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The Drell-Hearn sum rule and the lepton magnetic moment in the Weinberg model of weak and electromagnetic interactions
Volume 40B, number 3
PHYSICS LETTERS
THE DRELL-HEARN
10 July 1972
SUM RULE AND THE LEPTON MAGNETIC MOMENT
IN THE WEINBERG MODEL OF WEAK AND ELECTROMAGNETIC
INTERACTIONS
G. ALTARELLI Istituto di Fisica dell'Universitt2, Roma, Italy
N. CABIBBO Istituto di Fisica dell'Universita, Roma, Italy and Istituto Nazionale di Fisica Nucleare, Serione di R oma, Italy
and L. MAIANI lstituto Superiore di SaniRl, Roma, Italy Istituto Nazionale di Fisica Nucleare, Sottosezione Sanit?~, Roma, Italy
Received 27 April 1972
We verify that the fourth-order Compton amplitude for the scattering of a photon off a charged lepton is finite in Weinberg's model of weak and electro-magnetic interactions, and obeys the Drell-Hearn sum rule. We also compute the second-order weak corrections to the anomalous magnetic moment of a c_h~rgedlepton in the same model. For the case of the muon, we find the bounds: + 0.2 X 10- < ~ ~<+ 0.56 X 10- .
The model of weak and electromagnetic interactions proposed by Weinberg [1 ] in 1967 has been brought back to attention by the recent work of t'Hooft [2], who has given formal arguments to show that a class of theories including the Weinberg model are renormalizable. This proof is however invalidated in this particular model by the occurence of Ward identity anomalies [3], of the kind first discovered by Adler [4]. Although the original model is thus non renormalizable, suitable modifications can be devised to cure the Adler anomalies [3] and it is still an open question whether these modified models are really renormalizable without further anomalies. In a recent paper, Weinberg [5] has started an investigation of several physical processes in this model, illustrating directly some cancellation mechanisms which reduce the potentially high degree of divergence of the theory. In order to gain s o m e insight in the structure of the theory, we have followed the line of ref. [5], and carried on a study of some electromagnetic properties of the charged leptons in this model. We have restricted ourselves to low orders in perturbation theory, such that the Adler anomalies are not present, and therefore have considered the original version of the theory. In this paper we first study that particular Compton amplitude whose absorptive part appears in the Drell-Hearn sum rule (for "/-charged lepton scattering) [6]. We verify that the Drell-Hearn sum rule is indeed satisfied by the Compton amplitude computed in fourth order perturbation theory. The only feature of the Weinberg theory which is crucial to this result is the fact that the charged vector boson is given an anomalous magnetic moment g w = 1, i.e. a gyromagnetic ratiog w = 2. We also study the behaviour of the Pauli form factor F2(k 2) in the electromagnetic current of the charged lepton. We find that, to second order, F 2 is finite, so that this model gives a well determined correction, of order G m 2, to the anomalous magnetic moment of the lepton (a result already anticipated in ref. [5]). Unlike conventional renormalizable theories F2(k2 ) turns out to increase logarithmically at high k 2, so that, it does not satisfy an unsubtracted dispersion relation. 415
Voiume 40B, number 3
PHYSICS LETTERS
10 July 1972
We consider the Drell-Hearn sum rule for the scattering of a photon off an electron (or muon): oo
2n2aK2_( m2
Op(m)-- OA(m )
,Y
0
(1)
dco
co
K is the anomalous magnetic moment of the lepton (in units of its Bohr magneton), m is its mass, co is the photon laboratory energy and Op(OA) are the total cross-sections for a photon with spin parallel (antiparallel) to the lepton spin. We first consider the case of ordinary quantum electrodynamics. We expand both i¢ and the cross-sections in powers of a. The righthand side starts with terms of order a 2. The left hand-side may contain terms of order a, if the bare value of gyromagnetic ratio is different from two. In order for eq. (1) to be consistent this must not happen, and the bare value of K must be zero. This fact has been observed by Weinberg [7] who noticed that the validity of the Drell-Hearn sum rule for the scattering of a photon off any particle requires a bare gyromagnetic ratio equal to two. This condition is satisfied in Weinberg's model by the charged leptons (as in usual QED) as well as by the charged vector boson (for which gw = 2 implies an anomalous magnetic moment Kw = 1). To order o¢2 we have a further consistency condition. In fact, ifgbare = 2, the left-hand side of eq. (1) starts with terms of order a, so that to order a 2 the right-hand side has to vanish. To this order the only contribution to Op - o A comes, in QED, from elastic e - 7 scattering computed in Born approximation. We have found: O(p2} - e(A2) = -- (27ra2/mco) {(1 +re~co)In (1 +
2co~m) -- 211 + co2/(m + 26o)2]} .
(2)
All explicit integration shows that, indeed, 00
0(2)(00) dco = 0
°(2)(co)
(3)
03
0
We consider now the case of Weinberg's model of weak and e.m. interactions• For convenience we reproduce here the interaction Lagrangian, keeping only terms which are relevant to our purpose. Using the notations of ref. [5]: L 1-
(g2igg' +g ,2)~,_AvIWU(3 W+v+ 3vW+) W~+(~ W v - r v W ) +KWaU(W •
[-(1--75)
4 (g2~g,2)}ZVlTvL-~g
,2
+~
(1 +"/5) ,2 -2
(g _g2
)J
W~ -
WvW;)I + 7 7 ~ l - g , ~ l l
l+i~2W;v17u(l+T5)l+i~Wd7u(l+75)__2
,
2
v! + .
I and vI denote the fields of the charged lepton and o f its neutrino respectively, Wu represents the field of the charged intermediate vector boson, Z u and ¢ represent neutral vector and scalar boson fields. In eq. (4) we have left unspecified the value of the anomalous magnetic moment of W. The symmetry structure of this model requires Kw = 1. For fixed values of the masses, all coupling constants can be expressed in terms of a single one which can be chosen to be the electric charge e:
g= {Mz/x/2z - M2,}e ,
g'
=(Mz/Mw) e,
g~o= {rnMz/2Mwx/M2Z - M 2 }e.
Lowest order contributions to the anomalous magnetic moment of the charged lepton arise from the graphs reported in fig. 1 and are all of order u. 416
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Volume 40B, number 3
PHYSICS LETTERS
10 July 1972
Again the validity of the Drell-Hearn sum rule requires eq. (3) to be satisfied for the o~2 terms of the total lepton-photon scattering cross section. Contributions of lowest order to this cross section come from the Born approximation to the processes:
7 + l - ~ 7 + l-
(6)
7+l
(7)
-+¢ +/-
7 + l--~ Z + l-
(8)
7 + 1--+ W - + t,/.
(9)
The contribution from reaction (6) coincides with the one given in eq. (2). Reactions (7) and (8) give contributions very similar to eq. (2), which separately obey eq. (3). Reaction (9) gives a contribution to Op - o A which, for general values of Kw, does not vanish asymptotically: [o ( 2 ) - o(A2)lw ~(1¢ W - 1)2X const.
(10)
If •w 4: 1, reaction (9) gives thus a logarithmically divergent contribution to eq. (3), which violates the Drell-Hearn sum rule. If t¢W = I the asymptotic behaviour is instead:
1@2~- O(A2)1 ~ l lCa,)n
co
which makes the integral in eq. (3) convergent. Explicit calculation shows that also this contribution vanishes. We have thus verified that in the Weinberg model, where Kw = 1, the Drell-Hearn sum rule is obeyed up to order c~2. We note that the only feature of the model which is essential to this result is that all charged particles have the "normal" value for the bare gyromagnetic ratio, i.e. g = 2. The relations eq. (5) between masses and coupling constants are here irrelevant, and will probably play a role at higher orders. To proceed further, we consider the particular combination of forward Compton amplitudesJ2(co2), defined as
1 G(.~2)_fA(~O2)] f2(co 2) = ~--~-
(ll)
f p A being the amplitudes for spin parallel (antiparallel) to the lepton spin [6]. ""The above result lmphes that, to order ~ ,f~(oo ) obeys a non substracted dispersion relaUon, so that it could (2)(co) - o A (2) (co) which is its imaginary part (up to a constant factor). be computed directly from Op Alternatively f2(co2 ) could be computed from the fourth-order Feynman graphs. The fourth-order graphs can be classified in four groups according to the nature of the internal boson lines, which can be 7, ~P,Z or W lines. Most of the graphs give divergent contributions. However each group sum up to a finite result. The mechanism of cancellation of divergent terms can be illustrated by considering the case of those graphs contain internal W - lines (see fig. 2 where some graphs relevant to the computation off2 have been reported). Graph (a) represents a vertex correction. In general we can expand the vertex function P (p, p', k), where p, p' and k are the lepton and photon momenta around the mass-shell points: p2 =p,2 = m 2, k 2 __UO: -
•
2
2
Cla(p,p',k) =aT.+bo k u +(1)2-m 2),4 +(p'2-m2)A' +k2B +...
.
.
•
(12)
In eq. (12), a and b are constants, A., A, and Buare vectors formed with y-matrices and momenta. We find that a, is quadratically divergent but it is compensated by quadratically divergent terms in the fermion self-energy. As in QED this cancellation is guarenteed by charge conservation, b is related to the anamalous magnetic moment of the t lepton, and turns out to be finite, as we discuss later. The terms A . and A . are logarithmically divergent and. when r
417
Volume 40B, number 3
PHYSICS LETTERS
10 July 1972
inserted in graph (a), give rise to a contact term, i.e. a term with no fermion pole. This combines with similar contact terms arising from different vertex insertions and from self energy insertions (e.g. graph (b)) to give an overall contact term which cancels the logarithmically divergent part of the box diagrams (e.g. graph (c)), thus yielding a finite result [8]. We encounter here a difference between this theory and conventionally renormalizable theories, such as QED. In the latter case mass, charge and wave function renormalization not only yield finite S-matrix elements, but also finite off-shell Green's functions, such as the vertex function and propagators. In the present theory the vertex function is not finite even after renormalization. In fact charge and wave function renormalization can remove t the quadratically divergent coefficient a, but not the logarithmically divergent terms contained in A u, A u, and Bu. On the other hand, on the mass-shell Green's functions must be finite, since they are observable, either being directly related to S-matrix elements for physical processes, or, as in the case for the vertex function Fu, appearing as residues of one particle poles in such matrix elements. In the case of Pu' this latter requirement implies that b has to be fimte. Terms hke Au,A. or B (the latter containing the charge radms of the lepton) need not be fin te, provided that, for any physical pr~cess,'*they combine with similar terms from other diagrams, yielding a finite contribution to S-matrix elements. An explicit calculation from Feynman diagrams off2(6o2 ) shows that, in agreement with our previous result on its imaginary part, this amplitude contains a term proportional to (~w - 1)2, which grows logarithmically at high energy. If ~¢w = 1, f2(o02) vanishes at infinity and therefore obeys an unsubstracted dispersion relation. As discussed above, the finiteness of the anomalous magnetic moment of the lepton is a necessary condition for the theory to be renormalizable. We have computed the lowest order contributions to this quantity in the basis of the graphs of fig. 1. Graph (a) gives the well known Schwinger result ~¢ = od2rr. The contributions of the graphs (b), (c) and (d) can be considered as genuine weak interaction corrections. The contributions arising from W exchange has been computed by many authors, in particular by Brodsky and Sullivan [9] who give the result for any value ofK w. It is interesting to observe that a finite contribution is obtained only for ~¢w = 1, the value required in Weinberg's model. We have computed the contribution of this graphs, finding a result in agreement with ref. [9] as well as the contributions from graphs (c) and (d), which are separately finite. We find: •
Kb
-
5 Gm 2
12rr 2 V~-
"
'
'
.
( gc _ - -1 Gm 2 1 _ 6 M 2 67r2 V~-
-~
.
.
.
~W)
1
4-~-+
i
M2
Gm 4 In [___£ \m2]~
4rr2X/--2M2 t¢d=
"
Where we have neglected, in each case, terms which are smaller than those here recorded, at least by a factor (m/M) 2 In (M2/m2), M being the mass of the exchanged boson. Note that if Me >> m, the last term is negligible with respect to the other two and can be dropped. We have also used the relation:
a/x/2 = (e 2 /SMw) 2 M z/(M 2 z2 _ M 2 ) which fixes the value of the Fermi constant, so that the only unknown parameter in the leading total correction to the anomalous magnetic moment of the lepton is the ratioM2/M 2, which in this theory has to be smaller than or equal to one. For the case of the anomalous magnetic moment of the muon if we let M2w/M2 Z vary between zero and one, we get the bounds:
W,'
',W
,
(a)
, J
(b)
(c)
(d)
Fig. 1. Feynman graphs for the corrections of order e 2 to the anomalous magnetic moment of the charged lepton. 418
Volume 40B, number 3
PHYSICS LETTERS
(a)
(b)
10 July 1972
(c)
Pig. 2. Some of the graphs involving W exchange contributing to the amplitude f 2 ( ~ 2) to order e4.
+ 0.20 X 10 - 8 ~ g b ( # ) + go(//) ~ + 0.56 X 10 - 8 .
(13)
2 2 The m i n i m u m value corresponds to: Mw/M Z = ¼ a value noted by Salam and Strathdee [2], as that for which the Z-meson is coupled to the charged lepton with a pure axial coupling. This correction is rather small, it being about ten to twenty times smaller than the hadronic corrections (which for the muon have been estimated [I0] to be ~ 6 X 10-8). Our motivation foe doing this calculation was partly to investigate the large k 2 behaviour of the Pauli form factor F2(k2 ) (defined so that F2(0 ) = k). Ex~_licit calculation shows that, for large k 2, Im F2(k2 ~ ~ 0 for graphs (c) and (d), whereas Im F~b) (k 2) ~ const and ~2o) (k 2) ~ In k 2. The bad asymptoticbehavior o f h b ) can be traced back to those terms in graph (b) which are superficially logarithmically divergent . Due to the structure of the theory, these terms combine to give a finite amplitude, which however contains extra factors of the external momenta. It should be observed that the logaritmic increase o f F o ( k 2) does not necessarily imply any trouble with S-ma• processes such as e + e - ~/~ + // - ~ . In fact the second order F~b )(k 2 ) we have been considertrix elements for physical ing appears as one of a group of terms in the 4th order corrections to e+e - ~ / / + / / - which involve a pair of internal W lines.__Weinberg [5] has found that the complete second-order amplitude for e+e - ~ + / a - ) ~ W+W- behaves as 1/x/k 2 so that at least the absorptive part of the e+e - -÷//+/~- amplitude coming from this group of terms, has a behaviour at large energies compatible with unitarity. To complete the argument one would have to check explicitly whether the same compensation mechanism works for the whole amplitude. * Terms with higher divergences do not contribute to F 2.
References [1] S. Weinberg, Phys• Rev. Lett. 19 (1967) 1264. See also A. Salam, in Proc. Eight Nobel Syrup. (Almqvist and Wicksel, Stockholm 1968). [2] G. t'Hooft, Nucl. Phys. B35 (1971) 167. See also B.W. Lee, Renormalizable massive vector meson theory. Perturbation theory of the Higgs phenomenon, Stony Brook Preprint; A. Salam and J. Strathdee, A renormalizable gauge model of lepton interactions, Trieste Preprint IC/71/145 who specially discuss the application of t'Hoofts arguments to the model of ref. [1 ]. [3] D.J. Gross and R. Jackiw, The effect of anomalies in quasi renormalizable theories; C. Bouchiat, J. lliopoulos and P. Mayer, An anomaly free version of Weinberg's model, Orsay Preprint. The possible role of Ward identity anomalies for the renormalizability of this model was pointed out be several authors. See for example ref. [2 ]. [4] S. Adler, Phys. Rev. 177 (1969) 2426. [5] S. Weinberg, Phys. Rev. Lett. 27 (1971) 1688. [6] S.D. Drell and A. Hearn, Phys• Rev. Lett. 16 (1966) 908. [7] S. Weinberg, Lectures on elementary particles and quantum field theory, vol. 1, eds. S. Deser, M. Grisaru and H. Pendleton (MIT press, 1970, Cambridge, USA). [8] This cancellation mechanism is similar to the one described in T. Appelquist and H. Quinn, Divergence cancellations in a simplified weak interaction model, Harvard Preprint 1972. [9] S.J. Brodsky and J.D. Sullivan, Phys. Rev. 156 (1967) 1644 and references therein. [10] S.J. Brodsky, Proc. 1971 Internat. Symp. on Electron and photon interactions at high energy, Cornell, USA. 4 19
PHYSICS LETTERS
THE DRELL-HEARN
10 July 1972
SUM RULE AND THE LEPTON MAGNETIC MOMENT
IN THE WEINBERG MODEL OF WEAK AND ELECTROMAGNETIC
INTERACTIONS
G. ALTARELLI Istituto di Fisica dell'Universitt2, Roma, Italy
N. CABIBBO Istituto di Fisica dell'Universita, Roma, Italy and Istituto Nazionale di Fisica Nucleare, Serione di R oma, Italy
and L. MAIANI lstituto Superiore di SaniRl, Roma, Italy Istituto Nazionale di Fisica Nucleare, Sottosezione Sanit?~, Roma, Italy
Received 27 April 1972
We verify that the fourth-order Compton amplitude for the scattering of a photon off a charged lepton is finite in Weinberg's model of weak and electro-magnetic interactions, and obeys the Drell-Hearn sum rule. We also compute the second-order weak corrections to the anomalous magnetic moment of a c_h~rgedlepton in the same model. For the case of the muon, we find the bounds: + 0.2 X 10- < ~ ~<+ 0.56 X 10- .
The model of weak and electromagnetic interactions proposed by Weinberg [1 ] in 1967 has been brought back to attention by the recent work of t'Hooft [2], who has given formal arguments to show that a class of theories including the Weinberg model are renormalizable. This proof is however invalidated in this particular model by the occurence of Ward identity anomalies [3], of the kind first discovered by Adler [4]. Although the original model is thus non renormalizable, suitable modifications can be devised to cure the Adler anomalies [3] and it is still an open question whether these modified models are really renormalizable without further anomalies. In a recent paper, Weinberg [5] has started an investigation of several physical processes in this model, illustrating directly some cancellation mechanisms which reduce the potentially high degree of divergence of the theory. In order to gain s o m e insight in the structure of the theory, we have followed the line of ref. [5], and carried on a study of some electromagnetic properties of the charged leptons in this model. We have restricted ourselves to low orders in perturbation theory, such that the Adler anomalies are not present, and therefore have considered the original version of the theory. In this paper we first study that particular Compton amplitude whose absorptive part appears in the Drell-Hearn sum rule (for "/-charged lepton scattering) [6]. We verify that the Drell-Hearn sum rule is indeed satisfied by the Compton amplitude computed in fourth order perturbation theory. The only feature of the Weinberg theory which is crucial to this result is the fact that the charged vector boson is given an anomalous magnetic moment g w = 1, i.e. a gyromagnetic ratiog w = 2. We also study the behaviour of the Pauli form factor F2(k 2) in the electromagnetic current of the charged lepton. We find that, to second order, F 2 is finite, so that this model gives a well determined correction, of order G m 2, to the anomalous magnetic moment of the lepton (a result already anticipated in ref. [5]). Unlike conventional renormalizable theories F2(k2 ) turns out to increase logarithmically at high k 2, so that, it does not satisfy an unsubtracted dispersion relation. 415
Voiume 40B, number 3
PHYSICS LETTERS
10 July 1972
We consider the Drell-Hearn sum rule for the scattering of a photon off an electron (or muon): oo
2n2aK2_( m2
Op(m)-- OA(m )
,Y
0
(1)
dco
co
K is the anomalous magnetic moment of the lepton (in units of its Bohr magneton), m is its mass, co is the photon laboratory energy and Op(OA) are the total cross-sections for a photon with spin parallel (antiparallel) to the lepton spin. We first consider the case of ordinary quantum electrodynamics. We expand both i¢ and the cross-sections in powers of a. The righthand side starts with terms of order a 2. The left hand-side may contain terms of order a, if the bare value of gyromagnetic ratio is different from two. In order for eq. (1) to be consistent this must not happen, and the bare value of K must be zero. This fact has been observed by Weinberg [7] who noticed that the validity of the Drell-Hearn sum rule for the scattering of a photon off any particle requires a bare gyromagnetic ratio equal to two. This condition is satisfied in Weinberg's model by the charged leptons (as in usual QED) as well as by the charged vector boson (for which gw = 2 implies an anomalous magnetic moment Kw = 1). To order o¢2 we have a further consistency condition. In fact, ifgbare = 2, the left-hand side of eq. (1) starts with terms of order a, so that to order a 2 the right-hand side has to vanish. To this order the only contribution to Op - o A comes, in QED, from elastic e - 7 scattering computed in Born approximation. We have found: O(p2} - e(A2) = -- (27ra2/mco) {(1 +re~co)In (1 +
2co~m) -- 211 + co2/(m + 26o)2]} .
(2)
All explicit integration shows that, indeed, 00
0(2)(00) dco = 0
°(2)(co)
(3)
03
0
We consider now the case of Weinberg's model of weak and e.m. interactions• For convenience we reproduce here the interaction Lagrangian, keeping only terms which are relevant to our purpose. Using the notations of ref. [5]: L 1-
(g2igg' +g ,2)~,_AvIWU(3 W+v+ 3vW+) W~+(~ W v - r v W ) +KWaU(W •
[-(1--75)
4 (g2~g,2)}ZVlTvL-~g
,2
+~
(1 +"/5) ,2 -2
(g _g2
)J
W~ -
WvW;)I + 7 7 ~ l - g , ~ l l
l+i~2W;v17u(l+T5)l+i~Wd7u(l+75)__2
,
2
v! + .
I and vI denote the fields of the charged lepton and o f its neutrino respectively, Wu represents the field of the charged intermediate vector boson, Z u and ¢ represent neutral vector and scalar boson fields. In eq. (4) we have left unspecified the value of the anomalous magnetic moment of W. The symmetry structure of this model requires Kw = 1. For fixed values of the masses, all coupling constants can be expressed in terms of a single one which can be chosen to be the electric charge e:
g= {Mz/x/2z - M2,}e ,
g'
=(Mz/Mw) e,
g~o= {rnMz/2Mwx/M2Z - M 2 }e.
Lowest order contributions to the anomalous magnetic moment of the charged lepton arise from the graphs reported in fig. 1 and are all of order u. 416
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PHYSICS LETTERS
10 July 1972
Again the validity of the Drell-Hearn sum rule requires eq. (3) to be satisfied for the o~2 terms of the total lepton-photon scattering cross section. Contributions of lowest order to this cross section come from the Born approximation to the processes:
7 + l - ~ 7 + l-
(6)
7+l
(7)
-+¢ +/-
7 + l--~ Z + l-
(8)
7 + 1--+ W - + t,/.
(9)
The contribution from reaction (6) coincides with the one given in eq. (2). Reactions (7) and (8) give contributions very similar to eq. (2), which separately obey eq. (3). Reaction (9) gives a contribution to Op - o A which, for general values of Kw, does not vanish asymptotically: [o ( 2 ) - o(A2)lw ~(1¢ W - 1)2X const.
(10)
If •w 4: 1, reaction (9) gives thus a logarithmically divergent contribution to eq. (3), which violates the Drell-Hearn sum rule. If t¢W = I the asymptotic behaviour is instead:
1@2~- O(A2)1 ~ l lCa,)n
co
which makes the integral in eq. (3) convergent. Explicit calculation shows that also this contribution vanishes. We have thus verified that in the Weinberg model, where Kw = 1, the Drell-Hearn sum rule is obeyed up to order c~2. We note that the only feature of the model which is essential to this result is that all charged particles have the "normal" value for the bare gyromagnetic ratio, i.e. g = 2. The relations eq. (5) between masses and coupling constants are here irrelevant, and will probably play a role at higher orders. To proceed further, we consider the particular combination of forward Compton amplitudesJ2(co2), defined as
1 G(.~2)_fA(~O2)] f2(co 2) = ~--~-
(ll)
f p A being the amplitudes for spin parallel (antiparallel) to the lepton spin [6]. ""The above result lmphes that, to order ~ ,f~(oo ) obeys a non substracted dispersion relaUon, so that it could (2)(co) - o A (2) (co) which is its imaginary part (up to a constant factor). be computed directly from Op Alternatively f2(co2 ) could be computed from the fourth-order Feynman graphs. The fourth-order graphs can be classified in four groups according to the nature of the internal boson lines, which can be 7, ~P,Z or W lines. Most of the graphs give divergent contributions. However each group sum up to a finite result. The mechanism of cancellation of divergent terms can be illustrated by considering the case of those graphs contain internal W - lines (see fig. 2 where some graphs relevant to the computation off2 have been reported). Graph (a) represents a vertex correction. In general we can expand the vertex function P (p, p', k), where p, p' and k are the lepton and photon momenta around the mass-shell points: p2 =p,2 = m 2, k 2 __UO: -
•
2
2
Cla(p,p',k) =aT.+bo k u +(1)2-m 2),4 +(p'2-m2)A' +k2B +...
.
.
•
(12)
In eq. (12), a and b are constants, A., A, and Buare vectors formed with y-matrices and momenta. We find that a, is quadratically divergent but it is compensated by quadratically divergent terms in the fermion self-energy. As in QED this cancellation is guarenteed by charge conservation, b is related to the anamalous magnetic moment of the t lepton, and turns out to be finite, as we discuss later. The terms A . and A . are logarithmically divergent and. when r
417
Volume 40B, number 3
PHYSICS LETTERS
10 July 1972
inserted in graph (a), give rise to a contact term, i.e. a term with no fermion pole. This combines with similar contact terms arising from different vertex insertions and from self energy insertions (e.g. graph (b)) to give an overall contact term which cancels the logarithmically divergent part of the box diagrams (e.g. graph (c)), thus yielding a finite result [8]. We encounter here a difference between this theory and conventionally renormalizable theories, such as QED. In the latter case mass, charge and wave function renormalization not only yield finite S-matrix elements, but also finite off-shell Green's functions, such as the vertex function and propagators. In the present theory the vertex function is not finite even after renormalization. In fact charge and wave function renormalization can remove t the quadratically divergent coefficient a, but not the logarithmically divergent terms contained in A u, A u, and Bu. On the other hand, on the mass-shell Green's functions must be finite, since they are observable, either being directly related to S-matrix elements for physical processes, or, as in the case for the vertex function Fu, appearing as residues of one particle poles in such matrix elements. In the case of Pu' this latter requirement implies that b has to be fimte. Terms hke Au,A. or B (the latter containing the charge radms of the lepton) need not be fin te, provided that, for any physical pr~cess,'*they combine with similar terms from other diagrams, yielding a finite contribution to S-matrix elements. An explicit calculation from Feynman diagrams off2(6o2 ) shows that, in agreement with our previous result on its imaginary part, this amplitude contains a term proportional to (~w - 1)2, which grows logarithmically at high energy. If ~¢w = 1, f2(o02) vanishes at infinity and therefore obeys an unsubstracted dispersion relation. As discussed above, the finiteness of the anomalous magnetic moment of the lepton is a necessary condition for the theory to be renormalizable. We have computed the lowest order contributions to this quantity in the basis of the graphs of fig. 1. Graph (a) gives the well known Schwinger result ~¢ = od2rr. The contributions of the graphs (b), (c) and (d) can be considered as genuine weak interaction corrections. The contributions arising from W exchange has been computed by many authors, in particular by Brodsky and Sullivan [9] who give the result for any value ofK w. It is interesting to observe that a finite contribution is obtained only for ~¢w = 1, the value required in Weinberg's model. We have computed the contribution of this graphs, finding a result in agreement with ref. [9] as well as the contributions from graphs (c) and (d), which are separately finite. We find: •
Kb
-
5 Gm 2
12rr 2 V~-
"
'
'
.
( gc _ - -1 Gm 2 1 _ 6 M 2 67r2 V~-
-~
.
.
.
~W)
1
4-~-+
i
M2
Gm 4 In [___£ \m2]~
4rr2X/--2M2 t¢d=
"
Where we have neglected, in each case, terms which are smaller than those here recorded, at least by a factor (m/M) 2 In (M2/m2), M being the mass of the exchanged boson. Note that if Me >> m, the last term is negligible with respect to the other two and can be dropped. We have also used the relation:
a/x/2 = (e 2 /SMw) 2 M z/(M 2 z2 _ M 2 ) which fixes the value of the Fermi constant, so that the only unknown parameter in the leading total correction to the anomalous magnetic moment of the lepton is the ratioM2/M 2, which in this theory has to be smaller than or equal to one. For the case of the anomalous magnetic moment of the muon if we let M2w/M2 Z vary between zero and one, we get the bounds:
W,'
',W
,
(a)
, J
(b)
(c)
(d)
Fig. 1. Feynman graphs for the corrections of order e 2 to the anomalous magnetic moment of the charged lepton. 418
Volume 40B, number 3
PHYSICS LETTERS
(a)
(b)
10 July 1972
(c)
Pig. 2. Some of the graphs involving W exchange contributing to the amplitude f 2 ( ~ 2) to order e4.
+ 0.20 X 10 - 8 ~ g b ( # ) + go(//) ~ + 0.56 X 10 - 8 .
(13)
2 2 The m i n i m u m value corresponds to: Mw/M Z = ¼ a value noted by Salam and Strathdee [2], as that for which the Z-meson is coupled to the charged lepton with a pure axial coupling. This correction is rather small, it being about ten to twenty times smaller than the hadronic corrections (which for the muon have been estimated [I0] to be ~ 6 X 10-8). Our motivation foe doing this calculation was partly to investigate the large k 2 behaviour of the Pauli form factor F2(k2 ) (defined so that F2(0 ) = k). Ex~_licit calculation shows that, for large k 2, Im F2(k2 ~ ~ 0 for graphs (c) and (d), whereas Im F~b) (k 2) ~ const and ~2o) (k 2) ~ In k 2. The bad asymptoticbehavior o f h b ) can be traced back to those terms in graph (b) which are superficially logarithmically divergent . Due to the structure of the theory, these terms combine to give a finite amplitude, which however contains extra factors of the external momenta. It should be observed that the logaritmic increase o f F o ( k 2) does not necessarily imply any trouble with S-ma• processes such as e + e - ~/~ + // - ~ . In fact the second order F~b )(k 2 ) we have been considertrix elements for physical ing appears as one of a group of terms in the 4th order corrections to e+e - ~ / / + / / - which involve a pair of internal W lines.__Weinberg [5] has found that the complete second-order amplitude for e+e - ~ + / a - ) ~ W+W- behaves as 1/x/k 2 so that at least the absorptive part of the e+e - -÷//+/~- amplitude coming from this group of terms, has a behaviour at large energies compatible with unitarity. To complete the argument one would have to check explicitly whether the same compensation mechanism works for the whole amplitude. * Terms with higher divergences do not contribute to F 2.
References [1] S. Weinberg, Phys• Rev. Lett. 19 (1967) 1264. See also A. Salam, in Proc. Eight Nobel Syrup. (Almqvist and Wicksel, Stockholm 1968). [2] G. t'Hooft, Nucl. Phys. B35 (1971) 167. See also B.W. Lee, Renormalizable massive vector meson theory. Perturbation theory of the Higgs phenomenon, Stony Brook Preprint; A. Salam and J. Strathdee, A renormalizable gauge model of lepton interactions, Trieste Preprint IC/71/145 who specially discuss the application of t'Hoofts arguments to the model of ref. [1 ]. [3] D.J. Gross and R. Jackiw, The effect of anomalies in quasi renormalizable theories; C. Bouchiat, J. lliopoulos and P. Mayer, An anomaly free version of Weinberg's model, Orsay Preprint. The possible role of Ward identity anomalies for the renormalizability of this model was pointed out be several authors. See for example ref. [2 ]. [4] S. Adler, Phys. Rev. 177 (1969) 2426. [5] S. Weinberg, Phys. Rev. Lett. 27 (1971) 1688. [6] S.D. Drell and A. Hearn, Phys• Rev. Lett. 16 (1966) 908. [7] S. Weinberg, Lectures on elementary particles and quantum field theory, vol. 1, eds. S. Deser, M. Grisaru and H. Pendleton (MIT press, 1970, Cambridge, USA). [8] This cancellation mechanism is similar to the one described in T. Appelquist and H. Quinn, Divergence cancellations in a simplified weak interaction model, Harvard Preprint 1972. [9] S.J. Brodsky and J.D. Sullivan, Phys. Rev. 156 (1967) 1644 and references therein. [10] S.J. Brodsky, Proc. 1971 Internat. Symp. on Electron and photon interactions at high energy, Cornell, USA. 4 19