# Remarks on the strong asymptotic decrease of form factors

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## Summary

Wu and Yang have recently suggested on the basis of statistical considerations that the nucleon electromagnetic form factor*F(t)* (which may be either electric or magnetic) should decrease like exp [−*c*|*t*|^{1/2}],*c*>0 when the square of the invariant momentum transfer*t* tends to infinity. They have also shown that this behaviour is consistent with the experimental data on*F(t)* for large spacelike momentum transfers. We prove that if i)*F(t)* decreases faster than any inverse power of*t* when*t*→−∞, and ii) Im*F(t)*=*O* (exp [−*c*|*t*|^{1/2}]),*c*>0 when*t*→+∞, then*F(t)* is uniquely determined by the subtraction function*G(t)* associated with its dispersion relation and that this function must be an entire function of nonzero order. It is also shown that*F(t)* must increase faster than any power of*t* when*t*→∞ along some real or complex direction. The spectral function for*F(t)* is explicitly exhibited as the Fourier transform of an analytic function which is defined by a power series in a certain domain of its regularity. The power series involves only quantities which are determined by*G(t)*. Unitarity is not used in the proof so that the unitary structure of*F(t)* and hence the mass spectrum in the*t*-channel must be consequences of the properties of*G(t)* and of the assumed asymptotic behaviour of*F(t)*.

## Keywords

Form Factor Spectral Function Moment Problem Inverse Power Sulla Base## Riassunto

Wu e Yang hanno recentemente suggerito sulla base di considerazioni statistiche che il fattore di forma elettromagnetico del nucleone*F(t)* (che può essere elettrico o magnetico) dovrebbe decrescere come exp [−*c*|*t*|^{1/2}],*c*>0 quando il quadrato del momento trasferito invariante*t* tende all’infinito. Essi hanno anche dimostrato che questo comportamento concorda con i dati sperimentali per*F(t)* per grandi momenti spaziali trasferiti. Si dimostra che se i)*F(t)* decrease più rapidamente di ogni potenza inversa di*t* quando*t*→−∞, e se ii) Im*F(t)*=*O*(exp[−*ct*^{1/2}]),*c*>0 quando*t*→+∞, allora*F(t)* è determinata unicamente dalla funzione di sottrazione*G(t)* associata con la sua relazione di dispersione e che questa funzione deve essere una funzione interna di ordine non nullo. Si dimostra anche che*F(t)* deve decrescere più rapidamente di ogni potenza di*t* quando*t*→∞ lungo qualche direzione reale o complessa. Si scrive esplicitamente la funzione spettrale di*F(t)* come trasformata di Fourier di una funzione analitica definita da una serie di potenze in un certo dominio della sua regolarità. La serie di potenze coinvolge solo quantità che sono determinate da*G(t)*. Nella dimostrazione non si fa uso dell’unitarietà cosicchè la struttura unitaria di*F(t)* e quindi lo spettro di massa nel canale*t* devono essere conseguenze delle proprietà di*G(t)* e dell’ammesso comportamento asintotico di*F(t)*.

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## References

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*F(t)*=*O*[exp [*c*|*t*|^{1/2}]],*c*>0 when*t*→∞ along any ray. This is incompatible with our assumptions [except possibly in the case mentioned in footnote (14)] as the result β) implies that*G(t)*and hence*F(t)*increases at least like exp [*c*|*t*|] when*t*→∞ in some fashion.Google Scholar - (4).
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*Phys. Rev.*,**137**, B 177 (1965). Note that the proof given here can also be carried through with the assumptions that Im*F(s, z)*is*O*(exp [−*c*(θ)*s*^{1/2}]),*c*(θ)>0 and Re*F(s,z)*decreases faster than any inverse power of*s*when*s*→+∞. The minimal hypothesis ofKinoshita (9) perhaps corresponds to the situation where Im*F(s, z)∼A*exp [−*c*(θ)*s*^{1/2}] and Re*F(s, z)*decreases faster than any inverse power of*s*when*s*→+∞ and not to the situation where*F(s, z)∼A*exp [−*c*(θ)*s*^{1/2}] as suggested in this paper.ADSMathSciNetCrossRefGoogle Scholar - (9).
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*L*^{p}-functions which do not tend to zero when*t*(or*x*) increases such that it is always contained in a sequence of intervals [*t*_{n},*t*_{n}+△_{n}] where*t*_{n}→+∞ and △_{n}→0 when*n*→∞. If Re*I(t)*is a function of this sort, the limit*t*→+∞ is to be taken with*t*avoiding these intervals in the limiting process. It may happen that |*G(t)*|→∞ exponentially when*t*→+∞ only if*t*increases through the sequence of intervals [*t*_{n},*t*_{n}+Δ] while |*G(t)*| does not have such a fast increase when this limit is taken according to the prescription which makes Re*I(t)*→0. But, then, there exist circles of radii*r*_{n},*t*_{n}+△_{n}<*r*_{n}<*t*_{n+1}, on which |*G(t)*| acquires a value which is larger than its maximum value in the real interval [*t*_{n},*t*_{n}+Δ_{n}] by the maximum modulus theorem. Therefore, there exists an infinite sequence {*t*^{′}_{n}}, arg*t*^{′}_{n}≠0 or 2π for any finite*n*and |*t*^{′}_{n}|∞ when*n*→∞, such that when*t*approaches infinity through this sequence,*G(t)*approaches infinity exponentially. Then, if arg*t*_{n}+→0 or 2π when*n*→∞, the bound (2.26) ensures that*G(t)*dominates*F(t)*when*t*=*t*^{′}_{n}and*n*→∞ and*F(t)*also increases exponentially. We are unable to demonstrate the exponential increase of*F(t)*, however, for the case arg*t*^{′}_{n}→0 or 2π. This exceptional case has been ignored in the statement of the asymptotic properties of*F(t)*in the body of the paper. None of the other conclusions will of course be affected even if this exception turns out to physically be true.Google Scholar - (15).J. A. Shohat andJ. D. Tamarkin:
*The problem of Moments*(Providence, R. I., 1963), p. 28.Google Scholar - (16).Reference (15),J. A. Shohat andJ. D. Tamarkin:
*The problem of Moments*(Providence, R. I., 1963), p. 22.Google Scholar