JHEP03(2014)049 Springer March 11, 2014 January 6, 2014 : : February 16, 2014 : Received Published 10.1007/JHEP03(2014)049 Accepted doi: Published for SISSA by [email protected] (100) TeV, which naturally explains the , O c . 3 and Satoshi Shirai 1312.7854 a,b The Authors. c Supersymmetry Phenomenology

The discovery of the Higgs boson with a mass of around 125 GeV gives a strong , [email protected] Tokyo 113-0033, Japan Berkeley Center for Theoretical Physics,and Department Theoretical Physics of Group, Physics, LawrenceUniversity Berkeley of National California, Laboratory, Berkeley, CA 94720,E-mail: U.S.A. Department of Physics, NagoyaNagoya University, 464-8602, Japan Department of Physics, University of Tokyo, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: constraints on sfermion flavor violation.us a We valuable find knowledge that on proton-decayfrom sfermion experiments other flavor may violation, low-energy give and precisionsfermion by experiments, combining sector it we as with well can the as extract results the insights underlying to grand the unification model. structure of time, sizable flavor violationexperiments in when the sfermion mass masses125 GeV scale is Higgs is still mass. as allowed highand In by as branching the low-energy fractions presence precision of ofstudy the the proton effects sfermion decay of flavor sfermion can violation, flavor be structure however, on the drastically proton rates decay changed. and discuss In the experimental this paper, we Abstract: motivation for further studythis of framework, a the high-scale minimalheavy supersymmetry sfermions SUSY (SUSY) suppress SU(5) breaking the model. proton grand decay unification via In model color-triplet Higgs may exchanges. be At viable the same since Natsumi Nagata Sfermion flavor and protonin decay high-scale supersymmetry JHEP03(2014)049 18 18 7 15 – 1 – 23 18 19 28 9 7 15 27 25 21 28 3 5 27 24 5 27 3 22 6 -boson contribution 2 X 12 3.6.3 3.6.1 Threshold correction3.6.2 to Yukawa couplings Contribution from soft baryon-number violating operator 2.2.1 Meson mixing 2.2.2 EDM D.3 Pion/eta and charged lepton C.2 At electroweak scale D.1 Kaon and chargedD.2 lepton Pion and anti-neutrino C.1 At SUSY breaking scale 3.4 Flavor constraints3.5 from proton decay Uncertainty of3.6 decay rate Possible additional corrections 3.1 Minimal SUSY3.2 SU(5) GUT Dimension-five proton3.3 decay Results 2.1 Mass spectrum 2.2 Flavor constraints D Partial decay width B RGEs of the Wilson coefficients C Matching conditions 4 Summary and discussion A Input parameters 3 Proton decay with sfermion flavor violation 2 High-scale SUSY model Contents 1 Introduction JHEP03(2014)049 and GUT P ]. ¯ d/M 22 ¯ u ¯ e u , which relate to QQQL/M and ¯ † j ], seems to give the Q i ], which is somewhat 13 Q GUT – 7 8 , 6 (100) TeV sfermions, which O QQQL/M ], in which the sfermion mass scale 5 – 1 ]. In addition, heavy sfermions prevent too 20 – 2 – ]. ]. 19 26 – – 14 24 ] via the dimension-five operators 21 (1) coefficients result in too rapid proton decay even in the high-scale ]. This discrepancy clearly comes from the underlying assumptions of a O 23 with ], however, such effects are not considered. Since sizable flavor violation may P 22 The sfermion structure considerably affects the proton decay rate. In the previous Even if such a flavor symmetry actually exists and the dangerous dimension-five op- However, it was also pointed out that Planck-suppressed operators Such a scenario is also helpful for the construction of a grand unification theory (GUT). ¯ d/M ¯ u ¯ e ¯ in the minimal SU(5)proton GUT decay rate model is with drasticallygives high-scale changed strong depending SUSY. on constraints the It onfind sfermion is flavor the a structure, found flavor smoking-gun which that signature violationfuture the for in proton-decay resultant the the experiments. sfermion sfermion flavor In sector. violation, consequence, which Further, proton-decay may we be experiments will searched might in shed experimental constraints [ study [ be present in theflavor case violation of on high-scale proton SUSY,paper, decay it therefore, and is we to study important the examine to impact find it of out in the the proton-decay sfermion consequence flavor experiments. structure of on In the this proton decay This is because the flavorsoft charges sfermion of masses, non-holomorphic operators dependin like on the the sfermion underlying masses models.lation may can occur Therefore, be in large allowed some flavorlarger in flavor violation than the models. 100 high-scale TeV, even SUSY In the scenario; fact, maximal if such flavor the violation sizable sfermion may flavor be mass vio- consistent scale with is the much current flavor symmetry. The operatorssmall from Yukawa the couplings. color-triplet Higgs Thereduce exchanges flavor the are symmetry coefficients suppressed which of by realizes such Planck-suppressed the operators. Yukawa hierarchy may erators are well suppressed, the sfermion flavor structure is not necessary under control. SUSY SU(5) GUT wasexplain reexamined the and 125 GeV it Higgs was mass, shown can be that consistentu with the current constraints [ SUSY model [ level, since they form complete SU(5)a multiplets. sense, Indeed, as the unification the can thresholdcompared be corrections improved with in to the the gauge low-scalerapid couplings SUSY at proton the ones decay GUT [ [ scalegenerated from can the be color-triplet small Higgs exchanges. Recently, the proton decay in the minimal strongest motivation for the high-scalephenomenological SUSY aspects model. of For this suchafter reason, a the both framework Higgs theoretical have discovery and [ been further investigated, especially Decoupling sfermions does not affect the successful gauge coupling unification at one-loop A high-scale supersymmetry (SUSY) breaking modelis [ much higher thanview, the such weak as the scale, SUSY hasthe flavor/CP discovery many problems of and attractive the the features Higgs cosmologicaltoo boson from problems. heavy for with various In a a points weak-scale particular, mass minimal of SUSY of standard around model 125 (MSSM) GeV [ [ 1 Introduction JHEP03(2014)049 , . Rj X for e e , we 2 (2.4) (2.2) (2.3) (2.1) ∗ ij ) 3 R ˜ 2 e /M 2 | m ( X F ∗ Ri is almost the | e e couples to the c 0 − is charged under X = m , † Rj 2 0 2 e d X / X 3 m ij ) m R ˜ ]. Because of the charge 3 2 d α π 29 m 3 ., 4 ( – c − . ] or extra particles in some ∗ Ri 27 e [ d , 35 but allows = 2 + h − . In this case, ˜ / g d ∗ 3 X P Rj ) are the gauge couplings and the -term may be generated via H e m u M b ˜ g MSSM u 00 ij to the gauge supermultiplets and the c Φ ) ,M H ˜ 2 R 00 W, = X / ˜ 2 u c 3 ˜ H m B, + m † MSSM ( µ 2 d π Φ = ∗ Ri α 4 ], the gaugino masses are given by (1) parameter, which depends on the species. H e u -terms are also suppressed by a loop-factor and X u a – 3 – † O and 34 ( − = A and the soft – 2 X | a Lj ˜ W 2 2 XH H ∗ 30 e L / † c M µ is devoted to summary and discussion. 3 is an M ij X ) 4 m -term vacuum expectation value (VEV) of the field c 2 | is the Planck scale ∗ L 0 ,M 00 ˜ 2 L c F ∗ . 2 c 3 − M / 2 m 3 / + M ( 3) and 3 K , direct couplings of 2 the , ∗ m , and ∗ Li 3 − 2 m d X 1 -terms in this case arises from the anomaly mediation effects. e , L X ]. The trilinear α π /M A K − F 4 ,H 2 | 11 u 37 = 1 Lj , X 3 5 e Q F a ,H | 36 ( 0 ij = M c ) ˜ π L B ˜ 4 = 2 Q / M b = Φ 2 a m ( g ∗ Li ≡ e Q MSSM a − α = Next, we introduce our convention for the sfermion mass-squared matrices. The soft The supersymmetric Higgs mass This paper is organized as follows: in the next section, we introduce a high-scale SUSY is the cutoff scale of the theory. These terms give soft masses as ∗ soft thus we neglect them hereafter. mass terms of sfermions are given as L where gaugino masses, respectively. This massfrom relation the can SUSY be modified breakinghigher-energy scale via effects [ quantum by corrections the MSSM particles [ of the SUSY breaking filed superpotential can be forbiddenmasses by and the the symmetry. trilinear TheWith main pure contribution anomaly to mediation the effects gaugino [ which leads to where Φ M the MSSM scalars, with One of the naturalsame choices as of the gravitino mass following discussion. Suppose thatsome the symmetry. supersymmetry This suppresses breaking theMSSM field operators superfields. linear Especially, in the following terms in the K¨ahlerpotential can be present: experimental bounds on it. Section 2 High-scale SUSY model 2.1 Mass spectrum To begin with, let us briefly discuss a high-scale SUSY model which we consider in the the electroweak scale. model which we deal withexperimental in the constraints following on discussion, flavor and violationevaluate give in the a proton the brief decay review sfermion rates of sector. in the Then, the current presence in of section sfermion flavor violation and discuss the light on the structure of sfermion sector even when the SUSY scale is much higher than JHEP03(2014)049 . , we 3.1 (2.5) (2.6) ’s and . The 1 δ 0 m 2 TeV. We , − t DL and all 3 to 3 V = 300 GeV, and = 0 t / . In figure ) A 0 ˜ g ˜ β W L m ˜ m 2 L M M m = ( 0 ∗ m DL = 4 (green line). In this V = , L is favored in the high-scale = ˜ = 4 (green line) are also shown. Q 3 H     β R ˜ L µ f 3 ˜ = 600 GeV, 2 d ˜ Q 3 ˜ ˜ ˜ m B f f 13 23 δ δ M = 300 GeV, and 1 + ∆ and ˜ f 2 ˜ W ∗ † ˜ ˜ f QE 12 f 23 M δ δ V t ’s and ∆’s are zero. For the experimental ) 1 + ∆ δ R ˜ f 1 ˜ 2 e – 4 – ∗ ∗ m ˜ ˜ f f 12 13 ( 9 (black line) and ∆ 9 (black line) and ∆ δ δ . . 1 + ∆ QE for the observed Higgs mass. Red and blue bands show and all V = 0     = 0 = 600 GeV, 0 0 2 0 R = ˜ . We estimate the theoretical error by changing the scale A ˜ R u 13 m B m ˜ u δ 13 m δ † A M QU = = ˜ V 2 f = . In the minimal SU(5) GUT, there are relations among the = t L ) ˜ L m Q 13 are the GUT “CKM” matrices, which are defined in section 0 L R for the observed Higgs mass as a function of the sfermion mass δ e L ˜ ˜ 2 u Q 13 , m β δ m R DL ( e e V = , R QU e H in appendix d V 3 denote the generation indices. The squark mass matrices are defined , µ and as a function of , 1 R 2 = β e u , QE L , ˜ V 2 Q L , e = 1 Q m . tan . The red and blue bands show the experimental and theoretical uncertainties, QU 0 = 2 TeV. The cases of i, j V ˜ − m f As we will see, the proton decay rate has strong dependence on tan = ˜ g also show the casesestimation, of we use thegauginos) two-loop system renormalization and group theThis one-loop equations figure threshold (RGEs) illustrates effects in from thatSUSY the heavy scenario. a (SM sfermions relatively + and small higgsinos. value of tan scale respectively, for inputs, see table of matching between the MSSMgaugino and masses the are (SM+gauginos) set system to from be where In this paper, however, weto treat the these five above mass GUT matrices relation, independently, to without clarify restricted each effectshow on the proton predicted decay. tan with sfermion mass matrices at the GUT scale: in the so-called super-CKMand basis, in squarks which are the rotated up-typestructure in quark as parallel mass follows: to matrices are their diagonal superpartners. We further parametrize their Figure 1 the experimental and theoretical∆’s uncertainties, are respectively, set for toM be zero. The gaugino masses are setwhere to be JHEP03(2014)049 (2.7) (2.8) (2.9) ). The ’s. The δ , C.7 4 C 5 3 ' − ]. As we will see, 5 38 , , j i ) ) , the dominant SUSY q q , ˜ g , α Lj ,C α Rj q q Jj m Jj . The contribution to the Jj ) ) ) β β Ri Li R 2 L q q ˜ L q ˜ q ˜  q )(¯ )(¯ R R R , ( ˜ q ( ( β β Lj Lj A ˜ g iJ q q m iJ ˜ iJ ) O ) ) α α R Ri Ri A L L ˜ † q ˜ q q † q ˜ † q ˜ C R R R ( ( ( = (¯ = (¯ 3 =1 Ij Ij X Ij A 3 5 ) , ) ) ) R L R ˜ + q = 2 effective Hamiltonian, ˜ q ˜ q ’s from the meson mixings. The left (right) β Lj R δ A R q I J R F are less significant in the present model. ( ,O ,O ( ˜ ˜ ( q q µ ) ) O – 5 – iI 3 iI γ iI A ) ’s are unitary matrices defined in eq. ( ) ˜ β ) Lj β Rj C R β Li L C q q R R ˜ † q ˜ † q q ˜ † q β β Ri Li R ˜ R g 5 =1 )(¯ R q q and X A )( )( )( α Lj )(¯ )(¯ 2 q = RJ LJ ) and ˜ are defined as follows: LJ α α µ Lj Lj ˜ C ˜ 2 q 2 q ˜ 2 q y q q , γ A eff 3 ˜ α α α − Li Ri Ri , m , m O , m H q q q ,C x RI LI 2 ( RI ˜ ˜ 2 q 2 q ˜ 2 q / C = (¯ = (¯ = (¯ ) and m m m ( ( 1 2 4 ( A j i O O O H H x/y q q H O . In the large squark-mass limit, 2 2 3 3 2 3 L 3 α α α 36 36 11 11 ↔ , we show the constraints on ' − ' ' ) = log( 3 R 4 1 1 ˜ C C C = 2 meson mixings give strong constraints on the flavor violation x, y by ( . An example of the dominant diagram contributing to the meson mixings in the presence F H A ˜ O In figure where other Wilson coefficients panel illustrates the case where flavor violation occurs in either (both) chirality. To get the and contributions to the Wilson coefficients are approximately given by where the operators 2.2.1 Meson mixing The ∆ dominant contribution comes fromoscillation the is box represented diagram by of the following figure ∆ which are severely restrictedthe by flavor low-energy violation precision ofviolation not experiments squarks so [ much. can Inconstraints strongly the on rest affect of the the proton squark section, flavor decay, we mixing. briefly and review the the current slepton experimental flavor Figure 2 of the squark flavor mixing. 2.2 Flavor constraints The soft SUSY-breaking terms in general introduce new sources of flavor and CP violation, JHEP03(2014)049 0 4 = ¯ 10 On K ˜ B - 0 1 M K ) ) ) ) ) ) 2 TeV in 0 0 0 0 0 0 d s B B D D K K − ( ( ( ( ( ( | | | | | | L L L L L L ˜ ˜ ˜ ˜ ˜ ˜ = Q Q 13 23 Q Q 23 12 Q Q 12 23 δ δ δ δ δ δ | | | | | | 3 ˜ g ======10 | | | | | | R R L R L R ˜ ˜ ˜ d d M 13 23 ˜ ˜ u ˜ d 12 12 Q 13 Q 13 δ δ δ δ δ | | δ | | | | = = | | R R [TeV] ˜ ˜ u d 23 23 δ δ 0 | | m = = ) | | R R g (b) ˜ ˜ d 13 u 13 ( δ δ R | | 2 u γ 10 . (a): one chirality flavor violation = 300 GeV, and δ ˜ W R are greatly relaxed in the case of CP ˜ 1 M u 0 10 1 ¯ D 0.1 0.01 are stringently restricted from the - 0.001

0 R purbound Uppuer ˜ t R e d D 12 δ ]. The CP phase is chosen so that the strongest 4 – 6 – L 10 and ˜ t 41 = 600 GeV, and – 0 ˜ (10) times larger. L B ¯ 39 K e Q 12 - L O ) ) ) ) ) ) 2 TeV. M 0 0 0 0 0 0 d s δ 0 ˜ ˜ u g B B D D K K ( ( − ( ( ( ( | | | | K | | R R R R R R ˜ ˜ ˜ ˜ d d 13 23 ˜ ˜ u u d d 23 12 23 12 = δ δ δ δ δ δ | | | | | | 3 ˜ g = = TeV. In the absence of CP violation, these constraints get | | 10 R R 2 M ˜ ˜ d 13 u ’s to be 13 δ δ | | δ 10 L u [TeV] > 0 m 0 (a) m 2 10 = 300 GeV, and ˜ W . Upper-bound on the flavor violating mass terms . An example of the dominant diagram contributing to the EDMs and CEDMs of light M 1 10 1 EDMs of diamagnetic atoms, such as the EDM of mercury, also provide similar constraints on the squark

0.1

0.01 1 purbound Uppuer limits on the flavor mixingquantities in in the nature. sfermion masses, As thoughsensitive we the to shall EDMs the are see flavor-conserving squark below, the flavor dimension-five violation, proton which decay isflavor rate violation, constrained is which quite by are comparable the to neutron those from EDM. the neutron EDM within the theoretical uncertainty. less. Especially, constraints from conservation, which allows 2.2.2 EDM In the presence of CP violation, the electric dipole moments (EDMs) provide stringent the results of new physicsconstraint fits is of to refs. be [ obtained. Wethis set plot. It is foundmixing that even especially in the caseto of be high-scale SUSY. sizable when Other flavor-violating parameters are allowed Figure 4 quarks in the presence of the squark flavor mixing. constraints, we evolve the Wilson coefficients down to relevant hadronic scale and then use Figure 3 (b): both chirality flavor600 violation. GeV, We choose the “worst” case of the CP phases and take JHEP03(2014)049 , . ]. | 0 R e 45 d m 13 ] to , δ | (2.10) 44 50 , = | 49 L of up quark e Q 13 δ u | ˜ d , ), and thus can be | 2 3 are not constrained. R α e R ( u 13 e . In the calculation, d 23 δ O δ 0 | . In the figure, the ,  m 5 = ∗ and R = | ) (2.11) , e u 13 L ] might be comparable to that d L  δ and CEDM e ˜ e Q H 23 d ], we obtain constraints on Q 13 ∗ L 43 δ µ δ u R e | Q 13 59 45 e d u 13 . δ βδ L e + 0 Q 13 cot u ˜ βδ d e g ] to solve the strong CP problem, the 30 M . cot 2 TeV, and 42 H e g (0 − µ e  M = , we show an example of the diagrams which + H ˜ Im g 4 µ u The CP violating effects induced by squarks are  – 7 –  (1) flavor mixing results in constraints on the d M | 2 e g O 0 20 Im . t 4 0 m M 0 | ) TeV. m m 2 −  u d ln d (10 t eQ 4 0 , respectively, as functions of the sfermion mass scale | O 3 ]. The results are shown in figure 79 π m m . R = 300 GeV, α 4 e d 12 3 46 π δ 4 3 ˜ | α W 4 = 0 6 3 M n = cm [ d | ' − ' · L u u e ˜ Q e 12 d d δ | 26 − 10 , and = 600 GeV, The figure illustrates that | × ˜ R B the charge of up quark. Similar expressions hold for down and strange quarks. e u 4 12 9 . δ ]. M ]. At this moment, both methods have large uncertainty and no consensus has been reached yet. | u 2 48 47 = eQ < | When one imposes the Peccei-Quinn symmetry, the strange CEDM contribution to the neutron EDM However, the contribution of the dimension-six Weinberg operator [ These approximate formulae, in particular that for the EDM, do not work well as squark mass is taken L | 4 2 3 e n Q 12 d δ to be larger, though; inIn such a our case calculation, the we mixing effect include of the the effect CEDM by into the usingcompletely EDM the vanishes becomes in renormalization dominant [ the group case equations. This of may the indicate sum-rule that computation. theately. sum-rule In Therefore, fact, calculation the does contribution nottheory is expected [ include to the be strange-quark sizable contribution from the appropri- estimation based on the chiral perturbation clarify our notation and conventions used inof this EDMs and paper. CEDMs.neglected in In Just the the leading like present order case, calculation. the however, Georgi-Glashow the operator is induced at 3 Proton decay with3.1 sfermion flavor violation Minimal SUSYIn SU(5) GUT this section, we give a short review on the minimal SUSY SU(5) GUT [ we use to estimate the neutronrules EDM, [ which issfermion obtained mass by scale using as the high method as of the QCD sum the flavor mixing parameters from| the current experimental boundpurple, on blue, the red, neutron EDM, and| green lines showWe the take constraints on with Notice that bothlize the the left-handed enhancement and byrenormalization right-handed heavy group squark quark improved mixings masses. method are described required By in to evaluating ref. uti- the [ contribution with the that of top quark, flipare the approximately chirality. give For as instance, the EDM relevant effective operators of the lowestdipole mass moments dimension (CEDMs) of are light the quarks. EDMsincluded and into chromoelectric these two quantities.yield In the figure EDMs and theis CEDMs. given As by illustrated the in flavor-violating the processes, diagram, where the the dominant mass contribution terms of heavy quarks, especially the assumption of the Peccei-Quinn mechanism [ JHEP03(2014)049 ij h , and (3.1) | R , while = 5 and , are in i e u 12 i . Purple, δ 0 β e | m = . The field | L are called the i, j e Q 12 fields, Φ δ C | ¯ ¯ 5 , , are incorporated H | representation; the 2 TeV, tan d 4 R , − e d 13 10 H b ˆ δ 10 | = ¯ and H ˜ g ˆ a = ⊕ j C and | M ¯ 5 Φ L | | | | H b ˆ u e L L L L Q 13 , ˆ a i are in the ˜ ˜ ˜ ˜ Q Q Q Q 13 12 13 12 δ H δ δ δ δ | } i | | | | Ψ j , and singlet leptons, ¯ , = = = = L i | ij e | | | | R R R R R f ij Q ˜ ˜ ˜ ˜ d d 13 12 u u 12 13 e u 13 2 3 ) is the totally antisymmetric tensor δ δ δ δ | | | | δ ]. | 10 ˆ e √ ˆ d = 300 GeV, QE ˆ c 52 = − b V ˆ | ˜ ˆ ( a W ˆ e L  , e M Q 13 H j δ [TeV] ˆ d | u ˆ c j 0 ij m Ψ ) b ˆ – 8 – ˆ a i QU Ψ 2 ˆ e V ˆ 10 d ( , doublet quarks, ˆ c i b ˆ and doublet leptons = 600 GeV, , ˆ a i u i ˜  B ¯ Q d ij M h 3 { 1 4 i . To take the difference into account, we write the relation = i Ψ . The MSSM Higgs superfields, i 1 superfields and their SU(5) partners 10 1 ¯ 5 and Φ 0.1 Yukawa is symmetric with respect to the generation indices

0.01

i purbound Uppuer W ij h and ], the MSSM matter fields are embedded in a 5 51 = 1–5 represent the SU(5) indices; -bosons, and they acquire masses of the order of the GUT scale after the SU(5) singlet up-type quarks, ¯ = 1; , respectively. We take | ··· X L R e d 12 b, ˆ . Constraints on flavor mixing parameters as functions of sfermion mass scale . singlet down-type quarks δ is twelve. Among them, six is for quark mass eigenvalues and four is for the CKM representations, Ψ | 0 a, 12345 L ij m  = f These Yukawa terms are matched to the MSSM Yukawa couplings at the GUT scale. In the minimal SUSY SU(5) GUT, the Yukawa interactions originate from the following | 10 = L e Q 12 H δ Note that the generation basisSU(5) of superfields the Ψ MSSM superfieldsbetween the may be SU(5) components different from and that the of MSSM the superfields as where ˆ with re-definition of Ψ and Φand reveals that the numbermatrix of elements, the so physical we degrees have of two additional freedom phases in [ superpotential: into a pair of color-triplet Higgs multiplets. Thevector gauge multiplet. vector The multiplets new arecalled gauge the embedded fields into introduced an togauge adjoint group form is the broken adjoint into representation the are SM gauge group by the VEV of an adjoint Higgs boson. SU(5) model [ SU(2) the SU(2) the Figure 5 blue, red, and green lines| illustrate constraints on µ JHEP03(2014)049 (3.4) (3.5) (3.2) (3.3) ). In GUT ) denote the ), give rise d M k l ( H u d 3.5 · j a, b, c CKM L V ( i , e ≡ Ca ) is the CKM matrix ij ) H C V e . = 1l ) ˆ f j H GUT ( L Cc DL · M − ( H a i indices, and C ja jb , Q d , H L d ( ) ij CKM ij ia d ij ,V V ) )) u d )) H ) ˆ ]. The diagrams which induce the f · , abc ∗ a i )  GUT V GUT 54 GUT i ij j Q , ) M ( u e , ( M d M ij ˆ GUT } + ( 53 ( d ( f = 1 and ) ˆ j u f ∗ d c ˆ C M ∗ ˆ f L f P ( V ∗ H ij e P V ∗ ˆ CKM ) ) f – 9 – V b j k l V P ( ( Q L = ( = ( Q DL = · − V ) = − ij ij ( a i h a f C ja , QE Q representing the SU(2) i u ( H GUT ) d j u e M abc (  H ia α, β C 3 { d · ˆ ij u i f H ) ,V a i ij are unitary matrices, which play a similar role to the CKM Φ u ∗ ) ˆ Q f ( with P V P DL u ij C ( β ˆ ) = f V 1 2 H u B ˆ f α are diagonal and non-negative Yukawa matrices of the up-type quarks, − + ( QU A e V ˆ = ( f αβ , and  QE ≡ i i V ) , and Yukawa Q Q d , ˆ B f W · is a diagonal phase matrix with det . Supergraphs for color-triplet Higgs exchanging processes where dimension-five effective , QU u ˆ A V P f 3.2 Dimension-five protonNow decay we discuss theset proton of decay formulae via usedinteractions the in of color-triplet the color-triplet Higgs following Higgsto exchange calculation . multiplets, the of dimension-five which the We proton are proton first decay decay displayed give rate. operators in a eq. [ The ( Yukawa Here, ( color indices. Asthat it the can Yukawa couplings be of the seen up-type from quarks and the the above charged expression, leptons are we diagonalized. have chosen our basis so this basis, the Yukawa terms are written in terms of the MSSM superfields as where the down-type quarks, and the charged leptons, respectively, and where at the GUT scale. Then, we have the matching condition as follows: where matrix. In this paper, we take them as Figure 6 operators for proton decay are induced. Bullets indicate color-triplet Higgs mass term. JHEP03(2014)049 (3.7) (3.8) (3.6) . mode. The Wilson coef- kl , ¯ ν ., , ) ) c d l + . ˆ kl f L ) ∗ K · d ˆ f V k c ∗ ∗ → + h symmetry. A set of such operators Q , P V they must include at least two gen- p ( ( )( Y lc j b ij ij R d ) R ) 5 ijkl Q 5 ijkl u kb u,d · ˆ V f O U(1) u O u f i a H ˆ j P f ⊗ R e ( ( Q ijkl 5 L f ( B, ia are defined by C C C and u H H are given by 1 1 abc R g + W,  – 10 – 5 L ijkl abc M M 2 1 5 ijkl e R g, SU(2) . O L ijkl 5 O 5 ijkl θ θ  ⊗ C 2 2 B O C d d ) = + ) = + . By integrating out the color-triplet Higgs multiplets, and L ijkl Z Z 5 6 and L C 5 ijkl ≡ ≡ GUT GUT L = O ijkl 5 M M L R ( ( C 5 5 ijkl ijkl eff 5 L R O O L ijkl ijkl 5 5 C C ) are determined at the GUT scale. To evaluate the proton decay rate, 3.8 , sfermions decouple from the theory, and the dimension-five operators reduce 0 is the mass of color-triplet Higgs multiplets. Note that because of the totally m C . One-loop diagram which yields proton decay four-Fermi operators. The gray dot indi- H , an one-loop diagram which yields the four-Fermi operators is illustrated. Here, M 7 The dimension-five operators contain sfermions in their external lines. At the sfermion to the dimension-six four-Fermi operatorsfigure via the exchangethe of gray gauginos dot and indicates higgsinos. thethe dimension-five In effective mass interactions term and of the blackin exchanged dot an particles. represents invariant form The under four-Fermi the operators SU(3) induced here are written operators is accompanied byficients strange in quarks; eq. like ( the we need to evolve themcoefficients down are to low-energy presented regions in by appendix using the RGEs. Themass RGEs scale for the Here, antisymmetric tensor in the operators erations of quarks. For this reason, the dominant mode of proton decay induced by the and the Wilson coefficients where the effective operators operators are illustrated inwe figure obtain the effective Lagrangian Figure 7 cates the dimension-five effective interactionsparticles; and gauginos black or dot higgsinos. represents the mass term of exchanged JHEP03(2014)049 , → in em . p . 2 (3.9) (  (3.11) (3.10) B  ) A ) U(1) i  i ) ν ν i ⊗ c L ν c L C u s c L d )( )( b L )( b L s d . Let us express b R a L a L s d u a R (3) ijkl ( ( u = 2 GeV, where the , O ( abc abc 2  µ |   abc )  ) i  ) i ,  ¯ i ν ) ) i + . = 2 GeV multiplied by the K Llβ dsuν D udsν ( µ L ( indices for usdν → ( LL L LL p c Lkδ 4). Then, they are matched with , , ( C RL C , ) ) Q 3 . C + |A , + at the electroweak scale are listed in )( Ll Rl ]. Their values are listed in table 2 )  2 e  + L ) ,  ) i Rl 58 · i  b Ljγ LL c Rk e ν 2 p ) ν 2 K i u Q c L C c L c Lk ν = 1 m s c Rk m d )( c L Q u )( I a Liα u )( − b Lj ( )( )( b R b L and Q )( 1 d Q – 11 – ( s ) b Rj b Rj b R · a R I a L  , the effective operators are no longer invariant un- d d s γδ ( ijkl u RL u  p π Z a R ( a Li ( O a Ri a Ri C d m αβ u Q u m 32 ( abc abc  ( ( (   for =   abc abc abc abc abc ) )  i ) = symmetry; instead, they must respect the SU(3) i     µ  ) i as follows: ) I ¯ ν ijkl i Y ( = = = = 5 + ] C udsν usdν K ( 57 ( (3) (4) (1) (2) ijkl ijkl ijkl ijkl dsuν U(1) – ( O O O O RL LL → mode. The effective Lagrangian which yields the decay mode is ⊗ 55 C C L p RL ¯ ν C + + + Γ( are the masses of proton and kaon, respectively. The amplitude K K SU(2) ) = at the SUSY breaking scale. The matching conditions are summarized in i . For the hadron matrix elements of the effective operators, we use the → ¯ ⊗ m ν . Again, the coefficients are evolved down to the electroweak scale according . C + R p B ijkl 5 . By using the results, we can eventually obtain the partial decay width of the K mode as and C i A C.2 C.1 p ¯ ν → ]. + m p 57 and ( ) is given by the sum of the Wilson coefficients at i K By following a similar procedure, we can also evaluate the partial decay rates for other The Wilson coefficients are taken down to the hadronic scale Below the electroweak scale L ¯ ν We have slightly changed the labels of the operators as well as the order of fermions from those presented 5 L + ijkl → 5 corresponding hadron matrix elements. modes. The resultant expressions are presented in appendix in ref. [ p where K matrix elements of the effectivein operators appendix are evaluated. Theresults RGEs presented for by the the step latticeappendix are QCD given calculation [ Here, all ofcondition the for fermions the are Wilsonappendix written coefficients in terms of the mass eigenstates. The matching ators is the written down as follows: to the RGEs. The RGEs below the SUSY breakingder the scale SU(3) are also givenand in all appendix of theabove, fields the in dominant the mode operators of are proton to decay induced be by written the in dimension-five the effective mass oper- basis. As mentioned Here we explicitly write the way oftheir contracting Wilson the coefficients SU(2) by C appendix is summarized in refs. [ JHEP03(2014)049 . 0 m flavor (3.13) (3.12) 1, the  . L 0 e e g Q 13 gives rise δ M ∼ L e Q 13 L proton decay, δ e , , Q 13 ¯ ν δ 2 2 . By comparing

+ β ∗ ∗ L L K e e Q Q 13 13 δ δ → ) ) p ∗ ∗ cs cb . V V τ 1 2 cs cs V V 

, ν ud ud e g µ H V V . Here, the cross-mark indicates ν ( ( µ M 3 3

∗ e 8 d L iϕ iϕ β e Q 13 e e δ 1 e e g g 2 0 2 0 e b sin 2 m m M M  ) contributes to the 2 W 2 W b s e × g is most important; in particular m m m m 3 t t C.5 C C – 12 – L − H H e m m Q e t 10 M M ∗ ], we can see that the gluino contribution becomes × 3 3 β β L e 2 Q 13 α α in eq. ( 22 [ δ 2 2 e g e u α α | sin 2 sin 2 0 & 4 3 4 3 m

s ijkl (3) L -mark. C e ' Q 13 × δ ' − ' −

u d ) ) H ], the charged wino and higgsino exchange processes give rise to τ µ µ 59 udsν udsν ( ( LL LL C C mode is induced by the diagram in figure . Since only the 3 ¯ ν . Diagram which induces the dominant contribution in the presence of the α + K Before showing the results for the full computation, we briefly comment on the features → of other contributions. Therelatively wino small and bino gauge contributions couplings are in compared general with suppressed the by the gluino contribution. The higgsino and other Wilson coefficientsAs are we found have to mentioned be above,the sub-dominant. results the to Here, contribution the we strongly higgsino assume to depends contribution in on be the tan dominant minimal flavor when dominant violation when case, which is found contribution is evaluated as value of the flavor mixing in theto mass the matrix biggest of effects.p Let us estimate thethe significance. flavor The mixing. dominant contribution to When the the flavor violation is small but sizable, e.g., As discussed in ref.the [ dominant contribution toflavor the violation. dimension-five proton When decay thehand, in sfermion the not sector case only contains ofcontribute. sizable the the flavor charged minimal Especially, violation, fermions, the on but the gluino other also contribution becomes the significant neutral because gauginos of and the higgsinos large can Figure 8 mixing, which is denoted by 3.3 Results JHEP03(2014)049 GeV. ∆’s = 100 TeV, 16 0 m = 10 C + + H e µ 0 0 M π π ). We set → → p p 3.3 , and 0 (b) ’s. Red, blue, green, and yellow (d) m δ = + H ]. µ 61 , = 5, 60 β mode is not enhanced because of the same ¯ ν + – 13 – K mode, are considerably enhanced. We will discuss → , respectively. The color bands show the uncertainty from 2 TeV, tan + L p µ − e Q 23 0 δ = π ˜ g → M , and p R e u 13 + ¯ ν δ µ + , 0 L K e K Q 12 δ in the GUT Yukawa couplings defined in eq. ( → , → p L = 300 GeV, P p e Q 13 ˜ δ W (a) (c) M . Proton lifetime as functions of flavor mixing parameters ’s which are not displayed in the figure are set to be zero. Black dashed lines represent the As we will see below, the effects of the other mixing parameters are generally sub- δ = 600 GeV, ˜ B contribute to the protonsquarks decay feel in the such flavor violation,reason. a the case. In such In atheir case, addition, final on when states, the such only other asthe the hand, the feature right-handed the in decay more modes detail including below. a charged lepton in the most of the enhancementflavor mixing from in the sfermion third masses generation does Yukawa couplings. not Therefore, increase the thedominant. contribution any In more. particular,proton when decay the rate flavor is violation rarely occurs changed. only This in is because the the slepton gluino sector, exchange the process does not unknown CP phases M and experimental limits presented by Super-Kamiokande [ contribution has already exploited the flavor changing in the Yukawa couplings to make Figure 9 lines correspond to JHEP03(2014)049 01, . 0 gives 2 TeV, 2 TeV, L  − − ˜ Q 13 = 6 TeV, L δ channel in ) is shown ˜ = = ˜ Q 13 B δ ˜ g ˜ g + 3.3 M µ . M M 0 01, both gluino P . π 0 → , is also important. ∼ . The red bars show R p e d 13 L δ ˜ 10 Q 13 δ ). For small = 300 GeV, , and and = 300 GeV , illustrate the features of the 3.12 ˜ + W , while they can enhance the R ˜ W µ e u 13 10 0 M δ (3) ijkl M yields the most significant effects K ’s in figure can contribute to the decay rate. O δ L and we do not include the running e → Q 13 , while they become significant in the 6 (4) ijkl δ ¯ channel, while other decay modes get ν p O + , ). The red, blue, green, and yellow lines ¯ ν + K + e GeV, 2.5 0 = 600 GeV, K and = 600 GeV, π → 16 ˜ B scarcely affect the anti-neutrino decay modes GeV, while the green bars correspond to the ˜ , we show the proton lifetime as functions of p B → → – 14 – M (2) ijkl R 16 ]. From the figure, it is found that 9 e M d 13 p p = 10 O , as described in eq. ( δ L , respectively. A characteristic feature in this case , as can been seen from the formulae presented in , respectively. In this figure, the uncertainty coming 61 C ˜ ’s in eq. ( , Q 13 L δ = 10 9 H . δ e Q 23 (4) ijkl 60 and δ M C O (4) ijkl can be as heavy as the GUT scale in the case of the high-scale SUSY R H ; without flavor violation, the decay rates of these modes e u 13 O C but also δ M = 100 TeV, 9 H in the GUT Yukawa couplings defined in eq. ( = 100 TeV, 0 , and M , and 0 R P (3) 20 TeV. The bar charts in figure ijkl 0 m e u 13 01. In the region, the proton partial decay rate is approximately m − O δ . m decay mode given in the plot (a), gluino dressing parts become , and 0 , 0 = L ¯ ν e ˜ g Q 12 m  and = + + δ L M , which are induced by the operator , K ˜ H Q 13 ¯ L ν (1) ijkl δ µ = + e Q 13 + → O δ K H p µ . For this reason, = 5, → ]. D β p 20 = 5, = 3 TeV, and The sfermion flavor violation also alters the branching ratio. This can be again seen We also present the results for the Now we show the results. In figure β The color-triplet Higgs mass ˜ 6 W dimension-five proton decay discussedthe above; most in significant the decay casealso mode of viable is the once the minimal you flavor switch violation, on the flavor violation; scenario [ decay rates of selected protonthe decay case modes in for which various wetan take case where the gauginoM masses are ten times as large as the previous ones: charged lepton modes through from the plots (b–d)are in extremely figure small compared withpresence that of of sizable flavor violation. To see the feature more clearly, we show the partial the operators Notice that in theonly gluino contribute exchange to process the the operatorappendix right-handed squark flavorsuch violation as can the plots (b), (c),is and that (d) the in right-handedThis figure squark is flavor because violation, when such the as final state of proton decay includes a charged lepton, not only dominant when proportional to the fourth poweron of the other hand, thethus higgsino the dressing lifetime contribution dominates hardlyand the depends decay higgsino on amplitude, dressing and the contributionssignificant flavor are cancellation violation. comparable between to them, When depending each on other, the which GUT may CP result phases in a effects on the gauginopresented masses. by The Super-Kamiokande blackstrong [ dashed impacts lines on represent thecontribution the of proton experimental gluino lifetime limits exchange forcase processes each to of decay the the channel. proton decay It rates. results For from instance, the in the large selected flavor violating parameters correspond to from the unknown phases as color bands. Weand take tan JHEP03(2014)049 - X . This decay is 2 ¯ ν the proton + 7 K ), 5 ’s. Compared to → δ p , have sizable decay rates. To , charged leptonic decay is also ¯ ν . + ¯ ν π + 3.6.3 K → ) and the EDM (figure p → 3 , in the minimal flavor violation case, only p , suffer from the CKM suppression. We will + and 10 e 0 ¯ ν – 15 – + K K → p are relatively minor. Another important uncertainty → 1 . As a result, less (up-)quark EDM is predicted in the p in the minimal SU(5) GUT. L and ˜ mode is the severest, and thus it may offer a good prove R Q 13 . Hence, such decay modes can be regarded as characteristic e u ij δ enhances the decay rates of the charged lepton modes, rather + δ + µ R µ ' 0 ˜ u 0 13 π 3.6.3 L δ π e Q ij δ → → p -boson exchanging process. Since the process is induced by the gauge p X , we show the upper-bound on the size of flavor-violation 11 After all, in the presence of sfermion flavor violation, which can naturally be sizable in Now let us look for a specific signature of the proton decay associated with sfermion Notice that we expect 7 significant uncertainty comes fromprovides error a of factor 10 the uncertainty hadronparameter for matrix inputs the proton elements shown decay in in rate.comes table table from The effects the of short-distance the parameters. experimental In addition to the color-triplet higgsino mass large parameter space of the minimal SU(5) GUT model. 3.5 Uncertainty of decayHere rate we briefly discuss uncertainties of estimation of the proton decay rate. The most absence of observation of protonIn decay figure gives constraints onthe the constrains sfermion from flavor violations. thedecay meson stringently mixings constrains (figure minimal SU(5) GUT. In other words, future discovery of the quark EDM’s can exclude a way of investigating the structure of sfermion sector. 3.4 Flavor constraints fromAs proton we decay have seen above, thethe sfermion dimension-five flavor violations operators. accelerate the Therefore, proton decay in rate the from context of the minimal SU(5) GUT, the the sfermion sector. the high-scale SUSY scenario, a varietybe of proton probed decay in modes may future liemight proton in shed decay a region light experiments. which on In can SUSY consequence, even proton though decay it experiments is broken at a relatively high-scale, and provide see this feature in section of extra flavor violationconstraint on if the they arefor the actually sfermion observed. flavordetected violation. Among in future them, If experiments, the the it decay experimental may process suggest as the well existence as of sizable the flavor violation in distinguish the flavorcharged violating lepton contribution decay from modes.induced it, via therefore, As the we showninteractions, should in the section focus CKM matrix onboson is the exchange the contribution, only the source decayfinal for modes states, the which such include flavor as violation. different generations Thus, in in their the gives little contribution; than those of the anti-neutrino modes such as flavor violation. As onethe can anti-neutrino see decay from modes, figure on the proton decay rate, contrary to the flavor violation in slepton mass matrices, which JHEP03(2014)049 40 40 40 40 10 10 10 10 38 38 38 38 10 10 10 10 36 36 36 36 10 10 10 10 1 5 5 5 . . . . [year] [year] [year] [year] = 0 = 0 = 0 = 0 1 1 1 1 34 34 34 34 − − − − L L Γ Γ Γ Γ L 10 10 10 10 R ˜ ˜ ˜ ˜ L 13 Q Q u 13 12 12 δ δ δ δ (f) (h) 32 32 32 32 (b) (d) 10 10 10 10 is taken, while green bars correspond 30 30 30 30 9 10 10 10 10 + + + + + + + + + + + + ¯ ¯ ¯ ¯ ν ν ν ν + + + + µ µ µ µ e e e e ¯ ¯ ¯ ¯ ν ν ν ν + + + + µ µ µ µ e e e e + + + + + + + + 0 0 0 0 0 0 0 0 0 0 0 0 + + + + 0 0 0 0 K K K K ηµ ηe ηµ ηe ηµ ηe ηµ ηe K K K K K K K K π π π π π π π π π π π π 40 40 40 40 – 16 – 10 10 10 10 38 38 38 38 10 10 10 10 36 36 36 36 10 10 10 10 5 5 5 . . . [year] [year] [year] [year] ]. = 0 = 0 = 0 1 1 1 1 34 34 34 34 − − − − Γ Γ Γ Γ L 10 10 10 10 R R ˜ 61 ˜ e ˜ 13 u 13 Q 23 , δ δ δ 60 (g) (e) (c) 32 32 32 32 (a) Minimal FV 10 10 10 10 30 30 30 30 10 10 10 10 + + + + + + + + + + + + ¯ ¯ ¯ ¯ ν ν ν ν + + + + µ µ µ e e e µ e ¯ ¯ ¯ ¯ ν ν ν ν + + + + µ µ µ µ e e e e + + + + + + + + 0 0 0 0 0 0 0 0 . Dependence of the proton decay modes on the flavor structure. Red bars show the 0 0 0 0 0 0 0 0 + + + + K ηµ ηe K K K ηe ηµ ηe ηµ ηe ηµ K K K K K K K K π π π π π π π π π π π π Figure 10 case where a similarto set the of case parameters inrepresent to the which Super-Kamiokande those the constraints at in gaugino 90of % figure masses CL Hyper-Kamiokande are while [ yellow ten lines show times the future as prospects large as the previous ones. Black lines JHEP03(2014)049 GeV, . The 16 1 2 TeV. (b): = 10 − C H = ’s are obtained. M δ ˜ g ’s and ∆’s are zero. δ M mode. The SUSY mass ¯ ν + K (b) Heavy gauginos → from proton decay. Red, green, blue, p δ = 300 GeV, and ˜ , respectively. We take W R e u 13 M δ . Red region displays the uncertainty from the error , and – 17 – 9 L e 20 TeV. The shaded gray regions show the case that Q 23 δ − , . Blue represents uncertainty from the error of the input = 600 GeV, L = e 2 . The red region displays the uncertainty from the error Q 12 ˜ B ˜ g δ 9 , M M )) are taken so that the strongest bounds on L e Q 13 δ 3.3 . Green is the theoretical uncertainties. 1 = 5. (a): = 3 TeV, and β ˜ W , we show the uncertainties in the case of (a) Light gauginos (defined in eq. ( M 12 P and tan . Error estimation of the proton decay rate. We show (one-sigma) error bands. The . Upper-bound on the flavor violating mass terms 0 m , the proton decay is quite sensitive to the Yukawa and gauge couplings at the high- = 6 TeV, In figure = C ˜ H B H one-loop RGE. To estimate possiblewe contributions also from study higher order (incomplete) corrections two-loop we level ignore, RGEs. spectrum is same as thatof in the figure matrix elements, while blue represents that from the input parameters in table parameters shown in table M energy regions. Inthe our sfermions analysis, and however, GUT we do sector, not and include thus finite our threshold result effects cannot from achieve accuracy beyond the Figure 12 SUSY mass spectrum is sameof as that the in matrix figure elements shown in table GUT phases Figure 11 and purple lines correspondµ to M the proton decay rate conflicts the current experimental limits, even when all JHEP03(2014)049 (3.14) . Therefore, we expect -terms corresponding to A Rj u 2.2.2 . , we show an example of such ∗ R 3 13 ˜ u j δ L 3 ˜ Q i Rj δ ˜ u ∗ ˜ g β R j ˜ u 3 M δ R tan ∗ H ˜ t 2 0 µ H 3 α t πm L – 18 – ˜ t f 4 L 9 8 3 ˜ Q i δ Li ∼ ˜ ˜ g Q MSSM ). However, in the present parameter space, it is difficult to ij δf 3.4 Li ) uncertainty of estimation of the Yukawa couplings at the GUT ]. Q 62 GUT . An example of threshold corrections to Yukawa couplings. ) e f − d f (( Figure 13 O The minimal SUSY SU(5) GUT predicts the unification of down-type quark and lepton will be done elsewhere [ 3.6.2 Contribution fromUp soft to baryon-number violating now, operator wepersymmetric. only consider However, the through dimension-five the effective supergravity operators effects, which the are exactly su- achieve the successful Yukawa unification. ThisYukawa means couplings, that such we as omit those some corrections fromoperators to the induced the GUT-scale at particles the orthere Planck some is scale. higher-dimensional an With ourscale. ignorance It of may such significantly affect corrections, the we prediction expect of the proton decay rate. A detailed analysis tions to the Yukawa couplings.EDMs in However the note presence that ofthese similar threshold CP processes effects violation, may to as be also discussed smallthe give as in current rise long section limits as to from we consider the the EDM parameter experiments. region which evades Yukawa couplings as in eq. ( corrections. In this case, the size of the correction is roughly given by Therefore, large flavor violation in the sfermion sector possibly leads to significant correc- In the present analysis, wesfermions as ignore well the as the threshold GUT-scale corrections particlesdepending or to on some the the Planck suppressed Yukawa parameter, couplings operators. thesethreshold from However, corrections corrections may at get the significant. sfermion mass Let scale. us first In discuss figure the alter our present analysis in the subsequent subsection. 3.6 Possible additional corrections Here, we consider additional corrections which may be3.6.1 sizable in some particular Threshold cases. correction to Yukawa couplings green band shows theresults theoretical with uncertainty, the which one- we and regard two-loop as RGEs. the We difference will between discuss other contributions which may JHEP03(2014)049 . 14 (3.15) (3.16) .. c . , the dimension-four + h 2 θ 2 / ]. In this formalism, the ∗ Rl 3 e e 68 m – ∗ Rk . e u -boson, exchange processes to 65  τ X ∗ Rj ν R e d = 1+ 5 ijkl i ∗ Ri O e u Σ , should be of the third generation. h 2 R ijkl / 5 ∗ e d 3 L 14 e C τ Q 13 e δ ν m Hc + e b M R ijkl L 5 f C ijkl e actually vanishes. After all, the contribution g B O − – 19 – 14 ]. This can be readily understood by means of e e t b L ijkl Ll 5 ∗ ∗ -term VEV as L L e e e L Q Q 13 23 C 64 F δ δ ,  Lk e e u s 1 63 Σ e Q θ Lj 2 e d u d s Q Li Z e Q 2 ), compared to the usual one loop contribution. This effectively / 0 3 Hc m -term contribution is really suppressed in the presence of large flavor /m ˜ g M A 2 L ijkl M C )( 2 − π = -boson contribution (16 . Contribution of the soft terms for the dimension-five operators to proton decay, which / soft X 2 L g of the soft termscouplings, could and not thus can use be additional safely enhancement neglected3.6.3 by in the the present third calculation. generation Yukawa Next, we discuss the contribution of theproton SU(5) decay. gauge In boson, this case, the effective Lagrangian is expressed in terms of the dimension- exploited via the flavorTo violation. make Such the an most examplevertex, of for which the the is enhancement, process illustrated allNevertheless, is as the such shown a fields in a gray included figure dot vertextherefore in in the the is diagram figure effective forbidden presented interaction by in figure the antisymmetry of the color indices, and with the exchange of gauginos andfactor higgsinos. This contribution isresults suppressed in by a additional two-loop suppression factortrivial in whether the the case of anomaly-mediation.violation, However since it additional is not enhancement of the third generation Yukawa couplings can be Then, after the compensator getssoft-terms the are induced. The leading terms are given as These soft terms also generate the proton decay four-Fermi operators via two-loop diagrams the superconformal compensator formalism ofdimension-five supergravity operators [ should be accompanied by the compensator Σ as Figure 14 turns out to vanish. these operators are also induced [ JHEP03(2014)049 = e g M (3.21) (3.17) (3.19) (3.20) (3.22) (3.18) GeV, and other , -boson. Note that 16 , , = 300 GeV, 8 β l X L . f c W ijkl ijkl 6(1) 6(2) . ). Further, there is no
† l = 10 M α k C C e c Q X
3.2 ]. 3.20   † k G 3 M 8 3 8 3 u 69 g 2 G − − 3 , e g   , 2 3 3 B π π − 0 jl . is the mass of 6(2) ijkl , α α 4 4 g e ) , 2 3 ) ) ∗ = 600 GeV, O X 0 0 B − jl V 0 e e B δ ( m m g M b ijkl 6(2) ( ( 3) + 3) + 2 3 ik ik
M C e † j δ δ − − i i ( ( d ijkl ijkl 6(1) 6(2) bβ j + -boson exchanging process the CKM is the 2 2 iϕ iϕ π π C C a Q e e α α 4 4
X † i 2 2 X X aα i 2 2 6(1) 5 5 ijkl -boson mass to be + + u g g – 20 – ) = ) = Q O M M X 0 0   αβ − − m m αβ  ijkl 6(1) ( ( 11 15 23 15 = 100 TeV,  = = C − − 0 abc ijkl ijkl abc (1) (2)   = m C C ¯ ¯ ijkl ijkl 1 1 6(1) 6(2) θ  θ π π 2 2 α α 4 4 eff 6 = 5. From the figure, we see that the decay rates of the C C   L θd θd β 2 2 = = d d Z Z ijkl ijkl 6(2) 6(1) = = C C . Here, we set the d d dµ dµ , and tan 15 6(1) 6(2) ijkl ijkl 0 µ µ O O m = H is the unified gauge coupling constant and µ 5 g Now we evaluate the decay lifetime for various modes, which are summarized in the The coefficients are evolved down according to the one-loop RGEs, The two-loop RGEs for the Wilson coefficients are also given in ref. [ 8 2 TeV, as mentioned above. Thisonly is source because of in the the flavorroom violation, for which the can flavor mixing beIn effects seen this in from sense, the eq. sfermion the ( mass prediction matrices given to here modify is the robust. decay rates. bar chart in figure parameters are taken as− follows: modes that contain different generations in their final states are considerably suppressed, operators as The rest of the calculation is same as that carried out in section At the SUSY breaking scale, the coefficients are matched with those of the four-Fermi the proton decay rate. Moreover,on it the is new found phases thatthe the appearing overall resultant in phase. amplitude the does GUT not Yukawa depend couplings, since the factors only affect where the results do not suffer frombreaking the terms. model-dependence, In such as this the sense, structure the of SU(5) the gauge soft interactions SUSY provide a robust prediction for By integrating out the superheavy gauge bosons, we obtain the Wilson coefficients as where six effective operators: JHEP03(2014)049 ]. 70 = 5. Black = 100 TeV, 0 β m 40 ], and observations 10 73 , close to unity would be 25 GeV, and tan e Q 13 16 δ 38 ’s are close to unity. As for δ 10 ), P = 10 M X ( M O 36 , -boson exchange. We take 0 = 10 X m ]. C H = matters is to be constrained while that of 62 , their impacts are small. In terms of the [year] M H R ˜ d 1 µ 34 10 δ − Γ 10 – 21 – and 2 TeV, R − ˜ e ]. 32 δ = 10 , 61 e g , GeV. Even if L may be smoking-gun signature of sfermion flavor violation. ˜ L M 60 16 + δ µ ], gluino decay in collider experiments [ 0 π 30 ], we can extract insights to the structure of sfermion sector as 72 , 10 affects the proton decay rates most significantly. The constraint 74 → 71 p ˜ , Q 13 + + δ = 300 GeV, + ¯ ν + µ e ¯ ν + µ e + 0 0 + 26 + 0 0 is around 10 , f W K π K ηµ ηe K π π C 24 M H M . Lifetime of each decay mode induced by the We also have discussed possible corrections to the proton decay rates. These corrections Further we have found that the flavor violation changes the proton decay branch. The Other mixing patterns in the left-handed squarks, as well as those in the right-handed = 600 GeV, e is not. This may be consistent with observed large flavor mixing of neutrinos [ B ¯ of gravitational waves [ well as the underlying GUT model. are uncertain, unless wescope clarify of the this whole paper picture and of will the be GUT done model. elsewhere [ This is beyond the decay pattern of protonlepton reflects modes the such as sfermion flavorCombining structure. indirect probes In particular, ofmeasurements [ the sfermion charged sector via, e.g., the low-energy flavor and EDM up-type squarks also affectthe the proton other decay sfermion modes, violation, SU(5) if GUT these matters, the5 flavor violation of in the minimal SUSY SU(5)left-handed GUT squark model. Weon have it found from that the the protonHiggs flavor decay mass violation bound of is the strongerconfronted than with that the from current the experimental EDMs observations. when the triplet prospects of Hyper-Kamiokande [ 4 Summary and discussion In this paper, we have studied the impact of the sfermion flavor structure on proton decay Figure 15 M lines represent the Super-Kamiokande constraints at 90 % CL while yellow lines show the future JHEP03(2014)049 ]. 82 5 ) Z ) [GeV] m b ]. We adopt ) 18(3) ( . ]. 2 1184(7) 3 m . 4 78 ( a 58 0 b m 0.054(11)(9) 0.044(12)(5) 0.076(14)(9) 0.093(24)(18) ]. 0.042(13)(8) 0.036(12)(7) 0.111(22)(16) 0.098(15)(12) [GeV] 81 ] and CMS [ – 2 / 77 i − i − i − i i i − i 1 i ) [GeV] 13 75 82(16) p p p p p p , p [MeV] . p c | | | δ − | | | | | ) L L L L L L 227(61) 275(25) τ L 41 L m . . µ s d s u d u ( u u 1 1 246.21971 m c R L G R L R L 1776 R L ) ) ) ) ) ) 2 ) ) m √ us us ud ud ds ds us us ( ( ( ( ( ( ( ( ( | | | | | | | | 0 0 + + + + + + K K K K K K K K Matrix element Value (GeV h h h h h h h h 5) . 13 [MeV] θ – 22 – ) [GeV] [MeV] 2 5(2 . 1875(21) Z µ sin . We take an average of the top mass measured by . 00362(12) 93 . 2GeV s m m 1 91 0 105.6583692 m . Physical parameter inputs [ ] as the electroweak gauge boson masses. We use the PDG 0.103(23)(34) 0.146(33)(48) 0.133(29)(28) 0.188(41)(40) 0.015(14)(17) 0.088(21)(16) 79 23 ] and the Higgs mass by ATLAS [ [MeV] θ [GeV] [MeV] i − i 76 i − 367(7) i i i 70(20) . p p p e . p p | sin p | W Table 1 | | | | 04173(57) 4 L L 80 . 2GeV d L L m L L m d d 0 0.510998918 u u u u . Matrix elements obtained by the lattice simulation in ref. [ m R L R L R L ) ) ) ) ) ) ], we set the weak scale SM parameters. ud ud ud ud ud ud 83 ( ( ( ( | | ( ( | | | | 0 0 + + 0 0 Table 2 12 π π π π η η [MeV] Matrix element Value (GeV h h h h h h [GeV] [GeV] θ 40(45) 24(64) ] and Tevatron [ . . 15(15) . sin pole h pole t 22535(59) 75 2 . 2GeV u 125 173 0 m m m LHC [ the fitting result ofaverage Gfitter of [ the light quarkby masses using and estimate the the four-loopFollowing Yukawa ref. couplings RGEs [ for and the light three-loop quarks, decoupling effects from heavy quarks [ A Input parameters In this section, we list theparameters set are of summarized input in parameters which table. we use in our calculation. The SM Acknowledgments The work of N.N. ismotion of supported Science by for Research Young Fellowships Scientists. of the Japan Society for the Pro- JHEP03(2014)049 ). , 3.9
, (B.1) (B.2) (A.1)  0 ikjl (3) L C ijkl 5 C + 0 ], the proton l l ) . 58 kjil † (3) e jkl L 0  f i 5 C l e 0 C f l 0 + ) = 2 + 1 flavor lattice i i d ) f f + ( . † d † jikl d (3) l N i 0 f f C i p d L | 0 f ijk (2 5 i c 0 ) u 4) C + u 0 L R f k − † ijkl u 5 k † P ( u . f ) ) C f 2 π † u b d , , α 4 f d f (2 + d kjil (4) R k f + 0 ijkl ijkl jkl (1) (2) C + ( R 0 k i = 2 GeV. In the case of the other two 5 C C + ) CP 4) u   C † L ijkl (3) µ u ijkl − f 5 aT f ( † C 4) 4) u + u u C 1  f π ( − − f α 4 ( (  R 4) (2 2 3 ijkl 3 3 l 5 abc π π 0 g + (  − + α α 4 4 | C ( 8 R – 23 – 0 ijk kl 5 3 π 0  + + π L − α 4 ijkl C 2 3 (4) h ij 5 2 g 2   C C + g 8 = 0 9 2 9 2  6 j j i j 0 − − − 3) + 4) ]. j ) p − . In the table, we use an abbreviated notation like | 2 ) † 1 d   − − e L 2 g 1 62 f ( ( 2 2 2 f π π g u d 2 3 † 5 α α e π π 4 4 f 12 2 5 R f α α 4 4 ) − − + + + (2 + + † ud u   kl ( 0 f   |   R u 0 2 2 ij 5 1 5 6 5 11 10 23 10 f π π π C 1 1 h quarks are degenerate in mass respecting the isospin symmetry. The − − − − 16 16 + + ( d     1 1 1 1 π π π π α α α α 4 4 4 4 ) = ) =     and µ µ ( ( u = = = = ] R L ijkl ijkl 5 5 57 ijkl ijkl ijkl ijkl (4) (1) (2) (3) C C C C C C ∂ ∂ ∂µ ∂µ ∂ ∂ ∂ ∂ ∂µ ∂µ ∂µ ∂µ Next, we evaluate the RGEs for the coefficients of the four-Fermi operators in eq. ( We also need the hadron matrix elements for the calculation. In ref. [ µ µ µ µ µ µ Here we neglect theinclusion of contributions the of Yukawa interaction changes theanalysis the Yukawa proton will couplings. decay be rate done by In elsewhere about some 10 [ %. parameter Detailed region, We have [ in terms of thewave-function superpotential, renormalization the of renormalization eachnon-renormalization effects theorem. chiral are We superfield readily derive obtained them in at from the one-loop the operators, level as thanks to the B RGEs of theIn Wilson this coefficients section,violating we operators. present First, the we giveIn RGEs the this for RGEs of case, the the since Wilson dimension-five proton the coefficients decay theory operators. of is the supersymmetric baryon-number and the effective operators are written The first and secondThe parentheses matrix represent elements statistical are and evaluatedcombinations at systematic of the errors, chirality, scale the respectively. of matrixthe elements parity are transformation. derived from the above results through decay matrix elements are evaluated usingQCD, the where direct method with results are summarized in table. JHEP03(2014)049 , (B.6) (B.7) (B.8) (B.9) (B.3) (B.4) (B.5) , 1375 8004 = 2 GeV. −  430 µ 2001 π π − 29 29  23 23 π , π , 29 29 f 23 23 ) + ) +
, N b Z ) F µ m ) + m ) + +9∆ ( ( C ( 2 b f Z s b s C N α m 18 α m  ( + 2 (  2 s s ) f 42+4 2 s α . α π − N α  c 2047 1 11550 2 (4 b 3) N − 173 825  /   3 − 10 ) π 10  ) 0 77 µ 50 π π − + ∆ + µ ( ( 77 77 s 50 f 2 50 π c s ) 77 α 50 α N N 2 Z 2 4 9 b b ] as denotes the number of quark flavors, and ) + 3 m 34 b ( + ) + + is the quadratic Casimir invariant defined by f + 84 b − m (2 GeV) 1 C 1 ( N F 3 s m C 14 = πb – 24 – (2 GeV) + ( C πb α s s 4 2  4 (2 GeV) + ≡ α α s  +  L α 1 2 257 (for253 ∆ = 0) (for ∆ = b . . s 6 A  π 23 1 1 − α 4 , b 6  ). The solution of the equation is 23  4 ( ) ) f )  )  b Z 0 ) RL ) N = µ − b µ m Z ( m 2 C ( ( L s ( ( s s m s m − α ( 3 A ( α α α s ) = s c LL   µ α α N ( C 6  25 = C 11 6  25 ) ) 0 ∂ −  . By using the result, we can readily compute the long-distance factor µ ∂µ µ ) c ( ( b µ 3) for = ) N C / C b m 1 2 ( b defined by / 10 s m 3. Numerically, (2 GeV) ( 2 α 1) − s s / b (2 GeV) α α s − 10 is the strong coupling constant, = 3 is the number of colors and  α 2 c − c s and  = N α N 1 ( = L b Finally, we evaluate the long-distance QCD corrections to the baryon-number violating ≡ A L A F C Matching conditions Here, we present the matching conditions for the Wilson coefficients. for ∆ = for ∆ = 0, and as follows: where C with where ∆ = 0 (∆ = dimension-six operators below the electroweak scaleThey down are to calculated the at hadronic scale two-loop level in ref. [ JHEP03(2014)049 .  ) } 0 l 0  Jj

) L )

(C.5) (C.2) (C.3) (C.4) ijk . 5 ∗ (C.1) , d Ll kl 0 ) f C Jj } j . ) L 0 L  Q ¯ i − 5 d Ll } } 0 R ) R Ll

C R ( , ( ) T Jj Ll ( e jil ] ¯

e L J Ll 0 − ) ) L 0 f J 0

l ) l k 0 ∗ 5 , j ¯ e R e 0 T d ¯ ) e j i ) ( Kk

f 0 C . ) Jj f

R † L ) R ¯ ¯ L ( ) † Q L L † d d ( 0 ¯ u ( kj Jj 5 l R 1 2 Jj Q L ) R R R R ) ( L  ) 0 R ¯ C ( 0 ( d ( ( l ¯ † e R l ( ( ) Q J L ( ) Ii Ii R )+ ¯ 0 R † 0 e Kk Kk 1 2 K ¯ † e ) 0 l ) ( J R j ( 0 ) ) l 0 ) ∗ ( ) ¯ R 0 u u J k ¯ R j u ¯ Q 0 ( † d † L ) f J ( ) )+ j L R 0 Kk ¯ R † u ijk R ) 5 j ( Q R ) R † Q ( ¯ kl ( Jj ) † d ( Kk ( I ¯ u 0 R C ) 0 R ) R j K ( i ¯ † , Q K Ii L ( R d 0 0 ( R 0 ) T i − ) u ( 5 I Kk k ( Ii e k R ¯ 0 0 † u R Ii B f ) ) ) T i ) ( C | u Ii ( ) K ¯ ∗ ¯ jl u † ) u ¯ u u ) R 0 f 0 † Q J Ii Q ( − f ¯ 0 k L u ¯ † Q u R R ) R l j ik ) 5 R  ( ( ijkl 0 ( Q R (3) ) for the dimension-five operators R R ( Q ¯ R ) † u ( I ¯ C ( † d ( ( Ii K 0 R kj J C  I ( 0 L R i I 0 ) R ¯ { I 0 ( ) 2 d e 0 i R 0 ( 5 ) k i  ( ) ¯ i u R i ( L ) ) I ¯ + † K u ijkl ) 5 J  ) C 0 0 ,

¯ e † u  2 L R i ) ¯ † Q † ( u e ¯ , m † u k 2 Q R ) ( ) C e I L )  Ll ( R f W B I R R L R ¯ 2 e ¯ | | e ) e † 2 u Q ( 0 ( ¯ ( , m  2 e † Q e ( i

{ , m ) L )  R {  ) K I R ) ¯ ( ) ) † u Jj , m K , m R e ( ijkl ijkl and e L (3) (4) 2 Q , m 2 Q ) J J J ¯ (  e 2 u ¯ 2 e e { e ∗ K R ¯ ¯ B J 2 d 2 d e e ) e C C ) 2 Q Q L ¯ ( ¯ e 2 u 2 d e 0 J , m L L l , m  M R , m , , e , m ijkl ) 5 2 Q ) e + + ( e 2 L , m , m ( I B , m ∗ H , m , m † L K I I e e ¯ e 2 u J K C I e e H H g g F e 0 µ ¯ ¯ ¯ ∗ e e e e 2 u 2 u 2 u ∗ | | | | ¯ B e B 2 u M ) e – 25 – R , m j ( 2 Q , m 0 ( ( ) I , m M K F M e , m , m , m F 2 Q † Q ( , m ) ( ijkl ijkl ijkl ijkl e e ∗ lkj (1) (2) (3) (4) B , m 2 Q ) ∗ l 0 Kk e g I 0 0 H H R i e R F ) F ¯ i B C C C C e 2 u ( ∗ 5 ) 0 M , m represent the contribution of higgsino-, gluino-, wino-, ) µ µ M Q jil 0 ( , m j ( ( L e g 0 ( Ii 0 L C M kj 5 e ij ) k R , m ( 5 k ∗ H F B F F 0 0 F ( M ∗ ) C ) ) ) = ) = ) = ) = e g − Q µ ) C ( F 0 il j kl R j R 0 0 0 0 0 ( K 0 0 ) 0 ∗ ∗ 5 R 5 j 0 l − j M F k 0 li − 0 ( 0 k F lkj ( 0 ) m m m m 0 i l C C 0 I ) ) l 0 0 0 ( ( ( ( k il i 0 R 0 l R R 0 kli j F R , and i kli L 0 i ∗ R ∗ ∗ † 5 Q 5 5 0 − ∗ 0 ) − kj 5 ) ∗ 5 5 L jil j L L C 0 C C R ki 0 5 0 † Q C L ijkl ijkl ijkl ijkl f kj ijk C W (1) (2) (3) (4) 5 5 ij C ( k 0 5 li ( kj C l , 0 R −  − − C C C C C C 0 0 − − C R ( ) k ( e g R 0 k j il R R 0 d ∗ − 5 0  j ij L ∗ L − ∗ , 5 5 0 0 kl − ) Y e l 0 2 L L lk C 0 C e kl lkj C J 0 0 j R kl H 0 0 L l Y i 0 ( k kli ( R 0 R R e i 0 u i j R 2 Q 5 ∗ − ∗ ∗ 5 j L 5 5 0 L R ∗ R , m 5 L 0 i Y 5 e L i e C j 5 C Q C C  ijk 0 K 5 C ( ( Y ( ( C Y , m Y ( C 1 e π 2 Q L [ lk I C 2 2 R R (  R 0 2 2 α 4 ( (2 ) ) e i 1 Q u d u R 2 Q π 3 3 π π ∗ 2 2 5 1 1 π π 6 5 Y Y Y Y α 4 , m α α 4 4 ) ) C 6 5 1 1 , m (4 π (4 π − 4 4 3 3 ( f + + + + W f (4 (4 W − − = = − + + M ( M e e = = = = B B ( F | | e e g g 2 e e F π | | H H f W | | | α 4 2 ijkl ijkl π (3) (4) α 4 ijkl ijkl + C C (3) (4) ijkl ijkl (2) (1) ijkl (3) C C = C C C and bino-exchanging diagrams, respectively. They are computed as follows: where the subscripts At the sfermion mass scale,are the matched coefficients to those for the four-Fermi operators. The results are given as C.1 At SUSY breaking scale JHEP03(2014)049 ), L = 1 kjil 5 (C.8) (C.9) (C.6) (C.7) (C.10) C k − . = L ijkl 5 i . = 0 C , is the VEV of

)]  v k ( 2 2 2 ¯ e 2 u (3) kjil 0 m M , m i  − O ¯ e 2 u v/m ln 2 , m (3) kijl e g , M , ) 2 2 M 3 ) , ( − O e − 3 2 Q m ¯ , e 2 u ) F 2 2 2 2 (3) jikl , m m − , m m 2 = 0 2 O ) e 2 Q ¯ − j e 2 u − ¯ 2 d e + δβ 2   , m q , m 2 2 1 1 , m 1 e i αγ )( (3) ijkl ¯ e 2 Q 2 u ¯ m  e 2 u 2 1 M O m m m  +  , m ln e g − L iM αδ ijkl 5 2  2 – 26 – M q ( C γβ M 1 2 vanish; they become proportional to ( )( 2 1 = diag( = diag(  F 2 e g − )[ ¯ † u m | † Q − = 2 1 M R R t klij γδ R ijkl m ) L (3) ∗ − ) are unitary matrices which diagonalize the correspond- 5  (3) ijkl e  R 2 Q ¯ e C 2 C ¯ e u 2 2 2 O αβ q ¯ m ) d,  ( m − m ( q Q 2 L ¯ u, ¯ u 4 kjil 5 − M π R d ilkj R R 2 1 C , they again vanish in the degenerate mass limit. On the other ∗ 5 e g | m Z C − Q, L, ( = ≡ ijkl (4) L = ∝ ijkl 5 ) ) is a loop-function defined by C 2 2 e g f 2 2 C | ( ( , m f , m ijkl ∝ (4) 2 1 2 1 R C (3) ijkl O M, m M, m ( e g ( | F F ijkl (3) In the case of From the above expression, it is found that in the limit of degenerate squark masses or C Since charm quarkcomponents, is which heavier prove than toarguments proton, be can zero all be as we appliedresult, one to one have can can the to find see case that considerin from it of this is is the the limit. the the above bino charged expression. wino and and neutral-wino higgsino Similar contributions. contribution that does As remain a case, and thus they cansquarks. remain Their sizable contribution when to there the exists proton mass decay difference rate among turns right-handed out to be negligible, though. and the Fierz identities. hand, they may not vanish when there is no flavor-mixing in squark mass matrices; in this The last equality immediately follows from the identity the Higgs field) such as the left-right mixing terms inno flavor-mixing, sfermion the mass coefficients matrices. and thus and so on. In the calculation, we ignore the terms suppressed by The matrices ing sfermion mass matrices; for instance, Here, JHEP03(2014)049 ) . 3.10  (D.1) (D.2) (D.3) (D.4) mode ) (C.11) ¯ ν  Ri l ) + c R . Li , K l u , in eq. ( i i c L  p )( p | 2 | u | b R → LL L L ) s )( , u C u + i contribute to the b L p a R l L )] L s u ) 0 ) Z ( a L K and u , (3) us ijkl us m ( ( abc ( ( ) mode is given as | )] | O i  0 → 1 0 +  RL Z abc )  p i kj K C (3) K m  , µ ( h h ( ) C and ) ) + R i i i i e 11 − usul . |A j , (2) ( ) = (1) ijkl ) ) usul C Z usul + usul ( O Z Z + i ( , RR ( l 2 m | ) ( m m C ( ) LL ) + ( ( i Z LL RR 1 + i + i C Z + ji ki l C C l m , , 1 1 jk 0 0 (3)  ( m k j ) ) + (3) (3) i ) + + ( C 1 K Z Z i  K [ C C j i i 1 ) Ri 2 2 2 1 (3) l j p p m m k k k | | 1 (2) ( ( Li → → c R ) ) ) . C l , L L 2 C ji ji u p p ) ) c L u u [ j ( ) 2 12 11 u )( Z Z (1) (1) R R j L – 27 – ) ) b L CKM CKM CKM ) )( C C m m s V V V 1 2 ( ( |A b R us us ( ( ( j j i i a L  CKM s ( ( ) ) 1 1 1 u | | 2 j j j V a R ( CKM 0 0 ) ) ) 121 121 ( (1) (4)  u V ( K K C − C 2 p abc K 2 CKM CKM h h  ) ) V V  m . The result is abc i i CKM CKM CKM m ( ( ) ,  Z i V V V ) = ) = ( ) = ) =  i i i i − − ) − m i 1 usul usul ( ( = usul  ) = 0 ) = ) = ) = ( ) = ( ) = ( usul usul usul usul ( i i i i i i ( ( ( ( p µ π usul RL LR ( LR C C m 32 LL RL LR RR C RL C dsuν usdν udsν dsuν usdν udsν C C C ( ( ( ( ( ( C + ) = ) = ) = + + LL LL LL i i RL RL RL + i l l l C C C 0 0 C C C ) = 0 + i K K l K 0 → → mode. K → p p ¯ ν p ( ( → + . L R Γ( R p K A A ( A L → Notice that we have usedfor the parity transformation to obtain the hadron matrix elements Then, we obtain the partial decay width as where The matching condition for the Wilson coefficients is Here, we summarizedescribed the in expressions the text. for other decayD.1 modes than Kaon the andThe charged effective lepton Lagrangian which induces the From the equations, itp is found that only the operators D Partial decay width Next, we give the matching conditionsat for the the Wilson electroweak coefficients scale C.2 At electroweak scale JHEP03(2014)049 .  (D.9) (D.5) (D.6) (D.7) (D.8) ) (D.14) (D.11) (D.12) (D.13) (D.10) ,  Ri  l ) ) c R . Li Li l u . . i ν , , c L )( i p , c L i i , i | u p p  d b R p p  | L | | | 2 )( d 2 | L d )( L | L L b L ) a R ) u L b L u u u d ) u + i d L + i L L l L ( a L l ) ) ) a L 0 ) , 0 u ud , u π ( η ( abc ud ( )] ud ud | 2 ud  ( | ( ( ( |  Z + | | ) abc | → → ) 0 i abc 0 0  . 0 π i ¯  m ν π  η h p π η p (  h ) h ) ( h h ( + i ) ki i i ) ) ) ) i 1 i R i π R i i 11 j (3) udul j (2) ( |A C |A → C 1 udul uddν , udul ( k udul uddν udul , + udul p ( + RR ( ) ( ( ( ( ) ( 2 2 C ji ) + | LL | Z LL ) ) LL modes. In this case we have LL RR |A LL RR Z 11 (1) C + C m 2 + i + i C CKM C C C C l ( l C m  + i i  + 0 l V 0 ( 1 ) + 1 + i ( j + + 0 + + 2 p 2 π  η π j i 1 ) 1 ) Ri η 1  i i (3) i i j j p l m ) m 1 | p p (2) ) p p . C Li | | → → | | c R , L l 1 Li → C L L L L u ) ) − [ j d p p c L ν CKM ) is u u 1 u u ( ( p Z Z u 1 R )( c L V j – 28 – CKM L L R R ) R R ( ) d b L + i m m  ) ) )( ) ) V l d ( ( − )( b R |A p i i |A π 0 ud CKM a L  ud ud  ( ud ud d b R π | V ( ( u ( ( 2 2 m CKM 32 d | | | | a R 111 111 ( ( (1) (4) + 0 0 ) = ) = ( 0 0 V   a R u i i → − C C π η η ( π π 2 p 2 η 2 p 2 π u abc h h h h h ( p  ) ) = ) ) ) ) m m m i m  i i i i i abc ) ¯  ν ) = ) = ) = ) = ( abc i uddν uddν i i i i  −  − + ( ( )  i 1 π 1 ) udul udul udul udul uddν i LL RL ( ( ( ( (   udul udul udul udul udul C → C ( p π ( ( ( ( p π RL LR udul RL LR RL p ( m m 32 LR 32 C C uddν C C C LL RL LR RR ( Γ( C C C C RL C RL C + ) = ) = modes, the effective Lagrangian is given as ) = ) = ) = ) = ) = i C + + i i i + + + i i i + i ¯ ν l l l l l l ¯ ν 0 0 0 0 0 0 + ) = + η η η π π π π + i ) = π l i 0 ¯ ν → → → → → → → π → + p p p p p p p π ( ( p ( ( ( → L R Γ( L L R Γ( → p A A A A A ( The same interaction also induces the p L ( L with With the coefficients, the partial decay width is expressed as where We have the matching condition for the Wilson coefficients at the electroweak scale as where D.3 Pion/eta andThe charged effective lepton Lagrangian for the The partial decay width is then computed as For the and the matching condition for the Wilson coefficients is D.2 Pion and anti-neutrino JHEP03(2014)049 ] ] ]. ]. ]. 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