University of Wisconsin - Madison

arXiv:hep-ph/9203220v1 26 Mar 1992University of Wisconsin - MadisonMAD/PH/693FERMILAB-PUB-92/61-TFebruary 1992Test of the Dimopoulos-Hall-Raby Ansatzfor Fermion Mass MatricesV. Barger,∗M.

S. Berger,∗T. Han∗∗and M. Zralek†∗Physics Department, University of Wisconsin, Madison, WI 53706, USA∗∗Fermi National Accelerator Laboratory, P.O.

Box 500, Batavia, IL 60510, USA†Physics Department, University of Silesia, Katowice, PolandABSTRACTBy evolution of fermion mass matrices of the Fritzsch and the Georgi-Jarlskog forms from the supersymmetric grand unified scale, DHR obtainedpredictions for the quark masses and mixings. Using Monte Carlo methods wetest these predictions against the latest determinations of the mixings, the CP-violating parameter ǫK and the B0d- ¯B0d mixing parameter rd.

The acceptablesolutions closely specify the quark masses and mixings, but lie at the edges ofallowed regions at 90% confidence level.

One of the outstanding problems in particle physics is that of explaining the fermionmasses and mixings. In the Standard Model (SM) the 6 quark masses, 3 charged leptonmasses, the 3 quark mixings and the CP-violating phase of the Cabibbo-Kobayashi-Maskawa(CKM) matrix are introduced as phenomenological parameters.

Over the years various mod-els have been proposed to reduce the number of these free parameters [1], of which the bestknown is the Fritzsch model [2]. Recently Dimopoulos, Hall and Raby (DHR) have pro-posed an ansatz for fermion mass matrices [3] in the framework of minimal supersymmetric(SUSY) Grand Unified Theories (GUT).

The DHR approach is based on the observationthat some discrete symmetries present at the grand unification scale are broken in the low-energy theory. Thus some elements of the fermion mass matrices that vanish at the GUTscale are non-zero at the electroweak scale, and their low-energy values are calculable fromthe renormalization group equations.

The fermion masses and mixings at the electroweakscale can thereby be expressed in terms of a smaller number of input parameters at theGUT scale. DHR work in the massless neutrino limit and relate the 13 SM parameters anda SUSY parameter tan β (discussed below) to 8 input parameters, leading to 6 predictionsthat include an allowed range of 147–187 GeV for the top quark mass (mt).

In comparisonthe Fritzsch approach gives 77 ≤mt ≤96 GeV [1], which is nearly excluded in the SM bythe CDF experiment [4] at a 90% confidence level (C.L. ).The DHR quark mass matrices at the scale mt areMu =0C0CδuB0BAv sin β√2Md =0Feiφ0Fe−iφEδd00Dv cos β√2,(1)where all the parameters are real, tan β = v2/v1 in terms of the Higgs doublet vacuumexpectation values, and v = 246 GeV.

The charged lepton mass matrix Me is obtained fromthe above form of Md by the substitutions φ = 0, δd = 0, E →−3E′, D →D′, F →F ′.At the SUSY-GUT scale, the parameters δu and δd vanish and D = D′, E = E′, F = F ′,so the input mass matrix Mu is of the Fritzsch form [2] and Md and Me are of the Georgi-Jarlskog form [5], giving the GUT scale mass relations mb = mτ, ms ≃mµ/3, md ≃3me2

between quarks and leptons. The mass ratio prediction(md/ms)(1 −md/ms)−2 = 9(me/mµ)(1 −me/mµ)−2(2)holds at all scales.The Wolfenstein parameterization [6] of the CKM matrix determined from the unitarymatrices that diagonalize the DHR mass matrices can be expressed in terms of four angles(θi) and a complex phase (φ) as followsλ = (s21 + s22 + 2s1s2 cos φ)12 = |Vcd| = |Vus| ,(3a)λ2A = s3 −s4 = |Vcb| ,(3b)λ√ρ2 + η2 = s2 = |Vub/Vcb| =qmu/mc ,(3c)η = s1s2 sin φ/λ2 ,(3d)with si = sin θi, ci = cos θi (i = 1, 2, 3, 4), where θ2, θ3 are the angles that diagonalize thematrix Mu, and θ1, θ4 are those for Md [3]; only three of these angles are independent.These mixing angles are related to the quark masses and other parameters bys1 ≃qmd/ms ,s2 ≃qmu/mc ,s3 ≃|B/A| ,s4 ≃s3 −|Vcb| .

(4)The evolution based on the SUSY renormalization group equations (RGE) from the GUTscale to the appropriate fermion mass scales, taking all SUSY particles and the second Higgsdoublet degenerate at the scale of mt [3], gives the following relations,mt =mbmcmτ|Vcb|2xηbηcη1/2 ,ms −md = 13mµηsη1/2/x ,(5a)sin β = mtπvs3I2ηh1 −y12i−1/2 ,s3 = |Vcb|mbη1/2ηbmτx ,(5b)wherex = (αG/α1)1/6(αG/α2)3/2 ,y = x(mb/mτ)η−1/2η−1b,(6a)η(µ) =Yi=1,2,3(αG/αi)ci/bi ,I(µ) =MGZµη(µ′)d ln µ′ . (6b)3

The RGE parameters bi , ci are given in Ref. [3].

In these equations the couplings α1 and α2are evaluated at the scale mt. The mass parameters are defined as mq(µ = mq) for quarksheavier than 1 GeV, and the lighter quark masses ms, md, mu are calculated at the scaleµ = 1 GeV.Starting from the well-determined values [7], α1(MZ) = 0.016887, α2(MZ) = 0.03322,and evolving at one-loop level to their intersection determines the GUT scale MG = 1.1×1016GeV and the GUT coupling constant αG = 1/25.4.

Evolving backwards, the strong couplingconstant αs(MZ) = 0.106 is obtained, consistent with the LEP result αs = 0.118 ±0.008 [8].Also the values α1(mt) = 0.017 and α2(mt) = 0.033 are determined, as well as the factorsη(mt) = 9.7 and I(mt) = 110. We have used a top-quark threshold of 180 GeV in the RGE,consistent with our output determination.In evolution below the electroweak scale weinclude 3-loop QCD and 1-loop QED effects in the running masses to obtain the evolutionfactors ηb = 1.44, ηc = 1.80 and ηs = 1.95 where ηq = mq(mq)/mq(mt) for q = b, cand ηs = [ms(1 GeV)/mµ(1 GeV)].[ms(mt)/mµ(mt)].

Quark and lepton thresholds werehandled by demanding that the couplings and running masses be continuous. The numberof active flavors in the β-functions and in the anomalous dimensions was changed as eachsuccessive fermion was integrated out of the theory.Following DHR, we take the following 8 relatively better-known parameters as in-puts: me, mµ, mτ, mc, mb, mu/md, |Vcb| and |Vcd|.We generate random values forall inputs within 90% C.L.

(1.64σ) ranges.The input mass values [9,10] are mτ=1784.1 + 2.7−3.6 MeV, mc(mc) = 1.27 ± 0.05 GeV, mb(mb) = 4.25 ± 0.1 GeV, where 1σ er-rors are quoted. We also impose the theoretical constraint 0.2 ≤mu/md ≤0.7 [11].

Wenext calculate md, ms from Eqs. (2) and (5a) (obtaining md = 5.93 MeV, ms = 146.5 MeV),s1 and s2 from Eq.

(4), s3 from Eq. (5b) for the input of |Vcb|, s4 from Eq.

(4), and φ from|Vcd| of Eq. (3a).

Using these values we evaluate the magnitudes of all elements of the CKMmatrix. We retain only those Monte Carlo events that satisfy the following ranges from the1992 Review of Particle Properties [9],4

|VCKM| =0.9747–0.97590.218–0.2240.002–0.0070.218–0.2240.9735–0.97510.032–0.0540.003–0.0180.030–0.0540.9985–0.9995,(7)as well as the ratio 0.051 ≤|Vub/Vcb| ≤0.149 [9]. Figure 1 shows a scatter plot of sin βversus mt obtained from our Monte Carlo analysis; one sees that only a narrow wedge ofthe space is permissible.From sin β ≤1 in Eq.

(5), |Vcb| must satisfy the inequality|Vcb| >∼" xπvs3I2(1 −y12)mbmcmτηηbηc#1/2>∼0.053 ,(8)which is just at the edge of the 90% C.L. allowed range.

Calculating mt from Eq. (5a) andsin β from Eq.

(5b) and requiring that |Vcb| be within its allowed range, we find174 < mt < 183 GeV ,sin β > 0.954 (tan β > 3.2) . (9)This top quark mass determination is consistent with estimates from the electroweak radia-tive corrections [8] but is much more restrictive.

The predicted value of tan β is large, whichmay have significant phenomenological implications for Higgs boson searches at colliders [12].Next we include the constraints from the measured values|ǫK| = (2.259 ± 0.018) × 10−3 [9] ,rd = 0.181 ± 0.043 [13] ,(10)of the CP-violating parameter ǫK and the B0d- ¯B0d mixing parameter rd, The theoreticalformulas, including QCD corrections, can be found in Eqs. (2.1) and (2.10) of Ref.

[14]. Inour Monte Carlo analysis we allow variations of the bag-factors and B-decay constant overthe following ranges [14]:0.33 ≤BK ≤1.5 ,0.1 GeV ≤qBB fB < 0.2 GeV ,(11)taking fK = 160 MeV and ∆MK = 3.521 × 10−15 GeV.

The solutions so obtained closelyspecify the CKM matrix to be5

|VCKM| =0.9749–0.97590.2185–0.22300.0027–0.00320.2185–0.22300.9735–0.97450.0530–0.05400.0106–0.01090.0518–0.05290.9985–0.9986,(12)and predict the CP-violating phase to be in the range70◦< φ < 80◦. (13)The inclusion of ǫK and rd almost uniquely determines the values of |Vtd| and |Vts|.

Since|Vcb| is near its allowed upper limit, |Vub| is pushed to its lower end by the unitarity condition.The output value of the ratio0.051 < |Vub/Vcb| < 0.059(14)is at the low end of the allowed range. Improved experimental determinations of |Vcb| and|Vub/Vcb| will test the DHR ansatz.

In terms of the Wolfenstein parameters [6], we find0.2185 < λ < 0.2230 ,1.07 < A < 1.13 ,0.195 < η < 0.243 ,0.105 < ρ < 0.129 ,0.222 < √ρ2 + η2 < 0.275 . (15)The output values of the mass ratios of the light quarks are0.52 < mu/md < 0.70 ,md/ms = 0.0405 ,0.021 < mu/ms < 0.028 ,(16)giving 3.08 < mu < 4.15 MeV.

These light quark masses and their ratios are consistent withthose obtained in Refs. [10,11], but do not agree as well with some other recent studies [15],in which mu/md <∼0.3 was obtained.The heavy quark masses are now constrained to the narrow ranges1.19 < mc < 1.23 ,4.09 < mb < 4.20 .

(17)Another interesting result is restrictive ranges for the constants BK and fB0.33 < BK < 0.43 ,0.14

on which theoretical uncertainties have been problematic [16].We conclude with some brief remarks. From Eq.

(5a), mt is inversely proportional toηbηcη1/2, and the theoretical uncertainty in this quantity could somewhat enlarge or closethe window in mt (and correspondingly the window in |Vcb|). The DHR analysis assumesdominance of the top quark Yukawa couplings in the RGE evolution.

Since the output tan βmay be large, the effects of fully including λb and λτ in the evolution may not be negligible;this question deserves further study. Also two-loop renormalization group equations betweenMZ and MGUT should eventually be incorporated.

We have studied the charged Higgs bosoneffects on ǫK and rd. With MH± degenerate with mt, as assumed in the model, we found nosignificant changes in our results.

This is due to the fact that H± effects are smaller at largetan β for the K and B systems. In summary, the DHR ansatz for fermion mass matrices isconsistent with all current experimental constraints at 90% C.L..

It leads to almost uniquevalues for mt and quark mixings which make it an interesting target for future experiments.ACKNOWLEDGMENTSWe thank M. Barnett and K. Hikasa for advance information from the Particle DataGroup, G. Anderson and L. Hall for communications, and A. Manohar for comments onquark mass ratios. This work was supported in part by the U.S. Department of Energy undercontract No.

DE-AC02-76ER00881 and in part by the University of Wisconsin ResearchCommittee with funds granted by the Wisconsin Alumni Research Foundation. T. Han wassupported by an SSC Fellowship from the Texas National Research Laboratory Commissionunder Award No.

FCFY9116.7

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FIGURESFIG. 1.Scatter plot of sin β versus mt from our Monte Carlo analysis of the DHR model,imposing the constraints of input masses and present values of CKM matrix elements.10

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