The perturbative structure of spin glass field theory

Cubic replicated field theory is used to study the glassy phase of the short-range Ising spin glass just below the transition temperature, and for systems above, at, and slightly below the upper critical dimension six. The order parameter function is computed up to two-loop order. There are two, well-separated bands in the mass spectrum, just as in mean field theory. The small mass band acts as an infrared cutoff, whereas contributions from the large mass region can be computed perturbatively (d>6), or interpreted by the epsilon-expansion around the critical fixed point (d=6-epsilon). The one-loop calculation of the (momentum-dependent) longitudinal mass, and the whole replicon sector is also presented. The innocuous behavior of the replicon masses while crossing the upper critical dimension shows that the ultrametric replica symmetry broken phase remains stable below six dimensions.

mean field theory of the Ising spin glass by Parisi, see [2] and references therein, and -according to our present knowledge -it seems to persist below the upper critical dimension six [6]; possibly (as the aforementioned numerical simulations suggest) down to three dimensions. Nevertheless the details of the RSB phase of the short range finite-dimensional model differ in many ways from its mean field counterparts. Important examples for such discrepancies are the leading behavior of the order parameter function and momentum-dependent masses (or, equivalently, correlation functions) close to criticality. Moreover, these details depend on the space dimension d which can be well illustrated by the breakpoint x 1 of the order parameter function q(x), see Refs. [6][7][8][9]: x 1 is proportional to τ ∼ (T c −T )/T c in mean field theory (which is equivalent to the infinite-dimensional model), and this behavior persists down to d = 8, with possibly a logarithm of τ at exactly eight dimensions. For 6 < d < 8, x 1 ∼ τ d 2 −3 , whereas at exactly six dimensions: x 1 ∼ | ln τ | −1 . Below six dimensions x 1 becomes finite at T c , and renormalization group arguments show [6] that its critical value is universal. It was computed in first order in ǫ = 6 − d in Ref. [6], the present paper extends this calculation to second order, see Eq. (31). There is a trend of increasing x 1 with decreasing d, a clear sign that RSB becomes more dominant. This contradicts expectations that a replica symmetric (RS) glassy phase, which is characterized by the so-called "droplet" picture [10][11][12], enters in some low dimension, for which a possible scenario would be if x 1 decreased to zero. Well below d = 6 one expects x 1 to be of order unity, meaning that intervalley overlaps become as important as self-overlap [2].
As an alternative to numerical simulations, replicated field theory provides analytic results, and it has the advantage that space dimension can be chosen at will by defining the model on a ddimensional hypercubic lattice. In the present paper, spin glass field theory is studied below eight dimensions, also passing through the upper critical dimension six. (In this domain of dimensions, a simple cubic model with the coupling constant w is sufficient for obtaining critically relevant properties, the quartic coupling of the truncated model, for instance, which is dangerously irrelevant in higher dimensions, can now be neglected.) Our main purpose is to understand how the perturbative method works in this system whose mass spectrum consists of two separated bands: a large one dominating the behavior in the "near infrared" momentum range, and a small oneextending to zero -related to the "far infrared" sector. We extend former calculations of the order parameter function q(x) to two-loop order. This calculation needs computing some of the one-loop self-energy insertions; the results can be used to get the momentum-dependent longitudinal mass and the replicon band in one-loop order. The findings for the replicon band support the idea that stability of the RSB phase persists below six dimensions.
The paper is organized as follows: The model is defined and the equation of state for the order parameter function presented in Sec. II, while the properties of the free propagators are studied in Sec. III. The different terms contributing to the equation of state up to two-loop order are worked out in Sec. IV. Section V contains the main results for the order parameter function in the different dimensional regimes, namely 6 < d < 8, d = 6, and d 6. This section is divided into three subsections: V A is for the breakpoint x 1 , V B for the Edwards-Anderson order parameter, while subsection V C is devoted to q(x) with x < x 1 . The momentum-dependent mass (the inverse propagator) is studied in Sec. VI. In subsection VI A the longitudinal mass is computed, and its scaling behavior below six dimensions displayed and proved. The replicon sector is left to VI B.
The discussion of the results is included in the last section, i.e. in Sec. VII. Several results are summarized in listed and tabulated forms in the three appendices.

II. THE REPLICATED CUBIC FIELD THEORY AND MASS RENORMALIZATION
The replicated field theory (representing the Ising spin glass on a d-dimensional hypercubic lattice in zero external magnetic field) has its dynamical variables φ αβ = φ βα -with φ αα = 0 and replica indices α, β, . . . taking values from 1 to n where n is the replica number -and Lagrangian L(φ αβ ) which is invariant under any permutations of the replicas (a trivial outcome of the replica trick) and under the transformation φ ′ αβ = (−1) α+β φ αβ (expressing the extra symmetry coming with the vanishing external field [13]). Concentrating on the glassy phase just below the critical temperature, invariants of the above symmetry which are higher than cubic can be neglected when d < 8, and the Lagrangian takes the relatively simple form Momentum conservation is indicated by the primed summation. The number N of the Ising spins becomes infinite in the thermodynamic limit, rendering summations to integrals over the continuum of momenta in the diagrams of the perturbative expansion. A momentum cutoff Λ is always understood to block ultraviolet divergences. The two important parameters of the model are the reduced temperature τ and the coupling constant w. As we are working in the immediate vicinity of the critical point, τ is assumed to be much smaller than Λ 2 . The critical bare mass can be computed relatively easily yielding (up to second order and for n = 0): For studying the glassy phase, the replica symmetric Lagrangian above is converted to the replica symmetry broken one by the transformation φ αβ is the exact homogeneous order parameter matrix. The Lagrangian in (1) gets then the additional The new fields have, by definition, zero mean now, and this condition yields the equation of state for q αβ : where the exact propagator satisfies Dyson's equation: and Σ the self-energy. (4) From now on, n is set to zero (the spin glass limit), and an infinite-step ultrametric structure [2] is assumed for q αβ = q(x), with the overlap x = α ∩ β. Construction of the free propagator G in spin glass field theory is not a trivial task. One can be guided by lessons learnt from Refs. [7][8][9]: τ = wq 1 + . . . and q(x) ∼ x in leading order. To generate a perturbation theory where the small parameter is the breakpoint x 1 of the order parameter function, one can divide the mass as M = M 0 + M 1 , and q as q(x) = q 1q (r) = q 1 [r + ∆q(r)] with r ≡ x/x 1 . By definition, ∆q(1) = 0since q 1 and x 1 are exact quantities -, and for the free propagator we have M 1 , on the other hand, plays the role of a quadratic counter term, and the exact propagator has the following expansion: with

III. THE FREE PROPAGATOR
One can follow Ref. [8] step by step to construct the free propagator from M 0 in Eq. (5). The replicon masses are easily found and can be displayed in the usual parametrization as Due to the second term of the diagonal element in Eq. (5), this replicon band extends from zero to x 1 wq 1 , and it is necessary for having a positive mass spectrum. While the replicon eigenvalues are exact and of order ∼ x 1 wq 1 , the band of large masses is centered around 2wq 1 and has the expansion: One can see that the small parameter x 1 has a double role in the mass spectrum: Firstly, it is the ratio of the small to large masses which are then clearly separated and, secondly, the widths of both bands are proportional to x 1 .
Inversion of the mass operator is complicated, but feasible by the techniques of Ref. [8]. As an illustration of the behavior of the free propagator components as a function of the momentum, let us consider the combination entering the equation of state (3) 1 : Y has the following expansion in the large (l) and small (s) mass regimes: • Large mass regime, i.e. p 2 ∼ 2wq 1 : When p 2 < 2wq 1 , we have the following expansion of the G (l) functions (although not indicated, the g (l) coefficients are still functions of r): 10 + g • Small mass regime, i.e. p 2 ∼ x 1 × 2wq 1 : The expansion of the G (s) functions for p 2 > x 1 2wq 1 (the r-dependence of the coefficients not indicated): 0 which can be computed by the least effort using the "smallest block approximation" (SBA), meaning that the γ summation in (9) is restricted to the smallest ultrametric blocks of size x 1 around α and β: where the near infrared form of G xx 1 1 from Sec. 6 of Ref. [8] has been used. The calculation of G There is complete matching in the momentum range x 1 2wq 1 < p 2 < 2wq 1 which is satisfied by the following matching-conditions between the two types of coefficients: The last term which is quadratic in ∆q will be neglected in the present stage of approximation.
We will need the following derivatives: (12b)

B. The one-loop term
Inserting the free propagator [i.e. the leading term in (6)] and m (1) c into (3), and using the definition of Y in (9), we have where the dimensionless quantityX 1 depends only on the three dimensionless parameters of the theory, namelyḡ ≡ w 2 K d /Λ ǫ (the dimensionless coupling constant),Λ ≡ Λ/(2wq 1 ) 1/2 , and x 1 : In the above formula the sub-and superscripts of the F functions correspond to the expansion of the free propagator in Sec. III: thus (l) [(s)] refers to the large (small) mass regime, respectively.
Introducing the notations for the (dimensionless) "replicon" and "longitudinal" propagators, Eqs. (10) and (2) give F (l) 0 : For obtaining F (l) 1 , one must go beyond the near infrared approximation for G xx 1 1 , and also SBA for Y . A lengthy calculation gives Neglecting contributions which are irrelevant close to the critical temperature, one can write: The constant is singular in six dimensions, namely it can be written as Unfortunately, the small mass term F (s) 0 is too complicated to get it in closed form. Nevertheless, its singular behavior at d = 6 can be extracted, and one gets a well-defined limit of F (l) C. The one-loop results for the equation of state The first and second derivatives (with respect to r) of F (l) 0 and the leading parts of Eqs. (12a) and (12b), i.e. neglecting ∆q, give the equations between τ and q 1 on the one hand: and between the dimensionless quantities on the other hand, in leading order: The diagonal mass counterterm δM in (7) can now be computed by the help of Eqs. (2), (19a), and (19b): D. The ∆q insertion term to ∆q, see Eqs. (6) and (7), and applying the SBA (which is correct up to this order) one arrives at: We can now use Eqs. (19a) and (19b) in correction terms containing ∆q, and also letΛ go to infinity whenever possible (thus neglecting irrelevant contributions close to T c ). Somewhat surprisingly, terms with ∆q ′ and ∆q ′′ both disappear due to exact cancellations when the ∆q insertion term is added to the zero-loop results in Eqs. (12a) and (12b). The following simple results (correct up to the present second order calculation) are obtained: and

E. The remaining two-loop term
By the help of (6) and (7), the basic two-loop contribution to the equation of state in (3) can be written as where the last equation was obtained by the SBA, hereby neglecting terms which are smaller by a factor of x 1 . For making easier to display and analyze the following -rather complicatedformulae, it is useful to define the following special linear combinations of a generic ultrametric matrix (such as G or Σ, the former is used in the definitions below), see also footnote 1: Evaluating the matrix products by the SBA again, we get: The one-loop self-energy, denoted here simply by Σ, is given by the expression where the third and fourth terms inside the curly brackets, i.e. {q ↔ p − q}, are the same as the first and second ones, but with q and p − q transposed. The equation between q 1 and τ is obtained by the first derivative of Eq. (3) evaluating at x = x 1 : where Eq. (21a) and the definitions in (13) and (22) were used. The second derivative of (3) together with Eq. (21b) provides x 1 as: (26b) Using (21b) again, but now for generic r, and also (26b), one gets the correction to the leading, purely linear order parameter function: Applying the results of subsection IV B and Appendix C, one can write for x 1 in Eq. (26b): with some d-dependent constants K 1 , K 2 , and K 3 . Since one can use the leading behaviorΛ ǫḡ ∼ x 1 in the correction terms, these constants get contributions from both X 1 and X 2 . Rearranging this equation providesΛ ǫḡ in terms of x 1 andḡ: The left-hand side is a measure of the distance from the critical temperature:Λ ǫḡ ∼ τ −ǫ/2 , note that ǫ < 0, and the temperature dependence of x 1 is obtained by finding the root of the above equation. It can be easily seen that, when approaching the critical temperature, the ratio of the correction term of x 1 to the leading one is proportional toḡ. Although this is the usual behavior , is a peculiarity of the spin glass field theory.

At the upper critical dimension: d = 6
One can evaluate (27) at exactly ǫ = 0 by using Eq. (18) for the X 1 part, furthermore Eqs.
(C2), (C4), (C5), (C6), (C7), and (C8) from Appendix C for the X 2 part: where C d=6 = 6 has been used. 2 Evidently, the correction term blows up when approaching the critical temperature, demonstrating the well-known phenomenon that the simple perturbative method breaks down at the upper critical dimension. Nevertheless, one can make a comparison with the expanded form of the renormalization group result in Eq. (34) of Ref. [6]: where the equality, in leading order, of lnΛ ≡ ln Λ/ √ 2wq 1 with − ln(2τ /Λ 2 )/2 has been used.
The discrepancy in theḡ 2 lnḡ term arises from two factors: Firstly, from the fact that only the linear leading term of the scaling functionx 1 (x) was computed in [6] which is, however, sufficient for getting theḡ 2 ln τ contribution. Secondly, theḡ 2 lnḡ term in (29) comes from the small mass regime, and it is not expected to be controlled by renormalization around the critical theory.
2 The x 2 1 ln x1 contributions from (d 2 /dr 2 X1)r=1 and (d 2 /dr 2 X2)r=1 with the Σ xx R self-energy insertion cancel each other -a surprising effect whose origin is not understood. The −2/3 x 2 1 ln x1 above comes from the Σ xx L , Σ x 1 x 1 x , and Σ xx 1 1 self-energy components.
Considering that in Ref. [6] x 1 was computed by an expansion around the replica symmetric field theory -in contrast to the present fully RSB calculation -, the agreement found in theḡ 2 ln τ term is rather reassuring.
3. Below the upper critical dimension: d = 6 − ǫ One can revive perturbation theory below six dimensions by the method of the ǫ-expansion: the d-dimensional integrals are expanded in ǫ, and the coupling constantḡ must be specially chosen for having correctly exponentiating logarithms. In this second order calculation, one can stay in six dimensions when evaluating the correction terms. As a result, Eq. (29) is only slightly modified: where C d = 6 + O(ǫ 2 ) was used. The correction term becomes infinite at criticality, except for 2ḡ = ǫ + O(ǫ 2 ) which is just the fixed point condition in the first order of the perturbative renormalization group, see for instance [6]. 3 In this way, the equation that determines x 1 becomes independent of the temperature, and x 1 itself is nonzero and universal below six dimensions: In a generic dimension d < 6, there should exist a functionF(x 1 ,ḡ) such that x 1 is determined whereĝ is the special coupling as explained in footnote 3 .
B. The Edwards-Anderson order parameter After evaluating Eq. (26a), one gets: with some d-dependent constants K ′ 1 ,. . . K ′ 4 [see the remark below Eq. (27) which is appropriate here too]. The terms first and second order in the dimensionless coupling constantḡ in the last line are quite usual in the perturbative expansion of an order parameter above the upper critical dimension, whereas the nonanalytic last term has its origin in the far infrared regime, and is specific to the spin glass field theory with the two distinct bands of the mass spectrum.
[It is remarkable that the different terms for ln 2Λ and lnΛ in d dr X 2 r=1 all cancel each other, and the −2/3 ln x 1 comes from the Σ xx R component, all the other contributions are zero.] Putting these second order terms into (33), and evaluating the first order one at d = 6, one gets: where ln x 1 was substituted by ln(6ḡ). This equation gives τ versus q 1 for a fixed temperature below T c . For getting the behavior of q 1 for τ approaching zero for a given system (i.e. for some givenḡ), one must resum an infinite series: this is just done by the renormalization group, as in Ref. [6].
The first term in the second row of Eq. (33) must now be expanded up to O(ǫ), while the second and third ones can be evaluated at six dimensions. We get for τ : The first three terms in the curly brackets can be considered as part of the expansion of (1 − 4ḡ + . . . )Λ κ , provided that (1 − 4ḡ)κ = 2ḡ − 4ǫḡ − 12ḡ 2 and 1 2 κ 2 = ǫḡ, yielding the exponentiation conditionḡ =ĝ = 1 2 ǫ -which coincides with the requirement of vanishing lnΛ contribution to x 1 found in the previous subsection, see footnote 3 -, and κ = 2ĝ − 12ĝ 2 + O(ĝ 3 ). Identifying the critical exponent β as β −1 = 1 − 1 2 κ, it is obtained: Unfortunately only the leading behavior ofĝ is available at the moment, preventing us from computing β up to O(ǫ 2 ), and compare it with known results from renormalization in the symmetric (high-temperature) theory [14,15]. (See also Ref. [8] where this first order result for β has been presented from calculation in the glassy phase.) C. The correction to the order parameter function: ∆q(r) Eq. (26c) shows that ∆q(1) = 0, as it must be. Furthermore, one can conclude from this formula that terms linear (∼ r) and cubic (∼ r 3 ) in X 1 and X 2 give no contributions to ∆q(r). Using Eqs.
(13), (14), (16), and (17), it then immediately follows: • As far as a generic dimension 6 < d < 8 is concerned, one can compute the first part of (36) by use of Eq. (17): Only terms proportional to r 5 give contributions to the second part of (36) -these come from Σ xx R , Σ xx L , Σ x 1 x 1 x , Σ xx 1 1 , and δΣ xx 1 -, yielding and also a complicated nonanalytical contribution of order x , coming from integrals in the far infrared region, which is negligible for d > 6, but becomes more and more important when approaching d = 6.
One can conclude from this 2-loop calculation that the order parameter function takes the where the nonanalytical contribution is subleading in this dimensional regime: All the temperature dependence of q(x) is absorbed into q 1 and x 1 which are, as it follows from our scheme, the exact Edwards-Anderson order parameter and breakpoint of q(x).
Furthermore, although q(x)/q 1 should, in principle, depend on both x 1 andḡ, it proved to be, at least up to the order considered,ḡ-independent.
The emergence of the x 1 × d d r 4 term, coming solely from the one-loop graph X 1 , may seem somewhat surprising, although a similar x 4 contribution has been found in the mean field order parameter function [16]. (But see Sec. VII for confronting the results here for 6 < d < 8 with their mean field counterparts, i.e. d = ∞.) • In exactly six dimensions, (36) can be evaluated by use of Eqs. (18), (C2), (C4), (C5), and (C6); note that only terms proportional to r 5 and the sole r 6 contribution from X 1 give nonvanishing result:

VI. THE STUDY OF THE MOMENTUM-DEPENDENT MASS
By Dyson's equation, the inverse of the exact propagator is identical with the mass operator Γ(p), and stability of the RSB phase demands that the eigenvalues of Γ(p = 0) be all nonnegative.
Eqs. (4) and (7) give the elements of the mass operator as: and everything above is understood to be exact quantity (the self-energy, for instance, although the same notation is used as for its first order part throughout the paper). Instead of a full analysis, our study will be confined to the highest longitudinal eigenvalue and to the family of the replicon ones.

A. The longitudinal mass
Applying the results from Ref. [17], an eigenvector f αβ of the longitudinal subspace has the same ultrametric structure as the order parameter q αβ , and we can use a similar parametrization for it, i.e.
The eigenvalue equation can now be written as The SBA has been used in the terms with the one-loop self-energy components, which is correct at the present level of approximation. It is easy to check that λ = p 2 + 2wq 1 andf (r) = r are the zeroth order solutions. Due to the x 1 in front of the bilinear form, see (11) for its definition, it is sufficient to replaceq andf by r: {q(r),f (r)} ≈ {r, r} = r − r 3 /3. Setting r = 1 and using Eq.
(A2) yield the first order result for the longitudinal momentum-dependent mass: By the help of Eqs. (B8) and (B9), and the row for δΣ xx 1 in Table I, the (zero-momentum) longitudinal mass above six dimensions is as follows: Considering that the longitudinal momentum-dependent mass is the inverse of the longitudinal exact propagator, one can expect that its behavior below six dimensions is governed by the critical fixed point, and it has the following scaling form: One can check this scaling by evaluating the self-energy components in (38) at exactly six dimensions: and Inserting these expressions into (38) and using x 1 = 6ḡ, one gets: in full agreement with the ǫ-expansion results from calculations in the symmetric (high temperature) phase in Ref. [15], wheneverḡ is substituted byĝ = ǫ/2, namely η = − 1 3 ǫ, ν = 1 2 (1 + 5 6 ǫ), and β = 1 + 1 2 ǫ.
The zero-momentum limit of Γ long is the inverse longitudinal susceptibility. By use of Eq. (40), one gets its behavior: By means of Eqs. (B5) and (A1), it follows that where the final result was obtained by neglecting irrelevant, i.e. ∼Λ −2 , terms. The zero-momentum limit of (38) can now be written as: One can then conclude that γ/β = 1 +ḡ and, using (35), γ = 1 + 2ḡ. This yields -again in full agreement with Ref. [15] -γ = 1 + ǫ, after the special conditionḡ =ĝ for the coupling constant has been applied.

B. The replicon band of Γ(p)
It was shown in Refs. [8,17] that the replicon eigenvalues of any ultrametric matrix can be easily computed by direct substitution of the matrix elements into an expression such as Eq. (41) in [17]. Inserting the components of Γ(p) in Eq. (37) into this formula, and keeping terms up to first order in x 1 , a surprisingly simple result is obtained: where 0 ≤ r = x/x 1 ≤ r 1 = u/x 1 , r 2 = v/x 1 ≤ 1, and Eq. (A2) was applied. The middle term is just the zero-momentum replicon mass; marginal stability is clearly demonstrated. The u = v = x mode is known exactly [18] being a zero-(Goldstone)mode, and we can see this here perturbatively: One can suspect that this marginality persists, and is satisfied order by order in the perturbation expansion.

VII. DISCUSSION OF THE RESULTS, AND SOME CONCLUSIONS
The glassy phase of the replica field theory representing the Ising spin glass has a special dimensionless parameter, x 1 , not present in ordinary field theories, which is a characteristic of the RSB low-temperature phase. Just below T c and in systems above the upper critical dimension (more precisely for 6 < d < 8) x 1 is related to the other two dimensionless quantities, namelȳ Λ = Λ/(2wq 1 ) In the classical perturbative regime above the upper critical temperature the behavior of τ /(wq 1 ) and Γ long (p = 0)/(2wq 1 ) are displayed in Eqs. (33) and (39). They both are the sum of a regular and an anomalous term. The regular ones have the following common structure: i,j c ijḡ i x j 1 , and the c ij coefficients for i + j = L can be calculated by an L-loop calculation (higher loop terms do not change the result), notwithstanding that the free propagator itself is an infinite series in x 1 . On the other hand, the anomalous term is nonanalytic in x 1 : it is proportional to x 2−ǫ/2 1 and x 1−ǫ/2 1 in the two cases. It is argued in the following that this anomalous contribution (which comes always from far infrared integration) is nonperturbative in the sense that higher loop graphs yield similar anomalous terms. Let us look at, as an example, the following k + 1-loop contribution to the equation of state in (3), see also (6): where the SBA was used, and only the replicon contribution is considered here, see also (24). Since {. . . } xx R is an exact eigenvalue of any generic ultrametric matrix, we can write: By the help of Eqs. (7), (A2), (B8), (B9), and Table I, one gets for momenta in the far infrared, i.e.
p 2 the approximation meaning leading order in x 1 . For the replicon free propagator one has exactly: where Eq. (8) has been used for getting the last equality. Finally, using the classification of far infrared propagators at the end of Appendix A and Eq. (A4), X (k) has the following leading contribution in the small mass regime: where the last formula was obtained by use of (19b). The order of this formula, i.e. the power of x 1 , is independent of the number of loops k + 1, and τ /(wq 1 ) gets a nonanalytic contribution which is proportional to x 2−ǫ/2 1 in any order of the loop-expansion.
What has been learnt from the study of the classical perturbative regime is extensible to the case of d < 6. τ /(wq 1 ) and Γ long (p = 0) can be separated into a regular part (coming from integration in the near infrared), and an anomalous one originating from the small mass (far infrared) sector.
The regular part is under the control of the critical fixed point, and usual critical exponents can be computed, after the separation has been done, as in V B 3 and VI A. The dangerous infrared behavior is cut off by the small mass, and it is isolated into the anomalous part; see as an example theḡ 2 lnḡ term in Eq. (34).
∆q(r) = q(x)/q 1 − r, r = x/x 1 , has been assumed in Ref. [6] to be proportional to x 2 1 , just as in mean field theory. It turns out that this is true only in high spatial dimensions, namely for d > 10. A preliminary study suggests that ∆q(r) ∼ x d/2−3 1 in the dimensional range of 8 < d < 10.
In this paper, we have computed ∆q(r) in leading order, and it proved to be of order x 1 in the whole range of 6 < d < 8. Beside this regular term, far infrared integration provides a nonanalytic (anomalous) contribution of the form ∆q(r) ∼ x 1−ǫ/2 1 , which is again expected to emerge in higher order too. Another remarkable finding of the present calculation is that ∆q(r) depends only on x 1 , and not onḡ (note that d > 6): this may be a generic property of q(x) for d < 8.
Finally, we obtained an important result for the family of the (zero-momentum) replicon mass, This formula is in complete agreement with that of the truncated model of mean field theory [8,19], any effect of the short range interaction and the geometry of the hypercubic lattice is embedded in x 1 and q 1 . Stability of the ultrametric RSB phase below six dimensions is thus demonstrated along the same lines as in mean field theory.
As an important task in the near future, one should reanalyze the small momentum behavior of G xx 11 (p) (this is the object studied numerically in three dimensions in Refs. [20][21][22]), and finding out how it changes when crossing the upper critical dimension. Now this seems to be feasible by the knowledge collected for the first order self-energy in Appendix B.
Appendix A: Some results for the free propagators in the near and far infrared regimes Equations like (24) and (B1) to (B7) are complicated but manageable in the large mass regime where everything can be expressed in terms of the two propagators introduced in (15). Here the relevant propagator components 4 are listed in dimensionless form, and, for easing the notation, we keep the old notations for the dimensionless quantities: i.e. (2wq 1 ) G → G and p 2 /(2wq 1 ) → p 2 .
All the results below are taken from Ref. [8], see also (23): As an application of these formulae, the leading contribution of the replicon self-energy (B1) at zero momentum is easily derived as where the integrals obtained were substituted by the results in Eqs. (19b) and (20).
We can also use the leading large mass propagators in (24), and after inserting the expressions from (A1), we get the following result valid in the near-infrared (p 2 ∼ 2wq 1 ) regime: The free propagator components in the far infrared region (small mass regime) have the following leading term (restoring now the dimensional dependence of p 2 again): and using the matching condition (see Sec. III) between the near and far infrared regimes, one can infer the k exponent from the leading infrared power of the large mass form in (A1): The following classes are found: 4 More precisely, their leading terms.
• k = 2: G xx LA , x < x 1 . This is the most infrared divergent propagator of all.
Appendix B: The self-energy components appearing in Eq. (24) A generic component of the one-loop self-energy matrix is shown in (25). In the present calculation, we can use the SBA when computing the self-energy components occurring in (24). These components are linear combinations according to the rules in Eq. (23), now applied for the selfenergy matrix. For easing the notation, the momentum arguments are not displayed: as a general rule, the first G is always at momentum q, whereas the second one in a product is at p − q. The interchange {q ↔ p − q} means, just as in (25), the same terms but with interchanging q and p − q. After some replica algebra one obtains:  ) is exact, only the leading 1/ǫ term for σ (a) and f (l) is shown (ǫ = 6 − d). C d is the notation for 6 Γ(d/2 − 2)Γ(4 − d/2), whereas r ≡ x/x 1 throughout the paper.

(B9)
C d is always zero except for Σ xx R , Σ xx L , and Σ x 1 x 1 L ; in these casesC d = −4C −1 d /(d − 4) where C d ≡ 6 Γ(d/2 − 2)Γ(4 − d/2). Some properties of the self-energy components which are relevant for the present calculation are summarized in Table I. The classification of the far infrared propagators in the end of Appendix A makes it possible to compute the contribution of the small mass regime p 2 ∼ x 1 (2wq 1 ). One gets a dangerous term nonanalytical in x 1 , namely X 2 ∼ (wq 2 1 ) x 2−ǫ/2 1 , which yields a ln x 1 in six dimensions: Adding together the results of (i) and (ii), one finally gets the Σ xx R -insertion result in six dimensions: The two longitudinal terms in the right hand side of Eq. (A3) can be most conveniently written as (i) The Σ xx L − Σ x 1 x 1 L part produces, due to the σ (na) in Σ xx L , a dangerous nonanalytical term in X 2 , but theΛ −2ǫ andΛ −ǫ contributions are canceled by the subtraction: ).
The interplay between the analytical and nonanalytical terms in x 1 [see Eqs. (B8), (B9), and the entries for Σ xx L in Table I] produces the ln x 1 for d = 6: (ii) The Σ x 1 x 1 L − δM insertion yields the contribution to X 2 in a generic dimension just as in (C1).
In six dimensions, it becomes: