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PHYSICS LETTERS B
ELSEVIER
Physics Letters B 659 (2008) 640-650
www.elsevier.com/locate/physletb
Lepton flavour violation in the constrained MSSM with constrained sequential dominance
S. Antuscha *, S.F. Kingb
a Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München, Germany b School of Physics and Astronomy, University of Southampton, Southampton, SO171BJ, UK
Received 20 September 2007; received in revised form 8 November 2007; accepted 26 November 2007
Available online 8 December 2007
Editor: B. Grinstein
Abstract
We consider charged lepton flavour violation (LFV) in the constrained minimal supersymmetric Standard Model, extended to include the seesaw mechanism with constrained sequential dominance (CSD), where CSD provides a natural see-saw explanation of tri-bimaximal neutrino mixing. When charged lepton corrections to tri-bimaximal neutrino mixing are included, we discover characteristic correlations among the LFV branching ratios, depending on the mass ordering of the right-handed neutrinos, with a pronounced dependence on the leptonic mixing angle 613 (and in some cases also on the Dirac CP phase 8). © 2007 Elsevier B.V. All rights reserved.
1. Introduction
Over the past decade neutrino physics has revealed the surprising fact not only that neutrinos have mass, but also that lepton mixing must involve two large mixing angles, commonly referred to as the atmospheric angle 623 and the solar angle 612 [1]. The latest neutrino oscillation data [2] is consistent with tri-bimaximal lepton mixing [3]. Theoretical attempts to reproduce this structure typically produce tri-bimaximal mixing in the neutrino sector [4], with charged lepton mixing giving important corrections to the physical lepton mixing. For example, in the see-saw mechanism [5], sequential dominance (SD) [6] is well known to provide a natural explanation of hierarchical neutrino mass together with large neutrino mixing angles. When certain constraints are imposed on the neutrino Yukawa matrix elements then tri-bimaximal neutrino mixing can result from such a constrained sequential dominance (CSD) [7]. Charged lepton corrections can provide calculable deviations from tri-bimaximal mixing, resulting in predictive neutrino mixing sum rules [7-9] which may be proved with future long baseline neutrino experiments [10].
When neutrino mass models are combined with supersymmetry (SUSY) then lepton flavour violation (LFV) is an inevitable consequence [11-13]. In the constrained minimal supersymmetric standard model (CMSSM), in which the soft scalar mass matrices are described by a single universal soft high energy parameter m0, and a universal trilinear parameter A0, then the only source of LFV is due to RGE running effects, and in this case the connection between LFV processes and neutrino mass models has received a good deal of attention [14]. In the case of SD models it has been shown that LFV could reveal direct information about the neutrino Yukawa couplings in the diagonal charged lepton basis, depending on the particular nature of the SD, for example whether the dominant right-handed neutrino is the heaviest one or the lightest one [15,16]. For example if the dominant right-handed neutrino is the heaviest one, then large rates for t ^ ¡iy are expected [15,16]. However even in this case, the amount of information one can deduce is limited due to the large number of unconstrained Yukawa couplings.
* Corresponding author. E-mail addresses: antusch@mppmu.mpg.de (S. Antusch), sfk@hep.phys.soton.ac.uk (S.F. King).
0370-2693/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2007.11.092
In this Letter we consider LFV for the case of CSD, where the number of independent neutrino Yukawa couplings is reduced. In this case the LFV predictions are also sensitive to the charged lepton mixings, so some further assumptions are required in order to make predictions. In addition to tri-bimaximal mixing via CSD, we shall also additionally assume CKM-like charged lepton corrections. This will lead to interesting correlations in LFV muon and tau decays, independent of the SUSY mass parameters, and Yukawa couplings, providing quite specific predictions for LFV.
2. (Constrained) sequential dominance
Sequential dominance (SD) [6] represents classes of neutrino models where large lepton mixing angles and small hierarchical neutrino masses can be readily explained within the see-saw mechanism. To understand how sequential dominance works, we begin by writing the right-handed neutrino Majorana mass matrix MRR in a diagonal basis as
/ MA 0 0 \
Mrr = 0 Mb 0 . (1)
V 0 0 MCJ
We furthermore write the neutrino (Dirac) Yukawa matrix Xv in terms of (1, 3) column vectors Ai, Bi, C; as
Yv = (A B C), (2)
using left-right convention. The term for the light neutrino masses in the effective Lagrangian (after electroweak symmetry breaking), resulting from integrating out the massive right-handed neutrinos, is
„v (vjAi)(ATjvj) (vTBi)(BTvj) (vTCi)(Cfvj)
Leff = MA + MB + MC (3)
where v; (i = 1, 2, 3) are the left-handed neutrino fields. Sequential dominance then corresponds to the third term being negligible, the second term subdominant and the first term dominant: AiAj BiBj CiCj
AA » -B » CT1- (4)
MA M- MC In addition, we shall shortly see that small 613 and almost maximal 023 require that
|Ai|«|A2|«|A31- (5)
Without loss of generality, then, we shall label the dominant right-handed neutrino and Yukawa couplings as A, the subdominant ones as -, and the almost decoupled (sub-subdominant) ones as C. Note that the mass ordering of right-handed neutrinos is not yet specified. Again without loss of generality we shall order the right-handed neutrino masses as M1 < M2 < M3, and subsequently identify MA, MB, MC with M1, M2, M3 in all possible ways. LFV in some of these classes of SD models has been analysed in [16].
Writing Aa = |Aa , -a = —a |eг4-a, Ca = |Ca |eг^Ca and working in the mass basis of the charged leptons, under the SD condition Eq. (4), we obtain for the neutrino mixing angles [6]:
| A31'
tan 0J2 «-J-1-—, (6b)
C23 | -21 cos 4*2 - S23 | -3 | cos 03
^4+4-1-4A2) |-1|(A2-2 + A3-3) MA ei(44+*A1 -4a2^1!
13 [|A2|2 +1A312]3/2 M- V|A2|2 +1A312 , ( )
and for the masses
^ (|A2|2 + | A3 |2)^2
m3 "-MA-, (7a)
|B1|2 v2
m2 ^ 2 ,, , (7b)
tan 02v3 , (6a)
m1 « O(|C|2v2/Mc). (7c)
|2 v2/
As in [6] the PMNS phase 8 is fixed by the requirement that we have already imposed in Eq. (6b) that tan(012) is real and positive,
C23 | -21 sin 4*2 ^ S23 | -3 | sin 4>3, (8)
C231 -21 cos 42 - S231 -31 cos 4>3 > 0, (9)
<2 = <B2 - <B1 - < + 8, 4>3=<b3 - <B1 + <A2 - <a3 - < + 8. (10)
The phase < is fixed by the requirement (not yet imposed in Eq. (6c)) that the angle 613 is real and positive. In general this condition is rather complicated since the expression for 613 is a sum of two terms. However if, for example, A1 = 0 then < is fixed by:
<A2 - <B1 - Z (11)
Z = arg(A*B2 + A3B3). (12)
Eq. (12) may be expressed as
„ , IB2|s23s2 + |B3|c23s3
tan Z *-. (13)
|B2|i23C2 + IB3IC23C3
Inserting < of Eq. (11) into Eqs. (8), (10), we obtain a relation which can be expressed as
+ 8) * |B2|C23^2 -|B3|S23S3 . (14)
-IB2IC23C2 + IB3S23C3
In Eqs. (13), (14) we have written si = sin Zi, ci = cos Zi, where we have defined
Z2 = <B2 - <A2 , Z3 = <B3 - <A3 , (15)
which are invariant under a charged lepton phase transformation. The reason why the see-saw parameters only involve two invariant phases rather than the usual six, is due to the SD assumption in Eq. (4) that has the effect of effectively decoupling the right-handed neutrino of mass MC from the see-saw mechanism, which removes three phases, together with the further assumption (in this case) of A 1 = 0, which removes another phase.
2.1. CSD and tri-bimaximal neutrino mixing
Tri-bimaximal neutrino mixing [3] corresponds to the choice [7]:
IA11 = 0, (16)
IA2I = IA31, (17)
IB11 = | B21 = | B31, (18)
A!B = 0. (19)
This is called constrained sequential dominance (CSD) [7]. For example, a neutrino Yukawa matrix in the notation of Eq. (2), which satisfies the CSD conditions in Eqs. (16)-(19), may be taken to be:
where C is not constrained by CSD, since it only gives a sub-subdominant contribution to the neutrino mass matrix, so we have written it as C = (c1,c2,c3) above. CSD leads to tri-bimaximal mixing in the neutrino mass matrix mv, i.e., to
0 be c1
Fv = -ae^3 be c2
! { ae^ be c3
V273 1/V3 0
'vl ,tri '
V! tri = -!A/6 1/^3 1/V2 . (21)
1/V6 -1/V3 1/V2,
3. Charged lepton corrections
The form of the PMNS matrix will depend on the charged lepton Yukawa matrix whose diagonalisation will result in a charged lepton mixing matrix VeL which must be combined with VjL to form ^PMNS. The resulting lepton mixing matrix will therefore not be precisely of the tri-bimaximal form, even in theories that predict precise tri-bimaximal neutrino mixing. We consider here the case that CSD holds in a basis where the charged lepton mass matrix is not exactly diagonal, but corresponds to small mixing. This is a situation, often encountered in realistic models [7,9,17].
In the presence of charged lepton corrections, the prediction of tri-bimaximal neutrino mixing is not directly experimentally accessible. However, this challenge can be overcome when we make the additional assumption that the charged lepton mixing
matrix has a CKM-like structure, in the sense that VeL is dominated by a 1-2 mixing 0 = i.e. that its elements (V^)i3, (VeL)23, (VeL)31 and (VeL)32 are very small compared to (VeL)ij (i, j = 1,2). In the following, we shall take these elements to be approximately zero, i.e.,
-s0e-iX 0
v eL ^ ° ' -
VeL«P I se cee~ix 0 I , (22)
where ce = cos e, se = sin e, X is a phase required to diagonalise the charged lepton mass matrix [7], and P is a diagonal matrix of phases P = diag(eim1 ,ew>2 ,eim3) which are chosen to remove phases from the product VeL VjL to yield the physical ^PMNS. In the present case it is convenient to choose m1 = 0, &>2 = X, m3 = 0, to yield,
e -see-iX
VeL seeiX ce 0 I . (23)
With this choice, then by constructing ^PMNS and comparing to the standard PDG form of this matrix, one obtains, by comparing with Eq. (82) of [7],
X = 8 - n (24)
where 8 is the standard PDG CP violating oscillation phase. Also note that X « 822 - 812 where 8ij — arg Me.
We remark that the assumption that the charged lepton mixing angles are dominated by (1, 2) Cabibbo-like mixing arises in many generic classes of flavour models in the context of unified theories of fundamental interactions, where quarks and leptons are joined in representations of the unified gauge symmetries [7,9,17]. Under this assumption, it follows directly from Eq. (A.3) that №MNS)31, №MNS)32 and №mns)33 are independent of VeL, and depend only on the diagonalisation matrix V^ of the neutrino mass matrix. This leads to the parameterization-independent relations [10]:
|« )31M№mns)31|, (25a)
1« )32H№mns)32|, (25b)
l(VjL )33M(tfPMNs)33|. (25c)
In addition to the assumption that VeL is of the form of Eq. (22) for tri-bimaximal neutrino mixing the 1-3 mixing in the neutrino mass matrix is zero,
(Vl )13 = (26)
Using Eq. (26) and applying the standard PDG parameterization of the PMNS matrix (see e.g. [18]), Eq. (25a) leads to the sum rule [7-9]:
s23s v2 « |S23S12 - S13C23C12e^1 « S23S12 - S13C23C12cos(8), (27)
where the last step holds to leading order in s13. This sum rule can be used to test tri-bimaximal (0J2 = arcsin(-13)) structure of the neutrino mass matrix in the presence of CKM-like charged lepton corrections.
4. LFV in CSD with charged lepton corrections
When dealing with LFV it is convenient to work in the basis where the charged lepton mass matrix is diagonal. Let us now discuss the consequences of charged lepton corrections of the form of Eq. (23) for the neutrino Yukawa matrix with CSD. After re-diagonalising the charged lepton mass matrix, resulting in the assumed charged lepton mixing matrix in Eq. (23), Yv in Eq. (20) becomes transformed as:
Yv ^ Y'v = VeL Yv. (28)
In the diagonal charged lepton mass basis the neutrino Yukawa matrix therefore becomes:
/as0e-iXei^i b(ce - s0e-iX)ei^2 (c1ce - c2s0e-iX)\ Y I = (A' -' C) =1 -acee^ b(ce + seeiX)ei^ (c1SeeiX + c2ce) , (29)
V aebei^2 c3 )
where the column vectors A', -', C' are now defined in the diagonal charged lepton basis according to Eq. (29). Thus the results in Eqs. (6a)-(6c) with the redefined column vectors A', -', C' now yield the physical lepton mixing angles since these are equal to the neutrino mixing angles in the diagonal charged lepton basis of Eq. (29).
At leading order in a mass insertion (MI) approximation [11,12] the branching fractions of LFV processes are given by
a iT ,2 o
Bri7 = Br(li — ljY) f(M2,^,m-v),mL ^¡j tan2 /, (30)
G2f l ij
where li = e, l2 = /, l3 = t , and where the off-diagonal slepton doublet mass squared is given in the leading log approximation (LLA) of the CMSSM by
2(LLA) _ (3m2 + A00)
'Lij ~ 2
with the leading log coefficients given by
m ~ A) «- 0 2 0 Kj, (31)
K21 = a2 a1 * ln — + b2 b1 * ln — + c2 C * ln —,
21 21 ma 2 1 mb + 2 1 mc ,
K32 = A3 A2*ln MA + B3 B2*ln — + C3 C2*ln — K31 = A3A1*ln — + B3 B1*ln —B + c3 c1*ln MTtc (32)
The factors ^ j in Eq. (30) represent the ratio of the leptonic partial width to the total width, r(li — ljViVj)
= nu — all) (33)
Clearly f21 = 1 but f32 is non-zero and must be included for correct comparison with the experimental limit on the branching ratio for t — /Y. This factor is frequently forgotten in the theoretical literature.
If LFV is only induced by RG effects from Y'v on the soft breaking terms, as in the CMSSM, then in the LLog and MI approximation, the branching ratios for LFV charged lepton decays, like l i — ljY, are proportional to
Bri7 a|Kij|2 =|(A'A'% ln(A/MA) + (B'B'f)jj ln(A/MB) + (CC't)i. ln(T/Mc)|2. (34)
We have only assumed so far that the right-handed neutrino mass matrix has the diagonal form shown in Eq. (1), MRR = diag(MA,MB,MC) with the dominant right-handed neutrino labelled by A, the leading subdominant one labelled by B, and the decoupled one labelled by C. However the masses of the right-handed neutrinos are not yet ordered, and we have not yet specified which one is the lightest and so on. After ordering MA, MB, Mc according to their size, there are six possible forms of Y,^ obtained from permuting the columns, with the convention always being that the dominant one is labelled by A, and so on. In particular the third column of the neutrino Yukawa matrix could be A', B' or C' depending on which of MA, MB or Mc is the heaviest.
In hierarchical models, the (3, 3) elements of the Yukawa matrices describing quarks and charged leptons are amongst the largest elements in the Yukawa matrices. In unified models this will also be the case for the neutrino Yukawa matrix. If the heaviest right-handed neutrino mass is MA then the third column of the neutrino Yukawa matrix will consist of the A' column, and since Y3V3 = A3 and A1 ~ A2 ~ A3 ~ a then we conclude that all elements of A' must dominate over those of B', C', and hence all LFV processes will be determined approximately by (A'A't)ij. Similarly if the heaviest right-handed neutrino mass is MB then the third column of the neutrino Yukawa matrix will consist of the B' column, and since Y3V3 = B'3 and B1 ~ B2 ~ B3 ~ b then we conclude that all elements of B' must dominate over those of A', C', and hence all LFV processes will be determined approximately by (B'B't)ij. Finally if the heaviest right-handed neutrino mass is MC then the third column of the neutrino Yukawa matrix will consist of the C' column which contains the large element Y3V3 = C3. However in this case we cannot conclude that all elements of C must dominate over those of A', B' for the determination of LFV processes since the elements c1, c2 are undetermined by the see-saw mechanism and could even be set equal to zero. Nevertheless it is possible that in this case all elements of C could dominate over those of A', B' and hence all LFV processes could be determined approximately by (C'C't)ij. In the following we consider the LFV predictions arising from the three cases
M3 = MA, M3 = MB, M3 = MC, (35)
corresponding to the dominant Yukawa columns being A', B', C, respectively.
5. Predictions for the ratios of LFV branching ratios
After ordering MA, MB, MC according to their size, there are six possible forms of Y'v obtained from permuting the columns, with the convention always being that the dominant one is labeled by A', and so on. In particular the third column of the neutrino Yukawa matrix could be A', B' or C' depending on which of MA, MB or MC is the heaviest. If the heaviest right-handed neutrino mass is MA then the third column of the neutrino Yukawa matrix will consist of the (re-ordered) first column of Eq. (29) and
Fig. 1. The left panel shows the ratios of branching ratios B^; of LFV processes t
j y in CSD for M3 = Ma with right-handed neutrino masses Mi = 108 GeV,
M2 = 5 x 108 GeV and M3 = 1014 GeV. Here the solid lines show the (naive) prediction, from the MI and LLog approximation and with RG running effects for the other parameters neglected, while the dots show the explicit numerical computation (using SPheno2.2.2. [19] extended by software packages for LFV branching ratios and neutrino mass matrix running [20,21]) with universal CMSSM parameters chosen as m0 = 750 GeV, «1/2 = 750 GeV, A0 = 0 GeV, tan f = 10 and sing(|) = +1. The right panel shows the predictions (from full computation) for Br^e = Br(| ^ ey) in the CMSSM extended by the see-saw mechanism with CSD for the case M3 = Ma with e^ = 3° and 8 = 0. In this panel we have chosen the CMSSM parameters to satisfy A0 = 0 GeV, tan f = 10 and «1/2 = 5«0, which approximately corresponds to the successful stau co-annihilation region of LSP neutralino dark matter (DM) giving Qdm within the current WMAP limits.
assuming Y3'3 ~ 1 we conclude that all LFV processes will be determined approximately by the first column of Eq. (29). Similarly if the heaviest right-handed neutrino mass is M- then we conclude that all LFV processes will be determined approximately by the second column of Eq. (29). Note that in both cases the ratios of branching ratios are independent of the unknown Yukawa couplings which cancel, and only depend on the charged lepton angle e = ef2 (and in some cases on X), which in the case of tri-bimaximal neutrino mixing is related to the physical reactor angle by e13 = ef2/\/2 = e/V2 [7,9]. Also note that X = 8 - n where 8 is the Standard PDG CP violating oscillation phase. The predictions for these two cases will now be discussed in detail. We will also comment on the third case M3 = MC, which is less predictive, and give an explicit minimal example.
5.1. M3 = MA
In this case, assuming that the third column of the neutrino Yukawa matrix (associated with the heaviest right-handed neutrino and hence the largest Yukawa couplings) is the dominant column A' associated with the atmospheric neutrino of mass m3, one can read off from Eqs. (34) and (29) that the Brj = Br(li ^ ijY) now satisfy
-T, | 2 |2<.
B| a |a sece |
-r, I 2 |2 <.
BrTe a |a2S01
BrrM a | a
Note that 0 = 0f2 = V2013, so there is a direct (and simple) connection to the measurable lepton mixing angle 013 in neutrino oscillation experiments in this case. In particular, we predict
= (soco)2 ^ =
isin(2V20i3)
^ = (co)2 ^ = [cos(V20i3)]2 ^
BrTe ste ste
= (so)2 = [sin(V20i3)]2.
The predictions for the ratios of branching ratios as a function of e13 as well as for Br|e, for some sample choice of parameters, are shown in Fig. 1.
5.2. M3 = MB
In this case, assuming that the third column of the neutrino Yukawa matrix (associated with the heaviest right-handed neutrino and hence the largest Yukawa couplings) is the leading subdominant column -' associated with the solar neutrino of mass m2, one
Fig. 2. The left panel shows the ratios of branching ratios Brj of LFV processes Ii ^ ijy in CSD for M3 = Mb and for S = 0. The other parameters are chosen as in Fig. 1. The solid lines in the left panel show the (naive) prediction, from the MI and LLog approximation and with RG running effects neglected, while the dots show the explicit numerical computation. The right panel shows the predictions (from full computation) for Br^e = Br(/u, ^ ey) in the CMSSM extended by the see-saw mechanism with CSD for the case M3 = Mb with 913 = 3° and S = 0. The other parameters are chosen as in Fig. 1.
can read off from Eqs. (34) and (29) that the Brij = Br(li ^ Ijy) now satisfy
BrMe a \b2 (c9 - S9e-il)(c9 + S9eiX)f^e, Brte a \b2(c9 - S9e~lX)\2^te,
Br^ a \b2(ce + seeix) f^/
Since e = ee2 = V2ei3, there is again a connection to the measurable lepton mixing angle e13 in neutrino oscillation experiments. Furthermore, the branching ratios also depend on the phase X, which is related to the Standard PDG CP violating oscillation phase S by X = S - n. The ratios of branching ratios are predicted as
Br Br/
= \ce - see
iX\2 k± = \cos(V2ei3) + sin(V2ei3 )e-iS\2
ST/x ST/x
iX \2 S/e _ JS \2 S//e
= \ce + seeiX \ = \cos(V2ei3) - sin(V2ei3)e'S\2
BrTe STe STe
BrTe ce - see-iX 2 \\
BrT/ ce + - see'X
cos(V2ei3) + sin^V2ei3)e-iS
cos(V2ei3) - sin(V2ei3)eiS
Fig. 2 shows the predictions for the ratios of branching ratios as a function of 913, for the example S = 0, as well as the prediction for BrMe for some sample choice of parameters.
5.3. M3 = MC
In this case, assuming that the third column of the neutrino Yukawa matrix (associated with the heaviest right-handed neutrino and hence the largest Yukawa couplings) is the most subdominant column C associated with the lightest neutrino of mass m1, assuming that c3 « 1, one can see from Eqs. (34) and (29) that the Brj = Br (I i ^ Ijy) now depend on undetermined coefficients c1, c2. Hence we cannot make definite predictions. Moreover, in some cases, the subdominant column of Yukawa coupling also contributes at the same order as the dominant one. Nevertheless, charged lepton corrections also have an impact here. Let us therefore generalize VeL to include also a small 9|3 ^ 9e2. As a minimal case, let us furthermore consider
C = (0, 0,c)T
This may be viewed as minimal scenario regarding LFV, since typically (barring cancellations) the zeros are replaced by small entries and since, as mentioned above, the subdominant column of Y v cannot in general be neglected. For a more accurate treatment of this scenario with respect to the charged lepton corrections, the (typically) even smaller 9f3 ^ 9|3 can be included analogously. Including charged lepton corrections from 9xe2 and 9|3 (by Y v ^ VeL Y v) leads to approximately
cs23si2,cs23 ,c)
Fig. 3. The left panel shows the ratios of branching ratios Brj of LFV processes li — lj y for a minimal example with CSD and M3 = Mc described in the text. The solid lines in the left panel show the (naive) prediction, from the MI and LLog approximation and with RG running effects neglected, while the dots show the explicit numerical computation. The right panel shows the predictions (from full computation) for Br/e = Br(/ — eY) in the CMSSM extended by the see-saw mechanism with CSD for the case M3 = Mc in the scenario with m1 = 10-3 eV and 5 = 0. The other parameters are chosen as in Fig. 1.
and thus to the following relations for the branching ratios:
Br/e a |C2(43 fsufS/e, (50)
BrTe a |C2i|3ie2|2^Te, (51)
Brt/ a |c2s|3 |2^t/. (52)
As in the cases M3 = MA and M3 = MB, the relation 0^2 = V2013 holds under the considered assumption about the charged lepton corrections. For the ratios of the branching ratios we obtain
Br1 = K^f JT = [sin(V2013)S2e3]2 (53)
Br/e [ e ]2 ^/xe
Br = [s2^2 f, (54)
BrTe ?Te
BrTe = [^12]2 = [sin(V2013)]2. (55)
The predictions for the ratios of branching ratios as a function of d13 as well as for Br/e as a function of d13 and m1/2 (set equal to 5m0 as an example) are shown in Fig. 3. To give an explicit example, we have chosen s^3 = sin^C™) ^ 2.36° and other parameters as stated in the caption of Fig. 1. We would like to stress again that, in contrast to the cases M3 = MA and M3 = MB discussed above, the shown results are no definite predictions for the case M3 = MC , but rather order of magnitude examples for certain classes of models of CSD where the LFV branching ratios are significantly smaller than for CSD with M3 = MA and M3 = MB. As can be seen from Fig. 3, this scenario can be readily distinguished from the cases M3 = MA and M3 = MB.
6. Conclusions
We have considered charged lepton flavour violation (LFV) in the constrained minimal supersymmetric Standard Model, extended to include the see-saw mechanism with constrained sequential dominance (CSD), where CSD provides a natural see-saw explanation of tri-bimaximal neutrino mixing. When Cabibbo-like charged lepton corrections to tri-bimaximal neutrino mixing are included, this leads to characteristic correlations among the LFV branching ratios BrT/, Br/e and BrTe which may be tested in future experiments.
There are two main differences between the study here and that in [16] where predictions for LFV were also presented for the CMSSM with SD. The first difference is that here we have focused on the special case of CSD, corresponding to tri-bimaximal neutrino mixing, where the neutrino Yukawa couplings are very tightly constrained compared to the general SD case. The second difference is that we have considered the effect of charged lepton corrections, which were not included in [16]. In particular we have mainly considered Cabibbo-like charged lepton corrections, which when combined with CSD leads to a very tightly constrained scenario in which ratios of branching ratios depend on 013, which is related to the charged lepton mixing angle 0^2. The predictions also depend crucially on which column of the Yukawa matrix is associated with the heaviest right-handed neutrino M3, since this column will have the largest Yukawa couplings.
For the case M3 = MA, also known as Heavy Sequential Dominance (HSD) since the dominant right-handed neutrino is the heaviest one, we find the characteristic ratios in Fig. 1. Compared to the results in [16], the hierarchy between Br/e and BrT/ is much milder. This can be understood from the fact that in [16] it was assumed that |A1| ^ |A2| ^|A3|~ 1 (ignoring charged lepton corrections) which led to large BrT/ but small Br/e. However, including charged lepton corrections, we see that | A11 ~ | A2|~| A31~ 1, leading to both large BrT/ and large Br/e. In the present case we focus on tri-bimaximal neutrino mixing, which before charged lepton corrections are included implies that |A1| ^ |A21 = |A31 ~ 1, corresponding to the CSD explanation of tri-bimaximal neutrino mixing. Then, after Cabibbo-like charged lepton corrections are included, this leads to well defined predictions for the each of the couplings | A11, | A21, | A31, and hence rather precise predictions for ratios of BrT/, Br/e and BrTe, which depend on 613, as shown in Fig. 1. We reemphasize that, after charged lepton corrections are included, | A11 ~|A2|~| A31 ~ 1, and hence both BrT/ and Br/e are large in this case, unlike [16] where charged lepton corrections were ignored.
In the case M3 = MB, where the leading subdominant right-handed neutrino responsible for the solar neutrino mass is the heaviest one, the predicted ratios of branching ratios are even milder, corresponding to the fact that all the Yukawa coupling in this column are equal before the inclusion of charged lepton corrections, |B1| = |B2| = |B3|~ 1, again corresponding to the CSD explanation of tri-bimaximal neutrino mixing. When Cabibbo-like charged lepton corrections are included this again leads to characteristic predictions for ratios of BrT/, Br/e and BrTe, depending on 613 (shown in Fig. 2 for 5 = 0) as well as on the Dirac CP phase 5, as given in Eqs. (45)-(47).
The least predictive case is M3 = MC, which includes the case where the dominant right-handed neutrino is the lightest one known as light sequential dominance (LSD). In this case the generic prediction from [16] was that the BrT/ was generally quite small, typically of order Br/e, due to the small neutrino Yukawa couplings. In particular the neutrino Yukawa couplings of the third column were not considered relevant due to the large mass of the associated right-handed neutrino, which was assumed to exceed the GUT scale from which the RGEs were run down. Then the relevant Yukawa couplings were those from the second column, which all take similar (small) values leading to BrT/ ~ Br/e. This may also be the case here, since including charged lepton corrections will not change this result, and CSD will only strengthen this conclusion. However in other cases, for example if the RGEs are run from the Planck scale, the third column of the neutrino Yukawa matrix should not be ignored. In Fig. 3 we considered an example of this, in which the LFV arises solely from the third column, and the Yukawa couplings in this column are again determined from charged lepton corrections, assuming that C = (0,0,c)T, which may be approximately true in practice, but which is by no means guaranteed.
In summary, the results presented here once again confirm that BrT/, Br/e and BrTe are all expected to be observed in the (near) future. If they are observed in the ratios predicted here, for some value of 013, then this may be an indication of a high energy theory with the characteristics of the CMSSM extended to include the see-saw mechanism with CSD, corresponding to tri-bimaximal neutrino mixing corrected by Cabibbo-like charged lepton mixing angles.
Acknowledgements
S.A. would like to thank M.J. Herrero and E. Arganda for many very helpful discussions about LFV in SUSY see-saw scenarios and for providing their software packages on LFV muon and tau decays.
Appendix A. Conventions
In general, the mixing matrix in the lepton sector, the PMNS matrix ^pmns, is defined as the matrix which appears in the electroweak coupling to the W bosons expressed in terms of lepton mass eigenstates. With the mass matrices of charged leptons Me and neutrinos mv written as
L =-1lMeeR - 2VLmvvL + H.c., (A.1)
and performing the transformation from flavour to mass basis by
VeL Me VeR = diag(me,mM,mT), Vvl myV^ = diag(m1,m2,m3), (A.2)
the PMNS matrix is given by
^PMNS = VeL <. (A.3)
Here it is assumed implicitly that unphysical phases are removed by field redefinitions, and ^PMNS contains one Dirac phase and two Majorana phases. The latter are physical only in the case of Majorana neutrinos, for Dirac neutrinos the two Majorana phases can be absorbed as well.
The standard PDG parameterization of the PMNS matrix (see e.g. [18]) is:
c12c13 S12c13 S13e 'S
Upmns =| -c23S12 - S13S23c12eiS c23c12 - S13S23S12eiS S23c13 | ^Maj, (A.4)
S23S12 - S13c23cnelS -S23c12 - S13c23S12e'S c23c13
which is used in most analyses of neutrino oscillation experiments. Here S is the so-called Dirac CP violating phase which is in
.«i -01
principle measurable in neutrino oscillation experiments, and PMaj = diag(e! 2 ,e' 2 , 0) contains the Majorana phases a1,a2. In the following we will use this standard parameterization (including additional phases) also for vJl and denote the corresponding mixing angles by 9j, while the mixing angles 9ij without superscript refer to the PMNS matrix.
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