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Fran o Ferrari,1, ∗ Vakhtang G. Rostiashvili,2, † and Thomas A. Vilgis2, ‡ 1 Institute

of Physi s, University of Sz ze in,

ul. Wielkopolska 15, 70-451 Sz ze in, Poland 2 Max

Plan k Institute for Polymer Resear h,

10 A kermannweg, 55128 Mainz, Germany In this arti le we study from a non-perturbative point of view the entanglement of two dire ted polymers subje ted to repulsive intera tions given by a Dira δ−fun tion potential. An exa t formula of the so- alled se ond moment of the winding angle is derived. This result is used to provide a thorough analysis of entanglement phenomena in the lassi al system of two polymers subje ted to repulsive intera tions and related problems. No approximation is made in treating the onstraint on the winding angle and the repulsive for es. In parti ular, we investigate how repulsive for es inuen e the entanglement degree of the two-polymer system. In the limit of ideal polymers, in whi h the intera tions are swit hed o, we show that our results are in agreement with those of previous works.

I.

INTRODUCTION

The statisti al me hani s of two polymers with onstraints on their winding angle has been extensively studied in order to understand the behavior of physi al polymer systems, like for instan e biologi al ma romole ules of DNA [1℄ or liquid rystals omposed of sta ks of disk-shaped mole ules [2℄, see Refs. [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18℄. A detailed review on the subje t, together with interesting proposals of how to in lude in the treatment of topologi ally entangled polymer link invariants whi h are more sophisti ated than the winding number, an be found in [19℄. Up to now, however, despite many eorts, ∗

Ele troni address: ferrariuniv.sz ze in.pl

†

Ele troni address: rostiashmpip-mainz.mpg.de

‡

Ele troni address: vilgismpip-mainz.mpg.de

2 mainly ideal polymer hains or loops winding around ea h others have been onsidered, while the repulsive intera tions between the monomers have been treated approximatively or exploiting in a lever way s aling arguments integrated by numeri al simulations, as for instan e in [7℄. Here we on entrate ourselves on the ase of two dire ted polymers intera ting via a repulsive Dira δ−fun tion potential [20, 21℄. We are parti ularly interested in the average degree of entanglement of the system, whi h we wish to estimate by omputing the square average winding angle of the two polymers. This quantity is also alled se ond moment of the winding angle or simply se ond moment and is a spe ial example of the topologi al moments rst introdu ed in Ref. [22℄. To a hieve our goals, we develop an approa h, whi h

ombines quantum me hani al and eld theoreti al te hniques. With respe t to previous works, we are able to obtain exa t results even if the repulsive intera tions are not swit hed o. In prin iple, the average of any observable like the squared winding angle an be derived on e the partition fun tion of the system is known, but in our ase it turns out that the partition fun tion is simply too ompli ated to obtain any analyti al result. This happens essentially be ause the full δ−fun tion potential is not a entral potential, sin e it mixes both radial and angular variables. For this reason, the usual pro edure of going to polar

oordinates and then solving the dierential equation satised by the partition fun tion of the entangled polymers with the method of separation of variables [19℄, does no longer produ e simple formulas as in the situations in whi h only entral for es are present. To avoid these di ulties, one possibility is to approximate the δ−fun tion potential with some radial potential, like for instan e the hard ore potential of Ref. [7℄. However, here we shall adopt a dierent strategy, based on eld theories, whi h does not require any approximation. This strategy has been developed in [8, 12℄ (see also Ref. [19℄ for more details) to ope with ideal losed polymers whose traje tories are on atenated. Also su h systems are hara terized by a non- entral potential, whi h omes out as a onsequen e of the topologi al onstraints imposed on the traje tories. In the eld theoreti al formulation of Refs. [8, 12, 19℄ the omputation of the se ond moment is redu ed to the problem of

omputing some orrelation fun tions of a eld theory. A bonus is provided by the fa t that this omputation requires just a nite number of Feynman diagrams to be evaluated. In the present ase, due to the presen e of the δ−fun tion potential, the eld theory whi h we

3 obtain is no longer free as that of Refs. [8, 12, 19℄. Nevertheless, we will see that the theory is still linear and thus it an be exa tly solved on e its propagator is known. Lu kily, this propagator may be omputed exa tly using powerful non-perturbative te hniques developed in the ontext of quantum me hani s to deal with Hamiltonian ontaining δ−fun tion potentials, see Refs. [23, 24, 25, 26, 27, 28, 29, 30, 31, 32℄. Basi ally, starting from the Green fun tion of a parti le whose dynami s is governed by a given Hamiltonian H0 , these te h-

niques provide an algorithm to onstru t the Green fun tion of a parti le orresponding to a perturbed Hamiltonian H = H0 + Vδ , where Vδ is the δ−fun tion potential. One advantage

of these methods is that there is a long list of potentials for whi h the Green fun tions of the unperturbed Hamiltonians H0 are known. In this way, it is easy to generalize our treatment

in luding new intera tions, whi h ould be relevant in polymer physi s, like for instan e the Coulomb intera tion. The pri e to be paid is that the quantum-me hani al algorithm works when the Green fun tions are expressed as fun tions of the energy instead of the time. In

the polymer analogy, assuming that the ends of the polymers are atta hed to two planar surfa es perpendi ular to the z−axis and lo ated at the positions z = 0 and z = L, the role of time is played by the distan e L, while the energy orresponds to the hemi al potential

onjugated L. To re over the original dependen e on L, one needs to al ulate an inverse Lapla e transform of the eld propagator with respe t to the energy. In general, this is not a simple task. On e the propagator of the linear eld theory is known, the orrelation fun tions whi h enter in the expression of the se ond moment may be al ulated ontra ting the elds in all possible ways using the Wi k pres ription. At the end, we get in this way an exa t formula of the se ond moment as a fun tion of the energy, whi h, we remember, has here the meaning of the hemi al potential onjugated to the distan e L. In the L spa e, due to the problems of omputing the inverse Lapla e transform of the propagator mentioned above, only an approximated expression of the se ond moment will be given in the limit of large values of L and assuming that the strength of the δ−fun tion potential is weak enough to allow a perturbative approa h. Our results allow both a qualitative and quantitative understanding of the way in whi h the repulsive intera tions ae t the entanglement of two dire ted polymers. The orre tions introdu ed by these intera tions in the expression of the se ond moment of ideal polymers have been studied in some interesting limits. First of all, it has been examined the limit

4 of long polymer traje tories, in whi h we show that repulsive intera tion be ome parti ularly relevant. Moreover, we have investigated also the perturbative regime and the strong

oupling limit, whi h is important to re over the ex luded volume intera tions. While it is not a problem to take the strong oupling limit within our exa t treatment of the repulsive intera tions, it turns out that, in this ase, the expression of the se ond moment is parti ularly ompli ated from the analyti al point of view. For this reason, in the Con lusions we will dis uss the appli ation of a powerful perturbative method to study eld theories at strong oupling due to Kleinert [33, 34, 35℄. Finally, the onsisten y of our results with the previous ones has been he ked by studying the limit of ideal polymers. The material presented in this paper is divided as follows. In the next Se tion, the problem of omputing the se ond moment of the winding angle of two dire ted polymers intera ting via a δ−fun tion potential is briey illustrated using the path integral approa h. A onstraint on the winding angle is imposed by oupling the traje tories of the polymers with a suitable external magneti eld, following the strategy of previous works like for instan e [8, 14, 15, 19℄. In Se tion III the se ond moment is expressed in the form of a nite sum of amplitudes of a linear eld theory. These amplitudes may be omputed on e the propagator of the theory is onstru ted. In our ase, the propagator oin ides with the Green fun tion of a parti le diusing in a δ−fun tion potential. The derivation of this Green fun tion in the energy representation using non-perturbative te hniques developed in the ontext of quantum me hani s [23, 24, 25, 26, 27, 28, 29, 30, 31, 32℄ is the subje t of Se tion IV. The δ−fun tion potential is responsible of the appearan e of singularities in the propagator at short distan es, whi h have been regulated here with the introdu tion of a

ut-o. This pro edure is motivated by the fa t that in polymer physi s there is no point in

onsidering distan es whi h are smaller than the dimensions of a monomer. A omparison with the more rigorous method of renormalization is made, showing the onsisten y of the two pro edures. The propagator derived in Se tion (IV) has a parti ularly ni e form, in whi h the ontributions oming from the repulsive for es an be separated from the free part of the propagator, whi h is related to the random walk of ideal polymers. This splitting of the propagator is used in Se tion V to dis uss qualitatively and qualitatively the ee ts of the δ−fun tion intera tions on the entanglement of the system. The results of Se tions III and IV provide in prin iple all the ingredients of the se ond moment. However, the amplitudes of the linear eld theory derived in Se tion III should still be evaluated. In

5 this task one en ounters the typi al problems o

urring in the evaluation of the analyti al expressions of Feynman diagrams. In the ase of the se ond moment there are just tree diagrams, but still one has to perform ompli ated integrations over the spatial oordinates whi h are transverse to the z−axis. Even assuming that polymers are ideal, the analyti al evaluation of these integrations requires drasti approximations, see for instan e [8℄. To avoid these di ulties, we average the se ond moment with respe t to the positions of the endpoints of the two polymers. This averaged version of the se ond moment an be omputed without any approximation in the energy representation. This is done in Se tion VI. The expression of the averaged se ond moment in the L−spa e is provided instead only at the rst perturbative order in the strength of the repulsive potential and assuming additionally that the value of L is large. We give also an exa t formula of the se ond moment without performing any averaging pro edure as a fun tion of L. This formula is however expli it only up to the al ulation of the inverse Lapla e transform of the propagator derived in Se tion IV. In Se tion VII we onsider the situation in whi h the polymers are not intera ting in order to allow the omparison with previous results. Finally, the dis ussion of the obtained results and ideas for further developments are presented in the Con lusions.

II.

THE STATISTICAL MECHANICS OF TWO DIRECTED POLYMERS WITH CONSTRAINED WINDING ANGLE

Our starting point is the a tion of two dire ted polymers A and B :

A0 = where V (

r

A

−

r

B)

Z

L 0

r

d A dz c dz

!2

r

d B +c dz

!2

−V(

r

A

r

−

B)

is the potential:

V(

r

A

−

r

B)

= −v0 δ(

r

A

−

r

B)

(1)

(2)

v0 > 0

The sign of v0 has been hosen in su h a way that the intera tion asso iated to the potential

r

V ( ) is repulsive. The parameters c and L determine the average length of the traje tories of the polymers. The ends of the polymers are supposed to be xed on two surfa es perpendi ular to the z−axis and lo ated at the heights z = 0 and z = L. Both polymers have a preferred dire tion along the z dire tion. The ve tors

r

A (z)

and

r

B (z),

0 ≤ z ≤ L, measure

the polymer displa ement along the remaining two dire tions of the spa e.

6 The a tion of Eq. (1) resembles that of two quantum parti les in the ase of imaginary time z . To stress these analogies with quantum me hani s, the z−variable will be treated as a pseudo-time and renamed using from now on the letter t instead of z . In the system of the enter of mass:

r=r

A

−

R= r

r

B

A

+ 2

r

B

(3)

the a tion (1) be omes:

A0 =

Z

L 0

r!

2

d dt

c dt 2

R!

d + 2c dt

2

r

− V ( )

The motion of the enter of mass, whi h is a free motion des ribed by the oordinate

(4)

R(t),

will be ignored. We onsider the partition fun tion of the above two-polymer system with the addition of a onstraint on the entanglement of the traje tories:

Zm =

Z

r

D e−

RL 0

+V (r)] dt[ 2c ( dr dt )

(5)

δ(m − χ)

χ is the so- alled winding angle. Its expression is given by: χ= where

L

Z

0

A(r(t)) · dr(t)

A(r) is a ve tor potential with omponents: 1 x A (r) = ǫ 2π r

(6)

i

j

ij 2

In the above equation we have represented the ve tor i. e.

r = (x , x ). 1

2

(7)

i, j = 1, 2

r using artesian oordinates x , x . 1

2

Moreover, from now on, middle latin indi es i, j, . . . = 1, 2 will label the

dire tions whi h are perpendi ular to the t−axis. The denition of the partition fun tion

Zm is ompleted by the boundary onditions at t = 0 and t = L:

r(0) = r

0

r(L) = r

1

(8)

The quantity in Eq. (6) be omes a topologi al invariant if the polymer traje tories are losed. In the present ase, in whi h the traje tories are open, χ just ounts the angle with whi h one polymer winds up around the other. Thus, the partition fun tion Zm gives the formation probability of polymer paths winding up of an angle

∆θ = 2πm

(9)

7 Exploiting the Fourier representation of Dira δ−fun tions Z

δ(m − χ) =

dλ iλ(mχ) e 2π

+∞

−∞

(10)

Eq. (5) an be rewritten as follows:

Zm =

Z

+∞

−∞

where

Zλ =

Z

dλ imλ e Zλ 2π

(11)

r

(12)

D e−

RL 0

dtL

The Lagrangian L is that of a parti le immersed in the magneti potential asso iated to the ve tor eld (7):

c L= 2

r!

d dt

2

+ iλ

r A − V (r)

d · dt

(13)

The Fourier transformed partition fun tion Zλ is the grand anoni al version of the original partition fun tion Zm , in whi h the number m is allowed to take any possible value.

Zλ oin ides with the propagator Gλ (L;

S hrödinger equation:

"

r , r ), 1

0

#

∂ − H Gλ (L; ∂L

whi h satises the following pseudo-

r ,r ) = 0 1

(14)

0

H is the Hamiltonian of the system, omputed starting from the Lagrangian (13): H=

A

r

1 (∇ − iλ )2 + V ( ) 2c

(15)

Eq. (14) is ompleted by the boundary ondition at L = 0:

Gλ (0;

r , r ) = δ(r − r ) 1

0

1

(16)

0

The average degree of entanglement of the two polymers an be estimated omputing the topologi al moments of the winding angle hm2k i

fun tion is known, the hm2k i

rr

=

The quantities hm2k i

rr

2k

hm i

1, 0

rr

1, 0

1, 0

may be expressed as follows:

R +∞

1, 0

r r , k = 0, 1, 2, . . . [22℄. On e the partition

2k −∞ dm m Zm R +∞ −∞ dm Zm

=

R +∞

r r r r

R 2k +∞ dλ imλ e Gλ (L; 1 , 0 ) −∞ dm m −∞ R +∞ R +∞ dλ 2π imλ G (L; 1, 0) λ −∞ dm −∞ 2π e

depend on the boundary onditions

r ,r 0

1

(17)

and, of ourse, on the

parameters c and L. For pra ti al reasons, we will also onsider the following averaged topologi al moments: 2k

hm i =

R

d2 r0 d2 r1 dmm2k Zm R R R d2 r0 d2 r1 dmZm R

R

(18)

8

r , r of the endpoints. This is equivalent to an average over the positions of the endpoints r (t), r (t)

As Eq. (18) shows, the average is performed with respe t to the relative positions

0

1

A

B

at the instants t = 0 and t = L, be ause the oordinates of the enter of mass have been fa tored out from the partition fun tion and thus they do not play any role. The advantage of the averaged topologi al moments is that, a posteriori, it will be seen that their omputation is easier than that of the topologi al moments given in Eq. (17). From the physi al point of view, the averaged topologi al moments measure the entanglement of two polymers, whose ends at the instants t = 0 and t = L are free to move. Here we will be interested only in the se ond moment hm2 i

rr

1, 0

and in the averaged se ond

moment hm2 i, i. e. in the ase k = 1 of Eqs. (17) and (18). The se ond moment is in fa t

enough in order to estimate the formation probability of entanglement with a given winding angle and to determine how the winding angle grows with in reasing polymer lengths. In the following it will be useful to work in the so- alled energy representation, i. .e

onsidering the Lapla e transformed of the partition fun tion Gλ (L;

L:

r , r ) with respe t to 1

0

r ,r ) = (19) dLe G (L; r , r ) The new partition fun tion G (E; r , r ) des ribes the probability of two entangled polymers Gλ (E; λ

1

Z

0

1

+∞

−EL

0

λ

1

0

0

of any length subje ted to the ondition that the relative positions of the polymer end at the initial and nal instants t0 and t1 are given by the ve tors

r

0

and

r . With respe t to the 1

formulation in the L− spa e, however, the distan e t1 −t0 is no longer exa tly equal to L, but

is allowed to vary a

ording to a distribution whi h is governed by the Boltzmann-like fa tor

eEL . Thus, E plays the role of the hemi al potential onjugated to the end-to-end distan e of the polymer traje tories in the t−dire tion. It is worth to remember that, roughly speaking, small values of E orrespond to large values of L, while large values of E orrespond to small values of L. Starting from Eq. (14) and re alling the boundary onditions (16), it is easy to he k that Gλ (E;

r , r ) satises the stationary pseudo-S hrödinger equation: 1

0

[E − H] Gλ (E;

r , r ) = δ(r − r ) 1

where H is always the Hamiltonian of Eq. (15).

0

1

0

(20)

9 III.

CALCULATION OF THE SECOND MOMENT USING THE FIELD THEORETICAL FORMULATION

In this Se tion we wish to evaluate the expression of the se ond moment as a fun tion of the energy E using a eld theoreti al formulation of the polymer partition fun tion. The starting point is provided by the formula of the se ond moment in the L−spa e suitably rewritten in the following way:

r ,r ) (21) r ,r ) For onsisten y with Eq. (17), the numerator N(L; r , r ) and the denominator D(L; r , r ) hm2 i

rr

1, 0

N(L; D(L;

=

1

0

1

0

1

0

1

0

appearing in Eq. (21) must be of the form:

N(L;

r ,r ) = 1

and

D(L;

0

+∞

Z

−∞

r ,r ) = 1

Z

0

2

dm m

+∞

−∞

dm

+∞

Z

−∞

Z

+∞

−∞

r ,r )

dλ imλ e Gλ (L; 2π

dλ imλ e Gλ (L; 2π

1

0

r ,r ) 1

0

Using Eq. (19), it is now straightforward to ompute the Lapla e transform of N(L; and D(L;

r , r ): 1

(22)

(23)

r ,r ) 1

0

0

N(E;

r ,r ) = 1

0

Z

+∞

−∞

Z

2

dmm

+∞

∞

dλ imλ e Gλ (E; 2π

r ,r ) 1

0

(24)

dλ e G (E; r , r ) r ,r ) = (25) dm 2π On e the fun tions N(E; r , r ) and D(E; r , r ) are known, one an onstru t the ratio: N(E; r , r ) hm ir r (E) = (26) D(E; r , r ) D(E;

1

1

Z

0

+∞

Z

−∞

0

imλ

∞

1

2

+∞

λ

1

0

0

1, 0

1

0

1

0

whi h is nothing but the se ond moment of the winding angle expressed as a fun tion of the

hemi al potential E . We remark that the Green fun tion Gλ (E;

r , r ) is related to the Feynman propagator 1

0

of the spin− 12 Aharonov-Bohm problem in quantum me hani s. In prin iple, this Green

fun tion an be omputed exa tly starting from Eq. (20) [29℄, but its nal expression is too ompli ated for our purposes. Moreover, the method used in [29℄ to renormalize the singularities oming from the presen e of the δ−fun tion potential is valid only in a restri ted region of the domain of λ. This is in ompatible with our requirements, be ause, to derive

10 the se ond moment, one has to integrate Gλ (E;

r , r ) with respe t to λ over the whole real 1

0

line. For this reason, we prefer here to use a eld theoreti al representation of this Green fun tion. This is a hieved by noting that Gλ (E;

r , r ) oin ides with the inverse matrix 1

0

element of the operator E − H:

r , r ) = hr | E −1 H |r i

Gλ (E;

1

0

1

(27)

0

and may be expressed in a fun tional integral form in terms of repli a elds:

Gλ (E;

r , r ) = lim 1

0

n→0

r

Z

r

DΨDΨ∗ ψ1 ( 1 )ψ1∗ ( 0 )e−S(Ψ

(28)

∗ ,Ψ)

In the above equation Ψ∗ , Ψ are multiplets of repli a elds:

with a tion

S(Ψ∗ , Ψ) =

Z

(29)

Ψ∗ = (ψ1∗ , . . . , ψn∗ )

Ψ = (ψ1 , . . . , ψn )

d2 xΨ∗ ⋆ E −

A

1 (∇x − iλ )2 − v0 δ(x) Ψ 2c

(30)

The symbol ⋆ in Eq. (28) denotes summation over the repli a index. For example Ψ∗ ⋆ Ψ = Pn

σ=1

ψσ∗ ψσ . Below it will be used also the onvention Ψ∗ ⋆ Ψ = |Ψ|2 . The details of the

derivation of Eq. (28) an be found in previous publi ations on the subje t [12, 29℄ and will not be provided here. In order to pro eed, it will be onvenient to expand the a tion (30) in powers of λ: (31)

S(Ψ∗ , Ψ) = S0 (Ψ∗ , Ψ) + λS1 (Ψ∗ , Ψ) + λ2 S2 (Ψ∗ , Ψ) where we have put:

S0 (Ψ , Ψ) =

Z

1 dx |∇Ψ|2 + (E − v0 δ(x)) |Ψ|2 2c

S1 (Ψ∗ , Ψ) =

i 2c

Z

∗

2

A · [Ψ

d2 x

1 S2 (Ψ , Ψ) = 2c ∗

∗

Z

(32)

⋆ (∇Ψ) − (∇Ψ∗ ) ⋆ Ψ]

(33)

A |Ψ|

d2 x

2

(34)

2

At this point we ome ba k to the omputation of the quantities N(E;

D(E;

r ,r ) 1

0

and

r , r ) appearing in the expression of the se ond moment. Exploiting the new form of 1

0

the partition fun tion given by Eqs. (2834), together with the relation Z

+∞

−∞

dm mν eimλ = 2π(i)ν

∂ ν δ(λ) ∂λν

ν = 0, 1, . . .

(35)

11 and the fa t that Z±∞ = 0, it is possible to rewrite Eqs. (24) and (25) as follows [44℄:

N(E;

r , r ) = lim 1

0

n→0

Z

r

r

DΨ∗ DΨψ1 ( 1 )ψ1∗ ( 0 )[2S2 (Ψ∗ , Ψ) − (S1 (Ψ∗ , Ψ))2 ]e−S0 (Ψ

D(E,

r , r ) = lim 1

0

n→0

Z

r

r

DΨ∗DΨψ1 ( 1 )ψ1∗ ( 0 )e−S0 (Ψ

(36)

∗ ,Ψ)

(37)

∗ ,Ψ)

The right hand sides of Eqs. (36) and (37) represent va uum expe tation values of a eld theory governed by the a tion S0 (Ψ∗ , Ψ) of Eq. (32). In the formulation in terms of quantum operators we have:

N(E;

r , r ) = lim h0|ψ (r )ψ (r )2S (Ψ , Ψ)|0i −lim h0|ψ (r )ψ (r )(S (Ψ , Ψ)) |0i 1

0

n→0

1

1

∗ 1

D(E;

0

∗

2

n

1

n→0

r , r ) = lim h0|ψ (r )ψ (r )|0i 1

0

1

n→0

∗ 1

1

0

∗ 1

1

0

1

∗

2

n

(38) (39)

n

The orrelation fun tions have a subs ript n to remember that, a

ording to the repli a method, they should be omputed rst assuming that the number of repli as n is an arbitrary positive integer and then taking the limit for n going to zero. The above orrelators may be evaluated using standard eld theoreti al methods. One

ould be tempted to use a perturbative approa h assuming that the value of v0 appearing in the a tion S0 (Ψ∗ , Ψ) of Eqs. (36) and (37) is small, but this is not ne essary. As a matter of fa t, if it is true that S0 (Ψ∗ , Ψ) does not des ribe free elds be ause of the presen e of the

δ−fun tion potential, it is also true that it is just quadrati in the elds. As a onsequen e,

xy

one is allowed to dene a propagator G(E; , ) asso iated with this a tion. It is easy to

xy

he k that G(E; , ) satises the equation:

x

xy

xy

1 E − ∇2x − v0 δ( ) G(E; , ) = δ( , ) 2c

(40)

Using the above propagator, one an evaluate the amplitudes in Eqs. (38) and (39) exa tly by ontra ting the elds in all possible ways a

ording to the Wi k theorem. After straightforward al ulations, one nds:

r

r

lim h0|ψ1 ( 1 )ψ1∗ ( 0 )|0in = G(E;

n→0

r

r

r

1

(41)

0

r

r r) = I (r , r ) + I (r , r ) + I (r , r ) + I (r , r )

lim h0|ψ1 ( 1 )ψ1∗ ( 0 )S2 (Ψ∗ , Ψ)|0in = K( 1 ,

n→0

lim h0|ψ1 ( 1 )ψ ∗ ( 0 )(S1 (Ψ∗ , Ψ))2 |0in

n→0

r ,r )

1

1

0

2

1

0

0

3

1

0

4

1

0

(42) (43)

12 where

r r ) = 2c1 d xA (x)G(E; r , x)G(E; x, r ) (44) Z h i 1 d xd y A (x)G(E; x, r )(∇ G(E; y, x))A (y)(∇ G(E; r , y)) (45) I (r , r ) = − 2c Z i h 1 I (r , r ) = + d xd y A (x)(∇ G(E; x, r ))G(E; y, x)A (y)(∇ G(E; r , y)) (46) 2c Z h i 1 d xd y A (x)G(E; r , x)(∇ ∇ G(E; x, y))A (y)G(E; y, r ) (47) I (r , r ) = + 2c Z h i 1 d xd y A (x)G(E; r , x)(∇ G(E; x, y))A (y)(∇ G(E; y, r )) (48) I (r , r ) = − 2c

K( 1 ,

0

Z

2

2

1

1

0

2

2

1

0

2

3

1

0

2

4

1

0

2

1

2

2

2

2

2

2

2

2

0

i

i

1

i x

i x

1

i

0

i x

i

0

i x

j

j y

0

j

j y

0

j y

j

j

1

j y

1

From the physi al point of view, the above equations may be interpreted in the following

x

way. The elds Ψ( ) and Ψ∗ (x) ontain operators whi h, inside ea h repli a se tor, reate and annihilate segments of the two polymers, whose relative positions are given by the ve tor

x. The two polymer system has been proje ted in the two-dimensional plane perpendi ular to the t−axis. For this reason, there appear only the transverse oordinates x. The only remnant of the third dimension is the dependen e on the energy E . The orrelation fun tions (42) and (43) des ribe the u tuations of the two polymers immersed in the δ−fun tion potential and subje ted to the intera tions represented by the ve tor potential (7). We

re all that the origin of the latter intera tions is the presen e of the onstraint on the winding angle in the partition fun tion (5). To evaluate the orrelation fun tions (42) and (43), one needs to onsider only a nite number of Feynman diagrams, orresponding to the relevant pro esses with whi h the two polymers intera t together. The result, after the analyti al evaluation of these diagrams, is provided by Eqs. (4548). Let us note that in these equations the repulsive intera tions due to the δ−fun tion potential are hidden in the

xy

propagators G(E; , ). The Feynman diagrams related to the amplitudes of Eqs. (45) (48) are all topologi ally equivalent to the diagram of Fig. 1. The amplitude of Eq. (44) is related instead to the Feynman diagram of Fig. 2. The ve tors r1 and r0 denote the relative positions of the end points of the two polymers at the initial and nal instants, as already mentioned. The integration variables x and y appearing in Eqs. (4448) may be regarded as the ve tors whi h give the relative positions of the traje tories of the two polymers at the instants in whi h they intera t together via the external ve tor potential

A of Eq. (7).

There is no restri tion on the domain of integration of x and y, so that the omponents of these relative ve tors are allowed to take any value. This implies that the distan e between the polymer segments when the intera tion with

A o

urs an be arbitrarily large.

13 A

A

r

r1

0

FIG. 1: Feynman diagram orresponding to the amplitudes of Eqs. (45)(48). The two polymers

A and B start at a distan e |r0 | from ea h other and intera t twi e with the the external eld A. At the end, the relative position of the end points at the instant t = L is given by

r1 .

The

three-verti es appearing in the Figure are related to the intera tion des ribed by Eq. (33).

A

A

r

r1

0

FIG. 2: Feynman diagram orresponding to the amplitude of Eq. (44). The two polymers A and

B start at a distan e |r0 | from ea h other and intera t with the the external eld A. At the nal instant t = L, the relative position of the end points is given by r1 . The four-vertex appearing in this Figure is related to the intera tion des ribed by Eq. (34).

Now that the orrelation fun tions whi h are present in the expressions of N(E; r1 , r0) and D(E; r1, r0 ) given in Eqs. (38) and (39) have been evaluated, see Eqs. (4448), we may put everything together and give to the se ond moment of Eq. (26) a more expli it form: 2

hm ir1 ,r0

2K(r0 , r0 ) − 4ω=1 Iω (r0 , r0 ) = G(E; r0 , r0 ) P

(49)

In on lusion, the initial problem of omputing the se ond moment of the winding angle

hm2 i

rr

1, 0

has been redu ed to the evaluation of a nite number of integrals, whi h are given

in Eqs. (4448). Of ourse, to make these integrals really expli it, we still need to derive the

xy

propagator G(E; , ), whi h is so far the only missing ingredient. This will be done in the next Se tion.

14 IV.

GREEN FUNCTIONS IN THE CASE OF HAMILTONIANS WITH A

δ−FUNCTION

POTENTIAL

xy

Let G0 (L; , ) be the solution of the dierential equation:

xy

!

∂ − H0 G0 (L; , ) = 0 ∂L

(50)

xy

for a given Hamiltonian H0 . When L = 0, G0 (L; , ) satises the boundary ondition:

xy

x y

(51)

G0 (0; , ) = δ( − )

In the ase of a Hamiltonian H, obtained by adding to H0 a δ−fun tion potential as a perturbation:

x

x

x

(52)

H( ) = H0 ( ) − v0 δ( ) we onsider the analogous dierential problem: !

xy

(53)

x y

(54)

∂ − H G(L; , ) = 0 ∂L

xy

xy

G(0; , ) = δ( − )

xy

We wish to ompute G(L; , ) starting from the Green fun tion G0 (L; , ), whi h is sup-

xy

xy

posed to be known. It is possible to show that G(L; , ) and G0 (L; , ) are related by the integral equation [30, 31℄:

xy

xy

G(L; , ) = G0 (L; , ) − v0

Z

0

L

Z

ds

xz z

zy

d2 zG0 (L − s; , )δ( )G(s; , )

(55)

We see that in the right hand side of the above equation the presen e of the δ−fun tion

xy

xy

for es us to onsider the fun tions G0 (L; , ) and G(L; , ) evaluated at the points and/or

x=0

y = 0. Usually, at these points Green fun tions may be not well dened due to the

presen e of singularities. A on rete pro edure to remove these singularities will be indi ated later. For the moment, we go further with formal manipulations, assuming that some kind of onsistent regularization of the possible divergen es has been introdu ed. First of all, we perform the integration over d2 z in Eq. (55):

xy

xy

G(L; , ) = G0 (L; , ) − v0

Z

0

L

x

y

dsG0 (L − s; , 0)G(s; 0, )

(56)

15 The integral in ds appearing in the right hand side of Eq. (56) is a onvolution whi h an be better treated after a Lapla e transform. Thus, we transform both sides of this equation with respe t to L:

xy

xy

x

y

(57)

G(E; , ) = G0 (E; , ) − v0 G0 (E; , 0)G(E; 0, ) where

xy

Z

+∞

xy

Z

+∞

G(E; , ) = and

0

G0 (E; , ) =

0

xy

(58)

e−EL G(L; , )dL

xy

(59)

e−EL G0 (L; , )dL

xy

At this point, it is easy to extra t from Eq. (57) the expression of G(E; , ):

xy

xy

G(E; , ) = G0 (E; , ) −

x

y

G0 (E; , 0)G0 (E; 0, ) 1 + G0 (E; 0, 0) v0

(60)

The above formula may be used in order to solve Eq. (40). In this ase, H0 oin ides with the free a tion:

H0 =

xy

1 2 ∇ 2c

(61)

and the fun tion G0 (E; , ) is given by:

xy

G0 (E; , ) =

√ c K0 ( 2Ec| − |) π

x y

(62)

Here K0 (z) denotes the modied Bessel fun tion of the se ond kind of order zero. Clearly, we annot apply dire tly Eq. (60) without introdu ing a regularization. As a matter of fa t, if not treated, the naive denominator in the se ond term of the right hand side is equal to innity, i. e.

1 v0

+ G0 (E; 0, 0) = +∞. This is due to the fa t that K0 (z)

diverges logarithmi ally in the limit z → 0:

K0 (z) ∼ − log z

for z ∼ 0

(63)

A natural regularization is suggested by the fa t that, in polymer physi s, it has no sense to onsider lengths whi h are smaller than the size of the mole ules whi h ompose the polymers. Thus, it seems reasonable to regulate ultraviolet divergen es by introdu ing a

ut-o a at short distan es. The length a is omparable with the mole ular size. A

ording to this pres ription, by inserting the Green fun tion of Eq. (62) in Eq. (60), we obtain: √ √ 2 √ c c K0 ( 2Ec| |)K0 ( 2Ec| |) √ G(E; , ) ≡ K0 ( 2Ec| − |) − (64) 1 c π π 2Eca) + K ( 0 v0 π

xy

x y

x

y

16 The symbol ≡ means that the quantity in the left hand side of an equation has been repla ed in the right hand side with its regulated version. The above Green fun tion is what we need in order to evaluate expli itly the amplitudes of Eqs. (4143). The innities oming from the δ−fun tion potential should be treated with some are in order to avoid ambiguities. For this reason, we would like to ompare the naive pres ription used here to derive Eq. (64) with the more rigorous pro edure of renormalization. It is known in fa t that the renormalization of the innities oming from δ−fun tion intera tions produ es physi ally sensible results [32℄. The divergen es will be regulated introdu ing a

ut-o Λ in the momentum spa e. As a onsequen e, it will be onvenient to express the free Green fun tion of Eq. (62) in momentum spa e. To this purpose, we use the following formula:

x y

1 Z 2 eip·(x−y) dp 2 K0 (m| − |) = 2π + m2

(65) p To evaluate the Green fun tion at the singular point x = y = 0 we need to ompute the following divergent integral:

1 Z I(m) = 2π

d2 p 2 + m2

p

(66)

Using the above ut-o pres ription to eliminate the ultraviolet singularities we get, in the assumption Λ2 ≫ m2 :

I(m) ∼ log

Λ m

(67)

Now, a

ording to the spirit of renormalization, we subtra t the innities from the bare parameters of the theory. In our ase, after hoosing an arbitrary mass s ale µ, whi h gives the renormalization point, we renormalize the bare oupling onstant v0 . A tually, it will be better to all it vbare instead of v0 in order to distinguish it from the ee tive oupling

onstant v0 appearing in Eq. (64). The subtra tion of innities is performed in su h a way that the quantity:

1 vbare

− G0 (E; 0, 0) =

1 vren

c Λ2 + log 2π µ2

!

c m2 − log 2π µ2

!

(68)

be omes nite. We hoose a sort of minimal subtra tion s heme, in whi h the renormalized

oupling onstant vren is related to the bare oupling onstant vbare as follows:

1 vbare

c Λ2 + log 2π µ2

!

=

1 vren

(69)

17 Applying the last two above equations ba k to Eq. (60), we get as a result: √ √ 2 √ c c K0 ( 2Ec| |)K0 ( 2Ec| |) G(E; , ) = K0 ( 2Ec| − |) − c 2Ec 1 π π − log 2 vren 2π µ

xy

x

x y

y

(70)

Eqs. (64) and (70) are re ipro ally ompatible. In fa t, sin e a is very small, be ause it is the smallest possible length s ale in our polymer problem, one an use the following approximation (see Eq. (63)) in the denominator of the se ond term of Eq. (64):

√ 1 1 c c + K0 ( 2Eca) ∼ − log(2Eca) v0 π v0 2π

(71)

Comparing with the analogous denominator in Eq. (70), it is possible to relate a with the inverse of the mass µ:

1 (72) a2 Moreover, the ee tive oupling onstant v0 of Eq. (64) may be identied with the renorµ2 =

malized oupling onstant vren , whi h gives the strength of the repulsive intera tion (2) at distan e s ales of order a. Before on luding this Se tion, we make a small digression about the translational invarian e of the free Hamiltonian (61) and onsequently of the free Green fun tion (62). Clearly, this is not the same translational invarian e that was already present in the original a tion (1) due to the translational invarian e of the potential (2). This new invarian e is rather related to the fa t that the physi s of the two polymer system in the absen e of any intera tion does not hange if we modify the relative positions of the polymer ends at t = 0 and

t = L in a symmetri way. An example of su h transformations is the translation of both

a

ends of polymer A at the initial and nal points by a onstant ve tor :

r (0) = r r (L) = r

a (L) + a

A

A (0)

A

A

(73)

+

(74)

r

As a result of the translations (7374), the relative ve tor (t) of Eq. (3) at the instants

t = 0 and t = L hanges as follows:

r =r +a r =r +a ′ 0

0

(75)

′ 1

1

(76)

Clearly the propagator (62) is invariant under the above transformations. This kind of invarian e an be explained as follows. As far as the two polymers A and B do not intera t,

18 ea h of them may be treated as an independent system. If we translate for instan e both ends of polymer A at t = 0 and t = L in the symmetri al way shown by Eqs. (73) and (74), the number of ongurations of polymer A and onsequently the ongurational entropy of the whole system do not hange, be ause the transformation is equivalent to a translation of polymer A in the spa e. Of ourse, this invarian e disappears as soon as the two polymers start to intera t or if they are entangled together. Indeed, if one adds to the free Hamiltonian (61) a δ−fun tion potential, the propagator stops to be translational invariant as shown by the Green fun tion of Eq. (60), whi h does not depend on the dieren e

V.

x − y.

REPULSIVE FORCES AND WINDING ANGLES: QUALITATIVE AND QUANTITATIVE CONSIDERATIONS

In prin iple we have at this point all the ingredients whi h are ne essary to ompute the se ond moment of Eq. (26). In Eqs. (38) and (39), in fa t, the quantities N(E;

D(E;

r , r ) and 1

0

r , r ) are written as linear ombinations of the amplitudes of Eqs. (4143), whi h an 1

0

be expli itly evaluated using the propagator G(E, u, v) given in Eq. (64) [45℄ and the formulas of Eqs. (4448). The remaining task is to perform the integrations over the oordinates

x and y in Eqs. (4448). From the analyti al point of view, the evaluation of these integrals poses severe te hni al problems, whi h an be solved only with the help of drasti approximations. However, the di ulties be ome milder if we average the se ond moment over the endpoints of the polymers as shown in Eq. (18). In the energy representation, whi h we are using, this means that we have to onsider the following averaged version of the se ond moment in Eq. (26):

hm2 i(E) =

N(E) D(E)

(77)

where

N(E) = D(E) =

Z Z

d 2 r0 d 2 r0

Z

Z

r ,r ) d r D(E; r , r )

d2 r1 N(E; 2

1

K(E) =

r

and

Z

d 2 r0

1

0

(78)

1

0

(79)

r r ) and I (r , r ), ω = 1, . . . , 4 of

A

ordingly, we need to integrate the quantities K( 1 , Eqs. (4448) with respe t to

1

r . Putting:

0

ω

1

0

0

Z

r r)

d2 r1 K( 1 ,

0

(80)

19

Iω (E) =

Z

d 2 r0

Z

r r)

d 2 r1 Iω ( 1 ,

ω = 1, . . . , 4

0

(81)

we obtain from Eqs. (42) and (43) the following expressions of N(E) and D(E):

N(E) = 2K(E) − Z

D(E) =

4 X

(82)

Iω (E)

ω=1

d2 r0 d2 r1 G(E;

r ,r ) 1

0

(83)

It will also be onvenient to split the propagator G(E; u, v) of Eq. (64) into two ontributions:

G(E; u, v) = G0 (E; u, v) + G1 (E; u, v)

(84)

where G0 (E; u, v) is the free propagator of Eq. (62), whi h is invariant with respe t to the transformations (75) and (76), while

G1 (E; u, v) =

√ √ c λ(E)K0 ( 2Ec|u|)K0 ( 2Ec|v|) π

(85)

In the above equation we have isolated in the expression of G1 (E; u, v) the fa tor:

c λ(E) = − π

√ 1 c + K0 ( 2Eca) v0 π

−1

(86)

It is lear that the origin of the term G1 (E; u, v) in the propagator is due to presen e of the

δ−fun tion intera tion (2) in the polymer a tion (1). In fa t, if v0 = 0, this term vanishes identi ally. Thus, using the splitting of the propagator of Eq. (84), it is now possible to separate in Eqs. (4448) the ontributions to entanglement given by repulsive for es. It seems natural to expand the quantities D(E), K(E) and Iω (E) dened in Eqs. (79), (80) and (81) with respe t to λ(E) as follows:

D(E) = D (0) (E) + D (1) (E)

(87)

K(E) = K (0) (E) + K (1) (E) + K (2) (E)

(88)

Iω (E) = Iω(0) (E) + Iω(1) (E) + Iω(2) (E) + Iω(3) (E)

(89)

where the supers ript (n), with n = 0, 1, 2, 3, denotes the order in λ(E). There are no higher order terms with n ≥ 4, so the above expansions are exa t.

It is easy to show how K(E) and the Iω (E)'s depend on the pseudo-energy E . After a

res aling of the integration variables

r , r , x and y in Eqs. (78) and (79), one nds in fa t 1

0

that:

K (n) (E) = λn (E)E −2 K (n)

n = 0, 1, 2

(90)

20

Iω(n) (E) = λn (E)E −2 Iω(n)

n = 0, 1, 2, 3

(91)

where the fa tors K (n) 's and the Iω(n) 's are fun tions of the parameters a and c, but not of E or v0 . In fa t, the oupling onstant v0 appears only inside the powers of λ(E). Let us note in Eqs. (90) and (91) the presen e of the overall fa tor E −2 in Eqs. (90) and (91). Looking at Eq. (82), it is lear that the whole fun tion N(E) is hara terized by the leading s aling behavior N(E) ∼ E −2 . In the L−spa e, after an inverse Lapla e transform, this behavior

orresponds to the following s aling law, whi h is typi al of ideal polymers: N(L) ∼ L. The

powers of λ(E), appearing in the expressions of K (n) (E) and Iω(n) (E), introdu e orre tions to this leading behavior that are at most logarithmi in E . As a matter of fa t, if the

ondition 2Eca2 ≪ 1 is satised, we have that:

√ c −1 c v0 − log( 2Eca) λ(E) ∼ − π π

−1

(92)

Naively, the above seems the only logarithmi orre tion whi h is possible in the expressions of N(E) and D(E) when E is small. However, that this is not true. In fa t, in deriving Eqs. (90) and (91), we have not onsidered the divergen es whi h arise in some of the

xyr

integrations over the variables , ,

0

and

r . After regulating these divergen es with some 1

pres ription, as for instan e the ultraviolet ut-o a used in Eq. (64), we will see in Se tion VI that the naive res aling of variables exploited in order to obtain Eqs. (90) and (91) does no longer work and one should add extra logarithmi fa tors to these equations. Eqs. (90) and (91) may be also useful to study the ase of polymers in onned geometries. As a matter of fa t, for large values of E , one re overs the limit of small values of L, in whi h the region between the initial and nal height is very narrow. Looking at Eqs. (90) and (91), it is lear that the only interesting orre tions when E is large ome from the powers of λ(E). To evaluate these orre tions, one should note that the modied Bessel fun tion of the se ond kind K0 (z) goes very fast to zero for large values of z . As a onsequen e, already in the domain of parameters in whi h 2Eca2 ≥ 10, it is possible to make the very interesting approximation

c λ(E) ∼ − v0 (93) π Unfortunately, it turns out that the values of the energy for whi h the above equation is satised are not physi al, as it will be shown below. Other useful information on the inuen e of repulsive for es on the winding angle an be obtained studying the form of the fun tion G1 (E; u, v) of Eq. (85). We remember in

21 fa t that all the ee ts of the repulsive for es are on entrated in this omponent of the propagator. Supposing for example that the value of |u| is very large, i. e.:

|u| ≫ √

1 2Ec

(94)

we have the following approximate expression of G1 (E; u, v): √ √ c G1 (E; u, v) = √ λ(E)(2Ec)1/4 e− 2Ec|u| K0 ( 2Ec|v|) 2π

(95)

A relation analogous to Eq. (95) may be written also for the variable v. In pra ti e, Eq. (95) means that the repulsive intera tions do not play any parti ularly relevant role in polymer

ongurations in whi h the ends of the traje tories at some point are very distant. This is not a surprise. If the traje tories at some height t are very far from ea h other, they will have little or no han e to intera t together via the repulsive intera tions of Eq. (2), whi h are of short range. Eq. (95) gives the on rete law with whi h the ontributions of the repulsive for es are suppressed in ongurations of this kind. In parti ular, if the distan e between the traje tories is mu h greater than the hara teristi length s ale

√ lrep = 1/ 2Ec

(96)

the inuen e of the repulsive for es eases to be relevant. Of ourse, even if at some points the traje tories are very distant, polymers will always have a han e to get near enough to be able to intera t if they are su iently long. As a onsequen e, we expe t that the

hara teristi length lrep in reases with the in reasing of the lengths of the traje tories. It is easy to he k that this is exa tly the ase. To show that, let us onsider the dependen e of lrep on the polymer length. One parameter whi h determines this length is the distan e

L between the ends of the polymers along the t−axis. Indeed, a traje tory onne ting the two ends of a polymer must be very long if these ends are lo ated at very distant heights. In the energy representation, large values of L orrespond to small values of E . For example, in the limit E = 0, whi h orresponds to innite polymer lengths, we have that lrep = ∞,

onrming our intuitive expe tations. Another onrmation omes from Eq. (104) below, where a rough estimation of the behavior of lrep with respe t to the distan e L is given. The dependen e on L is not the whole story. As a matter of fa t, during their random walk in the t−dire tion from t = 0 to t = L, polymers are also allowed to wander in the remaining two dire tions. Loosely speaking, the variations in the length of the traje tories asso iated

22 to the u tuations in these transverse dire tions are taken into a

ount by the parameter

c. Smaller values of c orrespond to longer traje tories and vi e-versa, see [36℄. It is now easy to realize from Eq. (96) that the hara teristi length lrep in reases when c de reases as expe ted. Taking into a

ount all the above onsiderations, it is possible to on lude that repulsive for es give relevant ontributions to the se ond moment only in the ase of

ongurations of the system in whi h the traje tories of the two polymers are not too far from ea h other. As a matter of fa t, in the propagators appearing in the amplitudes of Eqs. (4448) all ongurations in whi h the distan e between the traje tories at some height in the t−axis is bigger than a few hara teristi lengths lrep are exponentially suppressed a

ording to Eq. (95). One may also add that this suppression be omes milder in the ase of long polymers, be ause we have seen that the hara teristi lengths lrep grows with the length of the polymers with a law whi h has been given in Eq. (104). In the rest of this Se tion we will analyze some interesting limiting ases, in whi h repulsive intera tions be ome parti ularly weak or strong. To this purpose, it would be appealing to onsider the quantity c−1 λ(E), where λ(E) has been given in Eq. (86), as an energy dependent ee tive or running oupling onstant of the repulsive intera tions. This ould be suggested by the expansions of Eqs. (8789) and by the fa t that the quantity c−1 λ(E) has the right dimension to be a oupling onstant. Indeed, we will see that there are ases in whi h the strength of λ(E) really determines the strength of the repulsive for es. However, this is not true in general, as it should be be ause λ(E) is just a parameter whi h has been fa tored out from the expression of G1 (E; u, v) and thus its meaning does not oin ide with that of a running oupling onstant. Keeping that in mind, we start to study the perturbative regime, in whi h v0 is very small. In the part of the propagator in whi h there are the

ontributions of the repulsive for es, i. e. the fun tion G1 (E; u, v), v0 is present only inside

λ(E). Expanding this quantity in powers of v0 , we obtain: √ c c −v0 + v02 K0 ( 2Eca) + . . . λ(E) ∼ π π

(97)

We see that, at the leading order in v0 , λ(E) is proportional to v0 and thus, as it ould have been expe ted, G1 (E; u, v) may be treated as a small perturbation with respe t to the free propagator G0 (E, u, v). Let's now go ba k to Eq. (93). In that equation it turns out that λ(E) has the same behavior as in the perturbative regime, even if Eq (93) has been derived in the hypothesis that 2Eca2 ≥ 10, but without supposing that v0 is small. Before

23 dwelling upon the physi al meaning of this oin iden e, let's see what is the signi an e of the ondition 2Eca2 ≥ 10. To this purpose, we make the following approximations:

L ∼ E −1

1 ∼a c

(98)

As mentioned before, it is quite reasonable to assume that the length L is proportional to the inverse of the energy E , while the se ond approximation implies that polymers are very exible. For example, in polyethylene the Kuhn length∼ 1/c is of the order of mole ular sizes. Exploiting Eq. (98), it turns out that the ondition 2Eca2 ≥ 10 is equivalent to the

ondition L ≤ a5 . This would mean that our system is squeezed in a volume whose height L is smaller than the size of a monomer. Clearly, this situation is not very physi al.

Sin e we have been able to ompute the exa t form of the propagator G(E; u, v), it is not di ult to study also the strong oupling limit v0 −→ ∞. As in the perturbative ase, the

only ae ted part of the propagator (64) is the fa tor λ(E) appearing in G1 (E; u, v). After a trivial al ulation, one nds that, in the strong oupling limit, the form of G1 (E; u, v) is given by:

−1 √ √ √ G1 (E; u, v) ∼ K0 ( 2Eca) K0 ( 2Ec|u|)K0 ( 2Ec|v|)

(99)

Assuming that polymers are very long, let us study the left hand side of the above equation. This is a ratio of modied Bessel fun tions of the se ond kind. Sin e a is a very small quantity and these fun tions have a logarithmi singularity if their argument is small, see Eq. (63), it is li it to suppose that

√ √ √ K0 ( 2Eca) > K0 ( 2Ec|u|)K0 ( 2Ec|v|)

(100)

unless |u| ∼ a and/or |v| ∼ a.

On the other side, we know from Eq. (94) that, if √ √ 1 |u|, |v| ≫ √2Ec , the produ t of modied Bessel fun tions K0 ( 2Ec|u|)K0 ( 2Ec|v|) de-

ays exponentially. In other words, in the left hand side of Eq. (99) the denominator will dominate over the numerator whenever the distan e between the polymer traje tories is not of the order of a few mole ular sizes. Thus, if v0 is large, the major ontributions to winding angle oming from the repulsive intera tions o

ur when the traje tories are very near to ea h other. This ould be expe ted from the fa t that, in the strong oupling limit, one re overs the ex luded volume intera tions. Finally, let us study the domain of the parameters E and c in whi h the ondition

2Eca2 ≪ 1

(101)

24 is veried. We will see that this domain is parti ularly interesting, be ause if ondition (101) is veried, the orre tions of the repulsive intera tions to the entropy dominated behavior of ideal polymers be ome relevant. It has been already shown that under the assumption made in Eq. (101), the parameter λ(E) is approximated as in Eq. (92). Even if it is not stri tly ne essary, we suppose here that v0 has some nite value, while polymers are so long that the following inequality is satised: √ c v0−1 ≪ − log 2Eca π

(102)

This further assumption is to eliminate the dependen e on v0 , whi h ould introdu e onfusion in the following dis ussion due to possible interferen es of ondition (101) with those of the perturbative and strong oupling regimes. In the L−spa e, Eq. (102) orresponds to the inequality e2π/cv0 ≪

L . 2a

Now G1 (E; u, v) may be approximated as follows:

G1 (E; u, v) ∼

c √

π log( 2Eca)

√ √ K0 ( 2Ec|u|)K0 ( 2Ec|v|)

(103)

As promised, the above equation does not ontain the parameter v0 . We see from the left hand side of Eq. (103) that the fun tion G1 (E; u, v) is logarithmi ally suppressed, due to √ the presen e of log( 2Eca) in the denominator. This suppression ee t is ounterbalan ed only at short distan es by the two modied Bessel fun tions of the se ond kind appearing in √ √ the numerator, whi h diverge logarithmi ally whenever 2Ec|u| = 0 and/or 2Ec|v| = 0. The total result of these opposite ee ts in the expression of the averaged se ond moment will be presented in Se tion VI after performing the expli it omputation of the amplitudes of Eqs. (4448). To on lude this Se tion, let us give some on rete values of the involved parameters. First of all, let us estimate the values of L, for whi h the two polymer system is in the regime (101). Using the approximations made in Eq. (98), we may on lude that, if the relation (101) is satised, the length L needs su h that L ≫ 2a, i. e. L is at least of the

order of hundred mole ular lengths or more: L > 100a. Moreover, it is possible to give a rough estimation of the maximum distan e of the end points, after whi h the two polymers are too far from ea h other to allow a relevant ontribution to the winding angle due to repulsive intera tions. Using Eq. (94), in fa t, it turns out that the repulsive intera tions are relevant only in the range of distan es:

|u| ≪

s

La ∼ lrep 2

(104)

25 Finally, the situation opposite to ondition (101) is not realisti , be ause it leads to the

onstraint L ≪ 2a. This would orresponds to the ase of a polymer whi h is shorter than

the size of the mole ules omposing it.

VI.

CALCULATION OF THE AVERAGED SECOND MOMENT

At this point we are ready to ompute the quantities N(E) and D(E) of Eqs. (78) and (79). We start with D(E). Using Eqs. (83), (87) and the splitting (84) of the propagator, one has at the zeroth order in λ(E): (0)

D (E) =

Z

2

d r0

Z

2

d r1 G0 (E;

r ,r ) = 1

0

Z

2

d r0

Z

√ c d2 r1 K0 ( 2Ec| π

r − r |) 1

0

(105)

After a shift of variables, the above equation gives:

D (0) (E) = S where S =

R

Z

√ c d2 r1 K0 ( 2Ec| 1 |) π

r

(106)

d2 r0 is the total surfa e of the system in the two dimensional spa e, whi h is

transverse to the t−axis. Using the identity Z

√ c 1 d2 r1 K0 ( 2Ec| 1 |) = π E

r

(107)

one nds: (108)

D (0) (E) = S/E

This expression of D (0) (E) has the following interpretation: We are performing here an average of the se ond moment with respe t to all possible initial and nal positions of the endpoints of the polymers and D(E) ounts the number of these ongurations. The

xy

omponent D (0) (E) of D(E) depends only on the free propagator G0 (E; , ), whi h is translational invariant in the sense dis ussed after Eq. (62). This invarian e explains why the number of ongurations grows proportionally to the surfa e S . The reason is that, for ea h onguration of the polymers, one an obtain other equivalently probable ongurations by the symmetri translation of their ends on the surfa e S at the initial and nal instants. Let us now apply to D (0) (E) an inverse Lapla e transform, in order to go ba k to the

L−spa e. After a simple al ulation we obtain: D (0) (L) = S

(109)

26 i. e. D (0) (L) does not depend on L. The next and last ontribution to D(E) is given by:

D (1) (E) =

Z

d2 r0 d2 r1 G1 (E;

r ,r ) = 1

0

Z

√ √ c d2 r0 d2 r1 λ(E)K0 ( 2Ec| 1 |)K0 ( 2Ec| 0 |) (110) π

r

r

Exploiting Eq. (107) to integrate out the variables r0 and r1 , we get:

π λ(E)E −2 c

D (1) (E) =

(111)

We remark that the above ontribution to D(E) vanishes in the limit v0 = 0. This ould be expe ted due to the fa t that D (1) (E) olle ts all ontributions oming from the repulsive intera tions. These intera tions break expli itly the translational invarian e of the free part of the a tion and, as a onsequen e, D (1) (E) is no longer proportional to the surfa e

S as D (0) (E). Unfortunately, it is not easy to ompute the inverse Lapla e transform of D (1) (E) without making some approximation. To this purpose, we assume that the repulsive intera tions are weak, i. e. v0 ≪ 1, and that the value of L is large. In this ase, sin e we

are in the domain of small E 's, it is possible to expand D (1) (E) up to the se ond order in

v0 as follows:

π D (E) ∼ c (1)

c −2 c E v0 − v0 π π

2

E

−2

√

log( 2Eca)

!

(112)

In order to obtain the above equation we have used both Eqs. (63) and (97). The inverse Lapla e transform of Eq. (112) gives:

D (1) (L) ∼ v0 −

√ c 2 C −1 v0 log( 2ca) + π 2

L+

c 2 v L log L 2π 0

(113)

where C ∼ 0.577215664 is the Euler onstant.

Putting Eqs. (108) and (111) together, we obtain:

D(E) = D (0) (E) + D (1) (E) = SE −1 +

π λ(E)E −2 c

(114)

This is an exa t result. An approximated expression of D(L) an be derived instead from Eqs. (109) and (113). Now we turn to the derivation of N(E). We start by omputing order by order in λ(E) the ontributions to the quantities K(E) and Iω (E) of Eqs. (88) and (89) respe tively. At the zeroth order we have for K(E):

K (0) (E) =

c 2π 2

Z

A ( x)

d2 x

2

Z

√ d2 r1 K0 ( 2Ec|

r − x|) 1

Z

√ d2 r0 K0 ( 2Ec|

r − x|) 0

(115)

27 After performing an easy integrations over the oordinates

K

(0)

r , r , one obtains: 0

1

A (x )

1 −2 Z 2 dx (E) = E 2c

The remaining integral with respe t to the

(116)

2

x oordinate is both ultraviolet and infrared

divergent and needs to be regulated. We have already seen that the singularities in the ultraviolet domain may onsistently be eliminated with the introdu tion of the small distan e ut-o a. A large distan e ut-o is instead motivated by the fa t that the size of a real system is ne essarily nite. Impli itly, we have already used this kind of infrared regularization in Eq. (106), where we have assumed that the total surfa e S of the system in the dire tions whi h are transverse to the t−axis is nite. Supposing that the shape of S is approximately a disk of radius R, so that S ∼ πR2 , we may write:

A (x) = 2π1

Z

d2 x

2

R

Z

a

dρ ρ

(117)

Substituting Eq. (117) in Eq. (116), one obtains the following expression of K (0) (E):

K

(0)

S 1 −2 E log 2 (E) = 8πc aπ

(118)

The inverse Lapla e transform of K (0) (E) gives:

K

(0)

S L log 2 (L) = 8πc aπ

(119)

We have now to ompute the quantities Iω(0) (E), with ω = 1, . . . , 4. The expressions of the Iω(0) (E)'s may be obtained from Eqs. (81) and (4548), by substituting everywhere the

xy

xy

propagator G(E; , ) with its free version G0 (E; , ). It is easy to show that: for ω = 1, . . . , 4

Iω(0) (E) = 0

(120)

This vanishing, whi h is a tually a double vanishing, is due to the fa t that ea h of the

Iω(0) (E)'s ontains an integral of a total divergen e together with an integral whi h is zero for symmetry reasons. For some values of ω , like for instan e when ω = 3, to isolate su h integrals it is ne essary to perform some integrations by parts. This is allowed be ause the

Iω(0) (E)'s are not ae ted by divergen es, ontrarily to K(E). (0)

As an example, we work out expli itly the ase of I1 (E). The rst vanishing integral is the following: Z

d2 r0 ∇jy G0 (E;

r , y) = πc 0

Z

√ d2 r0 ∇jy K0 ( 2Ec|

r − y|) 0

(121)

28 (0)

This is of ourse zero due to symmetry reasons. The se ond vanishing integral in I1 (E) is of the form:

I=

Z

2

dx

x

Z

d2 r1 Ai ( )G0 (E;

After performing the integration over

r

1

r , x)∇ G (E; y, x) i x

1

(122)

0

with the help of a shift of variables and of Eq. (107),

we have, apart from a proportionality fa tor:

I∝

x

Z

x

yx

(123)

d2 xAi ( )∇ix G0 (E; , )

x

Sin e Ai ( ) is a divergen eless ve tor potential, i. e. ∇ix Ai ( ) = 0, I an be rewritten as the

integral of a total divergen e:

√ cZ 2 i I= d x∇x Ai ( )K0 ( 2Ec| − |) π

x

y x

(124)

Clearly, the left hand side of the above equation is zero. This fa t an be also he ked passing to the Fourier representation. Exploiting Eq. (65) and the formula

x

Ai ( ) =

1 (2π)2 i

Z

p pp e

j ip·x 2

d2 ǫij

(125)

in Eq. (123), one obtains:

pp

1 Z 2 d I =− (2π)2 (

ǫij pi pj 2 + 2Ec)

p

(126)

2 (0)

(0)

Thus I = 0 be ause ǫij pi pj = 0. In an analogous way one shows that also I2 , I3

(0)

and I4

are identi ally equal to zero. We are now ready to ompute the ontributions to N(E), whi h are linear in λ(E). First of all, we treat the term K (1) (E), whi h is given by:

K (1) (E) =

1 2c

Z

d2 x

Z

d 2 r0

r

The integrations over

0

Z

and

d 2 r1

A (x) [G (E; r , x)G (E; x, r ) + G (E; r , x)G (E; x, r )] 2

1

1

0

0

0

1

1

0

(127)

r

1

may be easily performed using Eq. (107) and give as a result

a fa tor whi h is proportional to E −2 . After that, only the following integral in

x remains

to be done: Z

A (x)K (

d2 x

2

0

√

x

2Ec| |) ≡

1 (2π)2

Z

|x|≥a

d2 x

√ 1 K ( 2Ec| |) 0 | |2

x

x

(128)

Here the ultraviolet divergen e, whi h is present in the left hand side, has been regulated in the usual way with the introdu tion of the short distan es ut-o a. Infrared divergen es

29 are absent. Going to polar oordinates, the right hand side of the above equation be omes: √ Z Z √ 1 1 +∞ K0 ( 2Ecρ) 1 2 dρ d x 2 K0 ( 2Ec| |) = (129) (2π)2 |x|≥a | | 2π a ρ

x

x

Putting everything together, one arrives at the nal result:

K If the quantity

√

(1)

1 −2 E λ(E) (E) = 2πc

Z

+∞

a

√ K0 ( 2Ecρ) dρ ρ

(130)

2Eca is small, it is possible to derive the following asymptoti expression

of K (1) (E):

K (1) (E) ∼

√ 1 −2 E λ(E) log2 ( 2Eca) 4πc

(131)

To go from Eq. (130) to Eq. (131), we have used the asymptoti formula: √ Z +∞ √ K0 ( 2Ecρ) 1 dρ ∼ log2 ( 2Eca) (132) ρ 2 a √ whi h is valid for small values of 2Eca. We see from Eqs. (130) and (132) that the presen e of ultraviolet divergen es, together with the needed regularization, has modied the naive form of K (1) (E) as a fun tion of the pseudo-energy E given in Eq. (90). The √ R modi ation onsists in the appearan e of the fa tor a+∞ dρ K0 ( 2Ecρ), whi h exhibits a ρ √ square logarithmi singularity in the limit 2Eca = 0. The inverse Lapla e transformed of K (1) (E) an be derived only making some approximation. As in the ase of D (1) (E), we will work in the double limit, in whi h v0 is very small and L is very large. After a few al ulations we obtain:

K

(1)

(

v0 1 Z L (L) ∼ ds [log(L − s) + C] (log s + C) + 4π 2 4 0 1 1 L log2 (2ca2 ) + log(2ca2 ) [(C − 1)L − L log L] 4 2

(133)

At this point we have to ompute the expressions of the Iω(1) (E)'s, ω = 1, . . . , 4. It is possible to show that these ontributions vanish identi ally, i. e.:

Iω(1) (E) = 0

for ω = 1, . . . , 4

(134)

The motivations of this vanishing are similar to the motivations for whi h there are no

ontributions at the zeroth order: All terms whi h appear in the quantities Iω(1) (E) ontain at least one integral of a total divergen e or one integral, whi h is zero for dimensional

30 reasons. As in the ase of the Iω(0) (E)'s, there are some values of ω for whi h it is ne essary to perform an integration by parts in order to isolate these vanishing integrals. On e again, this is allowed be ause the Iω(1) (E)'s do not ontain divergen es. At the next order in λ(E), we have the last ontribution to K(E):

K

(2)

1 (E) = 2c

Z

2

dx

Z

After performing the integrations in

K (2) (E) = The integral in

2

d r1

r

1

Z

and

d 2 r0

r

0

A (x)G (E; r , x)G (E; x, r ) 2

1

1

A (x ) 2

2 √ K0 ( 2Ec| |)

x

x is divergent and needs a regularization. K (2) (E) ≡

π 2 λ (E)E −2 c

0

(135)

with the help of Eq. (107), Eq. (135) be omes:

Z π 2 λ (E)E −2 d2 x c

obtain the result:

1

Z

+∞

a

(136)

Going to polar oordinates, we

2 √ dρ K0 ( 2Ecρ) ρ

(137)

Also in this ase, we note that the presen e of the regularization modies the dependen e of K (2) (E) on the pseudo-energy E with respe t to the naive formula of Eq. (90). The 2 √ √ R K0 ( 2Ecρ) . In the limit 2Eca = 0, this fa tor

orre tion onsists in the fa tor a+∞ dρ ρ √ diverges as powers of log( 2Eca). To on lude the analysis of the ontribution to N(E) at the se ond order in λ(E), we show that the Iω(2) (E)'s are identi ally equal to zero. As a matter of fa t, it is easy to verify that for ω = 1, 2, 4 ea h Iω(2) (E) ontains terms of the kind:

√ B( ) = Ai ( )∇ix K0 ( 2Ec| |)

x

x

x

(138)

These terms vanish identi ally be ause of the following identity: √ √ xi ∂K0 ( 2Ec| |) 1 i ∇x K0 ( 2Ec| |) = √ ∂|x| 2Ec |x|2

x

x

(139)

Substituting Eq. (139) in Eq. (138) and using the expli it expression of the ve tor potential

x

Ai ( ) of Eq. (7), we get:

x

√ ǫji xi xj ∂K0 ( 2Ec| |) 1 B( ) = √ ∂|x| 2π 2Ec |x|4

x

(140)

Clearly, the right hand side of the above equation is zero be ause ǫji xi xj = 0. If ω = 3,

x

instead, the vanishing fun tion B( ) of Eq. (138) may be isolated in the expression of (2)

I3 (E) = 0 only after an integration by parts.

31 Finally, at the third order in λ(E) we have only the quantities Iω(3) (E)'s, sin e K(E) has at most quadrati powers of λ(E). It is easy to realize that: 4 X

(141)

Iω(3) (E) = 0

ω=1

be ause the following relations hold [46℄: (3)

(3)

(3)

(3)

(142)

I1 (E) = −I2 (E) = I3 (E) = −I4 (E)

As a onsequen e of Eq. (141), it is lear that there are no ontributions to N(E) at this order. Using Eqs. (118), (130) and (137), we arrive at the nal result for N(E): Z +∞ √ dρ 1 S 1 −2 −2 + λ(E)E E log 2 K0 ( 2Ecρ) N(E) = 4πc aπ πc ρ a Z +∞ √ 2 2π 2 dρ + K0 ( 2Ecρ) λ (E)E −2 c ρ a

(143)

We an now insert in the formula of the se ond moment of Eq. (77) the fun tions D(E) and N(E) given in Eqs. (114) and (143) respe tively. The out ome is:

E 2

hm i(E) =

−1

"

1 4πc

log

S a2 π

+ λ(E)

R +∞ a

√

dρ K0 ( ρ

2Ecρ)

πc

+λ

2

(E) 2π c

S + πc λ(E)E −1

R +∞ a

dρ ρ

2 √ K0 ( 2Ecρ)

#

(144)

In the L−spa e, the already mentioned di ulties with the omputation of the inverse Lapla e transform of D(E) and N(E) allow an analyti al result only in the double limit of weak oupling onstant v0 and of large values of L. At the rst order in v0 , the expression of hm2 i reads as follows: 2

hm i =

L 8πc

log

S a2 π

+ K (1) (L)

S + v0 L

(145)

where K (1) (L) has been given in Eq. (133). So far, we have onsidered the averaged se ond moment of Eq. (26), orresponding to the

ase in whi h the polymer ends are not xed. In the energy representation, we have seen that this version of the se ond moment an be exa tly omputed. To on lude this Se tion, we would like to show that it is possible to provide also an exa t expression of the se ond moment hm2 ir1 ,r0 in the L− spa e and with xed polymer ends up to an inverse Lapla e

transform of the propagator given in Eq. (64). The starting point is the exa t formula of

32 the se ond moment hm2 i

r r (E) of Eq. (49). All the ingredients of this formula are dened 1, 0

in Eqs. (26), (38)(39) and (41)(48). Looking at Eq. (49), it is lear that:

N(E;

r , r ) = 2K(r , r ) − 1

0

1

and

0

4 X

r r)

Iω ( 1 ,

ω=1

(146)

0

r , r ) = G(E; r , r )

D(E;

1

0

1

(147)

0

r r ) are all equal up to integrations by parts, whi h an

Let us note that the fun tions Iω ( 1 ,

0

shift the dierential operators ∇x and ∇y in Eqs. (44)(48). This fa t will be used in order

r , r ) in the L−spa e. To ompute the inverse Lapla e transforms of both N(E; r , r ) and D(E; r , r ), we use the

to simplify the expression of the inverse Lapla e transformed of N(E; 1

1

0

0

1

0

following property of the inverse Lapla e transform of the produ t of two fun tions f (E) and g(E): −1

L (f (E)g(E)) =

Z

L

0

(148)

dsf (L − s)g(s)

r r)

Applying Eq. (148) to evaluate the inverse Lapla e transforms of K(E) and of the Iω ( 1 ,

0

in Eqs. (146) and (147), we obtain after some al ulations:

r r

A x

r x xr) xr × ds ∂ ∂ G(s − s ; y, x)G(s ; r , y)A (x)A (y) D(L; r , r ) = G(L; r , r )

L 2 d2 x 2 ( ) N(L; 1 , 0 ) = dsG(L − s; 1 , )G(s; , c 0 Z Z Z L 2 − 2 d2 x d2 y G(L − s; , 1 ) 0 Zc

Z

s

′ i j x y

0

1

Z

0

1

′

′

0

i

0

(149)

j

(150)

0

The se ond term in the right hand side of Eq. (149) is the ontribution given by the fun -

r r ), ω = 1, . . . , 4, while the rst term omes from K(r , r ). Remembering the denition (21) of the se ond moment in terms of N(L; r , r ) and D(L; r , r ), we get: tions Iω ( 1 ,

0

1

1

2

hm i

rr 1,

0

= [G(L;

r , r )] 1

0

−1

"

2 c

Z

xr)

Z Z L 2 Z − 2 d2 x d2 y dsG(L − s; , c 0

1

2

A ( x)

d x Z

0

s

2

Z

L

0

ds′ ∂xi ∂yj G(s

0

dsG(L − s; ′

0

yx

1

0

r , x)G(s; x, r ) 1

0

′

− s ; , )G(s ;

r , y)A (x)A (y) 0

i

j

#

(151)

xy

If we knew how to ompute the propagator G(L; , ) starting from its Lapla e transformed (64), we ould evaluate expli itly the expression of the se ond moment in the L−spa e. Unfortunately, it is too ompli ated to perform the inverse Lapla e transform of the propa-

xy

gator G(E; , ). Due to this te hni al di ulty, Eq. (151) is only formal. Progress an be

33 made however in the limit v0 = 0, in whi h the propagator is given by the Green fun tion

xy

G0 (E; , ) of Eq. (62). This will be done in the next Se tion.

VII.

THE CASE OF IDEAL POLYMERS

In order to allow the omparison with previous results, this Se tion is dedi ated to the

ase of ideal hains in whi h v0 = 0. First of all, we dis uss the formula of the averaged se ond moment derived in the previous Se tion, Eq. (145). In the limit v0 = 0, Eq. (145) be omes:

L S hm i0 = (152) log 2 8πcS aπ The presen e of a geometri al fa tor like the surfa e S of the system in the expression

2

of hm2 i0 has been already related to the translational symmetry of Eqs. (75) and (76). Assuming that this surfa e has approximately the shape of a dis of radius R, we an put

S = πR2 as in Eq. (117). Eq. (152) predi ts that the average degree of entanglement s ales as follows with respe t to the distan e R:

hm2 i0 ∝

log R R2

(153)

The meaning of Eq. (153) is the following. We remember that the averaged se ond moment

hm2 i0 des ribes the entanglement of two losed polymers whose ends on the surfa es at

t = 0 and t = L are not xed. In this way, the polymers are allowed to move freely and it is natural to suppose that, the bigger will be the volume SL in whi h the polymers u tuate, the bigger will be also the average distan e between them. Thus, if the surfa e S in reases its area, the probability of entanglement must de rease. The exa t law of this de reasing is given by Eq. (153). On the other side, one would expe t that the probability of getting entangled is higher for long polymers than for short polymers. Eq. (152) gives a result whi h is in agreement with the above expe tation, be ause the se ond moment hm2 i0 s ales as follows with respe t

to the parameters L and c, whi h determine the polymer length:

hm2 i0 ∝

L c

(154)

In parti ular, one an show that the total length of a polymer in reases proportionally to

L and it is inversely proportional to the square root of c [36℄. A

ordingly, we see from Eq. (154) that hm2 i0 in reases proportionally to L and inversely proportional to c.

34 At this point we wish to study the se ond moment hm2 i0,

rr

of polymers with xed

1, 0

endpoints. The subs ript 0 has been added to the symbol of the se ond moment to remember that we are working in the limit v0 = 0. Sin e we are dealing with ideal polymers, we have

xy

to substitute everywhere in Eq. (151) the full propagator G(L; , ) with the free one. The result of this operation is: 2

hm i0,

rr

1, 0

= [G0 (L;

r , r )] 1

0

−1

"

xr)

2 Z 2 Z 2 ZL dx dy dsG0 (L − s; , c2 0

1

2 c

Z

Z

s

0

A (x)

d2 x

2

Z

ds′ ∂xi ∂yj G0 (s

L

0

r , x)G (s; x, r )−

dsG0 (L − s;

yx

′

1

′

− s ; , )G0 (s ;

0

0

r , y)A (x)A (y) (155) 0

i

#

j

We noti e that, as it ould be expe ted, Eq. (155) oin ides with the expression obtained in [8℄ for the se ond moment of one polymer winding up around an innitely long straight wire

r , r ) an be expli itly onstru ted upon omputing the inverse Lapla e transform of the propagator G (E; r ; r ) of Eq. (62):

lying along the z−axis. Lu kily, the propagator G0 (L;

1

0

1

0

xy

G0 (L; , ) =

0

c c 2 e 2L |x−y| 2πL

(156)

It is easy to he k that the se ond term in the right hand side of Eq. (155), whi h is

r r )'s, does not grow with in reasing

asso iated with the ontributions oming from the Iω ( 1 ,

0

values of L. As a matter of fa t, after a res aling of variables, the numerator of this term gives:

2 c2

Z

2

dx

Z

2

dy

Z

0

L

xr)

dsG0 (L − s; ,

1

Z

s

0

yx

ds′ ∂xi ∂yj G0 (s − s′ ; , )G0 (s′ ;

r , y)A (x)A (y) = 0

i

j

2 2 Z c ′ ′ 2 c c Z 2 ′Z 2 ′Z 1 1 − 2(1−t) |x′ − rL1 | t dt′ 1 ∂ e− 2(t−t ′ ) |x −y | dt d y d x e 3 ′ 4π L 1−t t − t ∂x′i ∂yj′ 0 0

1 − 2(tc′ ) |y′ − rL0 |2 e t′

(157)

In the limit L −→ ∞, the quantity in the right hand side of the above equation s ales as

AL−1 , where A is a onstant. Moreover, the propagator (156), whi h is in the denominator, s ales as L−1 . Thus, the ratio between the right hand side of Eq. (157) and the propagator (156) does not depend on L. This ompletes the proof of our statement. As a onsequen e of this statement, as far as the s aling of hm2 i0,

rr

1, 0

for large values of

L is on erned, it is possible to make the following approximation: 2

hm i0,

rr

1, 0

2 ∼ [G0 (L; c

r , r )] 1

0

−1

Z

2

A (x)

dx

2

Z

0

L

dsG0 (L − s;

r , x)G (s; x, r ) 1

0

0

(158)

35 Unfortunately, despite of the fa t that we are treating ideal polymers, the integral in d2 x appearing in the above equation is still ompli ated and requires some approximation to be evaluated analyti ally. We will apply to this purpose the strategy used in Ref. [8℄ to ompute the se ond moment of three dimensional polymers, adapting it to our two-dimensional ase. First of all, let us note that the integral in (158) is ultraviolet divergent. However, the infrared divergen es whi h appeared in the energy representation are absent. This is due

xy

to the behavior of the propagator G0 (L; , ), whi h is mu h milder at innity than the

xy

behavior of the Green fun tion G0 (E; , ). To regulate the singularities at small distan es, we pro eed as usual by introdu ing the ut-o a. After a res aling of all variables similar to that of Eq. (157), we get:

hm2 i0,

Z

√ |x′ |≥ a√ c L

d2 x′ 1 L x′2

rr

r r

2 ∼ [G0 (L; 1 , 0 )]−1 × c √c 2 1 ′ √c 2 1 ′ ds′ c 2 − 2(1−s ′ ) |x −r1 L| e− 2s′ |x −r0 L | e ′ ′ s (1 − s ) 2π

1, 0

Z

0

1

(159)

To go further, following [8℄, we assume that the relevant ontribution to the integral in d2 x′

omes from a narrow region around the singularity in x′ = 0. Thus, we may put Z √c 2 √c 2 1 1 ′ ′ d2 x′ − 2(1−s − 2(1−s ′ ) |x −r1 ′ ) |x −r0 L| L| e e √ a c ′2 ′ x |x |≥ √ L s √c 2 √c 2 L r r − 1 − 1 ∼ 2π log a e 2(1−s′ ) | 1 L | e 2(1−s′ ) | 0 L | c

(160)

After making the above rude approximation, we obtain: 2

hm i0,

rr

1, 0

c [G0 (L; ∼ πL

1 2 c 1 2 c 1 − 2(1−s ′ ) r1 L − ′ r0 L e e 2s ′ s

s

r r

1 2 c 1 2 c 1 L Z 1 ′ − 2(1−s −1 ′ ) r1 L − ′ r0 L a ds e log + e 2s 1 , 0 )] ′ c 1−s 0

(161)

In deriving the above equation we have used the simple relation now study the integral

1 s′ (1−s′ )

=

1 (1−s′ )

+ s1′ . Let us

1 2 c 1 2 c ds′ − 2(1−s ′ ) r1 L − ′ r0 L e e 2s (162) ′ 0 s The other integral in ds′ appearing in (161) an be treated in the same way after the hange

I˜ =

1

Z

of variables 1 − s′ = t. It is not to allowed to take in the right hand side of Eq. (162) the

limit L −→ ∞ be ause in this way the integral will not be onvergent due to the singularity

in s′ = 0. For this reason, we split the domain of integration as follows:

I˜ =

Z

0

u

1 2 c ds′ − 2(1−s 1 2 c ′ ) r1 L − ′ r0 L e e 2s ′ s

36

+

Z

1

u

1 2 c 1 2 c ds′ − 2(1−s ′ ) r1 L − ′ r0 L e 2s e ′ s

(163)

where 0 < u < 1. Clearly, the se ond integral onverges after performing the limit L −→ ∞ in the integrand and gives:

Z

1 ds′ = log s′ u

1

u

(164)

The rst integral instead diverges logarithmi ally with growing values of L. However, now 1 2 c − 2(1−s ′ ) r1 L

it is possible to expand the exponential e

in powers of its argument, be ause the

singularity in s′ = 1 lies outside the interval [0, u]. Keeping only the leading order term with respe t to L, we get:

2

!

ur c I˜ ∼ −Ei − 0 − log u 2L

(165)

where Ei(z) is the exponential-integral fun tion. When L is large, this fun tion may be approximated as follows: Ei(z) ∼ log(−z) and, as a onsequen e:

r2 c I˜ ∼ − log 0 2L

!

(166)

The se ond integral whi h we have left in Eq. (161) gives the same result. Putting everything together in the expression of the se ond moment of Eq. (161), we obtain the nal result: 2

hm i0,

rr

1, 0

s

L r 2 r 2 c2 a log 1 02 ∼ −2 log c 4L

!

∼ 2 (log L)2

(167)

This is exa tly the behavior of the se ond moment derived in Ref. [8℄.

VIII.

CONCLUSIONS

In this arti le we have studied the entanglement of two dire ted polymers from a nonperturbative point of view. Our formulas of the se ond moment, a quantity whi h measures the degree of entanglement of the two polymers, take into a

ount the repulsive for es a ting on the segments of the polymers and are exa t. The averaged se ond moment dened in Eq. (18), a version of the se ond moment orresponding to the situation in whi h the end points of the polymers are free to move, has been omputed in Eq. (144) as a fun tion of the

hemi al potential E onjugated to the distan e L between the end points in the t−dire tion. The ase of free ends is relevant in the treatment of nemati polymers and polymers in a nemati solvent [21℄. Let us note that also the expression of the se ond moment without any

37 averaging and in the L spa e an be omputed. This has been done in Eq. (151). However, this equation is expli it only up to the inverse Lapla e transform of the propagator (64), whi h is too hard to be obtained in losed form. Eq. (144) shows that the averaged se ond moment is of the form hm2 i(E) = E −1 f (E).

The overall fa tor E −1 oin ides with the s aling power law of two ideal polymers. The or-

re tion f (E) to this fundamental behavior due to the repulsive intera tions is a ompli ated fun tion of E , whose analysis would require numeri al methods. Nevertheless, it is possible to identify a dominan e of the repulsive intera tions in the domain of parameters in whi h √ the ondition 2Eca ∼ 0 is satised. This orresponds roughly speaking to the situation in

whi h polymers are very long. In this region, the s aling laws with respe t to the energy E of the numerator and denominator appearing in the right hand side of Eq. (144) are orre ted √ by fa tors whi h are logarithmi powers of log( 2Eca), see for instan e Eq. (132). One advantage of our approa h is that it is easy to separate within the expression of the se ond moment the ontributions of purely entropi origin whi h are typi al of free polymers from the ontributions oming from the presen e of the δ−fun tion potential in the polymer a tion. This is essentially due to the splitting (95) of the propagator G(E; u, v) appearing in the amplitudes (4448). The omponent G0 (E; u, v) of the propagator oin ides with the propagator of ideal polymers, while the omponent G1 (E; u, v) takes into a

ount the ee ts of the intera tions. Thanks to the splitting (95), it has been possible to study the way in whi h the repulsive for es ae t the average degree of entanglement of the two polymers. This has been done in Se tion V. Our results are in agreement with the intuition. The pre ise law with whi h the ee ts of the repulsive for es on the entanglement de rease when the distan e between the traje tories in reases is given by Eq. (95). In Se tion V it has been dis ussed also the strong oupling limit, whi h should be taken to re over the limit of ex luded volume intera tions. In our exa t approa h, it is not di ult to onsider the

ase in whi h the oupling onstant v0 is large. For instan e, the omponent G1 (E; u, v) of the propagator, whi h is responsible of the ee ts due to the repulsive intera tions, has been given in the strong oupling limit in Eq. (99). Studying the form of this omponent assuming that polymers are very long, it has been argued that, at strong oupling, the major

ontributions to the winding angle oming from the repulsive intera tions o

ur when the traje tories are very near to ea h other. Many other qualitative and quantitative hara teristi s of the behavior of the two polymer system under onsideration have been presented in

38 Se tion V. The ase of ideal polymers, in whi h v0 = 0, has been dis ussed at the end of Se tion VII in order to make omparison with previous works. The s aling of the averaged se ond moment for large values of L obtained in Eq. (154) is in agreement with the results of [7℄, if one takes into a

ount the fa t that, after the averaging pro edure of Eq. (18) and the infrared regularization of Eqs. (106) and (117). one is ee tively treating a system of polymers

onned in a ylinder of nite volume SL. In Se tion VII we have evaluated the se ond moment, always of two ideal polymers, using the approa h of Ref. [8℄. The out ome of this

al ulation, namely the s aling behavior of hm2 i0,

rr

1, 0

at the leading order in L, is reported

in Eq. (167). This result is in agreement with the square logarithmi behavior obtained in [8℄, but not with the logarithmi behavior predi ted in [7℄. However, this dis repan y an be expe ted due to the fa t that, in Se tion VII, we have assumed, following Ref. [8℄, that the most relevant ontribution to the se ond moment oming from the integral in Eq. (159) is on entrated in a narrow region near the singularity in x′ = 0. This lashes with the

assumptions of Ref. [7℄, in whi h instead it is argued that the main in rease in the winding angle does not o

ur when the polymer traje tories are near, but rather when they are far one from the other. Finally, there is also an apparent dis repan y between the linear s aling with respe t to L of the averaged se ond moment hm2 i0 and the square logarithmi s aling of the se ond moment hm2 i0,

r r. 1, 0

This disagreement is explained by the fa t that, in the

rst ase, the ends of the polymers are free to u tuate, while in the se ond ase they are

xed. It is therefore li it to expe t that two polymers with free ends are more likely to entangle than two polymers whose ends are onstrained. Con luding, we would like to dis uss possible further developments of this work, together with some problems whi h are still left open. First of all, the number of entangling polymers has been limited to two. To go beyond this restri tion, one should explore the possibility of

x

repla ing the external ve tor potential Ai ( ) of Eq. (7) with ChernSimons elds. Abelian Chern-Simons eld theories have been already su

essfully applied in order to impose topologi al onstraints to the traje tories of an arbitrary number of losed polymer rings in [37℄. We hope to extend those results also to the ase of dire ted polymers in a forth oming publi ation. Of ourse, if the polymer traje tories are open, the onstraints among them are no longer of topologi al nature as in [37℄, so that the appli ation of Chern-Simons eld theory to dire ted polymers should be onsidered with some are.

39 We have also not made any attempt to introdu e in the treatment of polymer entanglement more sophisti ated onstraints than those whi h an be imposed with the help of the winding angle. This is in ee t still an unsolved problem, despite the fa t that two powerful and strategies have been proposed for its solution [38, 39, 40℄. In the rst approa h, pioneered independently by Kleinert, Kholodenko and one of the authors [19, 38, 39℄. the

onstraints are expressed via the Wilson loop amplitudes of non-abelian Chern-Simons eld theories. Some progresses toward a on rete realization of this program in polymer physi s have been made in Refs. [41, 42℄. In the se ond approa h, developed by Ne haev and oworkers, see [40℄ and referen es therein. polymer traje tories are mapped on a omplex plane with pun tures. The link invariants ne essary to impose the onstraints are then onstru ted using the properties of onformal maps. Another possible development is the treatment of attra tive intera tions, in whi h the strength v0 in Eq. (2) takes negative values. In this ase, the δ−fun tion potential admits a bound state [32℄ and the propagator of Eq. (64) develops a singularity, in whi h λ(E) = ∞,

at the energy orresponding to this bound state. It would be extremely interesting to investigate how these fa ts ae t the polymers' entanglement. Another issue whi h deserves attention is that of hairpin turns. Hairpins are important in nemati solvents [21℄ and an be in luded with the help of eld theories [43℄. We note also that in our formalism it is also possible to study the entanglement of polymers in onned geometries. For example,

values of E whi h are near to a−1 (E ≤ a−1 ) orrespond roughly speaking to the situation in whi h polymers u tuate in a quasi two-dimensional environment, in whi h the height in

the t−dire tion is of the order of a few mole ular sizes. Finally, an open problem, whi h has not be dis ussed here be ause we were mainly

r ,r ) of Eq. (19). As anti ipated in the Introdu tion, it is not an easy task to ompute G (E; r , r )

interested in the se ond moment. is the derivation of the full partition fun tion Gλ (E; λ

1

0

1

0

be ause the repulsive potential of Eq. (2) is not entral. We note however that the expression of Gλ (E;

r , r ) oin ides with the Green fun tion of a spin 1/2 Aharonov-Bohm problem in 1

0

the imaginary time formulation of quantum me hani s. This Green fun tion has been already derived in [29℄ using sophisti ated te hniques developed in Refs. [27, 32℄, whi h bypass all the

di ulties of dealing with a non- entral potential. Thus, in prin iple, the expression of the partition fun tion Gλ (E;

r , r ) is known. Unfortunately, some of the onsisten y onditions 1

0

imposed on the parameters in Ref. [29℄ seem to be in ompatible with the requirements of

40 our physi al problem, as noted in Se tion III. For these reasons, the omputation of the full partition fun tion Gλ (E;

r , r ) is still a problem whi h needs further investigations. Lu kily, 1

0

the knowledge of the partition fun tion is not ne essary if one is interested to study the ex luded volume intera tions, whi h arise in the strong oupling limit. In fa t, in this ase it is possible to apply a powerful method due to Kleinert [33, 34, 35℄. This method turns the weak oupling expansion into a strong oupling expansion whi h is onvergent for large values of v0 and is able to a

ommodate also the anomalous dimensions of quantum eld theories. The onvergen e of this strong oupling expansion is mostly very fast, so that only a few oe ients of the weak oupling expansion must be known, see Refs. [19, 35℄ for more details. These oe ients an be easily omputed starting from the well known partition fun tion of the Aharonov-Bohm problem without the insertion of the δ−fun tion potential [19℄ and treating this potential as a small perturbation assuming that the value of v0 is small. The appli ation of Kleinert's method in order to omplete the brief analysis of the strong oupling limit made in this paper is work in progress.

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[44℄ For details, see Ref. [12℄, where a similar al ulation has been done in the ase of losed polymers. [45℄ Throughout this Se tion we will use a notation in whi h the ve tors

x, y

appearing in the

denition of the propagator of Eq. (64) are repla ed by the ve tors u, v. This is to avoid

onfusions with the notation of Eqs. (4448), where the same pair of ve tors

x, y

has been

exploited to denote dummy integration variables. (3)

[46℄ Let us stress the fa t that ea h of the Iω (E)'s is separately equal to zero, be ause these

43 quantities ontain terms of the kind given in Eq. (138).