Fitting formulae for photon spectra from WIMP annihilation

Abstract

Annihilation of different dark matter (DM) candidates into Standard Model
(SM) particles could be detected through their contribution to the gamma ray
fluxes that are measured on the Earth. The magnitude of such contributions
depends on the particular DM candidate, but certain imprints of produced photon
spectra may be analyzed in a model-independent fashion. In this work we provide
the fitting formulae for the photon spectra generated by WIMP annihilation into
quarks, leptons and gauge bosons channels in a wide range of WIMP masses.

Full text

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Fitting formulae for photon spectra from WIMP
annihilation
J.A.R.Cembranos1, A.de la Cruz-Dombriz1,2, A.Dobado1,
R.Lineros3and A.L.Maroto1
1Departamento de F´ ısica Te´ orica I, Universidad Complutense de Madrid, E-28040, Spain.
2Department of Mathematics and Applied Mathematics, University of Cape Town, South
Africa.
3IFIC, CSIC-Universidad de Valencia, Ed. Institutos, Apdo. correos 22085, E-46071, Spain.
E-mail: dombriz at fis.ucm.es
Abstract.
particles could be detected through their contribution to the gamma ray fluxes that are measured
on the Earth. The magnitude of such contributions depends on the particular DM candidate,
but certain imprints of produced photon spectra may be analyzed in a model-independent
fashion.In this work we provide the fitting formulae for the photon spectra generated by
WIMP annihilation into quarks, leptons and gauge bosons channels in a wide range of WIMP
masses.
Annihilation of different dark matter (DM) candidates into Standard Model (SM)
1. Introduction
According to present observations of large scale structures, CMB anisotropies and light nuclei
abundances, DM cannot be accommodated within the SM of elementary particles. Indeed, DM
is not only a required component on cosmological scales, but it is also needed to provide a
satisfactory description of rotational speeds of galaxies, orbital velocities of galaxies in clusters,
gravitational lensing of background objects by galaxy clusters and the temperature distribution
of hot gas in galaxies and clusters of galaxies. The experimental determination of the DM nature
will require the interplay of collider experiments and astrophysical observations. Concerning
the latter, DM searches use to be classified in direct or indirect ones (see [1] and references
in the introduction section in [2]). Concerning direct detection, the elastic scattering of DM
particles from nuclei should lead directly to observable nuclear recoil signatures although the
weak interactions between DM and the standard matter makes DM direct detection extremely
difficult.
On the other hand, DM might be detected indirectly, by observing their annihilation products
into standard model particles. Thus, even if WIMPs (Weakly Interacting Massive Particles) are
stable, two of them may annihilate into ordinary matter such as quarks, leptons and gauge
bosons. Their annihilation in different places (galactic halo, Sun, etc.) produce cosmic rays
which could be discriminated from the background through distinctive signatures. A cascade
process follows the WIMPs annihilation. In the end, the potentially observable stable particles
would be neutrinos, gamma rays, positrons and antimatter that may be observed through
different devices. Among them, neutrinos and gamma rays have the advantage of maintaining
their original direction due to their null electric charges.
arXiv:1012.4473v1 [hep-ph] 20 Dec 2010

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Concerning gamma rays, photon fluxes in specific (mainly SUSY) DM models are usually
obtained by software packages such as DarkSUSY and micrOMEGAs based on PYTHIA Monte
Carlo event generator [3] after having fixed a WIMP mass for the particular model under
consideration.
This communication precisely focuses on providing the fitting functions for the photon
production coming from quarks (including t quark), leptons and gauge bosons annihilations
channels. This would allow to apply the results to alternative DM candidates for which software
packages have not been developed. In addition, our investigation determine the dependence of
such spectra on the WIMP mass in a model independent way.
On the other hand, the information about channel contribution and mass dependence can
be very useful in order to identify gamma-ray signals for specific WIMP candidates and may
also provide relevant information about the photon energy distribution when SM particle pairs
annihilate.
Let us remind that the γ-ray flux from the annihilation of two WIMPs of mass M into two
SM particles coming from all possible annihilation channels (labelled by the subindex i) is given
by:
dΦDM
dEγ
γ
=
1
4πM2
?
i
?σiv?dNi
γ
dEγ
×
1
∆Ω
?
∆ΩdΩ
?
l.o.s.ρ2[r(s)] ds ,(1)
where ?σiv? holds for the thermal averaged annihilation cross-section of two WIMPs into two
(ithchannel) SM particles and ρ is the DM density. The integral is performed along the line of
sight (l.o.s.) to the target and averaged over the detector solid angle ∆Ω.
2. Procedure
We have used the particle physics PYTHIA software [3] to obtain our results. The WIMPs
annihilation is usually split into two separated processes: The first describes the annihilation of
WIMPs and its SM output. The second one considers the evolution of the obtained SM unstable
products. Due to the typical velocity dispersion of DM, we expect most of the annihilations to
happen quasi-statically. This fact allows to state that by considering different center of mass
(CM) energies for the obtained SM particles pairs from WIMP annihilation process, we are
indeed studying different WIMP masses, i.e. ECM? 2M. The procedure to obtain the photon
spectra is thus straightforward: For a given pair of SM particles which are produced in the
WIMP annihilation, we count the number of photons. The number of simulated collisions in
each bin was fixed in order to provide suitable statistics in the number of produced photons.
For instance, for the high energy bins many collisions are required to get a significant number
of photons, whereas for low-intermediate energy, many photons are usually produced even for a
small number of collisions.
3. Results
According to the extensive PYTHIA simulations performed for electroweak gauge bosons,
leptons and quarks channels, three different fitting functions were required to fit the photon
spectra dNi
γ/dEγ: one for light quarks and leptons, one for gauge bosons (W and Z) and a final
one for t quark. Those expressions depended on both WIMP mass dependent and independent
parameters. Concerning the WIMP mass independent parameters, their values nevertheless do
depend on the considered annihilation channel whereas for WIMP mass dependent ones, their
evolutions with WIMP mass are parametrized by continuous and smooth curves as seen in [2].
Figure 1 shows two of these fits. The resulting expressions for the fitting functions are the
following:

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3.1. Light quarks and leptons
For q¯ q (except the t¯t studied separately in section 3.3), τ+τ−and µ+µ−channels, the most
general formula needed to reproduce the dNi
γ/dEγsimulations may be written as:
dNγ
dx
=
a1
x1.5exp
?
−b1xn1− b2xn2−
c1
xd1+
c2
xd2
?
+ q ln
?
p(1 − xl)
?x2− 2x + 2
x
(2)
where the logarithmic term takes into account the final state radiation through a Weizs¨ acker-
Williams expression. Strictly speaking, the p parameter in the Weizs¨ acker-Williams term in
the previous formula is (M/mparticle)2where mparticleis the mass of the charged particle that
emits radiation. However in our approach, it will be a free parameter to be fitted since the
radiation comes from many possible charged particles, which are produced along the decay and
hadronization processes. Therefore all the bremsstrahlung effects were encapsulated in a single
Weizs¨ acker-Williams-like term by using p and q parameters. l parameter is equal to one except
for the µ+µ−channel.
In fact, for the µ+µ−channel, the above expression (2) becomes simpler since the exponential
contribution is absent. The total gamma rays flux for this channel is thus well fitted by:
dNγ
dx
= q ln
?
p(1 − xl)
?x2− 2x + 2
x
(3)
For both leptons channels the covered WIMP mass range was from 25 to 5·104GeV. Concerning
the mass dependent parameters, they are n1and p in expression (2) for τ+τ−channel (the rest
of parameters are mass independent) and p, q and l in expression (3) for µ lepton. For q¯ q
channels, no general rule about mass (in)dependence of parameters can be settled, but specific
results channel by channel are presented in [2].
Let us finally mention that the gamma rays from e−e+pairs, the only contribution is
that coming from bremsstrahlung. Therefore, the previous expression (3) is also valid with
q = αQED/π, p = (M/me−)2and l ≡ 1, parameters choice that obviously corresponds to the
well-known Weizs¨ acker-Williams formula.
3.2. W and Z gauge bosons
For the W+W−and ZZ channels, the parametrization used to fit the Monte Carlo simulation
is:
?
x1.5dNγ
dx
= a1exp
−b1xn1−
c1
xd1
??ln[p(j − x)]
lnp
?q
(4)
This expression differs from expression (2) in the absence of the additive logarithmic contribution
that acquires nonetheless a multiplicative behaviour. The exponential contribution is also quite
simple with only one positive and one negative power laws. Moreover, a1, n1and q parameters
are WIMP mass independent (but they have different values depending on the studied gauge
boson channel). The rest of parameters, i.e., b1, c1, d1, p and j turned out to be both channel and
mass dependent. The covered WIMP mass range was from 100 to 104GeV for both channels.
Nonetheless, at masses higher than 1000 GeV, no significant change in the photon spectra for
both channels was observed.
3.3. t quark
Unlike the rest of the q¯ q channels, the parametrization given by (2) is not valid when photon
spectra are studied in the t¯t channel. For this one, the parametrization used to fit the simulations

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1e-05
0.0001
0.001
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1
x1.5 d Nγ /d x
Eγ /MWIMP
Total fit
Monte Carlo simulation
0.1
1
0.001 0.01 0.1
x1.5 d Nγ /d x
Eγ /MWIMP
1e-05
0.0001
0.001
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1
x1.5 d Nγ /d x
Eγ /MWIMP
Total fit
Monte Carlo simulation
0.01
0.1
1
0.001 0.01 0.1
x1.5 d Nγ /d x
Eγ /MWIMP
Figure 1. Photon spectra for MWIMP = 1 TeV in the b¯b (left) and W+W−(right) channels.
Dotted points are PYTHIA simulations and lines correspond to the proposed fitting functions.
is:
x1.5dNγ
dx
= a1exp
?
−b1xn1−
c1
xd1−
c2
xd2
??
ln[p(1 − xl)]
lnp
?q
(5)
In this formula, the exponential contribution is more complicated than the one in expression (4),
with one positive and two negative power laws. Again, the additive logarithmic contribution
is absent and acquires a multiplicative behaviour. Notice the exponent l in the logarithmic
argument, required to provide correct fits in this channel.
The covered WIMP mass range was from 200 to 105GeV. Nevertheless, at masses higher than
1000 GeV no significant change in the gamma-ray spectra was observed. The mass dependent
parameters for this channel are b1, n1, c2, p, q and l whereas a1, c1, d1 and d2 are mass
independent.
4. Conclusions
We have presented the model-independent fitting functions for the photon spectra coming
from WIMPs pair annihilation into Standard Model particle-antiparticle pairs for all the
phenomenologically relevant channels.Explicit calculations for all studied channels [2] are
available at the website http://teorica.fis.ucm.es/∼PaginaWeb/downloads.html .
5. Acknowledgments
This work was supported by MICINN (Spain) project numbers FIS 2008-01323 and FPA 2008-
00592, FPA2008-00319, CAM/UCM 910309, and Consolider-Ingenio MULTIDARK CSD2009-
00064. AdlCD was also supported by the National Research Foundation (South Africa). RL was
also supported by the EC contract UNILHC PITN-GA-2009-237920 and PROMETEO/2009/091
(Generalitat Valenciana).
References
[1] K. Sigurdson and M. Kamionkowski, PRL 92, 171302 (2004); J. A. R. Cembranos et al. , PRL 90, 241301
(2003); J. A. R. Cembranos and L. E. Strigari, PRD 77, 123519 (2008); J. A. R. Cembranos, PRL 102,
141301 (2009).
[2] J. A. R. Cembranos, A. de la Cruz-Dombriz, A. Dobado, R. A. Lineros, A. L. Maroto, arXiv(hep-ph):
http://arxiv.org/abs/1009.4936
[3] T. Sjostrand, S. Mrenna and P. Skands, JHEP05 (2006) 026 (LUTP 06-13, FERMILAB-PUB-06-052-CD-T).