Introduction to particle physics Lecture 3 - Durham...
Transcript of Introduction to particle physics Lecture 3 - Durham...
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
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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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Electromagnetic waves
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Introduction to particle physics Lecture 3
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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
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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
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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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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
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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
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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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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.
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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