Search for Sphalerons in Proton-Proton Collisions

In a recent paper, Tye and Wong (TW) have argued that sphaleron-induced transitions in high-energy proton-proton collisions should be enhanced compared to previous calculations, based on a construction of a Bloch wave function in the periodic sphaleron potential and the corresponding pass band structure. Here we convolute the calculations of TW with parton distribution functions and simulations of final states to explore the signatures of sphaleron transitions at the LHC and possible future colliders. We calculate the increase of sphaleron transition rates in proton-proton collisions at centre-of-mass energies of 13/14/33/100 TeV for different sphaleron barrier heights, while recognising that the rates have large overall uncertainties. We use a simulation to show that LHC searches for microscopic black holes should have good efficiency for detecting sphaleron-induced final states, and discuss their experimental signatures and observability in Run 2 of the LHC and beyond. We recast the early ATLAS Run-2 search for microscopic black holes to constrain the rate of sphaleron transitions at 13 TeV, deriving a significant limit on the sphaleron transition rate for the nominal sphaleron barrier height of 9 TeV.


Introduction
Non-perturbative effects in the electroweak sector of the Standard Model are predicted to violate baryon (B) and lepton (L) conservation, violating the combination B + L while conserving B − L. The first example was provided by electroweak instantons [1], which yield ∆B = 3 transitions that are suppressed to unobservable levels by factors ∼ exp(−2π/α W ), where α W = g 2 W /4π is the SU(2) coupling strength. The second example was provided by sphalerons [2,3], which are classical solutions of the electroweak field equations that interpolate between vacua with different values of the Chern-Simons number, providing a potential barrier E Sph to ∆B = 3 transitions that is expected to be 9 TeV. It has been thought that experimental observation of sphaleron-induced transitions would also be unobservable for the foreseeable future [4][5][6][7][8][9][10][11], because ∆n = ±1 transitions would be suppressed by exp(O(−4π/α W )).
However, this longstanding consensus has been challenged in a bold recent paper [12] by S.-H. Henry Tye and Sam S. C. Wong (TW), who argue that sphaleron-induced transition rates could be much larger than had been estimated previously. They argue that an essential element in calculating the rate of ∆n = 0 transitions is the periodic nature of the effective Chen-Simons potential, which should be taken into account by constructing the corresponding Bloch wave function. Their approach leads to a band structure for transitions through the sphaleron barrier, resulting in a reduced suppression at energies < E Sph that disappears entirely at energies ≥ E Sph . As stressed in [12], this remarkable claim raises the possibility that sphaleron-induced transitions might be observable at the LHC and higher-energy proton-proton colliders. The experimental observation of such transitions would not only be a beautiful confirmation of profound theoretical insights, but would also have important cosmological implications, since sphalerons are thought to have played an essential rôle in generating the baryon asymmetry of the Universe [11,[13][14][15][16][17].
In this paper we follow up the suggestion of TW by calculating the energy dependence of the rates for sphaleron-induced transitions in proton-proton collisions, including the factors arising from quark parton distribution functions, and use simulations of their possible final states to study the possible signatures of such transitions. As stressed by TW, there are inevitable uncertainties in calculations of the rates for sphaleron-induced transitions, notably including the sphaleron barrier height E Sph , the coefficient inside the exponential suppression, and any possible prefactor. That said, our calculations encourage us to explore how the rate for sphaleron-induced transitions might be constrained by experiments at the LHC, possibly during its Run 2 that has now started. Accordingly, we simulate the final states of sphaleron-induced transitions, demonstrating that the searches for microscopic black holes that have already been designed would have good acceptance for sphaleron-induced final states, which would also possess additional distinctive signatures. As an illustration, we constrain sphaleron transition rates by recasting the results of the ATLAS Run-2 search for microscopic black holes using ∼ 3 fb −1 of data recorded at 13 TeV in 2015 [18]. We find that these data already exclude a pre-exponential transition rate factor of unity for the nominal sphaleron barrier height of 9 TeV .

Theoretical Background
It is argued in [12] that sphaleron transitions can be modelled by a one-dimensional Schrödinger equation of the form − 1 2m where m is an effective "mass" parameter for the Chern-Simons number n whose value was first calculated in [12], Q ≡ µ/m W where µ is defined implicitly by nπ = µ − sin(2µ)/2, and the effective potential is taken from [2]: Two evaluations of m were discussed in [12]: one based on [2] that yielded the estimate m = 17.1 TeV, and the other based on [19] that yielded the estimate m = 22.5 TeV. The final results for the rate of sphaleron-induced transitions were very similar, and here we follow [12] in adopting the [2]-based calculation that led to m = 17.1 TeV.
The sphaleron barrier height E Sph is given by In a pure SU(2) theory, one finds E Sph = 9.11 TeV, and it is estimated that incorporating the U(1) of the Standard Model reduces this by ∼ 1%. Here we follow [12] in assuming a nominal value of E Sph = 9 TeV, while presenting some numerical results for the alternative choices E Sph = 8, 10 TeV. Later, we also use a recast of early Run-2 searches for microscopic black holes to constrain the sphaleron transition rate as a function of E Sph . As was discussed in detail in [12], the Bloch wave function for the periodic potential (2) is straightforwardly obtained, and the corresponding conducting (pass) bands can be calculated, as well as their widths and the gaps between the bands. The lowest-lying bands are very narrow, but the widths increase with the heights of the bands. Averaging over the energies E 1,2 of the colliding quark partons yields a strong suppression at E 1 + E 2 E Sph , which corresponds to the exponential suppression found in a conventional tunnelling calculation. However, this suppression decreases as E 1 + E 2 → E Sph , and there is no suppression The result of the analysis in [12] can be summarized in the partonic cross-section where E is the centre-of-mass energy of the parton-parton collision, c ∼ 2 and the suppression factor S(E) is shown in Fig. 8 of [12]. As seen there, it rises from the value S(E) = −1 in the low-energy limit (E E Sph ) to S(E) = 0 for energies E ≥ E Sph , with very similar results being found in [12] for calculations based on the work of [2] and [19]. For the purpose of our numerical calculations, we approximate S(E) at intermediate energies by whereÊ ≡ E/E Sph and a = −0.005.
The overall magnitude of Eq. (4) is not given. We speculate that the relevant scale should be proportional to the non-perturbative electro-weak cross-section for q-q scattering, σ EW qq . Analogously to the fact that the inelastic p-p cross-section is given roughly by ∼ 1/m 2 π , we take σ EW qq ∼ 1/m 2 W . Our cross-section formula is, thus, given as where p is an unknown constant and dL ab dE is the parton luminosity function of the colliding quarks a and b, which are obtained from the parton distribution functions at a momentum fraction x, f a (x), evaluated at the appropriate energy scale E: where E CM is the centre-of-mass energy of the p-p collision and τ = E 2 /E 2 CM .

Cross-Section Calculations
We include in our calculations collisions of all quarks and antiquarks in the lightest two generations, namely u, d, s and c. We recall that only left-handed (SU(2) doublet) quarks are active in inducing sphaleron transitions, so that the usual unpolarized quark-quark parton collision luminosity functions must be reduced by a factor 4. Additionally, we expect that quarks in the same generation must collide in an antitriplet state, reducing the corresponding luminosity functions by another factor 3. In principle, one should also incorporate Cabibbo mixing, but this is unimportant compared with the uncertainties in the calculation.
The upper panel of Fig. 1 displays the relative contributions of the collisions of different quark flavours for the nominal case E CM = 14 TeV, E Sph = 9 TeV, c = 2, with the normalization corresponding to p = 1 in (6). We see that, as expected, the dominant contribution to the sphaleron cross section is due to uu collisions, with ud collisions being the second most important, and other processes contributing < 3% of the total. Sphaleron production by collisions involving d quarks are suppressed at 14 TeV because the u parton distribution function is much larger than that for the d quark at large momentum fraction x.
The lower panels of Fig The dot-dashed and dashed curves in Fig. 2 are for the cases E Sph = 8 and 10 TeV, which lie far outside the uncertainty in E Sph ∼ 1% quoted in [12]. It is clear that the LHC cross section is smaller for larger E Sph , and the energy dependence is steeper, whereas the opposite statements hold for smaller E Sph . However, whereas in the former case sphaleron-induced processes could be more visible in Run 2 of the LHC, even in the latter case increasing E CM  (6) with S given by (5). The contributions of different parton-parton collision processes are colour-coded as indicated. Lower panels: As above, for the cases E CM = 13, 33 and 100 TeV.
should be a priority for the LHC.
Looking beyond the LHC, Fig. 2 shows that the sphaleron transition rate would increase significantly at colliders with higher E CM . Specifically, for our nominal choices E Sph = 9 TeV,  ball-park, such higher-energy colliders would be veritable sphaleron factories. However, we emphasize again that the overall magnitude of the sphaleron transition rate is very uncertain.
One should, perhaps, instead regard Fig. 2 as showing that higher-energy collisions may provide sensitivity to sphaleron transitions for p 1.

Simulations of Sphaleron-Induced Processes
We turn now to the prospective observability of sphaleron-induced processes, the simplest possibility being ∆n = −1 processes that give rise to effective interactions involving one member of each electroweak doublet, i.e., e/ν e , µ/ν µ , τ /ν τ , and 3 colours of u/d, c/s and t/b, leading to transitions of the form qq →¯ ¯ ¯ qqqqqqq .
Since the dominant processes are induced by uu and ud collisions: the final states should contain a singleū/d antiquark, one antilepton from each generation, threec/s antiquarks and threet/b antiquarks, for a total of 10 final state particles. The initial and final states are constrained so that the total electric charge is conserved. We simulate the momenta of final state particles according to the phase space. We also simulate the decays of heavy particles (t, W and τ ). We accept only particles with p T > 20 and |η| < 2.5. Neutrinos are removed from the list of observable particles.
The  1 There are suggestions that the baryon and lepton number violating processes are enhanced if fermions are produced associated with many O(1/α W ) electroweak bosons [5,6,[20][21][22]. We leave the investigation of this possibility for future work. the red histograms are for the ∆n = −1 processes leading to 10-particle final states discussed above (8). On should also consider processes with other values of ∆n, the next simplest being the ∆n = +1 process that leads to 14-particle final states: q q → q q q q q q q q q q q , whose simulation yields the blue histograms in Fig. 4. The difference between the cases with one and two top quarks is due to the phase-space suppression of final states with more top quarks than bottom quarks, as is that between the cases with zero and three top quarks.
Additional properties of 10-particle sphaleron final states are shown in Fig

Analysis of ATLAS 201Data
The ATLAS Collaboration has recently published the (null) results of a search for microscopic black holes using ∼ 3 fb −1 of data at 13 TeV recorded in 2015 [18]. This analysis was based on measurements of the numbers of events in search regions (SRn jet ) defined by cuts in the number of jets, n jet ≥ 3 to 8, accompanied by cuts in H T 5 TeV. We now compare the ATLAS measurements with our simulations of the final states induced by sphaleron transitions.   We may therefore recast the ATLAS search as a relatively efficient search for ∆n = −1 sphaleron-induced transitions. For each value of E Sph , we select the SRn that is expected to yield the best limit, finding that SR8 is expected to be the most sensitive for E Sph 9.3 TeV whereas SR7 is the most sensitive for E Sph 9.3 TeV. The exclusion limit resulting from this recasting of the ATLAS black hole search is shown in the right panel of Fig. 7. We display the 95% CL constraint in the (E Sph , p) plane, which is quite insensitive to c ∈ [1,4]. We note that this preliminary result already excludes p = 1 for the nominal value of E Sph = 9 TeV.
Thus far, we have discussed ∆n = −1 sphaleron transitions in which two quarks collide to yield 3 antileptons and 7 antiquarks, and now we consider the next simplest possibility of a ∆n = +1 sphaleron transition in which two quarks collide to yield 3 leptons and 11 quarks. The left panel of Fig. 6 shows the simulated H T distribution for this possibility as a blue histogram, which is shifted to larger values than for the ∆n = −1 sphaleron transitions. Correspondingly, the acceptances in the ATLAS search regions are higher for ∆n = +1 transitions, as seen in the left panel of Fig. 8, reaching ∼ 0.8 for SR8 for the nominal   Fig. 6, but for ∆n = +1 sphaleroninduced transitions to 14-particle final states. Right panel: The exclusion in the (E Sph , p) plane, as in Fig. 7 but for sphaleron-induced transitions to 14-particle final states. E Sph = 9 TeV. Consequently, the 95% CL exclusion in the (E Sph , p) plane for ∆n = +1 transitions is correspondingly stronger than for ∆n = −1 transitions, as seen in the right panel of Fig. 8, excluding p 0.2 for the nominal E Sph = 9 TeV 2 .
Run 2 of the LHC is expected to yield ∼ 100 fb −1 of data at 13 TeV, which should enable the sensitivity to p to be improved to ∼ 0.01 for E Sph = 9 TeV, which could be improved with an optimized, targeted analysis of the final states in sphaleron-induced transitions.
For example, as was pointed out in [12], ∆n = −1 sphaleron-induced processes would yield final states with multiple positively-charged leptons: e + , µ + and/or τ + . In particular, 1/8 of the final states would contain the distinctive combination of all three positivelycharged leptons: e + + µ + + τ + . Also, every ∆n = −1 sphaleron-induced event would contain The sensitivity could be further improved by a factor ∼ 6 if the LHC could make collisions at 14 TeV, and by another factor of 30 with 3000 fb −1 of luminosity, pushing the sensitivity to p < 10 −4 for E Sph = 9 TeV. The sensitivity could be further improved to p ∼ 10 −11 for two experiments each with 20,000 fb −1 of luminosity at 100 TeV in the centre of mass. The fact that future searches at the LHC and a possible future collider have such interesting prospective sensitivities to sphaleron-induced transitions reinforces the importance of assessing the reliability of the TW estimate of the sphaleron transition rate. Both the exponential factor S(E) and the pre-exponential factor p need close scrutiny. Our exploratory study shows that this is not just an academic study, but could have exciting implications for future pp collider experiments.