Academic literature on the topic 'Triangle and Box Feynman Diagrams'

Different mathematical methods have been applied to obtain the analytic result for the massless triangle Feynman diagram yielding a sum of four linearly independent (LI) hypergeometric functions of two variables F4. This result is not physically acceptable when it is embedded in higher loops, because all four hypergeometric functions in the triangle result have the same region of convergence and further integration means going outside those regions of convergence. We could go outside those regions by using the well-known analytic continuation formulas obeyed by the F4, but there are at least two ways we can do this. Which is the correct one? Whichever continuation one uses, it reduces a number of F4 from four to three. This reduction in the number of hypergeometric functions can be understood by taking into account the fundamental physical constraint imposed by the conservation of momenta flowing along the three legs of the diagram. With this, the number of overall LI functions that enter the most general solution must reduce accordingly. It remains to determine which set of three LI solutions needs to be taken. To determine the exact structure and content of the analytic solution for the three-point function that can be embedded in higher loops, we use the analogy that exists between Feynman diagrams and electric circuit networks, in which the electric current flowing in the network plays the role of the momentum flowing in the lines of a Feynman diagram. This analogy is employed to define exactly which three out of the four hypergeometric functions are relevant to the analytic solution for the Feynman diagram. The analogy is built based on the equivalence between electric resistance circuit networks of types Y and Δ in which flows a conserved current. The equivalence is established via the theorem of minimum energy dissipation within circuits having these structures.

A detailed investigation about the 6D single axial box anomalous amplitude is presented. The superficial degree of divergence involved, in the one-loop perturbative calculations, is quadratic and the corresponding theory is nonrenormalizable. In spite of this, we show that the phenomenon of anomaly can be clearly characterized in a completely analogous way as that of 4D single axial triangle anomaly. The required calculations are made within the context of a novel calculational strategy where the amplitudes are not modified in intermediary steps. Divergent integrals are, in fact, not really solved. Adequate representations for the internal propagators are adopted according to the degree of divergence involved, so that when the last Feynman rule is taken (integration over the loop momentum) all the dependence on the internal (arbitrary) momenta are placed only in finite integrals. In the divergent structures emerging, no physical parameter is present and such objects are not really integrated. Only very general properties are assumed for such quantities which are universal (all space–time dimensions). The consistency of the perturbative calculations fixes some relations among the divergent integrals so that all the potentially ambiguous terms can be automatically removed. In spite of the absence of ambiguities, the emerging results allow us to give a clear and transparent description of the anomaly. The present investigation confirms the point of view stated by the same prescription for the well-known 2D axial-vector (AV) two-point and 4D single (AVV) and triple (AAA) axial-vector anomalies: the anomalous amplitudes need not be assumed as ambiguous quantities to allow an adequate description of the anomalies. We show also that a surprising, but natural, connection between the coupling of fermions with a pseudoscalar tensor field is found. In addition, we show that the crucial mathematical aspects of the problem are deeply related to a recently arisen controversy involving the evaluation of the Higgs Boson decay and the question of unicity in the dimensional regularization.

The neutrino asymmetry in the early universe plasma, [Formula: see text], is calculated both before and after the electroweak phase transition (EWPT). In the Standard Model, before EWPT, the leptogenesis is well known to be driven by the abelian anomaly in a massless hypercharge field. The generation of the neutrino asymmetry in the Higgs phase after EWPT, in its turn, has not been considered previously because of the absence of any quantum anomaly in an external electromagnetic field for such electroneutral particles as neutrino, unlike the Adler–Bell–Jackiw anomaly for charged left and right polarized massless electrons in the same electromagnetic field. Using the neutrino Boltzmann equation, modified by the Berry curvature term in the momentum space, we establish the violation of the macroscopic neutrino current in plasma after EWPT and exactly reproduce the nonconservation of the lepton current in the symmetric phase before EWPT arising in quantum field theory due to the nonzero lepton hypercharge and corresponding triangle anomaly in an external hypercharge field. In the last case, the nonconservation of the lepton current is derived through the kinetic approach without a computation of corresponding Feynman diagrams. Then, the new kinetic equation is applied for the calculation of the neutrino asymmetry accounting for the Berry curvature and the electroweak interaction with background fermions in the Higgs phase. Such an interaction generates a neutrino asymmetry through the electroweak coupling of neutrino currents with electromagnetic fields in plasma, which is [Formula: see text]. It turns out that this effect is especially efficient for maximally helical magnetic fields.

AbstractAs part of the Herschel Multi-tiered Extragalactic Survey we have investigated the rest-frame far-infrared (FIR) properties of a sample of more than 4800 Lyman Break Galaxies (LBGs) in the Great Observatories Origins Deep Survey North field. Most LBGs are not detected individually, but we do detect a sub-sample of 12 objects at 0.7 < z < 1.6 and one object at z = 2.0. The LBGs have been selected using color-color diagrams; the ones detected by Herschel SPIRE have redder colors than the others, while the undetected ones have colors consistent with average LBGs at z > 2.5. The spectral energy distributions of the objects detected in the rest-frame FIR are investigated using the code cigale to estimate physical parameters. We include far-UV (FUV) data from GALEX. We find that LBGs detected by SPIRE are high mass, luminous infrared galaxies. It appears that LBGs are located in a triangle-shaped region in the AFUV vs. LogLFUV = 0 diagram limited by AFUV = 0 at the bottom and by a diagonal following the temporal evolution of the most massive galaxies from the bottom-right to the top-left of the diagram. This upper envelop can be used as upper limits for the UV dust attenuation as a function of LFUV. The limits of this region are well explained using a closed-box model, where the chemical evolution of galaxies produces metals, which in turn lead to higher dust attenuation when the galaxies age.