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Wilson-Polchinski exact renormalization group equation for O(N ) systems: Leading and next-to-leading orders in the derivative expansion.?

arXiv:hep-th/0501087v2 24 Jan 2005

C. Bervillier

Service de physique th?orique, CEA/DSM/SPhT-CNRS/SPM/URA 2306 e CEA/Saclay, F-91191 Gif-sur-Yvette C?dex, France e and Laboratoire de Math?matiques et Physique Th?orique, CNRS/UMR 6083, Universit? e e e de Tours, Parc de Grandmont, 37200 Tours, France Abstract. With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson-Polchinski equation in the case of the N -vector model with the symmetry O (N ). As a test, the critical exponents η and ν as well as the subcritical exponent ω (and higher ones) are estimated in three dimensions for values of N ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of other O (N ) exact RG equations at second order. The possibility of varying N allows to size up the derivative expansion method. The values obtained from the resummation of high orders of perturbative ?eld theory are used as standards to illustrate the eventual convergence in each case. A peculiar attention is drawn on the preservation (or not) of the reparametrisation invariance.

PACS numbers: 05.10.Cc, 11.10.Gh, 64.60.Ak

Submitted to: J. Phys.: Condens. Matter

E-mail: bervil@spht.saclay.cea.fr

? Dedicated to Lothar Sch¨fer on the occasion of his 60th birthday a 1. Introduction The renormalization group (RG) theory is suitable to the study of many modern physical problems. Generically, every situation where the scale of typical physical interest belongs to a (wide) range of correlated or coupled scales may be (must be?) treated by RG techniques. Critical phenomena, which are characterized by one (or several) diverging correlation length(s), provide “the” didactic example [1]. Quantum ?eld theory, with its strongly correlated quantum ?uctuations, is not less famous since it has given rise to the early stages of the RG theory [2].

Wilson-Polchinski exact RG equation

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Thanks to a fortunate success (essentially due to an impressive diagrammatic calculation [3]) in estimating the critical behavior of some systems [4, 5], the perturbative framework has pushed into the background the undoubtedly nonperturbative character [6, 7, 8] of the RG theory. As a consequence, there has been relatively little interest in the development of nonperturbative RG techniques [9]. In particular the formulation of the RG theory via an in?nitesimal change of the scale of reference (running scale), designated by the generic expression “exact renormalization group equation”[1] although known since 1971 [10], has actually been actively considered only since the beginning of the nineties [9]. Because the variety of systems to which the exact RG formulation could be applied is large [9, 11] (see also [12]) and also because the perturbative framework is generally not well adapted to such studies [8, 13, 14], it is worthwhile making every endeavour to master, if possible, the exact RG framework. The exact RG equation is an integro-di?erential equation the study of which calls for approximations and/or truncations. Among the possible approximations, those based on expansions in powers of a small parameter such as ? = du ? d or 1/N (where the upper-dimension du = 4 for the N-vector model) are perturbative in essence. They, however, present the advantage of allowing analytic calculations but are attached to the smallness of quantities that are actually not small in the cases of physical interest. In some cases the perturbative framework may fail [14, 8]. The derivative expansion [15], of present interest here, is an expansion in powers of the derivative of the ?eld. It is not associated to a small parameter though it is expected to be rather adapted to the study of phenomena at small momenta ? (large distances) like critical phenomena for instance. The interest of the derivative expansion is that the physical parameters (like d and N) may take on arbitrary values. Hence, in the range of validity of the expansion (thus presumed to be in the large distance regime), the approach is actually nonperturbative. The drawback is the necessary recourse to numerical techniques [for studying coupled nonlinear ordinary di?erential equations (ODE)] that are not always well controlled. Consequently very few orders of the derivative expansion have really been explicitly considered. Many studies have been e?ectuated in the local potential approximation (LPA), i.e. at the leading order [O (? 0 )] of the derivative expansion [9, 11, 16]. Also, estimates of the critical exponents for Ising-like models (N = 1) have been obtained several times from full studies of the next-to-leading order § [i.e. O (? 2 )] [15, 21, 22, 23, 24, 13], and even from a full study of the third order [O (? 4 )] [25]. On the contrary, only two full

? A recent interesting attempt to adapt the derivative expansion to phenomena that include e?ects at larger momenta is made in reference [17]. § In incomplete studies some contributions to a given order of the derivative expansion are neglected. For example in reference [18], despite an estimation of the critical exponent η (which is exactly equal to zero at order ? 0 ), the order ? 2 has not been completely considered because the evolution equation of the wave function renormalization has been neglected. In reference [19], O (N ) systems are incompletely studied up to O ? 2 because one di?erential equation has been discarded. Also, the study of reference [20] at order ? 4 , though interesting, is incomplete since only three di?erential equations, among ?ve constituting actually the order ? 4 , have been treated.

Wilson-Polchinski exact RG equation

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studies have been e?ectuated up to O (? 2 ) for the O (N) vector model [26, 27]. Yet, this model provides us with the opportunity of varying N and of comparing the results with the best estimates of critical exponents obtained from six [28, 29] or seven [30] orders of the perturbative framework in a wide range of values of N. Interesting informations on the possible convergence of the derivative expansion are then reachable. In the present study, I consider the O (N) Wilson-Polchinski exact RG equation expanded up to order O (? 2 ) in the derivative expansion and I calculate the critical exponents. Beyond the speci?c calculation (which was lacking) the real aim is to try to clarify (and also to evaluate) the present status of the derivative expansion. Still, I must explain why I consider the Wilson-Polchinski equation. Indeed, there are several di?erent approaches and treatments of the exact RG equation so that it is not easy to really estimate the validity of the choice of the equation or/and of the calculations available in the litterature. Let me try to brie?y summarize the situation and to justify my choice. There are two families of exact RG equations (for a review see [9]). The ?rst family expresses the RG ?ow (ΛdSΛ /dΛ) of the microscopic action SΛ [φ] (hamiltonian) associated to a running momentum scale Λ which, in the circumstances, is a running ultra-violet cuto?. The second family expresses the RG ?ow of ΓΛ [M], the Legendre transform of SΛ [φ], in that case the running scale Λ e?ectively appears as an infra-red momentum cuto?. There is no fundamental di?erence between the two families since the object of the RG is the same in the two cases: accounting for all the correlated scales Λ ranging from 0 to ∞ (at criticality); only the physical meaning of the ?eld variable at hand has been changed: φ is related to a microscopic description (like a spin of the Ising model) while M is thought of as a macroscopic variable (like the magnetization). If one wants to calculate an equation of state or some correlation functions or some universal critical amplitude ratios, the second family of equations is better adapted. But if one only wants to estimate critical exponents (for example to illustrate the convergence of the derivative expansion), then considering the ?rst family is surely more e?cient. Actually, the set of ODE generated in the derivative expansion is much simpler when considered with the ?rst family than with the second . The ?rst family is indeed better adapted to the calculation of critical exponents for the same reason as in the ?eld theoretical approach to critical phenomena [4], the critical exponents are de?ned from renormalization functions that are introduced within the microscopic action SΛ [φ]. The Wilson [1] and Polchinski [33] exact RG equations belong to the ?rst family, they only di?er by the way the smooth cuto? function has been de?ned (a speci?c function for Wilson, an arbitrary one for Polchinski). Of course, the two equations are physically equivalent, but due to a misunderstanding in the introduction of the critical exponent

In the study of reference [25] of the Ising case up to O ? 4 , the consideration of an exact RG equation of the second family yields a set of ODE the writing of which requires 20 pages [31] while, even at order ? 6 , the ?rst family yields a set of equations that holds on a half of page [32]. The complexity of dealing with the second family is also well illustrated in the appendix of reference [26].

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η in the exact RG equation as formulated by Polchinski, it is only recently that the equivalence has been clearly established [34, 13]. If the coupled set of ODE generated by the derivative expansion is di?erent in the two families of exact RG equations, the treatments of the di?erential equations encountered in the litterature di?er also. For convenience, let me classify the studies in two groups according to whether the authors have adopted the conventional approach (de?ned below) or not. The conventional approach is characterized as follows: (i) the set of ODE is numerically studied as such (e.g., without considering any artefact such as an expansion in powers of the ?eld). (ii) the critical exponent η is introduced in a conventional way as de?ned in references [1, 35, 13]. (iii) the critical exponents are estimated from a set of eigenvalue equations linearized about a ?xed point solution of the ?ow equation. (iv) the reparametrization invariance is explicitly accounted for. To limit myself to the studies mentioned above that consider O (N) systems via equations of the second family developped up to O(? 2 ) [26, 27]: the study of reference [26] follows the conventional approach while that of reference [27] does not. In fact, in this latter work, except the ?rst point above, none of the other points is satis?ed. This is particularly important relatively to point (iv) because the reparametrization invariance induces a line of equivalent ?xed points along with η is constant [36]. In the case where the invariance is broken (it is generally the case within the derivative expansion, except in reference [26]) then the ?xed points along the line are no longer equivalents and the e?ective η (when introduced conventionally) varies with the global normalisation of the ?eld φ. Nevertheless, even if the invariance is broken, one expects that a vestige of this invariance can still be observed [36, 37, 15] via an extremum of η on varying the global normalization of the ?eld φ. The absence of explicit consideration of the reparametrization invariance in reference [27] is intringuing inasmuch as the estimation of the critical exponents is excellent (see section 3.3). By considering the Wilson-Polchinski RG equation for O(N) systems, my ?rst aim is to illustrate the conventional treatment as described above. Additionnal aims follow: (i) Morris and Turner [26] have imposed the reparametrization invariance by choosing a speci?c cuto? function and the resulting estimates of the critical exponents are not very good, especially for ω. Does the Wilson-Polchinski RG equation also produce such bad results at order O (? 2 )?. (ii) Is the Wilson-Polchinski RG equation able to produce estimates of critical exponents comparable to those obtained by Gersdor? and Wetterich [27]? (iii) The O (N) exact RG equation expanded up to order ? 2 involves three coupled ODE. Compared to the Ising case N = 1, it is somewhat an intermediate state between

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the order ? 2 which involves two coupled equations and the order ? 4 which involves ?ve coupled equations [25, 20]. (iv) Does one observe some signs of convergence of the derivative expansion already at order ? 2 ? Relative to this latter point, it is worth mentioning here the work of Litim [38] whose aim, though very interesting, di?ers from that of the present work. Litim focuses its attention almost exclusively on the second family of the exact RG equations and especially on the arbitrariness introduced by the regularization process (cut-o? procedure) but he does not account for the reparametrization invariance. From general arguments (independent from the derivative expansion), he provides us with a criterium for choosing the regulator which should optimize the convergence of the derivative expansion already at a very low order. The considerations are surely useful especially for studying sophisticated systems, such as gauge ?eld theory for example, for which already the leading order of the derivative expansion is di?cult to implement. However, Litim has not actually studied the convergence properties of the derivative expansion in itself but, in fact, has implicitly assumed that it converges (at least, the fact that the expansion could yield only asymptotic series has been excluded). Furthermore, Litim’s criterium of choice does not apply to the Wilson-Polchinski RG equation. In particular, at leading order of the derivative expansion, his optimization provides the same estimates of critical exponents as those obtained with the Wilson-Polchinski RG equation (see section 3.2) which, at this order, does not display any dependence on the regularization process. 2. Derivative expansion up to O(? 2 ) 2.1. Flow equations According to reference [13], after subtraction of the high temperature ?xed point N 1 ? 2 ?1 φa φa from the action, the Wilson-Polchinski exact RG equation a=1 q P ψ q ?q 2 satis?ed by an O (N)-symmetric action S [φa ] (with a = 1, · · · , N) reads as follows: ˙ S=

N

?

a=1

ψ′ δS ? φa dφ + 2q 2 + q · ?q q ψ δφa q q δS δS δ2S ? a a a δφa δφq ?q δφq δφ?q (1)

+

q

? ? ?P ? q2P ′ ψ2

˙ in which S stands for dS/dt = ?ΛdS/dΛ (hence exp (?t) = Λ/Λ0 in which Λ0 is some initial momentum scale of reference [13]), q is a dimensionless d-vector (d is the 2 ? dimension of the euclidean space and q = {qi , i = 1, · · · , d}) , q 2 = d qi , P (q 2 ) is i=1 ? a dimensionless cuto? function that decreases rapidly when q → ∞ with P (0) = 1 , ψ (q 2 ) is an arbitrary function (except the normalization ψ (0) = 1) introduced to test ? the reparametrization invariance, ? = 1 ? η/2 and dφ = d/2 + ?. A prime denotes a 2 ′ 2 ?′ ? derivative with respect to q : ψ = dψ/dq , P = dP /dq 2 and q·?q f (q) = d qi ?f /?qi . i=1

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The expansion up to order ? 2 consists in projecting equation (1) onto actions of the form: S[φ] = with ρ= 1 N α α 1 2 φ φ = φ 2 α=1 2 dd x V (ρ, t)+Z(ρ, t)(? φ)2 +Y (ρ, t)(φ? φ)2

Then the ?ow equations for V , Z and Y read as follows: ˙ V = I0 (NV ′ +2ρV ′′ ) + dV ? (d + 2?) ρV ′ ? 2?ρV ′2 + 2I1 (NZ + 2ρY )

′ ˙ Z = I0 (NZ ′ +2ρZ ′′ + 2Y ) ? 2 (? + 1) Z? (d + 2?) ρZ ′ ? 2ψ0 V ′ ′ ?′ ? 4? (V ′ Z + ρV ′ Z ′ ) ? (? ? 1) P0 + 2?ψ0 V ′2 ′ ˙ Y = I0 (NY ′ +2ρY ′′ +4Y ′ ) ? (d + 2 + 4?)Y ? (d + 2?) ρY ′ ? 2ψ0 V ′′

? 4? (V ′′ Z + 2V ′ Y + ρV ′ Y ′ + 2ρV ′′ Y ) ? ? 2 (? ? 1) P ′ + 2?ψ ′ V ′′ V ′ + ρV ′′2

0 0

(2)

in which a prime acting on V , Z or Y denotes this time a derivative with respect to ρ, ′ ?′ ? while ψ0 ≡ ψ ′ (0), P0 ≡ P ′ (0) and: I0 =

q

? ? ?P ? q2 P ′ ψ2 ,

I1 =

q

? ? q2 ?P ? q2P ′ ψ2 1 ? Y NI0

(3)

It is convenient to perform the following changes: ρ = NI0 ρ, ? ? V = NI0 V , ? Z = Z, Y = (4)

and then to consider the new set of functions: ? dV ? ? v1 = , v2 = Z, v3 = Y d? ρ Using these new notations and restoring the writing ρ ?→ ρ, the set of equations ? (2) becomes: 2 ′′ 2 ′ ′ 2 ′ v1 + ρv1 ? ev1 ? (d + e) ρv1 ? e v1 + 2ρv1 v1 v1 = 1 + ˙ N N 2 ′ ′ + P1 v2 + (v3 + ρv3 ) (5) N 2 ′′ 2 ′ ′ v2 = v2 + ρv2 + v3 ? (e + 2) v2 ? (d + e) ρv2 + 2uv1 ˙ N N ′ 2 ? 2e (v1 v2 + ρv1 v2 ) + P2 v1 (6) 2 ′′ 4 ′ ′ ′ v3 + ρv3 ? (d + 2 + 2e)v3 ? (d + e) ρv3 + 2uv1 v3 = 1 + ˙ N N ′ ′ ′ ′ ? 2e [v1 (v2 + 2ρv3 ) + v1 (2v3 + ρv3 )] + 2P2 (v1 + ρv1 ) v1 (7) in which:

′ u = ? ψ0 , I1 P1 = 2 , I0

e = 2? P2 = ? e ? ? 1 P ′ (0) + eu 2

Wilson-Polchinski exact RG equation 2.2. Fixed point equations

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The ?xed point equations correspond to the three simultaneous conditions vi = 0 for ˙ i = 1, 2, 3 which yield three coupled nonlinear ODE of second order each: 2 2 N ′ ′ ev1 (1 + v1 ) ? 1 + v1 ? P1 v2 + v3 2ρ N N N ′ ′ + (d + e + 2ev1 ) v1 ? P1 v3 (8) 2 2 N ′ 2 ′′ (e + 2 + 2ev1 ) v2 ? v2 ? v3 ? 2uv1 ? P2 v1 v2 = 2ρ N N ′ + [(d + e) +2ev1 ] v2 (9) 2 4 N ′ ′′ (d + 2 + 2e + 4ev1 ) v3 ? 1 + v′ + 2 (ev2 ? u ? P2 v1 ) v1 v3 = 2ρ N 3 N ′ ′ ′ + [(d + e + 2ev1 ) v3 + 2 (2ev3 ? P2 v1 ) v1 ] (10) 2 The di?erential system is of order six, thus the general solution depends on six arbitrary constants. Three of these constants are ?xed so as to avoid the singularity at the origin ρ = 0 displayed by the equations, hence the three following conditions:

′′ v1 = ′ v2 (0) = [e + 2 + 2ev1 (0)] v2 (0) ?

2 v3 (0) ? v1 (0) [2u + P2 v1 (0)] N N 2 ′ ′ v1 (0) = ev1 (0) [1 + v1 (0)] ? P1 v2 (0) + v3 (0) N +2 N N ′ {[d + 2 + 2e + 4ev1 (0)] v3 (0) v3 (0) = N +4 ′ +2 [ev2 (0) ? u ? P2 v1 (0)] v1 (0)}

(11) (12)

(13)

If η is a priori ?xed, then the general solution of the set of equations (8–10) depends on the three remaining arbitrary constants, e.g. the values vi (0) for i = 1, 2, 3. In general the corresponding solutions are singular at some varying ρ? (moving singularity), with: v1 (ρ) ∝ (ρ? ? ρ)?1 , v2 (ρ) ∝ (ρ? ? ρ)?2 , v3 (ρ) ∝ (ρ? ? ρ)?2

But, the equations (8–10) admit another kind of solution that goes to in?nity (ρ → ∞) without encountering any singularity and which behaves asymptotically for large ρ as follows: v1asy (ρ) = G1 ρθ1 + (1 + 2θ1 ) G2 ρ2θ1 + · · · 1 v2asy (ρ) = uG1 ρθ1 + G2 (1 + θ1 ) 1 + G2 ρθ2 + · · · v3asy (ρ) = uθ1 G1 ρθ1 ?1 + 2G2 θ1 1 + G3 ρθ3 + · · · 2 (d ? e) u + d P 0 2θ1 ?1 ρ 2 (d + e) (16) ?′ 2 (d ? e) u + (d + 2d ?′ e ) P0 2θ1 ρ (15) (14)

Wilson-Polchinski exact RG equation with:

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e+2 e , θ2 = ? , θ3 = θ2 ? 1 d+e d+e The values of the three constants {Gi ; i = 1, 2, 3}, correspond to some adjustment of the set {vi (0) ; i = 1, 2, 3} and vice versa. This nonsingular solution is the ?xed point solution which we are interested in. When η is a priori ?xed, the six arbitrary constants of integration are then determined, the di?erential system is balanced. If η is considered as an unknown parameter to be determined, then one of the three preceding quantities {vi (0)} or {Gi } must be promoted to the rank of a ?xed parameter chosen a priori. In general one chooses θ1 = ? v2 (0) = Z0 (17)

which corresponds to having ?xed to Z0 the value of the kinetic term in S [φa ] and is customarily associated with the arbitrary global normalization of the ?eld φ. One thus obtains a function η (Z0 ) which should be a constant if the reparametrization invariance of the exact RG equation was preserved by the derivative expansion presently considered (η should be a constant along a line of equivalent ?xed points generated by the variation of Z0 ). Since it is not the case, one actually obtains a nontrivial function η (Z0 ). Fortunately a vestige of the reparametrisation invariance is preserved and η (Z0 ) displays an extremum in Z0 . This provides us with an optimal value (η opt ) of η (and similarly for Z0 ) via the condition: dη opt =0 dZ0 (18)

Instead of using this condition to determine η opt , I use the fact that the line of equivalent ?xed points is associated with a redundant operator with a zero eigenvalue? [36, 37]. Hence, one may determine η opt by imposing that the ?xed point of interest be associated to a zero eigenvalue. This leads us to the consideration of the system of eigenvalue equations. 2.3. Eigenvalue equations The eigenvalue equations are obtained by linearization of the ?ow equations (5–7) about ? a ?xed point solution {vi , i = 1, 2, 3}:

? vi = vi + εeλt gi

? The fact that the eigenvalue takes on a de?nite value although it is associated with a redundant operator is not in con?ict with the work of Wegner (see reference [39] and references therein) which indicates that the eigenvalue of a redundant operator generally varies with the renormalization process. As shown in reference [36], the linear character of the renormalization of the ?eld in the process of generating the exact RG equation implies a de?nite eigenvalue (the reparametrization invariance is a direct consequence of this linearity). On the contrary, in the case of a nonlinear renormalization scheme, the eigenvalue in question no longer is constant but depends on the renormalization procedure [36] in accordance with [39].

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Keeping the linear contribution in ε, the following set of coupled ODE comes: 2 2 N ? ′ ′ ′′ (λ + e + 2ev1 ) g1 ? 1 + g 1 ? P1 g 2 + g 3 g1 = 2ρ N N N ? ′ ?′ ′ + [(d + e + 2ev1 ) g1 + 2eg1 v1 ] ? P1 g3 (19) 2 N 2 ′′ ? ′ ? ? g2 = (λ + e + 2 + 2ev1 ) g2 ? g2 ? g3 + 2 (ev2 ? u ? P2 v1 ) g1 2ρ N N ? ′ ?′ (20) + [(d + e + 2ev1 ) g2 + 2eg1 v2 ] 2 N 4 ′′ ? ′ ′ g3 = (λ + d + 2 + 2e + 4ev1 )g3 ? 1 + g3 ? 2ug1 2ρ N ′ ? ? ?′ ? ′ ?′ +2e (g1 v2 + 2g1 v3 ) ? 2P2 (g1 v1 + v1 g1 ) + 2ev1 g2 ] N ? ′ ′ ? ?′ ?′ ′ ?′ + [(d + e + 2ev1 ) g3 + 2e (2g1 v3 + g1 v3 + 2v1 g3 ) ? 4P2 g1 v1 ] (21) 2 Similar considerations as those relative to the determination of the six integration constants associated to the ?xed point equations (8–10) stand. For a given ?xed point ? solution (vi ), there remain six constants to be determined. Three of them are ?xed so as to avoid the singularity at the origin ρ = 0 displayed by the equations: 2 ′ ? g2 (0) = [λ + e + 2 + 2ev1 (0)] g2 (0) ? g3 (0) N ? ? + 2 [ev2 (0) ? u ? P2 v1 (0)] g1 (0) 2 N ? ′ ′ [λ + e + 2ev1 (0)] g1 (0) ? P1 g2 (0) + g3 (0) g1 (0) = N +2 N N ′ ? ′ g3 (0) = {[λ + d + 2 + 2e + 4ev1 (0)] g3 (0) ? 2ug1 (0) N +4 ′ ? ? + 2e [g1 (0) v2 (0) + 2g1 (0) v3 (0)]

?′ ? ′ ?′ ?2P2 [g1 (0) v1 (0) + v1 (0) g1 (0)] + 2ev1 (0) g2 (0)}

One is interested in the solution that is regular when ρ → ∞. For λ a priori ?xed, the three values {gi (0) , i = 1, 2, 3} at the origin ρ = 0 must be adjusted so that the solution reaches the following regular asymptotic behavior: g1asy = S1 ρχ1 + 2 (1 + θ1 +χ1 ) G1 S1 ρθ1 +χ1 + · · · g2asy = S2 ρχ2 + uS1 ρχ1 + · · · g3asy = S3 ρχ3 + uS1 χ1 ρχ1 ?1 + · · · with: λ+e+2 λ+e , χ2 = ? , χ3 = χ2 ? 1 d+e d+e and the value of the set of constants {Si , i = 1, 2, 3} entering the regular solution at large ρ corresponds to the value of {gi (0)} adjusted at the origin and vice versa. As in any eigenvalue problem, the global normalization of the eigenvector may be chosen at will so that, ?xing g1 (0) = 1 for instance, allows one to determine discrete χ1 = ?

Wilson-Polchinski exact RG equation

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values of λ. Positive values give the critical exponents, negative values are subcritical (or correction-to-scaling) exponents. The peculiar value λ = 0, if present, is associated to the vestige of the reparametrization invariance [37, 15]. Indeed this zero eigenvalue is associated to the redundant operator that generates the line of equivalent ?xed points in the complete theory [36, 37]. Conversely, if one considers together the ?xed point equations (8–10) with the eigenvalue equations (19–21) in which λ is set equal to zero (and the condition g1 (0) = 1 is maintained), then the condition (17) may be abandoned and v2 (0) adjusted so as to get a common solution to the set of six coupled ODE. Then, the resulting value of η nececessarily coincides with η opt as de?ned by equation (18) and the resulting value opt of v2 (0) gives Z0 . Though the number of di?erential equations has increased twofold this procedure of determining the optimized ?xed point is the most e?cient one when parameters (like N and some other ones, see following section) have to be varied. 2.4. The free parameters In order to perform an actual numerical study of the set of second order ODE described in the preceding section, I make the following choice + :

2 ? P q 2 = e?q 1 ψ q2 = 1 + bq 2

(22) (23)

Following the terminology of reference [39], the free parameter b is redundant and is intended to be used to optimize the numerical results of the derivative expansion. The introduction of b is linked to the general property of reparametrization invariance which is broken by the present derivative expansion. The normalization ψ (0) = 1 is chosen in order to distinguish the e?ect of simply changing the global normalization of the ?eld which induces a line of equivalent ?xed points (at ?xed b). That line is customarily associated to the arbitrariness of Z0 the value of Z (0) (≡ v2 (0)), i.e. the coe?cient of the kinetic term in S [φa ] (see preceding section). Changing the value of b in the complete theory would induce new (equivalent) lines of equivalent ?xed points. Though they are part of the same invariance, the two free parameters b and Z0 have e?ects of di?erent nature in the derivative expansion. As a global constant of normalization, one can expect that, at a given order of the derivative expansion, the variation of Z0 will still reveal a vestige of the invariance of the exact theory (see preceding section). On the contrary, the e?ect of b spreads out over di?erent orders of the derivative expansion. Consequently, one expects to observe a progressive restoration of the redundant character of b as the order of the expansion increases. Regarding

? In reference [13], I considered one additionnal parameter within the cuto? function P (named a), however the equations were invariant in the change b → b/a so that one of the two parameters was unnecessary.

+

Wilson-Polchinski exact RG equation

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the extremely low order considered here (O (? 2 )), one must not expect too much from varying b (see section 3.2). ? Notice that the cuto? function P (q 2 ) is essentially a regulator of the integrals (3) generated by the derivative expansion. Besides the arbitrary choice in the decreasing at large q, the other sources of arbitrariness of the cuto? function may be included within the arbitrary function ψ (q 2 ). This is why the choice (22) does not involve any free parameter. 3. Numerical study and results There are two di?erent methods for numerically studying systems of coupled nonlinear ODE as those described above: the shooting and the relaxation methods (see, for example reference [40]). Because it is the easiest to implement, only the shooting method is considered here (though it is less numerically stable than the relaxation method). 3.1. The shooting method Considering an initial point ρ1 where known conditions (initial conditions) are imposed and trial values are given to the remaining integration constants, one integrates the ODE system up to a second point ρ2 where the required conditions are checked. Using a Newton-Raphson algorithm, one iterates the test until the latter conditions are satis?ed within a given accuracy. In the present study, the two points ρ1 and ρ2 are either the origin ρ = 0 and a large value ρasy (shooting from the origin) or the reverse (shooting to the origin). In principle, shooting to the origin is technically better adapted to the present study. For example, in the case of the ?xed point equations (8–10) with η ?xed, one starts from ρasy with trial values for the three constants {Gi } and the initial values of the ′ three functions and of their ?rst derivatives {vi , vi } de?ned by (14–16). After integration up to the origin, one checks whether the three conditions (11–13) are ful?lled or not. The system is balanced and the values of the other interesting parameters are simply by-products of the adjustment. For example, the value of Z0 associated to the arbitrarily ?xed value of η is simply read as the value that v2 (0) takes on after achievement of the adjustment. If one wants instead to determine η (Z0 ), then η has to be considered as a trial parameter like {Gi } and the supplementary condition (17) with Z0 an arbitrary ?xed number, must be ful?lled at the origin. Unfortunately, equations (8–10) are singular at ρ = 0 so that it is impossible to shoot to the origin. In reference [41] (where the leading order LPA was studied for small values of N), the di?culty was circumvented by shooting to a point close to the origin. On the contrary, I have chosen to shoot from the origin because it is easy to control the equations starting from that point. Starting from the origin implies that, for a given value of η, the adjustable parameters no longer are the {Gi } but the initial values {vi (0)}, the initial values

Wilson-Polchinski exact RG equation

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Table 1. Critical exponent estimates (for d = 3) from the O (N ) Wilson-Polchinski exact RG equation expanded up to O ? 2 in the derivative expansion for two values of the free parameter b. N 1 2 3 4 5 6 7 8 9 10 11 12 15 20 b = 0.03 η ν 0.01006 0.00866 0.00721 0.00592 0.00486 0.00404 0.00342 0.00294 0.00256 0.00227 0.00204 0.00184 0.00142 0.00102 0.6223 0.6723 0.7238 0.7713 0.8111 0.8424 0.866 0.8849 0.8993 0.9107 0.9214 0.9293 0.9438 0.9596 ω 0.7755 0.7266 0.7132 0.7223 0.7437 0.7695 0.7946 0.8172 0.8368 0.8533 0.8711 0.8836 0.9060 0.9330 b = 0.11 η ν 0.02494 0.02263 0.02007 0.01739 0.01489 0.01275 0.01100 0.00960 0.00848 0.00757 0.00683 0.00621 0.00488 0.00358 0.5994 0.6388 0.6838 0.7313 0.7762 0.8138 0.8440 0.8673 0.8854 0.8995 0.9108 0.9198 0.9387 0.9563 ω 0.8740 0.7656 0.7156 0.7010 0.7088 0.7307 0.7576 0.7842 0.8081 0.8292 0.8463 0.8613 0.8942 0.9258

of the ?rst derivatives follow from (11–13). At the point ρasy only three conditions are needed in order to balance the number of adjustable parameters. Thus one must eliminate the {Gi } from the six equations (14–16) to obtain three anonymous conditions. Consequently the precise knowledge of the asymptotic behavior of the regular ?xed point solution is not necessary. ′′ It appears that imposing conditions like {vi (ρasy ) = 0} is su?cient to determine the solution of interest. I ?rst adjust the trial parameters so as to reach some not too large a value of ρasy , then I increase ρasy until the desired accuracy is reached on the trial parameters. 3.2. Numerical results At the leading order LPA, the function v1 is considered alone. Only one second order ODE de?nes the ?xed point (twice this number for the eigenvalue problem). The study has already been done in reference [41] for d = 3 and N = 1 to 4. I have extended the results (for d = 3) to larger values of N. The study is simple because: (i) the reparametrization invariance is automatically satis?ed since, by construction, the coe?cient of the kinetic term is supposed to be constant. (ii) the equation (5) with v2 = v3 = 0, does not depend on any free parameter like b. Consequently there is no ambiguity: to any value of N, corresponds a unique value of ν and ω while η=0. As already mentioned in reference [42], it is noteworthy that the values I obtain (see also [41]) for the critical and subcritical exponents (without optimization since there is no free parameters at hand) agree, “to all published digits”,

Wilson-Polchinski exact RG equation

Table 2. Same as table 1 (two other values of b). N 1 2 3 4 5 6 7 8 9 10 11 12 15 b = 0.17011 η ν 0.03553 0.03278 0.02980 0.02643 0.02300 0.01986 0.01719 0.01502 0.01327 0.01185 0.01068 0.00972 0.00779 0.5850 0.6176 0.6574 0.7033 0.7506 0.7939 0.8291 0.8565 0.8779 0.8944 0.9075 0.9180 0.9552 ω 0.9515 0.7917 0.7115 0.6806 0.6809 0.7040 0.7331 0.7641 0.7938 0.8194 0.8413 0.8596 0.9209 b = 0.25611 η ν 0.04979 0.04670 0.04371 0.03999 0.03565 0.03175 0.02772 0.02352 0.02150 0.01937 0.01734 0.01588 — 0.5677 0.5908 0.6242 0.6664 0.7175 0.7737 0.8224 0.8565 0.8928 0.9132 0.9357 0.9552 — ω 1.0906 0.8273 0.7288 0.6570 0.6538 0.6682 0.7137 0.7456 0.8037 0.8165 0.8505 0.8741 —

13

Table 3. Compared to LPA on the lhs of the arrows, the order ? 2 (rhs of the arrows) induces a splitting into two of the subcritical exponents of degree higher than ω (values for d = 3, N = 1 and b = 0.11). The LPA values may be found in [43]. ω 0.6557 → 0.8740 ω2 3.18 → 2.83 3.62 ω3 5.91 → 5.70 6.40 ω4 8.80 → 7.30 9.6

with the “optimized” values obtained in reference [43] from a study of an exact RG equation of the second family. Hence, for the numerical results I have obtained at order LPA for ν, ω and other subleading critical exponents ωn , the reader is referred to table 1 of reference [43]. At order ? 2 , once the arbitrariness of Z0 has been removed via the de?nition of η opt , the values of the critical exponents still depend on b in such a way that, for a given value of N, it is impossible to de?ne “preferred” values. For example η (from now on, η stands for η opt ) depends on b almost linearly. Due to the spreading out over several orders of the e?ects of b, the study of the convergence of the derivative expansion relies rather on the consideration of several orders. Now the order ? 2 is still too low to allow an appreciation of the convergence of the derivative expansion. Instead of presently producing one “best” estimate for each critical exponent at a given value of N, it is preferable to maintain the freedom of b in order to better emphasize the early beginnings of some criteria of convergence if any (see section 3.3). Tables 1 and 2 display the estimates of η, ν and ω as obtained for four values of b and N varying from 1 to 20 (while d = 3).

Wilson-Polchinski exact RG equation

14

For N = 1, the results of reference [13] (preferred value of η opt for b = 0.11) is obtained again. In this latter work I had proposed a criterium of choice of the value of b which gave a preferred value of η. It was based on the idea of Golner [15] of a global minimization of the magnitude of the function Z (φ) (i.e., v2 (φ)). However the extension of this criterion to N > 1 is not easy to implement because the function Z (φ) has been split into two parts (Z(ρ) and Y (ρ)). The further discussion of these results is left to section 3.3. Table 3 displays a comparison between the LPA and O(? 2 ) results for d = 3, N = 1 and b = 0.11 of the series of subcritical exponents ωn (with ω1 = ω). Compared to LPA, the order O (? 2 ) has increased the potential number of subcritical exponents (due to the supplementary terms in the truncated action considered). For instance, if one adopts the dimensional analysis of perturbation theory, then at order LPA a term like φ6 contributing to S[φ] is associated with the subcritical ω2 while a φ8 -term generates ω3 . But at order O (? 2 ), new terms involving two derivatives contribute to S[φ], and a term proportionnal to φ2 (?φ)2 induces a priori an in-between correction exponent. Despite important di?erences between the classical dimensions of the couples φ6 and φ2 (?φ)2 , table 3 clearly shows that, for N = 1, the order ? 2 induces a splitting of the LPA values of the subcritical exponents ωn for n > 1 (ω is, in the perturbative approach, associated to the unique φ4 coupling and is not, fortunately for the perturbative framework, subject to this splitting). This simple splitting is presumably not preserved for other values of N due to the supplementary contribution of Y (φ). 3.3. Comparison with other studies, discussions and conclusion Figure 1 shows the evolution of η with N from di?erent works. The results obtained from the resummation of six [29] and seven [30] orders of the perturbation ?eld theory serves the purpose of standards (other accurate estimates of the critical exponents, especially for N ≤ 4, exist in the litterature, for a review see [5]). One sees that the present study yields generally small values of η compared to the standards except for small N and for the highest values of b. The evolution of η with N is smoother than in the work of Morris and Turner [26] but, as in this latter work, the non-monotonic behavior of the standards (responsible for the maximum of η about N = 2 or 3) is not reproduced. Instead, the results of Gersdor? and Wetterich [27] are better. The present results are however not so bad if one keeps in mind that the ?rst estimate of η in the derivative expansion is given at order ? 2 . In particular, ?gure 1 shows also recent estimates from the resummation of three orders of the perturbative series using an e?cient method [44]. One sees that the present estimates withstand the comparison (except the monotonic evolution with N). Notice also, for N ?xed, the monotonic evolution of η with b already mentioned. The results for the critical exponent ν are more interesting to discuss because the order ? 2 provides its second estimate. Figure 2 shows the evolution with N of ν at order ? 2 compared to the results at order LPA (obtained in the present work), the standards

Wilson-Polchinski exact RG equation

15

[29, 30] are reproduced also. Again, one observes that the results for large N are not as good as for small values. However, for these latter values, one clearly sees that there is a range of values of b where the two estimates at orders LPA and ? 2 ?ank the standards and another range where the two present estimates are on the same side (with respect to the standards). This is a phenomenon often observed in convergent series the elements of which depend on a free parameter (like b) but the resumed series does not: on varying the free parameter, one may observe monotonic or alternate approaches to the limit. These features may be used to determine error bars. Presently one additionnal order would be necessary to propose such error bars. Figure 2 shows also that, when N increases the dependence of ν on b becomes non-monotonic. This is interesting since such extrema may indicate a vestige of the primarily independence on b of the exact RG equation. Why this e?ect does not occur at small values of N is not explained. Once more, at least one supplementary order of the derivative expansion would be necessary to understand this point. Figure 3 shows the results for ν coming from [26] and [27] compared to the present results for b = 0.11 and the standards [29, 30]. One observes that the present results are globally better than in [26] and that again the estimates of Gersdor? and Wetterich [27] are excellent (for small N the points almost coincide with the standards). Figure 4 shows the present results for ω at order LPA and ? 2 compared to the standards [28, 30]. This ?gure is the matching piece to ?gure 2 and the same kinds of remarks stand: monotonic and alternate approaches to the standards at ?xed N exist as well as non-monotonic dependences on b. The magnitude of these e?ects are larger than for ν and the accuracy is worse, but this is expected for a subleading eigenvalue: the accuracy decreases as the order of the eigenvalue increases. Figure 5 shows the results for ω coming from [26] compared to the present results for b = 0.11 and the standards [28, 30]. One observes that the present results are much better than in [26]. One may regret that Gersdor? and Wetterich [27], who obtained excellent values for ν and η, had not estimated ω. As already said, this study [27] does not follow the conventional approach de?ned in the introduction. In particular nothing is said on the way the reparametrization invariance is accounted for. In fact, instead of leaving free the value of Z(0) = Z0 to get a function η(Z0), the procedure followed in [27] was to attach the determination of η to the minimum of the potential. This condition ?xes Z0 and the arbitrariness carried by the reparametrization invariance is implicitly removed this way. Because Morris and Turner [26] have considered an equation of the same family and have obtained disappointing results, I think that this particular way of choosing η opt could be the main reason for the excellent estimates of the critical exponents obtained in [27]. It will be interesting to adapt it to the study of the Wilson-Polchinski RG equation. To conclude, the derivative expansion at order O(? 2 ) already displays a tendency to converge. This must be con?rmed by considering the next order which is in progress [32]. The study of reference [25] which for N = 1 follows the procedure of reference [27] and the optimization process of reference [38] is very encouraging. I think that the Wilson-

Wilson-Polchinski exact RG equation

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Polchinski RG equation, which is the simplest exact RG equation, is better adapted to the estimation of the critical exponents. Further studies should be undertaken with a view to better determine the status of the derivative expansion. Acknowledgments I am indebted to C. Bagnuls for discussions and encouragements all along this work. References

[1] Wilson K G and Kogut J 1974 Phys. Rep. 12C 77 [2] Gell-Mann M and Low F E 1954 Phys. Rev. 95 1300 [3] Nickel B G, Meiron D I and Baker G A Jr 1977 Compilation of 2-pt and 4-pt graphs for continuous spin models Guelph University preprint unpublished [4] Zinn-Justin J 2002 Euclidean Field Theory and Critical Phenomena Fourth edition (Clarendon Press, Oxford) [5] Pelissetto A and Vicari E 2002 Phys. Rep. 368 549 [6] Wilson K G 1976 Phase Transitions and Critical Phenomena Vol. VI ed C. Domb and M.S. Green (Acad. Press, N.-Y.) p 1 [7] Bagnuls C and Bervillier C 1988 Phys. Rev. Lett. 60 1464 [8] Delamotte B and Canet L 2004 What can be learnt from the nonperturbative renormalization group? Preprint cond-mat/0412205 [9] Bagnuls C and Bervillier C 2001 Phys. Rep. 348 91 [10] Wilson K G 1970 Irvine Conference (unpublished) [11] Berges J, Tetradis N and Wetterich C 2002 Phys. Rep. 363 223 Polonyi J 2003 Cent. Eur. J. Phys. 1 1 [12] Wiese K J 2003 Ann. Inst. Henri Poincar? 4 473 e Reuter M and Weyer H 2004 Phys. Rev. D 70 124028 Honerkamp C 2004 Functional renormalization group in the two-dimensional Hubbard model Preprint cond-mat/0411267 [13] Bervillier C 2004 Phys. Lett. A 332 93 [14] Bervillier C 2004 Phys. Lett. A 331 110 [15] Golner G R 1986 Phys. Rev. B 33 7863 [16] Bagnuls C and Bervillier C 2000 Cond. Matt. Phys. 3 559 [17] Blaizot J P, Galain R M and Wschebor N 2004 Non Perturbative Renormalization Group, momentum dependence of n-point functions and the transition temperature of the weakly interacting Bose gas Preprint cond-mat/0412481 [18] Tetradis N and Wetterich C 1994 Nucl. Phys. B 422 541 [19] Bohr O, Schaefer B J and Wambach J 2001 Int. J. Mod. Phys. A 16 3823 [20] Ballhausen H 2003 The e?ective average action beyond ?rst order Preprint hep-th/0303070 [21] Ball R D, Haagensen P E, Latorre J I and Moreno E 1995 Phys. Lett. B 347 80 Comellas J 1998 Nucl. Phys. B 509 662 Kubyshin Y, Neves R and Potting R 1999 The Exact Renormalization Group ed A Krasnitz et al (World Scienti?c, Publ. Co., Singapore) p 159 [22] Morris T R 1994 Phys. Lett. B 329 241 ——1995 Phys. Lett. B 345 139 Aoki K I, Morikawa K, Souma W, Sumi J I and Terao H 1998 Prog. Theor. Phys. 99 451 [23] Bonanno A and Zappal` D 2001 Phys. Lett. B 504 181 a Mazza M and Zappal` D 2001 Phys. Rev. D 64 105013 a [24] Seide S and Wetterich C 1999 Nucl. Phys. B 562 524

Wilson-Polchinski exact RG equation

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[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

Ballhausen H, Berges J and Wetterich C 2004 Phys. Lett. B 582 144 Canet L 2004 Optimization of ?eld-dependent nonperturbative renormalization group ?ows Preprint hep-th/0409300 Canet L, Delamotte B, Mouhanna D and Vidal J 2003 Phys. Rev. B 68 064421 Morris T R and Turner M D 1998 Nucl. Phys. B 509 637 Gersdor? G V and Wetterich C 2001 Phys. Rev. B 64 054513 Sokolov A I 1998, Phys. Status Solidi 40 1169 Antonenko S A and Sokolov A I 1995 Phys. Rev. E 51 1894 Kleinert H 1999 Phys. Rev. D 60 085001 Canet L 2004 Processus de r?action-di?usion : une approche par le groupe de renormalisation non e perturbatif Thesis Bagnuls C, Bervillier C and Shpot M in progress Polchinski J 1984 Nucl. Phys. B 231 269 Golner G R 1998 Exact renormalization group ?ow equations for free energies and N-point functions in uniform external ?elds Preprint hep-th/9801124 Fisher M E 1983 Critical Phenomena Lecture Notes in Physics ed F.J.W. Hahne (Springer-Verlag, Pub.) p 1 Bell T L and Wilson K G 1974 Phys. Rev. B 10 3935 ——1975 Phys. Rev. B 11 3431 Litim D F 2000 Phys. Lett. B 486 92 ——2001 Phys. Rev. D 64 5007 Wegner F J 1976 Phase Transitions and Critical Phenomena Vol. VI ed C. Domb and M.S. Green (Acad. Press, N.-Y.) p 7 Press W H, Flannery B P, Teukolsky S A and Vetterling W T 1986 Numerical Recipes. The Art of Scienti?c Computing (Cambridge University Press) Comellas J and Travesset A 1997 Nucl. Phys. B 498 539 Litim D F 2001 Int. J. Mod. Phys. A 16 2081 ——2002 Nucl. Phys. B 631 128 Kleinert H and Yukalov V I 2004 Self-similar variational perturbation theory for critical exponents Preprint cond-mat/0402163

Wilson-Polchinski exact RG equation

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0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 5 10 15 20

η

N

Figure 1. Critical exponent η as function of N (d = 3). Open and full circles represent the standards, they were obtained from the resummation of six ( from [29]) or seven ( from [30]) orders of the perturbation ?eld theory. Recent estimates of [44] from the same perturbative series at order three only are also represented (△). The other estimates are from the exact RG equation expanded up to O ? 2 in the derivative expansion: full triangle up from [27], ? from [26]. The points linked by ? straight lines are from the present work: for b = 0.03, full triangle down for b = 0.11, + for b = 0.17011, * for b=0.25611. See text for discussion.

?

?

Wilson-Polchinski exact RG equation

19

1.0

0.9

0.8

ν

0.7

0.6

5

10

15

20

N

Figure 2. Critical exponent ν as function of N (d = 3). Open and full circles represent the standards as in ?gure 1. The other estimates are from the present work: ? from order LPA, the points linked by straight lines are from O ? 2 in the derivative ? for b = 0.03, full triangle down for b = 0.11, + for b = 0.17011, * for expansion: b=0.25611. See text for discussion.

Wilson-Polchinski exact RG equation

20

1.0

0.9

ν

0.8

0.7

0.6 0 5 10 15 20

N

Figure 3. Critical exponent ν as function of N (d = 3). Open and full circles represent the standards as in ?gure 1. The other estimates are from the exact RG equation expanded up to O ? 2 in the derivative expansion: full triangle up from [27], ? from [26]. The estimates of the present study for b = 0.11 is reported (full ? triangles down linked by straight lines, see ?gure 2 for the other values of b). See text for discussion.

Wilson-Polchinski exact RG equation

21

1.0

0.9

ω

0.8

0.7

5

10

15

20

N

Figure 4. Subcritical exponent ω as function of N (d = 3). Open and full circles represent the standards as in ?gure 1 but the opencircles ( ) are from [28]. The other estimates are from the present work: ? from order LPA, the points linked by straight ? for b = 0.03, full triangle down lines are from O ? 2 in the derivative expansion: for b = 0.11, + for b = 0.17011, * for b=0.25611. See text for discussion.

?

Wilson-Polchinski exact RG equation

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1.0 0.9 0.8 0.7

ω 0.6

0.5 0.4 0.3 0 5 10 15 20

N

Figure 5. Subcritical exponent ω as function of N (d = 3). Open and full circles represent the standards as in ?gure 4. The other estimates are from the exact RG equation expanded up to O ? 2 in the derivative expansion: ? from [26]. The estimates ? of the present study for b = 0.11 is reported (full triangles down linked by straight lines, see ?gure 4 for the other values of b). See text for discussion.

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