Lepton flavor violating decays as probes of neutrino mass spectraand heavy Majorana neutrino masses

Wan-lei GuoInstitute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China

(Received 18 October 2006; published 29 December 2006)

We investigate the lepton flavor violating (LFV) rare decays in the supersymmetric minimal seesawmodel in which the Frampton-Glashow-Yanagida ansatz is incorporated. The branching ratio of �! e�is calculated in terms of the snowmass points and slopes (SPS). We find that the inverted mass hierarchy isdisfavored by all SPS points. In addition, once the ratio of BR��! ��� to BR��! e�� is measured, onemay distinguish the normal and inverted neutrino mass hierarchies, and confirm the masses of heavy right-handed Majorana neutrinos by means of the LFV processes and the thermal leptogenesis mechanism. It isworthwhile to stress that this conclusion is independent of the supersymmetric parameters.

DOI: 10.1103/PhysRevD.74.113009 PACS numbers: 14.60.Pq, 13.35.�r, 13.35.Hb

I. INTRODUCTION

Recent solar [1], atmospheric [2], reactor [3], and accel-erator [4] neutrino oscillation experiments have providedus with very robust evidence that neutrinos are massive andlepton flavors are mixed. The canonical seesaw mechanism[5] gives a very simple and appealing explanation of thesmallness of left-handed neutrino masses—it is attributedto the largeness of right-handed neutrino masses. Theexistence of neutrino oscillations implies the violation oflepton flavors. Hence the lepton flavor violating (LFV)decays in the charged-lepton sector, such as �! e� �,should also take place. They are unobservable in the stan-dard model (SM), because their decay amplitudes areexpected to be highly suppressed by the ratios of neutrinomasses (mi & 1 eV) to the W-boson mass (MW �80 GeV). In the supersymmetric extension of the SM,however, the branching ratios of such rare processes canbe enormously enlarged. Current experimental bounds onthe LFV decays �! e� �, �! e� �, and �! �� �are [6]

BR ��! e��< 1:2� 10�11;

BR��! e��< 1:1� 10�7;

BR��! ���< 6:8� 10�8:

(1)

The sensitivities of a few planned experiments [7] mayreach BR��! e�� & 1:3� 10�13, BR��! e�� �O�10�8�, and BR��! ��� �O�10�8�.

For simplicity, here we work in the framework of theminimal supergravity (mSUGRA) extended with twoheavy right-handed Majorana neutrinos. Then all the softbreaking terms are diagonal at high energy scales, and theonly source of lepton flavor violation in the charged-leptonsector is the radiative correction to the soft terms throughthe neutrino Yukawa couplings. In other words, the low-energy LFV processes lj ! li � � are induced by the

renormalization group equations effects of the sleptonmixing. The branching ratios of lj ! li � � are given by[8,9]

BR �lj ! li�� ��3

G2Fm

8S

�3m2

0 � A20

8�2v2sin2�

�2jCijj2tan2�; (2)

where m0 and A0 denote the universal scalar soft mass andthe trilinear term at �GUT, respectively. In addition, mS is atypical mass of superparticles, can be approximately writ-ten as [10]

m8S � 0:5m2

0M21=2�m

20 � 0:6M2

1=2�2 (3)

with M1=2 being the gaugino mass; and

Cij �Xk

�MD�ik�MD�jk ln

�GUT

Mk(4)

with �GUT � 2:0� 1016 GeV to be fixed in our calcula-tions. MD and Mi (for i � 1, 2) represent the Dirac neu-trino mass matrix and the masses of the heavy right-handedMajorana neutrinos, respectively.

To calculate the branching ratios of lj ! li � �, weneed to know the following parameters in the frameworkof the mSUGRA: M1=2, m0, A0, tan�, and sign���. Theseparameters can be constrained from cosmology (by de-manding that the proper supersymmetric particles shouldgive rise to an acceptable dark matter density) and low-energy measurements (such as the process b! s� � andthe anomalous magnetic moment of muon g� � 2). Herewe adopt the snowmass points and slopes (SPS) [11] listedin Table. I. These points and slopes are a set of benchmarkpoints and parameter lines in the mSUGRA parameterspace corresponding to different scenarios in the searchfor supersymmetry at present and future experiments.

Since the canonical seesaw models are usually pesteredwith too many parameters, specific assumptions have to bemade for MD or Mi so as to calculate the LFV rare decays[12]. In addition, the LFV processes also have been dis-cussed in the supersymmetric minimal seesaw model*Electronic address: guowl@itp.ac.cn

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(MSM) [13–16], in which only two heavy right-handedMajorana neutrinos are introduced [17,18]. In the MSM,all model parameters can in principle be fixed by use of theLFV rare decays and electric dipole moment of the elec-tron [13]. Ibarra and Ross discuss that the LFV processesmay constrain the masses of right-handed heavy majorananeutrinos [14]. In the Frampton-Glashow-Yanagida [17](FGY) ansatz, Raidal and Strumia have given analyticapproximations to the LFV rare decays for the normalhierarchy case [15]. As shown in the literature [19], theFGY ansatz only includes three unknown parameters: M1,M2, and the smallest mixing angle �z. In this paper, we aregoing to numerically compute the LFV processes in termsof �z for both normal and inverted hierarchies in the FGYansatz. In the following parts, we shall show that thebranching ratio of �! e� may distinguish the neutrinomass hierarchies.1 In addition, once the ratio of BR��!��� to BR��! e�� is measured, one may distinguish thenormal and inverted neutrino mass hierarchies and confirmthe masses of two heavy right-handed Majorana neutrinosby means of the LFV processes and the thermal lepto-genesis mechanism [20]. It is worthwhile to stress thatthis conclusion is independent of the mSUGRA parame-ters. The remaining parts of this paper are organized asfollows. In Sec. II, we briefly describe the main features ofthe FGY ansatz and the thermal leptogenesis in the mini-mal supersymmetric standard model. In Sec. III, thebranching ratio of �! e� is computed in terms of theSPS. In Sec. IV, we numerically calculated the ratio ofBR��! ��� to BR��! e��. Finally, a summary is givenin Sec. V.

II. THE FGY ANSATZ AND THE THERMALLEPTOGENESIS

In the supersymmetric extension of the MSM, two heavyright-handed Majorana neutrinos NiR (for i � 1, 2) areintroduced as the SU�2�L singlets. The Lagrangian relevant

for lepton masses can be written as [21]

�Llepton � �lLYlERH1 � �lLY�NRH2 �12

�NcRMRNR

� H:c:; (5)

where lL denotes the left-handed lepton doublet, while ER

and NR stand, respectively, for the right-handed charged-lepton and neutrino singlets.H1 andH2 (with hypercharges1=2) are the minimal supersymmetric standard modelHiggs doublet superfields. After the spontaneous gaugesymmetry breaking, one obtains the charged-lepton massmatrix Ml � v1Yl and the Dirac neutrino mass matrixMD � v2Y�. Here vi is the vacuum expectation value ofthe Higgs doublet Hi (for i � 1, 2). An important parame-ter� is defined by tan� � v2=v1 or sin� � v2=vwith v ’174 GeV. The heavy right-handed Majorana neutrino massmatrix MR is a 2� 2 symmetric matrix and MD is a 3� 2matrix. Without loss of generality, we work in the flavorbasis where Ml and MR are both diagonal, real, and posi-tive; i.e., Ml � Diagfme;m�;m�g and MR �

DiagfM1;M2g. The seesaw relation [5]

M� � �MDM�1R MT

D (6)

remains valid. Note that this canonical seesaw relationholds up to the accuracy of O�M2

D=M2R� [22]. Since MR

is of rank 2, M� is also a rank-2 matrix with j det�M��j �m1m2m3 � 0, where mi (for i � 1, 2, 3) are the masses ofthree light neutrinos. As for three neutrino masses mi, thesolar neutrino oscillation data have set m2 >m1 [1]. Nowthat the lightest neutrino in the MSM must be massless, weare then left with eitherm1 � 0 (normal mass hierarchy) orm3 � 0 (inverted mass hierarchy). With the best-fit values�m2

sun � m22 �m

21 � 8:0� 10�5 eV2 and �m2

atm �jm2

3 �m22j � 2:5� 10�3 eV2 [23], one can numerically

calculate the neutrino masses [24].In the FGY ansatz [17], MD is taken to be of the form

MD �

a1 0a2 b2

0 b3

0@

1A: (7)

With the help of Eqs. (6) and (7), one may straightfor-wardly arrive at

TABLE I. Some parameters for the snowmass points and slopes in the mSUGRA. The massesare given in GeV. � in the Higgs mass term has been taken as �> 0 for all SPS.

Point M1=2 m0 A0 tan� Slope

1a 250 100 �100 10 m0 � �A0 � 0:4M1=2, M1=2 varies1b 400 200 0 302 300 1450 0 10 m0 � 2M1=2 � 850 GeV, M1=2 varies3 400 90 0 10 m0 � 0:25M1=2 � 10 GeV, M1=2 varies4 300 400 0 505 300 150 �1000 5

1We may also distinguish the normal and inverted neutrinomass hierarchies through measuring �z, the Jarlskog parameterof CP violation (JCP) and the effective mass of neutrinolessdouble beta decay (hmiee) [19].

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M� � �

a21

M1

a1a2

M10

a1a2

M1

a22

M1�

b22

M2

b2b3

M2

0 b2b3

M2

b23

M2

0BBBB@

1CCCCA: (8)

Without loss of generality, one can always redefine thephases of left-handed lepton fields to make a1, b2, and b3

real and positive. In this basis, only a2 is complex and itsphase � arg�a2� is the sole source of CP violation in themodel under consideration. Because a1, b2, and b3 of MD

have been taken to be real and positive, M� may bediagonalized in a more general way

M� � �PlV�m1 0 00 m2 00 0 m3

0@

1A�PlV�T; (9)

where Pl � iDiagfei�; ei�; ei�g and V is lepton flavor mix-ing matrix [25] parameterized as

V �

cxcz sxcz sz�cxsysz � sxcye�i �sxsysz � cxcye�i sycz

�cxcysz � sxsye�i �sxcysz � cxsye�i cycz

0BB@

1CCA

�

1 0 0

0 ei� 0

0 0 1

0BB@

1CCA; (10)

with sx � sin�x, cx � cos�x, and so on. �x � �sun, �y ��atm, and �z � �chz hold as a good approximation [26]. Inview of the current experimental data, we have �x � 34�

and �y � 45� (best-fit values) as well as �z < 10� at the99% confidence level [23]. It is worth remarking that thereis only a single nontrivial Majorana CP-violating phase(�) in the MSM, as a straightforward consequence ofm1 �0 or m3 � 0.

Note that all six phase parameters (, �,, �, �, and �)have been determined in terms of r23 � m2=m3 � 0:18,�x, �y, and �z [19]:

� arccos�c2

ys2z � r

223s

2x�c

2xs

2y � s

2xc

2ys

2z�

2r223s

3xcxsycysz

�;

� �1

2arctan

� cxsy sin

sxcysz � cxsy cos

�;

� � �1

2arctan

�r2

23s2xc

2z sin2�

s2z � r2

23s2xc2z cos2�

�;

� � ��� arctan� cxcysz sin

sxsy � cxcysz cos

�;

� �1

2arctan

�s2z sin2�

r223s

2xc

2z � s

2z cos2�

�;

� �� �� arctan� sxcysz sin

cxsy � sxcysz cos

�

(11)

for the normal mass hierarchy (m1 � 0). Similar resultsalso can be obtained for the inverted mass hierarchy (m3 �0) [19], but we do not elaborate on them here. Becausej cosj 1 must hold, we find 0:077 sz 0:086 (m1 �0) and 0:0075 sz 0:174 (m3 � 0). Hence, a measure-ment of the unknown angle �z becomes crucial to test themodel.

In the flavor basis where the mass matrices of chargedleptons and right-handed neutrinos are diagonal, one cancalculate the CP asymmetry in the decays of the lighterright-handed neutrino and obtain [27]

"1 �3

8�v2sin2�

M1j�M��12j2j�M��23j

2 sin2

fj�M��11j2 � j�M��12j

2gj�M��33j; (12)

when M1 � M2, where we have used

a21 � M1j�M��11j; ja2j

2 � M1j�M��12j2=j�M��11j;

b23 � M2j�M��33j; b2

2 � M2j�M��23j2=j�M��33j:

(13)

Then "1 can result in a net lepton number asymmetry YL.The lepton number asymmetry YL is eventually convertedinto a net baryon number asymmetry YB via the nonper-turbative sphaleron processes [28,29], which is given by

YB �nB � n �B

s� �cYL � �c

�g"1; (14)

where c � 28=79 � 0:35, g � 228:75 is an effectivenumber characterizing the relativistic degrees of freedomwhich contribute to the entropy s of the early universe, and� accounts for the dilution effects induced by the lepton-number-violating washout processes. The dilution factor �can be figured out by solving the full Boltzmann equations[27], however, we take the following approximate formula[30]:

� � 0:3�10�3eV

~m1

��ln�

~m1

10�3eV

���0:6

; (15)

with ~m1 � �MyDMD�11=M1. It is clear that "1 and YB only

involve two unknown parameters: M1 and �z.2 A generous

range 8:5� 10�11 & YB & 9:4� 10�11 has been drawnfrom the recent Wilkinson Microwave Anisotropy Probeobservational data [32]. It has been shown thatM1 > 3:4�1010 GeV (m1 � 0) and M1 > 2:5� 1013 GeV (m3 � 0)are required by the current observational data of YB [21].Flavor effects in the mechanism of thermal leptogenesishave recently attracted a lot of attention [33]. Because the �Yukawa coupling are in thermal equilibrium for 109GeV &

M1 & 1012 GeV, the flavor issue should be taken into

2Note that "1 is inversely proportional to the mSUGRAparameter sin2�. Because tan� & 3 is disfavored (as indicatedby the Higgs exclusion bounds [31]), here we focus on tan� � 5or equivalently sin2� � 0:96. Hence, sin2� � 1 is a reliableapproximation in our discussion.

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account in our model for the m1 � 0 case. Using theanalytic approximate formulas in Ref. [34], one may cal-culate the CP-violating asymmetry "N1

�� and the corre-sponding washout parameter KN1

�� for the � lepton doubletin the final states ofN1 decays. We find "N1

�� � 0 andKN1�� �

0 because MD31 � 0, namely, the N1 decays involving the� lepton doublet do not contribute to the baryon asymmetryYB. On the other hand, the CP-violating asymmetry "N2

�� ,which is produced by the out-of-equilibrium decays of N2,vanishes. Therefore, such flavor effects may be negligiblein our paper.

Note that there is in general a potential conflict betweenachieving successful thermal leptogenesis and avoidingoverproduction of gravitinos in the MSM with supersym-metry [35] unless the gravitinos are heavier than�10 TeV[36]. If the mass scale of gravitinos is of O�1� TeV, onemust have M1 & 108 GeV. This limit is completely disfa-vored in the FGY ansatz. Such a problem could be circum-vented in other supersymmetric breaking mediationscenarios (e.g., gauge mediation [14]) or in a class ofsupersymmetric axion models [37], where the gravitinomass can be much lighter in spite of the very high reheatingtemperature. For simplicity, we choose YB � 9:0� 10�11

as an input parameter in this paper. Hence, one mayanalyze the dependence of M1 on �z from the successfulleptogenesis.

III. DISTINGUISH THE NEUTRINO MASSHIERARCHY

With the help of Eqs. (4) and (7), jCijj2 can be writtenexplicitly as

jC12j2 � ja1j

2ja2j2

�ln

�GUT

M1

�2; jC13j

2 � 0;

jC23j2 � jb2j

2jb3j2

�ln

�GUT

M2

�2:

(16)

Because of jC13j2 � 0, we are left with BR��! e�� � 0.

If BR��! e�� � 0 is established from the future experi-ments, it will be possible to exclude the FGYansatz. UsingEq. (13), we reexpress Eq. (16) as

jC12j2 � M2

1j�M��12j2

�ln

�GUT

M1

�2;

jC23j2 � M2

2j�M��23j2

�ln

�GUT

M2

�2:

(17)

As shown in Sec. II, M1 may in principle be confirmed byleptogenesis for given values of sin�z, but M2 is entirelyunrestricted from the successful leptogenesis with M2 �M1.

We numerically calculate BR��! e�� for differentvalues of sin�z by using the SPS points. The results areshown in Fig. 1. Since the SPS points 1a and 1b (or points 2and 3) almost have the same consequence in our scenario,we only focus on point 1a (or point 3). When sin�z !

0:077 or sin�z ! 0:086, the future experiment is likely toprobe the branching ratio of �! e� � in the m1 � 0case. The reason is that sin�z ! 0:077 (or sin�z ! 0:086)implies! ��=2 (or! 0). Furthermore, the success-ful leptogenesis requires a very large M1 due to "1 /sin2. It is clear that all SPS points are unable to satisfyBR��! e�� 1:2� 10�11 in the m3 � 0 case.Therefore, we can exclude the m3 � 0 case when the

0.078 0.08 0.082 0.084 0.08610

−16

10−15

10−14

10−13

10−12

10−11

10−10

sin θz

(a)

BR

(µ→

e γ

)

SPS 1a

SPS 3

SPS 4

SPS 5

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

10−12

10−10

10−8

10−6

sin θz

(b)B

R(µ

→ e

γ)

SPS 5SPS 4

SPS 1a

SPS 3

FIG. 1 (color online). Numerical illustration of the dependenceof BR��! e�� on sin�z: (a) in the m1 � 0 case; and (b) in them3 � 0 case. The black solid line and black dashed-dotted linedenote the present experimental upper bound on and the futureexperimental sensitivity to BR��! e��, respectively.

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SPS points are taken as the mSUGRA parameters. Whensin�z ’ 0:014, BR��! e�� arrives at its minimal value inthe m3 � 0 case. For the SPS slopes, larger M1=2 yieldssmaller BR��! e��. We plot the numerical dependenceof BR��! e�� on M1=2 in Fig. 2, where we have adoptedthe SPS slope 3 and taken 300 GeV M1=2 1000 GeV.We find that M1=2 � 474 GeV (or M1=2 � 556 GeV) canresult in BR��! e�� 1:2� 10�11 for sin�z � 0:014(or sin�z � 0:1). For all values of M1=2 between300 GeV and 1000 GeV, BR��! e�� is larger than thesensitivity of some planned experiments, which ought toexamine the m3 � 0 case when the SPS slope 3 is adopted.The same conclusion can be drawn for the SPS slopes 1aand 2. In view of the present experimental results on muong� � 2, one may get M1=2 & 430 GeV for tan� � 10 andA0 � 0 [38], implying that the m3 � 0 case should bedisfavored.

IV. DETERMINE THE HEAVY MAJORANANEUTRINO MASSES

In this paper, an important conclusion is the masses oftwo heavy right-handed Majorana neutrinos (M1 and M2)can be derived through the leptogenesis and the LFV raredecays. With the help of Eqs. (2) and (17), one can obtain

R �BR��! ���BR��! e��

�M2

2j�M��23j2�ln��GUT=M2��

2

M21j�M��12j

2�ln��GUT=M1��2 :

(18)

Since the successful leptogenesis can be used to fix M1, ameasurement of the above ratio will allow us to determineor constrain M2. On the other hand, we may calculate theratio R by inputting the appropriate M2. It is worthwhile toremark that the ratio R is independent of the mSUGRAparameters. To illustration, we show the numerical resultsof R as functions of sin�z in Fig. 3(a) for the m1 � 0 caseand Fig. 3(b) for them3 � 0 case, respectively. In them3 �0 case, when sin�z ! 0:0075, the phase ! 0 which

0.078 0.08 0.082 0.084 0.08610

2

104

106

108

1010

(a)

sin θz

BR

(τ→

µγ)

/ B

R(µ

→ e

γ)

M2 = 10 M

1

M2 = 100 M

1

M2 = 7.4 × 1015 GeV

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1610

2

103

104

105

(b)

sin θz

BR

(τ→

µγ)

/ B

R(µ

→ e

γ)

M2 = 10 M

1

M2 = 50 M

1

M2 = 7.4 × 1015 GeV

FIG. 3 (color online). Numerical illustration of the dependenceof BR��! ���=BR��! e�� on sin�z: (a) in the m1 � 0 case;and (b) in the m3 � 0 case.

300 400 500 600 700 800 900 100010

−13

10−12

10−11

10−10

10−9

M1/2

GeV

BR

(µ

→ e

γ)

sin θz = 0.014

sin θz = 0.1

FIG. 2 (color online). Numerical illustration of the dependenceof BR��! e�� onM1=2 for SPS slope 3 in them3 � 0 case. Theblack solid line and black dashed-dotted line denote the presentexperimental upper bound on and the future experimental sensi-tivity to BR��! e��, respectively.

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implies M1 and M2 may be larger than the grand unifiedtheory (GUT) scale �GUT � 2:0� 1016 GeV. So we de-mand that M1 and M2 must be less than the GUT scale.Below the scale �GUT,M2

2�ln��GUT=M2��2 obtain its maxi-

mum value at

M2 � �GUT=e � 7:4� 1015 GeV: (19)

Consequently, we can derive the maximum value of R withM2 � 7:4� 1015 GeV. In Fig. 3(b), the numerical resultsat the left side of the dashed-dotted lines are ‘‘nonphys-ical,’’ because these results are corresponding to M2 >7:4� 1015 GeV for the M2 � 10M1 (M2 � 50M1) case.We can obtain the ratio M2=M1 once R is measured, andfurthermore calculate the heavy neutrino mass M2. FromFig. 3, one may derive R< 2� 109 for the m1 � 0 caseand R< 8� 103 for the m3 � 0 case, respectively. If thefuture experiments prove R � 8� 103, the inverted masshierarchy case can be excluded. It is worthwhile to stressthat the above conclusions are independent of themSUGRA parameters.

Finally, let us comment on the M1 � M2 case [15]. Inthis case, the CP-violating asymmetry "2, which is pro-duced by the out-of-equilibrium decay of N2, may finallysurvive. In order to produce the positive cosmologicalbaryon number asymmetry (YB > 0), of Eq. (11) musttake ‘‘�’’ sign. The flavor effects may be neglected sincethe successful thermal leptogenesis requires M2 �1012 GeV (m1 � 0) and M2 � 4:4� 1013 GeV (m3 � 0).For M1 � 5M2, we calculate the branching ratio of �!e� and find the inverted hierarchy case is disfavored for allSPS points and slopes. For the normal hierarchy case, onemay distinguish the M1 � M2 case (R � 14) and theM1 � M2 case (R & 24) in terms of the ratio R. In addi-tion, the masses of heavy neutrinos may be quasidegener-ate [39]. In the M1 ’ M2 case, the so-called resonantleptogenesis may occur [40]. Such a scenario could allow

us to relax the lower bound on the lighter right-handedMajorana neutrino mass. Therefore, the branching ratios of�! e� and �! �� may be much less than the sensitiv-ities of a few planned experiments. R ’j�M��23j

2=j�M��12j2 can be derived from Eq. (18).

Furthermore, one may obtain 0:04 & R�1 & 0:07 (normalhierarchy) and 0:00045 & R�1 & 0:22 (invertedhierarchy).

V. SUMMARY

We have analyzed the LFV rare decays in the super-symmetric version of the MSM in which the FGY ansatz isincorporated. In this scenario, there are only three un-known parameters: �z, M1, and M2. The successful lepto-genesis may fix M1 for given values of �z. We havenumerically calculated the branching ratio of �! e� fordifferent values of sin�z by using the SPS points. Then onecan find the inverted mass hierarchy case is disfavored byall SPS points. For the SPS slopes, the planned experimentsmay measure �! e� in the m3 � 0 case. In addition,once the ratio of BR��! ��� to BR��! e�� is mea-sured, we may also distinguish the neutrino mass hierar-chies and fix the masses of two heavy right-handedMajorana neutrinos by means of the LFV processes andthe leptogenesis mechanism. It is worthwhile to stress thatthis conclusion is independent of the mSUGRAparameters.

ACKNOWLEDGMENTS

I am indebted to Z. Z. Xing and S. Zhou for stimulatingdiscussions and reading the manuscript. I am also gratefulto Y. L. Wu for helpful communication. This work wassupported in part by the National Nature ScienceFoundation of China (NSFC) under Grant No. 10475105and No. 10491306.

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