Theory of the neutron EDM

Transcription

1 1 Theory of the neutron EDM Ulf-G. Meißner, Univ. Bonn & FZ Jülich Supported by BMBF 05P15PCFN1 by DFG, SFB/TR-110 by CAS, PIFI by Volkswagen Stftung by HGF VIQCD VH-VI-417 Nuclear Astrophysics Virtual Institute

2 2 CONTENTS Introduction The baryon EDMs from chiral perturbation theory Finite volume effects for baryon EDMs The neutron EDM from 2+1 flavor lattice QCD Summary & outlook

3 3 Introduction

4 ELECTRIC DIPOLE MOMENT: FUNDAMENTALS 4 consider here neutrons, protons and hyperons an (permanent) electric dipole moment (EDM) violates time-reversal symmetry if CPT holds, this is equivalent to CP-violation In the Standard Model (SM) small CP-violation tiny nedm, pedm,... d n < e cm current limits: d n < e cm fig. d p < e cm [indir.] fig. from Wikipedia very sensitive to physics beyond the SM (also µ eγ, (g 2) µ,...) precision low-energy hadron physics might well outdeliver the LHC

5 ELECTRIC DIPOLE MOMENT: DETAILS 5 EDM of an subatomic particle: d = d σ Energy of a particle with magnetic moment µ and electric dipole moment d: H = µ σ B d σ E T : H = µ σ B + d σ E P: H = µ σ B + d σ E a non-vanishing permanent EDM of a subatomic particle is thus T - and P-violating Be aware: non-degenerate ground state, otherwise level mixing fig. from Wikipedia

6 EXPERIMENTS on the NEUTRON EDM 6 performed and projected experiments (UCN): Beams UCN d n [e cm] SUSY SM PSI MultiCell 3He UCN year of experiment QCD θ

7 STRONG CP VIOLATION 7 Many SM and BSM sources of EDMs concentrate on one: QCD θ-term QCD has non-trivial topological vacua: θ = n e i n θ n Consider strong CP-violation induced by the θ-vacuum QCD in the presence of strong CP-violation L QCD = 1 4 Ga µν Ga,µν + flavors q (id/ M) q + θ 0 g 2 32π 2 Ga µν G a,µν }{{} E a B a A non-vanishing vacuum angle θ 0 entails d n 0 (also d p 0) d N θ 0 e M 2 π m 3 N θ 0 e cm θ 0 = O(10 10 )

8 BARYON EDMs - BASIC DEFINITIONS 8 Baryon electromagnetic form factors in the presence of strong CP -violation [or other sources of it] p J ν em p = ū (p ) Γ ν ( q 2) u (p) Γ ν = γ ν F 1 ( q 2 ) q 2 = (p p) 2 i 2m B σ µν q µ F 2 ( q 2 ) 1 2m B σ µν q µ γ 5 F 3 ( q 2 ) +... q p p Dipole moments and radii: d B = F 3,B(0) 2m B, df r 2 3 (q 2 ) ed = 6 dq 2 q2 =0 Similar formulae for light nuclei see talks by Mereghetti, de Vries, Wirzba,... Calculation requires non-perturbative technique: EFT and/or LQCD

9 9 The baryon EDM in chiral perturbation theory Ottnad, Kubis, UGM, Guo, Phys. Lett. B 687 (2008) 42 Guo, UGM, JHEP 1212 (2012) 097

10 BARYON EDMs - CALCULATIONS 10 non-perturbative methods: baryon chiral perturbation theory and/or lattice QCD study the nucleon/baryon electric dipole form factors in baryon CHPT already quite a number of investigations (mostly the neutron) Crewther, Veneziano, Witten, Pich, de Rafael,..., Borasoy, Narison, Hockings, van Kolck, de Vries,... complete one-loop calculation Ottnad, Kubis, M., Guo, Phys. Lett. B 687 (2008) 42 based on L eff [φ a, φ 0, B] supplemented by power counting LQCD calculations for θ and CP-violating form factors become available must address quark mass and finite volume corrections fruitful interplay between LQCD and CHPT/EFT practitioners

11 11 EFFECTIVE LAGRANGIAN Effective Lagrangian based on U(3) L U(3) R symmetry: L eff = L[U, B], ( 2 U = exp 3 i F 0 η 0 } {{ } U(1) + 2i ) φ F φ }{{} SU(3) Herrera-Siklódy et al., Borasoy Power counting: Kaiser, Leutwyler,... p = O (δ), m q = O ( δ 2), 1/N c = O ( δ 2) Determination of the low-energy constants meson sector: meson masses, η-η mixing,... meson-baryon sector: em form factors, large N c + naturalness

12 12 EFFECTIVE LAGRANGIAN U(3) U(3) effective meson-baryon Lagragian Borasoy (2000) L φb = i Tr [ Bγ µ [D µ, B] ] m Tr[ BB] D/F 2 Tr[ Bγ µ γ 5 [u µ, B] ± ] ] + b0 Tr[ BB] Tr [ χ + ia(u U ) ] +b D/F Tr [ B [ χ + ia(u U ), B ] ± +4A w 10 ( 6 F 0 η 0 Tr[ BB] + i w 13/14 θ 0 + w 6 13/14 F 0 η 0 )Tr [ Bσ µν γ 5 [F µν +, B] ] ± +w 16/17 Tr [ Bσ µν [F µν +, B] ] λ ± + 2 Tr[ Bγ µ γ 5 B ] Tr[u µ ] tree-level LECs: w 13, w 14, w 13, w 14 loop LECs: w 10, λ [some get renormalized] } more later! further LECs are absorbed in masses, magnetic moments, etc

13 BARYON EDMS at ONE LOOP 13 Guo, UGM, JHEP12 (2012) 097 Consider the ground state baryon octet (N, Λ, Σ,, Ξ) not only interesting in itself, but also provides sufficient data to fix LECs tree-level contributions to baryon EDMs [ α = 144V (2) 0 V (1) 3 /(F 0 F π M η0 ) 2] : d(n) = d(λ)/2 = d(σ 0 )/2 = d(ξ 0 ) = 8e θ 0 ( αw13 + w 13) /3 d(p) = d(σ + ) = 4e θ 0 [ α (w13 + 3w 14 ) + w w 14] /3 d(σ ) = d(ξ ) = 4e θ 0 [ α (w13 3w 14 ) + w 13 3w 14] /3 Charged particles feature more loop contributions than neutral ones Neutron case: {π, p} and {K +, Σ } Proton case: {π +, n}, {π 0 (η 8, η 0 ), p}, {K +, Σ 0 (Λ)} and {K 0, Σ + }

14 14 FEYNMAN GRAPHS for F B 3 (q2 ) tree (a,b) and one-loop (c,d,e,f) graphs (complete one-loop) η 0 (a) (b) (c) (d) (e) (f) other loop graphs (see Ottnad et al.) mutually cancel CP-odd dim. 2 CP-even

15 NEUTRON LOOP CONTRIBUTIONS 15 one-loop contributions to the neutron EDM (other neutral baryons similar): F loop 3,n (q2 ) = V { (2) 0 e θ 0 (D + F ) (b 2m N π 2 Fπ 4 D + b F ) [ ] 1 ln M 2 π + σ µ 2 π ln σ π 1 σ π +1 + π (2M 2 π q2 ) arctan q 2 2m N q 2 2M π [ (D F ) (b D b F ) 1 ln M 2 K µ 2 + σ K ln σ K 1 σ K +1 ( + π 2M 2 K q2 q 2 2m N 8 (b D b F ) ( MK 2 M ) ) π 2 ]} [ ] q 2 arctan σ π(k) = 1 4Mπ(K) 2 /q2 2M K only known masses and couplings!

16 PROTON LOOP CONTRIBUTIONS 16 one-loop contributions to the proton EDM (other charged baryons similar): F loop 3,p (q2 ) = V { (2) 0 e θ 0 6(D + F ) (b 2m N 6π 2 Fπ 4 D + b F ) [ ] 1 ln M 2 π + σ µ 2 π ln σ π 1 σ π πM π 2m N + π (2M 2 π q2 ) arctan q 2 2m N q 2 2M π ( ) +4 (Db D + 3F b F ) 1 ln M 2 K µ 2 + σ K ln σ K 1 σ K +1 + πm K m N [ + 4π arctan q 2 (Db D +3F b F ) ( q 2 2M K 2m N 2M 2 K q 2) +8 ( MK 2 M ( ( ) π) 2 F b 2 D + 3b 2 F 2 3 Db D (b D 3b F ) ) ] + π m N [6(D F ) (b D b F ) M K + (D 3F ) (b D 3b F ) M η8 + 2F 2 π (2D 3λ) ( ] 2b F0 2 D + 3b 0 + 6w ) } 10 M η0 }{{} β

17 EDMS of NEUTRAL BARYONS 17 Pion mass dependence of the neutral baryons loop contributions: dn loop e Θ0 fm n Ξ 0 loop d 0 e Θ 0 fm LO NLO Π Shintani et al. (2008) Shintani et al. (2012) Π d loop eθ 0 fm Λ Σ 0 loop d 0 e Θ 0 fm M Π GeV Π preliminary data from Shintani et al just for illustration

18 EDMS of CHARGED BARYONS 18 Pion mass dependence of the proton loop contributions (β in 1/GeV): β = 1 dp loop eθ0 fm 0.08 β = LO β = M Π GeV Shintani et al. (2008) Shintani et al. (2012) other charged baryons show similar sensitivity calculate all!

19 MAKING PREDICTIONS 19 Close inspection of the tree and one-loop expressions reveals only two combinations of LECs appear at NLO in all baryon EDMs: w a (µ) αw 13 + w r 13 (µ) w b (µ) 3[αw 14 + w 14 r(µ)] + V (2) 0 β 4πF0 2F π 2m ave use this formalism to analyze lattice data next section there are relations free of LECs such as M η0 d Σ 0 + d Λ ( MK 2 M ( ) π) 2 F b 2 D + 2Db D b F + 3F b 2 F + O(δ 4 ) ( ) d n d Ξ 0 (Db D + F b F ) 2 ln M 2 K + π M π M K Mπ 2 m ave + 8π ( ( ) M K M 2 K Mπ) 2 Db 2 D + 2F b D b F + Db 2 F + O(δ 4 ) now lets apply this formalism!

20 20 Finite volume effects for baryon EDMs Akan, Guo, UGM, Phys.Lett. B 736 (2014) 163

21 CALCULATIONAL SCHEME 21 Lattice QCD operates in a finite volume must consider finite volume corrections LO nedm corrections known O Connell, Savage (2006) work in the limit of infinite time and large volumes L 3 momenta quantized p = 2π n/l mode sums: i d 4 p (2π) 4 i L 3 n dp 0 2π finite volume corrections: δ L [Q] = Q(L) Q( ) these are entirely generated by loops here: perform NLO calculations for all baryon EDMs Guo, UGM (2012)

22 RESULTS for the NEUTRON EDM 22 At LO, recover the results of O Connell and Savage Find sizeable NLO corrections must be included δ L [d n ]/d loop n δl NLO [d n ]/δl LO[d n] 3.0 L dn dn loop M Π 138 MeV M Π 200 MeV M Π 300 MeV M Π 400 MeV L dn L d n LO L fm L fm available also for the proton and the hyperons

23 RESULTS for the PROTON EDM 23 again large sensitivity to the LEC β 1 LO NLO 1 Β 1 GeV 1 LO loop L dp dp L dp dp loop NLO Β 0 GeV 1 L dp dp loop NLO Β 1 GeV L fm calcs for the ffs more difficult, in progress Tiburzi, Greil et al., Akan et al.

24 24 The neutron EDM from 2+1 flavor lattice QCD Guo, Horsley, UGM, Nakamura, Rakow, Schierholz, Zanotti, Phys. Rev. Lett. 115 (2015) [arxiv: ]

25 LATTICE FORMULATION: GENERALITIES 25 work around the SU(3) symmetry point, keeping the singlet mass m fixed: m = (m u + m d + m s ) Bietenholz et al., Phys. Rev. D84 (2010) keeps the kaon mass low for varying strange quark masses constrained polynomials fits works fine for N f = [m u = m d = m l ] experiment N(lll) Λ(lls) Σ(lls) Ξ(lss) sym. pt M NO /X N [Octet] 1.0 M [MeV] M 2 2 π /X π 0 π K η s ρ K* φ s N Λ Σ Ξ Σ* Ξ* Ω

26 LATTICE FORMULATION at FIXED TOPOLOGY The θ-term on the lattice [at fixed topological charge Q]: S = S 0 + S θ, S θ = i θ Q, Q = 1 64π ɛ 2 µνρσ a 4 G b µν Gb ρσ Z x 26 rotate the θ-term into the fermionic part of the action Baluni (1979) S θ = i 3 θ ˆma4 x (ūγ5 u + dγ 5 d + sγ 5 s ) ˆm 1 = 1 3 ( m 1 u + m 1 d ) + m 1 s = 1 3 ( 2m 1 l ) + m 1 s take the vacuum angle imaginary: θ = i θ [th y analytic at θ = 0] S θ = θ m l m s 2m s + m l a 4 x (ūγ5 u + dγ 5 d + sγ 5 s ) real action, vanishes at m l = 0 and m s = 0

27 LATTICE SET-UP 27 Wilson fermions with a clover term & Symanzik improved gluon action (SLiNC) S q = S q 0 + Sq θ = a4 x q ( D 1 4 c SW σ µν G µν + m q + λ ) 2a γ 5 q lattice set-ups (β = 5.50) # κ l κ s V M π [MeV] M K [MeV] λ a = 0.074(2) fm from the average baryon octet mass only ensembles 1 to 4 in the PRL publication

28 LATTICE SET-UP: TOPOLOGICAL CHARGE 28 keep the singlet mass m fixed at its physical value and vary δm q = m q m λ = θ 2a m l m s 2m s + m l, am q = 1 2κ q 1 2κ 0,c, κ 0,c = Ensembles carry topological charge, Q m topological charge distribution #4 average topological charge p(q) 0.1 Q Q θ

29 i EVALUATION 29 at non-vanishing θ, Dirac spinors pick up a phase α(θ) can be obtained from a ratio of two-point functions Tr[G θ NN (t; 0)Γ 4γ 5 ] Tr[G θ NN (t; 0)Γ 4] ᾱ( θ) sin 2α(θ) = i 1 + cos 2α(θ) m π = 465 MeV m π = 360 MeV θ flavor-singlet ps density interacts with the nucleon through quark-line disconnected diagrams only γ 5 γ µ Form factor F 3 (q 2 ) extracted from a ratio of 3-point and 2-point functions generalizing the method of Capitani et al., Nucl. Phys. Proc. Suppl. 73 (1999) 294

30 i FORM FACTOR RATIOS 30 Extracting F θ,n 3 (q 2 )/F θ,p 1 (q 2 ) 0 m π = 465 MeV, at finite θ: easier extrapolation to q 2 = 0 θ,p /F θ,n F M π = 465 MeV (aq) 2 θ,p /F θ,n F m π = 360 MeV, θ,p /F θ,n F m π = 310 MeV, PRELIMINARY M π = 360 MeV M π = 310 MeV (aq) 2 (aq) 2

31 RESULTS 31 Form factor as a fct of θ continue θ and F θ 3 expand around θ = 0 : F θ 3 d n = ef (1) 3 (0)θ/2m N (0) to real values (0) = F (1) 3 (0)θ +... chiral extrapolation with 2M 2 K + M 2 π =const. 3 (0) θ,n F R m π = 465 MeV m π = 360 MeV θ w a (µ = 1 GeV) = 0.04(1) GeV preliminary d n = (2)(9) [e fm θ] dn [e fm θ] θ proton under analysis m 2 π [GeV 2 ]

32 OTHER RESULTS 32 Neutron EDM using N f = twisted mass fermions d n = 0.045(6)(1) θ e fm, one M π = 373 MeV Alexandrou et al., Phys. Rev. D 93 (2016) Neutron EDM and tensor charges from LQCD d n < e cm in split SUSY w/ gaugino mass unification Bhattacharya et al., Phys. Rev. Lett. 115 (2015) other groups are close to announce numbers domain wall fermions N f = 2 + 1, low pion masses, various volumes Shintani et al., Phys. Rev. D 93 (2016) Shintani s talk nucleon EDM from gradient flow, tested in quenched QCD Shindler et al., Phys. Rev. D 92 (2015)

33 SUMMARY & OUTLOOK 33 Baryon EDMs evaluated in U(3) U(3) CHPT at NLO Chiral extrapolation formulae worked out Only two LECs at complete one-loop order Finite volume corrections available LQCD calculation of the neutron EDM for 2+1 flavors simulation at various pion masses & lattice volumes working with an imaginary θ [th y assumed to be analytic at θ = 0] using CHPT, a precise value of d n at the physical point emerges: d n = (2)(9) [e fm θ] much more results in the pipeline

34 34 SPARES

35 35 BSM OPERATORS at LOW ENERGIES translate (B)SM sources of P- and T-odd interactions into tailored EFTs Fig. courtesy of A. Wirzba 100GeV q g q, G, γ, H H H qedm qcedm gcedm 4q 10GeV q q, γ, G N, π, γ,... 1GeV N... EFT

36 CALCULATIONAL SCHEMES 36 Chiral perturbation theory (CHPT) and extensions thereof allow to study the quark mass dependence and finite volume representation of hadron and nuclear properties Lattice QCD allows for ab initio calculations at (un)physical quark masses Horsley et al., Shintani et al., Bhattacharya et al., Guo et al. Consider one exotic property of elementary particles and nuclei here: the baryon & light nuclei electric dipole moment θ-vacua of QCD (topology) strong CP-violation also sensitive to physics beyond the SM challenge for precision calculations

37 EDMS of CHARGED BARYONS cont d Σ Σ loop d eθ 0 fm loop d eθ 0 fm M Π GeV M Π GeV Ξ loop d eθ 0 fm M Π GeV

38