Nucleon Matrix Elements Research Articles

Neutral pion photoproduction on the proton near threshold and the chiral symmetry.

The recent precise measurements of the neutral pion photoproduction on the proton near threshold seems to be inconsistent with the predictions of the low-energy theorems on photoproduction of pions. In this note, we have sought to resolve this problem, in the framework of an approximate SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) symmetry of the pion-nucleon dynamics, by incorporating the contribution from the nucleon matrix elements of the equal-time commutator between the divergence of the axial-vector current and the electromagnetic current, which was introduced earlier by Furlan et al. We find that a mean value of the nonstrange quark mass m^=1/2(${m}_{u}$+${m}_{d}$)=(7\ifmmode\pm\else\textpm\fi{}2) MeV, leads to an estimate of the symmetry breaking terms compatible with experiment.

Nucleons with chiral loops

This article extends the analysis of the low-energy structure of Green functions in QCD to processes with one external nucleon. We calculate various nucleon matrix elements of quark currents and determine the elastic πN scattering amplitude T πN for off-shell pions to one loop. As an application of our representation of T πN we evaluate the difference Δ R between the amplitude D at the Cheng-Dashen point and the scalar form factor σ(2 M π 2 ). The obtained result is a factor of 50 too small to explain the difference between the value of σ extracted from elastic πN scattering and the value obtained from the baryon mass spectrum.

Nonpertubative QCD corrections to the Bjorken and Gross-Llewellyn-Smith sum rules

There are nonperturbative quantum chromodynamic (QCD) corrections to the Bjorken and Gross-Llewellyn-Smith sum rules coming from the nonsinglet and singlet nucleon matrix elements, respectively, of the two-quark, one-gluon operators, which appear in the operator product expansion of the bilocal product of the charged weak currents. These twist-four, spin-one QCD corrections can be calculated assuming some quark confinement model to evaluate the nucleon matrix elements. We have considered several plausible models and have found the resulting corrections to be quite model. dependent. However in all of the models these nonperturbative QCD corrections are rather small compared to the usual perturbative QCD corrections calculated in leading-log approximation.

Twist-four effects in electroproduction: model dependence

The twist-four, spin-two effects on the second Nachtmann moment in electroproduction have been calculated using the operator product expansion. To explore the model dependence of the nucleon matrix elements of the operators, proton and neutron wave functions from several different models of quark confinement were considered. While the twist-four, spin-two effects were indeed found to be substantially model-dependent, thery were small (⪕2%) in all cases, confirming the conclusions of Jaffe and Soldate, based on the MIT bag model.

Normalizing the renormalization group analysis of deep inelastic leptoproduction

The nucleon matrix elements of the leading twist non-singlet operators which arise in the standard QCD analysis of leptoproduction are computed in an improved version of the bag model. QCD radiative corrections are used to evolve the bag predictions which are applicable at a low value of Q 2 to a high Q 2 regime where they can be compared with moments of non-singlet structure functions. The agreement with data is good and suggests that higher twist effects are not large.

Current algebra of quarks confined to a cavity

Equal-time commutators of fields with charges are calculated in a cavity approximation to the MIT bag model, with N flavours of non-interacting quarks confined to a rigid spherical cavity and SU( N) symmetry arbitrarily broken by mass terms. It is proved that inside the cavity the algebra is identical with that of free field theory, whilst on the boundary quark fields commute with axial charges. Vector divergences and sigma commutators belong to a (N, N) + ( N, N) multiplet of chiral SU( N) × SU( N). Axial divergences contain additional surface terms which do not contribute to sigma commutators. A non-strange quark mass in the range 20–44 MeV is required to give a value 30–70 MeV for the nucleon matrix element of the sigma commutator relevant to pion-nucleon scattering.

Renormalization constants for scalar, pseudoscalar, and tensor currents

We calculate the renormalization constants describing nucleon and pion matrix elements of scalar, pseudoscalar, and tensor ($S, P, T$) current densities. For certain of the constants, expressions can be obtained using standard S${\mathrm{U}}_{3}$ and chiral S${\mathrm{U}}^{3}$\ensuremath{\bigotimes}S${\mathrm{U}}^{3}$ methods. To get the remaining constants, we employ the quark model with spherically symmetric quark wave functions to relate the $S$, $P$, $T$ renormalization constants to known parameters of the usual vector and axial-vector ($V,A$) currents. We also evaluate the renormalization constants using the MIT "bag" model quark wave functions. We summarize our results in tabular form, compare the results of the various calculational methods used, and attempt to estimate the accuracy of our predictions.

Sidewise Dispersion Relations for Scalar Densities

We show how to calculate by sidewise dispersion relations the matrix element between nucleon states of ${u}_{i}$, the scalar density belonging to the $(3,\overline{3})+(\overline{3},3)$ representation of $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$. In particular we find a vanishing nucleon matrix element for the combination ${u}_{0}\ensuremath{-}\sqrt{2}{u}_{8}$ in the limit of the pion mass approaching zero.

IsSU(2)⊗SU(2)a Better Symmetry than SU(3)?

Using the $\ensuremath{\pi}N$ phase shifts and fixed-$t$ dispersion relations we have calculated (to first order in symmetry breaking) the nucleon matrix element of the current algebra "sigma" term, and found a value of about 110 MeV. This is an order of magnitude larger than the prediction of the ${(3,{3}^{*})}^{\ensuremath{\bigoplus}}({3}^{*},3)$ model for chiral symmetry breaking and it indicates that $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ breaking is comparable to SU(3) breaking.

Charge commutator for any momentum

The nucleon matrix elements of the charge commutation relations are considered for arbitrary momentum. The resulting expression is exact. The high-momentum limit reduces to the Alder-Weisberger sum rule. For zero momentum one obtains the known low-energy result together with a closed expression for the correction.

Relations for theN−N*Weak Form Factors from Current Algebra in the Infinite-Momentum Limit

The nucleon matrix elements of an $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ subalgebra of ${U}_{6}\ensuremath{\bigotimes}{U}_{6}$ are considered in the infinite-momentum, nonzero-momentum-transfer limit. Only the $N$ and ${N}^{*}$ intermediate states are retained. In this approximation we reproduce the results of the dispersive approach while also obtaining Oehme's equation which was derived for the zero-momentum-transfer case.

Electromagnetic Form Factors of the Nucleon

The physical interpretation of the electromagnetic form factors is discussed with special reference to the gauge invariance of particular theories. A distinction is made between the condition that the one nucleon matrix element satisfy the equation of continuity (weak gauge invariance) and the stronger condition imposed by the generalized Ward identity (strong gauge invariance). The former is shown to be a consequence of covariance under the improper Lorentz transformations, and hence it has no new content concerning the functional behavior of the form factors. The latter implies restrictions on the current operator which may have an important effect on the results of calculations of form factors.In connection with the physical interpretation, it is noted that the moments of the charge and current distribution are determined by ${F}_{\mathrm{eh}}={F}_{1}\ensuremath{-}(\frac{{q}^{2}}{2M}){F}_{2}$ and ${F}_{\mathrm{mag}}=(\frac{1}{2M}){F}_{1}+{F}_{2}$. Specifically the second moment of the charge distribution, $\ensuremath{-}6{{F}_{\mathrm{ch}}}^{\ensuremath{'}}(0)$, is found, in the case of the neutron, to be directly measured by the neutron-electron interaction without the intervening subtraction of the Foldy term.These matters are investigated in detail by means of a specific model of the nucleon which is a covariant generalization of the fixed source static model having the property that it gives results identical with the static model in the limit $M\ensuremath{\rightarrow}\ensuremath{\infty}$. It is found that strong gauge invariance requires the addition of line currents which make significant contributions to the form factors in general and, in particular, to the proton charge radius even in the static approximation. This suggests that as a consequence of strong gauge invariance, important contributions to the charge radius must arise in any theory from intermediate states of large mass. The model also provides a means of consistently calculating recoil corrections to the static model. They are found to be large.