Abstract
It is now widely believed that hadrons are composites built of quarks and gluons whose interactions are governed by quantum chromodymamics (QCD). The nature of this internal structure is the key to an understanding of hadronic properties, both at short and long distances. However the connection between the hadrons and their constituents often seems vague in applications of perturbative QCD. If we are to push beyond perturbation theory, we require a conceptual framework within which these notions can be made precise. A particularly convenient and intuitive framework is based upon the Fock state decomposition of hadronic states which arises naturally in the ‘light-cone quantization’ of QCD. In this approach, a hadron is characterized by a set of Fock state wave functions, the probability amplitudes for finding different combinations of bare quarks and gluons in the hadron at a given ‘light-cone time’ τ = t + z. These wave functions provide the essential link between hadronic phenomena at short distances (perturbative) and at long distances (non-perturbative).
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References
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Here Veff = K/\({V_{eff}} = K/\sqrt {x\left( {1 - x} \right)y\left( {1 - y} \right)} \) where K is the irreducible scattering amplitude. Helicity dependence is implicit.(MATH TYPE)
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Such regulators exist for QCD (see L.D. Faddeev and A.A. Slavnov. Gauge Fields (Benjamin, 1980), but they are cumbersome to say the least. Similar results can be obtained more easily by dimensionally regulating the k⊥ integrations and setting µ = Λ, the cut-off desired. The former regulator is more intuitive, however, and we adopt it for our discussions here.
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Our analysis is mathematically similar to Jackiw’s analysis of the axial vector anomaly using ‘point splitting techniques’ (see S. Treiman, R. Jackiw and D. Gross, Lectures on Current Algebra and its Applications, Princeton University Press (Princeton, 1972). Of course PCAC provides the link between his analysis of the axial vector amplitude and our direct analysis of πo→γ.
By ‘radius’ we mean the slope of the form factor near Q2 ≃ 0: Fπ(Q2)=1-Q2R Rπ /6.
This gauge invariant amplitude is defined in other gauges by (3.2b) but with a path order ‘string’ operator P exp(∫o1ds ig A(sz)•z) between the ψ and ψ. The line integral vanishes in light-cone gauge because A•z=A+z¯= 0. This all orders definition of ϕ uniquely fixes the definition of TH which must itself then be gauge invariant.
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This amplitude is also given by (3.24) in gauges other than light-cone gauge, but with i∂+ → i∂+ —gA+.
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We are simplifying matters slightly here, and later. The coupling to gauge fields could also have the form g2 →A 2, and could proceed through multipole couplings other than El. The El coupling we assume here is typical of all the couplings, and frequently dominant among them.
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Lepage, G.P., Brodsky, S.J., Huang, T., Mackenzie, P.B. (1983). Hadronic Wave Functions in QCD. In: Capri, A.Z., Kamal, A.N. (eds) Particles and Fields 2. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3593-1_3
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