Il Nuovo Cimento A (1965-1970)

, Volume 42, Issue 4, pp 804–813 | Cite as

Remarks on the strong asymptotic decrease of form factors

  • A. P. Balachandran


Wu and Yang have recently suggested on the basis of statistical considerations that the nucleon electromagnetic form factorF(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 transfert tends to infinity. They have also shown that this behaviour is consistent with the experimental data onF(t) for large spacelike momentum transfers. We prove that if i)F(t) decreases faster than any inverse power oft whent→−∞, and ii) ImF(t)=O (exp [−c|t|1/2]),c>0 whent→+∞, thenF(t) is uniquely determined by the subtraction functionG(t) associated with its dispersion relation and that this function must be an entire function of nonzero order. It is also shown thatF(t) must increase faster than any power oft whent→∞ along some real or complex direction. The spectral function forF(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 byG(t). Unitarity is not used in the proof so that the unitary structure ofF(t) and hence the mass spectrum in thet-channel must be consequences of the properties ofG(t) and of the assumed asymptotic behaviour ofF(t).


Form Factor Spectral Function Moment Problem Inverse Power Sulla Base 
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Wu e Yang hanno recentemente suggerito sulla base di considerazioni statistiche che il fattore di forma elettromagnetico del nucleoneF(t) (che può essere elettrico o magnetico) dovrebbe decrescere come exp [−c|t|1/2],c>0 quando il quadrato del momento trasferito invariantet tende all’infinito. Essi hanno anche dimostrato che questo comportamento concorda con i dati sperimentali perF(t) per grandi momenti spaziali trasferiti. Si dimostra che se i)F(t) decrease più rapidamente di ogni potenza inversa dit quandot→−∞, e se ii) ImF(t)=O(exp[−ct1/2]),c>0 quandot→+∞, alloraF(t) è determinata unicamente dalla funzione di sottrazioneG(t) associata con la sua relazione di dispersione e che questa funzione deve essere una funzione interna di ordine non nullo. Si dimostra anche cheF(t) deve decrescere più rapidamente di ogni potenza dit quandot→∞ lungo qualche direzione reale o complessa. Si scrive esplicitamente la funzione spettrale diF(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 daG(t). Nella dimostrazione non si fa uso dell’unitarietà cosicchè la struttura unitaria diF(t) e quindi lo spettro di massa nel canalet devono essere conseguenze delle proprietà diG(t) e dell’ammesso comportamento asintotico diF(t).


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Copyright information

© Società Italiana di Fisica 1966

Authors and Affiliations

  • A. P. Balachandran
    • 1
  1. 1.Physics DepartmentSyracuse UniversitySyracuse

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