Infrared dynamics of minimal walking technicolor

We study the gauge sector of minimal walking technicolor, which is an $SU(2)$ gauge theory with $n_f=2$ flavors of Wilson fermions in the adjoint representation. Numerical simulations are performed on lattices $N_t\times N_s^3$, with $N_s$ ranging from 8 to 16 and $N_t=2N_s$, at fixed $\beta =2.25$, and varying the fermion bare mass $m_0$, so that our numerical results cover the full range of fermion masses from the quenched region to the chiral limit. We present results for the string tension and the glueball spectrum. A comparison of mesonic and gluonic observables leads to the conclusion that the infrared dynamics is given by an $SU(2)$ pure Yang-Mills theory with a typical energy scale for the spectrum sliding to zero with the fermion mass. The typical mesonic mass scale is proportional to and much larger than this gluonic scale. Our findings are compatible with a scenario in which the massless theory is conformal in the infrared. An analysis of the scaling of the string tension with the fermion mass toward the massless limit allows us to extract the chiral condensate anomalous dimension ${\gamma }_{*}$, which is found to be ${\gamma }_{*}=0.22\pm 0.06$.

Figure 1

Sketches of the spectrum of a mass-deformed IR-conformal theory (square root of the string tension, $0^{++}$ and $2^{++}$ glueballs, pseudoscalar and vector isovector mesons). In the left plot, the locking sets up at an intermediate value of the fermion mass, where dynamical fermion effects account for the physics of the system, but the pseudoscalar is not much lighter than the other particles in the spectrum. In the right plot, the locking sets up at a high value of the fermion mass, where the heavy-quark effective theory provides a good description of the relevant degrees of freedom. This case is realized close to the Banks-Zacks point, but is possible in principle also if a strongly coupled IR fixed point is present.

Figure 2

Some distributions of Polyakov loops in the temporal and spatial directions. The distributions have been symmetrized by hand. At fixed volume, the distribution of the spatial Polyakov loop shows a single peak at zero at the higher mass, and it develops two peaks at the lower mass in some of the spatial directions. At fixed mass, it shows a single peak on the larger lattice, and it develops double peaks on the smaller lattice in some of the spatial directions.

Figure 3

Histories and distributions of Polyakov loops in the temporal (with fermionic antiperiodic boundary conditions, ABC) and spatial directions (with fermionic periodic boundary conditions, PBC), for bare mass $-1.125$ on the $24\times 24\times {12}^2$ lattice. The distributions have been symmetrized by hand.

Figure 4

Absolute value of the Polyakov loop on the $16\times 8^3$ and $24\times {12}^3$ lattices.

Figure 5

Spatial and temporal string tensions at various lattice sizes as a function of $a{m}_0$. Also shown is the infinite-volume estimate.

Figure 6

Effective potentials at $\mathbf{R}=(4a,0,0)$ and $\mathbf{R}=(6a,6a,6a)$ computed on the $32\times {16}^3$ lattice at bare mass $a{m}_0=-1.15$. Plateaus have been chosen in the white regions. The red lines represent the values of the potential (with errors), obtained by fitting the effective potential with a constant in the white region. Errors have been computed with a bootstrap procedure.

Figure 7

Static potentials computed with the method Potential 2 (see Sec. 6) from Wilson loops with HYP smearing. Data for bare masses $-0.95$, $-1$, $-1.05$, $-1.075$, and $-1.1$ are obtained on a $24\times {12}^3$ lattice; data for bare mass $-1.125$ are obtained on a $24\times 24\times {12}^2$ lattice; data for bare mass $-1.15$ are obtained on a $32\times {16}^3$ lattice. Superimposed is the function $V(R)=\sigma R+\mu +c/R$, where $\sigma$ has been taken from Table  (the two curves correspond to $\sigma -\Delta \sigma$ and $\sigma +\Delta \sigma$), while $\mu$ and $c$ have been obtained with a fit in the region $R\ge 3a$.

Figure 8

Forces computed with the method Force 2 (see Sec. 6). Superimposed is the string tension from Table .

Figure 9

The mass of the $0^{++}$ glueball in lattice units, $a{M}_{0^{++}}$, measured at various values of bare quark mass $a{m}_0$ on a $16\times 8^3$ and on a $24\times {12}^3$ lattice.

Figure 10

The mass of the $2^{++}$ glueball in lattice units, $a{M}_{2^{++}}$, measured at various values of bare quark mass $a{m}_0$ on a $16\times 8^3$ and on a $24\times {12}^3$ lattice.

Figure 11

The spectrum of the theory as a function of the PCAC mass $am$. The mass of the vector is not shown, since on the scale of the figure this state appears to be degenerate with the PS.

Figure 12

The ratio $M_{\mathrm{PS}}/{\sigma }^{1/2}$ as a function of $M_{\mathrm{PS}}$.

Figure 13

Comparison of the ratio $M_{\mathrm{V}}/M_{\mathrm{PS}}$ as a function of $a{M}_{\mathrm{PS}}$ in the quenched and the dynamical theory. The dynamical ratio approaches the infinite-mass value of 1 at large $a{M}_{\mathrm{PS}}$ and develops a plateau (signaling hyperscaling) at $a{M}_{\mathrm{PS}}\le 1.25$.

Figure 14

Comparison of the masses of glueballs of the dynamical theory with the interpolating functions of the quenched theory. Among the plotted data, the dynamical points at $a{\sigma }^{1/2}=0.2405(58)$ and $a{\sigma }^{1/2}=0.1587(23)$ correspond to the lightest masses, respectively, on the $16\times 8^3$ and $24\times {12}^3$ lattices. Hence, the glueball masses for those string tensions are affected by the largest finite-size effects.

Figure 15

The ratio ${\sigma }_t^{1/2}/M_{\mathrm{PS}}$ as a function of $M_{\mathrm{PS}}{L}_s$ at various lattice sizes. In the shaded region the crossover between the S and the A phase takes place.

Figure 16

$a{\sigma }^{1/2}$ as a function of $am$. A fit of the data to Eq. (74) is also shown. In particular, the two lines represent the extremal values ${\gamma }_{*}=0.16$ and ${\gamma }_{*}=0.28$.