246
H. Wittig
function is integrated over the Euclidean time x 0 . A variant of the time-momentum
representation uses the time moments of G(x 0 ) [503]. Finally, there also exists
a Lorentz-covariant formulation in coordinate space [504] involving the point-topoint vector correlator G(x, y).
In order to meet the precision goal of sub-percent uncertainty, it is mandatory to
have good control over the infrared regime which makes a sizeable contribution to
a
hvp
μ . In the formulation of Refs. [500, 501] this implies that momenta corresponding
to Q 2 m 2
μ must be included, since this is where the convolution integral
receives its dominant contribution. Instead, in the time-momentum representation
or the Lorentz-covariant formulation one must constrain the long-distance regime
of the correlator sufficiently well. The statistical accuracy that one can attain
for a
hvp
μ is affected by the well-known noise problem encountered for the vector
correlator, i.e. the fact that the signal-to-noise ratio increases exponentially at large
distances. 24 Another limiting factor for the overall precision of a
hvp
μ
in lattice
QCD is the knowledge of the lattice scale [499, 505]. At first sight this may seem
surprising, given that a
hvp
μ is a dimensionless quantity. However, employing the
time-momentum representation, one easily sees that the lattice scale enters through
the combination (x 0 m μ ) 2 in the kernel function. Similar arguments exist for the
other representations of a
hvp
μ . Furthermore, at the level of sub-percent precision, it
is necessary to include the contributions from quark-disconnected diagrams and the
effects from isospin breaking (see Sect. 5.9.2). All of this is explained in great detail
in Ref. [494].
First exploratory calculations of a
hvp
μ in full QCD were published in 2008 [506],
and in the following years several studies appeared [497, 507–509], employing a
range of different discretisations of the quark action, which were mostly aimed
at investigating systematic effects. The most recent calculations are focussed on
reducing the overall uncertainties [495, 496, 498, 499, 510–515, 530]. A comparison
of recent estimates for a
hvp
μ from lattice QCD to results obtained via the dispersive
approach is shown in Fig. 5.26. As of now, current calculations cannot match
the accuracy of the dispersive approach, but efforts are under way to reduce the
uncertainties to a level that makes the lattice approach competitive with data-driven
methods [494, 516].
In order to determine the hadronic light-by-light scattering contribution, it is
necessary to formulate the problem in such a way that a hlbl
μ is expressed in terms
of quantities that can be computed on the lattice with affordable effort. Several
different strategies have been proposed and are currently being pursued:
In a first method, the matrix element of the electromagnetic current between
explicit muon initial and final states is computed is QCD+QED [517]. In order to
isolate the desired light-by-light scattering contribution, one has to perform a nonperturbative subtraction. While the method has produced estimates in the expected
24 This is similar to, but less severe, than the noise problem encountered in nucleon correlation
functions discussed in Sect. 5.9.4 of this review.
H. Wittig
function is integrated over the Euclidean time x 0 . A variant of the time-momentum
representation uses the time moments of G(x 0 ) [503]. Finally, there also exists
a Lorentz-covariant formulation in coordinate space [504] involving the point-topoint vector correlator G(x, y).
In order to meet the precision goal of sub-percent uncertainty, it is mandatory to
have good control over the infrared regime which makes a sizeable contribution to
a
hvp
μ . In the formulation of Refs. [500, 501] this implies that momenta corresponding
to Q 2 m 2
μ must be included, since this is where the convolution integral
receives its dominant contribution. Instead, in the time-momentum representation
or the Lorentz-covariant formulation one must constrain the long-distance regime
of the correlator sufficiently well. The statistical accuracy that one can attain
for a
hvp
μ is affected by the well-known noise problem encountered for the vector
correlator, i.e. the fact that the signal-to-noise ratio increases exponentially at large
distances. 24 Another limiting factor for the overall precision of a
hvp
μ
in lattice
QCD is the knowledge of the lattice scale [499, 505]. At first sight this may seem
surprising, given that a
hvp
μ is a dimensionless quantity. However, employing the
time-momentum representation, one easily sees that the lattice scale enters through
the combination (x 0 m μ ) 2 in the kernel function. Similar arguments exist for the
other representations of a
hvp
μ . Furthermore, at the level of sub-percent precision, it
is necessary to include the contributions from quark-disconnected diagrams and the
effects from isospin breaking (see Sect. 5.9.2). All of this is explained in great detail
in Ref. [494].
First exploratory calculations of a
hvp
μ in full QCD were published in 2008 [506],
and in the following years several studies appeared [497, 507–509], employing a
range of different discretisations of the quark action, which were mostly aimed
at investigating systematic effects. The most recent calculations are focussed on
reducing the overall uncertainties [495, 496, 498, 499, 510–515, 530]. A comparison
of recent estimates for a
hvp
μ from lattice QCD to results obtained via the dispersive
approach is shown in Fig. 5.26. As of now, current calculations cannot match
the accuracy of the dispersive approach, but efforts are under way to reduce the
uncertainties to a level that makes the lattice approach competitive with data-driven
methods [494, 516].
In order to determine the hadronic light-by-light scattering contribution, it is
necessary to formulate the problem in such a way that a hlbl
μ is expressed in terms
of quantities that can be computed on the lattice with affordable effort. Several
different strategies have been proposed and are currently being pursued:
In a first method, the matrix element of the electromagnetic current between
explicit muon initial and final states is computed is QCD+QED [517]. In order to
isolate the desired light-by-light scattering contribution, one has to perform a nonperturbative subtraction. While the method has produced estimates in the expected
24 This is similar to, but less severe, than the noise problem encountered in nucleon correlation
functions discussed in Sect. 5.9.4 of this review.