May.18,2010BasedonarXiv:1005.xxxx[hep-ph](wtihF.Sannino)@ChuoUniv. HidenoriFukano ConformalWindowofGaugeTheorieswithFour-FermionInteractionsandIdealWalking ..

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Conformal Window of Gauge Theories with Four-Fermion Interactions and Ideal Walking

Hidenori Fukano

Kobayashi-Maskawa Institute, Nagoya Universtiy.

May. 18, 2010

Based on arXiv:1005.xxxx[hep-ph](wtih F.Sannino)

@ Chuo Univ.

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Outline

. . ¹ . Introduction to Technicolor

. . ² . Walking TC

. . ³ . Conformal window in the gauged NJL and Ideal walking

. . ⁴ . Summary

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Intoroduction (Overview of the SM/BSM)

♠ Roles of the spin-0 field in the Standard Model (SM) Electroweak symmetry breaking (EWSB)

Higgs boson

W/Z bosons and quarks/leptons mass

♠ Problems

Naturalness problem

Origin of yukawa interaction

♠ Solutions to these problems

Supersymmtric model

Extra dimensional model

Technicolor model

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EWSB in the SM

♠ SM : L = L kin + L yukawa − V (φ)

Invariant under gauge symmetry : G EW = SU (2) L × U (1) Y

Massive gauge bososns = Higgs mechansim

♠ EWSB in the SM

select the unique vacuum

→ EWSB

SSB : G EW → U (1) em.

V (φ)

φ

♠ After EWSB, L yukawa gives fermions masses.

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In the SM, the EWSB is occurred by hand ...

This situation is the same as the chiral symmetry in 1950s-1960s

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Chiral symmetry in 50s-60s (Schwinger;Polkinghorne;Gell-Mann and Levy)

Let’s consider the so-called linear σ-model (LσM) as L = L kin. − √

2g _{N N π}

[ ψ ¯ _L M ψ _R + ¯ ψ _R M ^† ψ _L

] − V (M )

where V (M ) = λ 4

4

( tr[M M ^† ] − v ² ) 2

Fields : ψ = (p, n) ^T ,~ π and σ Recast : ψ _L(R) = 1 ± γ ₅

2 ψ and M ≡ 1

√ 2 (σ + i~ π · ~ τ ) ψ _L(R) → g _L(R) ψ _L(R) and M → g L M (x) g ^† _R

where g _L(R) ∈ SU (2) _L(R)

Invariant under the chiral symmetry : SU (2) L × SU (2) R

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Chiral symmetry breaking (χSB ) in the LσM

♠ M = (

iτ 2 φ ^∗ , φ )

where φ ≡ 1

√ 2

( iπ 1 + π 2

σ − iπ 3

)

= ( φ ⁺

φ ⁰ )

♠ V (M ) = V (φ) = λ ₄ (

| φ | ² − 1 2 v ²

) ₂

We can always choose the vacuum : h σ i = v 6 = 0 and h π i = 0

⇓

SU(2) L × SU (2) R breaks down to SU (2) V and

♦ m ² _σ = ∂ ² V

∂σ∂σ = 2λ ₄ v ² and m ² _π = ∂ ² V

∂π∂π = 0

♦ πs are massless Nambu-Goldstone (NG) bosons

♦ Yukawa term = − g _{N N π} h σ i ψψ ¯ → m _N = g _{N N π} v

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χSB in the LσM and EWSB

LσM (50s-60s) EW sector in the SM Lagrangian L kin + L yukawa − V (φ) L kin + L yukawa − V (φ)

σ hσi = v 6= 0 hσi = v EW 6= 0

Scale v ' 93 MeV v EW ' 250 GeV

SSB pattern SU(2)

× SU(2)

→ SU (2)

SU (2)

× U(1)

→ U(1)

Type global sym. breaking gauge sym. breaking π massless NG-bosons massive gauge bosons fermion mass m _N = g _{N N π} v m _f = y _f v _EW

Higgs sector in the SM is almost same as the LσM.

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LσM → QCD, Higgs sector → ???

Now, we know that the color dynamics causes χSB by h q ¯ L q R i 6 = 0.

⇓

The EWSB might be explained by unknown strong dynamics.

i.e. strong coupling thoery might give

h Q _L Q _R i 6 = 0 and v _EW ' 250 GeV.

→ This is the original idea of Technicolor(TC)

(Weinberg,1976; Susskind,1979)

old-fashioned TC = scale-up version of QCD How much bigger than QCD? : v _EW

v = 250 GeV

93 MeV ' 2600

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Problems in the QCD-like TC

TC is based on a QCD-like dynamics so we face with S-parameter problem

FCNC problem

· · ·

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S in TC model (Peskin et.al., 1990,1992)

. Estimation of S in TC model .

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perturbative : S _TC ^naive = 1 6π

N _f 2 d(R).

vector meson saturation : S TC = 8π [ f _V ²

M _V ² − f _A ² M _A ²

]

N

: # of matter which participate in dynamics

d(R) : dimension of matter which participate in dynamics

f

/M

: decay constant/mass of techni-(axial-)vector meson

e.g.] SU (3) TC with N _f = 2 (1-doublet TC, scale-up QCD) S _TC ^naive = 1

2π ' 0.16 < S TC ' 0.3. Too Large !!

c.f. S _exp. = − 0.21 ± 0.09

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FCNC in the extended TC (ETC) (Dimopoulos et.al.,1979; Eichten,1980)

♠ TC-quark and SM quark live in the same multilet under G _ETC

♠ After G _ETC → G _TC ...

q q

q q

ETC ⇒

s

d s

d

FCNC @ tree level

∆M _K : Λ ² _ETC & (10 ³ TeV) ²

q q

Q Q

ETC ⇒ X

hQQi 6= 0

quark/lepton mass term h QQ i TC ' (700 GeV) ³

c.f. hqqi

' (250 MeV)

QCD-like : h QQ i

' h QQ i

→ m

' h QQ i

Λ

∼ O (0.1 MeV)

Too Small !!

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Problems (summary) and solution

. TC is based on a QCD-like dynamics so we face with .

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S-parameter problem : S is too Large Scale-up QCD → S _TC ^naive < S _TC ' 0.3 FCNC problem : strange mass is too Small

QCD-like → ^γ

' 0 → h QQ i

h QQ i

' 1 → m

∼ O (0.1 MeV) The nature might require a new dynamics different from QCD....

⇓

A candidate of new dynamics is so-called walking dynamics.

TC based on a walking dynamics = Walking technicolor

(Holdom,1981; Yamawaki et.al.,1986, Appelquist et.al.,1986)

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“walking” ?

Walking means slowly running/near conformal gauge coupling.

µ α

near critical

EW broken

  Walking TC

would be controlled by Fixed Point.

need to give γ _m ∼ 1.

. Thanks to the walking dynamics (near critical) .

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S _TC is expected to be less than S _TC ^naive .

(Appelquist et.al.,1999; Harada et.al.2004,2006; Kurachi et.al.,2006)

γ _m ∼ 1 → h QQ i ETC

h QQ i TC

' Λ _ETC

Λ TC ∼ O (1000)

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IR-fixed point (IRFP) in the β-fn. (Banks et.al. 1982)

µ ∂α(µ)

∂µ = β(α) where β ^2-loop (α) = −bα ² (µ) − cα ³ (µ) N running : b > 0 & c > 0

α β

α

µ

= ⇒

N IRFP : b > 0 & c < 0 = ⇒ walking around α _∗

α

β α _∗

α

µ α _∗ α _∗ = − b

= ⇒ c

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Dynamics : Ladder Schwinger-Dyson equation (LSDE)

♠ Full SDE (full propagator of fermion : iS ^{− 1} = A(p ² )p / − Σ(p ² ))

= +

− 1 − 1

♠ Ladder approx. : gauge propagator and vertex correction → bare

= =

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Criticality

♠ Results : for running case α > α ⁰ _crit = π

3C 2 (r) → Σ 6= 0 : SχSB solution α < α ⁰ _crit = π

3C ₂ (r) → Σ = 0 : no SχSB

♠ For a case with IRFP

β(α) = − bα ² (µ) − cα ³ (µ) criticality :

α _∗ = − b

c V.S. α ⁰ _crit

α

µ

α _∗

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Example : SU (2) (Appelquist et.al.,1998, Sannino et.al.,2004)

. Criticality .

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.Comparison of α _∗ with α ⁰ _crit = π/[3C 2 (r)]

α

µ

N

= 7

N

= 8 N

= 9

α ⁰ _crit

fundamental repr.

Large # of flavor for IRFP.

→ S _TC ^naive is large (d(r) = 2)

c.f. S

= 1 6π

N

2 d(r)

α

µ

N

= 2 α ⁰ _crit

2-index symmetric repr : . Small # of flavor for IRFP.

→ S _TC ^naive is small (d(r) = 3)

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Phase diagram of SU (N ) (Dietrich et.al. 2006)

Below

the upper solid : AF the lower solid : no IRFP the dashed : SχSB Representation :

Black : Red : Blue : Green : adj.

2 4 6 8 10

5 10 15 20

N c

N f

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Change from gauge theory w/o four fermi

We would like to reconsider the conformal window in the gauged NJL (gNJL) model generalized to different matter representations.

gNJL : gauge theory adding the effects of four-fermion interactions.

Why ? : four fermion interactions naturally arise, as effective operators, at the electroweak scale when augmenting the TC model with ETC interactions.

Q Q

Q Q

ETC

⇒

Q

Q Q

Q

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The gNJL model (For review : Yamawaki hep-ph/9603293)

♠ Lagrangian for the gNJL : L = L kin. + G

N f d [r]

[ ( ¯ ψψ) ² + ( ¯ ψiγ 5 T ^a ψ) ² ]

− 1 4

N ∑

− 1 a=1

F _µν ^a F ^aµν .

♠ LSDE for the gNJL

= + +

− 1 − 1

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Critical line in the gNJL

♠ g = GΛ ²

4π ² and α ⁰ _crit = π 3C ₂ (r)

α g

α ⁰ _crit g _crit = 1

4 (

1 +

√ 1 − α

α ⁰ _crit ) 2

1

1 4

Sym.

SχSB

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Critical gauge coupling for the gNJL

For given g :

α crit (g) =

 

 

 

 

 



4 ( √ g − g )

× α ⁰ _crit for 1

4 < g < 1 , α ⁰ _crit for 0 < g < 1

4 .

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Critical # of flavor for the gNJL

♠ Criticality : α _∗ (N, N _f ) V.S. α crit (g) for given g where α _∗ (N, N _f ) = − b

c = − 4π 11N − 4N _f C(r)

34N ² − 2N _f C(r) [10N + 6C ₂ (r)]

♠ critical # of flavor

• For 1

4 < g < 1 :

N _f ^crit (N, g) = 34N( √ g − g) + 33C ₂ (r)

20N ( √ g − g) + 12[1 + ( √ g − g)]C ₂ (r) · N C(r)

• For 0 < g < 1

4 (is the same as the case w/o G-term) : N _f ^crit (N, g) = 17N + 66C 2 (r)

10N + 30C 2 (r) · N

C(r)

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Examples : SU (2)

loss of A.F.

no IRFP

0.25 0.5 0.75 1 g

6 8 10 12 14 N

the solid line : no G-term the curved dashed line : gNJL case

loss of A.F.

no IRFP

0.25 0.5 0.75 1 g

2 4 N

the solid line : no G-term

the curved dashed line :

gNJL case

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Anomalous dimension

♠ The anomalous dim. along the critical line (g = g _crit with 1

4 < g < 1 ) γ m (g = g crit ) = 1 + ω where ω =

√

1 − α α ⁰ _crit

♠ The dependence on (N, N _f ) :

γ _m (N, N _f ) = 1 + ω(N, N _f ) where

ω(N, N _f ) =

√

1 + 6C ₂ (r) [11N − 4N _f C(r)]

17N ² − N f C(r) [10N + 6C 2 (r)]

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Examples : SU (2)

lossofA.F.

noIRFP

4 6 8 10 12 14 Nf

1.2 1.4 1.6 1.8 2.

Γm

the dashed line : N _f ^crit for g = 0

the curved solid line : gNJL case

lossofA.F.

noIRFP

1 2 3 4 5Nf

1.2 1.4 1.6 1.8 2.

Γm

the dashed line : N _f ^crit for g = 0

the curved solid line : gNJL

case

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The modified diagram

2 4 6 8 10

5 10 15 20

N

N _f SU (N ) w/o G

γ _m ' 1 on the dashed the shaded : conformal the dashed : SχSB

2 4 6 8 10

5 10 15 20

N

N f gNJL with g = 0.75

γ m ' 1.73 on the dashed

the shaded : conformal

the dashed : SχSB

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Summary

The DEWSB is still an interesting candidate of the beyond the SM.

Walking TC is the most suitable model among TC models.

The effects of the four fermion interaction is important for the conformal window.

The effects of a strongly coupled ETC sector on the TC can generate the large top mass.

We may need to :

1] study a concrete ETC model building .

2] confirm our results by several tools (all-orders β-fn, lattice simulation, ....)

Thank you very much !!

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Summary

The DEWSB is still an interesting candidate of the beyond the SM.

Walking TC is the most suitable model among TC models.

The effects of the four fermion interaction is important for the conformal window.

The effects of a strongly coupled ETC sector on the TC can generate the large top mass.

We may need to :

1] study a concrete ETC model building .

2] confirm our results by several tools (all-orders β-fn, lattice simulation, ....)

Thank you very much !!

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S-parameter (Peskin et.al., 1990,1992)

. The relevant corrections from general DEWSB models .

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The relevant corrections due to the presence of new physics trying to modify the electroweak breaking sector of the SM appear in the vacuum polarizations of the electroweak gauge bosons.

(Sannino, arXiv:0911.0931)

S is represented as S ∝ Π ⁰ ₃₃ (0) − Π ⁰ _3Q (0) . S _exp. = − 0.21 ± 0.09 (PDG, 2008) for heavy Higgs.

Π(Q ² ) = NP

EW EW

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S in QCD (Peskin et.al., 1990,1992)

♠ S = 8π [ f _V ²

M _V ² − f _A ² M _A ²

]

In the QCD case, WSRs are given by

1st : f _V ² − f _A ² = f _π ² , 2nd : f _V ² M _V ² − f _A ² M _A ² = 0

= ⇒ f _V ² = M _A ² f _π ²

M _A ² − M _V ² , f _A ² = M _V ² f _π ² M _A ² − M _V ²

= ⇒ S = 4π [

1 + M _V ² M _A ²

] f _π ²

M _V ² ' 0.25

f π = 93 MeV , M V = 770 MeV , M A = 1260 MeV

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S ^naive (c.f. Peskin-Shcroeder)

1-loop calculation

with | m _E − m _N | ¿ m _N , m _E and m _N , m _E À M _Z TC fermion

V V

− ^{A A}

= ⇒ S ^naive = −4π[Π ⁰ _{V V} (0) − Π ⁰ _AA (0)] = N D

6π d(R)

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2-loop β-fn. and IRFP

β ^2-loop (α) = − bα ² (µ) − cα ³ (µ) where

b = 11N − 4N _f C(R)

6π , c = 34N ² − 2N _f C(R) [10N + 6C ₂ (R)]

24π ² and

tr[T ^a T ^b ] = C(R)δ ^ab , T ^a T ^a = C 2 (R) . Asymptotically freedom : b > 0 ⇐⇒ N _f < N ^I ≡ 11N

4C(R) . IRFP : c < 0 ⇐⇒ N f > N _f ^II ≡ 17N ²

C(R) [10N + 6C 2 (R)] .