Is an effective Lagrangian a convergent series? Ariel R. Zhitnitsky * Physics Department, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada V6T 1Z1 ~Received 14 February 1996! We present some generic arguments demonstrating that an effective Lagrangian L eff which, by definition, contains operators O n of arbitrary dimensionality in general is not convergent, but rather an asymptotic series. It means that the behavior of the far distant terms has a specific factorial dependence L eff ; ( n ( c n O n / M n ), c n ;n !, n @1. We explain the main ideas by using QED as a toy model. However we expect that the obtained results have a much more general origin. We speculate on possible applications of these results to various physical problems with typical energies from 1 GeV to the Planck scale. @S0556-2821~96!05220-4# PACS number~s!: 11.10.Ef, 12.38.Lg I. INTRODUCTION Today it is widely believed that all of our present realistic field theories are actually not fundamental, but effective theories. The standard model is presumably what we get when we integrate out modes of very high energy from some unknown theory, and like any other effective field theory, its Lagrangian density contains terms of arbitrary dimensional- ity, though the terms in the Lagrangian density with dimen- sionality greater than four are suppressed by negative powers of a very large mass M . Even in QCD, for the calculation of processes at a few GeV we would use an effective field theory with heavier quarks integrated out, and such an effec- tive theory necessarily involves terms in the Lagrangian of unlimited dimensionality. The basic idea behind effective field theories is that a physical process at energy E ! M can be described in terms of an expansion in E / M , see recent reviews @1–3#. In this case we can limit ourselves by considering only a few first leading terms and neglect the rest. In this paper we discuss not this standard formulation of the problems, but rather, we are interested in the behavior of the coefficients of the very high dimensional operators in the expansion. We shall dem- onstrate that these coefficients c n grow as fast as a factorial n ! for sufficiently large n . Thus the series under discussion is not a convergent, but an asymptotic one. Such a behavior raises problems both of a fundamental nature, concerning the status of the expansion and of practical importance, as to whether divergences can be associated with new physical phenomena. It means, first of all, that in order to make sense, such a theory should be defined by some specific prescrip- tion, for example, by Borel transformation. Let us note, that our remarks about the factorial depen- dence of the series for large n @1 is an absolutely irrelevant issue for the analysis of standard problems when we are in- terested only in the low-energy limit. We have nothing new to say about these issues. However, sometimes we need to know the behavior of a whole series when the distant terms in the series might be important. In this case the analysis of the large order terms in the expansion has some physical meaning. Such a situation may occur in a variety of different prob- lems as will be discussed in more detail later in the text. Now let us mention that, in general, it occurs when the energy scale E is close to M and/or when two or more intermediate, not well-separated scales, come into the game @4#. This paper is organized in the following way. In the next section we consider our basic QED example, where the fac- torial behavior of the coefficients in front of the high- dimensional operators is explicitly calculated. After that we argue that this property is a very general phenomenon of the effective field theories. 1 In conclusion, we make some speculations regarding pos- sible applications of the obtained results to different field theories with very different scales ~from QCD problems to the cosmological constant problem!. II. BASIC EXAMPLE: QED We begin our analysis with the following remark. An ef- fective field theory can be considered as a particular case of the more general idea of the Wilson operator product expan- sion ~OPE!. It has been demonstrated recently @6#, that the OPE for some specific correlation functions ~heavy-light quark system Q ¯ q ) in QCD is an asymptotic, and not a con- vergent series. The general arguments of the paper @6# have been explicitly tested in two-dimensional QCD ~QCD 2 ) ~where the vacuum structure as well as the spectrum of the theory is known! with the same conclusion concerning the asymptotic nature of OPE @7#. In both cases the arguments * Electronic address: arz@physics.ubc.ca 1 The generality of this phenomenon can be compared with the well-known property of the large-order behavior in a perturbative series @5#. As is known, a variety of different field theories ~gauge theories, in particular! exhibits a factorial growth of the coefficients in the perturbative expansion with respect to a coupling constant. This growth in perturbative expansion is very different from the phenomenon we are discussing, where the factorial behavior is re- lated to high-dimensional operators, and not to coupling constant expansion. However, in spite of the apparent difference of these phenomena, they actually have some common general origin. We shall discuss this connection later. PHYSICAL REVIEW D 15 OCTOBER 1996 VOLUME 54, NUMBER 8 54 0556-2821/96/54~8!/5148~5!/$10.00 5148 © 1996 The American Physical Society