Hadronic matrix elements for probes of CP violation ACFI, January 22-24 2015 Renormalization of CP-odd operators of dimension ≤ 5 Vincenzo Cirigliano Los Alamos National Laboratory
Outline • BSM-induced CP violation at dimension 5 ~ • Operator renormalization in RI-SMOM scheme, _ suitable for lattice implementation • Matching RI-SMOM and MS at one loop ~ • Future steps and conclusions Collaborators: Tanmoy Bhattacharya, Rajan Gupta, Emanuele Mereghetti, Boram Yoon, arXiv:1501.xxxx
BSM-induced CP violation at dimension 5
The CP-odd effective Lagrangian • L eff below weak scale, including leading (dim=6) Δ F=0 BSM effects: • Dim 4: CKM + “theta”-term • Dim 5: quark EDM and CEDM • Dim 6: gluon CEDM (Weinberg), 4-quark operators
The CP-odd effective Lagrangian • L eff below weak scale, including leading (dim=6) Δ F=0 BSM effects: • Focus on dim ≤ 5 operators: • Phenomenological relevance of quark EDM & CEDM • dim=6 operators not needed to define finite dim=5 operators
The CP-odd effective Lagrangian • L eff below weak scale, including leading (dim=6) Δ F=0 BSM effects:
The CP-odd effective Lagrangian • L eff below weak scale, including leading (dim=6) Δ F=0 BSM effects:
• After vacuum alignment (see Tanmoy Bhattacharya’s talk) The derivation assumes that quark mass is the dominant source of explicit chiral symmetry breaking
• After vacuum alignment (see Tanmoy Bhattacharya’s talk) • No PQ mechanism Both singlet and non-singlet m u 0 0 m s m d m u M = 0 m d 0 m ∗ = m s ( m u + m d ) + m u m d 0 0 m s couplings as
• After vacuum alignment (see Tanmoy Bhattacharya’s talk) • No PQ mechanism Both singlet and non-singlet Mixture of electric and m u 0 0 magnetic s.d. m s m d m u M = 0 m d 0 m ∗ = couplings m s ( m u + m d ) + m u m d 0 0 m s couplings as
• After vacuum alignment (see Tanmoy Bhattacharya’s talk) • Assume PQ mechanism Flavor structure controlled by [d CE ]
• After vacuum alignment (see Tanmoy Bhattacharya’s talk) • To compute d n,p (d E , d CE ), need nucleon matrix elements of t a represents a flavor diagonal n F × n F matrix • Need renormalization of P , E, and C in a scheme that can be implemented non-perturbatively, e.g. in lattice QCD
Operator renormalization ~ in RI-SMOM scheme
Renormalization: generalities • P: dim=3 quark bilinear, renormalizes multiplicatively • E: tensor quark bilinear x EM field strength. Neglecting effects of O( α EM ), E renormalizes multiplicatively (as tensor density) P , T Bochicchio et al,1995 Non-perturbative renormalization well known ... Aoki et al 2009
Renormalization: generalities • P: dim=3 quark bilinear, renormalizes multiplicatively • E: tensor quark bilinear x EM field strength. Neglecting effects of O( α EM ), E renormalizes multiplicatively (as tensor density) P , T • C: self-renormalization + mixing with E and P γ g Even richer mixing structure in subtraction schemes that involve off-shell quarks/gluons and non-zero momentum injection at vertex
Operator basis (I) _ • C = ig s Ψσ μν γ 5 G μν t a Ψ can mix with two classes of operators: Kuger-Stern Zuber 1975 Joglekar and Lee 1976 Deans-Dixon 1978 • O: gauge-invariant operators with same symmetry properties of C, not vanishing by equations of motion (EOM) • N: operators allowed by solution of BRST Ward Identities. Vanish by EOM, need not be gauge invariant. Needed to extract Z O , but do not affect physical matrix elements
Operator basis (II) • Flavor structure of operators: use “spurion” method _ • L QED + L QCD - i(g s /2) Ψσ μν γ 5 G μν [D CE ] Ψ invariant under Quark mass and charge matrices • Allow only invariant operators, and eventually set
Operator basis (III) • Dimension-3: 1 operator • Dimension-4: no operators if chiral symmetry is respected • Dimension-5: 10 + 4 operators
Operator basis (III) • Dimension-3: 1 operator • Dimension-4: no operators if chiral symmetry is respected • Dimension-5: 10 + 4 operators trace
Operator basis (III) • Dimension-3: 1 operator • Dimension-4: no operators if chiral symmetry is respected • Dimension-5: 10 + 4 operators
Operator basis (III) • Dimension-3: 1 operator • Dimension-4: no operators if chiral symmetry is respected • Dimension-5: 10 + 4 operators EOM
Mixing structure Valid in any scheme ⇐ dimensional analysis, momentum injection, EOM
Mixing structure Physically relevant block Z O
Mixing structure • To identify [Z O ] ij , need to study the following Green’s functions: g, γ (5) O 1 ≡ C ~ (5) O 5 ≡ mGG (5) O n n=2,3,6-10 O
Renormalization schemes _ • MS scheme: use dim-reg and subtract poles in 1/(d-4) • Simple, widely used in calculations of Wilson coefficients • Subtlety: need to specify scheme for γ 5 - NDR: { γ μ , γ 5 } = 0 ∀ μ - HV: { γ μ , γ 5 } = 0 for μ =0-3, otherwise [ γ μ , γ 5 ] = 0
Renormalization schemes _ • MS scheme: use dim-reg and subtract poles in 1/(d-4) • Simple, widely used in calculations of Wilson coefficients • Subtlety: need to specify scheme for γ 5 - NDR: { γ μ , γ 5 } = 0 ∀ μ - HV: { γ μ , γ 5 } = 0 for μ =0-3, otherwise [ γ μ , γ 5 ] = 0 • RI-SMOM class of schemes: fix finite parts by conditions on quark and gluon amputated Green’s functions in a given gauge, at non- exceptional momentum configurations, such as = tree level p 2 = p’ 2 = q 2 = - Λ 2 • Regularization independent: can be implemented on the lattice
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each)
~ ⇒ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • On the lattice, first subtract power-divergence (C ↔ P mixing) : C L → C = C L - Z C-P P = 0 Z C-P (a, Λ 0 ) ~1/a 2 γ 5 t a projection p 2 = p’ 2 = q 2 = - Λ 02
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • Conditions on C: amputated 2-pt functions Coefficients of = 0 7 spin-flavor structures** p 2 = p’ 2 = q 2 = - Λ 2 Λ ≠ Λ 0 ** γ 5 t a , σ μν γ 5 p μ p’ ν t a , q μ γ μ γ 5 M t a , q μ γ μ γ 5 Tr[M t a ], γ 5 M 2 t a , γ 5 t a Tr[M 2 ], γ 5 M Tr[M t a ]
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • Conditions on C: amputated 2-pt functions Coefficients of = 0 7 spin-flavor structures** p 2 = p’ 2 = q 2 = - Λ 2 1 condition for gluons, = 0 1 condition for photons p 2 = p’ 2 = q 2 = - Λ 2
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • Conditions on C: amputated 3-pt functions (q-q-gluon) Kinematics: = tree-level** s = (p+q) 2 u = (p-k) 2 t = (p-p’) 2 p 2 = p’ 2 = q 2 = k 2 = − Λ 2 s = u = t/2 = − Λ 2 ** 3 spin-flavor structures: σ μν γ 5 k ν t a , σ μν γ 5 (p-p’) ν t a , γ 5 (p+p’) μ t a
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • Conditions on C: amputated 3-pt functions (q-q-gluon) Kinematics: = tree-level** s = (p+q) 2 u = (p-k) 2 t = (p-p’) 2 p 2 = p’ 2 = q 2 = k 2 = − Λ 2 s = u = t/2 = − Λ 2 ~ S point: can’t have s=u=t = - Λ 2 but s=u = - Λ 2 and conditions on 2pt- function eliminate non-1PI diagrams
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • Conditions on C: amputated 3-pt functions (q-q-photon) Kinematics: γ = 0 ** s = (p+q) 2 u = (p-k) 2 t = (p-p’) 2 p 2 = p’ 2 = q 2 = k 2 = − Λ 2 s = u = t/2 = − Λ 2 ** 2 spin-flavor structures: σ μν γ 5 k ν t a , γ 5 (p+p’) μ t a
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • Conditions on (mGG): amputated 2-pt functions 1 spin-flavor structure: = 0 γ μ q μ γ 5 t a ~ p 2 = p’ 2 = q 2 = - Λ 2 = tree 1 condition p 2 = p’ 2 = q 2 = - Λ 2
~ RI-SMOM scheme ~ • Require conditions on C (14), mGG (2), O 2,3,6-10 (one each) • Conditions on E, (m ∂ A) 1,2 and (m 2 P) 1,2,3 : amputated 2-pt functions = tree 1 spin-flavor structure each ~ p 2 = p’ 2 = q 2 = - Λ 2 Conditions are equivalent to RI-SMOM conditions on A, P , T Aoki et al 2009
~ Matching RI-SMOM and ____ MS at one loop
One-loop calculations • Insertions of C Z 1n n= 2,6-10, 11-13 Z 15 g, γ g, γ Z 1n , n=1, 11-13 Z 1n , n=3,11-14
One-loop calculations ~ • Insertions of mGG Z 55 , Z 56 • Insertions of E~T, (m ∂ A) 1,2 and (m 2 P) 1,2,3 Z nn n=2,3, 6-10 A, P , T
One-loop calculations • Schematic form of all 1-loop results Depends on scheme adopted for γ 5 (HV, NDR) Time-consuming part of the calculation Work in covariant gauge: Landau gauge ( ξ =0) can be implemented on the lattice • Determine Z O
One-loop calculations • Schematic form of all 1-loop results Depends on scheme adopted for γ 5 (HV, NDR) Time-consuming part of the calculation Work in covariant gauge: Landau gauge ( ξ =0) can be implemented on the lattice • ξ -independent Determine Z O ξ -dependent
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