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The Maximal Analyticity Hypothesis in Effective Theories.
Alexander V. Vereshagin
St. Petersburg State University, St. Petersburg.
In the publications [1], [2] 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 [2]. 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 [3]. 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.,[6]).
References:
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).

e-mail: asf@asf.e-burg.ru