 Chapter 1 Interactions And Field

1.4 The Weak And Gravitational Interaction

Interactions and fields

Particle Physics

The Weak And Gravitational Interaction

Fig 1

The weak interaction affects all particles except the graviton (the carrier for the gravitational interaction), but its effect are often overwhelmed by the strong and electromagnetic interactions. Although the range of this force is not known accurately, it is thought to be short (about 10 -17m). The weak interaction, as its name suggests, is about 10-12 times weaker than the strong interaction. The hypothetical carrier particles of the weak force are usually referred to as bosons. Dayalbagh Educational Institute

Fig 2 (a) 3

Interactions and fields Particle Physics To account for the weak interactions, there are positive and negative W bosons and a neutral Z boson. The large mass of the W boson accounts for the short range of the weak force. Feynman diagrams for some weak interaction are shown in Fig 2. The weak interaction is important in any process that involves a neutrino (υ). The neutrino was originally hypothesized to explain the apparent violation of conservation of momentum, energy, and spin angular momentum when a nucleus decayed by the emission of an electron. Fig 2 (b) The Feynman diagram for the decay of a neutron is shown in fig. 2(b). The decay may be written n → p + e- + υe,

(1)

where n represents the neutron and υe the electron's antineutrino. The W boson's apparent violation of the conservation laws occurs in a time Δt within the limit of the uncertainty principle.

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Interactions and fields Particle Physics At a fundamental level neutron decay (equation (1)) must require the quark change d → u + e- + υe in which a down quark in the neutron changes flavour to become an up quark in the proton. The reader may be puzzled to see leptons apparently coupling to quarks, the fundamental particles involved in the strong interactions but the leptons can interact only via the weak interactions or, in the case of the charged leptons, via the electromagnetic interactions.

Fig 3 : Feynman diagram for the β decay of a d quark.

The solution to the apparent paradox understood with the aid of figure 3.

may

be

At the top vertex the incoming d quark in the neutron is absorbed and a u quark is emitted along with a weak gauge boson, in this case the W-. Dayalbagh Educational Institute

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Interactions and fields Particle Physics The amplitude for this process is proportional to √αw; in analogy with the strong and electromagetic processes √αw may be thought of as a ‘weak charge’. The W- then propagates to the bottom vertex where it is absorbed and an electron and anti-neutrino are emitted. In this two-stage process, the leptons couple to the weak gauge boson and not directly to the quarks. Implicit in this viewpoint is the assumption that the quarks carry a weak charge in addition to the electric charge and colour or strong charge. The amplitude for the overall process is proportional to αw . The mass of the W- is roughly 100 times the mass of the proton, implying that the range of the weak interaction is extremely short. Since the strength of the weak interaction is small, neutrinos are produced very feebly and interact very weakly with matter. Reines and Cowan showed that neutrinos have preferred orientations for their spin with respect to their direction of motion. Neutrinos are left handed and anti-neutrinos are right handed.

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Interactions and fields Particle Physics In the inverse process of a neutron decaying to a proton, electron, and antineutrino, an antineutrino plus a proton produces a neutrino and a positron (e +): Ď&#x2026;e + p â&#x2020;&#x2019; n + e+. The electron, positron, neutrino, and antineutrino are all weakly interacting particles belonging to the family designation as leptons. The appearance of the antineutrino in the neutron decay and in the inverse process led to the establishment of a new conservation law: the conservation of electron number Ne. Rule: Electron number is assumed to be conserved in all interactions. The fact that we always get neutrinos when positrons are products of decay and anti-neutrinos when electrons are products of a decay is another example of the conservation of electron number. Fortunately for all of us, the mass of a neutron confined to a nucleus is less than the mass of a free neutron (some of the mass appears as binding energy) and it does not decay. Next member of the lepton family is the muon, with a rest-mass approximately 207 times that of the electron.

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Interactions and fields Particle Physics In the early sixties, when intense sources of neutrinos were provided by accelerators at Brookhaven and at the European organization for Nuclear research (CERN), experiments with these accelerators suggested an additional conservation law that divides the leptons into particles with electron number N e and muon number Nμ.

Decay that violated no known conservation laws did not take place. For example the decay of a positive muon to a positron and a photon was not observed. Neutrons and anti-neutrons, but there are electron neutrinos (υe) and muon neutrinos (υμ). The need for υμ and υe was demonstrated by the fact that neutrino that resulted from a muon decay produced muons in subsequent interactions but never electrons. Negativity charged muons (μ -) and muon neutrons (υμ) are assigned muon number of Nμ = +1, the antimuon (μ+) and the muon antineutrino have Nμ = -1. In a typical mode of decay for μ-, μ-

e-

(Nμ = +1) →

+

υe

+

υμ,

(Ne = +1) + (Ne = -1) + (Nμ = +1),

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Interactions and fields

Particle Physics

The Feynman diagram for the decay of the μ+ is shown in fig 2(c). One of the greatest mysteries in particle physics is Nature’s need for both muons and electrons. Fig 2(c)

Despite the need for two quantum numbers Nμ and Ne describes their decay schemes, muons and electrons are alike in every other respect, except for mass. While the probabilities for the decay schemes are quite different, (this difference is explained in Fermi’s theory of weak interaction), a positive pion may decay either to an anti-muon or to an anti-electron through the reactions π+ →

μ+

+

υμ

or

π+ →

e+

+

υe.

In addition, the electron and the muon (and the hypothesized bosons) are the only nonzero-rest-mass particles that do not take part in the strong interaction. In “exotic” atoms, electrons are replaced by negatively charged muons, pions, or kaons. Dayalbagh Educational Institute

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Interactions and fields Particle Physics For exotic atoms, the “radii” of the orbits for the same value of n are inversely proportional to the mass, while the energy of the level is directly proportional to the mass. Since the mass of the muon is approximately 207 times the mass of the electron, the “radius” of a muonic atom in the ground state is approximately 1/207 times the “radius” of hydrogen for n = 1. The energy required to ionize the muonic atom is 207 times the energy required to ionize the hydrogen atom. The strength of the week interaction field appears to be approximately 10-6 that of the strong interaction. Since the process shown in fig. 1 takes place in two steps, the effective strength of the weak interaction is about 10-12. Because the strength of the interaction is proportional to the probability of emission and absorption of the carrier, a weak interaction takes about 10 12 times as long as the strong interaction, and about 10 10 times longer than electromagnetic interactions. While these numbers have no exact meaning, they indicate the enormous difference in the strength of the interactions. We compare the strength of the interaction by seeing how quickly they cause a decay.

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Interactions and fields

Particle Physics

Basic Weak Interactions:

Matrix 1

The basic transformation patterns in the weak interactions are the following two: (I)

Xâ&#x2020;&#x2019;YW

+

or

Where X stands for any particle of the first row in matrix 1 and Y stands for the corresponding particles of the second row. For example,

c â&#x2020;&#x2019;sW

+

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or

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Interactions and fields and

υμ → μ- W+

Particle Physics

or

Specially for the quarks, the combination of X and Y may contain any pair, as long as one quark is taken from the first row and the other one is taken from the second row. For instance, we could transformation pattern:

u →d W

+

have

the

following

or

(II)

A → A Z Dayalbagh Educational Institute

or

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Interactions and fields Particle Physics Where A stands for any particle, for example:

d → d Z

-

-

or

μ → μ Z

or

υμ → υ μ Z

or

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Interactions and fields

Particle Physics

The weak force is responsible for some of the most important phenomena: Decays of the muon and tau Leptons Neutrino interactions Decays of the Lightest mesons and baryons Radioactivity, nuclear fission and fusion Boson

Z0

Long Lifetimes 10-13 - 103 s

Mass Ge V/c2

80.4

91.2

Small cross sections 10-13 mb

charge, e

±1

0

spin

Characteristics of Weak Processes:

Weak Force is propagated by massive W+, W - and Z0 bosons The interactions of W± and Z0 are different (related by symmetry of the weak interaction) W± and Z0 can interact with each other W± and γ can interact (as W± bosons are charged)

Fig 4

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Interactions and fields

Particle Physics

Weak vertices QED

W-boson

mediated by the exchange of virtual photons

mediated by the exchange of W boson

acts on all charged particles acts on all quark and leptons coupling strength ∝ e ∝ √α

coupling strength ∝ gW ∝ √αW

propagator term: 1/(q2-mγ2)= 1/q2

propagator term: 1/(q2-mW2)

For many processes: M ∝ e2/q2

For many processes: M ∝ gW2/(q2-mW2)

Table 1

Recall: matrix element, M, is the amplitude of a process. Scattering cross section, σ ∝ M 2/ Decay width, Γ ∝ M 2

Interactions Of The W± Boson Known as “charged current interactions” W± boson interacts with all fermions (all quarks and leptons) Charged current changes the flavour of the fermion:

e.g. electron emitting an W - boson can’t remain an electron violates conservation of charge! Dayalbagh Educational Institute

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Interactions and fields

Particle Physics

Fig 5

an electron turns into a electron neutrino An up quark turns into a down quark and vice versa! Coupling strength at every vertex ∝ gW Propagator term describing the W-boson ∝ 1/(q2-mW2) q is the four-momentum transferred by the W-boson

Allowed Flavour Changes At a W-boson vertex: Lepton numbers: Le, Lμ and Lτ, is conserved: Allowed lepton flavour changes: e- ↔ υe

μ- ↔ υμ τ- ↔ υτ

Total Quark Number, Nq, is conserved Individual quark flavour numbers: Nu, Nd, Ns, Nc, Nb, Nt are not conserved Allowed quark flavour changes: (Q= +2/3 e quark) ↔ (Q= -1/3 e quark) (dsb)↔(uct)

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Interactions and fields

Particle Physics

Fig 6

Each of the nine possible quark flavour changes has a different coupling strength: e.g. gWVud for u to d quarks (Vs are terms in CKM matrix VCKM) Main quark flavour changes are within a generation: d↔u

s↔c

b↔t

Interactions of the Z0 boson Known as “neutral current interactions” Acts on all fermions - (all quarks and leptons) Neutral current conserves flavour of the fermion No allowed fermion flavour changes Propagator term ∝ 1/(q2-mZ2) Coupling strength depends on fermion flavour

Fig 7 (a), (b) and (c) Dayalbagh Educational Institute

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Interactions and fields Particle Physics Anywhere a photon could be exchanged a Z0 boson can be exchanged. (Almost vice-versa, except Z0 boson also has neutrino interactions too!) Electromagnetic and weak neutral current interactions are linked!

Feynman Rules for Weak Interaction How to calculate the matrix element, M, for a weak decay or scattering

e.g. decay of a muon μ-→e- υμ υ-e Fig 8

Draw the Feynman diagram for the process give a four momentum for each particle Check quantum numbers conservation at every vertex For both W and Z: Le, Lμ and Lτ,, Nq, Q For Z only: no change of quark or lepton flavour Is energy and momentum conserved? For decay: Σminitial > Σmfinal Write down the coupling at each vertex: gW (for W) Work out four-momentum transferred by boson: q = (p3 – p1) = (p4 + p2) Write down the propagator term for each boson: 1/q2m2boson M is proportional to product of vertex and propagator terms: M ∝ gw2 /(q2 – mW 2 ) Dayalbagh Educational Institute

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Interactions and fields

Particle Physics

Summary The weak force acts on all quarks and leptons. ● Two massive bosons propagate the weak interaction: W± and Z0. ●

Weak interactions are characterized by: ● Long lifetimes 10-13 - 103 s ● Small cross sections 10-13 mb ●

W±-boson interactions Z0-boson interactions changes fermion flavour conserve the flavour of the e- ↔ υe μ- ↔ υμ τ- ↔ υτ fermion (Q= +2/3 e quark) ↔ (Q= -1/3 e • Z0-boson propagator term: quark) 1/(q2-mZ2) (dsb)↔(uct) Z0-boson interactions are ± ● quark coupling at W strongly linked to the vertex: VCKM electromagnetic interaction ± ● lepton coupling at W vertex: gW ● W± propagator term: 1/(q2mW2) Table 2

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Interactions and fields

Particle Physics

Each of the Forces can mediate decays Weak Since the weak force is 'weak' it takes a “long time” to induce a decay e+ gW gW

μ+

W+

νe

νμ τ ( μ → e νe νμ) ~2.2 x 10-6s

Fig 9

Note that mesons can be viewed as decaying by way of “quark” decay Transition is a better word

e+

gW gW

c

D+

νe

s

K0 d

d

τ ( D+ → K0 e νe ) ~ 10-12s Electromagnetic

γ

d π0

Fig 10

α

γ d Dayalbagh Educational Institute

Fig 11 20

Interactions and fields

Particle Physics

d d quark annihilation (cf → e+ e- → γ) τ (π0 → γ γ) ~ 9 x 10-17s

γ α

s Σ0

s u d

u d

Λ0 Fig 12

Quarks change their orbital angular momentum by radiating a photon τ ( Σ0 → Λ0 γ) ~ 7 x 10-20s EM decay is much faster than weak decay. Strong = Colour = hadronic decay τ ( ρ0 → π+ π -) ~ 4 x 10-24s. |ρ〉 = |u u〉 + |d d〉

u

αs

Mixture

u

π-

d

ρ0 d

u Fig 13

u

π+

Quarks carry colour Multiple gluon exchange is supposed to indicate a non perturbative process. Dayalbagh Educational Institute

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Interactions and fields

παs

Particle Physics αs (Q2) ~ αs (M2π) ≥ 1 Very strong

ρ0

Very fast decay

π+

Fig 14

In particle and nuclear physics we can neglect influence of gravity. Strongest force dominates Force acting determines Interaction cross section σ (ν p → n e) Weak σ (e+ e- → e+ e-) Electromagnetic

σ (π p → π p π π) Strong (colour) Decay mechanisms n → p e+ νe Weak π0 → γ γ

Electromagnetic

ρ0 → π+ π - Strong (colour) Strongest force which conserves all relevant quantum numbers dominates.

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Interactions and fields

Particle Physics

The Gravitational Interaction

Fig 15

The gravitational interaction is very long range, but its strength is minuscule compared to the other interactions. Like the electromagnetic interaction, its strength decreases as the square of the distance between interacting particles, and its range is in principle, unlimited. Unlike the electromagnetic interaction, gravitational force is always attractive.

the

Because of the weakness of the gravitational interaction and the lack of quantization of mass, it is difficult to conceive of a quantum theory for gravitation. However, paralleling the schemes for the other interactions, physicists have suggested the existence of a carrier for a gravitational field. Dayalbagh Educational Institute

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Interactions and fields Particle Physics The gravitational interaction is estimated to be 10 -40 of the strong interaction. While we recognize that these numbers have no exact meaning, let us make a rough comparison of the rate of emission of photons by an atom in an electromagnetic interaction and the rate of emission of gravitons in the gravitational interaction. Recall that the rate of emission of the carrier particles is proportional to the strength of the field. A typical time for an atom to drop a lower energy state with the emission of one photon is 10 -8 s. Since the electromagnetic interaction is 10 38 times the gravitational, the rate of emission of gravitons would be about in 1 in 1030s or 3.2 x 1022 years! Since this is a time that is longer than the age of the universe, little hope is held for the observation of a graviton. There are two possible explanations of gravitational interactions that do not depend on a signal communicated between masses with a speed exceeding that of light. In the general theory of relativity there is no requirement of the transport of a signal. The acceleration of an object in a gravitational field results from the warping a space rather than a signal sent from one mass to another. In the quantum-mechanical description, the gravitation with zero rest-mass communicates the signal with the speed of light. Dayalbagh Educational Institute

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Interactions and fields

Particle Physics

Exercise 1 Draw a valid Feynman diagram for each of the following interactions. State the nature of the interaction (strong, weak or electromagnetic) and label the appropriate exchange boson. The brackets following a particle name denotes the quark content. (a) e+ + μ- → e+ + μ(b) Δ0 (udd) → π- (dū) + p(uud) (c) Ω- (sss) → Ξ0 (uss) + π- (ūd) (d) π- (dū) + p(uud) → Σ- (dds) + K+(us) (e) J/Ψ (cc) → μ+ + μ(f) τ- → μ- + υμ + υτ

Solution 1 (a) (a) e+ + μ- → e+ + μ-

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Interactions and fields

Particle Physics

Since the energies and momenta of the positron and muon are not given, it is not possible to tell which of the EM or weak processes will dominate. At lower energies the EM will be the most important, but as energies are ranged the weak process will become more and more important.

Solution 1 (b) (b) Δ0 (udd) → π- (dū) + p(uud)

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Interactions and fields

Particle Physics

Solution 1 (c) (c) Ω- (sss) → Ξ0 (uss) + π- (ūd)

Solution 1 (d) (d) π- (dū) + p(uud) → Σ- (dds) + K+(us)

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Interactions and fields

Particle Physics

Solution 1 (e) (e) J/Ψ (cc) → μ+ + μ-

In (e), whilst the virtual boson could also be a Z0, given the relatively low energy of a J / psi at rest (rest mass 3:1 GeV/c2) we know that the main contributing diagram will be the EM one, with a virtual photon.

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Interactions and fields

Particle Physics

Solution 1 (f) (f) τ- → μ- + υμ + υτ

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Interactions and fields

Particle Physics

Exercise 2 Draw distinct Feynman diagram that contribute to the following process in lowest order: Ď&#x2026;e Ď&#x2026;e elastic scattering

Solution 2

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Interactions and fields

Particle Physics

Exercise 3 Some of what we have learned about QED is applicable to the weak force. The weak force can be propagated by the W±-boson with mass mW = 80 GeV/c2. For example, nuclear beta decay can be described as d → u W −, followed by the decay of the W − into e- υe. Estimate the maximum range of the weak force propagated by the W-boson.

Solution 3 The masses of all the other particles involved as much smaller than the mass of the W-boson. So then the violation of energy, ΔE ≈ mWc2. The time that we can have the violation for is: Δt ≈ ħ / ΔE = ħ/(mWc2). The maximum speed of the W-boson is c. So, the maximum range of the force, R = ħ/(mWc). To evaluate this, we can write R = ħc/(mWc2) = 197 MeV fm/80,000 MeV ≈ 2.5×10−3fm = 2.5 am. The effective range of the weak force is only 2.5 attometres! What does the Yukawa potential look like for exchange of a W-boson? The coupling of the W-boson, is written as gW.

Note that the R is the same as the expression for the range of the force we calculated above. Dayalbagh Educational Institute

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Interactions and fields

Particle Physics

Exercise 4

The W- was discovered in 1983 at CERN, using proton/anti proton scattering: p + p â&#x2020;&#x2019; W- + X where X represents one or more particles. What is the most likely X, for this process? Draw a Feynman diagram for the reaction.

Solution 4 The pion seems most likely â&#x20AC;&#x201C; it requires only one weak vertex, with no generation crossing, and its light (i.e. kinematically favored).

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Particle Physics

Exercise 5 Calculate the ratio of the gravitational attraction to the electrical repulsion between two stationary electrons.

Solution 5 The gravitational force : The electrical repulsive force : Hence

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Particle Physics

Definitions and Keywords Weak interaction: The interaction responsible for β -decay and other decay processes involving leptons and quarks. Electroweak theory: This is a unification of quantized theories of the electromagnetic and weak interactions. Weinberg angle: This angle describes the degree of the mixing that generates from four gauge bosons, the observables bosons γ, W±, Z0. Higgs mechanism: The mechanism by which the existence of a spin-0 particle can give a gauge boson mass without breaking the gauge symmetry. Higgs boson: the predicted spin-0 boson which must if the Higgs mechanism is responsible for giving mass to the gauge bosons W±, Z0, and to other particles. There may be more than one Higgs boson.

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# Chapter 1.4

The Weak and Gravitaional Interaction

# Chapter 1.4

The Weak and Gravitaional Interaction