SU(2) gauge theory with two fundamental flavors: A minimal template for model building

We investigate the continuum spectrum of the SU(2) gauge theory with $N_f=2$ flavors of fermions in the fundamental representation. This model provides a minimal template which is ideal for a wide class of Standard Model extensions featuring novel strong dynamics that range from composite (Goldstone) Higgs theories to several intriguing types of dark matter candidates, such as the strongly interacting massive particles (SIMPs). We improve our previous lattice analysis [1] by adding more data at light quark masses, at two additional lattice spacings, by determining the lattice cutoff via a Wilson flow measure of the $w_0$ parameter, and by measuring the relevant renormalization constants nonperturbatively in the regularization-invariant momentum (RI’-MOM) scheme. Our result for the lightest isovector state in the vector channel, in units of the pseudoscalar decay constant, is $m_V/F_{\mathrm{PS}}\sim 13.1(2.2)$ (combining statistical and systematic errors). For the axial channel our result is $m_A/F_{\mathrm{PS}}\sim 14.5(3.6)$, which however does include a similarly sized additional systematic error due to residual excited-states contamination. In the context of the composite (Goldstone) Higgs models, our result for the spin-one resonances are $m_V>3.2(5)\mathrm{TeV}$ and $m_A>3.6(9)\mathrm{TeV}$, which are above the current LHC constraints. In the context of dark matter models, for the SIMP case our results indicate the occurrence of a compressed spectrum at the required large dark pion mass, which implies the need to include the effects of spin-one resonances in phenomenological estimates.

Figure 1

Chiral behavior of $w_0$ as a function of $y^2$ in units of the lattice spacing (left panel) and in units of $w_0^{\chi }$ (right panel) for $W_{\mathrm{ref}}=1$. The data at four lattice spacings are displayed.

Figure 2

${\mathrm{\Lambda }}_{X=P,V,A,P/S}$ as a function of $(ap{)}^2$ for the most chiral point at $\beta =2.0$ (left panel) and $\beta =2.2$ (right panel). The filled data points are obtained without twisted boundary conditions while the empty symbols denote the use of a nonvanishing $\theta$. The vertical lines indicate where $(ap{)}^2=1$.

Figure 3

${\mathrm{\Lambda }}_{X=P,V,A,P/S}$ at fixed $(ap{)}^2=1$ as a function of $(a{m}_{\mathrm{PCAC}}^{\mathrm{bare}}{)}^2$ for $\beta =2.0$ (left panel) and $\beta =2.2$ (right panel).

Figure 4

$Z_{X=P,V,A,P/S}$ as function of the renormalization scale $(w_0\mu {)}^2=(w_{0}p{)}^2$ for $\beta =2.0$ (left panel) and $\beta =2.2$ (right panel).

Figure 5

Effective masses of the pseudoscalar, vector and axial meson masses ($\beta =1.8$, $m_0=-1.157$, $L=24$).

Figure 6

Effective masses of the pseudoscalar, vector and axial meson masses ($\beta =2.0$, $m_0=-0.958$, $L=32$).

Figure 7

Effective masses of the pseudoscalar, vector and axial meson masses ($\beta =2.2$, $m_0=-0.76$, $L=48$).

Figure 8

Effective masses of the pseudoscalar, vector and axial meson masses ($\beta =2.3$, $m_0=-0.675$, $L=32$).

Figure 9

$F_{\mathrm{PS}}$ versus $m_{\mathrm{PS}}^2$ for the four lattice spacings. The curves correspond to the best-fit parameters obtained fitting only $\beta =2.0$, $\beta =2.2$ and $\beta =2.3$ (subset $S_2$) and are drawn for the corresponding lattice spacing. The black curve indicates the continuum results.

Figure 10

$m_{\mathrm{PS}}^2/m_{\mathrm{f}}$ versus $m_{\mathrm{PS}}^2$ for the four lattice spacings. The curves correspond to the best-fit parameters obtained fitting only $\beta =2.0$, $\beta =2.2$ and $\beta =2.3$ (subset $S_2$) and are drawn for the corresponding lattice spacing. The black curve indicates the continuum results.

Figure 11

Values of the chiral parameters $B$ and $F$ in units of the reference lattice scale $w_0^{\chi }$ as extracted using strategy II described in the text. In this plot $B$ has been rescaled by a factor of 20 for graphical convenience.

Figure 12

Combined chiral and continuum extrapolation of the vector-meson mass $m_V$. Our data for four lattice spacings is presented together with the best fit at each lattice spacing. The grey band is our result for the continuum extrapolation and its $1\sigma$ confidence region.

Figure 13

Combined chiral and continuum extrapolation of the axial-vector meson mass $m_A$. Our data for four lattice spacings is presented together with the best fit at each lattice spacing. The grey band is our result for the continuum extrapolation and its $1\sigma$ confidence region.

Figure 14

$F_{\mathrm{PS}}$ versus $m_{\mathrm{PS}}^2$ for four lattice spacings, using $(w_0^{\chi }p{)}^2=17$ for the renormalization scale. The curves correspond to the best-fit parameters obtained fitting only $\beta =2.0$, $\beta =2.2$ and $\beta =2.3$ (subset $S_2$) and are drawn for the corresponding lattice spacing. The black curve indicates the continuum results.

Figure 15

$m_{\mathrm{PS}}^2/m_{\mathrm{f}}$ versus $m_{\mathrm{PS}}^2$ for four lattice spacings, using $(w_0^{\chi }p{)}^2=17$ for the renormalization scale. The curves correspond to the best-fit parameters obtained fitting only $\beta =2.0$, $\beta =2.2$ and $\beta =2.3$ (subset $S_2$) and are drawn for the corresponding lattice spacing. The black curve indicates the continuum results.

Figure 16

Analogue of Fig. 11 obtained for reference momentum $(w_0^{\chi }p{)}^2=17$ .

Figure 17

History of the topological charge for the most chiral run at $\beta =2.0$ (left) and $\beta =2.2$ (right) as a function of the Monte Carlo time $t_{\mathrm{HMC}}$.

Figure 18

Histogram of the topological charge for the same run as in Fig. 17 for $\beta =2.0$ (left) and $\beta =2.2$ (right).