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Finite interaction from an evanescent coupling
Nuclear Physics B97 (1975) 522 526 North-Holland Publishing Company
FINITE INTERACTION FROM AN EVANESCENT COUPLING C.G. BOLLINI and J.J. GIAMBIAGI Departamento de F[siea de la Universidad Nacional de La Plata y Conse/o Nacional de Investigaciones Cient{ficas 3' Tgcnicas
Received 30 I)ccember 1974 Revised 3 June 1975) We compute tile first-order vertex radialive corrections in a theory in which tile only interaction is an evanescent one of the form {Ys, Y# }. An effective finite ;'s interaction is obtained.
1. Introduction In previous papers we have introduced the idea of evanescent couplings [ 1 , 2 ] . First in c o n n e c t i o n with the Adler a n o m a l y [1 ], and later in a more general way [2]. In order to define them we made essential use of the dimensional regularization method [3]. The evanescent couplings depend on the n u m b e r of dimensions u in such a way that they go to zero for u going to four. So, when one looks at them in four dimensions they are zero, but when inserted (for u 4= 4) into a divergent integral (which has a pole at u = 4) then the zero of the coupling compensates the pole giving a finite result. This shows that the limit of the theory for u -+ 4 is not equal to the t h e o r y at u = 4. It is possible to give m a n y examples [2] of such couplings, b u t in this paper we consider that particular one connected with the Adler anomaly, namely
{v5,7.). We are not interested for the m o m e n t in problems connected with renormalizability or locality of tile interaction, since our main aim is to show that it is possible to o b t a i n a finite coupling from an interaction which is apparently equal to zero in four dimensions. The above m e n t i o n e d fact about evanescent couplings, of being "zero" but generating finite results for divergent diagrams suggests the possibility of starting with a theory in which the only coupling is an evanescent one, so that if one considers it at u = 4 one has the free particle case, but if one computes higher-order corrections for u 4= 4 and after that takes the limit u -+ 4, one may have a finite and well-defined result. What we shall discuss now is the following theory: a spinor field obeying the Dirac equation coupled in an evanescent way to a "massless p i o n " field. We compute
C.G. Bollini, J..L G&mbiagi / Evanescent coupling
523
then the first radiative correction to the vertex and find a finite result which, for the spinor on the mass shell is equivalent to a 75 coupling. This means that for v = 4 the spinor-spinor scattering is equivalent (up to this approximation) to the exchange of a pseudoscalar boson between the spinor particles. We take the following coupling {Ts, T(P +P')} (k2) -½ , (l) where p, p' are the momentum of the incoming and outgoing particles and k = p p' is the momentum of the pion. The factor in the denominator is added for reasons of simplicity in the calculations. With it, the number of divergent integrals (the only ones which contribute) is greatly diminislaed. For 9"5, we take the definition given by Akyempong and Delbourgo [4], namely: 1
"T5" =4i T[al')'a2Ta3Ta4]"
(2)
where [ ] means the completely antisymmetric combination of the indices. We nov,, give the rules for computing in v dimensions the necessary matrix elements
YuYp + 7p3', = 2rlup ,
Y5T5
7~Y ~ = v ,
(3)
1 [a lya2Taaya4] (4!)2 T[atTa2Ta3Taa] T 1
= A o = - 4 ! v(v
1)(v
2)(v-3),
(4)
vug, sy u = ( v - 8 ) "Ys,
(5)
(. 3'57075 = d 13'u -
~
8)
AOTu
(6)
"
In general, when making the contractions between both 75 indices, we have:
T57 a 1 ")'a2 ... TaIIT 5 =AnTalTa
2
... Tan +B n ~---
TalTa2Ta3Ta4 ...Tan L_ _ d
+ En£ UL3
[__~
TalTa2 ... Tan L...A ~_ a2Ta3Ta4")'a5 __d
7a l')'a2 ya 3Taa"/asTa6 "[a 7 d{a8 "'" ")'all ' L__2 I I I 1 W~
L __J
t
... I
(7)
where one can show that the coefficients satisfy tire following recurrence relations An +Bn =An 2'
Bn +Cn =Bn-2 '
Cn +Dn =Cpl 2'
Dn +En =Dn 2 "
(8)
524
C G. Bollini, J.J. Giambiagi / Evanescent coupling
Further ( v - n + 1)A n = ( v - 8 ) An_ 1 - ( n - 1 ) A n _
(9)
2 .
These relations specify the values of the coefficients as functions of v. The £ means summation over all possible contractions u. The symbol u means the contraction (i.e. the replacement OfTa i ... 7a] by 7?aia] with a + or - sign according to the number of permutations necessary to bring the two 7's together, being even or odd respectively. For instance 7 a T b = l?ab , t_A
7aTu7 b = L__J
7a"),bTcT d = --Uac~bd ,
~?abTu ,
etc.
(1 O)
Also k[a 1 "Ya2"[a3"/a4]
=
3 [7 "k, 7S ] ,
(11)
k[a lVa27a3Pa41 = ½ {7-k, [Ts, Vp] }.
It is clear that in lowest order, as 3'5 anticommutes with any 3',~, at v = 4 the vertex is zero. So, we go to the first radiative correction. Using the interaction (1) we get for the corresponding matrix element M =
q-~fd4kD - 1 ( 7 5 , 7 ( 2 P l X(T(p 2
k)} (7(p l - k ) +m){7'5,7(Pl + P 2 - 2 k ) )
k)+m){T5,7(2P2
k)),
(12)
where D = [(ill -- k)2 - m2] [(P2- k)2 - m2]k4 " k is the momentum of the virtual pion, P2 and Pl the incoming and outgoing fermion momentum, k 4, in D, comes from the propagator of the boson and the assumed coupling [1 ]. q is the external pion momentum: q2 = (Pl P2) 2' The indices corresponding to the first and last 75 are to be contracted (as in form (7)) as they correspond to the vertices joined by the virtual pion. The indices of 75 are free, as they correspond to the free pion. For the explicit calculation of (12) one introduces the auxiliary Feynman variables, and proceed in the usual way. All convergent integrals disappear in the limit v ~ 4, due to the 3'5 anticommutators. The only remaining integrals are those usually called logarithmic divergences. In order to compute them for v dimensions, we use the formula fd
Vk
~k2"n
'
~t. ) _ ~v (k 2 + A ) m iP(½v)
A~v+n-m
P(m)
P(½v+n) P(m
n-½v)
(13)
C. G. Bollini, J.J. Giambiagi / Evanescent coupling
525
In our case n = 2, m = 4. In the actual calculation, the 75 are eliminated b y repeated use of form (7). For instance, a typical expression obtained from (7) is t
- A
'
75")'aTb75")'5 -
'
t
t
6"/a')'bT5 +B6(~ab~/5 + ½7a [')'b' ")'5 ] -- ½Tb [')'a' ")'5]) 1
t
+ 4C6 ([')'5' r a ] ' ~'b )"
(14)
After performing the integration for the auxiliary variables and taking the limit v -+ 4 we obtain, on the mass-shell (see also refs. [1,2] ) M = 210 L n 2 m 7 5 q
(15)
.
We have also computed the same vertex correction in two dimensions. In this case, the coupling used was (75 , 7(P + P ' ) ) ,
(16)
and the new definition of 75 is ....
75
=1
(17)
~'Y[al')'a2 ] "
We proceed in an analogous way as we did in the previous case, but care must be taken in some formulae due to the new definition o f 75. Now we have ' "/575 = 47[al'Ya2 I 7[a17 a21 = A 0' = - ½ v ( V - 1 ) '
(18)
7u75 7u = (v - 4) 3,5 ,
(19)
_v-4 v
(20)
757tL75
, A0"/u = A 1 7 u •
Form (7) and (8) are valid with D'n = E'n -- 0. We have also the following recurrence relation (v-n+l)A'
n = ( v - 4 ) A'n - 1 - ( n - 1 ) A '
n-2"
(21)
Taking into account these modifications, the final result for the vertex correction in two dimensions (v -~ 2), on the mass-shell is M=-28
inrnq, 5 .
These results show that in spite o f having started with a theory which for v = 4 (resp. v = 2) corresponds to a free Lagrangian, the presence of the evanescent coupling, together with the presence of divergent integrals generate a particle-particle interaction which, in principle could be detected in a scattering experiment, although we do not pretend that the present example is a realistic theory.
526
C G. BollinL J.J. Giambiagi / Evanescent coupling
It is r a t h e r a m u s i n g t h a t w i t h this m e t h o d o n l y the divergent integrals seem to play a relevant role. T h e c o n v e r g e n t o n e s d i s a p p e a r as t h e y carry a n intrinsic evanescence in t h e m .
References [1 ] C.G. Bollini and J.J. Giambiagi, Acta Phys. Austriaca 38 (1973) 211. [2] C.G. BoUini and J.J. Giambiagi, Proc. Brazilian Symposium of Theoretical Physics. [3] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; J.t:. Ashmore, ICTP 72/72; C.M. Cicuta and C. Montaldi, Nuovo Cimento Letters 4 (1973) 329; C.G. Bollini and J.J. Giambiagi, Phys. Letters 40B (1972) 566; Nuovo Cimento 12B (1972) 20. [4] D.A. Akyempong and R. Delbourgo, Nuovo Cimento lTA (1973) 578; 18A (1973) 94.
FINITE INTERACTION FROM AN EVANESCENT COUPLING C.G. BOLLINI and J.J. GIAMBIAGI Departamento de F[siea de la Universidad Nacional de La Plata y Conse/o Nacional de Investigaciones Cient{ficas 3' Tgcnicas
Received 30 I)ccember 1974 Revised 3 June 1975) We compute tile first-order vertex radialive corrections in a theory in which tile only interaction is an evanescent one of the form {Ys, Y# }. An effective finite ;'s interaction is obtained.
1. Introduction In previous papers we have introduced the idea of evanescent couplings [ 1 , 2 ] . First in c o n n e c t i o n with the Adler a n o m a l y [1 ], and later in a more general way [2]. In order to define them we made essential use of the dimensional regularization method [3]. The evanescent couplings depend on the n u m b e r of dimensions u in such a way that they go to zero for u going to four. So, when one looks at them in four dimensions they are zero, but when inserted (for u 4= 4) into a divergent integral (which has a pole at u = 4) then the zero of the coupling compensates the pole giving a finite result. This shows that the limit of the theory for u -+ 4 is not equal to the t h e o r y at u = 4. It is possible to give m a n y examples [2] of such couplings, b u t in this paper we consider that particular one connected with the Adler anomaly, namely
{v5,7.). We are not interested for the m o m e n t in problems connected with renormalizability or locality of tile interaction, since our main aim is to show that it is possible to o b t a i n a finite coupling from an interaction which is apparently equal to zero in four dimensions. The above m e n t i o n e d fact about evanescent couplings, of being "zero" but generating finite results for divergent diagrams suggests the possibility of starting with a theory in which the only coupling is an evanescent one, so that if one considers it at u = 4 one has the free particle case, but if one computes higher-order corrections for u 4= 4 and after that takes the limit u -+ 4, one may have a finite and well-defined result. What we shall discuss now is the following theory: a spinor field obeying the Dirac equation coupled in an evanescent way to a "massless p i o n " field. We compute
C.G. Bollini, J..L G&mbiagi / Evanescent coupling
523
then the first radiative correction to the vertex and find a finite result which, for the spinor on the mass shell is equivalent to a 75 coupling. This means that for v = 4 the spinor-spinor scattering is equivalent (up to this approximation) to the exchange of a pseudoscalar boson between the spinor particles. We take the following coupling {Ts, T(P +P')} (k2) -½ , (l) where p, p' are the momentum of the incoming and outgoing particles and k = p p' is the momentum of the pion. The factor in the denominator is added for reasons of simplicity in the calculations. With it, the number of divergent integrals (the only ones which contribute) is greatly diminislaed. For 9"5, we take the definition given by Akyempong and Delbourgo [4], namely: 1
"T5" =4i T[al')'a2Ta3Ta4]"
(2)
where [ ] means the completely antisymmetric combination of the indices. We nov,, give the rules for computing in v dimensions the necessary matrix elements
YuYp + 7p3', = 2rlup ,
Y5T5
7~Y ~ = v ,
(3)
1 [a lya2Taaya4] (4!)2 T[atTa2Ta3Taa] T 1
= A o = - 4 ! v(v
1)(v
2)(v-3),
(4)
vug, sy u = ( v - 8 ) "Ys,
(5)
(. 3'57075 = d 13'u -
~
8)
AOTu
(6)
"
In general, when making the contractions between both 75 indices, we have:
T57 a 1 ")'a2 ... TaIIT 5 =AnTalTa
2
... Tan +B n ~---
TalTa2Ta3Ta4 ...Tan L_ _ d
+ En£ UL3
[__~
TalTa2 ... Tan L...A ~_ a2Ta3Ta4")'a5 __d
7a l')'a2 ya 3Taa"/asTa6 "[a 7 d{a8 "'" ")'all ' L__2 I I I 1 W~
L __J
t
... I
(7)
where one can show that the coefficients satisfy tire following recurrence relations An +Bn =An 2'
Bn +Cn =Bn-2 '
Cn +Dn =Cpl 2'
Dn +En =Dn 2 "
(8)
524
C G. Bollini, J.J. Giambiagi / Evanescent coupling
Further ( v - n + 1)A n = ( v - 8 ) An_ 1 - ( n - 1 ) A n _
(9)
2 .
These relations specify the values of the coefficients as functions of v. The £ means summation over all possible contractions u. The symbol u means the contraction (i.e. the replacement OfTa i ... 7a] by 7?aia] with a + or - sign according to the number of permutations necessary to bring the two 7's together, being even or odd respectively. For instance 7 a T b = l?ab , t_A
7aTu7 b = L__J
7a"),bTcT d = --Uac~bd ,
~?abTu ,
etc.
(1 O)
Also k[a 1 "Ya2"[a3"/a4]
=
3 [7 "k, 7S ] ,
(11)
k[a lVa27a3Pa41 = ½ {7-k, [Ts, Vp] }.
It is clear that in lowest order, as 3'5 anticommutes with any 3',~, at v = 4 the vertex is zero. So, we go to the first radiative correction. Using the interaction (1) we get for the corresponding matrix element M =
q-~fd4kD - 1 ( 7 5 , 7 ( 2 P l X(T(p 2
k)} (7(p l - k ) +m){7'5,7(Pl + P 2 - 2 k ) )
k)+m){T5,7(2P2
k)),
(12)
where D = [(ill -- k)2 - m2] [(P2- k)2 - m2]k4 " k is the momentum of the virtual pion, P2 and Pl the incoming and outgoing fermion momentum, k 4, in D, comes from the propagator of the boson and the assumed coupling [1 ]. q is the external pion momentum: q2 = (Pl P2) 2' The indices corresponding to the first and last 75 are to be contracted (as in form (7)) as they correspond to the vertices joined by the virtual pion. The indices of 75 are free, as they correspond to the free pion. For the explicit calculation of (12) one introduces the auxiliary Feynman variables, and proceed in the usual way. All convergent integrals disappear in the limit v ~ 4, due to the 3'5 anticommutators. The only remaining integrals are those usually called logarithmic divergences. In order to compute them for v dimensions, we use the formula fd
Vk
~k2"n
'
~t. ) _ ~v (k 2 + A ) m iP(½v)
A~v+n-m
P(m)
P(½v+n) P(m
n-½v)
(13)
C. G. Bollini, J.J. Giambiagi / Evanescent coupling
525
In our case n = 2, m = 4. In the actual calculation, the 75 are eliminated b y repeated use of form (7). For instance, a typical expression obtained from (7) is t
- A
'
75")'aTb75")'5 -
'
t
t
6"/a')'bT5 +B6(~ab~/5 + ½7a [')'b' ")'5 ] -- ½Tb [')'a' ")'5]) 1
t
+ 4C6 ([')'5' r a ] ' ~'b )"
(14)
After performing the integration for the auxiliary variables and taking the limit v -+ 4 we obtain, on the mass-shell (see also refs. [1,2] ) M = 210 L n 2 m 7 5 q
(15)
.
We have also computed the same vertex correction in two dimensions. In this case, the coupling used was (75 , 7(P + P ' ) ) ,
(16)
and the new definition of 75 is ....
75
=1
(17)
~'Y[al')'a2 ] "
We proceed in an analogous way as we did in the previous case, but care must be taken in some formulae due to the new definition o f 75. Now we have ' "/575 = 47[al'Ya2 I 7[a17 a21 = A 0' = - ½ v ( V - 1 ) '
(18)
7u75 7u = (v - 4) 3,5 ,
(19)
_v-4 v
(20)
757tL75
, A0"/u = A 1 7 u •
Form (7) and (8) are valid with D'n = E'n -- 0. We have also the following recurrence relation (v-n+l)A'
n = ( v - 4 ) A'n - 1 - ( n - 1 ) A '
n-2"
(21)
Taking into account these modifications, the final result for the vertex correction in two dimensions (v -~ 2), on the mass-shell is M=-28
inrnq, 5 .
These results show that in spite o f having started with a theory which for v = 4 (resp. v = 2) corresponds to a free Lagrangian, the presence of the evanescent coupling, together with the presence of divergent integrals generate a particle-particle interaction which, in principle could be detected in a scattering experiment, although we do not pretend that the present example is a realistic theory.
526
C G. BollinL J.J. Giambiagi / Evanescent coupling
It is r a t h e r a m u s i n g t h a t w i t h this m e t h o d o n l y the divergent integrals seem to play a relevant role. T h e c o n v e r g e n t o n e s d i s a p p e a r as t h e y carry a n intrinsic evanescence in t h e m .
References [1 ] C.G. Bollini and J.J. Giambiagi, Acta Phys. Austriaca 38 (1973) 211. [2] C.G. BoUini and J.J. Giambiagi, Proc. Brazilian Symposium of Theoretical Physics. [3] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; J.t:. Ashmore, ICTP 72/72; C.M. Cicuta and C. Montaldi, Nuovo Cimento Letters 4 (1973) 329; C.G. Bollini and J.J. Giambiagi, Phys. Letters 40B (1972) 566; Nuovo Cimento 12B (1972) 20. [4] D.A. Akyempong and R. Delbourgo, Nuovo Cimento lTA (1973) 578; 18A (1973) 94.