Likely values of the Higgs vacuum expectation value

We make an estimate of the likelihood function for the Higgs vacuum expectation value (vev) by imposing anthropic constraints on the existence of atoms while allowing the other parameters of the standard model to also be variable. We argue that the most important extra ingredients are the Yukawa couplings, and for the intrinsic distribution of Yukawa couplings we use the scale-invariant distribution which is favored phenomenologically. The result is successful phenomenologically, favoring values close to the observed vev. We also discuss modifications that can change these conclusions. Our work supports the hypothesis that the anthropic constraints could be the origin of the small Higgs vev.

Figure 1

The anthropic constraints on $m_u$, $m_d$, $m_e$ in MeV units.

Figure 2

Quark and lepton masses, defined at the energy $\mu =M_W$, on a logarithmic scale. A scale-invariant weight corresponds to a uniform distribution on this scale. We use this feature to describe the distribution of Yukawa couplings.

Figure 3

The staircaselike curve shows the empirical distribution function, defined as the fraction of the nine observed Yukawa couplings that lie below a given $\Gamma$ value. The other curves show the cumulative distribution functions predicted by our power-law probability distribution Ansatz. All curves assume ${\Gamma }_{\min }=1.2$, and the straight lines correspond to the log-uniform $\delta =1$ case with different values of ${\Gamma }_{\min }$, the solid line having the best-fit value ${\Gamma }_{\min }=5\times {10}^{-7}$. From top to bottom, the bent curves with the same endpoints take the $\delta$ values 1.2, 1.1, 1.0, 0.9, and 0.8, respectively. The $\delta =1$ line is seen to exhibit good consistency with the observed data.

Figure 4

The probability that our model is consistent with the nine observed Yukawa couplings is shown as a function of the power-law index $\delta$, assuming the best-fit value ${\Gamma }_{\min }=5\times {10}^{-7}$. The V-shaped curve shows the maximal difference between the predicted and observed distribution functions from Fig. 3.

Figure 5

The probability that our model is consistent with the nine observed Yukawa couplings is shown as a function of the lower cutoff ${\Gamma }_{\min }$, assuming the scale-invariant power-law index $\delta =1$. The V-shaped curve shows the maximal difference between the predicted and observed distribution functions from Fig. 3.

Figure 6

Contour plot of the log likelihood as a function of the lower endpoint for the Yukawa couplings ${\Gamma }_{\min }$ and the exponent $\delta$. We normalize ${\Gamma }_{\min }$ relative to the lowest observed Yukawa coupling, ${\Gamma }_e$, by defining $x_{\min }={\Gamma }_{\min }/{\Gamma }_e$. The darkest area is the $1\sigma$ range, the second darkest area marks the $2\sigma$ range, etc.

Figure 7

The likelihood function $L$ for the Higgs vev as a function of $v/v_0$ ($v_0$ is the observed Higgs vev of 246 GeV), constructed with our favored scale-invariant weight. The normalization gives the fraction of simulations which satisfy the anthropic constraints.

Figure 8

This shows the same likelihood function $L$ as in Fig. 7, but with the vertical axis on a linear scale.

Figure 9

The likelihood function $L$ shown in Figs. 7, 8 is shown here with both the horizontal and vertical axes on a linear scale.

Figure 10

The likelihood function for different values of the lower endpoint in the Yukawa distribution, ${\Gamma }_{\min }=0.4{\Gamma }_e$ (solid line) and ${\Gamma }_{\min }=0.04{\Gamma }_e$ (dashed line).

Figure 11

The likelihood function resulting from a power-law weight of exponent $\delta =0.86$ without any lower endpoint ${\Gamma }_{\min }$.

Figure 12

The likelihood function allowing any number of light quarks in the anthropic window (dashed line), along with that allowing $u$, $d$, $e$ only (solid line).

Figure 13

The likelihood function resulting from a bi-unitary diagonalization of the Yukawa matrices allowing any number of light quarks in the anthropic window (dashed line), along with that allowing $u$, $d$, $e$ only (solid line).