8.324 Quantum Field Theory II. Problem Set 5 Solutions

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1 8.34 Quantum Fild Thory II Problm St 5 Solutions. W will us PS convntions in this Problm St. W considr th scattring of high nrgy lctrons from a targt, a procss which can b dscribd to lading ordr in α by th following gnric diagram:! p k q Y p k! X Th gray filld circl rprsnts th coupling of th virtual photon to th targt, taking th targt from an initial stat X to som final stat Y, which w incorporat into a scattring amplitud factor ˆ ν M (q), whr q = p p = k k. Th full matrix lmnt for th procss is: M = ( i)ū(p )γ µ u(p) ig µν ˆ ν (q) () q M Parts a) and b) of th problm analyz th structur of th ū(p )γ µ u(p) pic, th rsults of which ar usd in c), d), and ) to find an xprssion for th cross-sction. (a) Dotting with q µ : ū(p )γ µ u(p) = Aq µ µ + Bq µ + Cɛ + Dɛ µ () LHS = ū(p )/qu(p) = ū(p )(p/ p/)u(p) = (m m)ū(p )u(p) = RHS = Aq + Bq q whr th last lin follows from th fact that q ɛ i =. Thus, (3) B q =. (4) A q q In th ultra rlativistic limit for th lctrons th kinmatics for this problm ar as follows: In th small θ limit, w can calculat p = (E, E ẑ) p = (E, E sin θ,, E cos θ) q = p p = (E E, E sin θ,, E E cos θ) q = (q, q) = (E E, E sin θ,, E cos θ E). (5) q = EE ( cos θ) EE θ q q = (E E ) + E sin θ + (E E cos θ) (E E ). (6) ( ) W s that B = q will b at most O θ A q q, and so B can b nglctd in th rst of th analysis. As notd in th problm A is irrlvant bcaus of th Ward idntity q µ Mˆ µ (q) =, which is just th consqunc of th gaug invarianc of QED. W conclud that it is nough to dtrmin C and D to solv th forward scattring problm.

2 (b) W want to comput xplicitly ū(p )γ ɛ i u(p) for masslss lctrons, whr u(p) and u(p ) ar spinors of dfinit hlicity. I will abbrviat u(p) as u and ū(p ) as ū for rst of th problm. First, w will us mthod of spinor products dvlopd in Problm 5.3 of PS to valuat ū γ i u for both lft and right hlicity. For this w nd to choos a light-lik vctor K which is indpndnt of p and p. An obvious choic for K is (,,, ). u Lγ i u u R/p γ i L = pu / R (p K)(p K) = T r(p R k/ p/ γ i p/) (p K)(p K) i = (K p )p (p p )K i + (K p)p i iɛ µνiρ K µ p ν p ρ ]. (7) EE Similarily, u i Rγ i u R = (K (p + iɛ µνiρ K µ p ν p ρ ]. EE p )p p )K i + (K p)p i (8) Notic that only ɛ-tnsor trm changs sign. Othr hlicity combinations ū Lγ i u R and ū Rγ i u L vanish ithr by simpl calculation or by th consrvation of hlicity for masslss frmions. Applying kinmatics to abov xprssions w gt ū γ u L = EE θ ū γ u L = +i EE θ ū γ 3 u L = EE ( iθ) ū L γ u R = EE θ ū L γ u R = i EE θ ū L γ 3 u R = EE (9) R R R ( + iθ). W can work out what ɛ and ɛ nd to b by rquiring thy b prpndicular to q and q. Rquiring that ɛ b prpndicular to th plan of scattring and ɛ to b in th plan of scattring givs ɛ µ ɛ µ = = () E θ E E Thrfor ū = ū = E + E EE Lγ ɛ u L R γ ɛ u R θ () E E ū Lγ ɛ = ū = i EE u L R γ ɛ u R θ. () Scondly, w will us a mor dirct approach to gt th sam rsult. With th basis choic of PS for th Dirac matrics th dfinit hlicity stats of th lctrons and photon ar: ( ) σ γ µ = µ (3) σ µ E ur = E u L = (4) sin θ cos θ E θ u R = E u E L = (5) θ ɛ µ ɛ µ L = (ɛ µ iɛµ ) = R (ɛ µ + iɛµ ) = (6) i = i E θ E θ E E E E Not forgtting that th photon is outgoing (in th subprocss) w gt E ū ɛ ɛ EE Lγ Lu ū L = Rγ Ru R = θ (7) E E E ū ɛ ɛ EE Lγ Ru ū L = Rγ Lu R = θ. (8) E E

3 This rsult maks th parity invarianc of th thory manifst (i.. flipping all th hlicitis dosn t chang th rsult) and can b asly chckd to agr with th rsult of th spinor product mthod. (c) Th xpansion of th spinor product ū(p )γ µ u(p) = Aq µ + Bq µ + Cɛ µ + Dɛ µ µ works bcaus ū(p )γ µ u(p) is a 4-vctor, and q µ, q, ɛ µ, and ɛµ consitut a basis of four linarly indpndnt 4-vctors. Th cofficints ar just th projction of ū(p )γ µ u(p) onto ach of th basis vctors. From th argumnt in part (a) B = O(θ ) and in th small θ limit can b nglctd. Hnc our scattring matrix lmnt M can b writtn: M = q ū(p )γ µ u(p)mˆ µ(q) 3 (9) = q (Cɛµ + Dɛµ )Mˆ µ(q). () µ Hr th A trm drops out bcaus q Mˆ µ = by th Ward idntity. Th cofficints C and D ar just th projctions C = ū γ.ɛ u D = ū γ.ɛ u () so w can writ: ] M = ū(p )γ.ɛ q u(p)ɛ µ Mˆ µ(q) + ū(p )γ.ɛ u(p)ɛ µ Mˆ µ(q) ] = ū(p )γ.ɛ q u(p)mˆ + ū(p )γ.ɛ u(p)mˆ () whr w introduc th notation Mˆ ɛ µ Mˆ µ(q) and Mˆ ɛ µ Mˆ µ(q). Using th rsults from part (b), w can valuat M for th various hlicity cass: ( ) ) = E + E M( EE θmˆ + iθ R R EE Mˆ q E E ( ) E + E ) = EE θ ˆ iθ EE M( L L q E E M Mˆ M( ) = M( L R R L ) =. (3) Th spin-avragd (or hlicity-avragd) squard matrix lmnt is givn by ( ( ) ) E + E M = q 4 EE θ Mˆ E E + Mˆ. (4) spins (Not that this rsult sums ovr th outgoing photon polarizations, as Ms thmslvs wr sums ovr photon polarizations.) Th total scattring cross-sction in th CM fram, using quation (4.79) from PS, is: d 3 p σ = dπ Y E E X ( + v X ) (π) 3 E M (5) whr E X and v X ar th initial nrgy and vlocity of th targt, and dπ Y = d 3 k (π) 3 E (π) 4 δ 4 (p + Y k p k) is a phas spac intgral ovr th final stat Y of th targt. For simplicity, lt us introduc th notation dπ Y dπ Y. (6) EX ( + v X ) Rwriting th intgral ovr p, th cross-sction bcoms: πp dp dp z σ = dπ Y (7) E (π) 3 E M spins spins

4 whr p z = E cos θ and p = E sin θ. Th cross sction is dominatd by forward scattring, it has a collinar singularity that will b rgulatd in part (d). Hnc th dominant contribution coms from th domain, whr θ is btwn and som small θ max. In this limit p z E and p E θ, so dp z de and dp E dθ. Our intgral for σ can b xprssd as: de θmax σ dπ 4E Y (π) dθ E θ M spins de ( θmax E E θ 3 ( ˆ E + E ) ) ˆ = dπ 4E Y (π) dθ q 4 M + E E M (8) ( θmax ( ) ) de E + E = dπ 4E Y (π) dθ Mˆ + Mˆ θ E E whr in th last stp w usd th fact that q 4 ( EE θ ) in th small θ limit. Notic that th intgral is logarithmically divrgnt for θ. (d) To trat th divrgnc w rintroduc th lctron mass in th xprssion for q just as w did for PS Problm 5.5, q = (p p) = (EE pp cos θ) + m, (9) whr p = p and p = p. Sinc E, E m, w can approximat p = E m m p = E m m E E. (3) E E Plugging ths into th xprssion for q givs us, aftr som rarrangmnt, ( ) ] q EE θ + m E, (3) E 4 whr w hav kpt only th lading trms. Substituting this q into th intgral for σ in Eq. (8), w find: σ dπ 4E Y de ( θmax θ 3 ( ˆ E + E ) ) ˆ (π) dθ θ + m ( E ) ] M + E E M. (3) E If w lt E ( E ) E, thn th intgral ovr θ can b valuatd as: θmax θ 3 θmax dθ log(m θ + m /E) + log(θ max + m /E] = /E) (θ max + m /E) log(m /E) (33) whr w hav approximatd th intgral by th dominant logarithmic singularity in th small m limit. First, w argu hand-wavingy: sinc E dpnds on th momntum transfr in th scattring procss, and has dimnsions of nrgy squard, w can say that E O(s), whr th Mandlstam variabl s = (p + k) is th total initial four-momntum of th systm. Thus log(m /E) log(s/m ), and w can writ th cross-sction σ as: ( ) ( ( ) ) de s E + E σ dπ 8E Y (π) log Mˆ m + Mˆ E E. (34) W will stablish this rsult with mor rigor in part (), whr w tak th diffrnc btwn E and s mor sriously.

5 () W assum th targt cross-sctions ar indpndnt of th photon polarization, which implis ˆ M = ˆ ˆ M M. Lt us introduc th variabl x = (E E )/E, th fraction of th initial lctron nrgy carrid off by th photon. Thn W can also xprss E with x and s de ( E + E ) ( x) dx = and E E E = x. (35) s = (E + E + M X ) 4E X 4E (36) ( ) ( ) ( ) EE = = x E x E s (37) E E x x whr w assumd E M X. Th cross-sction of Eq. (34) with th propr tratmnt E bcoms: α ( ) ( E σ dx log + ( x) ) m x dπ Y M ˆ 8πE α ( s ) ( )] ( x dx log + log + ( ) x) dπ ˆ Y πE m x x M α ( s ) + ( x) dx log dπ Y ˆ, (38) 4πE m x M ( ) x whr w droppd an O() trm by nglcting log in th intgrand. Not that dπ ˆ Y = dπ Y ˆ M M σ γx (x) (39) xe E X (xe)( + v X ) is th cross-sction of a photon of incidnt nrgy xe hitting th targt. Our total cross-sction can b writtn:. (a) W will rprsnt th vrtx corrsponding to H d 3 λ h int = x ψψ, th coupling of th Higgs boson to th lctron, by: x ( ) σ α s + ( x) dx log σ γx (x) π m x = dx N γ (x)σ γx (x) (4) α +( x) whr N γ (x) = π x log ( ) s m. Thus σ ffctivly looks lik th cross-sction of a bam of photons with nrgis xe and distribution N γ (x) hitting th targt. N γ (x) is th probability to find a photon of longitudinal fraction x in th incidnt lctron, whn probing it in a collision with CM nrgy squard s. 5 = i λ. To find th ffct of th Higgs on th lctron g, w look at its contribution to th lctron-photon vrtx, W know that th total vrtx has th form Γ µ (p, p) = γ µ F (q ) + iσµν F (q ), and that g = F (). At on loop w don t hav to worry about divrgncs for F (), hnc w can do th Dirac algbra in 4 dimnsions and forgt about dimnsional rgularization. Thus to find th Higgs contribution to g, w q nd to rwrit ū(p )δγ µ (p, p)u(p) so as to xtract th cofficint of iσµν ν and valuat it at q =. q ν m m

6 6! p p! k k = k + q k q! p = i ū(p )δγ µ (p, p)u(p). Th full xprssion for ū(p )δγ µ (p, p)u(p) is: ( ) d 4 k i iλ i(k/ + m) ū(p )δγ µ (p, p)u(p) = (π) 4 (p k) m h ( + ) ) iɛū(p k m + iɛ γµ i(k/ + m) iλ k m u(p) + iɛ iλ d 4 k ū(p )(k/ + m)γ µ (k/ + m)u(p) = (π) 4 (p k) m h + iɛ]k m + iɛ]k m + iɛ] iλ d 4 k ū(p )k/ γ µ k/ + m γ µ + m(γ µ k/ + k/ γ µ )]u(p) = (4) (π) 4 (p k) m h + iɛ]k m + iɛ]k m + iɛ] Th way w valuat this xprssion is analogous to th algbra dscribd in Sction 6.3 of Pskin and Schrodr. W look individually at th dnominator and numrator of th intgrand. For th dnominator, w lt A = k m + iɛ, A = k m + iɛ, A 3 = (p k) m h + iɛ, and rwrit it using Fynman paramtrs as: = dx dy dz δ(x + y + z ) (4) A D 3 A A 3 whr D = xa + ya + za 3 = x(k m ) + y(k m ) + z(k p) m h ] + (x + y + z)iɛ = k + k (yq zp) + yq + zp (x + y)m zm h + iɛ (43) In th scond lin w hav usd th fact that x + y + z = and k = k + q. W shift k to complt th squar, introducing a nw variabl l = k + yq zp. With som work, using lctron momnta that ar on shll, D simplifis to D = l Δ + iɛ, whr Δ = xyq + ( z) m + zm h. Turning to th numrator, w would lik to xprss it in trms of l. To simplify th algbra, w rcall th usful rsults, µν d 4 l l µ d 4 l l µ l ν d 4 l l 4 g (π) 4 D 3 =, (π) 4 D 3 = (π) 4 D 3, (44) which allow our numrator to b writtn in th quivalnt form numrator = ū(p )k/ γ µ k/ + m γ µ + m(γ µ k/ + k/ γ µ )]u(p) ū(p ) γ µ l + (( y)/q + z/p)γ µ ( y/q + zp/) + m γ µ ] + mγ µ ( y/q + zp/) + (( y)/q + zp/)γ µ ] u(p). (45) This can b furthr simplifid using th idntitis /pγ µ = p µ γ µ /p, /pu(p) = mu(p), ū(p )p/ = ū(p )m, q = p p + m. (46)

7 7 Th trms involving slashd variabls can b rwrittn: ū(p )(( y)/q + zp/)γ µ ( y/q + zp/)u(p) = ū(p )γ µ (q xy m z(z )) mz(xp µ + yp µ )]u(p) = ū(p )γ µ (q xy m z(z )) mz(p + p ) µ (x + y) + mzq µ (x y)]u(p) (47) ū(p )mγ µ ( y/q + z/p) + (( y)/q + zp/)γ µ ]u(p) = ū(p )m γ µ m(xp µ + yp µ )]u(p) Plugging Eqs. (47) and (48) into Eq. (45), our numrator bcoms: = ū(p )m γ µ m(p + p ) µ (x + y) + mq µ (x y)]u(p). (48) numrator = ū(p ) γ µ ( l + q xy + m (3 + z z )) m(p + p ) µ (x + y)( + z) ] + mq µ (x y)( + z) u(p) = ū(p ) γ µ ( l + q xy + m ( + z) ) + iσµν q ν m ( z ) ] m + mq µ (x y)( + z) u(p). (49) In th scond lin w hav usd th Gordon idntity and th fact that x + y + z =. Not that th trm proportional to q µ is odd undr xchang of x and y and will intgrat to zro, which is rquird by th q gaug invarianc. Th part w ar intrstd in is th cofficint of iσµν ν m, which givs us th contribution to F. Putting this togthr with th rsults for th dnominator, w hav: δf (q ) = iλ d 4 l dx dy dz δ(x + y + z ) m ( z ) (π) 4 l Δ + iɛ] 3 λ m ( z ) = dx dy dz δ(x + y + z ) (4π) Δ λ m ( z ) = dx dy dz δ(x + y + z ) (4π) xyq + ( z) m + zm For q =, w can valuat th x and y intgrals: h λ m ( z ) δf () = dx dy dz δ(x + y + z ) (4π) ( z) m + zm h λ z m ( z ) = dz dy (4π) ( z) m + zm h λ m ( z )( z) = dz (4π) ( z) m + zm h (5) λ m ( z )( z) = (4π) m dz h z + ( z) m (5) This is a vry ugly intgral, although Mathmatica can do it. Not that sinc m m h, w might think about xpanding th intgrand and intgrating trm by trm. Th lading ordr trm in that xpansion is, howvr, singular (thr is an IR divrgnc). It s still much nicr, with vry small loss in accuracy to xprss th rsult as an xpansion in m/m h. I find bst way to do this is to valuat th intgral in Mathmatica with th assumptions M m /m h > and 4M <. Thn tak th rsult and sris xpand around M =. This givs ( )] λ m m h 7 m δf () = log (4π) m h m 6 + O (5) m h g This is th Higgs contribution to a h = = F (). m h

8 (b) W nd to chck whthr th valus λ = 3 6 and m h > 6 GV ar consistnt with th xprimntal constraint a h <. Plugging into Eq. (5), w find a h < 9 3 for m =.5 MV, λ = 3 6, m h > 6 GV (53) so ths valus for λ and m h ar not xcludd. ( a h is a monoton dcrasing function of m h.) For th muon, λ = 6 4, and th xprimntal constraint is a h < 3 8. Plugging into Eq. (5), w find a h < 8 4 for m = 5.6 MV, λ = 6 4, m h > 6 GV (54) so ths valus for λ and m h ar also not xcludd. iλ (c) For th axion, which coupls to th lctron according to H int = d 3 x aψγ 5 ψ, w do a calculation analogous to th on in part a). Th axion contribution to th lctron-photon vrtx looks lik: ( ) d 4 k i λγ 5 i(k/ + m) ū(p )δγ µ (p, p)u(p) = (π) 4 (p k) m a + iɛ ū(p ) k m ( ) + iɛ γµ i(k/ + m) λγ 5 k m u(p) + iɛ 8 iλ d 4 k ū(p )γ 5 (k/ + m)γ µ (k/ + m)γ 5 u(p) = (π) 4 (p k) m + iɛ]k m + iɛ]k m a + iɛ] = iλ d 4 k ū(p )γ 5 k/ γ µ k/ + m γ µ + m(γ µ k/ + k/ γ µ )]γ 5 u(p) (π) 4 (p k) m + iɛ]k m + iɛ]k m + iɛ] a (55) Th dnominator is idntical to th on from part a), but th numrator diffrs by a minus sign and xtra factors of γ 5. Sinc γ 5 anticommuts with γ µ, and γ 5 γ 5 =, w can rwrit th numrator as: numrator = ū(p )γ 5 k/ γ µ k/ + m γ µ + m(γ µ k/ + k/ γ µ )]γ 5 u(p) = ū(p )k/ γ µ k/ + m γ µ m(γ µ k/ + k/ γ µ )]u(p) ū(p ) γ µ l + (( y)/q + z/p)γ µ ( y/q + zp/) + m γ µ ] mγ µ ( y/q + z/p) + (( y)q/ + z/p)γ µ ] u(p). (56) Plugging Eqs. (47) and (48) into Eq. (56), our numrator bcoms: numrator = ū(p ) γ µ ( l + q xy + m ( + z z )) m(p + p ) µ (x + y)(z ) ] + mq µ (x y)(z ) u(p) = ū(p ) γ µ ( l + q xy + m ( z) ) iσµν q ν m ( z) ] m + mq µ (x y)(z ) u(p) (57) iσ µν q ν As abov, th part w ar intrstd in is th cofficint of m, which givs us th contribution to F. Putting this togthr with th rsults for th dnominator, w hav: δf (q ) = iλ d 4 l m ( z) dx dy dz δ(x + y + z ) (π) 4 l Δ + iɛ] 3 λ m ( z) = dx dy dz δ(x + y + z ) (4π) Δ λ m ( z) = dx dy dz δ(x + y + z ) (58) (4π) xyq + ( z) m + zm a

9 9 For q =, w can valuat th x and y intgrals: λ m ( z) δf () = (4π) dx dy dz δ(x + y + z ) ( z) m + zm a λ z m ( z) = (4π) dz dy ( z) m + zm a λ m ( z) 3 = (4π) dz ( z) m + zm a I(λ, m a ) (59) W can numrically valuat I(λ, m a ) to find th allowd rgion whr I(λ, m a ) <. Th rsults ar shown blow, as a log-log plot: l xcludd rgion 4 3 m a êm allowd rgion Hr is a sampl Mathmatica command to gnrat th plot: tabl = Tabl{ExpM], λ/. NSolvλ /(4 Pi) NIntgrat( x) 3/(( x) + ExpM] x), {x,, }] == ( ), λ]]]}, {M, 5, 5,.3}]; ListLogLogPlottabl, Joind Tru, AxsOrigin {, }] 3. (a) Th frmion slf nrgy contribution du to Yukawa intraction is givn by (w us dimnsional rgularization) ( φ λ ) d D k i(k/ + m ) i iσ (/p) = i (π) D k m (p k) m h (6) λ d D l E m + zp/ = i dz, (π) D (l + Δ + iɛ) E whr in th numrator w hav droppd th linar trm in l = k zp and Δ = ( z)m +zm h z( z)p. Prforming th intgral ovr l givs Thn w hav λ ( 4πµ ) ɛ/ ( ɛ ) (p /) = dz (m 3π + z/p) Γ Δ ( ( )) (6) λ 4πµ = dz (m 3π + z/p) ɛ γ + log Δ Σ φ dσ φ λ ( ( 4πµ )) z( z)( + z)m ] δ φ Z = /p=m = dz z d/p 3π ɛ γ + log + Δ Δ (6)

10 whr Δ = Δ (p = m ) = ( z) m + zmh. For Z, w hav to comput th contribution to th vrtx with an xchang of a Higgs. Fortunatly, w hav alrady don most of th work in Problm. Copying th numrator form (49), th dnominator from (5) and stting q = w gt ū(p )δ φ Γ µ u(p) = iλ 4 d l ( dx dy dz δ(x + y + z ) ū(p)γµ D D l + m ( + z) ) u(p) (π) 4 l Δ 3 ] ( ) = u(p) dz( z) l u(p)γ µ d D l E λ (π) D (l + Δ ) DD 3 E + ( + z) E whr onc again Δ = ( z) m +zmh and w payd attntion to D whn doing th Dirac algbra unlik in Problm, whr w wr calculating a finit quantity. Prforming th intgral ovr l E and taking D = 4 ɛ λ ( 4πµ ) ( + z) m ] δ φ Z = dz( z) 3π ɛ γ + log + (63) Δ Δ Thn th diffrnc is λ ( 4πµ )] ( + z)( z) ] δ φ (Z Z ) = dz ( z) 3π ɛ γ + log ( z) + m (64) Δ Δ Intgrating by parts (th boundary trms vanish), w gt ( ) µ h dz( z) log = dz z( z) ( z)m + m (65) Δ Δ Thus λ δ φ (Z Z ) = dz z( z) m + z( z)m h ( z)δ + m ( + z)( z) ] = (66) 3π Δ W can asily s that this rsult combins to zro. Th Ward idntity will b tru in this thory. This can b sn in many diffrnt ways. Firstly, th Yukawa trm bhavs th sam way as a frmion mass trm undr gaug transformation (assuming that φ is not chargd undr U()), hnc th gaug invarianc is maintaind. Th Ward idntity is th consqunc of gaug symmtry, hnc it must b tru with this nw intraction. Scondly, w not that th addition of a frmion-frmion-scalar vrtx dos not affct th diagrammatic proof of th Ward idntity at all, sinc th photon dos not coupl to φ. (b) By analogy with th QED lctron vrtx Γ µ (p, k) w dfin th xact Yukawa vrtx with two on-shll lctron lins and on off-shll scalar lin as ] λ ū(p ) i Γ(p, p) u(p) (67) and dfin th rnormalization factor as Z y = Γ(q = ). At tr lvl Z y =, at on loop lvl δz gts contributions from th diagrams with on virtual photon and scalar fild: p p m k p k k p k p p d 4 k ig µν i(k/ + m) i(k/ + m) ū(p )δγ(q = )u(p) = (π) 4 ū(p ) (k p) + iɛ µ ( iγν ) k m + iɛ k m + iɛ ( iγµ )u(p) ( ) ( ) ] i iλ i(k/ + m) i(k/ + m) iλ +ū(p ) (k p) m h + iɛ k m + iɛ k m u(p) (68) + iɛ

11 Making us of th Fynmann paramtrs whr δz y = dz( z) d 4 l E γ ν (/l E + zp/ + m) γ ν λ (/l E + z/p + m) ] (π) 4 (l + Δ ) 3 (l + Δ ) 3 (69) E E Δ = ( z) m + zµ (7) Δ = ( z) m + zm h (7) W ar instructd to calculat just th UV divrgnt parts, hnc w only kp th le trms in th numrator. Hnc, th intgrals ovr th momnta and z giv ( ) α λ δz y = + + finit trms. (7) π 3π ɛ Using th rsult obtaind for th lctron slf nrgy diagram in part (a) and th photon contribution from class ( ) α λ δz = 4π + 64π (73) ɛ Thus δz y = δz. Not howvr, that by including a countr trm for th Yukawa intraction (and for vry possibl trm in th Lagrangian allowd by symmtry) all th divrgncs can b mad to cancl. Z = Z y has nothing to do with gaug symmtry, which is rprsntd by Z = Z and hnc thr is no pathology (i.. anomalous symmtry) in this thory.

12 MIT OpnCoursWar Rlativistic Quantum Fild Thory II Fall For information about citing ths matrials or our Trms of Us, visit:

PHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS

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