Introduction nEDM Mixing Renormalization Conclusions Renormalization of Lattice Operators I Tanmoy Bhattacharya Los Alamos National Laboratory Santa Fe Institute Work done in collaboration with Vincenzo Cirigliano, Rajan Gupta, Emanuele Mereghetti, and Boram Yoon Jan 23, 2015 Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction nEDM Lattice QCD Mixing Euclidean Space Renormalization Conclusions Introduction Lattice QCD Lattice is a nonperturbative formulation of QCD. Lattice uses a hard regulator: ¯ pψ → ¯ ψ / ψ / W ( p ) ψ , where W ( p ) is a periodic function: W ( p ) = W ( p + 2 πa − 1 ) . Hard regulators introduce a scale and allow mixing with lower dimensional operators. Hard regulators are unambiguous: no renormalon problem. Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction nEDM Lattice QCD Mixing Euclidean Space Renormalization Conclusions Ultraviolet divergence regulated by the periodicity: � ∞ � π ( m +1) /a � π/a ∞ � dp = dp → dp −∞ π ( m − 1) /a − π/a m = −∞ Infrared controlled by calculating in a finite universe. � (2 π L ) f (2 πn � dpf ( p ) → + p 0 ) L n Real world reached by lim . L →∞ a → 0 Current calculations a ∼ 0 . 05 – 0 . 15 fm and L ∼ 3 – 5 fm . Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction nEDM Lattice QCD Mixing Euclidean Space Renormalization Conclusions Introduction Euclidean Space Lattice can calculate equal time vacuum matrix elements: To extract � n | O | n � , one starts with Tr e − βH ˆ ne − HT f Oe − HT i ˆ n † ne − HT f Oe − HT i ˆ e − βE s � s | ˆ n † | s � = n | n f � e − M f T f � n j | O | n i � e − M i T i � n i | ˆ − → n † | Ω � � Ω | ˆ β →∞ − → � n | O | n � e − M 0 ( T i + T f ) T i ,T f →∞ Vacuum and states chosen by the theory: can only calculate ‘physical’ matrix elements. Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Lattice Basics nEDM Topological charge Mixing Quark Electric Dipole Moment Renormalization Quark Chromoelectric Moment Conclusions State of the Art nEDM Lattice Basics We can extract nEDM in two ways. As the difference of the energies of spin-aligned and anti-aligned neutron states: d n = 1 2 ( M n ↓ − M n ↑ ) | E = E ↑ By extracting the CP violating form factor of the electromagnetic current. F 3 ( q 2 ) � n | J EM n q ν σ µν γ 5 n | n � ∼ ¯ µ 2 M n F 3 ( q 2 ) d n = lim 2 M n q 2 → 0 Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Lattice Basics nEDM Topological charge Mixing Quark Electric Dipole Moment Renormalization Quark Chromoelectric Moment Conclusions State of the Art ✟ Difficult to perform simulations with complex ✟ CP action ✟ Expand and calculate correlators of the ✟ CP operator: � � � � d 4 x ( L CP + L ✚ � C ✚ ✚ ✚ CP ( x, y, . . . ) � CP + ✚ CP ) = [ DA ] exp − ✚ CP × C ✚ ✚ CP ( x, y, . . . ) � � � � d 4 x L CP ≈ [ DA ] exp − � � � d 4 x L ✚ ✚ C ✚ ✚ CP CP ( x, y, . . . ) × 1 − � C ✚ ✚ CP ( x, y, . . . ) L ✚ ✚ CP ( p µ = 0) � CP = Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Lattice Basics nEDM Topological charge Mixing Quark Electric Dipole Moment Renormalization Quark Chromoelectric Moment Conclusions State of the Art nEDM Topological charge To find the contribution of ¯ d 4 xG ˜ � Θ , we note that G = Q , the topological charge. So, we need the correlation between the electric current and the topological charge. � � � � (2 uγ µ u − 1 � ¯ � � n 3 ¯ dγ µ d ) Q � n = � � 3 1 1 uγ µ u + ¯ � n uγ µ u − ¯ � n � � � � � � � � n � (¯ dγ µ d ) Q + n � (¯ dγ µ d ) Q 2 6 Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Lattice Basics nEDM Topological charge Mixing Quark Electric Dipole Moment Renormalization Quark Chromoelectric Moment Conclusions State of the Art nEDM Quark Electric Dipole Moment Since the quark electric dipole moment directly couples to the electric field, we just need to calculate its matrix elements in the neutron state. � d γ uσ µν u + d γ d ¯ dσ µν d � � � �� n u ¯ = d γ u + d γ + d γ u − d γ uσ µν u + ¯ � n uσ µν u − ¯ � n d � dσ µν d � d � dσ µν d � � � � � n � ¯ n � ¯ 2 2 Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Lattice Basics nEDM Topological charge Mixing Quark Electric Dipole Moment Renormalization Quark Chromoelectric Moment Conclusions State of the Art nEDM Quark Chromoelectric Moment nEDM from quark chromoelectric moment is a four-point function: � � (2 � uγ µ u − 1 � � � ¯ d ¯ dσ νκ d ) ˜ � d 4 x ( d G uσ νκ u + d G � n 3 ¯ dγ µ d ) u ¯ G νκ � n � � 3 Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Lattice Basics nEDM Topological charge Mixing Quark Electric Dipole Moment Renormalization Quark Chromoelectric Moment Conclusions State of the Art Four-point functions can be calculated using noise sources. No experience yet with these. Alternatively, we can simplify using Feynman-Hellmann Theorem: � � � � � d ¯ dσ νκ d ) ˜ d 4 x ( d G uσ νκ u + d G � � n � J µ u ¯ G νκ � n � � � � ∂ � � � d 4 x ( d G uσ νκ u + d G d ¯ dσ νκ d ) ˜ � � = n u ¯ G νκ � n � � ∂A µ � E where the subscript E refers to the correlator calculated in the presence of a background electric field. Needs dynamical configurations with electric fields. Can use reweighting. Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Lattice Basics nEDM Topological charge Mixing Quark Electric Dipole Moment Renormalization Quark Chromoelectric Moment Conclusions State of the Art nEDM State of the Art Neutron electric dipole moment from Topological charge: Limits exist from lattie calculations Quark Electric Dipole Moment: Same as the tensor charge of the nucleon Preliminary results available Quark Chromoelectric Dipole Moment: not yet calculated Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Pattern of mixing nEDM Electric dipole moment Mixing Vacuum Alignment and Phase Choice Renormalization Lower dimensional operator Conclusions Mixing Pattern of mixing Renormalization of the lattice operators can be performed non-perturbatively. Topological charge is well studied and understood. Electric current and Quark Elecric Dipole moment operators are quark bilinears: well understood renormalization procedure. Quark Chromoelectric Moment operator mixes with Quark Elecric Dipole moment: need to disentangle. Quark Chromoelectric Moment operator has divergent mixing with lower dimensional operators: need high precision. Also need to calculate the influence of Chromoelectric moment of the quark on the PQ potential for Θ . Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Pattern of mixing nEDM Electric dipole moment Mixing Vacuum Alignment and Phase Choice Renormalization Lower dimensional operator Conclusions Mixing Electric dipole moment ψσ µν ˜ The operator is ¯ F µν ψ . It is a CP-violating quark-quark-photon vertex. At lowest order in electroweak perturbation theory, does not mix with any lower-or-same-dimension operators. At one loop in electroweak, chirally unsuppressed power-divergent mixing with ¯ ψγ 5 ψ , ψσ µν ˜ log-divergent mixing with ¯ G µν ψ , and doubly chirally suppressed log-divergent mixing with ¯ ψγ 5 ψ At two loops (mixed strong and electroweak), chirally suppressed, power-divergent mixing with G ˜ G . Other mixings vanish onshell at zero four momentum, but not necessarily at zero three momentum. Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Pattern of mixing nEDM Electric dipole moment Mixing Vacuum Alignment and Phase Choice Renormalization Lower dimensional operator Conclusions If we handle electroweak perturbatively on the lattice, we can work at lowest order and avoid this mixing. In this case, all we need is the tensor charge. Depending on accuracy needed, continuum running from high scale to hadronic scales need to account for mixing. MS does not see power divergence. Tanmoy Bhattacharya Renormalization of Lattice Operators I
Introduction Pattern of mixing nEDM Electric dipole moment Mixing Vacuum Alignment and Phase Choice Renormalization Lower dimensional operator Conclusions Mixing Vacuum Alignment and Phase Choice CP and chiral symmetry do not commute. Outer automorphism: CP χ ≡ χ − 1 CP χ also a CP . = iγ 4 C ¯ = iγ 4 C ¯ ψ CP ψ T L and ψ CP ψ T R . L R ψ χ L = e iχ ψ L and ψ χ R = e − iχ ψ R ψ CP χ = e − 2 iχ iγ 4 C ¯ L and ψ CP χ = e +2 iχ iγ 4 C ¯ ψ T ψ T L R R In chirally symmetric theory, vacuum degenerate: spontaneously breaks all but one CP χ . Addition of explicit chiral symmetry breaking breaks degeneracy of vacuum. Tanmoy Bhattacharya Renormalization of Lattice Operators I
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