4/24/2014
ATOMIC PARITY VIOLATION
OUTLINE
• Overview of the Atomic Parity Violation
• Theory: How to calculate APV amplitude?
• Analysis of Cs experiment and implications for search
for physics beyond the Standard Model
• Neutron skin
• Nuclear spin-dependent APV effects and weak hadronic
interactions
• Future prospects
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Atomic Parity Violation
High energies
Instead of search for new
processes or particles directly
Determine weak charge QW from atomic
parity violation studies and compare the result
with Standard Model
prediction
Low energies
http://public.web.cern.ch/, Cs experiment, University of Colorado
Two sides of the atomic parity violation
NUCLEAR
SPINSPIN-INDEPENDENT
PNC:
SEARCHES FOR NEW
PHYSICS
BEYOND THE
STANDARD MODEL
e
q
e
Z0
q
NUCLEAR
SPINSPIN-DEPENDENT
PNC:
STUDY OF PNC
IN THE NUCLEUS
a
Nuclear anapole
moment
Weak Charge QW
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EXPERIMENTAL
PNC
STUDIES
Cesium: atom with single (valence)
electron outside a closed core.
Cs Z=55
core
1s2…5p6
Need heavy atom
for atomic PNC
6s
valence
electron
Cs
7s
E1
l=0 to l=0 electric dipole transition is
forbidden by parity selection rules
6s
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Atomic Parity Violation
r→ ─r
Cs
7s
Non-zero transition amplitude
PNC amplitude EPNC
E1
6s
e
q
Both 6s and 7s states acquire an
opposite-parity (np1/2 ) admixture
Z0 exchange: Laporte’s rule is violated!
Z0
e
q
Atomic Parity Violation
r→ ─r
Cs
7s
Non-zero transition amplitude
PNC amplitude EPNC
E1
6s
e
q
Both 6s and 7s states acquire an
opposite-parity (np1/2 ) admixture
Z0 exchange: Laporte’s rule is violated!
e
Z0
q
Note: it is really tiny effect !!! EPNC ~10-11 atomic units
E1 amplitude for 6s – 6p transition is 4.5 atomic units
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The most precise measurement
of PNC amplitude ( cesium )
C.S. Wood et al. Science 275, 1759 (1997)
F=4
7s
F=3
2
1
F=4
6s
F=3
0.35% accuracy
Im ( EPNC )
β
 −1.6349(80) mV cm
=
 −1.5576(77) mV cm
1
2
Stark interference scheme to measure ratio of the
PNC amplitude and the Stark-induced amplitude β
Analysis of Cs PNC experiment
NUCLEAR
SPINSPIN-INDEPENDENT
PNC
NUCLEAR
SPIN-DEPENDENT
SPIN
PNC
F=4
7s
7s
F=3
1
F=3
Average of 1 & 2
si
Im ( EPNC
)
β
F=4
6s
6s
2
= −1.5935(56) mV cm
Weak Charge QW
Difference of 1 & 2
sd
∆  Im E PNC
β 
=

34 − 43
− 0.077(11) mV cm
(
)
Nuclear anapole moment
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Analysis of Cs PNC experiment
NUCLEAR
SPINSPIN-INDEPENDENT
PNC
7s
6s
Average of 1 & 2
si
Im ( EPNC
)
β
= −1.5935(56) mV cm
Weak Charge QW
How to extract weak charge Qw from
Cs experiment?
e
e
q
Z0
q
Electron-quark parity violating interaction
(exchange of virtual Z0 boson)
HW =
GF
eγ µ γ 5 e C1u uγ µ u + C1d dγ µ d + ...
2
(
){
}
Neutron density function
(1)
Electronic sector: H PNC
G
= F QW γ 5 ρ ( r )
2 2
Extraction of weak the charge:
Measured
value
Theoretical calculation of
PNC amplitude
theory inferred
EPNC = EPNC
QW
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Calculation of PNC amplitude
1. Main part – Coulomb interactions
Electric-dipole matrix elements
∞
theory
EPNC
=
∑
n= 2
7 s d np1/ 2 np1/ 2 H PNC 6 s
E6 s − Enp1/2
Energies
∞
7 s H PNC np1/2 np1/2 d 6 s
n= 2
E7 s − Enp1/2
+∑
PNC matrix elements
Sum is separated to main part, n = 6 - 9 and the tail
2. Other small corrections:
Breit, QED, Neutron skin, e – e weak interaction
REDUCING THEORY UNCERTAINTY:
WHY IS IT SO DIFFICULT?
Ψ v = Ω Ψ v(0)
Exact wave function
Cs: 55 electrons
Dirac-Hartree-Fock
wave function (lowest order)
Many-body operator,
describes excitations from lowest-order
55-fold excitations to get
exact wave function
Even for 100 function basis set
10055
Approximate methods: perturbation theory does not converge well,
Need to use all-order methods (coupled-cluster method and
correlation potential method)
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Development of Theoretical Methods
• Derive the formulas
• Write the corresponding computer code … debug …
• Optimize the code so it will finish in “reasonable time”
• Carry out the numerical calculations
• Make sure your results (1) make sense
(2) are in fact correct
• Estimate how accurate are your final numbers
Derive the formulas
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Derive the formulas
Derive the formulas
Contract operators by Wick’s theorem
H
1 2 (0)
S2 | Ψ v >→: ai+ a +j al ak : am+ an+ ar+ as+ ad ac ab aa av+ :| 0c >
2
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TOO MANY TERMS !!!
Contract operators by Wick’s theorem
H
1 2 (0)
S2 | Ψ v >→: ai+ a +j al ak : am+ an+ ar+ as+ ad ac ab aa av+ :| 0c >
2
800 TERMS!
Solution: develope codes that write formulas, i.e.
do all the work for you!
Input: ASCII input of terms of the type
∑ ∑g
mnrab
ijkl
ρ mnrvab : ai† a †j al ak : : am† an† ar† ab aa av : Ψ v(0 )
ijkl
Output: final simplified formula in LATEX you can now code
and put in you paper
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STILL TOO MANY TERMS TO CODE!!!
Solution: automated formula and code generation!
Codes that write formulas
Codes that write codes
Input:
Output:
list of formulas to be programmed
final code (need to be put into a main shell)
Features: simple input, essentially just type in a formula!
Optimize the code so it will finish in “reasonable time”
Less then the lifetime of the
Universe
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Optimize the code so it will finish in “reasonable time”
Before your current grant proposal
runs out
Optimize the code so it will finish in “reasonable time”
Before your patience runs out
indieberries.blogspot.com
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• Carry out the numerical calculations
• Make sure your results (1) make sense
(2) are in fact correct
• Estimate how accurate are your final numbers
Codes that write formulas
Codes that write codes
Codes that analyse data
RELATIVISTIC ALLALL-ORDER METHOD
Sum over infinite sets of many-body perturbation theory
(MBPT) terms.
Calculate the atomic wave functions and energies
Scheme:
Calculate various matrix elements
H Ψv = E Ψv
Calculate “derived” properties
such as PNC amplitudes
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LOWESTLOWEST-ORDER ATOMIC WAVE FUNCTION
Cs: atom with single (valence)
electron outside of a closed core.
1s2…5p6
core
Lowest-order
wave function
Ψ
†
(0)
= av
v
valence
electron
6s
Creation operator
for state v
Ψ
Core
core
Core wave function
Coupled-cluster method ( CCSD )
Ψ v = Ω Ψ (0)
= exp( S ) Ψ (0)
v
v
DHF wave function
CCSD single-double
exp( S1 + S 2 ) coupled-cluster
S1 + S 2
LCCSD coupled-cluster
core excitations
core excitation
S1
S2
valence excitation
core - valence excitations
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LCCSD (ALL(ALL-ORDER)
ATOMIC WAVE FUNCTION
Lowest order
core
valence electron
any excited orbital
Core
Single-particle
excitations
Double-particle
excitations
LCCSD ATOMIC WAVE FUNCTION
Lowest order
Core
core
valence electron
any excited orbital
Ψ (0)
v
Single-particle
excitations
∑
ma
ρma am† aa Ψ v(0)
∑ ρmv am av
m≠v
†
Ψ (0)
v
Double-particle
excitations
1
2
∑
mnab
ρmnab am† an† ab aa Ψ v(0)
∑
mna
ρmnva am† an† aa av Ψ v(0)
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Calculation of the atomic wave functions
reduces to the calculations of
excitation coefficients ρ.
These are found by iterative solution
of a very large system of linear equations.
All-order method:
Correlation correction to ground state
energies of alkali-metal atoms
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RESULTS FOR ALKALIALKALI-METAL
ATOMS: E1 MATRIX ELEMENTS (A.U.)
Na
3p1/2-3s
K
4p1/2-4s
Rb
5p1/2-5s
Cs
Fr
6p1/2-6s 7p1/2-7s
All-order 3.531
4.098
4.221
4.478
4.256
Experiment 3.5246(23) 4.102(5) 4.231(3) 4.489(6) 4.277(8)
Difference 0.18%
0.1%
0.24%
0.24%
0.5%
Experiment Na,K,Rb: U. Volz and H. Schmoranzer, Phys. Scr. T65, 48 (1996),
Theory
Cs:
R.J. Rafac et al., Phys. Rev. A 60, 3648 (1999),
Fr:
J.E. Simsarian et al., Phys. Rev. A 57, 2448 (1998)
M.S. Safronova, W.R. Johnson, and A. Derevianko,
Phys. Rev. A 60, 4476 (1999)
Cs PNC: Comparison with the standard model
Standard Model [1] :
QWSM = −73.16(3)
Most current result for Cs PNC Expt/Theory:
Atomic physics [2] :
QWinferred = −73.16(29)expt (20) theory
No deviation from the Standard Model
[1] C. Amsler et al. (Partical Data Group), Phys. Lett. B 667, 1 (2008)
[2] S. G. Porsev, K. Beloy and A. Derevianko, PRL 102, 181601 (2009),
Phys. Rev. D 82, 036008 (2010)
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IMPLICATIONS FOR
PARTICLE PHYSICS
Confirms fundamental “running” (energy dependence) of the
electroweak force over energy span 10 MeV 100 GeV
Figure is from Bentz et al. Phys. Lett. B693, 462 (2010).
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Probing new physics
New physics can be phenomenologically described by weak
isospin - conserving S and isospin - breaking T parameters [1].
∆Q = QWinferred − QWSM = −0.800S − 0.007T
Present result [2]:
S < 0.45
Parameter S is important for indirect constraint on the mass
of Higgs particle [1].
[1] J.L. Rosner, PRD 65, 073026 (2002)
[2] S. G. Porsev, K. Beloy and A. Derevianko, PRL 102, 181601 (2009),
Phys. Rev. D 82, 036008 (2010)
Probing new physics: extra Z bosons
e
q
e
q
+
e
Z0
q
e
Z’
q
Atomic parity violation is uniquely sensitive to Z’
Z’χ in SO(10) GUT, Marciano & Rosner
inferred
W
∆Q = Q
SM
W
−Q
 0.736 TeV / c 2 
≈



M
Z
'

x

2
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Probing new physics: extra Z bosons
Z’χ in SO(10) GUT, Marciano & Rosner
∆Q = QWinferred − QWSM
 0.736 TeV / c 2 
≈



M
Z 'x


2
Cs result [1] implies M Z ' > 1.3TeV / c 2
x
Direct search at Tevatron collider [2]
M Z 'x > 0.82TeV / c 2
[1] S. G. Porsev, K. Beloy and A. Derevianko, PRL 102, 181601 (2009)
[2] T. Aaltonen et al., Phys. Rev. Lett. 99, 171802 (2007)
NEUTRON SKIN AND
ATOMIC PARITY VIOLATION
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Neutron skin correction to PNC amplitude
The relative neutron skin correction to the PNC amplitude is
independent from electronic structure and is given by:
∆E
3
2 ∆Rnp
= − (α Z )
EPNC
7
Rp
ns
PNC
Difference between rms radii of
neutron and proton distributions
rms radius of proton distribution
Neutron skin correction is 0.2% for Cs, but 0.6% for Fr and Ra+
Therefore, neutron skin can be extracted from APV studies
if their uncertainties are smaller than these values.
Pollock et al., PRC 46, 2587 (1992), Brown et al., PRC 79, 035501 (2009)
The neutron skin calculations, Skxs20
Different isotopes for a given element are connected by lines.
The filled circles show nuclei of interest for atomic PV.
Alex Brown, 2008 PREX workshop, Brown et al., PRC 79, 035501 (2009)
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Isotopic chain APV experiments
(1) APV isotopic chain experiments should allow to largely cancel
dependence on atomic theory.
(2) Errors due to neutron skin in two isotopes are correlated. This should
allows to extract new physics from such experiments
(3) The sensitivity to neutron skin effect is the largest for the lightest and
heaviest pair of isotopes in the chain. It may be possible to extract
neutron skin ∆Rnp from such experiments.
Atom
A
∆Rnp
ns
∆EPNC
EPNC
Yb (Z=70)
168
0.141(35)
-0.0031(8)
176
0.215(67)
-0.0046(14)
209
0.121(36)
-0.0038(11)
221
0.206(53)
-0.0064(16)
Fr (Z=87)
Brown et al., PRC 79, 035501 (2009)
NUCLEAR SPINSPIN-DEPENDENT
PARITY VIOLATION EFFECTS
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Parity violation in atoms
NUCLEAR
SPINSPIN-DEPENDENT
PNC:
STUDY OF PNC
IN THE NUCLEUS
The other part
of the story
a
Nuclear anapole
moment
Spin-dependent parity violation:
Nuclear anapole moment
a
Parity-violating nuclear moment
F=4
7s
F=3
1
6s
2
(a)
H PNC
=
GF
2
Valence
nucleon
density
κ a α ⋅ I ρv (r )
F=4
F=3
Anapole moment
Nuclear anapole moment is parity-odd, time-reversal-even
E1 moment of the electromagnetic current operator.
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Constraints on nuclear weak
coupling contants
W. C. Haxton and C. E. Wieman, Ann. Rev. Nucl. Part. Sci. 51, 261 (2001)
Nuclear anapole moment?
The constraints obtained from the Cs experiment
were found to be inconsistent with constraints
from other nuclear PNC measurements, which
favor a smaller value of the133Cs anapole moment.
Possible atomic calculation solution?
κ
= 0.117(16)
Incomplete correlation calculation of
spin-dependent PNC amplitude?
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More spin-dependent PNC effects
κ = κ a + κ 2 + κ hf
Weak-hyperfine
interference term
(Ve,AN)
interaction
This term does not reduce to the
same interaction but “effective”
constant κ hf can be calculated.
Same Hamiltonian as
anapole moment term
with κ a ⇒ κ 2
W.R. Johnson, M.S. Safronova and U.I. Safronova, Phys. Rev. A 67, 062106 (2003)
New all-order (CCSD) calculation
of spin-dependent PNC
Electric-dipole matrix elements
EPNC = A 1 ∑
( 2,a )
j≠v
7s d j
(2,a )
j H PNC
6s
E6 s − E j
+ A2∑
j≠w
(2, a )
7 s H PN
C j
j D 6s
E7 s − E j
PNC matrix elements
First four terms in the sums are replaced by
all-order matrix elements
1% accuracy is expected
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Nuclear anapole moment:
Test of hadronic weak interations
The constraints obtained from the Cs experiment
were found to be inconsistent with constraints
from other nuclear PNC measurements, which
favor a smaller value of the133Cs anapole moment.
All-order (LCCSD) calculation of spin-dependent PNC amplitude:
k = 0.107(16)* [ 1% theory accuracy ]
No significant difference with previous value k = 0.112(16) is found.
NEED NEW EXPERIMENTS!!!
Fr, Yb, Ra+
*M.S. Safronova, Rupsi Pal, Dansha Jiang, M.G. Kozlov,
W.R. Johnson, and U.I. Safronova, Nuclear Physics A 827 (2009) 411c
NEED NEW
EXPERIMENTAL
PNC
STUDIES
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EXPERIMENTAL
PNC
STUDIES
All-order
Correlation potential
CI+MBPT
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Conclusion
A: New analysis of atomic PNC experiment is Cs:
Nuclear spin-independent part:
(1) Provided most accurate to-date test of the low-energy electroweak
sector of the SM.
(2) Confirmed fundamental “running” (energy dependence) of the
electroweak force.
(3) Placed constraints are on a variety of new physics scenarios beyond
the SM.
B: New analysis of atomic PNC experiment is Cs:
Nuclear spin-independent part (anapole moment)
(1) New calculations, accurate to 1% - essentially the same result.
(2) Constraints on nuclear weak coupling constants are still
inconsistent with nuclear physics experiments.
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