Maximal T violation in matter

V01ume 226, num6er 3,4

PHY51C5 LE77ER5 13

10 Au9u5t 1989

M A X 1 M A L 7 V 1 0 L A 7 1 0 N 1N M A 7 7 E R 5.705HEV 1n5t1tute[0r Nuc1ear Re5earch and Nuc1ear Ener9y, 8u19ar1anAcademy 0f 5c1ence5, 1784 50f1a, 8u19ar1a Rece1ved 7 March 1989

we der1ve 51mp1eexact ana1yt1ca1f0rmu1ae f0r the m1x1n9an91e5and D1rac pha5e 6 m1n matter 1n the three-neutr1n0 05c111at10n ca5e. U51n9 the5e f0rmu1ae we 5h0w that under 50me we11 5pec1f1ed c0nd1t10n5 e1ectr0n num6er den51ty N~ 6 ex15t5 5uch that 51n-~6m(N~~) = 1 wh1ch 1mp11e5max1ma1 7v101at10n 1n matter.

7 h e neutr1n0 05c111at10n51n m a t t e r [ 1,2 ] ~ m a y d1ffer c0n51dera61y fr0m the 05c111at10n51n v a c u u m [4 ]. 7 h e parameter5 0 f t h e 05c111at10n5 are m0d1f1ed a5 a re5u1t 0 f t h e d1fferent c0herent 5catter1n9 0 f t h e f1av0ur neutr1n05 v,., v~ and v~ 0ffe1ectr0n5. 1n the 9enera1 ca5e 0f n 1ept0n 9enerat10n5 ( a n d n0 r19ht-handed current5) the n u m 6 e r 0 f parameter5 wh1ch a p p e a r 1n the neutr1n0 m1x1n9 matr1x U, up= 2 U~,u, (~=e,~t, 2 ; 1 = 1 , 2 .... , n ) ,

(1)

t

depend5 0n the type 0 f t h e neutr1n05 v, hav1n9 def1n1te ma55e5 m1. 1f% are D1rac neutr1n05 there are n ( n - 1 ) / 2 m1x1n9 an91e5 and (n - 1 ) ( n - 2 ) / 2 CP v101at1n9 pha5e5 (50met1me5 ca11ed D1rac pha5e5 ). 1n the M a j 0 r a n a ca5e add1t10na1 ( n - 1 ) M a j 0 r a n a ( CP v101at1n9) pha5e5 are pre5ent [ 5 ]. 7 h e d e p e n d e n c e 0 f the neutr1n0 m1x1n9 an91e5 0 m (0r, m0re prec15e1y, 51n220 m) 1n m a t t e r 0n the e1ectr0n n u m 6 e r den51ty Are 1n the 51mp1e5t ( n = 2 ) ca5e ha5 a re50nant character [2,3]. A den51ty m a y ex15t [ 1,6,2,3] (ca11ed the re50nance den51ty) 5uch that 51n220m= 1 (1.e., 0m= ~r/4) even 1f the neutr1n0 m1x1n9 an91e 1n vacu u m 0 15 very 5ma11.7h15 mean5 that re50nant amp11f1cat10n 0fneutr1n0 05c111at10n5 1n m a t t e r 15 p055161e [2,3 ] (M1kheyev-5m1rn0v-W01fen5te1n effect). Ana1090u5 re5u1t5 have 6een 06ta1ned a150 f0r n > 2 [ 7 - 9 ] . 0 n the 0ther 51de, 1t ha5 6een 5h0wn 1n ref. [ 10 ] f0r ar61trary n that the 05c111at10n pr06a6111t1e51n m a t t e r d0 n0t d e p e n d 0n the M a j 0 r a n a pha5e5. 7heref0re, we m a y c0nc1ude that the effect5 0f m a t t e r w1th re5pect t0 the5e tw0 type5 0 f p a r a m e t e r 5 (the m1x1n9 an91e5 and the M a j 0 r a n a pha5e5) are rather we11 under5t00d. A5 f0r the D1rac pha5e5 the 51tuat10n 15 50mewhat d1fferent. 5uch a pha5e appear5 f1r5t 1n the three-neutr1n0 m1x1n9 ca5e. An expre5510n f0r the D1rac pha5e 1n m a t t e r ha5 6een 06ta1ned 1n ref. [9]. Unf0rtunate1y, the a19e6ra turn5 0ut t0 6e rather 1mpenetra61e ana1yt1ca11y. U51n9 numer1ca1 meth0d5 the auth0r5 have c0nc1uded that the pha5e ••depend5 0n the m a t t e r e1ectr0n den51ty••. 1n th15 1etter we der1ve 51mp1e exact ana1yt1ca1 f0rmu1ae f0r the m1x1n9 an91e5 017 ((1j) = (2 3 ), ( 13 ), ( 12 ) ) and the D1rac pha5e 6"~ 1n m a t t e r 1n the three-neutr1n0 m1x1n9 ca5e. U51n9 the5e f0rmu1ae we 5h0w that under 50me we11 5pec1f1ed c0nd1t10n5 51n26 m a150 90e5 thr0u9h a re50nance and that, c0n5e4uent1y, a den51ty ex15t5 5uch that 6m= •+ 2r/2 wh1ch, 1n a 5en5e, mean5 max1ma1 7 v101at10n 1n matter. 1n 0ur appr0ach we w111 u5e f0r the neutr1n0 m1x1n9 matr1x U the parameter12at10n [ 1 1,7,8 ] at 5ee ref1 [3 ] f0r a c0mprehen51ve rev1ew. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. ( N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n )

335

V01ume 226, num6er 3,4

U=

PHY51C5 LE77ER5 8

{1 0 ° r1° °][c3 ° 3][c20 5 c23 --523

523

0

1 0 0 e 1a"

¢23JL0

0 --513

1 0 0 C13j

-5,2

c,2 0

10 Au9u5t 1989

•]

=023(023)0(8)0~3(013)012(012),

(2)

where c~5= c05 0~j > 0 ,

% = 51n 0~/> 0 .

(3)

7 h e n the ev01ut10n e4uat10n f0r the amp11tude5 A~.~(t) 0 f t h e pr06a6111t1e5p~ ~ ( t ) 51t10n5 v~. ~v~ (~, ~ = e, ~t, x) 1n matter can 6e wr1tten 1n the f0rm

1~A,~(t)1= atLA~(t)j ~Am~,0~( 823)0(6) 013(013)012(012)

+

R{ 0 011

R230

= 1A~,~(t) 12 f0r the tran-

0127(012)0137(013)

FA0(,)1

0 01

0 0JJ

(4)

LA~(t)3

w1th 1n1t1a1 c0nd1t10n5: A~(0)=1,

1f~---~,

A~(0)=0,

1f~£~.

(5)

7 h e f0110w1n9 n0tat10n5 have 6een u5ed:

R23=Am2t/ Am21,

R3(Ne)=2x~ 6vNep/Am~ ,

(6)

where Arn~=m 2 - m 2 (rn1 < m 2 < m 3 ) , 6F 15 the Ferm1 c0n5tant and p 15 the m 0 m e n t u m 0f the re1at1v15t1c neutr1n05. 7 h e 5ma11 d1fference [ 12 ] 1n the c0herent f0rward 5catter1n9 0f the v, and v~ neutr1n05 ha5 6een ne91ected ~2. 1n matter w1th c0n5tant e1ectr0n num6er den51ty the 501ut10n 0fe4. (4) 15 very 51mp1e:

A~,~(8) = ]=~1 U~J eXP k--1 ~ P ~ P

M3•) Ur~* ,

(7)

where U m 15 the neutr1n0 m1x1n9 matr1x 1n matter. 1t 15 91ven 6y

(8)

U m = 023 (823)0(6) V m , w1th V m 6e1n9 a 3 × 3 0rth090na1 matr1x wh1ch d1a90na112e5 the matr1x M:

M-~013( 013)012( 012)1~

823

0127(012)0137(013) "•}-

0

0 0

~R23C23522"~523dVR 3 R23C13C12512 C13513( 1 --R23522 )4 R23c~2 --R23513C12512 I1" ~-1 R23C13C12512 L C13513(1--R23522) --R23513C12512 R23523522-•FC213 J

(9)

1n (7) Mj 0•= 1, 2, 3 ) are the e19enva1ue5 0f M:

Vm*MVm=d1a9(M,, M2, M3) • ~2 5uch a d1fference may turn 0ut t0 6e 1mp0rtant, h0wever, 1n matter w1th very h19h den51ty (p > 10~°9/cm3) [ • 2 ]. 336

(10)

V01ume226, num6er 3,4

PHY51C5 LE77ER5 8

10 Au9u5t 1989

7he expre5510n f0r them: {M,;j=1, 2, 3}=

{

,~

[1~

[-2a3+9a6-27c•

~(a2-36)~-c05~5Larcc05~

2(~5~2~

)+2kJr

])

a -3;

k=1,2,3

}

(11)

c01nc1de5, up t0 50me 519n5, w1th the re5u1t 06ta1ned 1n ref. [ 6 ]. We have u5ed 1n ( 1 1 ) the n0tat10n5

a-=-(1+R3+R23),

D.-=R23+R3[R23(1-c~3512)+c13], c~--R3R23c13512.

(12)

7he c01umn5 0f the 0rth090na1 matr1x V m are the e19envect0r5 0f the matr1x M. A 11near a19e6ra the0rem ex15t5 that the e19envect0r5 V,:~ 0f a rea1 5ymmetr1c matr1x M are 91ven 6y ( V;3)2 =D,/(D,1 +D2j +D3j),

(13)

where 04115 the 5u6determ1nant 0fthe e1ement (M,1-Mj) 1n the f0110w1n9 determ1nant:

M11 -M1 M12 M12 M22-M1 [

M13

M23

M13 M23

,

(14)

M33 -- M1

U51n9 f0r V"• a parameter12at10n 51m11ar t0 (2):

Vm=023( 0P2n31)013(0'1~Y)012( 0~1r1~) ,

(15)

we der1ve the expre5510n5 51n20~ =D23/(D23 + D 3 3 ) ,

• ~ ,m = 0 1 3 / ( 0 1 3 51n-013

"~ 1rn 51n-012 ~---012(D13 + D 2 3 + 0 3 3 ) / [ •

(D12 +022

+D23 +D33)

+ 0 3 2 ) (023 + 0 3 3 ) ] •

(16)

7he neutr1n0 m1X1n9 an91e5 02j m and the D1raC pha5e d m 1n matter are def1ned 6y ~3

U"~=023( 0m3)0(8m)013( 0~3)0~2( 0~2) •

(17)

1t 15 ea5y t0 Ver1fy that 0~3=0~1~,

0~1n2=0~1~ ~.

(18)

When the den51ty Are chan9e5 fr0m 0 t0 1nf1n1ty 51n220~3 and 51n220~ exh161t, 1n 9enera1, the re50nant 6ehav10r we a1ready kn0w fr0m the tw0-neutr1n0 m1x1n9 ca5e. F0r examp1e, 1f013 and 0r2 are 5ma1151n220~ e4ua15 1 when N~.~ Am~/2x/-2 6vP (1.e., when R3~- 1, 5ee f19. 1 ). 7he expre5510n f0r 0~ 15 51n 20~ =51n2(023 + 0 ~ ) -- 1 51n 2023 51n 202~ ( 1 - c 0 5 d) .

(19)

F0r the D1rac pha5e 1n matter ~m we der1ve the f0rmu1a ( dm=d-arct9

t90 "] ( t96 ) 1 +t9 023/t9 0~ c05 0~// - a r c t 9 1 - c t 9 023/t9 0~ c05 0~ "

(20)

1t 5h0w5 that 51n2dm may 90 thr0u9h a re50nance, 1.e., that an e1ectr0n num6er den51ty N ~ m a y ex15t5 5uch that

51n2d~(N~ ~) = 1 .

(21)

1ndeed, when, f0r examp1e, t9 023 15 5ma11 we can wr1te e4. (20) 1n the f0rm 1~3 At th15 p01nt We U5e exp11C1t1ythe fact that 05C1112t10n pr06a6111t1e5 P~], ~,,~(t) d0 n0t depend 0n the pha5e fact0r5 Wh1Ch mU1t1p1y U m

fr0m the 1eft. 337

V01ume 226, n u m 6 e r 3,4

/•

PHY51C5 LE77ER5 8

10 Au9u51 1989

1

1

),

0, 9

,5.4

0~. • "~" ~

....

/"

•

F19. 1. 7hree-d1men510na1 p10t 0f the funct10n 51n~0~3= f ( R 2 3 , R3) where R23 =,~tn~1/2Xm~1, R 3 = 2 x / m ~ 2 2 6vN,.p/Am31, 51n220~2=0.4 and 51n-~20~3~-0.2.

+,~1~.,.

~2:3 •

1

dm= arct9 1 + t9 0 ~ y ~ 023 c05 (5] - a r c t 9

1 - c t 9 023/t9 0 ~ c05 6) "

(22)

06v10u51y, 1n th15 ca5e e4. (21 ) 15 fu1f111ed 1f (23)

t9 0~2~(N~ ~) ~ t 9 023 c05 d.

7he 5ec0nd term 0n the r19ht-hand 51de 0f e4. (22) 1ead5 0n1y t0 a 5ma11 d15p1acement 0f the re50nant p01nt. Ana1090u51y, when ct9 023 15 5ma11 the re50nance 15 def1ned 6y t9 0 ~ ( N ~ ~) ~ c t 9 023 c05 d.

(24)

7he exact re50nance p01nt5 are ,m R6 t9023(N~ ) = 2 , =

1 t9023c05d

t9023 + [ ( 1 c05d 9023-c05d

~0~)

2

]~/2 +4

,

(25)

where 12 ~1--< 1221.7hree d1fferent ca5e5 are p055161e: Ca5e(1)

1t90~(N~)[<12L[

VNc.

7he re50nance c0nd1t10n cann0t 6e 5at15f1ed. N0 re50nance 0ccur5. Ca5e (11)

3N~R,~.. t9 0 ~ ( N ~ 6 ) = 2 , ;

,m (Nc) 1 < 1).2 [ [t9 023

VN~.

A re50nance den51ty N ~ 0 eX15t5 5UCh that 51n2d~ = 1 When dE ( - - 9 / 2 , 9 / 2 ) 0r When de (9/2, 3 9 / 2 ) , 6Ut n0t 1n 60th Ca5e5. Ca5e (111)

~N~1R,~( t"= 1 , 2): t9

,m ( N e 1 R 6) = 2 1 , 023

•rn ( N ~R9; t9 023 )=),~ .-

0 n e 0f the re50nance p01nt5 c0rre5p0nd5 t0 the ca5e 6~ ( - 9 / 2 , 9 / 2 ) and the 0ther t0 6~ (M2, 3 9 / 2 ) . 1n any part1cu1ar ca5e the p05516111ty wh1ch w111 6e rea112ed 15 determ1ned 6y the ran9e 0f va1ue5 wh1ch t9 0 ~ take5 when N~ 90e5 fr0m 0 t0 (~. F r 0 m e4. (16) we 9et [t9 0 ~ J = (D23/D33) 1/2 338

(26)

PHY51C5 L E 7 7 E R 5 8

V01ume 226, num6er 3,4

10 Au9u5t 1989

1t 15 n0t very d1ff1Cu1t t0 5ee that t9 0 ~

,v<..0+ 0

(27)

wh11e 1t90~2"71 ~

1c,25,21{5,3[(Am],1Am~,)-572]}1.

(28)

Hence, 1t90~"~(00) 1 > 12t1 (1221) 15 the 5uff1c1ent c0nd1t10n f0r the appearance 0 f a re50nance (f0r ar61trary

c~).

What 15 the phy51ca1 mean1n9 0f the re50nance e4uat10n (21 )• 1n vacuum the pre5ence 0f the pha5e c~ 1n the neutr1n0 m1x1n9 matr1x U (e4. ( 2 ) ) 1ead5 t0 CP v101at10n. 06v10u51y, when 8 = 0 , 9 (e~<~= •+ 1 ) CP15 c0n5erved. 7heref0re, d = • 5r/2 (e~<~= 11) mean5, 1n a 5en5e, max1ma1 CP v101at10n 0r, a55um1n9 CP7c0n5ervat10n, max1ma1 7v101at10n. Hence, 51nce 51n28~(N~~) = 1 mean5 max1ma1 7v101at10n 1n matter. "~ 2 and 0n N0 (thr0u9h 1n f19. 2 we pre5ent a three-d1men510na1 p10t 0fthe dependence 0f51n2~ m 0n Am5~/Am3~ the parameter R3 = 2 , / 2 6FN~p/Am 21 ). 1t 15 ea5y t0 5ee that f0r 1ar9er ma55 h1erarchy the re50nant p01nt m0ve5 t0 h19her den51t1e5 (0r d15appear5). 1n a manner very 51m11ar t0 the vacuum ca5e [ 13 ], the5e 7 v101at1n9 effect5 can 6e mea5ured [ 14 ] u51n9 the d1fference 6etween the 05c111at10n pr06a6111t1e5 P ~ >~ (t) and P ~ . . . . (t). 1n the parameter12at10n (2) the d1fference 15 pr0p0rt10na1 t0 Pvm~.~ (t) -P~"~ ,~ (t)0c (c05 0 ~ 1 8 ) 51n 20m~ 51n 2 0 ~ 51n 207•., c05 d m .

(29)

*t4 1n the VaCUUmCa5e th15 ha5 6een n0ted 1n ref. [ 15 ]. "5 7he pr0pert1e5 0f the repha51n9 1nVar1ant5 1~~k have 6een d15CU55ed preV10U51y [ 16 ] 1n C0nneCt10n W1th the pr061em 0f CP V101at10n 1n the 4Uark 5eCt0r.

•..///••

//"

/x// ../

15

0,8

/

1

~

0, 6 4

0,4 1 -"• r

•3 ¢, 0,

..1/

,:2-::

f

1

F19. 2. 7hree-d1men510na1 p10t 0f the funct10n 51n28m=f~(R2~, R3) Where R23=Am~ffAm2~, R3=2~f26vA~%p/Am~1, 51n220~-~=0.4, 51n22023=0.3, 51n220~=0.2 and 51n2~=0.1 ( ~ ( r r /

2, 3~r12 ) ).

%1

....

- ~

....

~2, . :

----

....

~

.....

~:23

]

- / %1

~/~6.01

163

~----~-¢" 1

F19. 3.7hree-d1men510na1 p10t 0 f t h e funct10n 1m(1 m) =f~ (R23, R3) where R23 = Am2ffAm31,2 , R~=2x/F26vN~p/Am~1, 9 5 n-20j2=0 4, 51n2202~=0.3 51n22013=0.2 and 51n25=0.1 (•0r/ 2, 37r/2)).

339

v01ume 226, num6er 3,4

PHY51c5 LE77ER5 8

10 Au9u5t 1989

7 h e expre5510n 0 n the r19ht-hand 51de 0 f e4. ( 2 9 ) 15 e4ua1 ~4 t0 the 1ma91nary part 0 f the repha51n9 1nvar1ant5 1 ~ ~5:

1m (1~k) = 1m ( U~ U~ U~* U~7) = (c05 0%/8 ) 51n 20~3 51n 207~3 51n 20~2 c05 6 ~n ,

(30)

where ~, ~ , ~ and 1, j, k are cyc11c. 7he 1ma91nary part5 1m(1~k) 0f a11 5uch 1nvar1ant5 1n the three-neutr1n0 m1x1n9 ca5e5 are e4ua1 51nce U m 15 a un1tary matr1x. E45. ( 2 9 ) a n d ( 3 0 ) 5h0w that the r e 5 0 n a n t d e p e n d e n c e 5 0 f 51n220,~ a n d 0 f 51n26 m 0 n the e1ectr0n n u m 6 e r den51ty Are are, 1n fact, tw0 d1fferent man1fe5tat10n5 0 f the effect 0 f r e 5 0 n a n t amp11f1cat10n 0 f 7 v101at10n 1n matter. 1n f19. 3 we 5h0w the d e p e n d e n c e 0 f 1m (1~ k) 0 n the rat10 0 n A m 2 ~/ Am 23 ~ a n d 0 n Ne ( t h r 0 u 9 h the p a r a m e t e r R 3 = 2 x / 2 6 v N e p / A m ~ ). 51nce 1 m ( 1 ~ k ) 15 pr0p0rt10na1 51mu1tane0u51y t0 51n 2073, 51n 2072 a n d c05 6 ~, 1t ha5 a m a x 1 m u m at a p01nt where the c0rre5p0nd1n9 re50nance5 are c105e t0 each 0ther -Are ~ A m 2 / 2 , , f 2 6v p (1.e., R3 ~- 1 ) a n d A m ~ / A m ~ ~ 2 (f0r the p a r a m e t e r 5 we have ch05en) (5ee f195. 1 a n d 2). 1n 5 u m m a r y , we have der1ved 51mp1e exact ana1yt1ca1 f0rmu1ae f0r the m1x1n9 an91e5 a n d the D1rac pha5e 6m 1n matter. U51n9 t h e m we h a v e 5h0wn that u n d e r 50me we11 5pec1f1ed c0nd1t10n5 den51ty N ~ ~ ex15t5 5uch that 51n2~m = 1, wh1ch, 1n a 5en5e, m e a n 5 max1ma1 7v101at10n. 7 h e a u t h 0 r 15 9ratefu1 t0 Pr0fe550r A.Yu. 5 m 1 r n 0 v f0r very u5efu1 d15cu5510n5 a n d t0 the 0r9an12er5 0 f the M 0 r 1 0 n d W0rk5h0p f0r the k1nd h05p1ta11ty dur1n9 the f1na1 5ta9e 0f th15 w0rk.

Reference5 [ 1 ] L. W01fen5te1n,Phy5. Rev. D 17 (1978) 2369; D 20 (1979) 2634. [ 2 ] 5.P. M1kheyev and A.Yu. 5m1rn0v, Yad. F12. 42 ( 1985 ) 1441. [ 31 5.P, M1kheyev and A.Yu. 5m1rn0v, U5p. F12. Nauk 153 ( 1987 ) 3. [4] 5.M. 811enkyand 8. P0ntec0rv0, Phy5. Rep. 41 (1978) 225, and reference5 there1n. [5] 5.M. 811enkyet a1., Phy5. Lett. 8 94 (1980) 495; M. D01 et a1., Phy5. Lett. 8 102 ( 1981 ) 323. [ 6 ] V. 8ar9er et a1., Phy5. Rev. D 22 ( 1980 ) 2718. [ 7 ] 7.K. Ku0 and J. Panta1e0ne, Phy5. Rev. Len. 57 (1986) 1805. [8] 5.705hev, Phy5. Len. 8 185 (1987) 177. [9] H.W. 2a91auer and K.H. 5chwar2er, 2. Phy5. C 40 (1988) 273. [ 10] P. Lan9acker, 5.7. Petc0v, 6.5te19man and 5. 705hev, Nuc1. Phy5.8 282 (1987) 589. [ 11 ] L.L. Chau and W.Y. Keun9, Phy5. Rev. Len. 53 (1984) 1802; H. Fr1t25ch, Phy5. Rev. D 32 (1985) 3058. [ 12 ] F.J. 80te11a, C.-5. L1m and W.J. Marc1an0, Phy5. Rev. D 35 ( 1987 ) 896. [ 13 ] N. Ca61660, Phy5. Le1t. 8 72 (1978) 333; V. 8ar9er et a1., Phy5. Rev. Lett. 45 (1980) 2084; 5.M. 811enkyand F. N1dermayer, 50v. J. Nuc1. Phy5. 34 ( 1981 ) 606. [ 14 ] 7.K. Ku0 and J. Panta1e0ne, Phy5. Lett. 8 198 (1987 ) 406. [ 151 H.-Y. Chen9, Phy5. Rev. D 34 ( 1986 ) 2794. [16] C. Jar15k09, Phy5. Rev. Lett. 55 (1985) 1039; 0.W. 6reen6er9, Phy5. Rev. D 32 ( 1985 ) 1841; D.-D. Wu, Phy5. Rev. D 33 (1986) 860.

340