Light-quark contribution to the leading hadronic vacuum polarization term of the muon $g-2$ from twisted-mass fermions

We present a lattice calculation of the leading hadronic vacuum polarization (HVP) contribution of the light $u$- and $d$-quarks to the anomalous magnetic moment of the muon, $a_{\mu }^{\mathrm{HVP}}(ud)$, adopting the gauge configurations generated by the European Twisted Mass Collaboration (ETMC) with $N_f=2+1+1$ dynamical quarks at three values of the lattice spacing ($a\simeq 0.062,0.082,0.089\mathrm{fm}$) with pion masses in the range $M_{\pi }\simeq 210–450\mathrm{MeV}$. Thanks to several lattices at fixed values of the light-quark mass and scale but with different sizes we perform a careful investigation of finite-volume effects (FVEs), which represent one of main source of uncertainty in modern lattice calculations of $a_{\mu }^{\mathrm{HVP}}(ud)$. In order to remove FVEs we develop an analytic representation of the vector correlator, which describes the lattice data for time distances larger than $\simeq 0.2\mathrm{fm}$. The representation is based on quark-hadron duality at small and intermediate time distances and on the two-pion contributions in a finite box at larger time distances. After removing FVEs we extrapolate the corrected lattice data to the physical pion point and to the continuum limit taking into account the chiral logs predicted by Chiral Perturbation Theory (ChPT). We obtain $a_{\mu }^{\mathrm{HVP}}(ud)=619.0(17.8)\times {10}^{-10}$. Adding the contribution of strange and charm quarks, obtained by ETMC, and an estimate of the isospin-breaking corrections and quark-disconnected diagrams from the literature we get $a_{\mu }^{\mathrm{HVP}}(udsc)=683(19)\times {10}^{-10}$, which is consistent with recent results based on dispersive analyses of the experimental cross section data for $e^{+}{e}^{-}$ annihilation into hadrons. Using our analytic representation of the vector correlator, taken at the physical pion mass in the continuum and infinite volume limits, we provide the first eleven moments of the polarization function and we compare them with recent results of the dispersive analysis of the ${\pi }^{+}{\pi }^{-}$ channels. We estimate also the light-quark contribution to the missing part of $a_{\mu }^{\mathrm{HVP}}$ not covered in the MUonE experiment.

Figure 1

The effective mass $M_{\mathrm{eff}}^{(ud)}(t)$ in lattice units [see Eq. (10)] corresponding to the light-quark vector correlator $V^{(ud)}(t)$ for the ETMC gauge ensembles A80.24, B55.32, and D30.48, evaluated using either 40 (left panel) or 160 (right panel) stochastic sources per each gauge configuration.

Figure 2

Left panel: results for $a_{\mu }^{\mathrm{HVP}}(ud)$ obtained using Eq. (8) (with $T_{\text{data}}=t_{\max }-2a$) for all the ETMC ensembles of Table 1 versus the simulated pion mass $M_{\pi }$. Right panel: lattice data in the case of the four ensembles A40.XX with $XX=20$, 24, 32 and 40, corresponding to a pion mass $M_{\pi }\simeq 320\mathrm{MeV}$ and a lattice spacing $a\simeq 0.089\mathrm{fm}$. The (red) solid and (black) dashed lines correspond, respectively, to an exponential, $A(1-B{e}^{-M_{\pi }L})$, and a power-law, $A^{\prime }(1-B^{\prime }/(M_{\pi }L{)}^3)$, phenomenological fit.

Figure 3

Left panel: the squared time-like pion form factor $|F_{\pi }(\omega ){|}^2$ determined by the KLOE experiment [62] from the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ (dots). Right panel: the experimental values of the scattering phase shift ${\delta }_{11}$ obtained in Ref. [63] (squares) and in Ref. [64] (diamonds). The solid lines represent the results of the GS parametrization (16)–(18) corresponding to $M_{\pi }=0.135\mathrm{GeV}$, $M_{\rho }=0.775\mathrm{GeV}$, and $g_{\rho \pi \pi }=5.50$.

Figure 4

The vector correlator $V^{(ud)}(t)$ in physical units corresponding to the ETMC gauge ensembles specified in the inset, which share an approximate common value of the (renormalized) light-quark mass $m_{ud}\simeq 12\mathrm{MeV}$ and differ in the values of the lattice spacing $a$. The solid line represents the pQCD prediction in the massless limit (cf. Eq. (3.22) of Ref. [30]).

Figure 5

The vector correlator $V^{(ud)}(t)$ (in lattice units) in the case of the gauge ensemble A40.24 corresponding to a pion mass of $\simeq 320\mathrm{MeV}$ versus the time distance $t$ (in lattice units). The blue dotted and the red dashed lines represent respectively the contributions of the dual correlator $V_{\text{dual}}(t)$ and of the two-pion correlator $V_{\pi \pi }(t)$. The green solid line is the sum of the two contributions. In the left panel a logarithmic scale is used, while in the right panel the region of low values of $t$ is better highlighted using a linear scale. Errors are statistical only.

Figure 6

The same as in the left panel of Fig. 5, but in the case of the gauge ensembles B25.32 and D15.48 corresponding to $M_{\pi }\simeq 260$ and $\simeq 220\mathrm{MeV}$, respectively. Errors are statistical only.

Figure 7

The (connected) light-quark contribution to the muon HVP, $a_{\mu }^{\mathrm{HVP}}(ud)$, evaluated for all the ETMC gauge ensembles of Table 1. Empty markers correspond to Eq. (8), where the lattice data for the vector correlator $V^{(ud)}(t)$ are directly used. Full markers are the results of Eq. (27), where the analytic representation (26) is adopted. For the latter case the points have been shifted horizontally for a better readability.

Figure 8

Left panel: the dual parameters $R_{\text{dual}}$ and $E_{\text{dual}}$ (given in physical units) versus $M_{\pi }L$, appearing in Eq. (25), for the four ensembles A40.XX ($m_{ud}\simeq 17\mathrm{MeV}$ and $a\simeq 0.089\mathrm{fm}$). Right panel: the parameters $M_{\rho }$ (given in physical units) and $g_{\rho \pi \pi }/4$ versus $M_{\pi }L$, appearing in the two-pion contribution (14). The solid and dashed lines correspond to the exponentially-suppressed Ansatz (28) with $\alpha =3/2$ and $\alpha =0$, respectively. The dotted lines correspond to the power-suppressed Ansatz (29).

Figure 9

Lattice data in the case of the four ensembles A40.XX (red dots) versus $M_{\pi }L$. The blue squares and the green diamonds correspond respectively to the data corrected by FVEs according to Eqs. (33)–(34), evaluated by including either the $\pi \pi$ contribution only or the “$\text{dual}+\pi \pi$” terms and by using in Eqs. (31)–(32) the infinite-volume values $R_{\text{dual}}^{\infty }$, $E_{\text{dual}}^{\infty }$, $M_{\rho }^{\infty }$, $g_{\rho \pi \pi }^{\infty }$ and $M_{\pi }^{\infty }$. The dashed line is a constant fit to the green points. The black triangles represent the data corrected by the FVEs evaluated using in Eqs. (31)–(32) the values of $R_{\text{dual}}$, $E_{\text{dual}}$, $M_{\rho }$, $g_{\rho \pi \pi }$, and $M_{\pi }$ obtained at each lattice size $L$ (see text).

Figure 10

Elastic P-wave $\pi \pi$ scattering phase shift ${\delta }_{11}$ obtained in Ref. [67] (blue circles) and with our A40.XX ensembles (red curve) versus the dimensionless variable $\omega /M_{\rho }$. The lattice setup of Ref. [67] corresponds to $N_f=2+1$ clover fermions with $M_{\pi }\simeq 320\mathrm{MeV}$, $M_{\rho }\simeq 800\mathrm{MeV}$, $a\simeq 0.114\mathrm{fm}$, and $L\simeq 3.65\mathrm{fm}$. Our A40.XX setup corresponds to $N_f=2+1+1$ twisted-mass fermions in the infinite volume limit with $M_{\pi }^{\infty }\simeq 315\mathrm{MeV}$, $M_{\rho }^{\infty }\simeq 850\mathrm{MeV}$, and $a\simeq 0.089\mathrm{fm}$.

Figure 11

Left panel: the dual parameter $R_{\text{dual}}$ versus the renormalized light-quark mass $m_{ud}$ (in the $\overline{\mathrm{MS}}(2\mathrm{GeV})$ scheme) obtained for all the ETMC ensembles of Table 1. Right panel: the same as in the left panel, but for the dual parameter $(M_{\pi }/E_{\text{dual}}{)}^2$. The solid lines represent respectively the fitting functions (35) and (36) evaluated in the continuum and infinite volume limits. The full (orange) diamonds identify the values of the parameters at the physical pion point, namely $R_{\text{dual}}(m_{ud}^{\mathrm{phys}},0,\infty )=1.14(6)$ and $E_{\text{dual}}(m_{ud}^{\mathrm{phys}},0,\infty )=479(22)\mathrm{MeV}$.

Figure 12

The same as in Fig. 11, but for the two-pion parameters $(M_{\pi }/M_{\rho }{)}^2$ and $g_{\rho \pi \pi }$. The solid lines represent respectively the fitting functions (37) and (38) evaluated in the continuum and infinite volume limits, namely $M_{\rho }(m_{ud}^{\mathrm{phys}},0,\infty )=760(19)\mathrm{MeV}$ and $g_{\rho \pi \pi }(m_{ud}^{\mathrm{phys}},0,\infty )=5.30(5)$.

Figure 13

The (connected) light-quark contribution to the muon HVP, $a_{\mu }^{\mathrm{HVP}}(ud)$, evaluated for all the ETMC gauge ensembles of Table 1. Empty markers correspond to Eq. (8), where the lattice data for the vector correlator $V^{(ud)}(t)$ are directly used. Full markers are the results of the subtraction of FVEs by means of Eqs. (33)–(34). The physical muon mass is used in the left panel, while the ELM mass (40) is adopted in the right panel.

Figure 14

Values of the (connected) light-quark contribution to the muon HVP, $a_{\mu }^{\mathrm{HVP}}(ud)$, corrected by FVEs and evaluated using either $m_{\mu }=m_{\mu }^{\mathrm{phys}}=105\mathrm{MeV}$ (left panel) or $m_{\mu }=m_{\mu }^{\mathrm{ELM}}$ (right panel). The dashed lines represent the fitting function (47), which includes the NLO ChPT prediction, evaluated at each value of the lattice spacing of the ETMC ensembles. The solid lines represent the same fitting function in the continuum limit. The full (orange) diamonds are the values extrapolated at the physical pion point and in the continuum limit.

Figure 15

The same as in Fig. 14, but adopting the fitting function (48), which includes also the NNLO ChPT prediction.

Figure 16

The light-quark mass dependence of $a_{\mu }^{\mathrm{HVP}}(ud)$, in units of $10^{10}$, obtained in the continuum and infinite volume limits using the physical muon mass. The blue squares correspond to the predictions of our “$\text{dual}+\pi \pi$” analytic representation (31)–(32) evaluated using the values $R_{\text{dual}}(m_{ud},0,\infty )$, $E_{\text{dual}}(m_{ud},0,\infty )$, $M_{\rho }(m_{ud},0,\infty )$, $g_{\rho \pi \pi }(m_{ud},0,\infty )$, and $M_{\pi }(m_{ud},0,\infty )$ obtained from Eqs. (35)–(39). The green dots represent the results of the ChPT fit (48) taken in the continuum limit.

Figure 17

Left panel: values of the connected light-quark contribution to the muon HVP, $a_{\mu }^{\mathrm{HVP}}(ud)$, obtained at the physical pion point and in the continuum and infinite volume limits in the present work (52), and by HPQCD [20], CLS/Mainz [21], BMW [22], and RBC/UKQCD [23]. Right panel: values of the muon HVP $a_{\mu }^{\mathrm{HVP}}(udsc)$ obtained in the present work (53) and in Refs. [20, 21, 22, 23]. The result obtained in Ref. [23] using the R-ratio method (which turns out to be based on lattice points by $\simeq 30\%$ and on dispersive $e^{+}{e}^{-}$ data by $\simeq 70\%$) is also included as an orange dot. The results of the recent dispersive analysis of Refs. [48, 49, 50] are shown together with the value of $a_{\mu }^{HV}$ corresponding to a vanishing muon anomaly (labeled as “no New Physics”).

Figure 18

Left panel: light-quark vector correlator $V^{(ud)}(t)$, multiplied by the muon kernel $f(t)$, evaluated using our “$\text{dual}+\pi \pi$” representation (26) extrapolated at the physical pion point and in the continuum limit for three values of the lattice size $L$ (see text). The infinite volume limit, constructed as explained in the text, is also shown by the black solid line. Right panel: the (squared) pion form factor $|F_{\pi }(\omega ){|}^2$ corresponding to the GS parametrization (16)–(21) evaluated in the infinite volume limit (see text) versus the two-pion energy $\omega$. The full dots are located at the position of the energy levels satisfying the Lüscher condition (12) for each value of the lattice size $L$.

Figure 19

Values of ${\mathrm{\Delta }}_{\mathrm{FVE}}{a}_{\mu }^{\mathrm{HVP}}(ud)$ [see Eq. (34)], evaluated in the continuum limit according to our “$\text{dual}+\pi \pi$” representation at the physical pion point (red circles) and at a larger pion mass equal to $M_{\pi }=300\mathrm{MeV}$ (blue squares). The dotted line corresponds to the predictions of ChPT at NLO [47, 60].

Figure 20

Left panel: ratio of the time moments (54) evaluated at finite lattice size $L$ and in the infinite volume limit using our “$\text{dual}+\pi \pi$” analytic representation of the light-quark vector correlator taken in the continuum limit and at the physical pion point. Right panel: values of the first eleven moments (57) evaluated at the physical point using the $\pi \pi$ contribution (32) in the infinite volume limit (red circles), compared with the results of the dispersive analysis of the experimental cross section for the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ channels of Ref. [50]. Courtesy of the authors of Ref. [77].