A decomposition theorem for SU(n) and its application to CP-violation through quark mass diagonalisation
P P D I V A K A R A N a n d R R A M A C H A N D R A N Tata Institute of Fundamental Research, Bombay 400 005 MS received 24 September 1979
Abstract. It is proved that the group G = SU(n) has a decomposition G = FCF where F is a maximal abelian subgroup and C is an ( n - 1) * parameter subset of matrices. The result is applied to the problem of absorbing the maximum possible number of phases in the mass-diagonalising matrix of the charged weak current into the quark fields; i.e., of determining the exact number of CP-violating phases for arbitrary number of generations. The inadequacies of the usual way of solving this problem are discussed. The n = 3 case is worked out in detail as an example of the constructive procedure furnished by the proof of the decomposition theorem.
Keywords. Semisimple Lie algebras; Cartan decomposition; CP-violating phases;
Kobayashi-Maskawa matrix; decomposition theorem; quark mass diagonalisation.
1. Introduction
T o i n c o r p o r a t e C P - n o n i n v a r i a n c e within the s t a u d a r d g a u g e m o d e l o f w e a k inter- actions, t h e c u r r e n t l y f a v o u r e d p r o c e d u r e is t o write the c h a r g e d w e a k c u r r e n t as
= ur,, ½ (1--r~) ~D, (1)
where U = (u 1, u~ . . . u,,) a n d D = (d 1, d~ . . . d,) are, respectively, sets o f ' u p ' q u a r k s o f c h a r g e @ 2/3 a n d ' d o w n ' q u a r k s o f c h a r g e - - 1/3. These q u a r k fields are defined as the eigenvectors o f the q u a r k mass o p e r a t o r s . T h e u n i t a r y n × n m a t r i x c will i n c o r p o r a t e C P - v i o l a t i o n if its elements include c o m p l e x n u m b e r s w h o s e phases c a n n o t be e l i m i n a t e d by redefining the p h a s e o f i n d i v i d u a l q u a r k fields.
T h e m a t r i x c is t o be d e t e r m i n e d as follows : the Y u k a w a c o u p l i n g o f the g a u g e g r o u p d o u b l e t s (u °, d °) with t h e Higgs field(s) leads to the mass m a t r i c e s M ( U ) a n d M ( D ) in the bases U °, D o respectively. These matrices are d i a g o n a l i s e d by u n i t a r y m a t r i c e s vtl , v D respectively, w i t h
vuU°=
U, VDD° = D.T h e c h a r g e d c u r r e n t is t h e n
0 0 D O -
Y = y~ ½ (1 - - ~'5) ~ Uy~ ½ (1 - - Ys) vD, (2)
where v is, in general, a n n2-parameter u n i t a r y matrix. W e a r e t h e n t o factorise v in such a w a y as t o separate a n overall phase t o be a b s o r b e d i n t o t h e W to w h i c h 1
p 47
couples and to similarly absorb as many additional phases as possible into the indi- vidual q u a r k fields. It is the last problem (raised, especially, by the phrase ' a s m a n y as possible ') that is the concern o f this paper. W h a t is required is a decom- position o f a general unitary matrix in the f o r m
v = f c f ' exp (ix), (3)
w h e r e f a n d f ' are ' m a x i m a l ' diagonal unitary matrices, X is real and c is the matrix required in (1). Clearly, we may choose exp ( i x ) ~ det v. Since the maximal abelian subgroup o f SU(n) has (n--- 1) parameters, (3) will imply that c is a matrix with at least n ~ - 1 - - 2 ( n - - - l ) : ( n - - l ) 2 parameters. On the other hand if c were real, it will be an orthogonal matrix and so can have at most ½ n ( n - - 1) para- meters ( ' E u l e r angles') so that the general c will have at least ½ ( n - 1) ( n - - 2 ) ' p h a s e angles'. These angles are responsible for CP-violation.
F o r n - 3, a parametrisation o f c in terms o f these Euler angles and one phase angle was first written down by Kobayashi and M a s k a w a (1973). Following them, it has generally been taken, on insufficient grounds, that there are precisely ½ ( n - - 1) ( n - - 2 ) phase angles in the general case. It is obvious f r o m the above that this n u m b e r is only a lower bound on the number o f phase angles - simple counting is not sufficient to establish the exact number o f CP-violating phases. T o justify the count- ing, what is required is a p r o o f of the decomposition t h e o r e m (3) (a more explicit critique o f the usual incomplete argument as given, e.g., in two recent expository articles (Harari 1976; Ellis 1978) will be f o u n d in the concluding section). We supply such a p r o o f here. Our p r o o f is also constructive; it lets us write down syste- matically the matrix c as a function o f Euler and phase angles. The case n : 2 is o f course trivial. T h e n ~ 3 case is sufficiently complicated to illustrate fully the general p r o c e d u r e ; extension to n > 3, if and when more generations o f fermions are discovered, only costs more labour.
Because o f the nature of the problem, this p a p e r is mathematical in content and form. Its direct relevance to the description o f an i m p o r t a n t physical p h e n o m e n o n , that o f CP-violation in weak-electromagnetic gauge theories should, however, be clear f r o m the remarks above.
2. The general decomposition theorem
The basic mathematical tool we use is a decomposition o f a connected semi-simple Lie g r o u p with a finite centre into factors which are its one-parameter subgroups.
This decomposition itself follows f r o m the cartan decomposition o f semi simple Lie algebras (see, e.g., H e r m a n n 1966). The result we wish to arrive at is e q u a t i o n (3) or, m o r e precisely (after first factoring out det v : : exp (ix)), the following:
Theorem: The g r o u p SU(n) --- G has a decomposition G ~ FCF, where F i s a ( ( n - - 1) dimensional) maximal abelian subgroup and C is a ((n--1)2-parameter) subset o f
SO(n).
Our p r o o f o f this theorem proceeds by first working out a suitable decomposition into o n e - p a r a m e t e r subgroups followed by a reordering o f factors. F o r complete- heSS, a n d as a n aid t o easy understanding, we give below in a subsection a b r i e f
summary o f relevant standard general results without proofs (two books we have found useful are Helgason 1962 and H e r m a n n 1966). The subsequent subsections o f this section prove o u r t h e o r e m with all details given.
Notation: A capital letter (e.g. G) will denote a group and the same letter in bold face (G) the corresponding Lie algebra. Lower case letters stand for group (or, when in bold face, Lie algebra) elements (g and g respectively).
2.1. Summary o f relevant general theory
Cartan's fundamental t h e o r e m on decompositions of semi-simple Lie algebras is t h e starting point:
Theorem 1 : A semisimple Lie algebra G has a direct sum decomposition into a sub- algebra K and a vector subspace P satisfying (i) [K, K] C K (i.e., K is a subalgebra);
(ii) [K, P] c P (i.e., adK leaves P invariant)* ; and (iii) [P, P] c K.
A K satisfying these conditions is a symmetric subalgebra. The corresponding subgroup K o f G is a symmetric subgroup and the coset space G[K a symmetric space.
A maximal abelian subalgebra o f P is called a Caftan subalgebra and d e n o t e d , typically, by A.
The analogue of t h e o r e m 1 for Lie groups is
Theorem 2: Let G be a connected Lie group with finite centre whose Lie algebra is G, K the connected subgroup whose Lie algebra is K, and P the exponentiation o f P.
Then G has the decomposition G : KP.
K and P (and hence K and P ) may be defined by the action o f a linear a u t o m o r p h i s m o n G : ~ ( k ) : k f o r k E K , ~ ( p ) : - - p for p E P , ¢b 2 - - i d e n t i t y . U n d e r the exponential map, these conditions become 95 (k) : k, k E K and 95 (p) : 95 (exp p) - - exp [~ (p)] - : exp ( -- p) __p-l, p E P, on the a u t o m o r p h i s m 95 on G. A 95 (or ~ ) satisfying these properties, called a symmetric automorphism, always exists u n d e r the conditions stated. Finding a symmetric a u t o m o r p h i s m is a convenient practical way of carrying out a C a f t a n decomposition, a way we shall follow. The require- ment that G must have a finite centre gives one more reason for working with SU(n) rather than U(n).
One further result we need is
Theorem 3 : If A is a C a r t a n subalgebra and A' is any abelian subalgebra of P, then there exists a k E K such that Adk(A') c A.
Given a decomposition G : KP, we may decompose K and P further. I n the case o f K, since it is a subgroup, one simply carries the KP decomposition a step further. As for P, it follows from Theorem 3 that every element o f P, considered as a one-dimensional (abelian) subalgebra o f P, can be written as adk (a) for a in a fixed t a r t a n subalgebra A and some k E K. Applying the exponential map, we then have P : AdK (A) : K A K (i.e., p : k'ak '-1 for some k' E K, a E A), so that
G = K A K ~ K1A1KI,
is the first step in the required decomposition. Now let K~. be a symmetric subgroup o f K1 (and Az correspondingly). The next stage o f the decomposition is
G ~ K~A~K~AtK2A~K~,
*The definitions and some properties of the adjoint maps we need here are given in the Appendix, P.--4
and so on. Eventually we have a decomposition into one-parameter subgroups, the particular decomposition depending on particular choices o f symmetric subgroups at each level. The familiar Euler angle decomposition o f SO(3, R) is a simple appli- cation o f this procedure, as shown in H e r m a n n (1966).
2.2. The case G : SU(n)
For G = SU(n), consider the map ~ (g) = goggo x with g~ = identity. The choice*
(1 )
) m l a - - 1(4)
is the most convenient [even though go is not a n element o f SU (n) for even n, is always a n (outer-)automorphism]. It is easily checked that the set K - ~ -(k I¢(k) = k]- is a subgroup, in fact the group S [U ( 1 ) × U ( n - - l ) ] ; k E K has the general f o r m
exp (in) )
k = , ( s )
exp [ - - i a / ( n - - 1)] v~x where ~ is real a n d v~_x E SU (n q 1).
We now determine P = {p ] ~ (p) = - - p]- and verify at the same time that G = K P is a caftan decomposition. Firstly, write go = exp go with
(o )
o N ~ )
i~1,. x
and g = exp g. Then $ (g) = go exp (g g~X) == exp (go g g~l) = exp [~) (g)] (see the appendix). F o r ¢ (g), we use the formula
¢) (g) = go g ~;' --- exp go g exp ( - - go),
= g + [go, g] + ~. [go, [go, g]] + 1 .--
Now, from (5), a n y k E K which commutes with go has the f o r m
('° )
k . = , (7)
- - [ i a / ( n - 1)] vn-1
*Here and in the following, we indicate the dimensionality of matrices by subscripts whenever it is necessary to be explicit. All blank entries in matrices stand for zero.
with vn-x E SU ( n - - 1) and a real. Clearly, P, the orthogonal complement of K, consists of all skew-hermitian matrices of the form
B~_x )
P I I =
- - B + I
(s)
where Bn-1 is a ( n - - 1 ) dimensional row vector. It follows that [go, g] : irtp and using the formula for ~ (g) above, we get
_ (i,,),
• (g) = g + i,,p t - ~ . , p + ...
g - - p + p exp (i~r) = g ~ 2p.
When g = k, p = 0 and we have @ (k) : k; and when g : p, we have @ (p) : p - - 2p : - - p, thus verifying that what we have exhibited is indeed a cartan decomposition.
Finally, exponentiating the right side of (8), P is itself the set of all matrices of the form
cos fl
P~ = \ - - f l sin fl B.+x
BL1 sin fl '~,
1,..1 + bn_x (cos 3 - - I)/8'
/
where fl is the non-negative real number (B~_I B+~-t) 1/2 and b,_ x is the matrix B+_~ B~I.
Having found the decomposition G = KP =_ K 1 Px, we have to split K s and P1 further. K x can be further split by repeating our earlier procedure on the SU(n---1) submatrix. Choose
(1 )
o~0p L...1 =
1~_~
so that ks E K s is of the form
( e x p (i~) /
k.. == exp [--ia/(n--l)+i~]
exp [--i~/(n--l)--ifl/(n--2)] v,,-~
where
v~_, E SU(n---2), for the decomposition K1 = K2 P2.pt E P=
is obtained by exponentiatingPl "~-
(0)
0 B . - , •
- n L , o
Correspondingly,
Further choices of go are now obvious. The decomposition of Pi in the sequence is achieved by writing Pt E Pt in the form k' t at k't -1 for at E At, the maximal abelian subgroup of P~, and
k'i E Ki.
The maximal abelian subgroup of Px (in fact, of any Pt) is easily seen to be one- dimensional. A 1 is of the form
0 b
- - b 0
a 1
0 L :
On exponentiation, we have, then
a I ~
where r2 is the SO(2, R) matrix
r~(b) ~- (
c o s b
\ - - s i n b The decomposition into sequence.
g = k i P l,
= K ~ p 2 K I a l K I ,
. . . 1
03
sin:).
c o s one-parameter
. ~ r . t 1 . t --' 1
-~-K2K2a2K 2- KlalK l -
=k~a2k~-lk[alk[ -x
~ & a b t - - l b t a b ~ r a c b ~ t - l k P - 1
~ K 2 a 2 K 2 - K l a l x 2 a 2 K 2-
~ e t c .
, bread.
subgroups then goes in the following
Ck
s = k z ,(k~ =k~k~').
It is clear that for SU(n), the subscripts on the k" at either extreme will be (n--l) (i.e., they will be products of n--1 factors) and that each k,_ 1 is a diagonal one-parameter matrix. We have thus the result we wanted; the extreme factors are elements of the maximal abelian subgroup of SU(n). The factors in the middle recombine to form the
' C a b i b b o - K o b a y a s h i - M a s k a w a ' matrix. Concrete section m a k e matters even clearer.
examples treated in the next
3. Explicit construction for n = 2, 3
3.1 n----2
The trivial SU(2) example is already instructive: Write g E SU(2) as*
[ c(0) exp (i31) s(O) exp (i8~)
g = | ] ~ v 2 (0 ; 81, 82).
\ - - s ( O) exp (--i32) c( O) exp (--iSz) ] F o r our choice go = diag (1, - - 1 ) , we have
k 1 = diag (exp (i81), exp (--/81)); Pl = v~(O; O, 8'), ( 8 ' : ½ (33--31)).
A x in this case is SO(2) and Px has the decomposition
! I _ 1 l
Pz :klr=(O)kz , k 1 : diag (exp (i8'), exp (--i8')).
Hence, the general U(2) matrix decomposes as
exp (ix) vz (0; 81, 32) - - diag [exp (ix+i3), exp (ix--i3)]
× rz (0) diag [exp (--i8'), exp (i3') [8 = ½ (31+32) ].
Thus, the three phase parameters X, 8 and 8' occur only in the extreme factors and can be ' a b s o r b e d ' by redefining the individual quarks. Alternatively, the overall phase can be got rid o f by redefining the W ± field to which the current constructed by sandwiching the U(2) matrix couples and 81 and 82 by redefining the quarks. T h e essential point is that even though we have five non-hermitian fields, only three phases can be got rid o f by redefining them, the reason being that the maximal abelian subgroup o f SU(2) is one-dimensional--a point which is obscured in the usual discussions. A U(2) matrix has o f course only three phase parameters and so there is no CP-violation.
3.2 n = 3
F o r v 3 : g E SU(3), go is diag (1, - - i, - - 1), k a E K 1 is o f the f o r m
( exp (i~1) )
kl : , v 2 E SU(2),
exp ( - - iq~1/2) vg.
*To avoid or at least to shorten, whenever feasible, the explicit writing down of big matrices, we follow from now on some additional notational abbreviations: v2(O; 31, 32) stands for the general SU(2) matrix parametrised as above (v~(0; 0, 0) = r~(O), the rotation matrix), diag (x, y .... ) for the diagonal matrix with diagonal entries x, y ... and c(O) and s(O) for cos 0 and sin 0,
and Px E Px is generated by
0 b x b~ )
- - b *
which is a 4-parameter matrix. A 1 is one-dimensional, in fact SO(2), and we choose the parametrisation
r2 (01) )
a 1 -~-
1
t t I t - - 1 t
Px cart then be written Px ---- xlalxx , kl ~ SU (2) c Kx, i.e..
( 1 )
k x ---~
v2 (03; ~1, %)
. . . . . .
1 (exp l, )
Therefore, vz : glKlalg I :
exp (i$1/2) v 2 (03; ill, #3)
×
(r2 01' 1)(1 1., )
Now, each of the SU(2) matrices can be further factorised as in § 3.1 to give v3 = diag [exp (i$1), exp (--i$1/2)1 diag [exp (ifl), exp (--ifl)l
X
1 ] \ [1, exp (--//r), exp (i~')]
diag
rz (02)
/
×
r2 (01) 1 )
diag [1, exp (ia), exp (--ia)]
× 1 | diag [1, exp (--ia'), exp (ia')],
\
r~ ( - - 0 8 )
/
where ¢, --- ½ (al + as), a = ½ (/3, + as), a' ---- ½ (a~-- as), 3' ---- ½ (fll--Bs).
The eight parameters #1, 83, 08, a, fl, a', fl' and 41 characterise v3 fully. We n o w reorder some o f the factors in order to exhibit another one parameter matrix o n the right extreme. This can be done in m a n y ways: the one we choose is
v 8 = d i a g [exp (i$1), exp
(i3-- i$J2-- ifl'),
exp ( - -ifl-- i~1/2-- ifl')
× diag [1, 1, exp (i8)]
r 2 (09.) 1
× 1 ] diag [I,
(i¢--ia'), (in +
ia')],\
exp exp
r= ( - - 03)
1
where 8 = 2 ~ ' - - a).
The product o f the middle four matrices is the Kobayashi-Maskawa matrix which, in full glory, is
f c (00 s (0,) c (08) - - s (00 s (08)
c3
- - s (01) c (02)
s (01) s (00
c (01) c (02) c (08) + s (02) s (03) exp (i8) - - c ( o 0 s (03) c (08)
+ c (o2) s (08) exp (i8)
- - c (0O c (0~) s (08) + s (02) c (03) exp (i8) c ( 0 0 s (02) s (08)
+ c (o2) c (03) exp
(i03
F r o m the point o f view o f the general decomposition o f § 2, this is not quite what we were after: neither the matrix c3 nor the diagonal matrices to its left and right is a unimodular matrix, even though the product v z is (there are only eight parameters; 8 is not independent). Once we choose to exhibit c3 in the Kobayashi-Maskawa f o r m , the phase matrices will, in general, not be unimodular. There are alternative ways of decomposition which will have a different 8-dependence and be unimodular. In any case, what is required is a decomposition of U(n) and not SU(n). But since the general theorems are applicable to SU(n) and not to U(n), we were forced to factorise the determinant out. It can be restored at the end without any difficulty as we have seen.
4. Conclusions
As we stated in the introduction, there are a number o f discussions in the literature on the question dealt with in this paper, at least as far as the counting o f absorbable phases (which in turn gives the number of CP-violating phases) is concerned. T h e y can be summarised in one sentence: in general, the 2n quark fields on either side o f the unitary matrix arising f r o m the mass diagonalisation can all have their phases redefined
except for one overallphase,
giving 2n--1 absorbable phases. This argument is fallacious for a number o f reasons. Firstly, as we saw in the introduction, it only gives the m a x i m u m n u m b e r o f absorbable phases and the m i n i m u m number o fCP-phases. M o r e importantly, the phrase italicised above is misleading because if the issue was settled directly by the number of (non-hermitian) fields available, there would be 2nq- 1 absorbable phases: one overall (goes into W ±) and 2n for the quarks, which is obviously absurd. The point is that the overall phase is not 'except' but 'in addition', and the correct counting is not ( 2 n ) - - I b u t 1 -q- 2 ( n - - l ) , 1 overall and ( n - - l ) for the rank (the dimension of the maximal abelian subgroup) o f SU(n).
In a n y case, the constructive procedure we have described f o r n = 3 can be carried t h r o u g h for larger n in exactly the same way.
Acknowledgements
R R wishes to thank Aravind Banerjee and A m i t a b h a Mukherjee for helpful discus- sions. P P D acknowledges the hospitality of the Max-Planck-Institute f a r Physik und Astrophysik, Munich, where a part of this work was done.
Appendix
l f g 0 E G, the inner a u t o m o r p h i s m A d , o (g): g - ~ go ggo 1 is an analytic isomorphism o f G onto itself. We write Adg 0 for the differential o f this m a p near the identity, which is an a u t o m o r p h i s m o n the tangent space (i.e. G, the Lie algebra o f G) such that exp (Adg o (g)) = go exp g go 1. The set o f a u t o m o r p h i s m s Ad G form a group G L (dim G) and the m a p g0-+ Adg0 is a group h o m o m o r p h i s m o f G into G L (dim G).
The differential o f this map near the identity, a h o m o m o r p h i s m of G into G L (dim G), is written as adg0 and is given by adg 0 (g) : g -+ [go, g] (Helgason 1962).
Consider the a u t o m o r p h i s m given by q~oo (g) : go g g o z where go 2 -- 1 and g E G (which is o f the kind used in the text). For matrix groups (which is what concerns us in this paper) we will show that this defines (I)~0 (g) : go ggo I with g ~ G.
Writing g (t) = exp (tg) we have
~oo (g (t)) --- go exp (tg) g o 1
[ t2g -~ ]
=go 1 ~ - t g + ~ +... go z.
(For the case o f matrix groups the product go g is defined).
this close to t = 0 will define ~go (g) on the Lie algebra.
given by
I~o0 (g) = go ggo 1,
The differential m a p o f This is easily seen to be
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Helgason S 1962 Differential Geometry and Symmetric spaces (New York and London: Academic Press)
Hermann R 1966 Lie Groups for Physicists (New York: W A/3enjamin) Kobayashi M and Maskawa T 1973 Prog. Theor. Phys. 49 652
