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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
4
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)
16
Singles and doubles S diagrams
Using T2 and S2, we getapproximate triples..
17
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
19
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
5
[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/