Abstract
The Lagrangian for strong and electromagnetic interactions of quarks may be written as
where Q g is the charge of quark q in units of the proton charge, e.
“Is there any point to which you would wish to draw my attention?” “To the curious incident of the dog in the night-time.” “The dog did nothing in the night-time!” “That was the curious incident,” remarked Sherlock Holmes.
Arthur Conan Doyle, 1892
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References
The simple, but somewhat tedious procedure, is to apply unitarity equations (1.2.8) and (1.2.9) to the forward scattering to order a2.
More refined analyses may be found in the paper of Barnett, Dine, and McLerran (1980) and work quoted therein. Recent, higher energy data of PETRA (up to 45 GeV) and LEP (90 GeV) also agree very well with the theory.
A comprehensive discussion of this, in particular, for deep inelastic scattering, may be found in Bardeen et al. (1978).
The factors 1/2 in (4.3.2) are introduced to average over the spin of the initial nucleon and the “helicity” of the virtual photon.
We write all formulas for e scattering; for p scattering, they are identical. At very high energies, the interference of Z exchange and y exchange becomes important, but we will not consider this in our book. Only the charged-current processes will be discussed here for v scattering. 6 These fi are slightly different from the standard ones; to be precise, ,f1 = 2xF1 standard f3 = xF3s,. Our definition aims at unifying the QCD equations that will be written later. Throughout this section, we use z for the position vector to distinguish it from the Bjorken variable, x.
Actually, we could have z2 as large as we wished. However, this corresponds to z2 < 0 where, by locality, the commutator [J (z), J(0)] vanishes and we get no contribution except when z ~ zi.e., ~ 0.
For processes that involve nondiagonal matrix elements, gradient terms have to be taken into account. An example of the last situation will be found in Section 5.7.
The labels F/V denote fermion/vector boson singlet operators.
Note that as in Section 3.6, the quark or gluon fields entering into the N
In principle, the dispersion relation should be written with subtractions, but it may be seen that these alter nothing, provided that the integral in equation (4.5.19).below is convergent. For infor-mation on dispersion relations, see the treatise of Eden et al. (1966).
Besides, of course, using the second-order expression for Section 3.7, and taking into account the finite parts of the first-order diagrams. 14 The Furmanski—Petronzio result has recently been checked independently, so it is their result that should be considered valid.
Some of the C were calculated previously by De Fnjula, Georgi, and Politzer (1977a); CaIvo (1977); Altarelli, Ellis, and Martinelli (1978); Kubar—Andre and Paige (1979); Abad and Humpert (1978); Zee, Wilczek, and Treiman (1974); Kingsley (1973); Walsh and Zerwas (1973); Hinchliffe and Llewellyn Smith (1977); Floratos, Ross, and Sachrajda (1979); Witten (1976), etc. The values reported by Bardeen et al (1978) or Buras (1980) have all been checked by at least two independent calculations.
For the singlet part, the diagrams of Figure 4.7.2 would also have to be considered
The equations are somewhat modified for the two values r2 ± such that d_(n (n ±)+ 1 = 0, where the next-to-leading correction is not 0(as) but 0(a log as)
A simple way to obtain this theorem is to relate it to the standard one for Laplace transforms by changing variables, z —> log z Alternatively, one may integrate (4.8.5) to get (4.8.2). 19 This is easily understood if we recall our calculation of Equation (2.3.18) and compare it to (3.3.29) and (3.3.30): the entire contribution to Z g comes from the gluon propagator in this gauge.
The equation is normalized so that if y* were real, the scattering amplitude would be F(y* + G + e,,nAu
Equations (4.8.13a,b) already take into account the correct value of rI.
In general, we have to go to Q 2 —, ao because of the residual dependence on the interaction due to the Wilson coefficients.
Note that the v’s or e/ p’s only probe quarks so the experimentally measured functions is precisely To obtain fV directly, one requires probes that act on gluons. 24 Cf., Brodsky and Lepage (1980) and references therein.
For Reggeology, see, for example, Barger and Cline (1969). 26 The details of the following proofs may be found in Martin (1979) for the nonsinglet and in Lopez and Yndurain (1981) for both singlet and nonsinglet to first and second order. In these references, behaviors other than Regge-like ones are also discussed.
Particularly since one can argue that vON s 7-- vos 2–2.5,0 < < 1.
For a discussion of this, see the papers of De Rujula, Georgi, and Politzer (1977a,b).
Some of the applications may be found in the extensive and pioneering work of Shifman, Vainshtein, and Zakharov (1979a, b).
See Nachtmann (1973), Georgi and Politzer (1976), and Barbieri et al. (1976).
The same order of magnitude for the k’s (k1 12 0.1 to 0.3 GeV) was obtained in a calculation in the bag model by Jaffe and Soldate (1981).
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© 1993 Springer-Verlag Berlin Heidelberg
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Ynduráin, F.J. (1993). Perturbative QCD. I. Deep Inelastic Processes. In: The Theory of Quark and Gluon Interactions. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02940-4_4
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DOI: https://doi.org/10.1007/978-3-662-02940-4_4
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