1. Teoria della Relatività - V. Lubicz 2. Meccanica quantistica - V. Lubicz 3. Particelle elementari e forze fondamentali - C. Tarantino 4. Gravità quantistica - V. Lubicz

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1. Teoria della Relatività - V. Lubicz 2. Meccanica quantistica - V. Lubicz

3. Particelle elementari e forze fondamentali - C. Tarantino

4. Gravità quantistica - V. Lubicz

Fisica Teorica Contemporanea (parte III)

Particelle elementari e Forze fondamentali

(Teoria quantistica dei campi, Modello Standard,…)

Testo di riferimento:

Q. Ho-Kim, N. Kumar, C.S. Lam Invitation to Contemporary Physics

Prof.ssa Cecilia Tarantino

Lezione n° 1

Elementary constituents of matter

Ambition of humankind:

Establish order and regularity in our complex surroundings

• Ancient Greeks: four elements (air, fire, water, earth)

• Ancient Chinese: five elements (metal, wood, water, fire, earth)

…

…

• 1808, John Dalton: (unalterable and indestructible) atom

• Middle of the 19° century, Dmitry Mendeleyev: periodic table

(by observing the (still unexplained) regularity of chemical elements)

• 1897, Joseph John Thomson: discovery of electrons

• 1911, Ernest Rutherford: discovery of atomic nucleus

Destruction of Dalton’s atomic concept Planetary model of atoms

• Chemical elements are different since they have a different number of electrons

• Chemical properties are due to the electronic arrangement (explained by Quantum Mechanics)

(A chemical element may have several isotopes, characterized by identical chemical properties and different nucleus mass)

• 1932, James Chadwick: discovery of neutron

• Nuclei are made up of protons and neutrons (collectively referred as nucleons):

• Protons (charged) are as many as electrons (atoms are neutral)

• Neutrons (neutral) can be of different number in different isotopes

(a nucleus with too many neutrons becomes unstable by undergoing β-decay)

The elementary particles seemed to be (no more atoms but):

protons, neutrons and electrons

• 1930’s : new particles discovered in cosmic rays or in high energy accelerators (mainly hadrons, i.e. particles subject to nuclear forces, like nucleons)

There were many hadrons and they seemed to be unrelated (like chemical elements before Mendeleyev)

• 1960’s : Murray Gell-Mann and Yuval Ne’eman found some regularity (SU(3)-symmetry)

• 1964: Murray Gell-Mann and George Zweig discovered an inner hadron sctructure

Nucleon

The elementary particles became:

•  quarks (composing nucleons and other hadrons)

•  leptons (unaffected by nuclear forces, like electrons)

Intimate interplay between theory and experiment

Observed puzzles lead to new ideas, confirmed by experiments

• As substructers become smaller and smaller, larger and larger microscopes are needed

• Microscopes for Particle Physics are (huge) accelerators

• Due to their quantum nature, particles behave like waves with a De Broglie wave length λ=h/p

(in order to see small substructure, high energies are needed)

• Accelerator beams explore inner structures by destroying target particles (with the help of theory the original particles can be reconstructed)

The elementary constituents of matter

as we know them today

•  Quarks (composing nucleons and other hadrons)

•  Leptons (unaffected by nuclear forces, like electrons)

• Both quarks and leptons appear in three generations (or families)

• Constituents in different generations are almost clones of each other (except for mass that is larger for higher generations)

• Constituents of the 2° and 3° generations are unstable

(cannot be found naturally but can be produced by high energy collisions)

We do not know why generations are 3!

When in 1936 the muon (µ, 2° generation charged lepton) was discovered, Isidor Isaac Rabi expressed his astonishment (“Who ordered that?”)

80 years later we still do not know!

There are 2 quarks and 2 leptons in each of the 3 generations

• Neutrino (postulated by Wolfang Pauli in 1930)= little neutral in Italian

• Leptons have integral units of electric charge, while quarks have multiples of 1/3

(quarks come in 3 colors)

• Hadrons made up of 3 quarks are called baryons, those made up of a quark and an antiquark

are called mesons

I II III

Matter constituents are fermions.

They obey the Fermi-Dirac statistics and the Pauli exclusion principle.

For each matter particle there exist an antiparticle

• Antiparticles have the same mass but opposite electric charge as the particle (and all the opposite additive quantum numbers)

• When they encounter particles of the same kind, they annihilate each other

(since earth is made up of particles, no antiparticles can live for long close to us)

• The existence of antiparticles was predicted theoretically in 1929 by Paul Dirac (to explain the negative-energy solutions of his quantum-relativistic equation)

The Standard Model (SM, i.e. the present theory ) also requires bosons (characterized by integer spin and obeying the Bose-Einstein statistics):

• Gauge particles (s=1) to transmit forces

• Higgs boson (s=0) to generate masses

Discovered at the LHC (summer 2012)!

Complete list of the Standard Model (SM) particles

Fundamental Forces

Ambition of humankind:

Establish order and regularity in our complex surroundings

• What binds protons and neutrons together in a nucleus?

• And electrons and a nucleus into an atom?

• And atoms together into a molecule?

• And molecules together in ourselves?

• And us on the surface of this planet?

• …

Can we study the forces of Nature and classify them systematically?

At first glance this seems impossible but, as for particles, IT IS NOT!

It turns out that all the complicated forces we experience are very complex manifestations of 4 fundamental forces (in order of decreasing strength):

• Strong

• Electromagnetic

• Weak

• Gravitational

A different World with different Forces

If the electromagnetic force had a short range and were very weak

• Electrons could not be bound to nuclei

• There would be a plasma

(soup of unbounded electrons and nuclei)

If the electromagnetic force

were much stronger than the nuclear force

• The electrostatic repulsion between protons would overcome the nuclear attractive force

• There would be only Hydrogen (1 proton) atoms

In both scenarios it would be almost impossible to see anything With plasma:

Light cannot travel far being constantly absorbed and emitted by charged particles

With a very strong electromagnetic force:

There would be a virtual plasma due to electron-positron pairs created by light (according to the indetermination principle)

If the nuclear force were too weak We would not have

complex nuclei

If the nuclear force were very much stronger very heavy chemical elements

could be formed

If gravity were too weak we would all float away like an astronaut in orbit

The kind of world we have around depends very sensitively on what forces

are present and how strong they are!

The gravitational force

• The weakest fundamental force and the 1° to be discovered

• Disovered by Newton in the 17° century while trying to explain the Kepler’s laws

• Universal, all pervasive and ruled by an inverse-square law

• It keeps planets going around the sun and satellites going around the earth

• It holds us to the surface of the earth being M_earth large

• Among two of us it is too weak to be felt

• Among particles it is negligible w.r.t. to the other fundamental forces

N•m

/kg

General Relativity

• 1915: Albert Einstein formulated the Theory of General Relativity

(probably the deepest theory in the 20° century)

• Gravity is interpreted as a revelation of the curvature of the space-time

• Light must fall under Gravity just like particles (observations show that light curves

passing near a massive body)

• On day-to-day experience and planetary motions the effects of General Relativity cannot be

distinguished from those of Newton’s law

• General Relativity predicts gravitational waves emitted by accelerating objects like

accelerating charges emit electromagnetic waves (recently observed by LIGO, February 2016)

Quantum effects in gravity are not understood!

They become relevant on very small scales

(very high energies ~M

~10

GeV)

The electromagnetic force

• It is, other than gravity, the only force we encounter in our daily lives

• It is responsible for: electromagnetic waves (light, radio, television,…), electronics, binding electrons in atoms, binding atoms in molecules, binding molecules in a liquid or solid,…

• Electricity and magnetism were first thought to be unrelated, until Maxwell’s equations (Clerk Maxwell, 1873)

• The electric force, like gravity, follows an inverse-square law

(Coulomb’s law). It is proportional to the product of the electric charges

(attractive between opposite charges, repulsive between same-sign charges)

The weak and strong (or nuclear) forces

• Discovered in the 20° century

• Not detectable in our daily lives due to the very short range

(10^-17÷10^-18 m for weak forces , 10^-15 m for nuclear forces between nucleons)

• At distance short compared to their ranges, they obey the inverse-square law.

Beyond these distances they become extremely small

• The weak force is responsible for the instability of the neutron, that decays by radioactive

beta-decay (β-decay) within a long time (weak force) of ~15 minutes

• Neutrons inside nuclei remain stable being protected by nuclear and electromagnetic forces, if they are not too many

• With many neutrons, they are not all protected and a neutron is changed into a proton, if the released energy (m_n-m_p-m_e≈0.8 MeV) wins over the additional

electrostatic repulsion in the nucleus with the additional proton

• The strong force is the strongest of the four forces (in its range)

• An enormous amount of energy can be derived from a small quantitiy of fissionable material used in a nuclear power plant or in an atomic bomb

• Between two hadrons it is an inverse-square law force with a range of 10^-15 m

• Between quarks appear to be peculiar

• No matter how far apart two quarks are, there is always a constant force to pull them back. The energy needed to pull the two quarks apart is proportional to their separation.

An infinite energy is needed to separate and isolate them. It is more convenient to produce couples of quark-antiquark than isolating quarks (CONFINEMENT)

• Electromagnetic and gravitational forces have long ranges and obey the inverse-square law

• Weak and strong forces have short ranges

• What makes them different?

• Why some of the forces are attractive (e.g. gravity between same-sign masses) and others repulsive (e.g. electric force between same-sign charges)?

• These questions are unanswerable in classical Physics nor in non-relativistic Quantum Mechanics

• They become answerable when Special Relativity is incorporated into Quantum Mechanics, in the resulting Quantum Field Theory

Special Relativity tells us:

•  the total energy E=√(m² •c⁴ + c² •p²) is conserved while the mass is not

• A particle with mass m has an energy E=m•c² at rest

• If that amount of energy is available, a particle of that mass can be created (as long as no conservation law is violated)

(Quantum Mechanics) Heisenberg’s uncertainty relation tells us:

•  a closed system may fluctuate by an amount ΔE~ h/Δt during a time interval Δt, with h=6.6 x 10^-25 GeV·s (1 GeV = 10⁹ eV)

• Thus, at short time intervals, enough energies are available to create particles

• A created particle of mass m can last only a time interval Δt~h/mc²

• Such a particle is known as a virtual particle (or exchange or mediator particle)

This is how force arises in a Quantum Field Theory:

A virtual particle created by particle A at one location and annihilated by particle B at

another transmits a force between A and B

A

B

Theory of forces

• Field theories deal with physical systems described by assigning the value of one or more quantities (the fields) in each space point

• Within classical (i.e. non-quantum) Physics there are several field thoeries:

in Acoustics, Elasticity, Hydrodynamics, Meteorology,…there appear fields of pressure, of density, of speed, of elastic deformations,…

These fields correspond to an idealization of the real system, they do not represent physical properties of the space points but average properties of atoms and molecules

Field Theories

• Within classical Physics fields that correspond to real physical properties of the space are the electromagnetic and the gravitational fields

• The role of fields has become crucial in order to interpret the most fundamental phenomena, within a quantum relativistic theory

• The concept of particle turns out to be reabsorbed in the concept of field

(particles are excitations of the field like waves are excitations of the sea surface)

• The grounds of Quantum Field Theory were formulated in the 30’s of the 20° century mainly by Paul Dirac, Wolfgang Pauli, Sin-Itiro Tomonaga, Julian Schwinger, Richard Feynman and Freeman Dyson

• It was originally elaborated within particle Physics to solve the problems that arised when Quantum Mechanics was tried to be extended to relativistic systems

The (non-relativistic) Schrödinger equation of Quantum Mechanics

was modified to describe relativistic systems obtaining the Klein-Gordon (for scalar particles) and the Dirac (for spin=1/2 particles) equations.

Both equations are problematic, in particular they lead to solutions with negative energies going to –∞ (i.e. there is no fundamental state)

• These equations, like the Schrödinger equation, are written in terms of the wave function, which describes single particle states.

The hearth of the problem is that they cannot describe processes (allowed by Special Relativity) of particle-antiparticle annihilation and more in general of creation and annihilation of particles

• This problem is solved within a Quantum Field Theory described in terms of a field, which is a quantity existing in each space point and defining the probability of particle creation and

annihilation

Quantum Field Theories

non-relativistic kinetic energy

Quantum rules dramatically change the meaning of empty space (or vacuum) For a classical field the vacuum is the state where

the field is flat

In a quantum field, due to the indetermination relation, the vacuum is a noisy and densely energetic surface

This random motion (vacuum fluctuations) give rise to virtual particles

that can briefly and spontaneously appear from the vacuum and disappear again

Interacting Quantum Field Theory

• Quantum field theories are particularly suited to describe interactions

• Different parts of a system can not interact istantaneously, being c the maximum speed for a signal

• It is much more convenient to describe the interaction as the interaction of a particle with the field than an interaction between two particles through a force transmitted in a time t≠0

• The mechanism of the interaction has to be local, involving only infinitesimally near points (two far-away points that can’t be connected by light do not interact)

• Interacting theories are much more complicated than free field theories and typically require a perturbative approach to be solved

• The perturbative method consists in expressing the results of the Quantum Field Theory as a power series in small

parameters called coupling constants (e.g. the electric charge e)

• Quantum Electrodynamics (QED) was the first interacting Quantum Field Theory to be formulated, in the 50’s of the 20° century, mainly by Richard Feynman

Feynman diagrams

• The perturbative approach used in interacting teories uses the fundamental instrument of Feynman diagrams

• Feynman diagrams are pictorial representations of the contributions appearing in the perturbative series of physical quantities (probability amplitudes)

• They have the great advantage of providing an intuitive comprehension of the way the interaction influences the physical quantity

• There exist rules to identify Feynman diagrams representing a process and to assign a mathematical expression to each diagram

At higher order in the perturbaive

series

Lezione n° 2

Fisica Teorica Contemporanea (parte III)

Particelle elementari e Forze fondamentali

(Teoria quantistica dei campi, Modello Standard,…)

Testo di riferimento:

Q. Ho-Kim, N. Kumar, C.S. Lam Invitation to Contemporary Physics

Prof.ssa Cecilia Tarantino

Special Relativity tells us:

•  the total energy E=√(m² •c⁴ + c² •p²) is conserved while the mass is not

• A particle with mass m has an energy E=m•c² at rest

• If that amount of energy is available, a particle of that mass can be created (as long as no conservation law is violated)

(Quantum Mechanics) Heisenberg’s uncertainty relation tells us:

•  a closed system may fluctuate by an amount ΔE~ h/Δt during a time interval Δt, with h=6.6 x 10^-25 GeV·s (1 GeV = 10⁹ eV)

• Thus, at short time intervals, enough energies are available to create particles

• A created particle of mass m can last only a time interval Δt~h/mc²

• Such a particle is known as a virtual particle (or exchange or mediator particle)

This is how force arises in a Quantum Field Theory:

A virtual particle created by particle A at one location and annihilated by particle B at

another transmits a force between A and B

A

B

Theory of forces

Range and Mass

• The range of the force is related to the mass of the exchange particle

• Even moving at the speed of light, the virtual particle can travel only a distance Δx ~c•Δt~h/mc before it is absorbed

• So, the range of the force is of the order hc/mc²

• This argument, invented by Hideki Yukawa in 1935 led him to predict the existence of a new particle, the pion (π)

• At that time it was known the range of the nuclear force, about 1 fm=10^-15 m

• The rest energy of the associated virtual particle is approximately hc/1 fm =0.197 GeV

• After the discovery of the pion (m_π≈0.14 GeV) Yukawa was awarded a Nobel prize

• We know that the pion is not an elementary particle, but that does not affect its ability to transmit a force

Virtual cloud

• If there is no other particle nearby, a created virtual particle is reabsorbed in the same point. Consequently every particle is sur- rounded by a Yukawa cloud of virtual particles, with an extent hc/mc²

• The density of the cloud depends on how often the virtual particle is created and thus on the strength of the interaction

• For nuclear interactions the cloud is dense and with an extent of 1 fm

• For the other interactions the cloud is thin and the effective size of the particle is not much increased by the presence of the cloud

• The virtual particle transmitting the electromagnetic force is the photon (γ), transmitting the gravitational force is the graviton (G).

Both are massless so the range of these forces is infinite

• Weak forces are trasmitted by W^± and Z⁰ bosons (charge-current and

neutral-current interactions). The masses of these bosons are M_W=80 GeV/c², M_Z=91 GeV/c². Therefore, the range of the weak force is about 2•10^-18 m

• Strong interactions between quarks are trasmitted by (massless) gluons. Gluons are confined inside hadrons and can not propagate.

The Yukawa argument does not apply.

Range of strong interactions between quarks and between nuclei

• The range of nuclear interactions is ~1 fm, as predicted by Yukawa, that is nuclear interactions are short-ranged

• Strong interactions between quarks are independent of distance, so surely long-ranged

• The short-ranged nuclear force is presumably a result of cancelation between the long-ranged forces from the different quarks inside a nucleon

• Similarly, the force between neutral atoms (van der Waals force) has a range much shorter than the range of the Coulomb forces between electrons and/or protons.

This is because an electron in one atom sees both repulsive and attractive forces from the electrons and the protons of another atom.

These almost equal and opposite forces tend to cancel each other

• The argument fails if the emission is confined to a cylinder of constant radius ρ

• At a distance r<< ρ, the same argument as before leads to the inverse-square law.

At a distance r: ρ<<r<<R it is different.

The exchange particle is guided down the cylinder, where the cross-sectional area is constant and the force remains independet of r. This is what seems to happen for quarks, with ρ≈1fm

Inverse-Square Law versus Confinement

• A force becomes zero beyond its range.

At a small distance compared to its range a force follows an inverse-square law

• In fact, at a distance r from the emitting particle A, that is well within the range of the virtual particle, the surface area of a sphere is 4πr² and the exchange particle can be anywhere on this surface

• The probability of having the virtual particle

along some fixed direction is proportional to 1/r², which is the origin of the inverse square-law.

• This argument is still valid if the exchange particle is not emitted isotropically

|ρ

Spin and nature of forces

• The spin of the exchange particle conveys

important information about the force it transmits

• Spin is a quantum angular momentum with no classical analog.

It is the angular momentum of the particle at rest.

It is an intrinsic and unchangeable attribute of the particle

• Angular momentum is described classically by a three-dimensional vector, i.e. by its three components.

Quantum mechanically, due to the uncertainty principle, the three components are not simultaneously measurable. One can specify the magnitude of the

vector and one of the three components (conventionally the z-component) The length of the spin vector is h√s(s+1) and the z-component is m_sh, with s a non-negative integer (s=0,1,2,..) or half-integer (s=1/2, 3/2, …). For a given s, m_s may assume 2s+1 discrete values, from –s, -s+1,…, to s-1, s

Quarks, electrons, neutrinos, nucleons are s=1/2 particles.

Photons, gluons, W^± and Z⁰ are s=1 particles, the graviton is a s=2 particle. The Higgs boson H⁰ has s=0

• The nature of the force carried by an exchange particle depends on its spin

• Quantum Field Theory dictates that if s is an even integer, the force between identical particles is attractive.

If s is an odd integer, the force between identical particles is repulsive and the force between a particle and its own antiparticle is attractive

• Pions have s=0, gravitons have s=2. Nuclear and gravitation forces of two protons, for instance, are attractive

• Photons have s=1, so the Coulomb force between two protons is repulsive

• An exchange particle carries with it 2s+1 bits of information on its orientation. These different bits can trigger different forces, so the complexity of forces increases with s

• E.g.,the Yukawa nuclear force, transmitted by the s=0 pion, is a single attractive force

• The electromagnetic force, carried by the s=1 photon, is electric and magnetic

• The gravitational force, transmitted by the s=2 graviton, is even more complex

• However, unless particles move very fast, the electric-like force dominates (that is the Coulomb force in

electromagentism and the Newtonian force in gravity)

How the exchange of a virtual particle can possibly produce an attractive force?

The exchange of a virtual particle from A to B intuitively seems to knock A away from B.

One would argue that if A emits a virtual particle towards B, A must recoil away from B and that

when B absorbs this virtual particle (and its momentum), B must be knocked away from A

This argument is classical, it fails in Quantum Mechanics.

In Quantum Mechanics, with A and B at fixed distance, the momentum of the exchange particle is undetermined

due to the uncertainty principle.

It could be opposite to what it is classically,

thus allowing for attractive instead of repulsive forces

In summary: The Yukawa potential V(r)

• The relation between force range and mass of the exchange particle can be derived from the Yukawa potential

• The interaction of A and B, separated by a distance r, is determined by a potential energy V(r)

• A physical system likes to minimize its potential energy by moving its particles towards a smaller V(r)

• There corresponds a force F=-dV(r)/dr

(the sharper the variation of V(r), the faster the particles move)

• The Yukawa potential has the form V(r)~ f_s/r•e^-(mc/h)r

• The exponential factor provides the range R≈h/mc as demanded by the Yukawa theory

• The factor f_s depends on the spin s of the exchange particle. For the dominant force it is

negative for even s and positive for odd s. This sign determines wether the force is attractive or repulsive.

• The inverse-square law at short range follows from F=-dV(r)/dr (r<<R)

Quantum Numbers

We distinguish different particles by their special characteristics, usually expressed as numbers (quantum numbers)

Other than the mass, they are usually discrete numbers (often integers but not always). Spin, electric charge,…are examples Unless otherwise stated, quantum numbers are conserved, i.e.

the sum of the values of a quantum number in an isolated system remains the same at all times.

If they changed with time they would not be permanent features of a particle and they could not be used to label particles

What distinguishes elementary particles?

Additive and Vectorial Quantum Numbers

Additive and vectorial quantum numbers differ in how the sum is taken

• For additive quantum numbers it is just the arithmetic sum

• Examples of additive quantum numbers are: electric charge, quark number (number of quarks minus number of antiquarks), lepton number (number of leptons minus number of antileptons)

• They are conserved, as it can be seen for instance in β-decay: n->p e^- ν_e

• A vectorial quantum number is the quantum version of a conserved vector, such as angular momentum

• Classically, if a vector has d components, the addition of two vectors is equivalent to the addition of their components

• A quantum mechanical vector, due to the uncertainty principle, can usually be specified only by r<d components, with r the rank of the vectorial quantum number

• The quantum mechanical vector is different from a collection of r additive quantum numbers because the length and some other characteristics of the vector can be also specified, in general t additional discrete numbers (p₁, p₂,…,p_t)

• In most of physical interesting vectorial quantum numbers t=r

• The p_i do not add arithmetically

• Spin is the simplest example of a vectorial quantum number

• Classically spin is a vector S, specified by three components

• Quantum mechanically, because of the uncertainty principle, only one of the three components (say the z-component S_z) may be specified, together with the length of the vector (d=3, r=t=1)

• Both these numbers are quantized. The length of the spin vector is h√s(s+1) and the z-component is m_sh, with s a non-negative integer (s=0,1,2,..) or half-integer (s=1/2, 3/2, …). For a given s, m_s may assume

2s+1 discrete values, from –s, -s+1,…, to s-1, s

How to find the sum (s,m

) of two spins specified by (s

,m

) and (s

,m

)?

• m_s is and additive quantum number, therefore m_s=m₁+m₂

• s is not additive, it has to be discrete (integer or half-integer).

It turns out to take on the discrete values between s₁+s₂ and |s₁-s₂|, in integer steps

• For instance, with s₁=s₂=1/2, s=0 (m_s=0) and s=1 (m_s=1, 0, -1) (4 states)

• With s₁=s₂=1, s=0 (m_s=0) and s=1 (m_s=1, 0, -1) and s=2 (m_s=2,1,0,-1,-2) (9 states)

For more complicated vectorial quantum numbers Group Theory is needed to tell us what the non-additive numbers are and how they combine

+

Symmetry and Conservation Laws

• Time-translational invariance can be shown to lead to energy conservation

• Space-translational invariance can be shown to lead to momentum conservation

• Rotational symmetry leads to angular momentum conservation

There is a correspondence between symmetries and conservation laws

(Emmy Noether’s Theorem)

• Symmetries are studied in a branch of mathematics called Group Theory

• A symmetry is specified by a group (of transformations),

which tells the conserved quantum number and how to add them

• Additive quantum numbers typically correspond to U(1) group

• It is an abelian group, i.e. the order in performing transformations is irrelevant

• Vectorial quantum numbers correspond to non-abelian groups

(performing transformations in different order leads to different results)

• The spin angular momentum corresponds to the SU(2) group, which is the simplest of SU(n) (n≥2) groups

SU(2) Group

• The SU(2) conserved quantum number is specified by a three-dimensional vector, e.g. the spin vector

• The length of this vector is discrete, given by p₁=2s

• For a fixed p₁, the allowed orientation of the vector is determined by an additive quantum number m_s, which takes on discrete values between s and –s in integer steps

• These p₁+1=2s+1 possible states are said to form a multiplet

• The size (2s+1) of the multiplet is called its multiplicity

• Note that the sum of m_s over the multiplet is 0 (this is what S in SU(n) means)

• The smallest non-singlet multiplet, given by s=1/2, is called a fundamental multiplet

• The multiplet s=1 has a multiplicity 3 equal to the

dimension of the spin vector (called adjoint multiplet)

These properties can be generalized to SU(n)

…

The same symmetry can be applied to different physical quantities

• For example, charge conservation, quark number conservation and lepton number conservation all correspond to a U(1) symmetry

• Unlike energy, momentum and angular momentum, these symmetries have nothing to do with ordinary space-time

• They are referred to as internal symmetries

• Different members of an internal multiplet correspond to different elementary particles

Isospin

• It is an internal quantum number, mathematically identical to spin

• It is an SU(2) quantum number specified by two numbers (I,I₃), with the same mathematical properties as the spin numbers (s,m_s)

• The 2I+1 particles having the same I form an isospin multiplet

• Multiplets with I=0,1,2 are known as isosinglet, isodoublets, isotriplets

• Isospin was invented by Werner Heisenberg to describe neutron-proton similarity

• They have slightly different masses, only due to the electric charge of the proton

• They can be regarded as different states of the same nucleon

• The nucleon multiplet has I=1/2, with I₃=+1/2 for the proton and I₃=-1/2 for the neutron

• Isospin conservation allows us to relate pp, pn and nn interactions

• Similarly, the three pions (π⁺, π⁰, π^-) form an isotriplet with I₃=+1,0,-1

• Nucleons and pions are made up of u, d quarks and their antiquarks

• The (u,d) quarks form an isodoublet, with I₃=+1/2 for u and I₃=-1/2 for d

• The (d,u) antiquarks form an isodoublet, with I₃=+1/2 for d and I₃=-1/2 for u

The isospin discussed above is also known as strong isospin, being related to strong interactions (e.g. it does not apply to leptons)

• There is another SU(2) quantum number called weak isospin, which plays an important role in weak interactions

• The six quarks form three weak isodoublets, one for each generation

• The six leptons form three weak isodoublets, one for each generation

Color

• It is an internal quantum number

• It is an SU(3) vectorial quantum number

• It has nothing to do with the colors we see

• Classically, color is an 8-dimensional vector

• Quantum mechanically, these vectors are discrete and are labeled by two non-negative integers (p₁ p₂) (the counterparts of I in isospin)

• The size of the multiplet is determined by both p₁ and p₂

• The particles within each multiplet are labeled by two additive quantum numbers (the counterparts of I₃ in isospin)

• Only elementary particles carrying non-zero color have strong interactions

• Only quarks and gluons carry a non-singlet color

• Each of the six quarks u,d,c,s,t,b constitutes a color triplet 3=(10).

Each of the six antiquarks constitutes the conjugate color triplet 3*=(01).

Gluons belong to the adjoint multiplet 8=(11)

• All in all, there are 8 gluons, 3 u-quarks, 3 u-quarks,…

How the color quantum number came about?

• There is a (excited state) baryon made up of three u quarks (Δ⁺⁺=uuu)

• Its spin is 3/2, so the quarks can be all allined up | >

• It has no orbital angular momentum between quarks

• Its wave function is therefore symmetric under the interchange of any two quarks, because each of the isospin, spin and orbital parts of the wavefunction are symmetric

• A symmetric wave function (for three identical fermions) would be against the Pauli exclusion principle

• The simplest way out is to assume there are 3 varieties (3 different colors: red, green, blue) of u quarks

• Then there is no difficulty in constructing an antisymmetric wave function in color

Space reflection

• Before the mid-1950’s Physics was believed to be the same under spatial reflection (also called parity transformation), i.e. if a process can happen the process viewed in a mirror can happen as well, with the same probability

• The corresponding conserved quantum number is known as parity

• In 1957 Tsung Dao Lee and Chen Ning Yang discovered that while parity is conserved in strong and electromagnetic interactions, it is not in weak interactions

Helicity and chirality

• Helicity is the spin component along the direction of motion

• For quarks and leptons (s=1/2) these components are either m_s=1/2 (right-handed helicity) or -1/2 (left-handed helicity)

• Helicity is an important concept as the total angular momentum is conserved (for a system invariant under rotations) and, together with L • p=(r x p) • p =0, this leads to the conservation of helicity

(J • p=(L+S) • p=S • p)

• Chirality is defined as (twice the) helicity for a massless s=1/2 particle

• If the particle is massive the concept of chirality is mathematically well-defined (as a Lorentz invariant), but hard to be physically described

• Parity transformation reverses helicity and chirality

• Since the late 1950’s weak interactions are known to be left-handed, i.e. only quarks and leptons of left-handed chirality and only antiquarks and antileptons of right

handed chirality participate in charged current interactions

• This asymmetry violates parity

Charge conjugation

• Charge conjugation symmetry means that changing all particles to their antiparticles, without altering other quantum numbers, the new process occurs as well and with the same probability

• The preference of one handedness over another, that violates parity symmetry, also violates charge conjugation symmetry

• The weak interaction violates charge conjugation because interactions occur for left- handed quarks and leptons but not for left-handed anti-quarks and anti-leptons. It is the right-handed antiquarks and antileptons that participate in the interactions

• The combined CP operation changes left-handed quarks and leptons to right- handed antiquarks and antileptons

• Weak interactions are almost invariant under CP-transformations. Actually the CP symmetry is violated at a small level

Fisica Teorica Contemporanea (parte III)

Particelle elementari e Forze fondamentali

(Teoria quantistica dei campi, Modello Standard,…)

Testo di riferimento:

Q. Ho-Kim, N. Kumar, C.S. Lam Invitation to Contemporary Physics

Prof.ssa Cecilia Tarantino

Lezione n° 3

Standard Model (SM)

• The Standard Model (SM) is the current theory for strong, electromagnetic and weak interactions

• It has so far passed all the experimental tests

• It cannot be, however, the ultimate theory (at least) since it does not contain

quantum gravity. The theory of General Relativity works for macroscopic bodies.

Quantum gravity effects become relevant at very short distance

(or very high energy M_Planck~ 10¹⁹GeV).They have not yet been tested nor understood

• The three fundamental forces in the SM are all described by gauge theories

• Einstein’s theory of gravity can be also expressed as a kind of gauge theory

• A gauge theory is probably, at present, the most elegant formulation of a force theory

• The simplest and the oldest of these gauge theories is the Maxwell’s electromagnetism

Gauge Theories

• A gauge theory is a type of field theory  in which the Lagrangian  is invariant  under a continuous group  of local transformations

• Many powerful theories in physics are described by Lagrangians  that are invariant  under some symmetry transformation groups

• When they are invariant under a transformation identically performed at every point  in the spacetime  in which the physical processes occur, they are said to have a global symmetry 

•  Local symmetry, the cornerstone of gauge theories, is a stricter constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in

spacetime

Electromagnetic (Gauge) Theory

• Classical electricity and magnetism were unified by James Clerk Maxwell in the late 19° century

• Its quantum version, known as Quantum Electrodynamics (QED), was invented in the 1930’s

• QED calculations ran into a snag (presence of divergencies) solved only in the late 1940’s with the new concept of renormalization

• From the long range of the electromagnetic force

we deduce that the exchange particle must be massless

• It must be also electrically neutral, otherwise the charge of the electron would be altered after a photon emission

• As it transmits both an electric and magnetic field shows that it must carry a non-zero spin, in fact the photon has s=1

From what one observes and knows to the gauge theory

Yang-Mills theory

• There are also gauge theories with gauge particles coupled universally to vectorial quantum numbers

• Such a theory was constructed by Chen Ning Yang and Robert Mills in 1954 and is known as a non-abelian gauge theory or Yang-Mills theory

• As in the abelian gauge theory the masslessness of the gauge particles is realized by demanding local gauge invariance and thus the universal coupling to the conserved vectorial quantum number

• In the Yang-Mills theory the gauge group is SU(n), which is non-abelian

(i.e. with non-commuting generators)

• In SU(n) there are n²-1 gauge particles

• Non-abelian gauge particles, unlike the abelian U(1) photon, carry the quantum number and can emit therefore other gauge particles

Electroweak theory

• Weak interaction was discovered at the beginning of the 20° century in beta decay

• The first theory to describe it was formulated by Enrico Fermi in 1934, shortly after Pauli postulated the existence of the neutrino

• Fermi understood that the missing energy was due to the (elusive) antineutrino

• Fermi’s theory was successful in predicting the energy distribution of the electron but when parity violation was discovered it had to be modified to take into account parity and charge-conjugation violations

• The resulting theory is based on an SU(2) gauge theory

• On the one hand the gauge particles have to be massless to allow for gauge invariance, on the other hand weak forces are found to be extremely short-range and the mass of exchange gauge particles should be very large

• The two requirements seem to be incompatible.

The dilemma is solved by the Higgs mechanism

Higgs mechanism

(predicted in the 60’s and

discovered in 2012 at CERN)

• The Higgs mechanism enables a gauge particle to become massive

• It is based on the presence of a vacuum condensate of a scalar (s=0) particle

• The scalar particle has to carry a (weak) isospin so that the W gauge boson interacts with it

• As a result of this interaction the W slows down from the speed of light, in analogy to ligth traveling in a dielectric material

• Slowing down from the speed of light is equivalent to acquiring a mass and thus a longitudinal polarization

• The extra degrees of fredoom of the W⁺, W^- and W⁰ gauge bosons, are supplied by three extra scalars

• To that end four scalars are introduced: one for the vacuum

condensate corresponding to a new scalar particle (the Higgs boson) and three to supply the longitudinal polarizations

• Actually a refinement is needed: weak and electro-magnetic interactions are combined into a single electroweak theory and gauge bosons are mixed

The electroweak interaction

• In the SM electromagnetic and weak interactions are combined into a single electroweak theory

• The electroweak theory was formulated by Sheldon Glashow,

Abdus Salam and Steven Weinberg as a combined abelian and non- abelian gauge theory

• The non abelian gauge boson W^± is coupled to the weak isospin

• The abelian gauge boson B⁰ is coupled to an additive quantum number, called (weak) hypercharge, Y=2(Q-I₃) where I₃ is the 3°

component of weak isospin and Q is the electric charge of a particle

•  Y=2(Q-I₃) with I₃ the 3° component of weak isospin and Q the electric charge of a particle

• Fermions in the 2° and 3° generation have identical quantum numbers as the 1°

• Antifermions have the same I and opposite Q and I₃

• Gravitons and gluons do not participate in electroweak interactions

• Particles with left-handed chirality (L) do not have the same isospin and hypercharge as particles with right-handed (R) chirality. So parity is explicitely broken in this theory (but conserved in the electromagnetic part of the theory)

• Only L components are affected by charged-current weak interactions mediated by W.

• The right-handed neutrino ν_R is assumed to be absent. It has I=I₃=Q=Y=0 and no color. Except for gravity it would not be subject to any force (sterile neutrino)

• The electroweak theory is an SU(2)_L x U(1)_Y gauge theory

• SU(2)_L refers to the Yang-Mills theory with the W coupled to weak isospin

• L is to remind us that only left-handed fermions get coupled

• U(1)_Y refers to the abelian gauge theory with the B coupled to hypercharge

• After the Higgs condensate is formed isospin I and hypercharge Y are no longer conserved, because the condensate carries these quantum numbers

• Electric charge Q is still conserved because the condensate is neutral

• The surviving gauge theory, after the Higgs mechanism of spontaneous symmetry breaking, is QED or U(1)_EM with EM denoting electromagnetism

SU(2)

x U(1)

U(1)

Higgs mechanism of

spontenous symmetry breaking

Mixing and masses of neutral gauge bosons

• The coupling of B⁰ and W⁰ to the condensate induces a coupling between the two, which makes neither of them have a definite mass.

One gets a 2x2 mass matrix that has to be diagonalized

• In Quantum Mechanics, in fact, in presence of an interaction states originally having definite energies will no longer be so. States with the same quantum numbers will mix to form new energy eigenstates. In relativistic theories the same happens to mass eigenstates

• As a result of mixing, one of the two resulting mass eigenstates is massive: the Z⁰ boson

• The other one is massless: the photon γ, which couples only to Q=I₃+Y/2, as expected W and Z bosons gain a mass through the Higgs mechanism:

M_W=80.42 ± 0.04 GeV, M_Z=91.188 ± 0.002 GeV

The strong interaction

How do quarks/anti-quarks interact to bind them into nucleons and more in general into hadrons (e.g. pions)?

Strong interactions are described by a gauge theory:

• called Quantum Chromodynamics (QCD)

• based on the gauge symmetry group SU(3)

• coupled to the color conserved quantum number

Asymptotic freedom

• In the late 1960’s an important property of strong interactions was exposed by an experiment performed at the Stanford Linear Accelerator (SLAC)

• High energy electrons thrown at a nucleon target were detected at large angles (large momentum transfers). As a result of the collision energy is taken from the electron to produce other particles

• The probability of detecting an electron depends on the energy loss of the electron and on the angle at which it is detected

• James D. Bjorken had predicted that in the deep inelastic scattering region (i.e. for electrons that suffered a great loss in energy) the probability depends only on one combination of these two variables

• This Bjorken scaling was indeed observed in the experiment

• Richard Feynman realized that this (Bjorken)-scaling phenomenon could be explained if the nucleon was composed of many consituents, which he called partons, that do not interact among themselves nor radiate other particles when knocked

• Since nucleons are made up of quarks bound together by gluons, these partons must be quarks, anti-quarks and gluons

BUT, there are two puzzling features in the parton picture

• A nucleon is supposed to contain three quarks while Feynman’s idea requires that the nucleon contains many partons

• Quarks and gluons interact strongly while Feynman’s idea is that they do not interact or very weakly

1° puzzle: many partons

• Every particle has a Yukawa cloud around it.

The cloud around a quark or gluon is made up of quark-antiquark pairs and gluons

• If we examine the particle with a large probe (long wavelength) the particle is seen as a whole

with its full Yukawa cloud. It is called constituent quark . A nucleon is made up of three constituent quarks

• If the probe is small (short wavelength like very energetic electrons of deep inelastic scattering), it sees a small fraction of the cloud, e.g. a single quark or antiquark, which is the Feynman parton

2° puzzle: (almost) non-interacting partons

• A strongly interacting particle when bombarded in a deep inelastic scattering experiment becomes weak. This property is known as asymptotic freedom

• David Gross and Frank Wilczek discovered that there are theories with such a property.

These are non-abelian gauge theories

• In particular, the theory of strong interactions is an

SU(3)_C non-abelian gauge theory, with C standing for color

Standard Model: SU(3)

x SU(2)

x U(1)

gauge theory

Confinement of QCD

• Quarks have never been seen in isolation. QCD must have the confinement property

• Hadrons, like nucleons and mesons, exist (as isolated states) being color singlets made up of quarks and anti-quarks. They interact through pion (not gluon) exchange

• The mechanism behind confinement is not yet clear due to the difficulty to treat QCD unless the coupling is small (deep inelastic regime, where perturbation theory is valid)

• In the non-perturbative (large coupling) regime one can perform numerical simulations (Lattice QCD calculations) which indicate that confinement occurs in QCD

Renormalization

Has it ever occurred to you that it is a miracle we can discover any physical law at all?

• Newton discovered his three famous laws of mechanics without knowing any atomic or molecular physics…

• This is possible since microscopic details are not important in macroscopic physics

• It does not mean that there is no connection between macroscopic and microscopic physics. The mass of an object, the thermal conductivity, the tensile strength,…

are controlled by the details of atomic and nuclear physics

• Macroscopic theories, however, work ignoring this connection and treating these parameters as experimental measurements

• This is the world we are used to. The theory in such a normal world is known as a renormalizable theory

(a theory that works in spite of some ignorance at smaller scales. Ignored quantities can be measured)

• We can imagine a completely different world in which the microscopic details are required to describe macroscopic physics

• Such a world would be so complex that physical laws could never be discovered

• In our normal world, physics at different scales is described by different equations, e.g.:

-Newton laws for macroscopic mechanics, quantum mechanics in the microscopic world -Navier-Stokes equation for the (macroscopic) motion of water, Schrödinger equation for the motion of water molecules

• In principle the macroscopic motion of water could be described by 10²⁶ coupled Schrödinger equations but it would be too complicated

• Navier-Stokes equation contains some parameters, like viscosity. Viscosity can be measured (making the macroscopic theory self-contained) or calculated from

microscopic theory

• Renormalization is the procedure whereby we relate physics of one scale to physics of another scale

• In the water example the equations in the two scales are different (Navier-Stokes versus Schrödinger)

• In other systems the equations can be essentially the same but the parameters in one scale are different from the parameters in another. In these cases, the

task of renormalization is to determine how these parameters change with scale.

SM equations are of this type

Why do the SM parameters change with scale?

• Take for example an electron in QED, probed at a size r=hc/µ with µ an energy parameter

• Different µ see different amounts of the Yukawa cloud around the electron, hence a different mass, which is an effective mass m(µ)

• Similarly different probes see different electric charge, due to the Yukawa cloud that contains e⁺ e^- pairs that shield the electric field of

the bare electron at the center (similarly to an electron in a dielectric material)

• The effective charge e(µ) increases with µ (i.e. at smaller r)

• m(µ), e(µ),…are known as running parameters

• In an abelian gauge theory, as seen above for QED, e(µ) grows with increasing µ

• In a non-abelian gauge theory, say QCD, the cloud is no longer color neutral and there are two competing effects due to gluons and q-q pairs

• Increasing µ, i.e. penetrating the cloud, we are seeing less q-q shielding and this effect increases the coupling strength g_s

• On the other hand we are also seeing less gluons which are colored objects

• Calculations show that gluons represent the main contribution so that g_s(µ) is a decreasing function of µ (opposite to the abelian QED case)

• At large µ, g_s(µ) becomes quite small, leading to

asymptotic freedom observed in deep inelastic scattering

• Renormalization was discovered in the late 1940’s in QED

• Quantum field theory with interaction is such a difficult theory that it can be solved only approximately, by making use of the smallness of the QED coupling e to obtain an iterative (perturbative) solution

• At the first order in perturbation theory everything works well and the result agrees approximately with experimental results

• At the second order, one gets an infinite contribution (divergent integral)!!!

• After the second World War microwave equipements developed for radar became available for precise spectroscopic measurements

• Willis Lamb and Robert Retherford found an unexpected energy shift (Lamb shift) in an atomic hydrogen line that required the calculation of the second order in perturbation theory