Beyond Schiff: Atomic EDMs from two-photon exchange Satoru Inoue (work with Wick Haxton & Michael Ramsey-Musolf) ACFI Workshop, November 6, 2014

Transcription

1 Beyond Schiff: Atomic EDMs from two-photon exchange Satoru Inoue (work with Wick Haxton & Michael Ramsey-Musolf) ACFI Workshop, November 6, 2014

2 Outline Very short review of Schiff theorem Multipole expansion as an expansion in powers of R N /R A 2-photon exchange appears to give larger atomic EDM than Schiff moment by 2 orders of magnitude - first, naive derivation Full derivation - the answer is essentially the same Relativistic enhancement is larger for Schiff moment (in progress)

3 Schiff theorem Derivation relies on 3 assumptions: 1. Constituent particles are point-like 2. Non-relativistic dynamics 3. Only electrostatic interactions

4 Schiff theorem Derivation relies on 3 assumptions: Loopholes 1. Constituent particles are point-like Nuclear size - Schiff moment 2. Non-relativistic dynamics Relativistic electrons - paramagnetic systems 3. Only electrostatic interactions Electromagnetic currents - topic of this talk

5 Natural size of interactions Schiff theorem is the cancellation between these diagrams C 1 C 1 C 1 C 1 indicating that the photon couples to the nuclear EDM

6 Natural size of interactions Schiff theorem is the cancellation between these diagrams C 1 C 1 C 1 C 1 indicating that the photon couples to the nuclear EDM Electron-nucleus interaction here is V C1 = 4 Z d 3 x e(~x ) 3 x 2 Y 1 (ˆx) = 4 3 R N R 2 A C A 1 C N 1 Z d 3 yy N (~y )Y 1 (ŷ) lengths can be divided out

7 Natural size of interactions This comes from multipole expansion of Coulomb potential ZZ V Coul = d 3 xd 3 y e(~x ) N (~y ) ~x ~y Electron is usually outside the nucleus (x > y) 1 ~x ~y = X 4 y l xl (x y)+ (y x) 2l +1 xl+1 yl+1 lm = X apple 4 y l x l 2l +1 x l+1 + y l y l+1 x l+1 (y lm Green terms survive in limit of point-like nucleus Y lm(ˆx)y lm (ŷ) x) Y lm(ˆx)y lm (ŷ) l-th multipole interaction goes as 4 R A RN R A l

8 Natural size of interactions This comes from multipole expansion of Coulomb potential ZZ V Coul = d 3 xd 3 y e(~x ) N (~y ) ~x ~y Electron is usually outside the nucleus (x > y) 1 ~x ~y = X 4 y l xl (x y)+ (y x) 2l +1 xl+1 yl+1 lm = X apple 4 y l x l 2l +1 x l+1 + y l y l+1 x l+1 (y lm Penetration terms are the effects of nonzero nuclear size Y lm(ˆx)y lm (ŷ) x) Y lm(ˆx)y lm (ŷ) They are suppressed by nuclear vs atomic volume RN R A 3

9 Natural size of interactions Schiff moment contribution comes from penetration terms Schiff Schiff Schiff moment electron-nucleus potential looks like V Sch = 4 " 3 RN X RA 4 r 3 i (~x i )+ 3 ( ~x i ) # r i S ~ R A R A ~S i - nuclear Schiff moment operator with lengths divided out Volume factor - RN R A 2 times smaller than EDM potential

10 Breit interaction 1-γ exchange at LO in non-relativistic expansion ZZ " # V Breit = d 3 xd 3 e N 1 ~j e ~j N +(~j e ˆn)(~j N ˆn) y ~x ~y 2 ~x ~y Current-current interaction have their own multipoles Transverse magnetic - magnetic dipole, quadrupole, etc. Z l y h i Mlm N = d 3 y Y l (ŷ) ~j N (~y ) note the powers of y lm R N Transverse electric - E N lm = i p (l + 1)(2l + 1) Z d 3 y y R N l 1 h Y l 1 (ŷ) ~j N (~y ) i lm

11 Multipole interactions Dividing out length scales as we did for charge multipoles, we find the following natural sizes for multipole interactions V Cl,V Ml V El V Sch 4 RN R A RN 4 R A 4 R A R A l R A l 1 RN R A 3 Can we find an interaction that gives a larger EDM than Schiff? We need a PVTV effect, and as little suppression as possible

12 Symmetries PCTC PVTC PCTV PVTV C even odd M odd even E odd even MQMs violate P and T, but?? we have J=0 electronic ground state, so we can t use MQM Schiff really is the leading contribution for 1-γ exchange

13 Breit iterated PCTC PVTC PCTV PVTV C even odd M odd even E odd even We can go to 2-γ exchange E 2 E 1 + Combining E1 and E2 multipoles can give PVTV effect They can also be recoupled to total J=1 (E1-M1 also possible)

14 Siegert s theorem Transverse electric multipoles can be written using a gradient r Z " l l +1 E lm = ir N d 3 y r l ~ y Y lm# ~j N R N Partial integration makes ~r ~j N appear, which is equal to Final result is d N /dt E lm = by current conservation r l +1 R N [C N l lm,h N ] where H N is the nuclear Hamiltonian. This shows that these multipoles have no diagonal M.E. s

15 Transverse electric multipoles Transverse electric multipoles have no static nuclear moments E 2 E 1 nuclear excitation In order to compare the effects of E1-E2 combination with Schiff, we should compare using a nuclear energy denominator V e E1 E2 V E1 1 E 0 E n V E2

16 Naive comparison So the E1-E2 effective interaction has size V e E1 E2 V E1 1 E 0 E n V E2 4 R A 1 4 R N E N R A R A =(4 ) 2 R N E N R 3 A Compare this with Schiff moment interaction V Sch (4 ) R3 N R 4 A VE1 e E2 4 V Sch E N R N R A R N E1-E2 is enhanced compared to Schiff moment

17 Problems with iterating Breit 1. Crossed diagram is ignored If electrons are relativistic, this is not small 2. Derivation of Breit interaction takes the step i k 2 = Not correct for inelastic scattering, which we have k 2 0 i ~ k 2 i ~ k 2

18 Time-ordered perturbation theory Old-fashioned perturbation theory, solves BOTH problems 1. Start with a normal Feynman amplitude 2. TOPT rewrites it as sum over all time-orderings of vertices i 3. Propagators turn into energy denominators k 2 Denominators show which time-orderings are important Also allows direct connection to non-relativistic calculation (derivation in Sterman s QFT textbook)

19 Time-ordered perturbation theory Trivial example: scalar propagator p 1 p 2 k where q ~k = ~ k2 + m 2 i k 2 m 2 + i = 1 p 1 p 2 2 ~k = 1 2 ~k i i k 0 ~k + i k 0 + ~k i apple i p 0 1 (p ~k )+i + i p 0 2 (p ~k )+i

20 Time-ordered perturbation theory Trivial example: scalar propagator p 1 p 2 k i k 2 m 2 + i = 1 p 1 p 2 2 ~k = 1 2 ~k Those are energy denominators i i k 0 ~k + i k 0 + ~k i apple i p 0 1 (p ~k )+i + i p 0 2 (p ~k )+i corresponding to the 2 possible time orderings of vertices

21 2-photon exchange in TOPT There are 4 orderings each for box & crossed diagrams Some orderings are highly suppressed by particle mass by nuclear excitation We want 1. massive particles traveling forward in time 2. only 1 intermediate state with nuclear excitations

22 2-photon exchange in TOPT Breit interaction comes from Iterating Breit corresponds to assuming these 4 are leading Only 1 of these is leading (1 nuclear excited state) Adjust answer by factor of 1/4

23 2-photon exchange in TOPT There are actually 6 leading diagrams - top middle is in Breit Left and right diagrams sum to middle ones (accident?) Crossed diagrams double the result - get back factor of 2x2 = 4

24 What did we learn? This shows that the naive estimate earlier is essentially correct, despite the incorrect use of the Breit interaction But we do want to make a better comparison to Schiff result 1. Use Siegert s theorem and commutator trick for nuclear part 2. Relativistic enhancement near the origin

25 Full expression The E1-E2 interaction is V E1-E2 (4 )2 15 p 20 X n R N R 3 A X i This is to be compared with the Schiff moment interaction V Sch = 4 " 3 RN X RA 4 r 3 i (~x i )+ 3 ( ~x i ) # S ~ r i R A R A RN 3 i i RA Nuclear part and electronic part can be manipulated further x i 3 Y 1 (ˆx i ) 1 E 0 E n [h0 E N 1 ni hn E N 2 0i] 1 +[E N 1 $ E N 2 ]

26 Back to Siegert s theorem Use Siegert s theorem to rewrite transverse multipoles as commutator of charge multipoles and nuclear Hamiltonian X 1 [h0 E1m N E n n E 1 nihn E2m N 2 0i +(E1 N $ E2 N )] 0 = p X 3RN 2 (E n E 0 )[h0 C1m N 1 nihn C2m N 2 0i +(C1 N $ C2 N )] n = p 3R 2 N h0 C N 1m 1,H N,C N 2m2 0i Closure sum eliminates nuclear intermediate states

27 Commutator C1 and C2 can be written using nucleon coordinates; assuming 2-body interactions are momentum-independent, X h0 C N 1m 1,H N,C N 2m2 0ih10 1m1 2m 2 i m 1,m 2 = 5p 6 4 ~ 2 m N R 2 N h0 d N,z R N 0i This interaction couples to the nuclear EDM, not the Schiff moment

28 EDM & Schiff comparison TABLE I. Electrodipole, Schiff, and magnetic-quadrupole moments of nuclei. The parameter 71 is the coefficient in the T- and P-odd interaction Hamiltonian (19). The value given in the table for the neutron was obtained from 16b) by dividing by a). spherical nuclei deformed light Sushkov, Flambaum, Khriplovich 1984 Nuclear EDM not as well calculated as Schiff moment Note the units: d ~ 10 8 η e fm, S ~ 10 8 η e fm 3

29 Atomic enhancement Electron wavefunctions near the origin are important Solving Dirac equation with Coulomb potential of nucleus, f u njlm (~r )= njl (r) jlm (ˆr) g njl (r)( i~ ˆr) jlm (ˆr) with f,g N Z1/2 R 3/2 A 2Zr R A 1 p (j +1/2) 2 Z Note the Z 1/2 enhancement in normalization, and characteristic length becoming R A /Z

30 Atomic enhancement For Schiff moment, hs 1/2 r ~ 3 (~x ) p 1/2 i Z = d 3 x 3 (~x )(f s (x)fp(x)+g 0 s(x)g p(x))h 0 1/2,0 ˆx 1/2,1 i Z N d 3 x 3 (~x )x 2 2 h 1/2,0 ˆx 1/2,1 i This is a negative power - blows up at the origin Integral must be cut off at x = R N. Result is ~10 5 enhancement hs 1/2 ~ r 3 (~x ) p 1/2 i Z 2 R R A ( s p ) 3/2 h 1/2,0 ˆx 1/2,1 i R 10 for large Z

31 Atomic enhancement For E1-E2, ˆx hs 1/2 x 3 p 1/2i Z d 3 x = x 3 (f s (x)fp(x) 0 g s(x)g p(x))h 0 1/2,0 ˆx 1/2,1 i Z N dxx 2 3 h 1/2,0 ˆx 1/2,1 i = N[x 2 2 ] R max 0 Same negative power, same cutoff at x = R N. This time, there s only 1 power of Z ˆx hs 1/2 x 3 p 1/2i ZR (Z ) 2 ( s p ) 3/2 2 2 h 1/2,0 ˆx 1/2,1 i

32 Final comparison (preliminary) Before incorporating atomic enhancement, I had 2 overall factors 4 10R A RN R A 3 (4 ) 2 p 3 15 p 20 for Schiff moment, E N R 3 N R 3 A for E1-E2, with E1-E2 being larger by ratio of ~135. Atomic enhancement is larger for Schiff by ~Z. E1-E2 appears to give comparable atomic EDM as Schiff moment

33 Conclusions R N /R A counting is a way to estimate the sizes of EDM contributions 2-γ exchange between electrons and the nucleus allows transverse electric multipoles to generate atomic EDM, with less R N /R A suppression than the Schiff moment term E1-E2 combination results in a coupling to the nuclear EDM, rather than the nuclear Schiff moment Despite the smaller relativistic enhancement, the new contribution appears comparable to Schiff. Nuclear EDM calculations are needed

34 Backup slides

35 Approximate calculation (old)

36 Giant dipole approximation This expression resembles X (E n E 0 ) h0 C 1 ni 2 X (E n E 0 )h0 C 1 nihn C 2 0i n6=0 n6=0 which is known to be dominated by the giant dipole resonance. We make the ansatz that our expression is also GDR dominated X (E n E 0 )h0 C 1 nihn C 2 0i n6=0 E GDR h0 C 1 C 2 0i

37 Final expression With these approximations, our final expression is VE1 e E2 p 3 (4 ) 2 15 p 20 ZX RA i=1 E GDR R 3 N R 3 A We have a numerical factor, an electronic multipole, and x i 3 Y 1 (ˆx i ) C N 1,C N 2 1 a nuclear multipole, which is charge dipole and quadrupole recoupled to J=1.

38 Toy model calculation We compare this nuclear moment with Schiff in a toy model Shell model : nucleons of 15 N (J=1/2) in HO potential In order to get PVTV nuclear moments, we insert V CPV G p 2 2m ~ ~r core (~y) as a perturbation to the HO potential, so that h 0 Ô 0i = X n6=0 h0 Ô nihn V CPV 0i E 0 E n +c.c. is the final result.

39 Toy model calculation Schiff moment has overall 1/10 in the definition ~S = 1 Z d 3 y 2 ~y 5 y 2 dn ~ y 10 RN 3 3Z RN 3 N (~y ) so compare with 10 times S. Operators h0kôk0i (normalized) 10 S C1 N,C2 N Nuclear moments are the same order, as expected from the R N /R A argument.

40 Final comparison Comparing with V Sch = 4 10R A VE1 e E2 p 3 (4 ) 2 15 p 20 ZX RA RN R A 3 " R 4 A i=1 X i x i 3 Y 1 (ˆx i ) E GDR R 3 N R 3 A C N 1,C N 2 1 r i 3 (~x i )+ 3 ( ~x i ) r i # (10 ~ S) the ratio of the numerical factors is 4 p 15 E GDR R A 135 taking reasonable values of E GDR = 20 MeV, R A = m

41 Multipoles

42 Multipoles Full expansion of Breit interaction V Cl = V Ml = V El = V E 0 l = 4 (2l + 1)R A 4 (2l + 1)R A 4 (2l + 1)R A 4 (2l + 1)R A RN RN R A RN R A RN R A R A l C A l l M A l l 1 E 0 A l l+1 E A l C N l M N l E N l E 0 N l There are 2 kinds of transverse electric terms, but one is larger by R N /R A counting

43 Multipoles List of electronic multipole operators Z l+1 Clm A d 3 RA x e (~x )Y lm(ˆx) x Z l+1 Mlm A d 3 RA x ~Y l x lm(ˆx) ~j e (~x ) Z apple 1 Elm A R l+2 A d 3 x ~r x ~ l+1 Ylm(ˆx) l ~j e (~x ) Z apple E 0 A lm RA l d 3 1 x ~r 2(2l 1)x ~ l 1 Ylm(ˆx) l E is the relevant transverse electric operator ~j e (~x )

44 Multipoles List of nuclear multipole operators Z l y Clm N d 3 y N (~y )Y lm(ŷ) M N lm Z E N lm 1 R l 1 N E 0 N lm 1 R l+1 N R N y d 3 y Z d 3 y Z R N d 3 y l ~Y l lm(ŷ) ~j N (~y ) h i ~r y l Y ~ l lm (ŷ) apple ~r E is the relevant transverse electric operator ~j N (~y ) y l+2 Y 2(2l + 3) ~ lm(ŷ) l ~j N (~y )

45 Time reversal

46 Time reversal Multipoles are either parity even or odd even: total charge (monopole), magnetic dipole, etc. odd: EDM, magnetic quadrupole, etc. Properties of operators under time reversal is more subtle, especially when we consider products of operators.

47 Time reversal The starting point is the relation ht T i = h i Notice that the bra and ket get reversed. Consider EDM in T conservation limit: hjj C N 10 jji = TC N 10 jji (T jji) = C N 10i 2j j, ji (i 2j j, ji) =hj, j C N 10 j, ji Wigner-Eckart relates these by a minus sign -> no EDM Another way to say this: C N 1 does not behave like spin under T

48 Time reversal For J=1 operators, TV moments come from operators that behave in the same way under T and complex conjugation Ô T ÔT 1 Ô C N 1m ( 1) m C N 1, m ( 1) m C N 1, m M N 1m ( 1) m M N 1, m ( 1) m+1 M N 1, m E N 1m ( 1) m E N 1, m ( 1) m+1 E N 1, m {E 1,E 2 } N 1m ( 1) m {E 1,E 2 } N 1, m ( 1) m {E 1,E 2 } N 1, m [E 1,E 2 ] N 1m ( 1) m [E 1,E 2 ] N 1, m ( 1) m+1 [E 1,E 2 ] N 1, m {E 1,M 1 } N 1m ( 1) m+1 {E 1,M 1 } N 1, m ( 1) m+1 {E 1,M 1 } N 1, m [E 1,M 1 ] N 1m ( 1) m+1 [E 1,M 1 ] N 1, m ( 1) m [E 1,M 1 ] N 1, m

49 Time reversal Multipoles are either parity even or odd even: total charge (monopole), magnetic dipole, etc. odd: EDM, magnetic quadrupole, etc. Properties of operators under time reversal is more subtle, especially when we consider products of operators.

50 Time-ordered perturbation theory

51 Time-ordered perturbation theory Steps to the derivation: 1. Write the energy-conserving delta functions at each vertex as (p 0 in p 0 out) = Z 1 1 dt e i(p0 in p0 out )t If an internal line, k, starts at vertex with time t i and ends at vertex with time t j, then there is a factor e ip0 k (t j t i )

52 Time-ordered perturbation theory 2. This factor, e ip0 k (t j t i ), lets us do the integration over p 0 k by closing the contour at infinity. Contour integral picks up a pole from the propagator of this line, and the pole corresponds to the on-shell energy of the particle. Result: Z d 4 p (2 ) 4 Any function of k 0 in the numerator is evaluated Z d 3 p (2 ) 3 2 ~k with on-shell energy, since that gives the residue

53 Time-ordered perturbation theory 3. We still have time integrals that we introduced at the start. The integrand is now e i(e in E out )t with on-shell energies. But whether the particle is in or out depends on the time ordering of the vertices. Split up the integral into regions, which correspond to orderings. Then we can perform the integrals fully. Result: an energy denominator for each intermediate state

54 Time-ordered perturbation theory Why is it sufficient to consider Justification comes from TOPT

55 Time-ordered perturbation theory These have a pure atomic excitation as the last intermediate state Compare with

56 Time-ordered perturbation theory for example This diagram has massive particles always traveling forward, and only 1 nuclear excitation. But atomic excited state always appears with a photon, which makes the denominator not so small