- Email: [email protected]
The LSND results and three-flavour neutrino oscillations
Nuclear Physics B (Proc. Suppl.) 85 (2000) 172-176
ELSEVIER
PROCEEDINGS SUPPLEMENTS www.elsevier.nlllocate/npe
The LSND Result and Three-Flavour Neutrino Oscillations M. Barone a aUniversit~ degli Studi di Urbino Via S. Chiara 27, 1-61029 Urbino, Italy The results from the LSND experiment are examined in the framework of three-flavour neutrino oscillations together with all other available experimental information. A comparison with solar, reactor and accelerator experiments indicates that the results are compatible in the 2.5 x 10-~ < AM ~ < 3.0 eV ~ range. By assuming AM '~ = 3.0 eV ~, the corresponding mixing angles values are determined. For AM ~ = 3.0 eV ~, data on solar and atmospheric neutrinos combined with the limit on reactor experiments give a value for P~u that is consistent with the LSND observation.
1. I n t r o d u c t i o n In the three-flavour neutrino-mixing scenario the simplified U-matrix describing the weak interaction states can be written as: cr, cl3 --812C23
~
C12823S13
8 1 2 8 2 3 - - C12C23813
s~2c~3
s~3
S12823813
S23C13
- - C 1 2 8 2 3 -- 812C23813
C23C13
C12C23 --
with cq = cosSij and sij = s i n S i j , 0ij < 45 °. With the additional hypotesis of a natural mass hierarchy (rnl < < m2 < < m3), fast and slow oscillations are characterized by only two ~m 2, a large A M ~ = m 2 - m?~ ,,, rn~ - m~ and a small A m ~ = m 2 - m 2, respectively.
2. Experimental i n p u t s a n d r e s u l t s One of the two J m 2 can be fixed to a value of a b o u t 10 -3 eV 2, according to atmosferic neutrino d a t a [1,2]. On the basis of the LSND [3] observation of ( P e t ~ ) L S N D =- (3.1 + 0.09 4- 0.05) x 10 -3 at L i E = 0.7 m / M e V , J m 2 has to be identified with A m 2. This also defines two ranges L i E < 103 m / M e V and L i E > 103 m / M e V . In the first interval, the LSND region, the oscillation probabilities depend only on two angles, el3 and 023. Concerning solar neutrinos, in this work it has been considered t h a t there is no energy dependence of the solar neutrino flux suppression [4]. In order to determinate the above mixing angles, the LSND results and those from other experiments in the ue - v u channel [5,6] could be
)
used. These results identify a wide regiou ill ~:h~ A M 2, Ae, plane. On the other hand, in the framework or' ;he three-flavour, a tighter limit (see also [7]) ol~ u, - u~ oscillation can be obtained by taking m,.~ account results of the other oscillatio~ ':hannol:~. Ue - uz and u , - u~, Therefore, expermlental upper limits on the transition probabiliti~s (1 - Pee)l = ( P e , ) l (from reactor experiments) and ( P u r ) l (from accelerators) have been used. The subscript 1 indicates that the probabilities are considered in the range L i E < 103 m / M e V . Usually, two-flavour analyses present results in terms of countour plots in the sin'-'(2U), &r~: plane, In the L i E range above, the sin'2(26))'s relative to the transitions ue - uz, ue - u, and v, u. are, respectively, the three-flavour oscillatior~ ~ l ~ plitudes ( A e x ) l ----- 4813C13, 2 '2 ( A e u ) i =-= 48'~3~:;cii:~ .... ' and ( A u ~ ) l ~-= 4 8 223 C 223 C 143 . Maximum values for the two parameters, (4~:~ and /723, can be determined from exper'imcntal probabilities. These values have no errors. An upper limit contour of the oscillation amplitude ( A e # ) l as a function of A M 2 can then be determined. This curve is shown in Figure 1. The 99% C.L. region allowed by LSND result is also shown (dotted lines). There is compatibility in the range: 2.5 x 10 -~ < A M '~ < 3.0 eV '~. The angles 013 and 0 2 3 depend on A~.'I '2. They also depend on the amount by which tim limit
0920-5632/00/$ - see front matter © 2000 ElsevierScience B.V. All rights reserved. PII S0920-5632(00)00502-8
M. Barone/Nuclear Physics B (Proc. Suppl.) 85 (2000) 172-176
~10
. . . . . . . ![
4
173
........
~ 10 3
103~.
:oii
10 ~
[0
~
/
/
i ....................
10 3
--
J
10 2
10"I
........
1
I
10
........
J
,
10~
1o'[ ~M~=:.8eV~ P., :o2~ ~ e / :o~ / :-.___f~/Po°
10
~
,
t,,,,,i
10] 104 L/E ~m/Mev)
....
,
I03
IC~
:
1 -~_~L
.. .....
10 l
10 3 \
10-2
10-
/
3
........
if5
~
lff4
........
J
10 2
. . . . . . . .
10-2 10-1 (Ae0~=4 sin2e2s sinZOt~ cos2e~
Figure 1. The upper limit on the oscillation amplitude(Aeu): = 4 8 223 S 1 32C 1 3 2 a s a function of A M ~ (full line) compared with the 99% C. L. region allowed by the LSND result (dotted lines). Three typical values of A M 2 are indicated by dots. The square and the triangle represent two solutions recently proposed.
of Figure 1 is accepted to be violated in order to reach the LSND-allowed region. To avoid exceeding any limit, they are conservatively chosen to coincide with their upper limits. In the allowed mass range above, s~3 cannot have large variations because of the tight reactor limits, whilst s~3 can swing by as much as a factor of twentyfive. Within this range, three tipical solutions are considered, A M ~ = 3.0 eV 2, A M '~ = 1.8 eV 2, A M 2 = 0.25 eV 2. These are the values by which the oscillation ( A M 2 L / 4 E ) term becomes larger, equal and smaller than Ir/2, respectively. The value of the third angle, 912 can be determined from (Pee)2 = 0.50 4- 0.06 [4]. Transition probabilities corresponding to the three values of A M 2 are shown in Figure 2, as a function of L / E . The curves are averaged over a Gaussian L I E distribution with 30% width. Experiments on atmospheric neutrinos permit to define an U / D asymmetry of the U upward-
I
I0
2--30e:A
\
10-3
i01
Fe.
10-
10 3
I.tE (m/MeV)
P.,
10"
10 ~
Id
i ~
102
J
I01
I
....
a . . . . . . . ~ , ........~ . . . . . , ~ . I0 lff ICe I0" L/E (m/MeV)
Figure 2. Transition probabilities as a function of L / E for three typical values of A M 2 .
going events, - 1 < cos 0 < 0.2, and D downwardgoing events, 0.2 < cos0 < 1.0, where O is the zenith angle. The asymmetry is defined as A = (U - D ) / ( U + D). The values measured by Superkamiokande [1,2] are: Ae = -0.0036 4- 0.067, A t = -0.296 ± 0.048 The expected values of the A asymmetries can be calculated by using for U and D the asymptotic values of the probabilities in Figure 2. These values are reported in Table 1. No errors are indicated.
Table 1 Ae and A~ asymmetries calculated from the asyntotic values of the transition probabilities A M 2 (eV") 3.0 1.8 0.25 Ae 0.197 0.199 0.010 A~ -0.138 -0.142 -0.005
For the smallest mass, the transition probabilities imply a vanishing value of Ae, in agreement
M. Barone /Nuclear Physics B (Proc. Suppl.) 85 (2000) 172-176
174
........
I
........
I
'I
.......
........
I
.......
'
1.5
0 (b
1
' 5o
.......,......,....+.
0 0.5
,
•
e-like
o
p-like
, ,,.ild
.
10
i,,,nJ
.
10 2
, .lliti~
10 3
'
i''*uil
I
10 4
IIII
10 s
UF_,, ( k m / G e V )
Figure 3. Atmosferic data from Superkamiokande measurements of e (full circles) and # (empty circles) rates as a function of L/E. Data are normalized to Monte Carlo expectations.
the first and second plateau by considering statistical errors only. By minimizing a X2 consisting of seven equations, a general solution for the oscillation parameter determination has been found. Four equations are the rates from atmosferic neutrinos. One comes from the limit on reactors. One from solar neutrinos and one from the limit on accelerators. Four parameters have to be determined: 012,013, 023, A M 2. For the moment, the analysis has been limited to three typical values of A M 2 in the 2.5 x 10 -1 < A M 2 < 3.0 eV 2 range. Therefore, the problem is reduced to six equations and three parameters (the three angles). As a matter of fact, in the mass range above, the only relevant limit is that fixed by reactors. The X~ can be written as reported in the following: (I(xi)
- yi)
i
with Superkamiokande. On the other hand, for larger masses there is agreement for Ap but positive values for Ae, within a 3a deviation. The conclusion is that in the considered range of A M 2 there is not a favoured solution, at least when available data from reactors, solar and accelerators are considered. To see if there is a serious inconsistency with the LSND data, the Superkamiokande results have been included to determine the oscillation parameters. The Superkamiokande data are shown if Figure 3 [8] as a function of L/E. In order to avoid mass dependence and L/E energy resolution dependence, the experimental data in the 30 < L / E < 300 m / M e V region have not been considered. Four values of the e and # rates have been extracted from the plot shown in Figure 3:
(e>~ xp = 1.12 -4-0.12, (e>~xp = 0.96 4-0.10
<~)~xp = 1.07 4- 0.10, (#),~P = 0.59 + 0.06 These are typical values for Superkarniokande in
where the theoretical rates and probabilities are: / ( x i ) : (e)l, <~h, ( ~ h , (~)2, (Fee)l, (Peel') and the experimental ones are: / ~exp Yi:~eh ,(e).~~p, /x,t ,~\lel x p
,
/
\exp
\~12
,
(Pee\eXp / p 11
, \~
~exp
eel2
For each of the three A M 2 values, the minimum X2 has been shown in Table 2.
Table 2 Results of the X2 minimization. The X2 consists of seven equation, with 3 doe A M 2 (eV2) 3.0 1.8 0.25 X2 9.76 10.03 10.24 C.L. (%) 2.1 1.8 1.7
The X2 is weekly dependent on A M 2, but fully acceptable for the three considered values, the confidence level beeing of the order of 2%. This result does not allow to say that there is a preferred solution. In order to calculate the full mixing matrix, the solution with the smallest X2
M. Barone~Nuclear Physics B (Proc. Suppl.) 85 (2000) 172-176 A M 2 = 3.0 eV ~, has been considered. This gives the values for the three parameters 81~.1 ~13, ~3 reported in Table 3 as sin/9. The U-matrix is unique and includes errors on parameters. The correlation matrix is: p =
1.00 -0.36 0.05
-0.36 1.00 0.07
0.05 ) 0.07 1.00
104 . . . . . . . .
~,
........
,
175
..................
,
1031
.......
1
(A~)1---0.008 -~
,°2I
+- 1 o
f
li
10
1
L
10 -1
Table 3 Values of the oscillation parameters calculated for A M 2 = 3.0 eV 2. e and # rates and Aeu amplitude are also shown. A M 2 (eV e) 3.0 s~2 0.86 :t: 0.05 813 0.12 + 0.09 s93 0.39 + 0.06 (e)l 0.98 ± 0.03 (e)2 1.21 + 0.09 (~1[~)1 0 , 7 5 5= 0.06 (p)~ 0.66 5= 0.05 (Ae~)z 0.008 5= 0.013
Using these results, e and/z rates and the Aeu amplitude can be calculated. They are reported in Table 3 with their errors. The line corresponding to the value of (Aeu)l --- 0.008 is shown in Figure 4 togheter with the l a line. This result is compatible with the LSND observation. A complete solution for three-flavour neutrino mixing has been obtained. All available experimental data, but the LSND data have been used. 3. C o n c l u s i o n s In the framework of the three-flavour neutrino phenomenology, the LSND data have been compared with other experimental information. A comparison among solar, reactor and accelerator experiments restricts the value of A M 2 in the 2.5 x 10 -1 < A M ~ < 3.0 eV '2 range. Neutrino atmospheric data have been used in order to further limit the above range. However, there is no indication of a favoured solution in the mass range above.
10 -2
3'
10-4
10-3
10-2
10-1
1
(Aeu)l=4 sin2023 sin~}j3 c o s ~ 1 3
Figure 4. Upper limit on the oscillation amplitude (Aeu)l ---- 4s23s13c13 2 2 '2 as a function of A M '~ (full crooked line) compared with the 99% C. L. region allowed by the LSND result (dotted lines). The value of (A~u)l calculated for A M 2 = 3.0 eV ~ including all available experimental data (except for LSND) is also shown (straight continous line).
By assuming A M 2 = 3.0 eV 2, a complete solution for mixing angles has been found. A unique full mixing matrix with errors has then been determined by using values of the angles with their errors. Starting from solar and atmospheric data and using the limits on reactors, the predicted result is fully compatible with LSND. As a matter of fact, the value of the oscillation amplitude (Ae~)l for A M 2 = 3.0 eV 2 is approximately that one observed by LSND. Given the large elz coupling, Am 2 must be of the order of 10 -3 eV 2.
Acknowledgements This work has been accomplished with the cooperation of G.Conforto and C. Grimani.
176
M. Barone / Nudear Physics B (Proc. Suppl.) 85 (2000) 172-176
REFERENCES
1. The Super-Kamiokande Collaboration, Y. Fukuda et al., hep-ex/9807003. 2. T. Kajita, Proceedings of the 18th International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, June 4-9, 1998, Nuel. Phys. Proc. Suppl. 77, 123 (1999) 3. C. Athanassopoulos et al., Phys. Rev. Lett. 77, 3082 (1996) 4. G. Conforto et al., hep-ph/9807306. 5. L. Borodovsky et al., Phys. Rev. Lett. 68, 274 (1992) 6. T. Jannakos, Latest results from Karmen2 presented at the XXXIV rencontres de Moriond on Electroweak Interactions and Unified theories, Les Arcs, France, March 1320,1999 7. G. Conforto et al., Phys. Lett. B447, 122 (1999). 8. K. Schoelberg for the Super-Kamiokande Collaboration, 04/99, private communication.
ELSEVIER
PROCEEDINGS SUPPLEMENTS www.elsevier.nlllocate/npe
The LSND Result and Three-Flavour Neutrino Oscillations M. Barone a aUniversit~ degli Studi di Urbino Via S. Chiara 27, 1-61029 Urbino, Italy The results from the LSND experiment are examined in the framework of three-flavour neutrino oscillations together with all other available experimental information. A comparison with solar, reactor and accelerator experiments indicates that the results are compatible in the 2.5 x 10-~ < AM ~ < 3.0 eV ~ range. By assuming AM '~ = 3.0 eV ~, the corresponding mixing angles values are determined. For AM ~ = 3.0 eV ~, data on solar and atmospheric neutrinos combined with the limit on reactor experiments give a value for P~u that is consistent with the LSND observation.
1. I n t r o d u c t i o n In the three-flavour neutrino-mixing scenario the simplified U-matrix describing the weak interaction states can be written as: cr, cl3 --812C23
~
C12823S13
8 1 2 8 2 3 - - C12C23813
s~2c~3
s~3
S12823813
S23C13
- - C 1 2 8 2 3 -- 812C23813
C23C13
C12C23 --
with cq = cosSij and sij = s i n S i j , 0ij < 45 °. With the additional hypotesis of a natural mass hierarchy (rnl < < m2 < < m3), fast and slow oscillations are characterized by only two ~m 2, a large A M ~ = m 2 - m?~ ,,, rn~ - m~ and a small A m ~ = m 2 - m 2, respectively.
2. Experimental i n p u t s a n d r e s u l t s One of the two J m 2 can be fixed to a value of a b o u t 10 -3 eV 2, according to atmosferic neutrino d a t a [1,2]. On the basis of the LSND [3] observation of ( P e t ~ ) L S N D =- (3.1 + 0.09 4- 0.05) x 10 -3 at L i E = 0.7 m / M e V , J m 2 has to be identified with A m 2. This also defines two ranges L i E < 103 m / M e V and L i E > 103 m / M e V . In the first interval, the LSND region, the oscillation probabilities depend only on two angles, el3 and 023. Concerning solar neutrinos, in this work it has been considered t h a t there is no energy dependence of the solar neutrino flux suppression [4]. In order to determinate the above mixing angles, the LSND results and those from other experiments in the ue - v u channel [5,6] could be
)
used. These results identify a wide regiou ill ~:h~ A M 2, Ae, plane. On the other hand, in the framework or' ;he three-flavour, a tighter limit (see also [7]) ol~ u, - u~ oscillation can be obtained by taking m,.~ account results of the other oscillatio~ ':hannol:~. Ue - uz and u , - u~, Therefore, expermlental upper limits on the transition probabiliti~s (1 - Pee)l = ( P e , ) l (from reactor experiments) and ( P u r ) l (from accelerators) have been used. The subscript 1 indicates that the probabilities are considered in the range L i E < 103 m / M e V . Usually, two-flavour analyses present results in terms of countour plots in the sin'-'(2U), &r~: plane, In the L i E range above, the sin'2(26))'s relative to the transitions ue - uz, ue - u, and v, u. are, respectively, the three-flavour oscillatior~ ~ l ~ plitudes ( A e x ) l ----- 4813C13, 2 '2 ( A e u ) i =-= 48'~3~:;cii:~ .... ' and ( A u ~ ) l ~-= 4 8 223 C 223 C 143 . Maximum values for the two parameters, (4~:~ and /723, can be determined from exper'imcntal probabilities. These values have no errors. An upper limit contour of the oscillation amplitude ( A e # ) l as a function of A M 2 can then be determined. This curve is shown in Figure 1. The 99% C.L. region allowed by LSND result is also shown (dotted lines). There is compatibility in the range: 2.5 x 10 -~ < A M '~ < 3.0 eV '~. The angles 013 and 0 2 3 depend on A~.'I '2. They also depend on the amount by which tim limit
0920-5632/00/$ - see front matter © 2000 ElsevierScience B.V. All rights reserved. PII S0920-5632(00)00502-8
M. Barone/Nuclear Physics B (Proc. Suppl.) 85 (2000) 172-176
~10
. . . . . . . ![
4
173
........
~ 10 3
103~.
:oii
10 ~
[0
~
/
/
i ....................
10 3
--
J
10 2
10"I
........
1
I
10
........
J
,
10~
1o'[ ~M~=:.8eV~ P., :o2~ ~ e / :o~ / :-.___f~/Po°
10
~
,
t,,,,,i
10] 104 L/E ~m/Mev)
....
,
I03
IC~
:
1 -~_~L
.. .....
10 l
10 3 \
10-2
10-
/
3
........
if5
~
lff4
........
J
10 2
. . . . . . . .
10-2 10-1 (Ae0~=4 sin2e2s sinZOt~ cos2e~
Figure 1. The upper limit on the oscillation amplitude(Aeu): = 4 8 223 S 1 32C 1 3 2 a s a function of A M ~ (full line) compared with the 99% C. L. region allowed by the LSND result (dotted lines). Three typical values of A M 2 are indicated by dots. The square and the triangle represent two solutions recently proposed.
of Figure 1 is accepted to be violated in order to reach the LSND-allowed region. To avoid exceeding any limit, they are conservatively chosen to coincide with their upper limits. In the allowed mass range above, s~3 cannot have large variations because of the tight reactor limits, whilst s~3 can swing by as much as a factor of twentyfive. Within this range, three tipical solutions are considered, A M ~ = 3.0 eV 2, A M '~ = 1.8 eV 2, A M 2 = 0.25 eV 2. These are the values by which the oscillation ( A M 2 L / 4 E ) term becomes larger, equal and smaller than Ir/2, respectively. The value of the third angle, 912 can be determined from (Pee)2 = 0.50 4- 0.06 [4]. Transition probabilities corresponding to the three values of A M 2 are shown in Figure 2, as a function of L / E . The curves are averaged over a Gaussian L I E distribution with 30% width. Experiments on atmospheric neutrinos permit to define an U / D asymmetry of the U upward-
I
I0
2--30e:A
\
10-3
i01
Fe.
10-
10 3
I.tE (m/MeV)
P.,
10"
10 ~
Id
i ~
102
J
I01
I
....
a . . . . . . . ~ , ........~ . . . . . , ~ . I0 lff ICe I0" L/E (m/MeV)
Figure 2. Transition probabilities as a function of L / E for three typical values of A M 2 .
going events, - 1 < cos 0 < 0.2, and D downwardgoing events, 0.2 < cos0 < 1.0, where O is the zenith angle. The asymmetry is defined as A = (U - D ) / ( U + D). The values measured by Superkamiokande [1,2] are: Ae = -0.0036 4- 0.067, A t = -0.296 ± 0.048 The expected values of the A asymmetries can be calculated by using for U and D the asymptotic values of the probabilities in Figure 2. These values are reported in Table 1. No errors are indicated.
Table 1 Ae and A~ asymmetries calculated from the asyntotic values of the transition probabilities A M 2 (eV") 3.0 1.8 0.25 Ae 0.197 0.199 0.010 A~ -0.138 -0.142 -0.005
For the smallest mass, the transition probabilities imply a vanishing value of Ae, in agreement
M. Barone /Nuclear Physics B (Proc. Suppl.) 85 (2000) 172-176
174
........
I
........
I
'I
.......
........
I
.......
'
1.5
0 (b
1
' 5o
.......,......,....+.
0 0.5
,
•
e-like
o
p-like
, ,,.ild
.
10
i,,,nJ
.
10 2
, .lliti~
10 3
'
i''*uil
I
10 4
IIII
10 s
UF_,, ( k m / G e V )
Figure 3. Atmosferic data from Superkamiokande measurements of e (full circles) and # (empty circles) rates as a function of L/E. Data are normalized to Monte Carlo expectations.
the first and second plateau by considering statistical errors only. By minimizing a X2 consisting of seven equations, a general solution for the oscillation parameter determination has been found. Four equations are the rates from atmosferic neutrinos. One comes from the limit on reactors. One from solar neutrinos and one from the limit on accelerators. Four parameters have to be determined: 012,013, 023, A M 2. For the moment, the analysis has been limited to three typical values of A M 2 in the 2.5 x 10 -1 < A M 2 < 3.0 eV 2 range. Therefore, the problem is reduced to six equations and three parameters (the three angles). As a matter of fact, in the mass range above, the only relevant limit is that fixed by reactors. The X~ can be written as reported in the following: (I(xi)
- yi)
i
with Superkamiokande. On the other hand, for larger masses there is agreement for Ap but positive values for Ae, within a 3a deviation. The conclusion is that in the considered range of A M 2 there is not a favoured solution, at least when available data from reactors, solar and accelerators are considered. To see if there is a serious inconsistency with the LSND data, the Superkamiokande results have been included to determine the oscillation parameters. The Superkamiokande data are shown if Figure 3 [8] as a function of L/E. In order to avoid mass dependence and L/E energy resolution dependence, the experimental data in the 30 < L / E < 300 m / M e V region have not been considered. Four values of the e and # rates have been extracted from the plot shown in Figure 3:
(e>~ xp = 1.12 -4-0.12, (e>~xp = 0.96 4-0.10
<~)~xp = 1.07 4- 0.10, (#),~P = 0.59 + 0.06 These are typical values for Superkarniokande in
where the theoretical rates and probabilities are: / ( x i ) : (e)l, <~h, ( ~ h , (~)2, (Fee)l, (Peel') and the experimental ones are: / ~exp Yi:~eh ,(e).~~p, /x,t ,~\lel x p
,
/
\exp
\~12
,
(Pee\eXp / p 11
, \~
~exp
eel2
For each of the three A M 2 values, the minimum X2 has been shown in Table 2.
Table 2 Results of the X2 minimization. The X2 consists of seven equation, with 3 doe A M 2 (eV2) 3.0 1.8 0.25 X2 9.76 10.03 10.24 C.L. (%) 2.1 1.8 1.7
The X2 is weekly dependent on A M 2, but fully acceptable for the three considered values, the confidence level beeing of the order of 2%. This result does not allow to say that there is a preferred solution. In order to calculate the full mixing matrix, the solution with the smallest X2
M. Barone~Nuclear Physics B (Proc. Suppl.) 85 (2000) 172-176 A M 2 = 3.0 eV ~, has been considered. This gives the values for the three parameters 81~.1 ~13, ~3 reported in Table 3 as sin/9. The U-matrix is unique and includes errors on parameters. The correlation matrix is: p =
1.00 -0.36 0.05
-0.36 1.00 0.07
0.05 ) 0.07 1.00
104 . . . . . . . .
~,
........
,
175
..................
,
1031
.......
1
(A~)1---0.008 -~
,°2I
+- 1 o
f
li
10
1
L
10 -1
Table 3 Values of the oscillation parameters calculated for A M 2 = 3.0 eV 2. e and # rates and Aeu amplitude are also shown. A M 2 (eV e) 3.0 s~2 0.86 :t: 0.05 813 0.12 + 0.09 s93 0.39 + 0.06 (e)l 0.98 ± 0.03 (e)2 1.21 + 0.09 (~1[~)1 0 , 7 5 5= 0.06 (p)~ 0.66 5= 0.05 (Ae~)z 0.008 5= 0.013
Using these results, e and/z rates and the Aeu amplitude can be calculated. They are reported in Table 3 with their errors. The line corresponding to the value of (Aeu)l --- 0.008 is shown in Figure 4 togheter with the l a line. This result is compatible with the LSND observation. A complete solution for three-flavour neutrino mixing has been obtained. All available experimental data, but the LSND data have been used. 3. C o n c l u s i o n s In the framework of the three-flavour neutrino phenomenology, the LSND data have been compared with other experimental information. A comparison among solar, reactor and accelerator experiments restricts the value of A M 2 in the 2.5 x 10 -1 < A M ~ < 3.0 eV '2 range. Neutrino atmospheric data have been used in order to further limit the above range. However, there is no indication of a favoured solution in the mass range above.
10 -2
3'
10-4
10-3
10-2
10-1
1
(Aeu)l=4 sin2023 sin~}j3 c o s ~ 1 3
Figure 4. Upper limit on the oscillation amplitude (Aeu)l ---- 4s23s13c13 2 2 '2 as a function of A M '~ (full crooked line) compared with the 99% C. L. region allowed by the LSND result (dotted lines). The value of (A~u)l calculated for A M 2 = 3.0 eV ~ including all available experimental data (except for LSND) is also shown (straight continous line).
By assuming A M 2 = 3.0 eV 2, a complete solution for mixing angles has been found. A unique full mixing matrix with errors has then been determined by using values of the angles with their errors. Starting from solar and atmospheric data and using the limits on reactors, the predicted result is fully compatible with LSND. As a matter of fact, the value of the oscillation amplitude (Ae~)l for A M 2 = 3.0 eV 2 is approximately that one observed by LSND. Given the large elz coupling, Am 2 must be of the order of 10 -3 eV 2.
Acknowledgements This work has been accomplished with the cooperation of G.Conforto and C. Grimani.
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