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Dimensional renormalizations of polymer theory
Bertrand Duplantier
To cite this version:
Bertrand Duplantier. Dimensional renormalizations of polymer theory. Journal de Physique, 1986, 47
(4), pp.569-579. �10.1051/jphys:01986004704056900�. �jpa-00210237�
Dimensional renormalizations of polymer theory
B. Duplantier
Service de Physique Théorique, CEN-Saclay, 91191 Gif-sur-Yvette Cedex, France (Reçu le 25 octobre 1985, accepte le 20 dicembre 1985)
Résumé. - Nous montrons comment on peut effectuer systématiquement, à tous les ordres, deux renormalisations dimensionnelles différentes (ou soustractions minimales), directement en théorie des polymères dans le cas critique (pour d
=4 - 03B5, 03B5 ~ 0). L’un des schémas de renormalisation dimensionnelle coincide avec celui de la théorie des
champs et a été récemment justifié par M. Benhamou et G. Mahoux. L’autre, le schéma z de renormalisation dimen-
sionnelle, est nouveau et le paramètre z de Yamakawa de la théorie des polymères y est renormalisé de manière minimale. Les relations existant entre ces renormalisations et la « renormalisation directe » de J. des Cloizeaux
sont explicitées. Les indices critiques sont recalculés au second ordre en 03B5.
Abstract.
-We show how to perform systematically to all orders, two different dimensional renormalizations (or minimal subtractions) directly in polymer theory, in the critical case (for d
=4 - 03B5, 03B5 ~ 0). One of the dimensional renormalization schemes is that of field theory and has been justified recently by M. Benhamou and G. Mahoux.
The other one, the z-dimensional renormalization scheme is new and renormalizes minimally the Yamakawa parameter z of polymer theory. We explicit the relation of these renormalizations to the « direct renormalization » of J. des Cloizeaux. We recalculate the critical indices to order 03B52.
Classification Physics Abstracts
05.20
-05.40 - 11.10 - 61.40K - 64.70
1. Introduction.
The most useful model for polymer theory is the
continuous chain model, introduced by S. F.
Edwards [1]. There exist various renormalization methods for treating it, i.e. field theoretic ones using
the n - 0 analogy due to de Gennes or more direct
ones using only polymer diagrams [2]. (See also [3]
and references therein.) The direct renormalization method of J. des Cloizeaux [2] is in fact a physical
one, since one considers there physical quantities
which fix the scale of the Kuhnian chain. The study by L. Schafer and T. A. Witten of the renormalization of polymer theory was closer to field theory [4]. These
authors used the so-called massless scheme of field
theory and transferred it to polymers, obtaining,
within their scheme, a multiplicative renormalization of the polymer theory, including polydispersity.
However, this scheme is not easily applied directly
in polymer theory since it relies mostly on field theory (through the use of vertex functions and not of poly-
mer partition functions). On the other hand, there is in field theory a dimensional renormalization method, popularized by t’Hooft and Veltman [5], which has
certain minimal subtraction properties [5]. Our aim
is to present here how to do rigorously various dimen-
sional renormalizations directly in polymer theory.
The t’Hooft-Veltman renormalization already appear- ed in works about polymer systems [3], but without general proofs. The idea of using the minimal renor-
malization prescription of t’Hooft and Veltman, for proving the finiteness of the des Cloizeaux scheme, is
due to M. Benhamou, who remarked that the Laplace-
de Gennes transform of polymer partition functions
« commutes » with dimensional renormalization fac- tors (1,2). This shows [6] that in polymer theory finite perturbation expansions can be obtained by multi- plicative renormalization of the partition functions (the direct renormalization method [2]).
We consider here the continuous model for polymer
chains in a good solvent [1], i.e. the two-parameter model, in a space dimension d 4. We present two different dimensional renormalization schemes (I and II) for this polymer theory, showing their existence
from dimensional renormalization in field theory.
(’) J. des Cloizeaux, private communication.
(2) In their different (massless) scheme Schafer and Wit-
ten [4] used also the commutation of the corresponding
renormalization factors with generalizations of the Laplace
transform.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704056900
One of these schemes (II) is exactly the field theoretic
one and recovers the results used in reference [3] and justified in reference [6]. The other one is a dimen-
sional renormalization scheme proper to polymer,
since it renormalizes minimally the well-known dimen- sionless z-parameter of polymer theory. We perform explicit calculations with these dimensional renorma-
lizations directly in polymer theory. For completeness,
the des Cloizeaux’ method is considered and proven to give finite results to all orders. We calculate the finite renormalizations relating his method to the dimensional ones.
-
In the next article, we consider also the case of polymer chains near the tricritical theta point, and study the direct and dimensional renormalizations of
the associated three-parameter model [7]. Let us finally mention that we have also established in ano-
ther work [8] an explicit and convergent integral representation for the dimensionally renormalized
polymer partition functions. It permits to calculate directly (i.e. without renormalization factors) the
finite renormalized perturbation expansion of the polymer partition functions, in a particularly simple
way.
2. Polymer theory.
We consider here the continuous model for polymer chains, with the probability density for a configuration rj(sj) of N chains j
=1, ..., N
The Sj are the Brownian areas of the N chains. They
can be written in terms of a unique Brownian area S :
In the following we shall consider the general poly- disperse case, but the examples will be treated in the
simpler monodisperse case
The interaction parameter b is related to the dimen- sionless z parameter by [2]
with d
=4 - s. We work in dimensional regulariza-
tion : all the quantities we consider are analytical
continuations to d
=4 - 8 of the same quantities
which are defined for d 2 [9]. In particular, one
defines the set of connected regularized partition
functions of N polymer chains, in Fourier space [2] by :
where Po represents the probability density for
Brownian chains (b
=0). The connected total parti-
tion function of the N chains is then given by
By dimensional analysis, one has immediately [2] for Fl
identical chains :
where 3N (z) is a function of z alone, also depending on
the dimension d, and which is singular when d -+ 4.
Now we can define two different dimensional renormalization schemes, for polymer theory :
In the first one, we set (considering the monodis-
perse case for simplicity)
where Xz, Xb, Xi have the singular series expansion :
A is a pure number, entirely arbitrary (3). The fXn,j are
entirely independent of /L Substituting (I) into the
(3) It pl4ys the role of the well known t’Hooft’s mass p in
minimal prescription.
partition functions make their double expansions in
terms of ZR and s, and expressed as functions of
S’, £R, finite when s -+ 0. The particular form of equation (6) is the defmition of the dimensional renormalization or minimal subtraction scheme [5].
The renormalization constants have no fmite parts in 8 but only poles.
The relation between the regularized partition
functions (3) and the renormalized ones is then (for
the monodisperse case)
where iT[ has a finite limits for s -+ 0, when expressed
in terms of zR (and not z).
The second dimensional renormalization scheme we use is (for the polydisperse case)
where the renormalization constants Zb, Zm, Z have
also the minimal form (6). As we shall see, these
renormalization constants Zb, Zm, Z are exactly those of field theory in the limit of a number of components
n
=0. In this scheme, the relation between regularized
and renormalized partition function will be
where the fl’ R are other renormalized partition func- tions, depending on { Sj,R } bR, and it, and regular in s, when s - 0.
’
The two schemes (I) and (II) are not the same. The
difference is in the renormalizations of b or z
=(2 n)-d/2 bS 2 - d/2. That of b is the standard one in field theory. That of z is more convenient for polymer theory, since the variable z is a natural expansion
variable. This is the reason why we introduce it.
Incidentally, let us note that the convergent integral representation established by us for the dimensionally
renormalized polymer partition functions [8] corres- ponds to the scheme II of this article. This integral representation has nevertheless a quite simple expres- sion in terms of the z-parameter [8].
To prove the existence of these two schemes, we shall use field theory. We shall prove the existence of the scheme II first, and then of the other one I, which is
a little more subtle.
3. Field theory and proof of existence.
As it is well known [10], there is a correspondence
between the continuous polymer theory just described,
and the O(nN) field theory (n -+ 0) given by the
Hamiltonian
There are as many fields (p,, and masses mj, as there are polymer chains. If one defines in Fourier space
and the connected Green functions by
one has the exact correspondence
(The contour is taken on the right hand side of the singularities of the integrand.)
Now, we know that there exists [5] a (unique) dimen-
sional renormalization or minimal subtraction scheme
(for n - 0) :
where Zb, Z., Z have the same minimal polar struc-
ture as in (6). Substituting (11) makes the renormalized Green’s functions finite, when expressed in terms of bR, Mj2 ,R. The parameter p is the arbitrary mass scale [5],
which makes bR dimensionless. Two important facts
must be noted. First, the renormalization of the mass
is mass-independent as emphasized by previous
authors [5]. Second, in principle, for n # 0, there is
a mixing of the renormalizations of the masses [5] :
However, all the non-diagonal terms of the Zmi, j
matrix involve internal loops and vanish in the n - 0
limit, yielding a great simplification. Moreover all the
diagonal terms are equal (4). Incidentally, we note that
the same essential remark appears in reference [4] in
the different context of the massless scheme, and that
an argument similar to that of this section is used there.
The multiplicative renormalization of the Green’s functions reads then :
Now, by this change of variables (11), we know that gR has a finite limit when E - 0, once it is expressed in
terms of Mj2 ,R and bR only. Inserting (11), (12) into equation (10) and performing a change of variables
in the masses, it is easy to rewrite ff as
and where
The mj,R have become in (14) simple integration variables, The contour q’ still lies on the right-hand
side of the singularities of the integrand. gR being
finite for 8 -+ 0, 1l’ R is a regular function of s.
There are other quantities of interest in polymer theory, like the form factor of a single polymer chain [13]
This form factor is related to the Green’s functions of the field theory with insertions of cp2. One has exactly :
where the partition function with two insertions of q, - q,3 (2) [13] reads [11] :
g (2) is the Green function with two qJ2 insertions, defined by [ 11 ]
where m2(r) is allowed to depend on the position in
space.
Then, in the same way as before, we use the dimen- sional renormalization (11), (12) and (14) for obtaining immediately
where iTj2> is defined by
Now F(2) tends towards a finite limit when s --> 0.
Finally, owing to equations (13, 16), we find
(4) In terms of the usual renormalization factors Z, Z(2), Z(4) of the field p, of p2, and of the 4-point vertex function,
we have in the notation of Brezin et al. [12] Z,
=Z (4)lz 2,
zm
=Z(2/Z.
Thus h(q) is finite when expressed in terms of SR, bR as already used in [13].
We have thus recovered the renormalizability
to all orders of the theory in the scheme II, as also shown in reference [6]. We may also note at this stage that Schafer and Witten [4] established similar results in their different massless scheme. Since the latter and the dimensional one are related in field theory by
a finite (however complicated) renormalization, the
scheme II in polymer theory is, of course, also related
to that of reference [4] by a finite renormalization. Let
us now turn to the first renormalization scheme I.
From II, we have
Then we face immediately a difficulty. Zb, Zm are
minimal renormalization factors having the struc-
ture (6), with no constant terms. But Zb Z m e/2 is not
minimal, due to the occurrence of the 0(s) power. Thus from this it is not obvious that the minimal renorma-
lization constant Xz does exist. We are then led to the
following question : does it exist an 8-regular change
of variable b (zR,,6) which yields (2 7r)-d/2 bR Zb x Zm E/2(bR, 8)
=ZR X.(ZRI 8), where Xz is minimal ?
In this case we set A
=Jl2 SR.
A second requirement is that the other renormaliza- tion factors defined by
stay minimal. We show in the Appendix A that this is
actually possible and even that the change of variable bR(zR) is completely independent of s.
It follows then that the minimal renormalization I exists for the z variable, as II exists for the b variable.
Q.E.D.
In the following, we consider the des Cloizeaux’s scheme [2]. We also perform explicit calculations of these dimensional renormalizations, up to two-loop
order.
4. Link with the direct renormalization method.
We shall now consider physical quantities important
in polymer theory [2]. There are : the partition function
of one isolated chain
the Kuhnian size of the isolated polymer chain : R’ = I r(S) - r(O) 112 >= dX 2, where
the dimensionless second virial coefficient g [2] :
and finally the square radius of gyration
The previous analysis shows that all these quantities
are finite when expressed in terms of zR, SR, or bR, SR, using the dimensional renormalizations I or II. Let us now consider the « direct renormalization method » introduced by J. des Cloizeaux [2]. It states that the
new renormalized partition functions 3R defined in
that scheme by
are all finite when empressed in terms of the physical quantities X 2, g instead of S, z, when d -+ 4- or c - 0+. This can be proven to be true to all orders in e
and g, as a direct consequence of the multiplicative renormalizability described above. Using (7) in the
z-dimensional scheme, we find immediately :
Using (13) gives alternatively :
Thus 9 R " is a finite function when expressed in terms of S’, zR, h or in terms of SR, bR, M. Incidentally, let us
note that this is also true for the form factor h(q) (16).
Therefore for proving the « direct renormalization method » to work to all orders, it suffices to note, as stated above, that the changes of variables
or
are regular with respect to s, i.e. they contain no singularities when s ---> 0. This substitution is studied in
Appendix B. Of course, A or It cannot appear in the
physical quantities, which do not depend on them and
this requires that the variables { S’, zR, À. } or { SR, bR, p ) recombine for giving finally two variables { X 2, g }
in the physical quantities.
Let us note finally that, since the form factor h(q)
can be written as a regular function of bR, SR as in (17),
it has also a finite expression in terms of g, qZ X 2, staying well defined when s ---> 0. As a consequence,
h d.. I
.6 R 2
..I f the dimensional ratio N -
.2 G is a universal func- [R"
tion of g, finite order by order in g and s. These facts, intuitively obvious, were already checked in previous
works [13,14]. In Appendix C, we show in addition the
finiteness of the renormalization functions
and of the Wilson function ,
Iof the direct renormalization method.
’
5. Explicit calculations.
Let us now apply the general minimal schemes I and II introduced before, by performing calculations directly
in polymer theory, at the two-loop order. We shall in particular determine the renormalization factors Xz, Xo, xi and the renormalized zR, or equivalently Zbl Zm, Z and bR. For this we consider the four quantities (18-21) which have been previously computed directly
in polymer theory [2, 14]. One has [2]
and [14]
here
These expansions are very interesting since we
shall be able to calculate for instance zR, 3Co, 3Ci from
the three first ones, using the minimal renormalization I, and check that, for instance, they make N finite when expressed with zR. Using the form (I) and equa- tion (6) we find the simple results
from g :
from X 2 :
These relations go with the useful finite expressions (5) :
Inserting z(zR, 8) into N yields, as it must, the finite series expansion
If we use instead the dimensional renormalization scheme of the field theory II, and equation (6), and
insist on making finite the same quantities (22), (23), (24), we find naturally a new answer :
where
and for the area S :
and for the partition function :
These expressions go with fmite series expansions (6) :
which slightly differ from (29).
So, as expected, the two renormalization schemes
equations (26)-(28) and equations (31)-(33) differ at
the two-loop order. Incidentally we note that the expressions 31)-(32) appear in reference [15] which
uses the minimal subtraction scheme II, but without
general proof.
(5) For the last two equations of (29), we have not retained the A dependent terms, setting A
=1.
(6) For the last two equations of (34), we have not retained
the p dependent terms, which amounts to set 2 1tJ.l2 SR
=1.
Let us now calculate the finite renormalization functions which appear in the two schemes. The Wilson function is defined by
for the scheme (I), and by
in the field theoretic one (II).
Using (26) we find
with a fixed point value
The second scheme gives, using (31)
with a fixed point value
We note that both fixed point values (36), (38) yield,
when inserted respectively in (29) and (34), the same
fixed point value of the second virial coefficient of Kuhnian polymer chains :
in agreement with reference [2].
We also check that the Wilson function (37) of the
scheme II has exactly the form, as it must, given by the
Russian school [16] in the minimal renormalization scheme, for n
=0.
Both Wilson functions W’(zR) or W(bR) have also
the fundamental property of being of the form
where W4 is independent of s. This well-known form characterizes the dimensional renormalization.
Let us check for instance the validity of our z-dimen-
sional renormalization scheme I by computing the
critical indices. We define the functions
From equations (26)-(28) one gets the finite series expansions
which, at the fixed point zQ (36), have the values
in agreement with the well-known s-expansions for
n
=0 [2,12]. In the same way, we fmd for the universal ratio N (30)
in agreement with reference [14].
We are also in position to compare our method to
the « direct renormalization method » of reference [2]
(see Section 4). Consider the physical renormalization
constants :
and the second virial coefficient g (Eq. (20)). They
differ from those of the dimensional renormalizations
by a fmite renormalization. For instance, one has
immediately, from equations (27)-(29)
while g is given by (29).
The « direct renormalization method » is interest-
ing since it works directly with physical quantities.
The draw-back is related to the fact that it is not a
minimal renormalization. All the fmite renormaliza- tion functions for instance are more complicated. For instance, the associated Wilson function reads [2] :
with a fixed point value
as in (39). -
In conclusion, we have at our disposal three different
renormalization schemes for polymer theory. One
is the « direct renormalization » method of des
Cloizeaux [2] which uses physical quantities but is not
minimal. One may also use the z-dimensional renor-
malization proposed here which is minimal, and particularly convenient for polymer theory since it
uses the well-known z parameter, cherished by polymer physicists [17], but to renormalize it into zR.
It introduces « non physical » quantities S’, zR which
can be related to the physical quantities X 2, g,
However, from the point of view of the theorist, I
think it is much more fundamental with respect to the
structure of the theory of the Kuhnian chain. All the renormalization functions are then indeed very simple.
The last one, the field theoretic one, uses the minimal renormalization of the n
=0 field theory. It has the
same advantage as the z-scheme for the theorist, having included in itself the heart of the renormaliza- tion structure. But it is also « not physical ». In poly-
mer theory, it will lead to slightly more complicated
calculations. It has an advantage : one may use the field theory calculations, as we have shown in another
work [8].
Finally, it would be very interesting to try to find
«physical quantities » corresponding in polymer theory to the minimal zR, Xb, Xi. It would be also very
interesting, since now one can work directly in poly-
mer theory with dimensional renormalization, to try to find the hidden renormalized Edward’s type model it corresponds to.
Acknowledgments.
I thank F. David for very interesting discussions.
Appendix A.
PROOF OF EXISTENCE OF THE MINIMAL Z-SCHEME.
-The renormalization equations
are equivalent to the integral representations
where by definition :
In (A. 2) the lower value of the integration variable
is not precised (it can’t be zero), but this is natural since it corresponds to the arbitrariness of it.
Now, the minimal polar form of the renormalization
constants Zb, Zm (A. 2), (A. 3)
is equivalent to the fact that W has the well-known
particular form
where W4(t) is the value for d
=4, which is comple- tely independent of B. ym(t) itself is also independent
of s. This can be checked easily on equations (A. 2), (A. 3). Now we have by definition
which can be rewritten, owing to (A .1)
From now on we drop the factor (2n)-2 which is
reabsorbed into the defmition of b : (2 n)-2 b - b,
and we also forget about the (2 nJ.l2 SR)E/2 = À. e term,
of which the renormalization factors are necessarily independent. As we already said, the factor Zb Zm2/2
does not have the minimal structure (A. 5), with respect to bR, or, equivalently, Fb Zm E/2 does not have
the form (A. 2), (A. 6).
So we search for a change of variable
regular in E, which makes Fb Zme/2 minimal with
respect to the new variable zR.
We have, owing to (A. 2), (A. 3)
or changing of variable t(t’) ; bR(zR) :
With respect to the variable t’ or zR, the scheme will be minimal if we can rewrite F. (A. 8) in the form
where W’(t’) has the particular form (A. 6)
So we identify (A. 8), (A. 9), with the help of (A. 10)
It is sufficient to set
or by integrating
Since y.(t)
=0(t) for t small, the change of variable
.
(A. 11) has a regular series expansion in powers of t,
t’ = t + O( t 2). Thus we see that the change of variable
which is entirely independent of ~, gives a minimal
structure to Fz, and we can set
where xz has the property (A. 5). It is easy to check that the renormalization factors
which were minimal with respect to bR, are also left
minimal with respect to zR, in the change (A 12). So (A 12) gives a minimal subtraction scheme with respect to z. Q.E.D. We note that the present analysis applies identically for proving, in field theory, the
existence of a minimal subtraction scheme in the variable bm-E, instead of in b, a fact which apparently
has not been noticed before.
Appendix B.
REGULARITY OF THE SUBSTITUTION { ZR, SR, À. } - { g, X2}. - We shall prove this regularity in the
z-dimensional scheme. The same can be proven in the b-dimensional scheme in the same way.
Going back to equation (19) and using the funda-
mental equation (7), we have
By dimensional analysis, this yields
where Fo is a regular function of s.
On the other hand, the dimensionless second virial
coefficient g defined in (20) reads, owing to (7) and to
the preceding equation :
By dimensional analysis, this can certainly be written
as
where F2 is, again, a regular function of e. This achieves the proof of the regularity of the substitution
{ ZR, Si, Â. } --+ { g, X 2 }.
°In the same way, one shows that for the substitution
{bR, SR, A}_{g,X2}, there exist two functions Fo, F2 such that
which are regular with respect to 8.
Appendix C.
RENORMALIZATION FUNCTIONS AND CRITICAL INDICES.
-
We shall here consider the renormalization equa- tion (13)
and draw consequences from it, which have been assumed and used in reference [2]. We define the
renormalization functions
For { k }
={ 0 }, the F are the partition functions
and thus the aN are effective indices. The mass
is trivially kept fixed since ff does not depend on ,u.
But this proves useful in the following. We use the
fundamental equation (13) to get
But bR(b, M, s) does not depend on S. So trivially
and thus, since SR
=Z. S, one has also
So we immediately see that UN is finite in s, when
expressed in terms of renormalized quantities SR, bR.
Moreover, since we know that the substitution
{ SR, bR, ,u } - ( X 2, g } is regular, we see that the 6N
are finite functions (in s) of X and g. Q.E.D. They are
also fmite functions of (SR, zR) or (SR, bR), or (X, zR)
or (X, bR).
Let us also consider the Wilson function introduced in reference [2]
and prove it is a finite function of g and E. We write it trivially and use again the identity
Then, according to the analysis of Appendix B we have g
=F2(bR, ,u2 SR, s) and thus
Thus W" is a finite function (in s) of bR, ,u2 SR’ It is
therefore also finite when expressed in terms of g,
J.l2 X 2, but, since it does not depend on p, it is a finite function of g alone, Q.E.D.
We can also relate easily these effective indices (IN
to the usual critical indices of field theory and critical phenomena. Let us consider again (C .1) and apply
on both sides of it. Since trivially :
one gets, for the special case { k } = { 0}, and using
dimensional analysis, the Callan-Symanzyk like equa- tion for the connected partition function TN :
where the renormalization functions W, Ym, ’1 are defined in the usual way [12] by
In terms of the usual critical indices [12] one has the-
refore [12]
Using (C. 5), we obtain from (C. 7)
Now, at the infrared fixed point bR of the field theory
one has
and, in this limit :
o oUsing (C .9) we find
Thus we find, since by definition
1-
