Unitarity Boomerang

For the three family quark flavor mixing, the best parametrization is the original Kobayashi-Maskawa matrix, $V_{KM}$, with four real parameters: three rotation angles $\theta_{1,2,3}$ and one phase $\delta$. A popular way of presentation is by the unitarity triangle which, however, explicitly displays only three, not four, independent parameters. Here we propose an alternative presentation which displays simultaneously all four parameters: the unitarity boomerang.


Introduction
As is well known, there are different ways of parameterizing the Kobayashi-Maskawa [1] quark mixing matrix, V KM . For three generations of quarks, V KM is a 3 × 3 unitary mixing matrix with three rotation angles (θ 1 , θ 2 , θ 3 ) and one CP violating phase δ. The magnitudes of the elements V ij of V KM are physical quantities which do not depend on parametrization.
However, the value of δ does. For example, in the Particle Data Group (PDG) parametrization [2], adopted from Ref. [3], δ ∼ 70 • , whereas the phase in the original KM parametrization has a different value, δ ∼ 90 • . Care must be exercised in quoting a value of δ, as it depends on how the matrix is parameterized. For example, the statement made after Eq. (11.3) in the current edition of PDG is misleading, because it identifies, incorrectly, the phase δ of Ref. [1].
It can therefore be more useful to employ only physically-measurable quantities. To this end, it has long ago been suggested that a unitarity triangle (UT) be used [4] as a useful presentation for the quark flavor mixing, especially of CP violation [5]. Because of the unitary nature of the KM matrix, one has i V ij V * ik = δ jk and i V ji V * ki = δ jk , where the first and second indices of V ij take the values u, c, t, ... and d, s, b, ..., respectively. For three generations of quarks, when j = k, these equations form closed triangles in a plane, the UT s. Six UT s can be formed with all of them having the same area. A(UT ), which is equal to half of the value of the Jarlskog determinant [6] J, so that A(UT ) = 1 2 J. The inner angles of a given UT are therefore closely related to the CP violating measure J. When the inner angles are measured independently, their sum, whether it turns out to be consistent with precisely 180 • , provides a test for the unitarity of the KM matrix. The unitarity triangle is also a popular way, to present CP violation, with three generations of quarks.
A UT , however, does not contain all the information encoded in the KM matrix, V KM .
Although a UT has three inner angles and three sides, it contains only three independent parameters. The three parameters can be chosen to be two of the three inner angles and the area, or the three sides, or some combination thereof. One needs an additional parameter fully to represent the physics: this is hardly surprising, as the original UT idea of [4] involved only two, of the three, rows or columns of the 3 × 3 matrix, V KM , An improved presentation is thus rendered desirable, in order better to present the KM matrix, V KM , diagrammatically. In this Letter, we propose such a new diagram, the unitarity boomerang.
The unitarity boomerang contains information from a pair of UTs. The different ways of choosing the pair contain, of course, equivalent information. Nevertheless, the specific choice, in the next section, was made judiciously [7], such as to maximize the minimum vertex angle in the unitarity boomerang. This choice is, we believe, the most convenient.

Unitarity Boomerang
We indicate the KM matrix and its elements by V KM = (V KM ) ij , with i = u, c, t and The j = k and i = k cases form, respectively, the six possible different UT presentations for V KM in a convenient two-dimensional plane. There are, thus, a total of 18 inner angles in the six UT s. However, only 9 are different because, by Euclidean geometry, each angle, in any particular UT , must have its equal counterpart in another, different, UT . This coincides with the fact that there are 9 different phase expressions of the KM matrix for different parameterizations [8]. To understand this simple but crucial discussion consider the two UT s defined by The inner angles defined by UT (a), in Eq.
(1), are Correspondingly, the unitarity triangle, UT (b) in Eq. (1), defines another three inner angles Since all the six UT s have the same area J/2, not all the different 9 angles are independent.
It can be shown that only 4 independent parameters are needed to parameterize the six UT s, and two different UT s contain the needed (1), is almost a right triangle, by virtue of φ 2 . Numerically, the angles φ ′ 1 and φ ′ 3 are close to φ 1 and φ 2 , respectively. All the angles in the two UT s are sizable, making experimental determination of them merely challenging, while for the other four choices of UT there is always, at least, one small angle where measurement may be exceptionally difficult. It is therefore easiest to work with the two UT s, UT (a) and UT (b), for practical purposes. We now show that, by combining information from these two UT s, into the boomerang diagram 1 displayed in Fig. 1, all information needed to specify the KM matrix, V KM , can be extracted.
One can choose the area (J/2) of the triangles, two inner angles from one of the UT s (for example φ 1 and φ 2 ), and a third angle from the other UT (for example φ ′ 3 ) as the four independent parameters.

Original KM parametrization and Unitarity Boomerang
To show explicitly how the unitarity boomerang can provide all information needed to specify the quark flavor mixing, we work with a specific parametrization, V KM , originally given by Kobayashi and Maskawa[1] One can also work with other parameterizations, such as that adopted by the PDG. But we find an interesting feature of the original KM parametrization which turns out to be very convenient for the discussions of the unitarity boomerang.
The fact that φ 2 = (88 +6 −5 ) • implies δ ≈ 90 • . The approximate right angle at the top of the boomerang diagram may indicate that CP, from a deeper perspective, is maximally violated [10,11]. Kobayashi and Maskawa, with remarkable prescience, made an excellent choice of parametrization. We suggest that the original parametrization of Kobayashi-Maskawa matrix be used as the standard parametrization. A parametrization suggested by Fritzsch and Xing [10], which also has its phase close to φ 2 , is another alternative interesting parametrization. From the unitarity boomerang, one can easily obtain approximation solutions for the four physical parameters. One first notices that the relation in Eq.(5) allows one to read off the δ from the top angle in the diagram. Taking the ratio, of the two sides With c 1 and therefore s 1 known, the length of the sides AB and AC' then provide the values for s 2 and s 3 .
One can obtain more precise solutions by using the following information from four sides, AC = a, BC = b, AB = c and AB ′ = d of the unitarity boomerang: Using the above, one can express s 1,2,3 and δ as functions of a, b, c and d. The KM parameters can be determined. For example Solving for the roots of the above equations, the c 2 1 is determined up to four possible discrete solutions. Restricting to real positive solutions with magnitude less than 1, one can further limit the choices.
The other angles, and the phase, can be determined from the following relations After applying the constraint on c 2 2,3 , that they satisfy 0 ≤ c 2 2,3 ≤ 1, the solution is even more restricted. Putting in numerical values, for the sides, and comparing with the approximate solution above, we find that a unique solution survives.
and these numbers are self consistent.
One should be aware, that there remain errors, on the sides and angles of the boomerang.
This leads to distortion of the UB away from the true one. When constructing the UB, one can first use measurable quantities without assuming unitarity to form one of the UT , say, the UT defined by triangle ABC in Fig. 1. This can be achieved by using the measured α and β and also the length of side AB, c = |(V KM ) td (V KM ) * tb |. The major error comes from the uncertainty in | Assuming |(V KM ) tb | is almost one, then [2], |(V KM ) td | = (8.09 ± 0.6) × 10 −3 . One then uses information on the values of |(V KM ) ud | and |(V KM ) ub | to construct the sides AB ′ and AC ′ to complete the boomerang. The error in |(V KM ) td | will cause uncertainty in the side AB ′ of the UB with d = (7.88 ± 0.58) × 10 −3 . At present within error bars, one cannot be sure which side, AB or AB ′ , is longer. Further reduce the errors in |(V KM ) td (V KM ) * tb | can be achieved by better understanding of the bag factor in B d −B d mixing [2]. Another way to improve the situation is to note that the value |(V KM ) tb |/|(V KM ) ud | plays an important role which also determine the ratio of AC and AC ′ . Therefore precise measurement of |(V KM ) tb | is crucial in constructing an accurate UB. Future studies of top quark decay and single top quark production at colliders, such as the LHC, will provide useful information.
To give a quantitative feeling, we have carried out an estimate assuming that the errors in a, b, c and d are given by the current PDG data with Gaussian errors to obtain the resultant errors in the KM angles. We obtain ∆c 1 = 0.046, an error which is reasonably small. But errors on s 2,3 are large with ∆s 2 = 0.032 and ∆s 3 = 0.077. Such a larger error bolsters preference for the boomerang, to disentangle, most perspicuously, the quark flavor mixing.
Note that errors, on s 2,3 , are due to empirically-generated uncertainties on (V KM ) td , (V KM ) cb and (V KM ) ub .
Indeed, when we look more closely at Eq. (7), it does turn out that the quantity c enters that equation, only in a combination (c 2 /d 2 ), just so that (V KM ) td cancels out. If one takes into account, the errors are reduced to ∆c 1 = 0.032, ∆s 2 = 0.023 and ∆s 3 = 0.055.
If uncertainties on all four sides can be reduced, say by another factor of three, we project that errors can be reduced to ∆c 1 = 0.011, ∆s 2 = 0.076 and ∆s 3 = 0.018, thus illustrating how the chosen boomerang may, in the foreseeable future, return to increase human knowledge. Our proposal, to move from a single triangle to a boomerang combination, therefore reflects, more than anything else, the increase in precision which is justifiably anticipated from the high-energy experiments.

Discussion
The most popular way to present the flavor mixing for three generations of quarks is by a unitarity triangle which, however, explicitly displays only three of four independent parameters. To have a diagrammatical representation for the full four independent parameters, we have proposed improvement to the unitarity boomerang.
By studying the unitarity boomerang, one can obtain all the information enshrined in KM matrix. We find that the original parametrization by Kobayashi and Maskawa is particularly convenient for this purpose. The angle φ 2 in the boomerang diagram, to a good approximation, can be identified with the phase δ in the original KM parametrization [1].
The fact that φ 2 = (88 +6 −5 ) • implies δ ≈ 90 • , so that this parametrization may be the right one to study assiduously, in order to probe further the connection to the origin of, possibly maximal, CP violation. We, therefore, humbly submit that the original parametrization of KM matrix be kept as the standard, and that the unitarity boomerang shown in FIG.1 be used unambiguously to present the experimental information.