Higgs couplings in the Composite Higgs Models

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1 Higgs couplings in the Composite Higgs Models Aleksandr Azatov Dipartimento di Fisica, Università di Roma La Sapienza and INFN Sezione di Roma University of Padova 6/12/2012

2 Composite Higgs setup Models where Higgs is a composite state give natural solution to the hierarchy problem Higgs can be made naturally light if it is Pseudo Nambu Goldstone Boson(PNGB) (Georgi, Kaplan; Giudice,Grojean,Pomarol,Rattazzi) EWPT ρ requires that the symmetry breaking structure should be SU(2) L SU(2) R /SU(2) V The minimal construction with custodial symmetry is realized in SO(5) SO(4)(Contino, Agashe, Pomarol)

3 Fermions: Partial compositeness SM fermions mix only linearly with composite fermions (partial compositeness mechanism (Kaplan)) Fermion mass generation need separate composite partner for each SM fermion, M 10 TeV

4 The Model Higgs is a PNGB ( we will consider only SO(5)/SO(4) cosets) SM fermion masses are generated by partial compositeness mechanism

5 The Model Higgs is a PNGB ( we will consider only SO(5)/SO(4) cosets) SM fermion masses are generated by partial compositeness mechanism Outline hγγ and hgg couplings (AA, Galloway, arxiv: ) hz γ coupling AA,Di Iura,Contino,Galloway arxiv 1212:xxxx

6 Composite Higgs as a goldstone boson, SILH approach(giudice,grojean, Pomarol,Rattazzi) SILH Lagrangian, parametrize effects of new physics in terms of the higher dimensional operators, the operators relevant for the for the hgg, hγγ, interactions are O HW = i(d µ H) σ i (D ν H)Wµν, i O HB = i(d µ H) (D ν H)B µν O g = (HH )G µν G µν, O BB = (HH )B µν B µν O gg, O BB are contributing to the hgg, hγγ,, O HW, O HB contribute to the

7 Higgs couplings to the gluons/photons O g = (HH )G µν G µν, O BB = (HH )B µν B µν violate the Goldstone symmetry and must be suppressed They must be proportional to the Goldstone Symmetry breaking parameters: SM fermions Yukawa couplings, gauge couplings Only composite partners of the third generation can contribute to the O g, O B

8 Hgg/Hγγ and the β function (Ellis,Gaillard,Nanopoulos;Shifman,Vainshtein,Voloshin,Zakharov) hgg coupling is related to the gluon self energy in the background of the Higgs field 1 L = 1 4 gs 2(µ,h)G2 µν = 1 4 [ 1 g 2 s (µ) i bt i h 4π 2 v ] ln M i (h) ln v matching of the β functions is valid only for the fields heavier than the Higgs boson i ln M i(h) = 1 2 ln DetM(h)M (h) In the SM only top contributes to the Hgg coupling so the coupling is given by 1 v. Exact loop integral has very week dependence on the mass of the field f (m ψ = m h )/f (m ψ = 100 m h ) 1.07

9 Single 5 model We will use two site description with just one composite multiplet of SO(5) Q = 1 2 χ 5/3 + B 1/3 i(χ 5/3 B 1/3 ) T 2/3 + T 2/3 i(t 2/3 T 2/3 ) 2 T 2/3 L = λ q q L P q U Q R + λ t t R P t U Q L + Y f ( Q R Σ 0 )(Σ 0 Q L) + M 5 Q R Q L Σ 0 (0, 0, 0, 0, 1), U e iπ = exp[ 2T 4 h/f ]

10 Hgg in the model L = λ q q L P q U Q R + λ t t R P t U Q L + Y f ( Q R Σ 0 )(Σ 0 Q L) + M 5 Q R Q L log(det M t) v = 2 f cot ( ) 2v f 1 v SM (1 3 2 v SM = f sin v f ) v 2 SM f 2 No dependence on mixing parameters λ and the mass of top partner M 5! (Falkowski; Low,Vichi)

11 Hgg coupling Total contribution is the sum of the diagram with SM top and diagrams with t. What about SM top Yukawa?

12 Top Yukawa coupling Field redefinition Q CCWZ ξq, Σ = (0, 0, 0, sin( h f ), cos( h f )) L = λ q q L P q Q R + λ t t R P t Q L + Y f ( Q R Σ)(Σ Q L ) + M 5 Q R Q L If we promote λ couplings to spurions of SU(2) U(1) SO(5) we can see that, we can parameterize effects of the heavy composite states in terms of the operators ) ( ) ˆλ i O SU(2) L U(1) ˆλi Y G q L (ˆλq Σ Σˆλ t t R sin(2v/f ) q L q L ˆλ qσ 2, t R t R ˆλ t Σ 2

13 Top Yukawa coupling t t ( O(λH/(fM 5 )) 2) m m Zq Zt [ ( Z t 1 + λ t 2 sin 2 (v/f ) M5 [ 2 ytf 2 sin(2v/f ) 1 λt cos2 (v/f ) ( sin 2 (v/f ) (M 5+Yf ) 2 )] M cos2 (v/f ) (M 5+Yf ) 2 )] +... unlike Hgg, top Yukawa strongly depends on the masses of the top partners

14 Hgg vs y t in the model with 5 Top Yukawa coupling strongly depends on the masses of the composite partners Overall Hgg depends only on the v f parameter Contribution of the diagrams with t and t to Hgg sum up to give simple trigonometric rescaling of the Hgg coupling 2 f cos 2v f sin 2v f vs 1 v SM in SM Is it always the case? What about contributions of the b, χ

15 Determinant properties detm t λ q λ t No terms with higher powers of λ q,t λ q q L P q Q R + λ t t R P t Q L promote λ mixing parameters which explicitly break global SO(5) to the spurions which transform under U(1) EM SO(5) ˆλ i e iαq i ˆλ i G,

16 Determinant properties Determinant should be SO(5) U(1) EM invariant det M t = (Σ ˆλ t q)(ˆλ t Σ) P(M, Y, f ), v/f dependence is factorized, in the model with 5 we get Is it always the case? det M t (Σ ˆλt q )(ˆλ t Σ) = sin 2v f

17 Determinant properties t L (t R ) mixes more than with one operator λ q q L P q ξ Q R i λ (i) q q L P q ξ Q (i) R for example t L mixes with 10 and 5 and t R mixes with 10 ˆλ q (ˆλ (5) (10) q, ˆλ q ) ( det M = ( Σ (10) ˆλ q Σ ˆλ(10) q (ˆλ(10) ) ) ( t Σ P 1(M, Y ) + ( ) ) (ˆλ (10) t Σ sin(2v/f ) and v/f dependence is not factorized any more Σ λ (10) t Σ λ (10) t (ˆλ(5) ) ) q P 2(M, Y ), ) ) (ˆλ (5) q sin(v/f ).

18 Example Model 10,5 q L mixes with 10, 5 operators, t R - with 10 ( (ˆλ(10) ) ) t QL L 10+5 = t R SM tr + tr( Q ˆλ(10) R q )ql SM + ( T ˆλ(5) R q )ql SM det M t λ (10) t sin ( ) ( v 2fM10 Ỹ f 5λ (5) q cos ( v f ) ( ) ) f 2 Y 5 Ỹ 5 + fm 5Y 10 λ (10) q. if Ỹ 5 f = Y 5 f = Y 10 f = M 5 = M 10 ( log det M t v 1 v SM [1 + 6λ (10) q 2λ (5) q (5) 2λ q 2λ (10) q ) ] v 2 SM 6f, 2 If 2λ (10) q < λ (5) q < 3 2λ (10) q the coupling Hgg increases, otherwise it is reduced.

19 Model with 14 If t R mixes with 14 and t L mixes with 5 there are two types of Yukawa couplings ˆλ 14 t 14, ˆλ 5 q 5 (Σˆλ 14 t Σ)(Σˆλ 5 q) sin(v/f ) ( 1 5 cos 2 (v/f ) ), (Σˆλ 14 t ˆλ 5 q) sin(v/f )

20 t effects We can get very nontrivial modifications of the Hgg in the case when SM t L (t R ) quark mixes with different representations of the composite fermions If these operators are present in the light quark sectors, we have strong constraints from the Higgs mediated flavor violation d L s R H d R s L (Agashe,Contino)

21 b and χ effects What about χ 5/3 effects? b? Hgg coupling (H H)G µνg µν explicitly violates Goldstone symmetry, so it must be proportional to the explicit symmetry breaking parameters χ 5/3 does not mix with elementary fermions, so cannot contribute to this coupling. Contribution of the partners of the light generations will be suppressed. Naively contribution should be suppressed by the m b i.e. very small In the models where (t, b) L is fully composite this is not the case, we might expect large effects from b

22 Hgg from b Hgg Y M i >m H M i = log(det M b) v y b m b, detm b F (v/f ) ( y b m b F (v/f ) f + 1 v O λ 2 ) q v 2 log(det M b ) v = F (v/f ) f f 2 M 2 Hgg 1 v O ( λ 2 q v 2 f 2 M 2 If b L is fully composite λq M large, we will get large corrections to the Hgg coupling from b Modifications of the Hgg and y b couplings go in the opposite directions )

23 γγ signal modification m b' m h Figure: Contour plots for the rescaling of the σ(gg H) and σ(gg H) BR(H γγ).

24 summary of hgg and hγγ Only the composite partners of the third generation are important for the modifications of the hgg/hγγ couplings. hgg hγγ interactions in the most simple models are given by simple trigonometrical rescaling, independent of the mass of the lightest t These simple relation break down only in the models where top has some nonminimal mixing with composite sector and also if b L becomes fully composite.

25 Operators contributing to the hz γ O HW = i(d µ H) σ i (D ν H)W i µν, O HB = i(d µ H) (D ν H)B µν O BB = (HH )B µν B µν O HW and O HB are not suppressed by the goldstone symmetry, can get large correction interaction is controlled by c HB c HW, symmetry reason?

26 Symmetry properties of hz γ interaction Composite sector must be invariant under SU(2) L SU(2) R symmetry because of the ρ constraints SM B µ couples to the TR 3 of the composite sector. Z B W3 L, A B + W 3 we can introduce the spurious symmetry P LR under, which L R Z Z, A A, < H > < H > higgs vev < H > is invariant because it has vev along the (±1/2, 1/2) components, interaction violates P LR

27 Symmetry properties of hz γ interaction Composite sector must be invariant under SU(2) L SU(2) R symmetry because of the ρ constraints SM B µ couples to the TR 3 of the composite sector. Z B W3 L, A B + W 3 we can introduce the spurious symmetry P LR under, which L R Z Z, A A, < H > < H > higgs vev < H > is invariant because it has vev along the (±1/2, 1/2) components, interaction violates P LR SM yukawa couplings, gauging of SU(2) L and U(1) Y break P LR, is generated

28 P LR in the SILH approach We rewrite O HW, O HB using bidoublet notations O HW = Tr ( (D µ H) σ a D ν H ) W a L,µν, O HB = Tr ( D µ Hσ 3 (D ν H) ) B µν H (H, H ) under P LR : H H, W L B, O HW O HB, since is P LR violating interaction it must be proportional to the P LR violating combination of the couplings

29 Integrating out composite states SU(2) L and U(1) Y explicitly break P LR symmetry is it enough to generate P LR violating combination of the O HW, O HB by integrating out composite states? B couples TR 3 + T U(1) X to the composite states, U(1) X part of the coupling violates P LR however O HW = Tr ( (D µ H) σ a D ν H ) W a L,µν, O HB = Tr ( D µ Hσ 3 (D ν H) ) B µν both operators are sensitive only to the TR 3 part of the coupling, so we need to have some additional source of P LR breaking to generate from the composite sector

30 in CCWZ language CCWZ construction allows to write down lagrangian for nonlinearly realized symmetry breaking G/H Goldstone bosons of spontaneous symmetry breaking can be parametrized by the field U(Π) = e iπ, Π = ΠâT â and by the Maurer-Cartan form iu µ U = d âµt â + EµT a a d µ + E µ 2 d âµ = Aâµ + (D µ π)â + O(π 3 ) f Eµ a = A a µ i f 2 (π Dµ π) a + O(π 4 )

31 List of operators in CCWZ O(p 4 ) lagrangian (Rattazzi,Contino, Pappadopulo,Marzocca) O 1 = f 2 Tr(d µ d µ ) O H O ± µν 3 = Tr(E L E L,µν) ± Tr(E µν R E R,µν) O W, O B - S parameter O ± 4 = itr(e µν L ± Eµν)[d R µ, d ν ]) O HW, O HB interaction O(d 4 ) dim 8 operators P LR properties E L,R P LR E R,L P LR, P LR = Diag( 1, 1, 1, 1, 1) d P LR d P LR O 4 = itr(e µν L Eµν)[d R µ, d ν ]) the only operator that contributes to the

32 O 4 O 4 = itr(e L µν E R µν)[d µ, d ν ]) Higgs comes from the covariant derivative, µ h so this coupling will have no goldstone suppression No elementary composite mixing is needed!partners of the light fermions are important.

33 Zγ coupling in the SM This decay is generated by the loops of W ± and t ( ) 3 Γ(h Zγ) = 1 32π A 2 mh 2 1 m2 Z m A = αg h 2 4πm w (A F + A W ). The SM loop is dominated by the contribution of W, A W /A F 18

34 Zγ coupling in the SM This decay is generated by the loops of W ± and t ( ) 3 Γ(h Zγ) = 1 32π A 2 mh 2 1 m2 Z m A = αg h 2 4πm w (A F + A W ). The SM loop is dominated by the contribution of W, A W /A F 18 top contribution is suppressed because top coupling to Z is small T Z T 3 L 2q t sin 2 θ 0.2, so new fermions can be very important

35 vs hγγ What is the difference between hγγ and loops? Not all the fermions have the same couplings to Z Z can couple to two different mass eigenstates, so in the loop we can have two different fermions higgs derivative couplings are important

36 Derivative couplings/absence of log divergence L Zγ = [ ] µh λ h ab f ψ aγ µ ψ b + λ Z abz µ ψaγ µ ψ b + q ψ δ ab A µ ψaγ µ ψ b a,b from NDA is log divergent? Loop function is antisymmetric in µ, ν(higgs and Z indices) Amplitude Tr(λ h λ Z ) Tr(λ Z λ h ) = 0 we need at least one mass insertion no log divergence

37 Model with 5 under SU(2) L SU(2) R : 5 = (2, 2) + 1, the breaking of the SO(5) does not break P LR no in the absence of the elementary composite mixing top Yukawa coupling ( breaks down P LR so these effects will be λ suppressed by O 2 v 2 M 2 f ), however in the SM top contribution is 2 much smaller than the W contribution. A SM A W 20A top corrections from t in the loops will be of the order of 0.05 v 2 f 2 modification is dominated by the trigonometric rescaling of the W coupling, A 5 A SM 1 v 2 /f 2

38 Model with 10 under SU(2) L SU(2) R : 10 = (2, 2) + (3, 1) + (1, 3) Different masses and interactions of (3, 1) and (1, 3) respect SU(2) L SU(2) R but break P LR L = M y ef y o f (3, 1) (3, 1) Ignore elementary composite mixing L H+Z = 10( E)10 λ (1,3) 10P ( (2,2) d P (1,3) 10 ) λ (3,1) 10P ( (2,2) d P (3,1) 10 ).

39 Model with 10 In the limit M M the loop function for the amplitude simplifies and we get A y of ( f 2 y 2 e + 4M (fy e + fy o ) + 8M 2) M (fy e + 2M) (fy o + M) v f 2 4 M M Note that not only top partners contribute to the amplitude but also partners of the light generation! If we look at the ratio of the new physics effects to the contribution of the SM top we will get ( v ) 2 M ratio M M 15N gener f M v f 2

40 P LR vs Z bb constraints Large modification to the requires P LR breaking in the composite sector. In order to reproduce the top mass, electroweak doublet q L = (t L, b L ) must mix strongly with the composite sector. Z bb constraints require b L to mix strongly only with the operator which respects P LR (Agashe,Contino, Pomarol,DaRold) in MCHM5 (b L B(1/2, 1/2)))

41 P LR vs Z bb constraints Large modification to the requires P LR breaking in the composite sector. In order to reproduce the top mass, electroweak doublet q L = (t L, b L ) must mix strongly with the composite sector. Z bb constraints require b L to mix strongly only with the operator which respects P LR (Agashe,Contino, Pomarol,DaRold) in MCHM5 (b L B(1/2, 1/2))) model with 5 has an accidental P LR (Contino,Rattazzi,Pappadopulo,Marzocca) symmetry due to the fact that 5 = 1 + (2, 2) SO(5)/SO(4) breaking cannot split masses inside (2, 2)

42 Constructing realistic model with P LR breaking q L = (t, b) L must mix with 5 in order to be protected from Z bb

43 Constructing realistic model with P LR breaking q L = (t, b) L must mix with 5 in order to be protected from Z bb Take MCHM5 but mix b R with 10 instead of 5 Minimal P LR model λ 10 q q L q L fine λ 5 q 5 5 λ b b R m b λ 10 q λ b λ 10 q λ t t R m t λ 5 qλ t λ 5 q, Z bb is L mixing = λ 10 q q L P q 10 + λ 5 q q L P q 5 + λ b br P b 10 + λ t t R P t 5 Forbid mixing between 10 and 5 imposing different U(1) X charges

44 Model with 10, numerical calculation Figure: ratio of the A NP /A top for the model with 10 for one generation, red f = 500, blue f = 800 GeV Figure: ratio of the Γ NP /Γ SM in the model with 10 with three generations

45 Conclusion We reviewed the Higgs couplings in the composite models Modifications to the hgg/hγγ are almost always small and insensitive to the mass spectrum of the composite resonances We studied in the composite models, unlike hgg/hγγ it is not suppressed by the goldstone symmetry is sensitive to the mass spectrum of the partners of the light fermions In the most simplistic models the modification of the is small however in the case of the large P LR breaking in the strong sector we can have O(1) deviation from the SM value

46 L = i ψ 1 ψ 1 + i ψ 2 ψ 2 + iα µh( ψ 1 γ µ ψ 2 ψ 2 γ µ ψ 1 ) +m 1 ψ 1 ψ 1 + m 2 ψ 2 ψ 2 + T Z 12 ψ 1 Zψ 2 + T Z 21 ψ 2 Zψ 1 (1) A F (12) = 2α(m 2 m 1 ) π 2 Q ψ F (m 1, m 2 )T 12 Q ψ (2) 2F (m 1, m 2, m 2 h, m2 z ) = 1 2(m 2 h M2 z )2 [ (m 2 h M2 z ) {m 1 (2m 1 (m 1 + m 2 ) m 2 h + M2 z )C 0 (m 1, m 1, m 2 ) + (m 1 m 2 ) }] 1 + [2M 2 ] 2(m h 2 M2 z )2 z (m 1 + m 2 )B 0 (M2 z, m2 1, m2 2 ) 2M2 z (m 1 + m 2 )B 0 (m2 h, m 1, m 2 ) (m 1 + m 2 ) (m 2 h M2 z ) (3)

47 LHC and hz γ(low, Gainer, Schwaller,Keung)