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Non-linear renormalisation and the equivalence theorem
Nuclear Physics B266 (1986) 536-546 © North-Holland Publishing Company
NON-LINEAR RENORMALISATION AND THE EQUIVALENCE THEOREM Guy B O N N E A U and Francois D E L D U C
Laboratoire de Physique Thborique et Hautes Energies, Paris*, France Received 26 August 1985
The consequences of a field redefinition on the renormalisability character of a theory are thoroughly studied in the special example of the non-linear a model in two dimensions. Using a minimal-dimensional method, we show that, although multiplicative renormalisability is lost, the divergences of the new field theory can be reabsorbed through a field-dependent renormalisation of the fields and, for the physical parameters, the same multiplicative renormalisation constants as in the original theory, to all orders of perturbation theory.
1. Introduction The equivalence theorem [1] asserts that the S-matrix of a theory is left unchanged through a field redefinition. On the contrary, the renormalisability properties are generally modified*: as was shown in ref. [2] for the special example of the non-linear o model in two dimensions for O ( N + 1)/O(N), the multiplicative renormalisability is lost. In fact new divergences appear and their subtraction would involve new parameters (new normalisation conditions), a situation characteristic of non-renormalisable theories. However, the latter quantum field theory being physically as good as the former, we know that these parameters will be unphysical and harmless. Then one would like to attach to the transformed theory some renormalisability character. Using a minimal-dimensional method, we will show that the divergences of the new theory can be reabsorbed through a field-dependent renormalisation of the fields and, for the physical parameters, masses and coupling constants, the same multiplicative renormalisation constants as in the original one: we find it natural to call this theory renormalisable up to a field redefinition. * Postal address: Universit~ Paris VII, Laboratoire associ6 au CNRS, UA 280, 2, place Jussieu, Tour 14, 56me ~tage, 75251 Paris Cedex 05, France. * The equivalence theorem does not address this question as it (only) proves that in the redefinition ~ ~k= F(ff), Green functions of ~ fields in the original theory described by E0(q~) are the same as Green functions of ~b= F ( ~ ) in the transformed theory described by E0(~k) ~-Ez(qs). It tells us nothing about Green functions of q~ in the transformed theory EI(~). 536
G. Bonneau, F. Delduc / Non-linear renormalisation
537
Although only proved for one specific model, we are confident of the general validity of our result: a field theory equivalent of a multiplicatively renormalisable one is renormalisable up to a field redefinition. The new parameters needed to renornaalise the theory will be unphysical, thanks to the equivalence theorem. Since we know from pure gravity or supergravities (at least up to one- or two-loops) that a finite S-matrix can exist even in a non-renormalisable model, we hope that this extension of the notion of renormalisability can be of some use in less trivial examples than the one under study here. The paper is organized as follows: the lagrangian of the model and the main results of ref. [2] are recalled in sect. 2. The Ward identities that constrain the generating functionals of Green functions are derived in sect. 3 and we obtain their general solution depending on 2 physical parameters m 2 and g2 and an infinite number of unphysical ones b,, n = 0 . . . . . oo. Using a minimal-dimensional method the renormalisibility of the model is proved, and the Callan-Symanzik equation given, to all orders of perturbation theory (sect. 4).
2. The model
In the usual parametrisation of the coset space O(N + 1 ) / O ( N ) with a N-components field H i ( o = I/1 - II 2 ), the lagrangian density is:
E=-t 1(0fii)2+2
1-II 2
+m2 1-~--~
(1)
6v/1 - II 2 = -13. I I .
(2)
and the non-linear symmetry reads: 8ri = 13v/1 - II 2 ,
The multiplicative renormalisability has been proved for any N in ref. [3]: there are only two renormalisation constants, Z n and Z t (as Zm2 = Z t / Z ~ 2 ) . A one-loop calculation gives* t
Zn=I-N4~
-
-
t
e,
Zt=l-(U-1)-~
e
==~
Zm2-1
--
t ( U - Z ) 8~re
(3)
Through the following field redefinition (stereographic parametrisation introduced by Schwinger in ref. [4]): II l + ' l -2I' I~ ' I I V
2~ l + c P 2'
(4)
* W e use d i m e n s i o n a l regularisation: f d 2 k / ( 2 ~ ) 2 ~ f ( d O k / ( 2 e r ) O ) i t 2 - ° and e = 1 - ID . 1
538
G. Bonneau, F. Delduc / Non-linearrenormalisation
the lagrangian becomes (t = 4g 2) ±m2 ~2 ] = 2 1+q~2] e l + m 2 e 2 •
1 1 (O~cp)2 e=~-2 ~ ( 1 + ~ 2 ) 2
The non-linear symmetry that leaves
(5)
~1 invariant is now
8%=[1(1- ~2)3i,+%e%]Bj_l 1 ~t = ~B~ + ~R~j~OtBj
(6)
but it does not close: then an infinite number of successive operators will be involved in the renormalisability discussion! In spite of this, in ref. [2] the N = 2 case ( - CP(1)) has been shown to be multiplicatively renormalisable to all orders of perturbation theory. A one-loop calculation with dimensional regularisation gives the following expression for the necessary counter-term lagrangian density:
[ 4g2]e £+ke=
l+-~e]
492 1 [~ (?.q~)2
+ ( N - 2 ) ~ - ~ e 2g z
1+~2
(~ altO) 2 (1+~2) 2
(7)
A bare lagrangian is then obtained (refs. [2, 5]):
fd2x (e+ ae)=fd2xeo(g2,m~,wo),
[
qg0=cp 1 - ( N - 2 )
(8a)
]
(1+q~2) ,
(8b)
g2=g2[1-(N-1) 492]
(8c)
m2=mZ[1-(N-2)
(8d)
492 ]
As previously mentioned [2], the renormalisation constants Zg2 and Z,,2, related to the fl and y,, functions of the Callan-Symanzik equation, are the same as the corresponding ones in the lI, o parametrisation (eq. (3)). With quantization in mind, let us now make a systematic study of the non-linear symmetry (6) (for N > 2).
539
G. Bonneau, F. DeMuc / Non-linear renormalisation
3. Ward identities
The generating functional for the Green functions of model (5) is:
w--S°~fDl..°~OigNexp-~Sd2x { 1
(au~D) 2
2 (1 "}- {~D2)2
+JA ~1 - - -1 + ~ 2
qD2 i r a 2 i q._ (p2 -4- JIq) i
+J~l+g~2 t-
k=2
J(k)
eP6""%,
, (9)
as under the non-linear symmetry (eq. (6)): }IX i j ~ k ~ l ,
R~/=
8~al+ ~',~/ - ¢,~*',
(10)
the soft breaking mass term transformation closes in two steps: 3i I + 1])2 - 1 + ~2 ,
3~ 1 + 11)2
8ij 1 _ 1 .}_q)2 q)2
"
The Ward identities are readily obtained: (11)
6~lfiW= 0,
% f d=x
8 3 i 3 ji ~ + .._~._f~2ji + ± R k t l j 2, i j ~ ,Rikl - ½ ( m 2 + 2JA)~-~ + BSJ A 2g vo(2,
+ k=2 l k
J(i~j"i'~-'i~j(i~z].;,¢,_ , --O(k)
ij
~(k+l)
,k,)]
One then defines the generating functional for connected Green functions Z and the one for 1PI Green functions F through i W = exp --7 g - Z,
r = z - f d~x4'x#, ( x ) , where 3Z ~ , ( x ) = 34 %
540
G. Bonneau, F. Delduc / Non-linearrenormalisation
and one finally obtains:
6F fd2x
-½(m2+24(~))~
3F
3F
6F
1
+J~(~)3J~(x/
2 3¢(x)
3F
--'-ak' +4~(x)+;(x) + kE: 3 ½G'~; "''~ " 2 ij ~(pj(X) ~4k~
=
"-[- E llrlil'"ik-lJl~ikik+l 2'~(k) "'q
.~F .
3F
] = 0.
(12)
These Ward identities are now to be understood as defining the model. One then looks for the general solution of eq. (12) compatible with O(N) symmetry and power counting (the sources JA(X), J~(x), J(i~ji~(x) have canonical dimension 2 and transform respectively as a scalar, vector and k-symmetric tensor under the linear O(N) symmetry). The most general F, polynomial in the sources, fields and their derivatives and fulfilling these requirements is:
r= f d2~ ( a.~) 2/(~ 2 ) + ( O.~2)2g(~p2) + m2h(cp 2) +JA(x)A(~2) + JB(x)%B(~2) + k=2 ~-- J(~)'' 6T(~)i... ik((p)] , (13) where T ~ i ¢(~p) is a symmetric tensor of order k built with q~: E(k/2)
T(2iik(oP) = E
.k (op2 ). (3i,i:...Si:p_d2p)%:,÷,. ..1~ ,kJp
(14)
p=O
The solution of constraints (12) may be written as:
r = fd~x
K
(1 +~2)2
½~'m2
+~" JA ½ 1 + ~ 2
+
k=2
+ B1+~2
i ...ik^i ^ } (15) J~) r&...i k (~o),
G. Bonneau, F. Delduc / Non.finear renormalisation
541
where
ep,= ~p,b(~2),
b(~ 2) being an arbitrary function with b(0) ~ 0, r, h
are two arbitrary constants,
(16a) (16b)
E(k/2) T(~)'-.ik(~) =
E (~ili2""~i2p_,i2p)*i2,+l '' • ff ik ~*' p t ~ ^2') p=O
(16c)
and the functions ~k(y) are given by the following recursion relations: d
~.k
^
kjpk+l = (1 + y) ~yyJt; + ( k - 2p))~,
k>~2,
(N-2p)j~+l+(1-y)~_l+yj~_+l-]~_-~=Sk,2b(Y), A.1 Jo ---0,
^z
d
jd = ~ y ((1 +y)b(y)),
(17a) k>~2,
(17b)
db(y) (N - 2)j1: = 1 - b(y) - y ( 1 + y ) - dy (17c)
Eq. (17a) with p = 0 gives
]°k=(k
1 1)~ (d)kl ~
[(I+y)k-lb(Y)]'
k>~2.
(18)
Then eq. (17b) fixes .i~ when p 4: 0. The remaining constraints (17a) for p ~ 0 are then shown to be automatically satisfied V p <~E(½k). Finally solution (15) may be written as
1
=fd2x
(0,,,~) ~
2g 2 (1+~2) 2
1+
mo
+L ½
2g~ 1 + ~2
k=2
,~: 1+
G. Bonneau, F. Delduc / Non-linearrenorrnalisation
542 with
oo % = ffib(ff2),
b(y)=
E b.y",
n~O
bo*O,
1+~ 2 ^ JA = •JA + 2 71 "- ~- "- ~- ~- J,o)(tP), ^i = ~"JBi + (1 + ~ 2 ) . ~ ) ( f f ) JB ""ik ~
(2p+k)V
E
p=0 2p+k>~2
1
2gc2 = x,
/
-2Pp!k! - - - " ~ 2 p +Jpk ( ~ n 2~,w ~ J 'Ii l '(2p+k) " i 2 ° + k [ ~p
2= X
mc
)
teI~Irms ~,
k=0,1...~,
2
(20)
2---~-m .
Then the classical theory is to be defined by I', depending on two physical parameters g2 and m 2c and an infinite number o f - unphysical, thanks to the equivalence theorem [1] - parameters b, (n = 0,1... ). In ref. [2], as in sect. 2 of this paper, all b, were chosen to be zero (but for b 0 = 1): this does not survive renormalisation* as shown in eq. (8b). The Ward identity (12) takes the same form on I' when the fields and sources are changed to the corresponding hatted ones according to definitions (20) and rn2, g2 being replaced by m 2, gc2. Notice from eqs. (17) that the choice b ( y ) = b0/(1 + y ) makes all ]~(y) vanish (except j2). Then in eq. (19), which defines I', the sources J(k ) (k >/2) disappear. This is not surprising since the change ffi ~ qoi= bo#Pi/(1 + if2) leads back to the original H i, a parametrisation (see eq. (4)) whose transformations close (eq. (2)).
4. All-order analysis of the model As a consequence of the general study of the previous section and of the use of a regularisation that preserves the non-linear symmetry (10), one obtains the renormalisability of the theory with two renormalisation constants Zs2 and Z,,2 and a field dependent renormalisation of the field, to all orders of perturbation theory. The Callan-Symanzik equation also follows from the previous analysis. Indeed let us look for a complete basis of symmetric operators on Green functions A q:
[ ~ , Aq] =0, where H i is given by eq. (11). * We recall that we stay here in a minimal scheme,withoutexplicit normalisationconditions.
(21)
543
G. Bonneat~ F. Delduc / Non-linear renormalisation
They are readily found to be ~ 3 g2~0 + k~="=l f d2xJ'~i'*(x) M,~;'*(x)'
m2
[
---50 gm + fd2x &(x)s&(x )- +4(x)~ 0
02'
(22a)
,
n = 0 , 1 ....
(22b)
(22c)
in one to one correspondence with the parameters of the general solution (19). On the other hand, as it commutes with ~21f,,/~20/0#2 is a symmetric operator. As a consequence we can expand /~20/0/~2 on the previous complete basis and we obtain 2 /.t ~
3 ] f d2xJ,i~i"i*3jizj..i ~
2 0 + fl(g2) g ~g2 +
a
W(S,,jA,4, jai,k)=O"
(23)
n=O
The independence of fl and 3',, with respect to br results from the equivalence theorem and the use of a minimal scheme (c.f. the independence of fl with respect to the gauge parameter a in gauge theories*). In particular these functions are the same as the ones of the II, o parametrisation (choose b(y)= b0/(1 + y ) as indicated in the previous section). On ordinary proper Green functions (without any insertions), eq. (23) reduces to the usual form:
~+fl(g2)
g2
Ogz
-2N
+ym(g2)m 2
2
0 } 1 F 21v
+ ~ y,(br, g ) - ~ n=O
" One can mimic the proof given in ref. [7].
8m 2
7 i.... 6~tpl,...,p2N)=O.
(24)
544
G. Bonneau, F. Delduc / Non-linearrenormalisation
Let us show these results on a one-loop calculation with lagrangian
~(~i)-~(~i)-~"
_ _
2g 2 (1 + * 2 ) 2
m
2
_
1 +,2
,
(25)
where qo, = ffib(ff2). For the general two-dimensional model of a bosonic field ~
E(~p)=- ~l [ss~igij(~)cg~J-m2h(@) ]
(26)
where gij is a metric on the field manifold and h ( ~ ) a scalar function, the one-loop counterterm lagrangian can be written as (refs. [8, 9]):
L F'i r, jk 1 4rre X 4*-jki5 ]
-- - -
,,211
+ --4~re - --2g2
m2 k
O~'~iRij a~'Oj + -4-~g2g Dkh't
, (27)
where h d = Oh(~)/O,~ t and D k, Fjk, Rij are the usual covariant derivative, connection and Ricci tensor associated to the metric g~j(~) and g~jgjk = 8k. The second part of expression (27), being a scalar, is invariant under reparametrisations ~ --+ ffi and then in our case where the manifold is the coset O(N + 1 ) / O ( N ) (eq. (25)) one finds
Oiz~i Ol~J Rij({~ ) ~ 0 ~ i Ol.,~J R O ( , ) ,
(28a)
with
Rij(~ ) = - 8(N - 1 ) g i j ( , ) , gkt(~)Dk(~)h,t(~ ) = gkt(~)Dk(~)h t(, ) = - 8 N h ( * ) + 2N.
(28b)
Then the renormalisation constants for the physical parameters m 2 and g2 are the same as in sect. 2 (eqs. (8c), (8d))
4hg 2 Z,.~ = 1 - ( N - 2) 8ere
(29a)
4hg 2 Zg~ = 1 - ( N - 1) 4~re
(29b)
G. Bonneau, F. DeMuc / Non-linear renormalisation
545
The first part of expression (27) gives a non-linear field renormalisation: cp;=cp i 1 -
(N-2)(l+y)
l+2y-ff
- ( l + y ) 2 (N+2)~--+2Y--b--l] ]. (29c)
Defining b°(y) through 9~o-- ~ v ~ u d
I
(30)
one obtains
b°(y)=b(y)
-
(N-2)(l+y)(b+2yb')-(l+y)2[(N+2)b'+2yb"]].
(31) After expansion in powers of y, one finds the renormalisation of the parameters bn. Finally the Callan-Symanzik coefficients ~/, are given at one-loop by their generating function:
~,7.(b. gE)y"=y(y,b, g2),
(32a)
n
with hg 2 y ( y , br, g2) = - ~ [ ( N - 2 ) ( 1 + y ) ( b + 2 y b ' ) - ( 1 + y ) 2 [ ( N + 2)b'+ 2yb"]].
(328) Notice from eq. (31) that the choice b(y) = b0/(1 + y) leads to a linear renormalisation of the field: hg 2 b°(y) = b ( y ) 1 - 2N-~-~ e , or
in agreement with eq. (3). As previously mentioned, this is not surprising.
546
G. Bonneau, F. Delduc / Non-linear renormalisation
5. Concluding remarks We have shown how, after a field reparametrisation of a multiplicatively renormalisable theory, the divergences of the new one can be reabsorbed through a field dependant renormalisation of the field and, for the physical parameters, masses and coupling constants, the same multiplicative renormalisation constants as in the original one. Such a renormalisabihty with a non-linear field redefinition was also exhibited by Piguet and Sibold in their analysis of supersymmetric Yang Mills theories [10]. There, as in bosonic models in 2 space-time dimensions, the canonical dimension of the (super) field vanishes. But in that case, the theory is not a field redefinition of a multiplicatively renormalisable parametrisation. Rather, the arbitrary parameters can be shown to be gauge parameters and the non-linear field redefinition associated to a change of gauge. We like to speculate that this non-linear field renormalisation might also occur in some power counting non-renormalisable theory where all but a finite number of the infinites would be unphysical.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10]
M.C. Bergrre and Y.M.P. Lam, Phys. Rev. D13 (1976) 3247 and references therein G. Bonnean, Nucl. Phys. B221 (1983) 178 E. Br~zin, J.C. Le GuiUou and J. Zinn Justin, Phys. Rev. D14 (1976) 2615 J. Schwinger, Phys. Lett. 24B (1967) 473; Phys. Rev. 167 (1968) 1432 W.A. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D14 (1976) 985 Y.M.P. Lam, Phys. Rev. D6 (1972) 2145; D7 (1973) 2943 D.J. Gross, 1975 Les Houches Summer School, Methods in field theory, ed. R. Balian and J. Zinn Justin (North-Holland, Amsterdam, 1976) pp. 193-194 G. Ecker and J. Honerkamp, Nucl. Phys. B35 (1971) 481; D. Friedan, Phys. Rev. Lett. 45 (1980) 1057 G. Bonneau and F. Delduc, in preparation O. Piguet and K. Sibold, Nucl. Phys. B248 (1984) 301
NON-LINEAR RENORMALISATION AND THE EQUIVALENCE THEOREM Guy B O N N E A U and Francois D E L D U C
Laboratoire de Physique Thborique et Hautes Energies, Paris*, France Received 26 August 1985
The consequences of a field redefinition on the renormalisability character of a theory are thoroughly studied in the special example of the non-linear a model in two dimensions. Using a minimal-dimensional method, we show that, although multiplicative renormalisability is lost, the divergences of the new field theory can be reabsorbed through a field-dependent renormalisation of the fields and, for the physical parameters, the same multiplicative renormalisation constants as in the original theory, to all orders of perturbation theory.
1. Introduction The equivalence theorem [1] asserts that the S-matrix of a theory is left unchanged through a field redefinition. On the contrary, the renormalisability properties are generally modified*: as was shown in ref. [2] for the special example of the non-linear o model in two dimensions for O ( N + 1)/O(N), the multiplicative renormalisability is lost. In fact new divergences appear and their subtraction would involve new parameters (new normalisation conditions), a situation characteristic of non-renormalisable theories. However, the latter quantum field theory being physically as good as the former, we know that these parameters will be unphysical and harmless. Then one would like to attach to the transformed theory some renormalisability character. Using a minimal-dimensional method, we will show that the divergences of the new theory can be reabsorbed through a field-dependent renormalisation of the fields and, for the physical parameters, masses and coupling constants, the same multiplicative renormalisation constants as in the original one: we find it natural to call this theory renormalisable up to a field redefinition. * Postal address: Universit~ Paris VII, Laboratoire associ6 au CNRS, UA 280, 2, place Jussieu, Tour 14, 56me ~tage, 75251 Paris Cedex 05, France. * The equivalence theorem does not address this question as it (only) proves that in the redefinition ~ ~k= F(ff), Green functions of ~ fields in the original theory described by E0(q~) are the same as Green functions of ~b= F ( ~ ) in the transformed theory described by E0(~k) ~-Ez(qs). It tells us nothing about Green functions of q~ in the transformed theory EI(~). 536
G. Bonneau, F. Delduc / Non-linear renormalisation
537
Although only proved for one specific model, we are confident of the general validity of our result: a field theory equivalent of a multiplicatively renormalisable one is renormalisable up to a field redefinition. The new parameters needed to renornaalise the theory will be unphysical, thanks to the equivalence theorem. Since we know from pure gravity or supergravities (at least up to one- or two-loops) that a finite S-matrix can exist even in a non-renormalisable model, we hope that this extension of the notion of renormalisability can be of some use in less trivial examples than the one under study here. The paper is organized as follows: the lagrangian of the model and the main results of ref. [2] are recalled in sect. 2. The Ward identities that constrain the generating functionals of Green functions are derived in sect. 3 and we obtain their general solution depending on 2 physical parameters m 2 and g2 and an infinite number of unphysical ones b,, n = 0 . . . . . oo. Using a minimal-dimensional method the renormalisibility of the model is proved, and the Callan-Symanzik equation given, to all orders of perturbation theory (sect. 4).
2. The model
In the usual parametrisation of the coset space O(N + 1 ) / O ( N ) with a N-components field H i ( o = I/1 - II 2 ), the lagrangian density is:
E=-t 1(0fii)2+2
1-II 2
+m2 1-~--~
(1)
6v/1 - II 2 = -13. I I .
(2)
and the non-linear symmetry reads: 8ri = 13v/1 - II 2 ,
The multiplicative renormalisability has been proved for any N in ref. [3]: there are only two renormalisation constants, Z n and Z t (as Zm2 = Z t / Z ~ 2 ) . A one-loop calculation gives* t
Zn=I-N4~
-
-
t
e,
Zt=l-(U-1)-~
e
==~
Zm2-1
--
t ( U - Z ) 8~re
(3)
Through the following field redefinition (stereographic parametrisation introduced by Schwinger in ref. [4]): II l + ' l -2I' I~ ' I I V
2~ l + c P 2'
(4)
* W e use d i m e n s i o n a l regularisation: f d 2 k / ( 2 ~ ) 2 ~ f ( d O k / ( 2 e r ) O ) i t 2 - ° and e = 1 - ID . 1
538
G. Bonneau, F. Delduc / Non-linearrenormalisation
the lagrangian becomes (t = 4g 2) ±m2 ~2 ] = 2 1+q~2] e l + m 2 e 2 •
1 1 (O~cp)2 e=~-2 ~ ( 1 + ~ 2 ) 2
The non-linear symmetry that leaves
(5)
~1 invariant is now
8%=[1(1- ~2)3i,+%e%]Bj_l 1 ~t = ~B~ + ~R~j~OtBj
(6)
but it does not close: then an infinite number of successive operators will be involved in the renormalisability discussion! In spite of this, in ref. [2] the N = 2 case ( - CP(1)) has been shown to be multiplicatively renormalisable to all orders of perturbation theory. A one-loop calculation with dimensional regularisation gives the following expression for the necessary counter-term lagrangian density:
[ 4g2]e £+ke=
l+-~e]
492 1 [~ (?.q~)2
+ ( N - 2 ) ~ - ~ e 2g z
1+~2
(~ altO) 2 (1+~2) 2
(7)
A bare lagrangian is then obtained (refs. [2, 5]):
fd2x (e+ ae)=fd2xeo(g2,m~,wo),
[
qg0=cp 1 - ( N - 2 )
(8a)
]
(1+q~2) ,
(8b)
g2=g2[1-(N-1) 492]
(8c)
m2=mZ[1-(N-2)
(8d)
492 ]
As previously mentioned [2], the renormalisation constants Zg2 and Z,,2, related to the fl and y,, functions of the Callan-Symanzik equation, are the same as the corresponding ones in the lI, o parametrisation (eq. (3)). With quantization in mind, let us now make a systematic study of the non-linear symmetry (6) (for N > 2).
539
G. Bonneau, F. DeMuc / Non-linear renormalisation
3. Ward identities
The generating functional for the Green functions of model (5) is:
w--S°~fDl..°~OigNexp-~Sd2x { 1
(au~D) 2
2 (1 "}- {~D2)2
+JA ~1 - - -1 + ~ 2
qD2 i r a 2 i q._ (p2 -4- JIq) i
+J~l+g~2 t-
k=2
J(k)
eP6""%,
, (9)
as under the non-linear symmetry (eq. (6)): }IX i j ~ k ~ l ,
R~/=
8~al+ ~',~/ - ¢,~*',
(10)
the soft breaking mass term transformation closes in two steps: 3i I + 1])2 - 1 + ~2 ,
3~ 1 + 11)2
8ij 1 _ 1 .}_q)2 q)2
"
The Ward identities are readily obtained: (11)
6~lfiW= 0,
% f d=x
8 3 i 3 ji ~ + .._~._f~2ji + ± R k t l j 2, i j ~ ,Rikl - ½ ( m 2 + 2JA)~-~ + BSJ A 2g vo(2,
+ k=2 l k
J(i~j"i'~-'i~j(i~z].;,¢,_ , --O(k)
ij
~(k+l)
,k,)]
One then defines the generating functional for connected Green functions Z and the one for 1PI Green functions F through i W = exp --7 g - Z,
r = z - f d~x4'x#, ( x ) , where 3Z ~ , ( x ) = 34 %
540
G. Bonneau, F. Delduc / Non-linearrenormalisation
and one finally obtains:
6F fd2x
-½(m2+24(~))~
3F
3F
6F
1
+J~(~)3J~(x/
2 3¢(x)
3F
--'-ak' +4~(x)+;(x) + kE: 3 ½G'~; "''~ " 2 ij ~(pj(X) ~4k~
=
"-[- E llrlil'"ik-lJl~ikik+l 2'~(k) "'q
.~F .
3F
] = 0.
(12)
These Ward identities are now to be understood as defining the model. One then looks for the general solution of eq. (12) compatible with O(N) symmetry and power counting (the sources JA(X), J~(x), J(i~ji~(x) have canonical dimension 2 and transform respectively as a scalar, vector and k-symmetric tensor under the linear O(N) symmetry). The most general F, polynomial in the sources, fields and their derivatives and fulfilling these requirements is:
r= f d2~ ( a.~) 2/(~ 2 ) + ( O.~2)2g(~p2) + m2h(cp 2) +JA(x)A(~2) + JB(x)%B(~2) + k=2 ~-- J(~)'' 6T(~)i... ik((p)] , (13) where T ~ i ¢(~p) is a symmetric tensor of order k built with q~: E(k/2)
T(2iik(oP) = E
.k (op2 ). (3i,i:...Si:p_d2p)%:,÷,. ..1~ ,kJp
(14)
p=O
The solution of constraints (12) may be written as:
r = fd~x
K
(1 +~2)2
½~'m2
+~" JA ½ 1 + ~ 2
+
k=2
+ B1+~2
i ...ik^i ^ } (15) J~) r&...i k (~o),
G. Bonneau, F. Delduc / Non.finear renormalisation
541
where
ep,= ~p,b(~2),
b(~ 2) being an arbitrary function with b(0) ~ 0, r, h
are two arbitrary constants,
(16a) (16b)
E(k/2) T(~)'-.ik(~) =
E (~ili2""~i2p_,i2p)*i2,+l '' • ff ik ~*' p t ~ ^2') p=O
(16c)
and the functions ~k(y) are given by the following recursion relations: d
~.k
^
kjpk+l = (1 + y) ~yyJt; + ( k - 2p))~,
k>~2,
(N-2p)j~+l+(1-y)~_l+yj~_+l-]~_-~=Sk,2b(Y), A.1 Jo ---0,
^z
d
jd = ~ y ((1 +y)b(y)),
(17a) k>~2,
(17b)
db(y) (N - 2)j1: = 1 - b(y) - y ( 1 + y ) - dy (17c)
Eq. (17a) with p = 0 gives
]°k=(k
1 1)~ (d)kl ~
[(I+y)k-lb(Y)]'
k>~2.
(18)
Then eq. (17b) fixes .i~ when p 4: 0. The remaining constraints (17a) for p ~ 0 are then shown to be automatically satisfied V p <~E(½k). Finally solution (15) may be written as
1
=fd2x
(0,,,~) ~
2g 2 (1+~2) 2
1+
mo
+L ½
2g~ 1 + ~2
k=2
,~: 1+
G. Bonneau, F. Delduc / Non-linearrenorrnalisation
542 with
oo % = ffib(ff2),
b(y)=
E b.y",
n~O
bo*O,
1+~ 2 ^ JA = •JA + 2 71 "- ~- "- ~- ~- J,o)(tP), ^i = ~"JBi + (1 + ~ 2 ) . ~ ) ( f f ) JB ""ik ~
(2p+k)V
E
p=0 2p+k>~2
1
2gc2 = x,
/
-2Pp!k! - - - " ~ 2 p +Jpk ( ~ n 2~,w ~ J 'Ii l '(2p+k) " i 2 ° + k [ ~p
2= X
mc
)
teI~Irms ~,
k=0,1...~,
2
(20)
2---~-m .
Then the classical theory is to be defined by I', depending on two physical parameters g2 and m 2c and an infinite number o f - unphysical, thanks to the equivalence theorem [1] - parameters b, (n = 0,1... ). In ref. [2], as in sect. 2 of this paper, all b, were chosen to be zero (but for b 0 = 1): this does not survive renormalisation* as shown in eq. (8b). The Ward identity (12) takes the same form on I' when the fields and sources are changed to the corresponding hatted ones according to definitions (20) and rn2, g2 being replaced by m 2, gc2. Notice from eqs. (17) that the choice b ( y ) = b0/(1 + y ) makes all ]~(y) vanish (except j2). Then in eq. (19), which defines I', the sources J(k ) (k >/2) disappear. This is not surprising since the change ffi ~ qoi= bo#Pi/(1 + if2) leads back to the original H i, a parametrisation (see eq. (4)) whose transformations close (eq. (2)).
4. All-order analysis of the model As a consequence of the general study of the previous section and of the use of a regularisation that preserves the non-linear symmetry (10), one obtains the renormalisability of the theory with two renormalisation constants Zs2 and Z,,2 and a field dependent renormalisation of the field, to all orders of perturbation theory. The Callan-Symanzik equation also follows from the previous analysis. Indeed let us look for a complete basis of symmetric operators on Green functions A q:
[ ~ , Aq] =0, where H i is given by eq. (11). * We recall that we stay here in a minimal scheme,withoutexplicit normalisationconditions.
(21)
543
G. Bonneat~ F. Delduc / Non-linear renormalisation
They are readily found to be ~ 3 g2~0 + k~="=l f d2xJ'~i'*(x) M,~;'*(x)'
m2
[
---50 gm + fd2x &(x)s&(x )- +4(x)~ 0
02'
(22a)
,
n = 0 , 1 ....
(22b)
(22c)
in one to one correspondence with the parameters of the general solution (19). On the other hand, as it commutes with ~21f,,/~20/0#2 is a symmetric operator. As a consequence we can expand /~20/0/~2 on the previous complete basis and we obtain 2 /.t ~
3 ] f d2xJ,i~i"i*3jizj..i ~
2 0 + fl(g2) g ~g2 +
a
W(S,,jA,4, jai,k)=O"
(23)
n=O
The independence of fl and 3',, with respect to br results from the equivalence theorem and the use of a minimal scheme (c.f. the independence of fl with respect to the gauge parameter a in gauge theories*). In particular these functions are the same as the ones of the II, o parametrisation (choose b(y)= b0/(1 + y ) as indicated in the previous section). On ordinary proper Green functions (without any insertions), eq. (23) reduces to the usual form:
~+fl(g2)
g2
Ogz
-2N
+ym(g2)m 2
2
0 } 1 F 21v
+ ~ y,(br, g ) - ~ n=O
" One can mimic the proof given in ref. [7].
8m 2
7 i.... 6~tpl,...,p2N)=O.
(24)
544
G. Bonneau, F. Delduc / Non-linearrenormalisation
Let us show these results on a one-loop calculation with lagrangian
~(~i)-~(~i)-~"
_ _
2g 2 (1 + * 2 ) 2
m
2
_
1 +,2
,
(25)
where qo, = ffib(ff2). For the general two-dimensional model of a bosonic field ~
E(~p)=- ~l [ss~igij(~)cg~J-m2h(@) ]
(26)
where gij is a metric on the field manifold and h ( ~ ) a scalar function, the one-loop counterterm lagrangian can be written as (refs. [8, 9]):
L F'i r, jk 1 4rre X 4*-jki5 ]
-- - -
,,211
+ --4~re - --2g2
m2 k
O~'~iRij a~'Oj + -4-~g2g Dkh't
, (27)
where h d = Oh(~)/O,~ t and D k, Fjk, Rij are the usual covariant derivative, connection and Ricci tensor associated to the metric g~j(~) and g~jgjk = 8k. The second part of expression (27), being a scalar, is invariant under reparametrisations ~ --+ ffi and then in our case where the manifold is the coset O(N + 1 ) / O ( N ) (eq. (25)) one finds
Oiz~i Ol~J Rij({~ ) ~ 0 ~ i Ol.,~J R O ( , ) ,
(28a)
with
Rij(~ ) = - 8(N - 1 ) g i j ( , ) , gkt(~)Dk(~)h,t(~ ) = gkt(~)Dk(~)h t(, ) = - 8 N h ( * ) + 2N.
(28b)
Then the renormalisation constants for the physical parameters m 2 and g2 are the same as in sect. 2 (eqs. (8c), (8d))
4hg 2 Z,.~ = 1 - ( N - 2) 8ere
(29a)
4hg 2 Zg~ = 1 - ( N - 1) 4~re
(29b)
G. Bonneau, F. DeMuc / Non-linear renormalisation
545
The first part of expression (27) gives a non-linear field renormalisation: cp;=cp i 1 -
(N-2)(l+y)
l+2y-ff
- ( l + y ) 2 (N+2)~--+2Y--b--l] ]. (29c)
Defining b°(y) through 9~o-- ~ v ~ u d
I
(30)
one obtains
b°(y)=b(y)
-
(N-2)(l+y)(b+2yb')-(l+y)2[(N+2)b'+2yb"]].
(31) After expansion in powers of y, one finds the renormalisation of the parameters bn. Finally the Callan-Symanzik coefficients ~/, are given at one-loop by their generating function:
~,7.(b. gE)y"=y(y,b, g2),
(32a)
n
with hg 2 y ( y , br, g2) = - ~ [ ( N - 2 ) ( 1 + y ) ( b + 2 y b ' ) - ( 1 + y ) 2 [ ( N + 2)b'+ 2yb"]].
(328) Notice from eq. (31) that the choice b(y) = b0/(1 + y) leads to a linear renormalisation of the field: hg 2 b°(y) = b ( y ) 1 - 2N-~-~ e , or
in agreement with eq. (3). As previously mentioned, this is not surprising.
546
G. Bonneau, F. Delduc / Non-linear renormalisation
5. Concluding remarks We have shown how, after a field reparametrisation of a multiplicatively renormalisable theory, the divergences of the new one can be reabsorbed through a field dependant renormalisation of the field and, for the physical parameters, masses and coupling constants, the same multiplicative renormalisation constants as in the original one. Such a renormalisabihty with a non-linear field redefinition was also exhibited by Piguet and Sibold in their analysis of supersymmetric Yang Mills theories [10]. There, as in bosonic models in 2 space-time dimensions, the canonical dimension of the (super) field vanishes. But in that case, the theory is not a field redefinition of a multiplicatively renormalisable parametrisation. Rather, the arbitrary parameters can be shown to be gauge parameters and the non-linear field redefinition associated to a change of gauge. We like to speculate that this non-linear field renormalisation might also occur in some power counting non-renormalisable theory where all but a finite number of the infinites would be unphysical.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10]
M.C. Bergrre and Y.M.P. Lam, Phys. Rev. D13 (1976) 3247 and references therein G. Bonnean, Nucl. Phys. B221 (1983) 178 E. Br~zin, J.C. Le GuiUou and J. Zinn Justin, Phys. Rev. D14 (1976) 2615 J. Schwinger, Phys. Lett. 24B (1967) 473; Phys. Rev. 167 (1968) 1432 W.A. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D14 (1976) 985 Y.M.P. Lam, Phys. Rev. D6 (1972) 2145; D7 (1973) 2943 D.J. Gross, 1975 Les Houches Summer School, Methods in field theory, ed. R. Balian and J. Zinn Justin (North-Holland, Amsterdam, 1976) pp. 193-194 G. Ecker and J. Honerkamp, Nucl. Phys. B35 (1971) 481; D. Friedan, Phys. Rev. Lett. 45 (1980) 1057 G. Bonneau and F. Delduc, in preparation O. Piguet and K. Sibold, Nucl. Phys. B248 (1984) 301