Convergent Perturbation Theory for the lattice 4 -model Aleksandr - - PowerPoint PPT Presentation

Convergent Perturbation Theory for the lattice φ4-model

Aleksandr Ivanov1, Vasily Sazonov2, Vladimir Belokurov1, Eugeny Shavgulidze1

1Moscow State University, 2University of Graz

ivanov.as@physics.msu.ru

July 15, 2015

Motivation

We study convergent series for lattice φ4-model

◮ To check the method of the convergent series on the simple

example, allowing one a direct comparison with the Monte Carlo simulations. The method was developed for continuum scalar field theories [A. Ushveridze, Phys. Let. B, 1984] and recently reformulated for QCD [V. Sazonov, arXiv:1503.00739].

◮ To design new methods for lattice computations, which may

help to avoid the Sign problem.

◮ To compare results with the Borel resummation

Lattice φ4-model

Continuous theory in the Euclidean space-time S =

• dx

1 2(∂φ)2 + 1 2M2φ2 + 1 4!λφ4 Theory on the lattice S =

V

• n=0
• −1

2

• µ
• φnφn+µ + φnφn−µ − 2φ2

n

• + 1

2M2φ2

n + λ

4!φ4

n

• In the following we write the quadratic part of the action as

V

• n=0
• −1

2

• µ
• φnφn+µ + φnφn−µ − 2φ2

n

• + 1

2M2φ2

n

• ≡ φ2.

Calculations

We calculate the observable φ2

n ,

using the

◮ Monte Carlo method [M. Creutz, B. Freedman, Annals

• Phys. 1981]

◮ Borel resummation of the standard perturbation theory [Jean

Zinn-Justin arXiv:1001.0675v1 2010]

◮ Convergent series [A. Ushveridze, Phys.Let.B, 1984]

Another ways to obtain convergent series

◮ V. Belokurov, V. Kamchatny, E. Shavgulidze, Y.

Solovyov, Mod.Phys.Let. A, 1997

◮ Y. Meurice, arxiv.org/abs/hep-th/0103134v3, 2002

Ushveridze method. Main ideas

◮ New non-perturbed part of the action ◮ Positive determined series for the perturbation ◮ Interconnection between new and standard perturbation theory

Ushveridze method

Let’s split the action as S[φn] = N[φn] + P[φn] = N[φn] + (S[φn] − N[φn]) . Then the partition function can be calculated in the following way Z =

V

• n
• [dφn] e−S[φn] =

V

• n
• [dφn] e−N[φn]+(N[φn]−S[φn]) =

=

V

• n
• [dφn] e−N[φn]

∞

• l=0

(N[φn] − S[φn])l l! N[φn] ≥ S[φn]

Ushveridze method

The partition function after interchanging of integration and summation is Z =

∞

• l=0

V

• n
• [dφn] e−N[φn] (N[φn] − S[φn])l

l! . Let us choose the non-perturbed part of the action as N[φn] = φ2 + σφ4

How to find σ

The action and its non-perturbed part are S = φ2 +

V

• n=0

λ 4!φ4

n,

N = φ2 + σφ4 So, for σ we have N[φn] ≥ S[φn] ⇐ ⇒ σφ4 ≥

V

• n=0

λ 4!φ4

n =

⇒ = ⇒ σ ≥ λ 6M4

How to solve new initial approximation

The observable φ2

n is the sum of terms of the following type V

• n
• [dφn] φ2

n e−N[φn] (N[φn] − S[φn])l

l! = using the δ-function we change φ to a new variable t = ∞ dt exp

• − t2 − σt4 V
• n
• [dφn] φ2

n δ(t − φ)×

×(σt4 − V

n=0 λ 4!φ4 n)l

l!

How to solve new initial approximation

∞ dt exp

• − t2 − σt4 V
• n
• [dφn] φ2

n δ(t − φ)×

×(σt4 − V

n=0 λ 4!φ4 n)l

l! We rescale field φ as tφ, expand brakets (....)l and end up with the sum of the integrals like

• t − depending integral
• ·

V

• n
• [dφn] φn1...φnkδ(1 − φ)

1-dimensional results, 5 loops vs Monte Carlo

0.3 0.35 0.4 0.45 2 4 6 8 10 <n2>

• Borel

Monte-Carlo Perturbation Theory Ushveridze

Figure: Comparison of the results for the 1d case on the V = 100 lattice

Behavior of results in dependence on order

0.432 0.434 0.436 0.438 0.44 0.442 0.444 0.446 0.448 1 2 3 4 5 <n2> Order Monte-Carlo Borel Ushveridze Perturbation Theory

Figure: 1d case for λ = 0.1

Behavior of results in dependence on order

0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 1 2 3 4 5 <n2> Order Monte-Carlo Borel Ushveridze Perturbation Theory

Figure: 1d case for λ = 1

Behavior of results in dependence on order

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 2 3 4 5 <n2> Order Monte-Carlo Borel Ushveridze Perturbation Theory

Figure: 1d case for λ = 10

2-dimensional results, 5 loops vs Monte Carlo

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 2 4 6 8 10 <n2>

• Borel

Monte-Carlo Perturbation Theory Ushveridze

Figure: Comparison of the results for the 2d case on the V = 10 × 10 lattice

Behavior of results in dependence on order

0.246 0.248 0.25 0.252 0.254 0.256 1 2 3 4 5 <n2> Order Monte-Carlo Borel Ushveridze Perturbation Theory

Figure: 2d case for λ = 0.1

Behavior of results in dependence on order

0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 1 2 3 4 5 <n2> Order Monte-Carlo Borel Ushveridze Perturbation Theory

Figure: 2d case for λ = 1

Behavior of results in dependence on order

• 0.1

0.1 0.2 0.3 0.4 1 2 3 4 5 <n2> Order Monte-Carlo Borel Ushveridze Perturbation Theory

Figure: 2d case for λ = 10

Conclusions

◮ We have checked the convergent series method in the

application to the lattice φ4-model.

◮ The results of 5-loop calculations of φ2 n are in the good

agreement with Monte Carlo data in the wide range of the coupling constants.

◮ This supports the further utilization of this method for

continuum QFT (including Yang-Mills, QCD...)

◮ and opens new ways for the computations on the lattice,

which probably can help to avoid Sign problem.