A Short Introduction to Feynman Diagrams

A short Introduction to Feynman Diagrams J. Bijnens, November 2008 This assumes knowledge at the level of Chapter two in G. Kane, “Modern Elementary Particle Physics.” This note is more advanced than needed for FYTN04 but hopefully still useful. For more details see any field theory book. 1 Kinetic terms A quantum field theory is defined by a Lagrangian. The first step is to define the fields.1 such that they represent particles. That means that there should be no terms with only one field present and all vacuum expectation values should have been subtracted2 The kinetic terms, those containing exactly two fields, are assumed to be diagonalized. This means that all terms with two fields are of one of the forms, for a field corresponding to a particle with mass m Field Lagrangian Field strength 1 1 real scalar field φ ∂ φ∂µφ m2φ2 2 µ − 2 µ 2 complex scalar field Φ ∂ Φ∗∂ Φ m Φ∗Φ µ − Dirac fermion ψ ψiγµ∂µψ mψψ − 1 1 real vector field A F F µν + m2A Aµ F = ∂ A ∂ A µ −4 µν 2 µ µν µ ν − ν µ 1 µν 2 µ complex vector field W W ∗ W + m W ∗W W = ∂ W ∂ W µ −2 µν µ µν µ ν − ν µ 2 An aside on diagonalization Diagonal means that there are no terms that mix two different fields but it is only • the same field and its complex conjugate that show up in the kinetic terms. If mixed terms are present, they can always be removed by redefining fields. This • is true for both masses and the terms with partial derivatives as can be seen in the example below. As the example, take a theory with two real fields, φ1 and φ2, that has as Lagrangian, with 2 2 2 some constants,a, b, c, µ1, µ2 and µ12, a c 1 1 = ∂ φ ∂µφ + b∂ φ ∂µφ + ∂ φ ∂µφ µ2φ φ µ2 φ φ µ2φ φ L 2 µ 1 1 µ 1 2 2 µ 1 1 − 2 1 1 1 − 12 1 2 − 2 2 2 2 1I will sometimes refer to a generic field φ, that means all possibilities then. 2This can always be done by transformations of the type φ = v + φ′. 1 This has terms that mix φ1 and φ2 and is actually the most general form since φ1φ2 = φ2φ1 and the same for the terms with partial derivatives. We can rewrite this in the matrix notation µ 2 2 1 a b ∂ φ1 1 µ1 µ12 φ1 = ∂µφ1 ∂µφ2 µ φ1 φ2 2 2 L 2 b c ∂ φ2 − 2 µ µ φ2 12 2 a b The matrix is a real symmetric matrix and can thus be diagonalized with an b c a b λ orthogonal matrix O with the result = OT 1 O. We now define linear b c λ 2 combinations of φ1 and φ2 via φ φ φ φ 3 = O 1 or 1 = OT 3 φ φ φ φ 4 2 2 4 O is a constant matrix so the same relation is true for derivatives of the fields. The Lagrangian thus becomes 2 2 1 µ 1 µ 1 µ1 µ12 T φ3 = λ1∂µφ3∂ φ3 + λ2∂µφ3∂ φ3 φ3 φ4 O 2 2 O L 2 2 − 2 µ µ φ4 12 2 λ1 and λ2 must be positive, since kinetic energy must always be positive. We do thus one more redefinition φ5 = √λ1,φ3 and φ6 = √λ2 φ4 and obtain 1 µ 1 µ 1 φ5 = ∂µφ5∂ φ5 + ∂µφ6∂ φ6 φ5 φ6 B L 2 2 − 2 φ6 with B a symmetric matrix 1 µ2 µ2 1 B = √λ1 O 1 12 OT √λ1 , BT = B. 1 µ2 µ2 1 √λ2 ! 12 2 √λ2 ! m2 B can thus be diagonalized by an orthogonal matrix P with B = P T 7 P . A final m2 8 φ φ redefinition 7 = P 5 leads now to diagonal kinetic terms φ φ 8 6 1 1 1 1 = ∂ φ ∂µφ + ∂ φ ∂µφ m2φ2 m2φ2 L 2 µ 7 7 2 µ 8 8 − 2 7 7 − 2 8 8 So you see you can always diagonalize the kinetic terms. If there are interaction terms, you have to rewrite those also in terms of the final diagonalized fields (in this example φ7 and φ8). 2 3 Creation and annihilation operators As explained in Chapter two of Kane we expand the fields in the eigenmodes with creation and annihilation operators ip x ip x real scalar : φ(x) a e− · + a† e · ∝ ~p ~p ~p X I have not written out some normalization factors for the different modes. These are • taken care of correctly in the final rules as given. For a continuous system the sum over all three momenta ~p is really an integral. • The four vectors p =(p0, ~p ) have p0 = ~p 2 + m2 • | | a annihilates or destroys or removes ap particle/quantum corresponding to the field • ~p φ with four momentum p. a† creates or adds a particle/quantum with four momentum p. • ~p As the notation indicates, a† is the Hermitian conjugate of a for a real field. • ~p ~p There is a set of these operators for each field. They can be distinguished by adding • indices to indicate which field the operator corresponds to. For a complex scalar field the two terms have a different operator in the two terms since the total doesn’t have to be real (Hermitian) ip x ip x complex scalar : Φ(x) a e− · + b† e · ∝ ~p ~p ~p X a~p annihilates or destroys or removes a particle corresponding to the field φ with four • momentum p. b† creates or adds an anti-particle with four momentum p. • ~p As the notation indicates, b† is not the Hermitian conjugate of a for a complex field. • ~p ~p The Hermitian conjugate of the field ip x ip x complex conjugate scalar : Φ∗(x) b e− · + a† e · ∝ ~p ~p ~p X b~p annihilates or destroys or removes an anti-particle corresponding to the field φ • with four momentum p. a† creates or adds a particle with four momentum p. • ~p 3 I have used a~p here for both the real and complex scalar fields but remember that they • are really distinct operators and with different operators for each field that occurs. This can be taken care of by adding extra indices. For particles with spin, the different spin modes should be included as well. The sums are thus over ~p and spin s. ip x ip~x Dirac fermion ψ u a e− · + v b† e ∝ ~p,s ~p,s ~p,s ~p ~p,s X ip x ip~x Conjugate Dirac fermion ψ u a e− · + v b† e ∝ ~p,s ~p,s ~p,s ~p ~p,s X The spinors u~p,s and v~p,s are the positive and negative energy solutions of the Dirac equation 3 of spin s. a~p,s annihilates or removes a particle with momentum p and spin s while a†~p,s creates it. b~p,s annihilates or removes an anti-particle with momentum p and spin s while b†~p,s creates it. The same principle goes for spin one particles but with polarization vectors ε~p,i µ for a vector with momentum p and spin s instead of spinors for the Dirac case. ip x ip x real vector A ε a e− · + ε∗ a† e · µ ∝ ~p,s µ ~p,i ~p,s µ ~p,i ~p,s X ip x ip x Complex vector W ε a e− · + ε∗ b† e · µ ∝ ~p,s µ ~p,i ~p,s µ ~p,i ~p,s X ip x ip x Conjugate complex vector W ∗ ε b e− · + ε∗ a† e · µ ∝ ~p,s µ ~p,i ~p,s µ ~p,i ~p,s X The same type of comments as above apply here to the creation and annihilation operators. 4 What are now Feynman diagrams? Feynman diagrams are a technique to solve quantum field theory. Their main use is to calculate the amplitude (or rather i times the amplitude) for a state with specified incoming particles with momenta and spins specified to evolve to a different state with specified particles and their momenta and spins.4 We divide the Lagrangian into Kinetic terms: those with two fields as described in section 1. These terms produce • the propagators and give the lines that connect different points of a Feynman diagram. Internal lines must be summed over all momenta and spins. 3I have been a little bit sloppy here with the signs on the spins so be careful when comparing with field theory books. 4It is sufficient if all momenta are different, this requirement is there to have contributions from con- nected diagrams only. 4 Interaction terms: those terms with three or more fields. This part is usually called • . It provides connection points called vertices where three or more lines meet. LI At the vertices momentum is conserved: the sum over all incoming momenta must • be equal to the sum over all outgoing momenta at each vertex. You can check that for tree level diagrams, i.e. no closed loops, this means that all occurring momenta are specified in terms of the incoming and outgoing momenta. The previous point also leads to momentum conservation for the full diagram. The • sum over all outgoing momenta is equal to the sum over all incoming momenta. To construct the amplitudes for a process one now does the following 1.

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