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Multiplicative renormalization of continuous polymer theories, in good and θ solvents, up to critical dimensions
M. Benhamou, G. Mahoux
To cite this version:
M. Benhamou, G. Mahoux. Multiplicative renormalization of continuous polymer theories, in good and θ solvents, up to critical dimensions. Journal de Physique, 1986, 47 (4), pp.559-568.
�10.1051/jphys:01986004704055900�. �jpa-00210236�
559
Multiplicative renormalization of continuous polymer theories,
in good and 03B8 solvents, up to critical dimensions
M. Benhamou and G. Mahoux
Service de Physique Théorique, CEA-Saclay, 91191 Gif sur-Yvette Cedex, France (Reçu le 14 octobre 1985, accepté le 20 décembre 1985)
Résumé. - On montre que les théories standard de polymères continus, en bons solvants et en solvants 03B8, se
renormalisent selon le schéma de des Cloizeaux (renormalisation multiplicative). On utilise pour cela l’équivalence
de ces théories avec les théories du champ scalaire ~ à n composantes, en interactions ~4 ou ~6 avec lui-même, dans la limite où n s’annule, équivalence réalisée par la transformation de Laplace-de Gennes. On montre que la renormalisation dimensionnelle minimale de ’t Hooft et Veltman est particulièrement bien adaptée au problème,
car elle commute avec la transformation de Laplace-de Gennes.
Abstract. - It is shown that standard continuous theories of polymers, in good and in 03B8 solvents, renormalize
according to des Cloizeaux’ scheme (multiplicative renormalization). Use is made of the equivalence of these theories, through Laplace-de Gennes transforms, with ~4 and ~6 self-interacting n component scalar field theories,
in the limit of zero components. It is argued that the minimal dimensional renormalization procedure of ’t Hooft and Veltman is particularly well adapted to the problem, since it commutes with Laplace-de Gennes transforms.
J. Physique 47 (1986) 559-568 AVRIL 1986,
Classification Physics Abstracts
ll.lOG - 36.20 - 61.25H
1. Introduction.
It has been shown by Edwards [ 1] that polymers in
solution can be described as continuous chains with contact interactions. The theory has been developed through perturbation expansions in the coupling
constant, and it exhibits short-range divergences
which necessitate a regularization procedure (for example with a cut-off which may be related to the size of one link). A « renormalized » theory emerges in the limit where the regularization is removed. The situation is quite similar to that encountered in local quantum field theory. As a matter of fact, there is a complete equivalence [2] between the theory of
continuous chains with two-body (respectively two
and three body) interactions, and the theory of n
component scalar fields T, with ~4 (respectively qJ4
and qJ6) self interactions, in the limit of zero compo- nents. The mathematical bridge between both theories is the Laplace-de Gennes transform. Thanks to it, all
the renormalization machinery of the second theory
may be exploited to renormalize the first one. Our purpose here is precisely to show that this program may be completed, up to the critical dimension where the theory is just renormalizable, and to show that polymer partition functions renormalize multipli- catively.
Indeed, this has already been done for polymers
in good solvents [3], when the dimension is strictly
smaller than the critical one d = 4. In that case, the associated field theory is super-renormalizable, and only a mass renormalization is needed. Through the
inverse Laplace transform, this mass renormalization
(a shift in the square mass) turns into a divergent
factor : this is the multiplicative renormalization of
polymer theory. To be more specific, let Z(S, u, A) be
the partition function of an isolated polymer in a space of dimension d 4, where S is its « area », u is the
coupling constant, and is some regulator which
takes account of the divergences of the theory (1). Let G(k2, m2, u, A) be the regularized propagator of the associated Euclidean scalar cp4 theory in the same d
dimension (m is the bare mass, u the coupling cons- tant). Then the inverse Laplace-de Gennes transform reads :
All divergences of the propagator G can be reor-
(1) All these quantities will be defined in details in sec-
tion 2.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704055900
ganized in a shift of m2. More precisely, defining M’(u, A) by the equation
one shows [4] that G(O, m2, u, A) is a finite function of MI _ M2 and u :
Thus : O
where all divergences are collected in the exponential factor, and where the renormalized partition function ZR is given by
When the dimension is equal to 4, further diver- gences appear, which necessitate the renormalization of the coupling constant. Equation (3) is replaced by
where mR, uR and 0 are functions of m2, u and A, which diverge when A - oo, and GR is a finite function of
mi and uR.
Now, since in general both 0 and GR are functions
of m2 at fixed u and A, the factorization of the diver- gences of G in equation (6) does not entail any longer through the Laplace transform (1), the same desired property for Z. However, if it so happens that the
renormalization constant 0 does not depend on m2,
then it factorizes out in equation (1), and one recovers
the multiplicative renormalization of Z(S, u, A). As it
is well known, this favourable circumstance occurs
with the minimal dimensional renormalization pro- cedure of ’t Hooft and Veltman [5], which provides
mass independent renormalization constants. This is the clue of the problem, and in the following sections,
we show that it works for any polymer partition
function.
Of course, this result is not unexpected. Indeed, it
has been explicitly assumed in the literature [3, 6],
on physical grounds.
The initial motivation for proving the multipli-
cative renormalization property at critical dimension
was the following. Polymer s-expansion calculations
are usually performed (in the good solvent case) at
dimension d 4, and physical quantities are first
expressed in terms of the unrenormalized coupling
constant u. Of course, the expression one obtains diverge when s = 4 - d goes to zero. But if the variables u and S are eliminated to the benefit of more
physical quantities like g, the dimensionless second virial coefficient, and R 2, the end to end distance,
then it is observed [3, 7, 8] that the singularities at
E = 0 disappear. This is indeed expected for the
coherence of the theory. We will show that it is a
consequence of the divergence factorization property,
so that the singularities at 8 = 0 must disappear at any order of the perturbative expansion.
Section 2 deals with polymers in good solvents, that
is to say polymers that can be described as continuous chains with two-body contact interactions. In sec-
tion 3, we extend the analysis to polymers in 0-sol-
vents [9], that is to say continuous chains with two-
body and three body contact interactions.
2. Continuous chains with two body interactions.
2.1 POLYMER THEORY. - In the continuous model
[1, 3, 10], a polymer in good solvent is viewed as a
continuous chain with contact two-body interactions, in a space of dimension d ( = 4 - s). A point on the
chain is given by a vector r(s) (0 s S), where the parameter s, which has the dimension of a surface, is the coordinate of the point along the chain. S is the
total« area » of the chain.
Consider a set of N polymers, of areas S (a =1, ..., Fl ).
The statistical weight of a configuration of this set
is given by :
k/)
where
u is the contact interaction coupling constant, and
we call Po [r 1, . - -, r N I the weight for u = 0 (Gaussian weight).
With the notation of reference [3], we define the
mean value ( F ) of the functional F[rl,..., r,] by
Among all the partition functions that can be defined, only a few in a first step will be of interest for us,
561
those which are used to compute the few physical parameters characterizing polymers in good solvents. They are (2)
z 7(l)(S, u ; s) and Z(2)(81, S2, u ; E) are the partition
functions of one and two polymers; Z(’)(k 2; S, u ; s)
is the partition function of one polymer with two
insertions of wave vectors k and - k at both extre- mities of the chain; Z(1,2)(k 2 si, s2, S3, u ; s) has the
same definition, except that the wave vectors k and
- k are injected at two points of coordinates 81 and 81 + S2 (S = Sl + S2 + S3); Z(2)(k; Si, s2, u; 8),
where k stands for the collection {k 1 ... k4 }, is the partition function of two polymers with insertions of
wave vectors ki,..., k4 at the four extremities, and it is
defined for k 1 + ... + k4 = 0.
All these functions can be expanded with respect
to the coupling constant u. As a matter of fact, we are
interested in the contributions of the only connected diagrams, and from now on, the various Z functions defined above will represent connected partition
functions.
All of them are plagued with divergences, and they
have to be regularized. We choose the dimensional
regularization, so that it is e = 4 - d which plays
the role of the regulator.
2.2 FIELD THEORY. - To this polymer theory is
associated the Euclidean scalar field theory described by the Landau-Ginzburg-Wilson action. For a set of
N scalar fields with n components (p (a = 1, ..., N ;
i = l, ..., n), the bare action writes (3 )
Note that i) we have given different masses ma ..
(2) Notation : the upper indices of the partition function Z(N, L)(...) refer respectively to the numbers rBJ of chains and L of insertions which are not at the extremities of the chains. When the second index L is zero, it is omitted. Here N = 1 or 2, and L = 0 or 2.
(3) The factors 1/2 in front of (V(p,,i)2, and 1/8 in the
interaction term, are there to recover the usual Feynman
rules of polymer theory [11].
to the fields lpa’ ii) the interaction term is o(FIn) symmetric (Nn dimensional orthogonal group), but
the complete action is not, precisely because of the
mass terms.
To the above defined connected partition functions
are associated the following connected Green func- tions (4) :
i) the propagator G (2 )(k 2 ; m;, u ; 8) of the field lpa’
which in the limit of zero components (n = 0), depends
on the only mass ma ;
ii) the vertex function G (4) (k; m;, mi, u ; 8), Fourier
transform of the connected vacuum expectation value ((Jalx) ((JalY) (ppj(z) (p,j(t) > c; as previously k stands
for the collection { kl, ..., k4 }, and k, + ... + k4 = 0;
in the limit n = 0, G (4) depends on the only masses
ma and m,,;
iii) the connected Green function G(2,1)(k1 k2, q;
ma, MP2) u ; s), Fourier transform of the vacuum expec- tation value ((Jai(X) Tpj(y) (T,,,i wfl j) (z) >c of a product
of two T fields, and one composite field «((Jai ((Jpj) (z), always in the n = 0 limit; kl, k2 and q are respectively
the conjugate variables of x, y and z, and kl +k2 +q = 0.
iv) the connected Green function G (2,2)(k k2,
ql, q2 ; ma, 9 M02, m y 29 u ; s), Fourier transform of the
vacuum expectation value
kl, k2, ql and q2 are respectively the conjugate variables of x, y, z and t, and k, + k2 + q, + q2 = 0.
All these Green functions are dimensionally regu- larized.
2. 3 LAPLACE-DE GENNES TRANSFORMS. - The rela- tions (5) between polymer partition functions and
Green functions of the field theory read as follows (6) : (4) Notation : as usual, the upper indices of the Green function G(2 N,L)(...) refer respectively to the numbers 2 N of (p fields and L of (p(p fields involved in its definition as a vacuum expectation value. When the second index L is zero, it is omitted. Here N = 1 or 2, and L = 0, 1 or 2.
(5) An easy way to derive them is to use the diagrammatic
rules of both theories.
(6) More generally, Z N, L) is the inverse Laplace transform ofG(2N,L).
As for the partition functions Z(1){8, u; s) and
Z(21(SI, S2, u; E), they are given by equations (17)
and (18), for respectively k2 = 0 and k = 0.
In the above equations, integrals run over the usual
inverse Laplace transform contours, that is to say
contours that stay at the right of the rightest singu-
larities in the square masses complex planes.
2.4 DIMENSIONAL FIELD THEORY RENORMALIZATION WITH MINIMAL-PRESCRIPTION. - As explained in the
introduction, we use the minimal renormalization
prescription of’t Hooft and Veltman [5] to renormalize the above defined field theory at d = 4, which insures that the counters terms built at each order of the
perturbation expansion do not depend on the masses.
A first important consequence of this mass inde-
pendence is that in the renormalized action AR,
the O(Nn) symmetry of the interaction term is pre-
served, so that we have to introduce only one renor-
malized coupling constant uR : -
p is an arbitrary mass introduced in such a way that UR be dimensionless irrespective of d, and the maR’s are
the renormalized masses.
To prove the above assertion, one has to show that
the vertex counterterms built with the above renor-
malized action do possess the 0(fin) symmetry.
In fact any given vertex diagram, with two cpa and two qJp legs, contributes a counterterm of the form
c(uR, s) 91 9# if cx and c(uR, s) qJ:12 otherwise,
the last factor 1/2 coming from an extra S2 symmetry of the diagram when a = For all a and I fl’s, I I this gives
a total counterterm which is
0(N n) symmetric.
’ ’
The relations between bare and renormalized
quantities read [12], in the zero component limit :
where 03B8~, 03B8~2, 0m2 and 8u, functions of uR and B, are respectively the 9 field, composite TT field, masses
and coupling constant renormalization constants,
independent of the masses.
Now, the connected Green function G IM,N)( .. ),
vacuum expectation value of a product of M ~ fields and N 9T composite fields, renormalizes according
to the general formula :
The renormalized Green function GRM,N), as a
function of 8, is free from singularities at s = 0.
Of course, the renormalization of the mass, and that of the 99 composite field are not independent, and
this implies a relation between OM2 and 03B8~2. Let us
prove that
One knows that the function G (2,1)(k, - k, 0; ...)
with one 9T insertion of vanishing momentum is
related to the propagator by :
This entails for the corresponding renormalized functions the relation
Taking account of equation (23), one obtains
563
Due to the minimal prescription, the left hand-side,
00
as a function of s, has the structure 1 + E am(UR)/8m.
m=0
The right hand side is regular at 8 = 0. Consequently
both sides are equal to one.
2. 5 DIMENSIONAL POLYMER RENORMALIZATION WITH MINIMAL PRESCRIPTION. - It remains now to trans-
late equations (17) --+> (19) written for bare functions,
into relations between renormalized ones. The inverse
Laplace transforms have to be performed on the bare
variables ma, at fixed k2, k, u and s. Thanks to equa- tions (23) and (24), they can be changed into integrals
on the renormalized variables m.2,R with 0.2 as Jacobian,
at fixed k2, k, UR, 03BC and s. Since the 0’s do not depend
on the masses, they factorize out. Taking account of equation (26) one obtains :
where the renormalized partition function ZR’s, regular at c = 0, can be expressed as inverse Laplace transforms
of renormalized Green functions :
The renormalized polymer areas SaR and saR are defined by Sa m« = S R m,2,R, that is to say
and similarly for the S0152R’S (note that in Eq. (35), the
total renormalized area SR of the chain is given by SR = S1R + S2R + S3R). Areas are multiplicatively re- normalized, like the masses in equation (23).
Formulae for the partition functions Z(1)(8, u; s)
and Z(2)(81, S2, u ; e) are obtained simply by setting
k2 = 0 and k = 0 respectively in equations (30), (33)
and (31), (34).
Obviously the above treatment may be generalized
to any polymer partition function. The general result
which emerges is the following. Let Z(N, l){k; S1, ..., 8 N, ..., u; s) be any connected partition function of N
polymers, with Z insertions of wave numbers k.
Then
where the renormalized partition function ZR(N,L) is regular at s = 0. Thus each continuous chain contri- butes a short range divergent factor 8lfJ 8m2. (Note
that the divergent factor in Eq. (37) does not depend
on the number L of insertions, a result which was already pointed out in Ref. [3], see Eq. (5.13) there.)
This demonstrates that the continuous polymer theory is multiplicatively renormalizable up to dimen- sion d = 4 (in the good solvent case).
We now turn to the consequences for
2.6 POLYMER PHYSICAL PARAMETERS. - The mean
square end to end distance R Z, and the radius of
gyration R 2 are defined by
Using equations (30) and (32), one obtains
This expresses R 2 and RG as functions, finite up to d = 4, of the renormalized parameters SR and UR, and of the mass 11-
One also defines the dimensionless physical parameter g, which is essentially the second virial coefficient [3] :
The « effective » area eS is defined by R 2 = d eS. From equations (30) and (31) one obtains
which shows that g, as a function of SR, uR and p, stays finite when s -+ 0.
We can take advantage of equations (40) and (43)
to eliminate SR and uR to the benefit of eS and g.
In that way, SR P2 and uR become dimensionless functions of eS,u2, g and s :
Now let Q(q2, S, u ; s) be any dimensionless physical quantity, which renormalizes without multiplicative divergences, that is to say
where QR stays finite when 8 - 0, at fixed SR Q 2, SR J12
and uR. Thanks to equations (44) and (45), QR can be
expressed as a function of eSq2, eSj,l2, g and 8, regular
at 8 = 0. Since QR is a physical (measurable) dimen-
sionless quantity, it cannot depend at fixed eSq2 and g,
on the arbitrary mass p. Consequently, one can write
This result applies to the ratio RG/R 2, dimensionless
(unction of S, u and s. Thus
where p(g,8) has no singularities at 8 = 0, a result
which has been checked in a second order pertur- bative calculation [8].
Equation (47) also applies to the form factor defined
by
According to equations (30) and (32)
Thus the form factor can be written as a function
HR(eSq2, g, E) which is independent of the mass p,
and stays finite when e --> 0, at fixed ’Sq2and g.
Consider now a monodisperse polymer solution.
The osmotic pressure H and the concentration C, in terms of the fugacity f read
Defining the renormalized fugacity fR by
and using equation (37), one obtains the relations
which express n p and C as s = 0 regular functions of
the renormalized parameters fR, SR, uR and of the mass p, or equivalently of the variables fR, eS, g and p.
Eliminating fR, and arguing as previously, one
obtains that np/c is a dimensionless function, regular
at c = 0, of eSd/2 C and g.
Note that the multiplicative renormalization of the
565
fugacity implies an additive renormalization of the chemical potential.
Finally, let us give the relations between the two renormalization constants O(P and 0.,, and those
defined in reference [3], the swelling factor Xo = ’SIS
X2 = Z (1)(S’ U;,e) :
Our result (37) can now be compared to equa-
tion (5.12) of reference [3]. It appears that the renor-
malized partition functions defined by des Cloizeaux
differ from ours by a factor (regular at s = 0) which
is a product of ZR"’(SR lt2, UR; p),S.
Note also that the renormalization constant X2
of reference [3] is nothing but (}cp2 OM2-
3. Continuous chains with two and three-body inter-
actions.
3.1 POLYMER AND FIELD THEORIES, LAPLACE-DE GEN-
NES TRANSFORMS. - A set of N polymers is now
defined as a set of N chains, of areas Sa, with the
statistical weight [3, 9] :
where I and J are as in the previous section, and where the three body interaction K has the form :
The critical dimension is d = 3, so we set s = 3 - d.
Besides the partition functions that we define as in equations (11) ----> (15) but with the new weight (58), and
which now depend on the extra variable v, we need two new ones which are connected partition functions of three polymers :
This last function is defined for k 1 + ... + k6 = 0.
The associated Euclidean scalar field theory is defined by the bare action
At critical dimension d = 3, u and v have respectively mass dimensions 1 and 0 : u is a super-renormalizable coupling, v is just renormalizable.
Besides the Green functions introduced in the previous section, and which now depend on the extra variable
v, we need the six point connected Green function G(6)(k,. ma, ma, mY, u, v ; 8), Fourier transform of the vacuum
expectation value
/ 6 B
the ki’s, conjugate variables of the xi’s, are restricted by £ k; = 0 . G(6) is related to the polymer partition
B i=1 i /
function Z(3) by the Laplace-de Gennes transform :
This equation completes equations (17) --+ (19), which have to be written with the extra v dependence.
