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The sign of the B parameter and KM matrix parametrization
Nuclear Physics B (Proc. Suppl.) 13 (1990) 443--445 North-Holland
443
THE SIGN OF THE B PARAMETER AND KM MATRIX PARAMETRIZATION Dan-di
Insftute of High Energy Physics, P.O. Box 918, Bei]ing 100039, People'sRepublic of China The sign of the B parameter is related to signs of some CP violating parameters via a well defmed sign of mass difference and is crucial for a convenient phase convention of the KM matrix.
Since the sign of a quantity is just a question of alternatives, it is often omitted. However the number of possible choices becomes 2 n if the said quantity appears in n formulas. This talk is to clarify kind of sign ambiguity in pseudoscalar mixing systems, which, as we know, are in the center of many physical interests.
we are in a position to define the mess and width differences Am = m+ - m . , A~ = ~+ - ¥ . .
(3)
To study the signs of Am and A¥, we must start from the beginning. Assuming CPT invariance, and
The so called bag parameter B is de/'med as
taking CPIBd> = IBd> (this choice is not essential for the final result) we have
B = I n~ '128~pt 8 f2BmB
(z) B± -- PIBd> :1:qlBd>
n
(4)
where p and q satisfy a seqular equation.
where 8Cp = + or - when CP parity of the intermediate state is even or odd. Thus B is positive if CP even tntermediate states dominate in (I).
rewritten
Eq.(2) is
as
pa~. - qa7. The sign of mass difference may have a well
(5)
paT~ + qa~.
defined meaning, once the behaviour of the following CP violating parameter, which is a
with aT. = <7.lBd> ,~1~ --- • The determinant of the
resemblance of the 11+" for the K-coraplex 1,
seqular equation should vanish <7. p CP = + IB-> TI7.= < 7 . , C P r - + I B + >
(2) i * ( M I 2 _ ~ . F I 2 ~ ( M I 2 . ~i. F 1•2 ) = ~ .1( A m
.~.~,)2 i
(6)
is given, where B+ and B- are two physical eigenstates. One requires TIX to be proportional to sin8 (not necessarily smaller than 1) in the standard Kobayashi-Maskawa (KM) model 2 other than (sinS) "1 (for B+ and B- being misplaced in (2)).
Now
0920-5632/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
which tells the relative sign between Am and A¥ 3 Am AT = 4Re M12 1"12
(7)
444
D.D. Wu / B parameter and KM matrix parametr/zation
but not the sign of Am itself.
We need a formula in
violating effects are Am related 8 and Eq.(12) fixes
which Am appears linearly, and the seqular
the sign ambiguity of these effects, so is testable.
equation itself is just what we need
Truly, for the K-complex, Am is measured negative and B K calculated positive, meeting Eq.(12) exactly.
i P-= q
i)
M12.~-rl 2 _ ~(Am
-
i
-
m-~'a
(8)
* i * M12 - 'r12
So we conclude that the yet arguable long distance part of M12(K) is either small or sharing the same sign with the short distance part.
The second
~ituation seems feasible shown by a recent study of the long distance contribution to MI2(K) 9. The Combining (4) and (8) we have4
validity of (12) for the K-complex is essential for
1 (Ira V Z + ~'Am Y~) + i (21m U~.- I A y Y ~ ) ~.-
1
1
finding the sign of sin6 from the 11+- measurement. It is notable that the overlap o of B+ and B. is
+
(2Re UT, + ~'Am Y ~ ) - i (Re VT, + ~'AyY~)
very small
(9) Im MI2 r12
1
= ~- =
where
(13) 41M12 [2 + Ir1212 '
-. UT. = M12 ~ a -~. . . VT. ffi r12 a.Aa~., ~ + = t ~ 12 ± ~:ZI2
though 11k of Eq(9) could be larger than 1. ~ tells are all phase convention independent quantities and therefore should be expressed by phase
how much B+ and B. are mixed while the so called mixing parameter e does not because it is phase
convention independent quantities of the KM matrix 5
convention dependent where • is 1 - ¢
(10)
Aia = vii vk¥
l+e
•p
cl _- r
(14j
In a complex plane, r makes a loop with radius Irl I0. and does a big one with radius about
(ijk and a~y are cyclic). Note I n d i a = const. ~ L A brief observation of Eq.(9) tells that ~1k has the following behaviour
Exactly,
based on box calculation of M12 and r 1 2 6 2 Im Aia (11)
-B IAml Re A2j[3 + Am IBI ,Aj~12
1 i Im M12 + ~ ' I m r 12 = i 1 i Re M I 2 - ~ ' R e F 1 2 + ~-Am - ~ ' A ¥
(15)
Both Wu-Yang and Kobayashi-Maskawa have paid which gives ~lk ~ J only when7
particular attention to choose phase conventions in order to make e K small enough, so formulae like
AraB<0
(12) Am -- 2Re M12,6yffi 2Re r12 ,¢~ = R~ e
All calculations so far predict B > 0.
Taking it for
granted, we predict Am < 0 fzcm (12).
Many CP
(16)
D.D. Wu / B parameter and KM matrix parametrization may be used for the K-complex. It is tempting to choose a convention for the KM matrix to make e's small at least for the most interesting mixing systems K, Bd, Bs at the same time. One makes ¢ small by making M12 almost real in addition to choose a correct phase of ReM12 to maximize the denominator of (15).
Noting MI2(K) - -
BK(VcsVcd)2, MI2(B d ) ~ -BB(VtbVtd)2 and M12(B s --BB(VtbVts )2, it is proved impossible to make e°s small for three systems at the same time 11 due to unitarity and the special structure of the KM matrix explored by Wolfenstein. Fortunately, making two e's small is possible and changing from one phase convention to the other is easy, For instance
Vw--
Z
A2TIZSe'iB
+ A~lZ3e-i8
1 - 1Z2 .AZ2
AZ3(1 " vleiS) AZ2(I + ~lZ2ei8) 1
445
In conclusion, the signs of Am and Ay may be well defined after the value of a CP violating parameter ilk in Eq.(2) is given, as in Eq.(3). The sign of AmAy is given in F,q.(7). The sign of Am in the standard KM model is opposite to that of ~ B parameter, F-al.(12), so is calculable. On the other hand, Am appears in many CP violating formulas and is measurable. The calculated and measured signs of Am coincide for the K-complex. It is interesting to check whether they do for B systems. Choosing convenient phase of KM matrix for study of a specific mixing system is suggested, similar to choosing a proper coordinate before solving a mechanical problem. REFERENCES 1. T.T. Wu and C.N. Yang, Phys. Rev. Lett. 13 (1980) 380. 2. M. Kobayashi and T. Maskawa, Pmgr. Phys. 49 (1973) 652.
Tbeor.
3. D.D. Wu, Phys. Lctt. 9013 (1980) 452. 4. D.D. Wu, Phys. Rev. D33 (1986) 860.
(17) gives small eK and ¢Bs (and small eD' eTc )"
5. C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039, D.D. Wu, Ref. 4. 6. J.S. Hagelin, Phys. Rev. D20 (1979) 2893.
VW2 = ( 1 1 el8 )
V W ( 1 1 e-i~ )
7. D.D. Wu, BIHEP-TH-86-6, to appear in Phys. Rev.D.
gives small eK and eBd (as well as e D) and
8. e.g.I. Dunietz, J. Hauser and LL. Rosner, Phys. Rev. D35 (1987) 2166
WW3-- V w ( i B
9. C.R. Ching and X.Q.Li. ITP (Beijing) preprint,
1 1)
gives small eBd and eBs etc. It is recommended to choose a suitable phase convention before study of a specific mixing system so that not only (16) are valid, but also, for the Bd and Bs systems, 11~, -. -i Im az/Re aZ
if X is an eigenstate of strong interactions.
(18)
10. P.K. Kabir and D.D. Wu, University of Virginia preprint, to appear. 11. W. Cben and D.D. Wu, BIHEP-TH-88-38.
443
THE SIGN OF THE B PARAMETER AND KM MATRIX PARAMETRIZATION Dan-di
Insftute of High Energy Physics, P.O. Box 918, Bei]ing 100039, People'sRepublic of China The sign of the B parameter is related to signs of some CP violating parameters via a well defmed sign of mass difference and is crucial for a convenient phase convention of the KM matrix.
Since the sign of a quantity is just a question of alternatives, it is often omitted. However the number of possible choices becomes 2 n if the said quantity appears in n formulas. This talk is to clarify kind of sign ambiguity in pseudoscalar mixing systems, which, as we know, are in the center of many physical interests.
we are in a position to define the mess and width differences Am = m+ - m . , A~ = ~+ - ¥ . .
(3)
To study the signs of Am and A¥, we must start from the beginning. Assuming CPT invariance, and
The so called bag parameter B is de/'med as
taking CPIBd> = IBd> (this choice is not essential for the final result) we have
B = I n~ '
(z) B± -- PIBd> :1:qlBd>
n
(4)
where p and q satisfy a seqular equation.
where 8Cp = + or - when CP parity of the intermediate state is even or odd. Thus B is positive if CP even tntermediate states dominate in (I).
rewritten
Eq.(2) is
as
pa~. - qa7. The sign of mass difference may have a well
(5)
paT~ + qa~.
defined meaning, once the behaviour of the following CP violating parameter, which is a
with aT. = <7.lBd> ,~1~ ---
resemblance of the 11+" for the K-coraplex 1,
seqular equation should vanish <7. p CP = + IB-> TI7.= < 7 . , C P r - + I B + >
(2) i * ( M I 2 _ ~ . F I 2 ~ ( M I 2 . ~i. F 1•2 ) = ~ .1( A m
.~.~,)2 i
(6)
is given, where B+ and B- are two physical eigenstates. One requires TIX to be proportional to sin8 (not necessarily smaller than 1) in the standard Kobayashi-Maskawa (KM) model 2 other than (sinS) "1 (for B+ and B- being misplaced in (2)).
Now
0920-5632/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
which tells the relative sign between Am and A¥ 3 Am AT = 4Re M12 1"12
(7)
444
D.D. Wu / B parameter and KM matrix parametr/zation
but not the sign of Am itself.
We need a formula in
violating effects are Am related 8 and Eq.(12) fixes
which Am appears linearly, and the seqular
the sign ambiguity of these effects, so is testable.
equation itself is just what we need
Truly, for the K-complex, Am is measured negative and B K calculated positive, meeting Eq.(12) exactly.
i P-= q
i)
M12.~-rl 2 _ ~(Am
-
i
-
m-~'a
(8)
* i * M12 - 'r12
So we conclude that the yet arguable long distance part of M12(K) is either small or sharing the same sign with the short distance part.
The second
~ituation seems feasible shown by a recent study of the long distance contribution to MI2(K) 9. The Combining (4) and (8) we have4
validity of (12) for the K-complex is essential for
1 (Ira V Z + ~'Am Y~) + i (21m U~.- I A y Y ~ ) ~.-
1
1
finding the sign of sin6 from the 11+- measurement. It is notable that the overlap o of B+ and B. is
+
(2Re UT, + ~'Am Y ~ ) - i (Re VT, + ~'AyY~)
very small
(9) Im MI2 r12
1
= ~-
where
(13) 41M12 [2 + Ir1212 '
-. UT. = M12 ~ a -~. . . VT. ffi r12 a.Aa~., ~ + = t ~ 12 ± ~:ZI2
though 11k of Eq(9) could be larger than 1. ~ tells are all phase convention independent quantities and therefore should be expressed by phase
how much B+ and B. are mixed while the so called mixing parameter e does not because it is phase
convention independent quantities of the KM matrix 5
convention dependent where • is 1 - ¢
(10)
Aia = vii vk¥
l+e
•p
cl _- r
(14j
In a complex plane, r makes a loop with radius Irl I0. and does a big one with radius about
(ijk and a~y are cyclic). Note I n d i a = const. ~ L A brief observation of Eq.(9) tells that ~1k has the following behaviour
Exactly,
based on box calculation of M12 and r 1 2 6 2 Im Aia (11)
-B IAml Re A2j[3 + Am IBI ,Aj~12
1 i Im M12 + ~ ' I m r 12 = i 1 i Re M I 2 - ~ ' R e F 1 2 + ~-Am - ~ ' A ¥
(15)
Both Wu-Yang and Kobayashi-Maskawa have paid which gives ~lk ~ J only when7
particular attention to choose phase conventions in order to make e K small enough, so formulae like
AraB<0
(12) Am -- 2Re M12,6yffi 2Re r12 ,¢~ = R~ e
All calculations so far predict B > 0.
Taking it for
granted, we predict Am < 0 fzcm (12).
Many CP
(16)
D.D. Wu / B parameter and KM matrix parametrization may be used for the K-complex. It is tempting to choose a convention for the KM matrix to make e's small at least for the most interesting mixing systems K, Bd, Bs at the same time. One makes ¢ small by making M12 almost real in addition to choose a correct phase of ReM12 to maximize the denominator of (15).
Noting MI2(K) - -
BK(VcsVcd)2, MI2(B d ) ~ -BB(VtbVtd)2 and M12(B s --BB(VtbVts )2, it is proved impossible to make e°s small for three systems at the same time 11 due to unitarity and the special structure of the KM matrix explored by Wolfenstein. Fortunately, making two e's small is possible and changing from one phase convention to the other is easy, For instance
Vw--
Z
A2TIZSe'iB
+ A~lZ3e-i8
1 - 1Z2 .AZ2
AZ3(1 " vleiS) AZ2(I + ~lZ2ei8) 1
445
In conclusion, the signs of Am and Ay may be well defined after the value of a CP violating parameter ilk in Eq.(2) is given, as in Eq.(3). The sign of AmAy is given in F,q.(7). The sign of Am in the standard KM model is opposite to that of ~ B parameter, F-al.(12), so is calculable. On the other hand, Am appears in many CP violating formulas and is measurable. The calculated and measured signs of Am coincide for the K-complex. It is interesting to check whether they do for B systems. Choosing convenient phase of KM matrix for study of a specific mixing system is suggested, similar to choosing a proper coordinate before solving a mechanical problem. REFERENCES 1. T.T. Wu and C.N. Yang, Phys. Rev. Lett. 13 (1980) 380. 2. M. Kobayashi and T. Maskawa, Pmgr. Phys. 49 (1973) 652.
Tbeor.
3. D.D. Wu, Phys. Lctt. 9013 (1980) 452. 4. D.D. Wu, Phys. Rev. D33 (1986) 860.
(17) gives small eK and ¢Bs (and small eD' eTc )"
5. C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039, D.D. Wu, Ref. 4. 6. J.S. Hagelin, Phys. Rev. D20 (1979) 2893.
VW2 = ( 1 1 el8 )
V W ( 1 1 e-i~ )
7. D.D. Wu, BIHEP-TH-86-6, to appear in Phys. Rev.D.
gives small eK and eBd (as well as e D) and
8. e.g.I. Dunietz, J. Hauser and LL. Rosner, Phys. Rev. D35 (1987) 2166
WW3-- V w ( i B
9. C.R. Ching and X.Q.Li. ITP (Beijing) preprint,
1 1)
gives small eBd and eBs etc. It is recommended to choose a suitable phase convention before study of a specific mixing system so that not only (16) are valid, but also, for the Bd and Bs systems, 11~, -. -i Im az/Re aZ
if X is an eigenstate of strong interactions.
(18)
10. P.K. Kabir and D.D. Wu, University of Virginia preprint, to appear. 11. W. Cben and D.D. Wu, BIHEP-TH-88-38.