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 - N f 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 ( N f ) of massless fermions (Note : “large flavor” here does not mean N f → ∞ ) Here, we take N = 3 for concreteness, so we call it the large - N f QCD

What is interesting about the large N f 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 - N f QCD µ d dµ α ( µ ) = β ( α ) = − b α 2 ( µ ) − c α 3 ( µ ) RGE ( N c = 3 ) N f < 8 8 < N f < 16 . 5 16 . 5 < N f + + − 1 b = 6 π (33 − 2 N f ) + − − 1 c = 12 π 2 (153 − 19 N f )

Two-loop running coupling in the large - N f QCD µ d dµ α ( µ ) = β ( α ) = − b α 2 ( µ ) − c α 3 ( µ ) RGE ( N c = 3 ) N f < 8 8 < N f < 16 . 5 16 . 5 < N f + − + 1 b = 6 π (33 − 2 N f ) + − − 1 c = 12 π 2 (153 − 19 N f ) β α µ α ( ) ( ) α log µ µ Λ Λ = N f < 8

Two-loop running coupling in the large - N f QCD µ d dµ α ( µ ) = β ( α ) = − b α 2 ( µ ) − c α 3 ( µ ) RGE ( N c = 3 ) N f < 8 8 < N f < 16 . 5 16 . 5 < N f + + − 1 b = 6 π (33 − 2 N f ) + − − 1 c = 12 π 2 (153 − 19 N f ) β α α µ ( ) ( ) 6 α * 1 5 0 ✂✁✂ �✁� ✂✁✂ �✁� ✂✁✂ �✁� 4 −1 3 2 −2 1 −3 0 α 0 1 2 3 4 5 −15 −10 −5 0 5 10 15 log µ = Λ µ Λ 8 < N f < 16 . 5 ( α ∗ = − b/c )

Two-loop running coupling in the large - N f QCD µ d dµ α ( µ ) = β ( α ) = − b α 2 ( µ ) − c α 3 ( µ ) RGE ( N c = 3 ) N f < 8 8 < N f < 16 . 5 16 . 5 < N f + + − 1 b = 6 π (33 − 2 N f ) + − − 1 c = 12 π 2 (153 − 19 N f ) β α µ α ( ) ( ) 6 1 5 N f = 9 0 4 Walking N f = 10 −1 3 N f = 11 2 −2 N f = 12 1 −3 0 α 0 1 2 3 4 5 −15 −10 −5 0 5 10 15 log µ Λ Chiral restoration at N f = N cr f ≃ 12 ( α ∗ = α cr = π/ 4 )

What is interesting about the large N f 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 �� ( N f / 2) = − 4 π d S ˆ Π V V ( q 2 E ) − Π AA ( q 2 � � S ≡ E ) � dq 2 � q 2 E E =0 • Current-current correlator Π JJ : δ ab � � d 4 x e iqx � 0 | TJ a q µ q ν Π JJ ( q 2 ) = i � µ ( x ) J b q 2 − g µν ν (0) | 0 �  ψ ( x ) λ a ¯ V a µ ( x ) = 2 γ µ ψ ( x ) ,  J a µ ( x ) = ψ ( x ) λ a ¯ A a µ ( x ) = 2 γ µ γ 5 ψ ( x ) , 

How to calculate the current correlators We need p q + 2 ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� q • the three point vertex function χ ( J ) �✁�✁�✁� ✂✁✂✁✂✁✂ αβ : �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� p q − 2 � f ′ � λ a d 4 p � (2 π ) 4 e − ipr χ ( J ) δ j αβ ( p ; q, � ) i 2 f � ψ jf ′ d 4 xe iqx � 0 | T ψ αif ( r/ 2) ¯ = � µ β ( − r/ 2) J a µ ( x ) | 0 � • and the full fermion propagator :

How to calculate the current correlators We need p q + 2 �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ q • the three point vertex function χ ( J ) ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� αβ : ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ p − q 2 • and the full fermion propagator : ✂✁✂✁✂✁✂ �✁�✁�✁� ✂✁✂✁✂✁✂ �✁�✁�✁� �✁�✁�✁� ✂✁✂✁✂✁✂ Π JJ ( q 2 �✁�✁�✁� ✂✁✂✁✂✁✂ 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 SD equation IBS equation bare gauge boson bare gauge boson propagator propagator ✝✁✝✁✝ ✆✁✆✁✆ ✝✁✝✁✝ ✆✁✆✁✆ �✁�✁�✁� ✂✁✂✁✂✁✂ ✝✁✝✁✝ ✆✁✆✁✆ ☎✁☎✁☎✁☎ ✄✁✄✁✄✁✄ �✁�✁�✁� ✂✁✂✁✂✁✂ ✆✁✆✁✆ ✝✁✝✁✝ ☎✁☎✁☎✁☎ ✄✁✄✁✄✁✄ �✁�✁�✁� ✂✁✂✁✂✁✂ ✆✁✆✁✆ ✄✁✄✁✄✁✄ �✁�✁�✁� ✝✁✝✁✝ ☎✁☎✁☎✁☎ ✂✁✂✁✂✁✂ ✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎ ✂✁✂✁✂✁✂ �✁�✁�✁� ✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎ ✂✁✂✁✂✁✂ �✁�✁�✁� ✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎ Σ �✁�✁�✁� ✂✁✂✁✂✁✂ ☎✁☎✁☎✁☎ ✄✁✄✁✄✁✄ �✁�✁�✁� ✂✁✂✁✂✁✂ (x) = ✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎ ✂✁✂✁✂✁✂ �✁�✁�✁� ✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎ ✂✁✂✁✂✁✂ �✁�✁�✁� ✄✁✄✁✄✁✄ �✁�✁�✁� ☎✁☎✁☎✁☎ ✂✁✂✁✂✁✂ ☎✁☎✁☎✁☎ ✄✁✄✁✄✁✄ �✁�✁�✁� ✂✁✂✁✂✁✂ ☎✁☎✁☎✁☎ ✄✁✄✁✄✁✄ �✁�✁�✁� ✂✁✂✁✂✁✂ ✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎ ✂✁✂✁✂✁✂ �✁�✁�✁� ✞✁✞✁✞ ✟✁✟✁✟ ☎✁☎✁☎✁☎ ✄✁✄✁✄✁✄ �✁�✁�✁� ✂✁✂✁✂✁✂ ✞✁✞✁✞ ✟✁✟✁✟ ✄✁✄✁✄✁✄ ☎✁☎✁☎✁☎ ✞✁✞✁✞ ✟✁✟✁✟ ✟✁✟✁✟ ✞✁✞✁✞ ✞✁✞✁✞ ✟✁✟✁✟ p q running coupling running coupling full fermion propagator full fermion propagator

Numerical Result We calculate the S parameter of the large N f QCD in the range from N f ≃ 10 ( α ∗ ≃ 1 . 8) to N f ≃ N crit ≃ 12 ( α ∗ ≃ 0 . 9) f • N f ≃ N crit ≃ 12 : Walking regime ( Σ / Λ � 1 ) f = ⇒ The scale relevant for the determination of physical quantities is near the IR fixed point • N f ≃ 10 : QCD-like regime ( Σ / Λ ≃ 0 . 2 ) = ⇒ The scale relevant for the determination of physical quantities is farther from the IR fixed point