Matrix Elements Of Local Operators Research Articles

Form factors in finite volume I: Form factor bootstrap and truncated conformal space

We describe the volume dependence of matrix elements of local fields to all orders in inverse powers of the volume (i.e., only neglecting contributions that decay exponentially with volume). Using the scaling Lee–Yang model and the Ising model in a magnetic field as testing ground, we compare them to matrix elements extracted in finite volume using truncated conformal space approach to exact form factors obtained using the bootstrap method. We obtain solid confirmation for the form factor bootstrap, which is different from all previously available tests in that it is a non-perturbative and direct comparison of exact form factors to multi-particle matrix elements of local operators, computed from the Hamiltonian formulation of the quantum field theory. We also demonstrate that combining form factor bootstrap and truncated conformal space is an effective method for evaluating finite volume form factors in integrable field theories over the whole range in volume.

The B+− Bd0 lifetime difference beyond leading logarithms

We compute perturbative QCD corrections to the lifetime splitting between the charged and neutral B meson in the framework of the heavy quark expansion. These next-to-leading logarithmic corrections are necessary for a meaningful use of hadronic matrix elements of local operators from lattice gauge theory. We find the uncertainties associated with the choices of renormalization scale and scheme significantly reduced compared to the leading-order result. We include the full dependence on the charm-quark mass m c without any approximations. Using hadronic matrix elements estimated in the literature with lattice QCD we obtain τ( B +)/ τ( B 0 d )=1.053±0.016±0.017, where the effects of unquenching and 1/ m b corrections are not yet included. The lifetime difference of heavy baryons Ξ 0 b and Ξ − b is also briefly discussed.

Hadron structure functions in a chiral quark model: Regularization, scaling and sum rules

We provide a consistent regularization procedure for calculating hadron structure functions in a chiral quark model. The structure functions are extracted from the absorptive part of the forward Compton amplitude in the Bjorken limit. Since this amplitude is obtained as a time-ordered correlation function its regularization is consistently determined from the regularization of the bosonized action. We find that the Pauli-Villars regularization scheme is most suitable because it preserves both the anomaly structure of QCD and the leading scaling behavior of hadron structure functions in the Bjorken limit. We show that this procedure yields the correct pion structure function. In order to render the sum rules of the regularized polarized nucleon structure functions consistent with their corresponding axial charges we find it mandatory to further specify the regularization procedure. This specification goes beyond the double subtraction scheme commonly employed when studying static hadron properties in this model. In particular the present approach serves to determine the regularization prescription for structure functions whose leading moments are not given by matrix elements of local operators. In this regard we conclude somewhat surprisingly that in this model the Gottfried sum rule does not undergo regularization.

Next-to-leading order QCD corrections to the lifetime difference of Bs mesons

We compute the QCD corrections to the decay rate difference in the B s – B ̄ s system, Δ Γ B s , in the next-to-leading logarithmic approximation using the heavy quark expansion approach. Going beyond leading order in QCD is essential to obtain a proper matching of the Wilson coefficients to the matrix elements of local operators from lattice gauge theory. The lifetime difference is reduced considerably at next-to-leading order. We find ( ΔΓ/Γ) B s =(f B s /210 MeV) 2[0.006B(m b)+0.150B S(m b)−0.063] in terms of the bag parameters B, B S in the NDR scheme. As a further application of our analysis we also derive the next-to-leading order result for the mixing-induced CP asymmetry in inclusive b→u u ̄ d decays, which measures sin2 α.

Cut vertices and semi-inclusive deep inelastic processes

Cut vertices, a generalization of matrix elements of local operators, are revisited, and an expansion in terms of minimally subtracted cut vertices is formulated. An extension of the formalism to deal with semi-inclusive deep inelastic processes in the target fragmentation region is explicitly constructed. The problem of factorization is discussed in detail.

Hadro-Production of Quarkonia in Fixed Target Experiments

In this talk I review the recent progress made in the calculations of quarkonia production in fixed target experiments. NRQCD organizes the calculations in a systematic expansion in αs and v, the relative velocity between the heavy quarks. Within this formalism there are octet contributions which are not included in the color singlet model. These contributions depend upon unknown matrix elements of local operators which are fit to the data. Using these fits, there are several predictions which do indeed improve agreement with the data. However, the prediction for the polarization of the produced states as well as the ratio of the χ1 to χ2 cross sections differ substantially from the data for the case of pion beams. Possible large corrections from higher twist effects are discussed as is the issue of the the proper choice of masses.

Exact sum rule for transversely polarized DIS

We have shown how, working in a field-theoretic framework, one can derive expressions for the even moments of the valence parts of ${g}_{1,2}(x)$. These expressions cannot be written as matrix elements of local operators and do not coincide with the analytic continuation to $n=$ even integer of the OPE results. Just as for the OPE one can in some cases argue that the hadronic matrix elements should be small, leading to approximate sum rules for the moments of the valence parts of ${g}_{1,2}(x)$. But, most importantly, for the case $n=2$ we have proved rigorously that the hadronic matrix element vanishes, yielding an exact sum rule. We have argued that the convergence properties of this sum rule are good and are a further test of QCD.

OPE analysis of the nucleon scattering tensor: a review including weak interactions and finite mass effects

We perform a systematic operator product expansion of the most general form of the nucleon scattering tensorWμν including electro-magnetic and weak interaction processes. Finite quark masses are taken into account and a number of higher-twist corrections are included. In this way we derive relations between the lowest moments of all 14 structure functions and matrix elements of local operators. Besides reproducing well-known results, new sum rules for parity-violating polarized structure functions and new mass correction terms are obtained.

Form factors approach to current correlations in one-dimensional systems with impurities

We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasi-particles. Correlators follow then from matrix elements of local operators between multiparticle states, the “massless form factors”. Although an infinite sum of these form factors has to be considered in principle, we find that for current, spin, and energy operators, only a few (typically two or three) are necessary to obtain an accuracy of more than 1%, for arbitrary coupling strength, that is all the way from short to large distances. As examples we compute, at zero temperature, the frequency-dependent conductance in a Luttinger liquid with impurity, the spectral function in the double-well problem of dissipative quantum mechanics and part of the space-dependent susceptibility in the Kondo model.

OPE analysis of the nucleon scattering tensor: a review including weak interactions and finite mass effects

We perform a systematic operator product expansion of the most general form of the nucleon scattering tensor $W_{\mu \nu}$ including electro-magnetic and weak interaction processes. Finite quark masses are taken into account and a number of higher-twist corrections are included. In this way we derive relations between the lowest moments of all 14 structure functions and matrix elements of local operators. Besides reproducing well-known results, new sum rules for parity-violating polarized structure functions and new mass correction terms are obtained.

PQCD analysis of inclusive semileptonic decays of B mesons.

We develop the perturbative QCD formalism for inclusive semileptonic B meson decays, which includes Sudakov suppression from the resummation of large radiative corrections near the high end of charged lepton energy. Transverse degrees of freedom of partons are introduced to facilitate the factorization of B meson decays. Ambiguities appearing in the quark-level analysis are then avoided. A universal distribution function, arising from the nonperturbative Fermi motion of the b quark, is constructed according to the heavy quark effective field theory based operator product expansion, through which the mean and the width of the distribution function are related to hadronic matrix elements of local operators. Charged lepton spectra of the B\ensuremath{\rightarrow}${\mathit{X}}_{\mathit{ul}}$\ensuremath{\nu} decay are presented. We find 50% suppression near the end point of the spectrum. The overall suppression on the total decay rate is 8% for the free quark model, and is less than 7% for the use of smooth distribution functions. With our predictions, it is then possible to extract the Cabibbo-Kobayashi-Maskawa matrix element \ensuremath{\Vert}${\mathit{V}}_{\mathit{ub}}$\ensuremath{\Vert} from experimental data. We also discuss possible implications of our analysis when confronted with the rather small observed semileptonic branching ratio in B meson decays. \textcopyright{} 1996 The American Physical Society.

Effective Hamiltonian for B-->Xse+e- beyond leading logarithms in the naive dimensional regularization and 't Hooft-Veltman schemes.

We calculate the next-to-leading QCD corrections to the effective Hamiltonian for \Bsee in the NDR and HV schemes. We give for the first time analytic expressions for the Wilson Coefficient of the operator $Q_9 = (\bar s b)_{V-A}(\bar e e)_V$ in the NDR and HV schemes. Calculating the relevant matrix elements of local operators in the spectator model we demonstrate the scheme independence of the resulting short distance contribution to the physical amplitude. Keeping consistently only leading and next-to-leading terms, we find an analytic formula for the differential dilepton invariant mass distribution in the spectator model. Numerical analysis of the $\mt$, $\Lms$ and $\mu \approx {\cal O}(\mb)$ dependences of this formula is presented. We compare our results with those given in the literature.

Form factors for integrable lagrangian field theories, the sinh-Gordon model

Using Watson's and the recursive equations satisfied by matrix elements of local operators in two-dimensional integrable models, we compute the form factors of the elementary field φ( x) and the stress-energy tensor T μν ( x) of sinh-Gordon theory. Form factors of operators with higher spin or with different asymptotic behaviour can easily be deduced from them. The value of the correlation functions are saturated by the form factors with lowest number of particle terms. This is illustrated by an application of the form factors of the trace of T μν ( x) to the sum rule of the c-theorem.

Effective hamiltonians for ΔS = 1 and ΔB = 1 non-leptonic decays beyond the leading logarithmic approximation

We calculate the effective hamiltonians for ΔB = 1 and ΔS = 1 decays including next-to-leading QCD corrections. In particular we present the two-loop 6 × 6 anomalous dimension matrices describing the mixing of current-current and QCD penguin operators calculated in the dimensional regularization scheme with anticommuting γ 5 and in the 't Hooft-Veltman scheme. The renormalization scheme dependences and their cancellation for physical quantities are discussed in detail. We point out that beyond the leading order, matching between coefficients and hadronic matrix elements of local operators both in the renormalization scale μ and in the renormalization scheme for the operators has to be made. The next-to-leading corrections enhance substantially the Wilson coefficients of QCD penguin operators and these effects are stronger than the ones found for current-current operators. consequently, the contributions of QCD penguins to B-decays, the ΔI = 1 2 rule and ϵ′ / ϵ are enlarged. For μ ≈ 1 GeV and Λ MS ≈ 300 MeV , these effects amount to 20–30%. In particular we emphasize that for μ < 1 GeV the next-to-leading corrections to the coefficients of penguin operators are large, reaching 40% for μ ≈ 0.8 GeV. For such low values of μ the renormalization group improved perturbation theory cannot be trusted any longer. Moreover the results are very sensitive functions of Λ MS .

Parametrization scheme for effective interactions.

An algorithm is developed by which two nucleon effective interactions are constructed to fit on- and off-shell t- and/or g-matrix elements. The effective interaction is defined as plane-wave matrix elements of local operators that may have explicit energy and medium dependences. It comprises central, tensor, spin-orbit, quadratic spin-orbit, and angular momentum square operators, all with Yukawa form factors. As examples, the Paris and Bonn potentials are used to construct t matrices for projection onto chosen forms of effective interactions.

A lattice study of nucleon structure

We present the formalism necessary to compute matrix elements of local operators between nucleon states by numerical simulation of QCD on the lattice. We then compute the lowest two moments of the non-singlet structure function, the electric form factor of the proton and the σ-term.

One-loop operator matrix elements in the Unruh vacuum

We present the details of a study of the matrix elements of local operators in the Unruh vacuum for real massive scalar fields in arbitrary d-spatial dimensions (also the Casimir effects for an infinite plane conductor). This state produces negative expectations for renormalized operators such as φ 2 and T μν with the structure of thermal corrections where the temperature is a local Hawking temperature. We trace this to the lack of precise boundary conditions for the theory on the horizon. Though the Minkowski vacuum appears to be populated with Rindler particles, it produces the general coordinate transform of the Minkowski results as dictated by general covariance. A simple corollary is that broken symmetries are not seen to be restored at large acceleration; the dynamical interpretation of this fact by the accelerating observer is surprising.

Calculation of Weak Transitions in Lattice QCD

We propose the use of Monte Carlo simulations of QCD to evaluate hadronic matrix elements of local operators encountered in electroweak and grand-unified-theory transitions. Preliminary Monte Carlo estimates are made of the $\ensuremath{\Delta}S=2$ matrix elements responsible for the ${K}_{L}\ensuremath{-}{K}_{S}$ mass difference and the $\ensuremath{\Delta}S=1$ operators believed to explain the $\ensuremath{\Delta}I=\frac{1}{2}$ enhancement.

Elastic Scattering and Hadron Dynamics at Short Distances

Short distance behavior is investigated for elastic scattering of a boson by a target of composite particle. The source function of the boson is assumed to be a local operator given in terms of constituent particles, and the operator product expansion (OPE) is applied to the source functions. General expressions are given for matrix elements of local operators appear­ ing in OPE. A formula analogous to the ~-scaling in the deep inelastic processes is obtained. Regge lore for hadronic processes is examined in connection with constituent particle scattering. Scattering of composite particle by a target of composite particle is discussed. § 1. Introduction The operator product expansion (OPE)!) has been extensively used to in­ vestigate the inclusive leptonic processes and perturvative QeD is shown to realize experimental data reasonably weI1. 2 )-5) In OPE the structure functions are broken up into two contributions, the coefficient functions and matrix ele­ ments of local operators. The former is determined by the anomalous dimen­ sions of field operators of constituent particles and local operators appearing in OPE, and the latter involves dynamical effects of the constituents. 6 ),7) For the lepton-production processes OPE is equivalent to the O( 4) partial wave expan­ sion7)~9) and the matrix elements of local operators are written in terms of 0(4) partial waves for forward scattering of constituent particles, where the Wick rotation is assumed to be possible and momenta are taken to be the Euclidean four vectors. Analyzing inclusive electroproduction processes in the Regge region, we showed 4 ),5) that high energy forward amplitudes for the quark-quark and gluon­ gluon scattering are dominated by the Regge pole and that the Regge poles coincide with that of hadronic processes, that is, for quark-quark scattering aSQ( 0) -:-0-1 and a\s( 0) -:-0- 0.5 with aSQ and a.\S being the Regge poles for the singlet and non-singlet states in the q- (j channel respectively and for gluon-gluon scat­ tering a(;L( 0)-:-0-1. Theoretically, these results may be understood as follows: If there are resonance states in (j -q channel, one may get Regge trajectory for quark-quark scattering via the multi-peripheral model. For the Pomeron, the leading loga­ rithmic approximation 10 ) in QeD leads to the Pomeranchuk trajectory and no cut appears in the leading order. ll ) The Pomeron may also be interpreted in terms of glueball spectra in the extended QeD. 12 )

QCD and resonance physics. theoretical foundations

A systematic study is made of the non-perturbative effects in quantum chromodynamics. The basic object is the two-point functions of various currents. At large Euclidean momenta q the non-perturbative contributions induce a series in ( μ 2/ q 2) where μ is some typical hadronic mass. The terms of this series are shown to be of two distinct types. The first few of them are connected with vacuum fluctuations of large size, and can be consistently accounted for within the Wilson operator expansion. On the other hand, in high orders small-size fluctuations show up and the high-order terms do not reduce (generally speaking) to the vacuum-to-vacuum matrix elements of local operators. This signals the breakdown of the operator expansion. The corresponding critical dimension is found. We propose a Borel improvement of the power series. On one hand, it makes the two-point functions less sensitive to high-order terms, and on the other hand, it transforms the standard dispersion representation into a certain integral representation with exponential weight functions. As a result we obtain a set of the sum rules for the observable spectral densities which correlate the resonance properties to a few vacuum-to-vacuum matrix elements. As the last bid to specify the sum rules we estimate the matrix elements involved and elaborate several techniques for this purpose.