A note on SU(6) spin-flavor symmetry.

Transcription

1 A note on SU(6) spin-flavor symmetry. 1 3 j = 0,, 1,, 2,... SU(2) : 2 2 dim = 1, 2, 3, 4, 5 SU(3) : u d s dim = 1,3, 3,8,10,10,27... u u d SU(6) : dim = 1,6,6,35,56,70,... d s s

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3 SU(6) irrep 56 + (l=0) 70 - (l=1) 56 + (l=0 ) 56 + (l=2) 70 + (l=0 ) SU(3) f irrep J P S = 0 S= - 1 I = 0 I = 1 S = - 2 S = /2 + N(939) Λ(1116) Σ(1193) Ξ(1318) /2 + Δ(1232) Σ(1385) Ξ(1533) Ω(1672) 2 8 3/2 - N(1520) Λ(1690) Σ(1580)** Ξ(1820) 1/2 - N(1535) Λ(1670) Σ(1620)** 4 8 1/2 - N(1650) Λ(1800) Σ(1750) 5/2 - N(1675) Λ(1830) Σ(1775) 3/2 - N(1700)? Σ(1670) /2 - Δ(1620) 3/2 - Δ(1700) 2 1 1/2 - Λ(1405) 3/2 - Λ(1520) 2 8 1/2 + N(1440) Λ(1600) Σ(1660) /2 + Δ(1600) Σ(1690)** 2 8 3/2 + N(1720) Λ(1890)? 5/2 + N(1680) Λ(1820) Σ(1915) /2 + Δ(1910) 3/2 + Δ(1920) Σ(2080)** 5/2 + Δ(1905) 7/2 + Δ(1950) Σ(2030) 2 8 3/2 - N(1710) Λ(1800) Σ(1880)** 4 8 1/2 - N(1540)* /2 - Δ(1550)* 2 1 1/2 - N(2100)*

4 Contracted SU(2 N f ) c group and its relation with the usual SU(2 N f ) c group.. We start by considering a simpler situation: connection between E2 (Euclidean group in 2D) and SO(3) (rotations in 3D). SO(3) GROUP Parameters Rotations α, β, γ Euler angles 0 α, γ < π ; 0 β 2π compact Generators Rotations Lie Algebra [ L, L ] = i ε Casimir i j L, L, L ijk Lk L = L + L + L Parameters Rotations Translations b, b [ J, P ] = i ε P [ P, P ] = E2 GROUP b, b < ; 0 ϕ 2π non-compact Generators Rotations J Translations Lie Algebra k Casimir ϕ km P = P + P m P, P 1 2

5 We will see that one can go from SO(3) to E2 by group contraction. We can establish a correspondence between elements of this groups by the stereographic projections of points on the sphere onto the plane tangent to the north pole. The correspondence becomes simpler as the radius of the sphere becomes larger. In the limit R the group SO(3) contracts to E2 Note that b 1 / R = tanθ θ 2 2 as R with b 1 kept fixed. Thus, in that limit e e e iθ L ib L / R ib P Similar association can be done between L and P 1 2

6 Thus we identify the mapping L / R P ; L / R P ; L J We can check in the algebra the result of this correspondence [ P, P] [ L / R, L / R] = [ L, L ]/ R = il / R [ P, J] [ L / R, L ] = [ L, L ]/ R = il / R ip [ P, J] [ L / R, L ] = [ L, L ]/ R = il / R ip 2 1 Therefore, it is said that in the limit R the group SO(3) contracts to E2.

7 What about the group irreducible representations (irrep)? The irreps of SO(3) are wellknown. They are labeled by the Casimir eingenvalue l and are finite-dimensional with dim(l) = 2l +1. The usual set of basis vectors are chosen so that they also diagonalize L 3 with eigenvalue m and are denoted l m > E2 is a non-compact group. Thus, its irreps are, in general, infinite dimensional. They can be labeled by the Casimir eingenvalue p 2. A possible set of basis vectors can be chosen so that they also diagonalize J 3 with eigenvalue m and are denoted p m >. It is not difficult to show that the action of the generators on these basis vectors give J pm, >= m pm, > P pm, >= ip pm, ± 1 > where P = P± ip ± ± 1 2 From where it clearly results that the irreps are infinite dimensional (except the ones with p=0 which are 1 dim). This basis is called the angular momentum basis. Very interestingly this irrep basis p m > can be obtained form the irreps l m > of SO(3) if l = pr as R

8 One can construct an alternative basis using the method of induced representation. In this case one starts by considering (T2 the abelian translational group of E2) which is generated by Next one takes an standard eigenvector of these operators p P = ( P, P) 0 such that, 1 for 2 example, P 1 P 2 P p0 >= p p0 > p0 >= 0 p >= p p > One generates the full irreducible invariant space using the group operations that produce new eigenvalues of P, i.e. those associated with generators which do not conmute with it. In this case we have only R(φ)=exp(-I φ J). Therefore, the alternative basis states (so-called plane wave basis ) are given by p, φ > p>= R( φ) p > One can easily go from one basis to the other using the projection method m dφ imφ m imφ p, m i e p, φ, p, φ i e p, m 2π >= > >= > 0 m

9 Before going back to the large N c spin-flavor algebra let us consider the method of induced representations for E3, the Euclidean group in 3D. Let us recall that this group has two Casimir operators: P 2, PL i whose eigenvalues can be used to label the irreps. The plane wave basis will consist of eigenvectors of the operator 2 2 set { P, PLP i, }. The eigenvalues will be denoted { p, λ, p }. It suffices to label the eigenvectors { p, λ, pˆ } where pˆ = p / p. Thus 2 2 P p, λ; pˆ >= p p, λ; pˆ > P L p, λ; pˆ >= λ p, λ; pˆ > P p, λ; pˆ >= p p, λ; pˆ > p ˆ = (0,0,1) We consider now the subspace characterized by the standard vector 0. We have to look at the action of the rotation operators. We note, however, that this vector is invariant under rotations around the z-axis. These operations form the little group of ˆp 0 which is SO(2) in the present case. The irreps of SO(2) are 1 dim and labeled λ, the eigenvalue of L 3. Thus λ can only take the values λ=0,±1, Finally, the full vector space for the irrep of E3 can be generated with rotations which are not in the little group. Namely, ˆ ˆ0 p, λ; p>= R( φθ,,0) p, λ; p > As before, the spherical basis can be obtained from this plane wave basis using the projections method.

10 Coming back to our case, the relevant algebra is J, J = iεijk J ; J, X = iεijk X i j k i ia ka T, T = i fabc T ; T, X = i fabc X a b c a ib kc ia jb X, X = 0 The basis vectors can be obtained using the induced representation method. In this ia case the relevant reference vector is X 0 = δ ia and the associated little group is generated by K = I + J The basis of axial current baryon eigenstates (plane wave basis) is given by the ia states X0, K, K z >. These eigenstates are not, however, states of good spin and isospin as one expects for the physical baryons. To obtain them, one uses the projection method so as to go to the spherical basis J, J, I, I, K > where J and I should be consistent with a given value of K z z

11 For N c odd the lowest allowed baryon spin is J=1/2, Then, for each value of K we have the following towers K = 0 (1/2,1/2),(3/2,3/2),(5/2,5/2),... N Δ K = 1/2 (1/2,0),(1/2,1),(3/2,1),(3/2,2),(5/2,2),(5/2,3),... Λ Σ Σ * The low-spin states of each tower have the correct spin and isospin quantum numbers to be identified with the spin-1/2 octet and spin-3/2 decuplet baryon states of QCD is K = N s /2. K = 1 (1/2,1/2),(1/2,3/2),(3/2,1/2),(3/2,3/2),(3/2,5/2),... Ξ K = 3/2 (1/2,1),(1/2,2),(3/2,0),(3/2,1),(3/2,2),(3/2,3),... Ω Ξ * The other states correspond to states with exist for Nc but not for N c =3. Notice that the towers with K >0 contain extra states even for J= 1/2, 3/2

12 Having obtained the baryon states we can proceed to discuss the baryon static properties as e.g. masses, magnetic moments, etc. The idea is that any of the related operators admit a 1/N c expansion, and that the form of each term of the expansion can be determined using adequate consistency conditions. Les us consider for the example the mass operator in the 2 flavor case. It can be expressed as 1 M = N M + M + M + where M = operators 0 c i ϑ( Nc ) Nc We can now repeat the calculation of the πb scattering amplitude taking into account the effect of a baryon mass splitting ΔM. This splitting induces a change in the intermediate baryon propagator which we now reads i/(e π - ΔM). Expanding up to leading order in ΔM one obtains that in order to have an amplitude of ϑ(n c 0 ) one must have [ X,[ X, M ]] = 0 [ X,[ X, M ]] = 0 ia jb ia jb which only admit the solution M 1 = M 0 = I. Thus, as expected, M = m0 NcI +ϑ(1 / Nc)

13 To determine the 1/ N c corrections we have to consider the πb B π πscattering amplitude given by the diagrams + permutations This yields the constraint [ X,[ X,[ X, M ]] = 0 ia jb kc ia ia ia ia the only independent solution is J. Therefore 0 which has solutions M = J, I, X X. Since X X = const and J = I ( K = 0) we have that m M = m N + J c1 ϑ(1 / Nc) Nc

14 An alternative way to obtain the 1/N c expansion of physical operators is to use the so-called quark representation. This is based on the use of SU(2 N f ) algebra instead of the contracted one. As we have seen the GS baryons must be in the completely symmetric irrep of this group. For example, for N f =3 such irrep admits the SU(2)xSU(3) decomposition Note that for SU(2) f this leads to the tower (1/2,1/2), (3/2,3/2), (5/2,5/2),.. where, of course the only first two states appear for N c =3. On the other hand for N f =3 we can look into the isospin content of the SU(3) irreps corresponding to each spin. Defining K = N S /2 we have we see then that K = I + J implying that, in fact, as N c we recover the towers obtained with the contracted algebra.

15 The idea is, then, that any color singlet QCD operator can be represented by a series of composed operators acting in the completely symmetric irrep of SU(2N f ), and ordered according to powers of 1/N c O = c Φ ( n) ( n) QCD i i ni, n-body operator obtained as the product of n generators of SU(2 N f ), i.e. J i, T a, G ia Rules for N c counting Unknown coefficients to be fitted n-body operators need at least n quarks exchanging (n-1) gluons. According to usual rules it carries a suppression factor (1/ N c ) n-1 e.g a 3-body operator g 4 ~ N c -2 while J i and T a have matrix elements of ϑ(n c 0 ) those of G ia are, in general, of ϑ(n c 1 ) (remember that X 0 ia = ϑ(n c 0 ) = lim G ia /N c ) N c

16 As example we construct the three flavor GS baryon masses. Up to 1/N c contributions and leading order in the SU(3) flavor breaking ε, we get {, } J T G S T G M = a N + a + a + a + a + b T i a ia 0 c 1,1 1,2 1,3 1,4 2 1ε 8 Nc Nc Nc Nc It looks like there are many contributions up to this order. However, since for GS baryons the operators are acting on fully symmetric irrep of SU(6), several reduction formulae appear (Dashen, Jenkins, Manohar, PRD49(94)4713; D 51(95), 3697). To eliminate G 2 and T 2 In favor of J 2 To eliminate {T a, G ia } in favor of J i

17 Using these relations M simplifies to 2 J M = a N + a + b ε T 8 0 c 1 1 Nc Note that for SU(2) f this reduces to the result already obtained using consistency relations Proceeding in this way can verify the the most general mass operator containing up to 3-body contributions is We have 8 parameters to determine the 8 masses of the GS baryons. Therefore, if we set a parameter to zero we obtain a certain mass relation. For example, if we set d 3 =0 we obtain a mass relation which is valid up to ϑ(ε 3 /N c 2 ). Namely, Δ Σ + Ξ Ω= * * ( ε / N c ) To test how well they are satisfied empirically it is convenient to define err= L-R /(L+R), where in this case L= Δ+Ω and R=3(Σ *+ Ξ * )

18 In this way one obtains This provides an excellent illustration of the convergence of the combined 1/ N c and flavor breaking expansion. This type of analysis has been applied to analyze axial couplings and magnetic moments, etc. (Dai,Dashen,Jenkins,Manohar, PRD53(96)273, Carone,Georgi, Osofsky, PLB 322(94)227, Luty,March-Russell, NPB426(94)71, etc).

19 Quark operator analysis for excited baryons. Carone et al, PRD50(94)5793, Goity, PLB414(97)140; Pirjol and Yan, PRD57(98)1449; PRD 57(98)5434 proposed to extend these ideas to analyze low lying excited baryons properties. Take as convenient basis of states: multiplets of O(3) x SU(2 N f ) (approximation since they might contain several irreps of SU(2N f ) c Schat, Pirjol, PRD67(03)096009, Cohen, Lebed, PLB619(05)115) Excited baryon composed by Core (N c -1 quarks in S irrep) + Excited quark with l Coupled to MS or S irrep of spin-flavor N 1 c Low lying Excited Baryon For lowest states relevant multiplets are [0 +, 56 ], [1 -, 70], [2 +, 56]

20 In the operator analysis, effective n-body operators are now O = R Φ ( n) ( n) (n) where is an (3) operator and Φ an (2 ) tensor R O SU N f { a i ( n) 1 Φ = λ Λ... where =,, 1 c Λ λ t n c a i N Λ c c Tc c n 1 ia s g =, S, G ia c This approach has been applied to analyze the excited baryon masses of different excited multiplets (Carlson, Carone, Goity and Lebed, PLB438 (98) 327; PRD59 (99) Schat, Goity and NNS, PRL88(02)102002, PRD66 (02)114014, PLB564 (03) 83. Matagne, Stancu, PRD71(05)014010)

21 Analysis of masses of (1 -,70) states Goity, Schat, NNS, PRL88(02) Although O(3)xSU(6) symmetry is broken at O(N c0 ), the corresponding operators (e.g. spin-orbit) have small coefficients. Hyperfine operator S c S c /N c of O(N c -1 ) give chief spinflavor breaking. Λ(1520) Λ(1405) determined only by spin-orbit operator. For the rest of spin-orbit partners other operators appear. Important is l i t a G c ia /N c

22 S=0 S=-1 Green shaded boxes correspond to the experimental data, the black lines are the Isgur-Karl quark model and the hatched boxes are the 1/N c results. S=-2,-3 As for the case of the gs baryons parameter independent mass relations can be also obtained. Unfortunately, not enough empirical data to test them.

23 STRONG DECAYS We consider decays of the type B * M = π, η where B * are non-strange members of the [1 -,70], [0 +,56 ] and [2 +,56] multiplets. The decay widths are given by B = N, Δ For [1 -,70] partial waves are l M =S,D For [0 +,56] partial waves is l M = P For [2 +,56] partial waves are l M = P,F ( ) ( + )( + ) C ( k ) < J, I J, I, S > [ lm, im] [ lm, im] * * * n M lm, im k n M M B n 2 8π M * * B* 2I 1 2J 1 Γ = 2 where ( ) [ l () M, i l M ξ ] = Φ [ l, i ] [ l', i ] M M M Acts on orbital wf of excited quark Acts of on spin-flavor wf of the excited quark core system

24 Analysis of strong decays of non-strange (1 -,70) resonances Goity, Schat, NNS, PRD71(05) Clear dominance of 1B LO operator g ia in π-decays as in χqm NLO coefficents poorly determined due to large data error bars. More precise inputs needed to determine significance of NLO corrections N*(1535) ratio of decays to Nη/Nπ well reproduced.

25 PHOTOPRODUCTION AMPLITUDES The helicity amplitudes of interest are defined as 2πα = η, λ ˆ. ( ω ˆ), λ 1 ( ) * * Aλ B B e+ 1 J γ z N ωγ λ=1/2, 3/2 is helicity along γ-momentum (z-axis) ê +1 is γ-polarization vector η(b*) sign factor which depends on sign strong amplitude π N B*. When B* can decay through 2 partial waves (e.g. P or F ) undetermined sign (ξ =P/F=±1) J ( ω γ zˆ ) can be represented in terms of effective multipole baryonic operators with isospin I =0,1. Then, the electric and magnetic components of the helicity amplitudes can be expressed as L ML 3α N ω Aλ = η B g B λ I N λ I [ LI] [ ] ( ) [ ] * c γ [ L, I ] * ( ) n ;, 3 n ; 1, [10] 3 4ω γ m ρ ni, L 1 EL 3( L+ 1) α N ω Aλ = η B g B λ I N λ I + [ LI] [ ] ( ) [ ] * c γ [ L, I ] * ( ) n ;, 3 n ; 1, [10] 3 4(2L 1) ω γ m ρ ni, where ( ) [ LI ξ, ] = Φ [ LI, ] ( l) [ l', I] Acts orbital wf of excited q Acts of spin-flavor wf of the excited quark core system

26 Helicity amplitudes for [1 -,70]-plet non-strange resonances Goity, Matagne, NNS, Phys. Lett. B663 (08) 222 ξ = -1 and θ 3 =2.82 clearly favored by fits. Only this case is shown. 2B

27 SUMMARY ON LARGE N c 1/N c expansion useful and predictive for QCD hadrons and dynamics. It is a systematic expansion which yields model-independent results. Large-N c baryons have exact contracted spin-flavor symmetry SU(2N f ) c. 1/N c expansion gives a quantitative understanding of spin-flavor symmetry breaking for GS baryons. 1/N c expansion can be extended to the study of excited baryons. The analyses of masses show small order N c0 breaking of spin-flavor symmetry. This is dominated by the subleading hyperfine interaction. Other properties like e.g. strong decays, photoproduction amplitudes, etc have been studied.