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Physics Letters B

www.elsevier.com/locate/physletb

Flavor mixing democracy and minimal CP violation

Jean-Marc Gerarda, Zhi-zhong Xingb *

a Centre for Cosmology, Particle Physics and Phenomenology (CP3), Universite Catholique de Louvain, B-1348, Louvain-la-Neuve, Belgium b Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

ARTICLE INFO

ABSTRACT

Article history:

Received 9 March 2012

Received in revised form 12 May 2012

Accepted 18 May 2012

Available online 21 May 2012

Editor: T. Yanagida

We point out that there is a unique parametrization of quark flavor mixing in which every angle is close to the Cabibbo angle 9C ~ 13° with the CP-violating phase faq around 1°, implying that they might all be related to the strong hierarchy among quark masses. Applying the same parametrization to lepton flavor mixing, we find that all three mixing angles are comparably large (around n /4) and the Dirac CP-violating phase fai is also minimal as compared with its values in the other eight possible parametrizations. In this spirit, we propose a simple neutrino mixing ansatz which is equivalent to the tri-bimaximal flavor mixing pattern in the fai ^ 0 limit and predicts sin013 = 1/V2sin(fai/2) for reactor antineutrino oscillations. Hence the Jarlskog invariant of leptonic CP violation Ji = (sin fa)/12 can reach a few percent if 813 lies in the range 7° ^ 813 ^ 10°.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Within the standard electroweak model, the origin of CP violation is attributed to an irremovable phase of the 3 x 3 Cabibbo-Kobayashi-Maskawa (CKM) quark flavor mixing matrix [1] in the charged-current interactions:

Vud Vus Vub\ ld

Vcd Vcs Vcb s

Vtd Vts Vtb! \b

-4c = ^= (u c t)Ly^( Vcd Vcs Vcb)[s)W+ + h.c. (1)

The size of this nontrivial CP-violating phase depends on the explicit parametrization of the CKM matrix V. One may in general describe V in terms of three rotation angles and one CP-violating phase, and arrive at nine topologically different parametrizations [2]. If V takes the Cabibbo flavor mixing pattern [3]

1 /11 1 \

Vc = ^ 1 a a2 , (2)

V3 \1 a2 a J

where a = e'2n/3 is the complex cube-root of unity (i.e., a3 = 1), then one can immediately find that the CP-violating phases in all the nine parametrizations are exactly n/2. Hence VC characterizes the case of "maximal CP violation" in a parametrization-independent way, although it is not a realistic quark flavor mixing matrix. Among the nine parametrizations of V listed in Ref. [2], the one advocated by the Particle Data Group [4] is most popular and its CP-violating phase is about 65°. The idea of a "geometrical T violation" has been suggested in Ref. [5] to explain such a CP-violating phase around n/3. In comparison, the CP-violating phase is about 90° in the parametrization recommended in Ref. [6] or in the original Kobayashi-Maskawa representation [7]. Accordingly, the concept of "maximal CP violation" has sometimes been used to refer to a quark flavor mixing scenario in which the CP-violating phase equals n/2 for given values of the mixing angles [8-11].

Of course, the value of the CP-violating phase is correlated with the values of the mixing angles in a given parametrization of V. Indeed, the parametrization itself depends on the chosen flavor basis and only the moduli of the matrix elements Vij are completely basis-independent. Although all the parametrizations of V are mathematically equivalent, one of them might be phenomenologically

* Corresponding author.

E-mail addresses: jean-marc.gerard@uclouvain.be (J.-M. Gerard), xingzz@ihep.ac.cn (Z.-z. Xing).

0370-2693/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016Zj.physletb.2012.05.037

more interesting in the sense that it might either make the underlying physics of quark mass generation and CP violation more transparent or lead to more straightforward and simpler relations between the fundamental flavor mixing parameters and the corresponding observable quantities. It is therefore meaningful to examine different parametrizations of the CKM matrix V and single out the one which is not only phenomenologically useful but also allows us to have a new insight into the flavor puzzles and possible solutions to them.

In this Letter we pose such a question: is it possible to ascribe small CP-violating effects in the quark sector to a strongly suppressed CP-violating phase in the CKM matrix V in which all three mixing angles are comparably sizable? The answer to this question is actually affirmative as already observed in Ref. [12], and the details of such a nontrivial description of quark flavor mixing and CP violation will be elaborated in Section 2. We show that the CP-violating phase faq is only about 1°, while every quark mixing angle is close to the Cabibbo angle 6C ~ 13° in this unique parametrization of V, implying that they might all have something to do with the strong hierarchy of quark masses. We argue that this particular representation reveals an approximate flavor mixing democracy and "minimal CP violation". It also provides a simple description of the structure of the matrix V, which is almost symmetric in modulus about its Vud-Vcs-Vtb axis.

Applying the same parametrization to the lepton flavor mixing, we find that all three angles are comparably large (around n/4) and the Dirac CP-violating phase fa is also minimal as compared with its values in the other eight possible parametrizations. We start from this observation to propose a simple and testable neutrino mixing ansatz which is equal to the well-known tri-bimaximal flavor mixing pattern [13] in the fa ^ 0 limit. It predicts sin613 = 1/y2sin(fa/2) for reactor antineutrino oscillations, and its two larger mixing angles are consistent with solar and atmospheric neutrino oscillations. The Jarlskog invariant for leptonic CP violation turns out to be Ji = (sin fa)/12, which can reach a few percent if 613 lies in the range 7° < 613 < 10°.

2. Quark flavor mixing

The parametrization of the CKM matrix V, which assures an approximate flavor mixing democracy and nearly minimal CP violation in the quark sector, takes the form

0 sy Cx sx 0

1 0 -sx cx 0

0 cy 0 0 e -i0q 1

z + sysze' sxc y

-sxCz cx

Cz 0 Sz xcysz

1 0 0 Cz

-cxsycz + cvsze i^q

Cxsysz + CyCze-i^q

^x^y Lx->y->z + L yLzc

where cx = cos 6x and sx = sin 6x, and so on. Without loss of generality, we arrange the mixing angles to lie in the first quadrant but allow the CP-violating phase fa to vary between zero and 2n. Comparing Eq. (1) with Eq. (3), we immediately arrive at the relation cos6x = | Vcs\ together with

tan 9 y = tan 9z =

Vus Vcb

In this parametrization the off-diagonal asymmetries of V in modulus [14] are given as

AqL = | Vus|2 - | VCd 12 = I Vcb I 2 - I Vts | 2 = | Vtd | 2 - | Vub | 2 = s2 (s2 - s2), aR = | Vus|2 - | Vcb| 2 = | Vcd|2 - | Vts| 2 = |Vtb| 2 - | Vud| 2 = s2(c2 - s2).

Note that the other eight parametrizations listed in Ref. [2] are unable to express AqL and aR in such a simple way. Furthermore, the Jarlskog invariant for CP violation [15] reads

Jq = VudVcsV*sV*d) = Im(VusVcbV*bV*s) = ••• = Cxs2xCysyCzsz sin

We observe that choosing |Vcs |, Aq, AR and Jq as four independent parameters to describe the CKM matrix V is also an interesting possibility, because they determine the geometric structure of V and its CP violation in a straightforward and rephasing-invariant manner.

To see the point that 6x, 6y and 6z are comparable in magnitude, let us express them in terms of the well-known Wolfenstein parameters [16]. Up to the accuracy of O(X6), the Wolfenstein-like expansion of the CKM matrix V [17] is given as

-X[1 - A2X4(2 - p) + iA2X4n]

1 - 1 X2 - 8 (1 + 4A2)X4

AX3(p - in)\ AX2

V AX3[1 - (1 - 2^2)(p + in)] -AX2[1 - X2( 1 - p) + ix2n] 1 - 1A2X4 )

where X = 0.2253 ± 0.0007, A = 0.808+0 015, P = 0.135+0'o14 and n = 0.350 ± 0.013 extracted from a global fit of current experimental data on flavor mixing and CP violation in the quark sector [4]. Comparing Eq. (7) with Eq. (3), we arrive at the approximate relations

tan 9x ~ X

1 + 2 (1 + A2)X2

1 _ 1 x2- 1X4

tan 9y — Ak

tan 9z — Ak,

sin 4>q — k n

1 - 2 (1 - 2p)k

1 + -(1 + 2A2 - 2p)k2

which hold up to the accuracy of O(k5). Therefore, we obtain

9X~ 13.2°

ey —10.1°,

Qz~ 10.3°

— 1.1°

We see that the small difference between 6y and 6z signifies a slight off-diagonal asymmetry of the CKM matrix V in modulus about its V„d-Vcs-Vtb axis. Note that this tiny asymmetry is quite stable against the renormalization-group-equation (RGE) running effects from the electroweak scale to a superhigh-energy scale or vice versa. Indeed, only the Wolfenstein parameter A is sensitive to the RGE evolution [18] so that 0y and 6z run in the same way even at the two-loop level.1 In contrast, 6x and are almost insensitive to the RGE running effects. The striking fact that the CP-violating phase is especially small in this parametrization was first emphasized in Ref. [12]. Indeed, the other eight parametrizations listed in Ref. [2] all require > 60°. Moreover, the values

aL — 6.3 • 10

aR — 4.9 • 10-2, Jq — 3.0 • 10

indicate that 0y = 0z and = 0 might be two good leading-order approximations from the point of view of model building. In these two limits the CKM matrix V is real and symmetric in modulus. Consequently, the small off-diagonal asymmetry and the small CP-violating phase of V might come from some complex perturbations at the level of quark mass matrices.

Why may ~ k2 coexist with 0X ^ 0y ^ 6z ^ k? The reason is simply that Vub is the smallest CKM matrix element and only a small $>q guarantees a significant cancellation in Vub = -cxcysz + sycze-i^q to make |VUb| ~ O(k4) hold.2 The point that Vub strongly depends on motivates us to propose a phenomenological ansatz for quark flavor mixing in which Vub ^ 0 holds in the ^ 0 limit. In this case we find that the condition tan 6y = tan 6z cos 6x must be fulfilled and the CKM matrix reads

/ Sy /Sz V0 = -SxCz

^ SxCyS

SxSz Cz/Cy,

Of course, V0 can approximately describe the observed moduli of the nine CKM matrix elements. The relation tan 6y = tan 6z cos 6x implies that 6z must be slightly larger than 6y, and thus it has no conflict with the numerical results obtained in Eq. (9). Now the CP-violating phase is switched on and V0 is changed to

(c2 + S2e-i^q)Sy/Sz SxCy -SyCz(1 - e-i^q)

-CySz(c2 - e-i^q) -

(S2y + Cle-itq )Cz/Cy,

which predicts |VUb| = 2SyCz sin($q/2) — SyCz sin for very small . Comparing Eq. (12) with Eq. (7), we arrive at tan6x — k, tan6y — tan6z — Ak and sin— k2^/p2 + n2 in the leading-order approximation. We conclude that this ansatz is essentially valid, and it provides us with a good lesson for dealing with lepton flavor mixing in Section 3.

It has long been speculated that the small quark flavor mixing angles might be directly related to the strong quark mass hierarchies [19,20], in particular when the quark mass matrices possess a few texture zeros which can naturally originate from a certain flavor symmetry [21]. In this sense it is also interesting for us to consider possibly simple and instructive relations between quark mass ratios (mu/mC, mC/mt, md/mS and mS/mb) and flavor mixing parameters (6x, 6y, 6z and ) in the parametrization of V under discussion. In view of the values for the quark masses renormalized at the electroweak scale [22], we make the naive conjectures

■a md mu

sin 6x — J--1--,

sin 6y — sin 6z — , mS

sin 4>q —

Of course, these approximate relations are only valid at the electroweak scale, and whether they can easily be derived from a realistic model of quark mass matrices remains an open question. But a possible correlation between the smallness of the CP-violating phase and the smallness of quark mass ratios (e.g., sin— mS/mb as first conjectured in Ref. [12]) is certainly interesting and suggestive, because it might imply a common origin for the quark mass spectrum, flavor mixing and CP violation. We hope that such a phenomenological observation based on our particular parametrization in Eq. (3) may be useful to infer the presence of an underlying flavor symmetry from the experimental data in the near future.

1 We thank H. Zhang for confirming this point using the two-loop RGEs of gauge and Yukawa couplings.

2 Because of A — 0.808, p — 0.135 and n — 0.350, the true order of \Vub\ is k4 instead of k3. Following the original spirit of the Wolfenstein parametrization [16], one

may consider to take Vub = Ak4(p - in) by redefining two O(1) parameters p = p/k — 0.599 and n = n/k — 1.553.

3. Lepton flavor mixing

We proceed to consider the 3 x 3 Maki-Nakagawa-Sakata-Pontecorvo (MNSP) lepton flavor mixing matrix [23] in the weak charged-current interactions:

Ue1 Ue2 Ue3\ /VX'

U U | U |3 V2 Ut 1 UT2 UT3/ Wl

-Lcc = ^2 (e I r )lYIX

W - + h.c.

The MNSP matrix U can be parametrized in the same way as in Eq. (3):

ca -sa

'Ob 0 sb 0 1 0

- sb 0 cb 0 ' cacbcc + sbsce-ifal -sacc

,-casbcc + cbsce

sa 0 cc 0 -sc ca 0 0 1 0 P v

0 e-ifai / \sc 0 cc / sacb -cacbsc + sb€ce-ifal\

ca sa sc

-sasb casbsc + Wce-^1 )

where Pv = Diag{eip, eia, 1} denotes an irremovable phase matrix if the massive neutrinos are Majorana particles, ca = cosOa and sa = sin Oa, and so on. Current experimental data indicate that at least two lepton mixing angles are much larger than the Cabibbo angle 6C ~ 13° [4]. In particular, the tri-bimaximal flavor mixing pattern [13]

' —3

\ -L --L

x V6 V3

is quite consistent with the observed values for the solar and atmospheric neutrino mixing angles and can easily be derived from a number of neutrino mass models based on discrete flavor symmetries [24]. Comparing Eq. (15) with Eq. (16), we see that they become equivalent to each other if the conditions

0a ~ 54.7°

Ob = 45°

0c = 60°

fa = 0°

are satisfied. A particularly interesting point is that the relation tan Ob = tan Oc cos Oa exactly holds and thus the matrix element Ue3 = -cacbsc + sbcce-ifa automatically vanishes as fa approaches zero. This observation, together with the promising ansatz for the quark flavor mixing discussed in Eqs. (11) and (12), motivates us to consider the following lepton flavor mixing ansatz:

( 2-6(1 + 3e-fa) 1

V-2-6(1 - 3e-ifal)

- 2—2(1 - e-ifal )^ 1

2m(1+e-ifa) )

which reproduces the tri-bimaximal flavor mixing pattern U0 in the fa ^ 0 limit. In other words, the generation of nonzero Ue3 is directly correlated with the nonzero CP-violating phase fa (or vice versa). Similar to the case of quark flavor mixing, all three lepton mixing angles are comparably large in this parametrization. Hence it also assures the "minimal CP violation" in the lepton sector, although one has not yet observed CP-violating effects in neutrino oscillations.

One may similarly calculate the Jarlskog invariant of leptonic CP violation and off-diagonal asymmetries of U in modulus based on Eqs. (15) and (18). The results are

Jl = hn(Ue1 Ui2 Ue2 U^ = HUe2 UI Ue3 UU = " = ^l^c sin fa = ^ sin fa,

AL = |Ue2 |2 - |U|1|2 = |U|3|2 - |Ut2 |2 = |Ut 1|2 - |Ue3|2 = s^2 - s2) = +-

AR = | Ue2 |2 - |U|3|2 = |U |1|2 - |U t 2 |2 = |U t 3 |2 - |Ue1|2 = sj (c2 - s2) = -g .

It becomes obvious that the MNSP matrix U is more asymmetric in modulus than the CKM matrix V, and CP violation in the lepton sector is likely to be much larger than that in the quark sector simply because the lepton flavor mixing angles are not suppressed.

To see why the ansatz proposed in Eq. (18) is phenomenologically interesting in a more direct way, let us compare it with the standard parametrization of the MNSP matrix U [4]. In this case the neutrino mixing angles are predicted to be

1 fa sin O13 = — sin — ,

tan O12 =

tan O23 =

2 - 3sin2 O13

1 - 2sin2 O13

So 613 < 45° must hold for arbitrary values of &. Given the generous experimental upper bound 613 < 6C [4], the upper limit of & turns out to be & < 37.1°. A global analysis of current neutrino oscillation data seems to favor 613 — 8° [25], implying & — 22.7° together with 612 — 35.7° and 623 — 45.6°. These results are certainly consistent with the present experimental data. The resulting value of the leptonic Jarlskog parameter is Ji = (sin&)/12 — 3.2%, which should be large enough to be observed in the future long-baseline neutrino oscillation experiments. Furthermore, the CP-violating phase Si in the standard parametrization is found to be much larger than & in our ansatz:

■ „ V2 cos2 613

sin Si = t . (22)

2 - 3sin2 613

Therefore, we obtain Si — 84.4° for 613 — 8°. Note again that 613 < 45° holds, so Eq. (22) is always valid for the experimentally allowed range of 613.

The values of the charged-lepton masses at the electroweak scale have already been given in Ref. [22], from which we obtain me/m— 4.74 • 10-3 and m^/mT — 5.88 • 10-2. In view of the neutrino mass-squared differences extracted from current neutrino oscillation experiments [25], we get m2/m3 — 0.17 in the m1 — 0 limit for a normal mass hierarchy. A naive conjecture is therefore

sin & — /—, (23)

implying &i — 24.3° and thus 613 — 8.6°. Since 6a, 6b and 6C are all large, it seems more difficult to link them to the charged-lepton or neutrino mass ratios.

Finally, it is worth pointing out that one may propose similar ansatze of lepton flavor mixing based on some other constant patterns with Ue3 = 0. For example, we find that the mixing angles of the democratic [26], bimaximal [27], golden-ratio [28] and hexagonal [29] mixing patterns expressed in our present parametrization can also satisfy the condition tan 6b = tan 6C cos 6a, and thus the matrix element Ue3 = -CaCbSC + SbCCe-'& automatically vanishes in the & ^ 0 limit. Given such a constant pattern, a lepton flavor mixing ansatz analogous to the one proposed in Eq. (18) can similarly be discussed. Its salient feature is therefore the prediction

sin 613 = |Ue3|=2SbCC sin &, (24)

which directly links & to 613. Given 6b = 45° and 6C = 60° for the tri-bimaximal flavor mixing pattern, the first relation in Eq. (21) can then be reproduced from Eq. (24).

4. Summary

We have explored a unique parametrization of fermion flavor mixing in which the mixing angles are nearly democratic and the (Dirac) CP-violating phase is minimal. Within such a parametrization of the CKM matrix V we have shown that all three quark mixing angles are close to the Cabibbo angle 6C — 13° while the CP-violating phase &q is only about 1°. It also provides a simple description of the structure of V, which is almost symmetric in modulus about its Vud-VCS-Vtb axis. When the MNSP matrix U is parametrized in the same way, we find that the lepton mixing angles are comparably large (around n/4) and the Dirac CP-violating phase & is also minimal as compared with its values in the other eight possible parametrizations. These interesting observations have motivated us to propose a simple and testable neutrino mixing ansatz which is equal to the well-known tri-bimaximal flavor mixing pattern in the & ^ 0 limit. It predicts sin 613 = 1/V2sin(0i/2) for reactor antineutrino oscillations, and its two larger mixing angles are consistent with solar and atmospheric neutrino oscillations. The Jarlskog invariant of leptonic CP violation is found to be Ji = (sin &)/12, which can reach a few percent if 613 lies in the range 7° < 613 < 10°.

It is worth remarking that the unique parametrization discussed in this Letter provides us with a novel description of the observed phenomena of quark and lepton flavor mixings. Different from other possible parametrizations suggesting either a "geometrical" or a "maximal" CP-violating phase, it allows us to deal with a "minimal" one. Although it remains unclear whether such a new point of view is really useful in our quest for the underlying flavor dynamics of fermion mass generation and CP violation, we believe that it can at least help understanding the structure of flavor mixing in a phenomenologically interesting way.

Note added

Soon after this Letter appeared in the preprint archive (arXiv:1203.0496), the Daya Bay Collaboration announced their first ve ^ ve oscillation result: sin2 2613 = 0.092 ± 0.016(stat) ± 0.005(syst) (or equivalently, 613 — 8.8° ± 0.8°) at the 5.2a level [30]. We find that our expectations, such as 613 — 8.6° given below Eq. (23), are in good agreement with the Daya Bay observation.

Acknowledgements

Z.Z.X. would like to thank J.M.G. for warm hospitality at CP3 of UCL, where this work was done. He is also grateful to J.W. Mei and H. Zhang for useful discussions. The work of J.M.G. was supported in part by the Belgian IAP Program BELSPO P6/11. The work of Z.Z.X. was supported in part by the Ministry of Science and Technology of China under grant No. 2009CB825207, and in part by the National Natural Science Foundation of China under Grant No. 11135009.

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