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Many Body Studies of PNC/EDM as probes of physics beyond the Standard Model

Geetha Gopakumar (TMU, Japan)Prof. Bhanu Pratap Das (IIA, India)Prof. D. Mukherjee (IACS, India)Prof. Kimihiko Hirao (RIKEN, Japan)Prof. Hada (TMU, Japan)Minori Abe (TMU, Japan)

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Outline

• Parity (P) and Time-reversal (T) operations

• E1PNC (P Violation) and EDM (P and T Violation) • Sources of PNC/EDMs in atoms

• Computation of E1PNC/EDMs – Requirement of atomic many-body theory

• Coupled Cluster Method – IP, EE, lifetime, hyperfine constant and E1PNC• Present Limits and Implications for Particle Physics

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Standard Model in Particle Physics

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The Royal Swedish Academy of Sciences has decided to award the Nobel

Prize in Physics for 2008 with one half to

Yoichiro Nambu, Enrico Fermi Institute, University of Chicago, IL, USA

"for the discovery of the mechanism of spontaneous broken

symmetry in subatomic physics"

and the other half jointly to

Makoto Kobayashi, High Energy Accelerator Research Organization

(KEK), Tsukuba, Japan and Toshihide Maskawa, Yukawa Institute for

Theoretical Physics (YITP), Kyoto University, and Kyoto Sangyo University,

Japan

"for the discovery of the origin of the broken symmetry which

predicts the existence of at least three families of quarks in

nature"

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Fundamental transformation leading to symmetries

C Particle ---------------- antiparticle (Q > -Q)

Position --------------- inverted (r > -r)

T Time ----------------- reversed (t > -t )

Charge-Parity-Time Reversal – CPT theorem

This means that if any particle is replaced with its corresponding antiparticle, and the space coordinate and time are reversed, the physical laws are unchanged.

A Physical system/process can violate each of these symmetries individually as long as the combined CPT is conserved

A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is "preserved"

under some change

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P Violation – E1PNC State vector under parity transformation

Hamiltonian under parity transformation

ie.

[Parity Conservation ] Systems of Interest > Atom where chiral property arises from

the interactions between the constituents which favours one orientation with respect to other

Weak Interactions > Nucleons and Electrons (Parity non conserving)

|ψ> = |ψ(0)> + ʎ |ψ （１） > (mixing of opposite parity states)

　　　　（ even/odd) (odd/even)

(Interaction being weak considered perturbatively)

E1PNC = <ψα|D|ψβ> ≠ 0 α and β are of same parity

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P & T violation EDMMolecules of ammonia, water permanent EDMs due to degeneracy of states

EDMs of interest > P and T violations in non-degenerate systemsAny vector is aligned either parallel or anti-parallel to J (Projection Theorem)

Quantity P T

D - D + D 　 J 　　　　　 J 　　　 - J D = αJD = -αJD = -αJ

α = 0 ; implies P and T are not violatedα ≠ 0 ; implies P and T are violated

edm

P

T

J

J J

J

|ψ> = |ψ|ψ> = |ψ(0)(0) > > + ʎ |ψ + ʎ |ψ （１） （１） >>　　　　　（　　　　　（ even/odd) (odd/even) even/odd) (odd/even)

EDM = EDM = <<ψψαα|D|ψ|D|ψα α >> ≠ 0 ≠ 0

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Sources of PNC in atoms

Nuclear Weak Current Interaction between nucleus and the electrons (mediated by Z0 bosons

EM interaction between nuclear anapole moment and the electrons

np e

C1n

C1p

HNSIPNC

= GF/2 √ 2 Qwγ5ρN （ｒ）

GF = Fermi Coupling constant ~ 2.2 X 10-11 au

(measure of weakness of the interaction)

Qw = 2(C1p Z + C1n N) ∞ Z

ρN （ｒ） = nucleon number density ∞ Z

γ5= Dirac matrix >> σ.p ∞ velocity ∞ Z

E1PNC ∞ Z3 >>> HEAVY ATOMS

ρN （ｒ） ---- HPNC ≠ 0 only for electron wavefunctions with finite value at the nucleus hence connects only s(1/2) and p(1/2) orbital

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Sources of EDMs in atomsElectron EDM

Nuclear EDM

P and T violating interactions between electrons and nucleons

Closed shell atoms: Nuclear EDM and electron-nucleon tensor pseudo tensor interaction and Schiff moment

Eg. Yb, Hg (J ≠0)

Open Shell Atoms : Electron EDM and electron-nucleon scalar-pseudo scalar interaction

Eg. Cs, Fr with single free electron outside the core

He-N = ΣN GF2√2 CT βα.I ρN(r)

CT = (Z C T,p + N CT,n) ∞ Z

β, α = Dirac Matrix ∞ Z ; I – Spin

ρ N(r) =nucleon number density ∞ Z

EDM ∞ Z3 >>> HEAVY ATOMS

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= unperturbed Hamiltonian

Atomic Many Body Thoery to compute

E1PNC/EDM We need to know • Hamiltonian of the system• Accurate relativistic electron wave functions

H= Dirac Hamiltonian for a many-electron atom

In the presence of a (P, PT) violating interactions,

Ht = H + λHPNC (Parity Non Conserving interaction)

Ht = H + λ HPTV (Parity and Time Violating interaction)

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The Schrödinger equation for an exact atomic state is

Ht |where

| (0) λ (1)Unperturbed wave function

First-order perturbedwave function

(0)'s are obtained by solving the unperturbed Schrödinger equation,

(0)(0)(0)

The perturbed Schrödinger equation hence becomes

(H - E(1)PNCEDM(1)

The E1PNC / EDM is given by

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Many -body perturbation theory Configuration Interaction Coupled-cluster theory

E1PNC = <ψα |D |ψβ>/ √<ψα| ψα > √<ψβ| ψβ >

= < ψα1|D| ψβ

0> + < ψα0|D| ψβ

1>

√<ψα| ψα > √<ψβ| ψβ >

EDM = <ψα |D |ψα>/ <ψα| ψα >

= < ψα1|D| ψα

0> + < ψα0|D| ψα

1>

<ψα| ψα >

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Coupled Cluster Method (CCM)Many-electron wf in closed-shell CC is given

by

eT |Φ0>

T is the cluster operators which considers excitations

from core to virtuals

|Φ 0> N-1 electron closed shell DF reference state

Ha |

Subtracting <Φ 0|H

a |Φ

0> from both sides we get

HN |is correlation energy

Using the exponential form of and pre multiplying

by e-T, we get H-N|Φ

0> = |Φ

0>

<Φ 0gives< Φ

0H-

N |Φ0

Φ 0

*gives<Φ0H-

N |Φ0

H-N=

e-TH

NeT= (H

NeT)

C

linear : HN + (H

NT)

C

non linear : HN + (H

NT)

C +

(HNTT)

C + (H

NTTT)

C+

(HNTTTT)

C

T

1T

1, T

1T

2, T

2T

2,

T1T

1T

2, T

1T

1T

1T

1 - negligible

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Singles equation (T1) Doubles equation (T2)

non linear diagrams

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Coupled Cluster method for open shell systems

Single valence caseS considers excitations from valence to virtual

One electron is added to the kth virtual orbital

Using the similar techniques used in the closed shell case

( IP equ.)

The above equation is non-linear as IP itself depends on S

Excitation energy (EE) = IP (valence) – IP (appropriate orbital)

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Singles and doubles S diagrams

Using T2 and S2, we getapproximate triples..

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Ba+ ion E1PNC evaluation using sum over states

approach

Expt – Fortson et al (PRL,7383,1993)

Mixed parity approach

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Basis set Part numerical + part analytical

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Basis set

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IP and EE calculations (energy check)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

6s(1/2) 5d(3/2) 5d(5/2) 6p(1/2) 6p(3/2)

% o

f err

or in

IP

analytical

hybrid

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

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[6s(1/2)-5d(3/2)]

[6s(1/2)-5d(5/2)]

[6s(1/2)-6p(1/2)]

[6s(1/2)-6p(3/2)]

% o

f err

or in

EE

analytical

hybrid

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Check on Dipole and EE

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Check on Dipole and HPNC matrix elements

PNC and hyperfinematrix elementsdepends on the overlapof the orbital wave functionwith the nuclear region

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Probing physics beyond the SM

QW = E1PNC (expt)/ X (theory) ; E1PNC (expt) ~ Φ(M1/E2)Theory and Experiment should have to be found accurately

in order to test SM

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Present Limits

∆ QW

= QW

– QW

SM

if ∆ QW

≠ 0 , Physics beyond the Standard Model

Caesium (55) QW = -72.57 ± (0.29)

expt ±(0.36)

theo

SM Qw = -73.09(0.3)

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Implications to particle physics

Work being pursued..1. EDM in YbF /YbLi molecule

2. PNC in Ra+ ion

3. Photo association spectroscopy calculations in YbLi molecule

Atom smasher sets record energy levels : CERN http://uk.news.yahoo.com/