∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany

# Double and multiple bangs at tau neutrino telescopes

A novel probe of sphalerons with cosmogenic neutrinos

Guo-yuan Huange1,addr1

(Received: date / Accepted: date)

###### Abstract

In light of the exciting campaign of cosmogenic neutrino detection, we
investigate the double and multiple tau bangs detectable at future tau
neutrino telescopes. Such events are expected from the Standard Model (SM)
higher-order processes, which can be easily identified with broad techniques
anticipated at future tau neutrino telescopes. We find that SM perturbative
processes can already contribute observable double-bang events to telescopes
with a sensitivity of collecting $\mathcal{O}(100)$ cosmogenic neutrino
events. The detectable but suppressed rate in fact makes the double and
multiple bangs an excellent probe of SM unknowns and possible new physics
beyond. As a case study, the nonperturbative sphaleron process, which can
copiously produce multiple tau bangs, is explored.

###### Keywords:

Cosmogenic neutrino telescopes Neutrino interactions Sphaleron

††journal: Eur. Phys. J. C

##  1 Introduction

The energy frontier of particle collisions has been pushed with continuous
efforts. Other than artificial colliders, the free cosmic particle fluxes have
offered us another opportunity to study particle physics. Remarkably, many
cosmic rays and gamma rays at ultrahigh energies have been observed with the
ground-based arrays PierreAuger:2020qqz ; TelescopeArray:2021zox ;
Ivanov:2021mkn ; LHAASO:2021cbz ; LHAASO:2021crt ; LHAASO_nature . Ultrahigh
energy neutrinos, as the most elusive messenger associated with this frontier,
were also detected up to $\mathcal{O}(\rm PeV)$ energies by the IceCube
Observatory IceCube:2013cdw ; IceCube:2020acn ; IceCube:2020wum . Yet, the
cosmogenic neutrinos with energies as high as ${\rm EeV}\equiv 10^{3}~{}{\rm
PeV}$, which are guaranteed by the Greisen-Zatsepin-Kuzmin mechanism
Greisen:1966jv ; Zatsepin:1966jv ; Beresinsky:1969qj , have not been observed
IceCube:2018fhm ; PierreAuger:2019ens ; ANITA:2019wyx .

In the near future, robust observations of the EeV neutrino flux are highly
anticipated with a large number of experimental programs Huang:2021mki ;
Ackermann:2022rqc ; Abraham:2022jse ; Neronov:2016zou ; GRAND:2018iaj ;
Otte:2018uxj ; Otte:2019aaf ; ARA:2019wcf ; Abarr:2020bjd ; Anker:2020lre ;
RNO-G:2020rmc ; IceCube-Gen2:2020qha ; POEMMA:2020ykm ; Romero-Wolf:2020pzh ;
Wissel:2020fav ; Wissel:2020sec ; Hallmann:2021kqk ; Ogawa:2021dK ;
deVries:2021BA ; see Ref. Huang:2021mki  for a recent compilation. Among them,
a promising experimental class is the tau neutrino telescope Berezinsky:1975zz
; Domokos:1997ve ; Domokos:1998hz ; Capelle:1998zz ; Fargion:1999se ;
Fargion:2000iz ; LetessierSelvon:2000kk ; Feng:2001ue ; Kusenko:2001gj ;
Bertou:2001vm ; Cao:2004sd ; Baret:2011zz , which detects the decay products
of tau leptons converted from tau neutrinos. With this new energy frontier
being established, an unprecedented center-of-mass (COM) energy of more than
$43~{}{\rm TeV}$ for the neutrino-nucleon collision can be reached, much
higher than that has been achieved in laboratories. This guarantees a new
place to probe the Standard Model (SM) of particle physics and possible new
physics beyond. However, general issues of its particle physics potential are
the large SM background and systematic uncertainties of the initial neutrino
flux Jezo:2014kla ; Huang:2019hgs ; Denton:2020jft ; Soto:2021vdc ;
Huang:2021mki ; Valera:2022ylt ; Huang:2022pce ; Esteban:2022uuw . In such a
case, new physics effects mostly manifest themselves indirectly by affecting
the energy and angular distributions of tau events.

In this work, we explore a powerful topology at tau neutrino telescopes, the
double- and multi-bang ($\geq 3$) events. Such events can be generated from
higher-order (perturbative) processes in the SM, which by themselves are
observable with the upcoming experimental programs. Furthermore, the
relatively suppressed rate makes such events a distinctive probe of various
physics unknowns. As a concrete example, we study the production of the well-
motivated sphaleron, expected from SM nonperturbative processes, by the
neutrino-nucleon scattering with an EeV incoming neutrino flux.

Figure 1: Representative Feynman diagrams for the neutrino-nucleus scattering,
with the productions of a single tau at the leading order (a) as well as
double (b, c1 and c2) and triple taus (c3). The sphaleron process (d)
producing multiple taus is also illustrated. Note that not all diagrams with
the same final state are shown.

##  2 Distinctive double and multiple tau bangs

Tau neutrino telescopes are designed to observe the extensive air shower
produced by the decay of a tau lepton (a “single bang”), originated from the
charged-current (CC) deep-inelastic scattering (DIS) $\nu_{\tau}+N\to\tau+X$,
with $N$ being the nucleus and $X$ being the nuclear remnant. The extensive
air shower can either be directly captured Capelle:1998zz  or indirectly
observed in forms of radio emission, Cherenkov light, fluorescence and so on
Jelley1958erenkovRI ; 1953Nature ; Greisen1966 ; Kampert:2012vi ;
Schroder:2016hrv ; Ginzburg:1945zz ; Motloch:2015wca ; Revenu:2013lza ;
deVries:2021BA .

In those cases discussed, the extensive air shower is considered to be
produced from a single tau bang, and the sensitivities to the cosmogenic
neutrino flux are estimated based on this assumption Neronov:2016zou ;
GRAND:2018iaj ; Otte:2018uxj ; Otte:2019aaf ; ARA:2019wcf ; Abarr:2020bjd ;
Anker:2020lre ; RNO-G:2020rmc ; IceCube-Gen2:2020qha ; POEMMA:2020ykm ;
POEMMA:2020ykm ; Romero-Wolf:2020pzh ; Wissel:2020fav ; Wissel:2020sec ;
Hallmann:2021kqk ; Ogawa:2021dK ; deVries:2021BA . While this is true at the
leading order (LO), the novel double- and multi-bang events can arise from the
higher-order processes in the SM Zhou:2019vxt ; Zhou:2019frk ; Garcia:2020jwr
; Zhou:2021xuh ; Soto:2021vdc ; Seckel:1997kk ; Alikhanov:2014uja ;
Alikhanov:2015kla , which cannot be neglected providing the future detection
potential. In general, these processes (see Fig. 1) are represented by

$\displaystyle\nu_{\ell}+N\to n\,\tau+X\;,$ (1)

where $\ell\in\\{e,\mu,\tau\\}$ stands for the neutrino flavor and $n\geq 2$
is the number of taus produced in this reaction. Two or more taus then decay
subsequently to produce the double or multiple extensive air showers, within a
short time window. Besides the higher-order processes, multiple tau final
states are also predicted by the nonperturbative sphaleron process in the SM,
when a certain energy threshold is reached. Moreover, new physics scenarios
such as the microscopic black hole can induce similar outgoing states
Uehara:2001yk ; Alvarez-Muniz:2002snq ; Dutta:2002ca ; Kowalski:2002gb ;
Jain:2002kz ; Stojkovic:2005fx ; Illana:2005pu ; Anchordoqui:2006fn ;
Kisselev:2010zz ; Arsene:2013nca ; Reynoso:2013jya ; Mack:2019bps . Note that
our double-bang event from di-tau is different from the “double-bang” topology
at IceCube Learned:1994wg ; IceCube:2020abv , for which the first bang is
generated from $\nu_{\tau}\to\tau+X$ and the second bang is from $\tau\to{\rm
any}$. For tau neutrino telescopes that observe air showers, the scattering
vertex $\nu_{\tau}\to\tau+X$ inside matter is unobservable GRAND:2018iaj .

Figure 2: The schematic diagram (not to scale) of the detection of double and
multiple tau bangs, originated from the processes in Fig. 1. For illustration,
two prototypical telescopes, GRAND and Trinity, are taken.

In Fig. 1, we give several representative diagrams relevant for EeV cosmogenic
neutrinos. Diagram (a) is just the leading-order CC scattering, which
dominates the neutrino-nucleon cross section above a few TeV. Diagrams (b),
(c1), (c2) and (c3) are capable of producing two and even three taus in a
single collision. Among them, diagram (b) is initiated by a bottom parton in
the nucleon Soto:2021vdc , for which a top quark is generated and subsequently
decays into a tau. Whereas diagrams (c1), (c2) and (c3) are initiated by gauge
boson contents, i.e., $\gamma$, $Z$ and $W$, inside the nucleus. In general,
the cross section initiated by $\gamma$ is much larger than those by $W$ and
$Z$, and can also benefit from the coherent scattering at the nucleus level.
The last process, represented by diagram (d), takes place via a
nonperturbative sphaleron with the COM energy $\sqrt{s}>E_{\rm sp}\sim
9~{}{\rm TeV}$. We will come back to the detailed analyses of those processes
in later discussions.

How do the di-tau and multi-tau productions manifest themselves when
confronted with the actual experimental configurations? To see it, we take
GRAND GRAND:2018iaj  and Trinity Otte:2018uxj ; Otte:2019aaf ; Brown:2021tf ;
Wang:2021/M  as two representative ground-based telescopes, and discuss their
prospects in identifying the double and multiple bangs.

GRAND has a mountain-valley topography, where the radio detection arrays sit
on a mountain slope, facing towards another mountain over a valley (see Fig. 2
for a schematic picture). Radio arrays with $10{\rm k}$ antennas form a
projection screen of the area $100\times 100~{}{\rm km}^{2}$, with adjacent
antennas separated by a distance of $1~{}{\rm km}$. Providing 20 replicates of
such arrays (GRAND200k), an unprecedented sensitivity to the cosmogenic
neutrino flux $E^{2}\Phi\sim 10^{-10}~{}{\rm GeV\cdot cm^{-2}\cdot
sr^{-1}\cdot s^{-1}}$ can be achieved with a ten-year exposure GRAND:2018iaj .
In contrast, Trinity observes the Cherenkov light emitted from the extensive
air shower induced by tau decays. Instead of containing the whole shower
profile like GRAND, Trinity monitors the activity of only beamed photons but
with a wide field of view. An integrated sensitivity $E^{2}\Phi\sim 5.9\times
10^{-10}~{}{\rm GeV\cdot cm^{-2}\cdot sr^{-1}\cdot s^{-1}}$ is expected with
three stations and ten years of exposure Otte:2018uxj ; Otte:2019aaf .

A schematic diagram of the effect of the double and multiple bangs is
illustrated in Fig. 2. Two or three taus are first generated from a single
neutrino interaction in matter. After emerged from the mountain or Earth
surface, those taus propagate an average distance of $50~{}{\rm
km}\cdot(E_{\tau}/{\rm EeV})$ and then decay in the air. There is a large
randomness in tau decays, which allows the spatial separation of several
bangs. Finally, those extensive air showers will be observed within a short
coincidence window, and resolved if they are separated enough. Depending on
the actual experimental design, both timing and spatial information can be
utilized to resolve the double and multiple bangs. For the specific GRAND and
Trinity setups, some remarks are given as follows.

  * •

By design, GRAND antennas will have an excellent time resolution ($\sim
5~{}{\rm ns}$) GRAND:2018iaj . This is in fact the essential ingredient for
GRAND to reconstruct the arrival direction of the extensive air shower, by
looking at the timing difference in triggered antennas. As for double bangs,
the tau, which decays closer to the radio arrays, will trigger a more
concentrated zone. Whereas, the other tau will cover more antennas with the
same radio emission cone. For some mutual off-axis antennas in coverage of two
bangs, there will be characteristically two radio pulses arriving in sequence.
To set the scale of the typical timing difference, we assume the GRAND screen
to sit on a mountain slope inclined by $3^{\circ}$ and the distance between
adjacent antennas to be 1 km. Then, suppose that two taus, emerging from the
horizon, decay subsequently around $100~{}{\rm km}$ away from the arrays and
are separated by $5~{}{\rm km}$. Taking the Cherenkov cone to be
$1.5^{\circ}$, there will be around $400$ antennas triggered in total. For the
antenna near the edge of the Cherenkov cone, the timing difference of two
bangs is found to be $5.3~{}{\rm ns}$. Given the time resolution power of
GRAND, those two pulses should be resolvable. Furthermore, apart from the
timing information, the spatial radio footprints of two showers, which overlap
with each other, are also promising to resolve with dedicated simulations.
This can be seen from GRAND’s excellent power in measuring the column depth of
the shower maximum $X_{\rm max}$. A target precision of $\sigma(X_{\rm
max})\sim 20~{}{\rm g\cdot cm^{-2}}$ is already achievable with the radio
technique Buitink:2014eqa ; Corstanje:2021kik , which roughly corresponds to a
resolvable distance of $d=0.17~{}{\rm km}$ given the atmosphere density
$\rho_{\rm atm}\approx 1.2\times 10^{-3}~{}{\rm g\cdot cm^{-3}}$.

  * •

For Trinity, multiple separate tau bangs are both resolvable in direction and
in timing. Trinity has an excellent field of view around the Earth horizon,
and the photon sensitivity is not lost even far outside the typical Cherenkov
cone of $1.5^{\circ}$ for the EeV shower. Hence, we do not expect any obvious
obstacles for Trinity to identify two or more bangs. The only requirement is
from the intrinsic angular resolution $\sim 0.3^{\circ}$ of the camera
Otte:2018uxj . This corresponds to a resolvable trajectory distance of
$3~{}{\rm km}$ if the showers are typically $100~{}{\rm km}$ away from the
station and the off-axis angle of showers is $10^{\circ}$. The typical
probability distributions of the distance between two bangs are given in the
Appendix. The probability to have a separation less than $3~{}{\rm km}$
($30~{}{\rm km}$) is $\sim 10\%$ ($70\%$) for two taus of $0.5~{}{\rm EeV}$.
The timing information can also be utilized to distinguish such events from
the single bang at Trinity. For the above off-axis shower, the timing
difference of two bangs separated by $3~{}{\rm km}$ is found to be $157~{}{\rm
ns}$. If the shower is strictly horizontal, i.e., the telescope is on the axis
of the shower, we cannot distinguish several tau bangs. However, we note that
such events are very rare in the overall event sample, given the large
acceptance angle of Trinity Otte:2018uxj . The possible reinforcement of
trajectory-sensitive fluorescence detectors will further enhance the ability
to reconstruct such events.

Note that the air shower development occurs along a finite length and will
partly smooth away the sharp timing and spacing difference of several bangs.
In practice, a likelihood analysis may be necessary to single out such events.
In principle, other ground-based tau neutrino telescopes, e.g., Ashra-NTA
Ogawa:2021dK , BEACON Wissel:2020fav ; Wissel:2020sec  and TAMBO Romero-
Wolf:2020pzh , also have great potentials in identifying such events with
similar setups.

There are also space-borne tau neutrino telescopes, such as POEMMA, deploying
satellites in orbit with an altitude of $525~{}{\rm km}$ POEMMA:2020ykm .
POEMMA in Limb mode monitors the shower activity near the Earth horizon.
However, due to the large distance ($\sim 2640~{}{\rm km}$) from satellites to
the shower in the atmosphere, it will be challenging to clearly distinguish
several bangs with the decay length of $50~{}{\rm km}$ at EeV energies. Hence,
we do not consider such telescopes in this work.

We have seen the potential of ground-based telescopes in identifying the
double- and multi-bang events. We move on to estimating the event rates of
those processes as in Fig. 1.

##  3 Perturbative di-tau and tri-tau productions

Due to the relatively small flux of cosmogenic neutrinos at high energies, a
large detection volume is required for a positive observation. The detection
principle makes tau neutrino telescopes very sensitive to cosmogenic neutrinos
around EeV energy scales, where a bump of flux is usually expected. However,
as a tradeoff of the large detection volume, below
$E_{\nu}=\mathcal{O}(10~{}{\rm PeV})$ the signal becomes too weak to trigger
the detection threshold and the sensitivity of most telescopes falls off. In
such a case, the Glashow resonance from the neutrino-electron scattering at
$E_{\nu}=6.3~{}{\rm PeV}$ will be suppressed for the detection at most tau
neutrino telescopes; for relevant discussions, see Refs. Fargion:1999se ;
Fargion:2000iz ; Huang:2019hgs ; Soto:2021vdc . Thus, we only focus on the
neutrino-nucleus scattering in this work.

Figure 3: Cross sections of the neutrino-nucleus scattering including: the leading-order CC conversion (black curve), di-tau production initiated by b quark (purple curve), DIS $\gamma$ (red solid curve), coherent $\gamma$ (red dashed curve), or $Z$ (red dotted curve), as well as tri-tau production (blue curve). We take ${}^{16}{\rm O}$ to be the nucleus, and those cross sections are averaged over the nucleon number. Telescopes | 1$\,\tau$ | 2$\,\tau$ | 3$\,\tau$ | Sphaleron $n\,\tau$
---|---|---|---|---
Ashra-NTA Ogawa:2021dK  | 19 | 0.2 | 0.007 | 0.7 (0.5)
BEACON Wissel:2020fav ; Wissel:2020sec  | 137 | 1.6 | 0.062 | 7.1 (5)
GRAND GRAND:2018iaj  | 178 | 2.1 | 0.082 | 10 (7)
Trinity Otte:2018uxj ; Otte:2019aaf  | 16 | 0.2 | 0.006 | 0.6 (0.4)
TAMBO Romero-Wolf:2020pzh  | $>7$ | $0.1$ | $0.002$ |  $0.11$ (0.08)
Table 1:  The estimated numbers of single-, double- and triple-bang events
from perturbative calculations (first three columns) for several ground-based
tau neutrino telescopes with ten years of observation. For illustration, a
prediction of the cosmogenic neutrino flux, based on cosmic ray data of
Telescope Array Anker:2020lre , is taken as the input to obtain the rough
event number. For the sphaleron process, the event number with $n\geq 2$ (or
$n\geq 3$ for the number in parentheses) is shown, assuming
$\nu_{e}:\nu_{\mu}:\nu_{\tau}=1:1:1$ and $p_{\rm sp}=0.003$ with the $s$-wave
unitarity bound.

At the leading order, the DIS dominates the neutrino-nucleus cross sections
for $E_{\nu}>\mathcal{O}(10~{}\rm PeV)$. Going beyond that, the interest in
higher-order processes of the neutrino-matter scattering arises recently
Zhou:2019vxt ; Zhou:2019frk ; Garcia:2020jwr ; Zhou:2021xuh ; Soto:2021vdc ;
Seckel:1997kk ; Alikhanov:2014uja ; Alikhanov:2015kla . In particular, the
process $\nu_{e,\mu}+N\to\tau+\nu_{\tau}+X$ was found to contribute a
background of a few percent to the detection of cosmogenic tau neutrinos
Soto:2021vdc . However, such a contribution is hard to disentangle from the
intrinsic flux uncertainty of cosmogenic neutrinos.

The on-shell $W$ (or $Z$) production becomes kinematically viable when
$E_{\nu}>m^{2}_{W}/(2m_{N})\approx 3.4~{}{\rm TeV}$ for neutrino-nucleon
scatterings (as illustrated in Fig. 1), which is different from the suppressed
trident production usually discussed Altmannshofer:2014pba ; Magill:2016hgc ;
Ge:2017poy ; Ballett:2018uuc ; Altmannshofer:2019zhy . The $W$ production in
diagrams (c1) and (c2) of Fig. 1 can be initiated by both $\gamma$ and $Z$.
However, because of the collinear enhancement of the massless photon, the
major contribution comes from the photon-initiated process at relatively low
energy scales. Diagram (c1) has a tau propagator and is dominant below
$E_{\nu}\approx 0.6~{}{\rm EeV}$, whereas above that diagram (c2) with the
$W$-$\gamma$ fusion becomes the dominant channel. The $Z$ production in
diagram (c3) and additional diagrams with the same final state (not shown) is
suppressed by the heavy mass of $W$. We have calculated all possible diagrams
and summarize those cross sections in terms of the incoming neutrino energy in
Fig. 3. The branching ratios ${\rm Br}({W\to\tau\nu_{\tau}})\approx 11\%$ and
${\rm Br}({Z\to\tau\tau})\approx 3.4\%$ have been taken into account for our
di-tau and tri-tau productions. For the DIS, the parton distribution functions
(PDFs) from CT18qed set Xie:2021equ  have been used.

An estimation of the double- and multi-bang events is possible, considering
that higher-order processes only manifest themselves as perturbations to the
neutrino and tau fluxes. The event number can be roughly estimated by
comparing the respective cross section $\sigma_{n\tau}$ to the leading-order
one $\sigma_{\rm CC}$. Due to the tau decay and energy loss processes, the
largest length scale $L_{\tau}$ that a tau can travel in medium is around
$50~{}{\rm km}$ for all possible energies, which is much shorter than that of
a neutrino. Hence, tau events registered in the telescope are essentially
produced within a thin layer of $50~{}{\rm km}$ close to the matter surface.
As a result, the event numbers for both the leading-order and higher-order
processes roughly follow the relation

$\displaystyle N_{1\tau/n\tau}\propto\Phi^{\rm out}_{\nu}\cdot\sigma_{\rm
CC/n\tau}\cdot L_{\tau}\cdot P^{\rm det}_{1\tau/n\tau}\;,$ (2)

where $\Phi^{\rm out}_{\nu}$ is the exiting neutrino flux near the Earth
surface, which can be derived by solving the propagation equation inside
matter, and $L_{\tau}$ is the aforementioned largest length scale that a tau
can travel in the Earth medium, maximally around $50~{}{\rm km}$ and also
limited by the intrinsic matter thickness. Moreover, $P^{\rm
det}_{1\tau/n\tau}$ is the probability that the exiting tau can be accepted by
the detector, which depends on its energy and emerging angle. In principle,
this probability should be determined by Monte-Carlo simulations including the
effects of the tau decay, the shower development as well as the detector
response. However, we do not attempt to tackle this issue in the present work.
Instead, we only check the significance of the double- and multi-bang events
with the ready materials.

By canceling mutual terms for leading- and higher-order processes in Eq. (2),
we arrive at

$\displaystyle\frac{N_{n\tau}}{N_{1\tau}}\approx\frac{\sigma_{\rm n\tau}\cdot
P^{\rm det}_{n\tau}}{\sigma_{\rm CC}\cdot P^{\rm det}_{1\tau}}\;.$ (3)

For the single tau case, the detection probability reads $P^{\rm
det}_{1\tau}=\int\mathrm{d}s\;p_{\rm decay}(E_{\tau},s)\;p_{\rm
det}(E_{\tau},\Omega,s)$, where $s$ is the distance traveled by tau before the
decay, $\Omega$ is the incoming angle, $p_{\rm decay}$ is the decay
probability and $p_{\rm det}$ is the probability to trigger the detector. The
detector response to the single tau $P^{\rm det}_{1\tau}$ is almost saturated
for tau energies higher than $\mathcal{O}(1~{}\rm EeV)$. As an example, see
Fig. 24 of Ref. GRAND:2018iaj  for GRAND, where the dependence of the
effective detection area on $E_{\nu}$ becomes mild around $(1-10)~{}{\rm
EeV}$. For two or more taus, the detection probability $P^{\rm det}_{n\tau}$
further depends on the actual efficiency to distinguish such events from
single bang, which should not to be a fundamentally difficult issue as we
argued previously. If the efficiency is of order one and all final-state taus
are energetic ($>{\rm EeV}$), we should expect $P^{\rm det}_{n\tau}\sim P^{\rm
det}_{1\tau}$, which leads to the rough relation
${N_{1\tau}}/{N_{n\tau}}\sim\sigma_{\rm CC}/\sigma_{n\tau}$. However, we
should note that if the tau energy is too low to trigger the detector, this
simple relation will overestimate the double- and multiple-bang events. In
such a case, the actual number requires a dedicated simulation of the detector
response, which is, however, beyond the scope of the present work. In the
Appendix, we give the probability distributions of the energy of two taus, and
the probability that both taus have energies higher than $E_{\nu}/10$ can be
$50\%$.

Now, let us first estimate the single-bang events based on given differential
sensitivities of GRAND GRAND:2018iaj  and Trinity Otte:2018uxj ; Otte:2019aaf
. The sensitivities to the cosmogenic neutrino flux from the single bang are
already given for various ground-based telescopes. To be definite, we briefly
explain in the following how the differential sensitivities were obtained and
how to use them to get the actual events for a given input flux. For a
specific telescope, the all-flavor sensitivity, defined as the $90\%$
confidence level to observe a positive signal within an energy decade, is
recast as Reno:2019jtr

$\displaystyle F(E_{\nu})=\frac{2.44\times
N_{\nu}}{\mathrm{log}(10)\,E_{\nu}\,\langle A\Omega\rangle\,T}\;.$ (4)

Here, $2.44$ is the required expectation value of event number, $N_{\nu}=3$
represents three generations of neutrinos, $\log(10)$ sets the energy interval
on the logarithmic scale to be a decade, $\langle A\Omega\rangle$ is the
effective aperture of the telescope for tau neutrinos, and $T=10~{}{\rm yr}$
is the observational time. Note that the effective aperture can be obtained
reversely if the differential sensitivity is given.

For a given input of the cosmogenic neutrino flux
$E^{2}_{\nu}\mathrm{d}\Phi_{\nu}/\mathrm{d}E_{\nu}$, the differential event
for the single bang can be obtained with

$\displaystyle\frac{\mathrm{d}N_{1\tau}}{\mathrm{d}\log_{10}E_{\nu}}=2.44\,E^{2}_{\nu}\frac{\mathrm{d}\Phi_{\nu}}{\mathrm{d}E_{\nu}}\frac{1}{F(E_{\nu})}\;.$
(5)

When $E^{2}_{\nu}\mathrm{d}\Phi_{\nu}/\mathrm{d}E_{\nu}=F(E_{\nu})$, we
reproduce the definition of the differential sensitivity, i.e., $2.44$ events
per energy decade. The $n\tau$ event number can be roughly estimated by using
the relation in Eq. (3). Under the optimistic assumption of $P^{\rm
det}_{n\tau}\sim P^{\rm det}_{1\tau}$, we have

$\displaystyle\frac{\mathrm{d}N_{n\tau}}{\mathrm{d}\log_{10}E_{\nu}}\sim\frac{\mathrm{d}N_{1\tau}}{\mathrm{d}\log_{10}E_{\nu}}\frac{\sigma_{\rm
n\tau}(E_{\nu})}{\sigma_{\rm CC}(E_{\nu})}\;.$ (6)

The total event number can be obtained by integrating over the energy.

The obtained numbers of single-, double- and triple-bang events are listed in
Table 1, for several ground-based telescopes. Note that for TAMBO Romero-
Wolf:2020pzh , the acceptance to the neutrino flux with $E_{\nu}>0.1~{}{\rm
EeV}$ is not officially given; therefore the numbers shown there should be
interpreted as lower bounds. For demonstration, we take the cosmogenic
neutrino flux from Ref. Anker:2020lre , which was produced based on recent
cosmic ray data of Telescope Array Bergman:20197 . We have taken the median
flux (i.e., the middle of the flux range on the logarithmic scale) in the
allowed statistical range, which is well compatible with the existing
constraints on the cosmogenic neutrino flux IceCube:2018fhm ;
PierreAuger:2019ens ; ANITA:2019wyx . Unless very pessimistic flux predictions
are chosen, the neutrino events will not be affected by orders of magnitude by
adopting some popular flux choices; see, e.g., Ref. Valera:2022ylt . In
general, with an observation of $\mathcal{O}(100)$ standard single-bang
events, one would expect $\mathcal{O}(1)$ double-bang events, whereas the
triple-bang rate is rather suppressed and negligible in light of the
foreseeable programs.

The observation of double-bang events itself is important for our
understanding of how neutrinos interact at ultralight-high energy scales.
Moreover, because such events are very distinctive experimentally, the
detectable but suppressed rate actually makes them an excellent probe of
physics unknowns. One of the examples to be discussed is the sphaleron
process.

##  4 The sphaleron process and nonperturbative production of many

The baryon and lepton numbers are violated by nonperturbative processes in the
SM Adler:1969gk ; Bell:1969ts ; Bardeen:1969md , when a transition takes place
between vacua characterized by different integer Chern-Simons numbers.
Topologically distinct vacua are separated by an energy barrier of $E_{\rm
sp}\sim m_{W}/\alpha_{W}$ with $\alpha_{W}\approx 1/30$, associated with the
sphaleron configuration Dashen:1974ck ; Manton:1983nd ; Klinkhamer:1984di . At
low energy scales, the transition between different vacua is possible via the
quantum tunneling effect (instanton), which is however exponentially
suppressed. In the early Universe, the transition violating $B+L$ number
becomes very efficient beyond tunneling Linde:1981zj , when the temperature
can overcome the sphaleron energy barrier $E_{\rm sp}$. This possibility has
been thought to be a crucial ingredient for the baryon asymmetry in our
Universe Kuzmin:1985mm ; Fukugita:1986hr ; Rubakov:1996vz ; Trodden:1998ym ;
Buchmuller:2004nz ; Davidson:2008bu ; Morrissey:2012db ; Jaeckel:2022osh .

The sphaleron process might also be possible for two-particle collisions at
nearly zero temperature, when the COM energy can overcome the sphaleron
barrier. The formation of coherent sphaleron state from two-particle
collisions was thought to be suppressed Ringwald:1989ee ; Espinosa:1989qn ;
Porrati:1990rk ; Khlebnikov:1990ue ; Khoze:1990bm ; Mueller:1991fa ;
Rebbi:1996zx ; Bezrukov:2003er ; Bezrukov:2003qm ; Ringwald:2002sw ;
Ringwald:2003ns ; Khoze:2020paj . However, recent studies including the
sphaleron periodicity (through multiple sphaleron) suggest counter results
Tye:2015tva ; Tye:2017hfv ; Qiu:2018wfb . While this fundamental issue itself
is still under debate, one can instead turn to the high-energy experimental
frontiers for the answer Ellis:2016ast ; Ellis:2016dgb ; Brooijmans:2016lfv ;
Cerdeno:2018dqk ; Jho:2018dvt ; Anchordoqui:2018ssd ; Zhou:2019uzq ;
Papaefstathiou:2019djz . In particular, possible high-multiplicity events at
cosmic ray detectors have been studied Jho:2018dvt , with assumed background
rejection methods. At EeV energies, the primary sphaleron process can on
average produce $\mathcal{O}(400)$ initial pions in the air shower, compared
to the CC/NC one with only $\mathcal{O}(100)$ pions Jho:2018dvt .

Figure 4: Cross sections of the neutrino-nucleon scattering via the sphaleron
process. Without the $s$-wave unitarity requirement, the cross section for
$E_{\rm sp}=9~{}{\rm TeV}$ and $p_{\rm sp}=1$ is given by the red dashed
curve. The inclusion of unitarity leads to the red solid ($p_{\rm
sp}=10^{-3}$) and red dotted ($p_{\rm sp}=1$) curves, respectively.

In our case, the double- and multi-bang topology is very clean in identifying
the sphaleron events. Considering that the COM energy of the neutrino-nucleon
collision can be as high as $43~{}{\rm TeV}\gg E_{\rm sp}\sim 9~{}{\rm TeV}$
for an EeV incoming neutrino, the detection of cosmogenic neutrinos will be a
powerful probe of sphalerons. The process of our concern reads Ringwald:1989ee
; Espinosa:1989qn ; Gibbs:1994cw

$\displaystyle\nu_{\ell}+N\to 2\overline{l}+8\overline{q}+n_{\rm
B}\,W/Z/\gamma+X\;,$ (7)

where two anti-leptons $\overline{l}$ (can be $\tau$) have flavors different
from the incoming neutrino $\nu_{\ell}$, and $n_{\rm B}$ is the number of
gauge bosons produced. From the decay of $W/Z$ or top quark, one can have
multiple taus in the final state. The sphaleron reaction features a large
multiplicity in the final state, and the average number of radiated gauge
bosons is of the order of $n_{\rm B}\sim 1/\alpha_{W}\approx 30$
Ringwald:1989ee . In our analysis, we do not attempt to get into the sphaleron
dynamics of the process, for which a solid theoretical evaluation is lacking
thus far. Instead, we can take $n_{\rm B}\approx 30$ and further assume a flat
phase space for the final state, such that the average energy of each outgoing
particle reads $\overline{E}=E_{\nu}/(10+n_{\rm B})\approx 0.25~{}{\rm
EeV}\cdot E_{\nu}/(10~{}{\rm EeV})$. Moreover, the production ratio of
$W^{+}:W^{-}:Z:\gamma$ is assumed to be $1:1:\cos^{2}{\theta_{\rm
W}}:\sin^{2}\theta_{\rm W}$ Gibbs:1994cw .

Taking account of the branching ratios of $W\to\tau\nu_{\tau}$ and
$Z\to\tau\tau$, the probability to have $n\tau$ with $n\geq 2$ (or $n\geq 3$)
from the sphaleron process is high, around $p=83\%$ (or $61\%$) for
$\nu_{e,\mu}$ and $p=74\%$ (or $49\%$) for $\nu_{\tau}$ in the initial state.
The difference between the scenarios of $\nu_{e,\mu}$ and $\nu_{\tau}$ stems
from the fact that $\nu_{e,\mu}$ has a probability of producing a primary
$\tau$ through the $\Delta L_{\tau}=1$ sphaleron process while $\nu_{\tau}$
does not.

The parton-level cross section of the sphaleron process can be parameterized
as Gibbs:1994cw ; Ellis:2016ast

$\displaystyle\sigma_{0}(\hat{s})=\frac{p_{\rm
sp}}{m^{2}_{W}}\Theta(\sqrt{\hat{s}}-E_{\rm sp})\;,$ (8)

where $\sqrt{\hat{s}}$ is the total energy in the neutrino-parton COM frame,
$p_{\rm sp}$ controls the overall scattering strength, and the Heaviside
function is motivated by the consideration of the sphaleron barrier. Since the
sphaleron process is approximately pointlike and spherically symmetric, its
cross section should be ceiled by the $s$-wave unitarity requirement of the
scattering matrix, namely $\sigma_{\rm unitarity}=16\pi/\hat{s}$ for
$\sqrt{\hat{s}}>E_{\rm sp}$ Gibbs:1994cw , which is often ignored. For the
phenomenological study, a reasonable choice of the cross section is
$\sigma={\rm min}\\{\sigma_{0},\sigma_{\rm unitarity}\\}$ Gibbs:1994cw ;
Papaefstathiou:2019djz , implying that the higher-order corrections should
come to the rescue when the unitarity is saturated. The total neutrino-nucleon
cross section can then be obtained by convolving the parton-level one with
quark PDFs.

An illustration of cross sections of the sphaleron process is given in Fig. 4
by taking the sphaleron energy scale to be $E_{\rm sp}=9~{}{\rm TeV}$. The
most notable message is that the $s$-wave unitarity requirement sets an upper
bound to the sphaleron cross section, shown as the brown shaded region. A
value of $p_{\rm sp}\approx 0.002$ can already saturate the unitarity bound.
Without the unitarity ceiling, the sphaleron process can be even more
efficient than the standard CC conversion (not $s$-wave) for
$E_{\nu}>10^{8}~{}{\rm GeV}$. Imposing unitarity will considerably reduce the
sphaleron cross section, e.g., down to $2\%$ of the CC process at
$E_{\nu}=10^{9}~{}{\rm GeV}$. However, we note that all three neutrino flavors
can contribute to the sphaleron process while only $\nu_{\tau}$ contributes to
the CC conversion to tau.

As shown in Table 1, the combination of several telescopes, capable of
collecting $\mathcal{O}(300)$ cosmogenic neutrino events, can detect
$\mathcal{O}(10)$ double and multiple bangs originated from the sphaleron
process even with the unitarity bound. The anticipated signal is beyond the
background predicted by our perturbative calculations. However, we want to
emphasize that the results here more or less overestimate the sphaleron event
number because the final-state tau energy is degraded by orders of magnitude
compared to the initial neutrino. The actual reduction in the event number
depends on the shape of the cosmogenic neutrino flux, which can only be
answered along with a dedicated detector simulation.

##  5 Summary and outlook

As a flagship option towards the cosmogenic neutrino detection, tau neutrino
telescopes feature an unprecedented detection volume. Though great discovery
potentials are provided by such facilities, a broad class of event topologies
is lacking for the particle physics study.

We have explored the potential of the double and multiple bangs, which can be
induced by SM higher-order diagrams as well as nonperturbative processes. The
topology can also be applied to other interesting scenarios. It is worthwhile
to further investigate the systematic new physics scenarios with more
dedicated experimental simulations in a future work.

##  Appendix A The probability distributions of energy and separation distance
of di-tau

For the convenience of discussion, we distinguish different taus in the
process of di-tau production: the primary one as $\tau_{1}$ and the secondary
one generated from the $W$ decay as $\tau_{2}$.

Figure 5: The probability distributions of the energy of two taus produced
from different di-tau production processes, initiated by an EeV incoming
neutrino. The primary tau is represented by $\tau_{1}$ and the secondary one
generated from the $W$ decay by $\tau_{2}$. The probability $p$ is normalized
and can be interpreted as the probability to observe tau showers within the
energy bin. Figure 6: The probability distributions of the distance between
two tau bangs (red and blue histograms). For the single tau case, the
probability of the decay distance is shown.

Two taus feature distinct energy distributions. The energy distributions of
two taus can be easily calculated by using the ready differential cross
sections of related processes. For illustration, we show in Fig. 5 the
probability distributions of the energy for two taus initiated by an EeV
incoming neutrino. The probability is normalized and assigned to each bin in
the histogram plot. Different processes are shown separately, including the
DIS (in red) and coherent (in purple) processes initiated by $\gamma$ as well
as the DIS process initiated by b quark (in green). We notice that the DIS and
coherent $\gamma$-initiated processes feature similar distributions. For all
processes the phase space is dominated by the small momentum transfer regime
(compared to the center-of-mass energy). As a consequence, the primary tau
features a higher average energy, while the secondary one from $W$ decays is
lower. The integrated probability that the secondary tau has an energy higher
than $E_{\nu}/2$ ($E_{\nu}/10$) reads approximately $30\%$ ($50\%$).

Furthermore, the probability distributions of the separation distance between
$\tau_{1}$ and $\tau_{2}$ decays are shown in Fig. 6. Several benchmark
choices are illustrated, including $(E_{\tau 1},E_{\tau 2})=(0.5,0.5)~{}{\rm
EeV}$ shown in red and $(E_{\tau 1},E_{\tau 2})=(0.9,0.1)~{}{\rm EeV}$ shown
in blue. For comparison, the decay length of a single tau is illustrated as
the black histogram. The probability to have a separation distance larger than
$3~{}{\rm km}$ ($30~{}{\rm km}$) is as high as $90\%$ ($30\%$) for $E_{\tau
1}=E_{\tau 2}=0.5~{}{\rm EeV}$.

###### Acknowledgements.

The author would like to thank Shun Zhou for very helpful comments on the
manuscript. The Feynman diagrams in Fig. 1 are generated with the help of
JaxoDraw Binosi:2003yf . The cross sections are calculated in part by using
FeynCalc Shtabovenko:2020gxv  and CalcHEP Belyaev:2012qa  packages. This work
is supported by the Alexander von Humboldt Foundation.

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