Matrix Elements Of Four-quark Operators Research Articles

Heavy-flavor-conserving hadronic weak decays of charmed and bottom baryons

Three decades ago, heavy-flavor-conserving (HFC) weak decays of heavy baryons such as ${\mathrm{\ensuremath{\Xi}}}_{Q}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{Q}\ensuremath{\pi}$ and ${\mathrm{\ensuremath{\Omega}}}_{Q}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Xi}}}_{Q}\ensuremath{\pi}$ for $Q=c$, $b$ had been studied within the framework that incorporates both heavy quark and chiral symmetries. It was pointed out that if the heavy quark in the HFC process behaves as a spectator, then the $P$-wave amplitude of ${\mathcal{B}}_{\overline{3}}\ensuremath{\rightarrow}{\mathcal{B}}_{\overline{3}}+\ensuremath{\pi}$, with ${\mathcal{B}}_{\overline{3}}$ being an antitriplet heavy baryon, will vanish in the heavy quark limit. Indeed, this is the case for ${\mathrm{\ensuremath{\Xi}}}_{b}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{b}\ensuremath{\pi}$ decays. For ${\mathrm{\ensuremath{\Xi}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}\ensuremath{\pi}$ decays, they receive additional nonspectator contributions arising from the $W$-exchange diagrams through the $cs\ensuremath{\rightarrow}dc$ transition. Spectator and nonspectator $W$-exchange contributions to the $S$-wave amplitude of ${\mathrm{\ensuremath{\Xi}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}\ensuremath{\pi}$ are destructive, rendering the $S$-wave contribution even smaller. However, the nonspectator effect on the $P$-wave amplitude was overlooked in all the previous model calculations until a very recent investigation within the framework of a constituent quark model in which the parity-conserving pole terms were found to be dominant in ${\mathrm{\ensuremath{\Xi}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}\ensuremath{\pi}$ decays. Since the pion produced in the HFC process is soft, we apply current algebra to study both $S$- and $P$-wave amplitudes and employ the bag and diquark models to estimate the matrix elements of four-quark operators. We confirm that ${\mathrm{\ensuremath{\Xi}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}\ensuremath{\pi}$ decays are indeed dominated by the parity-conserving transition induced from nonspectator $W$-exchange and that they receive largest contributions from the intermediate ${\mathrm{\ensuremath{\Sigma}}}_{c}$ pole terms. We also show that the $S$-wave of ${\mathrm{\ensuremath{\Omega}}}_{b}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Xi}}}_{b}\ensuremath{\pi}$ decays vanishes in the heavy quark limit, while ${\mathrm{\ensuremath{\Omega}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Xi}}}_{c}\ensuremath{\pi}$ receive additional $W$-exchange contributions via $cs\ensuremath{\rightarrow}dc$ transition. The $P$-wave contribution to ${\mathrm{\ensuremath{\Omega}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Xi}}}_{c}\ensuremath{\pi}$ is enhanced by the ${{\mathrm{\ensuremath{\Xi}}}^{\ensuremath{'}}}_{c}$ pole, though it is not so dramatic as in the case of ${\mathrm{\ensuremath{\Xi}}}_{c}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}\ensuremath{\pi}$. The asymmetry parameter $\ensuremath{\alpha}$ is found to be positive, of order 0.70 and 0.74 for ${\mathrm{\ensuremath{\Xi}}}_{c}^{0}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\mathrm{\ensuremath{\Xi}}}_{c}^{+}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}_{c}^{+}{\ensuremath{\pi}}^{0}$, respectively. The predicted branching fraction is of order $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ for ${\mathrm{\ensuremath{\Omega}}}_{c}^{0}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Xi}}}_{c}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ for ${\mathrm{\ensuremath{\Omega}}}_{c}^{0}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Xi}}}_{c}^{0}{\ensuremath{\pi}}^{0}$ both with the asymmetry parameter close to $\ensuremath{-}1$.

Phenomenological study of heavy hadron lifetimes

The lifetimes of bottom and charmed hadrons are analyzed within the framework of the heavy quark expansion (HQE). Lifetime differences arise from the spectator effects such as weak W-annihilation and Pauli interference. Spectator effects originating from dimension-7 four-quark operators are derived. Hadronic matrix elements of four-quark operators are parameterized in a model-independent way. Using the dimension-6 bag parameters recently determined from HQET sum rules and the vacuum insertion approximation for meson matrix elements of dimension-7 operators, the calculated B meson lifetime ratios τ(B+)/τ(Bd0) = 1.074− 0.016+ 0.017 and τ(Bs0)/τ(Bd0) = 0.9962 ± 0.0024 are in excellent agreement with experiment. Likewise, based on the quark model evaluation of baryon matrix elements, the resulting baryon lifetime ratios τ(Ξb−)/τ(Λb0), τ(Ξb−)/τ(Ξb0), τ(Ωb−)/τ(Ξb−) and the Λb − B0 lifetime ratio τ(Λb0)/τ(Bd0) = 0.953 also agree well with the data. Contrary to the bottom hadron sector where the HQE in 1/mb works well, the HQE to 1/mc3 fails to give a satisfactory description of the lifetimes of both charmed mesons and charmed baryons. This calls for the subleading 1/mQ corrections to spectator effects. The relevant dimension-7 spectator effects are in the right direction for explaining the large lifetime ratio of τ(Ξc+)/τ(Λc+). However, the destructive 1/mc corrections to Γ(Ωc0) are too large to justify the validity of the HQE, namely, the predicted Pauli interference and semileptonic rates for Ωc0 become negative. Demanding these rates to be positive for a sensible HQE, we find the lifetime pattern τ(Ξc+) > τ(Ωc0) > τ(Λc+) > τ(Ξc0), contrary to the current hierarchy τ(Ξc+) > τ(Λc+) > τ(Ξc0) > τ(Ωc0). We conclude that the Ωc0, which is naively expected to be shortest-lived in the charmed baryon system owing to the large constructive Pauli interference, could live longer than the Λc+ due to the suppression from 1/mc corrections arising from dimension-7 four-quark operators. The new charmed baryon lifetime pattern needs to be tested in forthcoming experiments.

Neutral kaon mixing beyond the Standard Model with nf = 2 + 1 chiral fermions. Part 1: bare matrix elements and physical results

We compute the hadronic matrix elements of the four-quark operators relevant for $K^0-{\bar K^0}$ mixing beyond the Standard Model. Our results are from lattice QCD simulations with $n_f=2+1$ flavours of domain-wall fermion, which exhibit continuum-like chiral-flavour symmetry. The simulations are performed at two different values of the lattice spacing ($a\sim0.08$ and $a\sim 0.11 \, \fm $) and with lightest unitary pion mass $\sim 300\, \MeV$. For the first time, the full set of relevant four-quark operators is renormalised non-perturbatively through RI-SMOM schemes; a detailed description of the renormalisation procedure is presented in a companion paper. We argue that the intermediate renormalisation scheme is responsible for the discrepancies found by different collaborations. We also study different normalisations and determine the matrix elements of the relevant four-quark operators with a precision of $\sim 5\%$ or better.

Lifetimes and heavy quark expansion

Kolya Uraltsev was one of the inventors of the Heavy Quark Expansion (HQE), that describes inclusive weak decays of hadrons containing heavy quarks and in particular lifetimes. Besides giving a pedagogic introduction to the subject, we review the development and the current status of the HQE, which just recently passed several non-trivial experimental tests with an unprecedented precision. In view of many new experimental results for lifetimes of heavy hadrons, we also update several theory predictions: [Formula: see text], τ(Bs)/τ(Bd) = 1.001 ±0.002, τ(Λb)/τ(Bd) = 0.935 ±0.054 and [Formula: see text]. The theoretical precision is currently strongly limited by the unknown size of the non-perturbative matrix elements of four-quark operators, which could be determined with lattice simulations.

NNLO QCD corrections to the [formula omitted] matrix elements using interpolation in [formula omitted

One of the most troublesome contributions to the NNLO QCD corrections to B ¯ → X s γ originates from three-loop matrix elements of four-quark operators. A part of this contribution that is proportional to the QCD beta-function coefficient β 0 was found in 2003 as an expansion in m c / m b . In the present paper, we evaluate the asymptotic behaviour of the complete contribution for m c ≫ m b / 2 . The asymptotic form of the β 0 -part matches the small- m c expansion very well at the threshold m c = m b / 2 . For the remaining part, we perform an interpolation down to the measured value of m c , assuming that the β 0 -part is a good approximation at m c = 0 . Combining our results with other contributions to the NNLO QCD corrections, we find B ( B ¯ → X s γ ) = ( 3.15 ± 0.23 ) × 10 −4 for E γ > 1.6 GeV in the B ¯ -meson rest frame. The indicated error has been obtained by adding in quadrature the following uncertainties: Non-perturbative (5%), parametric (3%), higher-order perturbative (3%), and the interpolation ambiguity (3%).

THE B → ππ, πK PUZZLES: IMPLICATIONS FOR HADRON PHYSICS, NEW PHYSICS AND RARE DECAYS

The B-meson system is an interesting probe for the exploration of strong interactions, the quark-flavour sector of the Standard Model, and the search for new physics. In this programme, non-leptonic B decays, which are particularly challenging from the point of view of QCD, play a key rôle. After discussing strategies to deal with the corresponding hadronic matrix elements of four-quark operators and popular avenues for new physics to manifest itself in B decays, we focus on puzzling patterns in the B-factory data for B → ππ, πK decays; we explore their implications for hadron physics, new physics and rare K and B decays.

Short-distance analysis of [formula omitted] mixing

We study the Standard Model short-distance prediction for the mass and lifetime differences between the two neutral D meson mass eigenstates. We find that, despite α s / 4 π suppression, next-to-leading order (NLO) short-distance QCD corrections exceed the corresponding leading order (LO) amplitudes. For the lifetime difference, this stems from the lifting of helicity suppression of a light-quark intermediate state. We find y D is given by y NLO to a reasonable approximation but x D is greatly affected by destructive interference between x LO and x NLO . The net effect is to render y D ∼ x D ≃ 6 × 10 −7 . Our NLO short-distance results, still smaller than most long-distance estimates, depend on the same two nonperturbative matrix elements of four-quark operators as in leading order.

Non-factorization effects in heavy mesons and determination of |Vub| from inclusive semileptonic B decays

The effects of spectator light quark in decays of heavy mesons are considered, which vanish in the limit of factorization of matrix elements of four-quark operators over the mesons. These effects include the difference of the total widths as well as of the semileptonic decay rates between the D^0 and D_s mesons and also a contribution to inclusive semileptonic decay rates of B^0 and B^\pm into the channel X_u l nu related to determination of the weak mixing parameter V_{ub}. If the observed difference of the lifetimes between D_s and D^0 mesons is attributed to non-factorizable terms, such terms can naturally give rise to a significant difference in inclusive semileptonic decay rates of these mesons, and to a light-flavor dependent contribution to decays B -> X_u l nu. The latter contribution affects mostly the upper part of the inclusive spectrum of the invariant mass of the lepton pair, and may significantly exceed the previously claimed in the literature theoretical uncertainty in determination of |V_{ub}|^2 from that part of the spectrum.

Matrix elements of four-quark operators relevant to life time difference $\Delta \Gamma _{B_s}$ from QCD sum rules

We extract the matrix elements of four-quark operators $O_{L,S}$ relevant to the $B_s$ and ${\bar B}_s$ life time difference from QCD sum rules. We find that the vacuum saturation approximation works reasonably well, i.e., within 10%. We discuss the implications of our results and compare them with a recent lattice QCD determination.

The electroweak π+− π0 mass difference and weak matrix elements in the 1/ Nc expansion

The π +− π 0 mass difference generated by the electroweak interactions of the Standard Model is expressed, to lowest order in the chiral expansion and to leading order in α, in terms of an integral involving the same correlation function Π LR ( Q 2) which governs the well known electromagnetic π +− π 0 mass difference. We calculate this contribution within the framework of QCD in the limit of a large number of colours N c . We show how this calculation, which requires non-trivial contributions from next-to-leading terms in the 1/ N c expansion, provides an excellent theoretical laboratory for studying issues of long- and short-distance matching in calculations of weak matrix elements of four-quark operators. The electroweak π +− π 0 mass difference turns out to be a physical observable which is under good analytical control and which should, therefore, be an excellent testground of numerical evaluations in lattice-QCD and of model calculations in general.

Chiral properties of low-energy matrix elements of local four-quark operators

The chiral properties of low-energy matrix elements of four-quark operators describing the weak vertices in the effective Lagrangian of weak interactions are investigated. A constituent quark diagram technique for the derivation of “soft” pion low-energy theorems is proposed. The technique is applied to the calculation of the⌆-term operator correlator in the low-energy theorem of the penguin-operator matrix elements.

The effective action of four-quark operators and non-leptonic weak transitions

My talk at the Montpellier Workshop was a summary of recent work which has now appeared in print in quite a detailed form: [1] and [2]. Reference [1] describes the derivation, under a well defined set of approximations, of a low-energy effective chiral Lagrangian of the strong interactions from QCD. The effective Lagrangian includes only the Nambu-Goldstone modes (i.e., the octet of pseudoscalar particles) in the presence of external sources, with inclusion of the explicit chiral symmetry breaking due to the light quark masses M. The coupling constantsof the O( p 4) terms of the chiral Lagrangian are calculated first to leading order in the 1/ N c expansion but with O( α s ) terms and higher orders neglected. Next, the corrections O( α s ) to the couplings which remain in the chiral limit M → 0 are also taken into account. In reference [2], the matrix elements of four-quark operators of the effective weak Hamiltonian for ΔS = 1 and ΔS = 2 transitions are studied with two approaches. First inclusively : two-point functions of four-quark operators are studied. Their corresponding spectral functions at very short-distances are computed using perturbative QCD with inclusion of O( α s ) corrections; and at very long-distances, using the chiral effective realization of QCD in terms of Goldstoneparticles. A qualitative picture emerges which requires, for consistency between the two extreme behaviours, a large coupling constant for ΔI = 1 2 transitions. The other more direct approach described in ref. [2], consists in calculating the effective action of four-quark operators by integrating at the quark fields in a gluonic background and in the presence of a source term which triggers spontaneous chiral symmetry breaking. The procedure follows the method elaborated in ref. [1]. This way one can compute the coupling constants of the lowest order effective chiral Lagrangian with ΔS = 1(ΔI = 1 2 and 3 2 ) and ΔS = 2 (the so called B ̂ - parameter ). The calculations include the well known leading O( N c 2) contributions as well as the subleading O( N c ) with inclusion of the O( α s N c ) terms. The picture which emerges is very encouraging. Comparison with related work can also be found in the lists of references in [1] and [2].

Four-quark operators and non-leptonic weak transitions

Matrix elements of four-quark operators are studied with two approaches. First inclusively: we consider two-point functions of four-quark operators. We compute their corresponding spectral functions at very short distances using perturbative QCD with inclusion of O(α rms) corrections; and at very long distances, using the chiral effective realization of QCD in terms of Goldstone particles. A qualitative picture emerges which requires, for consistency between the two extreme behaviours, a large coupling constant for ΔI = 1 2 transitions. The other, more direct approach, consists of calculating the effective action of four-quark operators by integrating out the quark fields in a gluonic background and in the presence of a source term which triggers spontaneous chiral symmetry breaking. The procedure follows the method elaborated recently in ref. [1]. This way we compute the coupling constants of the lowest-order effective chiral lagrangian with ΔS = 1 (ΔI = 1 2 and 3 2 ) and ΔS = 2 (the so called B ̂ - factor .) The calculations include the well-known O( N c 2) contributions as well as the subleading O( N c) with inclusion of the O( α s N c) terms. The picture which emerges at this approximation is already very encouraging. Comparison with other approaches is also made.

KL,S → l+l− decays in the quark model of superconductivity type

The probabilities of KL,S → l+l− decays are calculated by considering only the dominant two-lepton channel KL,S →γ*γ* → l+l−. The transitions K0 → X, where X=π0,η,η′ orɛ (700) dominating in the KL,S →γ*γ*decay amplitudes, are described by the effective weak Lagrangian. The matrix elements of four-quark operators are calculated within the framework of the quark model of superconductivity type. It is shown that the leading contribution to matrix elements of K0 → X transitions comes from the penguin operator matrix elements.

Nonleptonic Interactions of Kaons in the Quark Confinement Model

Nonleptonic, electromagnetic decays of K-mesons and K°–K̄° transition are treated with the effective Hamiltonians with ΔS = 1, 2. Matrix elements of four-quark operators are calculated in the framework of Quark Confinement Model. The account of the intermediate scalar mesons turns out to allow to explain ΔI = 1/2 rule. The account of intermediate scalar and pseudo-scalar states leads to experimental value of mass difference of KL°–KS° mesons. The obtained values for nonleptonic and electromagnetic decay widths as well as mass difference ΔmLS are in good agreement with experimental data.

Exact dependence of multiquark operators on the number of colors.

We develop a method which enables us to calculate explicitly matrix elements of multiquark operators between two nonstrange baryon states, thus obtaining the exact dependence of these operators on N, the number of color degrees of freedom of quarks. The method employs permutation symmetry of the N-quark wave function and it is illustrated by evaluating ${g}_{A}$/${g}_{V}$ and the matrix elements of four-quark operators of the \ensuremath{\Delta}S=0 effective weak Hamiltonian. Implications of these results for low-energy phenomenology and for the interpretation of Skyrme-model results are discussed.