We calculate the coupling constant and the energy dependence of the scattering amplitudes for baryon- and lepton-number-violating processes in the context of the Standard Model, in the semiclassical approximation. Due to the chiral nature of the fermionic representations, the baryon and lepton currents have anomalies [SU(2)]2 and [U(1)]2 [1]. Thus, the rate of any process mediated by gauge field tunneling (for small g) is suppressed by an exponentially small factor exp(-l61r2 / g2).
Thus, naively, for large n, the latter factor can compensate for the smallness of the former and avoid exponential suppression. However, there is something peculiar about the previous reasoning: for a given small value of the coupling constant g, it seems that the scattering amplitude should be the largest for the largest n. In Chapter 4, we study fermionic zero modes in the bounded instanton field for nonzero Yukawa couplings.
We are interested in the Fourier transform in momentum space of Green's functions of the type. It is possible to prove (see Section 3.3) that the index of the operators DQ and DL (remember that index A = dim ker A - dim ker At) is independent of the Higgs field configuration and depends only on the Pontryagin index of the SU (2) gauge field as in the pure Yang-Mills case. It is not. so however, because finiteness of the Euclidean action also implies a non-trivial topology for the Higgs field [11].
The above result is a reflection of the fact that the anomalous divergence equation (2.5) is independent of the scalar Higgs field [12]. Using the completeness of the eigenfunctions of the Hermitian operator fJtjJ and the moment representation for the delta function inside the parentheses in the last formula, we can write In order to achieve the desired asymptotic behavior of the instanton, we must transform it into the so-called "singular gauge" (obtained from (3.8) with a gauge transformation singular in the origin n-1.
From (3.19) one can determine the proportionality constants in (3.18) and the asymptotic behavior of the terms in the correct perturbative expansion of the constrained inst an ton (containing logarithms of m). In this order the operators OA and OH play a role in the integration over the limited bosonic fluctuations (see (3.34)), but otherwise they are completely excluded from the calculation. Moving to momentum space, the integration over the position of the constrained instanton xo (which guarantees translational invariance) produces a total momentum-saving delta function.
In fact, in (3.30) A(q) and ry(k) are the Fourier transform of the gauge and the Higgs components of the bounded instanton respectively. In the next chapter we give the explicit forms of the zero modes of DQ and DL in the bounded instanton field. Their Fourier transforms, as well as the bounded instanton Fourier transform, are given in Chapter 5.
The expressions for the components eB, eA, vB of the zero mode of DL can read-.
Conclusions
Our result for the total unpolarized cross section for the process (5.11) in the extreme relativistic case is then. From (5.15) we see that the emission of an extra-gauge boson or Higgs particle in an anomalous scattering process costs an extra power of g2 in the cross section, and not an extra factor of g-2 as assumed in Ref. However, it is apparent from (5.15) that for sufficiently high s a can be large to the point of violating the unitarity constraint requirement.
Can this unphysical result be attributed to the breakdown of our approximations when s is very large. These should include corrections to the stationary point approximation of the path integral (2.22) in the one-instanton sector, as well as contributions from multi-instanton-anti-instanton configurations considered beyond the dilute gas approximation. The calculated cross section shows the unphysical growth behavior as a power of the total CM-energy squared s, as opposed to the unity limit.
We must therefore conclude that the perturbation theory in the instanton sector fails for large energies, although we have no idea what mechanism could be responsible for this. That the perturbation theory can be completely unreliable when high energies are involved was suggested in Ref. At most, if this were the case, we could expect our results to be valid only when vs is much smaller than the height of the potential barrier between topologically different vacuums, that is, the sphaleron energy Esp f rv'2v/g.
This breakdown of perturbation theory has nothing to do with the high multiplicity of particles that would be expected in a high-energy (B+L) non-conservative process: the bound unity is also violated in a purely fermionic anomalous process (n, m = 0) . However, our approach is limited by the condition (5.6) to small multiplicities of gauge and Higgs bosons and cannot be expected to apply to processes involving 8] that our calculation shows that fermions involved in an anomalous scattering event can be in arbitrary momentum states (consistent with four-momentum conservation), and therefore should carry the corresponding occupation factors that arise due to the Pauli exclusion. principle in a given physical situation.
The operators iJ, iJt do not have well-defined eigenvalue problems, since they change A-spinors into B-spinors and vice versa. The Hermitian operators !Jt!J and fJ !Jt, on the other hand, preserve spinor type and therefore have well-defined eigenvalue problems. In the diploma thesis, the 2x2 matrices, marked with the letters a and T, represent the same numerical matrices, but they act on spin and isospin spaces, respectively.
