Exploring the phases of Yang-Mills theory with adjoint matter - - PowerPoint PPT Presentation

Exploring the phases of Yang-Mills theory with adjoint matter through the gradient flow

Camilo L´

• pez

Friedrich Schiller University of Jena

with Georg Bergner and Stefano Piemonte

SIFT 2019, Jena

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Table of Contents

• 1. The gradient flow: introduction
• 2. The gradient flow and renormalisation
• 3. Thermal super Yang-Mills
• 4. (Near) conformal field theories
• 5. Gradient flow and RG flow
• 6. Mass anomalous dimension of adjoint QCD

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Table of contents

• 1. The gradient flow: introduction
• 2. The gradient flow and renormalisation
• 3. Thermal super Yang-Mills
• 4. (Near) conformal field theories
• 5. Gradient flow and RG flow
• 6. Mass anomalous dimension of adjoint QCD

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The Yang-Mills gradient flow

• A gradient flow (curve of steepest descent) in a linear space M is a curve

γ : R → M, such that for a functional S : M → R γ′(t) = −∇S(γ(t))

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The Yang-Mills gradient flow

• A gradient flow (curve of steepest descent) in a linear space M is a curve

γ : R → M, such that for a functional S : M → R γ′(t) = −∇S(γ(t))

• For a Yang-Mills (YM) field A one defines the functional as the action

S(A) = 1 2|dA|2 = 1 2|FA|2

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The Yang-Mills gradient flow

• A gradient flow (curve of steepest descent) in a linear space M is a curve

γ : R → M, such that for a functional S : M → R γ′(t) = −∇S(γ(t))

• For a Yang-Mills (YM) field A one defines the functional as the action

S(A) = 1 2|dA|2 = 1 2|FA|2

• And the gradient flow is given by the differential equations

∂tBµ = DνGνµ, Bµ|t=0 = Aµ:

flow of gauge fields

∂tχ = DµDµχ, χ|t=0 = ψ:

flow of fermion fields

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The Yang-Mills gradient flow

• A gradient flow (curve of steepest descent) in a linear space M is a curve

γ : R → M, such that for a functional S : M → R γ′(t) = −∇S(γ(t))

• For a Yang-Mills (YM) field A one defines the functional as the action

S(A) = 1 2|dA|2 = 1 2|FA|2

• And the gradient flow is given by the differential equations

∂tBµ = DνGνµ, Bµ|t=0 = Aµ:

flow of gauge fields

∂tχ = DµDµχ, χ|t=0 = ψ:

flow of fermion fields

• These evolve the fields to local minima of the action

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What does the flow imply for the quantum theory?

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Table of contents

• 1. The gradient flow: introduction
• 2. The gradient flow and renormalisation
• 3. Thermal super Yang-Mills
• 4. (Near) conformal field theories
• 5. Gradient flow and RG flow
• 6. Mass anomalous dimension of adjoint QCD

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The gradient flow in QFT

• The flow has a smoothening effect on the fields, which are Gaussian-like

smeared over an effective radius rt = √ 8t Bµ(t, x) =

• dDy Kt(x − y)Aµ(y) + non linear terms,

Kt(z) =

• dDp

(2π)D eipze−tp2 (at leading order)

• e−tp2 is UV cut-off for t > 0. It remains at all orders in perturbation

theory [L¨

uscher and Weisz,arXiv:1405.3180]

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The gradient flow in QFT II

Q: Are the correlators of flowed fields renormalised?

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The gradient flow in QFT II

Q: Are the correlators of flowed fields renormalised?

• D+1 dimensional QFT with flow time as spurious dimension

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The gradient flow in QFT II

Q: Are the correlators of flowed fields renormalised?

• D+1 dimensional QFT with flow time as spurious dimension
• t-propagator is the heat kernel K.

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The gradient flow in QFT II

Q: Are the correlators of flowed fields renormalised?

• D+1 dimensional QFT with flow time as spurious dimension
• t-propagator is the heat kernel K.
• BRS-Ward identities → no counter-terms for the gauge fieds

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The gradient flow in QFT II

Q: Are the correlators of flowed fields renormalised?

• D+1 dimensional QFT with flow time as spurious dimension
• t-propagator is the heat kernel K.
• BRS-Ward identities → no counter-terms for the gauge fieds
• Fermions get an extra multiplicative renormalisation

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The gradient flow in QFT III

Monomials renormalise according to the field content

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The gradient flow in QFT III

Monomials renormalise according to the field content Correlation functions of monomials of flowed bare fields are finite (almost) without additional renormalisation

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The gradient flow in QFT III

Monomials renormalise according to the field content Correlation functions of monomials of flowed bare fields are finite (almost) without additional renormalisation This method is regularisation-scheme independent → holds in the lattice!

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The gradient flow in QFT III

Monomials renormalise according to the field content Correlation functions of monomials of flowed bare fields are finite (almost) without additional renormalisation This method is regularisation-scheme independent → holds in the lattice! Facilitates computation of densities and currents, e.g. condensate, supercurrent, energy-momentum tensor...

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The gradient flow in QFT III

Monomials renormalise according to the field content Correlation functions of monomials of flowed bare fields are finite (almost) without additional renormalisation This method is regularisation-scheme independent → holds in the lattice! Facilitates computation of densities and currents, e.g. condensate, supercurrent, energy-momentum tensor... We used this to investigate the phase structure of SU(2) and SU(3) SYM

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Table of contents

• 1. The gradient flow: introduction
• 2. The gradient flow and renormalisation
• 3. Thermal super Yang-Mills
• 4. (Near) conformal field theories
• 5. Gradient flow and RG flow
• 6. Mass anomalous dimension of adjoint QCD

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Brief review of minimal SYM

LE = 1 4F 2 + 1 2 ¯ λ( / D + m˜

g)λ +

θ 32π2 ˜ FF

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Brief review of minimal SYM

LE = 1 4F 2 + 1 2 ¯ λ( / D + m˜

g)λ +

θ 32π2 ˜ FF

• Vector supermultiplet with one Yang-Mills field A and one Majorana

spinor λ in the adjoint representation

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Brief review of minimal SYM

LE = 1 4F 2 + 1 2 ¯ λ( / D + m˜

g)λ +

θ 32π2 ˜ FF

• Vector supermultiplet with one Yang-Mills field A and one Majorana

spinor λ in the adjoint representation

• Only supersymmetric theory without scalars and thus similar to QCD

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Brief review of minimal SYM

LE = 1 4F 2 + 1 2 ¯ λ( / D + m˜

g)λ +

θ 32π2 ˜ FF

• Vector supermultiplet with one Yang-Mills field A and one Majorana

spinor λ in the adjoint representation

• Only supersymmetric theory without scalars and thus similar to QCD
• Expected to have mass gap, confinement and spontaneous breaking of

chiral symmetry

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Brief review of minimal SYM

LE = 1 4F 2 + 1 2 ¯ λ( / D + m˜

g)λ +

θ 32π2 ˜ FF

• Vector supermultiplet with one Yang-Mills field A and one Majorana

spinor λ in the adjoint representation

• Only supersymmetric theory without scalars and thus similar to QCD
• Expected to have mass gap, confinement and spontaneous breaking of

chiral symmetry

• Low energy degrees of freedom: glueballs, meson-like states, baryon-like

(see Sajid Ali’s poster)

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At zero temperature

ZNc Centre symmetry ⋆ Not broken through adjoint fermions ⋆ Polyakov loop (PL) vev vanishes

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At zero temperature

ZNc Centre symmetry ⋆ Not broken through adjoint fermions ⋆ Polyakov loop (PL) vev vanishes Chiral symmetry

• Anomaly free Z2Nc symmetry
• Condensate < ¯

λλ >= 0 ⇒ Z2Nc → Z2

• A domain wall interpolates Nc degenerated vacua
• Chern-Simons theory on domain wall with deconfined quarks

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Thermal phase transitions

• At some Tdec

c

: Phase transition to broken centre symmetry

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Thermal phase transitions

• At some Tdec

c

: Phase transition to broken centre symmetry

• At some Tχ

c : Z2Nc symmetry restored, < ¯

λλ >→ 0

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Thermal phase transitions

• At some Tdec

c

: Phase transition to broken centre symmetry

• At some Tχ

c : Z2Nc symmetry restored, < ¯

λλ >→ 0

• From ’t Hooft anomaly matching: Tdec

c

≤ Tχ

c

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Thermal phase transitions

• At some Tdec

c

: Phase transition to broken centre symmetry

• At some Tχ

c : Z2Nc symmetry restored, < ¯

λλ >→ 0

• From ’t Hooft anomaly matching: Tdec

c

≤ Tχ

c

We computed the PL and the flowed condensate at different T → bound is saturated for SU(2). Evidence for SU(3)

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Computing the condensate

✗ Additive renormalisation constant needed because of the Wilsonian fermion discretisation λλR = Zλλ(β)(λλB − b0).

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Computing the condensate

✗ Additive renormalisation constant needed because of the Wilsonian fermion discretisation λλR = Zλλ(β)(λλB − b0).

• b0 can be fixed so that the condensate vanishes at T = 0

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Computing the condensate

✗ Additive renormalisation constant needed because of the Wilsonian fermion discretisation λλR = Zλλ(β)(λλB − b0).

• b0 can be fixed so that the condensate vanishes at T = 0
• BUT!: Information at zero temperature lost

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Computing the condensate

✗ Additive renormalisation constant needed because of the Wilsonian fermion discretisation λλR = Zλλ(β)(λλB − b0).

• b0 can be fixed so that the condensate vanishes at T = 0
• BUT!: Information at zero temperature lost
• One way out is to use chiral lattice fermions...

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Computing the condensate

✗ Additive renormalisation constant needed because of the Wilsonian fermion discretisation λλR = Zλλ(β)(λλB − b0).

• b0 can be fixed so that the condensate vanishes at T = 0
• BUT!: Information at zero temperature lost
• One way out is to use chiral lattice fermions...

Another way out is the gradient flow

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Gaugino condensate from the gradient flow

• No additive renormalisation constant necessary for the flowed condensate,

even with Wilson fermions

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Gaugino condensate from the gradient flow

• No additive renormalisation constant necessary for the flowed condensate,

even with Wilson fermions

• The flowed condensate is measured on the lattice through

χt(x) = −

• v,w
• tr

     K(t, x; 0, v)

• diff eq kernel

Dirac propagator

S(v, w) K(t, x; 0, w)†     

• 15 / 36

Camilo Lopez QFT phases from the gradient flow

Gaugino condensate from the gradient flow

• No additive renormalisation constant necessary for the flowed condensate,

even with Wilson fermions

• The flowed condensate is measured on the lattice through

χt(x) = −

• v,w
• tr

     K(t, x; 0, v)

• diff eq kernel

Dirac propagator

S(v, w) K(t, x; 0, w)†     

• ...The inversion and the fermion adjoint flow are the most expensive part
• f the numerics

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Results: SU(2)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 (ama−π)2 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 ¯ λλ

Non-vanishing condensate at zero temperature in the chiral / supersymmetric limit

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Results: SU(2)

5 × 10−5 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005 0.2 0.25 0.3 0.35 0.4 0.45 Chiral susceptibility T =

√t0 Nt

κ = 0.14925 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Tc = 0.2542(23) B4(PL) T =

√t0 Nt

κ = 0.14925

• Deconfinement critical temperature coincides with peak of chiral

susceptibility Deconfinement and chiral restoration phase transitions

• ccur at the same critical temperature T ∼ 0.25

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How can we understand/explain this observation?

Mathematically (Witten, 97): * studied configuration of branes in M-theory, which is in universality class of N = 1 SYM * showed that (QCD strings ↔ fundamental strings) can end in (domain walls ↔ D-branes)

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How can we understand/explain this observation?

Qualitatively (Rey): * Domain wall connects different θ-vacua

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How can we understand/explain this observation?

Qualitatively (Rey): * Domain wall connects different θ-vacua * Confinement: Monopole cond. (θ = 0), dyons (θ = 0)

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How can we understand/explain this observation?

Qualitatively (Rey): * Domain wall connects different θ-vacua * Confinement: Monopole cond. (θ = 0), dyons (θ = 0) * Domain wall colour charged when dyons pass through → confining string can end there

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How can we understand/explain this observation?

Qualitatively (Rey): * Domain wall connects different θ-vacua * Confinement: Monopole cond. (θ = 0), dyons (θ = 0) * Domain wall colour charged when dyons pass through → confining string can end there Wiese, Holland, Campos; 98: * EFT of PL and condensate with SU(3)

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How can we understand/explain this observation?

Qualitatively (Rey): * Domain wall connects different θ-vacua * Confinement: Monopole cond. (θ = 0), dyons (θ = 0) * Domain wall colour charged when dyons pass through → confining string can end there Wiese, Holland, Campos; 98: * EFT of PL and condensate with SU(3) * Witten’s observation holds only if chiral restoration and deconfinement occur simultaneously

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Preliminary results: SU(3)

0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 PL susceptibility T =

√t0 Nt

SU(3), β = 5.5, κ = 0.1673 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 Chiral susceptibility T =

√t0 Nt

SU(3), β = 5.5, κ = 0.1673

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Preliminary results: SU(3)

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Preliminary results: SU(3)

0.038 0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 < ¯ λλ > |PL| β = 5.5, κ = 0.1673

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Table of contents

• 1. The gradient flow: introduction
• 2. The gradient flow and renormalisation
• 3. Thermal super Yang-Mills
• 4. (Near) conformal field theories
• 5. Gradient flow and RG flow
• 6. Mass anomalous dimension of adjoint QCD

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Phases of a QFT

• Given a general QFT, it is interesting to study its behaviour at different

energy scales, i.e. its renormalisation group flow.

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Phases of a QFT

• Given a general QFT, it is interesting to study its behaviour at different

energy scales, i.e. its renormalisation group flow.

• IR phases:

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Phases of a QFT

• Given a general QFT, it is interesting to study its behaviour at different

energy scales, i.e. its renormalisation group flow.

• IR phases:
• I. Gapped, e.g 4d Yang-Mills (YM)
• II. Massless, e.g massless QCD
• III. Conformal, e.g. theories with IR fixed point (FP)
• IV. Non-trivially gapped, i.e. topological QFT, BPS states...

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Phases of a QFT

• Given a general QFT, it is interesting to study its behaviour at different

energy scales, i.e. its renormalisation group flow.

• IR phases:
• I. Gapped, e.g 4d Yang-Mills (YM)
• II. Massless, e.g massless QCD
• III. Conformal, e.g. theories with IR fixed point (FP)
• IV. Non-trivially gapped, i.e. topological QFT, BPS states...
• For a YM theory with fermions, one has different scenarios depending on

Nf and Nc:

24 / 36 Camilo Lopez QFT phases from the gradient flow

Phases of a QFT

• Given a general QFT, it is interesting to study its behaviour at different

energy scales, i.e. its renormalisation group flow.

• IR phases:
• I. Gapped, e.g 4d Yang-Mills (YM)
• II. Massless, e.g massless QCD
• III. Conformal, e.g. theories with IR fixed point (FP)
• IV. Non-trivially gapped, i.e. topological QFT, BPS states...
• For a YM theory with fermions, one has different scenarios depending on

Nf and Nc:

• 1. Small Nf: chiral symmetry breaking (IR massless)
• 2. Nl

f < Nf < N u f : Banks-Zaks (BZ) FP conformal window (IR

conformal)

• 3. Nf > N u

f : not asymptotically free

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Conformal window

[Desy-M¨ unster collaboration]

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IR conformal phase

• Gauge invariant operators obtain an anomalous scaling dimension γ as

they flow

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IR conformal phase

• Gauge invariant operators obtain an anomalous scaling dimension γ as

they flow

• γ freezes at the BZ fixed point

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IR conformal phase

• Gauge invariant operators obtain an anomalous scaling dimension γ as

they flow

• γ freezes at the BZ fixed point
• At the fixed point:

Particle interpretation fails Observables: correlation functions, operator dimensions

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IR conformal phase

• Gauge invariant operators obtain an anomalous scaling dimension γ as

they flow

• γ freezes at the BZ fixed point
• At the fixed point:

Particle interpretation fails Observables: correlation functions, operator dimensions

• Methods to compute observables: Lattice Monte Carlo (LMC), conformal

bootstrap, ...

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IR conformal phase

• Gauge invariant operators obtain an anomalous scaling dimension γ as

they flow

• γ freezes at the BZ fixed point
• At the fixed point:

Particle interpretation fails Observables: correlation functions, operator dimensions

• Methods to compute observables: Lattice Monte Carlo (LMC), conformal

bootstrap, ...

• Within LMC: take mass-deformed theory, i.e. away from the FP and

compute the anomalous dimensions from Mass spectrum of the theory Monte Carlo renormalisation group techniques Spectral density of Dirac operator (mode number) Recently: Gradient flow and RG flow [Carosso, Hasenfratz and Neil, PRL 121 no.20, 201601]

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IR conformal phase II

Motivation to study the IR phase of QFT on the lattice

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IR conformal phase II

Motivation to study the IR phase of QFT on the lattice

• In general, important to classify theories which become conformal at the IR

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IR conformal phase II

Motivation to study the IR phase of QFT on the lattice

• In general, important to classify theories which become conformal at the IR
• It is hard to analitically study non-susy theories.

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IR conformal phase II

Motivation to study the IR phase of QFT on the lattice

• In general, important to classify theories which become conformal at the IR
• It is hard to analitically study non-susy theories.
• Near conformal QFTs are important for phenomenology, e.g. technicolor

models

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IR conformal phase II

Motivation to study the IR phase of QFT on the lattice

• In general, important to classify theories which become conformal at the IR
• It is hard to analitically study non-susy theories.
• Near conformal QFTs are important for phenomenology, e.g. technicolor

models

• Being able to study RG flow through the GF opens up the possibility to

compute conformal data on the lattice

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Table of contents

• 1. The gradient flow: introduction
• 2. The gradient flow and renormalisation
• 3. Thermal super Yang-Mills
• 4. (Near) conformal field theories
• 5. Gradient flow and RG flow
• 6. Mass anomalous dimension of adjoint QCD

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Gradient flow vs RG flow

• GF similar to RG: smoothening of the fields ↔ elimination of high energy

modes

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Gradient flow vs RG flow

• GF similar to RG: smoothening of the fields ↔ elimination of high energy

modes

• YM GF is however not a complete RG transformation:

✗ Lack of scale transformation (dilatation) ✗ Lack of normalisation of the fields

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Gradient flow vs RG flow

• GF similar to RG: smoothening of the fields ↔ elimination of high energy

modes

• YM GF is however not a complete RG transformation:

✗ Lack of scale transformation (dilatation) ✗ Lack of normalisation of the fields

• On the lattice:

✓ Consider correlators at long distances ✓ Include renormalisation of the fields by using an exact conserved current (e.g. vector)

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Gradient flow vs RG flow

• GF similar to RG: smoothening of the fields ↔ elimination of high energy

modes

• YM GF is however not a complete RG transformation:

✗ Lack of scale transformation (dilatation) ✗ Lack of normalisation of the fields

• On the lattice:

✓ Consider correlators at long distances ✓ Include renormalisation of the fields by using an exact conserved current (e.g. vector)

• GF allows for blocked fields without having to know the blocked action

[Carosso, Hasenfratz and Neil]

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GF and RG flow

• Gradient Flow: φ → φt. Supression of high momentum modes

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GF and RG flow

• Gradient Flow: φ → φt. Supression of high momentum modes
• RG Transformation:

a → a′ = b a g → g′ m → m′ O(0)O(x0)g,m = b−2(dO+γO)O(0)O(x0/b)g′,m′

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GF and RG flow

• Gradient Flow: φ → φt. Supression of high momentum modes
• RG Transformation:

a → a′ = b a g → g′ m → m′ O(0)O(x0)g,m = b−2(dO+γO)O(0)O(x0/b)g′,m′

• RHS: Monte Carlo RG (MCRG)

O(0)O(x0/b)g′,m′ = Ob(0)Ob(x0/b)g,m

• Ob≡O(φb)

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GF and RG flow

• Gradient Flow: φ → φt. Supression of high momentum modes
• RG Transformation:

a → a′ = b a g → g′ m → m′ O(0)O(x0)g,m = b−2(dO+γO)O(0)O(x0/b)g′,m′

• RHS: Monte Carlo RG (MCRG)

O(0)O(x0/b)g′,m′ = Ob(0)Ob(x0/b)g,m

• Ob≡O(φb)
• Relate blocked and flowed fields through φb(xb) = bdφ+η/2φt(bxb) and

√ t ∝ b

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GF and RG flow

• Gradient Flow: φ → φt. Supression of high momentum modes
• RG Transformation:

a → a′ = b a g → g′ m → m′ O(0)O(x0)g,m = b−2(dO+γO)O(0)O(x0/b)g′,m′

• RHS: Monte Carlo RG (MCRG)

O(0)O(x0/b)g′,m′ = Ob(0)Ob(x0/b)g,m

• Ob≡O(φb)
• Relate blocked and flowed fields through φb(xb) = bdφ+η/2φt(bxb) and

√ t ∝ b Ot(0)Ot(x0) O(0)O(x0) = b

2∆O−2nO∆φ ,

∆i = di + γi (canonical + anomalous dim)

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GF and RG flow II

• Get rid of ∆φ through conserved operator V (γV = 0)

RO(t, x0) = O(0)Ot(x0) O(0)O(x0) V(0)V(x0) V(0)Vt(x0) nO/nV ∝ tγO/2+dO/2−dV /2

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GF and RG flow II

• Get rid of ∆φ through conserved operator V (γV = 0)

RO(t, x0) = O(0)Ot(x0) O(0)O(x0) V(0)V(x0) V(0)Vt(x0) nO/nV ∝ tγO/2+dO/2−dV /2

• The mass anomalous dimension of the operator O can be then defined as

γO(¯ t) = log(RO(t1)/RO(t2)) log (√t1/√t2)

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Table of contents

• 1. The gradient flow: introduction
• 2. The gradient flow and renormalisation
• 3. Thermal super Yang-Mills
• 4. (Near) conformal field theories
• 5. Gradient flow and RG flow
• 6. Mass anomalous dimension of adjoint QCD

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Low energy adjoint QCD from GF

Q1: For which number of flavours is SU(2) adjoint QCD (near-)conformal? Q2: What is the value of the anomalous dimension?

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Low energy adjoint QCD from GF

Q1: For which number of flavours is SU(2) adjoint QCD (near-)conformal? Q2: What is the value of the anomalous dimension? Compute RG flow of γ with the GF: V = lattice vector current O = pseudoscalar meson

33 / 36 Camilo Lopez QFT phases from the gradient flow

Low energy adjoint QCD from GF

Q1: For which number of flavours is SU(2) adjoint QCD (near-)conformal? Q2: What is the value of the anomalous dimension? Compute RG flow of γ with the GF: V = lattice vector current O = pseudoscalar meson ⇒ Look for freezing of γ in the RG flow ⇒ Extrapolate γ towards the deep IR

33 / 36 Camilo Lopez QFT phases from the gradient flow

Preliminary results

0.2 0.4 0.6 0.8 1 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 γPS µ =

1

√

8·(t1+t2)/2

Nf = 2 , β = 2.25 , κ = 0.13 , mpcac ∼ 0.04 Nf = 3/2 , β = 1.7 , κ = 0.134 , mpcac ∼ 0 Nf = 1 , β = 1.75 , κ = 0.175 , mpcac ∼ 0.03 34 / 36 Camilo Lopez QFT phases from the gradient flow

Conclusions

• The gradient flow method as smoothing operator:

* Correlators of flowed composite local operators are renormalised * This facilitates computation of densities and currents on the lattice

35 / 36 Camilo Lopez QFT phases from the gradient flow

Conclusions

• The gradient flow method as smoothing operator:

* Correlators of flowed composite local operators are renormalised * This facilitates computation of densities and currents on the lattice

• The gradient flow and RG flow:

* GF is part of RG transformation * In principle possible to compute CFT data like anomalous dimensions

• n the lattice

35 / 36 Camilo Lopez QFT phases from the gradient flow

Conclusions

• The gradient flow method as smoothing operator:

* Correlators of flowed composite local operators are renormalised * This facilitates computation of densities and currents on the lattice

• The gradient flow and RG flow:

* GF is part of RG transformation * In principle possible to compute CFT data like anomalous dimensions

• n the lattice
• Chiral and center symmetries intertwined in super Yang-Mills theory

* SU(2): second order phase transition * SU(3): first order phase transition

35 / 36 Camilo Lopez QFT phases from the gradient flow

Conclusions

• The gradient flow method as smoothing operator:

* Correlators of flowed composite local operators are renormalised * This facilitates computation of densities and currents on the lattice

• The gradient flow and RG flow:

* GF is part of RG transformation * In principle possible to compute CFT data like anomalous dimensions

• n the lattice
• Chiral and center symmetries intertwined in super Yang-Mills theory

* SU(2): second order phase transition * SU(3): first order phase transition

• Adjoint QCD with Nf = 1, 3/2, 2 is at least near-conformal

35 / 36 Camilo Lopez QFT phases from the gradient flow

Conclusions

• The gradient flow method as smoothing operator:

* Correlators of flowed composite local operators are renormalised * This facilitates computation of densities and currents on the lattice

• The gradient flow and RG flow:

* GF is part of RG transformation * In principle possible to compute CFT data like anomalous dimensions

• n the lattice
• Chiral and center symmetries intertwined in super Yang-Mills theory

* SU(2): second order phase transition * SU(3): first order phase transition

• Adjoint QCD with Nf = 1, 3/2, 2 is at least near-conformal
• Nf = 1 has large anomalous dimension → may be relevant for BSM

35 / 36 Camilo Lopez QFT phases from the gradient flow