Baryon Number Violating Rate as A Function of the Proton-Proton Collision Energy

The baryon-number violation (BV) happens in the standard electroweak model. According to the Bloch-wave picture, the BV event rate shall be significantly enhanced when the proton-proton collision center of mass (COM) energy goes beyond the sphaleron barrier height $E_{\rm sph}\simeq 9.0\,{\rm TeV}$. Here we compare the BV event rates at different COM energies, using the Bloch-wave band structure and the CT18 parton distribution function data, with the phase space suppression factor included. As an example, the BV cross-section at 25 TeV is 4 orders of magnitude bigger than its cross-section at 13 TeV.


I. INTRODUCTION
Matter-antimatter asymmetry is an important mystery in our Universe.The baryon-number violation (BV) via the instanton [1] in the Standard Electroweak Model observed by 't Hooft [2,3] provides a crucial avenue to understanding baryogenesis.Therefore, observing (confirming) such BV in the laboratory will be immensely valuable.
The underlying physics of the BV process can be reduced to a simple quantum mechanical system.With the Chern-Simons number Q (or n = m W Q/π) as the coordinate, one obtains the one-dimensional time-independent Schrödinger equation, with mass m ≃ 17 TeV [4]: where the sphaleron potential V (Q) is periodic, with minima at integer values of n and maxima at n + 1/2, with barrier height E sph = 9.0 TeV [5][6][7].Although this Schrödinger equation is well accepted, it is the interpretation of the underlying physics of V (Q)'s periodicity that needs clarification: whether the solution of this equation has a Bloch wave band structure or not.Let us first consider the SU (2) gauge theory without the fermions: in this case, all integer n states are physically identical; that is, n → n±1 simply goes back to itself (though in a different gauge).So there is no band structure, as is the case in the QCD theory.This is analogous to a rigid pendulum rotating by 2π via tunnelling [8].
Once left-handed fermions couple to the electroweak SU (2) gauge theory, different n state has different baryon (and lepton) numbers, so they are physically different: as we go from the n to the n + ∆n state, baryon number changes by 3∆n.As Q runs from −∞ to +∞, a band structure emerges.Changing Q is no longer exponentially suppressed within each band.For energies below the height of the sphaleron potential of 9.0 TeV, band gaps dominate over the bandwidths, so the BV crosssection σ BV is still small.As E increases, the bandwidths grow while the gaps between bands decrease.Once the energy goes above 9.0 TeV, bands take over, and the BV cross-section is no longer exponentially suppressed.This is in contrast to the QCD theory which has no bands.
The Large Hadron Collider (LHC) at CERN ran at proton-proton collision energy E pp = 13 TeV and is presently running at E pp = 13.6 TeV.Since the quarks and gluons inside a proton share its energy, the quarkquark energy E qq is only a fraction of the total E pp .It is important to see how σ BV grows as E pp increases.This is a simple kinematic issue.Reference [9] has estimated the growth of σ BV as a function of E pp .Here we like to dwell into the estimate in more detail by taking the band structure fully into account as well as an additional phase space factor: even if E qq > 9.0 TeV, not all energy goes to the BV process.That is, E qq has to be shared between baryon-number conserving (BC) scattering and BV scattering.In this note, we present for E pp above 13 TeV, the ratio A rough estimate simply assumes that σ BV is totally suppressed for E qq < 9.0 TeV and completely unsuppressed for E qq ≥ 9.0 TeV.As an exercise, we first present an analytical evaluation of η(E pp ).However, as we shall see, this estimate is not accurate enough.Using the particle distribution function (PDF) for the valence quarks from the CTEQ program [10], the estimate for η(E pp ) agrees with that in Ref. [9].Next, we take the band structure into account: σ BV is completely unsuppressed for E qq inside a Bloch band and totally suppressed for E qq in a band gap.It turns out this result is close to the above simple estimate if we choose the critical E qq = 9.1 TeV instead of 9.0 TeV.However, even inside a Bloch band, not all E qq goes to BV scatterings; some energies flow to the baryon-conserving (BC) channel.
We also perform estimates on the BV cross-section including this phase space suppression factor, again using parton distribution functions (PDFs) from the CTEQ program [10].Our final result is presented in Fig. 1.We see that σ BV (25 TeV) is 4 orders-of-magnitude bigger than σ BV (13 TeV).Including gluon + quark scattering has little effect on the result as gluon PDF is rather soft, as shown in Fig. 2.

II. ESTIMATE OF η(Epp)
Consider proton-proton (pp) collisions.In the center of mass (COM) frame, the proton momenta are P 1 = (E, 0, 0, E) and P 2 = (E, 0, 0, −E).where s = 4E 2 .So the quark-quark momentum is where x j is the fraction of momentum carried by quark q j .The invariant energy carried by the quark-quark system is v, where Let f q/p (x j , Q 2 ) be the PDF of quark q j inside a proton at the scale Q.So the BV cross-section σ BV (E pp ) is given by, before the inclusion of the phase space factor, where E pp ≡ √ s, and v = √ x 1 x 2 E pp .

A. Crude Estimate
We can make a rough analytical estimate to get some idea, even though the resulting numerical values need improvement.As a start, we consider a simple (toy) PDF for valence quarks q in a proton which is scaleindependent, where This PDF (5) allows an analytic discussion, but is only qualitatively valid.
If we do not care about species, we shall choose 1 0 dxf q/p (x) = 3, so A q = 180.The Bloch-wave picture indicates that the σBV (v) is exponentially enhanced when v ≳ E sph due to the overlap of high energy Bloch bands.Thus, for the purpose of estimating, we here simply take a cut-off model, where c = (E sph /E pp ) 2 .σ 0 is an overall normalization.We may assume that, for v ≫ 10 TeV, σ 0 ≲ σ total (pp), where σ total (pp) does not vary much.Since we are comparing the BV event rate between different E pp (1), the value of σ 0 is not important here.With this approximation, we could write where As a check, we have σ BV (c = 1) = 0.As a reasonable approximation, we take E sph = 9 TeV as a benchmark.For E pp = 13 TeV, c = (9/13) 2 = 0.479, while c = 0.413 for E pp = 14 TeV, etc.
So we have η(13.6TeV) = 1.80 and η(14 TeV) = 2.51.This indicates that only a factor of 2.5 gains in going from 13 TeV to 14 TeV.Compared to higher energies, we now have η(20 TeV) = 23.2 and η(25 TeV) = 39.6.About a factor of 20 gain from 13 to 20 TeV.For even higher energies, η(50 TeV) = 61.1 and η(100 TeV) = 62.8.One improves a little (1.03 gain) going from 50 TeV to 100 TeV, which is much less efficient compared to the improvement from 13 Tev to 25 TeV.This is due to the behavior at x → 0 which comes from x 2 suppression.That is, the enhancement is saturated.

B. Numerical Estimate with θ Phase Suppression
Eq. ( 6) is an oversimplification of the Bloch-wave solution.According to the Bloch-wave picture [4,11], we have σBV (v) = σ 0 if v falls inside a Bloch wave band and σBV (v) = 0 otherwise.The center energies of Bloch bands and their widths are shown in Table I.Here for those bands with energies above the first row in Table I, we consider them to be continuous due to the overlaps.So, for example, σBV (v > 9.113 TeV) = σ 0 for Manton potential.We neglect those bands with widths smaller than 10 −9 TeV.
The PDF Eq. ( 5) 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 GeV could be parametrized as where P q (y q ; a 3 , a 4 , • • • ) 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  3.A sketch of phase space for x1-x2.Gray shaded region is the integration region in Eq. ( 4), where v falls in the Bloch bands.
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 η(E pp ).
The PDFs morph for higher scale.f (x → 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.As we go to higher energies, one shall include more bands, 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 η(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., ∆n = 0.It is the presence of the BC direction that allows finite ∆n BV process to happen [12].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 θ phase suppression (PS).There are two orthogonal momentum directions in the phase space: the BC and BV directions.One could write down where E C(V) stands for the energy that goes into the baryon-number conserving (violating) direction.By introducing a parameter θ, which is a random number that differs for every collision, one could conclude that only E V = v sin θ 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 discontinuing bands as illustrated in Fig. 3.Here D(θ) is the shaded region, where E i is the center energy of i-th Bloch band and ∆ i is its width.For fixed E pp , one could see that smaller θ 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 θ = π/2 is equivalent to no suppression scenario.As shown in Fig. 4, smaller θ shall lead to huge suppression on the integration.The cross-section σBV depends on θ for every event.We average out θ according to its probability density P (θ), to compare the efficiency for different E pp in observing BV events.Thus, we have ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0.0 0.5 1.0 1.5 It is natural to assume that θ is sampled from a uniform distribution for every collision.Here we choose P (θ) = 2/π for θ ∈ [0, π/2] in the second line above.The summation shall run over all quark species that participate in the BV process.For simplicity, we consider only the dominating contribution from q, q ′ ∈ {u, d, s, ū, d, s} . (13) 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 cut-off (Eq.( 6)) and band structure model of σBV (v), in Table II we show the numerical result of σ BV (E pp )/σ 0 with various E sph chosen for cut-off model together with the band structure.Note that here they are both under the θ PS.As one can see, the η(E pp ) result with the band structure is equivalent to a simple cut-off with an effective Êsph ≃ 9.1 TeV, slightly higher than the actual E sph = 9.0 TeV.Also, one sees that the differences between Manton and AKY potentials are minor.
Figure 1 shows the enhancement on σ BV (E pp ) in Manton potential.For comparison, in AKY potential, one finds η(14 TeV) ≃ 6.51, η(20 TeV) ≃ 1.84 × 10 3 and η(25 TeV) ≃ 1.55 × 10 4 ; that is a 4 orders of magnitude enhancement going from 13 TeV to 25 TeV.However, going from 50 TeV to 100 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 θ is about 1 order of magnitude at the beginning, E pp ∼ 13 TeV, and decreases to only roughly 0.5 at E pp ∼ 100 TeV.

C. Numerical Estimate with K Phase Suppression
We consider another scenario, which simply introduces a suppression factor to the cross-section integral, named TABLE II.σ(Epp) with band structure and simple cut-off (Eq.( 6)).Here θ phase suppression is applied.The first three columns are cut-off models and the last two are band models.K phase suppression.This is

Epp
where the integration D = D(π/2) is the band structure consideration without θ suppression.Naturally, the phase 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 suppression Also, K(v) should be significantly enhanced when v ∼ E sph because the distinct scale in the BV process is E sph .Thus, we assume that Here we take a monotonically increasing function which is parameterized by α > 0 and β > 0. Note that β = 0 corresponds to no suppression.Adopting CT18 PDFs [10] and considering quark content Eq. ( 13) in the Bloch band picture, we numerically calculate σ BV (E pp ) in unit of σ 0 with various choice of α and β in Table III and IV.Minor differences between Manton and AKY potentials are observed and order-ofmagnitude 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 η(E pp ) in the Manton potential, which is similar to the AKY potential.As shown in Fig. 6, varying α has little impact on η(E pp ).For β = 10, one essentially changes the behavior of K(E sph ), which shall lead to a significant change on η(E pp ) and against our assumption in Eq. (15).For reasonable choices of α and β, one shall have ∼ 4 order enhancement on BV event rate going from E pp = 13 TeV to 25 TeV, and only about 1 order gain from E pp = 50 TeV to 100 TeV.

III. SUMMARY AND DISCUSSION
In this short note, we demonstrate the enhancement of the baryon-number violating event rate when the COM energy for the pp collider is increased.The estimate includes the Bloch band structure for unsuppressed BV scatterings and the phase space suppression from the baryon-number conserving direction.The Bloch band structure yields an effective cut-off of E qq ≃ 9.1 TeV, a   little above the simple cut-off of E qq ≃ 9.0 TeV 1 .The phase space suppression factor is formulated in two ways, θ and K phase suppression.θ scenario introduces a random parameter θ for every collision describing the energy budget of participating in the BV and BC process.We compare the event rate for different COM energy by integrating out θ, which is sampled from a uniform distribution.K PS scenario introduces a monotonic function that describes the suppression from phase space.For reasonable choices of parameters in K, we have similar results as that in the θ PS case.In summary, combining all scenarios considered above, we now have (η(13 TeV) = 1 by definition), up to two significant digits, η(13.6 TeV) ≃ 3.1-3.
The results indicate that increasing the COM pp energy from 13 TeV to 25 TeV will yield a huge enhancement to the event rate, while the enhancement will be more modest going from 50 TeV to 100 TeV.

Manton
FIG. 1. Solid curve is η(Epp) = σBV(Epp)/σBV(13 TeV) with the θ phase suppression (PS) as a function of Epp.The dashed curve is the η(Epp) without the PS for comparison.Bloch bands for the AKY potential produce a very similar result as bands for the Manton potential shown here.

7 E
FIG.2.The solid curve is η(Epp) without gluon contribution.The dashed curve is η(Epp) with gluon + quark scattering included for 50-100 TeV.Here they are calculated under the cut-off model with the effective cut-off at Êsph = 9.1 TeV and θ phase suppression.

FIG. 4 .
FIG.4.Iuu as a function of θ for Epp = 13 TeV.Here Manton and AKY potentials lead to very similar results.Other I qq ′ also gives a similar suppression behavior.

TABLE I .
[4]ch Wave Bands from Ref.[4].Here Ei is the band center energy and ∆i is the band width.Those bands with a width smaller than 10 −9 TeV are neglected for simplicity. 2FIG.

TABLE III .
σ(Epp) with K phase suppression factor in band model of Manton potential.Here E sph = 9 TeV.

TABLE IV .
σ(Epp) with K phase suppression factor in band model of AKY potential.Here E sph = 9 TeV.