Baryon asymmetry from leptogenesis with four zero neutrino Yukawa textures

0.28 MB·English

by  Biswajit Adhikary

Baryon asymmetry from leptogenesis with
four zero neutrino Yukawa textures
1 Biswajit Adhikarya,b , Ambar Ghosala and Probir Roya
1 ∗ † ‡
0
a) Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India
2

n b)Department of Physics, Gurudas College, Narkeldanga, Kolkata-700054, India
a
J

7
January 28, 2011
2

]
h
p
-
p Abstract
e
h
[ The generation of the right amount of baryon asymmetry η of the Universe from

2 supersymmetric leptogenesis is studied within the type-I seesaw framework with three
v
heavy singlet Majorana neutrinos N (i = 1,2,3) and their superpartners. We assume
5 i
3 the occurrence of four zeroes in the neutrino Yukawa coupling matrix Y , taken to be
ν
6
2 µτ symmetric, in the weak basis where Ni (with real masses Mi > 0) and the charged
.
9 leptons l (α = e,µ,τ) are mass diagonal. The quadrant of the single nontrivial phase,
α
0
allowed in the corresponding light neutrino mass matrix m , gets fixed and additional
0 ν
1 constraints ensue from the requirement of matching η with its observed value. Special
:
v attention is paid to flavor effects in the washout of the lepton asymmetry. We also
i
X comment on the role of small departures from high scale µτ symmetry due to RG
r
a evolution.
1 Introduction
Baryogenesis through leptogenesis [1, 2, 3] is a simple and attractive mechanism to explain
the mysterious excess of matter over antimatter in the Universe. A lepton asymmetry is first
∗biswajit.adhikary @saha.ac.in
†[email protected]
‡[email protected]
generated at a relatively high scale (> 109 GeV). This then gets converted into a nonzero
η, the difference between the baryonic and antibaryonic number densities normalized to the
photon number density (n n )n 1, at electroweak temperatures [4] due to B+L violat-
B − B¯ −γ
ing but B L conserving sphaleron interactions of the Standard Model. Since the origin of
−
the lepton asymmetry is from out of equilibrium decays of heavy unstable singlet Majorana
neutrinos [5], the type-I seesaw framework [6, 7, 8, 9, 10], proposed for the generation of light
neutrino masses, is ideal for this purpose. We study baryogenesis via supersymmetric lepto-
genesis [11] with a type-I seesaw driven by three heavy (> 109 GeV) right-chiral Majorana
neutrinos N (i = 1,2,3) with Yukawa couplings to the known left chiral neutrinos through
i
therelevant Higgsdoublet. There have been some recent investigations [12, 13, 14, 15] study-
ing the interrelation between leptogenesis, heavy right-chiral neutrinos and neutrino flavor
mixing. However, our angle is a little bit different in that we link supersymmetric leptogen-
esis to zeroes in the neutrino Yukawa coupling matrix. In fact, we take a µτ symmetric [16]
neutrino Yukawa coupling matrix Y with four zeroes [17] in the weak basis specified in the
ν
abstract.
There are several reasons for our choice. First, a seesaw with three heavy right chiral
neutrinosisthesimplest type-Ischeme yieldingasquareYukawacouplingmatrixY onwhich
ν
symmetries can be imposed in a straightforward way. Second, µτ symmetry [18] - [46] in the
neutrino sector provides a very natural way of understanding the observed maximal mixing
of atmospheric neutrinos. Though it also predicts a vanishing value for the neutrino mixing
angle θ , the latter is known from reactor experiments to be rather small. A tiny nonzero
13
value of θ could arise at the 1-loop level via the charged lepton sector, where µτ symmetry
13
is obviously broken, though RG effects if the said symmetry is imposed at a high scale [16].
Third, four has been shown [17] to be the maximum number of zeroes phenomenologically
allowed inY within the type-I seesaw framework in the weak basis described earlier. Finally,
ν
four zero neutrino Yukawa textures provide [47] a very constrained and predictive theoretical
scheme - particularly if µτ symmetry is imposed [16].
The beautiful thing about such four zero textures in Y is that the high scale CP violation,
ν
required for leptogenesis, gets completely specified here [17] in terms of CP violation that
is observable in the laboratory with neutrino and antineutrino beams. In our µτ symmetric
scheme [16], which admits two categories A and B, the latter is given in terms of just one
phase(foreachcategory)whichisalreadyquiteconstrainedbytheextantneutrinooscillation
data. Indeed, the quadrant in which this phase lies - which was earlier unspecified by the
same data - gets fixed by the requirement of generating the right size and sign of the baryon
2
asymmetry. Moreover, the magnitude of this phase is further constrained.
In computing the net lepton asymmetry generated at a high scale, one needs to consider not
only the decays of heavy right-chiral neutrinos N into Higgs and left-chiral lepton doublets
i
as well as their superpartner versions but also the washout caused by inverse decay processes
in the thermal bath. The role of flavor [48, 49, 50, 51] can be crucial in the latter. In the
Minimal Supersymmetric Standard Model (MSSM [52]), this has been studied [53] through
flavor dependent Boltzmann equations. The solutions to those equations demonstrate that
flavor effects show up differently in three distinct regimes depending on the mass of the
lightest of the three heavy neutrinos and an MSSM parameter tanβ which is the ratio
v /v of the up-type and down-type Higgs VEVs. In each regime there are three N mass
u d i
hierarchical cases : (a) normal, (b) inverted and (c) quasidegenerate. All these, considered
in both categories A and B, make up eighteen different possibilities for each of which the
lepton asymmetry is calculated here. That then is converted into the baryon asymmetry by
standard sphaleronic conversion and compared with observation. These lead to the phase
constraints mentioned above as well as a stronger restriction on the parameter tanβ in some
cases.
If µτ symmetry is posited at a high scale characterized by the masses of the heavy Majorana
neutrinos, renormalization group evolution down to a laboratory energy λ breaks it radia-
tively. Consequently, a small nonzero θλ , crucially dependent on the magnitude of tanβ,
13
gets induced. The said new restrictions on tanβ coming from η in some cases therefore cause
strong constraints on the nonzero value of θλ which we enumerate.
13
One possible problem with high scale supersymmetric thermal leptogenesis is that of the
overabundance of gravitinos caused by the high reheating temperature. For a decaying
gravitino, this can lead to a conflict with Big Bang Nucleosynthesis constraints, while for a
stablegravitino(darkmatter)thisposesthedangerofoverclosing theUniverse. Theproblem
can be evaded by appropriate mass and lifetime restrictions on the concerned sparticles, cf.
sec. 16.4 of ref [52]. Such is the case, for instance, with gauge mediated supersymmetry
breaking with a gravitino as light as O(KeV) in mass. In gravity mediated supersymmetry
breaking there are sparticle mass regions where the problem can be avoided – especially
withinaninflationaryscenario. Anillustrationisamodel [54], withagluino andaneutralino
that are close in mass, which satisfies the BBN constraints. Purely cosmological solutions
within the supersymmetric inflationary scenario have also been proposed, e.g. [55]. We feel
that, while the gravitino issue is one of concern, it can be resolved and therefore need not
3
be addressed here any further.
The plan of the rest of the paper is as follows. In section 2 we recount the properties
of the allowed µτ symmetric four zero Y textures. Section 3 contains an outline of the
ν
basic steps in our calculation of η. In section 4, η is computed in our scheme for the three
different heavy neutrino mass hierarchical cases in the regimes of unflavoured, fully flavored
and τ-flavored leptogenesis for both categories A and B. Section 5 consists of our results on
constraints emerging from η on the allowed µτ symmetric four zero Y textures. In section
ν
6 we discuss the departures - due to RG evolution down to laboratory energies - from µτ
symmetry imposed at a high scale min (M ,M ,M ) M . Section 7 summarizes
1 2 3 lowest
∼ ≡
our conclusions. Appendices A, B and C list the detailed expressions for η in each of the
eighteen different possibilities.
2 Allowed µτ symmetric four zero textures of Y
ν
The complex symmetric light neutrino Majorana mass matrix m is given in our basis by
ν
1
m = v2Y diag.(M 1,M 1,M 1)YT = Udiag.(m ,m ,m )UT. (2.1)
ν −2 u ν 1− 2− 3− ν 1 2 3
We work within the confines of the MSSM [52] so that v = vsinβ and the W-mass equals
u
1gv, g being the SU(2) semiweak gauge coupling strength. The unitary PMNS mixing
2 L
matrix U is parametrized as
1 0 0 c 0 s e iδD c s 0 eiαM 0 0
13 13 − 12 12
−
U = 0 c23 s23 0 1 0  s12 c12 0 0 eiβM 0,
− −
0 s23 c23 s13eiδD 0 c13  0 0 1 0 0 1
    
    (2.2)
where c = cosθ , s = sinθ and δ , α , β are the Dirac phase and two Majorana
ij ij ij ij D M M
phases respectively.
The statement of µτ symmetry is that all couplings and masses in the pure neutrino part of
the Lagrangian are invariant under the interchange of the flavor indices 2 and 3. Thus
(Y ) = (Y ) , (2.3a)
ν 12 ν 13
(Y ) = (Y ) , (2.3b)
ν 21 ν 31
(Y ) = (Y ) , (2.3c)
ν 23 ν 32
4
(Y ) = (Y ) (2.3d)
ν 22 ν 33
and
M = M . (2.4)
2 3
Eqs. (2.3) and (2.4), in conjunction with eq.(2.1), lead to a custodial µτ symmetry in m :
ν
(m ) = (m ) = (m ) = (m ) , (2.5a)
ν 12 ν 21 ν 13 ν 31
(m ) = (m ) . (2.5b)
ν 22 ν 33
Eqs. (2.5) immediately imply that θ = π/4 and θ = 0. With this µτ symmetry, it was
23 13
shown in Ref. [16] that only four textures with four zeroes in Y are allowed. These fall into
ν
two categories A and B - each category containing a pair of textures yielding an identical
form of m . These allowed textures may be written in the form of the Dirac mass matrix
ν
m = Y v /√2 in terms of complex parameters a , a , b , b .
D ν u 1 2 1 2
a a a a a a
1 2 2 1 2 2
(1) (2)
Category A : mDA =  0 0 b1 ,mDA =  0 b1 0 , (2.6a)
 0 b1 0   0 0 b1 
   
   
a 0 0 a 0 0
1 1
(1) (2)
Category B : mDB = b1 0 b2,mDB = b1 b2 0 , (2.6b)
b1 b2 0  b1 0 b2
   
   
The corresponding expressions for m , obtained via eq.(2.1), aremuch simplified by a change
ν
of variables. We introduce overall mass scales m , real parameters k , k , l , l and phases
A,B 1 2 1 2
¯
α¯ and β defined by
Category A :
a M a a
m = b2/M , k = 1 2, k = 2 , α¯ = arg 1. (2.7a)
A − 1 2 1 b sM 2 b a
(cid:12) 1(cid:12) 1 (cid:12) 1(cid:12) 2
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Category B : (cid:12) (cid:12) (cid:12) (cid:12)
a M b M b
m = b2/M , l = 1 2, l = 1 2, β¯ = arg 1. (2.7b)
Then the light nBeutr−ino2ma2ss ma1trix(cid:12)(cid:12)(cid:12)(cid:12)bfo2r(cid:12)(cid:12)(cid:12)(cid:12)seaMch1cate2gory(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)bc2a(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)nsbMe 1written as [5b32]
k2e2iα¯ +2k2 k k l2 l l eiβ¯ l l eiβ¯
1 2 2 2 1 1 2 1 2
mνA = mA k2 1 0 ,mνB = mBl1l2eiβ¯ l22e2iβ¯+1 l22e2iβ¯ . (2.8)
 k2 0 1  l1l2eiβ¯ l22e2iβ¯ l22e2iβ¯+1
   
5
We shall also employ the matrix
h = m†DmD (2.9)
which is identical for the two textures of Category A as well as for the two textures of
Category B. Indeed, it can be given separately for the two categories as
k2 x1/4k k e iα¯ x1/4k k e iα¯
1 1 2 − 1 2 −
hA = |mA|M1x1/4k1k2eiα¯ √x(1+k22) √xk22 , (2.10a)
x1/4k1k2eiα¯ √xk22 √x(1+k22) 
 
l2 +2l2 x1/4l e iβ¯ x1/4l e iβ¯
1 2 2 − 2 −
hB = mB M1x1/4l2eiβ¯ √x 0 , (2.10b)
| |
x1/4l2eiβ¯ 0 √x 
 
 
where
M2
x = 2=3. (2.11)
M2
1
¯
Restrictions on the parameters k , k , cosα¯ and l , l , cosβ from neutrino oscillation data
1 2 1 2
were worked out in ref. [16]. The relevant measured quantities are the ratio of the solar to
atmospheric neutrino mass squared differences R = ∆m2 /∆m2 and the tangent of twice
21 32
the solar mixing angle tan2θ . One can write
12
R = 2(X2 +X2)1/2[X (X2 +X2)1/2]−1, (2.12a)
1 2 3 − 1 2
X
1
tan2θ = . (2.12b)
12
X
2
The quantities X are given for the two categories as follows :
1,2,3
Category A :
X = 2√2k [(1+2k2)2 +k4 +2k2(1+2k2)cos2α¯]1/2, (2.13a)
1A 2 2 1 1 2
X = 1 k4 4k4 4k2k2cos2α¯, (2.13b)
2A − 1 − 2 − 1 2
X = 1 4k4 k4 4k2k2cos2α¯ 4k2. (2.13c)
3A − 2 − 1 − 1 2 − 2
Category B :
X = 2√2l l [(l2 +2l2)2 +1+2(l2 +2l2)cos2β¯]1/2, (2.13d)
1B 1 2 1 2 1 2
X = 1+4l2cos2β¯+4l4 l4, (2.13e)
2B 2 2 − 1
X = 1 (l2 +2l2)2 4l2cos2β¯. (2.13f)
3B − 1 2 − 2
6
We also choose to define
X = (X2 +X2 )1/2. (2.14)
A,B 1A,B 2A,B
At the 3σ level, tan2θ is presently known to be [56] between 1.83 and 4.90. For this range,
12
onlythe inverted mass ordering forthe light neutrinos, i.e. ∆m2 < 0, is allowed forCategory
32
A with the allowed interval for R being 4.13 10 2 eV2 to 2.53 10 2 eV2. In contrast,
− −
− × − ×
the same range of tan2θ allows only the normal light neutrino mass ordering ∆m2 > 0 for
12 32
Category B with R restricted to be between 2.46 10 2 eV2 and 3.92 10 2 eV2. A thin
− −
× ×
sliver is allowed [16] in the k k plane for Category A, while a substantial region with two
1 2
−
branches is allowed [16] in the l l plane for Category B. Finally, cosα¯ is restricted to the
1 2
−
¯
interval bounded by 0 and 0.0175, while cosβ is restricted to the interval bounded by 0 and
¯
0.0523. Thus, α¯, β could be either in the first or in the fourth quadrant. The interesting new
point in the present work is that the baryogenesis constraint leads to restrictions on sin2α¯
¯ ¯
and sin2β to the extent of removing the quadrant ambiguity in α¯ and β.
3 Basic calculation of baryon asymmetry
Armedwithµτ symmetryaswellaseqs. (2.8)and(2.10),wecantackleleptogenesisatascale
M . There are three possible mass hierarchical cases for N . Case (a) corresponds to a
lowest i
∼
normal hierarchy of the heavy Majorana neutrinos (NHN), i.e. M = M << M = M .
lowest 1 2 3
In case (b) one has an inverted hierarchy for N (IHN) with M = M = M << M .
i lowest 2 3 1
Case (c) refers to the quasidegenerate (QDN) situation with M M M M .
1 2 3 lowest
∼ ∼ ∼
Working within the MSSM [52] and completely neglecting possible scattering processes [53]
which violate lepton number, we can take the asymmtries generated by N decaying into a
i
doublet of leptons L and a Higgs doublet H as
α u
Γ(N LCH ) Γ(N L HC) 1 1
ǫα = i → α u − i → α u αf(x )+ α , (3.1)
i Γ(Ni → LCαHu)+Γ(Ni → LαHuC) ≃ 4πvu2hii Xj6=i"Iij ij Jij1−xij#
Iiαj = Im [(m†D)iα(mD)αjhij], (3.2)
Jiαj = Im [(m†D)iα(mD)αjhji], (3.3)
where
x = M2/M2 (3.4)
ij j i
7
and
2 1+x
ij
f(x ) = √x ln . (3.5)
ij ij
"1 xij − xij #
−
We note here that the α term does not contribute to ǫα in our scheme since it vanishes [16]
Jij i
on account of µτ symmetry. Further, contributions to ǫα from N decaying into sleptons and
i i
˜
higgsinos and fromsneutrinos N decaying into sleptons and Higgsas well as into leptons and
i
higgsinos have been included by appropriately choosing the x -dependence in the RHS of eq.
ij
(3.5). Observe also that α (and hence ǫα) gets an overall minus sign from Im(e iα¯,e iβ¯),
I1j 1 − −
whereas α, α (and hence ǫα ) get an overall plus sign from Im(eiα¯,eiβ¯). Except for being
I2j I3j 2,3
positive in the region 0.4 x < 1, the function f(x ) of eq.(3.5) is negative for all other
ij ij
≤
values of its argument. These signs are crucial in determining the sign of η and hence those
¯
of α¯, β.
The decay asymmetries ǫα get converted into a leptonasymmetry Yα = (nα n¯α)s 1, s being
i l − l −
the entropy density and nα (n¯α) being the leptonic (antileptonic) number density (including
l l
superpartners) for flavor α via the washout relation [53]
Yα = ǫα αg 1. (3.6)
i Ki ⋆−i
i
X
Ineq. (3.6), g istheeffectivenumber ofspindegreesoffreedomofparticlesandantiparticles
⋆i
at a temperature equal to M . Furthermore, when all the flavors are active, the quantity
i
α is given by the approximate relation [12, 51], neglecting contributions from off-diagonal
Ki
elements of A,
8.25 Aαα Kα 1.16
( α) 1 + | | i . (3.7)
Ki − ≃ Aαα Kα 0.2 !
| | i
In eq. (3.7), Kα is the flavor washout factor given by
i
Kα = Γ N → LαHuC = |mDαi|2 MPl , (3.8)
i (cid:16) H(M ) (cid:17) M 6.64π√g v2
i i ⋆i u
M being the Planck mass. This follows since the Hubble expansion parameter H(M ) at a
Pl i
temperature Mi is given by 1.66√g⋆iMi2MP−l1. Moreover, to the lowest order, Γ(Ni → LαHuC)
equals |mDαi|2Mi(4πvu2)−1. An additional quantity, appearing in eq. (3.7), is Aαα, a diagonal
element of the matrix Aαβ defined by
Yα = AαβYβ. (3.9)
L ∆
β
X
8
Here Yα = s 1(nα n¯α), nα being the number density of left-handed lepton and slepton
L − L − L L
doublets of flavor α and Yα = 1Y Yα, Y being the baryonic number density (normalized
∆ 3 B− B
to the entropy density s) including all superpartners. The precise forms for Aαβ in different
regimes of leptogenesis will be specified later.
One can now utilize the relation between Y = (n n )s 1 and Y = Yα, namely [57]
B B − B¯ − l α
P
8n +4n
F H
Y = Y , (3.10)
B l
−22n +13n
F H
where n (n ) is the number of matter fermion (Higgs) SU(2) doublets present in the
F H L
theory at electroweak temperatures. For MSSM, n = 3 and n = 2 so that eq. (3.10)
F H
becomes
8
Y = Y . (3.11)
B l
−23
The baryon asymmetry η = (n n )n 1 can now be calculated, utilizing the result [58]
B − B¯ −γ
that sn 1 7.04 at the present time, to be
−γ ≃
s
η = Y 7.04Y 2.45Y . (3.12)
B B l
n ≃ ≃ −
γ
Leptogenesis occurs at a temperature of the order of M and the effective values of Aαα
lowest
and α depend on which flavors are active in the washout process. This is controlled [53]
Ki
by the quantity Mlowest(1+tan2β)−1. There are three different regimes which we discuss
separately.
2 1 12
(1) Mlowest(1+tan β)− > 10 GeV.
In this case there is no flavor discrimination and unflavored leptogenesis takes place. Thus
Aαβ = δαβ and all flavors α can just be summed in eqs. (3.1). Thus ǫ = ǫα, α = 0,
− i i αJij
α = Im (h )2 and Y = ǫ g 1 with 1 = 8.25K 1 + (K /0.2)1.16 and
Iij ≡ αIij ij i i ⋆−i Ki Ki− i− P i P
Ki = PαKiα = hiiMPl(6.64π√g⋆iMiPvu2)−1. For the normal hierarchical heavy neutrino
(NHN) case (a), M may be ignored and the index i can be restricted to just 1, taking
P 2=3
g = 232.5. For the corresponding inverted hierarchical (IHN) case (b) M can be ignored
⋆1 1
and i made to run over 2 and 3 with g = 236.25, all quantities involving the index 2
⋆2=3
being identical to thecorresponding ones involving 3. Coming to the quasidegenerate (QDN)
heavy neutrino case (c), g = 240 and the contributions from i = 1 must be separately added
⋆
to identical contributions from i = 2,3.
2 1 9
(2) Mlowest(1+tan β)− < 10 GeV.
9
Here, all flavors are separately active and one has fully flavored leptogenesis. Now the A-
matrix needs to be taken as [53]
93/110 6/55 6/55
−
AMSSM =  3/40 19/30 1/30  (3.13)
−
 3/40 1/30 19/30
 − 
 
and eqs. (3.6) – (3.8) used for each flavor α. Once again, we consider the different cases
(a), (b) and (c) of heavy neutrino mass ordering. Ignoring M for case (a) and with
2=3
g = 232.5, we have η 1.05 10 2 ǫα α. Similarly, ignoring M for case (b) and
⋆1 ≃ − × − α 1K1 1
with g = 236.25, one gets η 1.04 10 2 (ǫα α + ǫα α). For case (c), g = 240
⋆2=3 ≃ − P× − α 2K2 3K3 ⋆
and η 1.02 10 2 (ǫα α +ǫα α +ǫα α).
≃ − × − α 1K1 2K2 3K3 P
P
9 2 1 12
(3) 10 GeV < Mlowest(1+tan β)− < 10 GeV.
In this regime the τ- flavor decouples first while the electron and muon flavors act indistin-
guishably. The latter, therefore, can be summed. Now effectively A becomes a 2 2 matrix
×
A˜ given by [53]
541/761 152/761
A˜ = − (3.14)
46/761 494/761!
−
and acting in a space spanned by e+µ and τ. Indeed, we can define e+µ and ˜τ by
Ki Ki
8.25 A˜11 (Ke +Kµ) 1.16
( e+µ) 1 = + | | i i , (3.15a)
Ki − A˜11 (Ke +Kµ) 0.2 !
| | i i
8.25 A˜22 (Kτ) 1.16
(˜τ) 1 = + | | i . (3.15b)
Ki − A˜22 Kτ 0.2 !
| | i
Now, for case (a) with g = 232.5, η 1.05 10 2[(ǫe +ǫµ) e+µ +ǫτ ˜τ]. Case (b) has
⋆1 ≃ − × − 1 1 K1 1K1
g = 236.25 and η 1.04 10 2 [(ǫe +ǫµ) e+µ +ǫτ ˜τ]. Finally, case (c), with
⋆2=3 ≃ − × − k=2,3 k k Kk kKk
g = 240, has η 1.02 10 2 [(ǫe +ǫµ) e+µ +ǫτ ˜τ].
⋆ ≃ − × − i i P i Ki iKi
P
4 Baryon asymmetry in the present scheme
(1) Regime of unflavored leptogenesis
As explained in Sec. 3, there is no flavor discrimination if Mlowest(1+tan2β)−1 > 1012 GeV.
The lepton asymmetry parameters ǫ can now be given after summing over α. Additional
i
10