Non-perturbative renormalization of operators in near-conformal systems using gradient flow Anna Hasenfratz University of Colorado Boulder with Andrea Carosso and Ethan Neil arXiv:1806.01385 � 1
1) Is gradient flow a renormalization group transformation? 2) Can we use GF to calculate anomalous dimensions? � 2
1) Is gradient flow a renormalization group transformation? 2) Can we use GF to calculate anomalous dimensions? 1) It is not, but it can be tricked: • normalize correctly • calculate appropriate quantities → GF acts like RG blocking with continuous scale change 2) Pilot study: N f =12 flavor SU(3), determine anomalous dimension of mass and baryon operators next talk: Andrea Carosso, Φ 4 model � 2
Wilson RG in a nutshell: Step 1: Introduce “blocked” fields and integrate out the original ones Step 2: rescale Λ cutoff → Λ cutoff /b (or a → b a) (1-f) Credit: Wilson-Kogut 1973,Ch.11 • The partition function is unchanged, S ( φ , g 0 ) → S ( φ , g ′ ) • The action changes • The RG flow runs along the renormalized trajectory either to the ξ =0 trivial or ξ = ∞ UVFP � 3
〈 O (0) O ( x 0 ) 〉 g , m Correlation function of An RG transformation of scale change b: S ( φ , g ) → S ( φ , g ′ ) − 2 Δ O 〈 O (0) O ( x b = x 0 / b ) 〉 ′ 〈 O (0) O ( x 0 ) 〉 g , m = b g , ′ m Δ O = d O + γ O scaling dimension and x 0 >> b S ( φ , g ') We do not need to simulate with — just use the principle of MCRG � 4
Monte Carlo Renormalization Group Swendsen PhysRevLett.42.859,1979 Action Configuration ensemble MC { φ } S ( φ , g 0 ) RG block MC S ( φ , g ′ ) { Φ b } RG transformed expectation values can be calculated without explicit knowledge of the blocked action 〈 O (0) O ( x b ) 〉 ′ m = 〈 O b (0) O b ( x b ) 〉 g , m g , ′ O b = O ( Φ b ) is the operator of the blocked fields � 5
Gradient flow could be “blocking” Luscher Comm.Math Phys 293, 899 (2010) GF is a continuous smoothing that removes short distance fluctuations Gauge flow: ∂ t V t = − ( ∂ S W [ V t ]) V t , V 0 = U Fermions evolve on the gauge background: ∂ t χ t = Δ [ V t ] χ t , χ 0 = ψ Luscher JHEP 04 123 (2013) (The flow action does not have to match the model) GF misses two important attributes of an RG transformation: – there is no rescaling Λ cut → Λ cut /b or coarse graining – linear transformation does not have the correct normalization Z φ = b − η /2 (wave function renormalization or ) Both issues can be solved � 6
Gradient flow could be “blocking” Luscher Comm.Math Phys 293, 899 (2010) GF is a continuous smoothing that removes short distance fluctuations Gauge flow: ∂ t V t = − ( ∂ S W [ V t ]) V t , V 0 = U Fermions evolve on the gauge background: ∂ t χ t = Δ [ V t ] χ t , χ 0 = ψ Luscher JHEP 04 123 (2013) (The flow action does not have to match the model) GF misses two important attributes of an RG transformation: – there is no rescaling Λ cut → Λ cut /b or coarse graining – linear transformation does not have the correct normalization Z φ = b − η /2 (wave function renormalization or ) Both issues can be solved GF does not flow to FP � 6
GF vs RG Original Φ fields Flowed Φ t fields GF RG transformation (b=2) – gradient flow : φ t ( φ ) Φ b = Z b φ t = b − η /2 φ t – blocked fields: – Coarse grain and rescale with b : x → x/b � 7
GF vs RG Original Φ fields Flowed Φ t fields GF RG transformation (b=2) – gradient flow : φ t ( φ ) Φ b = Z b φ t = b − η /2 φ t – blocked fields: – Coarse grain and rescale with b : x → x/b � 7
GF vs RG Original Φ fields Flowed Φ t fields GF 2-point functions do not care about decimation: 〈 O b ( Φ b (0)) O b ( Φ b ( x b )) 〉 g , m = b − η 〈 O ( φ t (0)) O ( φ t ( x b )) 〉 g , m At the level of expectation values GF is a proper RG transformation � 8
GF as RG Put it together − 2 Δ O 〈 O (0) O ( x b = x 0 / b ) 〉 ′ RG 〈 O (0) O ( x 0 ) 〉 g , m = b g , ′ m 〈 O (0) O ( x b ) 〉 ′ m = 〈 O b (0) O b ( x b ) 〉 g , m MCRG g , ′ 〈 O b ( Φ b (0)) O b ( Φ b ( x b )) 〉 g , m = b − η 〈 O ( φ t (0)) O ( φ t ( x b )) 〉 g , m GF � 9
GF as RG Put it together − 2 Δ O 〈 O (0) O ( x b = x 0 / b ) 〉 ′ RG 〈 O (0) O ( x 0 ) 〉 g , m = b g , ′ m 〈 O (0) O ( x b ) 〉 ′ m = 〈 O b (0) O b ( x b ) 〉 g , m MCRG g , ′ 〈 O b ( Φ b (0)) O b ( Φ b ( x b )) 〉 g , m = b − η 〈 O ( φ t (0)) O ( φ t ( x b )) 〉 g , m GF Ratio of flowed & unplowed correlators predict the anomalous dimension x 0 ≫ b 〈 O t (0) O t ( x 0 ) 〉 2 Δ O − 2 n O Δ φ 〈 O (0) O ( x 0 ) 〉 = b Δ O = d O + γ O Δ φ = d φ + η /2 � 9
Anomalous dimensions Calculate η by an operator that does not have an anomalous dimension: — vector or axial charge (A(x)) The super-ratio R ( t , x 0 ) = 〈 O t (0) O t ( x 0 ) 〉 〈 O (0) O ( x 0 ) 〉 ( 〈 A (0) A ( x 0 ) 〉 〈 A t (0) A t ( x 0 ) 〉 ) n O / n A = b γ O independent of x 0 >> b and predicts 𝛿 � 10
Anomalous dimensions Calculate η by an operator that does not have an anomalous dimension: — vector or axial charge (A(x)) The super-ratio R ( t , x 0 ) = 〈 O t (0) O t ( x 0 ) 〉 〈 O (0) O ( x 0 ) 〉 ( 〈 A (0) A ( x 0 ) 〉 〈 A t (0) A t ( x 0 ) 〉 ) n O / n A = b γ O independent of x 0 >> b and predicts 𝛿 - t and b are still independent! • Natural choice : b 2 ~ t - it is advantageous to flow only the source, not the sink - 𝛿 is universal at the FP only : set fermion mass to zero - t has to be large enough, and � 10
Anomalous dimensions Calculate η by an operator that does not have an anomalous dimension: — vector or axial charge (A(x)) The super-ratio R ( t , x 0 ) = 〈 O t (0) O t ( x 0 ) 〉 〈 O (0) O ( x 0 ) 〉 ( 〈 A (0) A ( x 0 ) 〉 〈 A t (0) A t ( x 0 ) 〉 ) n O / n A = b γ O γ O ∝ t /2 independent of x 0 >> b and predicts 𝛿 - t and b are still independent! • Natural choice : b 2 ~ t - it is advantageous to flow only the source, not the sink - 𝛿 is universal at the FP only : set fermion mass to zero x 0 ≫ 8 t - t has to be large enough, and � 10
Pilot study: N f =12 Low statistics study with staggered fermions - 24 3 x48 , 32 3 x64 volumes, m=0.0025 – mass anomalous dimension 𝛿 m =0.23-0.25 from perturbation theory, FSS numerical studies, Dirac eigenmodes – the gauge coupling walks very slow - substantial scaling violation effects are expected – baryon and tensor anomalous dimensions would be interesting where no non-perturbative prediction exists � 11
Ratio of ratios - pseudo scalar O ( x 0 ) = 〈 O (0) O t ( x 0 ) 〉 〈 O (0) O ( x 0 ) 〉 ( 〈 A (0) A ( x 0 ) 〉 〈 A (0) A t ( x 0 ) 〉 ) n O / n A = t γ O R t has no x 0 dependence if x 0 >> b pseudoscalar Oscillation is due to operator overlap ∝ 2 8 t —> limits max t flow time dependence of the plateau gives anomalous dimension � 12
Pseudo scalar Flow time dependence indicates slowly running gauge coupling � 13
Finite volume corrections − γ O R ( g , s 2 t , sL ) R ( g ', t , L ) = s − γ O ( R ( g , t , sL ) − R ( g , t , L ) ) + h.o. R ( g , s 2 t , s 2 L ) = R ( g , s 2 t , sL ) + s � 14
Pseudo scalar: γ m = 0.24(3) , t →∞ extrapolate to t → ∞ : error: systematic + statistical α 1 + d β t result consistent with other methods α 2 γ m ( β , t ) = γ 0 + c β t � 15
Nucleon channel nucleon - Lambda Anomalous dimension is small Minimal flow time dependence, 𝛿 N = 0.05(5) but limited x 0 range (perturbative: 𝛿 N = 0.09 ) � 16
Vector channel vector - tensor Oscillation pronounced but little flow time dependence Fit as − m 1 x 0 + B t e − m 2 x 0 −Δ mx 0 1 + B t / A t e A t e = A t − m 1 x 0 + Be − m 2 x 0 A −Δ mx 0 1 + B / Ae Ae 2 anomalous dimensions, from A t /A and B t /B both vanish within errors � 17
Summary & outlook – GF can describe an RG transformation • can aid our understanding of GF away from perturbation theory • determine anomalous dimension in conformal system (probably most promising method to get nucleon anomalous dim.) • determine renormalization factors in QCD (needs work) – Finite volume effects deserve more attention – Staggered fermions are a poor choice here (oscillations): DW is more promising – Anyone with existing conformal configurations can try the method (but need massless or nearly massless configs) – Beyond BSM: • Z factors in QCD need perturbative matching • 3D O(n) model: might not compete with FSS but can predict anomalous dimension of irrelevant operators (A. Carosso, next talk) � 18
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