Vacuum alignment in a composite 2HDM Chengfeng Cai, in - PowerPoint PPT Presentation

Intro to SM Composite model Summary Backup Vacuum alignment in a composite 2HDM Chengfeng Cai, in collaboration with H-H. Zhang and G. Cacciapaglia School of Physics, Sun Yat-Sen University Based on arXiv:1805.07619 CHEP2018 Conference 20-24 June 2018, Shanghai Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 1 / 18

Intro to SM Composite model Summary Backup The Standard Model What we know from the Standard Model (SM): Fermionic fields: quarks, leptons −→ Matter , Vector fields: photon, W ± , Z , gluons ⇝ Force , Scalar fields: Higgs boson ��� origin of mass . Not explained by SM: Why m h ≪ Λ GUT ? (Hierarchy problem), Dark energy, dark matters , Neutrino masses and oscillation, Matter − antimatter asymmetry, Strong CP problem,... New physics are needed! Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 2 / 18

Intro to SM Composite model Summary Backup Fundamental Composite Higgs Model 2 N f fermions ψ i charged under some gauge Group G TC . Global flavor symmetry G F = SU ( 2 N f ) or SU ( N f ) × SU ( N f ) , Non-abelian G TC , asymptotic freedom −→ ψ i condense in the IR, 〈 ψ i ψ j 〉 ∼ Σ i j ̸ = 0 ⇒ G F → H (1) where H is a subgroup of G F . ψ i : real reps. of G TC −→ SU ( 2 N f ) → SO ( 2 N f ) , ψ i : pseudo-real reps. of G TC −→ SU ( 2 N f ) → Sp ( 2 N f ) . ψ i : complex reps. of G TC −→ SU ( N f ) × SU ( N f ) → SU ( N f ) . pNGBs, coset space G F / H , ∑ U = e i Π ( φ ) , φ i X i Π ( φ ) = (2) i EW gauge group SU ( 2 ) L × U ( 1 ) Y ⊂ H , Higgs doublet ⊂ pNGBs. Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 3 / 18

Intro to SM Composite model Summary Backup Sp ( 2 N ) group Sp ( 2 N ) = Sp ( 2 N , C ) ∩ SU ( 2 N ) , 2 N × 2 N matrices U satisfy � � � N × N UEU T = E , E = , (3) − � N × N S a E + E ( S a ) T = 0 U = e i θ a S a , or (4) Choice of E is not unique,   J S a = OS a O − 1 , � � Σ 0 = OEO T , ˜ 0 1 ± J Σ 0 = J = (5)   ,  , S a ) T = 0 S a Σ 0 + Σ 0 ( ˜ ˜ − 1 0  ... SU ( 4 ) / Sp ( 4 ) : minimal model [E. Katz (2005), B. Gripaio (2009), M. Frigerio (2012), G. Cacciapaglia (2014)], SU ( 6 ) / Sp ( 6 ) : 2HDM. Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 4 / 18

Intro to SM Composite model Summary Backup SU ( 6 ) → Sp ( 6 ) composite model 6 left-handed Weyl spinors ψ , fundamental reps of G TC = SU(2). In the IR, 〈 ψ i ψ j 〉 ∼ Σ i j antisymmetric, SU ( 6 ) → Sp ( 6 ) . NGBs: d.o.f = 35 − 21 = 14 , decomposition: 14 Sp ( 6 ) → ( 2,2,1 ) ⊕ ( 2,1,2 ) ⊕ ( 1,2,2 ) ⊕ ( 1,1,1 ) ⊕ ( 1,1,1 ) (6) Case Higgs SU ( 2 ) L U ( 1 ) Y SU ( 2 ) L Y ψ 1 2 0 ( 2,2,1 ) T 3 2 + ξ T 3 A ψ 2 1 ± 1 / 2 SU(2) 1 3 [( 2,1,2 ) if ξ = 1 ] ψ 3 1 ± ξ/ 2 ψ 1 2 0 T 3 B ψ 2 2 0 SU(2) 1 + SU(2) 2 ( 2,1,2 ) + ( 1,2,2 ) 3 ψ 3 1 ± 1 / 2 Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 5 / 18

Intro to SM Composite model Summary Backup The pNGBs ∑ 14 i = 1 φ i X i , pNGBs: Σ ( φ ) = U ( φ ) Σ , U ( φ ) = exp [ i Π ( φ )] , Π ( φ ) = ( D µ Σ ( φ )) † · D µ Σ ( φ ) χ · Σ † ( φ ) + χ † · Σ ( φ ) � � � � χ = 2 BM † L ( p 2 ) = f 2 Tr − Tr , (7) ψ Before EW symmetry breaking, 〈 φ 4 〉 = 〈 φ 8 〉 = 0 ,   i σ 2 0 0 〈 ψ i ψ j 〉 ∼ Σ = Σ 0 = 0 − i σ 2 0 (8)   0 0 − i σ 2   S 1 H 1 H 2 i Π ( φ ′ ) · Σ 0 = 1 − H T S 2 G (9)   1 2 − H T − G T S 3 2 H 1 ∼ ( 2,2,1 ) , H 2 ∼ ( 2,1,2 ) −− 2HDM G ∼ ( 1,2,2 ) : neutral and charged singlets S 1,2,3 : ( 1,1,1 ) ⊕ ( 1,1,1 ) singlets pesudo-scalars Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 6 / 18

Intro to SM Composite model Summary Backup Vacuum misalignment and EWSB � � EW breaking, 〈 φ 4 〉 = v 1 , 〈 φ 8 〉 = v 2 , tan β = v 2 / v 1 , θ = v 2 1 + v 2 2 / 2 2 f Σ = Ω θ , β Σ 0 Ω T U ( φ ) θ , β = Ω θ , β U 0 ( φ ) Ω † Ω θ , β = R β Ω θ R † (10) θ , β , θ , β , β    cos θ sin θ  2 i σ 2 � 2 0 0 2 � 2 0 sin θ cos θ R β = cos β � 2 − sin β � 2 Ω θ = 2 i σ 2 (11) 0  , 2 � 2 0    sin β � 2 cos β � 2 0 0 0 � 2 Sp(6) Sp(6) SU(6) Ω θ , β SU(6) SU(2) L U(1) em × U(1) Y Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 7 / 18

Intro to SM Composite model Summary Backup Gauge bosons and fermions masses Gauge bosons’ masses and hV V coupling are generated by ( D µ Σ ( φ )) † · D µ Σ ( φ ) L ( p 2 ) ⊃ f 2 Tr � � m 2 � W = 2 g 2 f 2 sin 2 θ , m 2 m 2 W ⇒ 2 f sin θ ≈ 246 GeV Z = , v SM = 2 cos 2 θ W � 2 g 2 f sin θ cos θ = g SM g h 1 WW = g h 1 ZZ cos θ 2 (12) W = hWW cos θ Top Yukawa generated by y ′ y ′ � † α ( ψ T P α � † α ( ψ T P α � � t 1 Q L t c t 2 Q L t c (13) 1 ψ ) + 2 ψ ) Λ 2 R Λ 2 R t t in the θ vacuum: R β ( y t 1 P α 1 + y t 2 P α 2 ) R T β = Y t 1 P α 1 + Y t 2 P α 2 . Top mass and Yukawa coupling: � † � � � Q L t c Y t 1 Tr [ P α 1 · Σ ( φ ′ )] + Y t 2 Tr [ P α 2 · Σ ( φ ′ )] ⊃ L ( p 2 ) f R α � † − Y t 1 � i �� � † + h . c . � t L t c t L t c ∼ − f sin θ Y t 1 � c θ h 1 + � s θ η 2 R R 2 2 3 tt = g SM ⇒ m t = Y t 1 f sin θ , g h 1 ¯ tt cos θ (14) h ¯ Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 8 / 18

Intro to SM Composite model Summary Backup Three origins of the pNGBs potential Gauge loops 2 + 3 g 2 + g ′ 2 3 g 2 + g ′ 2 � g ′ 2 � h 1 V g = − C g f 4 c 2 θ − � s 2 θ + ... (15) 2 2 2 2 Fermions’ loops �� h 1 | Y t 1 | 2 + | Y b 1 | 2 � | Y t 1 | 2 + | Y b 1 | 2 � � V Yuk = − C t f 4 s 2 θ + � s 2 θ + 2 2 f 2 s θ + ϕ 0 h 2 � ℜ Y t 1 Y ∗ t 2 + ℜ Y b 1 Y ∗ � � ℜ Y t 1 Y ∗ t 2 − ℜ Y b 1 Y ∗ � � c θ � s θ 2 s θ + b 2 b 2 2 f 2 f � A 0 η 3 � � � � ℑ Y t 1 Y ∗ t 2 − ℑ Y b 1 Y ∗ ℑ Y t 1 Y ∗ t 2 + ℑ Y b 1 Y ∗ � c θ 2 s θ + � s θ 2 s θ (16) b 2 b 2 2 f 2 f , δ m R ≡ m R 1 − m R 2 , ∆ ≡ m L − m R Explicitly breaking mass term, ( M ≡ m L + m R m L + m R ) 2 2 � V m = − 8 B M ( 1 − ∆ + 2 c θ ) − δ m R c 2 β ( 1 − c θ ) � h 1 h 2 � � − � 2 M + δ m R c 2 β s θ + � δ m R s 2 β s θ 2 + ... (17) 2 2 f 2 f Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 9 / 18

Intro to SM Composite model Summary Backup Composite inert 2HDM A special vacuum: all fermions couple to the same SU ( 2 ) R ∂ V ∂ β = 0 + tadpoles vanish ⇒ y f 2 = Y f 2 = 0, β = 0, Y f 1 = y f 1 , (18) 16 BM / f 4 + 8 B δ m R / f 4 ∂ V ⇒ ∂ θ = 0 cos θ = (19) 2 C t ( | y t 1 | 2 + | y b 1 | 2 ) − C g ( 3 g 2 + g ′ 2 ) Higgs mass and couplings: C g h 1 = C t Z ) ∼ ( 125 GeV ) 2 ⇒ C t ∼ 2 m 2 4 m 2 16 ( 2 m 2 W + m 2 t − (20) g hX X = g SM hX X c θ , (21) SU ( 2 ) R 2 is only broken by gauging T 3 R 2 ∼ Y , to a remnant U ( 1 ) DM . η 0 ⊃ ( 1,2,2 ) ( H + , H 0 ) ⊃ ( 2,1,2 ) , η + , Q DM = 1 fields : (22) Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 10 / 18

Intro to SM Composite model Summary Backup Masses spectrum The neutral/charged components mass matrices in basis ( H 0 , η 0 ) / ( H ± , η ± )   ( 1 + c θ ) − 8 K δ s θ C t Y 2 t 1 f 2 Y 2 M 2 t 1 neut. = C g ( 3 g 2 + g ′ 2 )   ( 1 + c θ − 2 ∆ c θ ) − 4 ( 1 − ∆ ) K δ 8 s θ − ( 1 − ∆ ) c θ Y 2 C t Y 2 t 1 t 1 C g g ′ 2 f 2 � � ( 1 − c θ ) 0 M 2 charg. = M 2 neut. + 0 ( 1 + c θ ) 4 1 1 250 5 500 4 3 750 1000 Fixing 2 0 0 1250 0.5 C g = 1 3 C t , θ = 0.2 1500 1 0.2 � 0.1 - 1 - 1 0.05 ( 2 2 f = 1.2 TeV) Δ Δ - 2 - 2 - 3 - 3 - 5 - 4 - 3 - 2 - 1 0 1 2 - 5 - 4 - 3 - 2 - 1 0 1 2 K δ K δ Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 11 / 18

Intro to SM Composite model Summary Backup The most general vacuum alignment The vacuum Σ = Ω θ , β , γ · Σ 0 · Ω † γ · R † Ω θ , β , γ = R β · R γ · Ω θ · R † θ , β , γ , β   sin γ σ 1 cos γ � 2 0 � � � 2 γ S 14 = 2 β S 21 , 2 θ X 4 , R γ = e − i 2  (23) R β = e i 2 Ω θ = e i 0 0 � 2  − sin γ σ 1 0 cos γ � 2 � 2 f sin τ , sin τ 2 ≡ cos γ sin θ EW vev: v SM = 2 2 , Higgs mixing: � cos γ cos θ � � sin γ h 1 + cos γ cos θ � 1 1 h ′ 2 h 1 − sin γ ϕ 0 ϕ ′ 1 = 0 = 2 ϕ 0 (24) , cos τ cos τ 2 2 g h ′ g h ′ 1 WW 1 ZZ (25) = = cos τ g SM g SM hWW hZZ Top mass and Yukawa coupling, m t = 2cos γ 2 sin θ � Y t 2 sin γ 2 + Y t 1 cos γ 2 cos θ � = f sin ( τ ) Y top (26) 2 2 � � 1 Y t 2 sin γ sin θ 2 + Y t 1 cos θ Y top = (27) cos τ 2 2 Chengfeng Cai (SYSU) Vacuum alignment in a composite 2HDM June 2018 12 / 18