wykresx.eps Acta Polytechnica Vol. 51 No. 1/2011 Supersymmetry in the Quark-Diquark and the Baryon-Meson Systems S. Catto, Y. Choun Abstract A superalgebra extracted from the Jordan algebra of the 27 and 2̄7 dim. representations of the group E6 is shown to be relevant to the description of the quark-antidiquark system. A bilocal baryon-meson field is constructed from two quark-antiquark fields. In the local approximation the hadron field is shown to exhibit supersymmetry which is then extended to hadronic mother trajectories and inclusion of multiquark states. Solving the spin-free Hamiltonian with light quark masses we develop a new kind of special function theory generalizing all existing mathematical theories of confluent hypergeometric type. The solution produces extra “hidden” quantum numbers relevant for description of supersymmetry and for generating new mass formulas. Keywords: supersymmetry, relativistic quark model. A colored supersymmetry scheme based on SU(3)c × SU(6/1) Algebraic realization of supersymmetry and its experimental consequences based on supergroups of type SU(6/21) and color algebras based on split octonionic units ui (i = 0, . . . ,3) was considered in our earlier papers [1, 2, 3, 4, 5]. Here, we add to them the local color group SU(3)c. We could go to a smaller supergroup having SU(6) as a subgroup. With the addition of color, such a supergroup is SU(3)c × SU(6/1). The fun- damental representation of SU(6/1) is 7-dimensional which decomposes into a sextet and a singlet under the spin-flavor group. There is also a 28-dimensional representation of SU(7). Under the SU(6) subgroup it has the decomposition 28=21+6+1 (1) Hence, this supermultiplet can accommodate the bosonic antidiquark and fermionic quark in it, providedwe are willing to addanother scalar. Togetherwith the color symmetry,weare led to consider the (3, 28) representation of SU(3) × SU(6/1) which consists of an antidiquark, a quark and a color triplet scalar that we shall call a scalar quark. This boson is in some way analogous to the s quarks. The whole multiplet can be represented by an octonionic 7×7 matrix Z at point x. Z =u · ( D ∗ q iqT σ2 S ) (2) HereD∗ is a 6×6 symmetricmatrix representing the antidiquark,q is a 6×1 columnmatrix,qT is its transpose and σ2 is the Pauli matrix that acts on the spin indices of the quark so that, if q transforms with the 2 × 2 Lorentz matrix L, qT iσ2 transforms with L −1 acting from the right. Similarly we have Zc =u∗ · ( D iσ2q ∗ q† S∗ ) (3) to represent the supermultiplet with a diquark and antidiquark. The mesons, exotic mesons and baryons are all in the bilocal field Z(1)⊗ Zc(2) which we expend with respect to the center of mass coordinates in order to represent color singlet hadrons by local fields. The color singlets 56+ and 70− will then arise as shown in our earlier papers [1, 2, 3]. Now the (D̄q) system belonged to the fundamental representation of the SU(6/21) supergroup. But Z belongs to the (28) representation of SU(6/1)which is not its fundamental representation. Are there any fields that belong to the 7-dimensional representation of SU(6/1)? It is possible to introduce such fictitious fields as a 6-dimensional spinor ξ and a scalar a without necessarily assuming their existence as particles. We put ξ =u∗ · ξ, a =u∗ ·a, (4) 77 Acta Polytechnica Vol. 51 No. 1/2011 so that both ξ and a are color antitriplets. Let λ = ( ξ a ) , λc = ( ξ̂ a∗ ) (5) where ξ̂ =u · (iσ2ξ∗), a∗ =u ·a∗. (6) Consider the 7×7 matrix W = λλc† = ( ξξ̂† ξa aξ̂† 0 ) . (7) W belongs to the 28-dimensional representation of SU(6/1) and transforms like Z, provided the components of ξ are Grassmann numbers and the components of a are even (bosonic) coordinates. The identification of Z and W would give s= a×a=0, qα = ξα ×a, D∗αβ = ξα × ξβ (8) A scalar part in W can be generated by multiplying two different (7) representations. The 56+ baryons form the color singlet part of the 84-dimensional representation of SU(6/1) while its colored part consists of quarks and diquarks. Now consider the octonionic valued quark field qiA, where i = 1,2,3 is the color index and A stands for the pair (α, μ) with α = 1,2 being the spin index and μ = 1,2,3 the flavor index. (If we have N flavors, A =1, . . . , N). As before qA = uiq i A =u ·qA (9) Similarly the diquark DAB which transforms like a color antitriplet is DAB = qAqB = qBqA = ijku ∗ kq i Aq j B =u ∗ ·DAB We note, once again, that because qiA are anticommuting fermion operators, DAB is symmetric in its two indices. The antiquark and antidiquark are represented by q̄A =u ∗ · q̄A, (10) and D̄AB =u ·D̄AB , (11) respectively. If we have 3 flavors qA has 6 components for each color while DAB has 21 components. At this point let us study the system (qA, D̄BC) consisting of a quark and an antidiquark, both color triplets. There are two possibilities: we can regard the system as a multiplet belonging to the fundamental representation of a supergroup SU(6/21) for each color, or as a higher representation of a smaller supergroup. The latter possibility is more economical. To see what kind of supergroup we can have, we imagine that both quarks and diquarks are components ofmore elementary quantities: a triplet fermion f iA and a boson C i which is a triplet with respect to the color group and a singlet with respect to SU(2N) (SU(6)) for three flavors). The f iA is taken to have baryon number 1/3 while C has baryon number −2/3. The system qiA = ijkf̄ j AC̄ k (12) will be a color triplet with baryon number 1/3. It can therefore represent a quark. We can write qA =u ·qA =(u∗ · f̄A)(u∗ · C̄) (13) With two anti-f fields we can form bosons that have the same quantum numbers as antidiquarks: D̄AB =u · D̄AB =(u∗ · f̄A)(u∗ · f̄B) (14) In this case the basic multiplet is (fA,C) which belongs to the fundamental representation of SU(6/1) for each color component. The complete algebra to consider is SU(3)c × SU(6/1) and the basic multiplet corresponds to the representation (3,7) of this algebra. Let F = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ f1 f2 ... f6 C ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ , fA =u · fA, C =u ·C (15) 78 Acta Polytechnica Vol. 51 No. 1/2011 We write f1, f2, etc., as f1 = ⎛ ⎜⎜⎜⎜⎜⎝ f11 f12 ... f16 ⎞ ⎟⎟⎟⎟⎟⎠ , f 2 = ⎛ ⎜⎜⎜⎜⎜⎝ f21 f22 ... f26 ⎞ ⎟⎟⎟⎟⎟⎠ , . . . (16) Combining two such representations and writing X̄=F×FT we have X̄ =u∗ ·X̄= ( f1 C1 ) (f2T C2) − ( f2 C2 ) (f1T C1) (17) Further making the identifications u∗ ·D11 =2f11 f 2 1 , u ∗ ·D12 = f11 f 2 2 − f 2 1 f 1 2 , etc. (18) and u∗ · q̄1 = f11C 2 − f21 C 1, u∗ · q̄2 = f12 C 2 − f22 C 1, etc., (19) we see that X̄ has the structure X̄ =u∗ · X̄=u∗ · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ D11 . . . D16 q̄1 D12 . . . D26 q̄2 D13 . . . D36 q̄3 D14 . . . D46 q̄4 D15 . . . D56 q̄5 D16 . . . D66 q̄6 −q̄1 . . . −q̄6 0 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (20) The 27 dimensional representation decomposes into 21+6̄ with respect to its SU(6) subgroup. Consider now an antiquark-diquark system at point x1 = x − 1 2 ξ and another quark-antidiquark system at point x1 = x + 1 2 ξ, and take the direct product of X(x1) and X(x2). We see that we get (DAB(x1), q̄D(x1))⊗ (D̄EF(x2), qC(x2)) (21) consisting of the pieces H(x1, x2)= ( q̄D(x1)qC(x2) DAB(x1)qC(x2) q̄D(x1)D̄EF(x2) DAB(x1)D̄EF(x2) ) (22) The diagonal pieces are bilocal fields representing color singlet 1+35 mesons and 1+35+405 exotic mesons, respectively, with respect to the subgroup SU(3)c × SU(6) of the algebra. The off diagonal pieces are color singlets that are completely symmetrical with respect to the indices (ABC) and (DEF). They correspond to baryons and antibaryons in the representations 56 and 56 respectively of SU(6). We can define the baryonic components FABC as FABC = − 1 2 {DAB, qC} (23) and evaluate its components: DABqC = ( u∗1(q 2 Aq 3 B + q 2 Bq 3 A)+ u ∗ 2(q 3 Aq 1 B + q 3 Bq 1 A)+ u ∗ 3(q 1 Aq 2 B + q 1 Bq 2 A) ) × (u1q1C + u2q 2 C + u3q 3 C) (24) This becomes DABqC = −u∗0 ( (q2Aq 3 B + q 2 Bq 3 A)q 1 C +(q 3 Aq 1 B + q 3 Bq 1 A)q 2 C +(q 1 Aq 2 B + q 1 Bq 2 A)q 3 C ) (25) 79 Acta Polytechnica Vol. 51 No. 1/2011 Similarly the qC DAB part is qC DAB = −u0((q2Aq 3 B + q 2 Bq 3 A)q 1 C +(q 3 Aq 1 B + q 3 B q 1 A)q 2 C +(q 1 Aq 2 B + q 1 B q 2 A)q 3 C) (26) Since u0 + u ∗ 0 =1, we have FABC = − 1 2 {DAB, qC} =(q1Aq 2 B + q 1 Bq 2 A)q 3 C +(q 2 Aq 3 B + q 2 Bq 3 A)q 1 C +(q 3 Aq 1 B + q 3 Bq 1 A)q 2 C (27) which is completely symmetric with respect to indices (ABC), corresponding to baryons. In the limit x2 − x1 = ξ → 0, H can be represented by a local supermultiplet with dimension 2 × 56+ 2(1+35)+405 = 589 of the original algebra. This representation includes 56 baryons, antibaryons, mesons and q2q̄2 exotic mesons. It is empirically well known for many years that all light-quark hadrons lie upon linear Regge trajectories. Relation between linear trajectories, linear confinement and relativistic dynamics had been well studied. Massless quarks bound by a linear confinement potential generate a family of parallel Regge trajectories. Regge slopes and daughter spacings depend on the Lorentz nature and other properties of the interaction. These are well known also through numerical studies (exact solutions) to the massless spinless Salpeter wave equation and the quantized straight string. A generalized second order equationwith a normalizedwave function including quarkmass m is given by [6] for H2: H2 =4 [ (m + 1 2 br)2 + P2r + l(l +1) r2 ] ; P2r = − ∂2 ∂r2 − 2 r ∂ ∂r (28) and Schrödinger equation is ∂2ψ ∂r2 + 2 r ∂ψ ∂r + ( E2 4 − 1 4 b2 ( r + 2m b )2 − l(l +1) r2 ) ψ =0 (29) We have developed [7] a new solution for the normalized wave function (including the small mass m): ψnr ,nζ ,l,m̄(r, θ, φ) = ( (nr −1)!(2l +2)!(nr + l − 12)! √ π (2b)l+ 2 3(l +1)!(l + 12)! − m {nr−1∑ p=0 2l+2(p + l +1)! bl+2p! ( (nr −1)!(12)! (nr − p −1)!(p − nr + 32)! )2 + nr−1∑ n=0 (nr −1)!(nr + l − 12)!(n − 1 2)!(n + l)!(2n + l +1) n!(−12)!(nr − n −1)!(n + l + 1 2)! × nζ−1∑ k=n (−1)nr+k2n+l+2(−12)!(nζ − n −1)!(n + k + l +1)!(n + k + 1 2)! bn+l+2(k + 12)!(nζ − k −1)!(k + l +1)!(k + n − nr 3 2)! })−12 × rle− b 4(r+ 2m b )2 n∑ n=0 (−1)n(nr −1)!(nr + l − 12)! n!(nr − n −1)!(n + l + 12 ( 1 2 br2 )n × { 1+ mr nζ−n−1∑ k=0 (−1)k(n + l)!(2n + l +1)(nζ − n −1)!(n − 12)! 2(k + n + 12)!(nζ − k − n −1)!(k + n + l +1)! ( 1 2 br2 )k} × Y m̄l (θ, φ) (30) In the above equation n∑ n=0 (−1)n(nr −1)!(nr + l − 12)! n!(nr − n −1)!(n + l + 12 ( 1 2 br2 )n = F |α|n ( |α| = nr −1, γ = l + 3 2 ;z = 1 2 br2 ) (31) This last expression is the confluent hypergeometric function. For this reason we call Eq. (31) as Grand Confluent hypergeometric function. The nr is the radial quantum number. We give a name of nζ to an additional hidden radial quantum number that appears in our solution. In many of the special functions for 80 Acta Polytechnica Vol. 51 No. 1/2011 second order differential equation there is only one eigenvalue, but when we include the small mass m, two eigenvalues are created. We already know that one eigenvalue is equal to E2r =4b ( l +2nr − 1 2 ) (32) This was shown in our earlier paper [6]. The second one, caused by small mass m, is equal to E2ah =4b ( l +2nζ + 1 2 ) (33) Typical wave functions has only three quantum numbers. But this wave function has four quantum numbers which are nr, nζ, l, and m̄. When we let the small mass m equal to zero, then Eq. (31) simply becomes ψnr ,nζ ,l,m̄(r, θ, φ)= ( (nr −1)!(2l +2)!(nr + l − 12)! √ π (2b)l+ 2 3(l +1)!(l + 12)! )−12 rle− b 4 r 2 F |α|n Y m̄ l (θ, φ) (34) where F |α|n = F |α| n ( |α| = nr −1, γ = l + 3 2 ;z = 1 2 br2 ) Eq. (35) is exactly the same as our previous wave function when the small mass m is neglected. Our generalized solution gives rise to new mass formulas in remarkable agreement with experiments which will be described in our forthcoming publications. We believe that the new special functions we createdwill have great many uses in physics and other areas of sciences. Acknowledgement This work was supported in part by DOE contracts No. DE-AC-0276 ER 03074 and 03075. One of us (SC) thanks Dean Jeffrey Peck for the travel fellowship and Professor Cestmir Burdik for the kind invitation to this conference to present this paper. References [1] Catto, S., Gürsey, F.: Nuovo Cim. 86 A, 201 (1985). [2] Catto, S., Gürsey, F.: Nuovo Cim. 99A, 685 (1988). [3] Catto, S.: MiyazawaSupersymmetry. InH. Sakai, B. F.Gibson (Eds.): New Facets of Three Nucleon Force. (2008) AIP 1011. [4] Catto, S.: Scalar mesons, Multiquark States and Supersymmetry. In G. Rupp, B. Hiller, F. Kleefield (Eds.) Workshop on Scalar Mesons and Related Topics. (2008) AIP 1030. [5] Catto, S.: Effective SupersymmetryBased on SU(3)c × S Superalgebra. InV. Dobrev (ED.)Lie Theory and Its Applications in Physics. 2010. [6] Catto, S., Cheung, H. Y., Gürsey, F.: Mod. Phys. Lett. A 38 3485 (1991). [7] Catto, S., Choun, Y.: In preparation. Sultan Catto E-mail: scatto@gc.cuny.edu The Graduate School The City University of New York and Baruch College 17 Lexington Avenue, New York, NY 10010 Physics Theory Group The Rockefeller University 1230 York Avenue, New York, NY 10021-6399 Yoon Choun The Graduate School The City University of New York and Baruch College 17 Lexington Avenue, New York, NY 10010 81