Exploring the phases of Yang-Mills theory with adjoint matter through the gradient flow Camilo L´ opez Friedrich Schiller University of Jena with Georg Bergner and Stefano Piemonte SIFT 2019, Jena 1 / 36 Camilo Lopez QFT phases from the gradient flow
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 2 / 36 Camilo Lopez QFT phases from the gradient flow
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 3 / 36 Camilo Lopez QFT phases from the gradient flow
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 )) 4 / 36 Camilo Lopez QFT phases from the gradient flow
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 � 1 � 1 2 | dA | 2 = 2 | F A | 2 S ( A ) = 4 / 36 Camilo Lopez QFT phases from the gradient flow
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 � 1 � 1 2 | dA | 2 = 2 | F A | 2 S ( A ) = • And the gradient flow is given by the differential equations ∂ t B µ = D ν G νµ , B µ | t =0 = A µ : flow of gauge fields ∂ t χ = D µ D µ χ, χ | t =0 = ψ : flow of fermion fields 4 / 36 Camilo Lopez QFT phases from the gradient flow
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 � 1 � 1 2 | dA | 2 = 2 | F A | 2 S ( A ) = • And the gradient flow is given by the differential equations ∂ t B µ = D ν G νµ , B µ | t =0 = A µ : flow of gauge fields ∂ t χ = D µ D µ χ, χ | t =0 = ψ : flow of fermion fields • These evolve the fields to local minima of the action 4 / 36 Camilo Lopez QFT phases from the gradient flow
What does the flow imply for the quantum theory? 5 / 36 Camilo Lopez QFT phases from the gradient flow
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 6 / 36 Camilo Lopez QFT phases from the gradient flow
The gradient flow in QFT • The flow has a smoothening effect on the fields, which are Gaussian-like √ smeared over an effective radius r t = 8 t � d D y K t ( x − y ) A µ ( y ) + non linear terms , B µ ( t, x ) = � d D p (2 π ) D e ipz e − tp 2 K t ( z ) = (at leading order) • e − tp 2 is UV cut-off for t > 0 . It remains at all orders in perturbation theory [L¨ uscher and Weisz,arXiv:1405.3180] 7 / 36 Camilo Lopez QFT phases from the gradient flow
The gradient flow in QFT II Q: Are the correlators of flowed fields renormalised? 8 / 36 Camilo Lopez QFT phases from the gradient flow
The gradient flow in QFT II Q: Are the correlators of flowed fields renormalised? • D+1 dimensional QFT with flow time as spurious dimension 8 / 36 Camilo Lopez QFT phases from the gradient flow
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 . 8 / 36 Camilo Lopez QFT phases from the gradient flow
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 8 / 36 Camilo Lopez QFT phases from the gradient flow
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 8 / 36 Camilo Lopez QFT phases from the gradient flow
The gradient flow in QFT III � Monomials renormalise according to the field content 9 / 36 Camilo Lopez QFT phases from the gradient flow
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 9 / 36 Camilo Lopez QFT phases from the gradient flow
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! 9 / 36 Camilo Lopez QFT phases from the gradient flow
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... 9 / 36 Camilo Lopez QFT phases from the gradient flow
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 9 / 36 Camilo Lopez QFT phases from the gradient flow
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 10 / 36 Camilo Lopez QFT phases from the gradient flow
Brief review of minimal SYM L E = 1 4 F 2 + 1 θ ¯ 32 π 2 ˜ λ ( / D + m ˜ g ) λ + FF 2 11 / 36 Camilo Lopez QFT phases from the gradient flow
Brief review of minimal SYM L E = 1 4 F 2 + 1 θ ¯ 32 π 2 ˜ λ ( / D + m ˜ g ) λ + FF 2 • Vector supermultiplet with one Yang-Mills field A and one Majorana spinor λ in the adjoint representation 11 / 36 Camilo Lopez QFT phases from the gradient flow
Brief review of minimal SYM L E = 1 4 F 2 + 1 θ ¯ 32 π 2 ˜ λ ( / D + m ˜ g ) λ + FF 2 • 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 11 / 36 Camilo Lopez QFT phases from the gradient flow
Brief review of minimal SYM L E = 1 4 F 2 + 1 θ ¯ 32 π 2 ˜ λ ( / D + m ˜ g ) λ + FF 2 • 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 11 / 36 Camilo Lopez QFT phases from the gradient flow
Brief review of minimal SYM L E = 1 4 F 2 + 1 θ ¯ 32 π 2 ˜ λ ( / D + m ˜ g ) λ + FF 2 • 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) 11 / 36 Camilo Lopez QFT phases from the gradient flow
At zero temperature Z N c Centre symmetry ⋆ Not broken through adjoint fermions ⋆ Polyakov loop (PL) vev vanishes 12 / 36 Camilo Lopez QFT phases from the gradient flow
At zero temperature Z N c Centre symmetry ⋆ Not broken through adjoint fermions ⋆ Polyakov loop (PL) vev vanishes Chiral symmetry • Anomaly free Z 2 N c symmetry • Condensate < ¯ λλ > � = 0 ⇒ Z 2 N c → Z 2 • A domain wall interpolates N c degenerated vacua • Chern-Simons theory on domain wall with deconfined quarks 12 / 36 Camilo Lopez QFT phases from the gradient flow
Thermal phase transitions • At some T dec : Phase transition to broken centre symmetry c 13 / 36 Camilo Lopez QFT phases from the gradient flow
Thermal phase transitions • At some T dec : Phase transition to broken centre symmetry c • At some T χ c : Z 2 N c symmetry restored, < ¯ λλ > → 0 13 / 36 Camilo Lopez QFT phases from the gradient flow
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