A Model of Fermion Masses and Flavor Mixings with Family Symmetry SU(3)⊗U(1)
aa r X i v : . [ h e p - ph ] O c t A Model of Fermion Masses and Flavor Mixingswith Family Symmetry SU (3) ⊗ U (1) Wei-Min Yang * Qi Wang Jin-Jin Zhong
Department of Modern Physics, University of Science and Technology of China,Hefei 230026, P. R. China*E-mail: [email protected]
Abstract : The family symmetry SU (3) ⊗ U (1) is proposed to solve flavor problemsabout fermion masses and flavor mixings. It’s breaking is implemented by someflavon fields at the high-energy scale. In addition a discrete group Z is introducedto generate tiny neutrino masses, which is broken by a real singlet scalar field atthe middle-energy scale. The low-energy effective theory is elegantly obtained afterall of super-heavy fermions are integrated out and decoupling. All the fermionmass matrices are regularly characterized by four fundamental matrices and thirteenparameters. The model can perfectly fit and account for all the current experimentaldata about the fermion masses and flavor mixings, in particular, it finely predictsthe first generation quark masses and the values of θ l and J lCP in neutrino physics.All of the results are promising to be tested in the future experiments. Keywords : family symmetry; fermion mass; flavor mixing; neutrino physics
PACS : 12.10.-g; 12.15.Ff; 14.60.Pq 1 . Introduction
The precise tests for the electroweak scale physics have established plenty ofknowledge about the elementary particles [1]. The standard model (SM) has beenevidenced to be indeed a very successful theory at the current energy scale [2].However, there are some imperfections in the SM, among other things, a ugly defectis too many parameters exist in the Yukawa sector. As a result, fermion massesand flavor mixings seem intricate and ruleless. During the past decade a seriesof new experiment results about B physics and neutrino physics have told us agreat deal of information about flavor physics [3]. What deserves to be paid specialattention are some facts as follows. The mass spectrum of quarks and chargedleptons emerges a large hierarchy, which ranges from one MeV to a hundred GeVor so [1]. The neutrinos have been verified to have nonzero but Sub-eV masses[4], nevertheless, that their nature is Majorana or Dirac particle has to be furtheridentified by experiments such as 0 νββ [5]. On the other hand, the flavor mixing inthe quark sector is distinctly different from one in the lepton sector. The former hassmall mixing angles and its mixing matrix is close to an unit matrix [6], whereas thelatter has bi-large mixing angles and its mixing matrix is close to the tri-bimaximalmixing pattern [7]. In the lepton mixing, it is yet in suspense whether sinθ is zeroand the CP violation vanishes or not [8]. These impressive puzzles always attractgreat attention[9], and also are expected to be explained by new theories beyond theSM. The issues in the flavor physics possibly implicate great significance. They arenot only bound up with origin of matter in the universe [10], but also in connectionwith the genesis of the matter-antimatter asymmetry and the original nature of thedark matter [11].Any new theory beyond the SM has to be confronted with the various intractableissues mentioned above, however, some approaches and theoretical models have beenproposed to solve them [12]. For instance, the Froggatt-Nielsen mechanism with U (1) family symmetry can account for mass hierarchy [13]. The discrete familygroup A can lead to the tri-bimaximal mixing structure of the lepton mixing ma-trix [14]. The non-Abelian continuous group SU (3) is introduced to explain theneutrino mixing [15]. In [16], the model with the family group SO (3) successfullyaccommodates the whole experimental data of quarks and leptons. In addition,some models of grand unification (GUT) based on SO (10) symmetry group canalso give some reasonable interpretations for fermion masses and flavor mixings [17].Although these theories seem successful in explaining some of the flavor problems,it seems very difficult for them to solve the whole flavor problems all together. Itis especially hard for some models to keep the principle of the smaller number ofparameters. On all accounts, it now remains to be a large challenge for theoreticalparticle physicists to uncover these mysteries of the flavor physics.As remarked above, the flavor puzzle is very difficult and complex to solve com-pletely. In this works, we consider a new approach and construct a model withfewer parameters to try to understand these problems. On the one hand, we believe2hat there is some inherence relations among all kinds of fermion mass and mixingparameters. The family symmetry SU (3) F , which is a family symmetry group ofthe three generation fermions, is appropriate for seeking the relations. In addition,we introduce some super-heavy fermion and flavon fields which appear only at thehigh-energy scale. They communicate with the low-energy fermions of the SM byflavor interaction. The different super-heavy fermions and flavons are distinguishedby an appended Abelian group U (1) N . On the other hand, the model with twoHiggs doublets and three generation right-handed neutrino singlets is theoreticallywell-motivated extension of the SM[18]. On that basis, we introduce a real scalarsinglet and append a discrete group Z , under which these scalar and right-handedneutrino singlets all reverse sign. The model goes through three steps of breakings.Firstly, the family symmetry SU (3) F ⊗ U (1) N is broken at the flavon dynamics scale.It is accomplished by means of the flavon fields developing the vacuum structuresalong the specific directions. After all the super-heavy fermions are integrated outand decoupling, the low-energy effective theory is elegantly obtained. Secondly, thediscrete symmetry Z is broken at the middle-energy scale by the real scalar devel-oping non-vanishing vacuum expectation value (VEV). This directly causes that theeffective Yukawa couplings of neutrinos are drastically suppressed, thus far smallerthan ones of the other fermions. This also becomes a source of the neutrino tinymasses. Lastly the electroweak symmetry breaking is completed, all of quarks andleptons attain Dirac masses. All the fermion mass matrices are regularly given andcharacterized only by the four fundamental matrices and fewer parameters. Finally,the model can naturally and correctly give rise to the fermion mass spectrum andflavor mixing angles. All the numerical results are very well in agreement with thecurrent experimental data.The remainder of this paper is organized as follows. In Section II we outlinethe model. In Sec. III, the symmetry breaking procedures are introduced and thefermion mass matrices are discussed. In Sec. IV, we give the detailed numerical re-sults about the fermion masses and flavor mixings. Sec. V is devoted to conclusions. II. Model
The model is based on the symmetry group SU (3) C ⊗ SU (2) L ⊗ U (1) Y ⊗ SU (3) F ⊗ U (1) N ⊗ Z , among them, the first three subgroups are namely the SM symmetry atthe low-energy scale. The family symmetry at the high-energy scale is characterizedby the subgroups SU (3) F ⊗ U (1) N . The subgroup Z is a discrete symmetry at themiddle-energy scale. The model particle contents and their quantum numbers underthe family symmetry subgroups are listed in the following. The low-energy fermions3nd Higgs scalar fields consist of Q L ∼ (3 , , u R ∼ (3 , , d R ∼ (3 , ,L L ∼ (3 , , ν R ∼ (3 , , e R ∼ (3 , ,H ∼ (1 , , H ∼ (1 , − , φ ∼ (1 , . (1)The three generation of fermions are in representation of the family subgroup SU (3) F , and they have no charges of the subgroup U (1) N , and so on. Under theSM group, the representations of these fields are clear as usual but only ν R and φ are singlets.We introduce some super-heavy fermion fields as follows η u,ν ∼ (3 , , η u,ν ∼ (3 ,
53 ) , η u,ν ∼ (3 ,
43 ) , η u,ν ∼ (3 ,
23 ) ,η u,ν ∼ (3 ,
32 ) , η u,ν ∼ (3 , , η u,ν ∼ (3 ,
12 ) , ζ u ∼ (1 , , χ ν ∼ (3 , ,η d,e ∼ (3 , − , η d,e ∼ (3 , −
53 ) , η d,e ∼ (3 , −
43 ) , η d,e ∼ (3 , −
23 ) , ζ d,e ∼ (1 , − , (2)whose left-handed and right-handed fields are unified. The superscripts respectivelyindicate the corresponding right-handed fermions in (1), namely under the SM groupthe quantum numbers of the super-heavy fermions are the same as ones of the low-energy right-handed fermions. In comparison with the super-heavy quark fields,the super-heavy lepton fields have χ ν instead of ζ ν . These super-heavy quarks andleptons are possessed of the super-heavy masses, so they appear only in the veryhigh energy circumstances.We also introduce the super-heavy scalar flavon fields such as F ∼ (8 ,
13 ) , F ∼ (8 ,
53 ) , F ∼ (8 ,
23 ) , F ∼ (8 ,
12 ) ,T ∼ (3 , , T ∼ (3 , − . (3)All of them are singlets under the SM group, but under the family subgroup, F , · · · , F are hermitian octet representations, and T , T are complex triplet rep-resentations. In addition, they have different charges of U (1) N . These flavon fieldsare responsible for the family symmetry breaking.Finally, we define the discrete group Z as follows. Only the ν R and φ fields aretransformed as ν R −→ − ν R , φ −→ − φ , (4)and all of the other fields are uniformly transformed as themselves.4nder the model symmetry group, the gauge invariant Yukawa couplings in thequark sector are written as L q = Q L H η u R + η u L ( T ζ uR + F η u R + F η u R + F η u R ) + η u L F η u R + η u L F η u R + η u L F η u R + (cid:16) ζ uL T † + η u L F + η u L F + η u L F (cid:17) u R + Q L H η d R + η d L (cid:0) T ζ dR + F ∗ η d R + F ∗ η d R (cid:1) + η d L F ∗ η d R + (cid:16) ζ dL T † + η d L F ∗ + η d L F ∗ (cid:17) d R + h.c. . (5)For the sake of concision, we have left out the coupling coefficient at the front ofeach term in (5), which should be ∼ O (1). Easy to notice, the couplings in theup-type sector are different from ones in the down-type sector. Those F-type flavonfields in the down-type sector are complex conjugate form. The Yukawa couplingsin the lepton sector are similarly given as L l = L L H η ν R + η ν L ( F η ν R + F η ν R + F η ν R ) + η ν L F η ν R + η ν L F η ν R + η ν L F η ν R + (cid:0) η ν L F + η ν L F + η ν L F (cid:1) χ νR + χ νL φν R + L L H η e R + η e L ( T ζ eR + F ∗ η e R + F ∗ η e R ) + η e L F ∗ η e R + (cid:16) ζ eL T † + η e L F ∗ + η e L F ∗ (cid:17) e R + h.c. , (6)likewise, all the coupling coefficients are omitted. In comparison with the quarksector, the lepton sector has the exclusive terms related to χ ν instead of ζ ν . Thesedifferences play key roles in generating distinct masses and mixings for quarks andleptons. The (5) and (6) Lagrangian indicate that the flavor interactions amongthe super-heavy fermions are transmitted by means of the super-heavy flavon fields,and the effects is ultimately transferred to the low-energy fermions after the multipletransmissions. III. Symmetry Breakings and Fermion Mass Matrices
The model symmetry breakings go through three stages. The first step of thebreaking chain is that the subgroups SU (3) F ⊗ U (1) N break to nothing, namely thefamily symmetry vanishes. This is implemented by the flavon fields F , · · · , F , T , T developing VEVs along specified directions in the family space. The detailed vacuumstructures are as follows h F i Λ F ∼ ε λ , h F i Λ F ∼ ε √ (cid:16) λ − √ λ + √ λ (cid:17) , h F i Λ F ∼ ε λ + λ + λ − λ ) , h F i Λ F ∼ ε λ , h T i Λ F ∼ , h T i Λ F ∼ , (7)5here Λ F is the family symmetry breaking scale, that is the dynamics scale of thesuper-heavy fermions and flavons, which is usually close to Planck scale of 10 GeV.The only breaking parameter ε is a ratio of the F-type VEV to the T-type VEV.We consider that the former is one order of magnitude smaller than the latter, thus ε is ∼ O (0 . λ , λ , · · · , λ are the standard Gell-Mann matricesrepresenting the generators of SU (3). As before one coefficient of O (1) is impliedin the right formula of each wave notation in (7). Below the scale Λ F , all of theflavon fields develop the vacuum states with the structures, consequently the familysymmetry is broken. It can be seen from (7) that the breakings of T and T bringthe family symmetry down from SU (3) to SU (2). On the other hand, the breakingsof F and F occur along the direction of the subgroup S (2 ↔
3) in the family space,which is a permutation group between the second generation fermions and the thirdones, and the F and F breakings are respectively in the directions of the subgroup S (1 ↔
3) and S (1 ↔ F , all the super-heavy fermions are actually decoupling. After all of them are integrated out fromthe original Lagrangian, then an effective Yukawa Lagrangian at the low energy isderived as L effY ukawa = Q L H Y u u R + Q L H Y d d R + L L H Y e e R + L L H Y ν φ Λ F ν R + h.c. (8)with Yukawa coupling matrices Y u = y u R + y u ε R + y u ε R + y u ε R ,Y d = y d R + y d ε R + y d ε R ∗ ,Y e = y e R + y e ε R + y e ε R ∗ ,Y ν = y ν ε R + y ν ε R + y ν ε R , (9)where R = , R = − −√ −√ − ,R = 1 √ − i
01 + i i − i , R = . (10)In (9), y u , · · · , y ν are some effective coupling coefficients, which are left out before,now they are visibly retrieved and written out. These coefficients are mostly ∼ O (1),6 a ) Q L u R H T T † η u R η u L ζ uR ζ uL × × H F F Q L η u R η u L η u R η u L u R ( b ) × ×× ×× × Figure. 1. The graph of generating the effective Yukawa coupling matrices from thefamily symmetry breaking, (a) and (b) respectively give rise to the terms of R and R in Y u .we take them as real numbers without loss of generality, so all the Yukawa matricesare hermitian. We can illustrate this procedure, for instance, the terms of R and R in the Y u matrix are generated respectively by (a) and (b) in the Figure 1,and so on. The effective theory, which is valid until the scale Λ F , includes theSM fermions as well as three singlet right-handed neutrinos, moreover, has twodoublet and one singlet Higgs fields. Since the term involving ν R and φ in (8) isdrastically suppressed by Λ F , it is far smaller than the other terms in Lagrangian,furthermore, it is also non-renormalizable. It is very clear from (9) and (10) that thisset of Yukawa coupling matrices indeed have some regular and intrinsic relations.Several notable characteristics can be seen very easy. First, every Yukawa matrixis a linear combination from the four fundamental matrices R , · · · , R , and thecombination coefficients are expanded by a power series of ε . By virtue of suchstructures the elements of every Yukawa matrix show themselves large hierarchy. Asa result, the R and R terms, namely the ε and ε terms, will respectively dominatethe third and second generation fermion masses. The rest of the terms will makemain contributions to the first generation fermion mass. Second, in contrast with Y u , Y d , Y e , the leading term of R is no in Y ν . In view of these structure features ofthe Yukawa matrices, it is in the course of nature that the transformation matricesdiagonalizing Y u , Y d , Y e are all close to the unit matrix, whereas the transformationmatrix diagonalizing Y ν is approximately the tri-bimaximal mixing pattern. Thisdifference is the principal source of generating distinct flavor mixings for the quarksand leptons. Third, the imaginary elements of R are the only source of the C and CP violations in the Yukawa sector. To sum up, the four R -type matrices andthe ε parameter all together make up the skeleton frame of every Yukawa matrix,7herefore they play key roles in fermion masses and flavor mixings.The second step of the breaking chain is that the discrete subgroup Z is brokenby the real singlet scalar field φ developing VEV as follows h φ i Λ F = κ . (11)The breaking scale is considered as some intermediate value between the familybreaking scale Λ F and the electroweak breaking scale. If the parameter κ is about10 − or so, the effective Yukawa couplings of the neutrinos will be drastically sup-pressed owing to the κ factor, thus they will be far smaller than the Yukawa couplingsof the charged fermions. This possibly becomes a source of the neutrino tiny masses,of course, it is different from the usual see-saw mechanism [19].After the Z breaking, the model remaining symmetry is exactly the SM symme-try group. The last step of the breaking chain is namely the electroweak symmetrybreaking. It is accomplished by the doublet Higgs fields H and H developing VEVsas follows h H i v ew = (cid:18) cosβ (cid:19) , h H i v ew = (cid:18) sinβ (cid:19) , (12)where v ew is the electroweak scale and tanβ is a ratio of the up-type VEV to thedown-type VEV. After the electroweak breaking all of the quarks and leptons obtainDirac masses. The whole fermion mass terms are now given as − L mass = u L M u u R + d L M d d R + e L M e e R + ν L M ν ν R + h.c. (13)with the mass matrices M u = − v ew sinβ Y u , M ν = − κ v ew sinβ Y ν ,M d = − v ew cosβ Y d , M e = − v ew cosβ Y e . (14)Now three new parameters κ, v ew , tanβ are added into the model besides the ε and y-type parameters in (9). All of these quantities are undetermined except theelectroweak scale v ew . However, some parameters among them are in the formof product factors in the mass matrices, there are actually three non-independentparameters. We have some freedoms to remove the non-independent parameters, forexample, this three parameters y u , y d , y ν can be absorbed collectively by redefinitionsof the three parameters ε, tanβ, κ , so each of them will be equal to one instead offree parameters hereinafter. Therefore, the effective independent parameters in themodel are only thirteen in all. It can be seen from (14) that v ew dominates massscale of the quarks and the charged leptons, tanβ is responsible for mass split of theup-type and down-type fermions, and the factor κ causes that the neutrino massesare far smaller than ones of the charged fermions. Since this three parameters areonly some product factors in the mass matrices, anyway, they have no influence on8he flavor mixings. The mass hierarchy and the flavor mixings are still controlledmainly by the ε parameter and the R -type matrices. In a word, this set of massmatrices properly embody all of the information about fermion mass hierarchy, flavormixings and the CP violations.In virtue of the model’s intrinsic characteristics, all the fermion mass matricesare hermitian, therefore all of fermion mass eigenvalues are conveniently solved bydiagonalizing them as follows U † u M u U u = diag ( m u , m c , m t ) , U † ν M ν U ν = diag ( m , m , m ) ,U † d M d U d = diag ( m d , m s , m b ) , U † e M e U e = diag ( m e , m µ , m τ ) . (15)Because the R and R terms are respectively the leading and next-to-leading termsin the mass matrices, the second and third generation of the quark and chargedlepton masses can be calculated approximately such as m c ≈ v ew sinβ (cid:0) ε + ε (cid:0) y u − y u (cid:1)(cid:1) , m t ≈ v ew sinβ (cid:0) y u − ε (cid:1) ,m s ≈ v ew cosβ (cid:18) ε + ε (cid:18) y d + y d (cid:19)(cid:19) , m b ≈ v ew cosβ (cid:0) − y d + ε (cid:1) ,m µ ≈ v ew cosβ (cid:18) ε y e + ε (cid:18) y e y e + y e y e (cid:19)(cid:19) , m τ ≈ v ew cosβ (cid:0) y e − ε y e (cid:1) . (16)However, the first generation of the quark and charged lepton masses have no suchapproximate expressions since they depend on all the terms of every mass matrix.It can be seen from (16) that, in the leading approximation, there are the massrelations m c m t ≈ ε y u − ε , m s m b ≈ ε − y d + ε , m µ m τ ≈ ε y e y e − ε . (17)We can easily estimate values of some parameters from (16) and (17). Finally, theflavor mixing matrices for the quarks and leptons are respectively given by [20] U † u U d = U CKM , U † e U ν = U P MNS . (18)The mixing angles and CP -violating phases in the two unitary matrices of U CKM and U P MNS can be worked out by the standard parameterization in particle datagroup [1].
IV. Numerical Results
Now we present the model numerical results. As is noted earlier, altogether themodel parameters involve the ten y-type coefficients, the three breaking parameters ε, κ, tanβ , and the electroweak scale v ew . Once this set of parameters are chosen9s the input values, according to the model we can calculate the various outputvalues of the fermion masses and flavor mixings, moreover, all of the results can becompared with the current and future experimental data.The electroweak scale v ew is essentially determined in the gauge sector by weakgauge boson masses and gauge coupling constant. The accurate measures havegiven v ew = 174 GeV. The other thirteen parameters are really free parameters inthe Yukawa sector, however, all of them have to be fixed by fitting the experimentaldata of the fermion masses and flavor mixings. Because the number of the modelparameters is much less than the experimental values of the masses and mixings,and the majority of them have precisely been measured, the space of the modelparameters is constrained very narrow and the tuning scope of the parameters isindeed very small. Although the fit is a non-trivial and no easy one, in advance onecan find some parameter values by (16) and (17), and then the global fit can besuccessfully finished. We here give the values of the best fit instead of the detailednumerical analysis. The input values of the model parameters are elaborately chosenas follows ε = 0 . , κ = 1 . × − , tanβ = 12 . ,y u = 1 , y u = − , y u = − . , y d = − . , y d = 2 . ,y ν = 3 . , y ν = 7 . , y e = 0 . , y e = 0 . , y e = 0 . . (19)These values are reasonable and consistent with the previous estimates. Each of they-type parameters is dedicated to a certain impact on the fermion masses and flavormixings, for instance, y u has main impact on the quark sinθ q and CP -violatingphase, while y d makes main contribution to the quark sinθ q , and so forth.Finally, a variety of the numerical results calculated by the model are in detaillisted in the following. For the quark sector, all of mass eigenvalues and mixingangles are (mass in GeV unit) m u = 0 . , m c = 1 . , m t = 172 . ,m d = 0 . , m s = 0 . , m b = 4 . ,s q = 0 . , s q = 0 . , s q = 0 . , δ q = 0 . π ≈ . ◦ , (20)where s αβ = sinθ αβ , in addition, the Jarlskog invariant measuring the CP violationis calculated to J qCP ≈ . × − . (21)It is very clear that the above results are very well in agreement with the currentmeasures of the quark masses, mixing and CP violation [1]. Although the firstgeneration of the quark masses have not been accurately measured so far, theirvalues are finely predicted to be about the center values of the experimental limits.10or the lepton sector, the parallel results are m e = 0 .
511 MeV , m µ = 105 . , m τ = 1778 MeV ,m = 0 . × − eV , m = 0 . × − eV , m = 5 . × − eV ,s l = 0 . , s l = 0 . , s l = 0 . , δ l = − . π ≈ − . ◦ . (22)The charged lepton masses are completely identical with ones in the particle list [1].For the sake of comparison with the experimental data, the common used quantitiesin neutrino physics are explicitly calculated as follows △ m ≈ . × − eV , △ m ≈ . × − eV ,sin θ l ≈ . , sin θ l ≈ . , sin θ l ≈ . ,J lCP ≈ − . , (23)where △ m αβ = m α − m β . These results are excellently in agreement with the recentneutrino oscillation data [21]. In particular, the model predicts that the heaviestone of the three generation neutrinos is about 0 .
05 eV, the lepton mixing angle θ l is ∼ . ◦ but nonzero, in addition, the CP -violating effect is of the order of 10 − in thelepton sector, which is three order of magnitude larger than one in the quark sector.Since all the neutrinos in the model are Dirac-type rather than Majorana-type, theneutrinoless double beta decay is inevitably nought. Although these quantities havenot strictly measured by now, some running and coming neutrino experiments areon the way toward these goals [22]. We have confidence that all the predictions arepromising to be tested in the near future.To sum up the above numerical results, in fact, only with the thirteen parametersdoes the model accurately and excellently fit the total twenty values of the fermionmasses and flavor mixings. All the current measured values are exactly reproduced,meanwhile, all the non-detected values are finely predicted in the experimental lim-its. All of the results are naturally produced without any fine tuning. This fullyshow a strong prediction power of this model. In the case of the best fit, the sizesof the parameters y u and y u both are coincidently one, the reason about it is yetunknown and expected to research deeply. V. Conclusions
In the paper, we have suggested a new model to solve the fermion masses andflavor mixings, which is based on the family symmetry SU (3) F ⊗ U (1) N and thediscrete group Z . The family symmetry breaking is carried out by means of the in-troduced super-heavy fermion and flavon fields. After all of the super-heavy fermionsare integrated out and decoupling, the low-energy effective theory is obtained withthe regular Yukawa coupling matrices. The ε parameter and the four fundamental R -type matrices all together make up the skeleton frame of the Yukawa matrices. Infact they play leading roles in the model, namely they dominate the fermion mass11ierarchy and flavor mixing results. The discrete group Z is broken by the singletscalar field at the middle-energy scale. This leads that the Yukawa couplings of theneutrinos are drastically suppressed, and then gives rise to the tiny nature of theneutrino masses. That set of the fermion mass matrices derived from the modelsymmetries and their breakings are characterized only by the thirteen effective pa-rameters. The model successfully and perfectly fits all the current experimental dataabout the fermion masses and flavor mixings, in particular, it finely predicts the firstgeneration quark masses and the values of θ l , J lCP in neutrino physics. All of theresults are excellent and inspiring, and also fully show a great prediction power ofthe model. Finally, we expect all the results to be tested in future experiments onthe ground and in the sky. These experiments will undoubtedly provide us moreimportant information about the flavor physics, and then enlighten us to understandfinely the mystery of the universe. Acknowledgments
One of the authors, W. M. Yang, would like to thank his wife for large helps.This research is supported by chinese universities scientific fund.
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