Nontrivial Dark Matter Scenarios at the Energy Frontier

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ˆ

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Bib p+mmK 2tT2+iiBQM pHm2-s≡ ⟨S"⟩,

V_bQ7i = m²_Hu|Hu|²+m²_Hd|Hd|²+m²_S|S|² +

#

λAλSHuHd+1

3κAκS³+?X+X$

, URV r?2`2 λM/κ`2 i?2 >B;;b@bBM;H2i +QmTHBM; M/ bBM@

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+QKTQM2Mi A⁰₁ Q7 i?2 bBM;H2iS- M/ i?2 KBtBM; +QKTQ@

M2Mi Q7 i?2 M2ti iQ HB;?i2bi alau T`iB+H2 ULGaSV M/

i?2 GaS Qmi Q7 p`BQmb M2mi`HBMQ +QKTQM2MibX h?2 KBt@

BM; #2ir22M i?2 bBM;H2i M/ /Qm#H2i >B;;b Bb M2;HB;B#H2

bBM+2 Qm` bBM;H2i Kbb Bb ivTB+HHv O(1) :2o r?BH2 i?2

>B;;b Bb Rk8 :2o- bQ i?i i?2 Kbb Q7 A⁰₁ rQmH/ #2 m²_A1 ≃λ(Aλ+ 4κs)v²bBM2β

2s −3κsAκ. UkV

h?2 M2mi`HBMQ Kbb Ki`Bt r?B+? T`QpB/2b i?2 KBtBM;

M;H2b KQM; p`BQmb M2mi`HBMQb `2

Mχ!⁰ =

⎛

⎜⎜

⎜⎝

M1 0 −mWiMθW+Qbβ mWiMθWbBMβ 0

0 M2 mW+Qbβ −mWbBMβ 0

−mWiMθW +Qbβ mW+Qbβ 0 −µ₂₇₇ −λvu

mWiMθW bBMβ −mWbBMβ −µ277 0 −λvd

0 0 −λvu −λvd 2κs

⎞

⎟⎟

⎟⎠.

UjV

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M1/2≡2M1=M2= 1

3M3, U9V A0≡At=Ab =Aτ.

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RX *QHHBKi2/ JmQMb 7`QKA1 i i?2 G>*

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bi`i2;vX AM Qm` +b2 Q7 ?B;;BMQ LGaS TB` T`Q/m+iBQM- i?2`2 rQmH/ #2 i?`22 +QKT2iBM; /2+v T`Q+2bb2b Q7 χ˜<sup>0</sup><sub>2</sub> iQ i?2 /`F Kii2` χ!⁰₁ BM bbQ+BiBQM rBi? Z/h_aJ/A⁰₁- /2T2M/BM; QMλX

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H2TiQM +?MM2H (R3Ĝky) r?2`2 irQ H2TiQMb `2 7`QKZ #Q@

bQM M/ QM2 Bb 7`QKW<sup>±</sup>7`QK +?`;BMQ /2+v bBKBH`Hv

h#H2 RX _M;2b Q7 i?2 LJaaJ BMTmi T`K2i2`b b+MM2/ iQ Q#iBM χ!⁰₁ UGaSV rBi? Kbb #2HQr 2 :2o M/χ!⁰₂ULaGSV /QKBMi2/ #v i?2 ?B;;BMQ +QKTQM2MiX

S`K2i2` a+MM2/ `M;2

M0U:2oV 8yy Ĝ kyyy

M1/2U:2oV jyy Ĝ Ryyy

A0U:2oV y Ĝ 9yyy

µ277U:2oV Ryy Ĝ jyy

iMβ e Ĝ k8

λ yXyR Ĝ yX9

κ Ry⁻⁵ Ĝ Ry⁻¹

AλU:2oV y Ĝ 8yyy

AκU:2oV −100Ĝ y

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BM; i?2 +m``2Mi `2HB+ /2MbBiv- A⁰₁ /2+vb iQ ?B;?Hv +QHHB@

Ki2/ T`iB+H2b /m2 iQ i?2 H`;2 #QQbi Q7A⁰₁X

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b2b `2 `2[mB`2/ iQ iF2 +`2 Q7 i?2 Z*. #+F;`QmM/

2p2M r2 miBHBx2 i?2 ?2pv ~pQ` i;;BM; b ¯cc#2BM; `2@

;`/2/ b D2i /m2 iQ i?2 #QQbiX AM i?2 G>* /2i2+@

iQ`b r?2`2 r2 +M mb2 HH bm#@/2i2+iQ`b iQ i; KmQMb-

LQMi`BpBH .`F Jii2` a+2M`BQb i i?2 1M2`;v 6`QMiB2` Ĝ .QvQmMEBK·Jv2QM;?mMS`F N3R

6B;X kX U*QHQ` QMHBM2V h?2 "_Uχ!⁰₂ →Zχ!⁰₁V M/ i?2 "_

Uχ!⁰₂→A1χ!⁰₁V b 7mM+iBQMb Q7 i?2 T`K2i2`λX

A⁰₁ →µ⁺µ⁻ /2+v +?MM2H +M T`QpB/2 FBM2KiB+H BM@

7Q`KiBQM p2`v T`2+Bb2HvX "mi B7 r2 mb2 +QMp2MiBQMH KmQM i;;BM; K2i?Q/ r?2`2 MHvb2b `2[mB`2 M BbQH@

iBQM `QmM/ KmQM- r2 +M ?p2 QMHv QM2 `2+QMbi`m+i2/

KmQM r?B+? ?b i?2 #2bi χ² }iiBM; 7`QK i`D2+iQ`B2b BM p`BQmb bm# /2+i2+iQ`bX Ai KQiBpi2b mb iQ /Ti +QM@

+2Ti Q7 dzH2TiQM@D2iǴ (kk) iQ +Hmbi2` +QHHBKi2/ KmQM bvb@

i2K b bBM;H2 D2i@HBF2 Q#D2+iX 6Q` /2iBH2/ /Bb+mbbBQM QM i?2 G>* MHvbBb r2 `272` (Rd) M/ ?2`2 r2 BHHmbi`i2

?Qr iQ /2}M2 Bi +QM+2TimHHvX

RX *Hmbi2` +QHHBKi2/ KmQMb rBi? D2i@HBF2 Q#D2+i µ+QHQ7 i?2 +QM2 bBx2R= 0.4X q2 BKTQb2 i?2 BbQH@

iBQM +`Bi2`B-IbmK<3:2o- r?2`2IbmKBb i?2 b+H`

bmK Q7 i?2 i`Mbp2`b2 KQK2Mi Q7 HH //BiBQMH +?`;2/ i`+Fb rBi? `2[mB`2K2Mi Q7pT >0.5:2oX kX _2[mB`2 i?2 KBMBKmK i`Mbp2`b2 KQK2MimK Q7

µ_+QH, pT(µ_+QH)>50:2oX

#2M+?K`F TQBMi Bb BHHmbi`i2/ BM h#H2kM/ i?2 `2bmHi 7Q` i?2 HmKBMQbBiv 300 7#⁻¹ Q7 i?2 G>* 14h2o Bb bmK@

K`Bb2/ BM h#H2 jX b b22M BM Qm` `2bmHi 7`QK JQMi2

*`HQ bBKmHiBQM- rBi?Qmi M2r +QM+2Ti- dzH2TiQM@D2iǴ- Bi rQmH/ #2 /B{+mHi iQ T`Q#2 HB;?i /`F Kii2` b+2M`BQ i i?2 G>*X Pm` bim/v- i?2`27Q`2- b?2/b HB;?i QM i?2 .J b2`+? BM i?Bb `2;`/c b i?2 G>* +M T`Q/m+2 LG@

aSb KQ`2 i?M i?2 .J- B7 r2 /2bB;M T`QT2` MHvbBb K2i?Q/- r2 +M +Tim`2 i?2 +?`+i2`BbiB+ Q7 HB;?i /`F Kii2` T?vbB+b- r?BH2 Bi Bb MQi M 2bv rQ`F 7Q` +QMp2M@

iBQMH b2`+? K2i?Q/b r?B+? /Q MQi Tv ii2MiBQM iQ M BMi2`+iBQM ivT2 #2ir22M .J M/ aJ T`iB+H2bX

h#H2 kX Pm` #2M+?K`F TQBMib mb2/ 7Q` i?2 G>* MH@

vbBb

"S Jbb2b

m_χ!0

1 U:2oV RX9y3R

m_χ!0

2 U:2oV RdyXRj

m_χ!⁰

3 U:2oV −182.27

m_χ!±

1 U:2oV RedXdk

mA1 U:2oV kXN38e

mH2 U:2oV Rk8XdN

"`M+?BM; _iBQb

BR(χ!⁰₂→Zχ!⁰₁) yXeyj

BR(χ!⁰₂→A1χ!⁰₁) yXy3N

BR(χ!⁰3→Zχ!⁰1) yXdy9

BR(χ!⁰3→A1χ!⁰1) yXy3R

BR(A1→µ⁺µ⁻) yXy3d

h#H2 jX R9 h2o G>* rBi? HmKBMQbBiv Q7 300 7#⁻¹ HmKBMQbBivX ?2`2 a_w+ `2T`2b2Mib i?`22 H2TiQM +?MM2H BM (R3)X

SQBMi S/BBM MHvbBb b2MbBiBpBiv b(σ)BM MHvbBb 3ℓUa_w+ `2;BQMV µ+QH 3ℓUa_w+ `2;BQMV µ+QH

"S yX9je R8 kXy kd

AAAX G>* 6P_ .J *P@LLA>AGhAPL _1:APL

AM i?Bb b2+iBQM- r2 iF2 M 2tKTH2 Q7 .J #BMQ@;HmBMQ +Q@MMB?BHiBQM T`Q+2bb BM amT2`bvKK2i`B+ KQ/2Hb r?2`2 /`F Kii2` χ˜⁰₁ ?b KQbiHv rBi? #BMQ +QKTQM2Mi M/

LGaS Bb ;HmBMQ U˜gV- i?2 bmbv T`iM2` Q7 aJ ;HmQMX AM i?Bb +b2-˜g/2+vb BMiQχ˜⁰₁ rBi? irQ bQ7i D2ibX b Kbb /Bz2`2M+2 #2ir22M ;HmBMQ M/ /`F Kii2` #2+QK2b M2;HB;B#H2 +QKT`2/ iQ i?2 Kbb Q7 ;HmBMQ- D2ib 7`QKg˜ /2+v #2+QK2 iQQ bQ7i iQ #2 i;;2/- H2pBM; QMHv KQMQ@

D2iY̸ET MHvbBb 7Q` i?2 G>* bim/vX h?2 MHvb2b `2HvBM;

QM Aa_ D2i `2 MQi b2MbBiBp2 iQ /2iBHb Q7 i?Bb T`Q+2bb b i?2 ?`/M2bb Q7 Aa_ QMHv /2T2M/b QM i?2 2M2`;v b+H2 Q7 i?2 ?`/ b+ii2`BM;- BM Qm` +b2 i?2 Kbb Q7 ;HmBMQX h?mb Bi rQmH/ #2 M2+2bb`v iQ /2p2HQT K2i?Q/ iQ i;

bQ7i D2ib 7`QK ;HmBMQ /2+vBM; iQ mM/2`biM/ /2iBHb 7`QK r?B+? r2 +M BM72` i?2 Mim`2 Q7 T`Q+2bbX AM i?Bb b2+@

iBQM- r2 `2pB2r Qm` bim/v BM (kj) r?2`2 r2 T`QTQb2 K2i?Q/ iQ +Hmbi2` bQ7i D2ib #v miBHBxBM; H`;2 bBx2 Q7 D2iX

6B;X jX U*QHQ` QMHBM2V 1t+HmbBQMb 7`QK miBHBxBM; 7i@D2i i;;BM; QM b2`+? b2MbBiBpBiv Q72σU_2/ rBi? ;`B/ HBM2bV +QKT`2/ iQ `2bmHib 7`QK +QMp2MiBQMH hGa b2`+? BM BM KmHiB@D2i bB;MH `2;BQM U;`2vVX

RX C2i am#bi`m+im`2 J2i?Q/b iQ h; aQ7i S`iB@

+H2b

q?2M ;HmBMQ /2+vb BMiQ irQ [m`Fb M/ /`F Ki@

i2`- ˜g → q¯qχ˜<sup>0</sup><sub>1</sub> i?`Qm;? Qzb?2HH b[m`F- i?2 BMp`BMi Kbb Q7 irQ [m`Fbmqq¯HQ+i2 M2` i?2 T2FPqq¯Q7 i?`22

#Q/v BMp`BMi Kbb /Bbi`B#miBQM BM KQbi 2p2Mibc

Pqq¯ =

⎡

⎣m²_˜g+m²_χ˜0 1

3

⎛

⎝2− -.

./1 + 12m²_˜gm²_χ˜0 1

(m²_˜g+m²_χ˜0 1)²

⎞

⎠

⎤

⎦

1 2

∆m−→≪m˜g

∆m√ 2 ,

r?2`2∆mBb Kbb ;T #2ir22M ;HmBMQ M/ /`F Ki@

i2` ∆m=mg˜−mχ˜<sup>0</sup><sub>1</sub>X h?mb B7 r2 `2[mB`2 pT Q7 (q,q)¯ bvbi2K bp<sub>T(q,q)¯</sub> ≥∆m- r2 +M +Tim`2 irQ [m`Fb rBi?

bBM;H2 D2i Udz7i@D2iǴV Q7 H`;2 +QM2 bBx2R b R= 2M(q,q)¯

pT(q,¯q) ≥2

#∆m

√2

$ 1

∆m =√

2. U8V h?2 KDQ` +QM+2`M iQ miBHBx2 H`;2 bBx2 D2i iQ i;

bB;MH Bb iQ `2/m+2 aJ Z*. #+F;`QmM/b r?B+? rQmH/

#2 i?2 #B; Bbbm2 BM i?2 ?/`QM +QHHB/2`bX

AM Qm` +b2 i?2 KDQ` #+F;`QmM/ rQmH/ #2 Z(→ νν) +¯ jsX h?2 Kbb Q7 MQ`KH Z*. D2i ULGPV Bb T@

T`QtBKi2Hv iQ 2

< M_J²>LGP ≈0.2pJR . UeV q?2M r2 T`Q#2 i?2 +Q@MMB?BHiBQM `2;BQM Q7 ∆m ≃ 100 :2o- i?2 Kbb Q7 #+F;`QmM/ Z*. D2i #2+QK2b

3< M_J²> ∼ 30 :2o rBi? H`;2 +QM2 bBx2 R ≃ √ 2 r?BH2 i?2 Kbb Q7 bB;MH D2i M(q,¯q) ≃ 70 :2oX h?mb BM T`BM+BTH2 r2 +M bQ`i Qmi #+F;`QmM/ D2ib QMHv rBi? D2i KbbX "mi MQ`KH Z*. D2i ;2i //BiBQMH Kbb2b 7`QK p`BQmb +QMiKBMiBQMb BM+Hm/BM; bQ7i Z*. M/ mM/2`Hv@

BM; 2p2MibX h?Bb #2+QK2b KQ`2 b2`BQmb B7 QM2 mb2 H`;2 bBx2 D2iX hQ `2/m+2 bQ7i Z*. +iBpBiB2b BM i?2 Kbb Q7 D2i- p`BQmb ;`QQKBM; K2i?Q/b ?p2 #22M BMi`Q/m+2/

(k9Ĝkd)X AM Qm` MHvbBb r2 mb2 Jbb .`QT h;;2`

UJ.hV (k9) r?B+? rb BMp2Mi2/ iQ 2M?M+2 >B;;b /Bb@

+Qp2`v BMH →b¯bb Qm` bim/v TQBMi Q7 Kbb /Bz2`2M+2 Bb bBKBH` iQ i?2 Kbb Q7 >B;;b b ∆m ≃ O(100) :2oX q2 HbQ `2[mB`2 +mi 7`QK D2i bm#bi`m+im`2b iQ `2/m+2 Z*. #+F;`QmM/b 7m`i?2`X AM p`BQmb D2i bm#bi`m+im`2 p`B#H2b- r2 mb2ρp`B#H2

ρ=m²_j/4

p²_{T j}R²5 UdV bBM+2 7Q` bB;MH D2i r?B+? ?b irQ [m`Fb rBi? b?`@

BM; 2[mH 2M2`;v- ρ≃0.2 r?BH2 D2ib BM aJ #+F;`QmM/b r?B+? `2 BMBiBi2/ #v bBM;H2 [m`F ?p2 ρ≤0.1X "v miBHBxBM; D2i Kbb 7i2` J.h M/ i;;BM; bB;MH D2ib rBi? D2i bm#bi`m+im`2 p`B#H2ρ- r2 +M bmTT`2bb #+F@

;`QmM/b 2{+B2MiHvX hQ b22 i?2 Qp2`HH T2`7Q`KM+2 r2 +QKT`2 Qm` K2i?Q/ Q7 mbBM; 7i@D2i rBi? +QMp2MiBQMH MHvbBb BM hGa KmHiB@D2i bB;MH `2;BQM (k3) BM 6B;XjX b r2 b22 BM i?Bb THQi- mbBM; 7i@D2i BM b2`+? 7Q` .J +M 2M?M+2 b2`+? b2MbBiBpBivX q2 HbQ +M ;2i ?BMib 7Q` i?2 T`QT2`iB2b Q7 .J M/ `2Hi2/ "aJ T`iB+H2b b r2 +M mb2 BM7Q`KiBQM 7`QK p`BQmb FBM2KiB+H /Bbi`B@

#miBQMb #v i;;BM; T`iB+H2b BMpQHp2/ BM .J T`Q/m+iBQMX AM i?Bb `2;`/- +QHHB/2`b BM+Hm/BM; i?2 +m``2Mi G>* ?p2 p2`v mMB[m2 TQbBiBQM KQM; p`BQmb 2tT2`BK2Mib 7Q`

/`F Kii2` b2`+?2bX

AoX *PL*GaAPL

AM i?Bb `2pB2r- r2 /2KQMbi`i2 MQM@+QMp2MiBQMH b2`+? bi`i2;B2b i i?2 G>* iQ b2`+? 7Q` .J BM p`@

BQmb BMi2`2biBM; T`K2i2` bT+2- i?2 QM2 Bb 7Q` i?2 HB;?i /`F Kii2` M/ i?2 Qi?2` Bb 7Q` i?2 +Q@MMB?BHiBQM `2@

;BQMX h?2 +m``2Mi G>* rBi?13h2o +QHHBbBQM 2M2`;v +M T`Q#2 MQi QMHv /`F Kii2`UbV #mi HbQ i?2 K2+?MBbK Q7 /`F Kii2` T`Q/m+iBQMX h?Bb +M 2MHB;?i2M i?2 /`F b2+iQ` Q7 Qm` mMBp2`b2X

LQMi`BpBH .`F Jii2` a+2M`BQb i i?2 1M2`;v 6`QMiB2` Ĝ .QvQmMEBK·Jv2QM;?mMS`F N3j

*ELPqG1.:1J1Lha

.X EX M/ JX SX Bb bmTTQ`i2/ #v A"a mM/2` i?2 T`QD2+i +Q/2- A"a@_yR3@.RX .X EX Bb HbQ bmTTQ`i2/ #v i?2 "@

bB+ a+B2M+2 _2b2`+? S`Q;`K i?`Qm;? i?2 LiBQMH _2@

b2`+? 6QmM/iBQM Q7 EQ`2 UL_6V 7mM/2/ #v i?2 JBM@

Bbi`v Q7 a+B2M+2- :`Mi LQX kyR8_R*RRykyjd3jyX

_161_1L*1a

(R) aX *?M;- _X 1/2x?i?- CX >mi+?BMbQM M/ JX Gmiv- S?vbX _2pX .3N- yR8yRR UkyR9VX

(k) >X M- GX hX qM; M/ >X w?M;- S?vbX _2pX . 3N- RR8yR9 UkyR9VX

(j) uX "B M/ CX "2`;2`- CX >B;? 1M2`;v S?vbXRjRR- RdR UkyRjVX

(9) X .B6`MxQ- EX AX L;Q- X _D`KM M/ hX JX SX hBi- CX >B;? 1M2`;v S?vbXRjRR- yR9 UkyRjVX (8) PX "m+?Km2HH2`- JX CX .QHM M/ *X J+*#2- CX

>B;? 1M2`;v S?vbXR9yR- yk8 UkyR9VX

(e) :X "mbQMB- X .2 aBKQM2- CX :`KHBM;- 1X JQ`;Mi2 M/ X _BQiiQ- CX *QbKQHX bi`QT`iX S?vbX R9ye- yey UkyR9VX

(d) JX STm++B- X oB+?B M/ EX JX wm`2F- CX >B;?

1M2`;v S?vbXR9RR- yk9 UkyR9VX

(3) JX _X "m+FH2v- .X 62H/ M/ .X :QM+Hp2b- S?vbX _2pX .NR- yR8yRd UkyR8VX

(N) :X / 2i HX (hGa *QHH#Q`iBQM)- 1m`X S?vbX CX *d8- kNN UkyR8VX 1``imK, 1m`X S?vbX CX *d8- 9y3 UkyR8VX

(Ry) oX E?+?i`vM 2i HX (*Ja *QHH#Q`iBQM)- 1m`X S?vbX CX *d8- kj8 UkyR8VX

(RR) CX :QQ/KM- JX A#2- X _D`KM- qX a?2T?2`/- hX JX SX hBi M/ >X "X um- S?vbX _2pX . 3k- RReyRy UkyRyVX

(Rk) SX CX 6Qt- _X >`MBF- CX EQTT M/ uX hbB- S?vbX _2pX .38- y8eyRR UkyRkVX

(Rj) uX "B M/ hX JX SX hBi- S?vbX G2iiX " dkj- j39 UkyRjVX

(R9) GX JX *`T2Mi2`- X L2HbQM- *X a?BKKBM- hX JX SX hBi M/ .X q?Bi2bQM- S?vbX _2pX . 3d- yd9yy8 UkyRjVX

(R8) SX .`T2`- hX GBm- *X 1X JX q;M2`- GX hX qM; M/

>X w?M;- S?vbX _2pX G2iiXRye- RkR3y8 UkyRRVX (Re) CX hBz2M#2`; 2i HX (.JA* *QHH#Q`iBQM)-

`sBp,RjRyXee33 (bi`Q@T?XAJ)X

(Rd) *X >M- .X EBK- aX JmMB` M/ JX S`F- C>1S R8yd- yyk UkyR8VX

(R3) (hGa *QHH#Q`iBQM)- hGa@*PL6@kyRj@yj8X (RN) :X / 2i HX(hGa *QHH#Q`iBQM)- CX >B;? 1M@

2`;v S?vbXR9y9- ReN UkyR9VX

(ky) oX E?+?i`vM 2i HX (*Ja *QHH#Q`iBQM)- 1m`X S?vbX CX *d9- jyje UkyR9VX

(kR) .X :X *2`/2MQ- SX :?Qb?- *X "X S`F M/ JX S2B`Q- CX >B;? 1M2`;v S?vbX R9yk- y93 UkyR9VX

(kk) LX `FMB@>K2/ M/ LX q2BM2`- CX >B;? 1M2`;v S?vbXy3Rk- Ry9 Ukyy3VX

(kj) *X >M M/ JX S`F- S?vbX _2pX . N9- yRR8yk UkyReVX

(k9) CX JX "mii2`rQ`i?- X _X .pBbQM- JX _m#BM M/ :X SX aHK- S?vbX _2pX G2iiX Ryy- k9kyyR Ukyy3VX (k8) .X E`Q?M- CX h?H2` M/ GX hX qM;- CX >B;? 1M@

2`;v S?vbXRyyk- y39 UkyRyVX

(ke) aX .X 1HHBb- *X EX o2`KBHBQM M/ CX _X qHb?- S?vbX _2pX .3y- y8R8yR UkyyNVX

(kd) aX .X 1HHBb- *X EX o2`KBHBQM M/ CX _X qHb?- S?vbX _2pX .3R- yN9ykj UkyRyVX

(k3) :X / 2i HX(hGa *QHH#Q`iBQM)- CX >B;? 1M@

2`;v S?vbXR9yN- Rde UkyR9VX