Three-loop formula for quark and gluon contributions to the QCD trace anomaly

In the QCD energy-momentum tensor $T^{\mu\nu}$, the terms that contribute to physical matrix elements are expressed as the sum of the gauge-invariant quark part and gluon part. Each part undergoes the renormalization due to the interactions among quarks and gluons, although the total tensor $T^{\mu\nu}$ is not renormalized thanks to the conservation of energy and momentum. Recently it has been shown that, through the renormalization, each of the quark and gluon parts of $T^{\mu\nu}$ receives a definite amount of anomalous trace contribution, such that their sum reproduces the well-known QCD trace anomaly, $T^\mu_\mu= (\beta/2g)F^{\mu\nu}F_{\mu\nu}+ m (1+\gamma_m)\bar{\psi}\psi$, and the corresponding formulas have been derived up to two-loop order. We extend this result to the three-loop order, working out all the relevant three-loop renormalization structure for the quark and gluon energy-momentum tensors in the (modified) minimal subtraction scheme in the dimensional regularization. We apply our three-loop formula of the quark/gluon decomposition of the trace anomaly to calculate the anomaly-induced mass structure of nucleons as well as pions.Comment: 26 pages, text improved and references adde

Transverse-spin gluon distribution function

We introduce the spin-operator representation for the gluon as well as quark distribution functions as nucleon matrix element of the gauge-invariant bilocal light-cone operators in QCD. To identify the relevant spin operators for quarks and gluons in a unified manner, we rely on the transformation properties of the quark and gluon fields in the coordinate space under the action of the generator of the Lorentz group. In particular, this approach allows us to define the transverse-spin gluon distribution function $G_T(x)$, which is the genuine counterpart of the transverse-spin quark distribution function $g_T(x)$ relevant to the transverse-spin structure function $g_2(x, Q^2)$ in the deep inelastic scattering. We show that $G_T(x)$ is given by the sum of the chromoelectric and chromomagnetic correlators associated with helicity-flip by one unit, and the treatment of the latter correlator completes the classification of the collinear parton distribution functions up to twist three. We show that $G_T(x)$ receives the three-gluon and quark-gluon correlation effects and discuss the operator product expansion for $G_T(x)$. We also discuss the relevance of the first moment of $G_T(x)$ for the partonic decomposition of the transverse nucleon spin.Comment: 6 pages, Proceedings of the XXII. International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2014) 28 April - 2 May 2014, Warsaw, Poland. Reference: PoS DIS2014 (2014) 22

Operator Relations for Gravitational Form Factors

The form factors for the hadron matrix element of the QCD energy-momentum tensor not only describe the coupling of the hadron with a graviton as the ``gravitational form factors'', but also serve as unique quantities for describing the shape inside the hadron reflecting dynamics of quarks and gluons, such as the internal shear forces acting on the quarks/gluons and their pressure distributions. We consider the gravitational form factors for a hadron, in particular, for a (pseudo)scalar hadron and for the nucleon. We derive and clarify the relations satisfied by the gravitational form factors as direct consequences of the symmetries and the equations of motion in QCD, and connections to the generalized parton distributions. Our results reveal that the gravitational form factors are related to the higher-twist quark-gluon correlation effects inside the hadrons and also to QCD trace anomaly.Comment: 4 pages. To appear in the proceedings of 8th International Conference on Quarks and Nuclear Physics (QNP2018), November 13-17, 2018, Tsukuba, Japa