The $O(4)$ $\phi^4$ model as an effective light meson theory: a lattice-continuum comparison

We investigate the possibility of using the 4 dimensional $O(4)$ symmetric $\phi^4$ model as an effective theory for the sigma-pion system. We carry out lattice Monte Carlo simulations to establish the triviality bound in the case of explicitly broken symmetry and to compare it with results from continuum functional methods. In case of a physical parametrization we find that triviality restricts the possible lattice spacings to a narrow range, therefore cutoff independence in the effective theory sense is practically impossible for thermal quantities. We match the critical line in the space of bare couplings in the different approaches and compare vacuum physical quantities along the line of constant physics (LCP).

We investigate the possibility of using the 4 dimensional O(4) symmetric φ 4 model as an effective theory for the sigma-pion system. We carry out lattice Monte Carlo simulations to establish the triviality bound in the case of explicitly broken symmetry and to compare it with results from continuum functional methods. In case of a physical parametrization we find that triviality restricts the possible lattice spacings to a narrow range, therefore cutoff independence in the effective theory sense is practically impossible for thermal quantities. We match the critical line in the space of bare couplings in the different approaches and compare vacuum physical quantities along the line of constant physics (LCP).

I. INTRODUCTION
The φ 4 scalar model with an internal O(4) symmetry in 4 space-time dimensions has been long used as a model for spontaneous chiral symmetry breaking [1]. The direction of the symmetry breaking is associated with the sigma meson, while the pions are the Goldstonebosons emerging as a result of the spontaneous symmetry breaking. It is also widely known, that as a field theory it is trivial, it has no finite ultraviolet limit with non-zero coupling strength [2]. Although this property is still discussed (see e.g. [3]) we accept it as a fact and investigate what is the bound set by triviality to the quantitative applicability of the model. Based on a calculation carried out by Lüscher and Weisz in the same model applied to the Higgs particle [4] one can estimate the lowest lattice spacing that can be reached in a parametrization adjusted to light mesons. This turns out to be a LW min = 0.40(4) fm, which corresponds to a maximal cutoff in momentum representation to a few times 500 MeV. This foreshadows that a scaling region of physical quantities as a function of a on the lattice is unlikely to be found without getting too close to the triviality bound, and therefore cutoff-independence, even in the effective theory sense is not feasible.
The above estimate was derived in a specific renormalization scheme for the case without explicit symmetry breaking. It is interesting to see to what extent it changes when, compared to [4], a different renormalization scheme is employed in the case when the pions are massive. At the same time, experience shows [5] that the use of continuum functional methods is less restricted in the shadow of triviality and can retain some predictivity. To study this in more details we use two continuum methods: the functional renormalization group (FRG) [6,7] in the local potential approximation (LPA) and the 2-loop and O(g 2 0 ) truncations of the two-particle irreducible approach (2PI) [5,8,9]. Treating the model as a cutoff theory, we solve it using the same bare cou- * marko@achilles.elte.hu † szepzs@achilles.elte.hu plings as in the lattice version along the LCP. Then, to compare the values of physical quantities, we need the relation between the lattice spacing a and the cutoff Λ. This is determined by matching the critical line of the model at zero temperature with the one determined by Lüscher and Weisz in [2] using hopping parameter expansion. The paper is structured as follows. In Sec. II we introduce notations for the model and summarize the details of the lattice simulations. In Sec. III we define the LCP and describe how the triviality bound is obtained. We also discuss the immediate consequences of the value of the minimal lattice spacing. In Sec. IV we compare the lattice results with those obtained in the continuum approximations, and finally in Sec. V we summarize our findings.

II. GENERALITIES
We discuss the O(N ) symmetric, Euclidean φ 4 model specifically for N = 4, described in terms of bare quantities denoted by the subscript 0 by the continuum action (omitting the obvious x ≡ (t, x) dependencies) where φ 0 is the N = 4 component field, m 0 is the mass, g 0 is the quartic coupling. In the explicit symmetry breaking term the external field H 0 is chosen to point in the direction of the first component of the scalar field with a length of H 0 , independent of x. Discretization on a periodic, 4 dimensional cubic lattice consisting of N T × N 3 S sites (using a forward derivative), and rewriting in terms of the hopping parameter κ leads to the well known lattice action [2]: with a being the lattice spacing andμ is the usual 4 dimensional unit vector. The connection between the continuum and the lattice parameters are We use Monte Carlo integration with importance sampling to evaluate path integrals. Configuration generation is done by using a poor man's heat bath algorithm, in which each site is updated using 10 metropolis steps before its neighbors are updated in order to make the new field value at the chosen site practically independent of its initial value. Between two heat bath sweeps we also include two overrelaxation sweeps in order to sample a much larger part of the phase space using the same number of configurations.

III. LINE OF CONSTANT PHYSICS
A. Observables defining the LCP The explicitly broken O(4) symmetric φ 4 model has 3 parameters, the hopping parameter κ, the quartic coupling λ and the external field h. In order to define a continuum limit 1 we give two physical prescriptions, which restrict our parameter space to the LCP, along which the lattice spacing a tends to zero in physical units, at least in principle. The two prescriptions are where m σ,π are the respective pole masses andφ R is the expectation value (denoted by the bar) of the σ component of the renormalized field, which takes the role of the pion decay constant in the linear sigma model (LSM). We choose a lower sigma mass (300 MeV) than what is generally agreed upon (≈ 450 MeV) [10]. Our choice is limited on the one hand by the fact that higher sigma masses are barely reachable in approximate continuum solutions of the LSM [11,12] and on the other hand by the fact that we want to retain the kinematic possibility of the σ → 2π decay.
To obtain the pole masses we measure time slice correlators. Let us define a time slice as and then is the time slice correlator matrix for one configuration. The ensemble average of C ij (t) is the time slice correlator. By our choice of h the σ direction is i = 1, therefore C σ (t) ≡ C 11 (t) is dominated by m σ , while C ii (t) , i = 1 are all dominated by m π . We do a least squares fit using the function with parameters A , B and m to C σ (t) as well as to the average of the three pion directions 2 The average and error of the fit parameters and in particular the masses are obtained by a jackknife analysis.
The fit is carried out on each jackknife sample, leaving out the t = 0 point of the correlator from the data in order to lower the distortions caused by higher excitations.
In the case of the sigma mass, one must take care of the disconnected part of the correlator. The connected part of the correlator is where M 1 is the first component of the average field over one configuration To subtract the correlated errors from connected sigma correlator, instead of (12) we use an other prescription (the two definitions differ only in a constant), The definition (12) has a bad signal to noise ratio due to correlated errors which are cancelled in (14) leading to a better signal. We show the reduction of error achieved by using the definition in (14) in Fig. 1.  (12), while the greens using (14). The two definitions differ in a constant, but here they are shifted on top of each other for better comparison.
The measurement ofφ R goes as follows. The ensemble average of the first component (the sigma direction) of M is where the 0 index on the right hand side denotes that φ 0 is a bare fields, that is wave function renormalization is still needed. Then the renormalized vacuum expectation value isφ We obtain Z by prescribing the value of the zeromomentum inverse pion propagator to be the pion pole mass: Through a Ward identity [14] the inverse two-point function can be rewritten as which, in terms of lattice quantities and combined with the renormalization prescription, leads to The value of Z is slowly changing between 0.74 and 0.8 along the LCP in the measured range of a.

B. Determining the LCP
We search for each point of the LCP curve by fixing one of the parameters (usually h, but in the region where triviality strongly influences the LCP we fix λ) to a chosen value and then scan the 2 remaining dimensions, measuring the observable ratios appearing in (7a) and (7b) on the parameter grid. We then find the set of physical points by linear interpolation between grid points for both criteria and find the intersection of the two sets by fitting parabolas on them. The intersection is one point of the LCP corresponding to the h or λ where the grid was defined. The error is estimated by a bootstrap resampling using 10 4 samples, while the original observable ratios were obtained using 16 × 16 3 lattices with 10 5 field configurations.
Using the method outlined above, we obtain the points of the LCP shown in Fig. 2. The conversion of a to  (15) 0.763 (14) 1.02 (31) 0.631 (9) 0.595 (15) 0.574 (15) 0.564 (11) [2] is also shown for comparison. Bottom: The original bare self-coupling defined in (1) as a function of the lattice spacing. The second order perturbative β-function (21) is also shown (red line, standard deviation shaded), with the parameters g1 and a1 being fitted to the data. Using this functional form we can also estimate the triviality bound amin. The error was obtained using bootstrap resampling which also had samples having their poles around a = 0.55 fm causing the standard deviation growing enormously in that region. physical units is done using aφ R = a phys × 93MeV = a phys × 93/197.327 fm −1 . (20) In the top panel of Fig. 2 we see that the LCP follows the critical line in the κ − λ plane. In a theory with a proper continuum limit the LCP should run into the critical line at least at infinite coupling. Triviality appears here by seeing that even at λ → ∞ the LCP does not converge to the critical line, meaning that a remains finite. This means that the bare φ 4 coupling g 0 must have a pole as a function of a, at the minimal value of the lattice spacing. The results for g 0 are shown in the bottom panel of Fig. 2 and are in compliance with the generally accepted view on the triviality of the φ 4 model. Fitting g 1 and a 1 to the data shown in the bottom panel of Fig. 2 in the second order perturbative β-function with being the standard β-function coefficients [15], we estimate a min = 0.52 (2) fm. This leads to an estimate for the minimal value of the lattice σ mass, am σ = 0.79 (3). The result for the minimal lattice spacing can be compared to the one which can be given based on [4]. In the renormalization scheme of Lüscher and Weisz where g R is the renormalized quartic coupling, m R is the renormalized mass, which we identify with the sigma mass for the sake of the estimate, and v R ≡φ R takes the value of the pion decay constant as in our case, although the Z factor, which we do not need here, is defined differently. In [4] the renormalization trajectories are described, and taking the λ → ∞ limit in them yields a relation between m R a min and g R : where the number 1.9(1) is the result of a numerical calculation at a high order of the hopping parameter expansion. Plugging m R = 300 MeV and v R = 93 MeV into (23) and (24) yields a LW min = 0.4 fm already mentioned in Sec. I. We see that our result a min = 0.52 (3) is even more restrictive.
An important implication of the largeness of a min is that on the lattice the maximal temperature that can be simulated is T = (N t · a min ) −1 | Nt=1 ≈ 420 MeV. Furthermore if one is interested in a "continuum limit" in the effective theory sense the feasible temperature range is definitely below 50 MeV. This limits the comparison of the continuum methods practically to vacuum quantities.  The critical curve determined in the continuum theory using various approximations as compared to that obtained with the hopping parameter expansion to 14th order. In the perturbation theory (PT) and the 2PI approach c = 4.9, while in the FRG study c = 6.923. The points of the LCP corresponding to the fixed h data points shown in the top panel of Fig. 2 are also presented.

IV. COMPARISON WITH 2PI AND FRG RESULTS
Since according to the previous section, a comparison between lattice and continuum physical quantities is not feasible at finite temperature, we remain at T = 0 and using continuum functional methods we determine the masses along the LCP shown in Fig. 2. In order to compare to a lattice result determined at a fixed lattice spacing, we need to treat the continuum version of the model as a cutoff theory, hence we need the relation between the continuum cutoff Λ and the lattice spacing a, i.e. we need c = Λa. This relation was studied in [16,17] where the conversion factor c ≈ 4.9 was calculated analytically for the 4D hypercubic lattice and obtained also by fitting the perturbative continuum result (using (4) and (6)) to the critical line m 2 0,c (g 0 ) obtained by Lüscher and Weisz in [2]. The above equation comes from the condition of vanishing curvature mass at vanishing field value at second order in the perturbation theory. We can see in Fig. 3 that at O(g 2 0 ) it reproduces the LW critical line only at small values of the coupling. One expects that this behavior changes if one uses a more sophisticated approximation.
A. The critical line m 2 0,c (g0) in the 2PI framework In the 2PI framework, the curvature mass at vanishing field value is given at O(g 2 0 ) level of truncation of the effective action byM 2 φ=0 =M 2 φ=0 (K = 0), where the gap mass satisfies the self-consistent equation [18] The tadpole and setting-sun integrals involves the propa-gatorḠ(K) = 1/(K 2 +M 2 φ=0 (K)). The critical line is determined from the condition of vanishing curvature mass: M 2 φ=0 =M 2 φ=0 (K = 0) = 0. The nontrivial momentum dependence makes (26) rather hard to solve, however the solutionM 2 φ=0 (K = 0) can be approximated by using a localized propagator with momentum independent mass-gapM 2 = m 2 0 + N +2 . This approximation corresponds in fact to the two-loop 2PI truncation. In this approximation, the tadpole can be explicitly computed with a 4D cutoff Λ and the condition of vanishing curvature mass can be written as whereS(M 2 ) is a perturbative setting-sun integral at vanishing external momentum and we used tilde for a quantity scaled by appropriate powers of Λ. For a given g 0 , one then solves (27a) forM 2 and using this solution one has m 2 0,c from (27b). The critical line obtained in this way is shown in Fig. 3. It still deviates from the LW curve, but remains closer to it in a wider range of the coupling than the O(g 2 0 ) perturbative curve. We mention that (25) can be obtained by using first (27a) in the first term on the r.h.s. of (27b) and then taking theM 2 → 0 limit, in whichS(M 2 ) → 2/(4π) 4 . Now let us discuss the determination of the critical line by solving (26) without further approximation. m 2 0,c (g 0 ) could be obtained in principle by approaching it from the symmetric phase: fixing g 0 , the equation is solved for increasing values of |m 2 0 | and m 2 0,c (g 0 ) is obtained by extrapolating the determined values ofM 2 φ=0 (K = 0) to zero. As detailed in Appendix B, (26) is solved by treating the setting-sun S(K) as a double convolution: a convolution of the propagator with a bubble integral, where the later is itself a convolution of two porpagators. It turned out that the solution to (26) is lost for a value of m 2 0 whereM 2 φ=0 (K = 0) is nonzero (see Fig. 6). This loss of solution, which seems to be a feature of the O(g 2 0 ) 2PI gap equation, and was investigated in details in [18], prevents us from the direct determination of the critical line at this order of the 2PI truncation scheme and furthermore from a comparison along the LCP. As Fig. 3 shows the simpler, 2-loop approximation indeed has a critical line determined by the equations (27a) and (27b). Nevertheless a loss of solution can also happen in this approximation in the broken phase (that is at φ = 0) depending on the parameters [18]. We found that the usual iterative procedure to solve the broken phase 2-loop equations (which were written down and solved as detailed in [5] with little modifications to accomodate for the use of non-renormalized equations and approximating the T → 0 limit numerically) break down close to the critical line in comparison to where the points of the LCP are and therefore in the LCP points no solution exists and no comparison can be made. We checked that this loss of solution persists in the even simpler localized 2-loop approximation which we detailed in [18]. We conclude that in the considered approximations the 2PI formalism cannot be compared to the lattice LCP results.

B. Determination of observables using the FRG method
Another functional method from which one can calculate curvature masses along the LCP is the functional renormalization group method. The flow-equation describing the evolution of the scale-dependent average action Γ k from the ultraviolet (UV) scale k = Λ, where the microscopic theory is defined through the bare action, down to the deep infrared (IR) where the usual quantum effective action is obtained in the k → 0 limit is [6] where R k is a regulator function, that is in momentum space it suppresses the IR modes, while ∂ k R k regulates the integral in the UV. In the local potential approximation (LPA) the Ansatz is used, where ρ = φ 2 /2 is O(N )-invariant and it is customary to choose the LPA-optimized regulator [19] R k (q) = (k 2 − q 2 )Θ(k 2 − q 2 ) (q is the Euclidean 4momentum). Then, using ij with P L/T being the longitudinal/transverse projectors, the integral can be performed and, at zero temperature and d = 4, one obtains . This equation is solved numerically by integrating it down to k = 0 (in practice to some k end > 0, due to the flattening of the potential) starting at scale k = Λ, where the initial condition for the potential is given in terms of the couplings m 2 0 and g 0 as U k=Λ (ρ) = m 2 0 ρ + g 0 ρ 2 /6. In the so-called grid method U k (ρ) is discretized using N ρ grid points so that (30) transforms into a system of N ρ coupled ordinary differential equations. We solve this system using the Runge-Kutta-Fehlberg algorithm with adaptive step-size control provided by the GNU Scientific Library (GSL) [20]. We work in units of the cutoff, denoting with tilde a quantity scaled with the cutoff, and choose N ρ = 5000 equidistant values ofρ = ρ/Λ in the range between 0 toρ max = 0.026. The flow was stopped atk end = 1.28 · 10 −2 where all the monitored quantities became practically constants. At each point of the grid the 1st and 2nd order derivatives of the potential are calculated with O(∆ρ 4 ) finite difference formulas. The minimum of the potential is obtained with spline interpolation, while the curvature masses at the minimum are obtained fitting a 6th order polynomial to the potential in an appropriateρ interval which has the minimum as its left endpoint.
The transverse and longitudinal curvature masses are obtained at k end asM T = U k (ρ) andM L = (M 2 T + 2ρU k (ρ) 1/2 , whereρ =φ 2 /2 is the minimum of the potential. In the LPA they can be regarded as approximations to the pole masses due to the simplicity of the Euclidean propagator. In order to compareM T/L along the LCP with the value M σ = 300 MeV and M π = 138 MeV which in the lattice simulation are constant along the line, we need to know what is the relation between the cutoff scale Λ and the lattice spacing a. This relation is obtained by matching the critical curvem 2 0,c (g 0 ) determined in the FRG case to the one obtained by Lüscher and Weisz in [2] using hopping parameter expansion to 14th order. We determinem 2 0,c (g 0 ) working at fixed g 0 and using dichotomy onm 2 0 , as shown in Fig. 4, where the quantity that distinguishes between the broken and symmetric phases is U k (0)/k 2 .
Once we matchm 2 0,c and m 2 0,c a 2 at some value of g 0 , finding the relation aΛ ≈ 6.923, the entire critical curve determined using FRG agrees with the one obtained by Lüscher and Weisz, as shown in Fig. 3. The very good agreement of the two critical curves is in line with the findings of Ref. [21], where it was reported that in the one component φ 4 model the critical line, obtained in the LPA with Litim regulator and lattice discretization, compares well with the one determined with Monte Carlo simulations. Having obtained the relation between a and Λ, we can now solve the flow-equation (30) and determine the curvature masses for the fixed h data points of Fig. 2. The results shown in the first four raws of Table I in units of the cutoff can be used in two ways. In the first case, 0 | > |m 2 0,c | we are in the broken symmetry phase and U k (0)/k 2 → −1 as k → 0, while for |m 2 0 | < |m 2 0,c | we are in the symmetric phase and U k (0)/k 2 → ∞ as k → 0. The closer |m 2 0 | is to the critical value, the larger is ln(Λ/k) at which a curve steeply goes upwards or downwards.
shown in the last four columns of Table I, one can determine for each point of the LCP the value of the cutoff from the lattice spacing using (31). Thenφ is smaller than f π = 93 MeV by ∼ 8%,M T is 5 − 8% larger than M π = 138 MeV, whileM L is 15 − 20% larger than M σ = 300 MeV. In the second case one can requireφ to be f π . In this case, due to the larger value of the cutoff, one finds thatM L is 22 − 30% larger than the sigma values used to determine the LCP, whileM T is larger by around 10% than the pion mass. The deviation from the lattice results decreases for smaller a. With the chosen quartic potential at the initial value of the scale, k = Λ, the flow-equation (30) cannot be solved for m 2 0 < −Λ 2 < 0 due to a singularity in the equation. One could either change the initial condition by including higher order, perturbatively nonrenormalizable terms in the potential or, as we do it here following [22], try to circumvent the problem by modifying the flow-equation expanding in power series to some order N g the fractions appearing in the right hand side of (30) 1 where ξ = (M 2 0 −M 2 L/T (k))/(k 2 + M 2 0 ) with M 0 some large parameter, i.e. M 0 > Λ, which for numerical reasons has to be chosen appropriately. 3  TABLE I. Field and curvature mass values in units of the cutoff at the minimum of the potential of the LCP shown in Fig. 2 and Fig. 3. The points are denoted by Pi with i ∈ 1, ..., 10 in increasing order from left to right of the LCP. For the first six (fixed h) points the values comes from the direct numerical solution of (30), while for the last four (fixed λ) points the values come from the solution obtained using the expansion (32) with an extrapolation to Ng = ∞. First, keeping the numerical framework used so far, that is changing only the right hand side of (30) according to (32), we tested the method in a case where a direct solution to (30) exists and then we applied it for the fixed λ data points of the LCP shown in Fig. 2 (points P7-P10 in Table I). In the latter case the solution is regarded as an approximation to the solution of the original Wetterich equation (28), assumed to exist for an appropriate form of the effective action at scale Λ.
In case of point P1, it turns out that in order to reproduce the available direct solution of (30) with the expansion method, one has to go to rather high orders in the expansion. Also, for the method to work, the 1st and 2nd derivatives of the potential atρ max had to be kept fix as a function of k, however the chosen values were practically arbitrary. We fixed the derivatives to their values calculated at k = Λ.
At a given order of the expansion the deviation from the direct result increases with M 0 . Among the studied quantitiesM L , presented in Fig. 5, shows the slowest convergence rate with N g at a fixed value of M 0 . For M 2 0 = 2 and N g = 50 the deviation from the direct result is around 10%. To estimate the result of the curvature masses and the minimum of the potential we fitted with f (x) = a + b/(x − c) d the data obtained at various N g with the expansion method. For P1 one can practically recover the direct results from a data set obtained with up to N g 100 terms in the expansion, but as |m 2 0 |/Λ 2 increases we need larger M 2 0 and larger values of N g to maintain the quality of the fit. Eventually, numerical errors prevent us for going above a certain value of N g . All these features are illustrated in Fig. 5 and the results obtained with the expansion values are given in the last four rows of Table I. Based on the variation of the extrapolated results on the fitting N g -interval, one can estimate the error ofM L to be 1 − 2% for P7 and P8 and 5 − 10% for P9 and P10. For the other two quantities the error of the extrapolation to N g = ∞ is smaller.  Table I

V. CONCLUSION
We studied the four component Euclidean φ 4 model in four dimensions. In the presence of an explicit symmetry breaking term, we determined with Monte Carlo simulations the line of constant physics (LCP) in the bare parameter space of the model based on ratios involving the pion and sigma masses and the expectation value of the field. In this process we brought further evidence in support of the triviality of the model in a renormalization scheme which is different from the one usually used by the lattice community (see [23] for a recent study).
Using the bare couplings of the LCP, we solved the model with two continuum functional methods (the 2PI formalism and the FRG method) in an attempt to compare the vacuum masses and expectation value obtained with these continuum methods to the corresponding input values of the lattice study of the model. The manifestation of triviality prevented us from a meaningful comparison of finite temperature quantities. It turned out that the comparison at T = 0 can be done only with the FRG, since the 2PI is hindered by the loss of solution to the propagator equation. The needed relation between the lattice spacing and the cutoff, used in the lattice and continuum versions of the model, respectively, was obtained by matching the critical line of the parameter space determined originally by Lüscher and Weisz using hopping parameter expansion. is calculated in this way on a grid, then splined and used for the calculation of S(K). The iterative solution of (26) obtained for g 0 = 150 and m 2 0 /Λ 2 = −0.5275 using under-relaxation method [27] with parameter α = 0.1 is shown in Fig. 6. The upper part of the figure shows what happens if the solution obtained with DHT is used as an initial propagator in the solver that computes the convolutions using adaptive integration routines on a grid with 256 momentum values. We see thatM 2 φ=0 (K) obtained in the first iteration deviates by 5 − 8% from the solution obtained with DHT as a result of the fact that, as anticipated, the setting-sun calculated with DHT is not accurate. As the iteration progresses,M 2 φ=0 (K) departs even more from the used initial function and hence the converged solution is substantially different than the one obtained with DHT.
In the lower part of Fig. 6 we showM 2 φ=0 (K min ) as function of |m 2 0 | at four values of g 0 . The difference between the curves obtained with the two ways of treating the convolution increases with the value of the coupling. This is due to the fact that the numerical error made in computing the convolution with DHT is magnified when the setting-sun is mupltiplied with a larger coupling. More importantly, the shape of the curves is compatible with the fact that the solution of (26) is lost at some value of m 2 0 whereM 2 φ=0 (K min ) is still finite. As a result the critical line cannot be determined.