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 d D x T µµ φ ( x 1 ) ...φ ( x n ) � � � �� � ∂ = − k β k ∂ g k + n ( γ φ + d φ ) � φ ( x 1 ) ...φ ( x n ) � φ 4 -theory in 3D, m 2 0 < 0: toy model for theories with IR fixed point m 0 ✻ IR FP ✶ ✉ ❪ ✕ Gaussian FP ✲ ✉ � λ 0 � � � � � � � ✠ � η 0 1 / 15

Energy-momentum tensor and Ward identity Euclidean action � 1 � � 2( ∂ µ φ ) 2 + 1 0 φ 2 + λ 0 2 m 2 4! φ 4 d D x S = Energy-momentum tensor � � 1 ∂ σ φ∂ σ φ + 1 0 φ 2 + λ 0 � 2 m 2 4! φ 4 T µρ ( x ) = ∂ µ φ∂ ρ φ − δ µρ 2 σ Translation Ward identity � δ x ,ρ P � = −� P ∂ µ T µρ ( x ) � Local operator of translation δ P δ x ,ρ P = δφ ( x ) ∂ ρ φ ( x ) 2 / 15

Translation Ward identity on the lattice Lattice action � 1 ∂ µ φ ) 2 + 1 0 φ 2 + λ 0 � S = a D � ˆ 2(ˆ 2 m 2 4! φ 4 n Lattice regularisation breaks translation symmetry explicitly � � � ˆ δ x ,ρ ˆ P � = −� ˆ ∂ µ ˆ ˆ T µρ + ˆ � P R ρ Renormalised lattice TWI � T µρ ] + ˆ � � Z δ ˆ δ x ,ρ ˆ P � = −� ˆ ∂ µ [ ˆ ˆ ¯ P R ρ � Renormalised ˆ T µρ ( x ) � � [ ˆ � ˆ µρ − � ˆ T ( i ) T ( i ) T µρ ( x )] = c i µρ � i 3 / 15

Renormalisation of the EMT � � m 2 2 φ 2 + 1 ∂ σ φ + λ 0 T µρ ( x ) = ˆ ˆ ∂ µ φ ˆ � ∂ σ φ ˆ ˆ 0 4! φ 4 ∂ ρ φ − δ µρ 2 σ Possible mixing: D ≤ 3, Lorentz, φ → − φ , x → − x ∂ µ φ ˆ ˆ ∂ ρ φ, φ ˆ ∂ µ ˆ ∂ ρ φ, � � ∂ σ φ ˆ ˆ φ ˆ ∂ σ ˆ ∂ µ φ ˆ ˆ ∂ µ φ, φ ˆ ∂ µ ˆ φ 2 , φ 4 , φ 6 , � � δ µρ ∂ σ φ, ∂ σ φ, ∂ µ φ σ σ Perturbative analysis shows that divergencies are ∝ φ 2 � � T µρ ] = c c 2 2 φ 2 + c ′ ∂ σ φ + c 4 [ ˆ ∂ µ φ ˆ ˆ � ∂ σ φ ˆ ˆ 4! φ 4 ∂ ρ φ + δ µρ 2 2 σ 0 − 0 . 0215 λ 0 c 2 = − m 2 a 4 / 15

Wilson flow - gradient flow on the lattice Flow equation [ Monahan,Orginos 2014 ] ∂ t ϕ ( t , x ) = ˆ ∂ 2 ϕ ( t , x ) , ϕ ( t , x ) | t =0 = φ ( x ) t ✻ r ϕ ✂ ❇ ✂ ❇ ✂ ❇ ✂ ❇ ❜✉❧❦ ✂ ❇ ✬ ✩ ✂ ❇ ❏ ✂ ❇ ❏ ✂ ❇ ❜♦✉♥❞❛r② ✂ ❇ ❏ φ ❏ ✫ ✪ r ❏ ❏ ❏ ❏ ❏ √ Smoothing effect, radius r = 2 Dt 5 / 15

Renormalisation of the EMT using the Wilson flow Renormalised TWI � T µρ ] + ˆ � � Z δ ˆ δ x ,ρ ˆ P � = −� ˆ ∂ µ [ ˆ ˆ ¯ � P R ρ Renormalisation condition [ Del Debbio,Patella,Rago 2013 ] Choose probe ˆ P t : function of fields at t > 0, then: Coefficients c i can be tuned such that EMT is finite ˆ ¯ R ρ → 0 Z δ � ˆ δ x ,ρ ˆ � c i � ˆ P t ˆ ∂ µ ˆ T ( i ) P t � = − µρ ( x ) � i Determine Z δ separately, Z δ = 1 System of (at least) 4 equations with 4 different operators P ( k ) t V ( k ) = − � c i M ( k , i ) i 6 / 15

Phase diagram Interested in staying close to critical line, m 2 0 < 0 Lines of constant physics defined by ρ = λ 0 / m R 0.0 -0.1 -0.2 2 m 0 -0.3 ρ =1.5 ρ =3 -0.4 ρ =5 ρ =10 critical line -0.5 0.00 1.00 2.00 3.00 4.00 5.00 λ 0 7 / 15

Results - c 2 ˆ ∂ µ φ ˆ c 1 ∂ ρ φ , expected: c = 2 √ c ( t ) = 6 t / L ρ = 10 L=8 3.0 L=10 L=12 L=16 L=24 2.8 L=32 2.6 c 2.4 2.2 2.0 0.30 0.40 0.50 0.60 0.70 c(t) 8 / 15

Results - c 2 c 2 1 2 φ 2 δ µρ Expected from PT: c 2 / m 2 0 = 0 . 83 ρ = 10 6.0 L=8 L=10 5.0 L=12 L=16 L=24 L=32 4.0 2 c 2 / m 0 3.0 2.0 1.0 0.0 0.30 0.40 0.50 0.60 0.70 c(t) 9 / 15

Results - c ′ c ′ 1 σ ˆ ∂ σ φ ˆ � ∂ σ φδ µρ 2 Expected: c ′ = − 1 ρ = 10 0.0 -1.0 -2.0 c’ -3.0 L=8 L=10 L=12 -4.0 L=16 L=24 L=32 -5.0 0.30 0.40 0.50 0.60 0.70 c(t) 10 / 15

Results - c 4 c 4 1 4! φ 4 δ µρ Expected: c 4 /λ 0 = − 1 ρ = 10 -0.5 L=8 -0.6 L=10 L=12 L=16 L=24 -0.7 L=32 -0.8 c 4 / λ 0 -0.9 -1.0 -1.1 0.30 0.40 0.50 0.60 0.70 c(t) 11 / 15

Continuum limit EMT 2 ˆ ∂ µ φ ˆ c 1 ∂ ρ φ ρ = 10 L=8 3.0 L=10 L=12 L=16 L=24 2.8 L=32 2.6 c 2.4 2.2 2.0 0.30 0.40 0.50 0.60 0.70 c(t) 12 / 15

Continuum limit EMT c at c ( t ) = 0 . 49 along line of constant physics c(t)=0.49 2.7 2.6 2.01 c 2.5 1.99 1.97 2.4 0.0 c 2 (a/L) 2.3 2.2 MC data y=A+Bx, L=16->32 y=A+Bx, L=12->32 2.1 2 , L=12->32 y=A+Bx+Cx 2.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 2 (a/L) 13 / 15

Continuum limit EMT Extrapolated c for continuum limit of EMT at different t Quadratic fit 2.01 c 2.00 1.99 0.36 0.42 0.48 0.54 0.6 0.66 c(t) 14 / 15

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 15 / 15

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! 15 / 15