5 QCD on the Lattice
239
lattice result for f K + /f π + implies |V ud | 2 + |V us | 2 = 0.99986(46). Thus, first-row
unitarity can be probed with permil-level precision [247].
Heavy-Light Decay Constants and Form Factors The treatment of heavy quarks
on the lattice presents additional significant challenges: since the mass of the charm
quark is close to typical values of the inverse lattice spacing, which acts as the
ultraviolet cutoff, lattice results are prone to suffering from large discretisation
errors. Moreover, the mass of the bottom quark exceeds currently accessible values
of a −1 , and specially designed methods are required for a consistent treatment. This
has been discussed extensively in Sect. 5.7.2 of the original review.
The overall precision of lattice estimates for weak hadronic matrix elements
involving charm and bottom quarks has vastly improved over the past decade. As
shown in Table 2 of FLAG 2016 [247], the leptonic decay constants of the B and B s
mesons are now known at the level of 2%, while ratios such as f B s /f B have been
determined with even better accuracy [347, 366–373]. Since the 2016 edition of the
FLAG report, new results obtained with N f = 2+1+1 flavours of dynamical quarks
[343, 374, 375] have pushed the overall precision to the sub-percent level, which is
an impressive achievement. Also the estimates of the individual B-parameters ˆ
B B
and ˆ
B B s , their ratios and combinations with the leptonic decay constants are now
known with overall errors at the percent level [347, 370, 376, 377].
Results for form factors describing semi-leptonic decays of hadrons containing
b-quarks, such as B → (D, D ∗ ))ν, or even b → ppν have reached a
level of precision that is sufficient for competitive determinations of the CKM
matrix elements V cb and V ub from exclusive processes. An extensive discussion
is presented in the web update of the FLAG report.
5.9.4 Nucleon Matrix Elements
The understanding of the internal structure of the nucleon in terms of the fundamental interactions between its constituents, the quarks and gluons, has become a
major activity within the field of lattice QCD. Structural information is encoded
in quantities such as form factors, structure functions and (generalized) parton
distribution functions (PDFs). An open problem in this context is the decomposition
of the proton’s spin in terms of the spins of quarks and gluons, as well as their
angular momentum [378, 379]. Another important issue is the so-called “proton
radius puzzle” [380], which arises due to the observed discrepancy between the
proton radius extracted from the Lamb shift in muonic hydrogen [381, 382]
compared to the more traditional determinations from electron-proton scattering
[383] or the Lamb shift in electronic hydrogen [384]. Accurate knowledge of the
electromagnetic form factors of the proton are indispensable in order to resolve—or
corroborate—this puzzle.
The determination of quantities such as nucleon form factors in lattice QCD proceeds by calculating the corresponding hadronic matrix elements between nucleon
239
lattice result for f K + /f π + implies |V ud | 2 + |V us | 2 = 0.99986(46). Thus, first-row
unitarity can be probed with permil-level precision [247].
Heavy-Light Decay Constants and Form Factors The treatment of heavy quarks
on the lattice presents additional significant challenges: since the mass of the charm
quark is close to typical values of the inverse lattice spacing, which acts as the
ultraviolet cutoff, lattice results are prone to suffering from large discretisation
errors. Moreover, the mass of the bottom quark exceeds currently accessible values
of a −1 , and specially designed methods are required for a consistent treatment. This
has been discussed extensively in Sect. 5.7.2 of the original review.
The overall precision of lattice estimates for weak hadronic matrix elements
involving charm and bottom quarks has vastly improved over the past decade. As
shown in Table 2 of FLAG 2016 [247], the leptonic decay constants of the B and B s
mesons are now known at the level of 2%, while ratios such as f B s /f B have been
determined with even better accuracy [347, 366–373]. Since the 2016 edition of the
FLAG report, new results obtained with N f = 2+1+1 flavours of dynamical quarks
[343, 374, 375] have pushed the overall precision to the sub-percent level, which is
an impressive achievement. Also the estimates of the individual B-parameters ˆ
B B
and ˆ
B B s , their ratios and combinations with the leptonic decay constants are now
known with overall errors at the percent level [347, 370, 376, 377].
Results for form factors describing semi-leptonic decays of hadrons containing
b-quarks, such as B → (D, D ∗ ))ν, or even b → ppν have reached a
level of precision that is sufficient for competitive determinations of the CKM
matrix elements V cb and V ub from exclusive processes. An extensive discussion
is presented in the web update of the FLAG report.
5.9.4 Nucleon Matrix Elements
The understanding of the internal structure of the nucleon in terms of the fundamental interactions between its constituents, the quarks and gluons, has become a
major activity within the field of lattice QCD. Structural information is encoded
in quantities such as form factors, structure functions and (generalized) parton
distribution functions (PDFs). An open problem in this context is the decomposition
of the proton’s spin in terms of the spins of quarks and gluons, as well as their
angular momentum [378, 379]. Another important issue is the so-called “proton
radius puzzle” [380], which arises due to the observed discrepancy between the
proton radius extracted from the Lamb shift in muonic hydrogen [381, 382]
compared to the more traditional determinations from electron-proton scattering
[383] or the Lamb shift in electronic hydrogen [384]. Accurate knowledge of the
electromagnetic form factors of the proton are indispensable in order to resolve—or
corroborate—this puzzle.
The determination of quantities such as nucleon form factors in lattice QCD proceeds by calculating the corresponding hadronic matrix elements between nucleon