Neutrino and Graviton Rest Mass Estimations by a Phenomenological Approach

Neutrino and graviton rest mass estimations by a phenomenological approach

Dimitar Valev

Stara Zagora Department, Bulgarian Academy of Sciences, P.O. Box 73, 6000 Stara Zagora, Bulgaria

Abstract

The rest mass of the lightest free massive stable particle, acted upon by a particular interaction, is shown to be proportional to the coupling constant of the respective interaction at extremely low energy. The found mass relation supports a nonzero graviton mass and unifies the masses of four stable particles (proton, electron, electron neutrino and graviton), covering an extremely wide range of values, reaching to 45 orders of magnitude. The estimations of the neutrino and graviton masses have been made on the basis of this approach. The obtained -37 2 evaluations for the graviton and electron neutrino masses are mg ~ 1.2×10 eV/c 2 and mνe ~ 0.0002 eV/c , respectively. The masses of the heavier neutrino flavors νµ and ντ have been estimated by the results of solar and atmospheric neutrino experiments of 0.008 eV/c2 and 0.05 eV/c2, respectively, and support hierarchical neutrino models.

PACS numbers: 12.10.Kt; 14.60.Pq; 13.10.+q; 04.80.Cc

Key words: neutrino mass limit; graviton mass estimation; coupling constants; mass relation; hierarchical neutrino models

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E-mail: [email protected]

I. INTRODUCTION

Although the neutrino and the graviton belong to different particle kinds (neutral lepton and quantum of the gravitation, respectively), they have some similar properties. Both particles are not acted upon by the strong and the electromagnetic interactions, which makes their detection and investigation exceptionally difficult. Besides, both have masses that are many orders of magnitude lighter than the masses of the rest particles and they are generally accepted to be massless. Decades after the experimental detection of the neutrino [1], it was generally accepted that the neutrino mass (moν) is rigorously zero. In the Fermi’s theory of β-decay [2] as well as in the Electroweak theory [3,4] and hence, in the Standard model (SM), the neutrinos have been accepted massless. Despite this, attempts to determine the neutrino mass have been made as early as it was found. The recent experiments bound moν on the top and its upper limit has decreased millions of times in the latest experiments, as compared to the initial theoretical estimations [5]. The first experiment, hinting that the neutrino probably possesses a mass, is dated back to the 60-ies [6]. The total flux of neutrinos from the Sun is about 3 times lower than the one, predicted by theoretical solar models and thereby created the problem for the Solar Neutrino Deficit (SND). This discrepancy can be explained if some of the electron neutrinos transform into another neutrino flavor. Within the frame of the SM, however, there is no place for massive neutrinos and neutrino oscillations. As a result, the detection of neutrino oscillations appears crucial for the SM and it requires its extension in direction to the Grand Unified Theories (GUT), Super Symmetry, String Theory etc.

Later, the experimental observations showed that the ratio between the atmospheric ν µ and ν e fluxes is less than the theoretical predictions [7,8]. This discrepancy became known as the Atmospheric Neutrino Anomaly (ANA). Again it could be explained by the neutrino oscillations. The crucial experiments with the 50 kton neutrino detector Super- Kamiokande found strong evidence for oscillations (and hence - mass) in the atmospheric neutrinos [9]. The direct neutrino measurements allow to bound the neutrino mass. The upper limit for the mass of the lightest neutrino flavor ν e was obtained from experiments for measurement of the high-energy part of the tritium β-spectrum and recent experiments yield ~ 2 eV/c2

upper limit [10,11]. As a result of the recent experiments, the upper mass limits of ν µ and 2 2 ντ are 170 keV/c [12] and 18.2 MeV/c [13], respectively. The Solar and atmospheric 2 2 2 2 neutrino experiments allow to find the mass splitting ∆msol = mν 2 − mν1 and ∆ atm = 2 2 mν 3 − mν 2 , but not the absolute value of the neutrino masses. The astrophysical constraint 2 of the neutrino mass is mν < 2.2 eV/c [14]. The recent extensions of the Standard Model lead to non-zero neutrino masses, which are within the large range of 10-6 eV/c2 ÷ 10 eV/c2. Similarly to the case with the neutrino before 1998, the prevailing current opinion is that the quantum of the gravitation (graviton) is massless. This opinion is connected with Einstein’s theory of General Relativity, where the gravitation is described by a massless field of spin 2 in a generally covariance manner. The nonzero graviton mass leads to a finite range h of gravitation rg ~ D g = . There are 2 kinds of astrophysical methods for estimation of mg c

the upper limit of the graviton mass (or low limit of D g ) – static and dynamic methods. The

2 static methods are based on the search of difference between Yukawa potential for massive graviton and Newton potential for massless graviton. The Solar system measurements infer 15 -22 2 D g > 2.8×10 m, that is equivalent to mg < 4.4×10 eV/c [15]. Rich galactic clusters -29 −1 2 allow to estimate mg < 2×10 heV/c [16], where h ≈ 0.65 is a dimensionless Hubble constant. This is the lowest limit of the graviton mass, although it is less robust due to the uncertainty about the large scales matter distribution in the Universe. This value is used in the present paper. The dynamic methods are based on the differences of the emission and propagation of the gravitational waves from binary stellar systems in cases of massless and massive graviton. The possibilities of the astrophysical measurements to bound the graviton mass, including Laser Interferometer Space Antenna (LISA), are still of the order of the static tests magnitude in the Solar system [17,18]. A considerable improvement of the results is anticipated [19, 20]. The theoretical estimations of the graviton mass are most often based on the assumption that the Compton wavelength of the graviton D g is about the Hubble distance c/H ≈ 26 hH -33 2 1.43×10 m, which leads to the value of the graviton mass mg ~ ≈ 1.38×10 eV/c , c2 where H ≈ 65 km s −1Mpc−1 is the Hubble constant [21,22]. In this case, as a result of the Universe extension, the graviton mass should decrease by “the age of the Universe” ( H −1 ), which appears least probable.

II. NEUTRINO AND GRAVITON MASS ESTIMATIONS APPROACH

A. Determination of coupling constants at extremely low energy

Among the multitude of particles, several free particles are notable, which are stable or at least their lifetime is longer than the age of the Universe – the proton (p), electron (e), neutrino (ν) in three flavors, graviton (g) and photon (γ). Only free massive particles are examined in this paper. Quarks and gluons are bound in hadrons by confinement and they cannot be immediately detected in the experiments, and the photon is massless. Therefore, these particles are not a subject of this paper. A measure for the interaction strength is a dimensionless quantity - the coupling constant of the interaction (αi), which is determined from the cross section of the respective processes. It is known that the interactions coupling constants depend on the energy (“running of coupling constants”) [23,24]. With increase of the processes energy, the coupling constant of the strong interaction decreases and the rest coupling constants increase. Recent unified theories predict unification of the four interactions on Planck scale (E ~ 1019 GeV). Close to such energy, the four coupling constants merge. Since the modern experiments are performed with energy of hundreds GeV, a value of the weak coupling constant, close on the electroweak scale, approaches the coupling constant of the electromagnetic interaction. On the other hand, the strong coupling constant is too low at such high energy and reaches α s ≈ 0.11 at 189 GeV [25]. The purpose of this paper is by a unified approach to estimate the masses of the lightest stable particles, acted upon by the weak and gravity interactions, the electron neutrino and the graviton, respectively. That is why it is necessary to determine the coupling constants under conditions when the interactions (and the coupling constants) are differentiated to the maximum and, this is the case with the extremely low processes energy. This means that it is necessary to determine the minimum values of the weak and gravitation coupling constant and the maximum value of the strong coupling constant. Thus, each coupling constant obtains

3 unique asimptotical value at extremely low energy. A version of the approach suggested in the paper, was used for preliminary estimation of the electron neutrino mass [26].

The coupling constant of the electromagnetic interaction αe is known as the fine structure constant: 2 e -3 αe = ≡ α ≈ 7.30×10 (1), hc where e is the elementary electrical charge, h - Planck constant, c – the light velocity.

The coupling constant of the weak interaction αw is determined by the expression: 2 GF m c αw = 3 (2), h where GF is Fermi coupling constant, m – the interacting particle mass.

As substantiated above, it is necessary the coupling constants to be determined at extremely low energy of the processes. It is known that the lowest-energy process, occurring under the influence of the weak interaction, is the neutron β - decay. The process is running extremely slowly (τ ~ 880 s), and a minimum quantity of energy is released:

2 ∆E ≈ 0.78 MeV ~ mec = 0.511 MeV (3)

Therefore, m in expression (2) is substituted by the electron mass me and for the weak coupling constant at extremely low energy, the following value is obtained: 2 GF me c -12 αw ~ 3 = 3.00×10 (4) h The value, obtained for the coupling constant of the weak interaction practically coincides α with the one, accepted in [27], where w ~ 10-10. The value of the weak coupling constant, α obtained in (4) is minimal as a result of the minimal energy of the neutron β - decay. The rest processes involve the weak interaction, occurring with considerably higher energy and, as a result, the typical values of αw are in orders of magnitude higher than such energy. For this reason namely αw is usually determined by (2) replacing m with pion (or proton) mass. As it was mentioned, the unique value of αw at extremely low energy of the neutron β - decay is used in this paper. The coupling constant of the strong interaction is determined from expression (5): g 2 αs = (5), hc where g is a strong interaction constant, depending on the energy. It is known [28] that in cases of slow nucleons scattering (without angular momentum) the strong coupling constant has a maximum value αs ~ 10 ÷ 17. In [29] it has been accepted that αs ~ 15 and in [30] it has been accepted that αs ~ 14. The modern accelerators’ energy exceeds hundreds GeV and the value of the strong coupling constant with such energy is less than a unit. The calculations in this paper involve the maximum value of the strong coupling constant αs ~ 15. The coupling constant of the gravitational interaction is determined by the expression: 2 GN m αg = (6), hc where GN – universal gravitational constant.

-45 The replacement of m with the electron mass me yields αg ≈ 1.75×10 .

B. Results

The detection of the neutrino oscillations by the Super Kamiokande experiment proved unambiguously that the neutrino mass moν ≠ 0, although until then most physicians accepted the neutrino to be a massless particle. The theoretical models predict for the neutrino mass -6 2 moν > 10 eV/c . Analogously, it is very possible that the graviton has a negligible, yet a nonzero mass mog ≠ 0. Table I presents the above-obtained coupling constants of the interactions at extremely 2 low energy E ~ mec , as well as the masses of the lightest free massive stable particles acted upon by the respective interactions. The experimental upper limits of the electron neutrino and graviton masses are presented.

TABLE I: Coupling constants of interactions and masses of the lightest free massive stable particles acted upon by the respective interaction. Fundamental Coupling Lightest stable particle acted upon Experimental Mass interaction constant by a respective interaction (eV/c2) Strong ~ 15 Proton (p) 9.38×108 Electromagnetic 7.30×10-3 Electron (e) 5.11×105 -12 Weak 3.00×10 Electron neutrino (νe) 0 < m < 2 Gravitational 1.75×10-45 Graviton (g) < 3.1×10-29

Table I shows that with increasing the interaction strength (coupling constant), the mass mmin of the lightest stable particle acted upon by the respective interaction also increases. The data in Table I are presented in a logarithmic scale in Fig.1, which shows that the trend is clearly expressed.

FIG. 1: Dependence between the coupling constants of the interactions and the mass of the lightest free massive stable particle acted upon by the respective interaction. The red line presents the approximation (7) of e, p and the upper limit masses of g and νe. The blue line presents the approximation (8) of e, p and the lowest limit mass of νe from the models. The black line presents the strict linear approximation (S = 1).

The points in Fig. 1, corresponding to the electron and proton masses and to the upper limit masses of the electron neutrino and graviton, are approximated by the method of the least squares with a power function (7):

Log mmin = 0.825 Log α + 8.309 (7)

Although this approximation is only on 4 points, the correlation found is close and the correlation coefficient reaches r = 0.998, which supports the power function. The slope (S) is a little bit smaller than one and wherefore it can be said that the regression is close to a linear one. In addition, it should be reminded that instead of the electron neutrino and graviton masses, their upper limit values are used, which produce a certain underestimation of the S value. Analogously, the points, corresponding to the electron and proton masses and to the low limit of the electron neutrino mass from the models are approximated with a power function, too:

Log mmin = 1.194 Log α + 7.862 (8)

This approximation is too close to (7) and the correlation is close again (r = 0.999). Now the slope is a little bigger than one. The certain slope overestimation is due to the fact that in this case instead of the still unknown value of the electron neutrino mass, the lowest neutrino mass limit is used from the theoretical models. Approximations (7) and (8) produce at αg = 1.75×10-45 a value for the graviton mass in an interval from 2.3×10-46 eV/c2 to 3.1×10-29 eV/c2, i.e. the graviton possesses a negligible, yet a nonzero mass. These approximations show that the mass of the lightest stable particle, acted upon by a particular interaction, increases with the increase of the respective coupling constant by a power function with S ~ 1, i.e. close to the linear one. Thus, the experimental data direct to a linear dependence (S = 1) between the mass of the lightest stable particle and the coupling constants:

Log mmin = Log α + k0 (9),

where k0 is a constant.

Expression (9) transforms into (10):

k0 mmin = 10 α = kα (10)

In this way the experimental data and constraints suggest that the mass of the lightest free massive stable particle, acted upon by a particular interaction, is proportional to the coupling constant of the respective interaction at extremely low energy:

mi min = kαi (11),

where k is a constant, i =1, 2, 3, 4 - index for each interaction and the lightest massive stable particle acted upon by a respective interaction.

6 Constant k can be determined by the fine structure constant (α1 ≡ α) and the electron mass

me 7 2 (m1min ≡ me) because both are measured with very high precision k = ≈ 7.00×10 eV/c . α The substitution of this value in (11) yields the mass relation (12):

me 7 2 mimin = αi ≈ 7.00×10 αi eV/c (12) α Taking a logarithm of (12) produces:

Log mimin = Log αi + 7.845 (13),

which is close to the approximated expressions (7) and (8).

The found mass relation (12) can be examined by the strong interaction because the proton mass is measured with high precision. The application of the mass relation on the strong interaction predicts the lightest stable hadron mass:

9 2 mp ≈ 1.05×10 eV/c (14)

The proton mass value obtained by the mass relation is with 12 % higher than the experimental value of mp. This result confirms the reliability of the found mass relation and shows that this relation possesses heuristic power. The application of the mass relation on the weak interaction affords to evaluate the mass of the electron neutrino:

-4 2 mνe ≈ 2.1×10 eV/c (15)

This value is in close accordance with the prediction of the simple SO(10) model for the -5 2 -4 2 lightest neutrino mass mνe = 5×10 eV/c ÷ 5×10 eV/c [31]. Finally, the application of the mass relation on the gravitation interaction produces an estimation of the graviton mass:

-37 2 mg ≈ 1.2×10 eV/c (16)

The obtained estimation of the graviton mass is several orders of magnitude less than the upper limit of the graviton mass, obtained by astrophysical constraints. The negligible mass of the graviton can impede considerably the experimental determination of mg. The predicted masses of the four massive stable particles are presented in Table II. It is seen that the fitting of the predicted values and the experimental data is satisfactory.

TABLE II: Experimental and predicted values of four free massive stable particles. Free massive stable particle Experimental mass (eV/c2) Predicted mass (eV/c2) Proton 9.38×108 1.05×109 Electron 5.11×105 - Electron neutrino 0 < m < 2 2.1×10-4 Graviton < 3.1×10-29 1.2×10-37

Now it is not clear yet what the cause for the relationship between the free stable particles and the interaction coupling constants is but its existence is confirmed by the experimental data for the masses of four stable particles and the coupling constants of interactions at extremely low energy. Most probably the found mass relation represents an expression of a

7 universal symmetry, including free stable particles of most diverse kinds (hadron, charged lepton, neutral lepton and quantum of gravitation). The mass range of these particles reaches to 45 orders of magnitude. 2 The obtained value mνe ~ 0.0002 eV/c and the results from the solar and atmospheric neutrino experiments allow to estimate the masses of the heavier neutrino flavor - νµ and ντ. 2 The results from the Super Kamiokande experiment lead to square mass difference ∆m23 ~ 2.7×10-3 eV 2 [32]. Recent results on solar neutrinos provide hints that the large mixing angle (LMA) Mikheyev-Smirnov-Wolfenstein (MSW) solution of the SND is more probable 2 -5 2 than SMA MSW [33]. The LMA solution leads to ∆m12 ~ 7×10 eV [34] and the small 2 -6 2 mixing angle (SMA) MSW solution leads to ∆m12 ~ 6×10 eV [35]. In this way both MSW 2 solutions yield mντ ~ 0.05 eV/c . The most appropriate LMA MSW solution yields mνµ ~ 2 2 0.008 eV/c . SMA MSW solution yields mνµ ~ 0.0025 eV/c . Thus, the obtained values of the neutrino masses support the hierarchical neutrino models.

C. Discussions

The presence of an exceptionally small, yet nonzero mass of the graviton, involves Yukawa potential of the gravitational field: G M φ (r) = - N exp(-r/ ) (17), r D g

where D g - Compton wavelength of the graviton, determined by (18): h 30 26 D g = ≈ 1.67×10 m >> c/H ≈ 1.43×10 m (18) mg c

Since D g is considerably larger than Hubble distance (c/H), the deviation of Newton potential from Yukawa potential is manifested very weakly (|∆φ /φ| < 10-4) even at a distance r ~ c/H. As a result, the experimental determination of the graviton mass will be a serious challenge. Yet, it can be expected that appropriate astrophysical or laboratory experiments will be found for determination of the graviton mass. The nonzero mass of the graviton

involves a finite range of the gravitational interaction rg ~ D g >> c/H. The massive graviton might turn of considerable importance for the description of the processes in the nuclei of the active galaxies and quasars, the gravitational collapse as well as for the improvement of the cosmological models. It should be noted that even if the very high value for the coupling constant of the

GN mem p -42 gravitation αg = ≈ 3.22×10 is accepted, according to the mass relation (12) of hc -34 2 hH the graviton mass, the result obtained is mg ≈ 2.25×10 eV/c < . For this value of mg, c2 however, the deviation of Yukawa potential from Newton potential remains too small (|∆φ /φ| < 0.15). It is noticeable that the constant in the mass relation (12), possessing mass dimension, is about half a pion mass and it is very possible not to be an accidental coincidence. The massive graviton places other challenges before the modern Unified theories. Among them are the famous van Dam-Veltman-Zakharov (vDVZ) discontinuity [36,37] and the violation of the gauge invariance and the general covariance. Yet, there are already encouraging attempts to solve vDVZ discontinuity in anti de Sitter (AdS) background [38,39].

III. CONCLUSIONS

The interaction coupling constants are determined at extremely low energy when the interactions are completely different in their strength and are differentiated clearly from one another. It has been substantiated that the mass of the lightest stable particle, acted upon by a particular interaction, is proportional to the coupling constant of the respective interaction (at extremely low energy) mi min = kαi. This mass relation unifies the masses of four free stable particles (p, e, νe and g), covering an extremely wide range of values, reaching to 45 orders of magnitude. The suggested approach shows that the graviton mass is nonzero. The graviton and -37 2 electron neutrino masses are estimated by this approach to mg ~ 1.2×10 eV/c and mνe ~ 2 0.0002 eV/c , respectively. The masses of the heavier neutrinos νµ and ντ are estimated by the results of the Super Kamiokande and SND experiments to 0.008 eV/c2 and 0.05 eV/c2, respectively, and they support the hierarchical neutrino models.

Acknowledgements

I would like to thank I. Koprinkov for his useful comments and discussions.

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