Strangeness-changing Vector Research Articles

Analyticity constraints on the strangeness changing vector current and applications to τ→Kπντ and τ→Kππντ

We discuss matrix elements of the strangeness changing vector current using their relation, due to analyticity, with πK scattering in the P-wave. We take into account experimental phase-shift measurements in the elastic channel as well as results, obtained by the LASS collaboration, on the details of inelastic scattering, which show the dominance of two quasi-two-body channels at medium energies. The associated form factors are shown to be completely determined, up to one flavor symmetry breaking parameter, imposing boundary conditions at t=0 from chiral and flavor symmetries and at t→∞ from QCD. We apply the results to the τ→Kπντ and τ→Kππντ amplitudes and compare the former to recent high statistics results from B factories.

Light quark masses from scalar sum rules

In this work, the mass of the strange quark is calculated from QCD sum rules for the divergence of the strangeness-changing vector current. The phenomenological scalar spectral function which enters the sum rule is determined from our previous work on strangeness-changing scalar form factors [1]. For the running strange mass in the MS_bar scheme, we find m_s(2 GeV) = 99 +- 16 MeV. Making use of this result and the light-quark mass ratios obtained from chiral perturbation theory, we are also able to extract the masses of the lighter quarks m_u and m_d. We then obtain m_u(2 GeV) = 2.9 +- 0.6 MeV and m_d(2 GeV) = 5.2 +- 0.9 MeV. In addition, we present an updated value for the light quark condensate.

The strange-quark mass from QCD sum rules: An update

Three different ways of determining the strange quark mass using QCD sum rules are reviewed. First, from a QCD sum rule determination of the up and down quark masses, together with the current algebra ratio m s (m u + m d) . Second and third, from QCD sum rules involving the strangeness changing vector and axial-vector current divergences, respectively. Present results are encompassed in the value: m s (1 GeV) = 150±70 MeV. It is argued that until direct, and precision measurements of the relevant spectral functions become availabel, the above error is not likely to be reduced.

QCD determination of the strange-quark mass

The strange-quark mass is determined from finite energy and Laplace transform QCD sum rules for the two-point function involving strangeness changing vector currents and their divergences. Improved QCD input and experimental data in the I = 1 2 , S-wave Kπ amplitude are used to obtain: m ̂ s = 266 ± 29 MeV , or m s (1 GeV) = 194 ± 4 MeV , where m ̂ s and m are the invariant and the running quark masses, respectively.

K¯*0(890)decay density matrix elements and partial conservation of strangeness-changing vector current

We show that the zeros in the density matrix elements ${\ensuremath{\rho}}_{1,\ensuremath{-}1}^{t}$ and $\mathrm{Re}{\ensuremath{\rho}}_{10}^{s}$ for ${K}^{\ensuremath{-}}p\ensuremath{\rightarrow}{{\overline{K}}^{*}}^{0}n$ (${K}^{+}n\ensuremath{\rightarrow}{{K}^{*}}^{0}p$) are correlated and imply fundamental relationships among the Ball amplitudes following from partial conservation of strangeness-changing vector currents and the smoothness hypothesis.

Exact representation of meson-meson scattering amplitudes consistent with current-algebra constraints

All four-point Green functions connected to the vector and axial-vector currents by Ward identities are constructed, consistent with the constraints of an arbitrary current algebra. The structure of the four-point functions is separated into one-particle reducible and irreducible parts, without approximations. That is, no single-particle approximation is made, nor is it assumed that the strangeness-changing vector currents are conserved. The result is a representation of the meson-meson scattering amplitude that explicitly satisfies all the constraints of the current algebra. In particular, it satisfies: (i) threshold theorems, (ii) crossing symmetry and (iii) reduction to the usual tree-approximation in the single-particle approximation. Application is made to kaon-pion scattering, obtaining a representation of the scattering amplitude that is suitable for the investigation of the constraints of elastic unitarity.

Estimation of the vacuum SU(3) breaking in the [formula omitted] model

It is pointed out that the usual (3, 3) + ( 3,3) model of chiral symmetry breaking and recent mathematical analyses of the parameters in K ℓ3 decay restrict the possible value of Δ K (0), the generalized propagator of the strangeness-changing vector divergence at zero four momentum squared. The requirement of approximate SU(3) invariance of the vacuum reduces the allowed range of values still more. When this parameter is then evaluated using fits to the Kπ phase-shift data, it is found to fall outside available interval. Therefore it is concluded that, within this scheme, the Kπ scattering data is incompatible with the SU(3) symmetry of the vacuum. Possible ways of avoiding this conclusion, or of altering the scheme, are considered.

Model of Spontaneous Breakdown of SU(3) ⊗ SU(3) Symmetry. II

Strong interactions of spin-zero and spin-one mesons are studied in a model in which the Lagrangian is approximately invariant under coordinate-dependent $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ gauge transformations. The scalar and pseudoscalar nonets are assigned to the (3,3*) \ensuremath{\bigoplus} (3*,3) representation, whereas the vector and axial-vector octets are introduced as gauge fields belonging to the (1,8) \ensuremath{\bigoplus} (8,1) representation. Spontaneous symmetry breaking to the isospin-hypercharge level is introduced by giving nonzero vacuum expectation values to some scalar fields. The effective Lagrangian obtained after considering field mixings and renormalizations is used in a phenomenological manner to study the masses and couplings of various particles. With some of the experimental meson masses as input, the predicted masses of remaining observed mesons are in reasonable agreement with experiment. Some scalar-meson masses are also predicted (${\ensuremath{\mu}}_{{S}_{K}}=527$ MeV and ${\ensuremath{\mu}}_{{S}_{\ensuremath{\pi}}}=771$ MeV). The widths for $V\ensuremath{\rightarrow}\mathrm{PP}$ and $A\ensuremath{\rightarrow}\mathrm{VP}$ decays are calculated and compare well with experiment. The axial-vector currents and the strangeness-changing vector currents satisfy partial conservation equations. Decay constants for spin-zero mesons are calculated. We get $\frac{{F}_{K}}{{F}_{\ensuremath{\pi}}}=1.17$ and $\frac{{F}_{{S}_{K}}}{{F}_{\ensuremath{\pi}}}=\ensuremath{-}0.59$. The ${K}_{l3}$ form factors are calculated, the effect of the scalar kaon being included; we get ${f}_{+}(0)=0.86$, $\ensuremath{\xi}=\ensuremath{-}0.197$, ${\ensuremath{\lambda}}_{+}=0.023$, and ${\ensuremath{\lambda}}_{\ensuremath{-}}=0.013$.

Bounds for K ℓ3 decay parameters

The extremum of the integral ∫ ∞ f( x) ‖ F ( x)‖ 2 d x is determined for the case when the values taken by the function F ( x) 1 at three points are known. The result is applied to the K ℓ3 problem, the function F ( x) being replaced by the form factor f +( q 2)+ f −( q 2) q 2( m K 2− m π 2) of the divergence of the strangeness changing vector current.

Bounds onKl3-Decay Form Factors

We rederive certain bounds on ${K}_{l3}$ decay parameters found recently by Li, Pagels, and Okubo and obtain some new results with a simpler method based on a direct application of the maximum-modulus theorem for holomorphic functions. For an arbitrary value of the momentum-transfer variable $t$ in the complex cut $t$ plane, we find that the domain of values which can be taken by the form factor $d(t)$ of the divergence of the weak strangeness-changing vector current ${V}_{\ensuremath{\mu}}^{(K)}$ is bounded by a circle in the plane ($\mathrm{Re}d(t),\mathrm{Im}d(t)$). We express the radius and the position of the center of this circle in terms of ${f}_{+}(0)$ and of the propagator $\ensuremath{\Delta}(t)$ of the divergence of ${V}_{\ensuremath{\mu}}^{(K)}$.

Use of equal-time commutators involving the symmetric energy-momentum tensor for the derivation of sum rules

We emphasize the central role that second-order Schwinger terms in equal-time commutators involving the energy density T oo play in determining the scaling behaviour of local operators. Thereby a new derivation is given of the fact (recently derived by Chanowitz) that the time component of a conserved current has dimensions 3 if it has a dimension at all and if a state | A〉 exists such that 〈A| J o|A〉 ≠ 0. Assuming that the time component of the non-conserved strangeness changing vector current V μ K has a dimension we show by saturation of the matrix element 〈π + ( q) | [ K o , V o K (0) | K + ( q)〉 (with K μ the conformal charges) in the infinite momentum frame that the root mean square gravitational factor radii (i.e. the slope of the gravitational form factor G 1( t) at t = 0) of π and K are the same. In this case, Regge theory is used to justify the infinite momentum limit. Formal application of this limit also shows that the dimensions of ∂ μ V μ K and V k K (if any) should be 2 and 3, respectively. This result is less reliable since in this case Regge theory implies invalidity of the procedure. Analogous results hold under somewhat stronger assumptions for η too. We discuss the consequences of SU(3) and our results for tensor meson dominance and find that a (numericaly possible small) subtraction that depends on the SU(3) index M must be present in the pseudo-scalr-tensor meson form factor G 1 M( t). Our results are in at least approximate agreement with SU(3) for 2 + → 0 −0 − and allow us to predict - correct within the errors - a value for the width f → K K . They also agree with what is known about f → ηη and f′ → ηη. The assumptions include neglect of contributions from the K A, i.e. the doubtful Q-range 1240–1400 MeV.

Spectral Function Sum Rules, Symmetry Breaking andK*-Width

Sometime ago, Das, Mathur and Okubo1> derived and generalised Weinberg's spectral function sum rules1> for the vector and axial,vector currents, on the basis of the asymtotic properties of the propagation functions for the currents, as suggested by symmetry groups. It was noted, 3> however, that both the sum rules should not be used simultaneously as the results thus obtained were in disagreement with experiment. In this note we shall assume a modified form of the second sum rule corresponding to chiral SU (2) X SU (2) along with the first one in the original form. For SU(3) symmetry, we shall consider only the first sum rule. For the particular SU (2) X SU (2) subgroup of chiral SU (3) X SU (3) pertaining to the strangeness changing vector and axial-vector currents, the relations are:

Model of Spontaneous Breakdown ofSU(3)⊗SU(3)Symmetry. I

Strong interactions of spin-zero and spin-one mesons are studied in a model in which the Lagrangian is approximately invariant under coordinate-dependent $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ gauge transformations. The scalar and pseudoscalar nonets are assigned to the (3,3*) \ensuremath{\bigoplus} (3*,3) representation, whereas the vector and axial-vector octets are introduced as gauge fields belonging to the (1,8) \ensuremath{\bigoplus} (8,1) representation. Spontaneous symmetry breaking to the isospin-hypercharge level is introduced by giving nonzero vacuum expectation values to some scalar fields. The effective Lagrangian obtained after considering field mixings and renormalizations is used in a phenomenological manner to study the masses and couplings of various particles. With some of the experimental meson masses as input, the predicted masses of remaining observed mesons are in reasonable agreement with experiment. Some scalar-meson masses are also predicted (${\ensuremath{\mu}}_{{S}_{K}}=527$ MeV and ${\ensuremath{\mu}}_{{S}_{\ensuremath{\pi}}}=771$ MeV). The widths for $V\ensuremath{\rightarrow}\mathrm{PP}$ and $A\ensuremath{\rightarrow}\mathrm{VP}$ decays are calculated and compare well with experiment. The axial-vector currents and the strangeness-changing vector currents satisfy partial conservation equations. Decay constants for spin-zero mesons are calculated. We get $\frac{{F}_{K}}{{F}_{\ensuremath{\pi}}}=1.17$ and $\frac{{F}_{{S}_{K}}}{{F}_{\ensuremath{\pi}}}=\ensuremath{-}0.59$. The ${K}_{l3}$ form factors are calculated, the effect of the scalar kaon being included; we get ${f}_{+}(0)=0.86$, $\ensuremath{\xi}=\ensuremath{-}0.197$, ${\ensuremath{\lambda}}_{+}=0.023$, and ${\ensuremath{\lambda}}_{\ensuremath{-}}=0.013$.

Hard-meson current algebra analysis of the K [formula omitted]3 decay

A hard-meson calculation of the K l 3 , K ∗ and κ decays is given based upon (i) chiral SU(3) × SU(3) current algebra, (ii) pion and kaon PCAC, (iii) single meson saturation, (iv) approximation of meson vertices by low-order polynomials in momentum. No a priori assumption on the size of SU(3) or chiral breakdown, or type of chiral breakdown, is made. The possible existence of a κ-meson is included by (v) the PCVC condition on the strangeness changing vector current. If ƒ +(0) takes on its SU(3) symmetric value (ƒ +(0) ≈ 1.0) then large breakdown of chiral invariance exists, while the chiral symmetric value (ƒ +(0) ≈ 0.86) implies large breakdown of SU(3). A value of λ + consistent with the K l 3 data and yielding also the correct K ∗ width if one uses the chiral symmetric ƒ +(0) can be obtained, and implies the vanishing of the K ∗-K-π vertex asymptotically. If no κ-meson exists, the theory requires both |ξ| and λ − to be small. This is also generally the case if a κ meson exists and one assumes a (3 ∗, 3) + (3, 3 ∗) form of chiral breakdown. A large |ξ| and small λ − (which is consistent with both the polarization and the most recent branching ratio determinations) can only be achieved if both a κ-meson exists and one violates the (3 ∗, 3) + (3, 3 ∗) assumption. The theory than also predicts a κ width in agreement with recent data. It is further shown that the hard meson assumption are inconsistent with both a large |ξ| and a large λ −, and that the Callan-Treiman relation may be in error by as much as 50% (due to the presence of non-gentle terms in the K l 3 amplitude).

Mass of kappa meson from spectral function sum rules

A modified form of the second Weinberg sum rule is proposed for strangeness-changing vector and axial-vector currents. Together with the first sum rule, it is used to predict the kappa mass. The estimate is compared with the value obtained from chiral symmetry considerations of Glashow and Weinberg.

Structure of Three-Point Functions fromSU(3)⊗SU(3)Algebra

We construct three-point functions of vector and axial-vector currents consistent with the constraints of chiral $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ algebra. We make no single-particle approximations, nor do we assume that the strangeness-changing vector currents are conserved. We do, however, assume that the effects of symmetry breaking may be represented by a set of scalar and pseudoscalar fields transforming as a representation of the algebra. The results are presented in a form in which any approximations will be guaranteed to maintain the constraints of current algebra.

Behavior of Current Divergences underSU3×SU3

We investigate the behavior under SU3×SU3 of the hadron energy density and the closely related question of how the divergences of the axial-vector currents and the strangeness-changing vector currents transform under SU3×SU3. We assume that two terms in the energy density break SU3×SU3 symmetry; under SU3 one transforms as a singlet, the other as the member of an octet. The simplest possible behavior of these terms under chiral transformations is proposed: They are assigned to a single (3,3*)+(3*,3) representation of SU3×SU3 and parity together with the current divergences. The commutators of charges and current divergences are derived in terms of a single constant c that describes the strength of the SU3-breaking term relative to the chiral symmetry-breaking term. The constant c is found not to be small, as suggested earlier, but instead close to the value (-sqrt[2]) corresponding to an SU2×SU2 symmetry, realized mainly by massless pions rather than parity doubling. Some applications of the proposed commutation relations are given, mainly to the pseudoscalar mesons, and other applications are indicated.

On the lifetime of the kappa meson

The lifetime of a hypotheticalκ meson with low mass, introduced byGlashow andWeinberg in a model for simultaneous breaking ofSU(3) and chiral symmetry, is calculated. The coupling constant of theκ toKπ is obtained by assuming that the divergence of the strangeness changing vector current is dominated by theκ pole.

Algebra of Currents, Divergence of Strangeness-Changing Vector Currents, Unsubtracted Dispersion Relations, and theKl3form Factors

Algebra of Currents, Divergence of Strangeness-Changing Vector Currents, Unsubtracted Dispersion Relations, and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi><mml:mn>3</mml:mn><mml:mn /></mml:mrow></mml:msub></mml:mrow></mml:math>form Factors

A tadpole model for nonleptonic decays based on current-algebra and spectral-function sum rules

A tadpole model for the nonleptonic decays based on a scalar κ field proportional to the divergence of the strangeness-changing vector current-density, and on the pseudoscalar mesons π and K satisfying PCAC, is studied by use of the current-algebra and spectral-function sum rules. A reasonably good fit to the hyperon decay data is obtained.