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Subnuclear Physics in the
1970s
IFIC Valencia. 4-8 November 2013
Lecture 4
Measuring the weak mixing angle
Polarised electron-deuteron scattering
Parity violation in atoms
25-May-17
A. Bettini LSC, Padova University and INFN
1
The charged weak currents (CC)
CC: at the vertex incoming and outgoing charges are different, difference taken by W±
Space-time structure is V–A
Universality: all the couplings are equal (after Cabibbo rotation for quarks)
e–
g ne
m–
W–
ge µ 1  5 n e
g nm
W–
m 1  5 n m
t–
g nt
W–
t 1  5 n t
We can simplfy the expression of the currents:
 1  5  m  1  5 
m
ge  1  5 n e  2ge 
 
n e  2geL n eL
 2   2 
m
Weak charge  chirality: states coupled to W have negative chirality
25-May-17
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2
Electro-weak theory
In the EW theory the photon and the massive vector bosons of the weak interactions are
introduced as “gauge bosons”, initially mass-less
The symmetry spontaneously breaks down (Higgs) giving mass to W± and Z˚, leaving the
photon massless
The symmetry group is SU(2)U(1). Their fundamental representations contain respectively 3
and 1 objects: the gauge fields, two charged and two neutral
W = (W1, W2, W3) = corresponding to the (non Abelian) symmetry SU(2)
interact by means of the weak isospin, IW
B corresponding to the (Abelian) symmetry U(1)
interacts by means of weak ypercharge, YW=2 (Q–IW3)
Will see that W+ and W+ are (trivial) linear combinations of W1and W2, the photon and the Z are
linear combinations of W3 and B
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3
Weak isospin
The W±, which mediates CC, couples only to the Left components of leptons and quarks (all
of them)
Leptons. Each family has two Left leptons: one charged and its neutrino. They are lodged in
doublets of weak isospin IW=1/2

 IW3  1 / 2
 I  1 / 2
 W3
 
   n eL
  e
  L

,



n mL

=
 m–
 L

,



nt L

=
 t–
 L




Charged leptons are massive and have also a Right component. They are lodged in isospin
singlets (IW =0)
eR , m R , t R
Quark. Very similar, taking into account Cabibbo mixing.
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4
Electroweak mixing
Wµ = (Wµ1, W µ2, W µ3) is a space time four vector and an isovector (IW=1) in SU(2)
Interacts with the leptons charged current Jµ (four-vector and isovector) with the coupling
constant g
Bm is a four-vector isoscalar (IW=0)
Interacts with the leptons neutral current JµY (fourvector-isoscalar) trhough hypercharge with the
coupling constant g’
1

W1  iW2 
The fields of the physical bosons are W 
2
 Z0
 g g '  W3   cosW  sin W   W3 
1
  B    sin 
 A  
2
2 
g
'
g
cosW   B 

g  g'
W
W  tan1

g'
g
YW  2 Q  IWz
Weinberg angle

 J µY  2J µEM – 2J 3µ

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C.8 A. Bettini
5
Unification
L




g
g
J µWµ  J µWµ 
J µ3  sin 2 W J µEM Z µ  g sin W J µEM A µ
cosW
2
WI CC
WI NC
EM
Relationship with
the Fermi
constantè
GF
g2

2 8M W2
gsin W 
qe
 4
 0 hc
Electroweak unification
All the vector bosns interactions are
determined by the elementary electric
charge qe and W
Fermions both left and right are coupled to
the Z by the coupling constant
g
4
g
gZ 
IWz  Qsin 2 W 
IWz  Qsin 2 W 
cZ
cosW
sin W cosW
cosW



The Z-charges
are
25-May-17

cZ  IWz  Qsin2 W
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The neutral currents
NC have important differences from CC
•Couple a particle with itself only (ee, not em; uR uR, not uR uB, not uc, …)
•Are not V-A,  both Left and Right fields
The currents (for the 1˚ family)
j0,ne  gLne n e
1
1  5 n e  gLne n eLn eL
2
j0,e  gLe eL eL  gRe eR eR
j0,u  gLuuL uL  gRuuR uR
j0,d  gLd dL dL  gRudR dR
The 3x7=21 couplings are determined by two parameters = elementary electric charge and
weak angle sin2W
qe
The Z is universally coupled

I z  Qsin 2 W
sin W cosW
•couples to both left and right fermions
•couples to the Ws
•also couples to electrically neutral particles, provided they have Iz ≠0, such as neutrinos
•does not couples to states with both Q=0 and Iz =0, such as the  and itself.

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A. Bettini LSC, Padova University and INFN

7
Vector boson masses
Two constants to be determined by experiments: charge  and mixing angle sin2W
GF
g2

2 8M W2
EW unification+Fermi constant (from
beta decays etc.)
Experiments (examples to be discussed)
M W ; 80 GeV
1/2


F
MZ 
EW theory
25-May-17
 g2 2 
MW  
 8G 
1
37.3

GeV
2GF sin W sin W
MW
cosW
sin 2 W  0.232
M Z ; 90 GeV
modulo small higher order
corrections
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Measuring the weak mixing
The unification between electromagnetic and weak interactions appears mainly in the NC weak
processes. Where we measure the Z-charges that theory gives as functions of a sole parameter,
sin2
Experimentally verified in many processes, at different energy scales
in red to be discussed
•Parity violation in atoms (100 eV)
•Polarised electron deuteron scattering (GeV)
•Asymmetries in e+ e–  m+ m– (10 GeV - 200 GeV)
•Deeply inelastic scattering of nm on (several GeV)
•Elastic nm electron scattering (MeV)
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Inelastic ed scattering
Inelastic ed scattering gives information on sin2W
Need detecting interference between Weak NC (small)
and EM (large) processes
Quark level diagrams on the lefy
Experiment done at the LINAC of SLAC at
momentum transfer around q21.6 GeV2. Small
enough to have a pointlike effective weak interaction


4 2 gLe eL  eL  geReR  eR geL u L  u L  geRu R  u R  geL d L  d L  geRdR  d R

Hence
 8 2   

GF 
4 2  
2



1
4
sin

e

e

e


e
1
sin

u

u

1
sin

d

d

u


u

d


d




W 
 5 
W
W
5
5 





3
3


2
2

Parity violation comes from the product of terms of opposite parity
1.
vector electron current times axial quark current
2.
axial electron current times quarks vector currents
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10
ed
scattering
Parity violation appears as an asymmetry
Prescott et al. Prepared a circular polarized electron beam. The direction of polarization can
be inverted. The flux of electrons scattered by a liquid deuterium target at fixed q2 is
measured. The looked for asymmetry is the difference between the fluxes corresponding to the
two polarisations
X
e–(polarised) + d  e– + X
  
A 
Expected few
 
p.p.m. values
epol
Ein
Efin
counter
Calculation gives (considering only valence quarks)

GF q2 9  20 2 
2
A
1
sin

1
4
sin

K
y




W
W


 9

2 2 10 

A  1.62 10
4

2

Expected A = 10–4 q2 (GeV2)
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
y
 20 2 

2
q /GeV 1 sin W 1 4 sin W K y

 9
 
2

e

Ein  E fin
K y 
Ein
1 1 y
2
1 1 y
2
C.Y. Prescott et al. Phys. Lett. 77B (1978) 347; Phys. Lett. 84B (1979) 524

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11
The polarised source
Crucial technical development
A die LASER (pulsed in phase with the LINAC) goves light at l=710 nm linearly polarised
The orientation of the polarisation plane is defined by the direction of the axis of a calcite prism
The light beam enters a Pokels cell. It is a crystal with bi-refringence proportional to the square ot
the applied electric field – field intensity is chosen to have a l/4delay). Circular polarisation
(photons with helicity + or –) is obtained by rotating the calcite relative to the cell
Photons the enter a GaAs crystal, whose surface has been treated to have negative electron
affinity. They pump electrons from valence to conduction band, giving them their helicity
By rotating the light polarisation plane (the calcite) by an angle FP the electrons helicity varies as
cos(2FP)
Helicity can be inverted by inverting the electric field on the cell. This can be done often and
randomly, in order to minimize drifting effects on the asymmetry
N.B. A linear accelerator does not contain magnetic field gradients, hence it does nothing on
polarisation
25-May-17
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12
The Prescott experiment at SLAC
Measure beam polarisation
Møller Scattering (elastic e–
e– ) in a magnetised Fe foil
Pe=[37±1(stat)±0.2(syst)]
%
Spectrometer selects charged particles of
definite momentum (≈80%Ein)
Beam intensity= 4x1011 el/pulse
Intensity at detectors = 1000 el/pulse
Detectors (2 for cross-checks) on the beam
analysed by spectrometer
Flux is large, cannot count particles
Ni is the output signal for the pulse I of the PM
of the Cherenkov or the Lead-glass calorimeter
Qi is the measured charge of the beam pulse i = integrated beam current
Count I is defined as Yi=Ni/Qi
Average <Y+> for helicity + and <Y–> for helicity –
Aexp 
Y  Y
Y  Y
Check on non polarised beam Aexp=(–2.5±2.2) x 10–5
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13
Asymmetry. First method
The following values were chosen for the angle
FP between calcite and cell and of the sign of
DV on the cell
FP=0˚
DV>0
electron helicity +
DV<0
electron helicity –
FP=45˚
DV>0
electron helicity 0
DV<0
electron helicity 0
FP=90˚
DV>0
electron helicity –
DV<0
electron helicity +
Expected behaviour Aexpect=Pecos(2 FP)
Compatible with 0. OK
Opposite values. OK
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14
Asymmetry. Second method
Second method. Vary electron helicity
Electrons spins perform precession in the beam
transport magnetic structure (remember g–2≠0). The
total precession angle depends on the beam energy Ein
Ein GeV 
Ein g 2


 rad
bend
2 2
3.237
mec
E GeV 
Aexp  Pe Acos in

 3.237

Measured at dfferent beam energies
Ein = 16.2 - 22.2 GeV, q2 = 1. - 1.9 GeV2
The EM amplitude varies as 1/q2

Hence asymmetry A should e proportional to q2
Plot A/Peq2 vs energy
 prec 
 20


A  1.62 104 q2 1 sin 2 W 1 4 sin 2 W K y

 9

2
1 1 y
K y 
Ein  E fin
2
y
 0.21,
q 2  1.6 GeV2
1 1 y
Ein

25-May-17


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15
Systematics and result
Systematic effects can (do) induce spurious asymmetries
Drifts in the PM gain
effect is minimised by frequent inversions
Beam fluctuations
under control, residual effects small enough
Variations in beam parameters correlated with helicity changes (dangerous)
accurate monitoring of the beam parameters, register data, feedback corrections using
micro-processor
DEin/Ein<10–6D(A/q2)<0.26x10–5
A/q2=(–9.5±1.6) x 10–5 GeV–2
Final result
(theor.)
Total uncertainty = linear sum of
statistical
systematic on Pe
systematic on beam parameters
±9%
±5%
±3.3%
sin2W = 0.224±0.012 (stat) ±0.008 (syst.) ±0.010
Theoretical error from the uncertainty on the deuteron quark structure
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16
Atomic parity violation
Landmarks
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17
Stark-PV-interference technique
Invented by the Bouchats in the 1970s
Phys. Lett. 48B (1974) 111 and J. Physique 35 899–927
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Atomic parity violation
Electrons are bound to atomic nuclei by exchanging photons and Z˚
The latter contribution is too small to be observed as a shift of energy level, but
polarisation effects due to the interference between the two amplitudes can be observed
Sensitive to electron-quark coupling
Energy scale much different from accelerator experiment
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19
Atomic parity violation
Precision limiting factors
experimental accuracy
complexity of the atom system
theoretical uncertainties in connecting the measured quantity with the weak charge
Expected effect grows with the nuclear charge as Z3
Less uncertain theory for simple atoms  alkaline
Good candidate Cesium
Only one stable isotope 133Cs, Z = 55, N = 78, A= 133
Atomic level outside closed shell (valence electron) 6s
Atom angular momenta
J = L + S = 0 + 1/2 = 1/2
Nuclear spin
I = 7/2
Total angular momentum
F = 3 and 4
Energy difference between F=3 and F=4 9.19 GHz
S.L. Gilbert e C. E. Wieman P.R. A, 34 (1986) 792
M.C Noecker et al P.R.L. 61 (1988) 310
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20
Theory
Dominating contribution is the product of the electron axial current and the nucleon quarks vector
current
We can define the Z-charges of the nucleons as sums of those of their quarks
gVp  2gVu  gVd ,
gVn  gVu  2gVd ,
Considering that T3 and the charge Q are additive, the relations are the same as for quarks
1
gVnucleon  T3  Q sin 2
W
gAe  T3 
Also
2
The interesting interaction is



GF
e   5e n   n  1 4 sin 2 W p  p
2 2


The effective potential can be calculated consdering that
 to the momentum transfer is large compared to the nuclear
•the wavelength corresponding
radius
•the electron is non-relativistic, and its wave function becomes


x 

 r

r
 x      p
with  x  two component spinor
,

x


2m
e

 LSC,A.Padova
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Bettini University and INFN
A. Bettini
21
Parity violating effective potential
We substitute
Calculating effective potential we obtain for neutron and proton
p  
r r r
GF  r r r
GF 1 4 sin 2 W  r r r
r r r

Vn r  
i




r

i

r



     Vp r  

i   r  i r   
4 2me
4 2me
over the Z protons and N neutrons of the nucleus obtaining, the total weak potential
We sum
VNPC r  

 

GF
Z 4 sin 2 W 1  N i r   ir  
4 2me
Notice that the potential is pseudoscalar. As such it mixes levels of opposite parity.
We tink to the single valence electron (of the alkaline) as living in the average potential of all
the other components of the atom
The mixed levels are then S1/2 and P1/2
The matrix element must be calculated
S1/2 VNPC P1/2 



3i GF 
2
 dRL1 0 R 0
1
4
sin

Z

N
W
L0
 dr
16mec 2 

QW  1 4 sin 2 W Z  N
is the parity violating “weak charge”
There are uncertainties on the radial wave function at the origin R(0) and its gradient dR/dr
25-May-17

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22
Boulder Cs experiment. Scheme
The parity violating potential mixes S and P
states, making possible a small dipole (E1)
transition amplitude between states of the
same parity: APNC
We measured it by observing its interference
with a much larger parity conserving
amplitude AST
AST = electric dipole transition induced
through Stark effect by an external stationary
resonant excitation of transitions 6S7S in
electric field E, which mixes S and P.
the presence of perpendicular crossed
electric E and magnetic B fields
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23
Boulder Cs experiment. Continues
A(4,4;3,3)
B stationary (weak = 7 mT): separates m levels
The LASER polarised beam electromagnetic
field induces the wanted transitions
Cs target is an atomic beam, in order to reduce
substantially the Doppler broadening and
allowing working with narrow lines
Transitions to be induced
6S1/2  7S1/2
DF =1, Dm =±1
25-May-17
F m  F' m'
4 4 3 3
4 4  3 3
3 3 4 4
3 3  4 4
A. Bettini LSC, Padova University and INFN
Measure the
intensity of these
radiations to
monitor the rates
of the LASER
induced
transitions
24
Space structure of the observables
The observables whose matrix elements correspond to
P conserving processes are scalars, those that violate P
are pseudoscalar.
Transition to be observed are induced by the LASER
field must be proportional to the wave electric field 
and to its magnetic field [equivalently to k] and also
to E
The structures of all possible observables are obtained taking the scalar products of , k e
E with the availbale vectors (k, E) and axial vectors (B e Ek). The latter are by
construction parallel, and consequently equivalent (provided angles are perfectly 90˚)
Scalar quantities are
.k zero for EM wave (but not for diffuse radiation)
.E “scalar” polarizability. Gives zero contribution to transitions DF≠0
k.B magnetic dipole (small). Equivalent to k.µ (with µ electron magnetic moment)
E.B “vectorial” polarizability
One pseudoscalar quantitye
.B Parity violating potential. Equivalent to . (with  electron spin)
(the other expressions are zero or non-independent. Example k.E= E.k=kE. that is
not independent from B. 
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25
Transition amplitude
The electric dipole transitions AST are due to the interaction between the wave electric field 
and the dipole moment d induced by the stationary electric field E (Stark effect). these are the
matrisx elements of bE.B. Here b is the vector poralizability (spin-orbit) that must be
calculated
The dipole moment transition amplitudes AM1 are due to the interactions of the wave magnetic
field, directed as k   and the magnetic moment m directed as B. Hence: M1 k.µ (“strongly
forbidden” transition)
The parity non conserving amplitudes APNC due to VPNC .
For all are different from zero only transitions with m’=m and m’=m±1
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A. Bettini LSC, Padova University and INFN
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Interference
For the transition between two given levels m and m’
maximize
FF'
I mm'
 AST  AM1  APNC
2
minimize
 Interference between the large
amplitudes AST and AM1 is zero if B and
k are exactly perpendicular
The single interference terms between AST and the small APNC are independent on m and
m’. The resulting interference is the weighted sum of the contributions of the different m.
It is zero if the positions of the levels are not different.
Questa è la funzione del campo B
Control of the systematics is vital
Need to verify that the directions of the fields and of the beam are relly perpendicular
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A. Bettini LSC, Padova University and INFN
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What do we expect?
It is easily shown that the spectrum of the 6S(F=4) 7S(F=3) transition is made of 8 lines,
equidistant for weak magnetic fields (Zeeman)
The two extreme ones correspond to one transition, m =4  3 and m =–4  –3 respectively,
while the other ones are sum of one Dm=1 and one Dm=–1
Stark transitions induced by E
Spectrum due to the parity violating potential.
Enlarged
Expected spectrum
We must measure the asymmetry between the
intensities at m and –m (expected O(10–6))
frequency
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The weak magnetic field induces an asymmetry,
which can be accurately calculated
A. Bettini LSC, Padova University and INFN
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Parenthesis. A modern version
25-May-17
A. Bettini LSC, Padova University and INFN
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The experiment
We chose the pump LASER frequency to define the
levels between which the transition should happen
6S(F=4, m =4 ) 7S(F=3,m=3)
and (after 4 h)
6S(F=4, m =–4 ) 7S(F=3,m=–3)
The tuned Fabry Perot cavity effectively
amplifies the LASER power by 20
Main experimental points: 4 quasi
independent spatial inversion operations
Parity violating interference changes sign
for each inversion, while the (much larger)
Stark term does not
• Inverting E (0.2 Hz)
• Inverting B (0.02 Hz)
• Inverting circular polarisation. (2 Hz)
• Inverting the sign of m by changing the
LASER frequency (30 min.)
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Circular polarisation
can be inverted rapidly
Intensity = 1015
atoms cm–2s–1
Photodiode sensitive at 852/894 nm
A. Bettini LSC, Padova University and INFN
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Results
Measured quantity
VNPC/b
Bouchat et al. 1984
VNPC/b = –1.78±0.26(stat.)±0.12 (syst) mV/cm
Noecker et al. 1988
VNPC/b = –1.576±0.034(stat.)±0.008 (syst) mV/cm
Polarisability b known from computations b = 27.0 a0 with 5% uncertainty
Weak charge is
QW= Z(1–4sin2W)–N = 55 (1–4sin2W)– 78
for Cs
To extract it from the data wave functions and their uncertainties must be known. One obtains
(Noecker)
QW= –69.4±1.5 (sperim.)±3.8 (theor.)
sin2W = 0.219±0.007±0.018
Present values
QW= –73.20±0.35
sin2W = 0.2385±0.0015
Radiative corrections must be taken into account when comparing with other measurements of
sin2W
APV gives the most precise measurement of the coupling between vector weak electron current
and d- quark current (u-quark contribution is small)
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A. Bettini LSC, Padova University and INFN
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sin2W “runs”
sin2W as a function of energy scale. APV: atomic parity violation; APV : asymmetry in
polarised Møller scattering; Z-pole measurements; AFB: forward-backward
asymmetry at LEP2
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A. Bettini LSC, Padova University and INFN
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