Fundamental forces Symmetries

Introduction to particle physicsLecture 3

Frank Krauss

IPPP Durham

U Durham, Epiphany term 2009

F. Krauss IPPP

Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Outline

1 Fundamental forces

2 Symmetries

F. Krauss IPPP

Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Fundamental forces

The four forces in nature

F. Krauss IPPP

Introduction to particle physics Lecture 3

Fundamental forces Symmetries

How forces actForces act through exchange particles associated with them.

F. Krauss IPPP

Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Gravity

Most familiar force, but: Important only in macro-world!

Newton’s law: F = G Mmr2 (G = Newton’s constant)

(implicit: Gravitational = inertial mass)

General relativity: Gµν = 8πGTµν ,where Gµν = metric, and Tµν = energy-momentum tensor.

Quantum formulation of gravity tricky - seemingly notrenormalisable.

But: Gravity becomes quantum at roughly the Planck mass:

Mplanck =√

~c/G = 1.2209 · 1019 GeV = 2.17645 · 10−8 kg.

Carrier of gravity (hypothetical): massless spin-2 graviton

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Electromagnetism

Force that we understand best - both classically and quantum.(Because the coupling strength e, e2/(4π) = α ≈ 1/137 smallenough for perturbation theory!)

Coulomb force law for static charges: F = Kqqq2

r2 .

Equations governing the laws of classical electromagnetism:

Maxwell’s laws.

Structure: Two fields (~E and ~B - vector and axial vector), all firstderivatives, currents and charges −→ 4 equations:

One scalar, pseudoscalar, vector, axial-vector equation.

Interesting: the sources are asymmetric (no magnetic monopole!).

Classically: Electromagnetic waves - theory of light.

Carrier of electromagentism: massless spin-1 photon

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Electromagnetic waves

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

The first quantum field theory: QED

Quantum ElectroDynamics(QED) describes theinteractions of electricallycharged particles (electrons,. . . ) with the electromagneticfield.

It has only one interactionvertex, namely eeγ.

Propagators Vertex

Fermions (e±, . . . )

Photons

It was the first realistic quantum field theory and served as anexample for the construction of all other theories.

Important feature: Using gauge invariance (see later) as aconstruction principle to generate interactions.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Detour: How to draw Feynman diagrams

Rules for QEDFor simplicity: Fix a time-axis.

Fermions (electrons andpositrons) are represented withstraight lines and arrows andphotons with wavy lines.

If the arrow points(anti-)parallel to the time-axis:(anti-)particle. This is relevantonly for external particles.

Along a fermion line, “clashingarrows” are not allowed - theyare related to fermion number(charge) violation.

ExamplesBhabha-Scattering(e−e+ → e−e+)

Pair annihilation(e−e+ → γγ)

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

How (not) to draw Feynman diagrams - Counterexamples

Illegal vertex

Rule: Use only vertices thatare present in the theory.They exactly correspond tothe interaction terms in theHamiltonian.

In QED: Only eeγ vertex.

In example: Used an eeγγvertex.

Clashing arrows

Rule: Don’t revert thearrow along a fermion line(flow of quantum numbersand conservation laws).

In QED: fermion numberand electrical charge.

In example: globally okay,locally (at each vertex)wrong.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Electron’s anomalous magnetic moment in QED

“Classically”, intrinsic magnetic momentof a particle with charge q, mass m andspin s given by:

~µ = gq

2m~s with g = 2.

However, g is altered by quantumcorrections, necessitating renormalisationand also giving rise to finite corrections.

First calculation by Schwinger in 1948 (diagram above), yielding

g − 2

2=

α

2π≈ 0.0011614.

By now, one of the most precise number in physics:

g − 2

2

∣

∣

∣

∣

exp

= 0.00115965218085(76)

10 digits-agreement with theory =⇒ used for precise value of α.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Weak nuclear forceMost obvious “everyday” feature: triggers neutron decay.

Decay mode : n→ pe−νe , half-life around 10 minutes=⇒ force responsible for the neutron decay must be weak.

The same force (and typically also the same decay channel) isresponsible for radioactive β-decay.

Neutrinos ν only interact through the weak force, very hard (nearimpossible) to detect: Needs huge detectors.

Carriers of weak force: massive spin-1 electroweak gauge bosonsW± and Z 0.

Large mass of carriers limits range of force.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Aside: NeutrinosHypothesised by Pauli 1930:Famous letter to the “Dearradioactive ladies andgentlemen” (see left).

Reason: In β-decay only twoparticles seen, but continuousenergy spectrum of theelectron. Impossible intwo-body decays.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Strong nuclear force - QCD

Naive question: Nuclei made from protons and neutrons.Why do they stick together (despite electromagnetic repulsion)?

Answer: Existence of a strong nuclear force.

In nuclei, this is realised by exchange of pions as force carriers.

Later we will see that they are bound states of a quark-antiquarkpair (qq), while the nucleons are bound states of 3 quarks, qqq.

Strong force mediated by massless spin-1 gluons.

The charge related to it is called colour - quarks come,e.g. in three colours, while the 8 gluons carry a colour-anti-colour each.

Quantum ChromoDynamics (QCD)

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Symmetries in classical physics

Invariance and conservation lawsFrom classical physics it is known that invariance of a system undercertain transformations is related to the conservation ofcorresponding quantities.

Examples:

Invariance under Conserved quantityrotations ⇐⇒ angular momentumtime translations ⇐⇒ energyspace translations ⇐⇒ momentum

Formalised by Emmy Noether (thus: Noether’s theorem)

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Introduction to particle physics Lecture 3

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Invariance and conservation lawsThe same ideas work also in quantum physics:Invariances give rise to conservation laws.

There, however, internal symmetries also play a role.In fact, they are used to construct interactions in theories.

Example:

Invariance under phase transformations of the fields

ψ(x , t) → ψ′(x , t) = exp(iθ)ψ(x , t) ⇐⇒ |ψ|2 = |ψ′|2

yields conserved charges like, e.g., the electrical charge.The photon filed couples to this charge and is thus related to theinvariance under such phase transitions (later more).

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Mathematical formulation(Not examinable)

Ideas of symmetry are formalised in group theory.

Definition of groups:

Consider sets of elements S = {a, b, . . . } with operation “·”: a · b.Such sets are called groups, ifa · b ∈ S (closure)a · (b · c) = (a · b) · c (associativity)∃1 ∈ S: a · 1 = 1 · a = a ∀a ∈ S (neutral element)∀a ∈ S: ∃a

−1 ∈ S such that a · a−1 = a−1 · a = 1 (inverse element).

Examples: S = integer numbers, · = +, rotations with arbitraryangles, the set {1, 2, 3, . . . , p − 1} under multiplication modulo p, ifp is a prime number, . . .

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Discrete vs. continuous symmetries

Consider two slabs with quadratic and round cross section.

The quadratic one has a discrete symmetry w.r.t. rotation along itsaxis, while the round one enjoys a continuous symmetry.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Important discrete symmetries

A prime example: Parity

Parity is invariance under

ψ(~x , t)←→ Pψ(~x , t) = λψ(−x , t):

This translates into |ψ(~x , t)|2 = |ψ(−~x , t)|2.

Therefore, λ = ±1 corresponding to even (+) or odd (-) parity.

If a force (e.g. electromagnetism) respects parity, parity even statescannot emerge during interaction from odd states and vice versa.

Note: Electron orbits in an atom have a defined parity.

In addition, particles have intrinsic parity.

Parity is multiplicative.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Parity at work

Atomic physics example:Since orbitals have definite parity and because electrodynamicsrespects parity, not all transitions are allowed. This manifest itself in“missing” spectral lines (although the orbitals exist and thetransition is energetically allowed): Parity selects allowed transitions.

Particle physics example:A J/ψ is a vector particle, it therefore has P = −1. If it decays intotwo particles, they must always form an s-wave (even parity). To seethis consider the c.m. system of the decaying particle - the twooutgoing ones are back-to-back with no angular momentum.Therefore a decay of a vector into two vectors is forbidden.

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Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Further discrete symmetries

Time reversal: Invariance under

ψ(~x , t)←→ T ψ(~x , t) = ψ(x , −t)

Charge conjugation: Invariance under

ψe−(~x , t)←→ Cψe−(~x , t) = ψe+(x , t)

The CPT -theorem:This theorem assures that theories and physics are invariant underthe combined action of C, P and T , although individual forces maybreak invariance of individual discrete symmetries or even theirproducts.

F. Krauss IPPP

Introduction to particle physics Lecture 3

Fundamental forces Symmetries

Summary

Feynman diagrams as terms in the perturbative expansion of thetransition amplitude between states.

Short introduction of how to construct Feynman diagrams.

Briefly reviewed the four fundamental forces in nature:Gravitational, electromagnetic, weak and strong.

Introduced symmetries (continuous vs. discrete).

To read: Coughlan, Dodd & Gripaios, “The ideas of particlephysics”, Sec 5-6.

F. Krauss IPPP

Introduction to particle physics Lecture 3