Rephasing-invariant parametrization of the quark mixing matrix

Volume 208, number l

PHYSICS LETTERS B

7 July 1988

REPHASING-INVARIANT PARAMETRIZATION OF THE QUARK MIXING MATRIX G.C. B R A N C O i a n d L. L A V O U R A CERN, CH- 12 l 1 Geneva 23, Switzerland Received 14 October 1987

We suggest that within the context of the standard model with three generations, the most convenient way of parametrizing the Kobayashi-Maskawa matrix is through four independent moduli of its matrix elements. For three generations the strength of CP violation is completely determined from the knowledge of a set of independent moduli. We extend this parametrization to the case of four generations and analyze in detail the unitarity constraints on the quark mixing matrix. These unitarity constraints are expressed in terms of independent moduli of the quark mixing matrix and can be a useful tool in obtaining precise experimental tests of unitarity.

1. Introduction Recently, increasingly m o r e accurate d a t a on the quark mixing matrix have been accumulated ~. This wealth o f experimental results motivates the search for a suitable p a r a m e t r i z a t i o n o f the K o b a y a s h i - M a s k a w a ( K M ) matrix [ 2 ]. In this letter we suggest that the most convenient way o f parametrizing the K M matrix is through i n d e p e n d e n t m o d u l i o f its matrix elements. This p a r a m e t r i z a t i o n has the advantage o f being obviously invariant u n d e r a rephasing o f the quark fields, and furthermore its p a r a m e t e r s are directly and easily measurable quantities. This is not the case for t r a d i t i o n a l p a r a m e t r i z a t i o n s o f the K M matrix through generalized Euler angles and CP-violating phases. In general, one does not measure either Euler angles or K M phases, but rather certain rephasing-invariant products o f the K M matrix elements. We will show that in the standard m o d e l with three generations the knowledge o f four i n d e p e n d e n t m o d u l i completely determines the K M matrix, and in particular the strength o f C P violation. The reader m a y w o n d e r how is it possible to fix the strength o f C P violation through the knowledge o f non-CP-violating quantities like the moduli. We will show that this is possible as a consequence o f the unitarity o f the three-generation K M matrix. We will first present our p a r a m e t r i z a t i o n for the three-generation case, giving special emphasis to the unitarity constraints on the moduli. We then discuss the extension o f our p a r a m e t r i z a t i o n to four generations, pointing out the new, non-trivial p r o b l e m s that then arise.

2. Invariant parametrization for three generations The K M matrix arises from the bi-diagonalization o f the quark mass matrices, which leads to a non-diagonal charged weak current. U n d e r a change o f phases o f the quark fields, the diagonalized quark mass matrices rem a i n invariant, while the K M m a t r i x V t r a n s f o r m s as Vi/-- V o e x p [ i ( f li - o ~ i ) ],

(1)

Permanent address: INIC, Centro de Fisica da Mat6ria Condensada, 1699 Lisbon Codex, Portugal. ~ For a recent review see ref. [ 1]. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

123

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where ai, flj denote the arbitrary rephasings of the up and down quark fields. Only functions of the V,j which are invariant under the rephasing ( 1 ) are observable [ 3,4 ]. The simplest invariants are the moduli [ V~j[ and the quartic products V~j, Vkt, V~, V~j, which will be designated "quartets". It is easily shown that any invariant of higher order (V~jVklV,,n...Vr~Vi*tV~-n...V~ ) can be written as a product of quartets, divided by some squared moduli. These we denote by Uo = I V~j[2 An invariant parametrization of the KM matrix consists of a minimum set of"basic invariants" which completely determine all other invariants. Next we will show that for three generations a very convenient choice of basic invariants are four independent squared moduli. There are, of course, many possible choices for the independent U,s, and their choice should be dictated by the availability and precision of the experimental data. For definiteness, we first take as basic moduli Ul ~, Ul2, U21 and U22. The remaining moduli are trivially deduced from these through unitarity. The quartets too are related among themselves and with the moduli through unitarity [3,4]. For example, the unitarity relation VT3 V23 = -- VT1 V 2 1 - VT2 V22 implies V~ V23 VT3 V~l ~-- 1/11 V22 VT2 V~l - Ull U21, SO that out of the nine quartets only one, say I/"11V22 VT2 V~, is independent. In particular, the imaginary parts of all quartets are equal up to their sign [ 3,4 ] as a consequence of these relations among quartets. We now show that the real part of any quartet is a function of the moduli. Indeed, from unitarity we have I V33] = ] ( VII V22-- Vl2V21 ) l, and therefore, expressing U33 as a function of the basic moduli:

R=Re(Vll Va2VT2V~l):(I-UII-U22-UI2-U21--[-UI l U22.-~UI2U21)/2.

(2)

Since J = I m ( V,l V22 VT2 V~, ) = ( Ull U22 Ul2 U2~ - R 2 ) 1/2,

(3)

the value of the imaginary part of the quartets is determined from the four independent moduli with only an ambiguity in the sign of J remaining.

3. Unitarity constraints In our parametrization the quark mixing matrix is not manifestly unitary. It is therefore important to analyze the constraints on the moduli implied by unitarity. There are obvious constraints obliging 0 ~
(4)

One thus obtains 4 U l l U22 UI2 U21 ~ ( U13 U23 - Ull U21 -- Ul2 U22 ) 2

(5)

which in terms of the chosen basic invariants reads

4U11U22Ul2 U21 ~

(1 - UII - U22 - Ul2 - U21 + e l l U22 "~- UI2 U21 )2.

(6)

In a 3 × 3 unitary matrix one has six orthogonality relations which thus lead to six unitarity constraints analogous to eq. (5). However, when expressed in terms of a set of independent moduli, all these relations turn out to be equivalent. Thus eq. (6) is in fact the only non-trivial unitarity constraint. Remembering eq. (2), we find 124

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that eq. (6) is also obtained by requiring the real part of any quartet not to be greater than the modulus of that quartet. However, the above geometrical derivation generalizes more easily to the four-generation case, where the unitarity constraints are much more complex. We close this discussion by noting that: (i) the condition of eq. (6) further restricts the basic moduli, beyond the trivial restrictions arising from 0 ~< Uii~< 1; (ii) these trivial constraints, together with eq. (6), are not only necessary, but also sufficient conditions for unitarity of the KM matrix. Of course, equivalent unitarity conditions are obtained i f a different set of basic moduli is used.

4. Confrontingexperiment In this section we take as basic invariants UI2 , UI3, e21 and U23. Our present experimental knowledge of the moduli of the KM matrix elements is [ 1 ] 1V111=0.9747+0-0010, 1V211=0.207+0.024,

[V12[=0.220+0.002, 1V221=0.95+0.14,

IV131~ <0.010,

1v231=0.045+0.010.

(7)

Unitarity further restricts the allowed range for the basic moduli. First positivity of U31 and U32 constrains U2~ to the interval [ ( U j 2 - U23), ( U~2+ U~3) ]. Second, the unitarity constraint ofeq. (6) reads, when expressed in terms of the new set of basic moduli, P( U2~ ) =Cl U2j +c2 U2~ +c3 ~<0,

(8)

where the c, are functions of U~2, U~3 and U23:

cl = ( 1 - U , 3 ) 2, C2=2[U]2(U23--1)+U]3(UI2--U23 +UI2U23 +UI3U23) ], c3=[UI2(U23-1)"[-UI3U23] 2. (9) Since P( U2, ) is a quadratic polynomial in U2~, the inequality (8) enforces U2j to lie in the interval limited by the two roots of P. These are functions of U12, U~3 and U23. Given the experimentally allowed range for these moduli, it turns out that this interval for U2~ is contained in the interval previously derived from normalization of the rows and columns of V. In fig. 1 we give the allowed range of I V2,1. If one keeps I Vl21 fixed at 0.220, I V2,1 may vary between the solid lines if I V~3] =0.010, or between the dashed lines if I V~3] =0.003. When I V[31 =0, IV2,1 is completely determined by U]2 and U23: U2, = U , 2 ( 1 - U23),

(10)

which is marked as a dotted line in fig. 1. We kept I Vl21 fixed at 0.220 in fig. 1, but other values for I V,21 can be studied from that figure as well. The reason is that if one takes another value, say 0.220 +x, for I V121, the allowed range for [ V2]I is, to a great accuracy, simply displaced by x, provided this increment is small enough. For our choice of basic moduli, we have

J= +_(-P)'/2/2,

(11)

with P given in eqs. (8), (9). Clearly, for given values of U~2, U, 3 and U23 the maximum value of IJ I is reached when U2~ = -c2/2cj. It turns out that, due to the smallness of U~3, this value of U2~ coincides, to great accuracy, with the value given in eq. (10). If we keep U2~ harbored at -c2/2c~, and vary the other moduli within their experimentally allowed ranges, we easily find that the maximum value presently allowed for IJI is 1.08 × 10-4, which arises when I V121, I Vj31 and I V231 are all put equal to their upper limits. There is no lower bound for IJl: indeed, J vanishes in the boundaries of the unitarity regions, as given in fig. 1. 125

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I V211

0.2202

0.2200 0.2198 0.2196

0.2194

0.2192

0.2190 I

0.035

I

O.Ot~O

I

O.OL,5

t

0.050

L

0.055

I V231

Fig. 1. The values of J V2~I allowed by unitarity are plotted against [ V23J, for three values of [ V~3Iand with I VI21fixed at 0.220. The solid and dashed lines limit the unitarity regions for [ V~3[=0.010 and [ V~3[=0.003, respectively. The dotted line gives the value of ]V_7] when [ Vi31=0. 5. A minimal set of mass matrix invariants

We have seen that a set of four independent moduli of the KM matrix elements, together with the sign of J, and the six quark masses, exhaust the physical information contained in the quark mass matrices. This information can be directly deduced in any weak basis, without the need to diagonalize the matrices. To prove this, consider the following weak basis invariants [ 5 ]: tr(HuHD),

tr(H~HD),

tr(HuH~),

tr(H2H~),

(12)

where HU,D=Mu,DMb,D, together with the coefficients of the characteristic equations for Hu, HD. These latter invariants fix the quark masses. We now show that the invariants in (12) allow us to determine the Uu, thus fixing the KM matrix expect for the ambiguity in the sign of J. First notice that 3

tr(H~H{~)=

•

U~/D~Uu,

(13)

i,j= I

where a, b are integers and Ui, Di denote the squared masses o f up and down quarks respectively. Using unitarity to relate the different moduli, one may then re-write eq. ( 13 ) in terms of a set of four independent U u. One thus obtains a set of four linear equations for the four U u. This set of equations has a unique solution, provided the quark masses in the up and down sectors are separately non-degenerate. This theorem is trivially generalized to any number n of generations, and we find that the n 2 invariants t r ( H ~ H ~ ) , with a, b from 1 to ( n - 1 ), allow us to fix the Uu, provided the quark masses are non-degenerate. A crucial question is then whether the knowledge of the Uu completely fixes the K M matrix, for n greater than three. We will see that this is not a trivial question. 126

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6. Four generations Inspired by the three-generation case, we will try to apply the idea of parametrizing the KM matrix by independent moduli of its matrix elements to the case of four generations. Let us write the unitary 4 x 4 mixing matrix Vas

(14)

and adopt the convention that a capital letter stands for the squared modulus of the corresponding matrix element: A = 1a 12, B= 1b 12, etc. Our aim is on the one hand to derive non-trivial unitarity constraints on the moduli of the 4 x 4 KM matrix and on the other hand to find all quartets in terms of a set of nine independent moduli, say, those of the 3 x 3 upper left-hand submatrix: A, B, .... K. It will be seen that the task is rather complicated, but its importance should be stressed. The moduli of the KM matrix elements are the only direct experimental information one can obtain about that matrix. This makes our parametrization the most natural one from the experimentalist’s viewpoint. Furthermore the non-trivial unitarity constraints, expressed in terms of independent moduli, are crucial in order to obtain precise tests of the unitarity of the KM matrix. In order to address the above questions we start by noting that there are only four independent quartets. First, one can express all quartets involving matrix elements either of the last row or of the last column of V, in terms of quartets not involving such matrix elements, in a fashion similar to what was done in the three-generation case. One thus reduces to ( 3C,)2=9 the number of potentially independent quartets. A further reduction is achieved by noting that any of these quartets may be written as a product of two other quartets, divided by some moduli. For example, (bgc*f* ) = (a*f*be) ) (agc*e* ) / (AE). Therefore, our task would be completed if we could express the arguments of four independent quartets, say or=arg(afb*e*),

P=arg(agc*e*),

Y=arg(ajb*i*),

in terms of a set of nine independent

moduli.

6=arg(akc*i*),

Now, the orthogonality

(15) of the first two rows of Pleads to

Re( afb*e* +agc*e* + bgc*f*) = d, where d, = (DH-AEand uses the relations to eq. (16a):

BF- CC) 12. If one considers all similar equations for any pair of rows or columns of V among quartets and moduli implied by unitarity, one finds four more equations similar

Re(ajb*i*+akc*i+bkc*J*) Re( afb*e*+ajb*i*+

c7

cos(y-a)+~,

cl cosol+cd

=d,,

Re(ejf*i*+ekg*i*+jky~*)

e$*i*) =d4,

where the d, are functions quartets, one can translate cl coscr+cz

(16a)

cos(d-/?)+c,

cos y+c,

(16’w)

Re( agc*e*+akc*i* + ekg*i*) =d,,

(160)

of the moduli. If one now takes into account the algebraic relations among the nine eqs. ( 16) into equations involving only the phases of the independent quartets:

cos(/3-a)=d,,

cosp+c3

=d,,

c4 cosy+c,

COS~+C, cos(S-y)=d,,

cos(a+&P-y)=d3,

cos(y-cx)=d,,

c2 cos /3+c, cos 6+c, cos(6-/?)

(170) (17c)

=d5,

(IWe)

where the c,, d, are functions of the moduli only, e.g. c, = (AFBE) I/‘, c2= (AGCE)‘I’, etc. We will now address the following questions: (i) What conditions should the independent moduli satisfy in order for eqs. ( 17) to have a solution? (ii) For a given set of independent moduli satisfying the unitarity constraints, how to find a, p, Y, 6 and thus explicitly construct the KM matrix? (iii) For a given set of independent moduli, how many different solutions are there to that system of equations? The solutions of cqs. ( 17) have a trivial degeneracy: 127

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starting from a given solution, one may obtain another one by inverting all phases of all quartets simultaneously. The two solutions differ only in an overall sign of all the CP-violating amplitudes. We will show later that in some cases this is not the only degeneracy of the system ( 17 ). We now address question (i) and derive necessary conditions for a set of moduli to be compatible with unitarity, beyond the trivial constraints arising from normalization of all the rows and columns of V. From orthogonality of the first two rows it follows that the four complex numbers (Vii V~,), i = 1, 2, 3, 4, should form a quadrilateral in the complex plane. For this to be possible, the modulus of each one of these complex numbers should not exceed the sum of the moduli of the other three. This is easily shown to be equivalent to requiring that 4

E l~_2 E li2,17 -811121314 <<.0,

t= 1

i~j

(18)

where l~= I V~V*il 2. This is the generalization to four generations of the condition of eq. (4) for three generations. One can derive analogous necessary conditions for unitarity from orthogonality of other pairs of rows or columns of V. It is easily found that these conditions, together with the present experimental knowledge, imply that the matrix of the U o is roughly symmetric, and has all off-diagonal elements small, with the possible exception of a large mixture of the third and fourth generations [4,6]. If one expresses the l, in terms Of Cl, C2, C3 and d~, eq. (18) becomes:

d 7 - c l 2 - c 2 - c 2 <<-2x 1/2,

(19)

where x=2clc2cgdl

+ (cl C2)2-~ (C 1 C3)2-1- ( C 2 C 3 ) 2 • 0 ,

(20)

which turns out to be the conditions for eq. (17a) to have a solution, given some non-negative c~, c2, c3 and arbitrary d~. As noted before, the condition of eq. (18) is just the constraint which has to be satisfied in order for the V~i V~, to be able to form a quadrilateral in the complex plane. The complexity of the four-generation problem stems essentially from the fact that the lengths of the four sides of a quadrilateral do not fix its exact shape, unlike what happens with a triangle. It is clear that there are some more unitarity constraints beyond those of the type of eq. (18). For it is not sufficient to require that all quadrilaterals might be built; since the angles of the various quadrilaterals are related, it is also necessary that all quadrilaterals somehow "fit" into each other. This implies some extra conditions which we derive next. Consider eq. (17a), and let us solve for cos ft. One obtains a quadratic equation, whose coefficients are functions of c~, c> c3, d~ and cos a. The discriminant of this equation must be non-negative. This is found to be equivalent to the condition

c~ dL "JffC2C3 - - x l /2 ~ C21 COS o~ ~ cl d 1 "~ C2C 3 "~-X I /2.

(21)

This condition is important by itself. It means that a constraint on the phase c~ can be obtained just from an incomplete set of independent moduli, namely those of the first two rows of the KM matrix. Given the experimental values of these moduli, eq. (21) already constrains c~ to be quite close to 180 ° , just as it happened in the three-generation case (for two generations it is obvious that c~= 180 ° ). Now, an entirely analogous but independent constraint on c 2 cos ~ can be obtained from eq. (17d), namely: c l d 4 - t - c 4 c 7 -- y l / 2 <~C2 COS oz ~ cl d4"t-c4c 7 -t- y I/2,

(22)

with y=2c~ c 4 c 7 d 4 "1- (clc4)2"1 - (ClC7)2q - (c4c7) 2. Thus, eqs. (21), (22) imply that c21 cos e~ must be inside two different intervals. Obviously, these two intervals must have a non-empty intersection for unitarity to hold. In general, the condition for any two intervals [ a - b , a + b ] and [ c - d , c + d], with b and d positive, to intersect, is [ a - c l ~<(b + d). From eqs. (21 ), (22) one thus obtains the following unitarity constraint on the independent moduli: 128

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]CH+ D G - I N - Z ~ I / 2 <~( C H D G ) '/2 ..~ ( I N J M ) ~/2.

(23)

We see that from eqs. (17a), (17d), which arose from the orthogonality of the rows and columns entering in the quartet (afb*e*), we have derived a unitarity constraint on independent moduli which goes beyond those of the type of eq. (18). Obviously, similar conditions arise from other quartets in the KM matrix. Note however that these conditions are identical when they arise from two quartets built from complementary sets of both rows and columns of V. For example one ends up with the same condition, from consideration of the quartets G2 V43 V~'3V~2 and ~ ~ 4 V*4 V~. It is clear that in order for unitarity to hold it is necessary not only that conditions of the type ofeqs. ( 18 ) and (23) be satisfied but also that the whole set of eqs. (17) has a solution. Our experience in working numerically with the whole set of unitarity conditions for some given moduli, indicates that the conditions of the type ofeq. ( 18 ) are much more important than those of the type of eq. (23 ): it is seldom that the latter introduce any new constraints upon the moduli. Next we will give some examples which illustrate the practical interest of our results. Consider the following realistic choice of moduli: IG~12=0.95,

1V~2[2=0.0485,

IGsl2=0.0001,

1V2~12=0.049,

1v2312=0.003.

(24)

Then eq. (18) implies the bound 0.857< [ V2212<0.948. Let us choose I V221z=0.94, so that all independent moduli of the first two rows are fixed. Then equations of the type of eq. (21 ) can be used to obtain bounds on the phases of the various quartets involving elements of the first two rows of V. One thus obtains for example: -0.999923 < cos c~< -0.999890,

0.498729
(25)

but no useful constraint is obtained on ft. Let us now suppose that we knew all the moduli of the matrix elements in the first two rows, and in the first two columns, of V. Consider for instance the following matrix of squared moduli, which has its rows and columns already normalized:

IV'12=

t0.95 0.049 0.0008

0.0485 0.94 0.011

~0.0002 0.0005

0.0001 0.003 x (0.9969-x)

0.0014 0.008 / (0.9882-x)]'

(26)

(0.0024+x)/

with x = I V~3I. The unitarity constraints of the type of eq. (18 ) lead to the bound 0.090 < I V3312<0.449 thus implying a large mixing between the third and fourth generations. Let us now choose I V3312=0.1, and solve the whole system of equations ( 17 ). Using as input the nine independent moduli, we find two different solutions to that system of equations: c~=179.232312 °,

f l = - 7 4 . 7 1 0 1 5 9 °,

y = - 8 5 . 1 3 3 5 9 1 °,

,~=-173.150282 °,

c~=179.181612 °,

f l = - 9 3 . 8 6 0 9 2 1 °,

y = - 8 5 . 3 5 6 6 1 5 °,

~ = - 1 7 1 . 6 6 7 3 7 2 °.

(27)

We have omitted the trivial degeneracy of change of signs for all phases, by choosing 0 ° ~
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parameters of a given parametrization, from the knowledge of a set of i n d e p e n d e n t [ V,j [. The use of parametrizations through Euler angles and phases usually renders things not easier, but inextricably more complicated [9].

7. Conclusions We have presented a parametrization of the quark mixing matrix in terms of i n d e p e n d e n t moduli of the K M matrix elements. This is, from the experimentalist's viewpoint, the most natural way of looking at the K M matrix, and at the constraints that unitarity imposes on it. We have shown that in the standard model with three generations one can deduce the strength of CP violation from the knowledge of quantities like the moduli, which are not directly related to CP violation. At present, one can only derive an upper b o u n d on IJI, the i n v a r i a n t which controls the strength of CP violation for three generations. The d e t e r m i n a t i o n of a lower b o u n d on IJI is at present not possible, given the experimental uncertainties in I V211, I V221. The derivation of such a lower b o u n d would be a proof that charged gauge current interactions do indeed violate CP. For four generations, we have derived two sets of unitarity constraints expressed in terms of i n d e p e n d e n t moduli, namely those typified by eqs. ( 18 ), (23). Since these unitarity constraints are expressed in terms o f m o d u l i , they may be an i m p o r t a n t tool in obtaining precise experimental tests of the unitarity of the K M matrix. While this work was in progress, we have received an interesting preprint by Bjorken and Dunietz [ 10 ], where a different rephasing-invariant p a r a m e t r i z a t i o n of the K M matrix is suggested.

Acknowledgement We thank G u i d o Altarelli for a careful reading of the manuscript. We are grateful to the C E R N Theory Division for the kind hospitality extended to us during our visit.

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