Chaptr 3 A brif viw of Quantum Elctrodynamic Fw rfrncs: I.J.R Aitchison & A.J.G. Hy, Gaug thoris in particl physics, vol, Chaptrs 6,7,8, IoP 3. M.E. Pskin & D.V. Schrodr, An introduction to Quantum Fild Thory, Chaptr 4,5, Wstviw Prss Inc 995 With th first two chaptrs, w saw that th quantization of th solutions of th wav quations lads to quantum filds, a wll adaptd framwork to trat stats mad of many particls that can b cratd or annihilatd bcaus of intractions. This chaptr will brifly try to xplain how this can b adaptd to th intraction of lctrons with photons. I want to warn th radr: this cours is not a quantum fild cours. Th goal hr, is to giv th concpts without complt dmonstrations and prpar th radr to b abl to do simpl calculation of scattring procsss at th lowst ordr. 3. Th action and Lagrangians 3.. Th last action principl Classical cas: In th hamiltonian formulation of classical mchanics, th quations of motion ar dducd by minimizing a quantity calld th action S. Lt us rcall how it works using a simpl xampl. Considr a non-rlativistic particl moving in dimnsion x with a kintic nrgy T = m x and with a potntial nrgy V (that can b mgx for instanc). Th quantity: L = T V is calld a Lagrangian with th dimnsion of th nrgy and dpnds obviously on x and x. Th momntum and nrgy ar givn by: p E = p x L = @L @ x 63
64 A brif viw of Quantum Elctrodynamic Th quation of motion btwn th tim t and t is dducd by minimizing th action S dfind by: S = Z t t L(x, x)dt Th maning of minimization is that among all trajctoris x(t) that can b imagind starting from x at t and finishing at x at t, th on which will hav th lowst valu of th action will b th on adoptd by th particl. Lt us call x (t) th trajctory w want to find. Any trajctory can b writtn x(t) =x (t)+ x(t). Th action bcoms: S = Z t t L(x + x, x + d dt x)dt = Z t t L(x, x )+ @L @x x + @L @ x d dt x dt Intgrating by part: Z t t @L @ x d dt x dt = appl @L @ x x t t Z t t d dt @L @ x x dt = Z t t d dt @L @ x x dt whr x(t )= x(t ) = has bn usd (all trajctoris start from th sam point and finish at th sam point). Hnc th variation of th action is just: and thus: S = Z t t @L @x S =) @L @x d @L dt @ x x dt d @L dt @ x = which is th Eulr-Lagrang quation. Applying it to our simpl xampl: @L @x = d @L dt So, w find th usual Nwton s law. @V @x @ x = mẍ mẍ = @V @x = F Simpl quantum cas: Lt us call q r (r =,n) th gnralizd coordinats (was x bfor, with n = ). In quantum mchanics, w hav th additional constraint: [q r,p r ]=i rr (with ~ = ), q r and p r bing oprators. Th Lagrangian L(q,...,q n, q,..., q n ) is now an oprator and w still hav: p r H = @L @ q r =( P n r= p r q r ) L whr H th hamiltonian, givs th total nrgy of th systm. Sinc th hamiltonian is hrmtian (th nrgy is ral), th Lagrangian must b hrmtian too. Now, th Eulr-Lagrang quation is simply: @L d @L = @q r dt @ q r P.Paganini Ecol Polytchniqu Physiqu ds particuls
Fr lctron Lagrangian 65 Quantum fild cas: in th continuous cas i n!,thnq r oprators bcom a fild (~x, t ) dpnding on spactim coordinats, and th summation must bcom an intgral. Thrfor, instad of considring th Lagrangian L itslf, w considr th Lagrangian dnsity L: Z L = L d 3 ~x (3.) and hnc, th action bcoms: Z S = L d 4 x (3.) As bfor L dpnds on and but sinc is a continious function of spactim, L can also dpnd on @ /@x, @ /@y, @ /@z so that L is now L(, @ µ ). Th momntum p bcoms a momntum fild (x µ ): and th hamiltonian dnsity fild is: (x µ )= @L @ (x µ ) Th quantization gnralizs what w had in th prvious cas : (3.3) H = (x µ ) (x µ ) L (3.4) [ (x µ ), (x µ )] = i (3) (~x ~y ) (3.5) This is this procdur which yilds th quantizd fild of th prvious chaptr, th commutator btwn th cration and annihilation oprators bing a simpl consqunc of th prvious rlation. Finally, th Eulr-Lagrang quation bcoms: 3.. Fr lctron Lagrangian @L @ @ µ @L @(@ µ ) = (3.6) Whn th wav quation is known (in our cas th Dirac quation), it is not too di cult to find th Lagrangian. Th fr lctron Lagrangian, namly th Dirac Lagrangian dnsity, is: L D = (i/@ m) (3.7) Lt us chck that it givs th Dirac s quation. Th variabls that hav to b considrd ar th 4 componnts of th Dirac s fild, its adjoint fild (sinc is complx, w can vary and indpndntly) and th drivativs @ µ and @ µ with = {, 4}. Making xplicit th Lagrangian in trms of componnts: L = X, i µ @ µ m whr is th Krönckr symbol. Applying th Eulr-Lagrang quation for = : @L @ = P i µ @ µ m @L @(@ µ ) = ) X i µ @ µ m = on has to us anticommutator in cas of frmions as in th prvious chaptr. P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
66 A brif viw of Quantum Elctrodynamic Sinc this quation is valid for th 4 componnts, w do rcovr th Dirac s quation: i µ @ µ m =. Now, applying th Eulr-Lagrang quation for : @L @ = P m ) @L @(@ µ ) = P i µ m X @ µ i µ = whr putting togthr th 4 quations ( pr componnt), w find i@ µ µ +m =, th adjoint quation.5. 3..3 Fr photon Lagrangian As sn in th prvious chaptr, th fr photon obys th wav quation (.7) namly A µ @ µ (@ A ) = (bfor xploiting th gaug invarianc). Th corrsponding Lagrangian is thn: L = 4 F µ F µ (3.8) whr F µ is th usual lctromagntic tnsor. Lt us chck this by applying th Eulr-Lagrang quation. Rwriting first th Lagrangian: L = = = 4 (@ µa @ A µ )F µ = 4 (@ µa F µ + @ A µ F µ )= @ µa F µ @ µa (@ µ A @ A µ ) gµ g @ µ A (@ A @ A ) whr in th first lin w usd succssivly F µ = F µ and µ $. Thus: @L = @(@ µ A ) gµ g (@ A @ A ) gµ g @(@ A ) @ µ A @(@ µ A ) = (@µ A @ A µ n o ) @ A µ µ = (@µ A @ A µ ) {@µ A @ A µ } = (@ µ A @ A µ ) @(@ A ) @(@ µ A ) th dnoting th Krönckr symbol ( µ =ifµ = and othrwis.).th Eulr-Lagrang quation is thn: @L @L @ µ @A @(@ µ A ) =) +@ µ@ µ A @ µ (@ A µ )= A @ µ (@ A µ )= which is th wav quation of a fr photon. 3..4 Gaug invarianc consquncs In 97, Emmy Nothr publishd a thorm that stats that vry continuous symmtry yilds a consrvation law, and convrsly vry consrvation law is a sign of an undrlying symmtry. On of th famous consquncs of hr thorm is th fact that th momntum is consrvd for a systm which is invariant undr translations in spac. Similarly, for th consrvation of th angular momntum and th invarianc undr rotations. P.Paganini Ecol Polytchniqu Physiqu ds particuls
Gaug invarianc consquncs 67 Global phas invarianc: of th fild: whr q and Now, considr th Dirac s Lagrangian 3.7 and th transformation ar simpl ral numbrs. Th Lagrangian bcoms:! = iq (3.9) L D!L = +iq (i /@ m) iq = L D Thus, th Lagrangian is invariant undr this transformation. Such transformation has all th proprtis of a symmtry namly ) th product of transformations is also a transformation, ) thr xists an idntity transformation, 3) thr xists an invrs transformation and 4) thr is associativity: if R,,3 ar 3 transformations, R (R R 3 )=(R R )R 3. Th st of transformations 3.9 forms a group calld U() and th proprtis just mntiond ar actually th ons of a group. Morovr, w s that th ordr of transformations dosn t mattr, or in othr words, th U() group lmnts commut. Such group is calld Ablian. According to Nothr s thorm, thr must b a consrvation law. Considr an U() infinitsimal transformation of paramtr : = iq and imposing th Lagrangian to rmain unchangd L = L = @L @ + @L @(@ µ ) (@ µ )+ @L @ + (@ µ ) = @L @ iq @L @(@ µ ) iq @ µ + iq @L nh @L = iq @L @ @ @L µ @(@ µ ) @L @(@ µ ) + @ µ @L @(@ µ ) i @ + iq @ µ @L h @L @ @(@ µ ) @µ @L @(@ µ ) io + @ µ @L @(@ µ ) Th first two trms in th two brakts cancl bcaus of th Eulr-Lagrang quation applid to and, so that: n o @L = @ µ iq @L @L @(@ µ ) = @(@ µ ) and hnc th currnt j µ = iq @L @L @(@ µ ) is consrvd. Using th Dirac s Lagrangian @(@ µ ) @L 3.7, @(@ µ ) = i µ @L and =, thus w finally hav @(@ µ ) j µ = q µ (3.) which is th charg-currnt w introducd in quation.8! Th continuity quation @ µ j µ thn mans that th charg Q = R d 3 xj must b consrvd. Local phas invarianc: Now, lt us assum that th phas in 3.9 is a ral function dpnding on th spactim coordinats: How is th Dirac s Lagrangian transformd?! = iq (x) (3.) iq (x) L D!L = +iq (x) (i /@ m) = +iq (x) i µ ( iq@ µ (x)) iq (x) + iq (x) @ µ m = L D + q µ @ µ (x) = L D + j µ @ µ (x) P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
68 A brif viw of Quantum Elctrodynamic If w wish to hav a Lagrangian invariant undr this local transformation, w nd to add an xtra trm L int to th Dirac Lagrangian: L int = j µ A µ = q µ A µ (3.) which will compnsat th trm j µ @ µ (x). L int coupls th vctor A µ to th charg-currnt of th Dirac s fild j µ. Lt us s how th modifid Lagrangian now transforms: But j µ = q L D + L int!l = L D + j µ @ µ (x) j µ A µ µ = q µ = j µ,thus: If A µ transforms as: L D + L int!l = L D + j µ @ µ (x) A µ A µ! A µ = A µ + @ µ (x) (3.3) th nw Lagrangian L D + L int would b invariant. W hav alrady ncountrd th abov transformation: it is th sam as th on allowd by th gaug invarianc of th lctromagntic fild.73! 3..5 QED Lagrangian Lt us rcap: promoting th fr Dirac Lagrangian to b invariant undr a local phas transformation U(), rquirs an additional trm that coupls a vctor fild A µ, consistnt with th photon fild (having th sam transformations), to th lctron charg-currnt. Furthrmor, if w us th corrspondnc principl.46 as in th cas of th Dirac s quation in prsnc of an lctromagntic fild: @ µ! D µ = @ µ + iqa µ (3.4) th Dirac Lagrangian bcoms: L D! (i µ D µ m)!l D q µ A µ!l D + L int (3.5) whr L int is th sam as in 3.! D µ is calld th covariant drivativ. Conclusion: th local gaug invarianc rquirmnt, i th local phas invarianc, includd in th fr lctrons thory, has gnratd a nw fild, th photon, that coupls to th lctron, through th Lagrangian L int. This is an xampl of th gaug principl which lvats a global symmtry (U() hr), into a local on. Th gaug principl rquirs to rplac th normal drivativ by th covariant drivativ. W can now spcify th full QED Lagrangian: L = L D + L + L int = (i/@ m) 4 F µ F µ q µ A µ (3.6) It is th sum of th fr lctron Lagrangian (i th Dirac Lagrangian), th fr photon Lagrangian, and th intraction Lagrangian btwn lctrons and photons. Th rcip to obtain th QED Lagrangian is to start from th fr Lagrangians and rplac th drivativ by th covariant drivativ. Doing so, w obtain a QED Lagrangian that is locally U() gaug invariant 3 : th fild transformation (3.) bing absorbd by th A µ transformation (3.3). For lctrons q = has to b usd. 3 Th trm F µ bing obviously invariant undr transformation 3.3. P.Paganini Ecol Polytchniqu Physiqu ds particuls
Prturbation with intracting filds: th S-matrix 69 3. Prturbation with intracting filds: th S-matrix W saw in th prvious sction that lctron fild and photon fild intract. W thn xpct procsss whr for xampl, th lctron will b scattrd by a photon. Hnc, w nd to b abl to calculat masurabl quantitis as th cross-sction using tim-dpndnt prturbation thory as xposd in sction.6. This sction aims at making a connction btwn th filds and th matrix lmnt apparing in formulas such as.6. 3.. Schrödingr, Hisnbrg and Intraction rprsntations In th wll known Schrödingr rprnsntation, th tim volution of a systm is givn by: i @ @t s(t) =H s (t) (3.7) whr w us th subscript s to clarly rmind that w work in th Schrödingr rprsntation. If th hamiltonian H dosn t dpnd on tim, s can b computd for any tim t providd th stat is known at anothr tim lt us say t =via 4 : s(t) =U S (t, ) s () = ith s() (3.8) whr U S is unitary (U S = U S )sinch is hrmitian. Thus, th stat vctors carry th ntir tim dpndanc. Th fild can b xpandd in trms of cration and annihilation oprators at t =. Howvr, th problm with this rprsntation, is this particular tim: it is not manifstly Lorntz invariant. Now, not that th matrix lmnt of an oprator A btwn stat s and s can b stimatd at th tim t with: Z Z h s A s i = s (t) A s (t) = s () US (t, ) AU S (t, ) s () Now, this xprssion can b sn di rntly, with an oprator A H dpnding on tim: A H (t) =U S (t, ) AU S (t, ) and stats H = s (), H = s() without tim dpndanc, such as: h s A s i = h H A H H i This nw rprsntation is th Hisnbrg on (thus th subscript H) whr oprators dpnd on tim and stats do not. Th tim-dpndncy is ntirly transfrrd to oprators via th Hisnbrg quation: i d dt A H(t) =[A H (t),h] With this rprsntation, th volution of th cration and annihilation oprators apparing in th fild xpansion, is givn by th prvious quality, which rquirs th usag of th full hamiltonian. Whn this hamiltonian is th on of fr particls, w know how to do it. For mor complicatd cass, th full hamiltonian is not ncssarily known. Not that th Hamiltonian H 4 Th xponntial mans ith = P k= ( ith)k /k! P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
7 A brif viw of Quantum Elctrodynamic is idntical in th rprsntations sinc it commuts with itslf. Also, not that H can b writtn as: H = U S (t, ) s(t) = ith s(t) Finally, thr is th Intraction rprsntation which is a hybrid of th two prvious rprsntations. It is wll adaptd for th prturbativ calculations. In this rprsntation, both th stats and th oprators hav a tim dpndncy. Considr an hamiltonian: H = H + H int whr H dscribs a fr systm and H int th prturbation sourc of intractions. Intraction pictur, th stat I is dfind as: In th I(t) = ith s (t) (3.9) Notic th H instad of H as in th Hisnbrg rprsntation. Sinc (t) obys to 3.7, it is asy to conclud that I (t) obys to: i @ @t s(t) =H s (t) ) i @ @t ( ith I (t)) = (H + H int ) ith I (t) ) i d dt I(t) = ith H int ith I (t) =H I int (t) I (t) (3.) with: H I int (t) = ith H int ith Similarly as for th Hamiltonian, any oprator in th Intraction pictur can b xprssd from th corrsponding oprator in th Schrödingr pictur as: A I (t) = ith A ith (3.) which mans that its tim volution satisfis th Hisnbrg quation: i d dt A I(t) =[A I (t),h ] (3.) with H rplacd by H. Th fild oprators satisfy to this quation which is valid vn in th prsnc of an intraction. In othr words, th fild can still b xprssd as a linar combination of th fr fild solution. 3.. Dyson xpansion W wish to find th volution oprator U which allows to calculat th stat at any tim t knowing th stat at a givn tim t : I(t) =U(t, t ) I (t ) (3.3) Thus: but I also satisfis: d dt i d dt I(t) = du(t, t ) dt I(t ) I(t) =H I int (t) I (t) (3.4) P.Paganini Ecol Polytchniqu Physiqu ds particuls
Dyson xpansion 7 so that: and hnc: i du(t, t ) dt I(t )=H I int (t) I (t) =H I int (t) U(t, t ) I (t ) i d dt U(t, t )=H I int (t) U(t, t ) (3.5) From now on, w ar going to us rducd notation by forgtting th subscript I rfrring to th Intraction rprsntation. Equation 3.5 can b solvd by an itrativ procdur: U(t, t )=U(t,t ) But U(t,t ) is by dfinition (s 3.3), so that: i Z t t dt H int (t ) U(t,t ) U(t, t ) = i R t t dt H int (t ) U(t,t ) = i R h t t dt H int (t ) i R i t t dt H int (t ) U(t,t ) = i R t t dt H int (t )+( i) R t R t t dt t dt H int (t ) H int (t ) U(t,t ) Pursuing th sam approach at highr ordr, w finally gt: U(t, t )= X Z t ( i) n n= t dt Z t t dt Z tn t dt n H int (t ) H int (t ) H int (t n ) Now, w can r-arrang th intgrals to hav th sam rang of intgration [t,t]. Considr th intgrals of th scond ordr trm: Z t t dt Z t t dt H int (t ) H int (t )= Z t t dt Z t t dt (t t ) H int (t ) H int (t ) whr is th usual stp function, so th implicit condition t >t is imposd. Now, splitting in and changing th ordr of th intgrals in th scond trm, it coms: R t R t t dt t dt H int (t ) H int (t ) = R t R t t dt t dt (t t ) H int (t ) H int (t )+ R t R t t dt t dt (t t ) H int (t ) H int (t ) = R t R t t dt t dt (t t ) H int (t ) H int (t )+ R t R t t dt t dt (t t ) H int (t ) H int (t ) = R t R t t dt t dt (t t ) H int (t ) H int (t )+ (t t ) H int (t ) H int (t ) (3.6) whr in th last lin, th intgration labls wr intrchangd. Th trick now is to introduc th T -product (or tim-ordrd product) of oprators dfind as: T (H int (t ) H int (t )) = H int (t ) H int (t ) if t >t H int (t ) H int (t ) if t >t = H int (t ) H int (t ) (t t )+H int (t ) H int (t ) (t t ) (3.7) which for th product of n oprators A rads: T (A(t ) A(t ) A(t n )) = A(t i ) A(t i ) A(t in ) with t i >t i > >t in (3.8) Hnc, 3.6 bcoms: Z t Z t Z t Z t dt dt H int (t ) H int (t )= dt dt T (H int (t ) H int (t )) t t t t P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
7 A brif viw of Quantum Elctrodynamic For th highr ordr trms, on has to tak into account th numbr of prmutations in th T -product, and th final formula is: U(t, t )= X ( i) n n= n! Z t t dt Z t t dt Z t which can b simply writtn in th symbolic form: t T (H int (t ) H int (t ) H int (t n )) (3.9) U(t, t )=T i R t t d H int ( ) (3.3) whr again, H int is undrstood hr in th intraction rprsntation. Now lt us considr th following scattring procss: long bfor th intraction, say at t!, H int is ngligibl and according to 3.4, is a stady stat in th Intraction rprsntation. That s on of th main intrst of this rprsntation whr th kts volv with tim only whn thr is an intraction. Lt us dnot this initial stat ii = ( )i. Exprimntally, ii is typically th particls that ar supposd to collid in an acclrator. As tim gos by, th particls dscribd by ii may scattr and H int is not ngligibl anymor. Th tim volution of is thn givn by: (t) =U(t, ) ( ) Long aftr th intraction, for th sam rason, th systm potntially constitutd of many particls rsulting of th collision, is in anothr stady stat (+)i. Th amplitud of th probability for finding a givn stat fi is thn: Th matrix: S fi = hf (+)i = hf U(+, ) ( )i = hf U(+, ) ii S = U(+, ) =T i R + d H int( ) (3.3) is calld th scattring matrix (S-matrix). Its lmnt S fi givs th amplitud probability for having a transition i! f. According to 3.9 th S-matrix xprssion is thus: S = X S [n] (3.3) n= whr th n th ordr trm is: S [n] = ( i)n n! Z Z dt dt n T (H int (t ) H int (t n )) (3.33) 3..3 Connction with th gnral formula of transition rat W considr an initial stat ii mad of svral particls with dfinit momnta and similarly for th final stat fi. ii and fi ar th stats corrsponding to th asymptotic conditions at t = and t =+. Th probability of a transition from ii to fi is thn givn by: P i!f = hf S ii hi iihf fi (3.34) If thr is no raction at all, w must hav S = l (th idntity oprator), and i = f and so w do hav P =. Thrfor, w xpct S to b writtn as S =l + whr th dots ar an xprssion P.Paganini Ecol Polytchniqu Physiqu ds particuls
Connction with th gnral formula of transition rat 73 of th oprators for a non trivial intraction. W do not nd to hav an xplicit formulation of ths oprators (which could b horribly complicatd), only th matrix lmnt hf S ii for th initial and final stats nvisagd is rquird. How can w writ it? It is convnint to factor out a dlta function implmnting th 4-momntum consrvation 5 to dfin th Lorntz invariant scattring amplitud, M fi : hf S ii = hf ii +( ) 4 (4) (P f P i ) im fi whr P f and P i ar rspctivly th final and initial total momnta. Th factor ( ) 4 is convnintly factorizd so that im fi will b th rsult of a Fynman diagram calculation. Sinc w ar intrstd in non trivial intractions, w hav: hf S ii =( ) 4 (4) (P f P i ) im fi, i 6= f (3.35) Now lt us mak xplicit th trms of (3.34). Using th commutation ruls for bosons.4 (but w would find th sam rsult with frmions), w notic that: hp pi = h a p a p i = h a pa p i +( ) 3 E p (3) ( ~ p ~p )=( ) 3 E p (3) ( ~ p ~p ) whr a is th cration oprator of th corrsponding particls (boson or frmion). This rsult is known as th rlativistic normalization of momnta stats. Howvr, that mans that if w considr for simplicity an initial stat with just on particl with momntum p i,whav: hi ii =( ) 3 E pi (3) () which is an infinit quantity! If instad of plan wavs (with dfinit momnta) w had considrd localizd wav packts for th asymptotic initial and final stats, th rsult would hav bn finit. Th problm appars hr bcaus w implicitly us plan wavs in an infinit volum (th whol 3-D spac). Indd, according to formula.69, w hav: with (3) (p) = Z ( ) 3 ip.x d 3 x = lim L! L (p) L(p) = R L/ ( ) 3 L/ dx R ipxx L/ L/ dy R ipyy L/ L/ dz ipzz = L sin(pxl/) ( ) 3 p xl/ L sin(pyl/) p yl/ L sin(pzl/) p zl/ = V sin(p xl/) sin(p yl/) sin(p zl/) ( ) 3 p xl/ p yl/ p zl/ whr V = L 3 is th volum of 3-spac (a box with sids of lngth L). W conclud that L() V/( ) 3 sin x (sinc lim x! x = ) is quivalnt to (3) () in th limit V!and hnc hi ii is quivalnt to E pi V. W can thn do all our calculations with a finit volum V, and for physics calculations, th volum should cancl out so that th infinit volum limit can b takn safly. It is straightforward to gnraliz to n particls in th initial stat: hi ii = ny E k V = V n k= 5 All intractions ar assumd to consrv th 4-momnta. n Y k= E k P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
74 A brif viw of Quantum Elctrodynamic and similarly for n particls in th final stats: hf fi = V n Th last trm in 3.34 to b dtrmind is hf S ii. According to 3.35, it is: Y n k= E k hf S ii =( ) 4 (4) (P F P I )( ) 4 (4) (P F P I ) M fi Hr, w hav to valuat th squar of th dlta-function (4) (P F P I ). Bing applid twic, th first dlta-function imposs P F = P I into th scond and hnc (4) (P F P I ) (4) (P F P I )= (4) (P F P I ) (4) (). Th qustion is thus to stimat (4) () = (3) () () = V Procding as for our stimation of (E) = Z ( ) and hnc w conclud () T ( ) (3) (), w hav: ie.t dt = Z T/ ( ) lim dt iet = T! T/ ( ) lim T sin(et/) T! ET/ lading to: hf S ii =( ) 4 (4) (P F P I )VT M fi Putting all th pics togthr, w finally stablish: P i!f = V n n T ( ) 4 (4) (P F P I ) M fi n Y k= E k n Y E k= k or mor appropriatly in trms of transition rat (i.. probability pr unit of tim): i!f = V n n ( ) 4 (4) (P F P I ) M fi n Y k= E k n Y E k= k ( ) 3 (). Howvr, w forgot somthing: th final stats hav a continuous nrgy spctrum. Th particls in th final stat hav not an infinitly accurat momnta/nrgis but should rathr b dscribd by a st of final stats blonging to a givn phas spac lmnt (i.. blonging to p x ± p x,p y ± p y,p z ± p z ). Lt us considr an infinitsimal phas spac lmnt so that w can assum that M fi rmains constant for ths dn stats. Th infinitsimal transition probability pr unit tim is thn th sum of th probabilitis of ach individual stats: d i!f = V n n ( ) 4 (4) (P F P I ) M fi n Y k= E k n Y E k= k As wll known from quantum mchanics, in cas of a singl particl in a box of sids L, th componnts of its momntum ar quantizd as: p x,y,z =( /L) n x,y,z, and thrfor th numbr of stats in d 3 ~p = dp x dp y dp z is dn = dn x dn y dn z = Vd 3 ~p / ( ) 3. Thrfor for n particls in th final stat, dn bcoms dn = Q n k= Vd3 ~ p k /( )3 and thus:! dn d i!f = V n ( ) 4 (4) (P F P I ) M fi n Y k= E k n Y k= d 3 ~ p k ( ) 3 E k (3.36) P.Paganini Ecol Polytchniqu Physiqu ds particuls
Fynman ruls and diagrams 75 which is xactly th formula.6 w usd in th first chaptr to stablish th formulas of th dcay rat and cross-sction whr, as it should, th volum V cancls out. W hav sn in quation (3.35) that th matrix lmnt M fi is rlatd to hf S ii dfind at a ordr [n] by (3.33). Hnc: ( ) 4 (4) (P f P i ) im [n] fi = ( i)n n! Z dt dt n T (hf H int (t ) H int (t n ) ii) In ordr to mak xplicit th Lorntz invarianc, it is bttr to us th hamiltonian dnsity H (H = R d 3 xh(x)). Th final formula for th matrix lmnts thn bcoms: ( ) 4 (4) (P f P i ) im [n] fi = ( i)n n! Z dx dx n T (hf H int (x ) H int (x n ) ii) (3.37) whr, onc again, H int is th intraction hamiltonian givn in th Intraction rprsntation (maning that th product of filds constituting th hamiltonian ar thmslvs givn in th intraction rprsntation). 3.3 Fynman ruls and diagrams Lt us considr th Lagrangian of QED 3.6. W wish to find th xprssion of H int knowing that according to 3.4 and 3.3: H = @L (x µ ) L (3.38) @ (x µ ) Filds drivativ in 3.6 only ntrs in th two fr Lagrangians L D and L. Hnc, H int rducs to: H int = L int = q µ A µ (3.39) 3.3. Elctron-photon vrtx Considr th first ordr whr q has to b undrstood as th charg of th particl, not th on of th antiparticl 6 : Z Z S [] = i d 4 x H int (x) = iq d 4 x µ A µ Now, injct th 3 filds dvlopmnt for (.67), (.68), and A µ (.76): d 3 ~p d 3 ~q ( ) 3 E p ( ) 3 E p ( ) 3 E q Pr=, P r =, P 3 = S [] = iq R d 4 x R d 3 ~p i h i hc ~p r ūrp +ipx + d ~p r v rp ipx µ c ~p r u r p ipx + d ~p r v r p +ip x i h µq ~q iq.x + µq ~q +iq.x 6 For lctron-positron, q =. P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
76 A brif viw of Quantum Elctrodynamic Intgrating ovr x (using.69) and rarranging, w finally obtain: S [] = ( ) 4 R d 3 ~p d 3 ~p d 3 ~q ( ) 3 E p ( ) 3 E p c ~p r c ~p r ~q (ū rp ( iq µ ) u r p ) µq ( ) 3 E q Pr=, c ~p r d ~p r ~q (ū rp ( iq µ ) v r p ) µq d ~p r c ~p r ~q ( v rp ( iq µ ) u r p ) µq d ~p r d ~p r ~q ( v rp ( iq µ ) v r p ) µq c ~p r c ~p r ~q (ū rp ( iq µ ) u r p ) µq c ~p r d ~p r ~q (ū rp ( iq µ ) v r p ) µq d ~p r c ~p r ~q ( v rp ( iq µ ) u r p ) µq d ~p r d ~p r ~q ( v rp ( iq µ ) v r p ) µq P P 3 r =, = { (4) (p p q)+ (4) (p + p q)+ (4) (p + p + q)+ (4) (p p q)+ (4) (p p + q)+ (4) (p + p + q)+ (4) (q p p )+ (4) (p p + q) } (3.4) Thus, S [] is constitutd of 8 trms 7 having a similar form: product of 3 oprators, a chargcurrnt (product of an adjoint-spinor and a spinor sandwiching a µ matrix), a photon polarisation vctor and a dlta function. Looking at th oprators, on notics that th total charg is always zro: for xampl th first lin crats an lctron, annihilats an lctron and annihilats a photon, whil th scond crats an lctron and a positron and annihilats a photon. Lt us dtaild th first lin as an xampl. Clarly, it givs a contribution only if it is sandwichd btwn an initial stat containing a photon and an lctron i, i (that ar annihilatd) and a final stat containing an lctron f i (that is cratd): h f c ~p r c ~p r ~q i, i = h c ~p f r f c ~p r c ~p r ~q c ~p i r i ~q i = h c ~p f r f c ~p r c ~p r c ~p i r i ~q ~q i = h c ~p r c ~p f r f +( ) 3 E (3) f (~p f ~p ) rf r c ~p i r i c ~p r +( )3 E (3) i (~p i ~p ) ri r ~q ~q g ( ) 3 E (3) (~q ~q ) i = ( ) 3 E f (3) (~p f ~p ) rf r( ) 3 E i (3) (~p i ~p ) ri r g ( ) 3 E (3) (~q ~q ) h i Insrting this rsult into th S [] xprssion and prforming th intgrations and summations, w gt a trm 8 ( ) 4 (4) (p f p i q ) ū rf pf ( iq µ ) u ripi µq (3.4) Comparing with 3.37 w conclud in that cas: im = ū rf p f ( iq µ ) u ri p i µq (3.4) Th cration of th final lctron is thus associatd to an adjoint spinor ū whil th annihilation of th initial lctron is associatd a spinor u. Similarly, th annihilation of th photon is associatd to a vctor polarization. This yilds our first Fynman diagram shown on figur 3.. Th arrow of tim on th horizontal axis is going from th lft to th right whil th spacial dimnsion is vrtical. Th photon is symbolizd by a wavy lin, and th lctron (actually any frmion) by a solid lin. Th intrsction of th lins is calld a vrtx. Th dirction of th 7 3 filds with oprators giv 3 =8combinations. 8 Ral photons hav only transvrs polarisation, i =or. Thusg =. P.Paganini Ecol Polytchniqu Physiqu ds particuls
Elctron-photon vrtx 77 Figur 3.: A basic vrtx in QED. arrow on th lctron lin has a maning: for lctrons in th initial stat, it gos toward th vrtx whil for lctrons in th final stat it points away from th vrtx. In othr words, th arrow of a frmion is always orintd from th past to th futur. Th dvlopmnt w mad to th first ordr of prturbation lads to diagrams with only vrtx. Th numbr of vrtics rprsnts th ordr of th prturbation dvlopmnt. Th 7 othr trms of S [] dvlopmnt yild similar kind of diagrams. Thy ar all shown on th figur 3.. Positrons (or any antifrmions) ar rprsntd by th sam lin as lctrons. What di rs Figur 3.: Th vrtics of QED at first ordr of prturbation. is only th dirction of th arrow. For xampl, th scond diagram on th first row, dpicts a photon that annihilats in lctron (lin going up) and positron. Not that for th outgoing positron, th arrow gos back in tim. Similarly, in th fourth diagram of th first row, th incoming particl is a positron, th arrow going back in tim. On th photon, thr is no arrow sinc it is its own antiparticl. According to 3.4, following th sam procdur that yilds 3.4, w s that antifrmions in th initial stat ar associatd to adjoint antispinor v and antispinor v in th final stat. For photons, a polarisation vctor µ is associatd to an incoming photons and µ for an outgoing on. Looking back to formula 3.4 and 3.4 w s that all trms contain a iq µ contribution sandwichd by (anti-)spinors. This factor is logically associatd to th vrtx itslf in th diagram. Th charg q = is th coupling btwn lctrons or positrons and photons. Dvlopmnt at nth ordr would produc n vrtics, lading to a factor n in th amplitud. Th convrgnc of th prturbation sris dpnds on, which finally rprsnts th strngth of th prturbation. In natural units ' p 4 /37 '.3. Finally, Thr is a last trm that w havn t xploitd yt in 3.4: th dlta function. It clarly imposs th nrgy-momntum consrvation at th vrtx. Wll, it appars that non P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
78 A brif viw of Quantum Elctrodynamic of ths 8 trms ar physically possibl with fr particls! An obvious on: th third on th first row of figur 3.. Th nrgy consrvation imposd by (4) (p + p + q) mans that E + E + E + =) E = E = E + = which is impossibl (thr is at last th rst nrgy sinc lctrons ar massiv). Evn, mor convntional diagrams as th first on, ar impossibl. Both nrgy consrvation and momntum consrvation can t b satisfid at th sam tim (bcaus th photon mass is ). Th only possibility would b to nvisag that on of ths particls dos not hav its normal mass valu (or mor appropriatly that it dos not rspct th rlativistic rlation E = p + m ). Such a particl is calld a virtual particl and is allowd by th uncrtainty inhrnt in quantum mchanics: E ~/T Th rlation can b intrprtd as follow: th gratr th nrgy (or mass) is shiftd from its nominal valu, th shortr th particl will liv. In othr words, virtual particls (thus not satisfying E = m + p with m th usual mass), must liv a short tim: thy hav to b quickly rabsorbd (annihilatd) by anothr procss 9. Conclusion: anothr vrtx is ndd and on has to go to th scond ordr of th prturbation dvlopmnt. 3.3. Photon and lctron propagators Considr th scond ordr of prturbation: S [] = ( Z i) q d 4 xd 4 yt( (x) µ (x)a µ (x) (y) (y)a (y)) (3.43) If w dvlop all quantitis as in th prvious sction w hav 8 = 64 trms, and thus 64 diagrams. Two such diagrams ar shown in figur 3.3. In th first diagram, w s that a Figur 3.3: Exampl of QED diagrams at th scond ordr. (virtual) photon is cratd at th spactim point x (assuming t x <t y ) and annihilatd at y. Similarly, th diagram on th right-hand sid shows a virtual lctron bing xchangd. W can d-coupl th initial and final stats and considr that a virtual particl must b abl to b cratd from th vacuum and annihilatd. Looking at th formula 3.43, w s that, for th photon, th product A µ (x)a (y) has to b involvd. Indd, dvlopd in trms of cration-annihilation oprators, th combination ~q (x) ~q (y) appars, which will giv a nonzro contribution whn sandwichd with th vacuum i. Similarly, for th frmions, th only product giving non-zro trm involvd c ~p r (x)c ~p r (y) or d ~p r (x)d ~p r (y) which originats only from 9 In fact, all particls can b considrd as virtual: for instanc a photon mittd by a star is dtctd on arth by a photomultiplir which annihilats th photon. Its dgr of virtuality is howvr ridiculously tiny to b abl to liv during million yars! P.Paganini Ecol Polytchniqu Physiqu ds particuls
Summary of QED Fynman ruls 79 (x) (y). Conclusion: th two quantitis in 3.43 that can dscrib th propagation of a virtual particl from a point x to a point y (i its cration followd by its annihilation) ar: h T (A µ (x)a (y)) i, h T ( (x) (y)) i (3.44) Thy ar calld propagators (rspctivly of th photon and a frmion). Th calculation (not so complicatd) can b found in any quantum fild thory book (s for instanc [8, p. 6]). W simply giv hr th rsults in th momntum spac (i.. aftr having intgratd on all possibl position for x and y): Photon propagator: ig µ p + i (3.45) whr p is th 4-momntum of th (virtual) photon and is an infinitsimal ral positiv numbr to avoid singularitis whn p!. (In practic, is droppd). This formula corrsponds to a particular choic of gaug calld Fynman gaug. For th lctron, w hav: spin / propagator: i p /p + m m + i (3.46) whr again, p is th 4-momntum of th (virtual) frmion and m its mass (i th nominal mass usd in E = ~p + m ). Rmmbr that /p = µ p µ is a matrix. Hnc, diagrams, as th first of figur 3.43 corrspond mathmatically to an amplitud: im = v r + p + ( iq µ ig µ ) u r p p ū + i r p ( iq ) v r + p + (3.47) whr th first charg currnt j µ = v r + p + (q µ ) u r p corrsponds to th first prturbation trm btwn th lctron, positron and th virtual photon, ig µ /(p + i ) corrsponds to th propagation of th virtual photon, and th scond charg-currnt j = ū r (q ) v p r + p to th scond prturbation trm btwn th lctron, positron and th virtual photon, which + annihilats th virtual photon. Bfor concluding this sction, lt us xamin th lft diagram of figur 3.4. At first sight, it sms hard to intrprt in trm of a particl bing cratd and thn absorbd sinc both vrtics ar at th sam tim (tim gos from lft to right hr). But, considr th othr diagrams in 3.4. Whn t <t, in th cntr diagram, a virtual positron (= an lctron moving backward in tim on th diagram) is first cratd and thn absorbd whras whn t >t, in th right hand sid diagram, a virtual lctron is first cratd and thn absorbd. Sinc, in th dfinition of th propagators 3.44, thr is a T-product of fild oprators, th Fynman diagram on th lft-hand sid, includs th possibilitis dpictd by th cntr and right picturs! Both tim-ordrings ar always includd in ach propagator lin! It maks prfctly sns, sinc thr is no way to distinguish xprimntally th two modalitis. 3.3.3 Summary of QED Fynman ruls In this sction, w wish to summaris th rcip to calculat th amplitud of a procss.. Draw, at a givn ordr, all diagrams dscribing th transition btwn an initial stat (in practic, or particls) and a final stat (can contain many particls). A QED vrtx must always connct (anti-)frmions and photon. For ach vrtx: Gnrally, th positions of particls in a raction ar not known but thir momnta ar. P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
8 A brif viw of Quantum Elctrodynamic γ "!" γ " "!" γ "!" +"!" "!" γ "!" γ "!" γ " Figur 3.4: Lft:! Fynman diagram. Cntr and right: th corrsponding tim-ordrd diagrams. Enrgy-momntum is consrvd. Charg of th incoming particl(s) is always qual to th charg of th outgoing particl(s). QED can b sn as a flow of chargs (that s why th charg-currnt is involvd).. Calculat im of ach diagram by using th factors givn in tabl 3.. Incoming and outgoing particls ar connctd to vrtics xchanging virtual particls. Th corrsponding factors in 3. ar rspctivly th ons of th Extrnal lins, Vrtx and Propagators. 3. Combin th amplituds of all diagrams to gt th total amplitud of th procss. If diagrams di r only by th xchang of xtrnal idntical frmions ( outgoing, or incoming, or incoming frmion and outgoing anti-frmion, or incoming anti-frmion and outgoing frmion), subtract th amplituds. add all othr amplituds. For compltnss, w also giv th factors for massiv spin boson intracting with photons. Th procdur is similar to th frmion cas: from th wav quation, th fr Lagrangian is dducd, allowing th quantization of filds. Th propagator can thn b calculatd. Th intraction trm is thn obtaind, by rplacing in th Lagrangian, th drivativs by covariant drivativs 3.4. Th vrtx factors and xtrnal lins factors ar dtrmind accordingly. 3.4 QED and hlicity/chirality W hav sn that QED intractions involv th charg currnt. So trms as qū µ u, q v µ v, qū µ v, q v µ u ar xpctd. Now, w saw in sction.3.4.3, that any spinor can b dcomposd in lft-handd and right-handd componnts of chirality through th projctors.6. Hnc, th currnt can b dcomposd: qū µ u = q(ū L +ū R ) µ (u L + u R ) = qū L µ u L + qū R µ u R + qū L µ u R + qū R µ u L This can b undrstood as a consqunc of th fact that whn idntical frmions ar intrchangd, th wav function gts a minus sign. It is clarly rlatd to th spin-statistic thorm. P.Paganini Ecol Polytchniqu Physiqu ds particuls
QED and hlicity/chirality 8 Extrnal lins Spin Boson incoming Boson outgoing anti Boson incoming anti Boson outgoing Spin Frmion incoming u spinor Frmion outgoing ū spinor anti Frmion incoming v spinor anti Frmion outgoing v spinor Spin Photon incoming µ Photon outgoing µ Propagators Spin Boson i p m +i Spin Frmion i /p+m p m +i Spin Photon ig µ p +i Vrtx Spin iq(p µ + p µ ) Spin iq µ q = for ± Tabl 3.: Fynman ruls in QED. But: ū L µ u R = u ( 5 ) µ ( + 5 )u = 4 u ( 5 ) µ ( + 5 )u = 4 u ( + 5 )( 5 ) µ u = P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
8 A brif viw of Quantum Elctrodynamic And similarly: ū L µ u R =ū R µ u L = v L µ v R = v R µ v L = ū L µ v L =ū R µ v R = v L µ u L = v R µ u R = (rmmbr that v L = P R v). Now, at high nrgy in th ultra-rlativistic rgim whr E m, th hlicity stats and chirality stats ar th sam (s sction.3.4.3). W can thn conclud that: In scattring procsss (t-channl): only th combinations: ū L µ u L +ū R µ u R or v L µ v L + v R µ v R giv non-zro contribution. Th hlicity of th particl aftr th scattring is th sam as th on bfor. Hlicity is consrvd in th ultra-rlativistic limit by QED (and chirality is always consrvd). In annihilation/pair cration procsss (s-channl): only th combinations: ū L µ v R +ū R µ v L or v L µ u R + v R µ u L giv non-zro contribution. Th hlicity of th particl is th opposit of th on of th antiparticl. In our jargon, w r still talking about hlicity consrvation in th sns that only spcific hlicity configurations ar possibl. Hnc, in th cntr of mass fram of th pair, th spins of th particl and antiparticl must b alignd, lading to J z = ±. Th following figur summariz th situation. Figur 3.5: Th consrvation of Hlicity in QED. 3.5 Simpl xampls of graph calculation 3.5. Spin and polarisation summations: tracs thorms Considr th simplst QED graph µ! µ shown on figur 3.6. P.Paganini Ecol Polytchniqu Physiqu ds particuls
Spin and polarisation summations: tracs thorms 83 µ Figur 3.6: Lowst ordr diagram µ! µ µ Lt us dnot, p and k th 4-momnta of rspctivly and µ in th initial stat. And sam notation with a prim ( ) for th final stat. Th photon has a momntum q = p p = k k. Following Fynman ruls 3., th amplitud is: im =(ū k,r (i µ )u k,r ) ig µ q (ū p,s (i )u p,s )=i q (ū k,r µ u k,r )(ū p,s µu p,s ) Th cross sction formula is proportional to th probability of th procss and thus to M. Gnrally, th initial stat is not polarizd and th spin of th final stat is not masurd. Hnc, th masurd cross-sction is an avrag ovr th spins of th initial stat and a sum ovr th spins of th final stat. Lt us dnot: M = X s X X X r s r M = 4 q 4 Lµ (µ )L µ ( ) (3.48) with: Now, noting that: L µ (µ ) = Pr,r ūk,r µ u k,r ūk,r u k,r L µ ( ) = (3.49) Ps,s ūp,s µu p,s ūp,s u p,s (ū k,r µ u k,r ) = u k,r µ ū k,r = u k,r µ (u k,r ) =ū k,r µ u k,r L µ can b rwrittn: L µ (µ ) = Pr,r ūk,r µ u k,r ūk,r u k,r = P r ūk,r µ P r u k,r ū k,r u P k,r r ūk,r µ /k + m u k,r = whr w usd th compltnss rlation.66. Now, w writ xplicitly th matrix lmnt of L µ with labl, : L µ = P P =,, Pr ū( ) k,r h P r u( ) µ /k + m ( ) u( ) i µ k,rū( ) /k + m ( ) = P, [/k + m] ( ) µ /k + m ( ) = Tr (/k + m) µ /k + m On may wondr why w don t sum ovr th amplituds of th di rnt spin stats and tak th modulussquard of th sum. Th ky hr is that a particl with spin-up or spin-down dfins di rnt spin stats that ar in principl distinguishabl (vn if spin is not masurd). k,r P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
84 A brif viw of Quantum Elctrodynamic Tr dnoting th trac 3. W obtaind ths usful qualitis: L µ uu = Pr,r ūk,r µ u k,r ūk,r u k,r L µ vv = Pr,r vk,r µ v k,r vk,r v k,r = Tr (/k + m) µ /k + m = Tr (/k m) µ /k m (3.5) whr th rsult for anti-spinor was obtaind similarly. Following th sam approach, on can also show that: L µ vu = Pr,r vk,r µ u k,r vk,r u k,r = Tr (/k + m) µ /k m L µ uv = Pr,r ūk,r µ v k,r ūk,r v k,r = Tr (/k m) µ /k + m (3.5) Now rcall that: so that whn m = m as in our cas: Tr( A + B) = Tr(A)+ Tr(B), Tr(AB) =Tr(BA) L µ uu(m = m )= k k Tr( µ )+mk Tr( µ )+mk Tr( µ )+m Tr( µ ) (3.5) Thrfor, w nd to valuat tracs of products of matrics. Sinc, th s satisfis th Cli ord algbra.7 ({ µ, } = µ + µ =g µ ), l w conclud th following ruls: Tr( µ )=g µ Tr(l ) = 4g µ (3.53) In passing, w thus hav: Tr(/p/k) =4p.k (3.54) Now rcall 5 proprtis.55 and.58 (i ( 5 ) =, 5 µ = µ 5 ) Tr( µ µ n ) = Tr( µ µ n 5 5 ) =Tr( 5 µ µ n 5 ) =( ) n Tr( µ µ n 5 5 ) =( ) n Tr( µ µ n ) Hnc: Now, for 4 matrics: Tr(odd nb of ) = (3.55) Tr( µ ) = Tr( µ ) = Tr( µ )+g Tr( µ ) = Tr( µ )+8g g µ =Tr( µ ) 8g g µ +8g g µ = Tr( µ )+8g µ g 8g g µ +8g g µ Thus: Tr( µ ) = 4(g µ g g g µ + g g µ ) (3.56) 3 If A and B ar matrics, th product C = AB has lmnts C ij = P k A ikb kj and thus Tr(C) = P i Cii = P i,k A ikb ki. P.Paganini Ecol Polytchniqu Physiqu ds particuls
Elctron-muon scattring: µ! µ 85 In passing, w thus hav: Hnc, coming back to 3.5 so that: Tr( /p /p /p 3 /p 4 ) = 4(p.p p 3.p 4 p.p 3 p.p 4 + p.p 4 p.p 3 ) (3.57) L µ uu(m = m )= (4k k (g µ g g g µ + g g µ )+4m g µ ) L µ uu(m = m )=L µ vv (m = m ) = (k µ k + k k µ +(m k.k )g µ ) (3.58) For compltnss, w giv othr usful formulas: with µ = for µ,,, an vn prmutation of,, 3, 4, On can dduc th following proprtis of µ : µ µ =4 µ µ = µ µ =4g (3.59) µ µ = Tr( 5 µ )= Tr( 5 µ )= 4i µ (3.6) for odd prmutation, othrwis. µ = µ = µ = µ µ = µ = µ = µ µ µ = ( ) (3.6) Anothr quality which is worth mntioning is that for any 4-vctor p, whav/p = p µ p µ = p µ p (g µ µ )=p /p and thus: /p = p (3.6) Polarization P summation: for photons, thr is a similar summation as for frmions (i r u rū r = /k + m). Th rsult is [8, p. 59]: 3.5. Elctron-muon scattring: µ! µ 4X µ = g µ (3.63) = W can now complt th calculation of th amplitud of µ! µ. Coming back to 3.48 and insrting 3.58, w hav: M = 4 q 4 Lµ (µ )L µ ( ) L µ (µ )L µ ( ) = (k µ k + k k µ +(m µ k.k )g µ )(p µ p + p p µ +(m pp )g µ ) = 4 k.p k.p +k.p k.p + (m p.p )k.k + (m µ k.k )p.p + (m p.p )(m µ k.k )4 = 8 k.p k.p + k.p k.p m k.k m µp.p +m m µ (3.64) P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
86 A brif viw of Quantum Elctrodynamic Lt us considr th ultra-rlativistic limit nglcting trms with m and m µ. W can valuat th matrix lmnt in th cntr of mass fram and using th Mandlstam variabls: s =(k + p) =(k + p ) ' k.p ' k. p t =(p p ) =(k k) = q u =(p k ) =(p k) ' p.k ' p.k so that: M = 4 t (s + u ) (3.65) 3.5.3 Elctron-positron annihilation: +! µ + µ Th graph for (p) + (k)! µ + (k ) µ (p ) is shown in figur 3.7. Of cours, w can follow th µ + + Figur 3.7: Lowst ordr diagram +! µ + µ µ sam kind of calculation as in th prvious xampl, but w can do bttr using th crossing symmtry (and nglcting th masss). Comparing graph 3.7 and 3.6 w s that th t-channl of µ! µ is just th s-channl of +! µ + µ as illustratd in th figur 3.8. () (3) () ( ) µ + () ( ) µ + () µ (4) µ ( 3) + (4) µ ( 3) + Figur 3.8: Lft: th raction µ! µ considrd as th s-channl i.. s =(p + p ), t = (p p 3 ) u =(p p 4 ). Cntr: in ordr to obtain th t-channl of th lft diagram, on has to tak th opposit of p and p 3 (thus s $ t) and intrprt th rsult as anti-particls. Right: sam diagram as th cntr but rdraw in a mor convntional way (aftr horizontal strtching ). (4) µ Hnc, w just hav to rplac t $ s in th prvious matrix lmnt! That s a nic xampl of crossing. W gt: M!µ + µ = 4 s (t + u ) (3.66) Lt us go furthr by computing th di rntial cross-sction. Using.86, w hav: d d = 64 s 4 (t + u ) ~p f s ~p i P.Paganini Ecol Polytchniqu Physiqu ds particuls
Compton scattring:! 87 Insrting th fin structur constant 4 : = 4 (3.67) and dvloping th Mandlstam variabls in th cntr of mass fram whr ~p i = ~p f = p (bcaus th masss ar nglctd): s =(k + p) =4p t =(p k ) = p ( cos ) u =(p p ) = p ( + cos ) whr is th angl (in th cntr of mass fram) btwn in incidnt and th outgoing µ +. w hav: d d = (4p 4 ( cos ) +4p 4 (+cos ) s 6p 4 = 4s ( + cos ) Intgrating ovr th angl, w conclud: d d(cos ) = s ( + cos ) Exprimntally, th angl which is usd is gnrally th on btwn th particls,µ (which is qual to th on btwn +,µ + ). Thus! laving howvr unchangd th formula. Is this prdiction in agrmnt with th xprimntal data? Look at figur 3.9 on th top which shows th s d d cos distribution for th raction +! µ + µ. Th data wr collctd at an + collidr at p s = 9 GV (PEP collidr at SLAC in th 98 ). Th plot on th top is for +! µ + µ and th dashd lin dscribs QED prdictions but at highr ordr (but largly dominatd by th cos bhaviour). Comparing to th data point, th agrmnt is good but th solid lin givs a bttr fit. It corrsponds to QED and additional lctrowak corrctions whr +! µ + µ can occur via Z boson xchang. W will s this in th nxt chaptrs. To conclud this calculations, w can dtrmin th total cross-sction by intgration ovr cos : Z = d(cos ) s ( + cos )= 4 3s (3.68) which is in agrmnt with th data at th % lvl. 3.5.4 Compton scattring:! W ar going to calculat th cross-sction of th Compton scattring: (p) (k)! (p ) (k ) Thr ar diagrams, shown in figur 3., contributing to this procss at th lowst ordr: s-channl (lft-hand sid of 3.) and u-channl 5 (right-hand sid). Sinc photons ar bosons, 4 It s a dimnsionlss masur of th strngth of th lctromagntic intraction 5 It may not b obvious that th contributions ar s and u-channls. Lt us assign to th particls in (p) (k)! (p ) (k )thlabls()+()! (3) + (4). Thn th u-channl is obtaind by swapping () and (4) (namly th photons) in th lft-hand sid diagram of figur 3.. It givs th diagram on th right aftr strtching in th vrtical dirction. P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
th QED asymmtry is latd to b.% in Raction () and A, o =. 5/o in Raction (3). Systmatic rrors in th rlativ normalization of Ractions () and (3) to Raction () aris from A = 88 6 5 4 3 """= (a) 3 o b b (b) cross sctions, g ' =.3+.3. hav bn xpli.95 fittd cross s in Fig. (c). T (c) -. 75 primnts. FIG., Th diffrntial cross sctions for (a) Rac- hav ld 6x Th w computd in th valus quotd ur th product g ' only corrsp d d + + Figur 3.9: s d cos in (a), +! + in (b). In (c) th ratio ofwould vs cos for! µ µ d cos + + normalizd to th and Baction QED tion (b) () (3), with th xpctd QED contribution for th raction!. Dashd lin: QED contribution [] (c) Th ratio of th diffrntial cross sc- Th g(~3) QED prdiction. tion for Raction () to th xprimntal cross sction in (c) is normalizd to th lctrowak cross sction takn from a simultanous fit to all thr ractions. In (a) (c) th dashd lin corrsponds to th O(. ) QED cross sction and th solid lin to th fittd cross sction. th total amplitud is th sum of th two. Following thn Fynman ruls 3., th amplitud of th first diagram is: ) µ k i p / + k/ + m ( iq (p + k) m ) k up Thr is no ambiguity in th position of th di rnt trms in th quation: sinc th amplitud is a simpl complx numbr, th adjoint spinor has to b on th lft, and hnc th scond vrtx P.Paganini Ecol Polytchniqu '. ordr &' xcha mass dpndnc Figur 3.: Th lowst ordr compton diagrams µ '" mnt with th.75 cosa im = u p ( iq " If w assum +.6+. fro simultanous f into account th I. 5.9 prdiction. cr prdicts" A tions ar supr and (b). Th currnt is tst 4 8 ratio Ractions ( cross sction p maximum likli distributions, A brif viw of Quantum Elctrodynamic + =.3.7+ whr th first systmatic. ' T ground-, and Q lctrowak as full cosg intrv 4./o A, 5 Vl on th Physiqu ds particuls
Compton scattring:! 89 trms ar on th lft. Similarly, th scond amplitud is: im =ū p ( iq µ /p /k + m ) µk i (p k ) m ( iq ) k u p W can simplify a bit ths xprssions by using th fact that incoming/outgoing particls ar on shll so that w can us: p = p = m and k = k =. Hnc, th amplituds bcoms using q = : M = µk k ū p M = µk k ū p µ /p+/k+m p.k /p /k +m p.k whr w hav changd in M : µ $. Now, noting that /p = p = p ( +g )= /p +p and similarly /p µ = µ /p +p µ, it coms for th total amplitud: M = µk k ū p = µk k ū p = µk k ū p µ /p+p +/k + m p.k u p µ u p µ p +/k p µ /k µ p.k p.k u p µ p +/k + p µ /k µ s m u m u p µ /p+p µ /k µ + µ m p.k u p whr th idntity (/p m)u p = has bn usd in th scond lin for on-shll frmions and s =(p + k) = m +p.k ) p.k =(s m )/ and p.k = (u m )/ has bn usd in th last lin. W wish to calculat th un-polarizd cross-sction. Thus w hav to avrag ovr th transvrs polarizations of th incoming ral photon, and th spins of th lctron. And w hav to sum ovr th final polarization stats. Hnc: with: M = X =, X s X =, X M = 4 4(s m ) P h µk k ū p M = 4 4(u m ) P h µk k ū p M M = 4 4(s m ) (u m ) P h µk k ū p s M = M + M + M M + M M µ (p + /k )u p ih k k ū p (p µ + /k µ )u p ih k k ū p µ (p + /k )u p ih k k ū p (p + /k )u p i (p + /k )u p i (p + /k )u p i W ar just going to dtail th first calculation. No nd to calculat M sinc th scond graph is just th u-channl of th first: rplacing s $ u will giv th answr. For th intrfrnc trm, thr is no othr solution than doing th calculation (which is vry similar to th first trm). M = 4 4(s m ) X k k X X µk k s,s ūp µ (p + /k )u p ūp (p + /k )u p According to 3.63, P k k = g and P µk k = g µ and: ūp (p + /k )u p = u p(p + k ) up = u p(p + k ) u p = u p(p + /k ) u p =ū p (p + /k) u p P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
9 A brif viw of Quantum Elctrodynamic Hnc: M = = = = 4 P g 4(s m ) g µ s,s ūp 4 P 4(s m ) s,s ūp 4 µ (p + /k )u p ūp (p + /k) u p µ (p + /k )u p ūp (p + /k) µ u p Tr (/p + m) µ (p + /k )(/p + m)(p 4(s m ) + /k) µ 4 { Tr 4(s m ) /p µ (p + /k )/p(p + /k) µ +m Tr ( µ (p + /k )(p + /k) µ ) (using 3.55) Lt us considr th first trac: Tr /p µ (p + /k )/p(p + /k) µ = 4p Tr /p µ /p µ + Tr /p µ /k/p µ + Tr /p µ /p /k µ +Tr /p µ /k /p /k µ = 4m Tr /p µ /p µ +Tr /p µ /k µ +Tr /p µ /k /p /k µ = 4m Tr /p (p + k ) p Tr /p µ /k /k µ = 8m (4p (p + k )) + 4p Tr /p /k /k = 3m (p.p + p.k) + 4Tr /p /k/p/k = 3m (p.p + p.k) + 6(p.k p.k p.p k + p.k k.p) = 3( m p.p + p.k(p.k m )) whr w hav usd succssivly th proprtis /p = p = m, (3.59), (3.53), (3.57) and k =. Using a similar approach, on finds for th scond trac: Tr ( µ (p + /k )(p + /k) µ ) = 4p Tr( µ µ) + Tr( µ /p/k µ ) + Tr( µ /k/p µ )+Tr( µ /k /k µ ) = 6m Tr( ) l + 8p.kTr( ) l + 8k.pTr( ) l + 4Tr( µ /k µ) = 64m + 64p.k and thus, injcting th rsults of th two tracs: M = 4 4 (s m ) m p.p +p.k p.k m p.k +4m 4 +4m p.k Now xprssing th xprssion as function of th Mandlstam variabls s and u (rmmbring that s + t + u =m ): It finally coms: s =(p + k) = m +p.k ) p.k = s m t =(p p) =m p.p ) p.p = m t = s+u u =(k p) =(k p ) = m p.k ) p.k = m u appl M =4 4 m 4 (s m ) + m u m s m s m Th xprssion for M is obtaind by swapping s and u: appl M =4 4 m 4 (u m ) + m s m u m u m and for th intrfrnc (aftr a similar boring calculation): appl M M = 4 m u m + s m + 4m (s m )(u m ) P.Paganini Ecol Polytchniqu Physiqu ds particuls
Compton scattring:! 9 so that finally M = M + M + M M +(M M ) rads: M = 4 " u m s m + s m u m 4 m s m + m m!# u m + s m + m u m Lt us dnot p =(m,~), k =(!, ~ k), p =(E,~p ), k =(!, ~ k ) and th angl btwn ~ k and ~ k. W notic that: u = (p k ) = m m! ) u m = m! s = (p + k) ) s m =m! m = p =(p + k k ) = m k.k +p.k p.k = m!! ( cos )+m! m! )!! = m ( cos ) and hnc: h i M = 4!! +!! +4 m m!! +( m m!! ) h i = 4!! +!! + ( cos )+ ( cos ) = 4 h!! +!! sin i W can thn xprss th cross-sction using formula.76: d = 4 p (p.k) ( )4 (4) (p + k p k) M d 3 ~ p = 8m!( ) (4) (p + k p k) 4 h!! +!! sin h = (E +! m!) 4! 8m!( )! +!! sin h = 8m! (E +! m!) 4!! +!! sin d 3 k ~ ( ) 3 E ( ) 3! i d 3 p ~ d 3 k ~ i E!! d! d i E!! d! d(cos ) E R d 3~ p R d But: E = p m + ~p = q m +( ~ k ~ k ) = p m +! +!!! cos And thus: h i d d(cos ) =! 6m! 4! +!! sin! d! E ( p m +! +!!! cos +! m!) h =! 6m! 4! +!! sin h =! 6m! 4! +!! sin = 6m! 4 h!! +!! sin = 6m! 4 h!! +!! sin i! i i E!! cos E +!!! cos +E E = m +!!! m+!( cos ) cos = m(! i!! whr w usd th dlta proprty R + (g(x)) dx = P i @g @x (x i) to intgrat ovr!. Insrting =4, w obtain th formula:! ) appl d d(cos ) =! m! +!! sin!! P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
9 A brif viw of Quantum Elctrodynamic which is known as th Klin-Nishina formula. At low nrgy, whn!! (and thus!! = +! m ( cos )! ) d d(cos ) ' m sin = + cos m Z ) = + cos m = 8 3m In this xprssion, w rcogniz th classical radius of th lctron 6 r = /m = p 3 /(8 ) ' 3 3 cm (with ' 6 5 cm ). 3.6 Fw words about th rnormalization Lt us considr again th µ scattring 3.6: (p ) µ (p )! (p ) µ (p ) for which th amplitud is: im =ū p i µ ig µ u p q ū p i u p =4 ū p µ u p ig µ q ūp u p with = /4 is th usual lctromagntic fin structur constant. Suppos you want to masur with this procss. Lt us imagin, w hav a bam of and µ and w dtct th scattrd and µ. By playing with th bam nrgy and th scattring angl, w can count th numbr of rcordd vnts and so masur th cross-sction at a givn momntum transfr q. From th masurmnt, w would thn gt th valu of mas =. Now, thr ar highr ordr corrctions to this procss. On is shown on figur 3.. Th µ Figur 3.: A virtual photon fluctuating in + pair. µ virtual photon splits into a pair of virtual lctron-positron (th frmion loop). If th photon has a momntum q, thn on mmbr of th pair carris a momntum p and th othr q p, so that th nrgy-momntum is consrvd. howvr, thr is no constraint on th valu of p! Thrfor, all possibl valus hav to b takn into account. Th amplitud of this diagram can b calculatd but it rquirs additional ruls with rspct to th on givn in sction 3., namly a factor for th closd frmion loop 7, and th trac of th associatd photon vrtx must b usd. Justification for th trac can b found in [8, p. ], but a naiv xplanation is that, 6 Th classical radius of th lctron is th siz th lctron that is dtrmind assuming that only th lctrostatic potntial nrgy contributs to its mass. Namly, in natural unit: E = m = /(4 r ) ) r = /m. 7 Th lctron and positron in th loop can b intrchangd, thus th. P.Paganini Ecol Polytchniqu Physiqu ds particuls
Fw words about th rnormalization 93 o basically w hav a! procss which cannot dpnd on spinor indics. Th amplitud is thn givn by: ( ( ) 4 h i )Tr i µ p+m i / p m i p i / /q+m ig (p q) m q n im = ū p i µ igµµ u p q R d 4 p ū p i u p n = ū p i µ u R d 4 p p q 4 ū p i u p ( ) 4 ( Adding th amplituds, w finally hav: im =ū p i µ u p M =ū p µ g u µ p q whr: Z i [] µ (q )= d 4 p ( ) 4 ( h )(i ) Tr ig µ + i [] µ (q ) q q 4 + [] µ (q ) q 4 appl )(i ) Tr io p+m µi / p p m i / /q+m (p q) m ū p i u p ū p u p p µi / + m /p /q + m p m i (p q) m Hnc, w s that adding this contribution can b intrprtd as a modification of th initial photon propagator as: ig µ q! ig µ q + i [] µ (q ) q 4 W can considr this nw photon propagator in th graph calculation, but thr is a problm sinc th intgral in µ divrgs! Mor prcisly, on can show that µ can b rducd to th only contributing trm: [] µ (q )= with f(q ) a finit contribution function givn by: f(q )=6 applz q dp g µ 3 m p f(q ) + + O( ) Z dz z( z) log q z( z) m (3.69) Clarly R dp m divrgs as a logarithm. Lt us introduc a cut-o, namly a maximum valu p for th uppr bound of th intgral. Is it shocking to truncat th intgral this way? Wll, any thory must hav its domain of validity. W can s as a paramtrization of our ignoranc, abov which nw physics must appar. Thn, with this prscription, [] µ bcoms: [] µ (q )= q g µ log 3 m f(q ) + + O( ) and hnc, th amplitud bcoms (rwrittn in trms of ): M =ū µ g µ p u p q 4 3 log m + 3 f(q ) ū p u p Now in ordr to avoid confusion, w ar going to rnam th abov as : M =ū µ g µ p u p q 4 3 log m + 3 f(q ) ū p u p P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
94 A brif viw of Quantum Elctrodynamic Th labl rfrs to th fact that this is th that is usd in th cas of loop as at th bginning of this sction. is calld th bar constant. Th quation now is how can w intrprt th masurmnt of th cross sction w hav mad? Th actual scattring amplitud M is not supposd to dpnd on an arbitrary paramtr. If is changd, w don t want that our prdictd cross-sction (that w r going to compar to th masurd on) changs. In othr words, w would hav to chang itslf in ordr to compnsat th chang of!, th constant of th QED lagrangian is not a physical paramtr! Th ky of th rnormalization ida is to xprss what w masur with physical (masurd) paramtrs. Hr, with our scond ordr dvlopmnt, if our xprimnt is mad at q = µ, w would idntify: mas = 3 log m + 3 f(µ ) = 3 log m f(µ ) + O( ) (3.7) Our masurmnt was prformd at q = µ. To clarly point this out, I now us th nam mas (µ ). Lt s rvrt th prvious quation: log f(µ ) + O( ) = mas (µ )+ 3 = mas (µ )+ mas (µ ) 3 log m m f(µ ) + O( mas(µ )) Th scond quality is quivalnt to th first on, to th scond ordr usd hr. But what if instad of µ, w would hav chosn a arbitrary valu q? W would hav: mas (q ) = 3 log f(q ) + O( m ) = mas (µ )+ mas (µ ) 3 log f(µ ) m log f(q ) + O( mas(µ )) mas (µ ) 3 m = mas (µ )+ mas (µ ) 3 f(q ) f(µ ) + O( mas(µ )) Th good nws now, is that th arbitrary dosn t appar anymor! W can xprss any procss at any nrgy if w hav masurd mas at a givn rfrnc point. This is th miracl of th rnormalization. But it has a pric to pay: what w calld a coupling constant is not a constant! It dpnds on th nrgy scal. W manag to liminat at th scond ordr of prturbation. Th rnormalization works if w can liminat it at any ordr. In QED, it turns out to b th cas. Coming back to f(q ), whn th momntum transfr is larg with rspct to m (which is th cas in high nrgy physics xprimnts), by dvloping f(q ) at ordr O( m ), it is not q too di cult to show that: f(q )! log m q 5/3 = log. Not that q < with q 5/3 m scattring procsss. W can dfin th positiv quantity Q = q. In this approximation, mas (Q ) that w now simply dnot by (Q ) bcoms 8 : appl (Q )= (µ ) + (µ ) Q 3 log µ + O( ) Now, this rsult is valid at ordr. W can go at highr ordr by adding mor loops. Th dpndncy of with Q thn bcoms: (Q )= 8 µ is now undrstood as a positiv quantity (µ ) (3.7) (µ ) 3 log Q µ P.Paganini Ecol Polytchniqu Physiqu ds particuls
A major tst of QED: g- 95 Lt us intrprt this formula valid only in th rgim Q m : it givs th valu of th fin structur constant at a givn nrgy, if it has bn masurd at anothr nrgy scal. Its variation is only logarithmic with th scal. In Coulomb scattring, at Q ', =/37 whil at th Z pol, it is about /8. Th variation of th charg (or ) as function of th scal is a ral physical ct. It is du to th vacuum polarization. Imagin you want to tst th charg of an lctron by approaching a prob-charg. In th vicinity of th lctron, virtual chargd pairs can b cratd for a short amount of tim t ~/mc. Thy can sprad apart at a maximum distanc c t, crating a dipol. Th virtual positrons tnd to b closr to th lctron than th virtual lctrons. If th prob-charg dosn t approach nough (i d c t), it will s mainly th surrounding clouds of positron, yilding to a masurmnt undrstimating th ral charg of th lctron. In othr words, at small Q (i high distanc), th charg sms smallr. This phnomna is usually calld a scrning. On th contrary, at high Q, th prob charg pntrats th virtual cloud and can s th whol lctron charg. 9 In summary, th rnormalization is a procdur that allows to absorb all infinit quantitis by a rdfinition of a finit numbr of paramtrs. In QED, thy ar th charg of th lctron (or quivalntly th fin structur constant), th mass of th lctron (ys, th mass!) and th normalization of th filds (thus th nam rnormalization). 3.7 A major tst of QED: g- Thr ar many tsts that validat QED, both at high nrgy (in acclrator) and low nrgy. On of th most imprssing tst is probably th prdiction of th gyromagntic ratio g which ntrs in th magntic momnt of th lctron. 3.7. Prdiction with Dirac s quation Non rlativistic prdiction without spin: Lt us rcall first, th classical formula for th magntic momnt of chargd particl moving in a circular orbit. Classically, th magntic momnt is dfind as: µ =currnt ara For a circular orbit or radius r, th ara is r. And th currnt is th charg pr unit tim, namly th charg tims th frquncy of rotation (th vlocity dividd by th circumfrnc of th orbit). Hnc: µ = q v r r = q v r Exprssing as function of th angular momntum ~ L = ~r ~p = ~r m~v, whav: ~µ = g q m ~ L g = g = in th classical cas is calld th Landé g-factor or gyromagntic ratio. Elctrons placd in a magntic fild B ~ would thn hav an additional potntial nrgy du to th fild: E B = ~µ. ~ B = g q m ~ L. ~ B 9 Wll, it s not tru: virtual pairs of highr mass particls can b cratd. Thr is always a virtual cloud sn by th prob. P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
96 A brif viw of Quantum Elctrodynamic which can b confirmd by considring th non-rlativistic hamiltonian apparing in th Schrödingr quation is H = ˆp m In th prsnc of a magntic fild, w mak as usual th substitution ˆp! ˆp q ~ A, and thn: H = ˆp m q A.ˆp m ~ + q A m ' ˆp m q m ~ A.ˆp whr th trm in q is nglctd. Th additional nrgy is thn givn by: E B = q m ~ A.ˆp But, B ~ = r ~ A ~ and A ~ can b chosn satisfying th gaug r. ~ A ~ =. Thus, A ~ = B ~r and E B bcoms: q E B = m ( B ~ q ~r ).ˆp = m (~r ˆp). B ~ q = ˆ~L. B m ~ = ˆ~µ. B ~ whr w hav applid a scalar tripl product proprty. W finally find th xpctd xprssion for ~µ. Sinc lctrons hav an intrinsic angular momntum, th spin S, ~ w would xpct an additional nrgy: E B = g q S. m ~ B ~ with g = Howvr, at th tim of Dirac, th masurmnts for g xcludd th classical valu, and was compatibl with. Non rlativistic prdiction with spin: w can follow th sam procdur, starting from Dirac s quation instad of Schrödingr on. Hnc, starting from th momntum spac quation.3: (/p m)u =( µ p µ m)u = p µ!p µ qa µ! ( µ p µ q µ A µ m)u = ( E q ~. ~p + q~. ~ A m)u = In trms of -componnts, this quation rads: (E q m) ~. ~p + q~. ~ A ~. ~p q~. ~ A (E q + m)! ua u b = (~. ~p q~. A)u ~ b =(E q m)u a (~. ~p q~. A)u ~ a =(E q + m)u b Now, to b abl to asily idntify th magntic momntum as in th cas of th Schrödingr quation, w wish to find a rlation satisfid by th spinors in th non-rlativistic limit of th Dirac s quation. In this limit, E ' m, bute m = T and T m whr T is th kintic nrgy. For usual lctromagntic fild, th fild nrgy is ngligibl with rspct to mc. Thrfor, m q. and thus: (~. ~p q~. A)u ~ b =(T q )u a (~. ~p q~. A)u ~ a ' m u b P.Paganini Ecol Polytchniqu Physiqu ds particuls
Highr ordr corrctions 97 So that by multiplying th first quation by m and insrting th scond: () (~. ~p q~. A)(~ ~. ~p q~. A)u ~ h a =m(t q )u a (~. ~p ) + q (~. A) ~ q (~. ~p )(~. A)+(~ ~. A)(~ ~ i. ~p ) u a =m(t q )u a h ~p + q A ~ q ~p. A ~ + i~.(~p A)+ ~ A.~p ~ + i~.( A ~ i ~p ) u a =m(t q )u a h (~p q A) ~ iq~. ~p A ~ + A ~ i ~p u a =m(t q )u a whr in (), w hav usd th quality (~. ~a )(~. ~ b)=~a. ~ b+i~.(~a ~ b). Going back to th languag of wav functions (and thus rplacing u a by = u a ip.x and ~p by i r), ~ w hav: ( i r ~ q A) ~ q m m ~. ~r ( A ~ )+ A ~ r ~ =(T q ) ( i r ~ q A) ~ q m m ~. ~r A ~ =(T q ) h ( i r ~ q A) ~ q m m ~. B ~ i + q = T Th quantity in th brackt: H E'm = (~p q ~ A) m q m ~. ~ B + q corrsponds to th non-rlativistic hamiltonian of th Dirac s quation. Rcalling that th spin oprator for -componnts spinors is ~ S = ~,widntify: maning that: ~µ = q m ~ = q m ~ S g = Hnc, th non-rlativistic limit of th Dirac s quation dos imply a gyromagntic ratio of. It was on of th succsss of Dirac s prdiction. 3.7. Highr ordr corrctions Considr th lctron charg currnt involvd in th basic vrtx of th lft diagram of figur 3.. It can b dcomposd in parts (as known as Gordon dcomposition): j µ = µ f i = [ū f µ u i +ū µ f u i ] i(p i p f ).x = u ip.x = m [ū f p µ f u i +ū µ f p i u i ] i(p i p f ).x ū f (/p f m)=, (/p i m)u i = = mūf [p f (g µ µ )+ µ p i ] u i i(p i p f ).x µ h i + µ =g µ = mūf Now, using th matrix: p µ f + µ (p i p f ) u i i(p i p f ).x µ = i ( µ µ ) ) µ = g µ i µ so that w gt: j µ = h = mūf h i p µ f + gµ (p i p f ) i µ (p i p f ) u i i(p i mūf (p µ f + pµ i )u i + iū f i m µ (p i p f )u i i(p i p f ).x p f ).x (3.7) P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
98 A brif viw of Quantum Elctrodynamic Rfrring to th Fynman ruls of tabl 3., w s that th first trm in th brackt mūf (p µ f + p µ i )u i is analogous to th intraction of a chargd spin particl. W thn xpct th scond trm: j s µ = iū µ f (p i p f )u i i(p i p f ).x m to b rlatd to th intraction involving th spin of th lctron. To chck this assumption, considr, to th first ordr of prturbation, th amplitud givn by th intraction of such currnt with an lctromagntic fild A µ : S [] = Z i Z d 4 xj s µ A µ = d 4 x ū f m µ (p i p f )u i i(p i p f ).x A µ Lt us considr a static fild (coulomb scattring) for simplification. Thn A µ dos not dpnd on tim. W can prform th intgration ovr th tim: Z S [] = (E i E f ) d 3 x ū f m µ (p i p f )u i i(p i p f ).x A µ Bcaus of th Dirac-dlta function, th tim-lik componnt of th 4-momnta in th intgral has to b qual, and hnc cannot contribut to th amplitud. Only th spatial-lik componnts mattr, and aftr r-arrangmnt, it rads: Z S [] = i (E i E f ) d 3 x ū f m µj u i @ j i(p i p f ).x A µ In th xprssion abov, only th xponntial and A µ dpnd on spac coordinats (th spinors dpnd only on 4-momnta). Hnc, aftr intgration by part (th fild A µ vanishing at infinity): Z S [] = i (E i E f ) d 3 x ū f m µj u i i(p i p f ).x @ j A µ Figur 3.: Lft: simplst QED vrtx. Right: a corrction to th vrtx. As in th prvious sction, in ordr to s clarly th spin contribution, w ar going to considr th low nrgy limit, whr th mass dominats ovr th 3-momntum. Thn, only th uppr componnt of th spinors mattrs sinc in th lowr componnt, thr is th trm ~. ~p / (E + m) (s formula.4). In othr words, spinors simplify to: u = ua ~. ~p E+m u a! ua P.Paganini Ecol Polytchniqu Physiqu ds particuls
Highr ordr corrctions 99 In ths conditions, th trm ū µj f u i A µ = ū j f u i A +ū kj f u i A k ū kj f u i A k. Indd, according to th Dirac s rprsntation of th matrics: appl j = i l j j l j l j j = i l and hnc ū j f u i. Thus, only ū kj f u i givs a significant contribution to th amplitud: Z S [] = i (E i E f ) d 3 x ū kj f u i i(p i p f ).x @ j A k m Lt us xprss kj : kj = i appl k k j j j j = i ( k j j k ) P ( k j j k ) = i i l l kjl i P l l kjl = P l l kjl l k k j whr kjl is th usual antisymmtric tnsor ( kjl = for cyclic prmutation of 3, = anticyclic prmutation, othrwis). Hnc: ū kj f u i = X kjl (u l l uai a f, ) l l = X kjl u a l f u ai l l for Now, noticing for xampl that: ū j f u i @ j =ū f @ u i +ū f @ u i +ū 3 f @ 3 u i = P l l u a l f @ u ai + P l l u a l f @ u ai + P l 3l u a l f @ 3 u ai = u a f Pl l l @ + P l l l @ + P l 3l l @ 3 u ai = u a f 3 @ @ 3 u ai = u a f h~ ~ @i uai and similarly for th othr componnts. Thus, w conclud: ū f kj u i @ j A k = u a f h~ ~ @i k Ak u ai = u a f Hnc, sinc B ~ = @ ~ A, ~ w finally hav: Z S [] = i (E i E f ) d 3 xu a f whr w idntify th magntic momnt of th lctron ~µ = at th bginning of this sction, th currnt: j µ s = iū f m h ~ @ ~ i. Au ~ h i ai = u ~@ a f A ~.~ u ai ~B.~ uai i(p i p f ).x m µ q u i i(p i p f ).x m ~. Conclusion, as announcd P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé
A brif viw of Quantum Elctrodynamic with q = p i p f, yilds naturally th gyromagntic ratio. Now, considr th scond diagram on th right hand sid of figur 3.. Th computation of th corrction lads to a modification of th currnt in th low momntum transfr approximation [9, p. 6] (dropping th xponntial): n j µ = ū f m (pµ f + pµ i )+i n! ū f h+ µ q 3 log( m m m m µ q o u i 3 8 i + i m µ q o u i whr m is a cut-o to avoid an infinit trm du to th so calld infra-rd divrgnc. W can immdiatly dduc that adding th diagrams will yild a modification of th gyromagntic ratio: ~µ = m ~! + ~ ) a = g = m whr a, th dviation with rspct to th valu g =, is calld th anomalous magntic momnt. Numrically, w obtain a '. 6, to b compard to th most prcis xprimntal valu obtaind so far []: a xp = 59 65 8.73 (.8) (3.73) th digits in parnthss dnoting masurmnt uncrtainty in th last two digits at on standard dviation. Prtty clos! Such xprimntal accuracy pushs vry far th thory and rquirs a calculation at last at th 8th ordr (i 4 ). Th th ordr, which rprsnts 67 vrtx-typ Fynman diagrams (!!), is alrady partially valuatd []. Hadronic (vacuum polarization), lctrowak cts and small QED contributions from virtual muon and tau-lpton loops contributions hav also to b takn into account. Fortunatly, th diagrams ar now valuatd numrically and a rcnt computd valu of a is []: a th = 59 65 8.3 (.)(.37)(.)(.77) whr th first, scond, third, and fourth uncrtaintis com from th calculatd ighth-ordr QED trm, th crud tnth-ordr stimat, th hadronic and lctrowak contributions, and th fin structur constant, rspctivly. Both thory and xprimnt ar in vry good agrmnt sinc: a xp a th =.4 (.88) Th largst sourc of uncrtainty of a th is now th xprimntal valu usd for th fin structur constant (obtaind from Csium or Rubidium atom xprimnts), and not anymor an uncrtainty coming from th calculation itslf! Thrfor, it maks sns now to obtain from th thory and th masurd valu of a instad of th contrary. Th accuracy of such computation of th g-factor is probably on of th most imprssiv triumphs of th thory of quantum lctrodynamics. 3.7.3 Masurmnt of th g-factor W hav alrady sn that an lctron in a magntic fild gts xtra contribution to its nrgy with a trm ~µ. B ~ = g q ~ m S. B. ~ A fild on th Z-axis will yild g m ~m sb z = h s m s with aftr rnormalization. P.Paganini Ecol Polytchniqu Physiqu ds particuls
Masurmnt of th g-factor m s = ± and th spin frquncy s is rlatd to th cyclotron frquncy c with s = g c and c = Bz m. Th principl of th masurmnt is thus to masur both c and s or mor prcisly c and a = s c, th anomaly frquncy sinc: g = s =+ s c c c =+ a c Th xprimntal st-up [] is basd on a Pnning-trap which suspnds a singl lctron thanks to a strong magntic fild (5.36 T) along th z-axis and an lctrostatic quadrupol potntial as shown on figur 3.3. Th ct of th lctric fild is to confin th lctron in a potntial wll in which it maks small vrtical oscillation. Th Pnning trap is usd to artificially bind th lctron in an orbital stat, as if it was an lctron of an atom. Actually, th lctrostatic potntial shifts th cyclotron frquncy from c to c and thus a to a = s c. Th lowst nrgy lvl including th lading rlativistic corrction can b approximatd by []: E n,ms = h s m s + n + h c h (n + + m s) whr th quantity is of th ordr 9 c. According to this formula giving th nrgy lvls, th transitions n, m s i =, i!, i!, i!, i: E, E, E, E, E, E, = h( c 3 ) = h a = h( c ) allow to masur c, and a and hnc g/. It turns out that c 5 GHz whil a 74 MHz. Sinc c ' c is proportional to B, a high magntic fild is ncssary to incras th spacing btwn th cyclotron nrgy lvls. Th cavity of th Pnning-trap is maintaind at a Figur 3.3: Schma of a Pnning trap [3]. Th constant lctric fild (blu) is gnratd by a quadrupol (a and b). Th suprposd constant and homognous magntic fild (rd) is gnratd by a surrounding cylindr magnt (c). A particl, indicatd in rd (hr positiv) is stord in btwn caps of th sam polarity. Th particl is trappd insid a vacuum chambr. vry low tmpratur ( mk) to avoid transitions from th ground stat to othr stats du P.Paganini Ecol Polytchniqu Physiqu ds particuls avancé