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FLAVOR MIXING AND THE MASSES OF LEPTONS AND QUARKS
H. Fritzsch
To cite this version:
H. Fritzsch. FLAVOR MIXING AND THE MASSES OF LEPTONS AND QUARKS. Journal de
Physique Colloques, 1984, 45 (C3), pp.C3-189-C3-196. �10.1051/jphyscol:1984332�. �jpa-00224046�
JOURNAL DE PHYSIQUE
Colloque C3, suppl6ment au n03, Tome 45, mars 1984 page C3-189
FLAVOR MIXING AND THE MASSES OF LEPTONS AND QUARKS
H. Fritzsch
UnivezsitSt M g n c h e n , M P I f C r P h y s i k , F o e h d n g e r R i n g 6 , 0-8000 M iinchen, F.R .G.
A l l phenomena i n p a r t i c l e physics p r e s e n t l y known can be described w i t h i n the so- c a l l e d standard model, based on QCD
-
t h e oauge theory o f s t r o n g i n t e r a c t i o nphysics
-,
and on the SU(2) x U ( l ) theory-
t h e gauge theory o f t h e weak and e l e c t r c - magnetic i n t e r a c t i o n s . No doubt, the i n t r o d u c t i o n o f t h e standard model o f leptons and quarks i s a g r e a t achievement; a l l i n t e r a c t i o n s observed i n nature w i t h t h e exception o f g r a v i t y a r e described s u c c e s s f u l l y . Yet i t i s s u r p r i s i n g t h a t t h e theory gives very l i t t l e i n s i g h t i n t o t h e problem o f masses. I n t h e standard model t h e l e p t o n and quark masses a r e i n t r o d u c e d by hand. No e x p l a n a t i o n o f these masses o r o f mass d i f f e r e n c e s i s given. The masses o f t h e weak bosons W and Z a r e i n - troduced as f u n c t i o n s o f t h e vacuum e x p e c t a t i o n value o f a s c a l a r Higgs f i e l d @:M
-
e.
c d ~ l o > , ( e =J4aa').
Phenomenologically t h i s vacuum e x p e c t a t i o n value iF72etermined by the Fermi constant, i .e. by t h e s t r e n g t h o f t h e weak i n t e r a c t i o n amp1 i tudes a t low energies.During t h e l a s t t e n years an important progress has been made i n understanding the mass s c a l e s o f s t r o n g i n t e r a c t i o n physics. The t y p i c a l mass s c a l e o f phenomena i n s t r o n g i n t e r a c t i o n physics i s o f t h e o r d e r o f 1 GeV. This mass s c a l e i s a r e f l e c - t i o n o f t h e confinement phenomenon i n QCD. The l a t t e r i m p l i e s t h a t the quarks i n - s i d e a hadron a r e confined i n a f i n i t e r e g i o n o f space, whose extension i s described by a confinement parameter A ~ . A l l hadronic masses can be c a l c u l a t e d , a t l e a s t i n p r i n c i p l e , i n terms o f A,. Mass r a t i o s l i k e Mp/M a r e independent o f A and can be computed. Thus t h e theory o f quantum chromodynamycs allows us f o r t h e f i r s t time t o c a l c u l a t e t h e masses o f p a r t i c l e s .
O f course, QCD does n o t g i v e y i n f o r m a t i o n about t h e quark masses. The l a t t e r are i n t r o d u c e d as f r e e parametersfy, i .e. they p l a y a r61e s i m i l a r t o the one played by the e l e c t r o n
-
o r muon mass i n atomic physics o r i n quantum electrodynamics.The o r i g i n o f t h e e l e c t r o n
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o r quark masses i s s t i l l unknown. A t t h e present t i n e i t i s n o t p o s s i b l e t o reach a deep understanding o f t h e p r o p e r t i e s o f t h e l e p t o n - quark spectrum. However time has come t o speculate about c e r t a i n p r o p e r t i e s , f o r example about p o s s i b l e connections between the weak m i x i n g angles and t h e quark masses.The leptons and quarks can be c l a s s i f i e d i n t h r e e f a m i l i e s . Each f a m i l y has t h e charge s t r u c t u r e
i . e . i t i n c l u d e s a n e u t r i n o , a l e p t o n o f charge -1 ( i n u n i t s o f e), a quark o f charge 213 and a quark o f charge (-1/3) ( t h e t h r e e c o l o r s a r e denoted e x p l i c i t l y ) . The masses o f t h e charged fermions a r e as f o l l o w s ( i n MeV):
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984332
C3-190 JOURNAL DE PHYSIQUE
We s h a l l suppoze t h a t t h e t-quark e x i s t s . I t s mass must be g r e a t e r than about 22 GeV otherwise t h e tt ground s t a t e would have been observed i n t h e eie-- a n n i h i l a t i o n experiments2). The masses o f t h e quarks are, o f course, n o t measurable d i r e c t l y , b u t must be i n f e r r e d from observed p r o p e r t i e s o f t h e hadronic spectra. The mass values depend on t h e energy s c a l e used f o r t h e determination o f the quark masses.
lJe f o l l o w r e f . (1) and denote t h e quark masses a t an energy scale o f 1 GeV.
The eigenstates of t h e quark mass m a t r i x do n o t c o i n c i d e w i t h the weak i n t e r a c t i o n eigenstates. I t i s customary t o denote the weak i n t e r a c t i o n m i x i n g by i n t r o d u c i n g m i x t u r e s o f mass eigenstates f o r t h e charge -1/3
-
quarks d ' , s ' , b ' :where Vij i s a 3 x 3 u n i t a r y m a t r i x .
I n t h e formal l i m i t md = ms mb = 0 an a r b i t r a r y u n i t a r y t r a n s f o r m a t i o n o f t h e f i e l d s (d,s,b) can be made w i t h o u t changing t h e physics. I n p a r t i c u l a r t h e m a t r i x Vij can be diagonalized:
Thus t h e m a t r i x V i j has n o n t r i v i a l m a t r i x elements o n l y i f a t l e a s t one o f the masses md, m o r m b i s d i f f e r e n t from zero ( o r analogously one o f t h e masses mu, mc o r mt?. Therefore one may conclude: t h e quark masses and t h e weak m i x i n g angles a r e r e l a t e d . Any theory o f these masses must a t t h e same time g i v e i n - f o r m a t i o n about t h e weak m i x i n g angles, and v i c e versa. One may even discuss t h e hypothesis t h a t t h e weak m i x i n g angles are f u n c t i o n s o f the quark masses. This s i t u a t i o n a r i s e s provided thequark mass m a t r i x obeys c e r t a i n c o n s t r a i n t s , which are f u l f i l le d as a consequence o f u n d e r l y i n g d i s c r e t e o r continuous symmetries (see e.g. r e f . ( 3 ) ) .
Below I s h a l l discuss some p o s s i b l e connections between t h e weak m i x i n g angles and t h e quark masses. The observed p a t t e r n o f t h e quark masses e x h i b i t s c l e a r l y a w e l l
-
d e f i n e d h i e r a r c h y . The 1 ig h t e s t quarks (u,d) a r e much 1 ig h t e r than t h e quarks(c,s), which i n t u r n a r e s u b s t a n t i a l l y l i g h t e r than ( t , b ) . Before employing t h i s h i e r a r c h y f o r a s p e c i f i c purpose, l e t me mention a few f a c t s about t h e weak m i x i n g m a t r i x V .
..
1 J
The m i x i n g m a t r i x V i
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i s u s u a l l y w r i t t e n as a f u n c t i o n o f t h r e e m i x i n g angles ei ( i = 1,2,3) and a$base
parameter a:s1 C3 S3
(V. .) =
( "
- s 1 c 2 c 1 c 2 c 3 + s 2 s 3 e i6 c 1 c 2 s 3 - s 2 c 3 e 1 J-sl s2 c1 s2 c3
-
c s e c s s + c 2 c 3 e i a2 3 1 2 3
(c.= cosei, s.= s i n e . ) .
1 1 1
I n the l i m i t e2= eg= 6 = 0 t h e m i x i n g m a t r i x reduces t o t h e Cabibbo m i x i n g m a t r i x
(el: Cabibbo angle)
For small m i x i n g angles t h e m a t r i x V can be w r i t t e n approximately as f o l l o w s :
Since
m
>> m,
t h e b-quark decay can proceed o n l y v i a t h e m a t r i x elemente Vcb and Vub. l n t t h elhit
e2= e3= 0 the b-quark would be s t a b l e .Recently the l i f e time o f t h e b - f l a v o r e d mesons (quark composition i b , db) has been determined:
This l i f e time i s s u r p r i s i n g l y long. I f the m a t r i x elements Vcb ('dub) w r e o f the same o r d e r as t h e m a t r i x element Vus ( sz 0.22), one expects ~ ( b ) c 10- 15 s. Thus VUb and Vcb must be much l e s s than Vus.
Furthermore i t i s known t h a t t h e m a t r i x element Vub i s much l e s s than vcbz7)
Gathering a l l the informations about the weak m a t r i x elements (see e.g. r e f . ( 7 ) ) , one f i n d s a m i x i n g m a t r i x as f o l l o w s 8 ) :
(
0.9723.. .0.9737 0.228...
0.234 0.000...
0.008 (V..) = 1 J 0.228...
0.234 0.9704...
0.9726 0.042...
0.0670.003
...
0.016 0.041...
0.066 0.9979...
0.9991[ t h e complex phase parameter 6 has been omitted].
C3-192 JOURNAL DE PHYSIQUE
I t i s w o r t h w h i l e t o remark t h a t t h e submatrix
i s r a t h e r close t o t h e u n i t m a t r i x , i . e . n a t u r e seems t o be c l o s e t o t h e l i m i t where the heavy quark doublet (t,b) decouples from the o t h e r quarks. One i s i n v i t e d t o speculate about the reason f o r t h i s s u r p r i s i n g phenomenon. Below we s h a l l discuss two hypotheses about t h e quark mass m a t r i x , which a r e r e l a t e d t o t h i s "decoupling phenomenon".
I. Decoupl i n g . Hypothesis
Suppose o n l y the f i r s t two f a m i l i e s would be present. We consider the l i m i t mu = md = 0. I n t h i s case t h e masses of t h e quarks c and s a r e " i n f i n i t e l heavy c6mpared t o the (u,d)
-
masses, and one m i g h t expect t h a t i n t h i s l i m i t {he physics o f the (u,d)-
system decouples e n t i r e l y from the physics o f t h e (c,s)-
system. I n o t h e r words: the Cabibbo angle should vanish as mu, md j 0:9)I n r e f . (3) a s p e c i a l mass m a t r i x f o r the quarks was discussed, which l e d t o t h e re1 a t i o n :
where !3 i s an (unknown) phase parameter. This r e l a t i o n f u l f i l s t h e decoupling c o n d i t i o n .
I n t h e case o f s i x quarks t h e decoupling hypothesis i m p l i e s t h a t a l l weak mixing angles vanish as mu, md -> 0 and mc, ms -> 0:
i .e. the heavy quarks (t,b) decouple, and the b-quark i s s t a b l e . 11. S c a l i n g Hypothesis
Suppose we s t a r t from a c e r t a i n s e t o f weak eigenstates denoted by t h e doublets
I n t h i s b a s i s the mass matrixes w i l l n o t be diagonal. L e t m ( 2 / 3 ) t h e mass m a t r i x f o r (u,c,t), and m ( - 1 / 3 ) t h e mass m a t r i x f o r (d,s,b). We consider t h e s p e c i a l case w h e r e m ( 2 1 3 ) and (-113) a r e p r o p o r t i o n a l t o each o t h e r :
I n t h a t case a diagonal i z a t i o n o f m ( 2 / 3 ) leads a u t o m a t i c a l l y t o a diagonal i z a t i o n o f %I (-1/3). I n t h e b a s i s where t h e quarks o f charge (2/3) are mass eigenstates, t h e quarks o f charge (-113) w i l l a l s o be mass eigenstates. No weak i n t e r a c t i o n m i x i n g r e s u l t s , and t h e quark masses f u l f i l t h e r e l a t i o n s :
Suppose i t i s p o s s i b l e t o express t h e weak m i x i n g angles as functions o f the quark masses. I n t h i s case we f i n d i t n a t u r a l t o adopt t h e f o l l o w i n g s c a l i n g hypothesis A:
I f the masses o f t h e quarks o f charge (2/3) are p r o p o r t i o n a l t o t h e masses o f (-1/3)-charged quarks, a l l weak m i x i n g angles vanish.
The m i x i n g angles ei can o n l y be f u n c t i o n s o f t h e dimensionless r a t i o s o f quark masses m. / m. ( i , j = u,d,s ...). 'Jhatever the dynamics u n d e r l y i n g t h e weak m i x i n g phenomendn ma3 be, i t w i l l presumably be a dynamics i n v o l v i n g o n l y p r o p e r t i e s o f t h e quarks o f t h e same e l e c t r i c charge. I n o t h e r words: the m a t r i x elements o f t h e quark mass m a t r i x aa) ( 2 / 3 ) w i l l depend o n l y on p r o p e r t i e s o f the quarks o f charge 2/3, and w i l l n o t depend on p r o p e r t i e s o f the quarks o f charge (-1/3). We a r r i v e a t t h e second p a r t B o f t h e s c a l i n g hypothesis:
The weak m i x i n g angles depend o n l y on r a t i o s o f quark masses o f t h e same e l e c t r i c charge:
Below we discuss a few u s e f u l a p p l i c a t i o n s o f both t h e decoupling and s c a l i n g hypo- t h e s i s .
I f we apply t h e s c a l i n g hypothesis t o t h e s p e c i a l two-family-case discussed above (see r e f . (3) )
,
one f i n d s :As required, t h e m i x i n g angle depends o n l y on t h e r a t i o s of masses o f quarks w i t h the same e l e c t r i c charge. I n the l i m i t where (u,c) and (d,s) a r e p r o p o r t i o n a l , the angle vanishes.
I n r e a l i t y one has md / ms >> m / mc, i .e. one i s r a t h e r f a r from a p r o p o r t i o n a l i t y o f t h e quark masses. Taking theU quark masses as i n d i c a t e d above one f i n d s
ec 0.17. However, t h e values o f t h e u,d,s-quark masses a r e r a t h e r u n c e r t a i n . The measured value o f e, (ec 0.22) i s reproduced by t a k i n g e.g. mu = 12 MeV,
md = 15 MeV, ms = 150 MeV. We i n t e r p r e t t h e r e l a t i v e l y l a r g e value o f the Cabibbo angle as a consequence o f t h e l a r @ e d e v i a t i o n from t h e p r o p o r t i o n a l i t y case mu / mc = md / ms. I f t h i s were t r u e , t h e u-quark mass would be about e i g h t times l a r g e r than md. I n r e a l i t y mu i s l e s s than md, i m p l y i n g t h a t t h e neutron i s heavier than t h e proton.Thus t h e neutron- roto on mass d i f f e r e n c e and t h e r e l a t i v e l y l a r g e value o f 8, a r e i n t i m a t e l y r e l a t e d t o each o t h e r . Therefore t h e r e l a t i v e l y l a r g e value o f t h e Cabibbo angle has important consequences f o r low energy physics, e s p e c i a l l y f o r physics a t t h e I L L . I f ec were much s m a l l e r than observed, t h e neutron would be l i g h t e r than the proton, and n o t t h e neutron, b u t the p r o t o n would decay v i a t h e 6-decay process.
L e t us consider -the second and t h i r d generation. The u- and d- quark masses are very small compared t o a l l o t h e r quark masses. The l i m i t mu = md = 0 i s expected t o be a good approximation. Applying the decoupling c o n d i t i o n , one expects the (u,d) system t o be unmixed i n t h i s l i m i t , i .e. 8, = 0. The o n l y m i x i n g which can be present i s a m i x i n g between s and b, i . e . the s i x quark m i x i n g m a t r i x takes the form:
C3-194 JOURNAL DE PHYSIQUE
The (2 x 2 ) submatrix can be w r i t t e n i n terms o f one angle n:
The experimental data on t h e b-quark l i f e time imply t h a t t h e angle n i s very small: n
*
0.05. Thus one must be r a t h e r c l o s e t o t h e p r o p o r t i o n a l i t y case m t :m,
=mf,:
mg, i . e . mt = X ~ ? ' G ~ V . (Of course, t h i s value i s r a t h e r un- c e r t a i n , mostly due t o t h e u n c 8 r t a i n t x o f m ~ , which may vary between 150 and 200 MeV-
mt v a r i e s correspondingly between 35 and 47 GeV.)Applying t h e special mass m a t r i x discussed i n r e f . (3), one f i n d s
The c o n s t r a i n t n 0.05 given two p o s s i b l e s o l u t i o n s f o r mt. The a c t u a l value o f mt depends r a t h e r s e n s i t i v e l y on ms. I n t h e f o l l o w i n g t a b l e we g i v e t h e s o l u t i o n s f o r various values o f ms. F o r mc and mb (normalized a t E = 1 GeV) we adopt t h e values mc ( 1 GeV) = 1.35 GeV, mb ( 1 GeV) = 5.3 GeV (see r e f . ( 1 ) ) .
Sesides t h e values o f mt ( p = 1 BeV) we have a l s o denoted t h e value o f mt renormalized a t E = nt. (This i s t h e mass parameter t o be used i n e s t i m a t i o n s o f t h e spectrum o f t - p a r t i c l e s ) . The e x t r a p o l a t i o n o f mt (1 GeV) t o mt(mt) depends r a t h e r sensi- t i v e l y on t h e magnitude o f as (aS: s t r o n g i n t e r a c t i o n c o u p l i n g constant) o r r e - s p e c t i v e l y , on t h e s c a l e parameter A. Ne have used A = 100 MeV.
The f i r s t s o l u t i o n f o r mt i s compatible w i t h t h e experimental c o n s t r a i n t s f o r ms < 120 MeV, w h i l e t h e second s o l u t i o n gives r e l a t i v e l y l a r g e values f o r mt (above 40 GeV). We should l i k e t o s t r e s s t h e l a r g e d e v i a t i o n between mt ( 1 GeV), obtained from o u r considerations about m i x i n g angles, and mt(mt), which i s supposed t o be the quark mass value t o be used i n e s t i m a t i o n s o f the spectrum o f tt- and t-
f l a v o r e d p a r t i c l e s .
Independent o f whether t h e s p e c i a l mixing scheme discussed i n r e f . (3) i s c o r r e c t o r not, o u r hypotheses imply:
0.15 28 18 96 52
0.18 24 15 7 5 42 ms [GeV I
m t ( l GeV)
-
1. S o l u t i o n mt(mt)m t ( l GeV)
-
2. s o l u t i o n mt(mt).0.20 22 14 6 5 37
-
0.12 34 2 2 135 7 1
I f t h e determinant
vanishes, t h e s-b-mixing vanishes as w e l l . Nature seems t o be n o t f a r from t h i s case (see a l s o r e f . ( 1 0 ) ) . Nevertheless t h e t-quark mass depends s t i l l r a t h e r s e n s i t i v e l y on t h e a c t u a l value o f ms and on t h e s-b-mixing angle, and, o f course, on t h e s p e c i a l p r o p e r t i e s o f t h e m i x i n g mechanism. The t-quark mass can vary over a l a r g e range (see t h e t a b l e above).
U n f o r t u n a t e l y these observations about t h e s t r u c t u r e o f t h e quark mass m a t r i x d i s - cussed here g i v e very l i t t l e i n f o r m a t i o n about t h e l e p t o n mass m a t r i x . O f course, t h e l e p t o n i c weak m i x i n g angle can be s e t t o zero, i f a l l n e u t r i n o masses vanish,
i . e . we a r e d e a l i n g w i t h t h r e e unmixed weak doublets:
However as soon as a t l e a s t one o f t h e n e u t r i n o s acquires a non- zero r e s t mass, weak m i x i n g angles a r e expected t o be present.
He could discuss both t h e decoupling and s c a l i n g hypotheses i n t h e l e p t o n case as w e l l . However i t seems t h a t the s i t u a t i o n here i s d i f f e r e n t . The t h i r d charged l e p t o n T i s very heavy, b u t the associated n e u t r i n o v need n o t be much h e a v i e r than the o t h e r n e u t r i n o s v,, and ve. Thus one needs n o t be c l o s e t o the l i m i t
,, / mvT + me,Ft / m, -t 0, where t h e t h i r d d o u b l e t (v,,~) decouples from the ot.66; ones. I t seems t h a t t h e mechanism responsible f o r t h e generation o f n e u t r i n o masses ( i f t h e r e i s any), must d i f f e r q u a l i t a t i v e l y from t h e mechanism which generates t h e masses f o r t h e charged leptons and t h e quarks. The n e u t r i n o masses may w e l l show no h i e r a r c h i c a l p a t t e r n and be o f s i m i l a r o r d e r o f magnitude, i . e . mvp- m,
-
,, w N mv_, i n which case t h e m i x i n g among t h e n e u t r i n o s i s expected t o be large.,The search f o r non-zero n e u t r i n o masses ( f o r example by searching f o r neu- t r i n o o s c i l l a t i o n s ) remains an i m p o r t a n t task f o r t h e f u t u r e .I n t h i s t a l k I have discussed some ideas about t h e s t r u c t u r e o f t h e fermion mass m a t r i x , which have been developed i n c l o s e c o n t a c t w i t h experimental observations.
I have n o t mentioned t h e y e t unknown u n d e r l y i n g physics, which leads t o observed p a t t e r n o f masses and m i x i n g angles. Despite t h e r e g u l a r i t i e s i n t h i s p a t t e r n discussed above, the quark and l e p t o n mass spectrum i s r a t h e r complex. Probably a s o l u t i o n t o t h e problem o f masses can o n l y be found w i t h a theory, which takes the i d e a o f a s u b s t r u c t u r e o f l e p t o n s and quarks s e r i o u s l y l l ) . The most i n t e r e s t i n g p o s s i b i l i t y i s t o r e l a t e t h e f i n i t e range o f t h e weak i n t e r a c t i o n ( o f t h e o r d e r o f 10-l6 cm) t o t h e r a d i i o f leptons, quarks and weak bosons. I n t h i s case t h e l e p t o n and quarks masses are (perhaps) j u s t electromagnetic s e l f energies; t h e observed p a t t e r n o f masses and m i x i n g angles can be i n t e r p r e t e d t h a t way12). I n t h i s case t h e physics phenomena a t v e r y low energy, e.g. n e u t r i n o o s c i l l a t i o n s , t h e neutron- proton mass d i f f e r e n c e e t c . a r e i n t i m a t e l y r e l a t e d t o the s u b s t r u c t u r e o f t h e leptons and quarks, i .e. t o t h e physics a t energies o f t h e o r d e r o f 1 TeV o r l a r g e r
References and Footnotes
1. See e.g. : J. Gasser and H. Leutwyler, Physics Reports
8 7
(1982) 77.2. The present lower l i m i t on mt i s about 22 GeV (G. U o l f , p r i v a t e communication).
C3-196 JOURNAL DE PHYSIQUE
See e.g.: H. F r i t z s c h , Nucl. Phys. B 155 (1979) 189, and references t h e r e i n F o r r e c e n t reviews see: H. F r i t z s c h and P. Minkowski,
Physics Reports 73 (1981) 69.
C. J a r l s kog, Proceedings o f t h e I n t . Europhysics Conference on High Energy Physics, B r i g h t o n (UK)
,
1983.G. Hanson, Proceedings o f t h e I n t . Europhysics Conference, B r i g h t o n (UK), 1983 G. Chadwick, i b i d . , E. Fernandez e t al., SLAC
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p r e p r i n t 3154 (1983).See e.g.: G. A l t a r e l l i , Proceedings o f t h e I n t . NATO Summer I n s t i t u t e , Munich (1983).
F o r a review see: C. J a r l s k o g ( r e f . 4). See a l s o : K. Kleinknecht and B. Renk, t o appear i n Z. Phys. C (1983).
See a l s o : H. H a r a r i , Proceedings o f t h e I n t . NATO Summer I n s t i t u t e , Munich (1983.
F o r a d i f f e r e n t approach which f u l f i l s b o t h the decoupling and s c a l i n g hypotheses see: B. Stech, CERFI
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p r e p r i n t TH 3644 (1983).See e.g. : H. F r i t z s c h , Composite W-bosons and T h e i r Dynamics, MPI Munich p r e p r i n t (Nov. 1983).
To appear.in: Proceedings o f t h e I n t . School on Subnuclear Physics, E r i c e , Si c i 1 y (1983).
12. U. Baur and H. F r i t z s c h , t o appear i n : Phys. L e t t . B (1984).
