Earth-bound Milli-charge Relics

Maxim Pospelov1, 2 and Harikrishnan Ramani3, ∗

1School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA2William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,

University of Minnesota, Minneapolis, MN 55455, USA3Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA

Dark sector particles with small electric charge, or millicharge, (mCPs) may lead to a varietyof diverse phenomena in particle physics, astrophysics and cosmology. Assuming their possibleexistence, we investigate the accumulation and propagation of mCPs in matter, specifically inside theEarth. Even small values of millicharge lead to sizeable scattering cross sections on atoms, resultingin complete thermalization, and as a consequence, considerable build-up of number densities ofmCPs, especially for the values of masses of GeV and higher when the evaporation becomes inhibited.Enhancement of mCP densities compared to their galactic abundance, that can be as big as 1014,leads to the possibility of new experimental probes for this model. The annihilation of pairs of mCPswill result in new signatures for the large volume detectors (such as Super-Kamiokande). Formationof bound states of negatively charged mCPs with nuclei can be observed by direct dark matterdetection experiments. A unique probe of mCP can be developed using underground electrostaticaccelerators that can directly accelerate mCPs above the experimental thresholds of direct darkmatter detection experiments.

I. INTRODUCTION

Charge quantization is a century old mystery. Whileexplanations for quantization exist, the resultant pre-dictions of magnetic monopoles and/or manifestation ofgrand unification (GUT) have not been observed despitesystematic efforts. This has led to the more open-mindedapproach to charge quantization, and exploration of thepossible existence of non-quantized charges also referredto as milli-charge particles (mCPs). In recent yearsmCPs have received further theoretical and experimentalscrutiny (see e.g. a selection of papers on theoretical andexperimental efforts: [1–11]).

On the theoretical side, models with pure mCPs as wellas models where smallness of effective electric charge isachieved via photon mixing with a new nearly masslessgauge boson have been considered [12]. Since their sta-bility is guaranteed by their U(1) charge, a non-trivialrelic abundance surviving from the Big Bang can be ex-pected. Depending on their mass and charge, they couldexplain all or part of the observed dark matter, calledmilli-charge dark matter or mCDM with their abundanceset by the freeze-out or freeze-in. (Freeze-out refers tothe self-depletion through annihilation from the initiallyfully thermally excited abundance, while the freeze-inis a sub-Hubble-rate-induced population correspondingto smaller couplings.) Regardless of cosmological abun-dance of mCPs, there exists a smaller yet irreducibleabundance arising from the interaction of cosmic-rayswith intervening matter [7, 10].

Owing to the enhancement of mCP scattering crosssections at low momentum transfer, they have been in-voked recently as an explanation of certain low-energy

∗ hramani@stanford.edu

anomalies, such as enhanced absorption of CMB by 21cmabsorbers [13–15], and excess of the keV scale ionizationin the Xenon 1T experimental results [10, 16, 17].

Regardless of possible anomalous results explained bymCPs, there have been a plethora of efforts lookingfor mCPs in collider and beam-dump experiments, thatshould be viewed in a broader context of exploring thedark sectors [18]. mCP relics depending on their speedcould also be detected in dark matter direct detectionand neutrino experiments. In addition, there are stronglimits on mass vs coupling parameter space arising fromcosmology [19–21] and galactic astrophysics [22, 23] aswell as from stellar energy losses [24, 25].

Despite these efforts, there is a tantalizing windowof parameter space that current and future experimen-tal efforts cannot access. This window corresponds tomQ ≈ 10 MeV and heavier, where BBN bounds do notapply (notice, however that BBN can still limit such mod-els through the excess abundance of dark photons in cor-responding models, see e.g. [26]). Defining the mCPcharge as εe, with e the electric charge, ε . 0.1 are notdirectly limited by collider and beam dump experiments,for mQ ≈ 1 GeV and heavier. If these mCPs make upa fraction fQ of the DM, then for large enough charge,the atmospheric or rock overburden is enough to slowthem down to small values of kinetic energies and mak-ing them inaccessible to current direct detection (DD)experiments.

In this paper, the main point to be investigated andexploited for possible novel signatures is mCPs slowingdown inside the Earth, resulting in a dramatic increaseof their number densities at the locations of undergroundlaboratories. This paves the way to novel methods ofsearching for mCP that we also explore in this paper. Adirect consequence of the mCP’s precipitous slow-down isthat this mCP thermalizes with the atmosphere (earth)and for large enough mQ, does not possess a large enough

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velocity to escape the planet subsequently. Barring sub-sequent evaporation, this builds up through the age of theEarth t⊕ ∼ O(4× 109) years leading to terrestrial densi-ties of mCPs several orders of magnitude larger than thevirial density of weakly-interacting DM. If the incomingmCP flux makes up a fraction fQ of the incoming DM

flux, then terrestrial densities as high as nterrQ ≈ fQ

1014

cm3

can be obtained. Depending on the precise value of themass, this tremendous density increase may be concen-trated inside the Earth’s core, or be spread out throughthe whole Earth’s volume. Even in the case of heavymasses, the constant vertical downward drift of thermal-ized mCPs is slow, leading to the “traffic jam” effect thatwe have discussed earlier [27–29].

Previous literature has explored the build up of thislarge density of DM that has large cross-section withthe SM nuclei [30], as well as its consequences on Earth,stars [31], comets [32] and even exoplanets [33]. Howeverto our knowledge, the build up of milli-charges specifi-cally and its consequences has missed scrutiny. In thiswork, we explore various sources of mCP flux on Earthand the subsequent build-up terrestrially. It was shownpreviously that masses above a GeV sink to the centerof the Earth and below a GeV evaporate away leading toa narrow window of mass where this terrestrial accumu-lation is relevant for experiments near the surface [30].However, due to the massless-mediator nature of mCP-SM interactions, the mCP slow-down increases this cross-section leading to larger abundances expanding the mCPmasses that accumulate appreciably near the Earth’s sur-face.

The large enhancement in local density compared tothe virial density by extremely large factors (e.g. forsome parts of the parameter space by as much as 1014),opens up novel detection strategies. In this paper, wediscuss only a small subset of possible new phenomenaand strategies for mCP detection:

- Terrestrial mCPs with relatively large charge canbind with SM nuclei which can be looked for in ex-otic isotope searches. If the binding energy is largeenough, there is a chance of seeing negative mCP- atomic nucleus “recombination” inside a detector(cf. Refs. [34–36] for analogous ideas).

- For small enough ε, where binding is not allowed,annihilations of mCPs with anti-particles into SMcan be looked for in the large volume neutrino de-tectors, and specifically in super-Kamiokande.

- Milli-charged particles that have accumulated in-side electrostatic accelerators can be “accidentally”accelerated (for large enough Q) and gain energy.The subsequent scattering in the low-threshold di-rect detection experiments is capable of providingvery strong sensitivity to the mCPs. This is espe-cially relevant in light of new efforts to install MeV-scale accelerators (for the studies of rare nuclearreactions) in the underground laboratories [37].

The last idea on this list is somewhat reminiscent of theproposal [38] that seeks to perturb the flow of mCPs byEM fields, with subsequent detection of this perturbationin the adjacent spatial region. The proposal of Ref. [38]is aimed at smaller mass and smaller ε compared to thoseexplored in this paper.

The rest of this work is organized as follows. In Sec-tion. II, we explore the capture of virial mCDM, subse-quent evaporation and the resultant density near the sur-face. In Section. III, sources of high rigidity mCP fluxesare explored and their resultant density near the surfaceis calculated. This is followed by Section. IV which dealswith bound state formation between mCPs and SM parti-cles. Section. V deals with the detectable consequences ofmCPs annihilating inside volumes of neutrino detectors.Section. VI provides a novel proposal to detect mCPs inan electrostatic accelerator. We present concluding re-marks in Section. VII.

II. TOP-DOWN ACCUMULATION

The goal of this section is to consider, in broad strokes,the accumulation of virial mCDM with large enoughcharge such that thermalization on Earth is rapid.

In the literature, two types of mCPs have been con-sidered. The first type is minimal mCPs, truly milli-charged under the SM U(1), without photon-dark-photonmixing. These particles have properties identical to SMcharge at all length scales. This property results in highlycomplex dynamics for the mCP propagation through thegalaxy, through the solar system and on Earth. This isbecause these mCPs interact with the galactic magneticfield, with the solar wind and finally with the electricfield between the ionosphere and ground. We leave thisproblem for future work.

Here, we instead consider particles Q that are chargedunder a dark U(1) with charge gQ, with the dark photonkinetically mixing with the SM photon with mixing pa-rameter κ. At energy scales ω mA′ , the mass of thisdark photon, these particles Q act effectively as mCPswith charge ε = κgQ. A large enough mA′ can be chosenso as to turn off the long range effects described aboveand yet keep the milli-charge properties which lead totestable consequences that we highlight below. It is likelythat such a set-up would lead to tensions with cosmol-ogy via the increase of Neff [26], which perhaps could becircumvented at the cost of adding additional ingredientsto the evolution of primordial the universe.

A. Capture and evaporation

For the couplings we are interested in, all of the mCDMgets captured, or in other words, it is in the multiplescattering regime, σ× (natom`) 1, and will lose its ini-tial kinetic energy down to characteristic thermal energy.

3

[]

⟨⟩/

[

-]

=

=

=

[]

/[

-]

=

=

FIG. 1: Terrestrial density of virial mCDM accumulated on Earth with charges corresponding to 100% capture isshown. The left panel shows the density of accumulated mCDM averaged over Earth’s volume, after accounting forevaporation and the long accumulation time. The right panel estimates this density as a function of depth with the

effects of sinking included.

The total number density of dark matter captured is

〈ncapQ 〉 =

πR2⊕vvirt⊕

4/3πR3⊕

fQρDM

mQ

≈ 3× 1015

cm3

t⊕1010year

fQGeV

mQ(1)

Here fQ is the fraction of virialized DM in milli-charges,defined as

fQ =mQnQρDM

, (2)

where ρDM is the local dark matter density, ρDM '0.3 GeV/cm3.vvir refers to an average velocity of galactic mCPs. Eqn

(1) neglects gravitational focussing, and in the case of theEarth’s capture it is well-justified.

For smaller masses, dark matter that thermalizes withatmosphere (water, rock etc) has a thermal velocityvth > vesc. Therefore, there exists a “last scatteringsurface” somewhere in the atmosphere, from which themost velocitized mCPs can freely escape, i.e. evaporate.Adopting earlier results, see e.g. [30], the evaporationrate per one mCP particle can be estimated as,

Γloss ≈3vth

2π12R⊕

(1 +

v2es

v2th

)exp

(− v

2es

v2th

)(3)

The equilibrium density on Earth is given by,

〈nQ〉 = 〈ncapQ 〉

1− exp (Γlosst⊕)

Γlosst⊕(4)

This is plotted in Fig. 1, left panel. One can see thatabove 1 GeV the evaporation is no longer a factor, andthe captured number density steadily grows with time. Itis evident from this plot that the the accumulated density

through the lifetime of the Earth can be up to fifteenorders of magnitude larger than the galactic density ofthe mCPs.

At the same time, the regime of light mCPs experi-encing strong evaporation is more difficult to analyze.In particular, slow-down in the upper atmosphere andsubsequent diffusion and evaporation can be altered bymany effects including the macroscopic mass transport.Precise analysis of this regime (e.g. mQ ' 10−500 MeV)goes beyond the scope if this work and is excluded fromFig. 1.

B. Density near the surface

While the number density averaged over the Earth vol-ume is given in Eqn. 4, the equilibrium density profile asa function of depth depends on the mass of the mCDM.The presence of gravity as well as pressure and temper-ature gradients results in rearrangement with a densityprofile that is mass dependent. This profile was evaluatedin [30] for the resultant stable population of strongly in-teracting particles. The main conclusions of [30] werethat, for relevant cross-sections, the number density atthe surface njeans ≈ 〈nQ〉 at mQ ≤ 1 GeV, while therewas diminishing number density of dark matter near thesurface for mQ ≥ 1 GeV owing to sinking to a greaterdepth.

However this sinking is not immediate. Diffusion ratesand terminal velocities determine the net sinking of heav-ier dark matter to lower altitudes. To estimate theserates we need the transfer cross section in terrestrialmedium (which we will call “rock”). The transfer cross-section σT for thermalized dark matter with atoms is es-timated in Appendix A. In the perturbative regime, onecan get good estimates with simple models of the charge

4

distribution inside an atom. We find that to a very goodapproximation for both attractive and repulsive interac-tions,

σT ∼ Min

(2πZ2α2ε2

µ2rock,Qv

4th

,4π

µ2rock,Qv

2th

)(5)

In this formula, µ2rock,Q stands for the reduced mass of

an atom-mCP system, Z is the atomic number and vth

is the typical thermal velocity of a particle with mass µ .While all of the dark matter is captured in the atmo-

sphere or close to the solid Earth’s surface, the randomwalk due to thermal motion will cause DM to diffusedeeper into rock. The time taken to diffuse to depth h isgiven by,

tdiff(h) ∼ h2nrockσTvth

, (6)

assuming simple Brownian motion. Here nrock is a typicalnumber density of atoms. Hence the effective verticalvelocity at depth h is

vdiff(h) =vth

hnrockσT(7)

In addition, in the presence of gravity there is an addi-tional mechanism for vertical sinking of DM, through thegravitational pull.

Under the gravitational interactions mCPs can acquireterminal velocity that can be estimated as [39],

vterm =3mχgT

m2gasngas〈σT v3〉

mQ > matom

=mQg

3ngasT

⟨v

σT

⟩mQ < matom (8)

These two velocities can be added to an effective verticalvelocity vvert(h) ' vterm +vdiff(h), and in reality over thevast parameter space that we explore, one or the otherterm is dominant.

The enhancement in number density occurs because offlux conservation, as smaller velocity at a depth h trans-late to larger number densities:

η ≡ nQ(h)

nvir=

vvir

vvert(h)' vvir

Max(vterm, vdiff(h))(9)

This enhancement factor is the most relevant for theheavy dark matter that has its equilibrium position muchcloser to the center of the Earth than to its surface. How-ever, for the GeV-scale dark matter, this type of “trafficjam” enhancement can be still small compared to equi-librium Jeans-type contribution. Putting these two en-hanced populations together, finally, the number densityfor mQ ≥ 1 GeV at an underground laboratory can beestimated as,

nloc = Max (njeans, nQ(h)) (10)

while for mQ < 1 GeV it is simply given by njeans cal-culated in [30]. Fig. 1, right panel, illustrates this totalenhanced density, and shows it for three characteristicdepths. It is easy to see that at mQ > 1 GeV and depth of1 km, very significant number densities can be achieved.

III. BOTTOM-UP ACCUMULATION

The previous section dealt with top-down accumula-tion; mCPs rapidly thermalizing in the overburden fol-lowed by diffusion/gravity populating lower altitudes.However, mCPs with large enough rigidity (momentum

charge )

could penetrate the overburden and get deep into theEarth before thermalizing. They then diffuse throughrock and the atmosphere before finally evaporating. Thisdiffusion time effectively acts as the time of accumulationof these mCPs leading to moderate local density. Thislarge rigidity could arise either due to mCP possessinglarge momenta or small charge. An irreducible sourceof a fast flux occurs due to cosmic ray produced mesonswhich decay into mCPs. The flux for such mCPs wastreated in detail in [7, 10], and we do not repeat it here.An alternative source of the fast flux could be the cosmicray collisions with mCDM that accelerates mCDM par-ticles to higher velocity via Rutherford scattering [10].Finally, virial mCPs with small enough charge ε couldalso penetrate the overburden and diffuse subsequently.

A. Fast Flux

We next estimate the accumulation of mCPs due tothe atmospheric fast flux. In the absence of evaporation,and in the assumption that all mCPs generated in theatmosphere are captured and retained, we would have,

nloc =

∫ βγmax

d(βγ)dΦ

d(βγ)

πR2⊕

43πR

3⊕t⊕ (11)

where dΦd(βγ) is the incoming mCP flux per interval of βγ.

An mCP with mass mQ, charge ε and boost factor βγpenetrates a distance dpen(mQ, ε, βγ) in the rock, that weestimate using the Bethe-Bloch formula. We cut off theintegral approximately at βγmax, above which particlescannot be stopped by the entire column density of theEarth. βγmax is given by equating the penetration depthto the Earth’s radius: dpen(mQ, ε, βγmax) = R⊕. Wetake the estimate for the atmospheric flux from [7]. Thisquantity nloc is plotted in Fig. 2 (left panel) as a functionof ε and mQ. Thus, this plot gives an expected averagedensity created by cosmic rays, if evaporation can be ne-glected. Existing terrestrial bounds shown in gray areadapted from [10]. We find that for ε . 5×10−6, there isnegligible terrestrial accumulation since the mCP inter-acts feebly enough to penetrate the entire Earth withoutthermalization. As ε is increased, there is also a largerflux due to preferential meson decays resulting in larger

5

nQ=10/ccnQ=1/ccnQ=0.1/ccnQ=10

-2 /ccnQ=10

-3 /ccnQ=10

-4 /ccnQ=10

-5 /cc

10-3 10-2 0.1 1 1010-6

10-5

10-4

10-3

10-2

0.1

[]

ϵ

nQ=10-4 /cc

nQ=10-5 /cc

nQ=10-6 /cc

nQ=10-7 /cc

10-3 10-2 0.1 1 1010-6

10-5

10-4

10-3

10-2

0.1

[]

ϵ

FIG. 2: Accumulated terrestrial density of mCPs arising from decay of mesons produced by cosmic rays in theatmosphere. Left panel: number densities neglecting evaporation; right panel: realistic number densities upon

accounting for evaporation.

10-3 10-2 0.1 110-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

[]

ϵ

=

=

=-

=-

= -

10-3 10-2 0.1 110-10

10-9

10-8

10-7

10-6

10-5

[]

ϵ /

↑-↑

/

/

/

/

FIG. 3: Left: Existing DD limits on mCDM parameter space adopted from [40], Right: Contours of nQ/fQ arisingfrom accumulation due to virial mCDM density are plotted.

accumulation. We find that densities up to nQ ≈ 1cm−3

can be achieved barring evaporation.It is clear that this density will be diminished due to

evaporation, and the total local density will depend sen-sitively on the retention time. This can be thought asthe time taken for the mCP to diffuse out to the sur-

face (with subsequent evaporation determined by mQ) isgiven by the diffusion time tdiff(dpen) given in Eqn. 6.We approximate the total number of mCPs collected inthe infinitesimal shell with depth dpen to have been dis-tributed with linearly decreasing density in the shell ofthickness dpen. Thus we have for the local density,

nloc(h) ≈∫ βγmax

d(βγ)dΦ

d(βγ)

πR2⊕tdiff(dpen)

43π(R3⊕ − (R⊕ − dpen)3

) h

dpen≈∫ βγmax

d(βγ)dΦ

d(βγ)

h

vthλ(12)

This quantity is plotted in Fig. 2 (right panel). The effectof evaporation is severe for lighter masses, due to their su-perior thermal velocities which leads to shorter diffusiontimes. Above a GeV, evaporation is negligible and theleft and right panels present near identical densities. In

the region currently allowed by terrestrial bounds, den-

sities upto nQ ≈ 10−4

cm3 can be achieved. While this isseveral orders of magnitude smaller than the densitiesfound in Section II for mCDM, it is important to notethat this is an irreducible density with no assumptions

6

regarding the relic density of these mCPs.

B. Virial mCDM with Small ε

Alternatively, virial mCPs with lower charge ε couldalso reach significant depths before thermalizing. Thisthermalized mCP then diffuses outwards before eventu-ally evaporating. Unlike the model variation presentedin Section II, these mCPs can reach surface and under-ground detectors without significant slow-down, leadingto strong limits from existing DD experiments shown inFig. 3, left panel, which we adopt from Ref. [40]. How-ever, as seen in the figure, the limits relax for subcompo-nent mCDM, with no existing limits below fQ ≈ 10−8.

We next calculate the local thermalized density to ex-plore a complementary probe of this parameter space.

The local density can be calculated using Eqn. 12 withthe flux given by,

dΦ

d(βγ)≈ dΦ

dvQ= vQ

fQρDM

mQg(vQ) (13)

where the non-relativistic approximation βγ ≈ vQ isused, and g(vQ) is the Maxwell Boltzmann distributionboosted to the Earth frame given in Appendix B of [41].

We provide contours of the resultant mCDM densityat 1km depth in rock in Fig. 3 (Right panel). For smallenough ε the entire Earth is not enough to stop virialDM and hence there is no accumulation. Above the blackdashed line, mCDM stops in the overburden and we leavethe estimation of the number densities for top-down ac-cumulation to future work.

We note that for part of this parameter space (e.g. forε ∼ 10−6, mQ ∼ 100 MeV − 1 GeV and fQ ∼ 10−8 con-centrations of mCPs in the underground laboratories canreach 102 cm−3, which may still provide some basis forfuture detection, despite the smallness of ε, as discussedin Section VI.

IV. BOUND STATES

The thermalized dark matter now has kinetic energyon the order of kT ∼ 0.025 eV. The negatively chargedmCDM can now form bound states with atoms. At asufficiently large mass, the lowest orbit of such boundstates is inside the atomic K-shell if,

r =a0

Zε

me

µQ,N<a0

Z

ε >me

µQ,N(14)

Here µQ,N is the reduced mass of the mCDM-nuclear sys-tem, a0 is the Bohr radius, and Z is the atomic numberof the nucleus. If this condition is satisfied, the bind-ing energy will be on the order of EB ' Z2ε2µQ,N/2 '13.6 eV × Z2ε2(µQ,N/me).

One may worry that if the bound states form, the ex-isting atomic electron gains some positive energy due toeffective screening of the atomic nucleus, Z → Z−ε. Thetotal binding energy of an atom scales as 16eVZ

73 . This is

obtained by observing that to a good approximation, the

electrons Ze = Z in number at a distance a0Z− 1

3

N fromthe nucleus with effectively un-screened charge ZN ' Z.Substituting ZN → ZN − ε, the net binding energy is

∆EB =

(13.6ε2Z2µQ,N

me− 21.3εZ

43

)eV (15)

Requiring EB > 0 gives

ε >8× 10−4

Z23

GeV

µQ,N. (16)

This is always weaker than Eqn. (14) for Z > 1, andtherefore we take (14) as the main criterion for the boundstate formation by negatively charged mCPs.

These exotic bound objects have net charge ε. Thenucleus could attract more negative charged mCPs butit is unlikely because there are fewer mCPs than nuclei.They could form bound states with positively chargedmCPs as well.

The positive mCPs could either form bound states withfree electrons,

EB = 13.6 eVε2 > 0.025eV

ε > 0.042. (17)

or could form bound states with negatively chargedmCDM / atom-negative charge hybrids. This boundstate has energy,

E = 13.6 eVε4mQ

me> 0.025eV

ε > 0.2

(me

mQ

) 14

(18)

Notice, however, that the bound states of Q+Q− can oc-cur due to A′ exchange, and can be significantly deeperthan electrostatic bound states of the same particles. Theexistence of the Q+Q− binding can have profound con-sequences for cosmological abundances and late annihi-lation of mCP, as is discussed e.g. in Refs. [42–45].

In both Eqs. (17) and (18), we have required thatthe depth of these bound states exceed typical thermalenergy, because otherwise they can be easily broken upby thermal collisions. An approximate position of thecritical dividing lines for the bound states is shown in Fig.4. As is easy to understand, the electrostatic attractionbetween Q− and a nucleus is strongest, and the solidblack line is plotted for a typical nucleus with Z ∼ 30.

We briefly outline observable physics effects that canoccur due to a formation of (Q−N+) bound states. Thecross section leading to these bound states is not nec-essarily small. A typical formation of this bound state

7

will occur with an Auger-type ejection of an electron,and subsequent cascade of the bound state down to itsground state,

Atom +Q− → (Atom+nQ−) + ne+ (γ). (19)

The rate for such a process can be large, and we makea crude estimate of the cross section by accounting forthe relatively small size of the nucleus-Q− bound state,∼ π(a0/Z)2, the probability of an outer electron to bewithin that distance from the nucleus ∼ Z−2. The crosssection will contain 1/vQ, the inverse velocity of the in-coming particle, which will be made dimensionless by thetypical velocity of an electron inside the K-shell, ∼ Zα.This way, one get the following estimate for the crosssection of bound state formation,

σcapturevQ ∼παa2

0 × cZ3

∝ 10−23 cm2× c× (30/Z)3. (20)

This size of the cross section will ensure relatively rapidcapture of Q−, if the bound state formation is possible.The refinement of this estimate along the line of compu-tations performed in [34, 35] is possible.

The capture of Q− by the nuclei of light elements maylead to exotic concentrations of (HQ), (CQ) and (OQ).However, the search techniques that involve ionizationand mass spectrometry [46, 47], as well as “alternative”chemical history for the milli-charged bound states posescertain difficulties in applying such bounds.

A less uncertain approach to search for an mCP “re-combination” with an atom would consist in searchingfor heat/ionization provided by the process (19). For ex-ample, just below the (Q−N+) boundary in Fig. 4 therecombination with light elements is not possible but re-combination with atoms such as Xe or I happens readily.An ideal setup for such a probe would be the DM-Ice ex-periment [48] that utilizes NaI crystals shielded by ∼2.5km of ice. Assuming the range of parameters that doesnot allow the formation of bound states with H, O andelements in the atmosphere, one could still expect - fora right range of mQ, ε, the exothermic reaction of Q−

association with iodine atoms. Taking into account thatthe capture cross sections can be significant (20), all (ornearly all) of the negative mCP incident on NaI crystalmay undergo the capture process. In this case one shouldexpect that the counting rate is

Events

time∼ nQ(2.5 km)× vth ×Area

∝ 106Hz× nQ1 cm3

× Area

100 cm2×(

100 GeV

mQ

)1/2

. (21)

If the binding energy is between few keV to a 100keV, one should compare it with the counting ratesobserved by these experiments that do not exceedO(10) kg−1day−1keV−1, which for 10 kg crystals and twodecades in energy does not exceed the total counting rateof 0.1 Hz. Therefore, for 100 GeV particles it translatesto sensitivity to nQ at the level of 10−7 cm−3, and given

results of Fig. 1, to fQ as small as 10−19. Conversely,one can achieve some sensitivity to cosmic ray generatedmCP flux. Further gains in sensitivity can be achievedby exploiting that one and the same amount of energyis released in the formation of the bound states with agiven atom, which would then show as an “unidentified”line in the spectrum taken by a detector.

We finish this subsection by acknowledging the factthat the full exploration of the sensitivity to bound statesthroughout mQ, ε parameter space is difficult, as thebinding to lighter elements will change patterns of mCPaccumulation and distribution with depth. We leavemore detailed exploration of the bound state related ob-servables to forthcoming work.

-

-

-

-

-

[]

ϵ

-

-

+-

+-

-+

FIG. 4: Parameter space that allows for the boundstates of mCPs with nuclei, electrons and between

themselves. Curves for the direct detection sensitivityare shown for fQ = 1.

V. ANNIHILATION INSIDE LARGE VOLUMEDETECTORS

In this section we explore the possibility of mCDMbeing in equal amounts of particle anti-particle pairs.The presence of a large number of positive and nega-tive mCDM terrestrially can lead to annihilations. No-tice that the negative charge can be bound deep insideatoms, and the probability of annihilation may get signifi-cantly reduced due to the electrostatic repulsion. (A pos-itively charged mCP would not be able to approach theorbit of bound negatively charged mCP). On the otherhand, the A′-induced attraction between the two mCPparticles may actually overcome the Coulomb repulsion,and the annihilation may proceed even if the negativemCP is locked inside an atomic bound state. Unfortu-nately, reliably predicting the abundance of mCP at lo-cations of underground laboratories when negative mCPsare intercepted by atoms is extremely challenging. Forthis reason, we concentrate on the region of parameterspace where both the positive and negative mCDM areunbound by atoms.

8

Annihilation of mCPs can occur via a variety of dif-ferent channels. The largest cross section presumablyoccurs due to annihilation to two dark photons, QQ →A′A′. While subsequent conversion/oscillation of A′ intoA can occur, it is severely suppressed at small mass ofA′. Visible annihilation channels via an s-channel virtualphoton, QQ→ A∗ → SM, will occur at a rate suppressedby ε but will result in immediate release of energy in theform of SM charged particles.

The annihilation cross-section of QQ into e+e− is givenby,

σannv =πε2α2

3m2Q

∼ 2× 10−32ε2GeV2

m2Q

cm2 (22)

Next, let us estimate the number of annihilation eventsinside a volume that we will take to correspond to thefiducial volume of the Super-Kamiokande experiment.The event rate is given by,

n2QσannvV =400

( ε

10−5

)2(

nQ108/cm3

)2

(23)

× V

22000 m3

(GeV

mQ

)21

year,

which potentially may result in a very strong sensitivityto ε× nQm−1

Q .

0.5 1 5 10 50 10010-6

10-5

10-4

10-3

10-2

[]

ϵ

=-

=

-

=

-

=-

=- -

+

-

-

FIG. 5: Limits corresponding to more than 200 eventsin 22 kton year exposure at Super-K for QQ→ e+e−.

Fig. 5 shows the sensitivity contours for the differentinput values of fQ. We take O(200) events per year tobe roughly a limiting count rate. In calculating nQ, wealready use all the machinery developed in the previoussection. As a result, we can see that large regions of theparameter space can be probed by direct annihilation ofmCPs inside the Super-K volume.

VI. ELECTROSTATIC ACCELERATORS

Self-annihilation, enhanced by terrestrial accumula-tion, provides non-trivial limits on the parameter space

of the mCDM model. At the same time, quadratic scal-ing with abundance does not allow to probe very smallfQ. The smallest fQ where experiments like Super-Kwill have sensitivity is for fQ ∼ 10−7. In this section wepropose a novel strategy to test even smaller densities.

As alluded to in the Introduction, the local density ofmCPs could be accelerated in a large electric field andthe accelerated mCPs could then be detected. Owingto disparate charge to mass ratio compared to SM par-ticles, oscillating field accelerators will not be suitablefor mCPs. Instead electrostatic accelerators such as Vande Graaf generators and Cockcroft-Walton acceleratorswould be suitable. Modern accelerators with potentialdifference (∆V ) in the megavolt range are used in nu-clear physics experiments. Examples among these areLUNA (∆V = 3.5 MV) [49], JUNA (∆V = 0.4 MV) [50]and CASPAR (∆V = 1.1 MV) [51].

Since we are considering mCPs with a dark photon,we need to first determine the plasma masses so as toensure that the electric field is not shielded by mCPs.The plasma mass of the dark photon A′ in the presenceof a number density nQ of mCPs Q is given by,

Π = g2Q

nQmQ≈ (3× 10−5eV)2 nQ

1014/cm3

GeV

mQ(24)

Here gQ → 1 is set, to be conservative. The range of theelectric field is thus,

λ ≈ 7mm×

√1014/cm3

nQ

√mQ

GeV(25)

In other words, the screening length is larger than 1 meter

for nQ . 108

cm3 . If the concentrations of mCPs exceed thislevel, it is likely that even the acceleration of “normal”protons will get compromised.

We consider the accelerator field to be turned on, butwith the proton source inside the accelerator being “off”.While the mCPs outside the region with the electric fieldreceive no net acceleration, mCP particles on the insidemay get accelerated. Given a Ethr required for detection,it is clear that in order to have sensitivity one shouldrequire

εe∆V > Ethr, (26)

where ∆V is the accelerating voltage.

To create a more realistic description, we model theaccelerator tube as a 1mm radius 1 meter long tube sim-ilar to the LUNA setup [49]. The flux of mCPs seepinginto the pipe that gets accelerated to detectable energies

9

is given by

Φ[E > Ethr] = 2πrL

(1− Ethr

εe∆V

)fQηρQmQ

vth

×Min[1,rεe∆V

L√TEthr

]

= 2× 1021HzfQ

(GeV

mQ

) 52(

1− Ethr

εe∆V

)×Min[1,

rεe∆V

L√TEthr

] (27)

This expression is derived in detail in Appendix. B.In Fig. 6, we plot the rate for accelerated mCPs to

come out of this setup. The accelerated flux of the mCPsis relatively collimated, and could be detected with rel-atively compact dark matter detectors. To translate therate of accelerated mCPs to the counting rate we needto estimate the probability of generating one signal eventby a single accelerated mCP particle.

We assume a dark matter detector that completely cov-ers the geometric parameters of the mCP beam, but withan opening in the shielding that allows mCPs to enter.The probability for mCP to create a signal event can beestimated as

P ∼ σQ,atom × natoms`, if P < 1. (28)

where ` is taken to be ∼ 5 cm is a realistic size for asmall-to-medium size DM detector, natoms is the num-ber density of active targets in the detector. We takenatoms ' 4.5 × 1022 cm−3 to correspond to the num-ber density of germanium atoms in Ge crystal. Finally,σQ,atom is the cross section leading to atomic recoil orionization by the accelerated mCPs. Depending on themass of the mCP, and the type of detector, the relevantscattering is on the atomic electrons or elastic scatteringon a whole atom, with subsequent ionization created inthe inter-atomic collisions. In the latter case, the result-ing ionization energy is typically quenched by a factorof ∼ 0.1. The transfer of energy in elastic scattering onan atom is the most efficient if mQ ∼ matom. Using anon-pertubative estimate (Appendix A) for such a crosssection, σ ∝ 4π×(µvQ)−2 ∝ 10−23cm2 for a typical massof mCP in the 100 GeV range, and its kinetic energy inthe ∼ keV range, it is easy to see that for this size ofthe cross section, the probability of scattering within the5 cm detector is O(1). In addition, in order to maximizethe signal from such detectors in terms of mQ, it would beadvantageous to use devices with a wide range of atomicmasses, such as CaWO4 of CRESST [52].

The lowest thresholds for detection, and sensitivityto the lowest mQ can be achieved through scatteringon electrons/atomic ionization. Despite the fact thateven after the acceleration, the mCPs are relatively slow,one could apply perturbation theory for estimating thecross sections leading to ionization. Taking the outershell atomic electron to be localized within space region∼ a0, its interaction with an incoming mCP to scale

as U ∼ αεa−10 , perturbation theory is valid as long as

U ×∆t ∼ αεv−1Q < 1 [53]. Taking mQ below a few GeV,

ε in the 10−6 − 10−3 range and vQ ∼ (|εe∆V |/mQ)1/2 ∼ε1/2× (10−2− 10−1), we see that the perturbativity con-dition is satisfied, and the cross section can be estimatedto scale as πa2

0 × (αεv−1Q )2. Again, this estimate exceeds

1/(natom`) making P ∼ O(1). Thus we conclude thatthe rate of accelerated mCPs in Fig. 6 can indeed trans-late to a similar counting rate in a small-to-medium sizedetector intercepting the “beam” of mCPs.

Fig. 6 summarizes possible counting rates of acceler-ated mCPs for fQ = 10−8. Since the probability of de-tection can indeed be P ∼ O(1), these rates translateto possible counting rates inside a detector placed alongthe path of the accelerated particles. Given that it isrealistic to find detectors with background counts as lowas 10−3 Hz, it is clear that a dedicated search along thelines suggested in this section could probe fQ down toextremely small values. For mQ & 1 GeV, Fig. 6 (Left),achieving sensitivity to fQ as small as 10−20 looks re-alistic, with fQ ≈ 10−8 achievable all the way to verylarge mQ. (At large mQ 100 GeV one should be cog-nizant of the fact that the recoil energy of atoms drops asmatom/mQ, and ionization of atoms may be suppressed if

αεv−1Q becomes greater than 1, and the mCP-electron in-

teraction becomes adiabatic. This regime would requireadditional analysis of ionization efficiency, and would alsobenefit from detectors that are sensitive to energy release,e.g. phonons, spread between many atoms.) In Fig.6 (Right) we illustrate the rate sensitivity and energythresholds required to probe mCDM with masses below1 GeV that have accumulated via bottom-up mechanism.Given rapid advance in dark matter detectors, some partof the parameter space can be probed with existing tech-nology. It is also clear that if at some point in the fu-ture, the detection thresholds for DM-induced recoil canbe brought to a sub-eV level (see e.g. [54]), even ε’s assmall as 10−8 can be probed via accelerating mCPs inthe underground MV voltage electrostatic accelerators.

Also of notice is potential sensitivity to mCPs via theirgeneration by cosmic rays which results in an irreduciblepopulation on Earth. This irreducible number densitiesshown in Fig. 2 translate to acceleration rates plotted inFig. 7. We find that a maximum obtainable rate can beas high as 103 Hz, while a lot of new parameter spacecan be explored with counting rates reaching down to1 Hz and below.

In addition to novel probes discussed in this paper, wehave also examined a number of other ways of constrain-ing ε, nQ parameter space, summarized in Appendix C.None of them carry as much promise/sensitivity as thethree pathways outlined in this work.

10

1 10 100 103 104 105 106 10710-6

10-5

10-4

10-3

10-2

0.1

1

[]

ϵ

-

-

-

=-

0.01 0.1 110-8

10-7

10-6

10-5

[]

ϵ

=-

↑-

↑

FIG. 6: The rates of accelerated mCDM for fQ = 10−8 in an electrostatic accelerator. In the Left panel, the ratecorresponding to mQ & 1 GeV, where top-down accumulation is dominant is shown with the requirement for the finalenergy to exceed 10 eV, 100 eV, 1 keV. In the Right panel, the rate corresponding to mQ . 1 GeV, where bottom-up

accumulation is dominant is shown with the requirement for the final energy to exceed 100 meV, 1 eV, 10 eV.

0.01 0.1 1 1010-6

10-5

10-4

10-3

0.01

0.1

[]

ϵ

-

FIG. 7: The rates of accelerated mCPs produced inmeson decays in the atmosphere in an electrostatic

accelerator are shown with the requirement for the finalenergy to exceed 10 eV, 100 eV, 1 keV

VII. CONCLUSION

We have shown that the direct probes of milli-chargedparticles can be advanced using their accumulation in-side the Earth. Owing to a relatively large cross-sectionsand short free path inside dense media, the number den-sities of mCPs can indeed be many order of magnitudelarger than the cosmological abundances. In this paperwe have analyzed the main mechanisms of their accu-mulation, finding that when evaporation is impossible,strong enhancements of number densities are expected atthe locations of the underground laboratories. The en-hancement factors can approach 1015, and therefore evenvery subdominant fractions of the cosmological mCP DMcan be probed. Cosmic ray induced production of mCPscreates far less abundant concentration, which neverthe-less could reach up to 10−4cm3.

We pointed out three different methods that could leadto the most precise probes of mCP properties to date.Concentrations upward of 107cm−3 can be probed via theannihilations of mCP particles to the SM states insidelarge volume detectors, such as super-Kamiokande. Aminimal adjustment of already existing searches will beable to refine sensitivity in 1-to-10 GeV mass range (Fig.5). mCPs created by cosmic rays cannot be probed thisway, as the predicted densities are too small.

Another method for potentially probing even verysmall abundances of dark matter is via the formationof bound states of negatively charged mCPs and atomicnuclei. If mQ, ε parameters are right (i.e. just be-low the solid line on Fig. 4), the binding to heavy ele-ments may be possible, while the binding to light ele-ments is not. In this case, the dark matter experimentsthat use heavier elements (Xe or I) may be quite sensi-tive to the energy release accompanying the formation ofbound states. In particular, the DM-Ice experiment thatutilizes NaI shielded by Antarctic ice is a good exampleof a device that is extremely sensitive to such a scenario.We argue that nQ as small as 10−7cm−3 can be probed,that due to accumulation enhancement leads to the sen-sitivity to tiny fQ. (Detailed analysis of mQ, ε param-eter space is left for future work, as formation of boundstates throughout column density may significantly al-ter the predictions for the density profiles as function ofdepth.) We also note that for a fixed mCP mass andcharge, the amount of binding energy to a given type ofan atom is fixed, and therefore the formation of boundstates leads to a mono-energetic energy deposition. It iswell known that the recently observed Xenon 1T excessevents [16] is consistent with a monochromatic signal (seee.g. [55–58]). It is then possible to speculate that the ex-cess is maybe coming from Q−-Xe nucleus bound stateformation. However, more work needs to be done to un-derstand whether the required densities of the mCPs can

11

be realistically expected for the environment (Gran Sassolab) where the experiment is operating.

Finally, perhaps the most direct way of testing themCP particles is their acceleration in underground accel-erators (that see their primary application for measuringastrophysically relevant nuclear reaction cross sections).Even an “accidental” acceleration of mCPs may resultin their kinetic energy going up from thermal to accel-erated energies ∼ ε∆V . For MV-type electrostatic ac-celerators, and for a generous range of ε, 10−5 − 10−1,the resulting gained kinetic energy can be far above thethresholds of direct detection experiments at 10 eV (andpossibly lower in the near future). Therefore, combin-ing underground accelerations with the specially placeddark matter detectors along the mCP accelerated trajec-tory can bring significant new sensitivity, and indeed testlocal concentrations of mCPs down to unprecedented lowvalues. This way, many “physics targets” can be covered.A dedicated effort in this direction has a potential to ex-plore mCP densities created by cosmic rays, and accessthe region of parameter space consistent with the expla-nation of the EDGES anomaly.

VIII. ACKNOWLEDGEMENTS

We are indebted to Drs. A. Berlin and H. Liu for valu-able critical comments. We are also grateful to Drs. R.Harnik and R. Plestid for earlier collaboration on themCP project. M.P. would like to thank Dr. T. Bring-mann for earlier discussions of related ideas. M.P. is sup-ported in part by U.S. Department of Energy (Grant No.desc0011842).

Appendix A: mCP-atom scatterting using Hulthenpotential

For a mCP of mass M , and charge ε (with electroncharge 1), the perturbative expression for an atomic scat-tering can be written as

dσ

dq2= 4πZ2α

2ε2

v2q4F 2(q), (A1)

where F (q) is atomic form factor normalized to 1 at high-momentum transfer (which corresponds to elastic scat-tering on un-screened nucleus). The same cross sectioncan also be written as,

dσ

dΩ= 4Z2α

2ε2

q4F 2(q) (A2)

We will take the simplest ansatz for F (q) that neverthe-less captures the main physics regimes:

F (q) =a2zq

2

1 + a2zq

2(A3)

where

az =a0

4

(9π2

2Z

) 13

∼ 0.89a0

Z13

∼ 1

4.2Z13 keV

. (A4)

The momentum transfer cross-section is relevant to cal-culate the overburden required for thermalization as wellas the eventual terminal velocity. It is given by

σT =π

2(µv)4

∫ 2µv

0

dσ

dΩq2dq2. (A5)

Setting azµv = R, we get,

σT = 2πα2Z2ε2log(4R2 + 1

)− 4R2

4R2+1

µ2v4

= 2πa2zB2

(log(4R2 + 1

)− 4R2

4R2 + 1

)(A6)

Here B = Zαεazµv2

. It is easy to see that at R 1 the

cross section scales as v−4, while in the opposite regime,R 1, it does not depend on velocity.

The perturbative answer is valid only if eitherZεαazµ 1, which for Zrock ∼ 20 translates to,

ε 7.5× 10−5 GeV

µ(A7)

or if Zεαv 1, for Zrock ∼ 20,

ε 6.3v (A8)

For larger coupling, expressions are available in theclassical limit, µazv 1 which is not valid here,0.6√

µGeV ∼ 1.

We will instead use the expressions derived fromHulthen potential in Ref. [59].

σT =4π

µ2v2sin2 δ0 (A9)

Here,

δ0 = arg

(iΓ(i2µazvκ

)Γ(λ+)Γ(λ−)

)(A10)

λ± = 1 +iµazv

κ±

√Zεα2µ

κmφ− µ2a2

zv2

κ2attractive

= 1 +iµazv

κ± i

√Zεα2µ

κmφ+µ2a2

zv2

κ2repulsive

(A11)

12

We find that a very good approximation for both attrac-tive and repulsive interactions,

〈σT 〉th ∼ Min

(16πZ2α2ε2

µ2v4th

,4π

µ2rock,Qv

2th

)

〈σT v〉th ∼ Min

(8πZ2α2ε2

µ2v4th

× vth,2.2π

µ2rock,Qv

2th

× vth

)

〈σT v3〉th ∼ Min

(5πZ2α2ε2

µ2v4th

× v3th,

2.2π

µ2rock,Qv

2th

× v3th

)(A12)

These cross sections form the basis for our code that cal-culates the effective slow-down, sinking velocity, diffusioncoefficients, and ultimately nQ(h) in the main body of thetext.

Appendix B: Accelerator geometry

The dark matter is thermal and hence has velocity

vth =

√2T

mQ∼ 7× 10−6

√GeV

mQ(B1)

The differential angular flux coming in per infinitesimalpipe length is

dΦ

dld cos θdϕ= 4r

fQηρQmQ

vth (B2)

Here θ ∈ 0, π2 , is the angle between the incoming par-ticle velocity and the beam axis, while ϕ ∈ 0, π2 sub-tended by the velocity on the radial direction.

The time spent by the particles inside the pipe is,

τ(θ, ϕ) =2r

vth sin θ cosϕ(B3)

The maximum time the particles can spend (becauseof acceleration along the beam axis) is,

τmax =

√2l

a=√

2lL

√mQ

εe∆V(B4)

If a particle spends time τ = Min[τmax, τ(θ, ϕ)], inside,it is accelerated to,

EQ =1

2mQv

2f =

1

2mQa

2τ2 (B5)

now,

EQ[max] =1

2mQa

2τ2max =

l

Lεe∆V (B6)

Given a threshold for subsequent detection Ethr, thereis an lmin,

lmin =Ethr

εe∆VL (B7)

Furthermore, we require that the particles enter at thecorrect angle, this is satisfied if,

sin θ cosϕ <rεe∆V

L√TEthr

(B8)

Putting this together we get,

Φ[E > Ethr] =

∫ L

lmin

dl

∫d cos θdϕ

dΦ

dld cos θdϕ(B9)

×Θ

(rεe∆V

L√TEthr

− sinθ cosϕ

)(B10)

This can be simplified further to

Φ[E > Ethr] =2πrL

(1− Ethr

εe∆V

)fQηρQmQ

(B11)

× vthMin

[1,

rεe∆V

L√TEthr

]. (B12)

Appendix C: Other probes of terrestrialmilli-charged dark matter

There are several recent experimental limits which setconstraints on a local population of thermalized DM. Weconsider them in turn and show that none of them setconstraints on terrestrial mCPs.CryogensAnomalous heating of cryogens was considered in

[29, 30, 60]. However all of these experiments employshielding to reduce black-body radiation and hence willcool the mCDM to the shield temperature before theyhit the cryogen if the mean-free-path corresponding tothe transfer cross-section is smaller than the thickness ofthe shield. This can be evaluated using

λMFP =1

nshieldσT≈ 10−6cm

µ

GeV(C1)

Here we approximate nshield = 1022

cm3 and take the cross-

section in Eqn. 5 to be σT ≈ 4πµ2v2th

. This MFP is much

smaller than any realistic shielding. As a result mCDMwill not be constrained by cryogens.LHC beam lifetimeIn [30], limits were placed on accumulated DM with

contact interactions from LHC beam lifetime, and thesearguments can be generalized to other particle storagerings and accelerators. Rutherford scattering on mCPscan also take particles from the beam. This will comeon top of particle loss due to the residual scattering onatomic constituents of the residual gas molecules insidethe beam pipe. The scattering on mCP (that can betreated perturbatively in this problem) is at least ε2 timessmaller than scattering on electrons and nuclei. This way,one can estimate the “best case” sensitivity via compar-ing the beam losses on atoms vs mCPs

nQ ∼ natomsε−2 ∼ 1010cm−3 × (10−2/ε)2, (C2)

13

where we took a realistic residual gas density at 106 par-ticles per cm3. This may provide some additional sen-sitivity to mCP if fQ is large, but it is inferior to otherprobes discussed in this paper.

Stability of nuclear isomers: 180m Ta.

In [27, 28] the non-observation of the decay of isomericTantalum was used to set limit on terrestrial DM. Theform-factor to scatter with 180m Ta naturally picks mCPmasses in the hundreds of GeV and above. For positivelycharged mCDM, there is Coulomb repulsion with nuclei.The probability of overcoming this is given by the Gamowfactor.

Pg(T ) = e−EgT ≈ e

−11 ε2

v2th (C3)

where Eg = 2µ (παZTaε)2

and T is the ambient temper-ature. This factor evaluates to tiny values rendering thecross-section too small for [28] or any future projections

of a Tantalum experiment to set relevant limits. Neg-atively charged mCDM of this mass in the heavy massrange can induce 180m→ 180 g.s. transition. If the pa-rameter range is such that the formation of bound stateswith nuclei is possible, de-excitation of 180m isotope willnot provide competitive sensitivity to e.g. nucleus-mCPrecombination process discussed in section IV. The mainreason for that is relatively small isotopic abundance of180m that would not be competitive with many kilo-grams of I or Xe employed by direct detection experi-ments. If ε is very small (and the mass is in ∼ 100 GeVrange to ensure efficient de-excitation), then surface di-rect detection experiments will be more sensitive to Q−

than Tantalum 180m.Anomalous heat transport The heat conductivity

of Earth can be modified by the presence of an Earth-bound exotic species [30]. However, the heat conductivityis proportional to the mean-free-path which is extremelysmall in rock as explained above. As a result, there areno useful limits that we can derive from this effect.

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