### 16.3.2 Leading-order Lagrangian

The derivation of the SCET Lagrangian follows similar steps as described for HQET in Section 16.2.1. One begins by deriving the Lagrangian for a theory containing only a single collinear sector. Similar to HQET, one separates the full QCD field into two components, $q_n(x) = \psi_n(x) + \eta_n(x)$, where (with $n \cdot \bar{n} = 2$)

$$ \psi_n(x) = \frac{\not{q}\not{q}}{4} q_n(x), \quad \eta_n(x) = \frac{\not{q}\eta\not{q}}{4} q_n(x). \qquad (16.11) $$

The degrees of freedom described by the field $\eta_n$ are far off shell and can therefore be eliminated using its equation of motion. This gives

$$ \mathcal{L}_n = \bar{\psi}_n(x) \left[ i\not\cdot D + i\not D^\perp \frac{1}{i\bar{n} \cdot D} i\not D^\perp \right] \frac{\not q}{2} \psi_n(x). \qquad (16.12) $$

As a next step, one separates the large and small momentum components by decomposing the collinear momentum into a “label” and a residual momentum, $p^\mu = P^\mu + k^\mu$ with $n \cdot P = 0$. One then performs a phase redefinition on the collinear fields, such that $\psi_n(x) = e^{-iP \cdot x} \xi_n(x)$. Derivatives acting on the fields $\xi_n(x)$ now only pick out the residual momentum. Since unlike in HQET the label momentum in SCET is not conserved, one defines a label operator $\mathcal{P}^\mu$ acting as $\mathcal{P}^\mu \xi_n(x) = P^\mu \xi_n(x)$ [56], as well as a corresponding covariant label operator $i\mathcal{D}_n^\mu = \mathcal{P}^\mu + gA_n^\mu(x)$. Note that at leading order in power counting $i\mathcal{D}_n^\mu$ does not contain the soft gluon field. This leads to the final SCET Lagrangian [56,57,59,60]

$$ \mathcal{L}_n = \bar{\xi}_n(x) \left[ i\not\cdot D_n + gn \cdot A_s + i\not D_n^\perp \frac{1}{i\bar{n} \cdot D_n} i\not D_n^\perp \right] \frac{\not q}{2} \xi_n(x) + \dots, \qquad (16.13) $$

where we have split $in \cdot D$ into a collinear piece $in \cdot D_n = in \cdot \partial + gn \cdot A_n$ and a soft piece $gn \cdot A_s$. This latter term gives rise to the only interaction between a collinear quark and soft gluons at leading power in $\lambda$. The ellipses represent higher-order interactions between soft and collinear particles.

The Lagrangian describing collinear fields in different light-like directions is simply given by the sum of the Lagrangians for each direction $n$, i.e. $\mathcal{L} = \sum_n \mathcal{L}_n$. The soft gluons are the same in each individual Lagrangian. An alternative way to understand the separation between large and small momentum components is to derive the Lagrangian of SCET in position space [60]. In this case no label operators are required, but the collinear field operators are separated by a light-like distance. An important difference between SCET and HQET is that the SCET Lagrangian is not corrected by short distance fluctuations. The physical reason is that in the construction described above no high-momentum modes have been integrated out [60]. Such hard modes arise when different collinear sectors are coupled via some external current (e.g. in jet production at $e^+e^-$ or hadron colliders), or when collinear particles are produced in the rest frame of a decaying heavy object (such as in B decays). Short-distance effects are then incorporated in the Wilson coefficients of the external source operators.

### 16.3.3 Collinear gauge invariance and Wilson lines

An important aspect of SCET concerns the implementation of local gauge invariance. Because the effective field operators describe modes with certain momentum scalings, the effective Lagrangian respects only residual gauge symmetries. One of them satisfies the collinear scaling

$$ (\bar{n} \cdot \partial, n \cdot \partial, \partial^{\perp}) U_n(x) \sim Q(1, \lambda^2, \lambda) U_n(x), \qquad (16.14) $$

and one the soft scaling

$$ (\bar{n} \cdot \partial, n \cdot \partial, \partial^{\perp}) U_s(x) \sim Q(\lambda^2, \lambda^2, \lambda^2) U_s(x). \qquad (16.15) $$

While the soft gauge transformation is common for all fields, collinear fields in different directions each transform under their own collinear gauge transformation. In other words, each collinear sector has to be separately gauge invariant under its collinear gauge transformation. This requires the introduction of collinear Wilson lines [56]

$$ W_n(x) = P \exp \left[ -ig \int_{-\infty}^{0} ds \bar{n} \cdot A_n(s\bar{n} + x) \right], \qquad (16.16) $$

which transform under collinear gauge transformations according to $W_n \to U_n W_n$. Thus, the combination $\chi_n \equiv W_n^\dagger \psi_n$ is gauge invariant. In a similar manner, one can define the gauge-invariant gluon field $B_n^\mu = g^{-1}W_n^\dagger iD_n^\mu W_n$ [61, 62]. Collinear operators in SCET are typically constructed from such collinearly gauge-invariant building blocks.

### 16.3.4 Derivation of factorization theorems

One of the important applications of SCET is to understand how to factorize cross sections or decay rates involving energetic particles moving in different directions into simpler quantities that can either be calculated perturbatively or determined from data. For the purpose of this discussion, we focus on factorization in the context of collider physics. Note that factorization theorems have been around for much longer than SCET (see [63] for a review). However, the effective theory allows for a conceptually simpler understanding of certain classes of factorization theorems [61], since most simplifications happen already at the level of the Lagrangian. The discussion in this section is valid to leading order in the power counting of the effective theory.

As discussed in the previous section, the Lagrangian of SCET does not involve any direct couplings between collinear particles moving in different directions. A process involving highly energetic initial- and/or final-state particles moving along different directions $n_i$ is therefore described in SCET by operators involving gauge-invariant collinear building blocks for each direction, which only interact amongst each other via soft gluon exchange. At leading power, soft gluons couple to collinear quarks only through the term $\bar{\xi}_n gn \cdot A_s(\not q/2) \xi_n$ in the effective Lagrangian in (16.13), which is similar to the coupling of soft gluons to heavy quarks in HQET, see Section 16.2.4. This coupling, and the corresponding coupling to collinear gluons can be removed by means of the field redefinitions [57]

$$ \psi_n(x) = Y_n(x) \psi_n^{(0)}(x), \quad A_n^a(x) = Y_n^{ab}(x) A_n^{b(0)}(x), \qquad (16.17) $$

where the soft Wilson lines $Y_n$ and $Y_n^{ab}$ live in the fundamental and adjoint representations of SU(3), respectively. Each collinear building block gives rise to a soft Wilson line. After this decoupling, the different collinear sectors no longer interact with each other and the soft interactions are entirely encoded in the product of soft Wilson lines. This is the basis of factorization proofs in SCET.

The simplest applications of this formalism concern inclusive processes, such as the Drell-Yan production of electroweak bosons in the threshold region, $pp \to Z/H + X_s$, where $X_s$ is a final state containing only soft particles, and event shape distributions in $e^+e^-$ collisions in the two-jet limit. For Drell-Yan the factorization formula is given by [61, 64]

$$ \begin{aligned} \sigma(pp \to ZX_s) &= \sum_q H_q(\mu) [f_{q/p}(\mu) f_{\bar{q}/p}(\mu)] \otimes S_F(\mu), \\ \sigma(pp \to HX_s) &= H_g(\mu) [f_{g/p}(\mu) f_{g/p}(\mu)] \otimes S_A(\mu), \end{aligned} \qquad (16.18) $$

where the collinear matrix elements give rise to the parton distributions $f_{a/p}(\mu)$, the soft function are vacuum matrix element of the soft Wilson lines in the fundamental or adjoint representation, and the hard functions are the squares of the Wilson coefficients of the relevant SCET operators. The convolution symbol indicates a convolution over the longitudinal momentum fractions of the colliding partons. The objects in the factorization theorem depend on the renormalization scale $\mu$, whereas the cross sections are scale independent. Solving the renormalization group equations (RGE) allows to resum large logarithms in the cross section (see below).

For event shapes such as thrust $T$, no initial-state partons are present, and the factorization formula takes the form (with $\tau = 1 - T$) [65, 66]

$$ \frac{d\sigma}{d\tau} = H(\mu) [J_n(\mu) J_{\bar{n}}(\mu)] \otimes S_{n\bar{n}}(\mu), \qquad (16.19) $$

where the jet functions are vacuum matrix elements of the outgoing collinear quark fields, which are convoluted with the soft
function $S_{n\bar{n}}$ defined as another vacuum matrix element of the soft Wilson lines.

For a more general high-energy scattering process such as $pp \to N$ jets, where the incoming protons move along directions $n_{a,b}$ and the jets define directions $n_1, \dots, n_N$, the factorization theorem for the differential cross section can be written as

$$d\sigma \sim \sum_{ab} H_{ab...N}(\mu) [B_{a/p}(\mu) B_{b/p}(\mu)] \otimes [J_1(\mu) \dots J_N(\mu)] \otimes S_{ab...N}(\mu). \qquad (16.20)$$

where the beam functions are generalized parton distributions [67] and the jet functions depend on the algorithm used to define the jets. It should be mentioned that the most difficult part of traditional factorization proofs involves showing that so-called Glauber gluons do not spoil the above factorization theorem [68]. The description of Glauber effects within SCET [69] has been developed in [70], where a closed form for the effective Lagrangian describing these interactions was derived. The formalism has been extended to Glauber quarks in [71]. In this context, a proof of factorization requires demonstrating that this Lagrangian has no impact on a particular cross section. If an observable is non-global, in the sense that energetic radiation is restricted to certain regions in phase space, the above factorization theorem needs to be generalized [72–74].

### 16.3.5 Resummation of large logarithms

Many processes of interest depend on multiple scales which can be widely separated. The perturbative series of such processes depends logarithmically on large scale ratios. For high-energy processes involving soft and collinear particles, there can be up to two powers of large logarithms for each power of the coupling constant. These double logarithms are referred to as Sudakov logarithms and can spoil the perturbative expansion.

SCET can be used to sum large logarithms $L$ using renormalization-group equations. This reorganizes the perturbative series in such a way that the product $\alpha_s L$ is treated as an $O(1)$ parameter. In many cases, one can write a perturbative series for the logarithm of a cross section, such that

$$\sigma \sim \exp[L g_0(\alpha_s L) + g_1(\alpha_s L) + \alpha_s g_2(\alpha_s L) + \dots], \quad (16.21)$$

with functions $g_n(x)$ that need to be determined. Note that each term in this series in the exponent is one power of $\alpha_s$ suppressed relative to the previous term.

The important ingredient in achieving this resummation is the fact that SCET factorizes a given cross section into simpler pieces, each of which depends on a single physical scale. The only dependence on that scale can arise through logarithms of its ratio with the renormalization scale $\mu$. Thus, for each of the components in the factorization theorem one can choose a renormalization scale $\mu$ for which the large logarithmic terms are absent. Of course, the factorization formula requires a common renormalization scale $\mu$ in all its components, and one therefore has to use the renormalization group (RG) to evolve the various component functions from their preferred scale to the common scale $\mu$. A novel feature of RG equations in SCET, as opposed to other EFTs, is that the anomalous dimensions entering the evolution equations of the hard, beam, jet and soft functions in a factorization formula such as (16.20) contain a single power of the logarithm of the relevant energy scale. For example, the anomalous dimension $\gamma_H$ of the hard function has the form

$$\gamma_H(\mu) = c_H \Gamma_{\text{cusp}}(\alpha_s) \ln \frac{Q^2}{\mu^2} + \gamma(\alpha_s), \qquad (16.22)$$

where $c_H$ is a process-dependent coefficient and $\Gamma_{\text{cusp}}$ denotes the so-called cusp anomalous dimension [22, 75]. Collinear and soft functions have similar anomalous dimensions, which also involve a cusp and a non-cusp part. The non-cusp part $\gamma$ of the anomalous dimensions is process (and observable) dependent. The presence of a logarithm in the anomalous dimension is characteristic of Sudakov problems and arises since the perturbative series contains double logarithms of scale ratios.

Solving the RG equations one can systematically resum all large logarithms of scale ratios in the factorized cross section and express the functions $g_n(\alpha_s L)$ introduced above in terms of ratios of running coupling constants. In order to compute the first two terms $L g_0(\alpha_s L) + g_1(\alpha_s L)$ in $\ln \sigma$, corresponding to the next-to-leading logarithmic (NLL) approximation, one needs two-loop expressions for the cusp anomalous dimension and $\beta$ function, one-loop expressions for the non-cusp pieces in the anomalous dimensions, and tree-level matching conditions for all component functions at their characteristic scales. To calculate the next term $\alpha_s g_2(\alpha_s L)$ in the expansion, corresponding to NNLL order, one needs to go one order higher in the loop expansion, and so on.

### 16.3.6 Factorization and resummation in SCET$_{II}$

The effective theory SCET$_{II}$ contains collinear and soft particles with momenta scaling as $(p_{\bar{n}}, p_{\bar{n}}^{\perp}, p_{n}^{\perp}) \sim Q(1, \lambda^2, \lambda)$ and $(p_{\bar{s}}, p_{\bar{s}}^{+}, p_{\bar{s}}^{\perp}) \sim Q(\lambda, \lambda, \lambda)$. They have the same small virtuality $(p_{\bar{n}}^{\perp} \sim p_{\bar{s}}^{2} \sim Q^2 \lambda^2)$ but differ in their rapidities. An important class of observables, for which this scaling is relevant, contains cross sections for processes in which the transverse momenta of particles are constrained by external kinematics. The prime example are the transverse-momentum distributions of electroweak gauge bosons or Higgs bosons produced at hadron colliders. The parton transverse momenta are constrained by the fact that their vector sum must be equal and opposite to the transverse momentum $q_T$ of the boson. Standard RG evolution in the effective theory controls the logarithms arising from the fact that the virtualities of the collinear and soft modes are much smaller than the hard scale $Q$ in the process (the boson mass). However, additional large logarithms arise since the rapidities of collinear and soft modes are parametrically different, such that $e^{|y_c - y_s|} \sim 1/\lambda$. These logarithms can be traced to a new source of divergences and an unusual failure of dimensional regularization. They need to be factorized in the cross section and resummed by other means.

Two approaches exist for how to deal with the additional rapidity logarithms in SCET. In the approach of [76], they are interpreted as a consequence of a “collinear anomaly” of the effective theory SCET$_{II}$, resulting from the fact that a classical rescaling symmetry of the effective Lagrangian is broken by quantum effects. The extra large logarithms can be resummed by means of simple differential equations, which typically state that, to all orders in perturbation theory (and in an appropriate space), the logarithm of the cross section contains only a single extra logarithm of $\lambda \sim q_T/Q$ not contained in the hard function. Another approach to resum the rapidity logarithms uses the “rapidity renormalization group” [77], in which the relevant differential equations are obtained by considering a new type of scale variation in a parameter $\nu$, which separates the phase space for collinear and soft particles along a hyperbola in the $(p_-, p_+)$ plane. In contrast to the standard RG, there is no running coupling involved in the $\nu$ evolution, since the different contributions live at the same virtuality.

As an important application, consider the Higgs-boson production cross section in gluon fusion at the LHC, defined with a jet veto stating that no jet in the final state has transverse momentum above a threshold $p_T^{\text{veto}}$, which can be factorized in the form [78,79]

$$\sigma(p_T^{\text{veto}}) = H(m_H, \mu) \left(\frac{\nu_B}{\nu_S}\right)^{-2F_{gg}(R, p_T^{\text{veto}}, \mu)} S_{gg}(R, p_T^{\text{veto}}, \mu, \frac{\nu_S}{p_T^{\text{veto}}}) \times \int_{\tau}^{1} \frac{dz}{z} B_{g/P}(z, R, p_T^{\text{veto}}, \mu, \frac{\nu_B}{m_H}) B_{g/P}(\tau, R, p_T^{\text{veto}}, \mu, \frac{\nu_B}{m_H}), \qquad (16.23)$$

where $\tau = m_H^2/s$, and $\mu \sim p_T^{\text{veto}}$ is a common factorization scale. The beam functions $B_{g/P}$, the soft function $S_{gg}$ and the exponent $F_{gg}$ all depend on the jet radius $R$ as well as the jet clustering algorithm. The scale dependence of the hard function $H$ is controlled by standard RG evolution in SCET. The beam functions can be factorized further into calculable collinear kernels convoluted with parton distribution functions. In addition to the renormalization scale $\mu$, the beam and soft functions depend on
two rapidity scales $\nu_B \sim m_H$ and $\nu_S \sim p_T^{\text{veto}}$, respectively. In [78] the default values $\nu_B = m_H$ and $\nu_S = p_T^{\text{veto}}$ are used for these scales, and the soft function $S_{gg}$ is absorbed into the beam functions. In [79] the exponent $F_{gg}$ is called $-\gamma_\nu g/2$. The second factor on the right-hand side of the factorization formula (16.23), which resums large rapidity logarithms, implies that the logarithm of the jet-veto cross section contains a single large logarithm $\ln \sigma = -2F_{gg}(R, p_T^{\text{veto}}, \mu) \ln(m_H/p_T^{\text{veto}}) + \dots$ not contained in the hard function. Its coefficient can be calculated in fixed-order perturbation theory.

SCET$_{II}$ also plays an important role in the study of factorization for a variety of exclusive B meson decays, such as $\bar{B} \to \pi l \bar{\nu}_\nu$, $\bar{B} \to K^*\gamma$ and $\bar{B} \to \pi \pi$, for which the virtualities of energetic (collinear) final-state particles are of order $\Lambda_{QCD}$, which is also the scale for the soft light degrees of freedom contained in the initial-state B meson.

### 16.3.7 Applications

Most of the applications of SCET are either in flavor physics, where the decay of a heavy B meson can give rise to energetic light partons, or in collider physics, where the presence of jets naturally leads to collimated sets of energetic particles. For some of these applications alternative approaches existed before the invention of SCET, but the effective theory has opened up alternative ways to understand the physics of these processes. For many examples, however, SCET has allowed new insights and new applications.

The investigation of heavy-to-light form factors has been instrumental for understanding factorization in exclusive semileptonic B decays [80]. SCET has also provided a field-theoretic basis for the QCD factorization approach to exclusive, non-leptonic decays of B mesons [81]. Using SCET methods, proofs of factorization were derived for the color-allowed decay $B^0 \to D^+\pi^-$ [82], the color-suppressed decay $B^0 \to D^0\pi^0$ [83], and the radiative decay $\bar{B} \to K^*\gamma$ [84]. In recent years, the factorization approach has been extended to include QED effects [85, 86]. Further examples are factorization theorems and the resummation of endpoint logarithms for quarkonia production [87], the resummation of large logarithmic terms for the thrust [88] and jet broadening [89] distributions in $e^+e^-$ annihilation beyond NLL order, the development of new factorizable observables to veto extra jets [67, 90], all-order factorization theorems for processes containing electroweak Sudakov logarithms [91], and the resummation of threshold (soft gluon) logarithms in momentum space for several important processes at hadron colliders [64, 92, 93]. There has also been a lot of activity describing $p_T$-based resummation at hadron colliders. Prominent examples are the transverse-momentum distributions of electroweak bosons [76, 77, 94], which give access to the $p_T$-dependent parton distribution functions. Higher-order resummations of event shape distributions [88], combined with two-loop fixed-order expressions [95, 96] and including nonperturbative effects by means of a shape function, have been used for an accurate determination of the strong coupling constant $\alpha_s$ [97, 98]. A method based on N-jettiness ($T_N$) slicing [99, 100] has provided a new way of performing the IR integrations needed for precise fixed-order calculations, using the fact that the singular dependence on $T_N$ can be calculated using SCET at NNLO. This has been used to compute various processes with final states containing up to one hard, colored particle [101–105]. Calculations of the leading power corrections in $T_0/Q$ [106, 107] have helped to improved the numerical stability of the method. The N-jettiness slicing method has also been used in the combination of higher-order resummation with parton showers [108–110]. Finally, SCET has given new insights into the jet substructure methods (see [111] for a review).

A novel type of SCET factorization theorem is needed to describe non-global observables, in which energetic radiation is limited to certain regions of phase space, e.g. inside jets. Jet events play a prominent role in collider physics, because they closely mirror the underlying short-distance dynamics. In the theoretical description of such observables one encounters so-called “non-global” logarithms (NGLs) [112, 113], whose resummation cannot be performed using standard techniques. Using SCET, this problem has been reduced to the solution of a RG evolution equation derived from a novel type of factorization theorem, in which the

hard and soft functions are operators in color space and in the (infinite) space of parton multiplicities [72, 74] (see also [73, 114]). In $e^+e^-$ collisions, solving the RGE in the large-$N_c$ limit using parton-shower techniques [115] gives a result that agrees with the solution of the BSM integral equation for the NGLs [116]. The first application of this formalism was to the light jet mass distribution [117], and the extension of the resummation to NLL accuracy has been accomplished in [118, 119]. Non-global observables at hadron colliders are yet more intricate, since Glauber-gluon exchange between the colliding partons lead to a new type of double logarithmic corrections in high orders of perturbation theory, the so-called superleading logarithms (SLLs) [120, 121]. Based on the corresponding SCET factorization theorem, the all-order resummation of SLLs has been accomplished for arbitrary $pp \to n$ jet processes [122].

Recent years have seen a strong effort to push the applications of SCET factorization and resummation techniques to next-to-leading power (NLP) in the expansion in $\lambda$. The subleading SCET Lagrangian [60, 123, 124] and current operators arising in B-meson decays and their anomalous dimensions [59, 60, 125–127] have been studied a long time ago. For important collider processes, such as Drell-Yan or Higgs production, the general sets of operators have been identified [128–130], and several of their anomalous dimensions have been calculated [130, 131]. Resummed results at subleading power have been presented for event shapes [132] and the Drell-Yan process [133]. More applications of SCET can be found in Section 6.4 of the review article [134].

## 16.4 Open issues and perspectives

HQET has successfully passed many experimental tests, and there are not many open questions that still need to be addressed. One concept that has not been derived from first principles is the notion of quark-hadron duality, which underlies the application of HQET to the description of inclusive decays of B mesons. The validity of global duality (at energies even lower than those relevant in B decays) has been tested experimentally using high-precision data on semileptonic B decays and on hadronic $\tau$ decays. However, assigning a theoretical uncertainty due to possible duality violations remains a difficult task. Another known issue is that the measured values of the CKM elements $|V_{ub}|$ and $|V_{cb}|$ extracted from exclusive or inclusive decays of B mesons differ from each other by several standard deviations (see Section 76). This measurement relies on the heavy-quark limit, and the uncertainty quoted includes a theoretical estimate of the effect of power corrections arising from the finite b-quark mass. It remains an open question whether the discrepancy is due to underestimated theoretical or experimental uncertainties.

SCET, on the other hand, is still an active field of research, and new results are being obtained regularly. In particular, it has been realized that at NLP SCET factorization theorems generically suffer from endpoint-divergent convolution integrals [135–140]. To remove these divergences, one uses exact D-dimensional refactorization conditions, which relate different terms in the NLP factorization theorem in the singular limits, to rearrange these terms in such a way that the endpoint divergences are systematically removed. Successful applications of this technique include the NLP factorization theorem for the b-quark induced contribution to the diphoton decay amplitude of the Higgs boson [139, 141] and the $gg \to H$ production amplitude [142], the “off-diagonal gluon thrust” in $e^+e^-$ collisions [143], and the NLP factorization theorems for some exclusive [144, 145] and inclusive [146] B-meson decays. On the other hand, for some other processes, such as electron-muon backward scattering [147], the factorization of endpoint singularities is still a subject of ongoing research.

Another active field concerns the further study and applications of Glauber gluons and quarks in SCET and their relation to the BFKL equation familiar from small-$x$ physics [114, 148]. The theoretical formalism developed in [70, 71] sets the basis for a solid understanding of the impact of Glauber exchanges on factorization theorems. Moreover, using the rapidity renormalization group one can derive the BFKL equation in the presence of a single Glauber exchange describing forward scattering [70, 148] and for small $x$ resummation in deep inelastic scattering [149]. Glauber gluons play an important role in SCET-based analysis of
jet propagation in dense QCD media [150–152], which gives rise to the jet-quenching phenomenon in heavy-ion collisions. They are also a crucial ingredient in studies of violations of collinear factorization at hadron colliders [69, 153–157].

We close this short review by mentioning a particularly nice application combining the methods of heavy-particle EFTs such as HQET and non-relativistic QCD with SCET in the context of describing the interactions of heavy dark matter (with mass $M \gg v$) with SM particles. In [158] it was realized that the interactions of heavy, weakly interacting massive particles (WIMPs) with nuclear targets can be described in a model-independent way using heavy-particle EFTs. The WIMPs are charged under $SU(2)_L$ and can interact with electroweak gauge bosons and the Higgs boson. The WIMP EFT was later extended by describing the produced, highly energetic electroweak gauge bosons in terms of soft or collinear fields in SCET [159–161]. This allows one to systematically separate all relevant mass scales, resum electroweak Sudakov logarithms and disentangle the so-called Sommerfeld enhancement from the short-distance hard annihilation process.

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# 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.)

<table><tbody><tr><td>17.1</td><td>Lattice regularization of QCD</td><td>350</td></tr><tr><td>17.1.1</td><td>Gauge invariance, gluon fields and the gluon action</td><td>350</td></tr><tr><td>17.1.2</td><td>Lattice fermions</td><td>351</td></tr><tr><td>17.1.3</td><td>Heavy quarks on the lattice</td><td>352</td></tr><tr><td>17.1.4</td><td>QED on the lattice</td><td>353</td></tr><tr><td>17.1.5</td><td>Basic inputs for lattice calculations</td><td>353</td></tr><tr><td>17.1.6</td><td>Sources of systematic error</td><td>354</td></tr><tr><td>17.2</td><td>Methods and status</td><td>355</td></tr><tr><td>17.2.1</td><td>Monte-Carlo method</td><td>355</td></tr><tr><td>17.2.2</td><td>Two-point functions</td><td>356</td></tr><tr><td>17.2.3</td><td>Three-point functions</td><td>356</td></tr><tr><td>17.2.4</td><td>Scattering amplitudes and resonances</td><td>357</td></tr><tr><td>17.2.5</td><td>Status of LQCD simulations</td><td>357</td></tr><tr><td>17.3</td><td>Physics applications</td><td>357</td></tr><tr><td>17.3.1</td><td>Spectrum</td><td>358</td></tr><tr><td>17.3.2</td><td>Decay constants and bag parameters</td><td>358</td></tr><tr><td>17.3.3</td><td>Form factors (K → πℓν, D → Kℓν, B → πℓν, B → D(*)ℓν, etc.)</td><td>358</td></tr><tr><td>17.3.4</td><td>Strong gauge coupling</td><td>359</td></tr><tr><td>17.3.5</td><td>Quark masses</td><td>359</td></tr><tr><td>17.3.6</td><td>Other physical quantities of interest</td><td>360</td></tr><tr><td>17.3.7</td><td>Other applications of lattice QCD</td><td>360</td></tr><tr><td>17.4</td><td>Outlook</td><td>360</td></tr><tr><td>17.5</td><td>Acknowledgments</td><td>361</td></tr></tbody></table>

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 of electroweak decay constants and form factors, which are needed to extract Cabbibo-Kobayashi-Maskawa (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, $g_μ-2$.

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.

## 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 growth of 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 definition of 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 lattice spacing $a$, with quark fields placed on sites and gauge fields on the links between sites. The lattice spacing 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 reached by 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 non-perturbative calculations by numerical evaluation of the path integral that defines the theory.

Practical LQCD calculations are limited by the availability of computational resources and the efficiency 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, from the use of non-zero values of $a$. There are also different ways in which the QCD action can be discretized, and all must give consistent results in the continuum limit, $a \to 0$. It is the purpose of this review to provide an outline of the methods of LQCD, with particular focus on applications to particle physics, and an overview of the various sources of error. This should allow the reader to better understand the LQCD results that are presented in other reviews, primarily those on “Quark Masses,” “Quark Model,” “Quantum Chromodynamics,” “CKM quark-mixing matrix,” “$V_{ud}$, $V_{us}$, Cabibbo angle and CKM Unitarity,” “Leptonic Decays of Charged Pseudoscalar Mesons,” “$B^0-B^0$ Mixing,” and “Semileptonic b-Hadron Decays, Determination of $V_{cb}$ and $V_{ub}$.” For more extensive explanations the reader should consult the available textbooks or lecture notes, the most up-to-date of 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 preservation of gauge invariance leads to considerable simplifications, e.g., restricting the form of operators that can mix under renormalization.

The gauge transformations of lattice quark fields are just as in the continuum: $q(x) \to V(x)q(x)$ and $\bar{q}(x) \to \bar{q}(x)V^\dagger(x)$, 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. $x = a(n_1, n_2, n_3, n_4)$ for a hypercubic lattice, with the $n_j$ being integers. Quark bilinears involving different lattice points can be made gauge invariant by introducing the gluon field $U_μ(x)$. For example, for adjacent points the bilinear is $\bar{q}(x)U_μ(x)q(x+a\hat{\mu})$, with $\hat{\mu}$ the unit vector in the $\mu$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 $A_μ$ which takes values in the Lie algebra. The bilinear is invariant if $U_μ$ transforms as $U_μ(x) \to V(x)U_μ(x)V^\dagger(x+a\hat{\mu})$. The lattice gluon field is naturally associated with the link joining $x$ and $x+a\hat{\mu}$, and corresponds in the continuum to a Wilson line connecting these two points, $P \exp(i \int_x^{x+a\hat{\mu}} dx_\mu A_\mu^{\text{cont}}(x))$ (where P indicates a path-ordered integral, and the superscript on $A_μ$ indicates that it is a continuum field). The trace of a product of the $U_μ(x)$ around any closed loop is easily seen to be gauge invariant and is the lattice version of a Wilson loop.

<figure><img src="image_78.png" /><figcaption>**Figure 17.1:** Sketch of a two-dimensional slice through the $\mu-\nu$ plane of a lattice, showing gluon fields lying on links and forming either the plaquette product appearing in the gauge action or a component of the covariant derivative connecting quark and antiquark fields.</figcaption></figure>

The simplest possible gauge action, usually called the Wilson gauge action, is given by the product of gauge links around ele-
mentary plaquettes:

$$S_g = \beta \sum_{x, \mu < \nu} \left[ 1 - \frac{1}{3} \mathrm{ReTr} [U_{\mu}(x) U_{\nu}(x+a\hat{\mu}) U_{\mu}^{\dagger}(x+a\hat{\nu}) U_{\nu}^{\dagger}(x)] \right]. \quad (17.1)$$

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 $U_{\mu}(x) = \exp(i\alpha A_{\mu}(x))$. Keeping only the leading non-vanishing term, and replacing the sum with an integral, one finds the continuum form,

$$S_g \rightarrow \int d^4 x \frac{1}{4g_{\text{lat}}^2} \text{Tr}[F_{\mu\nu}^2(x)], \quad (17.2) \\ (F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} + i[A_{\mu}, A_{\nu}])$$

as long as one chooses $\beta = 6/g_{\text{lat}}^2$ for the lattice coupling. In this expression, $g_{\text{lat}}$ is the bare gauge 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 $\overline{\text{MS}}$ scheme (see Sec. 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 as much 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 compared to the lattice cutoff ($|p| \ll 1/a$) 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 corrections to the continuum terms come in at $O(a^2)$. They take the form $\sum_j a^2 c_j O_6^{(j)}$, with the sum running over all dimension-six operators $O_6^{(j)}$ allowed by the lattice symmetries, and $c_j$ unknown coefficients. Some of these operators violate Euclidean rotational invariance, and all of them lead to discretization errors of the form $a^2 \Lambda^2$ (up to $\log(a)$ corrections that will be discussed below), where $\Lambda$ 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 coefficients to 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 the four links needed for the original plaquette action, Eq. (17.1)) to define the $O(a^2)$ improved (or “Symanzik”) action [6]. In practical implementations, the improvement is either at tree-level (so that residual errors are proportional to $\alpha_s a^2$, where the coupling is evaluated at a scale $\sim 1/a$), or at one-loop order (errors proportional to $\alpha_s^2 a^2$). Another popular choice is motivated by studies of renormalization group (RG) flow. It has the same terms as the $O(a^2)$ improved action but with different 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 used is more extensive than for gauge fields. Recall that the continuum fermion action is $S_f = \int d^4 x \bar{q}[D_\mu \gamma_\mu + m_q]q$, where $D_\mu = \partial_\mu + iA_\mu$ is the gauge-covariant derivative. The simplest discretization replaces the derivative with a symmetric difference:

$$D_{\mu} q(x) \rightarrow \frac{1}{2a} [U_{\mu}(x) q(x + a\hat{\mu}) - U_{\mu}(x - a\hat{\mu})^{\dagger} q(x - a\hat{\mu})]. \quad (17.3)$$

The factors of $U_\mu$ ensure that $D_\mu q(x)$ transforms under gauge transformations in the same way as $q(x)$, so that the discretized version of $\bar{q}(x)\gamma_\mu D_\mu q(x)$ is gauge invariant. The choice in Eq. (17.3) leads to the so-called naive fermion action. This, however, suffers from the fermion doubling problem—in $d$ dimensions it describes $2^d$ 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 $\delta q = \epsilon\gamma_5 q$, $\delta \bar{q} = \epsilon\bar{\gamma}\gamma_5$ ($\epsilon$ is infinitesimal) without producing doublers. Naive lattice fermions do have chiral 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 a term proportional to $a\bar{q}\Delta q$ to the fermion action (the “Wilson term”—in which $\Delta$ is a covariant lattice Laplacian). This gives a mass of $O(1/a)$ to the doublers, so that they decouple in the continuum 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 of great practical importance as it brings the discretization error from the fermion action down to the same 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 numerical instabilities due to spurious near-zero modes of the lattice Dirac operator. There are, however, 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 of this approach is that all errors linear in $a$ are automatically removed (without the need for tuning of parameters) by a clever choice of twisted mass and operators [15]. A disadvantage is the presence of isospin breaking effects (such as a splitting between charged and neutral pion masses even when up and down quarks are degenerate), which, however, vanish as $a^2 \Lambda^2$ 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 neighboring sites [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 $O(a^2)$. Their disadvantage is that they retain some of the doublers (3 for $d=4$). The action thus describes four degenerate fermions in the continuum limit. These are usually called “tastes”, to distinguish them from physical 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 from the 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, suggest that the procedure is valid as long as one takes the continuum limit before approaching the light quark 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 $O(a^2)$ errors and, to lowest order in the quark speed $v/c$, $O(\langle am \rangle^4)$ 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 a continuum-like chiral symmetry without introducing unwanted doublers. The lattice Dirac operator $D$ for these fermions satisfies the Ginsparg-Wilson relation $D\gamma_5 + \gamma_5 D = aD\gamma_5 D$ [28]. In the continuum, the right-hand-side vanishes, leading to chiral symmetry. On the lattice, it is non-vanishing, but with a particular form (with two factors of $D$) that restricts the violations of chiral symmetry in Ward-Takahashi identities to short-distance terms that do not contribute to physical matrix elements [29]. In fact, one can define a modified chiral transformation on the lattice (by including dependence on the gauge fields) such that Ginsparg-Wilson fermions have an exact chiral symmetry 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 $O(a^2)$.

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 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, and various approaches are being used. In fact, it is possible to rewrite these approximations in terms of a five-dimensional formulation, showing that the DWF and Overlap approaches are essentially equivalent [33, 34]. Numerically, the five-dimensional approach appears to be more computationally efficient.

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 adjacent links 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 class of applications (including the calculation of the $\Delta I = 1/2$ amplitude for $K \to \pi\pi$ 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, $m_q \lesssim 1/a$. This is because there are discretization errors proportional to powers of $am_q$, and

if $am_q \gtrsim 1$ these errors are large and uncontrolled. Present LQCD simulations typically have cutoffs in the range of $1/a = 2–5$ GeV (corresponding to $a \approx 0.1–0.04$ fm). Thus, while for the up, down and strange quarks one has $am_q \ll 1$, for bottom quarks (with $m_b \approx 4.5$ GeV) one must use alternative approaches. Charm quarks ($m_c \approx 1.5$ 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 lattice spacing and to improve the fermion action so as to reduce the size of errors proportional to powers of $am_c$. This approach has been followed successfully using the HISQ, twisted-mass and domain-wall actions [24, 25, 27, 39–41]. It is important to note, however, that reducing $a$ increases the computational cost because an increased number of lattice points are needed for the same physical volume. One cannot reduce the spatial size below 2–3 fm without introducing significant finite volume errors. Present lattices have typical sizes of $\sim 64^3 \times 128$ (with the long direction being Euclidean time), and thus allow a lattice cutoff up to $1/a \sim 4–6$ 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 lattices of size up to $144^3 \times 288$ and lattice spacings down to $a \approx 0.03$ fm ($1/a \approx 6.6$ GeV). Extrapolation in $m_b$ 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 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 quark-mass 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 $1/m_b$, 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 $1/m_b$ corrections.

Another way of introducing the $1/m_b$ 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 described by 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 be treated without any extrapolation in $m_b$. A disadvantage is that some of the parameters in this effective theory are determined perturbatively (at tree-level or at one-loop [49]), which limits the precision of the final results. Although discretization effects can be controlled with good numerical precision for a range of lattice spacings, these artifacts cannot be extrapolated away by taking the lattice spacing to zero. This is because NRQCD is a non-relativistic effective field theory and so ceases to work when the cutoff $\pi/a$ becomes much larger than the heavy-quark mass. In practice these 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 can tune the parameters of an improved Wilson quark action so that the leading HQET corrections to the static quark limit are correctly accounted for. As the lattice spacing becomes finer, the action smoothly 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 the clover 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 tuning of 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 $m_c$, which can be simulated with a relativistic action, and extrapolates to $m_b$ incorporating the behavior predicted by HQET. The particular implementation relies on the use of ratios. As an example, consider the B meson decay constant $f_B$. According to HQET, this scales as $1/\sqrt{m_B}$ for $m_B \gg \Lambda_{QCD}$, up to a logarithmic dependence that is calculable in perturbative QCD (but will be suppressed in the following). Here $m_B$ is the B meson mass, which differs from $m_b$ by $\sim \Lambda_{QCD}$. One considers the ratio $y(\lambda, m_{b'}) \equiv f_{B''}\sqrt{m_{B''}}/f_{B'}\sqrt{m_{B'}}$ for fictitious B mesons containing b quarks with unphysical masses $m_{b'}$ and $m_{b''} = \lambda m_{b'}$. HQET implies that $y(\lambda, m_{b'})$ approaches unity for large $m_{b'}$ and any fixed $\lambda > 1$. The ratios are evaluated on the lattice for the sequence of masses $m_{b'} = m_c, \lambda m_c, \lambda^2 m_c$, all well below the physical $m_b$, and for each the continuum limit is taken. The form of the ratio for larger values of $m_{b'}$ is obtained by fitting, incorporating the constraints implied by HQET. The result for $f_B\sqrt{m_B}$ is then obtained as a product of $y$'s with $f_D\sqrt{m_D}$ [58].

### 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 $O(\alpha_{EM})$. Thus, for example, the proton mass is increased by $\sim 1$ MeV relative to that of the neutron due to its overall charge although this effect is more than compensated for by the $\sim 2.5$ MeV relative decrease due to the up quark being lighter than the down quark [59]. This example shows that once pure QCD, isospin-symmetric lattice calculations 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 been obtained for various quantities, e.g., light hadron masses and decay constants (see Ref. [60]), and simulations 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 essential change is that the quark must now propagate through a background field containing both gluons and photons. The gauge field $U_\mu$ that appears in the covariant derivative of Eq. (17.3) is extended from an SU(3) matrix to one living in U(3): $U_\mu \to U_\mu e^{iqaeA_\mu^{EM}}$. Here $A_\mu^{EM}$ is the photon field, $e$ the 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 version of 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 $A_\mu^{EM}$, which simplifies the QED part 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 can be shown to be equivalent to removing the zero-momentum mode from the photon field, an approach denoted QED$_L$ [61]. However, this modification leads to enhanced finite-volume dependence for physical quantities, scaling as $1/L^n$ with $n = 1, 2, \dots$. This should be compared to the $\exp(-m_\mu L)$ 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 note that 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 the last 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 $\alpha_{EM}$ about the QCD only case [72, 73], with the latter approach by now the most popular. Most calculations to date have included QED effects for the valence quarks but not the sea quarks (the “electro-quenched approximation”). [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 a hadron 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 been reported for leptonic pion and kaon decays [79–81]. An application to semi-leptonic decays has been developed [64, 82].

### 17.1.5 Basic inputs for lattice calculations

Since LQCD is nothing but a regularization of QCD, the renormalizability of QCD implies that the number of input parameters in LQCD is the same as for continuum QCD—the strong gauge coupling $\alpha_s = g^2/(4\pi)$, the quark masses for each flavor, and the CP violating phase $\theta$. The $\theta$ 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 the inverse lattice spacing $1/a$, and choosing the bare gauge coupling is equivalent to fixing the lattice spacing.

In principle, $a$ can be determined using any dimensionful quantity measured accurately by experiments. For example, using the mass of hadron $H$ one has $a = (am_H)^{\text{lat}}/m_H^{\text{exp}}$. 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 the physical 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 $\Xi$ and $\Omega^{-}$ baryons, and the pion decay constant $f_\pi$. Ultimately, all choices must give consistent results for $a$, and that this is the case provides a highly non-trivial check of both the calculational method and of QCD.

Many recent lattice calculations use intermediate length scales in place of a direct determination of the lattice spacing. These length scales, which we denote $\mathcal{R}$, have the advantage that they can be precisely, and relatively cheaply, computed numerically. Examples are $r_0$, derived from the heavy quark potential [83], and $t_0$ and $w_0$, 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 $a^dQ$ and $\mathcal{R}/a$ in a given lattice calculation, and forms the product $(a^dQ) \times (\mathcal{R}/a)^d = Q\mathcal{R}^d$. In a second step, one uses results available from previous dedicated lattice calculations that have determined $\mathcal{R}$ in physical units (i.e. fm) by relating them to physical quantities as discussed above. Then one obtains $Q = (Q\mathcal{R}^d)/\mathcal{R}^d$. 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 light quarks, in the sense that $m_q < \Lambda_{QCD}$. (The standard definition of $\Lambda_{QCD}$ is given in the “Quantum Chromodynamics” review; in this review we are using it only to indicate the approximate non-perturbative scale of QCD.) This condition is stronger than that used above to distinguish quarks with small discretization errors, $m_q < 1/a$. 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 $m_u = m_d \equiv m_\ell < m_s$, with charm-quark loops either ex-
cluded (referred to as “$N_f = 2+1$” simulations) or included (de-
noted “$N_f = 2+1+1$” simulations). As noted above, precision is
now reaching the point where isospin breaking effects must be in-
cluded. To do so without approximation requires simulating with
non-degenerate up and down quarks (leading to $N_f = 1+1+1$ or
$1+1+1+1$ simulations) as well as including electromagnetism
(as described above).

We now describe the tuning of $m_\ell$, $m_s$ and $m_c$ to their physical values. (For brevity, we ignore isospin violation in the following discussion.) The most commonly used quantities for these tunings are, respectively, $m_\pi$, $m_K$ and $m_{\eta_c}$. If the scale is being set by $m_\Omega$, then one adjusts the lattice quark masses until the ratios $m_\pi/m_\Omega$, $m_K/m_\Omega$ and $m_{\eta_c}/m_\Omega$ take their physical values. In the past, most calculations needed to extrapolate to the physical value of $m_\ell$ (typically using forms based on chiral perturbation theory [ChPT]), while simulating directly at or near to the physical values of $m_s$ and $m_c$. Present calculations are increasingly done with physical or near physical values of $m_\ell$, 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 ~ $\alpha_s(m_b)\Lambda_{QCD}^2/m_b^2$ and are likely to be very small. In the past, the same approximation has been made for the *c* quark, leading to errors ~ $\alpha_s(m_c)\Lambda_{QCD}^2/m_c^2$. (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 ($\Upsilon$) or heavy-light (*B*) meson masses to their 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 calculation in order for the result to be
a useful input to phenomenology. The statistical error is due
to the use 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 in lattice calculations, although the size of any given error
depends on the particular quantity under consideration 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 auto-correlations in the sequence
of lattice configurations generated by the Monte Carlo method
can lead to underestimates of statistical errors [90]. It is known
that the global topological charge of the gauge 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 $O(\alpha_s a \Lambda)$, $O(a^2 \Lambda^2)$, or $O(\alpha_s a^2 \Lambda^2)$, where $\Lambda$ 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 in current state-of-the-art calcu-
lations. Residual errors arise from the exclusion of sub-leading *a*
dependence 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 this
have been undertaken [96–98].

For many quantities the typical momentum scale in the system
is ~ $\Lambda_{QCD} \approx 300$ MeV. Discretization errors are expected to be
larger for quantities involving larger scales, for example form fac-
tors or decays involving particles with momenta or masses larger
than $\Lambda_{QCD}$.

17.1.6.2 Infinite volume limit

LQCD calculations are necessarily carried out in finite space-
time boxes, leading to departures of 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 $N_s$ points in the three
spatial directions and $N_t$ in the (Euclidean) temporal direction.
The spatial and temporal sizes in physical units are thus $L_s = aN_s$
and $L_t = aN_t$, respectively. (Anisotropic lattice spacings are also
sometimes used, as discussed below in Sec. 17.2.2.) Typically,
$L_t \ge 2L_s$, a longer temporal direction being used to allow excited-
state 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 in-
volving no more than a single strongly-interacting particle in ini-
tial 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 from virtual pions wrapping around the
lattice in the spatial directions. This effect is exponentially sup-
pressed as the volume becomes large, roughly as ~ $\exp(-m_\pi L_s)$,
and has been estimated using ChPT [99] or other methods [100].
These estimates suggest that finite volume shifts are sub-percent
effects when $m_\pi L_s \gtrsim 4$, and most large-scale simulations use lat-
tices satisfying this condition. (See also Sec. 2.1.1 of the FLAG
report [85] for more detailed discussion concerning the conditions
required to control finite volume effects.) This becomes challeng-
ing as one approaches the physical pion mass, for which $L_s \gtrsim 5$ fm
is required.

Finite volume errors are usually determined by repeating the
simulations on two or more different volumes (with other param-
eters 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 truncated at 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 $K \to \pi\pi$ amplitudes.
Finite volume effects are much larger in these cases, with power-
law terms (e.g., $1/L_s^3$) in addition to exponential dependence. In-
deed, 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 $m_\pi L_s \gtrsim 4$ and is thus more challenging than
for single-particle quantities.

17.1.6.3 Chiral extrapolation

Until recently, an important source of systematic error in LQCD
calculations was the need to extrapolate in $m_u$ and $m_d$ (or, equiv-
alently, in $m_\pi$). This extrapolation was usually done using func-
tional forms based on ChPT, or with analytic functions, with the
difference between different fits used as an estimate of the sys-
tematic 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 un-
certainties in the extrapolation, such that this error is subdomi-
nant.

17.1.6.4 Operator matching

Many of the quantities that LQCD can precisely calculate in-
volve hadronic matrix elements of operators from the electroweak
Hamiltonian. Examples include the pion and kaon decay con-
stants, semileptonic form factors and the kaon mixing parameter
$B_K$ (the latter defined in Eq. (17.13)). The operators in the lattice matrix elements are defined in the lattice regularization scheme. To be used 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 which such matching is not needed is if the operator is a conserved or partially conserved current. Similar matching is also needed for the conversion of lattice bare quark masses to those in the continuum $\overline{MS}$ scheme.

Several methods are used to calculate the matching factors: perturbation theory (usually to one- or 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 be approximately estimated) by statistical and systematic errors that can be determined reliably and systematically reduced.

An issue that arises in some of such calculations (e.g., for quark masses and $B_K$) 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 $\overline{MS}$ 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 at which 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 $\overline{MS}$ 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 $\overline{MS}$ 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 the path integral formulation. The Euclidean-space partition function is

$$Z = \int [dU] \prod_f [dq_f][d\bar{q}_f] e^{-S_g[U] - \sum_f \bar{q}_f(D[U] + m_f)q_f}, \quad (17.4)$$

where link variables are integrated over the SU(3) manifold, $q_f$ and $\bar{q}_f$ are Grassmann (anti-commuting) quark and antiquark fields of flavor $f$, and $D[U]$ is the chosen lattice Dirac operator with $m_f$ the quark mass in lattice units. Integrating out the quark and antiquark fields, one arrives at a form suitable for simulation:

$$Z = \int [dU] e^{-S_g[U]} \prod_f \det(D[U] + m_f). \quad (17.5)$$

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

$$\langle O(U, q, \bar{q}) \rangle = (1/Z) \int [dU] \prod_f [dq_f][d\bar{q}_f] \mathcal{O}(U, q, \bar{q}) e^{-S_g[U] - \sum_f \bar{q}_f(D[U] + m_f)q_f}. \quad (17.6)$$

If the operators depend on the (anti-)quark fields $q_f$ and $\bar{q}_f$, then integrating these fields out leads not only to the fermion determinant but also, through Wick's theorem, to a series of quark “propagators”, $(D[U] + m_f)^{-1}$, connecting the positions of the fields.

This set-up allows one to choose, by hand, the masses of the quarks in the determinant (the sea 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.

### 17.2.1 Monte-Carlo method

Since the number of integration variables $U$ is huge ($N_s^3 \times N_t \times 4 \times 8$), 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) distributed according to the probability measure $[dU]e^{-S_g[U]} \prod_f \det(D[U] + m_f)$. Once the configurations are generated, expectation values $\langle O(U, q, \bar{q}) \rangle$ 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 $1/\sqrt{N}$.

The most challenging part of the generation of gauge configurations is the need to include the fermion determinant. Direct evaluation of the determinant is not feasible, as it requires $\mathcal{O}((N_s^3 \times N_t)^3)$ computations. Instead, one rewrites it in terms of “pseudo-fermion” fields $\phi$ (auxiliary fermion fields with bosonic statistics). For example, for two degenerate quarks one has

$$\det(D[U] + m_f)^2 = \int [d\phi] e^{-\phi^\dagger ((D[U] + m_f)(D[U] + m_f)^\dagger)^{-1} \phi}. \quad (17.7)$$

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 is highly non-local since it includes the inverse Dirac operator $(D[U] + m_f)^{-1}$. Thus, the large sparse matrix $(D[U] + m_f)$ 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 the rational 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 $(D[U] + m_f)$ 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 class of algorithms, computational cost is determined by the condition number of the matrix, which is the ratio of maximum and minimum eigenvalues. For $(D[U] + m_f)$ the smallest eigenvalue is $\approx m_f$, 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 $(D[U] + m_f)$ are associated with long wavelength 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 iterative solver, and the whole procedure may be nested multiple times. Variants of such methods have been 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 the Markov 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 some uncertainty in the resulting statistical errors, as well as the possibility of insufficient equilibration.

The computational cost of gauge generation grows with the lattice volume, $V_{\text{lat}} = N_s^3 N_t$, as $V_{\text{lat}}^{1+\delta}$. Here $\delta = 1/4$ for the HMC algorithm [126] and can be reduced slightly using modern variants. Such growth with $V_{\text{lat}}$ provides a (time-dependent) limit on the largest lattice volumes that can be simulated. At present, the largest lattices being used have $N_s = 144$ and $N_t = 288$ or $N_s = N_t = 160$. Typically, one aims to create an ensemble of $\sim 10^3$ statistically independent configurations at each choice of parameters ($a, m_q$ and $V_{\text{lat}}$). For most physical quantities of interest, this is sufficient to make the resulting statistical errors smaller than or comparable to the 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 of measurements 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 of quantities, as partly covered by the following sections.

### 17.2.2 Two-point functions

One can extract properties of stable hadrons from two-point correlation functions, $\langle O_X(x)O_Y^\dagger(0) \rangle$. Here $O_{X,Y}(x)$ are operators that have non-zero overlaps with the hadronic state of interest $|H\rangle$, i.e. $\langle 0|O_{X,Y}(x)|H\rangle \neq 0$. On the lattice, the two-point correlation function can be constructed, using Wick's theorem, from the quark propagators $(D[U] + m_f)_{x_0}^{-1}$ from the insertion point of the source operator $O_Y(0)$ to the sink operator $O_X(x)$. An average over gauge configurations yields the estimate of $\langle O_X(x)O_Y^\dagger(0) \rangle$.

One usually Fourier transforms in the spatial directions and considers correlators as a function of Euclidean time:

$$C_{XY}(t; \mathbf{p}) = \sum_{\mathbf{x}} \langle O_X(t, \mathbf{x}) O_Y^\dagger(0) \rangle e^{-i\mathbf{p} \cdot \mathbf{x}}. \quad (17.8)$$

(Here and throughout this section all quantities are expressed in dimensionless lattice units, so that, for example, $\mathbf{p} = a\mathbf{p}_{\text{phys}}$.) By inserting a complete set of states having spatial momentum $\mathbf{p}$, the two-point function can be written as

$$C_{XY}(t; \mathbf{p}) = \sum_{i=0}^{\infty} \frac{1}{2E_i(\mathbf{p})} \langle 0|O_X(0)|H_i(\mathbf{p})\rangle \langle H_i(\mathbf{p})|O_Y^\dagger(0)|0\rangle e^{-E_i(\mathbf{p})t}, \quad (17.9)$$

where the energy of the $i$-th state $E_i(\mathbf{p})$ appears in the eigenvalue of the Euclidean time evolution operator $e^{-Ht}$, $e^{-E_i(\mathbf{p})t}$. The factor of $1/[2E_i(\mathbf{p})]$ is due to the relativistic normalization used for the states. For large enough $t$, the dominant contribution is that of the lowest energy state $|H_0(\mathbf{p})\rangle$:

$$C_{XY}(t) \xrightarrow{t \to \infty} \frac{1}{2E_0(\mathbf{p})} \langle 0|O_X(0)|H_0(\mathbf{p})\rangle \langle H_0(\mathbf{p})|O_Y^\dagger(0)|0\rangle e^{-E_0(\mathbf{p})t}. \quad (17.10)$$

One can thus obtain the energy $E_0(\mathbf{p})$, which equals the hadron mass $m_H$ when $\mathbf{p} = 0$, and the product of matrix elements $\langle 0|O_X(0)|H_i(\mathbf{p})\rangle \langle H_i(\mathbf{p})|O_Y^\dagger(0)|0\rangle$.

This method can be used to determine the masses of all the stable (in QCD) mesons and baryons by making appropriate choices of operators. For example, if one uses the axial current, $O_X = O_Y = A_\mu = \bar{d}\gamma_\mu\gamma_5 u$, then one can determine $m_{\pi+}$ from the rate of exponential fall-off, and in addition the decay constant $f_\pi$ from the coefficient of the exponential.

The expression given in (17.9) for the correlator $C_{XY}(t; \mathbf{p})$ shows how, in principle, one can determine the energies of the excited hadron states having the same quantum numbers as the operators $O_{X,Y}$, by fitting the correlation function to a sum of exponentials, including both ground-state 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, $\langle 0|O|H\rangle$, with the different states, and to adopt a variational approach, such as that of [130]. Use of an anisotropic lattices in which $a_t$, the lattice spacing in the time direction, is smaller than its spatial counterpart, $a_s$, allows more temporal samples of the correlation function at small times where there is a signal for excited states. Using these and other technical improvements, extensive excited-state spectra have been obtained [131–136].

A complication arises for states with high spins ($j \ge 4$ for bosons) because the spatial rotation group 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, have overlap 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 be used 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 a representative example, the $D \to K$ amplitude. As in the case of two-point correlation functions one constructs operators $O_D$ and $O_K$ having overlap, respectively, with the $D$ and $K$ mesons. We are interested in calculating the matrix element $\langle K|V_\mu|D\rangle$, with $V_\mu = \bar{c}\gamma_\mu s$ the flavor-changing vector current.

To obtain this, we use the three-point correlator

$$C_{KV_\mu D}(t_x, t_y; \mathbf{p}) = \sum_{\mathbf{x}, \mathbf{y}} \langle O_K(t_x, \mathbf{x}) V_\mu(0) O_D^\dagger(t_y, \mathbf{y}) \rangle e^{-i\mathbf{p} \cdot \mathbf{x}}, \quad (17.11)$$

and focus on the limit $t_x \to \infty$, $t_y \to -\infty$. In this example we set the $D$-meson at rest while the kaon carries three-momentum $\mathbf{p}$. Momentum conservation then implies that the weak operator $V_\mu$ inserts three-momentum $-\mathbf{p}$. Inserting a complete set of states between each pair of operators, we find

$$C_{KV_\mu D}(t_x, t_y; \mathbf{p}) = \sum_{i,j} \frac{1}{2m_{D_i} 2E_{K_j}(\mathbf{p})} e^{-m_{D_i} t_x - E_{K_j}(\mathbf{p})|t_y|} \times \langle 0|O_K(0)|K_j(\mathbf{p})\rangle \langle K_j(\mathbf{p})|V_\mu(0)|D_i(0)\rangle \langle D_i(0)|O_D^\dagger(0)|0\rangle. \quad (17.12)$$

The matrix element $\langle K_i(\mathbf{p})|V_\mu(0)|D_j(0)\rangle$ can then be extracted, since all other quantities in this expression 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 the limit 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 extraction of the ground state is not always straightforward, especially when a significant contribution is expected from states with an extra pion. An example relevant to $B \to \pi\ell\nu$ decay is discussed in Ref. [139].

From (17.12) the matrix element $\langle K_i(\mathbf{p})|V_\mu(0)|D_j(0)\rangle$ is obtained at several discretized values of momentum $\mathbf{p}$, given by a multiple of $2\pi/L$ in each spatial direction. Each momentum corresponds to a certain value of the momentum transfer
$q^2 = (m_D - E_K(p))^2 - p^2$. The restriction on the available 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 a general $U(1)$ phase upon translation by distance $L$. The choice of phase leads to a quark momentum that does not have to be a multiple of $2\pi/L$. The method has been applied in a number of lattice calculations.

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 (where by 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 for resonances coupled to two-particle final states in the elastic regime, starting from the seminal work of 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 continuing from 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 to $p_E^2 = -m^2$ 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 $\pi\pi$ (for $I=2, 1$ and $0$), $\bar{K}K$, $K\pi$, $\pi\omega$, $\pi\phi$, $KD$, $DD^*$, $B\pi$, and $N\pi$. 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 and non-degenerate, and including spin) [154], and has been applied in lattice calculations of $3\pi^+$, $2\pi^+ + K^+$, $\pi^+ + 2K^+$, 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 two particles below the inelastic threshold, e.g., $A(K \to \pi\pi)$ [158]. Results for both the $\Delta I = 3/2$ and $1/2$ amplitudes with physical quark masses have been obtained [35–37,159], the former now including a controlled continuum limit [160]. First results for the CP-violating quantity $e'$ have been 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 $\gamma\pi \to \rho$ amplitude, albeit for unphysically large quark masses [164, 165], as well as $\gamma K \to K^*$ [166]. Finally, the formalism for using LQCD to calculate electroweak

decays or transitions to three particles, e.g. $\gamma^* \to 3\pi$ and $K \to 3\pi$, 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, 167, 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 are inserted 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 approach has been recently applied to provide a lattice determination of the inclusive semileptonic $\tau$ decay rate [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 the basic 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 ($m_\ell \sim 50-100$ MeV), such that the extrapolation to the physical light quark masses was a source of large systematic errors [176]. During the 2000s, advances in both algorithms and computers allowed simulations to reach much smaller quark masses ($m_\ell \sim 10-20$ MeV) such that LQCD calculations of 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 have become standard, removing the need for a chiral extrapolation and thus significantly reducing the overall 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 $g_{\text{lat}} \to \infty$ [1]. At weaker couplings there are precise numerical calculations of the potential that clearly show that this behavior persists in the 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 the continuum 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 the chiral 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].

## 17.3 Physics applications

In this section we describe the main applications of LQCD that are both computationally mature and relevant for the determination of particle properties.

A general feature to keep in mind is that, since there are many different choices for lattice actions, 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 and often 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 relevance for a variety of phenomenologically interesting quantities
and provide averages of those results that pass 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 well as the intermediate scale-setting quantities discussed earlier. The most recent FLAG review is from 2021 [85] (see also older editions, Refs. [60, 188]). The interested reader can consult this review for very 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 of light quarks (i.e. $N_f = 2+1$ 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 be predicted. This includes hadrons composed of light ($u, d$ and $s$) quarks, as well as heavy-light and heavy-heavy hadrons. It also includes quark-model exotics (e.g., $J^{PC} = 1^{-+}$ mesons) and glueballs. Thus, in principle, LQCD calculations should be able to reproduce many of the experimental results compiled in the Review of Particle Properties. Doing so would test both that the error budgets of 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 are most straightforward for stable, low-lying hadrons. Calculations of the properties of resonances that can decay into only two particles are more challenging, but are becoming standard in the meson 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 masses of flavor singlet states (which can annihilate into purely gluonic intermediate states) than those of flavor non-singlets, although again algorithmic and computational advances have begun to make such 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 of the octet light baryons, while results with less than complete control are available for the decuplet baryon resonances, the vector meson resonances and the $\eta$ and $\eta'$. 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 incorporation of QED). There are also extensive results for heavy-light ($D$ and $B$ systems) and heavy-heavy ($J/\psi$ and $\Upsilon$ systems). All present results, which are discussed in the “Quark Model” review, are consistent with experimental values, and several predictions have been made. We refer the reader to that review for references to the relevant work.

### 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 $f_P$ of a meson $P$ is extracted from the weak matrix element involving the axial-vector current using the definition $\langle 0|A_\mu(x)|P(p)\rangle = f_P p_\mu \exp(-ip \cdot x)$, where $p_\mu$ is the momentum of $P$ and $A_\mu(x)$ is the axial-vector current. (In practice, results with the smallest errors are obtained using the pseudo-scalar density $P(x) = \bar{q}(x)\gamma_5 q(x)$ instead of $A_\mu(x)$.) 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 the pion and kaon decay constants now have sub-percent errors. The decay constants in the charm and bottom sectors, $f_D$, $f_{D_s}$, $f_B$, and $f_{B_s}$, 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 for charmed 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, $B_K$, 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

$$ \frac{8}{3} m_K^2 f_K^2 B_K(\mu) = \langle \bar{K}^0 | Q_{\Delta S=2}(\mu) | K^0 \rangle, \quad (17.13) $$

where $m_K$ is the kaon mass, $f_K$ is the kaon decay constant, $Q_{\Delta S=2} = \bar{s}\gamma_\mu(1-\gamma_5)d\bar{s}\gamma_\mu(1-\gamma_5)d$ is the four-quark operator of the effective electroweak Hamiltonian and $\mu$ is the renormalization scale. The short distance contribution to the electroweak Hamiltonian can be calculated perturbatively, but the hadronic matrix element parameterized by $B_K$ 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 $\overline{MS}$, as described in Sec. 17.1.6.4. Determinations with percent-level precision using different fermion actions and $N_f = 2+1$ 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 $\hat{B}_K = 0.7625(97)$ [85] for its renormalization group invariant definition (for original papers, see [11, 192–194]).

The bag parameters for $B$ and $B_s$ meson mixing are defined analogously to that for kaon mixing. The $B$ and $B_s$ 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 $B_s$ and $B$ mixing with $N_f = 2+1$, which are based on results from Refs. [87, 195, 197], are $f_{B_s} \sqrt{\hat{B}_{B_s}} = 274(8)$ MeV and $f_B \sqrt{\hat{B}_B} = 225(9)$ MeV, with their ratio (which is somewhat better determined) being $\xi = 1.206(17)$. FLAG also quotes an “average” for $N_f = 2+1+1$, which comes from a single calculation [196] and gives $f_{B_s} \sqrt{\hat{B}_{B_s}} = 256(6)$ MeV, $f_B \sqrt{\hat{B}_B} = 211(6)$ MeV, and $\xi = 1.216(16)$. These are consistent with the $N_f = 2+1$ results at the $2\sigma$ 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 operators that 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 lattice calculations, 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.

The results for mixing matrix elements are used in the reviews “The CKM Quark-Mixing Matrix,” and “$B^0 - \bar{B}^0$ Mixing.”

### 17.3.3 Form factors ($K \to \pi\ell\nu$, $D \to K\ell\nu$, $B \to \pi\ell\nu$, $B \to D(*)\ell\nu$, 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 pseudo-scalar 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

$$ \langle D(p_D) | V_{\mu} | P(p_P) \rangle = f_{+}(q^2)(p_D + p_P - \Delta)_{\mu} + f_{0}(q^2)\Delta_{\mu}, \quad (17.14) $$

where $q = p_D - p_P$, $\Delta_\mu = (m_D^2 - m_P^2)q_\mu/q^2$ and $V_\mu$ 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 $f_0(q^2)$ is suppressed when the final state lepton is light.
The form factor $f_+(0)$ for $K \to \pi\ell\nu$ 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 and has now matched the precision of the phenomenological estimates [203–210]. The present FLAG average (from $N_f = 2 + 1 + 1$ 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 $B \to \pi\ell\nu$ (and the similar decay $B_s \to K\ell\nu$) have been 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 lattice and 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 to obtain $|V_{ub}|$ without model dependence [229]. Analyticity and unitarity can be used to obtain further constraints, including extra information on susceptibilities relevant to the form factors in the time-like region [146, 230–233].

The semileptonic decays $B \to D\ell\nu$ and $B \to D^*\ell\nu$ (and the similar decays $B_s \to D_s\ell\nu$ and $B_s \to D_s^*\ell\nu$) can be used to extract $|V_{cb}|$ once the corresponding form factors are known. The lattice calculation is most precise at zero recoil since the bulk of the systematic error cancels for appropriate ratios between $B \to D^*$ and $B \to B$ or $D^*(\to D^*)$ [234]. The unquenched calculation of the $B \to D^*(\to D^*)\ell\nu$ form factor at zero recoil has been performed with various formulations for the heavy quark [235]. Calculations at non-zero recoil have also now been performed to constrain the functional form of the form factor, which can be used to extrapolate the experimental data to the zero-recoil point or to determine $|V_{cb}|$ directly at the non-zero recoil points [42, 236–241]. The range of $q^2$ 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 lattice calculations and experiments factors provides non-trivial cross checks. Semileptonic decays of the $\Lambda_b$ baryon can also be used to constrain $|V_{cb}|$ and $|V_{ub}|$ using lattice calculations of the relevant form factors [242].

The rare decays $B \to K\ell^+\ell^-$ involve matrix elements similar to those needed for semileptonic decays, Eq. (17.14), except that the vector current $V_\mu$ is replaced by the operators $\bar{s}\gamma^\mu(1 - \gamma_5)b$ or $\bar{s}\sigma^{\mu\nu}(1 + \gamma_5)b$. Lattice calculations of the corresponding form factors involve similar techniques to those 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 $c\bar{c}$ resonances are present 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 $b \to s\bar{c}\bar{c} \to s\ell^+\ell^-$ is ignored. Systematic errors including such long-distance contributions need to be carefully studied before drawing any definite conclusions. A similar study for baryonic decay mode $\Lambda_b \to \Lambda\ell^+\ell^-$ can be found in Ref. [245].

The results discussed in this section are used in the reviews “The CKM Quark-Mixing Matrix,” “$V_{ud}$, $V_{us}$, the Cabibbo Angle and

CKM Unitarity,” and “Semileptonic b-hadron decays, determination of $V_{cb}$, $V_{ub}$.”

### 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, $g_{\text{lat}}$, determines the lattice spacing $a$. If one then calculates $a$ as described in Sec. 17.1.5.1, one knows the strong gauge coupling in the bare lattice scheme at the scale $1/a$, $\alpha_{\text{lat}} = g_{\text{lat}}^2/(4\pi)$. This is not, however, useful for comparing to results for $\alpha_s$ obtained from other inputs, such as deep inelastic scattering or jet shape variables. This is because the latter results give $\alpha_s$ in the $\overline{\text{MS}}$ scheme, which is commonly used in such analyses, and the conversion factor between the lattice and $\overline{\text{MS}}$ schemes is known to converge extremely poorly in perturbation theory. Instead, one must use a method which directly determines $\alpha_s$ on the lattice in a scheme closer to $\overline{\text{MS}}$.

Several such methods have been used, all following a similar strategy. One calculates a short-distance quantity $K$ both perturbatively ($K^{\text{PT}}$) and non-perturbatively ($K^{\text{NP}}$) on the lattice, and requires equality: $K^{\text{NP}} = K^{\text{PT}} = \sum_{i=0}^{n} c_i \alpha_s^i$. Solving this equation one obtains $\alpha_s$ at a scale related to the quantity being used. Often, $\alpha_s$ thus obtained is not defined in the conventional $\overline{\text{MS}}$ scheme, and one has to convert among the different schemes using perturbation theory. Unlike for the bare lattice 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 $M_Z$).

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 through $O(\alpha_s^3)$ using the V-scheme defined through the heavy quark potential. The coefficients of even higher orders are estimated using the data at various values of $a$. In addition, this work obtains a result for $\alpha_s$ by matching with $\alpha_{\text{lat}}$ 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 $\overline{\text{MS}}$ coupling [248]. The method can be applied 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 $\alpha_s$ within 2+1 flavor QCD [257–262]. There is also a determination of $\alpha_s$ from a comparison of lattice data for the ghost-gluon coupling with that of perturbation theory [263, 264].

With a definition of $\alpha_s$ given using the Schrödinger functional, one can non-perturbatively control the evolution of $\alpha_s$ 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 $\alpha_s$ have significantly different sources of systematic error. The FLAG review [85] reports an estimate $\alpha_{\overline{\text{MS}}}^{(5)}(M_Z) = 0.1184(8)$, 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 $\overline{\text{MS}}$ scheme at a canonical scale such as 2 or 3 GeV. Ratios such as $m_c/m_s$ and $m_b/m_c$ are also useful as they are free from multiplicative renormalization (in a mass-independent scheme), and can be computed precisely in lattice QCD, with a precision of 0.2% for $m_c/m_s$ [44] and 0.3% for $m_b/m_c$ [77].

As discussed in Sec. 17.1.6.4, one must convert the lattice bare quark mass to that in the $\overline{\text{MS}}$ scheme. Older calculations did so
directly using low-order perturbation theory; most recent calcula-
tions use an intermediate NPR method (e.g., RI/MOM or RI/S-
MOM) 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 short distances calculated on the
lattice to those obtained in continuum perturbation theory in
the MS scheme [40, 77, 243, 253–255], or to use HQET mass re-
lations [273, 277]. This has allowed an accurate determination of
$m_c$ and $m_b$ [104, 247, 254].

The ratio method for heavy quarks (discussed earlier) can also
be used to determine $m_b$ [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 $\langle O_X(x)O_Y^\dagger(0) \rangle$ for 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 the evaluation of the Standard Model
contribution to the muon anomalous magnetic moment $g-2$, is the
hadronic vacuum polarization function $\Pi_{\mu\nu}(x) = \langle V_\mu(x)V_\nu(0) \rangle$
and its Fourier transform $\Pi_{\mu\nu}(q^2)$. Since the lattice is in Eu-
clidean space-time, only space-like momenta, $q^2 = -Q^2 < 0$, are
accessible. Nevertheless, this quantity is of significant interest.
It determines the running of the QED coupling constant $\alpha_{QED}$,
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 the cross section for $e^+e^- \to$ hadrons,
and is needed for a first-principles calculation of the “hadronic
vacuum polarization” contribution to the muon anomalous mag-
netic moment $a_\mu$. Following the pioneering 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 $m_\mu$, this
quantity is most sensitive to the low-energy region $Q^2 \simeq m_\mu^2$
of $\Pi_{\mu\nu}(-Q^2)$, where the long-range contribution of multi-body
states become relevant. The lattice calculation is challenging be-
cause of this and also because the necessary precision is high (be-
low 0.5%). Many systematic effects must be carefully studied
and controlled in order to achieve this precision, including finite
volume errors [286], errors from the chiral-continuum extrapolapa-
tions [285], isospin breaking [287, 288], quark-line disconnected
diagrams [288, 289], and QED corrections [288, 290, 291]. One
lattice calculation has achieved the required sub-percent preci-
sion [283], a level that is comparable to the determination of the
HVP contribution using dispersion relations based on $e^+e^-$ scat-
tering data (the “data-driven” approach). The lattice result is in
tension with the data-driven approach by $2.1\sigma$. In order to fa-
cilitate 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 the hadronic vacuum
polarization and focuses in the region where the lattice calculation
provides the best accuracy. The results from various lattice col-
laborations [283,291–293] are in good agreement but disagree with
the corresponding data-driven estimate at the level of $3.8\sigma$. 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 $a_\mu$
have been performed. These involve the calculations of four-point
correlation functions with various external momenta. Ingenious
methods to evaluate the contribution to $a_\mu$ have been developed
by two groups and the results are in good agreement [294].

There are other quantities for which lattice calculations can
make a significant contribution to establishing a quantitative un-

derstanding of hadronic processes. One example is the long-
distance contribution to the neutral kaon mass splitting, $\Delta M_K$.
This also requires the evaluation of a four-point function, con-
structed from the two-point functions described above by the in-
sertion of two electroweak Hamiltonians [295,296]. Another exam-
ple is the long-distance contribution to $\epsilon_K$, which provides a cor-
rection to the contribution of the mixing parameter $B_K$, and has
also be computed recently [297]. Rare kaon decays $K \to \pi\ell^+\ell^-$
and $K \to \pi\nu\bar{\nu}$ are also important processes for which first lattice
studies have appeared [298]. A related process is $\pi^0 \to e^+e^-$,
which has recently been studied on the lattice [299]. Radiative
leptonic decays $\pi \to \ell\nu\gamma^{(*)}$, $K \to \ell\nu\gamma^{(*)}$, and $D \to \ell\nu\gamma^{(*)}$ also
include two operator insertions, i.e. an electroweak Hamiltonian
and an electromagnetic current, and similar techniques developed
for the rare decays can be applied. First lattice results have ap-
peared [64, 300]. Finally, we note that similar lattice methods
allow the calculation of the $\gamma W$ box contribution to the radia-
tive corrections to semileptonic decays of pions and kaons [301].
The extension to neutron $\beta$ decay is an important next step, as
this promises to reduce the theoretical uncertainty in the extrac-
tion of $|V_{ud}|$ from neutron decays. Firs results have appeared
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 Par-
ticle Properties. We have not considered at all several other appli-
cations that are being actively pursued by simulations. Here we
list the major such applications. The reader can consult the afore-
mentioned texts [2–4] for further details, as well as the proceed-
ings of 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, and in particular to study how con-
finement and chiral-symmetry breaking are lost as $T$ and $\mu$ (the
chemical potential) are increased. For example, as $T$ is increased
at $\mu = 0$, It is found that, for the physical values of the quark
masses, the deconfinement and chiral-symmetry-restoration tran-
sitions are smooth crossovers, rather than phase transitions, and
that they occur together. This is of relevance to heavy-ion colli-
sions, the early Universe and neutron-star structure. In practice,
finite temperature simulations are computationally tractable and
relatively mature, while simulations at finite $\mu$ 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 el-
ements are calculable with a precision that is now starting to
rival that for some mesonic quantities. Of particular interest are
those of the axial current (leading to $g_A$) and of the scalar den-
sity (with $\langle N|\bar{s}s|N \rangle$ needed for dark matter searches), both of
which are reviewed by FLAG [85]. Other such matrix elements
provide information on the parton distribution functions (PDFs),
including their low moments. Recently, methods to directly ac-
cess 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 is similar to those of the me-
son 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 represen-
tations 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 an important role in constraining the
Standard Model, there are many calculations that could be done
in principle but are not yet mature due to limitations in computa-
tional resources. As we move to exascale resources ($10^{18}$ 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 and $\Lambda_b$ baryon form factors covering the full range of $g^2$; results for excited hadrons, including quark-model exotics, at close to physical light-quark masses; results for structure functions and related parton distribution functions; results for a variety of nucleon matrix elements; $K \to \pi\pi$ amplitudes (allowing a precise prediction of $\epsilon'/\epsilon$ from the Standard Model); hadronic vacuum polarization contributions to $g_\mu-2$, the running of $\alpha_{EM}$ and $\alpha_s$ (the status of the first of which was discussed in Sec. 17.3.6); $\pi \to \gamma\gamma$ and related amplitudes; long-distance contributions to $K \leftrightarrow K$ mixing; the light-by-light contribution to $g_\mu-2$; and determinations of long distance contributions to rare kaon decays such as $K \to \pi\nu\bar{\nu}$. There will also be steady improvement in the precision attained for the mature quantities discussed above. As already noted, for several of these quantities, attaining the desired 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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18. Structure Functions

Revised August 2023 by E.C. Aschenauer (BNL), R.S. Thorne (UCL) and R. Yoshida (ANL).

18.1 Deep inelastic scattering

High-energy lepton-nucleon scattering plays a key role in de-
termining the partonic structure of the proton. The process
ℓN → ℓ'X is illustrated in Fig. 18.1. The filled circle in this
figure represents the internal structure of the proton which can
be expressed in terms of structure functions.

<figure><img src="image_79.png" /><figcaption>**Figure 18.1:** Kinematic quantities for the description of deep inelastic scattering. The quantities $k$ and $k'$ are the four-momenta of the incoming and outgoing leptons, $P$ is the four-momentum of a nucleon with mass $M$, and $W$ is the mass of the recoiling system $X$. The exchanged particle is a $\gamma$, $W^{\pm}$, or $Z$; it transfers four-momentum $q = k - k'$ to the nucleon.</figcaption></figure>

Invariant quantities:

$\nu = \frac{q \cdot P}{M} = E - E'$ is the lepton's energy loss in the nucleon rest frame (in earlier literature sometimes $\nu = q \cdot P$). Here, $E$ and $E'$ are the initial and final lepton energies in the nucleon rest frame.

$Q^2 = -q^2 = 2(EE' - \vec{k} \cdot \vec{k}') - m_{\ell}^2 - m_{\ell'}^2$ where $m_{\ell}(m_{\ell'})$ is the initial (final) lepton mass. If $EE' \sin^2(\theta/2) \gg m_{\ell}^2, m_{\ell'}^2$, then

$\approx 4EE' \sin^2(\theta/2)$, where $\theta$ is the lepton's scattering angle with respect to the lepton beam direction.

$x = \frac{Q^2}{2M\nu}$ where, in the parton model, $x$ is the fraction of the nucleon's momentum carried by the struck quark. Beyond leading order the equation remains the definition of $x$, but this is no longer identical to nucleon momentum fraction.

$y = \frac{q \cdot P}{k \cdot P} = \frac{\nu}{E}$ is the fraction of the lepton's energy lost in the nucleon rest frame.

$W^2 = (P+q)^2 = M^2 + 2M\nu - Q^2$ is the mass squared of the system $X$ recoiling against the scattered lepton.

$s = (k+P)^2 = \frac{Q^2}{xy} + M^2 + m_{\ell}^2$ is the center-of-mass energy squared of the lepton-nucleon system.

The process in Fig. 18.1 is called deep ($Q^2 \gg M^2$) inelastic ($W^2 \gg M^2$) scattering (DIS). In what follows, the masses of the initial and scattered leptons, $m_\ell$ and $m_{\ell'}$, are neglected.

18.1.1 DIS cross sections

The double-differential cross section for deep inelastic scattering can be expressed in terms of kinematic variables in several ways.

$$
\frac{d^2\sigma}{dx dy} = x(s - M^2) \frac{d^2\sigma}{dx dQ^2} = \frac{2\pi M\nu}{E'} \frac{d^2\sigma}{d\Omega_{\text{Nrest}} dE'} . \quad (18.1)
$$

In lowest-order perturbation theory, the cross section for the
scattering of polarized leptons on polarized nucleons can be ex-
pressed in terms of the products of leptonic and hadronic tensors
associated with the coupling of the exchanged bosons at the upper
and lower vertices in Fig. 18.1 (see Refs. [1–4])

$$
\frac{d^2\sigma}{dx dy} = \frac{2\pi y \alpha^2}{Q^4} \sum_j \eta_j L_j^{\mu\nu} W_{\mu\nu}^j . \quad (18.2)
$$

For neutral-current processes, the summation is over $j = \gamma, Z$
and $\gamma Z$ representing photon and $Z$ exchange and the interference

between them, whereas for charged-current interactions there is
only W exchange, $j = W$. (For transverse nucleon polarization,
there is a dependence on the azimuthal angle of the scattered
lepton.) The lepton tensor $L_{\mu\nu}$ is associated with the coupling
of the exchange boson to the leptons. For incoming leptons of
charge $e = \pm 1$ and helicity $\lambda = \pm 1$,

$$
\begin{align}
L_{\mu\nu}^{\gamma} &= 2 (k_{\mu} k_{\nu}^{\prime} + k_{\mu}^{\prime} k_{\nu} - (k \cdot k' - m_{\ell}^{2}) g_{\mu\nu} - i \lambda \epsilon_{\mu\nu\alpha\beta} k^{\alpha} k^{\beta}), \nonumber \\
L_{\mu\nu}^{\gamma Z} &= (g_{V}^{e} + e \lambda g_{A}^{e}) L_{\mu\nu}^{\gamma}, \quad L_{\mu\nu}^{Z} = (g_{V}^{e} + e \lambda g_{A}^{e})^{2} L_{\mu\nu}^{\gamma}, \nonumber \\
L_{\mu\nu}^{W} &= (1 + e \lambda)^{2} L_{\mu\nu}^{\gamma}, \tag{18.3}
\end{align}
$$

$$
\text{where } g_V^e = -\frac{1}{2} + 2 \sin^2 \theta_W, \quad g_A^e = -\frac{1}{2}.
$$

Although here the helicity basis is adopted, an alternative ap-
proach is to express the tensors in Eq. (18.3) in terms of the
polarization of the lepton.

The factors $\eta_j$ in Eq. (18.2) denote the ratios of the correspond-
ing propagators and couplings to the photon propagator and cou-
pling squared

$$
\begin{align}
\eta_{\gamma} &= 1 ; \quad \eta_{\gamma Z} = \left( \frac{G_F M_Z^2}{2\sqrt{2}\pi\alpha} \right) \left( \frac{Q^2}{Q^2 + M_Z^2} \right); \nonumber \\
\eta_{Z} &= \eta_{\gamma Z}^2 ; \quad \eta_{W} = \frac{1}{2} \left( \frac{G_F M_W^2}{4\pi\alpha} \frac{Q^2}{Q^2 + M_W^2} \right)^2 . \tag{18.4}
\end{align}
$$

The hadronic tensor, which describes the interaction of the ap-
propriate electroweak currents with the target nucleon, is given
by

$$
W_{\mu\nu} = \frac{1}{4\pi} \int d^4 z \, e^{iq \cdot z} \langle P, S | [J_{\mu}^{\dagger}(z), J_{\nu}(0)] | P, S \rangle, \quad (18.5)
$$

where $J_\alpha$ is the hadronic contribution to the electromagnetic, or
weak current and $S$ denotes the nucleon-spin 4-vector, with $S^2 =
-M^2$ and $S \cdot P = 0$.

18.2 Structure functions of the proton

The structure functions are defined in terms of the hadronic
tensor (see Refs. [1–3])

$$
\begin{align}
W_{\mu\nu} = & \left(-g_{\mu\nu} + \frac{q_\mu q_\nu}{q^2}\right) F_1(x, Q^2) + \frac{\hat{P}_\mu \hat{P}_\nu}{P \cdot q} F_2(x, Q^2) \nonumber \\
& - i \epsilon_{\mu\nu\alpha\beta} \frac{q^\alpha P^\beta}{2P \cdot q} F_3(x, Q^2) \nonumber \\
& + i \epsilon_{\mu\nu\alpha\beta} \frac{q^\alpha}{P \cdot q} \left[S^\beta g_1(x, Q^2) + \left(S^\beta - \frac{S \cdot q}{P \cdot q} P^\beta\right) g_2(x, Q^2)\right] \nonumber \\
& + \frac{1}{P \cdot q} \left[\frac{1}{2} \left(\hat{P}_\mu \hat{S}_\nu + \hat{S}_\mu \hat{P}_\nu\right) - \frac{S \cdot q}{P \cdot q} \hat{P}_\mu \hat{P}_\nu\right] g_3(x, Q^2) \nonumber \\
& + \frac{S \cdot q}{P \cdot q} \left[\frac{\hat{P}_\mu \hat{P}_\nu}{P \cdot q} g_4(x, Q^2) + \left(-g_{\mu\nu} + \frac{q_\mu q_\nu}{q^2}\right) g_5(x, Q^2)\right] \tag{18.6}
\end{align}
$$

where

$$
\hat{P}_{\mu} = P_{\mu} - \frac{P \cdot q}{q^2} q_{\mu}, \quad \hat{S}_{\mu} = S_{\mu} - \frac{S \cdot q}{q^2} q_{\mu}. \quad (18.7)
$$

In [2],the definition of $W_{\mu\nu}$ with $\mu \leftrightarrow \nu$ is adopted, which changes the sign of the $\epsilon_{\mu\nu\alpha\beta}$ terms in Eq. (18.6), although the formulae given below are unchanged. Ref. [1] tabulates the relation between the structure functions defined in Eq. (18.6) and other choices available in the literature.

The cross sections for neutral- and charged-current deep inelas-
tic scattering on unpolarized nucleons can be written in terms of
the structure functions in the generic form

$$ \frac{d^2\sigma^i}{dxdy} = \frac{4\pi\alpha^2}{xyQ^2} \eta^i \left\{ \left(1 - y - \frac{x^2y^2M^2}{Q^2}\right) F_2^i + y^2xF_1^i \mp \left(y - \frac{y^2}{2}\right) xF_3^i \right\}, \quad (18.8) $$

where $i = \text{NC}$, CC corresponds to neutral-current ($eN \to eX$) or charged-current ($eN \to \nu X$ or $\nu N \to eX$) processes, respectively. For incoming neutrinos, $L_{\mu\nu}^W$ of Eq. (18.3) is still true, but with $e, \lambda$ corresponding to the outgoing charged lepton. In the last term of Eq. (18.8), the – sign is taken for an incoming $e^+$ or $\bar{\nu}$ and the + sign for an incoming $e^-$ or $\nu$. The factor $\eta^{\text{NC}} = 1$ for unpolarized $e^\pm$ beams, whereas

$$ \eta^{\text{CC}} = (1 \pm \lambda)^2 \eta_W \quad (18.9) $$

with $\pm$ for $\ell^\pm$; and where $\lambda$ is the helicity of the incoming lepton and $\eta_W$ is defined in Eq. (18.4); for incoming neutrinos $\eta^{\text{CC}} = 4\eta_W$. The CC structure functions, which derive exclusively from W exchange, are

$$ F_1^{\text{CC}} = F_1^W, \quad F_2^{\text{CC}} = F_2^W, \quad xF_3^{\text{CC}} = xF_3^W. \quad (18.10) $$

The NC structure functions $F_2^\gamma, F_2^{\gamma Z}, F_2^Z$ are, for $e^\pm N \to e^\pm X$, given by [5],

$$ F_2^{\text{NC}} = F_2^\gamma - (g_V^e \pm \lambda g_A^e) \eta_\gamma Z F_2^{\gamma Z} + (g_V^{e2} + g_A^{e2} \pm 2\lambda g_V^e g_A^e) \eta_Z F_2^{Z} \quad (18.11) $$

and similarly for $F_1^{\text{NC}}$, whereas

$$ xF_3^{\text{NC}} = -(g_A^e \pm \lambda g_V^e) \eta_\gamma Z xF_3^{\gamma Z} + [2g_V^e g_A^e \pm \lambda(g_V^{e2} + g_A^{e2})] \eta_Z xF_3^Z \quad (18.12) $$

The polarized cross-section difference

$$ \Delta\sigma = \sigma(\lambda_n = -1, \lambda_\ell) - \sigma(\lambda_n = 1, \lambda_\ell), \quad (18.13) $$

where $\lambda_\ell, \lambda_n$ are the helicities ($\pm 1$) of the incoming lepton and nucleon, respectively, may be expressed in terms of the five structure functions $g_1, \dots, g_5(x, Q^2)$ of Eq. (18.6). Explicitly,

$$ \begin{aligned} \frac{d^2 \Delta \sigma^i}{dxdy} = \frac{8\pi\alpha^2}{xyQ^2} \eta^i & \left\{ -\lambda_\ell y \left(2 - y - 2x^2y^2\frac{M^2}{Q^2}\right) xg_1^i \right. \\ & + \lambda_\ell 4x^3y^2\frac{M^2}{Q^2}g_2^i + 2x^2y\frac{M^2}{Q^2}\left(1 - y - x^2y^2\frac{M^2}{Q^2}\right)g_3^i \\ & \left. - \left(1 + 2x^2y\frac{M^2}{Q^2}\right)\left[\left(1 - y - x^2y^2\frac{M^2}{Q^2}\right)g_4^i + xy^2g_5^i\right]\right\} \end{aligned} \quad (18.14) $$

with $i = \text{NC}$ or CC as before. The Eq. (18.13) corresponds to the difference of antiparallel minus parallel spins of the incoming particles for $e^-$ or $\nu$ initiated reactions, but the difference of parallel minus antiparallel for $e^+$ or $\bar{\nu}$ initiated processes. For longitudinal nucleon polarization, the contributions of $g_2$ and $g_3$ are suppressed by powers of $M^2/Q^2$. These structure functions give an unsuppressed contribution to the cross section for transverse polarization [1], but in this case the cross-section difference vanishes as $M/Q \to 0$.

Because the same tensor structure occurs in the spin-dependent and spin-independent parts of the hadronic tensor of Eq. (18.6) in the $M^2/Q^2 \to 0$ limit, the differential cross-section difference of Eq. (18.14) may be obtained from the differential cross section Eq. (18.8) by replacing

$$ F_1 \to -g_5, \quad F_2 \to -g_4, \quad F_3 \to 2g_1, \quad (18.15) $$

and multiplying by two, since the total cross section is the average over the initial-state polarizations. In this limit, Eq. (18.8) and Eq. (18.14) may be written in the form

$$ \begin{aligned} \frac{d^2\sigma^i}{dxdy} &= \frac{2\pi\alpha^2}{xyQ^2} \eta^i [Y_+ F_2^i \mp Y_- xF_3^i - y^2F_L^i], \\ \frac{d^2\Delta\sigma^i}{dxdy} &= \frac{4\pi\alpha^2}{xyQ^2} \eta^i [-Y_+ g_4^i \mp Y_- 2xg_1^i + y^2g_L^i], \end{aligned} \quad (18.16) $$

with $i = \text{NC}$ or CC, where $Y_\pm = 1 \pm (1-y)^2$ and

$$ F_L^i = F_2^i - 2xF_1^i, \quad g_L^i = g_4^i - 2xg_5^i. \quad (18.17) $$

In the naive quark-parton model, the analogy with the Callan-Gross relations [6] $F_L^i = 0$, are the Dicus relations [7] $g_L^i = 0$. Therefore, there are only two independent polarized structure functions: $g_1$ (parity conserving) and $g_5$ (parity violating), in analogy with the unpolarized structure functions $F_1$ and $F_3$.

### 18.2.1 Structure functions in the quark parton model

In the naive quark-parton model [8, 9], contributions to the structure functions $F^i$ and $g^i$ can be expressed in terms of the quark distribution functions $q(x, Q^2)$ of the proton, where $q = u, \bar{u}, d, \bar{d}$ etc. The quantity $q(x, Q^2)dx$ is the number of quarks (or antiquarks) of designated flavor that carry a momentum fraction between $x$ and $x+dx$ of the proton's momentum in a frame in which the proton momentum is large. One of the most striking predictions of the quark-parton model is that the structure functions $F_i, g_i$ scale, i.e., $F_i(x, Q^2) \to F_i(x)$ in the Bjorken limit that $Q^2$ and $\nu \to \infty$ with $x$ fixed [10]. This property is related to the assumption that the transverse momentum of the partons in the infinite-momentum frame of the proton is small.

<figure><img src="image_80.png" /><figcaption>**Figure 18.2:** The proton structure function $F_2^p$ given at two $Q^2$ values (6.5 GeV$^2$ and 90 GeV$^2$), which exhibit scaling at the 'pivot' point $x \sim 0.14$. See the captions in Fig. 18.10 and Fig. 18.12 for the references of the data. The various data sets have been renormalized by the factors shown in brackets in the key to the plot, which were globally determined in a previous HERAPDF analysis [11]. The curves were obtained using the PDFs from the HERAPDF analysis [12]. In practice, data for the reduced cross section, $F_2(x, Q^2) - (y^2/Y_+)F_L(x, Q^2)$, were fitted, rather than $F_2$ and $F_L$ separately. The agreement between data and theory at low $Q^2$ and $x$ can be improved by a positive higher-twist correction to $F_L(x, Q^2)$ [13, 14] (see Fig. 8 of Ref. [14]), or small-$x$ resummation [15, 16].</figcaption></figure>
For the neutral-current processes $ep \to eX$,

$$ 
\begin{aligned}
[F_2^\gamma, F_2^{\gamma Z}, F_2^Z] &= x \sum_q [e_q^2, 2e_q g_V^q, g_V^{q2} + g_A^{q2}] (q+\bar{q}), \\
[F_3^\gamma, F_3^{\gamma Z}, F_3^Z] &= \sum_q [0, 2e_q g_A^q, 2g_V^q g_A^q] (q-\bar{q}), \\
[g_1^\gamma, g_1^{\gamma Z}, g_1^Z] &= \frac{1}{2} \sum_q [e_q^2, 2e_q g_V^q, g_V^{q2} + g_A^{q2}] (\Delta q + \Delta \bar{q}), \\
[g_5^\gamma, g_5^{\gamma Z}, g_5^Z] &= \sum_q [0, e_q g_A^q, g_V^q g_A^q] (\Delta \bar{q} - \Delta q),
\end{aligned}
\quad (18.18) $$

where $g_V^q = \pm \frac{1}{2} - 2e_q \sin^2 \theta_W$ and $g_A^q = \pm \frac{1}{2}$, with $\pm$ according to whether $q$ is a u- or d-type quark respectively. The quantity $\Delta q$ is the difference $q\uparrow -q\downarrow$ of the distributions with the quark spin parallel and antiparallel to the proton spin.

For the charged-current processes $e^-p \to \nu X$ and $\bar{\nu}p \to e^+X$, the structure functions are:

$$ 
\begin{aligned}
F_2^{W-} &= 2x(u + \bar{d} + \bar{s} + c \dots), \\
F_3^{W-} &= 2(u - \bar{d} - \bar{s} + c \dots), \\
g_1^{W-} &= (\Delta u + \Delta \bar{d} + \Delta \bar{s} + \Delta c \dots), \\
g_5^{W-} &= (-\Delta u + \Delta \bar{d} + \Delta \bar{s} - \Delta c \dots),
\end{aligned}
\quad (18.19) $$

where only the active flavors have been kept and where CKM mixing has been neglected. For $e^+p \to \bar{\nu}X$ and $\nu p \to e^-X$, the structure functions $F^{W+}$, $g^{W+}$ are obtained by the flavor interchanges $d \leftrightarrow u, s \leftrightarrow c$ in the expressions for $F^{W-}$, $g^{W-}$. The structure functions for scattering on a neutron are obtained from those of the proton by the interchange $u \leftrightarrow d$. For both the neutral- and charged-current processes, the quark-parton model predicts $2x F_1^i = F_2^i$ and $g_4^i = 2x g_5^i$.

Neglecting masses, the structure functions $g_2$ and $g_3$ contribute only to scattering from transversely polarized nucleons, and have no simple interpretation in terms of the quark-parton model. They arise from off-diagonal matrix elements $\langle P, \lambda'|[J_\mu^i(z), J_\nu(0)]|P, \lambda \rangle$, where the proton helicities satisfy $\lambda' \neq \lambda$. In fact, the leading-twist contributions to both $g_2$ and $g_3$ are both twist-2 and twist-3, which contribute at the same order of $Q^2$. The Wandzura-Wilczek relation [17] expresses the twist-2 part of $g_2$ in terms of $g_1$ as

$$ g_2^i(x) = -g_1^i(x) + \int_x^1 \frac{dy}{y} g_1^i(y). \quad (18.20) $$

However, the twist-3 component of $g_2$ is unknown. Similarly, there is a relation expressing the twist-2 part of $g_3$ in terms of $g_4$. A complete set of relations, including $M^2/Q^2$ effects, can be found in [18].

### 18.2.2 Structure functions and QCD

In QCD, there are perturbative corrections to the partonic cross sections defining structure functions. Also, the radiation of hard gluons from the quarks violates the assumption that the transverse momentum of the partons is small, leading to logarithmic scaling violations, which are particularly large at small $x$, see Fig. 18.2. The radiation of gluons produces the evolution of the partons and structure functions. As $Q^2$ increases, more and more gluons are radiated, which in turn split into $q\bar{q}$ pairs. This process leads both to the softening of the initial quark momentum distributions and to the growth of the gluon density and the $q\bar{q}$ sea as $x$ decreases.

For $Q^2 \gg M^2$, the structure functions satisfy the factorization theorem [19],

$$ F_i = \sum_a C_i^a \otimes f_a + \mathcal{O}(M^2/Q^2), \quad (18.21) $$

where $\otimes$ denotes the convolution integral

$$ C \otimes f = \int_x^1 \frac{dy}{y} C(y) f\left(\frac{x}{y}\right), \quad (18.22) $$

and where the coefficient functions $C_i^a$, where $a = g$ or $q$, are given as a power series in $\alpha_s$. The scale-dependent parton distribution $f_a(x, \mu^2)$ corresponds, at a given $x$, to the density of parton $a$ in the proton integrated over transverse momentum $k_t$ up to the factorization scale $\mu$. Typically, $\mu$ is the scale of the probe $Q$. For parton distributions $x$ always refers to the nucleon momentum fraction of the parton, whereas for structure functions it retains the definition in Sec. 18.1. The parton evolution in $\mu$ is described in QCD by a DGLAP equation (see Refs. [20–23]) which has the schematic form

$$ \frac{\partial f_a}{\partial \ln \mu^2} \sim \frac{\alpha_s(\mu^2)}{2\pi} \sum_b (P_{ab} \otimes f_b), \quad (18.23) $$

where the $P_{ab}$, which describe the parton splitting $b \to a$, are also given as a power series in $\alpha_s$. Although perturbative QCD can predict, via Eq. (18.23), the evolution of the parton distribution functions from a particular scale, $\mu_0$, these DGLAP equations cannot predict them a priori at any particular $\mu_0$. Thus they must be measured at a starting point $\mu_0$ before the predictions of QCD can be compared to the data at other scales, $\mu$. In general, all observables involving a hard hadronic interaction (such as structure functions) can be expressed as a convolution of calculable, process-dependent coefficient functions and these universal parton distributions, e.g. Eq. (18.21).

<figure><img src="image_81.png" /><figcaption>Figure 18.3: Kinematic domains in $x$ and $Q^2$ probed by fixed-target and collider experiments, where here $Q^2$ can refer either the literal $Q^2$ for deep inelastic scattering, or the hard scale of the process in hadron-hadron collisions, e.g. invariant mass or transverse momentum $p_T^2$. Some of the final states accessible at the LHC are indicated in the appropriate regions, where $y$ is the rapidity. The incoming partons have $x_{1,2} = (Q/14 \text{ TeV})e^{\pm y}$ where $Q$ is the hard scale of the process shown in blue in the figure. For example, open charm production [24] and exclusive $J/\psi$ and $\Upsilon$ production [25] at high $|y|$ at the LHC may probe the gluon PDF down to $x \sim 10^{-5}$.</figcaption></figure>