Composite Higgs

Composite Higgs

Florian Herren|February 11, 2016

Contents

1 Chiral Lagrangian

2 EWSB

3 Composite Higgs

4 Searches

QCD

Quantum Chromodynamics

describes interaction between quarks negativeβ-function

nonperturbative at low energies gauge group:SU(3)

L=

nf

X

i=1

Ψ_i iD/−^mi

Ψ_i −¹

4G^a_µνG^a,µν G^a_µν =∂_µA^a_ν−∂_νA^a_µ+gf^abcA^b_µA^c_ν

What happens at low energies?

cannot use QCD Lagrangian for perturbation theory anymore quarks and gluons form a plethora of hadrons

protons and neutrons form nuclei

all other hadrons decay via the weak interaction hadrons are much more massive than quarks

Pions

the three lightest mesons

masses: 134,98 MeV (π⁰), 139,57 MeV (π^±)

much lighter than all other resonances (m_η =547,86 MeV) pseudoscalar

Where does the proton/neutron mass come from?

proton and neutron form an isospin doubletΨ = p

n

in the massless case we can decompose left- and right-handed fields:

L=iΨL∂Ψ/ L+iΨR∂Ψ/ R

ΨL,R= ¹

2(1∓γ⁵)Ψ invariant underSU(2)_L⊗^SU(2)_R

Gell-Mann and Levi: generate mass through spontaneous breaking of chiral symmetry

Σ ^-Model

L=iΨ/∂Ψ−^gΨ_LΣΨ_R−^gΨ_RΣ^†Ψ_L+L(Σ)

Σtransforms likeLΣR^†underSU(2)_L⊗^SU(2)_R

L(Σ)invariant underSU(2)L⊗^SU(2)R⇒ L=f Tr[ΣΣ^†] linear ansatz:Σ =σ+iπ^aτ^awith simple symmetry breaking potential with VEVF_πand the Pauli matricesτ^a

nonlinear ansatz:Σ =ρexp(iπ^aτ^a/F_π) both models break the chiral symmetry chiral currentj₅^a_,µ=−(∂_µπ^a)F_π+O(φ²)

⇒ h^0had|^j5^a,µ|π^bi=iF_πp_µδ^ab

Mass sector

L^mass=M_W²W_µ⁺W^−µ+¹

2M_Z²Z^µZ_µ

−X

i,j

u⁽_Lⁱ⁾M_ij^uu⁽_R^j)+d⁽_Lⁱ⁾M_ij^dd_R^(j)+e⁽_Lⁱ⁾M_ij^ee⁽_R^j)

+h.c.

scattering of longitudinally polarizedW^±andZ bosons leads to violation of unitarity

rewrite boson masses by introducingΣ(x) =exp(iσ^aχ^a/v) Goldstone bosons interact with vector bosons trough

D_µΣ =∂µΣ−^igσ^a

2 W_µ^aΣ +ig^′Σσ³ 2 B_µ

underSU(2)_L⊗^U(1)_Y Σtransforms asΣ→^UL(x)ΣU_Y^†(x)

Mass sector

L^mass= ^v

2

4 Tr h

(D_µΣ)^†(D^µΣ)i

−√^v 2

X

i,j

u⁽_Lⁱ⁾d⁽_Lⁱ⁾

Σ λ^u_iju⁽_R^j) λ^d_ijd_R^(j)

! +h.c.

vis the Higgs VEV

in unitary gaugehΣi=1 this reproduces the former mass Lagrangian

ρ≡ _M^2MW2

Zcos²θW =1

Custodial Symmetry

L^massinvariant underSU(2)_L⊗^SU(2)_Rforg^′=0 andλ^u_ij^,d =0 SU(2)_C remains after EWSB

χ^atriplett underSU(2)C⇒^MW =M_Z g^′6=0⇒^MW =M_Zcos²θ_W

Yukawa couplings lead to small corrections toρ Extensions of the SM should respectSU(2)_C

SM Higgs Boson

introduceh(x)as a singlet underSU(2)_L⊗^SU(2)_R L^H= ¹

2(∂_µh)²+V(h) +^v

2

4 Tr h

(D_µΣ)^†(D^µΣ)i

1+2ah v +bh²

v² +O(h³)

−√^v 2

X

i,j

u⁽_Lⁱ⁾d⁽_Lⁱ⁾ Σ

1+ch

v +O(h²)

λ^u_iju_R^(j) λ^d_ijd_R^(j)

!

+h.c.

unitarizes scattering of Goldstone bosons fora=b =c=1 takes the standard form with:

H(x) = √¹

2exp(iσ^aχ^a/v) 0

v+h(x)

Why would one want another strong sector?

no mass corrections from above the compositness scale

⇒solves hierachy problem new resonances unitarize theory

possible connection to higher dimensional models

⇒new physics to explore

Ingredients

global symmetryG, broken down toH₁at a scalef

⇒ⁿ=dim(G)−dim(H₁)Goldstone bosons H₀⊂^Ggauged by external vector bosons H =H₁∩^H0unbroken gauge group

⇒ⁿ0 =dim(H₀)−dim(H)eaten up⇒ⁿ−ⁿ0survive

A minimal example

For the SMH₀ =SU(2)_L⊗^U(1)_Y

G=SO(5)⊗^U(1)_X broken down toSO(4)⊗^U(1)_X

⇒ⁿ=4

H₀⊂^SO(4)≃^SU(2)_L⊗^SU(2)_R

⇒ⁿ⁰=0

hypercharge generatorY =T^3R+X

(H,H^c)transforms as(2,2)underSU(2)_L⊗^SU(2)_R SU(2)_L⊗^U(1)_Y unbroken at tree level

Gexplicitly broken by couplings of SM particles to the strong sector

⇒fermions and gauge bosons generate Higgs potential m_h∼^gSMv,m_ρ∼^gρf

Back to the Σ ^-model

Σ = Σ₀exp −ⁱ√

2T^ˆah^ˆa(x)/f

Σ₀preservesSO(4)symmetry:Σ₀ = (0,0,0,0,1)

⇒Σ = ^sin(_f^h/f) h¹,h²,h³,h⁴,hcot(h/f)

consider the wholeSO(5)⊗^U(1)X is gauged, so we can writeLⁱⁿ momentum space:

L= ¹₂P_T^µν

Π^X₀(q²)X_µX_ν + Π₀(q²)Tr(A_µA_ν) + Π₁(q²)ΣA_µA_νΣ^T Σclassical background, derivative interactions not included expanding aroundΣ0one obtains

L= ¹₂P_T^µν

Π^X₀(q²)X_µX_ν + Π_a(q²)Tr(A^a_µA^a_ν) + Π_ˆa(q²)A^ˆa_µA^ˆa_ν , Π_a= Π₀,Π_ˆa= Π₀+^Π₂¹

Back to the Σ ^-model

from our discussion of pions we can deduce that P_T^µνΠ_ˆa(0) =

J_ˆa^µ(0)J_ˆa^ν(0)

=η^µνf₂² a similar discussion leads toΠ_a(0) =0

⇒Π₀(0) = Π^X₀(0) =0,Π₁(0) =f²

switching off the unphysical gauge fields and using our ansatz forΣ we obtain

L= ¹ 2P_T^µν

"

Π^X₀(q²) + Π₀(q²) + ^sin

2(h/f) 4 Π₁(q²)

B_µB_ν

+

Π0(q²) +^sin

2(h/f) 4 Π1(q²)

W_µ^aW_ν^a

+2 sin²(h/f)Π₁(q²) ˆH^†T^aLY_HAˆ ^aL

µB_ν

#

Let’s compare this to the SM

forq²≪^m2ρand aligning the Higgs VEV along theh³direction we obtain

L=P_T^µν

"

1 2

f²sin²(h^hi/f) 4

B_µB_ν+W_µ³W_ν³−^2Wµ³B_ν +

f²sin²(h^hi/f) 4

W_µ⁺W_ν⁻

+^q

2

2

Π^′₀(0)W_µ^aW_ν^a+ (Π^′₀(0) + Π^X₀^′(0))B_µB_ν +. . .

#

for the gauge couplings we obtain ¹

g² =−Π^′₀(0)and

1

g^′2 =−(Π^′₀(0) + Π^X₀^′(0))

the Higgs VEV is given byv =fsin^hh_fⁱ, defineξ ≡ ^v_f²²

Let’s compare this to the SM

expandingf²sin^{2 h}_f leads tov²+2v√

1−ξh+ (1−²ξ)h²whereh is now the physical Higgs field

w.r.t the SM the VVh and VVhh couplings are modified:

g_VVh=g_VVh^SM√

1−ξ,g_VVhh=g_VVhh^SM (1−²ξ) this meansa=√

1−ξandb= (1−²ξ)

for nonvanishingξthe Higgs only partly unitarizes the scattering of vector bosons

forξ=1f =vand we obtain a minimal Technicolor theory with a light scalar

What changes w.r.t Fermions?

things work different than in the boson sector

have to choose a representation ofSO(5)in which the fermions live spinorial representation (MCHM4): c=√

1−ξ fundamental representation (MCHM5):c= √¹⁻₁^2ξ

−ξ

How do observables change in the MCHM4?

fermionic and bosonic couplings scale by a factor of√ 1−ξ branching ratios remain the same

total width reduced by a factor 1−ξ the same for production cross-sections

in principle loop induced decays could be modified by new particles (e.g. top-partners)

How do observables change in the MCHM5?

fermionic and bosonic couplings scale differently

partial decay width for fermions and gluons reduced by ⁽¹⁻₁^2ξ)2

−ξ

partial decay width for vector bosons reduced by(1−ξ) Higgs coupling to photons more complicated, since there are fermion- and W-loops

gluon fusion andttHcross-sections reduced by ⁽¹₁^−2ξ)2

−ξ

How do branching ratios change in the MCHM5?

0 0.2 0.4 0.6 0.8 1

ξ 10^-1

10^-2

10^-3 1

bb^-

τ⁺τ^- gg

cc^- ZZ

WW γγ

Zγ

BR(H) M_H=120 GeV MCHM5

Figure:Espinosa, Grojean and M ¨uhlleitner [arXiv:1003.3251]