Multiplicative renormalization of continuous polymer theories in semi-infinite geometry

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Multiplicative renormalization of continuous polymer theories in semi-infinite geometry

M. Benhamou, G. Mahoux

To cite this version:

M. Benhamou, G. Mahoux. Multiplicative renormalization of continuous polymer theories in semi- infinite geometry. Journal de Physique, 1988, 49 (4), pp.577-586. �10.1051/jphys:01988004904057700�.

�jpa-00210732�

577

Multiplicative renormalization of continuous polymer theories in semi-

infinite geometry

M. Benhamou and G. Mahoux

Service de Physique Théorique, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, ^France (Requ ^{le 9} septembre 1987, accepté le 18 dgcembre 1987)

Résumé. ²⁰¹⁴ La conjecture ^{de des} Cloizeaux, ^selon laquelle ^les théories standard de polymères ^{continus se}

renormalisent multiplicativement jusqu’aux dimensions critiques, ^{est démontrée} pour des théories en

géométrie semi-infinie. Ceci correspond ^à des situations physiques ^{où les solutions de} polymères (en ^{bon ou 03B8} solvant) ^{sont en contact avec une} paroi plane impénétrable qui ^{exerce des forces sur les} polymères, ^{et où ces}

forces sont suffisamment fortes pour induire des phénomènes d’adsorption ^{sur la} paroi.

Abstract. ²⁰¹⁴ The conjecture ^{of des} Cloizeaux, which asserts that standard continuous polymer ^theories

renormalize multiplicatively up ^to critical dimensions, ^is proved for models with semi-infinite geometry. ^These models correspond ^to physical situations where polymer ^solutions (in good ^{or 03B8} solvents) ^{are in contact with a} plane impenetrable wall which exerts forces on the polymers, ^and where these forces are strong enough ^to

induce adsorption phenomena ^{on the wall.}

J. Phys. ^{France 49} (1988) ^{577-586 AVRIL} 1988,

Classification

Physics ^Abstracts

11.10G - 36.20 ^- 62.25H

1. Introduction.

In a previous paper [1], ^{we have} proved ^des

Cloizeaux’s conjecture

[2],

^{which asserts that stan-}

dard continuous polymer ^theories

(in

good ^{and 0} solvents) renormalize multiplicatively, up ^to critical dimensions. This ^was proved in infinite geometry models. In this paper, ^we extend the proof ^{to the}

case of semi-infinite geometry, ^{that is to} say ^to situations where the polymer ^solution occupies ^a half-space ^bounded by ^a plane impenetrable ^wall,

which exerts short range forces ^on the polymers. ^Of particular interest is the case where these forces are

strong enough ^{to induce a} transition to a phase

where polymers ^{are adsorbed on the wall.}

In 1969, ^{de Gennes}

[3]

argued that, under certain

conditions, the effect of the adsorptive ^{forces can be} summarized, ^{in the} limit where the range of these forces vanishes, by ^a simple boundary ^condition imposed ^on polymer partition functions.

If 5(r,

r’)

denotes the partition function of one polymer, ^the

extremities of which are fixed at r and r’, ^{de Gennes’}

condition writes :

where z is the distance to the wall

(normal

^compo-

nent of r), ^{and where the} phenomenological ^constant

k depends ^on temperature.

Then, he showed that ^a phase transition occurs

when k(T) changes sign :

k (T )

^is expected ^{to be} negative ^at high temperature, ^{and to} increase when the temperature decreases. As soon as it becomes

positive, polymers ^{are adsorbed on} the wall. The critical temperature ^is thus defined by ^the vanishing

of k(T).

Since then, de Gennes’ condition has been widely

used to define and study ^{models of} adsorption, ⁱⁿ

presence of excluded volume interactions

[4-10].

Here we consider two models of polymer solutions, in good ^{and 0} solvents. The first ingredient ^{to prove}

the multiplicative renormalizability ^{of these models,}

is Laplace-de Gennes transform [4, 6], which estab- lishes their equivalence ^with theories of n component scalar fields _cp in semi-infinite geometry, ^with

(cp 2)2 [or

(cp

2)2

^and (cp

2)’]

^bulk self-interactions and

cp 2

[or

cp 2 and (cp 2)2]

^surface self-interactions, ^{in the} limit of zero components. ^{Thanks to this} equival-

ence, all the renormalization machinery ^{of field}

theories _{may be} exploited ^to renormalize polymer

theories. The second

ingredient

^{is ’t Hooft and}

Veltman dimensional renormalization procedure,

which provides ^mass independent renormalization constants, ^or any ^other renormalization procedure

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904057700

with the same feature, ^{such as the massless renor-} malization scheme of Brezin et al. [11]. These pro- cedures are particularly ^well adapted ^{to the} problem,

since they ^{commute with} Laplace-de ^{Gennes trans-}

form.

In section 2, ^we define notations, ^{and extend} Laplace-de Gennes transform ⁱⁿ infinite geometry,

to situations where exists an external potential

U (r).

The extension is straightforward, ^{and the} proof ^follows closely ^{the «} analytic » proof ^{of refer-}

ence [12]. ^{This covers in} particular any situation where exist boundaries, ^and adsorptive ^{forces on}

these boundaries.

In section 3, ^the potential

flL (r)

^is particularized

to describe semi-infinite geometry situations : U

depends only ^on the distance z to the boundary, ^{it is}

infinite ^on one side, ^{and is a} short-range potential ^on

the other side. Before letting the range R go ^{to zero,}

we show that the « bare propagator » expresses

simply ^{in terms of the} « S matrix » of the one

dimensional Schrodinger equation ^with potential

U

(z).

^{We recover} the fact that a phase ^transition

occurs when a first bound state appears [3]. Indeed,

the above phenomenological parameter k(T) ^{is the}

square ^{root of} the ground ^state binding energy

(wave

number). ^{We show also} that, when the range ^R of the potential goes to zero, if there is a bound state of fixed energy, then the potential ^{has a limit more} singular ^{than a} Dirac distribution 6

(z).

^{As a matter}

of fact, ^{a 8}

(z )

potential in front of an impenetrable

wall (infinite potential) ^is completely ineffective, ^it disappears ^{in the} wall, ^{and cannot bind a state}

(1).

In sections 4 and 5, ^we prove the multiplicative renormalizability ^of polymer theories in respectively good ^{and 0 solvents.}

2. Equivalence ^{theorem for} polymers ^{in an external} potential.

Let us consider an ensemble of JY’ polymers ^{in a d}

dimensional space. ^A configuration C ^{of this ensem-}

ble is described by ^{a set :}

where the rj ^{’s are} d-dimensional vectors ; sj, ^which

has the dimension of an area, is the coordinate of a

point along ^the polymer j,

and Sj

is the total « area »

of the polymer j.

The action which governs ^the physics ^{of the} system contains three terms. The first one is the Brownian action :

The second one is the self-interaction ^{term. It} is a sum of contributions of two-body, three-body...

interaction potentials

(2) :

where

C (r)

^{is the} polymer density ^{in the} configur-

ation C :

Finally ^{comes the} interaction with the external

potential

U (r),

^{which we} denote U . C :

Let

Po f C) be

the Brownian weight ^associated

with the configuration C :

and let

d{C}

^{be the} measure on the configurations C, ^{which we normalize as follows :}

Then the mean value

F> 0

^{of a} functional

F{C}

^{is the} functional integral:

(1)

^{This is in} disagreement with reference [9], ^{where the} polymer ^action equation (2.1) is written with ^a 6(z) potential. ^{Such an} expression ^is appropriate ^{for a} pen- etrable wall. It is not for an impenetrable ^{one. Further-}

more, de Gennes’ parameter k(T) ⁱⁿ equation (2.2) ^{of this} reference, coincides with the coupling ^constant 60 in ^front

of the 6(z) potential. ^{This is} certainly ^{incorrect since,}

when 8 o ^{vanishes, that is to} say in absence of adsorbing potential, ^the boundary ^{condition on} the surface of an

impenetrable ^{wall is} Dirichlet’one, ^{and not Neuman’s one}

as it results from this equation (2.2).

(2)

^The potentials

V p (r l’ ..., r p)

depend only ^{on the}

differences (tri -

r,).

579 With these notations, ^the partition function of N polymers with fixed extremities reads :

By introducing ^{a source}

J (r)

coupled ^{to the} density

C (r),

^{one can} give ^this partition function the convenient form :

where

3 (r,

r’ ; S, J ) ^{is the} partition function of a Brownian chain in presence of the external ^source J, ^{that is}

to say (here ^{N =} 1) :

It satisfies the equation :

where

3o(r,

^{r’ ;}

S )

^{is the} partition function of a free Brownian chain :

Note that in

equation (2.10),

^{J is} given the value - U and not zero as in reference

[12].

We now introduce the Laplace ^{transform of 3 :}

and for a Brownian chain in presence of the external

source J :

Equation

(2.10)

^{entails :}

and g (r,

^{r’ ; a,} J ) satisfies the equation :

where

go (r, r’ ; a),

^the Laplace transform of

30 (r,

^{r’ ;}

S ),

^is given by :

From this formula and equation (2.17), ^one easily

derives the diffusion equation ^satisfied by

19 (r, r’ ; a, J) :

In terms of linear operators, the kernels of which

are 9 (r, r’ ; a, J )

^and

go (r, r’ ; a), (Eq. (2.17))

writes :

where J is the diagonal operator ^defined by ^the

kernel

J(r) 5d(r - r’).

It remains to prove that the g (2,W) ’s ^are Green functions of a field theory. ^{Let us} consider X scalar fields

q;> j (r)

^{with n} components

q;> jf (r).

^{We note 4, a}

configuration

of this ensemble of fields :

The action

A {cP}

^{we now} construct, ^here again

contains three terms :

Using ^{the notation}

cP jf (r) = (r I cP jf)

^{of refer-}

ence

[12], Âo {c/>}

^{is defined by :}

I

{K}

^{and flL . K are defined as in} equations (2.3)

and

(2.5a),

and K is the composite ^{field :}

Let

Qo { cp}

^{be the}

(free

fields) weight :

and let

d { p}

^{be the} measure on the configurations cp, ^{which we normalize as follows :}

Then the mean value

(F) 0

^{of a functional}

F {cp}

is the functional integral :

By construction, ^the propagator of the free field cP jt in presence of the external ^source J coincides with the function :

and from equation (2.16) :

Furthermore, the Green functions of the field

theory

governed

by ^{the total}

action A { cp },

^which

we note as expectation

values (...)

^{without zero} index, ^{are defined as follows :}

Now, ^the denominator of the right ^hand-side happens ^{to be}

equal

^{to 1 in} the limit where n, the number of components of the fields (pj, ^{goes to}

zero :

Then, ^from equations (2.29) ^and

(2.30)

^we

conclude that the Laplace transforms g (2 Y) ^{of the} polymer partition functions 3 (y) ^are Green functions of the field theory ^defined by the action

(2.22),

ⁱⁿ

the limit n -+ 0 :

This is the expected ^result.

It can be extended to more general polymer partition functions, ^{where not} only the extremities but also other points ^of the chains are fixed

[13].

^For

the sake of simplicity, ^{let us consider a} single ^chain (N ⁼ 1), ^{of « area » S, and let us fix the} point ^of

abscissa s, along ^{the chain at} position ^{R in the d}

dimensional space. With s2 ^defined by S = sl + s2, the partition function reads :

We introduce its Laplace transform :

We now need a field theory ^{with two n} component fields _cp and cp2, of square masses al

pndi

^a2,

constructed with the action

A { p}

^of equation

(2.22)

for X ⁼ 2. Then

In the right hand-side appears the composite quadratic ^field

cp 11 (R) cp 21 (R).

Following ^{the same}

581 lines of reasoning ^{as above, we arrive at the} following

result :

More generally, let 3 (-v, c) ^{be the} partition ^function

of N polymers, the extremities of which and also C other points ^are fixed in space. To avoid intricate notations, ^{we do not} specify ^{all the} arguments, which are 2 N + C points in the d-dimensional space, and N ⁺ C areas. Let g(2X,C) ^be the Laplace

transform of 3 (x, c) with respect ^to its last X + C arguments :

Let ^us construct as above a field theory ^with

N + C fields cP j’ ^of components cp l p and square

masses aj. Then g (2 V, E is a Green function of this

theory, expectation ^{value of a} product ^{of 2 X fields} cP jt ^{and C} quadratic composite ^fields _{cP jt} cP km’ in the limit n - 0.

3. Impenetrable adsorptive ^wall.

We are interested in the physical situation where the volume occupied by ^the polymers ^{is a} half-space ^V

limited by ^a plane impenetrable ^wall

(semi-infinite

geometry), and where this wall exerts short-range

forces on the polymers. Ultimately ^we shall take the limit where the range ^R of these forces goes ^{to zero.}

Such ^a situation is described by ^a particular

external potential U (r). ^{Let z} be the distance of a

point ^{r to the} wall, ^and p the d - 1 dimensional component ^of r, parallel ^to the wall: r ⁼

(p, z ).

Then U is a function of the only ^{variable z, and}

(see Fig. 1) :

The quadratic part of the action

(2.22)

^is

Ao { p}

^{+ ’11 . K. The} corresponding propagator ^for the field

a j

^{is the} function 9

(r,

_{r’; ai, -} U ) ^which

satisfies the diffusion equation :

We take advantage of the invariance of the

potential ^U under the translations parallel ^{to the} wall, ^to Fourier transform 9 :

with E = a +

k2/2 . g’B1 (z,

^{z’ ;}

E )

is solution of the

Schrodinger

equation (the

^« energy ^» is -

e) :

and it satisfies the boundary conditions

(3) :

In the region ^{where the} potential ^vanishes

(z

^and

z’ >

R), Gu(z,

^{z’ ;}

E )

^{takes a} simple ^form

[14] :

In this formula appears the «S matrix ^»

S ( E ).

Much more than the potential ^itself, it is the relevant

quantity ^{in our} problem, ^{and from now on, we shall}

focus our attention on it.

Let us first come back to the partition ^function

5(r,

r’ ; S, -

U ),

^{which we} Fourier transform :

3’B1 (k, z, z’ ; S ) can

^be reconstructed from

gu (z,

^{z’ ;}

E )

by ^{an inverse} Laplace transform :

3u(k, z, z’ ; ^{S) =}

r is a contour in the complex ^E plane,

parallel

^to

the

imaginary

^{axis, and situated at the right of the} rightmost singularity ^of

Gu (z, z’ ;

^E

^).

^{And as usual,}

the behaviour of

3u (k,

^{z, z’ ;}

S )

^{when S -+ oo}

(criti-

cal behaviour _{of very} long ^chains

[2])

is controlled

by

the rightmost singularity ^of 9,m

(z,

z’ ;

E ).

(3)

^Indeed Gu (z, ^{z’ ;} E ) ^vanishes identically ^{for z or}

z’,0,

We are thus led to look at the analyticity proper- ties of

S ( E ),

^{which are} determined by ^the spectrum of the

Schrodinger

operator

1/2 5 2/aZ2 _ U (Z):

^if

the

Schrodinger equation

^{has no bound state,}

S(E)

^is analytic ^{in the} complex plane ^cut along ^the negative ^real axis ; ^{if there are bound states of} binding energies E « > 0,

S (E )

^has simple poles ^at

e ⁼ E a . In the absence of bound states, the critical behaviour of

5,

is controlled by ^the singularity ^at

E ⁼ 0. As soon as there appears ^a bound state, which becomes the rightmost singularity ⁱⁿ place ^of

the origin ^{e =} 0, the critical behaviour of

5,,

^and

consequently ^that also of all polymer partition

functions 3 (N, r),

changes

abruptly :

this is the signa-

ture of a phase transition. In the new phase ^of

course, ^one expects polymers ^to be adsorbed on the wall at z ⁼ 0.

To describe in more details the analyticity proper- ties of S

(E ),

^it is convenient to map the ^{E cut} plane

onto the half-plane ^{Im K} > 0, ^where K, the « wave-

number », is defined by :

We note

S(K)

⁼

S (,E )

the S matrix as a function of K. The potential

’tL (z)

being short-range,

S(K)

enjoys ^the following properties

[14] :

1)

analyticity : ^we already know that it is

meromorphic in the upper half-plane, ^its singularities

there being simple poles ^{localized on the} imaginary

axis. It can be also analytically ^continued through

the real axis into a function meromorphic ^{in the}

entire plane.

2)

reality :

3) unitarity :

and

Indeed unitarity implies ^{that for any} complex ^{K :}

Then using equation

(3.10),

^{we obtain}

Both formula

(3.13)

^and

(3.14)

realize the analytic

continuation of

S ( K )

^through the real axis.

Let us now assume that there exists ^a neighbour-

hood 9J of the origin, symmetric ^with respect ^{to the} origin

(4)

^{where the only} singularity ^of

S (K ) is

^{a pole}

(4)

^{This condition insures, thanks to} equation (3.14),

that

S (1()

^has only ^{one zero in} 9J, ^{at x = ic.}

at K ^{= -} ic (c real). Then, ^{from the} above proper- ties, ^one easily shows that S(K) can ^be represented

in V as follows :

where

f(K 2)

^{is in V a real} analytic function of

K 2.

Equivalently :

We also assume that

S (K )

^{has no other}

singularity

on the positive imaginary ^{axis. Then, the} rightmost singularity ^of

S(E )

^{is E = 0 if} c > 0, ^{and E =}

c2/2

^if

c 0. This is a situation which describes the above discussed phase transition. The parameter ^{c can be} interpreted ^{as a « reduced} temperature ^»

[3] :

where Ta ^{is the} adsorption temperature. ^{If a} > 0, adsorption ^{occurs below} Ta.

Now, in the limit of very long ^chains, only ^is

relevant in the function

S(E)

^{its behaviour at the} rightmost singularity. So, ^as long ^{as c is small} enough

to satisfy the condition :

we can neglect

kf(kz)

ⁱⁿ equation

(3.15).

^{This leads}

us to the approximate ^one parameter expression ^of

S(e ) :

valid when first, ^the temperature ^{is close} enough ^to

the critical one, and second, ^{the chains are} very long.

Furthermore, ^{there is no} problem ^{now to take the}

limit where the range R of the adsorptive ^forces goes to zero. In this limit, ^the

expression (3.6)

^of

Gu(z,

^{z’ ;}

E )

^is valid for any

positive

^z and z’. Since

Gu(z,

^{z’ ;}

E )

^{vanishes for z or z’}

negative,

^{we can}

write for any ^z and z’.

We recognize ^the propagator ^{of a} scalar field cp in semi-infinite geometry, ^{with a surface} quadratic ^self- coupling

[15].

^More precisely, the action

.ae { cf>}

^of

the field theory which is in correspondence

through

the Laplace-de Gennes transform with the above defined model of adsorption, ^{writes as in}

equation (2.22),

^with

583

Note that the

integrations

^{run over the} half-space

V in

equations (3.21)

^and

(3.23),

^{and on the} adsorptive wall aV in equation

(3.22).

^{This last}

equation

corresponds

formally

^{to an external} poten- tial of the form cB

(z).

^However, in the limit where its range R vanishes, the potential

flL (z )

^{does not} go to the Dirac distribution 5

(z).

^{Indeed a 5}

(z)

poten- tial would be swallowed by ^the impenetrable wall,

and would be unable to bind a state. It is instructive to look at the square ^{well of}

depth

^U and range R,

which has a bound state of

binding

energy K2. In the limit where R - 0, ^{U of course becomes} infinite, ^and

one finds that :

Obviously,

the limit of this square well is ^more

singular ^than

ð (z ).

The propagator

(3.20)

^{and its inverse} Laplace

transform

(3.8) satisfy

^{the same} boundary condition :

It is the condition proposed by ^{de Gennes}

[3].

^At adsorption temperature, it becomes Neumann’s con-

dition. In the absence of potential U,

S(,E)

^is equal

to 1. It corresponds ^{to c -+ oo, and thus to} Dirichlet’s condition

(see

foot-note in

Sect. 1).

4. Renormalization of polymers ⁱⁿ good solvent ad-

sorption theory.

We now specify the form of the polymer ^self-

interaction

term I {C} .

It contains only contact two-

body ^forces

(excluded

^volume interaction), ^which

are the relevant forces for polymers ⁱⁿ good ^solvent [16] :

JOURNAL DE PHYSIQUE. - T. 49, N° 4, AVRIL 1988

The associated field theory ^{is a}

( cp 2)2

scalar field

theory in semi-infinite geometry, ^defined by equa- tions (3.21 ^to 3.23), ^{with a} self-coupling ^which

writes :

One knows [17-18] that the field

’Pjt(r)

^{does not}

renormalize the same way in the bulk and ^on the

boundary. ^It is thus necessary ^to distinguish ^the polymer points ^{which are} fixed in the bulk, from those fixed on the boundary. ^{Let us}

call 3 (Xb, -V,, Eb, EI)

the partition ^{function of a} system ^{of X} polymers, ^of

which JY’b (resp.

Xs)

extremities are fixed in the bulk

(resp.

^{on the} boundary)

(5).

^{Of course :}

Let 9 ^{Nb, Ns, Eb, Cs)} be the Laplace-de ^{Gennes trans-}

form

of 3 (Xb, X,, lb, C.,)’It

^{is a} field theory ^Green function, expectation ^{value of a} product ^of JY’b ^fields cP jf ^{in the} bulk, Xs ^{ones on the} boundary, Eb composite fields cP kf cp km in the bulk, and Cs ^same composite ^{fields on the} boundary.

In the renormalization equations ^{we are} going ^to

write for these different fields, ^the indices j, ^{f ...} play

no role. To simplify ^the writing, ^we suppress them,

but introduce the index s for fields ^on the boundary.

Thus g (Xb’ XS’ Cb, Cs)

is the expectation ^{value of a} product ^of Nb ^fields cP, Ns ^fields cP s’ Lb ^fields

cp 2

and Ls ^fields

cp;-.

Furthermore, ^{from now} on, the 9 and 3("’)s

will designate connected Green and partition ^func- tions, ^{which are the} objects of interest.

As in reference [1], ^to renormalize the field

theory, ^{we use} the minimal dimensional renormali- zation procedure of’t Hooft and Veltman

[19],

which provides ^mass independent renormalization

constants. A first consequence of this ^mass independ-

ence is that the

0 ( (X + C )n)

^{symmetry of the}

action is

preserved [1]

^{in the} renormalization _pro- cess, ^so that we have to introduce only ^{two renor-}

malized coupling ^constants uR and _{cR. At} dimension

(5)

^{At thermal} equilibrium, ^{the ratio of the numbers} Nb ^and X (or ^{rather of} their bulk and surface densities) ^is

a function of temperature, ^{which can be} calculated by introducing ^{in a} standard way ^a grand partition ^function

with convenient fugacities. Although ^{it is not our purpose}

in this paper ^{to do this} calculation, ^{let us indicate} that, ^due

to the fact that the fields do not renormalize the same way in the bulk and on the boundary, ^all divergencies ^{do not} disappear ^{in the} expression ^of the above ratio. The situation and its way ^{out are} quite ^{similar to those}

encountered when calculating ^the probability ^{of contact}

between two points ^{of a} polymer [23] (see ^for example

Ref. [12], p. 538 sq.).

38

d ⁼ 4, where the theory ^is just renormalizable, ^{u is} dimensionless and c is a mass

(inverse length).

^As usual, ^{we introduce an} arbitrary ^{scale mass 4, and}

write the renormalized interaction

IR (K)

^{as follows}

(e = 4 - d) :

in such ^a way that uR be dimensionless, irrespective

of d.

Let

cp R, cp R, (cp 2)R, ( cp S )R

^{and aR}

(without

^indi-

ces j,f ... )

^{be the} renormalized fields and masses.

Then, the renormalization equations ^write

[18] :

where the renormalization constant

Z cP’ Z CPs’ Z 9 29 Zcp;,

^{Za, Z, and} Zu depend only ^on the dimensionless

coupling ^constant uR and ^E

(Zgs

^{is usually written}

Z9 Zl).

^Indeed

Zcp, Zcp2, Za

^{and Zu} which renormalize bulk quantities, ^{do not} differ from what they ^{are in}

the infinite geometry model. Note also that there is

no term proportional ^to

c2

^{in the} right hand-side of

equation (4.5e),

^as would be allowed by ^a simple

dimensional counting argument : ^{the renormali-} zation of the square ^masses is not affected by ^the

presence of ^a boundary.

Furthermore, the connected Green functions re-

normalize according ^{to the} general ^formula

(6) :

The renormalized Green functions

GR( ... )

depend

on the renormalized parameters aR, UR, cR, the scale mass A and the dimension ^E. They ^are regular ^at

£=0.

The seven renormalization constants Z’s defined above are not all independent. ^{We have seen in}

reference [1] ^{that :}

a result which still holds here. Let us prove that ^one has also :

(6) ^{The dots stand for} position ^{and momenta variables.}

From equation

(3.20),

^one gets :

This formula implies ^that differentiating ^with respect ^{to c} any Green function, ^is equivalent ^to making ^an insertion with vanishing ^transverse

momentum, ^{of the}

quadratic

composite surface field

o/?

^S.

Renormalizing both sides gives :

Due to the minimal prescription, ^the left hand-

oo

side has the structure 1

+ L am(UR)/em.

^{The right}

m = 1

hand-side is regular ^{at e = 0.} Consequently ^both

sides are equal ^{to one.}

It now remains to write renormalization equations

for the polymer partition functions 3 ^...

),

by ^inverse Laplace transforming equation (4.6). ^{Thanks to} equation (4.5e), integrals ^{over the}

aj’s

^{can be}

changed ^into integrals ^over the renormalized

ajR’S,

with Za ^as Jacobian. Since the Z’s do not depend ^on

the _masses, they factorize out, and ^we get multiplica-

tive renormalization equations :

Here, s and sR stand for the collections of ^areas

Sj ^and

SjR (i

^{= 1, ..., JW + C}

^),

^{where the} SjR ^{’s are the}

renormalized areas :

and the

3k"’) ’s,

renormalized partition ^functions regular ^{at E =} 0, ^{are inverse} Laplace transforms of the renormalized Green functions

GR(...).

Note that, ^{if the} divergent factor in the right ^hand-

side of equation

(4.12)

^{does not} depend ^on Eb

[1],

^it

does depend ^on Cs, the number of polymer points

fixed on the adsorptive ^wall.

5. Renormalization of polymers in 8 solvent adsorp-

tion theory.

In 0 solvent, ^the relevant self-interactions ^are three-

body ^forces [20] :

585

The associated field theory ^{is a} (cp

2)3

scalar field

theory in semi-infinite geometry, ^{with the} following

bulk self-couplings :

The renormalization of this theory ^induces

( cp 2)2

surface counterterms. Consequently, from the very

begining, ^we write the action

A {0}

^{with a} quartic

surface term :

This is equivalent ^to saying ^{that the} polymer ^bulk three-body forces induce surface two-body ^forces.

As a result, ^the physical system ^{is now described} by

four

(unrenormalized)

parameters, ^{u, v, c and}

ul [21], ^{and to the} interactions

(5.1)

^{we add the}

surface term :

At the critical dimension d ⁼ 3, v and ul ^are

dimensionless, ^{and u and c are inverse} lengths. ^{The v}

and ul couplings ^are just renormalizable, ^{the u and c}

ones are

super-renormalizable.

^{In the} following,

convenient powers of the scale ^{mass tt} will be introduced in such a way that the renormalized

constants

VR and

ul R be dimensionless, ^and

UR and

CR have ^mass dimension, ^{all that} irrespective ^{of d.}

We set of course ^{e =} 3 - d.

The renormalization

equations

take the form :

The bulk quantities ^{cp ,}

q; 2,

^{a, u and v} renormalize

as in the infinite geometry ^model [17-18]. ^{As a}

consequence, first, ^the renormalization constants

Zq;, z lp 29

^Za,

Za,

^{Zu and Z, do not} depend ^{on the}

dimensionless constant ul R ; second, ^{as in the} previ-

ous section, there is ^no term proportional ^to

c2

^{in the} right hand-side of equation

(5.5e).

^Note

also the presence in equation

(5.5i)

of the last term

independent of ul R, which implies that ul does ^not vanish when U1R goes ^to zero, if v R =F ⁰

(7).

^{This is}

the above mentioned induction of two-body ^surface

forces by three-body bulk forces.

These equations generalize equations (65-69) ^of

reference [1]. They ^{are written} according ^{to the rules}

demonstrated for infinite geometry ^theories

[22],

extended in a straightforward way ^to semi-infinite geometry ^theories.

The connected Green functions renormalize as

follows :

By Laplace transforming ^this equation, ^{we obtain}

the multiplicative renormalization equations ^{of the} polymer partition functions :

where the renormalized areas are defined as in

equation (4.13).

(’)

^{We are thankful to the referee for} indicating ^our

initial awkward way ^to write equation (5.5i).