Renormalization of quark propagator, vertex functions, and twist-2 operators from twisted-mass lattice QCD at $N_f=4$

We present a precise nonperturbative determination of the renormalization constants in the mass independent RI’-MOM scheme. The lattice implementation uses the Iwasaki gauge action and four degenerate dynamical twisted-mass fermions. The gauge configurations are provided by the ETM Collaboration. Renormalization constants for scalar, pseudoscalar, vector and axial operators, as well as the quark propagator renormalization, are computed at three different values of the lattice spacing, two volumes and several twisted-mass parameters. The method we developed allows for a precise cross-check of the running, thanks to the particular proper treatment of hypercubic artifacts. Results for the twist-2 operator $O_{44}$ are also presented.

Figure 1

Pion mass for each $\beta =2.10$ (lhs) and $\beta =1.95$ (rhs) ensemble, before $\theta$ average. The $x$-axis is the renormalized quark mass $M_0=\sqrt{(Z_A{m}_{\mathrm{PCAC}}{)}^2+m_q^2}$ and the $y$-axis is the pion mass squared, both in lattice units. The difference between m/p ensembles illustrates the consequence of nonmaximal twist and $\mathcal{O}(a)$ effects. The result of the straight line fit using pion mass values computed after $\theta$ average is shown in dashed blue curve.

Figure 2

$\overline{d}\mathrm{\Gamma }u$ scalar (lhs) and pseudoscalar (rhs) vertex functions versus pion mass squared (in lattice units) for ensemble $3p$ ($\beta =2.10$) for several values of $a^2{\overrightarrow{p}}^2$ ($a{p}^0=\frac{\pi }{T}$ for all curves except the magenta one, for which $a{p}^0=\frac{21\pi }{T}$).

Figure 3

Coefficient of the $1/m_{\pi }^2$ term (lhs) and of the $m_{\pi }^2$ term (rhs) in the chiral fit [in (25), $C(p^2)$ and $B(p^2)$ resp.] as a function of $1/p^2$ in lattice units, for ensemble $3p$ ($\beta =2.10$). The green line serves as eye guidance mainly and represents a linear fit at large $p^2$.

Figure 4

Quark renormalization constant (lhs) and scalar renormalization constant (rhs) as a function of $p^2$ in lattice units. Both exhibit the typical fishbone structure induced by the breaking of the $O(4)$ rotational symmetry of the Euclidian space-time by the lattice discretization, into the hypercubic group $H(4)$.

Figure 5

lhs: Effect of hypercubic corrections on quark renormalization constant, as a function of $a^2{p}^2$, for ensemble $3mp$ ($\beta =2.10$, ${\mu }_{\text{sea}}=0.0046$, ${32}^3\times 64$ lattice). rhs: Renormalization constants as a function of $a^2{p}^2$, after removing $H(4)$ artifacts, still for ensemble $3mp$ ($\beta =2.10$, ${\mu }_{\text{sea}}=0.0046$, ${32}^3\times 64$ lattice).

Figure 6

Effect of hypercubic corrections on $Z_{44}$, as a function of $a^2{p}^2$, for ensemble $3mp$ ($\beta =2.10$, ${\mu }_{sea}=0.0046$, ${32}^3\times 64$ lattice).

Figure 7

Running of $Z_q$ for ensemble $3mp$ ($\beta =2.10$, $\mu =0.0046$, volume ${32}^3\times 64$) using different fitting formulas.

Figure 8

lhs: Running of $Z_S$ for ensemble $3mp$ ($\beta =2.10$, $\mu =0.0046$, volume ${32}^3\times 64$). The standard running formula is represented in solid blue line, the dashed cyan curve includes an $1/p^2$ and an $p^2$ term (in lattice units). This latter fit leads to $Z_S(10\mathrm{GeV})=0.869(4)$. rhs: Running of $Z_P$ with the standard running expression from (30) (solid blue curve), and adding a $1/p^2$ and a $p^2$ terms (in lattice units, dashed cyan curve). The modified running gives $Z_P(10\mathrm{GeV})=0.623(2)$.

Figure 9

$Z_P/Z_S$ for ensemble $3mp$ ($\beta =2.10$, $\mu =0.0046$, volume ${32}^3\times 64$). Lattice artifacts have been removed separately from $Z_S$ and $Z_P$. The ratio of these two RCs is compatible with a constant over the whole $p^2$ interval considered and $Z_P/Z_S=0.717(3)$.

Figure 10

Fits of the residual $p_{\text{latt}}^2$ dependence of $Z_V$ and $Z_A$ for ensemble $3mp$ ($\beta =2.10$, $\mu =0.0046$, volume ${32}^3\times 64$).

Figure 11

Running of $Z_{44}$ for ensemble $3mp$, $\beta =2.10$, $\mu =0.0046$, $L=32$, $T=64$. The black points are the data after hypercubic artifacts removal. The dashed blue curve is the standard running expression Eq. (32), and the solid red line includes $1/(p_{\text{latt}}^2)$ and $p_{\text{latt}}^2$ artifacts.

Figure 12

lhs: $N_f=4$ local RCs dependence with the pion mass. All RCs are given in the RI’-MOM scheme at 10 GeV. The straight dashed lines are constant fits for each $\beta$ values. The red points correspond to $\beta =2.10$, the black ones to $\beta =1.95$, and the blue ones to $\beta =1.90$. rhs: local RCs after chiral extrapolation, versus $\log a^2$. All RCs follow a linear dependence with $\log a^2$ to a very high accuracy.

Figure 13

lhs: $N_f=4Z_{44}$ dependence with the pion mass. $Z_{44}$ is given in the RI’-MOM scheme at 2 GeV. The straight dashed lines are constant fits for each $\beta$ values. The red points correspond to $\beta =2.10$, the black ones to $\beta =1.95$, and the blue ones to $\beta =1.90$.