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J. Hisano, H. Murayama, and T. Yanagida

arXiv:hep-ph/9207279v2 4 Aug 1992TU-400July, 1992Nucleon Decayin the Minimal SupersymmetricSU(5) Grand UnificationJ. Hisano, H. Murayama, and T. YanagidaDepartment of Physics, Tohoku UniversitySendai, 980 JapanAbstractWe make a detailed analysis on the nucleon decay in the minimal super-symmetric SU(5) grand unified model.

We find that a requirement of theunification of three gauge coupling constants leads to a constraint on a massMHC of color-triplet Higgs multiplet as 2×1013 GeV ≤MHC ≤2×1017 GeV,taking both weak- and GUT-scale threshold effects into account. Contraryto the results in the previous analyses, the present experimental limits onthe nucleon decay turn out to be consistent with the SUSY particles lighterthan 1 TeV even without a cancellation between matrix elements contributedfrom different generations, if one adopts a relatively large value of MHC(≥2×1016 GeV).

We also show that the Yukawa coupling constant of color-triplet Higgs multiplet does not necessarily blow up below the gravitationalscale (2.4×1018 GeV) even with the largest possible value of MHC. We pointout that the no-scale model is still viable, though it is strongly constrained.

1IntroductionThe hierarchy problem has been the most serious problem in the grand unified theory(GUT) [1]. At present, the only feasible solution to this problem is to introduce thesupersymmetry [2].

Furthermore, the supersymmetric (SUSY) SU(5) model [3] is nowstrongly supported phenomenologically by the sin2 θW measurement [4] made at the LEPexperiments [5]. Once we regard the SUSY-GUT as a serious candidate of the physicsbeyond the standard model, a natural question is how we can test the model.

The moststriking consequence of the grand unification is the instability of nucleons. However,the nucleon decay via exchanges of X and Y gauge bosons is strongly suppressed asτ −1n,p ∝M−4GUT in the SUSY-GUT because of the large unification scaleMGUT ∼2 × 1016 GeV.

(1.1)On the other hand, the nucleon decay via exchanges of color-triplet Higgs multiplet [6],which is suppressed only by M−2GUT, may still allow us to verify the model in the nearfuture.The main purpose of this paper is to study the implication of the present experi-mental limits on the nucleon decay in the minimal SUSY SU(5) GUT (MSGUT). Sim-ilar analyses have been carried out by Ellis, Nanopoulos, and Rudaz [7], and later byArnowitt, Chamseddine, and Nath [8, 9] rather thoroughly.

However, there has been nocriterion given on how heavy the color-triplet Higgs multiplet can be. In this paper, weexamine the experimental limits in the most conservative way, making the color-tripletHiggs multiplet as heavy as we can, allowed from the renormalization group (RG) analy-sis of the gauge coupling constant unification [10].

As a consequence, we find weaker con-straints than those given in the previous analyses. The authors of Refs.

[9] have claimedthat the data of nucleon-decay experiments at that time are already stringent enoughso that the SUSY particles below 1 TeV are excluded unless there is a delicate can-cellation between the proton decay matrix elements from second- and third-generationcontributions. On the contrary, we find that the present limits from the nucleon-decayexperiments are still consistent with the SUSY particles below 1 TeV even without such1

a cancellation. We also study a possible reach of the superKAMIOKANDE experiment.It will be shown that superKAMIOKANDE, together with LEP-II, is capable of coveringmost of the region with SUSY particles below 1 TeV, and hence it is highly expected toobserve the nucleon decay at superKAMIOKANDE.

It will be also stressed that moreprecise measurements on the gauge coupling constants, especially on that of QCD, willgive a strong impact on the determination of the color-triplet Higgs mass MHC.The paper is organized as follows.A brief review on the MSGUT is presentedin Sect. 2, to summarize our conventions.We critically re-examine the analysis ofRefs.

[8, 9] in Sect. 3.We find that the coefficients of the dimension-five operatorsare larger than theirs by a factor of 2.The decay rates for various modes are pre-sented.

As pointed out in Ref. [9], there may occur a cancellation between second- andthird-generation contributions.

We present the partial lifetimes in terms of unknownparameters ytK or ytπ which represent the ratios of the third- to the second-generationcontributions. In Sect.

4, we give an upper bound on the mass of color-triplet Higgsmultiplet, requiring that the gauge coupling constants are unified. Then the present ex-perimental limits are examined in Sect.

5. There it is explicitly shown that the presentdata still allow for SUSY particles below 1 TeV, even without the cancellation betweenmatrix elements mentioned above.

The reach of the superKAMIOKANDE and the LEP-II experiments is discussed in Sect. 6.

Sect. 7 is devoted to conclusions and discussions.An analysis on the dimension-six operators is presented in Appendix A.

Appendix Bsummarizes discussions on the renormalization effects on the dimension-five operators.We improve the analysis given in Ref. [7], but the difference turns out to be small.

Thechiral Lagrangian technique adopted to calculate the nucleon-decay matrix elements isdescribed in Appendix C.2Minimal SUSY SU(5) GUTIn this section, we review the minimal SUSY SU(5) GUT (MSGUT) [11], summariz-ing our conventions. We also clarify the origin of new CP-violating phases in Yukawa2

coupling constants of color-triplet Higgs multiplet to matter multiplets.There are quite a few multiplets in the MSGUT. An adjoint Higgs multiplet Σ(24)breaks the SU(5) GUT group down to SU(3)C ×SU(2)L ×U(1)Y , and a pair of quintetsH(5) and H(5∗) contain doublet Higgs multiplets Hf, Hf in the minimal SUSY standardmodel as well as their color-triplet partners HC, HC.

The superpotential of this modelisW=f3 TrΣ3 + 12fV TrΣ2 + λHα(Σαβ + 3V δαβ)Hβ+hij4 εαβγδǫψαβi ψγδj Hǫ +√2f ijψαβi φjαHβ,(2.1)where the Latin indices i, j = 1, 2, 3 refer to families, and the Greek ones α, β, γ · · ·represent the SU(5) indices. The chiral superfields ψ(10), φ(5∗) are matter multiplets.Contents of the Higgs multiplets areΣ=ΣaT a=Σ8Σ(3,2)Σ(3∗,2)Σ3+12√15200−3Σ24,(2.2)tH=(HC, HC, HC, H+f , H0f),(2.3)tH=(HC, HC, HC, H−f , −H0f),(2.4)and those of the matter multiplets areψ=1√20uc−ucud−uc0ucuduc−uc0ud−u−u−u0ec−d−d−d−ec0,tφ=(dc, dc, dc, e, −ν).

(2.5)where all the matter multiplets are written in terms of the chiral (left-handed) super-fields. The chiral superfields u and d contain left-handed up-type and down-type quarks,uc and dc the charge conjugations of right-handed up-type and down-type quarks, e and3

ν left-handed charged leptons and neutrinos, and ec the charge conjugations of righthanded charged-leptons. In the following, tQ ≡(u, d) and tL ≡(ν, e) will denote chiralsuperfields of weak-doublet quarks and leptons, respectively.The SU(5) GUT symmetry is broken by a vacuum expectation value of the Σ field,⟨Σ⟩= V222−3−3,(2.6)giving masses to X and Y gauge bosonsMV ≡MX = MY = 5√2g5V,(2.7)where g5 is the unified SU(5) gauge coupling constant.

The invariant mass parameterof H and H is fine-tuned to realize masslessness of Hf and Hf, while it keeps theircolor-triplet partners, HC and HC, superheavy asMHC = MHC = 5λV. (2.8)The components Σ8 and Σ3 acquire the same massMΣ ≡MΣ8 = MΣ3 = 52fV,(2.9)while the (physical) components Σ(3∗,2) and Σ(3,2) form superheavy vector multiplets ofmass MV together with the gauge multiplets.

The mass of the singlet component Σ24 is(1/2)fV .To analyze the dimension-five operators, we have to examine the Yukawa couplingsof H and H to matter multiplets. An important question is how many independentparameters we have in the Yukawa couplings [12].

The Yukawa coupling constants hijand f ij in Eq. (2.1) form a parameter space C6× C9, since hij is a symmetric matrix.The freedom of field re-definition is U(3)×U(3), corresponding to the choice of the basis4

of ψi and φi. Thus the physical degrees of freedom of the Yukawa coupling constants is(6 + 9) × 2 −9 × 2 = 12 = 3 + 3 + 4 + 2.

First two 3’s stand for the eigenvalues for up-and down-type mass matrices, 4 for the Kobayashi-Maskawa matrix elements, and 2 forthe additional phase degrees of freedom. We will parameterize the coupling matrices hijand f ij ashij=hieiϕiδij,(2.10)f ij=V ∗ijf j,(2.11)with Vij being the Kobayashi-Maskawa matrix.

Only two of the phases eiϕi are inde-pendent, and we can takeϕu + ϕc + ϕt = 0. (2.12)In this parameterization, the corresponding bases of the matter multiplets areψi∋tQi ≡(ui, d′i) = (ui, Vijdj),(2.13)ψi∋e−iϕiuci,(2.14)ψi∋Vijecj,(2.15)φi∋dci,(2.16)φi∋tLi ≡(νi, ei),(2.17)in terms of the mass eigenstates ui, di, uci, dci, νi, ei, eci.

Then the Yukawa couplings ofHiggs to matter multiplets are given byWY=hiQiuciHf + V ∗ijf jQidcjHf + fieciLiHf(2.18)+12hieiϕiQiQiHC + V ∗ijf jQiLjHC+hiVijuciecjHC + e−iϕiV ∗ijf jucidcjHC.It should be clear from the above expression that the phases eiϕi would be completelyirrelevant if HC and HC were absent. The phases appearing in the Yukawa couplings ofHC and HC cannot be absorbed by the field re-definition without affecting the couplingsof Hf and Hf.

As we will see later, these phases are important in the nucleon-decay5

amplitudes induced by the HC and HC exchanges. However, they are perfectly irrelevantto the nucleon decay caused by the X and Y gauge-boson exchanges.3Dimension-Five Operators and Decay RatesIn this section, we re-examine the previous analyses by Ellis, Nanopoulos, and Rudaz[7], and by Arnowitt, Chamseddine, and Nath [8, 9].

Several corrections to the formulaand numerical factors are made, and as a consequence nucleon-decay amplitude turnsout to be smaller than their result by a factor of 2.In the SUSY-GUT there are several baryon-number violating operators since it hasmany scalar bosons with color quantum numbers.Dimension-six operators inducedby the X and Y gauge-boson exchanges are suppressed by 1/M2GUT. These operatorscause unacceptably large nucleon-decay rates in the minimal non-SUSY SU(5) GUT[14], but there is no problem in the MSGUT since MGUT is much larger than in thenon-SUSY case.

In fact, we have analyzed the nucleon decays caused by the dimension-six operators, and found that they are always suppressed compared to those causedby the dimension-five operators. A brief discussion on the dimension-six operators ispresented in Appendix A.

The dimension-five operators are much more dangerous. Theseoperators are generated by HC and HC exchanges in the MSGUT, and is suppressed onlyby 1/MGUT [6].

The nucleon-decay amplitudes are obtained by dressing these operatorsby SUSY particle exchanges to convert scalar bosons to light fermions. Therefore, thenucleon-decay rates are sensitive to SUSY particle masses as well as the mass of HC andHC.

It has been also noted that there may be baryon-number violating dimension-fouroperators [13]. However, they can be forbidden by imposing the R-parity invariance,and we will not consider them in this paper.Let us now discuss the dimension-five operators that cause the nucleon decay.

Asupergraph is presented in Fig. 1.

The operators can be written explicitly asW5 =12MHChieiϕiV ∗klf l(QiQi)(QkLl) +1MHChiVije−iϕkV ∗klf l(uciecj)(uckdcl),(3.1)6

where the contraction of the indices are understood as(QiQi)(QkLl)=εαβγ(uαi d′βi −d′αi uβi )(uγkel −d′γk νl),(3.2)(uciecj)(uckdcl)=εαβγuciαecjuckβdclγ. (3.3)with α, β, γ being color indices.

Note that the total anti-symmetry in the color indexrequires that the operators are flavor non-diagonal (i ̸= k). Therefore dominant decaymodes in the MSGUT generally involve strangeness, like n, p →K¯ν [15, 7].The dimension-five operators will be converted to four-fermi operators at the SUSYbreaking scale, by exchanges of gauginos or doublet Higgsino.

However, the importantcontributions come only from the charged-wino dressing of the (QiQi)(QkLl) operators,and we will concentrate to this case.The exchanges of gluino, neutral gaugino and neutral Higgsino are small in general[9], since they are flavor diagonal and hence suppressed by the Yukawa coupling constantsof first and second generations appearing in the dimension-five operators. Though thegluino exchanges have stronger gauge coupling α3 than the wino exchanges, it will vanishcompletely in the limit where all the squark masses are degenerate [15].

Since the highdegeneracy is required to suppress the unwanted flavor changing neutral current,∗thegluino exchanges turn out to be always small.The charged-Higgsino exchanges arealso suppressed due to their small Yukawa coupling constants to the first or secondgenerations.† Thus the charged-wino exchanges give dominant contributions. Note thatcharged-wino dressing is impossible for the second operators (uciecj)(uckdcl), since all thefields involved are right-handed fields.

Though the left-right mixing due to the A-termsand superpotential |∂W/∂H|2 induces the wino dressing to the operators (uciecj)(uckdcl),their contribution is suppressed similarly to the charged-Higgsino exchanges since the∗A high degeneracy is required to suppress flavor-changing neutral currents, especially between firstand second generations. See Ref.

[16].†There are contributions from the third generation to the charged-Higgsino exchange amplitudes.However, the bottom-quark Yukawa coupling constant f b is smaller than the SU(2) gauge couplingconstant g2 unless tan βH in Eq. (3.5) is extremely large.

Therefore, the charged-Higgsino exchangesare most likely smaller than the charged-wino exchanges.7

mixing is proportional to the Yukawa coupling constants. In the following we refer tothe charged-wino simply as wino.By dressing the dimension-five operators with the wino exchanges, we will get four-fermi operators.

The results depend on the masses of charginos, squarks, and sleptons inthe loops. There is a mixing between wino and charged Higgsino, with the mass matrixMchargino =m ˜w√2mW cos βH√2mW sin βHµ.

(3.4)Here, m ˜w is a pure wino mass, µ a pure Higgsino mass, and βH a vacuum angle ofdoublet Higgs scalars, defined bytan βH = ⟨H0f⟩⟨H0f⟩. (3.5)The interaction of squarks and sleptons with wino is fixed byL = g2(˜u∗L ˜w+dL + ˜d∗L ˜w−uL + ˜ν∗L ˜w+eL + ˜e∗L ˜w−νL) + h.c.,(3.6)giving the triangle diagram factor [9]α22πf(u, d) ≡g22Zd4ki(2π)4 1m2˜u −k2!

1m2˜d −k2! 1Mchargino −̸k!11.

(3.7)We have taken only the (1, 1) component of the chargino propagator, since the nucleon-decay amplitudes are dominated by the pure wino component in the chargino states.Though the integral Eq. (3.7) depends on the mass eigenvalues and the mixing angles,we have found that it is well approximated by the pure wino exchange, and then it canbe given byα22πf(u, d) ≡α22πm ˜w1m2˜u −m2˜d m2˜um2˜u −m2˜wln m2˜um2˜w−m2˜dm2˜d −m2˜wln m2˜dm2˜w!.

(3.8)This approximation can be easily justified if m ˜w ≫mW, since then the off-diagonalelements in the mass matrix Eq. (3.4) can be neglected.

On the other hand, if m ˜w ∼mW,the off-diagonal elements cannot be neglected in general. However, if m ˜Q, m˜L ≫m ˜w,the triangle diagram factor f in Eq.

(3.7) is simply given byf ≃(Mchargino)11m2˜Q= m ˜wm2˜Q,(3.9)8

and hence the above approximation is again justified. The region m ˜w ∼m ˜Q ∼m˜L ∼µ ∼mW requires an exact treatment of the mixing, but this region turns out to bealready excluded, and is irrelevant to our analyses.Notice that the nucleon decay rates depend on the SUSY particle masses only throughthe function f. It is useful to see the dependence of the function f on m ˜w, m ˜Q, andm˜L.In the limit m ˜w ≪m ˜Q ∼m˜L, f behaves as in Eq.

(3.9).In the other limitm ˜w ≫m ˜Q ∼m˜L, it behaves asf ≃1m ˜wln m2˜wm2˜Q. (3.10)These behaviors will be used to put bounds on these masses in section 5.The resulting four-fermi operators can be written down explicitly asL =1MHCα22πhieiϕiV ∗jkf kεαβγ×h(uαi d′βi )(d′γj νk)(f(uj, ek) + f(ui, d′i)) + (d′αi uβi )(uγj ek)(f(ui, di) + f(d′j, νk))+(d′αi νk)(d′βi uγj )(f(ui, ek) + f(ui, d′j)) + (uαi d′βj )(uγi ek)(f(d′i, uj) + f(d′i, νk))i,(3.11)where the contraction of spinor indices are taken in each brackets ().

Here we haveassumed that the mixing between squarks is negligible. This is true in most of the su-pergravity models (for example, see Ref.

[17]) which ensure the absence of flavor-changingneutral current. Notice that Eq.

(3.11) is larger than that given in Refs. [8, 9] by a factorof 2.

We suspect that this difference arises from an inconsistency of the normalizationof their Yukawa coupling constants (See Eqs. (1.5), (1.6) in Ref.

[8]).We have to include three kinds of renormalization effects to perform quantitativeanalyses. First, the Yukawa coupling constants appearing in the Eq.

(3.1) are those eval-uated at the GUT-scale, and we have to calculate their magnitudes using the low-energyquark masses. Second, the dimension-five operators receive anomalous dimensions dueto the wave-function renormalizations of the external lines.

Third, the four-fermi oper-ators obtained after the gaugino-dressing will be further renormalized from the SUSYbreaking scale down to 1 GeV. These three effects are first discussed by Ellis, Nanopou-9

los, and Rudaz [7]. However, they dropped the SU(2) gauge interactions in estimatingthe first renormalization effects, which led to an overestimation by 50%.They alsoneglected the contributions from the top-quark Yukawa coupling.

Since the top-quarkmass is heavier than 90 GeV, these contributions, which appear in the wave-functionrenormalization of Higgs doublets, enhance the dimension-five operators by ∼30%.However, corrections on these two points give roughly the same result as theirs. Thedetails are explained in the Appendix B.After taking the renormalization effects into account, the nucleon-decay operatorsat 1 GeV are given byL = 2α22MHCmuimdkeiϕiV ∗jkm2W sin 2βHAS(i, j, k)AL×εαβγh(uαi d′βi )(d′γj νk)(f(uj, ek) + f(ui, d′i)) + (d′αi uβi )(uγj ek)(f(ui, di) + f(d′j, νk))+(d′αi νk)(d′βi uγj )(f(ui, ek) + f(ui, d′j)) + (uαi d′βj )(uγi ek)(f(d′i, uj) + f(d′i, νk))i,(3.12)where AS(i, j, k) represents the short-range renormalization effect between GUT- andSUSY breaking scales depending on the flavor i, j, k, and AL the long-range renormal-ization effect between SUSY scale and 1 GeV.

Here, the quark masses mui and mdk aredefined at 1 GeV in the MS scheme [7].We first show the prediction of nucleon-decay rates for the dominant modes, n, p →K¯νµ. The main contributions in the four-fermi operators Eq.

(3.12) come from the terms(i = c, j = u, k = s) (proportional to mcms), and (i = t, j = u, k = s) (proportional tomtms). The relevant terms are given byL=2α22MHCmsV ∗usALm2W sin 2βHεαβγ(dαuβ)(sγνµ) + (sαuβ)(dγνµ)×hAS(c, u, s)mceiϕcVcsVcd(f(c, µ) + f(c, d′))+AS(t, u, s)mteiϕtVtsVtd(f(t, µ) + f(t, d′))i.

(3.13)We have neglected the terms propotional to mu. Though the terms coming from the˜c-exchange can be computed precisely, the contribution of ˜t-exchange is ambiguous due10

to the unknown Kobayashi-Maskawa matrix elements for top quark [18]. In fact, theratio of the ˜t-contribution relative to the ˜c-one [8],ytK ≡mteiϕtVtsVtdAS(t, u, s)(f(t, µ) + f(t, d′))mceiϕcVcsVcdAS(c, u, s)(f(c, µ) + f(c, d′))(3.14)ranges between0.096 < |ytK| < 1.3,(3.15)if we take the triangle diagram factors f to be common, and the top-quark mass mt tobe 100 GeV.‡Note that eiϕt and eiϕc in Eq.

(3.14) are independent of each other.We cannotmeasure these phases from the present-day experiments, since they are irrelevant to anyobsevables as far as HC and HC are decoupled. Thus we are completely ignorant whetherytK is constructive or destructive to the ˜c-exchange amplitude.

We should regard thiscomplex parameter free in the present analyses.If the modes n, p →K¯νµ have a cancellation between ˜c- and ˜t-exchange amplitudes,we have to study the other possible decay modes.Next-leading modes are n, p →π¯νµ, suppressed by the Cabbibo angle sin2 θC compared to the K¯νµ modes. There are,similarly to the K¯νµ modes, contributions from ˜c- and ˜t-exchange to these modes, andtheir ratioytπ ≡mteiϕtV 2tdAS(t, u, s)(f(t, µ) + f(t, d′))mceiϕcV 2cdAS(c, u, s)(f(c, µ) + f(c, d′)),(3.16)ranges as0.041 < |ytπ| < 1.7,(3.17)with common f and mt = 100 GeV.

Though ytK and ytπ are correlated, we also regardytπ as an independent parameter because of too large uncertainties in the Kobayashi-Maskawa matrix elementsytπytK=VtdVcsVtsVcd(3.18)=0.22 −2.94.‡Note that mt in Eq. (3.14) is defined at the renormalization point 1 GeV.

For example, mt =270 GeV for mt = 100 GeV.11

We find that a perfect “double” cancellation in K¯ν and π¯ν modes is not possible,since ytK and ytπ have a non-vanishing relative phase coming from the CP-violating onein the Kobayashi-Maskawa matrix. Thus, we will not consider the “double” cancellationfurther in this paper, and take |1 + ytπ| = 1 throughout.

A “double” cancellation islogically possible only if the CP-violation is dominated by a non-KM mechanism. Evenin the presence of such a “double” cancellation, there are decay modes which do not havesuch ambiguities (i.e., p →K0µ+, η0µ+, π0µ+ and n →π−µ+).

The operators containingcharged leptons have only a contribution from the up-quark Yukawa coupling constant.Therefore, the decay rates do not suffer from the ambiguity of a possible cancellation.However, the decay rates are very small since the up-quark Yukawa coupling constantis tiny.To obtain matrix elements at the hadronic level from the operators written in termsof the quarks fields, we adopt a chiral Lagrangian technique [19, 20]. Details on thismethod is shown in appendix C. We present the results on the partial lifetimes ofnucleons in Tables 1–3, in terms of the parametersβ, MHC, AS, βH, ytK, ytπ,and the triangle factors f’s.

A parameter β is the hadron matrix element parameterused in the chiral Lagrangian technique [21, 22]βuL(⃗k) ≡ǫαβγ⟨0|(dαLuβL)uγL|p,⃗k⟩,(3.19)which ranges asβ = (0.003 – 0.03) GeV3,(3.20)depending on the methods of the theoretical estimation. Due to the uncertainty of anorder of magnitude, the predictions of nucleon partial lifetimes receive an ambiguity oftwo orders of magnitude.

A more precise determination of β is strongly desired.Table 1 summarizes predictions on the nucleon partial lifetimes for the dominantmodes n, p →K¯ν. One sees that the partial lifetimes of neutron are a little shorterthan ones of proton.

This is because the former has a larger chiral Lagrangian factor.12

Table 2 presents next-dominant modes n, p →π¯ν. These modes are suppressed by theCabbibo-angle sin2 θC.

Ratio of the decay rates into ¯νµ and ¯νe is simply the squaredratio of strange- and down-quark masses. The decay rates into charged lepton µ+ arelisted in Table 3.

These decay rates do not suffer from the ambiguity of a possiblecancellation. However, the partial lifetimes are very long because of the small Yukawacoupling constant of up quark.The dimension-five operators given by Eq.

(3.12) are larger than the expressionsgiven in Refs. [8, 9].

However, our final results shown in Tables 1–3 are smaller than theconclusion in Ref. [9] by a factor of 2 which may originate in an inconsistency betweentheir analytic formula Eq.

(2.3) and its numerical evaluations Eq. (2.7) in Ref.

[9].So far we have not discussed the decay rates of the modes containing ¯ντ. This isbecause we cannot make definite predictions for these decay modes since V ∗ub has a largeambiguity.

While the decay rates of ¯ντ modes can be as large as ones of ¯νµ mode if wetake the largest possible V ∗ub value, they can be also negligible with the smallest possibleV ∗ub value. In any case the decay rates can be only comparable to those into the ¯νµ, andhence the total decay rate into ¯ν is raised at most by a factor of two.

This gives onlya factor of√2 stronger constraint on the squark masses, and we will not include the¯ντ modes, hereafter. Once we know V ∗ub more precisely, it is easy to incorporate the ¯ντmodes into the present analyses.Finally, we note that dimension-five operators depend sensitively on quark masses.We use the central values of current quark masses at 1 GeV which are estimated fromthe chiral perturbation theory and QCD sum rule in Ref.

[23]. Quark masses of the firstand second generations have large ambiguities.

Especially, the strange-quark mass hasa large error-bar (i.e., ms(1 GeV) = 175 ± 55 MeV, given in the MS scheme). In ouranalyses we use ms(1 GeV) = 175 MeV.13

4Constraints on the GUT-scale Mass SpectrumSince the nucleon-decay rates due to the dimension-five operators are proportional tothe inverse square of the color-triplet Higgs multiplet mass M−2HC, it is very important todetermine it by some means. In the previous analyses [7, 8, 9], the authors have chosenMHC = (1 −2) × 1016 GeV ad hoc.

However, we have shown recently [10], that onecan obtain limits on the GUT-scale mass spectrum in the MSGUT, just by requiringthe unification of three gauge coupling constants. In particular, we have derived theupper bound on MHC without any theoretical prejudice.A theoretical requirementthat the Yukawa couplings remain perturbative below the gravitational scale MP /√8π(2.4 × 1018 GeV), poses further constraint on the GUT-scale mass spectrum.We first discuss the renormalization-group (RG) evolution of three gauge couplingconstants.

It was shown in Refs. [24, 25] that the simple step-function approximation isaccurate for supersymmetric theories, justified in the “supersymmetric regularization”DR scheme [26].

To illustrate how the GUT-scale spectrum receives constraints, we firstdiscuss the one-loop RG equations. After that we include the two-loop corrections.The running of three gauge coupling constants in the MSGUT can be obtained easilyat the one-loop level asα−13 (mZ)=α−15 (Λ) + 12π−2 −23Ngln mSUSYmZ+ (−9 + 2Ng) ln ΛmZ−4 ln ΛMV+ 3 ln ΛMΣ+ lnΛMHC),(4.1)α−12 (mZ)=α−15 (Λ) + 12π−43 −23Ng −56ln mSUSYmZ+ (−6 + 2Ng + 1) ln ΛmZ−6 ln ΛMV+ 2 ln ΛMΣ,(4.2)α−11 (mZ)=α−15 (Λ) + 12π−23Ng −12ln mSUSYmZ+2Ng + 35ln ΛmZ−10 ln ΛMV+ 25 lnΛMHC).

(4.3)Here, the scale Λ is supposed to be larger than any of the GUT-scale masses.Thenumber of generations Ng is three, and we have assumed a common mass mSUSY for all14

the SUSY particles and for the scalar component of one of the Higgs doublets. A massof the other doublet Higgs boson is taken at mZ.

By eliminating α−15from the aboveequations, we obtain simple relations:(3α−12−2α−13−α−11 )(mZ)=12π125 ln MHCmZ−2 ln mSUSYmZ,(4.4)(5α−11−3α−12−2α−13 )(mZ)=12π(12 ln M2V MΣm3Z+ 8 ln mSUSYmZ). (4.5)The Eqs.

(4.4), (4.5) imply that we can probe the GUT-scale mass spectrum from theweak-scale parameters (i.e., gauge coupling constants and mass spectrum of the SUSYparticles).§ Especially, MHC is determined independently of MV and MΣ. Eq.

(4.5)determines a combination of the vector and adjoint-Higgs masses (M2V MΣ)1/3, and wewill call it as “GUT-scale” MGUT = (M2V MΣ)1/3, hereafter.¶So far we have assumed a common mass mSUSY for the SUSY particles, but the masssplitting among the SUSY particles is also important to determine the GUT-scale massspectrum. To avoid unnecessary complications, we restrict ourselves to the minimalsupergravity model [17], where the SUSY-breaking mass parameters at the weak-scalecan be determined from a small number of parameters at the Planck scale, by using theRG equations [28].

Therefore, the squark and the slepton masses are given bym2˜u=m2 + 6.28M2 + 0.35m2Z cos 2βH,m2˜d=m2 + 6.28M2 −0.42m2Z cos 2βH,m2˜uc=m2 + 5.87M2 + 0.16m2Z cos 2βH,m2˜dc=m2 + 5.82M2 −0.08m2Z cos 2βH,(4.6)m2˜e=m2 + 0.52M2 −0.27m2Z cos 2βH,§It was claimed in Ref. [27] that the threshold corrections at the GUT-scale is so large that one cannotpredict the SUSY-breaking scale even if measurements on α3 become much more precise.

This highsensitivity on GUT-scale mass spectrum implies that one can probe it through precision measurementson the weak-scale parameters. Therefore, our result is consistent with their claim.¶This “GUT-scale” MGUT does not necessarily correspond to the scale where all three gauge couplingconstants meet.15

m2˜ν=m2 + 0.52M2 + 0.50m2Z cos 2βH,m2˜ec=m2 + 0.15M2 −0.23m2Z cos 2βH,in terms of the universal scalar mass m and the gaugino mass M at the GUT-scale. Wehave neglected the contributions from the Yukawa couplings to the renormalization ofthe particle masses.

Also, the gaugino masses at the weak-scale are given bym ˜B=α1(mZ)α5(MGUT)M,m ˜w=α2(mZ)α5(MGUT)M,(4.7)m˜g=α3(mZ)α5(MGUT)M,where m ˜B and m˜g represent masses of bino and gluino, respectively.The effect of the mass splitting can be taken into account by replacing ln(mSUSY /mZ)in Eqs. (4.4), (4.5) as−2 ln mSUSYmZ−→4 ln m˜gm ˜w+ Ng5 ln m3˜ucm2˜dcm˜ecm4˜Qm2˜L−85 ln m˜hmZ−25 ln mHmZ(4.8)in Eq.

(4.4), and8 ln mSUSYmZ−→4 ln m˜gmZ+ 4 ln m ˜wmZ+ Ng lnm2˜Qm˜ecm˜uc(4.9)in Eq. (4.5).

Two doublet Higgs bosons are assumed to have masses at mH and mZ,respectively. The symbol m˜h represents a mass of doublet Higgsino.

We have neglectedthe mixings among gauginos and doublet Higgsino.For the time being we will restrict ourselves to the case where the universal scalarmass dominates the SUSY breaking (i.e., m ≫M). The terms ln(m3˜ucm2˜dcm˜ec/m4˜Qm2˜L)in Eq.

(4.8) and ln(m2˜Q/m˜ecm˜uc) in Eq. (4.9) are negligibly small.

The term ln(m˜g/m ˜w)stays constant, since m˜g/m ˜w = α3/α2 ≃3.5. The dependence on mH in Eq.

(4.8) isweak due to its small coefficient, and we set mH = 1 TeV. Therefore, we find that MHCdepends mainly on the Higgsino mass m˜h, and MGUT on the product of gaugino masses16

m˜gm ˜w. We have also examined the constraint on MHC in the no-scale model [29] (i.e.,m = 0), and found that the difference is negligible.Now we are at the stage to derive the GUT-scale mass spectrum from the above RGanalysis.

In our numerical calculation, we use the one-loop RG equations for the weak-and the GUT-scale thresholds, and the two-loop ones between these two distant scales.The two-loop RG equations in the minimal SUSY standard model are [24]µ∂gi∂µ=116π2big3i +1(16π2)23Xj=1bijg2jg3i(4.10)where i, j = 1, 2, 3, andbi=0−6−9+222Ng +310120NHf ,(4.11)bij=0000−24000−54+38156588152514811153683Ng+9509100310720000NHf . (4.12)Here, NHf is the number of doublet Higgs multiplets (NHf = 2).

The threshold cor-rections at the two-loop level are expected to be small, since their mass splittings onlywithin the same order of magnitude do not produce large logarithms. As the inputparameters, we use the MS gauge coupling constants at the Z-pole given in Ref.

[30],α = 127.9 ± 0.2, sin2 θW = 0.2326 ± 0.0008, and α3 = 0.118 ± 0.007. However, theuse of the simple step-function approximation is only justified in the DR-scheme.

Sincewe employ the simple step-function approximation, we have to convert these couplingconstants at the Z-pole into the DR-scheme by1αDRi=1αMSi−Ci12π,(4.13)17

where C1 = 0, C2 = 2, and C3 = 3 [31].Combining all the above discussions, we find that MHC is constrained to the range2.2 × 1013 GeV ≤MHC ≤2.3 × 1017 GeV,(4.14)and the “GUT-scale” is tightly constrained as∥0.95 × 1016 GeV ≤(M2V MΣ)1/3 ≤3.3 × 1016 GeV,(4.15)for 100 GeV ≤m˜g ≤1 TeV.The allowed range of MHC is much less constrained,because of the small gauge-group representation for the Higgs multiplets. The largeambiguity of MHC comes mainly from the uncertainty in the strong coupling constant α3,and the prediction on the nucleon decay will be drastically improved if the uncertaintydiminishes.

In Fig. 2 we present the GUT-scale spectrum derived from the present gaugecoupling constants, and also that expected if the error-bar of α3 is reduced by a factorof 2 with the same central value.

The importance of more precise measurements on α3should be clear from the figure.When one uses these constraints, Eq. (4.14) and Eq.

(4.15), one needs to pay two at-tentions. First, we have taken only one standard deviation for gauge coupling constants.If we allow two standard deviations, allowed region of MHC spreads to both ends by twoorders of magnitude, loosing any practical limits.

On the other hand, the “GUT-scale”is still constrained tightly. Therefore, we restrict our analyses to only one standarddeviation as in Eqs.

(4.14), (4.15).∗∗Second, we have used the RG equations at thetwo-loop level. One may be concerned for whether three-loop corrections are important.The difference in MHC between the one-loop and the two-loop results is a factor of 30,which is a very small factor compared to the large ratio MGUT/mZ appearing in the∥The bound on MGUT quoted in Ref.

[10], 0.90 × 1016 GeV ≤MGUT ≤3.1 × 1016 GeV, includes aminor mistake, and should be replaced by Eq. (4.15).∗∗In other words, it is still possible to raise MHC up to near the gravitational scale if we allow twostandard deviations.

However, we tentatively take this one-sigma bound seriously to perform furtheranalyses. It should be noted that nucleon decay can be generated at observable rates for reasonablerange of parameters, even with the color-triplet Higgs of mass at the gravitational scale.18

solutions of the RG equations. Thus we expect that the three-loop corrections are muchless than O(1).Although we have concentrated to the RG analysis on the gauge coupling constantsto determine the GUT-scale mass spectrum, we will obtain further constraint on themass spectrum from the RG analysis on Yukawa coupling constants.As shown inEq.

(2.8), MHC is given by a Yukawa coupling constant λ between HC, HC and Σ,which is not known. On the other hand, the mass MV is determined by the SU(5)gauge coupling constant g5, whose strength is known by the RG analysis.A largemass splitting MV ≪MHC requires that the λ is very large compared to g5.

Thusthe applicability of the perturbation theory restricts the mass splitting to be not large.The same argument applies to the mass MΣ, which originates in a self-coupling of theadjoint-Higgs as seen in Eq. (2.9).A constraint arises by requiring that those Yukawa coupling constants do not blow-up below the gravitational scale, MP/√8π = 2.4×1018 GeV.

The running of the Yukawacoupling constants in Eq. (2.1) are described by the RG equations,µ∂λ∂µ=1(4π)2−985 g25 + 5310λ2 + 2140f 2 + 3(ht)2λ,(4.16)µ∂f∂µ=1(4π)2−30g25 + 32λ2 + 6340f 2f,(4.17)µ∂ht∂µ=1(4π)2−965 g25 + 125 λ2 + 6(ht)2ht,(4.18)µ∂g5∂µ=−3(4π)2g35,(4.19)where ht is the Yukawa coupling constant to top quark.

The conservative limit on λcan be obtained in the case f = ht = 0. A numerical study shows that the mass MHCis limited from above,MHC =λ√2g5MV < 2.0MV .

(4.20)A similar limit on MΣ can be obtained with λ = ht = 0,MΣ =f2√2g5MV < 1.8MV . (4.21)19

One may feel uneasy about the assumption that there is no new physics betweenthe GUT-scale and the gravitational scale. We have examined the above analysis againrequiring that the Yukawa coupling constants do not blow-up below 1017 GeV.

Thisrequirement relaxes the constraint Eqs. (4.20), (4.21) at most by a factor of 2.

Therefore,in the following calculation we use Eqs. (4.20), (4.21).We obtain constraints on MV and MΣ separately, combining the discussions above.The limits Eq.

(4.15) and Eq. (4.21) giveMV>0.78 × 1016 GeV,(4.22)MΣ<4.9 × 1016 GeV.(4.23)Eq.

(4.22) will be used to put limits on the dimension-six operators in the followingsections.5Present Limits on Dimension-Five OperatorsIn this section we combine the predictions obtained in the previous sections with theresults of the nucleon-decay experiments to see the present status of the MSGUT. Wefind that the present lower bounds on the nucleon partial lifetimes are still consistentwith the SUSY particles below 1 TeV if one adopts a relatively large value of MHC(≥2 × 1016 GeV).

In Table 4, we have listed the experimental lower limits on thepartial lifetimes of nucleon [18].The most dominant decay mode by dimension-five operators is n →K0¯νµ, as shownin Table 1. This mode dominates slightly over the similar decay mode, p →K+¯νµ,because of the chiral Lagrangian factor.

These decay modes, however, have an am-biguity in the parameter ytK, which is the relative ratio of the ˜t-exchange to the ˜c-exchange contributions. Furthermore, the parameter ytK contains an unknown phasefactor ei(φc−φt), which cannot be determined from any low-energy experiments.

In fact,Arnowitt, Chamseddine, and Nath [8] have shown a possible cancellation between thesecond- and third-generation contributions. However, if the combination |1 + ytK| de-20

creases, the modes n, p →K¯νµ cease to be dominant. Then the experimental limit onthe other modes n, p →π¯νµ become important.

This interchange occurs at|1 + ytK| = 0.40,(5.1)when |1 + ytπ| = 1.The parameters ytK and ytπ are correlated as clear from theirdefinitions (Eqs. (3.14), (3.16)).However, we find it impossible that |1 + ytK| and|1+ytπ| are canceled out simultaneously, as explained in section 4.

We take |1+ytπ| = 1throughout.In Fig. 3, we show the lower bound on MHC derived from the present nucleon-decayexperiments by varying |1 + ytK|.

We choose other parameters such that the nucleonlifetimes become as long as possible (i.e., m ˜Q = m˜L = 1 TeV, m ˜w = 45 GeV, andtan βH = 1). In this figure, the upper horizontal line corresponds to the maximum value(MHC = 2.3 × 1017 GeV) in Eq.

(4.14). There are two curves representing the lowerlimit on MHC obtained from the experimental limits on the nucleon lifetimes.

The uppercurve corresponds to the case of the hadron matrix element β = 0.03 GeV3, and thelower curve to the case of β = 0.003 GeV3. The smaller hadron matrix element β givesweaker constraint as expected.

Thus, the conservative lower bound on MHC from thenucleon-decay experiments isMHC ≥5.3 × 1015 GeV. (5.2)We illustrate how the lower bounds on MHC depend on the SUSY breaking param-eters, the wino mass m ˜w and the squark mass m ˜Q in Fig.

4, assuming m˜L ≃m ˜Q.†† Inthis figure the dashed line shows the dependence on m ˜w when we choose the most con-servative set of parameters, m ˜Q = m˜L = 1 TeV, tan βH = 1, |1 + ytK| < 0.4, AS = 0.67,and β = 0.003 GeV3. The lower bound on MHC rises linearly on m ˜w in the regionm ˜w < 1 TeV.

However, the lower bound decreases as m−1˜wbeyond 1 TeV, and we donot obtain an upper bound on m ˜w with this conservative choice of parameters. The††The situation does not change even if we allow mass splittings between sleptons and squarks (say,m˜L ≪m ˜Q), since the denominator in Eq.

(3.9) is dominated by the heavier mass, m ˜Q.21

dash-dotted line shows the dependence on m ˜Q again for the most conservative case,m ˜w = 45 GeV, tan βH = 1, |1 + ytK| < 0.4, AS = 0.67, and β = 0.003 GeV3. The lowerbound on MHC goes down as 1/m2˜Q, leading to the lower bound on m ˜Q,m ˜Q ≥150 GeV.

(5.3)We also show the dependence on tan βH in Fig. 5.

We have fixed m ˜Q = m˜L = 1 TeV,m ˜w = 45 GeV, β = 0.003 GeV3, and |1+ytK| = 1.0 or |1+ytK| < 0.4. The lower boundon MHC is proportional to1sin 2βH= 12 1tan βH+ tan βH!.

(5.4)When tan βH ≫1, the dependence is almost linear. We find a constraint on tan βH,tan βH ≤85,(5.5)which is, however, much weaker than tan βH < 40 obtained from the requirement thatthe Yukawa coupling constant f b for bottom quark remains in the perturbative regimebelow the “GUT-scale”.‡‡Taking MHC as heavy as possible given in Eq.

(4.14), we obtain limits on the massesm ˜w and m ˜Q. The allowed region is shown in Fig.

6. Here we have taken |1 + ytK| = 1and m˜L ∼m ˜Q.

The present experimental limits on the wino and the squark masses fromdirect-search experiments at LEP [32] and CDF [33] are shown for comparison. Sincethe decay rate behaves like (m ˜w/m2˜Q)2 in the region m ˜w ≪m ˜Q, the lower bound on m ˜Qbehaves like m1/2˜w .

In the other extreme, m ˜w ≫m ˜Q, the decay rate goes like 1/m2˜w,and around m ˜w ∼105 GeV the constraint on m ˜Q from the nucleon-decay experimentsbecomes weaker than that from the CDF experiments. We see that the “natural” massregion <∼1 TeV for the SUSY particles still survives the nucleon-decay experiments.Though the authors of Ref.

[9] claimed that the present limits on the nucleon decayare stringent enough to exclude the SUSY particles lighter than 1 TeV in the absence‡‡The tan βH has to be larger than 0.5, since otherwise the top-quark Yukawa coupling constant willblow up below the “GUT-scale” with mt ≥90 GeV.22

of a delicate cancellation between matrix elements of the dimension-five operators (i.e.,β ≃0.003 GeV3, |1 + ytK| ≃0.2), we see now that there is a wide allowed range. Thisis mainly because we use the maximum value of MHC (2.3 × 1017 GeV) given from theRG analysis while they chose MHC ≃MGUT just by hand (i.e., 2.0 × 1016 GeV).We show similar limits from the π¯νµ mode in Fig.

7. As discussed above, these decaymodes become dominant if |1+ytK| < 0.4.

The limit is weaker compared to the previouscase without the cancellation as shown in Fig. 6.We have taken the largest value of MHC in Eq.

(4.14) in Figs. 6,7.

This value requiresMV larger than 1 ×1017 GeV in order to satisfy the requirement Eq. (4.20), which leadsto MΣ smaller than 3 × 1015 GeV.

Though this case needs a mass splitting among theheavy particles, it is still within two orders of magnitude, and it is completely acceptablephenomenologically.In the minimal supergravity model, the SUSY particle masses are determined mainlyby the universal scalar mass and the gaugino mass.In models where the universalscalar mass dominates the SUSY breaking parameters, squarks and sleptons are almostdegenerate, that are larger than wino. Therefore, models of this type are preferred.

Inthe no-scale model the SUSY breaking is dominated by the gaugino mass [29], whichresults in a definite prediction of the SUSY particle masses (m ˜Q ∼m˜g ≃3m ˜w ∼3m˜L)[35]. Since all the mass parameters in the triangle factor f defined in Eq.

(3.8) areproportional to m ˜w, f behaves as m−1˜w . This enables us to derive the lower bound on m ˜wfrom the nucleon-decay experiments.

We have found that lower limit on m ˜w is 70 GeV(equivalently, m ˜Q > 210 GeV), in the most conservative case (i.e., β = 0.003 GeV3,MHC = 2.3 × 1017 GeV, tan βH = 1, and |1 + ytK| = 0.4).∗Thus, we conclude that theno-scale model is still surviving.†Finally, we briefly discuss the nucleon decay caused by the dimension-six operators,which come from the X and Y gauge-boson exchanges. We show the decay rates by the∗In the absence of the cancellation (i.e., |1 + ytK| = 1), the corresponding limit is m ˜w > 180 GeV.†The wino mass in this region is consistent with the limit m ˜w < 300 GeV [35] in the radiativebreaking scenario of the electroweak symmetry in the no-scale model.23

dimension-six operators in Appendix A. We find that dimension five operators alwaysdominate dimension-six operators, in agreement with the naive expectations.

The ratio(Rτ) of the partial lifetime of the decay n →K0¯νµ via the dimension-five operators tothat of the decay p →π0e+ via the dimension-six operators isRτ≡τ(n →K0¯νµ)τ(p →π0e+)(5.6)≤2.8 × 10−4 ×MHCMV4 1016 GeVMHC!2,where all parameters besides MHC were chosen so that Rτ becomes as large as possible.The ratio of MHC to MV is smaller than 2.0 from Eq. (4.20), and Rτ is always smallerthan160.

The MSGUT predicts that the dimension-five operators should be observedearlier than dimension-six operators.6Future Tests on the Minimal SUSY SU(5) GUTNow we examine how stringent constraint on MSGUT we can obtain from the nucleon-decay experiments in the near future. The superKAMIOKANDE experiment will pushup the lower bound of the nucleon lifetime by a factor of 30.

Meanwhile, the LEP-IIexperiment will be able to find wino below mZ. Since the larger m ˜w means faster nucleondecay as we discussed in previous sections, the combination of these two experimentscan put more stringent limits on the dimension-five operators.In Figs.

8 and 9, we show the expected limits on the dimension-five operators ifsuperKAMIOKANDE does not observe the K¯νµ and π¯νµ decay modes, and if LEP-IIdoes not discover wino below mZ. For the K¯νµ decay modes, superKAMIOKANDE andLEP-II will be able to exclude most of the region with the SUSY particles below 1 TeV,even with the smallest hadron matrix element, β = 0.003 GeV3.

In the case wherethe π¯νµ mode is dominant, the region below 1 TeV is almost closed leaving a littlewindow. Thus one can see that the LEP-II and the superKAMIOKANDE experimentsare extremely important for testing the MSGUT.24

One may be concerned for a little region below 1 TeV which may be left by theseexperiments. This window is open because of the large maximum value of MHC obtainedfrom the gauge coupling unification.

If the error-bar of α3 is reduced by a factor of 2 withthe same central value in the future, the maximum value of MHC becomes 6.1×1016 GeV,and one will be able to close the window completely. In Fig.

10 we demonstrate the casewhere the error-bar of α3 is reduced by a factor of 2. If one wishes to test the MSGUTcompletely, it is quite important to reduce the error-bar of α3.At the end of this section, we see whether the nucleon decay via the dimension-six operators can be observed in the near future.

From Eq. (4.22) and Eq.

(A.2), thetheoretical lower limit of the partial lifetime of the decay p →π0e+ is obtained asτ(p →π0e+) ≥4.1 × 1033 0.03 GeV3α!2years,(6.1)in terms of the hadron matrix element α (=0.003–0.03 GeV3, see Appendix A). SincesuperKAMIOKANDE is expected to reach up to τ(p →π0e+) ≃1034years [36], thereis a possibility to observe the nucleon decay via the dimension-six operators in theMSGUT.7Conclusions and DiscussionsWe have analyzed the nucleon decay in the minimal SUSY SU(5) GUT (MSGUT) indetails.

First, we have studied the GUT-scale particle spectrum using the RG analysis,and found a maximum value of MHC to be 2.3 × 1017 GeV. Then, we have studiedthe nucleon partial lifetimes with the largest possible MHC.

We have found that thepresent nucleon-decay experiments are still consistent with the MSGUT, even withouta cancellation between the matrix elements from exchanges of squarks in different gen-erations. We have emphasized the important role of precise measurement of the gaugecoupling constants, especially the QCD coupling constant α3 to determine the massMHC.

We have also stressed that the combined information of the lower bound on thechargino mass (LEP-II) and on the nucleon lifetimes (superKAMIOKANDE), will give25

a stringent constraint on the MSGUT.It deserves to mention that the nucleon decay prefers supergravity models wherethe SUSY breaking parameters are dominated by scalar masses, rather than by gauginomasses. For example, the no-scale model, where the SUSY breaking parameters comeonly from the gaugino masses, is strongly constrained from the nucleon-decay experi-ments, though it is still viable.

Increasing the limits on the gaugino masses will havestrong impact on the MSGUT phenomenology.It is important to see whether predictions on the dimension-five operators becomedrastically altered by modifying the MSGUT. In the MSGUT, there is a prediction ofmass relation, mb = mτ, ms = mµ, and md = me, at the GUT-scale.It is knownthat, though mb = mτ is consistent with observations if top quark is not too heavy [37],ms = mµ and md = me are not.

Therefore, the modification of the Yukawa couplingstructure is needed.Georgi and Jarlskog proposed that mb = mτ, 3ms = mµ, andmd =13me at the GUT-scale, introducing a 45 dimensional Higgs scalar [38]. Also,Kim and ¨Ozer proposed to make an “effective” 45 dimensional Higgs scalar by usinghigher dimension operators [39].

These modifications may produce more uncertaintiesin the Yukawa couplings of HC and HC to matter multiplets. However, we have checkedthat these models receive at most a few times stronger constraint, and hence the mainconclusion in the present analyses does not change qualitatively.It has been argued that the introduction of a Peccei-Quinn symmetry may be ableto eliminate the dimension-five operators [6].

However, the requirement of the couplingconstant unification does not allow us to introduce new multiplets of arbitrary masses.We have shown in a separate paper [40] that a suppression of the dimension-five operatorsby introducing the Peccei-Quinn symmetry cannot be stronger than in the MSGUT.Another interesting result of our RG analysis on the GUT-scale spectrum is that onecan raise the grand unification scale up to the gravitational scale, by lowering the MΣdown to the intermediate scale (1011−12 GeV). In this case the heavy vector multiplet aswell as HC and HC lie at the gravitational scale, if one allow for two standard deviationsin Eq.

(4.14). These light Σ8 and Σ3 require an extremely small Yukawa coupling fΣ3,26

leading to a very flat potential of Σ.The possibility that the three gauge couplingconstants meet at the gravitational scale has come out with a seemingly accidentalparameter tuning. However, this may suggest a completely different underlying physics(non-GUT) like the superstring theory [41].AcknowledgementWe would like to acknowledge K. Inoue for useful discussions.Note addedAfter completing this work, we have received preprints by R. Arnowitt and P. Nath [42].In these works, they also derive a constraint on MHC, requiring the Yukawa couplingconstants to remain in the perturbative regime.

The difference between their constraintMHC < 3MV and our Eq. (4.20) is due to different requirements.

Namely, they imposethat the Yukawa coupling constant λ should not blow up below 2MHC, while we imposethat below the gravitational scale. However, they are not aware of the possibility toraise MV by lowering MΣ, and hence obtain a smaller upper bound on MHC.

This leadsto the opposite conclusion on the no-scale model, where they claim it to be excludedwhile we have found it still viable.Appendix ADecay Rates via Dimension-six Oper-atorsIn this appendix we analyze the nucleon-decay rates via dimension-six operators. Thedimension-six operators are caused by exchanges of X and Y gauge-bosons or color-triplet Higgs scalars.

The color-triplet Higgs scalars interact with matter only by smallYukawa coupling constants, and also its mass should be larger than 5 × 1015 GeV (see27

Eq. (5.2)).

Thus the dimension-six operators induced by the color-triplet Higgs scalarexchanges are negligible.The dimension-six operators via the X and Y gauge-boson exchanges are dominatedby generation-diagonal decay modes, and the decay modes containing strangeness aresuppressed by the Cabibbo angle sin θC. We discuss here only the dominant decay modep →π0e+.

The amplitude by the X and Y gauge-boson exchanges isLX,Y=AReiφu g25M2Vǫαβγ(dαRuβR)(uγLeL) + (1 + |Vud|2)(dαLuβL)(uγReR). (A.1)The renormalization factor AR calculated by the authors in Ref.

[43] is AR = 3.6.Since the decay rate of this mode caused by the dimension-five operator is extremelysuppressed, the observation of this decay mode would suggest the presence of the X andY gauge-bosons exchanges. We calculate the decay rate using the chiral Lagrangiantechnique, asτ(p →π0e+) = 1.1 × 1036 ×MV1016 GeV4 0.003 GeV3α!2years.

(A.2)Here, α is the hadron matrix elementαuL(⃗k) ≡ǫαβγ⟨0|(dαRuβR)uγL|p,⃗k⟩,(A.3)whose absolute value is the same as β (i.e., |α| = |β|) [21]. It is straightforward tocompare it with the nucleon-decay rates via the dimension-five operators.

The result isgiven in Eq. (5.6).Appendix BRenormalization FactorsIn this appendix, we present several formulae necessary to compute the renormalizationfactors which have appeared in the section 3.

There are three kinds of renormalizationeffects. First, we have to derive the Yukawa coupling constants of HC and HC from theobserved quark masses.

Second, the dimension-five operators derived at the GUT-scalereceive anomalous dimensions from the wave-function renormalizations of the external28

fields. Third, the four-fermi operators dressed at the SUSY breaking scale should berenormalized down to 1 GeV.

In the text the short-range renormalization factor betweenthe SUSY breaking and the GUT-scales is denoted as AS, and the long-range renormal-ization factor between 1 GeV and the SUSY breaking scale as AL. Though AS and ALare estimated by the authors in Ref.

[7], we need minor corrections to their calculationof AS. We demonstrate the derivation of AS, and also comment on AL.Since the Yukawa coupling constants in Eq.

(3.1) are those given at the GUT-scale,we have to calculate the values of the Yukawa coupling constants by solving the RGequations from the SUSY breaking scale up to the GUT-scale. This was done in theRef.

[7]. Since the Yukawa couplings Hf and Hf are F-terms, what we have to computeis only the wave-function renormalizations of each chiral superfields thanks to the non-renormalization theorem of the F-terms [34].

The authors of Ref. [44] have given thefollowing general formula for the running of arbitrary F-terms.

Any F-terms can bere-written in the following form,W = 13!hklmφkφlφm,(B.1)where φ’s are chiral superfields, and the coupling constant hklm are supposed to becompletely symmetric under the interchange of the indices k, l, m. If the fields φk’sbelong to the representation Rk of certain gauge group i, the coupling constants hklmfollow the RG equationdhklmd ln µ =1(4π)2hθkk′hk′lm + θll′hkl′m + θmm′hklm′i,(B.2)where the constants θ’s are given byθkk′ = −2Ci2(Rk)g2i δkk′ + 12!hkpqh∗k′pq. (B.3)The gauge coupling constant is denoted by gi, and the second Casimir of the represen-tation Rk by Ci2(Rk).

If the coupling constant hklm can be neglected in the expressionof θ, the RG equation can be easily integrated to the formhklm(µ) = hklm(µ0)Yr=k,l,mYi=1,2,3 αi(µ0)αi(µ)!Ci2(Rr)/bi. (B.4)29

Here, the coefficient of the β-function bi is defined asdα−1id ln µ = −bi2π. (B.5)The explicit forms of bi’s in the minimal SUSY standard model are given in Eq.

(4.11).Thus, in our case, the Yukawa coupling constants of the lower generations will be renor-malized from the SUSY breaking to the GUT-scales byhu(MGUT)=hu(mZ) α1(mZ)α5(MGUT)!13/198 α2(mZ)α5(MGUT)!3/2 α3(mZ)α5(MGUT)!−8/9,f d(MGUT)=f d(mZ) α1(mZ)α5(MGUT)!7/198 α2(mZ)α5(MGUT)!3/2 α3(mZ)α5(MGUT)!−8/9,(B.6)f e(MGUT)=f e(mZ) α1(mZ)α5(MGUT)!3/22 α2(mZ)α5(MGUT)!3/2.We have assumed the SUSY breaking scale to be mZ. Here, hu, f d, f e are the Yukawacoupling constants of Hf and Hf to up-, down-type quark and charged-lepton multiplets.Note that the expressions differ from those given in Ref.

[7].‡The most importantdifference from the previous analysis is that the formula in Ref. [7] does not have thefactors of α2, and hence they overestimated the Yukawa coupling constants at the GUT-scale.Another ingredient which has not been included in Ref.

[7] is the large top-quarkYukawa coupling constant. Its effect appears in two points.

First, the large Yukawa cou-pling constant contributes to the wave-function renormalization of the Higgs multiplet.If top quark becomes heavy, its large Yukawa coupling constant cannot be neglected inthe RG equation of the Yukawa coupling constants of first and second generations. Itseffect is to enhance the Yukawa coupling constants at the GUT-scale , and hence en-hancing the dimension-five operators.

Second, the top-quark Yukawa coupling constant‡The authors in Ref. [7] seem to have considered the renormalization of the mass operators ratherthan the Yukawa coupling constants.

Since what determines the coefficient of the dimension-five op-erators is the Yukawa coupling constants themselves rather than the mass parameters induced by theSU(2)×U(1) breaking, it is clear that one should consider the running of the Yukawa coupling constantsup to the GUT-scale.30

itself become quite large at the GUT-scale, due to the wave-function renormalizationof the top-quark multiplet. This further enhances the dimension-five operators for thethird-generation contribution.

Since the RG equation cannot be solved analytically ifone includes the top-quark Yukawa coupling constant, we can only present the numericalresults.From the Yukawa coupling constants at the GUT-scale, we obtain the coefficientof the dimension-five operators in Eq. (3.1).

The dimension-five operators receive theanomalous dimensions, which we have to estimate to know their coefficients at theSUSY breaking scale. Here again, since the dimension-five operators are F-terms, allthe renormalization effects come from the wave-function renormalizations of the externallines.

We obtain the same result as that given in Ref. [7].

The QQQL operators includingonly first- and second-generation fields in Eq. (3.1) receive an enhancement factor α1(mZ)α5(MGUT)!−1/33 α2(mZ)α5(MGUT)!−3 α3(mZ)α5(MGUT)!4/3.

(B.7)The enhancement factor of the operators including the third-generation fields is a littlereduced by the effect of the top-quark Yukawa coupling.We show the numerical values of the coefficient AS(i, j, k), the short-range renormal-ization effect between the GUT- and the SUSY breaking scales. First, if the contributionof the top-quark Yukawa coupling constant is neglected, this value becomesAS= α1(mZ)α5(MGUT)!7/99 α3(mZ)α5(MGUT)!−4/9=0.59.

(B.8)Thus our short range renormalization factor AS is smaller than that in Ref. [7] by 23(their value is 0.91).

Next, we show the numerical values of the coefficient AS in Fig. 11for varying mt/√2 sin βH.

Since AS depend on the top-quark Yukawa coupling constantrather than mt, only a combination mt/√2 sin βH is relevant. The solid line representsAS for the dimension-five operators only with first- and second-generation fields, and thedash-dotted line for the operator (QtQt)(QcLµ).

The lower horizontal line is AS with31

the top-quark contribution neglected. One can see that the top-quark Yukawa couplingenhances the dimension-five operators.The factor AL in Eq.

(3.12) is the long-range renormalization factor due to the QCDinteraction between the SUSY scale and 1 GeV scale, and contain the renormalizationof Yukawa coupling constants and the anomalous dimension of four fermi-operators. Itis given in Ref.

[7],AL= α3(1 GeV)α3(mc)!−2/3 α3(mc)α3(mb)!−18/25 α3(mb)α3(mZ)!−18/23=0.22(B.9)Combining all the renormalization effects, the four-fermi operators can be written downas in Eq. (3.12).Appendix CChiral Lagrangian TechniqueIn this appendix we present a chiral Lagrangian technique for translating operators atthe quark level to those at the hadron level.

This technique has been developed in theRefs. [19, 20]The baryon number violating four-fermi operators derived from the dimension-fiveoperators areO(duuνi)=ǫαβγ(dαLuβL)(dγLνiL),O(dudei)=ǫαβγ(dαLuβL)(uγLeiL),O(sudνi)=ǫαβγ(sαLuβL)(dγLνiL),(C.1)O(dusνi)=ǫαβγ(dαLuβL)(sγLνiL),O(suuei)=ǫαβγ(sαLuβL)(uγLeiL),where quark and lepton fields are in mass eigenstates, and i denotes the generationindices.

There are also baryon number violating four-fermi operators derived from thedimension-six operators, and we will concentrate only on the operators relevant for the32

decay mode p →π0e+,˜O(1)=ǫαβγ(dαRuβR)(uγLeL),˜O(2)=ǫαβγ(dαLuβL)(uγReR). (C.2)The effective Lagrangian Lq at the quark level for each decay modes can be written asLq(n, p →π(η)¯νi)=C(duuνi)O(duuνi),Lq(n, p →π(η)e+i )=C(dudei)O(dudei),Lq(n, p →K¯νi)=C(sudνi)O(sudνi) + C(dusνi)O(dusνi),Lq(n, p →Ke+i )=C(suuei)O(suuei),Lq(n, p →πe+i )=˜C(1) ˜O(1) + ˜C(2) ˜O(2).

(C.3)The coefficients C’s are derived from Eq. (3.12), and ˜C’s from Eq.

(A.1),C(duuνi)=4α22MHCmcmdieiφcV ∗udiV 2cdALAS(c, u, di)m2W sin 2βH(1 + ytπ)(f(u, d) + f(u, e)),C(dudei)=2α22MHCmumdieiφuV ∗cdiVcdALAS(u, c, di)m2W sin 2βH(f(u, d) + f(d, ν)),C(sudνi)=C(dusνi)=2α22MHCmcmdieiφcV ∗udiVcdVcsALAS(c, u, di)m2W sin 2βH(1 + ytK)(f(u, d) + f(u, e)),C(suuei)=2α22MHCmumdieiφuV ∗cdiVcsALAS(u, c, di)m2W sin 2βH(f(u, d) + f(d, ν)),˜C(1)=eiφu g25ARM2V,˜C(2)=eiφu g25ARM2V(1 + |Vud|2). (C.4)We need to translate these effective Lagrangians at the quark level Lq to the operatorswritten in terms of the baryon and meson fields Lh .Let us review the chiral Lagrangian for baryons and mesons.

The Nambu-Goldstonebosons Φ associated with the spontaneous breaking of chiral SU(3)L×SU(3)R symmetrycan be written byU=exp 2iΦfπ! (C.5)33

where fπ is the pion decay constant, andΦ =q12π0 +q16ηπ+K+π−−q12π0 +q16ηK0K−¯K0−q23η. (C.6)Similarly, the baryon fields can be written in the matrix form,B =q12Σ0 +q16ΛΣ+p+Σ−−q12Σ0 +q16Λn0Ξ−Ξ0−q23Λ.

(C.7)Now the most general SU(3)L × SU(3)R invariant Lagrangian for strong interactions ofmesons and baryons isL0=18f 2πTr(∂U)(∂U†) + Tr ¯B(i̸∂−Minv)B+12iTr ¯Bγµ hζ(∂µζ†) + ζ†(∂µζ)iB+12iTr ¯BγµBh(∂µζ)ζ† + (∂µζ†)ζi−12i(D −F)Tr ¯Bγµγ5Bh(∂µζ)ζ† −(∂µζ†)ζi+12i(D + F)Tr ¯Bγµγ5hζ(∂µζ†) −ζ†(∂µζ)iB,(C.8)whereζ = exp"iMfπ#. (C.9)Since the (current) quark masses that break the chiral symmetry are small for up, down,and strange quarks, we can use L0 to estimate the lifetimes of nucleon.Now we translate the effective Lagrangians containing quark fields Lq to the onescontaining baryons and mesons Lh, by comparing the transformation properties underthe SU(3)L × SU(3)R symmetry.

The transformation properties of the baryon numberviolating operators given above are,O(duuνi), O(dudei)as(8, 1),34

O(sudνi), O(suuei)as(8, 1),O(sudνi)as(8, 1),˜O(1)as(3, 3∗),˜O(2)as(3∗, 3). (C.10)Thus, the four-fermi operators translating as (8,1) can be expressed in terms of thebaryon and meson fields with a dimensionful constant β,Lh(n, p →π(η)¯νi)=βC(duuνi)νdLTr[P1ζBLζ†] + h.c.,Lh(n, p →π(η)e+i )=βC(dudei)edLTr[P2ζBLζ†] + h.c.,Lh(n, p →K¯νi)=βC(sudνi)νdLTr[P3ζBLζ†] + βC(dusνi)νdLTr[P4ζBLζ†] + h.c.,Lh(n, p →Ke+i )=βC(suuei)edLTr[P5ζBLζ†] + h.c,(C.11)and the ones translating as (3, 3∗) or (3∗, 3) with a dimensionful constant α,Lh(n, p →πe+)=α ˜C(1)edLTr[P6ζBLζ] + α ˜C(2)edRTr[P7ζ†BRζ†] + h.c, (C.12)with the coefficients C’s and ˜C’s defined in Eq.

(C.4). In the above formulae, Pi areprojection operators,P1 =000000010,P2 =000000100,P3 =0000−10000,P4 =000000001,P5 =000−100000,P6 =000000100,P7=000000100.

(C.13)35

The constants α and β are the same as those defined in Eqs. (A.3), (3.19) [19].

Whenone estimates the decay rates of the nucleons, one needs to include the virtual baryonexchanges.For example, to estimate the lifetimes of decay modes n, p →K¯ν, oneshould add the contributions from diagrams with virtual Σ and Λ exchanges. We showthe chiral Lagrangian factors of each decay modes in Table 5.

We took mB ≡mΣ =mΛ = 1150 MeV, D = 0.81, F = 0.44 in Tables 1–3.Here we have some comments on the chiral Lagrangian factors in the decay rates ofthe nucleons. First, the ratio of K¯ν decay rates of neutron and proton isΓ(n →K0¯ν)Γ(p →K+¯ν)=2 + 2mnmB F21 + mpmB (D + F)2=1.8.

(C.14)This shows that the decay rate of the neutron is larger than that of the proton. Thesituation is different in π¯ν mode.

The decay rate of the mode p →π+¯ν is two timeslarger than that of n →π0¯ν.However, we use n →π0¯ν in section 5 because theexperimental lower bound on τ(n →π0¯ν) is longer than that on τ(p →π+¯ν).Second, the decay rates into η of dimension-five operators are as large as those intoπ0. For example, the ratio in ¯ν decay mode isΓ(n →η¯ν)Γ(n →π0¯ν)=(m2n −m2η)2m4n31 −13(D −3F)2|1 + D + F|2=0.35.

(C.15)Recall that the decay rates into η from dimension-six operators is negligible [19]. Thusthis mode may be interesting.The results in this appendix depend on the parameters α and β.

However, they aresensitive to hadron dynamics, and they differ for each hadron models. Even among thelattice calculations, the results vary from 0.03 GeV3 [45] to 0.0056 GeV3 [22].36

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Table 1The prediction of the nucleon partial lifetimes for the dominant decay modes, arisingfrom the dimension-five operators (QcQc)(QuLµ) and (QtQt)(QuLµ). This class of decaymodes depends on the parameter ytK.

The mass degeneracy m˜c = m˜u and m˜µ = m˜eis assumed. The function f is defined in Eq.

(3.8) and depends on the SUSY particlemasses. See the text for other variables.τ(p →K+¯νµ)=6.9 × 1031τ(p →K+¯νe)=1.4 × 1033τ(n →K0¯νµ)=3.9 × 1031τ(n →K0¯νe)=7.7 × 1032×0.003 GeV3β0.67ASsin 2βH1 + ytKMHC1017 GeVTeV−1f(u, d) + f(u, e)2yrsTable 2The prediction of the nucleon partial lifetimes for the next-leading decay modes, arisingfrom the dimension-five operators (QcQc)(QuLµ) and (QtQt)(QuLµ).

This class of modesdepends on the parameter ytπ. The mass degeneracy m˜c = m˜u and m˜µ = m˜e is assumed.The function f is defined in Eq.

(3.8) and depends on the SUSY particle masses. Seethe text for other variables.τ(p →π+¯νµ)=1.4 × 1032τ(p →π+¯νe)=2.9 × 1033τ(n →π0¯νµ)=2.9 × 1032τ(n →π0¯νe)=5.7 × 1033τ(n →η0¯νµ)=8.2 × 1032τ(n →η0¯νe)=1.6 × 1034×0.003 GeV3β0.67ASsin 2βH1 + ytπMHC1017 GeVTeV−1f(u, d) + f(u, e)2yrs41

Table 3The prediction of the nucleon partial lifetimes for the decay modes which depend neitheron the parameter ytK nor ytπ. The relevant dimension-five operator is (QuQu)(QcLµ)alone.

The mass degeneracy m˜c = m˜u and m˜νµ = m˜νe is assumed. The function f isdefined in Eq.

(3.8) and depends on the SUSY particle masses. See the text for othervariables.τ(p →K0µ+)=1.0 × 1035τ(p →π0µ+)=2.0 × 1035τ(p →η0µ+)=5.7 × 1035τ(n →π−µ+)=9.9 × 1034×0.003 GeV3β0.67ASsin 2βHMHC1017 GeVTeV−1f(u, d) + f(d, ν)2yrsTable 4The experimental lower bounds on the nucleon partial lifetimes at the 90% C.L.

[18].τ(p →K+¯ν)>1.0 × 1032yrsτ(n →K0¯ν)>8.6 × 1031yrsτ(p →π+¯ν)>2.5 × 1031yrsτ(n →π0¯ν)>1.0 × 1032yrsτ(n →η¯ν)>5.4 × 1031yrsτ(p →K0µ+)>1.2 × 1032yrsτ(p →π0µ+)>2.7 × 1032yrsτ(p →ηµ+)>6.9 × 1031yrsτ(n →π−µ+)>1.0 × 1032yrsτ(p →π0e+)>5.5 × 1032yrs42

Table 5Chiral Lagrangian factors in the nucleon-decay matrix elements. For notations, see thetext.Γ(p →K+¯νi)=(m2p −m2K)232πm3pf 2πC(sudνi) 2mp3mBD + C(dusνi)1 + mp3mB(D + 3F)2Γ(n →K0¯νi)=(m2p −m2K)232πm3pf 2πC(sudνi)1 −mn3mB(D −3F)+ C(dusνi)1 + mp3mB(D + 3F)2Γ(p →π+¯νi)=mp32πf 2π|C(duuνi) [1 + D + F]|2Γ(n →π0¯νi)=mn64πf 2π|C(duuνi) [1 + D + F]|2Γ(n →η¯νi)=(m2n −m2η)264πm3nf 2π3C(duuνi)1 −13(D −3F)2Γ(p →K0e+i )=(m2p −m2K)232πm3pf 2πC(suuei)1 −mpmB(D −F)2Γ(p →π0e+i )=mp64πf 2π|C(dudei) [1 + D + F]|2Γ(p →ηe+i )=(m2p −m2η)264πm3pf 2π3C(dudei)1 −13(D −3F)2Γ(n →π0e+i )=mn32πf 2π|C(dudei) [1 + D + F]|2Γ(p →π0e+)=mp64πf 2π ˜C(1)2 + ˜C(2)2[1 + D + F]243

Figure CaptionsFig. 1 A supergraph contributing to the dimension-five operators of the nucleon decay.Fig.

2 Allowed ranges on the color-triplet Higgs mass MHC and the “GUT-scale” MGUT ≡(M2V MΣ)1/3 obtained from the renormalization group analysis (thick lines), byvarying m˜h and m˜g between 100 GeV and 1 TeV. MHC depends only on m˜h, andMGUT only on m˜g.

We use the gauge coupling constants at the weak-scale given inthe text. Also shown are the ranges with an improved measurement on the strongcoupling constant, α3 = 0.118 ± 0.0035 (thin lines).Fig.

3 Lower bound on MHC derived from the nucleon-decay experiments. The horizontalaxis represents |1+ytK|, the sum of the second- and third-generation contributionsnormalized by the second-generation one.

The vertical axis corresponds to MHC.The shaded region is excluded. The upper curve corresponds to the hadron matrixelement β = 0.03 GeV3, the lower one to β = 0.003 GeV3.

The experimental limitscome from the mode n →K0¯νµ for |1 + ytK| > 0.4, and from the mode n →π0¯νµfor |1 + ytK| < 0.4 and |1 + ytπ| = 1. The short-range renormalization factor ASis taken to be AS = 0.67.

The maximum value on MHC(= 2.3 × 1017 GeV) fromthe renormalization-group (RG) analysis requiring gauge coupling unification (seesection 4) is also shown.Fig. 4 The dependence of the lower bound of MHC on the parameters m ˜Q and m ˜w.

Thedashed line shows the dependence on m ˜w taking m ˜Q = 1 TeV. The dash-dotted lineshows the dependence on m ˜Q when m ˜w = 45 GeV.

In both curves we have takenthe most conservative set of parameters, tan βH = 1, |1 + ytK| < 0.4, |1 + ytπ| = 1,AS = 0.67, and β = 0.003 GeV3. We have assumed m˜L ≃m ˜Q.

The maximumvalue on MHC(= 2.3 × 1017 GeV) from the renormalization-group (RG) analysisrequiring gauge coupling unification (see section 4) is also shown.Fig. 5 The dependence of the lower bound of MHC on tan βH.

The upper curve is obtainedwith |1 + ytK| = 1, and the lower curve with |1 + ytK| < 0.4 and |1 + ytπ| = 1. We44

have taken m ˜Q = m˜L = 1 TeV, m ˜w = 45 GeV, AS = 0.67 and β = 0.003 GeV3.The maximum value on MHC(= 2.3 × 1017 GeV) from the renormalization-group(RG) analysis requiring gauge coupling unification (see section 4) is also shown.Fig. 6 The limits on m ˜w and m ˜Q from the KAMIOKANDE nucleon-decay experiments, inthe absence of the cancellation between second- and third-generation contributions(i.e., |1 + ytK| = 1).

The most conservative parameters, MHC = 2.3 × 1017 GeV,tan βH = 1, β = 0.003 GeV3, and AS = 0.67, are used. We have assumed m ˜Q ≃m˜Lfor simplicity.

The shaded region is excluded. Also shown are the limits from thedirect search experiments on wino and squarks at LEP and CDF.Fig.

7 The same as in Fig. 6, but allowing the cancellation between second- and third-generation contributions (i.e., |1 + ytK| < 0.4, |1 + ytπ| = 1).Fig.

8 The same as in Fig. 6, but with an improved constraint by a factor of 30 expectedat superKAMIOKANDE.

The expected limit on m ˜w from the LEP-II experiment(m ˜w > 90 GeV) is also shown.Fig. 9 The same as in Fig.

7, but with an improved constraint by a factor of 30 expectedat superKAMIOKANDE. The expected limit on m ˜w from the LEP-II experiment(m ˜w > 90 GeV) is also shown.Fig.

10 The same as in Fig. 9.

We have assumed that the error-bar of α3 is reduced bya factor of 2 with the same central value, leading to a stronger upper bound onMHC(< 6.1 × 1016 GeV) (see Fig. 2).Fig.

11 The renormalization factor AS vs. mt/√2 sin βH. The solid line represents the ASfor the dimension-five operators only with first- and second-generation fields, anddash-dotted line for the operator (QtQt)(QcLµ).

The upper horizontal line is ASderived by the authors in Ref. [7].

The lower horizontal line is AS which does notcontain the contribution of the top-quark Yukawa coupling (i.e., mt = 0).45

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출처: arXiv:9207.279 • 원문 보기