U FIFT -H EP-05-6
M C T P-05-65
arXiv:hep-ph/0503222v1 22 Mar 2005
Cabibbo Haze in Lepton M ixing
A seshK rishna D atta1;2,Lisa Everett1,and Pierre R am ond1
1. Institute for Fundam entalT heory,D epartm entofPhysics
U niversity ofFlorida,G ainesville,FL,32611,U SA
2. M ichigan C enter for T heoreticalPhysics,D epartm entofPhysics
U niversity ofM ichigan,A nn A rbor,M I,48109,U SA
A bstract
Q uark-lepton uni cation suggests C abibbo-sized e ects in lepton m ixings, w hich we call
C abibbo haze. W e give sim ple exam ples and explore possible W olfenstein-like param etrizations ofthe M N SP m atrix w hich incorporate leptonic C abibbo shifts. W e nd that the size
ofthe C H O O Z angle is not alwayscorrelated w ith the observability ofC P violation.
1
Introduction
W ith recentexperim entalprogressin the neutrino sector,a quantitative picture ofthe M aki-N akagawa-SakataPontecorvo (M N SP) lepton m ixing m atrix [1,2]has begun to em erge. In contrast to the C abibbo-K obayashiM askawa (C K M ) quark m ixing m atrix,w hich is approxim ately the identity m atrix up to e ects oforder the
C abibbo angle
sin c 0:22,the M N SP m atrix contains two large angles , ,and one sm allangle 13:
0
1
cos
sin
sin 13
A ;
UM N S P ’ @ cos
sin
cos
cos
sin
(1)
sin
sin
sin
cos
cos
forw hich com bined data from the solar(w ith SN O -salt[3]and Super-K am iokande (SK )[4]),atm ospheric (from
the latest SK data [5]), reactor (K am LA N D [6], C H O O Z and Palo Verde [7]), and accelerator experim ents
(K 2K ) [8]indicates at 3 the follow ing constraints on the atm ospheric and C H O O Z angles [9]:1
= 45
+ 10
10
;
13
< 13 ;
(2)
w hile the solar angle is constrained to a rather precise range of
= 32:5
+ 2:4
2:3
(3)
;
asindicated by the SN O -saltdata w hen com bined w ith the data from the K am LA N D reactorexperim ent. N ote
that the lim iting value of 13 is approxim ately equalto c.
U nderstanding the origin ofthe M N SP m ixing anglesprovidesa new perspective and challengesforaddressing the avorpuzzles ofthe Standard M odel(SM ).In this letter,we work in the contextofquark-lepton grand
uni cation 1
[2], for w hich allavailable quark and lepton sector data can be synthesized in the quest toward
a credible theory of avor. T he qualitative di erences between the quark and lepton m ixing m atrices,w hile
perhaps surprising,are not inconsistent w ith grand uni cation. T he seesaw m echanism [
13]introduces a new
unitary m atrix into the M N SP m atrix,w ith no analogue in the quark sector [14], that can provide a source
for the discrepancy. Explaining the observed pattern oftwo large and one sm alllepton m ixing angles w ithout
ne-tuning in a three-fam ily m odelis the A chilles’heelfor m ost m odel-building attem pts.
W olfenstein’s param etrization ofthe C K M m atrix [15]as an expansion in powers of not only has wellknow n practicaladvantages for phenom enology,but also provides a theoreticalfram ework for exam ining the
quark sector in the ! 0 lim it. O ne aim of this letter is to seek a sim ilar param etrization of the M N SP
m atrix. W e begin by dem onstrating that grand uni cation im plies that C abibbo-sized e ects are expected in
the lepton as wellas the quark sector. Such e ects create a C abibbo haze in lepton avor m ixing,keeping in
m ind that unlike the C K M m atrix,the M N SP m atrix is not know n in the ! 0 lim it,except that it contains
two large angles. Leptonic C abibbo-sized perturbations can then shift the atm ospheric and solar angles from
their unknow n initialvalues by
c,provide the dom inant source for the C H O O Z angle,and dictate the size
ofC P-violating e ects. A lthough the data is not yet precise enough to pinpoint a particular W olfenstein-like
param etrization,regularities m ay em erge upon m ore precise m easurem ents ofthe M N SP param eters.
2
Seesaw and G rand U ni cation
In the SM , Iw = 1=2 electroweak breaking generatesD irac m assesforthe quarksand charged leptonsthrough
Yukawa couplings. T hese D irac m ass m atrices are diagonalized by biunitary transform ations,in w hich U2=3 ,
U 1=3 ,and U 1 are the rotation m atricesforthe left-handed states,w hile V2=3,V 1=3 ,and V 1 are the rotation
m atrices for the right-handed states. Physicalobservables depend only on the C K M m atrix
UC K
M
y
= U2=3
U
1=3
:
(4)
In the original form ulation of the SM , all three neutrinos are m assless and lepton m ixing is unobservable.
N onvanishing neutrino m assesrequireoneto go beyond theSM ,and add e.g. oneright-handed neutrino foreach
1 Signi cant im provem ents in constraining
13 (dow n to 3 or below ) are foreseen at the planned reactor neutrino experim ents,
superbeam s and/or neutrino factories [10];not m uch im provem ent in the atm ospheric sector is foreseen [11].
2
(0)
fam ily. Electroweak breaking then also generates D irac m ass term s for the neutrinos,w ith M D irac = U0D 0V0y.
If there is no seesaw , the D irac m ass eigenvalues m Di (D 0 = diag(m Di )) are the physical neutrino m asses.
H owever,large Iw = 0 m assterm sforthe right-handed neutrinosarise naturally,asthey are unsuppressed by
(0)
gauge quantum num bers. T his M ajorana m ass m atrix M M aj has entries w hich can be m uch larger than the
electroweak scale. A fter the seesaw ,
M
(0)
S eesaw
= M
(0)
D irac
1
M
(0)
M aj
(0)T
D irac
M
;
(5)
w hich can be reexpressed [14]as
M
(0)
S eesaw
=
=
(6)
1
D 0 V0y
M
V0 D 0 U0T ;
(0)
M aj
M
U0 C U0T :
C is the centralm atrix
C
1
U0 D 0 V0y
(7)
V0 D 0 ;
(0)
M aj
diagonalized by the unitary m atrix F
C = F D FT ;
(8)
w here D is the diagonalm atrix ofthe physicalneutrino m assesm 1,m 2,and m 3. T he M N SP m atrix can then
im m ediately be w ritten in the suggestive form
UM
= U y1 U0 F :
N SP
(9)
Eq.(9) highlights the di erences between UM N S P and UC K M ,and provides the basis for our discussion.
G rand uni cation suggests connections between the M N SP and C K M m atrices. T he sim plest H iggs structures lead to the follow ing relations
:
M
( 1=3)
SO (10):
M
(2=3)
SU (5)
( 1)T
M
M
(0)
D irac
:
;
(10)
(11)
w hich im ply
U
V
1=3
1
;
U0;
U2=3
so that
UM
N SP
= V T1=3 U
1=3
UCy K
M
F :
(12)
M odelscan then be categorized according to the structure ofthe charge 1=3 Yukawa couplingsand the num ber
oflarge angles in F . A particularly illustrative exam ple [16]is the class ofm odels w ith sym m etric M 1=3 ,for
w hich
U 1=3 = V 1=3 ;
w hich im plies that the M N SP and C K M m atrices are sim ply related
UM
N SP
= UCy K
M
F :
(13)
In this case, F m ust contain two large m ixing angles
and
. A s we w ill discuss in the nextpsection,
Eq.(13)then im plies
t
hatt
he
s
ol
arangl
e
exper
i
encesa
C
abi
bbo
shift
cos
= 2,and
p
s
i
n
=
2
due
t
o
t
he
O
(
)
1
2
m
i
xi
ng
i
n
U
.
13
CK M
T he above class of m odels provides well-m otivated exam ples of leptonic C abibbo shifts, but it is by no
m eans the only theoretical possibility. In the context of grand uni cation, the data also can hint that the
m ixing m atrix is initially bim axim al( =
= 45 ),w ith the solar angle shifted by a full-strength C abibbo
shift:
avor theories, one should
c [17,18,19,20]. W hile these exam ples can be m otivated by
keep in m ind that the data is not yet precise enough to select particular scenarios and the values ofthe large
angles are not know n in the ! 0 lim it (ifindeed that lim it is m eaningfulfor theory). H ence,we w illnow
explore param etrizations ofthe M N SP m atrix w hich incorporate such leptonic C abibbo e ects purely from a
phenom enologicalstandpoint.
3
3
System atics
T he W olfenstein param etrization ofthe C K M m atrix is based on the idea that the observed hierarchicalquark
m ixing angles can be understood as powers ofthe C abibbo angle ,w ith U C K M = 1 in the ! 0 lim it:
UC K
= 1 + O ( ):
M
(14)
T he m ixing angles ofthe quark sector (w hich have been very precisely determ ined using direct m easurem ents
and unitarity constraints)have the follow ing hierarchicalstructure [15]:
0
1
2
1 2
A 3(
i)
2
A + O ( 4 ):
UC K M = @
(15)
1 2
A 2
A 3 (1
i)
A 2
1
In the above, and A are wellknow n ( = 0:22,A ’ 0:85),but and are lessprecisely determ ined [21]. T he
C P-violating phase at lowest order is in the m ost hierarchically suppressed 1 3 m ixing,but any other choice
leads to the sam e rephasing invariant m easure ofC P violation. A s m easured by the JD G W invariant [22,23],
C P violation in the quark sector is sm all
(C K M )
’ A2 6 ;
(16)
JC P
due to the sm allm ixing angles,not the C P-violating phase,w hich is the only (possibly) large angle in UC K
To seek a sim ilar param etrization for the M N SP m atrix,we assum e a expansion ofthe form
UM
N SP
= W + O ( );
M
.
(17)
in w hich the starting m atrix W has two large angles
and
, corresponding to the \bare" values of the
atm ospheric and solar angles,and a vanishing C H O O Z angle:
1
1 0
0
cos
sin
0
1
0
0
A @ sin
(18)
cos
0A :
sin
W = R 1 ( )R 3( ) @ 0 cos
0
0
1
0
sin
cos
W isthen perturbed by a unitary m atrix V( ). U nlike the case ofthe quarks,the perturbation m atrix doesnot
generically com m ute w ith the starting m atrix
[W ;V( )] 6
= 0;
(19)
resulting in three possible types ofC abibbo shifts (see also [16,18,19,20]):
R ightC abibbo Shifts:
UM
N SP
= W V( )
(20)
UM
N SP
= V( )W
(21)
LeftC abibbo Shifts:
M iddle C abibbo Shifts:
UM
N SP
= R 1 ( )V( )R 3 ( )
(22)
G iven
and ,one can choose V( ) to shift the atm ospheric and solar angles into the range allowed by the
data,and study the resulting im plicationsfor 13 in the three scenarios.A novelfeature ofourparam etrizations
is that the JD G W invariant depends not only on the type ofC abibbo shift,but also on how the C P-violating
(C K M )
(M N S P )
due to the larger
is typically expected to be m uch larger than JC P
phase is introduced in V( ). JC P
m ixing angles,and can be as large as
sin (for 13
O (1)).
c and
To illustrate these ideas,letus considerm odels forw hich Eq.(13) holds,w hich are left C abibbo shifts w ith
F given by Eq.(18) and V( )= U Cy K M . T he shifts in the angles are given to O ( 2 ) by
=
=
13
=
1
sin 2
4
cos
;
(A +
sin
:
4
) 2;
(23)
(24)
(25)
U sing the data to constrain F ,assum ing the centralvalues
= 45 and
= 32:5 for sim plicity,
’ 48
(the shift of 3 is a typicalO ( 2 ) correction),
’ 41 ,and 13 = sin 1 ( sin ) ’ 9:4 . T he JD G W
invariantis given by
1
(M N S P )
cos sin 2 sin 2 A 3 + O ( 4 );
(26)
JC P
=
4
w hich is 10 3 (setting the C K M param eter
0:4). A s the M N SP m atrix has two O (1) and one O ( )
(C K M )
(M N S P )
.
islargerthan JC P
m ixing angles,the e ective M N SP phase issuppressed by O ( 2 ),even though JC P
4
P aram etrizations
W e now analyze W olfenstein-like param etrizations ofthe M N SP m atrix w hich incorporate the di erent types
ofleptonic C abibbo shifts. A s C abibbo shifts are at m ost
c (for O ( ) perturbations),the bare angles
and
can e.g. be in the approxim ate ranges 15 <
< 45 ,30 <
< 60 . O ne should also keep in m ind
that the errorbars on
and the bound on 13 are roughly O ( ),w hile the errorbars on
are ofO ( 2 ).
G iven
and ,the perturbation m atrix V( ) m ustshift the two large anglesin the range allowed by the
data. W hether the C H O O Z angle is shifted by
c or a subleading contribution depends on the details ofthe
O ( ) perturbations in V( ) and the type ofC abibbo shift,leading to three basic categories:
Single shift m odels. T hese m odels have only one O ( ) perturbation in V( ),w hich can be either in the
1 2 m ixing V12
,the 2 3 m ixing V23
,or the 1 3 m ixing V13
.
D ouble shift m odels. T hese m odels have two O ( ) perturbations. T here are three possibilities: V12
V23
,V12 V13
,or V13 V23
.
Triple shiftm odels. T hese m odels have V12
V23
V13
.
Focusing for the m om ent on single shifts,there are severalbroad classes ofparam etrizations:
C abibbo-shifted atm ospheric angle:
In this class ofm odels,the perturbation V( ) shifts the atm ospheric angle by
0
1
1
0
0
V( )= @ 0
1
a A + O ( 2 );
0
a
1
w ith a O (1). T he solarangle rem ainsunshifted (
depend on the type ofshift scenario.R ight shifts
=
N SP
is that
(27)
). T he shiftsin the atm ospheric and C H O O Z angles
0
UM
c. O ne possibility
1
= R 1( )R 3 ( )@ 0
0
0
1
a
1
0
a A
1
(28)
lead to
13
’
’
+ a cos
a sin ;
;
(29)
D ue to the form ofEq.(28),the size ofC abibbo shifts are -dependent. Left shifts
0
1
1
0
0
UM N S P = @ 0
1
a A R 1 ( )R 3( )
0
a
1
and m iddle shifts
0
UM
N SP
1
= R 1( )@ 0
0
5
0
1
a
1
0
a A R 3( )
1
(30)
(31)
give the sam e results (to this order in ):
+a ;
’
0:
’
13
(32)
T he atm ospheric angle isnow shifted by a full-strength C abibbo e ect,and the C H O O Z angle isa higherorder
e ect. If,however,
0
1
1
0 a
V( )= @ 0
1 0 A + O ( 2 );
(33)
a 0 1
right shifts lead to
a sin
’
a cos
’
13
;
(34)
;
w hile the left and m iddle shifts leave the atm ospheric angle unchanged at this order.
C abibbo-shifted solar angle:
In this case,V( ) shifts the solar angle by
c. O ne possibility
0
1
a
0
V( )= @
a
1
0
is that
1
0
0 A + O ( 2 );
1
(35)
just as in the C K M .T he atm ospheric angle is unchanged to this order in for allscenarios.For rightshifts
0
1
1
a 0
UM N S P = R 1( )R 3 ( )@ a
1 0A
(36)
0
0 1
and m iddle shifts
0
UM
N SP
1
a
0
= R 1( )@
1
0
0 A R 3( )
1
a
1
0
(37)
the leading order shifts in the solar and C H O O Z angles are
+a ;
’
13
In contrast,left shifts
0:
’
0
UM
N SP
= @
1
a
0
a
1
0
(38)
1
0
0 A R 1 ( )R 3( )
1
(39)
yield
+ a cos
’
13
’
a sin
;
(40)
:
Left shifts also works ifV( ) is given by Eq.(33),in w hich case
a ;
’
13
N ote that here the C abibbo shifts are sized by
’
a cos
:
-dependent factors.
6
(41)
U nshifted solar and atm ospheric angles:
In this case, the starting values of the large angles are very close to their central values. T here are two
possibilities: V( ) has no entries linear in ,or V( ) is given by Eq.(33),in w hich case m iddle C abibbo shifts
0
1
1
0 a
UM N S P = R 1( )@ 0
1 0 A R 3( )
(42)
a 0 1
lead to
13
= a and unshifted large angles.
A sim ilar analysis can be carried out for the double and triple C abibbo shift scenarios. In analogy w ith the
single shift m odels, for w hich either one or two of the three m ixing angles receive an O ( ) shift, either two
or allthree ofthe m ixing angles can receive large C abibbo shifts in double shift m odels. Triple shifts lead to
O ( ) shifts in all three m ixing angles. In Table 1, results are show n for double and triple shifts w hich are
param etrized using one V( ) m atrix (w ith two or three O ( ) entries) and right,left,or m iddle shifts.
W e do not discuss m ore com plicated ways to introduce O ( ) e ects in detail, as they add little to our
qualitative conclusions. For exam ple, double and triple shift m odels can be constructed using two or three
single shifts (w ith V( ) given by Eq.(27),Eq.(33),or Eq.(35)) and com binations ofright,left,and m iddle
shifts. Triple shift m odels can also incorporate com binations ofdouble and single shifts. T hese m odels lead to
shifts w ith sim ilar order ofC abibbo suppression,though their sizes can be di erent. A n illustrative exam ple is
a double shift obtained by a left shift w ith V12 a and a m iddle shift w ith V13 a0 (a; a0 O (1)),w hich
yields
=
+ acos
, =
,and 13 = (a0 + asin ) .
T he various param etrizations can also be classi ed in term s ofthe predictions for 13. Perturbations w ith
V13
O ( ) (such as in Eq.(33)) always lead to 13 oforder c. It is also possible to obtain a shift in 13 of
that size through O ( ) entries in 1 2 m ixing w ith left shifts and 2 3 m ixing w ith right shifts.
Finally,we note thatbased on thisleading orderanalysis,m any m odelscan be constructed by specifying the
bare angles ;
and including subleading perturbations. W e choose not to do this at this stage,given the
w ide range ofpossibilities consistent w ith current experim entaldata. Particular param etrizations m ay em erge
as potentially com pelling from the standpoint of avor theory. Im proved data, particularly for the C H O O Z
angle,w illcertainly be invaluable in narrow ing the range ofpossible param etrizations.
5
C P violation
C lassifying the param etrizations to leading order in is not su cient for addressing C P violation,since the
JD G W invariants are given by the product of allthe entries ofUM N S P . W ithin each of the basic classes of
m odels,exam ples can be constructed by specifying the details ofthe subleading perturbations. To illustrate
these points,let us consider two representative exam ples:
Single shifts w ith C abibbo-shifted atm ospheric angles,in w hich V( ) is given by
0
1
1
O ( 3) O ( 2 )
V( ) @ O ( 3 )
1
a A :
O ( 2)
a
1
D ouble shifts w ith C abibbo-shifted atm ospheric and solar angles,in w hich V( ) is given by
0
1
1
a
O ( 2)
V( ) @
a
1
a0 A :
2
0
O( )
a
1
(43)
(44)
To include the e ects ofC P violation,we allow for a phase ofa prioriunknow n size (though we w illassum e
it is O (1)) that can enter either w ith the dom inant or subleading term s ofV( ).
T he results,w hich are presented in Table 2,dem onstrate explicitly that the C P-violating invariantsdepend
not only on the details ofthe subleading perturbations but also on the type ofC abibbo shift,since the m ixing
7
angles depend on the particular shift scenario. U nlike the quark sector, the leptonic JD G W invariants also
depend on w hether the C P-violating phase is introduced in the leading or subleading perturbations.
T he param etrizations can be classi ed according to their predictions for the JD G W invariants. A s anticipated,the JD G W invariants
(M N S P )
4
(
)sin
(45)
JC P
6
are m uch largerthan in the quark sector(w hich is
). JD G W invariantsoforder
sin are m ore com m on
for double (and triple) shift scenarios,though single shift m odels can also predict such large e ects.
O ne generic and novelfeature of these param etrizations is that the C P-violating invariants can be m uch
sm aller than naive expectations based on the size of the lepton m ixing angles, because the e ective M N SP
phase can be additionally suppressed even for
O (1) (as opposed to the C K M phase,w hich is O (1)). W e
previously discussed one exam ple, the class of m odels based on grand uni cation w hich satisfy Eq.(13), for
w hich the JD G W invariantis O ( 3 ) rather than O ( ),as show n in Eq.(26).
A nother illustrative exam ple is the single shift scenario ofEq.(43),w ith right C abibbo shifts. In this case,
O ( ) (see Eq.(29)), and hence the JD G W invariant is expected to be O ( ) if the
13 is predicted to be
e ective M N SP phase is O (1). A s show n in Table2,this in fact occurs ifthe O (1) phase is introduced in
(M N S P )
sin . H owever,if is introduced in the m osthierarchically
the dom inant2 3 m ixing,for w hich JC P
suppressed perturbations (as in the quark sector),V( ) is given by
0
V( )= @
1
ab
( 2 + cei )
b 2
(
3
ab
2
+ ce i )
2
1 2
a
1
b 2
A + O ( 4)
a
3
1
(46)
2
2
(M N S P )
invariant:JC P
4
(a; b; c O (1)). R ightshiftspredicta suppressed JD G W
sin . In thiscase,a larger
C H O O Z angle does not lead to large C P violation because the e ective M N SP phase is suppressed by O ( 3 ).
Param etrizations w hich predict a suppressed e ective M N SP C P-violating phase abound in Table2 and
appearto bequitegeneric,re ecting theintriguing possibility thatthesizeoftheC H O O Z angleisnotnecessarily
correlated w ith the m agnitude ofC P violation.
6
C onclusions
W e are beginning to read the new lepton data, but there is m uch work to do before a credible theory of
avor is proposed. In the m eantim e, we have found it illustrative to exam ine the lepton sector through the
lens ofquark-lepton uni cation,and investigate param etrizationsofthe M N SP m atrix w hich include C abibbosized perturbations. T hese W olfenstein-like param etrizations have severalnovelfeatures,including the generic
possibility thatthe size ofthe C H O O Z angle isnotnecessarily correlated w ith the observability ofC P violation.
O ur approach em phasizes the need for precision m easurem ents ofthe M N SP m atrix,as the present data is
not su cient to single out a particular param etrization. Should the lim it ofzero C abibbo m ixing prove to be
m eaningfulfor theory,w ith im proved data we m ay be able to see avor patterns through the C abibbo haze.
A cknow ledgm ents
T his work issupported by the U nited States D epartm entofEnergy undergrantD E-FG 02-97ER 41209.A .D .is
also supported by the U S D epartm ent ofEnergy and M ichigan C enter for T heoreticalPhysics.
R eferences
[1] Z.M aki,M .N akagawa and S.Sakata,Prog.T heor.Phys.28,870 (1962).
[2] B .Pontecorvo,Sov.Phys.JET P 6,429 (1957)[Zh.Eksp.Teor.Fiz.33,549 (1957)].
[3] T he SN O C ollaboration,S.N .A hm ed etal.,Phys.R ev.Lett.92 (2004)181301.
8
[4] T he Super-K am iokande C ollaborations,S.Fukuda etal.,Phys.Lett.B 539 (2002)179.
[5] T he Super-K am iokande C ollaborations,S.Fukuda et al., Phys.R ev.Lett.81 (1998) 1562;ibid 85 (2000)
3999.
[6] T he K am LA N D C ollaboration,K .Eguchietal.,Phys.R ev.Lett.90 (2003)021802.
[7] T he C H O O Z C ollaboration, M . A pollonio et al., Phys.Lett. B 338 (1998) 383; ibid 420 (1998) 397;
ibid 466 (1999) 415; the Palo Verde collaboration, F. B oehm et al., Phys. R ev. Lett. 84, 3764 (2000)
[hep-ex/9912050]. For im proved treatm ents leading to signi cant relaxation of the upper bound see M .
A pollonio etal.,Eur.Phys.J.C 27 (2003) 331.
[8] T he K 2K C ollaboration,M .H .A hn et al.,Phys.R ev.Lett.90 (2003)041801.
[9] M .M altonietal.,N ew J.Phys.6 (2004)122;M .C .G onzalez-G arcia and M .M altoni,hep-ph/0406056;S.
G oswam i,A .B andyopadhyay and S.C houbey,hep-ph/0409224.
[10] M .Freund,P.H uber and M .Lindner,N ucl.Phys.B 615,331 (2001)[hep-ph/0105071].
P.H uber,M .Lindnerand W .W inter,N ucl.Phys.B 645 (2002)3;M .A pollonio etal.,hep-ph/0210192;P.
H uberetal.,N ucl.Phys.B 665 (2003)487;S.Pakvasa and J.W .F.Valle,Proc.Indian N atl.Sci.A cad.70A ,
189 (2004) [hep-ph/0301061]; V .B arger,D .M arfatia and K .W hisnant, hep-ph/0308123 and references
therein;\W hite Paper R eport on U sing N uclear R eactors to Search for a Value of 13",K .A nderson et
al.,January,2004,docum ent available at http://www.hep.anl.gov/minos/reactor13/white.html
[11] T .K ajita,talk given at N O O N -2004,Tokyo,Feb.2004.
[12] J.C .Patiand A .Salam ,Phys.R ev.D 10,275 (1974).
[13] M .G ell-M ann, P.R am ond,and R .Slansky in SanibelTalk,C A LT -68-709,Feb 1979 retropreprinted as
hep-ph/9809459,and in Supergravity (N orth H olland,A m sterdam 1979).T .Yanagida,in Proceedings ofthe
W orkshop on U ni ed T heory and Baryon N um ber ofthe U niverse,K EK ,Japan,Feb 1979.P.M inkow ski,
Phys.Lett.B 67,421 (1977).
[14] A .D atta,F.S.Ling and P.R am ond,N ucl.Phys.B 671,383 (2003)[hep-ph/0306002].
[15] L.W olfenstein,Phys.R ev.Lett.51,1945 (1983).
[16] P.R am ond,hep-ph/0401001;hep-ph/0405176;N ucl.Phys.Proc.Suppl.137,317 (2004)[hep-ph/0411009];
hep-ph/0411010.
[17] H . M inakata and A . Y . Sm irnov, Phys. R ev. D 70, 073009 (2004) [hep-ph/0405088]; A . Y . Sm irnov,
hep-ph/0402264;M .R aidal,Phys.R ev.Lett.93,161801 (2004);P.H .Fram pton and R .N .M ohapatra,
JH EP 0501,025 (2005);S.T .Petcov and W .R odejohann,hep-ph/0409135.
[18] C .G iuntiand M .Tanim oto,Phys.R ev.D 66,053013 (2002)[hep-ph/0207096].
[19] P. H . Fram pton, S. T . Petcov and W . R odejohann, N ucl. Phys. B 687, 31 (2004) [hep-ph/0401206];
W .R odejohann,Phys.R ev.D 69,033005 (2004);
[20] K .C heung,S.K .K ang,C .S.K im and J.Lee,hep-ph/0503122.
[21] S.Eidelm an etal.,Phys. Lett. B 592,1 (2004).
[22] C .Jarlskog,Phys.R ev.Lett.55,1039 (1985).
[23] I.D unietz,O .W .G reenberg and D .d.W u,Phys.R ev.Lett.55,2935 (1985).
9
D ouble Shifts
V12
V23
R ight
=
a
a0
=
13
V12
V13
a
a0
+a
+ a0 cos
= a0sin
=
=
13
a
a0
a00
+ acos
=
+ a0
13 = asin
+a
a0 cos
= a0cos
=
13
=
+ (a0 cos
asin )
= (acos + a0sin )
Triple Shifts
V12
V23
V13
=
=
a
a0
13
V13
V23
Left
+ (acos
a0sin )
=
= (asin
+ a0cos )
=
13
R ight
asin
=
+ a0
= acos
Left
=
+a
=
+ (a cos
a00sin )
00
0
a sin )
13 = (a cos
=
0
13
+ (acos
a00sin )
0
=
+a
= (a00cos + asin )
M iddle
=
=
13
=
13
+a
+ a0
= 0
+a
=
= a0
=
+ a0
= a
=
13
M iddle
=
=
13
+a
+ a0
= a00
Table 1: Leading order shifts in the M N SP m ixing angles for double and triple shifts (a; a0; a00
O (1)).
Table 2: JD G W invariants for a representative single shift scenario w ith V( ) given by Eq.(43) and double
shift scenario w ith V( ) given by Eq.(44). In colum n 1,the labeldenotes the placem ent ofthe phase in the
subblock ofthe perturbing m atrix. a (a0),b,and c are O (1) param eters associated w ith the O ( ),O ( 2 ),and
O ( 3 ) perturbations (see e.g. Eq.(46)).
Single
12
13
23
R ight
ab
16 (sin 3
b
4 cos
a
4 sin
D ouble
12
13
23
4
7sin )sin 2
sin
2
sin 2 sin 2
sin
sin 2 sin 2
sin
Left
c
4
3
sin 2 sin sin 2
sin
2
cos sin 2 sin 2
sin
ab
3
+ cos3 )sin 2
sin
16 (7cos
b
4
R ight
aa0
16 (sin 3
b
4 0cos
a
4 sin
2
7sin )sin 2
sin
2
sin 2 sin 2
sin
sin 2 sin 2
sin
M iddle
ac
8 sin 2
b
4 sin 2
ab
2 cos2
Left
a
4 sin 2
b
4 cos
aa0
16 (sin 3
10
sin sin 2
sin
2
sin 2 sin 2
sin
2
7sin )sin 2
sin
sin 2
sin 2
sin 2
4
2
sin
sin
3
sin
M iddle
aa0
8 sin 2
b
4 sin 2
aa0
8 sin 2
sin 2
sin 2
sin 2
2
sin
sin
2
sin
2