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 on New Frontiers in Physics, 27 August 2019

Model with sgoldstino ◮ Supersymmetric model with low-scale SUSY breaking in hidden sector. Goldstone theorem ⇒ goldstino G , sgoldstino φ . ◮ Heavy fields are integrated out. Effective theory with MSSM fields, goldstino multiplet and gravitino. ◮ Sgoldstino can decay to SM particles (R-even). Interactions in effective lagrangian: sgg , shh , sW + W − , sZZ , s γγ , sZ γ . ◮ Aim: consider the possibility of increase in di-Higgs production cross section at LHC due to processes with sgoldstino gg → s → hh . ◮ Main SM diagrams for di-Higgs production.

MSSM Lagrangian ◮ sum over all gauge superfields W α → kinetic terms of gauge fields 1 � d 2 θ Tr W α W α + h . c ., � (1) 4 α ◮ K¨ ahler potential (sum over all matter superfields Φ k and Higgs fields) � d 2 θ d 2 ¯ � Φ † k e g 1 V 1 + g 2 V 2 + g 3 V 3 Φ k , θ (2) k ◮ superpotential � � D H j d 2 θǫ ij µ H i U + Y L ab L j a E c b H i D + � b H j + Y D ab Q j a D c b H i D + Y U ab Q i a U c + h . c . (3) U

Spontaneous SUSY breaking Goldstone theorem ⇒ goldstino fermion G , its superpartner sgoldstino φ . Chiral superfield √ 2 θ G + F φ θ 2 , where F φ is an auxiliary field. Φ = φ + Its nonzero VEV, � F φ � ≡ F � = 0, breaks the supersymmetry. The lagrangian of goldstino supermultiplet: �� � � d 2 θ d 2 ¯ θ Φ † Φ − d 2 θ F Φ + h . c . L Φ = (4)

Lagrangian of the model with SUSY breaking L = L K + L superpotential + L gauge + L Φ , (5) 1 − m 2 � � � d 2 θ d 2 ¯ � Φ † F 2 Φ † Φ k k e g 1 V 1 + g 2 V 2 + g 3 V 3 Φ k , L K = θ (6) k � �� � µ − B d 2 θǫ ij D H j H i L superpotential = F Φ U + ab + A L ab + A D � � � � Y L ab L j a E c b H i Y D ab Q j a D c b H i + F Φ D + F Φ D + ab + A U � � � b H j Y U ab Q i a U c + F Φ + h . c ., (7) U L gauge = 1 � � 1 + 2 M α � Tr W α W α + h . c ., � d 2 θ (8) 4 F α

Potential of scalar fields p.1 V = V 11 + V 12 + V 21 + V 22 , (9) � − 1 � V 11 = g 2 � 1 + M 1 1 h † F ( φ + φ ∗ ) d h d − h † u h u − 8 �� 2 − φ ∗ φ � m 2 d h † d h d − m 2 u h † u h u , (10) F 2 � − 1 � V 12 = g 2 � 1 + M 2 h † 2 F ( φ + φ ∗ ) d σ a h d + h † u σ a h u − 8 �� 2 − φ ∗ φ � d h † m 2 d σ a h d + m 2 u h † u σ a h u . (11) F 2 Here g 1 , g 2 are coupling constants of the groups U (1) Y , SU (2) L , √ M 1 , M 2 are soft masses, corresponding to gauginos, F is a scale of supersymmetry breaking, σ a are Pauli matrices.

Potential of scalar fields p.2 � − 1 1 − m 2 u h u − m 2 d h d − m 4 u h u − m 4 � F 2 h † F 4 φ ∗ φ h † F 2 h † u d F 4 φ ∗ φ h † u d V 21 = d h d 2 F − m 2 u + m 2 � u + h − h + � � B � ��� µ − B � − h 0 d h 0 d φ ∗ � � � F + F φ , (12) � � F 2 � 2 � V 22 = µ 2 � � � µ − B F 2 | φ | 2 � � � m 2 u h † d h d + m 2 d h † h † d h d + h † � � + F φ . u h u u h u � � � (13) � h 0 � h + � � d Higgs doublets h d = , h u = , h − h 0 u µ is a real parameter of higgsino mixing from superpotential.

Some notation v u ≡ v sin β , v d ≡ v cos β , v = 174 GeV The expansion of fields around electroweak vacuum u = v u + 1 i h 0 √ ( h cos α + H sin α ) + √ A cos β, (14) 2 2 d = v d + 1 i h 0 √ √ ( − h sin α + H cos α ) + A sin β. (15) 2 2 Extract scalar s and pseudoscalar p from sgoldstino field 1 √ φ = ( s + ip ) . (16) 2 Introduce masses Z ≡ g 2 1 + g 2 m 2 v 2 , m 2 A ≡ m 2 u + m 2 d + 2 µ 2 . 2 (17) 2

Trilinear couplings before mixing sHH term. − v 2 1 � g 2 1 M 1 + g 2 � � √ 2 M 2 (2 cos 2 α cos 2 β − sin 2 α sin 2 β + 1) + 4 2 F � m 2 � 1 − sin 2 β � �� − µ 2 A + µ sin 2 α . (18) 2 sin 2 α shh term. � v 2 1 g 2 1 M 1 + g 2 � � √ (2 cos 2 α cos 2 β − sin 2 α sin 2 β − 1) − 2 M 2 4 F 2 � m 2 � � �� 1 + sin 2 β − µ 2 A − µ sin 2 α . (19) 2 sin 2 α Other sgoldstino-Higgs vertices: shH , sAA , sh + h − , pAH , pAh , ssH , ssh , ppH , pph .

Rotation towards mass basis p.1 Mass terms have the following form:   m 2 0 Y / F � H � 1 H m 2 2 ( H h s ) h 0 X / F +   h m 2 s Y / F X / F s � m 2 � � � + 1 Z / F A 2 ( A p ) A , (20) m 2 p Z / F p where � g 2 1 M 1 + g 2 2 M 2 v 2 cos 2 β sin ( α + β ) + µ m 2 X = v A sin 2 β sin ( α − β )+ 2 m 2 A − 2 µ 2 � � � + µ cos ( α + β ) (21) � g 2 1 M 1 + g 2 2 M 2 v 2 cos 2 β cos ( α + β ) + µ m 2 Y = − v A sin 2 β cos ( α − β )+ 2 2 µ 2 − m 2 � � � + µ sin ( α + β ) (22) A � m 2 A − 2 µ 2 � Z = µ v . (23)

Rotation towards mass basis p.2 Mixing angles, formulae are valid in approximation θ, ψ, ξ ≪ 1 X Y Z θ = s ) , ψ = s ) , ξ = p ) . F ( m 2 h − m 2 F ( m 2 H − m 2 F ( m 2 A − m 2 (24) Old fields via new ones (mass matrix is diagonal in new basis) Y H = ˜ H − ˜ s s ) , (25) F ( m 2 H − m 2 X h = ˜ h − ˜ s ) , (26) s F ( m 2 h − m 2 Y X s = ˜ s ) + ˜ H h s ) + ˜ s , (27) F ( m 2 F ( m 2 H − m 2 h − m 2 Z A = ˜ A − ˜ p p ) , (28) F ( m 2 A − m 2 Z p = ˜ A p ) + ˜ p . (29) F ( m 2 A − m 2

New trilinear couplings Order Vertex Example of coefficient hhh , HHH , hhH , hHH , 0 C hhh , C HAA hh + h − , Hh + h − , hAA , HAA sHH , shh , shH , sAA , 1 C sHH / F , C pAh / F sh + h − , pAH , pAh C ssH / F 2 , C pph / F 2 2 ssH , ssh , ppH , pph s ˜ H ˜ � � ˜ H X Y C sHH − C hHH − 3 C HHH , (30) m 2 m 2 h − m 2 H − m 2 F s s s ˜ h ˜ ˜ � � h X Y C shh − 3 C hhh − C hhH . (31) m 2 m 2 h − m 2 H − m 2 F s s

Numerical computation of sgoldstino production cross section Main process is the gluon fusion, gg → s . Tree-level — due to 1 M 3 F sG µν G µν . vertex sgg in sgoldstino lagrangian: − √ 2 2 Integrate gluon distribution functions by momentum fraction: � 1 d x s ) g ( τ x g ( x , m 2 x , m 2 σ ( pp → s ) = σ 0 τ s ) , (32) τ σ 0 = π M 2 τ = m 2 √ 3 s 32 F 2 , S , S = 13 TeV . (33) Table CTEQ6L PDF for calculations in leading order. Loop QCD contributions: K-factor ≃ 1.6.

Sgoldstino production cross section for different centre-of-mass energies

Numerical computation of cross section pp → s → hh Fix tan β = 10, µ , m A = 5 TeV, m s = 1 TeV, M 1 = 1 TeV, √ M 2 = 1 TeV, M 3 = 3 TeV, F = 20 TeV. Using them find α , m 2 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 ) = σ prod ( pp → s ) Br ( s → hh ) , (34) Br ( s → hh ) = Γ ( s → hh ) . (35) Γ tot ( s ) Sgoldstino decay channels s → hh , s → gg , s → WW , s → ZZ , s → γ Z , s → γγ Given the mixing with h , H , we compute widths and Br

Sgoldstino decay widths [Zwirner et al., 2000] M 2 Γ ( s → gg ) = 1 1 C ˜ s ˜ h ˜ F 2 m 3 3 h s , Γ ( s → hh ) = F 2 , (36) 4 π 8 π m s 1 1 � 2 m 3 M 1 cos 2 θ W + M 2 sin 2 θ W � Γ ( s → γγ ) = s , (37) F 2 32 π � 3 ( M 2 − M 1 ) 2 � 1 − m 2 1 cos 2 θ W sin 2 θ W m 3 Z Γ ( s → γ Z ) = . (38) s F 2 m 2 16 π s C sZZ T = − M 1 sin 2 θ W − M 2 cos 2 θ W , C sWW T = − M 2 , (39) √ √ 2 2 C ˜ sWW L = − sin( β − α ) − cos( β − α ) = 2 C ˜ sZZ L , (40) v v � C 2 � � m 4 6 − 4 m 2 + m 4 √ 1 2 C sWW T sWW T W s s Γ ( s → WW ) = − 6 C ˜ sWW L × m 2 m 4 16 π F 2 m s F W W �� � � � � 1 − 4 m 2 m 2 3 − m 2 m 4 s + C 2 s s W × 1 − + , (41) ˜ sWW L 2 m 2 m 2 4 m 4 m 2 W W W s � C 2 � � m 4 6 − 4 m 2 + m 4 √ Γ ( s → ZZ ) = 1 2 C sZZ T sZZ T Z s s − 3 C ˜ sZZ L × 4 F 2 m 2 m 4 8 π m s F Z Z �� � � � � 1 − m 2 3 − m 2 + m 4 1 − 4 m 2 s + C 2 s s Z (42) × . sZZ L ˜ 2 m 2 m 2 4 m 4 m 2 s Z Z Z

Experimental searches for scalar resonances

Br for main sgoldstino decay channels Figure: Branching ratio of sgoldstino.

Dependence of di-Higgs production cross section on tan β

Dependence of di-Higgs production cross section on M 3

Curves Br ( s → hh ) = 0 . 125 √ Fixed parameter values M 1 = M 2 = 1 TeV, F = 20 TeV. (a) Curves in plane (tan β, µ ) (b) Curves in plane ( M 3 , µ ) (c) Curves in plane ( m s , µ ) (d) Curves in plane ( m A , µ )

Upper limit on M 3 / F from experimental data Regime 1:2:1 Br ( s → hh ) = Br ( s → ZZ ) = 0 . 25, Br ( s → WW ) = 0 . 5. For fixed m s σ max hh , σ max WW , σ max are upper limits on hh , WW , ZZ production ZZ cross section in resonant scalar decays. Then σ prod < σ max hh / Br ( s → hh ) etc. 3 / F 2 ⇒ σ prod ∼ M 2 � σ max � � max σ max � M 3 / 3 TeV prod XX √ = ≤ prod Br ( s → XX ) . (43) σ ′ σ ′ F / 20 TeV) 2 ( prod

Upper limit on M 3 / F from experimental data at 95%CL

√ Lower limit on F assuming M 3 ≥ 1 . 9 TeV