Moose Moose and and Holographic Holographic Higgsless Higgsless Models Models Roberto Casalbuoni Dept. of Physics, INFN and GGI - Florence

Milano, March 13, 2008

1

Outline of the talk (Based on papers by J. Bechi, R.C., F. Coradeschi, S. De Curtis, D. Dolce, D. Dominici, R. Gatto)

• Motivations for Higgsless models • Example of breaking the EW symmetry without Higgs (BESS) • Linear moose • Unitarity bounds • Delocalizing fermions •The continuum limit • The holographic approach • Summary and conclusions 2

Problems of the Higgs sector Consider the Higgs potential

2 µ V(ϕ ) = -µ 2 | ϕ |2 + 2 | ϕ |4 v

µ2 2 2 λ = 2 , m H = 2λv v

The evolution of the coupling (neglecting gauge fields and fermions contributions) shows up a Landau pole at MLp

1 λ(M) = 1 3 M2 - 2 log 2 λ(m H ) 4π mH

M Lp = m H e

4π 2 v 2 /3m2H

● Or MLp pushed to infinity, but then λ goes to 0, triviality! ● Or there is a physical cutoff at a scale M < MLp. From this we get a bound on mH

3

2

4π 2 1 m < v M 3 log mH 2 H

1023

3 M log 2 2 4π mH

λ 2 δm = 2 M 8π 2 H

MPlanck

1017

with M of the order of MPlanck, or MGUT. The naturalness problem follows and to avoid it the quadratic divergence should cancel (SUSY).

MGUT

1014 1011 140

2

If the cutoff is big λ(mH) is small (~ 0.18). The theory is perturbative, but the Higgs mass acquires a mass of order

M(GeV)

1020

λ(m H ) =

1

160

180

200

220

240

260

mH(GeV)

4

If we keep the cutoff of order of TeV, λ(mH) is large (>>1). The theory is nonperturbative, but there are no problems with the quadratice divergence, since mH ~ Μ.

M(GeV)

2800. 2700. 2600. 2500.

1) λ > 1 ⇒ new particles around 1 TeV In the following the second option will be considered: new strong physics at the TeV scale

2400.

2300.

2200.

900

1000

1100

1200

mH(GeV)

5

Symmetry Breaking ● Since we are considering a strongly interacting theory: an effective description of the SB

● We need to break SU(2)LxU(1) down to U(1)em. The SB sector should be

G ⇒ H, G ⊃ SU(2) L ⊗ U(1), H ⊃ U(1)em

of the type

● In the SM the SB sector is the Higgs sector with

G = SU(2) L ⊗ SU(2) R , H = SU(2) V ● If the SB sector is strongly interacting one can describe it at low

energies making use of a general σ model of the type G/H

6

● For instance, in the case SU(2)LxSU(2)R/SU(2)V the model can be

described in terms of a field Σ in SU(2) transforming as

Σ → g L Σg , g L ∈ SU(2) L , g R ∈ SU(2) R † R

● The breaking is produced by

〈 0 | Σ | 0〉 = 1

2

v µ † iπ·τ/v L = ( ∂ µ Σ∂ Σ ) , Σ = e , 4

π =0

● In this way one could describe also an explicit

breaking SU(2)LxSU(2)R to U(1)em through an explicit SU(2)V breaking term (the ρ-parameter is the standard one)

2

2

2 v v µ † † µ L = ( ∂ µ Σ∂ Σ ) + ( ρ − 1) ⎡⎣Tr(T3Σ ∂ Σ) ⎦⎤ 4 2 7

● The model can be easily gauged to SU(2)xU(1) introducing the gauge covariant derivatives

Dµ Σ = ∂ µ Σ + igWµ Σ - ig′ΣBµ ● With W and B the gauge fields of SU(2) and U(1) respectively. Notice

that with respect to the strong dynamics described by the σ model, the interactions with W and B are to be considered as perturbations.

● The strong dynamics is completely characterized by the

transformation properties of the field Σ which can be summarized in the following moose diagram.

8

The σ model can be obtained as the formal limit of the SM in the limit of MH to infinity. 2 H 2

2

1 M ⎛1 µ † † 2⎞ L = Tr ( Dµ MD M ) TrM M v ⎜ ⎟ + 4 8v ⎝ 2 ⎠ 1 1 µν + TrFµν (W)F (W) + TrFµν (B)Fµν (B) 2 2 Fµν (W) = ∂ µ Wν − ∂ ν Wµ + g[Wµ , Wν ] Fµν (B) = ∂ µ Bν − ∂ ν Bµ i i Wµ = - Wµ · τ, Bµ = - Yµ τ 3 2 2 Through the definition

M = SΣ

Where S is a singlet field with a vev fixed to v in the limit of large λ or Higgs mass. 9

The field Σ describe the Goldstones giving mass to W and Z. Therefore they are related to the longitudinal modes. The interesting amplitude is then WLWL to WLWL which is strictly related to the physics of the Goldstone bosons. This can be seen easily by separating the different contributions and examining how the cancellation of the bad high-energy behaviour happens within the SM which is renormalizable and therefore satisfying unitarity:

● The quartic divergence is cancelled by the gauge contributions ● The quadratic part is cancelled by the Higgs boson contribution

10

γ

Z

Tgauge

s ≈ −g 2 MW 2

2 1 s s 2 2 2 2 2 (g v) ≈ g , s >> M , M H W 2 4 2 s - MH MW MW

1 2 M 2H 1 | a 0 |= ≤ g 2 16π M W 2

2 M T = gauge + Higgs ≈ g 2 2H MW

⇓ 2 M M 2H ≤ 8π 2W = 8πv 2 ≈ (1.25TeV) 2 g

11

An important result is the Equivalence theorem (Cornwall, Levin & Tiktopoulos, 1974; Vayonakis, 1976), stating the for E>>MW the previous scattering amplitude can be evaluated by replacing the longitudinal vector bosons with the corresponding Goldstone bosons

+

∼

TGoldstones

γ, Ζ

∼

+

+

2

2

2 M M ⎛ MH ⎞ ⎛ MH ⎞ 2 H ≈λ+λ ≈ = g = g ⎟ ⎜ ⎟ ⎜ s - M 2H ⎝ v ⎠ ⎝ M W ⎠ M 2W

For E >> MW, MH

2 H

TGoldstones = Tgauge+Higgs 12

In the limit s 0, is of opposite sign with respect to the vector boson V. The result is 2

1⎛ g ⎞ b ε3 = ⎜ ⎟ 4 ⎝ g1 ⎠ 2

23

2

1⎛ g ⎞ b ε3 = ⎜ ⎟ 4 ⎝ g1 ⎠ 2 If there are no fermions, for g1 ~ g - 5g, we get

g1 = g ⇒ ε3 = 0.25, g1 = 5g ⇒ ε3 = 10

-2

Since experimentally (we are speaking of the contribution from new physics)

ε3 ≈ 10−3 we would need an unnatural value of g1 bigger than 16g. If a direct coupling of fermions to V is present we can improve the situation but at the expenses of some fine tuning since b should be fixed at the level of 2x10-3. 24

Unitarity Unitarity bounds bounds for for the the linear linear moose moose (Chivukula, He; Papucci, Muck, Nilse, Pilaftis, Ruckl; Csaki, Grojean, Murayama, Pilo, Terning)

●

The worst high-energy behavior comes from the scattering of longitudinal vector bosons. For s >> MW2 the amplitude can be evaluated using the equivalence theorem. We do this introducing the GB’s associated to W and Z GG if π⋅τ /2fi2 i

Σ =e

• In the high-energy limit 4

A π+ π− →π+ π−

f =− 4

⎛1 1⎞ ⎜ 6 + 6 ⎟u ⎝ f1 f 2 ⎠

, i = 1, 2

Unitarity limit

u f1 = f 2 → A = - 2 4v Λ moose = 2Λ HSM ≈ 2.5TeV 25

• By taking into account all the vector bosons

and using the

equivalence theorem:

(Σ = e i

G G iπi ⋅τ /2 fi

)

A π+ π− →π+ π− i

i

i

i

u →− 2 4f i

• The unitarity limit is determined by the smallest link coupling. By taking f1 = f2 (see also Chivukula, He, 2002) Unitarity limit

f1 = f 2 → A → -

u 2v 2

Λ moose = 2Λ HSM ≈ 1.8TeV g1 M V < Λ moose , but M V = 2 M W g ⇓ g1 g1 2 M W < 1.8TeV ⇒ < 10 g g

Not compatible with electro-weak experimental constraints without direct fermionic couplings 26

Breaking Breaking the the EW EW Symmetry Symmetry without without Higgs Higgs Fields Fields ● Let us now generalize the moose construction.

● General structure given by many copies of the gauge group G intertwined by link variables Σ.

● Condition to be satisfied in order to get a Higgsless SM before gauging the EW group is the presence of 3 GB and all the moose gauge fields massive.

● Simplest example:

Gi = SU(2). Each Σi describes three scalar fields. Therefore, in a connected moose diagram, any site (3 gauge fields) may absorb one link (3 GB’s) giving rise to a massive vector field. We need:

# of links = # of sites +1 27

● Example: ● The model has two global symmetries related to the beginning and to the end of the moose, that we will denote explicitly by GL and GR . † GL

Σ1

Σ3

Σ2 G1

G2

.....

ΣK-1 Σ K Σ K+1 GK-1 G K

Σ1 ⇒ U L Σ1U1

GR

Σi ⇒ U i-1Σi U †i Σ K+1 ⇒ U K Σ K+1U †R

● The SU(2)L x SU(2)R global symmetry can be gauged to the standard

SU(2)xU(1) leaving us with the usual 3 massive gauge bosons, W and Z, the massless photon and 3K massive vectors. Prototypes of this theory are the BESS model, K = 1, (R.C, De Curtis, Dominici, Gatto, 1985) and its generalizations (R.C, De Curtis, Dominici, Gatto, Feruglio, 1989). Notice that SU(2)V is a custodial symmetry. 28

Electro -weak corrections Electro-weak corrections for for the the linear linear moose moose

S moose

K +1 ⎛ K 1 i µν i 2 † ⎞ = ∫ d x ⎜ − ∑ 2 Tr ⎡⎣ Fµν F ⎤⎦ + ∑ f i Tr ⎡⎣ (D µ Σ i )(D µ Σ i ) ⎤⎦ ⎟ i =1 ⎝ i =1 2g i ⎠ 4

●

If the vector fields are heavy enough one can derive a low-energy effective theory for the SM fields after gauging

SU( 2) L ⊗ SU( 2) R ⇒ SU( 2) ⊗ U(1) One has to solve finite difference equations along the link-line in terms of the SM gauge fields at the two ends (the boundary)

29

●

From vector mesons saturation one gets

g2 ε3 = 4

K ⎛ g2 g2 ⎞ (1- y i )y i 2 2 2 -2 2 ⎜⎜ nV - nA ⎟⎟ = g g g f f (M ) = g ∑ ⎜⎝ m4 m4 ⎟⎠ ∑ g2 1 K 1 K+1 2 1K n i=1 n n i

f2 yi = ∑ x i , x i = 2 , fi j=1 i

● ●

Since

Example:

K+1 1 K+1 1 = ∑ 2 ⇒ ∑ xi = 1 2 f i=1 f i i=1

0 ≤ yi ≤ 1 ⇒ ε3 ≥ 0 fi = fc , gi = gc

(follows also from positivity of M2-1)

1 g 2 K(K + 2) ⇒ ε3 = 6 g c2 K +1

• Notice that ε3 increases with K (more convenient small K) 30

● Possible solution: Cut a link, with f i = 0, M 2 becomes block diagonal and (M -22 )1K = 0

ε3 = 0 Add a Wilson line:

U = Σ 1Σ 2

Σ K Σ K +1

Groups connected through weak gauging Particular example: D-BESS model (R.C. et al. 1995,1996) The theory has an enhanced symmetry ensuring ε3 = 0 (Inami , Lim, Yamada, 1992)

[SU(2) L ⊗ SU(2) R ]

2 31

Unitarity Unitarity bounds bounds for for the the linear linear moose moose (Chivukula, He; Papucci, Muck, Nilse, Pilaftis, Ruckl; Csaki, Grojean, Murayama, Pilo, Terning)

●

We proceed as for the BESS case (K=1) and evaluate the scattering of longitudinal gauge bosons using the equivalence theorem, that is using the amplitude for the corresponding GB’s.

● We choose the following parametrization for the Goldstone fields (we are evaluating WLWL-scattering)

Σi = e

GG if π⋅τ / 2fi2

●The resulting 4-pion amplitude is given by 32

A π+ π− →π+ π−

f4 =− 4

K +1

u f4 + ∑ 6 4 i =1 f i

K

−1 −1 L (u − t)(s − M ) + (u − s)(t − M ) ( ∑ ij 2 ij 2 ij )

i, j=1

⎛1 1 ⎞⎛ 1 1 ⎞ Lij = g i g j ⎜ 2 + 2 ⎟ ⎜ 2 + 2 ⎟ ⎜ ⎟ ⎝ f i f i +1 ⎠ ⎝ f j f j+1 ⎠

• In the low-energy limit, mW