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Il Nuovo Cimento A (1971-1996)

, Volume 12, Issue 2, pp 331–340 | Cite as

Recoil form factor for the deuteron

  • J. M. Namysłowski
Article

Summary

Making an eikonal approximation in two relative momenta in a 3-body system, keeping trace of all recoil terms, and assuming that the ranges of forces responsible for πp and πn elastic scattering are small, we find the following form factor for the double-scattering term in πd → πd:
$$\left[ {1 + \left( {\frac{{\Delta \mu }}{{4kM}}} \right)^2 } \right]^{\frac{3}{2}} \left\langle {\frac{1}{{\varrho ^2 }}\exp \left[ {i\varrho \left[ {\frac{{\Delta ^2 }}{{8k}}\left( {1 + \frac{\mu }{{2M}}} \right) - \frac{{\mu B}}{k}} \right]} \right]\cos \left( {\varrho \frac{{\Delta ^2 \mu }}{{8kM}}} \right)} \right\rangle _d ,$$
, whereϱr[1+(Δμ/4kM)2]−1/2,B is the binding energy of the deuteron,μ, M are pion and nucleon masses, respectively,Δ is the momentum transfer in the πd → πd process. The above form factor should be compared with the ordinary one of Glauber
$$\left\langle {\frac{1}{{r^2 }}} \right\rangle _d $$
and with the form factor of Blankenbecler and Gunion, and Gottfried
$$\left\langle {\frac{1}{{r^2 }}\exp \left[ {ir\eta } \right]} \right\rangle _d ,$$
, whereηΔ2/8k.

Форм-фактор отдачи для деитрона

Реэуме

Исполяэуя зиконаляное приблизение по двум относителяным импулясам в трех-частичнои системе, сохраняя след всех членов отдали и предполагая, что области деиствия сил, ответственных эа πр и πп упругое рассеяние, являутся малыми по сравнениу с расстояниями мезду р и n в деитроне, мы получаем следуушии форм-фактор для члена двукратного рассеяния в πd→πd
$$\left[ {1 + \left( {\frac{{\Delta \mu }}{{4kM}}} \right)^2 } \right]^{\frac{3}{2}} \left\langle {\frac{1}{{\varrho ^2 }}\exp \left[ {i\varrho \left[ {\frac{{\Delta ^2 }}{{8k}}\left( {1 + \frac{\mu }{{2M}}} \right) - \frac{{\mu B}}{k}} \right]} \right]\cos \left( {\varrho \frac{{\Delta ^2 \mu }}{{8kM}}} \right)} \right\rangle _d ,$$
, где В естя знергия свяэи деитрона, μ, М соответственно массы пиона и нуклона, а Л естя передаваемыи импуляс в процессе πd→πd. Полученныи выще форм-фактор долзен бытя сравнен с обычным форм-фактором Глаубера
$$\left\langle {\frac{1}{{r^2 }}} \right\rangle _d $$
, и с форм-фактором Бланкенбеклера и Гуниона и Готтфрида
$$\left\langle {\frac{1}{{r^2 }}\exp \left[ {ir\eta } \right]} \right\rangle _d ,$$
, где η ≡ Δ2/8k.

Riassunto

Eseguendo un’approssimazione iconale in due impulsi relativi in un sistema di 3 corpi, prendendo la traccia di tutti i termini di rinculo e supponendo che i raggi d’azione delle forze responsabili dello scattering elastico πp e πn siano piccoli, si trova il seguente fattore di forma per il termine di scattering doppio nel processo πd → πd:
$$\left[ {1 + \left( {\frac{{\Delta \mu }}{{4kM}}} \right)^2 } \right]^{\frac{3}{2}} \left\langle {\frac{1}{{\varrho ^2 }}\exp \left[ {i\varrho \left[ {\frac{{\Delta ^2 }}{{8k}}\left( {1 + \frac{\mu }{{2M}}} \right) - \frac{{\mu B}}{k}} \right]} \right]\cos \left( {\varrho \frac{{\Delta ^2 \mu }}{{8kM}}} \right)} \right\rangle _d ,$$
, doveϱr[1+(Δμ/4kM)2]−1/2,B è l’energia di legame del deutone,μ, M sono rispettivamente le masse del pione e del nucleone,gD è l’impulso trasferito nel processo πd → πd. Il suddetto fattore di forma deve essere confrontato con quello ordinario di Glauber
$$\left\langle {\frac{1}{{r^2 }}} \right\rangle _d $$
, e con il fattore di forma di Blankenbecler e Gunion e di Gottfried
$$\left\langle {\frac{1}{{r^2 }}\exp \left[ {ir\eta } \right]} \right\rangle _d ,$$
, doveηΔ2/8k.

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References

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    B. R. Karlsson andJ. M. Namysłowski:Nuovo Cimento,8 A, 520 (1972);The eikonal expansion in the Faddeev 3-body theory, Warsaw University preprint, IFT/71/10, unpublished.ADSCrossRefGoogle Scholar
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Copyright information

© Società Italiana di Fisica 1972

Authors and Affiliations

  • J. M. Namysłowski
    • 1
  1. 1.Institute of Theoretical PhysicsWarsaw UniversityWarsawPoland

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