Light flavor-singlet scalars and walking signals in $N_f=8$ QCD on the lattice

Based on the highly improved staggered quark action, we perform lattice simulations of $N_f=8$ QCD and confirm our previous observations, both of a flavor-singlet scalar meson (denoted as $\sigma$) as light as the pion and of various “walking signals” through the low-lying spectra, with higher statistics, smaller fermion masses $m_f$, and larger volumes. We measure $M_{\pi }$, $F_{\pi }$, $M_{\rho }$, $M_{a_0}$, $M_{a_1}$, $M_{b_1}$, $M_N$, $M_{\sigma }$, $F_{\sigma }$, $⟨\overline{\psi }\psi ⟩$ (both directly and through the Gell-Mann-Oakes-Renner relation), and the string tension. The data are consistent with the spontaneously broken phase of the chiral symmetry, in agreement with the previous results: Ratios of the quantities to $M_{\pi }$ monotonically increase in the smaller $m_f$ region towards the chiral limit similarly to $N_f=4$ QCD, in sharp contrast to $N_f=12$ QCD where the ratios become flattened. We perform fits to chiral perturbation theory, with the value of $F_{\pi }$ found in the chiral limit extrapolation reduced dramatically to roughly $2/3$ of the previous result, suggesting the theory is much closer to the conformal window. In fact, each quantity obeys the respective hyperscaling relation throughout a more extensive $m_f$ region compared with earlier works. The hyperscaling relation holds with roughly a universal value of the anomalous dimension, ${\gamma }_m\simeq 1$, with the notable exception of $M_{\pi }$ with ${\gamma }_m\simeq 0.6$ as in the previous results, which reflects the above growing up of the ratios towards the chiral limit. This is a salient feature (walking signal) of $N_f=8$, unlike either $N_f=4$, which has no hyperscaling relation at all, or $N_f=12$ QCD, which exhibits universal hyperscaling. The effective ${\gamma }_m\equiv {\gamma }_m(m_f)$ of $M_{\pi }$ defined for each $m_f$ region has a tendency to grow towards unity near the chiral limit, in conformity with the Nambu-Goldstone boson nature, as opposed to the case of $N_f=12$ QCD where it is almost constant. We further confirm the previous observation of the light $\sigma$ with mass comparable to the pion in the studied $m_f$ region. In a chiral limit extrapolation of the $\sigma$ mass using the dilaton chiral perturbation theory and also using the simple linear fit, we find the value consistent with the 125 GeV Higgs boson within errors. Our results suggest that the theory could be a good candidate for walking technicolor model, having anomalous dimension ${\gamma }_m\simeq 1$ and a light flavor-singlet scalar meson as a technidilaton, which can be identified with the 125 GeV composite Higgs in the $N_f=8$ one-family model.

Figure 1

A schematic picture of the gauge coupling of massless large-$N_f$ QCD as a walking gauge theory in the $\mathrm{S}\chi \mathrm{SB}$ phase near the conformal window. $m_D$ is the dynamical mass of the fermions generated by the $\mathrm{S}\chi \mathrm{SB}$. The effects of the bare mass of the fermion $m_f$ would be qualitatively different depending on the cases: case 1: $m_f\ll m_D$ (the red dotted line), which is well described by ChPT, and case 2: $m_f\gg m_D$ (blue dotted line), which is well described by the hyperscaling, with a possible nonuniversal exponent for $M_{\pi }$.

Figure 2

Histories of the plaquette (left) and chiral condensate $\mathrm{\Sigma }$ (right) for the three indicated ensembles.

Figure 3

Histories and histograms (with a Gaussian fit) of the topological charge $Q$ for the three indicated ensembles. The first shows frozen behavior, while the latter two show good coverage of topological sectors.

Figure 4

The effective mass $M^{\mathrm{eff}}(t)$ of the NG pion calculated with ${\tilde{G}}_{PS}(t)/{\tilde{G}}_{PS}(t+2)$ for $N_f=8$ $m_f=0.03$, $24^3\times 32$ (left) and $m_f=0.012$, $42^3\times 56$ (right), as typical examples. Red lines show the $t$ range of the global fit, the central value of the mass from the fit, and the jackknife error band.

Figure 5

The effective mass of ${\pi }_{SC}$ (left) and its staggered parity partner $a_0$ (right) for $N_f=8$ $m_f=0.012$, $42^3\times 56$.

Figure 6

The effective mass of ${\rho }_{PV}$ (left) and its staggered parity partner $a_1$ (right) for $N_f=8$ $m_f=0.012$, $42^3\times 56$.

Figure 7

The effective mass of $N$ (left) and its staggered parity partner $N_{\mathbf{1}}^{*}$ (right) for $N_f=8$ $m_f=0.012$, $42^3\times 56$.

Figure 8

The lattice volume dependence of $M_{\pi }$ (top), $F_{\pi }$ (middle) and $M_{\rho }$ (bottom) for various fermion masses $m_f$.

Figure 9

The finite-volume effects $\delta M_{\pi }(L)$ (top) and $\delta F_{\pi }(L)$ (bottom) defined in Eq. (16) as a function of $L{M}_{\pi }$ for various fermion masses $m_f$.

Figure 10

The lattice volume dependence of $M_{\pi }$ (top) and $F_{\pi }$ (bottom) at $m_f=0.02$. The black dashed line represents the ChPT motivated fit given by Eqs. (17) and (18).

Figure 11

The spectra for the NG pion and its taste partners, $M_{{\pi }_{\xi }}$ for $N_f=8$.

Figure 12

The spectra for the NG pion and its taste partners, $M_{{\pi }_{\xi }}$ for $N_f=4$.

Figure 13

The taste symmetry breaking $(M_{{\pi }_{\xi }}^2-M_{\pi }^2)/M_{\pi }^2$ for $N_f=8$, where $M_{\pi }$ and $M_{{\pi }_{\xi }}$ represent the mass of the NG pion and its taste partner with an index $\xi$, respectively.

Figure 14

The taste symmetry breaking $(M_{{\pi }_{\xi }}^2-M_{\pi }^2)/M_{\pi }^2$ for $N_f=4$, where $M_{\pi }$ and $M_{{\pi }_{\xi }}$ represent the mass of the NG pion and its taste partner with an index $\xi$, respectively.

Figure 15

$F_{\pi }/M_{\pi }$ (top) and $M_{\rho }/M_{\pi }$ (bottom).

Figure 16

$F_{\pi }/M_{\pi }$ (top) and $M_{\rho }/M_{\pi }$ (bottom) in $N_f=4$ QCD.

Figure 17

Edinburgh-type plots: $M_N/M_{\rho }$ versus $M_{\pi }/M_{\rho }$ (top), $M_N/F_{\pi }$ versus $M_{\pi }/F_{\pi }$ (middle), and $M_N/F_{\pi }$ versus $M_{\rho }/F_{\pi }$ (bottom).

Figure 18

Edinburgh-type plot, $M_N/M_{\rho }$ versus $M_{\pi }/M_{\rho }$, for $N_f=12$ with $\beta =4.0$. The spectra obtained with the largest volume are selected at each $m_f$.

Figure 19

Edinburgh-type plots: $M_N/M_{\rho }$ versus $M_{\pi }/M_{\rho }$ (top) and $M_N/F_{\pi }$ versus $M_{\rho }/F_{\pi }$ (bottom) in $N_f=4$ QCD.

Figure 20

$F_{\pi }$ as a function of $m_f$. Curves are fit results with polynomial function, Eq. (19). “quad” and “linear” denote quadratic and linear fit results, respectively. Each fit result in the chiral limit is expressed by a colored symbol. The square and triangle are shifted in the horizontal axis for clarity. The diamond symbol has two error bars: The outer represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical. The systematic error is discussed in Sec. 4d.

Figure 21

$F_{\pi }$ as a function of $m_f$ on three volumes in $N_f=4$ QCD at $\beta =3.7$. The solid curve is a quadratic fit result with the largest volume data for each $m_f$.

Figure 22

$M_{\pi }^2/m_f$ (top) and $M_{\pi }^2$ (bottom) as a function of $m_f$. Solid lines are fit results using polynomial functions. Quad and linear denote quadratic and linear fit results, respectively. In the top panel, each fit result in the chiral limit is expressed by a colored symbol. The square and triangle are shifted to the negative direction on the horizontal axis for clarity. The diamond symbol has two error bars: The outer represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical. The systematic error is discussed in Sec. 4d.

Figure 23

$M_{\pi }^2/m_f$ as a function of $m_f$ on three volumes in $N_f=4$ QCD at $\beta =3.7$. The solid curve is the result of a linear fit using the largest volume data in each $m_f$.

Figure 24

$⟨\overline{\psi }\psi ⟩$, $\mathrm{\Sigma }=F_{\pi }^2{M}_{\pi }^2/4m_f$, and ${\mathrm{\Sigma }}^{\prime }=F{F}_{\pi }{M}_{\pi }^2/4m_f$ as a function of $m_f$, as defined by Eqs. (23), (24), and (25), respectively.

Figure 25

$⟨\overline{\psi }\psi ⟩$ as a function of $m_f$. The solid curve is a quadratic fit result.

Figure 26

${\mathrm{\Sigma }}^{\prime }=F{F}_{\pi }{M}_{\pi }^2/4m_f$ as a function of $m_f$. The solid curves are polynomial fit results. Quad and linear denote quadratic and linear fit results, respectively. Each result in the chiral limit is expressed by a colored symbol. Square and diamond represent results for $⟨\overline{\psi }\psi ⟩$ and GMOR relation in the chiral limit, respectively. They are shifted to the negative direction on the horizontal axis for clarity. The square symbol has two error bars: The outer represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical. The systematic error is discussed in Sec. 4d.

Figure 27

$\mathrm{\Sigma }=F_{\pi }^2{M}_{\pi }^2/4m_f$ as a function of $m_f$. The solid and dashed curves are a quadratic fit result and expected result, respectively, from each fit of $F_{\pi }$ and $M_{\pi }$.

Figure 28

$M_{\rho }$ and $M_{a_0}$ as a function of $m_f$, together with $M_{\pi }$. Solid and dashed lines express the linear fit results for $M_{\rho }$ and $M_{a_0}$, respectively.

Figure 29

As in Fig. 28, but for $M_{a_1}$ and $M_{b_1}$. Triangle and diamond symbols are shifted to the positive direction on the horizontal axis for clarity.

Figure 30

The same as Fig. 28, for $M_N$ and $M_{N_{\mathbf{1}}^{*}}$.

Figure 31

Mass ratio $M_{a_1}/M_{\rho }$ as a function of $m_f$. The solid line expresses a linear fit result. The star symbol represents the ratio in usual QCD.

Figure 32

Left: The hyperscaling fit (40) for $F_{\pi }$, $M_{\pi }$ or, $M_{\rho }$ data. The fit range is set to $m_f=0.012-0.03$; the shaded region has been excluded. Right: The mass anomalous dimension $\gamma$ obtained by the hyperscaling fit (40) for various observables (red squares). The fit range is set to $m_f=0.012-0.03$. See also Table 12. For comparison, we have quoted the results reported by the LSD Collaboration (blue circles): fit results for the mass range of 0.015–0.03 with $\mathrm{d}.\mathrm{o}.\mathrm{f}.=2$ in Table 10 in Ref. [33].

Figure 33

The effective mass anomalous dimensions ${\gamma }_{\mathrm{eff}}$ obtained by the hyperscaling fit (40) for $N_f=8$ (left; see Table 32 for details) and 12 (right; see Tables 33 and 34 for details) spectrum data. In the right panel, the open (filled) symbols represent the results with $\beta =3.7$ (4.0). The fit range is slid by keeping the fit degrees of freedom 1. The value of $m_f$ in the horizontal axis is the average of the maximum and minimum among three fermion masses.

Figure 34

The FSHS test for $N_f=8$: The normalized mass spectra $L{F}_{\pi }$ (upper left), $L{M}_{\pi }$ (upper right), $L{M}_{\rho }$ (lower left), and $L{M}_N$ (lower right) are plotted as a function of the scaling variable $X=L{m}_f^{1/(1+\gamma )}$ for selected $\gamma$.

Figure 35

The FSHS test for $N_f=12$ spectra with $\beta =4.0$; the normalized mass spectra $L{F}_{\pi }$ (upper left), $L{M}_{\pi }$ (upper right), $L{M}_{\rho }$ (lower left), and $L{M}_N$ (lower right) are plotted as a function of the scaling variable $X=L{m}_f^{1/(1+\gamma )}$ for selected $\gamma$. The spectral data are found in Appendix pp7.

Figure 36

The FSHS test for $N_f=4$; the normalized mass spectra $L{F}_{\pi }$ (left), $L{M}_{\pi }$ (middle), and $L{M}_{\rho }$ (right) are plotted as a function of the scaling variable $X=L{m}_f^{1/(1+\gamma )}$ for selected $\gamma$.

Figure 37

The mass anomalous dimension $\gamma$ obtained by FSHS fits for various observables with the ansatz (43). See also Table 35.

Figure 38

The simultaneous FSHS fit of combined data for $\{F_{\pi },M_{\pi },M_{\rho }\}$. For $\gamma$ obtained from the fits (see Table 15), the data $\{F_{\pi }(\text{red boxes}),M_{\pi }(\text{green circles}),M_{\rho }(\text{blue triangles})\}$ are plotted in the $X-Y$ plane, where $X=L{m}_f^{1/(1+\gamma )}$ and $Y$ is defined by Eq. (45), (47), or (54) depending on the fit ansatz. For details, see the text. The simultaneous fit line (solid purple) is given by $Y=X$. Left: Naive FSHS (43). Middle: FSHS with power-law correction (46) for the fixed exponent of $\alpha =1$. Right: FSHS with RG-motivated correction (50).

Figure 39

The strength of the correction term $R^{M_H}$ defined in Eq. (57). Left: In the case of FSHS with power-law correction $(\alpha =1)$ in $N_f=8$ QCD. Middle: In the case of FSHS with RG-motivated correction in $N_f=8$ QCD. Right: In the case of FSHS with RG-motivated correction in $N_f=12$ QCD with $\beta =3.7$.

Figure 40

The Wilson loop potential $\mathcal{V}(r,t)$ given in Eq. (60) for $m_f=0.012$ with $L=42$ as a function of time $t$ for selected spatial extensions $r=3$ (left), 8 (middle), and 15 (right). The green band represents the results of the fit over the plateau.

Figure 41

The static fermion potential (left) and the Creutz ratio (right) as a function of spatial distance $r$ for $m_f=0.012$ with $L=42$. In the left panel, the solid green line represents the fit line using data at $r=3-15$. Substituting the obtained $(\alpha ,s)$ into Eq. (64), we have obtained the solid green line in the right panel.

Figure 42

The square root of the string tension in lattice units as a function of fermion mass. The inner error bars represent the statistical uncertainty. In the outer error bars, the statistical and systematic uncertainties are added in quadrature. The solid light blue and dashed light green lines show, respectively, the fit results with the quadratic and conformal ansatz described in the text. In the fits, only the statistical errors are taken into account. The gray (slightly shifted) symbols were not used in the fit but are shown to confirm that the finite-volume effects are negligible.

Figure 43

Connected $-C(t)$ and disconnected correlators $2D(t)$ for $L=30$, $m_f=0.02$.

Figure 44

Effective mass for $L=30$, $m_f=0.02$, from correlators using the projection explained in the text.

Figure 45

Effective mass for $L=36$, $m_f=0.015$ (left) and $m_f=0.020$ (right), from correlators using the projection explained in the text.

Figure 46

The fermion mass dependence of the mass of the flavor-singlet scalar $M_{\sigma }$. Masses of the NG pion $\pi$ and vector meson $\rho (PV)$ mass are also shown. The outer error represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical.

Figure 47

Edinburgh-type plots: $M_{\sigma }/F_{\pi }$ as a function of $M_{\rho }/F_{\pi }$. The outer error represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical.

Figure 48

NG-pion mass dependence of the mass of the flavor-singlet scalar. The outer error represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical. Results of the chiral extrapolation by the DChPT are plotted by the solid line and full circle.

Figure 49

Fermion mass dependence of the mass of the flavor-singlet scalar. Masses of the NG-pion $\pi$ and vector meson [$\rho (PV)$] are also shown. The solid and dashed lines represent results of linear and hyperscaling fits, respectively (fit range $m_f=0.012-0.03$). The outer error represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical.

Figure 50

Left: Scalar decay constant of the flavor-singlet scalar. Right: Dilaton decay constant from a semidirect method. The blue and black lines show the chiral fits using linear and quadratic polynomials, respectively, in $m_f$, with the lightest four and five data points used. The outer error represents the statistical and systematic uncertainties added in quadrature, while the inner error is only statistical.

Figure 51

${\sigma }_0[{\mathrm{cm}}^2]$ as a function of $M_N[\mathrm{GeV}]$. The results for $m_f=0.030$, 0.020, 0.015, 0.012, and the chiral limit are shown from upper right to lower left. Both the statistical and systematic errors are included. The experimentally allowed region is below the plotted window. The current experimental bound for ${\sigma }_0$ approximately is ${10}^{-45}[{\mathrm{cm}}^2]$ in this mass range under the assumption that the dark matter interacting in the detector is a thermal relic.

Figure 52

The solid curve is the result for $F$ as a function of the matching point $m_f^c$. The dashed and dotted lines represent $m_f^c$, where $\mathcal{X}=1$, and the polynomial fit result tabulated in Table 4, respectively.

Figure 53

As in Fig. 52, but for $B$. The polynomial fit result corresponds to $C_0/2$ in Table 4.

Figure 54

The expansion parameter of ChPT $\mathcal{X}$ as a function of $m_f^c$. The dotted line represents $\mathcal{X}=1$.

Figure 55

As in Fig. 52, but for $B{F}^2/2$. The polynomial fit result is tabulated in Table 6.

Figure 56

Left: The mass anomalous dimension $\gamma$ obtained by the simultaneous fit for combined data $\{F_{\pi },M_{\pi },M_{\rho }\}$ as a function of the second exponent $\omega$. Right: The corresponding ${\chi }^2/\mathrm{dof}$.

Figure 57

The mass dependence of $M_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 58

The mass dependence of $F_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 59

The mass dependence of $M_{\rho }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 60

The mass dependence of $M_N$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 61

The dimensionless ratio of $F_{\pi }/M_{\pi }$ versus $M_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 62

The dimensionless ratio of $M_{\rho }/M_{\pi }$ versus $M_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 63

The dimensionless ratio of $M_N/M_{\pi }$ versus $M_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 64

The dimensionless ratio of $M_{a_0}/M_{\pi }$ versus $M_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 65

The dimensionless ratio of $M_{a_1}/M_{a_0}$ versus $M_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.

Figure 66

The dimensionless ratio of $M_{a_1}/M_{\rho }$ versus $M_{\pi }$ at $\beta =4$ (left) and $\beta =3.7$ (right) in $N_f=12$.