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PHL424: Feynman diagrams

In 1940s, R. Feynman developed a diagram technique to describe particle interactions in space-time.

Feynman diagram example

Richard Feynman

time

 Particles are represented by lines • Particles go forward in time

Antifermion𝑓𝑓 • go backwards in time , , 𝑓𝑓̅ 𝛾𝛾 𝑍𝑍 𝑊𝑊

Boson

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 PHL424: Feynman diagrams

In 1940s, R. Feynman developed a diagram technique to describe particle interactions in space-time.

Feynman diagram example

Richard Feynman

time

Main assumptions and requirements:

 Time runs from left to right (convention)  Particles are usually denoted with solid lines, and gauge bosons - with helices or dashed lines  Arrow directed towards the right indicates a particle, otherwise –  Points at which 3 or more particles meet are called vertices  At any vertex, , angular momentum and charge are conserved (but not )

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 PHL424: Feynman diagrams

Feynman diagrams are like circuit diagrams – they show what is connected to, but length and angle of momentum vectors are not relevant.

space A particle moving (instantaneously) from Richard Feynman one point to another

A particle moving forward in time and space

A particle at rest

time Fermion

Antifermion𝑓𝑓 , , 𝑓𝑓̅ 𝛾𝛾 𝑍𝑍 𝑊𝑊 Gluon

Boson

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Vertices

 Lines connect into vertices, which are the building blocks of Feynman diagrams

 Charge, number and baryon number as well as momentum are always conserved at a vertex.

Compton

A scatters from an producing a photon and an electron in the final state

Lowest order diagram has two vertices

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Feynman Diagrams

Each Feynman diagram represents an Amplitude (M) 2 In lowest order Fermi’s Golden Rule: = × M is the Fourier transformation of the potential. “Born Approximation” 𝜋𝜋 2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑀𝑀 𝑝𝑝푝𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Differential for two body scatteringℏ (e.g. ) in the CM system: q = final state momentum 1 𝑝𝑝𝑝𝑝 → 𝑝𝑝𝑝𝑝 f = 2 vf = speed of final state particle 4 𝑓𝑓 𝑑𝑑𝜎𝜎 𝑞𝑞 2 vi = speed of initial state particle 2 𝑖𝑖 𝑓𝑓 𝑀𝑀 The decay rate (Γ) for a two body decay𝑑𝑑 Ω(e.g. 𝜋𝜋 𝜈𝜈 𝜈𝜈 ) in the CM system: 0 + − m = mass of parent = 𝐾𝐾 → 𝜋𝜋 𝜋𝜋 p = momentum of decay particle 𝑆𝑆 ∙ 𝑝𝑝⃗ 2 S = statistical factor (/bosons) Γ 2 𝑀𝑀 8𝜋𝜋�𝑚𝑚 𝑐𝑐

In most cases cannot be calculated exactly. Often M is expanded𝟐𝟐 in a power series. Feynman diagrams𝑴𝑴 represent terms in the series expansion of M.

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Feynman Diagrams

= , , , , + 𝜈𝜈 −𝑖𝑖𝑔𝑔𝜇𝜇𝜇𝜇 𝜇𝜇 −𝑖𝑖𝑖𝑖 𝑢𝑢� 𝑝𝑝3 𝜎𝜎3 𝑖𝑖𝑖𝑖𝛾𝛾 𝑣𝑣 𝑝𝑝4 𝜎𝜎4 2 𝑣𝑣̅ 𝑝𝑝2 𝜎𝜎2 𝑖𝑖𝑖𝑖𝛾𝛾 𝑢𝑢 𝑝𝑝1 𝜎𝜎1 𝑝𝑝1 𝑝𝑝2

𝜇𝜇 𝜈𝜈 for massive particle: 𝜇𝜇𝜇𝜇 𝑝𝑝 𝑝𝑝 −𝑖𝑖 𝑔𝑔 − 𝑚𝑚2 2 2 𝑝𝑝 −𝑚𝑚

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Feynman Diagrams

 A (multiplication factor) is associated with each vertex.  Value of coupling constant depends on type of interaction

= (in units of electron charge) 𝑄𝑄𝑓𝑓 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐� 𝑓𝑓 𝛾𝛾 = 2 Electromagnetic 𝑓𝑓 𝑒𝑒 𝑄𝑄𝑓𝑓 ∙ 𝛼𝛼 𝑄𝑄𝑓𝑓 ∙ 4𝜋𝜋 𝑓𝑓 → 𝑓𝑓𝑓𝑓  Example: of an electron

1 = 137

2 1⁄2 1⁄2 � 𝛼𝛼 𝛼𝛼 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 ∝ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ∝ 𝛼𝛼 2 2 4 𝜎𝜎 ∝ 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 ∝ 𝛼𝛼 ∝ 𝑒𝑒

. Total four-momenta conserved at a vertex . Can move particle from initial to final state by replacing it into its antiparticle

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 𝑓𝑓 → 𝑓𝑓𝑓𝑓 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓̅ → 𝛾𝛾 Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Real processes

For a real process there must be energy conversation it has to be a combination of virtual processes →

1⁄2 1⁄2 𝛼𝛼 𝛼𝛼

1⁄2 Electron𝛼𝛼-electron scattering, single photon exchange1⁄2 𝛼𝛼

Any real process receives contributions from all possible virtual processes

1⁄2 1⁄2 𝛼𝛼 𝛼𝛼

1⁄2 1⁄2 Two-photon𝛼𝛼 exchange𝛼𝛼 contribution

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Real processes

1⁄2 1⁄2 𝛼𝛼 𝛼𝛼

1⁄2 1⁄2 Two-photon𝛼𝛼 exchange𝛼𝛼 contribution

 Number of vertices in a diagram is called its order  Each vertex has an associated probability proportional to a coupling constant, usually denoted as “α“. In the electromagnetic processes this constant is 1 = 4 2 137 𝑒𝑒 𝛼𝛼𝑒𝑒𝑒𝑒 ≈  For the real processes, a diagram of the𝜋𝜋𝜀𝜀 0order n gives a contribution of order 𝒏𝒏 𝜶𝜶  Provided that α is small enough, higher order contributions to many real processes can be neglected.

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Real processes

 From the order of diagrams one can estimate the ratio of appearance rates of processes: = + − 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑒𝑒 𝑒𝑒 → 𝛾𝛾𝛾𝛾𝛾𝛾 𝑅𝑅 ≡ + − 𝑂𝑂 𝛼𝛼 This ratio can be measured experimentally;𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑒𝑒 𝑒𝑒 → it 𝛾𝛾𝛾𝛾appears to be R = 0.9·10-3, which is -3 smaller than αem = 7·10 , but the equation above is only a first order prediction.

Diagrams are not related by time ordering

For nucleus, the coupling is proportional to Z2α, hence the rate of this process is of the order of Z2α3.

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Exchange of a massive boson

Exchange of a massive particle X

In the rest frame of particle A: , , + ,

where𝐴𝐴 𝐸𝐸0 𝑝𝑝0 =→ 𝐴𝐴 ,𝐸𝐸 𝐴𝐴 𝑝𝑝⃗ = 𝑋𝑋0,0𝐸𝐸,0𝑋𝑋 − 𝑝𝑝⃗

0 𝐴𝐴 0 =𝐸𝐸 +𝑀𝑀 𝑝𝑝, = + 2 2 2 2 𝐴𝐴 𝐴𝐴 𝑋𝑋 𝑋𝑋 From this one can estimate𝐸𝐸 the𝑝𝑝 maximum𝑀𝑀 distance𝐸𝐸 over𝑝𝑝 which𝑀𝑀 X can propagate before being absorbed: = + This energy violation can exist only for , the interaction range is ∆𝐸𝐸 𝐸𝐸𝑋𝑋 𝐸𝐸𝐴𝐴 − 𝑀𝑀𝐴𝐴 ≥ 𝑀𝑀𝑋𝑋 / ∆𝑡𝑡 ≈ ℏ⁄ ∆𝐸𝐸 𝑟𝑟 ≈ 𝑅𝑅 ≡ ℏ𝑐𝑐 𝑀𝑀𝑋𝑋 Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Exchange of a massive boson

 For a massless exchanged particle, the interaction has an infinite range (e.g. electromagnetic)  In case of a very heavy exchanged particle (e.g. a W boson in ), the interaction can be approximated by a zero-range, or point interaction

Point interaction as a result of MX → ∞ 197.3 10 = = 80.4 = 2 10 80.4 −18 2 −18 𝑊𝑊 𝑊𝑊 ∙ Considering𝑅𝑅 particleℏ𝑐𝑐⁄𝑀𝑀 X asℏ 𝑐𝑐an� electrostatic𝐺𝐺𝐺𝐺𝐺𝐺⁄𝑐𝑐 potential V(r), the Klein≈ ∙-Gordon𝑚𝑚 equation for it will look like 1 = = 2 𝜕𝜕 2 𝜕𝜕𝜕𝜕 2 𝛻𝛻 𝑉𝑉 𝑟𝑟 2 𝑟𝑟 𝑀𝑀𝑋𝑋 ∙ 𝑉𝑉 𝑟𝑟 𝑟𝑟 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 (1935)

For nuclear with a range ~10 , Yukawa hypothesis predicted a spinless quantum of mass: −15 𝑅𝑅 𝑚𝑚 = 100 2 The𝑀𝑀𝑐𝑐 pionℏ observed𝑐𝑐⁄𝑅𝑅 ≈ in 1947𝑀𝑀𝑀𝑀𝑀𝑀 had = 140 , = 0 and strong nuclear interactions. 2 𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀⁄𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

Nowadays: exchange still accounted for the longer-range part of nuclear potential.

However, full details of interaction are more complicated.

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Electroweak Interactions – β--Decay

Beta decay: + + − 𝑛𝑛 → 𝑝𝑝 𝑒𝑒 𝜈𝜈�𝑒𝑒

Mediated by charged W exchange:

The charge that goes into the vertex must equal the charge that comes out of it.

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018

At particle level the interaction is with the

Photons mediate the between protons and

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018

Gluons hold protons and together and are responsible for the Strong force between them

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018 Use of Feynman Diagrams

Although they are used pictorially to show what is going on, Feynman Diagrams are used more seriously to calculate cross sections or decay rates.

 Draw all possible Feynman Diagrams for the process:

Propagator ~ 1 2 2 𝑝𝑝 +𝑚𝑚

 Assign values to each part of the diagram: free particle

Vertex ~ charge

 Calculate the amplitude by multiplying together.

 Add the amplitudes for each diagram (including interference).

 Square the amplitude to get the intensity/probability (cross section or decay rate).

Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2018