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Flipped angles and phases: a systematic study
Physics Letters B 308 (1993) 70-78 North-Holland
PHYSICS LETTERS B
Flipped angles and phases: a systematic study John Ellis Theorettcal Phystcs Dlvtslon, CERN, CH-1211 Geneva 23, Swttzerland
Jorge L. Lopez, D.V. Nanopoulos Center for Theoretwal Physics, Department of Phystcs, Texas A&M Umverslty, College Statlon, TX 77843-4242, USA Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77382, USA
and Keith A. Olive School of Phystcs and Astronomy, Umverslty of Minnesota, Mmneapohs, MN 55455, USA Received 31 March 1993 E&tor: R. Gatto
We discuss systematically the fermion mass and mixing matrices m a generic field-theoreUcal flipped SU (5) model, with particular applications to neutrino and baryon number-changing physics. We demonstrate that the different quark flavour branching ratios in proton decay are related to the Cabibbo-Kobayashl-Maskawa angles, whereas the lepton flavour branching ratms are undetermined. The hght neutrino mixing angles observable via oscillation effects are related to the heavy conjugate (right-handed) neutrino mass matrix, which also plays a key role m cosmological baryogenesls. The ratios of neutrino and charged lepton decay modes in baryon decay may also be related to neutrino oscillation parameters. Plausible Ansatze for the generation structure of couphng matrices motivate additional relations between physical observables, and yield a satisfactory baryon asymmetry.
1. Introduction One o f the most welcoming avenues leading beyond the Standard Model is that leading to G r a n d Unified Theories ( G U T s ) . By unifying the three known partlcle gauge interactions within a simple group G, one may understand why baryon n u m b e r is conserved to a good approximation, but not perfectly, and why neutrino masses may be very small, but non-zero. In so doing, G U T s also offer scenaria for cosmological baryogenesis and the nature o f hot dark matter. Interest in G U T s has been further whetted by the close consistency o f the measured values of the SU (3), SU (2) and U ( 1 ) gauge couplings with m i n i m a l supersymmetric G U T s [ 1 ], which high-precision LEP data have rendered even more impressive [2]. W i t h all these phenomenologlcal motivations, it 70
was natural that string model-builders should seek to emulate GUTs. However, it was soon realized that there was a sizeable roadblock to deriving a G U T from string: gauge symmetry breaking and various other phenomenological constraints in G U T s generally require adjoint or larger Higgs representations, and these are not obtainable using conventional modelbuilding technology based on k = 1 K a c - M o o d y currents on the world-sheet [3]. Hence the revived interest m an SU (5) × U ( 1 ) G U T , flipped SU (5) [4,5], which only required 5- and 10-dimensional Higgs fields and could be derived from string. F l i p p e d S U ( 5 ) is the closest homage string can pay to the simple G U T s o f old. Naturally, there has been considerable discussion within flipped S U ( 5 ) o f the "classic" new phenomena that m o t i v a t e d so much work on GUTs, namely
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B V. All rights reserved.
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baryon decay [6] and neutrino masses [5,7-11]. However, there has not been a systematic investigation of all the fermxon mass matrices and m~xing angles that enter into baryon decay branching ratios and neutrino oscillations, and their possible relations to the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix and cosmological baryogenesis. In th~s paper we seek to remedy this lack m the hterature, working in the general framework of mimmal fieldtheoretical flipped SU(5), supplemented at the end by some (to us) plausible general hypotheses about mass matrices. It may serve a useful purpose to recall first some important features of minimal field-theoretical flipped SU (5) [ 5 ]. The first is that baryon decay via dimension-5 operators is very strongly suppressed, and also dimension-6 Higgs exchanges are presumably neghgible compared to the dimension-6 massive vector boson exchange operators expected to dominate. In minimal SU (5), it was possible to relate the mixing angles in the corresponding dimension-6 operators for proton decay to the CKM angles, modulo two additional complex phases that would be difficult to measure [12]. The relation ~s not so direct in flipped SU (5), because the flipping of the particle assignments within 5 and 10 representations, as well as the assignment of conjugate charged leptons to singlet representations, means the branching ratios into different lepton species are independent of the quark mixing angles. It has recently been reahzed, on the other hand, that the dominant mechanism for baryogenesls in flipped SU(5) may be the decay of massive conjugate ("right-handed") neutrinos [ 13 ], producing a lepton asymmetry which is subsequently reprocessed [14] into a baryon asymmetry by nonperturbative electroweak interactions [ 15 ]. This raises the possibdity that the cosmological baryon asymmetry might be related to observable parameters in the light neutrino mass and mixing matrices, which could also show up m baryon decay. In this paper, we investigate these possibilities systematically, starting from an adaptation in section 2 of the analysis of ref. [ 12 ] to diagonalize the fermion mass matrices and determine the independent umtary flavour rotation matrices. Next, in section 3 we apply these results to derive interrelations between observables in conventional weak decays, proton decay, neutrino oscillations and cosmological baryogenesis.
24 June 1993
Then, in section 4 we postulate plausible Ansatze for the fermion mass matrices which lead to certain additional quantitative relations between the different flavour rotatxon mamces and hence observable quantrees. Finally, m section 5 we summarize our conclusions and mention directions for future study.
2. Flipped mixing matrices Let us first remmd the reader of the various couphng matrices that appear in the minimal field-theoretical version of the fhpped SU(5) model. The superpotentlal that characterizes its Yukawa couphngs is [5]
w = 3`~F, Fjh + 3`TF,f j h + ,l t33 f , l j Ch + 24HHh + 2 5 H H h + 2Z6aFzH$a + 27hh~b0 + ].labCa~b ,
(1)
where the F,, f , , l~ (t = 1, 2, 3) are the three generations of 10, g and singlet representations of SU(5) that comprise the light matter particles of the Standard Model, H and H are 10 and 10 Higgs representations, h and h are 5 and 5 Higgs representations, and the $0, ~ba (a = 1,2, 3) are auxiliary singlet fields. The first three terms in the superpotential ( 1 ) give masses to the charge 2/3 quarks u,, charge - 1 / 3 quarks d, and charged leptons l, respectively, the next two terms spht the light Hxggs doublets from their heavy colour triplet partners in a natural way, the sixth term provides a large element in the see-saw neutrino mass matrix, the product 3.7 (~b0) gives the traditional Higgs mixing parameter, and the last term is the auxiliary slnglet mass matrix [7]. Our task m this section is to diagonalize the couphng matrices 3.1,2,3,6 and the corresponding mass matrices mu,d,t, identifying in the process the needed unitary rotation matrices whose observability we will explore in section 3. Our analysis of the dmgonalizataons is modelled on that made for minimal SU (5) in ref. [ 12 ]. We start by diagonalizlng the 3.2 coupling matrix wxth the unitary transformations
f ' = -fU*uc,
F' = U*uF,
(e)
which yield 71
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f22F
=
PHYSICS LETTERS B
-f'Uu~22UuF'
f'2fF'
=
which we now diagonahze by the new transformations
:
,~ = u..~2u..
(3)
2'~ = U~2~Uu.
(4)
We now diagonalize this by a further unitary transformation on the fields F ' :
=
U4F':
= uT~DIu4.
2',
(5)
As in ref. [ 12 ], it is convenient to separate phase factors in the matrix U4: denoting elements o f this matrix by e'%u,j where q,s and u,j are both real, we decompose
U4 = U5UU6 :
(6)
U7 = U 2,
=-f'5'UT :
III]Dlcl
5"~3 "
-f'u~,
F"
=
U6F',
(8)
leaving the diagonahzed matrix 22° unchanged. In this basis, the first term in eq. ( 1 ) becomes F
ttT
T
D
U U72~UF
tt
.
:
,~D = U i U 6 U u c 2 3 U l c
Cr#q~ = ¢ , r # ~ ,
(0'= U~O,
(13)
:
I t° = Uf#U~,,
FT,~6~ = F"T2~6¢':
(7)
and we impose a phase convention, det U7 = 1. We can then absorb the U6 phases into the fields =
(12)
for the dlagonallzed mass matrix. A further novel feature beyond the analysis of ref. [ 12] is the dlagonalization of the light and heavy neutrino mass matrices. We start by diagonahzmg the ~b mass matrix:
(F~ +.
--f"
~ £
l c' = Utcl ,
(14)
2~6 = U~U#26U(~.
(15)
The u ¢ = F4s mass matrix is of the see-saw form in the ( F ~ , ~b') basis, and so can be written in the form
This leads to the representation
2', = u 6 u T u 7 2 D u u 6 :
f5
which leads to the representation
U5 = diag(e~na),
U6 = diag(e 'm~ ) e -'~1~ •
--tit
leading to
As a result, the 2~ coupling term becomes
Fr21F = F'T2'IF':
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(9)
The charge - 1 / 3 quark mass matrix may finally be cast in real and dmgonal form by the redefinitions
~T
t ~rtl
) m~c tF45 +
):
m',c = 2~T(#°)-~2~ (V) 2,
(16)
where (V) = (10). We recall that the transformations (10) applied to the coloured components o f F in order to dlagonahze the quark mass matrices have not been applied to the u c = F45 components. We now define F~'
rrt t~uc~w,, 45,
(17)
in terms of which the u c mass matrix is diagonal: F4ttt T
D tit 5 rnucF~5 "
F'~; = UF'~
for 1 ~ . , p ~< 4,
F2Vp = U7F')~
for 1 ~< ~,fl ~< 3,
muD
= U ~T m ~tU ~
= U~2~T(#°)-12~ (V) 2 U.~
(10)
of relevant components o f the 10 representations F . In contrast to the conventional S U ( 5 ) case discussed in ref. [ 12 ], the charged lepton mass matrix is not directly related to the quark mass matrices. Starting from the third term in the superpotential ( 1 ) and making the above transformations of eqs. (2), (8) of the 5 fields f , we find
The light neutrino mass matrix is also of the see-saw form, this time in the (u = f 4 , F4s) basis. We diagonahze it in terms of the diagonalized u c masses rn~°~ ( 18 ). In terms of the unitary transformations already made:
7;t3l ~ = 7 " ( U6 u~¢z3)l ~ ,
74,,~ 2F45 = J"Utt~D ...... 4 A 2 t~'ucr45 ,
72
( 11 )
X (uguT~6u~)
('~')2 U v c .
(18)
(19)
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yielding
4, , m u J74, , r :
~ - ' U~c22 7" ~ , m , = ~ U ~ c (m~c)
(20)
which we must dlagonalize by the further transformation 7~"
-7,,,,*.
m ~ = U,[AzD U~,~(m,,c) D -1 T D U/,~22 ]U~"~
(21)
for the light neutrino mass eigenstates. After this profusion (confusion?) of dmgonalizations and unitary matrices, we take mercy on the reader by summarizing the final mass e~genstates (represented by suffices F ) :
UF = (U6U2)FI<~a~3,5, t * u~e = -f- l<.~,<.3(U'~U~ ),
(22)
(23)
l-Y = lsUMsw-UF.
(26)
Given the previous results for the mass elgenstates 1r (24) and us (25) we find quite simply UMSW = UlU~.
(27)
-~d = -UF UcKMdF .
(28)
Using now the expressions (22) and (23) for the mass eigenstates Us and ds we can write
ls = 7~ (u'.~sl * *u;),
leF = Ut~l ,c ,
In parhcular, we wdl be interested in identifying the mixing matrix for the MSW mixing and the origin of the necessary CP-violatlng phase for the production of the cosmic lepton asymmetry. We will also identify the Cablbbo-Kobayashi-Maskawa (CKM) matrix and its role, as well as those of other matrices of interest in proton decay processes. We begin with the MSW mixing matrix. MSW solar neutrino mixing will result if there is a mismatch between the neutrino states produced in the charged current interactions and the neutrino mass eigenstates. Therefore, we can define the MSW mixing matrix by
Similarly, the CKM matrix is the charged current mismatch in the quark sector,
dg = (UU6U2)FI<~a<.g.3,4, d~. = (U7UU6U2)FI
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(24) UCWM= U ~ •
//F = 7 4 ( u ; t~ v ; *u 2 ) ,
b'~r = (UlutcV682 )f45.
(25)
3. Specific processes As we mentioned earlier, among the motivations for Grand Unified Theories is the possibility for baryogenesls and small neutrino masses. One of the remarkable features of the flipped SU (5) model is its ability to provide cosmologically-interesting neutrino masses (i.e., for u~), while at the same time allowing for observable Mikheyev-Smirnov-Wolfensteln (MSW) [16] neutrino oscillations and baryogenesis via leptogenesis and subsequent sphaleron reprocessing. All three of these highly desirable features are related to the s a m e neutrino mass matrix [ 10,13 ]. Given the complete decomposition of mass eigenstates in the flipped SU (5) model above, we are now in a position to examine in detail the interrelationships between the various mixing matrices and phases of interest.
(29)
A priori, we see no relationship between the quark and neutrino mixing matrices. Next we study the effective Lagranglan for baryon decay in the flipped SU(5) model, which has been discussed previously in ref. [ 171. As has already been recalled, the dominant mechanism for baryon decay is expected to be dimension-6 vector boson exchange. Using the analysis of the previous section, as was done for conventional SU(5) in ref. [12], we find the following massive vector boson couplings: Cx = ( g / v ~ ) X ? .
[e'Jk--dckFUTy"PLdj~
+-U,eTUPRv c] + h.c., £ r = ( g / x / 2 ) Y, - u[e ' s kd~ k v U v U y PLUj~
+-ff,r TUPRl c] + h.c.,
(30)
where we have worked in the mass eigenstate basis for the quarks, denoted by subscripts F, the lepton fields are in the f " basis of the previous section, the colour 73
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indices (t, j, k ) are noted explicitly, and we have indicated the handednesses of all fermion fields by using the projection operators PL and P~. The exchanges of X and Y bosons (assumed as usual to have indistmgmshable masses) therefore give rise to the following effective Lagrangian for baryon decay:
£.~a~O = (g2/2M2)
F (p --+ e + n o ) = ½cos 20c{ Ut,, 121" (P --* -flit + ) = COS20c I Oil I [2F (n --~ -flito),
F ( n ~ e+n - ) = 21"(p --+ e+it°), F ( n ~ i~+n - ) = 2F(p ~ #+ito), 1"(p ~ ~+ n °) = ½cos20clUl,~I2F(p -~ -fin + )
x [(¢'Jkd~FU7yUPtdjF)(ur~yuPtv) + h.c. t~,yk-2c U7U~gPLUjF)(uTF~uPLI) + h.c.]
(35)
= COS20clUllzlz1"(n ---, -flit0) .
(31)
As in the case of conventional SU (5), we see from this expression that two additional CP-vlolatmg phases appear in the quark parts of the AB # 0 operators [i.e., from U7 = Us2 in eq. (6) and detU7 = 1], beyond the single phase in the CKM matrix: these are in principle measurable via loop diagrams. Since the only quarks of relevance for baryon decay at the tree level are u, d and s, and not more than one of the latter, we can write the relevant parts of the AB # 0 Lagranglan (above) as
£,~a~o = (g2/2M~) yUpLdj )(Ur, yuPLU~) + h.c.
Thus it is possible m principle to correlate baryon decay branching ratios with the CKM angles, and to measure elements of the charged-lepton mixing matrix Ut, though not o f the neutrino mixing matrix U,. Moreover, we note that the decay branching ratios above are characteristically different from those m conventional SU (5) [ 17 ]. However, it lies beyond the scope o f this paper to calculate the total baryon decay rate in flipped SU(5). We now examine the mixing matrix and the necessary CP-violating phase which can provide for a net lepton asymmetry, and subsequently a net baryon asymmetry via sphaleron reprocessmg [ 14]. As described m ref. [ 13 ], the lepton asymmetry is produced by the out-of-equihbrium decay of the mass e~genstate u~ = F~' via the mass term f4.s22F45. In terms of mass eigenstates the neutrino mass term becomes
+ [e 'Jk (d~,e 2'"~' cos 0~ + ~ke 2'n21 sin Oc)YUPLUj]
--tit D tit f4,5 Uv,122 UvcF45 •
× (Ur~yuPLll) + h.c.]
It is then natural to introduce the lepton-numberviolating couphng
(32)
where we recall that VL and lL are not in the mass eigenstate basis, so that
UL = ueU~,
IL
= IFUI.
(33)
The phase factors in (32) are not measurable, nor can one distinguish between the different neutrino flavours, so the following are the predictions that can be made on the basis of (32):
F ( B ~ l + + X[strange) = tan z O~ ~ 1 F (B -~ l + + XJnon-strange) 2-0"
(36)
D 2L ~ Uv,I22 Uuc.
(37)
We expect the dominant contribution to the lepton (baryon) asymmetry to be that due to decays of the lightest u ~ mass eigenstates, which we expect to be that associated with ue.u and call v~,2. The CP asymmetry in the decay of u~ into second and third generation particles is g~ven by 1
F ( B --, (Xu) + Xlstrange) = 0, F ( B ~ ( X v ) + X[non-strange)
~J(34)
Comparing decays to neutrinos and charged leptons requires knowledge of specific hadronic matrix elements. Following ref. [ 17 ], we find 74
24 June 1993
2n(2~2L)11
x E
(Im[ (2*z2L)O]2) g(ME/M~) ,
(38)
J
where
g ( x ) = 4x/-xln l + X x
( ~- 4 for x >> 1) ~
'
(39)
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where (38) is the supersymmetrlc expression for the C P asymmetry [18]. Analogous expressions can be written for the corresponding asymmetries e2.3 in the decays ofu~. 3. Using the definition (31 ) of 2L we find
where 0e,u,#r a r e the usual MSW mixing angles. It ts interesting to note that the baryon decay branching fractions in eq. (35) depend in this approximation on UII l
/~fL,~L = u'ct ( &D) 2 u.~.
(40)
This depends only on U,~ and, what is more, the C P violating phase we are interested in is a priori unrelated to the CKM phase or MSW mixing. In general, we could expect (2tL2L) t t " (2~33) 2 (the largest entry in 22O ), and if it were the case that Ml ,.~ ME << M3 we would estimate f ~
21n2 ~D 7~
2X
(41)
~'233 ~ ,
where ~ is the phase associated with the imaginary part of (2tL2L)12 in (40). (This is slightly larger than our previous esumate [13].) A satisfactory baryon asymmetry would result for ~ > l0 -2. In the next section we will revise this generic estimate in the l i g h t of a specific Ansatz for neutrino masses.
1,
=
U62 = 012,
(45)
giving F(p
~ e + n °) = i cos z G F (p --, -fin + )
= COS2 G F ( n
(46)
---, -fin°).
Note that our general Ansatze about mixing angles indicate that the baryon decay branching fractions into muons should be suppressed relative to those into electrons. Also, R. remains undetermined this way since only the sum over all branching fractions into neutrino final states can be observed. On the other hand, eqs. (20), (21) allow us to obtain U. once U.c and m.~ are given. Since the latter two appear in the baryogenesis parameter e (eqs. 38, 40), it is interesting to correlate all these parameters. Starting from eq. (21) and writing U. = I + R . and U,c = 1 + R,c we obtain m,, = (22°), (m,,c), o -1 (22D),,
4. Phenomenological Ansfitze In this section we propose some plausible forms for the various matrices appearing above, and obtain correlations among the various observables of interest. Clearly, the unitary matrices U are not expected to be equal to the identity matrix. However, experience with the CKM and MSW mixing matrices indicates that they probably should not differ too much from unity either. We therefore write (42)
U, = 1 + R , ,
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(47)
which leads to the previously advocated phenomenologically interesting neutrino mass ratios [ 10 ]. Moreover, from (rn.),j D = 0 for i # J we find R. in terms of R.~, (R~),l _
-1
(2f), (2f)j
rn.j - m . ,
x
( R . ~ ) , : ~ j - (RT.~),j
,
(48)
where M, = (m.~),,. In particular
for all matrices U, defined above, with R, =
0 -0;~ o
0h 0
0 ) 0~3 , o
O ~ ( z - m,, u
(43)
such that U, U,~ = I through order 0, since R~ = - R , . By analogy with the C K M matrix, we have neglected the far off-diagonal entries in the matrices R,. For the MSW matrix UMsw = UtUt, we have RMsw = R t + R~ = R t - R~, and o,. =
- 01%
= oh-
(44)
m,e
).u ~.~,cM2
(49)
The last expresmon holds in the limit Ma >> MI. Also,
--J.c2t
O~3 -- m , , - m , u
2c ~,~c M3
.c 1 023
M33
_
_.uc 1 ] 023 M22J
(50)
with the final expression valid in the hmit 343 >> ME. 75
As is plausible for any matrix with a hierarchy of eigenvalues, such as a see-saw mass matrix, we make the Ansatz //¢
o7~ _~ x/-M~M2,
023 -~ V/--~2/M3,
(51 )
which gives
012 ~ (2~/2c) V/-~z/M,,
0~3 ~- (~/,~,) v/-M[3/M2.
(52)
If for the moment we neglect t h e 0~2,23 contributions to the neutrino mixing angles [see eq. (44) ], the prediction for Ou~ in eq. (52) is exactly what was proposed in ref. [10] on purely phenomenologlcal grounds. This relation (with M3/M2 ~ 10) and the ratios of neutrino masses in eq. (47) have been shown [ 10] to lead to Interestingly observable uu-u~ oscdlatlons and a z neutrino mass large enough to provide an interesting amount of astrophysical hot dark matter. The other relation in eq. (52) should reproduce the present fits to the solar neutrino data based on the MSW mechanism. These require Oeu = (3.2-6.1) × 10 -2 [19] and therefore M2/MI = 64-225. (Note: the present expression for 0eu differs from that in ref. [ 10 ].) Turning now to the baryogenesls parameter c, from eq. (40) we have (to order 0)
+(R~).
2:2M1 x ~, = ~ c ~ , 2 ,
),j6.
[(,~),~ -- (2 o2 ) J2] -
Thus lm t2t t L2 L )~2 tt
~
(53)
I m t~.2 tL2 L /~2 ' 3 ~--" 0, and
t 2 ~_ ~4 .,iv e 2COS2¢12, Im (2L2L),2 .~c ut2
am (2~;tL)~3 ~- 2410~12 c o s 2¢23,
where
76
(57)
n L l ( m ~ )
10-7-~32 d23
'/2
10-8d23,
(58)
~2 ~qvc 2
c t-'12 ,
a21ovcl2 (~flL)].L)22 .,., ~2 + ~tlt"23t ,
We finally get
623 = COS2¢23.
For the decay of the first and lightest generation v[, this result is a factor of (2c/2t) 2 (M,/In 2M2) ,,~ 10 -6 smaller than that obtained in the generic case in eq. (41) above. However, we see that for the decay of the second generation v~ the result is only suppressed by (M2/ln 2343) ~ 7, and is therefore large enough a priori to produce a satisfactory lepton, and hence baryon, asymmetry. Normally, it is the llghtest generation which produces a net asymmetry [20]. This is because the lightest generation will typically wipe out any prior asymmetries (e.g., via inverse decays) before producing its own. If we considered a standard out-of-equllibrmm decay mechanism, we would have to require that both first- and second-generation S ' s be far out-ofequlhbrlum at the time of their decay. This would impose the r e s t r i c t i o n s (,~L/~L)I1,22 ~,~ 103M,,E/Mv, conditions which would be difficult to satisfy given our Ansatze. However, invoking a mechanism proposed in ref. [21 ] we assume here that the v ° s are produced in the decay of the inflaton, r/, subsequent to inflation. Then u c decays occur immediately and out-of-equilibrium at T << M,,, and no destruction of an asymmetry can occur. Thus our only requirement IS that MI,2 < m e. This is satisfied for m, ~ few x 101'GeV as inferred from the COBE result [22] on density fluctuations [ 18,13 ]. Our final lepton and hence baryon asymmetry now becomes nB
(,~;tL)33 ~- ,~2.
(56)
(54)
where Cu = arg(0ff). Analogously, the denominator factors are given by (/~L~L)ll .,., ~2 +
2,2M2x ~2 = ~ , ~ 2 3 ,
4~2M2 e3 = ~.~t ~33 In M~ [0~12d23,
6, 2 = COS2¢,2,
=
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(55)
where A ~ 10 -3 iS an entropy dilution factor which accounts for the entropy produced during breaking of SU (5) × U ( 1 ) [2 3,13 ] This IS encouragingly large: recall the phase 623 is not related to the standard (CKM) source of CP violation, and is not expected to be partIcularly suppressed. Note that there could be extra
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sources o f entropy which would further suppress the estimate (58), so c~23 may not need to be very small.
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in part by DOE grant DE-AC02-83ER-40105, and by a Presidential Young Investigator Award. J.L.L. would like to thank the CERN Theory Division for its kind hospitality while part o f this work was carried out.
5. Summary and outlook We have made in this paper a systematic study o f the mass matrices and mixing angles in the m i n i m a l field-theoretical version o f flipped SU (5). We have identified the mixing angles that appear in observable processes such as neutrino oscillations and proton decay, as well as the C K M angles in the charged electroweak interactions, and related them as far as possible in a independent way. We have also discussed a scenario for producing the baryon a s y m m e t r y w a the out-of-equilibrium decays o f heavy singlet conjugate neutrinos, which produce a lepton asymmetry that is subsequently reprocessed into a baryon asymmetry by sphaleron transitions. Additional plausible Ansatze for the mixing angles lead to a n u m b e r of further relations, and a satisfactory baryon asymmetry. We were m o t i v a t e d to study flipped SU (5) because it is the only G U T that can be derived from string theory, and it was precisely the baryon- and leptonnumber-violating processes discussed above that motivated the derivation o f a G U T from string. However, the m i n i m a l field-theoretical flipped SU(5 ) model analyzed above is not what one obtains from string theory. On the one hand, flipped SU (5) models derived from string contain more states, thus complicating the analysis, but on the other hand they can cast light on plausible forms for the mass matrices and mixing angles. Therefore it would be desirable to complement this general analysis with some more modeldependent studies. We believe that flipped S U ( 5 ) can give us m a n y insights into the exciting new era o f massive neutrino physics that solar neutrino experiments and models o f the formation o f large-scale astrophysical structures suggest m a y be opening up before us.
Acknowledgement The work o f J.L.L. has been supported by an SSC Fellowship. The work of D.V.N. was supported in part by DOE grant DE-FG05-91-ER-40633 and by a grant from Conoco Inc. The work o f K.A.O. was supported
References [1] U. Amaldi et al., Phys. Rev. D 36 (1987) 85, G. Costa et al, Nucl. Phys. B 297 (1988) 244. [2] J Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 249 (1991) 441 [3] See e g. J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375, A.M. Font, L.E. Ibdfiez and F. Quevedo, Nucl. Phys. B 345 (1990) 389. [4]S. Barr, Phys. Lett. B 112 (1982) 219, Phys. Rev D 40 (1989) 2457, J. Derendlnger, J Klm and D.V Nanopoulos, Phys. Lett. B 139 (1984) 170. [5] I. Antonladls, J. Ellis, J. Hagelln and D.V. Nanopoulos, Phys. Lett. B 194 (1987) 231. [6] J Elhs, J L. Lopez and D.V. Nanopoulos, Phys. Lett. B 252 (1990) 53; G. Leontarls and K. Tamvakls, Phys. Lett. B 260 (1991) 333, G. Leontans, Z. Phys. C 53 (1992) 287. [7] I. Antomadis, J Ellis, J. Hagehn and D.V. Nanopoulos, Phys. Lett. B 208 (1988) 209. [8] G. Leontans, Phys. Lett, B 207 (1988) 447; G. Leontans and D.V. Nanopoulos, Phys. Lett. B 212 (1988) 327; G. Leontans and K. Tamvakls, Phys. Lett. B 224 (1989) 319; S. Abel, Phys. Lett. B 234 (1990) 113; I. Antoniadls, J. Rlzos and K. Tamvakis, Phys. Lett. B 279 (1992) 281. [9] J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 359 [10] J. Ellis, J.L. Lopez and D V. Nanopoulos, Phys. Lett. B 292 (1992) 189. [ 11 ] G Leontans and J. Vergados, Ioannma preprlnt IOA290 (1993). [12] J. Ellis, M.K Galllard and D V. Nanopoulos, Phys. Lett. B 88 (1979) 320. [ 13 ] J. Ellis, D V Nanopoulos and K.A. 0live, Phys Lett. B 300 (1993) 121. [ 14 ] M. Fukugita and T. Yanaglda, Phys. Lett. B 174 (1986) 45. [15] G. 't Hooft, Phys. Rev Lett. 37 (1976) 8, V. Kuzmm, V. Rubakov and M. Shaposhnikov, Phys. Lett. B 155 (1985) 36. [16] L Wolfensteln, Phys. Rev D 17 (1978) 2369, D 20 (1979) 2634, S.P. Mlkheyev and A.Yu. Smirnov, Yad. Flz. 42 (1985) 1441, Nuovo Clmento 9C (1986) 17. 77
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[17] J Ellis, J.S. Hagehn, S. Kelley and D V. Nanopoulos, Nucl. Phys. B 311 (1988/89) 1. [ 18 ] B. Campbell, S. Davldson, and K A. Olive, University of Minnesota preprmt UMN-TH-IlI4/92 (1992), Nucl. Phys B 399, m press. [19] GALLEX Collab., P. Anselmann et al, Phys. Lett. B 285 (1992) 390. [20] J.N. Fry, K.A. Olive and M.S. Turner, Phys. Rev. Lett. 45 (1980) 2074.
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[21 ] A.D. Dolgov and A.D. Lmde, Phys. Lett. B 116 (1982) 329, D.V. Nanopoulos, K.A. Ohve and M. Sredmekl, Phys. Lett. B 127 (1983) 30. [22] F.L. Wright et al, Ap. J Lett 396 (1992) L13 [23] B. Campbell, J Ellis, J.Hagelln, D.V. Nanopoulos and K A. Olive, Phys. Lett B 197 (1987) 355, B 200(1988) 483; J. Ellis, J.Hagehn, D V. Nanopoulos and K.A Olive, Phys. Lett. B 207 (1988) 451.
PHYSICS LETTERS B
Flipped angles and phases: a systematic study John Ellis Theorettcal Phystcs Dlvtslon, CERN, CH-1211 Geneva 23, Swttzerland
Jorge L. Lopez, D.V. Nanopoulos Center for Theoretwal Physics, Department of Phystcs, Texas A&M Umverslty, College Statlon, TX 77843-4242, USA Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77382, USA
and Keith A. Olive School of Phystcs and Astronomy, Umverslty of Minnesota, Mmneapohs, MN 55455, USA Received 31 March 1993 E&tor: R. Gatto
We discuss systematically the fermion mass and mixing matrices m a generic field-theoreUcal flipped SU (5) model, with particular applications to neutrino and baryon number-changing physics. We demonstrate that the different quark flavour branching ratios in proton decay are related to the Cabibbo-Kobayashl-Maskawa angles, whereas the lepton flavour branching ratms are undetermined. The hght neutrino mixing angles observable via oscillation effects are related to the heavy conjugate (right-handed) neutrino mass matrix, which also plays a key role m cosmological baryogenesls. The ratios of neutrino and charged lepton decay modes in baryon decay may also be related to neutrino oscillation parameters. Plausible Ansatze for the generation structure of couphng matrices motivate additional relations between physical observables, and yield a satisfactory baryon asymmetry.
1. Introduction One o f the most welcoming avenues leading beyond the Standard Model is that leading to G r a n d Unified Theories ( G U T s ) . By unifying the three known partlcle gauge interactions within a simple group G, one may understand why baryon n u m b e r is conserved to a good approximation, but not perfectly, and why neutrino masses may be very small, but non-zero. In so doing, G U T s also offer scenaria for cosmological baryogenesis and the nature o f hot dark matter. Interest in G U T s has been further whetted by the close consistency o f the measured values of the SU (3), SU (2) and U ( 1 ) gauge couplings with m i n i m a l supersymmetric G U T s [ 1 ], which high-precision LEP data have rendered even more impressive [2]. W i t h all these phenomenologlcal motivations, it 70
was natural that string model-builders should seek to emulate GUTs. However, it was soon realized that there was a sizeable roadblock to deriving a G U T from string: gauge symmetry breaking and various other phenomenological constraints in G U T s generally require adjoint or larger Higgs representations, and these are not obtainable using conventional modelbuilding technology based on k = 1 K a c - M o o d y currents on the world-sheet [3]. Hence the revived interest m an SU (5) × U ( 1 ) G U T , flipped SU (5) [4,5], which only required 5- and 10-dimensional Higgs fields and could be derived from string. F l i p p e d S U ( 5 ) is the closest homage string can pay to the simple G U T s o f old. Naturally, there has been considerable discussion within flipped S U ( 5 ) o f the "classic" new phenomena that m o t i v a t e d so much work on GUTs, namely
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B V. All rights reserved.
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baryon decay [6] and neutrino masses [5,7-11]. However, there has not been a systematic investigation of all the fermxon mass matrices and m~xing angles that enter into baryon decay branching ratios and neutrino oscillations, and their possible relations to the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix and cosmological baryogenesis. In th~s paper we seek to remedy this lack m the hterature, working in the general framework of mimmal fieldtheoretical flipped SU(5), supplemented at the end by some (to us) plausible general hypotheses about mass matrices. It may serve a useful purpose to recall first some important features of minimal field-theoretical flipped SU (5) [ 5 ]. The first is that baryon decay via dimension-5 operators is very strongly suppressed, and also dimension-6 Higgs exchanges are presumably neghgible compared to the dimension-6 massive vector boson exchange operators expected to dominate. In minimal SU (5), it was possible to relate the mixing angles in the corresponding dimension-6 operators for proton decay to the CKM angles, modulo two additional complex phases that would be difficult to measure [12]. The relation ~s not so direct in flipped SU (5), because the flipping of the particle assignments within 5 and 10 representations, as well as the assignment of conjugate charged leptons to singlet representations, means the branching ratios into different lepton species are independent of the quark mixing angles. It has recently been reahzed, on the other hand, that the dominant mechanism for baryogenesls in flipped SU(5) may be the decay of massive conjugate ("right-handed") neutrinos [ 13 ], producing a lepton asymmetry which is subsequently reprocessed [14] into a baryon asymmetry by nonperturbative electroweak interactions [ 15 ]. This raises the possibdity that the cosmological baryon asymmetry might be related to observable parameters in the light neutrino mass and mixing matrices, which could also show up m baryon decay. In this paper, we investigate these possibilities systematically, starting from an adaptation in section 2 of the analysis of ref. [ 12 ] to diagonalize the fermion mass matrices and determine the independent umtary flavour rotation matrices. Next, in section 3 we apply these results to derive interrelations between observables in conventional weak decays, proton decay, neutrino oscillations and cosmological baryogenesis.
24 June 1993
Then, in section 4 we postulate plausible Ansatze for the fermion mass matrices which lead to certain additional quantitative relations between the different flavour rotatxon mamces and hence observable quantrees. Finally, m section 5 we summarize our conclusions and mention directions for future study.
2. Flipped mixing matrices Let us first remmd the reader of the various couphng matrices that appear in the minimal field-theoretical version of the fhpped SU(5) model. The superpotentlal that characterizes its Yukawa couphngs is [5]
w = 3`~F, Fjh + 3`TF,f j h + ,l t33 f , l j Ch + 24HHh + 2 5 H H h + 2Z6aFzH$a + 27hh~b0 + ].labCa~b ,
(1)
where the F,, f , , l~ (t = 1, 2, 3) are the three generations of 10, g and singlet representations of SU(5) that comprise the light matter particles of the Standard Model, H and H are 10 and 10 Higgs representations, h and h are 5 and 5 Higgs representations, and the $0, ~ba (a = 1,2, 3) are auxiliary singlet fields. The first three terms in the superpotential ( 1 ) give masses to the charge 2/3 quarks u,, charge - 1 / 3 quarks d, and charged leptons l, respectively, the next two terms spht the light Hxggs doublets from their heavy colour triplet partners in a natural way, the sixth term provides a large element in the see-saw neutrino mass matrix, the product 3.7 (~b0) gives the traditional Higgs mixing parameter, and the last term is the auxiliary slnglet mass matrix [7]. Our task m this section is to diagonalize the couphng matrices 3.1,2,3,6 and the corresponding mass matrices mu,d,t, identifying in the process the needed unitary rotation matrices whose observability we will explore in section 3. Our analysis of the dmgonalizataons is modelled on that made for minimal SU (5) in ref. [ 12 ]. We start by diagonalizlng the 3.2 coupling matrix wxth the unitary transformations
f ' = -fU*uc,
F' = U*uF,
(e)
which yield 71
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f22F
=
PHYSICS LETTERS B
-f'Uu~22UuF'
f'2fF'
=
which we now diagonahze by the new transformations
:
,~ = u..~2u..
(3)
2'~ = U~2~Uu.
(4)
We now diagonalize this by a further unitary transformation on the fields F ' :
=
U4F':
= uT~DIu4.
2',
(5)
As in ref. [ 12 ], it is convenient to separate phase factors in the matrix U4: denoting elements o f this matrix by e'%u,j where q,s and u,j are both real, we decompose
U4 = U5UU6 :
(6)
U7 = U 2,
=-f'5'UT :
III]Dlcl
5"~3 "
-f'u~,
F"
=
U6F',
(8)
leaving the diagonahzed matrix 22° unchanged. In this basis, the first term in eq. ( 1 ) becomes F
ttT
T
D
U U72~UF
tt
.
:
,~D = U i U 6 U u c 2 3 U l c
Cr#q~ = ¢ , r # ~ ,
(0'= U~O,
(13)
:
I t° = Uf#U~,,
FT,~6~ = F"T2~6¢':
(7)
and we impose a phase convention, det U7 = 1. We can then absorb the U6 phases into the fields =
(12)
for the dlagonallzed mass matrix. A further novel feature beyond the analysis of ref. [ 12] is the dlagonalization of the light and heavy neutrino mass matrices. We start by diagonahzmg the ~b mass matrix:
(F~ +.
--f"
~ £
l c' = Utcl ,
(14)
2~6 = U~U#26U(~.
(15)
The u ¢ = F4s mass matrix is of the see-saw form in the ( F ~ , ~b') basis, and so can be written in the form
This leads to the representation
2', = u 6 u T u 7 2 D u u 6 :
f5
which leads to the representation
U5 = diag(e~na),
U6 = diag(e 'm~ ) e -'~1~ •
--tit
leading to
As a result, the 2~ coupling term becomes
Fr21F = F'T2'IF':
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(9)
The charge - 1 / 3 quark mass matrix may finally be cast in real and dmgonal form by the redefinitions
~T
t ~rtl
) m~c tF45 +
):
m',c = 2~T(#°)-~2~ (V) 2,
(16)
where (V) = (10). We recall that the transformations (10) applied to the coloured components o f F in order to dlagonahze the quark mass matrices have not been applied to the u c = F45 components. We now define F~'
rrt t~uc~w,, 45,
(17)
in terms of which the u c mass matrix is diagonal: F4ttt T
D tit 5 rnucF~5 "
F'~; = UF'~
for 1 ~ . , p ~< 4,
F2Vp = U7F')~
for 1 ~< ~,fl ~< 3,
muD
= U ~T m ~tU ~
= U~2~T(#°)-12~ (V) 2 U.~
(10)
of relevant components o f the 10 representations F . In contrast to the conventional S U ( 5 ) case discussed in ref. [ 12 ], the charged lepton mass matrix is not directly related to the quark mass matrices. Starting from the third term in the superpotential ( 1 ) and making the above transformations of eqs. (2), (8) of the 5 fields f , we find
The light neutrino mass matrix is also of the see-saw form, this time in the (u = f 4 , F4s) basis. We diagonahze it in terms of the diagonalized u c masses rn~°~ ( 18 ). In terms of the unitary transformations already made:
7;t3l ~ = 7 " ( U6 u~¢z3)l ~ ,
74,,~ 2F45 = J"Utt~D ...... 4 A 2 t~'ucr45 ,
72
( 11 )
X (uguT~6u~)
('~')2 U v c .
(18)
(19)
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PHYSICSLETTERSB
yielding
4, , m u J74, , r :
~ - ' U~c22 7" ~ , m , = ~ U ~ c (m~c)
(20)
which we must dlagonalize by the further transformation 7~"
-7,,,,*.
m ~ = U,[AzD U~,~(m,,c) D -1 T D U/,~22 ]U~"~
(21)
for the light neutrino mass eigenstates. After this profusion (confusion?) of dmgonalizations and unitary matrices, we take mercy on the reader by summarizing the final mass e~genstates (represented by suffices F ) :
UF = (U6U2)FI<~a~3,5, t * u~e = -f- l<.~,<.3(U'~U~ ),
(22)
(23)
l-Y = lsUMsw-UF.
(26)
Given the previous results for the mass elgenstates 1r (24) and us (25) we find quite simply UMSW = UlU~.
(27)
-~d = -UF UcKMdF .
(28)
Using now the expressions (22) and (23) for the mass eigenstates Us and ds we can write
ls = 7~ (u'.~sl * *u;),
leF = Ut~l ,c ,
In parhcular, we wdl be interested in identifying the mixing matrix for the MSW mixing and the origin of the necessary CP-violatlng phase for the production of the cosmic lepton asymmetry. We will also identify the Cablbbo-Kobayashi-Maskawa (CKM) matrix and its role, as well as those of other matrices of interest in proton decay processes. We begin with the MSW mixing matrix. MSW solar neutrino mixing will result if there is a mismatch between the neutrino states produced in the charged current interactions and the neutrino mass eigenstates. Therefore, we can define the MSW mixing matrix by
Similarly, the CKM matrix is the charged current mismatch in the quark sector,
dg = (UU6U2)FI<~a<.g.3,4, d~. = (U7UU6U2)FI
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(24) UCWM= U ~ •
//F = 7 4 ( u ; t~ v ; *u 2 ) ,
b'~r = (UlutcV682 )f45.
(25)
3. Specific processes As we mentioned earlier, among the motivations for Grand Unified Theories is the possibility for baryogenesls and small neutrino masses. One of the remarkable features of the flipped SU (5) model is its ability to provide cosmologically-interesting neutrino masses (i.e., for u~), while at the same time allowing for observable Mikheyev-Smirnov-Wolfensteln (MSW) [16] neutrino oscillations and baryogenesis via leptogenesis and subsequent sphaleron reprocessing. All three of these highly desirable features are related to the s a m e neutrino mass matrix [ 10,13 ]. Given the complete decomposition of mass eigenstates in the flipped SU (5) model above, we are now in a position to examine in detail the interrelationships between the various mixing matrices and phases of interest.
(29)
A priori, we see no relationship between the quark and neutrino mixing matrices. Next we study the effective Lagranglan for baryon decay in the flipped SU(5) model, which has been discussed previously in ref. [ 171. As has already been recalled, the dominant mechanism for baryon decay is expected to be dimension-6 vector boson exchange. Using the analysis of the previous section, as was done for conventional SU(5) in ref. [12], we find the following massive vector boson couplings: Cx = ( g / v ~ ) X ? .
[e'Jk--dckFUTy"PLdj~
+-U,eTUPRv c] + h.c., £ r = ( g / x / 2 ) Y, - u[e ' s kd~ k v U v U y PLUj~
+-ff,r TUPRl c] + h.c.,
(30)
where we have worked in the mass eigenstate basis for the quarks, denoted by subscripts F, the lepton fields are in the f " basis of the previous section, the colour 73
Volume 308, number 1,2
PHYSICS LETTERS B
indices (t, j, k ) are noted explicitly, and we have indicated the handednesses of all fermion fields by using the projection operators PL and P~. The exchanges of X and Y bosons (assumed as usual to have indistmgmshable masses) therefore give rise to the following effective Lagrangian for baryon decay:
£.~a~O = (g2/2M2)
F (p --+ e + n o ) = ½cos 20c{ Ut,, 121" (P --* -flit + ) = COS20c I Oil I [2F (n --~ -flito),
F ( n ~ e+n - ) = 21"(p --+ e+it°), F ( n ~ i~+n - ) = 2F(p ~ #+ito), 1"(p ~ ~+ n °) = ½cos20clUl,~I2F(p -~ -fin + )
x [(¢'Jkd~FU7yUPtdjF)(ur~yuPtv) + h.c. t~,yk-2c U7U~gPLUjF)(uTF~uPLI) + h.c.]
(35)
= COS20clUllzlz1"(n ---, -flit0) .
(31)
As in the case of conventional SU (5), we see from this expression that two additional CP-vlolatmg phases appear in the quark parts of the AB # 0 operators [i.e., from U7 = Us2 in eq. (6) and detU7 = 1], beyond the single phase in the CKM matrix: these are in principle measurable via loop diagrams. Since the only quarks of relevance for baryon decay at the tree level are u, d and s, and not more than one of the latter, we can write the relevant parts of the AB # 0 Lagranglan (above) as
£,~a~o = (g2/2M~) yUpLdj )(Ur, yuPLU~) + h.c.
Thus it is possible m principle to correlate baryon decay branching ratios with the CKM angles, and to measure elements of the charged-lepton mixing matrix Ut, though not o f the neutrino mixing matrix U,. Moreover, we note that the decay branching ratios above are characteristically different from those m conventional SU (5) [ 17 ]. However, it lies beyond the scope o f this paper to calculate the total baryon decay rate in flipped SU(5). We now examine the mixing matrix and the necessary CP-violating phase which can provide for a net lepton asymmetry, and subsequently a net baryon asymmetry via sphaleron reprocessmg [ 14]. As described m ref. [ 13 ], the lepton asymmetry is produced by the out-of-equihbrium decay of the mass e~genstate u~ = F~' via the mass term f4.s22F45. In terms of mass eigenstates the neutrino mass term becomes
+ [e 'Jk (d~,e 2'"~' cos 0~ + ~ke 2'n21 sin Oc)YUPLUj]
--tit D tit f4,5 Uv,122 UvcF45 •
× (Ur~yuPLll) + h.c.]
It is then natural to introduce the lepton-numberviolating couphng
(32)
where we recall that VL and lL are not in the mass eigenstate basis, so that
UL = ueU~,
IL
= IFUI.
(33)
The phase factors in (32) are not measurable, nor can one distinguish between the different neutrino flavours, so the following are the predictions that can be made on the basis of (32):
F ( B ~ l + + X[strange) = tan z O~ ~ 1 F (B -~ l + + XJnon-strange) 2-0"
(36)
D 2L ~ Uv,I22 Uuc.
(37)
We expect the dominant contribution to the lepton (baryon) asymmetry to be that due to decays of the lightest u ~ mass eigenstates, which we expect to be that associated with ue.u and call v~,2. The CP asymmetry in the decay of u~ into second and third generation particles is g~ven by 1
F ( B --, (Xu) + Xlstrange) = 0, F ( B ~ ( X v ) + X[non-strange)
~J(34)
Comparing decays to neutrinos and charged leptons requires knowledge of specific hadronic matrix elements. Following ref. [ 17 ], we find 74
24 June 1993
2n(2~2L)11
x E
(Im[ (2*z2L)O]2) g(ME/M~) ,
(38)
J
where
g ( x ) = 4x/-xln l + X x
( ~- 4 for x >> 1) ~
'
(39)
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PHYSICS LETTERS B
where (38) is the supersymmetrlc expression for the C P asymmetry [18]. Analogous expressions can be written for the corresponding asymmetries e2.3 in the decays ofu~. 3. Using the definition (31 ) of 2L we find
where 0e,u,#r a r e the usual MSW mixing angles. It ts interesting to note that the baryon decay branching fractions in eq. (35) depend in this approximation on UII l
/~fL,~L = u'ct ( &D) 2 u.~.
(40)
This depends only on U,~ and, what is more, the C P violating phase we are interested in is a priori unrelated to the CKM phase or MSW mixing. In general, we could expect (2tL2L) t t " (2~33) 2 (the largest entry in 22O ), and if it were the case that Ml ,.~ ME << M3 we would estimate f ~
21n2 ~D 7~
2X
(41)
~'233 ~ ,
where ~ is the phase associated with the imaginary part of (2tL2L)12 in (40). (This is slightly larger than our previous esumate [13].) A satisfactory baryon asymmetry would result for ~ > l0 -2. In the next section we will revise this generic estimate in the l i g h t of a specific Ansatz for neutrino masses.
1,
=
U62 = 012,
(45)
giving F(p
~ e + n °) = i cos z G F (p --, -fin + )
= COS2 G F ( n
(46)
---, -fin°).
Note that our general Ansatze about mixing angles indicate that the baryon decay branching fractions into muons should be suppressed relative to those into electrons. Also, R. remains undetermined this way since only the sum over all branching fractions into neutrino final states can be observed. On the other hand, eqs. (20), (21) allow us to obtain U. once U.c and m.~ are given. Since the latter two appear in the baryogenesis parameter e (eqs. 38, 40), it is interesting to correlate all these parameters. Starting from eq. (21) and writing U. = I + R . and U,c = 1 + R,c we obtain m,, = (22°), (m,,c), o -1 (22D),,
4. Phenomenological Ansfitze In this section we propose some plausible forms for the various matrices appearing above, and obtain correlations among the various observables of interest. Clearly, the unitary matrices U are not expected to be equal to the identity matrix. However, experience with the CKM and MSW mixing matrices indicates that they probably should not differ too much from unity either. We therefore write (42)
U, = 1 + R , ,
24 June 1993
(47)
which leads to the previously advocated phenomenologically interesting neutrino mass ratios [ 10 ]. Moreover, from (rn.),j D = 0 for i # J we find R. in terms of R.~, (R~),l _
-1
(2f), (2f)j
rn.j - m . ,
x
( R . ~ ) , : ~ j - (RT.~),j
,
(48)
where M, = (m.~),,. In particular
for all matrices U, defined above, with R, =
0 -0;~ o
0h 0
0 ) 0~3 , o
O ~ ( z - m,, u
(43)
such that U, U,~ = I through order 0, since R~ = - R , . By analogy with the C K M matrix, we have neglected the far off-diagonal entries in the matrices R,. For the MSW matrix UMsw = UtUt, we have RMsw = R t + R~ = R t - R~, and o,. =
- 01%
= oh-
(44)
m,e
).u ~.~,cM2
(49)
The last expresmon holds in the limit Ma >> MI. Also,
--J.c2t
O~3 -- m , , - m , u
2c ~,~c M3
.c 1 023
M33
_
_.uc 1 ] 023 M22J
(50)
with the final expression valid in the hmit 343 >> ME. 75
As is plausible for any matrix with a hierarchy of eigenvalues, such as a see-saw mass matrix, we make the Ansatz //¢
o7~ _~ x/-M~M2,
023 -~ V/--~2/M3,
(51 )
which gives
012 ~ (2~/2c) V/-~z/M,,
0~3 ~- (~/,~,) v/-M[3/M2.
(52)
If for the moment we neglect t h e 0~2,23 contributions to the neutrino mixing angles [see eq. (44) ], the prediction for Ou~ in eq. (52) is exactly what was proposed in ref. [10] on purely phenomenologlcal grounds. This relation (with M3/M2 ~ 10) and the ratios of neutrino masses in eq. (47) have been shown [ 10] to lead to Interestingly observable uu-u~ oscdlatlons and a z neutrino mass large enough to provide an interesting amount of astrophysical hot dark matter. The other relation in eq. (52) should reproduce the present fits to the solar neutrino data based on the MSW mechanism. These require Oeu = (3.2-6.1) × 10 -2 [19] and therefore M2/MI = 64-225. (Note: the present expression for 0eu differs from that in ref. [ 10 ].) Turning now to the baryogenesls parameter c, from eq. (40) we have (to order 0)
+(R~).
2:2M1 x ~, = ~ c ~ , 2 ,
),j6.
[(,~),~ -- (2 o2 ) J2] -
Thus lm t2t t L2 L )~2 tt
~
(53)
I m t~.2 tL2 L /~2 ' 3 ~--" 0, and
t 2 ~_ ~4 .,iv e 2COS2¢12, Im (2L2L),2 .~c ut2
am (2~;tL)~3 ~- 2410~12 c o s 2¢23,
where
76
(57)
n L l ( m ~ )
10-7-~32 d23
'/2
10-8d23,
(58)
~2 ~qvc 2
c t-'12 ,
a21ovcl2 (~flL)].L)22 .,., ~2 + ~tlt"23t ,
We finally get
623 = COS2¢23.
For the decay of the first and lightest generation v[, this result is a factor of (2c/2t) 2 (M,/In 2M2) ,,~ 10 -6 smaller than that obtained in the generic case in eq. (41) above. However, we see that for the decay of the second generation v~ the result is only suppressed by (M2/ln 2343) ~ 7, and is therefore large enough a priori to produce a satisfactory lepton, and hence baryon, asymmetry. Normally, it is the llghtest generation which produces a net asymmetry [20]. This is because the lightest generation will typically wipe out any prior asymmetries (e.g., via inverse decays) before producing its own. If we considered a standard out-of-equllibrmm decay mechanism, we would have to require that both first- and second-generation S ' s be far out-ofequlhbrlum at the time of their decay. This would impose the r e s t r i c t i o n s (,~L/~L)I1,22 ~,~ 103M,,E/Mv, conditions which would be difficult to satisfy given our Ansatze. However, invoking a mechanism proposed in ref. [21 ] we assume here that the v ° s are produced in the decay of the inflaton, r/, subsequent to inflation. Then u c decays occur immediately and out-of-equilibrium at T << M,,, and no destruction of an asymmetry can occur. Thus our only requirement IS that MI,2 < m e. This is satisfied for m, ~ few x 101'GeV as inferred from the COBE result [22] on density fluctuations [ 18,13 ]. Our final lepton and hence baryon asymmetry now becomes nB
(,~;tL)33 ~- ,~2.
(56)
(54)
where Cu = arg(0ff). Analogously, the denominator factors are given by (/~L~L)ll .,., ~2 +
2,2M2x ~2 = ~ , ~ 2 3 ,
4~2M2 e3 = ~.~t ~33 In M~ [0~12d23,
6, 2 = COS2¢,2,
=
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(55)
where A ~ 10 -3 iS an entropy dilution factor which accounts for the entropy produced during breaking of SU (5) × U ( 1 ) [2 3,13 ] This IS encouragingly large: recall the phase 623 is not related to the standard (CKM) source of CP violation, and is not expected to be partIcularly suppressed. Note that there could be extra
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sources o f entropy which would further suppress the estimate (58), so c~23 may not need to be very small.
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in part by DOE grant DE-AC02-83ER-40105, and by a Presidential Young Investigator Award. J.L.L. would like to thank the CERN Theory Division for its kind hospitality while part o f this work was carried out.
5. Summary and outlook We have made in this paper a systematic study o f the mass matrices and mixing angles in the m i n i m a l field-theoretical version o f flipped SU (5). We have identified the mixing angles that appear in observable processes such as neutrino oscillations and proton decay, as well as the C K M angles in the charged electroweak interactions, and related them as far as possible in a independent way. We have also discussed a scenario for producing the baryon a s y m m e t r y w a the out-of-equilibrium decays o f heavy singlet conjugate neutrinos, which produce a lepton asymmetry that is subsequently reprocessed into a baryon asymmetry by sphaleron transitions. Additional plausible Ansatze for the mixing angles lead to a n u m b e r of further relations, and a satisfactory baryon asymmetry. We were m o t i v a t e d to study flipped SU (5) because it is the only G U T that can be derived from string theory, and it was precisely the baryon- and leptonnumber-violating processes discussed above that motivated the derivation o f a G U T from string. However, the m i n i m a l field-theoretical flipped SU(5 ) model analyzed above is not what one obtains from string theory. On the one hand, flipped SU (5) models derived from string contain more states, thus complicating the analysis, but on the other hand they can cast light on plausible forms for the mass matrices and mixing angles. Therefore it would be desirable to complement this general analysis with some more modeldependent studies. We believe that flipped S U ( 5 ) can give us m a n y insights into the exciting new era o f massive neutrino physics that solar neutrino experiments and models o f the formation o f large-scale astrophysical structures suggest m a y be opening up before us.
Acknowledgement The work o f J.L.L. has been supported by an SSC Fellowship. The work of D.V.N. was supported in part by DOE grant DE-FG05-91-ER-40633 and by a grant from Conoco Inc. The work o f K.A.O. was supported
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