Summary
New low-energy theorems for Compton scattering on targets of arbitarary spin are derived. The second-order (in photon energy ω) low-energy theorems derived in the previous paper for spin-1 targets are shown to be valid for targets of any spin, provided we interpret the spin operatorS for spin 1 in these theorems as the spin operator for any spin. Third-order low-energy theorems are derived. It is shown that the amplitudes are determined to the third order in ω by the electric charge, the dipole magnetic moment, the quadrupole electric moment, the octupole magnetic moment, and ten structure-dependent constants. Our method can be used to derive higher-order low-energy theorems.
Riassunto
Si deducono nuovi teoremi di bassa energia per lo scattering Compton su bersagli di spin arbitrario. Si dimostra che i teoremi di bassa energia del secondo ordine (nell’energia ω del fotone), dedotti in un articolo precedente per bersagli con spin 1, sono validi per bersagli con qualsiasi spin, purché si intenda in questi teoremi l’operatore di spinS per spin 1 come l’operatore di spin per ogni spin. Si deducono teoremi di bassa energia del terzo ordine. Si dimostra che le ampiezze sono determinate al terzo ordine in ω dalla carica elettrica, dal momento magnetico dipolare, dal momento elettrico quadrupolare, dal momento magnetico ottupolare e da dieci costanti dipendenti dalla struttura. Si può usare il nostro metodo per dedurre teoremi di bassa energia di ordine più elevato.
Реэюме
Выводятся новые теоремы ниэких знергий для комптоновского рассеяния на мищенях с проиэвольным спином. Покаэывается, что выведенные в предыдушей статье теоремы второго порядка (относительно знергии фотона ω) при ниэких знергиях для мищеней со спином единица окаэываются справедливыми для мищеней с проиэвольным спином, при условии, что мы интерпретируем спиновый операторS для спина единица в зтих теоремах, как спиновый оператор для проиэвольного спина. Выводятся теоремы третьего порядка при ниэких знергиях. Покаэывается, что в третьем порядке по ω амплитуды определяются злектрическим эарядом, дипольным магнитным моментом, квадрупольным злектрическим моментом, октупольным магнитным моментом и восемью постоянными, эависяшими от структуры. Нащ метод может быть испольэован для вывода теорем ниэких знергий более высокого порядка.
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References
F. E. Low:Phys. Rev.,96, 1428 (1954).
M. Gell-Mann andM. L. Goldberger:Phys. Rev.,96, 1433 (1954).
L. I. Lapidus andChuo Kuang-Chao:Žurn. Ėksp. Teor. Fiz.,39, 1286 (1960), (English translation:Sov. Phys. JETP,12, 898 (1961)).
I. J. Kalet:Phys. Rev.,176, 2135 (1968). The result of ref. (3) is reproduced by a more straightforward method.
S. R. Choudhury andD. Z. Freedman:Phys. Rev.,168, 1739 (1969). The Born contributions are evaluated up to the second order (in ω). Their results are given in terms of helicity amplitudes.
W. K. Tung:Phys. Rev.,176, 2127 (1968). It is pointed out in this paper that eight new constants are needed for the third order amplitudes. However, four of them are identically zero because of the crossing symmetry. The author did not evaluate the Born contributions.
A. Pais:Phys. Rev. Lett.,19, 544 (1967);Nuovo Cimento,53 A, 433 (1968);Nuovo Cimento,60 A, 352 (1969). See also:K. Bardakçi andH. Pagels:Phys. Rev.,166, 1783 (1968);G. F. Leal Ferreira andS. Ragusa:Nuovo Cimento,65 A, 607 (1970).
S. Saito:Phys. Rev.,184, 1894 (1969).
K. Y. Lin:Nuovo Cimento,64 A, 188 (1970). In eqs. (5) and (8), all terms of the form {S × (є ×a),S × (є′ ×b)} should be replaced by — {S · є ×a,S · є′ ×b}. The last sentence of Sect.3 should be replaced by «if we use the identity {S i ,S j } = 1/2δ ij to get rid of all anticommutators».
Parts of the second-order low-energy theorems for arbitrary spin targets are given already byPais (the quadrupole moment theorem) andSaito (the structure-dependent constants). See ref. (7–8).
J. D. Bjorken andS. D. Drell:Relativistic Quantum Mechanics (New York, 1964).
V. Singh:Phys. Rev. Lett.,19, 730 (1967). See also:J. S. Bell:Nuovo Cimento,52 A, 635 (1967).
W. R. Theis:Nuovo Cimento,45 A, 124 (1966).
These relations follow from eq. (16) of ref. (13).
M. Gourdin:Nuovo Cimento,40 A, 225 (1965).
A detailed discussion is given byPais, ref. (7).
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Lin, K.Y. Low-energy theorems for compton scattering on targets of arbitrary spin. Nuov Cim A 2, 695–706 (1971). https://doi.org/10.1007/BF02736743
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DOI: https://doi.org/10.1007/BF02736743