5. Feynman Diagrams. Particle and Nuclear Physics. Dr. Tina Potter. Dr. Tina Potter 5. Feynman Diagrams 1
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1 5. Feynman Diarams Partile an Nulear Physis Dr. Tina Potter 2017 Dr. Tina Potter 5. Feynman Diarams 1
2 In this setion... Introution to Feynman iarams. Anatomy of Feynman iarams. Allowe verties. General rules Dr. Tina Potter 5. Feynman Diarams 2
3 Feynman Diarams Rihar Feynman 1965 Noel Prize The results of alulations ase on a sinle proess in Time-Orere Perturation Theory (sometimes alle ol-fashione, OFPT) epen on the referene frame. The sum of all time orerins is frame inepenent an provies the asis for our relativisti theory of Quantum Mehanis. A Feynman iaram represents the sum of all time orerins + = time time time Dr. Tina Potter 5. Feynman Diarams 3
4 Feynman Diarams Eah Feynman iaram represents a term in the perturation theory expansion of the matrix element for an interation. Normally, a full matrix element ontains an infinite numer of Feynman iarams. Total amplitue M fi = M 1 + M 2 + M Total rate Γ fi = 2π M 1 + M 2 + M ρ(e) Fermi s Golen Rule But eah vertex ives a fator of, so if is small (i.e. the perturation is small) only nee the first few. (Lowest orer = fewest verties possile) Example: QED = e = 4πα 0.30, α = e2 4π Dr. Tina Potter 5. Feynman Diarams 4
5 Feynman Diarams Perturation Theory Calulatin Matrix Elements from Perturation Theory from first priniples is umersome so we ont usually use it. Nee to o time-orere sums of (on mass shell) partiles whose proution an eay oes not onserve enery an momentum. Feynman Diarams Represent the maths of Perturation Theory with Feynman Diarams in a very simple way (to aritrary orer, if ouplins are small enouh). Use them to alulate matrix elements. Approx size of matrix element may e estimate from the simplest vali Feynman Diaram for iven proess. Full matrix element requires infinite numer of iarams. Now only nee one exhane partile, ut it is now off mass shell, however proution/eay now onserves enery an momentum. Dr. Tina Potter 5. Feynman Diarams 5
6 Anatomy of Feynman Diarams Feynman evise a pitorial metho for evaluatin matrix elements for the interations etween funamental partiles in a few simple rules. We shall use Feynman iarams extensively throuhout this ourse. Topoloial features of Feynman iarams are straihtforwarly assoiate with terms in the Matrix element Represent partiles (an antipartiles): Spin 1/2 Quarks an Leptons Spin 1, W ±, An eah interation point (vertex) with a Eah vertex ontriutes a fator of the ouplin onstant,. Dr. Tina Potter 5. Feynman Diarams 6
7 Anatomy of Feynman Diarams External lines (visile real partiles) Spin 1/2 Partile Antipartile Inomin Outoin Inomin Outoin Spin 1 Partile Inomin Internal lines (propaators; virtual partiles) Outoin Spin 1/2 Partile/antipartile Spin 1, W ±, Eah propaator ives a fator of 1 q 2 m 2 Dr. Tina Potter 5. Feynman Diarams 7
8 Verties A vertex represents a point of interation: either EM, weak or stron. The strenth of the interation is enote y EM interation: = Qe Weak interation: = W Stron interation: = α s (sometimes enote as Q α, where α = e 2 /4π) A vertex will have three (in rare ases four) lines attahe, e.. e + Qe Qe Qe Qe e + At eah vertex, onserve enery, momentum, anular momentum, hare, lepton numer (L e = +1 for, ν e, = 1 for e +, ν e, similar for L µ, L τ ), aryon numer (B = 1 3 (n q n q )), straneness (S = (n s n s )) & parity exept in weak interations. Dr. Tina Potter 5. Feynman Diarams 8
9 Allowe Verties EM must involve a photon, an hare partiles ouplin strenth Qe Q=hare µ τ e + u µ + τ + t ū s t Triple Gaue Vertex s W + Dr. Tina Potter 5. Feynman Diarams 9
10 Allowe Verties Weak must involve a aue vetor oson or W ± ouplin strenth W tip: if you see a ν or ν, it must e a weak interation with W ± µ τ ν e ν µ ν τ s Same family quarks are Caio favoure ū t s s Cross one family Caio suppresse ū t Dr. Tina Potter 5. Feynman Diarams 10
11 Allowe Verties Weak must involve a aue vetor oson or W ± ouplin strenth W tip: if you see a ν or ν, it must e a weak interation with W ± Cross two families Douly Caio suppresse ū t Also, Triple/Four Gaue Vertex W + W + W + W + W + W + W + Dr. Tina Potter 5. Feynman Diarams 11
12 Allowe Verties with Weak Same as iarams, ut also verties with ν µ τ i.e. f e + ν e µ + ν µ τ + ν τ f ν e u ū ν µ s s ν τ t t Not Allowe: Flavour Chanin Neutral Currents (FCNC) Dr. Tina Potter 5. Feynman Diarams 12 s
13 Allowe Verties Stron must involve a luon an/or quark q ouplin strenth α s onserve straneness, harm et u t ū s t s Also, Triple Gaue Vertex Dr. Tina Potter 5. Feynman Diarams 13
14 Forien Verties q X l W ± Dr. Tina Potter 5. Feynman Diarams 14
15 Examples Eletromaneti Qe q Stron q αs αs p Qe p q q M (e)2 q 2 M ( α s ) 2 q 2 u V u W Weak W u u ν e M V u 2 W q 2 m 2 W Dr. Tina Potter 5. Feynman Diarams 15
16 Drawin Feynman Diarams A Feynman iaram is a pitorial representation of the matrix element esriin partile eay or interation a a + + To raw a Feynman iaram an etermine whether a proess is allowe, follow the five asi steps elow: 1 Write own the initial an final state partiles an antipartiles an note the quark ontent of all harons. 2 Draw the simplest Feynman iaram usin the Stanar Moel verties. Bearin in min: Similar iarams for partiles/antipartiles Never have a vertex onnetin a lepton to a quark Only the weak CC vertex hanes flavour within enerations for leptons within/etween enerations for quarks Dr. Tina Potter 5. Feynman Diarams 16
17 Drawin Feynman Diarams Partile satterin If all are partiles (or all are antipartiles), only satterin iarams involve e.. a + + a Initial State Final State If partiles an antipartiles, may e ale to have satterin an/or annihilation iarams e.. a + + (Manelstam variales s, t, u) a p 1 p 3 a p 1 p 3 p 2 p 4 t-hannel, q 2 = t = (p 1 p 3 ) 2 = (p 2 p 4 ) 2 p 2 p 4 s-hannel, q 2 = s = (p 1 + p 2 ) 2 = (p 3 + p 4 ) 2 Dr. Tina Potter 5. Feynman Diarams 17
18 Drawin Feynman Diarams Iential Partiles If we have iential partiles in final state, e.. a + + may not know whih partile omes from whih vertex. Two possiilities are separate final Feynman iarams: a p 1 p 3 a p 1 p 3 p 4 p 2 p 4 t-hannel, q 2 = t = (p 1 p 3 ) 2 = (p 2 p 4 ) 2 p 2 u-hannel, q 2 = u = (p 1 p 4 ) 2 = (p 2 p 3 ) 2 Crossin not a vertex Dr. Tina Potter 5. Feynman Diarams 18
19 Drawin Feynman Diarams Bein ale to raw a Feynman iaram is a neessary, ut not a suffiient onition for the proess to our. Also nee to hek: 3 Chek that the whole system onserves Enery, momentum (trivially satisfie for interations, so lon as suffiient KE in initial state. May fori eays) Chare Anular momentum 4 Parity Conserve in EM/Stron interation Can e violate in the Weak interation 5 Chek symmetry for iential partiles in the final state Bosons ψ(1, 2) = +ψ(2, 1) Fermions ψ(1, 2) = ψ(2, 1) Finally, a proess will our via the Stron, EM an Weak interation (in that orer of preferene) if steps 1 5 are satisfie. Dr. Tina Potter 5. Feynman Diarams 19
20 Summary Feynman iarams are a ore part of the ourse. Make sure you an raw them! Feynman iarams are a sum over time orerins. Assoiate topoloial features of the iarams with terms in matrix elements. Verties ouplin strenth etween partiles an fiel quanta Propaator for eah internal line (off-mass shell, virtual partiles) Conservation of quantum numers at eah vertex Up next... Setion 6: QED Dr. Tina Potter 5. Feynman Diarams 20
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