slides - Institute for Particle Physics Phenomenology

Higgs Beyond the 

SM and SUSY 

Adam Martin 

CERN/Notre Dame 

(adam.martin@cern.ch) 

YETI Winter School, IPPP Durham UK, 2013 

Wednesday, January 9, 2013

Outline 

Lecture #1 

• Where are we now and where do we go from here 

• Did it have to be a Higgs 

• EFT for beyond the SM 

• Composite Higgs (Higgs as a Goldstone Boson) 

Lecture #2 

• more Composite Higgs (Higgs as a Goldstone Boson) 

• where & how to look at LHC 


Where are we now 

• massive W ± /Z 0 fermions, massless γ, g: 

‘electroweak symmetry is broken’ 


Where are we now 

• massive W ± /Z 0 fermions, massless γ, g: 

‘electroweak symmetry is broken’ 

• As of July 4th, 2012... 

we have a new boson X, 

with mass ~125 GeV 

likely spin-0, 

likely CP-even 

0 

Local p 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

1 

-1 

-2 

-3 

-4 

-5 

-6 

-7 

-8 

-9 

-10 

-11 

ATLAS 2011 - 2012 

s = 7 TeV: 

s = 8 TeV: 

-1 

"Ldt = 4.6-4.8 fb 

-1 

"Ldt = 5.8-5.9 fb 

Obs. 

Exp. 

±1 ! 

110 115 120 125 130 135 140 145 150 

m H 

[GeV] 

0! 

1! 

2! 

3! 

4! 

5! 

6! 

Wednesday, January 9, 2013 

X is observed in channels and with rates similar 

Figure 9: The observed (solid) local p 0 as a function of m H in the 

low mass range. The dashed curve shows the expected local p 0 under 

the hypothesis of a SM Higgs boson signal at that mass with its ±1σ 

band. The horizontal dashed lines indicate the p-values corresponding 

to significances of 1 to 6 σ. 

to what we expect for a SM Higgs boson

What now 

• To what extent are EWSB and X related 

Meaning what are the similarities & 

differences between X and SM Higgs 

• What is the bigger picture 


Is there a bigger picture 

• LOTS we don’t know.. many things we observe are 

not in the SM. 

Dark Matter 

baryon vs. anti-baryon asymmetry 

neutrino masses 

# generations, charge assignments 

• Scale/properties of these other phenomena is 

unknown. 

• Hope that understanding EWSB ( & connection w/ X 

boson) will shed some light on these other topics 


Did it have to be Higgs 

Was finding a Higgs-like boson inevitable Is that 

the only path in nature for what we see 





NOPE 





NOPE 

we know how Higgs works 

(see lectures by Mück,Englert,Duehrssen) 

H ∈ (2, 1/2) of SU(2) w ⊗ U(1) Y 

|D µ H| 2 − λ(H † H − v2 

2 )2 



Σ(x) =exp 

i 2π a τ a 

v 

 

Instead lets add a field Σ(x): 

Pauli matrices 

= 1 2×2 +2i π a τ a 

v 

− 2 π aπ b τ a τ b 

v 2 + ··· 

3 “pion” fields EW scale 



Σ(x) =exp 

i 2π a τ a 

v 

 

Instead lets add a field Σ(x): 

Pauli matrices 

= 1 2×2 +2i π a τ a 

v 

− 2 π aπ b τ a τ b 

v 2 + ··· 

3 “pion” fields EW scale 

expanding out, we get a (2x2) matrix 

cos (ˆπ/v)+iˆπ3 sin (ˆπ/v) i(ˆπ 

Σ = 

1 − iˆπ 2 )sin(ˆπ/v) 

i(ˆπ 1 + iˆπ 2 )sin(ˆπ/v) cos(ˆπ/v) − iˆπ 3 sin (ˆπ/v) 

ˆπ = π 2 a, 

 

ˆπ a = π a /ˆπ 

remember, the SM Higgs doublet 

H(x) = 1 √ 

2 

 

h1 + ih 2 

h 0 + ih 3 

 

has 4 fields, our 

Σ has 3 



under SU(2)w x U(1)Y: 

UL Σ U † R 

D µ Σ = ∂ µ Σ − ig W µ Σ + ig Σ B µ τ 3 

how is this different than H (forget U(1) Y for now..) 

UL is a 2 x 2 matrix: U L =exp(iα a (x)τ a ) 

acting on the Higgs vs. on Σ: 

U L H(x) : 

U L Σ(x) : 

mixes up the 4 components h i 

ex.) h 3 → h 3 + i h 0 α₃- i h 1 α₁ + i h 2 α₂ 

shifts the πₐ fields, π 3 → π 3 + α 3 



out of Σ, we have: 

L Σ = v2 

4 tr(Dµ Σ D µ Σ † )+··· 

more derivatives, 

etc. 

by a gauge transformation: 

U L = Σ † , Σ → Σ † Σ = 1 , 

iD µ Σ =(g W µ − g B µ ) 

L Σ = v2 

4 (g W µ − g B µ ) 2 = m 2 W W + µ W µ− + m 2 Z Z µ Z µ 

so with only 3 degrees of freedom, we can give 

mass to W ± /Z (fermions too: y t vQΣ u ∗ R , etc.). 

‘Higgsless’ EWSB 

we haven’t explained what dynamics gives Σ ... 

but no explanation for V(H) in SM. 


So what’s the difference 

Look at WL WL → WL WL scattering in the H & Σ theories 

in the Σ theory: 

(at tree level) 

A ∼ s 

v 2 

amplitude grows 

with energy 

in the SM: 

extra contributions coming from Higgs 

exchange. amplitude → constant 

A ∼ m2 H 

s 

hW + W - , etc. couplings set by gauge invariance 


So what 

Unitarity imposes relations on QM amplitudes 

Im(A) 

Re(A) 2 + (Im(A)-1/2) 2 = 1/4 

log(s/m 2 h) 

½ 

Re(A) 

tree level amplitude is real, at loop level A gets an imaginary part 

Σ theory: A ~s/v 2 , Re(A) grows with energy 

bigger Re(A) is, bigger Im(A) must get to keep unitary relation. 

If Im(A) = Re(A) then 1-loop is same size as tree level 

= loss of perturbativity = theory is strongly coupled 

SM theory: A ~ m 2 H/s, perturbativity retained for all s 


So what 


Im(A) 

Re(A) 2 + (Im(A)-1/2) 2 = 1/4 

log(s/m 2 h) 

½ 

Re(A) 








So what 


Im(A) 

Re(A) 2 + (Im(A)-1/2) 2 = 1/4 

log(s/m 2 h) 

½ 

Re(A) 








So what 

we’ve seen strong coupling before: QCD in the 

~ 100 MeV - 1 GeV range 

π π → π π scattering 

 

 

ρ-meson 

 

 

strong coupling manifests in 

exchange of resonances: 

ρ, a₁, ρ’, etc., other q̅q bound 

states 

in QCD, strong coupling tells us that above a certain 

energy there is a transition and quarks & gluons are the 

right degrees of freedom 


But... 

L Σ has massive particles, but no new scalar state (X, 

or h) ... so it needs to be augmented 

L Σ = 1 2 (∂ µh) 2 + v2 

 

4 tr(D µΣ D µ Σ † ) 1+a 2 h 

v + b h2 

··· 

v 2 + 

1+c + y ij vQ i Σ u ∗ h ··· 

Rj 

v + + h.c. 

− m2 H 

2 h2 

+ analogous terms for other fermions, + terms with more derivatives 

Looks familiar a = b = c = 1 → SM Higgs Lagrangian, 

where three πₐ fields + h combine to the H doublet 

but that is just a special case, LΣ is more general: 

triplet of states eaten by W ± /Z 0 + real scalar 

= EFT for LHC Higgs 


Higgs EFT 

What is the meaning of a, b, c ≠ 1 

A ∼ 

(s + t) 

v 2 

m 

2 

(1 − a)+O H 

s 

for a ≅ 1, energy growth 

in A is suppressed 

pert. lost when Re(A) ≈ 1/2 

 

partial 

A 0 = E2 (1 − a) 

32π v 2 = 1 wave A l 2 

above this scale (& without other terms), strong dynamics 

E lim = 4√ πv 

√ 1 − a 

• a=0, strong dynamics ~ TeV scale (Technicolor) 

• a=1, theory stays perturbative (SM Higgs) 

• a≅1, strong dynamics scale pushed higher 


= s + t 

v 2 

h 

 

1 − a 

2 + O 

m 

2 

h 

E 2 

Higgs . EFT 

there are processes other than WLWL → WLWL that can grow 

with energy 

introduced a new particle in the theory, we have to check that also the 

els involving h are unitarized. The χχ → hh scattering (equivalent to 

t high energy), is perturbatively unitarized for b = a 2 : 

W + L 

W - L 

h 

h 

W + L 

W - L 

A(χ + χ − → hh) = s v 2 

b − a 

2 + O 

W + L 

ψ 

W + L 

m 

2 

h 

E 2 

→ ψ ¯ψ scattering (equivalent to W L W L → ψ ¯ψ at high energy) is unita- 

1 

h 

h 

A(W + L W − L 

. 

ψ 

→ hh) ∼ s 

m 

2 

v 2 (b − a2 )+O H 

s 

b≠1, c≠1 also lead to 

ill-behaved amplitudes 

 

ms showed W - Lin present section, 

ψ̅ 

dashed W - L and solid lines denote respectively 

ψ̅ 

the fields χ 

olid lines with an arrow denote fermions. 

A(χ + χ − → ψ ¯ψ) = m √ 

ψ s 

A(W + L W − L 

→ v 2 ¯ψψ) ∼ m ψ 

9 

(1 − ac)+O 

m 

2 

h 

E 2 

√ s . m 

2 

(1 − ac)+O H 

s 

= c = 1 the EWSB sector is weakly interacting (provided the scalar h 

example a = 1impliesastrongWW → WW scattering with violation 

unitarity at energies √ s ≈ 4πv/ √ 1 − a 2 Wednesday, January 9, 2013 

, and similarly for the other 

v 2

Higgs EFT 

there are existing, indirect constraints on a,b,c 

χ 1 

χ 2 (χ 1 ) 

W 3 B 

χ 1 (χ 2 ) χ 1 (χ 2 ) 

ex.) loop level contributions 

to 

S,T parameters 

χ 2 

h 

here χ a = W a L 

B 

h 

W 3 B 

χ 3 χ 3 

∆S =+ 1 

Λ 

2 

12π (1 − a2 )log 

m 2 H 

3 

Λ 

∆T = − 

16π cos 2 (1 − a 2 2 

)log 

θ W 

χ 3 

Figure 8: Logarithmically divergent contributions to S (left diagrams) and T ( 

grams) from loops of would-be NG Goldstones χ’s (upper row) and of the Higgs bos 

row). In the SM the Higgs divergent contribution exactly matches that from the χ 

a finite result. At scales below m h , the upper left diagram contributes to the runn 

coefficient of the operator Tr Σ † W µν ΣB µν , see eq.(32). Similarly, the upper right 

m 2 H 

 

contributes to the running of the coefficient of Tr T 3 Σ † D µ Σ 2 . See Ref. [18]. 

which leads to a constraint on the mass of the lightest spin-1 resonance m ρ . 1 

Concerning ∆ρ (or equivalently the Peskin-Takeuchi T parameter), the t 

correction due to the exchange of heavy spin-1 resonances identically vanish 

SO(5)/SO(4) model as a consequence of the custodial symmetry of the stron 

a 

roughly constrains 

0.75 ≤ a ≤ 1.5, 

depending on assumptions 

similarly, off-diagonal cij strongly constrained by flavor physics 

diagonal cii, b are less constrained 


B

Higgs EFT in action 

UPDATED SANITY CHECK 

several groups (theory & experiment) are already 

looking at LHC Higgs data in the a,b,c space 

c = 

c F 

2.0 

1.8 

1.6 

1.4 

1.2 

CMS Preliminary 

s = 7 TeV, L = 5.1 fb 

s = 8 TeV, L = 5.3 fb 

-1 

-1 

20 

18 

16 

14 

12 

-2 ! ln L 

2.0 

1.5 

m h 125 GeV 

68 CL 

90 CL 

95 CL 

SM 

CM 

1.0 

10 

0.8 

8 

0.6 

6 

0.4 

4 

0.2 

2 

0.0 

0.0 0.5 1.0 1.5 

c V 

0 

= a 

c 

1.0 

many subtleties! care 

0.5 

required in 

interpretation 

(talk by Duehrssen) 

0.0 

0.0 0.2 0.4 


Recap 

• to fit observation, we need massive W ± /Z 0 /fermions 

+ extra scalar 

• general setup: LΣ .. contains LSM in special 

a = b = c = 1 limit 

• for a,b,c≠1, LΣ becomes strongly coupled at some 

energy Elim ... expect (from QCD experience) some 

new dynamics to enter at that scale 

• useful framework for LHC Higgs data 

• BUT: What UV dynamics actually leads to LΣ What 

else (other states, couplings) is present in those 

scenarios 


Light scalar fields 

m 2 H 

2 h2 

masses of scalar fields are sensitive to the highest 

scales of a theory: δm²H ∼ Λ² 

Having a scalar mass

who cares about natural 

UK, lead by new player Higgs, 

defeats Germany in World Cup Final 

1000 - 0 

theoretically possible, but 

hard to imagine within the 

rules we trust... 

either Higgs is unlike the other 

particles/players we know, or 

there is more going on 


Light scalar fields 

m 2 H 

2 h2 

masses of scalar fields are sensitive to the highest 

scales of a theory: δm²H ∼ Λ² 

Having a scalar mass light scalar (Higgs) 


Higgs as a pseudo-Goldstone Boson 

For starters: let’s study a simpler setup, 2 flavor QCD. As 

we’ll see, the pion of QCD is a pNGB, so many lessons from 

Lπ will carry over to LH. 

At high energy, QCD is quarks and gluons 

this theory is invariant under rotations of LH, 

RH quarks among themselves 

u 

 

L 

d 

 

L 

 

= U L 

 

uL 

d L 

u 

 

R 

d 

 

L 

 

= U R 

 

uR 

d R 

 

global symmetry is SU(2)L ⊗ SU(2)R 



at E ~ 1 GeV, the strong force becomes confining, 

quarks & antiquarks get bound together.. only color 

singlet states allowed 

color-singlet condensates form: ¯Q L Q R =0 

under global symmetry: 

¯Q L Q R →¯Q L U † L U R Q R 

only invariant when UL = UR, the ‘vectorial’ subgroup 

so, as a result of the strong dynamics, symmetry has 

been broken: SU(2)L ⊗ SU(2)R → SU(2)V 

Goldstone’s theorem: for each generator of a spontaneously 

broken, continuous, global symmetry 

→ a massless scalar (a Goldstone boson) 



at E ~ 1 GeV, the strong force becomes confining, 

quarks & antiquarks get bound together.. only color 

singlet states allowed 

color-singlet condensates form: ¯Q L Q R =0 

under global symmetry: 

¯Q L Q R →¯Q L U † L U R Q R 

only invariant when UL = UR, the ‘vectorial’ subgroup 

so, as a result of the strong dynamics, symmetry has 

been broken: SU(2)L ⊗ SU(2)R → SU(2)V 

+3 NGB 

Goldstone’s theorem: for each generator of a spontaneously 

broken, continuous, global symmetry 

→ a massless scalar (a Goldstone boson) 



the interactions of the NGB can be described by the 

‘chiral lagrangian’ 

introduce: U =exp 

2 iπ 

f π 

 

π = π a τ a 

pion decay constant = 93 MeV 

fix U → UL U U † R 

then U has the same transformation 

properties as 

UU † = 1, so terms in Lπ must involve derivatives 

L π = f 2 π 

4 tr(∂ µU ∂ µ U † )+c 1 tr(∂ µ U ∂ µ U † ) 2 + ··· 

other 4-deriv. terms 

expanded out: 

L π = 1 2 (∂ µπ a ) 2 + ··· 

multiple-π interactions 

(π ∂µπ)², etc. 



Look familiar it should! same setup as triplet of fields in 

LΣ, but with v → fπ. Setup is the same because the 

symmetry (& symmetry breaking) is the same 

but remember: our goal is to have a setup where 

triplet PLUS h are ALL NGBs.. needs more work 

First, some more observations of L π & QCD: 

• number of πₐ is set by the amount of symmetry broken 

• the transformation properties of πₐ under unbroken 

symmetry (SU(2)V) also set by pattern of symmetry 

breaking 

• all interactions of πₐ involve ∂μ 



more observations of L π & QCD: 

• there is more to QCD than just πₐ... 

There are other bound states of quarks = resonances like ρ, 

a₁, ω. These resonances have various JPC, & interact strongly 

with the πₐ. 

M ρ =770MeV,J PC =1 −− 

M ω =782MeV,J PC =1 −− 

M a1 =1230MeV,J PC =1 ++ 

M η =539MeV,J PC =0 −+ 

... 

They are not contained in Lπ.. no first-principles way to 

include them, instead must use phenomenological models 



adding electromagnetism: 

• U(1)em is part of the SU(2)V that remains unbroken (LH, 

RH quarks have the same EM charge). What if we turn 

on this gauge interaction 

∂ µ U → D µ U, D µ U = ∂ µ U + ieA µ ˆQU 

ˆQ = 

2 

3 

− 1 3 

 

we get new interactions of pions and photons, some of 

which have no derivatives 

loops of photons generate V(π) 

A 

A 

+ + + ··· 

A 


Figure 7: 1-loop contribution of the SM gauge fields to the Higgs potential. A grey blob 

represents the strong dynamics encoded by the form factor Π .


mass term: 

∼ c 2 

e 2 

cutoff: typically Λ = O(4πf) 

+ + + ··· 

coeff. 

16π 2 Λ2 π 2 a 

loop factor 

other terms in V(π) generated similarly. 

Figure 7: 1-loop contribution of the SM gauge fields to the Higgs potential. A grey 

represents the strong dynamics encoded by the form factor Π 1 . 

Above is just an estimate... loops of strongly coupled 

particles involved. For more rigorous calculation, see: 

Contino 1005.4269 + ref. therein 

section 3.3, as we are now ready to derive the Coleman-Weinberg potential fo 

composite Higgs. 

We will concentrate on the contribution from the SU(2) L gauge fields, neglectin 

smaller correction from hypercharge. The contribution from fermions will be d 

V(π) 

in section 3.4. The 1-loop Coleman-Weinberg potential resums the class of diag 

in Fig. 7. From the effective action (48), after the addition of the gauge-fixing te 

With full calculation, can show V(π) has a min at π = 0 


L GF = − 1 

2g 2 ζ 

π + , π - get mass, π 0 , γ stay 

massless 

 

∂ µ A a L 2 

µ , 

it is easy to derive the Feynman rules for the gauge propagator and vertex: 

π


mass term: 

∼ c 2 

e 2 

+ + + ··· 

coeff. 

cutoff: typically Λ = O(4πf) 

16π 2 Λ2 π 2 a 

loop factor 

is there a mπ “hierarchy problem”: unnatural for mπ ≪ Λ 


Figure 7: 1-loop contribution of the SM gauge fields to the Higgs potential. A grey 

represents the strong dynamics encoded by the form factor Π 1 . 

NO...we can’t take Λ arbitrarily high 

section 3.3, as we are now ready to derive the Coleman-Weinberg potential fo 

composite Higgs. 

We will concentrate on the contribution from the SU(2) L gauge fields, neglectin 

smaller correction from hypercharge. The contribution from fermions will be d 

in section 3.4. The 1-loop Coleman-Weinberg potential resums the class of diag 

in Fig. 7. From the effective action (48), after the addition of the gauge-fixing te 

above a certain scale, π description no longer 

makes sense, the π falls apart into its quark 

constituents 

L GF = − 1 

2g 2 ζ 

 

∂ µ A a L 

µ 

2 

, 

it is easy to derive the Feynman rules for the gauge propagator and vertex:


thought experiment: what if π interacted with fields other 

than the photon (some fermions, other gauge interactions, 

etc.) 

these other interactions would also affect V(π). Shape of V(π) 

no longer set... 

What if V(π) developed a non-trivial minima 

V(π) 

potential minimized at 

= v π 

π 

vπ 

fπ 



V(π) 

potential minimized 

at = v π 

vπ 

then ≠ 0 breaks U(1)em 

fπ 

π 

e 2 A µ A µ π + π − → e 2 v 2 π A µ A µ 

photon gets a mass 

scale of the breaking is vπ < fπ 

clearly this is not a situation we want for QCD! 



V(π) 

potential minimized 

at = v π 

vπ 

then ≠ 0 breaks U(1)em 

fπ 

π 

e 2 A µ A µ π + π − → e 2 v 2 π A µ A µ 

photon gets a mass 

scale of the breaking is vπ < fπ 

clearly this is not a situation we want for QCD! 

BUT: shows that NGB interactions lead to a potential, 

and can lead to breaking of symmetries the strong 

dynamics respected. New scale vπ develops 



based on our QCD analogy, we have a recipe for a Goldstone 

Higgs scenario: 

• assume some strong dynamics at a high scale f. 

EWS should remain unbroken: G → H ⊃ SU(2)w ⊗ U(1)Y 

• that dynamics generates a bunch of Goldstone bosons, 

including the Higgs (4-plet of particles), as a doublet. 

• interactions exterior to strong dynamics lead to V(h). 

Choose interactions such that V(h)min is at h ≠ 0. 

Instead hmin = v 

• Higgs is a composite particle → no hierarchy problem 

• v < f means EWSB happens at a scale lower than the 

strong dynamics 





step 1: 















step 1: 








step 2: 





Higgs Beyond the 

SM and SUSY 

lecture #2 

Adam Martin 

CERN/Notre Dame 

(adam.martin@cern.ch) 

YETI Winter School, IPPP Durham UK, 2013 


Outline 

Lecture #1 

• Where are we now and where do we go from here 

• Did it have to be a Higgs 

• EFT for beyond the SM 

• Composite Higgs (Higgs as a Goldstone Boson) 

Lecture #2 

• more Composite Higgs (Higgs as a Goldstone Boson) 

• where & how to look at LHC 


ecap from yesterday: 

idea behind composite Higgs scenario: 

assume there exists some new strong 

dynamics in the multi-TeV range 

H 

SU(2)⊗U(1) 

composite objects, 

including H, are formed 

electroweak symmetry unbroken 

mH = 0 at tree level. Potential V(h) generated by loops 

involving gauge/Yukawa interactions 

V(h) minimized 

at h = v ≪ f 

V(π) 

electroweak symmetry broken 

vπ 

f 

π 


ecap from yesterday: 

pions of QCD are a well-known example of similar physics: 

SU(2)L ⊗ SU(2)R → SU(2)V + 3 NGB 

U(1)em ∈ SU(2)V generates mπ (more generally, V(π)) 

pions in U: 

U = 1 2×2 exp 

2 iπ 

f π 

 

, U → UL U U † R 

CH setup will get us: 

• a naturally light Higgs: no hierarchy problem since the 

Higgs falls apart into constituents above some scale 

• Higgs couplings in the LΣ (a,b,c) form 

• separation of EW physics (W/Z/h, etc.) from the strong 

dynamics by f/v 



















step 1: 















step 1: 








step 2: 






step 1: 

the art in these models is picking the right 

pattern of symmetry breaking... 

the (global) group left unbroken by the strong dynamics: 

• must contain SU(2) ⊗ U(1)... 

• actually SU(2)⊗ SU(2) ≅ SO(4) works better (T parameter) 



step 1: 






starting group must be bigger than ending group: 

choice that works: SO(5) 

SO(5) 

SO(4) 

: 

10 generators → 6 generators 

= 4 broken generators = 4 NGB 

just enough! 



step 1: 






starting group must be bigger than ending group: 

choice that works: SO(5) 

SO(5) 

SO(4) 

: 

10 generators → 6 generators 

= 4 broken generators = 4 NGB 

just enough! 

assemble: 

Σ = exp 

2i χa T a 

f 

strong scale 

Σ 0 

broken generator 

NGB 

symmetrybreaking 

‘vev’ 



Huh 

 

Σ ex =exp i ⎝ 

f 

use SU(3)/(SU(2)⊗U(1)) as an explicit example: 

⎛ 

⎞ 

χ 4 − iχ 5 

χ 6 − iχ ⎠ 7 

χ 4 + iχ 5 χ 6 + iχ 7 

⎛ 

 

⎝ 

0 

0 

1 

⎞ 

⎠ 

Σ 0 



Huh 

 


f 


⎛ 

⎞ 

χ 4 − iχ 5 

χ 6 − iχ ⎠ 7 

χ 4 + iχ 5 χ 6 + iχ 7 

⎛ 

 

⎝ 

0 

0 

1 

⎞ 

⎠ 

SU(2)⊗ U(1) correspond to these (unbroken) generators 

Σ 0 



Huh 

 


f 


⎛ 

⎞ 

χ 4 − iχ 5 

χ 6 − iχ ⎠ 7 

χ 4 + iχ 5 χ 6 + iχ 7 

⎛ 

⎝ 

 

0 

0 

1 

⎞ 

⎠ 


Σ 0 

NGB come with broken generators. 

The set of 4: χ₄,₅,₆,₇ form a doublet under the SU(2)w ⊗ U(1)Y 



Huh 

 


f 


⎛ 

⎞ 

χ 4 − iχ 5 

χ 6 − iχ ⎠ 7 

χ 4 + iχ 5 χ 6 + iχ 7 

 

⎛ 

⎝ 

0 

0 

1 

⎞ 

⎠ 


Σ 0 

NGB come with broken generators. 

The set of 4: χ₄,₅,₆,₇ form a doublet under the SU(2)w ⊗ U(1)Y 

L Σ = f 2 

4 tr(Dµ Σ ex D µ Σ † ex)+··· 

contains interactions of χ₄,₅,₆,₇ with 

W/Z/γ and each other 



Now for the real thing: SO(5)/SO(4) 

L Σ = f 2 

4 tr(Dµ Σ D µ Σ T )+··· 

after LOTS of ugly group theory & algebra 

L Σ = (∂ µh) 2 

2 

+ g2 f 2 

4 

sin 2 h 

f 

 

W µ + W −µ + g2 f 2 h 

 

8cos 2 θ sin2 f 

Z 0 µZ 0µ 



Now for the real thing: SO(5)/SO(4) 

L Σ = f 2 

4 tr(Dµ Σ D µ Σ T )+··· 

after LOTS of ugly group theory & algebra 

L Σ = (∂ µh) 2 

2 

+ g2 f 2 

4 

sin 2 h 

f 

 

W µ + W −µ + g2 f 2 h 

 


Z 0 µZ 0µ 

ASSUMING: ≠0 (have to justify later with V(h) ) 

set h → h + in above, and expand 

define: 

v = f sin 

 

f 

 

EW scale v < scale of 

strong dynamics f 



Keep L Σ expanding: = (∂ µh) 2 

+ g2 f 2 

2 4 

f 2 sin 2 h 

f 

 

sin 2 h 

f 

 

W µ + W −µ + g2 f 2 h 

 


= v 2 +2vh 1 − ξ + h 2 (1 − 2ξ) + ··· 

where: ξ = v2 

f 2 

Z 0 µZ 

recall our Higgs EFT: 

m 2 W 

 

1+a 2 h ··· 

v + bh2 v 2 + 

∴ in the SO(5)/SO(4) composite Higgs model 

a = 

 

1 − v2 

f 2 

, 

b =1− 2 v2 

f 2 



bad behavior in WLWL amplitudes delayed by a factor of 

(1 − a) −1/2 ∼ f 2 

v 2 

eventual strong dynamics... 

∴ expect resonances at scale ~f 

in analogy with to QCD 

recall: precision EW bounds a require 

v 

f 0.5 

so strong coupling scale pushed to ~ 10 TeV (at least) 

other patterns of symmetry breaking would have different 

values for a,b,c, as well as more states 

ex.) SO(6)/SO(5) has 5 NGBs, 

4 ∈ H + 1 extra scalar η 

many other 

possibilities 


Higgs potential 

How do we get ≠ 0 in the first place 

right now our hi interact with gauge fields, but we 

know from QCD experience that V(h) generated from 

these interactions alone has a minimum at h = 0 

Solution: Yukawa couplings 

even forgetting V(h), our theory was incomplete, 

because it did not explain how fermion masses arise 

we can write 

yfQ L Σ u ∗ R + h.c. 

but why does such a term exist for gauge bosons, 

gauge invariant demanded W, Z talk to h. No such reason 

for the fermion mass term 


Higgs potential 

Also: If we imagine Σ is a bound state of more 

fundamental fermions (the things that the composite Higgs 

is made of), analogous to QCD pion = 

yfQ L Σ u ∗ R → y (Q Lu ∗ )( ¯ψψ) R 

Λ 2 

previous term is dimension-6, 

suppressed by some scale 

need Λ low to make fermion masses big enough 

(mt!), but low Λ would cause large flavor 

problems 

i.e.) 

(Q L d ∗ R )2 

Λ 2 

leads to large K 0 - K̅0, 

B 0 -B̅0 mixing, etc. 

So, try a different approach: ‘partial compositeness’ 


Partial Compositeness 

• new strong dynamics makes mesons (including the h), so it 

can make ‘baryons’ = composite fermions too 

• composite baryons are massive even without EWSB, just as 

proton would have mass even without quark masses. 

• proton interacts strongly with QCD pion ∴ composite 

fermions will interact strongly with composite higgs. 









• by mixing the SM fermions with the composite fermions, 

the SM fermions can acquire mass 









• by mixing the SM fermions with the composite fermions, 

the SM fermions can acquire mass 

• the price we pay for massive SM fermions is new states, 

the composite fermions. New states -> new LHC signals 



in practice (schematically) 

composite fermions 

mass terms for composites 

L F = ∆ L Q L Q R + ∆ R t R T L + M Q Q L Q R + M T T L T R + Y T Q L Σ T R + h.c. 

SM fields 

composite + higgs couplings 

blue terms: come from strong sector & therefore obey 

same SO(5) symmetry 

there are several choices for what representations 

composite fermions sit in (4, 5, 10, etc.), leading to slightly 

different structure of the interactions 

Note: 

Q L , Q R , 

etc. must be colored particles 


Undo the mixing 


L F = ∆ L Q L Q R + ∆ R t R T L + M Q Q L Q R + M T T L T R + Y T Q L Σ T R + h.c. 

 

QH 

q L 

 

= 

 

∆L 

M Q 

0 0 

 

QL 

Q L 

 

+ similar for t R T L,R 

yields 

Q L =cos(φ L ) Q H + sin (φ L ) q L 

Q L = − sin (φ L ) Q H + cos (φ L ) q L 

(q L h t ∗ R) Y T sin (φ L )sin(φ R ) 

SM fermions get mass by 

mixing with composites 

expanding h about 

vev, find c*: 

c = 

 

1 − v2 

f 2 

+ similar for t R T L,R 

Q L 

h 

*depends on 

representation of 

Q L , Q R ,etc. 

t R 


About that potential 

within this setup, we can calculate the V(h) from loops of 

fermions, in same fashion as done before 

+ + + ··· 

from gauge interactions 

t L 

+ 

V g (h) ∼ = c 1 sin 2 h/f 

from top interactions 

t R 

t L 

t R 

t R 

+ ··· 

t L 

Figure 12: 1-loop contribution of the SM top and bottom quark to the Higgs potential. 

Upper row: diagrams where the same elementary field, either q L =(t L ,b L ) or t R , circulates 

in the loop with a propagator i/(p Π 0 ). A grey blob denotes the form factor pΠ 1 . Lower row: 

diagrams where both t L and t R circulate in the loop with a Higgs-dependent propagator ∼f (see 

text). In this case a grey blob denotes the form factor M1 u. 

V t (h) ∼ = −c 2 cos h/f − c 3 sin 2 h/f 


About that potential 

Vmin at h≠0 is possible, requires delicate balance 

between +ve and -ve contributions to obtain v ≪ f = 

requires some ‘tuning’ of parameters 

what’s in these blobs 

= + + ... 

know π 

become strongly 

coupled at high 

energy (A ∼ s) 

parameterize as 

+ + 

L ⊃ Π µν (q 2 ) A µ A ν πa 

2 

Π μν encodes all the strong dynamics effects. Some aspects like Π(0), Π(∞) 

are set by symmetries, details require model 

ex.) large N, extra dimensions 

7: 1-loop contribution of the SM gauge fields to the Higgs poten 

ts the strong dynamics encoded by the form factor Π 1 . 


Review 

high energies: constituents rather than composites are 

relevant d.o.f, analog of q, g of QCD 

O(f)-O(4πf) 

v 


Review 



O(f)-O(4πf) 

strong dynamics kick in, constituents confined, breaks 

SO(5) → SO(4). EWS unbroken 

not: 

¯q L q R 

instead: 

ij ψ i ψ j 

v 


Review 



O(f)-O(4πf) 



not: 

¯q L q R 

instead: 

ij ψ i ψ j 

NGB + massless gauge fields, described by L Σ 

loop-level gauge, Yukawa interactions generate V(h) 

v 


Review 



O(f)-O(4πf) 



not: 

¯q L q R 

instead: 

ij ψ i ψ j 

NGB + massless gauge fields, described by L Σ 

loop-level gauge, Yukawa interactions generate V(h) 

v 

V(h) has nontrivial minima,≠0, EWSB 

bigger separation: f ≫ v, more SM-like 


E 

Review 

O(f)-O(4πf) 

v 

t 

W ± /Z 0 

b 


E 

Review 

O(f)-O(4πf) 

new vector resonances: other composites of strong 

dynamics with different spin, parity. 

 

W’, Z’, W’’, ... 

... 

Analog of ρ, a 1 , etc. of QCD 

v 

t 

W ± /Z 0 

b 


E 

Review 

O(f)-O(4πf) 

new vector resonances: other composites of strong 

dynamics with different spin, parity. 

 

W’, Z’, W’’, ... 

... 

Analog of ρ, a 1 , etc. of QCD 

v 

... 

t 

 

fermion resonances: ‘baryons’ that mix with 

SM fermions 

T’, B’, etc. 

T̅’ L H T’ R interactions + mixing → SM mass 

terms 

W ± /Z 0 

b 


1.) Higgs couplings: a, b, c 

LHC signals 

study all possible Higgs production and decay process to extract a, c 

intricate process, as different production mechanisms scale differently 

with a, c, and contribute differently to each final state 

ct 

a 


LHC signals 

1.) Higgs couplings: a, b, c 

relevant NOW 

study all possible Higgs production and decay process to extract a, c 

intricate process, as different production mechanisms scale differently 

with a, c, and contribute differently to each final state 

ct 

a 


LHC signals 

different composite Higgs models → different a, c, possibly 

even extra Higgs decay modes from new particles 

ex.) 

gg → h → W μ 

+ 

W μ 

- 

∼ a c t /Γ H 

VBF pp → h → τ + τ - ∼ a c τ /Γ H 

gg→ h → τ + τ - ∼ c t c τ /Γ H 

BUT, careful: H + jj is not VBF alone, H+0j is not just gg → H 

also, Γ H knows about all c i 


LHC signals 

3A. COMPOSITE (GOLDSTONE) HIGGS 

compiling Higgs data (prior to HCP) 

[More minimal models: Four Goldstones+Custodial symmetry] 

CMS Likelihoods 

m h 125 GeV 

SO(5)/SO(4) with 68 CLSM fermions in spinor (“MCHM4”): 

SM 

90 CL 

95 CL 

a = c = 1 − v 2 /f 2 

SO(5)/SO(4) with SM fermions in fundamental (“MCHM5”): 

c 

2.0 

1.5 

1.0 

0.5 

SO(5)/SO(4) 

model 

a = 1 − v 2 /f 2 

c = 1 − 2v2 /f 2 

 

MCHM4 

1 − v2 /f 2 

MCHM5 

 

 

alternate SO(5)/SO(4) 

model w/ different 

representation for fermions 

Realizes a fermiophobic limit; 

0.0 

studies exist, more ongoing... 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 

decreasing v/f 

a 

(Gabrielli et al, 1202.1796) 

[from J. Galloway] 


LHC signals 

coupling b is trickier.. requires studying multi-Higgs 

production in detail 

h 

low cross section and b must be 

disentangled from other 

hh production diagrams 

h 

see ex.) [Dolan, Englert, Spannowsky 

1206.5001,1210.8166] 

W + W - h 3 signal .. from further 

expansion of f sin²(h/f) 

doesn’t exist at tree level in the SM 

h 

h 

h 


LHC signals 

coupling b is trickier.. requires studying multi-Higgs 

production in detail 

h 

h 

low cross section and b must be 

disentangled from other 

hh production diagrams 

high-luminosity needed! 

see ex.) [Dolan, Englert, Spannowsky 

1206.5001,1210.8166] 

W + W - h 3 signal .. from further 

expansion of f sin²(h/f) 

doesn’t exist at tree level in the SM 

h 

h 

h 


2.) production of new particles 

LHC signals 

we’ve seen that composite Higgs models generically have 

new vector (spin-1: W’, Z’,W’’) and fermion (T’, B’) resonances 

• interactions of Higgs with W/Z set by choice of symmetry 

breaking... much more model dependence for resonances 

& their interactions 

• additional complication: we know the theory is strongly 

coupled.. no obvious ‘best’ way to proceed. 

One idea: rescale properties of QCD resonances 

ρ → W , 

M W = 

 

3 

N C 

f 

M ρ 

f π 

not very quantitative 

∼ 4 TeV for f = 2 v 


Extra dimensional models on a slide 

One way to model the strong dynamics is with an extra dimension 

4 dim 

∂ 2 φ = φ + ∂ 2 zφ 

4-d deriv. deriv. along ẑ 

φ + X 2 φ 

‘massive’ field 

φ 

‘massless’ field 

ẑ 

couplings among fields = integral over ẑ of 

overlapping profiles 

5D field = whole tower of 4D fields of increasing 

mass, interpret as SM field + resonances 

L 

masses, interactions, Π(q²) set by few parameters: 

size, geometry of 5th dimension 

but not fundamental, just a model 

0 

dz φ 1 (z)φ 2 (z)φ 3 (z) 


spin-1 resonances: 

LHC signals 

slight mixing between W’, Z’ and W, Z, means new 

resonances produced most easily in ŝ-channel 

q 

q̅’ 

W W’ 

+ similar for 

Z’, Z’’, etc. 

may look like usual W’, Z’ 

BUT: W’,Z’ couple strongest to other 

strong-sector states, like the longitudinal 

W, Z & h (even t). Big couplings mean Γ W’ , 

etc. can be big. 

usual W’, Z’ LHC searches assume zero (or very small) 

W’WZ interactions... these need to be reinterpreted for 

particles w/ strong interactions with W, Z, etc. 


BR [pb] 

" ! 

LHC signals 

cleanest signal for W/Z decay products is the fully leptonic 

mode: W’ → WZ → 3 + ν 

10 

1 

$ 

16 8 Summary 

7 

CMS s = 7 TeV 

"(pp # W’ # WZ) 

ATLAS 

Obs. "(pp Limit # % , a # WZ) 

T T Exp. Limit 

Expected Limit 

Exp. Expected ± 1# Limit Exp. ± 2# 

10 1 

# (LO) 

# 

Expected ± 1" 

W' 

Expected ± 1" W' 

(NNLO) 

CMS W' ! 3 l + % 



Observed limit 

Observed limit 

-1 

10 

s= 7 TeV 

s= 7 TeV 

1 

-2 

-1 

10 

-1 -1 

Limit 

Ldt = 1.02 fb 

$ 

Ldt = 1.02 fb 

$ 

W' 

(LO) = 889 GeV 

L dt = 5.0 fb 

Limit W' 

(NNLO) = 938 GeV 

200 300 700 400 800 900 500 1000600 1100 1200 700 1300 800 1400 1500 

m W’ [GeV] 

m % [GeV] M WZ 

(GeV) 

T 

ATLAS 

(a) 

1.02 fb -1 CMS s = (b) 7 TeV 

ATLAS 

200 300 400 500 600 700 800 900 1000 

IG. 3: The observed and expected limits on σ × BR(W ′ → WZ)forW ′ → WZ (a) and pp → ρ T ,a T → WZ (b). The 

eoretical prediction is shown with a systematic uncertainty of 5% due to the1 

Exp. ± 1# Exp. ± 2# 

choice of PDF and is estimated by comparing 

e differences between the predictions of the nominal PDF set MRST2007LO ∗ # (LO) # 

and the ones given RS by MSTW2008 RS 

(NLO) 

LO PDF 

sing the LHAPDF framework. The green and yellow bands represent respectively the 1σ and 2σ uncertainty on the expected 

mit. 

[GeV] 

m &T 

500 

400 

BR [pb] 

" ! 

leptonic modes have small BR .. combined with small 

production cross section, rate 10 -1 

will be a problem as mW’ 

500 

increases 

ATLAS 

-2 

Limit 

10 

RS 

(LO) = 803 GeV 

-1 -1 

Ldt = 1.02 fb 

400 $ $ 

L dt = 5.0 fb 

Limit RS 

(NLO) = 879 GeV 

ATLAS 

-1 

$ 

Ldt = 1.02 fb 

m aT 

= 1.1 m % 


T 

[GeV] 

m &T 

B (W' ! WZ) )(pb) 

# (pp ! W') " 

! ZZ) )(pb) 

KK 

B (G 

) " 

KK 

# (pp ! G 

m aT 

>> m % 

CMS 5 fb -1 

Obs. Limit 

Exp. Limit 

800 T 1000 1200 1400 1600 1800 2000

LHC signals 

semi-leptonic modes ( + jj, ν+jj) look swamped by 

background (W/Z + jets) ... 

but we can use the fact that W, Z from a ~few TeV 

W’ will be highly boosted 

angular sepn. of W→jj ~ 2 m W /p T ~ 0.3 for p T ~ 500 GeV .. both 

decay products fall into the same ‘jet’ 

boosted W 

QCD 

jet ‘substructure’ will be an essential tool for 

uncovering such signals 

tutorial session at work! 

[Butterworth et al ’08, 

Kaplan et al ’08, ...] 


s, 

se 

rs 

ts 

to 

g 

ce 

II 

el 

t, 

ls, 

e, 

u- 

in 

y 

ns 

in 

ns 

ke 

≡ 

s, 

te 

2 M 2 − m 2 + δ 2 

θ r = 1 

 

2 δ M 

2 tan−1 M 2 − m 2 − δ 2 

LHC signals (4) 

fermionic m t = cos θresonances l cos θ r (m + δ tan = θ l new + M tan heavy θ l tanfermions 

θ r ) 

m T = cos θ l cos θ r (M − δ tan θ r + m tan θ l tan θ r ) 

In exactly a more useful what parametrization, states are present the angles and & their couplings 

may of the be expressed composite as a function fermions of m(masses t ,m T and M, 

masses depends on 

details representations) 

η = δ/M . 

simple tan θexample, r = m T 

tanto θ l show some general (5) features: 

m t 

t R mixing with composites T, T c ∈ (3,1) 2/3 

θ l = 1 

2 sin−1 η 2m tm T 

m 2 T − m2 t 

y t Q 3 Ht c + M TT c + δ T t c + h.c. 

L ⊃ y 1 Q 3 Ht c + δ Tt c + MTT c 

where Q 3 is the (SM) third generation elec 

blet (t, b). Note that an additional term Q 

allowed under all symmetries. It can, howe 

nated by rotating t c and T c and consequen 

δ and M. 

The mass eigenstates are combinations 

t c and T c . The full mass matrix, including n 

vacuum expectation value (vev) is given by 

 

t 

c 

T c , 

(6) 

In order to study the collider phenomenology of 

this minimal setup we introduce four component Dirac 

t1 

t2 

spinors t D = and T D = . The non-diagonal 

t c† 

1 

t c† 

2 

interactions of heavy top can then be recast in the form 

L ⊃ m t cos 2 θ l 

v 

h ¯T D (tan θ r P L + tan θ l P R ) t D 

+ g 2 sin θ l cos θ l 

 

Z µ ¯TD γ µ P L t D + ¯t D γ µ 

P L T D (7) 

2 cos θ W 

+ g 2 sin θ 

√ l 

 

W 

+ 

µ ¯TD γ µ P L b D + Wµ − ¯b D γ µ 

P L T D , 2 

where P L,R are the usual projectors. 

Within the minimal model, the top-partner can only 


m 0 

t T 

δ M 

where m = √ y 1 

2 

v and v is the Higgs vev. In 

mass matrix is not symmetric and can be di 

a bi-unitary transformation which rotates th 

and the right handed quarks by different a 

such a rotation, the quark mass eigenstate 

to the quarks in Eq. (1) in the following wa

LHC signals 

new interaction 

q 3 

T t c 

× 

H 

Branching ratio, up to small 

corrections, 

set by Goldstone equivalence: 


LHC signals 


q 3 

T t c 

× 

H 


corrections, 


T decay modes 

T 

t 

h 

t 

Z 

b 

W 

∼ 25% ∼ 25% ∼ 50% 


LHC signals 


q 3 

T t c 

× 

H 


corrections, 


T decay modes 

T 

t 

h 

t 

Z 

b 

W 

∼ 25% ∼ 25% ∼ 50% 

in large mass limit, 

only parameter is MT 


LHC signals 


q 3 

T t c 

× 

H 


corrections, 


T decay modes 

T 

t 

h 

t 

Z 

b 

W 

∼ 25% ∼ 25% ∼ 50% 

in large mass limit, 

only parameter is MT 

extra ‘chiral’ quarks 

(4th generation) 

only have this decay mode 


LHC signals 

both pair-production (T T̅ ) and single production 

possible. Pair production dominates for MT ≲ 1 TeV 

q 

T 

b 

T 

W 

q̅ 

Probably, stronger constraints could be found in mixed modes. 

T̅ 

q 

q’ 

• fairly large cross section 

• lots of W,Z,h, b in the final state 

Probably, stronger constraints could be found in mixed modes. 

The most interesting targets are 

The most interesting targets are 

T T → tZ 0 bW − + c.c. 

T T → tZ 0 bW − + c.c. 

→ bjj + − b + (W ) 

→ bjj + − b + (W ) 

where (W ) = ν or boosted hadronic W 

where (W ) = ν or boosted hadronic W 

T T → tZ 0 th 0 + c.c. 

→ bjj + − bνbb 

where the lowest-mass (bb) combination is in the light Higgs 

window. 

where the lowest-mass (bb) combination is in the light Higgs 

window. 


T T → tZ 0 th 0 + c.c. 

→ 

bjj + − bνbb 

FIG. 5: Total cross sections for T ¯T production (dashed) and T +jet production (solid and dotte 

via t-channel W -exchange versus mass M T at the LHC. The solid line is for the couplings λ 1 = λ

existing limits on T’, B’: 

LHC signals 

assume 100% BR into one mode: T→ W b (4th gen), T → t Z 

B’→ t W 

CMS 1109.4985 

also 

ATLAS 1204.1265 

recent 

counterexamples 

Rao, Whiteson 

1204.4504, 1203.6642 

De Simone et al 

1211.5663 

so, limits need to be reinterpreted. Mixed modes likely to yield 

stronger constraints. 

substructure could be useful for identifying hadronic W/Z 


more exotic possibilities: 

LHC signals 

In some scenarios, Q3L doublet is extended & includes 

higher charge states: X5/3, or X7/6 

g 

g 

T 5/3 

W − 

l + 

ν 

t 

W + 

l + 

ν 

b 

g 

cascade decays of 

X → t W→ W b 

gives like-sign W − 

dileptons 

g 

B 

¯q q ′ l + ν 

t 

W + 

b 

g 

¯t 

W − 

¯b 

g 

low SM background 

¯B 

W + 

¯t 

W − 

¯b 

[Contino, Servant ’08] 

W + l + q ′ 

¯T 5/3 

q ′ ¯q 

¯q q ′ 

ν 

¯q 

Figure 1: Pair production of T 5/3 and B to same-sign dilepton final states. 


LHC signals: summary 

• the era of precision Higgs: composite-ness of the Higgs is 

encoded in deviations of couplings from their SM values 

mass scale of new particles ~f is constrained via Higgs 

coupling measurements. More SM-like couplings → 

smaller v/f → heavier new states 

• direct production of new particles: 

both spin-1 and fermionic resonances have large 

couplings to W/Z/h/t: W/Z/h/t-rich final states 

W’/Z’/T’ have different properties than LHC searches 

usually assume -- care required in interpreting limits 


Conclusions: 

• immediate LHC focus: how SM Higgs-like is X(125) 

LΣ EFT is a good framework to use, look for deviations a,b,c≠1 

a,b,c≠1 indicate strong coupling enters at some scale 

• Composite Higgs: Higgs as a Goldstone boson. 

UV setup that gives light Higgs + a,b,c≠1 

gauge and Yukawa interactions generate nontrivial V(h) 

and lead to EWSB. tuning of different contributions to get 

v ≪ f 

other composites (spin-1, fermions) in spectra, possible 

targets for LHC searches. 

different than ‘generic’ W’Z’/T’: large couplings to W/Z/h/t

slides - Institute for Particle Physics Phenomenology

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