H. S. Nataraj · 2008-07-28 · Nataraj et al. Phys. Rev. Lett. 101, 033002 (2008) Most dominant...

24 July 2008, UW Seattle INT082A H. S. Nataraj

Applications of relativistic coupledcluster theory to Applications of relativistic coupledcluster theory to P & Tviolating interactions in atomic systems P & Tviolating interactions in atomic systems

H. S. Nataraj (with Bijaya Kumar Sahoo)

Non-Accelerator Particle Physics Group Indian Institute of Astrophysics

Bangalore – 560034, India


➢ INTRODUCTION TO ATOMIC EDM

➢ PRINCIPLE OF ATOMIC EDM EXPERIMENTS

➢ METHOD OF CALCULATION

➢ RESULTS AND DISCUSSIONS

➢ IMPLICATIONS FOR NEW PHYSICS

OUTLINE OF THE TALKOUTLINE OF THE TALK


INTRODUCTIONINTRODUCTION

➢ The Electric Dipole Moment for a pair of equal and opposite charges is defined as the magnitude of the charge times the distance between them.

➢ The electron being a point charge can have intrinsic EDM, as predicted by

many particle physics models, as can be understood from field theory, it is surrounded by a virtual cloud of particle-antiparticle pairs.

➢ For an electron, the EDM is given by, and it aligns either parallel or anti-parallel to the total angular momentum of the atom.

➢ The non-degerate atomic system will have a non-vanishing EDM only when the relativistic effects are considered (Sandars 1965).

➢ The atomic EDM arising from the e-EDM will scale as,

dr

d = q r

Da d

e

2 Z 3

de= d

e


Permanent EDM of a particle VIOLATES both P - & T - invariance.

d

d

J

J

⟨E , j , m∣d ∣E , j , m⟩ = cE , J⟨E , j , m ∣J ∣E , j , m⟩

⇒ d = 0

Naive Demonstration of P & T- Violation


Sources of Atomic EDMs

Elementary

Particles Nucleon Nucleus Atomic EDM

de (Cs, Tl, Fr)

e – e Da (paramagnetic)

e - q e - n e - N

dq dn dN Da (diamagnetic)

q - q dn, n - n (Xn, Hg, Ra)

In the present talk our interest is limited to the e-EDM (de) contribution to atomic EDM (Da).


The Standard Model prediction of de is ~10-12 orders of magnitude smaller than that predicted by many supersymmetric models which are in the reach of current experiments.

e- EDM: Various Predictions


EXPERIMENTS ON ATOMIC EDMEXPERIMENTS ON ATOMIC EDM

. . . Principle of Measurement

2

EB1 =

−2⋅B − 2 Da⋅E

ℏ

2 =−2⋅B 2 D

a⋅E

ℏ

= 2 −1 =4 D

a⋅Eℏ

HI= − D

a⋅E − ⋅B

EB

1

If the atomic EDM, andDa≈ 10−26 e cm E = 105V /m ; ≈ 10−7 Hz


A Long Journey . . .

( Courtesy: Yasuhiro Sakemi, RCNL, Japan )


Future e-EDM Experiments

Cesium Fountain EDM Experiment

( Harvey Gould et al., Lawrence Berkely National Laboratory, California, USA )

EDM Experiment of Cs and Rb in 1-D Optical Lattice

( David Weiss et al., Penn. State University, USA )

Cesium EDM Experiment using Optical Dipole Force Traps

( Heinzen et al., University of Texas, Austin, USA )

Francium EDM Experiment using Optical Dipole Force Traps

( Yasuhiro Sakemi et al., RCNL, Osaka University, Japan )

Projected accuracy de 10−28 e cm

Projected accuracy de≈ 5 × 10−30 e cm

Projected accuracy de≈ 1× 10−29 e cm

Projected accuracy de≈ 10−29 e cm


How do we get e-EDM ?

Daexpt

Da /d eth

de

Measured Computed

We need high precision both in the measurement and in the theory to

obtain a reliable limit on the electron EDM.


. . . Dirac - Fock Theory

For a relativistic N-particle system, we have a Dirac-Fock equation given by,

H 0=∑i

{c i⋅pi i−1 m c2 V N ri U HF r i}

The single particle wave functions ’s can be expressed in Dirac form as,

0=

1

N ! ∣

1x

1

1x

2

1x

3 ⋯

1x

N

2 x

1

2x

2

2x

3 ⋯

2x

N

⋯ ⋯ ⋯ ⋯ ⋯

Nx

1

Nx

2

Nx

3 ⋯

Nx

N∣

We represent the ground state wave function as an N×N Slater determinant,

a=

1

r Par

a,m

a

iQar

−a, m

a

METHOD OF CALCULATION METHOD OF CALCULATION

V res r ij=∑i j

V ri j−∑i

U HF riResidual Coulomb interaction, is considered as perturbation.

H 0 ∣0 ⟩ = E0 ∣0 ⟩

where,


∣0 ⟩ = eT 0 ∣ 0 ⟩

∣v⟩ = eT 0

{1Sv

0 }∣

v⟩

. . . Coupled Cluster Theory

The coupled cluster wave function for a closed-shell atom is given by,

For the open-shell paramagnetic atoms with a valence electron, it reduces to,

T 0= T 10 T 2

0 S v0= S v1

0 S v2

0For CCSD, and

The RCC operator amplitudes can be solved in two steps; first we solve for closed-

shell amplitudes using the following equations:

H 0 = e−T 0 H 0 eT 0 ⟨0 ∣ H 0 ∣0 ⟩ = Eg

⟨0∗ ∣ H 0 ∣0 ⟩ = 0and Where,


The total atomic Hamiltonian in the presence of EDM as a perturbation is given by,

The effective ( one-body ) perturbed EDM operator is given by,

Thus, the modified atomic wave function is given by,

⟨v∣ H

op{1S

v0 }∣

v⟩ = −E

v

⟨v∗ ∣ H

op{1S

v0 }∣

v⟩ = −E

v⟨

v∗ ∣ {S

v0}∣

v⟩

The open-shell operators can be obtained by solving the following two equations :

Where, is the negative of the ionization potential of the valence electron v.Ev

HEDMeff= 2 i c d

e 5 p2

H = H0 H

EDM

∣v' ⟩ = e

T 0 deT 1{1 S

v0 d

eS

v1}∣

v⟩

T = T 0 deT1 Sv = Sv

0 de Sv

1Then the cluster operators become: and


⟨v∗∣ H

N0− E

v S

v1 H

N0 T 1 H

EDMeff {1 S

v0}∣

v⟩ = 0

The perturbed cluster amplitudes can be obtained by solving the following two

equations self consistently :⟨

0∗ ∣ H

N0 T 1 H

EDMeff ∣

0⟩ = 0

The expectation value of atomic EDM for the state is then given by, ∣v' ⟩

Where, HN= H

0− ⟨

0∣ H

0∣

0⟩

The e-EDM enhancement factor is given by, R = Da/ d

e

R =⟨

v∣{{1S

v

0†} D0 {T 1T 1S

v

0S

v

1} {S

v

1†S

v

0†T 1†

T 1†

} D0 {1Sv

0}}∣

v⟩

⟨v∣ eT 0

†

eT 0 S

v

0†eT 0†

eT 0 Sv

0∣

v⟩

⟨Da⟩ =

⟨v' ∣ e z ∣

v' ⟩

⟨v∣

v⟩

Nataraj et al. Phys. Rev. Lett. (2008), J. Phys. Conf. Ser. (2007)

eT0 †

D eT0


Results and DiscussionsResults and Discussions


Details on the Basis sets:Details on the Basis sets:

No. of radial basis functions in each Symmetry for the RCCSD(T) calculations for RbRb

103 (15 14 14 12 12 10 10 8 8)

95 (15 14 14 10 10 9 9 7 7)

92 (14 13 13 10 10 9 9 7 7)

109 (15 14 14 13 13 11 11 9 9)

82 (12 10 10 9 9 9 9 7 7)

86 (12 11 11 10 10 9 9 7 7)

S1/2 P1/2 P3/2 D3/2 D5 /2 F5/2 F7 /2 G7/2 G9 /2Basis

We use kinetically balanced Gaussian basis functions forgenerating the single particle orbitals.


No. of radial basis functions in each Symmetry for the RCCSD(T) calculations for CsCs

100 (12 12 12 11 11 11 11 10 10)

106 (14 13 13 12 12 11 11 10 10)

S1/2 P1 /2 P3 /2 D3 /2 D5/2 F5 /2 F7 /2 G7 /2 G9 /2Basis


Term-wise contributions of dominant RCC terms to EDM Term-wise contributions of dominant RCC terms to EDM

enhancement factorsenhancement factors

Norm -0.45 -2.85

Total 25.74 120.53

RCC term Rubidium Cesium

T 1† 1D0

S1† 1D0

S2† 1D0

S1† 1D0S1

0

S1† 1D0S2

0

S2† 1D0S1

0

DF

1.18 5.73

27.30 131.65

-1.10 -6.52

-0.43 -2.17

-0.98 -5.92

-0.07 -0.51

19.55 94.19


Convergence of results for Rb: Size of the RCC model spaceConvergence of results for Rb: Size of the RCC model space

Nataraj et al. Phys. Rev. Lett. 101, 033002 (2008)

Most dominant correlation

terms like pair-correlation,

core-correlation and core-

polarization etc. are

studied. Their variation with

respect to the basis size

are plotted. Our results

show full convergence.

a D0 Sv1 vs. Basis size

vs. Basis size

vs. Basis size

vs. Basis sized R

b D0 T 1

c Sv0†D0Sv

1


Atom Dirac-Fock RCC Others Method Reference

Rubidium 19. 55 25.74 24.6 MBPT Johnson et al. 1986 25.68 MCDF+MBPT Shukla et al. 1994 24.0 Semi-emp. Sandars 1966

Cesium 94.19 120.53 114.9 MBPT Johnson et al. 1986 130.5 MCDF+MBPT Das 1988 114 MBPT Hartley et al. 1990 119 Semi-emp. Sandars 1966

The EDM enhancement factors due to e-EDMThe EDM enhancement factors due to e-EDM

The measured value of EDM of the ground state of Cs:

Combining with our enhancement factor, we obtain:

dCs = −1.8±6.7±1.8×10−24 ecm

d e= −1.5 ± 3.2 × 10−26 ecmNataraj et al. Phys. Rev. Lett. (2008)

Murthy et al. Phys. Rev. Lett. 63, 965 (1989)


Atomic properties of relevance : Estimating the error bar on Atomic properties of relevance : Estimating the error bar on RR

⟨ Da⟩ =

4 c de

ℏ ∑I≠

⟨IO ∣z ∣

O⟩ ⟨

O ∣ i5 p2 ∣IO⟩

EO− E

IO

≈ A2S1/2⋅A2P1/ 2The error in EDM matrix elements is

Differences between the Computed and the Measured results give individual errors.

By adding the errors in quadrature we estimate a maximum of 1% error in our results.

At. property Transition/ State Our work Expt. Reference

Excitation energy 12579.87 12578.95 NIST Database

E1 Tr. Amplitude 4.26 4.23 Volz et al 1996

Hyperfine Constant 1009.33 1011.91 Vanier et al. 1974

Hyperfine Constant 117.71 120.64 Das et al. 2006

5 2S1/2 5 2P1/25 2S1/2 5 2P1/2

5 2S1 /25 2P1/2


The current best limit on the electron EDM comes from Tl :

Experiment : Regan et al. Phys. Rev. Lett. 88, 071805 (2002) ;

Theory : Liu and Kelly Phys. Rev. A 45, R4210 (1992)

FUTURE IMPROVEMENTS POSSIBLEFUTURE IMPROVEMENTS POSSIBLE

The EDM enhancement factors due to e-EDM for The EDM enhancement factors due to e-EDM for TlTl

Limitations of Liu and Kelly's work (R = -585) :

* Linearized Coupled-cluster theory

* Included triples only to unperturbed singles equation

* Freezed the core up to 4S

de = 6.9 ± 7.4 × 10−28 e cm ≡ 1.6 × 10−27 ecm

Our preliminary result for Thallium is, R = -395 obtained using RCCSD(T);

We are still analyzing the results.


EDMs: Probe for Physics Beyond Standard ModelEDMs: Probe for Physics Beyond Standard Model

The proposed e-EDM experiments could observe or obtain tight limits on de by

lowering their sensitivities to 2-3 orders below the present limits, which may unveil a new arena of physics beyond the Standard Model.

( Courtesy: Yasuhiro Sakemi, RCNL, Japan )


We have calculated the electron EDM enhancement factors for the two paramagnetic

atoms; Rb and Cs arising from the intrinsic electron EDM contribution to atomic EDM.

We have sown the convergence of our results with the basis size. By the detailed

analysis of the closely related atomic properties and their discrepancies with respect

to accurately known experimental results we estimate an error bar of about 1%.

Several Atomic (Rb, Cs, Fr etc.) EDM experiments are underway. Results of Rb and

Cs experiments when completed could in combination with our enhancement factors

give improved limits for the electron EDM.

A non-zero EDM is, a background free signal of CP violation beyond the SM.

We can constrain various models of CP violation, Seesaw parameters in neutrino

sector, observable baryon asymmetry, supersymmetric parameters.

ConclusionsConclusions


In collaboration with,

Bhanu Pratap Das (IIAP, Bangalore, India)

Debashis Mukherjee (IACS, Kolkata, India)

Rajat Kumar Chaudhuri (IIAP, Bangalore, India)


THANK YOUTHANK YOU

Special Thanks to INT OrganizersSpecial Thanks to INT Organizersand Indian Institute of Astrophysicsand Indian Institute of Astrophysics


“ Electric dipole moments seem to me to offer one of the most

exciting possibilities for progress in particle physics ”

- Steven Weinberg

“ The Limit from the neutron EDM measurements already produces

what is called strong CP problem and even if that is ignored the

measurements are close to producing trouble for supersymmetry

models . . . Originally I wanted to be the first person to discover an

EDM, but now at least I want to know the answer. I have therefore

personally established the time limited Ramsey prize of $5000, for

the first person, or group, during my lifetime, to announce the

convincing discovery of a non-zero dipole moment for any

elementary particle or atomic nucleus ”

- N. F. Ramsey (ICAP, 2006)

Quotes ............

H. S. Nataraj · 2008-07-28 · Nataraj et al. Phys. Rev. Lett. 101, 033002 (2008) Most dominant...

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