^{1}

^{1}

^{2}

^{3}.

We analyze the low-

At the fundamental level, the electroweak form factors of hadrons originate from the dynamics of the constituents of quantum chromodynamics (QCD), namely, quarks and gluons. While a wealth of precision data exists for the electromagnetic form factors of the proton and, to a lesser extent, of the neutron (see, e.g., Refs.

In this article, based on the results of Ref.

In terms of the up-quark and down-quark fields,

The axial-vector current matrix element between nucleon states can be parametrized as

The result of older experiments has been somewhat under debate (see Table II of Ref.

Introducing the spherical tensor notation

Because of its very short lifetime of the order of

Note that

Even though a description in terms of stable states does not exist, we use Dirac’s bra-ket notation, with the understanding that the relevant amplitude is extracted at the complex pole. The Lorentz structure of the reduced matrix element may be written as

In the following, it is always understood that the “tensor”

Equations

Here, we recall an SU(6) spin-flavor quark-model relation, which will be applied in the subsequent calculations. In the static quark model, the operator

Using

At lowest order in the quark-mass and momentum expansion, the relevant interaction Lagrangian for nucleons reads

The expansion of the chiral vielbein in the pion fields yields

On the other hand, the second term results in the pseudovector pion-nucleon interaction,

For the nucleon-to-delta transition the lowest-order Lagrangian is given by [see Eq. (4.200) of Ref.

Masses and coupling constants.

The vector mesons

The Lagrangian for the interaction of the

The lowest-order Lagrangian for the interaction of the

The contribution to the invariant amplitude for the axial-vector transition induced by

Note that, due to a typo, Eqs. (46) and (47) of Ref.

In essence, the loop diagrams play no role in the one-loop calculation of the axial form factor

Axial masses reported by recent (quasi)elastic neutrino and antineutrino scattering experiments.

Including the

Figure

We would like to thank U.-G. Meißner for providing the data in the form of a table.

The fits are performed for different values of the maximal squared momentum transfer,Strictly speaking, because of a loop correction to the threshold electric dipole amplitude

Comparison of the axial masses, mean-square axial radii, and mean-quartic axial radii obtained from the dipole expression of the form factor

Figure

Note that the dipole form shows this behavior.

Motivated by the observation that the dipole fit and theComparison of the parameters

At order

Induced pseudoscalar form factor

In order to discuss the

As in the nucleon case, the loop contributions to the low-

At this point, we make use of the quark-model relation of Eq.

Axial form factor of the nucleon

By analogy with Eq.

Using the values of Table

Axial nucleon-to-delta transition form factor

We analyzed the low-

The purpose of the present investigation was to identify the

Y. Ü. and A. K. would like to thank the Collaborative Research Center 1044 of the German Research Foundation for financial support during their stay in Mainz. The work of Y. Ü. was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).

For the normalization of spinors and states, we follow Appendix A of Ref.

We define the pion-nucleon-