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Not the no-renormalisation theorem?
Volume 258, number 3,4
PHYSICS LETTERS B
l l April 1991
Not the no-renormalisation theorem? I. Jack Theory Divtston, CERN, CH-1211 Geneva23, Switzerland
D.R.T. Jones DAMTP. Untversttyof Ltverpool, Ltverpool L69 3BX, UK
and P. West Department of Mathemattcs, Kmg's College, London WC2R 2LS, UK Received 8 December 1990
We show that if the lagranglan of a supersymmetnc theory contains massless fields then the superpotentml suffers fimte renormahsatlons m general
1. Introduction T h e n o - r e n o r m a l l s a t i o n ( N R ) t h e o r e m [ 1 ] has played a central role m N = 1 s u p e r s y m m e t r y , particularly with regard to p h e n o m e n o l o g l c a l a p p l i c a t i o n s Briefly, the c o n t e n t o f the t h e o r e m is as follows: In a r e n o r m a h s a b l e s u p e r s y m m e t r l c theory, all c o n t r i b u t i o n s to the effective action f r o m loop d m g r a m s can be written in the f o r m
Ac;= ~ d40d4xl .d4xn G(xl ..... x,,, O, O) ,
(1 1 )
and no c o n t r t b u t l o n s can artse o f the f o r m o f the superpotentlal, W
(1 2)
A , = j- dZ0d4x W ( x , 0) + c . c .
T h e r e is, however, an i m p o r t a n t e x c e p t t o n to the a b o v e assertion. W h e n massless fields are present ~1, the following f o r m o f A~, can arise:
A,; : f d 4 0 d 4 x l D 2 W ( ~ ) + c c
,
(1.3)
where ~ is a chlral superfield, d e f i n e d by the c o n d i t i o n I ) q s = 0. D, 1)are the usual c o v a r l a n t s p l n o r d e r i v a t i v e s R e p l a c i n g fd40--,fd20 02 and using I)2D2qb= I-1~, we h a v e at o n c e that
( 1.4)
A<, : f d20 d4x 1 ~ ( ~ ) + c . c . , d
~1 That massless fields pose a possible problem for the NR theorem was not unknown see for example ref [2] 382
0370-2693/91/$ 03 50 © 1991 - Elsevier Science Pubhshers B V ( North-Holland )
Volume 258, number 3,4
PHYSICS LETTERS B
11 April 1991
which is manifestly o f the form (1.2) and thus constitutes a renormahsatlon o f the superpotential W. Elementary power counting shows that such a term will be finite, if it exists: but even a finite term would be a vlolaUon o f the N R theorem as usually stated. In ref. [3 ] it was in fact noted that for a whole class o f graphs just such a violation o f the N R theorem does take place. In a recent paper [4], one o f the authors has given a more detailed examination of these graphs and has indicated the loopholes m both previous proofs of the N R theorem whxch allow radiative corrections to the effective potenUal. In this paper we demonstrate explicitly that breakdown o f the N R theorem occurs at the two-loop level in the sxmplest possible four-dlmensmnal theory: the massless W e s s - Z u m m o model [ 5 ].
2. The W e s s - Z u m i n o model
The lagrangian for the W e s s - Z u m i n o ( W Z ) model is
L= ½0,,O~'O*+½iq/~gt- ¼22(00")2 + I20~PL q/+ ½20" ~TPR~//,
(2.1)
where 0 is a complex scalar field and gt is a M a j o r a n a fermlon. Let us examine the I P I (q~Vq/) Green's function. The presence of the PLR projection operators means that there is no one-loop contribution, at the two-loop level we encounter the graphs shown in figs. 1 and 2. F r o m fig. l we obtain a contribution A~,
25 f d4kd4qPL(l((I-]~)(¢+(i-]~)(¢+l~-]~) k2q2(p_q)2(q_k)2(s+q_k)2(p+s_k)2,
(2.2)
z l l - (2zr)8
and from fig. 2 a contribution A2,
A2-
~5 f
d4kd4qpL~!
(2.3)
k2q2(k_q)2(p_k)2(s+p_q)2.
(2zr)8
The evaluation o f A~ and A2 is a non-trivial problem. However if we set the f o u r - m o m e n t u m o f the external scalar, s, to zero, then we find A=AI
25
f
d 4 k d n q ~/~
+A2= ( ~ - ~ Pc J k 2 q 2 ( k - q ) 2 ( p - q ) 2 ( p - k ) 2"
(2.4)
Note the cancellaUon o f the potentially divergent terms, and the fact that no spurious infra-red divergences have been introduced by setting s = 0. F r o m (2.4),
P
\
l III s
S
.
.
.
.....
42
.
.
.
.
.
.
.
Fig 1 Two-loop c o n m b u t m n s / q to the Yukawa vertex renormahsaUon in the W e s s - Z u m m o model Dashed hnes are scalars, sohd lines fermlons
Fig 2 Two-loop contribution z12 to the Yukawa vertex renormallsatmn m the W e s s - Z u m m o model Dashed and sohd hnes as in fig 1
383
Volume 258, number 3,4
PHYSICS LETTERS B
25PL A-½P 2 (27r)s J
3 ( ( 3 ) 2spL - (16~2)2 •
d4kd4q f k2q2(k-q)2(p-q)2(p-k)2
l 1 April 1991
(2 5)
Eq. (2.5) is the key to the breakdown of the NR theorem If the propagators were massive, then the factor p2 arising from the Dlrac algebra would guarantee that A could not be a renormalisatmn of the ~'~'0 vertex; but of course in the massless case the hmit p ~ 0 leaves afinlte result, and we thus obtain a finite renormahsatlon of the ~'qJ0 1PI Green's function. Thus in the effective action, ½20q/PL~'~
2+ (16n2)~+
. 0qTPLgt.
(26)
But of course renormallsatlon of this vertex corresponds precisely to a renormahsation of the superpotentml, of the same theory written m superfield form. Thus as promised we see that the NR theorem ~s vmlated by finite terms, calculable in perturbation theory. We have used the component form of the actmn in our calculation, ~t ~s straightforward to repeat using superfields, when the sole graph has the topology of fig 1
3. Naturalness In this section we consider the lmphcatlons of this result for model building It is clear that the considerations of the previous secUon are relevant whenever there are graphs contributing to 4 3 Green's functions with massless fields. Gauge fields are massless, as are the quark and lepton superfields in the supersymmetrlc standard model, it is clearly the case. therefore, that much previous work will require reexamination. For example, undesirable terms are someUmes omitted from the superpotential W, even when not forbidden by some symmetry, and the NR theorem revoked to forbid their generatmn through radiative correcuons [6]. This procedure is clearly not vahd In general One interesting consequence is the following' ~t ~s generally accepted that ff a theory has a s u p e r s y m m e m c ground state in the tree approximation, then supersymmetry is unbroken, at least m perturbatmn theory. It ~s apparently possible, however, that the generatmn of terms omitted from W by the mechamsm of section 2 will lead to spontaneous supersymmetry breaking via the O'Raffeartalgh [7] mechamsm. One could, of course, restore such a term at the tree level and fine-tune the total c o n t n b u t m n Including radiative corrections to zero: a somewhat unaesthetlc procedure, however. Low energy supersymmetry ~s popular because it resolves the naturalness problem the questmn of why the mass-scale of electroweak phenomena, provided ultimately by the Hxggs mass, does not undergo radiative correctmns p r o p o m o n a l to heawer masses perhaps even the Planck mass Th~s problem is resolved within supersymmetry because the relevant radmtlve corrections are only logarithmically d~vergent The finite renormahsatlon of the superpotentml described here does not affect this argument, and hence does not detract m general from the deslrablhty of low energy supersymmetry Nevertheless, there is much in prevmus work that reqmres re-examinat~on It is probably necessary m most cases to construct superpotentlals w~th the same philosophy that we use to construct lagranglans, include every possible term consistent with the symmetries of the theory.
References [ 1 ] J Wessand B Zumlno, Phys Lett B 49 (1974) 52,
J lhopoulosand B Zummo, Nucl Phys B 76 (1974)310, S Ferrara, J lhopoulosand B Zumlno, Nucl Phys B 77 (1974)413, B Zummo, Nucl Phys B 89 (1975) 535 384
Volume 258, number 3,4
PHYSICS LETTERS B
11 April 1991
P West, Nucl Phys B 106 (1976) 219, S Ferrara and O Plguet, Nucl Phys B 93 (1975) 261, M T Grlsaru, W Siegel and M Ro~ek, Nucl Phys B 159 (1979) 429 [2 ] D.R T Jones, Supersymmetry, in. The standard model and beyond, Proc 1986 BUSSTEP, ed W Zakrewskl (Adam Hllger, Bristol, 1987) [3] P S Howe and P West, Phys Lett B 227 (1989) 379 [4] P West, Phys Lett. B 258 ( 1991 ) 375 [ 5 ] J. Wess and B Zumlno, Nucl Plays B 70 (1974) 39 [6] For reviews, see for example H P Nllles, Plays Rep. 110 (1984) 1, P C West, Supersymmetry. a decade of development (Adam Hilger, Bristol, 1986), S Ferrara, Supersymmetry (World Scientific, Singapore, 1987) [7] L O'Ralfeartaigh, Nucl Phys B 96 (1975) 331
385
PHYSICS LETTERS B
l l April 1991
Not the no-renormalisation theorem? I. Jack Theory Divtston, CERN, CH-1211 Geneva23, Switzerland
D.R.T. Jones DAMTP. Untversttyof Ltverpool, Ltverpool L69 3BX, UK
and P. West Department of Mathemattcs, Kmg's College, London WC2R 2LS, UK Received 8 December 1990
We show that if the lagranglan of a supersymmetnc theory contains massless fields then the superpotentml suffers fimte renormahsatlons m general
1. Introduction T h e n o - r e n o r m a l l s a t i o n ( N R ) t h e o r e m [ 1 ] has played a central role m N = 1 s u p e r s y m m e t r y , particularly with regard to p h e n o m e n o l o g l c a l a p p l i c a t i o n s Briefly, the c o n t e n t o f the t h e o r e m is as follows: In a r e n o r m a h s a b l e s u p e r s y m m e t r l c theory, all c o n t r i b u t i o n s to the effective action f r o m loop d m g r a m s can be written in the f o r m
Ac;= ~ d40d4xl .d4xn G(xl ..... x,,, O, O) ,
(1 1 )
and no c o n t r t b u t l o n s can artse o f the f o r m o f the superpotentlal, W
(1 2)
A , = j- dZ0d4x W ( x , 0) + c . c .
T h e r e is, however, an i m p o r t a n t e x c e p t t o n to the a b o v e assertion. W h e n massless fields are present ~1, the following f o r m o f A~, can arise:
A,; : f d 4 0 d 4 x l D 2 W ( ~ ) + c c
,
(1.3)
where ~ is a chlral superfield, d e f i n e d by the c o n d i t i o n I ) q s = 0. D, 1)are the usual c o v a r l a n t s p l n o r d e r i v a t i v e s R e p l a c i n g fd40--,fd20 02 and using I)2D2qb= I-1~, we h a v e at o n c e that
( 1.4)
A<, : f d20 d4x 1 ~ ( ~ ) + c . c . , d
~1 That massless fields pose a possible problem for the NR theorem was not unknown see for example ref [2] 382
0370-2693/91/$ 03 50 © 1991 - Elsevier Science Pubhshers B V ( North-Holland )
Volume 258, number 3,4
PHYSICS LETTERS B
11 April 1991
which is manifestly o f the form (1.2) and thus constitutes a renormahsatlon o f the superpotential W. Elementary power counting shows that such a term will be finite, if it exists: but even a finite term would be a vlolaUon o f the N R theorem as usually stated. In ref. [3 ] it was in fact noted that for a whole class o f graphs just such a violation o f the N R theorem does take place. In a recent paper [4], one o f the authors has given a more detailed examination of these graphs and has indicated the loopholes m both previous proofs of the N R theorem whxch allow radiative corrections to the effective potenUal. In this paper we demonstrate explicitly that breakdown o f the N R theorem occurs at the two-loop level in the sxmplest possible four-dlmensmnal theory: the massless W e s s - Z u m m o model [ 5 ].
2. The W e s s - Z u m i n o model
The lagrangian for the W e s s - Z u m i n o ( W Z ) model is
L= ½0,,O~'O*+½iq/~gt- ¼22(00")2 + I20~PL q/+ ½20" ~TPR~//,
(2.1)
where 0 is a complex scalar field and gt is a M a j o r a n a fermlon. Let us examine the I P I (q~Vq/) Green's function. The presence of the PLR projection operators means that there is no one-loop contribution, at the two-loop level we encounter the graphs shown in figs. 1 and 2. F r o m fig. l we obtain a contribution A~,
25 f d4kd4qPL(l((I-]~)(¢+(i-]~)(¢+l~-]~) k2q2(p_q)2(q_k)2(s+q_k)2(p+s_k)2,
(2.2)
z l l - (2zr)8
and from fig. 2 a contribution A2,
A2-
~5 f
d4kd4qpL~!
(2.3)
k2q2(k_q)2(p_k)2(s+p_q)2.
(2zr)8
The evaluation o f A~ and A2 is a non-trivial problem. However if we set the f o u r - m o m e n t u m o f the external scalar, s, to zero, then we find A=AI
25
f
d 4 k d n q ~/~
+A2= ( ~ - ~ Pc J k 2 q 2 ( k - q ) 2 ( p - q ) 2 ( p - k ) 2"
(2.4)
Note the cancellaUon o f the potentially divergent terms, and the fact that no spurious infra-red divergences have been introduced by setting s = 0. F r o m (2.4),
P
\
l III s
S
.
.
.
.....
42
.
.
.
.
.
.
.
Fig 1 Two-loop c o n m b u t m n s / q to the Yukawa vertex renormahsaUon in the W e s s - Z u m m o model Dashed hnes are scalars, sohd lines fermlons
Fig 2 Two-loop contribution z12 to the Yukawa vertex renormallsatmn m the W e s s - Z u m m o model Dashed and sohd hnes as in fig 1
383
Volume 258, number 3,4
PHYSICS LETTERS B
25PL A-½P 2 (27r)s J
3 ( ( 3 ) 2spL - (16~2)2 •
d4kd4q f k2q2(k-q)2(p-q)2(p-k)2
l 1 April 1991
(2 5)
Eq. (2.5) is the key to the breakdown of the NR theorem If the propagators were massive, then the factor p2 arising from the Dlrac algebra would guarantee that A could not be a renormalisatmn of the ~'~'0 vertex; but of course in the massless case the hmit p ~ 0 leaves afinlte result, and we thus obtain a finite renormahsatlon of the ~'qJ0 1PI Green's function. Thus in the effective action, ½20q/PL~'~
2+ (16n2)~+
. 0qTPLgt.
(26)
But of course renormallsatlon of this vertex corresponds precisely to a renormahsation of the superpotentml, of the same theory written m superfield form. Thus as promised we see that the NR theorem ~s vmlated by finite terms, calculable in perturbation theory. We have used the component form of the actmn in our calculation, ~t ~s straightforward to repeat using superfields, when the sole graph has the topology of fig 1
3. Naturalness In this section we consider the lmphcatlons of this result for model building It is clear that the considerations of the previous secUon are relevant whenever there are graphs contributing to 4 3 Green's functions with massless fields. Gauge fields are massless, as are the quark and lepton superfields in the supersymmetrlc standard model, it is clearly the case. therefore, that much previous work will require reexamination. For example, undesirable terms are someUmes omitted from the superpotential W, even when not forbidden by some symmetry, and the NR theorem revoked to forbid their generatmn through radiative correcuons [6]. This procedure is clearly not vahd In general One interesting consequence is the following' ~t ~s generally accepted that ff a theory has a s u p e r s y m m e m c ground state in the tree approximation, then supersymmetry is unbroken, at least m perturbatmn theory. It ~s apparently possible, however, that the generatmn of terms omitted from W by the mechamsm of section 2 will lead to spontaneous supersymmetry breaking via the O'Raffeartalgh [7] mechamsm. One could, of course, restore such a term at the tree level and fine-tune the total c o n t n b u t m n Including radiative corrections to zero: a somewhat unaesthetlc procedure, however. Low energy supersymmetry ~s popular because it resolves the naturalness problem the questmn of why the mass-scale of electroweak phenomena, provided ultimately by the Hxggs mass, does not undergo radiative correctmns p r o p o m o n a l to heawer masses perhaps even the Planck mass Th~s problem is resolved within supersymmetry because the relevant radmtlve corrections are only logarithmically d~vergent The finite renormahsatlon of the superpotentml described here does not affect this argument, and hence does not detract m general from the deslrablhty of low energy supersymmetry Nevertheless, there is much in prevmus work that reqmres re-examinat~on It is probably necessary m most cases to construct superpotentlals w~th the same philosophy that we use to construct lagranglans, include every possible term consistent with the symmetries of the theory.
References [ 1 ] J Wessand B Zumlno, Phys Lett B 49 (1974) 52,
J lhopoulosand B Zummo, Nucl Phys B 76 (1974)310, S Ferrara, J lhopoulosand B Zumlno, Nucl Phys B 77 (1974)413, B Zummo, Nucl Phys B 89 (1975) 535 384
Volume 258, number 3,4
PHYSICS LETTERS B
11 April 1991
P West, Nucl Phys B 106 (1976) 219, S Ferrara and O Plguet, Nucl Phys B 93 (1975) 261, M T Grlsaru, W Siegel and M Ro~ek, Nucl Phys B 159 (1979) 429 [2 ] D.R T Jones, Supersymmetry, in. The standard model and beyond, Proc 1986 BUSSTEP, ed W Zakrewskl (Adam Hllger, Bristol, 1987) [3] P S Howe and P West, Phys Lett B 227 (1989) 379 [4] P West, Phys Lett. B 258 ( 1991 ) 375 [ 5 ] J. Wess and B Zumlno, Nucl Plays B 70 (1974) 39 [6] For reviews, see for example H P Nllles, Plays Rep. 110 (1984) 1, P C West, Supersymmetry. a decade of development (Adam Hilger, Bristol, 1986), S Ferrara, Supersymmetry (World Scientific, Singapore, 1987) [7] L O'Ralfeartaigh, Nucl Phys B 96 (1975) 331
385