PPPC 4 DMν: a Poor Particle Physicist Cookbook for Neutrinos from Dark Matter annihilations in ...

27 Pages·2014·5.24 MB·English

Home Search Collections Journals About Contact us My IOPscience
PPPC 4 DMν: a Poor Particle Physicist Cookbook for Neutrinos from Dark Matter
annihilations in the Sun
This content has been downloaded from IOPscience. Please scroll down to see the full text.
JCAP03(2014)053
(http://iopscience.iop.org/1475-7516/2014/03/053)
View the table of contents for this issue, or go to the journal homepage for more
Download details:
IP Address: 131.114.71.156
This content was downloaded on 16/05/2016 at 13:00
Please note that terms and conditions apply.
J
C A P
ournal of osmology and stroparticle hysics
An IOP and SISSA journal
PPPC 4 DMν: a Poor Particle
Physicist Cookbook for Neutrinos
from Dark Matter annihilations in
J
C
the Sun
A
P
Pietro Baratella,a Marco Cirelli,b Andi Hektor,c,d Joosep Pata,c 0
Morten Piibelehtc and Alessandro Strumiac,e 3
(
aScuola Normale Superiore and INFN,
Piazza dei Cavalieri 7, Pisa, 56126 Italy 2
bInstitut de Physique Th´eorique, CNRS URA 2306 & CEA-Saclay, 0
Gif-sur-Yvette, 91191 France
1
cNational Institute of Chemical Physics and Biophysics,
4
Ravala 10, Tallinn, Estonia
dHelsinki Institute of Physics, )
P.O. Box 64, Helsinki, FI-00014 Finland
0
eDipartimento di Fisica dell’Universit`a di Pisa and INFN,
5
Largo Buonarroti 2, Pisa, Italy
3
E-mail: [email protected], [email protected], [email protected],
[email protected], [email protected], [email protected]
Received January 8, 2014
Accepted March 18, 2014
Published March 27, 2014
Abstract. We provide ingredients and recipes for computing neutrino signals of TeV-scale
Dark Matter (DM) annihilations in the Sun. For each annihilation channel and DM mass
we present the energy spectra of neutrinos at production, including: state-of-the-art energy
losses of primary particles in solar matter, secondary neutrinos, electroweak radiation. We
then present the spectra after propagation to the Earth, including (vacuum and matter)
flavor oscillations and interactions in solar matter. We also provide a numerical computation
of the capture rate of DM particles in the Sun. These results are available in numerical form.
Keywords: dark matter theory, ultra high energy photons and neutrinos
ArXiv ePrint: 1312.6408
ArticlefundedbySCOAP3. Contentfromthisworkmaybeused
underthetermsoftheCreativeCommonsAttribution3.0License. doi:10.1088/1475-7516/2014/03/053
Anyfurtherdistributionofthisworkmustmaintainattributiontotheauthor(s)
andthetitleofthework,journalcitationandDOI.
Contents
1 Introduction 1
2 The DM annihilation rate 3
2.1 The DM capture rate 4
3 The neutrino energy spectra at production 7
3.1 Neutrino spectra with MonteCarlo codes 8
3.1.1 Pythia 8 J
3.1.2 Geant 9 C
3.1.3 Results and comparisons 10
A
3.2 Adding electroweak bremsstrahlung 11
3.3 Spectra at production: results 13 P
0
4 Neutrino propagation 13
3
4.1 Formalism 13
4.2 Transition probabilities 16 (
4.3 Earth crossing 18
2
5 Final result 19 0
1
6 Conclusions 19
4
1 Introduction )
0
The Dark Matter (DM) particles that constitute the halo of the Milky Way have a small but
finite probability of scattering with a nucleus of a massive celestial body (e.g. the Sun or the 5
Earth)iftheirorbitpassesthroughit. Iftheirvelocityafterthescatteringissmallerthanthe 3
escapevelocityfromthatbody, theybecomegravitationallyboundandstartorbitingaround
the body. Upon additional scatterings, they eventually sink into the center of the body and
accumulate, building up a local DM overdensity concentrated in a relatively small volume.
There they annihilate into Standard Model particles, giving origin to fluxes of energetic
neutrinos [1–4]. These neutrinos are the only species that can emerge, after experiencing
oscillations and interactions in the dense matter of the astrophysical body. The detection
of high-energy neutrinos from the center of the Sun or the Earth, on top of the much lower
energyneutrinofluxduetonuclearfusionorradioactiveprocesses, wouldarguablyconstitute
oneofthebestproverbialsmokinggunsforDM,astherearenoknownastrophysicalprocesses
able to mimic it (except possibly for the flux of neutrinos produced in the atmosphere of the
Sun, which however are expected to have a different spectral shape [5, 6]).
While the above idea is rather old, recently it has gained more interest since large neu-
trinotelescopes(Antares, IcecubeanditsplannedextensionPingu, SuperKamiokande
and its proposed upgrade HyperKamiokande, the future Km3Net) are reaching the sen-
sitivity necessary to probe one of the most interesting portions of the parameter space, in
competition with other DM search strategies.
Along the years there have been several computations of the ingredients necessary to
predict the signal in a detector, in particular the DM capture rate [7–13] and the spectra of
– 1 –
neutrinos [14–21]. In [18] a subset of the authors of this paper had computed the neutrino
spectra: their production in solar matter and their propagation subject to interactions and
flavor oscillations. Similar results, but in an event-based approach, had been subsequently
obtained by Wimpsim [19].
In this paper we improve the computation of the neutrino energy spectra with respect
to previous computations and in particular to [18] in various ways:
• we consider a full list of two-body SM annihilation channels;
• we implement a better description of the energy losses of the primary particles in solar
matter, including secondary neutrinos produced by scattered solar matter;
J
• we extend to low energy the neutrino spectra in order to include the large flux of
C
neutrinos generated by stopped µ,π,K (an effect pointed out in [20, 21]);
A
• we take into account EW bremsstrahlung;
P
• weupdatetheneutrinooscillationparametersaddingtherecentlydiscoveredθ ≈ 8.8◦
13 0
and the two possible mass hierarchies of neutrinos (normal or inverted).
3
Letusnowreviewthemainstepsofthecomputationandpresenttheresultsweprovide.
(
The final energy spectrum of the neutrino flux at the detector location is written as
2
dΦ Γ dN
ν ann ν
= (1.1)
dE 4πd2 dE 0
ν ν
wheredistheSun-Earthdistance,Γ isthetotalDMannihilationrateintheSun,dN /dE 1
ann ν ν
is the energy spectrum of νe,µ,τ and ν¯e,µ,τ produced per DM annihilation after taking into 4
account all effects. The computation proceeds as follows:
)
1. The total rate of DM annihilations Γ in the Sun is computed in section 2 in terms
ann 0
of the spin-independent and spin-dependent DM cross sections with nucleons.
5
2. In section 3 we compute the energy spectra of ν and ν¯ at production for DM annihila-
3
tions into
e+e−, e+e−, µ+µ−, µ+µ−, τ+τ−, τ+τ−, ν ν¯ , ν ν¯ , ν ν¯ ,
L L R R L L R R L L R R e e µ µ τ τ
(1.2)
qq¯, cc¯, b¯b, tt¯, γγ, gg, W+W−, W+W−, Z Z , Z Z , hh
L L T T L L T T
where q = u,d,s denotes a light quark, the subscripts L,R on leptons denote Left or
Right-handed polarizations, the subscript L,T on massive vectors denote Longitudinal
or Transverse polarization. h denote the higgs boson, for which we assume a mass of
125GeV.
Decays, hadronization, energy losses in matter and secondary neutrinos generated by
matter particles scattered by DM decay products (an effect ignored in previous com-
putations) are computed running Geant4 [22, 23] using the EU Baltic Grid facilities.
As a check, the first three effects are also independently computed with Pythia [24],
modified by us to include energy losses in matter.
3. As described in section 3.2, we correct the spectra at production including the dom-
inant electroweak radiation effects (not included in MonteCarlo codes) enhanced by
logarithms of M /M , as described in [25]. EW corrections depend on the polarisa-
DM W
tion of the SM particles, which is why we need to specify it in eq. (1.2).
– 2 –
4. As described in section 4, we propagate the fluxes of ν and ν¯ at production, around
the center of the Sun, to the Earth taking into account oscillations, matter effects,
absorption and regeneration from collisions with matter using the neutrino density
matrix formalism as in [18].
The final results are presented and briefly discussed in section 5. In section 6 we
conclude.
The main numerical outputs of the computation are given on the PPPC 4 DM ID
website (‘DMν’ section).
J
2 The DM annihilation rate
C
In this section we compute the DM solar annihilation rate, which sets the overall normal-
A
ization of the expected neutrino flux. We essentially revisit the calculations in [7–13]. For
P
definiteness, we focus on the case of the Sun, although we mention in passing the modifica-
tions needed for the Earth. 0
The number N of DM particles accumulated inside the Sun varies with time under the
3
action of different competing processes: 1) DM particles are captured from the halo (N is
(
thus increased by one unit) via the multiple scattering processes discussed above, 2) DM can
pair annihilate (hence N decreases by two units) and 3) DM particles can be ejected (N 2
decreases by one unit) by a hard scattering on a hot nucleus of the interior of the body, i.e.
0
the inverse process with respect to capture. In formulæ:
1
dN
4
= Γ −2Γ −Γ (2.1)
capt ann evap
dt
)
where Γ is the capture rate, Γ is the DM annihilation rate and Γ is the DM
capt ann evap 0
evaporation (or ejection) rate.
5
The evaporation process is important only for DM lighter than a few GeV,1 so that we
will neglect it for all practical purposes in the following. 3
The annihilation rate is proportional to N2: two DM particles annihilate (hence the
square). It is given by
(cid:90)
1 1
Γ = d3xn2((cid:126)x)(cid:104)σv(cid:105) = C N2, (2.2)
ann ann
2 2
where (cid:104)σv(cid:105) is the usual annihilation cross section averaged over the initial state2 and n((cid:126)x) is
the number density of DM particles at position (cid:126)x inside the Sun, such that the total number
of DM particles is N = (cid:82) d3xn((cid:126)x). After capture, subsequent scatterings thermalize the DM
1This can be easily understood [7–13]: if one requires that the speed of a DM particle thermalized with
(cid:112)
the interior of the Sun v ∼ 2T /M be smaller than the escape velocity from the center of the Sun
DM (cid:12) DM
v (cid:39) 1387 km/s, one obtains the condition M (cid:38) 0.15GeV. A more realistic computation takes into
(cid:12)esc DM
accountthatDMparticlesactuallyfollowa(Maxwellian)velocitydistributionandcomparesthetimeittakes
to deplete the high-velocity tail (above the escape velocity) of such distribution with the age of the Sun,
finding M (cid:38)5GeV. Yet more refined calculations find typically M (cid:38)4GeV [26].
DM DM
2If DM is a real particle (e.g. a Majorana fermion) this is the usual definition of σ and the factor 1/2
takes into account the symmetry of the initial state. If DM is a complex particle (e.g. a Dirac fermion) then
n≡n +n (hereassumedtobeequal)andtheaverageoverinitialstatesisσ≡ 1(2σ +σ +
DM DM 4 DMDM DM DM
σ ). In many models, only DMDM annihilations are present, so that σ=σ /2.
DMDM DMDM
– 3 –
particles to the solar temperature T , such that their density n((cid:126)x) acquires the spherically
(cid:12)
symmetrical Boltzmann form
n(r) = n exp[−M φ(r)/T ] (2.3)
0 DM (cid:12)
where n is the central DM number density and φ(r) = (cid:82)rdrG M(r)/r2 is the Newtonian
0 0 N
gravitational potential inside the Sun, written in terms of the solar mass M(r) enclosed
within a sphere of radius r. Taking for simplicity the matter density in this volume to be
constant and equal to the central density ρ , all the integrals can be explicitly evaluated .
(cid:12)
One finds that DM particles are concentrated around the center of Sun,
(cid:18) 3T (cid:19)1/2 (cid:114)100GeV
n(r) = n0e−r2/rD2M, with rDM = (cid:12) ≈ 0.01R(cid:12) . (2.4) J
2πG ρ M M
N (cid:12) DM DM
C
Within this approximation, one obtains from eq. (2.2)
(cid:18)G M ρ (cid:19)3/2 A
N DM (cid:12)
C = (cid:104)σv(cid:105) . (2.5)
ann 3T P
(cid:12)
Here ρ = 151 g/cm3 and T = 15.5 106 K are the density and the temperature of matter 0
(cid:12) (cid:12)
aroundthecenteroftheSun. Thesameexpressionwouldholdforotherastrophysicalbodies, 3
adapting these two quantities.
(
Neglecting Γ and solving eq. (2.1) with respect to time one finds
evap
2
(cid:18) (cid:19)
Γ = Γcapt tanh2 t t(cid:29)(cid:39)τ Γcapt (2.6)
ann 0
2 τ 2
(cid:112) 1
where τ = 1/ Γ C is a time-scale set by the competing processes of capture and
capt ann
annihilation. At late times t (cid:29) τ one can approximate tanh(t/τ) = 1. In the case of the 4
Sun, the age of the body (∼4.5 Gyr) and the typical values of the parameters in τ indeed
)
satisfy this condition (in the case of the Earth this is not generally the case). Therefore one
0
attains the last equality of eq. (2.6). Physically, this means that the fast (compared to the
age of the Sun) processes of capture and annihilation come to an equilibrium: any additional 5
captured particle thermalizes and eventually is annihilated away.
3
2.1 The DM capture rate
As a consequence of the attained equilibrium, the computation of the annihilation rate Γ
ann
crucially depends on the computation of the capture rate Γ , which acts as a bottleneck.
capt
The computation of the latter proceeds by summing the contribution to capture of all the
individual shells of matter located at position r within the massive body. The result is [7–13]
Γ = ρDM (cid:88)σ (cid:90) R(cid:12)dr 4πr2 n (r)(cid:90) ∞dv 4πv2f (v)v2+v(cid:12)2esc℘ (v,v ). (2.7)
capt i i (cid:12) i (cid:12)esc
M v
DM 0 0
i
Its derivation is lengthy: we will only sketch it in the following by illustrating the individual
pieces of this equation.
• ρ /M corresponds just to the local number density of DM particles at the location
DM DM
of the capturing body.
• The sum runs over all kinds of nuclei i with mass m and number density n (r), to be
i i
integratedoverthevolumeoftheSun. Wetakethestandardsolarelementalabundances
from [27–30].3 The factor σ is the low-energy DM cross section on nucleus i, assumed
i
3Results available in numerical form from A. Serenelli’s webpage (AGSS09 model).
– 4 –
to be isotropic. In terms of the standard Spin Independent and Spin Dependent cross
sections normalized on a single nucleon at zero momentum transfer4 (denoted σ and
SI
σ )wetakeσ = σ +σ forHydrogenandσ (cid:39) 44σ (m +M )2/(m +M )2
SD H SI SD He SI p DM He DM
forthespin-lessHelium. Analogously, forheaviernucleiofmassnumberAonecantake
σ (cid:39) A2σ (m /m )2 (m +M )2/(m +M )2 (spin independent contribution) +
i SI A p p DM A DM
σ (cid:39) A2σ (m +M )2/(m +M )2 (spin dependent contribution). This latter
i SD p DM A DM
SD formula should be complemented by weighting factors due to the spin content in
the different nuclei. In practice, only Hydrogen is relevant for our case, so that a more
careful evaluation is not necessary.
• f (v) is the angular average of the DM velocity distribution with respect to the solar
(cid:12) J
restframeneglectingthegravitationalattractionoftheSun. Thisquantityisconnected
C
to the f(v) with respect to the galactic rest frame as
A
1 (cid:90) +1 (cid:113)
f (v) = dc f( v2+v2 +2vv c) (2.8) P
(cid:12) 2 (cid:12) (cid:12)
−1
0
in terms of the solar velocity v ≈ 233km/s (here c is the cosine of the angle be-
(cid:12) 3
tween v and v ). Observations and numerical simulations suggest that f(v) can be
(cid:12)
parameterized as [31–36] (
2
(cid:20) (cid:18)v2 −v2(cid:19) (cid:21)k
f(v) = N exp eskcv2 −1 θ(vesc−v) (2.9) 0
0
1
with normalisation (cid:82)∞dv 4πv2f(v) = 1 and parameters
0 4
220km/s < v < 270km/s, 450km/s < v < 650km/s, 1.5 < k < 3.5. )
0 esc
(2.10)
0
Here v is the escape velocity from the Galaxy (not to be confused with v , the
esc (cid:12)esc
5
escape velocity from the Sun) at the location of the solar system. For k = 0, f(v)
reduces to a Maxwell-Boltzmann distribution with a sharp cut-off at v < v : f(v) = 3
esc
N × e−v2/v02θ(vesc − v). For k > 0 the cut-off gets smoothed. The resulting f(cid:12)(v)
are normalised such that (cid:82)∞dv 4πv2f (v) = 1. The actual choice of the DM velocity
0 (cid:12)
distributionturnsouttohavearatherlimitedimpact, seee.g.[37]forarecentanalysis.
• The gravity of the Sun is taken into account by v (r), which is the escape velocity
(cid:12)esc
(fromtheSun)atradiusr,suchthatv2+v2 isthesquaredvelocitythataDMparticle
(cid:12)esc
acquires when arriving at r from a very distant point. One here neglects the effect of
other bodies in the solar system, which presumably is a good approximation [38–44].
• The probability that a DM particle, with velocity v far from the Sun, gets captured
when scattering on a nucleus located where the escape velocity is v is given in first
(cid:12)esc
approximation by
(cid:18) (cid:19)
∆ −∆
max min
℘ (v,v ) = max 0, (2.11)
i (cid:12)esc
∆
max
where∆ = 4m M /(M +m )2 and∆ = v2/(v2+v2 )arethemaximaland
max i DM DM i min (cid:12)esc
minimalfractionalenergyloss∆E/E thataparticlecansufferinthescatteringprocess,
4Weassumethattheyareequalforprotonsandneutronsandthattheydonotdependonthemomentum
transfer nor on the DM velocity.
– 5 –
SpinIndependentDMcapturerateintheSun SpinDependentDMcapturerateintheSun
1030 1029
tot
tot
pb(cid:42) 1029 O He Fe pb(cid:42) 1028 vve0sc(cid:61)(cid:61)255500kkmmsseecc
c c
1se (cid:144) Ne Si 1se 1027(cid:144) k(cid:61)1 (cid:144)
n n
ip 1028 S ip (cid:144)
(cid:71)Σcapt (cid:144) Mg (cid:71)Σcapt 1026(cid:144) N
ate 1027 ate
r r 1025
e e
ptur v0(cid:61)250kmsec ptur J
Ca 1026 vesc(cid:61)550kmsec N Ca 1024 H C
k(cid:61)1
(cid:144)
A
(cid:144)
1025 1023
1 10 100 1000 104 1 10 100 1000 104 P
DMmassM GeV DMmassM inGeV
DM DM 0
(a) (b)
3
Figure 1. DM capture rate in the Sun Γ of DM particles with mass M assuming a Spin-
capt DM (
Independent (left) or a Spin-Dependent (right) DM cross section on matter. We normalize assuming
a cross section on protons σ = 1pb and we adopt the indicated values for the parameters of the 2
p
galactic DM velocity distribution. We show the total as well as the contributions from the most
0
relevant individual elements in the Sun. The dashed line is the analytical approximation valid when
the DM is much heavier than the nuclei. 1
4
provided that it is captured. I.e., the probability is computed as the ratio of the size of
)
the interval in energy losses leading to capture (∆E < ∆E < ∆E ) relative to the
min max
0
whole possible interval (0 < ∆E < ∆E ), assuming a flat distribution of the scatter-
max
ing cross section in energy. In general, however, one needs to introduce the form fac- 5
tors F (∆E) that take into account the nuclear response as function of the momentum
i 3
transfer. Explicitly, |Fi(∆E)|2 = e−∆E/E0, with E0 = 5/2miri2 for spin-independent
and E = 3/2m r2 for spin-dependent scattering (here r ∼ (cid:112)3/5A1/31.23 fm (cid:39)
0 i i i i
0.754 ·10−13cm(m /GeV)1/3 is the effective radius of a nucleus with mass number A
i i
and mass m ). The numerator of the ‘ratio of sizes’ becomes then an integral of the
i
form factor over the energy loss ∆E:
1 (cid:90) E∆max
℘ (v,v ) = d(∆E)|F (∆E)|2, (2.12)
i (cid:12)esc i
E∆
max E∆min
eq.(2.11)and(2.12)meanthatthefractionofscatteringsthatleadtocaptureislargest
for nuclei with mass m comparable to the DM mass M (∆ is maximized) and
i DM max
for DM particles that are slow (small v) and in the central regions of the body (large
v ).
(cid:12)esc
Figure 1a shows the capture rate in the Sun having assumed a Spin Indipendent cross
section σSI = 1pb on protons. One sees that several elements contribute figure 1b shows the
p
Spin Dependent capture rate, with the corresponding assumption σSD = 1pb on protons.
p
OnlyHydrogenmattersforthiskindofcapture,withaverysmallcontributionfromNitrogen.
– 6 –
The dotted lines in figure 1 are simple approximations valid in the limit of heavy DM,
M (cid:29) m . In such a limit DM can be captured only if it is very slow,
DM i
MDM(cid:29)mi (cid:112)
v < 2v m /M . (2.13)
(cid:12)esc i DM
Thereby the capture rate is proportional to 1/M2 and can be approximated as
DM
Γ MDM(cid:39)(cid:29)mi ρDM 4πf (0)(cid:88)m σ I (2.14)
capt M2 (cid:12) i i i
DM i
where
J
(cid:34) (cid:35)
(cid:90) R(cid:12) 1 (cid:18)E (cid:19)2 E (cid:18)E (cid:19)
Ii = 4πr2ni(r) 2 m0i − m0ie−2miv(cid:12)2esc(r)/E0i 2m0i +v(cid:12)2esc(r) dr (2.15) C
0 i i i A
In the limit of negligible form factors, E0i (cid:29) mi, the term in square brackets simplifies P
to v4 .
(cid:12)esc 0
The integrals I are adimensional in natural units, and their values are given in table 1
i
3
for the main capturing elements. Inserting their values we find
(
3.28·10−16 (cid:32) ρ (cid:33)(cid:18)100GeV(cid:19)2(cid:32)270 km(cid:33)3 1.8·1046σ +2.16·1049σ
Γ (cid:39) DM sec SD SI. 2
capt sec 0.3 GcmeV3 MDM v0eff pb 0
(2.16)
We here parameterised f (0) = (cid:0)1/π1/2veff(cid:1)3, such that the parameter veff coincides with 1
(cid:12) 0 0
the parameter v for a Maxwellian distribution with no cut-off and neglecting solar motion. 4
0
We also provide a numerical function which gives the annihilation rate Γ = Γ /2
ann capt )
in terms of M , the standard DM cross sections on a single nucleon σ and σ (as defined
DM SD SI
0
above) and the parameters of the velocity distribution v , v , k.
0 esc
5
3 The neutrino energy spectra at production 3
In this section we discuss the computation of the neutrino fluxes emerging from the DM
annihilation process. Neutrinos are produced at several stages of the hadronic and leptonic
cascadesoriginatingfromtheprimaryparticlesproducedbyannihilations,andthesecascades
develop within the dense matter of the astronomical body (the Sun). Such an environment
has important consequences in determining the spectra. For example, some byproducts of
the cascade can be absorbed by the surrounding matter; some others will be just ‘slowed
down’; others are negligibly affected. It is therefore important to model in the best possible
way all these effects. We adopted and compared two strategies:
1) As described in section 3.1.1, we modified the public MonteCarlo code Pythia to
include the effects of cascading energy losses with matter and absorption. The disad-
vantage is that in this way we cannot include the neutrinos emitted by matter particles
scattered by products of DM annihilation.
2) As described in section 3.1.2, we run the Geant4 code, dedicated to particle interac-
tions with matter. The disadvantage is that Geant4 is much more time-consuming
than Pythia: to reach a reasonable statistics we employ the computational resources
of the Baltic grid.
– 7 –
SD capture H 1.60·1047 N 2.42·1043
SI capture He 2.00·1046 N 2.84·1043
O 7.34·1043 Ne 1.03·1043
Mg 3.01·1042 Si 1.98·1042
S 5.73·1041 Fe 8.87·1040
Table 1. Value in natural units of I , for the main elements of the Sun.
i
In section 3.1.3 we compare the two results, interpret them and decide to adopt the Geant4
result. In section 3.2 we describe how we subsequently add electroweak bremsstrahlung J
corrections, which are not included in any Monte Carlo code.
C
3.1 Neutrino spectra with MonteCarlo codes A
3.1.1 Pythia P
WeheredescribehowwemodifiedthePythiapubliccodetoincludetheeffectofcascadingin 0
matterinanevent-by-eventbasis. Weimprovedwithrespecttoourpreviouscomputation[18] 3
where we had only included averaged energy losses in matter. Moreover, we here use Pythia
(
version 8.176 (version 6.2 was used in [18]).
2
The process of particles with mass m and initial energy E (cid:29) m decaying with life-time
0
τ, while at the same time losing energy due to interactions with the surrounding matter, is 0
described by a differential equation for their energy E and their number n:
1
dE dn nm 4
= −α−βE, = − ; (3.1)
dt dt τ E
)
that is: the energy loss of particles in matter can by approximated as a constant term α plus
0
a term β proportional to their energy, while their number n is just governed by τ (the factor
5
m/E takes into account time dilatation). Therefore
3
dn dn/dt nm
= = (3.2)
dE dE/dt τE(α+βE)
is solved by
(cid:18)β +α/E (cid:19)m/τα
0
n(E) = n(E ) . (3.3)
0
β+α/E
We thereby instruct Pythia to perform the decay of particles by first replacing their initial
energy E with a lower energy E randomly chosen according to the above distribution pro-
0
duced by energy losses, eq. (3.3), and then letting them decay. This equation takes different
specific forms according to the particle considered, as we discuss now.
Dominant β: stopping of hadrons. The stopping of hadrons is well approximated by
α = 0. In such a case eq. (3.3) simplifies to
(cid:20) (cid:21)
n(E) E E
E(t) = E e−βt, = exp cr − cr = ex0−x (3.4)
0
n(E ) E E(t)
0 0
where E = m/βτ is a critical energy below which energy losses are negligible. The distribu-
cr
tionofenergiesatdecayisexponentialinx = E /E (withx ≡ E /E ). Inthecenterofthe
cr 0 cr 0
– 8 –