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Upper bound on baryogenesis scale from neutrino masses
Volume
246, number
PHYSICS
1, 2
Upper bound on baryogenesis
LETTERS
B
23 August
scale from neutrino
1990
masses
Ann E. Nelson aYb~1~2 and S.M. Barr a,’ ’ Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA b Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA ’ Bartol Research Institute, University of Delaware, Newark, DE 19716, USA Received
4 June 1990
We examine the constraints on baryogenesis if anomalous weak baryon violation is in thermal equilibrium at high temperatures. If neutrinos have Majorana masses, there is an upper bound on the scale of baryogenesis: T,s lo’* GeV( 1 eV/m,)‘, where M, is the mass of the lightest neutrino, and no baryon number is generated at temperatures below T,
It has been argued that anomalous weak baryon and lepton number violating processes are in thermal equilibrium at high temperatures [l-3], and could wash out any baryon number created at high temperatures in the early universe, in disagreement with the observed excess of baryons over anti-baryons in our part of the universe. There are several ways of remedying this catastrophe. The old idea of baryogenesis at very high temperatures in the early universe [4] is still viable, because anomalous processes conserve the difference between baryon and lepton numbers B - L. In fact B - 3e, B - 3~, and B -3~ are all conserved by the anomaly, where e, p, and r are the electron, muon, and tau lepton quantum numbers respectively. If non-zero values are generated at a high temperature T, for B, e, p, or T, and the effective theory below T, is simply the standard model, the baryon number today will be B =&j[ (B - 3e) + (B -3/1)+(B-3r)]. It is also possible that the observed baryon number was made at relatively low temperatures during the weak phase transition [3,5-81 or in some other manner [9]. The basic constraint on baryogenesis is that the effective theory below the baryogenesis scale has to have at least one conserved quantum number which is a linear combination of baryon number and any other U(1) symmetry.
If neutrinos get Majorana masses from the seesaw mechanism [lo], in general lepton number will not be a good symmetry, and it is possible that thermal processes will drive the baryon and lepton numbers to their equilibrium values of zero “. There will be some constraint on the neutrino mass matrix from the condition that at least one combination of B, e, p and r number violating interactions must be out of thermal equilibrium at all temperatures below To. Alternatively an upper bound on the baryogenesis scale T, can be derived if the mass matrix is known. There is no limit on To if one of the weak eigenstate neutrinos is massless, if the neutrino mass matrix conserves a linear combination of e, /* and T numbers, or if the low energy effective theory contains some new quantum number, such as technibaryon number, which is only violated by the weak anomaly. In deriving the upper bound on the baryogenesis scale To we assume the effective theory below T, is just the three family standard model with the additional dimension-five operator L H21Ti a,m,l+ o2 where
m,
h.c.,
is the
neutrino
(1) mass
matrix
at zero
” A different analysis of lepton and baryon number equilibra’ On leave from Stanford University, Stanford, ’ DOE Outstanding Junior Investigator. 0370-2693/90/$
03.50 @ 1990 - Elsevier
CA 94305, USA.
Science
Publishers
tion in the presence of neutrino masses reaches different conclusions [ 111.
B.V. (North-Holland)
has been done, which
141
Volume 246, number 1, 2
PHYSICS LETTERS B
temperature, H is the Higgs field, the I's are the usual left-handed weak leptonic doublets and v is the Higgs v a c u u m expectation value o f 246 GeV. It is convenient to work in a basis where the charged lepton mass matrix is diagonal. We also assume a n o m a l o u s weak a n o m a l o u s baryon violation is in thermal equilibrium up to a temperature --g8kg,1/2mo~ ~ 10 u GeV ( g . is the effective n u m b e r of degrees o f freedom), and require that the b a r y o n to entropy ratio not be driven below a value of ~ 4 × 10 i1, the lowest value compatible with nucleosynthesis [12]. Let m,x be the smallest o f the three diagonal entries in m~. In principle x could be either e, I~ or r. If rex, a n d the off-diagonal entries mx~ are sufficiently small, then B - 3 x violating interactions from (1) may be out of thermal equilibrium below To, and if a non-zero B - 3 x is generated it will not be washed out. Above the weak symmetry restoration temperature, the processes which change x - n u m b e r are lepton annihilation into Higgses, lepton n u m b e r violating l e p t o n Higgs scattering, and a n o m a l o u s weak processes (which conserve B - 3x). If no other quantum numbers are a p p r o x i m a t e l y conserved, the rate of change of y = - ( B - 3 x ) / s (where s is the entropy) is about
d
dtY-~6.0
(2o'~+
)
)~ o-~ T3y.
z,~x
(2)
Here %z is the cross section for x- and z-type charged leptons to annihilate, which can be c o m p u t e d from (1) to be 2
mx.~
(3)
23 August 1990
it is safe to assume that 1
Y0 < - - g.
10-2.
(7)
Below the weak phase transition temperature T o - v it is possible that a n o m a l o u s baryon number violation goes out o f equilibrium, but unless baryons are made during or after the transition we must have y(T~) ~>4 x 10 -~1,
(8)
which when c o m b i n e d with eqs. (7) and (6) yields an u p p e r b o u n d on the baryogenesis scale: - v 4 l n [ 4 x 10 9(10-Z/y0)]
To~< 2 x 1017 GeV (rn~, + l ~ z ~ m~z) ~<1012GeV
(
leV2 2 +1
\mxx
2
)
,
(9)
~ z , ~ x m~z
for To> v. In conclusion, it is conceivable that if neutrino masses are large enough, all the neutrino masses will be measured and determined to be of the lepton n u m b e r violating M a j o r a n a type. If all the diagonal elements of the mass matrix are heavier than 0.1 eV, then baryon n u m b e r generation below the G U T scale is mandatory. This research was s u p p o r t e d in part by the N S F under PHY-89-04035, s u p p l e m e n t e d by funds from the National Aeronautics and Space Administration; A.N. was s u p p o r t e d in part by DOE contract # D E FG03-88ER40472, N S F contract #PHY-88-01426 and by D O E contract # D E - F G 0 3 - 9 0 E R 4 0 5 4 6 .
O[~cz : 8,j3./24"
Eq. (2) may be written as
d
d~Y-~3.5
( 2oi~.~+ ~ o~x) g,l/Zmmy,
References
(4)
z~ x
where T is the temperature / 0 . 3 m p l \ ~/2
This equation may be integrated to yield Y( T)
Yo
exp(0.01, ( 2 m ~ + X~ ""/)4rn~z) rap,( T - To).) ,
(6) where Yo is the value of y at a temperature of To, and 142
[1] N. Christ, Phys. Rev. D 21 (1980) 1591. [2] N. Manton, Phys. Rev. D 28 (1983) 2019; F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212. [3] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36. [4] A.D. Sakharov, JETP Lett. 6 (1967) 24; S. Dimopoulos and L. Susskind, Phys. Rev. D 18 (1978) 4500; M. Yoshimura, Phys. Rev. Len. 41 (1978) 281; D. Toussaint, S. Treiman, F. Wilczek and A. Zee, Phys. Rev. D 19 (1979) 1036; S. Weinberg, Phys. Rev. Lett. 42 (1979) 850; D.V. Nanopoulos and S. Weinberg, Phys. Rev. D 20 (1979) 2484;
Volume 246, number 1, 2
[5]
[6] [7]
[8] [9]
PHYSICS LETTERS B
S.M. Barr, G. Segr~, and H.A. Weldon, Phys. Rev. D 20 (1979) 2494; for an alternative with baryon number violation in equilibrium see A.G. Cohen and D.B. Kaplan, Nucl. Phys. B 308 (1988) 913; Phys. Lett. B 199 (1987) 251. M.E. Shaposhnikov, Nucl. Phys. B 287 (1987) 757; B 299 (1988) 797; A.I. Bochkarev, S.Yu. Khlebnikov and M.E. Shaposhnikov, Nucl. Phys. B 329 (1990) 490. L. McLerran, Phys. Rev. Lett. 62 (1989) 1075. A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Lett. B 245 (1990) 561; preprint NSF-ITP-90-85, UCSD/PTH 90-09, BUHEP-90-15. A. Dannenberg and L.J. Hall, Phys. Lett. B 198 (1987) 411. B. Holdom, Phys. Rev. D 28 (1983) 1419; M. Claudson, L.J. Hall and I. Hinchlitte, Nucl. Phys. B 241 (1984) 309; S. Dimopoulos and L.J. Hall, Phys. Lett. B 196 (1987) 135.
23 August 1990
[10] T. Yanagida, Prog. Theor. Phys. B 135 (1978) 66; M. Gell-Mann et al., in: Supergravity, eds. P. van Nieuwenhuizen and D. Freedman (North-Holland, Amsterdam, 1979). [11] M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45; Kyoto University preprint RIFP-847 (~1990); J.A. Harvey and M. Turner, preprint Fermilab-PUB90/49-A, EFI 90-33 (1990). [12] E. Kolb and M. Turner, The early universe (AddisonWesley, Reading, MA, 1990), and references therein; K.A. Olive, preprint UMN-TH-819/90 (1990).
143
246, number
PHYSICS
1, 2
Upper bound on baryogenesis
LETTERS
B
23 August
scale from neutrino
1990
masses
Ann E. Nelson aYb~1~2 and S.M. Barr a,’ ’ Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA b Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA ’ Bartol Research Institute, University of Delaware, Newark, DE 19716, USA Received
4 June 1990
We examine the constraints on baryogenesis if anomalous weak baryon violation is in thermal equilibrium at high temperatures. If neutrinos have Majorana masses, there is an upper bound on the scale of baryogenesis: T,s lo’* GeV( 1 eV/m,)‘, where M, is the mass of the lightest neutrino, and no baryon number is generated at temperatures below T,
It has been argued that anomalous weak baryon and lepton number violating processes are in thermal equilibrium at high temperatures [l-3], and could wash out any baryon number created at high temperatures in the early universe, in disagreement with the observed excess of baryons over anti-baryons in our part of the universe. There are several ways of remedying this catastrophe. The old idea of baryogenesis at very high temperatures in the early universe [4] is still viable, because anomalous processes conserve the difference between baryon and lepton numbers B - L. In fact B - 3e, B - 3~, and B -3~ are all conserved by the anomaly, where e, p, and r are the electron, muon, and tau lepton quantum numbers respectively. If non-zero values are generated at a high temperature T, for B, e, p, or T, and the effective theory below T, is simply the standard model, the baryon number today will be B =&j[ (B - 3e) + (B -3/1)+(B-3r)]. It is also possible that the observed baryon number was made at relatively low temperatures during the weak phase transition [3,5-81 or in some other manner [9]. The basic constraint on baryogenesis is that the effective theory below the baryogenesis scale has to have at least one conserved quantum number which is a linear combination of baryon number and any other U(1) symmetry.
If neutrinos get Majorana masses from the seesaw mechanism [lo], in general lepton number will not be a good symmetry, and it is possible that thermal processes will drive the baryon and lepton numbers to their equilibrium values of zero “. There will be some constraint on the neutrino mass matrix from the condition that at least one combination of B, e, p and r number violating interactions must be out of thermal equilibrium at all temperatures below To. Alternatively an upper bound on the baryogenesis scale T, can be derived if the mass matrix is known. There is no limit on To if one of the weak eigenstate neutrinos is massless, if the neutrino mass matrix conserves a linear combination of e, /* and T numbers, or if the low energy effective theory contains some new quantum number, such as technibaryon number, which is only violated by the weak anomaly. In deriving the upper bound on the baryogenesis scale To we assume the effective theory below T, is just the three family standard model with the additional dimension-five operator L H21Ti a,m,l+ o2 where
m,
h.c.,
is the
neutrino
(1) mass
matrix
at zero
” A different analysis of lepton and baryon number equilibra’ On leave from Stanford University, Stanford, ’ DOE Outstanding Junior Investigator. 0370-2693/90/$
03.50 @ 1990 - Elsevier
CA 94305, USA.
Science
Publishers
tion in the presence of neutrino masses reaches different conclusions [ 111.
B.V. (North-Holland)
has been done, which
141
Volume 246, number 1, 2
PHYSICS LETTERS B
temperature, H is the Higgs field, the I's are the usual left-handed weak leptonic doublets and v is the Higgs v a c u u m expectation value o f 246 GeV. It is convenient to work in a basis where the charged lepton mass matrix is diagonal. We also assume a n o m a l o u s weak a n o m a l o u s baryon violation is in thermal equilibrium up to a temperature --g8kg,1/2mo~ ~ 10 u GeV ( g . is the effective n u m b e r of degrees o f freedom), and require that the b a r y o n to entropy ratio not be driven below a value of ~ 4 × 10 i1, the lowest value compatible with nucleosynthesis [12]. Let m,x be the smallest o f the three diagonal entries in m~. In principle x could be either e, I~ or r. If rex, a n d the off-diagonal entries mx~ are sufficiently small, then B - 3 x violating interactions from (1) may be out of thermal equilibrium below To, and if a non-zero B - 3 x is generated it will not be washed out. Above the weak symmetry restoration temperature, the processes which change x - n u m b e r are lepton annihilation into Higgses, lepton n u m b e r violating l e p t o n Higgs scattering, and a n o m a l o u s weak processes (which conserve B - 3x). If no other quantum numbers are a p p r o x i m a t e l y conserved, the rate of change of y = - ( B - 3 x ) / s (where s is the entropy) is about
d
dtY-~6.0
(2o'~+
)
)~ o-~ T3y.
z,~x
(2)
Here %z is the cross section for x- and z-type charged leptons to annihilate, which can be c o m p u t e d from (1) to be 2
mx.~
(3)
23 August 1990
it is safe to assume that 1
Y0 < - - g.
10-2.
(7)
Below the weak phase transition temperature T o - v it is possible that a n o m a l o u s baryon number violation goes out o f equilibrium, but unless baryons are made during or after the transition we must have y(T~) ~>4 x 10 -~1,
(8)
which when c o m b i n e d with eqs. (7) and (6) yields an u p p e r b o u n d on the baryogenesis scale: - v 4 l n [ 4 x 10 9(10-Z/y0)]
To~< 2 x 1017 GeV (rn~, + l ~ z ~ m~z) ~<1012GeV
(
leV2 2 +1
\mxx
2
)
,
(9)
~ z , ~ x m~z
for To> v. In conclusion, it is conceivable that if neutrino masses are large enough, all the neutrino masses will be measured and determined to be of the lepton n u m b e r violating M a j o r a n a type. If all the diagonal elements of the mass matrix are heavier than 0.1 eV, then baryon n u m b e r generation below the G U T scale is mandatory. This research was s u p p o r t e d in part by the N S F under PHY-89-04035, s u p p l e m e n t e d by funds from the National Aeronautics and Space Administration; A.N. was s u p p o r t e d in part by DOE contract # D E FG03-88ER40472, N S F contract #PHY-88-01426 and by D O E contract # D E - F G 0 3 - 9 0 E R 4 0 5 4 6 .
O[~cz : 8,j3./24"
Eq. (2) may be written as
d
d~Y-~3.5
( 2oi~.~+ ~ o~x) g,l/Zmmy,
References
(4)
z~ x
where T is the temperature / 0 . 3 m p l \ ~/2
This equation may be integrated to yield Y( T)
Yo
exp(0.01, ( 2 m ~ + X~ ""/)4rn~z) rap,( T - To).) ,
(6) where Yo is the value of y at a temperature of To, and 142
[1] N. Christ, Phys. Rev. D 21 (1980) 1591. [2] N. Manton, Phys. Rev. D 28 (1983) 2019; F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212. [3] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36. [4] A.D. Sakharov, JETP Lett. 6 (1967) 24; S. Dimopoulos and L. Susskind, Phys. Rev. D 18 (1978) 4500; M. Yoshimura, Phys. Rev. Len. 41 (1978) 281; D. Toussaint, S. Treiman, F. Wilczek and A. Zee, Phys. Rev. D 19 (1979) 1036; S. Weinberg, Phys. Rev. Lett. 42 (1979) 850; D.V. Nanopoulos and S. Weinberg, Phys. Rev. D 20 (1979) 2484;
Volume 246, number 1, 2
[5]
[6] [7]
[8] [9]
PHYSICS LETTERS B
S.M. Barr, G. Segr~, and H.A. Weldon, Phys. Rev. D 20 (1979) 2494; for an alternative with baryon number violation in equilibrium see A.G. Cohen and D.B. Kaplan, Nucl. Phys. B 308 (1988) 913; Phys. Lett. B 199 (1987) 251. M.E. Shaposhnikov, Nucl. Phys. B 287 (1987) 757; B 299 (1988) 797; A.I. Bochkarev, S.Yu. Khlebnikov and M.E. Shaposhnikov, Nucl. Phys. B 329 (1990) 490. L. McLerran, Phys. Rev. Lett. 62 (1989) 1075. A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Lett. B 245 (1990) 561; preprint NSF-ITP-90-85, UCSD/PTH 90-09, BUHEP-90-15. A. Dannenberg and L.J. Hall, Phys. Lett. B 198 (1987) 411. B. Holdom, Phys. Rev. D 28 (1983) 1419; M. Claudson, L.J. Hall and I. Hinchlitte, Nucl. Phys. B 241 (1984) 309; S. Dimopoulos and L.J. Hall, Phys. Lett. B 196 (1987) 135.
23 August 1990
[10] T. Yanagida, Prog. Theor. Phys. B 135 (1978) 66; M. Gell-Mann et al., in: Supergravity, eds. P. van Nieuwenhuizen and D. Freedman (North-Holland, Amsterdam, 1979). [11] M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45; Kyoto University preprint RIFP-847 (~1990); J.A. Harvey and M. Turner, preprint Fermilab-PUB90/49-A, EFI 90-33 (1990). [12] E. Kolb and M. Turner, The early universe (AddisonWesley, Reading, MA, 1990), and references therein; K.A. Olive, preprint UMN-TH-819/90 (1990).
143