Phase structure of the Higgs-Yukawa
model in the strong-coupling regime
arXiv:1111.4544

C.-J. David Lin
National Chiao-Tung University
KMI Nagoya
20/03/2012

The collaboration• Germany– NIC, DESY Zeuthen and Humboldt University BerlinJohn Bulava (CERN), Philipp Gerhold (→ d-Fine), Karl Jansen, Attila Nagy.
• Japan– Kobayashi-Maskawa Institute, Nagoya University
Kei-Ichi Nagai.
• Taiwan– National Chiao-Tung University, Hsinchu
C.-J.David Lin, Kenji Ogawa.
– National Taiwan University, Taipei
George W.-S. Hou, Bastian Knippschild (Mainz →), Brian Smigielski(→ U.S.).Motivation
Heavy fermions beyond SM3?
• Not much is known for strong (non-perturbative) Yukawa theory.
• Heavy extra generation of fermions may– enhance CP violation.
G.W.S. Hou, 2008

– offer an alternative way to break EW symmetry dynamically and induces bound states to unitarise W W scattering.
B. Holdom, 2007

– UV stablise the SM.
P.Q. Hung, C. Xiong, 2009

Outline• Goals, general issues and recent developments.
• Simultation setup.
• The phase structure.
• Exploratory numerical studies.
– VEV.
– Susceptibility and critical exponents.
• Future plan.

Targets for the bare strong-Yukawa regime• The nature of the phase transitions.
⇒ Connection to the continuum world (next slide).
• Possible bound states.
⇒ Computation of the spectrum.
• Possible new mechanism for dynamical symmetry breaking.
⇒ Heavy scalar with fermion condensate?

General issues and strategy• The triviality (Landau-pole) problem.
⇒ Non-trivial to take the lattice spacing to zero.
• Look for 2nd-order phase transitions via ”scanning simulations”.
⇒ ξ → ∞.
• Problem: Finite-volume effects.
⇒ Phase transitions are washed out.
⇒ Severe near the critical points since L = L̂a.
• Chiral fermions required. Challenging to simulate chiral gauge theories.

New ingredients in current work
• Previous studies (circa 1990):

Lee, Shigemitsu, Shrock; Bock et al.,. . .

– Use fermions without exact chiral symmetry.
⇒ Ambiguity in defining chiral fermions.
– Small (∼ 83 × 16) volumes and no L → ∞ limit taken.
• Current new-generation simulations:– Use the overlap fermion (exact chiral symmetry).
– Several large volumes and L → ∞ limit taken.
⇒ Test finite-size scaling behaviour.
⇒ Determine the order of the phase transition.

Reminder: Notaion for scalar field theory• The discretised scalar action (a = 1)
X 1X1(2d + m20)ϕαx ϕαx + λ0 (ϕαx ϕαx)2 .
Sϕ = −
ϕαxϕαx+µ̂ +24xx,µ• ϕ=√λ̂, λ0 = κλ̂2m20 = 1−2κXX
α α
φαxφαx + λ̂(φαxφαx − 1)2 ,Sφ = −2κ
φx φx+µ̂ +2κφ,xx,µZφ =Z Y

dφαx exp(−Sφ) =x,αdµ(φαx) = dφαx expZ Ydµ(φαx ) expx,α2α αα α−φx φx − λ̂(φx φx − 1) .

2κX

φαxφαx+µ̂x,µ• “staggered symmetry”: κ → −κ and φαx → (−1)x1+x2+...+xd φαx.

!

,Fermions and the Yukawa couplings• Use the overlap Dirac operator with exact lattice chiral symmetry.
• The Yukawa terms SHY =Px y(t̄x , b̄x )L Φx bx,R + y(t̄x , b̄x )L Φ̃x tx,R + h.c..

– Φ is a complex scalar doublet and Φ̃ = iτ2Φ∗ .
• Results presented in this talk are from 83 × 16, 123 × 24 and 163 × 32.

Phase diagram of the H-Y model (qualitative)
0.1`
Fixed Λ<Φ>=0<Φ>s=0<Φ>=0<Φ>s=0<Φ>¹0, <Φ>s=0
FM HBrokenL

Κ

0.0

SYM

-0.1

SYM

<Φ>=0-0.2<Φ>=0<Φ>s¹0AFM-0.3
05<Φ>s¹0<Φ>¹0<Φ>s¹01015

AFM
20y∗ From earlier work using Wilson fermions.
⇒ Controversy from staggered-fermion calculations.

2530

Evidence of a symmetric phase at large yConsistent with recent results in P. Gerhold and K. Jansen, 2007.

The bare scalar vev at large Y

Finite-size scaling of susceptibility• Susceptibility: χ = V4hφ2i − hφihφi .

• The scaling behaviour from solving the RGE,−γ/ν– Universal function χLs1/ν

∼ g(t̃Ls ), where t̃ = (y/ycrit − 1).

– critical exponents γ and ν.
– Modelling the scaling violation fromM. Fisher and M. Barber, 1972−γ/ν

⇒χLs

1/ν

∼ g(tLs ), where t = (y/(ycrit − A4/Lbs) − 1).

– Fit all the data to the (partly empirical) function at fixed κK. Jansen and P. Seuferling, 1990n o−γ/2b 2−2/νχ = A1 Ls.
+ A2,3 y − ycrit − A4/Ls

Finite-size fit of susceptibility
Susceptibility, κ=0.00

12Fit range:

Susceptibility, κ=0.06

L=8
L=1210 L=16
χ (y)8χ (y)

9
L=88 L=127 L=1665
4321014 15

64
2016

17y

18

19y = 14.5 ∼ 19.5

20

10 12 14 16 18 20 22 24yy = 14.0 ∼ 22.0

Finite-size scaling of susceptibility
rescaled Susceptibility, κ=0.06
L=8L=120.08 L=16(-γ/ν)

0.10.04

χ(y)*L

χ(y)*L(-γ/ν)rescaled Susceptibility, κ=0.00
0.06 L=8
L=120.05 L=160.03
0.02

0.04
0.02

0.01
0-10

0.06

-50t*L

(1/ν)

5100-10-50t*L

(1/ν)

510

Probing the phase structure using susceptibility
κ = 0.00
κ = 0.06
O(4) scalar model
ycrit 16.57 ± 0.06 18.11 ± 0.06
N/Aγ
1.02 ± 0.02 1.08 ± 0.01
1ν
0.57 ± 0.03 0.66 ± 0.02
0.5
b2.05 ± 0.20 2.04 ± 0.20?
• Quoted errors are statistical, from uncorrelated fits with χ2/dof ∼ 0.001.
• Estimate systematics by changing the fit range in y.
• Systematic effects– ycrit is very stable.
– γ can change by ∼ 2%.
– ν can vary by ∼ 8%. ⇒ Different from O(4) scalar model?

Outlook• Improving results by– running at large lattices, 243 × 48. (finishing soon.)– studying the scaling behaviour of Binder’s cummulant.

• More information:– Compute three renormalised couplings to “trade” with κ, λ̂ and y.
– Study the spectrum in the strong Yukawa regime.
A lot more to do and to understand.