Scholarly article on topic 'Quasi-degenerate neutrino masses in terms of mass-squared differences'

Quasi-degenerate neutrino masses in terms of mass-squared differences Academic research paper on "Physical sciences"

CC BY
0
0
Share paper
Academic journal
Physics Letters B
OECD Field of science
Keywords
{}

Abstract of research paper on Physical sciences, author of scientific article — E.M. Lipmanov

Abstract The absolute neutrino masses are obtained in terms of the atmospheric and solar mass-squared differences within the framework of low energy phenomenology by suggestion of an analogy between the hierarchies of the neutrino and charged lepton mass ratios. It points to a nearly degenerate three neutrino mass pattern with the neutrino mass scale mν≅Δm2 atm / (2 2 Δm2 sol ) likely located in the range 0.1–0.3 eV, and the best-fit value m ν ≅0.18–0.20 eV. Restrictions on the neutrino mass scale from the WMAP data are considered.

Academic research paper on topic "Quasi-degenerate neutrino masses in terms of mass-squared differences"

Available online at www.sciencedirect.com

SCIENCE ^DIRECT8

Physics Letters B 567 (2003) 268-272

www. elsevier. com/locate/npe

Quasi-degenerate neutrino masses in terms of mass-squared

differences

E.M. Lipmanov

40 Wallingford Road #272, Brighton, MA 02135, USA Received 23 May 2003; received in revised form 30 June 2003; accepted 30 June 2003

Editor: H. Georgi

Abstract

The absolute neutrino masses are obtained in terms of the atmospheric and solar mass-squared differences within the framework of low energy phenomenology by suggestion of an analogy between the hierarchies of the neutrino and charged lepton mass ratios. It points to a nearly degenerate three neutrino mass pattern with the neutrino mass scale mv =

Am^tm/y(2sfl Am^ol) likely located in the range 0.1-0.3 eV, and the best-fit value mv = 0.18-0.20 eV. Restrictions on the neutrino mass scale from the WMAP data are considered. © 2003 Elsevier B.V. All rights reserved.

1. Introduction

The known sharp contrast between the neutrinos and charged leptons (CL) is the very large difference of their mass scales. The CL masses me, mM and mT are well known [1]. Two large mass ratios and a large hierarchy of these mass ratios characterize the mass pattern of the CL:

mß/Me » 1,

(mT/mß)2 = (mß/me)V2.

The discovery of the finite neutrino masses in the neutrino oscillation experiments [2-4] does raise the question: what is the neutrino mass pattern and what relation is there between the two mass patterns if any?

E-mail address: e.lipmanov@verizon.net (E.M. Lipmanov).

This problem is widely discussed [5] in the contexts of different basic extensions of the SM with higher mass and energy scales. There is no definite answer to this question since the exact absolute values of the neutrino masses are unknown as yet, while the neutrino oscillation data give only neutrino mass-squared differences.

In this Letter, an attempt is made to answer the above question in the framework of low energy phenomenology guided by the neutrino oscillation data against the background of a virtual broken lepton mass eigenstate symmetry (flavor problem). In spite of the disparity of the mass scales, an analogy between the neutrino and CL mass ratio hierarchies is suggested and described by an extension of the condition (1), taking into consideration the factual violation of the lepton mass state symmetry. This analogy relates the three absolute neutrino masses to the two oscillation mass-squared differences.

0370-2693/$ - see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2003.06.052

2. Two opposite lepton mass ratio patterns

By definition, the sequence of the lepton masses (CL or neutrinos) let be

mi < m2 < m3. (2)

In view of two basic experimental facts—the CL mass ratio hierarchy in (i) and the hierarchy of the atmospheric and solar neutrino mass-squared differences—we suggest an approximate unifying nonlinear relation of the lepton mass ratios x2 = m3/m2 and xi = m2/mi at a common low scale,

(x2 - l)2 = (xi - 1)V2, (3)

to be accurate to within a few percent. If supported by data, relation (3) hints at a nonlinear generic feature of the seemingly opposite mass patterns of the neutrinos and CL.

The dimensionless quantities (xn — i), n — i, 2, are the basic physical quantities here. They should have a deeper physical meaning than the mass ratios themselves. These quantities measure the deviations from the mass eigenvalue degeneracy, and so they estimate the virtual violation of the lepton mass eigenstate symmetry.

Eq. (3) for the lepton mass ratios has two dual extreme solutions with respectively very large and very small violations of the lepton mass symmetry:

(1) A solution with large mass ratios: xi > i, x2 > i, xi > x2. Relation (3) shows that if one mass ratio x2 is large, the other one xi must be much larger. It is appropriate for the CL with xi — m^/me and X2 — mt /mM, see (i), and can be represented in an exponential form

m^/nig = exp /,

mr/mM = V2exp//2, /»1, (1')

with one unknown parameter x. In this solution, the violation of the mass (lepton flavor) symmetry is a large effect, / = 5, =

mxlmtl = V2exp5/2 to within a few percent [1].

(2) A solution with near to unity mass ratios: xi — i,

x2 — i, (xi — i) « (x2 — i), a hidden mass

hierarchy. Relation (3) shows that if one mass ratio

x2 is close to unity, the other one xi must be

much closer to unity. The violation of the lepton

mass symmetry is a small effect here. This other type of solution for the lepton mass ratios can be appropriate only for the neutrinos with a nearly degenerate mass pattern [6],

(m2/mi) — i, (m3/m2) = i,

[(m3/m2) - l]2 = [(m2/mi) - l]V2. (4)

It is a probable solution for the neutrinos. With two equations for the atmospheric and solar mass-squared differences and the Eq. (4), there is a full set of three equations for three unknown absolute neutrino masses.

With the definition of the neutrino mass sequence (2), two different cases (A) and (B) are possible for the neutrino solution. Case (A) is as stated in (4). In the other case (B) the ratios (m3/m2) and (m2/mi) are interchanged. All estimations below are presented in case (A). They remain intact in case (B).

The neutrino solution (4) can be represented in an exponential form

m3/m2 = exp(V2g2), m2/m\ = exp(V2g4).

It contains only one unknown real dimensionless parameter g in the exponents, bound by the consistency condition

g2 « i. (6)

The relation between the exponents of the two mass ratios in the neutrino solution (5) reflects the nonlin-earity feature of Eq. (3), unlike the CL solution (i').

With solution (5), the atmospheric and solar neutrino mass-squared differences are given by

Am2m = Am2 = (mj - mj) = 2^2/m2, (7)

Am2ol = Amj = (m2 ~ m\) = (8) As a result, it follows

m2>>km2^ m\>>Am2„l, (9)

Am2tm/Am2ol = (m2/m2)(i/g2). (i0) Since (m2/m2) — i, relation (i0) is simplified

Am^tm/Am^oi — i/g2 (ii)

It should be noted, that large ratio of the atmospheric and solar mass-squared differences, Am^lm/

Am2ol > 1, is a positive result of the neutrino oscillation experiments [2-4]. With (11), this experimental result renders strong evidence in favor of the condition (6) above, and therefore it supports the nearly degenerate neutrino mass ratio pattern (5) and (6).

The absolute neutrino masses follow from (7), (8) and (11):

m2 = 7(Am2m/^22V2) = Am2atJ J{2^2 Am2J,

m3 = m2 + Am^tm/2m2, (13)

m1 = m2 — Am^ol/2m2. (14)

The neutrino mass scale is determined here only by two of the neutrino oscillation data: Am^tm and Am2ol. Relation (12) can be rewritten in another form

(Amljmlf^lVliAm^/ml), (12')

where mv = m2 is the mass scale of the quasi-degenerate neutrinos. The hierarchy of the dimension-less-made mass squared differences for quasi-degenerate neutrinos in (12') is analogous to the hierarchy of the CL mass squared ratios.

In fact, the statement (12)-(14) for the absolute neutrino masses is a motivated by analogy eigenvalue ansatz for the neutrino mass matrix, to be probed with accurate neutrino mass and oscillation data.

With the best-fit value of the atmospheric neutrino oscillation mass-squared difference [2,7],

Am2tm = 2.5 x 10—3 eV2, (15)

and the best-fit one for the favored LMA MSW solar neutrino oscillation solution [4,8],

Am2o[ = 5.5 x 10—5 eV2, (16)

the ratio in (11) is given by

Am2tm/Am2ol = 45, g2 = 1/45. (17)

With another estimation of the best-fit solar neutrino mass-squared difference [9],

Am2ol = 7 x 10—5 eV2, (18)

the ratio in (11) is

Am2tm/Am2ol = 36, g2 = 1/36. (19)

The inputs (15) and (16) lead to the estimation of the neutrino mass scale (12),

m2 = 0.20 eV. (20)

With (15) and (18), the estimation of this scale is m2 = 0.18 eV. (21)

With the solar input (18) and the allowed 3 a range from a global analysis [7,9] of the atmospheric neutrino data, instead of (15),

1.2 x 10-3 eV2 < Arn2tm < 4.8 x 10-3 eV2, (22)

the estimation for the neutrino mass scale is given by

0.09 eV <m2 < 0.34 eV. (23)

Though the neutrino mass estimations above are dependent on the exact data values of both the atmospheric and solar neutrino mass-squared differences, they are much more sensitive to the atmospheric data than to the solar ones.

In the discussion above, the dimensionless parameter g2 plays a crucial role. It determines the neutrino mass ratios (5) and the ratio of the atmospheric neutrino and solar neutrino mass-squared differences (11). As a coincidence, the estimated in (19) value of g2 is close to the semiweak coupling constant squared g2 = g\l\Tt = GFni^-Jl/n = 0.034,

Am2tm/Am2ol = 30, mv = 3.26^ Am2im, with the input Am2tm = (2.5-3) x 10-3 eV2, it follows mv = (0.16-0.18) eV and Am2ol = (8.3-10) x 10-5 eV2. Also, to within the same accuracy, there is a noticeable connection between the exponents / and g2, namely g2 = /exp(-/) = 5^2(melm,x), i.e., m3/m2 = exp(10me/m^), m2/mi = exp[(10me/mM)2/V2]. These approximate coincidences come out into view at the level of the lepton mass ratio quantities (xn) in a quasi-degenerate neutrino scenario if considering the neutrino mass-squared differences in terms of the primary quantities (xn - 1). The basic physical meaning of the parameters x and g2 in the mass ratios (5) and (1') is outside the scope of the present Letter.

As an important test to date, the above estimations of the absolute neutrino masses obey the recent cosmological limit mv < 0.23 eV from the WMAP measurements of cosmic microwave background anisotropy [10], what is a powerful tool for

constraining the neutrino mass scale in the quasi-degenerate scenario. With neutrino mass scale (12), this upper limit on the neutrino mass leads to a restriction,

Am2tm/(2^Am2ol)1/2<0.23eV. (24)

According to a subsequent more conservative analysis [11], the restriction is

Am2tm/(2v/2Am2ol)1/2<0.33eV. (25)

These restrictions are compatible with the best-fit values of the atmospheric and solar mass-squared differences in (15), (16) and (18). With the data range (22) for the atmospheric mass-squared difference, a significant inference from the restriction (24) is that the LMA MSW solar neutrino oscillation solution is the only one compatible with the present phenomenology of the neutrino mass ratios.

3. Conclusions

An analogy between two basic experimental facts in lepton mass physics—large hierarchy of the CL mass ratios, and large hierarchy of the atmospheric and solar neutrino mass-squared differences—is described by the nonlinear phenomenological equation (3), an extension of the observed relation (i) for the CL data mass ratios. Two exponential solutions of Eq. (3), with large and small exponents, conform respectively to the mass ratio patterns of the CL and neutrinos, Eqs. (i') and (5). Approximate quantitative relations between these exponents are noted. The main results for the absolute neutrino masses are:

(1) The special quasi-degenerate neutrino mass pattern (5) and (6) is supported by the neutrino oscillation data: (Am;;tm/Am2ol)exp > i. Three absolute neutrino masses are expressed in terms of two neutrino mass squared differences, as a motivated eigenvalue ansatz for the still unknown exact form of the neutrino mass matrix. The three eigenvalues of the neutrino mass matrix are given in (i2), (i3) and (i4);

(2) The neutrino mass scale (i2): mv — Am^lm/

y/(2\/2 A/jr^). it is much more sensitive to the atmospheric neutrino data than to the solar

ones. By comparison with the available neutrino oscillation data, this neutrino mass scale is located likely within the range 0.i-0.3 eV, with the best-fit value mv — 0.i8-0.20 eV; (3) The estimated neutrino mass scale is compatible with the recent constraints on the absolute neutrino mass from the WMAP data [i0,ii], with the LMA MSW solution being the only acceptable solar neutrino oscillation solution.

There are no free parameters in the neutrino mass scale (i2) to adjust. As a physical statement it is consistent with the relevant neutrino data to date, and should be confronted with new data. More stringent bounds on the neutrino mass from the coming satellite WMAP measurements (or other relevant data) in combination with more accurate values of Am^lm and Am2ol from the neutrino oscillation experiments will test definitely this neutrino mass scale.

Acknowledgement

I would like to thank my daughter Janna Kaplan for her invaluable support.

References

[1] Particle Data Group, Phys. Rev. D 66 (2002) 0i000i.

[2] Super-Kamiokande Collaboration, Y. Fukuda, et al., Phys. Lett. B 433 (i998) 9;

Super-Kamiokande Collaboration, Y. Fukuda, et al., Phys. Lett. B 436 (i998) 33;

Super-Kamiokande Collaboration, Y. Fukuda, et al., Phys. Rev. Lett. 82 (i999) 2644.

[3] SNO Collaboration, Q.R. Ahmad, et al., Phys. Rev. Lett. 89 (2002) 0ii30i;

SNO Collaboration, Q.R. Ahmad, et al., Phys. Rev. Lett. 87 (200i) 07i30i.

[4] KamLAND Collaboration, K. Eguchi, et al., Phys. Rev. Lett. 90(2003) 02i802.

[5] E.g., review by R.N. Mohapatra, hep-ph/02ii252, and references therein.

[6] The nearly degenerate neutrino mass pattern was first considered by D.O. Caldwell, R.N. Mohapatra, Phys. Rev. D 48 (i993) 3259;

A.S. Joshipura, Phys. Rev. D 5i (i995) i32i;

It is widely discussed in the literature. For a few recent

references, V. Barger, S.L. Glashow, D. Marfatia, K. Whisnant,

Phys. Lett. B 532 (2002) i5;

Z. Xing, Phys. Rev. D 65 (2002) 077302;

K.S. Babu, E. Ma, J.F.W. Valle, Phys. Lett. B 552 (2003) 207; The recent experimental indications for the neutrinoless double beta decay, H.V. Klapdor-Kleingrothaus, et al., Mod. Phys. Lett. A 16 (2001) 2409, if borne out, can point to a nearly degenerate neutrino mass pattern.

[7] M. Maltoni, T. Schwetz, M.A. Tortola, J.W.F. Valle, Phys. Rev. D 67 (2003)013011;

G. Fogli, et al., Phys. Rev. D 66 (2002) 093008.

[8] SNO Collaboration, Q.R. Ahmad, et al., Phys. Rev. Lett. 89 (2002)011302;

G.L. Fogli, et al., Phys. Rev. D 66 (2002) 053010; V. Barger, D. Marfatia, K. Whisnant, B.P. Wood, Phys. Lett. B 537 (2002) 179;

J.N. Bahcall, M.C. Gonzalez-Garcia, C. Pena-Garay, JHEP 0207 (2002) 057;

P.C. de Holanda, A.Yu. Smirnov, Phys. Rev. D 66 (2002) 113005;

A. Bandyopadhyay, et al., Phys. Lett. B 540 (2002) 14; P. Aliani, et al., Phys. Rev. D 67 (2003) 013006; A. Strumia, et al., Phys. Lett. B 541 (2002) 327. [9] M.C. Gonzales-Garcia, C. Pena-Garay, hep-ph/0306001; G.L. Fogli, et al., hep-ph/0212127; S. Pakvasa, J.W.F. Valle, hep-ph/0301061.

[10] D.N. Spergel, et al., astro-ph/0302209;

A. Pierce, H. Murayama, hep-ph/0302131.

[11] S. Hannestad, JCAP 05 (2003) 004; S. Pastor, hep-ph/0306233.