17. Lattice Quantum Chromodynamics

1

17. Lattice Quantum Chromodynamics
Revised August 2023 by S. Hashimoto (KEK) and S.R. Sharpe (U. Washington), written September
2011 with J. Laiho (Syracuse U.)
17.1

Lattice regularization of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.1 Gauge invariance, gluon fields and the gluon action . . . . . . . . . . . . . . .
17.1.2 Lattice fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.3 Heavy quarks on the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.4 QED on the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1.5 Basic inputs for lattice calculations . . . . . . . . . . . . . . . . . . . . . . . .
17.1.6 Sources of systematic error . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 Methods and status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.1 Monte-Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.2 Two-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.3 Three-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2.4 Scattering amplitudes and resonances . . . . . . . . . . . . . . . . . . . . . .
17.2.5 Status of LQCD simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3 Physics applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.1 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.2 Decay constants and bag parameters . . . . . . . . . . . . . . . . . . . . . . .
17.3.3 Form factors (K → π`ν, D → K`ν, B → π`ν, B → D(∗) `ν, etc.) . . . . . . .
17.3.4 Strong gauge coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.5 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.6 Other physical quantities of interest . . . . . . . . . . . . . . . . . . . . . . .
17.3.7 Other applications of lattice QCD . . . . . . . . . . . . . . . . . . . . . . . .
17.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Many physical processes considered in the Review of Particle Properties (RPP) involve hadrons.
The properties of hadrons—which are composed of quarks and gluons—are governed primarily
by Quantum Chromodynamics (QCD) (with small corrections from Quantum Electrodynamics[QED]). Theoretical calculations of these properties require non-perturbative methods, and Lattice
Quantum Chromodynamics (LQCD) is a tool to carry out such calculations. It has been successfully applied to many properties of hadrons. Most important for the RPP are the calculation ofelectroweak decay constants and form factors, which are needed to extract Cabbibo-KobayashiMaskawa (CKM) matrix elements when combined with the corresponding experimental measurements. LQCD has also been used to determine other fundamental parameters of the standard
model, in particular the strong gauge coupling and quark masses, as well as to predict hadronic
contributions to the anomalous magnetic moment of the muon, gµ −2.
This review describes the theoretical foundations of LQCD and sketches the methods used to
calculate the quantities relevant for the RPP. It also describes the various sources of error that must
be controlled in a LQCD calculation. Results for hadronic quantities are found in the corresponding
dedicated reviews.
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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17. Lattice Quantum Chromodynamics

17.1 Lattice regularization of QCD
Gauge theories form the building blocks of the Standard Model. While the SU(2) and U(1)parts have weak couplings and can be studied accurately with perturbative methods, the SU(3)component—QCD—is only amenable to a perturbative treatment at high energies. The growthof the gauge coupling in the infrared—the flip-side of asymptotic freedom—requires the use of
non-perturbative methods to determine the low energy properties of QCD. Lattice gauge theory,
proposed by K. Wilson in 1974 [1], provides such a method, for it gives a non-perturbative definitionof vector-like gauge field theories such as QCD. In lattice regularized QCD—commonly called lattice
QCD or LQCD—Euclidean space-time is discretized, usually on a hypercubic lattice with latticespacing a, with quark fields placed on sites and gauge fields on the links between sites. The latticespacing plays the role of the ultraviolet regulator, rendering the quantum field theory finite. The
continuum theory is recovered by taking the limit of vanishing lattice spacing, which can be reachedby tuning the bare gauge coupling to zero according to the renormalization group.
Unlike dimensional regularization, which is commonly used in continuum QCD calculations,the definition of LQCD does not rely on the perturbative expansion. Indeed, LQCD allows nonperturbative calculations by numerical evaluation of the path integral that defines the theory.
Practical LQCD calculations are limited by the availability of computational resources and theefficiency of algorithms. Because of this, LQCD results come with both statistical and systematic
errors, the former arising from the use of Monte-Carlo integration, the latter, for example, fromthe use of non-zero values of a. There are also different ways in which the QCD action can bediscretized, and all must give consistent results in the continuum limit, a → 0. It is the purpose ofthis review to provide an outline of the methods of LQCD, with particular focus on applications toparticle physics, and an overview of the various sources of error. This should allow the reader tobetter understand the LQCD results that are presented in other reviews, primarily those on “Quark
Masses,” “Quark Model,” “Quantum Chromodynamics,” “CKM quark-mixing matrix,” “Vud , Vus ,Cabibbo angle and CKM Unitarity,” “Leptonic Decays of Charged Pseudoscalar Mesons,” “B 0 -B̄ 0
Mixing,” and “Semileptonic b-Hadron Decays, Determination of Vcb and Vub .” For more extensive
explanations the reader should consult the available textbooks or lecture notes, the most up-to-dateof which are Refs. [2–4].
17.1.1 Gauge invariance, gluon fields and the gluon action
A key feature of the lattice formulation of QCD is that it preserves gauge invariance. This is
in contrast to perturbative calculations, where gauge fixing is an essential step. The preservationof gauge invariance leads to considerable simplifications, e.g., restricting the form of operators thatcan mix under renormalization.
The gauge transformations of lattice quark fields are just as in the continuum: q(x) −→V (x)q(x) and q̄(x) −→ q̄(x)V † (x), with V (x) an arbitrary element of SU(3). The only difference is that the Euclidean space-time positions x are restricted to lie on the sites of the lattice,i.e. x = a(n1 , n2 , n3 , n4 ) for a hypercubic lattice, with the nj being integers. Quark bilinears involving different lattice points can be made gauge invariant by introducing the gluon field Uµ (x).
For example, for adjacent points the bilinear is q̄(x)Uµ (x)q(x+aµ̂), with µ̂ the unit vector in theµ’th direction. (This form is used in the construction of the lattice covariant derivative.) This is
illustrated in Fig. 17.1. The gluon field (or “gauge link”) is an element of the group, SU(3), in
contrast to the continuum field Aµ which takes values in the Lie algebra. The bilinear is invariant
if Uµ transforms as Uµ (x) → V (x)Uµ (x)V † (x+aµ̂). The lattice gluon field is naturally associatedwith the link joining x and
x+aµ̂, and corresponds in the continuum to a Wilson line connecting
R x+aµ̂
these two points, P exp(i xdxµ Acontµ (x)) (where P indicates a path-ordered integral, and thesuperscript on Aµ indicates that it is a continuum field). The trace of a product of the Uµ (x)31st May, 2024

17. Lattice Quantum Chromodynamics

3

around any closed loop is easily seen to be gauge invariant and is the lattice version of a Wilson
loop.

Figure 17.1: Sketch of a two-dimensional slice through the µ − ν plane of a lattice, showing gluon
fields lying on links and forming either the plaquette product appearing in the gauge action or acomponent of the covariant derivative connecting quark and antiquark fields.
The simplest possible gauge action, usually called the Wilson gauge action, is given by the
product of gauge links around elementary plaquettes:
Sg = β1[1 − ReTr[Uµ (x)Uν (x+aµ̂)Uµ† (x+aν̂)Uν† (x)]] .
3
x,µ<ν
X

(17.1)

This is illustrated in Fig. 17.1. For small a, assuming that the fields are slowly varying, one can
expand the action in powers of a using Uµ (x) = exp(iaAµ (x)). Keeping only the leading nonvanishing term, and replacing the sum with an integral, one finds the continuum form,
Sg −→

Z

d4 x

1
2
2 Tr[Fµν (x)] ,4glat(17.2)(Fµν = ∂µ Aν − ∂ν Aµ + i[Aµ , Aν ])2 for the lattice coupling. In this expression, g
as long as one chooses β = 6/glat
lat is the baregauge coupling in the lattice scheme, which can be related (by combining continuum and lattice
perturbation theory) to a more conventional gauge coupling such as that in the MS scheme (seeSec. 17.3.4 below).
In practice, the lattice spacing a is non-zero, leading to discretization errors. In particular,
the lattice breaks Euclidean rotational invariance (which is the Euclidean version of Lorentz invariance) down to a discrete hypercubic subgroup. One wants to reduce discretization errors asmuch as possible. A very useful tool for understanding and then reducing discretization errors is
the Symanzik effective action: the interactions of quarks and gluons with momenta low comparedto the lattice cutoff (|p|  1/a) are described by a continuum action consisting of the standard
continuum terms (e.g., the gauge action given in Eq. (17.2)) augmented by higher dimensional
operators suppressed by powers of a [5]. For the Wilson lattice gauge action, the leading correcP(j)tions to the continuum terms come in at O(a2 ). They take the form j a2 cj O6 , with the sum
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17. Lattice Quantum Chromodynamics(j)running over all dimension-six operators O6 allowed by the lattice symmetries, and cj unknown
coefficients. Some of these operators violate Euclidean rotational invariance, and all of them leadto discretization errors of the form a2 Λ2 (up to log(a) corrections that will be discussed below),
where Λ is a typical momentum scale for the quantity being calculated. These errors can, however,be reduced by adding corresponding operators to the lattice action and tuning their coefficientsto eliminate the dimension-six operators in the effective action to a given order in perturbation
theory or even non-perturbatively. This is the idea of the Symanzik improvement program [5]. In
the case of the gauge action, one adds Wilson loops involving six gauge links (as opposed to thefour links needed for the original plaquette action, Eq. (17.1)) to define the O(a2 ) improved (or“Symanzik”) action [6]. In practical implementations, the improvement is either at tree-level (sothat residual errors are proportional to αs a2 , where the coupling is evaluated at a scale ∼ 1/a), orat one-loop order (errors proportional to αs2 a2 ). Another popular choice is motivated by studies ofrenormalization group (RG) flow. It has the same terms as the O(a2 ) improved action but withdifferent coefficients, and is called the RG-improved or “Iwasaki” action [7].
17.1.2 Lattice fermions
Discretizing the fermion action turns out to involve subtle issues, and the range of actions
being Rused is more extensive than for gauge fields. Recall that the continuum fermion action is
Sf = d4 xq̄[Dµ γµ + mq ]q, where Dµ = ∂µ + iAµ is the gauge-covariant derivative. The simplest
discretization replaces the derivative with a symmetric difference:Dµ q(x) −→1[Uµ (x)q(x + aµ̂) − Uµ (x − aµ̂)† q(x − aµ̂)] .
2a(17.3)

The factors of Uµ ensure that Dµ q(x) transforms under gauge transformations in the same way asq(x), so that the discretized version of q̄(x)γµ Dµ q(x) is gauge invariant. The choice in Eq. (17.3)
leads to the so-called naive fermion action. This, however, suffers from the fermion doublingproblem—in d dimensions it describes 2d equivalent fermion fields in the continuum limit. The
appearance of the extra “doubler” fermions is related to the deeper theoretical problem of formulating chirally symmetric fermions on the lattice. This is encapsulated by the Nielsen-Ninomiya
theorem [8]: one cannot define lattice fermions having exact chiral symmetry of the standard form
δq = γ5 q, δ q̄ = q̄γ5 ( is infinitesimal) without producing doublers. Naive lattice fermions do havechiral symmetry but at the cost of introducing 15 unwanted doublers (for d = 4).
There are a number of different strategies for dealing with the doubling problem, each with
their own theoretical and computational advantages and disadvantages. Wilson fermions [1] add aterm proportional to aq̄∆q to the fermion action (the “Wilson term”—in which ∆ is a covariant
lattice Laplacian). This gives a mass of O(1/a) to the doublers, so that they decouple in thecontinuum limit. The Wilson term, however, violates chiral symmetry at non-zero lattice spacing,and introduces discretization errors linear in a. A commonly used variant that eliminates the O(a)discretization error is the O(a)-improved Wilson (or “clover”) action [9]. In this application of
Symanzik improvement, methods have been developed to remove O(a) terms non-perturbatively
using auxiliary simulations to tune parameters [10]. Such “non-perturbative improvement” is ofgreat practical importance as it brings the discretization error from the fermion action down to thesame level as that from the gauge action.
The advantages of Wilson fermions are their theoretical simplicity and relatively low computational cost. Their main disadvantage is the lack of chiral symmetry, which makes them difficult
to use in cases where mixing with wrong chirality operators can occur, particularly if this involves
divergences proportional to powers of 1/a. A related problem is the presence of potential numericalinstabilities due to spurious near-zero modes of the lattice Dirac operator. There are, however,
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studies that successfully ameliorate the problems due to the lack of chiral symmetry and increase
the range of quantities for which Wilson fermions can be used (see, e.g., Refs. [11–13]).
Twisted-mass fermions [14] are a variant of Wilson fermions in which two flavors are treated
together with an isospin-breaking mass term (the “twisted mass” term). The main advantage ofthis approach is that all errors linear in a are automatically removed (without the need for tuning ofparameters) by a clever choice of twisted mass and operators [15]. A disadvantage is the presence ofisospin breaking effects (such as a splitting between charged and neutral pion masses even when up
and down quarks are degenerate), which, however, vanish as a2 Λ2 in the continuum limit. Strange
and charm quarks can be added as a second pair, with a term added to split their masses [16, 17].
Staggered fermions are a reduced version of naive fermions in which there is only a single
fermion Dirac component on each lattice site, with the full Dirac structure built up from neighboringsites [18]. They have the advantages of being somewhat faster to simulate than Wilson-like fermions,of preserving some chiral symmetry, and of having discretization errors of O(a2 ). Their disadvantage
is that they retain some of the doublers (3 for d = 4). The action thus describes four degeneratefermions in the continuum limit. These are usually called “tastes”, to distinguish them fromphysical flavors, and the corresponding SU(4) symmetry is referred to as the “taste symmetry”.
The preserved chiral symmetry in this formulation has non-singlet taste. Practical applications
usually introduce one staggered fermion for each physical flavor, and remove contributions fromthe unwanted tastes by taking the fourth-root of the fermion determinant appearing in the path
integral. The validity of this “rooting” procedure is not obvious because taste symmetry is violated
for non-zero lattice spacing. Theoretical arguments, supported by numerical evidence, suggestthat the procedure is valid as long as one takes the continuum limit before approaching the lightquark mass region [19]. Additional issues arise for the valence quarks (those appearing in quark
propagators, as described in Sec. 17.2 below), where rooting is not possible, and one must ignore
the extra tastes, or account for them by including appropriate factors [20], which can be nontrivial
in applications involving baryons [21].
Just as for Wilson fermions, the staggered action can be improved, so as to reduce discretization
errors. The Asqtad (a-squared tadpole improved) action [22] was used until recently in many large
scale simulations [23]. More recent calculations use the HISQ (highly improved staggered quark)action, introduced in Ref. [24]. At tree-level it removes both O(a2 ) errors and, to lowest order in thequark speed v/c, O([am]4 ) errors. It also substantially reduces effects caused by taste-symmetry
breaking. This makes it attractive not only for light quarks, but means that it is also quite accurate
for heavy quarks because it suppresses (am)n errors. It is being used to directly simulate charm
quarks and to approach direct simulations of bottom quarks (for early studies, see, e.g., [25–27]).
There is an important class of lattice fermions, “Ginsparg-Wilson fermions,” that possess acontinuum-like chiral symmetry without introducing unwanted doublers. The lattice Dirac operator
D for these fermions satisfies the Ginsparg-Wilson relation Dγ5 + γ5 D = aDγ5 D [28]. In thecontinuum, the right-hand-side vanishes, leading to chiral symmetry. On the lattice, it is nonvanishing, but with a particular form (with two factors of D) that restricts the violations of chiralsymmetry in Ward-Takahashi identities to short-distance terms that do not contribute to physicalmatrix elements [29]. In fact, one can define a modified chiral transformation on the lattice (byincluding dependence on the gauge fields) such that Ginsparg-Wilson fermions have an exact chiralsymmetry for on-shell quantities [30]. The net result is that such fermions essentially have the same
properties under chiral transformations as do continuum fermions, including the index theorem [29].
Their leading discretization errors are of O(a2 ).
Two types of Ginsparg-Wilson fermions are currently being used in large-scale numerical simulations. The first is Domain-wall fermions (DWF). These are defined on a five-dimensional space,in which the fifth dimension is fictitious [31]. The action is chosen so that the low-lying modes are
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17. Lattice Quantum Chromodynamics

chiral, with left- and right-handed modes localized on opposite four-dimensional surfaces. For an
infinite fifth dimension, these fermions satisfy the Ginsparg-Wilson relation. In practice, the fifth
dimension is kept finite, and there remains a small, controllable violation of chiral symmetry. The
second type is Overlap fermions. These appeared from a completely different context and have an
explicit form that exactly satisfies the Ginsparg-Wilson relation [32]. Their numerical implementation requires an approximation of the matrix sign function of a Wilson-like fermion operator, andvarious approaches are being used. In fact, it is possible to rewrite these approximations in termsof a five-dimensional formulation, showing that the DWF and Overlap approaches are essentiallyequivalent [33,34]. Numerically, the five-dimensional approach appears to be more computationallyefficient.
The various lattice fermion formulations are often combined with the technique of link smearing. Here one couples the fermions to a smoother gauge link, defined by averaging with adjacentlinks in a gauge invariant manner. Several closely related implementations are being used. All
reduce the coupling of fermions to the short-distance fluctuations in the gauge field, leading to an
improvement in the numerical stability and speed of algorithms. One cannot perform this smearing too aggressively, however, since the smearing may distort short distance physics and enhance
discretization errors.
As noted above, each fermion formulation has its own advantages and disadvantages. For
instance, domain-wall and overlap fermions are theoretically preferred as they have chiral symmetry
without doublers, but their computational cost is greater than for other choices. If the physics
application of interest and the target precision do not require near-exact chiral symmetry, there
is no strong motivation to use these expensive formulations. On the other hand, there is a classof applications (including the calculation of the ∆I = 1/2 amplitude for K → ππ decays [35–37]
and the S-parameter [38]) where chiral symmetry plays an essential role and for which the use of
Ginsparg-Wilson fermions is strongly favored.
17.1.3 Heavy quarks on the lattice
The fermion formulations described in the previous subsection can be used straightforwardly
only for quarks whose masses are small compared to the lattice cutoff, mq / 1/a. This is because
there are discretization errors proportional to powers of amq , and if amq ' 1 these errors arelarge and uncontrolled. Present LQCD simulations typically have cutoffs in the range of 1/a =2–5 GeV (corresponding to a ≈ 0.1–0.04 fm). Thus, while for the up, down and strange quarks
one has amq  1, for bottom quarks (with mb ≈ 4.5 GeV) one must use alternative approaches.
Charm quarks (mc ≈ 1.5 GeV) are an intermediate case, allowing simulations using both direct
and alternative approaches, although increasingly the direct approach is being used.
For the charm quark, the straightforward approach is to simultaneously reduce the latticespacing and to improve the fermion action so as to reduce the size of errors proportional to powersof amc . This approach has been followed successfully using the HISQ, twisted-mass and domainwall actions [24, 25, 27, 39–41]. It is important to note, however, that reducing a increases thecomputational cost because an increased number of lattice points are needed for the same physicalvolume. One cannot reduce the spatial size below 2–3 fm without introducing significant finite
volume errors. Present lattices have typical sizes of ∼ 643 × 128 (with the long direction being
Euclidean time), and thus allow a lattice cutoff up to 1/a ∼ 4–6 GeV.
This approach can, to some extent, be extended to the bottom quark, by the use of simulations
with small lattice spacings [26,42]. This has been pursued with the HISQ action [43], using latticesof size up to 1443 × 288 and lattice spacings down to a ≈ 0.03 fm (1/a ≈ 6.6 GeV). Extrapolationin mb is still needed [44], however, and this makes use of the mass dependence predicted by Heavy
Quark Effective Theory (HQET). Small lattice spacings are also helpful for the simulations of
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quarkonium for which a different heavy-quark mass scaling is expected. See Ref. [45] for a recent
study.
Alternative approaches for discretizing heavy quarks are motivated by effective field theories.
For a bottom quark in heavy-light hadrons, one can use HQET to expand about the infinite quarkmass limit. In this limit, the bottom quark is a static color source, and one can straightforwardly
write the corresponding lattice action [46]. Corrections, proportional to powers of 1/mb , can be
introduced as operator insertions, with coefficients that can be determined non-perturbatively using
existing techniques [47]. This method allows the continuum limit to be taken controlling all 1/mb
corrections.
Another way of introducing the 1/mb corrections is to include the relevant terms in the effective
action. This leads to a non-relativistic QCD (NRQCD) action, in which the heavy quark is describedby a two-component spinor [48]. This approach has the advantage over HQET that it can also be
used for heavy-heavy systems, such as the Upsilon states. Moreover, the bottom quark can betreated without any extrapolation in mb . A disadvantage is that some of the parameters in thiseffective theory are determined perturbatively (at tree-level or at one-loop [49]), which limits theprecision of the final results. Although discretization effects can be controlled with good numericalprecision for a range of lattice spacings, these artifacts cannot be extrapolated away by taking thelattice spacing to zero. This is because NRQCD is a non-relativistic effective field theory and soceases to work when the cutoff π/a becomes much larger than the heavy-quark mass. In practicethese effects are accounted for in the error budget.
This problem can be avoided if one uses HQET power counting to analyze and reduce discretization effects for heavy quarks while using conventional fermion actions [50]. For instance, one cantune the parameters of an improved Wilson quark action so that the leading HQET corrections tothe static quark limit are correctly accounted for. As the lattice spacing becomes finer, the actionsmoothly goes over to that of a light Wilson quark action, where the continuum limit can be taken
as usual. In principle, one can improve the action in the heavy quark regime up to arbitrarily
high orders using HQET, but so far large-scale simulations have typically used clover improved
Wilson quarks, where tuning the parameters of the action corresponds to including all corrections
through next-to-leading order in HQET. Three different methods for tuning the parameters of theclover action are being used: the Fermilab [50], Tsukuba [51] and Columbia [52] approaches. An
advantage of this HQET approach is that the c and b quarks can be treated on the same footing.
Parameter tuning has been done perturbatively, as in NRQCD, or using non-perturbative tuningof some of the parameters [53, 54]. One can improve the effective theory including the terms beyond the next-to-leading order. The Oktay-Kronfeld action that includes dimension-six and -seven
operators has been constructed [55] and used in large-scale numerical calculations [56].
Another approach is the “ratio method” introduced in Ref. [57]. Here one uses quarks with
masses lying at, or slightly above, the charm mass mc , which can be simulated with a relativistic
action, and extrapolates to mb incorporating the behavior predicted by HQET. The particular
implementation relies on the use of ratios. As an example, consider the B meson decay constant fB .
√According to HQET, this scales as 1/ mB for mB  ΛQCD , up to a logarithmic dependence that is
calculable in perturbative QCD (but will be suppressed in the following). Here mB is the B meson√√
mass, which differs from mb by ∼ ΛQCD . One considers the ratio y(λ, mb0 ) ≡ fB 00 mB 00 /fB 0 mB 0
for fictitious B mesons containing b quarks with unphysical masses mb0 and mb00 = λmb0 . HQET
implies that y(λ, mb0 ) approaches unity for large mb0 and any fixed λ > 1. The ratios are evaluatedon the lattice for the sequence of masses mb0 = mc , λmc , λ2 mc , all well below the physical mb , andfor each the continuum limit is taken. The form of the ratio for larger values of mb0 is obtained by√fitting, incorporating the constraints implied by HQET. The result for fB mB is then obtained as√a product of y’s with fD mD [58].
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17.1.4 QED on the lattice
Quarks in nature are electrically charged, and the resultant coupling to photons leads to shifts
in the properties of hadrons that are generically of O(αEM ). Thus, for example, the proton mass isincreased by ∼ 1 MeV relative to that of the neutron due to its overall charge although this effectis more than compensated for by the ∼ 2.5 MeV relative decrease due to the up quark being lighter
than the down quark [59]. This example shows that once pure QCD, isospin-symmetric latticecalculations reach percent-level accuracy, further improvement requires the inclusion of effects due
to both electromagnetism and the up-down mass difference. This level of accuracy has in fact beenobtained for various quantities, e.g., light hadron masses and decay constants (see Ref. [60]), andsimulations including isospin-breaking and QED effects are becoming more common.
One approach for including QED is to treat electromagnetic gauge fields in a similar fashion to
those of QCD. This extension is straightforward, although some new subtleties arise. The essentialchange is that the quark must now propagate through a background field containing both gluons
and photons. The gauge field Uµ that appears in the covariant derivative of Eq. (17.3) is extended
EM
is the photon field, e thefrom an SU(3) matrix to one living in U(3): Uµ → Uµ eiaqeAµ . Here AEMµ
electromagnetic coupling, and q the charge of the quark, e.g., q = 2/3 for up and −1/3 for down
and strange quarks. The lattice action for the photon that is typically used is a discretized versionof the continuum action Eq. (17.2), rather than the form used for the gluons, Eq. (17.1). This
“non-compact” action has the advantage that it is quadratic in AEMµ , which simplifies the QEDpart of the generation of configurations.
One subtlety that arises is that Gauss’ law forbids a charged particle in a box with periodic
boundary conditions. This can be overcome by including a uniform background charge, which canbe shown to be equivalent to removing the zero-momentum mode from the photon field, an approach
denoted QEDL [61]. However, this modification leads to enhanced finite-volume dependence forphysical quantities, scaling as 1/Ln with n = 1, 2, · · · . This should be compared to the exp(−mπ L)
dependence expected for many quantities (as discussed in Sec. 17.1.6.2 below). Methods to remove
the leading few powers of 1/L have been developed [62, 63], and there is a promising alternative
approach in which QED effects are calculated analytically in infinite volume [64]. Finally we notethat there is an alternative approach using so-called C ∗ boundary conditions that avoids the Gauss’
law issue altogether, although the numerical application is still at an early stage [65, 66].
Substantial progress on including QED and isospin-breaking effects has been made over thelast few years. The direct approach of using different up and down-quark masses and simulating
QED has been successfully carried out for a range of quark masses approaching the physical values[59, 67–69]. Alternative approaches have also been used: reweighting the QCD fields a posteriori[70], using a massive photon [71], and keeping only the linear term in an expansion in αEM about
the QCD only case [72,73], with the latter approach by now the most popular. Most calculations todate have included QED effects for the valence quarks but not the sea quarks (the “electro-quenchedapproximation”). [41, 74–77].
The QED corrections to processes including leptons, such as the leptonic and semileptonic
decays of hadrons, involve additional diagrams in which a photon propagator bridges between ahadron and a lepton. Such diagrams induce infrared divergences that cancel against soft photon
radiation (Bloch-Nordsieck theorem [78]). Methods have been developed to implement this cancellation in lattice calculations, treating the soft photon analytically [62], and results have beenreported for leptonic pion and kaon decays [79–81]. An application to semi-leptonic decays has
been developed [64, 82].
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917. Lattice Quantum Chromodynamics

17.1.5 Basic inputs for lattice calculations
Since LQCD is nothing but a regularization of QCD, the renormalizability of QCD implies thatthe number of input parameters in LQCD is the same as for continuum QCD—the strong gauge
coupling αs = g 2 /(4π), the quark masses for each flavor, and the CP violating phase θ. The θ
parameter is usually assumed to be zero, while the other parameters must be determined using
experimental inputs.
17.1.5.1 Lattice spacing
In QCD, the gauge coupling is a function of scale. With lattice regularization, this scale is theinverse lattice spacing 1/a, and choosing the bare gauge coupling is equivalent to fixing the latticespacing.
In principle, a can be determined using any dimensionful quantity measured accurately by
experiments. For example, using the mass of hadron H one has a = (amH )lat /mexp
H . One chooses
quantities that can be calculated accurately on the lattice, and that are only weakly dependent on
the quark masses. The latter property minimizes errors from extrapolating or interpolating to thephysical light quark masses or from mistuning of other quark masses.
Commonly used choices are the spin-averaged 1S-1P or 1S-2S splittings in the Upsilon system,
the masses of the Ξ and Ω − baryons, and the pion decay constant fπ . Ultimately, all choices must
give consistent results for a, and that this is the case provides a highly non-trivial check of boththe calculational method and of QCD.
Many recent lattice calculations use intermediate length scales in place of a direct determinationof the lattice spacing. These length scales, which we denote R, have the advantage that they can be
precisely, and relatively cheaply, computed numerically. Examples are r0 , derived from the heavy
quark potential [83], and t0 and w0 , determined from the gradient flow of the gauge field [84]. These
scales are used in the following manner, explained here in the context of calculating a quantity Q
with mass dimension d (e.g. a decay constant for which d = 1). In the first step, one calculates
the dimensionless quantities ad Q and R/a in a given lattice calculation, and forms the product(ad Q) × (R/a)d = QRd . In a second step, one uses results available from previous dedicatedlattice calculations that have determined R in physical units (i.e. fm) by relating them to physicalquantities as discussed above. Then one obtains Q = (QRd )/Rd . The use of intermediate quantities
is reviewed in the latest edition of the Flavor Lattice Averaging Group (FLAG) report [85].
17.1.5.2 Light quark masses
In LQCD simulations, the up, down and strange quarks are usually referred to as the lightquarks, in the sense that mq < ΛQCD . (The standard definition of ΛQCD is given in the “Quantum
Chromodynamics” review; in this review we are using it only to indicate the approximate nonperturbative scale of QCD.) This condition is stronger than that used above to distinguish quarks
with small discretization errors, mq < 1/a. Loop effects from light quarks must be included in
the simulations to accurately represent QCD. At present, most simulations are done in the isospin
symmetric limit mu = md ≡ m` < ms , with charm-quark loops either excluded (referred to as“Nf = 2 + 1” simulations) or included (denoted “Nf = 2 + 1 + 1” simulations). As noted above,
precision is now reaching the point where isospin breaking effects must be included. To do so
without approximation requires simulating with non-degenerate up and down quarks (leading to
Nf = 1 + 1 + 1 or 1 + 1 + 1 + 1 simulations) as well as including electromagnetism (as describedabove).
We now describe the tuning of m` , ms and mc to their physical values. (For brevity, we ignore
isospin violation in the following discussion.) The most commonly used quantities for these tunings
are, respectively, mπ , mK and mηc . If the scale is being set by mΩ , then one adjusts the latticequark masses until the ratios mπ /mΩ , mK /mΩ and mηc /mΩ take their physical values. In the past,
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most calculations needed to extrapolate to the physical value of m` (typically using forms basedon chiral perturbation theory [ChPT]), while simulating directly at or near to the physical valuesof ms and mc . Present calculations are increasingly done with physical or near physical values ofm` , requiring at most only a short extrapolation or interpolation.
17.1.5.3 Heavy quark masses
The b quark is usually treated only as a valence quark, with no loop effects included. The
errors introduced by this approximation can be estimated to be ∼ αs (mb )Λ2QCD /m2b and are likely
to be very small. In the past, the same approximation has been made for the c quark, leading
to errors ∼ αs (mc )Λ2QCD /m2c . (See Ref. [86] for a quantitative estimate of the effects of including
the charm quark on some low energy physical quantities, and Ref. [87] for similar estimates for
B-meson matrix elements.) For high precision, however, dynamical charm quarks are necessary,and some of the most recent simulations now include them.
The b quark mass can be tuned by setting heavy-heavy (Υ ) or heavy-light (B) meson masses totheir experimental values. Consistency between these two determinations provides an important
check that the determination of parameters in the heavy quark lattice formulations is being done
correctly (see, e.g., Ref. [26,88,89]). For instance, the b quark masses obtained from heavy-light [44]and heavy-heavy [77] mesons are in excellent agreement within the precision of 0.5%.
17.1.6 Sources of systematic error
Lattice results have statistical and systematic errors that must be quantified for any calculationin order for the result to be a useful input to phenomenology. The statistical error is due to theuse of Monte Carlo importance sampling to evaluate the path integral (a method discussed below).
There are, in addition, a number of systematic errors that are always present to some degree inlattice calculations, although the size of any given error depends on the particular quantity underconsideration and the parameters of the ensembles being used. The most common lattice errors
are reviewed below.
Although not strictly a systematic error, it is important to note that the presence of long autocorrelations in the sequence of lattice configurations generated by the Monte Carlo method can leadto underestimates of statistical errors [90]. It is known that the global topological charge of thegauge fields decorrelates very slowly with certain algorithms [90, 91]. The effect of poorly sampling
topological charge is expected to be most significant for the pion mass and related quantities [92–94].
This issue becomes more relevant as the precision of the final results improves. The problem of slow
decorrelation of topology can be mitigated using open boundary conditions, for which the global
topological charge is not quantized [95].
17.1.6.1 Continuum limit
Physical results are obtained in the limit that the lattice spacing a goes to zero. The Symanzik
effective theory (SET) determines the scaling of lattice artefacts with a. Most lattice calculations
use improved actions with leading discretization errors of O(αs aΛ), O(a2 Λ2 ), or O(αs a2 Λ2 ), whereΛ is a typical momentum scale in the system. Knowledge of the scaling of the leading discretization
errors allows controlled extrapolation to a = 0 when multiple lattice spacings are available, as incurrent state-of-the-art calculations. Residual errors arise from the exclusion of sub-leading adependence from the fits, either that due to prefactors containing powers of log(a), or from higher
powers of a. The former can, in principle, be understood using the SET, and first studies of thishave been undertaken [96–98].
For many quantities the typical momentum scale in the system is ∼ ΛQCD ≈ 300 MeV. Discretization errors are expected to be larger for quantities involving larger scales, for example form
factors or decays involving particles with momenta or masses larger than ΛQCD .
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1117. Lattice Quantum Chromodynamics

17.1.6.2 Infinite volume limit
LQCD calculations are necessarily carried out in finite space-time boxes, leading to departuresof physical quantities (masses, decay constants, etc.) from their measured, infinite volume values.
These finite-volume shifts are an important systematic that must be estimated and minimized.
Typical lattices are asymmetric, with Ns points in the three spatial directions and Nt in the(Euclidean) temporal direction. The spatial and temporal sizes in physical units are thus Ls = aNs
and Lt = aNt , respectively. (Anisotropic lattice spacings are also sometimes used, as discussed
below in Sec. 17.2.2.) Typically, Lt ≥ 2Ls , a longer temporal direction being used to allow excitedstate contributions to correlators to decay. This means that the dominant impact of using finite
volume is from the presence of a finite spatial box.
The highest-precision LQCD calculations are of quantities involving no more than a single
strongly-interacting particle in initial and final states. For such quantities, once the volume exceeds
about 2 fm (so that the particle is not “squeezed”), the dominant finite-volume effect comes fromvirtual pions wrapping around the lattice in the spatial directions. This effect is exponentiallysuppressed as the volume becomes large, roughly as ∼ exp(−mπ Ls ), and has been estimated using
ChPT [99] or other methods [100]. These estimates suggest that finite volume shifts are sub-percent
effects when mπ Ls & 4, and most large-scale simulations use lattices satisfying this condition. (Seealso Sec. 2.1.1 of the FLAG report [85] for more detailed discussion concerning the conditions
required to control finite volume effects.) This becomes challenging as one approaches the physicalpion mass, for which Ls & 5 fm is required.
Finite volume errors are usually determined by repeating the simulations on two or more different volumes (with other parameters fixed). If different volumes are not available, the ChPT
estimate can be used, often inflated to account for the fact that the ChPT calculation is truncatedat some order.
In the future, LQCD calculations involving more than a single hadron will become increasingly
precise. Examples include the calculation of resonance parameters and the K → ππ amplitudes.
Finite volume effects are much larger in these cases, with power-law terms (e.g., 1/L3s ) in addition
to exponential dependence. Indeed, as will be discussed in Sec. 17.2.4, one can use the volume
dependence to indirectly extract infinite-volume quantities such as scattering lengths. Doing so,
however, requires a set of lattice volumes satisfying mπ Ls & 4 and is thus more challenging thanfor single-particle quantities.
17.1.6.3 Chiral extrapolation
Until recently, an important source of systematic error in LQCD calculations was the needto extrapolate in mu and md (or, equivalently, in mπ ). This extrapolation was usually doneusing functional forms based on ChPT, or with analytic functions, with the difference between
different fits used as an estimate of the systematic error, which was often substantial. Increasingly,
however, calculations work directly at, or very close to, the physical quark masses. This either
removes entirely, or greatly reduces, the uncertainties in the extrapolation, such that this error is
subdominant.
17.1.6.4 Operator matching
Many of the quantities that LQCD can precisely calculate involve hadronic matrix elements ofoperators from the electroweak Hamiltonian. Examples include the pion and kaon decay constants,
semileptonic form factors and the kaon mixing parameter BK (the latter defined in Eq. (17.13)).
The operators in the lattice matrix elements are defined in the lattice regularization scheme. To beused in tests of the Standard Model, however, they must be matched to the continuum regularization
scheme in which the corresponding Wilson coefficients have been calculated. The only case in whichsuch matching is not needed is if the operator is a conserved or partially conserved current. Similar
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17. Lattice Quantum Chromodynamics

matching is also needed for the conversion of lattice bare quark masses to those in the continuum
MS scheme.
Several methods are used to calculate the matching factors: perturbation theory (usually to oneor two-loop order), non-perturbative renormalization (NPR) using Landau-gauge quark and gluon
external states [101], NPR using gauge-invariant methods based on the Schrödinger functional [102],
NPR using gauge-invariant short-distance hadron correlators [103], and NPR using gauge-invariant
heavy-heavy correlators [27, 104]. The NPR methods replace truncation errors (which can only beapproximately estimated) by statistical and systematic errors that can be determined reliably andsystematically reduced.
An issue that arises in some of such calculations (e.g., for quark masses and BK ) is that, using
NPR with Landau-gauge quark and gluon external states, one ends up with operators regularized
in a MOM-like scheme (or a Schrödinger-functional scheme), rather than the MS scheme mostly
used for calculating the Wilson coefficients. To make contact with this scheme requires a purely
continuum perturbative matching calculation supplemented by the operator product expansion(OPE). (The importance of power corrections is emphasized in [105].) The resultant truncation
error of perturbative expansion and OPE can be minimized by pushing up the momentum scale atwhich the matching is done using step-scaling techniques as part of the NPR calculation [106].
It should also be noted that this final step in the conversion to the MS scheme could be avoided
if continuum calculations used a MOM-like scheme or if one imposes a renormalization condition for
quantities that are calculable both in the MS scheme and in LQCD, such as the hadron correlators
at short distances (see, e.g., Ref. [107]).

17.2 Methods and status
Once the lattice action is chosen, it is straightforward to define the quantum theory using thepath integral formulation. The Euclidean-space partition function is
Z

Z=

[dU ]

Y

[dqf ][dq̄f ]eP−Sg [U ]−fq̄f (D[U ]+mf )qf,(17.4)fwhere link variables are integrated over the SU(3) manifold, qf and q̄f are Grassmann (anticommuting) quark and antiquark fields of flavor f , and D[U ] is the chosen lattice Dirac operator
with mf the quark mass in lattice units. Integrating out the quark and antiquark fields, one arrives
at a form suitable for simulation:ZZ=[dU ]e−Sg [U ]Y

det(D[U ] + mf ) .

(17.5)

f

The building blocks for calculations are expectation values of multi-local gauge-invariant operators,
also known as “correlation functions”,
Z

hO(U, q, q̄)i = (1/Z) [dU ]Y[dqf ][dq̄f ]O(U, q, q̄)e−Sg [U ]−Pfq̄f (D[U ]+mf )qf.

(17.6)f

If the operators depend on the (anti-)quark fields qf and q̄f , then integrating these fields out
leads not only to the fermion determinant but also, through Wick’s theorem, to a series of quark“propagators”, (D[U ] + mf )−1 , connecting the positions of the fields.
This set-up allows one to choose, by hand, the masses of the quarks in the determinant (thesea quarks) differently from those in the propagators (valence quarks). This is called “partial
quenching”, and is used by some calculations as a way of obtaining more data points from which
to extrapolate both sea and valence quarks to their physical values.
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17.2.1 Monte-Carlo method
Since the number of integration variables U is huge (Ns3 × Nt × 4 × 8), direct numerical integration is impractical and one has to use Monte-Carlo techniques. In this method, one generates a
Markov chain of gauge configurations (a “configuration” being the set of U ’s on all links) distributedQ
according to the probability measure [dU ]e−Sg [U ] f det(D[U ] + mf ). Once the configurations are
generated, expectation values hO(U, q, q̄)i are calculated by averaging over those configurations.
In this way the configurations can be used for many different calculations, and there are several
large collections of ensembles of configurations (with a range of values of a, lattice sizes and quark
masses) that are publicly available through the International Lattice Data √Grid (ILDG) [108]. As
the number of the configurations, N , is increased, the error decreases as 1/ N .
The most challenging part of the generation of gauge configurations is the need to include thefermion determinant. Direct evaluation of the determinant is not feasible, as it requires O((Ns3 ×Nt )3 ) computations. Instead, one rewrites it in terms of “pseudo-fermion” fields φ (auxiliary fermion
fields with bosonic statistics). For example, for two degenerate quarks one has
2det(D[U ] + mf ) =Z†† −1 φ[dφ]e−φ ((D[U ]+mf )(D[U ]+mf ) ).

(17.7)

By treating the pseudo-fermions as additional integration variables in the path integral, one obtains
a totally bosonic representation. The price one pays is that the pseudo-fermion effective action ishighly non-local since it includes the inverse Dirac operator (D[U ] + mf )−1 . Thus, the large sparse
matrix (D[U ] + m) has to be inverted every time one needs an evaluation of the effective action.
Present simulations generate gauge configurations using the Hybrid Monte Carlo (HMC) algorithm [109], or variants thereof. This algorithm combines molecular dynamics (MD) evolution [110–112] in a fictitious time (which is also discretized) with a Metropolis “accept-reject”
step [113]. It makes a global update of the configuration, and is made exact by the Metropolis step.
In its original form it can be used only for two degenerate flavors, but extensions (particularly therational HMC [114]) are available for single flavors. Considerable speed-up of the algorithms has
been achieved over the last two decades using a variety of techniques.
All these algorithms spend the bulk of their computational time on the repeated inversion of(D[U ] + m) acting on a source (which is required at every step of the MD evolution). Inversions
are done using a variety of iterative algorithms, e.g., the conjugate gradient algorithm. In this classof algorithms, computational cost is determined by the condition number of the matrix, which is
the ratio of maximum and minimum eigenvalues. For (D[U ] + m) the smallest eigenvalue is ≈ m,and the cost is inversely proportional to the quark mass. This is a major reason why simulations
at the physical light quark masses are challenging.
Algorithmic improvements have significantly reduced this problem. The main idea is to separate
different length scales [115, 116]. Since the low eigenvalues of (D[U ] + m) are associated with longwavelength quark modes, one may project the problem onto that of a coarse-grained lattice by
averaging the field within a block of sublattices and carrying out the inversion on this coarse lattice.
The result is then fed back to the original lattice as an efficient preconditioner for the iterativesolver, and the whole procedure may be nested multiple times. Variants of such methods havebeen implemented, specifically domain-decomposition [117, 118], deflation [119–122] and multigrid[123, 124]. They are increasingly used in large-scale lattice simulations.
A practical concern is the inevitable presence of correlations between configurations in theMarkov chain. These are characterized by an auto-correlation length in the fictitious MD time.
One aims to use configurations separated in MD time by greater than this auto-correlation length.
In practice, it is difficult to measure this length accurately, see, e.g., [125], and this leads to someuncertainty in the resulting statistical errors, as well as the possibility of insufficient equilibration.
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The computational cost of gauge generation grows with the lattice volume, Vlat = Ns3 Nt , as
Here δ = 1/4 for the HMC algorithm [126] and can be reduced slightly using modern
variants. Such growth with Vlat provides a (time-dependent) limit on the largest lattice volumes
that can be simulated. At present, the largest lattices being used have Ns = 144 and Nt = 288
or Ns = NT = 160. Typically, one aims to create an ensemble of ∼ 103 statistically independentconfigurations at each choice of parameters (a, mq and Vlat ). For most physical quantities ofinterest, this is sufficient to make the resulting statistical errors smaller than or comparable tothe systematic errors. Recently, the master-field approach is being investigated [13, 127, 128]. It
aims to create a large-volume ensemble and repeat the calculation of physical quantities in different
small patches of the entire lattice. Translational invariance guarantees that the correct average is
obtained with fewer configurations than in the traditional approach, with the advantage of reduced
finite-volume effects [129].
In the past, the cost of generating gauge configurations was larger than that of performing“measurements” on those configurations. However, as the number of quantities being calculated
and their complexity has increased, the balance has shifted to the point that the total cost ofmeasurements exceeds that of generation. Sharing the configurations, e.g. through the ILDG, has
become more common, and a number of groups are utilizing them to compute a wide variety ofquantities, as partly covered by the following sections.
1+δ
Vlat
.

17.2.2 Two-point functions
One can extract properties of stable hadrons from two-point correlation functions, hOX (x)OY† (0)i.
Here OX,Y (x) are operators that have non-zero overlaps with the hadronic state of interest |Hi,
i.e. h0|OX,Y (x)|Hi 6= 0. On the lattice, the two-point correlation function can be constructed,using Wick’s theorem, from the quark propagators (D[U ] + mf )−1
x0 from the insertion point of thesource operator OY (0) to the sink operator OX (x). An average over gauge configurations yields
the estimate of hOX (x)OY† (0)i.
One usually Fourier transforms in the spatial directions and considers correlators as a functionof Euclidean time:X(17.8)CXY (t; p) =hOX (t, x)OY† (0)ie−ip·x .
x(Here and throughout this section all quantities are expressed in dimensionless lattice units, so
that, for example, p = apphys .) By inserting a complete set of states having spatial momentum p,
the two-point function can be written as
CXY (t; p) =∞X

1
h0|OX (0)|Hi (p)ihHi (p)|OY† (0)|0ie−Ei (p)t ,
2E
(p)
i
i=0

(17.9)

where the energy of the i-th state Ei (p) appears in the eigenvalue of the Euclidean time evolution
operator e−Ht , e−Ei (p)t . The factor of 1/[2Ei (p)] is due to the relativistic normalization used forthe states. For large enough t, the dominant contribution is that of the lowest energy state |H0 (p)i:t→∞

CXY (t) −→1
h0|OX (0)|H0 (p)ihH0 (p)|OY† (0)|0ie−E0 (p)t .
2E0 (p)(17.10)One can thus obtain the energy E0 (p), which equals the hadron mass mH when p = 0, and the
product of matrix elements h0|OX (0)|Hi (p)ihHi (p)|OY† (0)|0i.
This method can be used to determine the masses of all the stable (in QCD) mesons andbaryons by making appropriate choices of operators. For example, if one uses the axial current,¯ µ γ5 u, then one can determine mπ+ from the rate of exponential fall-off, andOX = OY = Aµ = dγ
in addition the decay constant fπ from the coefficient of the exponential.
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17. Lattice Quantum Chromodynamics

15

The expression given in (17.9) for the correlator CXY (t; p) shows how, in principle, one candetermine the energies of the excited hadron states having the same quantum numbers as theoperators OX,Y , by fitting the correlation function to a sum of exponentials, including both groundstate and excited-state contributions. It is important to do this in order to obtain accurate results
for the ground-state parameters. To reliably identify excited-state parameters in practice, one
often needs to use a large basis of operators with a variety of overlaps, h0|O|Hi, with the different
states, and to adopt a variational approach, such as that of [130]. Use of an anisotropic lattices
in which at , the lattice spacing in the time direction, is smaller than its spatial counterpart, as ,allows more temporal samples of the correlation function at small times where there is a signal for
excited states. Using these and and other technical improvements, extensive excited-state spectra
have been obtained [131–136].
A complication arises for states with high spins (j ≥ 4 for bosons) because the spatial rotationgroup on the lattice is a discrete subgroup of the continuum group SO(3). This implies that lattice
operators, even when chosen to lie in irreducible representations of the lattice rotation group, haveoverlap with states that have a number of values of j in the continuum limit [137]. For example
j = 0 operators can also create mesons with j = 4. Methods to overcome this problem in practice
are available [131, 138] and have been used successfully.
The two-point function (17.8) contains, in principle, information on all possible states having
the quantum numbers of the operator O. Thus, if O is chosen to be a vector current, it can beused to estimate the hadronic vacuum polarization function contribution to the muon anomalous
magnetic moment, as discussed in Sec. 17.3.6.
17.2.3 Three-point functions
Hadronic matrix elements needed to calculate semileptonic form factors and neutral meson
mixing amplitudes can be computed from three-point correlation functions. We discuss here, as arepresentative example, the D → K amplitude. As in the case of two-point correlation functions
one constructs operators OD and OK having overlap, respectively, with the D and K mesons. We
are interested in calculating the matrix element hK|Vµ |Di, with Vµ = c̄γµ s the flavor-changing
vector current.
To obtain this, we use the three-point correlator
CKVµ D (tx , ty ; p) =†hOK (tx , x)Vµ (0)OD(ty , y)ie−ip·x ,X(17.11)x,yand focus on the limit tx → ∞, ty → −∞. In this example we set the D-meson at rest while thekaon carries three-momentum p. Momentum conservation then implies that the weak operator Vµinserts three-momentum −p. Inserting a complete set of states between each pair of operators, we
find
CKVµ D (tx , ty ; p) =1
−m t −E (p)|ty |e Di x Kj
2m
2E
(p)
Di
Kj
i,j

X

†
× h0|OK (0)|Kj (p)ihKj (p)|Vµ (0)|Di (0)ihDi (0)|OD(0)|0i. (17.12)

The matrix element hKi (p)|Vµ (0)|Dj (0)i can then be extracted, since all other quantities in thisexpression can be obtained from two-point correlation functions. Typically, one is interested in
the weak matrix elements of ground states, such as the lightest pseudo-scalar mesons. In thelimit of large separation between the three operators in Euclidean time, the three-point correlation
function yields the weak matrix element of the transition between ground states. The extractionof the ground state is not always straightforward, especially when a significant contribution is
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17. Lattice Quantum Chromodynamics

expected from states with an extra pion. An example relevant to B → π`ν decay is discussed inRef. [139].
From (17.12) the matrix element hKi (p)|Vµ (0)|Dj (0)i is obtained at several discretized values ofmomentum p, given by a multiple of 2π/L in each spatial direction. Each momentum corresponds
to a certain value of the momentum transfer q 2 = (mD − EK (p))2 − p2 . The restriction on theavailable momentum can be relaxed by introducing twisted boundary conditions [140–142], which
are a generalization of (anti-)periodic boundary conditions in the spatial directions that allows ageneral U (1) phase upon translation by distance L. The choice of phase leads to a quark momentum
that does not have to be a multiple of 2π/L. The method has been applied in a number of latticecalculations.
To interpolate in q 2 , the so-called z parametrization [143–146] is often employed. It takes
account of the analytic structure of the form factor in the complex q 2 plane, and a polynomial
expansion in z is justified.
17.2.4 Scattering amplitudes and resonances
The methods described thus far yield matrix elements involving single, stable particles (whereby stable we mean here absolutely stable to strong interaction decays). Most of the particles
listed in the Review of Particle Properties are, however, unstable—they are resonances decaying
into final states consisting of multiple strongly interacting particles. LQCD simulations cannot
directly calculate resonance properties, but methods have been developed to do so indirectly forresonances coupled to two-particle final states in the elastic regime, starting from the seminal workof Lüscher [147].
The difficulty faced by LQCD calculations is that, to obtain resonance properties, or, more generally, scattering phase-shifts, one must calculate multi-particle scattering amplitudes in momentum space and put the external particles on their mass-shells. This requires analytically continuingfrom Euclidean to Minkowski momenta. Although it is straightforward in LQCD to generalize
the methods described above to calculate four- and higher-point correlation functions, one necessarily obtains them at a discrete and finite set of Euclidean momenta. Analytic continuation top2E = −m2 is then an ill-posed and numerically unstable problem. The same problem arises for
single-particle states, but can be largely overcome by picking out the exponential fall-off of the
Euclidean correlator, as described above. With a multi-particle state there is no corresponding
trick, except for two particles at threshold [148], although recent ideas using smeared correlators
and advanced spectral-reconstruction methods offer hope for future progress [149, 150].
What LQCD can calculate are the energies of the eigenstates of the QCD Hamiltonian in a
finite box. The energies of states containing two stable particles, e.g., two pions, clearly depend on
the interactions between the particles. It is possible to invert this dependence and, with plausible
assumptions, determine the scattering phase-shifts at a discrete set of momenta from a calculation of
the two-particle energy levels for a variety of spatial volumes [147]. This is a challenging calculation,
but it has been carried through in several channels with quark masses approaching physical values.
Channels studied include ππ (for I = 2, 1 and 0), K̄K, Kπ, πω, πφ, KD, DD∗ , Bπ, and N π.
For recent comprehensive reviews see [151, 152]. Extensions to nucleon interactions are being
actively studied [153]. The formalism has been generalized to three particles (both identical andnon-degenerate, and including spin) [154], and has been applied in lattice calculations of 3π + ,2π + + K + , π + + 2K + , and 3K ∗ systems [155] as well as to the I = 1 three-pion channel [156]. For
recent reviews, see [152, 157].
It is also possible to extend the methodology to calculate electroweak decay amplitudes to twoparticles below the inelastic threshold, e.g., A(K → ππ) [158]. Results for both the ∆I = 3/2
and 1/2 amplitudes with physical quark masses have been obtained [35–37, 159], the former now
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1717. Lattice Quantum Chromodynamics

including a controlled continuum limit [160]. First results for the CP -violating quantity 0 havebeen obtained [35, 36].
Partial extensions of the formalism above the elastic threshold have been worked out, in particular for the case of multiple two-particle channels [161]. Another theoretical extension is to allow
the calculation of form factors between a stable particle and a resonance [162] , and between two
resonances [163]. The former has been used to calculate the γπ → ρ amplitude, albeit for unphysically large quark masses [164, 165], as well as γK → K ∗ [166]. Finally, the formalism for using
LQCD to calculate electroweak decays or transitions to three particles, e.g γ ∗ → 3π and K → 3π,
has recently been worked out [167, 168].
While a systematic extension to decays with many multi-particle channels, e.g., hadronic B
decays, has, however, yet to be formulated, some interesting new ideas have been proposed [150,
169, 170], including a method to compute inclusive decay rates or cross sections [171, 172]. The
latter represents the inclusive processes using the Compton amplitude, in which two currents areinserted and all possible final states are created between the two [150]. The relative weight among
different final states depends on the Euclidean time separation between the current, from which one
can extract the inclusive rates. First calculations have been reported [173,174]. A related approachhas been recently applied to provide a lattice determination of the inclusive semileptonic τ decayrate [175].
17.2.5 Status of LQCD simulations
Until the 1990s, most large-scale lattice simulations were limited to the “quenched” approximation, wherein the fermion determinant is omitted from the path integral. While much of thebasic methodology was developed in this era, the results obtained had uncontrolled systematic errors and were not suitable for use in placing precision constraints on the Standard Model. During
the 1990s, more extensive simulations including the fermion determinant (also known as simulations with “dynamical” or “sea” quarks) were begun, but with unphysically heavy quark masses(m` ∼ 50−100 MeV), such that the extrapolation to the physical light quark masses was a source oflarge systematic errors [176]. During the 2000s, advances in both algorithms and computers allowed
simulations to reach much smaller quark masses (m` ∼ 10 − 20 MeV) such that LQCD calculationsof selected quantities with all sources of error controlled and small became available. Their results
played an important role in constraints on the CKM matrix and other phenomenological analyses.
In the last decade, simulations directly at the physical isospin-symmetric light quark masses havebecome standard, removing the need for a chiral extrapolation and thus significantly reducing theoverall error. The present frontier, as noted above, is the inclusion of isospin breaking. This will
be needed to push the accuracy of calculations below the percent level.
On a more qualitative level, analytic and numerical results from LQCD have demonstrated
that QCD confines color and spontaneously breaks chiral symmetry. Confinement can be seen
as a linearly rising potential between heavy quark and anti-quark in the absence of quark loops.
Analytically, this can be shown in the strong coupling limit glat → ∞ [1]. At weaker couplings there
are precise numerical calculations of the potential that clearly show that this behavior persists inthe continuum limit [177–179].
Chiral symmetry breaking was also demonstrated in the strong coupling limit on the lattice[18, 180], and there have been a number of numerical studies showing that this holds also in thecontinuum limit. The accumulation of low-lying modes of the Dirac operator, which is the analog
of Cooper pair condensation in superconductors, has been observed, yielding a determination of thechiral condensate [181–187]. Many relations among physical quantities that can be derived under
the assumption of broken chiral symmetry have been confirmed by a number of lattice groups [188].
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17.3 Physics applications
In this section we describe the main applications of LQCD that are both computationally matureand relevant for the determination of particle properties.
A general feature to keep in mind is that, since there are many different choices for latticeactions, all of which lead to the same continuum theory, a crucial test is that results for any given
quantity are consistent. In many cases, different lattice calculations are completely independent andoften have very different systematic errors. Thus, final agreement, if found, is a highly non-trivial
check, just as it is for different experimental measurements.
The number, variety and precision of the calculations has progressed to the point that an international collaboration, FLAG, has been formed, which aims to collect all lattice results of relevancefor a variety of phenomenologically interesting quantities and provide averages of those results thatpass appropriate quality criteria. The averages attempt to account for possible correlations between
results (which can arise, for example, if they use common gauge configurations). The quantities
considered are those we discuss in this section, with the exception of the hadron spectrum, as wellas the intermediate scale-setting quantities discussed earlier. The most recent FLAG review is from2021 [85] (see also older editions, Refs. [60, 188]). The interested reader can consult this review forvery extensive discussions of the details of the calculations and of the sources of systematic errors.
We stress that the results we quote below are those obtained using the physical complement oflight quarks (i.e. Nf = 2 + 1 or 2 + 1 + 1 simulations).
17.3.1 Spectrum
The most basic prediction of LQCD is of the hadron spectrum. Once the input parameters
are fixed as described in Sec. 17.1.5, the masses or resonance parameters of all other states can bepredicted. This includes hadrons composed of light (u, d and s) quarks, as well as heavy-light andheavy-heavy hadrons. It also includes quark-model exotics (e.g., J P C = 1−+ mesons) and glueballs.
Thus, in principle, LQCD calculations should be able to reproduce many of the experimental resultscompiled in the Review of Particle Properties. Doing so would test both that the error budgetsof LQCD calculations are accurate and that QCD indeed describes the strong interactions in the
low-energy domain. The importance of the latter test can hardly be overstated.
What is the status of this fundamental test? As discussed in Sec. 1.2, LQCD calculations aremost straightforward for stable, low-lying hadrons. Calculations of the properties of resonancesthat can decay into only two particles are more challenging, but are becoming standard in themeson sector, with the frontier being decays involving baryons. As noted above, the formalism
for resonances decaying to three particles exists, but has yet to be applied to resonant channels
other than in pioneering calculations. It is also more technically challenging to calculate massesof flavor singlet states (which can annihilate into purely gluonic intermediate states) than those offlavor non-singlets, although again algorithmic and computational advances have begun to makesuch calculations accessible, including first calculations that reach physical quark masses [189].
The present status for light hadrons is that fully controlled results are available for the masses ofthe octet light baryons, while results with less than complete control are available for the decuplet
baryon resonances, the vector meson resonances and the η and η 0 . This is discussed in the “Quark
Model” review—see, in particular, Fig. 15.9. In addition, it has been possible to calculate the isospin
splitting in light mesons and baryons (due to the up-down mass difference and the incorporationof QED). There are also extensive results for heavy-light (D and B systems) and heavy-heavy(J/ψ and Υ systems). All present results, which are discussed in the “Quark Model” review, areconsistent with experimental values, and several predictions have been made. We refer the reader
to that review for references to the relevant work.
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17.3.2 Decay constants and bag parameters
The pseudo-scalar decay constants can be determined from two-point correlation functions
involving the axial-vector current, as discussed in Sec. 17.2.2. The decay constant fP of a meson P
is extracted from the weak matrix element involving the axial-vector current using the definitionh0|Aµ (x)|P (p)i = fP pµ exp(−ip · x), where pµ is the momentum of P and Aµ (x) is the axial-vector
current. (In practice, results with the smallest errors are obtained using the pseudo-scalar density
P (x) = q̄(x)γ5 q(x) instead of Aµ (x).) Since they are among the simplest quantities to calculate,decay constants provide good benchmarks for lattice methods, in addition to being important inputs
for flavor physics phenomenology in their own right. Results from several lattice groups for thepion and kaon decay constants now have sub-percent errors. The decay constants in the charmand bottom sectors, fD , fDs , fB , and fBs , have also been calculated using the various heavy quark
formulations on the lattice outlined in Sec. 17.1.3. The precision has reached sub-percent errors forcharmed mesons, and percent-level errors for bottom mesons. Lattice results for all of these decay
constants are discussed in detail in the review “Leptonic Decays of Charged Pseudoscalar Mesons.”
Another important lattice quantity is the kaon bag parameter, BK , which is needed to turn the
precise measurement of CP -violation in kaon mixing into a constraint on the Standard Model. It
is defined by
8 2 2
0(17.13)m f BK (µ) = hK |Q∆S=2 (µ)|K 0 i,
3 K K
where mK is the kaon mass, fK is the kaon decay constant, Q∆S=2 = sγµ (1−γ5 )dsγµ (1−γ5 )d is thefour-quark operator of the effective electroweak Hamiltonian and µ is the renormalization scale. The
short distance contribution to the electroweak Hamiltonian can be calculated perturbatively, but thehadronic matrix element parameterized by BK must be computed using non-perturbative methods.
In order to be of use to phenomenology, the renormalization factor of the four-quark operator must
be matched to a continuum renormalization scheme, e.g., to MS, as described in Sec. 17.1.6.4.
Determinations with percent-level precision using different fermion actions and Nf = 2 + 1 light sea
quarks are now available using DWF [190], staggered fermions [191], DWF valence on staggered
sea quarks [192], and Wilson fermions [11]. The results are all consistent, and the present FLAG
average is B̂K = 0.7625(97) [85] for its renormalization group invariant definition (for original
papers, see [11, 192–194]).
The bag parameters for B and Bs meson mixing are defined analogously to that for kaon mixing.
The B and Bs mesons contain a valence b-quark so that calculations of these quantities must use
one of the methods for heavy quarks described above. Calculations have been done using NRQCD[195,196], the Fermilab formalism [87], and static heavy quarks [197]. All results are consistent. The
FLAG averages for the quantities relevant for Bs and B mixing withq
Nf = 2+1, which are based onqresults from Refs. [87,195,197], are fBs B̂Bs = 274(8) MeV and fB B̂B = 225(9) MeV, with theirratio (which is somewhat better determined) being ξ = 1.206(17). FLAG alsoq
quotes an “average”
for Nf = 2 + 1 + 1, which comes from a single calculation [196] and gives fBs
q

B̂Bs = 256(6) MeV

fB B̂B = 211(6) MeV, and ξ = 1.216(16). These are consistent with the Nf = 2 + 1 results atthe 2σ level. Errors for quantities involving b quarks are typically larger than those for quantities
involving only light quarks, although the difference has steadily decreased in recent years.
For the K, D and B systems, one can also consider the matrix elements of four-fermion operatorsthat arise in beyond-the-standard-model (BSM) theories, which can have a different chiral structure.
Knowledge of these matrix elements allows one to constrain the parameters of the BSM theories,
and is complementary to direct searches at the LHC. Reliable results are now available from latticecalculations, and are reviewed by FLAG in the case of kaon mixing [85]. Complete results for D
and B mixing are presented in Ref. [198, 199] and Ref. [87, 200], respectively.
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The results for mixing matrix elements are used in the reviews “The CKM Quark-Mixing
Matrix,” and “B 0 − B̄ 0 Mixing.”
17.3.3 Form factors (K → π`ν, D → K`ν, B → π`ν, B → D(∗) `ν, etc.)
Semileptonic decay rates can be used to extract CKM matrix elements once the semileptonic
form factors are known from lattice calculations. For example, the matrix element of a pseudoscalar meson P undergoing semileptonic decay to another pseudo-scalar meson D is mediated by
the vector current, and can be written in terms of form factors as
hD(pD )|Vµ |P (pP )i = f+ (q 2 )(pD + pP − ∆)µ + f0 (q 2 )∆µ ,(17.14)

where q = pD − pP , ∆µ = (m2D − m2P )qµ /q 2 and Vµ is the quark vector current. For K and D meson
decays, the shapes of the form factors are well determined by experiment and the value of f+ (q 2 )at some reference value of q 2 is needed from the lattice in order to extract CKM matrix elements.
Lattice calculations of the form factor shapes may provide further consistency checks. Typically,f+ (q 2 ) dominates the decay rate, since the contribution from f0 (q 2 ) is suppressed when the final
state lepton is light.
The form factor f+ (0) for K → π`ν decays is highly constrained by the Ademollo-Gatto theorem[201] and chiral symmetry. Old estimates using chiral perturbation theory combined with quark
models quote sub-percent precision [202], though they suffer from some model dependence. Utilizing
the constraint from the vector current conservation that f+ (0) is normalized to unity in the limit
of degenerate up and strange quark masses, the lattice calculation can be made very precise andhas now matched the precision of the phenomenological estimates [203–210]. The present FLAG
average (from Nf = 2 + 1 + 1 simulations) is f+ (0) = 0.9698(17), based on Refs. [210].
Charmed meson semileptonic decays have been calculated by different groups using methods
similar to those used for charm decay constants, and results are steadily improving in precision[211–216]. Charmed baryon decays have also be computed using a similar method [217, 218].
For semileptonic decays involving a bottom quark, one can use HQET or NRQCD to control
the discretization errors of the bottom quark. HQET is also used to constrain the extrapolation of
the lattice results obtained in the lower heavy-quark mass region using relativistic lattice fermion
actions.
The form factors for the semileptonic decay B → π`ν (and the similar decay Bs → K`ν) havebeen calculated in unquenched lattice QCD by a number of groups [219–228]. These B semileptonic
form factors are difficult to calculate at low q 2 , i.e. when the mass of the B-meson must be balanced
by a large pion momentum, in order to transfer a large momentum to the lepton pair. The low
q 2 region has large discretization errors and very large statistical errors, while the high q 2 region
is much more accessible to the lattice. For experiment, the opposite is true. To combine latticeand experimental results it has proved helpful to use the z-parameter expansion [143–146]. This
provides a theoretically constrained parameterization of the entire q 2 range, and allows one toobtain |Vub | without model dependence [229]. Analyticity and unitarity can be used to obtainfurther constraints, including extra information on susceptibilities relevant to the form factors in
the time-like region [146, 230–233].
The semileptonic decays B → D`ν and B → D∗ `ν (and the similar decays Bs → Ds `ν andBs → Ds∗ `ν) can be used to extract |Vcb | once the corresponding form factors are known. The
lattice calculation is most precise at zero recoil since the bulk of the systematic error cancels forappropriate ratios between B → D(∗) and B → B or D(∗) → D(∗) [234]. The unquenched calculationof the B → D(∗) `ν form factor at zero recoil has been performed with various formulations for theheavy quark [235]. Calculations at non-zero recoil have also now been performed to constrain thefunctional form of the form factor, which can be used to extrapolate the experimental data to the
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zero-recoil point or to determine |Vcb | directly at the non-zero recoil points [42,236–241]. The rangeof q 2 that can be handled for B-meson decays in lattice QCD is growing as more accurate results
become available on finer lattices. Comparison of the shape of the form factors between latticecalculations and experiments factors provides non-trivial cross checks. Semileptonic decays of theΛb baryon can also be used to constrain |Vcb | and |Vub | using lattice calculations of the relevant
form factors [242].
The rare decays B → K`+ `− involve matrix elements similar to those needed for semileptonic
decays, Eq. (17.14), except that the vector current Vµ is replaced by the operators s̄γ µ (1 − γ5 )b ors̄σ µν (1 + γ5 )b. Lattice calculations of the corresponding form factors involve similar techniques tothose for the semileptonic form factors. The range of q 2 for which lattice calculations can be done
is being extended, as above, but here there is an intermediate q 2 region where cc̄ resonances arepresent and this must be avoided. Recent lattice calculations [223, 243, 244] suggest some tension
with the experimentally observed decay rate. In these calculations, the long-distance contribution
from decay chain b → scc̄ → s`+ `− is ignored. Systematic errors including such long-distancecontributions need to be carefully studied before drawing any definite conclusions. A similar study
for baryonic decay mode Λb → Λ`+ `− can be found in Ref. [245].
The results discussed in this section are used in the reviews “The CKM Quark-Mixing Matrix,” “Vud , Vus , the Cabibbo Angle and CKM Unitarity,” and “Semileptonic b-hadron decays,
determination of Vcb , Vub .”17.3.4 Strong gauge coupling
As explained in Sec. 17.1.5.1, for a given lattice action, the choice of bare lattice gauge coupling,glat , determines the lattice spacing a. If one then calculates a as described in Sec. 17.1.5.1, one
2 /(4π). Thisknows the strong gauge coupling in the bare lattice scheme at the scale 1/a, αlat = glat
is not, however, useful for comparing to results for αs obtained from other inputs, such as deep
inelastic scattering or jet shape variables. This is because the latter results give αs in the MS
scheme, which is commonly used in such analyses, and the conversion factor between the latticeand MS schemes is known to converge extremely poorly in perturbation theory. Instead, one mustuse a method which directly determines αs on the lattice in a scheme closer to MS.
Several such methods have been used, all following a similar strategy. One calculates a shortdistance quantity K both perturbatively (K PT ) and non-perturbatively (K NP ) on the lattice, andP
requires equality: K NP = K PT = ni=0 ci αsi . Solving this equation one obtains αs at a scale related
to the quantity being used. Often, αs thus obtained is not defined in the conventional MS scheme,
and one has to convert among the different schemes using perturbation theory. Unlike for the barelattice scheme, the required conversion factors are reasonably convergent. As a final step, one uses
the renormalization group to run the resulting coupling to a canonical scale (such as MZ ).
In the work of the HPQCD collaboration [246, 247], the short-distance quantities are Wilson
loops of several sizes and their ratios. These quantities are perturbatively calculated throughO(αs3 ) using the V -scheme defined through the heavy quark potential. The coefficients of evenhigher orders are estimated using the data at various values of a. In addition, this work obtains a
result for αs by matching with αlat in a tadpole-improved scheme that improves convergence.
Another choice of short-distance quantities is to use current-current correlators. Appropriate
moments of these correlators are ultraviolet finite, and by matching lattice results to the continuum
perturbative predictions, one can directly extract the MS coupling [248]. The method can beapplied for light meson correlators [249–252] as well as heavy meson correlators [40, 247, 253–256].
Yet another choice of short-distance quantity is the static-quark potential, where the lattice result
for the potential is compared to perturbative calculations; this method was used to compute αs
within 2+1 flavor QCD [257–262]. There is also a determination of αs from a comparison of lattice
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data for the ghost-gluon coupling with that of perturbation theory [263, 264].
With a definition of αs given using the Schrödinger functional, one can non-perturbatively
control the evolution of αs to high-energy scales, such as 100 GeV, where the perturbative expansion
converges very well. This method developed by the ALPHA collaboration [106] has been applied
to 2+1-flavor QCD in [265–267].
The various lattice methods for calculating αs have significantly different sources of systematic(5)
error. The FLAG review [85] reports an estimate αMS (MZ ) = 0.1184(8), based on Refs. [247, 252,
254, 262, 265, 267–269]. A comparison to other phenomenological determinations can be found in
the “Quantum Chromodynamics” review.
17.3.5 Quark masses
Once the quark mass parameters are tuned in the lattice action, the remaining task is to convert
them to those of the conventional definition. Since the quarks do not appear as asymptotic states
due to confinement, the pole mass of the quark propagator is not a physical quantity. Instead,one defines the quark mass after subtracting the ultra-violet divergences in some particular way.
The conventional choice is again the MS scheme at a canonical scale such as 2 or 3 GeV. Ratios
such as mc /ms and mb /mc are also useful as they are free from multiplicative renormalization (ina mass-independent scheme), and can be computed precisely in lattice QCD, with a precision of
0.2% for mc /ms [44] and 0.3% for mb /mc [77].
As discussed in Sec. 17.1.6.4, one must convert the lattice bare quark mass to that in the MS
scheme. Older calculations did so directly using low-order perturbation theory; most recent calculations use an intermediate NPR method (e.g., RI/MOM or RI/SMOM) which is then converted
to the MS scheme using three-loop perturbation theory (see, e.g., [41, 190, 270–274]).
Alternatively, one can use a definition based on the Schrödinger functional, which allows one to
evolve the quark mass to a high scale non-perturbatively [275, 276], where perturbative matching
to the MS scheme can be done accurately.
Other approaches available for heavy quarks are to match current-current correlators at shortdistances calculated on the lattice to those obtained in continuum perturbation theory in the MS
scheme [40,77,247,253–255], or to use HQET mass relations [273,277]. This has allowed an accurate
determination of mc and mb [104, 247, 254].
The ratio method for heavy quarks (discussed earlier) can also be used to determine mb [278].
Results are summarized in the review of “Quark Masses.”
17.3.6 Other physical quantities of interest
In some physics applications, one is interested in the two-point correlation function hOX (x)OY† (0)ifor all values of the separation x, not just its asymptotic form for large separations (which is used
to determine the hadron spectrum as sketched in Sec. 17.2.2). A topical example, related to theevaluation of the Standard Model contribution to the muon anomalous magnetic moment g − 2,is the hadronic vacuum polarization function Πµν (x) = hVµ (x)Vν (0)i and its Fourier transformΠµν (q 2 ). Since the lattice is in Euclidean space-time, only space-like momenta, q 2 = −Q2 < 0,are accessible. Nevertheless, this quantity is of significant interest. It determines the running of
the QED coupling constant αQED , for which non-perturbative effects arise from quark loops. First
lattice calculations appeared recently [279, 280]. It is also related by a dispersion relation to thecross section for e+ e− → hadrons, and is needed for a first-principles calculation of the “hadronic
vacuum polarization” contribution to the muon anomalous magnetic moment aµ . Following thepioneering work or Ref. [281], there have been many lattice calculations of this contribution. The
status as of March 2020 is summarized in the White Paper of the g − 2 Theory Initiative [282].
More recent calculations are those of Refs. [283–285].
Since the relevant scale is set by the muon mass mµ , this quantity is most sensitive to the
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low-energy region Q2 ' m2µ of Πµν (−Q2 ), where the long-range contribution of multi-body states
become relevant. The lattice calculation is challenging because of this and also because the necessary
precision is high (below 0.5%). Many systematic effects must be carefully studied and controlled inorder to achieve this precision, including finite volume errors [286], errors from the chiral-continuum
extrapolations [285], isospin breaking [287, 288], quark-line disconnected diagrams [288, 289], andQED corrections [288,290,291]. One lattice calculation has achieved the required sub-percent precision [283], a level that is comparable to the determination of the HVP contribution using dispersion
relations based on e+ e− scattering data (the “data-driven” approach). The lattice result is in tension with the data-driven approach by 2.1σ. In order to facilitate the comparison among different
lattice calculations with controlled systematic errors, the so-called “intermediate window quantity”
was introduced [288]. It essentially eliminates short-distance and long-distance contributions to thehadronic vacuum polarization and focuses in the region where the lattice calculation provides thebest accuracy. The results from various lattice collaborations [283, 291–293] are in good agreement
but disagree with the corresponding data-driven estimate at the level of 3.8σ. This discrepancy
between the lattice and data-driven results needs to be understood before drawing any conclusion
about new-physics contribution to the muon g − 2.
Calculations of the light-by-light scattering contribution to aµ have been performed. These involve the calculations of four-point correlation functions with various external momenta. Ingenious
methods to evaluate the contribution to aµ have been developed by two groups and the results arein good agreement [294].
There are other quantities for which lattice calculations can make a significant contribution toestablishing a quantitative understanding of hadronic processes. One example is the long-distancecontribution to the neutral kaon mass splitting, ∆MK . This also requires the evaluation of afour-point function, constructed from the two-point functions described above by the insertion of
two electroweak Hamiltonians [295, 296]. Another example is the long-distance contribution toK , which provides a correction to the contribution of the mixing parameter BK , and has also
be computed recently [297]. Rare kaon decays K → π`+ `− and K → πν ν̄ are also important
processes for which first lattice studies have appeared [298]. A related process is π 0 → e+ e− , which
has recently been studied on the lattice [299]. Radiative leptonic decays π → `νγ (∗) , K → `νγ (∗) ,and D → `νγ (∗) also include two operator insertions, i.e. an electroweak Hamiltonian and anelectromagnetic current, and similar techniques developed for the rare decays can be applied. First
lattice results have appeared [64, 300]. Finally, we note that similar lattice methods allow thecalculation of the γW box contribution to the radiative corrections to semileptonic decays of pions
and kaons [301]. The extension to neutron β decay is an important next step, as this promises toreduce the theoretical uncertainty in the extraction of |Vud | from neutron decays. Firs results haveappeared recently [302].
17.3.7 Other applications of lattice QCD
In this review we have concentrated on applications of LQCD that are relevant to the quantities
discussed in the Review of Particle Properties. We have not considered at all several other applications that are being actively pursued by simulations. Here we list the major such applications.
The reader can consult the aforementioned texts [2–4] for further details, as well as the proceedingsof recent lattice conferences [303], and several recent white papers [304]
LQCD can be used, in principle, to simulate QCD at non-zero temperature and density, andin particular to study how confinement and chiral-symmetry breaking are lost as T and µ (thechemical potential) are increased. For example, as T is increased at µ = 0, It is found that, for thephysical values of the quark masses, the deconfinement and chiral-symmetry-restoration transitions
are smooth crossovers, rather than phase transitions, and that they occur together. This is of
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relevance to heavy-ion collisions, the early Universe and neutron-star structure. In practice, finite
temperature simulations are computationally tractable and relatively mature, while simulations at
finite µ suffer from a “sign problem” and are at a rudimentary stage.
Another topic under active investigation is nucleon structure and inter-nucleon interactions.
The simplest nucleon matrix elements are calculable with a precision that is now starting to rivalthat for some mesonic quantities. Of particular interest are those of the axial current (leading togA ) and of the scalar density (with hN |s̄s|N i needed for dark matter searches), both of which arereviewed by FLAG [85]. Other such matrix elements provide information on the parton distribution functions (PDFs), including their low moments. Recently, methods to directly access PDFs
have been developed (see Ref. [305] for a recent review). Also, neutrino-nucleon scattering matrix
elements play a crucial role in the estimate of the expected event rate in the current and future
neutrino experiments, such as T2K and DUNE. The lattice computation of the matrix elements issimilar to those of the meson semi-leptonic decays and the lattice results are reaching the required
precision as summarized in [306].
Finally, we note that there is much recent interest in studying QCD-like theories with more
fermions, possibly in other representations of the gauge group (see, e.g., [307]). The main interest
is to find nearly conformal theories which might be candidates for “walking technicolor” models.

17.4 Outlook
While LQCD calculations have made major strides in the last decade, and are now playing animportant role in constraining the Standard Model, there are many calculations that could be donein principle but are not yet mature due to limitations in computational resources. As we moveto exascale resources (1018 floating point operations per second), the list of mature calculations
will grow. Examples that we expect to mature in the next few years are results for B meson andΛb baryon form factors covering the full range of q 2 ; results for excited hadrons, including quarkmodel exotics, at close to physical light-quark masses; results for structure functions and relatedparton distribution functions; results for a variety of nucleon matrix elements; K → ππ amplitudes(allowing a precise prediction of 0 / from the Standard Model); hadronic vacuum polarization
contributions to gµ −2, the running of αEM and αs (the status of the first of which was discussed
in Sec. 17.3.6); π → γγ and related amplitudes; long-distance contributions to K ↔ K mixing; thelight-by-light contribution to gµ −2; and determinations of long distance contributions to rare kaon
decays such as K → πν ν̄. There will also be steady improvement in the precision attained for themature quantities discussed above. As already noted, for several of these quantities, attaining thedesired precision will require the inclusion of isospin-breaking and electromagnetic effects.

17.5 Acknowledgments
We are grateful to Jack Laiho for his collaboration on previous editions of this review, and to
Christine Davies, Takashi Kaneko, and Stefan Meinel for comments and suggestions.
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