The fourth Standard Model (SM) family quarks and weak iso-singlet quarks The
fourth Standard Model (SM) family quarks and weak iso-singlet quarks predicted
by ${\rm E}_{6\text{}}$ GUT are considered. The spin-average of the
pseudoscalar $\eta_{4}(n^{1}S_{0})$ and vector $\psi_{4}(n^{3}S_{1})$
quarkonium binding masses of the new mesons formed by the fourth Standard Model
(SM) family and iso-singlet ${\rm E}_{6\text{}}$ with their mixings to ordinary
quarks are investigated. Further, the fine and hyperfine mass splittings of the
these states are also calculated. We solved the Schr\"{o}dinger equation with
logarithmic and Martin potentials using the Shifted large-${\rm N}$ expansion
technique. Our results are compared with other models to gauge the reliability
of the predictions and point out differences.Comment: 13 page

Faustov R. N.

RAMAZAN SEVER

SAMEER M. IKHDAIR

English

arXiv

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BOUND ENERGY MASSES OF MESONS CONTAINING THE FOURTH GENERATION AND ISO-SINGLET QUARKS

The fourth Standard Model (SM) family quarks and weak iso-singlet quarks predicted by E-6 GUT are considered. The spin-average of the pseudoscalar eta(4)(n(1)S(0)) and vector psi(4)(n(3)S(1)) quarkonium binding masses of the new mesons formed by the fourth Standard Model (SM) family and iso-singlet E-6 with their mixings to ordinary quarks are investigated. Further, the fine and hyperfine mass splittings of the these states are also calculated. We solved the Schrodinger equation with logarithmic-and Martin potentials using the shifted large-N expansion technique. Our results are compared with other models to gauge the reliability of the predictions and point out differences

Ikhdair, Sameer M.

Sever, Ramazan

OpenMETU (Middle East Technical University)

arXiv:hep-ph/0507209v1  18 Jul 2005Bound Energy Masses of Mesons Containing the FourthGeneration and Iso-singlet QuarksSameer M. Ikhdair∗ and Ramazan Sever†∗Department of Electrical and Electronic Engineering, Near EastUniversity, Nicosia, North Cyprus†Department of Physics, Middle East Technical University, 06530Ankara,TurkeyPACS NUMBER(S): 12.39.Pn, 14.40-n, 12.39.Jh, 14.65.-q, 13.30.GdKey Words: Fourth family, Bound state energy, Logarithmic potential, Martin potential,Shifted large N-expansion method, Quarkonia, Mesons(March 1, 2019)Typeset using REVTEX∗Electronic address (internet): sameer@neu.edu.tr†Electronic address (internet): sever@metu.edu.tr1AbstractThe fourth Standard Model (SM) family quarks and weak iso-singlet quarkspredicted by E6 GUT are considered. The spin-average of the pseudoscalarη4(n1S0) and vector ψ4(n3S1) quarkonium binding masses of the new mesonsformed by the fourth Standard Model (SM) family and iso-singlet E6 withtheir mixings to ordinary quarks are investigated. Further, the fine and hy-perfine mass splittings of the these states are also calculated. We solved theSchrödinger equation with logarithmic and Martin potentials using the Shiftedlarge-N expansion technique. Our results are compared with other models togauge the reliability of the predictions and point out differences.I. INTRODUCTIONThe toponium quark does not form hadronic states because of its large mass value (mt ≈ 175GeV ) and full strength of tbW vertex. On the other hand, there are strong reasons that thefourth Standard Model (SM) family should exist [1,2]. The flavor democracy (DemocraticMass Matrix approach) [2] favors the existence of the nearly degenerate fourth SM family,whereas the fifth SM family is disfavored both by the mass phenomenology and precision testsof the SM [3]. The fourth SM family fermions and also the isosinglet quarks production atµ+µ− colliders have been investigated [3]. Thus, SM may be treated as an effective theory offundamental interactions rather than fundamental particles. The multi-hundreds GeV fourthgeneration up-type quark (u4), if exist, will be produced at the CERN Large Hadron Collider(LHC) [3,4] via gluon-gluon fusion [4]. Hence, the observation of the fourth SM family quarkin ATLAS has been considered in [4,5]. It is expected that the masses of the fourth family2quarks lies between 300 ≤ mu4 ≤ 700 GeV with preferable value mu4 = 4gwη = 8mw∼= 640GeV [3]. The partial-wave unitarity leads to mQ ≤ 700GeV = 4mt and in general mt ≪mu4 ≪ m5. The fourth generation up-type quark would predominatly decay via u4 → Wb,with an expected event topologies are similar to those for t−quark pair production. Thebest channel for observing it will be [5]: u4u4 →WWbb→ (lν)(jj)(bb), where oneW decaysleptonically and the other hadronically. The mass resolution is estimated to be 20(40) GeVfor mu4 = 320(640) GeV. The pseudoscalar quarkonium state η4(1S0) formed by the SMfourth generation quarks is the best candidate among the fourth generation quarkonia to beproduced at the LHC and VLHC [6]. On the other hand, new heavy quarks known as weakiso-singlet quarks are also predicted by various extensions of SM and by E6 GUT [7] whichis favored by superstring theory [8]. For down-type quark (d4), the decay mode is: d4 → Wtand the final state contains four W bosons: d4d4 → tW−tW+ → bW+W−bW−W+.The small inter-family mixings [9] leads to the formation of the fourth family quarkonia.The most promising candidate for LHC is the pseudoscalar quarkonium state, η4, whichwill be produced reasonantly via gluon-gluon fusion particularly through decay channel:η4 → ZH [10].In spite that the masses of new quarks are larger than mt, they can form new hadrons be-cause of the small inter-family mixings leads to the formation of the fourth family quarkoniabetween new heavy and ordinary quarks (u, d, s, c, b). Indeed, according to the parametriza-tion of mass matrices given in [9] mixing between the fourth and third family quarks ispredicted to be |Vqu4| ≈ 10−3. Similar situation is expected for iso-singlet down-type quarks.The condition for forming (QQ) quarkonia states with new heavy quarks is [11]mQ ≤ (125 GeV ) |VQq|−2/3 . (1)where q = d, s, b for Q = u4 and q = u, c, t for Q = d4. Differing from t quark, fourth familyquarks will form quarkonia because u4 and d4 are almost degenerate and their decays aresuppressed by small mixings [2,9,12].One of the important goals of the present work is to extend the shifted large-N expansion3technique (SLNET) developed for the Schrödinger wave equation [13, 14, 15] and then appliedto semirelativistic wave equations [16,17] to reproduce the spectroscopy of the fourth SMgeneration up-type (u4) quark and weak iso-singlet down-type (d4) quark with their smallmixings to the ordinary type (u, d, s, c, b) quarks.Here, we study the present status of the new heavy mesons formed by new quarks inthe framework of nonrelativistic potential model and give some predictions for their boundenergy masses. In section II, we present the solution of the Schrödinger equation using theflavor-independent logarithmic and Martin potentials for the the self- and non-self conjugateqiqi and qiqj mass spectra, respectively. A brief conclusion appear in section III.II. NEW HEAVY MESONSThe fourth SM family and E6 isosinglet quarks have formed hadron states if their mix-ing with ordinary known quarks is sufficiently small. In the fourth family quarks, theparametrization given in [9] satisfies condition (1), whereas new hadrons are not formedin the case of paramerization given in [1]. Concerning E6 iso-singlet quarks, we have nosimilar parametrization (one deals with 6 × 6 mass matrix) and one can make qualitativeestimations. For example, if the lightest isosinglet quark has the mass md4∼= 0.5 TeV , newheavy hadrons are formed for |Vqd4| < 0.09. In order to calculate the binding masses of newmesons we investigate two potentials: (1) Logarithmic potential of the form [14,16,18]V (r) = −0.6631 + 0.733 ln(r × 1GeV ), (2)where r is the interquark distance which is singular at r = 0. (2) Martin’s potential of theform [14,16,19]V (r) = −8.093 + 6.898r0.1, (3)which behaves in some respects, as its power approaches to zero, as the logarithmic potential.Hence, these potential types are stimulated by the approximate equality of mass splittings[18]4M(Υ′(bb))−M(Υ(bb)) ≈M(ψ′(cc))−M(ψ(cc)), (4)which is independent upon reduced mass. These static quarkonium potentials are monotonenondecreasing, and concave functions satisfying the condition [14]V ′(r) > 0 and V ′′(r) ≤ 0. (5)For two particles system, we shall consider the N−dimensional space Schrödinger equa-tion for any spherically symmetric potential V (r). If ψ(r) denotes the Schrödinger’s wavefunction, a separation of variables ψ(r) = Yℓ,m(θ, φ)u(r)/r(N−1)/2 gives the following radialequation (~ = c = 1) [13,14,15,16,17]{−14µd2dr2+[k − (1− a)][k − (3− a)]16µr2+ V (r)}u(r) = En,ℓu(r), (6)where µ =(mqimqj)/(mqi + mqj ) is the reduced mass for the two quarkonium compositeparticles. Here, En,l denotes the Schrödinger binding energy of meson, and k = N + 2ℓ− a,with a representing a proper shift to be calculated later on and l is the angular quantumnumber. We follow the shifted 1/N or 1/k expansion method [13,14,15] by definingV (x(r0)) =∞∑m=0(dmV (r0)drm0)(r0x)mm!Qk(4−m)/2, (7)and also the energy eigenvalue expansion [14,15]En,l =∞∑m=0k(2−m)QEm, (8)where x = k1/2(r/r0 − 1) with r0 is an arbitrary point where the Taylor’s expansions isbeing performed about and Q is a scale parameter to be set equal to k2at the end of ourcalculations. Following the approach presented by Ref.[14], we give the necessary expressionsfor calculating the binding energies:E0 = V (r0) +Q16µr20, (9)E1 =Qr20[(nr +12)ω −(2− a)8µ], (10)5E2 =Qr20[(1− a)(3− a)16µ+ α(1)], (11)E3 =Qr20α(2), (12)where α(1) and α(2) are two useful expressions given by Imbo et al [13] and also the scaleparameter Q is defined by the relationQ = 8µr30V′(r0). (13)Thus, for the N = 3 physical space, the Schrödinger binding energy to the third order is[14]En,ℓ = V (r0) +12r0V′(r0) +1r20[(1− a)(3− a)16µ+ α(1) +α(2)k+O(1k2)]. (14)where the shifting parameter, a, is defined bya = 2− (2nr + 1)[3 +r0V′′(r0)V ′(r0)]1/2, (15)and the root, r0, is being determined via1 + 2l + (2nr + 1)[3 +r0V′′(r0)V ′(r0)]1/2=[8µr30V′(r0)]1/2, (16)where nr = n − 1 is the radial quantum number and n is the principal quantum number.Once r0 is found via equation (16), then the Schrödinger binding energy of the qiqj systemin (14) becomes relatively simple and straightforward. Hence, the bound state mass of theqiqj system is written asM(qiqj)nl = mqi +mqj + 2En,l. (17)The expansion parameter 1/N or 1/k becomes smaller as l becomes larger since the param-eter k is proportional to n which it appears in the denominator in higher-order correction.Since systems that we investigate in the present work are often considered as nonrela-tivistic system, then our treatment is based upon Schrödinger equation with a Hamiltonian6Ho = −▽22µ+ V (r) + VSD, (18)where VSD is the spin-dependent term taking the simple form (cf. Ref.[14] and the referencestherein)VSD −→ VSS =32παs9mq1mq2δ3(r)s1.s2. (19)The spin dependent potential is simply a spin-spin part and this would enable us to makesome preliminary calculations of the energies of the lowest two S-states of the new quarksand their small mixings with the ordinary quarks. The potential parameters in this sectionare all strictly flavor-independents and are fitted to the low-lying energy levels of cc and bbsystems. The strong coupling constant αs is fitted to the observed charmonium hyperfinesplitting of 117 MeV. The numerical values of αs for the two potential types have beenadjusted in Ref.[14] asα(L)s (cc) ≃ 0.220 and α(M)s (cc) ≃ 0.251, (20)which are found to be dependent on the potential type. Baldicchi and Prosperi [20] usedthe standard running QCD coupling expressionαs(Q) =4π(11− 23nf)ln(Q2Λ2) . (21)with nf = 4 and Λ = 0.2 GeV cut at a maximum value αs(0) = 0.35, to give the rightJ/ψ − ηc splitting of cc quarkonium and to treat properly the infrared region. Detail ontheir numerical works are given in Ref. [20]. Whereas Brambilla and Vairo [21] took in theirperturbative analysis 0.26 ≤ αs(µ = 2GeV ) ≤ 0.30.The potential parameters in (2) and (3) together with the quark masses are obtainedfrom the experimental SAD mass: M(ψ(1S)) (cf. e.g. Refs.[14,16]). In our calculationswe used mu = md = 0.367 GeV, ms = 0.561 GeV whereas mc and mb masses are given byRef.[14]. Further, the fourth SM family up-type quark and E6 isosinglet down-type quarkmasses are taken as mu4 = 638.6 GeV [9,22] and md4 = 0.5 TeV [22], respectively.7Firstly, we calculate the masses of bound states of cc and bb in Table 1 for the twopotential types. It is seen that our results are in good agreement with the experimentalresults [23,24]. In Table 2, we present the masses of bound states of quarkonia and mesonsformed by the fourth SM family up-type quark and E6 isosinglet quark, respectively. InTable 3, the predicted mass splittings is compared with Ref.[22]. In Table 4, the hyperfinemass splittings ψ4−η4 for the first two S−states are also given. The best mechanism for theproduction of heavy mesons formed by u4 and d4 quarks is the resonance formation of 3S and4S quarkonia at lepton colliders with subsequent decay into corresponding meson-antimesonstates.III. CONCLUSIONWe have produced the mesons formed by new heavy quarks. The mass splitting of thelogarithmic potentials is found to be independent on the reduced mass and is constant inmagnitude for any chosen state for all studied quarkonium families. However, it is seen inMartin’s potential that the mass splittings are equal for any chosen state in the fourth family,u4, quark and isosinglet quark with the same type of ordinary quark mixing. Further, thehyperfine splitting mass is found to be much smaller compared to the large mass of the newmesons. Therefore, the pseudoscalar η4(n1S0) and the vector ψ4(n3S1) have nearly samemasses. On the other hand, the Cornell potential fails to produce the masses of the u4u4and d4d4 quarkonium famileis and their hyperfine mass splittings properly because of thesmall values of the roots r0 for the lowest states where the Coulombic part of the potentialgoes to high order fas we take higher order derivatives in the SLNET method.ACKNOWLEDGMENTSS. M. Ikhdair thanks his wife, Oyoun, and his son, Musbah, for their assistance, patienceand love.8REFERENCES[1] A. Datta and S. Raychaudhuri, Phys. Rev. D 49 (1994) 4762.[2] A. Celikel, A.K. Çiftçi and S. Sultansoy, Phys. Lett. B 342 (1995) 257.[3] S. Sultansoy, [arXiv:hep-ph/0004271]; A.K. Çiftçi, R. Çiftçi and S. Sultansoy, Phys.Rev. D 65 (2002) 055001.[4] ATLAS Detector and Physics Performance Technical Design Report CERN/LHCC/99-15 (1999) p.663; E. Arik et al., ATLAS Internal Note ATL-PHYS-98-125 (1999); E.Arık et al., [arXiv:hep-ph/0109037].[5] E. Arık et al., Phys. Rev. D 58 (1998) 117701.[6] E. Arık et al., [arXiv:hep-ph/0208169].[7] F. Gursey, P. Ramond and P. Sikivie, Phys. Lett. B 60 (1976) 177.[8] J.L. Hewett and T.G. Rizzo, Phys. Reports 183 (1989) 193.[9] S. Atağ et al., Phys. Rev. D 54 (1996) 5745.[10] E. Arık et al., ATLAS Internal Note ATL-PHYS-99-061 (1999).[11] I. Bigi et al., Phys. Lett. B 181 (1986) 157.[12] R. Çiftçi et al., [arXiv:hep-ph/0203083].[13] T. Imbo, A. Pagnamenta and U. Sukhatme, Phys. Rev. D 29 (1984) 1669; H. Chris-tiansen et al., Phys. Rev. A 40 (1989) 1760.[14] S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. A 18 (2003) 4215.[15] S.M. Ikhdair and R. Sever, Z. Phys. D 28 (1993) 1.[16] S.M. Ikhdair and R. Sever, Int. J. Mod. Phys. A 19 (2004) 1771; ibid. A 20 (2005)[arXiv:hep-ph/0403280]; ibid. [arXiv:hep-ph/0406005] (in press).9[17] S.M. Ikhdair et al., Turkish J. Phys. 16 (1992) 510; ibid. 17 (1993) 474; S.M. Ikhdairand R. Sever, Hadronic Journal 15 (1992) 375; S.M. Ikhdair et al., ibid. 16 (1993) 57;S.M. Ikhdair and R. Sever, Z. Phys. C 56 (1992) 155; ibid. 58 (1993) 153; S.M. Ikhdairet al., Hadronic J. 17 (1994) 151; ibid. Hadronic J. Supp. 19 (1994) 417; A. Bekmezci,S.M. Ikhdair, M.A. Magdy and R. Sever, ibid. 16 (1993) 339.[18] C. Quigg and J.L. Rosner, Phys. Lett. 71 B (1977) 153; ibid. Phys. Rep. 56 (1979) 167.[19] M. Martin, Phys. Lett. B 93 (1980) 338; ibid. B 100 (1980) 511.[20] M. Baldicchi and G.M. Prosperi, Phys. Lett. B 436 (1998) 145; ibid. Phys. Rev. D 62(2000) 114024; ibid. Fiz. B 8 (1999) 251.[21] N. Brambilla and A. Vairo, Phys. Rev. D 62 (2000) 094019.[22] H. Çiftçi and S. Sultansoy, Mod. Phys. Lett. A18 (2003) 859.[23] D.E. Groom et al., The Eur. Phys. J. C 15 (2000) 1; K. Hagiwara et al., Particle DataGroup Collaboration, Phys. Rev. D66, 010001 (2002).[24] R.N. Faustov et al., Int. J. Mod. Phys. A 15 (2000) 209.[25] K. Abe et al., (the Belle Collaboration), Phys. Rev. Lett. 89, 142001 (2002).[26] A.M. Badalian and B.L.G. Bakker, Phys. Rev. D67, 071901 (2003).10TABLESTABLE I. The pseudoscalar, vector and the spin-averaged masses together with hyperfinesplittings for the first two-states of cc and bb states (in MeV ).States cc bb cc bb cc bbLogarithmic: Martin: Experiment [23,24]:1S 3068 9444 3068 9445 3068± 2 9448± 513S1 3097 9460 3097 9461 3097 946011S0 2980 9395 2980 9397 2980 −∆1S 117 65 117 64 117 10017± 52S 3654 10030 3670 10018 3663± 5 1002323S1 3668 10037 3685 10030 368621S0 3614 10008 3625 9994 3622± 12a∆2S 54 30 60 33 57± 8b3S 3976 10352 4021 10351 10350± 54S 4200 10575 4272 10590 105805S 4370 10746 4469 107771P 3505 9881 3505 9861 3525± 1 9900± 12P 3878 10254 3907 10243 10261± 1aHere we cite Ref.[25].bHere we cite Ref.[26].11TABLE II. Predicted Spin-averaged masses of the bound states formed by the fourth SM familyu4 and isosinglet d4 quarks (in GeV ).SAD Mass u4u4 u4u u4s u4c u4b d4d4 d4u d4s d4c d4bLogarithmic:M(1S) 1275.05a 639.30 639.34 639.92 642.89 997.94b 500.70 500.74 501.32 504.29M(2S) 1275.64 639.88 639.92 640.50 643.47 998.53 501.28 501.32 501.90 504.88M(3S) 1275.96 640.21 640.24 640.82 643.80 998.85 501.61 501.64 502.22 505.20M(4S) 1276.18 640.43 640.47 641.05 644.02 999.07 501.83 501.87 502.45 505.42M(1P ) 1275.49 639.73 639.77 640.35 643.33 998.38 501.13 501.17 501.75 504.73M(2P ) 1275.86 640.11 640.15 640.73 643.70 998.75 501.51 501.55 502.13 505.10M(3P ) 1276.11 640.36 640.39 640.97 643.95 999.00 501.76 501.79 502.37 505.35Martin:M(1S) 1274.82 638.77 638.80 639.62 642.64 997.69 500.17 500.20 501.02 504.04M(2S) 1275.28 639.39 639.42 640.21 643.20 998.15 500.79 500.82 501.61 504.60M(3S) 1275.54 639.76 639.78 640.55 643.52 998.42 501.16 501.18 501.95 504.92M(4S) 1275.73 640.02 640.03 640.79 643.75 998.61 501.42 501.43 502.19 505.15M(1P ) 1275.15 639.22 639.25 640.05 643.04 998.03 500.62 500.65 501.45 504.44M(2P ) 1275.46 639.64 639.66 640.44 643.41 998.33 501.04 501.06 501.84 504.81M(3P ) 1275.67 639.93 639.94 640.70 643.67 998.55 501.33 501.34 502.11 505.07aFourth family system scale offset by 2.05 GeV to agree with Ref.[22].bIso-singlet system scale offset by 1.96 GeV to agree with Ref.[22].12TABLE III. Predicted fine-splittings of S and P levels (in MeV ).Mass splittings u4u4 [22] u4u [22] u4s [22] u4c u4b d4d4 [22] d4u [22] d4s [22] d4c d4bLogarithmic:2S − 1S 59 59 58 59 58 59 58 58 59 59 58 59 58 59 58 593S − 2S 32 32 33 33 32 32 32 33 32 32 33 33 32 32 32 324S − 3S 22 22 22 22 23 23 23 22 22 22 22 22 23 23 23 222P − 1P 37 − 38 − 38 − 38 37 37 − 38 − 38 − 38 373P − 2P 25 − 25 − 24 − 24 25 25 − 25 − 24 − 24 25Martin:2S − 1S 46 62 62 59 56 46 62 62 59 563S − 2S 26 37 36 34 32 27 37 36 34 324S − 3S 19 26 25 24 23 20 26 25 24 232P − 1P 31 42 41 39 37 30 42 41 39 373P − 2P 21 29 28 26 26 22 29 28 27 26TABLE IV. Hyperfine mass splittings (in MeV ).Level u4u4 u4u u4s u4c u4b d4d4 d4u d4s d4c d4b cc bbLogarithmic:∆1S 5.69 0.39 0.48 0.78 1.39 6.43 0.49 0.61 0.99 1.78 117(117)a 65(86)∆2S 2.62 0.18 0.22 0.36 0.64 2.96 0.23 0.28 0.46 0.82 54(57± 8)b 30(35)Martin:∆1S 4.09 0.45 0.54 0.89 1.38 4.70 0.57 0.69 1.13 1.76 117 64∆2S 2.09 0.23 0.28 0.45 0.71 2.41 0.29 0.35 0.58 0.90 60 33aThe quantity in bracket is the experimental or taken from other references.bHere we cite Ref.[26].13

International Journal of Modern Physics A

Bound energy masses of mesons containing the fourth generation and iso-singlet quarks

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Bound Energy Masses of Mesons Containing the Fourth Generation and Iso-singlet Quarks

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