On the Non-triviality of the Model4λϕ

Loganantham Kuppan

Liew Voon Hooi

Foong See Kit

Content

• Is the model trivial?• Our Approach

– In the continuum– On the lattice– Computation and curve fitting– Approaching continuum limit

• Result – Verification in 2-dimensions– Results in 4-dimensions

• Conclusion & Possible Work

4λϕ

d > 4 : trivial

( ) ⎥⎦

⎤⎢⎣

⎡++∂= ∫ 42

22

!4221][ ϕλϕϕϕ μ

mxdS dE

4λϕIs the model trivial?

model4λϕ

d < 4 : non-trivial

d = 4 : no conclusive answer!

m2 and λ : bare mass and bare coupling constant

1. Physical coupling constant λR non-zero2. Renormalized propagator shows interaction

Two ways to determine triviality status:

Summary of Past Findings

Triviality of λϕ4

model in 4‐dimensions? ‐

Mixed conclusions

Trivial Non-Trivial

1.

Aizenmann

(1983)2.

Bender & Jones (1988)

3.

Butera

& Comi

(2001)4.

C B Lang (1986)

5.

Freedman (1982)6.

Kuti

(1988)

7.

Luscher

& Weisz

(1987, 88)

1.

W H Huang (1994)2.

Pederson (1992)

3.

Stevenson (1986)4.

J F Yang & J H Ruan

(2002)

5.

Consoli

(1988, 89)

4λϕIs the model trivial?

Others’

approach

• Smaller scale simulations, thus  require more extrapolation. 

• Restriction on certain parameters‐

Coupling constant λ

→∞

‐

Fixed field value φ

• Analyze only on the renormalized  coupling constant  λR

• Continuum limit taken  L → finite or 

a

→ 0

Our approach

• Simulation of larger scale is  possible.

• No restriction is imposed on λ and φ to allow a general 

conclusion.

• Analyze the two‐point function as  well λR

• More rigorous continuum limit  approach. 

L →∞ AND a

→ 0

Their Limitations

4λϕIs the model trivial?

The Renormalized 2-point connected correlation function

Renormalized propagator

B. DeWitt proposed anzatz for the propagator which we adopt:

proposed, in the broken symmetry phase, the renormalized propagator takes the form

The exact form is not known butM. Luscher and P. Weisz, [ Nucl. Phys. B295, 65 (1988)]

β : measures strength of interaction (deviation from free field propagator)

2effRm physical mass square

Our Approach – In the continuum

Our Approach – On the Lattice

computed on the supercomputer

Curve fitting with lattice version - DeWitt’s ansatz

Z2effRμ α[ ]22222

,

)()(~

kk

Z

effReffR

kkrr

rr

κμακμ +++=Γ −

On lattice / discretize

Monte Carlo method

222 ameffReffR =μ

Z : field renormalization constant

222

lim effRc

αμβμμ →

=

• Monte Carlo method is employed - Embedded Wolff algorithm

Computation and curve fitting- Computation of the propagator

>><<−>=<Γ ''', αααααα φφφφ rrrrrr

The lattice expectation value

[ ]22222

,

)()(~

kk

Z

effReffR

kkrr

rr

κμακμ +++=Γ −

Fourier Transformation

kkrr

−Γ ,~

• Curve fitting - Values of the parameters Z, , α2effRμ

Parameter:

Computation and curve fitting2effRμParameter: α

Definition:

Assuming that a shrinks and L expands by same factor

Approaching continuum limit

∞→N

02 →effRμ222 ameffReffR =μ 0→aas

L=Na

Parameter:

Approaching continuum limit2effRμ

Approaching continuum limit

effReffR NN μμ

'' =

Content

• Is the λϕ4 model trivial?• The Approach

– In the continuum– On the lattice– Computation and curve fitting– Approaching continuum limit

• Result – Verification in 2 dimensions– Results in 4 dimensions

• Conclusion & Possible Work

Result - Verification in 2 dimensions

2effRμ 0

α diverges

0417.024 −==Λβ 0417.024 −==Λβ

0548.0240 −==Λβ

Result - Verification in 2 dimensions

Content

• Is the λϕ4 model trivial?• The Approach

– In the continuum– On the lattice– Computation and curve fitting– Approaching continuum limit

• Result – Verification in 2 dimensions– Results in 4 dimensions

• Conclusion & Possible Work

4D λ = 120

-0.005

-0.0045

-0.004

-0.0035

-0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0-8.5 -8 -7.5 -7 -6.5 -6

N=8

N=10

4D λ = 120

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-7.2 -7.1 -7 -6.9 -6.8 -6.7 -6.6 -6.5 -6.4 -6.3 -6.2 -6.1 -6 -5.9

μ2

μ eff

R2

N=4

N=6

N=8

N=10

Results in 4 dimensions – parameters from curve fitting

Simulation results fitted with DeWitt’s Ansatz revealed divergence of α

4D

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0

lambda = 120

lambda = 240

lambda = 57600

2 2 2effR effRm aμ =

2R*

lim effβ αμ=

Results in 4 dimensions

Simulation results fitted with DeWitt’s Ansatz reveal non-zero of β

2R

2R

0lim effμ

β αμ→

=

Results in 4 dimensions

Non‐zero renormalized coupling constant λR

4D Analysis of renormalized coupling constant

0

5

10

15

20

25

30

35

40

45

-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0

lambda = 57600

lambda = 240

lambda = 120

Li (l bd

2 2

2

2

6limc

RR

Rμ μ

μλφ→

= −

Results in 4 dimensions

Content

• Is the λϕ4 model trivial?• The Approach

– In the continuum– On the lattice– Computation and curve fitting– Approaching continuum limit

• Result – Verification in 2 dimensions– Results in 4 dimensions

• Conclusion & Possible Work

Conclusions & Possible Work

• DeWitt’s Ansatz produce consistent result in 2- dimensions

• It also shows signs that the model is non trivial in 4-dimensions

4λϕ

To study the implication of our results for the Higgs model.

Possible work:

Thank you

To study the triviality status of the model in d=4 4λϕIntroduction – DeWitt’s proposal

Summary of Past FindingsII. Literature Review

Three main approaches

Analytical NumericalRenormalization Group

M. Aizenmann

(Norbert Wiener, Prize Princeton University) 

‐‐

Phys. Rev. Lett. 47, 1(1981)

J. Frohlich

(Max Planck Medal 2001, ETH Zurich

) 

‐‐

Nuc. Phys. B 200 [FS4] 281 (1982)

Callaway

(Rockefeller University) ‐‐

Physics Reports 167, 5 (1988) 241.

Cristiane

Aragao

& Carneiro

‐‐

Phys. Rev. D 68, 065010(2003)

J.F. Yang & J.H. Ruan

‐‐

Phys. Rev. D 65, 125009(2002) 

Consoli, Cea, Cosmai

(INFN, Italy), Stevenson 

‐‐

Mod. Phys. Lett. A, 14, 1673 (1999)

The Approach – Approaching continuum limit

Parameter: Z

Computation and curve fitting

From graph: This means, by letting

Continuum limit is approached

is finite

Approaching continuum limit

effReffR NN μμ

'' =