The invisible majoron

Phy51c5 Letter5 8 266 ( 1991 ) 408-412 N0rth-H011and

PHY51C5 LE77ER5 8

7he 1nv15161emaj0r0n W. F15ch1er, 6 . F . 61ud1ce, R . 6 . Le19h 1, 5. P a 6 a n 2 a n d 5 . 7 h 0 m a 5 7he0ry 6r0up, Department 0f Phy51c5, un1ver51ty0f 7exa5, Au5t1n, 7 x 78712, u5A Rece1ved 10 June 1991

We 91vea 5urvey0f c0up11n95and p055161e60und5 0n 51n91etmaj0r0n5, a550c1atedw1th 5p0ntane0u51y6r0ken 1ept0n num6er. We f1nd that the5e 60und5 can depend 5tr0n91y0n 5h0rt-d15tancephy51c5,6ut that, 1n any ca5e, maj0r0n5 rema1n e1u51ve.

dence f0r a 17 keV neutr1n0 [ 6 ] w1th c05m01091ca1 60und5.

1.1ntr0duct10n

1n the 1a5t few year5 a p055161e 501ut10n t0 the 501ar neutr1n0 pr061em ha5 emer9ed [ 1,2 ]. 7h15 501ut10n pred1ct5 a re50nant c0nver510n 0f e1eCtr0n-type neutr1n051nt0 mu0n-type neutr1n05. 7he data 50 far 5eem t0 6e c0n515tent w1th Maj0rana ma55e5 f0r the neutr1n05 0f 0rder 10-3-10 -5 eV. 7h15 ma9n1tude f0r the ma55e5 15 what 15 expected [ 3 ] fr0m the d1men510nf1ve 1ept0n num6er v101at1n9 0perat0r 1 (9L= ~ t (1H) 2 ,

(1)

where M = (X/~6Fmv) - t 15 0f 0rder 10 16 6 e V t0 1019 6eV, and mv 15 the Maj0rana ma55 0f the neutr1n0. 1n th15 paper we w111enterta1n the 1dea that 1ept0n num6er 15 6r0ken 5p0ntane0u51y [4] at a 5ca1e fx, a55umed t0 6e 9reater than the weak 1nteract10n 5ca1e 6 y ~/2. We w111f1nd 60und5 0n parameter5 a550c1ated w1th th15 5ymmetry 6reak1n9, 1n part1cu1ar the decay c0n5tant fx. 70 d0 50 we w111exam1ne the pr0pert1e5 0f the (51n91et) maj0r0n, 2, the 601d5t0ne 6050n a550c1ated w1th th15 5p0ntane0u51y 6r0ken 5ymmetry. Recent1y, 1t ha5 a150 6een 5u99e5ted [ 5 ] that th15 type 0f 5cenar10 may 6e 1nv01ved 1n rec0nc111n9 the ev1¢r Re5earch 5upp0rted 1n part 6y the R06ert A. we1ch F0undat10n and N5F 6rant PHY 9009850. Addre55after 1 5eptem6er 1991: 5anta Cru2 1n5t1tutef0r Part1c1ePhy51c5,Un1ver51ty0f Ca11f0rn1a,5anta Cru2, CA 95064, U5A. 2 Fu16r19htFe110w. 408

2. L0w ener9y phy51c5 and maj0r0n5

We w111a55ume that at 5ca1e5 6e10wfx 6ut a60ve the weak 1nteract10n 5ca1e the 119ht de9ree5 0f freed0m are th05e 0f the 5tandard m0de1 p1u5, 1n add1t10n, the maj0r0n. 8ecau5e 0fth15 5ymmetry 6reak1n9 there are, at ener91e5 6e10wfx, 1ept0n v101at1n9 0perat0r5 a5 we11 a5 0perat0r5 1nv01v1n9 the maj0r0n. 7he 10we5t d1men510n 0perat0r5 v101at1n9 1ept0n num6er are (9+ = 1 (1,H#+1#H,~)2,

(2)

(9- = M (1~H#-1#H")2~

(3)

where ~ are the 5tandard 1ept0n d0u61et5 and H 15 (are) the 5tandard H1995 d0u61et(5). 0~, f1.... are 5UL(2) 1nd1ce5 and 9enerat10na1 1nd1ce5 are 5uppre55ed. (1fthere 15 0n1y 0ne H1995 d0u61et, (9+ and (9• are n0t 1ndependent. ) 8e10w the weak 5ca1e, the5e 0perat0r51ead t0 Maj0rana ma55e5 f0r the neutr1n05. 7he 0perat0r5 1nv01v1n9 2, the maj0r0n, mu5t have the pr0perty that the 2er0 m 0 m e n t u m c0mp0nent 0f th15 f1e1d dec0up1e5. 1n 9enera1, the 10we5t d1men510na1 0perat0r5 1nv01v1n9 2 are 1

(4)

0370-2693/91/$ 03.50 • 1991 E15ev1er5c1encepu6115her58.V. A11r19ht5re5erved.

V01ume 266, num6er 3,4

1 0 4 y v ffY1,~

(9r= ~ ~. 4 n ~ 2 . ~

PHY51C5 LE77ER5 8

,

(5)

where F ~ , 15 the 5UL (2) f1e1d 5tren9th and F ~, 15 the weak hyperchar9e f1e1d 5tren9th. At 5ca1e5 a60ve the weak 5ca1e there are n0 0ther d1men510n-f1ve 10ca1 0perat0r5 c0up11n9 the maj0r0n t0 119ht f1e1d5. 7h15 15 6ecau5e 1n the 11m1t 0f 1nf1n1te f~ there 15 c1a551ca11y an exact c0n5erved 1ept0n num6er carr1ed 6y the u5ua1 1ept0n5 and 50 there 15 n0 601d5t0ne 6050n c0up11n9 t0 the 119ht 1ept0n current. 7 h e current t0 wh1ch the maj0r0n d0e5 c0up1e ha51t51ead1n9 c0ntr16ut10n fr0m the f1e1d5 that ac4u1re 1ar9e 5 U L ( 2 ) × U r ( 1 ) 1nvar1ant ma55e5 after 1ept0n num6er 5ymmetry 6reak1n9. 1nte9rat1n9 the heavy f1e1d5 at the 5ca1e M w111 then, 1n 9enera1, 1ead t0 the f0110w1n9 5UL(2) × U y ( 1 ) 1nvar1ant d1men510n-5even 0perat0r5:

C7 ~ ~

1

0~2

~

(1.Ha +1pH~.)*a~( HHP +1PH~) , (6)

1 0~,2(1,~Hp~1pH,~).a~,(1,~Hp~1pH~)

(7) 51nce the5e 0perat0r5 have a much 5ma11er effect *~ at 10w ener91e5 than the d1men510n-f1ve 0perat0r5 90_+, (9w and (~r, we w111c0ncentrate 0n the 1atter. 7 h e 6e5t 60und5 w111 c0me fr0m the c0up11n9 0f the maj0r0n t0 ph0t0n5 thr0u9h the 0perat0r 1

a

~

Cv= ~ X ~ F~,~Fvu~ ,

(8)

29 Au9u5t 1991

m e n t [ 8 ] wh1Ch re11e5 0n a re50nant effect that ex15t5 f0r ma551e55 p5eUd05Ca1ar5 W1th the C0Up11n9 (8) t0 ph0t0n5. 7 h e maj0r0n 15 1ndeed Very C105e t0 6e1n9 ma551e55; 1t5 C0mpt0n wave1en9th 15 mUCh 1ar9er than the pre5ent h0r120n. (A1th0u9h the 1ept0n1C Current 15 an0ma10U5, 1t 15 0n1y Weak1y an0ma10U5, theref0re n0n-pertUr6at1ve effect5 [9 ] W1119enerate an exp0nent1a11y 5ma11 ma55, w1th an add1t10na1 5uppre5510n due t0 ferm10n1c 2er0 m0de5 that carry 6ary0n num6er. ) 7he exper1ment 1n 4ue5t10n c0n51der5 the 1nduced 61refr1n9ence 0f the vacuum pr0duced 6y the m1x1n9 0f the ma551e55 p5eud05ca1ar w1th the ph0t0n 1n the pre5ence 0f a ma9net1c f1e1d. 7he5e exper1ment5 pre5ent1y 91ve a 10wer 60und [8 ] 0f ~ 103 6 e V f0r the 1nver5e m a j 0 r 0 n - p h 0 t 0 n c0up11n9. 0 n e a150 m19ht w0nder 1f there are 60und5 fr0m acce1erat0r exper1ment5. 0 n e pr0ce55 that we have c0n51dered a5 an examp1e, 5h0wn 1n f19. 1, 15 the 6rem55trah1un9 0f a maj0r0n 6y an 0n-5he11 2 °. 7 h e 0ff-5he11 2 ° then decay5 t0 tw0 1ept0n5 0r tw0 jet5, 91v1n9 an event 519nature w1th an exce55 0f m1551n9 ener9y. 5earche5 f0r 5uch event5 w1th a 2 ° 6ranch1n9 rat10 at the 1eve10f ~ 10 -6, w11191ve a 60und 0 f a 6 0 u t 0n1y 20 6 e V 0n the 1nver5e 2 - 2 ° - 2 ° c0up11n9. A 51m11ar re5u1t may 6e 06ta1ned f0r the 2 - 2 ° - 7 c0up11n9 [10]. A5 f0r c05m0109y, there 15 n0 upper 60und here fr0m c0herent 05c111at10n5 a5 f0r the ax10n, 51nce there 15 n0 p0tent1a1 f0r the maj0r0n. 7here w111, h0wever, 6e 9106a1 5tr1n95 f0rmed after 1ept0n num6er 15 5p0ntane0u51y 6r0ken. 6106a1 5tr1n95 are expected t0 reach

where F~,, 15 the e1ectr0ma9net1c f1e1d 5tren9th.

3. 8 0 u n d 5

A5tr0phy51C5 pr0v1de5 900d 60Und5 0n the C0Up11n9 0f 119ht 6050n5 t0 ph0t0n5, a5 1n the Ca5e 0 f t h e ax10n. 7he 6e5t 60und C0me5 fr0m the he11Um 6urn1n9 11fet1me 0f10w ma55 5tar5 [7 ], wh1ch re4u1re5 the 1nver5e m a j 0 r 0 n - p h 0 t 0 n c0up11n9 ~2fx t0 6e 1ar9er than ~ 107 6eV. 7here are a150 5evera1 terre5tr1a1 1a60rat0ry exper1ment5 ava11a61e. F0r examp1e, there 15 an exper1*~ 7h15 a55umpt10n 154ue5t10na61e 0n1y1fM 15n0t much 1ar9er than the e1ectr0weak 5ca1eand neutr1n0 D1rac ma55e5are re1at1ve1y1ar9e.

,2 7he 60und5 that we 91ve1n th15 5ect10n0 n fx a55ume that the 0perat0r5 Cv, etc., appear 1n the ham11t0n1anw1th un1t c0eff1c1ent. 1n a11ca5e5,we are 60und1n9an effect1vedecay c0n5tant, wh1ch may d1fferfr0m 0perat0r t0 0perat0r.

.2 t

£~, 4

£7,4 F19. 1. Pr0ce551ead1n9t0 a 60und 0n the X-2°-2° c0up11n9. 409

v01ume 266, num6er 3,4

PHY51c5 LE77ER5 8

a 5ca11n9 d15tr16ut10n 1n a manner 51m11ar t0 9au9e 5tr1n95 [ 11 ] and 50 can 5urv1ve unt11 the pre5ent. F0r 9106a1 5tr1n95 the 6e5t upper 60und 0n the 5tr1n9 ten510n/t 15 expected t0 c0me fr0m the 065erved 150tr0py 0f the m1cr0wave 6ack9r0und, wh1ch 91ve5 6 N # < 0 . 5 × 10 -5 [1 1,12]. 7he 5tr1n9 ten510n f0r a 9106a1 5tr1n9 15 1t~nf~ 1nfxR, where R 15 re1ated t0 the typ1ca1 1nter-5tr1n9 d15tance, r0u9h1y the h0r120n 512e. 7h15 91ve5 an upper 60und 0f ~ 10 ~5 6 e V f 0 r fx. 7h15 60und c0u1d 6e av01ded 1f an 1nf1at10n 0ccurred after the 5tr1n95 f0rmed. 7here are 0ther p055161e 60und5 0n the maj0r0n 0f a5tr0phy51ca1 0r191n5 [ 13 ]. 1ndeed, 0ne can 100k at the c0up11n95 0f maj0r0n5 t0 e1ectr0n5 and nuc1e0n5. 7he5e m19ht 6e c0mpet1t1ve 1n the ca5e where the maj0r0n-ph0t0n c0up11n9 15 5uppre55ed.

29 Au9u5t 1991 <46>

H

"2 7 J/ < ~N

F19. 2. Feynman 9raph that pr0duce5 6+~. H

"~~1,N

4. 5h0rt d15tance phy51c5 7he5e 60und5 0nf~ can 6e weakened c0n51dera61y. 7h15 may 6e ach1eved 1f the heavy ferm10n5, wh05e ma55 0r191nate51n the 6reak1n9 0f1ept0n num6er, d0 n0t 9enerate an an0ma10u5 v101at10n 0f the 1ept0n current. 70 111u5trate the dra5t1ca11y d1fferent 6ehav10ur at 10w ener91e5, we w1116r1ef1y de5cr16e tw0 51mp1e examp1e5.7hey have d1fferent f1e1d c0ntent at the 1ept0n 6reak1n9 5ca1e and 1ndeed have a very d1fferent 6ehav10ur at 10w ener91e5. 1n the f1r5t examp1e, where a u5efu1 60und 0n f~ 15 06ta1ned, c0n51der 1n add1t10n t0 the 5tandard m0de1 an 5U (2) tr1p1etferm10n .~# that ha5 n0 weak hyperchar9e. 7he new ren0rma112a61e term5 1n the 1a9ran91an 1nc1ude

~ = ~9~.~#(1,~H# +1#H,) + 9~ ~,~t3 ~ # +h.c. - V(H, ~),

(9)

where the p0tent1a1 V(H, 4~) 15 5uch that 1t 9enerate5 ( H ) = (2x/~ 6v) -1/2 and ( ~ ) =fx. 1n th15 ca5e the 1ept0n current ha5 a c0ntr16ut10n t0 the 5UL(2) an0ma1y fr0m the tr1p1et. At 10w ener91e5 th15 1ead5 t0 the maj0r0n-ph0t0n c0up11n9 0f e4. (8). 0 n e theref0re expect5 that the 60und5 06ta1ned 1n the 1a5t 5ect10n app1y t0 f2 d1rect1y. 7h15 m0de1 a150 9enerate5 the d1men510n-f1ve 0perat0r (9+ and the d1men510n5even 0perat0r (97.7h15 can 6e 5een 6y 1nte9rat1n9 0ut the heavy tr1p1et ~ # at tree 1eve1 (5ee f195. 2 and 3). 410

H "

~N9

F19. 3. Feynman 9raph 1ead1n9t0 ~ 0r (;~. A5 a 5ec0nd examp1e c0n51der the add1t10n 0f an

8UL(2) XUy( 1 ) 51n91et [4] t0 the 5tandard m0de1. 7he add1t10na1 term5 1n the 1a9ran91an are

~N = 9NN (1H ) + 9 ~ dp(NN) + h . c . - V ( H, 40), (10) where N15 an 8UL(2) XUy( 1 ) 51n91et. 1n th15 ca5e at ener91e5 6e10w the weak 5ca1e there 15 5t111a maj0r0nph0t0n 1nteract10n due t0 the weak an0ma1y 0f the 5tandard 1ept0n current (5ee f19. 4); h0wever, 1t 15 5uppre55ed. 1ndeed, 1nte9rat1n9 0ut the heavy 51n91et at tree 1eve1 9enerate5 the 0perat0r (9% 0fe4. (7); 5ee f19. 3 . 7 h e effect1ve term 1n the 1a9ran91an 15 then 0/ ~2 mv 2

~a~ 4tc 47r M fx Fvu~F~uv~

( 11 )

where M ~ 9 ~ ( ~ ) 15 n0w the heavy 51n91et ma55. N0te the extra 5uppre5510n fact0r (0t2/4~)m,,/M, where mv 15 the 119ht Maj0rana neutr1n0 ma55 that ar15e5 fr0m the d1men510n-f1ve 0perat0r 0f e4. ( 1 ). 1n add1t10n t0 the ar9ument 91ven ear11er u51n9 the 11m1t 0ff2~00, th15 extra 5uppre5510n can 6e under5t00d 6y a 5119ht1yd1fferent 5ymmetry ar9ument. 80th ar9ument5 u5e 11m1t5where the heavy and 119ht f1e1d5

V01ume 266, num6er 3,4

PHY51C5 LE77ER5 8

29 Au9u5t 1991

J

W

W

4

J80r J L ~

~,~x,,t~x..t~ W

4

W

1

6

F19. 5.7hree-100p d1a9ram c0up11n9tree-1eve1current5 t0 c010ur F/v.

mv/M.

........~ V 7 F19. 4. Feynman 9raph 1ead1n9t0 the X-7-~/amp11tude, and 1t5 ana109 at 5ca1e5far 6e10w the W-ma55. 7he x-v-v c0up11n9repre5ented 6y the dark c1rc1e15pr0duced 6y the 9raph 0ff19. 3. dec0up1e. 1n the tw0 m0de15 a60ve, 1n the 11m1t that 9~ (0r 90e5 t0 2er0, there 15 an exact c1a551ca1 Uh(1)×Ud1) 5ymmetry (at 5ca1e5 a60ve fx) 1nv01v1n9 5eparate pha5e r0tat10n5 0f the heavy and 119ht f1e1d5. 7 h e 601d5t0ne 6050n a550c1ated w1th the 6reak1n9 0f th15 5ymmetry c0up1e5 0n1y t0 the heavy f1e1d. 1n the ca5e 0f the 51n91et, the maj0r0n can c0up1e t0 9au9e f1e1d5 0n1y 1f9N#0. 1n 0ther w0rd5, the m a j 0 r 0 n - p h 0 t 0 n c0up11n9 15 the remnant 0f an 0perat0r 0f d1men510n h19her than f1ve and 15 5up-

9N)

pre55ed 6y a fact0r C0nver5e1y, 1n the ca5e 0f the tr1p1et ferm10n, Uh( 1 ) ha5 an 5UL(2) an0ma1y, and thu5 there 15 a c0up11n9 0f the maj0r0n t0 ph0t0n5 un5uppre55ed 6y 9~. N0t1ce that e4. ( 11 ) d0e5 n0t mean that there 15 an e1ectr0ma9net1c an0ma1y f0r the 1ept0n current carr1ed 6y the 5tandard 119ht 1ept0n5, wh1ch w0u1d v101ate the Ad1er-8ardeen the0rem [ 14 ]. 7he 5ame 155ue ar15e5 1n the 5tandard m0de1 where 1t 1nv01ve5 91u0n5, w1th c0n5e4uence5 that w0u1d 6e dramat1c. D0e5 the 1ept0n 0r 6ary0n current have a Q C D an0ma1y 9enerated at three 100p5 (5ee f19. 5)• 7 h e an5wer 15 06v10u51y n0. What happen5 15 that 1n the pr0ce55 0f 1nte9rat1n9 m0de5 w1th 5h0rt wave1en9th5, 0perat0r m1x1n9 0ccur5. 1n part1cu1ar, the u5ua1 1ept0n 0r 6ary0n current m1xe5 thr0u9h a 6 1 M 60x w1th the weak 6ary0n current (5ee f19. 6). A150 the 0perat0r 6u11t 0ut 0 f 5 U ( 2 ) f1e1d 5tren9th5 m1xe5 w1th 1t5 Q C D c0unterpart thr0u9h a 4uark 100p (f19. 7). 1n 0ther w0rd5 a5 5h0rt d15tance m0de5 are 1nte9rated 0ut there 15 a 5ma11 adm1xture 0f 6ary0n cur-

Fa.u~,au~ f

W

J8 0r

4

J L ~

>

~

J8

J8 0~ JL W

F19. 6. D1a9ram5a550c1atedw1th the m1x1n90f tree-1eve1current5. 411

v01ume 266, num6er 3,4

PHY51C5 LE77ER5 8

w

6

5 w

4 6

F19. 7.61M 60x d1a9ram a550c1ated w1th the m1x1n9FF 0f 5u~.( 2 ) w1th c010ur FF. rent 1n the 1ept0n c u r r e n t a n d v1ce ver5a, pre5erv1n9 1n effect 6 0 t h 51de5 0 f the an0ma1y e4uat10n 1n acc 0 r d a n c e w1th the A d 1 e r - 8 a r d e e n t h e 0 r e m . 1n c0nc1u510n, we h a v e f 0 u n d t h a t the 5tandard 51n91et m a j 0 r 0 n 15 1ndeed 1nv15161e 6ecau5e 0 f the 5uppre5510n fact0r m,,/M 1n a11 amp11tude51nv01v1n9 the m a j 0 r 0 n . 7h15 can a150 6e t h 0 u 9 h t 0 f a5 9enerat1n9 an effect1ve decay c0n5tant fefr~ (M/mv)fx f0r the m a j 0 r 0 n . 6ener1ca11y, h 0 w e v e r , 5 u p e r h e a v y 1ept0n5 w111 9 e n e r a t e an an0ma10u5 v101at10n 0 f 1ept0n n u m 6 e r , a n d the 60und5 0n the5e m0de15 are t h e n 51m11ar t0 60und5 0n ax10n5. 1n any ca5e 1t w111 6e h a r d t0 k n 0 w w h e t h e r 0r n0t 1ept0n n u m 6 e r 15 5p0ntane0u51y 6r0ken.

Ackn0w1ed9ement We w0u1d 11ke t0 t h a n k V. Kap1un0v5ky f0r u5efu1 d15cu5510n5.

412

29 Au9u5t 1991

Reference5 [ 1 ] L. w01fen5te1n, Phy5. Rev. D 17 (1978) 2369; 5.P. M1kheev and A.Yu. 5m1rn0v, 50v, J. Nuc1. Phy5. 42 (1985) 913. [2] H.A. 8ethe, Phy5. Rev. Lett. 56 (1986) 1305; J. 8ahca11 and H.A. 8ethe, Phy5. Rev. Lett. 65 (1990) 2233. [3] 5. we1n6er9, 5ummary ta1k Xx111 1ntern. C0nf. 0n H19h ener9y phy51c5 (8erke1ey, CA, 1986); 1ntern. J. M0d. Phy5. A2 (1987) 301. [4 ] Y. Ch1ka5h19e, R.N. M0hapatra and R.D. Pecce1, Phy5. Lett. 898 (1981) 265. [5] 5.L. 61a5h0w, Phy5. Lett. 8 256 ( 1991 ) 255. [6] J.J. 51mp50n and A. H1me, Phy5. Rev. D 39 (1989) 1825; A. H1me and J.J. 51mp50n, Phy5. Rev. D 39 (1989) 1837; A. H1me and N.A. Je11ey, Phy5. Lett. 8 257 ( 1991 ) 441; 8.5ur et a1., Phy5. Rev. Lett. 66 ( 1991 ) 2444. [7] D.A. D1Cu5, E.W. K016, V.L. 7ep11t2 and R.V. Wa90ner, Phy5. Rev. D 18 (1978) 1829; M. Fuku91ta, W. Watamura and M. Y05h1mUra, Phy5. Rev. Lett. 48 (1982) 1522; 6 . 6 . Raffe1t and D.5.P. Dear60rn, Phy5. ReV. D 36 (1987)

2211. [8] 6. Raffe1t and L 5t0d015ky, Phy5. Rev. D 37 (1988) 1237; Y. 5emert21d15 et a1., Phy5. Rev. Lett. 64 (1990) 2988. [9] 6. •t H00ft, Phy5. Rev. Lett. 37 (1976) 8. [ 10] J.E. K1m and U.W. Lee, Phy5. Lett. 8 233 (1989) 496. [11] A. V11enk1n and 7. VaCha5pat1, Phy5. Rev. D 35 (1987) 1138. [ 12] F.R. 80UChet, D.P. 8ennett and A. 5te661n5, Nature 335 (1988) 410. [ 13] 6.6. Raffe1t, Phy5. Rep. 198 (1990) 1. [ 14 ] 5.L. Ad1er and W.A. 8ardeen, Phy5. Rev. 182 (1969) 1517.