Chiral-odd structure of the $N\to \Delta$ transition: tensor form factors from QCD light-cone sum rules

Abstract

We report the first calculation of the tensor transition form factors of the $N\to \Delta$ transition within QCD light-cone sum rules, and use them to construct the transverse multipole structure of this chiral-odd transition. The matrix element of the tensor current between the nucleon and $\Delta$ states requires six independent Lorentz structures, a number fixed by the dimension of the parity-reduced helicity amplitude space and reproduced by the same counting in the electromagnetic, axial and gravitational $N\to \Delta$ channels. The six form factors organize into transverse multipoles whose rank ceiling is the sum of the spin-transition rank and the rank of the probing operator. For a spin-$1/2$ target the ceiling is two, so the diagonal tensor sector and the electromagnetic $N\to \Delta$ transition both stop at a quadrupole, whereas the chiral-odd transition reaches rank three. The resulting octupole has no counterpart in the diagonal matrix elements of the vector, axial-vector and tensor currents, nor in the electromagnetic transition. Each multipole is extracted from its own sum rule rather than by combining separately fitted form factors, improving the stability against the continuum threshold by an order of magnitude. Using nucleon distribution amplitudes for two sets of light-cone parameters, we obtain the monopole and dipole in both isospin channels and for both quark flavors, and the quadrupole in the isovector channel. The multipoles fall with rank, the dipole about half the monopole and the quadrupole about a fifth. A flavor decomposition sorts the form factors into two families according to whether the Lorentz structure they multiply contains a $\gamma$ matrix, and the transverse dipole is carried entirely by the $d$ quark. The results provide model-independent input for analyses of chiral-odd $N\to \Delta$ transition GPDs, to be checked against lattice QCD.

27 pages, 5 Tables, 3 figures