PHY401: Nuclear and Particle Physics · Fundamental Forces From weak interaction processes at low energies (e.g. decay), we can estimate that mW ˇ80 GeV/c2. Can compare the magnitude

PHY401: Nuclear andParticle PhysicsLecture 20,Monday, October 12, 2020

Dr. Anosh JosephIISER Mohali

Elementary Particles -Properties and Interactions

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali

Elementary Particles - Properties andInteractions

1932: We knew about only three elementary particles.

Electron (e−), proton (p) and neutron (n).

An elementary particle - an object without anysubstructure.

A point particle.

However, structure can be probed only up to any givenscale...

... that is limited by the available energy.



Our definition of what is elementary or fundamental isalways tentative.

For example, to examine the structure of matter atlength scales of ∆r . 0.1 fm, requirestransverse-momentum transfers (∆pT ) at least of theorder

∆pT ≈ h

∆r=

hc(∆r)c

≈ 197 Mev fm0.1 fm c

≈ 2000 MeV/c. (1)



To be sensitive to small length scales, ...

...the energy of the particles used as probes must bevery high.

Because of this need, the study of elementary particleshas also come to be known as high-energy physics.


Fundamental Forces

Classical forces: electromagnetic and gravitationalforces.

Investigations of nuclear phenomena, tells us...

... that there are two more forces that have importancein the subatomic domain.

Strong force: responsible for the binding of nucleonsinside a nucleus.


Fundamental Forces

Weak force: Appears in processes such as β decay ofnuclei.

These two forces have no classical analogs.

They are also exceedingly short ranged.

It seems that we can point to four fundamental forcesin nature.


Fundamental Forces

They are:

1. Gravitation,

2. Electromagnetism,

3. Weak force,

4. Strong force.


Fundamental Forces

Forces can be distinguished through the strengths oftheir interaction.

Can estimate the relative magnitudes of these fourforces in a heuristic way...

... by considering their effective potentials.

Although such potentials are fundamentallynon-relativistic in concept,...

... they provide a useful guide for rough comparison.


Fundamental Forces

Consider two protons separated by a distance r.

Magnitudes of the Coulomb and of the gravitationalpotential energies for the two particles are

Vem(r) =e2

r, (2)

Vgrav(r) =GNm2

r. (3)

GN : Newton’s constant.

m: mass of the proton.


Fundamental Forces

More instructive to write the potential energies in theFourier transformed momentum space.

Except for an overall normalization, they take the form

Vem(q) =e2

q2, (4)

Vgrav(q) =GNm2

q2. (5)

q: magnitude of momentum transfer thatcharacterizes the interaction.


Fundamental Forces

Ratio of Vem and Vgrav is in fact independent of themomentum scale.

We can evaluate this ratio as

VemVgrav

=e2

GNm2=

(e2

hc

)1

(mc2)2 hc × c4

GN

≈(

1137

)1

(1 GeV)2)1039 GeV2

6.7≈ 1036. (6)


Fundamental Forces

In the above we have substituted 1 GeV/c2 for themass of the proton.

We used the value of

α =e2

hc=

1137

(7)

for the electromagnetic fine-structure constant.

Thus for charged elementary particles, thegravitational force is inherently much weaker than theelectromagnetic force.


Fundamental Forces

Since both the strong and the weak forces areshort-ranged, they can be describedphenomenologically by Yukawa potentials of the form

Vstrong =g2sr

exp(−

mπc2r hc

), (8)

Vweak =g2Wr

exp(−

mW c2r hc

). (9)


Fundamental Forces

gs and gW : coupling constants (effective charges) forthe strong and the weak interactions.

mπ and mW : masses of the force-mediating (orexchanged) particles.


Fundamental Forces

We can transform the above potentials to momentumspace.

Except for an overall normalization constant we get

Vstrong =g2s

q2 + m2πc2, (10)

Vweak =g2W

q2 + m2W c2. (11)


Fundamental Forces

Values of coupling constants can be estimated fromexperiments.

They are

g2s hc

≈ 15, (12)gW hc

≈ 0.004. (13)

Can think of π meson (mπ ≈ 140 MeV/c2) as themediator of the strong nuclear force.


Fundamental Forces

From weak interaction processes at low energies (e.g.β decay), we can estimate that mW ≈ 80 GeV/c2.

Can compare the magnitude of the Coulomb potentialenergy to that for the strong and the weakinteractions.

However, there appears to be an explicit dependenceon momentum scale in the ratio.


Fundamental Forces

Since we are considering the interaction of twoprotons...

...it is natural to choose the momentum scale tocorrespond to that of the proton mass.

Thus, choosing

q2c2 = m2c4 = (1 GeV)2 (14)

we get

VstrongVem

=g2s hc

hce2

q2

q2 + m2πc2=

g2s hc

hce2

m2c4

m2c4 + m2πc4

≈ 15× 137× 1 ≈ 2× 103. (15)


Fundamental Forces

Also

VemVweak

=e2

hc

hcg2W

m2c4 + m2W c4

m2c4

≈ 1137

10.004

(80)2 ≈ 1.2× 104. (16)

This shows once again that...

... the strong force is stronger than theelectromagnetic force, which in turn...

... is stronger than the weak force, and thatgravitation is the weakest of all the forces.


Fundamental Forces

For larger momentum scales of order ≈ mW , the weakand electromagnetic energies and strengths becomemore comparable.

Suggesting interesting possibility for a unification ofthe two forces at very high energies.


Elementary Particles

Before it was fully appreciated that quarks were thefundamental constituents of nuclear matter,...

... all the known elementary particles were groupedinto four classical categories...

... that depended on the nature of their interactions.



Particle Symbol Range of Masses

Photon γ . 2× 10−16 eV

Leptons e−, µ−, τ−,

νe, νµ, ντ . 3 eV to 1.78 GeV

Mesons π±, π0, K±, K0,

ρ±, ρ0 135 MeV/c2 to few GeV

Baryons p, n, Λ0, Σ±,

Σ0, ∆++, ∆0, Ω− . 938 MeV to few GeV



All particles participate in gravitational interactions.

Photon can interact electromagnetically with anyparticle that carries electric charge.

All charged leptons participate both in the weak andelectromagnetic interactions.

Neutral leptons have no direct electromagneticcoupling.



Leptons do not sense the strong force.

All hadrons (mesons and baryons) respond to thestrong force...

... and appear to participate in all the interactions.



All the particles in nature can be classified as eitherbosons or fermions, with the basic difference betweenthem being the statistics that they obey.

Bosons obey Bose-Einstein statistics whereasfermions satisfy Fermi-Dirac statistics.

This is reflected in the structure of their wavefunctions.



For example, the quantum mechanical wave functionfor a system of identical bosons is symmetric underthe exchange of any pair of particles.

That is,

ΨB(x1, x2, · · · , xn) = ΨB(x2, x1, · · · , xn). (17)

xi : denote, collectively, space-time coordinates as wellas internal quantum numbers of particle i.



On the other hand, under similar assumptions,...

...the quantum mechanical wave function for a systemof identical fermions is antisymmetric under theexchange of any pair of particles.

We have

ΨF (x1, x2, · · · , xn) = −ΨF (x2, x1, · · · , xn). (18)



Pauli exclusion principle is therefore automaticallybuilt into the antisymmetric fermionic wavefunction,...

... thereby forbidding a pair of identical fermions tooccupy the same quantum state.

This follows because, for x1 = x2, the wave function inEq. (18) would equal its negative value, and wouldtherefore vanish.



Can be shown from fundamental principles that...

... all bosons have integer values of spin angularmomentum, while fermions have half integral spinvalues.

Experimentally we determine spins of elementaryparticles.

Studies show that the photon and all mesons arebosons, whereas the leptons and all baryons arefermions.

Also, every known particle has a correspondingantiparticle.



The antiparticle has the same mass as the particle,but otherwise opposite quantum numbers.

Eg: positron e+ is the antiparticle of the electron, andcarries a negative lepton number and a positivecharge.

Antiproton (p̄) has one unit of negative charge and oneunit of negative baryon number,...

... in contrast to the proton which is positively chargedand has a positive baryon or nucleon number.



Certain particles cannot be distinguished from theirown antiparticles.

Example: π0, which has no electric charge, is its ownantiparticle.

It is clear that for a particle to be its own antiparticle,it must, at the very least, be electrically neutral.



However, not all electrically neutral particles are theirown antiparticles.

Eg: neutron has no electric charge, yet theantineutron is distinct because of its negative baryonnumber and opposite sign of its magnetic moment.

Similarly, the K0 meson, although charge neutral, hasa distinct antiparticle.



Still unknown whether the neutrino is distinct from itsantiparticle.

Antiparticles are denoted by the same symbol as theparticles, but with a bar over that symbol.

Except where it is redundant, or where there is aspecial symbol.


Quantum Numbers

To formulate general principles, we must deduce fromexperiments the type of quantum numbers that areconserved...

... and the conservation laws that are appropriate foreach of the interactions of the elementary particles.


Baryon Number

Experiments suggest that there is some conservationprinciple that forbids certain types of decays.

Can account for this simply by asserting that baryonscarry an additive and conserved quantum number(baryon or nucleon number) that equals B = 1 for allbaryons...

B = −1 for anti-baryons.

B = 0 for photons, leptons, and mesons.


Baryon Number

Consequently, if baryon number is conserved in allphysical processes,...

...then the proton, being the lightest baryon, shouldnot decay.


Lepton Number

Similarly, we can postulate a quantum number forleptons.

Assert that all leptons carry lepton number L = 1,whereas the photon and hadrons carry no leptonnumber.

Introduction of a lepton quantum number isnecessitated by many experimental observations.

Experimental also suggest that there must be differentkinds of lepton numbers within the family of leptons.


Lepton Number

Le Lµ Lτ L = Le + Lµ + Lτ

e− 1 0 0 1

νe 1 0 0 1

µ− 0 1 0 1

νµ 0 1 0 1

τ− 0 0 1 1

ντ 0 0 1 1


Lepton Number

Note: Although proton decay p→ e+ + π0 violates bothbaryon number and lepton number,...

... the combination B − L is conserved in the process.

This interesting feature should be incorporated intoany physical theory.


Strangeness

Early studies of cosmic-ray showers, showed thatcertain particles,...

... which have since been identified with K mesonsand the Σ and Λ0 baryons, were produced strongly.

That is, with large cross-sections of the order ofmillibarns.

But they had lifetimes characteristic of weakinteractions, namely as ≈ 10−10 s.


Strangeness

Gell-Mann and Pais proposed that these particlescarried a new additive quantum number,...

... which they called strangeness,...

... which is conserved in strong production processes,but violated in weak decays.


Strangeness

All the ordinary mesons and baryons (as well as thephoton) were assumed to be non-strange (S = 0).

Thus, in a reaction with the initial state having nostrangeness, the total strangeness of the particles inthe final-state must also add up to zero.


Strangeness

If we arbitrarily choose

S(K0) = 1, (19)

it follows thatS(K+) = S(K0) = 1, (20)

and that

S(Λ0) = S(Σ+) = S(Σ0) = S(Σ−) = −1. (21)


Isospin

Proton and neutron are baryons with spin 12 .

They are essentially degenerate in their mass.

They are quite similar in their nuclear properties,...

... except that the proton has a positive chargewhereas the neutron is electrically neutral.

Correspondingly, their electromagnetic interactionsare quite different.

Their magnetic dipole moments also have oppositesign.


Isospin

It has been known for a long time that the strong forcedoes not depend on the charge of a particle.

In fact, studies of mirror nuclei have demonstratedthat the strong binding force between p − p, n − n andp − n is essentially the same.

Scattering experiments have revealed that,...

... if we correct for electromagnetic effects, the crosssection for the scattering of two protons is the same asthat for two neutrons.


Isospin

Thus, the strong interactions do not distinguishbetween a proton and a neutron.

Consequently, if we imagine a world where only thestrong force is present,...

... and the weak and electromagnetic forces are turnedoff,...

... then in such a world a proton would beindistinguishable from a neutron.


Isospin

In such a world, we can think of the proton and theneutron as two orthogonal states of the sameparticle...

... that we can call the nucleon, and write the statesfor the neutron and proton as

p =

(1

0

), n =

(0

1

)(22)


Isospin

This language is very similar to that used indiscussing the “spin up” and the “spin down” states ofa spin 12 particle,...

.. which are also indistinguishable in the absence ofany interaction that breaks rotational symmetry (e.g.,a magnetic field).


Isospin

The two spin states will be degenerate in energy untilwe apply an external magnetic field,...

... which picks out a preferred direction in space, andremoves the degeneracy of the two states.

Can think of the proton and the neutron as beingdegenerate in mass because of some symmetry of thestrong force,...

... and we call this symmetry the isotopic-spin orisospin symmetry.


Isospin

In reality,...

... the presence of electromagnetic and weak forcesbreaks this symmetry,...

... lifts the degeneracy in the masses, and allows us todistinguish between a neutron and a proton.

Thus, it appears that the strong force does notdistinguish between different kinds of π mesons.


Isospin

Therefore, in the absence of electromagnetic and weakforces,...

... we can think of the three π mesons ascorresponding to different states of one particle, the πmeson.

We can represent the pion states as

π+ =

100

, π0 =01

0

, π− =00

1

. (23)


Isospin

These three states are degenerate in mass in ourhypothetical world.

Analogy with spin now corresponds to the three spinprojections of a J = 1 particle...

... that are degenerate in energy for a rotationallyinvariant Hamiltonian.


Isospin

Similarly,...

(K+, K0) doublet, (K̄0, K−) doublet and (Σ+,Σ0,Σ−)triplet,...

... each correspond to states that can be considered asdifferent manifestations of single particles, the K, K̄and Σ, respectively.

This discussion can be extended to all the knownhadrons, which can be classified into multiplets...

... corresponding to some quantum number verymuch like the spin quantum number.


Isospin

We will refer to this quantum number as the strongisotopic spin or strong isospin.

Its conservation suggests the invariance of the strongHamiltonian under isospin transformations.


Isospin

These transformations correspond to rotations...

... very much like those that occur for spin, but...

... the rotations are in an internal Hilbert space andnot in space-time.

The isospin quantum number (or I-spin) is found to beconserved in strong interactions.

It is a symmetry of the strong force.


Isospin

However, I-spin does not appear to be conserved inelectromagnetic or weak processes.

In the table below we summarize the strong isospinquantum numbers of different hadrons, as determinedfrom scattering experiments.

The assignment for the third-component, or projectionof the isospin chosen such that, in any given isospinmultiplet, a particle with a larger positive charge has ahigher value of the isospin projection.


Isospin

We have also denoted the projection as I3 instead ofthe conventional notation Iz, in order to emphasizethat isospin is not a space-time symmetry.

We cannot assign unique strong isospin quantumnumbers to leptons or to the photon, because...

...isospin transformations are a symmetry of only thestrong-interaction Hamiltonian,...

... and the photon and the leptons do not participatein strong reactions.


Isospin

Hadron Mass (MeV/c2) I I3

p 938.3 1212

n 939.6 12 −12

π+ 139.6 1 1

π0 135.0 1 0

π− 139.6 1 −1

K+ 494.6 1212

K0 497.7 12 −12

K̄0 497.7 1212

K− 494.6 12 −12


References

I A. Das and T. Ferbel, Introduction To Nuclear AndParticle Physics, World Scientific (2003).


End


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PHY401: Nuclear and Particle Physics · Fundamental Forces From weak interaction processes at low energies (e.g. decay), we can estimate that mW ˇ80 GeV/c2. Can compare the magnitude