Geodesic
Sum rules for the ratio of the $\pi^{\pm} p$-scattering amplitudes
Teoretičeskaâ i matematičeskaâ fizika, Tome 5 (1970) no. 2, pp. 235-243

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Analyticity, unitarity, and crossing symmetry are used to obtain an exact integral relationship between $|f_{+}(E)/f_{-}(E)|$ and the difference of the phases of $f_{+}(E)$ and $f_{-}(E)$ and the analogous relationship between $|f_{+}(E)f_{-}(E)|$ and the sum of the phases of these amplitudes [$(f_{\pm}(E)$ are the $\pi^{\pm}-p$-forward scattering amplitudes]. Restrictions on $\displaystyle\int_{1}^{E}\ln\biggl|\frac{f_{+}(E')}{f_{-}(E')}\biggr|\, \frac{dE'}{\sqrt{{E'}^2-1}}$ are found. 
@article{TMF_1970_5_2_a5,
     author = {V. Z. Baluni and Yu. S. Vernov},
     title = {Sum rules for the ratio of the $\pi^{\pm} p$-scattering amplitudes},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {235--243},
     publisher = {mathdoc},
     volume = {5},
     number = {2},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1970_5_2_a5/}
}
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V. Z. Baluni; Yu. S. Vernov. Sum rules for the ratio of the $\pi^{\pm} p$-scattering amplitudes. Teoretičeskaâ i matematičeskaâ fizika, Tome 5 (1970) no. 2, pp. 235-243. http://geodesic.mathdoc.fr/item/TMF_1970_5_2_a5/