Threshold Corrections to Baryon Number Violating Operators in Supersymmetric SU(5) GUTs

The nucleon decay is a significant phenomenon to verify grand unified theories (GUTs). For the precise prediction of the nucleon lifetime induced by the gauge bosons associated with the unified gauge group, it is important to include the renormalization effects on the Wilson coefficients of the dimension-six baryon number violating operators. In this study, we have derived the threshold corrections to these coefficients at the one-loop level in the minimal supersymmetric SU(5) GUT and the extended one with additional SU(5) vector-like pairs. As a result, it is found that the nucleon decay rate is enhanced about 5% in the minimal setup, and then the enhancement could become smaller in the vector-like matter extensions.


Introduction
The supersymmetric grand unified theories (SUSY GUTs) are attractive extensions of the Standard Model (SM). The three gauge groups of the SM are unified into one, and the SM fermions are embedded into the fields charged under the unified gauge group in the GUT. The minimal candidate for the gauge symmetry is SU (5), and we may understand the origin of the hypercharge assignment according to the group structure of SU (5). SUSY also plays a crucial role in the gauge coupling unification as well as the natural explanation of the gauge hierarchy problem, and we are looking forward to the discovery of the SUSY particles at the LHC experiment. In 2012, it was reported that a scalar particle, which may be consistent with the SM Higgs boson, was discovered around 126 GeV [1,2]. The SM is firmly established and we expect that new physics predicted by the SUSY GUT is also discovered near future, although it has not been found yet at the LHC [3][4][5][6][7][8][9].
On the other hand, it is true that there are several issues which should be carefully studied in the SUSY GUTs. One of the issues is how to achieve the 126-GeV scalar boson. The low-energy effective field theory (EFT) for the SUSY GUT is considered to be the minimal supersymmetric standard model (MSSM). It is known that the MSSM predicts the upper bound on the Higgs mass, and the observed Higgs mass may require high-scale SUSY [10][11][12], or very specific SUSY mass spectrums [13], unless the MSSM is further extended, for instance, introducing extra vector-like fields [14].
Another big issue is from the experimental constraints on baryon number violation, such as nucleon decay. The GUTs unify quarks and leptons, so that the baryon-number-violating processes are introduced through the gauge interaction. The processes are strongly suppressed by the GUT scale, but it is possible to test the models through the nucleon decay search. The current status of the nucleon decay experiments is as follows: the partial lifetime limit on p → π 0 e + is τ (p → π 0 e + ) > 1.4 × 10 34 years [15,16], and the partial lifetime limit on p → K + ν is τ (p → K + ν) > 5.9 × 10 33 years [17]. The prediction of the GUT depends on the scenario between the electroweak (EW) and the GUT scale (∼ 10 16 GeV). In the minimal SUSY SU (5) GUT, the color-triplet Higgs exchange induces dangerous dimension-five operators to cause baryon number violation [18,19]. It is a serious problem, if the SUSY scale is close to the EW scale. If the SUSY scale is much higher, the constraint from the color-triplet Higgs becomes mild and the dominant decay mode p → K + ν may be detected at the future detectors [20,21]. Furthermore, the heavy gaugino masses make the GUT scale lower, so that the decay rate for p → π 0 e + , induced by a massive gauge boson (X boson), may be also large enough to be detected at the future detectors [22]. If we introduce additional SM-charged fields, the gauge coupling constants would become larger at the GUT scale since the extra fields contribute to the running of the gauge coupling constants [14]. Then the nucleon decay through the X-boson exchange is enhanced [23]. Note that the lifetime of proton is very sensitive to the X-boson mass, because the decay width is suppressed by the fourth power of the X-boson mass. This means that we need careful analysis to draw the constraint on the X boson.
In this paper we derive the threshold corrections to the Wilson coefficients of the baryonnumber violating dimension-six operators induced by the X boson in the minimal setup of the SU (5) GUT and the extended one with extra SU (5) vector-like pairs. The two-loop order corrections to the dimension-six operators have been investigated, including the long-distance effect [24] and the short-distance effect [25]. However, the threshold corrections to the dimensionsix operators at the GUT scale have never been discussed. The correction will not be non-negligible, especially when the gauge coupling constants at the GUT scale are large. We evaluate the corrections at the one-loop level analytically. This paper is organized as follows: in Section 2, we introduce the minimal SUSY SU (5) GUT to summarize our notations. In Section 3, we show the radiative corrections such as the wave function renormalizations, vertex corrections, and box-like corrections, using supergraph techniques. The definition of covariant derivatives on superfields in this paper is the same as in Ref. [26] though we use the metric signature η µν = diag(1, −1, −1, −1). We adopt the DR scheme [27] for the gauge coupling constants while we impose the on-shell condition to the X boson mass M X . For simplicity, we choose the Feynman gauge (ξ = 1) through this paper. In the next section, we estimate the threshold corrections to the Wilson coefficients of the dimension-six operators at the GUT scale, and we evaluate the numerical results for these finite corrections in the minimal SUSY SU (5) GUT and its vector-like matter extensions. Finally, we summarize our paper in Section 5. We introduce the gauge interactions relevant to our analysis in Appendix A. Our explicit results on the one-loop corrections are shown in Appendix B, and the renormalization group equations (RGEs) of gauge couplings, Yukawa couplings and the Wilson coefficients for dimension-six operators are discussed in Appendix C.

SUSY SU (5) GUTs
In the SUSY extensions of the SM, it is useful to use the superfield formalism in order to describe the fundamental interactions. Matter fields, Higgs fields, and their superpartners are embedded in chiral superfields and their conjugation. Gauge bosons and gauginos are described by vector superfields.
In the SUSY extension [28] of the minimal SU (5) GUT [29], the matter fields are given by the 5 and 10 representational superfields which are denoted by Φ and Ψ as follows: where A, B, · · · = 1, 2, · · · , 5 are the indices of the SU (5), α, β, · · · = 1, 2, 3 and r, s, · · · = 1, 2 are the indices of the SU (3) C and SU (2) L , respectively. i, j = 1, 2, 3 denote the generations. All the chiral superfields include the left-handed fermions in the flavor basis. ϕ i and V ij correspond to additional phases in the minimal SUSY GUT and the CKM matrix with the constraint i ϕ i = 0. Q and L denote the weak-doublet chiral superfields for left-handed quarks and left-handed leptons, respectively: where U, D, E, and N are the chiral superfields for left-handed up-type and down-type quarks, and left-handed charged and neutral leptons, respectively. U C , D C , and E C denote the chiral superfields for the charge-conjugation of right-handed up-type and down-type quarks, and righthanded charged lepton, respectively. In the Higgs sector, there are 5, 5, and 24 representational superfields, where K MSGUT and W MSGUT are the Kähler potential and the superpotential, respectively. g 5 denotes the unified gauge coupling constant. The field strength chiral superfield W α consists of vector superfield Here, D and D denote the covariant derivatives on superspace. The vector superfield V 5 is decomposed in terms of the SM gauge group: G, W , and B are the vector superfields for SU (3) C , SU (2) L , and U (1) Y , respectively, and they are defined as using the generators T a and t a of SU (3) C and SU (2) L , respectively. X is the vector superfield for the X boson, which induces baryon-number violating operators. It acquires the heavy mass by eating the Nambu-Goldstone (NG) modes, Σ (3,2) and Σ (3 * ,2) , after the SU (5) symmetry breaking. M X denotes the mass for the X boson in this paper.
In the minimal SUSY SU (5) GUT, the Kähler potential and the superpotential in the flavor basis of matter superfields are written as (2.8) y denotes the cubic coupling constant of the adjoint Higgs multiplet and v Σ is the VEV of the adjoint Higgs multiplet. y i u and y i d denote the diagonalized Yukawa matrices. The adjoint Higgs multiplet and the color-triplet Higgs multiplets acquire heavy masses through the interactions in the superpotential. The doublet-triplet splitting is achieved by tuning µ 0 in the minimal SUSY SU (5) GUT. M H C (= 5y H v Σ ) denotes the mass of the colortriplet Higgs multiplets. The masses of the adjoint Higgs multiplets are also split after the SU (5) symmetry breaking. The triplet Σ 3 and the octet Σ 8 have a common mass denoted as M Σ (= 5 2 yv Σ ), and the mass for Note that y and y H should be large, if the color-triplet Higgs and adjoint Higgs multiplets are much heavier than the X boson.
In the minimal setup of the SUSY SU (5) GUT, the X-boson interactions with the matter superfields are given by the following terms, (2.12) and the baryon-number violating operators are effectively induced by integrating out the X boson at the low energy. The effective dimension-six operators are written as follows at the tree level; 1 (2.14) Below, we investigate the one-loop correction to the 4-Fermi interactions and especially estimate how large the threshold correction is according to the heavy particles decoupling around the 1 Notice that the propagators of the vector superfields differ from those of canonically normalized gauge bosons by a factor 1/2 under our conversion for the kinetic terms of the vector superfields.
GUT scale. We focus on the operators relevant to nucleon decay in not only the minimal SUSY SU (5) GUT but also its vector-like extensions, where SU (5) vector-like chiral superfields are additionally introduced. In the later case, we simply assume that the vector-like pairs have supersymmetric masses without the mixing between the extra fields and the MSSM fields. We only discuss the gauge interactions in our calculation. The gauge interactions in the minimal SUSY SU(5) GUT, which are relevant to the evaluation of the threshold correction to the baryon-number violating operators, are summarized in Appendix A. For simplicity, we omit the generation indices (i, j · · · ) below.

Radiative Correction to the Baryon-Number Violating Operators
In the supersymmetric theories, effective Kähler potentials are useful to derive the radiative corrections. In order to evaluate the corrections to the baryon-number violating dimension-six operators induced by the X boson, we discuss the effective Kähler potentials at the one-loop level, and evaluate the threshold corrections to the operators. First of all, let us discuss a general effective supersymmetric action Γ[Φ, Φ † ], which is the function of chiral superfield Φ, antichiral superfield Φ † , and their derivatives. The general form of the effective supersymmetric action would be as follows, where D A is the superspace covariant derivative which consists of ∂ µ , D α , and Dα. Here, we do not include vector superfields for simplify. The perturbative corrections appear only in the D term due to the non-renormalization theorem. The effective supersymmetric Lagrangian L eff is divided into two parts under ∂ µ Φ = 0, where K(Φ, Φ † ) is the effective Kähler potential and F(D α Φ, the effective auxiliary potential. While some diagrams may generate the terms including superfields on which more than three covariant derivatives act, we may always obtain the above form by using algebra of super-covariant derivatives (D algebra). The effective auxiliary potential vanishes in the limit that D α Φ = 0 and DαΦ † = 0, so that the effective Kähler potential is identified by taking the limit. Below, we study the threshold corrections to the baryon-number violating dimension-six operators at the GUT scale with the effective Kähler potential. First, we calculate the effective actions for constant fields in both full and effective theories at the one-loop level with the supergraph technique [30]. We adopt the modified dimensional reduction (DR) scheme [27] as the renormalization scheme of the gauge coupling constants while we impose the on-shell condition for the X boson mass. We also introduce the IR cut off in order to control fictitious IR singularities. Then, we identify the effective Kähler potential for the baryon-number violating operators by taking D α Φ = 0 and DαΦ † = 0 together with the D algebra. By matching the effective Kähler potentials in full and effective theories, we derive the one-loop threshold corrections to the Wilson corrections of the dimension-six operators.

Radiative Corrections in the Full Theory
In this subsection, we show the radiative corrections to the baryon-number violating dimensionsix operators in the full theory, where the X boson is activated. The radiative corrections consist of the wave function renormalization of quarks and leptons, the vacuum polarization of the X boson, the vertex correction, and the box-like corrections. In this section, we show only the results of the supergraph calculation. Details of the calculations are given in Appendix B.

Two-Point Functions for Matter Fields
First we study two-point functions for matter fields at the one-loop level. The functions generally include UV divergences which are renormalized by the wave function renormalization factors. We estimate the factors in the DR scheme, ignoring the contributions from the Yukawa interactions. The radiative corrections to the two-point functions via the gauge interactions are determined by the gauge groups, in the both of the full theory and the EFT.
In general, the one-loop renormalized two-point function for chiral superfield Φ is defined as The wave function renormalization constant for the matter superfield Z Φ absorbs the UV divergent terms proportional to 1/ǫ ′ in the DR scheme: 2 denote the wave function renormalization factors in the full and the effective theories, respectively. g 3 , g 2 , and g 1 are the gauge couplings of SU (3) C , SU (2) L , and unified U (1) Y gauge symmetries. c Φ 5 and c Φ n (n = 3, 2, 1) are the quadratic Casimir of Φ in SU (5), SU (3) C , SU (2) L , and GUT normalized U (1) Y gauge symmetries. 3 Then, the one-loop renormalized two-point function in the full theory is given by is the tree-level two-point function, and a Φ and b Φ are the constants obtained from the one-loop calculations, We set the mass of the MSSM vector superfields to be a non-zero value which is denoted by µ IR in order to regularize the IR divergence, as mentioned above. The function f in Eq. (3.4) is defined as  where µ denotes the renormalization scale in the DR scheme. The two-point function in the effective theory is derived by removing the X boson contribution in Eq. (3.4) when g 5 = g 3 = g 2 = g 1 .

Vacuum Polarization
Next, we estimate the radiative corrections to the propagator for the X boson. Not only the MSSM fields but also the GUT-scale fields such as the SU (5)-adjoint field contribute to the vacuum polarization of the X boson. The chiral superfields have three kinds of the contributions which are described in Fig. 1. The diagrams (a) and (b) are induced by the supergauge interaction Φ † V Φ and Φ † V 2 Φ, respectively. The diagram (c) is generated by the SU (5)-breaking adjoint Higgs superfield, which has interactions Σ † V 2 Σ and Σ † V 2 Σ after acquiring the VEV.
For the gauge sector, we have the four-type diagrams to contribute to the two-point function of the X boson. The diagrams (a) and (b) in Fig. 2 arise from the self interactions of the vector superfields. If the internal vector superfields in the diagram (b) are massless, the diagrams have no contribution to the two-point function in the DR scheme. The diagrams (c) and (d) show the ghost loop contribution.
Finally, the two-point function of the X boson is in the form as below: where Σ X (k 2 ) is the renormalized vacuum polarization for X boson. The UV divergence in the one-loop corrections is absorbed by the wave function factor (Z X ) and mass (M X ) of the X boson. In this paper the on-shell condition for the X boson mass is imposed so that this leads the equation Σ X (M 2 X ) = 0. This is because heavy particles are decoupled from Σ X (0) under the on-shell condition, if they have SU (5) symmetric masses much larger than the X boson mass. 4 Σ X (0) will appear in the threshold correction to the baryon-number violating operators. The counter term δZ X is determined to absorb the UV divergence which arise from the gauge contributions and the matter contributions such as Figs. 1,2. We obtain As expected, δZ X is proportional to the one-loop beta function for the SU (5) gauge coupling constant. In the SUSY SU (5) GUT models with 5 + 5 vector-like matter superfields and 10 + 10 vector-like matter superfields, we find where N f , n 5 , and n 10 are the number of generations, 5 + 5 and 10 + 10 vector-like pairs, respectively.
In the SUSY SU (5) GUT with extra vector-like matters, the vacuum polarization Σ X ( (3.10) where N r (r = 5, 5, 10, 10) denotes the number of the massless superfields in r representation. The loop functions A and B are defined as is defined. The first and second lines in Eq. (3.10) correspond to the contributions of the massless and massive fields in Fig. 1(a). The third line is for diagram (c) in Fig. 1, in which the VEV of the adjoint Higgs multiplet is included in the vertices. In the fourth line, we show the contributions Figure 3: Diagrams for vertex correction from the gauge sector: The first term in the forth line is induced by the three-vector interactions ( Fig. 2(a)), while the second term corresponds to the ghost diagrams ( Fig. 2(c)). The p 2independent terms come from the diagrams Fig. 1 Fig. 2(b), and Fig. 2(d).
We finally obtain the full one-loop corrections to the two-point function by summing of the contributions from the chiral superfields, the vector superfield, and the ghost superfields. The resumed propagator D XX (p 2 ) of X superfield in terms of the superfield notation is given by After the spontaneous symmetry breaking of the GUT gauge symmetry, the baryon-number violating dimension-six operators are induced by the X boson, and the coefficients are proportional to 1/M 2 X . In order to match the full and the effective theories at the one-loop level, we need to take into account the one-loop corrections to the propagator of the X boson. Since the momenta of external fields in the baryon-number violating dimension-six operators are negligible compared with the X boson mass, we may set the momentum of internal X boson zero.

Vertex Corrections
Next, we show the one-loop vertex corrections to the X boson interactions with quarks and leptons. The tree-level interactions are given in Eq. (2.9).
Several one-loop diagrams in Fig. 3 contribute to the vertex corrections. Since the supersymmetric gauge interactions in terms of the superfield formalism have the form Φ † e 2gV Φ (Φ is a matter chiral superfield, and V and g are a vector superfield and its gauge coupling, respectively), there exist diagrams which do not appear in component calculation. The diagram (a) has only the vertex 2gΦ † V Φ, and the diagrams (b) and (c) include the vertex 2g 2 Φ † V 2 Φ. The diagrams (d) and (e) include the three-point self interactions of vector superfields. Since the external vector superfield is for the broken gauge symmetry, two internal vector superfields must be massive and massless ones. The contribution from the diagram (f) is vanishing due to the suparspace integral.
Thus, we calculate the contributions from the diagrams (a)-(e) in Fig. 3. The momenta of all the external superfields are set to be p 2 = 0, for simplicity. In some diagrams, since they contain IR divergent contributions in this momentum assignment, the non-zero masses of the MSSM vector superfields (µ IR ) are introduced as IR regulators. Under this momentum assignment, we carry out loop momentum integrals and Grassmann integrals, and we discard the auxiliary terms. We expand the one-loop Kähler terms around p 2 = 0, and then we extract the dominant contributions around p 2 = 0. The vertex corrections to the gauge interactions between the MSSM matter fields and the X boson are as follows: correspond to the correction from the diagram (a), and the ones from the diagrams (b) and (c) in Fig. 3, respectively. After the loop momentum and superspace integrals, we find that C  2 defined in Appendix B in the limit that p 2 vanishes. Now, we determine the renormalization constants for the vertices. One-loop renormalized vertex functions are given by (3.14) When K Vn (n = 1, 2, 3) are described as K Vn = C Vn O Vn with the operators O Vn and the Wilson coefficients C Vn , Z C Vn are defined to renormalize the UV divergences in C Vn . Then we find 15) which are consistent with the one-loop beta function for the gauge coupling.

Box-like Corrections
The box-like diagrams contribute to the radiative corrections of the dimension-six operators. Fig. 4 shows all type of the box-like diagrams; we refer to the diagram (a) as the box diagram,  Fig. 4. In these figures, one of two internal gauge superfield lines must be massive since we focus on the baryon-number violating operators. As is the case in the vertex corrections, we set all momenta of the external superfields to be p 2 = 0 and the fictitious masses of the MSSM vector superfields to be µ IR , and we remove the auxiliary terms. For the momentum assignment, we find that the box diagram (a) vanishes while the crossing box diagram (b) and the triangle diagram (c) are given by the following functions: (3.16) These loop functions correspond to the coefficients in the effective Kähler potential K cross and K triangle defined in Appendix B in the limit that p 2 vanishes. In SUSY SU (5) GUTs, the baryon-number violating dimension-six operators are generated at the tree level in Eq. (2.13). The one-loop radiative corrections from the box-like diagrams are written by C cross and C triangle : (3.17)

Radiative Corrections in EFT
Now we consider the radiative correction to the higher-dimensional Kähler terms in the EFT.
There are three kinds of contributions to the radiative correction. The first one is the diagram (a) in Fig. 5, where a vector superfield is attached to two chiral superfields or two antichiral We adopt the same momentum assignment which we used in the full theory. After the loop momentum and the superspace integrals, we derive the one-loop corrections as (3.18) Here, the diagram (a) vanishes while the diagrams (b) and (c) are given by C EFT 1 (µ IR ) and C EFT 2 (µ IR ), respectively: These functions correspond to the coefficients defined in Eq. (B.17) in the limit: p 2 vanishes. The effective Kähler potentials up to the one-loop level are described as (3.20) The logarithmic divergences are absorbed by the counter terms of C A , and then we have (3.21) These are consistent with the results of Ref. [33]. In the next section, we determine the threshold corrections for the wave functions and the Wilson coefficients of the dimension-six baryonnumber violating operators by matching the full and effective theories.

Threshold Corrections of the Dimension-Six Operators
In the previous section, we have shown the radiative corrections to two-, three-, and four-point vertex functions in the SUSY SU (5) GUTs and we have shown also the radiative corrections to the Wilson coefficients of the dimension-six operators in the EFT. Now, we determine the threshold corrections by matching the amplitudes in the EFT and those in the full theory. First, let us discuss the threshold corrections to the two-point functions for matter superfields. As we have seen in Eq. (3.4), the one-loop two-point functions are divided into two parts: one is linear to f (M 2 X ) and the other is linear to f (µ 2 IR ). The latter is the contribution from the MSSM gauge interactions, and the former is the contribution from the broken gauge interaction in SU (5). On the other hand, the two-point functions in the EFT at the GUT scale have the form; Here, the chiral superfield in the EFT is given by (1 − λ Φ /2)Φ (Φ is in the full theory). λ Φ is determined so as to match the two-point function in the EFT and that in the full theory: where (λ Q ,λ U ,λ D ,λ L ,λ E ) = (3, 4, 2, 3, 6) is defined. Next, we determine the threshold corrections for the baryon-number violating dimension-six operators. The two-point functions of the matter superfields in the full theory and the EFT are matched above, and we have determined the threshold corrections to the renormalizable kinetic terms. For a matter superfield Φ, the renormalizable kinetic term has the form (1 − λ Φ )Φ † Φ in the EFT. The finite corrections to the two-point functions in the EFT appear in the correction to the Wilson coefficients of higher-dimensional operators. The Wilson coefficients of higherdimensional operators themselves also include the finite corrections. Thus, we redefine the effective Kähler potentials K eff I (I = 1, 2) as the ones with threshold corrections up to the one-loop level as follows: In the full theory (the SUSY SU (5) GUTs), we have computed the effective Kähler potential for the dimension-six operators at the one-loop level, (4.4) The first terms include the vacuum polarization of the X boson Σ(0) and the one-loop effective couplings C V 1 , C V 2 , and C V 3 which are defined in Eq. 3.14.
There are IR divergences in K full I and K eff I (I = 1, 2), which are represented by µ IR . The divergences are absorbed by the operators O (I) . 5 Then, we divide the effective Kähler potentials into the coefficients C I and the renormalized operator O (I) r : The one-loop coefficients in the full theory are given by (4.7) We assume that the matching scale is µ = M GUT (≃ M X ), where the unification g 1 = g 2 = g 3 = g 5 is achieved. By comparing the amplitudes obtained in the full and effective theories, we determine the threshold corrections to the Wilson coefficients of dimension-six operators λ 1 and λ 2 at the one-loop level: (4.8) We find that the corrections to the wave function for the matter field and the vertex of the X boson are cancelled with each other as expected from the Ward identity and that the threshold corrections come from the corrections to the vacuum polarization and the box-like contributions. Now, we give numerical results of the short-range renormalization factor including threshold corrections in the minimal SUSY SU (5) GUT and its vector-like extension. In the minimal SUSY SU (5) GUT, the X multiplet, the color-triplet Higgs multiplets, and the adjoint Higgs multiplet acquire heavy mass through the VEV of the adjoint Higgs multiplet. First, we set the masses of the GUT particles to be degenerate in mass 2.0 × 10 16 GeV since they are model-dependent parameters. The dependence of the threshold correction on the GUT scale mass spectrum is shown later. The threshold corrections in the minimal SUSY SU (5) GUT are divided into two parts: the one comes from the vacuum polarization of the X boson as another one comes from the box-type diagram: We have also evaluated the short-range renormalization factor to the partial decay rate (p → e + + π 0 ). We define the ratio of the short-range renormalization factor with and without the threshold correction to the Wilson coefficients of the dimension-six operators as where the denominator and numerator correspond to the short-range enhancement factor of the nucleon decay rate without and with threshold corrections, respectively. V ud denotes (1, 1) component of the CKM matrix. We obtain R = 0.948 in the minimal SUSY SU (5) GUT, that is, there is about 5% suppression compared with the short-range renormalization factor without threshold corrections. and then, we obtain R = 0.938. In Fig. 6, we describe the heavy mass dependence on the ratio of the short-range renormalization factor in the minimal SUSY SU (5) GUT. Here, we set the mass of the component fields of the adjoint Higgs multiplet to be degenerate in M Σ , that is, we set M Σ 24 = M Σ , for simplicity. The left panel of Fig. 6 shows the color-triplet Higgs mass (M H C ) dependence of the ratio with the fixed adjoint Higgs mass M Σ = 2.0 × 10 16 GeV. The right panel of Fig. 6 shows the adjoint Higgs mass (M Σ ) dependence of the ratio with the fixed color-triplet Higgs mass M H C = 2.0 × 10 16 GeV. Since, in a large M H C region, the vacuum polarization behaves as Σ ∼ 1 2 M 2 X ( 1 2 − ln M 2 H C /µ 2 ), the decay rate of proton is slightly enhanced in this region. In the SUSY SU (5) GUT with light vector-like matter scenario, the threshold corrections to the Wilson coefficients of the dimension-six operators are enhanced since the unified gauge coupling becomes large. This large unified coupling leads to the large renormalization effect to the Wilson coefficients of the dimension-six operators.
In Fig. 7, we show the ratio of the short-range renormalization factors in the vector-like matter scenario. The horizontal line and the vertical line present the mass scale of the vectorlike matters and the ratio of the short-range renormalization effect, respectively. The solid lines correspond to the case that the number of 5 + 5 vector-like matters is set to be n 5 = 1, · · · , 4 from top to bottom without 10+10 vector-like matter. In this estimation, we assume the masses of the heavy multiplets and the GUT scale are set to be 2.0 × 10 16 GeV. If the mass (number) of the vector-like superfields is sufficiently light (large), the unified gauge coupling at the GUT scale becomes larger, and then the threshold correction makes the proton lifetime longer.
In Fig. 8, we show the partial proton lifetime (p → π 0 + e + ) in the minimal SUSY SU (5) and its vector-like extension. In this evaluation, we assume the masses of the GUT spectrum are set to Figure 7: Ratio of short-range renormalization effects with and without threshold effect in the minimal SUSY SU (5) GUT with light vector-like matters. We take n 5 = 1, · · · , 4 in solid lines from top to bottom. The case of the minimal SUSY SU (5) with no light vector-like matter is shown in dotted line.
be the same mass (2.0×10 16 GeV), especially the X-boson mass is set to be M X = 2.0×10 16 GeV. We use the two-loop RGEs of the Wilson coefficients of the dimension-six operators as shortdistance [25,34] and as long-distance [24]. We also use the hadron matrix elements evaluated with the lattice calculation [35]. The deep gray region is corresponding to the present lower bound on this decay mode by the Super-Kamiokande (τ (p → π 0 + e + ) > 1.4 × 10 34 years). The gray region, on the other hand, corresponds to the future sensitivity on this decay mode by the Hyper-Kamiokande (τ (p → π 0 + e + ) > 1.0 × 10 35 years). Due to the extra fields, the lifetime is suppressed since the unified coupling becomes large at GUT scale. However, the threshold correction lifts the partial proton lifetime up.

Conclusion and Discussion
In this study, we have derived the threshold corrections to the Wilson coefficients which cause proton decay (p → π 0 + e + ) at the GUT scale in SUSY SU (5) GUTs. We find that the threshold correction makes the proton decay rate suppressed about 5% in the minimal SUSY SU (5) GUT. Furthermore, we also have investigated the threshold effect on the partial proton decay rate in the extended SUSY SU (5) GUT with additional vector-like pairs, motivated by the achievement of the 126 GeV Higgs boson. In these models, we find that the decay rate is O(10)% suppressed due to the large unified gauge coupling. In our study, we neglect the threshold corrections induced by the Yukawa interactions, because the Yukawa interactions involving light quarks and leptons are negligibly small at the GUT scale.
The matrix elements relevant to nucleon decay have been evaluated with the lattice QCD and they have 30% uncertainties at present [35]. We expect that the uncertainty would be Finally, we note the application of our work to the other SUSY GUTs. We only have investigated the threshold effects in the minimal SUSY SU (5) GUT and the extra vector-like matter extensions in this paper. When, however, we apply our formulae for the extension of the SUSY SU (5) GUTs, for instance the missing-partner model [36], we only have to evaluate additional contributions to the vacuum polarization for the X boson. That is remaining as one of our future work.

A.1 Interactions of Vector Superfields
In super-Yang-Mills theories, the renormalizable Lagrangian is written as where the field strength chiral superfield is given in Eq. 2.5. The Lagrangian is expanded in the vector superfield V as The decomposition of the SU (5) vector superfield V 5 is given by Eq. 2.6. As mentioned in text, we denote SU (3) C , SU (2) L , and U (1) Y vector superfields in the MSSM with G, W , and B. The kinetic terms of the vector superfields in the SU (5) GUTs are given into the following form; where X denotes the massive vector superfield associated with the broken SU (5) generators. Here, P T (≡ D α D 2 D α /(8 )) is the projection operator to the transverse mode (P 2 T = P T ). From the second term of Eq. A.2, the three-point interaction terms between X and MSSM vector superfields are obtained as Here, spinor indices are contracted like α α orαα. The four-point self interaction of X is given as (A.6)

A.2 Vector-Ghost Interactions
The Lagrangian for the massless Fadeev-Popov ghost chiral superfields, which are denoted by b and c, are given as where L A B is the Lie derivative (L A B ≡ [A, B]). Therefore, the kinetic terms for ghost fields in the SU (5) GUTs are obtained as where the ghost multiplets are decomposed in a similar way to the gauge multiplets as After spontaneously breaking of the GUT group by the adjoint Higgs chiral superfield, there exist kinetic mixing terms between X and the Nambu-Goldstone chiral superfields Σ (3,2) and Σ (3 * ,2) . By using the supersymmetric R ξ -gauge [37], we remove the kinetic mixing terms, and we find the mass terms for the ghost chiral superfields [37] as: We note that the terms such as b X c X and b † X c † X vanish by the superspace integral since these are chiral (or antichiral) superfields. Then, the propagator for massive ghost superfields is modified as In the evaluation of the self energy of X, we need interaction terms for X and the massive ghosts. In general, three-point and four-point interaction terms of ghost superfields and vector superfields are obtained from Eq. (A.7) as follows, Then, the interaction terms between X and the ghosts are given by: and (A.14) Here, we define δ βδ αγ ≡ δ β α δ δ γ and δ tr su ≡ δ t s δ r u . In the three-point interactions, we define the term (K a bcV ) rα sβ as:

A.3 Gauge Interactions of Matter Superfields
Now, we summarize the gauge interactions of the matter and Higgs multiplets in SUSY SU (5) GUTs. The renormalizable Kähler potential in the SU (5) GUTs is given as: (A.16) The three-point gauge interaction of the 5 representation matter field Φ is given as For the four-point vertices, we only use the interactions which include only one X, Here, (A · B) ≡ ǫ rs A r B s . We also obtain the relevant gauge interactions from the 10 representation matter field Ψ, There are also the three-and four-point interactions with Higgs multiplets of X. One of those comes from the interaction of the anti-fundamental Higgs superfield H = (H C , H d ), (A.21) Another one comes from the fundamental Higgs superfield H = (H C , H u ), The adjoint Higgs superfield is decomposed as In our calculation, we need the interaction terms with the adjoint Higgs superfield of X, After symmetry breaking of GUT, there exist the three-point interaction terms between MSSM vector superfields, Nambu-Goldstone multiplet, and X with VEV v Σ of the adjoint Higgs multiplet.

B Radiative Corrections at One-loop
In this appendix, we give the explicit formulae of the loop integrals in terms of supergraphs. All the external momenta of the chiral (antichiral) superfields are set to be p, and the masses of the MSSM vector superfields are set to be µ IR in order to regularize the IR divergence. For simplicity, we set all coupling constants to be 1 through this appendix. For the corrections to the three-point vertex functions and the box-like corrections, the loop integrals in text are the coefficients of Kähler potentials in the limit that the external momenta p 2 vanishes.

Radiative Corrections to Two-Point Functions for Matter Superfields
The correction to the self energy of the chiral and antichiral matter superfields in the first generation is induced by the gauge interactions. The one-loop contribution is given as where p is external momentum and M is the mass for the internal vector superfield. δ ij denotes the δ-function for the Grassmann valuable, δ ij ≡ (δ i − δ j ) 2 (δ i − δ j ) 2 . The renormalized one-loop two-point function of matter superfields in the SU (5) GUTs are given as where function f (M 2 ) is defined in Eq. 3.6. c Φ 5 and c Φ n (n = 3, 2, 1) are the quadratic Casimir defined in text. In the MSSM, we also obtain Here, we do not write the external momenta of external superfields for simplicity since we set them to be the same momentum p. As mentioned above, we set the mass of massless vector superfields to be µ IR as IR regularization. Γ box (p; M ) vanishes at the point with p 2 = 0, as mentioned in text. The contribution of the crossing-box diagram ( Fig. 4(b)) is given by (B.13) Here, we define the mnemonic symbol (σ µ DD) ≡ (σ µ )α α DαD α . This correction has the auxiliary terms. The corresponding Kähler term is given by removing the auxiliary terms as iK cross (p; M ) = 1 4 (B.14) Finally, we show the contribution from the triangle diagram in Fig. 4(c). The correction from the triangle diagram is obtained as follows: Since auxiliary terms are not included in the radiative corrections Γ box and Γ triangle , the corresponding Kähler terms are just written by these corrections as Γ n = d 4 θ K n (n = box, triangle). The diagram in Fig. 4(d) vanishes as mentioned in text.

One-loop Corrections in EFT
In the last of this appendix, we show the radiative corrections in EFT presented in Fig. 5. We obtain the one-loop effective vertex functions Γ EFT 1 , Γ EFT 2 , and Γ EFT 3 which correspond to the diagram Fig. 5 (b), (c), and (a), respectively, as follows: The momentum assignment is the same as in calculation of the box-like diagrams. The corresponding Kähler terms are given by removing the auxiliary terms as

Wilson Coefficients of D = 6 Baryon-Number Violating Operators
In Ref. [25], they have derived the two-loop RGEs for the Wilson coefficients of the following dimension-six baryon-number violating operators in the SUSY invariant theories, where i = 1, · · · 3, and the coefficients are given as (C. 16) Here, b i (i = 1-3) is given in Eq. C.2.