auto_contractor.operators — Hadronic Operator Construction¶
Source: qlat/auto_contractor/operators.py
Note: Update this document when updating the source file.
Outline¶
Overview¶
operators builds symbolic hadronic operators for lattice QCD correlation
functions. It provides factory functions for mesons, baryons, currents, and
four-quark operators (including \(\Delta S = 1\) four-quark operators \(Q_1\)–\(Q_{10}\)
in both standard and \((8,1)\) representations). All operators are constructed
using Qb, Qv, G, and Bfield from the wick module and support the
is_dagger flag for Hermitian conjugation.
Pi and K operators follow Eq.(103,122) of Christ et al., Phys. Rev. D 101 (2020) 014506.
Quark Bilinears¶
Low-level bilinear constructors used by higher-level operators.
Function |
Definition |
Description |
|---|---|---|
|
\(\bar q_1 q_2(p)\) |
Scalar bilinear |
|
\(\bar q_1 \gamma_5 q_2(p)\) |
Pseudoscalar bilinear |
|
\(\bar q_1 \gamma_\mu q_2(p)\) |
Vector bilinear |
|
\(\bar q_1 \gamma_\mu\gamma_5 q_2(p)\) |
Axial-vector bilinear |
|
\(i\bar q_1 \gamma_5 q_2(p)\) |
Meson bilinear (with \(i\)) |
Meson Operators¶
Pseudoscalar Mesons¶
Function |
Particle |
Definition |
|---|---|---|
|
\(\pi^0\) |
\(\frac{i}{\sqrt{2}}(\bar u\gamma_5 u - \bar d\gamma_5 d)\) |
|
\(\pi^+\) |
\(i\bar u\gamma_5 d\) |
|
\(\pi^-\) |
\(-i\bar d\gamma_5 u\) |
|
\(K^+\) |
\(i\bar u\gamma_5 s\) |
|
\(K^-\) |
\(-i\bar s\gamma_5 u\) |
|
\(K^0\) |
\(i\bar d\gamma_5 s\) |
|
\(\bar K^0\) |
\(-i\bar s\gamma_5 d\) |
|
\(\eta_l\) |
\(\frac{i}{\sqrt{2}}(\bar u\gamma_5 u + \bar d\gamma_5 d)\) |
|
\(\eta_s\) |
\(i\bar s\gamma_5 s\) |
Scalar Mesons¶
Function |
Particle |
Definition |
|---|---|---|
|
\(a_0^0\) |
\(\frac{1}{\sqrt{2}}(\bar u u - \bar d d)\) |
|
\(a_0^+\) |
\(\bar u d\) |
|
\(a_0^-\) |
\(\bar d u\) |
|
\(\sigma\) |
\(\frac{1}{\sqrt{2}}(\bar u u + \bar d d)\) |
|
\(\kappa^+\) |
\(\bar u s\) |
|
\(\kappa^-\) |
\(\bar s u\) |
|
\(\kappa^0\) |
\(\bar d s\) |
|
\(\bar\kappa^0\) |
\(\bar s u\) |
Vector Mesons¶
Function |
Definition |
|---|---|
|
\(\bar u \gamma_\mu s\) |
|
\(\bar s \gamma_\mu u\) |
|
\(\bar d \gamma_\mu s\) |
|
\(\bar s \gamma_\mu d\) |
Two-Meson Operators¶
Pi-Pi¶
Isospin decompositions for two-pion operators:
Function |
Isospin |
Description |
|---|---|---|
|
\(I=2, I_z=+2\) |
\(\pi^+\pi^+\) |
|
\(I=2, I_z=+1\) |
\(\frac{1}{\sqrt{2}}(\pi^+\pi^0 + \pi^0\pi^+)\) |
|
\(I=2, I_z=0\) |
\(\frac{1}{\sqrt{6}}(2\pi^0\pi^0 + \pi^-\pi^+ + \pi^+\pi^-)\) |
|
\(I=1, I_z=+1\) |
\(\frac{1}{\sqrt{2}}(\pi^+\pi^0 - \pi^0\pi^+)\) |
|
\(I=1, I_z=0\) |
\(\frac{1}{\sqrt{2}}(\pi^+\pi^- - \pi^-\pi^+)\) |
|
\(I=0\) |
\(\frac{1}{\sqrt{3}}(-\pi^0\pi^0 + \pi^-\pi^+ + \pi^+\pi^-)\) |
K-K¶
Function |
Isospin |
Description |
|---|---|---|
|
\(I=1, I_z=+1\) |
\(K^+\bar K^0\) |
|
\(I=1, I_z=0\) |
\(\frac{1}{\sqrt{2}}(-K^0\bar K^0 + K^+K^-)\) |
|
\(I=0\) |
\(\frac{1}{\sqrt{2}}(K^0\bar K^0 + K^+K^-)\) |
|
— |
\(K^0\bar K^0\) |
K-Pi¶
Function |
Isospin |
Description |
|---|---|---|
|
\(I=1/2\) |
\(\frac{1}{\sqrt{3}}K^0\pi^0 + \frac{\sqrt{2}}{\sqrt{3}}K^+\pi^-\) |
|
\(I=1/2\) |
\(\frac{\sqrt{2}}{\sqrt{3}}K^0\pi^+ + \frac{1}{\sqrt{3}}K^+\pi^0\) |
|
\(I=3/2\) |
\(K^0\pi^-\) |
|
\(I=3/2\) |
\(-\frac{\sqrt{2}}{\sqrt{3}}K^0\pi^0 + \frac{1}{\sqrt{3}}K^+\pi^-\) |
|
\(I=3/2\) |
\(-\frac{1}{\sqrt{3}}K^0\pi^+ + \frac{\sqrt{2}}{\sqrt{3}}K^+\pi^0\) |
|
\(I=3/2\) |
\(K^+\pi^+\) |
Two-meson functions accept is_sym=True to symmetrize over momentum arguments.
Currents¶
Function |
Definition |
|---|---|
|
\(\bar d\gamma_\mu\gamma_5 u\) |
|
\(\bar s\gamma_\mu\gamma_5 u\) |
|
\(-\bar u\gamma_\mu\gamma_5 s\) |
|
\(\frac{1}{\sqrt{2}}(\bar u\gamma_\mu\gamma_5 u + \bar d\gamma_\mu\gamma_5 d)\) |
|
\(\bar s\gamma_\mu\gamma_5 s\) |
|
\(\bar d\gamma_\mu u\) |
|
\(\bar s\gamma_\mu u\) |
|
\(\frac{2}{3}\bar u\gamma_\mu u - \frac{1}{3}\bar d\gamma_\mu d - \frac{1}{3}\bar s\gamma_\mu s\) |
|
\(\frac{2}{3}\bar u\gamma_\mu u - \frac{1}{3}\bar d\gamma_\mu d\) |
|
\(-\frac{1}{3}\bar s\gamma_\mu s\) |
|
\(\frac{1}{\sqrt{2}}(\bar u\gamma_\mu u + \bar d\gamma_\mu d)\) (I=0) |
|
\(\frac{1}{\sqrt{2}}(\bar u\gamma_\mu u - \bar d\gamma_\mu d)\) (I=1) |
|
\(\bar u\gamma_\mu d\) (I=1) |
|
\(-\bar d\gamma_\mu u\) (I=1) |
Four-Quark Operators¶
Basic Structures¶
Function |
Structure |
|---|---|
|
\((\bar f_1\gamma_\mu f_2)(\bar f_3\gamma_\mu f_4)\) |
|
\((\bar f_1\gamma_\mu f_2)(\bar f_3\gamma_\mu\gamma_5 f_4)\) |
|
\((\bar f_1\gamma_\mu\gamma_5 f_2)(\bar f_3\gamma_\mu f_4)\) |
|
\((\bar f_1\gamma_\mu\gamma_5 f_2)(\bar f_3\gamma_\mu\gamma_5 f_4)\) |
|
\((\bar f_1 f_2)(\bar f_3 f_4)\) |
|
\((\bar f_1 f_2)(\bar f_3\gamma_5 f_4)\) |
|
\((\bar f_1\gamma_5 f_2)(\bar f_3 f_4)\) |
|
\((\bar f_1\gamma_5 f_2)(\bar f_3\gamma_5 f_4)\) |
|
VV - VA - AV + AA |
|
VV + VA - AV - AA |
\(\Delta S = 1\) Operators (\(Q_1\)–\(Q_{10}\))¶
Function |
Description |
|---|---|
|
Subtraction operator |
|
Standard \(\Delta S = 1\) four-quark operators |
|
\((8,1)\) representation operators |
Baryon Operators¶
Function |
Description |
|---|---|
|
Generic baryon from three quarks |
|
Proton (\(uud\)) |
|
Neutron (\(ddu\)) |
|
Spin-3/2 baryon |
|
Omega baryon (\(sss\)) |
spin values: "u", "d" for spin-1/2; "u3", "u1", "d1", "d3" for spin-3/2.
baryon_type: "std", "pos" for spin-1/2; "std3", "pos3" for spin-3/2.
Examples¶
import qlat as q
q.begin_with_mpi([[1, 1, 1, 4]])
from qlat.auto_contractor.operators import mk_pi_p, mk_pi_m, mk_k_0, mk_Q1
from qlat.auto_contractor.wick import contract_expr, simplified
# Build a kaon-to-pion weak matrix element operator product
expr = mk_pi_p("x1", is_dagger=True) * mk_Q1("x") * mk_k_0("x2")
c_expr = contract_expr(expr)
c_expr = simplified(c_expr)
print(c_expr.show())
q.end_with_mpi()