Hadronic EDMs from Dyson-Schwinger: Rho-Meson & Nucleon Mario - PowerPoint PPT Presentation

Hadronic EDMs from Dyson-Schwinger: Rho-Meson & Nucleon Mario Pitschmann Institute of Atomic and Subatomic Physics, Vienna University of Technology January 22 nd , 2015 / ACFI

Introduction Introduction Part A: The Theoretical Framework Part B: The ρ Meson Part C: The Nucleon

Introduction: The Energy Scale & Effective EDM Operators for dim ≥ 4 at scale ∼ 1 GeV Calculation of hadronic EDMs naturally splits into 2 parts Calculation of Wilson coefficients 1 by integrating out short distances Switching from perturbative quark-gluon description to 2 non-perturbative treatment – (much harder and larger uncertainties) Effective EDM Operators for dim ≥ 4 at scale ∼ 1 GeV Θ g 2 = − i ¯ L 1 GeV 32 π 2 G a s µν ˜ G a µν M − i q σ µν γ 5 q F µν − i q 1 � � ˜ 2 λ a σ µν γ 5 q G a d q ¯ 2 g s d q ¯ µν 2 q = u , d q = u , d + i K Λ 2 ε ij (¯ Q i d )(¯ Q j γ 5 u ) + · · ·

Part A: The Theoretical Framework

1. Dyson-Schwinger Equations

Dyson-Schwinger Equation Non-perturbative continuum approach to any QFT A shift in the integration variable ( ϕ ( x ) → ϕ ( x ) + λ ( x ) ), does not change the path integral for suitable b.c., i.e. D [ ϕ ] δ � δϕ f [ ϕ ] = 0 Application to the generating functional Z [ J ] yields � � − δ S � d 4 x J ϕ = 0 e − S + � D [ ϕ ] δϕ + J � d 4 x L . This can be rewritten as with the action S = � δ � − δ S � � + J Z [ J ] = 0 δϕ δ J

Dyson-Schwinger Equation In QCD the fermion propagator is obtained by derivation of � δ � δ S η, − δ δη, δ � � − + η ( x ) Z [ η, ¯ η, J ] = 0 δ ¯ ψ ( x ) δ ¯ δ J µ with respect to η leading after several formal manipulations to the Gap Equation for the quark propagator S F ( p ) − 1 = i / p Z 2 + m q ( µ ) Z 4 d 4 q λ i � ( 2 π ) 4 g 2 D µν ( k − p ) γ µ + Z 1 2 S F ( k )Γ ν ( k , p ) − 1 − 1 − =

Dyson-Schwinger Equation Gap equation contains the full vertex Γ µ and full gluon 1 propagator D µν ( k − p ) , each satisfies it’s own DSE DSE for the full vertex Γ µ contains the four-point vertex , 2 which has it’s own DSE . . . = ⇒ DSE is an infinite tower of equations relating all correlation functions DSE are exact relations and are the quantum Euler-Lagrange equations for any QFT Perturbative Expansion yields standard perturbative QFT

2. Bethe-Salpeter Equations

Bethe-Salpeter Equations Bethe-Salpeter equation is the DSE describing a bound 2 body system Obtained by four derivatives of the generating functional and several formal manipulations d 4 q � q + P q − P � � � � Γ( k ; P ) = ( 2 π ) 4 K ( q , k ; P ) S F Γ( q ; P ) S F 2 2 Solutions for discrete set P 2 yield mass spectra =

Rainbow-Ladder Truncation A symmetry-preserving truncation of the infinite set of DSEs which respects relevant (global) symmetries of QCD is the rainbow-ladder truncation in combination with the impulse approximation 1. In BSE kernel µν ( q ) λ a 2 γ µ ⊗ λ a K ( p , p ′ ; k , k ′ ) → −G ℓ ( q 2 ) D free 2 γ ν 2. In gap equation µν ( q ) λ a ν ( k , p ) → G ℓ ( q 2 ) D free Z 1 g 2 D µν ( q )Γ a 2 γ ν R-LT is first term in systematic expansion of q ¯ q scattering kernel K ( p , p ′ ; k , k ′ )

Gluon Propagator DSE and unquenched QCD lattice studies show that the Full gluon propagator µν ( p ) = δ ab G ℓ ( p 2 ) � δ µν − p µ p ν � D ab p 2 p 2 is IR finite , i.e. p 2 → 0 D ab µν ( p ) = finite lim - the gluon has dynamically generated mass in the IR EM Observables in the static limit ( q µ → 0 ) probe gluon propagator for small transversed momenta = ⇒ Point-like vector ⊗ vector contact interaction 4 πα IR g 2 D ab µν ( p ) = δ ab δ µν m 2 G

Contact Interaction Model This implies Non-renormalizable theory Introduce proper-time regularization Λ uv = 1 /τ uv cannot be removed but plays a dynamical role and 1 sets the scale of all dimensioned quantities Λ ir = 1 /τ ir implements confinement by ensuring the absence of 2 quark production tresholds Scale m G , is set in agreement with observables In the static limit q 2 → 0 results "indistinguishable" from any other however sophisticated DSE approach For q 2 � M 2 dressed deviations are expected from other experimental values

Part B: 1 The ρ Meson 1M. P ., C. Y. Seng, M. J. Ramsey-Musolf, C. D. Roberts, S. M. Schmidt and D. J. Wilson, Phys. Rev. C 87 (2013) 015205

The ρ Meson "Per se" from an experimental point of view uninteresting Short lifetime ( ∼ 10 − 24 s ) makes EDM measurements hard (or rather impossible) Simplest system possibly providing EDM and hence perfect prototype particle Results available in QCD sum rules and other techniques Profile I G ( J PC ) = 1 + ( 1 −− ) 1 m = 775 . 49 ± 0 . 34 MeV, 2 Γ = 149 . 1 ± 0 . 8 MeV Primary decay mode ( ∼ 100 % ): ρ → ππ 3

The ρ -Meson in Impulse Approximation Impulse Approximation q k − + k ++ Γ µ Γ α Γ β p ′ p k −− d 4 k � Γ ( u ) � Γ ρ ( u ) � S ( k ++ )Γ ( u ) µ S ( k − + )Γ ρ ( u ) αµβ ∝ ( 2 π ) 4 Tr CD S ( k −− ) α β EDM sources induce CP violating corrections to the q γ q vertex 1 Bethe-Salpeter amplitude 2 Propagator 3

The Magnetic Moment Results for M ( 0 ) in units e / ( 2 m ρ ) DSE - CIM 2 . 11 DSE - RL RGI-improved 2 . 01 DSE - EF parametrisation 2 . 69 LF - CQM 2 . 14 LF - CQM 1 . 92 1 . 8 ± 0 . 3 QCD sum rules point particle 2

The Θ -Term Only CP violating dimension 4 operator Θ g 2 L eff = − i ¯ µν ˜ 32 π 2 G a s G a µν No suppression by heavy scale (strong CP problem) U ( 1 ) A anomaly allows to rotate it into complex mass for evaluation (effective propagator correction) 0 . 7 × 10 − 3 e ¯ DSE Θ / 1 GeV 4 . 4 × 10 − 3 e ¯ QCD sum rules Θ / 1 GeV

The Θ -Term Only CP violating dimension 4 operator Θ g 2 L eff = − i ¯ s µν ˜ 32 π 2 G a G a µν No suppression by heavy scale (strong CP problem) U ( 1 ) A anomaly allows to rotate it into complex mass for evaluation (effective propagator correction) 0 . 7 × 10 − 3 e ¯ Θ / 1 GeV DSE 4 . 4 × 10 − 3 e ¯ QCD sum rules Θ / 1 GeV

The Θ -Term Only CP violating dimension 4 operator Θ g 2 L eff = − i ¯ s µν ˜ 32 π 2 G a G a µν No suppression by heavy scale (strong CP problem) U ( 1 ) A anomaly allows to rotate it into complex mass for evaluation (effective propagator correction) 0 . 7 × 10 − 3 e ¯ Θ / 1 GeV DSE 4 . 4 × 10 − 3 e ¯ QCD sum rules Θ / 1 GeV

The Θ -Term Only CP violating dimension 4 operator Θ g 2 L eff = − i ¯ s µν ˜ 32 π 2 G a G a µν No suppression by heavy scale (strong CP problem) U ( 1 ) A anomaly allows to rotate it into complex mass for evaluation (effective propagator correction) 0 . 7 × 10 − 3 e ¯ Θ / 1 GeV DSE 4 . 4 × 10 − 3 e ¯ QCD sum rules Θ / 1 GeV

The Θ -Term Only CP violating dimension 4 operator Θ g 2 L eff = − i ¯ s µν ˜ 32 π 2 G a G a µν No suppression by heavy scale (strong CP problem) U ( 1 ) A anomaly allows to rotate it into complex mass for evaluation (effective propagator correction) 0 . 7 × 10 − 3 e ¯ Θ / 1 GeV DSE 4 . 4 × 10 − 3 e ¯ QCD sum rules Θ / 1 GeV

The Θ -Term Only CP violating dimension 4 operator Θ g 2 L eff = − i ¯ s µν ˜ 32 π 2 G a G a µν No suppression by heavy scale (strong CP problem) U ( 1 ) A anomaly allows to rotate it into complex mass for evaluation (effective propagator correction) 0 . 7 × 10 − 3 e ¯ Θ / 1 GeV DSE 4 . 4 × 10 − 3 e ¯ QCD sum rules Θ / 1 GeV

The Quark-EDM The intrinsic EDM of a quark itself L eff = − i � d q ¯ q σ µν γ 5 q F µν 2 q = u , d Effective q γ q vertex correction 0 . 79 ( d u − d d ) DSE - CIM DSE 0 . 72 ( d u − d d ) Bag Model 0 . 83 ( d u − d d ) 0 . 51 ( d u − d d ) QCD sum rules Non-relativistic quark model 1 . 00 ( d u − d d )

The Chromo-EDM The Intrinsic Chromo-EDM of a quark itself L eff = − i q 1 � ˜ 2 λ a σ µν γ 5 q G a d q ¯ 2 g s µν q = u , d Effective q γ q vertex correction − 0 . 07 e ˜ d − − 0 . 20 e ˜ DSE - q γ q d + − 0 . 12 e ˜ d − + 0 . 11 e ˜ DSE - BSA d + 1 . 35 e ˜ d − − 0 . 60 e ˜ DSE - Propagator d + 1 . 16 e ˜ d − − 0 . 69 e ˜ DSE d + − 0 . 13 e ˜ QCD sum rules d −

The Chromo-EDM The Intrinsic Chromo-EDM of a quark itself L eff = − i q 1 � ˜ 2 λ a σ µν γ 5 q G a d q ¯ 2 g s µν q = u , d Effective q γ q vertex correction − 0 . 07 e ˜ d − − 0 . 20 e ˜ DSE - q γ q d + − 0 . 12 e ˜ d − + 0 . 11 e ˜ DSE - BSA d + 1 . 35 e ˜ d − − 0 . 60 e ˜ DSE - Propagator d + 1 . 16 e ˜ d − − 0 . 69 e ˜ DSE d + − 0 . 13 e ˜ QCD sum rules d −

The Chromo-EDM The Intrinsic Chromo-EDM of a quark itself L eff = − i q 1 � ˜ 2 λ a σ µν γ 5 q G a d q ¯ 2 g s µν q = u , d Effective q γ q vertex correction − 0 . 07 e ˜ d − − 0 . 20 e ˜ DSE - q γ q d + − 0 . 12 e ˜ d − + 0 . 11 e ˜ DSE - BSA d + 1 . 35 e ˜ d − − 0 . 60 e ˜ DSE - Propagator d + 1 . 16 e ˜ d − − 0 . 69 e ˜ DSE d + − 0 . 13 e ˜ QCD sum rules d −