Parity and Time Reversal Violations in Atoms: Present Status and Future Prospects. Bhanu Pratap Das

Transcription

1 Parity and Time Reversal Violations in Atoms: Present Status and Future Prospects Bhanu Pratap Das Non-Accelerator Particle Physics Group Indian Institute of Astrophysics Bangalore , India

2 Outline of the talk Introduction to Atomic EDMs Theory of Atomic EDMs Measurement of Atomic EDMs Atomic EDM results and future improvements Conclusions

3 D = c J D = 0 Permanent EDM of a particle VIOLATES both P - & T invariance. T-violation implies CP-violation via CPT theorem.

4 EDM and Degeneracy : Consider the degeneracy of opposite parity states in a physical system = a e b o D = e r 0 EDM can be nonzero for degenerate states. Examples : EDMs of Water, Ammonia etc. P and T violations in non-degenerate systems implies nonzero EDM.

5 Sources of Atomic EDM Elementary Coupling Particles Nucleon Nucleus constant Atomic d e d e D a (open shell) C s D a (open shell) e-q e-n e-n C T D a (closed shell) d q d n d N Q D a (closed shell) q-q d n, n-n d N Q D a (closed shell)

6 P and T Violating Interactions in Atoms 1. Atomic EDM from Electron EDM H PTV = d e E I (Non-relativistic) H PTV = d e E I (Relativistic) : ( Sandars 1968) E I = { i V N r i i j V C r ij } H PTV can be expressed as an effective one-body Hamiltonian H PTV = j 2i c d e 5 p j 2 (Das 1988) 2. Atomic EDM from e N scalar-pseudoscalar (S-PS) interaction H PTV = G F 2 J S J PS H PTV = i G F C S A 2 i 5 N r i C s is the e N S-PS coupling constant

7 3. Atomic EDM from e N Tensor-Pseduotensor (T-PT) interaction H PTV = i 2 G F C T i I N r i C T is the e N T-PT coupling constant 4. Atomic EDM from P and T violating interaction from the nucleus Total charge density of the nucleus 0 r Normal nuclear charge density; r = 0 r r r Correction due to P- and T-violating interactions EDM of the Nucleus D N = e r N r d 3 r P- and T-violating Nuclear electrostatic potential Interaction of an electron with the above potential Q : Schiff Moment ; depends on R = 4 Q R H PTV = 4 e Q R 0, and related quantities

8 P and T Violating Coupling Constants All the P- and T-violating interactions scale as Z 3 or Z 2. The atomic EDM (D a ) experiments are therefore done on heavy atoms (Cs, Tl, Hg, etc). The measured value of D a in combination with the calculated value of D a /C PTV will give the C PTV. C PTV is the P- and T-violating coupling constant : d e, C S, C T and Q Atomic experiment and theory are BOTH needed to extract the above coupling constants

9 EXPERIMENTS ON ATOMIC EDM... Principle of Measurement H I = D a E B B E B E 1 = 2 B 2 D a E ħ 2 = 2 B 2 D a E ħ 1 2 = 1 2 = 4 D a E ħ If the atomic EDM D a ~ e-cm and E = 10 5 V/cm; ~ 10-5 Hz Major source of error: B m = v E c 2

10 Theory of Atomic EDMs The relativistic atomic Hamiltonian is, H a = i {c i p i i m c 2 V N r i } i j e 2 r ij Treating H EDM as a first-order perturbation, the atomic wave function is given by = 0 C PTV 1 The atomic EDM is given by D a = D D a C PTV = 0 D 1 1 D 0 This ratio is calculated by relativistic atomic many-body theory Unique many-body problem involving the interplay of the long range Coulomb interaction and short range P- and T-violating interactions. The accuracy depends on the precision to which 0 and 1 are calculated.

11 Relativistic Wavefunctions of Atoms Atoms of interest for EDM studies are relativistic many-body systems; Wavefunctions of these atoms can be written in the mean field approximation as 0 = Det { 1 2 N } (Relativistic Dirac-Fock wavefunction) T 1 = a, p 0 T 1 0 T 2 0 t ap a p a a T 2 = a, b, p,q t abpq a p a q a b a a Relativistic Coupled-cluster (CC) wavefunction; 0 = exp T 0 T = T 1 T 2 CC wfn. has electron correlation to all-orders of perturbation theory for any level of excitation. In presence of EDM, = 0 C PTV 1 = exp {T C PTV T 1 } 0 First-order EDM Perturbed RCC wfn. satisfies : H 0 E 0 1 = H PTV 0

12 ATOMIC EDM RESULTS D a / C PTV Has been calculated by semi-empirical and ab initio methods. The calculation of this quantity by RCC theory. D a /d e : Liu & Kelly Phys. Rev. A Rap. Comm D a /d e, D a /C s : Nataraj et al. Phys. Rev. Lett D a /C s : Sahoo et al. Phys. Rev. A Rap. Comm D a /C T, D a /Q : Latha et al. (Unpublished) (Da/de) Atom Our Work Others Rubidium (Shukla et al 1994) Cesium (Das 1988), 114 (Hartley et al 1990) Thallium - 395* (Liu & Kelly 1992) The best limit for de is from Thallium EDM experiment (Regan et al. Phys. Rev. Lett ) and theory (Liu and Kelly) d e e cm * Our work is in progress to improve the above limit.

13 (Da/C s ) in units of 10^{-18} e-cm Atom Our Work Others Rubidium ± (Shukla et al 1994) Cesium ± ± 0.10 (Martensson & Lindroth 1991) (Venugopal 1990) Thallium ± ± 2 (Martensson & Lindroth 1991) The current best limit for C T is by combining our calculated Da/C T for Hg (Latha et al. 2008) and Hg EDM experiment (Romalis et al. Phys. Rev. Lett. 2001) at 95% C.L e cm Da/Q has been calculated by Dzuba et al. Phys. Rev. A 2002 by using a hybrid method involving relativistic configuration interaction and relativistic many-body perturbation theory. Combining this with Hg EDM experiment : Q e fm 3

14 Limits for Hadronic T-violating coupling constants The above limits have been obtained by using the limit for the Schiff moment (Q) from atomic physics and combining with nuclear structure and QCD calculations.

15 Experiments : Ongoing Atomic EDM Experiments and Theory Rb: Cs: Fr: Ra*: Hg: Xe: Yb: Rn: Ra: Weiss, Penn State Gould, LBNL ; Heinzen, UT, Austin; Weiss, Penn State Sakemi, Tohoku Jungmann, KVI, Netherlands Fortson, UW, Seattle Romalis, Princeton Takahashi, Univ. of Kyoto, Natarajan, IISc, Bangalore Chupp, Univ. of Michigan Holt and Lu, ANL Theory : Flambaum, UNSW, Sydney ; Das, IIA, Bangalore; Angom, PRL, Ahmedabad Improved accuracies in experiments and relativistic many-body theory for Rb, Cs and Fr could give new limits for d e and C s New limits for C T and Q are expected from Hg in the near future.

16 Molecular EDMs The electron EDM enhancement factors in certain molecules can be several orders of magnitude larger than those in atoms. Some of the current molecular EDM experiments that are underway are : YbF : Hinds, Imperial College, London PbO * : DeMille, Yale University HfF + : Cornell, JILA, Colorado The sensitivities of these experiments could be 2-3 orders of magnitude better than that of the current limit from Tl. Molecular EDM calculations are currently in their infancy : Petrov et al. Phys. Rev. A 72, (2005) Meyer et al. Phys. Rev. A 73, (2006) Nayak et al. Phys. Rev. A 75, (2007)

17 Limits on d e : Past, Present and Future 2010 Rb, Cs, Fr, YbF, PbO*, HfF +

18 Conclusions Atomic EDMs could serve as excellent probes of physics beyond the standard model. Atoms are a rich source of T (or CP) violation : Can provide information on leptonic, semi-leptonic and hadronic T violations. The current best atomic EDM limits come from Tl (d e, C s ) and Hg (C T, Q) Several Atomic ( Hg, Rb, Cs etc. ) and Molecular ( YbF, PbO*, HfF +, etc ) EDM experiments are underway. Results of some of these experiments could in combination with relativistic many-body calculations give improve the limits for the T violating coupling constants.

19 Co-Workers: H. S. Nataraj (IIAP, Bangalore, India) Bijaya Kumar Sahoo (KVI, Groningen, Netherlands) Rajat Kumar Chaudhuri (IIAP, Bangalore, India) Debashis Mukherjee (IACS, Kolkata, India) K. V. P. Latha (NCTS, Taiwan) Dilip Angom (PRL, Ahmedabad, India) Malaya Nayak (BARC, Mumbai, India)

20 Aside

21 METHOD OF CALCULATION... Dirac - Fock Theory For a relativistic N-particle system, we have a Dirac-Fock equation given by, H 0 = I {c I p I I 1 m c 2 V N r I } I J We represent the ground state wave function as an N N Slater determinant, x1 1 x2 1 x3 1 x N 0 = 1 2 x 1 2 x 2 2 x 3 2 x N N! N x 1 N x 2 N x 3 N x N 1 The single particle wave functions s expressed in Dirac form as, a = 1 r P a r a, m a iq a r a,m a e 2 r IJ

22 ... Coupled Cluster Theory The coupled cluster wave function for a closed shell atom is given by, 0 = e T 0 0 Since the system considered here has only one valence electron, it reduces to v = e T 0 {1 S 0 } v Where, T 0 = T 1 0 T 2 0 and S 0 = S 1 0 S 2 0 The RCC operator amplitudes can be solved in two steps; first we solve for closed shell amplitudes using the following equations: 0 H 0 0 = E and g 0 H 0 0 = 0 Where, H 0 0 T = e H 0 e T 0

23 The open shell operators can be obtained by solving the following two equations : v H op {1 S v 0 } v = E v v H op {1 S v 0 } v = E v v {S v 0 } v Where, E v is the negative of the ionization potential of the valence electron v. The total atomic Hamiltonian in the presence of EDM as a perturbation is given by, v = e T 0 d e T 1 {1 S 0 d e S 1 } v The effective ( one-body ) perturbed EDM operator is given by, v H op {1 S v 0 } v = E v Thus, the modified atomic wave function is given by, v = e T 0 d e T 1 {1 S 0 d e S 1 } v

24 The perturbed cluster amplitudes can be obtained by solving the following equations self consistently : 0 H 0 N T 1 eff H EDM 0 = 0 v H 0 N E v S 1 v H 0 N T 1 eff H EDM {1 S v 0 } v = 0 Where, H N = H 0 0 H 0 0 The atomic EDM is given by, D a = v D a v v v

25 Particle Physics Model Electron EDM (e-cm) Standard Model < Super-symmetric Model Left-Right Symmetric Model Multi-Higgs Model

26 ATOMIC EDM DUE TO THE ELECTRON EDM ( NON-RELATIVISTIC CASE ) The interaction between the electron spin and internal electric field exerted by the nucleus and the other electrons gives, d e E I where, E I = { i V N r i i j V C r ij } The total atomic Hamiltonian is then, H = i 2 { p i 2m Z e r i } i j e 2 r ij d e i i E i I Using perturbation theory, we can write, where, H O = i 2 { p i 2m Z e r i } i j e 2 r ij ; H / = d e i H = H O H / i E i I ; H O O = E O O

27 When there is an external electric field, induced electric dipole moment arises. Total electric dipole moment operator of an atom is then given by, The atomic EDM is, Using perturbation theory, e r D a = i {d e i e r i } D a = D a = O d e 1 d e 2 2 As d e is small, d e term can be neglected.

28 Assume, the applied field is in the z direction. d e z parity, where as, e z term is odd under parity. is even under and O 1 are of opposite parity, then the non-vanishing terms of the EDM are: D a = d e O i zi O d e e{ O i z i 1 1 i z i 0 } D O D 1 From the Time-independent Non-degenerate perturbation theory, we have, 1 = O I O H / O I I E O O E I H / = d e E i

29 D a = D O D 1 D 1 = d e O i zi O D O = d e O i z i O D a = 0 ( Sandars 1968 ) Hence, in the non-relativistic scenario, even though the electron is assumed to have a tiny EDM, when all the interactions in the atom are considered, the total atomic EDM becomes zero.

30 ATOMIC EDM DUE TO THE ELECTRON EDM ( RELATIVISTIC CASE ) The total atomic Hamiltonian, including intrinsic electron EDM is, H = i {c i p i i 1 mc 2 Z e H 0 r i } i j e 2 r ij d e i i i E i I H / The expectation value of atomic EDM in the presence of applied electric field is given by, D a = d e O i i zi O d e e{ O i z i 1 1 i z i 0 } D O D 1

31 Finally, the expression for Atomic EDM reduces to, D a = 4c d e ħ I O z I O I O i 5 p 2 O E O E I O D a 0 Hence, in the relativistic scenario, the total atomic EDM is non-zero. ( Sandars 1968 )