Search for Sphalerons: IceCube vs. LHC

We discuss the observability of neutrino-induced sphaleron transitions in the IceCube detector, encouraged by a recent paper by Tye and Wong (TW), which argued on the basis of a Bloch wave function in the periodic sphaleron potential that such transitions should be enhanced compared to most previous calculations. We calculate the dependence on neutrino energy of the sphaleron transition rate, comparing it to that for conventional neutrino interactions, and we discuss the observability of tau and multi-muon production in sphaleron-induced transitions. We use IceCube 4-year data to constrain the sphaleron rate, finding that it is comparable to the upper limit inferred previously from a recast of an ATLAS search for microscopic black holes at the LHC with $\sim 3$/fb of collisions at 13 TeV. The IceCube constraint is stronger for a sphaleron barrier height $E_{\rm Sph} \gtrsim 9$ TeV, and would be comparable with the prospective LHC sensitivity with 300/fb of data at 14 TeV if $E_{\rm Sph} \sim 11$ TeV.


Introduction
The recent discovery of the Higgs boson with a mass of 125 GeV by ATLAS and CMS [1,2] completes the electroweak sector of the Standard Model. Ongoing measurements of interactions of Higgs and gauge bosons find good agreement with perturbative predictions of the Standard Model with its spontaneously broken SU (2) × U (1) gauge group [3][4][5][6][7][8]. A direct consequence of the SU (2) gauge group, beyond the perturbative regime, is the existence of non-perturbative topological effects. Field configurations with finite Euclidean action are classified by an integer topological winding number, namely the Chern-Simons number N CS .
Topologically distinct ground states are separated by an energy barrier, and the sphaleron [9] is an extremal saddle point on top of the barrier with half-integer N CS and an energy E Sph 9 TeV. Owing to the Adler-Bell-Jackiw anomaly, transitions through sphaleron configurations would lead to striking (B+L)-violating processes. Direct observation of such interactions can be of capital importance in explaining the mechanism underlying the cosmological baryon asymmetry [10][11][12][13][14], which might arise from the transmutation of a primordial lepton asymmetry.
While the energy of the sphaleron is directly linked to the shape of the effective potential for the Chern-Simons number and is thus rather undisputed, its production rate is subject to large theoretical uncertainties [15][16][17]. Over the years, there have been many estimates of the rate of sphaleron transitions in high-energy collisions, most of them with discouraging results for the prospects for experimental searches. However, a new approach [9], exploiting the periodicity of the Chern-Simons potential, has challenged this paradigm and argued in favour of enhanced sphaleron-induced transition rates.
The authors indicate that, since the effective Chern-Simons potential is periodic, it is appropriate to use a Bloch wave function to estimate the sphaleron-induced transition rate.
They find that, although the transition rate is still strongly suppressed at energies below the sphaleron threshold energy, the rate may not be suppressed at higher energies above E Sph . Motivated by this suggestion, two of us (JE and KS) have analyzed the observability of sphaleron-induced transitions at the LHC and possible future higher-energy pp colliders, and recast a recent ATLAS search for microscopic black holes using ∼ 3/fb of data at 13 TeV in the centre-of-mass as a search for sphalerons [18]. In this way, expanding on previous proposals [19,20], we were able to establish for the first time a significant direct experimental constraint on sphaleron-induced transitions, assess the potential improvement in sensitivity of future LHC runs with higher integrated luminosities and possibly energy, and preview the corresponding prospects for possible future higher-energy pp colliders.
In this connection, it is important to remember that cosmic rays provide collisions with centre-of-mass energies beyond those attainable with the LHC. However, the low fraction of ultra-high-energy pp collisions that would produce sphaleron transitions, combination with the limitations of the Auger experiment [21] for extracting detailed information from air showers renders challenging this avenue in searches for sphalerons [22].
Better prospects may be offered by ultra-high-energy neutrino events [23][24][25][26][27], which could be observed in a cubic-kilometer neutrino telescopes, such as IceCube [28,29] and KM3NeT [30]. As we discuss in more detail below, the estimates of TW suggest that the rate of sphaleron-induced transitions in the highest-energy cosmic-ray collisions observed by IceCube could be comparable to the conventional neutrino cross section, and might even dominate the interactions of cosmogenic neutrinos [31] produced by the Greisen-Zatsepin-Kuzmin (GZK) process [32,33]: p + γ CMB → π + → ν.
Accordingly, in this paper we extend the analysis of ES to ultra-high-energy neutrino interactions, estimating the upper limit on sphaleron transitions obtainable from present IceCube results and considering the implications for GZK neutrinos. Remarkably, we find that the present IceCube sensitivity is very similar to that of the first LHC data at 13 TeV.
If the rate of sphaleron transitions were to saturate the present LHC limit, the rate of GZK neutrino interactions would be significantly higher than is conventionally estimated, improving significantly the prospects for their future detection with IceCube or a cubickilometre detector.

Neutrino-Nucleon Cross-Section Calculations
The analysis of [9] is based on the idea that sphaleron transitions changing the Chern-Simons number n can be modelled by considering a Bloch wave function for an effective one-dimensional Schrödinger equation where m is an effective "mass" parameter variously estimated to be ∼ 17. The sphaleron barrier height E Sph is the maximum value of the effective potential V (Q), which is E Sph = 9.11 TeV in a pure SU (2) theory, and is estimated to be reduced by ∼ 1% when the U (1) of the Standard Model is included. Following [9], we assume E Sph = 9 TeV as a nominal value, but present some numerical results for E Sph ∈ [8,11] TeV. Ref. [9] found that the pass-band structure in the Bloch wave function approach reproduced the expected (near-exponential) tunnelling suppression of sphaleron transitions in collisions of quark partons with subprocess centre-of-mass energies √ŝ E Sph , but found that there is no suppression for √ŝ ≥ E Sph . They found that the rates of sphaleron-induced transitions were similar for the two values of m, and adopted m = 17.1 TeV for definiteness.
The result of the TW analysis can be expressed as partonic cross-section where c ∼ 2 and the suppression factor S( √ŝ ) is shown in Fig. 8 of [9]. As discussed in [18], In the absence of a reliable calculation of the overall magnitude of Eq. (2.3), following [18] we parametrize the partonic cross section for the sphaleron-induced neutrino-quark collision for √ŝ > E Sph and apply the suppression factor (2.4) for √ŝ < E Sph . Our numerical results are relatively insensitive to the form of this suppression factor. The overall factor p in (2.5) depends in general onŝ [9]. However, our result is also not very sensitive to such an energy dependence, since (as we discuss below) the interaction is dominated by subprocess energies near the threshold √ŝ E Sph , due to the sharply-falling cosmogenic neutrino flux. The cross section for sphaleron transitions in neutrino-nucleon collisions is given by where f q (x, µ) is a parton distribution function for the quark flavour q and m N is the mass of nucleon. The neutrino-nucleon centre-of-mass energyÊ = √ 2m N E ν , neglecting the m 2 N term, and the neutrino-parton subprocess centre-of-mass energy   The cross section estimates in Fig. 1 can be convoluted with the cosmogenic neutrino flux, , to calculate the event rates. We use the cosmogenic neutrino flux estimated in [36] throughout this paper. The event rate in the IceCube detector also depends on the energy-dependent effective neutrino detection area, A eff (E ν ), which has been evaluated by the IceCube collaboration [37] using conventional neutrino-nucleon interaction. Assuming the same detection efficiency, we estimate the sphaleron-induced IceCube event rate as where E thres ν is the energy threshold of incoming cosmogenic neutrinos. In the second integral we take into account only neutrinos coming from the upper hemisphere of IceCube, since  the neutrinos from the lower hemisphere will be absorbed by the interaction with the Earth.

Events/Year
In Fig. 2   inside the nucleon targets in the ice. As was to be expected, interactions with u and d quarks dominate, followed by interactions with antiquarks and heavy flavours. In the right panel we show the corresponding distributions in the reduced neutrino-quark subprocess centreof-mass energies √ŝ , which are sharply peaked at the sphaleron energy E Sph , taken here to have its nominal value of 9 TeV. This peaking implies that our results would not be affected strongly by a possible energy dependence in the overall factor p, but depend essentially only on the value of p at the sphaleron threshold energy.

Leptons in Sphaleron-Induced Transitions
In the IceCube detector [29], neutral current interaction and charged current interaction of electron neutrinos leave a shower-like signature, whilst high energy muons and very high energy taus (E τ > 10 7 GeV) leave a track-like signature. IceCube expects to be able to see a 'double-bang' signature for τ leptons with energies ∈ [10 6 , 10 7 ] GeV.
We simulate distributions of leptons (µ and τ ) produced by the sphaleron-induced neutrino-quark collision events in parton level. We consider the simplest possibility of such events: qν → 8q2¯ induced by the gauge invariant (qqq) 1 (qqq) 2 (qqq) 3    This separation typically requires the opening angle of the leptons to be 2 · 10 −2 . The right panel of Fig. 4 displays histograms of the − opening angles in the laboratory frame, colour-coded for the different combinations of primary and secondary leptons, and again normalised to a single sphaleron-induced event. We see that the opening angles are in all cases much smaller than the IceCube angular resolution, so we do not expect multiple lepton tracks to be distinguished.

IceCube Constraints on Sphaleron-Induced Transitions
In the absence of a distinctive leptonic signature, we use the generic IceCube search for which detection efficiency is encoded in the effective neutrino detection area given in [37].
Moreover, we assume in the absence of a detailed simulation of the IceCube efficiency for detecting sphaleron-induced final states that it is the same as that for conventional final states, and that the neutrino spectrum keeps falling at energies above 10 11 GeV. Fig. 5 compares the upper limit on the overall cross-section factor p obtained in this way from IceCube 4-year [38] (solid red lines) with the upper limits derived in [18] from recasting the ATLAS Run 2 search for microscopic black holes with ∼ 3/fb of data at 13 TeV [39] (solid blue lines). The upper panel of Fig. 5 is for ∆n = −1 transitions, which yield final states with 10 energetic particles at the LHC, and the lower panel is for ∆n = +1 transitions, which yield 14-particle final states at the LHC.
The LHC constraints in Fig. 5 are different for the 10-and 14-particle final states, and quite sensitive to the assumed value of the sphaleron energy E Sph . (We recall that our nominal value is E Sph = 9 TeV, but we display results for E Sph ∈ [8,11] TeV.) This is because the rate for sphaleron-induced transitions at the LHC with a centre-of-mass energy of 13 TeV is quite sensitive to E Sph . In contrast, the IceCube 4-year limit is quite insensitive to E Sph over the range studied, because of the larger range of neutrino energies. Within our assumptions, the IceCube efficiencies and hence limits for ∆n = ±1 transitions are the same, whereas the LHC limits are stronger for ∆n = +1 transitions, for which we estimated in [18] a greater detection efficiency.
By a remarkable coincidence, we see that the LHC and IceCube constraints are almost identical for the nominal value E Sph = 9 TeV, but the IceCube limits are stronger for larger E Sph , becoming some 3 orders of magnitude stronger for E Sph = 11 TeV.
We also display in Fig. 5 the prospective future LHC exclusion sensitivities for higher integrated luminosities (dashed and dot-dashed blue lines) and energy (dashed and dotdashed green lines). We see that 300/fb of luminosity at 14 TeV would be needed for the LHC sensitivity to surpass the IceCube constraint for E Sph = 11 TeV. We anticipate that the IceCube sensitvity will also be improved by longer operating time and/or effective size, and note that an order-of-magnitude improvement in the IceCube sensitivity would make it highly competitive with the LHC with 3000/fb for E Sph = 11 TeV, with both being able to reach p 10 −2 . On the other hand, for the nominal value E Sph = 9 TeV, the LHC would have a greater reach than IceCube, down to p < 10 −4 .

Summary and Conclusions
We have shown that IceCube could have a sensitivity to spahaleron-induced transitions that is comparable to that of the LHC. For a cross-section prefactor p = 1, the rate of such transitions in neutrino collisions would exceed the sum of conventional charged-and neutralcurrent interactions for E ν 2×10 8 GeV, as seen in Fig. 1, yielding a larger number of events for a neutrino threshold energy above E ν = 10 7 GeV, as seen in Fig. 2 with a limited fraction of 'double-bang' τ events, as seen in the left panel of Fig. 4, and multilepton bundles that are probably not resolvable, as seen in the right panel of Fig. 4.
Remarkably, the prospective IceCube constraints on sphaleron-induced transitions are comparable to those from the LHC, as seen in Fig. 5, with IceCube having an advantage for large sphaleron energies E Sph and the LHC at small E Sph . The crossover is currently close to the nominal value E Sph = 9 TeV. Our estimates need to be validated by dedicated experimental simulations for IceCube as well as for the LHC, but our results indicate that both have interesting sensitivities for sphaleron-induced transitions, able to probe significantly below p = 1, and hence able to test or constrain the suggestion by Tye and Wong [9] that sphaleron transitions may be much less suppressed than commonly thought previously.