EDMs, CP-odd Nucleon Correlators & QCD Sum Rules

Transcription

1 Hadronic Matrix Elements for Probes of CP Violation - ACFI, UMass Amherst - Jan 2015 EDMs, CP-odd Nucleon Correlators & QCD Sum Rules Adam Ritz University of Victoria Based on (older) work with M. Pospelov, see e.g. the review M. Pospelov & AR, Ann. Phys. 318, 119 (2005) [hep-ph/ ] (plus some updates)

2 Experimental EDM Limits H = de S S EDMs are powerful (amplitude-level) probes for new (T,P) violating sources, motivated e.g. by baryogenesis. Best current limits from neutrons, para- and dia-magnetic atoms and molecules Neutron EDM dn < 3 x e cm [Baker et al. 06] Diamagnetic EDMs dhg < 3 x e cm [Griffith et al 09] dxe < 4 x e cm [Rosenberry & Chupp 01] Paramagnetic EDMs ΔE ThO/ℇext < 3 x e cm [Baron et al. 13] ΔEYbF/ℇext < 1.4 x e cm [Hudson et al. 11] Negligible SM (CKM) background - contribution is (at least) 4-5 orders of magnitude below the current neutron sensitivity, and lower for the atomic EDMs 2

3 Summary of the bounds log(d [e cm]) Real sensitivity to underlying sources of CP violation depends on significant enhancement and suppression factors ~ dq and dq from the neutron ~ dq from Hg de from ThO impact of recent order of magnitude improvement in paramagnetic EDM sensitivity -32 The generic sensitivity to new physics follows from taking df mf -34 3

4 Multi-scale calculational scheme CP violation Model-dependent (e.g. perturbative) QCD scale Nucleon matrix elements (focus of this meeting), nucleon EDMs, pionnucleon, nucleon-nucleon couplings nuclear/atomic scale Nuclear scale, e.g. Schiff moment, magnetic quadrupole Observable EDMs Atomic/Molecular EDM 4

5 Multi-scale calculational scheme CP violation Model-dependent (e.g. perturbative) Significant uncertainties for nucleon, nuclear and diagmagnetic EDMs QCD scale Nucleon matrix elements (focus of this meeting), nucleon EDMs, pionnucleon, nucleon-nucleon couplings nuclear/atomic scale Nuclear scale, e.g. Schiff moment, magnetic quadrupole Observable EDMs Atomic/Molecular EDM 5

6 EDM Sensitivity to (short distance) CP-violation Energy Fundamental CP phases TeV QCD electron EDM semi-leptonic qqee θ-term, quark EDMs, CEDMs etc. nuclear µ EDM semi-leptonic NNee pion-nucleon πnn and NNNN EDMs of nuclei and ions (deuteron, etc) Nucleon EDMs (n,p) atomic EDMs of paramagnetic atoms and molecules (Tl,YbF, ThO, HfF +,...) Atoms in traps (Rb,Cs,Fr) solid state EDMs of diamagnetic atoms (Hg,Xe,Ra,Rn,...) 6

7 EDM Sensitivity to (short distance) CP-violation Energy Fundamental CP phases TeV QCD electron EDM semi-leptonic qqee θ-term, quark EDMs, CEDMs etc. nuclear µ EDM semi-leptonic NNee pion-nucleon πnn and NNNN EDMs of nuclei and ions (deuteron, etc) Nucleon EDMs (n,p) atomic EDMs of paramagnetic atoms and molecules (Tl,YbF, ThO, HfF +,...) Atoms in traps (Rb,Cs,Fr) solid state EDMs of diamagnetic atoms (Hg,Xe,Ra,Rn,...) 7

8 CP-odd operator expansion (at ~1GeV) (Flavor-diagonal) CP-violating operators at ~1GeV L e = X n c n d 4 O(n) d [ hadronic sector discussed in detail in Jordy s talk] L dim 4 s G G = 0 ArgDet(M u M d ) 0 q NB: (i) Basis at 1 GeV is simpler than at EW scale, after integrating out W,Z,h etc. (ii) Use of QCD dofs assumes that the new physics scale is above 1 GeV) 8

9 CP-odd operator expansion (at ~1GeV) (Flavor-diagonal) CP-violating operators at ~1GeV L e = X n c n d 4 O(n) d L dim 4 L dim 6 L dim 6 s G G X q=u,d,s wg 3 sgg G d i cy i v d q qf 5 q + d q qg 5 q + X l=e,µ 2 d l lf 5 l 9

10 CP-odd operator expansion (at ~1GeV) (Flavor-diagonal) CP-violating operators at ~1GeV L e = X n c n d 4 O(n) d L dim 4 s G G L dim 6 X d q qf 5 q + d q qg 5 q + X d l lf 5 l q=u,d,s l=e,µ L dim 6 wg 3 sgg G + C ff ( f f ) LL ( f f ) RR f,f, Schematic form of a few special 4-fermion operators, requiring no Higgs insertion - suppressed without new UV sources of LR mixing 10

11 CP-odd operator expansion (at ~1GeV) (Flavor-diagonal) CP-violating operators at ~1GeV L e = X n c n d 4 O(n) d L dim 4 s G G L dim 6 X d q qf 5 q + d q qg 5 q + X d l lf 5 l q=u,d,s l=e,µ L dim 6 wg 3 sgg G + C ff ( f f ) LL ( f f ) RR f,f, L dim 8 q, C qq q q q i 5 q + C qe q qē i 5 e + C ij cy i Y j v

12 CP-odd operator expansion (at ~1GeV) (Flavor-diagonal) CP-violating operators at ~1GeV L e = X n c n d 4 O(n) d L dim 4 L dim 6 s G G X q=u,d,s d q qf 5 q + d q qg 5 q + X l=e,µ d l lf 5 l L dim 6 wg 3 sgg G + f,f, C ff ( f f ) LL ( f f ) RR L dim 8 q, C qq q q q i 5 q + C qe q qē i 5 e + NB: Relative importance of different operators is very model-dependent, and the expansion can be misleading. E.g. for the SM (and SUSY and 2HDM regimes at large tanbeta), these 4-fermion sources are dominant 12

13 CP-odd operator expansion (at ~1GeV) (Flavor-diagonal) CP-violating operators at ~1GeV L e = X n c n d 4 O(n) d L dim 4 s G G L dim 6 X d q qf 5 q + d q qg 5 q + X d l lf 5 l q=u,d,s l=e,µ L dim 6 wg 3 sgg G + C ff ( f f ) LL ( f f ) RR f,f, L dim 8 q, C qq q q q i 5 q + C qe q qē i 5 e + 13

14 CP-odd operator expansion (at ~1GeV) (Flavor-diagonal) CP-violating operators at ~1GeV L e = X n c n d 4 O(n) d L dim 4 s G G L dim 6 X d q qf 5 q + d q qg 5 q + X d l lf 5 l q=u,d,s l=e,µ L dim 6 wg 3 sgg G + C ff ( f f ) LL ( f f ) RR f,f, L dim 8 q, C qq q q q i 5 q + C qe q qē i 5 e + nucleon/nuclear scales d (n,p) NF 5 N +ḡ (1) NN N 0 N +ḡ (0) NN N N + (4 nucleon) + d e ēf 5 e + C (0) S NNēi 5 e + 14

15 EFT hierarchy Energy Fundamental CP phases TeV QCD nuclear µ EDM pion-nucleon couplings ( ) EDMs of nuclei and ions (deuteron, etc) Nucleon EDMs (n,p) atomic EDMs of paramagnetic atoms and molecules (Tl,YbF, ThO, HfF +,...) Atoms in traps (Rb,Cs,Fr) solid state EDMs of diamagnetic atoms (Hg,Xe,Ra,Rn,...) 15

16 Energy TeV EFT hierarchy Fundamental CP phases focus of the rest of this talk! QCD nuclear µ EDM pion-nucleon couplings ( ) EDMs of nuclei and ions (deuteron, etc) Nucleon EDMs (n,p) atomic EDMs of paramagnetic atoms and molecules (Tl,YbF, ThO, HfF +,...) Atoms in traps (Rb,Cs,Fr) solid state EDMs of diamagnetic atoms (Hg,Xe,Ra,Rn,...) 16

17 The QCD scale Chiral EFT (chiral constraints) L = L(, (K),N, ) = i 2 N(d n + d p + )F 5 N N(ḡ (0) NN low energy { constants [ Emanuele s talk] a a +ḡ (1) NN d N (, d q, d q,w,c ij,...) ḡ (0,1) NN (, d q,c ij,...) 0 )N + [Crewther et al 79; Hisano & Shimizu 04; Stetcu et al 08, de Vries et al 11,12; An et al 12; Guo & Meissner 12, Bsaisou et al 14 ] 17

18 The QCD scale Chiral EFT (chiral constraints) L = L(, (K),N, ) = i 2 N(d n + d p + )F 5 N N(ḡ (0) NN low energy { constants LEC s related by IR loops (chiral logs) still need input to fix counterterms a a +ḡ (1) NN d N (, d q, d q,w,c ij,...) ḡ (0,1) NN (, d q,c ij,...) 0 )N + d n = e 4 2 m n g NN ḡ (0) NN ln m + C ct need UV threshold corrections 18

19 The QCD scale Chiral EFT (chiral constraints) L = L(, (K),N, ) = i 2 N(d n + d p + )F 5 N N(ḡ (0) NN low energy { constants LEC s related by IR loops (chiral logs) still need input to fix counterterms Simplest option is NDA - a a +ḡ (1) NN d N (, d q, d q,w,c ij,...) ḡ (0,1) NN (, d q,c ij,...) 0 )N + had/f g s (µ) 4 m q, av m 2 / had q d q dq d n ḡ (0) NN em q 2 had m q f O(1) O( ) eg s 4 2 had f 19

20 Why do better than NDA? (Our) pre-historic motivations... back in the 1990 s, when SUSY (MSSM) was just around the corner, there was a focus on combining multiple EDM contributions to search for cancelations to ameliorate the SUSY CP problem. This requires a systematic procedure to add contributions from multiple CP-odd sources, with relative signs! At the time, there weren t many viable approaches, and we utilized QCD sum rules Current (and more generic) motivations... disentangle CP-violating sources, given one (or more) detections indicate possible enhancement/suppression factors (cf. NDA) still require a systematic procedure able to handle multiple CPodd sources 20

21 (CP-odd) nucleon correlators Consider the two-point function of the nucleon interpolating current in the presence of CP-odd sources Features: multiple interpolating currents with lowest dimension j n =2 abc (d T a C 5 u b )d c + 2 abc (d T a Cu b ) 5 d c unphysical parameter New - chiral ambiguity of the nucleon current 0 j n n =( )e i 5/2 v CP-odd sources introduce an unphysical phase in the coupling of the nucleon current and the physical state, which can (unphysically) mix d and µ 21

22 (CP-odd) nucleon correlators Another (related) new feature - the nucleon current can now mix with CP-conjugate currents, j n = CPj n CP Need to account for mixing by re-diagonalizing at linear order in the source j n j n + i CP j n So the correlator is correspondingly rotated jj F, /CP + i /C P j j + j j F + i 2 h j n j n hj n j n j n j n i CP / j n j n i 22

23 QCD sum rules Work in a general basis of CP sources, as a cross-check Account for CP-odd current mixing to linear order (as above) Isolate EDM from a chirally-invariant structure in off-shell dipole form-factor (2-pt function, to first order in F) 1(p) F {F 5,/p} d 2 n m n (p 2 m 2 n) 2 + A p 2 m 2 n + + Isolate tensor structure independent of unphysical phase α (avoids mixing of d and µ structures) 3-pt mixing with excited states cannot be exponentially suppressed with a Borel transform, due to lack of positivity in dispersive integral for 3-pt correlators. Must include mixing coefficient A explicitly in the fit. effective 3-pt vertex allows mixing with excited states (introduces new fitting parameter A) 23

24 1(p) F {F 5,/p} QCD sum rules Work in a general basis of CP sources, as a cross-check Account for CP-odd current mixing to linear order (as above) Isolate EDM from a chirally-invariant structure in off-shell dipole form-factor (2-pt function, to first order in F) d 2 n m n (p 2 m 2 n) 2 + A p 2 m 2 n + + Isolate tensor structure independent of unphysical phase α (avoids mixing of d and µ structures) 3-pt mixing with excited states cannot be exponentially suppressed with a Borel transform, due to lack of positivity in dispersive integral for 3-pt correlators. Must include mixing coefficient A explicitly in the fit. Perform a self-consistent fit for the EDM, using other CPeven nucleon sum rules (mass, σn, etc) to determine {mn,λ,a} FAC criterion fixes β=1, i.e. to optimize convergence of the OPE 24

25 Neutron EDM schematic structure of the OPE [Pospelov & AR 99-00] 1(p) F {F 5,/p} d 2 n m n (p 2 m 2 n) 2 + A p 2 m 2 n + + depends on vacuum condensates, e.g. implicit dependence of condensates on the CP-odd sources determined via χpt, and saturation with π and η exchange. (vacuum realignment ) 25

26 Neutron EDM Results: d n ( ) = (1 ± 0.5) qq (225 MeV) e cm If the axion relaxes θ, the CEDM sources shift the minimum of the axion potential V(θ) away from zero [Bigi & Uraltsev] Sensitive only to ratios of light quark masses (via GMOR relation, given dq ~ mq etc.) [Pospelov & AR 99, 00; Hisano et al 12] at this order, s-quark CEDM contribution cancels under axion relaxation (appears accidental) 26

27 Neutron/Proton EDM Results: d n ( ) ecm d p ( ) ecm If the axion relaxes θ, the CEDM sources shift the minimum of the axion potential V(θ) away from zero d (PQ) p (0.4 ± 0.2)[4d u d d 5.3e( d u d d )+ ]+O(d s,w,c qq ) d (PQ) n (0.4 ± 0.2)[4d d d u +2.7e( d d +0.5 d d )+ ]+O(d s,w,c qq ) Appearance of the same relative coefficients as the NQM appears accidental, as it depends (at ~ 30%) on the choice of β [0,1] 27

28 Neutron EDM Precision? numerical coefficients are consistent with NDA, NQM (for dq), and the chiral log (for θ) another test for dn(dq) via (LQCD) nucleon tensor charge N 1 2 d q q F q N = 1 2 d q F µ N µ N = 1 2 gq T d q N F N =) d n (d q )=g u T d d + g d T d u 0.8d d 0.25d u [e.g. Falk et al 99] In the isospin-symmetric limit, inserting (connected) LQCD results [Hagler 09; Bhattacharya et al 11, 13] (NB: Recent extractions from transversity slightly lower) 28

29 Neutron EDM Precision? numerical coefficients are consistent with NDA, NQM (for dq), and the chiral log (for θ) another test for dn(dq) via (LQCD) nucleon tensor charge N 1 2 d q q F q N = 1 2 d q F µ N µ N = 1 2 gq T d q N F N =) d n (d q )=g u T d d + g d T d u 0.8d d 0.25d u sum-rules fixes (dn ~ <qq>/λ 2 ), so the normalization of the nucleon coupling matters GeV ± 0.01 GeV 3 from analysis of CP-even sum rules for mn, σn, etc (or lattice result for tensor charge above) [Pospelov & AR 99, 00] [e.g. Falk et al 99] from LQCD [Y. Aoki et al 08] run down from 2 GeV, *BUT* <qq> is also larger with LQCD values for mq so may be consistent [Hisano et al 12, Fuyuto et al 12] 29

30 Neutron EDM Precision? numerical coefficients are consistent with NDA, NQM (for dq), and the chiral log (for θ) another test for dn(dq) via (LQCD) nucleon tensor charge N 1 2 d q q F q N = 1 2 d q F µ N µ N = 1 2 gq T d q N F N =) d n (d q )=g u T d d + g d T d u 0.8d d 0.25d u sum-rules fixes (dn ~ <qq>/λ 2 ), so the normalization of the nucleon coupling matters GeV ± 0.01 GeV 3 from analysis of CP-even sum rules for mn, σn, etc (or lattice result for tensor charge above) [Pospelov & AR 99, 00] higher order dependence on s-quark EDM? [e.g. Falk et al 99] from LQCD [Y. Aoki et al 08] run down from 2 GeV, *BUT* <qq> is also larger with LQCD values for mq so may be consistent [Hisano et al 12, Fuyuto et al 12] 30

31 Pion-nucleon couplings Can follow a similar approach for the pion-nucleon couplings focus on the isovector coupling [Pospelov 01] (1) 31

32 Pion-nucleon couplings Can follow a similar approach for the pion-nucleon couplings focus on the isovector coupling [Pospelov 01] (1) { cancelation in vacuum 32

33 Pion-nucleon couplings Using QCD sum rules Z 2 2ḡ d 4 xe ip x NN m N j n (x),j n (0) dq H q /p (p 2 m 2 + A N )2 p 2 m 2 N + + isolate chirally invt structure ḡ (1) NN ( d q ) (2 12)GeV qq (225 MeV) 3 ( d u dd )+O( d s,w) ḡ (0) NN ( d q ) ( 1 3)GeV qq (225 MeV) 3 ( d u + d d )+O( d s,w) [Pospelov 01] normalization again consistent with NDA, but larger errors due to cancelations between direct & rescattering terms [result slightly smaller than estimates using LETs: Falk et al 99; Hisano & Shimizu 04] dependence on quark EDMs suppressed by αem 33

34 EFT hierarchy Energy Fundamental CP phases TeV QCD nuclear µ EDM pion-nucleon couplings ( ) EDMs of nuclei and ions (deuteron, etc) Nucleon EDMs (n,p) atomic EDMs of paramagnetic atoms and molecules (Tl,YbF, ThO, HfF +,...) Atoms in traps (Rb,Cs,Fr) solid state EDMs of diamagnetic atoms (Hg,Xe,Ra,Rn,...) 34

35 Resulting Bounds on fermion EDMs & CEDMs ThO EDM YbF EDM Tl EDM [±20%] d e + e(26 MeV) 2 d e + e(21 MeV) 2 3 C ed m d + 11 C es m s +5 C eb m b d e + e(26 MeV) 2 3 C ed m d + 11 C es m s +5 C eb < e cm m b < ecm 3 C ed m d + 11 C es m s +5 C eb m b < e cm n EDM [±50%?] Hg EDM [±O(few)?] e d d du + O(d e, d s,c qq,c qe ) < e cm Generic scaling: See also recent compilation of limits: [Engel, Ramsey-Musolf, van Kolck 13 ] 35

36 Concluding Remarks EDM computations require a multi-scale approach: d exp (C atomic (C nuclear (C QCD (C new physics )))) Reviewed the sum rules approach to QCD-scale calculations of CP-odd nucleon matrix elements nucleon correlators illustrate new sources of mixing in CPviolating backgrounds improving precision is hard without further input on (i) excited state mixing, and (ii) interpolating current ambiguity Examples that can benefit from lattice input: d N (hn qf 5 q Ni) via tensor charges d N (hn G G Ni),d N (hn qg 5 q Ni), ḡ NN (hn qg s G q m 2 0 qq Ni) s-quark matrix elements CP-even, relevant for all nuclear EDMs [ see also Emanuele s talk]

37 Extra slides 37

38 Further operators Weinberg operator: d n µ n N O CP N m n Ni 5 N ḡ (1) NN =ḡ(1) NN (w) 3g s m 2 0 µ n 32 2 w ln(m 2 /µ 2 IR) e GeVw(1 GeV) [Demir, Pospelov, AR 02] suppressed by light quark masses 4-quark (factorizable) operators: d n (C qq ) (few) 10 2 GeV C qq [Khatsimovsky et al 88; Hamzaoui & Pospelov 99; An, Ji & Xu 09] ḡ (1) NN (C ij) =C ij q i q i 2f N q j q j N via PCAC, vacuum saturation [Demir et al 03] 38