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Polarized radiation and the Emergence of Biological Homochirality on Earth and Beyond

It has been proposed that spin-polarized cosmic radiation can induce asymmetric changes in helical biopolymers that may account for the emergence of biological homochirality. The parity violation in the weak interaction has direct consequences on the transport of polarization in cosmic ray showers. In this paper, we show that muons retain their polarization down to energies at which they can initiate enantioselective mutagenesis. Therefore, muons are most likely to succeed in establishing the connection between broken symmetries in the standard model of particle physics and that found in living organisms. We calculate the radiation doses deposited by primary and secondary cosmic rays at various prime targets for the searches of life in the solar system: Mars, Venus, Titan, icy moons and planetesimals, and discuss the implications for the enantioselective mutagenesis proposed as to be the driver of homochiralization. Earth is unusual in that spin-polarized muons dominate the cosmic radiation at its surface.


INTRODUCTION
The origin of life is still a puzzle. However, it is a chiral puzzle: living organisms use only one class of enantiomer (molecule that has a non-superimposable mirror image) in the construction of proteins and nucleic acids, a fundamental property of life known as homochirality (see Avnir 2020, for a recent review). Homochirality allows biopolymers to adopt stable helical structures (Bada 1995;Nanda et al. 2007), a natural conformation for biological macromolecules to adopt and one that plays a key role in fundamental biological processes (Totsingan et al. 2012). While helical biopolymers have not been found, yet, in extraterrestrial environments, enantiomeric excesses (e.e.) of a few percent have been reported in the amino acids content of some meteorites (Pizzarello 2006;Burton & Berger 2018), but their relation to biological homochirality remains a mystery.
An important clue pertaining to the emergence of homochirality comes from studies of low energy, spin-polarized electrons which can promote enantioselective chemistry (e.g., Dreiling & Gay 2014;Rosenberg 2019). In addition, the Chirality-Induced Spin Selectivity, CISS, effect shows that helical biopolymers, specifically double strand DNA and α-helical peptide, can act as spin filters for sub-eV electrons (Naaman & Waldeck 2012). This may be related to biological function and may play a role in the emergence of homochirality.
Cosmic rays, play a major role in generating mutation and promoting evolution on Earth. They create pions in the upper atmosphere which decay into highly polarized muons which comprise the most prevalent high energy particles at ground level. Muons decay into electrons and positrons which are also polarized. In Globus & Blandford (2020), henceforth GB20, it was proposed that this polarization could impose a small chiral preference on the development of one set of biological enantiomers over the other which could evolve to homochirality over millions or even billions of generations. Some mechanisms for expressing this preference were discussed, supplying a causal connection from a fundamental physics asymmetry to an equally fundamental asymmetry in biology.
There are three stages to this scheme: I. Understanding cosmic ray polarization in different environments, II. Estimating the chiral preference present in the mutation rate with a given irradiation and III. Translating this into an influence on biological evolution. In GB20, the focus was on stage II and some semi-classical models of the interaction between cosmic rays and a chiral distribution of electrons in simple biopolymers were discussed. A quantity called lodacity, L =μ ·v, was introduced because it is the cosmic ray magnetic moment, µ, not the spin, that mediates this interaction. The effects were small; larger effects may be exhibited in a quantum mechanical description. However it was also argued that the effects could have been sufficient to induce a enantioselective biological response and establish homochirality during stage III, especially if enantiomeric conflict is present.
In this paper, we focus on Stage I. We are most concerned with cosmic rays on Earth today. However, there are many quite different environments present within the solar system and evidence for life could be found here too. Or not. Either way, it is of considerable interest to understand the evolution of cosmic ray polarization under different circumstances and so we also consider Mars, Titan, Venus, icy moons, and smaller bodies as well as Earth.
In Section 2, we explain the origin of the asymmetry in cosmic radiation. In Section 3, we discuss the depolarization effects for muons and electrons in water, before presenting the different environments we consider in Section 4, and the doses in section 5. We discuss the biological implications in Section 6.

ORIGIN OF POLARIZATION ASYMMETRY IN SECONDARY COSMIC RADIATION
Before discussing the results of detailed shower simulations, it is helpful to outline the development of the cosmic rays and their polarization from relativistic to the atomic energies that may be of most interest biologically. The cosmic rays observed in the solar system predominantly consist of hydrogen and ionized, typically stable nuclear isotopes (see the Particle Data Group (2020) and Gaisser et al. (2016) for reviews). If the total flux of cosmic ray nuclei is converted into a flux of nucleons, one finds that only 15% are neutrons and the rest protons. This charge asymmetry is conserved in the fluxes of secondary hadrons once protons interact with matter, since on average more π + and K + mesons are produced in the projectile fragmentation region. Hence, particles that emerge from hadronic processes and reach a "fount" (which we define as a location where trans-biotic polymers can develop, to distinguish it from a "habitat" where living organisms are well-established) will carry a net positive charge whereas particles from electromagnetic cascades have a charge ratio of one above the γ → e + e − threshold. At Earth, today, most particles at ground level are muons, which are tertiary cosmic rays predominantly produced through twobody decays of charged pions and kaons, e.g. π + → µ + ν µ . Since neutrinos are effectively massless, for our purposes, and produced with left-handed helicity and in the pion's rest frame its zero spin is conserved, the muon is produced fully polarized with the opposite helicity of the neutrino. Due to its large mass, roughly 1/3 of the muons would have positive helicity and 2/3 would have negative helicity if they originate from relativistic pions (p π > m π ). These two factors, charge asymmetry and parity violation of the weak interaction, imply that helicity state and charge have to be treated separately, resulting in four muon components µ + L , µ + R , µ − L and µ − R (Volkova & Zatsepin 1965;Dermer 1986;Lipari 1993), where L (R) label correspond to negative (positive) helicity. Since we are only interested in the magnetic moment polarization,μ ·v, we denote by as the sum of doses deposited by muons with µ + L + µ − R (i.e.μ ·v < 0) and as the sum of doses deposited by µ + R + µ − L (i.e.μ ·v > 0).The muons decay primarily into positrons and electrons together with two additional neutrinos. These also retain a significant spin polarization and lodacity L, which we define to be the average ofμ ·v and is always of the same sign (GB20).

Muons
Hadronic cascades produce muons which are subject to ionization loss and decay into electrons and positrons. With a vertical atmospheric grammage ∼ 1000 g cm −2 and a scale height ∼ 10 km, the surface muon particle flux is broad in energy but peaks at a few GeV. Lower energy muons mostly lose energy and decay before reaching the ground. Fewer high energy muons are produced and they are subject to extra radiative and pair losses.
For illustration, we consider underwater fate of few GeV muons and this elucidates what happens in other environments. Muons lose energy in water though ionization loss on a timescale where the kinetic energy T G, M, k ≡ T/(1 GeV, 1 MeV, 1 keV). Muon decay can be ignored on these timescales and once they become subrelativistic, muon ranges decrease ∝ T and so they essentially stop on the spot.
We show in Appendix A that, for a particle that slows through ionization loss, the muon scattering is relatively unimportant. This implies that muons continue in the same direction and that they remain polarized because their spins are much less deflected in encounters than their velocities. The muon lodacity will decrease according to L ∝ exp(−Kλ), where the "lethargy" λ = sech −1 (v/c) and the depolarization factor K = 0.037 for muons in water (see Eq. A4 in Appendix A). This implies a lodacity reduction by a factor of only ∼ 0.83 as the muons cool to a kinetic energy T k ∼ 10. 1 Down to few keV, the depolarization is ignorable. Below a few keV, there is a significant difference between positive and negative muons. Positive muons capture electrons with atomic cross sections (e.g. Kulhar 2004) when their speeds are comparable with those of valence electrons, T k ∼ 10, to form muonium, Mu. This reduces the muon polarization by by a half (Nagamine 2003). There will be further scatterings and exchanges, leading to further depolarization as the Mu thermalizes. . The spectrum and composition of the primary cosmic rays are the one observed at Earth at the present epoch. In the left panels, the total µ + + µ − spectra at different depths are figured by the solid lines; the polarized muons by dashed lines. The fraction of polarized muons is close to unity since they mostly come from pion decay. In the right panels, the total e + + e − spectra, i.e. those produced by all lepto-hadronic processes and also from bremsstrahlung pairs, are shown by solid lines at different depths. The polarized e + + e − from decay of muons are shown by dashed lines.
Negative muons, by contrast, continue to cool through atomic interactions mostly through excitation as they are too slow to ionize valence electrons. However, their deflections remain small and they retain their polarization until they reach atomic energies, T k ∼ 0.03. At this point, they form muonic atoms and rapidly cascade into high energy ground states very close to the host nuclei before decaying. The capture cross section is spin-dependent and may be sensitive to the chirality M of the molecule (GB20). Modest depolarization will occur within the atom through spin-orbit interaction. For instance, negative muons stopped in carbon retain about 15% of their initial polarization in the K-shell of spinzero nuclei before they decay (Garwin et al. 1957). The deexcitation cascade to the ground state is accompanied by the emission of Auger electrons and circularly polarized X-rays which can also transfer some polarization.
To summarize, muons retain almost complete polarization until they form Mu or muonic atoms. They transport polarization efficiently from high to low energies where they can have a biological impact. This is not the case for the electrons, as we discuss in the next section.

Electrons
In Fig. 1, we show the underground fluxes of muons and the accompanying electrons as a function of depth (in meter water equivalent), on Earth and on a moon without an atmosphere. While the muons are highly polarized, the electron polarization is negligible. The basic reason is that electrons scatter faster than they cool (Appendix B). This implies that polarized electrons made in muon decays in the atmosphere are rapidly depolarized and do not reach the surface. Electrons produced in muon decays below the surface are also unimportant.
However, the muons cool by creating unpolarized, knockon electrons, an unavoidable "entourage" of high energy electrons that must accompany the muons. High-energy muons (T >> GeV) that dominate at larger depths emit photons through bremsstrahlung µ ± → µ ± + γ and initiate electromagnetic sub-cascades through γ → e + + e − . These contribute to the "entourage" of unpolarized electrons, hence diluting effects induced by the high-energy muon polarization. A given biological molecule will evolve in the presence of a spectrum of both particles. While we do not understand their separate effects, a first step is to estimate their relative densities. This is discussed more in Appendices B and C.

FOUNTS
Now we have explored the effect of depolarization and dilution, we will calculate the polarized radiation doses at various environments: Mars, Titan, Venus, and bodies without an atmosphere (moons, asteroids). We are interested in founts, that we defined in the introduction as places where transbiotic polymers can develop. The search for life in our solar system has been focused on finding water, one of the key prerequisites for life as we know it, in the form of subsurface lakes or oceans. In some environments, geologic and hydrologic activity could have enabled physical processes necessary for the growth of the first helical biopolymers (see Camprubí et al. 2019, for a review).

Worlds with a thin or negligible atmosphere
Mars is a frozen world punctuated by periods of hydrologic activity (Cabrol 2018). Recently, a 20-km-wide lake of liquid water has been detected underneath solid ice in the Planum Australe region, at approximately 1.5 kilometer depth (Orosei et al. 2018). Although there is still no evidence for life on Mars, the red planet has once deployed environments characteristics of wet-dry cycling, a process thought to have enabled the production of the first helical biopolymers (Ross & Deamer 2016); the Mars Exploration Rover Spirit reported 3.65-billion-year-old hot spring deposits, approximatively of the same age as the Dresser hot springs on Earth that showed evidence for microbial life (Van Kranendonk et al. 2017;Damer & Deamer 2020). If microbial life started in similar hot springs, it is likely that after Mars' geological death and the loss of its atmosphere, microbial life would have only be able to survive underground, since Martian subsurface seems to have the key ingredients to support life for hundreds of millions of years (Checinska Sielaff & Smith 2019).
Evidence has accumulated that subsurface liquid regions (lakes or oceans) exist beneath the surface of the icy moon of Jupiter, Europa, and the icy moon of Saturn, Enceladus (Thomas et al. 2016). Mass spectrometers aboard the Cassini spacecraft found that the plumes of Enceladus contain water, salts, ammonia, carbon dioxide, and small and large organic molecules, suggesting that Enceladus' ocean may entertain an organic chemistry (Khawaja et al. 2019). In these environments, the atmosphere is negligible or absent (as in asteroids, comets or meteroids). Like on Mars, we will see that in these worlds where the atmosphere is absent, the spin-polarization transport is different than on Earth.

Worlds with a very dense atmosphere
We also consider worlds with atmospheres denser than the present Earth. Titan, Enceladus and Europa are among the most geologically active worlds in our solar system but in contrast to Enceladus and Europa, Titan possess a very dense atmosphere. The essential chemical building blocks for life are present in Titan's atmosphere (Yung et al. 1984). Titan is also thought to have a subsurface ocean of water (Iess et al. 2012) For Venus, the habitable region lies in the upper atmosphere (Morowitz & Sagan 1967). Recently, there has been a lot of debate on the possible detection of the simplest achiral amino acid, glycine, in the atmosphere of Venus but these results still need confirmation (Manna et al. 2020).

POLARIZED RADIATION DOSES
The current absorbed dose rates from cosmic radiation in different environments, are presented in Figure 2. It is an interesting coincidence that for Earth, at ground, 85% of ionizing cosmic radiation is due to the polarized muonic component that increases to almost 100% two meter water equivalent (m.w.e.) below the surface.
In other bodies of the solar system, the presence or absence of atmosphere leads to different results. For Mars (middle panel of Fig. 2), cosmic rays are only slightly attenuated by the thin Martian atmosphere and the polarized muons become the dominant source of cosmic radiation at ∼ 10 m.w.e. with a dose of ∼ 0.01 mGy/yr, much lower than the 0.2 mGy/yr on Earth. For planetesimals without an atmosphere or smaller bodies of the solar system (bottom right panel of Fig. 2), the polarized muons component dominates at ∼ 15 m.w.e. and with only one tenth of the intensity. In a dense, solid medium, the interaction length is much shorter than the decay length, leading predominantly to (re-)interaction of unstable hadrons rather than their decay. This suppresses the production of high energy muons and leads a swiftly decreasing polarized muon component as a function of depth.
In Titan and Venus polarized muons dominate the radiation at altitudes 40, 50 km. The surface irradiation, comparable to that below ∼ 400 m of rock is negligible.
The terrestrial polarized muonic dose has a maximum of ∼ 0.2 mGy/yr at the current epoch at ground level, but could have been larger during the Hadean period where a young sun could have accelerated cosmic rays beyond ∼ 10 GeV. If the cutoff of stellar cosmic rays arises below that energy (Stellar proton events (SPEs) or a young T Tauri cosmic ray spectrum as modeled by Padovani et al. (2018), this would only change the radiation dose by a factor ∼ 2, as we show in Fig. 3. All environments harbor other natural sources of radiation that may exceed the dose deposited by cosmic rays.  Top and Middle panels: Absorbed dose rate due to cosmic ray radiation at Earth and Mars. The black curves are obtained by summing the ionization losses from all charged particles. Red curves ( ) are the sum of doses deposited by muons with µ + L + µ − R whereas the blue ( ) are those with µ + R + µ − L . The purple curve is the total muon component. At Earth polarized muons constitute the dominant source of cosmic radiation at sea level and shallow underground depths. On Mars, polarized muons become the main source of cosmic radiation at ∼ 10 meters water equivalent (m.w.e). To estimate the underground fluxes in rock, the x-axis of the right-hand side panels can be multiplied with ρ rock /(g/cm 3 ). Bottom left panel: Same calculation for the dense atmospheres of Titan and Venus. Bottom right panel: Same calculation for planetesimals, like Earth's moon, or asteroids, protoplanets or comets, with a negligible atmosphere. "Large" signifies bodies with size much larger than the depth for which up-going particles can be neglected. In "small" bodies, the radiation deposited is roughly isotropic and shown for the center of a sphere with radius equal to the depth. The additional unpolarized, prompt muons, shown in green, originate from three-body decays of short-lived charmed and unflavored mesons and from electromagnetic pair production. Since at large depths low energy muons stop or decay, E µ > GeV and radiative losses dominate, emitting high-energy bremsstrahlung. These gamma rays convert through pair production into a part of the electronic "entourage" and dilute the dose deposited by polarized muons. Note that the dose is plotted logarithmically for clarity. T Tauri (Young Sun) Supernova @ 50pc after 100yr Figure 3. Absorbed dose at Earth for the presently observed cosmic ray flux according to the GSF model (Dembinski et al. 2017) as in Fig. 2, a phase in which the young Sun is modeled with a T Tauri star spectrum from Fig. 10 in Padovani et al. (2018), and for a nearby supernova 100 years after explosion at 50pc (Melott et al. 2017). Despite the high proton densities assumed in the Taurosphere, the impact on the underground fluxes is negligible since most protons have energies below the particle production threshold and hence do not produce additional muons. In the nearby supernova scenario the upper edge represents case A from Melott et al. (2017) and the lower case B, respectively. Shown is the maximal fluxes expected 100 years after explosion that drop significantly within a few kyr.
About 3.5 billion years ago, life was likely exposed to a background radiation field of ∼ 6 mGy/yr (Karam & Leslie 1999). However, more energetic transient events such a nearby supernova could increase the doses due to cosmic muons up to ∼ 100 mGy/yr at ground level.We show in Fig. 3 the absorbed dose at Earth 100yrs after the explosion of a nearby supernova (Melott et al. 2017) compared to the actual doses.
Since the muons retain their lodacity, higher radiation doses from cosmic muons could have altered enantioselectively the evolutionary trajectories in shorter timescales due to higher mutation rates.

DISCUSSION
So far, scenarios for the emergence of biological homochirality involved two processes. The first one is the generation of an initial enantiomeric excess (e.e.) in biologically relevant but still simple chiral monomers, such as amino acids or sugars. Various ways to achieve an initial e.e. have been proposed: parity-violating energy differences at the molecular level (Szabó-Nagy & Keszthelyi 1999, and references therein), asymmetric destruction due to irradiation by beta particles in radioactive decay (Vester et al. 1959), differential absorption of circularly polarized light (D'hendecourt et al. 2019), asymmetric adsorption on chiral mineral surfaces like clays (Hazen et al. 2001). Because the resulting enantiomeric excesses are very small (only up to few percent, see review by Burton & Berger 2018), a second necessary process is the amplification of the initial e.e. from low to high values. Enantioselective chemistry might be able to amplify the e.e. up to 99.5% (Soai et al. 2000) but the validity of such chemical processes in natural environments is still debated.
In contrast, the scenario proposed by GB20 only involves enantioselective mutagenesis, for which the timescale depends on the mutation rates of the first trans-biotic polymers and the magnitude of the bias. In order for the particular enantiomeric choice observed in living systems on Earth to be causally related to the choice expressed by the weak interaction and mediated by cosmic rays, we argued in GB20 that there needs to be a coupling between the lodacity, L, of the cosmic rays and the molecular chirality, M, of the molecular structure. The mutation rate has to include a term proportional to the product of these two pseudoscalars. The possibilities include an electromagnetic interaction in which the scalar product of the electric and magnetic dipoles or, equivalently E · B had a preferred sign. There is no good reason for this to occur in an isolated atom. However, it is just what is expected in a segment of a helical molecule containing a conduction or a magnetization current. In this case < E · B > is negative (positive) for a right-(left-) handed helix, independent of the polarity.
We conclude with two thoughts. Firstly, Earth is the only body in the solar system with an atmosphere just the right density and scale height to ensure that muons dominate the cosmic radiation at ground level and for ∼ 10 m.w.e. underground. It is thick enough to stop the other constituents, thin enough for the muons to survive. For worlds with thin or negligible atmosphere, such as Mars, polarized muons are the dominant source of cosmic radiation at depths of 10 m.w.e. while for worlds with dense atmospheres, like Venus or Titan, muons dominate below ∼40 km altitude but the radiation doses are negligible at the surface. Now, muons are the one constituent of cosmic radiation that is created with and retains high polarization as it decelerates to speeds αc when it can promote mutation of trans-biotic biopolymers. Might this be connected with the fact that Earth is the only body known to harbor life?
The second thought is that the relative importance of the muons and their electronic entourage for mutation is not un-derstood. In GB20, ionization was taken as a proxy for mutation. If this is the case, then collisional ionization by electrons is probably more frequent than muon capture, diluting the chiral effect of the cosmic rays. Muons have advantages of better depth-dose distributions. However, even ionization may be reparable and not so conducive to genetic alteration. There could be an appreciable chiral contribution to the Hamiltonian for the formation of a muonic atom or Mu in a helical molecule. The fact that negative muons retain their polarization down to the energies where they can undergo a spin dependent capture, may also have an effect on enantioselective mutagenesis. A single muonic atom decay will produce polarized X-rays and Michel electrons 3 and will surely fracture the molecule. Even the pickup of an elec-tron in Mu formation could be quite disruptive. Perhaps it is muons, not electrons, that dominate mutation of trans-biotic biopolymers, thereby increasing the chiral transfer and accelerating evolution to homochirality.
These matters and the prospect of performing experimental investigations will be discussed in a future paper.
Next multiply Eq. B5 by T and integrate over dT . We find the approximate solution In other words, the knock-on electron energy current in energy space at low energy is roughly the same as the energy current for the muons at high energy.
In this paper we are most interested in the muon and electrons spectra which can then contribute to the biological evolution of trans-biotic polymers and introduce a chiral bias. Many processes contribute to the full picture as we shall discuss elsewhere. For the moment, as an example, let us consider ∼ 10 keV electrons which are known to induce mutation in DNA. These are produced by muons with energy T 1 MeV. As the energy current in muons with energy ∼ 3 GeV is roughly the same as the energy flowing through ∼ 10 keV as electrons, we can conclude that the number current of ∼ 10 keV electrons is roughly 3 GeV/10 keV ∼ 3 × 10 5 times the conserved number current of muons. The implications of this for biological processes then depend on combining these rates with the relevant cross-sections.
C. COMPUTATION OF PARTICLE FLUXES We compute differential particle fluxes Φ(E, X), where E is the kinetic energy and X(h) = ∞ h d ρ( ) is the slant depth in g/cm 2 , using the one-dimensional cascade equation solver MCEq 4 (Fedynitch et al. 2015) that is optimized for calculations of inclusive atmospheric lepton fluxes at high energies above a few GeV (Fedynitch et al. 2019). We have extend the kinetic energy range down to 10 MeV and for computations in different media, such as carbon-dioxide and water by using hadronic cross sections and particle production tables from the DPMJET-III-19.1 5 event generator (Roesler et al. 2001). Electromagnetic cross sections have been computed using the numerical routines from Meighen- Berger & Li (2019). Ionization losses dE/dX are based on tables from (Particle Data Group 2020; Berger et al. 2017) and are tracked for each charged particle whereas energy deposition by tertiary electromagnetic cascades from fast neutrons are neglected. MCEq adapts the notation and formulae for muon polarization in the relativistic approximation from Lipari (1993) that we verified to be valid for E kin,µ > 10 MeV.
The flux of cosmic rays in the solar system is considered to be isotropic and represented by the Global Spline Fit (GSF) (Dembinski et al. 2017), which is an modern parameterization of cosmic ray fluxes at Earth between rigidity of a few GV and the highest observed energies at Earth. The rigidity cutoff due to the planetary magnetic field, the location within the solar system and the Sun's level of activity affects the low energy cosmic rays below a few GV/nucleon. The omission of magnetic fields and solar modulation is considered as a source of uncertainty of this calculation but is not expected to qualitatively change the result, in particular not at larger depths or underground.
The absorbed dose rate D in Gy/s is calculated from the differential fluxes Φ with the default units (GeV cm 2 s sr) −1 using where θ max = max(π/2, π−arcsin (R/(R + h)), h the altitude above ground and R the radius of the planet. The index p iterates over particle species. The expression for cos θ max takes into account that at higher altitudes, particle cascades can develop upwards relative to the horizon. The models of the planetary atmospheres are the Standard Atmosphere (1976) for Earth, derived from Huygens Atmospheric Structure Instrument measurements for Titan (Fulchignoni et al. 2005), from the Soviet VeGa-2 probe (Linkin et al. 1986; Lebonnois & Schubert 2017) for Venus, and from a fit of piece-wise barometric formulae to Mars Global Surveyor data Underground fluxes are computed for a water target with the ionization loss computed computed with ESTAR and PSTAR (Berger et al. 2017). Within the precision of this calculation and depths < 1 km, the difference in stopping power compared to a dedicated calculation in standard rock is negligible for muons.
In Figs. C1 and C2 MCEq is compared with FLUKA 4.0 6 (Battistoni et al. 2015) simulations in water for 1 TeV proton projectiles. The technical low energy cutoff in MCEq is 10 MeV, which is sufficient to capture a good description for the total number of muons (blue curve in Fig. C1 and right panel of Fig. C2). At tens of meters in depth, the differences between codes become more pronounced since the particle cascade is increasingly affected by multiple subsequent interactions that amplify differences in production and absorption cross sections between the models. To simulate water targets with MCEq it was necessary to include ionization losses for all charged particles and obtain inelastic cross sections for water from the hadronic interaction models. None the less the agreement is remarkably good since MCEq has not yet been employed for simulations down to low energies and water targets. Vertical 1 TeV proton shower in ice.

MCEq muons
MCEq electrons FLUKA muons FLUKA electrons Figure C1. Particle number in a particle cascade initiated by a 1 TeV proton in ice. In MCEq the hadronic interaction model is DPMJET-III-19.1 and PEANUT is used as default model in FLUKA 4.0. The full colored triangles represent results for a lower integration cutoff at E kin,e ± |µ ± > 10 MeV whereas the fainter triangles are obtained for cutoff value of 10 keV. The difference between the faint and full markers can be used to estimate the fraction particles lost due to the lower grid boundary of 10 MeV in MCEq.
10 4 10 3 10 2 10 1 10 0 10 1 10 2 10 3 Kinetic energy (GeV)  Figure C2. Comparison of electron + positron (left) and muon (µ + + µ − ) (right) spectra between MCEq and FLUKA using the same settings as in Fig. C1. For muon fluxes, the origin of the differences is mostly related to the hadronic interaction model DPMJET-III-19.1 in MCEq vs PEANUT in FLUKA. The hadronic model also affects the electromagnetic cascades, but differences at energies below 100 MeV may arise from incomplete partially the case For electromagnetic cascades the differences are partly related to, For very low energy muons, MCEq tends to predict lower fluxes than FLUKA. This is likely related to using DPMJET-III at energies close to particle production threshold, where it is known to be incomplete. agreement is good except at around the cascade maximum. Since the majority of electromagnetic energy is of hadronic origin through the π 0 → γγ process, the hadronic model differences between DPMJET-III-19.1 and PEANUT/FLUKA have impact. An estimate for the fraction of electrons that are lost due to the 10 MeV cutoff in MCEq can be made from the comparison between the pale and dark red triangles in Fig. C1. Since the gradient of the electromagnetic component is very steep after the maximum, the differences between MCEq and FLUKA do not qualitatively alter our conclusions from the main text with respect to the depth at which polarized muons constitute the dominant source of cosmic radiation.