Lattice tests of beyond standard model dynamics

Over the last few years lattice techniques have been used to investigate candidate theories of new physics beyond the standard model. This review gives a survey of results from these studies. Most of these investigations have been of systems of gauge fields and fermions that have slowly running coupling constants. A major portion of the review is a critical discussion of work in this particular subfield, first describing the methods used and then giving a compilation of results for specific models.

Figure 1

The vector meson spectrum vs quark mass for different $N_c$. The data points are crosses for $N_c=2$, diamonds for $N_c=3$, octagons for $N_c=4$, squares for $N_c=5$, fancy crosses for $N_c=6$, fancy squares for $N_c=7$, and bursts for $N_c=17$. From [29].

Figure 2

Conjectured phase diagram for nonsupersymmetric theories with fermions in the (i) fundamental representation (top shaded band, blue), (ii) two-index antisymmetric representation (next lower one, purple), (iii) two-index symmetric representation (next lower one, red), and (iv) adjoint representation (lowest band, green) as a function of the number of flavors $N_f$ and the number of colors $N$. The shaded areas depict the corresponding conformal windows from their calculation. The dashed curve represents the change of sign in the second coefficient of the beta function. From [97].

Figure 3

Artist’s conception of a beta function for a walking theory—a negative beta function that approaches the origin, then turns away.

Figure 4

The quantity $f_q^{(B)}$ defined in Eq. (42) vs the squared pseudoscalar to vector meson mass ratio $(m_{PS}/m_V{)}^2$. Data shown include quenched $SU(5)$ and $SU(7)$ (diamonds, octagons), quenched and dynamical $SU(3)$ (squares), and dynamical $SU(4)$ AS2 (crosses). From [78]. Also plotted are results from [13] for quenched fundamental $SU(4)$, for bare gauge coupling $\beta =11.5$ (fancy diamonds) and $\beta =12.0$ (fancy crosses).

Figure 5

Running coupling constant from Eq. (58) for a set of initial couplings: $N_c=3$, $N_f=3$. The dash-dotted line is a line of constant $1/g^2$.

Figure 6

Running coupling constant from Eq. (58) for a set of initial couplings: $N_c=3$, $N_f=12$. The dash-dotted line is a line of constant $1/g^2$.

Figure 7

Pseudoscalar and vector meson masses, in lattice units, from quenched $SU(3)$ gauge theory. Octagons and squares are data from a weaker gauge coupling simulation; crosses and diamonds from stronger coupling. From [85].

Figure 8

Squared peudoscalar and vector meson masses, in lattice units, from quenched $SU(3)$ gauge theory. Octagons and squares are data from a weaker gauge coupling simulation; crosses and diamonds are from stronger coupling. The data are identical to what are shown in Fig. 7.

Figure 9

Pseudoscalar and vector meson masses, in lattice units, from $SU(2)$ gauge theory coupled to $N_f=2$ adjoint fermions. Octagons and squares are data from a weaker gauge coupling simulation; crosses and diamonds are from stronger coupling. These data are a $12^3\times 24$ volume lattice at weaker coupling and a $24^3\times 64$ volume at stronger coupling. From [46].

Figure 10

Rescaled discrete beta function for $SU(2)$ gauge theory with $N_f=2$ adjoints, calculated in two loops for various scaling factors $s$. Also shown are the one- and two-loop beta functions; the rescaled DBF approaches the two-loop beta function when $s\to 1$. Top to bottom, the curves are in the order shown in the legend. From [81].

Figure 11

SF coupling $1/g^2$ vs $L/a$ (on a logarithmic scale), for $SU(2)$ gauge theory coupled to $N_f=2$ adjoint fermions. Data are at lattice gauge coupling (from the top) $\beta =3.0$, 2.5, 2.0, 1.75, 1.5, and 1.4. The lines through the data points are fits to the data at each $\beta$ of the form $1/g^2(L)=a+b\ln (L/a)$. The dotted line has the slope $2b_1/(16{\pi }^2)$ as given by the lowest-order beta function, Eq. (24). From [81].

Figure 12

Continuum extrapolation of the Schrödinger functional coupling for $SU(2)$ with $N_f=2$ adjoints. From [212].

Figure 13

Continuum extrapolation of the gradient flow coupling for $SU(2)$ with $N_f=2$ adjoints. From [212].

Figure 14

Sketch of the RG flow around a fixed point (FP) with one relevant operator. The coupling pair $(K,K^{\prime })$ indicates matched couplings whose correlation lengths differ by a factor of $s$. The line labeled RT is the renormalized trajectory. From [141].

Figure 15

Pseudoscalar renormalization constant $Z_P$. From the top, data are from $\beta =3.0$, 2.5, 2.0, 1.75, 1.5, and 1.4. Lines are fits to $\ln Z_P(L)=-{\gamma }_m\ln (L/a)+\text{const}$ for each $\beta$. From [81].

Figure 16

Curve collapse in $SU(3)$ gauge theory with $N_f=2$ symmetric-representation fermions. Plots of ${\xi }_L/L$ vs $m_q{L}^{y_m}$ at $\beta =5.2$ for four choices of $y_m$: (a) $y_m=1.0$, (b) $y_m=1.4$, (c) $y_m=1.8$, and (d) $y_m=3.0$. Plotting symbols are for different simulation volumes, diamonds, $12^3\times 6$ ($L=6$); octagons, $12^3\times 8$ ($L=8$); squares, $16^3\times 8$ ($L=8$); crosses, $12^4$ ($L=12$); and bursts, $16^4$ ($L=16$). Curve collapse seems to be best in (b) and (c). From [74].

Figure 17

[57]’s best curve collapse fit, combining their data and that of the LH and LatKMI Collaborations ([114]; [6]).

Figure 18

Exponent $y_m$ for the pseudoscalar mass, vector mass, and pseudoscalar decay constant, not including the nonleading coupling. The abscissa is the bare gauge coupling. From [57].

Figure 19

Exponent $y_m$ for the pseudoscalar mass, vector mass, and pseudoscalar decay constant, including the nonleading coupling. From [57].

Figure 20

Continuum running for $SU(3)$ gauge theory coupled to $N_f=12$ fundamentals Results shown for running from below the infrared fixed point (triangles) are based on ${\overline{g}}^2(L_0)\equiv 1.6$. Also shown is continuum backward running from above the fixed point (squares), based on ${\overline{g}}^2(L_0)\equiv 9.0$. From [16].

Figure 21

Raw lattice data for $SU(3)$ gauge theory coupled to $N_f=12$ fundamentals. Lines connect data with the same bare couplings. The line at the bottom is the slope expected from one-loop running. From [16].

Figure 22

Raw lattice data for $SU(3)$ gauge theory coupled to $N_f=8$ fundamentals. Lines connect data with the same bare couplings. The line at the bottom is the slope expected from one-loop running. From [16].

Figure 23

$N_f=10$ beta function for $u=g^2$, along with perturbative expectations. From [146].

Figure 24

$N_f=8$ spectroscopy with fits motivated by chiral symmetry breaking. From [14].

Figure 25

$N_f=8$ spectroscopy with fits to power laws, Eq. (61). From [14].

Figure 26

The ratio $⟨\overline{\psi }\psi ⟩/F^3$ of $N_f=6$ to $N_f=2$, $R_m\equiv [M_m^2/2m{F}_m{]}_{6f}/[M_m^2/2m{F}_m{]}_{2f}$, vs the average fermion mass $\overline{m}$. From [20].

Figure 27

$S$ parameter for $N_f=2$ (diamonds and band) and $N_f=6$ (circles and band). For each of the solid points, $M_{P}L>4$. From [21].

Figure 28

Beta function for $SU(2)$ with $N_f=2$ adjoints. This is based on comparisons of two $L$ values related by a scale factor $s=2$. The horizontal dash-dotted line is the one-loop beta function, and the curved line is the two-loop beta function. From [81].

Figure 29

Mass anomalous dimension ${\gamma }_m(g^2)$. The horizontal bar at the top marks our location (with its uncertainty) for the critical coupling. The crosses are the data of [47], analyzed with the same fit. The diagonal line is the lowest-order perturbative result. From [81].

Figure 30

${\gamma }_m$ from simulations with two flavors of symmetric-representation fermions, as a function of $\lambda =g^{2}N$, where $g^2$ is the Schrödinger functional coupling. The line is the lowest-order large-$N$ prediction. From [82].

Figure 31

The beta function from simulations with two flavors of adjoint fermions. The lines are the beta functions at one and two loops. From [81], [84].

Figure 32

My attempt at a synthesis of lattice results for the vacuum structure of various systems. The labels are “C” for confined and chirally broken, “D” for deconfined, chirally restored, and probably conformal in the massless limit, and “?” for unknown. The rightmost symbols for $N_c=2$ and 3 are S2 (equivalent to adjoint for $N_c=2$) and the leftmost ones are for fundamentals.

Figure 33

Two-loop beta function for $SU(3)$ with (from the bottom up) $N_f=3$, 6, 9, 12, and 15 flavors of fundamental representation fermions. $N_f=9$ has a zero at $g^2=66$, far to the right off the plot.