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Lecture Notes Particle Physics II Quantum Chromo Dynamics 6

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Lecture Notes Particle Physics II Quantum Chromo Dynamics 6 Lecture notes Particle Physics II Quantum Chromo Dynamics 6. Feynman rules and colour factors Michiel Botje Nikhef, Science Park, Amsterdam December 7, 2012 Colour space We have seen that quarks come in three colours i = (r; g; b) so • that the wave function can be written as (s) µ ci uf (p ) incoming quark 8 (s) µ ciy u¯f (p ) outgoing quark i = > > (s) µ > cy v¯ (p ) incoming antiquark <> i f c v(s)(pµ) outgoing antiquark > i f > Expressions for the> 4-component spinors u and v can be found in :> Griffiths p.233{4. We have here explicitly indicated the Lorentz index µ = (0; 1; 2; 3), the spin index s = (1; 2) = (up; down) and the flavour index f = (d; u; s; c; b; t). To not overburden the notation we will suppress these indices in the following. The colour index i is taken care of by defining the following basis • vectors in colour space 1 0 0 c r = 0 ; c g = 1 ; c b = 0 ; 0 1 0 1 0 1 0 0 1 @ A @ A @ A for red, green and blue, respectively. The Hermitian conjugates ciy are just the corresponding row vectors. A colour transition like r g can now be described as an • ! SU(3) matrix operation in colour space. Recalling the SU(3) step operators (page 1{25 and Exercise 1.8d) we may write 0 0 0 0 1 cg = (λ1 iλ2) cr or, in full, 1 = 1 0 0 0 − 0 1 0 1 0 1 0 0 0 0 0 @ A @ A @ A 6{3 From the Lagrangian to Feynman graphs Here is QCD Lagrangian with all colour indices shown.29 • µ 1 µν a a µ a QCD = (iγ @µ m) i F F gs λ j γ A L i − − 4 a µν − i ij µ µν µ ν ν µ µ ν F = @ A @ A 2gs fabcA A a a − a − b c We have introduced here a second colour index a = (1;:::; 8) to label the gluon fields and the corresponding SU(3) generators. If we multiply-out the field tensor contraction F µνF a , we see all • a µν the elements of a QCD Feynman diagram in the Lagrangian: ψ¯ (iγµ∂ m)ψ quark propagator i µ − i (∂µAν ∂ν Aµ)(∂ Aa ∂ Aa ) gluon propagator a − a µ ν − ν µ ¯ a µ a gs ψiλijψjγ Aµ quark-gluon vertex g (∂µAν ∂ν Aµ)f Ab Ac 3-gluon vertex s a − a abc µ ν 2 µ ν d e gs fabcAb Ac fadeAµAν 4-gluon vertex 29Summation over repeated indices is implied, irrespective of their position (upper or lower); the colour indices are just placed wherever the Lorentz indices leaves room for them. 6{4 Feynman rules of QCD To each element of a Feynman diagram corresponds a mathemati- • cal expression which, together with the prescription how to assem- ble a scattering amplitude, make up the set of Feynman rules. Deriving these rules is the subject of Quantum Field Theory and is beyond the scope of these lectures. Let us mention at this point that our QCD Lagrangian is not complete and that so-called ghost fields must be introduced to make the theory consistent, but this is also beyond the scope of these lectures. The full set of QCD Feynman rules (without ghosts) can be found in Griffiths p.287{8. In the following we will calculate so-called colour factors for • the leading order qq¯ qq¯ and qq qq diagrams. Here is the ! ! dictionary that we will need for that calculation: uc incoming quark uc¯ † outgoing quark vc¯ † incoming antiquark vc outgoing antiquark ab i gµν δ gluon propagator − q2 a ig λ γµ quark-gluon vertex − s 2 6{5 Colour factor for qq¯ qq¯ ! v¯2 c2† v4 c4 λb ig γµ − s 2 g δab i µν − q2 λa ig γν − s 2 u1 c1 u¯3 c3† a λ ν i =¯u c† ig γ u c − M 3 3 − s 2 1 1 g δab i µν × − q2 b λ µ v¯ c† ig γ v c × 2 2 − s 2 4 4 g2 = s (¯u γµu )(¯v γ v ) M − q2 3 1 2 µ 4 1 a a c† λ c c† λ c × 4 3 1 2 4 colour factor 6{6 Colour factor for qq qq ! u2 c2 u¯4 c4† λb ig γµ − s 2 g δab i µν − q2 λa ig γν − s 2 u1 c1 u¯3 c3† a λ ν i =¯u c† ig γ u c − M 3 3 − s 2 1 1 g δab i µν × − q2 b λ µ u¯ c† ig γ u c × 4 4 − s 2 2 2 g2 = s (¯u γµu )(¯u γ u ) M − q2 3 1 4 µ 2 1 a a c† λ c c† λ c × 4 3 1 4 2 colour factor 6{7 Colour factors We see that the leading order QCD amplitudes for qq¯ qq¯ and • 2 ! qq qq scattering can be written as (αs g =4π) ! ≡ s QCD = αs colourfactor QED M × × M This means that we can take the QED result (from PP-I for in- stance), provided that we replace the fine structure constant α by the strong coupling constant αs, and multiply by the colour factor. This correspondence does not only hold for tree diagrams (di- • agrams without loops) but for any diagram where photons can be replaced by gluons. However, it is not true that QCD colourfactor QED because • ≡ × there are QCD diagrams which do not exist in QED. 6{8 Bookkeeping I a a Exercise 6.1: [0.2] Show that cyλ cj is the matrix element λ , • i ij sometimes also written as i λa j . h j j i To calculate the colour factors it is handy to use the identity • 1 a a 1 1 4 λijλkl = 2 δilδjk 6 δijδkl f(ijkl) a − ≡ X The colour factor f(ijkl) thus depends on whether pairs of colours • are the same or not and we get, from the identity above, 1 if i = j and k = l −6 1 if i = l and j = k f(ijkl) = 8 2 > 1 > if i = j = k = l <> 3 0 otherwise > > :> Exercise 6.2: [0:8] Use the λ matrices on page 1{24 to check the • colour factor f(ijkl). Hint: Restrict yourself to two colours (red and green, say) so that i; j; k; l can only take the values 1 and 2. After this it is easy to generalise to the other combinations red-blue and green-blue. 6{9 Bookkeeping II qq¯ qq¯ qq qq ! ! k l l k j i j i 1 λa λa 1 λa λa 4 ij kl f(ijkl) 4 ij kl Using the rule of `walking against the arrows' we see how for a • given f(ijkl) the indices i; j; k; l are assigned to the incoming and outgoing particles in qq¯ qq¯ and qq qq scattering. ! ! Now we can list all allowed combinations of colour `x' and `y': • q q¯ q q¯ q q q q ! ! j k i l j l i k f( ijkl ) ! ! x x x x x x x x f(xxxx) = 1 ! ! 3 x x y y x y y x f(yxxy) = 1 ! ! 2 x y x y x y x y f(xxyy) = 1 ! ! −6 x y y x x x y y f(yxyx) = 0 ! ! In fact, the table above reflects colour flow through the diagram, • as is shown on the next page for the combination red and green. 6{10 Colour flow qq¯ qq¯ and qq qq diagrams showing colour exchange and the • ! ! possible cc¯ combinations carried by the exchanged gluon. r¯ r¯ r¯ g¯ g¯ g¯ rr,¯ gg,¯ b¯b rg,¯ rg¯ rr,¯ gg,¯ b¯b r r r g r r r r g r g g rr,¯ gg,¯ b¯b rg,¯ rg¯ rr,¯ gg,¯ b¯b r r r g r r Exercise 6.3: [1:0] Here are some colour flow diagrams for (a) • qq¯ qq¯ and (b) qq qq interactions. Please draw all possible ! ! colour flow topologies. (a) (b) 6{11 Gluon colour states The gluon carries one unit of colour and one unit of anticolour and • because 3 3¯ = 8 1 (see page 1{27) we take the singlet as our ⊗ ⊕ principal axis in colour-anticolour space and then construct a basis by orthogonalisation. The invariant singlet state is obviously 0 1 (r¯r+ g¯g+ bb)¯ : j i ≡ 3 By simple orthogonalisationq we find a fully orthonormal basis as30 r¯g rb¯ g¯r gb¯ b¯r b¯g r¯r g¯g bb¯ 0 = 1 0 0 0 0 0 0 1 1 1 > j i 3 j 1 = q 1 0 0 0 0 0 0 0 0 > j i j 2 = 0 1 0 0 0 0 0 0 0 > j i j 3 = 0 0 1 0 0 0 0 0 0 > j i j 4 = 0 0 0 1 0 0 0 0 0 > j i j 5 = 0 0 0 0 1 0 0 0 0 > j i j 6 = 0 0 0 0 0 1 0 0 0 > j i j 7 = 1 0 0 0 0 0 0 1 1 0 > j i 2 j − 8 = q1 0 0 0 0 0 0 1 1 2 > j i 6 j − A singlet gluonq would exist as a free particle and would be as • common as the photon and give rise to a long-range strong force.
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