Fundamental Forces David Morrissey Key Concepts, March 15, 2013
Not a fundamental force...
Also not a fundamental force...
What Do We Mean By Fundamental? Example: Electromagnetism (EM) electric forces magnetic forces Van der Waals forces radio waves rainbows... These different phenomena are all manifestations of EM. EM is said to be a fundamental force.
The Four Fundamental Forces * 1. Electromagnetism (EM) binds atoms, light, shocks 2. Strong holds nuclei together 3. Weak source of nuclear decays 4. Gravity why you re sitting here * Force = way for particles to interact
Four Fundamental Forces * 1. Electromagnetism (EM) binds atoms, light, shocks 2. Strong holds nuclei together 3. Weak source of nuclear decays 4. Gravity why you re sitting here The Cold Hard Truth: These forces might not actually be fundamental. There may be more (or less) than four. * Force = way for particles to interact
How Do We Measure Forces? 1. Pull two particles apart: how much energy V(r) does this take? Need really good tweezers... 2. Scatter particles: Stronger Force Weaker Force Stronger Force More Scattering
Forces and Scattering dσ d(cosθ) (p 1,p 2 ) = differential scattering cross-section = prob. for particles to scatter with angle θ A 2 A is the quantum mechanical amplitude. p 1, p 2 are the initial momenta of the particles. p 1 p 1 θ p 2 p 2
For non-relativistic scattering, A d 3 x e i q x V( x) Ṽ( q) where q = ( p p ) is the momentum transfer. Ṽ( q) is the Fourier Transform of the potential. Also: V( x) = d 3 q (2π) 3e i q x Ṽ( q). Scattering experiments teach us about forces!
Electromagnetism (Relativistic) Scattering experiments yield A Ṽ(p) = Q 1Q 2 e 2 p 2, where p = (E, p) is the transferred 4-momentum, (Q 1 e) and (Q 2 e) are the electric charges of the particles. Fourier transforming (in the non-relativistic limit) gives V( x) = Q 1Q 2 e 2 4π 1 r. Science works (...)!
In relativistic quantum mechanics, A = 1 p 2 m 2, p2 = E 2 p 2 is the amplitude for a particle of mass m to propagate with momentum p. Interpret the electromagnetic force as being mediated by a massless particle - the photon. e e γ e e The photon travels at the speed of light. In fact, the photon is a particle of light (or EM radiation).
Feynman Diagram electron scattering: e e γ e e Feynman Diagram Compton scattering: γ γ e e e
Electromagnetism has a U(1) em gauge symmetry. The Hamiltonian for EM is invariant under: ψ(x) e iqα ψ(x) (charged particle wavefnctn) φ(x) φ(x) 1 α (scalar EM potential) e t A(x) A(x) 1 α (vector EM potential) e for any function α(x). This symmetry COMPLETELY fixes how the photon couples to charged matter. all of electromagnetism follows from this simple gauge symmetry principle!
Aside: General Structure of Ṽ(p) We ll see that Ṽ(p) has the same general structure for all forces we will look at: Ṽ(p) = g 2 S 1 p 2 m 2 Here, g = dimensionless coupling strength of the force S = dependence on particle spins 1 p 2 m 2 = propagation of the force mediator I won t say much at all about S today.
The Strong Force Binds quarks into baryons and mesons, holds nuclei together. baryon = qqq bound state e.g. p = (uud), n = (udd) meson = q q bound state e.g. π 0 = (uū, d d), K + = (u s) A ZX nucleus = [Zp + (A-Z)n] bound state e.g. 4 2 He = 2p+2n, 16 8 O = 8p+8n Of the elementary particles we have discovered, only quarks and gluons feel the strong force.
Elementary Particles of the Standard Model: Fermions Bosons u c t γ d s b g ν e e ν µ ν µ τ τ W Z + 0 h
In stuffed toy form:
Scattering experiments tell us that: Ṽ(p) = g 2 s (p2 ) p 2, p 2 GeV 2, quark scattering gnnπ 2 p 2 m 2, p 2 GeV 2, nucleon scattering π Why do we think both come from the same basic force? Why don t we see quarks at low energies?
Start with quark scattering (p GeV): Ṽ(p) g2 s p 2 1/p 2 massless mediator the gluon. g s describes the strength of the strong force. It depends on p : g s 1 GeV At p 1GeV the coupling blows up! This confines quarks and gluons into baryons and mesons: V( x) 1 r +Λ2 r, Λ GeV fm 1. p
At lower energies, look at nucleon scattering (p GeV) Ṽ(p) = g2 NNπ p 2 m 2 π The force is mediated (mostly) by pions. g NNπ is the residue of the strong force after confinement. (Like van der Waals forces between neutral atoms.) N N π N N
Fourier transforming Ṽ(p) gives V( x) = g2 Nππ 4π 1 r e m πr Yukawa force with range r 1/m π 1fm. This is the typical separation between nucleons in nuclei! (Yukawa proposed the pion based on the range of the force.)
The strong force is based on a SU(3) c gauge symmetry. U(1) = 1 1 unitary matrices = phase transformations. SU(N) = N N unitary matrices with (determinant = 1). SU(3) c interchanges the 3 colour charges carried by quarks. strong force = quantum chromodynamics = QCD This symmetry COMPLETELY fixes the strong force! Gluons also carry colour charge. (Photons have no EM charge.)
The Weak Force Allows decays forbidden by the EM and strong forces: n p ν e e π ν µ µ b c ν e e µ ν µ ν e e (d u ν e e at the quark level) (dū ν µ µ at the quark level) These decays are very slow compared to EM or strong, but they are the only ones that mix flavours. The weak force is much more interesting above 100GeV.
At lower energies, p 100GeV, scattering gives Ṽ(p) constant G F g2 w m 2 W with g w 0.6, m W 80GeV. Fourier transforming gives V( x) G F δ (3) ( x) zero range point interaction
The party starts at higher energies, p 100GeV: Ṽ(p) g 2 w p 2 m 2 W, g 2 w p 2 m 2 Z with g w 0.65, m W 80.4GeV, m Z 91.2GeV. For p m W this reduces to what we had before. Looks like a force mediated by particles with masses m W, m Z. W, Z W ± and Z 0 spin 1 bosons were discovered in the 1980 s.
Electroweak Unification A gauge symmetry principle joins the weak and EM forces into a single electroweak force. The symmetry group is SU(2) L U(1) Y, contains U(1) em. Most of this symmetry is hidden at low energies. Only the U(1) em subgroup of EM remains unhidden. Hiding the symmetry means: W ± and Z 0 gauge bosons acquire masses the weak force has a finite range m 1 W the weak force is much weaker than EM for p m W
SU(2) L U(1) Y has coupling constants g and g. They are related to g w and e by g w = g, e = gg / g 2 +g 2. A spin 0 Higgs boson particle is thought to induce this electroweak symmetry breaking. We re trying really hard to find it, but no luck so far.
More Forces, and not so Fundamental The Higgs is also thought to generate fermion masses. If it does, there are also new Higgs forces. For scattering of two fermions with masses m 1 and m 2, Ṽ(p) = (m 1m 2 /v 2 ) p 2 m 2 h with m h 100GeV and v = 174GeV. This is a new Yukawa-type force: V( x) = (m 1m 2 /v 2 ) 4π 1 r e m hr. The coupling strength to a fermion of mass m is m/v.
Strong and Electroweak Couplings: g s > g > g. This is at p 100GeV. (g s 1, g 0.65, g 0.35) All three couplings depend on the scattering energy: g s decreases going to higher energies g, g increase going to higher energies Does the strong force get weaker than the weak force? Maybe depends on what new physics is around.
With no new physics (except maybe a little supersymmetry): g s g? g 1 GeV 10 GeV It looks like the couplings all meet at a point! Maybe the strong and EW forces have a common origin? 16 p SU(3) c SU(2) L U(1) Y SU(5), SO(10), E 6,... gauge unification into a single force with coupling g U? Symmetry breaking would split them into components.
Gravity Much weaker than the other three fundamental forces. almost always negligible in laboratory experiments Scattering of masses m 1 and m 2 gives (p M Pl ) Ṽ(p) m 1m 2 M 2 Pl 1 p 2, with M Pl 2.4 10 18 GeV = 1/ 8πG N. This gives Yay! V( x) = G Nm 1 m 2 r
Interpret gravity as being mediated by a graviton. massless spin 2 particle The graviton coupling strength to matter is m/m Pl. Graviton couplings are fixed by a gauge symmetry. Symmetry Group = Local Coordinate Transformations x x (x) This reproduces General Relativity at the classical level. We don t know what gravity does at energies above M Pl, where quantum corrections become important.
Summary Fundamental Forces The 4FF are all based on gauge symmetries. But we think there are more forces out there. And the fundamental forces might not be fundamental. We hope to learn much more at the LHC!