Elsevier

Nuclear Physics B

Volume 741, Issue 3, 8 May 2006, Pages 390-403
Nuclear Physics B

Repairing Stevenson's step in the 4d Ising model

https://doi.org/10.1016/j.nuclphysb.2006.02.026Get rights and content

Abstract

In a recent paper Stevenson claimed that analysis of the data on the wave function renormalization constant near the critical point of the 4d Ising model is not consistent with analytical expectations. Here we present data with improved statistics and show that the results are indeed consistent with conventional wisdom once one takes into account the uncertainty of lattice artifacts in the analytical computations.

Introduction

One of the apparently simplest quantum field theories in four dimensions is the ϕ4 theory with n components. Conventional wisdom (CW) holds that the theory is trivial for all n1.2 Unfortunately there is presently no rigorous proof. One relies heavily on the validity of renormalization group (RG) equations for physical quantities [1] together with boundary conditions at finite cutoff provided by non-perturbative methods [2], [3].

Apart from its purely theoretical interest it is of phenomenological relevance since for the case n=4 it constitutes the pure Higgs sector of the Minimal Standard Model (MSM). The fact that ϕ44 is (probably) trivial does not invalidate (renormalized) perturbative computations for amplitudes at energies well below the physical cutoff where the MSM may be a good effective theory.

In the past triviality has been invoked to propose upper bounds on the mass of the Higgs boson (see e.g. Refs. [4], [5], [6], [7], [8]). It must be stressed that these bounds are non-universal, they depend on the particular regularization. But for a given regularization it is conventionally accepted that such a bound can be given a precise meaning. In recent papers Cea, Cosmai, and Consoli (CCC) [9], [10] claim that triviality itself cannot be used to place upper bounds on the Higgs mass even for a given regularization. They assert that standard predictions of the RG analysis for the behavior of some quantities near the critical line are not valid. If true this would indeed be rather important because it would reveal a serious flaw in our conventional theoretical understanding of the pure ϕ44!

In Ref. [11] Duncan, Willey and the present authors explained why critiques of the standard picture raised in Ref. [9] were not relevant. Nevertheless one must admit that the unconventional picture of CCC cannot be ruled out by present numerical simulations. Recently two papers appeared, the first by CCC [10] and the second by Stevenson [12], again claiming finer but significant discrepancies between quantitative (standard) analytic predictions and numerical data in the 4d Ising model.3

It is the purpose of this paper to reply to these challenges and to demonstrate that they are too weak to seriously cast doubt on CW. The main objection by CCC and one by Stevenson are rather easy to dismiss. A second objection by Stevenson is more difficult. It concerns a certain difference Δ between the wave function renormalization constants below and above the critical pointΔ=ZˆR(κ=0.074)ZˆR(κ=0.0751), which we have called “Stevenson's step” in the title. The κ values chosen here are about the closest to the critical point where one can presently obtain good statistics with some (but not unreasonable) computational effort.4 Stevenson claims that “theoretical predictions for Δ cannot be pushed above 0.05 well short of the “experimental” value ΔMC=0.071(6)". It is of course debatable whether such a small discrepancy indicates a potential problem, however we decided it merited more careful investigation.

It is however clear that one is here addressing few percent effects, and to clarify the situation we need both data and analyses which are precise to this level. The main analytic sources of error concern the treatment of higher order cutoff effects in the framework of renormalized PT. The main sources of error in the numerical side are the determinations of the zero momentum mass mR; apart from the statistical errors one has the systematic errors in the procedures to extract mR from the (finite volume) data particularly in the symmetry broken phase.

In the next section we present a summary of the available raw data in both the symmetric and broken phases. We have performed simulations in both phases and in particular increased the statistics at previously measured κ values in the broken phase by a factor of ∼10.

We then discuss various determinations of ZˆR from the data and compare with theoretical expectations. Next we show that these data are not in contradiction with conventional wisdom. The reason for this conclusion differing from that of Stevenson has two main origins. Firstly, unfortunately our central value of ZˆR at κ=0.0751 in [11] is about one standard deviation lower than that obtained from the present run. In this connection we remark that in those runs we were not aiming at high precision but only sufficient to reach our goal to present evidence that ZˆR is not increasing logarithmically as one approaches κc. Secondly, we point out that there is a quantitative uncertainty on the O(a2) lattice artifacts which are of course increasingly relevant as one goes away from the critical point. This is of course not at all new, however sometimes forgotten and perhaps underestimated in standard RG analyses.

Section snippets

Ising MC simulation and results

We work on hypercubic lattices of volume L4 with periodic boundary conditions in each direction and with standard action. In this paper we adopt the notations and definitions in Ref. [11], and will generally not repeat them here.

If one just wanted to obtain Stevenson's step, measurements at only two κ values are required. However precise simulations at these points are CPU expensive and hence it is useful to compute also at points with smaller correlation length to observe the approach to

Determination of mR and derived quantities

The usual expressions for the wave function renormalization constant and renormalized couplings involve the infinite volume zero momentum mass. In particular a precise determination of ZˆR (as required to discuss Stevenson's step) requires an equally reliable determination of mR. Firstly, in considering the r-point functions one can verify using the formulae given in Ref. [17] that finite volume effects coming from tunneling are negligible for our lattices. We then adopted two fitting

Reply to some other critiques

In Ref. [10] the authors reconsider the quantity v2χ in the broken phase which according to CW behaves asv2χ=a1(2527ln)+a2+, where |ln(κκc)| witha1=9C2232π2,a2a1=lnC3+2lnC11.6317. A fit of the expression (4.1) to the data gave [11] a1=1.267(14) and a2=2.89(8) whereas the theoretical prediction based on the results quoted in [3] for the values of the non-perturbative constants C1, C2 and C3 is a1=1.20(3) and a2=1.6(5). The authors of [10] claim that such a comparison “shows that the

Conclusions

An experimental observation contradicting a prediction of an until then accepted theory is always an exciting event. It invalidates the theory as it stands and inevitably leads to progress in our understanding. Similarly finding mismatches between theoretical predictions and numerical simulations in the ϕ44 theory as claimed in Refs. [10], [12] would be a serious blow if they withheld scrutiny. We hope to have convinced the reader in this paper that conventional wisdom concerning this

Acknowledgements

We thank the Leibniz-Rechenzentrum where part of the computations were carried out. This investigation was supported in part by the Hungarian National Science Fund OTKA (under T049495 and T043159) and by the Schweizerischer Nationalfonds.

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1

On leave from Eötvös University, HAS Research Group, Budapest, Hungary.

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