Signals of a light dark force in the galactic center

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

Recent evidence for an excess of gamma rays in the GeV energy range about the Galactic Center have refocused attention on models of dark matter in the low mass regime (m ^sub [chi]^ m ^sub Z^ /2). Because this is an experimentally well-trod energy range, it can be a challenge to develop simple models that explain this excess, consistent with other experimental constraints. We reconsider models where the dark matter couples to dark photon, which has a weak kinetic mixing to the Standard Model photon, or scalars with a weak mixing with the Higgs boson. We focus on the light (1.5 GeV) dark mediator mass regime. Annihilations into the dark mediators can produce observable gamma rays through decays to [pi] ^sup 0^, through radiative processes when decaying to charged particles (e ^sup +^ e ^sup -^ , [mu] ^sup +^ [mu] ^sup -^ , . . .), and subsequent interactions of high energy e ^sup +^ e ^sup -^ with gas and light. However, these models have no signals of ...... production, which is kinematically forbidden. We find that in these models, the shape of resulting gamma-ray spectrum can provide a good fit to the excess at Galactic Center. We discuss further constraints from AMS-02 and the CMB, and find regions of compatibility.

Published for SISSA by Springer

Received: July 1, 2015 Accepted: July 18, 2015 Published: August 12, 2015

Jia Liu,a,b Neal Weinerb and Wei Xuec

aPRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics, Johannes Gutenberg University,55099 Mainz, Germany

bCenter for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, U.S.A.

cCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: Recent evidence for an excess of gamma rays in the GeV energy range about the Galactic Center have refocused attention on models of dark matter in the low mass regime (m[notdef] [lessorsimilar] mZ/2). Because this is an experimentally well-trod energy range, it can be a challenge to develop simple models that explain this excess, consistent with other experimental constraints. We reconsider models where the dark matter couples to dark photon, which has a weak kinetic mixing to the Standard Model photon, or scalars with a weak mixing with the Higgs boson. We focus on the light ([lessorsimilar] 1.5 GeV) dark mediator mass regime. Annihilations into the dark mediators can produce observable gamma rays through decays to 0, through radiative processes when decaying to charged particles (e+e, [notdef]+[notdef], . . .), and subsequent interactions of high energy e+e with gas and light.

However, these models have no signals of p production, which is kinematically forbidden. We nd that in these models, the shape of resulting gamma-ray spectrum can provide a good t to the excess at Galactic Center. We discuss further constraints from AMS-02 and the CMB, and nd regions of compatibility.

Keywords: Beyond Standard Model, Electromagnetic Processes and Properties

ArXiv ePrint: 1412.1485

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP08(2015)050

Web End =10.1007/JHEP08(2015)050

Signals of a light dark force in the galactic center

JHEP08(2015)050

Contents

1 Introduction 1

2 A new dark force 3

3 Fitting the data 43.1 The role of prompt photons 43.2 The role of ICS and Bremsstrahlung 8

4 Constraints 104.1 Constraints on [epsilon1] 104.2 Constraint from AMS-02 124.3 Constraint from CMB 14

5 Summary and conclusions 15

A Branching ratios 16

B Photon spectrum in the lab frame 19

C Photon spectrum from di erent nal states 21C.1 FSR 21C.2 [notdef] and [notdef] radiative decay 23C.3 n-body nal states 24C.4 Photons from individual channels 25

D Electron spectrum calculation 26

E CMB limits on thermal cross-section 28

1 Introduction

The search for dark matter (DM) remains one of the cornerstone components in the search for physics Beyond the Standard Model (BSM). While arguments of naturalness, both of the weak scale and the QCD -parameter point us to new physics, DM remains unique in being an experimental indication of new physics, and likely of a particle type.1 DM

appears within many BSM scenarios, with candidates such as the axion and the WIMP well explored in their potential signals. If DM is one of these candidates, these signals make the prospect of discovering the particle nature not only exciting, but possible.

1Neutrino physics also provides an experimental motivation for new physics, but with the most natural scale for the new physics near the GUT scale, at least with our current understanding.

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A great e ort has been undertaken to do this, especially for the broad WIMP and WIMP-like particles, with masses in the 11000 GeV range, and with interaction strengths

characterized by the weak scale. The standard set of searches - nuclear recoil, missing energy, cosmic ray - have shown a diverse set of anomalies [16] which have been interpreted as various DM candidates. For many of these anomalies, systematics have shown up [1, 2], others have stayed, but with strong alternative hypotheses [35], while others persist with neither clear resolution, nor viable alternatives [6].

Of late, a particular candidate signal has been growing in signicance - both statistically and systematically. Originally argued by Hooper and Goodenough [7], a component of the gamma ray signal from the vicinity of the Milky Ways center could be explained by DM. While the candidates have varied somewhat (from a 7 GeV WIMP annihilating

to

to a 35 GeV WIMP annihilating to b

b), the signal has been relatively persistent, peaking in E2dN/dE near 2 GeV [817].

Hooper et al. [15] argue for and explanation of a 35 GeV WIMP annihilating to bb, claiming that such a scenario is quite simple. Moving beyond this narrative to simplied models provides more information [1821]. However, UV-complete models that respect the low energy constraints from direct detection and colliders (e.g., [22]) are often more complicated and constrained than these simple descriptions would suggest. Moreover, other indirect detection constraints should be considered here [2328]. Recent studies of anti-proton constraints [23, 24] would show that these hadronic models are already under serious pressure by the data, although we note a conicting interpretation of the anti-proton data [29]. This has prompted an explosion of models with a variety of features [22, 3057]. Recently, [58] have argued that the uncertainties also admit heavier models.

There is an exceedingly simple framework to explain the excess that manifestly avoids a number of constraints [59], and helps us understand why the scale of these models may be low, and yet so far elusive. The idea builds on the idea of DM with cascade annihilations into a dark force carrier [6065]. In these scenarios DM is charged under a dark U(1) [6668], which kinetically mixes with the SM, or if DM couples to a dark scalar, which mixes with the Higgs. DM annihilates via [notdef][notdef] ! [notdef][notdef] followed by [notdef] ! SM, yielding

signicant cosmic ray signals are possible, without immediate constraints from colliders. Instead, the terrestrial constraints come from low energy, high luminosity experiments, such as APEX [69], MaMi [70], broad constraints from BaBar [71], CLEO [72], and future experiments [73, 74].

In this paper, we will revisit this scenario, focusing on the light mediator window(i.e., m [lessorsimilar] 1.5GeV) proposed in [59], which is less constrained than the case with heavier mediators, which has also been explored elsewhere [40, 7577]. In this window, gamma rays from the Galactic Center can come either from prompt photons (from 0s in the decay of the ) or radiatively (from nal state radiation or internal bremsstrahlung in e.g., ! e+e), or from subsequent interactions (such as ICS, Inverse Compton Scattering).

In section 2, we will restate the model. In section 3, we discuss the parameter space where the dark mediator can explain the Galactic Center excess. In section 4, we discuss connections to other experiments and in section 5, we conclude.

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2 A new dark force

The class of models we consider in this article consists of a DM particle [notdef] and a dark force with a mass MeV < m < GeV, which is lighter than the DM mass, m[notdef]. The DM has a dominant annihilation process, [notdef]+[notdef] ! +, followed by cascade decays of the dark force

to the Standard Model particles. We consider the dark force to be either a gauge eld [notdef] or

a scalar eld 0. Generically, we will use to denote mediator without regard to its spin.

With a U(1)D gauge eld as a dark force, the models are quite simple. With a dark photon eld strength strength [notdef] , we have kinetic mixing with Standard Model hyper-charge Y[notdef] ,

~

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[epsilon1]2[notdef] Y [notdef] . (2.1)

At low energy, the mixing occurs with the EM eld strength, and the cascade decay is triggered by the coupling of dark force and the Standard Model currents

Lint[similarequal] ~[epsilon1] cos w[notdef]J[notdef]em = [epsilon1][notdef]J[notdef]em (2.2)

where [epsilon1] ~

[epsilon1] cos w to simplify the notation.

For a detectable signal, we must have a present day annihilation rate of [angbracketleft]v[angbracketright]

1026cm3s1. For a vector dark force, we take the DM to be a Dirac fermion.2 The cross section for DM-DM annihilation is s-wave,

v[notdef][notdef]! [similarequal]

g4X 16m2[notdef]

(1 x)3/21 x2

2 , (2.3)

where gX is the gauge coupling of the dark force, and x = m2/m2[notdef].

In the case of a scalar dark force, we can take a real scalar to be the dark force () and a complex scalar as DM ([notdef]). The potential for the scalar dark force is

Vint = gX1[notdef] [notdef] + gX22 2[notdef] [notdef] + 1 [notdef]H[notdef]2 + 22 [notdef]H[notdef]2 (2.4)

+ m2

2 2 +

2 4 [notdef]2 [notdef]H[notdef]2 +

2 [notdef]H[notdef]4

We neglect the Higgs portal term [notdef][notdef] [notdef]H[notdef]2, which can a ect the relic abundance and

direct detection signals, but could be absent if the theory arises from a SUSY theory at a higher scale, or if the sectors are sequestered, such as via an extra dimension. We assume that DM carries some quantum number (e.g. a Z2 charge, or hidden global charge). The singlet will acquire a mixing term via the trilinear when the Higgs gets a vev.3 We assume the mixing is small, so as to avoid a sizable direct detection cross section.

2Alternatively, we can consider a pseudo-Dirac fermion, in which case the thermal cross section is naturally a factor of two larger [angbracketleft]v[angbracketright] 6 [notdef] 1026cm3s1. See the discussion in [78].

3The singlet could also acquire a vev spontaneously, and mix without a trilinear term. We will not pursue this possibility here, because of the possibility of domain walls and the subsequent cosmological issues. For our purposes the phenomenology is the same.

3

The DM annihilation to is s-wave with the following form,

2g2X12m2[notdef] m2 [parenrightBigg]2

E2

dr2DM(r, ) (3.2)

Jf is calculated by taking a 5 cone from GC, to match the data from [15], which is taken as 268.7 for the generalized NFW prole ( = 1.26). For each parameter point

{mDM, m[notdef], the BR for each channel and photon spectrum

dNidE are xed. We scan over

BF to minimize the [notdef]2 for each point. The tting resulting from a consideration only prompt photons for annihilations into dark photons are shown in the left panel of gure 1. The gray scale indicates the BF from the [notdef]2 tting. To count the uncertainty in the error estimation, we show the contour plot with double error-bar of the [15]. For the moment,

4

v[notdef][notdef]! [similarequal]

qm2[notdef] m264m3[notdef] gX2 +

R[circledot]2[circledot] 8m2DM

. (2.5)

While we have considered the scalar DM case, one can also consider a fermionic scenario. The principle obstacles to this is that for a fermion the annihilation of [notdef][notdef] to is p-wave suppressed. This can be evaded if the annihilation is into a complex scalar. In this case, either the pseudoscalar would be massless (and thus would be an additional relativistic degree of freedom), or could mix with the Higgs via a CP-violating mixing term eiQ [notdef]H[notdef]2 + h.c.. Our points below do not depend crucially on these details, however.

3 Fitting the data

The branching ratios and photon spectra are complicated, but straightforward. We refer the reader to the appendices for details. In the appendix, we calculate the branching ratio of the dark mediator decay in section A. In section B, we show how to calculate photon spectrum in lab frame, with the assumption that the spectra from each daughter particle are known. In section C, we briey interpret how we calculate the photon spectra from each channels. In section D, we introduce how we calculate the electron spectra in a same way as for photon spectra.

3.1 The role of prompt photons

With the BR information and photon spectrum from each decay channel, we can calculate the prompt photon ux as below.

E2 d Prompt

dE = Jf [notdef] [angbracketleft]v[angbracketright] [notdef] BF [notdef]

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Xi

BRi dNi

dE , (3.1)

where R[circledot] is 8.5 kpc, the distance to the GC; the [circledot] is the local DM density, 0.4GeVcm3;

and the [angbracketleft]v[angbracketright] is the annihilation cross-section taken as 3 [notdef] 1026cm3s1. BF stands for the

boost factor of the cross section, and Jf is the standard dimensionless factor for the l.o.s. integration with the following expression,

Jf( ) = 1

R[circledot]2[circledot] [integraldisplay]

los

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Figure 1. The 2 and 3 tting contours for the dark photon by prompt only (left panel) and including ICS and Bremsstrahlung (right panel). The red triangle is the best t point for the model. The gray scale indicates the BF from the [notdef]2 tting. We use twice the error-bar of [15].

we focus only on prompt photons from the decays of the , and do not include additional contributions from ICS and bremmstrahlung.

The best prompt photon t for dark photon is [notdef]5.7GeV, 0.59GeV[notdef] for DM mass and

mediator mass respectively, shown as red triangle in the plot. We plot 2 and 3 contours for the parameter space. The color bar shows the BF for each point, after minimizing the [notdef]2. We can see the best regions are around 5.5 9GeV for DM mass and 0.2 0.8GeV

for mediator mass. In these regions, the BR of e+e, [notdef]+[notdef] and + channels dominate in the decay. We plot the prompt photon spectra for each channels with di erent mediator mass in the left panel of gure 2 and gure 3. Interestingly, the best t for prompt photon spectra are dominated by e+e, 0 and . The latter two have small BR but high photon yield, because the number of hard photons in e+e goes as /, while the 0 and channels have O(1) number of photons. For mediator mass smaller than 0.4GeV, the photon spectrum is dominated by radiative processes arising from e+e. However, for heavier mediator around 1GeV, the contribution comes from meson channels like KK, 0 ,

+0 and +00. It shows that including meson channels is quite important in the light mediator analysis. The BF in these regions are around O(1), which means the tting is quite reasonable.

For the dark scalar, we show the tting by prompt photon in the gure 4. The best t point for dark scalar is [notdef]16.4GeV, 0.25GeV[notdef] for DM mass and mediator mass re

spectively. The best regions are separated as three regions. The rst region is around 5.5 7.5GeV for DM mass and 0.0 0.2GeV for mediator mass, where e+e channel

dominates. The next region is around 12.5 22GeV for DM mass and 2m[notdef] for mediator

mass, where [notdef]+[notdef] channel dominates due to mediator mass opens for [notdef]+[notdef] channel but not for pions. The best t is also in this region, and we plot the prompt photon spectra for each channels in left panel of gure 5. One can see the best t is dominated by mediators where the photons arise from radiative processes involving [notdef]+[notdef]. For these points, however, the BF is quite large, about 10, due to the small number of photons from these

5

Dark Photon (Prompt Only )

Dark Photon (with ICS+Brem )

5.10

5.10

m =7.5GeV, m =0.28GeV, BF=1.95

m =7.5GeV, m =0.28GeV, BF=0.71

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ICS Brem Prompt Total

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m =9.0GeV, m =0.52GeV, BF=0.83

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E

- 0.1 0.5 1 5 10 50

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E (GeV)

Figure 2. Left Panel: the prompt photon spectra from FSR and IB for the dark photon scenario with di erent DM mass and mediator mass. The ICS and regular Bremsstrahlung are assumed to be negligible. The dashed green is the total prompt photon spectrum, while the other color lines correspond to decay channels for dark photon in the gure 10. Right Panel: the photon spectra including the ICS and regular Bremsstrahlung. (see text).

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Dark Photon Global Best Fit (Prompt Only )

Dark Photon (with ICS+Brem )

5.10

5.10

mDM=6.0GeV, m

=0.6GeV, BF=1.06

mDM=6.0GeV, m

=0.6GeV, BF=0.68

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=0.8GeV, BF=0.36

mDM=6.5GeV, m

=0.8GeV, BF=0.33

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=1.4GeV, BF=0.16

mDM=7.5GeV, m

=1.4GeV, BF=0.15

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(GeV)

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(GeV)

Figure 3. Same as gure 2, but for m values that produce 0 contributions, when prompt photon signals are dominant.

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Figure 4. The 2 and 3 contour plot for the dark scalar with prompt photon only. The red triangle is the best t point for the model. We use twice the error-bar of the [15].

radiative processes. For mediator mass between 2m 1GeV, there is no good t because

the 00 provides too many hard photons. The third region is 5.3 8.5GeV for DM mass

and 1.1 1.5GeV for mediator mass, where Kaon channels dominates over the pion chan

nels. Although the Kaon decays to 0, yielding copious photons, since it is cascade decay the photon spectra are generally softer than 00 channel. We plot the prompt photon spectra for the 1.2GeV scalar mediator in the gure 5 as an example. It is interesting that although the BR of 00 channel and are smaller than Kaon channels, but they still dominate in the photon spectrum. The BF is quite small here, around 0.1, due to the high photon yields from those meson channels.

For these points in parameter space, we can ask about alternative indirect constraints. Gamma rays from dwarf galaxies are a natural constraint [2527]. In scenarios where the ICS (Inverse Compton Scattering) component is negligible , we expect the dwarf constraints are similar to those for comparable models (such as annihilation). In scenarios where the ICS is signicant, it will be weaker, with no starlight or conning magnetic elds to trap the electrons near the dwarfs to produce a comparable signal. Searches for p are clearly not relevant, as they are kinematically forbidden, and are an important distinguishing feature of these models. CMB constraints from WMAP is not sensitive to our scenario currently, but the updated Planck constraints may put new limits on the dark photon model [7981]. We will return to the AMS constraints on positrons shortly.

3.2 The role of ICS and Bremsstrahlung

Models that produce copious e+e pairs can produce secondary photons from interactions with the surrounding medium (gas, starlight, cosmic rays). These components can contribute to the total signal [8284]. In particular, we nd that for very light dark mediators m [lessorsimilar] 0.5GeV, these can be the dominant component in the central region. For heavier mediators, it can be an O(1) change to the spectral shape at low energies, while for the heaviest mediators m [greaterorsimilar] 1GeV, which have 0s, it is a small e ect.

Bremsstrahlung is perhaps the hardest to model, because it has a prole that is tightly correlated to the gas, and thus to the disk. However, not all of this will be absorbed into

8

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m =7.25GeV, m =1.2GeV, BF=0.066

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Figure 5. Left Panel: the prompt photon spectra from FSR and IB for the dark scalar scenario with di erent DM mass and mediator mass. The ICS and regular Bremsstrahlung are assumed to be negligible. The dashed green is the total prompt photon spectrum, while the other color lines correspond to decay channels for dark scalar in the gure 11. Right Panel: the photon spectra including the ICS and regular Bremsstrahlung. (see text).

9

the disk model. To account for this, we calculate the contributions from bremsstrahlung by masking out the disk region 1 < b < 1 . These plots should be understood to be the

contributions to the signal in the inner galaxy region, where 1 < [notdef]b[notdef] < 20 and [notdef]l[notdef] < 20 .

We see in right panel of gures 2 and 3 that for light mediators, where the dominant contribution is IB (Internal Bremsstrahlung) and FSR (Final State Radiation), that the ICS and Bremsstrahlung signals contribute at a sizable level, while for heavier mediators, the e ect can be merely to add additional soft gamma, or to have a marginal e ect. Interestingly, once taking into account the e ects of these secondary photons, no point in parameter space requires a boost factor much larger than 1. In the gure 5, the dark scalar also has similar story.

Furthermore, this raises the prospect, however, if at some point we have an accurate map of this signal, to look for deviations in the spectral shape as we move from the inner region to the outer, where these secondary gammas are less prevalent. Indeed, this may lead to a more rapid fallo in the size of the signal that would have been expected from the DM prole alone, simply because these secondary photons become less signicant in the outer region.

4 Constraints

Constraints on this scenario can be grouped into constraints on the signals of the DM, itself, or on the dark mediator.

4.1 Constraints on [epsilon1]

The constraints on the mediator are strongest when it is a dark photon, and come mainly related to its mixing parameter with Standard Model, [epsilon1]. These limits are derived from searches in beam-dump experiments, xed target experiments, and e-e collisions, among others. For a given DM mass m[notdef] and DM coupling to the dark photon, gX, a constraint can also be derived from DM direct detection searches. These constraints are summarized in gure 6.

Figure 6 shows the parameter space for dark photon. The beam dump experiments, such as E141 [86], E137 [87], E774 [88], etc used the displaced decay vertex covering the lower left corner of the parameter space. The xed target experiments, the anomalous magnetic moment measurement and e+e and hadronic collisions give the constraints on the upper part of the space. Much of the high mass range has been explored by the BaBar experiment [71]. There is much parameter space left for the dark photon search in the dark photon mass from 10 MeV to a few GeV, although this is now being probed by MaMi [70], APEX [69], HPS [73], and DarkLight [89, 90], among others.

We display the constraints from direct detection on this plot as well. DM-nucleus scattering arises via dark photon exchange. The DM-proton scattering cross section is

p [similarequal]

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[epsilon1]2 g2X e2

[notdef]2[notdef]p

Q2 + m2A[prime]

2 [similarequal] 1 [notdef] 1043cm2 [parenleftBig]

gX 0.1

[epsilon1]1 [notdef] 108

2 0.1GeV mA[prime]

4(4.1)

2

[parenleftbigg]

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Figure 6. Parameter space for Dark Photon. Diagonal lines: contours of spin-independent direct detection constraints for di erent DM mass from LUX and superCDMS. Backgrounds shows current dark photon constraints from other dark photon search [85]. These limits do not apply to the scalar mediator, or pseudo-Dirac DM case.

where [notdef][notdef]p is the DM and proton reduced mass; and Q is the monmentum transfer Q = p2mN Er, which is related to the nuclei mass mN and the recoil energy Er.

In the second equality of (4.1), we assume the dark photon mass is larger than the t-channel momentum transfer of the scattering process. The dark photon mass should be larger than O(10)MeV for this assumption to be valid. For smaller dark photon masses,

this breaks down and the t-channel momentum transfer becomes important. To clarify this e ect and the limits of validity of our curves, we have inserted a momentum transfer Q into the propagator, in which Q = 35 MeV for LUX, Q = 5 MeV for CDMSlite and Q = 17 MeV for superCDMS. This changes the behavior of the limits in gure 6, and we have changed color into lighter ones in this regions where it occurs, in which case these limits are only approximate.

With the DM mass given, we can x gX through the relic density constraint, (e.g. for m[notdef] = 10GeV, gX = 0.06). In gure 6, superCDMS [91] and LUX [92] are considered, which are currently the best constraints of spin-independent cross section in the DM mass range of 5 GeV - 30 GeV.

Importantly, is that these limits are only present if the dark matter is a Dirac fermion. If the DM is split into a pseudo-Dirac state after U(1) breaking, then the scattering is

11

inelastic and can be kinematically suppressed [93], leaving no appreciable constraint on these models.

Finally, these constraints are on the dark photon model. For the dark scalar, with its weaker interaction with ordinary matter, both the production and direct detection constraints are weaker.

4.2 Constraint from AMS-02

AMS-02 precisely measured the smooth electron, positron spectrum and the positron ratio. We can turn these smooth data into a constraint on light DM [9496]. If the light DM annihilates to electrons and positrons and this cross section is large enough, after the transportation of the electrons and positrons, a bump feature would expect to be seen in the AMS-02 positron ratio data. Since we have not seen this bump yet, the current measurement is able to put stringent constraints on light DM models.

We revisit the study of [94] on the limit of DM annihilation from AMS-02, and consider more channels and the systematic uncertainties from solar modulation and magnetic elds. Our limits are not as stringiest as those in [94], and so we list the major di erences here:

we use 2 parameters (m[notdef] and [angbracketleft]v[angbracketright]) to compute the relevant regions for [notdef]2, while [94]

use 1 parameter to do so. Furthermore, we plot a 3 sigma contour, and [notdef]2 = 11.83, while [94] plots 90%CL.

we consider the uncertainties of solar modulation, while [94] consider specic values

of solar modulation parameters

we choose one plain di usion model, but test the uncertainties from the parameters

in the cosmic ray di usion. It turns out that the variation of the magnetic eld or the e ect of the energy loss inuence the AMS-02 constraints most.

we set [circledot] = 0.4GeV/cm3 to be consistent with our Galactic Center analysis, while

in [94], the minimum density is [circledot] = 0.25GeV/cm3First of all, instead of simulating the astrophysical background, we apply polynomial functions to t the AMS-02 electron spectrum and positron ratio separately from 1 GeV. After obtaining the two functions, we derive the positron spectrum, and recheck the t to AMS-02 positron data. Secondly, we compute the positron or electron ux from DM annihilation propagating in our galaxy, by using a public cosmic ray code DRAGON [97].

Before propagation, the positron spectrum is delta function for the process of [notdef] + [notdef] !

e+ + e, dNe

dx (2e) = (1 x) by neglecting fragmentation. For this process with one step

cascade decay, [notdef]+[notdef] ! + and ! e++e, the spectrum is a box-like function dNedx (4e) =

2 (1 x). After propagation, the di usion and energy loss make the positron ux softer.

We compare the cross section limits by choosing di erent magnetic elds and considering the variation of the solar modulation or not in gure 7. The magnetic eld is modeled as two main components, regular one and the turbulent one [98, 99], but little is known for the size of magnetic eld. The total magnetic eld we choose at Sun is B[circledot] = 15[notdef]G. In

the left panel of gure 7, the solid line is B[circledot] = 15[notdef]G, while the dotted lines corresponds

12

JHEP08(2015)050

Figure 7. Exclusion curves for di erent DM models and for di erent assumptions of cosmic ray propagation. In the left panel, the process of [notdef] + [notdef] ! 2e and [notdef] + [notdef] ! 4e are considered. The

solid lines take into account of the uncertainties from solar modulation, and choose large magnetic elds B[circledot] = 15[notdef]G. The dashed lines choose the solar modulation = 0.5 GeV, and the dotted lines

consider a smaller magnetic elds B[circledot] = 7.5[notdef]G. In the right panel, the exclusion limit of various

dark force mass assuming a dark photon model are included.

to B[circledot] = 7.5[notdef]G. In addition, the solid line considers the variation of the solar modulation,

while the dashed line xes the solar modulation potential by = 0.5 GeV. The limits di er by a factor of 2 for DM mass smaller than 10GeV. In the right panel of gure 7, we plot the exclusion limit for di erent mass of dark force mediator.

The implication for result is that for 10 GeV DM, if the branching ratio of [notdef] + [notdef] !

e+ +e or [notdef]+[notdef] ! 2e+ +2e is larger than 5% and the cross section is the thermal cross

section 3[notdef]1026cm3/s, the model has tension with AMS-02. In other words, if the branch

ing ratio is 100% to 2e and 4e, the cross section should be smaller than 12[notdef]1027cm3/s.

For the dark photon models, the branching ratio to 4e is generally about 30%, except in the resonance region. In the resonance region (e.g. m 0.8GeV), 4e channel is suppressed

and AMS constraint could be satised. In the non-resonance region, one needs either a small BF by large 0 production in heavy dark photon region or a large ICS and Bremsstrahlung contribution in the light dark photon region, to alleviate the AMS constraint.

We see that most of the light dark photon mediator models would appear to be constrained. For instance, for the light mediators, we require a cross section 2[notdef]1026cm3s1,

while the limits are 2 3 [notdef] 1027cm3s1. However, for heavier mediators, this is less of a

problem. For a 1.4 GeV mediator, for instance, we need a cross section 4.5[notdef]1027cm3s1,

while the limit is 5 [notdef] 1027cm3s1, comparable to the cross section we need. For a

0.8 GeV mediator, the limits are around 1026cm3s1, again comparable to the cross section we need. For dark scalar models, the constraint is generally much weaker, because e+e channel has much smaller BR than dark photon by Yukawa coupling. We note that since the Fermi signal arises from the central galaxy, while AMS is from more local annihilation, a somewhat steeper prole than what we take here could lead to alleviations in the remaining tensions.

13

JHEP08(2015)050

0.8

0.6

0.4

0.2

0.0 200 400 600 800 1000 1200 1400 1600

m

Figure 8. fe for the dark photon and dark scalar.

4.3 Constraint from CMB

DM annihilation can inject energy into the CMB, which distort its temperature and polarization power spectra [100, 101]. The anisotropy of CMB can constrain the DM annihilation [79, 80, 102]. In 2015 Planck data [103], it shows very strong constraint on low mass DM annihilation. To calculate the constraint the annihilation to dark mediators, we start with the e ciency factor fe , which describes the fraction of the energy injected into the gaseous background. Following the data in ref. [81, 104], we assume fe are 0.6, 0.2, 0.16 and 0.62, for dark mediator decay channels e+e, [notdef]+[notdef], + and respectively.

fe has some mild dependence on the dark matter mass m[notdef], but since we consider a small range of m[notdef] around 10 GeV, we neglect it. For other particles, we can build up their fe through decay branching ratio and decay products. For 0, we assume its fe is the same with . After some calculation, fe for K[notdef], K0L, K0S and are 0.18, 0.37, 0.42 and 0.54 respectively. We calculate fe for the dark photon and dark scalar, according to their decay branching ratios, in gure 8.

Planck can constrain the annihilation cross-section [angbracketleft]v[angbracketright]rb at recombination times the

e ciency parameter fe [103]. We assume the boost factor for annihilation at recombination is the same as today. To derive the constraints on the light dark force scenario, we apply the annihilation cross-section from the [notdef]2 t, which is the thermal cross-section times the BF from right panel of gure 1 and gure 4.

We plot the constraints on the light dark scenario in gure 9 in m[notdef] m plane. Inside

the black contour, it is the 3 best t region for dark photon and dark scalar. We can see that most of the best t region for GCE are excluded, as indicated by light red shaded region for dark photon and light blue shaded region for dark scalar, except when dark mediator is heavier than 1 GeV. Those region survive because their needed cross-section are quite small due to direct photon contribution from meson decay. This is true for both dark photon and dark scalar. Moreover, if we weaken our signal by a factor of 50%, signicant parameter

14

dark photon dark scalar

f eff

JHEP08(2015)050

[MeV]

JHEP08(2015)050

Figure 9. The CMB constraints on the DM annihilation in m~ m plane for dark photon (left

panel) and dark scalar (right panel). Inside the solid black contours are the 3 best t region for dark photon and dark scalar in gure 1 and 4. The light (dark) color shaded regions are excluded by CMB, assuming best t cross-section for GCE times [notdef]100% ([notdef]50%).

space opens for m < 1 GeV. It means if we allow a partial t to GCE, more parameter space could survive. In summary, the GCE excess from dark mediator interpretation can still survive signicant parameter space, e.g. m > 1 GeV or if we allow a partial interpretation for GCE. We also plot the contours of excluded annihilation cross-section at freeze-out from Plank as a function of m[notdef] and m in gure 14 in appendix E. We assume

BF at freeze-out and recombination are the same. It shows DM with thermal cross-section 3 [notdef] 1026cm3/s in the dark mediator models should be larger than 20 GeV.

5 Summary and conclusions

While the nature of dark matter has remained elusive, tremendous progress has been made in constraining its nature. The recent evidence of a GeV excess from the Galactic Center, arising from analysis of data from the FGST Galactic Center [15, 17] invites interpretations as being of a DM origin.

We have revisited the proposal of DM annihilating into a light mediator as an explanation for these signals. We have carefully studied the decay branching ratios of the light mediator and the various meson channels which produces the gamma-rays. We have scanned the best t region for the dark force scenario, both with dark photons and dark scalars. The result shows that for mediator masses [lessorsimilar] 1.5GeV and DM mass [lessorsimilar] 10GeV, lepton nal states or combination with meson nal state could give a very good t for the

GeV excess, which is in agreement with [59].

We note that what we have discussed here should be considered simplied models for this scenario. Annihilations [notdef][notdef] ! [notdef]h, where h is the Higgs eld for the dark photon can

occur at a parametrically similar rate for the Dirac DM case. There may be multiple dark

15

photons (i.e., as in [62]), leading to more complicated cascade spectra. And, if there are additional scalars in the dark sector, there could be an intermediate step in the cascade as well. Thus, the spectral shape may vary as these complications are present, which may lead to changes in interpretation. Much of these can be considered as combinations of the dark photon and dark scalar spectra presented.

While the prompt photon spectrum is typically dominant, the contributions from Bremsstrahlung and ICS can change the picture. For light mediators, it can be an O(1) component of the total signal in the GC, while for heavier mediators it becomes less important. As the lightest mediator models are more tightly constrained by AMS, it is unlikely that these secondaries are the dominant sources of the gamma rays we observe if DM is in the mass range we consider. However, it still may be important and lead to spectral changes going from the GC to the inner Galaxy regions.

Since we lack understanding about the detailed nature of the di usion of cosmic rays near the GC, there are important systematic uncertainties in calculating the ICS contribution to the gamma ray signal. Still, it is clear that the ICS from DM-induced electrons and positrons gives contributions to the gamma-ray spectrum, especially at slightly lower energy than the prompt photons. Interestingly, in some di usion models, the morphology for ICS is similar to the one of the GeV excess, while in other models it is di erent. Finally, while these uncertainties are present, it is essential to understand its e ects on GeV gamma-ray excess, both in the change of spectrum and of morphology, especially to do detailed comparisons of models and data.

Ultimately, while the nature of the gamma ray excess remains unclear, we do see here that annihilations into dark sector cascades provide a good explanation of the data. Upcoming searches, both terrestrial and astrophysical, may shed light on whether such a weakly coupled light sector exists in nature.

Note added. As this work was being completed, [105] appeared, which considers the ICS signals from somewhat heavier DM candidates. Our results are in good agreement on the consequences of ICS for these signals.

Acknowledgments

We thanks Tracy Slatyer, Jesse Thaler, Alfredo Urbano, Daniele Gaggero, Satyanarayan Mukhopadhyay for useful discussion. NW is supported by the NSF under grants PHY-0947827 and PHY-1316753. JL is supported by the PRISMA Cluster of Excellence and the DFG Grant KO4820/1-1.

A Branching ratios

In both dark photon and dark scalar scenarios, will decay to leptons and mesons. In order to obtain the photon spectrum from the decays, we will rst derive the branching ratios of their decaying channels. For the dark photon, a data driven method is employed, and for dark scalar, a theoretical analysis is provided.

16

JHEP08(2015)050

1

ee

K+K-

+-

K0K0

+-0

0

2

0

+

+-00

0

+-+-

0.100

0.010

BR

0.001

10-4

10-5

200 400 600 800 1000 1200 1400 1600

Dark Photon Mass (MeV)

JHEP08(2015)050

Figure 10. The decay branching ratios for dark photon.

In the dark photon scenario, DM annihilation to dark photons is followed by decay of the on-shell dark photons to SM particles. Since the kinetic mixing between dark photon and photon, the dark photon decay can be analyzed using the measurements of e+e !

hadrons at di erent Center of Mass (C.M.) energies. Suppose the dark photon mass is the same as the C.M. energy of the e+e collision, the ratio of the cross-section of the di erent nal states reveals the branching ratio of the dark photon decay products. When the mass of dark photon is above 2GeV, the perturbative QCD is valid from the observation that

the energy dependence of R(s) = (

+[notdef]) matches with the QCD prediction [106]; hence the underlying processes are ! qq and ! l

l. At the C.M. energy below 2GeV,

there are rich structure of resonance, such as , !, and , and di erent exclusive channels are measured separately. We obtain the branching ratio of the channels from the exclusive cross-sections at di erent C.M. energies [107, 108]. We have included all the two body nal states shown in gure 10. For multiple particle nal states, we only include three pion and four pion nal states and neglect others like K+K0, as well as ve pion and six pion states, because these have subdominant contribution to the photon yield.4 As a caveat, in the !0 channel, we only include the nal states when ! decays into 0 . The ! dominantly decays into three pions, but it is already considered in the three and four pion nal states. However, in the K+K and K0K0 channel, their cascade decays includes four pion nal states, which are not included in the four pion channel in gure 10. Thus we calculate the spectrum of KK and 4 states separately.

In the dark scalar mediator scenario, the DM annihilates into a pair of dark scalars, which, through their mixing, subsequently decay into SM fermions. The dark scalars coupling to SM fermions is proportional to the fermion mass, and suppressed by the mixing term [epsilon1], while the heavy fermions (c,b,t) are decoupled and will inuence the low energy hadronic process by coupling to gluons. Hence we are able to write down the e ective

4Only for ++ channel, there is measurement at 3GeV, while for other channels the highest measurement is around 2.4GeV.

17

e+e!hadrons)

(e+e![notdef]

Lagrangian in the following form

Le = [epsilon1] v

0

@

Xq=u,d,s

mq qq + sNH12 Ga[notdef] G[notdef] a1

A

, (A.1)

where v is the Higgs vev, and NH = 3 is the number of the heavy quarks. Introducing the trace of energy momentum tensor [notdef][notdef] can relate the quark level interaction to the hadronic process. First, [notdef][notdef] illustrates the anomaly of the conformal symmetry, which contains the terms proportional to QCD beta function and the terms proportional to the mass of the light quarks,

[notdef][notdef] =

2gs Ga[notdef] G[notdef] a +

Xq=u,d,s

mq qq . (A.2)

On the other hand, [notdef][notdef] is related to the hadronic process, and at the leading order,

+[notdef] [notdef][notdef][notdef]0[angbracketrightbig]= s + 2m2 + O(p4). (A.3)

From the rst order of the chiral Lagrangian, we are able to derive the other hadronic matrix element,

< +[notdef]Xq=u,d,s

mq qq[notdef]0 >[similarequal] m2 . (A.4)

After replacing the Ga[notdef] G[notdef] a term by [notdef][notdef] and

Pq mq qq in the e ective Lagrangian eq. (A.1), the decay width of the dark scalar is computed by combining the two matrix elements in eq. (A.3), (A.4),

( ! +) =

[epsilon1]2m3

JHEP08(2015)050

324v2 1

4m2 m2

!1/2

1 + 11m2 2m2

!2. (A.5)

Due to the isospin symmetry, the ratio of charged states (e.g. +) to neutral states (e.g. 00) is just 2 : 1. The decay width to K K and are similar with pion by adding a statistical factor of 4/3 and 1/3 respectively [109] and substituting the pion mass by Kaon mass and Eta mass.5 We also list the decay width to leptons here.

( ! [lscript]+[lscript]) =

"2m2[lscript]

8v2 m

3/2

1 4m2[lscript] m2

(A.6)

The decay width to two photons are the same as the Standard Model Higgs, except the mixing factor. We explicitly list the width formula for photons in the following,

( ! ) =

"2 2EM 2563

4m2W

m2

[parenrightbigg][vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]2, (A.7)

5For the light mass Higgs, there are debates about the ratio BR(+)/BR() (see [110] and references therein). Our result are insensitive to such debate, because the photon spectrum from muon pair nal states is similar to charged pion pair nal states.

m3

v2

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

Xi QiCQiF1/2

4m2i

m2

[parenrightbigg]+ F1

18

1

0.1

ee

+-

00

K+K-

K0K0

0.01

BR

0.001

10-4

10 0 200 400 600 800 1000 1200 1400

-5

dark scalar mass [LParen1]MeV[RParen1]

Figure 11. The decay branching ratios for dark scalar mediator.

where the v is the Higgs vev. i runs over all the fermions in the SM. QC is the color factor and Q is the charge of the fermion. F1 and F1/2 are the well known functions,

F1(x) = 2 + 3x + 3x (2 x) f(x) (A.8)

F1/2(x) = 2x [1 + (1 x) f(x)] .

The function f(x) is the following,

f(x) =

8

<

:

sin1

JHEP08(2015)050

p1/x

2, x 1

14 [bracketleftBig]

(A.9)

ln

1+p1x 1p1x [parenrightBig]

i

i2, x < 1

Since we are dealing with very light scalar mass, the quark mass will have signicant inuence on the width. Here we take the current quark mass. We plot the decay branching ratios for dark scalar mediator in the gure 11.

B Photon spectrum in the lab frame

We present how we calculate the photon spectrum in the cascade decays. We generally follow the notation and procedure in the [64]. The di erence is we take into account the nite mass of the mother particle and daughter particles, however, in [64] the daughter particles are treated as massless to simplify the calculation. In our case, since we want to scan for dark photon and dark scalar mass, there are regions where their mass are close to the threshold of the daughter particle, thus taking account the nite mass into boost calculation makes the photon spectrum more accurate. We take into account the dark photon and scalar mass, and also the various meson mass in their cascade decays.

To show the boost calculation quantitatively, we assume a process where mother particle A decays to daughter particles Bi, where the i is the ith daughter particle.

A !

Xi

Bi (B.1)

19

The number density distribution of photons from particle Bi in the Bi center frame is denoted as dNBi/dxBi. The distribution from FSR and radiative decay are described in detail in section C.1 and C.2. The xBi is dimensionless quantity dened as

xBi

2Ei

mBi , (B.2)

where Ei is the energy of photon from particle Bi in the Bi center frame and mBi is the mass of particle Bi. If the Bi decays directly to photons, for example 0, then the total number of hard photons NBi in the Bi center frame is about O(1). However, if the photons from Bi are from initial and nal state radiation, then NBi is about O( EM). This means once Bi decays directly to photons, then the spectrum dNBi/dxBi are usually determined by the direct photons. The mesons 0, ! and can directly decay to photons, which are quite important.

The Kaon mesons also makes O(1) number of photons, because their decay usually contains 0. There are various decay channels for those mesons, we only calculate the leading photon source in the cascade decay. To be concrete, take the decay to 0+ as an example, we only account the photons from 0. The photons from cascade decay in [notdef] into muon and nally electron are subdominant. The only exception is when dark photon decays into four pion, two charged and two neutral pions, we account both photon from neutral and charge pions. A detailed description of leading contribution for each channel is in section C.4.

With the dNBi/dxBi in hand, we want to know the photon distribution in the center frame of mother particle A, the dNA/dxA, where the xA is 2E

mA and E is the energy of photon in the A center frame. Suppose the momentum of particle Bi has an isotropic spherical distribution in the A center frame and Bi has energy EBi in A center frame, then the connection between the two distribution is,

dNA/dxA = [integraldisplay]

Min[1,xA[notdef] mAmBi [notdef]

JHEP08(2015)050

"Bi 1

r1" 2Bi

] dxBi dNBi

dxBi

1

2xBi

mA mBi

"Bi

xA[notdef] mAmBi [notdef]

"Bi 1+

r1" 2Bi

q1 "2Bi, (B.3)

EBi . Sometimes, the number of daughter particles is larger than 2, so EBi is not xed by two body nal state. In the multi-particles nal state like three pion and four pion, the pions do not have a denite energy as in the two body decay. We assume those pions have isotropic spherical distribution in momentum direction, and their energy distribution satisfy the natural phase space distribution. The natural phase space distribution means the momentum satisfy the phase space constraints, assuming the matrix element is a constant. The calculation of momentum distribution is in section C.3. With the distribution in hand, we can average dNA/dxA over EBi with proper possibility function.

We use this method to trace back the number distribution of photons level by level, until to the lab frame and take fully account the mass of all the daughter and mother particles. We only omit the daughter mass in the last step, when boosting the photon back into lab frame, DM + DM ! . The is dark photon or dark scalar. The last boost can be seen

as a hypothetical particle with mass of twice DM mass and decay into two . We assume the mass is negligible to this this hypothetical particle and set it to zero. In this case,

20

where "Bi =

mBi

# photons per annihilation (m =1GeV)

ee

K K

K K

2

+

Prompt

xdN/dx

x (=E /m )

# photons per annihilation (m =1GeV)

xdN/dx

ee

K K

K K

Prompt

x (=E /m )

# photons per annihilation (m =1.2GeV)

ee

K K

K K

2

+

Prompt

xdN/dx

x (=E /m )

# photons per annihilation (m =1.2GeV)

xdN/dx

ee

K K

K K

Prompt

x (=E /m )

JHEP08(2015)050

Figure 12. The photon distribution x2dN/dx for dark photon (left panel) and dark scalar (right panel) in the lab frame. The prompt photon means summing all the channels according to BR.

the equation (B.3) can be simplied as

dNA/dxA =

[integraldisplay]

1xBi , (B.4)

where xA = E/mDM and E is the photon energy in the lab frame. This simplication will not change the accuracy of the photon spectrum signicantly, because in our region of interest, the dark photon has mass around O(1) GeV, while twice DM mass is around O(10) GeV.

We plot the photon distribution x2dN/dx for dark photon and dark scalar in the lab frame in gure 12. It is clear that those channels with direct photons are dominant. In the channel on the left panel, there is a kink structure from direct photon and continuous photon from decay. In the Kaon channel, one can see that the photon spectrum for 1GeV is di erent from 1.2GeV, because two Kaon mass is close to 1GeV and have mass threshold e ect in the equation (B.3). The other channels like are not a ected by the mass di erence.

C Photon spectrum from di erent nal states

Here we will present the photon spectrum from Final State Radiation (FSR), three-body, four-body nal states, etc.

C.1 FSR

FSR from charged fermionic pairs and charged bosonic pair should be treated separately. For decay to the bosonic eld, as an example of + + , the composite structure of

21

1 dxBi dNBi

dxBi

[notdef] brings the FSR computation some theoretical uncertainties [111, 112], which will be neglected here. Hence, the scalar QED is employed to derive FSR spectrum. To study fermionic elds or other bosons, the mass of the particles should replace the pion mass m,

and other changes needed is written below.

In general, the FSR spectrum is divided into three parts: the spectrum from hard photon H, an exponential part taking into account the soft mutli-photon emission Bx1+B

and the virtual photon correction to the soft photon emission V +S. The photon spectrum from boson FSR dN

dx0 and fermion FSR dNfdx0 in the rest frame of are written as follows,

dN

dx0 = H(x) + 1 + V +S [parenrightbig]

xB(x)1B(x) (C.1)

dNf

dx0 = Hf(x) + [parenleftBig]

JHEP08(2015)050

1 + V +Sf

[parenrightBig]

xB(x)1B(x) (C.2)

where

H(x) =

2x [prime]

3 (C.3)

Hf(x) =

2x

3 2

[prime]

1 +1 [prime] ln [parenleftbigg]1 + [prime]1 [prime] [parenrightbigg][bracketrightbigg]

(C.4)

B(x) =

2(1 x) [prime]

"1 + [prime]22 [prime] ln [parenleftbigg]1 + [prime] 1 [prime]

[parenrightbigg]

1

[bracketrightBigg]

(C.5)

V +S =

2 + 2

ln [parenleftbigg]1 + 1

[parenrightbigg] 2 2 ln

1 24[parenrightbigg] 1 + 2 2

ln

1 + 1

[parenrightbigg]

[notdef] ln

(1 + ) 2[parenrightbigg]+ ln

1 + 2

ln

1 2

[parenrightbigg]+ 2Li2[parenleftbigg]2 1 +

[parenrightbigg]

+ 2Li2

1 2

[parenrightbigg] 2 32[bracketrightbigg]

(C.6)

V +Sf = V +S

1

2 ln [parenleftbigg]

1 + 1

[parenrightbigg]

, (C.7)

p1 4m2/s is the pion velocity without photon radiation, [prime] =

p1 4m2/((1 x)s). Notice that the soft-virtual part V +S taking into account the one-loop correction to ! + + , does not depends on x.

Boosting the spectrum dN0

dx at reference give the photon spectrum at the frame of DM. In the limit of m[notdef] m, the spectrum is

dN1 dx =

[integraldisplay]

where =

1 dx0 x0

dN0dx0 (C.8)

The above formula are derived from QED or scalar QED, which is tted well to the analysis of dark photon. In the case of dark scalar, the chiral perturbation theory complicates the situation, but due to the other uncertainties, such as branching ratio, this is a good approximation as well.

22

x

C.2 and [notdef] radiative decay

[notdef] is close to 100% decaying to [notdef] + [notdef]; besides that, there is 0.2% possibility that the radiative decay [notdef] ! [notdef][notdef] + [notdef] + happens. Inner Bremsstrahlung from the weak decays

as the dominant process contributing to the radiative decay are considered here, while the other decay processes from virtual hadronic are neglected since they are subdominant [106]. At the rest frame of [notdef], the photon spectrum is

dN dx1

=

2

(r 1)2 (x 1)x [bracketleftbig](2

+ x)2 + 4r(x 1)

1

[bracketrightbig]

(r + x 1)

, 0 x (1 r) (C.9)

where x is in the range of 0 x 1 r, and r = (m[notdef]/m)2. Since m[notdef] is not quite small

relative to m, we cannot assume r [similarequal] 0 to boost the spectrum. Under the assumption

that m m and m[notdef] m, the spectra in frame and DM frame have analytical

solutions, and in any frame, the spectrum has the same range 0 x 1 r. The photon

spectrum in the dark photon frame, dN dx0 , and in the DM rest frame, dN dx1 , from the process of ! + + + can be derived,

dN dx0 =

2(1 + r)2x

JHEP08(2015)050

+(x 1) 2r2 + 2rx + x2 2x + 2

[parenrightbig]

ln 1 x

r

2 (2 + 2r x) (1 + r + x) + 4r2x tanh1(1 2r)

+ tanh1(1 2x)

[parenrightbig]

(2x 4rx) ln

1 r

x +

2 + 2r2 + x rx + x2

+2(1 + r)x ln x] ln

r1 x

+ 2(1 + r)x [Li2(r) Li2(1 x)]

(C.10)

dN dx1 =

12(1 + r)2x

24(1 + r)2 +

42(1 + r) 2(1 + r + 2r2)

[bracketrightbig]

x 18x2

+ 24r2x tanh1(1 2x) 12 ln

r1 x

24r2x tanh1(1 2r)(1 + ln x)

+2 3(1 + r)x ln2(1 r)(1 + r + ln r) + 3 2r2 + 3x + x2

[parenrightbig]

ln r

1 x

ln x + 3(1 + r)(1 + r ln r)x ln2 x

+x ln(1 r)(2(1 + r) 6r + 6(1 + 2r) ln x)[bracketrightbigg]

+ 6x

+x 2(1 + r) + 6r + 3(1 + r) ln r

[parenrightbig]

[bracketleftbigg][parenleftbigg]

1 + r + 2r2

+2(1 + r) ln

1 r

x

Li2(r) + (1 + r + 2r2)Li2(x) + 2(1 + r) (Li3(1 r)

Li3(x))

[bracketrightbigg]

(C.11)

The pion radiative decay formula can apply to Kaon directly, but its gamma ray spectrum from radiative decay is negligible due to 0 from Kaon decay.

If the nal states are [notdef]+ + [notdef], the Branching ratio of [notdef] ! e

e [notdef] is (1.4 [notdef] 0.4)%,

which is one order magnitude larger than the branching ratio of [notdef] radiative decay. The

23

photon spectrum in di erent frame are listed as follows, dNdx1

=

3

1 x

x

[parenleftBigg][parenleftBigg]3

2x + 4x2 2x3

[parenrightbig]

ln 1

r +

[bracketleftbigg]

17

2 +

23

101

12 x2 +

6 x

55 12x3

+ 3 3x + 4x2 2x3

[parenrightbig]

ln(1 x)

[bracketrightbigg]

(C.12)

dN dx0 =

3

1 x

[parenleftbigg]

3 + 23x 6x2 + 3x3

2

3x4 + 5x ln x[parenrightbigg]

ln 1

r +

[bracketleftbigg]

17

2

32x +

191

12 x2

23

3 x3 +

74x4 + [parenleftbigg]

3 + 23x 6x2 + 3x3

2

3x4[parenrightbigg]

28

3 x ln x

ln(1 x)

+5x ln(1 x) ln x + 5xLi2(1 x)

[bracketrightbigg]

JHEP08(2015)050

(C.13)

139

dN dx1 =

3

1 x

[parenleftbigg]

3

18 x + 6x2

3 2x3 +

2 9x4

2

3x ln x

52xln2 x[parenrightbigg]

ln 1

r +

[bracketleftbigg]

19

2 +

+

2735108

29 5 (3)

x 74336 x2 +16136 x3 71108x4 + [parenleftbigg]3 13918 x + 6x2 3 2x3

+29x4[parenrightbigg]

ln(1x)+

92x526 x

ln x+ 143 x ln2 x23xLi2(x)+5xLi3(x)

[bracketrightbigg](C.14)

where r = m2e

m2 1, and the range of x is (0, 1) which does not depends on r since r is

negligible.

C.3 n-body nal states

Here we study the energy spectrum from the process of decay to n particles. As n = 2, the photon spectrum is a delta function, which is determined by kinematics. Whereas n 3, the phase space integral and matrix elements will inuence the shape of spectrum.

The energy spectrum for n-body nal states can be easily applied to ! +0, ! +00 and ! ++.

The n-body phase space integration Rn(s) is computed by a recursion relation [113, 114], and assuming matrix element constant, the energy spectrum can be computed by the phase space integral, The recursion relation of Rn is written as,

Rn(s) = (4)n1 [notdef] [integraldisplay]

(psmn)2

(m1+...+mn1)

2 dM2n1 [radicalBig]

(s, M2n1, m2n)

8s

[notdef]

[integraldisplay]

(Mn1mn1)2

(m1+...+mn2)

2 dM2n2 [radicalBig]

(M2n1, M2n2, m2n1)

8M2n1

[notdef] [notdef] [notdef] [notdef] [notdef]

[integraldisplay]

(M3m3)2

(m1+m2)2 dM22 [radicalbig] (M

2

3 , M22, m23)

8M23

p (M22, m21, m22) 8M22

(C.15)

where the angular integration is equal to the prefactor (4)n1 due to the assumption of constant matrix amplitude, and the Lorentz invariant function (x, y, z) x2 + y2 + z2

2xy 2yz 2zy. The energy spectrum of the n-th nal states can be derived,

dN dx =

1 Rn

dRn dx =

s Rn

dRn dM2n1

(C.16)

24

In the study of dark photon decaying to 3 or 4, we did not take the limit of m to zero, since the O(1) GeV dark photon mass is close to pion mass, but if we set the masses of all

the nal states to zero, eq. (C.16) has an analytical solution,

dNdx = (n 1) (n 2) (1 x)n3 x . (C.17)

C.4 Photons from individual channels

The photon spectra are computed channel by channel. We will briey mention the method to obtain the spectrum for the di erent channels. With no explicit mention of the dark force , we refer to both dark photon and dark scalar.

! e, photon from electron FSR are considered

! [notdef]

[notdef], photon from muon FSR and radiative decay

[notdef] ! +, from pion FSR including hard photon spectrum H(x) in eq. (C.1) and

[notdef] radiative decay. In the radiative decay, the form factor are neglected.

0 ! +, photon from pion FSR not including hard photon spectrum H(x) in

eq. (C.1) and [notdef] radiative decay. No including the hard photon spectrum is due to the fact that it mainly comes from the interaction term A[notdef]A[notdef]+, not for scalar

mediator.

0 ! 00. 98.82 % of pion cascade decays to 2 . The photon spectrum of the 0

decay in di erent frames are written as,

dN dx1

= 2 (1 x) (C.18) dNdx0 =

2

JHEP08(2015)050

p1 [epsilon1]20, 1

p1 [epsilon1]202 < x <1 +

p1 [epsilon1]20 2

[parenrightBigg]

(C.19)

dNdx1 = 8

>

>

<

>

>

:

2

p1[epsilon1]20ln 2x 1+

p1[epsilon1]20,

1p1[epsilon1]202 < x < 1+p1[epsilon1]202[parenrightbigg]

2

p1[epsilon1]20ln 1+p1[epsilon1]20 1p1[epsilon1]20,

0 < x < 1p1[epsilon1]202[parenrightbigg](C.20)

where [epsilon1]0 = 2m0m , and [epsilon1]1 = mm~ [similarequal] 0.

! K+K. 20.66 % of kaon decaying to hadronic modes K+ ! + + 0 are major

contribution. Due to the small branching ratio of ! K+K, this process is the

only one considered here. In the leptonic channel, K+ ! 0e+ e and K+ ! 0[notdef]+ [notdef]

are suppressed by the smaller branching ratio and three-body phase space.

! K0

K0, or we can think it as decays to CP even K0S and CP odd K0L. For K0S, the photon yield originates from the modes of 0. K0S ! 00 with the branching ratio 30.69 %. For K0L, K0L ! 000, 19.52 %, K0L ! +0, 12.54 %. Photon

from [notdef] are not included here.

25

! !0 ! 20 + . The second ! means that we consider one modes of the !

decay. Due to some experimental reason, the mode of ! ! +0, with 89.2 % BR

are included in the 4 nal states. Since these process are the process with two body nal states, we can use kinematics to derive the photon spectrum.

! +0. Following C.3, We assume the scattering matrix element is constant

and the photon from 0 are considered.

! +00. Assume the scattering matrix element is constant and the photon

from 0 are considered.

! ++. Assume the scattering matrix element is constant and the photon

from [notdef] radiative decays are considered.

! 0 . Two body nal states.

! . ! , 39.31 %, ! 000, 32.56 %, ! +0, 22.73 %. For the

three body nal states decay of , constant matrix element are assumed, and photon from [notdef] are neglected.

! . The photon from decay is the same as the treatment in the process of

! . With the photon in the frame, we can boost it to the and DM frame.

D Electron spectrum calculation

The electron spectra are calculated channel by channel. We start with the electron spectrum for muon at rest. In SM, the unpolarized muon has the following electron spectrum in muon rest frame,

dNe[notdef] /dx = 2x2(3 2x) (D.1)

where x 2Ee/m[notdef]. We have neglect the electron mass in the spectrum. As long as we

know the electron spectrum in daughter particle frame, we do boost accordingly to get the spectrum in the lab frame, similar as in photon spectrum. For example, the dark matter annihilating directly into a pair of muon, the electron spectrum in lab frame is

dNlab:2[notdef]e[notdef]/dx2 =

19 8x32 + 27x22 30Log(x2) 19

[parenrightbig]

(D.3)

Then we briey introduce how we get the electron spectrum for other particles. For +,

the decay to [notdef]+ + [notdef] is about 99.9877%, while the rest is to e+ + e. We boost the electron from muon and also add the electron from the direct decay into the electron spectrum. For

26

JHEP08(2015)050

13 4x32 9x22 + 5

[parenrightbig]

, (D.2)

where x2 Elabe/mDM. The calculation uses the boost formula in equation (B.4). If we

neglect the daughter particle mass at each step, we can have analytic expression for the cascade decay to four muon.

dNlab:4[notdef]e[notdef]/dx2 =

# e per annihilation (m = 1GeV)

ee

K K

K K

Total

dN/dx

x (=E /m )

# e per annihilation (m = 1GeV)

ee

K K

K K

Total

dN/dx

x (=E /m )

# e per annihilation (m = 1.2GeV)

ee

K K

K K

Total

dN/dx

x (=E /m )

# e per annihilation (m = 1.2GeV)

ee

K K

K K

Total

dN/dx

x (=E /m )

JHEP08(2015)050

Figure 13. The electron distribution dN/dx for dark photon (left panel) and dark scalar (right panel) in the lab frame.

Figure 14. The contours of excluded annihilation cross-section at freeze-out from Plank as a function of m~ and m. The left side of the contour is excluded. [notdef]1, 2, 3 denotes annihilation

cross-section in units of 1026cm3/s.

0, the decay to e+e is quite small, about 1.17%. We neglect electron from 0, because in most of the decay channels, 0 are produced with [notdef] at similar rate or even smaller. For K[notdef], there are seven decay channels relevant for electron spectrum, with [notdef], 0, [notdef][notdef] and e[notdef] in the nal states. We properly boost all the electron from the daughter particles, except 0 which is neglected in the calculation. For K0 and , the calculation is the same as K[notdef]. For 3 and 4 nal states, we use the natural phase space and only count the electrons from [notdef].

27

We plot the electron distribution dN/dx for dark photon and dark scalar in the lab frame in gure 13. The Kaon channel has di erent electron spectrum for 1GeV and 1.2GeV, due to dark mediator mass is close to two Kaon mass. The electron spectrum mainly comes from e+e at high energy for dark photon, but not for dark scalar. The dark scalar has smaller electron spectrum than dark photon due to small e+e BR.

E CMB limits on thermal cross-section

We plot the contours of excluded annihilation cross-section at freeze-out from Plank as a function of m[notdef] and m in gure 14. The contours are calculated following the formula,

hv[angbracketright] fe [notdef]Planck(m[notdef]) /fe (m) = 3 [notdef] 1026cm3/s , (E.1)

where [angbracketleft]v[angbracketright] fe [notdef]Planck(m[notdef]) is the Planck excluded [angbracketleft]v[angbracketright] fe and fe is the e ciency factor

for dark mediator model. It shows DM with thermal cross-section 3 [notdef] 1026cm3/s in the

dark mediator models should be larger than 20 GeV.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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