- Email: [email protected]
New tools for analyzing quark mixing
Physics Letters B 284 ( 1992 ) 390-393 North-Holland
PHYSICS LETTERS B
New tools for analyzing quark mixing Alexander Kusenko
1,2
Institute for TheoreticalPhysics, State University of New York, Stony Brook, NY 11794-3840, USA Received 10 March 1992; revised manuscript received 14 April 1992
We present some new mathematical tools which help to derive information about the quark mass matrices directly from experimental data and to elucidate the structure of these mass matrices.
One hopes that a successful choice o f a mass matrix help to suggest an a p p r o p r i a t e dynamical model explaining the quark mass spectrum. In any case, one has to be able to write a mass matrix consistent with the experiment, regardless of what particular d y n a m i c s are responsible for generating quark masses. Over the years, a n u m b e r o f proposals have been a d v a n c e d for quark mass matrices, such as those in refs. [ 1-6 ]. In this paper we would like to present some useful m a t h e m a t i c a l tools for the study of quark mass matrices and the connection with quark mixing. The starting point o f our study is an observation which has so far not been exploited: since the C a b i b b o - K o b a y a s h i - M a s k a w a ( C K M ) quark mixing matrix V is an element o f the Lie group U ( 3 ) , it is in one-to-one correspondence with an element H o f the Lie algebra u (3). H is thus the infinitesimal generator for V. It is o f interest to d e t e r m i n e the H corresponding to the empirically measured V. As we shall show, one can also use H to facilitate the d e t e r m i n a t i o n of families of quark mass matrices which are consistent with experiment. (Here we use the fact [ 7 ] that there are three light neutrinos, hence three usual SU ( 2 ) doublets ofleptons, and, as required, e.g., by a n o m a l y cancellation, thus also three SU ( 2 ) doublets of quarks, so that V~ U (3). ) We denote the quark mass matrices for the up and down sectors M , and Md correspondingly. By means o f a bilinear t r a n s f o r m a t i o n one can diagonalize Mu and Md:
Uu,LMuUtu,R= d i a g ( m u ,
me,
mr)-Do,
Ud,LMd U~,R = d i a g ( m d , ms, mb)=--Dd,
(1)
where Uu,L, Ud,L, Uu,R, Ud,R are unitary matrices. Below we shall suppress the subscript L. In the s t a n d a r d m o d e l with one left-handed charged weak currents, the matrix UR is not a measurable quantity; and therefore one may choose the Uu,R and Ud,R in such a way that Mu and Md are both hermitian. Indeed, an arbitrary complex 3 × 3 matrix is defined by 18 real parameters. In order to make it hermitian, one has to impose 9 constraints to eliminate 9 degrees o f freedom. To do that one may use 9 degrees o f freedom associated with each o f the matrices Uu,R and Ud,R. Below, we assume that both Mu and Md are hermitian. The experimental data [ 7] provides one with the absolute values o f the matrix elements of the C K M mixing matrix V= L~U*d (taken to be 3 × 3). One also obtains certain phase information from experiment which, for a given p a r a m e t r i z a t i o n , can b e t r a n s l a t e d into a d e t e r m i n a t i o n o f the phases of the elements of V. Suppose the pair o f h e r m i t i a n mass matrices Mu and Md,
M~=U*~DoUu,
Md = UtdDd Ud ,
(2)
Supported in part by NSF contract PHY-89-08495. -~ Email address: [email protected]. 390
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 284, number 3,4
PHYSICS LETTERS B
25 June 1992
corresponds to some particular matrix V. Then one can see that for every unitary matrix Uo there exists another pair of hermitian matrices m'u=(UuUo)tDu(UuUo),
m~ = (Ud L~)+D~(L~ Uo) ,
(3)
that corresponds to the same matrix V. One can also see that the family of matrices M'u and M~j as a function of an arbitrary unitary matrix Uo exhausts all possible solutions for the mass matrices. This proves that for every mixing matrix V~ U ( 3 ) there exists a 9-parameter family of matrices that might be chosen to be mass matrices. Empirically, the data constrains [ VI, the matrix of magnitudes of each of the elements of V, to be close to the identity matrix. Therefore, it is useful to represent Vas V= exp (it+H) = Uu U+~,
(4)
where H is some hermitian matrix and c+ is a real number. One may choose H to have its dominant (largest, in absolute value) eigenvalue to be of the order of 1, yet smaller than 1. Then for c~ consistent with the data [ 7 ], we find that I c~[ ~ 0.3. Below we will use the fact that c+< 1. Suppose that V is known. Six parameters, the phases of the quark fields, are arbitrary. We use three of them to rephase Vto make it be close to the 3 × 3 identity matrix. This is possible since I VI ++ 1. Then one can calculate the matrix c~H using Sylvester's theorem [ 8 ]: 3 l-[,++k( V - v, X 1) ic~H= ~ ln(v,) k=, Fi**k( v~ - v , )
(5)
The latter is valid as long as all the eigenvalues of V are distinct. If the matrix V has two or three degenerate eigenvalues, then In (V) is not uniquely defined, and there is a family of matrices H that satisfy (4). This family o f solutions is explicitly shown in ref. [ 8]. However, this mathematical subtlety is not of physical importance since one can always avoid the degenerate case by some small change in the phases of quark fields that removes any initial degeneracy in eigenvalues. The theorem ( 5 ) is particularly powerful since it gives an exact result in a closed form. We will demonstrate how one could construct a set of mass matrices starting from the mixing matrix V. In particular, we choose to describe those mass matrices that can be diagonalized by a unitary transformation U~:o generated by an element in the u ( 3 ) algebra that commutes with H. In terms of the matrix H, V={ + i t c H - ½c~2H2+ . . . + ( 1 / n ! ) ( i t c H ) " + ....
(6)
One can also decompose an arbitrary UI or U_,, that commutes with H, in a similar series whose coefficients are proportional to certain real numbers: {x},u) } and {x~ ~) } where n = 1, 2, 3 .... : Uu =~ +ic+xl~)H - Ix~U) oLZHZ'k-...-+- ( 1/n!)x},">(io+H)"+ .... Ud =~ +ie+xld)H - ½X~zd>c~2HZ + . . . + ( 1/n! )x~fl> (ic+H)"+ ....
(7)
This series is convergent if sup n=
[x,,I < ~ v .
1+2+3 ....
The constraints UuU+~= V, U+~U~ =+ and U+dUd----Mare satisfied simultaneously up to the third order in c~ if the coefficients are chosen as follows: L~=~+io+Hx-+o+2H'-x2+ ....
L ~ = ' ~ + i o + H ( x - 1 ) - ½ 0 z 2 H ' - ( x - 1 ) 2 + ....
(8)
where x is some real parameter. Then, to the second order in 0~, one may express Mo and Md as functions of x: M~=U*~Dug~=Du+iOex[Du, H] -~o~ , 2x 2 [ [ D . , H ] , H ] +
....
(9) 391
Volume 284, number 3,4
PHYSICS LETTERSB
Md = U*dDdbh =Dd + i a ( x - - 1 )[Dd, H] - l a 2 ( x -
25 June 1992
1 )2[ [Dd, H], H] + ....
(9 cont'd)
We see that, in our parametrization, one gets a good approximation of the M as a function of one real parameter. Thus, from Vand the resultant H, we have obtained a family of (experimentally acceptable) mass matrices depending on a parameter x. In passing, we note the following technical point: for rapid convergence of the series (9), V should be close to ~, the identity. The three phases of the diagonal elements of V are not physical and they may be changed by rephasing the quark fields. Such a change in the overall phase of V corresponds to shifting a by a real number. Clearly, ] c~] is the smallest for V~ ~, which yields rapid convergence for both (6) and (9). The family of solutions (9) for the "up" and "down" quark mass matrices (accurate to second order in a ) is Mu ~Ou + i a x [ D u , m ] - J2a2x2[ [ Du, H], H] , Ma ~Do + i a ( x -
1) [Do, m ] - ½ a 2 ( x - 1)2[ [Da, H], H] .
(10)
To illustrate the method with a simple numerical example let us take / 0.97525 o V=~-0.22099exp(i0.0359 ) \ 0 . 0 0 6 6 6 5 7 e x p ( - i 2 8 . 4 0 °)
0.22104 0.97430 exp(i 0.00184 o) -0.043403 exp(i 0.9485 °)
0.0050614 exp ( - i 40.00° )'~ 0.043619 J 0.99904
(11)
(consistent with current data [ 7 ] ). Note that this gives [ Vub[ / [ ~%b] = 0.11 and, for the parametrization-invariant CP violation quantity [9 ] J = Im{ VI~ ~2 V~*2V~ } the value J = 3.1 × 10- 5 First, we calculate the eigenvalues of V: v~=0.97415+0.22587i, v2=0.97443-0.22468i, v3=0.999999+ 0.00122i. Next, we substitute these into ( 5 ) to calculate H and obtain 0 aH= 0.223i \-0.0032+0.001i
-
0.223i 0 0.044i
- 0.0032- 0.001i'~ -0.0044i ).
(12)
We now choose the quark masses to be equal to their running masses evaluated at Qo = 1 GeV and with A = 200 MeV in the MS renormalization scheme. From the analysis of ref. [ 10]: r~u(Qo) -- (5.1 + 1.5) MeV, r~d(Qo) = (8.9 _+2.6) MeV, t~(Qo) = ( 175 _+55) MeV, rh~(Qo) = ( 1.35 +0.05) GeV, mb(Qo) = (5.9+0.1) GeV. We also use the value t~, (Qo) = 220 GeV corresponding to a physical mass of m t phys = 130 GeV, consistent with current constraints. Specifically, we choose the diagonal mass matrices to be D~=diag(5MeV,-1.35GeV, 220GeV),
Dd=diag(8.9MeV,-175MeV, 5.9GeV).
(13)
Suppose x = ½. In this case, one retains a symmetry between the "up" and "down" sectors in ( 15 ). For these values of quark masses the first order approximation of the "up" and "down" sector mass matrices gives the following: { -5.09×10-5 M , = 2 2 0 G e V ~ (6.77X10-4)exp(i3 °) \ ( 2 . 3 7 X 1 0 - 3 ) e x p ( - i 4 2 °) ( M~ =5.9 GeV
1"49X10-3 -3.47×
10 - 3
\ ( 1 . 7 8 X 1 0 - 3 ) e x p ( i 6 6 °)
( 6 . 7 7 × 1 0 - 4 ) e x p ( - i 3 °) - 5 . 5 8 × 1 0 -3 -0.0221 --3.47X10-3 -0.0293 0.0226
(2.37X10-3)exp(i42°)'~ -0.0221 ), 0.99951
(1.78X10-3)exp(--i66°)) 0.0226 . 0.99951
(14)
(15)
Event though this is only a second order approximation it still shows what is the relative value of different terms in the mass matrices for this particular case of x = ½. For instance, the matrix Vcalculated back from Mu a n d M d is different from the original Vonly in the third significant digit. It is of interest to express the CP violation quantity J directly in terms of H. Recall that 392
Volume 284, number 3,4
PHYSICS LETTERS B
25 June 1992
1
J=
-
--det(C), 2Fuk~
16)
where F,=(mt-mc)(mt-mu)(m,.-mu), C=-i[mu,Md]
17)
Fa=(mb--ms)(mb--ma)(ms--ma),
18)
.
Now ell~ = U~D. U~ = U~ V*Du VCd = Ud*(D~ + i a [Du, H] - ½~2[ [D~, H ] , HI + ...)Ud,
M~ = Ud*Dd Ud,
19)
so that up to the first n o n - v a n i s h i n g order in c~, i C - [Mu, aid] = i [og-+-0(o~2)] U*d[ [H, D , I , Dd] Ud.
(20)
Using (20), we thus find J - - [o/+0(o~ 2 ) ] 3 R e [ H , 2 H 2 3 H 3 t ]
.
(21)
This result allows one to calculate the parameter J up to the small corrections of the order of c~2. The higher order lerms can be obtained by retaining the corresponding higher order terms in (20). For example, for the numerical illustration ( 11 ) - ( 15 ), (21) yields J-~ 3.2 X 10- 5, very close to the exact value J = 3.1 X 10-5. These results thus establish a useful connection between the M matrices, the unitary transformations which diagonalize them, and the actual quark mixing matrix V. Furthermore, they introduce an interesting object, the matrix H in Lie algebra u (3) corresponding to V in the Lie group U (3). We have shown how with our tools, one can obtain a family of experimentally acceptable mass matrices directly, given knowledge of V. This technique complements the earlier typical one of hypothesizing some ansatz for a mass matrix and then checking to see if it fits experiment. The author would like to thank Professor Robert E. Shrock for many helpful discussions and comments.
References
[l ] Fritzsch, Phys. Len. B 70 (1977) 436; B 73 (1978) 317: Nucl. Phys. B 155 (1979) 189. [2 ] P. Kauss and S. Meshkov, Mod. Phys. Len. A 3 ( 1988 ) 1251- Phys. Rev. D 42 1990) 1863. [3] C.H. Albright and M. Linder, Z. Phys. C 44 (1989) 673; C.H. Albright, Phys. Len. B 246 (1990) 451. [4] G.C. Branco and L. Lavoura, Phys. Left. B 208 (1988) 123: G.C. Branco, Phys. Rev. D 44 ( 1991 ) R582. [5] S.N. Gupla and J.M. Johnson, Phys. Rev. D 44 (1990) 2110. [6] R.E. Shrock, Phys. Rev. D 45 (1992) RI0; Intern. J. Mod. Phys. A, in preparanon. [ 7 ] Particle Data Group, J.J. Hernfindezet al., Reviewof particle properties, Phys. Lett. B 239 (1990) 1; M. Danilov, in: Proc. 1991 Intern. Lepton-photon Conf. (Geneva), in press. [8] F.R. Gantmakher, Teoriya Matrits (Nauka, Moscow, 1966). [9] C. Jarlskog, Phys. Rev. ken. 55 (1985) 1039; P.H. Frampton and C. Jarlskog, Phys. Len. B 154 (1985) 421. [ 10] J. Gasser and H. Leutwyler,Phys. Rep. 87 ( 1982 ) 77, and referencestherein; see also H. Leutwyler,Nucl. Phys. B 337 (1990) 108.
393
PHYSICS LETTERS B
New tools for analyzing quark mixing Alexander Kusenko
1,2
Institute for TheoreticalPhysics, State University of New York, Stony Brook, NY 11794-3840, USA Received 10 March 1992; revised manuscript received 14 April 1992
We present some new mathematical tools which help to derive information about the quark mass matrices directly from experimental data and to elucidate the structure of these mass matrices.
One hopes that a successful choice o f a mass matrix help to suggest an a p p r o p r i a t e dynamical model explaining the quark mass spectrum. In any case, one has to be able to write a mass matrix consistent with the experiment, regardless of what particular d y n a m i c s are responsible for generating quark masses. Over the years, a n u m b e r o f proposals have been a d v a n c e d for quark mass matrices, such as those in refs. [ 1-6 ]. In this paper we would like to present some useful m a t h e m a t i c a l tools for the study of quark mass matrices and the connection with quark mixing. The starting point o f our study is an observation which has so far not been exploited: since the C a b i b b o - K o b a y a s h i - M a s k a w a ( C K M ) quark mixing matrix V is an element o f the Lie group U ( 3 ) , it is in one-to-one correspondence with an element H o f the Lie algebra u (3). H is thus the infinitesimal generator for V. It is o f interest to d e t e r m i n e the H corresponding to the empirically measured V. As we shall show, one can also use H to facilitate the d e t e r m i n a t i o n of families of quark mass matrices which are consistent with experiment. (Here we use the fact [ 7 ] that there are three light neutrinos, hence three usual SU ( 2 ) doublets ofleptons, and, as required, e.g., by a n o m a l y cancellation, thus also three SU ( 2 ) doublets of quarks, so that V~ U (3). ) We denote the quark mass matrices for the up and down sectors M , and Md correspondingly. By means o f a bilinear t r a n s f o r m a t i o n one can diagonalize Mu and Md:
Uu,LMuUtu,R= d i a g ( m u ,
me,
mr)-Do,
Ud,LMd U~,R = d i a g ( m d , ms, mb)=--Dd,
(1)
where Uu,L, Ud,L, Uu,R, Ud,R are unitary matrices. Below we shall suppress the subscript L. In the s t a n d a r d m o d e l with one left-handed charged weak currents, the matrix UR is not a measurable quantity; and therefore one may choose the Uu,R and Ud,R in such a way that Mu and Md are both hermitian. Indeed, an arbitrary complex 3 × 3 matrix is defined by 18 real parameters. In order to make it hermitian, one has to impose 9 constraints to eliminate 9 degrees o f freedom. To do that one may use 9 degrees o f freedom associated with each o f the matrices Uu,R and Ud,R. Below, we assume that both Mu and Md are hermitian. The experimental data [ 7] provides one with the absolute values o f the matrix elements of the C K M mixing matrix V= L~U*d (taken to be 3 × 3). One also obtains certain phase information from experiment which, for a given p a r a m e t r i z a t i o n , can b e t r a n s l a t e d into a d e t e r m i n a t i o n o f the phases of the elements of V. Suppose the pair o f h e r m i t i a n mass matrices Mu and Md,
M~=U*~DoUu,
Md = UtdDd Ud ,
(2)
Supported in part by NSF contract PHY-89-08495. -~ Email address: [email protected]. 390
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 284, number 3,4
PHYSICS LETTERS B
25 June 1992
corresponds to some particular matrix V. Then one can see that for every unitary matrix Uo there exists another pair of hermitian matrices m'u=(UuUo)tDu(UuUo),
m~ = (Ud L~)+D~(L~ Uo) ,
(3)
that corresponds to the same matrix V. One can also see that the family of matrices M'u and M~j as a function of an arbitrary unitary matrix Uo exhausts all possible solutions for the mass matrices. This proves that for every mixing matrix V~ U ( 3 ) there exists a 9-parameter family of matrices that might be chosen to be mass matrices. Empirically, the data constrains [ VI, the matrix of magnitudes of each of the elements of V, to be close to the identity matrix. Therefore, it is useful to represent Vas V= exp (it+H) = Uu U+~,
(4)
where H is some hermitian matrix and c+ is a real number. One may choose H to have its dominant (largest, in absolute value) eigenvalue to be of the order of 1, yet smaller than 1. Then for c~ consistent with the data [ 7 ], we find that I c~[ ~ 0.3. Below we will use the fact that c+< 1. Suppose that V is known. Six parameters, the phases of the quark fields, are arbitrary. We use three of them to rephase Vto make it be close to the 3 × 3 identity matrix. This is possible since I VI ++ 1. Then one can calculate the matrix c~H using Sylvester's theorem [ 8 ]: 3 l-[,++k( V - v, X 1) ic~H= ~ ln(v,) k=, Fi**k( v~ - v , )
(5)
The latter is valid as long as all the eigenvalues of V are distinct. If the matrix V has two or three degenerate eigenvalues, then In (V) is not uniquely defined, and there is a family of matrices H that satisfy (4). This family o f solutions is explicitly shown in ref. [ 8]. However, this mathematical subtlety is not of physical importance since one can always avoid the degenerate case by some small change in the phases of quark fields that removes any initial degeneracy in eigenvalues. The theorem ( 5 ) is particularly powerful since it gives an exact result in a closed form. We will demonstrate how one could construct a set of mass matrices starting from the mixing matrix V. In particular, we choose to describe those mass matrices that can be diagonalized by a unitary transformation U~:o generated by an element in the u ( 3 ) algebra that commutes with H. In terms of the matrix H, V={ + i t c H - ½c~2H2+ . . . + ( 1 / n ! ) ( i t c H ) " + ....
(6)
One can also decompose an arbitrary UI or U_,, that commutes with H, in a similar series whose coefficients are proportional to certain real numbers: {x},u) } and {x~ ~) } where n = 1, 2, 3 .... : Uu =~ +ic+xl~)H - Ix~U) oLZHZ'k-...-+- ( 1/n!)x},">(io+H)"+ .... Ud =~ +ie+xld)H - ½X~zd>c~2HZ + . . . + ( 1/n! )x~fl> (ic+H)"+ ....
(7)
This series is convergent if sup n=
[x,,I < ~ v .
1+2+3 ....
The constraints UuU+~= V, U+~U~ =+ and U+dUd----Mare satisfied simultaneously up to the third order in c~ if the coefficients are chosen as follows: L~=~+io+Hx-+o+2H'-x2+ ....
L ~ = ' ~ + i o + H ( x - 1 ) - ½ 0 z 2 H ' - ( x - 1 ) 2 + ....
(8)
where x is some real parameter. Then, to the second order in 0~, one may express Mo and Md as functions of x: M~=U*~Dug~=Du+iOex[Du, H] -~o~ , 2x 2 [ [ D . , H ] , H ] +
....
(9) 391
Volume 284, number 3,4
PHYSICS LETTERSB
Md = U*dDdbh =Dd + i a ( x - - 1 )[Dd, H] - l a 2 ( x -
25 June 1992
1 )2[ [Dd, H], H] + ....
(9 cont'd)
We see that, in our parametrization, one gets a good approximation of the M as a function of one real parameter. Thus, from Vand the resultant H, we have obtained a family of (experimentally acceptable) mass matrices depending on a parameter x. In passing, we note the following technical point: for rapid convergence of the series (9), V should be close to ~, the identity. The three phases of the diagonal elements of V are not physical and they may be changed by rephasing the quark fields. Such a change in the overall phase of V corresponds to shifting a by a real number. Clearly, ] c~] is the smallest for V~ ~, which yields rapid convergence for both (6) and (9). The family of solutions (9) for the "up" and "down" quark mass matrices (accurate to second order in a ) is Mu ~Ou + i a x [ D u , m ] - J2a2x2[ [ Du, H], H] , Ma ~Do + i a ( x -
1) [Do, m ] - ½ a 2 ( x - 1)2[ [Da, H], H] .
(10)
To illustrate the method with a simple numerical example let us take / 0.97525 o V=~-0.22099exp(i0.0359 ) \ 0 . 0 0 6 6 6 5 7 e x p ( - i 2 8 . 4 0 °)
0.22104 0.97430 exp(i 0.00184 o) -0.043403 exp(i 0.9485 °)
0.0050614 exp ( - i 40.00° )'~ 0.043619 J 0.99904
(11)
(consistent with current data [ 7 ] ). Note that this gives [ Vub[ / [ ~%b] = 0.11 and, for the parametrization-invariant CP violation quantity [9 ] J = Im{ VI~ ~2 V~*2V~ } the value J = 3.1 × 10- 5 First, we calculate the eigenvalues of V: v~=0.97415+0.22587i, v2=0.97443-0.22468i, v3=0.999999+ 0.00122i. Next, we substitute these into ( 5 ) to calculate H and obtain 0 aH= 0.223i \-0.0032+0.001i
-
0.223i 0 0.044i
- 0.0032- 0.001i'~ -0.0044i ).
(12)
We now choose the quark masses to be equal to their running masses evaluated at Qo = 1 GeV and with A = 200 MeV in the MS renormalization scheme. From the analysis of ref. [ 10]: r~u(Qo) -- (5.1 + 1.5) MeV, r~d(Qo) = (8.9 _+2.6) MeV, t~(Qo) = ( 175 _+55) MeV, rh~(Qo) = ( 1.35 +0.05) GeV, mb(Qo) = (5.9+0.1) GeV. We also use the value t~, (Qo) = 220 GeV corresponding to a physical mass of m t phys = 130 GeV, consistent with current constraints. Specifically, we choose the diagonal mass matrices to be D~=diag(5MeV,-1.35GeV, 220GeV),
Dd=diag(8.9MeV,-175MeV, 5.9GeV).
(13)
Suppose x = ½. In this case, one retains a symmetry between the "up" and "down" sectors in ( 15 ). For these values of quark masses the first order approximation of the "up" and "down" sector mass matrices gives the following: { -5.09×10-5 M , = 2 2 0 G e V ~ (6.77X10-4)exp(i3 °) \ ( 2 . 3 7 X 1 0 - 3 ) e x p ( - i 4 2 °) ( M~ =5.9 GeV
1"49X10-3 -3.47×
10 - 3
\ ( 1 . 7 8 X 1 0 - 3 ) e x p ( i 6 6 °)
( 6 . 7 7 × 1 0 - 4 ) e x p ( - i 3 °) - 5 . 5 8 × 1 0 -3 -0.0221 --3.47X10-3 -0.0293 0.0226
(2.37X10-3)exp(i42°)'~ -0.0221 ), 0.99951
(1.78X10-3)exp(--i66°)) 0.0226 . 0.99951
(14)
(15)
Event though this is only a second order approximation it still shows what is the relative value of different terms in the mass matrices for this particular case of x = ½. For instance, the matrix Vcalculated back from Mu a n d M d is different from the original Vonly in the third significant digit. It is of interest to express the CP violation quantity J directly in terms of H. Recall that 392
Volume 284, number 3,4
PHYSICS LETTERS B
25 June 1992
1
J=
-
--det(C), 2Fuk~
16)
where F,=(mt-mc)(mt-mu)(m,.-mu), C=-i[mu,Md]
17)
Fa=(mb--ms)(mb--ma)(ms--ma),
18)
.
Now ell~ = U~D. U~ = U~ V*Du VCd = Ud*(D~ + i a [Du, H] - ½~2[ [D~, H ] , HI + ...)Ud,
M~ = Ud*Dd Ud,
19)
so that up to the first n o n - v a n i s h i n g order in c~, i C - [Mu, aid] = i [og-+-0(o~2)] U*d[ [H, D , I , Dd] Ud.
(20)
Using (20), we thus find J - - [o/+0(o~ 2 ) ] 3 R e [ H , 2 H 2 3 H 3 t ]
.
(21)
This result allows one to calculate the parameter J up to the small corrections of the order of c~2. The higher order lerms can be obtained by retaining the corresponding higher order terms in (20). For example, for the numerical illustration ( 11 ) - ( 15 ), (21) yields J-~ 3.2 X 10- 5, very close to the exact value J = 3.1 X 10-5. These results thus establish a useful connection between the M matrices, the unitary transformations which diagonalize them, and the actual quark mixing matrix V. Furthermore, they introduce an interesting object, the matrix H in Lie algebra u (3) corresponding to V in the Lie group U (3). We have shown how with our tools, one can obtain a family of experimentally acceptable mass matrices directly, given knowledge of V. This technique complements the earlier typical one of hypothesizing some ansatz for a mass matrix and then checking to see if it fits experiment. The author would like to thank Professor Robert E. Shrock for many helpful discussions and comments.
References
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