Problem 9.1: Scalar QED

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Charles Boone
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1 Problm 9.1: Scalar QED This problm concrns th thory of a complx scalar fild φ intracting with th lctromagntic fild A µ. Th Lagrangian is L 1 4 F µν + D µ φ D µ φ m φφ φ, 1 whr D µ µ + ia µ is th usual gaug-covariant drivativ. a Us th functional mthod of Sction 9. to show that th propagator of th complx scal fild is th sam as that of a ral fild: p i p m φ + iɛ Also driv th Fynman ruls for th intractions btwn photons and scalar particls; you should find µ ν p µ ip + p µ p i g µν 3 b Comput, to lowst ordr, th diffrntial cross sction for + φφ. Ignor th lctron mass but not th scalar particl s mass, and avrag ovr th lctron and positorn polarizations. Find th asymptotic angular dpndnc and total cross sction. Compar your rsults to th corrsponding formula for + µ + µ. c Comput th contribution of th chargd scalar to th photon vacuum polarization, using dimnsional rgularization. Not that thr ar two diagrams. To put th answr into th xpctd form, Π µν q g µν q q µ q ν Πq, it is usful to add th two diagrams at th bginning, putting both trms ovr a common dnominator bfor introduction a Fynman paramtr. Show that, for q m, th chargd boson contribution to Πq is xactly 1/4 that of a virtual lctron-positron pair. 9.1 a Lt us start with th action for th thory S d 4 x 1 4 F µν + D µ φ D µ φ m φφ φ d 4 x 1 4 F µν + µ φ µ φ ia µ [φ µ φ µ φ φ] + A µ A µ φ φ m φφ φ d 4 x 14 F µν φ + m φφ ia µ [φ µ φ µ φ φ] + A µ φ IBP S γ + Sφ + S int. 4 1

2 Th fr thory gnrating functional is givn by Z [J m, J s, Js ] DA Dφ Dφ isγ +i d 4 x A µj µ m +isφ +i d 4 x {J s φ+φ J s} Z [,, ] i d 4 x d 4 y 1 J µ m xdµνx yj ν m y+j s xdx yjsy. 5 Hr, w assum that th Faddv and Popov procdur has bn prformd to rstrict th path intgral ovr A to physically uniqu fild configurations. W hav also ignord th ovrall infinit constant factor that this procdur gnrats rcall that w ar usually intrstd in ratios. Expanding th xponntial with th intraction trm w obtain th full gnrating functional as a sris in Z [J m, J s, Js ] DA 1 + Dφ Dφ Z [J m, J s, Js ] 1 + i d 4 x A µx φx + O 4 d 4 z A µ z [φ z µ φz µ φ zφz] + O. 6 W can now rad off th propagator and intraction vrtics. Th scalar fild propagator is Tφx 1 φ d 4 i x Dx 1 x iɛi x1 x π 4 p m φ + 7 whil th photon propagator is TA µ x 1 A ν x D µν x 1 x Fourir transforming th filds in th trm, d 4 z A µ [φ z µ φz µ φ zφz] i yilds th vrtx d 4 p π 4 d 4 d 4 z π 4 d 4 p d 4 p π 4 d 4 p π 4 d 4 i π 4 g µν 1 ξ µ ν iɛ. 8 Ãµ φ p φpp µ + p µ iz p p π 4 Ãµ p p φ p φp ip µ + p µ 9 p µ ip + p µ 1 p whil th trm, i d 4 x A µx φx, yilds th vrtx µ ν i g µν. 11 whr th coms from symmtry in th two A filds.

3 9.1 b To lowst ordr + φφ is givn by th diagram: + Applying th Fynman Ruls w obtain p p q φ φ im + φφ. 1 ig µν im + φφ vp iγ µ up p + p i + ν i vp + up p + p. 13 Th spin avragd matrix lmnt is 1 M + φφ p + p Tr [ p m m p + m m ] spin 4 p + p 4 p [p p ] + p [p p ] + p p [ ] + m [ + ]. 14 In th cntr of mass fram p p p + p E p, and + E,. This mans that E p E E. W also assum that p m so that E p. W nd th dot products p p E p E p cos θ E E cos θ 15 p p E p E + p cos θ E E + cos θ 16 p p E p + p E 17 E + E + 18 With ths rlations and taing m, th spin avragd matrix lmnt squard bcoms 1 M + φφ E E cos θ m φ E spin 8π α sin θ, 19 whr 1 m φ /E is th vlocity of th particls and α 4π is th fin structur constant. Sinc this is a two particl raction w can us th simplifid cross sction formula 4.84 dσ + φφ dω 1 1 E 16π E 4 M + φφ spin α 3 4E sin θ. 3

4 Intgrating ovr th polar coordinat yilds th total cross sction σ + φφ πα 3 Comparing our rsults to thos of th + µ µ + dσ + µ µ + dω α 4E [ σ + µ µ + 4πα 3E 1 + m µ E 3E m µ E + 1 m µ E cos θ ], w s that th angular dpndnc of th diffrntial cross sctions ar vry diffrnt. Th scalar particls hav a ar mor lily to mov prpndicular to th lctron bam axis whil th muons ar mor lily to mov along th lctron bam axis. 9.1 c Th scalar contribution to th photon vacuum polarization is givn by th following diagrams q µ ν im µν q q 1 3, and µ ν im µν q q. 4 Now applying th Fynman ruls in d-dimnsions w obtain im µν 1 im µν d d i q µ i i q ν i π d m φ iɛ q m φ + iɛ, d d π d q µ q ν m φ iɛ q m 5 φ + iɛ, d d i g µν i π d m φ iɛ, d d g µν π d m 6 φ iɛ. 4

5 Following th hint w add ths diagrams togthr bfor introducing Fynman paramtrs iπ µν im µν 1 + im µν d d i q µ i i q ν i π d m φ iɛ q m φ + iɛ + i g µν i m φ iɛ d d q µ q ν g µν q m φ π d m φ iɛ q m φ + iɛ d 4 4 µ ν µ q ν ν q ν + q µ q ν g µν q + q m φ π d m φ iɛ q m φ + iɛ. 7 Introducing th Fynman paramtrs x, y w hav 1 iπ µν dx 1 dy δx + y 1 d d π d 4µ ν µ q ν ν q ν + q µ q ν g µν q + q m φ [x ] + y y q + yq x + ym φ iɛ 1 dx d d π d 4µ ν µ q ν ν q ν + q µ q ν g µν q + q m φ [ ]. 8 x q + xq m φ iɛ By ltting l xq w complt th squar in th dnominator iπ µν dx dx dx dx dx d d l 1 π d [l ] 4l + xq µ l + xq ν l + xq µ q ν l + xq ν q µ + q µ q ν g µν l + xq l + xq q + q m φ, d d l 1 π d [l ] 4l µ l ν + xl [µ q ν] + x q µ q ν l µ q ν + xq µ q ν l ν q µ + xq ν q µ + q µ q ν g µν l + xq l + x q l q + xq + q m φ, d d l 1 π d [l ] 4l µ l ν + x q µ q ν xq µ q ν xq ν q µ + q µ q ν g µν l + x 1 q m φ, d d l π d 4l µ l ν g µν l + x 1 q µ q ν g µν x 1 q + g µν m φ [l ], d d l π d 4 d gµν l + x 1 q µ q ν g µν x 1 q + g µν m φ [l ], 9 5

6 whr w hav ignord trms linar in l and x1 xq + m φ. Intgrating ovr l yilds iπ µν 1 1 [ dx 4 g µν I 1 + ] x 1 q µ q ν g µν x 1 q + g µν m d φ I, [ dx 4 g µν I 1 + g µν m d φi + x 1 q µ q ν x 1 g µν q ] I, 3 whr I I 1 d d l 1 π d l i Γ d/ 4π d/ d/ Γ, 31 d d l l π d l i d 4π d/ Γ1 d/ 1 d/ Γ d d I. 3 Thus, iπ µν 1 [ d 4 dx d + m φ g µν + x 1 q µ q ν x 1 g µν q ] I, 1 dx [ x1 xq g µν + x 1 q µ q ν x 1 g µν q ] i 4π ɛ 1 dx [ x 1 q g µν q µ q ν + x 1g µν q ] ɛ i 4π 16π Γɛ, iα 1 dx [ x 1 q g µν q µ q ν + x 1g µν q ] [ 1 4π γ 4π ɛ + log E Γɛ ɛ Γ, ]. 33 For th last trm in th first bracts w chang variabls y x 1. Thn 1 x y whil y 1 4 q m φ. Sinc is vn in y whil th last trm in th first bracts is linar th intgral is odd and vanishs. Thus, iπ µν iα 4π iα 4π q g µν q µ q ν 1 [ 1 4π dx 1 x γ ɛ + log E q g µν q µ q ν 1 [ 1 4π dx 1 x γ ɛ + log E ], ], q g µν q µ q ν iπq, 34 whr iπq α 4π 1 [ dx 1 x 1 ɛ + log m φ ] 4π γ E. 35 x1 xq Th physically rlvant part, iˆπ, is iˆπq iπq iπ α 4π 1 dx 1 x log m φ m φ x1 xq. 36 6

7 Problm 9.: Quantum statistical mchanics a valuat th quantum statistical partition function Z Tr[ H ] using th stratgy of sction 9.1 for valuating th matrix lmnts of iht in trms of functional intgrals. Show that onc again on finds a functional intgral, ovr functions dfind on a domain that is of lngth and priodically connctd in th tim dirction. Not that th Euclidan form of th Lagrangian appars in th wight. b Evaluat this intgral for a simpl harmonic oscillator, L E 1 ẋ + 1 ω x, by introducing a Fourir dcomposition of xt: xt n x n 1 πint/. Th dpndnc of th rsult on is a bit subtl to obtain xplicitly, sinc th masur of th intgral ovr xt dpnds on in any discrtization. Howvr, th dpndnc on ω should b unambiguous. Show that, up to a possibly divrgnt and -dpndnt constant th intgral rproducs xactly th familiar xprssion for th quantum partition function of an oscillator. [You many find th idntity sinh z z 1 + z nπ usful.] c Gnraliz this construction to fild thory. Show that th quantum statistical partition function for a fr scalar fild can b writtn in trms of a functional intgral. Th valu of this intgral is givn formally by [ dt + m ] 1/ whr th oprator acts on functions of Euclidan spac that ar priodic in th tim dirction with priodicity. As bfor, th dpndnc of this xprssion is difficult to comput dirctly. Howvr, th dpndnc on m φ is unambiguous. Mor gnrally, on can usually valuat th variation of a functional dtrminant with rspct to any xplicit paramtr in th Lagrangian. Show that th dtrminant indd rproducs th partition function of rlativistic scalar particls. d Now lt ψt, ψt b two Grassmann-valud coordinats, and dfin a frmionic oscillator by writing th Lagrangian L E ψ ψ + ω ψψ. This Lagrangian corrsponds to th Hamiltonian H ω ψψ, with { ψ, ψ} 1; that is, to a simpl two-lvl systm. Evaluat th functional intgral, assuming that th frmions oby anti-priodic boundary conditions: ψt + ψt. Why is this rasonabl? Show that th rsult rproducs th partition function of a quantum-mchanical two-lvl systm, that is, of a quantum stat with Frmi statistics. 7

8 Dfin th partition function for th photon fild as th gaug-invariant functional intgral Z DA xp d 4 1 x E 4 F µν ovr vctor filds A µ that ar priodic in th tim dirction with priod. Apply th gaugfixing procdur discussd in Sction 9.4 woring, for xampl, in Fynman gaug. Evaluat th functional dtrminants using th rsult of part c and show that th functional intgral dos giv th corrct quantum statistical rsult including th corrct counting of polarization stats. 9. a Lt th systm b dscribd by th gnralizd coordinats {q i } and momnta {p i } whr i 1,,..., n. Th quantum mchanical partition function is givn by Z Tr [ H] d n q q H q 37 To valuat 37 w split th tmpratur intrval,, into N qual slics of siz ɛ w will vntually ta th ɛ or quivalntly th N limit. Th partition function bcoms Z d n q q } ɛh. {{.. ɛh } q N tims d n q d n q 1... d n q N 1 q ɛh q 1 q 1... q N 1 q N 1 ɛh q d n q 1... d n q N 1 q 1 ɛh q 1 q... q N 1 q N 1 1 ɛh q + Oɛ 38 W insrt unity in trms of a complt st of momntum stats, 1 dp p p, 39 N tims to gt Z d n q... d n q N 1 d n p... d n p N 1 q p p 1 ɛh q 1 q 1... q N 1 q N 1 p N 1 p N 1 1 ɛh q + Oɛ, d n q... d n q N 1 d n p... d n i N 1 n pn qn p N 1 π n/n 1 p 1 ɛh q 1 q 1... q N 1 p N 1 1 ɛh q + Oɛ, 4 At this point th drivation has bn gnral. Lt us spcify to th standard form of th Hamiltonian, Ĥ ˆp, ˆq 1 ˆp ˆp + V ˆq. 41 m 8

9 Insrtion into 41 yilds Z d n q... d n q N 1 d n p... d n p N 1 i N 1 n pn qn π n/n 1 p q 1 1 ɛhp, q 1 q 1... q N 1 p N 1 q 1 ɛhp N 1, q + Oɛ, d n q... d n q N 1 d n p... d n p N 1 ipn 1 q q1 π n... π n ɛhp,q1... ɛhp N 1,q + Oɛ, d n q... d n q N 1 d n p... d n p N 1 ip q q1 ip q q1 ɛp /m+v q1 d n q... π n d n q N 1 d n p... ip q q1 ɛp /m π n ipn 1 q q1 ɛp N 1 /m+v q... π n + Oɛ, d n p N 1 ipn 1 q q1 ɛp... π n N 1 /m ɛ N 1 V q + Oɛ. 4 Th momntum intgrals can b prformd, dp π N ip q q +1+iɛ p /m m n/ m ɛ q q +1, 43 πɛ to yild m nn 1/ Z πɛ m nn 1/ πɛ d n q... d n q... d n q N 1 N 1 m q q +1 ɛ ɛv q + Oɛ, d n q N 1 ɛ N 1 m q q +1 ɛ +V q + Oɛ. 44 This is of cours th discrtizd form of th path intgral with a Euclidan action, Uq, q N, i Dqi SE[q], 45 whr S E [q] p dτ m + V q, 46 and rprsnts th fact that th path intgral is rstrictd to paths which start and nd in th sam plac. Sinc th initial and final paths diffr by a tim,, w rquir that th paths in th trac b priodic in i.., q q. 9. b W want to valuat th intgral 46 for th simpl harmonic oscillator. Insrting th Fourir dcomposition now xists bcaus from part a w showd that th paths in th trac ar priodic into th Euclidan Lagrangian w obtain, L E 1 n m πin πim x n x m + ω πiτn+m/. 47 9

10 Lts first valuat th action S E [xτ] 1 dτl E dτ 1 n,m n,m πin πim x n x m + ω πiτn+m/ πin πim + ω δ n, m 4π n x n x n + ω 1 n 1 4π x n n + ω. n ω 4π n x + + ω x n. 48 W hav usd that fact that th xpansion of x implis: x n x n rality condition of x. Thus, th path intgral, Z, is a Gaussian [ Z Dxτ xp ω 4π n ] x + ω x n. 49 Hr, th path intgral is undrstood to b ovr th Fourir cofficints of x, Z dx drx 1... drx dimx 1... dimx [ ω 4π n Rxn xp x + + ω + Imx n ], ] dx xp [ ω [ 4π n ] x drx n xp + ω Rx n [ 4π n ] dimx n xp + ω Imx n, π π ω, 4π n + ω π π ω 4π n, 1 + ω π ω π 4πn 4πn ω 4π n ω 1 sinh ω/, ω/ π n 1 N sinh ω/. 5, 1

11 Now th ovrall constant N is not wll dfind, howvr, nithr was th intgration masur. Dividing by in ach intgral will gt rid of th xtra powrs of in N. Part c From part a th fild thortic gnralization follows radily: Z φ x H φ x D φ S E[φ] 51 whr th φ ar priodic, φx x φx x. Th Euclidan action is obtaind from th Minowsi action by Wic rotating th tim componnt of x, x ix. For th fr ral scalar fild, th Euclidan action is S E i d ix d 3 x 1 E µφ mφ 1 d 4 x E µφ + 1 m φφ. 5 To valuat th statistical path intgral w xpand th fild, φ, in its Fourir mods, as w did for th Harmonic oscillator of part b, φx n n πint/ 1 i x φ n, V πint/ 1 i x φ n, 53 V whr w hav tan th limit that spac is a finit volum so that th ar discrtizd. Th Fourir cofficints, φ, may b complx. Howvr, sinc φx is ral, thy must satisfy th rality condition φ n, φ n,. Insrting th mod xpansion of φ into th action, w obtain S E 1 d 4 x E µ φ µ E φ + m φφ 1 d 4 x 1 E V µ πimt/ i x φ m, µ E πint/ i x φ n, mn +m φ πimt/ i x φ m, πint/ i x φ n, 1 d 4 x 1 πimt/ πint/ i x i x φ m, φ n, 4π mn V + m φ mn 1 4π m φ m, φ m, + + m φ m 1 πm φ m, + E m 1 φ, + πm φ m, + E. 54 m> 11

12 Substituting th abov into th statistical path intgral w gt { Z drφ n, dimφ n, xp 1 φ, πm } φ m, + E m> m>,> N N 1 sinhe / E / 1 E 55 which is just th rlativistic partition function for E + m φ. To rlat th partition function to a functional dtrminant, rcall that for som oprator Ô th functional dtrminant is dfind by D φ d 4 1 xφôφ const dtô. 56 Intgrating th Euclidan action by parts w gt that th partition function is { 1 Z D φ xp d 4 xφ Eφ + m φφ } 1 const dt. 57 E φ + mφ φ Part d W ar givn th Lagrangian for a frmionic harmonic oscillator, Th action is givn by S E L E ψ ψ + ω ψψ. 58 dτ ψ ψ + ω ψψ whr th Grassman filds, ψ, ψ ar anti-priodic ψτ + ψτ, ψτ + ψτ. W can xpand th anti-priodic Grassman fild as and th complx conjugat fild as ψτ ψτ n n 59 ψ n πin 1/τ/ 6 ψ n πin 1/τ/ 61 whr ψ n and ψ ar Grassman numbrs. It is asy to s that this xpansion is indd anti-priodic ψτ + n ψ n πin 1/τ/ πin iπ ψτ. 6 Aftr substituting th xpansion for th Grassman filds th Euclidan action bcoms S E n ψ n ψ n [ πin 1/ 1 ] + ω. 63

13 Th functional intgral is thn, Z D ψdψ n d ψ n ψ n nψ n[ πin 1/ +ω] dψ n ψ nψ n[ πin 1/ +ω] [ ] d ψ n dψ n 1 ψ πin 1/ n ψ n + ω n [ ] d ψ πin 1/ n dψ n 1 + ψ n ψn + ω n [ ] πin 1/ + ω n [ ] πi + ω [ ] πin 1/ [ ] πi n + 1/ + ω + ω [ ] [ πi + ω πn ] 1/ + ω. 64 This can b simplifid by using th infinit product dfinition of cosh: cosh x With this, th partition function bcoms x 1 + π n 1/. 65 Z cosh ω/ ω/ + ω/. 66 Th last quation is just th partition function for a two-lvl systm. Part W ar givn th Euclidan Lagrangian for th photon fild, L E 1 4 F µν 1 µa ν µ A ν µ A ν ν A µ 1 µ A ν µ A ν A ν ν A µ + 1 A ν g µν µ ν A µ 67 whr A is priodic in τ with a priod of. Th Euclidan action is and th partition function is Z S E 1 [ D A xp 1 d 4 x E A ν g µν µ ν A µ 68 ] d 4 x E A ν g µν µ ν A µ

14 W must us th FP procdur to impos gaug invarianc, so that w ar only intgrating ovr uniqu fild configurations. Lt GA b a function of th photon fild that whn st to zro ncods th gaug condition. W thn insrt unity into th functional intgral δga Z D A D α δ GA α α dt SE[A]. 7 δα whr A α µ A µ + 1 µα. W chang variabls from A to A α to gt δga Z D A α D α δ GA α α dt S E[A α]. 71 δα Now w choos th gnralizd Lornz gaug for G, GA µ A µ x ωx 7 for any scalar function ω. Nxt not that th dtrminant is indpndnt of A, δ µ A α µ ωx dt dt. 73 δα Thrfor th functional intgral bcoms Z dt D A α D α δ µ A α µx ωx S E[A α ] dt D α D A δ µ A µ x ωx SE[A]. 74 Nxt w intgrat ovr Z with a Gaussian wight cntrd at ω, to gt Nξ D ω d 4 x ω ξ Z Nξdt D α D A D ω δ µ A µ x ωx d 4 ω x E ξ 1 d4 x E A νg µν µ ν A µ Nξdt D α D A d 4 µ Aµ x E ξ 1 d4 x E A νg µν µ ν A µ Nξdt D α D A 1 d4 x E A νg µν 1 1 ξ µ ν A µ 75 whr Nξ is a function of that normalizs th Gaussian intgral and th divrgnt intgral ovr th fild α will cancl in th ratios that dfin corrlation functions. Thrfor w simply writ Z dt D A 1 d4 x E A νg µν 1 1 ξ µ ν A µ. 76 Choosing th Fynman gaug ξ 1 th functional intgral bcoms Z dt D A 1 d4 x E A ν A ν dt D A 1 d4 x E A A D A 1 1 D A 1 d4 x E A A D A 3 1 dt [ ] 4 1 dt d4 x E A 1 A 1 d4 x E A 3 A 3 1 dt 77 14

15 W can now us th rsults of part c in to valuat th functional dtrminant: 1 dt lim 1 m dt + m lim N E / m 1 E > N /

PHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS

PHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS PHYSICS 489/489 LECTURE 7: QUANTUM ELECTRODYNAMICS REMINDER Problm st du today 700 in Box F TODAY: W invstigatd th Dirac quation it dscribs a rlativistic spin /2 particl implis th xistnc of antiparticl