Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 3 : Interaction by Particle Exchange and QED. Recap

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1 Prtile Physis Mihelms Term 2011 Prof Mrk Thomson g X g X g g Hnout 3 : Intertion y Prtile Exhnge n QED Prof. M.A. Thomson Mihelms Rep Working towrs proper lultion of ey n sttering proesses lnitilly onentrte on: e + e + + q q q q In Hnout 1 overe the reltivisti lultion of prtile ey rtes n ross setions M 2 x (phse spe) flux In Hnout 2 overe reltivisti tretment of spin-hlf prtiles Dir Eqution This hnout onentrte on the Lorentz Invrint Mtrix Element Intertion y prtile exhnge Introution to Feynmn igrms The Feynmn rules for QED Prof. M.A. Thomson Mihelms

2 Intertion y Prtile Exhnge Clulte trnsition rtes from Fermi s Golen Rule where is perturtion expnsion for the Trnsition Mtrix Element For prtile sttering, the first two terms in the perturtion series n e viewe s: f f j sttering in potentil sttering vi n intermeite stte i i Clssil piture prtiles t s soures for fiels whih give rise potentil in whih other prtiles stter tion t istne Quntum Fiel Theory piture fores rise ue to the exhnge of virtul prtiles. No tion t istne + fores etween prtiles now ue to prtiles Prof. M.A. Thomson Mihelms One possile spe- piture of this proess is: spe V ji x V fj i j f Initil stte i : Finl stte f : Intermeite stte j : The orresponing term in the perturtion expnsion is: (strt of non-exminle setion) Consier the prtile intertion whih ours vi n intermeite stte orresponing to the exhnge of prtile This -orere igrm orrespons to emitting x n then soring x refers to the -orering where emits x efore sors it Prof. M.A. Thomson Mihelms

3 Nee n expression for in non-invrint mtrix element Ultimtely iming to otin Lorentz Invrint ME Rell is relte to the invrint mtrix element y g x where k runs over ll prtiles in the mtrix element Here we hve is the Lorentz Invrint mtrix element for + x The simplest Lorentz Invrint quntity is slr, in this se is mesure of the strength of the intertion + x Note : the mtrix element is only LI in the sense tht it is efine in terms of LI wve-funtion normlistions n tht the form of the oupling is LI Note : in this illustrtive exmple g is not imensionless. Prof. M.A. Thomson Mihelms Similrly x Giving g The Lorentz Invrint mtrix element for the entire proess is Note: refers to the -orering where emits x efore sors it It is not Lorentz invrint, orer of events in epens on frme Momentum is onserve t eh intertion vertex ut not energy Prtile x is on-mss shell i.e. Prof. M.A. Thomson Mihelms

4 But nee to onsier lso the other orering for the proess spe i j f This -orere igrm orrespons to ~ ~ emitting x n then soring x ~ x is the nti-prtile of x e.g. e e W W+ The Lorentz invrint mtrix element for this orering is: In QM nee to sum over mtrix elements orresponing to sme finl stte: Energy onservtion: Prof. M.A. Thomson Mihelms Whih gives From 1 st orering giving g (en of non-exminle setion) After summing over ll possile orerings, is (s ntiipte) Lorentz invrint. This is remrkle result the sum over ll orerings gives frme inepenent mtrix element. Extly the sme result woul hve een otine y onsiering the nnihiltion proess Prof. M.A. Thomson Mihelms

5 Feynmn Digrms The sum over ll possile -orerings is represente y FEYNMAN igrm spe spe In Feynmn igrm: the LHS represents the initil stte the RHS is the finl stte everything in etween is how the intertion hppene It is importnt to rememer tht energy n momentum re onserve t eh intertion vertex in the igrm. The ftor is the propgtor; it rises nturlly from the ove isussion of intertion y prtile exhnge Prof. M.A. Thomson Mihelms The mtrix element: epens on: The funmentl strength of the intertion t the two verties The four-momentum,, rrie y the (virtul) prtile whih is etermine from energy/momentum onservtion t the verties. Note n e either positive or negtive. Here t-hnnel For elsti sttering: q 2 < 0 terme spe-like Here In CoM: s-hnnel q 2 > 0 terme -like Prof. M.A. Thomson Mihelms

6 Virtul Prtiles spe Time-orere QM spe Feynmn igrm Momentum onserve t verties Energy not onserve t verties Exhnge prtile on mss shell Momentum AND energy onserve t intertion verties Exhnge prtile off mss shell VIRTUAL PARTICLE Cn think of oservle on mss shell prtiles s propgting wves n unoservle virtul prtiles s norml moes etween the soure prtiles: Prof. M.A. Thomson Mihelms Asie: V(r) from Prtile Exhnge Cn view the sttering of n eletron y proton t rest in two wys: Intertion y prtile exhnge in 2 n orer perturtion theory. Coul lso evlute the sme proess in first orer perturtion theory treting proton s fixe soure of fiel whih gives rise to potentil V(r) f i Otin sme expression for using p YUKAWA V(r) potentil In this wy n relte potentil n fores to the prtile exhnge piture However, sttering from fixe potentil is not reltivisti invrint view Prof. M.A. Thomson Mihelms

7 Quntum Eletroynmis (QED) Now onsier the intertion of n eletron n tu lepton y the exhnge of photon. Although the generl ies we pplie previously still hol, we now hve to ount for the spin of the eletron/tu-lepton n lso the spin (polriztion) of the virtul photon. (Non-exminle) The si intertion etween photon n hrge prtile n e introue y mking the miniml sustitution (prt II eletroynmis) In QM: Therefore mke sustitution: where The Dir eqution: (here hrge) Prof. M.A. Thomson Mihelms Comine rest mss + K.E. Potentil energy We n ientify the potentil energy of hrge spin-hlf prtile in n eletromgneti fiel s: (note the A 0 term is just: ) The finl omplition is tht we hve to ount for the photon polriztion sttes. e.g. for rel photon propgting in the z iretion we hve two orthogonl trnsverse polriztion sttes Coul eqully hve hosen irulrly polrize sttes Prof. M.A. Thomson Mihelms

8 Previously with the exmple of simple spin-less intertion we h: g g In QED we oul gin go through the proeure of summing the -orerings using Dir spinors n the expression for. If we were to o this, rememering to sum over ll photon polriztions, we woul otin: = = Intertion of with photon Mssless photon propgtor summing over polriztions Intertion of with photon All the physis of QED is in the ove expression! Prof. M.A. Thomson Mihelms The sum over the polriztions of the VIRTUAL photon hs to inlue longituinl n slr ontriutions, i.e. 4 polristion sttes n gives: n the invrint mtrix element eomes: This is not ovious for the moment just tke it on trust (en of non-exminle setion) Using the efinition of the joint spinor This is remrkly simple expression! It is shown in Appenix V of Hnout 2 tht trnsforms s four vetor. Writing showing tht M is Lorentz Invrint Prof. M.A. Thomson Mihelms

9 Feynmn Rules for QED It shoul e rememere tht the expression hies lot of omplexity. We hve summe over ll possile orerings n summe over ll polriztion sttes of the virtul photon. If we re then presente with new Feynmn igrm we on t wnt to go through the full lultion gin. Fortuntely this isn t neessry n just write own mtrix element using set of simple rules e + Bsi Feynmn Rules: Propgtor ftor for eh internl line (i.e. eh internl virtul prtile) Dir Spinor for eh externl line (i.e. eh rel inoming or outgoing prtile) Vertex ftor for eh vertex Prof. M.A. Thomson Mihelms Externl Lines Bsi Rules for QED inoming prtile outgoing prtile spin 1/2 inoming ntiprtile outgoing ntiprtile inoming photon spin 1 outgoing photon Internl Lines (propgtors) spin 1 photon spin 1/2 fermion Vertex Ftors spin 1/2 fermion (hrge - e ) Mtrix Element = prout of ll ftors Prof. M.A. Thomson Mihelms

10 e.g. Whih is the sme expression s we otine previously e.g. e + Note: At eh vertex the joint spinor is written first Eh vertex hs ifferent inex The of the propgtor onnets the inies t the verties Prof. M.A. Thomson Mihelms Summry Intertion y prtile exhnge nturlly gives rise to Lorentz Invrint Mtrix Element of the form Derive the si intertion in QED tking into ount the spins of the fermions n polriztion of the virtul photons: We now hve ll the elements to perform proper lultions in QED! Prof. M.A. Thomson Mihelms