We develop a general procedure for the location of possible zeroes of the pion form factor. which relies on interpolation theory for analytic functions. The zeroes are confined (in the unit disk) to regions bounded by (real) roots of algebraic equations and by algebraic curves. These regions depend both on the interpolation data and the class of functions. which is suitable for the physical problem.
We discuss the question of zeroes of the pion form factor by a method which allows us to "quantify" the unavoidable assumptions concerning the high energy behaviour. The conclusion is that all the present data are consistent with the absence of zeroes inside the cut t-plane. If zeroes do exist, then they are excluded from certain regions around the data, whose boundaries are given in the text. (E.g. zeroes on the negative real t-axis are confined to the left of to = -6.5 (GeV/c)2.)The question has been raised many times whether the Omnes representation for the pion form factor F~(t) needs the polynomial part in order to accommodate the existing experimental data. Some of the papers published so far [1] try to draw conclusions about the existence of zeroes of F~(t) by referring essentially to the general representation of functions holomorphic in the cut t-plane**
F(t) =B(t) E(t).( 1) In (1) the factor B(t) stands for a function which has unit modulus on the cut (inner function in unit disk terminology [2]) and incorporates all zeroes of F(t); E(t) is a function without zeroes in the cut t-plane and with IF(t)l = IE(t)l on the cut. Then the maximum modulus theorem for B(t) states that IB(t)l< 1 at t=0 and B(t) = -1 if B(0)= 1. From the normalization condition F(0)=I one concludes that E(0)>I if B(t) is present at all. The condition E(0)> 1 can be expressed * Supported by Bundesministerium fiir Forschung und Technologic; on leave of absence from the
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