Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: http://www.ndana.edu/~sst/ndex.html 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 )
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 0 +++ f + (0,, 4,...) ++ 0 + f (,, 5,...) ++ 0 +! (,, 5,...) 0 +! + (0,, 4,...) ++ (0,, 4,...) ++ a (,, 5,...) ++ + + (,, 5,...) ++ + (0,, 4,...) 0 ++ + (0,, 4,...) + 0 + + (,, 5,...) + 0 h (,, 5,...) + 0 ++ 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. [].
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 s @ A = @ q q A @ 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
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 6 5 4 Ê w Ù a Ú f p Ø b Ï h h ÙÚ Ê Ù Ê Ï ÚÙ Ê ØÚ ÏÙ 0 0 4 5 6 7 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 0 + + 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 + + =! + + + 4
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 0 0 0 B /4 /4 /4 /4 /4 A B BB C @ B 4 A = / 0 / A B / 0 / 0 / C BA C @ / 0 / A @ 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
(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
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