Lattice operators

Theory, part 3 of 3. Previous: The hypercubic group · Clebsch-Gordan coefficients

Operators and their irreps

We can now turn to the problem of building all the operators that are relevant for physics applications, in particular for the moments of parton distribution functions. A generic operator of interest has the form

$$\mathcal{O}^{X}_{\nu \mu_1 \cdots \mu_n} = \bar\psi \, \Gamma^X_\nu \, \overleftrightarrow{D}_{\mu_1} \cdots \overleftrightarrow{D}_{\mu_n} \, \psi , \tag{1}$$

where ${\overleftrightarrow{D} = \overrightarrow{D} - \overleftarrow{D}}$ is the covariant derivative acting to the right minus the one acting to the left (a factor ${\tfrac{1}{2}}$ is also common: constant factors play no role here), and ${n}$ is the number of derivatives. The Dirac structure ${\Gamma^X}$ is one of those in the table below, and ${\nu}$ stands for its Lorentz indices: none for S and P, one for V and A, and the two indices of ${\sigma_{\nu_1 \nu_2}}$ for T.

X	Dirac structure ${\Gamma^X}$	Irrep of ${\Gamma^X}$	Code label	C-parity ${\eta_X}$ of ${\bar\psi \Gamma^X \psi}$
S	${\mathbf{1}}$	${\tau^{(1)}_1}$	(1, 1)	${+1}$
P	${\gamma_5}$	${\tau^{(1)}_4}$	(1, 4)	${+1}$
V	${\gamma_\nu}$	${\tau^{(4)}_1}$	(4, 1)	${-1}$
A	${\gamma_\nu \gamma_5}$	${\tau^{(4)}_4}$	(4, 4)	${+1}$
T	${\sigma_{\nu_1 \nu_2} = \tfrac{i}{2} [\gamma_{\nu_1}, \gamma_{\nu_2}]}$	${\tau^{(6)}_1}$	(6, 1)	${-1}$

To each covariant derivative ${\overleftrightarrow{D}_{\mu_i}}$ we associate the irrep ${\tau^{(4)}_1}$. The operators of Eq. (1) therefore span the tensor product of the irrep of ${\Gamma^X}$ with ${n}$ copies of ${\tau^{(4)}_1}$, and the Clebsch-Gordan coefficients of this product, ${c_{\nu \mu_1 \cdots \mu_n}}$, give the operators that transform according to definite irreps:

$$\mathcal{O} = \sum_{\nu, \mu_1, \dots, \mu_n} c_{\nu \mu_1 \cdots \mu_n} \, \mathcal{O}^{X}_{\nu \mu_1 \cdots \mu_n} , \tag{2}$$

one operator for each column of a Clebsch-Gordan block. For example, the vector operators with one derivative, ${\mathcal{O}_{\mu_1\mu_2} = \bar\psi \gamma_{\mu_1} \overleftrightarrow{D}_{\mu_2} \psi}$, are those of the explicit example of the previous page.

The complete catalogue of these operators, for S, P, V, A and T with up to three derivatives (1360 operators), is bundled with the library. It can be accessed with get_OperatorList, get_OperatorDict and Operator_from_database (see Operator database), or browsed as a PDF (see Operator catalogue). Operators with more derivatives are not precomputed, but they can be computed with the techniques outlined above, with make_operator_database.

For every operator, h4lat also gives its index symmetry (op.symm), whether its traces vanish (op.tr), its C-parity (op.C) and its kinematic factor (op.K), described below.

Kinematic factor

The kinematic factor ${K}$ of an operator is the tree-level factor that relates the nucleon matrix element of the operator to the corresponding moment. To compute it, h4lat first builds the Dirac matrix obtained by replacing every derivative with a component of the nucleon momentum,

$$M(p) = \sum_{\nu, \mu_1, \dots, \mu_n} c_{\nu \mu_1 \cdots \mu_n} \, \Gamma^X_\nu \, p_{\mu_1} \cdots p_{\mu_n} , \tag{3}$$

and then

$$K(p) = \frac{1}{2 E(p)} \, \frac{\mathrm{Tr}\left[ \Gamma_{\text{pol}} \left( -i \gamma_\rho p_\rho + m_N \right) M(p) \left( -i \gamma_\rho p_\rho + m_N \right) \right]} {\mathrm{Tr}\left[ \Gamma_{\text{pol}} \left( -i \gamma_\rho p_\rho + m_N \right) \right]} , \tag{4}$$

where:

• the ${c_{\nu \mu_1 \cdots \mu_n}}$ are the Clebsch-Gordan coefficients of the operator, Eq. (2);
• the ${\gamma_\mu}$ are the Euclidean gamma matrices, Hermitian, with ${\{\gamma_\mu, \gamma_\nu\} = 2\delta_{\mu\nu}}$, and the repeated index ${\rho}$ is summed over;
• ${p_\mu = (p_1, p_2, p_3, iE(p))}$ is the Euclidean four-momentum of the nucleon, with ${E(p) = \sqrt{m_N^2 + p_1^2 + p_2^2 + p_3^2}}$, so that ${p_\rho p_\rho = -m_N^2}$;
• ${\Gamma_{\text{pol}} = \tfrac{1}{2}(1 + \gamma_4)(1 - i \gamma_a \gamma_b)}$ is the polarisation projector: ${\tfrac{1}{2}(1 + \gamma_4)}$ projects onto positive energies, and the pair ${(a, b)}$ selects the spin polarisation. The default is ${(a, b) = (1, 2)}$; the six choices are listed in Gamma_pol_list, and op.set_polarization_matrix(k) switches an operator to choice k.

With this normalisation ${K = 1}$ for the vector charge ${\bar\psi \gamma_4 \psi}$ (operator 6 of the database), and ${K = m_N / E(p)}$ for ${\bar\psi \psi}$. Each derivative contributes only the momentum component ${p_\mu}$ to ${M(p)}$: constant factors per derivative, such as the ${i}$ in ${\partial_\mu e^{ipx} = i p_\mu e^{ipx}}$, are not included in ${K}$ (op.evaluate_K_real adds the factor ${i^n}$). ${K}$ is computed in exact arithmetic, as a SymPy expression in ${m_N}$, ${E(p)}$ and ${p_1, p_2, p_3}$.

C-parity

Under charge conjugation ${\mathcal{C}}$, the operators of Eq. (1) transform as

$$\mathcal{C} \, \mathcal{O}^{X}_{\nu \mu_1 \cdots \mu_n} \, \mathcal{C}^{-1} = \eta_X \, (-1)^n \, \mathcal{O}^{X}_{\nu \mu_n \cdots \mu_1} ,$$

i.e. charge conjugation reverses the order of the derivative indices and leaves those of ${\Gamma^X}$ in place. The sign ${\eta_X}$ is the C-parity of the local operator ${\bar\psi \Gamma^X \psi}$ (see the table above), and each derivative adds a factor ${-1}$. Since h4lat chooses the copies of every irrep even or odd under the reversal of the derivative indices (see Clebsch-Gordan coefficients), every operator of the bundled database has a definite C-parity, ${C = \pm 1}$ (op.C). For more context on C-parity we refer the reader to Göckeler et al. (1996).

In October 2026 the C-parity of the P operators was corrected: the pseudoscalar density ${\bar\psi \gamma_5 \psi}$ is C-even (as the ${\pi^0}$), so C_parity now gives ${\eta_X (-1)^n}$ with ${\eta_P = +1}$. P operators had not been bundled before, so no bundled operator changed.

Mixing under renormalisation

The classification by irrep and C-parity is important because of renormalisation. On the lattice, an operator can mix under renormalisation only with operators of equal or lower dimension (the dimension of ${\bar\psi \, \Gamma \, D^n \psi}$ is ${3 + n}$) that transform according to the same irrep and have the same C-parity. If the pair (irrep, C-parity) of an operator appears only once among the operators ${\bar\psi \, \Gamma \, D \cdots D \, \psi}$ of equal and lower dimension, the operator cannot mix with any of them, and its renormalisation is multiplicative. (Flavour-singlet operators can, in addition, mix with gluon operators.) Mixing with operators of lower dimension is particularly harmful, since its coefficients diverge as inverse powers of the lattice spacing.

The operators relevant for the moments of the unpolarised, helicity and transversity PDFs are those with ${\Gamma = \gamma_\mu, \gamma_\mu \gamma_5, \sigma_{\mu\nu}}$ (V, A and T). With the complete catalogue provided, it is easy to assess which irreps are free of mixing and to obtain explicit expressions for their operators:

• decomposition_analysis(X, n_der) lists the irreps of the product for the Dirac structure X with n_der derivatives, with their multiplicities, and marks an irrep as safe from mixing when it does not appear among the operators with fewer derivatives, for any of S, P, V, A and T. It does not take the C-parity into account, nor the operators of the same dimension.
• get_op_selection(n_der) returns a curated selection of V, A and T operators free of mixing, for one to three derivatives.
• get_paper_operators returns the operators used in published works (see Operators from the literature).

References

• M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. E. L. Rakow, G. Schierholz and A. Schiller, Lattice operators for moments of the structure functions and their transformation under the hypercubic group, Phys. Rev. D 54, 5705–5714 (1996), doi:10.1103/PhysRevD.54.5705, arXiv:hep-lat/9602029.

All the references of the wiki are collected on the References page.