Short range correlation between elements of a long polymer in a good solvent

HAL Id: jpa-00209237

https://hal.archives-ouvertes.fr/jpa-00209237

Submitted on 1 Jan 1980

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Short range correlation between elements of a long polymer in a good solvent

J. Des Cloizeaux

To cite this version:

J. Des Cloizeaux. Short range correlation between elements of a long polymer in a good solvent.

Journal de Physique, 1980, 41 (3), pp.223-238. �10.1051/jphys:01980004103022300�. �jpa-00209237�

Short range correlation between elements of a long polymer ^{in a} good ^solvent

J. des Cloizeaux

C.E.A., DPh-Service de physique théorique, ^C.E.N. Saclay, ^{B.P. n°} 2, ^{91190 Gif sur} Yvette, France (Reçu ^{le 9 août} 1979, révisé le 31 octobre 1979, accepté ^{le 5 novembre} 1979)

Résumé.

Certaines propriétés de corrélation d’un polymère ^{en bon solvant sont} étudiées à l’aide de techniques

de renormalisation. La distribution de probabilité ^d’un vecteur r, joignant, ^{dans un} espace de dimension d, les extrémités d’un segment ^de polymère, ^{formé de N maillons est une fonction} Ps,N(r) ^{où s est un indice qui se}

réfère à des situations diverses. Cas s

0 : le segment ^{coincide avec le} polymère ^{lui-même. Cas s}

^{1 : le} segment constitue l’une des extrémités d’un polymère infini. Cas s

2 : le segment ^est situé dans la partie centrale d’un

polymère infini. On montre que les distributions de probabilité obéissent à des lois d’échelle de la forme

que, pour ^x petit, fs(n) ^{~ x03B8s et} que les indices 03B80, 03B81’ 03B82 ^sont donnés par la renormalisation de ^vertex simples.

Les développements ^{de ces indices au} deuxième ordre en 03B5

4 2014 d ont été calculés. Les résultats sont

Des estimations de ces indices pour d

^{3 donnent}

Ces résultats montrent que la probabilité ^{de contact entre} les extrémités d’un segment ^{formé de N} maillons, appar- tenant à la partie centrale d’un polymère ^{infini est} proportionnelle à N-v(d+03B82)

N-2,18. Cette conclusion s’accorde

avec le fait prévu par l’auteur que, pour N ~ 1, les termes dominants de l’énergie ^{E d’un} polymère ^{formé de N}

monomères sont de la forme E

aN + b en bon solvant.

Abstract.

Correlation properties ^{of a} long polymer ^{in a} good ^{solvent are studied with the} help of renormalization

techniques. ^The probability distribution of the vector r joining, ^{in a} space of dimension d, ^{the end} points ^{of a} polymer segment ^{made of N} links, ^{is a function} Ps,N(r), where s is an index referring ^to various situations. Case

s

0 : the segment coincides ^{with the} polymer itself. Case s

1 : the segment ^{is at one} extremity ^{of an infinite} polymer. ^{Case s}

^{2 : the} segment is located in the central part ^{of an infinite} polymer. ^It is shown that the pro-

bability distributions obey scaling ^laws of the form Ps,N(r)

N-03BDd fs (r/N03BD), that, ^{for small x,} fs(x) ^{~ x03B8s and that}

the indices 03B80, 03B81, 03B82 ^are given by ^the renormalization of simple vertices. The second order expansions ^{of these}

indices with respect ^{to 03B5}

4 2014 d have been calculated. The results are :

Estimations of these indices for d

3 give 03B80

0.273, 03B81

0.46, 03B82

0.71. These results show that the pro-

bability ^{of contact of the end} points ^{of a} segment ^{made of N} links, belonging ^to the central part ^{of an infinite} poly-

mer is proportional ^to N-03BD(d+03B82)~ N-2.18. This conclusion agrees with the fact, predicted by ^the author, that for N~ 1, the dominant terms of the energy ^E of a polymer ^{made of N} monomers, ^are of the form E

aN + b,

in a good ^solvent.

Classification

Physics ^Abstracts

05.40

61.40K - 64.90

82.70

1. Introduction.

A polymer ^can be considered

as a chain of ^monomers and, ^{in a} polymer solution,

the position ^{of the monomers} belonging ^{to the same} polymer ^{or to different} polymers, ^are strongly ^corre-

lated. The properties ^{of the} polymer ^solution depend

very much ^on these correlations. Obviously, ^{it is} difficult ^{to make} good ^{theories and to} interpret ^the

results correctly ^without knowing precisely ^{the nature}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004103022300

of the short _range correlations. Thus, ^{this matter} deserves investigation.

In the present article, ^we study only very dilute solutions i.e. isolated polymers ^{and we show that}

even, in this simple case, correct ^answers cannot be

given by simple arguments. ^In fact, ^{the short} range correlations depend, ^{in a crucial way,} on renorma-

lization properties ^{of the} polymer. ^Our aim here is to study ^these properties.

A polymer may be considered as a chain made of

No links. The position ^{of an} articulation point ^on

such a chain is given by ^{a vector} rj, where j ^{is an} integer ( j

1, ^..., No ; ^see Fig. 1). ^{It is assumed that}

the chain with excluded volume belongs ^{to a} space of dimension d.

Fig. ^1.

The chain has No links with No = N2 ^{+ N +} Nl.

We want to know how the points ^{A and B are} correlated.

We are interested in the probability distribution of the vector r joining ^two points belonging ^{to the chain.}

Assuming ^that

we define this probability distribution (see Fig. 1), by setting :

In a preceding ^article [1], ^{the author has} studied the

properties ^{of the} probability distribution Po,N(r) ^{of a}

chain made of N links

For large ^{values of} N, ^this function is given by ^a scaling ^law

and it has been shown that for small values of _x, the function fo(x) ^{behaves as a} power of x

where 00 ^{is a} critical index.

The index 00 ^{can be} expressed ^{in terms of the}

current critical indices _y and ^v

These indices depend on d, and, for, d

3, ^{we have} [2]

In the present article, ^{we shall} generalize ^{these results} by considering ^other types of correlations (see Fig. 2).

The most fundamental ones are given by ^the following probability distributions

Fig. ^2.

Correlations of various type (0, 1, 2).

It will be shown that the probabilities distributions

Pj,N(r), ^{where s}

0, 1, 2, ^{have similar} properties.

For large N, they obey scaling ^laws

For small x, we shall find that

and the expansions ^{of the indices} 60, 81, 82 in ^terms

of 8

4 - d > 0, ^{will be} given. ^{In this} way, ^{we can} also find the dependence ^{of the} probability ^{of contact}

of two monomers, in terms of their monomeric distance N along ^{the chain} (for N > 1).

In section 2, ^after introducing ^a convenient Lagran- gian theory ^of polymers, ^we express the distribution

probabilities ^{in terms} of Green’s functions. Principles

of renormalization theory ^are recalled in section 3 and in the Appendix. ^The scaling properties ^{of the}

distribution probabilities ^are established in section 4.

The short range behaviour of the probability ^distri-

butions is examined in section 5. The most interesting

discussions ^are contained in sections 6 and 7 which deal with the determination and the calculation of the critical indices 03B63 and C’ 4 from which 81 ^and 03B82 ^{can be}

deduced. The results are summarized and commented in section 8.

2. Lagrangian ^{formalism and} polymer theory. -

The Euclidean field theory adapted ^{to our} problem

can be defined by ^a Lagrangian ^{of the form}

A cut-off is needed and the critical value for both fields is by definition a

b

ae.

We may ^assume in the beginning ^{that n is an} integer (n > 1). However, ^the diagrams ^will correspond ^to

the limit n ---> 0. More explicitly, ^{this means that closed}

solid lines will not appear in the diagrams [3, 4].

We note that, ^{for n}

0, ^{if one} type ^{of field} only (components ^of 9 ^or components ^of 03C8) appears in a

mean value, ^{the other} _type of field does not play any role in the calculation of this quantity.

The fields will be represented by ^functions qJ j(x)

and qlj(y) ^or their Fourier transforms ffJ j(k) ^and W

I

In the following, ^we shall also use the simplified

notation

The following ^{Green’s functions} correspond ^to

the types of correlations which have been described in section 1.

The expansion ^{of these functions can} be calculated

by applying ^the usual rules and typical diagrams ^are represented ^on figure ^{3. We note that two different}

Fig. ^3.

Diagrams contributing ^to

components of 0, namely 03C81 ^and t/J2’ ^{have been used}

in order to eliminate all the diagrams containing ^closed

solid lines; ⁱⁿ particular, ^the zero-momentum compo- nents 03C81 (0) ^and tfr 2(0) correspond ^{to the} free ends of the polymers.

As will be shown later, ^the denominators which appear in the definitions of G1(x, y) ^and G2(x, y)

have been introduced in order to insure the conver-

gence of these quantities ^{in the} limit b --+ _ac.

The space dimension d is assumed to belong ^to

the range 2 d 4 and for a # ac, b =1= ac, ^one

subtraction only in the self energy is necessary ^to make all the current diagrams ^finite. Thus ac ^{is the} only ^current quantity ^{which cannot be calculated}

without introducing ^{a cut-off.}

As usual [4], ^{we may} express the Green’s functions in terms of (normalized) numbers of configurations

of polymers. ^Let Z(x, y; N2, N, Nl) ^{be the number}

of chains of No ^links (No

N2 ^{+ N +} Nl) ^which

have two fixed points ^defined by ^{the conditions} rN2 = x, rN2+N = ^y.

The total number of chains of No ^links starting

from a fixed point (x) ^is

and we have

where I is an arbitrary length.

(The product a,, 12 ^is dimensionless.) ^More precisely,

for large No, ^we expect ^{the behaviour}

The correspondence existing ^{between the Green’s}

functions and Z(x, ^{y ;} NI, N, N2) ^{can be read on the} diagrams. ^{In this} way, ^we find the relations

On the other hand, ^the probability distributions

Po,N(y - x) ^are directly ^{related to} Z(x, y, N2, N, Nl)

The definitions of P 1,N(Y - ^{x) and} P2,N(y - ^{x) can}

be simplified by remarking ^that

Thus, ^we obtain

In order to describe situations in which N, >> ^N

and N2 > N, ^{we need to know the} behaviour of the Green’s functions in regions ^where

L

Accordingly, ^we may perform ⁱⁿ Eqs. (2.7) ^the following substitution

We find in this _way, a relationship ^{between the Green’s}

functions and the distribution probabilities.

In the following, ^{the notation 12 will mean that the}

index _may be 1 or 2.

Using ^this notation, ^we may write

The probabilities remain well defined when No - ^oo

and the integrals ^are convergent. Consequently, ^we

find that the Green’s functions remain also well defined when b - _ac. This result is of course essential;

it shows that the definitions (2.3) of the Green’s functions are adequate. ^{Still, we note} that these definitions have no meaning ^{for b}

ac but only ⁱⁿ

the limit b - _ac.

However, ^{we face a} difficulty. ^We remark that in

the absence of _any interaction and for 2 d 4,

the Green’s function

is not uniformly convergent ^{and that it is} strictly divergent ^if y

^x. Thus, ^it depends ^{on an} ultraviolet cut-off. On the other hand, ^the (normalized) ^func-

tion Z(x, y ; 0, N, 0) is well defined in the absence of any cut-off for small N. This anomaly ^{can be under-}

stood by considering ^the equation

where in the absence of interaction

We see that for _y

x, the integral ^is divergent ^for

small N when d > ^2. This property, ^which applies

also to GI(x, y) ^and G2(x, y), ^{remains true when an}

interaction is present and is the origin ^{of the} difficulty.

Still, ^the scaling ^{laws can be} applied ^without any

problem, ^when Eqs. (2.12) ^are replaced by ^{the fol-} lowing ^{set of} equations, ^{which are} always ^{valid, in}

the absence of _any ultraviolet cut-off, when p ^is large enough

3. Renormalization of fields and insertions.

For

large N, ^the main characteristics of the probability

distributions depend fundamentally ^{on the renorma-}

lization properties ^{of the} associated field theory.

Here, ^{we have to deal with two different zero} compo- nent fields qJ j(x) ^{and t/J,(X),} and the known results of the one-field Lagrangian theory ^{cannot be} directly applied ^without discussion, ^{but the} reader, may refer to the basic concepts recalled in the Appendix.

3.1 NOTATION. -- We are interested in Green’s functions of the form

where Fj(x) is the value at point ^{x of one} component of ₉ or 4/ ^and 01(y) ^the product of the values at y of several components of the fields _T or 03C8.

The Green’s functions can be expanded ^{in terms}

of the interaction and we define connected Green’s functions as usual.

The renormalization properties ^{of these functions} (i.e. their behaviour in the critical domain) ^are directly given by those of the vertex functions. In

the following, ^{the vertex} corresponding ^{to the one-}

irreducible Green’s function

will be noted

Its Fourier transform will be

3.2 RENORMALIZATION.

We start from a situa- tion where a = b # ^{c and we} consider the other situations as perturbations of this initial state. We stay ^{in the} vicinity of the critical point.

In this _way, we have only ^{one relevant} scaling length ^{m-1 where m} is defined by

A renormalization factor can be associated with each marginal composite ^{field O and the renorma-}

lization constant is a function of mgo 1/ E (See Appendix).

The dominant part ^{of 0 can} be renormalized by setting

where OR(X) ^is the renormalized composite ^{field and} zo(mgo 118) ^a renormalization factor which is deter- mined by renormalization conditions (See Appendix).

Thus for instance, ^we may ^set

and we may always ^define z2(mgo l/e) by ^{the condition} (See (A. 14) ^{and the} following discussion)

3.3 CRITICAL INDICES. - For small values of

mgo 1/e

where Co is the critical index associated with 0.

Thus, ^{the indices} corresponding ^{to zi} and Z2 ^are

given ^{in terms of the current indices v, y} and ti by

and, incidentally Eq. (A. 19) gives ^also

Expansions ^{of these} indices with respect ^{to E are} known [6] ^{and we have}

For n

0 and E

1, ^a direct calculation made by

Le Guillou and Zinn-Justin [2] ^has given

4. Scaling properties ^{of the} probability distributions.

- The fact that for large N, ^the probability ^distri-

butions obey scaling ^{laws is a} simple consequence of the renormalization properties ^{of the} system.

Here, ^{we have two fields} qJ and 03C8. However, ^the fact that each of them has zero components ^{leads to}

significant simplifications. Thus when a

b, any

product of the form qJJ(x), ^{qJf(x) or} Tj(x) ^{T,(x) is}

renormalized exactly ^{in the same way as the sum}

Fig. ^4.

Diagrams contributing ^{to the} renormalization of qJi(X) qJ/x) ^and [qJ2(X) ⁺ .p2(X)J. Diagram (a) ^{does not contribute}

to the renormalization of 9 1 (x) qJj(x). ^{If n}

^{0, diagram (b) does not}

contribute either to the renormalization of [(p’(x) ⁺ .p2(X)J ^but

it does if n # ^0.

This fact can be easily understood by looking ^{at the} diagrams displayed ^on figure ^4. However, ^{as was}

shown by ^{Amit and} Goldschmidt [7], ^this property is not true when, ⁱⁿ contradiction with the present situation, ^{the number of} components of’the fields do not vanish. Thus, ^{for non} vanishing ^{values of} (y - x)

and a = b i= ac, ^we may write according ^to Eqs. (2.3)

where _z, and _Z2 are the renormalization factors of the one-field theory ^{for n}

0. In the asymptotic limit, ^the renormalized Green’s functions GO,R(XI ^y)

and G12,R(X, ^y) depend only on x - y m.

Let us now consider the case a # ac, b --+ ^{ac. This}

situation can be obtained by adding ^{to the} symmetric

Lagrangian ^{with a}

b, ^a perturbation

where 0 ^x 1 and by taking ^{the limit x -+ l.}

Each term appearing ^{as a numerator or a deno-}

minator in the expressions giving G12(x, y) ^{can be} expanded ^with respect ^{to the} perturbation. ^This perturbation ^{can be written as follows}

As each insertion

of f ifJi.. introduces a factor m- 2,

we see immediately ^{that each term of the} expansion

behaves exactly like the first term.

Thus, ^{for 0 x} 1, Eq. (4.1) remains valid. On the other hand, ^{as was shown} previously a a ap P ^laP aP ^{G12(X, Y)} have finite limits when x - 1. Therefore, Eqs. (4 .1 )

can be considered as valid for a

a,,, b

^a,,.

The renormalized Green’s functions can be written

as follows

Putting ^these expressions ⁱⁿ (4.1) ^and considering

the asymptotic ^{limit m - 0. We obtain}

in the other hand, ^from Eqs. (3.4) ^and (3 . 5) ^we get

As _{go is} fixed, ^we may ^now forget ^the dependence

with respect ^to go and we obtain

Thus, ^{we obtain the} scaling ^laws

or using Eqs. (3.6)

Let us now return to Eqs. (2.11). ^The partition

function Z(N) ^{and the} distribution probabilities Ps,N(Y - x) (where s

0, 1, 2) ^can be deduced from

Go(x, y) ^and G12(x, y) by ^inverse Laplace ^transfor-

mations. Consequently, ^{the fact that} Go(x, y) ^and G12(X, y) obey scaling laws entails that Z(N) ^and Ps,N(Y - ^{x) possess similar} properties.

From a comparison ^between Eqs. (2.11) ^and (4.9),

we deduce the expected scaling laws valid for N > 1.

with s

5. Short range behaviour of the probability ^distri-

butions.

New divergences appear when y - x I

goes ^{to zero.} This fact is well known in renorma-

lization theory ^{and can} be related to the fact that the probability distributions vanish in this limit [1].

Thus, for small values of r, ^we may write

where 8S ^{is an} index which has to be determined. For small values of r as compared ^to NV, ^{we need a cut-} off and accordingly ^{we have} PN(0) N PN(l ) ^{where 1}

is a constant length.

This gives ^the dependence ^{of the} quantities P.1,N(o)

with respect ^{to N}

Let us now return to the definitions (2.12), (2.6) :

we obtain

These integrals ^are generally divergent ^{and there-}

fore meaningless. ^{We have to use} Eqs. (2.15) and, if p ^is large enough, ^these equations remain valid

when I y - x I -). ^{0. In a} corresponding way, ^we may write

In the absence of interaction, ^these equations ^are meaningful ^{for d}

⁴ and near d ⁼ 4, ^{if and} only if p > 2, ^{and this} property ^{remains true when inter-} actions are present

Therefore, ^the indices 0. define also the dependence

of the derivatives of the Green’s functions with respect ^to (a - a,,). ^{These functions are determined} by ^the equalities

and the indices es ^can be calculated by studying ^their

behaviour in the frame of renormalization theory.

We consider the case a :A ac, b -- ^{ar. As was done}

in the preceding ^{section, we reach} this limit by adding

to the symmetric Lagrangian (with a

b), ^a pertur- bation

11

and taking ^the limit x ---> l.

Changing x amounts to introducing ^{in the dia-}

grams, insertions proportional ^to (a - ac) 03C82. ^These

insertions do not change ^the general properties ^of

the Green’s functions. On the other hand, any diffe- rentiation of Green’s functions with respect ^{to «a »}

amounts to an insertion of ^{[cp2 + (1}

^{x) cp2] in}

these Green’s functions. All these insertions are

marginally divergent ^{for d}

^{4 and there is no} mixing

of components ^{of (p or} 0. Thus, ^these insertions can

be renormalized exactly ^like cp2 ^{in a} q>4 ^theories.

We have also to consider the insertion of compo- site fields of the form (p2(X) 03C81(x) ^and

These fields _may be split ^into parts ^{which must be} renormalized in a different way. It is clear that, only

one part ^will be relevant. Therefore, ^we may simply

set

and later which shall determine the relevant indices C’ 3

and (§ ^which correspond ^to z3 ^and z’ 4’

Thus, starting ^{from the} definitions (2.3), ^we may write in a formal _way

By differentiating ^with respect ^{to a, we find}

The canonical dimension of _any function Gs(x, x)

is (d - 2). Thus, using ^the asymptotic ^form (3. 5),

we obtain

Forgetting ^{now the} dependence ^with respect ^{to go} and using Eq. (4.7), ^{we obtain}

A comparison ^{of these} equations ^with Eqs. (5. 5) gives

Finally, using the relations (3.6), ^{we find the result.}

which has a rather simple interpretation. ^{We have}

now to determine the relevant indices 03B63 ^and 03B64..

-

6. Determination and calculation of C’ 3 near d = 4.

The index 03B63 corresponds ^{to the} insertion of

The diagrams ^of type ¹ which contain this insertion contain also _p insertions of q>2 which result from differentiations with respect ^{to a. We have seen in} section 5, ^{that to eliminate} spurious divergencies, ^we

have to take p > 2, ^{and we} check that the zeroth order diagram corresponding ^{to our} problem ^is

convergent ^for p > ^{2 but} divergent for p

^{0 or}

p = 1 (see Fig. 5).

Fig. ^5.

Correlations of type ^1. The zeroth order diagram (a) with one q>2 insertion is divergent but the zeroth order diagram (b)

with two q>2 intersections is convergent.

Thus, ^the calculation of the scaling properties ^of

2013 6’i(x, P ^{x) near} the critical point amounts to cal-

laP ^{t ) p}

culating ^the critical index associated with the three-

leg ^{vertex. The index} C’ 3 ^{can be} given by ^{the renor-}

malization of the ^vertex

Different masses correspond ^to the fields _T and 03C8.

However, ^{it is} possible ^{to use the} technique ^of

renormalization at the critical point [8]. ^A simpli-

fication occurs because, ^{in this} case, ^we need only

one field. The renormalization can be made at the symmetry point.

The four momenta kl, k2, k3, k4 ^{of the} symmetry

point ^{are defined} by ^{the relations}

The renormalization conditions at the critical

point ^{can be written}

I

The index (§ ^is given by ^the renormalization of

The calculation can be made by using ^{the method}

which has been developed by ^{Brezin, Le Guillou}

and Zinn-Justin [8] ^{for the} calculation of more cur-

rent indices. The relevant diagrams, up ^to second order, ^are given ^on figure ^6.

Fig. ^6.

Diagrams used for the calculation of C3 ^{and their} sym- metry ^factors.

The function fl(u) which determines the fixed point

is [8]

and its zero is

The expansion of the renormalization constant (§(u)

which respect ^{to u is} given by

The critical index (§ ^is given by ^the equation

Thus, ^we get

Replacing ’1 by ^its expansion (see Eq. (3.8), ^we

obtain

7. Determination and calculation Of C4 ^{near d}

^4.

The index (4 corresponds ^{to the insertion of}

The diagrams ^of type ² which contain this insertion contain also p insertions

of ^{cp2. In this case, the}

zeroth order diagram ^is convergent ^for p > ^{2 but} divergent for p ^{= 0 or} p

1 (see Fig. 7).

Fig. ^7.

Correlations of type ^2. The zeroth order diagram (a) with one q>2 insertion is divergent but the zeroth order diagram (b)

with two q>2 insertions is convergent.

Thus the calculation of the scaling properties ^of

near the critical point amounts to the calculation of the critical index associated with a four-leg ^vertex.

This vertex is obtained by opening ^{the closed line in}

the diagrams ^of type ^{2. More} precisely, ^{in a vertex}

of type 2, ^the contribution of the four-leg ^vertex

can be isolated by removing ^{the smaller} irreducible

subdiagrams containing ^the cp2 insertions (see Fig. 8).

The index C’ 4 corresponds ^{to the} renormalization of the vertex

Fig. ^8.

By removing ^the irreductible subdiagrams containing

the qJ2 insertion, ^{we extract from a} diagram (a) ^of type 2, the contri-

bution of the four-leg ^vertex (b).

and we remark that again, ^{we may use the} technique

of renormalization at the critical point ^{in the frame}

of the one-field Lagrangian theory.

Thus, ^the index C’ 4 ^is given by ^the renormalization of

where kl, k2, k3, k4 obey Eq. (6. 1).

At this point, ^{the reader} should remember that the renormalization of a four-leg insertion is not a

trivial matter because each composite operator ^is

coupled through ^the renormalization _process to all operators ^which have, ^{for d}

4, ^an equal ^{or smaller}

canonical dimension [5]. Thus, ^{as was shown} by Brezin, ^De Dominicis and Zinn-Justin [9] ^{at the}

critical point, ^{near d}

^{4, there is a} coupling ^between

the composite operators [10]

Here, ^{we have also a} four-leg insertion which produces

similar effects. This means that, ^{it is also} coupled

with the operators g-’ OvCP OvCP, j1-B al92 ^and P2 -, cp2.

However, this fact does not play ^{any role here as}

will be explained ^{now. The} operator mixing ^{is related}

to the fact that divergent subdiagrams, ^after regu- larization by ^{means of a} cut-off, ^can play ^{the role of}

an insertion. Here, this situation cannot occur, because fundamentally ^{we are} interested only ^{in the}

calculation of

ap G2(X, x) and that the insertions laP

of cp2, in the closed loop ^{of the} diagrams ^of type 2,

remove the divergences.

As an illustration of this fact, ^we may examine the effect which would result from the coupling ^of

our four-leg ^{vertex and of a} composite operator ^of the form M2-, 92.

The critical index associated with ¡.,t2-B cp2 ^is

If, ⁱⁿ Eq. (5.12), ^we replace (’ by 03B64a, ^{we obtain the}

contribution

This means that G1(x, x) ^{contains a finite constant} which results from the divergence ^{of the} diagrams

of type ^{2 when} p ⁼ 0. Thus, concerning this contri-

bution, Eq. (7.2) ^{should read}

In a similar way, ^the renormalization of 0292 ^cor- responds ^{to the index}

C’4b = 8 + C2 = 4 - d + C2 - (7.4)

Like C’,, ^and for similar _reasons, this index does not correspond ^to any interesting contribution, ^and

actually, here, ^{it does not} correspond ^to any contri- bution

at ^{all, because a zero momentum insertion}

of a2p2 always ^vanishes.

Brezin, ^De Dominicis and Zinn-Justin [9] ^have

shown the existence of another index

where m is the index which determines the behaviour of corrections to scaling [11]

However, ^{this index can be} forgotten because is not

dominant, ^{which means} that, ^{for n}

0, ^and (at least) small values of s, we have

as will be checked below.

The vertex

can be renormalized, by using ^{the method described}

in the preceding section. The relevant diagrams up to second order are given ^on figure ^9.

Fig. ^9.

Diagrams used for the calculation of (§ ^{and their} sym- metry ^factors.

The expansion of the renormalization constant with respect ^{to u is} given by

The critical index C’ 4 ^is given by

Thus, ^we get

Replacing ’1 by ^its expansion (see ^Eq. (3.8)), ^we

obtain

For n

0 and small _s, we verify ^the inequality C’ 4 C’ 4c*

Thus C’ 4 is dominant.

8. Summary, ^comments and conclusion. - The results can be summarized as follows. For large N, the probability distributions P,,,N(r) ^{with- s}

^{0, 1, 2}

obey scaling ^laws

For small values of x

and the corresponding ^contact probabilities ^are given by

The critical indices 80, 81 ^and 02 ^are given ^{in terms}

of an anomalous dimensions of fields (see Eq. (5.14))

and the indices (1’ 03B62,03B63,03B64 ^{have the} following expansions ^with respect ^{to s}

^{4 -} d (see Eqs. (3 . 8), (6.8) ^and (7.10))

Thus we obtain

For e

1, ^a precise ^{value of} 00 ^{can be obtained} by using the values of C, and’ calculated by ^{Le Guillou}

and Zinn-Justin [2] (see Eqs. (3.9)). The result is

On the other hand, ^{the values of} 01 and 03B82 ^{for s}

¹

can be roughly ^estimated by starting ^from the pre-

ceding expansions. ^{We know that the series} repre-

senting the critical indices are divergent; Brezin,

Le Guillou and Zinn-Justin [12] have shown indeed that these series can be written

where

(a, b, ^{c are} constants).

Therefore, the inverse Borel transforms of these series are convergent for small e, and it not unrea-

sonable to represent ^them by ^Pade approximants.

Consequently ^a reasonable approximation ^{of an}

index 0 given by ^{the series}

can be obtained by writing the series in the form

The numerical results are

(the ^{errors in} 81 ^and 02 ^have been obtained by ^com- paring ^several approximations).

More precise determinations of 81 and 03B82 ^are possible ^but require ^{more work.}

We note that for small values of B, the indices

03B80, 01, 02 ^are nearly proportional ^to the numbers l, 2, 4.

Such a result could be expected. ^{When two} points belonging ^{to a chain are} brought ^{near to each} other, they drag ^{with them a} train of other points ^{and the}

trains attached to the points under consideration

repel each other. If the point ^{is one end of the chain,}

it has a single train ; ^{if it is taken inside the} chain,

it has a double train (see Fig. 10). Thus, ^we may guess that the repulsion produced by ^a double train is about twice as large ^{as the} repulsion produced by

a simple ^train. However, this result is not valid when the repulsion ^is strong.

Fig. ^10.

^{Points on a chain and their} simple ^or double trains.

The preceding ^results give ^{also the} probability ^of

contact between two points ^{of the chain. If the} points

are inside the chain this probability ^is

where B 2 ^{is a constant.}

This expression ^{can be used to determine the} dependence ^{of the mean} energy E ^{of a chain with} respect ^{to N when N - oo. The} average number of contact C between elements of a chain can be roughly

estimated by writing

on the other hand, for small ^e

This seems to indicate that for d 4, ^{we have}

Indeed, ^{for d}

3, ^our evaluations give

This shows that, ⁱⁿ Eq. (8.13), ^{the sums with} respect

to n are convergent, ^{when N - oo.} Consequently,

for large N, ^we may write

As the interaction _energy of a chain is proportional

to e, ^{we see that for} large ^{N, the total} energy E is

given by ^an expression of the form

One might argue that the effects of the end points

have not been properly ^{taken into account but a}

more detailed discussion would not modify ^our

conclusion.

This result _agrees completely ^with previous ^inves- tigations of the author on the same subject [13]. ^The

fact that no Flory ^term appears in the expression

of the _energy is therefore confirmed.

Thus, the existence of the indices 00, 0, and 03B82

appears ^as very fundamental and it would be inte-

resting ^to study the effect of the correlations defined

by ^{these indices, on the} properties ^{of dilute or semi-}

dilute solutions. In fact, any description ^{of these}

solutions which would ignore ^these correlations cannot be _very realistic.

Apparently, ^{the indices} 03B81 ^and 02 ^{are not} directly

related to the current indices v and _y. This fact is

surprising; ^it shows, ^at least, ^{that the} notion of contact in a polymer solution is more subtle than

expected.

At present time, ^{there is no} experimental ^evidence

of the existence of the indices 00, 01 ^and 03B82 ^{but we}

may guess that the experimental importance ^{of these}

indices will be recognized, ^if experiments ^{are made}

to determine the detailed structure of the polymer

solutions.

Acknowledgments.

^The author has been sti- mulated by discussions with M. Bergere, ^{E. Brezin}

and J. Zinn-Justin.

APPENDIX

We study ^the renormalization of the vertices and of the Green’s functions associated with the Lagran- gian

The Lagrangian ^is regularized by ^{a cut-off.}

1. Divergences ^{of the} diagrams.

^{The vertex func-}

tions which are considered here represent physical quantities. They ^are always finite because there is

always ^{a natural} cut-off and because we never reach

exactly ^{the critical} point. ^{Still, their} properties depend

on the ultraviolet divergences which appear ^when the cut-off disappears ^{and on the infrared} divergences

which occur when a or b _goes to ac.

In momentum space, the superficial divergence ^of

a p-leg diagram of order s contributing ^{to a vertex} f(...) ^is (d ⁼ space dimension)

where 60 ^{is the degree of the} composite field. If 0 is the product ^of f field components and t momenta, its degree ^is

and therefore

(an ^insertion of 9’ ^or f/l2 corresponds ^to 62, ^{an inser-}

tion of q>f/l2 ^to b3, ^{and so on} ...). ^{If 6 a} 0, ^the diagram

is ultraviolet divergent ; ^{if 6} 0, ^{it is} infrared diver- gent.

The value d

4 is marginal ^and especially impor-

tant ; in this case, ^we have

First, ^{let us consider the} diagrams ^{which have}

ultraviolet divergences (6 a 0). ^We verify easily ^that

for 4 d 2, ^{there is} only ^one divergent ^vertex (p

2, q ⁼ 0) ^which may play ^{the role of an insertion}

in diagrams. ^This divergence ^can be eliminated by

one subtraction i.e. compensated by ^a counterterm which is contained in _ac. However, diagrams ^with

external insertions _may remain ultraviolet divergent.

New divergences ^also appear because the Fourier transforms, ^{in the} ordinary space, of the Green’s functions of the momentum space, do not always

exist. Such divergences ^{can be} eliminated when needed, ^either by subtraction or by differentiation with respect ^to a>> (see for instance Eqs. (2.15)).

Infrared divergences, ^{on the} contrary ^{occur when} 6 5 ^{0. Thus if we were} applying ^{the usual renorma-} lizability ^{criterion to the infrared} divergences, ^we

would _say that for 4 > d > 2, ^the theory ^{is unrenor-}

malizable because, ⁱⁿ diagrams ^of high ^{orders, diver-}

gences of higher ^and higher ^{order occur.} However, it is believed that these divergences ^add up ^to form

simple divergences characterized by critical indices.

The fundamentally divergent part, ^{in the infrared} region, appear at d

4, ^{when the} theory ^is just

renormalizable.

Thus, ^{let us} consider for d ⁼ 4 a vertex with p-legs, S2 two-leg ^vertex insertions, S3 three-leg ^vertex

insertions and S, four-leg ^vertex insertions. The cor-

responding 6 ^is

We find 6

0, ^for (p

2, S2

l, S3 S4

0) ^for ( p

3, S3

1,82

S4

0)andfbr(/? = 4,64

1, S2 S3

0). ^{This is the} hallmark of ^new diver- gences which imply ^the necessity ^of renormalizing

the vertices of order 2, 3 ^{and 4 for d 4.}

2. Renormalization conditions and renormalization factors.

We start by assuming ^{that a}

^{b and we}

stay ^{in the} vicinity ^{of the critical} point. ^{In this} way,

we have only ^one scaling length and infrared diver- gences ^{occur at zero momentum} when a

_{b -+ ac.}

In the classical texts on renormalization [5, 6], ^{it is}

shown that each diagram ^{can be made} convergent by

subtractions which are equivalent ^{to the} introduction of counterterms in the Lagrangian.

In statistical mechanics, ^a slightly ^different point

of view must be adopted. ^{We are not allowed to}

introduce real counterterms in the Lagrangian, ^we

can only ^consider parts ^{of the initial} Lagrangian ^as

counterterms. On the other hand, ^the bare Green’s functions are the functions of physical interest and the diagrams ^are always convergent ^{in the infrared} and in the ultraviolet domain. Renormalization aims

only ^at insuring ^{that all the} parts ^{of a} diagram ^scale

in the same way. Actually, ^a part ^{of a} diagram ^can

be considered as a black box which is itself a sum of

subdiagrams. ^{The sum of these} subdiagrams ^{can be}

infrared divergent. ^{In this case, the} dependence ^{of the}

black box with respect ^to its external momenta is anomalous and is not given by ^the canonical dimen- sion. This effect is corrected by renormalization of all the diagrams ^and subdiagrams. ^{As a} consequence the

scaling properties ^{of the} renormalized quantities ^are always given by their canonical dimensions.

The scale is defined by ^{a mass m} which is the inverse of a correlation length ^{and which} is small with respect

to gå/e (where E = 4 - d). ^{This mass} is defined by ^the equation

The only dimensionless parameter ^is (a - a.) gü2/£

and the approach ^{to the critical} point depends ^{on it.}

The dimensionless quantity mgo ’/’is ^a function of this parameter, ^and when a = b - ac, then m - 0.

The renormalization constants are functions of mgü ^{1/£. To find the} physical properties ^{of the} system

at the critical point, ^{we have} only ^to determine the

asymptotic behaviour of these constants when m - 0.

A renormalization factor can be associated with each

marginal composite ^{field. A} composite ^{field 0 is} marginal, if the vertices containing 0 ^and possessing

the same external structure as 0, ^{at zero} momentum, have a superficial divergence 6 which vanishes for d = 4.

The dominant part ^{of 0 can} be renormalized by setting

when OR(x) ^{is the} renormalized composite ^{field and} zo(mg-’I’) a renormalization factor which is deter- mined by renormalization conditions.

For instance, ^{we write}

Another renormalization constant must be intro- duced for the renormalization of 6 simple ^external legs. ^{In a} symbolic way, ^we may write

The constant zi ^can be considered as the renorma-

lization constant of qJ j(x) ^{and this} interpretation ^looks

obvious if instead of renormalizing ^vertex functions,

we try ^to renormalize one-irreducible Greens’ func- tions. Thus, ^me may write

The renormalization constants are determined by introducing renormalization conditions. Thus _{z, is} determined by setting

Simultaneously, ^we may write

where u is a dimensionless quantity which has a

finite limit u* at the critical point. ^{The constant} z2

can be determined by ^{the condition}

where c

constant but we may also set

This other definition of _z2 can be justified ^{as follows.}

An insertion

of f ^{(P 2 in a Green’s functions G is}

equivalent ^{to a} derivative with respect ^{to « a » and we} may write

or in a symbolic ^way

Let us introduce new renormalized Green’s functions.

As f (P ^{has the same canonical dimension as m - 2,}

we may write

From Eqs. (A. 15) ^and (A. 16), ^we get

on the other hand, ^the renormalization equation gives

Thus, combining ^the preceding equations, ^{we find}

that _{z2 N} m2/(a - ac) and this result shows _{that z2}

can be defined by Eq. (A. 14).

It is also _easy to show that z 1 (mgo l/f) is related to

z3(mgo lIt) by ^{the relation}

but this result is not used here.

3. Renormalization of insertions.

Renormalizing

the insertion of composite ^{fields is not} always simple

because an insertion _may create new divergent ^sub- diagrams ^{which have also to be} renormalized. A

composite ^{field can} be considered as made of several parts ^{which must} be renormalized in different _ways and which are characterized by different critical indices. The details can be understood by examining

the renormalization _process.

Following Bogoliubov [14], ^{we define a subdia-}

gram ^{as a} diagram ^{made of a connected subset of}

vertices of the original diagram, and of all the lines which join the vertices of this subset, ^{in the} original diagram. ^The dangerous diagrams ^are those which

are divergent ^{for d}

^{4. The most} important ^{ones are}

the marginal diagrams ^which diverge ^{for d}

^{4 and}

converge for d

4 - 0. The more divergent ^contri-

butions can generally ^be separated by subtraction and they ^{are not} always important ^{from a} physical point of view. The insertion of a composite ^{field 0}

may be dangerous ^{at zero momentum or for small}

momenta. It does not produce only ^a finite number of divergent diagrams ^{for the} calculation of which, ^a cut-off is needed; ^{it creates also} dangerous ^subdia-

grams, which _appear as insertions in general ^dia-

grams ; this situation implies ^the necessity ^{of a renor-}

malization. An insertion 0 is renormalized by ^insert- ing simultaneously ^the composite ^{field 0 and counter-}

terms AO. The renormalized field can be written

and we note that renormalization is important ^for

low momentum insertions.

In the simplest ^{case, the} counterterms called by Bogoliubov generalized vertices have at low momen-

tum the same structure as 0.

In this _case, at low momentum, ^we may write 0

as the sum of two parts

The second term in this sum plays ^{the role of a coun-}

terterm and, ^{we determine z} by ^{the condition}

Thus, ^{we have}

Inserting OR’ ^means inserting ^{0 and the} counterterms which are necessary for preserving scaling. ^{The inser-}

tion of ⁰ in a diagram may also generate ^subdia-