Statistics Of Multicomponent Polymer Stars

We analyze a polymer network made of chemically diierent polymer species. Considering the star-like vertices constituting it in order to describe their scaling properties we introduce a new set of critical exponents. In the case of network made of two species of polymers we call them copolymer star exponents. By means of mapping our theory on appropriate Lagrangean eld theory we calculate these exponents as well as the exponents governing scaling properties of star of mutually avoiding walks in the third order of perturbation theory. In the case of homogeneous stars we recover previously obtained values of star exponents. Our results agree as well with the previous studies of special cases which were done to the second order of the "-expansion. We found consistency and stability of the results in d = 2 and d = 3 with expected growing of deviations for large number of arms of one star. The same methods were applied previously to the problem of uniform star polymers and have led to results in good agreement with Monte Carlo simulations. We hope our present calculations might also stimulate Monte Carlo studies of the copolymer star problem. The re-summed values of the exponents for stars of mutually avoiding walks are in fair agreement with an exact result previously conjectured at d = 2. The study performed for d = 2 could give some insight to the problem of mapping our theory to a two dimensional conformal eld theory. By studying the convexity properties of the spectrum of copolymer star exponents we show that they are a good candidate for nding application in the theory of multifractal spectra generated by the harmonic diiusion near the absorbing fractal.


Introduction
Polymers and polymer solutions are among the most intensively studied and interesting objects in condensed matter physics.The behaviour of multicomponent solutions containing polymers of di erent species is especially rich 1].Of special experimental and technical interest are systems of polymer chains of di erent species which are linked chemically together.These systems may be block copolymers of two chains of di erent species linked c C.von Ferber, Yu.Holovatch, 1997 ISSN 0452{9910.Condensed Matter Physics 1997  at their endpoints or more generally any number of chains may be linked in the form of stars or networks of any topology (see Figs. 1 -3).For homogeneous systems of one species the scaling properties of such polymer networks have been extensively studied (for a review see 2]).Star polymers as the most simple polymer networks may be produced by linking together the endpoints of polymer chains at some core molecule (Fig. 1).In the same way more general networks of a given topology are formed (Fig. 2).Randomly linked polymer networks on the other hand are obtained as result of a vulcanization process randomly linking nearby monomers of di erent chains to each other.
The asymptotic properties of homogeneous systems of linear chain molecules in solution are universal in the limit of long chains.For each system there exists a so called temperature at which the two point attractive and repulsive interactions between the di erent monomers compensate each other and as a result the polymer chains may be described by random walks (up to higher order corrections): The mean square distance between the chains endpoints hR 2 i scales with the number of monomers N like hR 2 i N.
Above the temperature the e ective interaction between the monomers is repulsive resulting in a swelling of the polymer coil which is universal in the asymptotics: hR 2 i N 2 for N ! 1 with (d = 3) 0:588; (1.1) d being the dimension of space.The number of con gurations Z of a polymer chain of N monomers scales with N like Z e N N ?1 (1.2) with a non -universal fugacity e .In the early 70-ies following the work of de Gennes 3] the analogy between the asymptotic properties of long polymer chains and the long distance correlations of a magnetic system in the vicinity of the 2nd order phase transition was recognized and elaborated in detail (see 1,4].This mapping allows us to receive the above de ned exponents and as m !0 limits of the correlation length exponent and the magnetic susceptibility critical exponent of the O(m) -symmetric model.
On the other hand, if polymers of di erent species are present in the same solution the scaling behavior of the observables may be much more rich.Let us consider a solution of two di erent species of polymers in some solvent, a so called ternary solution.Depending on the temperature the system may then behave as if one or more of the inter-and intra-chain interactions vanish in the sense described above 5{9].This will lead to asymptotic scaling laws that may di er from those observed for each species alone 10].
Very interesting new systems are obtained when linking together polymers of di erent species.The most simple system of this kind is a so called block copolymer consisting of two parts of di erent species.They are of some technical importance e.g.serving as surfactants 11].For our study they give the most simple example of a polymer star consisting of chains of two di erent species (Fig. 3a) which we will call here a copolymer star.For the homogeneous polymer star the asymptotic properties are uniquely de ned by the number of its constituting chains and the dimension of space 2,12].For the number of con gurations Z f of a polymer star of f chains each consisting of N monomers one nds: Z f e Nf N f ?1 , N !1: (1.3) The exponents f , f = 1; 2; 3; : : : constitute a family of star exponents, which depend on the number of arms f in a nontrivial way.The case of linear polymer chains is included in this family with the exponent = 1 = 2 de ned in (1.2).For general numbers of arms f the star exponents f have no physical counterparts in the set of exponents describing magnetic phase transitions.Nevertheless they can be related to the scaling dimensions of composite operators of traceless symmetry in the polymer limit m !0 of the O(m) symmetric m vector model 13,14].These exponents have been calculated analytically in perturbation theory 12,13,15,16], by exact methods in two dimensions 12,17], and by Monte Carlo simulations 18{20].
It has been shown that the scaling properties of polymer networks of arbitrary but xed topology are uniquely de ned by its constituting stars 2], as long as the statistical ensemble respects some conditions on chain length distribution 13].Thus the knowledge of the set of star exponents allows to obtain the power laws corresonding to (1.3) also for any polymer network of arbitrary topology.In this article we address a somewhat more complex problem: What happens to the scaling laws if we build a polymer star or general network of chains of di erent species?In view of the above introduced ternary solutions, one may thus study sytems of polymer networks in which some of the intra and inter chain interactions vanish.For example one may descibe a copolymer star in solution consisting of say f a chains of species a and f b chains of species b (see Fig. 3b) in the situation that there are no interactions among the chains of each species alone but only between chains of di erent species a and b.This situation in turn may also be interpreted as a number f b of random walks which end at the core of a star of f a chains with the constraint that the f b random walks avoid the chains of the star.One may thus describe di usion phenomena with some complicated boundary condition and nd a tractable case of the more complex growth phenomena in a Laplacian eld 21,22].
The setup of our article is as follows.In section 2. we introduce notation and relate the polymer model to a Lagrangean eld theory.This eld theoretical formalism will be used throughout the paper.In section 3. we de ne the renormalization group procedures.We present two alternative approaches: zero mass renormalization together with "expansion see (e.g.23]) and massive renormalization at xed dimension 24].Section 4. is devoted to the study of the renormalization group ow of the ternary model and its xed points.Series for critical exponents governing the scaling behavior of copolymer stars and stars of mutually avoiding walks are obtained in section 5..In section 6. we discuss the problem of resummation of the asymptotic series arising in this context.Numerical results are presented in section 7..We close with concluding remarks and an outlook on possible applications of the theory in section 8. and give some calculational details in appendices.Some of our principal results have previously been announced in a Letter 25].

Model and Notations
Let us rst take a look at the model we use to describe polymers.In a rst discrete version we will describe a con guration of the polymer by a set of positions of segment endpoints: Con gurationfr 1 ; : : : ; r N g 2 IR d N : Its statistical weight (Boltzmann factor) with the Hamiltonian H divided by the product of Boltzmann constant k B and temperature T will be given by exp ? 1  k B T H] = expf? 1 4`2 0 N X j=1 (r j ?r j?1 ) 2 ?`d 0 The rst term describes the chain connectivity, the parameter `0 governing the mean segment length.The second term describes the excluded volume interaction forbidding two segment end points to take the same position in space.The parameter gives the strength of this interaction.The third parameter in our model is the chain length or number of segments N.
The partition sum Z will be calculated as an integral over all con gurations of the polymer divided by the system volume , thus dividing out identical con gurations just translated in space 26]: dr i exp ? 1 k B T Hfr i g]: (2.2) This will give us the `number of con gurations' of the polymer (1.2).We will do our investigations by mapping the polymer model to a renormalizable eld theory making use of well developed formalisms (see 1,4] for example).To this end we introduce a continuous version of our model as proposed by Edwards 27,28] generalizing it to describe a set of f polymer chains of varying composition possibly tied together at their end points.
The con guration of one polymer is now given by a path r a (s) in d -dimensional space IR d parametrized by a surface variable 0 s S a .The relation of the `Gaussian surface' S a of the chain a to the number of segments N a in the discrete model is S a = N`2 0 .We now allow for a symmetric matrix of excluded volume interactions u ab between chains a; b = 1; : : : ; f.
The Hamiltonian H is then given by with densities a (r) = R Sa 0 ds d (r ?r a (s)) .Now the partition sum has to be calculated using a functional integral: here the symbol D r a (s)] includes normalization such that ZfS a g = 1 for all u ab = 0. To make the exponential of -functions in (2.4) and the functional integral well-de ned in the bare theory a cuto s 0 is to be introduced such that all simultaneous integrals of any variables s and s 0 are cut o by j s ?s 0 j> s 0 .Let us note here that equation (2.4) describing a system of continuous chains may be understood as a limit of discrete self-avoiding walks, when the length of each step is decreasing while the number of steps N a is increasing keeping the mean square size of chain xed.Thus one can relate the Gaussian surface S a of each path to the notion of steps by N a = S a =s 0 : Now the continuous chain model (2.3) can be mapped onto a corresponding eld theory by a Laplace transform in the Gaussian surfaces S a to conjugate chemical potentials (\mass variables") a : where the Laplace-transformed partition function Zf f a g can be expressed as the m = 0 limit of the functional integral over vector elds a ; a = 1; : : : ; f with m components a ; = 1; : : : ; m : (2.8) The limit m = 0 in (2.6) can be understood as a certain rule to calculate the diagrams appearing in the perturbation theory expansions and can be easily checked diagrammatically.A rigorous proof based on the application of Stratonovich-Hubbard transformation to linearize terms in (2.3) is given for the multicomponent case in 10].
The one particle irreducible vertex function ? (L) (q i ) can be de ned by: ( X q i )? (L) (q i ) = Z e iqiri dr 1 : : : dr L h a1 (r 1 ) : : : aL (r L )i L 1PI : (2.9) Averaging in (2.9) is done with respect to the Lagrangean (2.7) while keeping only those contributions which correspond to one-particle irreducible graphs.
The partition function Z f fS a g of a polymer star consisting of f polymers of di erent species 1; : : : ; f constrained to have a common end point is obtained from (2.4) by introducing an appropriate product of -functions ensuring the \star-like" structure.It reads: Z f fS a g = 1 Z D r a ] expf?(2.10) The vertex part of its Laplace transformation may be de ned by: (p + X q i )? ( f) a1:::a f (p; q 1 : : : q f ) = Z e i(pr0+qiri) d d r 0 d d r 1 : : : d d r f h a1 (r 0 ) : : : a f (r 0 ) a1 (r 1 ) : : : a f (r f )i L 1PI ; (2.11) where all a 1 ; : : : a f are distinct.When only species a is present one can also de ne ?f as the m = 0 component limit of (p + X q i )? ( f) (q; p 1 : : : p f ) = Z e i(qr0+piri) d d r 0 d d r 1 : : : d d r f hN 1 ::: f a 1 (r 0 ) : : : a f (r 0 ) a 1 (r 1 ) : : : a f (r f )i L 1PI ; (2.12) where is index of the eld component and tensor N 1;::: f has traceless symmetry 14]: X N 3 ::: f = 0: (2. 13) In what follows below we will be mainly interested in the case when only two species of polymers are present, with interactions u 11 , u 22 between polymers of the same species and u 12 = u 21 between polymers of di erent species.
Thus we will present here results for the case of a ternary polymer solution.
Nevertheless the results obtained are easily generalized to the case of higher numbers of polymer species.
Let us give the expressions to the third order for those vertex functions we are going to deal with in our study.They involve the loop integrals D 2 ; I 1 ?I 8 which are given in the Appendix A together with their correspondence to the graphs of perturbation theory.The bare Gamma functions read: @ @k 2 ? (2)  (aa) = 1 ? 1 9 I 2 u 2 aa + 4  27 I 8 u 3 aa ; a = 1; 2; (2.14) ? (4)   (aaaa) = u aa ?
As a special case we may derive the vertex function ? (4)  1122 for the u 12 interaction using the relation ? ( 22)= @=@u 12 ? (4)1122 which is obvious from the perturbation theory (see 31] for instance): ? (4) 1122 = Z du 12 ? ( 22): With the same formalism we can also describe a star of f mutually avoiding walks 29].In this case all interactions on the same chain v aa vanish and only those u ab with a 6 = b remain: ? (f) MAW = ? (f) j u ab =(1? ab )u12 : (2.18) In this case each term with indices a 1 : : : a k aquires a factor f k k!.
As it is well known ultraviolet divergencies occur when the vertex functions (2.14) -(2.16) are evaluated naively 30].In the next section the eld theoretical renormalization group approach is to be applied to study this problem.

Renormalization
We apply renormalization group (RG) theory to make use of the scaling symmetry of the systems in the asymptotic limit to extract the universal content and at the same time remove divergencies which occur for the evaluation of the bare functions in this limit 23,30,31].We pass from the theory in terms of the initial bare variables to a renormalized theory.This can be achieved by a controlled rearrangement of the series for the vertex functions.Serveral asymptotically equivalent procedures serve to this purpose.Here we will use two somewhat complementary approaches: zero mass renormalization (see 23] for instance) with successive "-expansion 32] and the xed dimension RG approach 24].The rst is de ned directly for the critical point but on the other hand has to make use of the " = 4 ?d expansion in order to give results for critical exponents at physically interesting dimensions d = 2 and d = 3 32{35].The second approach renormalizes for non zero mass and leads to results for critical exponents directly in space dimensions d = 2, d = 3 36,37] but can not be performed at the critical point itself though leading to quantitative results for preasymptotic critical behaviour 38,39].Most authors tend to prefer one method and to exclude the other for non obvious reasons.The application of both approaches will enable us in particular to check the consistency of approximations and the accuracy of the results obtained.
Let us formulate the relations to obtain a renormalized theory in terms of corresponding renormalization conditions.Though being di erent in principle for the two procedures, we may formulate them simultaneously using the same expressions.Note that the polymer limit of zero component elds leads to essential simpli cation.Each eld a , mass m a and coupling u aa renormalizes as if the other elds were absent.First we introduce renormalized couplings g ab by: u aa = " Z 2 a Z aa g aa ; a = 1; 2 Here, is a scale parameter (equal to the mass at which the massive scheme is evaluated and giving the scale of external momenta in the massless scheme).The renormalization factors Z a ; Z ab are de ned as power series in the renormalized coupling which ful ll the following RG conditions: Z a (g aa ) @ @k 2 ? (2)aa (u aa (g aa )) = Z aa (g aa )? (4) aaaa (u aa (g aa )) = " g aa (3.4) Z 12 (g ab )? (4) 1122 (u ab (g ab )) = " g 12 (3.5)Evaluating these formulas perturbatively the corresponding loop integrals are evaluated in the massive approach for zero external momenta and in the massless approach for external momenta at the scale of as given in the appendix A. In the rst case the RG condition for the vertex function ? (2) reads ?(2) aa (u aa (g a a))j k 2 =0 = m 2 a ; a = 1; 2 in the second case of massles renormalization the corresponding condition reads In order to renormalize the star vertex functions we introduce renormaliza- In the same way we de ne the appropriate renormalization of the mutually avoiding walks (MAW) vertex function: The powers of absorb the engeneering dimensions of the bare vertex functions.These are given by f = f("=2 ? 1) + 4 ?": (3.10) The renormalized couplings g ab de ned by the relations (3.1),(3.2) depend on the scale parameter .By their dependence on g ab also the renormalization Z -factors depend implictly on .This dependence is expressed by the renormalization group functions de ned by the following relations: d The function a describes the pair correlation critical exponent, while the functions f1f2 and MAW f (g ab ) de ne the set of exponents for copolymer stars and stars of mutually avoiding walks.Explicit expressions for the and functions will be given in the next section together with a study of the RG ow and the xed points of the theory.

Renormalization Group Flow and the Fixed Points: "expansion and pseudo-"-expansion
Here we want to discuss the RG ow of the theory presented in section 3..In particular we want to nd appropriate representations for the xed points of the ow for both approaches used here.In a study devoted to ternary polymer solutions the RG ow has previously been calculated 10] in the frames of massless renormalization and given to third loop order in the "-expansion.
Note that for the diagonal coupling g aa the corresponding expressions are also found in the polymer limit m = 0 of the O(m)-symmetrical 4 model.
They are known in even higher orders of perturbation theory 40].To third loop order the expressions read: " gaa = ?"gaa + 1 3 (4 + 2" + 2" 2 )g 2 aa ?Here the Riemann -function with (3) 1:202 and the constant J 0:7494 occur.The index " at " is used to distinguish the -functions obtained in massless renormalization with successsive "-expansion from m obtained in massive eld theory.Similarlily, performing renormalization in the massive scheme, as described in the previous section we obtain the corresponding functions m .
In the three dimensional space of couplings g 11 ; g 22 ; g 12 the above discussed xed points are placed at the corners of a cube deformed in the g 12 direction    Let us pass on to study the expressions (4.3), (4.4) directly at xed dimension.The usual way of dealing with -functions of models containing several couplings obtained in this scheme consists in numerical solution of the system of equations for xed points.Being asymptotic, the series in the coupling constants are represented in the form of the corresponding resummed expressions res 44].However, the numerical solution of the Table 1.The xed points of ternary solutions as obtained in the " 3 expansion in the massless renormalization scheme 10].g G , g U , g S are given by (4.6), (4.7), (4.8).
G g 11 = 0 g 22 = 0 g 12 = g G U g 11 = g S g 22 = 0 g 12 = g U U 0 g 11 = 0 g 22 = g S g 12 = g U S g 11 = g S g 22 = g S g 12 = g S and look for the xed point solutions as series in .The resulting series for the xed points then can be either resummed (when the numerical value of the xed points is needed) or they can be substituted into the other expansions arising in theory.In the nal results we substitute = 1.
27 i 5 8 ? 9i 7 Looking for the stability of above described xed points one nds that only the xed point S is stable and thus in the excluded volume limit of in nitely long chains the behavior of a system of two polymer species is described by the same scaling laws as a solution of only one polymer species.Nevertheless studying polymer mixtures at di erent temperatures and also taking into account that real polymer chains are not in nitely long one in fact may study crossover phenomena in the system which are governed by the unstable xed points as well.The knowledge of the total ow allows for a description of the crossover phenomena in whole region 10].
However, for the purpose of our study we are interested only in the values of xed points and properties of the star vertex functions in these xed points.

Resummation
As is well known the series appearing in the perturbation theory of the eld theoretic RG scheme are not convergent.Already the data given in tables 3 and 4 indicate such a behavior for the exponents f1f2 , when the series is summed up without further analysis.
The growth of the coe cients of perturbation theory series may be estimated using information such as the growth of the combinatorial multiplicity of diagrams.It was proven that the series for the function of the O(m) symmetric 4 model with one coupling g has the following asymptotic behavior 46,47]: A k = ck b0 (?a) k k! 1 + O(1=k)] ; k ! 1 (6.2) the quantities a; b 0 ; c were calculated in 46,48].Such a behavior is also expected and may be proven for the critical exponents as series in terms of the coupling.These results also show the divergence of the " and pseudo " expansions used here and indicate their Borel summability 49].This procedure of resummation takes into account the asymptotic growth of the coe cients and allows to map the asymptotic series to a convergent series with the same limiting value.The function aa considered here (3.11) coinsides with the O(m) symmetric function (6.1) in the polymer limit m = 0.So its asymptotic behavior is known from this study.The asymptotic behaviour of the o diagonal function 12 was found by instanton analysis (see 50,51]) in 10].
Let us introduce the techniques for resummation of the series based on the knowlege of the asymptotic behavior.Her we make use of Pad e-Borel resummation and a resummation extended by a conformal mapping.The rst way of resummation is applicable only for alternating series, while the second is more universal.
The resummation procedures are as follows.For the asymptotic series for an exponent given as a series in the expansion parameter " (") = X j (j) " j ; (6.3) one de nes the Borel-Leroy transform B (") of the series by: B (") = X j (j) ?(j + b + 1) (") j ; (6.4) (b being the t parameter).Then the value of the initial series may be calculated from (") = Z 1 0 dtt b e ?tB ("t): (6.5) Evaluating this for the truncated series as calculated from the perturbation theory and substituting instead of B ("t) in (6.5) its analytic continuation in the form of Pad e approximant this procedure constitutes the Pad e Borel resummation.
The conformal mapping technique postulates in addition the knowledge of the constant a entering (6.2).Assuming the behavior (6.2) holds also for the expansion of (") in ", one concludes that the singularity of the transformed series B (") closest to the origin is located at the point (?1=a) and one can map the " plane onto a circle with a mapping leaving the origin invariant: w = (1 + a") 1=2 ? 1 (1 + a") 1=2 + 1 ; " = 4 a w (1 ?w) 2 : Thus one obtains an expression for B (") convergent in the whole cut plane and, as a result, the expression for the resummed function res .In order to weaken a possible singularity on w-plane the corresponding expression is multiplied by (1 ?w) and thus one more parameter is introduced.In the resummation procedure the value of a is taken from the known largeorder behavior of "-expansion series while was chosen in our calculations as a t parameter de ned by the condition of minimal di erence between resummed 2nd order and 3rd order results.The resummation procedure was seen to be quite insensitive to the parameter b introduced by the Borel-Leroy transformation (6.4).
For the resummation of the rexponents f1f2 we take into account in addition the knowledge of the combinatorial factors which multiply each contribution according to the numbers of chains f 1 and f 2 .This leads to an additional factor (f 1 + f 2 ) k for the kth order contributions.It is taken into account by multiplying the constant a by (f 1 + f 2 ).For resummation of the series at the xed points S,G and U the following values of a = a S ; a G ; a U are used 46,10]: a S = a G = 3=8 and a U = 27=64 (6.6)By analogy we use the same procedures as developed for the " expansion also for the -expansion which we assume to have the same asymptotic behavior as it is in the same way collecting contributions of the same loop order.
Let us note here that the conventional resummation of the -functions in the massive approach leads to a severe inconsistency which is the reason for us to take the pseudo-" or -expansion method.
The distinct feature of the ab -functions introduced here is that they are functions of di erent numbers of variables which leads to ambiguities in their analytical continuation via Pad e approximants or rational approximants of several variables (see 52]).Let us illustrate this on the example of the two-loop approximation, when the corresponding expressions read: vaa = ?(4?d)v aa f vaa (v aa ); a = 1; 2;  In spite of the fact that the rational approximant (6.9) holds projection properties of the initial series (6.8), i. e. putting any pair of variables fv 11 ; v 22 ; v 12 g to zero in (6.9) one gets the appropriate 1=1] Pad e approximant for the remaining variable, the \global" symmetry is now not preserved.Due to di erent analytical continuations for the Borel transforms of the series (6.7) on one hand and (6.The reason is that substituting numerical values of xed point coordinates v 11 ; v 22 into (6.10)we loose information about contributions into xed point value from di erent separate orders of the perturbation theory series.So it appears quite natural to restore this information by generalizing the pseudo-" expansion 45] to the case of several couplings as described in section 4..

Numerical Results
In this chapter we give results for the exponents G f1f2 , U f1f2 and MAW f .The exponent S f1f2 in the symmetrical xed point S describes a uniform star and coincides with f1+f2 de ned in this context 2] (see formulas (1.3),(5.5) of this article as well).Results for the symmetrical case are to be found in "-expansion in 13] and in pseudo-" expansion in 56].Even though the non-resummed results di er to great extend for higher numbers of arms resumation shows the consistency of the approaches on the numerical level.Further applying the resummation procedure based on the conformal mapping technique as described in the previous section we get the results given in tables 5 6.Table 5.Values of the copolymer star exponent G f1f2 at d = 3 obtained by "-expansion ( G " ) and by xed dimension technique ( G 3d ).Comparing the numerical values listed in the above tables it is convincing that the two approaches and the di erent resummation procedures all lead to results which lie within a bandwidth of consistency, which is broadening for larger values of number of chains.This is not surprising as we have seen in section 5. that our expansion parameters are multiplied by the number of chains.Rather it is remarkable that even for a total number of chains of Table 6.Values of the copolymer star exponent U f1f2 at d = 3 obtained by "-expansion ( U " ) and by xed dimension technique ( U 3d ). the order of 10 (see tables 5, 6) we still receive results which are comparable to each other.Also it seems remarkable that at least for low numbers of chains (f 1 + f 2 4) the non resummed -expansion seems to give results which do not di er essentially from the resummed values.However, applying the conformal mapping technique to the series in -expansion does not improve the Pad e-Borel results.

d = 2
While star polymers up to now have not found an experimental realization in two dimensions, their study is of some theoretical interest.It has been shown that the scaling dimensions of two-dimensional uniform polymer stars belong to a limiting case of the so-called conformal Kac Exact results for exponents of two dimensional systems which are described by a conformal eld theory with central charge c < 1 may be taken from the Kac table of scaling dimensions: h p;q (m) = (m + 1)p ?mq] 2 ? 1 4m(m + 1) ( where p; q are integers in the minimal block 1 p m ?1; 1 q p (7.2) The exact result for the star exponents of uniform stars in two dimensions is received in the limiting case of m = 2 (which means c = 0) for half integer values of p: x f = 2h f=2;0 = (9f 2 ?4)=48: (7.4) The scaling dimension x f is related to the exponent f by: x f = 1 2 F(d ? 2 + ) ? f (7.5)For the exponents of the star of MAW the following result was conjectured for d = 2 29]: The qualitative behavior of the exponent G f1f2 in the Gaussian xed point is shown in Fig. 5.The steps in the ` ying carpet' correspond to the di erence of the results of the two RG approaches.Note that the curvature of the surface along the diagonal in the f 1 ; f 2 plane has opposite sign to that along each of the axes.>From this curvature it is obvious that the dependence of the exponent on f 1 ; f 2 may not be described by a simple second order polynomial.The best t we could nd to our resummed data Note that the right hand side of (7.7) vanishes if f 1 or f 1 is zero according to our perturbative results.This might be a defect of the perturbation theory as a nite result may be expected in d = 2 as in equations (7.4),(7.6)evaluated for f = 0.

Conclusions and Outlook
We have extensively studied the spectrum of exponents governing the scaling properties of stars of walks taking into account the self and mutual interactions of a system of two species of polymers.This study was performed in the frames of eld theoretical RG theory using two complementary approaches: The renormalization at zero mass in conjunction with " expansion and massive renormalization at xed dimension with numerical evaluation of loop integrals.We have formulated the problem of nding the scaling exponents of stars of interacting and non interacting walks in terms of the determination of the scaling dimensions of composite eld operators of Lagrangean eld theory.On the one hand this allows for the application of well developped formalisms and methods for analysing the scaling properties.On the other hand this de nes these new families of exponents which extends previous sets in the framework of Lagrangean eld theory.Our results agree with the previous studies of special cases which were in part done only to 2nd order of the " expansion 29].We have here considered the general case of a star of two mutually avoiding sets of walks, the walks of each set either interacting or not.Also we have studied the case of a star of mutually interacting walks.All calculations were performed to third order of perturbation theory.The set of exponents are given in "-expansion (formulas (5.8) -(5.10)) and the results of the massive approach are given in terms of a pseudo-"-expansion (5.11) -(5.13)) .The latter has proven to be a most suitable tool to evaluate this massive theory containing serveral couplings.We have shown that the convetional way of direct solution even of the resummed expressions for the xed points of the theory would lead to severe problems in this case.We have evaluated the series obtained in both approaches for space dimensionality d = 2 and d = 3. Numerical values are produced by careful resummation of the asymptotic series using the results of an instanton analysis of the three coupling problem 10].For comparison we have also given the results of naive summation as well as standard Pad e Borel resummation for selected cases.
We have found remarkable consistency and stability of the results in d = 2 and d = 3 with expected growing of deviations for large number of arms of one star.The same methods were applied previously to the problem of uniform star polymers and have led to results 13,56] in good agreement with Monte Carlo (MC) simulations 18{20].We hope our present calculations might also stimulate MC studies of the copolymer star problem.
The study we performed for two dimenensions might have no direct application to the physics of real polymers but it could perhaps give some insight to the problem of mapping our theory to a two dimensional conformal eld theory.The resummed values of the exponents for stars of mutually avoiding walks are in fair agreement with an exact result previously conjectured 29].The exponents for the case of stars of two mutually avoiding sets of walks on the other hand show a dependence on the numbers of walks which may not be described by a second order polynomial as derived from the general Kac formula 57{59].This may be seen already qualitively from the fact that the curvature of the function f1f2 (see Fig. 5) of the two variables f 1 ; f 2 along each of the axes in the f 1 f 2 -plane has the opposite sign to the curvature along the diagonal f 1 = f 2 .It is this fact on the other hand, which shows that the series of exponents ( f1f2 ) f2 with f 1 xed is a good candidate for nding its application in the theory of multifractal (MF) spectra 60].The MF spectrum describing the moments of a fractal probability measure ful lls exact conditions of convexity.Deriving such a MF spectrum on the other hand from the scaling dimensions of a series of composite eld operators is only possible if the scaling dimensions show the appropriate convexity 61].This in fact is given for our case and the series of exponents may be related to the MF spectrum generated by harmonic di usion near an absorbing fractal 21].This relation and the calculation of the MF spectrum on the basis of the results presented here, is subject of a separate publication 22].

A Loop Integrals. Graphs, "-expansion, Numerical Values
This appendix is devoted to the contributions to perturbation theory, their representation in terms of Feynman graphs and corresponding loop integrals, and the evaluation of these integrals for the two RG approaches.Fig. 6 shows the Feynman graphs up to the three loop order representing the contributions to the functions ? (2), and ? (4)(we keep labeling of 42]).As it was noted in the section 2. the vertex function ? (f) of homogeneous star (2.12) can be determined as the m = 0 component limit of product of f elds (a) with traceless symmetry re ected by a tensor N a1:::a f (2.13).It is easy to see that only graphs with one N a1:::a f (a1) : : : (a f ) insertion give nonzero contribution to ? ( f) : the rest will contain some trace of N a1:::a f and thus will vanish.Relevant graphs can be obtained from the usual four-point graphs 2-U2 -12-U4 by considering each vertex in turn to be of traceless symmetry.In the three-loop approximation considered here in ? (f) there appear two more graphs which can not be produced in this manner.They are labelled as 13 and 14.In table 10 we show the correspondence between the numerical values of the loop integrals and appropriate Feynman graphs.
A diagramm with L loops is to be multiplied by S L with S = 2 d=2 1 (2 ) d ?(d=2) ; but this renormalization factor can be absorbed into rede nition of the coupling constants g ab !g ab =S.

Figure 1 .
Figure 1.Star polymer of f arms linked together at point r 0 with extremities placed at points r 1 : : : r f .

Figure 3
Figure 3. a: Block copolymer consisting of two polymer chains of di erent species (shown by solid and dashed curves) linked at their endpoints.b: Copolymer star consisting of f a arms of species a and f b arms of species b tied together at their endpoints.

(4. 8 )
Being interested in numerical values of the xed points of -functions obtained in the massive scheme (4.3), (4.4) (as well as in the other quantities of the theory) one has several alternatives in analysing the equations (4.3),(4.4).The rst possibility is to introduce the "-expansions for the loop integrals.For massive renormalization these are known for the one-and two-loop integrals (see 31]):

(6. 7 )
v12 = ?(4?d)v 12 f v12 (v 11 ; v 22 ; v 12 ) (6.8) with obvious expressions for f v11 ; f v22 ; f v12 .Now in order to obtain the analytical continuation of the Borel transformed functions one one variable f vaa (v aa ) (6.7) one can make use of the 1=1] Pad e approximant.Solving the correponding non linear equations numerically we get for the non-trivial xed point S: v 11 = v 22 = 1:1857 53].In order to apply a similar resummation technique to the function f v12 (v 11 ; v 22 ; v 12 ) (6.8) one can make use of a generalization of Pad e approximants to the case of several variables, i.e. represent the Borel transform F v12 of f v12 in the form of a rational approximant of three variables:
rst the case d = 3. Applying the Pad e-Borel resummation technique one obtains the results given in the tables 3 and 4 for the "and expansions in comparison with non-resummed results.

Figure 5 .
Figure 5. Exponent G f1f2 in the `Gaussian' xed point at d = 2 obtained in "-expansion and in xed d scheme.Steps on the \ ying carpet" correspond to the di erence of the results of the two renormalization group approaches.
21 (u 2 a1a1 + u 2 a2a2 ) Here D 21 = (4 ?d)=4.This value has been substituted into the results for the beta functions and xed points.D 21 does not enter expressions which are independent of the RG scheme, such as the resulting exponents.The integrals can be either "-expanded (see formulas (4.9) from this article for instance) or numerically calculated at arbitrary space dimensions42,43].
10] (see gure 4).Their numerical values in "-expansion are given here give the xed point values of ternary solutions in the massless and massive renormalization schemes and are the main results to be used in the subsequent calculations.

Table 4 .
Values of the copolymer star exponent f1f2 obtained in the one-( ), two-( 2 ) and three ( 3 ) loop approximation (power of corresponds to the number of loops) in 3d theory in Gaussian (G) and unsymmetrical (U) xed points for di erent values of f 1 ; f 2 .res stands for the results obtained by Pad e-Borel resummation of three-loop series.
table 57{59].They have also been calculated exactly by Couloumb gas techniques 12,17].An exact relation has also been proposed for stars of mutually avoiding walks 29].But it is still an open question if exact results for the copolymer star system may be derived in this formalism.

Table 8 .
Values of the copolymer star exponent G

Table 9 .
Values of the copolymer star exponent U