01The path integral

How do we do field theory?

a brief intro to lattice QCD

Fields
Correlators
Masses
Matrix elements

01The path integral

Give the fields a grid

μ labels a direction; μ̂ is its unit vector.

Quark fields ψ live on sites; gauge links U connect neighbouring sites.

The link brings the neighbouring field into the same local colour frame before subtraction.

$\begin{gathered}\text{Ordinary forward difference}\\[5pt]\displaystyle\partial_\mu^{+}\psi(x)=\frac{\psi(x+a\hat\mu)-\psi(x)}{a}\end{gathered}$ $\begin{gathered}\text{Gauge-covariant forward difference}\\[5pt]\displaystyle D_\mu^{+}\psi(x)=\frac{{\color[HTML]{A64C32}U_\mu(x)}\psi(x+a\hat\mu)-\psi(x)}{a}\end{gathered}$

a is the resolution. L is the size of the box.

Gattringer & Lang · textbook Lüscher · lattice derivatives

01The path integral

A weighted average over fields

$S_E=\underbrace{S_g[U]}_{\text{gluons}}+\underbrace{a^4\sum_{x,f}\bar\psi_f(x)\bigl(D_f[U]\psi_f\bigr)(x)}_{\text{quarks and their coupling to gluons}}$ $\langle O\rangle=\frac{1}{Z}\int\mathcal{D}U\,\mathcal{D}\bar\psi\,\mathcal{D}\psi\;O\,e^{-S_E}$

01FieldsU, ψ and ψ̄

02Actioncombine gluon and quark terms

03Weightuse the action in the path integral

D_f contains the quark mass and gauge-covariant differences. Choose an observable O to ask a physics question.

Z is the same integral without O; it normalizes the average.

The path integral defines the observable before any sampling begins.

Gattringer & Lang · textbook Lüscher · lattice derivatives Lüscher · computational methods

01The path integral

From quark fields to a probability

$\int\mathcal{D}\bar\psi\,\mathcal{D}\psi\;e^{-S_F}\;\propto\;\prod_f\det D_f[U]$

S_F is the quark part of the action. Its Gaussian integral gives determinants.

$P[U]=\frac{1}{Z}\,e^{-S_g[U]}\prod_f\det D_f[U]$

Normalize the remaining gauge-field weight. We assume a nonnegative measure.

$\langle O\rangle=\int\mathcal{D}U\,P[U]\,O[U]$

Quark fields in O become propagators. Call the resulting function O[U].

Integrate the quarks analytically; sample the gauge fields numerically.

Lüscher · computational methods

01The path integral

Monte Carlo: sample, measure, average

$\langle O\rangle=\int\mathcal{D}U\,P[U]O[U]\;\simeq\;\frac{1}{N}\sum_{k=1}^{N}O[U_k],\qquad U_k\sim P$

One-dimensional Gaussian illustration · not lattice-QCD data

After integrating out the quarks, generate gauge configurations from the QCD probability P[U].

Importance sampling: regions with more probability receive more samples.

Measure O on each configuration, then take the arithmetic mean.

Sampling by probability turns the integral into an ordinary average.

Lüscher · computational methods

02A pion observable

Build the pion correlator from quark propagators

$O_\pi(x)=\bar d(x)\gamma_5u(x)$ $\begin{gathered}S_f(x,y;U)=\langle\psi_f(x)\bar\psi_f(y)\rangle_{\psi\mid U}\\[7pt]\displaystyle [D_f[U]S_f](x,y)=\delta_{xy}\,\mathbf{1}\end{gathered}$

Solve from source y on each gauge field U.

The trace closes the colour and spin indices; the brackets average over U.

$C_{2}=\langle O_\pi(x)O_\pi^\dagger(y)\rangle=\Big\langle\operatorname{Tr}_{c,s}\!\left[\gamma_5 S_u(x,y;U)\gamma_5 S_d(y,x;U)\right]\Big\rangle_U$

Solve for quark propagation, contract the indices, then average over gauge fields.

Gattringer & Lang · textbook Lüscher · computational methods

03Importance sampling

Evaluate the path integral

$\widehat C_2(t)=\frac{1}{N}\sum_{k=1}^{N}C_2[U_k](t)$

Schematic configurations

Generate gauge configurations distributed according to P[U].

Measure the contracted pion correlator on each saved configuration.

The sampling frequency already accounts for the weight.

Lüscher · computational methods

03Importance sampling

A correlator with statistical uncertainty

Each point is an ensemble mean; bars show its statistical uncertainty.

Below: percentage deviation from the exact toy curve, with the same errors.

Real Markov chains have autocorrelations; nearby times are correlated too.

More configurations improve statistics, not every systematic error.

Lüscher · computational methods

04Masses & physical limits

Euclidean time separates energies

$C_2(t)=\sum_n A_n e^{-E_n t}$

Illustrative two-state model · forward propagation only

Insert energy eigenstates between source and sink. Each contributes an exponential.

Higher energies decay faster with Euclidean separation. At p = 0, the lowest energy is the mass.

The long-time slope reveals the ground-state energy.

Gattringer & Lang · textbook

04Masses & physical limits

Visualizing data

$a m_{\mathrm{eff}}(t)=\log\!\frac{C_2(t)}{C_2(t+a)}$

Exact two-state toy model · $a m_\pi$ = 0.23

A single exponential gives a constant effective mass.

The approach to that constant exposes contributions from higher energies.

A plateau is a useful diagnostic; it is not the whole analysis.

Gattringer & Lang · textbook

04Masses & physical limits

Making sure it's right

01Change the fit windowvary the earliest included time

02Change the modelallow additional energy levels

03Change the overlapuse improved or multiple operators

The same energy should survive reasonable changes to the extraction.

Keep statistical correlations and fit uncertainty in the analysis.

Excited-state control is an analysis, not a time cut alone.

Gattringer & Lang · textbook Dürr et al. · hadron spectrum

04Masses & physical limits

Which quark masses did we simulate?

Illustrative quark-mass dependence · no measured data

The light-quark mass is a simulation input; $m_\pi$ helps locate the physical point.

Use physical-mass ensembles, or controlled interpolation/extrapolation from nearby masses.

If $m_\pi$ is a tuning input, reproducing it is not a prediction.

PACS-CS · physical-point tuning Dürr et al. · hadron spectrum

04Masses & physical limits

SI → natural units → lattice units

$\renewcommand{\arraystretch}{1.35}\begin{array}{l|ccc}&\text{SI}&\hbar=c=1&\substack{\text{Lattice units}\\\text{(dimensionless)}}\\\hline\text{Mass }m&\mathrm{kg}&\mathrm{GeV}&am\\\text{Energy }E&\mathrm{J}&\mathrm{GeV}&aE\\\text{Length }L&\mathrm{m}&\mathrm{GeV}^{-1}&L/a\\\text{Time }t&\mathrm{s}&\mathrm{GeV}^{-1}&t/a\end{array}$

$e^{-mc^2t/\hbar}\;\longrightarrow\;e^{-mt}\;=\;e^{-(am)(t/a)}$

The decay exponent is dimensionless in every convention.

$a^{-1}=\frac{M_{\mathrm{ref}}^{\mathrm{phys}}}{(aM_{\mathrm{ref}})_{\mathrm{latt}}},\qquad m^{\mathrm{phys}}=(am)_{\mathrm{latt}}\,a^{-1}$

Match a reference mass to fix the scale, then convert other lattice masses.

The simulation gives dimensionless numbers; scale setting gives them physical units.

PDG · physical constants PACS-CS · physical-point tuning

04Masses & physical limits

Remove the lattice spacing

Illustrative a² extrapolation · scaling is action dependent

Repeat at several lattice spacings, matching physical masses and volumes.

Fit the discretization dependence and propagate the extrapolation uncertainty.

a → 0 removes the regulator; it does not enlarge the box.

Dürr et al. · hadron spectrum

04Masses & physical limits

The box is another approximation

01Finer griddecrease a at fixed physical L

02Larger boxincrease L at fixed a

03Physical targetcontrol both effects

For a stable pion, compare volumes or use controlled finite-volume corrections.

The dimensionless product $m_\pi L$ is a useful guide, not a universal accuracy guarantee.

A physical result needs controlled mass, spacing and volume dependence.

Dürr et al. · hadron spectrum Gattringer & Lang · textbook

05Three-point functions

Insert a current between source and sink

$C_{3,\mu}(t,T)=\langle O_f(T)J_\mu(t)O_i^\dagger(0)\rangle$

Prepare pion quantum numbers at 0, insert J_μ at t, and detect them at T.

The current probes the hadron; it is not an emitted real photon.

Two propagation intervals surround one matrix element.

Boyle et al. · pion form factor

05Three-point functions

Isolate the matrix element

$C_3^\mu\ \propto\ e^{-E_f(T-t)}\langle\pi_f|J^\mu|\pi_i\rangle e^{-E_i t}$

Illustrative excited-state terms on both sides of the current

Two-point fits determine energies and overlaps. Three-point fits determine the current amplitude.

Both t and T − t must suppress unwanted states.

A plateau in the insertion time alone is not sufficient.

Boyle et al. · pion form factor

05Three-point functions

One current, one pion form factor

$\langle\pi(p_f)|J^\mu_{\mathrm{em}}|\pi(p_i)\rangle=(p_f+p_i)^\mu F_\pi(q^2)$

01Known kinematicsthe factor (p𝒻 + pᵢ)μ

02Hadron structurethe function Fπ(q²)

03Charge checkFπ(0) = 1 for π⁺

The form factor contains the QCD information left after the kinematic factor is removed.

Compare different momenta to determine its momentum-transfer dependence.

A matrix element becomes a function of a physical invariant.

Boyle et al. · pion form factor

06Kinematics

Choose the spatial momentum

$\mathbf p=\frac{2\pi}{L}(n_x,n_y,n_z),\qquad n_x,n_y,n_z\in\mathbb Z$

$\widetilde\psi(t,\mathbf p)=\sum_{\mathbf x}e^{-i\mathbf p\cdot\mathbf x}\psi(t,\mathbf x)$

Fourier-transform the quark field over space at a fixed time.

$\widetilde O_\pi(t,\mathbf p)=\sum_{\mathbf x}e^{-i\mathbf p\cdot\mathbf x}\underbrace{\bar d(t,\mathbf x)\gamma_5u(t,\mathbf x)}_{O_\pi(t,\mathbf x)}$

To select the pion’s total momentum, project the pion operator.

We choose spatial momentum; the state supplies its energy.

Boyle et al. · pion form factor

06Kinematics

The states fix both four-momenta

$\begin{aligned}p_i&=(E_{\pi,i},p_{i,x},p_{i,y},p_{i,z}),\\p_f&=(E_{\pi,f},p_{f,x},p_{f,y},p_{f,z}),\\E_{\pi,r}&=\sqrt{m_\pi^2+p_{r,x}^2+p_{r,y}^2+p_{r,z}^2}\quad(r=i,f)\end{aligned}$

Two-point functions give the energies; the continuum relation completes both four-momenta.

$\widetilde J_\mu(t,\mathbf q)=\sum_{\mathbf y}e^{-i\mathbf q\cdot\mathbf y}J_\mu(t,\mathbf y),\quad\mathbf q=\mathbf p_i-\mathbf p_f$

Fourier-project the current in three spatial dimensions at fixed time t.

$\begin{aligned}q^2&=(p_f-p_i)^2\\&=(E_{\pi,f}-E_{\pi,i})^2-|\mathbf p_f-\mathbf p_i|^2\end{aligned}$

The states fix the energy transfer; there is no independent energy to choose.

Choose two spatial momenta → determine two energies → obtain q².

Boyle et al. · pion form factor

06Kinematics

π–J–π: momenta → form factor

$L=6\,\mathrm{fm},\quad m_\pi=0.140\,\mathrm{GeV},\quad\mathbf p_{i,f}=\frac{2\pi}{L}\mathbf n_{i,f},\quad\mathbf n_i=(1,0,0)$

Illustrative form-factor values and error bars · calculated q² positions

$\begin{array}{c|r}\mathbf n_f&q^2\;[\mathrm{GeV}^2]\\\hline \color[HTML]{A64C32}(0,0,0)&-0.031\\[5pt]\color[HTML]{196B62}(1,0,0)&0.000\\[5pt]\color[HTML]{497DA0}(0,1,2)&-0.202\end{array}$

Fix the moving source; vary the sink.

The momenta fix q². The matrix element supplies the form factor.

Equal-mass, on-shell pions give q² ≤ 0 in every frame.

Boyle et al. · pion form factor

06Kinematics

B–J–π: same momenta, new q²

$L=6\,\mathrm{fm},\quad m_B=5.28\,\mathrm{GeV},\quad\mathbf n_B=(1,0,0),\quad E_B=\sqrt{m_B^2+|\mathbf p_B|^2}$

Illustrative form-factor values and error bars · calculated q² positions

$\begin{array}{c|r}\mathbf n_f&q^2\;[\mathrm{GeV}^2]\\\hline \color[HTML]{A64C32}(0,0,0)&26.418\\[5pt]\color[HTML]{196B62}(1,0,0)&25.346\\[5pt]\color[HTML]{497DA0}(0,1,2)&22.796\end{array}$

Keep the spatial momenta; replace the initial pion by B.

The vector current has two form factors; here we show the vector form factor.

The heavier initial state puts these same spatial choices at positive q².

Bailey et al. · B → π

07Nonleptonic decays

Two pions change the final-state problem

$\begin{aligned}H_W(t)&=a^3\sum_{\mathbf x}\mathcal H_W(t,\mathbf x),\\C_W(t,T)&=\langle O_{\pi\pi}(T)\,H_W(t)\,O_B^\dagger(0)\rangle\end{aligned}$

Hadronic schematic · weak vertex followed by strong rescattering

No external probe or lepton pair carries away four-momentum.

The weak interaction acts inside the hadronic process through four-quark operators.

We still insert the weak Hamiltonian in the lattice correlator.

The physical ππ final state must carry the full B four-momentum.

RBC/UKQCD · K → ππ Hansen–Sharpe · coupled channels

07Nonleptonic decays

The decay fixes the pion kinematics

$\begin{gathered}p_B=p_1+p_2,\qquad(p_1+p_2)^2=m_B^2\\[5pt]\text{B rest frame:}\quad E_1=E_2=\frac{m_B}{2},\quad |\mathbf p_1|=|\mathbf p_2|=\sqrt{\frac{m_B^2}{4}-m_\pi^2}\simeq2.64\,\mathrm{GeV}\end{gathered}$

01Physical decaythe ππ invariant mass equals the B mass

02Finite boxonly discrete interacting energy levels

03Long Euclidean timeselects the lowest state with overlap

Changing the frame does not change the required ππ invariant mass.

A low-energy ππ level is not the physical B decay. We need a level at the decay energy.

Match the decay energy, then relate the box matrix element to an outgoing amplitude.

Lellouch–Lüscher · finite-volume amplitudes Hansen–Sharpe · coupled channels

08Conclusions

From QCD to precision flavour physics

01CalculateCorrelators give masses and matrix elements

02ControlCheck statistics, quark masses, spacing and volume

03ConnectForm factors link measured rates to CKM parameters

Lattice QCD supplies the strong-interaction input that flavour measurements need.

Together, lattice calculations and experiment turn decay measurements into tests of the Standard Model.

From quark fields to hadrons — and from hadrons to precision flavour physics.

Boyle et al. · pion form factor Bailey et al. · B → π