Abstract
We further develop the massive constructive theory of the Standard Model and use it to calculate the amplitude and squared amplitude for all two-body decays, a collection of weak three-body decays, as well as Higgs decay to four neutrinos. We compare our results with those from Feynman diagrams and find complete agreement. We show that in all the cases considered here, the amplitudes of massive constructive theories are significantly simpler than those resulting from Feynman diagrams. In fact, a naive counting of the number of calculations required for a matrix-element generator to compute a phase-space point is orders-of-magnitude smaller for the result coming from the constructive method suggesting that these generators might benefit from this method in the future, even in the case of massive weak amplitudes. We also anticipate that our simpler expressions will produce numerically more stable expressions.
Highlights
We further develop the massive constructive theory of the Standard Model and use it to calculate the amplitude and squared amplitude for all two-body decays, a collection of weak three-body decays, as well as Higgs decay to four neutrinos
We have calculated many two- and threebody decays without the emission of a gluon or photon and one four-body decay of the Standard Model (SM) using the massive constructive techniques described in [10] and the three-point vertices of [12]. We have squared these amplitudes and compared with the expressions coming from Feynman diagrams and found complete agreement
This reduction is necessary, as we discuss in that section, because the constructive amplitude method is a purely on-shell formalism for constructing an amplitude
Summary
Before we begin the calculations, we will describe some of the steps used to square the amplitude and reduce the amplitude to a suitable form but reserve a full treatment for our Appendixes. To replace the resulting spinor chains with an expression that only involves masses and traditional kinematic variables such as pi · pj , we derive a set of identities involving the product of two spinors with their spin indices “contracted.” This will be when two indices are the same and summed over with one up and one down in analogy with the Lorentz indices of four-vectors One of these identities is already well-known as it gives the momentum when one spinor is an angle spinor while the other is a square spinor. This allows both the external and internal lines to be on-shell, while satisfying momentum conservation This complexification is only done during intermediate steps, and the momenta are constrained to be real again at the end of the calculation. It is important to note that if we are able to completely remove the momenta from the numerator using on-shell identities for the spinors and momenta, in every case we have studied, we find exact agreement with Feynman diagrams. The Schouten identity is well-known, we derive generalizations of it that are useful for the reduction of these spinor chains in Appendix B and describe a mnemonic for remembering the generalized Schouten identity that we find very useful in our calculations
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