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P 622

by: Gay Boyle

P 622 PHYS

Gay Boyle
GPA 3.57

Radovan Dermisek

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Radovan Dermisek
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This 0 page Class Notes was uploaded by Gay Boyle on Sunday November 1, 2015. The Class Notes belongs to PHYS at Indiana University taught by Radovan Dermisek in Fall. Since its upload, it has received 39 views. For similar materials see /class/233470/phys-indiana-university in Physics 2 at Indiana University.


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Date Created: 11/01/15
Quantum Field Theory PHYSP 622 Radovan Dermisek Indiana University Notes based on M Srednicki Quantum Field Theory ilhnpters 3 I4 I ZI 2628 5 Sim6844 33 6974 3032 8486 75 87 89 29 Review of scalar eld theory Srednicki 5 9 IO The LSZ reduction formula based on 55 In order to describe scattering experiments we need to construct appropriate Unrul and 97ml sum and calculate suturing mwpidude Summary of free theory one particle state alk0 alk id31 alma x vacuum state is annihilated by all a s kllol 0 00 1 then one particle state has normalization klc 27r3 2w 63k k39 w k2 m2 l2 normalization is Lorentz invariant ak alk 27732w 63 k 7 k see eg Peskin amp Schroeder p23 Let s de ne a timeindependent operator ai E d3 f1kal k f1k EX expldk 102402 wave packet wnth Width 8 that creates a particle localized in the momentum space near k1 and localized in the position space near the origin go back to position space by Furier transformation all l0 is a state that evolves with time in the Schrodinger picture wave packet propagates and spreads out and so the particle is localized far from the origin in at t gt ioo alaElO for k1 y k2 is a state describing two particles widely separated in the past In the interacting theory alk is not time independent a1 A guess for a suitable initial state I73 t 125nm a tgta lttgtlogt we can normalize the wave packets so that 1 Similarly let s consider a nal state m t 133100 aluminum where again k 1 7E k 2 and 1 The scattering amplitude is then ltfigt lt0a1roltgta2i ooailt ooa oogtlogt A useful formula awk id3 eikxaH MI 00 deco e aieoo dt aoailttgt 1 1319 mm 14 80eik 590gt 4 dale f1k 14 ei mwg mm 2 z d3k f1k 024 ammo k2 MW 2 2 12 39 wk m iid3k f1kd4x e kmwg V2 m2goa 139 d3k mk 14 61quot 68 32 m2ltoltmgt Integration by parts 2 id3k f1kd4 1 e 39 62 m2ltpw surface term 0 ll Partide IS localized is 0 in free theory but not in interacting one wave acket needed P Eg 19w3 gt 62m2wgw2 Thus we have ai oo aim 239 dale f1k d4z amt 62 mmm or its hermitian conjugate 00 a1 oo i dsk f1k 014 Erika 62 m24px The scattering amplitude we put in time ordering without changing anything figt OITW00a23900ai00u 00l0 is then given as generalized to n i and n fparticles f1 k 63k 7 k1 figt in d4m1 eik11 a m2 a a 0 14161 6 ikllxll3 m2 x 0Tww1 ltpw 1 0gt in ll d4331 eik1m1 0 m2 14333 8 1131 63 m2 gtlt Ochpa1ltp 1l0gt LehmannSymanzikZimmermann formula LSZ Note initial and nal states now have deltafunction normalization multiparticle generalization of klk39 27r3 2w 63k k39 We expressed scattering amplitudes in terms of correlation functions Now we need to learn how to calculate correlation functions in interacting quantum eld theory Comments we assumed that creation operators of free eld theory would work comparably in the interacting theory acting on ground state lt0iealtzi0 lt0Ie iPzea0eiP I0gt P l0 0 lt0lltP0l0 c Is a L0I entz lIWallrllit numbel we want 0ltpz0 0 so that ultimo IO is a single particle state otherwise it would create a linear combination of the ground state and a single particle state we can always shift the field by a constant Lpz39 11 so that 0ltpz0 0 Q one particle state ltp90 v0 Fla WMOFHHIO 6 ipmlplw0l0 is a Lorentz invariant number we want plt000 1 since this is what it is in free eld theory Quito creates a correctly normalized one particle state we can always rescale renormalize the eld by a constant so that pltp00 1 Q multiparticle states pinltPl0 I Pinl6lP ltP0eiP 0 6 i ltpnlltp00gt is a Lorentz invariant number in general ant creates some multiparticle states One can show that the overlap between a oneparticle wave packet and a multiparticle wave packet goes to zero as time goes to in nity see the discussmn In Srednlckl p404l By waiting long enough we can make the multiparticle contribution to the scattering amplitude as small as we want Summary Scattering amplitudes can be expressed in terms of correlation functions of elds of an interacting quantum eld theory W i quotquot d4cc1 MIN a m2 d4sc391 6 Way1 012 m2 gtlt 0Tltpw1ltp 1 l0 LehmannSymanzikZimmermann formula LSZ provided that the elds obey 0l90 vl0 0 plltp0l0 1 these conditions might not be consistent with the original form of lagrangian Consider for example 1 1 2 2 1 3 I 28 cp6ltp 5m so 5990 After shifting and rescaling we will have instead Path integral for interacting eld based on 59 Let s consider an interacting phicubed QFT I Z p6l ltp8ltp me2ltp2 Zggltp3 Ygo with elds satisfying Olsomlo o klwml0gt 010 1 kllk 27032196304 k we want to evaluate the path integral for this theory ZJ E 00J up 61 j39d4m50c1m L Z 08 lt08ltp me2ltp2 Zgggo3 Yltp it can be also written as 2U eifd4m 1m up eifd4m oJcp X eifd4m 1m sa Z0 epsilon trick leads to additional factor to get the correct normalization we require 20 2 1 and for the path integral of the free eld theory we have found Z0J exp gfd lm 143339 JaAa x39v 2U 2 eifd4m 16J39Ex Dcp eifd4m oJltp x Biff 519 200 assumes 0 awaw m2ltp2 thus in the case of L igzwawaw me24p2 Zggzp3 Yga the perturbing Iagrangian is 1 Zggltp3 Let Lct Z P16 Lp6ltp e Zm71m2ltp2 Ygo countertel39m Iagmngian in the limit 9 gt O we expect Y gt 0 and Zi 1 we will nd Y0g and Zi10g2 Let s look at ZJ ignoring counterterms for now De ne ZJ ex 32 d4 iii 21 1 p699 mum 0 exponentials de ned by series expansion m a g 22699 6143 mm v V0 x f d4yd4zJyAy zJltzP P0 210 1 let s look at a term with particular values of P propagators andV vertices number of surviving sources after taking all derivatives E for external is E 2P 3V 3V derivatives can act on 2P sources in 2P 2P3V different ways eg forV 2 P 3 there is 6 different terms V2E0P3 Wwae 1 a dwwwwr W wzfzawgmy 21ltz2d4y3d4z31lty1Altygnglt211 3132222 131 132 2663222 1 1 1 1 dx1dxgiZgg2 EACcl 7 2A1 7 2A1 7 2 symmetry factor V2E0P3 W amp HWWWM WWWW WWWM 3132312 216631222 1 1 1 1 g d031d032 iZgg2 gAWl 1EA1 7962 A1 1 symmetry factor Feynman diagrams 1 Q a line segment stands for a propagator 7AL39 y Q vertex joining three line segments stands for iZggfd4a Q a lled circle at one end of a line segment stands for a source 239 f d4zv Jm egforVE 0 What about those symmetry factors symmetry factors are related to symmetries of Feynman diagrams Symmetry factors we can rearrange three derivatives we can rearrange three vertices without changing diagram ii 63 00 X 20 Efd kyd lz JyAy zJzP we can rearrange tWO sources we can rearrange propagators this in general results in overcounting of the number of terms that give the same result this happens when some rearrangement of derivatives gives the same match up to sources as some rearrangement of sourcesthis is always connected to some symmetry property of the diagram factor by which we overcounted is the symmetry factor 21 9 823 S2x3l Figure 91 All connected di ms with E 0 and V 2 the endpoints of each propagator can be swapped and the effect is duplicated by swapping the two vertices propagators can be rearranged in 3 ways and all these rearrangements can be duplicated by exchanging the derivatives at the vertices 22 WG 6 524 823 S24 S23x3 Figure 92 All connected diagrams with E 0 and V 4 S 2 Figure 93 All connected diagrams with E 1 and V 1 s 22 s 22 S 23 Figure 94 All connected diagrams with E 1 and V 3 24 Figure 95 All connected diagrams with E 2 and V 0 0 2 5 2 822 Figure 96 All connected diagrams with E 2 and V 2 25 E S 22 823 523 3 52 s22 823 S2 1 322 822 Figure 97 All connected diagrams with E 2 and V 391 26 53 Figure 98 All connected diagrams with E 3 and V 1 Age Figure 99 All connected diagrams with E 3 and V 3 823 Figure 910 All connected diagrams with E 4 and V 2 28 R 822 Figure 911 A11 connected diagrams with E 4 and V 4 29 All these diagrams are connected but ZJ contains also diagrams that are products of several connected diagrams eg forV 4 E O P 6 in addition to connected diagrams we also have G O 9 and also and also IN 30 All these diagrams are connected but ZJ contains also diagrams that are products of several connected diagrams eg forV 4 E O P 6 in addition to connected diagrams we also 9 9 A general diagram D can be written as the number of given C in D DHMW D I additional Symmetry faCtO39 particular connected diagram not already accounted for by symmetry factors of connected diagrams it is nontrivial only if D contains identical C s SD HnI I 31 Now Z1I is given by summing all diagrams D ZlJ x 2 D nzl 2 H 01quot m I 139 x H 2 i 01 I m0 quotI39 olt Hexp CI I 4x exp 21 CI thus we have found that Zl is given by the exponential of the sum of connected diagrams any D can be labeled by a set of n s imposing the normalization Z10 1 means we can omit vacuum diagrams those with no sources thus we have 2M expiiW1Ji mm 2 Z 0 17440 MW 0 vacuum diagrams are omitted from the sum 32 If there were no counterterms we would be done ZJ ZlJ in that case the vacuum expectation value of the eld is 0 lt I0 142 J 0 quot 7 mm 1 6 W J0 W1J J0 only diagrams with one source contribute 522 522 S23 the source is removed by the derivative and we nd ammo a my AxyAyy 0g3 we USGd Zg 1 since we know Zg 1092 which is not zero as required for the LSZ so we need counterterm gt 33 Including Ygo term in the interaction lagrangian results in a new type of vertex on which a line segment en s corresponding Feynman rule is iY 1411 at the lowest order of g only H contributes S 1 omzno w iamimm My iAw y was in order to satisfy 0pt0gt 0 we have to choose Y igA0 093 We 1 0 W Note A0 must be purely imaginary so thatY is real and in addition the integral over k is ultraviolet divergent to make sense out of it we introduce an ultraviolet cutoff A gt m and in order to keep Lorentztransformation properties of the propagator we make the replacement Ax y dilk eikz y A2 2 27f4 k2 l m2 716 k2 l A2 736 the integral is now convergent i A 0 1611392 we will do this type of calculations later A2 and indeed A0 is purely imaginary after choosingY so that 0I90xl0 0 we can take the limit A gt 00 Y becomes in nite we repeat the procedure at every order in g 35 eg at 093 we have to sum up and add toY whatever 093 term is needed to maintain 0l 0l0 0 this way we can determine the value on order by order in powers of g AdjustingY so that 0l90m0 0 means that the sum of all connected diagrams with a single source is zero In addition the same in nite set of diagrams with source replaced by ANY subdiagram is zero as well Rule Ignore any diagram that when a single line is cut k fall into two parts one of which has no sources tadpoles all that is left with up to 4 sources and 4 vertices is H a DOG Figure 913 All comlected diagrams without tadpoles with E 1 1 and V E 4 37 nally let s take a look at the other two counterterms 1 Zggtp3 Eat Eat Zp 18 ltpapltp Zm 1m202 ch weget Azzw 1 Bzzm 1 ZI exp 1 Mg A63 312 210 it results in a new vertex at which two lines meet the corresponding vertex factor or the Feynman rule is 239 f 1 A6 Bm2 for every diagram with a propagator there is additional one with this vertex we used integration by parts Summary we have calculated Z J in P3 theory and expressed it as Z J expliWJl whereW is the sum of all connected diagrams with no tadpoles and at least two sources 38 Scattering amplitudes and the Feynman rules based on Sl0 We have found ZJ for the phicubed theory and now we can calculate vacuum expectation values of the time ordered products of any number of elds Let s de ne exact propagator A1 12 E 01T I1w20gt short notation39 6 1 6 39 J i 61 OITvw1ltJm20 l zzmlm 61622 WJlJ0 61239WJlJO 62iWJJ0 21 expiWJ 6162iWJlJO 5jWJlJo 0lltPjl0 0 W contains diagrams WI ast two sources H thus we nd gal 2 Am1 m2 092 H 4point function OlT90199x29913 04l0 51525354ZU 61626364239W 61621W6364i v 6163iWX5264iW 5154iW6253iW we have dropped tel ms that contain 0lnpzl0 0 does not correspond to any interactionwhen plugged to LSZ no scattering happens Let s de ne connected correlatlon functions OlT P1 PEl0c E 515EiWIIJ and plug these into LSZ formula 0 0Tltpm1sow2s0w 1ltpw 2l0c 515251395239239W Jzo at the lowest order in g only one diagram contributes derivatives remove sources in 4 possible ways and label external legs in 3 distinct ways I 1 l 1 1 1 1 2 2 I gtlt 2 2 2 2 each diagram occurs 8 times which nicely cancels the symmetry factor 41 General result for tree diagrams no closed loops each diagram with a distinct endpoint labeling has an overall symmetry factor Let s nish the calculation of 0T90x1lt02 190 2lolc 616261I62r WlJ0 l l l 1 l l H I gtlt 2 2 2 2 2 2 putting together factors for all pieces of Feynman diagrams we get 0T90m190m290xi90w 0c 29 95 my d42Ay z x Am yAm2 yAxa 2Ama z Am1 yAx391yAm2 2A z Accl yAxg yAm2 2Ax1 z 094 42 For two incoming and two outgoing particles the LSZ formula is i4d4131 14332 d lx l 1422 eik11k21 2 kiz 1 k x 2 4 quot mix8 m2 a m2 a m2 XOlT P1 P2SO Bi90 0 and we have just written 0T90m190w290391lt 2l0C 515251395239inJ20 in terms of propagators The LSZ formula higth simpli es due to 33 m2A39L y 5413239 y We nd m2 2399 i f 142 d42Ay z eiikly 2y kiz k z eik1yk2z k ly k 2z eik1ykzz k iz k 2y 094 43 W 2399 G szMy z eilt wm 1zk zgt eik1yk227k lyik z eik1ykzz k lzik 2y 094 dye Bimyrz My 7 z 2w4 k2 m2 fie d4k 1 ltf1lQ2Wk2 39m2 is X 21rquot64k1k2k 2n454kk5k 2w464k1 k1k 2w464k 2 k2k 2n464k1 k 2k 2 64k1 k2k 094 1922vr464k1k2 k rk gt 1 1 1 X mm m2 Tk1 k 2 m2 k1 k 2 m392J C94 fourmomentum is conserved in scattering process W i922w46 k1k2 kik 1 1 x k1k22 m2 kl k3 m2 kl kg m2 09 Let s de ne fli 2w464km kouti7 scattering matrix element From this calculation we can deduce a set of rules for computing 1T 45 39 2 l 1 7 1 7 1 l 2T 19 Hum 1712 kleka vmi h zV39FmH l 0939 Feynman rules to calculate 2T for each incoming and outgoing particle draw an external line and label it with fourmomentum and an arrow specifying the momentum flow draw all topologically inequivalent diagrams Q for internal lines draw arrows arbitrarily but label them with momenta so that v momentum IS conserved In each vertex 9 assign factors I for each external line rms m2 7 i6 iZgg for each internal line with momentum k for each vertex sum over all the diagrams and get iT Additional rules for diagrams with loops a a diagram with L loops will have L internal momenta that are not xed integrate over all these momenta with measure mi 27r4 divide by a symmetry factor Q include diagrams with counterterm vertex that connects two propagators each with the same momentum k the value of the vertex is iAk2 Bm2 A ZED 7 1 B Zm 7 1 now we are going to use 2739 to calculate cross section LehmannKall n form of the exact propagator based on S 3 What can we learn about the exact propagator from general principles Let s de ne the exact propagator All y 410 Tvimivlyllm The eld is normalized so that 090w0 0 klsovl0 8quot Normalization of a one particle state in ddimensions kk 2 H2 5d 1k k39 I 7r w wk2m212 The ddimensional completeness statement 817 kk 11 identity operator in oneparticle subspace dd 1k dk 27rd 12w Lorentz invariant phasespace differential 48 Let s also de ne the exact propagator in the momentum space ddk k 7 3 2 Aa y2wde Ak In free eld theory we found 1 A k2 k2 m2 is it has an isolated pole at k2 m2 with residue one What about the exact propagator in the interacting theory 49 Alr y E W T wvyl0 Let s insert the complete set of energy eigenstates between the two elds for m0 gt yo we have ground state 0 energy one particle states A k2 m212 fdklt0ltpzlkksoy10 2 87clto ltzlkngtltknlsoyiogt Tl multiparticle continuum of states E speci ed by the total three m momentum k and other to k2M212 parameters relative m M 2 2m momenta denoted symbolically by n Olsovl0gt 0 0wvsoyl0 0l ml00ltpyl0 olsosclkgtltklsoltylogt 23 01mm nknlsoyl0 klcp1l0 6 37 9901 expiPmp990expiprp k ns00 6 ikzk n9000 k0 2 k2 Mz12 0s0xsoy0 ceikltx y 2 e k m ylknlso0l0l2 lt0ltpxsoyl0 ew 2 eik x ylknlw0lolz Let s de ne the spectral density 93 E Z lk lJltgt0l02 58 M2 524m2 slt4m2 115 2 0 95 0 kn k2m212 kn k2s12 then we have owmnm eikw Ms ew owmno ekltwgt fidspm eikr w v lm similarly lt0wywwl0 imsy W dspm aw and we can plug them to the formula for timeordered product OlTsaIs0yl0 9ID y 0lwwwyl0gt 9yu m 0lwywzl0 ddk eikz y 6 of o l 17c why 9 07 o dk ika y 270d k2m2 ie 1z y e 1lty x e was your homework we get ilt0T z lO ddquot aimy 1 w as s 1 lo lay M 2quotl k2m2 ie 4m2 p k2s i or in the momentum space Ak2 1 c ds s 1 k2m27i5 4m2 p k2si6 LehmannKall n form of the exact propagator it has an isolated pole at k2 39m2 with residue one


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