Murali Sharma, officemate and friend, who was always there to clear up any confusing issues that arose; and finally I would like to thank my parents, Werner and Maria Haas, for their support, emotional and otherwise. QCD interactions involving a heavy quark with energy much less than its mass can be understood in the context of an effective field theory in which the velocity of the heavy quark is kept fixed while its mass is brought to infinity. Nonleptonic decays of hadrons containing a heavy quark are further simplified when the exchanged gluons carry small momenta compared to the heavy quark mass.
The reliability of these assumptions is tested by calculating first-order, one-loop QCD corrections, assuming reasonable momentum distributions for the quarks within the light mesons.
Introduction
At the other end of the spectrum, QCD possesses rough symmetries when one or more quarks possess masses much larger than the QCD scale. The second and most powerful symmetry is an internal SU(2) spin symmetry associated with each of the heavy quarks. However, the complex interactions of light quarks in a bound system containing a heavy quark will not affect the motion of the system if mQ.
Under appropriate conditions, the inherent effects of the air in determining the flight of the ball can be neglected. The most important application of heavy quark effective field theory is semileptonic decay, e.g. Predictions regarding decays of this type are critical in determining the Cabbibo-Kobayashi-Maskawa matrix elements.3> Characterization of the corresponding transition matrix elements.
If we assume that this kind of "factorization" holds, physical predictions can be made. 4> Intuitively, factorization is a reasonable assumption if virtual gluons are produced. 4> Intuitively, factorization is a reasonable assumption if virtual gluon exchanges between the light quarks and heavy quarks are "soft", i.e. their momenta are small compared to the heavy quark. Chapter 2 reviews the properties of heavy quark effective field theory, followed by a discussion of its applications in Chapter 3.
Heavy Quark Effective Field Theory
V associated with incoming/outgoing heavy quark lines will always appear setting the peak in the effective theory. Alternatively, the effective theory can be developed by considering the Lagrangian of the full theory. 6>. Because there is no pair creation in the effective theory, there is a global symmetry U(l) associated with heavy quark conservation.
More importantly, the effective theory contains an SU(2) symmetry associated with spin conservation of heavy quarks.7> The lack of gamma matrices in the top of the heavy quark gluon (13) makes this symmetry easily clear. Finally, since the heavy quark masses do not appear in the effective theory (mQ .. oo), there is a taste symmetry between heavy quarks moving at the same speed. More generally, the new symmetry of the effective theory of heavy quarks can be used to establish properties for this kind of decay.
In this section, renormalization in the full theory of QCD is reviewed and then compared with renormalization in the effective theory. While the axial and vector currents do not require renormalization in the full theory, they do in the effective theory. Integrating (79) and using the matching condition (77) gives. The coefficient relating the axial current in the full theory, ~ y v y 5 Qi, to that in the effective theory, ~Yv y 5h~i) + numerator terms, is also given by (80).
The absence of gamma matrices in (95, 96) further extends the SU(2N) symmetry of HQEFf to include the intema1SU(2) spin symmetry of the light quarks. The Heavy Quark Effective Field Theory together with the complimentary notion of factorization developed in the previous chapters allows us to make definitive predictions about the non-leptonic decay of heavy mesons. Spinor induces on the quark fields and SU(3) color operators in the above operators are suppressed. c element of the Cabbibo-Maskawa matrix.
Instead, the QCD corrections to this non-leptonic weak decay are included in the hard scattering amplitudes, T?' p) • The pawn decay constant, f1t, is defined by. Since mrt :~=mP, the argument of the universal function must differ in the two transitions we consider. The large uncertainty in the experimental value of R makes it difficult to establish the reliability of the zero-order calculation.
In the first case, the corrections are independent of the fraction of momentum carried by the light quarks and thus fall out of the relation. Indeed, the symmetries of the effective theory of heavy quarks naturally relate heavy quark mesons with spin 0 and spin 1 of the same flavor, as discussed in the second chapter. The theory of the effective field of heavy quarks together with the complementary notion of factorization predicts that the zeroth order in the coupling a8(m~ the.
What remains is to relate these amplitudes, calculated in the full theory QCD, to the expansion for the heavy meson transition in terms of the hard scattering amplitude (rJs) (equation (112)).
Application of HQEFf
Factorization
Furthermore, the action of factors of(};+1) from the vertex in (11) and in the numerator of propagators produces factors of 2 when they act on spinors on the scale. The h!Q) part of Q is then used as a suitable approximation of the fundamental field in the mQ-oo limit. One might also be interested in the renormalization of compound operators, Green's functions of which may not be finite.
The procedure for calculating ZQ differs from the renormalization of wave functions in full theory, because Green's functions contain a logrhythmic dependence on the quark masses. to the choice of counterterms in (65) than in the ordinary theory. To compare matrix elements calculated in the effective theory with experimentally measured quantities, one must relate operators in the effective theory to those in the full theory QCD. The Heavy Quark Effective Field Theory, developed in the last two chapters, is especially useful in calculating decay rates for semileptonic processes such as.
It is possible to justify (85) for all orders of perturbation theory in a particular kinematic limit.3> In particular, in the case where the light quarks are almost collinear and the heavy quark masses are taken to infinity, diagrams with "soft" gluon exchanges. do not contribute to the matrix element in question. In the second case, they are absorbed in the definition of /ft , /p and the corresponding wave functions. We have also ignored any imaginary part arising from the terms in that flx branch of the logarithms; these terms do not contribute to the fee ratio in the order we are working.
These corrections are actually smaller than those discussed in the previous section and come from the kinemetric differences in the two decays. Calculation of rr> is carried out in the same way as in the scalar case.
Conclusion
It is reassuring to find that this prediction resists the one-cycle corrections a.s(m,) calculated in this thesis, although the AQCn / mb corrections may spoil this picture. It appears that the strong interactions, especially the non-leptonic decays discussed here, are well approximated by an effective field theory in which the heavy quark masses are taken to infinity (keeping their ratios fixed) and the matrix elements between the hadronic final states are factored. However, until improvements are made in the experimental uncertainty for the decay rates, the reliability of this approach cannot be definitively ascertained.
The incoming/outgoing spinors, the quark/gluon propagators, and quark-gluon-quark vertex factors follow from the Feynman rules for ordinary QCD. We would like an expression in which only factors of Yv(l-y s) occur between the heavy quark and light quark spinors. We are now in a position to work on the integrals, or at least re-express them in a more manageable form.
Fortunately, the infinite part of (Al2) is independent of x, the momentum fraction carried by the up quark, so it doesn't contribute to the ratio of the velocities (remember, the hard scattering amplitude is integrated. Similar; y, the flnite terms in (Al2) can be ignored because all x-dependence is in. The amplitudes corresponding to the remaining three diagrams in Figure 1 are calculated in a similar way.
Although we have calculated the amplitudes for each diagram separately, the gluon exchange between the light quark and the charm quark is completely analogous to the bottom quark case if we allow. We now use the approximations of the heavy quark effective theory and re-express the pion (or pho) momentum in terms of the heavy quark momentum:. Ignoring the piece y 5 (this contributes to rr> , the strong scattering amplitude for the excited heavy meson decays) we have.
For downward gluon exchange, the massing of the amplitude (A15) into the form corresponding to equation (112) is only slightly more involved. Wise, New Symmetries of the Strong Interactions, Lake Louise Winter Institute Lectures (1991), preprint CALT-68-1721.
