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Mirror fermions and the neutrino counting process
Volume 235, number 3,4
PHYSICS LETTERS B
I February. 1990
MIRROR FERMIONS AND THE NEUTRINO COUNTING P R O C E S S H. CZYZ 1,2
INFN. Sezione di Bologna, 1-40126 Bologna, Italy Received 13 November 1989
The influenceof the existenceof mirror fermions on the neutrino counting process (e+e - --'7+ missingneutrals ) is investigated. Because of mixing the contribution to the cross section from the generation where both mirror and normal neutrino are present is preferably smaller than the contribution from the normal generation. Besides. at least in principle, it is possible to distinguish Dirac and Majorana neutrinos even in the massless case.
Mirror fcrmions - fcrmions which couple to the W boson by the V + A current - appear naturally in many theoretical models. They were used to cancel a n o m alies and play an essential role in the regularization in lattice theories (for a review of this and many other subjects connected with mirror fermions see ref. [ 1 ] ). However, the question whether they really exist still remains open. There are m a n y ways of looking for them in present and future experiments [ 1 ] but at least one of them has not been studied theoretically yet, i.e. e + e - - . y + m i s s i n g neutrals. I would like to show how the presence of mirror neutrinos influcnces this well known proccss, which will be used for counting the n u m b e r of neutrino Species. The differential cross section has a ve~, simple form in the case we take the model given in rcf. [ 1 ] :
do
G20es(l-x)[(l
d x dy ×
K i = K'li "t- K2i + K 3 i , KI t = /
(a 2v, + v2v,)
=a~i
for Dirac n e u t r i n o s , for Majorana n e u t r i n o s ,
x2i =6)( l _ 4 z ) x / ~ _ n z
[½ ( a v2 i + v 2 ) _ ( 2 a v2 , - v v,2i ) z l
for Dirac n e u t r i n o s , = 6 ) ( 1 - 4 z ) ( 1 - 4z)3/2a~i for Majorana n e u t r i n o s ,
i 2y 2 ] - ~l x ) 2 + ~x
6zrZx( 1 _ y 2 )
K3i =ANir)( 1 - z ) [ 1 - 2 x z (
(a 2 + v 2) Y_,x,
1-z) ]
for Dirac a n d / o r Majorana n e u t r i n o s ,
[1 - s ( l - - x ) I M ~ ] 2 + F U M ~
+ terms coming from W-exchange d i a g r a m s , (1) where Gv is the Fermi constant, a ( M 2) = 1/ 128.5 is t On leave of absence from Department of Theoretical Physics, University of Silesia, PL-40007 Katowice, Poland. 2 Supported in part by Polish Minist~, of Education Grant CPBP 01.03.
322
the fine structure constant, s is the Mandelstam variable, x = 2E~,/~f~, and y = cos 0r, with E r the energy of the proton and 0v the photon angle with respect to the incident beam direction, Mz, Fz arc the mass and the total width of the Z, and ae = - ~ cos 20e, v~= - 21+ 2 sin20w, with 0w the Wcinberg angle, the sum is over the neutrino species, and
with z = r n 2 d s ( 1 - x ) , mN~ the mass of the ith mirror neutrino, O the Heaviside function, and avj = cos 2~,, b'vi=b'Ni= 1, aNi = --COS 2rpi, A N i = s i n 2~0i, where 0,, ~0~are the mixing angles (see ref. [ 1 ] ). The first term in the sum comes from the process e + e - - . 7 + v V (v is the normal n e u t r i n o ) , the second term from the process e + e - ~ 7 + N l q (N is the mirror n e u t r i n o ) and the third term from the processes e+e - -,7+v1'2," and e + e - - , y + N 9 . These processes give thc same signal assuming that the mirror neu-
0370-2693/90/$ 03.50 © ElsevierScience Publishers B.V. ( North-Holland )
Volume 235, number 3,4
PHYSICS LETTERS B
trino lifetime is long enough to decay outside a detector. In may analysis I neglected diagrams with W exchange as they are estimated to give small contributions to the process studied in the investigated beam energy region [2 ]. Besides, I assumed that the normal neutrino is massless (in case it is massive ic,, should be changed into the form o f ~c2~,where the subscript N is changed into the subscript v ). In the numerical calculations I put 0¢ = 0. In the case 0 ~ 0 , the cross section differs from that o f 0 , = 0 roughly by a factor cos220~ (as v~-~0), which decreases the cross section about 20% in the case we take 0~= 13.3 ° from the fit o f ref. [ 1 ]. It is a very, big effect which in principle could be observed in experiments (SLC, L E P ) as a m e a s u r e m e n t o f the n u m b e r o f neutrino species being less than three in the standard fit, when for example only the e + e - - - , 7 + v 9 channel is open. In the case o f no mixing between normal and mirror neutrinos, there is no difference between their contributions to the cross section (cf. ref. [ 3 ] ). So in that case in the reaction e+e - - - , 7 + m i s s i n g neutrals we measure the n u m b e r o f light neutrino species and there is no way to distinguish the m i r r o r neutrino from the normal one even if the a d d i t i o n a l neutrino is massive. However, in the case when the mixing does a p p e a r a lot o f interesting features a p p e a r as well. As we can see from eq. ( 1 ) and the form o f x2~, there exists a contribution to the cross section even in the case ran,> ½Mz as the channels vlq and 9N are open up to m N i .=-M z .
To c o m p a r e contributions to the total cross section from the generation which mixes with m i r r o r leptons with the massless neutrino contribution, I used the ratio R = aT (e + e - ~Yv9 + 7Nlq + yvlCq+ YNg) • o v ( e + e - ~'¢v9) ,
(2)
after integrating over lYl ~<0.94 and taking E y > 2 GeV, R has the form R = 2A + B + 1 + ( - A - 2 B - ½+ C) sin22~o~, for Dirac neutrinos = 2 A + 4 B + 1 + ( - - 2 A - 4 B - - 1 + C ) sin22~0i, for M a j o r a n a neutrinos
(3)
1 February 1990
where A, B, C are functions o f the m i r r o r neutrino mass and b e a m energy (see table 1 ). There are only small differences in A, B, C between v / s = 96 GeV (table 1 ) and n / s = 104 GeV. It is interesting that, in the presence o f the mixing, contributions from Dirac and Majorana neutrinos are different even in the massless (A=½, B = 0 , C = i ) case, quite opposite to the situation where there is no mixing and no way to distinguish between the two kinds o f neutrino even in principle [4]. In the case when the neutrinos are M a j o r a n a neutrinos, the contribution to the cross section from the generation in which both m i r r o r and normal neutrinos are present [wide generation ( W G ) ] could be o f the o r d e r o f the contribution o f one normal neutrino (R = 2-sin22~o,) when there is a big mixing between normal and mirror neutrinos. Such a mixing is still not rejected by experiment ( 0 ~ t p ~ 9 0 ° for the third ( t ) family [ 1,5 ] ). If neutrinos are Dirac and massless the contribution from the W G is R = 2 and does not d e p e n d on the mixing. The second interesting region o f neutrino masses is r n ~ > ½Mz ( A = B = 0 ) . If the mixing does not occur, the contribution from the W G is the same as that
Table 1 The values of the funelions A, B, C for x/s=96 GeV and Mz= 91.9 GeV (see the text ); A = B = 0 for m > Mz. m (GeV)
A
B
C
0 5 l0 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
0.50 0.50 0.49 0.47 0.45 0.42 0.38 0.32 0.24 0.09
0.000 -0.003 -0.011 -0.025 -0.043 - 0.063 -0.081 -0.094 -0.092 -0.045
1.0 1.0 0.99 0.97 0.95 0.92 0.88 0.84 0.79 0.74 0.68 0.62 0.55 0.48 O.4O 0.32 0.23 0.14 0.04 0.00
323
Volume 235, number 3,4
PHYSICS LETTERS B
from the normal generation ( N C ) : However, if there is a mixing R = 1 + ( C - ½) sin22~pi for Dirac n e u t r i n o s , = 1 + ( C - 1 ) sin22q~, for M a j o r a n a n e u t r i n o s .
(4)
As we can scen from table 1 the contribution from thc W G is bigger from the N G only in the mass range ½Mz< mr~ < 63 GeV for Dirac neutrinos and smaller in the whole mass range for M a j o r a n a neutrinos. In the mass range when even the 9N and vlq channels are closed (A = B = C = 0) R = 1 - ½ sin22q~i
1 Februar3' 1990
ments o f total decay widths of W and Z [ 1 ]. We can see that even if we find "\)~-3 with relatively small experimental error it will not mean that m i r r o r neutrinos do not exist as it is easy to fit Nv~-3 choosing a p p r o p r i a t e mixings and masses allowed by experiment. In that case searching for mirror fermions would need further investigations. Unfortunately the signal Nv< 3 is not unique for the investigated model and for example the model investigated in ref. [6] could give the same signature as well. However, one can distinguish between these two models by a detailed analysis o f the gauge boson decay modes and possibly also o f the massive neutrino decay modes which are different in the frameworks of both models (cf. refs. [ 1,6] ). I would like to thank Dr. M. Caffo for drawing my attention to ref. [6] and for carefully r e a d i n g the manuscript.
for Dirac n e u t r i n o s , = cos22{9i for Majorana neutrinos,
(5)
SO that in both cases, if the mixing occurs, the contribution from the W G is smaller than from the NG. As we can see, the possible signal o f existence o f mirror neutrinos is finding the n u m b e r o f neutrinos (Nv) to be less than three in the standard fit to the neutrino counting reaction. There is no chance to obtain the masses and mixings o f neutrinos from the reaction e+e - - + ' / + m i s s i n g neutrals only. However, combining this measurement with the other ones it is possible, at least in principle, to distinguish Dirac and M a j o r a n a neutrinos. In practice it is possible only when the mixing angles differ substantially from zero (still not excluded by e x p e r i m e n t ) . A n o t h e r source o f information about the existence of m i r r o r fermions could come for example from the measure-
324
References [ 1] J. Maalampi and M. Roos, University of Helsinki preprint HU-TFT-88-17, Phys. Rep. C, to be published, and references therein. [ 2 ] L. Bento, J.C. Ramao and A. Barroso, Phys. Rev. D 33 (1986) 1488. [ 3 ] A.D. Dolgov, L.B. Okun and V.I. Zakharov, Nucl. Phys. B 41 (1972) 197; E. Ma and J. Okada, Phys. Rev. Lett. 41 (1978) 287; K.J.F. Gaemers, R. Gastmans and F.M. Renard, Phys. Rev. D 19 (1979) 1650; G. Barbiellini, B. Richter and J.L. Siegrist, Phys. Lett. B 106 (1981) 414; H. Czy2, K. Kolodziej and M. Zratek, Phys. Lett. B 207 (1988) 511. [4] B. Kayser, Phys. Rcv. D 26 (1982) 1662. [5] K. Enqvist, K. Mursula and M. Roos, Nucl. Phys. B 226 (1983) 121. [6] S.L. Glashow, Phys. Len. B 187 (1987.) 367.
PHYSICS LETTERS B
I February. 1990
MIRROR FERMIONS AND THE NEUTRINO COUNTING P R O C E S S H. CZYZ 1,2
INFN. Sezione di Bologna, 1-40126 Bologna, Italy Received 13 November 1989
The influenceof the existenceof mirror fermions on the neutrino counting process (e+e - --'7+ missingneutrals ) is investigated. Because of mixing the contribution to the cross section from the generation where both mirror and normal neutrino are present is preferably smaller than the contribution from the normal generation. Besides. at least in principle, it is possible to distinguish Dirac and Majorana neutrinos even in the massless case.
Mirror fcrmions - fcrmions which couple to the W boson by the V + A current - appear naturally in many theoretical models. They were used to cancel a n o m alies and play an essential role in the regularization in lattice theories (for a review of this and many other subjects connected with mirror fermions see ref. [ 1 ] ). However, the question whether they really exist still remains open. There are m a n y ways of looking for them in present and future experiments [ 1 ] but at least one of them has not been studied theoretically yet, i.e. e + e - - . y + m i s s i n g neutrals. I would like to show how the presence of mirror neutrinos influcnces this well known proccss, which will be used for counting the n u m b e r of neutrino Species. The differential cross section has a ve~, simple form in the case we take the model given in rcf. [ 1 ] :
do
G20es(l-x)[(l
d x dy ×
K i = K'li "t- K2i + K 3 i , KI t = /
(a 2v, + v2v,)
=a~i
for Dirac n e u t r i n o s , for Majorana n e u t r i n o s ,
x2i =6)( l _ 4 z ) x / ~ _ n z
[½ ( a v2 i + v 2 ) _ ( 2 a v2 , - v v,2i ) z l
for Dirac n e u t r i n o s , = 6 ) ( 1 - 4 z ) ( 1 - 4z)3/2a~i for Majorana n e u t r i n o s ,
i 2y 2 ] - ~l x ) 2 + ~x
6zrZx( 1 _ y 2 )
K3i =ANir)( 1 - z ) [ 1 - 2 x z (
(a 2 + v 2) Y_,x,
1-z) ]
for Dirac a n d / o r Majorana n e u t r i n o s ,
[1 - s ( l - - x ) I M ~ ] 2 + F U M ~
+ terms coming from W-exchange d i a g r a m s , (1) where Gv is the Fermi constant, a ( M 2) = 1/ 128.5 is t On leave of absence from Department of Theoretical Physics, University of Silesia, PL-40007 Katowice, Poland. 2 Supported in part by Polish Minist~, of Education Grant CPBP 01.03.
322
the fine structure constant, s is the Mandelstam variable, x = 2E~,/~f~, and y = cos 0r, with E r the energy of the proton and 0v the photon angle with respect to the incident beam direction, Mz, Fz arc the mass and the total width of the Z, and ae = - ~ cos 20e, v~= - 21+ 2 sin20w, with 0w the Wcinberg angle, the sum is over the neutrino species, and
with z = r n 2 d s ( 1 - x ) , mN~ the mass of the ith mirror neutrino, O the Heaviside function, and avj = cos 2~,, b'vi=b'Ni= 1, aNi = --COS 2rpi, A N i = s i n 2~0i, where 0,, ~0~are the mixing angles (see ref. [ 1 ] ). The first term in the sum comes from the process e + e - - . 7 + v V (v is the normal n e u t r i n o ) , the second term from the process e + e - ~ 7 + N l q (N is the mirror n e u t r i n o ) and the third term from the processes e+e - -,7+v1'2," and e + e - - , y + N 9 . These processes give thc same signal assuming that the mirror neu-
0370-2693/90/$ 03.50 © ElsevierScience Publishers B.V. ( North-Holland )
Volume 235, number 3,4
PHYSICS LETTERS B
trino lifetime is long enough to decay outside a detector. In may analysis I neglected diagrams with W exchange as they are estimated to give small contributions to the process studied in the investigated beam energy region [2 ]. Besides, I assumed that the normal neutrino is massless (in case it is massive ic,, should be changed into the form o f ~c2~,where the subscript N is changed into the subscript v ). In the numerical calculations I put 0¢ = 0. In the case 0 ~ 0 , the cross section differs from that o f 0 , = 0 roughly by a factor cos220~ (as v~-~0), which decreases the cross section about 20% in the case we take 0~= 13.3 ° from the fit o f ref. [ 1 ]. It is a very, big effect which in principle could be observed in experiments (SLC, L E P ) as a m e a s u r e m e n t o f the n u m b e r o f neutrino species being less than three in the standard fit, when for example only the e + e - - - , 7 + v 9 channel is open. In the case o f no mixing between normal and mirror neutrinos, there is no difference between their contributions to the cross section (cf. ref. [ 3 ] ). So in that case in the reaction e+e - - - , 7 + m i s s i n g neutrals we measure the n u m b e r o f light neutrino species and there is no way to distinguish the m i r r o r neutrino from the normal one even if the a d d i t i o n a l neutrino is massive. However, in the case when the mixing does a p p e a r a lot o f interesting features a p p e a r as well. As we can see from eq. ( 1 ) and the form o f x2~, there exists a contribution to the cross section even in the case ran,> ½Mz as the channels vlq and 9N are open up to m N i .=-M z .
To c o m p a r e contributions to the total cross section from the generation which mixes with m i r r o r leptons with the massless neutrino contribution, I used the ratio R = aT (e + e - ~Yv9 + 7Nlq + yvlCq+ YNg) • o v ( e + e - ~'¢v9) ,
(2)
after integrating over lYl ~<0.94 and taking E y > 2 GeV, R has the form R = 2A + B + 1 + ( - A - 2 B - ½+ C) sin22~o~, for Dirac neutrinos = 2 A + 4 B + 1 + ( - - 2 A - 4 B - - 1 + C ) sin22~0i, for M a j o r a n a neutrinos
(3)
1 February 1990
where A, B, C are functions o f the m i r r o r neutrino mass and b e a m energy (see table 1 ). There are only small differences in A, B, C between v / s = 96 GeV (table 1 ) and n / s = 104 GeV. It is interesting that, in the presence o f the mixing, contributions from Dirac and Majorana neutrinos are different even in the massless (A=½, B = 0 , C = i ) case, quite opposite to the situation where there is no mixing and no way to distinguish between the two kinds o f neutrino even in principle [4]. In the case when the neutrinos are M a j o r a n a neutrinos, the contribution to the cross section from the generation in which both m i r r o r and normal neutrinos are present [wide generation ( W G ) ] could be o f the o r d e r o f the contribution o f one normal neutrino (R = 2-sin22~o,) when there is a big mixing between normal and mirror neutrinos. Such a mixing is still not rejected by experiment ( 0 ~ t p ~ 9 0 ° for the third ( t ) family [ 1,5 ] ). If neutrinos are Dirac and massless the contribution from the W G is R = 2 and does not d e p e n d on the mixing. The second interesting region o f neutrino masses is r n ~ > ½Mz ( A = B = 0 ) . If the mixing does not occur, the contribution from the W G is the same as that
Table 1 The values of the funelions A, B, C for x/s=96 GeV and Mz= 91.9 GeV (see the text ); A = B = 0 for m > Mz. m (GeV)
A
B
C
0 5 l0 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
0.50 0.50 0.49 0.47 0.45 0.42 0.38 0.32 0.24 0.09
0.000 -0.003 -0.011 -0.025 -0.043 - 0.063 -0.081 -0.094 -0.092 -0.045
1.0 1.0 0.99 0.97 0.95 0.92 0.88 0.84 0.79 0.74 0.68 0.62 0.55 0.48 O.4O 0.32 0.23 0.14 0.04 0.00
323
Volume 235, number 3,4
PHYSICS LETTERS B
from the normal generation ( N C ) : However, if there is a mixing R = 1 + ( C - ½) sin22~pi for Dirac n e u t r i n o s , = 1 + ( C - 1 ) sin22q~, for M a j o r a n a n e u t r i n o s .
(4)
As we can scen from table 1 the contribution from thc W G is bigger from the N G only in the mass range ½Mz< mr~ < 63 GeV for Dirac neutrinos and smaller in the whole mass range for M a j o r a n a neutrinos. In the mass range when even the 9N and vlq channels are closed (A = B = C = 0) R = 1 - ½ sin22q~i
1 Februar3' 1990
ments o f total decay widths of W and Z [ 1 ]. We can see that even if we find "\)~-3 with relatively small experimental error it will not mean that m i r r o r neutrinos do not exist as it is easy to fit Nv~-3 choosing a p p r o p r i a t e mixings and masses allowed by experiment. In that case searching for mirror fermions would need further investigations. Unfortunately the signal Nv< 3 is not unique for the investigated model and for example the model investigated in ref. [6] could give the same signature as well. However, one can distinguish between these two models by a detailed analysis o f the gauge boson decay modes and possibly also o f the massive neutrino decay modes which are different in the frameworks of both models (cf. refs. [ 1,6] ). I would like to thank Dr. M. Caffo for drawing my attention to ref. [6] and for carefully r e a d i n g the manuscript.
for Dirac n e u t r i n o s , = cos22{9i for Majorana neutrinos,
(5)
SO that in both cases, if the mixing occurs, the contribution from the W G is smaller than from the NG. As we can see, the possible signal o f existence o f mirror neutrinos is finding the n u m b e r o f neutrinos (Nv) to be less than three in the standard fit to the neutrino counting reaction. There is no chance to obtain the masses and mixings o f neutrinos from the reaction e+e - - + ' / + m i s s i n g neutrals only. However, combining this measurement with the other ones it is possible, at least in principle, to distinguish Dirac and M a j o r a n a neutrinos. In practice it is possible only when the mixing angles differ substantially from zero (still not excluded by e x p e r i m e n t ) . A n o t h e r source o f information about the existence of m i r r o r fermions could come for example from the measure-
324
References [ 1] J. Maalampi and M. Roos, University of Helsinki preprint HU-TFT-88-17, Phys. Rep. C, to be published, and references therein. [ 2 ] L. Bento, J.C. Ramao and A. Barroso, Phys. Rev. D 33 (1986) 1488. [ 3 ] A.D. Dolgov, L.B. Okun and V.I. Zakharov, Nucl. Phys. B 41 (1972) 197; E. Ma and J. Okada, Phys. Rev. Lett. 41 (1978) 287; K.J.F. Gaemers, R. Gastmans and F.M. Renard, Phys. Rev. D 19 (1979) 1650; G. Barbiellini, B. Richter and J.L. Siegrist, Phys. Lett. B 106 (1981) 414; H. Czy2, K. Kolodziej and M. Zratek, Phys. Lett. B 207 (1988) 511. [4] B. Kayser, Phys. Rcv. D 26 (1982) 1662. [5] K. Enqvist, K. Mursula and M. Roos, Nucl. Phys. B 226 (1983) 121. [6] S.L. Glashow, Phys. Len. B 187 (1987.) 367.