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HIGGS SECTOR IN THE SUPERSYMMETRIC EXTENSION OF THE STANDARD MODEL - - PowerPoint PPT Presentation
HIGGS SECTOR IN THE SUPERSYMMETRIC EXTENSION OF THE STANDARD MODEL - - PowerPoint PPT Presentation
HIGGS SECTOR IN THE SUPERSYMMETRIC EXTENSION OF THE STANDARD MODEL WITH LIGHT SGOLDSTINOS Sergey Demidov, Dmitry Gorbunov, Ekaterina Kriukova Lomonosov Moscow State University, Institute for Nuclear Research of RAS 8th International Conference
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MSSM Lagrangian
◮ sum over all gauge superfields Wα → kinetic terms of gauge
fields 1 4
- α
- d2θTrWαW α + h.c.,
(1)
◮ K¨
ahler potential (sum over all matter superfields Φk and Higgs fields)
- d2θd2¯
θ
- k
Φ†
keg1V1+g2V2+g3V3Φk,
(2)
◮ superpotential
- d2θǫij
- µHi
DHj U + Y L abLj aE c b Hi D+
+Y D
abQj aDc bHi D + Y U abQi aUc bHj U
- + h.c.
(3)
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Spontaneous SUSY breaking
Goldstone theorem ⇒ goldstino fermion G, its superpartner sgoldstino φ.
Chiral superfield
Φ = φ + √ 2θG + Fφθ2, where Fφ is an auxiliary field. Its nonzero VEV, Fφ ≡ F = 0, breaks the supersymmetry. The lagrangian of goldstino supermultiplet: LΦ =
- d2θd2¯
θΦ†Φ −
- d2θFΦ + h.c.
- (4)
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Lagrangian of the model with SUSY breaking
L = LK + Lsuperpotential + Lgauge + LΦ, (5) LK =
- d2θd2¯
θ
- k
- 1 − m2
k
F 2 Φ†Φ
- Φ†
keg1V1+g2V2+g3V3Φk,
(6) Lsuperpotential =
- d2θǫij
- µ − B
F Φ
- Hi
DHj U+
+
- Y L
ab + AL ab
F Φ
- Lj
aE c b Hi D +
- Y D
ab + AD ab
F Φ
- Qj
aDc bHi D+
+
- Y U
ab + AU ab
F Φ
- Qi
aUc bHj U
- + h.c.,
(7) Lgauge = 1 4
- α
- d2θ
- 1 + 2Mα
F
- TrWαW α + h.c.,
(8)
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Potential of scalar fields p.1
V = V11 + V12 + V21 + V22, (9) V11 = g2
1
8
- 1 + M1
F (φ + φ∗) −1 h†
dhd − h† uhu−
−φ∗φ F 2
- m2
dh† dhd − m2 uh† uhu
2 , (10) V12 = g2
2
8
- 1 + M2
F (φ + φ∗) −1 h†
dσahd + h† uσahu−
−φ∗φ F 2
- m2
dh† dσahd + m2 uh† uσahu
2 . (11) Here g1, g2 are coupling constants of the groups U(1)Y , SU(2)L, M1, M2 are soft masses, corresponding to gauginos, √ F is a scale
- f supersymmetry breaking, σa are Pauli matrices.
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Potential of scalar fields p.2
V21 =
- 1 − m2
u
F 2 h†
uhu − m2 d
F 2 h†
dhd − m4 u
F 4 φ∗φh†
uhu − m4 d
F 4 φ∗φh†
dhd
−1
- F +
- −h0
dh0 u + h−h+ B
F − m2
u + m2 d
F 2 φ∗
- µ − B
F φ
- 2
, (12) V22 = µ2 F 2 |φ|2 m2
uh† dhd + m2 dh† uhu
- +
- µ − B
F φ
- 2
h†
dhd + h† uhu
- .
(13) Higgs doublets hd = h0
d
h−
- , hu =
h+ h0
u
- ,
µ is a real parameter of higgsino mixing from superpotential.
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Some notation
vu ≡ v sin β, vd ≡ v cos β, v = 174 GeV The expansion of fields around electroweak vacuum h0
u = vu + 1
√ 2 (h cos α + H sin α) + i √ 2 A cos β, (14) h0
d = vd + 1
√ 2 (−h sin α + H cos α) + i √ 2 A sin β. (15) Extract scalar s and pseudoscalar p from sgoldstino field φ = 1 √ 2 (s + ip). (16) Introduce masses m2
Z ≡ g2 1 + g2 2
2 v2, m2
A ≡ m2 u + m2 d + 2µ2.
(17)
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Trilinear couplings before mixing
sHH term. 1 F √ 2
- −v2
4
- g2
1 M1 + g2 2 M2
- (2 cos 2α cos 2β − sin 2α sin 2β + 1) +
+µ sin 2α m2
A
2
- 1 − sin 2β
sin 2α
- − µ2
- .
(18) shh term. 1 F √ 2 v2 4
- g2
1 M1 + g2 2 M2
- (2 cos 2α cos 2β − sin 2α sin 2β − 1) −
−µ sin 2α m2
A
2
- 1 + sin 2β
sin 2α
- − µ2
- .
(19) Other sgoldstino-Higgs vertices: shH, sAA, sh+h−, pAH, pAh, ssH, ssh, ppH, pph.
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Rotation towards mass basis p.1
Mass terms have the following form: 1 2 (H h s) m2
H
Y /F m2
h
X/F Y /F X/F m2
s
H h s
- +
+ 1 2 (A p)
- m2
A
Z/F Z/F m2
p
A p
- ,
(20) where X = v g 2
1 M1 + g 2 2 M2
2 v 2 cos 2β sin (α + β) + µm2
A sin 2β sin (α − β)+
+
- m2
A − 2µ2
µ cos (α + β)
- (21)
Y = −v g 2
1 M1 + g 2 2 M2
2 v 2 cos 2β cos (α + β) + µm2
A sin 2β cos (α − β)+
+
- 2µ2 − m2
A
- µ sin (α + β)
- (22)
Z = µv
- m2
A − 2µ2
. (23)
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Rotation towards mass basis p.2
Mixing angles, formulae are valid in approximation θ, ψ, ξ ≪ 1 θ = X F(m2
h − m2 s ),
ψ = Y F(m2
H − m2 s ),
ξ = Z F(m2
A − m2 p).
(24) Old fields via new ones (mass matrix is diagonal in new basis) H = ˜ H − ˜ s Y F(m2
H − m2 s ),
(25) h = ˜ h − ˜ s X F(m2
h − m2 s ),
(26) s = ˜ H Y F(m2
H − m2 s ) + ˜
h X F(m2
h − m2 s ) + ˜
s, (27) A = ˜ A − ˜ p Z F(m2
A − m2 p),
(28) p = ˜ A Z F(m2
A − m2 p) + ˜
p. (29)
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New trilinear couplings
Order Vertex Example of coefficient hhh, HHH, hhH, hHH, hh+h−, Hh+h−, hAA, HAA Chhh, CHAA 1 sHH, shh, shH, sAA, sh+h−, pAH, pAh CsHH/F, CpAh/F 2 ssH, ssh, ppH, pph CssH/F 2, Cpph/F 2 ˜ s ˜ H ˜ H F
- CsHH − ChHH
X m2
h − m2 s
− 3CHHH Y m2
H − m2 s
- ,
(30) ˜ s˜ h˜ h F
- Cshh − 3Chhh
X m2
h − m2 s
− ChhH Y m2
H − m2 s
- .
(31)
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Numerical computation of sgoldstino production cross section
Main process is the gluon fusion, gg → s. Tree-level — due to vertex sgg in sgoldstino lagrangian: −
1 2 √ 2 M3 F sG µνGµν.
Integrate gluon distribution functions by momentum fraction: σ(pp → s) = σ0τ 1
τ
dx x g(x, m2
s )g(τ
x , m2
s ),
(32) σ0 = πM2
3
32F 2 , τ = m2
s
S , √ S = 13 TeV. (33) Table CTEQ6L PDF for calculations in leading order. Loop QCD contributions: K-factor ≃ 1.6.
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Sgoldstino production cross section for different centre-of-mass energies
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Numerical computation of cross section pp → s → hh
Fix tan β = 10, µ, mA = 5 TeV, ms = 1 TeV, M1 = 1 TeV, M2 = 1 TeV, M3 = 3 TeV, √ F = 20 TeV. Using them find α, m2
H, X, Y , θ, ψ.
Small mixing angles
Consider only points of parameter space where θ < 0.3, ψ < 0.3.
Narrow width approximation
σ (pp → s → hh) = σprod(pp → s)Br(s → hh), (34) Br(s → hh) = Γ (s → hh) Γtot (s) . (35)
Sgoldstino decay channels
s → hh, s → gg, s → WW , s → ZZ, s → γZ, s → γγ Given the mixing with h, H, we compute widths and Br
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Sgoldstino decay widths [Zwirner et al., 2000]
Γ (s → gg) = 1 4π M2
3
F 2 m3
s ,
Γ (s → hh) = 1 8πms C˜
s˜ h˜ h
F 2 , (36) Γ (s → γγ) = 1 32π 1 F 2
- M1 cos2 θW + M2 sin2 θW
2 m3
s ,
(37) Γ (s → γZ) = 1 16π (M2 − M1)2 F 2 cos2 θW sin2 θW m3
s
- 1 − m2
Z
m2
s
3 . (38) CsWWT = −M2, CsZZT = −M1 sin2 θW − M2 cos2 θW , (39) C˜
sWWL = −
√ 2 v sin(β − α) − √ 2 v cos(β − α) = 2C˜
sZZL,
(40) Γ (s → WW ) = 1 16π m4
W
ms C 2
sWWT
F 2
- 6 − 4 m2
s
m2
W
+ m4
s
m4
W
- − 6
√ 2 CsWWT F C˜
sWWL×
×
- 1 −
m2
s
2m2
W
- + C 2
˜ sWWL
- 3 − m2
s
m2
W
+ m4
s
4m4
W
1 − 4m2
W
m2
s
, (41) Γ (s → ZZ) = 1 8π m4
Z
ms C 2
sZZT
4F 2
- 6 − 4 m2
s
m2
Z
+ m4
s
m4
Z
- − 3
√ 2 CsZZT F C˜
sZZL×
×
- 1 − m2
s
2m2
Z
- + C 2
˜ sZZL
- 3 − m2
s
m2
Z
+ m4
s
4m4
Z
1 − 4m2
Z
m2
s
. (42)
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Experimental searches for scalar resonances
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Br for main sgoldstino decay channels
Figure: Branching ratio of sgoldstino.
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Dependence of di-Higgs production cross section on tan β
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Dependence of di-Higgs production cross section on M3
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Curves Br(s → hh) = 0.125
Fixed parameter values M1 = M2 = 1 TeV, √ F = 20 TeV.
(a) Curves in plane (tan β, µ) (b) Curves in plane (M3, µ) (c) Curves in plane (ms, µ) (d) Curves in plane (mA, µ)
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Upper limit on M3/F from experimental data
Regime 1:2:1
Br(s → hh) = Br(s → ZZ) = 0.25, Br(s → WW ) = 0.5. For fixed ms σmax
hh , σmax WW , σmax ZZ
are upper limits on hh, WW , ZZ production cross section in resonant scalar decays. Then σprod < σmax
hh /Br(s → hh) etc.
σprod ∼ M2
3/F 2 ⇒
- M3/3 TeV
( √ F/20 TeV)2 max =
- σmax
prod
σ′
prod
≤
- σmax
XX
σ′
prodBr(s → XX).
(43)
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Upper limit on M3/F from experimental data at 95%CL
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Lower limit on √ F assuming M3 ≥ 1.9 TeV
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Comparison with numerical results
Figure: Analytically and numerically obtained branchings.
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