The Maximal Analyticity Hypothesis in Effective Theories.
Alexander V. Vereshagin
St. Petersburg State University, St. Petersburg.
In the publications ,  the general method allowing one to fix the values of the low-energy coefficients (LEC’s) appearing in ChPT series was developed. The method is based solely on the commonly accepted postulates of Effective Field Theory (EFT) plus certain analyticity requirements (maximal analyticity and the polynomial boundedness) which are not guaranteed by the inner structure of EFT. Besides, we use the quark-hadron duality concept allowing us to work in terms of hadron fields.
In those papers we rely on the existence of a special parameterization allowing one to rewrite (without loss of generality) the Hamiltonian of EFT in a form containing only those vertices which survive on the mass shell. This simplifies considerably the process of computations. However, the price to be paid for simplicity is that we can work with S -matrix elements only; the generality happens lost if the method is used to study Green’s functions.
The maximal analyticity requirement is formulated as follows: the tree-level amplitude of an arbitrary scattering process must be a meromorphic function of the invariant kinematical variables (pair energies and momentum squares). This is by no means a trivial requirement because the presence of unlimited number of derivatives in the EFT Lagrangian results in the expression of a tree-level amplitude which takes a form of the infinite series expansion in powers of kinematical variables. Thus, even on the first—tree-level—step one meets a problem of convergence. This problem is closely related with the expected analytic structure of tree-level amplitudes because the divergence of a series expansion mirrors the presence of a singularity of the amplitude in question. Since the loop expansion machinery automatically produces all the necessary singularities (namely, those required by the unitarity condition), we conclude that the only way allowing one to prevent the generating of unnecessary ones in the process of loop calculations consists of imposing the formulated above condition of maximal analyticity on the structure of tree-level amplitudes or, the same, on the structure of allowed resonance spectrum and corresponding couplings.
The maximal analyticity requirement, by itself, is too general to fix a theory: in particular, it does not make any difference between strong and weak forces. This is the reason for attracting of one more requirement which takes account of the well known feature of strong interactions - the Regge asymptotic behavior of high energy amplitudes at fixed value of the momentum transfer. Mathematically, it is formulated as the polynomial boundedness requirement for the tree-level binary amplitudes, the bounding polynomial degrees being dictated by the values of Regge intercepts (on a given stage we consider only small values of the momentum transfer). It should be also stressed that the polynomial boundedness of a meromorphic function is understood precisely as in complex analysis (the boundedness on the infinite system of closed contours in a complex plane).
At first glance, the imposing of polynomial boundedness restriction on the tree-level amplitude might look a bit strange because the true high energy behavior cannot be computed if one neglects the loop contributions. However, there is another argument in favor of this requirement: the polynomial boundedness of trees at zero values of corresponding variables (momentum transfers) is a necessary condition providing the generalized renormalizability of EFT (the detailed proof of this statement would require much space; it will be given elsewhere). Thus, in fact, we need only to fix the specific values of the bounding polynomial degrees.Our choice looks preferable from the phenomenological standpoint . It is shown that it results also in a correct value of the experimentally known relation.
By construction (we do not consider a problem of anomalies), EFT is a renormalizable theory (in a general sense – see . Next, by the very meaning of a renormalization procedure, one has to keep fixed the numerical values of the essential parameters appearing in the bare Lagrangian - this is a matter of the renormalization prescription. If we call as symmetry any kind of relations between the (essential) bare parameters, we can say that the renormalization should respect symmetry requirements. In our case the symmetry requirements happen to be highly nontrivial: they appear in the form of an infinite system of bootstrap conditions connecting among themselves the values of bare Lagrangian parameters. One extremely interesting property of those relations is that they connect the parameters of boson spectrum with those of the baryon one, in other words, they demonstrate certain features of a (very complicated) supersymmetry.
As well known, supersymmetry strongly restricts the possible divergences, in certain cases it leads to finite theories. The other - no less important feature of the bootstrap conditions is that they are based on the same principles as those used as the basis for string theories.
Altogether, the above notes give a hope that the solution of bootstrap equations does exist and the corresponding spectrum parameters can be explained in terms of superstrings. Perhaps, this very circumstance explains the reason for interest to the models of low-energy effective action based on certain properties of a string theory (see, e.g.,).
1. Vladimir V. Vereshagin, ``Tree-level p ,K amplitude and analyticity", Phys. Rev. D55, 5349 (1997).
A. Vereshagin "Maximal analyticity requirements in effective field theories." Proc. of the St. Petersburg's Young Scientists Seminar for 1998 Physics and Astronomy Grant competition winners. A.F.Joffe Phys.Tech.Inst., Rus. Acad. of Sci., St. Petersburg, 1999. Pp. 7 - 8
2. Alexander V. Vereshagin, and Vladimir V. Vereshagin, ``Effective theories with maximal analyticity", Phys. Rev. D59, 016002 (1999).
3. S. Weinberg, ``The Quantum Theory of Fields" (Cambridge University Press, Cambridge, 1996), V. 1.
4. J. Alfaro, A. Dobado, and D. Espriu, ``Chiral Lagrangians and the QCD strings", Physics Letters, B460, 447 (1999).