Behavior of the S parameter in the crossover region between walking - - PowerPoint PPT Presentation

Behavior of the S parameter in the crossover region between walking and QCD-like regimes of an SU(N) gauge theory Masafumi Kurachi

C.N. Yang Institute for Theoretical Physics, SUNY at Stony Brook Refs.

• M. Kurachi and R. Shrock, hep-ph/0605290
• M. Kurachi and R. Shrock, Phys. Rev. D74:056003 (2006)
• Nov. 1, 2006, Joint Meeting of Pacific Region Particle Physics Communities

Outline

• 1. Introduction
• 2. Large -Nf QCD
• 3. Methods
• 4. Numerical results
• 5. Summary

Introduction

Possible scenarios for EWSB

• With fundamental Higgs · · · · · · SM, etc.
• Without fundamental Higgs
• Perturbative

· · · · · · · · · Higgsless models, etc.

• Non-perturbative

· · · · · · Technicolor models, etc.

Introduction

Possible scenarios for EWSB

• With fundamental Higgs · · · · · · SM, etc.
• Without fundamental Higgs
• Perturbative

· · · · · · · · · Higgsless models, etc.

• Non-perturbative

· · · · · · Technicolor models, etc.

The last one is the least hypothetical

We know dynamical symmetry breaking actually occurs in the real world (i.e., chiral symmetry breaking in QCD)

The S parameter in Technicolor models

• Perturbative calculations are not reliable

We can use the knowledge of QCD if we assume that technicolor is a just a scaled-up version of QCD

• Phenomenological difficulties...

Possibly large contribution to the S parameter from the strong dynamics (EW precision measurements require S ∼ O(0.1))

We investigate the large flavor SU(N) gauge theory as an example of Non-QCD-like dynamics What is the large flavor SU(N) gauge theory? SU(N) gauge theory with an arbitrary number (Nf) of massless fermions

(Note : “large flavor” here does not mean Nf → ∞)

Here, we take N = 3 for concreteness, so we call it the large -Nf QCD

What is interesting about the large Nf QCD?

Existence of the IR fixed point makes the theory quite different from QCD

• Walking behavior of the running coupling

(which is nice for solving the FCNC problem and providing large enough fermion masses)

• Chiral restoration

Two-loop running coupling in the large -Nf QCD

RGE µ d

dµα(µ) = β(α) = − b α2(µ) − c α3(µ)

(Nc = 3) Nf < 8 8 < Nf < 16.5 16.5 < Nf b =

1 6π (33 − 2Nf)

+ + −

c =

1 12π2 (153 − 19Nf)

+ − −

Two-loop running coupling in the large -Nf QCD

RGE µ d

dµα(µ) = β(α) = − b α2(µ) − c α3(µ)

(Nc = 3) Nf < 8 8 < Nf < 16.5 16.5 < Nf b =

1 6π (33 − 2Nf)

+

+ −

c =

1 12π2 (153 − 19Nf)

+ − −

log µ Λ

α µ β

α α

Λ µ

( ) ( )

=

Nf < 8

Two-loop running coupling in the large -Nf QCD

RGE µ d

dµα(µ) = β(α) = − b α2(µ) − c α3(µ)

(Nc = 3) Nf < 8 8 < Nf < 16.5 16.5 < Nf b =

1 6π (33 − 2Nf)

+ + −

c =

1 12π2 (153 − 19Nf)

+ − −

α*

= µ Λ

α µ β

α α

log µ Λ

( ) ( )

1 2 3 4 5 6 −15 −10 −5 5 10 15

−3 −2 −1 1 1 2 3 4 5

8 < Nf < 16.5 (α∗ = −b/c)

Two-loop running coupling in the large -Nf QCD

RGE µ d

dµα(µ) = β(α) = − b α2(µ) − c α3(µ)

(Nc = 3) Nf < 8 8 < Nf < 16.5 16.5 < Nf b =

1 6π (33 − 2Nf)

+ + −

c =

1 12π2 (153 − 19Nf)

+ − −

log µ Λ

β

α α

α µ

( )

Nf = 9 Nf = 10 Nf = 11 Nf = 12

( )

Walking

−3 −2 −1 1 1 2 3 4 5

1 2 3 4 5 6 −15 −10 −5 5 10 15

Chiral restoration at Nf = N cr

f ≃ 12 (α∗ = αcr = π/4)

What is interesting about the large Nf QCD?

Existence of the IR fixed point makes the theory quite different from QCD

• Walking behavior of the running coupling
• Chiral restoration

Contribution to the S parameter might be small compared to the QCD-like theory

How to calculate the S parameter

ˆ S ≡ S (Nf/2) = −4π d dq2

E

• ΠV V (q2

E) − ΠAA(q2 E)

• q2

E=0

• Current-current correlator ΠJJ :

δab

qµqν q2 − gµν

• ΠJJ(q2) = i
• d4x eiqx 0|TJa

µ(x)Jb ν(0)|0

Ja

µ(x) =

   V a

µ (x) =

¯ ψ(x) λa

2 γµψ(x),

Aa

µ(x) =

¯ ψ(x) λa

2 γµγ5ψ(x),

How to calculate the current correlators

We need

• the three point vertex function χ(J)

αβ :

p

2

q + p

2

q −

q

δj

i

λa 2 f′

f

• d4p

(2π)4e−iprχ(J)

αβ (p; q, )

= µ

• d4xeiqx0|T ψαif(r/2) ¯

ψjf′

β (−r/2) Ja µ(x) |0

• and the full fermion propagator :

How to calculate the current correlators

We need

• the three point vertex function χ(J)

αβ :

p

2

q + p

2

q −

q

• and the full fermion propagator :

ΠJJ(q2

E)

How to calculate the three-point vertex We numerically solve the inhomogeneous BS equation and the SD equation simultaneously with the improved ladder approximation

fermion propagator propagator full bare gauge boson

Σ

p q

(x) =

running coupling

SD equation

running coupling full fermion propagator bare gauge boson propagator

IBS equation

Numerical Result We calculate the S parameter of the large Nf QCD in the range from Nf ≃ 10 (α∗ ≃ 1.8) to Nf ≃ N crit

f

≃ 12 (α∗ ≃ 0.9)

• Nf ≃ N crit

f

≃ 12 : Walking regime (Σ/Λ 1)

= ⇒ The scale relevant for the determination of physical quantities is near the IR fixed point

• Nf ≃ 10 : QCD-like regime

(Σ/Λ ≃ 0.2)

= ⇒ The scale relevant for the determination of physical quantities is farther from the IR fixed point

Numerical Result

α *

N f N f

10

~

α * = 1.8

α cr

12

~

S

^ S ^

0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8

Numerical Result

α *

N f N f

10

~

α * = 1.8

α cr

12

~

S

^ S ^

0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8

ˆ S decreases significantly as one moves from the QCD-like to walking regime by about 40% in the range investigated here

Summary

• We calculated the S parameter in the large Nf QCD

by solving the SD and IBS equations with the improved ladder approximation

• We found that contribution from ladder diagrams to

ˆ S decreases significantly as one moves from the QCD-like to walking regime

• This results motivate us to do further investigations
• f walking gauge theories as candidates for the origin
• f the EWSB