Baryon number violating scatterings in laboratories

Earlier estimates have argued that the baryon number violating scattering cross section in the laboratory is exponentially small so it will never be observed, even for incoming 2-particle energy well above the sphaleron energy of 9 TeV. However, we argue in Ref. [1] that, due to the periodic nature of the sphaleron potential, the event rate for energies above the sphaleron energy may be high enough to be observed in the near future. That is, there is a discrepancy of about 70 orders of magnitude between the two estimates. Here we argue why and how the multisphaleron processes are crucial to the event rate estimate, a very important “resonant tunneling” property that has not been taken into account before. We also summarize the input assumptions and arguments adopted in our estimate when compared to the earlier estimates.

Figure 1

An example of how two hard incoming W bosons can give up their energy to a sphaleron. In the “few-to-many” process, each additional W boson costs a power of the gauge coupling $g$ while its coupling to the left sphaleron (denoted by the shaded disc) gains back a factor of $g^{-1}$. For a W boson propagating between 2 sphalerons, we gain a factor of $1/g^2$ for an off-shell boson and a factor of $1/g^4$ for an on-shell boson. The solid lines stand for the fermions.

Figure 2

The magnitude of $F_0=-{\alpha }_W\log (T)/2\pi$, where $T$ is the transmission coefficient, as a function of the simple harmonic oscillator frequency $\omega$ and the energy $E$. There is no tunneling suppression when $F_0=0$. The blue curve is for the $a=1,\omega \to \infty$ case, where $F_0=0$ at $E=E_{\text{sph}}$, which roughly coincides with the $a=0,\omega \to 0$ case. The orange curve is for the $a^2=2,\omega \to 0$ case where $F_0=0$ at $E=3E_{\text{sph}}$ [28]. The solid green curve is for the $a^2=5,\omega \to 0$ case where $F_0=0$ at $E=6E_{\text{sph}}$. The red dash curve and black dot-dashed lower bound are from the numerical calculation in Ref. [15].

Figure 3

A sketch of the extended Brillouin zone scheme for $E(k,K_1)$ versus $p=v(k+K_1)$. The height of the sphaleron potential is $E_{\text{sph}}=9.1\mathrm{TeV}$ (before turning on the $U(1)$ hypercharge coupling). Note that the bands follow the parabolic curve $\mathcal{E}=p^2/2m$ for $E>E_{\text{sph}}=9.1\mathrm{TeV}$. Turning on the $U(1)$ hypercharge will lower the sphaleron mass from 9.1 to 9.0 TeV.

Figure 4

The 3 regions, $A$, $B$ and $C$, separated by two identical potential barriers. The dashed lines are 2 of the possible paths in the path integral going from $A$ to $C$. If, for the right energy, the extra round trip in region $B$ attains a $2\pi$ phase for the top path, there are infinitely many paths with multiple round trips in region $B$ contributing to a coherent sum. This is the resonant tunneling phenomenon, which can overcome the exponential tunneling suppression to yield a transmission coefficient $T\to 1$ in QM.