Symmetries and conservation Laws: Part 2

Baryon and Lepton numbers

• The Baryon number B=1 is assigned to all baryons and B=-1 is assigned to all anti baryons: all other particles have B=0.
• The lepton number L_e=1 is assigned to electrons and electron neutrino, L_e=-1 to their anti particles ; all other particles have L_e=0
• L_μ=1 for muon and μ-neutrino and L_τ=1 for tau lepton and its neutrino.
• Significance of these numbers is that , in every process of whwtever kind , the total valuse of B, L_e,L_μ,L_τ separately remains constant.
• Conservation of leptons has a signefence for strond interactions.
• Another property that is conserved only in strong interactions is isospin.

Strangeness

• A number of particles were discovered that behsve so unexpectedly that they were called strange particles.
• They were created in pairs, and decay only in certain ways but not in others that were allowed by existing conservation laws.
• To clarify the observation Gell-Mann and Nishijina indepensently introduced the strangeness number.
• For photon , π⁰ and η⁰, B, L_e,L_μ, L_τ and S are zero . There is no way to distinguish between them and their antiparticles, and they are regarded as their own anti particles.
• Strangeness number is conserved in all processes mediated by strong and electromagnetic interactions.
• The multiple creation of particles with S≠0 is the result of this conservation principle
• S can change in an event mediated by the week interaction. Decays that proceed via a week interaction are relatively slow, a billion tomes slower than the interactions proceeded via strong interactions.
• Week interactions does not allow S to change by more than ±1 in a decay. For example,

  Ξ^- decays in two steps Ξ^- →Λ⁰+π^-→η⁰+π⁰

Isospin

• There are number of hadron families whose numbers have similar masses but different charges. Thes families are called multiplets. Member of multiplet represents different charged states of a single fundemental entity.
• Each multiplet according to number of charge states exhibits a number I such that the multiplicity of state is given by 2I+1.
• Isospin can be represented by vector I in an abstract iso space whose component in any specific direction is governed by the quantum number denoted by I₃.
• Possible values of I₃ varies from I, I-1 to -I. The charge of a baryon is related to its baryon numberB, its strangeness number S and the component I₃ of its isotopic spin by the formula

  $Q=e(I_3+\frac{B}{2}+\frac{S}{2})$

Conservation of statistics

• The interchange of identicle particles in a system is a type of symmetry operation which leads to the preservation of the wave
• Conservation of statistics signifies that no process occuring within an isolated system can change the statistical behaviour of the system.

Hypercharge

• Hypercharge is defined as Y=S+B
• Classification system for hadrons encompasses many short lived particles as well as relatively stable hadrons
• This scheme cillects isospin multiplets into submultiplets whose members have the same spin but different in isospin and a quantity called hypercharge.

Symmetries and conservation Laws: Part 1

• Every conservation principle corresponds to symmetry in nature
• A symmetry of a particular kind exists when a certain operation leaves something unchanged.
• There is an intimate connection between symmetry and so called conserved quantities.
• Well known conserved quantity is energy and corresponding symmetry in this case is time translation.

Momentum Conservation

• Holds for all type of interactions
• Related to the invariance of physical laws under translation in space.
• Thus laws of interaction do not depend on the place of measurement so the space is homogeneous.
• This transnational uniformity of space leads to the conservation of linear momentum.
• Particle at rest have no momentum. If it  decays into two less massive particles , momentum conservation requires that the two particles travel away in exactly opposite directions.

Conservation of Energy

• Holds for all type of interactions.
• related to the invariance of physical laws under translations along the time axis i.e., homogeneity of time.
• laws of interaction do not depend on the time of measurement

Angular momentum conservation

• In addition to transnational symmetry , space also has a rotational symmetry.
• This symmetry of space gives rise to another conserved quantity , angular momentum.
• This law is also of general validity for all types of interactions.
• It is related to the invariance of the physical laws under rotation (isotropy of space).
• The orbital and spin angular momentum may be separately conserved.

Parity Conservation

• Holds for strong, nuclear and electromagnetic interactions but is violated in case of week interactions.
• related to the invariance of the physical laws under inversion of space co-ordinates. x,y,z are replaced by -x.-y,-z.
• is equivalent to combined reflection and rotation.
• physical laws do not depend on the right handedness of co-ordinate system.
• Parity operation symmetry represents discrete symmetry (reflection and rotation through 180 degree)
• Every particle with non zero mass has an intrinsic parity  π which can either be +1(even) or -1 (odd). Thus total parity of a system of n particles is the product of their intrinsic parities and the orbital parity (-1)^l.
Thus, π_tot=π₁π₂π₃.......π_n(-1)^l
• Intrinsic parity of pions is odd.

Conservation of charge

• Conservation of electric charge is related to gauge transformations which are shifts in the zeros of the scalar and vector electromagnetic potentials V and A
• Gauge transformations leave E and B unaffected since the latter are obtained by differentiating potentials , and this invariance leads to charge conservation.
• Charge and baryon number are conserved in all interactions.

Baryons

• There is another whole class of unstable particles known as 'hyperons' , whose masses are each greater then that of protons.
• The first hyperon was found in cosmic rays named as Λ⁰ hypron , a neutral decaying particle.
• Charged particles seen in the decay were identified as proton and π^- meson, indicating a process

  Λ⁰ →p+π^-

• Anti-Λ⁰ hypron decay to an anti proton and π⁺-meson.
• The family of hyprons wit greatedt number of members is Σ-family.
• First if it to be observed is Σ⁺ with mass about 2328m_e , and two prominent decat schemes are

  Σ⁺→p+π⁰ ; Σ⁺→n+π⁺

• Σ^- has just one set of decay products

  Σ^-→n+π^-

  its mass being slightly greater than Σ⁺ and is 2341 m_e.

• Neutral Σ hyperon decays as

  Σ⁰→Λ⁰+γ

  its mass is 2328m_e

• The Σ hyperon form a triplet Σ⁺,Σ^- and Σ⁰. A corresponding triplet of anti Σ hyperon also exists.
• The anti particle of Σ⁺ can not be Σ^- because the two have slightly different masses, whereas particle anti particle must have exactly same masses.
• Another group of members belonging to hyperon family is Ξ hyperon originally called cascade particles.
• Theit negative and neutral forms have been observed with decay processes

  Ξ^-→Λ⁰+π^-

  Ξ⁰→Λ⁰+π⁰

  Their masses are about 2582 m_e

• Anti Ξ hyperons have been detected.
• Togather with nucleons (p and n) , the hyperons form the family of baryons.

Baryons(B=+1, L_e=L_μ=L_τ=0)

Particle	Symbol	mean life (s)	spin	S	Y	I	I3	Mass MeV/c2
Nucleon	n	stable	1/2	0	+1	1/2	-1/2	938.3
	p	886					+1/2	936.6
Lambda	Λ0	2.6×10-10	1/2	-1	0	0	0	1116
Sigma	Σ+	8.0×10-11	1/2	-1	0	1	+1	1189
	Σ0	6×10-20					0	1193
	Σ-	1.5×10-10					-1	1197
Xi	Ξ0	2.9×10-10	1/2	-2	-1	1/2	+1/2	1315
	Ξ-	1.6×10-10					-1/2	1321
Omega	Ω-	8.2×10-11	3/2	-3	-2		0	1672

• There is a sequence of decay for Ω^- baryon

  Ω^-→Ξ⁰+π^-→Λ⁰ +π⁰ ....

  Final result of decay is proton , two electrons and two photons.

Mesons

• Mesons are particles with zero or integral spin so they are Bosons.
• The lightest meson is pion or π-meson, with other meson masses ranging beyond proton mass.
• All mesons are unstable and decays in various ways.

π-mesons

• This particle is to transmit nuclear forces, it must interact strongly with nuclei, and therefore it should be scattered and absorbed quickly by matter through which it passes.
• π mesons are thus hypothetical particles responsible for the nuclear forces and had properties pridicted by Yukawa.
• Protons and neurtons can be transferred into one anotherby emitting or absorbing one of these particles.
• There are three kinds of pions π⁺, π^- and π⁰. π^- is the anti particle of π⁺.
• These new particles can be thought of making bonds between (n,n) , (p,p) and (n,p) or (p,n)

K mesons

• These are heavier unstable particles and have a great variety of different decay modes.
• THere are six different ways that K⁺ mesons commonly decay, in each case giving two or three less massive particles

  K⁺→π⁺+π⁰

  K⁺→μ⁺+ν_μ

  K⁺→π⁺+π⁺+π⁰

  K⁺→π⁺+π⁰+π⁰

  K⁺→e⁺+ν+π⁰

• Mass of K⁺ is 966m_e.
• K^- are anti particles of K⁺ mesons and have the same decay modes with appropriate exchange of decay products for their anti particles.
• K⁰ and $\overline{K^0}$are anti particles.

Mesons (B=0) Bosons

Particle	Symbol	Mass MeV/c2	Mean life (s)	spin	S	Y	I	I3
Pion	π+	140	2.6×10-8	0	0	0	1	+1
	π0	135	8.7×10-17					0
	π-	140	2.6×10-8					-1
Kon	K+	494	1.2×10-8	0	+1	+1	1/2	+1/2
	K0	498	9×10-11
	K-	494	5×10-8					-1/2
Eta	η0	549	6×10-19	0	0	0	0	0

Leptons

• particles which are untouched by strong forces and which participates in weak interactions are called leptons.
• All these particles (leptons) also have their corresponding anti particles.
• Unlike baryons and mesons no heavier leptons have been detected.
• Electrons with its anti particle positron along with its associated neutrino and anti nutrino are leptons.
• Muon with their respective neutrino and anti particles of these are also leptons. Another pair of leptons known as tau,τ and its associated neutrino.
• All tau's are charged and decay into electrons, muons or pions along with appropriate neutrino.
• The breakdown of parity conservation in weak interaction has important consequence for leptons.
• The particles are left handed and anti particles are right handed.
• In normal weak interactions the particles (electrons, negative muons, neutrinos) behaves as if they were left handed screws; i.e., observer think they spin clockwise when they are travelling towards him.
• Anti particles are right handed screws; an observer thinks they are spinning counter clockwise as they approaches him.
• Nature of weak interactions, with its voilation of such established symmetries as parity, charge conjunction, isotopic spin and strangeness is physics greatest problems.
• Muons were first discovered in decay of charged pions

  Charged pion decay:

  π⁺→μ⁺+ν_μ

  π^-→μ^-+$\overline{{\nu }_{\mu }}$

  Neutral pion decay:

  π⁰→γ+γ

• Muon decay

  μ⁺→e⁺+ν_e+$\overline{{\nu }_{\mu }}$

  μ^-→e^-+ν_μ+$\overline{{\nu }_e}$

Leptons have (B=0) (fermions) and their properties are

Particle	Symbol	Spin	Le	Lμ	Lτ	Mass (in MeV)	Mean Life
electron	e-	1/2	+1	0	0	0.511	stable
muon	μ-	1/2	0	+1	0	106	2.2×10-6
tau	τ-	1/2	0	0	+1	1784	3.4×10-25
electron neutrino	νe	1/2	+1	0	0	0	stable
μ neutrino	νμ	1/2	0	+1	0	0	stable
τ neutrino	ντ	1/2	0	0	+1	0	stable

Photons

• It only participates in EM interactions.
• Strong and weak interactions are not in photon domain of experience.
• When particle annihilate with anti particles the end product is often protons.
• Photon is its own anti particle.
• Under some circumstances it can disappear and can create particle -anti particle pair.
• Rest mass of photon is zero.
• Photon is a boson with angular momentum equal to 1.
• Spin of photon is 1

elementary particles and fundamental interactions

• Elementary particles are those microscopic elementary constituents out of which all matter in this universe is made of.
• Bound neutron is stable but unbound neutron is unstable and it decays according to equation

  $n\to p+e^{-}+\overline{{\nu }_e}$ (anti-nutrino)

  half life of free neutron is 14 min 49 sec.

Fundamental interactions

There are four fundamental interactions between particles

(1) strong

(2) electromagnetic

(3) week

(4) gravitational

Interaction	Particles affected	range	relative strength	particles exchanged	Role in universe
(1) Strons	Quarks	∼10-15m	1	gluons	Holds quarks togather to form nucleus
	Hadrons			Mesons	Holds nucleons togather to form atomic nuclei
(2) Electromagnetic	charged particles	infinite	∼10-2	photons	determine structure of atoms, molecules , solids etc., Important factor in astronomical universe.
(3) Weak	Quarks and leptons	∼10-17m	∼10-5m	Intermediate bosons	mediates transformations of quarks and leptons; helps determine composition of atomic nuclei
(4) Gravitational	all	infinite	∼10-39m	gravitons	Assemble matter into planet , galaxies and stars

• Elementary particles can be divided into four groups

  (1) photons

  (2) leptons

  (3) mesons

  (4) baryons

Anti particles

• A particle identical with proton except for negative charge, i.e., negative proton or antiproton was created by bombarding protons in a target with 6 GeV protons thereby inducing the reaction

  p+p+energy(6 GeV)→p+p⁺

• Particle and anti particle annihilate each otherto give rise to a form of energy.
• positron is the anti particle of electron.
• There must be an anti oarticle corresponding to each particle.
• From the collection of anti particles a world of anti matter could be created.

Reletionship between particle and anti particles is

property	relationship
(1) mass	same
(2) spin	same
(3) magnetic monemt	of opposite sign but same magnitude
(4) charge	of opposite sign but same magnitude
(5) mean life in free decay	same
(6) annihilation	in pair
(7) creation	in pair
(8) total isotopic spin	same
(9) intrinsic parity	same for bosons but opposite for fermions
(10) strangeness number	of opposite sign but same magnitude

Beta decay

Beta decay can involves three processes, in all the three processes atomic number of nucleus becomes one unecay dit greater or smaller but the mass number remains the same.

Problems in explaining beta decay

(1) non conservation of energy

(2) non conservation of angular momentum

To explain this Pauli suggested that a second particle is emitted with beta particles simultaneously but the sum of kinetic energies of two particles must be always equal to the energy difference between the parent and daughter nuclei. Thus the principle of conservation of energy is not voilated even the beta particle do not carry same energy.

The maximum beta particle energy is equal to the energy difference between the parent and daughter nuclei.

Beta decay Theory (Fermi's Theory):-

Electron, positron and ν does not as such exists inside the nucleus but are formed at the time of the decay. Fermi assumed that β-decay results from some form of nutrino, electrons and nucleus. This type of interaction is known as weak interaction. Constant required to express its strength is g=10^-47erg cm³ . It is because of the weakness of this interaction that the β-decay does not take place instantly in the case when it is energitically possible.

This theory must include some relationship between the particles of initial and final nuclei. The relationship is expressed by means of matrix element. It involves the wave function of initial and final nuclear states and hence their spin, parities and the arrangement of the nucleons. When two states are very different from each other |M_if|² becomes smaller. The total available β-decay energy E₀ , which is the energy difference between initial and final states, can be divided between electrons and ν in large number of ways which affects the shape of beta ray spectrum.

The number of ways of distributing total available energy between electron and ν per unit total energy E₀ is dN/dE₀.

Now , number of ways in which electron may be given volume V and having momentum between p_e and p_e+dp_e is given by

${\mathrm{dn}}_e=4\pi p_e^{2}V\frac{d{p}_e}{h^3}$

Similarly , number of ways in which neutrino can be arranged between volume V having momentum between p_ν and p_ν+dp_ν is

${\mathrm{dn}}_{\nu }=4\pi p_{\nu }^{2}V\frac{d{p}_{\nu }}{h^3}$

Number of ways in which β-decay can lead to an electron having momentum between p_e and p_e+dp_e and ν having momentum between p_ν and p_ν+dpν is

dN=dn_e.dn_ν

$\mathrm{dN}=\frac{16{\pi }^2{V}^2}{h^6}{p}_e^2{p}_{\nu }^{2}d{p}_{e}d{p}_{\upsilon }$

Relativistic momentum of particle of rest mass m is given by

$p=\frac{{\left[E\right(E+2m{c}^2]}^{1/2}}{c}$

since mass of neutrino is almost zero

p=$\frac{E}{c}$

Therefore momentum of neutrino is

$p_{\nu }=\frac{E_{\nu }}{c}=\frac{E_0-E_e}{c}$

and hence,

${\mathrm{dp}}_{\nu }=\frac{d{E}_0}{c}$ for given E

This shows that ,$\frac{\mathrm{dN}}{d{E}_0}=\frac{16{\pi }^2{V}^2}{h^6{c}^3}{(E_0-E_e)}^2{p}_e^{2}d{p}_e$

We shall now also consider the role of coulomb's barrier in letting out electron and positron against beta decay. The coulomb barrier aids the escape of positron but hinders the escape of electrons. The effect of Coulomb barrier depends on the atomic number Z and the energy of electron or positron. The fermi factor represented by F(Z,E_e) is a complex function.

Now Fermi's theory finally gives the probability of decay with the emission of an electron having a given momentum p_e by the expression which involves the nucleon-beta-neutrino force constant g , the matrix element |M_if|² and the function of Fermi factor.