We will try to understand the Delta I Equals One Half pattern of non-leptonic weak Kaon decays. We also draw attention to a number of surprising results that we arrive at during the calculation. In the time since it was introduced, quantum chromodynamics, or QCD, has become widely accepted as the best available theory of strong interactions.
In this thesis we present a calculation of the weak matrix elements relevant to the empirical (j.J = 1/ 2 rule observed to hold for the non-leptonic weak decay of Kaons. Of course, the most spectacular feature of the K° K0 system is that it is the only example in nature for which we could observe CP violation. In the long wavelength continuum limit, we identify these fields as the two quarks with the same names.
The only symmetry operation we will need explicitly is that of interchanging the fields of the two generations, denoted by T 1. Much of this degeneracy is accidental and unrelated to any symmetry of the Hamiltonian. A useful discussion of many of the methods that have been used in the context of lattice gauge theories is presented in [14].
Graph Construction
Graph Construction and Computational Equivalence
As mentioned in the previous section, the main reason why the strong coupling expansion has not been widely used for practical calculations is the higher order complexity. This suggests the possibility of writing a specialized computer algebra package to aid in the calculations. A final computer program designed for strong coupling calculations would simply take as input the initial and final states, and perhaps the form of the weak interaction Hamiltonian, and then grind until (perhaps in almost infinite time!) all the relevant plots have been produced and then evaluated .
For simplicity, much of our discussion will be limited to the pure partner situation. The second, and technically more difficult, feature we need is the ability to recognize when two different configurations are arithmetically equivalent, that is, they evaluate to the same number. This is important because computing an individual graph can be an expensive thing to do for the SU(N) and there are many computationally equivalent graphs.
In comparison, we wouldn't have to worry so much about gauge theory Z 2 (in the Euclidean formulation) because evaluating graphs is much easier.
Time Ordering and Complex Conjugation
The general factor of half arises because we can adjust the orientation of the initial or final state if one wishes. Most of the work involved in joining the two graphs together comes from constructing the time series and evaluating the energy denominators. Calculating the excluded volume coefficients, although straightforward in simple cases, becomes very difficult at higher orders.
Such algorithms will be of the same level of complexity as those used for the construction of the basic graphs. Most of the work is in the evaluation of the expected values of SU(N) linkage variables. The evaluation of the expectation values of a group of fermionic operators at one location is almost trivial.
The price we pay for this choice is the phases of the now distributed gamma matrices and the absence of nuclear isospin symmetry.
Processing the Amplitudes
Testing
If we have a lot of courage, we can use this method to correct the normalizations of the Cle bsch -Gordon coefficients. The results were quite disappointing, as most of the calculations in the literature had errors. Before attempting to construct the operators for the weak Hamiltonian, we will construct the wave functions for the initial and final states.
First, for pedagogical purposes we will construct the single-pion state, then the Kaons, and finally the two-pion wave function. To construct the fermion bilinear operators and the quartic operators in Chapter 6, we use the following method. Such a factor is not a problem in practice, except for the sign confusion it can cause.
For the purpose of constructing four-fermion operators, we exploit this ambiguity to construct operators for which the transformation law under space inversion is obvious. To find a(i), write the continuum bilinearly with respect to the fields u, d, and then in terms of off, g spinors, as discussed in Sect. We do not need to use all grid operations to determine cxr(i).
With these rules, the collection of fermionic bilinears we will need is presented in Appendix A. The trick is to remember that all bilinears of this form belong to the SU(4) adjoint representation. The prescription we use is to construct an appropriate fermionic bilinear operator using the method described in Section 5.1., and then use this operator as an interpolation field for the state we want to achieve.
We can immediately obtain the wave functions, 1T 1 and 1Tz (corresponding to the flavor generators T1 and Tz) by cyclic permutations and then construct the charged pions as As we go to the continuum limit, the contributions of operators spanning more than one connection, together with baryon-antibaryon pairs (which we explicitly rejected by using trivial operators) will be mixed into the true wave function.
The Initial State : The Kaons
Hamiltonian,n in the limit of strong coupling, and have the same quantum numbers as pions. The trivial operators serve the very convenient purpose of suppressing the effect of baryon-antibaryon pairs. We note that the Xz field resides only on the even faces of the lattice and the Xl field on the odd faces.
It is important to note now that we can no longer use chiral symmetries because the vacuum breaks them. Similarly, we can construct wavefunctions using T matrices to form charged mesons; but we won't need any of these wave functions. If we are to use (5.3.3) to represent K0, there is an important physical issue to address.
To get rid of the unwanted uc part, we might be inclined to construct the operator c u + s d, or 1f!a151/la, and then take the difference of the two operators to eliminate the cu. However, in the strong coupling limit, these two states have different energies because the point splitting operators traverse one and three bonds, respectively. We will choose (5.3.3), instead of1f;a-y51f;a, to be our representation of the K0, because in the limit of m5 ~ 0 this wave function and the pions degenerate.
You might expect cu components to give us a lot of trouble; however, in the matrix elements we will calculate, the cu part will not contribute and can simply be ignored. Another major concern is that the wave function (5.3.2) does not have IN = 1/2 because there are no nuclear isospin symmetries in this model.
The Final State : Two Pions
In a more elaborate calculation than we entertain here, one IJ?.ight tries to mix these conditions together in the strong coupling limit using the methods of [18]. Which of these wave functions we choose to calculate with should be arbitrary; after all, the 6.IN = 1/2 rule is supposed to be a general feature of non-leptonic weak decays. 5.4.2) best reproduces the way in which the wave function of the two pions spreads over space; however, it does not do such a good job of handling final state interactions, at least at low order in strong coupling perturbations.
34; the direction of these states on the grid is defined by the axis along which the contractions over color index are taken.
The Strange Quark Mass
At (or slightly below) the scale of weak interactions, which we take as Mw. We can now run the effective weak Hamiltonian up to the quark mass scale c (and s for this calculation). The form of the Hamiltonian quoted in (6.1.2) assumes that there are only four quark flavors.
Note that we use operator forms that are clearly eigenstates of the equality. We must be able to adjust the relative normalizations between the constituent parts of the weak lattice Hamiltonian. Finally, we come to the actual calculation of the matrix elements we are interested in.
The following arguments are independent of the explicit forms of the wave functions and operators. Note that we cannot make any use of the now-broken chiral symmetries to reduce the matrix elements. These results are independent of the shape of the operators and are preserved in the grid.
There is an important simplification at this level since all the graphs derived from the operators 0 1 (2 ) and 0 2( 2 ) apparently vanish. The first evaluates to - 1/12; the second graph will not appear due to the disappearance of the energy denominator. Within this part of the calculation there are two contributions that we want to isolate; the "Penguin" graphs, and the rest.
In the following, we present the expected values of operators using three representative values of the coefficient of insignificant operators;. Banks is credited with this concept of prescribing the values of the coefficients of trivial operators. 1] The result we found most surprising was the disappearance of weak matrix elements for the first two orders of strong coupling perturbation theory.
In the following two tables we describe, in the continuum language, all the four-fermion operators we need for this calculation.
SU(2) Clebsch-Gordan Coefficients
The important point to note is that the ratio changes sign and passes through zero between 0.10 < A 1 < 0.12.
DECAY AMPLITUDE
