MODIFIED COUPLINGS FOR A LIGHT COMPOSITE HIGGS. Roberto Contino Università di Roma La Sapienza

MODIFIED COUPLINGS FOR A LIGHT COMPOSITE HIGGS Roberto Contino Università di Roma La Sapienza

Part 1 Effective Lagrangians for a light Higgs

In absence of a direct observation of new particles, our ignorance of the EWSB sector can be parametrized in terms of an effective Lagrangian The explicit form of the Lagrangian depends on the assumptions one makes If new particles are discovered they can be included in the Lagrangian in a bottom-up approach 3

In absence of a direct observation of new particles, our ignorance of the EWSB sector can be parametrized in terms of an effective Lagrangian The explicit form of the Lagrangian depends on the assumptions one makes If new particles are discovered they can be included in the Lagrangian in a bottom-up approach I WILL ASSUME: 1) SU(2)LxU(1)Y is linearly realized at high energies 2) h(x) is a scalar (CP even) and is part of an SU(2)L doublet H(x) 3) The EWSB dynamics has an (approximate) custodial symmetry global symmetry includes: SU(2)LxSU(2)R SU(2)V 3

In absence of a direct observation of new particles, our ignorance of the EWSB sector can be parametrized in terms of an effective Lagrangian The explicit form of the Lagrangian depends on the assumptions one makes If new particles are discovered they can be included in the Lagrangian in a bottom-up approach POWER COUNTING: each extra derivative costs a factor 1/Λ each extra power of H(x) costs a factor g /Λ 1/f 4

In absence of a direct observation of new particles, our ignorance of the EWSB sector can be parametrized in terms of an effective Lagrangian The explicit form of the Lagrangian depends on the assumptions one makes If new particles are discovered they can be included in the Lagrangian in a bottom-up approach POWER COUNTING: each extra derivative costs a factor 1/Λ each extra power of H(x) costs a factor g /Λ 1/f For a strongly-interacting Higgs: Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 1 f 1 Λ 4

The list of dim=6 operators of the effective Lagrangian has been known in the literature since long time Buchmuller and Wyler, NPB 268 (1986) 621 In the following I will follow the SILH parametrization proposed in: Giudice, Grojean, Pomarol, Rattazzi JHEP 0706 (2007) 045 L SILH = c H 2f 2 µ H H µ H H + c T 2f 2 c 6λ f 2 H H 3 + + ic W g 2m 2 ρ cy y f f 2 H D µ H H D µ H H H f L Hf R + h.c. H σ i D µ H (D ν W µν ) i + ic Bg 2m 2 ρ H D µ H ( ν B µν ) + ic HW g m 2 ρ g 2 ρ 16π 2 (Dµ H) σ i (D ν H)W i µν + ic HBg m 2 ρ g 2 ρ 16π 2 (Dµ H) (D ν H)B µν + c γg 2 m 2 ρ g 2 ρ 16π 2 g 2 g 2 ρ H HB µν B µν + c gg 2 S m 2 ρ g 2 ρ 16π 2 y 2 t g 2 ρ H HG a µνg aµν 5

The list of dim=6 operators of the effective Lagrangian has been known in the literature since long time Buchmuller and Wyler, NPB 268 (1986) 621 In the following I will follow the SILH parametrization proposed in: Giudice, Grojean, Pomarol, Rattazzi JHEP 0706 (2007) 045 probe Higgs strong coupling g L SILH = c H 2f 2 µ H H µ H H + c T 2f 2 c 6λ f 2 H H 3 + + ic W g 2m 2 ρ cy y f f 2 H D µ H H D µ H H H f L Hf R + h.c. H σ i D µ H (D ν W µν ) i + ic Bg 2m 2 ρ H D µ H ( ν B µν ) + ic HW g m 2 ρ g 2 ρ 16π 2 (Dµ H) σ i (D ν H)W i µν + ic HBg m 2 ρ g 2 ρ 16π 2 (Dµ H) (D ν H)B µν + c γg 2 m 2 ρ g 2 ρ 16π 2 g 2 g 2 ρ H HB µν B µν + c gg 2 S m 2 ρ g 2 ρ 16π 2 y 2 t g 2 ρ H HG a µνg aµν 5

The list of dim=6 operators of the effective Lagrangian has been known in the literature since long time Buchmuller and Wyler, NPB 268 (1986) 621 In the following I will follow the SILH parametrization proposed in: Giudice, Grojean, Pomarol, Rattazzi JHEP 0706 (2007) 045 probe Higgs strong coupling g L SILH = c H 2f 2 µ H H µ H H + c T 2f 2 c 6λ f 2 H H 3 + + ic W g 2m 2 ρ cy y f f 2 H D µ H H D µ H H H f L Hf R + h.c. H σ i D µ H (D ν W µν ) i + ic Bg 2m 2 ρ H D µ H ( ν B µν ) the request of custodial invariance forbids this operator + ic HW g m 2 ρ g 2 ρ 16π 2 (Dµ H) σ i (D ν H)W i µν + ic HBg m 2 ρ g 2 ρ 16π 2 (Dµ H) (D ν H)B µν + c γg 2 m 2 ρ g 2 ρ 16π 2 g 2 g 2 ρ H HB µν B µν + c gg 2 S m 2 ρ g 2 ρ 16π 2 y 2 t g 2 ρ H HG a µνg aµν 5

The list of dim=6 operators of the effective Lagrangian has been known in the literature since long time Buchmuller and Wyler, NPB 268 (1986) 621 In the following I will follow the SILH parametrization proposed in: Giudice, Grojean, Pomarol, Rattazzi JHEP 0706 (2007) 045 probe Higgs strong coupling L SILH = c H 2f 2 µ H H µ H H + c T 2f 2 c 6λ f 2 H H 3 + + ic W g 2m 2 ρ cy y f f 2 g H D µ H H D µ H H H f L Hf R + h.c. H σ i D µ H (D ν W µν ) i + ic Bg 2m 2 ρ H D µ H ( ν B µν ) the request of custodial invariance forbids this operator probe mass scale Form factors Λ + ic HW g m 2 ρ + c γg 2 m 2 ρ g 2 ρ 16π 2 (Dµ H) σ i (D ν H)W i µν + ic HBg m 2 ρ g 2 ρ 16π 2 g 2 g 2 ρ H HB µν B µν + c gg 2 S m 2 ρ g 2 ρ 16π 2 y 2 t g 2 ρ g 2 ρ 16π 2 (Dµ H) (D ν H)B µν H HG a µνg aµν 5

Probes of Higgs strong interaction c H 2f 2 µ H H µ H H c y y ψ f 2 H H ψ L Hψ R + h.c. c 6 λ 4 f 2 H H 3 Parametrize corrections to tree-level Higgs couplings: c v2 c SM f 2 = g v 2 Λ 6

Probes of Higgs strong interaction c H 2f 2 µ H H µ H H c y y ψ f 2 H H ψ L Hψ R + h.c. c 6 λ 4 f 2 H H 3 Parametrize corrections to tree-level Higgs couplings: c v2 c SM f 2 = g v 2 Λ Energy Λ s = Λ 4πΛ g ch domain of validity of the effective theory m h 6 m V

Probes of Higgs strong interaction c H 2f 2 µ H H µ H H c y y ψ f 2 H H ψ L Hψ R + h.c. c 6 λ 4 f 2 H H 3 Parametrize corrections to tree-level Higgs couplings: c v2 c SM f 2 = g v 2 Λ Energy Λ s = Λ 4πΛ g ch to keep the theory perturbative new physics must come in before Λ s to regulate the scattering amplitudes domain of validity of the effective theory m h 6 m V

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance 7 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance Use equations of motions: D µ V µν V ν h =(m 2 V V ν + ψγ ν ψ)v ν h 7 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance Use equations of motions: D µ V µν V ν h =(m 2 V V ν + ψγ ν ψ)v ν h subleading correction to tree-level couplings c V m 2 W Λ 2 v f 2 7 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance Use equations of motions: D µ V µν V ν h =(m 2 V V ν + ψγ ν ψ)v ν h subleading correction to tree-level couplings c V m 2 W Λ 2 v f 2 contact correction to three-body decays δa A SM m 2 W Λ 2 inclusive WW,ZZ observables 7 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance Use equations of motions: D µ V µν V ν h =(m 2 V V ν + ψγ ν ψ)v ν h subleading correction to tree-level couplings c V m 2 W Λ 2 v f 2 Form factor effects contact correction to three-body decays δa A SM m 2 W Λ 2 inclusive WW,ZZ observables 7 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance Use equations of motions: D µ V µν V ν h =(m 2 V V ν + ψγ ν ψ)v ν h subleading correction to tree-level couplings c V m 2 W Λ 2 v f 2 Form factor effects contact correction to three-body decays δa A SM m 2 W Λ 2 16π2 g 2 exclusive WW,ZZ observables and Zγ 8 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance Use equations of motions: D µ V µν V ν h =(m 2 V V ν + ψγ ν ψ)v ν h subleading correction to tree-level couplings c V m 2 W Λ 2 v f 2 Form factor effects contact correction to three-body decays δa A SM v 2 16π 2 f 2 g 2 exclusive WW,ZZ observables and Zγ 9 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

c W g 2Λ 2 H σ a i D µ H (D ν W µν ) a c B g 2Λ 2 H i D µ H ( ν B µν ) D µ W + µνw ν h µ Z µν Z ν h µ γ µν Z ν h one linear combination fixed due to (accidental) custodial invariance Use equations of motions: D µ V µν V ν h =(m 2 V V ν + ψγ ν ψ)v ν h Notice: LEP already puts strong bounds on these operators Ŝ =(c W + c B ) m2 W Λ 2 10 3 effects expected to be small 10 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

ic HW g Λ 2 (D µ H) σ a (D µ H) W a µν ic HB g Λ 2 (D µ H) (D µ H) B µν c γ g 2 Λ 2 B µν B µν H H W + µνw µνh, Z µν Z µν h, γ µν γ µν h, Z µν γ µν h one linear combination starts at dim=8 c g g 2 S Λ 2 G µν G µν H H G µν G µν h 11

ic HW g Λ 2 (D µ H) σ a (D µ H) W a µν ic HB g Λ 2 (D µ H) (D µ H) B µν c γ g 2 Λ 2 B µν B µν H H W + µνw µνh, Z µν Z µν h, γ µν γ µν h, Z µν γ µν h one linear combination starts at dim=8 c g g 2 S Λ 2 G µν G µν H H G µν G µν h These operators imply couplings of photon and gluons to neutral particles and give corrections to the gyromagnetic ratios c γzh (c HW c HB ) (g 2) W (c HW + c HB ) 11

g 2 16π 2 g 2 16π 2 ic HW g Λ 2 (D µ H) σ a (D µ H) W a µν g 2 16π 2 ic HB g Λ 2 (D µ H) (D µ H) B µν c γ g 2 Λ 2 B µν B µν H H W + µνw µνh, Z µν Z µν h, γ µν γ µν h, Z µν γ µν h one linear combination starts at dim=8 g 2 16π 2 c g g 2 S Λ 2 G µν G µν H H G µν G µν h These operators imply couplings of photon and gluons to neutral particles and give corrections to the gyromagnetic ratios They cannot be generated by integrating out heavy states at tree-level in a minimally coupled gauge theory c γzh (c HW c HB ) (g 2) W (c HW + c HB ) [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ] 11

g 2 16π 2 g 2 16π 2 ic HW g Λ 2 (D µ H) σ a (D µ H) W a µν g 2 16π 2 ic HB g Λ 2 (D µ H) (D µ H) B µν c γ g 2 Λ 2 B µν B µν H H W + µνw µνh, Z µν Z µν h, γ µν γ µν h, Z µν γ µν h one linear combination starts at dim=8 g 2 16π 2 c g g 2 S Λ 2 G µν G µν H H G µν G µν h Corrections to h WW, ZZ rates: δa A SM m 2 W Λ 2 g 2 16π 2 Form factor 12 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

g 2 16π 2 g 2 16π 2 ic HW g Λ 2 (D µ H) σ a (D µ H) W a µν g 2 16π 2 ic HB g Λ 2 (D µ H) (D µ H) B µν c γ g 2 Λ 2 B µν B µν H H W + µνw µνh, Z µν Z µν h, γ µν γ µν h, Z µν γ µν h one linear combination starts at dim=8 g 2 16π 2 c g g 2 S Λ 2 G µν G µν H H G µν G µν h Corrections to h WW, ZZ rates: δa A SM m 2 W Λ 2 g 2 16π 2 Form factor Corrections to h WW, ZZ angular distributions and h γz rate: δa A SM v 2 f 2 test Higgs strong interactions 12 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

g 2 16π 2 g 2 16π 2 ic HW g Λ 2 (D µ H) σ a (D µ H) W a µν g 2 16π 2 ic HB g Λ 2 (D µ H) (D µ H) B µν c γ g 2 Λ 2 B µν B µν H H W + µνw µνh, Z µν Z µν h, γ µν γ µν h, Z µν γ µν h one linear combination starts at dim=8 g 2 16π 2 c g g 2 S Λ 2 G µν G µν H H G µν G µν h In principle h γγ, gg rates also probe strong dynamics... δa A SM v 2 f 2 13

g 2 16π 2 λ 2 g 2 g 2 16π 2 ic HW g Λ 2 (D µ H) σ a (D µ H) W a µν g 2 16π 2 ic HB g Λ 2 (D µ H) (D µ H) B µν c γ g 2 Λ 2 B µν B µν H H W + µνw µνh, Z µν Z µν h, γ µν γ µν h, Z µν γ µν h one linear combination starts at dim=8 λ 2 g 2 g 2 16π 2 c g g 2 S Λ 2 G µν G µν H H G µν G µν h In principle h γγ, gg rates also probe strong dynamics... δa A SM v 2 f 2 λ2 g 2 Form factor For a png boson Higgs additional suppression follows from breaking the shift symmetry 13 [ Giudice, Grojean, Pomarol, Rattazzi, JHEP 0706 (2007) 045 ]

Effective Lagrangian in the unitary basis L = 1 2 µh µ h 1 2 m2 hh 2 1 3m 2 + d h 3 h 3 +... 6 v m 2W W µ W µ + 12 m2z Z µ Z µ h 1+2c V v +... ψ=u,d,l m ψ (i) ψ (i) ψ (i) h 1+c ψ v +... + α em 8π 2 cww W + µνw µν + c ZZ Z µν Z µν +2c Zγ Z µν γ µν + c γγ γ µν γ µν h v + α s 8π c gg G a µνg aµν h v + c W (Wν D µ W + µν + h.c.) h v + c Z Z ν µ Z µν h v c W + c Z Z ν µ γ µν h sin θ W cos θ W tan θ W v +... 14

Effective Lagrangian in the unitary basis L = 1 2 µh µ h 1 2 m2 hh 2 1 3m 2 + d h 3 h 3 +... 6 v m 2W W µ W µ + 12 m2z Z µ Z µ h 1+2c V v +... ψ=u,d,l m ψ (i) ψ (i) ψ (i) h 1+c ψ v +... + α em 8π 2 cww W + µνw µν + c ZZ Z µν Z µν +2c Zγ Z µν γ µν + c γγ γ µν γ µν h v + α s 8π c gg G a µνg aµν h v + c W (Wν D µ W + µν + h.c.) h v + c Z Z ν µ Z µν h v c W + c Z Z ν µ γ µν h sin θ W cos θ W tan θ W v +... The same effective Lagrangian describes a generic scalar h (custodial singlet) with SU(2)LxU(1)Y non-linearly realized Each term can be dressed up with Nambu-Goldstone bosons and made manifestly SU(2)LxU(1)Y invariant 14 RC, Grojean, Moretti, Piccinini, Rattazzi JHEP 1005 (2010) 089 Azatov, RC, Galloway JHEP 04 (2012) 127

Effective Lagrangian in the unitary basis L = 1 2 µh µ h 1 2 m2 hh 2 1 3m 2 + d h 3 h 3 +... 6 v m 2W W µ W µ + 12 m2z Z µ Z µ h 1+2c V v +... ψ=u,d,l m ψ (i) ψ (i) ψ (i) h 1+c ψ v +... + α em 8π 2 cww W + µνw µν + c ZZ Z µν Z µν +2c Zγ Z µν γ µν + c γγ γ µν γ µν h v + α s 8π c gg G a µνg aµν h v + c W (Wν D µ W + µν + h.c.) h v + c Z Z ν µ Z µν h v c W + c Z Z ν µ γ µν h sin θ W cos θ W tan θ W v +... The only predictions of SILH (for single Higgs processes) are: (1) The deviation of each coupling from its SM value must be small ex: c V =1+ c H 2 v 2 f 2 15 (2) The following relation holds: c Zγ = c WW sin(2θ W ) c ZZ 2 cot(θ W ) c γγ 2 tan(θ W )

Effective Lagrangian in the unitary basis L = 1 2 µh µ h 1 2 m2 hh 2 1 3m 2 + d h 3 h 3 +... 6 v m 2W W µ W µ + 12 m2z Z µ Z µ h 1+2c V v +... ψ=u,d,l m ψ (i) ψ (i) ψ (i) h 1+c ψ v +... + α em 8π 2 cww W + µνw µν + c ZZ Z µν Z µν +2c Zγ Z µν γ µν + c γγ γ µν γ µν h v + α s 8π c gg G a µνg aµν h v + c W (Wν D µ W + µν + h.c.) h v + c Z Z ν µ Z µν h v c W + c Z Z ν µ γ µν h sin θ W cos θ W tan θ W v +... Large deviations from SM couplings do not necessarily disprove a Higgs doublet (e.g. non-linearities can be large) In that case a test of doublet/pngb Higgs can come from double (and triple) Higgs processes 16 RC, Grojean, Moretti, Piccinini, Rattazzi, JHEP 1005 (2010) 089 RC, Grojean, Pappadopulo, Rattazzi, Thamm, work in progress

SO(5)/SO(4) non-linear sigma-model Enlarge SO(4)/SO(3) to SO(5)/SO(4): [ Agashe, RC, Pomarol, NPB 719 (2005) 165 ] four NG bosons form an SU(2)L doublet H(x): the Higgs is the fourth NG boson resums powers of H/f (Higgs non-linearities), while still assuming expansion in /Λ Validity: E Λ 4πf 17

SO(5)/SO(4) non-linear sigma-model RULES: Dress up all operators with NG bosons and build the most general Lagrangian invariant under local SO(4) transformations Building blocks: CCWZ variables U(π) =e iπ(x)/f iu(π) D µ U(π) dâµ T â + E a µ T a d µ D µπ f +... E µ A µ + 1 f 2 π D µ π +... d µ h(π,g) d µ h 1 (π,g) E µ h(π,g) E µ h 1 (π,g)+i [ µ h(π,g)]h 1 (π,g) 18 h SO(4)

SO(5)/SO(4) non-linear sigma-model O(p 4 ) Lagrangian SILH Lagrangian f 2 Tr(d µ d µ ) O H O + 1 =[Tr(d µd µ )] 2 O + 2 =[Tr(d µd ν )] 2 dim=8 O ± 3 =Tr (E L µν) 2 ± (E R µν) 2 O W,O B O ± 4 =Tr (E L µν ± E R µν) i[d µ,d ν ] O HW,O HB O 5 =[Tr(T a L [d µ,d ν ])] 2 [Tr(T a R [d µ,d ν ])] 2 dim=8 19 [ RC, Marzocca, Pappadopulo, Rattazzi JHEP 1110 (2011) 081; Azatov, RC, Di Iura, Galloway, work in progress ]

SO(5)/SO(4) non-linear sigma-model O(p 4 ) Lagrangian SILH Lagrangian f 2 Tr(d µ d µ ) O H O + 1 =[Tr(d µd µ )] 2 O + 2 =[Tr(d µd ν )] 2 dim=8 O ± 3 =Tr (E L µν) 2 ± (E R µν) 2 O W,O B O ± 4 =Tr (E L µν ± E R µν) i[d µ,d ν ] O HW,O HB O 5 =[Tr(T a L [d µ,d ν ])] 2 [Tr(T a R [d µ,d ν ])] 2 dim=8 Not included [require SO(5) breaking]: B µν B µν H H G a µνg aµν H H (H H) 3 19 [ RC, Marzocca, Pappadopulo, Rattazzi JHEP 1110 (2011) 081; Azatov, RC, Di Iura, Galloway, work in progress ]

O + 1 =[Tr(d µd µ )] 2 O 2 + =[Tr(d µd ν )] 2 Operators with +(-) O 3 ± =Tr (Eµν) L 2 ± (Eµν) R 2 superscript are even (odd) under a LR parity O ± 4 =Tr (E L µν ± E R µν) i[d µ,d ν ] O 5 =[Tr(T a L [d µ,d ν ])] 2 [Tr(T a R [d µ,d ν ])] 2 PLR: πî πî πˆ4 + πˆ4 î =1, 2, 3 d µ P RL d µ P LR E L µ P RL E R µ P RL P LR = 1 1 1 +1 +1 20

O + 1 =[Tr(d µd µ )] 2 O 2 + =[Tr(d µd ν )] 2 Operators with +(-) O 3 ± =Tr (Eµν) L 2 ± (Eµν) R 2 superscript are even (odd) under a LR parity O ± 4 =Tr (E L µν ± E R µν) i[d µ,d ν ] O 5 =[Tr(T a L [d µ,d ν ])] 2 [Tr(T a R [d µ,d ν ])] 2 PLR: πî πî πˆ4 + πˆ4 î =1, 2, 3 d µ P RL d µ P LR E L µ P RL E R µ P RL P LR = 1 1 1 +1 +1 Notice: PLR: is an accidental invariance at O(p 2 ) 20 L (2) = f 2 4 Tr(d µd µ )

Part 2 The decay h Zγ in composite Higgs models [ Based on: Azatov, RC, Di Iura, Galloway, work in progress ]

Why looking at h Zγ Mediated by two operators of the SILH Lagrangian: O HW = ic HWg 16π 2 f 2 (Dµ H) σ a (D µ H) W a µν O HB = ic HBg 16π 2 f 2 (Dµ H) (D µ H) B µν O BB = c BB g 2 16π 2 f 2 B µνb µν H H c Zγ αem 2π γ µν Z µν h v c Zγ = c HB c HW 4 sin(2θ W ) 2 tan θ W c BB 22

Why looking at h Zγ Mediated by two operators of the SILH Lagrangian: only one breaks the Higgs shift symmetry O HW = ic HWg 16π 2 f 2 (Dµ H) σ a (D µ H) W a µν O HB = ic HBg 16π 2 f 2 (Dµ H) (D µ H) B µν O BB = c BB g 2 16π 2 f 2 B µνb µν H H c Zγ αem 2π γ µν Z µν h v c Zγ = c HB c HW 4 sin(2θ W ) 2 tan θ W c BB 22

Why looking at h Zγ Mediated by two operators of the SILH Lagrangian: only one breaks the Higgs shift symmetry O HW = ic HWg 16π 2 f 2 (Dµ H) σ a (D µ H) W a µν O HB = ic HBg 16π 2 f 2 (Dµ H) (D µ H) B µν O BB = c BB g 2 16π 2 f 2 B µνb µν H H c Zγ αem 2π γ µν Z µν h v c Zγ = c HB c HW 4 sin(2θ W ) 2 tan θ W c BB Unlike h γγ, gg, the rate h γz does not carry the extra spurion suppression: Γ(h Zγ) v 2 =1+O Γ(h Zγ) SM f 2 tests Higgs strong interactions 22

Why looking at h Zγ Mediated by two operators of the SILH Lagrangian: only one breaks the Higgs shift symmetry O HW = ic HWg 16π 2 f 2 (Dµ H) σ a (D µ H) W a µν O HB = ic HBg 16π 2 f 2 (Dµ H) (D µ H) B µν O BB = c BB g 2 16π 2 f 2 B µνb µν H H c Zγ αem 2π γ µν Z µν h v c Zγ = c HB c HW 4 sin(2θ W ) 2 tan θ W c BB In the SO(5)/SO(4) Lagrangian there is one operator at O(p 4 ): O HW O HB O4 =Tr (Eµν L Eµν) R i[d µ,d ν ] c Zγ = g 2 c 4 v f 2 23

Why looking at h Zγ Mediated by two operators of the SILH Lagrangian: only one breaks the Higgs shift symmetry O HW = ic HWg 16π 2 f 2 (Dµ H) σ a (D µ H) W a µν O HB = ic HBg 16π 2 f 2 (Dµ H) (D µ H) B µν O BB = c BB g 2 16π 2 f 2 B µνb µν H H c Zγ αem 2π γ µν Z µν h v c Zγ = c HB c HW 4 sin(2θ W ) 2 tan θ W c BB In the SO(5)/SO(4) Lagrangian there is one operator at O(p 4 ): O HW O HB O 4 ν h/f =Tr (Eµν L Eµν) R i[d µ,d ν ] γµν Z µ (v/f) c Zγ = g 2 c 4 v f 2 23

By integrating out heavy modes with mass M one expects: c 4 (m h)=c 4 (M)+ b 16π 2 log M m h 24

By integrating out heavy modes with mass M one expects: c 4 (m h)=c 4 (M)+ b 16π 2 log M m h at 1-loop: 1 16π 2 24

By integrating out heavy modes with mass M one expects: c 4 (m h)=c 4 (M)+ b 16π 2 log M m h at 1-loop: 1 16π 2 IR running due to light modes 24

By integrating out heavy modes with mass M one expects: c 4 (m h)=c 4 (M)+ b 16π 2 log M m h at 1-loop: 1 16π 2 IR running due to light modes Naively the IR contribution would dominate over the UV one 24

By integrating out heavy modes with mass M one expects: c 4 (m h)=c 4 (M)+ b 16π 2 log M m h at 1-loop: 1 16π 2 IR running due to light modes Naively the IR contribution would dominate over the UV one However: O 4 =Tr (E L µν E R µν) i[d µ,d ν ] is odd under PLR Accidental PLR invariance of the O(p 2 ) Lagrangian forbids any running due NG bosons 24

By integrating out heavy modes with mass M one expects: c 4 (m h)=c 4 (M)+ b 16π 2 log M m h at 1-loop: 1 16π 2 IR running due to light modes Naively the IR contribution would dominate over the UV one However: O 4 =Tr (E L µν E R µν) i[d µ,d ν ] is odd under PLR Accidental PLR invariance of the O(p 2 ) Lagrangian forbids any running due NG bosons c 4 entirely comes from UV threshold contributions 24

By integrating out heavy modes with mass M one expects: c 4 (m h)=c 4 (M)+ b 16π 2 log M m h at 1-loop: 1 16π 2 IR running due to light modes Naively the IR contribution would dominate over the UV one However: O 4 =Tr (E L µν E R µν) i[d µ,d ν ] is odd under PLR If PLR is an exact invariance of the strong dynamics, generating costs an additional weak spurion factor: O 4 25 c 4 1 16π 2 λ2 g 2 back to the estimate of γγ, gg

Example #1: tree-level exchange of a heavy spin1 Consider the case of one spin-1 (3,1) of SU(2)LxSU(2)R lighter than the other resonances ρ µ h(π,g)ρ µ h 1 (π,g)+[ µ h(π,g)] h 1 (π,g) ρ Λ = g f m ρ m h,m W 26

Example #1: tree-level exchange of a heavy spin1 Consider the case of one spin-1 (3,1) of SU(2)LxSU(2)R lighter than the other resonances ρ µ h(π,g)ρ µ h 1 (π,g)+[ µ h(π,g)] h 1 (π,g) ρ Λ = g f m ρ m ρ g ρ f g ρ <g m h,m W 26

Example #1: tree-level exchange of a heavy spin1 Consider the case of one spin-1 (3,1) of SU(2)LxSU(2)R lighter than the other resonances ρ µ h(π,g)ρ µ h 1 (π,g)+[ µ h(π,g)] h 1 (π,g) ρ Λ = g f m ρ m ρ g ρ f g ρ <g m h,m W Assumption: the derivative expansion of the Lagrangian is controlled by ( /Λ ): L = f 2 4 dâµdâµ 1 4gρ 2 ρ a L µν ρ a L µν + m2 ρ L 2gρ 2 L ρ a L µ E a L 2 µ + α i Q i + i 26

Example #1: tree-level exchange of a heavy spin1 Consider the case of one spin-1 (3,1) of SU(2)LxSU(2)R lighter than the other resonances ρ µ h(π,g)ρ µ h 1 (π,g)+[ µ h(π,g)] h 1 (π,g) ρ Λ = g f m ρ m ρ g ρ f g ρ <g m h,m W Assumption: the derivative expansion of the Lagrangian is controlled by ( /Λ ): L = f 2 4 dâµdâµ 1 4gρ 2 ρ a L µν ρ a L µν + m2 ρ L 2gρ 2 L ρ a L µ E a L 2 µ + α i Q i + i Q 1 =Tr(ρ µν i[d µ,d ν ]) Q 2 =Tr ρ µν f µν + leading operators in ( /Λ) Ex: α1( µ ρ µ ) 2 has α1 m2 ρ gρλ 2 2 1 g 2 26

ρ can be integrated out by solving the e.o.m at lowest order ρ µ = E µ + O( 2 /m 2 ρ) L eff = 1 g 2 ρ E L 2 + α 2 Tr µν +(α 1 α 2 )Tr Eµν L i[d µ,d ν ] 27

ρ can be integrated out by solving the e.o.m at lowest order ρ µ = E µ + O( 2 /m 2 ρ) L eff = 1 g 2 ρ E L 2 + α 2 Tr µν +(α 1 α 2 )Tr Eµν L i[d µ,d ν ] O 4 generated if α i are non-vanishing ν h h γ µν γ µν Z µν α 1 α 2 Z µ 27

ρ can be integrated out by solving the e.o.m at lowest order ρ µ = E µ + O( 2 /m 2 ρ) L eff = 1 g 2 ρ E L 2 + α 2 Tr µν +(α 1 α 2 )Tr Eµν L i[d µ,d ν ] O 4 generated if α i are non-vanishing ν h h γ µν γ µν Z µν α 1 α 2 Z µ non-minimal couplings (arise at loop level) 27

The coefficients α i can be estimated by assuming the following criterion: the strength of any interactions of the resonance does not exceed up to the cutoff scale g [ Partial UV completion ] RC, Marzocca, Pappadopulo, Rattazzi JHEP 1110 (2011) 081 28

The coefficients α i can be estimated by assuming the following criterion: the strength of any interactions of the resonance does not exceed up to the cutoff scale g [ Partial UV completion ] RC, Marzocca, Pappadopulo, Rattazzi JHEP 1110 (2011) 081 Ex: A(ππρ (T ) ) g ρ a 2 ρ + α 1 E 2 f 2 A g for E Λ α 1 1 < 1 g g ρ gρ 2 28

The coefficients α i can be estimated by assuming the following criterion: the strength of any interactions of the resonance does not exceed up to the cutoff scale g [ Partial UV completion ] RC, Marzocca, Pappadopulo, Rattazzi JHEP 1110 (2011) 081 Ex: A(ππρ (T ) ) g ρ a 2 ρ + α 1 E 2 f 2 A g for E Λ α 1 1 < 1 g g ρ gρ 2 By saturating the bound: α 1,2 1 g 2 ρ g2 ρ 16π 2 28

The coefficients α i can be estimated by assuming the following criterion: the strength of any interactions of the resonance does not exceed up to the cutoff scale g [ Partial UV completion ] RC, Marzocca, Pappadopulo, Rattazzi JHEP 1110 (2011) 081 Ex: A(ππρ (T ) ) g ρ a 2 ρ + α 1 E 2 f 2 A g for E Λ α 1 1 < 1 g g ρ gρ 2 By saturating the bound: α 1,2 1 g 2 ρ g2 ρ 16π 2 In general: c 4 =(αl 1 α R 1 ) (α L 2 α R 2 ) 1 16π 2 28

Example #2: one loop of heavy fermions Let us neglect the mixings with elementary fields and work in the CCWZ basis: L = χ (γ µ i µ m) χ + ζ χγ µ id µ χ 29

Example #2: one loop of heavy fermions Let us neglect the mixings with elementary fields and work in the CCWZ basis: L = χ (γ µ i µ m) χ + ζ χγ µ id µ χ µ + i(e L µ + E R µ ) LR symmetric 29

Example #2: one loop of heavy fermions Let us neglect the mixings with elementary fields and work in the CCWZ basis: L = χ (γ µ i µ m) χ + ζ χγ µ id µ χ µ + i(e L µ + E R µ ) LR symmetric arbitrary coupling 29

Example #2: one loop of heavy fermions Let us neglect the mixings with elementary fields and work in the CCWZ basis: L = χ (γ µ i µ m) χ + ζ χγ µ id µ χ µ + i(e L µ + E R µ ) LR symmetric arbitrary coupling E α Only one diagram generates : O 4 O 4 =Tr (E L µν E R µν) i[d µ,d ν ] d µ d ν 29

Example #2: one loop of heavy fermions Let us neglect the mixings with elementary fields and work in the CCWZ basis: L = χ (γ µ i µ m) χ + ζ χγ µ id µ χ µ + i(e L µ + E R µ ) LR symmetric arbitrary coupling E α Only one diagram generates : O 4 O 4 =Tr (E L µν E R µν) i[d µ,d ν ] d µ d ν SO(5) explicit breaking from gauging entirely subsumed into the external legs 29 d µ = Z µ (v/f)+ µ π/f +...

Example #2: one loop of heavy fermions Let us neglect the mixings with elementary fields and work in the CCWZ basis: L = χ (γ µ i µ m) χ + ζ χγ µ id µ χ µ + i(e L µ + E R µ ) LR symmetric arbitrary coupling E α no SO(5) explicit breaking into the loop: fermions can be taken as SO(4) eigenstates Only one diagram generates : O 4 O 4 =Tr (E L µν E R µν) i[d µ,d ν ] d µ d ν SO(5) explicit breaking from gauging entirely subsumed into the external legs 29 d µ = Z µ (v/f)+ µ π/f +...

MODEL 1 [MCHM5]: Composite fermions in 5 s of SO(5) χ =(2, 2) (1, 1) = {χ 4, χ 1 } L = χ 4 (γ µ i µ m 4 ) χ 4 + χ 1 (γ µ i µ m 1 ) χ 1 + ζ χ 4 γ µ id µ χ 1 + h.c. 30

MODEL 1 [MCHM5]: Composite fermions in 5 s of SO(5) χ =(2, 2) (1, 1) = {χ 4, χ 1 } L = χ 4 (γ µ i µ m 4 ) χ 4 + χ 1 (γ µ i µ m 1 ) χ 1 There is an accidental PLR invariance in the spectrum and in the couplings + ζ χ 4 γ µ id µ χ 1 + h.c. Noticed in: Mrazek et al. NPB 853 (2011) 1 30

MODEL 1 [MCHM5]: Composite fermions in 5 s of SO(5) χ =(2, 2) (1, 1) = {χ 4, χ 1 } L = χ 4 (γ µ i µ m 4 ) χ 4 + χ 1 (γ µ i µ m 1 ) χ 1 + ζ χ 4 γ µ id µ χ 1 + h.c. There is an accidental PLR invariance in the spectrum and in the couplings Noticed in: Mrazek et al. NPB 853 (2011) 1 E L α E R α (2, 2) - (2, 2) (2, 2) (2, 2) d µ (1, 1) d ν d µ (1, 1) d ν 30

MODEL 1 [MCHM5]: Composite fermions in 5 s of SO(5) χ =(2, 2) (1, 1) = {χ 4, χ 1 } L = χ 4 (γ µ i µ m 4 ) χ 4 + χ 1 (γ µ i µ m 1 ) χ 1 + ζ χ 4 γ µ id µ χ 1 + h.c. There is an accidental PLR invariance in the spectrum and in the couplings Noticed in: Mrazek et al. NPB 853 (2011) 1 E L α same coupling E R α (2, 2) - (2, 2) (2, 2) (2, 2) d µ (1, 1) d ν d µ (1, 1) d ν 30

MODEL 1 [MCHM5]: Composite fermions in 5 s of SO(5) χ =(2, 2) (1, 1) = {χ 4, χ 1 } L = χ 4 (γ µ i µ m 4 ) χ 4 + χ 1 (γ µ i µ m 1 ) χ 1 + ζ χ 4 γ µ id µ χ 1 + h.c. There is an accidental PLR invariance in the spectrum and in the couplings Noticed in: Mrazek et al. NPB 853 (2011) 1 E L α same coupling E R α (2, 2) - (2, 2) (2, 2) (2, 2) c 4 =0 d µ (1, 1) d ν d µ (1, 1) d ν 30

MODEL 2 [MCHM10]: Composite fermions in 10 s of SO(5) χ =(2, 2) (3, 1) (1, 3) = {χ 4, χ 3L, χ 3R } L = χ i (γ µ i µ m i ) χ i i=4,3l,3r + ζ L χ 4 γ µ id µ χ 3L + ζ R χ 4 γ µ id µ χ 3R + h.c. 31

MODEL 2 [MCHM10]: Composite fermions in 10 s of SO(5) χ =(2, 2) (3, 1) (1, 3) = {χ 4, χ 3L, χ 3R } L = χ i (γ µ i µ m i ) χ i i=4,3l,3r + ζ L χ 4 γ µ id µ χ 3L + ζ R χ 4 γ µ id µ χ 3R + h.c. E L α E L α E L α (3, 1) (3, 1) + (2, 2) (2, 2) + (2, 2) (2, 2) ζ L ζ L d µ (2, 2) d ν d µ (3, 1) d ν d µ (1, 3) d ν 31

MODEL 2 [MCHM10]: Composite fermions in 10 s of SO(5) χ =(2, 2) (3, 1) (1, 3) = {χ 4, χ 3L, χ 3R } L = χ i (γ µ i µ m i ) χ i i=4,3l,3r + ζ L χ 4 γ µ id µ χ 3L + ζ R χ 4 γ µ id µ χ 3R + h.c. E L α E L α E L α (3, 1) (3, 1) + (2, 2) (2, 2) + (2, 2) (2, 2) ζ L ζ L d µ (2, 2) d ν d µ (3, 1) d ν d µ (1, 3) d ν E R α - (1, 3) (1, 3) - ζ R d µ (2, 2) ζ R d ν same diagrams as above with ( ), have the same value L R 31

In general: c 4 1 Λ 16π 2 (ζl 2 ζr) 2 2 log m 3L m 3R +(ζ 2 L + ζ 2 R) log m2 3R m 2 3L + ζ2 L m2 3L ζ2 R m2 3R m 2 4 32

In general: c 4 1 Λ 16π 2 (ζl 2 ζr) 2 2 log m 3L m 3R +(ζ 2 L + ζ 2 R) log m2 3R m 2 3L + ζ2 L m2 3L ζ2 R m2 3R m 2 4 P RELIMINARY A(h Zγ) hzγ A(h hzγ top Zγ) top 4 2 Blue: Red: f = 800 GeV f = 500 GeV 0.6 0.4 0.2 0.2 0.4 0.6 m 3L m 3R m 3L + m 3R M 3,1 M 1,3 M 3,1 M 1,3 2 Azatov, RC, Di Iura, Galloway work in progress 4 32 FIG. 1. Ratio of the hzγ 10 hzγ top only for the one generation of the composite fermions. Blue (red) points correspond to the values of f = 800(500)GeV. During the scan we varied M (3,1),M (1,3),M (2,2) for ζ L = ζ R =1varying m 4,m 3L,m 3R between [2f,10f] between [2f,10f]. 1. Same in the two site basis

Conclusions Effective Lagrangian for a light Higgs useful to interpret current data and guide future searches 33

Conclusions Effective Lagrangian for a light Higgs useful to interpret current data and guide future searches Most important is to have power counting rules to estimate impact of new operators on physical observables 33

Conclusions Effective Lagrangian for a light Higgs useful to interpret current data and guide future searches Most important is to have power counting rules to estimate impact of new operators on physical observables Decay h Zγ can in principle be a probe of Higgs strong interactions, contrary to h γγ,gg 33

Conclusions Effective Lagrangian for a light Higgs useful to interpret current data and guide future searches Most important is to have power counting rules to estimate impact of new operators on physical observables Decay h Zγ can in principle be a probe of Higgs strong interactions, contrary to h γγ,gg In practice h Zγ is protected by PLR at leading order, same estimate as for γγ,gg follows in (motivated) models where strong dynamics is LR symmetric 33

Conclusions Effective Lagrangian for a light Higgs useful to interpret current data and guide future searches Most important is to have power counting rules to estimate impact of new operators on physical observables Decay h Zγ can in principle be a probe of Higgs strong interactions, contrary to h γγ,gg In practice h Zγ is protected by PLR at leading order, same estimate as for γγ,gg follows in (motivated) models where strong dynamics is LR symmetric 33 Yet scenarios with O(1) enhancements in h Zγ exist, and an experimental measurement would be important