Calculations of neutralino-stau coannihilation channels and the cosmologically relevant region of MSSM parameter space

John Ellis, Toby Falk, Keith A. Olive, Mark Srednicki

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Abstract

Assuming that the lightest supersymmetric particle (LSP) is the lightest neutralino χ̃, we present a detailed exploration of neutralino-stau (χ̃-τ̃) coannihilation channels, including analytical expressions and numerical results. We also include χ̃ coannihilations with the ẽ and μ̃. We evaluate the implications of coannihilations for the cosmological relic density of the LSP, which is assumed to be stable, in the constrained minimal supersymmetric extension of the Standard Model (CMSSM), in which the soft supersymmetry-breaking parameters are universal at the supergravity GUT scale. We evaluate the changes due to coannihilations in the region of the MSSM parameter space that is consistent with the cosmological upper limit on the relic LSP density. In particular, we find that the upper limit on mχ̃ is increased from about 200 GeV to about 600 GeV in the CMSSM, and estimate a qualitatively similar increase for gauginos in the general MSSM.

Original languageEnglish (US)
Pages (from-to)181-213
Number of pages33
JournalAstroparticle Physics
Volume13
Issue number2-3
DOIs
StatePublished - May 2000

Bibliographical note

Funding Information:
The work of K.O. was supported in part by DOE grant DE–FG02–94ER–40823. The work of T.F. was supported in part by DOE grant DE–FG02–95ER–40896, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. The work of M.S. was supported in part by NSF grant PHY–97–22022. Appendix A This section contains simplified formulae for the χ, , τ ̃ τ ̃ ∗ , τ ̃ τ ̃ τ ̃ τ ̃ ℓ ̃ and τ ̃ ℓ ̃ ∗ annihilation amplitudes, in the m τ ̃ →0 limit. Expressions for the ℓ ̃ ℓ ̃ ∗ , ℓ ̃ ℓ ̃ and χ ℓ ̃ amplitudes can be obtained by taking τ →ℓ in the χ, τ ̃ τ ̃ ∗ , τ ̃ τ ̃ τ ̃ formulae below, and μ ̃ e ̃ and e ̃ μ ̃ ∗ by taking τ → e ,ℓ→ μ in the τ ̃ ℓ ̃ and τ ̃ ℓ ̃ ∗ expressions. The a and b coefficients can be simply numerically extracted from the amplitudes, as described in Section 2 of the text. A.1 τ ̃ τ ̃ ∗ →W + W − I. s-channel H annihilation II. s-channel h annihilation V. s-channel Z annihilation VI. s-channel γ annihilation f , 1 =(−g 2 m W cos (β−α))(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W ) f , 2 =(−g 2 m W sin (β−α))(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W ) f , 5 =(−g 2 sin 2 θ W / cos θ W )(g 2 cos θ W ) f , 6 =e 2 , T I × T I =(12m W 4 −4m W 2 s+s 2 )/(4m W 4 (m H 2 −s) 2 ) , T II × T II =(12m W 4 −4m W 2 s+s 2 )/(4m W 4 (m h 2 −s) 2 ) , T V × T V =(128m τ ̃ R 2 m W 4 m Z 4 s−32m τ ̃ R 2 m W 2 m Z 4 s 2 −32m W 4 m Z 4 s 2 +8m W 2 m Z 4 s 3 +12m W 4 m Z 4 t 2 −4m W 2 m Z 4 st 2 +m Z 4 s 2 t 2 −24m W 4 m Z 4 tu+8m W 2 m Z 4 stu−2m Z 4 s 2 tu+12m W 4 m Z 4 u 2 −4m W 2 m Z 4 su 2 +m Z 4 s 2 u 2 )/(4m W 4 m Z 4 (m Z 2 −s) 2 ) , T VI × T VI =(128m τ ̃ R 2 m W 4 s−32m τ ̃ R 2 m W 2 s 2 −32m W 4 s 2 +8m W 2 s 3 +12m W 4 t 2 −4m W 2 st 2 +s 2 t 2 −24m W 4 tu+8m W 2 stu−2s 2 tu+12m W 4 u 2 −4m W 2 su 2 +s 2 u 2 )/(4m W 4 s 2 ) , T I × T II =(12m W 4 −4m W 2 s+s 2 )/(4m W 4 (m H 2 −s)(m h 2 −s)) , T I × T V =(−12m W 4 m Z 2 t+m Z 2 s 2 t+12m W 4 m Z 2 u−m Z 2 s 2 u)/(4m W 4 m Z 2 (m H 2 −s)(m Z 2 −s)) , T II × T V =(−12m W 4 m Z 2 t+m Z 2 s 2 t+12m W 4 m Z 2 u−m Z 2 s 2 u)/(4m W 4 m Z 2 (m h 2 −s)(m Z 2 −s)) , T I × T VI =−(−12m W 4 t+s 2 t+12m W 4 u−s 2 u)/(4m W 4 (m H 2 −s)s) , T II × T VI =−(−12m W 4 t+s 2 t+12m W 4 u−s 2 u)/(4m W 4 (m h 2 −s)s) , T V × T VI =(−128m τ ̃ R 2 m W 4 m Z 2 s+32m τ ̃ R 2 m W 2 m Z 2 s 2 +32m W 4 m Z 2 s 2 −8m W 2 m Z 2 s 3 −12m W 4 m Z 2 t 2 +4m W 2 m Z 2 st 2 −m Z 2 s 2 t 2 +24m W 4 m Z 2 tu−8m W 2 m Z 2 stu+2m Z 2 s 2 tu−12m W 4 m Z 2 u 2 +4m W 2 m Z 2 su 2 −m Z 2 s 2 u 2 )/(4m W 4 m Z 2 (m Z 2 −s)s) (A.1) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 5 2 T V × T V +f 6 2 T VI × T VI +2f 1 f 2 T I × T II +2f 1 f 5 T I × T V +2f 1 f 6 T I × T VI +2f 2 f 5 T II × T V +2f 2 f 6 T II × T VI +2f 5 f 6 T V × T VI A.2 τ ̃ τ ̃ ∗ →ZZ I. s-channel H annihilation II. s-channel h annihilation III. t-channel τ ̃ exchange IV. u-channel τ ̃ exchange V. point interaction f , 1 =(−g 2 m Z cos (β−α)/ cos θ W )(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W ) f , 2 =(−g 2 m Z sin (β−α)/ cos θ W )(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W ) f , 3 =(−g 2 sin 2 θ W / cos θ W ) 2 f , 4 =(−g 2 sin 2 θ W / cos θ W ) 2 f , 5 =−2g 2 2 sin 4 θ W / cos 2 θ W , T I × T I =(12m Z 4 −4m Z 2 s+s 2 )/(4m Z 4 (m H 2 −s) 2 ) , T II × T II =(12m Z 4 −4m Z 2 s+s 2 )/(4m Z 4 (m h 2 −s) 2 ) , T III × T III =(m τ ̃ R 8 −4m τ ̃ R 6 m Z 2 +6m τ ̃ R 4 m Z 4 −4m τ ̃ R 2 m Z 6 +m Z 8 −4m τ ̃ R 6 t+4m τ ̃ R 4 m Z 2 t+4m τ ̃ R 2 m Z 4 t−4m Z 6 t+6m τ ̃ R 4 t 2 +4m τ ̃ R 2 m Z 2 t 2 +6m Z 4 t 2 −4m τ ̃ R 2 t 3 −4m Z 2 t 3 +t 4 )/(m Z 4 (m τ ̃ R 2 −t) 2 ) , T IV × T IV =(m τ ̃ R 8 −4m τ ̃ R 6 m Z 2 +6m τ ̃ R 4 m Z 4 −4m τ ̃ R 2 m Z 6 +m Z 8 −4m τ ̃ R 6 u+4m τ ̃ R 4 m Z 2 u+4m τ ̃ R 2 m Z 4 u−4m Z 6 u+6m τ ̃ R 4 u 2 +4m τ ̃ R 2 m Z 2 u 2 +6m Z 4 u 2 −4m τ ̃ R 2 u 3 −4m Z 2 u 3 +u 4 )/(m Z 4 (m τ ̃ R 2 −u) 2 ) , T V × T V =(12m Z 4 −4m Z 2 s+s 2 )/(4m Z 4 ) , T I × T II =(12m Z 4 −4m Z 2 s+s 2 )/(4m Z 4 (m H 2 −s)(m h 2 −s)) , T I × T III =(−6m τ ̃ R 4 m Z 2 −20m τ ̃ R 2 m Z 4 −6m Z 6 +m τ ̃ R 4 s+2m τ ̃ R 2 m Z 2 s+5m Z 4 s+8m τ ̃ R 2 m Z 2 t+8m Z 4 t−2m τ ̃ R 2 st−2m Z 2 st−2m Z 2 t 2 +st 2 +4m τ ̃ R 2 m Z 2 u+4m Z 4 u−4m Z 2 tu)/(2m Z 4 (m H 2 −s)(m τ ̃ R 2 −t)) , T I × T IV =(−6m τ ̃ R 4 m Z 2 −20m τ ̃ R 2 m Z 4 −6m Z 6 +m τ ̃ R 4 s+2m τ ̃ R 2 m Z 2 s+5m Z 4 s+4m τ ̃ R 2 m Z 2 t+4m Z 4 t+8m τ ̃ R 2 m Z 2 u+8m Z 4 u−2m τ ̃ R 2 su−2m Z 2 su−4m Z 2 tu−2m Z 2 u 2 +su 2 )/(2m Z 4 (m H 2 −s)(m τ ̃ R 2 −u)) , T I × T V =(12m Z 4 −4m Z 2 s+s 2 )/(4m Z 4 (m H 2 −s)) , T II × T III =(−6m τ ̃ R 4 m Z 2 −20m τ ̃ R 2 m Z 4 −6m Z 6 +m τ ̃ R 4 s+2m τ ̃ R 2 m Z 2 s+5m Z 4 s+8m τ ̃ R 2 m Z 2 t+8m Z 4 t−2m τ ̃ R 2 st−2m Z 2 st−2m Z 2 t 2 +st 2 +4m τ ̃ R 2 m Z 2 u+4m Z 4 u−4m Z 2 tu)/(2m Z 4 (m h 2 −s)(m τ ̃ R 2 −t)) , T II × T IV =(−6m τ ̃ R 4 m Z 2 −20m τ ̃ R 2 m Z 4 −6m Z 6 +m τ ̃ R 4 s+2m τ ̃ R 2 m Z 2 s+5m Z 4 s+4m τ ̃ R 2 m Z 2 t+4m Z 4 t+8m τ ̃ R 2 m Z 2 u+8m Z 4 u−2m τ ̃ R 2 su−2m Z 2 su−4m Z 2 tu−2m Z 2 u 2 +su 2 )/(2m Z 4 (m h 2 −s)(m τ ̃ R 2 −u)) , T II × T V =(12m Z 4 −4m Z 2 s+s 2 )/(4m Z 4 (m h 2 −s)) , T III × T IV =(m τ ̃ R 8 +12m τ ̃ R 6 m Z 2 +38m τ ̃ R 4 m Z 4 +12m τ ̃ R 2 m Z 6 +m Z 8 −4m τ ̃ R 4 m Z 2 s−24m τ ̃ R 2 m Z 4 s−4m Z 6 s+4m Z 4 s 2 −2m τ ̃ R 6 t−14m τ ̃ R 4 m Z 2 t−14m τ ̃ R 2 m Z 4 t−2m Z 6 t+4m τ ̃ R 2 m Z 2 st+4m Z 4 st+m τ ̃ R 4 t 2 +2m τ ̃ R 2 m Z 2 t 2 +m Z 4 t 2 −2m τ ̃ R 6 u−14m τ ̃ R 4 m Z 2 u−14m τ ̃ R 2 m Z 4 u−2m Z 6 u+4m τ ̃ R 2 m Z 2 su+4m Z 4 su+4m τ ̃ R 4 tu+16m τ ̃ R 2 m Z 2 tu+4m Z 4 tu−4m Z 2 stu−2m τ ̃ R 2 t 2 u−2m Z 2 t 2 u+m τ ̃ R 4 u 2 +2m τ ̃ R 2 m Z 2 u 2 +m Z 4 u 2 −2m τ ̃ R 2 tu 2 −2m Z 2 tu 2 +t 2 u 2 )/(m Z 4 (m τ ̃ R 2 −t)(m τ ̃ R 2 −u)) , T III × T V =(−6m τ ̃ R 4 m Z 2 −20m τ ̃ R 2 m Z 4 −6m Z 6 +m τ ̃ R 4 s+2m τ ̃ R 2 m Z 2 s+5m Z 4 s+8m τ ̃ R 2 m Z 2 t+8m Z 4 t−2m τ ̃ R 2 st−2m Z 2 st−2m Z 2 t 2 +st 2 +4m τ ̃ R 2 m Z 2 u+4m Z 4 u−4m Z 2 tu)/(2m Z 4 (m τ ̃ R 2 −t)) , T IV × T V =(−6m τ ̃ R 4 m Z 2 −20m τ ̃ R 2 m Z 4 −6m Z 6 +m τ ̃ R 4 s+2m τ ̃ R 2 m Z 2 s+5m Z 4 s+4m τ ̃ R 2 m Z 2 t+4m Z 4 t+8m τ ̃ R 2 m Z 2 u+8m Z 4 u−2m τ ̃ R 2 su−2m Z 2 su−4m Z 2 tu−2m Z 2 u 2 +su 2 )/(2m Z 4 (m τ ̃ R 2 −u)) (A.2) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +f 4 2 T IV × T IV +f 5 2 T V × T V +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 1 f 4 T I × T IV +2f 1 f 5 T I × T V +2f 2 f 3 T II × T III +2f 2 f 4 T II × T IV +2f 2 f 5 T II × T V +2f 3 f 4 T III × T IV +2f 3 f 5 T III × T V +2f 4 f 5 T IV × T V A.3 τ ̃ τ ̃ ∗ →Zγ I. t-channel τ ̃ exchange II. u-channel τ ̃ exchange III. point interaction f , 1 =e(−g 2 sin 2 θ W / cos θ W ) f , 2 =e(−g 2 sin 2 θ W / cos θ W ) f , 3 =2eg 2 sin 2 θ W / cos θ W , T I × T I =(−2m τ ̃ R 6 +4m τ ̃ R 4 m Z 2 −2m τ ̃ R 2 m Z 4 +2m τ ̃ R 4 t+8m τ ̃ R 2 m Z 2 t−2m Z 4 t+2m τ ̃ R 2 t 2 +4m Z 2 t 2 −2t 3 )/(m Z 2 (m τ ̃ R 2 −t) 2 ) , T II × T II =(−2m τ ̃ R 6 +4m τ ̃ R 4 m Z 2 −2m τ ̃ R 2 m Z 4 +2m τ ̃ R 4 u−8m τ ̃ R 2 m Z 2 u−2m Z 4 u+2m τ ̃ R 2 u 2 +4m Z 2 u 2 −2u 3 )/(m Z 2 (m τ ̃ R 2 −u) 2 ) , T III × T III =3 , T I × T II =(6m τ ̃ R 6 +36m τ ̃ R 4 m Z 2 +6m τ ̃ R 2 m Z 4 −2m τ ̃ R 4 s−24m τ ̃ R 2 m Z 2 s−2m Z 4 s+4m Z 2 s 2 −7m τ ̃ R 4 t−12m τ ̃ R 2 m Z 2 t−m Z 4 t+2m τ ̃ R 2 st+4m Z 2 st+m τ ̃ R 2 t 2 +m Z 2 t 2 −7m τ ̃ R 4 u−12m τ ̃ R 2 m Z 2 u−m Z 4 u+2m τ ̃ R 2 su+4m Z 2 su+8m τ ̃ R 2 tu+2m Z 2 tu−2stu−t 2 u+m τ ̃ R 2 u 2 +m Z 2 u 2 −tu 2 )/(m Z 2 (m τ ̃ R 2 −t)(m τ ̃ R 2 −u)) , T I × T III =(−2m τ ̃ R 4 −15m τ ̃ R 2 m Z 2 −3m Z 4 +m τ ̃ R 2 s+5m Z 2 s+2m τ ̃ R 2 t+3m Z 2 t−st+2m τ ̃ R 2 u+4m Z 2 u−2tu)/(2m Z 2 (m τ ̃ R 2 −t)) , T II × T III =(−2m τ ̃ R 4 −15m τ ̃ R 2 m Z 2 −3m Z 4 +m τ ̃ R 2 s+5m Z 2 s+2m τ ̃ R 2 t+4m Z 2 t+2m τ ̃ R 2 u+3m Z 2 u−su−2tu)/(2m Z 2 (m τ ̃ R 2 −u)) (A.3) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 2 f 3 T II × T III A.4 τ ̃ τ ̃ ∗ →γγ I. t-channel τ ̃ exchange II. u-channel τ ̃ exchange III. point interaction f , 1 =e 2 f , 2 =e 2 f , 3 =−2e 2 , T I × T I =(4m τ ̃ R 4 +8m τ ̃ R 2 t+4t 2 )/(m τ ̃ R 2 −t) 2 , T II × T II =(4m τ ̃ R 4 +8m τ ̃ R 2 u+4u 2 )/(m τ ̃ R 2 −u) 2 , T III × T III =4 , T I × T II =(36m τ ̃ R 4 −24m τ ̃ R 2 s+4s 2 −12m τ ̃ R 2 t+4st+t 2 −12m τ ̃ R 2 u+4su+2tu+u 2 )/((m τ ̃ R 2 −t)(m τ ̃ R 2 −u)) , T I × T III =(−12m τ ̃ R 2 +5s+4u)/(2(m τ ̃ R 2 −t)) , T II × T III =(−12m τ ̃ R 2 +5s+4t)/(2(m τ ̃ R 2 −u)) (A.4) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 2 f 3 T II × T III A.5 τ ̃ τ ̃ ∗ →Zh[H] I. t-channel τ ̃ exchange II. u-channel τ ̃ exchange III. s-channel Z annihilation f , 1 =(−g 2 sin 2 θ W / cos θ W )(−g 2 m Z sin 2 θ W sin [− cos ](α+β)/ cos θ W ) f , 2 =−(−g 2 sin 2 θ W / cos θ W )(−g 2 m Z sin 2 θ W sin [− cos ](α+β)/ cos θ W ) f , 3 =(−g 2 sin 2 θ W / cos θ W )(−g 2 m Z sin [ cos ](β−α)/ cos θ W ) , T I × T I =(m τ ̃ R 4 +(m Z 2 −t) 2 −2m τ ̃ R 2 (m Z 2 +t))/(m Z 2 (m τ ̃ R 2 −t) 2 ) , T II × T II =(m τ ̃ R 4 +(m Z 2 −u) 2 −2m τ ̃ R 2 (m Z 2 +u))/(m Z 2 (m τ ̃ R 2 −u) 2 ) , T I × T II =(m τ ̃ R 4 +m Z 4 +m τ ̃ R 2 (6m Z 2 −t−u)+tu−m Z 2 (2s+t+u))/(m Z 2 (m τ ̃ R 2 −t)(m τ ̃ R 2 −u)) , T I × T III =(t(t−u)+m τ ̃ R 2 (−8m Z 2 −t+u)+m Z 2 (2s−t+u))/(2m Z 2 (m Z 2 −s)(m τ ̃ R 2 −t)) , T II × T III =((t−u)u+m τ ̃ R 2 (8m Z 2 −t+u)+m Z 2 (−2s−t+u))/(2m Z 2 (m Z 2 −s)(m τ ̃ R 2 −u)) , T III × T III =(−16m τ ̃ R 2 m Z 2 +4m Z 2 s+(t−u) 2 )/(4m Z 2 (m Z 2 −s) 2 ) (A.5) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 2 f 3 T II × T III A.6 τ ̃ τ ̃ ∗ →γh[H] I. t-channel τ ̃ exchange II. u-channel τ ̃ exchange f , 1 =(e)(−g 2 m Z sin 2 θ W sin [− cos ](α+β)/ cos θ W ) f ,) 2 =−(e)(−g 2 m Z sin 2 θ W sin [− cos ](α+β)/ cos θ W , T I × T I =−2(m τ ̃ R 2 +t)/(m τ ̃ R 2 −t) 2 , T I × T II =−(−6m τ ̃ R 2 +2s+t+u)/((m τ ̃ R 2 −t)(m τ ̃ R 2 −u)) , T II × T II =−2(m τ ̃ R 2 +u)/(m τ ̃ R 2 −u) 2 (A.6) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +2f 1 f 2 T I × T II A.7 τ ̃ τ ̃ ∗ →ZA I. s-channel h exchange II. s-channel H exchange f , 1 =(g 2 cos (α−β)/(2 cos θ W )(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W ) f , 2 =(g 2 sin (α−β)/(2 cos θ W )(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W ) , T I × T I =(m A 4 +(m Z 2 −s) 2 −2m A 2 (m Z 2 +s))/(m Z 2 (m h 2 −s) 2 ) , T II × T II =(m A 4 +(m Z 2 −s) 2 −2m A 2 (m Z 2 +s))/(m Z 2 (m H 2 −s) 2 ) , T I × T II =(m A 4 +(m Z 2 −s) 2 −2m A 2 (m Z 2 +s)/(m Z 2 (m H 2 −s)(m h 2 −s))) (A.7) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +2f 1 f 2 T I × T II A.8 τ ̃ τ ̃ ∗ →τ τ ̄ III. s-channel Z annihilation IV. s-channel γ annihilation V. t-channel χ exchange f , 3c =(−g 2 sin 2 θ W / cos θ W )(g 2 (1−4 sin 2 θ W )/(4 cos 2 θ W )) f , 3d =(−g 2 sin 2 θ W / cos θ W )(−g 2 /(4 cos 2 θ W )) f , 4c =e 2 K=g , 1 N i1 / 2 K′=g , 1 N j1 / 2 , T III × T III =2(−4f 3c 2 m τ ̃ R 2 s−4f 3d 2 m τ ̃ R 2 sf 3c 2 s 2 +f 3d 2 s 2 −f 3c 2 t 2 −f 3d 2 t 2 +2f 3c 2 tu+2f 3d 2 tu−f 3c 2 u 2 −f 3d 2 u 2 )/(m Z 2 −s) 2 , T III × T IV =2f 3c f 4c (4m τ ̃ R 2 s−s 2 +t 2 −2tu+u 2 )/((m Z 2 −s)s) , T IV × T IV =2f 4c 2 (−4m τ ̃ R 2 s+s 2 −t 2 +2tu−u 2 )/s 2 , T III × T V =−2(f 3c +f 3d )K 2 (4m τ ̃ R 2 s−s 2 +(t−u) 2 )/((m Z 2 −s)(m i χ ̃ 2 −t)) , T IV × T V =2(f 4c )K 2 (4m τ ̃ R 2 s−s 2 +(t−u) 2 )/(s(m i χ ̃ 2 −t)) , T V × T V =(16K 2 K′ 2 (m τ ̃ R 4 −tu))/((m i χ ̃ 2 −t)(−m j χ ̃ 2 +t)) (A.8) | T | 2 = T III × T III + T IV × T IV +2 T III × T IV + ∑ i,j=1 4 ( T I × T V + T II × T V + T III × T V + T IV × T V )/2+ T V × T V . A.9 τ ̃ τ ̃ ∗ →f f ̄ III. s-channel Z annihilation IV. s-channel γ annihilation f , 3c =(−g 2 sin 2 θ W / cos θ W )(g 2 (−2T 3 f +4Q f sin 2 θ W )/(4 cos 2 θ W )) f , 3d =(−g 2 sin 2 θ W / cos θ W )(g 2 (2T 3 f )/(4 cos 2 θ W )) f , 4c =−e f e 2 , T III × T III =2(16f 3d 2 m τ ̃ R 2 m f 2 −4f 3c 2 m τ ̃ R 2 s−4f 3d 2 m τ ̃ R 2 s−4f 3d 2 m f 2 s+f 3c 2 s 2 +f 3d 2 s 2 −f 3c 2 t 2 −f 3d 2 t 2 +2f 3c 2 tu+2f 3d 2 tu−f 3c 2 u 2 −f 3d 2 u 2 )/(m Z 2 −s) 2 , T III × T IV =2f 3c f 4c (4m τ ̃ R 2 s−s 2 +t 2 −2tu+u 2 )/((m Z 2 −s)s) , T IV × T IV =2f 4c 2 (−4m τ ̃ R 2 s+s 2 −t 2 +2tu−u 2 )/s 2 (A.9) | . T | 2 =( T III × T III + T IV × T IV +2 T III × T IV )(×3 for quarks ) A.10 τ ̃ τ ̃ ∗ →t t ̄ I. s-channel H annihilation II. s-channel h annihilation III. s-channel Z annihilation IV. s-channel γ annihilation f , 1a =(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W )(−g 2 m t sin α/(2m W sin β)) f , 2a =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−g 2 m t cos α/(2m W sin β)) f , 3c =(−g 2 sin 2 θ W / cos θ W )(g 2 (−1+4Q t sin 2 θ W )/(4 cos 2 θ W )) f , 3d =(−g 2 sin 2 θ W / cos θ W )(g 2 /(4 cos 2 θ W )) f , 4c =−e t e 2 , T I × T I =2f 1a 2 (−4m t 2 +s)/(−m H 2 +s) 2 , T II × T II =2f 2a 2 (−4m t 2 +s)/(−m h 2 +s) 2 , T III × T III =2(16f 3d 2 m τ ̃ R 2 m t 2 −4f 3c 2 m τ ̃ R 2 s−4f 3d 2 m τ ̃ R 2 s−4f 3d 2 m t 2 s+f 3c 2 s 2 +f 3d 2 s 2 −f 3c 2 t 2 −f 3d 2 t 2 +2f 3c 2 tu+2f 3d 2 tu−f 3c 2 u 2 −f 3d 2 u 2 )/(m Z 2 −s) 2 , T I × T II =2f 1a f 2a (−4m t 2 +s)/((−m H 2 +s)(−m h 2 +s)) , T I × T III =4f 1a f 3c m t (t−u)/((m H 2 −s)(−m Z 2 +s)) , T II × T III =4f 2a f 3c m t (t−u)/((m h 2 −s)(−m Z 2 +s)) , T I × T IV =4f 1a f 4c m t (t−u)/((m H 2 −s)s) , T II × T IV =4f 2a f 4c m t (t−u)/((m h 2 −s)s) , T III × T IV =2f 3c f 4c (4m τ ̃ R 2 s−s 2 +t 2 −2tu+u 2 )/((m Z 2 −s)s) , T IV × T IV =2f 4c 2 (−4m τ ̃ R 2 s+s 2 −t 2 +2tu−u 2 )/s 2 (A.10) | ( . T | 2 =3 T I × T I + T II × T II + T III × T III + T IV × T IV +2 T I × T II +2 T I × T III +2 T I × T IV +2 T II × T III +2 T II × T IV +2 T III × T IV ) A.11 τ ̃ τ ̃ ∗ →hh I. s-channel h annihilation II. s-channel H annihilation III. point interaction IV. t-channel τ ̃ exchange V. u-channel τ ̃ exchange f , 1 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−3g 2 m Z cos 2α sin (α+β)/(2 cos θ W )) f , 2 =(g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(g 2 m Z ( cos 2α cos (α+β)−2 sin (2α) sin (α+β))/(2 cos θ W )) f , 3 =−g 2 2 cos 2α sin 2 θ W /(2 cos 2 θ W ) f , 4 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W ) 2 f , 5 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W ) 2 , T I × T I =(m h 2 −s) −2 , T II × T II =(m H 2 −s) −2 , T III × T III =1 , T IV × T IV =(m τ ̃ R 2 −t) −2 , T V × T V =(m τ ̃ R 2 −u) −2 , T I × T II =1/((m H 2 −s)(m h 2 −s)) , T I × T III =1/(m h 2 −s) , T I × T IV =1/((m h 2 −s)(m τ ̃ R 2 −t)) , T I × T V =1/((m h 2 −s)(m τ ̃ R 2 −u)) , T II × T III =1/(m H 2 −s) , T II × T IV =1/((m H 2 −s)(m τ ̃ R 2 −t)) , T II × T V =1/((m H 2 −s)(m τ ̃ R 2 −u)) , T III × T IV =1/(m τ ̃ R 2 −t) , T III × T V =1/(m τ ̃ R 2 −u) , T IV × T V =1/((m τ ̃ R 2 −t)(m τ ̃ R 2 −u)) (A.11) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +f 4 2 T IV × T IV +f 5 2 T V × T V +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 1 f 4 T I × T IV +2f 1 f 5 T I × T V +2f 2 f 3 T II × T III +2f 2 f 4 T II × T IV +2f 2 f 5 T II × T V +2f 3 f 4 T III × T IV +2f 3 f 5 T III × T V +2f 4 f 5 T IV × T V A.12 τ ̃ τ ̃ ∗ →hA[HA] I. s-channel Z annihilation f , 1 =(−g 2 sin 2 θ W / cos θ W )(g 2 cos [ sin ](α−β)/(2 cos θ W )) , T I × T I =(t−u) 2 /(m Z 2 −s) 2 (A.12) | . T | 2 =f 1 2 T I × T I A.13 τ ̃ τ ̃ ∗ →W + H − I. s-channel H annihilation II. s-channel h annihilation f , 1 =(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W )(−g 2 sin (α−β)/2) f , 2 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−g 2 cos (α−β)/2) , T I × T I =(m H + 4 +(m W 2 −s) 2 −2m H + 2 (m W 2 +s))/(m W 2 (m H 2 −s) 2 ) , T I × T II =(m H + 4 +(m W 2 −s) 2 −2m H + 2 (m W 2 +s))/(m W 2 (m H 2 −s)(m h 2 −s))) , T II × T II =(m H + 4 +(m W 2 −s) 2 −2m H + 2 (m W 2 +s))/(m W 2 (m h 2 −s) 2 ) (A.13) | . T | 2 =f 1 2 T I × T I +f 2 2 T I × T I +2f 1 f 2 T I × T II A.14 τ ̃ τ ̃ ∗ →AA I. s-channel H annihilation II. s-channel h annihilation III. point interaction f , 1 =(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W )(g 2 m Z cos 2β cos (β+α)/(2 cos θ W ) f , 2 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−g 2 m Z cos 2β sin (β+α)/(2 cos θ W ) f , 3 =−g 2 2 cos 2β sin 2 θ W /(2 cos 2 θ W ) , T I × T I =(m H 2 −s) −2 , T II × T II =(m h 2 −s) −2 , T III × T III =1 , T I × T II =1/((m H 2 −s)(m h 2 −s)) , T I × T III =1/(m H 2 −s) , T II × T III =1/(m h 2 −s) (A.14) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 2 f 3 T II × T III A.15 τ ̃ τ ̃ ∗ →hH I. s-channel H annihilation II. s-channel h annihilation III. point interaction IV. t-channel τ ̃ exchange f , 1 =(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W )(g 2 m Z (2 sin 2α cos (β+α)+ sin (β+α) cos 2α)/(2 cos θ W )) f , 2 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−g 2 m Z (2 sin 2α sin (β+α)− cos (β+α) cos 2α)/(2 cos θ W )) f , 3 =−g 2 2 sin 2α sin 2 θ W /(4 cos 2 θ W ) f , 4 =(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W )(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W ) , T I × T I =(m H 2 −s) −2 , T I × T II =1/((m H 2 −s)(m h 2 −s)) , T I × T III =1/(m H 2 −s) , T I × T IV =1/((m H 2 −s)(m τ ̃ R 2 −t)) , T II × T II =(m h 2 −s) −2 , T II × T III =1/(m h 2 −s) , T II × T IV =1/((m h 2 −s)(m τ ̃ R 2 −t)) , T III × T III =1 , T III × T IV =1/(m τ ̃ R 2 −t) , T IV × T IV =(m τ ̃ R 2 −t) −2 (A.15) | . T | 2 = T I × T I + T II × T II + T III × T III + T IV × T IV +2 T I × T II +2 T I × T III +2 T I × T IV +2 T II × T III +2 T II × T IV +2 T III × T IV A.16 τ ̃ τ ̃ ∗ →HH I. s-channel H annihilation II. s-channel h annihilation III. point interaction IV. t-channel τ ̃ exchange V. u-channel τ ̃ exchange f , 1 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−3g 2 m Z cos 2α cos (α+β)/(2 cos θ W )) f , 2 =(g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(g 2 m Z ( cos 2α sin (α+β)+2 sin 2α cos (α+β))/(2 cos θ W )) f , 3 =g 2 2 cos 2α sin 2 θ W /(2 cos 2 θ W ) f , 4 =(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W ) 2 f , 5 =(g 2 m Z sin 2 θ W cos (α+β)/ cos θ W ) 2 (A.16) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +f 4 2 T IV × T IV +f 5 2 T V × T V +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 1 f 4 T I × T IV +2f 1 f 5 T I × T V +2f 2 f 3 T II × T III +2f 2 f 4 T II × T IV +2f 2 f 5 T II × T V +2f 3 f 4 T III × T IV +2f 3 f 5 T III × T V +2f 4 f 5 T IV × T V The T I × T I ⋯ are the same as for τ ̃ τ ̃ ∗ →hh , with ( m h ↔ m H ). A.17 τ ̃ τ ̃ ∗ →H + H − I. s-channel H annihilation II. s-channel h annihilation III. s-channel Z annihilation IV. s-channel γ annihilation V. point interaction f , 1 =(−g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−g 2 (m W cos (β−α)−m Z cos 2β cos (β+α)/(2 cos θ W ))) f , 2 =(g 2 m Z sin 2 θ W sin (α+β)/ cos θ W )(−g 2 (m W sin (β−α)−m Z cos 2β sin (β+α)/(2 cos θ W ))) f , 3 =(−g 2 sin 2 θ W / cos θ W )(−g 2 cos 2θ W /(2 cos θ W )) f , 4 =−e 2 f , 5 =−g 2 2 cos 2β sin 2 θ W /(2 cos 2 θ W ) , T I × T I =(m H 2 −s) −2 , T I × T II =1/((m H 2 −s)(m h 2 −s)) , T I × T III =(t−u)/((m H 2 −s)(m Z 2 −s)) , T I × T IV =(t−u)/(−(m H 2 s)+s 2 ) , T I × T V =1/(m H 2 −s) , T II × T II =(m h 2 −s) −2 , T II × T III =(t−u)/((m h 2 −s)(m Z 2 −s)) , T II × T IV =(t−u)/(−(m h 2 s)+s 2 ) , T II × T V =1/(m h 2 −s) , T III × T III =(t−u) 2 /(m Z 2 −s) 2 , T III × T IV =−((t−u) 2 /((m Z 2 −s)s)) , T III × T V =(t−u)/(m Z 2 −s) , T IV × T IV =(t−u) 2 /s 2 , T IV × T V =(−t+u)/s , T V × T V =1 (A.17) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +f 3 2 T III × T III +f 4 2 T IV × T IV +f 5 2 T V × T V +2f 1 f 2 T I × T II +2f 1 f 3 T I × T III +2f 1 f 4 T I × T IV +2f 1 f 5 T I × T V +2f 2 f 3 T II × T III +2f 2 f 4 T II × T IV +2f 2 f 5 T II × T V +2f 3 f 4 T III × T IV +2f 3 f 5 T III × T V +2f 4 f 5 T IV × T V A.18 →ττ τ ̃ τ ̃ I. t-channel χ exchange II. u-channel χ exchange K=g , 1 N i1 / 2 K′=g , 1 N j1 / 2 , T I × T I =(16K 2 K′ 2 m i χ ̃ m j χ ̃ s)/((m i χ ̃ 2 −t)(m j χ ̃ 2 −t)) , T II × T II =(16K 2 K′ 2 m i χ ̃ m j χ ̃ s)/((m i χ ̃ 2 −u)(m j χ ̃ 2 −u)) , T I × T II =(16K 2 K′ 2 m i χ ̃ m j χ ̃ s)/((m i χ ̃ 2 −t)(m j χ ̃ 2 −u)) (A.18) | . T | 2 = ∑ i,j=1 4 ( T I × T I + T II × T II +2 T I × T II ) A.19 τ ̃ ℓ ̃ ∗ →τ ℓ ̄ I. t-channel χ exchange K=g , 1 N i1 / 2 K′=g , 1 N j1 / 2 , T I × T I =−16K 2 K′ 2 (m τ ̃ R 2 m ℓ ̃ R 2 −tu)/((m i χ ̃ 2 −t)(m j χ ̃ 2 −t)) (A.19) | . T | 2 = ∑ i,j=1 4 T I × T I A.20 →τℓ τ ̃ ℓ ̃ I. t-channel χ exchange K=g , 1 N i1 / 2 K′=g , 1 N j1 / 2 , T I × T I =(16K 2 K′ 2 m i χ ̃ m j χ ̃ s)/((m i χ ̃ 2 −t)(m j χ ̃ 2 −t)) (A.20) | . T | 2 = ∑ i,j=1 4 T I × T I A.21 χ→Zτ τ ̃ I. s-channel τ annihilation II. t-channel τ ̃ exchange f , 1 =−(g 1 N j1 / 2 )(−g 2 /(2 cos θ W )) f , 2 =(g 1 N j1 / 2 )(−g 2 sin 2 θ W / cos θ W ) , T I × T I =2(2 sin 2 θ W ) 2 (m χ ̃ 4 s−m τ ̃ R 4 s+s(−m Z 4 +m Z 2 s+m Z 2 t−st−m Z 2 u)+m τ ̃ R 2 (2m Z 4 −2m Z 2 s+s 2 +st+su)−m χ ̃ 2 (2m Z 4 −2m Z 2 s+s(t+u)))/(m Z 2 s 2 ) , T II × T II =2(m χ ̃ 2 −t)(m τ ̃ R 4 +(m Z 2 −t) 2 −2m τ ̃ R 2 (m Z 2 +t))/(m Z 2 (m τ ̃ R 2 −t) 2 ) , T I × T II =−(2 sin 2 θ W )(m τ ̃ R 4 (s+t−u)+m χ ̃ 2 (−(m Z 2 s)+m Z 2 t+st−t 2 +m τ ̃ R 2 (8m Z 2 −s+t−u)−5m Z 2 u+tu)+(m Z 2 −t)(m Z 2 s−s 2 +m Z 2 t−t 2 −m Z 2 u+u 2 )+m τ ̃ R 2 (2m Z 2 s−s 2 −2m Z 2 t−st−2t 2 −2m Z 2 u+tu+u 2 ))/(m Z 2 s(m τ ̃ R 2 −t)) (A.21) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +2f 1 f 2 T I × T II A.22 χ→γτ τ ̃ I. s-channel τ annihilation II. t-channel τ ̃ exchange f , 1 =−(g 1 N j1 / 2 )(e) f , 2 =(g 1 N j1 / 2 )(e) , T I × T I =4(m χ ̃ 4 −m τ ̃ R 4 −su−m χ ̃ 2 (t+u)+m τ ̃ R 2 (s+t+u))/s 2 , T II × T II =−4(m χ ̃ 2 −t)(m τ ̃ R 2 +t)/(m τ ̃ R 2 −t) 2 , T I × T II =(s 2 +t 2 −u 2 +m τ ̃ R 2 (−s+3t+u)+m χ ̃ 2 (−8m τ ̃ R 2 +s−t+5u))/(s(m τ ̃ R 2 −t)) (A.22) | . T | 2 =f 1 2 T I × T I +f 2 2 T II × T II +2f 1 f 2 T I × T II A.23 χ→τh[H] τ ̃ II. t-channel τ ̃ exchange f , 2 =−(g 1 N j1 / 2 )(−g 2 m Z sin 2 θ W sin [− cos ](α+β)/ cos θ W ) , T II × T II =2(m χ ̃ 2 −t)/(m τ ̃ R 2 −t) 2 (A.23) | . T | 2 =f 2 2 T II × T II

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