Recent Developments in the relativistic many body theory of time-reversal and parity violations in diamagnetic atoms and molecules and their implications for hadronic electric dipole

moments Bhanu Pratap Das

Department Physics Tokyo Institute of Technology (Tokyo Tech), Tokyo, Japan

Akitada SakuraiTokyo Tech and NII

Yashpal SinghPRL, India LBNLTakeshi Tsutsui

Tokyo Metropolitan University

Bijaya SahooPRL, India

Minori AbeTokyo Metropolitan University

Koichiro AsahiRIKEN and Tokyo Tech

Outline of talk

✤ Diamagnetic Atoms

✤ Sources of EDMs due to Parity (P) & Time reversal (T) violations in diamagnetic atoms.

✤ Applications of Relativistic Many-Body Theory to EDMs of diamagnetic atoms with focus on Relativistic Coupled Cluster methods.

✤ Results for 129Xe and 199Hg EDM.

✤ Diamagnetic Molecules

✤ Sources of EDMs due to P&T violations in diamagnetic molecules.

✤ Results for T (or CP) violating quantities in TlF.

Determination of the limits of the CP violating quantities using atomic EDMs

✤ Using atomic relativistic many-body theories, we can calculate the ratio ( ! )of atomic EDM to P&T violating parameter ( ! ).

✤ Experiments measure the atomic EDM.

✤ Combining theoretical and experimental results, it is possible to get reliable constraints on CP violating quantities.

ℛλP&T

ℛλP&T = (datomλP&T )

λP&T <dexp

ℛλP&T

+ Experimental Result ( dexp )

Our Work

CP violating quantities in high energy regimes

CP-odd quantities in high energy regimes

Why 129Xe and 199Hg EDM are important

✤ The dominant P &T violating interactions scale as Z2. (Z : atomic number):

✤ The diamagnetic atoms like Xe and Hg are sensitive to hadronic (NSM) and semi-leptonic (T-PT) T or CP violation.

✤ Xe experiments, in 3 different laboratories in the world.

✤ Result for 199Hg EDM is most accurate to date.

Energy Scale

NuclearAtomic (~eV) Hadron and QCD (~GeV)

Diamagnetic atomic EDM

(129Xe, 199Hg, Ra)

Paramagnetic atomic EDM(Fr, Cs, Tl)

S-PS (Cs)

T-PT (CT)

P&T-odd Interaction

θQCDNuclear EDM

Nuclear Schiff Moment (S) q-cEDM

q-EDM

eEDM

!"#$$(&)

Nuclear QCD

()(*,,)

(-(*,,)

!"#$$(.)

Z(Xe) = 54 Z(Hg) = 80

Recent improvements in measured and calculated values

of Xe EDM: Current measured value:

1.5 (Munich) and 7 (Mainz) times improvement over

previous result.

Parity and Time-reversal violating interactions in diamagnetic atoms

✤We treat the P and T violating interactions as perturbations in the Dirac-Coulomb Hamiltonian.

✤Two important P&T violating interactions in diamagnetic atoms.

✤Nuclear Schiff Moment (NSM) Interaction between e- and Nucleus.

✤Tensor and PseudoTensor (T-PT) Interaction between e- and nucleons.

HDC =N

∑i

c ⃗p i ⋅ ⃗α + mc2β + Zeri− ∑

i

Nuclear Schiff Moment (NSM)

Φ( ⃗R ) = Φ( ⃗R )0 + δΦ( ⃗R )Nuclear potential :

HNSMe→N = − eΦNSM = e3 ⃗S ⋅ ⃗R

BρN( ⃗R )

P&T Hamiltonian in the atom

Nuclear Schiff Moment (NSM) : S =e

10 [⟨rr2⟩ − 53Z ⟨r⟩⟨r2⟩]

δΦ( ⃗R ) = ∫ d3 ⃗reδρN(r)

| ⃗R − ⃗r |+

1Z

( ⃗d N ⋅ ∇)∫ d3 ⃗rρN(r)

| ⃗R − ⃗r |Density distribution Nuclear EDM

The nuclear charge density : Nucleus has the EDM(N EDM) : dN

ρ( ⃗r ) = ρ0( ⃗R ) + δρ( ⃗r )P&T violating in the nuclear region

⃗S = S ⃗I| I |

The P&T violating interaction between electrons and nuclear (NSM) is arising from P&T-odd (CP-odd) quantities in nuclear region.

P QR

r

z

x

y

|" − $|

Nucleus

ATOM

Atomic Polarizability and EDM Calculations are Related✤ The EDM ratio ( ! ) cannot be measured

by experiments.

✤ To confirm the precision of our calculations, we also calculate the atomic polarizability which has been measured to high accuracy by experiment.

✤ The polarizability is the second-order energy shift due to the electric field.

✤ The dipole operator has similarities with the P&T violating operators.

ℛ

Operator Dipole TPT NSM

Rank 1 1 1

Parity -1 -1 -1

Nucleus

Electron Clouds

⃗E

⃗d = α ⃗E

Polarizability

αXe = 27.815(27)[a.u.]αHg = 33.91(34)[a.u.]

Relativistic Many Body Calculation of Atomic EDM

✤ The total atomic Hamiltonian consists of DC Hamiltonian and P&T violating interaction Hamiltonian.

✤ The ratio ( ) of atomic EDM to T&P violating parameter

ℛλP&T = ( dλP&T ) = ⟨Ψ(0)0 |D |Ψ

(1)0 ⟩ + ⟨Ψ

(1)0 |D |Ψ

(0)0 ⟩

(λP&T)

(HDC + λP&THP&T)( |Ψ(0)0 ⟩ + λP&T |Ψ(1)0 ⟩) = E(0)0 ( |Ψ(0)0 ⟩ + λP&T |Ψ(1)0 ⟩)The unperturbed state First-order perturbed state

Unperturbed Hamiltonian :

HA =N

∑i

c ⃗p i ⋅ ⃗α + mc2β + VN(ri) − ∑i

Dirac-Fock Method (Relativistic Mean Field Method)✤ The basic idea of the Dirac Fock (DF) approximation

is taking the mean approximation of the Coulomb Interaction which is the two-body operator.

✤ In the DF method, atomic state consists of the Slater determinant ( ) consisting of each electron orbital states ( ). The DF Potential (Mean field Potential) is written as

✤ Dirac-Fock Equations for each orbital.

VDF |ϕi(1)⟩ =N

∑j

(⟨ϕj(2) | 1r12|ϕj(2)⟩ |ϕi(1)⟩ − ⟨ϕj(2) |

1r12

|ϕi(2)⟩ |ϕj(1)⟩)

̂t(1) |ϕk(1)⟩ +N

∑i=1

(⟨ϕi(2) | e2

r12|ϕi(2)⟩ |ϕk(1)⟩ − ⟨ϕi(2) |

e2

r12|ϕk(2)⟩ |ϕi(1)⟩) = ϵk |ϕk(1)⟩

Coulomb Interaction

Mean Field Interaction

ϕi|Φ0⟩

HDC = T + VCoulomb HDF = T + VDF

Nucleus

Nucleus

electron

The ratios of atomic EDM to P&T violating parameter at DF level✤ In order to solve the DF equation, we have used the matrix method in which

four component single electron wave functions ( ! ) are expanded by Gaussian functions ( ! ) using even-tempered parameters.

✤ The results of DF for polarizability and EDM ratios due to TPT and NSM for 129Xe and 199Hg.

|ϕi⟩|Gn⟩

Even-Tempered (ET)

G(r) = NLrke−αir2|ϕi⟩ =N

∑n=1

Cn |Gn⟩ αi = α0βi−1 (i = 1,2,3,⋯, N )Gaussian Functions

DF 26.865 0.466 0.289

Exp 27.815(27)[1] — —

α ℛSℛCT

DF and experimental results of 129Xe

DF 40.95 -2.39 -1.2

Exp 33.91 (34) — —

α ℛSℛCT

DF and experimental results of 199Hg

(#) T-PT in 10−20CT⟨σN⟩ [ |e |em] (##) NSM in 10−17 [S⟨/ |e | fm3]

(#) (##)

[1] Hohm U, and Kerl K, Mol. Phys, 69, 819 (1990).

Post Dirac-Fock Methods✤ For accurate calculation of the atomic wave function, the mean field

approximation is not enough. In this case, it is important to include corrections from many-body effects .

✤ The exact atomic state can be represented by the DF ground state plus a complete set of states representing particle-hole excitations 
( ) with reference to the DF state.

Hatom = T + VDF + (V − VDF)Residual interaction

|Ψ0⟩ = |Φ0⟩⏟

+ ∑i,a

cai |Φai ⟩ + ∑

i, j,a,b

cabij |Φabij ⟩ + ⋯

|Φai ⟩, |Φabij ⟩, ⋯

Hres = V − VDF =12 ∑

p

⟨pq |v |rs⟩a†pa†qasar − ∑i

⟨p |VDF |q⟩a†paq

Two-Body One-Body

DF levelMany-body corrections

i

a

|Φai ⟩

Particle

|Φ0⟩

Fermi seaHole

Relativistic Coupled Cluster Method (RCCM)

|Ψ0⟩ = |Φ0⟩ + ∑ia

tai |Φai ⟩ +

12! ∑

ijab

tai taj |Φ

abij ⟩ + ∑

ijab

tabij |Φabij ⟩ + ⋯

= |Φ0⟩ + T1 |Φ0⟩ +12!

T21 |Φ0⟩ + T2 |Φ0⟩ + T1T2 |Φ0⟩ + ⋯

= eT |Φ0⟩

✤ In RCCM, an atomic wave function is written as

Cluster amplitudes :

T = T1 + T2 + T3⋯

= ∑ia

tai a†aai + ∑

ijab

tabij a†aa†b ajai + ∑

ijkabc

tabcijk a†aa†b a

†c akajai + ⋯

Cluster Operators :

RCCSD approximationtai , t

abij , t

abcijk , ⋯

✤ RCC amplitude equations :

Relativistic Coupled Cluster Method (RCCM)

HDC |Ψ0⟩ = E0 |Ψ0⟩ ⇒ HDCeT |Φ0⟩ = E0eT |Φ0⟩

✤ RCC ground state and RCC equations

✤ The ground state energy in RCCM :

E0 = ⟨Φ0 |e−THDCeT |Φ0⟩

⟨Φ* |e−THDCeT |Φ0⟩ = 0

( |Φ*⟩ : |Φai ⟩, |Φabij ⟩, ⋯)

Atomic EDM in RCCM

✤ The atomic cluster wave function with the first order perturbation due to P&T violating interaction :

✤ The ratio (R ) in RCCM

ℛ = ⟨Ψ(0) |D |Ψ(1)⟩ + ⟨Ψ(1) |D |Ψ(0)⟩

= 2⟨Φ0 | (eT(0)†DeT (0)T(1))con |Φ0⟩

= 2⟨Φ0 |D + DT(1) + T(0)†DT(1) + T(0)†D + T(0)†DT(1) + ⋯ |Φ0⟩

|Ψ0⟩ = eT(0)+λT (1) |Φ0⟩ ≈ eT

(0) |Φ0⟩ + T(1)eT(0) |Φ0⟩

|Ψ(1)0 ⟩|Ψ(0)0 ⟩

Non-terminating series

Non-terminating problem in CCM & Self - Consistent (SC) CCM

✤ The formulas of the expectation values of the CCM such as polarizability and EDM ratios do not terminate. In the present study, we have used a Self-Consistent (SC) Coupled Cluster Method. In this method, combined power of T0 and T0 dagger in the EDM ratio is systematically increased till the result converges.

ℛ = ⟨Ψ(0) |D |Ψ(1)⟩ + ⟨Ψ(1) |D |Ψ(0)⟩

= 2⟨Φ0 | (eT(0)†DeT (0)T(1))con |Φ0⟩

= 2⟨Φ0 |D + DT(1) + T(0)†D + T(0)†DT(1) + T(0)†DT(1) + ⋯ |Φ0⟩

Normal Coupled Cluster Method (Arponen 1983 )✤ In order to avoid the non termination problem in the CCM, we have

used a new type of the CCM, called Normal Coupled Cluster Method (NCCM). In NCCM, the bra state is written like:

✤ The de-excitation operators :

⟨Ψ̃ 0 | = ⟨Φ0 | (1 + T̃ )e−T⟨Ψ0 | = ⟨Φ0 |eT†

Cluster operatorsDe-excitation operators

CCM : NCCM :

T̃ = ∑ia

t̃ai a†i aa +

12! ∑

ijab

t̃abij a†i a

†j abaa +

13! ∑

ijkabc

t̃abij a†i a

†j a

†k acabaa + ⋯

RNCCSD approximation

⟨ Ψ̃0 |A |Ψ0⟩ = ⟨Φ0 | (1 + T̃ )e−T AeT |Φ0⟩

Terminating series

Expectation value :

RCCM vs RNCCM

⟨Da⟩ = ⟨Ψ |D |Ψ⟩c= ⟨Φ0 | D̄T1 |Φ0⟩ + ⟨Φ0 |T1

†D̄ |Φ0⟩

|Ψ0⟩ = eT0 |Φ0⟩

⟨Ψ0 | = ⟨Φ0 |eT0†

RCCM RNCCM

T0,1 = ∑I

t0,1I C†I

|Ψ⟩ = eT0eλT1 |Φ0⟩

H |Ψ⟩ = E |Ψ⟩ ⟨Ψ |H = ⟨Ψ |E

D̄ = eT0†DeT0

H |Ψ⟩ = E |Ψ⟩ ⟨Ψ̃ |H = ⟨Ψ̃ |E

⟨Da⟩ = ⟨Ψ̃ |D |Ψ⟩

= ⟨Φ0 | D̄T1 |Φ0⟩ + ⟨Φ0 | T̃1D̄ |Φ0⟩

−⟨Φ0 | T̃0D̄T1 |Φ0⟩

T̃0,1 = ∑I

t̃ 0,1I C−I{

|Ψ0⟩ = eT0 |Φ0⟩

⟨Ψ̃ 0 | = ⟨Φ0 | (1 + T̃0)e−T0

{|Ψ⟩ = eT0eλT1 |Φ0⟩

⟨Ψ̃ | = ⟨Φ0 | (1 + T̃0 + λT̃1)e−T0e−λT1

Expectation value of Dipole Expectation value of Dipole

Atomic EDM in RNCCM

✤ The atomic normal cluster wave function with the first-order perturbation due to P&T violating interaction :

✤ The ratio (R) in RCCM

⟨Ψ̃ 0 | = ⟨Φ0 | (1 + T̃(0) + λT̃(1))eT(0)+λT (1) = ⟨Ψ̃ (0)0 | + λ⟨Ψ̃

(1)0 |

ℛ = ⟨Φ0 |(T̃(1)e−T (0)DeT (0) + (1 + T̃(0))e−T (0)DeT (0)T(1)) |Φ0⟩Terminating series

⟨Ψ̃ (1)0 | = ⟨Φ0 | T̃(1)e−T (0) − ⟨Φ0 | (1 + T̃(0))e−T

(0)T(1)⟨Ψ̃ (0)0 | = ⟨Φ0 | (1 + T̃

(0))e−T (0)0th order :

1st order :

Results of polarizability ( ! ) and TPT ( ) and NSM ( ) of 129Xe

αℛCT ℛS

[1] A. Sakurai, B. K. Sahoo and B. P. Das, Phys. Rev. A 97, 062510 (2018).

[2] A. Sakurai, B. K. Sahoo, K. Asahi and B. P. Das, Phys. Rev. A Rap. Comm. (2019 In Print)

[3] Y. Singh, B. K. Sahoo, and B. P. Das, Phys. Rev. A 89, 030502(R) (2014).

Our Work Others[3]

DF 26.87 0.45 0.29 26.87 0.45 0.29

CCSD (SC) 28.12

[1] 0.48[2] 0.32[2] 28.13 0.47 0.33

NCCSD 27.52[1] 0.49[2] 0.32[2] - - -

Exp - - - 27.815 - -

α ℛSℛCT α ℛSℛCT

(#) T-PT in 10−20CT⟨σN⟩ [ |e |em] (##) NSM in 10−17 [S⟨/ |e | fm3]

The CP Violating Quantities obtained by 129Xe Calculation

ℛλP&T = (datomλP&T ) (λP&T = S, CT)+ dXe < 1.0 × 10−27 [e cm]

S < {3.0 × 10−10[e cm3]

3.2 × 10−10[e cm3] NCCSDCCSD

CT < {4.2 × 10−74.4 × 10−7 NCCSDCCSDUpper limits of NSM and TPT Coupling in Nuclear sector

Upper limits of CP violating quantities in QCD sector

C(u,d)T | d̃(u) − d̃(d) | < 1.4 × 10−23 θQCD < 2.8 × 10−7

In University of Mainz ( 2018 )

+ Nuclear Physics and Particle Physics

Our Work

Results of polarizability ( ! ) and TPT ( ) and NSM ( ) of 199Hg

αℛCT ℛS

[1] B. K. Sahoo and B. P. Das, PRL 120 203001 (2018)

Our Work Others

DF 40.95 -2.39 -1.20 - -2.0 -1.19

MBPT - - - - - -2.8

RPA - -5.80 -2.94 - -6.0 -2.8

CCSD (SC) 34.51[1] -3.17[1] -1.76[1] - - -2.46

NCCSD 34.22[1] -3.30[1] -1.77[1] - - -

Exp - - - 33.91(34) - -

α ℛSℛCT α ℛSℛCT

(#) T-PT in 10−20CT⟨σN⟩ [ |e |em] (##) NSM in 10−17 [S⟨/ |e | fm3]

Results for NSM ( ) of 199Hg (by others)

ℛS

✤RDF=1.2
(Sahoo and Das, PRL 120, 203001 (2018))

✤RCCM(SD)=1.76 (SC inclusion of CC amplitudes)

(Sahoo and Das, PRL 120, 203001 (2018) )

✤RNCCM(SD)=1.77
(Sahoo and Das, PRL 120, 203001 (2018))

✤RCI+MBPT=2.8
(V.A. Dzuba et al, Phys. Rev. A 80, 032120 (2009).)

✤RCCM(SD)=2.46
( Latha, et al, PRL 103, 083001 (2009) 115, 059902(E) (2015))

✤ RMCDF=2.22
(L. Radziute et al, Phys. Rev. A 93, 062508 (2016))

The CP Violating Quantities obtained by 199Hg Calculation

ℛλP&T = (datomλP&T ) (λP&T = S, CT)+ dHg < 7.4 × 10−30 [ |e | cm]

S < 4.18 × 10−13[e cm3] NCCSD CT < 6.73 × 10−10 NCCSDUpper limits of NSM and TPT Coupling in Nuclear sector

Upper limits of CP violating quantities in QCD sector

| d̃(u) − d̃(d) | < 5.5 × 10−27[ |e |cm]dn < 2.2 × 10−26[ |e |cm]dp < 2.1 × 10−25[ |e |cm]

B. Graner, Y. Chen, E. G. Lindahl, and B. R. Heckel, Phys. Rev. Lett. 116, 161601 (2016).

+ Nuclear Physics and Particle Physics

Our Work

Why multiple candidates ?

✤ In hadronic sector, NSM ( ! ) is written as combination of coupling constants of P&T-odd interactions between nucleons.

S

S = gπnn(a0g̃(0)πnn + a1g̃(1)πnn + a2g̃(2)πnn)= B g̃(0)πnn + C g̃(1)πnn

✤ In oder to solve the above equation, we need at least three atomic EDM results.

da = A CT + B g̃(0)πnn + C g̃(1)πnn

✤ Atomic EDM

Parity and Time-Reversal Interaction in Diamagnetic Molecules

✤ A general expression of Parity and Time-Reversal violating interactions in diamagnetic molecules (Hinds and Sandars, PRA,1980)

HPTV = − d ⃗σ N ⋅ ⃗λ ! : P & T-odd interaction coupling constant ! : Nuclear spin ! : Unit vector along the internuclear axis

d⃗σ N⃗λ

Tl F⃗λ

⃗σ N

Sources of P and T Violation in 205TlF

✤ One of the leading candidates for P and T violation studies in diamagnetic systems is 205TlF.

✤ Approximately 30 times more sensitive to proton EDM and NSM, compared to 199Hg. This can improve by one or two orders of magnitude with laser-cooling.

✤ For 205TlF, there are three sources of P&T violation.

(1) : Proton EDM(2) : Parity and Time-reversal interaction between nucleons(3) : Tensor-PseudoTensor e-N interaction

Proton EDM (dp) & Nuclear Shift Moment

HVolPTV = − dv ⃗σ N ⋅ ⃗λ

R = ⟨r2N⟩AV − ⟨r2N⟩3S

dv = dp R X

X = ∑j

Xj = ∑j

2π3 [ ⃗∇ Ψ†j ( ⃗r )Ψj( ⃗r )] ⃗r=0

✤ Proton EDM

✤ Nuclear Shift Moment ( ! )Q

HMagPTV = − dM ⃗σ N ⋅ ⃗λVolume Effect : Magnetic Effect :

dM = − 2 2dp(μN2

+1

2mNe)

M = ∑j

Mj = ∑j

1

2⟨Ψj |

( ⃗α × ⃗l)jr3

|Ψj⟩

Proton EDM: Volume effect and Magnetic effect.

HQPTV = − 6 Q X

How to estimate Proton EDM by diamagnetic molecules

✤ The energy shift ( ! ) of the ground state of a molecule with P&T-odd perturbation is written as the expectation values of P&T-odd Hamiltonians.

ΔE

⟨HVolMag⟩ ∝ ⟨M⟩

⟨HVolPTV⟩, ⟨HQPTV⟩ ∝ ⟨X⟩

ΔE = ⟨HVolPTV⟩ + ⟨HMagPTV ⟩

dp

+ ⟨HQPTV⟩

Q

✤ Our calculation of ! and ! calculation by Petrov et al (2002) combined with measurement of Cho et al (1991) gives limit for ! .

⟨X⟩ ⟨M⟩dp

Expectation value of X using RCCM

✤ The expectation value of X

⟨X⟩ =⟨Φ0 |eT

†XeT |Φ0⟩⟨Φ0 |eT

†eT |Φ0⟩= ⟨Φ0 |eT

†XeT |Φ0⟩c

⟨X⟩ =⟨Ψ |X |Ψ⟩

⟨Ψ |Ψ⟩✤ In RCCM, ! |Ψ⟩ = eT |Φ0⟩

⟨X⟩ = ⟨Φ0 | (1 + T†1 + T†2 )X(1 + T1 + T2) |Φ0⟩c

✤ In linearized RCCSD approximation Exact result for closed shell systems

✤ Leading correlation contributions come from linear terms for many properties of closed shell systems.

! and ! are proper RCCSD amplitudesT1 T2

Derivative Approach to First Order Properties

✤ The Hamiltonian with first order perturbation.

✤ First order change in energy can be calculated as an expectation value or as a first derivative of energy.

Perturbation theory : !E = E0 + λE1 + λ2E2 + ⋯

Taylor series expansion : !E = E0 + λ∂E∂λ λ=0

+λ2

2∂2E∂λ2 λ=0

+ ⋯

E1 =∂E∂λ λ=0

E1 = ⟨Ψ0 |H1 |Ψ0⟩

✤ In coupled cluster theory there are two approaches to calculate derivatives - numerical and analytical.

! and !H = H0 + λH1 H0 |Ψ0⟩ = E0 |Ψ0⟩

Finite Field Coupled Cluster Theory

✤ The Hamiltonian with first order perturbation

(H0 + λH1)eT(λ) |Φ0⟩ = E(λ)eT(λ) |Φ0⟩e−T(λ)(H0 + λH1)eT(λ) |Φ0⟩ = E(λ) |Φ0⟩

In our work we have six point central difference

E(λ) = ⟨Φ0 |e−T(λ)H0eT(λ) |Φ0⟩ + λ⟨Φ0 |e−T(λ)H1eT(λ) |Φ0⟩

✤ ! amplitudes can be calculated for a given ! and therefore ! can be foundT(λ) λ E(λ)

∂E∂λ λ=0

=E(λ) − E(−λ)

2λ : Two point central difference

✤ Advantage of this approach is that the expression for energy in coupled cluster theory terminates, so higher orders effects are properly included.

! and !H = H0 + λH1 |Ψ⟩ = eT(λ) |Φ0⟩

Results for X of TlF

MethodDF 9059 8747

RCCSD-LE 6946 -

RCCSD-FF 7161 -

RCCSD(T)-FFCC 6856 -

RCC-ECP - 7635

✤ All 90 electrons in TlF were excited. ✤ X was found to be very sensitive to choice of single particles basis

functions. Even tempered basis functions were used.

✤ Using experimental result of Cho at al + our calculation + Petrov magnetic calculation:

dp = (−1.9 ± 3.2) × 10−23[ |e |cm] Q = (7.1 ± 12) × 10−11[ |e | fm3]

Conclusions

✤ Using our results for different versions of relativistic coupled cluster theory and the recent measurements of Xe and Hg EDMs, we have shown that these two atoms have the potential to be important probes of new physics beyond the Standard Model.

✤ Applied two different relativistic coupled theories to get new results for T or CP violating quantities for TlF.

✤ We have got new limits for the Nuclear Schiff Moment and the proton EDM based on our theoretical work on Xe, Hg and TlF. The proton EDM from Hg EDM is among the best to date.

Thank you for your attention