Detecting interactions between dark matter and photons at ...yzhxxzxy.github.io/slides/1308_DM_pho_eecld.pdf · Motivations. . . . . Sensitivity. . Beam polarization. . . . . Unitarity

. . . .Motivations

. . . . .Sensitivity

. .Beam polarization

. . . . .Unitarity bounds

. . .Conclusions

. . . . . . .Backups

Detecting interactions between dark matter andphotons at high energy e+e− colliders

Zhao-Huan YU (余钊焕)Institute of High Energy Physics, CAS

with Qi-Shu YAN and Peng-Fei YIN

arXiv:1307.5740

Dalian, August 23, 2013

Zhao-Huan YU (IHEP) Detecting interactions between DM and photons at e+e− colliders Aug 2013 1 / 27

. . . .Motivations




. . .Conclusions

. . . . . . .Backups

DM-photon interaction

In general, dark matter (DM) are not luminous⇓

DM particles (χ) should not have electric chargeand not directly couple to photons

However, DM particles may couple to photons via loop diagrams

χ

χ

γ

γ For nonrelativistic DM particles, thephotons produced in χχ → γγ would bemono-energetic

⇓A γ-ray line at energy ∼ mχ

(“smoking gun” for DM particles)


. . . .Motivations




. . .Conclusions

. . . . . . .Backups





χ

χ

γ

γ

For nonrelativistic DM particles, thephotons produced in χχ → γγ would bemono-energetic




. . . .Motivations




. . .Conclusions

. . . . . . .Backups





χ

χ

γ

γ For nonrelativistic DM particles, thephotons produced in χχ → γγ would bemono-energetic




. . . .Motivations




. . .Conclusions

. . . . . . .Backups

A γ-ray line from the Galactic center region?

0

5

10

15

20

25

30

35

40

Counts

p-value=0.85, χ 2red=14.3/21

Signal counts: 53.4 (4.26σ) 80.5 - 208.5 GeV

Reg3 (ULTRACLEAN), Eγ =129.6 GeV

100 150 200

E [GeV]

-10

0

10

Counts - Model

Weniger, 1204.2797

Residual map

180 90 0 -90 -18000

-90

-45

0

45

90

00

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

keV cm

-2 s-1 sr -1

Su & Finkbeiner, 1206.1616

Using the 3.7-year Fermi-LAT γ-ray data, several analyses showed thatthere might be evidence of a monochromatic γ-ray line at energy∼ 130 GeV, originating from the Galactic center region (about 3− 4σ).It may be due to DM annihilation with

σannv∼ 10−27 cm3 s−1.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

(GeV)χm10 210

)-1 s3

95%

CL

Lim

it (c

mγγ

v>σ<

-3010

-2910

-2810

-2710

-2610

-25103.7 year R41 NFW Profile

Observed Upper LimitExpected LimitExpected 68% ContainmentExpected 95% Containment

Eve

nts

/ 5.0

GeV

0

10

20

30

40

50

60

70 = 133.0 GeVγP7_REP_CLEAN R3 2D E

= 17.8 evtssign

σ = 3.3 locals

= 276.2 evtsbkgn

= 2.76bkgΓ

(c)

Energy (GeV)60 80 100 120 140 160 180 200 220

)σR

esid

. (

-4-2024

Recently, the Fermi-LAT Collaboration has released its official spectralline search in the energy range 5− 300 GeV using 3.7 years of data.They did not find any globally significant lines and set 95% CL upperlimits for DM annihilation cross sections.Their most significant fit occurred at Eγ = 133 GeV and had a localsignificance of 3.3σ, which translates to a global significance of 1.6σ.

Fermi-LAT Collaboration, 1305.5597Zhao-Huan YU (IHEP) Detecting interactions between DM and photons at e+e− colliders Aug 2013 4 / 27

. . . .Motivations




. . .Conclusions

. . . . . . .Backups

DM-photon interaction at e+e− colliders

χ

χ

γ

γ

⇒γ

e−

e+

χ

χ

γ

The coupling between DM particles and photons that induce theannihilation process χχ → γγ can also lead to the process e+e−→ χχγ.Therefore, the possible γ-ray line signal observed by Fermi-LAT may betested at future TeV-scale e+e− colliders.

DM particles escape from the detector⇓

Signature: a monophoton associating with missing energy (γ+ /E)


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Effective operator approachIf DM particles couple to photons via exchanging some mediators whichare sufficiently heavy, the DM-photon coupling can be approximatelydescribed by effective contact operators.

For Dirac fermionic DM, consider OF =1

Λ3 χ iγ5χFµν Fµν :σannvχχ→2γ ≃

4m4χ

πΛ6 , σ(e+e−→ χχγ)∼ s2

Λ6

Fermi γ-ray line signal ⇐⇒ mχ ≃ 130 GeV, Λ∼ 1 TeV

For complex scalar DM, consider OS =1

Λ2χ∗χFµν Fµν :

σannvχχ∗→2γ ≃

2m2χ

πΛ4 , σ(e+e−→ χχ∗γ)∼ s

Λ4

Fermi γ-ray line signal ⇐⇒ mχ ≃ 130 GeV, Λ∼ 3 TeV


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

In the γ+ /E searching channel, the main background is e+e−→ ννγ:

e

Z0

e−

e+

ν

ν

γ

W

e

e−

e+

ν

ν

γ

W

W

e−

e+

ν

γ

ν

· · ·

Minor backgrounds: e+e−→ e+e−γ, e+e−→ τ+τ−γ, · · ·Simulation: FeynRules → MadGraph 5 → PGS 4

ILD-like ECAL energy resolution:∆E

E=

16.6%pE/GeV

⊕ 1.1%

Future e+e− colliders: ps = 250 GeV (“Higgs factory”),ps = 500 GeV (typical ILC), ps = 1 TeV (upgraded ILC & initial CLIC),ps = 3 TeV (ultimate CLIC)


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

dσ

/ θ

γ

(fb

/ d

eg

ree

)

θγ

e+e

− collider, √s = 500 GeV, γ + E ⁄

e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

dσ

/ p

Tγ (

fb /

Ge

V)

pTγ (GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 50 100 150 200 250

dσ

/ m

mis

s

(fb

/ G

eV

)

mmiss = [ ( pe− + pe

+ − pγ )2 ]1/2

(GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 100 200 300 400 500

Z0 pole

Cut 1 (pre-selection):Require a photon with Eγ > 10 GeVand 10 < θγ < 170Veto any other particle

Benchmark point: Λ = 200 GeV, mχ = 100 (50) GeV for fermionic (scalar) DM

Cut 2: Veto 50 GeV< mmiss < 130 GeV

Cut 3: Require 30 < θγ < 150

Cut 4: Require pγT >p

s/10


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

dσ

/ θ

γ

(fb

/ d

eg

ree

)

θγ

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

dσ

/ p

Tγ (

fb /

Ge

V)

pTγ (GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 50 100 150 200 250

dσ

/ m

mis

s

(fb

/ G

eV

)


+ − pγ )2 ]1/2

(GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 100 200 300 400 500

Z0 pole






s/10


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

dσ

/ θ

γ

(fb

/ d

eg

ree

)

θγ

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

dσ

/ p

Tγ (

fb /

Ge

V)

pTγ (GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 50 100 150 200 250

dσ

/ m

mis

s

(fb

/ G

eV

)


+ − pγ )2 ]1/2

(GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 100 200 300 400 500

Z0 pole






s/10


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

dσ

/ θ

γ

(fb

/ d

eg

ree

)

θγ

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0° 20° 40° 60° 80° 100° 120° 140° 160° 180°

dσ

/ p

Tγ (

fb /

Ge

V)

pTγ (GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 50 100 150 200 250

dσ

/ m

mis

s

(fb

/ G

eV

)


+ − pγ )2 ]1/2

(GeV)

e+e


e+e

− → νν−γ

e+e

− → e

+e

−γFermionic DM

Scalar DM

10-2

10-1

100

101

102

0 100 200 300 400 500

Z0 pole






s/10


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Cross sections and signal significances after each cutννγ e+e−γ Fermionic DM Scalar DMσ ( fb) σ ( fb) σ ( fb) S/

pB σ ( fb) S/

pB

Cut 1 2415.2 173.0 646.8 12.7 321.4 6.3Cut 2 2102.5 168.6 646.8 13.6 308.2 6.5Cut 3 1161.1 16.8 538.0 15.7 255.9 7.5Cut 4 254.5 1.9 520.7 32.5 253.9 15.8


Most of the signal events remaine+e−→ ννγ background: reduced by almost an order of magnitudee+e−→ e+e−γ background: only one percent survives

(ps = 500 GeV, 1 fb−1)


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Λ (G

eV

)

mχ (GeV)

Fermionic DM

√s = 250 GeV

√s = 500 GeV

√s = 1 TeV

√s = 3 TeV

102

103

104

5 50 500 10 100 1000

Fermi γ -ray line signal

Λ (G

eV

)

mχ (GeV)

Scalar DM

√s = 250 GeV

√s = 500 GeV

√s = 1 TeV

√s = 3 TeV

102

103

104

5 50 500 10 100 1000


< σ

an

nv >

(c

m3 s

-1)

mχ (GeV)

Fermionic DM

√s = 250 GeV

√s = 500 GeV

√s = 1 TeV

√s = 3 TeV

Fermi 3.7-year upper limit

10-35

10-34

10-33

10-32

10-31

10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

10-22

5 50 500 10 100 1000


< σ

an

nv >

(c

m3 s

-1)

mχ (GeV)

Scalar DM

√s = 250 GeV

√s = 500 GeV

√s = 1 TeV

√s = 3 TeV


10-31

10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

10-22

5 50 500 10 100 1000


Solid lines: 100 fb−1; dot-dashed lines: 1000 fb−1 (S/p

B = 3)ILC luminosity: 240− 570 fb−1/year [ILC TDR, Vol. 1, 1306.6327]


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Beam polarizationFor a process at an e+e− collider with polarized beams,σ(Pe− , Pe+) =

14

(1+ Pe−)(1+ Pe+)σRR+ (1− Pe−)(1− Pe+)σLL

+(1+ Pe−)(1− Pe+)σRL+ (1− Pe−)(1+ Pe+)σLR

Positro

n p

ola

rization P

e+

Electron polarization Pe−

σ (e+e

− → νν

−γ ) [fb]

-1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

8000

6000

4400

3000

2000

1200

750

500

500

Positro

n p

ola

rization P

e+


σ (e+e

− → χχ

−γ ) [fb], fermionic DM

-1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

250

250

400

400

540

540

640

640700

700

800

800

940

940

1100

1100

Positro

n p

ola

rization P

e+


σ (e+e

− → χχ*γ ) [fb], scalar DM

-1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

120

120

200

200

270

270

320

320350

350

400

400

470

470

550

550

(Pe− , Pe+) = (0.8,−0.3) can be achieved at the ILC[ILC technical design report, Vol. 1, 1306.6327]


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

< σ

an

nv >

(c

m3 s

-1)

mχ (GeV)

Fermionic DM

Unpolarized, 200 fb-1


(Pe−, Pe

+) = (0.8, −0.3), 200 fb-1

(Pe−, Pe

+) = (0.8, −0.3), 2000 fb-1


10-33

10-32

10-31

10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

5 50 500 10 100 1000


√s = 1 TeV

< σ

an

nv >

(c

m3 s

-1)

mχ (GeV)

Scalar DM



(Pe−, Pe

+) = (0.8, −0.3), 100 fb-1

(Pe−, Pe

+) = (0.8, −0.3), 1000 fb-1


10-31

10-30

10-29

10-28

10-27

10-26

10-25

10-24

10-23

5 50 500 10 100 1000


√s = 3 TeV

(S/p

B = 3)

Using the polarized beams is roughly equivalent to increasing theintegrated luminosity by an order of magnitude.

For fermionic DM (scalar DM), a data set of 2000 fb−1 (1000 fb−1)would be just sufficient to test the Fermi γ-ray line signal at an e+e−collider with ps = 1 TeV (3 TeV).


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

S-matrix unitarity

For quantum scattering theories,S-matrix unitarity (S†S = 1) ⇔ conservation of probability

A process violate the unitarity in a non-renormalizable effective theory⇓

The theory is invalid for this process⇓

A UV-complete theory may be needed for a full description

The effective operator treatment for DM searches at collidersshould be carefully checked by verifying the S-matrix unitarity.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Unitarity conditions

The 2→ 2 amplitude M(cosθ ) can be expanded as partial waves:

M(cosθ) = 16π∑

j(2 j+ 1)a j Pj(cosθ ), a j =1

32π

∫ 1−1

d cosθ Pj(cosθ)M(cosθ)

Unitarity condition for 2→ 2 elastic scattering:.

......Re ael

j

≤ 1

2, ∀ j

Unitarity condition for 2→ 2 inelastic scattering:.

......

ainelj

≤ 1

2pβ f

, ∀ j

(β f is the velocity of either of the final particles)


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

S†S = 1, S = 1+ iT ⇒ −i(T − T †) = T †T⇓

−i(Mα→β −M∗β→α) =∑γ

∫dΠγM∗β→γMα→γ(2π)4δ(4)(pα − pγ)

⇓2 ImMel(cosθαβ ) =

∫dΠγel

M∗β→γel

Mα→γel(2π)4δ(4)(pα − pγel

)

+∫

dΠγnM∗β→γn

Mα→γn(2π)4δ(4)(pα − pγn

) + other inelastic terms≥ 1

32π2

∫dΩk1

M∗el(cosθβγ)Mel(cosθαγ) +∫

dΠγnM∗β→γn


)

⇓Im ael

j ≥ |aelj |2+ |binel

j |2,

|binelj |2 ≡ 1

64π

∫d cosθαβ Pj(cosθαβ)

∫dΠγn

M∗β→γnMα→γn

(2π)4δ(4)(pα − pγn)

⇓Unitarity condition for any 2→ n inelastic scattering:

.

......binel

j

≤ 1

2, ∀ j


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Unitarity bounds: 2→ 2 vs 2→ 3

Λ

(Ge

V)

mχ (GeV)

Fermionic DM, √s = 3 TeV

102

103

5 50 500 10 100 1000

γ γ → χχ− unitarity violation region

e+e

− → χχ−γ unitarity violation region


Given the same ps, unitarity bounds for 2→ 2 scattering are muchmore stringent than those for 2→ 3 scattering.However, here the relevant bounds are those for 2→ 3 scattering.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Λ

(Ge

V)

mχ (GeV)

e+e

− collider, √s = 3 TeV, γ + E ⁄ (fermionic DM)

Unitarity bound

3σ reach (100 fb-1

)

3σ reach (1000 fb-1

)

102

103

104

5 50 500 10 100 1000


Unitarity violation region

Λ

(Ge

V)

mχ (GeV)

e+e

− collider, √s = 3 TeV, γ + E ⁄ (scalar DM)

Unitarity bound

3σ reach (100 fb-1

)


)

101

102

103

104

5 50 500 10 100 1000



Meaningful searching region Meaningful searching region

All the experimental reaches we obtained lie far beyond the unitarityviolation regions.

From the viewpoint of S-matrix unitarity, our effective operatortreatment do not exceed its valid range.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Λ

(Ge

V)

mχ (GeV)

e+e

− collider, √s = 3 TeV, γ + E ⁄ (fermionic DM)

Unitarity bound

3σ reach (100 fb-1

)


)

102

103

104

5 50 500 10 100 1000



Λ

(Ge

V)

mχ (GeV)

e+e

− collider, √s = 3 TeV, γ + E ⁄ (scalar DM)

Unitarity bound

3σ reach (100 fb-1

)


)

101

102

103

104

5 50 500 10 100 1000



Meaningful searching region Meaningful searching region

All the experimental reaches we obtained lie far beyond the unitarityviolation regions.

From the viewpoint of S-matrix unitarity, our effective operatortreatment do not exceed its valid range.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Conclusions and discussions

...1 In this work, we explore the sensitivity to the effective operatorsof DM and photons at TeV-scale e+e− colliders.

...2 With a 100 fb−1 dataset, the potential Fermi γ-ray line signal forthe fermionic DM can be tested at a 3 TeV collider, though thescalar DM searching would be challenging.

...3 Using the polarized beams is roughly equivalent to collecting 10times of data.

...4 In order to check the validity of the effective operator approach, wederive a general unitarity condition for 2→ n processes. Theexperimental reaches we obtained are valid since they lie farbeyond the unitarity violation regions.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

110210310410510 1 10 210 310 410 510

July 2013

s Quartic Coupling limits @95% C.L. Channel Limits L γγAnomalous WW

LEP L3 limitsD0 limits

limitsγCMS WW WW limits → γγCMS

-2 TeV2Λ/W0a

-2 TeV2Λ/WCa

-4 TeV4Λ /T,0f

γWW 0.20 TeV-1[- 15000, 15000] 0.43fb

WW→ γγ 1.96 TeV-1 [- 430, 430] 9.70fb

γWW 8.0 TeV-1 [- 21, 20] 19.30fb

WW→ γγ 7.0 TeV-1[- 4, 4] 5.05fb

γWW 0.20 TeV-1[- 48000, 26000] 0.43fb

WW→ γγ 1.96 TeV-1 [- 1500, 1500] 9.70fb

γWW 8.0 TeV-1 [- 34, 32] 19.30fb

WW→γγ 7.0 TeV-1 [- 15, 15] 5.05fb

γWW 8.0 TeV-1 [- 25, 24] 19.30fb

[CMS PAS SMP-13-009]

γ

q

q

γ

W−

W+

W

q

q′

γ

Z

W

...5 The unitarity condition for 2→ n scattering can be also applied toother interesting processes, e.g., the WWγ and W Zγ productioninduced by anomalous quartic gauge couplings.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Thanks for your attentions!


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Backup slides


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Note that our unitarity conditionbinel

j

≤ 1

2is derived without any

approximation.

Through an approximate method, a unitarity bound on the 2→ ninelastic cross section σinel(2→ n) can be derived to be

σinel(2→ n)≤ 4π

s.

[Dicus & H. -J. He, hep-ph/0409131]

We have compared the results given by these two formulas and find thattheir differences are rather small for the processes considered here.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

α(p1, p2)→ β(q1, q2) p1 p2 →q1

q2

θαβ

α(p1, p2)→ γel(k1, k2) p1 p2 →

k1

k2

θαγ

β(q1, q2)→ γel(k1, k2)

q1

q2

θαβ→

k1

k2

θβγ

Unitarity condition in terms of amplitudes:−i(Mα→β −M∗β→α) =

∑γ

∫dΠγM∗β→γMα→γ(2π)4δ(4)(pα− pγ)

For the elastic process 1+ 2→ 1+ 2, consider the transitions of state:


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

Since Mα→β =M∗β→α =Mel(cosθαβ), the unitarity condition becomes

2 ImMel(cosθαβ)

=

∫dΠγel

M∗β→γelMα→γel

(2π)4δ(4)(pα− pγel) + inelastic terms

≥ β1

32π2

∫dΩk1

M∗el(cosθβγ)Mel(cosθαγ),

where β1 ≡p

1− 4m21/s and dΩk1

= dϕk1d cosθαγ.

In terms of partial waves:

Im aelj ≥

β1

8π

∑k,l

(2k+ 1)(2l + 1)ael∗k ael

l

∫d cosθαβdΩk1

×Pj(cosθαβ)Pk(cosθβγ)Pl(cosθαγ)


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

x

y

z

~k1~p1

~q1

φk1

θαγθαβ

θβγ

The addition theorem for Legendre polynomials:Pk(cosθβγ) = Pk(cosθαβ)Pk(cosθαγ)

+2∑l

m=1(l−m)!(l+m)! Pm

k (cosθαβ)Pmk (cosθαγ) cos mϕk1

Carrying out all the integrations, we haveIm ael

j ≥ β1|aelj |2,

which is equivalent to

(Re aelj )

2+

Im aelj −

1

2β1

2≤ 1

(2β1)2.

For the scattering of massless particles,β1 = 1, and it implies

.

......

Re aelj

≤ 1

2, ∀ j.


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

α(p1, p2)→ β(q1, q2) p1 p2 →q1

q2

θαβ

α(p1, p2)→ γn(k3, · · · , kn+2) p1 p2 →k3

k4

· · ····

kn+2

β(q1, q2)→ γn(k3, · · · , kn+2)

q1

q2

θαβ→

k3

k4

· · ····

kn+2

For 2→ n inelastic scattering, consider the transitions of state:


. . . .Motivations




. . .Conclusions

. . . . . . .Backups

The unitarity condition becomes2 ImMel(cosθαβ ) =

∫dΠγel

M∗β→γelMα→γel

(2π)4δ(4)(pα − pγel)

+∫

dΠγnM∗β→γn


) + other inelastic terms≥ β1

32π2

∫dΩk1

M∗el(cosθβγ)Mel(cosθαγ)+∫

dΠγnM∗β→γn

Mα→γn(2π)4δ(4)(pα− pγn

).

Introducing a new quantity|binel

j |2 ≡ 164π

∫d cosθαβ Pj(cosθαβ)

∫dΠγn

M∗β→γnMα→γn

(2π)4δ(4)(pα − pγn),

we have Im aelj ≥ β1|ael

j |2+ |binelj |2. Thus

|binelj |2 ≤

1

4β1− β1

(Re ael

j )2 +

Im aelj −

1

2β1

2≤ 1

4β1.

For massless incoming particles,.

......binel

j

≤ 1

2, ∀ j.


Documents

Detecting interactions between dark matter and photons at ...yzhxxzxy.github.io/slides/1308_DM_pho_eecld.pdf · Motivations. . . . . Sensitivity. . Beam polarization. . . . . Unitarity