 ln(M2/m 2 -- 51
,
(11)
•
A factor 3 has been introduced to take into account the color degrees of freedom of quarks. Finite corrections, proportional to the ratio m2/M 2 have been ignored everywhere. For the neutral vector boson, we find analogously: xleptons 2 -1 4 2 2 2 = --(3Cw) (4S w - 2Sw + 1) [ln(M0/m~) - s] , VMo 6quarks r~ 2 . - l r . 1 6 4 2 ' 2 2 4 4 2 Mo = --I.JCw) lU~Sw -- 4s w + 2) In(M6/m u) + (gSw - 2s w + l)
(12)
hl(M2/m2) -
1(710 4 + 10S2 - 5] -~sw
(13)
The contribution of light fermions is quite large and it increases the boson masses. From the Higgs particle we have:
1 m2 6H = ~ ~
3
m2
M2
1 m2
-+H(M), 4 m2H- --- M 2 l n -m~H
8~o-~-
3
~
m2
M2+7~H(Mo)
gm~TM21n mH 2
cw
'
(14)
where the first two terms come from the self-energy corrections to muon decay and the function H(M) comes from the vector boson self-energies on the mass shell, 31 1 m4 H(M) =-i-~ + 12 M 4
1 m2 + ( 1 m4 2 M2 12 M 4 (
1 m6+__ 1 m4 12 M 6 3 M 4
+ 0(4M2 - m2)
( + O(m2 -
l m6+ lm4
4M2)
12 M 6
3 M4
1 m2
,
m2
4M 2 )m~m2 1 [ l l n m2
M2 ]
-~+\
(4M2~m2) 1/2 m2 arctg
(~4M 2\\ m2m2Hll/2/.j
(15,
m2]{M2 ! -~ \ In ~m2H 1- - ~l(m2--4M211/2~HH ! In l l + [[(m ( m 2 - 4 M4M2)/m 2 ) / m 2 ]2l /]1/2 2}--
In the limiting case of very large and very small Higgs mass, we have: 65
1 m2
H(M) m~>>MZ72 H(M)
~-
31
m2
8M2
51n m2 ~:~ 6
1 M2
1M 2
6m2
3m2
1 m2 2M 2
3 m2 1 m2H 1 M \ mH 4M 2 n~-~---2~---arctg~)2~H H- .
In
m2 H M2 '
(16)
As one can see, the terms proportional to the ratio m2/M 2 exactly cancel, in the limit of very large Higgs mass, the corresponding ones in eq. (14). This is exactly the screening theorem of ref. [4],; No singularity arises in the limit of very small Higgs mass. The expressions for 6~/, 8W° can be decomposed as follows: = R M + IIW , R1 = - 6 - 6 c 2
RM=R1-R2-R3, M2
M20_M2
lnM2+c2wlnM2+l 7 M2 M2 4 1
+(4S4w- 20s 2 + 10) 4s--~wlnMM~,
92
(17)
8<
In--
2+
Jw
-
(18a)
Volume 91B, number 1
R2=--3
22 sw'
2 2 R3=-3Cw-3
1 +
4s2 -1F --L-4C2 4s 2
4C2 3 (4S2w 8S4)7 +-] ' 4s 2 - 1 4c 2
W
[56 2 llW = ~ 2 [-~-Sw - - - +
1 12c2
+1 c4
W
s2-s+
W
24c 2
W
1 [_4s24 c2\
25 3c 2
24 March 1980
PHYSICS LETTERS
2 29 Sw 3 c2
Analogously we find: 2 2
6w° =RM + (Sw/Cw)R3 + Hw
(18b,c)
W
ln-M2
W
1
)
( 4M2
12C2w \
M02 1/2 arctg ||-[[4M2--7 M 2 ) 1/2 )
1
M2
!
\\
-
(19)
M(~
'
(20)
2 w 2 +17 1 13 In M~22 + 22 - 58 2 + 8SwC 12c 2 9-3 -cw 36 c 2
ii w _
(4M2-M2) 1/2 + \
M2
((4M2-M2)l/2)I arctg \ \
M~
4+29 2 2 -- 1 ) . \ - ~- ~ - c2w - 4SwCw 12c2w
(21)
In table 1 we show the uncorrected and corrected values of the vector boson masses for a range of S2w. The main theoretical uncertainty in our O(g 2) results lies in the choice of the light quark (u, d, s) masses. In table 1 we have used the values suggested by current algebra [5]. The uncertainties coming from the choice of the presently unknown t and H masses are very small. We note that the leading O(g 4) corrections, proportional to ln2(Mw/mf) can be easily derived, using renormalization group techniques, starting from the results given here. We have been informed that similar calculations have been done independently by M. Veltman. The authors are deeply indebted to Profs. N. Cabibbo and L. Maiani for their encouragement and invaluable help.
Table 1 Uncorrected (M, Mo) and corrected (M, Mo) values of the vector boson masses. Masses are given in GeV. For quarks we have used the following values: m u = 0.01, m d = 0.01, m c = 1.75, m s = 0.25, m t = 20, m b = 15. We have also assumed m H = 10 [6]. For the Fermi coupling constant we assume the value [7]: GF = 1.1663 X 10 -5 GeV -2 . S2w
M
~r
Mo
-Mo
0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27
85.529 83.363 81.354 79.484 77.737 76.100 74.562 73.114 71.748
90.418 87.886 85.554 83.398 81.395 79.530 77.787 76.153 74.618
95.032 93.203 91.531 89.998 88.589 87.293 86.097 84.994 83.974
100.202 98.028 96.053 94.255 92.613 91.110 89.731 88.465 87.300
93
Volume 91B, number 1
PHYSICS LETTERS
References [1] S. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Proc. 8th Nobel Syrup. (Almqvist and Wiksell, Stockholm, 1968); S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [2] R.E. Behrends, R.J. Finkelstein and A. Sirlin, Phys. Rev. 101 (1956) 866; S.M. Berman, Phys. Rev. 112 (1958) 267; T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652. [3] A. Ross, Nucl. Phys. B51 (1973) 116; A. Sirlin, Nucl. Phys. B71 (1974) 29. [4] M. Veltman, Acta Phys. Pol. B8 (1977) 475; Phys. Lett. 70B (1977) 253. [5] H. Leutwyler, Phys. Lett. 48B (1974) 45; M. Testa, Phys. Lett. 56B (1975) 53; S. Weinberg, Rabi Festschrift (New York Academy of Science). [6] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; S. Weinberg, Phys. Rev. D7 (1973) 2887; J. Ellis, M.K. Galliard, D.V. Nanopoulos and C.T. Sachrajda, CERN preprint TH. 2634. [7] D.H. Wilkinson, Nature 257 (1975) 189; D.H. Wilkinson and D.E. Alburger, Phys. Rev. C13 (1976) 2517.
94
24 March 1980