Can one detect zeroes of the pion form factor?

Abstract

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

Implications of the recent high statistics determination of the pion electromagnetic form factor in the timelike region

“…The results on the zeros reported in the literature [18, 42-44, 47, 48] are rather controversial. The best results for the regions free of zeros were obtained in [42,47,48], by a method related to ours. However, since the experimental information at that time was poor, the authors were forced to make some ad-hoc assumptions, especially on the modulus on the timelike axis.…”

“…Different analytic representations, either as standard dispersion relations [17], phase (Omnès-type) [18,46,50,52,53,65] or modulus [52] representations, as well as expansions based on conformal mappings [18,46,51] or Padé-type approximants [64], have been constructed in order to correlate the low-and highenergy properties of the form factor. Of special interest is the issue of the zeros of the form factor, investigated by means of dispersive sum-rules [18,[43][44][45]52] or by the more powerful techniques of analytic optimization theory [42,47,48]. In [61][62][63]66] similar functional-analytic techniques were applied for deriving bounds on the expansion coefficients at t = 0, from an weighted integral of the modulus squared along the cut, known from unitarity and dispersion relations for a related QCD correlator.…”

Dispersive representation and shape of the Kl3 form factors: robustness

“…The possibility that the form factors have one or several zeros can, however, not be completely overruled and has been studied in the literature for example for the pion form factor in Ref. [46]. Let us therefore discuss it here for the scalar form factor.…”