ELECTRON-MUON SCATTERING

Transcription

1 ELECTRON-MUON SCATTERING ABSTRACT Th lctron charg is considrd to b distributd or xtndd in spac. Th diffrntial of th lctron charg is st qual to a function of lctron charg coordinats multiplid by a four-dimnsional diffrntial volum lmnt. Th four-dimnsional intgral of this function is rquird to qual th lctron charg in all Lorntz frams. Th muon is tratd similarly. Th S-matrix for xtndd lctronxtndd muon scattring is calculatd. Th rsult is that th S- matrix of th xtndd lctron thory is a product of th S-matrix of th point lctron thory multiplid by an lctron form factor and a muon form factor. I. INTRODUCTION In th rst fram of an lctron charg distribution, lt x ν r = (x 0 r, x 1 r, x 2 r, x 3 r ) dnot a spactim charg point, and lt x ν r = (x 0 r, x 1 r, x 2 r, x 3 r) dnot th cntr of th charg distribution. Somtims th suprscript on th four-vctor (not th componnts) will b omittd, and w will writ Dat: May 14,

2 2 ELECTRON-MUON SCATTERING x r = (x 0 r, x 1 r, x 2 r, x 3 r ) and x r = (x 0 r, x 1 r, x 2 r, x 3 r). Introduc x r = x r x r or quivalntly x ν r = x ν r x ν r. In a fram of rfrnc in which th lctron charg distribution movs with a spd β in th +x 3 dirction, lt x m = (x 0 m, x 1 m, x 2 m, x 3 m) dnot a spactim charg point, and lt x m = (x 0 m, x 1 m, x 2 m, x 3 m) dnot th cntr of th charg distribution. Introduc x m = x m x m. A Lorntz transformation yilds x 1 r = x1 m, x 2 r = x 2 m, x3 r = γ( x 3 m β x0 m ), and x0 r = γ( x 0 m β x3 m ) whr γ = 1/ 1 β 2. Dnot this Lorntz transformation by x r = L( x m ). In th rst fram, th lctron charg is qual to ρ r ( x r )δ( x 0 r)d 4 x r whr ρ r ( x r ) is th charg dnsity in th rst fram and δ dnots th dlta function. 1 In th m fram, th lctric charg is qual to ρr (L( x m ))δ[γ( x 0 m β x 3 m)]d 4 x m. 2 So an lmnt of charg d m in th m fram is givn by d m = ρ r (L( x m ))δ[γ( x 0 m β x3 m )]d4 x m. (1) Th nxt sction is a rviw of point lctron scattring by a point muon. Th third sction will study scattring of an xtndd lctron by an xtndd muon. Th S-matrix of th xtndd lctron thory is found to b th product of th S-matrix of th point lctron thory multiplid by an lctron form factor and multiplid by a muon form

3 ELECTRON-MUON SCATTERING 3 factor. Th fourth sction will introduc a nw convntion for kping track of th imaginary i s. A short discussion follows. II. ELECTRON-MUON SCATTERING Th calculation for lctron-muon scattring will follow th calculation of lctron-proton scattring in Bjorkn and Drll whr th proton is tratd as a point particl of spin 1/2. 3 For th point lctron, th S matrix lmnt is approximatd by S fi = i d 4 x φ f (x)(γ ν )A ν (x)φ i (x). (2) This quation can b usd to study lctron-muon scattring sinc th muon is tratd hr as a point particl of spin 1/2. Th initial xact lctron wav function is approximatd by th plan wav solution to th Dirac quation. Th plan wav solution, which is normalizd to unity in a box of volum V, is φ i (x) = m E i V u i xp( ip i x) (3) whr and c hav bn st qual to 1, m is th lctron rst mass, u i is a four-componnt spinor, which dpnds on th initial spin and on p i = (p 0 i = E i, p 1 i,p 2 i,p 3 i), th initial four-momntum, and γ ν ar th

4 4 ELECTRON-MUON SCATTERING four Dirac matrics, which ar lablld by ν = 0, 1, 2, 3. Th final lctron wav function is φ f (x) = m E f V u f xp( ip f x), (4) whr p f is th final lctron four-momntum, E f is th final lctron nrgy, u f is th final lctron spinor, and φ f = φ f γ0. Th vctor potntial is givn by A ν (x) = d 4 y D F (x y)j ν (y) (5) whr D F (x y) is th photon propagator, and J ν (y) is th muon currnt. Th photon propagator is D F (x y) = d 4 q (2π) xp[ iq (x y)] 1 4 q 2 + iɛ (6) whr q is th four-momntum of th photon. For a ngativly chargd muon, th muon currnt is idntifid as φ F (y)γ ν φ F (y) whr φ F (y) is th final muon wav function and φ I (y) is th initial muon wav function. Th initial muon wav function is approximatd by th plan wav φ I (y) = M E I V u I xp( ip I y) (7)

5 ELECTRON-MUON SCATTERING 5 whr M is th muon mass, u I is a four componnt spinor, which dpnds on th initial muon spin and on p I = (p 0 I = E I, p 1 I, p2 I, p3 I ), th initial muon four-momntum. Th final muon wav function is φ F (y) = M E F V u F xp( ip F y) (8) whr p F is th final muon momntum four-vctor,e F is th final muon nrgy, and u F is th final four-componnt spinor of th muon. Thus, th S-matrix for lctron-muon scattring is S fi = i d 4 xd 4 y φ f (x)(γ ν )φ i (x)d F (x y) φ F (y)(γ ν )φ I (y). (9) Substituting Eqs. (3), (4), (6), (7), and (8) into Eq. (9) yilds S fi = +i2 mm(ū f γ ν u i )(ū F γ ν u I ) (2π) 4 V 2 E i E f E I E F d 4 xd 4 y d 4 q xp[i(p f p i q) x] xp[i(p F p I + q) y]. (10) q 2 + iɛ Prform th following intgrations: xp(i(p f p i q) x)d 4 x = (2π) 4 δ 4 (p f p i q); (11)

6 6 ELECTRON-MUON SCATTERING xp(i(p F p I + q) y)d 4 y = (2π) 4 δ 4 (p F p I + q); (12) δ 4 (p f p i q)δ 4 d 4 q (p F p I + q) q 2 + iɛ = δ4 (p F + p f p I p i ) ; (p f p i ) 2 (13) and find S fi = +i 2 mm V 2 (2π) 4 δ 4 (p f + p F p i p I ) (ū fγ ν u i )(ū F γ ν u I ). E i E f E I E F (p f p i ) 2 (14) III. EXTENDED ELECTRON-MUON SCATTERING In th prvious sction, x was th argumnt of th lctron wav function and th lctron charg spactim point in an arbitrary Lorntz fram. Tak th lctron to b initally moving with a spd β in th +x 3 dirction. This prviously was calld th m fram. So now x m is th argumnt of th wav function and also th cntr of th lctron charg distribution. Suppos that th muon is initially and unralistically at rst in th m fram. Thn, lt y r = (y 0 r, y 1 r, y 2 r, y 3 r ) dnot a spactim muon charg point, and lt y r = (y 0 r, y 1 r, y 2 r, y 3 r) dnot th cntr of th muon

7 ELECTRON-MUON SCATTERING 7 charg distribution in th m fram and also th argumnt of th muon wav function. Introduc ỹ r = y r y r. Th subscript r is attachd to y, y, and ỹ to mphasiz that th muon is at rst in th m fram. Eq. (9) will b modifid to tak into account th spatial charg distribution of th lctron and also th muon. Th intraction taks plac at charg points, so rplac D F (x y) by D F (x m y r ). In addition, th lctron charg is rplacd by th four-dimnsional intgral of d m whr d m is givn by Eq. (1). Sinc th muon is at rst in th m fram, rplac th muon charg by th four-dimnsional intgral of d µr = ρ µr (ỹ r ))δ(ỹ 0 r)d 4 ỹ r. (15) Hr ρ µr is th muon charg dnsity in th rst fram of th muon. For th xtndd lctron and th xtndd muon scattring in th m fram, th S-matrix is S FI = i d 4 x m d 4 y r φf (x m )(d m )γ ν φ i (x m ) D F (x m y r) φ F (y r )(d µr )γ ν φ I (y r ). (16) So now th S-matrix is

8 8 ELECTRON-MUON SCATTERING S FI = i d 4 x m d 4 y m φf (x m )(γ ν )φ i (x m )ρ r (L( x m ))δ[γ( x 0 m β x 3 m)]d 4 x m D F (x m y r ) φ F (y r )(γ ν )φ I (y r )ρ µr (ỹ r )δ(ỹ 0 r )d4 ỹ r. (17) Us D F (x m y r) = D F (x m y r ) xp( i x m q m ) xp(+iỹ r q m ) to show S FI = i d 4 x m d 4 y r φf (x m )(γ ν )φ i (x m )D F (x m y r ) φ F (y r )(γ ν )φ I (y r ) xp( i x m q m ) ρ r(l( x m )) δ[γ( x 0 m β x3 m )]d4 x m xp(+iỹ r q m ) ρ µr(ỹ r )) δ(ỹ 0 r)d 4 ỹ r. (18) Th first intgral is idntifid as S fi (s Eq. (9) and Eq. (14)). By Eq. (11) and Eq. (12), q m = p f p i = p I p F whr th momnta ar masurd in th m fram. Finally, S FI = S fi F(q)F µ (q) (19) whr th lctron form factor F(q) = xp( i x m q m ) ρ r(l( x m )) δ[γ( x 0 m β x 3 m)]d 4 x m, (20) and th muon form factor

9 ELECTRON-MUON SCATTERING 9 F µ (q) = xp(+iỹ r q m ) ρ µr(ỹ r )) δ(ỹ 0 r)d 4 ỹ r. (21) Thus th S-matrix for xtndd lctron and xtndd muon scattring is th S-matrix for point lctron-muon scattring tims an lctron form factor tims a muon form factor. As prviously shown, 2 th form factor is invariant, so F(q) = xp( i x r q r ) ρ r( x r ) δ( x 0 r)d 4 x r. (22) To gt a rough ida of how siz and structur affct scattring, pick th lctron charg to b uniformly distributd on sphrical shll of radius a in th rst fram. So ρ r ( x r ) = 4πa 2 δ( ( x 1 r) 2 + ( x 2 r) 2 + ( x r ) 2 a). (23) In sphrical coordinats, ( r r ) 2 = ( x 1 r )2 + ( x 2 r )2 + ( x 3 r )2, so that d 3 x r = ( r r ) 2 sin θ r d θ r d φ r d r r. Thn F(q) = δ( r a) 4πa 2 xp(+iq r r)ˆr 2 sin θ d θ d φd r = j 0 ( q r a) (24) whr q r 2 = (q 1 r) 2 +(q 2 r) 2 +(q 3 r) 2 = (q 1 m) 2 +(q 2 m) 2 +γ 2 (q 3 m βq 0 m) 2 and j 0 is th sphrical Bssl function of ordr 0. Hr q r is xprssd in

10 10 ELECTRON-MUON SCATTERING trms of th componnts of q m = p f p i, sinc it is p f and p i which ar masurd in th m or lab fram. Similarly, pick th muon charg distribution to b a a sphrical shll of radius a µ in th rst fram. Thn, ρ rµ (ỹ r ) = δ( (ỹ 4πa r) (ỹr) (ỹr) 3 2 a 2 µ ). (25) µ Introduc sphrical coordinats again. Procd similarly with Eq. (21), and find j 0 ( q m a µ ) whr q m = p f p i = p f 2 + p i 2 2p f p i. IV. NEW CONVENTION Th following convntion has bn adoptd to kp track of th imaginary i s in mor complicatd situations: multiply ach charg by i; and multiply th photon propagator by i. 4 Thus, Eq. (9) bcoms S fi = d 4 xd 4 y φ f (x)( iγ µ )φ i (x)id F (x y) φ F (y)( iγ µ )φ I (y). (26) V. DISCUSION Th rsult is that th S-matrix of th xtndd particl thory is th product of th S-matrix of th point particl thory multiplid by form

11 ELECTRON-MUON SCATTERING 11 factors of ach of th xtndd particls. Th charg dnsitis usd in sction III. wr chosn for thir mathmatical simplicity. Th charg dnsitis should b calculatd from th xprimntally dtrmind form factors. On th othr hand, if thr ar no xprimntal dviations from th point lctron thory, this sts an uppr limit on th radius of th xtndd particls. S th soon to b postd papr on Møllr scattring for stimats of th lctron radius and th muon radius. 5 ACKNOWLEDGEMENTS Th author thanks Ogdn, Su, Ruth, and Rob for thir hlp in gtting m startd on this projct. I spcially thank Bn for his many improvmnts to th papr. Rfrncs 1 S. Winbrg, Gravitation and Cosmology: Principls and Applications of th Gnral Thory of Rlativity (John Wily & Sons, Inc. Nw York, 1972), pp Scattring and Byond 3 J.D. Bjorkn and S. D. Drll, Rlativistic Quantum Mchanics (McGraw-Hill, Nw York, 1964), pp J.D. Bjorkn and S. D. Drll, Rlativistic Quantum Mchanics (McGraw-Hill, Nw York, 1964), p Scattring