19 Quantum electrodynamics

Transcription

1 Modern Quantum Feld Theory 77 9 Quantum electrodynamcs 9. Gaugng the theory Consderng the Drac Lagrangan L D m we observed the presence of a U( symmetry! e, assocated wth the Noether current j µ = e( µ. Consderng now to be a functon of space-tme = (x, we note (x e so t s no more nvarant. Nevertheless we may restore the symmetry by addng an nteracton term of the form A µ j µ A µ s the vector potental of an electromagnetc feld, transformng as A µ (x! A µ (x µ (x. Ths leads to the followng (bare Lagrangan: L = (@/ ea/ m 4 F µ F µ, (33 whch s the Lagrangan of quantum electrodynamcs (QED. Ths Lagrangan s nvarant under the gauge transformaton: (x! e (x (x; A µ (x! A µ (x µ (x. (33 Ths can be most convenently verfed by defnng a covarant dervatve: D µ +ea µ, such that D µ transforms covarantly under the gauge transformaton, namely D µ (x! e (x D µ (x Whle the nteracton n ths theory s dctated by gauge symmetry, we have already seen that n order to do calculatons n ths theory we need to fx the gauge (ths was shown n the free Maxwell theory, but holds n any nteractng gauge theory. Specfcally, n order to have a consstent path-ntegral formulaton (see eq. (35 above we ntroduce a gauge fxng term: L = (@/ ea/ m 4 F µ (@ µ A µ F µ. (33 9. Renormalzaton n QED The QED Lagrangan nvolves dmensonless constants n 4 space-tme dmensons, so we expect t to be renormalzble. Moreover, the Lagrangan already ncludes all terms allowed by symmetres (note that nvarance under party excludes µ F µ F. Thus all we need to renormalze the theory s to ntroduce renormalzaton constants n front of each term. Let us splt the full Lagrangan as usual, nto the free-theory Lagrangan L, and the nteracton terms L such that L = L + L L m 4 F µ (@ µ A µ F µ (333a L = e A/ ( m m ( 3 4 F µ F µ (333b The generatng functonal correspondng to the free feld s known: t s gven by the product of the one computed above for free fermons (3 tmes the one computed above for free photons (36. The correspondng free propagators are gven, respectvely, by (3 and (38. The latter, as emphaszed already, depends on the gauge parameter, ed F (k = k + g ( k k k dependence whch s expected to cancel out n physcal quanttes. 9.3 The renormalzed propagator Summng any number photon-self-energy (vacuum polarzaton nsertons, (334 Consderng the renormalzed theory, let us examne frst the full propagator. In analogy wth the scalar case, eq. (39, we defne the self-energy functon (k as the sum of all one-partcle rreducble Feynman dagrams

2 Modern Quantum Feld Theory 78 wth two external photon lnes (not ncludng the external propagators. We then obtan the full propagator through a seres representng any number of self-energy nsertons n a sequence: e µ (k = e D µ (k + e D µ (k (k e D (k + e D µ (k (k e D (k (k e D (k + O(e 6, (335 In order to sum up ths seres t s useful to treat separately transverse and longtudnal components. We wrte the propagator (354 as ed F (k = k T + k k + k, (336 T µ g µ k µ k k We have already seen that T µ acts as a transverse projecton operator, namely that and that k µ T µ = T µ T = T µ. The crucal property whch we shall now use wthout proof, and then check below at the one-loop level, s that the self energy µ (k s transverse,.e. that k µ µ (k = and therefore µ (k =k (k T µ (k. (337 Ths s expected f µ (k tself should be a physcal quantty (and ndeed t s, owng to the Ward dentty, but ths s beyond the materal of ths course. Interested students are encouraged to read Secton 7.4 n Peskn and Schröder. The extra factor n each term of (335 (a propagator tme s of the form: (k D e (k =k (k T (k T k (k+ k k + k = k (k k T (k + we used the propertes of T. The nterpretaton s that because s transverse the nserton of an extra tmes a propagator s also transverse. Snce T s a projecton operator ths remans true for any number of teratons, e.g. for n teratons we get: n k (k k T (k. + It s further noted that the transverse projecton acts n a smlar way on the free propagator to the left so the second term n (335 s T k µ (k+ kµ k + k k (k k T (k = + k k (k + k T µ (k + Ths s the n = term n our geometrcal seres. Smlarly any n> term would take the form: n k k (k + k T µ (k + However, there remans the frst term n (335 whch has two components: ed µ (k = T k µ (k+ kµ k + k, the frst, transverse component, fts the structure of the rest of the geometrcal seres, provdng the n = term there, whle the second, longtudnal component remans as an addtve term whch receves no correctons. We thus conclude that the sum of the seres s: whch may also be wrtten as e µ (k = e µ (k = k + k " T µ (k k (k k + T µ (k k (k + + k + # + kµ k k (338 k µ k k (339

3 Modern Quantum Feld Theory 79 k q+k q Fgure 9: The one-loop dagram contrbutng to the QED vacuum polarzaton (the photon self energy One-loop calculaton of the vacuum polarzaton Let us now evaluate µ (k at one-loop order. There s one dagram contrbutng, shown n fgure 9. As before we use dmensonal regularzaton to deal wth the expected ultravolet dvergence. The expresson correspondng to ths dagram s: bubble µ (k =, accordng to (333b, each vertex contrbutes ( ndces. Here we have used the notaton d d q ( d Tr ( e µ S(q( e S(k + q S(q = e µ and the fermon loop amounts to mnus trace over spnor (q/ + m q m + for the free fermon propagator. Substtutng ths we obtan: bubble d d q µ (k = ( d Tr (q/ + m e µ q m + whch may be wrtten as bubble µ (k = e N µ (q, k =Tr d d q ( d N µ µ(q/ + m (k/ + q/ + m e (k/ + q/ + m (k + q m + (34 q + m (k + q + m (34 =4g µ (m q (k + q + 8q µ q + 4(q k µ + q µ k Next, usng Feynman parametrzaton bubble µ (k = e d d q ( d N µ (q, k x( (k + q + m +( x( q + m = e d d q ( d N µ (q, k q + m xk q xk = e d d l ( d N µ (l xk, k l + m x( xk n the last stage we completed the square and defned a new ntegraton varable l = q + xk. In terms of ths varable the numerator takes the form: N µ (l xk, k =4g µ (m (l xk (l +( xk + 8(l xk µ (l xk + 4((l xk k µ +(l xk µ k =4g µ (m l + x( xk +8l µ l 8k µ k x( x + terms that are lnear n l Here we dropped all the terms that are lnear n l: these vansh owng to the fact that the rest of the ntegral s sphercally symmetrc. We therefore conclude that bubble µ (k = e d d l 8k µ k x( x+4g µ (m l + x( xk +8l µ l ( d (343 l + m x( xk (34

4 Modern Quantum Feld Theory 8 A new feature we encounter n (343 as compared to the calculatons we have done n the scalar theory, s the need to evaluate tensor ntegrals. In partcular, we shall need the result: d d l l µ l ( d ( l + M n = gµ d d l l d ( d ( l + M n (344 The numerator becomes N µ! N µ =4 g µ (m l + x( xk + d l g µ k µ k x( x Performng now Wck rotaton to Eucldean space we get: µ d d bubble (k = l E g µ (m + x( xk ( 4e d g µ le k µ k x( x ( d (345 l E + m x( xk We now defne the scale M = m x( xk Next, usng known ntegrals (see (97 d d d l E ( d (le + M n = (n (4 d/ (n M d n (346 and d d l E ( d le (le + M n = d (4 d/ (n d (n we can evaluate all the d-dmensonal momentum ntegrals gettng: " µ d d bubble (k = l E g µ (m + x( xk k µ k x( x 4e ( d l E + M M d + n (347 d d l E ( d g µ le ( d l E + M # = 4e (4 d/ ( d " g µ (m + x( xk k µ k x( x ( d g µ d ( d ( M d # M d (348 Some smplfcaton yelds: bubble (k = 4e d (4 d/ µ M d " g µ (m + x( xk k µ k x( x g µ m x( xk # (349 We note that the m terms cancel and the result s µ bubble (k = g µ k k µ k 8e d (4 d/ whch ndeed has a transverse Lorentz structure, as expected. x( x M d ( Exercses. Show that the expresson for the bubble dagram (34 may be wrtten as: bubble µ (k = e d d q ( d N µ q + m (k + q + m (35 N µ (q, k =4g µ (m q (k + q + 8q µ q + 4(q k µ + q µ k Gudance: Use the followng formulae to compute the trace over spnor ndces: Tr( µ =4g µ, Tr( µ = 4(g µ g g µ g + g µ g (35

5 Modern Quantum Feld Theory 8 The expresson for the numerator s: N µ (q, k =Tr. Startng wth (35 verfy eq. (343. µ(q/ + m (k/ + q/ + m = m Tr( µ + 4(g µ g g µ g + g µ g q (k + q =4m g µ +4q µ (k + q + 4(k + q µ q 4g µ q (k + q =4g µ (m q (k + q + 8q µ q + 4(q k µ + q µ k 3. Verfy that eq. (344 holds by contractng both sdes wth the metrc tensor. 4. Startng wth (345, verfy eq. (358.

6 Modern Quantum Feld Theory 8 Loop correctons and the beta functon n QED In the prevous lecture we started to study QED at the loop level. We have wrtten the Lagrangan densty as L = L + L L m 4 F µ (@ µ A µ F µ (353a L = e A/ ( m m ( 3 4 F µ F µ (353b such that the generatng functonal of the free theory (J,, may be computed explctly wth the momentum space propagators beng: DF (k = k g ( k k + k. (354 for the photon, and S (x y =h T (x (y d 4 p = ( 4 e p (x y (p/ + m p m + for the fermon. The nteracton and counterterms n (353b can then be taken nto account by applyng the functonal dervatves on (J,,.. Renormalzng the photon propagator Our frst step was to compute the renormalzed photon propagator, n whch radatve correctons are contaned n the one-partcle rreducble two pont functon, the photon self-energy, whch s expected to have a transverse Lorentz structure: µ (k =k (k g µ k µ k. Consequently the full propagator takes the form: e µ (k = k T µ (k k (k + k + k + At one-loop we have a sngle dagram, a fermon bubble, and the counterterm, gvng: µ (k = bubble µ (k+k for the bubble we found µ bubble (k = g µ k k µ k 8e d (4 d/ eµ g µ + k µk k (355 k µ k k (356 ( 3 (357 x( x m x( xk d whch ndeed has a transverse structure. Here the factor eµ s ntroduced (as conventonal n dmensonal regularzaton such that the couplng e remans dmensonless. Usng d = 4 we have bubble (k = 8 ( eµ (4 x( x m x( xk the fne structure constant = e /(4. Expandng the counterterm 3 =+ ( 3 + O(, we get from (357: (k = bubble (k ( 3 + O( (36 Expandng the result n and defnng µ =4 eµ e E we get: (k 8 ( = (4 x( x m x( xk ( 3 + O( = apple µ x( x +ln m x( xln x( x k m = µ 3 +ln m + x( xln x( x k m ( 3 + O( ( 3 + O( (358 (359 (36

7 Modern Quantum Feld Theory 83 k p p k p Fgure : The one-loop dagram contrbutng to the QED electron self energy and the correspondng counterterm. we performed the ntegral over the Feynman parameter x n the frst term. At ths pont we need to choose the renormalzaton condton. We shall renormalzed on shell, such that (k = =. (36 Ths yelds: ( 3 = 3 (k = µ +ln m x( xln x( x k m (363 + O( (364. Renormalzng the electron propagator Next consder the electron propagator. Frst, t s useful to wrte the free propagator as d 4 p S(x y = ( 4 e p (x y (p/ + m d 4 p p m = + ( 4 e p (x y p/ m + (365 Consderng now the Källen-Lehmann representaton of the full propagator we have: es(p/ = p/ m + + m th ds p/ (s p (366 s + (s s the spectral-densty functon. There s however a d culty here: whle sngle-partcle pole occurs at m (the electron on-shell, or pole mass also the threshold m th = m: a state wth a soft photon (or, n fact any number of soft photons and an electron may have a mass as low as the electron mass. Because the sngle-partcle state s not solated, the loop-corrected propagator would be nfrared dvergent on-shell. Ths s an example of an nfrared sngularty. More generally, t s worthwhle notng that also scatterng ampltudes n QED are, n general, nfrared dvergent. These are called soft sngulartes and they are related to the possblty to emt soft (zero energy photons. If we further take the electron mass to be zero there would also be collnear sngulartes. In any case, cross sectons would stll be fnte upon summng up vrtual and real emsson dagrams. To compute ndvdual contrbutons to the cross secton (whch are separately dvergent we would need to ntroduce an nfrared cuto. One way to do so s to assgn the photon a mass m. Eventually, when computng a physcal quantty, the lmt m! should be taken, and then t wll e ectvely be replaced wth some expermental resoluton (a smlar problem has been seen n the massless lmt of the scalar theory, exercse 3 n lecture 4. An alternatve s to use dmensonal regularzaton wth d>4, whch would render all logarthmcally nfrared-dvergent ntegrals fnte, generatng extra poles n. In ths case care should be taken to separate ultravolet sngulartes, whch are removed through the renormalzaton process, from nfrared ones, whch are not, and only cancel at the level of the cross secton. Consderng an nfrared-regulated photon propagator DF (k = k m g ( k k + k. (367 s su cent for dealng wth all nfrared dvergences. Havng done that we can stll use the on-shell scheme. Returnng to the electron propagator, we ntroduce the electron self-energy functon (p/, as the one-partcle rreducble two-pont functon. In terms of ths functon, the geometrc seres correspondng to any number of nsertons may be computed, gvng: es(p/ =. (368 p/ m + (p/+ Comparng ths expresson to the Källen-Lehmann representaton (366 we see that on-shell renormalzaton smply corresponds to (p/ = m =, (p/ = m =. (369

8 Modern Quantum Feld Theory 84 These two condtons guarantee that the electron mass m s the poston of the physcal pole, and that the resdue s, as n the free theory. As shown n fgure, at one loop there s just one dagram and a counterterm: (p/ = e loop (p/+( p/ ( m m (37 e loop (p/ = Insertng the free fermon propagator e S(p e loop (p/ = d d k ( d ( e µ e S(p k( e D µ F (37 k and the regularzed photon propagator (367 we get: d d k ( d ( e (p/ k/ + m µ (p k m ( e + k m + g µ ( kµ k k (37 It s most convenent at ths pont to pck the Feynman gauge =, we get: e loop (p/ = e d d k µ (p/ k/ + m ( d (p k m + µ k m + (373 Here we need to use a couple of relatons nvolvng the Drac matrces n d dmensons: µ µ = d, µ µ =( d (374 whch yeld e loop (p/ = e d d k (p/ k/( d+md ( d (p k + m k + m (375 Usng now Feynman parametrzaton we get: e loop (p/ = e d d k ( d = e (p/ k/( d+md x( (p k + m +( x( k + m d d l (( xp/ l/( d+md ( d l p x( x+xm +( xm (376 n the second lne we completed the square n the denomnator and changed varables to l = k note that the ntegral over l/ vanshes. Wck rotatng to Eucldean space we get: e loop (p/ = e d = e ( (4 d/ = e ( (4 d d l E (( xp/( d+md ( d le p x( x+xm +( xm h (( xp/( d+md h (( xp/( + +m(4 p x( x+xm +( xm d p x( x+xm +( xm px. Wenow n the second lne we performed the loop momentum ntegral usng (346 and n the thrd we substtuted d =4. Let us now examne the convergence of the ntegral (377 the e ectve scale s: M = p x( x+xm +( xm. The branch pont for m! s located at p = m, such that the ntegral s well defned for p <m and has a cut for p >m, an electron can be n the state of an electron plus any number of real photons. The power M n the ntegral guarantees convergence for!, however, n order to renormalze on shell, we need not only the functon tself but also t frst dervatve evaluate at p/ = m. We wll see that for the frst dervatve to be well defned, we need to keep the photon mass n place. If m = the photon mass becomes the only scale n the problem; n ts absence we would get scale-less ntegral. Whle such ntegrals are defned to be zero n dmensonal regularzaton, ths zero hdes ultravolet and nfrared senstvty. (377

9 Modern Quantum Feld Theory 85 Fgure : The frst two terms contrbutng to the QED vertex functon V µ (or form factor accordng to (38. In the followng we dentfy the ultravolet sngular terms. In ths calculaton we take m 6= and set m = (note that the / terms we obtan are of ultravolet orgn. Infrared sngulartes only emerge upon performng the x ntegral. Expandng n we get: h (( xp/ +4m e loop (p/ = 4 = h p/ +4m + p/ m 4 Includng the counterterms as n (37 we get: (p/ = We see that h p/ +4m + p/ m 4 ( = + (( xp/ m ( (( xp/ +4mln(M /µ ( (( xp/ +4mln(M /µ (378 ( (( xp/ +4mln(M /µ +( p/ ( m m + O( ( fnte, ( m = +fnte. (38 We can now mplement the on-shell renormalzaton condtons (369 to obtan (p/ n ths scheme. In dong ths the nfrared regulator m needs to be kept..3 Renormalzng the QED vertex functon Fnally, we turn to the QED vertex functon. We defne the one-partcle rreducble three pont functon V µ (p,p correspondng to the e µ A µ nteracton, the ncomng electron momentum s p, the outgong one s p and the photon momentum s k = p p. The expanson takes the form: µ loop (p,p= V ee V µ (p ee loop,p= e µ +Vµ (p,p+... (38 d d l ( d ( e e S(p + l( e µ e S(p + l( e D F (l (38 The calculaton proceeds along smlar lnes to the one of the electron self-energy. The resultng dvergent term n the one-loop renormalzaton constant s ( = + fnte ( The QED beta functon at one loop The us apply the consderatons of the renormalzaton group ntroduce n lecture 5 to the case of the QED couplng. Recall that the nteracton term n the Lagrangan may also be wrtten n terms of bare couplng e and bare felds as follows: L I = e eµ A = e A (384 Recall also the relatons between the bare feld and the renormalzed felds whch read: = 3 A = A (385a (385b Usng these n (384 we get that eeµ = e 3 = e = eeµ 3 = = eµ 3 (386

10 Modern Quantum Feld Theory 86 Takng a logarthm and d erentatng wth respect to the scale the lhs vanshes and we get: = d ln d ln µ + + d ln( d 3 d ln d ln µ (387 Ths allows to compute the QED beta functon. Let us demonstrate t usng the one-loop calculatons we have done above: 3 = 3 +fnte, (388a = 4 +fnte, (388b = 4 +fnte (388c We notce that = = 3 = 3, (389 ths equalty can be seen to be general, namely t perssts to all loop orders as a consequence of the QED Ward dentty. As a result the relaton (387 above smplfes to = d d ln µ apple d ln 3 d (39 whch yelds at one-loop order: = d d ln µ apple + 3 = d d ln µ = 3 + O( = O( 3 {z } ( (39 Ths s the well known result for the QED beta functon: t s nfrared free, meanng that the couplng s weak at low energy scales (µ! and t s ncreasng for hgh scales. Interestngly, an opposte stuaton occurs n non-abelan gauge theores such as QCD (provded tehre aren t too many fermons..5 Exercses. Reproduce the calculaton of the electron self energy at one-loop, determne the and m renormalzaton constants n the on-shell scheme as well as the renormalzed electron self-energy functon n that scheme. In partcular: (a Usng the d-dmensonal relatons n (374 to show that (373 takes the form n eq. (375. (b Verfy eq. (376. (c Verfy eq. (377. (d Implement the on-shell renormalzaton condtons (369 and determne (p/ n the on-shell scheme. (e Repeat the calculaton n a general covarant gauge startng from eq. (37, demonstrate that (p/ s n general, gauge-dependent, and then show that the n the on-shell lmt,.e. for (p/ = m, there s no gauge-dependence.