E - Academia Sinica

Double Action Dark Matter, PAMELA and ATIC Kingman Cheung, Po-Yan Tseng and T.-C. Yuan arXiv:0902.4035 [hep-ph]

IoP, Academia Sinica Lunch Seminar April 8 (2009) 2009年4月8日星期三

Outline Brief Review of Dark Matter Physics Anomalies Seen at PAMELA, ATIC Dark Matter Interpretation Theoretical Calculations and Chi Square Fits for PAMELA and ATIC Summary

2009年4月8日星期三

Rotation Curves of Galaxies Observed rotation curves exhibit flat behavior at large distances 292Suggest

v(r) =

!

M (r) = 4π

Bertone etprofile al. / Physics Reports 405 (2005) 279 – 390 universal haloG.density

GM (r) r !

r

ρ(r! )r!2 dr!

√

v ∼ 1/ r

(Expected)

v ∼ Const

(Observed)


Temperature Fluctuations δT (θ, φ) = T

∞ m=+l ! !

l=1 m=−l

alm Ylm (θ, φ)

1 Cl ≡ "|alm | # ≡ 2l + 1 2

WMAP

m=+l !

m=−l

|alm |2

Ωb h = 0.0224 ± 0.0009 and ΩM h = 2


2

+0.008 0.135−0.009

Big Bang Nucleosynthesis G. Bertone et al. / Physics Reports 405 (2005) 279 – 390

287

0.018 < Ωb h < 0.023

g Bang nucleosynthesis predictions for the abundances of light elements as a function of the baryon over photon ratio 2009年4月8日星期三

2

Thermal WIMP & Relic Abundance ΩCDM h2 (WMAP) = 0.113+0.016 −0.018

Ωh ∼ 0.1 × 2


!

#σv$freeze−out 3 × 10−26 cm3 s−1

"−1

the order of 3 " 10#8 . The cosmic-ray interactions is igible. Therefore an oban unequivocal signature niverse. ow geomagnetic cut-off pparatus, combined with e low momenta, allows r and earth physics to be

energy measurement of the calorimeter for high-energy electrons. Silicon sensors also measure ionization losses, allowing absolute particle charge to be determined for Zo6.

bined electron and positron allow the contribution of ion to be investigated. ows. Section 2 provides an includes a brief description f the acquisition and trigger -flight operations. In-flight e shown in Section 3, where o achieve the science goals

paratus

ption of the apparatus is following section where sed.

PAMELA is shown in cal characteristics of each 2009年4月8日星期三

Fig. 1. Schematic overview of the PAMELA apparatus. The detector is approximately 1.3 m high, has a mass of 470 kg and an average power consumption of 355 W. The magnetic field lines inside the spectrometer cavity are oriented along the y direction. The average value of the magnetic field is hBi ¼ 0:43 T.

PAMELA Flight Data


673z0:0014 {0:0013 ) 07 6 0.0012) 83 6 0.0011) 51 6 0.0012) 50 6 0.0012) 02 6 0.0014) 48 6 0.0016) 83 6 0.0018) 29 6 0.0023) 546z0:0029 {0:0028 ) 585z0:0030 {0:0031 ) 590z0:0040 {0:0041 ) 46 6 0.0059) 31 6 0.0093) 106z0:022 {0:023 ) 137z0:048 {0:043 )

ion and proton ections 2 and 3. The as the physical sumed to be identical harge dependence on d east–west effects e to the spectrometer on software. No ction efficiency above o particles (secondary re that are scattered re-enter the e), are significant o reach the Earth from t-off for the PAMELA 52009年4月8日星期三 GV for equatorial

Fraction of energy along the track, F

w(ez ) ed ðw(ez )zw(e{ )Þ op of payload

>25

1.0 0.9 0.8

20

0.7 0.6

15

0.5 10

0.4 0.3

5

0.2 0.1 0 –1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 Deflection (GV–1)

0.4

0.6

0.8

1.0

0

Figure 1 | Calorimeter energy fraction, F . The fraction of calorimeter energy deposited inside a cylinder of radius 0.3 Molie`re radii, as a function of deflection. The number of events per bin is shown in different colours, as indicated in the colour scale. The axis of the cylinder is defined by extrapolating the particle track reconstructed by the spectrometer. The Molie`re radius is an important quantity in calorimetry, as it quantifies the lateral spread of an electromagnetic shower (about 90% of the shower energy is contained in a cylinder with a radius equal to 1 Molie`re radius), and depends only on the absorbing material (tungsten in this case). The events were selected requiring a match between the momentum measured by the tracking system and the total detected energy and requiring that the electromagnetic shower starts developing in the first planes of the calorimeter. The particle identification was tuned to reject 99.9% of the protons, while selecting .95% of the electrons or positrons.

NATURE | Vol 458 | 2 April 2009

PAMELA Positron Excess [Nature 07942 (2009)]

300 GeV), hence pointing t of the electron spectrum w

0.4 0.3

Received 28 October 2008; acc 1.

Positron fraction, !(e+) / (!(e+) + !(e–))

0.2

2. 3.

0.1

0.02

0.01 10−1

4. 5. 6. ref. 1 PAMELA Aesop (ref. 13) HEAT00 AMS CAPRICE94 HEAT94+95 TS93 MASS89 Muller & Tang 19875,6 1

7. 8. 9. 10. 11. Energy (GeV)

10

102

Figure 2 | PAMELA positron fraction with other experimental data and with secondary production model. The positron fraction measured by the PAMELA experiment compared with other recent experimental data (see refs 5–7, 11–13, 30, and references within). The solid line shows a calculation1 for pure secondary production of positrons during the propagation of cosmic rays in the Galaxy without reacceleration processes. Error bars show 1 s.d.; if not visible, they lie inside the data points.

12. 13. 14.

15. 16. 17.


account when interpreting potential dark matter signals. A pulsar

Moskalenko, I. V. & Strong, positrons and electrons. As Atoian, A. M., Aharonian, F. cosmic rays. Phys. Rev. D 52 Heinz, S. & Sunyaev, R. Cos the CR spectrum. Astron. A Jungman, G., Kamionkowsk Rep. 267, 195–373 (1996). Golden, R. L. et al. Measurem above 5 GeV. Astrophys. J. Barwick, S. W. et al. Measu 50 GeV. Astrophys. J. 482, Aguilar, M. et al. Cosmic-ra with AMS-01. Phys. Lett. B 6 Chang, J. et al. An excess o Nature 456, 362–365 (200 Picozza, P. et al. PAMELA — light-nuclei astrophysics. A Delahaye, T. et al. Galactic Æhttp://arXiv.org/abs/080 Boezio, M. et al. The cosmic during solar minimum activ Alcaraz, J. et al. Leptons in Clem, J. & Evenson, P. in Pro al.) 477–480 (Universidad Aharonian, F. et al. First dete cosmic source: HESS discov L43–L47 (2006). Berezhko, E. G., Ksenofonto accelerated cosmic rays. A Serpico, P. On the possible c fraction. Phys. Rev. D 79, 02 Komatsu, E. et al. Five-year Cosmological interpretation

PAMELA Anti-proton/proton Ratio [Phys. Rev. Lett. 102, 051101 (2009)] -3

0.4

!10

0.35 0.3

Donato 2001 (D, " =500MV) Simon 1998 (LBM, " =500MV) Ptuskin 2006 (PD, " =550MV) PAMELA

p/p

0.25 0.2 0.15 0.1 0.05 0 1

10 kinetic energy (GeV)

102

Leptonphilic DM?

e antiproton-to-proton flux ratio obtained in this work compared with the 2009年4月8日星期三

PAMELA versus Background LETTERS PAMELA Positron Fraction

1.00 Table 1 | Summary of positron fraction results

Positron Fraction

0.50

0.20

No Peak Structure!

0.10 0.05

0.02

Solar activity! 1

5

10 Ee!


50 !GeV"

Rigidity at spectrometer (GV)

Mean kinetic energy at top of payload (GeV)

1.5–1.8 1.8–2.2 2.2–2.7 2.7–3.3 3.3–4.1 4.1–5.0 5.0–6.1 6.1–7.4 7.4–9.1 9.1–11.2 11.2–15.0 15.0–20.0 20.0–28.0 28.0–42.0 42.0–65.0 65.0–100.0

1.64 1.99 2.44 2.99 3.68 4.52 5.43 6.83 8.28 10.17 13.11 17.52 24.02 35.01 53.52 82.55

w(ez ) ðw(ez )zw(e{ )Þ at top of payload

Extrapolated

(0:0673z0:0014 {0:0013 ) (0.0607 6 0.0012) (0.0583 6 0.0011) (0.0551 6 0.0012) (0.0550 6 0.0012) (0.0502 6 0.0014) (0.0548 6 0.0016) (0.0483 6 0.0018) (0.0529 6 0.0023) (0:0546z0:0029 {0:0028 ) z0:0030 (0:0585{0:0031 ) (0:0590z0:0040 {0:0041 ) (0.0746 6 0.0059) (0.0831 6 0.0093) (0:106z0:022 {0:023 ) z0:048 (0:137{0:043 )

The errors are one standard deviation. Details concerning particle selection and proton background determination can be found in Supplementary Information sections 2 and 3. The detection efficiencies for electrons and positrons are assumed to cancel, as the physical processes that these species undergo in the PAMELA detectors can be assumed to be identic across the energy range of interest. Possible bias arising from a sign-of-charge dependence o the acceptance due to the spectrometer magnetic field configuration and east–west effects caused by the Earth’s magnetic field were excluded as follows. Effects due to the spectromete magnetic field were studied using the PAMELA Collaboration’s simulation software. No significant difference was found between the electron and positron detection efficiency abov 1 GV. East–west effects, as well as contamination from re-entrant albedo particles (secondar particles produced by cosmic rays interacting with the Earth’s atmosphere that are scattered upward but lack sufficient energy to leave the Earth’s magnetic field and re-enter the atmosphere in the opposite hemisphere but at a similar magnetic latitude), are significant around and below the lowest permitted rigidity for a charged cosmic ray to reach the Earth from infinite distance, known as the geomagnetic cut-off. The geomagnetic cut-off for the PAMEL orbit varies from less than 100 MV for the highest orbital latitudes to ,15 GV for equatorial regions. In this work, only events with a measured rigidity exceeding the estimated vertical (PAMELA z-axis) geomagnetic cut-off by a factor of 1.3 were considered. This reduced east–west effects and re-entrant particle contamination to a negligible amount. The vertical

100

500

1000

r the electron will be omitted as before.

About Background Fluxes

ckground fluxes Believed mainly due to Supernova shock and diffuse outward (primary) and from collisions among cosmic ray nuclei and intersellar medium (secondary)

kgroundSimple electron/positron fluxes from sources are belie parameterization for electron/positron (Baltzastrophysical and Edsjo, astro-ph/9808243) Normalization used in our fitsand is 0.7 are commonly parametrized as [18] to Supernova explosions bkdg Φe+ bkdg Φe−

4.5E 0.7 = , 2.3 4.2 1 + 650E + 1500E bkdg,prim bkdg,sec = Φe− + Φe− 0.16E −1.1 0.7E 0.7 = + , 0.9 2.15 1.5 4.2 1 + 11E + 3.2E 1 + 110E + 580E

in unit of GeV and the unit for the flux is GeV−1 cm−2 s−1 sr−1 .

on of 0.7 so that the flux calculation is consistent with the ATIC data i

ge of 20 − 70 GeV. 2009年4月8日星期三

The ATIC Balloon Payload


140

trument. plotted as a hted energy and the The shower

2009年4月8日星期三 nate of the

LETTERS

ATIC-1 and ATIC 2 Ee3.0dN/dEe (m−2 s−1 sr−1 GeV2)

1,000

McMurdo, Antarctica

100

10

10

100 Energy (GeV)

1,000

Figure 2 | ATIC-1 and ATIC-2 spectra at balloon altitude, showing good agreement with each other. The measured primary electron flux (scaled by E3) at flight altitude is shown for ATIC-1 (open squares) and ATIC-2 (filled circles). The errors are one standard deviation. Both balloon flights were from McMurdo, Antarctica, and circumnavigated that continent. ATIC-1 was a test flight in 2000–01 and the usable data correspond to an exposure of 0.61 m2 sr days. ATIC-2 was a science flight in 2002–03 with an exposure of 2.47 m2 sr days. To eliminate edge effects, we restrict the incident zenith angle to be less than ,37u (cos h $ 0.8), use only the central 80% of the SiM

instrument. ATIC becomes background limited for electrons only above several teraelectronvolts.

ATIC Energy Spectrum[Nature 07477 (2008)]

e indicates a nearby source result of an astrophysical erved in a variety of astroa remnant17, pulsar wind mediate-mass black hole). would need a very steep 21.4) with a high-energy overproduce teraelectrono-quasar could produce a but such an object would d active relatively recently. s have observed numerous 2009年4月8日星期三

1,000

Ee3.0dN/dEe (m−2 s−1 sr−1 GeV2)

ow an excess electron flux trum drops rapidly, with a 00 GeV. In particular, over rve 210 electrons, whereas cess of about six standard method for determining ughly four standard devian 4). the Polar Patrol Balloon though of lower statistical orimeter16 also indicate a results (see Fig. 3), giving his feature is real. ers in the GALPROP code o 700 GeV. This required a eproduce the drop in flux electrons above 1 TeV by a ng the well-measured data

100

10

10

100 Energy (GeV)

1,000

Figure 3 | ATIC results showing agreement with previous data at lower energy and with the imaging calorimeter PPB-BETS at higher energy. The electron differential energy spectrum measured by ATIC (scaled by E3) at the top of the atmosphere (red filled circles) is compared with previous observations from the Alpha Magnetic Spectrometer AMS (green stars)31, HEAT (open black triangles)30, BETS (open blue circles)32, PPB-BETS (blue crosses)16 and emulsion chambers (black open diamonds)4,8,9, with uncertainties of one standard deviation. The GALPROP code calculates a power-law spectral index of 23.2 in the low-energy region (solid curve)14. (The dashed curve is the solar modulated electron spectrum and shows that modulation is unimportant above ,20 GeV.) From several hundred to ,800 GeV, ATIC observes an ‘enhancement’ in the electron intensity over the GALPROP curve. Above 800 GeV, the ATIC data returns to the solid line. The PPB-BETS data also seem to indicate an enhancement and, as discussed in Supplementary Information section 3, within the uncertainties the emulsion chamber results are not in conflict with the ATIC data. 363

ATIC Energy Spectrum vs. KK Dark Matter | | NATURE Vol 456 20 November 2008

[Nature 07477 (2008)] 1,000

Ee3.0dN/dEe (m−2 s−1 sr−1 GeV2)

including pulsar wind ectra that fall as E22 with lt region3,17. This implies s of teraelectronvolts or he current electron data. eting the source requireremnant, whose potenhas been modelled4,18. a is about a factor of 60 plementary Information t discussed here have the nd there may well be a is accelerating the elec-

100

10 10

hilation of dark matter retical work on the prexy as well as on the products of dark matter re predicted as products suggested as dark matter particles from supersymparticles resulting from mensions—the ‘Kaluza– of supersymmetric and oceed through different ectron–positron pairs or ation section 6). The sigease in electron intensity 2009年4月8日星期三

100 Energy (GeV)

1,000

Figure 4 | Assuming an annihilation signature of Kaluza–Klein dark matter, all the data can be reproduced. The GALPROP general electron spectrum resulting from sources across the galaxy is shown as the dashed line. The dotted curve represents the propagated electrons from the annihilation of a Kaluza–Klein particle. The dotted curve assumes an isothermal dark matter halo of 4-kpc scale height, a local dark matter density of 0.43 GeV cm23, a Kaluza–Klein mass of 620 GeV, and an annihilation cross section rate of 1 3 10223 cm3 s21, which implies a boost factor of ,200. The sum of these signals is the solid curve. Here the spectrum is multiplied by E3.0 for clarity. The solid curve provides a good fit to both the magnetic spectrometer data30,31 and calorimeter data16,32 and reproduces all of the measurements from 20 GeV to 2 TeV, including the cut-off in the observed excess. All error bars are one standard deviation.

ATIC versus Background ATIC energy spectrum Peak

0.100

Hump E3 ! !GeV2 cm"2 s"1 sr"1 "

0.050

Sharp decline

0.020

0.010

0.005

0.002

0.001 10

20

50

100 Ee


200 !GeV"

500

1000

2000

Dark Matter Interpretation

Dark Matter Annihilation Dark Matter Decay eXciting Dark Matter (XDM) Inelastic Dark Matter (iDM,CiDM) etc


χχ → SM SM χ → 2 or 3 SMs

indexAnnihilation labeling variousthe channels to avoid notation in th ming a steady stateMatter condition whilethe solving diffusion equation for cluttering the positron A. Dark 210 M. Cirelli et al. / Nuclear Phys

Semi-analytical Calculation of Positron Flux

function canEarth be parametrized by into the following semiD ) at ersed across the universe, itsI(λ flux can be casted

equal to c in the' regimes ourpositron interest) a Assuming steady state condition solving diffusion equation for ofthe [Hisano eta al, PRD73, 055004 (2006); while Delahaye etsentially al, the PRD77, (2008)] %063527 & 2

f (t, x!, E) = dN obeys the diffusion-loss b1 − l (l − b2 ) e e+ /dE, I(λflux tanh exp(− )+ as it traversed across the universe, v its ataEarth be casted intoa2the following semiD) = 0 + a1can ! " + ∂ ∂f c1 2 c2 e − K(E) · ∇ f − b(E)f = Q, (1) analytical form [15, 16]Φe+ (E) = 4π fe+ (E) , ∂t ∂E

l form [15, 16]

Flux:

! ve+ the with l = log10 (λD /kpc) and diffusion length λ (E, E ) is given by D (E/GeV with diffusion coefficient K(E) = K 0 Φe+ (E) = fe+ (E) , (1)

closeAnnihilation: to the velocity of light c and the function fe+ (E) is given by16 4π (GeV τ' ( d E ) with τE = 10 s. They respectively

Number density per unit kinetic energy ! δ−1 δ−1 (E /GeV) − (E/GeV) netic fields and energy loss due to synchrotron 2the$ function fe+ (E) is given by !the with ve+ close to velocity c and ! of light "2 # E max λ = 4K . Emax0 τE 1 + galactic starlight. E 1 dN e Halo function CMB! photons and on! infrared !ρdm ! D ! δ−1 , + Qann (EdE = B η !I(λ !σv" I(λ (E, E )) (2) fe+ (E) = B dE (E, E )) ) D e D ! E max shape of a solid flat cylinder that sandwiches the g b(E) 1 Mdm dEe+ E ! b(E) E ! ! fe+ (E) = B dE I(λD (E, Eand )) Q )= 20 kpc in the r direction [19]. Th ann (E radius R b(E) E a0,1,2,3 , b1,2 , c1,2 and δ, K0 can be found respectively in T The constants with Emax = Mdm in the case of annihilating DM, Mdmx! and ρdm and theboundary densitycondi = (r$ , z$are ) = the (8.5mass kpc, 0). The

3"2 # loss rate ! $ Eishes on the b(E) surfaceinof the cylinder, outside of Ref. [17]. The energy function Eq.(2) is max + 1 ρ dN dm e 3 of the dark matter respectively, and the overall constant B is! the boost! factor. In Eq.(2) we

dE ofI(λ E )) , (2) Values theD (E, propagation ! parameters δ, K0 and L b(E) Mdm dE2e+ E and modelizations. We E adopt the sets discussed in have expressed the source term Qann according to b(E) = mass and the density with Emax DM, M dm and ρdm are the(GeV Boost factor for= Mdm in the case of annihilating × τEin)kpc2 /Myr L in k ! "2 # Model δ K 0 ρdm dNe+ clumpy profile of the dark matter respectively, is the, boost factor. In Eq.(2) we (3) Qann =and η the overall constant !σv"e+ B min 1 16 Mdm dEe+ (M2)we 0.55 where τE = 10 seconds. In our analysis, simply0.00595 employ the monoc have expressed the source term Qann according to med 0.70 0.0112 4 max (M1) 0.46is over 0.0765 15 where η = 1/2(1/4) spectrum for Majorana or Dirac The summation all possible ! "particle. 2 # ρdm dNe+ ! " + Qann = η !σv" , (3) + dN 1 E e e + + channels that can produce positron in theMfinal state, and dN /dE denotes the spectrum Finally, the source term due to DM dE(E) dm e+ e = e δ 1− . DM ann dE Mdm density x ) is Mdm e+ ρ(! of thewhere positron perforannihilation thatparticle. particular Weis$ have η =energy 1/2(1/4) Majorana orinDirac Thechannel. summation oversuppressed all possible the # 2 % Analogous formulas for the electron will here. d 1 be ρomitted + /dE produce positron the final state, and dN denotes index channels labelingthat the can various channels toinavoid notation cluttering ine+the equations. Q e= finjthe , spectrum fThe %σ v&k inj = halo 2 MDM k of the positron energy per annihilation in that particular channel. We have suppressed the function I(λD ) can be parametrized by 2009年4月8日星期三 = B

η

!σv"e+

(Source)

Monochromatic Spectrum:

Semi-analytical Calculation of Positron Flux ng Dark Matter (Cont.) where τ is the lifetime of the DM and dN /dE is the positro e+

dm

e+

ermDecay: for a decaying dark matter in a particular channel is decay of the DM. The function fe+ (E) is now given by where ! τdm"is the lifetime of the DM and dNe+ /dEe+ is the positron e + 1 the ρDM dNedN ! Emax dm and + is the here τdm is the lifetime of positron energy spectrum per e 1 Qdec = decay of the DM. The ,e+ /dE (8) function fe+ (E) is now given by (Source) dE ! I(λD (E, E ! )) Qdec (E ! ) τdm Mdm dEfee++ (E) = cay of the DM. The function fe+ (E) is now given b(E) by ! EEmax 1 " # ! Emax ! ! ! 1 1 ρ + f (E) = dE I(λ (E, E )) Q (E ) dm D dec e ! ! ! Emax = dE I(λ (E, E )) b(E) E D 1 ! b(E) τdec " !M # ! EEmax 4 dm(E dm fe+ (E) = dE ! I(λD (E, E )) Q ) 1 1 ρdm dN ! ! b(E) E = dE I(λD (E, E )) " # ! b(E) τdm M dE E dm matter Emax with /2 for the decaying dark and summation ov dm + 1 E1max =ρM dN dm e = dE ! I(λD (E, E ! )) , (9) ! b(E) dEe+and summation over with τEdm =MM for the decaying dark matter explicit. The flux is Ethe same as in Eq.(1). dm max dm /2

fluxmatter iswe theuse same as in In our The analysis, either a Eq.(1). (i) monochromatic th Emax = Mdm /2 for the explicit. decaying dark and summation over all decaydecaying channelsspect is

" decaying # spectru analysis, we use either a (i) monochromatic plicit. The flux is the same In as our in Eq.(1). dNe+ 2 2E (E) = δ 1− # , " dEee++ M Mdm dN 2 dm 2E In our analysis, we use either a (i) monochromatic decaying spectrum:

Monochromatic decaying spectrum:

dEe+

(E) =

Mdm

δ 1−

Mdm

,

" spectrum# or (ii) decaying + dNevarying 2 2E (E) = decaying δ 1spectrum − , (10) or (ii) varying " # dEe+ Mdm MdN dme+ 2E 3 80E (E) =80E2 " 1 −2E #3 . Varying decaying spectrum: + dN dEee+(E) = Mdm 1 − Mdm . (ii) varying decaying spectrum 2

dEe+


Mdm

Mdm (Justified later)

The exact form of the"varying decaying spectrum is not crucial in o #3 2E dN + 80E

where η = 1/2(1/4) for Majorana or Dirac particle. The summation function I(λ D ) can be param index labeling the various channels to avoid notation clutteri f the positron energy per annihilation in that particular channel. We have suppresse abeling the various avoid positron notationincluttering in the channelschannels that can to produce the final state, andequations. dNe+ /dEe+Th de ndex labeling the various channels to avoid notation cluttering in the equations. Th function I(λ ) can be parametrized by [Cirelli, NPB 800in(2008) D Franceschinni, n I(λD ) can of bethe parametrized by per Strumia, I(λD ) =We a0 h+ positron energy annihilation that204] particular channel. unction I(λD ) can be parametrized%by % & '( & ' index labeling the various channels to avoid notation e 2 l cluttering in the b − (l − b1%− l & ' (l − 1b2 ) ( 2 + (λ I(λfunction a exp(− ) a with l = log /kpc) and t I(λ ) = a + a tanh exp(− D ) = a0 + a1 tanh 2 3a D b − l (l − b ) D 0 1 2 10 1 2 be parametrized c1 I(λD ) I(λ = aD0 )+can a1 tanh a2byexp(− c2 c1 ) + a3 c1 % & c'2 2 b (l − b ) !1 − l 2 λ2! = 4 = log10 (λwith /kpc) and the diffusion length λ (E, E ) is given by D D I(λ ) = a + a tanh(E, E ! ) is given a2 exp(− ))+is a D D 0 1 l = log (λ /kpc) and the length λ (E, E D D with l = log10 (λD /kpc) and by 10 the diffusion length λDdiffusion c1 c2 ' ( δ−1 ( '! /GeV)δ−1 − (E/GeV) ' (E ! δ−1 2 ! δ−1 δ−1! (E /GeV) − (E/GeV) =λlog τ . withλDl = (λ /kpc) and the diffusion length λ (E, Ea0,1,2,3 )− is (E/GeV given (E /GeV) The constants , b1,2 ,by c1,2 24K D ED τ 1004K 2 = . 0 E D −δ 01−τE λD = δ4K ' 1 ( − 1 ! Ref. [17]. δ−1 The δ δ−1 energy loss rat (E /GeV) − (E/GeV) 2 λD =be4K . 0 τE nstants aDiffusion b1,2 ,, cblength: and δ, K can found respectively in Table 2 and Eq.( 0,1,2,3 1,2 0 The constants a,0,1,2,3 , c and δ, K can be found respectively in Table 2 and Eq.( 1,2 1,2 0 δ−1

Halo Fit Function

The constants a0,1,2,3 , b1,2 , c1,2 and δ, K0 can be found respecti

7]. The loss rate function b(E) in Eq.(2) is is Ref. [17]. energy The energy loss rate function b(E) in Eq.(2) The constants a0,1,2,3 , b1,2 , c1,2 and δ, K0 can be found respectively in Tab

Ref. [17]. The energy loss rate2 function b(E) in Eq.(2) is 2 EE Ref. [17]. The energy loss rate function b(E) in Eq.(2) is16 where τE = 10 seconds. In o = = Energy loss function: b(E)b(E) 2 (GeV (GeV × τ×Eτ)E ) E 2 E spectrum b(E) = b(E) = 16 16 where τE = seconds. 10 seconds. In analysis, our analysis, simply employ themonochromatic monochromatic annihi (GeV ×(GeV τE ) × τE ) annihi In our we we simply employ the E = 10

pectrum m

16 16 where τ = 10 seconds. In In ourour analysis, we simply employ the monochr ! analysis, " we simply E where τE = 10 dN seconds. employ th ! " + 1 E e + dN 1 E e (E) = δ 1 − . Analogous formulas for the e 2009年4月8日星期三 spectrum

shape of a solid flat cylinder that sandwiches the galactic plane, with height 2L in the z and radius R = 20 kpc in the r direction [19]. The location of the solar system corres x! = (r$ , z$ ) = (8.5 kpc, 0). The boundary conditions impose that the positron densit [Cirelli, Franceschini NPB800 (2008) 204-220] ishes on the surfaceand of Strumia, the cylinder, outside of which positrons freely propagate and Values of the propagation parameters δ, K0 and L are deduced from a variety of cosmic and modelizations. We adopt the sets discussed in [20]:

Propagation Models (for Electron/Positron)

Model

δ

K0 in kpc2 /Myr

L in kpc

min (M2) 0.55 0.00595 1 med 0.70 0.0112 4 max (M1) M. Cirelli et al. /0.46 Nuclear0.0765 Physics B 800 (2008)15 204–220

212

Finally, the source term due to DM DM annihilations in each point of the halo w Table 2 density ρ(! x ) is Fit parameters for the expression in Eq. (17) function I (λD ) that encodes the astrophysics of the production # for the$halo 2 % dNek+ density and the propagation of positrons1 in theρgalactic halo Q= finj , finj = %σ v&k , dEb2 Halo model Propagation a0 2 M a1DM a2 a3 b1 c1 c2 k

NFW

Moore

isoT

min (M2) 0.774 where k0.500 runs over all the−0.448 channels 0.649 with med 0.502 0.621 0.688 0.806 averaged cross sections σ v.1.533 max (M1) 0.502 0.756 0.672

0.096 0.211 positrons in 192.8 the final state, with33.88 the respective 0.891 1.205

0.721 0.799

0.143 0.155

0.071 0.067

med max (M1)

0.912 M $2 1.230 & DM

0.762 0.811

0.162 0.135

0.055 0.061

0.193 0.379 '

0.296 0.231

One assumes steady state conditions, so that the first term of Eq. (10) vanishes, and min (M2)tion for0.500 0.791flux −0.448 0.636 0.096in a useful 192.8 semi-analytical 0.211 33.86 the positron at Earth can be written form [20,21

min (M2) med max (M1)

0.503 0.503

0.826 0.889

0.938 1.778

0.610 # 0.571

v e + 1 ρ$ ) = B −0.449 Φ0.500 e+ (E, r!$0.903 0.557 4πb(E) 2 M DM 0.495 0.499

0.629 0.695

0.137 0.677

0.784 0.721

! " ' finj (E ) · I 0.210 λD (E, E ) , 0.096dE 192.8 33.91 0.766 E 1.092

'

'

0.550 0.951

where B ! 1 is an overall boost factor discussed below, λD (E, E ) is the diffusion len energy E ' to energy E: ( ' δ−1 ' δ−1 − (E /GeV) (E/GeV) Especially at energies justλbelow the DM mass M, positrons are dominantly produced close to 2 = 4K τ 0 E D 2009年4月8日星期三

Plot of HaloM.Function Cirelli et al. / Nuclear Physics B 800 (200

[Cirelli, Franceschinni, Strumia, NPB 800 (2008) 204]

Peak at r! ∼ 8.5 kpc

Positron created near home

Thin diffusion zone (L ~ 1 kpc)


Particle Physics Parameters

Dark Matter Mass --

MDM

Annihilation Cross Section -Decay Lifetime --

τDM

χ (σanni v, τDM , MDM ) = 2

! (theoryi − datai )2 i


< σanni v >

2 σi

TABLE I: Fitting with PAMELA data only. When we fix τdm = 1040 s, the contribution from decaying is negligible. Similarly, when we fix !σv" = 10−40 cm3 s−1 the contribution from annihilation is negligible. The “mono” means monochromatic spectrum, whereas “var” means varying spectrum for the decaying contribution. Prop.

τdm

model

(s)

(cm3 s−1 )

(GeV)

1040 (fixed)

10−23

0.45 · 1027 0.55 · 1027

NFW M2

NFW Med

NFW M1


χ2 /# d.o.f.

Comments

300 (fixed)

79.6/7

mono

10−40 (fixed)

212

18.1/6

var

0.77 · 10−24

200 (fixed)

10.1/6

var

0.15 · 10−23

250 (fixed)

3.0/7

mono

fixed

0.21 · 10−23

300 (fixed)

2.9/7

mono

at

0.37 · 10−23

400 (fixed)

2.8/7

mono

1040

0.83 · 10−23

600 (fixed)

2.7/7

mono

0.15 · 10−22

800 (fixed)

2.7/7

mono

0.22 · 10−22

1000 (fixed)

2.6/7

mono

fixed

0.59 · 10−24

200 (fixed)

3.6/7

mono

at

0.36 · 10−23

500 (fixed)

2.2/7

mono

1040

0.70 · 10−23

700 (fixed)

2.0/7

mono

0.14 · 10−22

1000 (fixed)

1.9/7

mono

B !σv"

Mdm

not show such effects. The less diffuse propagation model “M2” fits better than the “Med” and “M1” models. TABLE II: Fitting with ATIC data only. Other details are the same as Table I. Prop.

τdm

B!σv"

Mdm

model

(s)

(cm3 s−1 )

(GeV)

1040 (fixed)

0.51 · 10−23

0.36 · 1027

NFW M2

NFW Med

NFW M1


χ2 /# d.o.f.

Comments

536

39.5/19

mono

10−40 (fixed)

1072

39.5/19

mono

1.0 · 1026

10−40 (fixed)

3190

25.3/19

var

0.65 · 1027

0.80 · 10−23

744

23.7/18

var

1040 (fixed)

0.78 · 10−23

745

27.4/19

mono

0.32 · 1027

10−40 (fixed)

1490

27.4/19

mono

0.25 · 1029

0.78 · 10−23

745

27.4/18

var

1040 (fixed)

0.68 · 10−23

740

34.2/19

mono

0.36 · 1027

10−40 (fixed)

1470

34.2/19

mono

0.11 · 1027

10−40 (fixed)

4420

37.6/19

var

9

TABLE III: Fitting with PAMELA and ATIC data. Other details are the same as previous tables.

τdm

model

(s)

(cm3 s−1 )

(GeV)

1040 (fixed)

0.31 · 10−23

1040 (fixed)

NFW M2

NFW Med

NFW M1


B !σv"

Prop.

χ2 /# d.o.f.

Comments

400 (fixed)

297/28

mono

0.47 · 10−23

500 (fixed)

281/28

mono

1040 (fixed)

0.70 · 10−23

600 (fixed)

269/28

mono

1040 (fixed)

0.94 · 10−23

700 (fixed)

275/28

mono

1040 (fixed)

0.12 · 10−22

800 (fixed)

284/28

mono

1040 (fixed)

0.16 · 10−22

1000 (fixed)

343/28

mono

1040 (fixed)

0.55 · 10−23

535

267/27

mono

1040 (fixed)

0.27 · 10−23

400 (fixed)

96.8/28

mono

1040 (fixed)

0.42 · 10−23

500 (fixed)

74.3/28

mono

1040 (fixed)

0.60 · 10−23

600 (fixed)

59.1/28

mono

1040 (fixed)

0.80 · 10−23

700 (fixed)

59.3/28

mono

1040 (fixed)

0.10 · 10−22

800 (fixed)

63.2/28

mono

1040 (fixed)

0.15 · 10−22

1000 (fixed)

102/28

mono

0.16 · 1028

0.62 · 10−23

745

55.9/26

mono

0.13 · 1028

0.54 · 10−23

643

41.9/26

var

1040 (fixed)

0.34 · 10−23

500 (fixed)

55.3/28

mono

1040 (fixed)

0.48 · 10−23

600 (fixed)

41.1/28

mono

1040 (fixed)

0.64 · 10−23

700 (fixed)

38.8/28

mono

1040 (fixed)

0.83 · 10−23

800 (fixed)

40.0/28

mono

1040 (fixed)

0.99 · 10−23

900 (fixed)

66.9/28

mono

1040 (fixed)

0.12 · 10−22

1000 (fixed)

66.7/28

mono

0.38 · 1028

0.61 · 10−23

745

38.5/26

mono

0.33 · 1029

0.54 · 10−23

643

38.6/26

var

Mdm

Not Good!

NFW Med

3

2

-2 -1

-1

E Φ (GeV cm s sr )

ATIC data Decay+Anni Anni. only Decay only Bkgd

0.01 Mdm = 643 GeV τdm = 1.3 1027 s σv = 5.4 10-24 cm3 s-1

100

1000 Ee (GeV)

FIG. 1: The spectrum for the ATIC data. The dark matter mass is taken to be 643 GeV, with a monochromatic spectrum for DM DM → e+ e− annihilation and a soft decaying spectrum of 2 for DM → e+ e− X decay. dN/dE ∼ E(1 − 2E/Mdm )3 /Mdm 2009年4月8日星期三

1

Positron fraction

PAMELA data Decay+Anni Anni. only Decay only Bkgd

NFW Med

Mdm = 643 GeV 27

τdm = 1.3 10

-24

σv = 5.4 10

s 3 -1

cm s

0.1

0.01 1

10

100

1000

Ee+ (GeV)

FIG. 2: The positron fraction for the PAMELA data. The dark matter mass is taken to be 643 GeV, with a monochromatic spectrum for DM DM → e+ e− annihilation and a soft decaying 2 for DM → e+ e− X decay. spectrum of dN/dE ∼ E(1 − 2E/Mdm )3 /Mdm 2009年4月8日星期三

ATIC energy spectrum 0.100

E3 ! !GeV2 cm"2 s"1 sr"1 "

Moore Med del “Med” in the figures. From these two plots it is interesting 0.050

nd ATIC data can be fitted simultaneously by the double-action to justify 0.020this fact more quantitatively using the technique of 0.010

LA data 0.005 ay+Anni nni. only cay only 0.002 Bkgd

0.001 10


NFW Med

Mdm = 643 GeV 27

τdm = 1.3 10

-24

σv = 5.4 10

20

s 3 -1

cm s

50

100 200 Ee !GeV"

500

1000

2000

o justify this fact more quantitatively using the technique of

PAMELA Positron Fraction 1.00

Positron Fraction

A data 0.50 +Anni . only y only Bkgd 0.20

NFW Med

Mdm = 643 GeV

Moore Med

τdm = 1.3 1027 s σv = 5.4 10-24 cm3 s-1

0.10 0.05

0.02

1

5

10

100 Ee+ (GeV)


10 Ee!

50 100 !GeV"1000

500

1000

ATIC energy spectrum 0.100

IsoT Med

E3 ! !GeV2 cm"2 s"1 sr"1 "

model 0.050 “Med” in the figures. From these two plots it is interesting

A and ATIC data can be fitted simultaneously by the double-action

ant to 0.020 justify this fact more quantitatively using the technique of 0.010

0.005 MELA data Decay+Anni Anni. only Decay only 0.002 Bkgd 0.001 10


NFW Med

Mdm = 643 GeV 27

τdm = 1.3 10

-24

σv = 5.4 10

20

50

s 3 -1

cm s

100 200 Ee !GeV"

500

1000

2000

o justify this fact more quantitatively using the technique of

PAMELA Positron Fraction 1.00

IsoT Med

Positron Fraction

A data 0.50 +Anni . only y only Bkgd0.20

NFW Med

Mdm = 643 GeV τdm = 1.3 1027 s σv = 5.4 10-24 cm3 s-1

0.10 0.05

0.02

1

5

10

100 Ee+ (GeV)


10 Ee!

50 100 !GeV"1000

500

1000

2

with the rate givenThe by annihilate3(a))and decay. model for canNeutrinos be described by and the following in A Leptonphilic Model Masses

an [13, 14]

σvrel

Lint =

4 g1e

1 = 64π s −

(MN2

! 1

−1

"

Double Action Dark Matter

dx

s2 (1 + βN x)2 s2 (1 − βN x)2 + s 2 2 2 4(MN − MS2 − 2 (1 − βN x)) 4(MN2 − MS22 − 2s (1 + βN x))2

[Cheung and Seto, PRD69, 113009 (2004)] # 2

MS22

2MN s s 2 2 − 2s (1 − β+ (1 + βN x)) N x))(MN − MS2 − + 2

T− fαβ Lα Ciτ2 Lβ S1 1/2

where βN = (1 − 4MN2 /s)

+ g1α N1 S2 $αR +

+ g2α N2 S2 $αR

(18)

+ H.c.

. As βN → 0, the above annihilation rate vanishes. This is

+ M N T CN + M N T CN − V (S , S )

expected for theNannihilation rate 1 for identical N2 Majorana 2fermions is P-wave 1 2suppressed [19]. 1 1 2 √ When the center of mass energy s is slightly above the threshold of 2MN , the annihilation

α,β

energy spectrum is almost a monochromatic one. A naive order estimate and lαRelectron/positron are the lepton doublet and+singlet respectively with α, β den −

V (S , S ) ⊃ !S S + H.c.

4 gives σvrel ∼ g1e 10−29 cm3 s1−1 with 2 the current 1relative 2 velocity vvel ∼ 10−3 . Therefore, a

boost factortwo of order O(10 ) is needed to neutrinos, fit the data in thisC model. dices, N1,2largeare the right-handed is the charge-conjugat 5

d V (S1 , S2 ) is the scalar potential for e−

N

11 + S2

N the

two complex + S2

! + S1

+

N

fields S1 an

e+

νµ

e (a)

− e scalar

(b)

FIG. 3: Feynman diagrams for (a) annihilation N N → e− e+ and (b) decay N → e− e+ νµ . 2009年4月8日星期三

In Refs. [13, 14], the N is assumed stable by imposing a Z2 parity. Here we introduce a

Σ#NN " e# e$ $%v & #1031 %g 1 e 4 $ !cm3 %s$1 "

Annihilation Cross Section for NN " e# e$

vrel !c ! 2! ΒN ! 10#3

5.0

M S2 $ M N % ∆

2.0 ∆ $ #100

P-wave suppressed!

1.0 ∆$0 0.5

∆ $ 100 ∆ $ 300

∆ $ 200

0.2

200


400

600 MN !GeV"

800

1000

4 gives σvrel ∼ g1e 10−29 cm3 s−1 with the current relative velocity vvel ∼ 10−3 . Therefore, a

Rough Estimate of Lifetime of N

large boost factor of order O(105 ) is needed to fit the data in this model.

N N

ΓN ∼

e−

2 2 2 g+1e f12 ! MN /MS42

S2

e

g1e + S2

! + S1

+

e

f12

Typical N values:

+

νµ

e

−1 (a)

g1e ∼ 10

−

, f12 ∼ 10

−2

(b)

, MN ∼ MS2 ∼ TeV

FIG. 3: Feynman diagrams for (a) annihilation N N → e− e+ and (b) decay N → e− e+ νµ .

!

2

"2

eV ∼ 10 stable by imposing a Z2 parity. Here we introduce a In Refs. [13, 14], the NτN is assumed " 26

small violation of this parity by adding a term %S1+ S2− + H.c. to the scalar potential, where % ∼ (1 eV)2 , which is of the order of the square of neutrino mass. The decay of N can then

If ! is of order of neutrino mass square, 12 τN is of order 1026 second!


energy spectrum that we have used in our analysis given in the previous section. Note that the exact form is not crucial in the fits as long as the spectrum is soft. In this model, the

Energy Spectrum vs.leptonic. oneIt used inenhancement fittings) decay and annihilation of the dark(Exact matter are pure will not give to the p¯ flux. 4.5

+

-

N→e e ν fitted spectrum 80 x (1-2x)3

4 3.5

dN/dx

3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

x = E/Mdm

FIG. 4: Normalized energy spectrum 1/ΓdΓ/dx with x = E/Mdm for the decay N → e− e+ νµ and the approximation 80x(1 − 2x)3 .


Summary Dark matter of ~ 640 GeV with a monochromatic annihilation spectrum and a soft decaying spectrum is the best simultaneous fit to both PAMELA and ATIC Typically, one finds σanni v ∼ 10−23 cm3 /s, τDM ∼ 1027 s, and MDM ∼ 700 GeV Large boost factor and/or NP Sommerfeld enhancement

If the ATIC data structure can be sustained in the future by PAMELA, construct more realistic dark matter models that can lead to the above result may be interesting DAMA, INTEGRAL, WMAP haze ... Unified Theory of DM? Primary source from nearby pulsars Galactic gamma ray spectrum from FERMI satellite Good direction for students and young postdocs 2009年4月8日星期三

1,000 Ee3.0dN/dEe (m−2 s−1 sr−1 GeV2)

1,200 1,000

Events

800 600

100

400 200 0 0

20 40 60 80 100 120 BGO1 + BGO2 + FBGO7 + FBGO8

140

Figure 1 | Separation of electrons from protons in the ATIC instrument. Candidate electron events (162,000) with energy over 50 GeV are plotted as a histogram with the horizontal axis showing the sum of the ‘weighted energy fraction’ (F values as defined below) in the last two BGO layers and the shower width (root mean squared, r.m.s.) in the first two layers. The shower width is calculated as 2

hr:m:s:i ~

n X i~1

2

Ei ðXi { Xc Þ =

n X

Ei

i~1

where Xc is the coordinate of the energy centre, Xi is the coordinate of the centre of the ith crystal and Ei is the energy deposited in the ith crystal. The F value is calculated as Fn ~ðEn =SumÞ hr:m:s:i2 where En is the energy deposit in BGO layer n, Sum is the total energy deposit in all BGO layers and Ær.m.s.æ refers to layer n (ref. 12). Each event is also fitted to an electromagnetic cascade profile to estimate the starting point and the depth of the cascade maximum. An event is accepted if the cascade starts above the first BGO layer, which eliminates many protons (,75%) but passes most electrons (,90%). Next a diagonal cut in r.m.s. and F is determined for each energy bin and used to isolate the electrons. This removes most of the protons (2 in 104 remain) and retains 84% of the electrons12. The selected electrons are shown as the dotted histogram. 2009年4月8日星期三

10

10

Figure 2 | ATIC-1 and ATIC-2 s agreement with each other. Th E3) at flight altitude is shown fo circles). The errors are one stan from McMurdo, Antarctica, and was a test flight in 2000–01 and t 0.61 m2 sr days. ATIC-2 was a sc 2.47 m2 sr days. To eliminate ed angle to be less than ,37u (cos h and eliminate events in the out limits, the electron detection ef independent of energy. The effe function of particle energy cons reconstruction efficiency and th acceptance of the instrument in 0.15 m2 sr for E . 60 GeV. Abov were observed, with the highest background is also shown in th combination of unresolved pro secondary electrons produced i instrument. ATIC becomes bac several teraelectronvolts.

5

Dark Matter Candidates

e+

_ p

γ

Figure 1. Illustration of the volumes in the solar neigbourhood entering the calculation of the average boost factor in the dark matter halo. Here we have in mind a dark matter particle of mass around 100 GeV annihilating into, from left to right, positrons, antiprotons, and gamma-rays. The difference in size for antiprotons and positrons depends on the different energy loss properties, as positrons at these energies radiate through synchrotron and inverse Compton emission much faster than do antiprotons.

Bergstr¨om, arXiv:0903.4849

the influence of baryons could give an enhanced density through adiabatic contraction processes). 2009年4月8日星期三