Right Unitarity Triangles, Stable CP-violating Phases and Approximate Quark-Lepton Complementarity

Right Unitarity Triangles, Stable CP-violating Phases and Approximate Quark-Lepton Complementarity

Abstract

Current experimental data indicate that two unitarity triangles of the CKM quark mixing matrix are almost the right triangles with . We highlight a very suggestive parametrization of and show that its CP-violating phase is nearly equal to (i.e., ). Both and are stable against the renormalizaton-group evolution from the electroweak scale to a superhigh energy scale or vice versa, and thus it is impossible to obtain at from at . We conjecture that there might also exist a maximal CP-violating phase in the MNS lepton mixing matrix . The approximate quark-lepton complementarity relations, which hold in the standard parametrizations of and , can also hold in our particular parametrizations of and simply due to the smallness of and .

pacs:
PACS number(s): 12.15.Ff, 12.38.Bx, 12.10.Kt, 13.10.+q, 14.60.Pq, 25.30.Pt

I Introduction

In the standard model (SM) of electroweak interactions, it is the Cabibbo-Kobayashi-Maskawa (CKM) matrix that provides an elegant and consistent description of the observed phenomena of quark flavor mixing and CP violation (1). Unitarity is the only but powerful constraint, imposed by the SM itself, on the CKM matrix . This constraint can be expressed as two sets of orthogonality-plus-normalization conditions:

 ∑α(VαiV∗αj)=δij,    ∑i(VαiV∗βi)=δαβ, (1)

where the Greek subscripts run over the up-type quarks and the Latin subscripts run over the down-type quarks . The six orthogonality relations correspond to six triangles in the complex plane, the so-called unitarity triangles. Among them 1, the triangle is most popular because both its three inner angles (defined as , and in Fig. 1) and its three sides can well be determined at the -meson factories (3). The counterpart of is the unitarity triangle (as shown in Fig. 1), which will be measured and reconstructed at the LHC- (4) and (or) the super- factory (5). Note that one of the inner angles of is equal to the inner angle of . Current experimental data (3) tell us that these two triangles are approximately congruent with each other. For example, the inner angles and of are very close to and of :

 ξ−β≈γ−ζ≈λ2η≈1∘, (2)

where and are the well-known Wolfenstein parameters (6) in an -expansion of the CKM matrix (7). A more striking result is obtained by the CKMfitter Group (8) and the UTfit Collaboration (9). If holds exactly, then both and will be the right triangles.

The possibility of was actually conjectured a long time ago in an attempt to explore the realistic texture of quark mass matrices (10), and it has recently been remarked from some different phenomenological points of view (11); (12); (13). Here we are interested in the following questions and possible answers to them:

• What is the immediate consequence of on the CKM matrix and its four independent parameters?

• Could result from an underlying but more fundamental CP-violating phase in the quark mass matrices or in the CKM matrix?

• Is the result or stable against quantum corrections, for instance, from the electroweak scale to a superhigh-energy scale (such as the scale of grand unified theories or the scale of neutrino seesaw mechanisms)? In other words, is at possibly a natural low-energy consequence of at due to the renormalization-group running effect?

• Could the Maki-Nakagawa-Sakata (MNS) neutrino mixing matrix (14) contain a similar maximal CP-violating phase ?

We shall point out that simply implies . Given a very suggestive parametrization of advocated in Ref. (15), we show that its CP-violating phase is nearly equal to ; i.e., . But we find that both and are rather stable in the renormalizaton-group evolution from up to or vice versa, and thus it is impossible to obtain at from at by attributing the tiny difference to radiative corrections. We shall briefly discuss the approximate quark-lepton complementarity relations both in the standard parametrizations of and and in our particular parametrizations of and , and then make a conjecture of the maximal CP-violating phase for the MNS matrix at the end of this paper. We hope that some of our points, which might be helpful for building phenomenological models, can soon be tested with more accurate experimental data on quark and lepton flavor mixing parameters.

Ii Implications of α=90∘

Let us define the Jarlskog invariant of CP violation for the CKM matrix (16):

 Im(VαiVβjV∗αjV∗βi)=Jq∑γϵαβγ∑kϵijk, (3)

where the Greek and Latin subscripts run over and , respectively. All six unitarity triangles of have the same area which amounts to . Triangles and in Fig. 1 correspond to the orthogonality relations

 VudV∗ub+VcdV∗cb+VtdV∗tb = 0, VtbV∗ub+VtsV∗us+VtdV∗ud = 0. (4)

If holds (i.e., both and are right triangles), then we have as a straightforward consequence; namely, the rephasing-invariant quartet is purely imaginary. Hence implies a certain correlation between the parameters of in a specific parametrization. Let us illustrate this point by taking two well-known parametrizations of the CKM matrix .

• In the Wolfenstein parametrization of (6), we have

 Re(VtbVudV∗tdV∗ub)≈A2λ6[ρ(1−ρ)−η2]. (5)

So coincides with in this parametrization. Taking as an example, we obtain . Such typical values of and are certainly consistent with current experimental data (3).

• In the standard parametrization of recommended by the Particle Data Group (3),

 Re(VtbVudV∗tdV∗ub)=cos2θ12cos2θ13sinθ13cos2θ23(tanθ12tanθ23cosδ−sinθ13). (6)

Then leads to . Given , and for example (3), the CP-violating phase turns out to be . This result is also consistent with the approximate relation and the present experimental measurement of (3).

In both cases, however, we see nothing suggestive behind .

We proceed to consider a different parametrization of (15), which is more convenient to explore the underlying connection between quark masses and flavor mixing angles:

 V = Missing or unrecognized delimiter for \left (7) = ⎛⎜⎝susdc+cucde−iϕsucdc−cusde−iϕsuscusdc−sucde−iϕcucdc+susde−iϕcus−sds−cdsc⎞⎟⎠,

where , , and . The merits of this particular parametrization in understanding quark mass generation and studying heavy flavor physics are striking (15): (1) it directly follows the chiral expansion of up- and down-type quark mass matrices, and thus it can naturally accommodate the observed hierarchy of quark masses; (2) its three mixing angles are simply but exactly related to the precision measurements of -meson physics, , and ; (3) the physical meaning of its mixing angles and can well be interpreted in a variety of quark mass models (see Ref. (2) for a review with extensive references) with the interesting predictions and ; and (4) its CP-violating phase is closely associated with the light quark sector, in particular with the mass terms of and quarks 2. Using Eq. (7) to calculate the inner angle of and , we arrive at

 sinα=sinϕ[1−(tanθutanθdcosθcosϕ+12tan2θutan2θdcos2θ+⋯)] (8)

with higher-order terms of and having been omitted. It is clear that holds to a good degree of accuracy. Taking account of , and for example (17), we obtain either from or from . The result is interesting in the sense that current experimental data might imply at a superhigh energy scale and at the electroweak scale , if radiative corrections happen to compensate for the tiny discrepancy between and . We shall examine whether this point is true or not in the next section.

Is more fundamental than in describing the phenomenon of CP violation in the quark sector? The answer to this question should be affirmative if the textures of up- and down-type quark mass matrices ( and ) are parallel and originate from the same underlying dynamics (18). In this case, can be decomposed into a product of two unitary matrices: , where and are responsible respectively for the diagonalizations of and (i.e., and ) and take the following forms:

 Vu = ⎛⎜⎝e−iϕx000cxsx0−sxcx⎞⎟⎠⎛⎜⎝cu−su0sucu0001⎞⎟⎠, Vd = ⎛⎜⎝e−iϕy000cysy0−sycy⎞⎟⎠⎛⎜⎝cd−sd0sdcd0001⎞⎟⎠, (9)

where and are defined. It is obvious that and hold. Hence measures the phase difference between up- and down-type quark mass matrices and is the only source of CP violation in the quark sector. Let us make a new phenomenological conjecture of the relationship between (or ) and (or ):

 θx = −Quθ,    ϕx=−Quϕ; θy = −Qdθ,    ϕy=−Qdϕ, (10)

where and are the electric charges of up- and down-type quarks, respectively. Given the experimental values of , , and , it is then possible to determine and by using Eqs. (9) and (10). The reconstruction of and from and is straightforward, because the values of six quark masses are all known (19). If both and are taken to be Hermitian or symmetric in a particular flavor basis, then they can directly be reconstructed from quark masses and flavor mixing parameters.

Iii RGE effects on ϕ and α

The one-loop renormalization-group equations (RGEs) of the CKM matrix elements, together with the RGEs of gauge couplings and the RGEs of Yukawa couplings of quarks and charged leptons, have already been calculated by several authors (20). Here we focus on the RGE running behaviors of (for and ) by taking account of and , where and stand respectively for the eigenvalues of the Yukawa coupling matrices of up- and down-type quarks. In this excellent approximation, we simplify the results of Ref. (20) and arrive at

 16π2ddt⎡⎢ ⎢⎣|Vud|2|Vus|2|Vub|2|Vcd|2|Vcs|2|Vcb|2|Vtd|2|Vts|2|Vtb|2⎤⎥ ⎥⎦ (11) = 2Cy2b⎡⎢ ⎢ ⎢ ⎢ ⎢⎣|Vud|2|Vub|2|Vus|2|Vub|2−|Vub|2(1−|Vub|2)|Vtd|2|Vtb|2−|Vud|2|Vub|2|Vts|2|Vtb|2−|Vus|2|Vub|2−|Vcb|2(|Vtb|2−|Vub|2)−|Vtd|2|Vtb|2−|Vts|2|Vtb|2|Vtb|2(1−|Vtb|2)⎤⎥ ⎥ ⎥ ⎥ ⎥⎦ + 2Cy2t⎡⎢ ⎢ ⎢⎣|Vud|2|Vtd|2|Vub|2|Vtb|2−|Vud|2|Vtd|2−|Vub|2|Vtb|2|Vcd|2|Vtd|2|Vcb|2|Vtb|2−|Vcd|2|Vtd|2−|Vcb|2|Vtb|2−|Vtd|2(1−|Vtd|2)−|Vts|2(|Vtb|2−|Vtd|2)|Vtb|2(1−|Vtb|2)⎤⎥ ⎥ ⎥⎦,

where , in the SM and in the minimal supersymmetric SM (i.e., MSSM). Therefore,

 16π2ddtln|Vub|2|Vcb|2=−2Cy2b(1−|Vtb|2), 16π2ddtln|Vtd|2|Vts|2=−2Cy2t(1−|Vtb|2), 16π2ddtln|Vtd|2+|Vts|2|Vtb|2=−2C(y2b+y2t). (12)

Combining Eqs. (7) and (12), we immediately obtain

 16π2ddtlntanθu=−Cy2bsin2θ, 16π2ddtlntanθd=−Cy2tsin2θ, 16π2ddtlntanθ=−C(y2b+y2t); (13)

or equivalently,

 16π2dθudt=−12Cy2bsin2θusin2θ, 16π2dθddt=−12Cy2tsin2θdsin2θ, 16π2dθdt=−12C(y2b+y2t)sin2θ. (14)

Let us stress that the simplicity of RGEs of three quark mixing angles is naturally expected for our particular parametrization of , just because its matrix elements involving and quarks are very simple and exactly consistent with the - and -dominance approximations taken for the RGEs of (21).

We proceed to derive the RGE of the CP-violating phase from

 16π2ddt|Vud|2 = 2C|Vud|2(y2b|Vub|2+y2t|Vtd|2) (15) = Csin2θ(2sin2θusin2θdcos2θ+2cos2θucos2θd+sin2θusin2θdcosθcosϕ) ×(y2bsin2θu+y2tsin2θd).

Note that the derivative of can be given in terms of the derivatives of , , and as follows:

 ddt|Vud|2 = [sin2θu(sin2θdcos2θ−cos2θd)+cos2θusin2θdcosθcosϕ]dθudt (16) + [sin2θd(sin2θucos2θ−cos2θu)+sin2θucos2θdcosθcosϕ]dθddt − [sin2θusin2θdsin2θ+12sin2θusin2θdsinθcosϕ]dθdt − [12sin2θusin2θdcosθsinϕ]dϕdt.

Substituting Eqs. (14) and (15) into the right- and left-hand sides of Eq. (16), respectively, we simply arrive at

 16π2dϕdt=0. (17)

This result implies that the CP-violating phase is stable against radiative corrections at the one-loop level and in the approximation of quark mass hierarchies (i.e., and ). With the help of Eqs. (14) and (17), a straightforward calculation of the derivative of given in Eq. (8) leads to

 16π2dαdt=0. (18)

Hence the RGE running effect of is also negligibly small, implying that the low-energy result essentially keeps unchanged even if holds. In other words, it is impossible to get at from at through the one-loop RGE evolution.

Such a conclusion remains valid at the two-loop level. By using the two-loop RGEs of the CKM matrix elements (22), we have carried out a numerical analysis of the running behaviors of and from to (or vice versa) in both the SM and the MSSM 3. Here are our main observations: (1) the RGE running effect of or is too small (less than from to ) in the SM or in the MSSM with ; and (2) it cannot compensate for the small phase difference no matter how we adjust the energy scale (and the value of in the MSSM case).

Iv Quark-lepton complementarity

Compared with the parametrization of the CKM matrix given in Eq. (7), a similar parametrization of the MNS matrix is also convenient for the description of lepton flavor mixing and CP violation 4:

 U = ⎛⎜⎝clsl0−slcl0001⎞⎟⎠⎛⎜⎝e−iφ000cs0−sc⎞⎟⎠⎛⎜⎝cν−sν0sνcν0001⎞⎟⎠P (19) = ⎛⎜⎝slsνc+clcνe−iφslcνc−clsνe−iφslsclsνc−slcνe−iφclcνc+slsνe−iφcls−sνs−cνsc⎞⎟⎠P,

where , , and ; and is a diagonal phase matrix containing two nontrivial CP-violating phases when three neutrinos are Majorana particles. Although the form of in Eq. (19) is apparently different from that of the standard parametrization of (3), their corresponding flavor mixing angles () and () have quite similar meanings in interpreting the experimental data on solar and atmospheric neutrino oscillations. In the limit , one can easily arrive at and . Note that the tri-bimaximal neutrino mixing pattern (24), which is well consistent with a global fit of current neutrino oscillation data (25), does coincide with this interesting limit (i.e., , and ). Therefore, three mixing angles of can simply be related to those of solar, atmospheric and reactor neutrino oscillations in the leading-order approximation (21); i.e., , and as a natural consequence of very small .

The above comparison between our parametrization and the standard one indicates that both of them might be suitable for describing the approximate quark-lepton complementarity (QLC) relations (26). The latter means the following empirical observations in the standard parametrizations of the CKM and MNS matrices:

 θ12+ϑ12≈45∘,    θ23+ϑ23≈45∘, (20)

where and (for ) represent quark and lepton mixing angles, respectively. Eq. (20) is actually consistent with the present experimental data within error bars (3). Turning to our parametrizations of the CKM and MNS matrices in Eqs. (7) and (19), we find that similar QLC relations can approximately hold within error bars:

 θd+ϑν≈45∘,          θ+ϑ≈45∘. (21)

This result seems to be somewhat contrary to the expectation that the QLC relations are convention-dependent and may only hold in a single parametrization for and (27). We believe that the exact QLC relations can only be realized (or assumed) in a unique parametrization for and , but the approximate ones are possible to show up in different parametrizations. The reason for the latter point is quite simple: the smallest elements of the CKM and MNS matrices are both at their up-right corner (i.e., and ), and thus the flavor mixing between the first and second families is approximately decoupled from that between the second and third families. In other words, it is the smallness of (or ) and (or ) that assures the approximate QLC relations in Eqs. (20) and (21) to hold simultaneously.

Note again that the approximate QLC relations, similar to , are extracted from current experimental data at low energies. One may wonder whether such empirical relations are stable against radiative corrections, or whether they can be exact at a specific energy scale far above . Because quark and lepton flavor mixing angles obey different RGEs in their evolution from to (or vice versa) (28), we should have and in general (29); (30). This observation is also true for our parametrizations of the CKM and MNS matrices (see Ref. (21) for the explicit RGEs of , , and ), no matter whether neutrinos are Dirac particles or Majorana particles.

Finally, let us conjecture that holds in the lepton sector. This possibility can actually be realized in some specific neutrino mass models (e.g., was first obtained in the so-called “democratic” neutrino mixing scenario (31)). While is rather stable against quantum corrections from one energy scale to another, as already shown in Eq. (18), is in general sensitive to the RGE effects (21). Does imply that a pair of the leptonic unitarity triangles are right or almost right? The answer to this question depends on the value of (or equivalently in the standard parametrization of ), which has not been fixed by current neutrino oscillation experiments. For illustration, we consider the leptonic unitarity triangle defined by the orthogonality relation in the complex plane (2). Denoting the inner angle and taking the maximal CP-violating phase , we find

 sinαl=1−12c2ls2lc−2νs−2νc−2(s2ν−c2νc2)2+⋯, (22)

where higher-order terms of have been omitted. Then can be obtained from Eq. (22) with the typical inputs , and . It is easy to see that the value of approaches when approaches zero, but in the limit of there will be no CP violation (i.e., becomes trivial and can be rotated away from by rephasing the electron field) and all the leptonic unitarity triangles of must collapse into lines. This example illustrates that implies the existence of two nearly right unitarity triangles ( and its counterpart defined by the orthogonality relation ) in the lepton sector, similar to the case in the quark sector.

V Summary and concluding remarks

In view of the experimental indication that two unitarity triangles of the CKM matrix are almost the right triangles with , we have explored its possible implications on the phenomenology of quark flavor mixing and quark-lepton complementarity. Taking account of a very suggestive parametrization of , we have shown that its CP-violating phase is nearly equal to (i.e., ). Both and are stable against the renormalizaton-group evolution from the electroweak scale to a superhigh energy scale or vice versa, and thus it is impossible to obtain at from at . We have conjectured that there might also exist a maximal CP-violating phase in our parametrization of the MNS matrix . The approximate quark-lepton complementarity relations, which hold in the standard parametrizations of and (i.e., and ), can also hold in our particular parametrizations of and (i.e., and ). We have pointed out that the reason for this interesting coincidence simply comes from the smallness of and .

At this point, it is worthwhile to remark that the phenomenological ansatz proposed in Eq. (10) can be elaborated on so as to obtain an explicit texture of quark mass matrices. A similar ansatz can be made for the lepton sector by adopting the parametrization of advocated in Eq. (19) and decomposing it into in a way exactly analogous to Eqs. (9) and (10) with and . For simplicity, here we only illustrate how to reconstruct the Hermitian quark mass matrices and by using Eqs. (9) and (10). After taking account of the smallness of three mixing angles and the hierarchy of six quark masses, we approximately arrive at

 Mu ≈ ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λu+λcθ2u−λcθue+i60∘−23λcθuθe+i60∘−λcθue−i60∘λc+49λtθ2−23λtθ−23λcθuθe−i60∘−23λtθλt⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, Md ≈ ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λd+λsθ2d−λsθde−i30∘+13λsθdθe−i30∘−λsθde+i30∘λs+19λbθ2+13λbθ+13λsθdθe+i30∘+13λbθλb⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (23)

where (for and ), , and . Such a parallel texture of up- and down-type quark mass matrices is certainly suggestive and may serve as a phenomenological starting point of model building. For instance, setting leads to two interesting relations and .

Although different parametrizations of the CKM matrix are mathematically equivalent, one of them might be able to make the underlying physics of quark flavor mixing more transparent and to establish simpler connections between the observable quantities and the model parameters. We find that our parametrization of in Eq. (7) does satisfy the above criterion. We expect that the similar parametrization of the MNS matrix in Eq. (19) is also useful in describing lepton flavor mixing. Needless to say, much more experimental, phenomenological and theoretical attempts are desirable in order to solve three fundamental flavor puzzles in particle physics — the generation of fermion masses, the dynamics of flavor mixing and the origin of CP violation.

The author would like to thank H. Zhang and S. Zhou for their helps in running the two-loop RGE program. This work was supported in part by the National Natural Science Foundation of China under grant No. 10425522 and No. 10875131, and in part by the Ministry of Science and Technology of China under grant No. 2009CB825207.

Footnotes

1. Here we follow Ref. (2) to name each CKM unitarity triangle by using the flavor index that does not manifest in its three sides.
2. It is also worth pointing out that this parametrization is just Euler’s three-dimension rotation matrix if the CP-violating phase is switched off (and a trivial sign rearrangement is made).
3. H. Zhang and S. Zhou did this numerical exercise for me. Their RGE program has also been used to evaluate the running masses of quarks and leptons at different energy scales (19).
4. This parametrization may naturally arise from the parallel (and probably hierarchical) textures of charged-lepton and neutrino mass matrices. It is phenomenologically possible to obtain together with a suggestive relationship (23), where and are the neutrino masses corresponding to and flavors. Furthermore, can be decomposed into in a way similar to Eqs. (9) and (10) with and .

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