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Some aspects of the exotic interactions of neutrinos with an extremely light scalar
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291
PROCEEDINGS SUPPLEMENTS www.elsevieronl/locate/npe
Some aspects of the exotic interactions of neutrinos with an extremely light scalar R. Horvat "Rudjer Bo~koviC Institute, P.O.Box 1016, 10001 Zagreb, Croatia We discuss a string-inspired scheme in which an extremely light (or a strictly massless) scalar particle, coupled very weakly to both neutrinos and to a constituent of the matter, gives rise to neutrino masses within a medium. We also discuss how exotic properties of neutrinos such as possible violations of the equivalence principle may produce neutrino oscillations in such a scheme. The identification of the scalar with Quintessence is possible if one could avoid a moderate fine-tuning present here as a constraint on the scalar-neutrino coupling constant from Quintessential cosmology.
1. I n t r o d u c t i o n We suggest here that the intervening matter [ordinary m a t t e r as well as the cosmic neutrino background (CNB)] may induce appreciable masses for neutrinos, if they couple to an extremely light (or a strictly massless) scalar particle. Such a large effect may occur even if the scalar is assumed to interact with neutrinos only gravitationally (or even weaker) [1], i.e. via coupling constants suppressed by the inverse of the Planck mass. By inducing a nonzero mass squared difference for neutrinos within a medium, such exotic interactions may produce neutrino oscillations even for degenerate-in-mass neutrinos [1,2]. It is necessary for this to occur that the coupling constants representing interactions at the Planck scale are not universal but rather species-dependent, thereby violating the equivalence principle (VEP). It is important to stress that the masslessness of the scalar and V E P (the ingredients necessary for the efficient generation of neutrino masses and oscillations), are at the same time a basic outcome of a scenario of Damour and Polyakov [3] in string theory, in which the string dilaton field may remain massless in the low-energy world and violates the equivalence principle. One should not confuse the above scheme with another one, where the neutrino mixing is generated by a V E P induced by a breakdown of universality in the gravitational coupling strength between the conventional spin-2 patti-
cles and the neutrinos, which had been proposed first in [4] and discussed afterwards in several papers over the past ten years (see e.g. [5] and the references therein). We should mention t h a t ideas for generation of neutrino masses in m a t t e r along the similar lines as here were presented also in [6,7]. It is interesting to observe that the scalar field fitting the above scheme, may be identified (under certain circumstances to be discuss below) with Quintessence, a special sort of scalar fields, which, as the current d a t a [8,9] indicate, presumably play a fundamental role in cosmology. The plan of the paper is as follows. In Sec. II we derive a constraint on the scalar-neutrino coupling constant from Quintessential cosmology and suggest how to cope with the problems of fine-tuning that arise from this constraint. In Sec. III we show how (sizeable) neutrino masses may be generated within a medium. In Sec. IV we discuss neutrino oscillations which include V E P as a basic ingredient. We discuss our results in Sec. V.
2. C o n s t r a i n t o n t h e s c a l a r - n e u t r i n o c o u p l i n g c o n s t a n t f r o m Q u i n t e s s e n t i a l cosmology and fine-tuning problems The basic way [10] to constrain certain parameters of the effective scalar potential V(¢) from Quintessential cosmology is to observe that Quintessence is generated if, at the present epoch,
0920-5632/00/$ - see front matter © 2000 ElsevierScience B.V All rights reserved. PII S0920-5632(00)00519-3
R. Horvat/Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291
288
the potential obeys the flatness conditions
M p [V'/V] << 1
(1)
M 2 [V"/V[ << 1,
(2)
where M p is the reduced Planck scale. Let us concentrate to the mass term in V(¢) and consider an interaction of this smoothly distributed component in the Universe with another predominately smooth component, the CNB. In order to obtain an order-of-magnitude estimate for this effect we may always neglect inhomogeneities in the neutrino component, as numerical simulations suggest [11] that at present only 10 - 20% of the dark matter neutrinos are in galactic halos while the rest are distributed more smoothly. From a viewpoint of Thermal Field Theory (TFT), such an interaction is represented by the scalar self-energy He at finite temperature (T~ 2K), where also real relic neutrinos are running inside the loop. One can show that this interaction would induce an effective mass squared m~ + II¢(0) in V(¢) [12], where II¢(k0 = 0, k --+ 0) -
For neutrino components with rn~ << T~, we find at the one loop level that [13] n
(0)
- -
(3)
12
In the opposite and more realistic case, m~ >> T~, we find [13] He(0) ~ -0.07 g2 m~ Tv,
(4)
where the parameters in Eqs. (3) and (4) refer to the heaviest neutrino from the background, and we suppose that Quintessence couples to neutrinos with a Yukawa strength g~ (the vacuum mass term for neutrinos is of the Dirac type). Assuming no fine-tuned cancellations between various contributions to the slope of V, we are allowed to apply the flatness conditions directly to the shifted-mass term, obtaining for ¢ ~ MR [14] g~ << 10 -2s
(m~ << T~),
g~ << 4 x 10 -30 (0"07 \ ~ ] eV~] 1/2
(5) (m~>>T~).(6)
The limits (5) and (6) are to be compared with the most stringent limits on g~, that is with those set by the observations of neutrinos from SN 1987A. By making a claim of absence of large scattering of supernova neutrinos from dark matter neutrinos, one obtains, 9~ < 10 -3 [15]. Moreover we show that the powerful bound as given by Eq.(6), for neutrinos having masses in the eV range, is even more restrictive than the corresponding limit for ordinary matter coming from conventional solar-system gravity experiments. The present experimental data give upper limits of order 10 -3 for a possible admixture of a scalar component to the relativistic gravitational interaction, Be2xt < 10 -3 [16]. By adjusting g~ to be of gravitational origin only, ~, - x/~Mp(g~,/m~,), one obtains from EQ.(6) that ~2 << 4 x 10 -6
\m~/
.
(7)
It is to be noted however that for m~ -~ 1 eV, Eq.(7) represents a moderate fine-tuning in V(¢). Indeed, from a traditional viewpoint, the expected values for the /~ are of order of unity since they represent interactions at the Planck scale. The possibility to suppress such a coupling by imposing symmetries is viable only in pseudo-Goldstone boson models of Quintessence [17]. Here we give two possible solutions to the fine-tuning problem. The first solution is in agreement with the fact that the current data favor those models of structure formation with a nonvanishing cosmological constant over the mixed cold + hot dark matter models. In this respect, the presence of hot dark matter is no longer necessary (and eV neutrinos are not needed to provide this component) [18]. If we set m~ --~ 0.04 eV, a value consistent with the Super-Kamiokande experiment, t h e n / ~ < < 0.3, and sure enough there are ways to achieve this value without suppression by some symmetry. The second solution is a /east coupling principle introduced originally by Damour and Polyakov [3] in string theory. It is based on a mechanism which provides that a (universal) coupling of the scalar (the string dilaton field in their example) with the rest of the world, being dependent on the VEV of ¢ and hence time
R. Horvat/Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291 dependent, has a minimum close to the present value of the ¢'s VEV. Therefore, the present-day value for ~ can naturally be much less than unity. We have to assume however that the mechanism is also operative for Quintessence since in their paper Damour and Polyakov dealt with the string dilaton, which, as a recent analysis [19] shows, cannot provide us with the negative equation of state, and therefore is useless for the dynamical component of Quintessence. One can also show that the above bounds remain unchanged in the case in which the vacuum mass terms for neutrinos are of the Majorana type. 3. N e u t r i n o
masses
From a viewpoint of T F T , neutrino masses within a medium are generated in the above scheme via a tadpole graph. It consists of a medium-induced loop (in which relic neutrinos a n d / o r ordinary-matter particles are circulating) and also of a one-loop resummed propagator for the scalar. Using the real-time version of T F T [20], we find a contribution to the induced neutrino mass as
ind
=
m~,
--STrGNflV fib rav mY
n~(o)
f ×
d3k
(27r)3Ek
(8)
n F ( V / ~ + m2) ,
where GN is the Newton's gravitational constant, n f is the distribution function for a fermion, and the subscript b refers to a background particle. Eq.(8) assumes that the thermal mass squared for ¢ is always larger than the bare mass squared. One obtains for the neutrino mass induced by means of the CNB which is nonrelativistic at present,
which are nonrelativistic today, still need to be assigned a distribution function relevant for massless particles, as their total number density is fixed at about 100 cm -3 in the uniform nonclustered background. This means that we have had to set m b = 0 in the argument of nF in
Eq.(S). The contribution to the induced neutrino mass as given by (8) and (9) reveals several salient features which we are now going to discuss. First of all, it depends only on the ratio of couplings but not on their absolute values. From a viewpoint of perturbation expansion, this means that apparently of order O(g~), it is actually of order O(1) Since the only information on f ' s is the weak upper limit (7) and the weak upper limit from solar-system gravity experiments [16], it is quite obvious that the ratio is practically unconstrained - this gives us an opportunity to accommodate practically any value for the neutrino mass in the problem very easily (see below). 4. N e u t r i n o o s c i l l a t i o n s In the rest of the paper we shall be concerned with the case where f ' s from the neutrino sector are not universal but rather speciesdependent, thereby violating the equivalence principle. Specifically this means that ~ ~ ~ , where i is the /-type neutrino (we have already employed a separate mark for the background particles in the above expressions). In the presence of nonuniversal f ' s the mechanism will produce neutrino oscillation; as a special feature we stress that even for degenerate-in-mass neutrinos the mechanism will still produce oscillations. In the following we assume, for the sake of demonstration, a complete degeneracy for the oscillatory neutrinos i = 1, 2, i.e.
m~,, * = mo + mi,,?d , "
-
\fl~,,,I \m2~, ] m~,, (T, < < my,) ,
(9)
where m , b and fl~b refer to the highest-mass neutrino from the background. Note that Eq.(9), together with Eq.(4), represent a specific nonequilibrium situation where massive neutrinos,
289
(10)
where m0 is the common mass from the vacuum. If, for the sake of simplicity, one deals with the case where the mass and the gravitational eigenstates are identical, then it is easy to show that the evolution equation as well as the expression for flavor survival probability as a function of distance L (for say two flavors with mixing angle 8)
290
R. Horvat/Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291
have precisely the same form as in the standardoscillation mechanism,
( LAm*2 ~ P ( v e - + v e ) = l - s i n 2 ( 2 0 ) s i n 2 ~, ~ ] , (11) ,2 - rn,~ ,2 and E is the neutrino where A m .2 -- m~: energy. 5. D i s c u s s i o n Let us consider first A m .2 induced by means of the CNB. Using Eqs.(10) and (9) and keeping only terms to the first order in T~2/m~,b, 2 one obtains
where A~ = ~-2 - ~ 1 characterizes a VEP. By setting m~ b ,-~ 1 eV, T~ ~_ 50K as to account for a possibility that at least some percentage of the density of the halo of our Galaxy is in the form of light neutrinos, and Am .2 ~_ 10 - l ° eV as to explain the solar neutrino data via oscillation in vacuum, one finds by using (7) a limit on Aft imposed by the current solar data,
a z < < 3 × 10 -9 (eV 2 kin0 /
(13)
One finds for neutrinos of mass ,~ 1 eV (such a model for neutrino masses where the three known neutrinos have nearly the same mass, of about 1 eV, was presented in Ref.[21]) that the righthanded side of (13) is not as good as the most severe limit for ordinary matter [22]. It is however better than the limit obtained by comparing neutrinos with antineutrinos from SN 1987A [23]. The next question is whether the above mechanism may account for the MSW effect in the body of the Sun. For that purpose we need Am .2 induced by the solar plasma [1],
where T is the solar temperature, m v is the proton mass and ]~p is the scalar-proton coupling constant. Although one may expect that a more sophisticated treatment is needed to study the resonant condition for neutrinos propagating through
the Sun (as now A m 2 is temperature dependent), one can show that the overall behavior of the survival curves is consistent with those for the preferred MSW models. We find thus that for I AB/~p IN 10 - 104 MSW transitions will take place somewhere in the Sun. Finally, we mention the atmospheric neutrino problem. There is a common contribution for the neutrinos from above and for upmoving neutrinos, with A~/]3, b ~ 10 - 102. There is however an additional contribution in matter for the upmoving neutrinos with A~/~p ,,~ 10 -1 - 1. Hence, we see that the above oscillation scheme reveals an extra possibility that for two flavors the effect may in fact depend not only on the distance but on the intervening matter as well. It is important to mention that the oscillation length depends linearly on the neutrino energy in [1] (as in the standard-oscillation scenario), a feature recently confirmed by a detailed analysis of the Super-Kamiokande data [24] which also include the observed angular distribution of the high energy upward going muons. On the other hand, the models of Ref. [4,5] appear to be disfavored by the present data as there the neutrino oscillation length is proportional to the inverse of the neutrino energy. Acknowledgments. The author acknowledges the support of the Croatian Ministry of Science and Technology under the contract 1 - 03 068. REFERENCES 1. R. Horvat, Phys. Rev. D 58, 125020 (1998). 2. A. Halprin and C. N. Leung, Phys. Lett. B416 (1998) 361. 3. T. Damour and A. M. Polyakov, Gen. Rel. Grav. 26 (1994) 1171; Nucl. Phys. B 423 (1994) 532. 4. M. Gasperini, Phys. Rev. D38, (1988) 2635; D39, (1989) 3606. 5. A. Halprin, C. N. Leung and J. Pantaleone, Phys. Rev. D53, (1996) 5365. 6. R . F . Sawyer, Phys. Lett. B448 (1999) 174. 7. G . J . Stephenson, T. Glodman and B. H. J. McKellar, Int. J Mod. Phys. A13 (1998) 2765; Mod. Phys. Lett. A12 (1997) 2391.
R. Horvat/Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291
8. S. Perlmutter et al., Ap. J. 483 (1997) 567; Nature 391 (1998) 51. 9. A.G. Riess et al., astro-ph/9805201. 10. C. Kolda and D. H. Lyth, hep-ph/9811375. 11. A. L. Melott, Phys. Rev. Lett. 48, (1982) 894. 12. See e.g., M. E. Carrington, Phys. Rev. D 45, (1992) 2933. 13. R. Horvat, hep-ph/9904451. 14. I. Zlatev, L. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82, (1999) 896. 15. E. W. Kolb and M. Turner, Phys. Rev. D 36, (1987) 2895; A. Manohar, Phys. Lett. B 192, (1987) 217. 16. R. D. Reasenberg et al, Astrophys. J. 234 (1979) L219. 17. S. M. Carroll, Phys. Rev. Lett. 81, (1998) 3067. 18. Relic Neutrino Workshop, Sep 98, Trieste; A. Dighe, S. Pastor and A. Yu. Smirnov, hepph/9812244. 19. P. Binetruy, hep-ph/9810553. 20. A. J. Niemi and G. W. Semenoff, Nucl. Phys. B 230, (1984) 181; R. L. Kobes and G. W. Semenoff, Nucl. Phys. B 260, (1985) 714; Nucl. Phys. B 272, (1986) 329. 21. D. O. Caldwell and R. N. Mohapatra, Phys. Rev. D 48, (1993) 3259. 22. B. R. Heckel et al, Phys. Rev. Lett. 63, (1989) 2705; E. Adelberger et al, Phys. Rev. D 42, (1990) 3267. 23. M. J. Longo, Phys. Rev. Lett. 60, (1988) 173; L. M. Krauss and S. Tremaine, Phys. Rev. Lett. 60, (1988) 170; J. M. LoSecco, Phys. Rev. D 38, (1988) 3313; S. Pakvasa, W. Simmons and T. J. Weiler, Phys. Rev. D 39, (1989) 1761. 24. G. L. Folgi, E. Lisi, A. Marrone and G. Scioscia, hep-ph/9904248.
291
Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291
PROCEEDINGS SUPPLEMENTS www.elsevieronl/locate/npe
Some aspects of the exotic interactions of neutrinos with an extremely light scalar R. Horvat "Rudjer Bo~koviC Institute, P.O.Box 1016, 10001 Zagreb, Croatia We discuss a string-inspired scheme in which an extremely light (or a strictly massless) scalar particle, coupled very weakly to both neutrinos and to a constituent of the matter, gives rise to neutrino masses within a medium. We also discuss how exotic properties of neutrinos such as possible violations of the equivalence principle may produce neutrino oscillations in such a scheme. The identification of the scalar with Quintessence is possible if one could avoid a moderate fine-tuning present here as a constraint on the scalar-neutrino coupling constant from Quintessential cosmology.
1. I n t r o d u c t i o n We suggest here that the intervening matter [ordinary m a t t e r as well as the cosmic neutrino background (CNB)] may induce appreciable masses for neutrinos, if they couple to an extremely light (or a strictly massless) scalar particle. Such a large effect may occur even if the scalar is assumed to interact with neutrinos only gravitationally (or even weaker) [1], i.e. via coupling constants suppressed by the inverse of the Planck mass. By inducing a nonzero mass squared difference for neutrinos within a medium, such exotic interactions may produce neutrino oscillations even for degenerate-in-mass neutrinos [1,2]. It is necessary for this to occur that the coupling constants representing interactions at the Planck scale are not universal but rather species-dependent, thereby violating the equivalence principle (VEP). It is important to stress that the masslessness of the scalar and V E P (the ingredients necessary for the efficient generation of neutrino masses and oscillations), are at the same time a basic outcome of a scenario of Damour and Polyakov [3] in string theory, in which the string dilaton field may remain massless in the low-energy world and violates the equivalence principle. One should not confuse the above scheme with another one, where the neutrino mixing is generated by a V E P induced by a breakdown of universality in the gravitational coupling strength between the conventional spin-2 patti-
cles and the neutrinos, which had been proposed first in [4] and discussed afterwards in several papers over the past ten years (see e.g. [5] and the references therein). We should mention t h a t ideas for generation of neutrino masses in m a t t e r along the similar lines as here were presented also in [6,7]. It is interesting to observe that the scalar field fitting the above scheme, may be identified (under certain circumstances to be discuss below) with Quintessence, a special sort of scalar fields, which, as the current d a t a [8,9] indicate, presumably play a fundamental role in cosmology. The plan of the paper is as follows. In Sec. II we derive a constraint on the scalar-neutrino coupling constant from Quintessential cosmology and suggest how to cope with the problems of fine-tuning that arise from this constraint. In Sec. III we show how (sizeable) neutrino masses may be generated within a medium. In Sec. IV we discuss neutrino oscillations which include V E P as a basic ingredient. We discuss our results in Sec. V.
2. C o n s t r a i n t o n t h e s c a l a r - n e u t r i n o c o u p l i n g c o n s t a n t f r o m Q u i n t e s s e n t i a l cosmology and fine-tuning problems The basic way [10] to constrain certain parameters of the effective scalar potential V(¢) from Quintessential cosmology is to observe that Quintessence is generated if, at the present epoch,
0920-5632/00/$ - see front matter © 2000 ElsevierScience B.V All rights reserved. PII S0920-5632(00)00519-3
R. Horvat/Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291
288
the potential obeys the flatness conditions
M p [V'/V] << 1
(1)
M 2 [V"/V[ << 1,
(2)
where M p is the reduced Planck scale. Let us concentrate to the mass term in V(¢) and consider an interaction of this smoothly distributed component in the Universe with another predominately smooth component, the CNB. In order to obtain an order-of-magnitude estimate for this effect we may always neglect inhomogeneities in the neutrino component, as numerical simulations suggest [11] that at present only 10 - 20% of the dark matter neutrinos are in galactic halos while the rest are distributed more smoothly. From a viewpoint of Thermal Field Theory (TFT), such an interaction is represented by the scalar self-energy He at finite temperature (T~ 2K), where also real relic neutrinos are running inside the loop. One can show that this interaction would induce an effective mass squared m~ + II¢(0) in V(¢) [12], where II¢(k0 = 0, k --+ 0) -
For neutrino components with rn~ << T~, we find at the one loop level that [13] n
(0)
- -
(3)
12
In the opposite and more realistic case, m~ >> T~, we find [13] He(0) ~ -0.07 g2 m~ Tv,
(4)
where the parameters in Eqs. (3) and (4) refer to the heaviest neutrino from the background, and we suppose that Quintessence couples to neutrinos with a Yukawa strength g~ (the vacuum mass term for neutrinos is of the Dirac type). Assuming no fine-tuned cancellations between various contributions to the slope of V, we are allowed to apply the flatness conditions directly to the shifted-mass term, obtaining for ¢ ~ MR [14] g~ << 10 -2s
(m~ << T~),
g~ << 4 x 10 -30 (0"07 \ ~ ] eV~] 1/2
(5) (m~>>T~).(6)
The limits (5) and (6) are to be compared with the most stringent limits on g~, that is with those set by the observations of neutrinos from SN 1987A. By making a claim of absence of large scattering of supernova neutrinos from dark matter neutrinos, one obtains, 9~ < 10 -3 [15]. Moreover we show that the powerful bound as given by Eq.(6), for neutrinos having masses in the eV range, is even more restrictive than the corresponding limit for ordinary matter coming from conventional solar-system gravity experiments. The present experimental data give upper limits of order 10 -3 for a possible admixture of a scalar component to the relativistic gravitational interaction, Be2xt < 10 -3 [16]. By adjusting g~ to be of gravitational origin only, ~, - x/~Mp(g~,/m~,), one obtains from EQ.(6) that ~2 << 4 x 10 -6
\m~/
.
(7)
It is to be noted however that for m~ -~ 1 eV, Eq.(7) represents a moderate fine-tuning in V(¢). Indeed, from a traditional viewpoint, the expected values for the /~ are of order of unity since they represent interactions at the Planck scale. The possibility to suppress such a coupling by imposing symmetries is viable only in pseudo-Goldstone boson models of Quintessence [17]. Here we give two possible solutions to the fine-tuning problem. The first solution is in agreement with the fact that the current data favor those models of structure formation with a nonvanishing cosmological constant over the mixed cold + hot dark matter models. In this respect, the presence of hot dark matter is no longer necessary (and eV neutrinos are not needed to provide this component) [18]. If we set m~ --~ 0.04 eV, a value consistent with the Super-Kamiokande experiment, t h e n / ~ < < 0.3, and sure enough there are ways to achieve this value without suppression by some symmetry. The second solution is a /east coupling principle introduced originally by Damour and Polyakov [3] in string theory. It is based on a mechanism which provides that a (universal) coupling of the scalar (the string dilaton field in their example) with the rest of the world, being dependent on the VEV of ¢ and hence time
R. Horvat/Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291 dependent, has a minimum close to the present value of the ¢'s VEV. Therefore, the present-day value for ~ can naturally be much less than unity. We have to assume however that the mechanism is also operative for Quintessence since in their paper Damour and Polyakov dealt with the string dilaton, which, as a recent analysis [19] shows, cannot provide us with the negative equation of state, and therefore is useless for the dynamical component of Quintessence. One can also show that the above bounds remain unchanged in the case in which the vacuum mass terms for neutrinos are of the Majorana type. 3. N e u t r i n o
masses
From a viewpoint of T F T , neutrino masses within a medium are generated in the above scheme via a tadpole graph. It consists of a medium-induced loop (in which relic neutrinos a n d / o r ordinary-matter particles are circulating) and also of a one-loop resummed propagator for the scalar. Using the real-time version of T F T [20], we find a contribution to the induced neutrino mass as
ind
=
m~,
--STrGNflV fib rav mY
n~(o)
f ×
d3k
(27r)3Ek
(8)
n F ( V / ~ + m2) ,
where GN is the Newton's gravitational constant, n f is the distribution function for a fermion, and the subscript b refers to a background particle. Eq.(8) assumes that the thermal mass squared for ¢ is always larger than the bare mass squared. One obtains for the neutrino mass induced by means of the CNB which is nonrelativistic at present,
which are nonrelativistic today, still need to be assigned a distribution function relevant for massless particles, as their total number density is fixed at about 100 cm -3 in the uniform nonclustered background. This means that we have had to set m b = 0 in the argument of nF in
Eq.(S). The contribution to the induced neutrino mass as given by (8) and (9) reveals several salient features which we are now going to discuss. First of all, it depends only on the ratio of couplings but not on their absolute values. From a viewpoint of perturbation expansion, this means that apparently of order O(g~), it is actually of order O(1) Since the only information on f ' s is the weak upper limit (7) and the weak upper limit from solar-system gravity experiments [16], it is quite obvious that the ratio is practically unconstrained - this gives us an opportunity to accommodate practically any value for the neutrino mass in the problem very easily (see below). 4. N e u t r i n o o s c i l l a t i o n s In the rest of the paper we shall be concerned with the case where f ' s from the neutrino sector are not universal but rather speciesdependent, thereby violating the equivalence principle. Specifically this means that ~ ~ ~ , where i is the /-type neutrino (we have already employed a separate mark for the background particles in the above expressions). In the presence of nonuniversal f ' s the mechanism will produce neutrino oscillation; as a special feature we stress that even for degenerate-in-mass neutrinos the mechanism will still produce oscillations. In the following we assume, for the sake of demonstration, a complete degeneracy for the oscillatory neutrinos i = 1, 2, i.e.
m~,, * = mo + mi,,?d , "
-
\fl~,,,I \m2~, ] m~,, (T, < < my,) ,
(9)
where m , b and fl~b refer to the highest-mass neutrino from the background. Note that Eq.(9), together with Eq.(4), represent a specific nonequilibrium situation where massive neutrinos,
289
(10)
where m0 is the common mass from the vacuum. If, for the sake of simplicity, one deals with the case where the mass and the gravitational eigenstates are identical, then it is easy to show that the evolution equation as well as the expression for flavor survival probability as a function of distance L (for say two flavors with mixing angle 8)
290
R. Horvat/Nuclear Physics B (Proc. Suppl.) 85 (2000) 287-291
have precisely the same form as in the standardoscillation mechanism,
( LAm*2 ~ P ( v e - + v e ) = l - s i n 2 ( 2 0 ) s i n 2 ~, ~ ] , (11) ,2 - rn,~ ,2 and E is the neutrino where A m .2 -- m~: energy. 5. D i s c u s s i o n Let us consider first A m .2 induced by means of the CNB. Using Eqs.(10) and (9) and keeping only terms to the first order in T~2/m~,b, 2 one obtains
where A~ = ~-2 - ~ 1 characterizes a VEP. By setting m~ b ,-~ 1 eV, T~ ~_ 50K as to account for a possibility that at least some percentage of the density of the halo of our Galaxy is in the form of light neutrinos, and Am .2 ~_ 10 - l ° eV as to explain the solar neutrino data via oscillation in vacuum, one finds by using (7) a limit on Aft imposed by the current solar data,
a z < < 3 × 10 -9 (eV 2 kin0 /
(13)
One finds for neutrinos of mass ,~ 1 eV (such a model for neutrino masses where the three known neutrinos have nearly the same mass, of about 1 eV, was presented in Ref.[21]) that the righthanded side of (13) is not as good as the most severe limit for ordinary matter [22]. It is however better than the limit obtained by comparing neutrinos with antineutrinos from SN 1987A [23]. The next question is whether the above mechanism may account for the MSW effect in the body of the Sun. For that purpose we need Am .2 induced by the solar plasma [1],
where T is the solar temperature, m v is the proton mass and ]~p is the scalar-proton coupling constant. Although one may expect that a more sophisticated treatment is needed to study the resonant condition for neutrinos propagating through
the Sun (as now A m 2 is temperature dependent), one can show that the overall behavior of the survival curves is consistent with those for the preferred MSW models. We find thus that for I AB/~p IN 10 - 104 MSW transitions will take place somewhere in the Sun. Finally, we mention the atmospheric neutrino problem. There is a common contribution for the neutrinos from above and for upmoving neutrinos, with A~/]3, b ~ 10 - 102. There is however an additional contribution in matter for the upmoving neutrinos with A~/~p ,,~ 10 -1 - 1. Hence, we see that the above oscillation scheme reveals an extra possibility that for two flavors the effect may in fact depend not only on the distance but on the intervening matter as well. It is important to mention that the oscillation length depends linearly on the neutrino energy in [1] (as in the standard-oscillation scenario), a feature recently confirmed by a detailed analysis of the Super-Kamiokande data [24] which also include the observed angular distribution of the high energy upward going muons. On the other hand, the models of Ref. [4,5] appear to be disfavored by the present data as there the neutrino oscillation length is proportional to the inverse of the neutrino energy. Acknowledgments. The author acknowledges the support of the Croatian Ministry of Science and Technology under the contract 1 - 03 068. REFERENCES 1. R. Horvat, Phys. Rev. D 58, 125020 (1998). 2. A. Halprin and C. N. Leung, Phys. Lett. B416 (1998) 361. 3. T. Damour and A. M. Polyakov, Gen. Rel. Grav. 26 (1994) 1171; Nucl. Phys. B 423 (1994) 532. 4. M. Gasperini, Phys. Rev. D38, (1988) 2635; D39, (1989) 3606. 5. A. Halprin, C. N. Leung and J. Pantaleone, Phys. Rev. D53, (1996) 5365. 6. R . F . Sawyer, Phys. Lett. B448 (1999) 174. 7. G . J . Stephenson, T. Glodman and B. H. J. McKellar, Int. J Mod. Phys. A13 (1998) 2765; Mod. Phys. Lett. A12 (1997) 2391.
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