Nonleptonic Amplitudes Research Articles

The influence of dimension eight operators on weak matrix elements

I describe recent work with V. Cirigliano and E. Golowich on the effect of dimension eight operators on weak nonleptonic amplitudes. The basic mesage is that there is an inconsistency in the way that many calculations are traditionally performed. If one calculates matrix elements involving only physics below some scale μ, then one needs dimension eight operators explicitly in the weak OPE. On the other hand if one wants to use dimensional regularization throughout, then one needs all scales to be included within the matrix element, and this results in the same net effect. A numerical estimate indicates that this is important below μ = 2 GeV, and this calls into question many of the models that have been used to predict ε′/ε.

Charming-penguin enhanced B decays

Long-distance contributions of operators of the effective non-leptonic weak Hamiltonian containing charmed currents have been recently studied. Penguin-like contractions of these operators, denoted as charming penguins, have been shown to be relevant for several B decays into two pseudoscalar mesons. In particular, they are expected to give large enhancements to processes which would be otherwise Cabibbo-suppressed. Their contributions easily lead to values of BR( B + → K 0 π +) and BR( B d → K + π −) of about 1 × 10 −5, as recently found by the CLEO collaboration. In this paper, we show that such large branching fractions cannot be obtained without charming penguins. This holds true irrespective of the model used to compute the non-leptonic amplitudes and of the relevant parameters of the CKM mixing matrix. We use the experimental measurements of the B + → K 0 π + and B d → K + π − decay rates to constrain the charming-penguin amplitudes and to predict BR's for a large set of other two-body decay channels where their contributions are also important. These include several pseudoscalar-vector and vector-vector channels, such as B → ϱK and B → ϱK ∗ , which have not been measured yet.

Reanalysis of the decay KL--> pi 0e+e-

The decay ${\mathit{K}}_{\mathit{L}}$\ensuremath{\rightarrow}${\mathrm{\ensuremath{\pi}}}^{0}$${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ is searched for as a signal of direct \ensuremath{\Delta}S=1 CP violation. We provide a thorough updating of the analysis of the three components of the decay: (1) direct CP violation, (2) CP violation through the mass matrix, and (3) CP-conserving (two photon) contributions. First the chiral calculation of the ${\mathit{K}}_{\mathit{S}}$\ensuremath{\rightarrow}${\mathrm{\ensuremath{\pi}}}^{0}$${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ rate, due to Ecker, Pich and, de Rafael, is updated to include recent results on the nonleptonic amplitude. Then we systematically explore the uncertainties in this method. These appear to be so large that they will obscure direct CP violation unless it is possible to measure the ${\mathit{K}}_{\mathit{S}}$\ensuremath{\rightarrow}${\mathrm{\ensuremath{\pi}}}^{0}$${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ rate. The CP-conserving amplitude remains somewhat uncertain, but present indications are such that there may be a sizable CP-violating asymmetry in the ${\mathit{e}}^{+}$,${\mathit{e}}^{\mathrm{\ensuremath{-}}}$ energies from the interference of the CP-conserving and CP-violating amplitudes and this may potentially be useful in determining whether direct CP violation is present.

Anatomy of a weak matrix element

Although the weak nonleptonic amplitudes of the Standard Model are notoriously difficult to calculate, we have produced a modified weak matrix element which can be analyzed using reliable methods. This hypothetical nonleptonic matrix element is expressible in terms of the isovector and axialvector spectral functions ϱ v ( s) and ϱ A( s), which can be determined in terms of data from tau lepton decay and e + e − annihilation. Chiral symmetry and the operator product expansion are used to constrain the spectral functions respectively in the low energy limits. The magnitude of the matrix element thus determined is compared with its “vacuum saturation” estimate, and in the future may be accessible with lattice calculations.

The rule of discarding 1/ Nc in inclusive weak decays (I)

We continue to investigate the effects of the (dominant) subleading operator of the σH type in the 1/ N c parts of the weak nonleptonic amplitudes. If the previous work concentrated on exclusive decays now we analyse inclusive widths in a certain kinematic limit. Deviations from the parton model predictions are found due to soft-gluon color exchange. In the interference (i.e. 1/ N c) terms they are stronger than in the non-interferene terms. The sign is negative so that the 1/ N c parts tend to cancel. The absolute value of the effect is of order 1 in D-mesons and of order 0.25 in B-mesons.

| ΔS| = 1 non-leptonic amplitudes and the 1/ Nc expansion

The application of the 1/ N c expansion to the estimate of weak non-leptonic amplitudes is studied. The formulation is renormalization-group invariant and consistent with chiral symmetry. The problem of matching the short- and long-distance descriptions is properly displayed and evidence is shown for the need of more information beyond that known in these regimes. The matching problem first appears at O(1/ N c), but is important because in the 1/ N c approach the relevant octet enhancement (Δ1 = 1 2 rule) requires large contributions from these O(1/ N c) terms.

On summation of the gluonic corrections to the weak non-leptonic amplitudes

Leading logarithmic contributions to the effective | ΔS| = 1, |ΔT| = 1 2 weak lagrangian are calculated at the two-loop level. The terms of the kind α S 2ln 2 m W, α S 2ln m Wln m c, α S 2ln 2 m c are taken into account. The results agree with those obtained in refs. [2,3] by the renormalization group technique and disagree with the statement of refs. [4,5]. The cause of the disagreement is exposed.

QCD non-leading corrections to weak decays as an application of regularization by dimensional reduction

We compute in QCD with massless quarks the two-loop anomalous dimensions of the four-fermion operators relevant to weak non-leptonic amplitudes and discuss the physical implications for strange and charm particle decays. Our main goal was to derive the complete order α corrections to the inclusive decay width of a charm (or heavier) quark. We find that the correction terms in the effective hamiltonian show the same enhancement-suppression pattern as the leading terms. Also, for a free c quark, the non-leptonic decay width is further increased (and the semileptonic branching ratio further depleted) with respect to the leading approximation. Implications for D (especially D +) decays are discussed. The calculation was completely done by a recently proposed variant of dimensional regularization called dimensional reduction. Much of this paper is devoted to exposing and resolving (within two loops) the field theoretic subtleties associated with this method which was particularly suited to this computation.

The Weinberg-Salam model and weak radiative decays of K L mesons

Weak radiative decays K L → 2 γ and K L → π + π − γ are considered in the Weinberg-Salam model for weak interactions and in the framework of asymptotically free gauge theory (AFGT) for strong interactions (SI). The problem of calculation of the decays at short and large distances is discussed. It is shown that only the large distance contribution is essential for each decay. The estimates of these contributions being rough, it proves impossible to compare them with experiment. However, using anomalous PCAC theory we derive the relation connecting the contributions for these two decays and, hence, the decay amplitudes via the known constants. Experimental data confirm the obtained result. The important result of the paper is that the matrix elements of the effective Hamiltonian specifying the decays at short distances are not dynamically enhanced, unlike the case of weak non-leptonic amplitudes.

Spontaneously Broken SU(2) Gauge Invariance and the I = 1/2 Rule

A model of nonleptonic weak interactions is proposed which is based on spontaneously broken SU(2) gauge invariance. The SU(2) group is taken analogously to the U-spin. To this scheme, the source of nonleptonic decays consists of only neutral currents, and violation of strangeness stems from weak vector boson mixings. The model can provide a natural explanation of the ΔI = 1/2 rule and of the bulk of the ΔI = 1/2 nonleptonic amplitude. As a consequence, a picture is obtained that weak interactions originate in spontaneously broken gauge invariance under orthogonal SU(2) groups. Finally, a possibility of unifying weak and electromagnetic interactions is indicated.

The [formula omitted] rule with heavy quarks and right-handed currents

The effect of strong interactions on a two-fermion operator is calculated in an asymptotically free theory, thereby completing the calculation of the logarithmic exponents for the operators contributing to non-leptonic amplitudes. The case of models with both left-and right-handed weak currents is discussed and the implications for the ΔI= 1 2 rule and other non-leptonic decays are deduced.

The structure of parity violating strangeness conserving weak non-leptonic amplitudes in an asymptotically free theory

We study the structure of parity violating ΔS = 0 weak non-leptonic amplitudes in a gauge theory of strong, weak and electromagnetic interactions based on the Weinberg-Salam theory with three quartets of fractionally charged quarks. It is shown, that, similarly to what happens for ΔS = 1 non-leptonic amplitudes, a definite pattern of enhanced terms emerges. The enhanced operators have isospin zero and one, while isospin-two components are suppressed. The enhancement factors are comparable to those found in the same model for ΔS = 1 transitions and can account for at least a part of the present disagreement between experimental findings and theoretical estimates for the photon polarization in the process n + p → d + γ at thermal energies.

Unified Lepton-Hadron Symmetry and a Gauge Theory of the Basic Interactions

An attempt is made to unify the fundamental hadrons and leptons into a common irreducible representation $F$ of the same symmetry group $G$ and to generate a gauge theory of strong, electromagnetic, and weak interactions. Based on certain constraints from the hadronic side, it is proposed that the group $G$ is SU(4\ensuremath{'}) \ifmmode\times\else\texttimes\fi{} SU(4\ensuremath{''}), which contains a Han-Nambu-type SU(3\ensuremath{'}) \ifmmode\times\else\texttimes\fi{} SU(3\ensuremath{''}) group for the hadronic symmetry, and that the representation $F$ is (4,4*). There exist four possible choices for the lepton number $L$ and accordingly four possible assignments of the hadrons and leptons within the (4,4*). Two of these require nine Han-Nambu-type quarks, three "charmed" quarks, and the observed quartet of leptons. The other two also require the nine Han-Nambu quarks, plus heavy leptons in addition to observed leptons and only one or no "charmed" quark. One of the above four assignments is found to be suitable to generate a gauge theory of the weak, electromagnetic, and SU(3\ensuremath{''}) gluonlike strong interactions from a selection of the gauges permitted by the model. The resulting gauge symmetry is $\mathrm{SU}{({2}^{\ensuremath{'}})}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{({3}^{\ensuremath{'}\ensuremath{'}})}_{L+R}$. The scheme of all three interactions is found to be free from Adler-Bell-Jackiw anomalies. The normal strong interactions arise effectively as a consequence of the strong gauges $\mathrm{SU}{({3}^{\ensuremath{'}\ensuremath{'}})}_{L+R}$. Masses for the gauge bosons and fermions are generated suitably by a set of 14 complex Higgs fields. The neutral neutrino and $\ensuremath{\Delta}S=0$ hadron currents have essentially the same strength in the present model as in other $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ theories. The mixing of strong- and weak-gauge bosons (a necessary feature of the model) leads to parity-violating nonleptonic amplitudes, which may be observable depending upon the strength of SU(3\ensuremath{''}) symmetry breaking. The familiar hadron symmetries such as SU(3\ensuremath{'}) and chiral $\mathrm{SU}{({3}^{\ensuremath{'}})}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{({3}^{\ensuremath{'}})}_{R}$ are broken only by quark mass terms and by the electromagnetic and weak interactions, not by the strong interactions. The difficulties associated with generating gauge interactions in the remaining three assignments are discussed in Appendix A. Certain remarks are made on the question of proton and quark stability in these three schemes.

Violation of the [formula omitted] rule in non-leptonic decays

Tree graph contributions to the non-leptonic decays are calculated. They give the correct order of magnitude for the |Δ I| = 3 2 amplitude; their contribution is, however, too large by roughly a factor 2. Since this result appears consistently in all accessible non-leptonic decays we conclude that prominent parts of the non-leptonic amplitude as given by the tree-graph contributions are partially cancelled or suppressed by an unknown mechanism.

Formula omitted] baryon resonances and nonleptonic hyperon decays

The relative importance of J P = 1 2 ± baryon resonances in nonleptonic hyperon decays is investigated within a dispersion relation formulation of the current algebra-PCAC method with a view to establishing which resonances can be neglected. The dispersion relation formulation is reviewed, equal-time commutator and octet baryon pole terms are calculated, and then pion-octet baryon intermediate states are included in the calculation of the imaginary parts of the amplitudes for the dispersion relations. The imaginary parts of the amplitudes then involve pion-octet baryon scattering amplitudes for which a resonance approximation is used and weak nonleptonic amplitudes which are approximated by the equal-time commutator and octet baryon pole terms. An evaluation of the resulting dispersion integrals shows that it is safe to neglect resonances with J P = 1 2 + (J P = 1 2 − ) in S-wave ( P-wave) decays and, also, that the P-wave amplitude of Σ + → n + π + may be substantially enhanced by the resonance N′(1470).

Divergences of weak nonleptonic amplitudes and the breaking ofSU 3×SU 3

Divergences of weak nonleptonic amplitudes and the breaking ofSU 3×SU 3

Symmetries and Dynamics of Nonleptonic Decays

It is shown that the strength of the ${K}_{1}\ensuremath{\rightarrow}\mathrm{vacuum}\mathrm{vertex}$, arising solely through medium-strong $\mathrm{SU}(3)$-breaking interactions, is large and can provide a natural explanation for the octet enhancement of the parity-violating nonleptonic amplitudes. Furthermore, the sum rule $\ensuremath{-}A(\ensuremath{\Lambda}\ensuremath{\rightarrow}p+{\ensuremath{\pi}}^{\ensuremath{-}})+2A({\ensuremath{\Xi}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\Lambda}+{\ensuremath{\pi}}^{\ensuremath{-}})=\sqrt{3}A({\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}p+{\ensuremath{\pi}}^{0})$ for parity-violating hyperon decays, derived on the basis of the ${\ensuremath{\lambda}}_{6}$ transformation, also holds if the amplitudes are dominated by any or all of the following: the baryon octet, baryon decuplet, or scalar- or vector-octet pole terms, arising through the ${K}_{1}$ tadpole. This is true even though the latter transforms like ${\ensuremath{\lambda}}_{7}$. Thus it is concluded that neither the forbiddenness of the ${K}_{1}$ tadpole in the limit of $\mathrm{SU}(3)$ nor the existence of the sum rule for the parity-violating decays on the basis of the ${\ensuremath{\lambda}}_{6}$ transformation provide any argument against a possible dynamical picture of octet enhancement in nonleptonic transitions. The dynamics considered in the present note should be important regardless of whether the octet transformation has a dynamical or primary origin. Comments are also made on some other directly related problems: (i) the possible effect of symmetry violation on an otherwise forbidden transition (especially ${K}_{1}\ensuremath{\rightarrow}2\ensuremath{\pi}$ decay), (ii) the meaning of enhancement of nonleptonic rates compared with leptonic ones, and (iii) the ratio of $\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{0})$ to $\ensuremath{\Gamma}({K}_{1}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{\ensuremath{-}})$.

Properties of Weak Amplitudes in BrokenŨ(12)

Weak nonleptonic amplitudes are discussed in the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}(12)$ scheme on the assumption of octet dominance. The contributions to hyperon-decay $S$ waves of spurions of representations higher than 143 of the current \ensuremath{\bigotimes} current product are discussed in detail. In addition to Gell-Mann's relations for hyperon $S$ waves we find that ${\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}n+{\ensuremath{\pi}}^{+}$ is forbidden in $S$ waves apart from minor kinematical corrections. Possible modifications from a kinetic-energy spurion are also examined. Some remarks concerning consequences of $\mathrm{CP}$ invariance and concerning ${K}_{^{2}\ensuremath{\pi}}$ decays are included.