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Advances in High Energy Physics
Volume 2016, Article ID 9496758, 11 pages
http://dx.doi.org/10.1155/2016/9496758
Research Article

Octant Degeneracy and Quadrant of Leptonic CPV Phase at Long Baseline Experiments and Baryogenesis

1Department of Physics, Gauhati University, Guwahati 781014, India
2Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India

Received 30 May 2016; Revised 22 July 2016; Accepted 8 August 2016

Academic Editor: Abhijit Samanta

Copyright © 2016 Kalpana Bora et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Abstract

In a recent work by us, we have studied how CP violation discovery potential can be improved at long baseline neutrino experiments (LBNE/DUNE), by combining with its ND (near detector) and reactor experiments. In this work, we discuss how this study can be further analysed to resolve entanglement of the quadrant of leptonic CPV phase and octant of atmospheric mixing angle , at LBNEs. The study is done for both NH (normal hierarchy) and IH (inverted hierarchy), HO (higher octant), and LO (lower octant). We show how baryogenesis can enhance the effect of resolving this entanglement and how possible values of the leptonic CP violating phase can be predicted in this context. With respect to the latest global fit data of neutrino mixing angles, we predict the values of for different cases. In this context we present favoured values of ( range at ≥2σ) constrained by the latest updated BAU range and also confront our predictions of with an up-to-date global analysis of neutrino oscillation data. We find that some region of the favoured parameter space lies within the best fit values around . A detailed analytic and numerical study of baryogenesis through leptogenesis is performed in this framework within the nonsupersymmetric SO models.

1. Introduction

Today, physics is going through precision era; this is more so for neutrino physics. With the measurement of reactor mixing angle [17] precisely by reactor experiments, the unknown quantities left to be measured in neutrino sector are leptonic CP violating phase [813], octant of atmospheric angle [1420], mass hierarchy, nature of neutrino, and so forth. Long baseline neutrino experiments (LBNE [21, 22], NOA [23], T2K [24], MINOS [25], LBNO [26], etc.) may be very promising, in measuring many of these sensitive parameters.

Measuring leptonic CP violation (CPV) is one of the most demanding tasks in future neutrino experiments [27]. The relatively large value of the reactor mixing angle measured with a high precision in neutrino experiments [15] has opened up a wide range of possibilities to examine CP violation in the lepton sector. The leptonic CPV phase can be induced by the PMNS neutrino mixing matrix [2830] which holds, in addition to the three mixing angles, a Dirac-type CP violating phase in general as it exists in the quark sector and two extra phases if neutrinos are Majorana particles. Even if we do not yet have significant evidence for leptonic CPV, the current global fit to available neutrino data manifests nontrivial values of the Dirac-type CP phase [3134]. In this context, possible size of leptonic CP violation detectable through neutrino oscillations can be predicted. Recently, [8], we have explored possibilities of improving CP violation discovery potential of newly planned long baseline neutrino experiments (earlier LBNE, now called DUNE) in USA. In neutrino oscillation probability expression () relevant for LBNEs, the term due to significant matter effect changes sign when oscillation is changed from neutrino to antineutrino mode, or vice versa. Therefore in the presence of matter effects, CPV effect is entangled and hence one has two degenerate solutions, one due to CPV phase and another due to its entangled value. It has been suggested to resolve this issue by combining two experiments with different baselines [35, 36]. But CPV phase measurement depends on value of reactor angle , and hence precise measurement of plays crucial role in its CPV measurements. This fact was utilized recently by us [8], where we have explored different possibilities of improving CPV sensitivity for LBNE (USA). We did so by considering LBNE with(1)its ND (near detector),(2)reactor experiments.

We considered both appearance () and disappearance () channels in both neutrino and antineutrino modes. Some of the observations made in [8] are as follows:(1)CPV discovery potential of LBNE increases significantly when combined with near detector and reactor experiments.(2)CPV violation sensitivity is more in LO (lower octant) of atmospheric angle , for any assumed true hierarchy.(3)CPV sensitivity increases with mass of FD (far detector).(4)When NH is true hierarchy, adding data from reactors to LBNE improves its CPV sensitivity irrespective of octant.

Aim of this work is to critically analyse the results presented in [8], in context of entanglement of quadrant of CPV phase and octant of , and hence study the role of baryogenesis in resolving this entanglement. Though in [8] we studied effect of both ND and reactor experiments on CPV sensitivity of the LBNEs, in this work we have considered only the effect of ND. But similar studies can also be done for the effect of reactor experiments on LBNEs as well. The details of LBNE and ND are same as in [8]. Following the results of [8], either of the two octants is favoured, and the enhancement of CPV sensitivity with respect to its quadrant is utilized here to calculate the values of lepton-antilepton symmetry. This is done considering two cases of the rotation matrix for the fermions, CKM only and CKM + PMNS. Then, this is used to calculate the value of BAU within the nonsupersymmetric SO model [37], characterized by the presence of an intermediate mass scale where both the lepton number conservation and quark-lepton symmetry are broken. In the supersymmetric case, the two effects occur at the unification scale, and no intermediate scale is present.

This is an era of precision measurements in neutrino physics. We therefore consider variation of range at its CL versus range at ≥2σ over the corresponding distribution of -minima from Figure 2. We calculate baryon to photon ratio and compare with its experimentally known best fit value. As constrained by the latest updated BAU limits, , we plot range at its CL [6] from its central value versus range at ≥2σ over -minima distribution and find that for IH and LO case the allowed has a varied range altering within to in the upper quadrant ( to ) and to in the lower quadrant (0 to ). Similarly to IH and HO case the allowed has a varied range differing within to . As shown in our results in Section 4, in IH and LO case the spectrum of is mostly concentrated in the region for around to , , to , and to . Also exists for around to , to , . survives for around, , to , to in the higher quadrant ( to ). Similarly as shown in our results in Section 4, as allowed by the updated BAU limits in IH and HO case, the parameter space of in the lower quadrant (0 to ) demands to be around , for , , , and . There also exists which constrains to be around , , , , and . In the upper quadrant ( to ) for the present updated BAU constraint, the allowed region of parameter space becomes constrained with , for around , to , , , and . Also the BAU constraint requires to be equal to , for around , to , , , and . Also survives for around , , , , , and . A part of the allowed parameter space is found to lie within the best fit values of . As constrained by the current BAU bounds we present the 3D variation of the favoured range of parameters: range within its C.L, range at its CL, and range at ≥2σ varied within in Figure 2.

As can be seen from the results presented in Section 4, from Figures 3(a) and 3(b), we find that BAU can be explained most favourably for the following possible cases: , IH and LO of ; , IH and LO of ; , IH and LO of ; , IH and HO of ; , IH and HO of ; , IH and HO of ; , IH and HO of ; , IH and HO of ; , NH and HO of . It is worth mentioning that the value of and is close to the central value of from the recent global fit result [31, 34, 3842]. It is fascinating to notice that a nearly maximal CP-violating phase has been reported by the T2K [43], NOA [44], and Super-Kamiokande experiments [45], even if the statistical significance of all these experimental results is below level. This accords with one of our calculated favoured solutions which exactly holds with the current BAU constraints. Moreover, such hints of a nonzero were already present in global analyses of neutrino oscillation data, such as the one in [3133]. Our main aim in this work is to carry out a detailed analysis of the breaking of the entanglement of the quadrant of leptonic CPV phase and octant of by using the current data of mixing parameters and identify the CPV phase and spectrum required to get the breaking favourable with the current BAU constraint. These results could be important keeping in view that the quadrant of leptonic CPV phase and octant of atmospheric mixing angle are yet not fixed. Also, they are significant in context of precision measurements on neutrino oscillation parameters.

The paper is organized as follows. In Section 2, we discuss entanglement of quadrant of CPV phase and octant of . In Section 3, we present a review on leptogenesis and baryogenesis. In Section 4 we show how the baryon asymmetry (BAU) within the SO model, by using two distinct forms for the lepton CP asymmetry, can be used to break the entanglement. Finally in Section 5, we present our conclusions.

2. CPV Phase and Octant of

As discussed above, from Figure  3 of [8], we find that by combining with ND and reactor experiments, CPV sensitivity of LBNE improves more for LO (lower octant) than HO (higher octant), for any assumed true hierarchy. In Figure 1 we plot CP asymmetry,as a function of leptonic CPV phase , for . CP asymmetry also depends on the mass hierarchy. For NH, CP asymmetry is more in LO than in HO. For IH, CP asymmetry is more in LO than in HO. In this work we have used above information to calculate dependence of leptogenesis on octant of and quadrant of CPV phase. From Figure 1 we see that

Figure 1: CP asymmetry versus at DUNE/LBNE, for both the hierarchies. Red and green solid (dotted) lines are for NH (IH) with types of curve to distinguish HO and LO as the true octant, respectively.
Figure 2: In (a) and (b) versus sensitivity corresponding to CP discovery potential at LBNEs, for both the hierarchies and octant shown.
Figure 3: Allowed region constrained by the present BAU bounds, for , for the case when matrix consists of both and . The regions are obtained by varying range at ≥2σ over the corresponding minima distribution from Figure 2 and with its experimental values varied within . In Figure (a) (Figure (b)) we show the plot for the IH and LO case (IH and HO case). The blue (cyan) horizontal line represents around which the best fit values of CPV phase are assumed to lie.

For a given true hierarchy, there are eight degenerate solutions This eight-fold degeneracy can be viewed asentanglement. Out of these eight degenerate solutions, only one should be true solution. To pinpoint one true solution, this entanglement has to be broken. We have shown [8] that sensitivity to discovery potential of CPV at LBNEs in LO is improved more, if data from near detector of LBNEs or from reactor experiments is added to data from FD of LBNEs as shown in Figure  3 of [8]. Therefore 8-fold degeneracy of (3) gets reduced to 4-fold degeneracy, with our proposal [8]. Hence, following this 4-fold degeneracy still remains to be resolved:

The possibility of , that is, HO of , is also considered in this work. In this context the degeneracy is

In this work, we propose that leptogenesis can be used to break above-mentioned 4-fold degeneracy of (5) and (6). It is known that observed baryon asymmetry of the universe (BAU) can be explained via leptogenesis [4650]. In leptogenesis, the lepton-antilepton asymmetry can be explained, if there are complex Yukawa couplings or complex fermion mass matrices. This in turn arises due to complex leptonic CPV phases, , in fermion mass matrices. If all the other parameters except leptonic phase in the formula for lepton-antilepton asymmetry are fixed, for example, then observed value of BAU from experimental observation can be used to constrain quadrant of , and hence 4-fold entanglement of (5) and (6) can be broken. An experimental signature of CP violation associated with the Dirac phase , in PMNS matrix [51] can in principle be obtained, by searching for CP asymmetry in flavor oscillation. To elucidate this proposal, we consider nonsupersymmetric SO models, in which BAU arises due to leptogenesis, and this lepton-antilepton asymmetry [52] is generated by the out of equilibrium decay of the right handed, heavy Majorana neutrinos, which form an integral part of seesaw mechanism for neutrino masses and mixing. We consider type I seesaw mechanism, just for simplicity.

3. Leptogenesis and Baryogenesis Using Type I Seesaw in SO Models

In grand unified theories like SO, one right-handed heavy Majorana neutrino per generation is added to standard model and they couple with left-handed via Dirac mass matrix . When the neutrino mass matrix is diagonalised, we get two eigenvalues, light neutrino and a heavy neutrino state . This is called type I seesaw mechanism (it may be noted that type I seesaw may be obtained in other theories). Here, decay of the lightest of the three heavy RH Majorana neutrinos, , that is , will contribute to asymmetry [5358] (for leptogenesis), that is, . It may be noted that some results on leptogenesis in the context of SO models have been discussed earlier in [5965]. In the basis where RH mass matrix is diagonal, the type I contribution to is given by decay of where means decay rate of heavy Majorana RH of mass to a lepton and Higgs. We assume a normal mass hierarchy for heavy Majorana neutrinos. In this scenario the lightest of heavy Majorana neutrinos is in thermal equilibrium while the heavier neutrinos, and , decay. Any asymmetry produced by the out of equilibrium decay of and will be washed away by the lepton number violating interactions mediated by . Therefore, the final lepton-antilepton asymmetry is given only by the CP-violating decay of to standard model leptons () and Higgs (). This contribution is [66]where is the vev of the SM Higgs doublet that breaks the SM gauge group to . is a complex orthogonal matrix with the property that . can be parameterized as [67]where is the matrix of neutrino Yukawa couplings. is the vacuum expectation value of Dirac masses. In the flavour basis, where the charged lepton Yukawa matrix, , and gauge interactions are flavour-diagonal, , where . is the PMNS matrix and is the RH neutrino Majorana scale. On the basis of right-handed neutrinos, , where . Equation (8) relates the lepton asymmetry to both the solar () and atmospheric () mass squared differences. Thus the magnitude of the matter-antimatter asymmetry can be predicted in terms of low energy oscillation parameters, , , and a CPV phase. In SO models, the right-handed neutrino is generated from the Yukawa coupling of right-handed neutrinos with the Higgs field that breaks the unification or intermediate symmetry down to the standard model [68]. When such a Higgs field takes a VEV, the right-handed neutrinos get a Majorana mass. This happens because lepton number is broken at that scale. It has been discussed in [65] that in the supersymmetric case the mass scale of the right-handed neutrino is similar to the unification scale,  GeV, while in the nonsupersymmetric case the scale of is about the intermediate scale,  GeV [69, 70], being the scale of the quark-lepton symmetry [71].

As has been discussed in [66], the lepton-antilepton asymmetry gets connected to both the solar and the atmospheric mass difference square. Following [72], in this work we choose a basis where the complex orthogonal matrix takes the form, , where is the CKM matrix containing the quark mixing angles, and is the PMNS matrix containing neutrino mixing parameters. In equations and of [72], if is taken as quark mixing matrix (where, in SO theories, as quarks and leptons appear in same representation, neutrino mixing matrix can be taken to be same as quark mixing matrix at high scales) and is taken as PMNS matrix at low energies, then we get the relation . This assumption can also be justified, as it is well known that quark mixing CKM phase alone is not sufficient to explain the BAU, and leptonic CPV phase is needed to generate the observed BAU. Here, the matrix is orthogonal since (because ). Thus both the quark sector (quark mixing angles, phase) and the neutrino sector ( mixing angles and the leptonic CPV phase) appear in the expression for and in that case can be parameterized as Here, , , , and so forth represent the cosine of atmospheric mixing angle, sine of solar mixing angle, and cosine of reactor mixing angle, respectively. Similarly , , and are the quark mixing angles. and are the leptonic CPV phase and quark CPV phase, respectively. Here the equation presumably holds at the GUT scale but weak scale values can be used in the calculations since it is well known that quark mixing parameters do not change much under renormalization group (RG) evolution in hierarchical mass spectrum of SO theories. Hence CKM matrix at high scales can be used at low energies also and is taken as PMNS matrix at low energies. We also consider , where matrix consists of mixing angles and the leptonic CPV phase. Thus when left-right symmetry is broken at high intermediate mass scale in SO theory, CP asymmetry in this case is given by where The neutrino oscillation data used in our numerical calculations are summarised as follows [31, 3842]: For , , and , the quantities inside the bracket correspond to inverted neutrino mass hierarchy and those outside the bracket correspond to normal mass hierarchy. The errors are within the range of the oscillation parameters. It may be noted that some results on neutrino masses and mixing using updated values of running quark and lepton masses in SUSY SO have also been presented in [73]. Though we consider 3-flavour neutrino scenario, 4-flavour neutrinos with sterile neutrinos as fourth flavour are also possible [74]. It is worth mentioning that masses and mixing can lead to charged lepton flavor violation in grand unified theories like SO [75].

The origin of the baryon asymmetry in the universe (baryogenesis) is a very interesting topic of current research. A well known mechanism is the baryogenesis via leptogenesis, where the out of equilibrium decays of heavy right-handed Majorana neutrinos produce a lepton asymmetry which is transformed into a baryon asymmetry [72] by electroweak sphaleron processes [7679]. Lepton asymmetry is partially converted to baryon asymmetry through B + L violating sphaleron interactions [80]. As proposed in [65], a baryon asymmetry can be generated from a lepton asymmetry. The baryon asymmetry is defined aswhere , , and are number densities of baryons, antibaryons, and photons, respectively, is the entropy density, is the baryon to photon ratio, and (95% CL) [81]. The lepton number is converted into the baryon number through electroweak sphaleron process [7679]: where is the number of families and is the number of light Higgs doublets. In case of SM, and . The lepton asymmetry is as follows: is a dilution factor and in the standard case [65] is the effective number of light degrees of freedom in the theory. The dilution factor [65] is for and for , and , respectively, where the parameter [65] is ; here is the Planck mass. We have used the form of Dirac neutrino mass matrix from [37].

4. Calculations, Results, and Discussion

For the purpose of calculations, we use the current experimental data for three neutrino mixing angles as inputs, which are given at C.L, as presented in [31, 3842]. Here, we perform numerical analysis for both the hierarchies and octants. We explore the baryon asymmetry of the universe within nonsupersymmetric SO models [37] using (7)–(16) of the two hierarchies (NH and IH), two octants LO and HO, w ND, w/o ND (with and without near detector), and range at ≥2σ over the corresponding distribution of -minima (for maximum sensitivity from Figures 2(a) and 2(b), for which the CP discovery potential of the DUNE is maximum). For our purpose, we shall carry out general scanning of the parameters: range at ≥2σ (from Figures 2(a) and 2(b)), at its CL, and at its CL using the data given by the oscillation experiments [16, 31, 3842]. In this calculation, we have chosen  GeV. Also we note that, for values of  GeV, the baryon asymmetry becomes lower than the observed value. We scan the parameter space for IH and HO/LO in the light of recent ratio of the baryon to photon density bounds, (CMB) [81] in the following ranges: Similarly constrained by the present BAU bounds we perform random scans for the following range of parameters in NH, HO/LO case: We find that the updated BAU limit [81] together with a large [17] puts significant constraints on the parameter space in the IH and LO case. As can be seen from Figure 3(a), a part of the parameter space survives for in the IH and LO case as allowed by the current BAU constraint (CMB), corresponding to around . This leads to the conclusion that the parameter space for the best fit values of is allowed by the present BAU constraint. The allowed regions in Figure 3(a) for the lower quadrant (0 to ) require spectra, that is, to be equal to for around to , , to , and to . Almost continuous values of ranging from to are allowed for , to . For around , the values of mostly favoured are , , , , , , , and . The allowed region in the upper quadrant ( to 2) necessitates to be around , for around to , to , and as allowed by the current BAU bounds. Also exists for around , to , and to . Almost continuous ranging from to are allowed for around .

For IH and HO case in Figure 3(b), a part of the parameter space exists for constrained by the current BAU limits for around , , , , , , , , and . Thus the parameter space for the best fit values of is allowed by the present BAU constraint [81]. Figure 3(b) reveals that the allowed regions for the lower quadrant (0 to ) require to be around , for , , , and . There also exists which constrains to be around to . Also survives for around , , , , and . Almost continuous values of ranging from to are allowed for varying from to . In the upper quadrant ( to 2) for the present updated BAU constraint, the allowed region of parameter space becomes constrained with , for around , to , , , and . Also the BAU constraints necessitate to be around , for around , to , , , and as allowed by the current BAU bounds. Also survives for around , , , , , and . Almost continuous ranging from to are allowed for around and .

The constraints imposed on the and parameter in NH, HO/LO space are found to be more severe as compared to IH and HO/LO space. For, NH and LO only a particular value of CP violating phase, corresponding to , is consistent with the BAU constraint. From our analysis we find that for NH and HO case we are unable to resolve the entanglement of the quadrant of and octant of since no point in the parameter space () is consistent with the recent ratio of baryon to photon density bounds, . Therefore, this indicates that IH is the most favoured hierarchy for breaking the 4-fold degeneracy of (5) and (6). All of the analysis presented above is for the case when matrix consists of both and .

No points in the parameter space, consistent with the BAU constraint, are able to break the entanglement of the quadrant of and octant of , when matrix consists of only.

In Figures 4 and 5 we display the allowed 3D space (, , and ) for breaking the 4-fold degeneracy of (5) and (6) by varying the leptonic phase in all of its possible range at ≥2σ (from Figure 2), the mixing parameter within its level, and at its C.L [6].

Figure 4: Allowed region constrained by the present BAU bounds, for , , and for the case when matrix consists of both and . The regions are obtained by varying range at ≥2σ over the corresponding minima distribution from Figure 2 and with its experimental values varied within and at its C.L. The results of our calculation are presented for IH and LO case.
Figure 5: Allowed 3D region constrained by the present BAU bounds, for , , and for the case when matrix consists of both and . The regions are obtained by varying range at ≥2σ over the corresponding minima distribution from Figure 2, with its experimental values varied within from its central values, and at its CL. The results of our calculation are presented for IH and HO case.

From Figure 4 one can easily see the favoured values of , , and for IH and LO case, allowed by the updated recent ratio of photon density to baryon density bounds, (shown in Table 1).

Table 1: The summary of our calculated values of , , and in case of IH and LO for elements of matrix comprising and .

For IH and HO case, the results of our numerical analysis are shown in Figure 5 which shows allowed (, , ) space as allowed by the current BAU bounds. The values of , , and which are favoured simultaneously consistent with constraints, [81], are shown in Table 2. For NH case we get only one point as shown in

Table 2: The summary of our calculated values of , , and in case of IH and HO for elements of matrix comprising and .

5. Conclusion

Measuring CP violation in the lepton sector is one of the most challenging tasks today. A systematic study of the CP sensitivity of the current and upcoming LBNE/DUNE is done in our earlier work [8] which may help a precision measurement of leptonic phase. In this work, we studied how the entanglement of the quadrant of leptonic CPV phase and octant of atmospheric mixing angle at LBNE/DUNE can be broken via leptogenesis and baryogenesis. Here, we have considered the effect of ND only in LBNE, on sensitivity of CPV phase measurement, but similar conclusions would hold for the effect of reactor experiments as well. This study is done for both the octants and hierarchies. We considered two cases of fermion rotation matrix, PMNS only and CKM + PMNS. Following the results of [8], the enhancement of CPV sensitivity with respect to its quadrant is utilized here to calculate the values of lepton-antilepton symmetry. Then, this is used to calculate the value of BAU. This is an era of precision measurements in neutrino physics. We therefore considered variation of and within its 3 range from their central values. We calculated baryon to photon ratio and compared with its experimentally known best fit value.

It has been found [65] that, in SUSY theories, does not lie in the allowed range of the order of . Nonsupersymmetric SO model is favoured for baryogenesis with respect to the supersymmetric model. In the nonsupersymmetric model [37], we find that the baryon asymmetry is around the required order of magnitude . We have made a complete numerical analysis of the 3-dimensional parameters, , , and , that encode the breaking of the entanglement of the quadrant of CPV phase and octant of in the presence of the latest constraints on , , by taking neutrino oscillation mixing and mass scales as indicated by the experiments. By allowing range to vary within interval at ≥2σ over the -minima distribution from Figure 2, we have studied the absolute values of both and parameters in order to break the 4-fold degeneracy of (5) and (6).

The current data shows a preference of towards . From our analysis, one of the leptonic CPV phases determined in IH and LO of case is or corresponding to , and which is near the preferred data reported in [4345]. The current analysis reveals that our calculated value of of IH and HO of case is close to the best fit value of for inverted ordering from global fit results [31, 3842] and also of NH and HO of case is favoured with the recent hint of for normal hierarchy [31, 3842].

The main results of this work are presented in Tables 1 and 2 and (19) which show that leptonic CPV phases in all the four quadrants are allowed which lie within the constraints of present BAU. These values also contain the best fit values of leptonic CPV phase as discussed earlier.

These results could be important, as the quadrant of leptonic CPV phase and octant of atmospheric mixing angle are yet not fixed experimentally. Also, they are significant in context of precision measurements of neutrino oscillation parameters, specially the leptonic CPV phase, , and the reactor angle .

Future experiments like DUNE/LBNEs and Hyper-Kamiokande [82] looking for the leptonic CPV phase together with an improvement in the precision determination on the mixing angles would certainly provide worthy information to support or rule out the scenario presented in this work for breaking the entanglement of quadrant of CPV phase and octant of .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

Gayatri Ghosh would like to thank UGC, India, for providing RFSMS Fellowship to her, during which this work was done. Kalpana Bora thanks DST-SERB, Govt. of India, for financial support through a project. Debajyoti Dutta acknowledges the support from DAE Neutrino Project at HRI.

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