Dilatation invariance and spontaneous symmetry breakdown in dimensional renormalization

Nuclear Phv.sics B274 t1986) 121-138 North-Holland. Amsterdam

DILATATION INVARIANCE AND SPONTANEOUS SYMMETRY BREAKDOWN IN DIMENSIONAL RENORMALIZATION C. BECCHI. A. BLASI and R. COLLINA Diparttmento di Fisica dell'Untrerstt~ Gent~'a. lstituto Na:.ionale dt Fi.stca Nu¢lettre. S~':.ione dr Genol'a, Italy

Received 15 Janua~ 1986

Thi,,, paper completes a sy,,,tematic study of the renormalization of scale invariant models ~ith spontaneotl.', symmetry breakdown discu~,,,in~, an effective scheme for the con,,,truction of Fevnm,m amplitude,',. From a general point of view the original scale invariance of the m~xtclx is accounted for by constraining the dilatation anomalie,,, with a minimality criterion anab,zcd in a prcviou.', paper. This criterion is ha.,,cd on a k~'al Ward identity where the anomalie~ appear ~ouplcd to a ~,purion field. I lore the general framework is adapted to ;. dimcn~,ional renormalizalion ,.chcm¢ ;.rod it is shown to bc equivalent to a scale invariant choice of all cokmlertcrlll.'i and. a~, it h;.tppcns ira minimal scheme.,,, to the mass independence of the rcnormaliz:~tion con,,,tant.~, Ihcrcbv allowing a derivation of a rcnormalization ~,rt~llp equation with standard (mar,~. independent) ct,cfl'icicnt.,.. The whole analy~.i~ rcfcr.s a~. an example to the .,,implc~,t bosonic tn~,.Icl ~.'~mtainin~ all the fcalttrcs conllnon tt~ SCZlIc invariant theories with xpontitncotts .,.vnnuctry brcakdt~wn.

I. Intr~xluetion One of the phenomena described in rcnormalized quantum field theory is tile spontaneous symmetry breakdown in classically scale invariant systems. The model where this tucch.'mism first .',ppcarcd was proposed in 1973 by S. Coleman and E.J. Weinberg [11; since then many papers discussed the subject and in particular the difficulties related to the extension of this class of models to arbitrary orders in covariant perturbation theory. One such difficulty is due to the presence of two different scale parameters: the strength of a field vacuum expectation value and the renormalization scale, which could be mixed by the radiative corrections. A regularization independent solution to this problem is suggested in ref. [2]. These authors identify a local Ward identity (Wi) which, at the classical level, describes the spontaneously broken dilatation symmetry (SBI)S) and, at the quantum level, enforces the recursive condition that the trace :momaly be given by a scale invariant operator. The second main difficulty in obtaining a complete quantum extension of SBDS theories is related to the fact that they are necessarily accompanied by an infrared O550-3213/'X6/$¢)3.50 " El.~cvicr Science Puhli.~hcrs P,.V. INorti~- Ilolland Phv.~ic.~ Puhli.,,hing Divixion)

t22

c

B~.c.~ht t.t uL // D t l a t u t t o n

tt,tt'tart,~ltt~l"

instability brought about by radiative mass generation. Indeed. classically SBDS cannot be triggered by a quadratic term with imaginary mass which is forbidden by scale invariance; it only takes place if the vacuum is an indifferent equilibrium state for any field translation. The picture is modified by the quantum fluctuations which select a stable equilibrium state by introducing linear restoring forces corresponding to a radiative mass term and a consequent infrared sickness which, as shown in ref. [3]. can be completely cured by inserting the radiative mass directly into the propagator. The resulting perturbative development is still a formal power series in h (or in the relevant coupling constant) with In h corrections, and the ordering of the diagrams is compatible, although not coincident, with the loop ordering. To a given perturbative order (in h) contribute diagrams with a finite number of loops. In this framework one sees clearly the connection between the necessity of having a local scale invariant WI. and the spontaneous breakdown of the symmetry. Indeed a rigid scale transformation in the shifted fields produces non-integrable vertices in the theory built with the naive, massless, propagators. In terms of the modified Feynman rules, these contributions mix in a pathological way the orders of the perturbative series. An approach alternative to the one discussed above, which has been widely employed at least up to the one-loop level, is based on the cffective potential. ust,ally identified with the vertex functional computed in the constant field limit after extracting a space-time voh|tne factor [I,4,51. This method has bccn nltlch discussed [4,6 l, in particular for the gauge n;odcls, where there seemed to exist problcms, by now completely solved, with the gat,gc independence of the physicM i'~aramctcrs [4, 7, 81. On the other hand, who,1 discussing the higher order radiative corrections, in SBDS models the effective potenti;d appro;,ch encounters inst, pcrable difficulties, in particular N.K. Nilscn [7] h,ls remarked that in the ('olcmanWcinbcrg model there are diagrams with different loop number which contribtltc to the same perturbative order in the cot, piing const:mt e. One can easily verify th;tt in the Landau gauge all diagrams of the type in fig. 1 appear at the same order (eS), independently of the number of loops. For this reason the effective potcnti:tl beyond e ~' is not computable in terms of a finite number of diagrams. Therefore this approach is unapplicable to the construction of the complete renormalizcd theory which has to rely on the previously otitlincd strategy of a local scale identity.

I:ig.

1

C. Becchi et al. / Ddatutum mvurlam'e

123

Even if complete formal proof of renormalizability has been given in this framework [2], the problem of identifying - starting from a suitable regularization an effective scheme of computing Feynman amplitudes remains open. The natural choice, also in view of analyzing gauge models, is to adopt a dimensional renormalization scheme t. Hence it is important to have a clear definition of scale invariance for generic space-time dimensions d. Following the lines proposed in ref. [2]. the behaviour of a scalar field ~b(x) under an infinitesimal local scale transformation

x . --. x. + X,,( x )

(l .l )

(i.e. a general coordinate transformation) is specified by assigning a Weyi [9] weight % such that

,~q,(x ) = x~., a.,t,(x ) + % a.X.(x ) ~,(.~ ).

(1.2)

A regularized version of this transformation property is identified by letting % depend on the space-time dimension d as:

%(all=

d- 2 2,1

(1.3)

This however implies that polynomial interaction lagrangians in general do not yield invariant classical actions, in fact ~4(x) has weight ( 2 d - 4)/d instead of I, as necessary for scale invariance. The way out of this difficulty is to introduce, already at the classical level, a spurion rich.! S(x) with non-v:mishing vacuum expectation value (VEV) and carrying a Weyl weight ( 4 - d)/2d. The scale invariance of the theory is then described by a WI, with d dependent coefficients, involving the spurion field; we shall see, taking into account the muhiplicative nature of the renormalization procedure, that at the renormalized level this produces tile anomalous Wi discussed in ref. [2]. Other constraints come from the spontaneous symmetry breakdown; indeed the fields which acquire a non-vanishing VEV develop tadpoles which must be compensated for in the construction of the bare theory- that is for all space-time dimensions d - and before subtraction of the ultraviolet divergencies. In nondiagramatic language, this request means that the vacuum minimizes at all perturbatire orders and for every value of d, the effective action of the theory. Let us emphasize that it is the choice of these counterterms which discriminates among the possible quantum extensions of the classical theory, Indeed the spurion " Notice that although the dimensional rcgularization procedure respects the BRS symmetry a.ss~iated ,.,.ith gauge tran.,,formations, when there is SBDS this approach is not sufficient to guarantee the independence of the physical amplitudes fronl the gauge fixing parameters.

124

C. Be~~ht et aL / Dtlatutt,m tm'urttt.ct'

field appears in the theor,~ as a tt~l to define the minimality constraint of the breaking of the local dilatation Wl (trace anomaly) and hence to impose the scale invariance of all counterterms, in particular the one of the tadpole. Thus the tadpole counterterm has to be generated by scale invariant vertices through a field translation: it is only this last requirement that is needed for practical computations. It is also quite important to realize, in order that the coefficient of the tadpole counterterm possesses a perturbative development compatible with the loop ordering. that it has to be computed with the Feynman rules modified bv the presence of the radiative mass [31. To simplify our presentation, we shall postpone the discussion of gauge models t o a forthcoming paper, and our analysis here will be carried out taking as an example a model with a doublet of scalar fields and with spontaneous breaking of parity'. In sect. 2 we introduce the local, spontaneously broken, dilatation Wi in dimensions d for models with only scalar fields. The complete bare lagrangian for our reference model is identified, tip to a field redefinition, and the tadpole st, btractiou is discussed. We explicitly give the first-order valt, e of the related coefficient together with the expression for the radiative mass at the same order. Sect. 3 is devoted to the analysis of tile renormalization of the bare theory. In particular we show that the bare model can be muhiplicatively renormalized Ihrough au itcrative, spuriou dependent, procedure which co,npcnsatcs the divergencies at d = 4. A further, spt, rhm depeudcnt, finite multiplicative remwmalization rcdt, ces the Wl to the c:monical anomalous form of rcf. [21. This rcnormalizatio,I procedure, althot,gh non-minimal, is shown to satisfy a property peer, liar of minfimal scheme~,, n:,mcly the mass independence of the re,~ormalizatio,i COllMantx. A summary of results and their applicability to other (g:,t, gc) models is co,ltaincd in the concluding section.

2. I.ocal dilatation invariance in d = 4 - 2v dimension,, ()ur aim in this section is to write a local scale invariance W! which c:m be considered as a,1 extension to dimensions d = 4 - 2v of the WI discussed in ref. [21 for d = 4. We shall illustrate the general framework adopting a reference model built with a two-comptment scalar field ~ = (~'t.~-) whose classical action is characterized by the invariance under a simultaneous scale transform:|tion of ,#t and g,_, + t' :,nd under the inversion ~t "" - ~ t . This is the simplest non-trivial model where a non-vanishing VEV of a scalar field (t,) produces at the tree approximation a spontaneous breakdown of scale invariance. Let t,s first recall the strategy of rcf. [2]: aftcr thc introductio,l of aq external metric field +"~ and a connectitm %. one requires that the complete action be invarkmt under the followi,'ag Weyl representation [9] of the general coordinate

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C. Be~'chtet aL / Dtlatatum tm'ur,ant'e

transformations:

a,,(.,~) = x.(.~) a.¢,,(~) + ' o . X . ( x ) ( , , ( x )

+ ~,,).

~= (0.~,).

;=1,2.

(2.1a)

8 ~ . ' ( x ) = x . ( x ) a,~.'(.,:) + ~ a,x.(x) ~ . ' ( x )

~ " ( x ) - 0, (x) ~"(x)

-a,X"(x) ~,,.,,,(.,~) = x,(.,:)

a,,,,,(x) +

(2.lb) (2.1c)

a,,x~'(x) ,,,~,(x) - ~a, a~,x~'(.,:).

The invariance condition for the classical action of the system is written as a functional differential equation (WI) which in the flat limit (,~"'(.~)---, ~1"" the flat metric and %,(x) -, 0) turns out to be:

(~(.~)r~')~,

[

= o,,,(x) ~ r 8,~,(x) - ~ 0. a. 8

, ,

~'- , (,,(x) + v,)

~,~.~-)

l-~l + 2 a,.

(,,~".(.~) ~

~,~,(.,-) I "~t

j - -

4 ~A-~)

1 "'~

) (2.25

=0,

whose general solution is:

~-,= - fd".~ { ~"'(x)(,~. + ..(x))(~,(x)+

,,,)(,9, + -.( ~)5(~,(x 5 + ,,,)

+,,,t,4t(x) +

.,t,-~tx)(,bz(X)

+ t')- / , (2.3)

where the condition 81"~1

~ , (x)

~,-o = 0

(2.4)

defining the classical vacuum of the theory has been taken into account. From the analysis of ref. [2] we know that eq. (2.2) is not stable under radiative corrections; at the first non-trivial order instead of (2.2) we have

(%(x) r),:,. = a,l(x).

(2.55

The anom:dy I ( x ) to the Ward identity is really due to an instability of the Weyl scheme and can be characterized in the fiat limit" by the "minimality" requirement * OuL~id¢ of the flat limit thc.~e anomalies have been analyzed in rcf. [10].

t26

(7. Beech* et al. / Dtlatatum tm'artan(e

that / ( x ) be a local polynomial of canonical dimension equal to 4 in the translated fields and which cannot be written as a four-divergence. The anomaly is then taken into account by coupling it to a spurion field _V(x) with transformation law

8z(x)

= x q x ) a~_."(x) - h o~,x,(x )

(2.6)

and eq. (2.5) becomes 8

()~;(~)r)_ ~

6

+a~z(~)*-v-iTv-,r+h ~z(x)

a~ ~o_t.,.) t = 0 .

(2.7)

The radiative corrections generate a mass for the classically' massless field ~_, so that the perturbation series is to be modified [31 as recalled in the introduction. The renormalizability of eq. (2.7) is discussed in ref. [2] within a scheme independent framework, where it is proven that the renormalized theory is completely identified. except for some physically irrelevant couplings all involving derivatives of the spurion v. by the parameters specifying the classical action in the flat limit and for Z" -- 0. The resulting effective lagrangian contains couplings with arbitrary powers of the Y fiehJ which are recursively identified throt, gh eq. (2.7) itself in terms of the X-independent ones. We now come to the regularized approach within which we propose an extension of the theory to dimensions d = 4 - 2 v which reproduces, in the limit v = 0. the resuhs st, mmarized above. To achieve this we have at ot, r disposal the Weyl weights in eq. (2.1) which are now written as d- 2

a,t,, ( x ) = x'( x ) a,,,t,, ( x ) +

2¢1

a x,'(x ) ( ,t,,( ., ) + ~, "',,).

(2.sa)

2

- a , x , ( x ) U"( x ) - a,X"(x ) ~ (

x ). d - 2

,s,,,.(x) = X'Ix) o,,,,,(x / + o X~(x) ,,,,(x) - ~--57-dO~, a , X , ( x ) .

(2.Sb) (2.8c)

where the mass-scale parameter /, has been introduced to maintain the mass dimension of t, independent of the space-time dimensionality. Concerning the new action invariant under the extended d-dimensional coordinate tr:msformations, it turns out that in order that polynomial solutions exist outside of the free field case one has to introduce an external spurion field V(x) compens:tting for the missing weight corresponding to a quartic interaction. Hence

127

C. Beccht et al. / D t l a t a t t o n mt'artunce

we choose for S ( x ) the transformation law:

a Z ( x ) = X,(x) a , Z ( x ) +

4-d

2d a M ( x ) ( S ( x ) + I).

(2.9)

The new WI in the flat limit is:

8

d- 2

d

-

2

~$

[

,:l]]

8

2d a. [(~,(xl+~-',,,)~r

d~,~,(x) ~r"a%(x)

[ "~t+ga~

+ a,,_"(x) - - r " ' az(x)

2,t

(

- -

a

r~., a."

a. (".'(.~) + 1)

-

-

a_"(.,-)

a

r,.,)

J

=0

(2.1o)

and eq. (2.10) can be thought to hold for any complex value of the parameter ~,. The general solution of eq. (2.10) appears as tile obvious extension, to include the spurion fickl v of the naive classical action in cq. (2.3)

r,'= - f d".,-( ~e"(o,,,,r~,,,

+ o.(,: +

~, ,,,,)o,(,, + ~, ,,,))

+,~:~(t + z)-~, + :,,-~,- ~1 + "):~,](¢. + ~, ",,): (x).

(2.11a)

plus :t finite number of terms explicitly dependent on tile derivatives of the spurion field and which can be reduced to the canonical form*

I+S

+8

+~

(l + "):

(1 + z) ~

+ n~-'" ~x'(ax" a.S) + pP--"" ( a.S a.'...') z (1 + z') s

(1 + ,_.')"

(2.11b)

* Th¢~,c lcrm~, corrcxpond c,~actly to the '" physically irrelevant'" coupling.,, analysed in rcf [21.

128

C. Beccht et aL / Ddatatum tnt'artame

Here D = O,, + % is the covariant derivative and

= ~ = ~ z - ~( ~.. a~,~" + ~.~, a°~,'.-~o°a.~.o + s,~o} a o is the covariant d'alembertian. Notice that in eq. (2.11a) the contribution pt:'(l + ")-'('~z + ~- "v) a is not included since the configuration e/,, = 0 defines the classical vacuum of the model. Now the classical vacuum condition must be maintained at the higher orders as a necessary constraint for the proper vertex functional and the desired subtraction should correspond to a lagrangian counterterm containing a linear term in the relevant scalar field. There are many possible choices of such a counterterm; the natural request that the tadpole subtraction should take place without affecting eq. (2.10). selects the coupling (2.12)

p.2"(1 + " ) : ( ~ b 2 + p. "~')'~

as the only candidate*. As remarked in the introdt, ction the choice of this term discriminates among the possible quantt, m extension of the classical model in eq. (2.1 la). Tht, s, the complete bare lagrangian obeying eq. (2.10) is, up to a multiplicarive field redefinition.

'_7'~-_9"~+~c

h'A

v,a

-

(1 +-v)e(,k,+~

.I,

"~,)~,

(2.13)

~sherc ff[~ is given by the sum of the expression in eqs. (2.11a), (2.11b) and the coefficient A ( z,, a( u / v )>, h(l~/v )~ ) is a mcromorphic function of +, computed order

by order iq terms of tadpole Feymnan diagrams. "I'o summarize, we take as a bare. dimensionally regularized model with Sill).";. one whose vertex functional 1"h obeys in the flat limit the idcntit)

8 1

-

v

8

(

8

d a, a, 8,~,(.,.------)t',, + 2 a, ~Sr;~(x-------~ 1",

+a z'(.,.) ~ t ' ~ -

I

8~ 8

d 8~(x ) t'~,

7/a (_,'(x) + 1 ) ~ 1 ' ~

--o.

)

(2.14)

with the further vacuum constraint (eq. (2.4)) on 1"h. * Terms, of the type tL:(Z ~ 112 "(~, ~ p."vl: arc m~t allov, cd at the cla.,,~,ical I¢',cl ~,mc,.' thin, hay,.: an c,,~,¢ntial ,,inDdarity at v - 0 and do not provide a regularized vcr.,,ion of (he da~,~i,.'ai modal. Furthcrm,,rc they v.'ill ab, o never appear a~, c~mrltcrtcrm.,, for tt~c da,,sk.'al action in ¢ q (2. i 1} dcpcn,.b. ~m IL" and it~, radiative corrcgtiom, cannot generate p2 contribution.,..

129

C. Becchi et al. / Ddatatum inuartunce ,o a"

*l L

, •

oO I

Fig. 2

Let us remark that the dimensional extension we are proposing acts as a regularization only for the UV divergencies: the IR problems, related to the radiative mass generation, must be taken care of. already at the bare level, as outlined in the introduction [12]. In our reference model an easy computation yields the following one-loop values for the coefficient A and the radiative mass m~: ,'

A't' v,

km /

b ~

(

=

,,,t,

4(4~.)" 4~'~z~"

F(~,-l).

bZ"" (4rrP'2,/~/''(-j'-+ ! )

(2.15)

(2.16)

where wc have set m'- = b"v z and i" is the Euler function. The new Feynman rules [31 ,low include the I-loop radiative mass h(m~:)~l~ in the ~, propagator and it is only in terms of these new rules that the coefficient A possesses the loop-compatible pcrturbative development mentioned in the introduction. As an example the di:tgram in fig. 2 is easily seen to have a contribution h~ln h; the dot stands for the trilinear (~b.,) ~ vertex proportional to A~t~. l'-'inally let us remark that any renormalization procedure which, order by order in the perturbation expansion, removes at least the ~, poles from the bare amplitudes is going to alter the WI in eq. (2.14) since its coefficients are functions of u. We shall in the following discuss the class of Wis which are obtained from eq. (2.14) through a multiplicative renormalization procedure. Let us anticipate that the class of renormalized theories so identified coincides with the one analyzed in ref. [21.

3. Renormalization of the local dilatation Wi In this section we propose a renormalization procedure under which the local dilatation WI in eq. (2.14) transforms into the anomalous form in eq. (2.7). We shall meet two major difficulties. On the one hand the spurion field v is dimensionless, therefore one may expect that the radiative corrections require the introduction of local counterterms with arbitrary powers of it. For instance one easily checks at the one-loop level that for

[ 30

C. Bt'c(ht et aL / Dtlatat.,n tnrurtan( t"

instance the Green functions r~t~ -h _,"~,,'- l ~t),_,._,e~,:,whose corresponding couplings are not present in the bare lagrangian in eq. (2,13). have a simple pole for u = 0, On the other hand the WI in eq. (2.14) has u-dependent coefficients and it will possibly not su~ive unaltered a renormalization procedure which removes the v singularities of the bare Green functions. The first difficulty will be solved by showing that the counterterms needed to subtract the S field dependent divergencies are generated from the bare action by a multiplicative renormalization of the fields and of the parameters (counterterms with coefficients depending on the Z" field), Concerning the second problem we are going to see that the combined effect of a -".'-dependent multiplicative renormalization and of the v-dependent terms in the W! generates the anomalous contribution appearing in eq. (2.7). For convenience we shall proceed in the following way: we first discuss the renormalized version of the dilatation Wl assuming that there exists a muhiplicatire. S-dependent renormalization procedure which renders perturbatively finite the Green functions. The validity of this assumption will be proved at the end of this section. It is apparent that a S-dependent muhiplicativc renormalization of the fields such

(3.1a)

~b, + tt "v, -. (;,( X ) ( ~, + it "v, ),

h""-, H(_")h,".

(3.1h)

% -., l.( "')%.

(3.1c)

-. M(v) = v f ( S )

with/(0) ~ 0

(3.1d)

affects the original WI which, in the fk, t limit, bcco,nes 8

l-u •

l-v

,/ u

[

8 '

,I C (

1

]

8 +O~S(x)

I'

(~

iL 0 , L ( " ) 8 % ( x )

[l+i(")( i'('-')

~ I '

r

+ 2 o~,

(;,'('_')

8h,',(x~ I'

d 81PAx) r

=

. (3.2)

In eq. (3.2) the field h ;'~ is defined by ~j~'"= ~ ' " + h"', the prime denotes tile

131

C. 8ecc'ht et uL / Ddututum im'urmnce

derivative w . r . t . S . Every. single term in the above identity has a finite v = 0 limit: this is evident for the first three terms. Concerning the remaining ones. the possible singular contributions at the first non-trivial order satisfy

(3.3)

where the subscript s denotes the singular part at ~, = 0 and

1

d-21

"~= H

.s

2d

Jr'=

L "

~' I + M d

M"

t, 1 + M G"

,A/'' =

d

M"

(3.4)

G,

Now one can show that the four expressions in square brackets in eq. (3.3) contribute to linearly independent functionals and hence their coefficients vanish separately. Indeed referring, e.g.. to a scalar field (qh in our model) and isolating the terms with three derivatives on ~t one sees imnaediatcly that the first, second and fourth term in cq. (3.3) contribute to the independent expressions O,,,#l Oq, t - ~ 0~, a,¢t i/'~,l,

i/~, q,t O@t + 2 a~//~fft 9"¢,i + 0~,O~1.

a~,'/'l O@t + '#t i)~,O @t and the third term contributes only with one derivative acting on ¢ht. namely a , ( f f t )4. Thus the coefficients in eq. (3.4) are all finite at v = 0. in particular we c:m write the coefficient ..gt' in eq. (3.1d) as v

J[(S) = ~ [I + r(v. S, h) + hs(v. "-, h)],

(3.5)

where the v singular part of all contained in s(v, v, h). Recalling that ~H has a finite v -- 0 limit, we set s(v, S, h) = a / v + fl + O(h, S), and we obtain

1

lira

l, I + M

-h,-od

--=~a+O(h M' = re(Y),

S)

(3.6)

C Bec¢'hl et uL /

132

Dtlatutton lnt'artance

which also implies that ~4~ ' = O(h) and hence 1 - lim J t " ( X ) = n'(_v). h

(3.7)

J,~t)

At this point we know that if the theory is made finite by a multiplicative X-dependent renormalization, the bare WI is transformed into the renormalized one:

o,,,(.~) ~r-

~o~ (~,,(.,-) ÷,,,)~t"

+ o..,'(.~) ~ t "

I"

-t,~,

8."

.,~v-..,t~,(x)+,,,)~t" 8

8

"

=o.

I"

(3.s)

Now the coefficients ,.~, c..~, m, n, can be reabsorbed by the finite, -".'-dependent multiplicative renormalizations: (¢,, + ,,,) = ~1 ("-")(~',

+ ,';),

(3.9a)

,,,, = Z(-"),,,;,,

(3.9b)

h:: = h ( ,_")h2',

(3.9c)

x-- r'(,_"),

(3.9d)

with the choice I(X') = limC.C~(F(X')),

(3.10a)

v~ 0

~,(,2') = lira ~(f("_")).

(3.lOb)

F ( -~'') = m(F(X')),

(3.lo~)

A;(-") =.,(r('-")). A,(X')

(3.10d)

with F, A, determined up to irrelevant integration constants. With these redefinitions we find, in terms of the primed quantities, the anomalous Wi in eq. (2.7}. To complete the proof we still have to show that the bare theory is made finite through the renormalization exhibited in eqs. (3.1). In the framework of dimensional

C. Betchi et aL

/

Ddatution int'ariance

133

renormalization and subtraction prescription (counterterms) leading to a quantum extension of the theory will be collected in the effective lagrangian £',u of the model. We want to show that .-~',fr can be obtained, from the bare lagrangian £#b in eq. (2.13) by a S-dependent renormalization of fields and classical parameters. The proof is based on the consideration that at the first non-trivial order (typically one loop), the vertex functional of the bare theory /"b has a singular part at v = 0. denoted by (/'b)~. which is a local functional constrained by the classical scale invariance condition in 4 dimensions:

+28,

8h,7(.r)

(r,),+o,_~(x)~(r,)~=o.

4 8h~(,)

(3.11)

The formal power series solution of the above equation is given, in the flat limit, by

tub). = ,,,(E) ~ ,~,.~t a.~, + az(,_') ! a,~.. a,'~.. + ,,.,( ":)(~,)' + , , ( "-'),/,] ( ,/,: + ~,)" + , , ( -")( ,/,, + ~,)'

+,,.I "-'1

(

+

+ ,,/;I }

+ a~( Z)OS( o~,Z o~"S) + at,,( S)[ o~.S o~"X] z .

(3.12)

where all a , ( S ) coefficients are formal power series in the S field. Now it is crucial to observe that the expression in eq. (3.12) corresponds, except for the as(S)(e#," + t, )'~1_,-o term, to the variation of the bare action under the infinitesimal transformations q', = ~,, + O , ( S ,

~,)~,.

(3.13a)

a'=a+

ll,(V.~),

(3.13b)

b'=h+

l l h ( S , e ).

(3.13c)

~', = ~ + . ~ ( v v),

(3.13d)

134

C Beccht er al. / Dtlatutmn mrarlam e

where O,. [I v. H h. =- are singular at v = 0 " a n d for instance 6 ) t ( - ,v ) = ~ . a l ( S ) ,

6),(v) = !az(X').

(3.14a)

1

- ( 5 " ) - 2b,A,~,[ v ) as( V).

(3.14b)

Thus a renormalization prescription where the counterterms are generated by the transformations inverse to those in eqs. (3.13) subtracts in least order all divergent terms in the bare action except possibly the one proportional to 16z + e) a. Now the presence of this last divergence is excluded since all tadpoles are compensated in the bare theory to all perturbative orders and therefore they cannot contribute to any amplitude: thus the multiplicative renormalizability at I loop is proved. After this renormalization procedure the vertex functional obeys a new local dilatation WI corresponding to eq. (3.8) with coefficients ~.g', ,.,'U, ./{, ~ which can be chosen finite, as o n e c a n show by an analysis similar to the one performed after tile eq. (3.2). We now examine the next perturbative contribution repeating tile previous argument. Indeed the singular part (Fh), is again a local functional which obeys eq. (3.1 l) and therefore has the general structure given in eq. (3.12). It follows that the transforn~ations inverse to those in eqs. (3.13) can be chosen so as to c o n l p c n s a t e (/'b), and the resuhing Wi is, at this order too, of the kind given in cq. (3.8) with fiqite coefficients. Thus the procedure iterates and we have a E-dependent multiplicative reuormalization which makes the theory finite to all perturbative orders and which mai,ltains the Wl in eq. (3.8). This completes the proof siqcc we have already shown the equivalence of the multiplicatively renormalized theory with the oqe determined by the anomalous WI in eq. (2.7) which, as shown in ref. [21. guarantees the minimality of the breaking introduced by the renormalizatio,1 scale in SBI)S. Finally we prove that the discussed renormalization procedure enjoys a property typical of the minimal subtraction schemes, namely the mass independence of remwmalization constants. Let us briefly recall that in ref. [11] it is shown that in a completely massive field theory which has no IR problems and where the only subtractions are the UV ones, the divergent parts of the amplitudes, corresponding to any diagram where all proper subdiagrams have been minim:ally re,mrmalized, are polynomials in the masses and the external momenta. It readily follows that when the minimal subtraction scheme corresponds to a muhiplicative renormalization process, the related constants are mass independeqt. in our model the situation is somewhat different for two main reasons: first, the presence of tt, dpoles whose compensation, already in the bare theory, is implicitly " Notice thai the external fields %,, It+,,, need not b¢ transformed: the inclusion in {l't,). gi',cn in the flat limit in eq (3.11) of these fields is dictated by the covariantizatitm prt~.'edure and does nt,t intrt,duc¢ any additional parameters

C. Beccht et aL / Ddatatmn tncartance

135

Fig. 3

Fig, 4

required by the definition of the vertex functional: second, the phenomenon of radiative mass generation which requires a new propagator containing the one-loop value of the radiatively generated mass and for consistency the introduction of an IR extra subtraction which deletes the one-loop contribution to the propagator corrections at zero external momentum. Concerning tile tadpole subtraction, we observe that it can bc automatically implemented by a suitable alteration of the Feynman rules. As an example, consider the one-loop ~., t:|dpole diagram, represented in fig. 3. in our model which must be compensated for by tile counterterm bZ~.'~A~t~(l + Z')-'(q+,+p. "t,) "+. If we now examine the one-loop #,t-propagator corrections, we find, before tadpole subtraction. the diagrams in fig. 4 and the subtraction amounts to neglecting the last (tadpole) graph. The same reasoning for the two-loops +, propagator corrections shows that the tadpole subtr:|ction compensates the diagrams of fig. 5c and modifies the numerical coefficient of those in fig. 5b. leaving the rest unchanged. Now if we subtract minimally as in ref. [11] all the subdiagrams, we have that each divergent part of tile surviving graphs is a polynomial in the masses and external momenta and consequently the renormalization parameters are still mass independent. It remains the problem of the radiative mass generation which surety induces, from three-loops on in the divergent part of the single diagrams, terms proportional to the logarithm of mass ratios. However, with the same mechanism suggested in ref. [12] in the case of IR subtractions, the logarithmic terms vanish in the sum of all the diagrams contributing to a given Green function at a fixed perturbative order; thus no mass dependence, in the renorm:|lization constants, is introduced by this mechanism too. As an example of this compensation the q~ propagator correction corresponding to tile diagram a in fig. 6 contains a divergent contribution of order h j proportional to I n ( m / # ) which is cancelled by analogous terms from diagrams b and c.

136

C. Beccht et al. / Dtlatatton

. - . c v > .... © - -

mt'artume

-0,, --

..... C ) - c) o

S

...~_..

0

. . . . . Q .... / > . . .

b

C Fig. 5

_M..J ;

i

b

il

c

Fig, 6

We have thus .shown that tile renormalization procedure leading to eq. (3.8) is indeed minimal and the corresponding const:mts are mass independent. It is straightforward to verify directly, repeating the a,lalysis at the beginning of this section, that even the residual finite, ".'-dependeqt muhiplicative renormalizatio,, needed to reconstruct the anomalous W[ in eq. (2.7) has const:mts which are still mass independent. This result allows a completely standard derivation of a renormalization group equation with mass independent coefficients. In our reference model this equation, written for the vertex functional l'(a, b, ~t, m, q~,) at v = 0 is

~. a , r +/3, O,,l' + ,8,, a,,/" +/~,.m a,,,r + v,N,I" = 0,

(3.~5)

where N, is the counting operator for the field q,,. Arranging tile coupling constants in a two-component vector h t = a , X, = b . the corresponding vector fit =fl,,, f12 = ,Sh is given in terms of the renormalization matrix

( Z~. Z.I,) Z =

0

Z~h

c Becchl et uL / Dilatation mr'artam'e

137

as

,8 = - lim ~,(R-~),,X,c(j).

(3.16)

g,,=8,, + [O,,(tnZ),,lx,.

(3.17)

with c(1) = 2c(2) = 2 and

while ft,,. y, are expressed in terms of the mass and the wave function renormalization constants, respectively

Z , , = ZhhY" : ,

~'=

(-~t 0

0 )

~:

"

as

/3,, = lira u~lc(/)( R - I ) , t aa, In Z,,,.

(3.18a)

I,, ~ 1)

-),,-- lira

,,X,c(I)( R t),,Ox,

(In .%)'),.

(3.18b)

4. Conclusions This paper c()mpletes, from the systematic pohlt of view. the analysis of theories with spontaneously broken I~al dilatation symmetry. in the first stage of the analysis [21 the natural request of minimality for the breaking of the dilatation invariance induced by the rcnormalization process was formalized by means of a local W I involving a spurion field. The rcnormalization of this identity was also discussed in a very general context. The necessity of translating the above scheme in .a more effective procedure is here answered in the framework of dimensional regularization, identifying the particular renormalization which embodies the mentioned minimality criterion. The working rule which derives from this approach can be summarized without reference to the spurion field S and amounts to a purely muhiplicative rcnormalization after the subtraction of the tadpole with a quartic coupling in the translated field. A related aspect is the presence of a radiative mass for the translated field and the ensuing IR problems, which must be cured by modifying the propagator. This automatically generates a new perturbation series for the vertex function:d whose development is compatible, but not coinciding, with the loop ordering and contains logarithmic corrections in the expansion parameter. Finally, as a natural consequence of the minimality constraint, we find that renormalization is achieved through a muhiplicative procedure with constants which are mass independent. This same scheme, where illustrated on a model containing only scalar fields, can be directly employed to characterize to all orders of the loop expansion a quantum

C. Bec'c'h¢ et at. / Dtlutattmt t m ' u r t a n t e

extension of gauge models with spontaneous breaking of the dilatation symmetry. In particular we shall discuss in a forthcoming paper, the model introduced by Coleman and Weinberg in ref. [I]. References [i] S. Coleman and E.J. Wcinbcrg. Ph,.'s. Re,,. D7 (1073). 18,~8 [2] (;. Bandclloni. C. Becchi. A. Bla.,d and R. Collina. Nucl. Phys. Blq7 (1982) 347 [3] (;. Bandelloni. C. Becchi. A. Bla.,,i and R. Collina. Comm. Math. Ph,.s. 67 (L978) L47: (;. Bandclloni. A. Blasi and R. Collina. Nucl. Phys. BIT6 (1980) 500 J4] R. Jackiw. Phv>. Re,,'. D9 (1974) 1686 [5J J.S Kang. Phv',. Roy. DIO (L974) 3455: J Cornwall. R. Jackiv~' and E. Tombot, li~.. Phys. Rcv. DIO (1974) 242,~ [b] J. lliopoulo~,.. C. hzvkson and A Martin. Re,.. Mc,d Ph,.s. 47 (IO75) 165: J. lliopoulos and N. Papanicolaou. Nucl. Phys. BL05 (1'476) 77: Y. Ftoimolo. L. O'Raifcartaigh and (;. Parravicini. Nucl. Ph',~,. B212 (1083) 26g; T. Murphy and L. O'Raifcartaigh. Nucl. Ph~. I]218 (L'483) 48J, [7] N K . Nielsen. Nucl. Phys. BIOI {L975) 173 IN] I.JR. Aitchison and C.M. Fra~,er. Ann of Ph,,~,. 156 (1'484) L [O] It. Wcvl. Ann der Phys. (Leipzig) 59 (1'410) 101; Space-lime matter, tram.latcd b,, IlL. Brt,,,e (Mcthucn. l.or.don. L'422): I. Sinolm. Nucl I'll~, BL6() (1'47'4) 253 Ill) I ( ; II,ultlclhmi. Nllt,vo Chu. ,~S (1'485) l. 35 II li ( ; Ih,nilcau. Ntis21. Pllvs 11167 (l'48(i) 26L: 11171 (l'4~(i).147: I I, Ilreilcnlohn~:r and I ) Maid,on. ('Ollml M,ith. Ph'~',, 57 (1'477) 11 1121 I'. Ilrcill:nlohncr and I ) Mai~on. (.'Ollml ,Xlath. llhv~, 5 "~ 11'477) 55