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References
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Forn≥4, the following tensor bases are not all independent. Most bases can be written by linear combination of ten independent bases. Thus we can retain only nine suitably chosen independent bases in the summation\(\mathop \Sigma \limits_{i \geqslant j} \). We assume in the following that this task has been done. In eq. (1), we must take the so-called semi-disconnected part which is relevant to inclusive reactions and the totally connected or disconnected part must be discarded. We indicated this operation by the symbol “s.d.”. For a detailed prescription for taking discontinuities, seeH. P. Stapp:Phys. Rev. D,3, 1303 (1971).
We use noncovariant normalization of state.V denotes normalization volume.
The number of independent variables is 3n−2. It is convenient for our discussion to chooseq 2, ω,x α (α=2,...,n) and suitably chosen2n−3 tij's. The remainingt ij's can be expressed by nonlinear functions of 3n−2 independent variables. However we can see by physical considerations that there exist regions where our limit (denoted by\(\widehat{\lim }\) below) applies. We may work simply in the laboratory frame ofp 1 and keepp α finite and letq 2→∞. Ourx α reduces exactly to usual Feymannx F when the «mass» of the virtual photon −q 2 is fixed. If we letq 2→∞ along withs→∞, ourx α can be regarded as a generalization of Feymann'sx F in the off-mass-shell region.
The apparent fractional-power behaviour at largev 1 ofW ij (i≠j) does not bother us. In terms ofq 2, it isq 2Wij which scales.
Fractional derivatives of δ(x 2) are possibly present. However just as thev 1 term, these can also be shown to make a vanishing contribution in our\(\widehat{\lim }\) to the scaling function. The following form was deduced from the Jost-Lehman-Dyson representation for the matrix element of the commutator of currents.R. Jost andH. Lehman:Nuovo Cimento,5, 1598 (1957);F. J. Dyson:Phys. Rev.,110, 1460 (1958).
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Here we assume that\(\widehat{\lim }\) andη i integrations (below) commute. That this assumption of uniform convergence ofη i integrations is necessary is the same situation as in ref. (7) The apparent fractional-power behaviour at largev 1 ofW ij (i≠j) does not bother us. In terms ofq 2, it isq 2Wij which scales.
Among the vectors (x, p i), it is convenient to choose as independent scalarsx 2 and (xp i) and suitably chosen2n−3 t ij's.
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In order that this sum rule be meaningful, the integral in eq. (17) must of course converge. Forn=1, Regge arguments suggest that it diverges at ω=0. Forn≥2, however, we have no argument at present concerning its convergence or divergence properties.
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Fukuda, R. Scaling law for general inclusive reactions induced by current. Lett. Nuovo Cimento 7, 69–75 (1973). https://doi.org/10.1007/BF02728273
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DOI: https://doi.org/10.1007/BF02728273