Chapter 3. Building Hadrons from Quarks

Transcription

1 P570 Chapter Bilding Hadrons from Qarks Mesons in SU() We are now ready to consider mesons and baryons constrcted from qarks. As we recall, mesons are made of qark-antiqark pair and baryons are made of three qarks. Consider mesons made of and d qarks first. ψ = ψ (space) ψ (spin) ψ (flavor) ψ (color) It is straight-forward to write down the spin wave fnction of a meson. Since both q and q have spin ½, the qq can form either a spin-triplet or a spin-singlet state:, =, 0 = ( + ) spin triplet, = 0,0 = ( ) spin 0 singlet For the flavor wave fnction, we have an analogos sitation. The, d qarks are isospin doblet, and so are d,. However, a slight complication arises since, d transforms in the complex conjgate representation. Now = Ii d d and Ii, I j = i ε ijki k taking complex conjgate, we get I, I = i ε I * * * i j ijk k * * * or Ii, - I j = i εijk ( Ik)

2 P570 * I i transforms, d into, d : * = ( Ii ) d d (Eq. ) write ot * I i explicitly: 0 = 0 * ( I ) 0 i = i 0 * ( I ) 0 = 0 * ( I ) One can easily verify that * I i is related to Ii by a nitary transformation 0 * 0 ( Ii ) = I 0 0 i One can mltiply (Eq. ) by 0 0 on the left 0 0 * 0 0 = ( Ii ) 0 d (Eq. ) d I i (Eq. ) becomes or 0 0 = Ii 0 d 0 d d d = Ii Therefore, d transforms jst like and is the isospin doblet for d, d.

3 P570 Combining the singlet: d d with the, one obtains isospin triplet and isospin I, I =, = d z, 0 = ( dd ) I =, = d 0,0 = ( dd ) + I = 0 The total spin of the mesons can be S = 0 or. The lightest mesons have orbital anglar momentm L = 0. Therefore, their total anglar momentm J = 0 or. J = 0, I = : π +, π o, π - J =, I = : ρ +, ρ o, ρ - J =, I = 0 : ω o (mass ~ 40 MeV) (mass ~ 770 MeV) (mass ~ 780 MeV) (We will postpone the discssion of J = 0, I = 0 meson ntil later, since its wave fnction contains non-, d components.) One can readily combine the flavor and spin parts of the wavefnction. For example: o ρ = = + = + ( Jz 0) ( dd) ( ) ( d d d d ) π + = = = ( Jz 0) d ( ) ( d d ) Baryons in SU() Now, consider baryons made of, d qarks. Let s start with the flavor part.

4 P570 4 First, combining two qarks:, :, 0 :, : dd ( d d) + I = triplet 0,0 : ( d d) I = 0 singlet Note that I = is symmetric with respect to the interchange of the two qarks, while I = 0 singlet is anti-symmetric. We therefore label the symmetry property of the representations as = S A where the dimension of the representation is I+ (For example, I=/ ; I= ; I=0 ). Adding another /d qark, we have First consider A : ( ) ( ) ( ) = + = S A S A d ( d d) d d d ( d ) ( d ) This is an isospin doblet with mixed permtation symmetry (anti-symmetric WRT the interchange of the first two qarks). Therefore, we denote = A M A where M signifies mixed-symmetry while the sbscript A reminds s the antisymmetry between the first two qarks.

5 P570 5 We now consider the remaining part, S : ( d + d) d dd ( I I I I ) = = = = This simply corresponds to the prodct of an isospin triplet and an isospin doblet. The reslting isospin can have I = and I =. I = : d + ( d + d) ( d + d + d) = ( d d) d dd ( dd + dd + dd) + + ddd ddd I = : d ( d + d) ( ) d d d 6 = ( d d) d dd ( dd + dd dd + ) 6 In constrcting the I = and I = mltiplets, we simply se the Clebsch- Gordon coefficients to combine the I Iz I I z states into z I I states.

6 P570 6 I = mltiplet is totally symmetric WRT interchange of any pair, while I = mltiplet has mixed-symmetry and is symmetric WRT the interchange of the first two qarks. Therefore, we write S = 4S M S Collecting previos reslt, we now have = 4 S M M The spin wave fnctions for three qarks are constrcted exactly the same way o The flavor-spin wave fnctions of the,,, qartet are constrcted by combining the I = (flavor) with the J = (spin) wave fnction. S A : (4S, 4 S ) flavor spin ++ For example ( J Z ) + while ( J Z ) = is simply ( )( ) = is = ( d + d + d) ( + + ) = ( ) d + d + d + permtations Note that the flavor-spin wave fnction is totally symmetric with respect to exchange of any pair of qarks. Since qarks are fermions, the wave fnction shold be anti-symmetric overall. We now believe the color wave fnction is antisymmetric (in order to become a color-singlet state), making the fll wave fnction anti-symmetric. The proton and netron, being spin-½ isospin-½ particles, are constrcted with the following combination to make the overall flavor-spin wave fnction symmetric.

7 P570 7 ( M, M ) + ( M, M ) S S A A flavor spin flavor spin As an example, consider proton with J Z = ½ : ( ) ( ) ( Z = = i ) p J d d d 6 6 ( ) + d d ( ) = d ( ) + d ( ) 8 + d ( ) = + 8 ermtations { d d d p } Note that p (J Z = - ½) can be obtained from p (J Z = + ½) by interchanging with. Similarly, n (J Z = ½) can be obtained from p (J Z = ½) by interchanging with d qarks. SU() Adding strange qark, we extend SU() to SU(): d d s SU(N) has N - generators, and the fndamental representation has matrices of dimension N. The eight generators for SU() satisfy the Lie algebra:

8 P570 8 T, T = i f T i j ijk k where the strctre constants are f =, f47 = f 46 = f57 = f45 = f56 = f67 =, f 458 = f678 = The strctre constants are anti-symmetric with respect to interchange of any two indices. The fndamental representations for SU() are: T T T T = = i = i = 0 0 i 0 i 0 T T T T i 0 = i = = = Mesons in SU(), d, s qarks belong to the fndamental representation of SU(), denoted by., d, s belong to the conjgate representation. Combining qark and antiqark together:

9 P570 9 = 8 In terms of the weight diagram, the and representations are: Y Y d(- ½, ⅓) (½, ⅓) s (0, ⅔) ⅓ -½ ½ I I S(0, - ⅔) (-½, -⅓) d(½, -⅓) Where Y = B + S, B is the baryon nmber and S is the strangeness. is obtained by sperimposing the triplet on top of each site of the qark triplet. One obtains Y Y K o ds s K + π - π o π + I I d η d η (958) K - s sd K o 8 The SU() singlet mst contain s,, d qarks eqally and has the wave fnction ( + dd + ss ) The other two mesons occpying the I = 0, Y = 0 location are the SU() triplet state

10 P570 0 ( dd ) and the SU() singlet state 6 ( + dd ss ) + The lightest meson octet consists of K ( s) K ds, ( d ) o, ( ) π +, o π ( dd ), π ( d ) and η ( + dd ss ). Together with the 6 SU() singlet (η ), they form the psedoscalar (0 - ) nonet. The J = meson octet consists of: 0 * K * K + ρ - ρ o ρ + ω * K * K o and the J = singlet is φ. The ω and φ mesons are fond to be mixtres of the SU() states ω 8, φ ϕ ω8 + ϕ = ss ω 8 ω + ϕ = + ( dd ) Evidence for φ to be nearly a pre ss state is from the pecliar behavior of φ decays.

11 P570 φ has a width of only 4. MeV and its relatively long lifetime is de to the fact that the main decay channels for φ are + o o ϕ K K, K K (Branching Ratio 8%) L s The phase space for φ (00) decaying into a pair of kaons is greatly sppressed relative to the φ π + π - π o (Branching Ratio 6%) However, the φ π decay is sppressed by the Zweig rle if φ is mainly a ss state. The φ KK decays, which are not affected by the Zweig rle, became the dominant decay channels. The Zweig rle (or OZI rle, de to Okbo, Zweig, and Iizka) states that decay rates for processes described by diagrams with nconnected qark lines are sppressed. Ths, φ π + π - π o decay diagram π + s d d π - s π o shows that the π qark lines are completely disconnected from the initial ss + qark lines. In contrast, the ϕ K K decay diagram has some qark lines connected, and it is not sppressed by the Zweig rle. s s K - s K + s

12 P570 The three colors (R, B, G) of qark transform nder the SU() exact symmetry. Since mesons are made of q q, their color wave fnction shold be an SU() c singlet. The singlet state in c c = 8c c is simply ( RR + BB + GG) Therefore, the color wave fnctions of all mesons have the same form as the flavor wave fnction of an SU() singlet meson (η ). As discssed earlier, the color contents of glons are (color x color ). Recall the qark-glon copling diagram: q R q B G RB for a R qark changing into a B qark and the glon carrying RB color. The glons therefore have eight different color combinations: RG, RB, GR, GB, BR, BG, RR GG, RR GG BB 6 ( ) ( + ) Note that the color contents of the 8 glons are analogos to the flavor contents of the meson octet. Baryons in SU() The spin wave fnctions for baryons have been discssed earlier: ( ) = = 4 s M M For the flavor wave fnction, we first consider S A

13 P570 d d s s gives 9 basis states:, d, s, d, dd, ds, s, sd, ss These basis states can be rearranged to form basis states with specific permtation symmetry. In particlar, the basis states ψ iϕ j( i =, d, s; j =, d, s) can be expressed as ψ iϕ j = ( ψiϕ j + ψ jϕi) + ( ψiϕ j ψ jϕi) = Sij + Aij symmetric anti-symmetric It is clear that the symmetric S ij contains 6 elements, while the anti-symmetric A ij contains elements = 6 In terms of the SU() weight diagram s A Y x x x x I z = x x x x x 6 x x x x = + x x x x x

14 P570 4 Now = ( 6s A) = ( 6s ) ( A ) First consider 6 s Sijψ k = Sijψk + Sjkψi + Skiψ j + Sijψk Sjkψi Skiψ j ( ) ( ) The first part of the decomposition is totally symmetric and contains 0 elements, while the second part contains 8 elements and has mixed symmetry (symmetric only for i j interchange). Therefore 6s = 0s 8M S Next consider A Aijψ k = Aijψk + Ajkψi + Akiψ j + Aijψk Ajkψi Akiψ j ( ) ( ) The first term has element and is totally anti-symmetric, while the second term has 8 elements with mixed symmetry (anti-symmetric only for i j interchange). Therefore = 8 A M A Finally we have = 0s 8M 8M A A The explicit flavor wave fnctions for 0 s, 8 M, 8, and S MA A are listed on the next two pages. S A

15 P570 5 SU() Baryons Flavor: = 0s 8M 8M A ++ 0 s : : + : ( d + d + d) o : ( dd + dd + dd ) : ddd * + ( s s s) Σ : + + * o ( ds sd ds ds sd sd) Σ : * ( dds dsd sdd ) Σ : + + * o ( ss ss ss) Ξ : + + * ( ssd sds dss) Ξ : + + Ω * : sss 8 M : p : ( d d d) S 6 n : ( dd + dd + dd ) 6 + Σ : ( s s s) 6 o Σ : ds sd ds + ds sd sd Σ : ( dds dsd sdd ) 6 o Λ : ( sd + sd sd ds) o Ξ : ( ss + ss ss) 6 Ξ : ( sds + dss ssd ) 6 ( ) S A

16 P M : p : ( d d) A n : ( dd + dd ) + Σ : ( s s) o Σ : ( sd + ds sd sd ) Σ : ( dsd sdd ) o Λ : ds ds sd ds sd + sd o Ξ : ( ss ss) Ξ : ( dss sds) ( ) A : ( ds sd + ds ds + sd sd) 6 Note 8M S is obtained by the following seqence: starting from q q q a) symmetrize in positions and b) anti-symmetrize in positions and c) symmetrize in positions and * For Λ o, orthogonality to Σ o o, Σ, and A are imposed to obtain its wave fnction. For 8M A, the procedre is analogos, bt with anti-symmetrization in steps a) and c) and symmetrization in step b). Also, Σ o is obtained by reqiring orthogonality to Λ o * o, Σ and A.

17 P570 7 Spin: = 4 4 s :, : s M M, :, :, : S ( + + ) A ( + + ) S 6, : 6 + : M, : ( ) ( ) A, : : M, : ( ) ( ) Baryon decplet: (0 s, 4 s ) 8, + 8, Baryon octet: ( M M ) ( M M ) S S A A We can write down explicitly the flavor-spin wave fnctions of the baryon octet: { permtations} p = d d d + 8 n = d d + d d + d d + permtations 8 + Σ = { s s s + permtations} 8 o Σ = { d s d s d s + permtations} 6 Σ = { d d s d d s d d s + permtations} 8 o Ξ = { s s + s s + s s + permtations} 8 { }

18 P570 8 { s s d s s d s s d permtations} Ξ = o Λ = { d s d s permtations} + Note the following for the flavor-spin wave fnctions of the baryon octet: ) For p, n, Σ +, Σ -, Ξ o, Ξ -, they contain two identical qarks and there are three permtations, i.e. { p = d d d 8 + d d d + d d d } For Σ o and Λ, they have three different qarks, and there are six permtations, i.e. { o Λ = d s d s + d s d s + s d s d + s d s d + s d s d + d s d s } ) Once one knows the proton wave fnction, one can obtain the wave fnctions for other baryons by replacing (interchanging) the qark flavors. The reslting wave fnction is correct within an overall factor. p n ( d) p Σ + (d s) Σ + Σ - ( d) n Ξ o (d s) Ξ o Ξ - ( d) ) The total spin of an identical qark pair is always in an s = (symmetric) state. This is reqired since the overall wave fnction is anti-symmetric for this qark pair, and the color wave fnction is anti-symmetric. 4) For Σ o, the d have s =, while s = 0 for d in Λ o. Also, for Λ o the baryon s spin only resides on the strange qark.

19 P570 9 Magnetic Dipole Moments of Hadrons We have obtained the flavor-spin fnctions of the lightest mesons and baryons. Can this simple constitent qark model describe the static properties of these hadrons, sch as mass, spin, parity, magnetic dipole moment, lifetime, decay modes, etc.? The spin and parity of the mesons and baryons natrally emerge from the qark model. It trns ot that the magnetic dipole moments and the masses of hadrons are also well described by the qark model. First, we consider the magnetic dipole moments. For point spin-½ charged particles, sch as e ±, µ ±, the Dirac theory tells s that g =, where g is the gyromagnetic ratio: µ = Qe gs mc µ is the magnetic dipole moment, m is the mass of the point particle, and s is the spin of the particle. One gets, for example, for electron µ e e e mc σ = i i = mc σ e e magnitde of µ e is µ e = e mc e Similarly, for spin-½ qarks and anti-qarks, the magnetic dipole moments are e e µ = = mc mc µ d e e = = mc 6mc d d µ µ µ = µ = d d

20 P570 0 In the qark model, the magnetic dipole moment operator for a hadron is expressed as the vector sm of the magnetic dipole moment operator of the constitent qarks n µ H = µ i i= The magnetic dipole moment is defined as the expectation vale of the z- component of µ for a hadron with spin along the z direction. The baryon H magnetic dipole moment is therefore µ B µ ( σ ) B i z i i= = B As an example, for proton, we have { permtation} P = d d d + 8 (Note that the - pairs are always in an s = state, since are identical particles and their spin wave fnction has to be symmetric to prodce an overall antisymmetric wave fnction after the color wave fnction is inclded.) µ p = 4 ( µ + µ µ d) + ( µ µ + µ d) + ( µ µ + µ d) 8 = ( 8µ 4µ d + µ d) = ( 4µ µ d) 6 For netron s magnetic dipole moment, we simply interchange d and obtain µ n = 4 µ d µ ( ) An interesting prediction of the qark model can be obtained if one assmes that the p and down qarks have the same mass. Then, µ d = µ

21 P570 and µ µ n Experimentally, µ p p 4µ d µ µ = = = 4µ 9 µ d µ e e =.79 mc and µ n =.9 p mc p µ µ = 0.685, very close to the qark model calclation of ⅔. n p From the proton s magnetic dipole moment, one can even obtain an estimate of the, d qark mass. e ( 4 ) e µ p µ µ d µ = = = = mc mc bt µ p is measred to be.79 e mc. p e e Therefore.79 = mc m p = mp = 6 MeV mc.79 For the Λ-baryon magnetic dipole moment, we have { d s d s permtation} Λ = + (Note that the, d pair have spin = 0, and the spin of Λ is carried by the s qark alone.) µ = Λ [ µ µ d µ s µ µ d µ s] 6 µ s = From the experiment, µ 0.6 e = Λ mc p

22 P570 This shold be eqal to µ s according to the qark model. ( e ) e Therefore, µ Λ = 0.6 = µ s mc p mc s and ms = mp = 5 MeV 0.6 One can also predict the magnetic dipole moments for Σ ±, Σ o, Ξ -, Ξ o sing the qark model. Reasonably good agreements with the experimental data (within 0%) are obtained. Masses of Hadrons We now consider the masses of hadrons in the qark model. For hadrons with zero orbital anglar momenta, the interaction between qarks can depend on the relative orientation of their spins. An empirical expression for the masses of mesons and baryons are given as σ iσ M (meson) = m+ m + a mm M (baryon) m m m a σ iσ i j = i> j mm i j The σ ii σ j term is the QCD analog of the hyperfine interaction in QED, which originates from magnetic force. This term is called color-magnetic interaction. m, m, m correspond to the masses of the qarks. It is relatively straight forward to calclate the meson mass. Note that σ iσ = 4 i = 4 + ( s s ) ( s s ) s s s( s ) s ( s ) s ( s ) = = s( s+ )

23 P570 = for s =, or for s = 0 where s = s + s Using m = m d = 0 MeV, m s = 48 MeV, a = 60 MeV (m ) one obtains the following meson masses (in MeV): π ± ρ K K* η φ calclated observed For the mass of baryon, the qark model calclation is slightly more complicated. M (baryon) m m m a σ iσ σ iσ σ iσ = mm mm mm a) For baryons made of, d qarks only (n, p, ) or of s qarks only (Ω), m = m = m and the color-magnetic term becomes a ( i + i + i ) m σ σ σ σ σ σ ( 9 ) + + = = + s = s + s + s where σ iσ σ iσ σ iσ ( s s s s ) s( s ) s is the total spin of the baryon ( ) b) For J = baryon decplet 4 σ ( ) + σ = s+ s = s s = Similarly for s and s Therefore, σ+ σ = ( s s s ) = ( ) = Similarly, for σi σ, σi σ c) For J = ½ baryon octet Σ + (s), Σ - (dds), Ξ o (ss), Ξ - (ssd), Σ o (ds)

24 P570 4 All have m = m = m In addition, the first two qarks cople to s = σiσ σ iσ σiσ σiσ σiσ σiσ Therefore + + = + + mm mm mm m mm mm ( s s s ) σi σ = + ( σiσ + σiσ+ σi σ) m mm mm ( ) = s ( s ) s( s ) m mm mm d) For Λ o (ds), it is similar to c), except that s = 0. Nclei are made of protons and netrons. In certain sense, they are analogos to hadrons which are made of qarks and anti-qarks. However, protons and netrons are color-less objects, nlike the qarks and anti-qarks. This has interesting implications which can be illstrated by a consideration of the mass- nclear system H and He. H consists of (pnn) while He is made of (ppn). (p, n) are isospin doblet, jst like (d, ) form and isospin doblet. If one sbstittes (p, n) by (, d), then (pnn) (dd) and (ppn) (d). H n He p In the previos sections, we discssed the wave fnctions and magnetic moments of p, n in terms of the constitent qarks (, d). Can one extend sch an approach to describe H and He? Let s consider the magnetic moments of H and He. We recall that the magnetic moment of proton is expressed in terms of the magnetic moment of p, down qarks: µ p = ( 4 µ µ d) 4 Similarly, for netron: µ n = ( µ d µ )

25 P570 5 If the wave fnctions for He, H are analogos to those of p, n, then one cold express the magnetic moments of He, H in terms of the magnetic moments of p and n µ He 4 µ µ = ( p n ) µ = ( µ n µ p) H 4 Since µ p =.79 N.M, µ n = -.9 N.M., the above eqations give µ = 4.6 N.M. µ =.48 N.M. He H This reslt is in total disagreement with the experimental data of µ =. N.M. µ =.98 N.M. He H In fact, even the signs are wrong! Since proton and netron are color-netral objects, the color wavefnctions are symmetric. Therefore, one shold calclate the H and He magnetic moments assming that their color wave fnctions are symmetric (and hence the flavor-spin wave fnctions are anti-symmetric). The expressions for H and He s magnetic moments are now very different. We obtain µ = µ.9 N.M. He n = µ = µ.79 N.M. H p = These reslts are mch closer to the experimental vales of µ =. N.M. and He µ =.98 N.M. H This example illstrates the important role played by color for describing the hadrons. For He, H, the nderlying ncleons are colorless, reslting in very different magnetic moments from what wold have been expected assming a complete analogy between ( He, H) and (p, n). Another interesting example is the magnetic dipole moment of deteron. If one assmes a simple wave fnction for deteron: d = p n (deteron has total spin = )

26 P570 6 This is analogos to vector mesons, whose magnetic moments are given as the sm of the q and q s magnetic moments. One expects d µ d = µ + µ =.79.9 N.M. = 0.88 N.M. p n Experimentally, µ d = 0.86 N.M. The small discrepancy between the simple model prediction and the data reflects a small contribtion from the D state component in the deteron wave fnction. Other Hadrons We have so far only discssed the lightest mesons and baryons. While these hadrons are among the most important ones to be familiar with, clearly other hadrons can be formed by orbital or radial excitation of one or more qarks, or by replacing the /d/s qarks by the heavy charm (c) or beaty (b) qarks. We now consider mesons with total spin J, orbital anglar momentm L, and qark spin S J = L + S S = 0, The parity operator P applied to a qq meson gives P ( ) L ηη ( ) = = q q L+ where (-) L reflects the effect of space inversion on the orbital anglar momentm. η and η are the intrinsic parity of q and q, respectively. The opposite intrinsic q q parity of fermion and anti-fermion introdces a factor of. The operation of charge conjgation changes a particle into an anti-particle. For netral mesons which are their own anti-particles, they are eigenstates of the charge conjgation operator C. The C operation may be considered eqivalent to the exchange of q with q. On exchange, a factor of (-) L comes from the orbital anglar momentm, another factor of (-) from the opposite intrinsic parity of q and q, and finally a factor of (-) S+ comes from the symmetry property of the spin wave fnction.

27 P570 7 The C-parity for a self-conjgate meson is C = (-) (-) L (-) S+ = (-) L+S Since P = (-) L+ C = (-) L+S PC = (-) S+ Hence, for S = 0 mesons PC = - (P, C have opposite signs) for S = mesons, PC = + (P, C have same signs) The J PC of ordinary qq mesons are as follows: S = 0 S = L = L = , ++, ++ L = -+ --, --, -- L = +- ++, ++, It is important to note that ordinary qq mesons cannot have J PC = 0 --, 0 +-, -+, +-, -+,..... These are exotic qantm nmbers, which cold only be associated with non- qq exotic mesons sch as gg (gleball), qqq (hybrid), qq, etc. Note that K o, o K are netral mesons, bt not eigenstates of C-parity. Photon is an eigenstate of C. Since EM field changes sign nder C, photon has C = -. C is conserved in strong and electromagnetic interactions. Similarly, o π C=+ o ρ C = γγ is allowed, while π o π o o π o ρ π o η γγγ

28 P570 8 The crrent naming scheme for mesons considers two separate cases: a) Netral-flavor mesons ( S = C = B = 0) qq content S+ L J = (L even ) J (L odd ) J (L even ) J (L odd ) J d, dd, d (I = ) π b ρ a dd + and/or ss (I = 0) η, η h, h ω, φ f, f cc η c h c ψ χ c bb η b h b Υ χ b b) Mesons with non-zero S, C, B strange meson: K (I = ½ ) charm meson: D S (I = 0), D (I = ½ ) beaty meson: B S (I = 0), B C (I = 0), B (I = ½ ) K, D, B withot sbscript indicates that they also contain a light /d or / d.

29 P570 9 The following table from the most recent PDG lists the qark-model assignments for the known mesons:

30 P570 0 The SU(4) 6-plets for psedoscalar and vector mesons are shown as follows: Note that the netral mesons at the center of the center plane are mixtres of, dd, ss and cc states. Also note the 4-fold symmetry in the SU(4) 6-plets. o + + o For example, one can form an octet plane containing D, DS, K, K, DS, D (an SU() octet made of, s, c). Similarly octet planes made of (d, s, c) and (, d, c) can also be envisaged. Analogos SU(4) 6-plets can be constrcted for (, d, s, b) and for any orbitally or radially excited mesonic states.

31 P570 Baryon Excited States For baryons with L = 0, the space wave fnction is symmetric and the flavor-spin wave fnction is totally symmetric too. For baryons with one qark excited to the P state, both symmetric and mixed symmetry space wave fnction can be formed. Recall the SU(), SU() (flavor, spin) mltiplets: Symmetric: (0, 4) + (8, ) = 56 Mixed Symmetry: (0, ) + (8, 4) +(8, ) + (, ) = 70 Anti-symmetric: (8, ) + (, 4) = 0 For baryon grond states with L = 0: (0, 4) : J P = / + (8, ) : J P = / + For baryon states with one qark excited to L =, the space-symmetric state vanishes if the origin is chosen to be the center-of-mass of the three qark system (see F. Close s book, page 8). Therefore, only mixed symmetry states are allowed: L = : (L = ) (0, ) J = / -, / - (L = ) (8, 4) J = 5/ -, / -, / - (L = ) (8, ) J = / -, / - (L = ) (, ) J = / -, / - P The following table lists the (, N ) D L for many of the known (, d, s) baryons. D is the dimensionality of the SU() x SU(), L is the total qark orbital anglar momentm, and P is the total parity. N is the total nmber of qanta of excitation (both radial and orbital). Note that baryon states with total spin p to / + have been observed.

32 P570

33 P570 Baryon Naming Scheme The naming scheme for baryons containing, d, s, c, b qarks is relatively straightforward. The symbols N,, Λ, Σ, Ξ, and Ω originally sed for the baryons made of light qarks (, d, s) are now generalized for heavy baryons too. /d : N (Isospin = ½), (I = /) /d : Λ (I = 0), Σ (I = ) If the third qark is a c or b qark, its identity is given by a sbscript, i.e. Λ b, Λ c, Σ b, Σ c /d : Ξ. They have I = ½. If one or both of the non-/d qarks are heavy, then sbscripts are added: Ξ c, Ξ cc, Ξ b, Ξ bc, Ξ bs, etc (Note that Ξ c contains (/d) s c where s is not specified in the sbscript. No /d : Ω. They all have I = 0. The heavy qark contents are again shown explicitly in the sbscripts, sch as Ω ccc, Ω cc, Ω c, Ω bbb, Ω bbc, Ω bcc, etc. A baryon that decays strongly has its mass as part of its name, sch as (), Σ(85), etc. There are eleven known charm baryons, each with one c qark. The charmedbaryon spectrm is shown here in comparison with the light baryons:

34 P570 4 For b-baryons, both Λ b and Ξ b have been fond. Charmed Baryons For baryons containing, d, s, c qarks, the SU() needs to be extended to SU(4): = The SU() symmetric decplet is now extended to SU(4) totally symmetric 0- plet. The SU() mixed symmetry octet now becomes SU(4) mixed-symmetry 0- plet. These two SU(4) 0-plets are shown as follows:

35 P570 5 (a) mixed-symmetry 0-plets (b) symmetric 0-plets Note the 4-fold symmetry in these diagrams. The mixed-symmetry 0-plets contain 4 SU() octets made of (, d, s), (, d, c), (, s, c) and (d, s, c). Similarly, the symmetric 0-plets contain 4 SU() decplets. Exotic Mesons and Baryons While qq and qqq states are the simplest colorless hadrons one can constrct, there are clearly other colorless states which can be made ot of q, q, and glons. Extensive experimental searches for sch exotic hadrons have not yet provided any clear evidence for their existence. Some examples of the exotic hadrons are: a) Gleballs. These are colorless objects made of a pair of glons. Lattice gage calclations sggest that light gleballs have J PC = 0 ++ and ++ (~ 600 MeV for 0 ++

36 P570 6 and ~ 0 for ++ ). So far, the best candidate for gleball is the f o (500) 0 ++ state. These gleballs can mix with ordinary qq states, making their identification a difficlt task. b) Hybrids. These are exotic mesonic states consisting of qqg, where a glon combines with the qq. The presence of the extra glon allows some exotic qantm nmbers for the hybrids. Evidence for sch an exotic hybrid with J PC = -+ has been reported for the π (400) prodced in the π - p ηπ - p reaction, where π (400) coples to ηπ -. More extensive stdies for the hybrids are planned in the ftre at the Jefferson Laboratory. c) Dibaryons. These are six-qark states. The most celebrated example is the H particle made of (ddss). Extensive search has shown no evidence for sch particle.