240
H. Wittig
initial and final states. A strong motivation for computing such quantities is provided
by the fact that fundamental interactions are often probed in scattering experiments
involving nuclear targets. For instance, probing the neutrino sector requires accurate
knowledge of the scattering cross sections of neutrinos with nuclear targets. Similar
considerations apply to the search for dark matter candidates. Therefore, precise
determinations of the corresponding nucleon matrix elements are indispensable for
exploring the limits of the SM.
The past decade has seen a huge rise in the number of publications describing
lattice calculations of nucleon matrix elements. Quantities that have been studied
include
• the electromagnetic form factors of the nucleon, G E (Q 2 ) and G M (Q 2 ), which
give access to the electric and magnetic charge radii of the nucleon and its
magnetic moment [385–397];
• the iso-vector axial charge of the nucleon, g A , which is a measure of the strength
of weak interaction in neutron β-decay [386, 387, 389, 392, 397–414], as well as
the scalar and tensor charges, g S and g T [386, 393, 404–406, 411, 412, 414–419];
• axial and induced pseudoscalar form factors of the nucleon [397, 407, 409, 420,
421], as well as the strange electromagnetic and axial form factors [421–426,
529] which probe the quark sea inside the nucleon;
• the pion-nucleon σ -term σ πN [412, 427–438] and the strange content of the
nucleon σ s [412, 429–431, 435–444]. These σ -terms are proportional to nucleon
matrix element of the flavour-diagonal scalar density, qq, which parameterizes
the rate of change in the nucleon mass due to a non-zero value of the corresponding quark mass.
Recent reviews, presented at the annual conference on lattice field theory, can be
found in Refs. [445–447]. Some results on nucleon form factors and other matrix
elements are reviewed in section 3.2.5 of [448], and a dedicated chapter has been
prepared for the 2019 edition of the FLAG report. In addition, there has been a
community effort in the form of a white paper [449] in which lattice results are used
to reduce the overall uncertainties in polarized and unpolarized proton PDFs and
their moments.
The relevant nucleon hadronic matrix elements are extracted from suitable
three-point correlation functions of quark bilinears between interpolating operators
representing the initial and final-state nucleons. Examples of the corresponding
diagrams, with the initial-state nucleon placed at Euclidean time t = 0 (the source),
the final-state nucleon at time t s (the sink) and the operator insertion at time t,
are shown in Fig. 5.24. In addition to the quark-connected diagram, in which the
operator is inserted on a valence quark line, there are also quark-disconnected
diagrams in which the operator probes the quark sea. The latter class of diagrams
must be computed to determine, for instance, iso-scalar quantities, the strangeness
form factors and the σ -terms.
Precise determinations of nucleon matrix elements with controlled statistical and
systematic errors are particularly challenging. This is a consequence of the fact that
the noise-to-signal ratio in three-point correlation functions corresponding to the
H. Wittig
initial and final states. A strong motivation for computing such quantities is provided
by the fact that fundamental interactions are often probed in scattering experiments
involving nuclear targets. For instance, probing the neutrino sector requires accurate
knowledge of the scattering cross sections of neutrinos with nuclear targets. Similar
considerations apply to the search for dark matter candidates. Therefore, precise
determinations of the corresponding nucleon matrix elements are indispensable for
exploring the limits of the SM.
The past decade has seen a huge rise in the number of publications describing
lattice calculations of nucleon matrix elements. Quantities that have been studied
include
• the electromagnetic form factors of the nucleon, G E (Q 2 ) and G M (Q 2 ), which
give access to the electric and magnetic charge radii of the nucleon and its
magnetic moment [385–397];
• the iso-vector axial charge of the nucleon, g A , which is a measure of the strength
of weak interaction in neutron β-decay [386, 387, 389, 392, 397–414], as well as
the scalar and tensor charges, g S and g T [386, 393, 404–406, 411, 412, 414–419];
• axial and induced pseudoscalar form factors of the nucleon [397, 407, 409, 420,
421], as well as the strange electromagnetic and axial form factors [421–426,
529] which probe the quark sea inside the nucleon;
• the pion-nucleon σ -term σ πN [412, 427–438] and the strange content of the
nucleon σ s [412, 429–431, 435–444]. These σ -terms are proportional to nucleon
matrix element of the flavour-diagonal scalar density, qq, which parameterizes
the rate of change in the nucleon mass due to a non-zero value of the corresponding quark mass.
Recent reviews, presented at the annual conference on lattice field theory, can be
found in Refs. [445–447]. Some results on nucleon form factors and other matrix
elements are reviewed in section 3.2.5 of [448], and a dedicated chapter has been
prepared for the 2019 edition of the FLAG report. In addition, there has been a
community effort in the form of a white paper [449] in which lattice results are used
to reduce the overall uncertainties in polarized and unpolarized proton PDFs and
their moments.
The relevant nucleon hadronic matrix elements are extracted from suitable
three-point correlation functions of quark bilinears between interpolating operators
representing the initial and final-state nucleons. Examples of the corresponding
diagrams, with the initial-state nucleon placed at Euclidean time t = 0 (the source),
the final-state nucleon at time t s (the sink) and the operator insertion at time t,
are shown in Fig. 5.24. In addition to the quark-connected diagram, in which the
operator is inserted on a valence quark line, there are also quark-disconnected
diagrams in which the operator probes the quark sea. The latter class of diagrams
must be computed to determine, for instance, iso-scalar quantities, the strangeness
form factors and the σ -terms.
Precise determinations of nucleon matrix elements with controlled statistical and
systematic errors are particularly challenging. This is a consequence of the fact that
the noise-to-signal ratio in three-point correlation functions corresponding to the