Unstable Fermions Research Articles

Decay of entangled fermion pairs with post-selection

We consider a pair of unstable fermions in a spin-entangled state. After the decay of one fermion, a spin measurement is performed on the surviving partner, with a Stern-Gerlach experiment or similar. The measurement not only projects the spin of the surviving fermion, but is also physically equivalent to a spin projection for the decayed one — even when it no longer exists. This post-selection effect would be experimentally accessible using muon pairs in a maximally-entangled state, produced either in the decay of a scalar particle, or in e+e− collisions at wide angles.

Considerations concerning the generalization of the Dirac equations to unstable fermions

We discuss the generalization of the Dirac equations and spinors in momentum space to free unstable spin-$1/2$ fermions taking into account the fundamental requirement of Lorentz covariance. We derive the generalized adjoint Dirac equations and spinors, and explain the very simple relation that exists, in our formulation, between the unstable and stable cases. As an application of the generalized spinors, we evaluate the probability density. We also discuss the behavior of the generalized Dirac equations under time reversal.

All-order renormalization of propagator matrix for unstable Dirac fermions

We consider a system of unstable Dirac fermions in a general parity-nonconserving theory with intergeneration mixing and explain how to renormalize its propagator matrix to all orders in perturbation theory. We work in the pole scheme, in which the squares of the renormalized masses are identified with the complex pole positions and the wave-function renormalization (WFR) matrices are adjusted according to the Lehmann-Symanzik-Zimmermann reduction formalism. The unit-residue property is explicitly verified for the renormalized dressed propagator matrix. Closed analytic expressions for the pole-mass counterterms and WFR matrices in terms of the self-energy functions are presented. We identify residual degrees of freedom in the WFR matrices and propose an additional renormalization condition to exploit them. We demonstrate that, in the presence of instability, the WFR matrices of the in and out states bifurcate in the sense that they are no longer related by pseudo-Hermitian conjugation. The well-known one- and two-loop results for stable fermions are recovered. The all-order renormalized propagator of a single unstable fermion takes a particularly compact form. We also briefly discuss Dirac spinors for unstable fermions.

All-Order Renormalization of the Propagator Matrix for Fermionic Systems with Flavor Mixing

We consider a mixed system of Dirac fermions in a general parity-nonconserving theory and renormalize the propagator matrix to all orders in the pole scheme, in which the squares of the renormalized masses are identified with the complex pole positions and the wave-function renormalization (WFR) matrices are adjusted in compliance with the Lehmann-Symanzik-Zimmermann reduction formalism. We present closed analytic all-order expressions and their expansions through two loops for the renormalization constants in terms of the scalar, pseudoscalar, vector, and pseudovector parts of the unrenormalized self-energy matrix, which is computable from the one-particle-irreducible Feynman diagrams of the flavor transitions. We identify residual degrees of freedom in the WFR matrices and propose an additional renormalization condition to exhaust them. We then explain how our results may be generalized to the case of unstable fermions, in which we encounter the phenomenon of WFR bifurcation. In the special case of a solitary unstable fermion, the all-order-renormalized propagator is presented in a particularly compact form.

General Analysis of B Meson Decay into Two Fermions

We study how to measure the current structure of the process that $B$ meson decays into two unstable fermions $\bar f_a$ and $f_b$ in model independent way. We use the momentum distributions of subsequent decay products affected by $\bar f_a f_b$ spin correlation. We have found the following: (1) We can extract the absolute values of two effective coupling constants from the opening angle between the particles decayed from $\bar f_a$ and $f_b$ (2) We can extract the real part of the interference from the energy distribution of one of the decayed particles from $\bar f_a$ or $f_b$. (3) No new information can be obtained from the energy distribution of two decayed particles from $\bar f_a$ and $f_b$. (4) The imaginary part of interference is extracted from the azimuthal angle asymmetry of final-state decay products. (5) If only one of two fermions is unstable, we can extract the real part of interference from each of the energy distribution and opening angle distribution. We show several simple examples.

Pole mass, width, and propagators of unstable fermions

The concepts of pole mass and width are extended to unstable fermions in the general framework of parity-nonconserving gauge theories, such as the standard model. In contrast with the conventional on-shell definitions, these concepts are gauge independent and avoid severe unphysical singularities, properties of great importance since most fundamental fermions in nature are unstable particles. General expressions for the unrenormalized and renormalized dressed propagators of unstable fermions and their field-renormalization constants are presented.

A new renormalization scheme for the fermion field in the standard model

In order to obtain proper wavefunction renormalization constants for unstable fermions and be consistent with Breit–Wigner formula in the resonant region, we have assumed an extension of the LSZ reduction formula for unstable fermions and adopted an on-shell mass renormalization scheme in the framework without field renormalization. The comparison of the gauge dependence of physical amplitude between on-shell mass renormalization and complex-pole mass renormalization has been discussed. After obtaining the fermion wavefunction renormalization constants, we extend them to two matrices in order to include the contributions of off-diagonal two-point diagrams at fermion external legs for the convenience of calculations of S-matrix elements.

Global positioning of spin GPS scheme for half-spin massive spinors

We present a simple and flexible method of keeping track of the complex phases and spin quantisation axes for half-spin initial- and final-state Weyl spinors in scattering amplitudes of Standard Model high energy physics processes. Both cases of massless and massive spinors are discussed. The method is demonstrated and checked numerically for spin correlations in $\tau \bar\tau$ production and decay. Its application is in our work of combining effects due to multiple photon emission (exponentiation) and spin, embodied in the Monte Carlo event generators for production and decay of unstable fermions such as the $\tau$ lepton, t-quark and hypothetical new heavy particles. In particular, the recurrent problem of combining, for such unstable fermions, one author's calculation of production and another author's calculation of decay, in the presence or absence of multiple photon effects, is there given a practical solution, both for Weyl spinor methods and for the traditional Jacob-Wick helicity methods. Moreover, for massive fermions we give a simple representation of the amplitude for $n(\gamma)$ emission ideally suited for numerical evaluation. No other method is known to us which for arbitrary n has been realized numerically for unstable, massive fermions. Our paper can contribute also, to the discussion on design principles of the phenomenology work for the future accelerators such as LHC or NLC.

Relativistic bound state equation for unstable fermions and the toponium width

The bound state problem for a fermion-antifermion system is considered taking into account a finite decay width of the constituents. We propose an exactly solvable relativistic zero order equation similar to that of Barbieri and Remiddi, but including a constant width. We focus especially on the t t system for which we reconsider our recent calculation of the bound state corrections to the toponium width, which was performed in the narrow width approximation and needed the use of second order Bethe-Salpeter perturbation theory. We show that one obtains the same result already in first order BS perturbation theory if one uses our present approach. Furthermore the large cancellations of gauge dependent terms is demonstrated to be a consequence of a Ward identity. This cancellation mechanism is shown to be valid for general fermion-antifermion systems.