A proposal on the search for the hybrid with IG(JPC) = 1 (1 +) − − in the process J/ψ ρωππ at upgraded BEPC/BES ∗ → De-Min Lia†, Hong Yua,b and Qi-Xing Shena,b 0 0 0 aInstitute of High Energy Physics, Chinese Academy of Sciences, 2 n P.O.Box 918 (4), Beijing 100039,China‡ a J bInstitute of Theoretical Physics, Chinese Academy of Sciences, Bejing 100080,China 1 1 February 1, 2008 2 v 3 6 0 1 Abstract 0 0 0 ThemomentexpressionsforthebosonresonancesX withspin-parityJPXC =0++,1−+, / X h p 1++, and 2++ possibly produced in the process J/ψ ρX, X b1(1235)π, b1 ωπ are → → → - p givenintermsofthegeneralizedmomentanalysismethod. TheresonancewithJPXC =1−+ e X h : canbedistinguishedfromotherresonancesbymeansofthesemomentsexceptforsomerather v Xi special cases. The suggestion that the search for the hybrid with IG(JPC) = 1−(1−+) can r a be performed in the decay channel J/ψ ρωππ at upgraded BEPC/BESis presented. → PACS numbers: 13.20.Gd, 14.40.Cs Key words: J/ψ decay, Moment analysis, Hybrid mesons ∗The project is supported by the National Natural Science Foundation of China under Grant No. 19991487, and Grant No. LWTZ-1298 of the Chinese Academy of Sciences †E-mail: [email protected] ‡Mailing address 1 I. Introduction Apart from the ordinary qq¯ mesons, the new hadronic states such as glueballs (gg/ggg), hybrids (qq¯g) and four-quark states (qqq¯q¯) also exist according to the predictions of QCD. Discovery and confirmation of any one of these new hadronic states would bethe strong support to the QCD theory. Therefore, the search for and identifying these new hadronic states is a very excited and attractive research subject both theoretically and experimentally. These new hadronic states can have the same quantum number JPC as the ordinary qq¯ mesons, what’s more, they can also have the exotic quantum number JPC which are not al- lowed in the quark model such as 1−+, and thus they can not mix with the ordinary mesons. Experimentally, GAMS collaboration [1], E179 collaboration at KEK [2], VES group [3], E852 collaboration at BNL [4] and Crystal Barrel [5] all claimed that the evidence for the exotic state with JPC = 1−+ was observed. The observed ρπ, ηπ and η′π couplings of this state qualita- tively support the hypothesis that it is a hybrid meson, although other interpretations cannot be eliminated [6]. In terms of the predictions of the lattice QCD, thelowest lying glueball with JPC = 1−+ has a higher mass than J/ψ [7]. Bag model calculations [8] predict that the lowest lying qqq¯q¯states do not carry exotic quantum numbers and form nonets carrying the same quantum numbers as qq¯nonets, and that mostqqq¯q¯states can fall apart into two mesons and thus have a decay width in the order of their mass, which leads to that most qqq¯q¯states are expected to be essentially unconfined and will not be observed as resonance peaks with reasonably narrow widths. There- fore, the search for the gluballs and four-quark states with JPC = 1−+ at BEPC/BES could be disappointing. However, lattice QCD predicts that the mass of the hybrid with JPC = 1−+ is 1.2 2.5 GeV [8]. In addition, the naive estimate of pQCD predicts that the J/ψ hadronic ∼ decay processes are favorable to the production of hybrids. So, if the hybrids exist, the search for hybrid with JPC = 1−+ at BEPC/BES should be fairly hopeful. H. Yu and Q.X. Shen [9] have already discussed the possibility of the search for the hybrid with JPC = 1−+ in the process J/ψ ρX, X ηπ (η′π, ρπ). For the decay modes of the → → 2 hybrid with IG(JPC) = 1−(1−+), according to the symmetrization selection rule [10,11], the ηπ, η′π modes are strongly suppressed (A possible mechanism to explain why the 1−+ state was observed in the above suppressed decay channels is planned for separated publication). The ρπ mode is allowed, but this is a P-wave mode and thus the ρπ mode should not be a dominant decay mode. The dominant decay mode should be the b1(1235)π [11]. Therefore, the probability of discovering the hybrid with JPC = 1−+ in the process J/ψ ρX, X b1π, in → → ′ principle, should be higher than that in the process J/ψ ρX, X ηπ (η π, ρπ). Also, since → → the dominant decay mode of b1 is ωπ, compared to the study on the two-step two-body decay process of J/ψ in Ref. [9,12], the study on the three-step two-body decay process J/ψ ρX, → X b1π, b1 ωπ perhaps could present more information to the experimentists. In this work, → → we shall consider the process J/ψ ρX, X b1π, b1 ωπ. → → → This work is organized as follows. In Sect. II, we give the moment expressions for the resonances X with the above spin-parity in the process J/ψ ρX, X b1π, b1 ωπ in terms → → → of the generalized moment analysis method [9,12,13]. , and in Sect. III, we discuss how to identify the resonances X with different spin-parity. Our conclusion is reached in Sect. IV. II. Moment analysis We consider the process + − e + e J/ψ ρ + X, X b1 + π, b1 ω + π. (1) → → → → The S matrix element of the process (1) can be written as + − + − hρλρωλωππ|S −1|er er′i∝ hψλJ|T|er er′ihρλρXλX|T1|ψλJihb1λb1π|T2|XλXihωλωπ|T3|b1λb1i, (2) where hψλJ|T|e+r e−r′i∝ eλµJ∗(p~J)v¯r(p~+)γµur′(p~−); (3) hρλρXλX|T1|ψλJi ∝ AJλXρ,λXDλ1∗J,λρ−λX(0,θρ,0); (4) hb1λb1π|T2|XλXi ∝ BλJbX1DλJXX,∗λb1(φ1,θ1,−φ1); (5) 3 1∗ hωλωπ|T3|b1λb1i ∝ CλωDλb1,λω(φ2,θ2,−φ2); (6) ′ And λJ, λρ, λX, λb1 and λω are the helicities of J/ψ, ρ, X, b1 and ω, respectively; r and r are the polarization indexes of the positron and electron, respectively; p~J, p~+, p~− are the momenta ofJ/ψ, positronandelectron inthec.m. systemof e+e−, respectively; AJX ,BJX andC are λρ,λX λb1 λω the helicity amplitudes of the processed J/ψ ρX, X b1π and b1 ωπ, respectively; θρ is → → → the polar angle in the c.m. system of e+e− in which z axis is chosen to be along the direction of the incident positron and the vector meson ρ lies in x z plane; (θ1,φ1) describes the direction − of the momentum of b1 in the rest frame of X where the z1 axis is chosen to be along the direction of the momentum of X in the c.m. system of e+e−; Similarly, (θ2,φ2) described the direction of the momentum of the vector mesons ω in the rest frame of b1 where the z2 axis is along themomentum of b1 in therestframeof X; Thefunction DmJ,n is the(2J+1)-dimensional representation of the rotation group. Owing to the parity conservation for the process (1), these helicity amplitudes satisfy the following symmetry relations [14]: AJX =P ( 1)JXAJX , −λρ,−λX X − λρ,λX BJX = P ( 1)JXBJX, −λb1 X − λb1 C−λω = Cλω, (7) where P is the parity of X. X The angular distribution for the process (1) is W(θρ,θ1,φ1,θ2,φ2) ∝ I AJX AJX∗ BJXBJX∗C C∗ λXJ,λ′JλXX,λ′XλbX1,λ′b1λXρ,λω λJ,λ′J λρ,λX λρ,λ′X λb1 λ′b1 λω λω 1∗ 1 ×DλJ,λρ−λX(0,θρ,0)Dλ′J,λρ−λ′X(0,θρ.0) ×DλJXX,∗λb1(φ1,θ1,−φ1)DλJ′XX,λ′b1(φ1,θ1,−φ1) 1∗ 1 ×Dλb1,λω(φ2,θ2,−φ2)Dλ′b1,λω(φ2,θ2,−φ2), (8) where the density matrix elements I is λJ,λ′J 1 + − + − ∗ 2 IλJ,λ′J ≡ 4 XhψλJ|T|er er′ihψλ′J|T|er er′i ∝ 2|p~+| δλJ,λ′JδλJ,±1. (9) r,r′ 4 The moments for the process (1) can be defined by M(j,L,M,ℓ,m) = Z dθρsinθρdθ1sinθ1dφ1dθ2sinθ2dφ2W(θρ,θ1,φ1,θ2,φ2) ×D0j,−M(0,θρ,0)DML,m(φ1,θ1,−φ1)Dmℓ ,0(φ2,θ2,−φ2). (10) Eq. (10) can be reduced to M(j,L,M,ℓ,m) ∝ AJX AJX∗ BJXBJX∗C C∗ λJX=±1λXX,λ′XλbX1,λ′b1λXρ,λω λρ,λX λρ,λ′X λb1 λ′b1 λω λω ′ 1λ j01λ 1(λ λ )j( M)1(λ λ ) J J ρ X ρ X ×h | ih − − | − i ′ ′ J λ LM J λ J λ Lm J λ ×h X X | X Xih X b1 | X b1i ′ 1λ ℓm 1λ 1λ ℓ01λ , (11) ×h b1 | b1ih ω | ωi where j1m1j2m2 j3m3 is Clebsch-Gordan coefficients. h | i In the process X b1π, if we restrict ℓf 1, where ℓf is the relative orbital angular → ≤ momentum between b1 and π, the quantum number IG(JXPXC) of X allowed by the parity- isospin conservation law in the process (1) are 1−(1−+), 1−(0++), 1−(1++), and 1−(2++). For the resonances X with JPXC = 0++, 1−+, 1++, and 2++, the nonzero moment expressions X derived from Eq. (11) are shown in Appendix A, B, C, and D. There are four, twenty-one, sixteen, and thirty-one nonzero moment expressions for JPXC = X 0++, 1−+, 1++, and 2++, respectively. In the following section, we shall discuss how to identify the X with the above JPXC. X III. Discussion Since the helicity amplitudes C0 2 and C1 2 are independent of the spin-parity of the reso- | | | | nanceX, wefindthat if C0 2 = C1 2 themoment expressionshave thefollowing characteristics: | | 6 | | For JPXC = 0++, the moments is always equal to zero in the case L > 0 or M > 0 or m > 0; X For JPXC = 1++, the nonzero moments with L =0,1,2, M = 0,1,2 and m = 0,2 exist but the X 5 moments are zero in the case m = 1; For JPXC = 1−+, the nonzero moments with L = 0,1,2, X M = 0,1,2 and m = 0,2 exist, the nonzero moments with m = 1 also exist; For JPXC = 2++, X apart from the nonzero moments with L = 0,1,2, M = 0,1,2 and m = 0,1,2, the nonzero moments with L = 3,4 exist. Therefore, from these characteristics, we can easily identify the resonances X with JPXC= 0++, 1−+, 1++ and 2++ experimentally. X However, if C0 2 = C1 2,someoftheprecedingcharacteristics disappear,whichleadstothat | | | | the situations in the case C0 2 = C1 2 are more complex than those in the case C0 2 = C1 2. | | | | | | 6 | | We will turn to the special case C0 2 = C1 2 below. | | | | (A) C0 2 = C1 2, B01 2 = B11 2 and 3B02 2 = 4B12 2 | | | | | | 6 | | | | 6 | | In this case, only for JPXC = 2++, there are four nonzero moments with L = 4, so the X resonancewithJPXC = 2++ canbedistinguishedfromotherresonances. Then,forJPXC = 0++, X X there are only two nonzero moments with L = 0, and for J = 1, there are four nonzero X moments with L = 2, in addition to two nonzero moments with L =0, hence the resonance with J = 0 can also be distinguished from that with J = 1. Finally, to distinguish the resonance X X with JPXC = 1−+ from that with JPX = 1++, we consider the following moment expression X X 1 5 5 25 H M(00000) M(02000) M(20000)+ M(22000) and find that the H satisfies ≡ 8 − 4 − 4 2 0, (JPXC = 1++),  X H ∝  −247(|A100|2|B01|2−2|A111|2|B01|2−2|A100|2|B11|2)|C1|2, (JXPXC = 1−+). (12)  Using Eq. (12), we can still distinguish the resonance X with JPXC = 1−+ from that with X JPXC = 1++. X (B) C0 2 = C1 2, B01 2 = B11 2 and 3B02 2 = 4B12 2 | | | | | | 6 | | | | | | Inthiscase,comparedtothecase(A),thenumbersofthenonzeromomentsforJPXC = 0++, X 1−+ and 1++ remain unchange, but for JPXC = 2++, the moments with L = 4 disappear, There X are still only two nonzero moments with L = 0 for JPXC = 0++ and six nonzero moments with X L = 0, 2 not only for J = 1 but also for J = 2. In this case, owing to Eq. (12) remains X X unchange and 15 H ∝ − 2 |A200|2|B12|2|C1|2 < 0 (JXPXC = 2++), (13) 6 the crucial point is to distinguish the resonance with JPXC = 1−+ from that with JPXC = 2++. X X 1 5 We also find H1 ≡ 4M(02000)− 2M(22000) satisfies  3(|A200|2+|A211|2)|B12|2|C1|2 > 0, (JXPXC = 2++), H1 ∝  95|A111|2|B11|2|C1|2 > 0, (JXPXC = 1++), (14)  −95(|A100|2−|A111|2)(|B11|2−|B01|2)|C1|2, (JXPXC = 1−+). So, if it is determined experimentally that H > 0 or H1 ≤ 0, from Eq.(12)∼(14), the JXPXC of X must be 1−+. However, if H < 0 or H1 > 0, we can not distinguish the resonance with JPXC = 2++ from that with JPXC = 1−+. X X (C) C0 2 = C1 2 and B01 2 = B11 2 | | | | | | | | In this case, there are only two nonzero moments with L = M = ℓ = m = 0 both for JPXC = 0++ and JPXC = 1−+, there are two nonzero moments with L = 0 and four nonzero X X moments with L = 2 for JPXC = 1++, and there are two nonzero moments with L = 0 and X at least four nonzero moments with L = 2 for JPXC = 2++. Therefore, the resonances with X JPXC = 1++ and 2++ can be distinguished from the resonance with JPXC = 0++ ( or 1−+ X X ). But it is almost impossible to distinguish the resonance with JPXC = 1−+ from that with X JPXC = 0++ except in the radiative J/ψ decay process. Because for the radiative J/ψ decay X process e++e− J/ψ γX, X b1π, b1 ωπ, A000 = A001 = A100 = A101, we find → → → → 0, (JPXC =0++),  X M(00000)−10M(20000) ∝  108|A111|2|B11|2|C1|2, (JXPXC =1−+). (15)  Obviously, using Eq. (15) we can distinguish the 0++ state from the 1−+ state in the radiative J/ψ decay process. From the above discussions, we get that if C0 2 = C1 2 we can easily identify the resonances | | 6 | | X withJXPXC=0++, 1−+, 1++ and2++,butif |C0|2 = |C1|2 and|B01|2 = |B11|2 (or |C0|2 = |C1|2 and 3B2 2 = 4B2 2 ), the identification of the resonances X with JPXC=1−+ and 0++ ( or 2++ | 0| | 1| X ) is very difficult. However, we also want to note the following two points: 1) Since the ratio of the helicity amplitudes for the process b1(1235) ωπ, C0 and C1 , can be measured → | | | | experimentally in other process such as J/ψ b1π, b1 ωπ. The measurement of the ratio → → 7 of C0 and C1 can be first performed in order to confirm whether C0 2 is equal to C1 2 or | | | | | | | | not; 2) Even though C0 2 = C1 2, one could expect that the probability of the simultaneous | | | | appearance of C0 2 = C1 2 and B01 2 = B11 2 ( or C0 2 = C1 2 and 3B02 2 = 4B12 2 ) would | | | | | | | | | | | | | | | | be fairly small. Itisworth pointingoutthattheabove momentexpressionsandthediscussionsarealsovalid for the process J/ψ γX, X b1π, b1 ωπ provided A000 = A001 = A100 = A101 = A200 = → → → A2 = 0. 01 IV. Conclusion The twenty-one nonzero moment expressions for JPXC = 1−+ show the possibility of the X resonance X with JPXC = 1−+ produced in the process (1) exists. At the same time, we X can easily distinguish it from other resonances except for some rather special cases. Therefore, generally speaking, if the 50 million J/ψ events in the upgraded BEPC/BES are obtained, the search for the hybrid with JPC = 1−+ in the process J/ψ ρX, X b1π, b1(1235) ωπ is → → → feasible. Appendix A: The nonzero moments for JPXC = 0++ X 0 2 0 2 0 2 2 2 M(00000) 2(A00 +2A10 )B0 (C0 +2C1 ), ∝ | | | | | | | | | | 4 0 2 0 2 0 2 2 2 M(00020) (A00 +2A10 )B0 (C0 C1 ), ∝ 5 | | | | | | | | −| | 2 0 2 0 2 0 2 2 2 M(20000) (A00 A10 )B0 (C0 +2C1 ), ∝ −5 | | −| | | | | | | | 4 0 2 0 2 0 2 2 2 M(20020) (A00 A10 )B0 (C0 C1 ). ∝ −25 | | −| | | | | | −| | Appendix B: The nonzero moments for JPXC = 1 + X − 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(00000) 2(A00 +2A01 +2A10 +2A11 )(B0 +2B1 )(C0 +2C1 ), ∝ | | | | | | | | | | | | | | | | 4 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(00020) (A00 +2A01 +2A10 +2A11 )(B0 B1 )(C0 C1 ), ∝ 5 | | | | | | | | | | −| | | | −| | 8 4 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(02000) (A00 A01 +2A10 A11 )(B0 B1 )(C0 +2C1 ), ∝ 5 | | −| | | | −| | | | −| | | | | | 4 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(02020) (A00 A01 +2A10 A11 )(2B0 + B1 )(C0 C1 ), ∝ 25 | | −| | | | −| | | | | | | | −| | 12 1 2 1 2 1 2 1 2 1 1∗ 2 2 M(02021) (A00 A01 +2A10 A11 )Re(B1B0 )(C0 C1 ), ∝ 25 | | −| | | | −| | | | −| | 12 1 2 1 2 1 2 1 2 1 2 2 2 M(02022) (A00 A01 +2A10 A11 )B1 (C0 C1 ), ∝ 25 | | −| | | | −| | | | | | −| | 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(20000) (A00 A01 A10 +2A11 )(B0 +2B1 )(C0 +2C1 ), ∝ −5 | | −| | −| | | | | | | | | | | | 4 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(20020) (A00 A01 A10 +2A11 )(B0 B1 )(C0 C1 ), ∝ −25 | | −| | −| | | | | | −| | | | −| | 6 1 1∗ 1 1∗ 1 1∗ 2 2 M(21121) Im(A01A00+A10A11)Im(B1B0 )(C0 C1 ), ∝ 25 | | −| | 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(22000) (2A00 + A01 2A10 2A11 )(B0 B1 )(C0 +2C1 ), ∝ −25 | | | | − | | − | | | | −| | | | | | 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 M(22020) (2A00 + A01 2A10 2A11 )(2B0 + B1 )(C0 C1 ), ∝ −125 | | | | − | | − | | | | | | | | −| | 6 1 2 1 2 1 2 1 2 1 1∗ 2 2 M(22021) (2A00 + A01 2A10 2A11 )Re(B1B0 )(C0 C1 ), ∝ −125 | | | | − | | − | | | | −| | 6 1 2 1 2 1 2 1 2 1 2 2 2 M(22022) (2A00 + A01 2A10 2A11 )B1 (C0 C1 ), ∝ −125 | | | | − | | − | | | | | | −| | 6 1 1∗ 1 1∗ 1 2 1 2 2 2 M(22100) Re(A01A00 A11A10)(B0 B1 )(C0 +2C1 ), ∝ −25 − | | −| | | | | | 6 1 1∗ 1 1∗ 1 2 1 2 2 2 M(22120) Re(A01A00 A11A10)(2B0 + B1 )(C0 C1 ), ∝ −125 − | | | | | | −| | 18 1 1∗ 1 1∗ 1 1∗ 2 2 M(22121) Re(A01A00 A11A10)Re(B1B0 )(C0 C1 ), ∝ −125 − | | −| | 18 1 1∗ 1 1∗ 1 2 2 2 M(22122) Re(A01A00 A11A10)B1 (C0 C1 ), ∝ −125 − | | | | −| | 6 1 2 1 2 1 2 2 2 M(22200) A01 (B0 B1 )(C0 +2C1 ), ∝ −25| | | | −| | | | | | 6 1 2 1 2 1 2 2 2 M(22220) A01 (2B0 + B1 )(C0 C1 ), ∝ −125| | | | | | | | −| | 18 1 2 1 1∗ 2 2 M(22221) A01 Re(B1B0 )(C0 C1 ), ∝ −125| | | | −| | 18 1 2 1 2 2 2 M(22222) A01 B1 (C0 C1 ). ∝ −125| | | | | | −| | Appendix C: The nonzero moments for JPXC = 1++ X 1 2 1 2 1 2 1 2 2 2 M(00000) 8(A01 + A10 + A11 )B1 (C0 +2C1 ), ∝ | | | | | | | | | | | | 8 1 2 1 2 1 2 1 2 2 2 M(00020) (A01 + A10 + A11 )B1 (C0 C1 ), ∝ −5 | | | | | | | | | | −| | 4 1 2 1 2 1 2 1 2 2 2 M(02000) (A01 2A10 + A11 )B1 (C0 +2C1 ), ∝ 5 | | − | | | | | | | | | | 9 4 1 2 1 2 1 2 1 2 2 2 M(02020) (A01 2A10 + A11 )B1 (C0 C1 ), ∝ −25 | | − | | | | | | | | −| | 12 1 2 1 2 1 2 1 2 2 2 M(02022) (A01 2A10 + A11 )B1 (C0 C1 ), ∝ 25 | | − | | | | | | | | −| | 4 1 2 1 2 1 2 1 2 2 2 M(20000) (A01 + A10 2A11 )B1 (C0 +2C1 ), ∝ 5 | | | | − | | | | | | | | 4 1 2 1 2 1 2 1 2 2 2 M(20020) (A01 + A10 2A11 )B1 (C0 C1 ), ∝ −25 | | | | − | | | | | | −| | 2 1 2 1 2 1 2 1 2 2 2 M(22000) (A01 2A10 2A11 )B1 (C0 +2C1 ), ∝ 25 | | − | | − | | | | | | | | 2 1 2 1 2 1 2 1 2 2 2 M(22020) (A01 2A10 2A11 )B1 (C0 C1 ), ∝ −125 | | − | | − | | | | | | −| | 6 1 2 1 2 1 2 1 2 2 2 M(22022) (A01 2A10 2A11 )B1 (C0 C1 ), ∝ 125 | | − | | − | | | | | | −| | 6 1 1∗ 1 2 2 2 M(22100) Re(A11A10)B1 (C0 +2C1 ), ∝ −25 | | | | | | 6 1 1∗ 1 2 2 2 M(22120) Re(A11A10)B1 (C0 C1 ), ∝ 125 | | | | −| | 18 1 1∗ 1 2 2 2 M(22122) Re(A11A10)B1 (C0 C1 ), ∝ −125 | | | | −| | 6 1 2 1 2 2 2 M(22200) A01 B1 (C0 +2C1 ), ∝ −25| | | | | | | | 6 1 2 1 2 2 2 M(22220) A01 B1 (C0 C1 ), ∝ 125| | | | | | −| | 18 1 2 1 2 2 2 M(22222) A01 B1 (C0 C1 ). ∝ −125| | | | | | −| | Appendix D: The nonzero moments for JPXC = 2++ X 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 M(00000) 2(A00 +2A01 +2A10 +2A11 +2A12 )(B0 +2B1 )(C0 +2C1 ), ∝ | | | | | | | | | | | | | | | | | | 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 M(00020) (A00 +2A01 +2A10 +2A11 +2A12 )(B0 B1 )(C0 C1 ), ∝ 5 | | | | | | | | | | | | −| | | | −| | 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 M(02000) (A00 + A01 +2A10 + A11 2A12 )(B0 + B1 )(C0 +2C1 ), ∝ 7 | | | | | | | | − | | | | | | | | | | 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 M(02020) (A00 + A01 +2A10 + A11 2A12 )(2B0 B1 )(C0 C1 ), ∝ 35 | | | | | | | | − | | | | −| | | | −| | 4√3 2 2 2 2 2 2 2 2 2 2 2 2∗ 2 2 M(02021) (A00 + A01 +2A10 + A11 2A12 )Re(B1B0 )(C0 C1 ), ∝ 35 | | | | | | | | − | | | | −| | 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 M(02022) (A00 + A01 +2A10 + A11 2A12 )B1 (C0 C1 ), ∝ 35 | | | | | | | | − | | | | | | −| | 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 M(04000) (3A00 4A01 +6A10 4A11 + A12 )(3B0 4B1 )(C0 +2C1 ), ∝ 63 | | − | | | | − | | | | | | − | | | | | | 8 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 M(04020) (3A00 4A01 +6A10 4A11 + A12 )(3B0 +2B1 )(C0 C1 ), ∝ 315 | | − | | | | − | | | | | | | | | | −| | 4√10 2 2 2 2 2 2 2 2 2 2 2 2∗ 2 2 M(04021) (3A00 4A10 +6A10 4A11 + A12 )Re(B1B0 )(C0 C1 ), ∝ 105 | | − | | | | − | | | | | | −| | 10