t=4m s=0 u=0 t=0 u=4m s=4m 2

Transcription

1 Chapter 15 Appendix A: Variable The Mandeltam In Chapter. we already encontered diæerent invariant kinematical qantitie. We generalize Fig...1 inofar that we leave open which particle are incoming and which otgoing. Th we conider the kinematic itation given in Fig. A.1. 1 t m p 1 m p 1 Figre A.1 Deænition of ; t;. 84

2 In the dicion of Chapter., we already deæned and t and fond them to be efl variable. Here we deæne a third,, althogh all three, ; t;, are no longer independent. = èp 1 + p è = èp p 0 è =,èp 1 + p èèp p 0 è t = èp 1 + p 0 1è = èp + p 0 è =,èp 1 + p 0 1èèp + p 0 è = èm 1 + p 1 p 0 1è = èm + p p 0 è = èp 1 + p 0 è = èp + p 0 1è =,èp 1 + p 0 èèp + p 0 1è è15.1è The phyical igniæcance of thee variable can be expreed in two way: èaè i the qare of the c:m: energy if p 1 and p èor p 0 1 and p 0 è are incoming. t i the qare of the c:m: energy if p 1 and p 0 1 èor p and p 0 è are incoming. i the qare of the c:m: energy if p 1 and p 0 èor p and p 0 1è are incoming. Thi i a rather artiæcial decription i deæned by another proce. The three procee, in which ; t; are the qared c:m: energie are called the "; t; channel," repectively. Example: Let p 1 ;m 1 decribe a pion, and p ;m ancleon, then:, channel mean : : ç + N,! ç + N : ç + N ç,! ç + N ç t, channel mean : : ç + ç,! N + N ç : N + N ç,! ç + ç, channel mean : : ç + N,! ç + N : ç + ç N,! ç + ç N è15.è èbè If one decribe the meaning of ; t; in a deænite proce, e.g., the - channel, then i the qared c:m: energy. t i the qared for-momentm tranfer. èin particlar, it redce to the qared three-momentm tranfer in the Breit ytem.è ha no imple phyical meaning, ince there i no Lorentz ytem where it redce to anything obvio. A ; t; are not independent, it follow from è15.1è that + t + = 4m 1 + m + p 1 èp + p 0 + p 0 1è 85

3 = m 1 + m : è15.3è Let anticipate the notion of a cattering amplitde, i.e., the complex fnction which completely decribe the cattering proce. It will be a fnction of two variable, bt can be written a fnction of ; t; and having in mind that one of them i redndant, i.e., T è; t; è ç cattering amplitde : è15.4è One can prove, independent of pertrbation theory, that thi fnction i an analytic fnction of any two of the variable if thee are conidered to be complex. There are then certain domain in the complex t èor or tè plane, in which thee variable became real and have "phyical" meaning. Thee region are diconnected and belong to diæerent phyical procee. That T è; t; è i an analytic fnction of any two complex variable ot of ; t; mean that the "phyical cattering amplitde" i the bondary vale of that general fnction when ; t; take on phyical vale. In other word, the "phyical cattering amplitde" i obtained in any channel from the general fnction imply by pecializing to the "phyical vale" of ; t; for that channel. Here we hall not go into the "analytical trctre of the cattering amplitde," bt only explain the graphical repreentation of the variable ; t; and exhibit their "phyical region." Let conider the following triangle, b g b P c g c g a a Figre A. Geometric Conideration. 86

4 and remember from elementary geometry that if from any point p the three ditance G a ;G b ;G c to the ide a; b; c are taken, then ag a + bg b + cg c = ah a = bh b = ch c = F ; è15.5è where F denote the rface of the triangle. Here h a ;h b ;h c are the three height perpendiclar on a; b; c, repectively. Taking ch c and dividing by c give If we compare thi with è15.6è, we ee that we can identify a c G a + b c G b + G c = h c : è15.6è a c G a := ; b c G b := ; G c = t ; h c = m 1 + m è15.7è to have the relation between ; t; flælled. Th any three coordinate axe interecting ch that they form a triangle with h c = m 1 + m can erve to repreent ; t; in a plane. Of core, one chooe particlar triangle, where the repreentation become imple. The bet choice eem to be a = b = c; h = m 1 + m. Thi i very ymmetrical bt ha one diadvantage: The bondarie of the "phyical region" will be given in form of eqation between and t. 87

5 t t t=4m =0 =0 =4m t=0 =4m Figre A.3 Phyical region of ; t; channel in ymmetrical repreentation for m 1 = m. Every point in the plane atiæe + t + = 4m. The crve are actally eaier to draw in a rectanglar coordinate ytem, and we chooe b = c = a p = h = m 1 + m. 88

6 t t 4m =0 =0 t=0 4m =4m Figre A.4 Phyical region of ; t; channel in the carteian t-plane for m 1 = m. Every point in the plane atiæe + t + = 4m. ènote that the nit along the axi i maller by a factor p 1 compared to the t and axi.è We now ænd the "phyical region" of ; t; in the three poible channel of Fig. A.4. 89

7 1 1 1 p p p p p p channel t-channel -channel Figre A.5 The three diæerent channel. For m 1 6= m there i one ymmetry, namely that t i the momentm tranfer in both, the and the channel. It can th be expected that the phyical region in the and channel map on each other if and are interchanged. èthi i the famo croing ymmetryè. If the mae are eqal èm 1 = m è, then there i more ymmetry: Going from to t channel! keep it meaning. Going from to channel! t keep it meaning. Going from t to channel! keep it meaning. The phyical region are, therefore, mapped on each other if one èaè interchange $ t and keep. 90

8 èbè interchange $ and keep t. ècè interchange t $ and keep. Thee are in the ymmetrical repreentation of Fig. A., the reæection of the entire plane with repect to the three ymmetry axe of the baic triangle ABC. In Fig. A.3 the diæerent cale along the axe make the ægre apparently le ymmetric, bt one can eaily tranlate the phyical region from Fig. A. to Fig. A.3. Thi ymmetry allow to dic the -channel only. All conideration here are retricted to the cae m 1 = m. We have in the -channel èc:m: ytemè: = èp 1 + p è = èp p 0 è = èeè = 4èm + q è t = èp 0 1, p 1 è = èp 0, p è = q èco ç c:m:, 1è è15.8è where q i the c:m: momentm of all for particle. Hence, the "phyical region" in the -channel i given by 4m ç t min ç t ç 0 t min =,4q = 4m, : è15.9è With + t + = 4m, one ænd + t min + = 4m = + t min. Hence, the bondary t min = 4m, i identical with the line = 0. The phyical region of the -channel i, therefore, given by the two condition t ç 0 ç 0 : è15.10è Thi region i hown in Fig. A. and A.3 haded and marked by. The correponding region for the two other channel follow from the above ymmetry condition. 91