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the grand unified theories ( guts ) , which embed the standard model ( sm ) gauge group into a large single gauge group , are quite attractive , and so a variety of models of the theories are proposed since the earliest work based on the su(5 ) symmetry group was presented by georgi and glashow in 1974 @xcite . among them , the supersymmetric grand unified theories ( susy guts ) are considered to be promising candidates since they realize the gauge coupling unification with great accuracy @xcite as well as solving the hierarchy problem in the guts @xcite . the supersymmetric version of the georgi - glashow su(5 ) gut , which is the simplest among the susy guts , is called the minimal susy su(5 ) gut @xcite . one of the most distinctive features which guts predict is the existence of baryon - number violating interactions , such as proton decay . in the susy guts with @xmath6 parity , for example , proton decay is induced by two different processes : the colored - higgs and the @xmath7-boson exchanging processes . the colored higgs triplets are introduced for the higgs doublets in the minimal supersymmetric standard model ( mssm ) to be incorporated into the su(5 ) multiplets . the colored - higgs exchange yields baryon - number violating dimension - five operators , which give rise to the dominant contribution to proton decay in the minimal susy su(5 ) guts @xcite . on the other hand , @xmath7 bosons are the gauge bosons in the unified su(5 ) gauge group , and they induce proton decay through the dimension - six operators . since there has been no experimental signal of proton decay so far , strong limits are imposed on these interactions , especially on the former one . super - kamiokande gives the bounds on the proton lifetime : @xmath8  yrs with 219.7 kt - yr of data @xcite and @xmath9  yrs with 172.8 kt - yr of data at 90% confidence level @xcite . these bounds are so stringent that the minimal su(5 ) susy gut is excluded since the colored - higgs - boson exchange process yields too short lifetime in the @xmath10 channel @xcite . this prediction is , however , quite model - dependent . in fact , many models are proposed in order to allow them to evade the experimental constraints . such attempts @xcite are accomplished by pushing the mass of the colored higgs much heavier than the gut scale and/or suppressing the dimension - five operators by a certain symmetry , _ e.g. _ , the peccei - quinn ( pq ) symmetry @xcite in the four - dimensional guts . in the extra - dimensional models , the @xmath6 symmetry is introduced to suppress the dimension - five proton decay @xcite . with the extensions , the @xmath7-boson exchanging interactions become dominant and the proton lifetime turns out to be long enough to avoid the current limit . in this work , we assume such mechanism works and concentrate on the proton decay via the @xmath7-boson exchange .    nowadays , the weak - scale supersymmetry itself is severely constrained by the experiments at the large hadron collider ( lhc ) . since there has been no signal of susy particles , the atlas and the cms collaborations give severe limits on their masses , especially those of colored particles @xcite . moreover , the collaborations have recently announced that they detected signatures of higgs boson for the first time @xcite . according to their results , the mass of higgs boson is about 125 gev . the result is welcome to the mssm , which predicts the light higgs boson , while the substantial radiative corrections are required in order to raise the mass of higgs boson up to 125 gev . since the corrections are increased as the susy particles become heavy , this situation implies that the susy scale might be higher than the electroweak scale @xcite , and challenges natural solutions to the hierarchy problem . these results give constraints on models of the susy - breaking mediation . the low - energy gauge mediation model may be required to introduce several messenger multiplets so that the susy particles are heavy enough @xcite . on the other hand , there have been several alternatives to explain the 125 gev higgs boson in an extension of the mssm . introducing vector - like supermultiplets to the mssm is one of the simplest way to accomplish the purpose @xcite . in this case , the quantum effects by the extra multiplets help to increase the higgs - boson mass . with the multiplets being a representation of the grand unified group , the perturbative gauge coupling unification is still preserved with great accuracy . the mssm with vector - like multiplets and high - scale susy models modifies the proton decay rates . in this work , we examine it in detail . we will find that vector - like matters enhance the proton decay rate , while heavy sfermions extend its lifetime . this investigation indicates that the proton - decay experiments might provide a hint for the low - energy structure of the susy guts . we assume that the susy guts are realized in the four - dimensional spacetime , and not consider the extra - dimensional susy guts in which the gauge symmetry is broken by the boundary conditions . in the latter models , the main decay modes of proton depend on configuration of quarks and leptons in the extra - dimensional space @xcite . however , the decrease / enhancement of the proton lifetime we show are almost universal even in those models . this paper is organized as follows . we begin by reviewing the susy su(5 ) guts in sec . [ msgut ] , and discuss the proton decay via the @xmath7-boson exchange in the subsequent section . then , we evaluate the lifetime of proton decay in the case of the mssm with vector - like matters and heavy sfermions in sec . [ vectorlike ] and sec . [ heavy_susy ] , respectively . section [ conclusion ] is devoted to conclusion . in appendix , we present formulae for evaluating the long - distance part of the renormalization factors . first , we review the susy su(5 ) guts in order to illustrate the notation and conventions which we use in this article . just like the georgi - glashow su(5 ) model @xcite , we assume that the sm fermions , as well as their superpartners , are embedded in a @xmath11 representation . the multiplets @xmath12 and @xmath13 , which are the matter fields of @xmath14 and @xmath15 representations , respectively , are identified as @xmath16 where all of the component fields are expressed in terms of the left - handed chiral superfields , and primes show that the above fields are the gauge eigenstates . the super / sub scripts of @xmath17 , @xmath18 , @xmath19 , and @xmath20 denote a color index with @xmath21 . the chiral superfields @xmath22 and @xmath23 in @xmath12 as well as @xmath17 and @xmath18 in @xmath13 form the su(2)@xmath24 doublets , respectively , @xmath25 while @xmath19 , @xmath20 , and @xmath26 are the su(2)@xmath24 singlets . the su(5 ) gauge theory contains the 24 gauge bosons and each of them corresponds to a component of a vector superfield , @xmath27 , with @xmath28 indicating the gauge index . by exploiting the fundamental representation of the su(5 ) generators , @xmath29 , we define a @xmath30 matrix of the vector superfields : @xmath31 . the components of the matrix are written as @xmath32 where each component is expressed by the same symbol as that used for the corresponding gauge field . we collectively refer to @xmath33 , @xmath34 , and their hermitian conjugates as @xmath7 and @xmath35 bosons , and use the following notation for them : @xmath36 here @xmath37 denote the isospin indices . the higgs superfields in the mssm , on the other hand , are incorporated into a pair of fundamental and anti - fundamental fields as @xmath38 where the last two components are corresponding to the mssm higgs superfields , @xmath39 and the new higgs superfields @xmath40 and @xmath41 are called the higgs color triplet superfields . the superpotential for the yukawa couplings of quarks and leptons is given as @xmath42 where @xmath43 indicate the generations and @xmath44 represent the su(5 ) indices . the yukawa couplings @xmath45 and @xmath46 in eq . ( [ superpotential ] ) have redundant components and much of the degree of freedom is eliminated through the field re - definition of @xmath13 and @xmath12 @xcite . we parametrize the couplings according to ref .  @xcite as @xmath47 with @xmath48 the kobayashi - maskawa matrix . the phase factors @xmath49 are subject to a condition : @xmath50 with the parameters , we express the matter fields in terms of the mass eigenstates as follows : @xmath51 @xmath52 in the following discussion , we express interactions in the basis of mass eigenstates unless otherwise noted .    here , we do not take specific assumptions for symmetry breaking of su(5 ) and the mass generation of the colored higgs . they are related to suppression of the proton decay induced by dimension - five operators . next we discuss the proton decay rate induced by the @xmath7 and @xmath35 boson exchange . the couplings between the gauge bosons and the sm fermions are given as @xmath53~,\end{aligned}\ ] ] where @xmath54 is the su(5 ) gauge coupling constant and @xmath55 indicates the charge conjugation . @xmath56 and @xmath57 denote the second and third rank totally antisymmetric tensors , respectively . this interaction lagrangian causes proton decay . in the present case , the dominant decay mode is @xmath58 , and this decay process is induced by the following effective lagrangian : @xmath59~. \label{effective_lagrangian}\end{aligned}\ ] ] where @xmath60 is the mass of @xmath7 and @xmath35 bosons and @xmath61 . the renormalization effects resulting from the anomalous dimensions of the operators are represented by @xmath62 and @xmath63 , which we will evaluate below . the hadron matrix elements of the operators are evaluated by using the ordinary chiral lagrangian method  @xcite : @xmath64p_lu^\gamma     = \frac{\alpha_{\rm h}}{\sqrt{2}f_\pi}(1+d+f)p_lu_p(\bm{p},s)~,\ ] ] and @xmath65p_ru^\gamma     = \frac{\alpha_{\rm h}}{\sqrt{2}f_\pi}(1+d+f)p_ru_p(\bm{p},s)~,\ ] ] where @xmath66 is the pion decay constant , and @xmath67 is defined by the equation @xmath68p_lu^\gamma     = \alpha_{\rm h}p_lu_p(\bm{p},s)~,\ ] ] with @xmath69 the charge conjugation matrix . the value of @xmath67 is computed in ref .  @xcite as @xmath70 at the renormalization scale @xmath71 gev . by using the matrix elements , we obtain the partial decay width @xmath72 induced by the effective lagrangian in eq . ( [ effective_lagrangian ] ) : @xmath73~ , \label{decay_width}\end{aligned}\ ] ] where @xmath74 and @xmath75 are the masses of proton and the neutral pion , respectively , and @xmath76  . in the following calculation , we take @xmath77 , @xmath78 , and @xmath79 as in ref .  @xcite .    now , in order to evaluate the proton decay rate , all we have to do is to determine the unified gauge coupling constant @xmath80 , the @xmath7 boson mass @xmath60 , and the renormalization factors @xmath81 and @xmath82 . they are dependent on each gut model . the unified gauge coupling constant @xmath80 is computed by solving the renormalization group equations ( rges ) for the gauge coupling constants @xmath83 ( @xmath84 ) in the sm gauge interactions . in this article , we consider the gauge coupling running up to the two - loop level , and exploit the @xmath85 renormalization scheme in order to respect the supersymmetry @xcite . furthermore , we adopt the definition of @xmath80 as @xmath86 with @xmath87  gev .    in the mssm , the two - loop renormalization group equations ( rges ) for the gauge coupling constants are given as @xcite @xmath88~,\ ] ] where @xmath89 and @xmath90 with @xmath91 the yukawa couplings . here , we use the su(5 ) normalization for the u(1 ) hypercharge . since the yukawa couplings enter into the two - loop level contributions to the gauge coupling rges , it is sufficient to consider the rges for the yukawa couplings at one - loop level . they are given as @xmath92 , \nonumber \\ \mu\frac{\partial}{\partial \mu}y_b & = \frac{1}{16\pi^2}y_b \biggl[6y_b^2+y_t^2+y_\tau^2   -\frac{7}{15}g^2_1 - 3g_2 ^ 2-\frac{16}{3 }   g_3 ^ 2\biggr ] , \nonumber \\   \mu\frac{\partial}{\partial \mu}y_\tau & = \frac{1}{16\pi^2}y_\tau \biggl[3y_b^2 + 4y_\tau^2 -\frac{9}{5}g^2_1 - 3g_2 ^ 2\biggr ] .\end{aligned}\ ] ]    in the sm , on the other hand , the coefficients for the gauge coupling beta functions are  @xcite @xmath93 and @xmath94 the running of the yukawa couplings in this case is given as follows : @xmath95 , \nonumber \\ \mu\frac{\partial}{\partial \mu}y_b&=\frac{1}{16\pi^2}y_b \biggl[\frac{3}{2}y_t^2+\frac{9}{2}y_b^2 + y_\tau^2   -\frac{1}{4}g^2_1-\frac{9}{4}g_2 ^ 2 - 8   g_3 ^ 2\biggr ] , \nonumber \\    \mu\frac{\partial}{\partial \mu}y_\tau&=\frac{1}{16\pi^2}y_\tau \biggl[3y_t^2 + 3y_b^2+\frac{5}{2}y_\tau^2   -\frac{9}{4}g^2_1-\frac{9}{4}g_2 ^ 2\biggr ] .\end{aligned}\ ] ] modifications in the coefficients in particular models are mentioned to in the following sections . now we deal with the renormalization factors , @xmath81 and @xmath82 . the factors are expressed as the product of the long- and short - distance factors , _ i.e. _ , @xmath96 where @xmath97 and @xmath98 represent the long- and short - distance factors , respectively . the long - distance contribution @xmath97 is common to @xmath62 and @xmath63 , and independent of the high - energy physics . its value is evaluated as @xmath99 at two - loop level . details of the calculation are given in appendix . the short - distance factors are , on the other hand , model - dependent quantities . for instance , if there is no threshold between the electroweak and the gut scales they are evaluated at one - loop level in refs .  @xcite as follows : @xmath100^{-\frac{\gamma_3}{b_3^{(1)}}}\biggl [ \frac{\alpha_2(m_z)}{\alpha_2(m_{\rm gut } ) } \biggr]^{-\frac{\gamma_2}{b_2^{(1)}}}\biggl [ \frac{\alpha_1(m_z)}{\alpha_1(m_{\rm gut } ) } \biggr]^{-\frac{\gamma_1^{(1)}}{b_1^{(1)}}}~ , \nonumber \\ a_s^{(2)}&=\biggl [ \frac{\alpha_3(m_z)}{\alpha_3(m_{\rm gut } ) } \biggr]^{-\frac{\gamma_3}{b_3^{(1)}}}\biggl [ \frac{\alpha_2(m_z)}{\alpha_2(m_{\rm gut } ) } \biggr]^{-\frac{\gamma_2}{b_2^{(1)}}}\biggl [ \frac{\alpha_1(m_z)}{\alpha_1(m_{\rm gut } ) } \biggr]^{-\frac{\gamma_1^{(2)}}{b_1^{(1)}}}~,\end{aligned}\ ] ] where , in the mssm , @xmath101 while in the sm , @xmath102 the extension of the result to each case is referred to in the subsequent sections .    for reference , we evaluate the proton - decay lifetime assuming the susy scale to be 1  tev , _ i.e. _ , all the superparticles are assumed to have masses of @xmath103  tev . the result is @xmath104 here we neglect the possible effects of the threshold corrections from particles whose masses are around the gut scale . we also neglect them in the following calculations , since the effects are completely model - dependent . in this section , we discuss the grand unified models in which extra vector - like matters are added into the susy su(5 ) guts . we assume that there exist @xmath105 and @xmath106 pairs of chiral supermultiplets which transform as @xmath107 and @xmath108 representations , respectively , and evaluate the proton decay rate for the cases . for brevity , all of the multiplets are assumed to have the same mass , @xmath109 . as mentioned to in the introduction , such models are motivated by the symptoms of higgs boson with its relatively heavy mass , @xmath110  gev , reported recently at the lhc @xcite ; if the additional multiplets with mass around weak scale couple to the mssm higgs bosons , the higgs mass @xmath111 can be raised on account of the quantum corrections @xcite . the gauge mediation requires the vector - like matters as the susy - breaking messenger . thus , we take a wide parameter range for @xmath109 . the existence of the extra vector - like multiplets modifies the rges for the gauge coupling constants , as well as the short - distance renormalization factors . the beta functions of the gauge coupling constants in the mssm receive additional contributions @xcite : @xmath112 here @xmath113 and @xmath114 are the corrections to the leading and the next - to - leading order contributions for the beta functions of the gauge coupling constants , @xmath115 and @xmath116 , which are defined in eq . ( [ mssm_beta ] ) . here , we ignore the contribution from the yukawa couplings of the vector - like matters at two - loop level for simplicity . from the one - loop contribution in eq . ( [ beta_vec ] ) , it is found that addition of vector - like matters equally changes the running of the gauge couplings , and thus maintains the gauge coupling unification . the corrections to the one - loop contribution , @xmath113 , also modify @xmath117 and @xmath118 . to be concrete , the renormalization effects of the energy scale above @xmath109 are given as @xmath119^{-\frac{4}{3 [ b_3^{(1)}+ \delta b_3^{(1)}]}}\biggl [ \frac{\alpha_2(m)}{\alpha_2(m_{\rm gut } ) } \biggr]^{-\frac{3}{2[b_2^{(1)}+\delta b_2^{(1)}]}}\biggl [ \frac{\alpha_1(m)}{\alpha_1(m_{\rm gut } ) } \biggr]^{-\frac{11}{30[b_1^{(1)}+\delta b_1^{(1)}]}}~ , \nonumber \\   a_s^{(2)}&=\biggl [ \frac{\alpha_3(m)}{\alpha_3(m_{\rm gut } ) } \biggr]^{-\frac{4}{3 [ b_3^{(1)}+ \delta b_3^{(1)}]}}\biggl [ \frac{\alpha_2(m)}{\alpha_2(m_{\rm gut } ) } \biggr]^{-\frac{3}{2[b_2^{(1)}+\delta b_2^{(1)}]}}\biggl [ \frac{\alpha_1(m)}{\alpha_1(m_{\rm gut } ) } \biggr]^{-\frac{23}{30[b_1^{(1)}+\delta b_1^{(1)}]}}~.\end{aligned}\ ] ]    these modifications result in alternations of the proton - decay lifetime . in order to parametrize the alternations , we define the following ratio : @xmath120 where @xmath121 represents the proton - decay lifetime with ( without ) vector - like matters . as seen form eq . ( [ decay_width ] ) , this ratio does not depend on the @xmath7-boson mass but only on @xmath80 , @xmath117 and @xmath118 : @xmath122_{\rm w / o } } { \alpha_5 ^ 2\bigl[\bigl(a^{(1)}_r\bigr)^2 + \bigl(a^{(2)}_r\bigr)^2(1+|v_{ud}|^2)^2 \bigr]_{\rm w/}}~.\ ] ] here , w/ ( w / o ) again implies that the factor is for the case with ( without ) vector - like matters .     as functions of vector - like matter mass . left and right graphs show @xmath123 and @xmath124 in solid lines from top to bottom , respectively . light ( dark ) shaded region is excluded by the current experimental limit , @xmath125  years at 90% confidence level @xcite , in the case of @xmath126  gev ( @xmath127  gev ) . susy scale is set to be 1  tev.,height=245 ]    in fig . [ fig : vectorlike ] , we plot the ratio @xmath6 as functions of the masses of the vector - like multiplets . in the analysis , the susy scale is set to be @xmath128  tev . each solid line in the left graph corresponds to the number of @xmath107 multiplets @xmath129 from top to bottom without @xmath130 multiplets . in the right graph , on the other hand , the solid lines represent the cases with @xmath108 multiplets ( @xmath131 , from top to bottom ) and @xmath132 . the light ( dark ) shaded region is excluded by the current experimental limit , @xmath8  years at 90% confidence level @xcite in the case of @xmath133  gev ( @xmath127  gev ) . on the whole parameter region in this figure , the unified gauge coupling constant , @xmath80 , is less than @xmath134 , thus , perturbativity is maintained . in order to clarify the effects of the renormalization factors on the enhancement of proton decay rate , we also plot the ratio of the factors , @xmath135_{\rm w/}/ [ ( a^{(1)}_r)^2 + ( a^{(2)}_r)^2(1+|v_{ud}|^2)^2 ] _ { \rm w / o}$ ] , as functions of the masses of the vector - like multiplets in fig . [ fig : renormalization ] . here we set @xmath123 from bottom to top and the susy scale to be 1  tev . the renormalization factor in the absence of the vector - like multiplets is calculated as @xmath136_{\rm w / o}=40 $ ] . these results indicate that the mssm with vector - like matters might cause proton decay fast enough to be detected by the current or future experiments . in particular , if there are three @xmath107 vector - like matters or is a @xmath137 multiplet at the tev scale , the susy su(5 ) guts are excluded even by the present limit for @xmath126 gev .     from bottom to top . susy scale is set to be 1  tev.,height=245 ]     as functions of the masses of the multiplets . solid lines represent @xmath138 multiplets of @xmath139 representations from top to bottom . light ( dark ) shaded region is excluded by the current experimental limit , @xmath125  years at 90% confidence level @xcite in the case of @xmath126  gev ( @xmath127  gev ) . susy scale is set to be 1  tev.,height=264 ]    in addition , we calculate the ratio in the case where @xmath140 pairs of @xmath141 multiplets are introduced . this set of representations is the same as those of one generation of chiral matter fields and their corresponding complex representations . in this case , the beta functions are equal to those of @xmath142 in eq . ( [ beta_vec ] ) . after carrying out a similar calculation , we plot the ratio @xmath6 against the masses of the multiplets in fig .  [ fig : gen ] . here , solid lines represent @xmath138 multiplets of @xmath143 from top to bottom . we again set the susy scale to be 1  tev . from this figure , it is found that a pair of extra generations which have masses below @xmath144  gev yield too large proton decay rate to be excluded by the current experimental limit in the case of @xmath126  gev . although the threshold corrections around the gut scale might alter the predicted values of proton lifetime , a growing tendency in the proton decay rate is independent of particular models . therefore , the proton - decay experiments are extremely promising for constraining such scenario . in turn , we discuss the susy gut with sfermions having masses much larger than the tev scale . such a mass spectrum might be realized when the supersymmetry is broken via the anomaly - mediation mechanism @xcite , and there have been a lot of works which examine the scenario @xcite . in this scenario , although a high degree of fine - tuning is inevitable , the susy flavor and cp problems are relaxed owing to heavy masses of sfermions @xcite . the thermal relic scenario of dark matter in the universe is still achieved  @xcite , and the dark matter is to be directly detected in the future experiments @xcite . in addition , this heavy scale susy scenario is also suggested by the recent lhc results since there has been no signal of superparticles and the atlas and cms collaborations provide stringent limits on the masses of colored particles @xcite . moreover , in this case the sm - like higgs boson mass easily reaches @xmath145 gev as the radiative corrections are enhanced due to the heavy colored particles .    in the following discussion , we assume that the masses of squarks and sleptons are around the scale of @xmath146 , which is much higher then the electroweak scale , while the masses of gauginos , higgsinos , and higgs bosons except the lightest one are @xmath147  tev , which is denoted by @xmath148 hereafter . therefore , between @xmath148 and @xmath146 , the latter fields are to be added to the rge analysis . furthermore , we assume that their contributions to the gauge - coupling running through the yukawa couplings , as well as those with sfermions running in the loops , are negligible . hence , the beta functions of the gauge couplings in this region are given as @xmath149 with @xmath150 the same as the sm ones . since each one - loop contribution @xmath115 in the above equation differ from that in eq . ( [ mssm_beta ] ) by the same number ( 2 in this case ) , the perturbative gauge coupling unification is again preserved in the present case .    . solid , dashed , and dotted lines correspond to the cases where @xmath148 is set to be 1 , 3 , and 10  tev , respectively . @xmath7-boson mass is taken to be @xmath151  gev . , height=264 ]    in fig . [ fig : heavysusy ] , we plot the lifetime of proton against @xmath146 . solid , dashed , and dotted lines correspond to the cases where @xmath148 is set to be 1 , 3 , and 10  tev , respectively . the @xmath7-boson mass is taken to be @xmath152  gev in this figure . it is found that the proton - decay lifetime is slightly extended , although the enhancement factor is less than two . the contribution of the renormalization factors to the alternation of the proton lifetime is less significant in this case than that in the case discussed in sec . [ vectorlike ] ; it is at most a few % on the whole parameter region in this figure . compared with the case discussed in the previous section , the present situation does not so much change the proton decay rate . hence , searching for the proton decay is still stimulating even for the heavy susy scenario . we have studied the proton decay rate under the two situations ; one is the mssm with vector - like multiplets , and the other is the mssm with heavy sfermions . it is found that the proton lifetime is significantly reduced in the former case , while in the latter case it is slightly prolonged . in any case , the proton - decay experiments are , together with the lhc experiment and other precision measurements , expected to shed light on the supersymmetric grand unified models , as well as the supersymmetry itself . we thank yasumichi aoki for useful discussion . this work is supported by grant - in - aid for scientific research from the ministry of education , science , sports , and culture ( mext ) , japan , no . 20244037 , no . 20540252 , no . 22244021 and no.23104011 ( jh ) , and also by world premier international research center initiative ( wpi initiative ) , mext , japan . the work of nn is supported by research fellowships of the japan society for the promotion of science for young scientists . in this appendix , we demonstrate the evaluation of the long - distance contribution to the renormalization factors , _ e.g. _ , @xmath97 in eq . ( [ ar ] ) . first , we write down the two - loop renormalization group equations for the strong coupling constant below the electroweak scale : @xmath153 with @xmath154 the strong coupling constant and @xmath155 where @xmath156 denotes the number of quark flavors in an effective theory .    the long - distance factor , @xmath97 , is determined by the ratio of the coefficients for the effective operators at the scale of @xmath157 and 2  gev : @xmath158 with the coefficient @xmath159 satisfying the following rge at two - loop level  @xcite : @xmath160c(\mu)~.\ ] ] the solution of the equation is @xmath161^{-\frac{2}{b_1 } } \biggl [ \frac{4\pi b_1+b_2\alpha_s(\mu ) } { 4\pi b_1+b_2\alpha_s(\mu_0 ) } \biggr]^{\bigl(\frac{2}{b_1}-\frac{6 + 2n_f}{9 b_2}\bigr)}~,\ ] ] with @xmath162 and @xmath163 given in eq . 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in the supersymmetric grand unified theories ( susy guts ) , gauge bosons associated with the unified gauge group induce proton decay . 
 we investigate the proton decay rate via the gauge bosons in the susy guts under the two situations ; one is with extra vector - like multiplets , and the other is with heavy sfermions . 
 it is found that the proton lifetime is significantly reduced in the former case , while in the latter case it is slightly prolonged . 
 determination of the particle contents and their mass spectrum below the gut scale is important to predict the proton lifetime . 
 the proton decay searches have started to access to the @xmath0  gev scale . 
 ipmu12 - 0074    1.35 cm    1.2 cm    junji hisano@xmath1 , daiki kobayashi@xmath2 , and natsumi nagata@xmath3 + 0.4 cm _ 
 @xmath2department of physics , nagoya university , nagoya 464 - 8602 , japan _ + _ 
 @xmath4ipmu , todias , university of tokyo , kashiwa 277 - 8568 , japan _ + _ @xmath5department of physics , university of tokyo , tokyo 113 - 0033 , japan _