lqcd-physics-spectrum

name

lqcd-physics-spectrum

description

Lattice QCD physics reasoning skill. Starting from a correlator definition, derives its spectral decomposition: completeness relation, overlap factors, backward-state structure, and fit function templates for lqcd-analysis. Assumes the operator and correlator have already been identified, typically by lqcd-physics-correlator. Trigger on: spectral decomposition, excited-state contamination, two-point or three-point fit models, backward-propagating states, or "what fitting functions should I use?".

LQCD Physics Reasoning

Purpose

Given a physics observable (hadron mass, decay constant, form factor, ...), derive the complete chain:

Correlator(s) → Spectrum(s) → Fitting function(s)

so that downstream tools (analysis pipeline) know exactly what to compute.

Use this skill after the operator content and correlator definition are already known. If those are still missing, first use lqcd-physics-correlator.

Spectral decomposition

Once the correlator has been specified, one should understand its time dependence by inserting complete sets of energy eigenstates. This determines the fit function that the analysis pipeline (lqcd-analysis) will use.

Completeness relation

An interpolating operator $\mathcal{O}$ with quantum numbers $J^{PC}$ couples to all eigenstates carrying those quantum numbers:

$$\langle 0 | \mathcal{O} | n, \vec{p} \rangle = Z_n(\vec{p})$$

where $|n, \vec{p}\rangle$ is the $n$-th eigenstate (ordered by energy, $n = 0$ being the ground state) with three-momentum $\vec{p}$.

On a finite lattice with spatial volume $V = L^3$ and relativistic state normalization $\langle n, \vec{p} | m, \vec{q} \rangle = 2 E_n V, \delta_{\vec{p}\vec{q}},\delta_{nm}$, the completeness relation reads:

$$\mathbf{1} = |0\rangle\langle 0| + \sum_{n \geq 1}\sum_{\vec{p}} \frac{1}{2 E_n(\vec{p}),V};|n, \vec{p}\rangle\langle n, \vec{p}|$$

Two-point function

Insert completeness between $\mathcal{O}$ and $\mathcal{O}^\dagger$ in the two-point function $C_2(\vec{p};,t) = \langle \mathcal{O}(\vec{p},t),\mathcal{O}^\dagger(\vec{p},0)\rangle$.

At zero temperature ($T \to \infty$):

$$C_2(\vec{p};,t) = \sum_n \frac{|Z_n(\vec{p})|^2}{2 E_n(\vec{p})};e^{-E_n(\vec{p}),t}$$

(An overall volume factor may appear depending on whether one or both operators are momentum-projected; in practice it is absorbed into $Z_n$.)

At finite temporal extent $T$, backward-propagating contributions appear. Their sign depends on the boundary condition of the composite state:

• Mesons — composed of two anti-periodic quarks → effective periodic BC ($(-1)^2 = +1$):

$$C_2^\text{meson}(t) = \sum_n A_n\left(e^{-E_n t} + e^{-E_n(T-t)}\right)$$

• Baryons — composed of three anti-periodic quarks → effective anti-periodic BC ($(-1)^3 = -1$). With parity projector $P^+ = (1+\gamma_4)/2$, the forward state has positive parity and the backward state has negative parity (the opposite-parity partner):

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

Here the fit amplitude is defined as

$$A_n \equiv \frac{|Z_n|^2}{2 E_n}$$

absorbing all convention-dependent normalization factors into $Z_n$. The amplitude $A_n$ is always positive for physical states; its magnitude encodes how strongly the operator couples to the $n$-th state.

Key physics: smeared sources enhance $|Z_0|$ relative to excited-state overlaps $|Z_{n \geq 1}|$, producing a cleaner plateau in the effective mass. The spectral decomposition makes this quantitative: the excited-state contamination in the effective mass is proportional to $(A_1/A_0), e^{-\Delta E, t}$ where $\Delta E = E_1 - E_0$.

Three-point function

For a three-point function with current insertion $J$ at Euclidean time $\tau$ ($0 < \tau < t_\text{sep}$):

$$C_3(\tau,,t_\text{sep}) = \langle \mathcal{O}_f(\vec{p}f,,t\text{sep});J(\vec{q},,\tau);\mathcal{O}_i^\dagger(\vec{p}_i,,0)\rangle$$

Insert completeness on both sides of the current:

$$C_3(\tau,,t_\text{sep}) = \sum_{n,m} \frac{Z_n^f,(Z_m^i)^*}{4,E_n,E_m};\langle n | J | m \rangle;e^{-E_n(t_\text{sep}-\tau)},e^{-E_m,\tau}$$

The crucial factorization: each coefficient separates into three independent factors —

$$\underbrace{Z_n^f}{\text{sink overlap}} ;\times; \underbrace{\langle n | J | m \rangle}{\text{matrix element}} ;\times; \underbrace{(Z_m^i)^*}_{\text{source overlap}}$$

The overlap factors $Z_n$ are the same as those in the two-point function. This is what enables the simultaneous fit in lqcd-analysis:

• $C_2$ determines $E_n$ and $Z_n$ (or equivalently $A_n$)
• $C_3$, sharing $E_n$ and $Z_n$, determines the matrix elements $\mathcal{M}_{nm} \equiv \langle n | J | m \rangle$
• The ground-state matrix element $\mathcal{M}_{00}$ is the physics target

Thermal effects

For $t_\text{sep} \ll T$, backward-propagating contributions to the three-point function are exponentially suppressed and usually negligible. When $t_\text{sep}$ is not small compared to $T$, additional thermal terms appear and must be included in the fit model.

Summary: spectral decomposition → analysis handoff

Given the operator choice and boundary conditions, the spectral decomposition fully determines the fit function template:

Correlator	Fit function	Free parameters
$C_2^\text{meson}(t)$	$\sum_n A_n(e^{-E_n t} + e^{-E_n(T-t)})$	${E_n,,A_n}$
$C_2^{P^+\text{baryon}}(t)$	$\sum_n A_n^+ e^{-E_n^+ t} - \sum_n A_n^- e^{-E_n^- (T-t)}$	${E_n^+,,A_n^+,,E_n^-,,A_n^-}$
$C_3(\tau,t_\text{sep})$	$\sum_{n,m} B_{nm},e^{-E_n(t_\text{sep}-\tau)} e^{-E_m\tau}$	${E_n,,B_{nm}}$ with $B_{nm} \propto Z_n,\mathcal{M}_{nm},Z_m$

The analysis skill (lqcd-analysis) takes these templates as its fit models, using the energy-gap parametrization $E_n = \sum_{k=0}^{n}\Delta E_k$ with $\Delta E_k > 0$ to ensure proper state ordering.

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Common pitfalls

1. Periodic vs anti-periodic BC: Fermions use anti-periodic temporal boundary conditions, but the composite state's BC depends on the number of quarks: Mesons: (-1)² = +1 → C(t) ∝ e^{-mt} + e^{-m(T-t)} (cosh-like) Baryons: (-1)³ = -1 → backward state has opposite parity See the Spectral Decomposition section above for the fit function templates.