- Email: [email protected]
Comment on the article by D. Borah and B. Karmakar “Linear seesaw for Dirac neutrinos with A4 flavour symmetry”, Phys. Lett. B 789 (2019) 59–70, arXiv:1806.10685
Physics Letters B 798 (2019) 134979
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
Comment
Comment on the article by D. Borah and B. Karmakar “Linear seesaw for Dirac neutrinos with A 4 flavour symmetry”, Phys. Lett. B 789 (2019) 59–70, arXiv:1806.10685 V.V. Vien a,b , H.N. Long c , A.E. Cárcamo Hernández d,∗ a
Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi Viet Nam d Universidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 March 2019 Accepted 24 September 2019 Available online 27 September 2019 Editor: A. Ringwald
D. Borah and B. Karmakar [1] have proposed an A 4 flavoured linear seesaw model to realise light Dirac neutrinos. In this comment article, we show that some neutrino Yukawa interactions were missed in the model Ref. [1], thus implying that a different formula would be needed to determine the effective neutrino mass matrix, with significantly different results. Our result shows that, unlike stated in Ref. [1], that the inverted neutrino mass spectrum is not ruled out. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
A recent study made by D. Borah and B. Karmakar [1] has proposed an A 4 flavoured linear seesaw model to realise light Dirac neutrinos. That model predicts in terms of neutrino mass hierarchy, the leptonic Dirac CP phase, octant of atmospheric mixing angle as well as absolute neutrino masses. In the A 4 flavoured linear seesaw model of Ref. [1], the Yukawa † + † ¯ terms N¯L N R ( y ξ3 ξ + + y s3 φ + S + y a3 φ S ) and S L S R ( y ξ4 ξ + y s4 φ S + †
ya4 φ S ) must exist in the neutrino sector since they are invariant under all symmetries of the model. Therefore, the two corresponding contributions have to be added in Eq. (22) of Ref. [1] and the mixing between the heavy neutrinos N L −N R and S L −S R are generated at renormalizable level, with the corresponding mass matrices M N N , M S S given by:
⎛
MNN
=⎝ ⎛
MSS = ⎝
x3 0 s3 − a3 x4 0
s4 − a4
⎞
0 x3 0
s3 + a3 ⎠, 0 x3
0 x4 0
s4 + a4 ⎠, 0 x4
⎞
where
DOI of original article: https://doi.org/10.1016/j.physletb.2018.12.006. Corresponding author. E-mail addresses: [email protected] (V.V. Vien), [email protected] (H.N. Long), [email protected] (A.E. Cárcamo Hernández).
*
x3 = y ξ3 v ξ , s3 = y s3 v s , a3 = ya3 v s ,
In this case, the full neutrino mass matrix has the following form
⎛
0 M eff = ⎝ mν N M ν S
⎞
Mν S 0 MN S ⎠ ≡ M TD MSS
mν N MNN M N S
M D = (mν N M ν S ), M TD =
mν N M ν S
, MR =
MD MR
(3)
,
MNN M N S
MN S MSS
, (4)
which implies that the light Dirac neutrinos mass matrix can be written as [3]:
⎛
(1)
(2)
x4 = y ξ4 v ξ , s4 = y s4 v s , a4 = ya4 v s .
A1 + A2 1 T ⎝ 0 mν = − M D M − M = D R B3 + B4
0 A 0
⎞
B1 + B2 ⎠, 0 A3 + A4
(5)
where the different entries are given by:
A = (a bx1 + ab x2 − bb x3 − aa x4 )/(x1 x2 − x3 x4 ), A 1 = (a bx1 + ab x2 − bb x3 − aa x4 ) [(a1 − s1 )(a2 + s2 )
−(a3 − s3 )(a4 + s4 ) − x1 x2 + x3 x4 ] /α , A 2 = aa (a4 + s4 ) − a b(a1 + s1 ) + ab (a2 + s2 ) − bb (a3 + s3 )
https://doi.org/10.1016/j.physletb.2019.134979 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
2
V.V. Vien et al. / Physics Letters B 798 (2019) 134979
× [(a2 − s2 )x1 + (a1 − s1 )x2 − (a4 − s4 )x3 − (a3 − s3 )x4 ] /α ,
A 3 = (a bx1 + ab x2 − bb x3 − aa x4 ) [(a1 + s1 )(a2 − s2 )
−(a3 + s3 )(a4 − s4 ) − x1 x2 + x3 x4 ] /α , A 4 = b(−a1 a + a3 b + a s1 − b s3 ) + a(a4 a − a2 b + b s2 −a s4 ) × [(a2 + s2 )x1 + (a1 + s1 )x2 − (a4 + s4 )x3 −(a3 + s3 )x4 ] /α , B 1 = a b(a1 + s1 ) + ab (a2 + s2 ) − bb (a3 + s3 ) − aa (a4 + s4 ) × [(a1 + s1 )(a2 − s2 ) + (a3 + s3 )(−a4 + s4 ) − x1 x2 +x3 x4 ] /α , B 2 = (a bx1 + ab x2 − bb x3 − aa x4 ) [(a2 + s2 )x1 + (a1 + s1 )x2
−(a4 + s4 )x3 − (a3 + s3 )x4 ] /α ,
(6)
α = (a2 x1 − s2 x1 + a1 x2 − s1 x2 − a4 x3 + s4 x3 − a3 x4 + s3 x4 ) × [(a2 + s2 )x1 + (a1 + s1 )x2 − (a4 + s4 )x3 − (a3 + s3 )x4 ] + [(a1 + s1 )(a2 − s2 ) + (a3 + s3 )(−a4 + s4 ) − x1 x2 + x3 x4 ] × [(a1 − s1 )(a2 + s2 ) + (−a3 + s3 )(a4 + s4 ) − x1 x2 + x3 x4 ] .
The absolute values of neutrino masses as well as the neutrino mass ordering is still unknown, however, we can use the neutrino oscillation experimental data for normal hierarchy given in Ref. [2] to find: m2 ∼ 0.0087 eV. In this case, the parameters A , B and the other neutrino masses are explicitly given as A = 1.29304 × 10−3 eV, B = −1.29265 × 10−3 eV, m1 = 6.245 × 2 10−4 eV, m3 = 5.08497 × 10− eV. Then, the resulting sum of neumν ∼ 0.06017 eV, which is consistrino masses takes the form tent with the current cosmological constraints mν < 0.17 eV given in Ref. [4]. On the other hand, in the inverted neutrino mass spectrum [2], m221 = m22 − m21 = 7.53 × 10−5 eV2 and m232 = m23 − m22 = −2.56 × 10−3 eV2 , we obtain: A = −0.00131765 eV + m22 , B = 0.00124235 eV. Then, in the scenario of inverted neutrino mass ordering, we find m2 = 0.051 eV, with the parameters A , B and the other neutrino masses given as A = 1.28335 × 10−3 eV, B = 1.24235 × 10−3 eV, m1 = 5.02563 × 10−2 eV, m3 = 6.40312 × 10−3 eV. Thus, we find that the sum of neutrino masses takes the form mν ∼ 0.107659 eV, which is consistent with the current cosmological constraints of Ref. [4]. This result is completely different from the one obtained in Ref. [1] where the inverted neutrino mass hierarchy is not allowed.
(7) Here mν = aI, mν
= a I, M ν S
= bI, M ν S
= b I and x3,4 , a3,4 , s3,4 are
given by Eq. (2) and all the other parameters (e.g., x1,2 , a1,2 , s1,2 ,...) are the same as in Ref. [1]. To diagonalise the light active neutrino mass matrix mν in Eq. (5), we define a Hermitian matrix M, given by
⎛
m11 ⎝ 0 M = m+ ν mν = m∗13
0 m22 0
⎞
m13 0 ⎠, m33
(8)
where
m11 = ( A 1 + A 2 )( A ∗1 + A ∗2 ) + ( B 3 + B 4 )( B ∗3 + B ∗4 ), m22 = | A |2 , m33 = ( A 3 + A 4 )( A ∗3 + A ∗4 ) + ( B 1 + B 2 )( B ∗1 + B ∗2 ), m13 = ( A ∗1 + A ∗2 )( B 1 + B 2 ) + ( A 3 + A 4 )( B ∗3 + B ∗4 ),
(9)
with A , A 1,2,3,4 and B 12 are defined in Eqs. (6) and (7). The mass matrix M is diagonalized by the rotation matrix U 13 given by Eq. (21) of Ref. [1] and the light active neutrino masses m21,2,3 are given by
m21,3
=
1 2
m11 + m33 ±
m22 = m22 .
(m11 − m33
)2
+ 4|m13
|2
≡ A ± B, (10)
We see that the leptonic mixing matrix in this case is the same as in Ref. [1], however, the expressions of neutrino masses are different from each other. For instance, in the case of normal neutrino mass hierarchy, by taking the best fit values on neutrino mass squared differences given in Ref. [2], m221 = m22 − m21 = 7.53 × 10−5 eV2 and m232 = m23 − m22 = 2.51 × 10−3 eV2 , we find the solution: A = 0.00121735 eV + m22 , B = −0.00129265 eV.
1. Conclusion We have shown that some neutrino Yukawa terms must be added in the A 4 flavoured linear seesaw model of Ref. [1]. When these terms are included, the obtained results are significantly different than the ones reported in Ref. [1]. Namely, in the normal neutrino mass spectrum, the leptonic mixing matrix in the case where the missed Yukawa terms are added is the same as in [1], however, the expressions for neutrino masses are different from each other. On the other hand, in the inverted neutrino mass spectrum, our result is completely different from the one obtained in Ref. [1] and the sum of neutrino masses satisfies the current cosmological constraints, thus implying, unlike stated in Ref. [1] that this scenario is not ruled out. Acknowledgements This research has been financially supported by FONDECYT (Chile), Grants No. 1170803, CONICYT PIA/Basal FB0821, the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2017.341. H.N.L. acknowledges the financial support of the Vietnam Academy of Science and Technology under grant NVCC05.01/19-19. References [1] D. Borah, B. Karmakar, Phys. Lett. B 789 (2019) 59–70, arXiv:1806.10685 [hepph]. [2] M. Tanabashi, et al., Particle Data Group, Phys. Rev. D 98 (2018) 030001. [3] M. Chakrabortya, H. Zeen Devib, A. Ghosal, Phys. Lett. B 741 (2015) 210–216, arXiv:1410.3276 [hep-ph]. [4] F. Couchot, et al., Astron. Astrophys. 606 (2017) A104, arXiv:1703.10829 [hepph].
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
Comment
Comment on the article by D. Borah and B. Karmakar “Linear seesaw for Dirac neutrinos with A 4 flavour symmetry”, Phys. Lett. B 789 (2019) 59–70, arXiv:1806.10685 V.V. Vien a,b , H.N. Long c , A.E. Cárcamo Hernández d,∗ a
Theoretical Particle Physics and Cosmology Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi Viet Nam d Universidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 March 2019 Accepted 24 September 2019 Available online 27 September 2019 Editor: A. Ringwald
D. Borah and B. Karmakar [1] have proposed an A 4 flavoured linear seesaw model to realise light Dirac neutrinos. In this comment article, we show that some neutrino Yukawa interactions were missed in the model Ref. [1], thus implying that a different formula would be needed to determine the effective neutrino mass matrix, with significantly different results. Our result shows that, unlike stated in Ref. [1], that the inverted neutrino mass spectrum is not ruled out. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
A recent study made by D. Borah and B. Karmakar [1] has proposed an A 4 flavoured linear seesaw model to realise light Dirac neutrinos. That model predicts in terms of neutrino mass hierarchy, the leptonic Dirac CP phase, octant of atmospheric mixing angle as well as absolute neutrino masses. In the A 4 flavoured linear seesaw model of Ref. [1], the Yukawa † + † ¯ terms N¯L N R ( y ξ3 ξ + + y s3 φ + S + y a3 φ S ) and S L S R ( y ξ4 ξ + y s4 φ S + †
ya4 φ S ) must exist in the neutrino sector since they are invariant under all symmetries of the model. Therefore, the two corresponding contributions have to be added in Eq. (22) of Ref. [1] and the mixing between the heavy neutrinos N L −N R and S L −S R are generated at renormalizable level, with the corresponding mass matrices M N N , M S S given by:
⎛
MNN
=⎝ ⎛
MSS = ⎝
x3 0 s3 − a3 x4 0
s4 − a4
⎞
0 x3 0
s3 + a3 ⎠, 0 x3
0 x4 0
s4 + a4 ⎠, 0 x4
⎞
where
DOI of original article: https://doi.org/10.1016/j.physletb.2018.12.006. Corresponding author. E-mail addresses: [email protected] (V.V. Vien), [email protected] (H.N. Long), [email protected] (A.E. Cárcamo Hernández).
*
x3 = y ξ3 v ξ , s3 = y s3 v s , a3 = ya3 v s ,
In this case, the full neutrino mass matrix has the following form
⎛
0 M eff = ⎝ mν N M ν S
⎞
Mν S 0 MN S ⎠ ≡ M TD MSS
mν N MNN M N S
M D = (mν N M ν S ), M TD =
mν N M ν S
, MR =
MD MR
(3)
,
MNN M N S
MN S MSS
, (4)
which implies that the light Dirac neutrinos mass matrix can be written as [3]:
⎛
(1)
(2)
x4 = y ξ4 v ξ , s4 = y s4 v s , a4 = ya4 v s .
A1 + A2 1 T ⎝ 0 mν = − M D M − M = D R B3 + B4
0 A 0
⎞
B1 + B2 ⎠, 0 A3 + A4
(5)
where the different entries are given by:
A = (a bx1 + ab x2 − bb x3 − aa x4 )/(x1 x2 − x3 x4 ), A 1 = (a bx1 + ab x2 − bb x3 − aa x4 ) [(a1 − s1 )(a2 + s2 )
−(a3 − s3 )(a4 + s4 ) − x1 x2 + x3 x4 ] /α , A 2 = aa (a4 + s4 ) − a b(a1 + s1 ) + ab (a2 + s2 ) − bb (a3 + s3 )
https://doi.org/10.1016/j.physletb.2019.134979 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
2
V.V. Vien et al. / Physics Letters B 798 (2019) 134979
× [(a2 − s2 )x1 + (a1 − s1 )x2 − (a4 − s4 )x3 − (a3 − s3 )x4 ] /α ,
A 3 = (a bx1 + ab x2 − bb x3 − aa x4 ) [(a1 + s1 )(a2 − s2 )
−(a3 + s3 )(a4 − s4 ) − x1 x2 + x3 x4 ] /α , A 4 = b(−a1 a + a3 b + a s1 − b s3 ) + a(a4 a − a2 b + b s2 −a s4 ) × [(a2 + s2 )x1 + (a1 + s1 )x2 − (a4 + s4 )x3 −(a3 + s3 )x4 ] /α , B 1 = a b(a1 + s1 ) + ab (a2 + s2 ) − bb (a3 + s3 ) − aa (a4 + s4 ) × [(a1 + s1 )(a2 − s2 ) + (a3 + s3 )(−a4 + s4 ) − x1 x2 +x3 x4 ] /α , B 2 = (a bx1 + ab x2 − bb x3 − aa x4 ) [(a2 + s2 )x1 + (a1 + s1 )x2
−(a4 + s4 )x3 − (a3 + s3 )x4 ] /α ,
(6)
α = (a2 x1 − s2 x1 + a1 x2 − s1 x2 − a4 x3 + s4 x3 − a3 x4 + s3 x4 ) × [(a2 + s2 )x1 + (a1 + s1 )x2 − (a4 + s4 )x3 − (a3 + s3 )x4 ] + [(a1 + s1 )(a2 − s2 ) + (a3 + s3 )(−a4 + s4 ) − x1 x2 + x3 x4 ] × [(a1 − s1 )(a2 + s2 ) + (−a3 + s3 )(a4 + s4 ) − x1 x2 + x3 x4 ] .
The absolute values of neutrino masses as well as the neutrino mass ordering is still unknown, however, we can use the neutrino oscillation experimental data for normal hierarchy given in Ref. [2] to find: m2 ∼ 0.0087 eV. In this case, the parameters A , B and the other neutrino masses are explicitly given as A = 1.29304 × 10−3 eV, B = −1.29265 × 10−3 eV, m1 = 6.245 × 2 10−4 eV, m3 = 5.08497 × 10− eV. Then, the resulting sum of neumν ∼ 0.06017 eV, which is consistrino masses takes the form tent with the current cosmological constraints mν < 0.17 eV given in Ref. [4]. On the other hand, in the inverted neutrino mass spectrum [2], m221 = m22 − m21 = 7.53 × 10−5 eV2 and m232 = m23 − m22 = −2.56 × 10−3 eV2 , we obtain: A = −0.00131765 eV + m22 , B = 0.00124235 eV. Then, in the scenario of inverted neutrino mass ordering, we find m2 = 0.051 eV, with the parameters A , B and the other neutrino masses given as A = 1.28335 × 10−3 eV, B = 1.24235 × 10−3 eV, m1 = 5.02563 × 10−2 eV, m3 = 6.40312 × 10−3 eV. Thus, we find that the sum of neutrino masses takes the form mν ∼ 0.107659 eV, which is consistent with the current cosmological constraints of Ref. [4]. This result is completely different from the one obtained in Ref. [1] where the inverted neutrino mass hierarchy is not allowed.
(7) Here mν = aI, mν
= a I, M ν S
= bI, M ν S
= b I and x3,4 , a3,4 , s3,4 are
given by Eq. (2) and all the other parameters (e.g., x1,2 , a1,2 , s1,2 ,...) are the same as in Ref. [1]. To diagonalise the light active neutrino mass matrix mν in Eq. (5), we define a Hermitian matrix M, given by
⎛
m11 ⎝ 0 M = m+ ν mν = m∗13
0 m22 0
⎞
m13 0 ⎠, m33
(8)
where
m11 = ( A 1 + A 2 )( A ∗1 + A ∗2 ) + ( B 3 + B 4 )( B ∗3 + B ∗4 ), m22 = | A |2 , m33 = ( A 3 + A 4 )( A ∗3 + A ∗4 ) + ( B 1 + B 2 )( B ∗1 + B ∗2 ), m13 = ( A ∗1 + A ∗2 )( B 1 + B 2 ) + ( A 3 + A 4 )( B ∗3 + B ∗4 ),
(9)
with A , A 1,2,3,4 and B 12 are defined in Eqs. (6) and (7). The mass matrix M is diagonalized by the rotation matrix U 13 given by Eq. (21) of Ref. [1] and the light active neutrino masses m21,2,3 are given by
m21,3
=
1 2
m11 + m33 ±
m22 = m22 .
(m11 − m33
)2
+ 4|m13
|2
≡ A ± B, (10)
We see that the leptonic mixing matrix in this case is the same as in Ref. [1], however, the expressions of neutrino masses are different from each other. For instance, in the case of normal neutrino mass hierarchy, by taking the best fit values on neutrino mass squared differences given in Ref. [2], m221 = m22 − m21 = 7.53 × 10−5 eV2 and m232 = m23 − m22 = 2.51 × 10−3 eV2 , we find the solution: A = 0.00121735 eV + m22 , B = −0.00129265 eV.
1. Conclusion We have shown that some neutrino Yukawa terms must be added in the A 4 flavoured linear seesaw model of Ref. [1]. When these terms are included, the obtained results are significantly different than the ones reported in Ref. [1]. Namely, in the normal neutrino mass spectrum, the leptonic mixing matrix in the case where the missed Yukawa terms are added is the same as in [1], however, the expressions for neutrino masses are different from each other. On the other hand, in the inverted neutrino mass spectrum, our result is completely different from the one obtained in Ref. [1] and the sum of neutrino masses satisfies the current cosmological constraints, thus implying, unlike stated in Ref. [1] that this scenario is not ruled out. Acknowledgements This research has been financially supported by FONDECYT (Chile), Grants No. 1170803, CONICYT PIA/Basal FB0821, the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2017.341. H.N.L. acknowledges the financial support of the Vietnam Academy of Science and Technology under grant NVCC05.01/19-19. References [1] D. Borah, B. Karmakar, Phys. Lett. B 789 (2019) 59–70, arXiv:1806.10685 [hepph]. [2] M. Tanabashi, et al., Particle Data Group, Phys. Rev. D 98 (2018) 030001. [3] M. Chakrabortya, H. Zeen Devib, A. Ghosal, Phys. Lett. B 741 (2015) 210–216, arXiv:1410.3276 [hep-ph]. [4] F. Couchot, et al., Astron. Astrophys. 606 (2017) A104, arXiv:1703.10829 [hepph].