Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics

Подождите немного. Документ загружается.

8.7 Lepton Families 199

300 m

Protons

(400 GeV)

Target

Horn

Reflector

Decay tunnel

Absorber

Detectors

Collimator

Momentum

Absorber

(~ 400 m)

Fig. 8.9 Layout of neutrino beams at the CERN SPS: (a) narrow-band beam and (b) broad-band

beam

(Sect. 5.5.3). The electron and electron neutrino, as with the muon and muon

neutrino, have been grouped into different lepton families. A third family (the tau

and tau neutrino) was successively identiﬁed. As we shall see in Sect. 9.7, from the

precision measurements obtained in the LEP experiments, we know that there are

only three families of leptons.

The ﬂavor lepton number conservation is respected in every decay or interaction

process in Nature (and in all equations presented in this book). In Chap. 12, we

shall discover a new phenomenon which involves neutrino ﬂavor changes during

their propagation, the neutrino oscillations. This imposes a modiﬁcation in the

application of the ﬂavor lepton number conservation law.

Muon neutrino beams and experiments. Many experiments were carried

out with 



beams: they were used to test the quark composition of protons

and neutrons (Chap. 10), or to study particle ﬂavor oscillations (Chap.12).

To produce intense beams of high energy muon neutrinos, protons from a

synchrotron are accelerated to the maximum available energy (for example,

450 GeV at CERN SPS and 900 GeV at the Fermilab Tevatron). At the end of

the acceleration cycle, the protons are extracted and sent against a beryllium

target about one meter long where they produce  and K mesons. The mesons

are collimated and pass in a 300 m long vacuum tube where they decay

(Fig. 8.9). After the vacuum tube, there is a region with absorbing material.

In the absorber, photons and electrons are stopped ﬁrst, followed by hadrons

and ﬁnally muons. Only neutrinos proceed and reach the experimental area

where different detectors are installed.

The neutrino beam is not monochromatic; it can be either narrow-band or

broad-band, depending on the focusing of the charged hadrons produced in

200 8 Weak Interactions and Neutrinos

Fig. 8.10 Typical neutrino

(full line) and antineutrino

(dashed lines)ﬂuxfor10

incident 400 GeV protons at

the CERN SPS:

(a) broad-band beam and

(b) narrow-band beam, after

selection of 200 GeV/c 

and K

(a)

(b)

(GeV)

50 100 200 250 300150

ν (ν) flux / GeV per 10

incident protons

the primary collision. To obtain a narrow-band neutrino beam, quadrupole

magnets and magnetic deﬂectors are placed immediately after the target.

Abeamof and K mesons with selected momentum and wide enough

bandwidth (p=p ' 10%) is formed. The selected beam passes through a

quadrupole system to form a collinear beam optimized for the decay in the

vacuum tube (see Fig. 8.9a). In a broad-band neutrino beam, the focusing

system is optimized to produce the maximum intensity of  and K mesons

(see Fig. 8.9b). The narrow-band neutrino beam has an energy distribution

with a “two boxes” shape, while the broad-band has a maximum at a few

tens of GeV (see Fig. 8.10). The intensity of a narrow-band beam is obviously

smaller than that of a broad-band beam.

Neutrino detectors. Since the neutrino cross-section is small, the target must

be very massive to obtain a reasonable interaction rate. The mean free path

in iron of 10 GeV neutrinos is  D 2:6  10

km. This means that only a very

small fraction 3  10

13

of 10 GeV neutrinos interact in a meter of iron. With

a ﬂux of 10

neutrinos (for 10

accelerated protons incident on the target),

there are only 0.3 interactions in one meter of iron. In general, the target

corresponds to the detector sensitive volume, particularly if one is interested

in studying the hadronic system X produced in 



N !  C X interactions.

In the past, large bubble chambers were intensively used; alternatively, large

sampling calorimeters are used as target and as electronic detectors, often

8.8 Parity Violation in ˇ Decays 201

followed by a separated muon detector. It is important to identify muons

to separate the charged current interactions from the neutral current ones.

Electronic detectors used at the CERN SPS and Fermilab consisted of a target

calorimeter (to measure the total energy of hadrons and photons) and of

muon chambers (which also measured their momentum). These experiments

(CDHSW, CHARM, CHORUS and others) were used to study the properties

of neutrinos, the structure functions of nucleons (Sect. 10.5), and neutrino

oscillations. As described in Sect. 12.6, neutrino oscillations can be observed

if neutrinos travel a long distance L from the production point to the

detection point. For this reason, there are (2011) three long baseline beams

(see Sect. 12.8.1) where the neutrino is produced in a laboratory (CERN in

Europe, Fermilab in USA, Jpark in Japan) and detected in distant experiments

(OPERA at Gran Sasso for the CERN beam, MINOS at the Soudan laboratory

for the Fermilab beam, both located 730 km from the accelerator; Super-

Kamiokande in Japan, which is 250km away from Tsukuba).

8.8 Parity Violation in ˇ Decays

The idea (quite original at that time) that parity was not conserved in weak

interactions came up in 1955. It originated from the evidence that the K meson

decays into two states of opposite parity. In 1956, T.D. Lee and C.N. Yang

(Nobel prize in 1957) made a critical analysis of the results obtained from the

study of weak processes and concluded that the dependence of the interaction

on pseudoscalar terms was never experimentally studied. A pseudoscalar quantity

changes sign under a parity transformation. Examples of pseudoscalar quantum

mechanics operators are the helicity (Appendix A.4), that is, the scalar product of

the spin with the particle momentum (  p

) or with the electric ﬁeld (  E). All

the eigenvalues of these operators are pseudoscalar quantities.

Lee and Yang observed that the Hamiltonian of weak interaction processes

was expressed as a superposition of Fermi and Gamow–Teller terms, which are

separately invariant under parity.

They proposed a more general formulation. The

inclusion in the Hamiltonian of a term whose eigenvalues change sign under parity

transformation (as, for instance, the terms  E,   p) would result in the violation

of parity in the interaction (see Table 6.3). The term that Lee and Yang suggested

was connected with the longitudinal polarization   p. They also proposed some

experiments to evidence the possible parity violation in weak decays. Two of

them, the decay of polarized nuclei and the decay of the muon, were performed

The explanation of these concepts will be formalized in the theory presented in Sect.8.16.

202 8 Weak Interactions and Neutrinos

Fig. 8.11 Parity violation in ˇ decay. Initially (upper panel), the

Co nucleus spin (indicated

by the arrow) with J D 5 is aligned by the presence of a magnetic ﬁeld along the solenoid axis.

Two identical detectors are placed at the ends of the solenoid. In the middle panel, detector 2

counts events: they are electrons with spin aligned as shown. From momentum conservation, the

antineutrinos travel to detector 1 (which of course does not count anything); from the angular

momentum conservation, their spin is parallel to the direction of the momentum. The bottom

panel represents what you would see in a mirror, cutting the middle panel along the dashed

line on the right and approaching that side near the mirror. Unlike before, counter 2 now counts

nothing! In fact, taking into account the properties of angular momentum in the reﬂection, the

right-handed electrons should reach counter 2 because a left-handed antineutrino is going to

counter 1. However, the left-handed antineutrinos have never been observed, and counter 2 counts

nothing. The reﬂection in the mirror of this physical process gives a different result from the real

experiment!

in the following months and clearly showed that parity is not conserved in weak

interaction.

Here, we schematically describe the results obtained by Madame Wu and

collaborators on the decay of polarized cobalt nuclei. In Sect. 8.10, we shall analyze

the second experiment which involved the measurement of the     e decay

at the Nevis cyclotron at Columbia University by R. L. Garwin, L. M. Lederman

and M. Weinrich. The articles relating to both experiments were published in 1957

in the same issue of Physical Review Letters (Vol. 105, No. 4, pp. 1413–1414 and

1415–1417).

In the ˇ decay of

Co (see Fig. 8.11), a term 

p

that changes sign under the

application of the parity operator was considered ŒP .

 p

/ D.

 p

/;

it produces a pseudoscalar quantity.

Co polarization can be achieved by an

experimental setup at a very low temperature (0.01 K). The nuclear magnetic

moments of

Co, which are parallel to the nuclear spins, are oriented in a strong

magnetic ﬁeld B. The emitted electrons were counted in the directions parallel and

8.8 Parity Violation in ˇ Decays 203

024681012141618

TIME IN MINUTES

0.70

0.80

0.90

1.00

1.10

1.20

COUNTING RATE

< COUNTING RATE >

WARM

β ASYMMETRY

(AT PULSE

HEIGHT IOV)

EXCHANGE

GAS IN

Fig. 8.12 Measured asymmetry of electrons emitted in

Co decay in the Wu et al. experiment.

The upper counter measures different frequency counts when the magnetic ﬁeld (denoted as H) is

aligned towards the bottom (upper curve)ortop(lower curve). As time (abscissa) increases, the

Co sample warms up, the spins tend to lose alignment, and the asymmetry disappears

antiparallel to the nuclear magnetization: the two numbers were different, which

unambiguously demonstrated that parity is violated in WI.

The

Co .J D 5/ decays into an excited state of



.J D 4) through a pure

GT transition. The lifetime is seven and a half years, and the available energy is

D 0.32 MeV



J D 5







J D 4







: (8.31)

From the angular momentum conservation law, the electron and antineutrino are

emitted with spin parallel to the

Cospin, oriented in the direction of the magnetic

ﬁeld.

The intensity of the electrons emitted in the direction of the magnetic ﬁeld

and in the opposite direction (Fig.8.11) was measured using scintillation counters.

Counters 1 and 2 play a symmetric role: if Nature and its image in a mirror are the

same (parity conservation), counter 1 should measure the same rate as counter 2,

regardless of the orientation of the magnetic ﬁeld (and thus of the orientation of the

cobalt nuclear spin). The measurements have shown that when the magnetic ﬁeld is

inverted, the electron count changes. Electrons prefer to be emitted with momentum

in the opposite direction of the nuclear spin, and with the spin oriented in the same

direction of the nuclear spin. This corresponds to the same electron spin-momentum

assignment in nuclear decays of Fig. 8.8b. The right-left asymmetry in the electron

counts depends on the degree of the magnetization of the source (Fig. 8.12). The

intensity of the emitted electrons as a function of the angle  between the direction

204 8 Weak Interactions and Neutrinos

of electron emission and the polarization of the sample (emitted electron angular

distribution) has the form

I./ D 1 C

˛

 p

D 1 C˛

cos  (8.32)

where p

are the momentum and energy of the electron, and 

is the

Co spin.

Data are consistent with ˛ D1. The counts for the upward and downward

polarization are different and therefore parity is violated. In fact, in (8.32), under

inversion of coordinates (parity P transformation), the pseudoscalar term 

 p

changes sign: P.

 p

/ D

 p

This experiment showed for the ﬁrst time that electrons are prevalently emitted

longitudinally polarized in the opposite direction of its momentum (the electron has

negative helicity, it is left-handed). The charged current weak processes involve left-

handed e



and right-handed e

. The helicity  (see Appendix A.4) is measured as

the net longitudinal polarization of a set of particles. Using (8.32), one has

 D

 I



C I



D ˛

(8.33)

where I

ed I



are the relative intensities of the component with spin parallel and

antiparallel to the momentum p.Thevalueof˛ was found to be equal to 1 for e



. Dv=c) and equal to C1 for e

. DCv=c/.

8.9 The Two-Component Neutrino Theory

For a massless neutrino, v D c and (8.33) implies that the particle must be

completely polarized, i.e.,  DC1 or  D1. An experiment performed by

M. Goldhaber et al. in 1958 demonstrated that the neutrino helicity is negative, i.e.,

the neutrino spin 



is antiparallel to its momentum p



: schematically, .p



"+ 



The antineutrino is right-handed .p



"* 



/; for example, the spin pattern in

polarized neutron decay is

n ! p C e



C 

***

where LH means left-handed, i.e., with the spin antiparallel to the momentum

direction ."+/; RH stands for right-handed, which means that the spin and

momentum are in the same direction ."*/. As a result of many experiments, the

helicity assignment of leptons and antileptons is (see also Appendix A.4)

particle 





helicity probability 1 C 1  v=c C v=c: (8.34)

8.10 Charged Pion Decay 205

–

NaΙ(Tl)

Magnetized

iron

152

Sm*

152

–

capture

NaΙ(Tl)

Fig. 8.13 (a) Diagram of energy levels in the

152

Eu and

152



decays. (b) Schematic apparatus

used to measure the neutrino helicity. (c) Spin directions

These considerations lead to the two-component neutrino theory, i.e., that the

neutrino is left-handed while the antineutrino is right-handed.

Neutrino Helicity. [8G10] The process studied by Goldhaber was the electron

capture in the

152

Eu nucleus, that is,



152

Eu !

152



C 

152

Sm C: (8.35)

The excited samarium state decays with the emission of a 960 keV  (see diagram

of Fig.8.13a). The  rays are emitted in the same direction as the recoiling

152



and are immediately detected after a 90

scattering in a samarium target. Only the

 emitted in the direction opposite to the 

have an energy slightly higher than

960 keV and therefore sufﬁcient to give rise to a resonant scattering in the samarium

target, i.e.,  C Sm ! Sm



! Sm C  . For a neutrino with negative helicity, the

spin conﬁguration is shown in Fig.8.13c. If the electrons in the magnetized iron

are polarized in the same direction as the photon, the photon can go through; if the

photon has spin opposite to that of the electron, it can be absorbed in an interaction

that reverses the electron spin. The helicity of the photon is then measured by the

scattering in magnetized iron (reverting B, the polarization can be determined by

the change in the counting rate). It can be concluded that the neutrino is left-handed

(i.e., the spin has an opposite direction with respect to the momentum direction).

8.10 Charged Pion Decay

The second crucial experiment that conﬁrmed the nonconservation of parity, the

neutrino LH helicity and the antineutrino RH helicity is based on the study of



meson decays. Figure 8.14a shows the Feynman diagrams of the 

meson

206 8 Weak Interactions and Neutrinos

{

J =0

–

Fig. 8.14 Feynman diagrams (a)forthe

! 





and 

! e



decays, and (b)forthe



! e





decay

abc

Fig. 8.15 Illustration of the momentum and spin of the particles in the decays (a) 

! 





(b) 

! e





,and(c) 

! e



decays: 

! 





, 

! e



.The

meson is made of ud quarks;

it annihilates into a virtual W

vector boson which then decays into 







. The momentum and spin assignments of the 

and of the 



in the 

rest frame is shown in Fig. 8.15a. The 



and 

have equal momentum with

opposite directions: jp





jDjp



jDjpj.If



is LH, that is, with helicity

 D1, its spin is antiparallel to the momentum, as shown in Fig. 8.15a. From

angular momentum conservation, also the 

must have negative helicity. In the

following 

! e





decay in the 

rest frame, the momentum and spin

conﬁgurations must be as shown in Fig. 8.15b for the limit case where the 

and





are emitted in the same direction. The experiments carried out in 1957 veriﬁed

that the angular distribution of the positron is in agreement with that predicted

by the two-component neutrino theory with the spin conﬁguration shown in

Fig. 8.15b.

In the following, we will study the ratio of branching ratios (see Sect. 4.5.2)







! e





=





! 







; (8.36)

which gives important information on the mathematical structure of weak interac-

tion currents. On the basis of the Sargent rule (8.18), the decay  ! e should be

favored due to the larger energy available in the ﬁnal state (m



 m

). However,

the decay  ! e

is disfavored by a factor 10

compared to  ! 



.The

reason is that the emitted charged lepton has, in both cases, the wrong helicity.

Neutrinos have negative helicity . D1/ and antineutrinos have positive helicity

8.10 Charged Pion Decay 207

. DC1/; as a consequence, e

;

must have helicity as shown in Fig. 8.15a,

c. According to (8.34), antiparticles have spin parallel to the momentum with

a probability P

D v=c (Appendix A.4). The conﬁgurations of Fig. 8.15a,c

with negative helicity both for the 

and for the e

are disfavored conﬁgurations,

corresponding to a probability P

D .1  v=c/ since P

C P

D 1.The

quantity P

is much smaller for the positron, as its mass is small and the velocity

approaches c. The heavier 

is rather nonrelativistic. In the case of 

! 





the decay fraction is proportional to



!

' .helicity factor/.phase space factor/ '



1 





(8.37)

where p is the momentum of the 

and of the 



, and the total energy is E



D p C

C m



(units with c D 1). Consequently, one has

p D



 m





 m







 m







;



C m







;1





C m



and then



!



1 









C m







 m







C m







1 





: (8.38)

In the case of 

! e



, the decay fraction 

!e

is similar to (8.38) with the

change m



! m

. The ratio R between the decay fractions is

R D



!e



!



1 







1 











1 





' 1:27  10

4

; (8.39)

which is in good agreement with the experimental results. The key ingredient of

prediction (8.39) is the helicity assignments for the neutrinos and antileptons which

is formalized in the so-called V  A theory (Sect. 8.16).

208 8 Weak Interactions and Neutrinos

8.11 Strange Particle Decays

Hadrons and leptons are both subject to weak interactions. WIs are usually classiﬁed

as follows: (1) processes only involving leptons (leptonic WI) [8B84]; (2) processes

involving only hadrons (nonleptonic WI) [8P84]; (3) processes involving hadrons

and leptons (semileptonic WI) [8V84].

leptonic W 

! e





;



! 



(8.40)

semileptonic W S D 0



n ! pe





p ! ne

S D 1



! 



! 





(8.41)

nonleptonic W S D 0 fNN ! NN S D 1





! p



! 



(8.42)

Furthermore, in semileptonic or nonleptonic processes, decays in which strange

particles are not involved (S D 0), or strangeness violating decays with the

selection rule S D 1 are observed. The weak decay of strange particles

presents some anomalies, whether leptonic, semileptonic or nonleptonic processes

are considered.

Leptonic decays. Let us examine the case of some purely leptonic decays:

Strangeness Lifetime

Decay Change s BR D 

=





! 







ud ! W



! 







S D 02:610

8

100%



! 







us ! W



! 







S D 11:27 10

8

63.5%

The K



lifetime calculation follows the same procedure as that presented in the

previous paragraph. Using (8.37) and the Fermi constant G

, the computed K



lifetime is about 20 times smaller than the measured one.

To explain this new “strange” behavior, one can imagine that the effective WI

coupling constant to be used depends on the quark ﬂavor. In the case of  decay, the

d quark is involved. The related weak coupling constant should be G

' G

since

the  lifetime calculated with the Fermi constant G

gives an almost correct result.

In the case of K decay, the s quark is involved. The effective coupling constant to

be used must be such that G

in order to obtain the measured K lifetime.

From the measurements of the ; K lifetimes, and from the calculation of the

helicity and phase space factors, one obtains

' 0:05: (8.43)