Summer Workshop on the Reaction Theory Exercise sheet 8. Classwork

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1 Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: June June To be dscussed on Tuesday of Week-II. Classwok. Deve all the quantum numbes I G J PC n the t channel of the followng eactons (a)! and K K! K K (b) N! N, N! N and KN! KN (c) N! N and N! N (d)! Notaton: =( +,, 0 ); =( +,, 0 ) ; K =(K +,K 0 ) ; N =(p, n).. Assume that the Regge exchange fom a SU() octet and a SU() snglet wth the couplng fo the octet and the snglet beng dffeent. Consde a vecto and a tenso nonet (octet plus snglet). Fom the dualty hypothess and the absence of double chage meson, fnd the combnaton of octet-snglet tenso that decouples fom. Use the SU() Clebsch-Godan coeffcents fom Rev.Mod.Phys. 6 (964) 005. What ae the quak content and the K K couplngs of these states?. Assumng deal mxng fo the vecto and tenso, deve the exchange degeneacy elatons comng dualty and the absence of esonance n the followng eactons (a)! (b) K K! K K (c) KN! KN (d)! (and! ) 4. Deve a Loentz-covaant bass, the sospn decomposton and the cossng popetes fo the followng eactons (a) N! N and KN! KN (b) NN! NN (c)!! and B! J/ K (d)! (e) N! N and N! N (use F µ = µ k k µ )

2 Soluton. The lst of exchanges havng only I =0, s pesented on Table. Notaton: sgnatue =( ) J and natualty = P ( ) J. In the quak model, P =( )`+ and C =( )`+S, hence 0, (,, 5,...) + and (0,, 4,...) + ae fobdden n the quak model. Let s efe to these quantum numbes as exotc". Table : Regge Tajectoes I G J PC I G J PC f + (0,, 4,...) f (,, 5,...) ! (,, 5,...) 0 +! + (0,, 4,...) ++ (0,, 4,...) ++ a (,, 5,...) (,, 5,...) ++ + (0,, 4,...) (0,, 4,...) (,, 5,...) + 0 h (,, 5,...) h + (0,, 4,...) (0,, 4,...) + + (,, 5,...) + + b (,, 5,...) b + (0,, 4,...) + (a) fo : G =+, =+and ( ) I =+(Bose symmety) ) f + and. fo K K: =+and ( ) I =+) f +,,! and. (b) fo : G =, =+, I =; fo NN: I =0, and no exotc ). fo K K: =+; fo NN: I =0, and no exotc ) f +,,! and. (c) fo and 0 : C = ; fo NN: I =0, and no exotc. )! ±, ±, b and h. fo + : I =; fo NN: I =0, and no exotc. ) a ±, ±, b and. (d) fo : G = )a ±, ±,! ± and h ± Table : Exchanges (a) +! + f + ± 0 0! 0 0 f + K + K! K + K f + ±! + ± K + K 0! K 0 K + (b) p! n p p! 0 n + p! p f + ± + n! n f + + K p! K p 0 p p ( + a+ ) K + n! K 0 p ( a+ ) K p! K p f + ± + ±! K n! K n f + ±! (c) p! p (! + )+(h + b +! ) p! 0 p (! + )+(h + b +! ) p! + n ( + )+(b a ) n! p ( )+b a ) (d) + 0! 0 + ( + )+( + ) + +! + + (! + h + )+(! + h + ) + +! + + f +. Fo a geneal teatment of exchange degeneacy usng goup theoy, see Ref. [].

3 We assume that the esdues obey a SU() symmety: R ac(t) /h8y a I a I a ; 8 Yc I c Ic 8 Y R I R I R, () whee Y = Y and I = I. The hypechage Y s the stangeness and I s the sospn pojecton. The Clebsch-Godan coeffcents fo SU() ae lsted n Ref. []. Note the exta mnus sgn fo the + and K. The fou couplngs ae 8V, 8T, V and T fo the octet/ snglet fo the tenso and vecto tajectoes. The absence of sospn meson n + K +! K + + and n + +! + + lead to + K +! K + + : + +! + + : 8 0 T s T + 5 8T s 8T 6 8T s 8T 8V s 8V =0 (a) 8V s 8T =0 (b) We combne them to get 8V = 8V = T and (/5) 8T =(/8) T, I choose by conventon 8 T = Let us defne the octet-snglet mxng f cos T f 0 = sn T 5 V. () sn T cos T f8 f (4) The states ae f 8 = 8; 000 and f = ; 000. The notaton s R; YII. Let us mpose that the f 0 couplng to + + vanshes sn T 5 8T! + cos T 8 T! =0. (5) Wth the elaton between the couplngs, we obtan sn T = p cos T o tan T =/. The quak content ae then 0 uū+d d 0 q q 0 p uū+d d p s A q q 6 uū+d s s d+s s A (6) p The couplngs ae f = f K + K + = f 0 K + K + = 5 8T cos T + 8 T sn T = 5 8T (7a) 0 8T cos T + 8 T sn T = 5 8T (7b) 0 8T sn T + 8 T cos T = p 5 8T (7c). In ths secton, I only wote the elatve sgn, not the elatve magntude gven by SU() and SU() Clebsch-Godan coeffcents. In the + +! + + case we obtan 0= f + (t)s f + (t) (t)s (t). (8) Snce ths elaton s vald n a ange of s and t, we obtan (t) = f+ (t) and (t) = f + (t). Fo patcles wth spn, one can epeat the agument wth specfc combnaton of helcty ampltudes

4 havng good natualty. Hence we obtan EXD elatons between exchanges wth the same natualty. In the case of + +! + + case we obtan fo the natual exchanges 0=! (t)s! (t) (t)s h + (t)s h + (t) (t)s (t) (9a) =(! a (t)) s N h+ (t) (t) s EN(t), (9b) and fo the unnatual exchanges 0=! + (t)s! + (t) a (t)s a h (t)s h (t) + (t)s + (t) (9c) =! a (t)+ h (t) s U (t), (9d) In the eacton + +! + +, the exchanges pck up a sgn equal to PC, we obtan 0=(! a (t)) s N h+ (t) (t) s EN(t) 0=! a (t) h (t)+ + (t) s U (t) (9e) (9f) Thee ae then EXD elaton between exchanges wth same natualty, same PC, same G paty and opposte sgnatue. The Regge tajectoes ae ndcated on Fg.. The exchange degeneacy elatons ae summazed n Table and n Fg. 7 J Ê w Ù a Ú f p Ø b Ï h h ÙÚ Ê Ù Ê Ï ÚÙ Ê ØÚ ÏÙ M HGeV L Fgue : Regge tajectoes.the sold lnes ae N (t) =0.9(t m )+and U (t) =0.7(t m )+0. Table : Exchange degneacy elaton + +! + + f+ = f + = + + K + K 0! K 0 K + a+ = = K + K K + K 0 K + K +! K + K + f+ =! K + K + f + =! K + K + K + n! K + n a+ = pp = pp pp! K + p! K + p f+ =! f + pp = + +! + + h+ = h = ! + + h = + h + + = a+ =! + + =! + + a =!+ a + + =!

5 4. (a) Fo pon-nucleon scatteng, the covaant bass s [] h hn j (p 4 ) b (p ) N (p ) a (p ) =ū(p 4 ) A ba j +(p/ + p/ )Bj ba u(p ) (0a) A ba j = ba ja (+) + abc ( c ) j A ( ) (0b) A (±) (,t)=±a (±) (,t) (0c) B (±) (,t)= B (±) (,t) (0d) The cossng vaable s =(s u)/ wth s =(p + p ) and u =(p p 4 ). To deve the cossng elaton, use C nvaance T = C TC, v = Cū T, v =ū T C, C µc = T µ and C 5C =+ 5 T and take the tanspose complexe conjugate. Fo kaon-nucleon scatteng, the covaant bass s h hn j (p 4 )K l (p ) N (p )K k (p ) =ū(p 4 ) A lk j +(p/ + p/ )Bj lk u(p ) () and the sospn decomposton s A (0) and A () have sospn 0 and n the t A lk j = lk ja (0) +( a ) j ( a ) kl A () () channel. The cossng elatons ae A (0) (,t)=+a (0) (,t) B (0) (,t)= B (0) (,t) (a) A () (,t)= A () (,t) B () (,t)=+b () (,t) (b) (b) In nucleon-nucleon scatteng thee ae fve ndependent Loentz stuctues. One possble soluton s to use a t channel base 5X hn j (p 4 )N l (p ) N (p )N k (p ) = (A n ) lk A A j ū n u ū n u 4 (4) The ndex A s a collectve epesentaton of Loentz ndces. The tenso stuctues ae n= = = 5 µ = µ µ 4 = 5 µ µ 5 = [ µ, ] (5a) In ths base, the scala ampltudes A n have good t channel quantum numbes. One could also use a s channel base 5X hn j (p 4 )N l (p ) N (p )N k (p ) = (B n ) lk A A j ū n u ū n u 4 (6) Fetz denttes elate the two bass. The tansfomaton s B /4 /4 /4 /4 /4 A B BB B 4 A = / 0 / A B / 0 / 0 / C BA / 0 / A 4 A B 5 /4 /4 /4 /4 /4 A 5 The sospn decomposton s the same as n KN scatteng and the cossng elatons ae n= (7) A (0,,) (,t)=+a (0,,) (,t) (8a) A (0,,) (,t)=+a (0,,) (,t) (8b) A (0,,) (,t)= A (0,,) (,t) (8c) A (0,,) 4 (,t)= A (0,,) 4 (,t) (8d) A (0,,) 5 (,t)=+a (0,,) 5 (,t) (8e) A and A 4 pck up a mnus sgn because they coespond to negatve sgnatue exchanges (vecto and axal-vecto exchange). That s a good coss-check of the method. 5

6 (c) The eactons nvolve a vecto wth momentum p V and polazaton tenso µ (p V, ) and thee pseudoscala wth momenta p,,. We need a Lev-Cvta tenso fo paty (an odd numbe of unnatual paty mesons) f paty s conseved. If paty s not conseved (weak decay) thee ae two addtonal stuctues. In the paty consevng decay!! the Loentz stuctue s h a (p ) b (p ) a (p )!(p V, ) = A abc (,t)" µ (p V, )p p µ p. (9) The only sospn stuctue s A abc (,t)=" abc A(,t). Two pons ae always n sospn. The scala functon s odd unde cossng (p,! p, f t =(p V p ) ), A(,t)= A(,t), snce only vecto ae allowed. The decay B! J/ K can volate paty. Thee ae then thee Loentz stuctues h (p )K(p )J/ (p V ) B(p, ) = A (,t)" µ (p V, )p p µ p + A (,t) µ (p V, )(p p ) µ + A (,t) µ (p V, )(p + p ) µ (0) Isospn s not conseved, so the sospn stuctue s elevant. Ths base s elavant to study cossng unde p,! p,. We obtan A (,t)= A (,t), A (,t)=+a (,t) and A (,t)= A (,t). So the exchanges (o esonances) n the channel ae ( =+, = ) n A, ( =+, =+)n A and ( =, = ) n A. (d) Thee ae fou ndependent stuctues. Wth the notaton P =(p + p )/, (k, (k, ), they ae h d (p ) c (k, ) a (p ) b (k, ) = A abcd (,t) The sospn decomposton s A abcd = ac bda (0) + ab cd ) and + A abcd (,t) P P + A abcd (,t)[k P + P k ] + A abcd 4 (,t) k k. () ad bc A () + ab cd + ad bc A () () The Loentz and sospn bases ae chosen to have good popetes unde cossng the two pons (o the two s). We obtan, fo =,,, 4 A (0,) (,t)=+a (0,) (,t) A () (,t)= A () (,t) () (e) The momenta ae ( ) (k)+n(p )! (q)+n(p ) and p =(p +p )/. Use F µ = µ k k µ. Paty eques a 5 o an " µ. The matx element s hn j (p ) a (q) (k)n (p ) = X n (A n ) a j M n (4) We found n the notaton of Ref. [4] M = 5 µ F µ (5a) M = 5 q µ p F µ (5b) M = 5 µ q F µ (5c) M 4 = µ q F µ (5d) M 5 = 5 µ k F µ (5e) M 6 = 5 q µ k F µ (5f) 6

7 The last M 5,6 ae zeo fo photopoducton. The sospn decomposton s (A n ) a j = A(+) a j + A ( ) [ a, ] j + A (0) a j (6) Fnally the cossng popetes ae A (0,+) (,t)=+a (0,+) (,t) A ( ) (,t)= A ( ) (,t) =,, 4 (7a) A (0,+) (,t)= A (0,+) (,t) A ( ) (,t)=+a ( ) (,t) (7b) Refeences [] J. Mandula, J. Weyes and G. Zweg, Ann. Rev. Nucl. Pat. Sc. 0, 89 (970). [] P. S. J. McNamee and F. Chlton, Rev. Mod. Phys. 6, 005 (964). [] G. F. Chew, M. L. Goldbege, F. E. Low and Y. Nambu, Phys. Rev. 06, 7 (957). [4] F. A. Beends, A. Donnache and D. L. Weave, Nucl. Phys. B 4, (967). 7