Bar yon Resonanes and Str ong QCD Eb erhard Klempt Institut - PDF document

Bar yon Resonan es and Str ong QCD Eb erhard Klempt Institut f � ur Str ahlen{ und Kernphysik der Universit� at Bonn D-53115 BONN In tro du tion � Mo dels of bary on sp e tros op y � Regge tra je tories � A mass form ula for bary on resonan es � A new in terpretation of strong QCD � Con lusions � 1

Intr odu tion: why bar yon spe tr os opy Bary ons ha v e pla y ed a de i iv e role in the de- v elopmen t of the quark mo del and of SU(3). The o tet bary ons (total spin J = 1/2): S O tet 6 p n u u � A � A � S Singlet A � 6 A � A � � � A � 1 A � A - I - I u m ` u u u 3 3 A � A + 0 � � � � � A � A � A � A � A � A � u u 0 � � � The de uplet bary ons (total spin J = 3/2): 0 + ++ � � � � S = 0 � 0 + � � � S = - 1 � 0 � � S = - 2 � � S = - 3 � 2

Bary ons (with 3 quarks): 3 �a v ours x 2 spins. 6 � 6 � 6 = 56 � 70 � 70 � 20 M M 4 2 56 = 10 � 8 2 4 2 2 70 = 10 � 8 � 8 � 1 2 4 20 = 8 � 1 The 56-plet on tains N 's with spin 1/2 � � 's with spin 3/2 � The 70-plet on tains N 's with spin 1/2 and with spin 3/2 � � 's with spin 1/2 � (8 ) ha v e a mixed �a v our symmetry , the 10 M m ultiplet is symmetri , the 1 an tisymmetri in �a v our spa e. 4

Experiment al St a tus The P arti le Data Group lists: O tet N � � � De uplet � � � � Singlet � **** 11 7 6 9 2 1 *** 3 3 4 5 4 1 ** 6 6 8 1 2 2 * 2 6 8 3 3 0 No J - - 5 - 8 4 T otal 22 22 26 18 11 4 � 100 bary on resonan es � � 85 bary on resonan es of kno wn spin parit y � � 50 w ell established bary on resonan es � of kno wn spin parit y 9

Basi s of Bar yon Spe tr os opy The bary on w a v e fun tion: j qqq > = j olour > � j spa e ; spin ; �a v our > A S O ( 6 ) SU ( 6 ) The total w a v e fun tion m ust b e an tisymmet- ri w.r.t. the ex hange of an y t w o quarks. The olour w a v e fun tion is an tisymmetri , hen e the spa e-spin-�a v our w a v e fun tion m ust b e symmetri . W e no w onstru t w a v e fun tions. Sp a tial Symmetr y Ja ob ean o ordinates: r � r 1 2 r + r � 2r 1 2 3 r + r + r 1 2 3 Tw o relev an t separable motions � System is b ound ) � Tw o harmoni os illators � 11

Multiplet-stru ture of harmoni os illator (Hey and Kelly , Ph ys. Rep. 96 (1986) 71). O ( 6 ) ! O ( 3 ) � O ( 2 ) P O ( 6 ) O ( 3 ) O ( 2 ) ( D ; ) N L � N + ( 56 ; ) 0 1 1 1 0 � 0 1 6 3 2 ( 70 ; 1 ) � � 1 1 + + ( 5 + 1 ) ( 70 ; ) , ( 70 ; ) 2 20 2 2 0 � 2 2 2 + 5 1 ( 56 ; 2 ) � 2 + ( 20 ; ) 3 1 1 � 2 + 1 1 1 ( 56 ; 0 ) � 2 ( 7 + 3 ) ( 56 ; ) , ( 20 ; ) , ( 56 ; ) , ( 20 ; ) 3 50 2 3 3 1 1 � � � � � 3 3 3 3 3 ( 70 ; ( 70 ; ( 70 ; ( 7 + 5 + 3 ) 2 3 ) , 2 ) , 1 ) � � � � 1 3 3 3 ( 70 ; ) 6 3 2 1 � � 1 3 ( 70 ; + ( 70 ; + ( 70 ; + 4 105 ( 9 + 5 + 1 ) 2 4 ) , 2 ) , 0 ) � 4 4 4 4 + + + + ( 9 + + + 3 ) ( 70 ; ) , ( 70 ; ) , ( 70 ; ) , ( 70 ; ) 7 5 2 4 3 2 1 � 2 4 4 4 4 ( 56 ; + ( 56 ; + ( 56 ; + ( 9 + 5 + 1 ) 1 4 ) , 2 ) , 0 ) � 4 4 4 + + ( 7 + 5 ) ( 20 ; ) , ( 20 ; ) 1 3 2 � 4 4 ( 70 ; + ( 70 ; + 20 ( 5 + 1 ) 1 2 ) , 0 ) � 4 4 + ( 20 ; ) 3 1 1 � 4 ( 56 ; + 5 1 2 ) � 4 + ( 56 ; ) 1 1 1 0 � 4 12

Theoreti al models and resul ts Assume quarks mo v e in an e�e tiv e on�ne- � men t p oten tial generated b y a v ery fast olour ex hange b et w een quarks (an tisymmetrising the total w a v e fun tion) Assume the ligh t quarks a quire e�e tiv e mass � b y sp on tanous symmetry breaking Assume residual in tera tions � { One gluon ex hange quark mo del (Capsti k and Rob erts) r elativize d OGE �xed to HFS (N-�) ~ ~ L � S large, in on trast to data Set to zero ~ ~ ( omp. b y L � S from Thomas pre . ?) { Goldstone (pion) ex hange (Gloszman and Risk a) { Instan ton in tera tions Relativisti quark mo del with instan ton-indu ed for es (Krets hmer, L� oring, Mets h, P etry) Solv e equation of motion � (using w a v e fun tions of the harmoni os il- lator) 13

� N resonan es with inst anton indu ed f or es 3000 2700 ** 2600 *** 2500 2250 2220 2200 **** ** 2190 **** 2090 2100 2080 **** Mass [MeV] * 16 * ** 2000 1990 2000 1986 S ** ** 1900 1897 1895 ** S S 1720 1710 1700 1680 1675 **** 1650 *** *** **** **** **** 1535 1520 1500 **** **** 1440 **** 1000 939 **** J π 1/2+ 3/2+ 5/2+ 7/2+ 9/2+ 11/2+ 13/2+ 1/2- 3/2- 5/2- 7/2- 9/2- 11/2- 13/2- L P P F F H H K S D D G G I I 2T 2J 11 13 15 17 19 1 11 1 13 11 13 15 17 19 1 11 1 13

Many pr oblems still unsol ved: ! Whi h mo del is righ t ? ! Is it true that one in tera tion dominates ? ! Lo w mass of Rop er, � ( 1600 ) ... 3 = 2 + ! Lo w mass of negativ e-parit y � 's at 1950 MeV � ! Missing resonan es ! De a y prop erties of resonan es 24

Phenomenologi al appr o a h by Regge traje tories ] 2 [GeV 9 8 2 M 7 f (2510) 6 a (2450) 6 6 (2330) ρ 5 5 f (2044) a (2020) 4 4 4 3 (1667) ω (1691) ρ 3 3 2 a (1318) f (1275) 2 2 1 (782) ω ρ (770) 0 0 1 2 3 4 5 6 7 8 L Mesons with J = L + S lie on a Regge tra je tory 2 with a slop e of 1.142 GeV . 25

] 2 M[GeV 9 (2950) ∆ + 15/2 8 7 f (2510) 6 a (2450) 6 (2300) ∆ 6 + 11/2 (2330) ρ 5 5 f (2044) 4 a (2020) 4 (1950) ∆ 4 + 7/2 (1667) ω 3 3 (1691) (1232) ∆ ρ + 3 3/2 f (1275) 2 2 a (1318) 2 (782) ω 1 (770) ρ 0 3 7 11 15 J 2 2 2 2 � 's with L ev en and J = L + 3 = 2 ha v e the same � slop e as mesons. 26

Spin-orbit ouplings N N N N ∆ ∆ ∆ ∆ + + + + + + + + 7/2 5/2 3/2 1/2 7/2 5/2 3/2 1/2 (1950) (1895) (1935) (1895) (1990) (2000) (1900) (2100) 3 [a] 2 M ∆ 2 1 N N N ∆ ∆ - - - - - 3/2 1/2 5/2 3/2 1/2 (1700) (1620) (1675) (1700) (1650) 0 � and N resonan es assigned to sup erm ulti- plets with de�ned orbital angular momen tum. A t 2a: ~ ~ ~ + + + + L ( 2 ) + S ( 3 = 2 ) = J ( 7 = 2 ; 5 = 2 ; 3 = 2 ; 1 = 2 ) . A t 1a: ~ ~ ~ + + � with L ( 1 ) + S ( 1 = 2 ) = J ( 5 = 2 ; 3 = 2 ) ~ ~ ~ + + + N with L ( 1 ) + S ( 3 = 2 ) = J ( 7 = 2 ; 5 = 2 ; 3 = 2 ) 27

] 2 [GeV 9 (2950) ∆ + 15/2 2 8 M 7 6 (2300) ∆ + 11/2 5 (2220) ∆ - 7/2 4 (1950) ∆ + 7/2 3 (1720) ∆ - 3/2 2 (1232) ∆ + 3/2 1 0 0 1 2 3 4 5 6 7 8 L � 's with o dd L and J = L + 1 = 2 fall on the � same tra je tory . 28

] 2 [GeV 9 (2950) ∆ + 15/2 2 8 M 7 6 (2300) ∆ + 11/2 N (2250) 5 - 9/2 4 N (1990) + (1950) 7/2 ∆ + 7/2 3 N (1675) - 5/2 2 (1232) ∆ + 3/2 1 0 0 1 2 3 4 5 6 7 8 L N 's with in trinsi spin 3/2 fall on the same � tra je tory . 29

] 2 [GeV 9 (2950) ∆ + 15/2 8 2 M 7 6 (2300) ∆ + 11/2 N (2240) (2220) 5 ∆ - - 9/2 7/2 (1950) N (1990) ∆ 4 + + 7/2 7/2 (1720) ∆ - 3/2 3 N (1675) - 5/2 2 (1232) ∆ + 3/2 1 0 0 1 2 3 4 5 6 7 8 L The lo w est � (with spin 1/2 and 3/2) and the � N 's with in trinsi spin 3/2 and J = L + 3 = 2 fall � on the same Regge tra je tory . 30

� Wha t is about with intrinsi spin N = 1 = 2 S ? 9 ] 2 [GeV N (2700) 8 + 13/2 2 M N (2600) - 7 11/2 6 N (2220) + 9/2 5 N (2190) - 7/2 4 N (1720) N (1650) + + 3/2 5/2 3 N (1520) 2 N (1535) - 3/2 - 1/2 1 N (939) + 1/2 0 0 1 2 3 4 5 6 7 8 J The N masses (with in trinsi spin S = 1 = 2) lie � b elo w the standard Regge tra je tory . They are 2 smaller b y ab out 0.6 GeV for N in the 56-plet, � 2 and b y 0.3 GeV for N in the 70-plet. � 31

Radial ex it a tions 9 ] 2 [GeV (2750) 2.89 ∆ ≈ - 13/2 8 2 M 7 ∆ (2390) ∆ (2400) ≈ 2.47 + (2350) ∆ 5/2 - 6 - 9/2 5/2 (2350) ∆ - 5/2 5 (2200) 2.23 ∆ ≈ + ∆ (2150) 5/2 - 1/2 4 (1900) ∆ (1930) ∆ (1930) ≈ 1.95 ∆ - - - 5/2 3/2 1/2 3 ∆ (1600) ≈ 1.63 + 3/2 N (1440) 1.43 ≈ 2 + 1/2 1 0 0 1 2 3 4 5 6 7 8 L Radial ex itations ha v e masses larger than the lo w er mass state b y one ~ ! (not 2 ~ ! ). 32