A multiple commutator formula for the sum of Feynman diagrams

Lam, C. S.; Liu, K. F.

doi:10.1016/s0550-3213(96)00548-2

Cited by 16 publications

(42 citation statements)

References 33 publications

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“…Each U n can be decomposed into sums of products of irreducible components C m via the formula [1] U n = {m} ξ(m 1 m 2 · · · m k )C m 1 C m 2 · · · C m k ,…”

Section: Decomposition Formulamentioning

confidence: 99%

Time ordering, energy ordering, and factorization

Lam

2000

Journal of Mathematical Physics

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Relations between integrals of time-ordered product of operators, and their representation in terms of energy-ordered products are studied. Both can be decomposed into irreducible factors and these relations are discussed as well. The energyordered representation was invented to separate various infrared contributions in gauge theories. It is shown that the irreducible time-ordered expressions can be used to accomplish the same purpose. Besides, it has the added advantage of being factorizable.

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“…Each U n can be decomposed into sums of products of irreducible components C m via the formula [1] U n = {m} ξ(m 1 m 2 · · · m k )C m 1 C m 2 · · · C m k ,…”

Section: Decomposition Formulamentioning

confidence: 99%

Time ordering, energy ordering, and factorization

Lam

2000

Journal of Mathematical Physics

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“…Each A[σ] is given by the product of the vertex factor V [σ] = V σ 1 V σ 2 · · · V σn , and the product P [σ] of all the propagators. If the initial source particle is off-shell, there are n propagators like (1). If it is on-shell, then there are only n − 1 propagators, but it is convenient to include also an explicit on-shell δ-function factor −2πiδ(p ′ · p ′ − m 2 ) into A n , where p ′ = p + n i=1 k i is the momentum of the initial source particle.…”

Section: B Decomposition and Factorizationmentioning

confidence: 99%

“…The origin of this simplification is a decomposition formula [1] for the tree amplitude A n of n identical bosons. This formula allows A n to be decomposed into a sum of products of irreducible amplitudes I m , where m runs from 1 to n, labeling the number of bosons in it.…”

Section: Introductionmentioning

confidence: 99%

Small Recoil Approximation

Lam

1999

Int. J. Mod. Phys. A

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In this review we discuss a technique to compute and to sum a class of Feynman diagrams, and some of its applications. These are diagrams containing one or more energetic particles that suffer very little recoil in their interactions. When recoil is completely neglected, a decomposition formula can be proven. This formula is a generalization of the well-known eikonal formula, to non-Abelian interactions. It expresses the amplitude as a sum of products of the irreducible amplitudes, with each irreducible amplitude being the amplitude to emit one, or several mutually interacting, quasiparticles. For Abelian interaction a quasiparticle is nothing but the original boson, so this decomposition formula reduces to the eikonal formula. In non-Abelian situations each quasiparticle can be made up of many bosons, though always with a total quantum number identical to that of a single boson. This decomposition enables certain amplitudes of all orders to be summed up into an exponential form, and it allows subleading contributions of a certain kind, which is difficult to reach in the usual way, to be computed. For bosonic emissions from a heavy source with many constituents, a quasiparticle amplitude turns out to be an amplitude in which all bosons are emitted from the same constituent. For high-energy parton-parton scattering in the near-forward direction, the quasiparticle turns out to be the Reggeon, and this formalism shows clearly why gluons reggeize but photons do not. The ability to compute subleading terms in this formalism allows the BFKL-Pomeron amplitude to be extrapolated to asymptotic energies, in a unitary way preserving the Froissart bound. We also consider recoil corrections for Abelian interactions in order to accommodate the Landau-Pomeranchuk-Migdal effect. * E-mail: Lam@physics.mcgill.ca 3435 Int. J. Mod. Phys. A 1999.14:3435-3449. Downloaded from www.worldscientific.com by THE UNIVERSITY OF OKLAHOMA on 02/04/15. For personal use only. Int. J. Mod. Phys. A 1999.14:3435-3449. Downloaded from www.worldscientific.com by THE UNIVERSITY OF OKLAHOMA on 02/04/15. For personal use only.

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“…Fortunately these questions can be answered and difficulties overcome, if we use the nonabelian cut diagrams introduced elsewhere [15][16][17][18] instead of the conventional Feynman diagrams. Reggeons and Pomerons emerge naturally, and phase shifts can be calculated using only the leading-log approximation.…”

Section: Introductionmentioning

confidence: 99%

“…In Sec. 3, nonabelian factorization formula [15,16] and nonabelian cut diagrams [17,18] for tree amplitudes are reviewed; nonabelian cut diagrams are simply Feynman diagrams with the factorization formula built in. It is important to note that this factorization has nothing to do with the usual factorization of hard physics from soft physics [9].…”

Section: Introductionmentioning

confidence: 99%

Unitarized diffractive scattering in QCD and its application to virtual photon total cross sections

1999

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The problem of restoring the Froissart bound to the Balitskiǐ-Fadín-Kuraev-Lipatov ͑BFKL͒ Pomeron is studied in an extended leading-log approximation of QCD. We consider the parton-parton scattering amplitude and show that the sum of all Feynman-diagram contributions can be written in an eikonal form. In this form, dynamics is determined by the phase shift, and subleading-logs of all orders needed to restore the Froissart bound are automatically provided. The main technical difficulty is to find a way to extract these subleading contributions without having to compute each Feynman diagram beyond the leading order. We solve that problem by using non-Abelian cut diagrams introduced elsewhere. They can be considered as color filters used to isolate the multi-Reggeon contributions that supply these subleading-log terms. An illustration of the formalism is given for amplitudes and phase shifts up to three loops. For diffractive scattering, only phase shifts governed by one and two Reggeon exchanges are needed. They can be computed from the leading-log-Reggeon and the BFKL Pomeron amplitudes. In applications, we argue that the dependence of the energygrowth exponent on virtuality Q 2 for ␥*P total cross section observed at DESY HERA can be interpreted as the first sign of a slowdown of energy growth towards satisfying the Froissart bound. An attempt to understand these exponents with the present formalism is discussed. ͓S0556-2821͑99͒06513-3͔

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A multiple commutator formula for the sum of Feynman diagrams

Cited by 16 publications

References 33 publications

Time ordering, energy ordering, and factorization

Time ordering, energy ordering, and factorization

Small Recoil Approximation

Unitarized diffractive scattering in QCD and its application to virtual photon total cross sections

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