Neutron Electric Dipole Moment in the Standard Model and beyond from Lattice QCD

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1 Neutron Electric Dipole Moment in the Standard Model and beyond from Los Alamos National Laboratory Santa Fe Institute The 30 th International Symposium on Lattice Field Theory June 24 29, 2012

2 1 Dipole Moments Sakharov Conditions for Baryogenesis Standard Model CP Violation Effective Field Theory 2 Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment 3 Renormalization State of the Art Needed Calculations Outlook

3 Dipole Moments Dipole Moments Sakharov Conditions for Baryogenesis Standard Model CP Violation Effective Field Theory Unaligned spin precesses in Electric and Magnetic Fields. Precession Frequency depends on E through EDM d. H = d E S µ B S Change in Precession Frequency on flipping E measures EDM.

4 Dipole Moments Sakharov Conditions for Baryogenesis Standard Model CP Violation Effective Field Theory µ B S is even under C, P, and T B, S are parity even: B + B S + S B, S are time reversal odd: B B S S B, S are charge conjugation even: µ B +µ B S + S d E S term violates P, T, and CP violates Parity: E E S + S violates Time reversal: E + E S S conserves Charge conjugation: d E +d E S + S

5 Sakharov Conditions for Baryogenesis Dipole Moments Sakharov Conditions for Baryogenesis Standard Model CP Violation Effective Field Theory CP violation needed in the universe. Observed baryon asymmetry: n B /n γ = WMAP + COBE 2003 Without CP violation, freezeout ratio: n B /n γ Kolb and Turner, Front. Phys. 69 (1990) 1. Either asymmetric initial conditions or baryogenesis! Sakharov Conditions Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32. Baryon Number violation C, CP and T violation Out of equilibrium evolution

6 Standard Model CP Violation Dipole Moments Sakharov Conditions for Baryogenesis Standard Model CP Violation Effective Field Theory Two sources of CP violation in the Standard Model. Complex phase in CKM quark mixing matrix. Too small to explain baryon asymmetry Gives a tiny ( e-cm) contribution to nedm Dar arxiv:hep-ph/ Effective ΘG G interaction from QCD instantons Effects suppressed at high energies nedm limits constrain Θ Crewther et al., Phys. Lett. B88 (1979) 123. Contributions from beyond the standard model Needed to explain baryogenesis May have large contribution to EDM

7 Effective Field Theory Dipole Moments Sakharov Conditions for Baryogenesis Standard Model CP Violation Effective Field Theory Parameterize BSM contributions using an effective field theory at the weak scale. Two important dimension six operators are the Electric and Chromoelectric dipole moments of the quark. CP Even S = SQCD i Θ g2 16π 2 d 4 x G µν Gµν + i e dγ u Qσ Λ 2 µν γ 5 F µν i e d HU γ + d Qσ BSM Λ 2 µν γ 5 F µν HD BSM + i g 3 d G u Λ 2 BSM +. Qσ µν γ 5 λ A G µν A HU + i g 3 d G d Λ 2 BSM Qσ µν γ 5 λ A G µν A HD

8 Dipole Moments Sakharov Conditions for Baryogenesis Standard Model CP Violation Effective Field Theory The two quark dipole moments are generated at 3-loops in the standard model and give tiny nedm ( e-cm). They are generated at one loop in BSM. Expected contribution is around experimental limit e-cm. Baker et al., Phys. Rec. Lett. 97 (2006)

9 Model expectations Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment Model analysis estimate of the neutron electric dipole moment: [ ( ) d n 8π2 Mn 3 2m qσq F qgσq d Θ G q + g s 3 F 2 qq m q + qq 3 ( 4 d γ + g s qgσq 6 qq d dγ u ) ( 4 d G d dσd F F d G u ūσu F F ( 4 3 dγ d 1 ) 3 dγ u 2e qq ( 2 M n fπ 2 3 dg d + 1 ) 3 dg u, assuming the first term vanishes by Peccei-Quinn mechanism. )] Pospelov and Ritz, Ann. Phys. 318 (2005) 119

10 Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment Numerically, d n ( Θ) qq (1 ± 0.5) (225MeV) Θ 3 ( e-cm) d n (d γ,g q ) d n ( Θ = Θ ind ) + qq [ (1 ± 0.5) 1.1 (d G (225MeV) 3 d d G u ) e (d γ d 0.25 dγ u) ], where Θ ind ( cm) 1 d G q m 2 0 m q /MeV (0.8GeV) 2 is the minimum of the PQ potential. Pospelov and Ritz, Ann. Phys. 318 (2005) 119

11 Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment Note that the quark dipole moments violate chirality, and, hence, are expected to be of the scale κ q = m q 16π 2 M 2 Λ = e-cm m ( ) q 1T ev 2. 1MeV M Λ Rough estimates can also be made of the other dimension 6 operators: Weinberg Operator: ( ) w(1gev) d n (w) ( e-cm) 1TeV 2 Four-quark Operators: ( ) d n (C) ( Cbd (m b ) + C db (m b ) e-cm) 1TeV 2 Pospelov and Ritz, Ann. Phys. 318 (2005) 119

12 Lattice Basics Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment 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. n Jµ EM n F 3(q 2 ) n q ν Σ µν γ 5 n 2M n F 3 (q 2 ) d n = lim q 2 0 2M n

13 Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment Difficult to perform simulations with complex CP action Expand and calculate correlators of the CP operator: [ ] C CP (x, y,...) CP+ = [DA] exp d 4 x(l CP + L CP ) C CP (x, y,...) [ ] [DA] exp d 4 xl CP ( ) d 4 xl CP C CP (x, y,...) = C CP (x, y,...) L CP (p µ = 0) CP

14 Topological charge Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment To find the contribution of Θ, we note that d 4 xg G = Q, the topological charge. So, we need the correlation between the electric current and the topological charge. n (2 3ūγ µu 1 3 dγ µ d)q n = 1 n (ūγ µ u + 2 dγ µ d)q n + 1 n (ūγ µ u 6 dγ µ d)q n

15 Quark Electric Dipole Moment Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric 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. n d γ u ūσ µν u + d γ d dσ µν d = d γ u + d γ d 2 n ūσ µν u + dσ µν d n + d γ u d γ d 2 n ūσ µν u dσ µν d n

16 Quark Chromoelectric Moment Model expectations Lattice Basics Topological charge Quark Electric Dipole Moment Quark Chromoelectric Moment The effect of the quark chromoelectric moment can be simplified using the Feynman-Hellmann Theorem. n J µ = A µ dσ νκ d) G νκ n d 4 x(d G u ūσ νκ u + d G d n d 4 x(d G u ūσ νκ u + d G d dσ νκ d) G νκ n E where the subscript E refers to the correlator calculated in the presence of a background electric field.

17 Renormalization Renormalization State of the Art Needed Calculations Outlook 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 the Topological charge; need to disentangle. Also need to calculate the influence of Chromoelectric moment of the quark on the potential for Θ.

18 State of the Art Renormalization State of the Art Needed Calculations Outlook Neutron electric dipole moment from Topological charge: Limits exist from lattie calculations Quark Electric Dipole Moment: Same as the tensor charge of the nucleon Connected diagrams calculated Quark Chromoelectric Dipole Moment: not yet calculated

19 Needed Calculations Renormalization State of the Art Needed Calculations Outlook Preliminary calculations needed before one can estimate errors and resource requirements. For preliminary calculations Use previously generated staggerred lattices Use Clover valence quarks Study Statistical signal Chiral behavior Dependence on lattice spacing Excited state contamination

20 Outlook Renormalization State of the Art Needed Calculations Outlook The connected diagram for Quark Electric Dipole Moment will soon reach 20% precision. The calculation of the disconnected diagrams, and the other operators need more study. Remaining systematic errors not expected to be major. nedm not overly sensitive to neglected EM and isospin-breaking Modern calculations include dynamical charm