Scale and interaction in field theory

Scale and interaction in field theory

ANNALS OF PHYSICS 209, 75-96 (1991) Scale and Interaction in Field Theory M. C. BERG~RE Service de Physique Theorique de Sacla). * * F-91 191...

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209, 75-96


Scale and Interaction

in Field Theory

M. C. BERG~RE Service

de Physique


de Sacla). * * F-91 191 Gifsur-




Cede.\-, France

12. 1990

We construct for a (fj*)‘+” Interaction (in a zero dimension tield theory, for the sake of simplicity) an expansion in powers of 6 which is scale invariant order by order. The notion of “natural” scale emerges from the construction. %‘ 1991 Academic Press. Inc.



This publication is a note on the notion of scale in interacting field theory. We wish to develop here this notion on the very simple example of a scalar field theory defined in a space reduced to a single point (space of dimension zero). Later, we will generalize this work to a field theory defined over N points, over a d-dimensional rectangular lattice, etc. we will perform the thermodynamic and the continuous limits. Any observable in a physical system possesses a “natural” scale which is a weighted geometrical average of the “natural” scales of its constituents (the weights being dependent of the observable). The “natural” scale of each constituent is a function of all the parameters describing the physical system. A physical effect will appear more or less important to an observer which belongs to the system according to the magnitude of the effect measured in term of the “natural” scale of the system. Such a magnitude is scale invariant and does not depend upon a standard unit since the “natural” scale is constructed with the parameters of the system itself. The above assertions my look deep or trivial or both. This first publication aims at illustrating some of these assertions on the simplest example we may think of. A typical interacting scalar field theory can be described by a potential $2p (or a sum of such potentials); following C. M. Bender et al. [l-9] we choose to write this potential under the form (4’)’ + a so that the Green functions at dimension zero read as follows:

where N is a positive or nul integer; in this text, 4” means (4’Yv”. * Laboratoire

de la Direction

des Sciences de la Mat&e

du Commissariat

$ 1’Energie


15 OOO3-4916/91


Copyright :? 1991 by Academrc Press, Inc. All rlghts 01 reproduction m any lorm reserved




For 6 = 0, Eq. (I.1 ) provides us with Green functions of a Gaussian field of squared mass (m’+ 2g), that is to say, a linear field theory. As 6 increases, the non-linear features of the theory appear. We wish to explore the structure of the expansion of T,(m2, g) in powers of 6. In contradistinction with the perturbation series in powers of g which is violently divergent, the expansion in powers of 6 is convergent (for 161~ 1 if m2 = 0). This is so because g = 0 is a singular point of r,,,, since the integral (1.1) is divergent for g < 0 while 6 = 0 is a perfectly regular point. In other words, whatever small g is, the non-linear features of the theory appear as an abrupt transition from g = 0 to g > 0; on the contrary, when 6 becomes positive, the non-linearity of the theory appears smoothly in a continuous way. Although the expansion in powers of 6 is obviously convergent, it is not so easy to introduce it coherently, even in zero dimension. The striking feature of a naive expansion of r,,, in powers of 6 is that each order contains three related diseases due to the presence of powers of ln(m2 + 2g): -

the logarithm



the dimension

of m2 and the dimension

the logarithm


is not dimensionless, of g are different,

is not scale invariant.

The dimensionless quantity mN+ ‘TN(m2, g) is scale invariant. The number 6 being dimensionless, it is not satisfactory for the expansion of mN+ ‘rN(m2, g) in powers of 6 to be, order by order, dependent of the choice of units. The origin of this difficulty is due to the a-dependent dimension of g like a (mass)2+26. For the sake of an illustration, let us look at the special case where m = 0. In this case, the integral (1.1) is integrable: rN(“,

and the 6 expansion

TN(t), g) = g- [W







reads as





(~)lng+rtt(~)}+o(61)] (1.3)

(the complete expansion is given in the Appendix). Clearly enough, the expansion (1.3) is not acceptable; in addition to the presence of powers of In g, the overall multiplicative factor gmccN+- ‘)12) does not have the







dimension r,,, (by continuity arguments, rN has the dimension like a (mass))‘N+” while the dimension of g is like a (mass)26+2 ). However, the result (1.2) shows that all powers of In g can be summed up, through the complete 6 expansion, to restore the correct dimensional analysis. We say that the system develops a “natural” scale defined by L= and that <, is 6 independent z-,(0,



+ 6)

while g is not. Then, g)=-.-

1 (,“‘I

1 1+8


The magnitude of rN(O, g) measured in terms of its “natural” scale is the dimensionless, scale invariant quantity (l/( 1 + s))r[ (N + 1)/2( 1 + S)]; this quantity has an order by order scale-invariant expansion, in powers of 6 which reads:

(I.61 Let us say at once that the definition (1.4) for the “natural” could as well have chosen (see Section IV): g=ap+“’ I,


scale is not unique; we


where CI is any real positive number independent of 6, g, or any physical quantity. Once, such a number LX had been chosen, by continuity arguments of the integral(I.l) (see Section IV), this same number CI would appear everywhere (including field theory over N points, over lattices, etc.). The main reason for our conventional choice cc=1

(I.8 1

made once and for all, is purely technical and aesthetical: for c( # 1, the function T(x) and its derivatives which appear in the 6 expansions of the type (1.6) would be replaced by the function rm(x) = r(x)cci


and its derivatives in regards to x. One purpose of this publication is to show that the system (1.1) for the massive case behaves exactly in the same way as for the massless case. All powers of



ln(m2+2g) sum up themselves through the complete “natural” scale as the solution of the following equation m2 + 2g{i2’

constant y (i.e., g is measured in (1.11)



to define a (1.10)

= 2{:.

Consequently if we define an intrinsic coupling terms of the “natural” scale) by


then m2 y=”



The intrinsic coupling constant y is zero for Gaussian field theory and is one for massless field theory. Here again, y and 5, are considered as 6 independent quantities while g is not. The Green functions TN(m2, g) measured in terms of the “natural” scale are the scale invariant quantities pN(y, 6) defined by 5 ,“+ ‘r,(m2,

g) = pNb, 6)


and where (1.14) To illustrate Eqs. (Ill)-(1.12) where 6 = 1; we obtain

let us solve the system in the case of the 4” theory


y=&(Jm-m2)’ tt=

m2 + Jm4 + 16g




We note on this example that for g > 0, nothing prevents m2 from being negative; the “natural” scale remains well defined and the fact that y > 1 is not a problem for the convergence of (1.14) as long as 6 > 0. However, the convergence of the a-expansion might be affected since 6 =0 becomes a singular point of the integral which is divergent, in that case, for 6 < 0. If we eliminate the natural scale between (I.1 1) and (1.12), we obtain (1.17) It has been pointed out to us [lo]

that the relation

(1.17) presents some analogy



with the situation at and below dimension four in continuous scalar field theory where a renormalized coupling constant (here y) may be defined from renormization group techniques. If we define from (1.17) a so-called b-function by (1.18) we find OGyd

P(r)= -2(1+6)$3



which displays an infrared stable fixed point (massless theory) at y = 1 and an ultraviolet stable fixed point (large mass theory) at y = 0. We prove in Section IV that the Green functions (1.1) expressed in terms of m2 and y satisfy the differential equation


a a(mV)


1) T,(m2,y)=0,



where /3(y) is given in (1.19). In Section II, we develop the unacceptable 6 expansion in the massive case. In Section III, we perform a scale transformation, which cures the diseases related to dimensionality but which is not yet scale invariant. In Section IV, we define the “natural” scale for the massive case and we show its validity for all systems (1.1) corresponding to different values of 6; we also develop the formalism from the point of view of differential equations. In the conclusion we generalize the result to an interaction of the type P(@), where P is a polynomial. We also comment on the physical meaning of “natural” scales and on the analogy with the renormalization group techniques. We finally quote some results about field theory defined over several points [ 111 and about the notion of “natural” scales for constituents of a physical system. Let us mention already here that “natural” scales survive the thermodynamic and the continuous limits; the ultimate goal of this approach is to construct a better understanding of the role of scales in four-dimensional, continuous, renormalized quantum field theory. II. MASSIVE SCALAR FIELD THEORY ON A POINT AND 6 EXPANSION The general technique to obtain the 6 expansion is to write the integral under the form


(IT.1 )




and to expand the exponential containing the 6 factor in powers of g; the term proportional to (-g)“/n! is of order 6” and involves higher orders. This expansion is performed at all orders in the Appendix. The change of variable 4’ =x in (X.1) allows us to write (11.2) where the function F, (11,v, 6) is defined in (A.1 ). The 6 expansion of F, is calculated in Appendix A; we obtain for the three first terms

I-” (y)}

+ O(S3)].


The higher order terms are similar and calculable from the Appendix.

III. SCALE PROPERTIES OF THE GREEN FUNCTIONS In the integral (1.1) the field rj has the dimension of a (mass))‘, g has the dimension of a (mass)‘+ 26,and f, has the dimension of a (mass)-““+‘). Once we have extracted from TN a term of dimension (mass)-‘N+ I), the expansion in 6 should be dimensionlesssince 6 has no dimension. This, in fact, is not true in (11.3) because the dimension of g is 6 dependent. Let us perform in (I.1 ) the change of variable 4 + qS/& where r has the dimension of a mass. Then rN(m2, g)=

gf!-2(‘+6), 6 . >








This operation is equivalent to having chosen a unit of mass r (say gram or kilogram etc.). The expansion (11.3) becomes

fN(m2, g)=(


+ 2&









This expansion now has the right dimensions and the logarithms are dimensionless; unfortunately, it is not an expansion in powers of 6. Of course, if we try to express (111.2) as an expansion in powers of 6, 5 disappears and we get back (11.3). We notice that in the above analysis, we have not introduced for mass scales either m or g1i2C’+6) which are the intrinsic parameters of the system with the dimension of a mass. Let us express these parameters in terms of the unit r arbitrarily chosen, (111.3)

in which a and b are obviously dimensionless. The expansion (111.2) becomes TN(m2, g) = recN+ I)

1) 1 .yib


(111.4) Now, (111.4) is a 6 expansion built with dimensionless logarithms. Unfortunately, it is not scale invariant since the terms of the expansion depend on whether we choose gram or kilogram as the unit of mass. Indeed, suppose that we change the unit of mass r; then a and b are going to change in such a way that the ratio b +=-



remains unchanged. Then, the combination (a + 2b) is going to change in a 6 dependent way and this is not acceptable; choosing the unit 5 arbitrarily (thus, independent of the relative scales developed by the system itself like m or g”2(‘+6)) makes the number b in (111.3) a-dependent: b= g<-2-26. This difficulty

presently encountered


comes from the fact that the system (1.1)

M. C.



involves two scales, m and gi/*(l+ ‘) (which may be considered as S-independent if g becomes d-dependent accordingly), but none of them is actually the “natural” scale of the system since m and g may vanish independently while the system (1.1) still exists in that case. In fat, the “natural” scale of the system is a clever combination of m and g that we now introduce.

IV. A SCALE INVARIANT Let us define a “natural”



scale {,(m, g) as the solution of the equation (IV.1

m2+ 2gti2’ = 25:

and let us write (IV.2)

g= yg+*c Then, the expansion (111.2) becomes

rN Cm*,g) = 5; W+l+-(+,~+?)



and more generally, from (II.2) and (A.16) we obtain rN(m2y




where the function G, (y, 6) is defined in (A.1 1). The expansion (IV.3) does not contain any logarithm of y; y appears as a “natural” dimensionless measure of the interaction and all the diseases of the expansion (11.3) have been summed up into the “natural” scale 5,. The expansion of GCN+ i&y, 6) is now manifestly scale invariant order by order in 6. We note the interesting fact that the “natural” scale is fixed by simply looking at the first order of the expansion in powers of 6; this is a general feature for all systems. The system ( can be rewritten as

$(l-r,C’ (IV.5) In (IVS), 5, and y are considered 6 independent while g is not. Instead of (IV. l )-(IV.2) we could have imposed m2 + 2gr’,-” = 2~x5;~ g



+ 26,





where o! is a given positive 6 independent systems. We would have obtained





number chosen once and for all physical

(N+*,,* [qy)+?f

fNb2, g) = (cc-



which, after all, also displays the properties of scale invariance order by order in 6. Consequently (IV.6) defined a class of “natural” scales. The privilege of the “natural” scale (IV.1))(IV.2) is only aesthetical. We prefer to deal with the function T(x) and its derivatives rather than with the function T(x) r.--’ and its derivatives with respect to x (this property being related to the fact that the interaction strengh y satisfies for any m and g, 0 Q y < 1, while we would have 0 < y’ < a). Moreover, the introduction of c(# 1 would be a useless complication, since all powers of logarithms of LXsummed up to give an LXindependent function f N. We wish to understand at this point, whether the “natural” scale 5, is a perculiarity of the expansion around 6 = 0, or, on the contrary, if this scale is valid for any system generated by the different values of 6. We consider the system (1.1) around the value 6 = 6* and we perform the expansion in powers of (6 - 6*). We write

so that

rN(m’, g)=fiv*(m’, g)-g{G+2+26(m2T g)-G+2+26*(m2, g)>

+$ CG+3+46(m23 g)-2~,*+,+2,+2s*(~~2, 8) + G+4+4a* (m2, g,} + c?q6-6*)X. The Green functions at 6 =6* certainly write

-g 1

~~+z+zs(Y*~ (5,*)“‘2’26


are not known but, using (1.13).-(1.14), we can



C+2+2s*(y*, (p)N+2+26*


+o(6-6*)2 > (IV.10) 1



where i;,* is a scale invariant

object while t,* and y* satisfies m2 + 2gt,*-28*

= 2(,*



We denote by t,* and y* the solution of the system of Eqs. (IV.1 1) at 6 = 6* and for a given g. The expansion around 6 = 6* is rNw3

1 g)= (5y+’


caY*Y 6*)+

x {G+2+26*

2g(6 - a*) (tn*),+26*



(IV.12) where FL* means (a/aZV) r,$ ; we note, for instance, that at 6* = powers of ln[(m’+ J-)/4]. This expansion around 6 = 6* can be easily obtained at all orders the same diseases as the expansion around 6 = 0. Again, we prove that logarithms may be summed through the complete (6 - 6*) expansion; perform the change of variable 4 --t I$/,/%, we have r,(m’/%‘,

1, we obtain and contains all unwanted if in (1.1) we



The expansion (IV. 12) becomes 1

F*cy*(e) [ N

fN(m2, g) = eN+l[<:(e)]N+l

d*1+2ge-2-2”(-*) ’ cwr+2~*

(IV.14) where m2 ,,-tzge-


--26* = 2(,*(e)

(IV.15) ge-2-2”=y*(e)[5,*(e)]2+26*.

Clearly, the choice for a “natural” scale 8,, which sums up all logarithms expansion around 6 = 6*, is defined by the solution of the equation 5: 69 = 1

of the (IV.16)







which implies m2 + 2g0,;‘”

= 28:

(IV.17) g = y*(e,)

es+ 2*.

The solution to (IV.17) is 8, = 5,


Y*(e,) = Y, where 5, and y were defined in (I.lO)-( I.1 1). The expansion (IV.14) becomes T,(m’,

g) = - f,, 5,





and this proves the stability of the “natural” scale for all physical systems defined along the 6 path. We close this section by presenting the point of view of the differential equations. First of all, field equations for the Green fonctions defined in (1.1) read m2f,+2(6)+2g(1+6)rN+2+26(~)=(N+1)rN(6).


This equation can also be transformed into a differential equation






(IV.21 )

Another useful differential equation is

acm -= as



This differential equation can be iterated several times and, for instance, we find (IV.23) etc. These relations taken at 6 = 6* describe the structure of the different terms of the expansion in powers of (6 - 6*) (IV.3), (IV.19)). Equation (IV.22) also explains the presence of unwanted logarithms in the 6 expansion; for instance, (ar,(s)/%)l s=0 is related to (a/aN) TN+ 2 (0) but the dimensionality of rN+ 2(O) generates, by differentiation with respect to N, logarithms of dimensioned quantities. Let us write Lwr




where 5 is S-independent and has the dimension sionless; then (IV.22) becomes am -=




of a mass while p,,,(6) is dimen-





which shows the emergence of unwanted logarithms of massive terms. This disease can be cured if we give to the coupling constant a 6 dependence. In that case, (IV.22), must be changed into

arm -=as



so that (IV.25) becomes

(IV.27) We define the b-independent


scale 5, as the solution of




g= y<;+26.


that is


(IV.27) recovers the form Cm -----=-2y a6

aL+,+,,m aN

for the dimensionless quantity r,,,(6). To obtain constant y, we note that (IV.20) becomes $L+2 n Taking


(IV.30) the arbitrary



6 = 0 in (IV.31 ) and noting that (IV.32)







we obtain m2

3 + 2y = 2.


The relations (IV.33) and (IV.29) are the relations (IVS), solution to (IV.l). Finally, if we wish to express r, (m’, g, 6) as a function r, (m’, y, 61, the differential equation (IV.21 ) becomes


a m2----P(y)~+(N+I)lT,(m’,y,6)=0,


where p(y) is bien in (1.19). The function /3(y) can be seen on Fig. 1 for 6 =O, 1, and 2. V.


First, we wish to extend our result to any polynomial define P(d2, 8,) = i






gpb0 (V.1)

gp, 6) = i gp(42)‘+P6. p=l


The function

P(d2). Let us


for 6 = 0, 1, and 2



Then, the Green functions are defined as follows rN(m2,









We then apply the general techniques to obtain the scale invariant perform the change of variable 4 -+ d//5 and substract the polynomial note that

p(4’/5*, gp, 6) = P(d’, gp5-2(1+pb), 6).


6 expansion: at 6 = 0. We




The following result is now obvious: we define the “natural” of the equation m2+2



scale as the solution



that is to say, gp





Then rN(m2,

where ?;, is a scale invariant of 6,

gp, 6) = - 13,, 5”



function, order by order, in the expansion in powers


where c=





The extension to polynomials

with odd powers of 4 and negative couplings g, and








m2 does not present any diffkulty as long as the integral (V.2) converges; the convergence of the 6 expansion might not be true in some cases, however. Let us extend the result to a finite number of uncorrelated points. We define Green functions as (V.10) Clearly enough, each point i develops its own “natural” mf+2 Defining the dimensionless

scale [,,, defined by

f gi.p[;12ps=2
coupling constants gi,p=yiJ;(f+pd’

and given the partition



function z = T{O, .. O}>


then the observables V.14) can be written as V.15) In (V.15), F{,,) is scale invariant

order by order in powers of 6, N=xN,


and the “natural” scales tn.(N,J of the physical system are weighted geometrical averages of the “natural” scales of the constituents tn.




the weights being characteristic of the observable. Of course, if all points i have the same mass mj and the same polynomial of interaction Pi, all “natural” scales 5,,i are equal and there exists a unique “natural” scale for the physical system whatever observable is considered. It is the purpose of Ref. [ll] to prove that the results (V.15)-(V.17) are valid



when interactions occur between all points; however, in that case, the system (V.11) which determines the “natural” scales of the constituents satisfies a set of coupled equations, and each “natural” scale 5,,i becomes a function of all the parameters describing the physical system. What is the physical meaning of these “natural” scales? Although, it is certainly too early to understand fully the importance of these scales, let us try. “Natural” scales are certainly not a standard unit to measure masses (like gram or kilogram); on the contrary, grams and kilograms are rather emanations of our local “natural” scale. “Natural” scales are internal scales developed by physical systems submitted to non-Gaussian interactions (6 > 0). Without non-Gaussian interactions, how can we guess that the system

@“I@ e-“(~L-)2)2-(m2/2)(~:+):) rN,.N2= s&,dd2 has a “natural”


scale 5, given by + 4A) rf= m2(m2 2(m2 + 22) .


It is only after adding to (V.18) a non-Gaussian interaction of the type + ($:)““] and after performing the 6 expansion (first order in 6 is dMY +s enough) that [,(m, g) appears; then, taking the limit g -+ 0 we obtain (V.19). At this point, let us be clear on the terminology. The observables (dy’dp) are dimensoned quantities: (mass) N1+ Nz. If we choose a standard unit of mass p (gram or kilogram), pN1 + N2(&%#p) is a dimensionless number which is scale dependent since a change of the unit p changes the number. Special choices of p related to internal parameters of the system like p = m, p = &, p = gl/*(’ +6), p = r, define dimensionless scale independent numbers, but clearly

Z5 ,“’ +Nyqs~(b~).


From the four above quantities, only <,N’+N2(4;N1dp) is scale invariant, order by order, in an expansion in powers of 6. A local observer which belongs to the system of points (1.2) introduced in (V.18) and which measures (4:‘4?) = rN,,N2/r0,0, necessarily using its own “natural” scale c,, can only determine the dimensionless number tf’ + “‘( ~+4;“‘4?); consequently, he will only be able to measure the angle 0 defined conventionally by sin 28 = ~



0 < 8 < 7114


without any knowledge of m and 2 separately. It must take an external observer to







measure 1,‘~’ and m/p separately (where p is the “natural” scale of the external observer), but this requires the existence of a third point and the presence of interactions between point 1 and point 3, point 2 and point 3; that is, a physical system which differs from (V.18). In the limit A -+ 0, the “natural” scale 5, = m/J? (or m/& if we forget the universal convention c(= 1 of Section IV seems trivially unavoidable. Again, for the system po,





4” e-‘m2/2M2


~ CL,

the internal observer, which only has at his disposal the mass m to organize his measures, will obtain

(t,,)N(4N>(o, =

(V.23) N even

and cannot get any knowledge of his m. It takes an external observer to distinguish between the different rn: of different systems (V.22). The situation is similar to the observation of different bodies with free linear motions at different velocities, while each body feels at rest in its own frame of reference. We now come back to the system (1.1). The linear Gaussian system

where the “natural”

scales satisfies m2 + 2gr,;26 = 25:


appears as a tangent free field for the interaction g(+2)’ + ‘. If we wish to compare observables for the system (1.1) for two different values of 6, say 6, and 6,, we should not calculate the dimensionless number (V.26) at fixed g, which has an unacceptable order by order scale dependent expansion in power of 6, and 6,. On the contrary, we should sum all unwanted logarithms into “natural” scales and calculate the dimensionless number (V.27)

92 which has all (V.25) with 6 coincide only different, while


the required properties of scale invariance (t, in (V.27) is given by equal to 6, and a,, respectively). We note that (V.26) and (V.27) when we consider g in the system 6, and g in the system 6, to be tN is 6 independent

with m2 y=(l-YE,



g(6,) = yt;; + 262. As a special case, if we wish to compare observables in the system (1.1) and in the tangent Gaussian field, we must calculate

(v30) ’

By extension, if we wish to compare observables in two different physical systems, we must look for the dimensionlessnumber (V.31) The situation is similar to general relativity where systems are described in tangent spacesvia the g”‘“(x) metric and where it takes the notion of parallel displacements and the introduction of Christoffel’s symbols to compare different systems. This notion of tangent free field has always been a difficult problem along the history of quantum field theory. Very early [ 131, it was understood that in a Fock space of particles of mass m, the unitary operator describing the states of particles of mass p #m was an “improper” unitary operator (all elements being null due to the infinite number of degrees of freedom) and the existence of inequivalent representations of the commutation relations were recognized. Later, when the interacting fields d(x) were supposed to have for limit at time f co two free fields din,OUt(x), the difficulties were threefold: first, the existence of a “proper” unitary operator between 4(X, t) and $,,, ..,(X, t) was abandoned (Haag’s Theorem [ 141). Second, the mass of the asymptotic free fields was different from the parameter m in the Lagrangian (mass renormalization) (the situation (V.l),







(V.22), and (V.24) looks relatively similar). Finally, the LSZ theorem [ 151 and the reduction formulas told us that lim f -+ T Co4(X) = J;#irl,



where z is the wave function renormalization and where the limit is understood as a “weak” limit, that is, only in the sense of expectation values and not in the sense of an operator limit. In the exploration of quantum electrodynamics it has been dramatically observed that the electron mass renormalization and the wave function renormalization were infinite, order by order, in powers of the coupling constant. It was quickly understood that the solution to the regularization and renormalization program necessarily introduced an arbitrary scale, either as a cutoff to make the integrals convergent, or, in a space of dimension d = 4 - E, as a scale to measure the coupling constant which becomes dimensioned. The invariance in regards to the arbitrary scale was satisfied because of the existence of the renormalization group [ 161. One expression of this invariance is the so-called Callan-Symanzik [ 161 differential equation which is of the form (IV.34). We believe that “natural” scales, properly defined in continuous field theory from lattice field theory via thermodynamic and continuous limit, might help in better understanding the physical meaning of the renormalization group techniques in field theory. Let us end this publication with a technical remark. Suppose that had we chosen in (IV.6) a convention c(# 1, the j?(y) function would have been (V.33)

B(y)= -2(1+6)@&), showing two fixed points y* = 0 and y* = c(. As expected, the derivatives

p’(y=y*)= li’ +s) i

at at

y*=O Y*=UVS


are a-independent.


We define the function

for p and v > 0 (possibly p < 0 if v and 6 > 0). This function is analytic in 6 for s > 0 and p > 0 V’6 and for s > 0 and 6 > - 1 for



p = 0. The expansion in 6 is a convergent series (for 161< 1 if p = 0). Let us first mention two special cases: J-,(/i, 0,6) = T(s) ,Lp3


(A.2.a) (A.2.b)

TO describe the expansion in 6, we write F,(p, v, +s,:




m (-v)” 3o(jx xs~1e~IP+Y).Xl+a-x)P. J-,(/4 v, 6)= c b p=o P! s0


Each term of the above expansion can be calculated, F’P’(p, v, 6) = y3

p 2





and is of order P and more. The expansion in 6 contains powers of ln(p+ v); let us give some of the first terms of the expansion

Fs(p,v,~)=(p+v)-sT(s)-v T(s+1+6) { ( (p+v)‘+6

T(s+l) -m


rls+2+26)-2T(S+2+6)+T(s+2) (p+v)2+6

~,(P,v,~)=(p+v)-”[ r(s)+&





+ v*a2 {Z7s+2)ln2(~+v) 2(P + v)’





The general term of the expansion can be found since

i c;(-)“-qq’=(i q=o


for for










with r!

arp= (n,) C n,! ... HP!’

a00 -1 -


i a,>,,0 = 0,

the nb being strictly positive integers and (A.9) Consequently,

and the powers of ln(p + v) are generated by the derivatives in regards to S. We define for Odv
In that case, the expansion (A.6) simplifies since all logarithm obtain G,y(v, 6)=I-(s)-vW’(s+


terms vanish; we




and, more generally, G,(v,d)=


f’ !? i s(-v)T”‘(s+~). r=O r! p=. P!

Finally, a change of variable x + X/U in (A. 1) gives

If a is chosen in such a way that (l+a)=



then we may write F,(p,

v, 6)=a-“G,(va-(‘+“),

where a is the function a(~, v, 6) solution of (A.15). 595/209/ 1-7





We thank C. Bervillier for his careful reading of the manuscript and for his interesting remarks about the analogy with scale considerations in continuous scalar field theory below and at dimension four (renormalization group techniques, existence of a B function, etc.).

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