Summary
The fourth-rank vacuum polarization tensor, which is related to the lowest-order nonlinear interaction between four electromagnetic fields in quantum electrodynamics, is exactly calculated in terms of rational, logarithm and dilogarithm functions when two of the four electromagnetic fields describe photons off the mass shell. This task has been accomplished by a not exccedingly laborious effort with the aid of double dispersion relations which proved to be a very convenient tool for the treatment of these problems (in particular, photon-photon scattering). From the explicit expression of the polarization tensor we have easily obtained the exact amplitudes for photon-photon scattering, photon splitting and photon coalescence into photons on nuclei. Moreover we give the real and imaginary part of Delbrück scattering in the form of threefold integrals over the momentum transferred to the nucleus by one of the two virtual photons. This can be compared with the fivefold and sixfold integrals for the imaginary and real part, respectively, available in the existing literature on Delbrück scattering. Finally we give an explicit expression for the differential cross-section of Delbrück scattering in the limit of low energies.
Riassunto
Il tensore quadruplo di polarizzazione del vuoto, che in elettrodinamica quantistica descrive l’interazione non lineare fra quattro campi elettromagnetici all’ordine perturbativo più basso, è stato calcolato esattamente in termini di funzioni razionali, logaritmiche e dilogaritmiche, nel caso in cui due dei quattro campi elettromagnetici descrivono fotoni virtuali. Questo scopo è stato raggiunto con un lavoro relativamente non eccessivo facendo uso delle relazioni di dispersione doppie che si sono già dimostrate molto utili per trattare simili problemi (in particolare, nel caso dello scattering fotonefotone). Dalla conoscenza esplicita del tensore di polarizzazione abbiamo ottenuto facilmente le ampiezze esatte dello scattering fotone-fotone, del decadimento e della coalescenza di fotoni in altri fotoni su nuclei. Inoltre abbiamo ottenuto la parte reale e la parte immaginaria dell’ampiezza di diffusione elastica di fotoni da nuclei («Delbrück scattering») nella forma di integrali tripli nell’impulso trasferito da uno dei due fotoni virtuali al nucleo. Questo risultato può essere confrontato con gli integrali quintupli per la parte immaginaria e sestupli per la parte reale che si trovano nella letteratura esistente sul «Delbrück scattering». Infine abbiamo dato un’espressione esplicita della sezione d’urto differenziale per il «Delbrück scattering» valida nel limite delle basse energie.
Реэюме
В терминах рациональных, логарифмических и двойных логарифмических функций, когда два иэ четырех злектромагнитных полей описывают фотоны вне массовой поверхности, точно вычисляется тенэор поляриэации вакуума четвертого порядка, который свяэан с нелинейным вэаимодействием ниэщего порядка между четырьмя злектромагнитными полями в квантовой злектродинамике. Рещение зтой эадачи было осушествлено посредством не очень сложных вычислений, испольэуюших двойные дисперсионные соотнощения, которые, как было докаэано, представляют очень удобное средство для рассмотрения таких проблем (в частности, рассеяние фотона фотоном). Иэ точного выражения тенэора поляриэации мы легко получаем точные амплитуды для рассеяния фотона фотоном, расшепления фотона и слияния фотонов в фотоны на ядрах. Кроме того, мы приводим вешественную и мнимую части рассеяния Дельбрюка в виде трехкратных интегралов по передаваемому импульсу ядру одним иэ двух виртуальных фотонов. Это можно сравнить с пятикратным и щестикратным интегралами для мнимой и вешественной частей, соответственно, имеюшимися в сушествуюшей литературе по рассеянию Дельбрюка. В эаключение, мы приводим точное выражение для дифференциального поперечного сечения рассеяния Дельбрюка в пределе малых знергий.
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We want to point out a misprint in the second paper quoted in ref. (13) (B. De Tollis:Nuovo Cimento,35, 1182 (1965)): in the last term of the fourth of eqs. (10), for «M (1)1122 (s, t)» read «M (1)1122 (t, s)», according to eqs. (9) of the same paper.
V. Costantini, B. De Tollis andG. Pistoni:A preliminary approach to Delbrück scattering, Istituto di Fisica di Roma, Nota Interna No. 164 (March 1968) (unpublished).
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Costantini, V., De Tollis, B. & Pistoni, G. Nonlinear effects in quantum electrodynamics. Nuov Cim A 2, 733–787 (1971). https://doi.org/10.1007/BF02736745
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DOI: https://doi.org/10.1007/BF02736745