Renormalisation of the scalar energy-momentum tensor with the Wilson - - PowerPoint PPT Presentation

Renormalisation of the scalar energy-momentum tensor with the Wilson flow

Susanne Ehret

In collaboration with Francesco Capponi, Luigi Del Debbio, Masanori Hanada, Andreas J¨ uttner, Roberto Pellegrini, Antonin Portelli, Antonio Rago, Francesco Sanfilippo, Kostas Skenderis

Lattice 2016 - Southampton

Motivation

EMT relates to β-function

• dDx Tµµ φ(x1)...φ(xn)

= −

• k βk

∂ ∂gk + n(γφ + dφ)

• φ(x1)...φ(xn)

φ4-theory in 3D, m2

0 < 0: toy model for theories with IR fixed point

✻ ✲

• ✠

λ0 m0 η0

✉ ✉

Gaussian FP IR FP

✶ ✕ ❪

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Energy-momentum tensor and Ward identity

Euclidean action S =

• dDx

1 2(∂µφ)2 + 1 2m2

0φ2 + λ0

4! φ4

• Energy-momentum tensor

Tµρ(x) = ∂µφ∂ρφ − δµρ

• 1

2

• σ

∂σφ∂σφ + 1 2m2

0φ2 + λ0

4! φ4

• Translation Ward identity

δx,ρP = − P ∂µTµρ(x) Local operator of translation δx,ρP = δP δφ(x) ∂ρφ(x)

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Translation Ward identity on the lattice

Lattice action ˆ S = aD

n

1 2(ˆ ∂µφ)2 + 1 2m2

0φ2 + λ0

4! φ4

• Lattice regularisation breaks translation symmetry explicitly

ˆ δx,ρ ˆ P = − ˆ P

• ˆ

∂µ ˆ Tµρ + ˆ Rρ

• Renormalised lattice TWI

Zδ ˆ δx,ρ ˆ P = − ˆ P

• ˆ

∂µ[ ˆ Tµρ] + ˆ ¯ Rρ

• Renormalised ˆ

Tµρ(x) [ ˆ Tµρ(x)] =

• i

ci

• ˆ

T (i)

µρ − ˆ

T (i)

µρ

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Renormalisation of the EMT

ˆ Tµρ(x) = ˆ ∂µφˆ ∂ρφ − δµρ

• m2

2 φ2 + 1 2

• σ

ˆ ∂σφˆ ∂σφ + λ0 4! φ4

• Possible mixing: D ≤ 3, Lorentz, φ → −φ, x → −x

ˆ ∂µφˆ ∂ρφ, φˆ ∂µ ˆ ∂ρφ, δµρ

• φ2, φ4, φ6,
• σ

ˆ ∂σφˆ ∂σφ,

• σ

φˆ ∂σ ˆ ∂σφ, ˆ ∂µφˆ ∂µφ, φˆ ∂µ ˆ ∂µφ

• Perturbative analysis shows that divergencies are ∝ φ2

[ ˆ Tµρ] =c 2 ˆ ∂µφˆ ∂ρφ + δµρ

• c2

2 φ2 + c′ 2

• σ

ˆ ∂σφˆ ∂σφ + c4 4!φ4

• c2 = −m2

0 − 0.0215λ0

a

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Wilson flow - gradient flow on the lattice

Flow equation [Monahan,Orginos 2014] ∂tϕ(t, x) = ˆ ∂2ϕ(t, x), ϕ(t, x)|t=0 = φ(x)

✻ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏

t ❜✉❧❦ ❜♦✉♥❞❛r②

r ϕ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✬ ✫ ✩ ✪

φ r

Smoothing effect, radius r = √ 2Dt

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Renormalisation of the EMT using the Wilson flow

Renormalised TWI Zδ ˆ δx,ρ ˆ P = − ˆ P

• ˆ

∂µ[ ˆ Tµρ] + ˆ ¯ Rρ

• Renormalisation condition [Del Debbio,Patella,Rago 2013]

Choose probe ˆ Pt: function of fields at t > 0, then: Coefficients ci can be tuned such that EMT is finite ˆ ¯ Rρ → 0

Zδ ˆ δx,ρ ˆ Pt = −

• i

ci ˆ Pt ˆ ∂µ ˆ T (i)

µρ(x)

Determine Zδ separately, Zδ = 1 System of (at least) 4 equations with 4 different operators P(k)

t

V (k) = −

• i

ci M(k,i)

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Phase diagram

Interested in staying close to critical line, m2

0 < 0

Lines of constant physics defined by ρ = λ0/mR

0.00 1.00 2.00 3.00 4.00 5.00

λ0

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.0

m 0

2 ρ=1.5 ρ=3 ρ=5 ρ=10 critical line

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Results - c

c 1

2 ˆ

∂µφˆ ∂ρφ, expected: c = 2 c(t) = √ 6t/L

0.30 0.40 0.50 0.60 0.70

c(t)

2.0 2.2 2.4 2.6 2.8 3.0

c

L=8 L=10 L=12 L=16 L=24 L=32 ρ = 10

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Results - c2

c2 1

2φ2δµρ

Expected from PT: c2/m2

0 = 0.83

0.30 0.40 0.50 0.60 0.70

c(t)

0.0 1.0 2.0 3.0 4.0 5.0 6.0

c2 / m0

2 L=8 L=10 L=12 L=16 L=24 L=32 ρ = 10

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Results - c′

c′ 1

2

• σ ˆ

∂σφˆ ∂σφδµρ Expected: c′ = −1

0.30 0.40 0.50 0.60 0.70

c(t)

• 5.0
• 4.0
• 3.0
• 2.0
• 1.0

0.0

c’

L=8 L=10 L=12 L=16 L=24 L=32 ρ = 10

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Results - c4

c4 1

4!φ4δµρ

Expected: c4/λ0 = −1

0.30 0.40 0.50 0.60 0.70

c(t)

• 1.1
• 1.0
• 0.9
• 0.8
• 0.7
• 0.6
• 0.5

c4 / λ0

L=8 L=10 L=12 L=16 L=24 L=32 ρ = 10

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Continuum limit EMT

c 1

2 ˆ

∂µφˆ ∂ρφ

0.30 0.40 0.50 0.60 0.70

c(t)

2.0 2.2 2.4 2.6 2.8 3.0

c

L=8 L=10 L=12 L=16 L=24 L=32 ρ = 10

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Continuum limit EMT

c at c(t) = 0.49 along line of constant physics

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

(a/L)

2 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7

c

MC data y=A+Bx, L=16->32 y=A+Bx, L=12->32 y=A+Bx+Cx

2, L=12->32

c(t)=0.49

0.0

(a/L)

2

1.97 1.99 2.01

c

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Continuum limit EMT

Extrapolated c for continuum limit of EMT at different t

0.36 0.42 0.48 0.54 0.6 0.66

c(t)

1.99 2.00 2.01

c

Quadratic fit

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Summary

We studied the non-perturbative renormalisation of EMT The Wilson flow provides a new way to implement Ward identities free from contact terms We are able to define a properly renormalised EMT on lattice

Find coefficients of renormalised EMT at finite a Each t gives a different definition of the EMT Reproduce correct continuum limit of EMT

Next step: study scaling behaviour in IR

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Summary

We studied the non-perturbative renormalisation of EMT The Wilson flow provides a new way to implement Ward identities free from contact terms We are able to define a properly renormalised EMT on lattice

Find coefficients of renormalised EMT at finite a Each t gives a different definition of the EMT Reproduce correct continuum limit of EMT

Next step: study scaling behaviour in IR Thank you!

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