University of Wisconsin Madison
University of Wisconsin Madison
arXiv:hep-ph/9209232v1 14 Sep 1992University of Wisconsin MadisonMAD/PH/711September 1992Supersymmetric Grand Unified Theories: Two LoopEvolution of Gauge and Yukawa CouplingsV. Barger, M. S. Berger, and P. OhmannPhysics Department, University of Wisconsin, Madison, WI 53706, USAABSTRACTWe make a numerical study of gauge and Yukawa unification in super-symmetric grand unified models and examine the quantitative implicationsof fermion mass ans¨atze at the grand unified scale.Integrating the renor-malization group equations with α1(MZ) and α2(MZ) as inputs, we findα3(MZ) ≃0.111(0.122) for MSUSY = mt and α3(MZ) ≃0.106(0.116) forMSUSY = 1 TeV at one-loop (two-loop) order.Including b and τ Yukawacouplings in the evolution, we find an upper limit mt∼<200 GeV from Yukawaunification.
For given mt∼<175 GeV, there are two solutions for β, one withtan β > mt/mb, and one with sin β ≃0.78(mt/150 GeV). Taking a popularansatz for the mass matrices at the unified scale, we obtain a lower limit onthe top quark mass of mt∼>150(115) GeV for α3(MZ) = 0.11(0.12) and an up-per limit on the supersymmetry parameter tan β∼<50 if α3(MZ) = 0.11.
Theevolution of the quark mixing matrix elements is also evaluated.
I. INTRODUCTIONThere is renewed interest in supersymmetric grand unified theories (GUTs) [1] to explaingauge couplings, fermion masses and quark mixings [2–9]. Recent measurements of the gaugecouplings at LEP and in other low energy experiments [10,11] are in reasonably good accordwith expectations from minimal supersymmetric GUTs with the scale of supersymmetry(SUSY) of order 1 TeV or below [2].
Supersymmetric GUTs are also consistent with the non-observation to date of proton decay [12]. In addition to the unification of gauge couplings[13], the unification of Yukawa couplings has been considered to predict relations amongquark masses [14–16].
With equal b-quark and τ-lepton Yukawa couplings at the GUT scale,the mb/mτ mass ratio is explained by SUSY GUTs [4,15]. With specific ans¨atze for theGUT scale mass matrices (e.g.
zero elements, mass hierarchy, relations of quark and leptonelements), other predictions have been obtained from quark masses and mixings that areconsistent with measurements [4,6,7,17,18]. The consideration of fermion mass relationshipshas a long history [19,20] and includes single relations and mass matrices (“textures”) withoutevolution [21,22], and single relations and mass matrices with evolution [23].Our approach is to explore supersymmetric GUTs first with the most general assump-tions, and then proceed to add additional GUT unification constraints to obtain more pre-dictions at the electroweak scale.
The renormalization group equations (RGEs) used hereare for the supersymmetric GUTs [24,25] with the minimal particle content above the super-symmetry scale and the standard model RGEs [26] below the supersymmetry scale. In §IIwe explore the running of the gauge couplings in the supersymmetric model at the two-looplevel and compare the results to those obtained at the one-loop level.
Rather than try topredict the scale of supersymmetry (MSUSY ) which may be sensitive to unknown and modeldependent effects like particle thresholds at the GUT scale, we choose two values of MSUSYto illustrate the general trends that occur. We also investigate the effects of the Yukawacouplings on the gauge coupling running which enter at two loops [17] and have often beenneglected in the past.
In §III we explore the unification of Yukawa coupling constants. First2
we consider the one-loop analytic solutions which can be obtained by neglecting the bottomquark and tau Yukawa couplings λb and λτ relative to λt in the RGEs. This serves as a usefulstandard for comparison with the two-loop results for smaller values of tan β (<< mt/mb),and many of the general features of the solutions to the RGEs are already present at thisstage.
We then investigate the two-loop RGE evolution of the Yukawa couplings includingthe effects of λb, λτ, and λt. Analytic solutions are not available for the two-loop evolution,so we integrate the RGEs numerically.
In §IV we investigate relations between Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and the ratios of quark masses. We investigatetwo popular ans¨atze [6,7,16] for Yukawa coupling matrices at the GUT scale.
Both of theseans¨atze agree with all existing experimental data, and this agreement is preserved at thetwo-loop level. We also integrate the two-loop evolution equations for certain CKM matrixelements and quark mass ratios in §IV.
The two loop RGEs for both the minimal supersym-metric model and the standard model are given in the appendix.II. GAUGE COUPLING UNIFICATIONA consistent treatment to two loops in the running of the gauge couplings involves thegauge couplings gi and the largest Yukawa couplings λt, λb and λτ.
From general expressions[24,25] that are summarized in the appendix, we obtain the evolution equationsdgidt =gi16π2big2i +116π23Xj=1bijg2i g2j −Xj=t,b,τaijg2i λ2j,(1)dλtdt =λt16π2" −Xcig2i + 6λ2t + λ2b!+116π2 X cibi + c2i /2g4i + g21g22 + 13645 g21g23 + 8g22g23+λ2t65g21 + 6g22 + 16g23+ 25λ2bg21−n22λ4t + 5λ2tλ2b + 5λ4b + λ2bλ2τo !#,(2)dλbdt =λb16π2" −Xc′ig2i + λ2t + 6λ2b + λ2τ!3
+116π2 X c′ibi + c′2i /2g4i + g21g22 + 89g21g23 + 8g22g23+45λ2tg21 + λ2b25g21 + 6g22 + 16g23+ 65λ2τg21−n22λ4b + 5λ2tλ2b + 3λ2bλ2τ + 3λ4τ + 5λ4to !#,(3)dλτdt =λτ16π2" −Xc′′i g2i + 3λ2b + 4λ2τ!+116π2 X c′′i bi + c′′2i /2g4i + 95g21g22+λ2b−25g21 + 16g23+ λ2τ65g21 + 6g22−n3λ2tλ2b + 9λ4b + 9λ2bλ2τ + 10λ4τo !#,(4)The various coefficients in the above expressions are also given in the appendix. The variableis t = ln(µ/MG) where µ is the running mass scale and MG is the GUT unification mass.The renormalization group equations of dimensionless parameters like the gauge couplingsand Yukawa couplings are independent of the dimensionful soft-supersymmetry breakingparameters.We begin with the recent values of αem and sin2 ˆθW at scale MZ = 91.17 GeV given inthe 1992 Particle Data Book [11,27](αem)−1 = 127.9 ± 0.2 ,(5a)sin2 ˆθW = 0.2326 ± 0.0008 ,(5b)where ˆθW refers to the weak angle in the modified minimal subtraction MS scheme [28].These values correspond to electroweak gauge couplings ofα1(MZ)−1 = 58.89 ± 0.11 ,(6a)α2(MZ)−1 = 29.75 ± 0.11 ,(6b)For simplicity we initially set the supersymmetric scale MSUSY equal to the top quark massmt and set all Yukawa contributions in Eq.
(1) to zero. Then evolving α1 and α2 from scaleMZ up to scale mt, we have4
α1(mt)−1 = α1(MZ)−1 + 5330π ln(MZ/mt) ,(7a)α2(mt)−1 = α2(MZ)−1 −116π ln(MZ/mt) ,(7b)We use the value MZ = 91.17 GeV, neglecting its experimental uncertainty.Next, for a grid of αG and MG values, we evolve from the GUT scale down to the chosenmt scale and retain those GUT scale inputs for which Eqs. (6) and (7) are satisfied.
We usethe two-loop SUSY GUT unification condition αG = α1(MG) = α2(MG). For the acceptableGUT inputs we also evolve the strong coupling α3(MG) = αG down to scale mt and then use3-loop QCD to further evolve it to scale MZ.
The three-loop expressionα3(µ)−1 = −b02 ln µ2Λ2!+ b1b0ln ln µ2Λ2!−2b21b30"ln ln µ2Λ2!− b0b2b21−1!# ln µ2Λ2!−1, (8)with the bi given in Ref. [29], is iteratively solved to find Λ from α3(mt).
Eq. (8) is thenevaluated for µ = MZ to obtain α3(MZ).
The resulting values for Λ for two representativevalues of α3(MZ) are given in Table 1.5
α3(MZ) Λ(5)Λ(4)Λ(3)0.11129.1 188.3 225.00.12233.4 320.2 360.0Table 1: The QCD parameter Λ(nf ) in MeV,where nf is the number of active flavors.We also investigate the effects of taking a supersymmetry scale higher than mt. Be-low MSUSY , the RGE are similar to the non-supersymmetric standard model.A linearcombination of the Higgs doublets is integrated out of the theory at MSUSY leaving theorthogonal combination Φ(SM) = Φd cos β + ˜Φu sin β coupled to the fermions in a way thatdepends on tan β [4,30,31]; this combination results from the assumption that the threesoft-supersymmetry breaking parameters in the Higgs potential can be equated to MSUSY .We use the two-loop RGEs [26] for the standard model, matching the couplings at MSUSY .Taking a single SUSY scale is an idealized situation since in general the supersymmetricparticle spectrum is spread over a range of masses [9].
Without further assumptions wecannot predict this spectrum. Given that such uncertainties exist, the predicted range forα3 should be taken to be representative only.The ranges of α−1G and MG parameters obtained from the procedure outlined above arepresented in Fig.
1 for one-loop and two-loop evolution with the choices MSUSY = mt andMSUSY = 1 TeV. The shaded regions denote the allowed GUT parameter space.
The two-loop values obtained for αG and MG are higher than the one-loop values and consequentlyα3(MZ) is higher for the two-loop evolution. Note that raising the SUSY scale from mt to 1TeV lowers MG and αG; hence α3(MZ) decreases as well.Figure 2 shows the corresponding results of the two-loop evolution over the full range ofµ.
We find the ranges for α3(MZ) with mt = 150 GeV shown in Table 2.6
MSUSYone-looptwo-loopmt = 150 GeV 0.1112 ± 0.0024 0.1224 ± 0.00331 TeV0.1065 ± 0.0024 0.1161 ± 0.0028Table 2: Ranges obtained for α3(MZ) from theinput values αem and sin2 ˆθW.The two-loop values of α3(MZ) are about 10% larger than the one-loop values. The effect ofthe higher SUSY scale is to lower α3(MZ) by about 5%.Inclusion of Yukawa couplings in the two-loop evolution also lowers the value of α3(MZ)somewhat.
For example setting λt = λb = λτ = 1 at the GUT scale, we obtain a two-loopvalue of α3(MZ) = 0.1189 ± 0.0031 for MSUSY = mt.The effects on the gauge couplings of including the Yukawa couplings in the evolutionare rather small for Yukawa couplings in the perturbative regime, justifying their neglect inmost previous analyses; for large values of tan β the changes in the gauge couplings due toinclusion of Yukawa couplings can be a few percent.The experimental situation regarding the determination of α3 is presently somewhatclouded [10], with deep inelastic scattering determinations in the range of the one-loopcalculations in Table 2 and LEP determinations similar to the two-loop results of Table 2.There are other uncertainties not taken into account here, due to threshold correctionsfrom the unknown particle content at the heavy scale [32–34], which can also change the α3values obtained above. These corrections are model-dependent so we have not attemptedto include such contributions.However recent analysis have shown that the constraintsfrom non-observation of proton decay greatly reduce the potential uncertainties from GUTthresholds [17,35].7
III. YUKAWA UNIFICATIONA.
One-loop analytic resultsThe unification of Yukawa couplings first introduced by Chanowitz, Ellis and Gaillard [14]has been reconsidered recently [4,6,7,17,30]. The GUT scale condition λb(MG) = λτ(MG)leads to a successful prediction for the mass ratio mb/mτ provided that a low energy super-symmetry exists [4].
The b to τ mass ratio is given bymbmτ= ηbητRb/τ(mt) ,(9)whereRb/τ(mt) ≡λb(mt)λτ(mt) = mb(mt)mτ(mt) ,(10)is the b to τ ratio of running masses at scale mt andηf = mf(mf)mf(mt)ifmf > 1GeV ,(11)ηf = mf(1GeV)mf(mt)ifmf < 1GeV ,(12)is a scaling factor including both QCD and QED effects in the running mass below mt. Wehave determined the ηf scaling factors to three-loop order in QCD and one-loop order inQED.
The QCD running of the quark mass is described bymq(µ) = ˆmq (2b0α3)γ0/b0 h1 + γ1b0−γ0b1b20!α3+12 γ1b0−γ0b1b20!2+ γ2b0+ γ0b21b30−b1γ1 + b2γ0b20!α23 + O(α33)i,(13)where the anomalous dimensions γ0, γ1 and γ2 are given in Ref. [36].
The scale-invariantmass ˆmq cancels in the ratio in Eq. (11).
The one-loop QED running from scale µ′ to scaleµ introduces modificationsmf(µ) = mf(µ′) α(µ)α(µ′)!γQED0/bQED0,(14)8
where the QED beta function and anomalous dimension are given by [33]bQED0= 433XQ2u + 3XQ2d +XQ2e,(15)γQED0= −3Q2f ,(16)and the sums run over the active fermions at the relevant scale. The dependence of the QCD-QED scaling factors η on α3(MZ) is shown in Figure 3; these factors increase as α3(MZ)increases.We note that the physical top mass is related to the running mass by [37]mphyst= mt(mt)1 + 43πα3(mt) + O(α23).
(17)The effects of the top quark Yukawa λt can be studied semi-analytically at one-loopneglecting the effects of the bottom and tau Yukawa couplings λb and λτ in Eqs. (1) and(2), which is a valid approximation for small to moderate tan β (i.e.
tan β∼<10). FollowingRef.
[6] we find [38]mbmτ= yη1/2xηbητ,(18)where x(µ), y(µ), η(µ) defined byx(µ) = (αG/α1(µ))1/6(αG/α2(µ))3/2 ,(19)y(µ) = exp−116π2MGZµλ2t(µ′)d ln µ′,(20)η(µ) =Yi=1,2,3(αG/αi(µ))ci/bi ,(21)are to be evaluated at µ = mt in Eq. (18).
Henceforth x, y, η shall be understood as beingevaluated at scale mt when an argument is not explicitly specified. Typical values of thesequantities obtained in Ref.
[18] are x = 1.52, y = 0.75 −0.81, η = 10.3 for a bottom massgiven by the Gasser-Leutwyler (GL) QCD sum rule determination mb = 4.25 ± 0.1 GeV [39]taken within its 90% confidence range and α3 = 0.111. The quantity y gives the scaling fromMG to mt that arises from a heavy quark, beyond the scaling due to the gauge couplings.9
The factor y(mt) is constrained to lie in a narrow range of values by Eq. (18).
The integralin Eq. (20) is crucial in explaining the mb/mτ ratio.
In fact if λt is neglected then y = 1 andthe mb/mτ ratio is found to be too large.For a given value of mt, there exist two solutions for tan β. This fact can be understoodqualitatively by studying the one-loop RGE for Rb/τ ≡λb/λτ.dRb/τdt= Rb/τ16π2−Xdig2i + λ2t + 3λ2b −3λ2τ.
(22)For small tan β the bottom and tau Yukawas do not play a significant role in the RGE, andany particular value for mb/mτ is obtained for a unique value of λt(mt), which correspondsto a linear relationship between mt and sin β,mtsin β = v√2λt(mt) = πvs2η3Ih1 −y12i1/2 ,(23)where v = 246 GeV andI =MGZmtη(µ′)d ln µ′ . (24)The numerical value for I from Ref.
[18] is 113.79 for mt = 170 GeV. For large tan β, wherethe effects of λb and λτ on the running Yukawa couplings can be substantial, an increasein λb can be compensated in the RGE by a decrease in λt.
Hence, for increasing tan β, thecorrect prediction for mb/mτ is obtained for decreased values of the top quark Yukawa. Thusthere is a second solution to the RGE for Rb/τ with a large value of tan β.
The inclusion ofthe two-loop effects does not alter these observations.The one-loop RGE for Rs/µ ≡λs/λµdRs/µdt= Rs/µ16π2−Xdig2i. (25)is similar to Eq.
(22), except that it receives no contribution from the dominant Yukawacouplings λt, λb, and λτ. When the value Rs/µ(MG) = 1/3 is assumed at the GUT scale, theprediction at the electroweak scale ismsmµ= 13η1/2xηsηµ.
(26)10
Notice that this equation does not include the scaling parameter y because the top quarkYukawa does not affect the running of the second generation quarks and leptons.Thisrelation for ms/mµ is in good agreement with the experimental values, but it is not asstringent as the mb/mτ relation due to the sizable uncertainty in the strange quark mass. Theresult ms/mµ = 1.54 was obtained in Ref.
[18], to be compared with the GL determination[39] ms/mµ = 1.66 ± 0.52.A popular strategy is to relate the mixing angles in the CKM matrix to ratios of quarkmasses, taking into account the evolution from the GUT scale in non-SUSY [40] or SUSY[41] models. For example, one popular GUT scale ansatz is |Vcb| ≈qmc/mt which requiresa GUT boundary condition on Rc/t ≡λc/λt ofqRc/t(MG) = |Vcb(MG)| ,(27)The one-loop SUSY RGE for Rc/t isdRc/tdt= −Rc/t16π2"3λ2t + λ2b#.
(28)The corresponding one-loop SUSY RGE for the running CKM matrix element |Vcb| is [41],d|Vcb|dt= −|Vcb|16π2"λ2t + λ2b#,(29)The pure gauge coupling parts of the RGEs are not present in Eqs. (28) and (29) since Rc/tand Vcb are ratios of elements from the up quark Yukawa matrix and the down quark Yukawamatrix.Neglecting the non-leading effects of λb, the one-loop results of Ref.
[6] at the electroweakscale obtained from evolution are|Vcb(mt)| = |Vcb(MG)|y−1Rc/t(mt) = Rc/t(MG)y−3 ,(30)or equivalently using Eq. (27)|Vcb(mt)| =s ymcηcmt.
(31)11
Since y is already well constrained by the b-mass relation of Eq. (18) (for the one-loopvalue of α3(MZ) = 0.111), Eq.
(31) requires that mt must be large in order that |Vcb| fallsin the experimentally allowed range 0.032 −0.054 (and even then |Vcb| is found to be at theupper limit of its allowed range). If, however, we use a larger value of α3(MZ) indicatedby the two-loop equations, say 0.12, then ηc increases by about 14%, as shown in Figure3.
Furthermore the increased values of the scaling parameters η and ηb require about a 9%decrease in y to explain the mb/mτ ratio in Eq. (18).
The resulting |Vcb| is reduced by about12% and is then closer to its central experimental value. Of course, a consistent treatmentat the two-loop level requires the two-loop generalization of Eq.
(31) obtained by solving thefull set of RGEs. One of the questions we will address subsequently is for what values of mtand tan β can the |Vcb| and the mb/mτ constraints be realized simultaneously.The predictions above are all based upon the assumption that the couplings remain inthe perturbative regime during the evolution from the GUT scale down to the electroweakscale.
Otherwise it is not valid to use the RGEs which are calculated order by order inperturbation theory. One can impose this perturbative unification condition as a constraint.For mb at the lower end of the GL QCD sum rule range 4.1−4.4 GeV the top quark Yukawacoupling at the GUT scale, λt(MG), becomes large, as can be demonstrated from analyticsolutions to the one-loop RGEs in the approximation that λb and λτ are neglected comparedto λt (valid for small to moderate tan β).The top quark Yukawa at the GUT scale is given byλt(MG)2 = 4π23I" 1y12 −1#.
(32)Taking [18] α3(MZ) = 0.111 and mb = 4.25 GeV and mt = 170 GeV gives λt(MG) = 1.5.Larger values of α3(MZ) lead to increased ηb via Eq. (11) giving smaller y in Eq.
(18) anda correspondingly larger value of λt(MG). The quantity λt(MG) is plotted versus α3(MZ) inFigure 4.
Larger values of α3(MZ) ≈0.12 can yield λt(MG)∼>3 that cast the perturbativeunification in doubt. Keeping the gauge couplings fixed and varying mb, one sees that smallervalues of mb also yield larger values of λt(MG).12
The scaling parameter y is manifestly less than one by Eq. (20) since λ2t > 0 in the regionmt < µ < MG.
This implies an upper limit on mb in Eq. (18) ofmbmτ ∼<η1/2xηbητ,(33)B. Two-loop numerical resultsWhen the two-loop RGEs are considered, analytic solutions must be abandoned, butthe same qualitative behavior is found in the numerical solutions.
Furthermore, there isnow the possibility that the bottom quark Yukawa coupling at the GUT scale becomesnon-perturbative for large values of tan β. In our analysis we solve the two-loop RGEs ofEqs.
(1-4) numerically [42], retaining all Yukawa couplings from the third generation.First we choose a value of α3(MZ) that is consistent with experimental determinationsand the preceding one-loop or two-loop evolution of the gauge couplings in the absence ofYukawa couplings. Specifically we take α3(MZ) = 0.11 or α3(MZ) = 0.12, to bracket theindicated α3(MZ) range.
For each particular α3(MZ) we consider a range of values for tan βand mb(mb). For each choice of α3(MZ), tan β, mb we choose an input value of mt.
TheYukawa couplings at scale mt are then given byλt(mt) =√2mt(mt)v sin β,λb(mt) =√2mb(mb)ηbv cos β,λτ(mt) =√2mτ(mτ)ητv cos β,(34)and the αi(mt) are determined by Eqs. (7) and (8) from the central values in Eq.
(6) We take[11] mτ = 1.784 GeV. The running of the vacuum expectation value v between the fermionmass scales is negligible for the range of fermion masses considered here [5].
Starting at thescale mt, we integrate the RGEs to the GUT scale, defined to be the scale at which α1(µ)and α2(µ) intersect. We then check to see if the equality λb(MG) = λτ(MG) holds to within0.01%.
If the b and τ Yukawas satisfy this condition, the solution is accepted. If not, wechoose another value of mt and repeat the integration.
Since our primary motivation hereis to study the influence of the α3(MZ) value on the Yukawa couplings, we do not enforcethe requirement that α3(MG) is equal to α1(MG) and α2(MG). Nevertheless the equality ofα1, α2, and α3 at MG is typically violated by ∼< 4% (2%) for α3(MZ) = 0.11 (0.12).
Suchdiscrepancies could easily exist from threshold effects at the GUT scale [34,35].13
We also explore the effects of taking the SUSY scale above mt. We proceed as describedabove, integrating the following two-loop standard model RGEs numerically from the topmass to the SUSY scale:dgidt =gi16π2bSMig2i +116π23Xj=1bSMij g2i g2j −Xj=t,b,τaSMij g2i λ2j,(35)dλtdt =λt16π2"h−XcSMig2i + 32λ2t −32λ2b + Y2(S)i+116π2 1187600 g41 −234 g42 −108g43 −920g21g22 + 1915g21g23 + 9g22g23+22380 g21 + 13516 g22 + 16g23λ2t −4380g21 −916g22 + 16g23λ2b+52Y4(S) −2λ3λ2t + λ2b+32λ4t −54λ2tλ2b + 114 λ4b+Y2(S)54λ2b −94λ2t−χ4(S) + 32λ2!#,(36)dλbdt =λb16π2"h−Xc′SMig2i + 32λ2b −32λ2t + Y2(S)i+116π2 −127600g41 −234 g42 −108g43 −2720g21g22 + 3115g21g23 + 9g22g23−7980g21 −916g22 + 16g23λ2t +18780 g21 + 13516 g22 + 16g23λ2b+52Y4(S) −2λλ2t + 3λ2b+32λ4b −54λ2bλ2t + 114 λ4t+Y2(S)54λ2t −94λ2b−χ4(S) + 32λ2!#,(37)dλτdt =λτ16π2"h−Xc′′SMig2i + 32λ2τ + Y2(S)i+116π2 1371200 g41 −234 g42 + 2720g21g2214
+38780 g21 + 13516 g22λ2τ + 52Y4(S) −6λλ2τ+32λ4τ −94Y2(S)λ2τ −χ4(S) + 32λ2!#,(38)dλdt =116π2" ( 94 325g41 + 25g21g22 + g42−95g21 + 9g22λ + 4Y2(S)λ −4H(S) + 12λ2)+116π2 −78λ3 + 1835g21 + 3g22λ2 +−738 g42 + 11720 g21g22 + 1887200 g41λ+3058 g62 −867120g21g42 −1677200 g41g22 −34111000g61−64g23(λ4t + λ4b)−85g21(2λ4t −λ4b + 3λ4τ] −32g42Y2(S) + 10λY4(S)+35g21−5710g21 + 21g22λ2t +32g21 + 9g22λ2b+−152 g21 + 11g22λ2τ−24λ2Y2(S) −λH(S) + 6λλ2tλ2b+20h3λ6t + 3λ6b + λ6τi−12hλ4tλ2b + λ2tλ4bi !#. (39)HereY2(S) = 3λ2t + 3λ2b + λ2τ ,(40)Y4(S) = 13h3XcSMig2i λ2t + 3Xc′SMig2i λ2b +Xc′′SMig2i λ2τi,(41)χ4(S) = 943λ4t + 3λ4b + λ4τ −23λ2tλ2b,(42)H(S) = 3λ4t + 3λ4b + λ4τ ,(43)and the coefficients aSM, bSM and cSM are given in the appendix along with the full matrixstructure.15
The initial values for α3(MZ), mb and mt are chosen as before; in addition we are requiredto specify the initial value of the quartic Higgs coupling λ at scale mt. The Yukawa couplingsat scale mt areλt(mt) =√2mt(mt)v,λb(mt) =√2mb(mb)ηbv,λt(mt) =√2mτ(mτ)ητv,(44)and the αi(mt) are given by Eqs.
(6)-(8). After integrating to the SUSY scale we requirethat the matching conditionλ(M−SUSY ) = 1435g21(M+SUSY ) + g22(M+SUSY )cos2 2β ,(45)is satisfied to within 0.1%.
This condition [4,31] results from integrating out the heavy Higgsdoublet at MSUSY . Below this scale only a Standard Model Higgs remains with its quarticcoupling given by Eq.
(45). We also apply the matching conditionsgi(M−SUSY ) = gi(M+SUSY ) ,(46)λt(M−SUSY ) = λt(M+SUSY ) sin β ,(47)λb(M−SUSY ) = λb(M+SUSY ) cos β ,(48)λτ(M−SUSY ) = λτ(M+SUSY ) cos β .
(49)If Eq. (45) is not satisfied we choose another input value of λ(mt) and repeat the process.
Weallow tan β to span a wide grid of values. After obtaining a satisfactory value of λ that meetsthe boundary condition above, we integrate the two-loop SUSY RGEs to the GUT scale,defined by the equality α1(MG) = α2(MG).
At the GUT scale we require λb(MG) = λτ(MG)to within 0.1%. If this condition is not met, we repeat the entire process, choosing otherinitial values for mt and λ.The parameter β also runs in going from the SUSY scale to the electroweak scale [31].However this effect is small and we neglect it here.In Figure 5 the resulting contours of constant mb are given in the mt, tan β plane [4,17]for the choices of α3(MZ) = 0.11 and 0.12 and the supersymmetry scales MSUSY = mt and 1TeV.
The contours shown are mb = 4.1, 4.25, 4.4 GeV (corresponding to the central value of16
mb and its 90% confidence range from the GL QCD sum rule determination) and mb = 5.0GeV (representing a typical constituent b-quark mass value). For a given mb and mt∼<175GeV, there is a high solution and a low solution for tan β as anticipated in §IIIa.
Thus, oncemt is experimentally known and the choice of mb resolved by other considerations (such asthe CKM matrix elements addressed subsequently), the assumption of Yukawa unificationat the GUT scale will select two possible values for tan β. For example for mt = 150 GeVand mb = 4.25 GeV, the solutions with α3(MZ) = 0.11 aretan β = 1.35ortan β = 56 .
(50)For mt∼<175 GeV the low solution is well-approximated bysin β = 0.78mt150 GeV. (51)Such knowledge of tan β would greatly simplify SUSY Higgs analyses [43].
Without imposingany other constraints, the top quark mass mt can be arbitrarily small.The plots rise very steeply for the maximal value of mt. This results because the linearrelation exhibited in Eq.
(23) and in the plot in Ref. [18] between mt and sin β is mappedinto a vertical line for sufficiently large tan β (∼>10).
The deviation of these contours frombeing strictly vertical results from the contributions of λb and λτ to the Yukawa couplingevolution.An upper limit on mt is determined entirely by the mb/mτ ratio. We find the mt upperlimits shown in Table 3 for the two choices of α3(MZ).
It is interesting that the predictedupper limit for mt coincides with that allowed by electroweak radiative corrections [11].α3(MZ)MSUSY 0.11 0.12mt187 1931 TeV 192 19917
Table 3: Maximum values of mt(mt) in GeV consistent withthe 90% confidence levels of the mb(mb) values of GL.Our contours of mb/mτ in Fig. 5 have about a 10% higher mb than those given in Ref.
[17]presumably because they employed the one-loop QCD results for the scaling factors ηf withthe two-loop expression for α3 rather than the three-loop QCD for both ηf and α3 that weuse here.As α3(MZ) gets larger, smaller values of y are needed to obtain obtain the correct mb/mτratio. In turn larger values of λt(µ) are needed via Eq.
(20). For α3(MZ)∼>0.12 and mb∼<4.2GeV, the value of λt(µ) near the GUT scale can be driven into the nonperturbative regime.In Figure 6 we show the values of λt(MG) and λb(MG) obtained for the solutions in Fig.5.
Fixed points in the quark Yukawa couplings exist at λ ≈1, so a Yukawa coupling onlyslightly larger than the fixed point at the scale mt can diverge as it is evolved to the GUTscale. For large values of the Yukawa couplings the two-loop contributions to the RGEscontribute a fraction x of the one-loop contributions whenλt =s6(16π2x)22≈6.5√x ,(52)λb =s7(16π2x)28≈6.3√x ,(53)as can be deduced from Eqs.
(2) and (3). When x ≈1 we are clearly in the nonperturbativeregime.
If we adopt the criteria that the two-loop effects always be less than a quarter ofthe one-loop effects, then λt and λb are nonperturbative when they remain below 3.3 and3.1 respectively all the way to the GUT scale. This is true for all of the curves presented inFigure 6, except for the mb = 4.1 GeV contours for α3(MZ) = 0.12; hence the exact positionof this contour cannot be predicted with accuracy.In Figure 7 we show the evolution of the Yukawa couplings from the SUSY scale to theGUT scale.
The nonperturbative regime for the case discussed above occurs only near theGUT scale.18
In some SO(10) GUT models the top quark Yukawa coupling λt is unified with the λb andλτ at the GUT scale. Imposing this constraint selects a unique value for tan β and mt.
Thissolution is given by the intersection of λt(MG) and λb(MG), which occurs for large tan β∼>50:see Fig. 6.One could also consider the unification of the Yukawa couplings at some scale other thanthat at which the gauge couplings unify [4,17].
Since Rb/τ increases as it evolves from theGUT scale to the electroweak scale, Yukawa unification at a scale larger than the gaugecoupling unification scale gives a larger mb/mτ ratio.The authors of Ref. [4] predict the light physical Higgs mass rather precisely.
Howeverthis prediction is related to their assumption (and the one we use here) that the heavy Higgsdoublet is integrated out at MSUSY . This means that the heavy physical Higgs bosons havemasses MH ≈MA ≈MH± ≈MSUSY >> MZ, which requires that the light Higgs mass isclose to its upper limit.
The relation of sin β to mt then fixes the one-loop corrections to thelight Higgs mass.IV. FERMION MASS ANSATZBy assuming an ansatz for Yukawa matrices at the GUT scale and evolving these matricesdown to the electroweak scale, predictions can be obtained for the quark and lepton massesand the CKM matrix elements [4,6,7,16].
Much work has been done on individual relationssuch as |Vud| ≈qms/md and |Vcb| ≈qmc/mt which are imposed at the GUT scale asdescribed in §III. Recently interest has been revived in models that involve several suchrelations, leading to a number of predictions for quark masses and CKM matrix elements atthe electroweak scale [4,6–8].
The relations evolve according to RGEs, and the main effectsare determined by the largest couplings.For moderate values of tan β (i.e.tan β∼<10),these are the gauge couplings gi and the top quark Yukawa coupling λt. For large values oftan β(≈mt/mb) the effects of λb and λτ can also be significant.
Various individual relationsat the GUT scale such as |Vcb| ≈qmc/mt can be satisfied for certain choices of these Yukawa19
couplings. The remarkable aspect of these fermion mass ans¨atze is that many relations canbe made to work at one time.
We shall concentrate in this section on two predictive waysof generating mixing between the second and third generations which put those mixingcontributions entirely in the up quark Yukawa matrix [4,6,16] or entirely in the down quarkYukawa matrix [7].20
A. The HRR/DHR ModelHarvey, Ramond and Reiss [16] proposed that the Yukawa matrices at the GUT scalehave the formU =0C0C0B0BAD =0Feiφ0Fe−iφE000D,(54)E =0F0F−3E000D.
(55)These matrices incorporate both Fritzsch zeros [20] and the Georgi-Jarlskog relation [21]between down quark and charged lepton matrix elements. This relative factor of three hasbeen realized in Higgs models with certain vacuum breaking patterns.
HRR obtained theabove ansatz using a 10 and three 126 Higgs multiplets in an SO(10) GUT model to obtainvarious relationships between CKM matrix elements and quark masses. The GUT ansatz ofEqs.
(??) and (55) is also the basis for the recent RGE analysis by Dimopoulos, Hall andRaby [6].
Henceforth we shall refer to this ansatz as the HRR/DHR model. It yields therelation |Vcb| =qλc/λt at the GUT scale.Renormalization group evolution generates non-zero entries in the above Yukawa matricesand also splits B1 ≡U23 and B2 ≡U32 to give the matrices at the electroweak scale of theformU =0C0CδuB10B2AD =0Feiφ0Fe−iφEδd00D,(56)E =0F ′0F ′−3E′000D′.
(57)21
The quantities A, D and D′ are equivalent to λt, λb and λτ respectively up to subleadingcorrections in the mass matrix diagonalization. The one-loop solutions [6] to leading orderin the hierarchy can be obtained analytically neglecting λb and λτ.
The one-loop results forthe CKM elements at the scale mt are|Vus| ="ηsmdηdms+ ηcmuηumc+ 2sηsηcmumdηdηumsmccos φ#1/2,(58)|Vcb| =s ymcηcmt,(59)VubVcb =sηcmuηumc,(60)where ηi(mt) is defined by Eq. (11) and y(mt) by Eq.
(20). The angle φ is ´a priori arbitrary.The down-type quark masses are related to the corresponding lepton masses bymd = η1/2xηdηe3me ,(61)ms = η1/2xηsηµmµ3 ,(62)mb = yη1/2xηbητmτ .
(63)Using the general expressions for the two-loop RGEs given in the appendix and keepingonly terms unsuppressed by the hierarchy, one obtains Eqs. (1)–(4) as well asdB1dt = B116π2" −Xcig2i + 6λ2t + λtλbδdB1!+116π2 X cibi + c2i /2g4i + g21g22 + 13645 g21g23 + 8g22g23+λ2t65g21 + 6g22 + 16g23+ 25λtλbδdB1g21−(22λ4t + 5λ2tλ2b + λtλbδdB15λ2b + λ2τ) !#,(64)dB2dt = B216π2" −Xcig2i + 6λ2t + λ2b!+116π2 X cibi + c2i /2g4i + g21g22 + 13645 g21g23 + 8g22g23+λ2t65g21 + 6g22 + 16g23+ 25λ2bg21−n22λ4t + 5λ2tλ2b + 5λ4b + λ2bλ2τo !#,(65)22
dδudt =δu16π2" −Xcig2i + 3λ2t + 3λtB1B2δu+ λbδdB2δu!+116π2 X cibi + c2i /2g4i + g21g22 + 13645 g21g23 + 8g22g23+λ2t45g21 + 16g23+ λtB1B2δu25g21 + 6g22+ 25λbδdB2δug21−(9λ4t + 3λ2tλ2b + λtB1B2δu13λ2t + 2λ2b+ λbδdB2δu5λ2b + λ2τ) !#,(66)dδddt =δd16π2" −Xc′ig2i + 6λ2b + λ2τ + λtλbB1δd!+116π2 X c′ibi + c′2i /2g4i + g21g22 + 89g21g23 + 8g22g23+λ2b25g21 + 6g22 + 16g23+ 65λ2τg21 + 45λtλbB1δdg21−(22λ4b + 5λ2tλ2b + 3λ2bλ2τ + 3λ4τ + λtλbB1δd5λ2t) !#. (67)Notice that since 1/B2dB2/dt= 1/λtdλt/dt, the ratio B2/λt is constant over all scales and isin particular equal to its value at the GUT scale (B2G/λt(MG)).With these RGEs we can include the additional experimental constraints from the charmmass mc and the CKM matrix element |Vcb| to determine the allowed region of the HRR/DHRmodel in the mt, tan β plane.
An analysis at the one-loop level neglecting λb and λτ relativeto λt was presented in Ref. [18].The Yukawa matrices are diagonalized by unitary matrices V Lu , V Ru , V Ld , V Rdso thatUdiag = V Lu UV R†uand Ddiag = V Ld DV R†d .
The CKM matrix is then given by VCKM =V Lu V L†d . We define a “running” CKM matrix by diagonalizing the Yukawa matrices U andD at any scale t. We find that λc/λt and |Vcb| are described in terms of the Yukawa matricesbyRc/t ≡λcλt= B1B2λ2t−δuλt!,(68)|Vcb| = B1λt−δdλb,(69)23
withmcmt= ηcRc/t(mt) . (70)To leading order in the mass hierarchy, the ratio Rc/t is given by the ratio of eigenvaluesof the 2 × 2 submatrix of U in the second and third generations while Vcb is given by thedifference in the rotation angles needed to rotate away the upper right hand entry in thesubmatrices of U and D. Given that the mass hierarchies exist, there is a simple iterativenumerical procedure for diagonalizing the mass matrices U and D and obtaining the CKMmatrix.We have checked that the corrections to the above formulas from contributionssubleading in the mass hierarchy are small.It is straightforward to derive the resulting renormalization group equations fromEqs.
(64)-(67)dRc/tdt= −Rc/t16π2" 3λ2t + λ2b+116π2λ2t25g21 + 6g22+ 25λ2bg21 −13λ4t + 2λ2tλ2b + 5λ4b + λ2bλ2τ #, (71)d|Vcb|dt= −|Vcb|16π2" λ2t + λ2b+116π245λ2tg21 + 25λ2bg21 −5λ4t + 5λ4b + λ2bλ2τ #,(72)The corresponding evolution equations in the Standard Model are given bydRc/tdt= −Rc/t16π2" 32λ2t −32λ2b+116π2 λ2t22380 g21 + 13516 g22 + 16g23−λ2b4380g21 −916g22 + 16g23−2λ(3λ2t + λ2b)−214 λ4t + 174 λ2tλ2b −132 λ4b + 94λ2tλ2τ −54λ2bλ2τ !#,(73)d|Vcb|dt= |Vcb|16π2" 32λ2t + 32λ2b+116π2 λ2t7980g21 −916g22 + 16g23+ λ2b4380g21 −916g22 + 16g23+2λ(λ2t + λ2b)24
−132 λ4t + 112 λ2tλ2b + 132 λ4b + 54λ2tλ2τ + 54λ2bλ2τ !#,(74)The evolution equations in Eqs. (73)-(74) are obtained from the two-loop RGEs of thestandard model given in Ref [26] and in the appendix.In the supersymmetric model |Vcb| increases with the running from the GUT scale to theelectroweak scale [41]; this is evident at the two-loop level in Eq.(72).
The opposite behavioroccurs in Eq. (74) for the nonsupersymmetric Standard Model where |Vcb| decreases as therunning mass decreases [40].
Fig. 8 shows the running of |Vcb| for the cases MSUSY = mtand 1 TeV.
In contrast to |Vcb| the ratio Rc/t increases monotonically as the running massdecreases in both the Standard Model and supersymmetric model cases.We stress that Eqs. (71) and (74) are the correct evolution equations regardless of thefermion mass ansatz at the GUT scale.
Changing the ansatz just changes the boundaryconditions at the GUT scale (terms subleading in the mass hierarchy differ between models,but this is a negligible effect). In a model for which the relationship |Vcb| =qλc/λt holds (asin the HRR/DHR model), this boundary condition isqRc/t(MG) = |Vcb(MG)|.
In Giudice’smodel, to be described below, the mixing between the second and third generations arisesin the down quark Yukawa matrix alone, and so in his model Rc/t and |Vcb| are unrelated atthe GUT scale.In our analysis of the CKM constraints we proceed as in the discussion of the calculationfor Figure 5. We numerically solve the two-loop RGEs as given by Eqs.
(1)-(4),(71)-(72) forthe case MSUSY = mt. As before, we consider the representative choices α3(MZ) = 0.11 andα3(MZ) = 0.12.
For each α3(MZ) choice, we consider a grid of tan β values, holding |Vcb(mt)|and mc fixed. We then choose input values for mt and mb (given α3(MZ), tan β, |Vcb|, mc)in terms of which all running parameters are uniquely specified at mt: λt(mt), λb(mt) andλτ(mt) are given by Eq.
(34), αi(mt) are determined by Eqs. (7) and (8) using the centralvalues in Eq.
(6) Rc/t is given by Eq. (70), and |Vcb| at scale mt is an input.
After integratingthe RGEs from mt to MG we check the constraintsλb(MG) = λτ(MG) ,(75a)25
qRc/t(MG) = |Vcb(MG)| . (75b)If either of these conditions is not satisfied to within 0.2%, we choose another input valuefor mt and mb and repeat the integration.We also carry out the RGE calculations with a SUSY scale at 1 TeV.
This is doneexactly as described in the previous section. In addition to the other parameters, we choosean input value for the quartic Higgs coupling λ at scale mt.
We then integrate the two-loop standard model RGEs to the SUSY scale and require that Eq. (45) hold to within0.1%.
For such solutions we apply the other appropriate boundary conditions (given byEqs. (46)-(49)) and integrate the two-loop SUSY RGEs to the GUT scale, where we requirethat λb(MG) = λτ(MG) andqRc/t(MG) = |Vcb(MG)| to within 0.2%.
In our calculation werequire that mb, mc and |Vcb| be within the experimentally determined 90% confidence levelsof the quark mass determinations of GL (4.1 < mb < 4.4 GeV, 1.19 < mc < 1.35 GeV) andthe recent Particle Data Book value [11] for |Vcb| (0.032 < |Vcb| < 0.054).In Fig. 9 the contours of constant |Vcb| are shown in the mt, tan β plane for a fixedmc = 1.27 GeV.
In Figs.10 and 11 we show the contours obtained by applying onlythe constraint in Eq. (75a) as in Fig.
5 along with the contours obtained by applying bothEqs. (75a) and (75b) for fixed mc as in Fig.
9. In Fig.
10 the value of mc is fixed at 1.27 GeVand contours of |Vcb| are shown. In Fig.
11 |Vcb| is fixed at its maximum allowed experimentalvalue of |Vcb| = 0.054 (at 90% C.L.) and three values of mc are plotted (corresponding to thecentral mc value and the 90% C.L.
values from GL).For large tan β the effects of including λb and λτ in the RGEs increase |Vcb|. In orderto satisfy |Vcb| < 0.054, the maximum allowed value of tan β for α3(MZ) = 0.11 is about50(60) for MSUSY = mt(1TeV); see Fig.
11. For this value of α3(MZ) the HRR/DHR modelpredicts that |Vcb| still lies at the upper end of its allowed 90% confidence level range whenthe effects of λb and λτ at large tan β are included in the two-loop RGEs; see Fig.
10.Allowing mb to become larger than the narrow window mb = 4.1 −4.4 GeV requires bigger|Vcb| which is unacceptable. The higher b mass contour mb = 5 GeV is not consistent with26
the GUT scale ansatz for α3(MZ) = 0.11. The largest consistent values of mb are given inTable 4.α3(MZ)MSUSY 0.11 0.12mt4.56 5.281 TeV 4.70 5.33Table 4: Maximum values of mb(mb) in GeV consistent withthe 90% confidence levels of |Vcb| and mc(mc).With α3(MZ) = 0.12, |Vcb| can be much closer to its central value, enhancing the plausi-bility of the HRR/DHR model, with the only caveat being that low mb (∼<4.2 GeV) valuesproduce λt(MG) values which are close to being non-perturbative for most values of tan β:see Figs.
6b, 6d. Notice that the dominant effect of taking the larger value of α3(MZ) indi-cated by two-loop evolution is to increase the QCD-QED scaling factor ηc, thereby allowing|Vcb| to be smaller and in better agreement with experiment.Imposing the constraints on mb, mc and |Vcb| also gives the lower limits on the top quarkmass since the |Vcb| contours in the smaller tan β region are steeper and eventually cross themb/mτ contours [18].
These lower limits on mt are summarized in Table 5.α3(MZ)MSUSY0.110.12mt155 (1.45) 118 (0.75)1 TeV 151 (1.16) 116 (0.64)Table 5: Minimum values of mt(mt) (tan β) in GeV consistent withthe 90% confidence levels of mb(mb), |Vcb| and mc(mc).27
The constraints on mb/mτ, |Vcb| and mc completely determine the allowed region in themt, tan β plane of the HRR/DHR model. Other constraints such as the ǫ parameter for CPviolation in the neutral kaon system, B mixing or the lighter quark masses affect only theother parameters in the model [18].If the Yukawa unification is assumed to occur at a scale higher than the gauge couplings,then the predicted value for |Vcb| will be lower [4] and easier to reconcile with the experimentaldata.B.
The Giudice ModelGiudice has proposed a different Yukawa mass ansatz [7] of the formU =00b0b0b0aD =0feiφ0fe−iφdnd0ndc,(76)E =0f0f−3dnd0ndc(77)This model uses a geometric mean relation m2c = mumt at the GUT scale to eliminate oneparameter in the up quark Yukawa matrix. The down quark Yukawa matrix must thengenerate the mixing between the second and third generations to get a value for |Vcb| thatagrees with experiment.
Giudice sets the parameter n in the above mass matrices to be two.We see no ´a priori reason to suppose that this parameter must be an integer and treat it asa free parameter.We find the generalized one-loop solutions (neglecting λb and λτ in the RGEs)|Vus| = 3s memµ 1 −252memµ+ 4n29mµmτητηµ!,(78)|Vcb| = n3ymµmτητηµ 1 −memµ+ (n2 −3)9mµmτητηµ!,(79)|Vub| = y2ηcmcmt,(80)28
mu = y3ηuη2cm2cmt,(81)md = η1/2xηdηe3me 1 −8 memµ+ 4n29mµmτητηµ!,(82)ms = η1/2xηsηµmµ3 1 + 8 memµ−4n29mµmτητηµ!,(83)mb = yη1/2xηbητmτ . (84)Notice that at one-loop level to leading order in the mass heirarchy the running |Vcb| isrelated to the strange and bottom Yukawa couplings by|Vcb(µ)| = nRs/b(µ) ≡nλs(µ)λb(µ) .
(85)Eqs. (78)-(84) can be compared to Eqs.
(58)-(63) for the HRR/DHR model, except thatwe have retained the highest non-leading order corrections only for the Giudice model. Whenn = 2 the predicted value of |Vcb| agrees well with the experimental value.
On the otherhand |Vus| is just at the lower limit of its 90% confidence level. The overall situation can beimproved somewhat by allowing n to be slightly larger than two.The leading term in Eq.
(78) can be recognized as the Oakes relation [19] between theCabibbo angle and the quark masses, tan θc ≈qmd/ms, supplemented by the Yukawaunification relation md/ms = 9me/mµ.Notice that this relation involving the first andsecond generations does not run, so the prediction of the Cabibbo angle is insensitive to thesize of the gauge and Yukawa couplings. The two-loop effects for the most part increaseα3 and hence the QCD scaling factors ηq.
The influence of two-loop contributions in therunning of the Yukawas is small.For tan β∼<10, λb and λτ can be neglected in the RGEs; then the relation for mu inEq. (81) implies an upper limit on mt [7].
However further solutions for mb/mτ are possiblewith large tan β, as can be seen in Figure 5. In the allowed mb/mτ band at large tan β thepredicted value for mu from Eq.
(81) is still satisfactory, since mt is in the same range asfound for the small tan β solutions.The CP-violating phase is not very well constrained in the Giudice model since the29
phase does not enter in the well-measured CKM elements; in fact the phase can assumealmost any non-zero value within its zero to 2π range. Correspondingly CP asymmetriesto be measured in B decays are not very constrained in the model [44].
In contrast, theCP-violating phase in the HRR/DHR model is almost uniquely determined by |Vus| and theCP-violating asymmetries are predicted precisely. This remain the case at the two-loop level.In the HRR/DHR scheme the dependence on α3(MZ) cancels out in quark mass ratios, andsince the constraint on the phase arises from the first and second generation mixing angles,there is no dependence of the phase on λt.V.
CONCLUSIONWe have investigated unification scenarios in supersymmetric grand unified theories usingthe two-loop renormalization group equations. Our primary conclusions are the following:(1) Given the experimentally determined values for α1 and α2 at MZ, the RGEs pre-dict α3 ≃0.111(0.122) at one-loop (two-loop) for MSUSY = mt and α3 ≃0.106(0.116) forMSUSY = 1 TeV.
Including the Yukawa couplings in the two-loop evolution of the gauge cou-plings decreases α3(MZ) by only a few percent. Thus the values of α3(MZ) ≃0.12 obtainedexperimentally at LEP II are also theoretically preferred if GUT scale thresholds effects orintermediate scales are not important.
(2) For any fixed value of α3(MZ) and mb there are just two allowed solutions for tan βfor a given top mass if mt∼<180 GeV; the larger solution has tan β > mt/mb and the smallersolution is sin β ≃0.78(mt/150GeV). Allowing for some uncertainty in α3(MZ), mb andMSUSY , these unique solutions for tan β at given mt become a narrow range of values.
Formt ≈180 −200 GeV the value of tan β changes rapidly with mt. (3) With λb, λτ unification we find an upper limit mt∼<200 GeV on the top quark mass byrequiring the successful prediction of the mb/mτ ratio; we also obtain lower limits mt∼>150GeV (115 GeV) for α3(MZ) = 0.11(0.12) from evolution constraints on mb, mc and |Vcb|.These lower limits are only mildly sensitive to MSUSY .30
(4) The effects of raising MSUSY is to decrease both αG and MG and to decrease the valuesof α3(MZ) that yields successful unification. Also the allowed band for the mb/mτ ratio inthe mt, tan β plane is shifted towards slightly higher top masses.
This in turn slightly reducesthe prediction for |Vcb| in models that utilize the relationqλc(MG)/λt(MG) = |Vcb(MG)|. (5) In the HRR/DHR model we find an upper limit on the supersymmetry parametertan β∼<50(60) for MSUSY = mt(1TeV) if α3(MZ) ≃0.11; for α3(MZ) = 0.12 the solutions atlarge tan β extend into the region for which λb(MG) is non-perturbative.
(6) For the value α3(MZ) ≃0.12 indicated by the two-loop RGEs, the agreement ofthe |Vcb| prediction of the HRR/DHR ansatz with experiment is improved.In fact forα3(MZ) = 0.12 and MSUSY = 1 TeV the central values for |Vcb| and the mass ratio mb/mτalmost coincide in the mt, tan β plane; see Fig. 10d.
This result is more general than theHRR/DHR ansatz, and occurs for any model with the GUT scale relation |Vcb| =qλc/λt. (7) With α3(MZ) ≃0.12 a large top Yukawa coupling is needed to achieve the correctmb/mτ ratio, and the theory is in some jeopardy of having a non-perturbative λt(MG) if mbis smaller than about 4.2 GeV.
(8) GUT unification of λτ, λb and λt can be realized for tan β∼>50. (9) The predictions for the CP asymmetries in the HRR/DHR model are largely unaf-fected by our two-loop analysis.
(10) We have found new solutions to the Giudice model for large tan β. These resultsrequire the inclusion of λb and λτ in the RGEs, and therefore could not be obtained inGiudice’s analytic treatment at one-loop.ACKNOWLEDGMENTSOne of us (VB) thanks Pierre Ramond for a discussion.
One of us (MB) thanks GregAnderson for a discussion. We thank Tao Han for his participation in the initial stages ofthis project.
This research was supported in part by the University of Wisconsin ResearchCommittee with funds granted by the Wisconsin Alumni Research Foundation, in part by31
the U.S. Department of Energy under contract no. DE-AC02-76ER00881, and in part bythe Texas National Laboratory Research Commission under grant no.
RGFY9173. Furthersupport was also provided by U.S. Department of Education under Award No.
P200A80214.PO was supported in part by an NSF Graduate Fellowship.VI. APPENDIXTo consider a specific ansatz for Yukawa matrices at the GUT scale at the two-loop levelrequires knowledge of the RGEs.
These can be derived from formal expressions that existin the literature [25]. For the supersymmetric model with two Higgs doublets, the one- [24]and two-loop RGEs can be written for general Yukawa matrices asdgidt =gi16π2big2i +116π23Xj=1bijg2i g2j −Xj=U,D,Eaijg2i Tr[YjY†j],(86)with YU ≡U, etc.dUdt =116π2"h−Xcig2i + 3UU† + DD† + Tr[3UU†]i+116π2 X cibi + c2i /2g4i + g21g22 + 13645 g21g23 + 8g22g23+(25g21 + 6g22)UU† + 25g21DD† + (45g21 + 16g23)Tr[UU†]−9Tr[UU†UU†] −3Tr[UU†DD†] −9UU†Tr[UU†]−DD†Tr[3DD† + EE†] −4(UU†)2 −2(DD†)2 −2UU†DD†!#U ,(87)dDdt =116π2"h−Xc′ig2i + 3DD† + UU† + Tr[3DD† + EE†]i+116π2 X c′ibi + c′2i /2g4i + g21g22 + 89g21g23 + 8g22g23+(45g21 + 6g22)DD† + 45g21UU† + (−25g21 + 16g23)Tr[DD†] + 65g21Tr[EE†]−9Tr[DD†DD†] −3Tr[DD†UU†] −3Tr[EE†EE†] −3UU†Tr[UU†]−3DD†Tr[3DD† + EE†] −4(DD†)2 −2(UU†)2 −2DD†UU†!#D ,(88)32
dEdt =116π2"h−Xc′′i g2i + 3EE† + Tr[3DD† + EE†]i+116π2 X c′′i bi + c′′2i /2g4i + 95g21g22+6g22EE† + (−25g21 + 16g23)Tr[DD†] + 65g21Tr[EE†]−9Tr[DD†DD†] −3Tr[DD†UU†] −3Tr[EE†EE†]−3EE†Tr[3DD† + EE†] −4(EE†)2!#E ,(89)wherebi = (335 , 1, −3) ,(90)ci = (1315, 3, 163 ) ,(91)c′i = ( 715, 3, 163 ) ,(92)c′′i = (95, 3, 0) ,(93)di = c′i −c′′i ,(94)bij =19925275885952524115914,(95)andaij =265145185662440. (96)These equations agree with those in the last paper in Ref.
[25] for the case where the Yukawamatrices are diagonal, if the following minor corrections are made: (1) b31 should be decreasedby a factor three; (2) the parenthesis in the second term of γ(2)H2 should come before the α22;(3) the first term of γ(2)τshould have a factor α21 instead of α22.The two-loop RGEs for the standard model are [26]33
dgidt =gi16π2bSMig2i +116π23Xj=1bSMij g2i g2j −Xj=U,D,EaSMij g2i Tr[YjY†j],(97)dUdt =116π2"h−XcSMig2i + 32UU† −32DD† + Y2(S)i+116π2 1187600 g41 −234 g42 −108g43 −920g21g22 + 1915g21g23 + 9g22g23+22380 g21 + 13516 g22 + 16g23UU† −4380g21 −916g22 + 16g23DD†+52Y4(S) −2λ3UU† + DD†+32(UU†)2 −DD†UU† −14UU†DD† + 114 (DD†)2+Y2(S)54DD† −94UU†−χ4(S) + 32λ2!#U ,(98)dDdt =116π2"h−Xc′SMig2i + 32DD† −32UU† + Y2(S)i+116π2 −127600g41 −234 g42 −108g43 −2720g21g22 + 3115g21g23 + 9g22g23−7980g21 −916g22 + 16g23UU† +18780 g21 + 13516 g22 + 16g23DD†+52Y4(S) −2λUU† + 3DD†+32(DD†)2 −UU†DD† −14DD†UU† + 114 (UU†)2+Y2(S)54UU† −94DD†−χ4(S) + 32λ2!#D ,(99)dEdt =116π2"h−Xc′′SMig2i + 32EE† + Y2(S)i+116π2 1371200 g41 −234 g42 + 2720g21g22+38780 g21 + 13516 g22EE† + 52Y4(S) −6λEE†+32(EE†)2 −94Y2(S)EE† −χ4(S) + 32λ2!#E ,(100)34
dλdt =116π2" ( 94 325g41 + 25g21g22 + g42−95g21 + 9g22λ + 4Y2(S)λ −4H(S) + 12λ2)+116π2 −78λ3 + 1835g21 + 3g22λ2 +−738 g42 + 11720 g21g22 + 1887200 g41λ+3058 g62 −867120g21g42 −1677200 g41g22 −34111000g61−64g23Tr[(UU†)2 + (DD†)2]−85g21Tr[2(UU†)2 −(DD†)2 + 3(EE†)2] −32g42Y2(S) + 10λY4(S)+35g21−5710g21 + 21g22Tr[UU†] +32g21 + 9g22Tr[DD†]+−152 g21 + 11g22Tr[EE†]−24λ2Y2(S) −λH(S) + 6λTr[UU†DD†]+20Trh3(UU†)3 + 3(DD†)3 + (EE†)3i−12TrhUU†(UU† + DD†)DD†i !#,(101)wherebSMi= (4110, −196 , −7) ,(102)cSMi= (1720, 94, 8) ,(103)c′SMi= (14, 94, 8) ,(104)c′′SMi= (94, 94, 0) ,(105)Y2(S) = Tr[3UU† + 3DD† + EE†] ,(106)Y4(S) = 13h3XcSMig2i Tr[UU†] + 3Xc′SMig2i Tr[DD†] +Xc′′SMig2i Tr[EE†]i,(107)χ4(S) = 94Tr3(UU†)2 + 3(DD†)2 + (EE†)2 −23UU†DD†,(108)H(S) = Tr[3(UU†)2 + 3(DD†)2 + (EE†)2] ,(109)bSMij=19950271044591035612111092−26,(110)35
andaSMij=17101232323212220. (111)These renormalization group equations are those given in the classic papers of Machacek andVaughn after replacing H →U†, FD →D†, FL →E†, and making the following correctionsto Eq.
(101) mentioned in the paper of Ford, Jack and Jones [26]: (1) The λg22 term in theone-loop beta function has a coefficient 9 instead of 1. (2) The λg21g22 term in the two-loopbeta function has a coefficient +117/20 instead of −117/20.
(3) The λg41 in the two-loopbeta function has a coefficient +1887/200 instead of −1119/200.36
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FiguresFig. 1.
Allowed GUT parameter space for mt = 150 GeV with (a) MSUSY = mt (one-loop RGE) (b) MSUSY = mt (two-loop RGE) (c) MSUSY = 1 TeV (one-loop RGE) (d)MSUSY = 1 TeV (two-loop RGE) versus the running mass scale µ.The shaded regiondenotes the range of GUT coupling and mass consistent with the 1σ ranges of α1(MZ) andα2(MZ); the curves for α3(µ) represent extrapolations from the GUT parameters. We haveomitted the contributions from Yukawa effects here which depend on tan β.Fig.
2. Gauge coupling unification with two-loop evolution for (a) MSUSY = mt (b) MSUSY =1 TeV taking mt = 150 GeV and neglecting Yukawa couplings; µ is the running mass scale.Fig.
3. The QCD-QED scaling factors ηf of Eq.
(11) are shown for f = s, c, b versus α3(MZ),assuming running quark masses mf(mf) of mt = 170 GeV, mb = 4.25 GeV, mc = 1.27 GeV.Fig. 4.
The top Yukawa coupling at the GUT scale determined at the one-loop level isplotted versus α3(MZ) for mt = 170 GeV and mb = 4.25 GeV.Fig. 5.
Contours of constant mb in the mt, tan β plane obtained from the RGEs with (a)MSUSY = mt, α3(MZ) = 0.11; (b) MSUSY = mt, α3(MZ) = 0.12; (c) MSUSY = 1 TeV,α3(MZ) = 0.11; (d) MSUSY = 1 TeV, α3(MZ) = 0.12. The shaded band corresponds tothe 90% confidence level range of mb from Ref.
[39] (mb = 4.1 −4.4 GeV); the dotted curvecorresponds to mb = 5.0 GeV. The curves shift to higher mt values for increasing α3(MZ) orincreasing MSUSY .43
Fig. 6.
The Yukawa couplings λt(MG) and λb(MG) = λτ(MG) at the GUT scale with (a)MSUSY = mt, α3(MZ) = 0.11; (b) MSUSY = mt, α3(MZ) = 0.12; (c) MSUSY = 1 TeV,α3(MZ) = 0.11; (d) MSUSY = 1 TeV, α3(MZ) = 0.12. The Yukawa couplings become largerfor higher α3(MZ) or higher MSUSY .
The perturbative condition λ∼<3.3 from Eq. (52) issatisfied except for the lowest b mass value mb = 4.1 GeV for α3(MZ) = 0.12.
The solid dotsdenote λτ = λb = λt unification.Fig. 7.
Two-loop evolution of the Yukawa couplings (a) λt(µ) (b) λb(µ), λτ(µ) from lowenergies to the GUT scale for the case α3(MZ) = 0.12 and MSUSY = 1 TeV. We take tan β =20 and the values of mt = 198, 197, 196, 181 GeV specified by the mb = 4.1, 4.25, 4.4, 5.0GeV contours in Fig 5d.Fig.
8. Two-loop evolution of the quark Yukawa ratio Rc/t ≡λc/λt and the CKM matrixelement |Vcb| for (a) MSUSY = mt and (b) MSUSY = 1 TeV.
We have taken α3 = 0.11, tan β =5 and have chosen the top and bottom quark masses such thatqRc/t(MG) = |Vcb(MG)| andmc = 1.27 GeV: (a) |Vcb(mt)| = 0.054, mt = 180 GeV, mb = 4.33 GeV; (b) |Vcb(mt)| = 0.050,mt = 189 GeV, mb = 4.14 GeV.Fig. 9.
Contours for constant |Vcb| at fixed mc = 1.27 GeV in the mt, tan β plane obtainedfrom the RGEs with (a) MSUSY = mt, α3(MZ) = 0.11; (b) MSUSY = 1 TeV, α3(MZ) = 0.12.Fig. 10.
Comparison of contours for constant |Vcb| and constant mb in the mt, tan β planefrom the RGEs, taking mc = 1.27 GeV, for (a) MSUSY = mt, α3(MZ) = 0.11; (b) MSUSY =mt, α3(MZ) = 0.12; (c) MSUSY = 1 TeV, α3(MZ) = 0.11; (d) MSUSY = 1 TeV, α3(MZ) =0.12. The shaded band indicates the region where the 90% confidence limit is satisfied for44
mb. The right-most contours are discontinued when λt(MG) exceeds 6.Fig.
11. Comparison of contours for constant mc and constant mb in the mt, tan β plane fromthe RGEs, taking |Vcb| equal to its upper limit 0.54, for (a) MSUSY = mt, α3(MZ) = 0.11;(b) MSUSY = mt, α3(MZ) = 0.12; (c) MSUSY = 1 TeV, α3(MZ) = 0.11; (d) MSUSY = 1TeV, α3(MZ) = 0.12.
The shaded band indicates the region where 90% confidence limitsare satisfied for all three constraints: mb, mc and |Vcb|. An X marks the lower limit of thisshaded band and corresponds to the values in Table 5.45
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