Baryon number violating rate as a function of the proton-proton collision energy

We can make a rough analytical estimate to get some idea, even though the resulting numerical values need improvement. This subsection estimates $\eta(E_{pp})$ using an unrealistic PDF and without consideration of band structure and phase space suppression. The analytical calculations below give us a general sense of what $\eta(E_{pp})$ looks like. As a start, we consider a simple (toy) PDF for valence quarks $q$ in a proton which is scale-independent,

where $\int_{0}^{1}dxf_{u/p}(x)=2$ and $\int_{0}^{1}dxf_{d/p}(x)=1$. This PDF (6) allows an analytic discussion, but is only qualitatively valid.

If we do not care about species, we shall choose $\int_{0}^{1}dxf_{q/p}(x)=3$, so $A_{q}=180$. The Bloch-wave picture indicates that the $\hat{\sigma}_{\rm BV}(v)$ is exponentially enhanced when $v\gtrsim E_{\rm sph}$ due to the overlap of high energy Bloch bands. Thus, for the purpose of estimating, we here simply take a cutoff model,

where $c=(E_{\rm sph}/E_{pp})^{2}$. $\sigma_{0}$ is an overall normalization. We may assume that, for $v\gg 10\,{\rm TeV}$, $\sigma_{0}\lesssim\sigma_{\rm total}(pp)$, where $\sigma_{\rm total}(pp)$ does not vary much. Since we are comparing the BV event rate between different $E_{pp}$ (2), the value of $\sigma_{0}$ is not important here. With this approximation, we could write

where

As a check, we have $\sigma_{\rm BV}(c=1)=0$.

As a reasonable approximation, we take $E_{\rm sph}=9\,{\rm TeV}$ as a benchmark. For $E_{pp}=13\,{\rm TeV}$, $c=(9/13)^{2}=0.479$, while $c=0.413$ for $E_{pp}=14\,{\rm TeV}$, etc.

So we have $\eta(13.6\,{\rm TeV})=1.80$ and $\eta(14\,{\rm TeV})=2.51$. This indicates that only a factor of $2.5$ gains in going from $13\,{\rm TeV}$ to $14\,{\rm TeV}$. Compared to higher energies, we now have $\eta(20\,{\rm TeV})=23.2$ and $\eta(25\,{\rm TeV})=39.6$. About a factor of $20$ gain from $13$ to $20\,{\rm TeV}$. For even higher energies, $\eta(50\,{\rm TeV})=61.1$ and $\eta(100\,{\rm TeV})=62.8$. One improves a little (1.03 gain) going from $50\,{\rm TeV}$ to $100\,{\rm TeV}$, which is much less efficient compared to the improvement from $13\,{\rm TeV}$ to $25\,{\rm TeV}$. This is due to the behavior at $x\to 0$ which comes from $x^{2}$ suppression. That is, the enhancement is saturated.

Equation (7) is an oversimplification of the Bloch-wave solution. According to the Bloch-wave picture [4, 12], we have $\hat{\sigma}_{\rm BV}(v)=\sigma_{0}$ if $v$ falls inside a Bloch wave band and $\hat{\sigma}_{\rm BV}(v)=0$ otherwise. The center energies of Bloch bands and their widths are shown in Table 1. “Manton” [5, 6] and “AKY” [7] refer to two different parametrizations of the Sphaleron potential. Here for those bands with energies above the first row in Table 1, we consider them to be continuous due to the overlaps. So, for example, $\hat{\sigma}_{\rm BV}(v>9.113\,{\rm TeV})=\sigma_{0}$ for Manton potential. We neglect those bands with widths smaller than $10^{-9}\,{\rm TeV}$.

The PDF Eq. (6) used in the last subsection is also too crude. Here we use realistic PDFs from the CTEQ program. According to CT18 [10], the PDFs at the initial scale $Q_{0}=1.3\,{\rm GeV}$ could be parametrized as

where $P_{q}(y_{q};a_{3},a_{4},\cdots)$ and $y_{q}(x)$ are the polynomial functions that have different forms for each species. For PDFs at a higher energy scale, one could compute them by using renormalization equations. Details for those parameter values, polynomial forms, and higher scale evolution are included in Ref. [10]. Here we take the results from Ref. [10] to estimate the $\eta(E_{pp})$. We extract PDFs from the CT18NNLO dataset using LHAPDF. To have a consistent precision, we take discrete values of PDFs with the step $\delta x=10^{-4}$. Thus, for all numerical results, we take only $4$ significant digits below.

The PDFs morph for higher scale. $f(x\to 0,Q^{2})$ will usually becomes larger for higher $Q$ for every species. Also, the contribution from sea quarks and valance quarks shall be comparable for small $x$. One should include more bands as collision energy goes higher, and the integration region in $x_{1}$–$x_{2}$ phase space grows to include the smaller $x$ region This leads to the enhancement of the $\eta(E_{pp})$ for higher energies.

So far we have neglected the baryon-number conserving (BC) direction. Recall that different $n$ states have different numbers of baryons and leptons and so their ground states have slightly different energies. The resulting effective sphaleron potential is a slightly tilted periodic potential. In quantum mechanics, this alone will suppress the BV process, i.e., $\Delta n=0$. It is the presence of the BC direction that allows finite $\Delta n$ BV process to happen [11]. For our purpose here, we do not consider the tilted potential and take that including the BC direction in the phase space will further suppress the BV cross section.

Here we consider a simple scenario, named $\theta$ phase space suppression (PSS). There are two orthogonal momentum directions in the phase space: the BC $\vec{p}_{\rm C}$ and BV $\vec{p}_{\rm V}$ directions. One can write down

In the relativistic limit, it could be converted to $E_{qq}^{2}\equiv v^{2}=E_{\rm C}^{2}+E_{\rm V}^{2}$, where $E_{\rm C(V)}$ stands for the energy that goes into the baryon-number conserving (violating) direction. By introducing a parameter $\theta$, which is a random number that differs for every collision, one could conclude that only $E_{\rm V}=v\sin{\theta}$ shall participate in the BV process. Thus, the cross section is given by

where

Since we are considering Bloch bands here, such integration is performed over discontinuous bands as illustrated in Fig. 3. Here $D(\theta)$ is the shaded region,

where $E_{i}$ is the center energy of $i$th Bloch band and $\Delta_{i}$ is its width. For fixed $E_{pp}$, one could see that smaller $\theta$ indicates that one has to integrate over lower bands region in the phase space, where the band gaps are relatively huge and widths are exponentially smaller. Thus an extra suppression factor appears. Note that setting $\theta=\pi/2$ is equivalent to no suppression scenario. As shown in Fig. 4, smaller $\theta$ shall lead to huge suppression on the integration.

The cross section $\tilde{\sigma}_{\rm BV}$ depends on $\theta$ for every event. We average out $\theta$ according to its probability density $P(\theta)$, to compare the efficiency for different $E_{pp}$ in observing BV events. Thus, we have

It is natural to assume that $\theta$ is sampled from a uniform distribution for every collision. Here we choose $P(\theta)=2/\pi$ for $\theta\in[0,\pi/2]$ in the second line above.

The summation runs over all quark species that participate in the BV process. For simplicity, we consider only the dominating contribution from

The gluons do not participate in weak interactions and so contribute to the BV process only indirectly. So their contributions are not included here.

As a comparison between the simple cutoff [Eq.(7)] and band structure model of $\hat{\sigma}_{\rm BV}(v)$, in Table 2 we show the numerical result of $\sigma_{\rm BV}(E_{pp})/\sigma_{0}$ with various $E_{\rm sph}$ chosen for cutoff model together with the band structure. Numbers in Table 2 are all obtained under $\theta$ PSS for the sake of comparison. As one can see, the $\eta(E_{pp})$ result with the band structure is equivalent to a simple cutoff with an effective ${\hat{E}}_{\rm sph}\simeq 9.1\,{\rm TeV}$, slightly higher than the actual $E_{\rm sph}=9.0\,{\rm TeV}$. Also, one sees that the differences between Manton and AKY potentials are minor.

Figure 1 shows the enhancement on $\sigma_{\rm BV}(E_{pp})$ in Manton potential. For comparison, in AKY potential, one finds $\eta(14\,{\rm TeV})\simeq 6.508$, $\eta(20\,{\rm TeV})\simeq 1.842\times 10^{3}$ and $\eta(25\,{\rm TeV})\simeq 1.550\times 10^{4}$; that is a $4$ orders of magnitude enhancement going from $13\,{\rm TeV}$ to $25\,{\rm TeV}$. However, going from $50\,{\rm TeV}$ to $100\,{\rm TeV}$ will only give us roughly $1$ order-of-magnitude improvement in the event rate. Note that the size of phase space suppression from the random $\theta$ is about $1$ order of magnitude at the beginning, $E_{pp}\sim 13\,{\rm TeV}$, and decreases to only roughly $0.5$ at $E_{pp}\sim 100\,{\rm TeV}$.

We consider another scenario, which simply introduces a suppression factor to the cross section integral, named $K$ phase space suppression. This is

where the integration $D=D(\pi/2)$ is the band structure consideration without $\theta$ suppression.

Naturally, the phase space suppression factor $K(v)$ shall interpolate from $0$ to $1$. This is because when the energy is small, one shall expect little budget for BV. Meanwhile, when the energy is high enough, sphaleron potential could be neglected, and then the phase space suppression should vanish. Also, $K(v)$ should be significantly enhanced when $v\sim E_{\rm sph}$ because the distinct scale in the BV process is $E_{\rm sph}$. Thus, we assume that

Here we take a monotonically increasing function

which is parametrized by $\alpha>0$ and $\beta>0$. Note that $\beta=0$ corresponds to no suppression.

Adopting CT18 PDFs [10] and considering quark content Eq. (14) in the Bloch band picture, we numerically calculate $\sigma_{\rm BV}(E_{pp})$ in unit of $\sigma_{0}$ with various choice of $\alpha$ and $\beta$ in Table 3 and 4. Minor differences between Manton and AKY potentials are observed and order-of-magnitude behavior is the same. $K$ factor suppression is strong at low $E_{pp}$ and becomes weak when $E_{pp}$ go higher as anticipated. Figure 5 and Fig. 6 show the enhancement factor $\eta(E_{pp})$ in the Manton potential, which is similar to the AKY potential. As shown in Fig. 6, varying $\alpha$ has little impact on $\eta(E_{pp})$. For larger $\beta>3.4$, one essentially changes the behavior of $K(E_{\rm sph})$, which shall lead to a significant change on $\eta(E_{pp})$ and against our assumption in Eq. (16). For reasonable choices of $\alpha$ and $\beta$, one shall have $\sim 4$ order enhancement on BV event rate going from $E_{pp}=13\,{\rm TeV}$ to $25\,{\rm TeV}$, and only about $1$ order gain from $E_{pp}=50\,{\rm TeV}$ to $100\,{\rm TeV}$.