1 A. Bettini LSC, Padova University and INFN Subnuclear Physics in the 1970s 15-Oct-15 IFIC...

1A. Bettini LSC, Padova University and INFN

Subnuclear Physics in the 1970s

20 Apr 2023

IFIC Valencia. 4-8 November 2013Lecture 4

Measuring the weak mixing anglePolarised electron-deuteron scattering

Parity violation in atoms


The charged weak currents (CC)

20 Apr 2023

CC: at the vertex incoming and outgoing charges are different, difference taken by W±

Space-time structure is V–AUniversality: all the couplings are equal (after Cabibbo rotation for quarks)

e– e

W–

–

W–

–

W–

geγµ 1−γ5( )e γα 1−γ 5( )ν μ γα 1−γ 5( )ν τ

g g g

geγ 1−γ5( )e =2ge 1+γ52

⎛

⎝⎜

⎞

⎠⎟γ 1−γ5

2⎛

⎝⎜

⎞

⎠⎟e =2geLγ

eL

Weak charge chirality: states coupled to W have negative chirality

We can simplfy the expression of the currents:


Electro-weak theory

20 Apr 2023

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


Weak isospin

20 Apr 2023

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

IW3=−1/ 2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

eL

eL

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟, =

L

L–

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟, =

L

L–

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Charged leptons are massive and have also a Right component. They are lodged in isospin singlets (IW =0)

eR−, R

−, R−

Quark. Very similar, taking into account Cabibbo mixing.


Electroweak mixing

20 Apr 2023C.8 A. Bettini

Wµ = (Wµ1, W µ

2, 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 gB is a four-vector isoscalar (IW=0)

Interacts with the leptons neutral current JµY (fourvector-isoscalar) trhough hypercharge with the

coupling constant g’

Z 0

A

⎛

⎝⎜⎞

⎠⎟=

1

g2 +g'2g −g'

g' g⎛⎝⎜

⎞⎠⎟

W3

B

⎛

⎝⎜⎞

⎠⎟=

cosθW −sinθW

sinθW cosθW

⎛⎝⎜

⎞⎠⎟

W3

B

⎛

⎝⎜⎞

⎠⎟

€

θW = tan−1 g'g

Weinberg angle

The fields of the physical bosons are W ±=12

W1 ±iW2( )

YW =2 Q−IWz( ) ⇒ J µY =2J µ

EM –2J 3µ


Unification

20 Apr 2023

L = g2

J µ−W+

µ+ J µ+W−

µ( )+

gcosθW

J µ3 −sin2θWJ µ

EM( )Z

µ+gsinθWJ µEM Aµ

WI CC WI NC EM

All the vector bosns interactions are determined by the elementary electric charge qe and θW

Fermions both left and right are coupled to the Z by the coupling constant

Relationship with the Fermi constantè

GF

2=

g2

8MW2

gsinθW =

qe

ε0hc= 4πα

Electroweak unification

gZ ≡g

cosθW

IWz −Qsin2θW( ) =4πα

sinθW cosθW

IWz −Qsin2θW( ) =g

cosθW

cZ

cZ ≡IWz −Qsin2θWThe Z-charges are


The neutral currents

20 Apr 2023

NC have important differences from CC•Couple a particle with itself only (ee, not e; uR uR, not uR uB, not uc, …)•Are not V-A, both Left and Right fields

The currents (for the 1˚ family)

The 3x7=21 couplings are determined by two parameters = elementary electric charge and weak angle sin2θW The Z is universally coupled•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 γ and itself.

−qe

sinθW cosθW

I z − Qsin2 θW( )

jα,e

0 =gLeeγα

121−γ5( )e =gL

eeLγαeL


Vector boson masses

20 Apr 2023

GF

2=

g2

8MW2

MW =g2 28GF

⎛

⎝⎜⎞

⎠⎟

1/2

=πα2GF

1sinθW

=37.3

sinθW

GeV

M Z =MW

cosθW

EW unification+Fermi constant (from beta decays etc.)

EW theory

Experiments (examples to be discussed) sin2θW ≈0.232

MW ; 80 GeV M Z ; 90 GeV

modulo small higher order corrections

Two constants to be determined by experiments: charge α and mixing angle sin2θW


Measuring the weak mixing

20 Apr 2023

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, sin2θ

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– + – (10 GeV - 200 GeV)

•Deeply inelastic scattering of on (several GeV)

•Elastic electron scattering (MeV)


Inelastic ed scattering

20 Apr 2023

Inelastic ed scattering gives information on sin2θW

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 q21.6 GeV2. Small enough to have a pointlike effective weak interaction

€

4 2 gLe e Lγα eL + gR

e e Rγα eR[ ] gLe u Lγα uL + gR

e u Rγα uR + gLe d Lγα dL + gR

e d Rγα dR[ ]Hence

−GF

2 21− 4sin2 θW( )eγα e+ eγαγ 5e⎡⎣ ⎤⎦ 1−

8

3sin2 θW

⎛

⎝⎜

⎞

⎠⎟uγ α u + −1+

4

3sin2 θW

⎛

⎝⎜

⎞

⎠⎟dγ α d +uγ αγ 5u − dγ αγ 5d

⎡

⎣⎢

⎤

⎦⎥

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


ed scattering

20 Apr 2023

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

€

A =σ + − σ −

σ + +σ −Expected few p.p.m. values

Calculation gives (considering only valence quarks)

epol

e

X

counter

€

A = −GFq2

2 2πα9

101−

209

sin2 θW ⎛ ⎝ ⎜

⎞ ⎠ ⎟1− 4sin2 θW( )K y( )

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

A = −1.62 ⋅10−4 q2/GeV2( ) 1−

209

sin2 θW ⎛ ⎝ ⎜

⎞ ⎠ ⎟1− 4sin2 θW( )K y( )

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

K y( ) =1− 1− y( )

2

1+ 1− y( )2

Ein Efin

€

y =Ein − E fin

Ein

C.Y. Prescott et al. Phys. Lett. 77B (1978) 347; Phys. Lett. 84B (1979) 524

e–(polarised) + d e– + X

Expected A = 10–4 q2 (GeV2)


The polarised source

20 Apr 2023

Crucial technical developmentA die LASER (pulsed in phase with the LINAC) goves light at =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 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 P the electrons helicity varies as cos(2P)

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


The Prescott experiment at SLAC

20 Apr 2023

Measure beam polarisationMøller Scattering (elastic e– e– ) in a magnetised Fe foilPe=[37±1(stat)±0.2(syst)]%

Spectrometer selects charged particles of definite momentum (≈80%Ein)

Beam intensity= 4x1011 el/pulseIntensity at detectors = 1000 el/pulse

Detectors (2 for cross-checks) on the beam analysed by spectrometerFlux 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

Aexp =Y+ − Y−

Y+ + Y−

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 –

Check on non polarised beam Aexp=(–2.5±2.2) x 10–5


Asymmetry. First method

20 Apr 2023

The following values were chosen for the angle P between calcite and cell and of the sign of V on the cell

P=0˚

V>0 electron helicity +V<0 electron helicity –

P=45˚

V>0 electron helicity 0V<0 electron helicity 0

P=90˚

V>0 electron helicity –V<0 electron helicity +

Expected behaviour Aexpect=Pecos(2 P)

Opposite values. OK

Compatible with 0. OK


Asymmetry. Second method

20 Apr 2023

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

€

θprec =Ein

mec2

g− 22

θbend =Ein GeV( )

3.237π rad

Measured at dfferent beam energiesEin = 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

€

Aexp = Pe AcosEin GeV( )

3.237π

⎡

⎣ ⎢

⎤

⎦ ⎥

€

A = −1.62 ⋅10−4 q2( ) 1−

209

sin2θW ⎛ ⎝ ⎜

⎞ ⎠ ⎟1− 4sin2θW( )K y( )

⎡

⎣ ⎢ ⎤

⎦ ⎥

y =Ein −E fin

Ein

=0.21, q2 =1.6 GeV2


Systematics and result

20 Apr 2023

Systematic effects can (do) induce spurious asymmetriesDrifts in the PM gain

effect is minimised by frequent inversionsBeam fluctuations

under control, residual effects small enoughVariations in beam parameters correlated with helicity changes (dangerous)

accurate monitoring of the beam parameters, register data, feedback corrections using micro-processor

Ein/Ein<10–6(A/q2)<0.26x10–5

A/q2=(–9.5±1.6) x 10–5 GeV–2

Total uncertainty = linear sum ofstatistical ±9%systematic on Pe ±5%systematic on beam parameters ±3.3%

Final result sin2θW = 0.224±0.012 (stat) ±0.008 (syst.) ±0.010 (theor.)

Theoretical error from the uncertainty on the deuteron quark structure


Atomic parity violationLandmarks

20 Apr 2023


Stark-PV-interference technique

20 Apr 2023

Invented by the Bouchats in the 1970s

Phys. Lett. 48B (1974) 111 and J. Physique 35 899–927


Atomic parity violation

20 Apr 2023

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 observedSensitive to electron-quark couplingEnergy scale much different from accelerator experiment


Atomic parity violation

20 Apr 2023

Precision limiting factorsexperimental accuracycomplexity of the atom systemtheoretical 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 alkalineGood 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


Theory

20 Apr 2023 A. Bettini

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 = 2gV

u + gVd , gV

n = gVu + 2gV

d ,

Considering that T3 and the charge Q are additive, the relations are the same as for quarks

gVnucleon =T3 −Qsin2θW Also

€

gAe = T3 =

12

The interesting interaction is

€

GF

2 2−e γα γ5e( ) −n γα n + 1− 4sin2θW( ) p γα p[ ]

The effective potential can be calculated consdering that•the wavelength corresponding to the momentum transfer is large compared to the nuclear radius•the electron is non-relativistic, and its wave function becomes

ψ x( ) =

ϕ x( )rσ ⋅

rp

2me

ϕ x( )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟, with ϕ x( ) two component spinor


Parity violating effective potential

20 Apr 2023

Vn r( ) =

−GF

4 2me

−irσ ⋅

r∇δ r

r( )−iδ rr( )

rσ ⋅r∇⎡⎣ ⎤⎦

Vp r( ) =

GF 1−4sin2θW( )4 2me

−irσ ⋅

r∇δ r

r( )−iδ rr( )

rσ ⋅r∇⎡⎣ ⎤⎦

€

rp ⇒

r ∇

We sum over the Z protons and N neutrons of the nucleus obtaining, the total weak potential

€

VNPC r( ) = −GF

4 2meZ 4sin2 θW −1( ) + N[ ] −i

r σ ⋅

r ∇δ

r r ( ) − iδ

r r ( )

r σ ⋅

r ∇[ ]

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 atomThe mixed levels are then S1/2 and P1/2

The matrix element must be calculated

S1/2 VNPC P1/2 = 3i16πmec

GF

21−4sin2θW( )Z−N⎡⎣ ⎤⎦

dRL=1

dr0( )RL=0 0( )

There are uncertainties on the radial wave function at the origin R(0) and its gradient dR/dr

€

QW = 1− 4sin2 θW( )Z − N is the parity violating “weak charge”

We substitute Calculating effective potential we obtain for neutron and proton


Boulder Cs experiment. Scheme

20 Apr 2023

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 electric field E, which mixes S and P.

resonant excitation of transitions 6S7S in the presence of perpendicular crossed electric E and magnetic B fields


Boulder Cs experiment. Continues

20 Apr 2023

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

€

6S1/2 ⇒ 7S1/2

Transitions to be induced

A(4,4;3,3)

Measure the intensity of these radiations to monitor the rates of the LASER induced transitions

€

F m ⇒ F' m'

4 4 ⇒ 3 3

4 −4 ⇒ 3 −3

3 3⇒ 4 4

3 −3⇒ 4 −4F =1, m =±1


Space structure of the observables

20 Apr 2023

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 ε and to its magnetic field [equivalently to εk] and also to εE

The structures of all possible observables are obtained taking the scalar products of ε εk e εE with the availbale vectors (k, E) and axial vectors (B e Ek). The latter are by construction parallel, and consequently equivalent (provided angles are perfectly 90˚)Scalar quantities are

εk zero for EM wave (but not for diffuse radiation)εE “scalar” polarizability. Gives zero contribution to transitions F≠0εk.Bmagnetic dipole (small). Equivalent to εk.µ (with µ electron magnetic moment)εE.B “vectorial” polarizability

One pseudoscalar quantitye ε.B Parity violating potential. Equivalent to ε.σ (with σ electron spin)

(the other expressions are zero or non-independent. Example εk.E= Eεk=kε that is not independent from B. ε


Transition amplitude

20 Apr 2023

The electric dipole transitions AST are due to the interaction between the wave electric field ε and the dipole moment d induced by the stationary electric field E (Stark effect). these are the matrisx elements of εE.B. Here 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 ε and the magnetic moment directed as B. Hence: M1 εk.µ (“strongly forbidden” transition)

For all are different from zero only transitions with m’=m and m’=m±1

The parity non conserving amplitudes APNC due to VPNC ε.σ


Interference

20 Apr 2023

€

Imm 'FF ' = AST + AM1 + APNC

2

maximize

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

For the transition between two given levels m and m’

Control of the systematics is vital

Need to verify that the directions of the fields and of the beam are relly perpendicular


What do we expect?

20 Apr 2023

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

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 m=1 and one m=–1

The weak magnetic field induces an asymmetry, which can be accurately calculated


Parenthesis. A modern version

20 Apr 2023


The experiment

20 Apr 2023

Photodiode sensitive at 852/894 nm

Intensity = 1015 atoms cm–2s–1

The tuned Fabry Perot cavity effectively amplifies the LASER power by 20

Circular polarisation can be inverted rapidly

We chose the pump LASER frequency to define the levels between which the transition should happen

Main experimental points: 4 quasi independent spatial inversion operationsParity 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.)

6S(F=4, m =4 ) 7S(F=3,m=3)

6S(F=4, m =–4 ) 7S(F=3,m=–3)and (after 4 h)


Results

20 Apr 2023

Measured quantity VNPC/Bouchat et al. 1984 VNPC/ = –1.78±0.26(stat.)±0.12 (syst) mV/cm

Noecker et al. 1988 VNPC/ = –1.576±0.034(stat.)±0.008 (syst) mV/cm

Polarisability known from computations = 27.0 a0 with 5% uncertainty

Weak charge is

QW= Z(1–4sin2θW)–N = 55 (1–4sin2θW)– 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.)

sin2θW = 0.219±0.007±0.018Present values

QW= –73.20±0.35 sin2θW = 0.2385±0.0015Radiative corrections must be taken into account when comparing with other measurements of sin2θW

APV gives the most precise measurement of the coupling between vector weak electron current and d- quark current (u-quark contribution is small)

Measured quantity VNPC/Bouchat et al. 1984 VNPC/ = –1.78±0.26(stat.)±0.12 (syst) mV/cm

Noecker et al. 1988 VNPC/ = –1.576±0.034(stat.)±0.008 (syst) mV/cm

Polarisability known from computations = 27.0 a0 with 5% uncertainty

Weak charge is

QW= Z(1–4sin2θW)–N = 55 (1–4sin2θW)– 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.)

sin2θW = 0.219±0.007±0.018Present values

QW= –73.20±0.35 sin2θW = 0.2385±0.0015Radiative corrections must be taken into account when comparing with other measurements of sin2θW

APV gives the most precise measurement of the coupling between vector weak electron current and d- quark current (u-quark contribution is small)


sin2θW “runs”

20 Apr 2023

sin2θW 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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1 A. Bettini LSC, Padova University and INFN Subnuclear Physics in the 1970s 15-Oct-15 IFIC...

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