Cosmic Antiproton and Gamma-Ray Constraints on Effective Interaction of the Dark matter Authors: Kingman Cheung, Po-Yan Tseng, Tzu-Chiang Yuan Physics Department, NTHU Physics Division, NCTS Institute of Physics, Academia Sinica arxiv:1011.2310[hep-ph],jcap01(2011)004 arxiv:1104.5329v1 2011/05/03, NCTS JC
Introduction The present mass density of cold DM by WMAP collaboration is: Ω = 0.1099 ± 0.0062 h 2 CDM ΩCDM is the mass density of CDM normalized by the critical density, is the Hubble constant in units of. 100 km / s / Mpc If the DM was produced thermally in the early Universe, the DM annihilation cross section is about the order of Weak interaction. 2 0.1pb Ω X h, σ v 0.91pb σ v X σ v σ v is the DM particle, is the annihilation cross section, is relative velocity, is the thermal average. ( 1) h ( 2)
We know the gravitational nature of DM, but we know a little about its particle nature. In this work, we use the effective interaction to describe the interactions between DM and SM particles. DM exists in a hidden sector and interact with SM particle via a heavy mediator. For example, the interactions between a fermionic DM and the light quarks q (u,d,s,c,b) can be described by χ Γ χ qγ ' q, where Γ Γ = σ σ γ γ γ γ γ µν µν 5 µ µ 5 5, ',,,,,1 There have been some works to constrain these effective interactions by present and future collider experiments, and gammaray experiments. χ ( )( )
In this work, WMAP experiment can give us the lower bound of DM annihilation cross section σ v. Hence, it give us the lower bound of the strength of DM effective interactions. Otherwise, there will be too many DM in our universe. We also use the cosmic antiproton flux from PAMELA experiment to give us the upper bound of the strength of DM effective interactions.
Effective Interactions The effective interactions of Dirac fermion DM and light quarks via a (axial) vector-boson or tersor-type exchange are described by the dimension 6 operator: C Li = 1 6 = Oi = 1 6 = 2 ( χγ1χ )( qγ 2q) Λ i µ µ 5 µν µν 5 where Γ with, and C is coupling 1,2 = γ, γ γ, σ, σ γ σ µν i ( γ µ γ ν γ ν γ µ ) / 2 constant, is the cutoff scale. O ( 1 ) Λ i Dirac fermion DM via (pseudo) scalar-boson-type exchange: Cmq Li = 7 10 = Oi = 7 10 = χγ 3 1χ qγ 2q Λi 5 where Γ 1,2 = 1 or iγ, mq are the light quarks mass. ( )( )
Dirac DM couples to gluon field: Cα ( 2mχ ) Li 11 12 Oi 11 12 3 4 ( ) µν = = = = χγχ G G Λ s a a µν i ( ) ( ) Cα 2mχ µν Li = 13 14 = Oi = 13 14 = χγχ G G 3 4Λi α G G ρσ µν = ε µνρσ / 2 Complex scalar DM via vector boson exchange: C µ Li = 15,16 = Oi = 15,16 = 2 ( χ µ χ )( qγ Γq) Λ s a * a µν 5 where Γ = 1 or iγ, s are the strong coupling constant at scale, * 2m χ. 5 where Γ = 1 or γ and χ χ = χ ( χ ) ( χ ) χ µ µ µ. Complex scalar DM via scalar boson exchange: Cmq Li = 17,18 = Oi = 17,18 = 2 ( χ χ )( qγq ) Λ 5 where Γ =1 or iγ. i i
Complex scalar DM couple to gluon field: ( 2m ) ( χ χ ) Cα L O G G s χ aµν a i= 19 = i= 19 = 3 µν 4Λi ( 2m ) ( χ χ ) icα L O G G s χ aµν * a i= 20 = i= 20 = 3 µν 4Λi
Annihilation Cross Sections Around the Freeze-Out The WMAP results give us the DM annihilation cross section σ v in the early universe is: σ v 0.91pb If there were some other nonthermal source of DM in the early universe, the constrain of WMAP is revised into: σ v > 0.91pb O = We use the effective interaction i 1 20 to calculate σ v in the early universe and give upper limit of cutoff. Λ i χχ or χ χ qq or gg v 0.3c We assume DM velocity at around freeze-out time. We also included the DM annihilation into light quarks and top quark.
( m χ ) In the, Λ plane, the solid lines are the contours with σ v. The regions below the lines are the allowed values for. Λ 0.91pb O1 6 Figure 1: operators.
Figure 2: operator O7 14. The shaded area is Λ < m χ 2 π where the effective theory approach is not trustworthy.
Figure 3: operator O15 20.
Anitproton Flux In the present universe, the DM in our Galaxy halo will annihilate into quarks or gluons via the effective operators O i = 1 20, and quarks and gluons will fragment into cosmic antiproton flux. Operator χχ + Fragment qq p X In this calculation, we only consider the DM annihilate into light quarks (u,d,s,c,b).
The antiprotons produced from the Galaxy halo need propagate to our earth. There are magnetic field in the Galaxy will change the energy spectra of antiproton. This phenomena is described by the diffusion equation: ψ 2 1 i = Q( r, p) + ( Dxx ψ ) + p D ψ [ pψ ] pp 2 t p p p p where ( r ψ = ψ, p, t ) is the density of anti-p, D xx is the spatial diffusion coefficient, D pp is the diffusion coefficient in momentum space, Q is the i source term, is the momentum loss rate. p dp dt The source term from the DM annihilation is: ρ dn CDM p Qann = η σ v p M CDM dtp where η = 1 2 ( 1 4) for (non)-identical initial state, and Tp is the kinetic energy of antiproton. 2
We calculate the antiproton energy spectrum dn dt p p from DM annihilation and put the source term Q into the computer program GALPROP. GALPROP can solve the diffusion equation and output the cosmic antiproton energy spectrum in earth. We compare PAMELA antiproton data (data point above 4GeV) with 2 the GALPROP output and calculate the χ. 3σ = 9 2 2 We find the ( χ χ bkgd ) limit on the cutoff of each effective operators O i = 1 20 for different DM mass m 50,100,200,400GeV. χ = Λ i The PAMELA antiproton data give the lower limit of cutoff. Λ i
1 = Λ ( )( ) µ Figure 4: operator O χγ χ qγ µ q,dm mass=200gev. 1 2
Using WMAP data( DM thermal relic density ) and PAMELA data ( antiproton flux ) can give a valid range of cutoff Λ i for each effective operator O i = 1 20. For example, Dirac DM with (axial) vector interactions require 1.6TeV < Λ 1,3 < 3TeV for mχ = 200GeV. The best limit is from the Dirac DM with tensor interactions O 5,6 have 1.9TeV < Λ < 3.6TeV for m = 200GeV. 5,6 χ O 1,3 O O7 14 2,4 have velocity suppression. (Dirac DM with scalar-boson exchange) give weak limit because of the m q in the coupling constant. 11 14 (gluonic interaction) give weak limit because of the in the coupling constant. O 15,16 (complex scalar DM with vectorboson exchange) give weak limit because of the derivative bring down a factor of momentum. 1 O α s 10
Table 3: m 50,100, 200, 400GeV χ =
Velocity Dependence in the Nonrelativistic Limit The current velocity of DM near the sun is 3 about v 300 km / s 10 c. For example, the annihilation cross section of operator (Dirac DM with a scalar boson exchange) would be suppress by the 2 factor v. Operator O1 to O6, the relevant part of the annihilation amplitude of the Dirac DM is p Γ p ψ ( ) ψ ( ) 2 1 O 7 µ µ 5 µν µν 5,where Γ = γ, γ γ, σ, σ γ.
In the nonrelativistic limit, the Dirac spinor of DM and antiparticle are: 1 0 ψ = ξ, ψ η ( ε, 1) γ ε =,where ε = O( v / c). In Dirac representation, the gamma matrices are given by 0 I 0 0 i σ i 5 0 I γ =, γ =, γ = 0 I σ i 0 I 0 ( ), where σ i i = 1, 2,3 are Pauli matrices. µ In nonrelativistic limit, the space-like and time-like part of ψγ ψ are 0 ψγ ψ + 2εη ξ 2 ( 1+ ) i + ψγ ψ ε η σ iξ The time-like part is suppressed by v / c, but the space-like part are not.
In nonrelativistic limit, the space-like and time-like part of are ψγ 0 γ 5 ψ 1 ε 2 + + η ξ i 5 ψγ γ ψ ( ) + 2εη σ ξ,where space-like part are suppressed by v / c. i µ 5 ψγ γ ψ χχ qq In the calculation of annihilation cross section, we can consider the space-like and time-like parts of ψγψ separately, when it is squared, traced, and contracted with the trace of the light quark leg 5 γ µ γ µ γ. q q or q q 5 After being squared and traced, the time-like part of γ µ γ µ γ gives a value close to zero, and the space-like part of it gives a 2 value of order. m χ q q or q q
µ 5 Therefore, ψγ ψ multiplied to qγ µ q or qγ µ γ q will not be µ 5 5 suppressed, while ψγ γ ψ multiplied to qγ µ q or qγ µ γ q will be suppressed. Hence, the limits on O1 and 3 are stronger than those on O and O4. The O5 and O6 have unsuppressed components in µν = 0i. O 2 In nonrelativistic limit, the DM part of operators O to O are + ψψ εη ξ 5 ψγ ψ + η ξ O7 14 Therefore, the limits on O8 and O10 are stronger than those on and O 9. The limits on O12 and 14 are stronger than those on and. O 13 O 7 O O11
In O15 to O20, only O 15,16 are suppressed by v / c, others are not. Because there is the differential operator in, and this µ O15,16 will bring down a vertex factor. p p Therefore in, when χ O 15,16 µ χ multiplied to the quark µ µ 5 leg qγ q or qγ γ q, they will be suppressed by v / c. 0 i p µ m χ v / c
Fermi-LAT Gamma-Ray Data We use the Gamma-ray spectrum data from the mid-latitude area of 0 0 0 0 our galaxy skymap ( 10 b 20,0 l 360 ). We have clearly understanding with the material and point sources. We can calculate the Gamma ray background more accurately. There are more DM accumulate in the Galaxy Center. However, Galaxy Center may has supermassive black hole, and some unknown source.
Background The Background Gamma ray mainly comes from the interaction between high energy cosmic ray particle with the interstellar medium and radiation field of charged particle. 0 π 1. decay: The high energy cosmic ray ( proton ) collide with the 0 nucleons of the interstellar medium. Then, the π are produced 0 and most of π will decay into two photons. 2. Inverse Compton (IC) : The high energy collide with the photons of interstellar medium or CMB photons. 3. Bremsstrahlung: The high energy are deflected by the Coulomb field of the interstellar medium. e ± e ±
4. Extragalactic background (EGB) : It is expected to be isotropic. The sources of EGB come from unresolved point sources, diffuse emission of large scale structure formation, etc. Because there are large uncertainty in the EGB, it is determined lastly among the other components of Gamma-ray background. The Fermi-LAT measured the Gamma-ray and fitted the EGB by 2 dφ E E = A de 0.281GeV,where γ = δ 2 = 2.41± 0.05 is power-law index and + A 0.95 0.18 10. = GeV cm s sr ( ) 6 2 1 1 0.17 δ
We use Galprop (web version), which is a computer program and can calculate the cosmic ray flux, to do the theoretical estimation of the background Gamma-ray flux. Galprop can give the estimations of background 1, 2, and 3. 0 π Background 1 ( decay ) are mostly determined by the cosmic ray proton flux. Background 2 ( IC ) and 3 ( bremss ) are mostly determined by cosmic ray electron flux. Because Fermi-LAT and AMS experiments already measured the flux of cosmic electron and proton. Then, the Gamma-ray flux 0 background of π decay, IC, and bremss are almost determined.
We added background EGB into background from Galprop and fitted the Fermi-LAT mid-latitude data. Then, we obtained the flux of EGB is 0.36 2 dφ 6 E 2 1 1 ( 0.99 10 ) E = GeV cm s sr de 0.281GeV 2, where the power-law index is γ = 2.36. The χ = 2.435 for 7 d.o.f in our fitting. It is close to the EGB from Fermi-LAT paper. We can see that the theoretical spectrum fits with the Fermi-LAT data very well.
Gamma-ray flux bkgd:
Dark Matter Annihilation Via these effective operators, the DM in the galaxy halo will annihilate with each other and produce photons by the process:, where q = u, d, c, s, b. χ χ π + γ + 0 qq or gg X 2 X DM and anti-dm annihilate into quark and anti-quark ( or gluon pair ), 0 then quark and anti-quark will fragment into many π. About 99% 0 of will further decays into 2γ. π
We used two methods to obtain the photon energy spectrum of the DM annihilation process. dn de γ + * 1. Using the process e e Z qq in Pythia to simulate the process 0 χ χ qq π + X 2γ + X + * Using the process e e Z qq with the initial radiation turned off in Pythia. The e + e annihilate into a Z boson, and the Z boson go into quark and anti-quark. The center mass energy of e + e is 2m χ. Then, we extracted the energy spectrum of the final state photon. 0 π Most photons come from decay of, which is produced by the fragmentation of quark and anti-quark. Very small photons come from bremsstralung of the quark leg.
2. Considering the process: χ χ qq Using the fragmentation function code 0 ( B.A.Kniehl,G.Kramer,B.Potter ) of q, q, g into π, then we have 0 energy spectrum of π in the process: 0 χ χ qq. or gg π + X 0 0 Boosting the γ energy spectrum dn de γ ( π γγ ) into π 0 0 frame and convolute dn de γ ( π γγ ) with π spectrum. We can have dn de γ in this process χ χ π + γ + 0 qq or gg X 2 X The photon energy spectra dn de γ from the above two methods are consensus with each other.
DM mass=200gev. Photon energy spectrum from Pythia and Fragmentation code.
With the effective operators ( O11 to O14 and O19, O20 ) that DM will annihilate into gluon pair, we used the 2 nd method to extract the photon energy spectrum. The cosmic Gamma-ray flux can be calculate via this equation: Φ = σ v dn 1 π 2 2 deγ 4 mχ line of sight ds 2 ρ ( s, ψ ), where ρ m χ is the DM number density, σ v is the annihilation cross section, and dn de γ is the photon energy spectrum per annihilation. s is the parameter along the line of sight, and ψ is the angle from the direction of the Galaxy Center. The factor 2 in the denominator comes from the particle and anitparticle annihilation.
Adding the Gamma-ray contribution of DM annihilation into the background from Galprop and fitted EGB. We compared the theoretical Gamma-ray flux with Fermi-LAT data and calculate 2 the. χ = 9 2 2 2 We choose 3σ deviation ( χ χ χ bkgd ) from the Fermi-LAT data and give the lower limit of the cutoff Λ i as well as the upper limit of DM annihilation cross section σv for each operators.
Figure 1. For O, and m = 200GeV. Cutoff Λ = 1.5TeV 1 χ. The pick of spectrum from DM annihilation is about 0.1m χ.
Constraints from Fermi-LAT The lower limit of cutoff Λ O 1 TeV for the unsuppressed operators ( O1, O3, O5, O6 ) for m = 50, and the lower χ 500GeV limit of cutoff Λ O( 0.01 0.1) TeV for the suppressed operators by small velocity, light quark mass, or strong coupling constant. ( )
Table 2.Lower limit of cutoff Λ i from Fermi-LAT Gamma-ray
Antiproton and Gamma-ray Flux Comparison Once we have the lower limit of cutoff Λ for each DM mass, we can calculate the upper limit of σ v ( χ χ qq ) DM annihilation cross section as a function of DM mass m χ. ( ) The upper limit of σ v χ χ qq are approximately independent of the operators with quark and anti-quark final state. These operators produce similar shape of the Gamma-ray spectra from DM annihilation.
Figure 4.
In figure 4, the x-axis is at the value of σ v 0.91pb 3 10 cm s which gives the correct DM relic density, if DM was produced thermally in the early universe. 26 3 1 The two curves in figure 4 are the upper limit of the DM annihilation cross section σ v ( χ χ q q ) in the present universe. These two curves are obtained from the Fermi-LAT Gamma-ray data and PAMELA antiproton data, respectively. For lighter DM mass ( 50-300GeV ), PAMELA antiproton data can give more severe constraints on effective operators. For heaver DM mass ( 300-500GeV ), Fermi-LAT Gamma-ray data gives stronger constraints than PAMELA data.
Direct Detection Experiments Constraint CDMS and XENON100 give strong constraints on the DM spinindependent ( SI ) cross section. SI 44 2 The upper limit from CDMS is: σ χ N 3.8 10 cm at m χ = 70GeV The upper limit from XENON100 is: SI 44 2 σ χ N 0.7 10 cm at m χ = 50GeV For Dirac DM operators, only operators O1 and O7 ( Dirac DM with vector or scalar boson exchange) will give the DM spinindependent cross section.
( )( ) 3 For operator O = m χχ qq Λ, the SI cross section is: 7 q 7 2 µ SI χ N N 2 σ χ N = Gs π, where µ χ N is the reduced mass of the DM and nucleon, and N : N mq m N ftq light quarks Gs = N qq N, N qq N 3 = 2 N q Λ mq f : 27 Tg heavy quarks f N N f Tg Using the values of Tq and Tg in the DarkSUSY and taking SI average between the neutron and proton for. We have SI σ χ N m πλ 4 N 6 7 ( 0.3769) σ χ N SI 44 2 Using the limit σ χ N < 10 cm for m 50 200GeV, we have This limit for cutoff is better than that from Gamma-ray constraints ( Λ7 O ( 10) GeV ). χ Λ 7 > 330GeV Λ 7 2
( )( ) 2 µ For operator O1 = χγ χ qγ µ q Λ1, the SI cross section is: 2 µ SI χ N 2 σ χ N = bn 256π 3 V V V 2, where bn ( α 2 u + αd ), and α u, d = 1 Λ1 are the coefficients in front of the operator O 1. We have 2 SI 9 mn σ χ N 4 256π Λ SI 44 2 Using the limit σ χ N < 10 cm, we obtain Λ 1 > 4.4TeV It is about 2-3 times larger than the limit of cutoff from Gamma-ray constraints ( Λ O TeV ). ( ) 1 1 1 Λ 1
Discussion and Conclusions We use the effective interaction approach to study the interactions between DM and light quarks. Using WMAP data (CDM relic density) give the upper limit of cutoff and PAMELA antiproton data (Fermi-LAT Gamma-ray data) give the lower limit of cutoff, we can have the valid regions for each cutoff. Λ i The lower limit of cutoff is about 1-2 TeV, for the unsuppressed operators. It is about 0.01-0.1 TeV, for the suppressed operators.
Only the operators O1 (vector exchange) and O7 (scalar exchange) for the Dirac DM contribute to the spin-independent cross section. CDMS and XENON100 give very strong constraints for these two operators. These constrains for effective operators of DM and SM light quarks give useful information for collider searches and direct detection.
Thank You!
Table 1: