Bounds on New
Physics from APV
in cesium
R. Casalbuoni
(S. De Curtis, D. Dominici and R. Gatto)
hep-ph/9905568
CERN July 21 1999
Experiment 1999
_ (0.28)exp +_ (0.34) theor
+
Q W (133
72.06
Cs
=
)
55
= - 72.06 +_ 0.44 (0.6 %)
Standard Model
_ 0.13 (light Higgs)
+
Q W (133
-73.24
Cs
=
)
55
(hadronic loops)
Deviation
exp
QW
_ Q theor = 1.18 +_ 0.46
W
2.57 SD
disfavored at 99% C.L.
LEP physics puts heavy constraints on new
physics. But which kind of new physics is
detectable at LEP?
~
new particles
Z'
~
New physics invisible at LEP may come
from new 4-fermi interactions
E << M
Z'
Z'
contact interaction
Main physical eects of contact interactions is
to change the eective couplings of the gauge
bosons to the fermions. Although LEP puts
heavy constraints on the couplings it is not
really capable of measuring the couplings to
the light quarks. A good possibility are experiments in
Atomic
Parity
Violation
Measure the combinations veaq , aevq . In particular from the atomic cesium one gets
c1u1d = ;8 ae v1u1d
(see g.)
c 1u
_
c 1d
0
Cs (1999)
-0.2
Cs (1991)
-0.4
s
eD
2
θ
SM
-0.6
-0.8
eC
-1
0.12
0.14
0.16
0.18
0.2
c 1u + c 1d
Summary
Discussion of the experiment on atomic cesium
Implications of APV for new physics
{ New Vector Bosons
{ Extra-dimensions
{ Composite Models
{ Lepto-quarks
Conclusions
Atomic Parity Violation
Within the SM the relevant 4-fermi PV interaction between charged leptons and quarks is
given by
X
G
F
PV
Le = p2 (`5`)
c1q q q+
q=ud
+(``)
X
where
q=ud
c2q q 5q
c1q = ;8a`vq = ;(T3q ; 2s2 Qq )
c2q = ;8v`aq = ;T3q (1 ; 4s2 )
1
af = ; 2 T3fL
1
f
2
f
vf = 2 T3L ; 2s Q
In terms of nucleons
G
F
LPV
e = ; p2 (`5`)
X
N =pn
+(``)
X
N =pn
N +
c1N N
5 N
c2N N
where
c1p = ;2c1u ; c1d c1n = ;c1u ; 2c1d
In the non-relativistic limit one gets, for a
single nucleon,
G
HPV = p F c1N ~` p~ 3(~r)]++
2 2m
`
+c2N ~N p~ 3(~r)]+ ; ic2N (~` ^ ~N ) p~ 3(~r)]+
Finally, for a point-like nucleus
G
HPV = p F QW (Z A)~` p~ 3(~r)]++
4 2m
`
+2(c2pS~p+c2N S~n) p~ 3(~r)]+
;2i~` ^ (c2pS~p+c2nS~n) p~ 3(~r)]+
The weak charge of the nucleus is dened as
h
QW (Z A) = 2 c1pZ + c1nN
i
For large values of Z , the term in QW is dominant. There is a coherence eect (Bouchiat
and Bouchiat 1974-75)
hHPV i / Z 2QW (Z A) / Z 3
In fact, since
HPV / QW ~` p~ 3(~r)]+
one gets one Z from p~ and one Z from the wave
function at the origin. Whereas, for large Z ,
spin terms tend to cancel.
The general idea is to consider two parity eigenstates mixed by HPV , with + of the same
nominal parity as 0.
ψ+
∆E
ψ_
R+
R_
ψ0
R _+ = Decay probabilities
with
j+i ! j+i + j;i
h
j
H
j
i
;
PV
+
=
E
and construct the quantity
!1=2
2
2
jh
+
j
ij
;
jh
+
j
i
j
R
0
0
P
A = jh + jij2 + jh + ji j2 2 R ;
0
0
P
+
h
j
H
j
i
;
PV
+
=
E
To get big A requires
h+jHPV j;i large:
{ coherence
{ large overlap of with the nucleus
E small
R;=R+ large, R+ suppressed transition
Large overlap
It could be realized in nuclei
h+jHPV j;i V1
Since
h
j
H
j
i
;
PV
+
=
E
we get
atoms Z 3 1 1 r3E
r 3 E
atoms
nuclei
nuclei
!3
Enuclei Z 3(10;4)3106 Z 3 rrnuclei Eatoms
atoms
Z 3 10;6
For an atom like cesium, Z = 55 (Z 3 2 105)
we get
nuclei 1
atoms
In presence of parity violation there are two
physical eects related to the light propagation, due to the dependence of the refraction
index on the circular polarization of photons
traveling in a medium of atomic vapor density
D
q
i
h
2
D
n = 1 ; / R+ 2Im() R+R; f (! !0)
h
For a photon of frequency !, close to a resonant frequency !0, f (! !0) describes the lineshape of the resonance.
Optical rotation: The dierence of the real
parts of n gives rise to a rotation of the
polarization plane of a linearly polarized
laser beam tuned to !0
q
Im() R+R;
Circular dichroism: The resonant absorption of circularly polarized photons depends
on the polarization
v
u
u
t
; ; 2Im() R;
= + +
R+
+ ;
The amounts of uorescence light depends
on the handedness of the radiation
Last technique has been used in conjunction
with a forbidden transition R+, in atomic cesium, giving rise to 10;4 10;3, in Paris
(1982, 1984) and in Boulder (1985, 1988, 1997).
To overcome the background the interference
with a large electro-induced (Stark) transition
has been used.
Experiment ;! hHPV i QW PV .
The atomic form-factor PV must be evaluated theoretically.
I nucleus = 7/2
7S
F=4
F=3
γ
J=3/2
6P
540 nm
852 nm
890 nm
6S
F=4
F=3
J=1/2
γ
Status of the theoretical
evaluation of P V
Although the cesium is a relatively simple atom
since it is well described by a single valence
electron outside a spherically symmetric core,
the evaluation is not from rst principles
Many-body perturbative theory with HartreeFock potential
{ Exp. values of PC observables agree
within a given error
{ Stability within a given error against variation of the parameters
{ Dierent approximation scheme gives the
same result within the given error
Taken into account
{ Nuclear distribution
{ Nuclear spin-dependent eects
{ Z -exchange among the electrons
(< 0:03%)
Many auxiliary variables have been computed
Allowed E 1 transitions rates and excited
states lifetimes (test at large r)
Energies and ne structure splittings
Hyperne structure splittings (test near
r = 0), very important for APV
Stark-induced E 1 amplitudes
Theoretical error in 1992
1%
In 1999 down to 0:4%!!!!
The main reason is that after new measurements of relevant quantities in cesium the agreement with the theoretical calculation is much
better. Furthermore a problem of the previous calculations, when applied to sodium and
lithium, leading to 1% discrepancy in the lifetimes, it has now disappeared after new experiments.
=
=
6S HFS
7S HFS
6P1 2 HFS
7P1 2 HFS
=
=
Quantity
measured
!7S dc Stark shift
6P1 2 lifetime
6P3 2 lifetime
S
S
P
P
=
=
=
=
Calculation
Dierence (103 )
tested
Dzuba, et al. Blundell et al.
h7P jj D jj 7S i
;3.419]
;0.722]
h6S jj D jj 6P1 2i
;4.2;8]
4.31]
h6S jj D jj 6P3 2i
;2.6;41]
7.9;31]
h7S jj D jj 6P1 2 i, and
h7S jj D jj 6P3 2i
;
;1.4
same as ;
;0.8
6 (r = 0)
1.8
;3.1
7 (r = 0)
;6.0
;3.4
3
h1=r i6
;6.1
2.6
h1=r3i7
;7.1
;1.5
0.2
0.2
0.5
;
3.2
3.0
E xpt
1.04]
1.043]
2.322]
Many Body Perturbation Theory
Decompose the hamiltonian as
X
H = HC + HBr = Hfree + Vnucl: + j~r ; ~r j
i j
HBr = spin part
then introducing one-body potential P U (~ri)
HC = H0 + VC
H0 = Hfree + Vnucl: +
X
X
U (~ri)
X
;
U
(
~
r
)
i
j~ri ; ~rj j
since H0 is separable one has to solve one-body
problem, and then use this wave functions to
construct the Fock space. The Coulomb part
is dened in the Fock space as
X
X
1
y
y
VC = 2 gijklai aj alak ; Uij ayi aj
VC =
ij
gijkl = j~r ;1 ~r0j iy(~r)k (~r)jy(~r0)l(~r0)
Z
The fundamental state of the cesium is taken
as
jvi = ayv j0C i
that is a valence electron creation operator
acting upon the ground-state conguration of
the xenon. The one-body potential is taken
as the frozen-core V N ;1. One solves HF for
Cs+ in the conguration of the ground-state
of the xenon. Then keeping these orbitals
xed one solves for the valence orbitals. A
further improvement is obtained by expanding
the ground state wave function as
X
y
y
ji 1 + ijklai aj akal + tripleterms j0C i
and solving in the self-consistent HF potential
for the coecients in the expansion.
An alternative calculation is to start in a parity
mixed basis by putting HPV in the one-body
hamiltonian. Agreement within a few per mill.
The data on APV
In 1988 (Boulder) gets QW at a level of 2.5%
133
QW 55 Cs = ;71:04 (1:58)exp (0:88)theor
to be compared with the SM result
SM
133
QW 55 Cs = ;73:24 0:13 mH 100 GeV
leading to
theor = 2:2 1:81 (1:21 SD)
Qexp
;
Q
W
W
In 1999 the same group, with uncertainty 0:6%
133
QW 55 Cs = ;72:06 (0:28)exp (0:34)theor
theor = 1:18 0:46 (2:57 SD)
Qexp
;
Q
W
W
SM disfavored at 99 % CL
For increasing mH , QW decreases and the discrepancy increases
Let us parameterize
QW = ;72:72 0:13 ; 102rad
3 + N QW
rad
3 comes from radiative corrections in the
SM and depends on mH and mt. For mt =
175 GeV
;3
mH = 100 GeV rad
3rad = 5:110 10;3
mH = 300 GeV 3 = 6:115 10
;
3
mH = 1000 GeV rad
=
6
:
65
10
3
LEP and SLC physics constrains strongly deviations from the SM (see the pulls). For instance, considers new physics contributing to
the Z self-energy (oblique corrections)
contributes to QW as rad
3
N QW (oblique) = ;102 3N
Tampere 1999
Measurement
Pull
mZ [GeV]
91.1871 ± 0.0021
.07
ΓZ [GeV]
2.4944 ± 0.0024
-.53
σhadr [nb]
0
41.544 ± 0.037
1.78
Re
20.768 ± 0.024
1.15
Afb
0.01701 ± 0.00095
.96
Ae
0.1483 ± 0.0051
.35
Aτ
0.1425 ± 0.0044
-.91
sin θeff
0.2321 ± 0.0010
.51
mW [GeV]
80.350 ± 0.056
-.48
Rb
0.21642 ± 0.00073
.83
Rc
0.1674 ± 0.0038
-1.27
0,b
0.0984 ± 0.0020
-2.15
Afb
0,c
0.0691 ± 0.0037
-1.15
Ab
0.912 ± 0.025
-.90
Ac
0.630 ± 0.026
-1.45
0,e
2 lept
Afb
2 lept
sin θeff
2
0.23109 ± 0.00029 -1.71
sin θW
0.2255 ± 0.0021
1.09
mW [GeV]
80.448 ± 0.062
1.15
mt [GeV]
174.3 ± 5.1
.13
(5)
∆αhad(mZ)
Pull
-3 -2 -1 0 1 2 3
0.02804 ± 0.00065
-.10
-3 -2 -1 0 1 2 3
To compensate the discrepancy one would need
3N = (;11:6 4:5) 10;3
whereas
rad + 3N = (4:19 1) 10;3
exp
=
3
3
and for a light Higgs
3N = (;0:92 1) 10;3
almost an order of magnitude too small.
We need new physics not constrained by
LEP.
Bounds on N QW
theor (mH )
QW = Qexp
;
Q
W
W
QW = 0:66 + 1023(mH ) ; N QW 0:46
At a given CL we have the bound
b; N QW b+
b = 0:66 + 1023(mH ) 0:46 c
where c depends on the CL. For a light Higgs
95%CL 0:28 N QW 2:08
99%CL 0 N QW 2:37
The positive lower bound implies strong restrictions on new physics. For increasing mH
both bounds increase.
Models of New Physics
Extra-U(1) Models
Models with a further massive neutral vector
boson, Z 0 coupled to ordinary fermions with a
current
h
i
f
0
0
JZ = f vf + 5af f
0
e
-
u, d
Z'
e
-
u, d
The couplings depend on the angle 2
2
3
s
1
1
5
0
4
ae = 4 s ; 3 c2 + 3 s25 c2 = cos 2
2
s
3
vd0 = 14 s 4c2 + 53 s25
vu0 = 0
The corrections at QW , for Z 0 non mixed to Z ,
is
h
i M2
N QW = 16a0e (2Z + N )vu0 + (Z + 2N )vd0 M 2Z
Z
Bounds on N QW ! 95 % CL bounds on MZ
0
0
mH(GeV) = 100 1500
M Z' (GeV)
1250
1000
CDF
m H (GeV) = 300
750
500
250
-1.5
-0.5
-1
0.5
1
1.5
θ 2 (rad)
Lower positive bound on N QW ;! upper bound
on MZ . Excluded region ;! N QW 0.
Direct search at Tevatron ;! gives approximately MZ (GeV ) 600.
0
0
LR models
LR symmetric models with fermionic
couplings
0 = ;aSM
SM
a0evud
e vud
giving
2 SM
M
Z
N QW = ; M 2 QW > 0
Z
The bound is (for light Higgs)
540 MZ (GeV ) 1470
The limit from Tevatron is MZ 630 GeV
0
0
0
Notice that a simple scaled Z 0 model would
give the opposite sign
2 SM
M
Z
N QW = M 2 QW < 0
Z
Excluded at more than 99% CL
0
Extra-dimension Models
bulk (5 - dim)
γ
Z W g
fermions
wall (4 - dim)
= infinite tower of KK-resonances in 4 dim
The extra-dimensions are supposed to be compactied with a compactication radius R (>
than a few TeV). The KK excitations of the
SM gauge bosons have a mass
2
n
2
MKK = R2
The couplings of the KK modes to fermions
are
p
KK couplings = 2 SM couplings
If there were only Z -like KK modes, the correction to QW would be
1
2
p 2X
R
MZ2 n2 QW
N QW = ( 2)
n=1
However the W -like KK modes give a correction to GF implying (MZ2 > MW2 )
1
2
p 2X
R
2
2
N QW ! ( 2)
(MZ ; MW ) n2 QW < 0
n=1
The change of s also gives N QW < 0.
Extra-dimension models are disfavored at more
than 99% CL for any compactication radius.
The result does not change in presence of mixing terms as long as sin < 0:707 (N QW < 0).
Z Physics (E mZ << M )
2
X
charged
2
2
Leff =;~g(1 ; s~c X ) Ji W i
i=1
2
2 X
~
g
; 2m2 X Ji J i
Z i=1
e J Z Z (1 ; s2 X ) ; eJ emA
Lneutral
=
;
eff
~s~c 2
2
e
e
Z
Z
; 2~s2~c2m2 XJ J ; 2m2 XJemJ em
Z
Z
2 m2
X = 3 MZ2
=
!
2
~s2 = s2 1 + c~c 2
2
2
2
4
~c (1 ; s ) ; s X
m2W
m2Z
~e = e
2
e
G
F
p = 8s2c2m2
2
Z
!
2
s
2
= c 1 ; c rW
2
2
2
4
rW = ~c 1 ; 2s ; s X
Low-energy Physics (E << mZ )
X
2
G
F
low
;
en
Leff =;4 p2
Ji J i
i=1
+(1 + s2 (1 ; s2 )2X )JZ J Z
Deviations from the SM
Z-pole
2
s
2
2
1N = ;c X 1 + s c2 (1 + c2 )]
2N = ;c2 X 3N = ;2c2 s2 X
Low-energy
2c2
s
QW = QW 1 + s2 X (s2 ; 1)2] ; 4 c Z 2
=
2
2
2
4
~c (1 ; s ) ; s X
95% CL Bounds on Extra Dimension Models
from low-energy data
2 X f (~n2) 2 m2
m
Z (for 1 ED)
X = 2 MZ2
!
2
3 M2
~n ~n
0.05
X
0
0.2
0.4
0.6
0.04
1
0.03
APV
CHARM
0.01
0.02
0.01
95% CL
0
0.2
0.05
RCDHS
ν
0.04
Rν
0.02
0
CCRF
NuTeV
0.03
0.8
0.4
0.6
0.8
1
0
sβ
95% CL Bounds on Extra Dimension Models
from APV and high-energy data
2 X f (~n2) 2 m2
m
Z (for 1 ED)
!
X = 2 MZ2
2
3 M2
~n ~n
X.103
0.0
15
10
0.2
0.4
0.6
0.8
95% CL
1.0
15
APV
10
m H = 100 GeV
m H = 300 GeV
5
5
High-energy
0
0.0
0.2
0.4
0.6
0.8
1.0
sβ
0
95% CL Bounds on Extra Dimension Models
M(GeV)
0
4000
0.2
0.4
0.6
0.8
1
3500
95% CL
3500
3000
3000
EW
2500
2500
2000
2000
1500
NuTeV
1000
500
4000
0
0.2
0.4
R CHARM
ν
APV
1500
1000
500
0.6
0.8
1
sβ
R CDHS
CCRF
ν
95% CL Bounds on Extra Dimension Models
from APV and high-energy data
M(GeV)
0.0
0.2
4000
95% CL
0.4
0.6
0.8
1.0
High-energy
3000
4000
3000
m H = 300 GeV
2000
2000
m H = 100 GeV
APV
1000
0.0
0.2
0.4
0.6
0.8
1.0
sβ
1000
Other Contact Interactions
Using
h
QW = 2 c1pZ + c1nN
i
and
c1p = ;2c1u ; c1d c1n = ;c1u ; 2c1d
we nd
QW = ;2 (2Z + N )c1u + (Z + 2N )c1d]
The contact interactions modify the coecients c1u1d
c1u1d ! c1u1d + C1u1d
C 0 ;! N QW 0
leads to exclusion at more than 99 %CL
Composite Models
2
g
L = 2 e ; e q; q
with ; = 1;25 .
c1u1d ! c1u1d + C
In these models the contact interaction is generated by a strong interaction, then we expect
g2 4
p2
C = G 2
F
The negative sign in L is excluded, whereas for
the positive sign we have the 95 % CL bound
12:1 (TeV ) 32:9
Lepto-quarks
e-
q
MS
q
e-
2
2
L
R
L = 2M 2 eL ; eL uL ; uL + 2M 2 eR ; eR uR ; uR
S
S
with eL = 1;25 e.
p 2
2LR
c1u ! c1u + C C = 8G M 2
F S
From 0 ! `+`; ! L 0 or R 0.
If R 6= 0, C > 0 ;! N QW < 0 (excluded).
If R = 0 (L 6= 0) the 95 % CL bound is
1:7 MS (TeV ) 4:5
L
or, assuming a weak interaction, L2 4
em
0:5 MS (TeV ) 1:4
Conclusions
The new determination of QW could be affected by several errors
Statistical uctuations
Bigger errors on hHPV i ;! peculiar modications of , in view of the agreement with
PC quantities, testing all possible distances
Overlooked SM corrections to the atomic
quantities
Otherwise:
clear indication of new physics
Then our main conclusions are
The SM is disfavored at 99 % CL
Only new physics with no measurable effects at LEP is possible
QW puts lower and upper bound on new
scales
In particular models
{ Extra-U (1), the region ;0:66 2 0:25 is excluded, upper bounds on MZ
1 1:5 TeV (similar for LR)
0
{ Extra-dimension disfavored at more than
99% CL (sin < 0:707)
{ Composite models, 12 (TeV ) 33
{ Lepto-quarks, 0:5 MS (TeV ) 1:4
References
M.A. Bouchiat and C.C. Bouchiat, Phys. Lett.
B48 (1974) 111.
M.A. Bouchiat, J. Guena, L. Hunter and L.
Pottier, Phys. Lett. B117 (1982) 358.
M.C. Noecker, B.P. Masterson and C.E. Wieman, Phys. Rev. Lett. 61 (1988) 310.
C.S. Wood, S.C. Bennett, D. Cho, B.P. Masterson, J.L. Roberts, C.E. Tanner and C.E.
Wieman, Science 275 (1997) 1759.
S.C. Bennett and C.E. Wieman, Phys. Rev.
Lett. 82 (1999) 2484.
V. Dzuba, V. Flambaum, P. Silvestrov and O.
Sushkov, Phys. Lett. A141 (1989) 147.
S.A. Blundell, W.R. Johnson and J. Sapirstein,
Phys. Rev. Lett. 65 (1990) 1411.