Particle Physics Lecture Notes. Priv.-Doz. Dr. S. Schätzel

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1 Particl Physics Lctur Nots Priv.-Doz. Dr. S. Schätzl Mastr s Dgr Cours MKEP Ruprcht-Karls-Univrsität Hidlbrg 5

2 Vrsion as of 4 May 5 (Chaptr ) ii

3 Contnts Thortical basis and QED. Lagrangian fild thory Natural units Rlativistic notation Lagrang quations Continuity quation and consrvd charg Particl cration and annihilation Dirac quation Drivation Solutions for particls at rst Gnral solutions Spin Spinor normalisation Transformation proprtis of spinors Quantum lctrodynamics Lagrang dnsity Elctromagntic currnt dnsity and consrvd charg Local phas transformation Th photon fild Principl of gaug thoris Elctron-photon vrtx Antiparticls Th unphysical photon vrtx Virtual photons Photon propagator Units of lctric charg Elctromagntic scattring Goldn rul for scattring scattring Cross sction for µ µ Mott and Ruthrford scattring Mandlstam variabls Crossing symmtry µ + µ Bhabha scattring Møllr scattring Dpndnc on scattring angl Hlicity iii

4 CONTENTS CONTENTS.4. Chirality (Handdnss) Chirality in QED Parity consrvation in QED qq Adding amplituds vs. adding cross sctions R had Highr ordr corrctions Loop diagrams Rnormalisation Rsolving small structurs Vacuum polarisation Summary iv

5 Chaptr Thortical basis and QED. Lagrangian fild thory.. Natural units This sction follows chaptr 6. of []. Formulas in particl physics oftn contain th spd of light c and Planck s constantħ h π : E = (mc ) + (pc) S z = ħ (.) For clarity w drop ths factors from th formulas: E = m + p S z = (.) so that momntum and mass ar masurd in units of nrgy (for which w us th lctronvolt, V) and spin is masurd in units of Planck s constant. Ths units ar th natural units for particl physics bcaus thy mak th physics contnt of formulas stand out much clarr. For xampl: th nrgy of a particl is givn by its mass and its momntum, addd in quadratur. Th natural units of othr quantitis, lik cross sctions and liftims, ar not straightforward and not always asy to undrstand. To convrt from natural to ordinary units, w insrt into th formula factors of c andħ. Th following convrsion factor is vry convnint and worthwhil to rmmbr: ħc = 97 MV fm (.3) In this txt, thr-vctors ar dnotd by a bold-fac lttr, for xampl: p.

6 .. LAGRANGIAN FIELD THEORY CHAPTER. THEORETICAL BASIS AND QED Exampl. (Thomson scattring cross sction) Thomson scattring is th low-nrgy limit (E γ m ) of Compton scattring (γ+ γ+) and th cross sction is in natural units givn by σ= 8π 3 α m (.4) With m =.5 MV, th cross sction has th dimnsion of a squard invrs nrgy. To gt th cross sction in ordinary units, i.., with dimnsion of an ara, w multiply by (ħc) : σ= 8π 3 α m (ħc) = 8π 3 α m (97 MV fm) (.5) = cm =.665 b (.6) In th final stp w hav usd th high nrgy physics unit for cross sctions, th barn, which is dfind as b 4 cm = 8 m (.7) Exampl. (Positronium liftim) Th liftim of th + ground stat ( S ) is τ= α m in natural units. To gt th liftim in sconds w multiply byħc and divid by c: τ= α 97 MV fm m 3 8 m =.4 s (.8) s.. Rlativistic notation This sction follows chaptr. of []. Th usual 4-vctors ar usd in this txt: spac-tim coordinats: x = x µ = (t, x, y, z)=(t,x) nrgy and momntum: p = (E, p) Th mtric tnsor g µν = diag(,,, )= = g µν (.9) rlats covariant and contravariant vctors: a µ = 3 g µν a ν g µν a ν (.) ν= a µ = g µν a ν (.) in which th Einstin summation convntion was usd which implis summation ovr indics that appar twic.

7 CHAPTER. THEORETICAL BASIS AND QED.. LAGRANGIAN FIELD THEORY Exampl.3 a = g ν a ν = g a + g a + g a + g 3 a 3 = a (.) }{{}}{{}}{{} a i = a i (i =,,3) (.3) For th summation convntion Grk indics run from to 3, Roman indics from to 3 (spatial indics). Lorntz transformations ar rotations in Minkowski spac-tim rprsntd by th rotation matrix Λ Λ is a ral 4 4 matrix and satisfis th condition x µ = Λ µ ν x ν (.4) { Λ µ ν Λµ τ = δ ν τ ν=τ = ν τ (.5) Scalar products ar dfind as Exampl.4 x y x µ y µ = x y + x i y i = x y x i y i = x y x y (.6) p p = E E p p = E E p p cosθ (.7) Scalar products ar invariant undr Lorntz transformations: x y = x µ y µ = Λµ νx ν Λµ τ y τ= x ν Λ µ ν Λµ τ y τ = x ν y ν = x y (.8) }{{} δν τ 4-vctor drivativs ar dnotd by with, for xampl, th zro-componnt µ φ x µ φ (.9) x = t = t (.)..3 Lagrang quations This sction follows chaptr. of []. For a st of filds φ r (r =,,..., N ) th Lagrang dnsity L (φ r, µ φ r ) is a function of th filds and thir drivativs. Th action S in a spac-tim rgion Ω is givn by th intgral S= d 4 xl (.) Ω 3

8 .. LAGRANGIAN FIELD THEORY CHAPTER. THEORETICAL BASIS AND QED Equations of motion ar obtaind from th variational principl which postulats that variations of th filds φ r (x) φ r (x) = φ r (x)+δφ r (x) lav S unchangd if δφ r vanishs on th surfac that ncloss Ω (th boundary Ω): δφ r (x Ω)= (.) Th variation of S is givn by th intgral of th total drivativ of L : = δs= d 4 x δl (φ r, µ φ r ) (.3) Ω [ ] = d 4 L x δφ r + L φ r ( µ φ r ) δ( µφ r ) }{{} (.4) = Ω Ωd 4 x r + r r [ ( L L µ φ r ( ) L µ ( µ φ r ) δφ r }{{} a µ r (x) r ( µ φ r ) µ δφ r )] δφ r (.5) (.6) Annotation. (*) δφ=φ φ δ( µ φ)= µ φ µ φ= µ (φ φ)= µ δφ ( ( ) L L (**) hr N zros hav bn addd: = µ )δφ r µ δφ r ( µ φ r ) ( µ φ r ) Th divrgnc µ a µ r is quivalnt to th surfac intgral of a r ovr Ω (4-dimnsional divrgnc thorm). Howvr, th 4-vctor a r vanishs on th surfac of Ω bcaus of (.), so th trms in (.6) vanish. From (.5) it thn follows that, for δs to vanish for arbitrary δφ r, th following N Eulr-Lagrang quations nd to hold: ( ) L L = µ φ r ( µ φ r ) r =,,..., N (.7) Ths ar th quations of motions for th filds φ r. Adding to th Lagrang dnsity a trm which dos not dpnd on a fild or its drivativ will lav th quations of motion unchangd...4 Continuity quation and consrvd charg This sction follows chaptr.4 of []. In this sction w xamin th rol of symmtry transformations. Ths transformations act on th filds φ r and chang thm by th amount δφ r : φ r φ r = φ r + δφ r If this rplacmnt lavs th Lagrang dnsity invariant (or adds only a constant trm) thn th quations of motion ar unchangd and th systm posssss a symmtry. 4

9 CHAPTER. THEORETICAL BASIS AND QED.. LAGRANGIAN FIELD THEORY An invariant L implis a consrvd quantity, as w can s by looking at th total drivativ =δl = L δφ r + L φ r ( µ φ r ) δ( µφ r ) (.8) }{{} ( ) L = µ δφ r + L ( µ φ r ) ( µ φ r ) µ(δφ r ) (.9) ( ) L = µ ( µ φ r ) δφ r µ f µ (.3) Annotation. (*) according to th Eulr-Lagrang quations (.7) this trm is qual to µ ( Th currnt dnsity f µ ( ) f µ L = ( µ φ r ) δφ r fulfils th continuity quation L ( µ φ r ) ) (.3) µ f µ = (.3) which is quivalnt to t f = f= div f (.33) Upon intgration ovr a spac-volum V w obtain t F t d 3 x f = d 3 x div f= da f (.34) V V V so that th tmporal chang of th zro-componnt of f insid volum V is givn by th flux outward through th surfac A of that volum. Th currnt dnsity f (oftn rfrrd to simply as currnt) is rlatd to th filds φ r. If w ar looking at a configuration whr all filds ar rstrictd to a crtain rgion in spac and lt V ncompass all of that spac thn th flux through th surfac of V will b zro. In this cas th spac-intgral of th zro-componnt of f in that rgion is constant in tim: t F = (.35) A symmtry of th Lagrang dnsity hnc corrsponds to a consrvd quantity which is usually idntifid with a charg. This is an xampl of Nothr s Thorm...5 Particl cration and annihilation This sction follows chaptr.4 of []. W will us th xampl of an U () symmtry to illustrat how particls ar cratd and annihilatd in Quantum Fild Thory (QFT). W will us this concpt latr whn discussing Fynman diagrams in which particls ar cratd and dstroyd at ach vrtx. From th dfinition of th currnt dnsity (.3), th zro-componnt is givn by f = L ( φ r ) δφ r = L φ r δφ r r 5 π r δφ r (.36)

10 .. LAGRANGIAN FIELD THEORY CHAPTER. THEORETICAL BASIS AND QED with π r = L φ r (conjugat momntum) (.37) As an xampl w considr th global U () phas transformation which acts on a complx scalar fild φ. Th fild φ has two dgrs of frdom, th ral scalar filds φ a and φ b : φ= ( φa + iφ b ) (.38) Instad of using φ a and φ b, w work with φ and its Hrmitian conjugat φ and trat thm as two indpndnt filds: φ = ( ) φa iφ b (.39) This is possibl bcaus φ and φ ar linar combinations of φ a and φ b and working with th formr pair instad of th lattr corrsponds to choosing a diffrnt st of coordinats. Our st of filds {φ r } is givn by {φ,φ } (r =,) and th corrsponding conjugat momnta ar {π,π }. Th filds ar transformd by U () as follows: from which w find th changs φ φ = iε φ=(+iε)φ (.4) φ φ = iε φ = ( iε)φ (.4) δφ = φ φ=iεφ (.4) δφ = iεφ (.43) so that f = r [ π r δφ r = π δφ+π δφ = iε π φ π φ ] (.44) and [ F = d 3 x f = iε d 3 x π φ π φ ] (.45) W assum that th U () transformation lavs L invariant. This implis that F (and vry product of it) is constant in tim. W dfin a nw quantity Q q ε F (.46) which is also constant in tim. Th quantity q is a ral numbr and ε is th sam as in (.4) and (.4). For th qustion of how Q should b intrprtd, w chang from a classical pictur to QFT. In QFT, th filds φ r and th conjugat momnta π r bcom oprators which (in gnral) do not commut. Th following commutation rlation holds []: [ φr (x, t), π s (x, t) ] = iδ r s δ(x x ) (.47) In this formula, r and s tak th valus and. Not that φ r and π s ar takn at th sam tim t. Th Hrmitian conjugat is dfind as th transposd complx conjugat. Hr it simplifis to th complx conjugat bcaus φ is a scalar: φ = (φ ) T = φ. 6

11 CHAPTER. THEORETICAL BASIS AND QED.. LAGRANGIAN FIELD THEORY W lt th oprator Q act on th fild φ and tak Q and φ at th sam tim t (w hnc drop th tim from th following formula to mak it mor radabl): [ ] Q φ(x) = i q d 3 x π(x ) φ(x ) π (x ) φ (x ) φ(x) (.48) W will now mov φ(x) from th right to th lft of th intgral. Th filds φ r commut, and also φ and π commut bcaus of th δ r s in (.47) (φ corrsponds to r = and π to s= ) but π(x ) φ(x)=φ(x) π(x ) iδ(x x ) (.49) This rsults in Q φ(x) [ ] = i q φ(x) d 3 x π(x ) φ(x ) π (x ) φ (x ) iδ(x x ) (.5) [ ] = i q φ(x) d 3 x π(x ) φ(x ) π (x ) φ (x ) q φ(x) d 3 x δ(x x ) (.5) = φ(x) Q qφ(x) (.5) which is a commutation rlation (at qual tims) btwn Q and φ: [ Q, φ ] = q φ (.53) W assum that Q is an ignstat of Q with ignvalu Q : Q Q = Q Q }{{}}{{} ignvalu ignstat (.54) and lt Q act on φ Q : Q ( φ Q ) = (φq+ [Q,φ]) Q =φ ( Q q ) Q (.55) }{{} qφ So φ Q is an ignstat with ignvalu Q q. W intrprt ths rsults as follows: Q is th charg oprator φ annihilats a particl with charg q similarly: φ crats a particl with charg q = φ ( Q q ) Q = ( Q q ) φ Q (.56) This xampl shows on of th basic faturs of QFT: th filds ar oprators which crat and annihilat particls. 3 This concpt is crucial for undrstanding th calculations bhind Fynman diagrams which w will discuss latr on. Exampl.5 Q : stat with lctrons Q : charg of th stat = q φ Q : stat with charg Q q = q, corrsponding to only on lctron Th oprator φ has dstroyd on lctron. 3 A mor in-dpth tratmnt of QFT is byond th scop of this lctur and th intrstd radr is rfrrd to th ddicatd lcturs hld at this univrsity or txtbooks lik []. 7

12 .. DIRAC EQUATION CHAPTER. THEORETICAL BASIS AND QED As a summary: th charg Q is consrvd ifl is invariant undr th U () transformation (.4), (.4). This charg can b th ordinary lctric charg (U () EM ) or hyprcharg (U () Y ). Similarly, consrvation of momntum, nrgy, and angular momntum follows from th invarianc of L undr continuous transformations as follows: transformation U () translation in spac translation in tim rotation in spac L invarianc implis consrvation of charg momntum nrgy angular momntum (and spin) Qustions to Sction.. Which factor is usd to convrt HEP (High Enrgy Physics) units to ordinary units?. What is k in th following quations (a µ is a 4-vctor)? a = k a a i = k a i 3. Complt th sntnc: Invarianc of th Lagrangian L undr symmtry transformations lads to a continuity quation for of which is consrvd in tim (Nothr s thorm). 4. Which quantity is consrvd if L is invariant undr U () transformations? 5. Which rol do filds play in Quantum Fild Thory?. Dirac quation This sction follows chaptr 7. of []... Drivation Th Dirac quation is th quation of motion for spin ½ frmions. It is consistnt with but first ordr in tim. Whn applying th usual quantum mchanical substitutions in (.57) on arrivs at p = p µ p µ = E p = m (.57) p i (.58) E i t = i t (.59) p µ i µ (.6) ) ( t + φ=m φ (Klin-Gordon) (.6) 8

13 CHAPTER. THEORETICAL BASIS AND QED.. DIRAC EQUATION which is th Klin-Gordon quation for spin particls which w will not discuss hr. Dirac notd that (.57) can b turnd into an quation first ordr in tim by writing it as follows: = p µ p µ m = ( γ µ p µ + m ) ( γ ν p ν m ) (.6) Th quantitis γ µ ar four 4 4 matrics. For th quation to b tru, (at last) on of th two xprssions in parnthss nds to b zro and th convntional choic is th lattr on, which whn applying th rplacmnt (.6), lads to th Dirac quation: ( iγ µ µ m ) ψ= (Dirac) (.63) Not that summation ovr µ is implid hr and that th mass m is multiplid by th 4 4 unit matrix. For bttr radability w do not show th unit matrix in quations. Th quantity ψ is a spinor: ψ= ψ ψ ψ 3 ψ 4 (.64) It is not a 4-vctor bcaus it dos not transform lik Λx undr Lorntz transformations, cf. (.4). Th γ matrics (also rfrrd to as Dirac matrics) oby th following ruls: γ µ = g µν γ ν (.65) ( γ ) = γ γ = (.66) (γ i) = (.67) {γ µ,γ ν } γ µ γ ν + γ ν γ µ = (ν µ) (.68) Not that dnots th 4 4 unit matrix. Th xprssion {γ µ,γ ν } is calld anti-commutator of γ µ and γ ν. From (.65) follow γ = γ (.69) γ i = γ i (.7) A fifth gamma matrix is dfind as with γ 5 iγ γ γ γ 3 (.7) {γ µ,γ 5 }=γ µ γ 5 + γ 5 γ µ = (.7) W us th following rprsntation: ( ) ( ) ( ) σ i γ =, γ i = σ i, γ 5 (whr dnots th unit matrix) with th Pauli matrics ( ) ( ) ( ) i σ = σ x =, σ = σ y =, σ 3 = σ z = i (.73) (.74) 9

14 .. DIRAC EQUATION CHAPTER. THEORETICAL BASIS AND QED.. Solutions for particls at rst For a particl at rst (p=) th rplacmnt (.6) implis x ψ== y ψ= z ψ (.75) and th Dirac quation simplifis to ( iγ m ) ψ= (.76) W dfin ψ A and ψ B as th uppr two and lowr two componnts of ψ: ψ A ( ψ ψ ), ψ B ( ψ3 ψ 4 ) (.77) so that (.76) rads ( )( ) t ψ A = i m t ψ B ( ψa ψ B ) (.78) which sparats to th following quations t ψ A = i m ψ A (.79) t ψ B = i m ψ B (.8) with solutions ψ A = ψ A () i mt (.8) ψ B = ψ B () i mt (.8) Th tim-dpndnc of a quantum stat follows from th Schrodingr quation Hψ=i t ψ (.83) and is givn by ψ=ψ i Et (.84) In our cas E = m and ψ A bhavs as xpctd. Howvr, ψ B sms to corrspond to a particl with ngativ nrgy E = m (w considr masss to b positiv). Th solution for ψ B is rintrprtd to dscrib an antiparticl with positiv nrgy (cf. Sction.3.7).

15 CHAPTER. THEORETICAL BASIS AND QED.. DIRAC EQUATION Th solutions for th four spinor componnts ar (up to a multiplicativ factor): ψ ψ ψ 3 ψ 4 = = i mt i mt = i mt = i mt lctron, spin up (.85) lctron, spin down (.86) positron, spin down (.87) positron, spin up (.88) Ths ar th only indpndnt solutions. Thr is, for xampl, no solution ψ= i mt (.89) bcaus th lft hand sid of (.76) dos not vanish for such a ψ...3 Gnral solutions For particls with non-vanishing momntum w mak th following Ansatz of a plan wav: ψ= a i k x u(k) (.9) in which a is a normalisation factor, x is th ordinary spac-tim 4-vctor, and u is a spinor that dpnds on th wav 4-vctor k. Th drivativ of ψ is givn by µ ψ= i k µ ψ (.9) so that it follows from th Dirac quation (.63) that ( iγ µ µ m ) ψ = ( γ µ k µ m ) a i k x u(k) = i k x and a (.9) ( γ µ k µ m ) u(k) = (.93)

16 .. DIRAC EQUATION CHAPTER. THEORETICAL BASIS AND QED With th following rlation (drivation lft as an xrcis to th radr): ( ) k k σ γ µ k µ = k σ k in which ( k σ= k x σ x + k y σ y + k z σ z = (.93) can b writtn in componnts: = ( γ µ k µ m ) u(k)= k z k x + i k y k σ k m ) k x i k y k z ( k )( ) m k σ u A u B (.94) (.95) (.96) whr w usd u A for th two uppr componnts of u and u B for th two lowr componnts. From (.96) w find corrsponding to Using (.) in (.99) yilds From (.95) it follows that so that (.) lads to = (k m)u A k σ u B (.97) = k σ u A (k + m)u B (.98) u A = u B = from which w find th following rlation to th 4-momntum p: k σ k m u B (.99) k σ k + m u A (.) u A = (k σ) (k ) m u A (.) (k σ) = k (.) k = k m (.3) k = m = p (.4) Th rlativ sign btwn k and p dtrmins th tim-dpndnc of th spinor ψ (cf. (.9)): k = { +p : tim dpndnc i Et (particls) p : tim dpndnc i Et (antiparticls) Thr xist four indpndnt solutions: ( ) ( ) u A ()= χ and (cf. (.)) u B ()= p z E+ m p x + i p y ( ) ( ) u A ()= χ and (cf. (.)) u B ()= px i p y E+ m p z ( ) u B (3)=χ and (cf. (.99)) u A (3)= p z E+ m p x + i p y ( ) u B (4)=χ and (cf. (.99)) u A (4)= px i p y E+ m p z (.5) (k = p) (.6) (k = p) (.7) (k = p) (.8) (k = p) (.9)

17 CHAPTER. THEORETICAL BASIS AND QED.. DIRAC EQUATION Solutions (.6) and (.7) ar th particl solutions with k = p. Solutions 3 (.8) and 4 (.9) ar th antiparticl solutions with k = p. Writtn as spinors, th solutions ar u = N p z, u = N, u 3 = N E+m p x +i p y E+m with a normalisation factor N. p x i p y E+m p z E+m p z E+m p x +i p y E+m v, u 4 = N p x i p y E+m p z E+m v (.) Particls ar dscribd by Antiparticls ar dscribd by ψ = a i p x u, ψ = a i p x u (.) ψ 3 = a i p x v, ψ 4 = a i p x v (.) Th quantity a is a factor to kp units consistnt. For p, th solutions approach th ons for th particls-at-rst cas ((.85) (.88))...4 Spin Th spin oprator S is givn by ( ) S= σ Σ with Σ= σ Th z-componnt is ( ) S z = σz Σ z = = σ z diag(,,, )= (.3) (.4) Th spinors u, u, v, v ar in gnral no ignstats of S z. As an xampl, considr u : S z u = S z N p z = N p z (.5) E+m E+m p x +i p y E+m p x+i p y E+m Th sign of th lowst componnt is flippd and so u is no ignstat. Th spinors can b mad ignstats by choosing th z-axis along th dirction of motion of th particl. Thn p x = = p y and th spinors bcom p z E+m p z E+m u = N p z, u = N, v E+m = N, v = N (.6) p z E+m 3

18 .. DIRAC EQUATION CHAPTER. THEORETICAL BASIS AND QED and S z u = u spin up (.7) S z u = u spin down (.8) S z v = v spin down (.9) S z v = v spin up (.) Plan-wav solutions to th Dirac quation ar ignstats of S z only if p=(,, p z ). On can construct ignstats of th hlicity oprator Σ p/ p (cf. Sction.4.) from th plan-wav solutions in th gnral cas (i.., th momntum dos not hav to b along z)...5 Spinor normalisation By convntion [, 3], spinors ar normalisd so that W will calculat N for u in (.6), i.., with p x = = p y : u u= E (.) u p z = N ( ) (.) E+ m = p = E m = (E+ m)(e m) (.3) p z pz (E+ m) ) (.4) (E+ m)(e m) (E+ m) = E m E+ m (.5) E E E = u u= N (+ p z (E+ m) = N = + E m E+m = E+m+E m E+m = E+ m (.6) N = E+ m (.7) As an xampl, u from (.6) is givn by E+ m u = E m for p= (.8) p z..6 Transformation proprtis of spinors This sction follows chaptr 7.3 of []. It can b shown that th spinor ψ viwd from a systm moving with spd β along th x-axis is givn by a + a ψ a + a = S ψ with S= (.9) a a + a a + 4

19 CHAPTER. THEORETICAL BASIS AND QED.. DIRAC EQUATION in which a ± =± (γ±) with γ= β (.3) From this w find that ψ ψ is not Lorntz-invariant: ψ ψ = (Sψ) (Sψ)=ψ S S }{{} S ψ ψ ψ (.3) bcaus S (.3) as is vidnt from looking at th -componnt: (S ) = a+ + a = (γ+)+ (γ )=γ (.33) W dfin an adjoint spinor: which is a row spinor. ψ ψ γ = (ψ ψ ψ 3 ψ 4 ) (.34) = (ψ ψ ψ 3 ψ 4 ) (.35) Th quantity ψψ is a scalar. It is Lorntz-invariant as shown in th following: ψ ψ = ψ γ ψ = ψ S γ S }{{} ψ=ψψ (.36) γ Th parity transformation can b shown to corrspond to multiplication by γ : W find that ψψ has vn parity: Pψ=γ ψ (.37) P(ψψ)=γ ψγ ψ=(γ ψ) γ γ ψ=ψ γ ψ=ψψ (.38) }{{}}{{} γ Th quantity ψγ 5 ψ is a Lorntz-invariant scalar but it has odd parity: It is thrfor calld a psudo-scalar. P(ψγ 5 ψ)= ψγ 5 ψ (.39) On finds that ψγ µ ψ Lorntz-transforms lik a vctor and has odd parity, whras ψγ µ γ 5 ψ transforms lik an axial vctor (psudo-vctor). Th transformation proprtis of ths bilinar covariants ar listd in th following tabl. 5

20 .3. QUANTUM ELECTRODYNAMICS CHAPTER. THEORETICAL BASIS AND QED bilinar covariant ψψ ψγ 5 ψ ψγ µ ψ ψγ µ γ 5 ψ transformation proprty scalar psudoscalar vctor (V) axial vctor (A) Latr w will discuss th V-A structur of th wak intraction which is calld lik that bcaus of th occurrnc of trms lik ψ(γ µ γ µ γ 5 )ψ=ψγ µ ( γ 5 )ψ which is th diffrnc btwn a vctor (V) and an axial vctor (A). Qustions to Sction.. Which particls ar dscribd by th Dirac quation?. What is th diffrnc btwn th four solutions to th Dirac quation? 3. How do th following bilinar covariants transform undr a combind Lorntz and parity transformation? ψγ µ ψ ψγ µ γ 5 ψ.3 Quantum lctrodynamics In this sction w ar going to build th Lagrangian of Quantum Elctrodynamics (QED). Our basis is th Dirac quation which dscribs th propagation of fr lctrons and positrons. As first stp w construct a Lagrangian L from which th Dirac quation follows as on of th Eulr-Lagrang quations. This is th fr Lagrangian. Intractions btwn lctrons and positrons ar mdiatd by th photon and w will introduc it into L in a scond stp. In th construction of L w will ncountr a symmtry which corrsponds to th consrvation of lctric charg..3. Lagrang dnsity This sction follows chaptr 4. of []. W will show that th following Lagrang dnsity lads to th Dirac quation: L = ψ[iγ µ µ m]ψ (.4) Th Eulr-Lagrang quations (.7) ar writtn in trms of th filds φ r, of which thr ar two hr: φ = ψ and φ = ψ=ψ γ. For th Eulr-Lagrang quation for ψ w first assmbl th pics: L ψ L ( µ ψ) ( ) L µ ( µ ψ) 6 = ψm (.4) = ψiγ µ (.4) = ( µ ψ)iγ µ (.43)

21 CHAPTER. THEORETICAL BASIS AND QED.3. QUANTUM ELECTRODYNAMICS with which th Eulr-Lagrang quation for ψ rads ψm= ( µ ψ)iγ µ i ( µ ψ)γ µ + mψ= (.44) For th Eulr-Lagrang quation for ψ w find: L ψ = (iγµ µ m)ψ (.45) L = (.46) ( µ ψ) ( ) L µ = (.47) ( µ ψ) so that (iγ µ µ m)ψ= which is th Dirac quation (.63). Th dnsity L is invariant undr th U () transformation ψ ψ = iαq ψ=( iαq)ψ (.48) ψ ψ = iαq ψ γ (.49) ψ ψ = iαq ψ (.5) Th quantity q is a ral numbr and will b idntifid with th charg of th particl. Proof of invarianc: L = ψ [iγ µ µ m] ψ = iαq ψ [iγ µ µ m] iαq ψ (.5) = iαq iαq ψ [iγ µ µ m] ψ=ψ [iγ µ µ m] ψ=l (.5).3. Elctromagntic currnt dnsity and consrvd charg As discussd in Sction..4, th invarianc of L lads to a currnt dnsity f µ which fulfils th continuity quation (.3). Th currnt dnsity is givn by (.3): f µ = = L ( µ φ r ) δφ r L ( µ ψ) }{{} (.4) δψ+ L δψ (.53) ( µ ψ) }{{} = (.46) = ψiγ µ δψ=αqψγ µ ψ (.54) in which w hav usd (.48): W dfin a nw currnt dnsity which fulfils δψ=ψ ψ= iαq ψ (.55) j µ = f µ /α= qψγ µ ψ (.56) µ j µ = (.57) 7

22 .3. QUANTUM ELECTRODYNAMICS CHAPTER. THEORETICAL BASIS AND QED Th spac-intgral of th zro-componnt of th currnt dnsity is constant in tim (consrvd) and is givn by Q = d 3 x j = q d 3 x ψγ ψ (.58) }{{} ψ γ γ =ψ = q d 3 x ψ ψ (.59) Classically (i.., whn not working with oprators), ψ = ψ and ψ ψ= ψ(x) is th probability dnsity of finding th particl at x, so that d 3 x ψ = and Q = q, th charg of th particl. In QFT, th filds ar oprators and Q is th charg oprator. For QED th charg is th lctric charg..3.3 Local phas transformation Th transformation (.48) (.5) is calld a global phas transformation bcaus th paramtr α is indpndnt of th spac-tim coordinat x: µ α=(q is a ral numbr that also dos not dpnd on x). W ar now going to invstigat whthr L is also invariant undr a local phas transformation in which α dpnds on x: ψ ψ = iα(x)q ψ (.6) W rlabl L as L to indicat that this is th Lagrangian for a fr fild ψ, i.., this fild dos not intract with anothr particl (th Dirac quation is th quation of motion of a frly propagating particl). Th transformd L is givn by L = ψ [iγ µ µ m] ψ = iα(x)q ψ [iγ µ µ m] iα(x)q ψ (.6) = iα(x)q ψ iγ µ µ iα(x)q ψ iα(x)q ψm iα(x)q ψ (.6) Th scond trm simplifis bcaus iα(x)q is just a numbr which can b movd to th lft sid of ψ: iα(x)q ψm iα(x)q ψ= iα(x)q iα(x)q ψmψ=ψmψ (.63) In th first trm, both iα(x)q and ψ dpnd on x so whn applying th drivativ, w gt from th product rul Thrfor µ iα(x)q ψ= i q( µ α) iα(x)q ψ+ iα(x)q µ ψ (.64) L = q( µ α)ψγ µ ψ+iψγ µ µ ψ ψmψ (.65) = ψ [iγ µ µ m] ψ+q( µ α)ψγ ψ }{{} (.66) L W s that L is not invariant. Th trm that dstroys th invarianc is q( µ α)ψγ µ ψ. A diffrnt Lagrangian can b constructd which is invariant. To do this, w rplac th ordinary drivativ µ with th covariant drivativ D µ = µ + i q A µ (x) (.67) 8

23 CHAPTER. THEORETICAL BASIS AND QED.3. QUANTUM ELECTRODYNAMICS in which A µ is a nw fild which w postulat to transform as follows: Th nw Lagrangian is in which L I is th intraction Lagrangian A µ = A µ+ µ α (.68) L = ψ [iγ µ D µ m] ψ=l +L I (.69) L I = qψγ µ ψa µ (.7) namd in this way bcaus it introducs a coupling of ψ to A µ (s blow). L I transforms to L I = qψ γ µ ψ A µ (.7) = q iαq ψγ µ iαq ψ(a µ + µ α) (.7) = qψγ µ ψa µ qψγ µ ψ( µ α) (.73) }{{} L I So also L I is not invariant. But th trm that dstroys th invarianc is th ngativ of th trm that dstroyd th invarianc of L in (.66). Th full Lagrangian which is th sum of L and L I is thrfor invariant: L = L +L I = L +L I = L (.74) W constructd a Lagrangian L that is invariant undr local U() transformation of th mattr filds ψ,ψ by introducing a nw gaug fild A µ with crtain transformation proprtis. Th gaug fild coupls to th mattr filds. For compltnss: th gaug fild A µ can also propagat frly and th corrsponding Lagrangian is with th fild tnsor L γ = 4 F µν F µν (.75) F µν = ν A µ µ A ν (.76) (Th fild tnsor is not rlatd to th spac-intgratd currnt dnsity (.34), thy ar just rfrrd to by th sam symbol.) Th full Lagrangian of QED is givn by L = L +L I +L γ (.77).3.4 Th photon fild In classical lctrodynamics, potntials φ and A ar introducd such that th magntic and lctric filds ar givn by B=rot A= A=curl A (.78) E= grad φ t A= φ t A (.79) Th potntials ar not uniqu bcaus th sam filds B and E ar obtaind for φ = φ+ t α (.8) A = A α (.8) 9

24 .3. QUANTUM ELECTRODYNAMICS CHAPTER. THEORETICAL BASIS AND QED W dfin th 4-potntial ( ) ( ) φ φ A µ =, A µ = A A (.8) and th componnts (µ =,,,3) of this 4-potntial transform lik (.8) and (.8), and ths transformations can b writtn in 4-vctor notation as in (.68). Th 4-potntial (.8) has th xact transformation proprtis that ar rquird for th gaug fild A µ in Sction.3.3. W thrfor idntify th gaug fild with th 4-potntial and rfr to it as th photon fild bcaus of its rlation to th lctromagntic filds. Th labl gaug fild stms from th fact that (.8) and (.8) ar calld gaug transformations. A thory that is basd on th introduction of a gaug fild is calld a gaug thory..3.5 Principl of gaug thoris Du to th succss of QED in dscribing all lctromagntic (EM) phnomna to xtrmly high prcision, th thoris of th wak and strong intractions wr modlld aftr it as gaug thoris. Gaug thoris ar built according to th following rcip. Th starting point is a fr-fild Lagrangian L which is invariant undr global phas transformation of th mattr filds (ψ,ψ in our xampl). Global mans that th phas is changd by th sam amount at vry spac-tim point. Th invarianc of L implis a consrvd quantity or svral consrvd quantitis (dpnding on th transformation). At this point w dmand invarianc undr local phas transformation, i.., L should not chang vn if th transformation of th fild phass varis from spac-tim point to spac-tim point. This is achivd through th introduction of on or mor gaug filds that coupl to th mattr filds (and may coupl among thmslvs and gaug filds of othr transformations). Ths gaug filds provid th intraction btwn th mattr filds. Exampl.6 (QED) L follows from Dirac quation global phas transformation: U (), iαq, µ α= consrvd quantity: lctric charg local phas transformation: U (), iα(x)q, µ α gaug fild: A µ (photon) intractions of mattr particls with photon: L I.3.6 Elctron-photon vrtx Th lctromagntic currnt dnsity j µ is dfind by (.56) so that th QED intraction Lagrangian (.7) is givn by th ngativ of th product of th currnt dnsity and th photon fild: L I = j µ A µ = j A (.83) Th Fynman diagram of th lctron-photon vrtx is shown in Figur.. Th currnt dnsity j µ = ψ qγ µ ψ is in this cas calld th lctron currnt (it is undrstood that it rally is a dnsity). Th spinor ψ rprsnts an lctron and q is th lctric charg of th lctron (q = ). This charg is consrvd in th lctron-photon intraction: th charg in th initial stat is and

25 CHAPTER. THEORETICAL BASIS AND QED.3. QUANTUM ELECTRODYNAMICS j μ tim γ (A μ ) Figur.: Elctron-photon vrtx. Th tim axis points from lft to right. th charg in th final stat is also. In th thory, th rason for charg consrvation is that L is invariant undr th U () transformation. Fynman diagrams ar graphical rprsntations of intraction amplituds. Th diagram in Figur. is not a physical amplitud as w will s blow but w will us it to illustrat th building principl of ths diagrams (Fynman ruls). Th spinor ψ in th currnt is an oprator which dstroys an lctron at th photon vrtx. qγ µ is th vrtx factor and q is th coupling strngth btwn th lctron and th photon. Th spinor ψ crats th outgoing lctron..3.7 Antiparticls Th tim dpndnc of a quantum stat is givn by i Et (cf. (.84)) with positiv nrgy and tim (E >, t > ). In Fynman diagrams w symbolis such a stat as follows (in all Fynman diagrams in this txt th tim volvs from lft to right): t i Et This is a particl with positiv nrgy going forward in tim. A stat that bhavs lik i ( E)( t) is a particl with ngativ nrgy that is going backward in tim and it is symbolisd by i ( E)( t) Th two ar quivalnt bcaus i ( E)( t) = i Et and thrfor i Et E -E E = -E i ( E)( t) QED intractions dpnd on th currnt (.56). For an lctron with 4-momntum p µ = (E, p) th currnt j µ can b shown,.g., by using xplicit solutions to th Dirac quation, to b proportional to th product of th lctron charg and th momntum: j µ q pµ = q + ( p µ ) (.84) which corrsponds to th charg of a positron multiplid by th 4-momntum p µ = ( E, p). W now look again at th diagram of th photon vrtx and writ th nrgy and momntum of th particl nxt to its lin. A positron going backward in tim is symbolisd by

26 .3. QUANTUM ELECTRODYNAMICS CHAPTER. THEORETICAL BASIS AND QED + + (E, p) i E( t) This corrsponds to a stat bhaving lik i E( t). It is quivalnt to a stat bhaving lik i ( E)t so that th prvious diagram is quivalnt to th following in which w turnd th arrows on th positron lin: + + (-E,- p) i ( E)t ( ) E j µ + q + p This corrsponds to an lctron with positiv nrgy and momntum bcaus w chang th charg and th sign of th 4-momntum vctor simultanously so that th currnt is unchangd: - - (E, p) j µ q ( E p ) Th thr prvious diagrams ar thrfor quivalnt: + + (E, p) + + (-E,- p) = = - - (E, p) Comparing th lft-most diagram and th right-most on, w s that an antiparticl going backward in tim corrsponds to th corrsponding particl going forward in tim (with th sam nrgy and momntum). In th sam way, th following two diagrams ar quivalnt: + + (E, p) - - (E, p) = which mans that an antiparticl going forward in tim corrsponds to th corrsponding particl going backward in tim (with th sam nrgy and momntum). In th following, w will assum all particls and antiparticls to hav positiv nrgy and momntum (and thrfor not writ nrgy and momntum nxt to th lin in th Fynman diagram). In this txt w draw Fynman diagrams according to th following convntion, which is usd for xampl also in []:

27 CHAPTER. THEORETICAL BASIS AND QED.3. QUANTUM ELECTRODYNAMICS Th frmion lins in Fynman diagrams corrspond to particls (not to antiparticls). As th frmion lins ar always particls, w simply writ instad of. Th positron-photon vrtx is hnc symbolisd by.3.8 Th unphysical photon vrtx Th procss f f γ (whr f stands for any lctrically chargd frmion) is kinmatically forbiddn. f f In th rst fram of th initial frmion, th total initial stat nrgy E is givn by th mass of th frmion: E = m f bcaus th momntum is zro (rst fram, p f = ). Th total final stat nrgy E is givn by E = E f + E γ > m f = E (.85) }{{}}{{} >m f > Th nrgy of th photon is largr than zro and th final stat frmion has non-zro momntum bcaus it has to balanc th photon momntum (rcoil). Thrfor, E > E in violation of nrgy consrvation. Similarly forbiddn is th procss f f γ: f f W choos th f f cntr-of-mass systm to analys th procss (and us a script * to dnot variabls in th cntr-of-mass fram). Th total momntum of th initial stat is zro: p = i. But a photon always as non-zro momntum: p γ =E γ >. So momntum consrvation is violatd in this procss. A scond vrtx has to b introducd in ths diagrams to modl physical procsss..3.9 Virtual photons W considr lctron-muon scattring, µ µ, through photon xchang: 3

28 .3. QUANTUM ELECTRODYNAMICS CHAPTER. THEORETICAL BASIS AND QED γ μ μ Th arrow nxt to th photon lin rprsnts th photon 4-momntum which w choos to point away from th lctron-photon vrtx (this is an arbitrary choic and th rsults do not dpnd on it). For th 4-momnta at th vrtics w hav from nrgy and momntum consrvation (final stat quantitis ar primd): p = p γ + p (.86) p γ + p µ = p µ (.87) which whn combind yild p + p µ = p + p µ (.88) Th mass of a photon is zro, thrfor on xpcts Instad w find p γ = E γ p γ = m γ = (.89) p γ = (p p ) = p }{{} m = m [ E E p p ] p p + p }{{} m (.9) (.9) Th scalar product of a 4-vctor is Lorntz-invariant. W can valuat it in any coordinat systm. For th following analysis w choos th cntr-of-mass systm (CMS) of th initial stat µ. In this systm, th incoming and outgoing lctron nrgis and momnta ar qual bcaus th collision is lastic (s xrcis 3 on sht ): Thrfor p γ = p γ E (.9) p = p (.93) = m [ E E = m E }{{} = p m +p p ] p p [ cosθ ] (.94) cosθ (.95) (.96) in which θ is th angl in th CMS btwn th dirctions of th incoming lctron and th outgoing lctron: θ * μ μ 4

29 CHAPTER. THEORETICAL BASIS AND QED.3. QUANTUM ELECTRODYNAMICS From (.96) w s that p γ dpnds on th scattring angl θ : θ cosθ pγ no scattring ( and µ do not intract) 8 4p back-scattring <θ 8 4p pγ < scattring For our thory to dscrib scattring w nd pγ <. Th xchangd photon is not a ral photon (for which th mass vanishs). Instad it is a virtual particl. Virtual particls hav a squard 4-momntum that is diffrnt from th squar of th rst mass and ar thrfor also said to b off thir mass shll. QED dscribs lctromagntic intractions vry wll and th concpt of virtual particls provs to b succssful. Th quantity pγ spcifis how far th photon is off its mass shll. It thrfor quantifis how virtual th photon is. Bcaus pγ is ngativ but valus of p γ nar zro man low virtuality, on introducs a nw variabl Q as th ngativ of pγ : Q p γ (.97) Larg valus of Q corrspond to high virtuality and on thrfor calls Q th virtuality of th photon..3. Photon propagator Th photon in µ scattring travls from on vrtx to th othr: γ μ μ Whn calculating th probability amplitud of a givn Fynman diagram, th trm that stands for such a propagation of an xchang particl btwn two vrtics is calld propagator trm. In gnral, propagator q m (.98) in which q is th 4-momntum of th xchangd particl and m is th ral or on-shll mass of th xchang particl. Th usag of q to dnot th momntum is customary but it unfortunatly clashs with our prvious usag of q for th charg. Howvr, it should b clar from th contxt whthr a charg is mant or a momntum. For th photon, th mass m vanishs and photon propagator q (.99) For othr xchang particls lik th W and Z bosons, th mass is non-zro and has to b considrd in (.98). Using th nomnclatur of Sction.3.9, q = p γ and Q = q. 5

30 .3. QUANTUM ELECTRODYNAMICS CHAPTER. THEORETICAL BASIS AND QED Fynman ruls ar rcips to calculat transition amplituds from diagrams. Ths ruls follow from th xpansion of th scattring matrix and dtails can b found in books on QFT, for xampl in []. W will us som Fynman ruls hr to illustrat th concpt and to dvlop an undrstanding of how cross sctions ar rlatd to th coupling strngth btwn particls. Th transition amplitud M in µ scattring is givn by im = currnt at vrtx propagator currnt at vrtx (.) ( = u i g γ ν) i g νβ ( u q u µ i g γ β) u µ (.) in which g µν is th mtric tnsor (.9) and g is th lctric coupling. Th summation ovr β lads to M = u ( i g γ ν) u q u µ ( i g γ ν ) uµ (.) = g q u γ ν u u µ γ ν u µ (.3) Th spinor u dstroys th incoming lctron at vrtx and th adjoint spinor u crats th outgoing lctron. Th spinors u µ and u µ play similar rols for th muons (hr th subscript µ spcifis th particl associatd with th spinor, it is not a Grk indx of a 4-vctor). A short-hand notation oftn ncountrd is to writ for th lctron spinor, tc., so that (.3) rads M = g q γ ν µγ ν µ (.4) Th coupling g is rlatd to th lctromagntic constant α as follows: g = q 4πα (.5) in which q is th charg of th particl (not antiparticl!) at th vrtx in units of th positron charg. Exampls ar, µ, τ : g = 4πα u-typ quark: g = 3 4πα d-typ quark: g = 3 4πα Th cross sction is proportional to th absolut squar of th amplitud (Goldn Rul, s nxt sction), so that th dpndnc of th cross sction on α is σ M α.3. Units of lctric charg Diffrnt txtbooks us diffrnt units of lctric charg. Hr is an ovrviw which shows how ths units ar rlatd. systm SI Gauss Havisid-Lorntz unit of charg Coulomb lctrostatic units (.s.u.).s.u. ( statcoulomb ) α (= 37 ) SI 4πε G positron charg SI = 4παε G = α.85 HL = 4πα.3 6 HL 4π

31 CHAPTER. THEORETICAL BASIS AND QED.4. ELECTROMAGNETIC SCATTERING For th lctric coupling g, th charg of th particl and th positron charg hav to b takn in th sam systm. For xampl, [, 3] us th Havisid-Lorntz systm of charg and g is givn by g = q HL HL 4πα= qhl (usd in [, 3]) (.6) In [], th Gauss systm is usd and g = q G G 4πα= qg 4π (usd in []) (.7) Qustions to Sction.3. What is th signatur consrvd quantity of QED? Why is it consrvd?. Why is th photon calld a gaug boson? 3. What is th principl of Gaug Thoris? 4. What dtrmins th strngth of th QED coupling? 5. What is a virtual particl? 6. What is th photon propagator?.4 Elctromagntic scattring.4. Goldn rul for scattring This sction follows chaptr 6.. of []. Th cross sction σ for th scattring of two particls with givn 4-momnta p and p which producs many particls in th final stat n is givn by S σ= 4 (p p ) m m M (π) 4 δ 4 (p + p p 3 p 4... p n ) n j=3 π δ(p j m j ) θ(p j ) d 4 p j (π) 4 (.8) Th intgral is ovr th outgoing particl momnta. Th first dlta function nsurs nrgy and momntum consrvation btwn th initial and final stat. Th scond dlta function nsurs that th outgoing particls ar ral, i.., on thir mass-shll. Th thta function lads to positiv outgoing particls nrgis. Th dynamics of th scattring ar containd in M. S is a statistical factor which accounts for idntical particls in th final stat. For ach group g i of idntical final stat particls, S contains a factor g i! (prmutation factor). 7

32 .4. ELECTROMAGNETIC SCATTERING CHAPTER. THEORETICAL BASIS AND QED Exampl.7 Considr in which 3 and 4 ar idntical particls. Intgration ovr d 4 p 3 and d 4 p 4 yilds th cross sction for th sam procss twic (doubl counting). This can b sn for xampl by looking at th scattring angls: w ar intgrating ovr all angls θ π for both particl 3 and 4 but w should us only θ 3 < π for 3 and π θ 4 π for 4 bcaus th final stat is physically idntical undr 3 4. Thrfor w nd to us S=. W bring (.8) into a form mor suitabl for th calculations of th following sctions. W rwrit th dlta function which nsurs ral outgoing particls: δ(p m )=δ(e p m )=δ(e (p + m )) (.9) W xploit th following proprty of th dlta function: and us δ(x a )= [δ(x a)+δ(x+ a)] for a> (.) a x = E = p (.) a= p + m > (.) so that (.9) bcoms δ(p m )= δ(e p + m )+δ(e+ p + m ) (.3) p + m }{{} (.4) Th trm (*) dos not contribut to th intgral in (.8) bcaus θ(p ) nsurs that E >. W can thrfor rplac th dlta function δ(p j m ) in th intgral with j so that S σ= 4 (p p ) m m p j + m j δ(e j p j + m j ) M (π) 4 δ 4 (p + p p 3 p 4... p n ) n j=3 π p j + m j W carry out th intgrations ovr p j = E j and obtain S σ= 4 (p p ) m m δ(e j p j + m j ) θ(p j ) d 4 p j (π) 4 (.5) M (π) 4 δ 4 (p + p p 3 p 4... p n ) n j=3 p j + m j d 3 p j (π) 3 (.6) with positiv outgoing particl nrgis E j = p j + m in M and th dlta function. j 8

33 CHAPTER. THEORETICAL BASIS AND QED.4. ELECTROMAGNETIC SCATTERING.4. scattring This sction follows p. 9 and following in []. As a spcial cas of th gnral Goldn Rul for scattring introducd in Sction.4., w considr scattring: As a first stp, w calculat (p p ) m m which is a Lorntz-invariant scalar that w can valuat in any coordinat systm. In th cntr-of-mass systm, p = p (.7) and p p = ( ) E T ( ) E p p = E E + p (.8) m i = E i p (.9) m m = E E E p p E + p 4 (.) (p p ) = E E + E E p + p 4 (.) (p p ) m m = E p + p E + E E p (.) = p ( E + E + E E ) (.3) = p ( E + E ) (.4) With this rsult, (.6) bcoms S σ= 64π (E + M δ 4 (p E ) p + p p 3 p 4 ) d 3 p 3 p 3 + m 3 d 3 p 4 p 4 + m 4 Th 4-dimnsional dlta function sparats into an nrgy part and a momntum part: (.5) δ 4 (p + p p 3 p 4 )=δ(e + E E 3 E 4 ) δ3 ( p 3 p 4 ) }{{} δ 3 (p 3 +p 4 ) (.6) and th d 3 p 4 intgration in (.5) lads to p 4 = p 3 (.7) W thrfor gt S σ= (8π) (E + E ) p M δ(e + E p p 3 + m m 3 p p 3 + m m 4 ) d 3 p 3 (.8) To solv th intgral w introduc sphrical coordinats sinθ cosφ p 3 = p 3 sinθ sinφ (.9) cosθ r p 3 (.3) dω = dφ d(cosθ ) (.3) d(cosθ ) dθ = sinθ (.3) 9

34 .4. ELECTROMAGNETIC SCATTERING CHAPTER. THEORETICAL BASIS AND QED so that d 3 p 3 = r dr dω (.33) Th diffrntial cross sction dσ dω is dfind as σ= dω dσ dω (.34) and is givn by dσ dω = S (8π) (E + E ) p M δ(e + E r + m 3 r + m4 ) r + m3 r + m4 r dr (.35) W chang from r to th variabl u: u r + m 3 + r + m4 (.36) du dr = r r + m3 + r = r + m4 r u r + m 3 r + m 4 (.37) so that th intgral in (.35) is givn by M δ(e + E u) r du (.38) u Upon intgration, th dlta-function snds u to th cntr-of-mass nrgy of th collision: From (.36) it follows thn aftr som algbra that u= E + E = E CM (.39) r = E CM E 4 CM + m4 3 + m4 4 m 3 m 4 E CM (m 3 + m 4 )= p 3 p f (.4) which is by dfinition (.3) th final stat momntum p 3 = p 4 that is consistnt with nrgy and momntum consrvation and which w labl p f. In summary, th cross sction for th procss is givn by dσ dω = S (8π) E CM p f p i M (.4) which in th cas of lastic scattring ( p f = p ) simplifis to i dσ dω = S (8π) E CM M (for lastic scattring) (.4) If thr ar no idntical particls in th final stat, th prmutation factor S=. Enrgy-momntum consrvation implis that th only fr paramtrs ar th two angls θ and φ which spcify th flight dirction of particl 3, cf. (.9). In gnral, M dpnds on ths angls and thn so dos th cross sction dσ d dω = σ dφ d(cosθ ). As a consistncy chck w analys th numbr of variabls and constraints, starting from th four 4-momnta p, p, p 3, p 4 (6 unknowns). From nrgy-momntum consrvation, p + p = p 3 + p 4, w hav four constraints. W know th particl masss m i, which giv anothr four constraints. Th xprimntal stup dtrmins th thr-momnta p and p (six constraints). In total w gt 4 constraints, laving two unknown variabls. 3

35 CHAPTER. THEORETICAL BASIS AND QED.4. ELECTROMAGNETIC SCATTERING.4.3 Cross sction for µ µ This sction follows in part chaptrs of []. W ar going to calculat th cross sction for lctron-muon scattring in QED using Fynman ruls. Th goal of this sction is to dmonstrat how cross sctions ar calculatd using first principls. It is not xpctd that such a calculation can b prformd in th xam. Th procss undr study is (p, s )+µ (p, s ) (p 3, s 3 )+µ (p 4, s 4 ) (.43) in which th p i dnot th 4-momnta and th s i th spin configurations of th particls. Th Fynman diagram looks as follows: p p 3 γ q μ p p 4 μ Th arrows nxt to th frmion and photon lins dnot th dirctions of th particl momnta. Th QED Fynman ruls to calculat im ar as follows. Follow th arrows on th frmion lins and writ th following trms from right to lft: xtrnal lins: incoming particl ( ): u incoming antiparticl ( ): v outgoing particl ( ): u outgoing antiparticl ( ): v vrtx factor: i g γ µ with g = q q 4πα and th lctric charg of th particl in units of th positron charg photon propagator: i g µν q nrgy-momntum consrvation: (q is hr th 4-momntum of th photon) for ach vrtx writ (π) 4 δ 4 (k +k +k 3 ) in which th k i ar th 4-momnta coming into th vrtx intgrat ovr intrnal momnta: d 4 q (π) 4 drop th ovrall δ-function (π) 4 δ 4 (p + p p 3 p 4 ) Following this rcip w gt im = u (s3) (p 3 ) ( i g γ µ) u (s) (p ) (π) 4 δ 4 (p p 3 q) }{{} i g µν q u (s 4) (p 4 ) ( i g γ ν) u (s ) (p ) (.44) (π) 4 δ 4 (p + q p 4 ) }{{} d 4 q (π) 4 (.45) 3

36 .4. ELECTROMAGNETIC SCATTERING CHAPTER. THEORETICAL BASIS AND QED Th δ-function * snds q to p p 3 and ** bcoms (π) 4 δ 4 (p + p p 3 p 4 ) which w drop according to th last Fynman rul. W thrfor obtain im = i g [ ][ ] (p p 3 ) u (s3) (p 3 )γ µ u (s) (p ) u (s4) (p 4 )γ µ u (s) (p ) (.46) Each of th brackts ncloss a numbr and M is a numbr which can b calculatd whn th momnta p i and spin configurations s i ar spcifid. Spin avraging A situation frquntly ncountrd in high nrgy physics is that particl bams ar unpolarisd and th dtctor dos not distinguish btwn spin stats. In that cas, th masurd cross sction corrsponds to a combination of diffrnt spin configurations. An unpolarisd bam mans that th probability of having th incoming lctron spin in th up stat is 5% and th probability of having it in th down stat is also 5%. Th sam is tru for th incoming muon spin. To obtain th unpolarisd cross sction on thrfor has to avrag ovr th four initial stat spin configurations. For th final stat, th fact that th dtctor dos not distinguish btwn th spin stats up and down for th outgoing particls mans that what is masurd ar all possibl combinations of spin final stats, i.., th sum of th procsss that lad to (up,up), (down,up), (up,down), and (down, down). W ar daling with probabilitis hr, so th cross sctions nd to b addd. Th only part of th cross sction that dpnds on th spin is M (cf. Goldn Rul), and w avrag ovr th initial spin configurations (s, s ) and sum ovr th final spin configurations (s 3, s 4 ) of (.46): M = 4 s s s 3 s 4 }{{}}{{} avraging summing M (s, s, s 3, s 4 ) (.47) W now calculat M and us a simplifid notation in which u(i ) stands for u (s i ) (p i ). M = MM = = g 4 (p p 3 ) 4 [ u(3)γ µ u() ][ u(4)γ µ u() ][ u(3)γ ν u() ] [ u(4)γ ν u() ] g 4 (p p 3 ) 4 [ u(3)γ µ u() ][ u(3)γ ν u() ] [ u(4)γ µ u() ][ u(4)γ ν u() ] (.48) (.49) whr w could r-ordr th bracktd trms bcaus ach of thm corrsponds to a numbr. Casimir s trick W ncountr hr twic th gnric form G = [ u(a)γ u(b) ][ u(a)γ u(b) ] = [ u(a)γ u(b) ][ u(a)γ u(b) ] (.5) (.5) 3

37 CHAPTER. THEORETICAL BASIS AND QED.4. ELECTROMAGNETIC SCATTERING in which Γ stands for a γ matrix. W xamin th scond brackt: [ u(a)γ u(b) ] = [ u (a)γ Γ u(b)] (.5) With this dfinition of Γ, th gnric form rads = [ γ Γ u(b) ] u(a) (.53) = u (b)γ }{{} γ u(a) (.54) γ = u (b)γ γ Γ }{{} γ u(a) (.55) = u(b)γ Γ }{{ γ } u(a) (.56) Γ (.57) G = [u(a)γ u(b)] [u(b)γ u(a)] (.58) Summing ovr th spin orintations of particl b, ( ) G = u(a)γ u(b)u(b) Γ u(a) (.59) s b s b }{{} Th xprssion * corrsponds to u (sb) (p b )u (sb) (p b )=γ µ p b,µ + m b (compltnss rlation) (.6) s b = This rlation is provn in xrcis 4 on sht. W introduc th short-hand slash notation γ µ p µ /p (.6) so that with a 4 4 matrix Q. G = u(a) Γ ( /p b + m b )Γ s b }{{} Q u(a) (.6) W now sum ovr th spin configurations of particl a: s a,s b G = s a u(a) Q u(a) (.63) W writ it in componnts so w can r-ordr th trms: s a,s b G = 4 s a = µ,ν= u (s a) µ (p a ) Q µν u (s a) ν (p a ) (.64) = Q µν u (s a) ν (p a ) u (s a) µ (p a ) µ,ν s a }{{}}{{} 4 4 = µ,ν Q µν [ s a u (s a) (p a ) u (s a) (p a ) ] νµ (.65) (.66) 33

38 .4. ELECTROMAGNETIC SCATTERING CHAPTER. THEORETICAL BASIS AND QED which, bcaus of (.6), corrsponds to G = [ Q µν /p a + m a ]νµ s a,s b µ,ν (.67) = [ Q ( /p a + m a ) ] µµ = Tr( Q ( /p a + m a ) ) µ (.68) in which Tr ( Q ( /p a + m a ) ) stands for th trac of th 4 4 matrix Q ( /p a + m a ). Insrting th dfinitions of G (.5) and Q (.6) w hav just provn th following rlation which is known as Casimir s trick: [ u(a)γ u(b) ][ u(a)γ u(b) ] ( ) = Tr Γ ( /p b + m b )Γ ( /p a + m a ) (.69) s a,s b Calculation of th spin-avragd cross sction To apply Casimir s trick to our cas, w us Γ = γ µ (.7) Γ = γ ν (.7) Γ = γ γ ν γ = γ ν (xrcis) (.7) and with (.49), (.47) bcoms M = 4 s,s,s 3,s 4 g 4 (p p 3 ) 4 [ u(3)γ µ u() ][ u(3)γ ν u() ] [ u(4)γ µ u() ][ u(4)γ ν u() ] (.73) = g 4 4(p p 3 ) 4 Tr( γ µ ( /p + m )γ ν ( /p 3 + m 3 ) ) Tr ( γ µ ( /p + m )γ ν ( /p 4 + m 4 ) ) (.74) This corrsponds to quation (7.6) in [] and for th following w adopt th notation usd thr, which is to dnot th lctron mass with m and th muon mass with M: m = m 3 = m m (.75) m = m 4 = m µ M (.76) In calculating th tracs in (.74), w mploy trac thorms, which ar, for xampl givn in [] on pags 5 and 53. Thos ar 6 ruls for taking tracs of combinations of γ matrics, of which w will quot only thos that ar ndd for our purpos. First w will calculat th lctron trac Tr ( γ µ ( /p + m)γ ν ( /p 3 + m) ) = Tr ( (γ µ /p γ ν + mγ µ γ ν )( /p 3 + m) ) (.77) = Tr ( γ µ /p γ ν /p 3 + mγ µ γ ν /p 3 + mγ µ /p γ ν + m γ µ γ ν) (.78) = Tr ( γ µ /p γ ν ) ( /p 3 + m Tr γ µ γ ν ) /p }{{ 3 +m Tr ( γ µ /p } γ ν) +m Tr ( γ µ γ ν) }{{} (.79) whr w usd th known rlations from matrix algbra: Tr(A+ B)=Tr(A)+Tr(B) and Tr(αA)= α Tr(A) (for a scalar α). Th scond and third trac in (.79) vanish on account of rul (using th numbring in []) which stats that th trac of a product of an odd numbr of γ matrics vanishs. Rmmbr that thr is a γ matrix hiddn in th slash notation: /p = γ λ p λ. 34

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