Simple non-perturbative resummation schemes beyond mean-field: case study for scalar $\phi^4$ theory in 1+1 dimensions

I present a sequence of non-perturbative approximate solutions for scalar $\phi^4$ theory for arbitrary interaction strength, which contains, but allows to systematically improve on, the familiar mean-field approximation. This sequence of approximate solutions is apparently well-behaved and numerically simple to calculate since it only requires the evaluation of (nested) one-loop integrals. To test this resummation scheme, the case of $\phi^4$ theory in 1+1 dimensions is considered, finding approximate agreement with known results for the vacuum energy and mass gap up to the critical point. Because it can be generalized to other dimensions, fermionic fields and finite temperature, the resummation scheme could potentially become a useful tool for calculating non-perturbative properties approximately in certain quantum field theories.


Introduction
There exist two main avenues to analytically calculating general observables in quantum field theories. These are based on our ability to exactly solve free (non-interacting) quantum field theories and certain gauge theories in the limit of a large number of colors and infinite interaction strength using the conjectured gauge/gravity duality.
Perturbations around the exactly solvable cases of zero and infinite interaction strength may also be calculated, but strict perturbative series expansions prove to be divergent in most cases. Thus, quantitatively reliable results away from zero and infinite interaction strength either require the use of non-perturbative resummations, or a-posteriori justification through either experimental or numerical methods such as lattice field theory 1 .
However, there are several physics cases where one would want to have access to quantitatively accurate results for the case of intermediate coupling strengths, such as transport coefficients. This begs the question of whether it could be possible to develop new, non-perturbative techniques for quantum field theories that allow the calculations of observables at arbitrary coupling strength at least approximately.
An example of a non-perturbative resummation scheme is the so-called mean-field approximation where an infinite number of perturbative terms are resummed into an interaction-strength dependent mass term. While wildly successful, the mean-field approximation suffers from the problem that certain interesting types of observables, such as order/disorder transitions or transport coefficients, are determined by contributions which are formally higher order in perturbation theory, and hence are not captured by the mean-field approach.
The present work is meant as a step towards developing new, non-perturbative resummations schemes that could allow one to access interesting observables at intermediate couplings approximately rather than exactly, say to within 30 percent accuracy.
To keep the discussion simple, for this work I selected one theory (single component scalar theory with quartic interaction) and one particular setup in which mean-field theory fails (symmetry breaking at zero temperature in 1+1 dimensions). However, the methods presented here are straightforward to implement in other dimensions, for fermions, at finite temperature and for other observables 2 .

Setup
I consider the path integral formulation of φ 4 theory in two Euclidean dimensions given by where it should be noted that λ here has mass dimension 2 and I assume m 2 > 0 here 3 . The theory is formulated at finite temperature T by having one of the Euclidean directions (say x 0 ) be compactified on a circle of radius β ≡ T −1 . (Note that only the zero temperature limit will be considered in this work.) Introducing two auxiliary fields σ, ζ using 1 = Dσδ(σ − φ 2 ) = DσDζe i ζ(σ−φ 2 ) and subsequently integrating out σ leads to an alternative representation of the partition function as which is well-known. In this form, it is possible to integrate out the original field φ exactly, but the resulting determinant of the operator is in practice too unwieldy to handle for the subsequent integration over ζ, thus preventing an exact solution to the field theory. Except for the global zero-mode of ζ.
where V is the "volume" of the Euclidean direction x 1 , one finds Note that neglecting the contribution S I in the action, the resulting approximation becomes exactly solvable because the zero-mode contribution ζ 0 only enters in the diagonal of the operator L, thus effectively taking the role of an effective mass contribution 4 . Despite being extremely simple, the resulting approximation thus retains non-perturbative character at non-zero value of the interaction strength λ, because it corresponds to a resummation of an infinite number of diagrams in the standard perturbative approach. Therefore, I consider (2.4) with S I = 0 the starting point of the sequence of resummation schemes discussed in this work, and will refer to it as "Resummation Level Zero" ("R0") in the following.

Resummation Level Zero ("R0")
Using standard methods in field theory [1], evaluating the partition function in the R0 approximation is no more difficult than the corresponding tree-level calculation in standard perturbation theory. In the zero temperature limit one finds is the MS scale parameter. In the large volume V → ∞ limit, the remaining integral over ζ 0 can be evaluated exactly from the saddle point of the action iζ 0 = z * given by z * R0 = 4λI 0 m 2 + z * R0 , where I 0 (α) = 2 dJ 0 (α) dα 2 . The two-point function in the R0 approximation is similarly easy to compute, and one finds [1] (2.9) where K = (k 0 , k), K · X ≡ k 0 x 0 + kx 1 and here and in the following Since the pole mass M of the two-point function G(X, z * ) is an observable in principle, the sum m 2 + z * R0 must be finite and hence in the MS scheme I am led to introduce the renormalized mass m R at R0 level as such that which has the form of a non-perturbative "gap-equation". In order to make contact with other non-perturbative studies of φ 4 in 2 dimensions I adopt the scheme choicē µ 2 = m 2 R such that a solution of the above R0-level gap equation is m 2 + z * R0 = m 2 R . Note that the R0 approximation coincides with the leading-order approximation for φ 4 theory with a large number of components, cf. Ref. [2].

Resummation Level One ("R1") a.k.a. Mean-Field Resummation
In order to improve on the R0-level approximation, contributions from the fluctuation field ζ in the determinant of the operator (2.3) need to be taken into account. Expanding e −S I perturbatively in powers of ζ would be an option in principle, except that it would ruin the non-perturbative character of the approximation. An alternative approach is to re-write the theory on the level of the action by formally performing the replacement in (2.4) with an unknown constant mass parameter ν. To fix ν, it is sufficient to calculate the two-point function φ(X)φ(Y ) to first non-trivial order in the fluctuation field ζ and the counter-term ν 2 , finding which in turn suggests Similarly to the R0 approximation discussed above, the effective mass term ν 2 is determined non-perturbatively through a gap-equation, effectively resumming an additional infinite number of diagrams in the standard perturbative approach. Therefore, I consider Eq. (2.4) with the re-organization (2.13) the first correction to R0 and will refer to it as "Resummation Level One" ("R1") in the following. It is not hard to verify that the R1 approximation is nothing else but the familiar mean-field resummation. The partition function in the R1 approximation is obtained by including contributions to first non-trivial order in the fluctuation field ζ and the counter-term ν 2 , finding As before, in the large volume limit the integral can be evaluated exactly through its saddle point iζ 0 = z * given by where it should be noted that the contribution ∂ν 2 ∂iζ 0 originating from the function J 0 in the exponent exactly cancels against that from 2λG 2 R1 (0, iζ 0 ) = ν 4 32λ . The two-point function in the R1 approximation given by then gives rise to a finite pole mass M in the R1 approximation if such that (2.20) With the additional scheme choiceμ 2 = m 2 R this corresponds to the renormalization condition adopted in the lattice studies of φ 4 in 2 dimensions, cf. Refs. [3][4][5].

Resummation Level Two ("R2")
Going beyond the mean-field resummation R1 requires resumming contributions originating from higher powers of the fluctuation field ζ in the determinant of the operator (2.3). Taking the case of the R1-level resummation as an example, a further refinement of the R1 scheme can be defined by introducing dynamical propagators for both the φ, ζ fields through rewriting the action (2.4) as with unknown self-energies Π(X), Σ(X). To fix the self-energies, and thus the two-point functions G(X), D(X), I calculate the φ(X)φ(Y ) and ζ (X)ζ (Y ) to first non-trivial order in the fluctuation field ζ and the counter-terms Σ, Π, finding where I dropped the argument iζ 0 in G(X, iζ 0 ) for clarity of notation. Note that if one sets Σ(X) = 0 and Π(X) = ν 2 δ(X) one recovers the R1 level equation (2.14) for G(X, iζ 0 ). Using straight lines to depict propagators G(X) and wiggly lines to denote D(X), the above equations suggest the solutions where it should be noted that all propagators are themselves fully resummed, e.g. in momentum spacẽ . (2.25) The equations (2.24), (2.25) correspond to resummations of a class of diagrams beyond the mean-field resummation R1, hence they will be referred to as "Resummation Level Two" ("R2") in the following. Note that in theD(K) propagator for the ζ field, this reduces to the well-known RPA approximation, whereas in R2 this RPA approximation is coupled self-consistently with the approximation for the φ field. The partition function in the R2 approximation is obtained by including contributions to first non-trivial order in the fluctuation field ζ and the counter-terms Σ, Π, finding As before, in the large volume limit the integral over ζ 0 can be evaluated through its saddle at iζ 0 = z * given by such that the renormalization condition (2.19) is unchanged from the R1 level approximation.

Resummation Level Three ("R3")
Since propagators in the action (2.4) have been fully resummed in R2, going beyond the R2 level approximation requires resummation of vertices. Taking the lessons learned from the R1 and R2 level approximations, the R2 scheme can be further refined by introducing vertex corrections δΓ for the interaction term ζ φ 2 by rewriting (2.4) as where Γ = 1+δΓ and S 0 , G(X, iζ 0 ), D(X) are formally identical to the R2 expressions in Eqns. (2.21) and Γ = 1+δΓ. In order to fix the unknown vertex correction δΓ(X, Y, Z), I calculate the connected three point function ζ (X)φ(Y )φ(Z) to first non-trivial order in the fluctuation field ζ and counter-term δΓ. Expressing Γ in Fourier-space through the two incoming momenta on the φ propagator lines asΓ(P, K), this leads to where a "blob" represents a fully resummed vertex. The unknown functions Π, Σ are in turn fixed by calculating the two point functions φφ , ζ ζ to second non-trivial order in the fluctuation field ζ , second order in counter-terms Π, Σ and first order in the counter-term δΓ. This leads to where contributions have conspired in such a way that one vertex is fully resummed, whereas the other vertex is bare. The equations (2.25), (2.31), (2.32) correspond to the resummation of a class of diagrams beyond the R2 level approximation, hence they will be referred to as "Resummation Level Three" ("R3") in the following. The partition function in the R3 approximation is obtained by including contributions to second nontrivial order in the fluctuation field ζ and counter-terms Π, Σ, and first order in the vertex counter-term δΓ, finding where I used the R3 equivalent of (2.28) to rewrite S eff R3 . Using the form of the resummed vertex (2.31) it is possible to rewrite the formally three-loop contribution to the partition function as a one-loop integral as (2.34) The large volume limit of the partition function is again determined by the saddle point iζ 0 = z * given by such that the renormalization condition (2.19) is unchanged from the R1 and R2 level approximation.

Beyond R3
How to go beyond the R3-level approximation scheme? On the one hand, it is possible to improve upon the results found in the R0-R3 schemes by performing a power series expansion of S I . Taking the example of R0, this would imply performing an expansion, not around free field theory, but around a field theory with a non-perturbative mass parameter iζ 0 . This approach will require the calculation of diagrams beyond oneloop order and I suspect that it will run into the same kind of difficulties that beset ordinary perturbation theory, such as divergent series character, results oscillating with truncation order of the series, and so on. On the other hand, it is conceivable that the present schemes may be generalized by noting that the type of diagrams not fully included in the R3-level approximations are those that correspond to resummed 4-vertices. This suggests that a potential R4level approximation scheme may be constructed by adding and subtracting an effective 4-vertex Γ (4) to the action (2.4) and calculating the connected 4-point functions to first non-trivial order in the auxiliary field and counter-terms. A similar procedure would then be employed for R5 (5-point functions), R6 (6-point functions) and so on. Progressing in this fashion, it is conceivable that one could generalize the R-level schemes to higher accuracy and retain attractive features such as requiring only the calculation of one-loop integrals.

Results
The pole mass M (defined by the pole of the two-point function ( φφ (K 2 = −M 2 )) −1 = 0) is amenable to precision calculation in 1+1 φ 4 theory and hence is an interesting quantity to test the present resummation scheme. After renormalization, the pole mass can be expressed as a function of the dimensionless coupling g ≡ λ m 2

R
, and the theory becomes gap-less at a critical coupling g c with critical exponent c 1 as Similarly, at the critical coupling the two-point function scales as where the exact results are c 1 = 1, c 2 = 1 8 [6]. Another interesting observable is the vacuum energy, defined as Using the scheme choiceμ 2 = m 2 R , the pole mass in the R0, R1 approximations simply becomes M 2 = m 2 R , hence it is not interesting to discuss these approximation schemes here.
For the R2 scheme, the pole mass M is formally defined through the inverse twopoint function in Fourier space (2.29) as The R2 scheme is amenable to efficient numerical evaluation using Gauss-Legendre integration (details for the numerical evaluation can be found in appendix A). In this scheme, values for δΠ are calculated and stored for positive K 2 , and linear extrapolation is used to obtain δΠ(K 2 = −M 2 ). Results for M and Λ as a function of g are shown in Fig. 1, where also comparison to recent precision calculations of the same quantities from Ref. [7,8] are shown. As can be seen from Fig. 1, R2 results for the mass gap and vacuum energy are qualitatively similar to those obtained from Refs. [7,8], but show clear quantitative deviations. Moreover, the R2 scheme predicts a phase transition close to g c 1.89 with a critical exponent c 1 that is not consistent with the exact value c 1 = 1. (Curiously, the R2 value c 2 = 0.12(5) obtained numerically at g c 1.89 seems to be close to the exact value). Similar behavior has been reported in the RPA approximation in Ref. [9], and thus the behavior for R2 seen in Fig. 1 is thus not too surprising given that the R2 approximation employs an RPA approximation in the ζ propagator.
The R3 approximation is considerably closer to the results from precision calculations for the vacuum energy and mass gap, and indeed seems to provide a quantitatively reliable approximation at intermediate coupling. However, there are marked quantitative differences between the precision calculations and the R3 approximation, notable concerning the critical coupling g c . While recent works such as Refs. [4,5,7,8] consistently find g c 2.8, the critical coupling in the R3 approximation is approximately g c 2.2. Curiously, both the vacuum energy and the critical exponent c 2 = 0.12 (5) in the R3 approximation are numerically close to the corresponding results from precision calculations.
For convenience, the source for the R2 and R3 level algorithms described here is publicly available at [10].

Summary and Conclusions
In this work, I have presented a series of non-perturbative approximation schemes. These schemes are simple to implement, because they only require evaluation of oneloop integrals.
To test the quality of these approximation schemes, I considered scalar φ 4 theory in 1+1 dimensions at zero temperature, finding approximate agreement for the vacuum energy and mass gap with precision results from other methods at intermediate values of the interaction. Since the well-known mean-field resummation scheme utterly fails at reproducing these observables, it seems that the resummation schemes R2 and R3 presented here encode information about the quantum field theory at intermediate interaction strength approximately.
Because the resummation schemes R0-R3 are all approximate in nature, they will not be useful for calculating quantities where other, in particular exact, methods are available, such as in the calculation of critical exponents in φ 4 theory [11]. However, R0-R3 may serve as a general purpose tool for calculating properties of quantum field theories approximately at intermediate interaction strength that are inaccessible by other methods. In particular, this could be the case for calculating properties of scalar φ 4 in 2+1 and 3+1 dimension, finite-temperature properties (including transport), or properties for theories with similar Lagrangians, such as the Gross-Neveu model [12] or maybe even gauge theories along the lines of [13].

Acknowledgments
This work was supported in part by the Department of Energy, DOE award No DE-SC0017905. I would like to thank Marcus Pinto for helpful discussions and urging me to write this up note and Philipp Stanzer and Mark Watson for spotting typos. I also would like to thank Joan Elias Miro and Gabriele Spada for providing the data points from Refs. [7,8], respectively.

A Details of the Numerical Algorithm for the R2 and R3 schemes
In this appendix, details on the numerical evaluation of the R2 and R3 approximation schemes are given. For notational brevity, I choose to describe the R3 scheme and note which steps to omit in order to get the R2 approximation. As pointed out in the main text, the numerical schemes require the evaluation of one-loop integrals, e.g. of the form and whereΓ(Q, P ) = 1 (δΓ(Q, P ) = 0) in the R2 scheme. Renormalization is explicitly performed by replacing cf. Eq. (3.2). Using mass units m R = 1, I find it advantageous to compactify the integration domain by using q 0 = tan πx 2 such that for instance Integrals such as this one are evaluated numerically as quadratures with N stencils x 1 , x 2 , . . . , x N and weights w 1 , w 2 , . . . w N . I find it convenient to use Gauss-Legendre quadrature where the stencils are given by the roots of the Legendre polynomial of order N and standard weights (A.5) I find that tabulated values for x i can be efficiently obtained to high precision for N up to N 2000. In practice, I find that N 200 is typically sufficient for obtaining results of percentage precision, and I chose N odd in order to include the point x i = 0 in the set of stencils. To be explicit, for the results shown in Fig. 1, I used N = 501 for R2 and N = 201 for R3.
Once the number of stencils has been chosen, integrals such as (A.4) become and I used modified weights Evaluation ofG(x ij ) (and similarlyT (x ij )) is done by discretizing the self-energies δΠ(K), Σ(K), which at zero temperature only depend on K 2 . While it would be possible to re-use the discretization K 2 → K 2 i = tan πx i 2 from above, I find it advantageous to only store self-energies up to a certain maximal value of K 2 max , because for a fixed number of Legendre stencils, numerical errors quickly accumulate for K 2 3 4 tan πx N 2 . Hence I use the discretization δΠ(K 2 ) → δΠ i = δΠ (tan (x i arctan(K 2 max ))). In practice, I find that for K 2 max > 1000m 2 R , results no longer depend on the specific value of K 2 max . For the vertices (not needed in R2), I use a discretization that storesΓ(P, K) as a function of P 2 i = tan πx i 2 , K 2 i = tan πx i 2 and the relative angle φ between these two vectors, cos φ = P ·K √ P 2 K 2 . For the angular discretization, I use Fourier-stencils φ → φ l = lπ N φ with l = 0, 1, . . . 2N φ − 1. In practice, I find that N φ = 48 leads to acceptable accuracy in the numerical evaluations, and this choice was also used for the R3 scheme in Fig. 1.
Integrals such as (A.1) also require evaluation of self-energies and vertices inbetween stencils, for which I use linear interpolation. Also, results for K 2 > K 2 max are needed for which I set δΠ, δΓ to zero, while for Σ I use the analytic result when neglecting the internal self-energy and vertex correction The algorithm to obtain observables in the R3 (R2) scheme is then as follows: such that (2.33) becomes S eff R3 [z * ] = C Π + C V + C Σ + C Σ , , where the modified self-energy δΣ(K) from (2.34) is Note that δΣ vanishes for R2 as does the contribution C Σ . Only the contribution C V contains divergencies, while all the other ones are finite and are calculated using the quadrature scheme discussed above. Using m 2 + 3z * = m 2 R + δm 2 from (A.3) and the renormalization condition (2.19), C V can be calculated as (A.10) Renormalizing this contribution by demanding that C V (λ = 0) = 0, this leads to Summing up the contributions C ren V , C Π , C Σ , C Σ leads to the results for the vacuum energy shown in Fig. 1.
For convenience, the source for the R2 and R3 level algorithms described here is publicly available at [10].