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OHSTPY-HEP-T-04-002 PreprinttypesetinJHEPstyle-HYPERVERSION sin 2 Quark Mass Textures and β 4 0 0 2 n a J 2 Hyung Do Kima,b, Stuart Rabya and Leslie Schradina 2 aDepartment of Physics, The Ohio State University, 1 174 W. 18th Ave., Columbus, Ohio 43210, USA v 9 bSchool of Physics, Seoul National University, 6 Seoul, 151-747, Korea 1 1 0 E-mail: hdkim,raby,[email protected] 4 0 / h p Abstract: Recent precise measurements of sin2β from the B-factories (BABAR and - p BELLE) and a better known strange quark mass from lattice QCD make precision e tests of predictive texture models possible. The models tested include those hierar- h : chical N-zero textures classified by Ramond, Roberts and Ross, as well as any other v i hierarchical matrix Ansatz with non-zero 12 = 21 and vanishing 11 and 13 elements. X r We calculate the maximally allowed value for sin2β in these models and show that all a the aforementioned models with vanishing 11 and 13 elements are ruled out at the 3σ level. While at present sin2β and V /V are equally good for testing N-zero texture ub cb | | models, in the near future the former will surpass the latter in constraining power. Keywords: Quark Mass, Mixing Angles, CP Violation, Texture Zeros. Contents 1. Introduction 1 2. Data 3 2.1 CKM elements 3 2.2 Masses 3 3. Setup 4 4. Analysis 7 4.1 2 2 light quark matrices 7 × 4.2 3 3 quark matrices 7 × 4.2.1 Models with 11 = 13 = 0 and 12 = 21 = 0 8 6 4.2.2 Models with non-zero 13 elements 11 5. Conclusion 12 A. Loop Rephasing Method 13 1. Introduction The problem of understanding the origin of fermion masses has persisted for more than twenty years [1, 2, 3]. Models predicting relations between fermion masses and mixing angles may give insights into possible solutions to this problem. On the other hand, testing theories of fermion masses requires precision data; the accuracy of which has been severely limited by theoretical uncertainties inherent in QCD. In particular, light quark masses and some CKM elements have been difficult to measure with precision. RecentresultsfromB-factoriescombinedwithadvancesinthetheoryofheavyquarksas well as in lattice QCD have reduced the errors considerably for these observables [4, 5]. Current measurements of V and V have errors at the 2% level [4]. sin2β is now us cb | | | | known to 6.5% from experiments on the asymmetry in B decays [5]. Even V /V , ub cb | | whoseerrorslargelycomefromnon-perturbativeQCDeffects, hasnowbeendetermined to about 10% [4]. In the mass sector, the most important improvement has been in m , s – 1 – whose uncertainty has decreased from 50% to 12% over the last ten years. Moreover, lattice QCD results with light dynamical quarks indicate that the strange quark mass is much lighter than previously thought [6]. In a pioneering work Hall and Rasin [7] showed that the relation V /V = m /m (1.1) ub cb u c | | p is obtained for any hierarchical texture with vanishing 11, 13, 31 elements. Roberts et al. [8] then re-analyzed these textures, using more recent data, and concluded that such textures were disfavored, i.e. disagreeing with data at about 1σ (see also [9]). Whereas the addition of a small 13 31 element gave good fits to the data. They also studied ≈ non-hierarchical asymmetric textures, with vanishing 13, 31 elements but satisfying the “lopsided” relation 32 33. Good fits to the data were also obtained in this case. ∼ The strongest constraint, in their analysis, came from the observable V /V , which ub cb | | at the time had an uncertainty (22%).1 On the other hand, the value of sin2β was O not well known and, in fact, the central value was much lower than it is now. However, with symmetric textures with non-zero 13 = 31 elements, they predicted sin2β to be near its present experimental value. With the significant improvement in the data, we feel that it is once again a good time to re-analyze quark mass textures. In this paper we study hierarchical textures satisfying 12 = 21. We find that such textures, with vanishing 11, 13, 31 elements, are now excluded by 3σ. In the present study, constraints from both V /V and sin2β ub cb | | are equally strong. However, in the near future sin2β may provide the most stringent constraint. For example, it is expected that the experimental precision of sin2β will greatly improve (by a factor of 2), perhaps by 2006 when an expected 500 fb−1 of data will have been tabulated by both BaBar and Belle [10]. On the other hand, the experimental uncertainties associated with V /V (limited by uncertainties in V ) ub cb ub | | | | may require a Super-B factory (perhaps by 2010 [11]) to obtain a similar factor of 2 reduction [12]. InSection2wetabulatethelatestdataonquarkmassesandmixingangles. Thenin Section 3 we present the relevant approximations used when diagonalizing hierarchical fermion mass textures. In Section 4.1, as a warm-up, we consider the oldest successful texture describing the lightest two quark families. We then obtain our main result in Section 4.2.1 where we study hierarchical 3 3 quark mass textures satisfying 12 = 21 × with 11 = 13 = 31 0. Finally in Section 4.2.2 we show that good fits to the data ≡ can be obtained with non zero 13, 31 elements. In particular we focus on two 5-zero texture models considered by Ramond et al. [13]. 1Although Roberts et al. assumed an uncertainty half this size [8]. – 2 – 2. Data In this Section we tabulate the present data for CKM elements and quark masses used in our analysis. 2.1 CKM elements We take the CKM element values from [4] and sin2β from [5]. V = 0.2240 0.0036 (2.1) us | | ± V = (41.5 0.8) 10−3 cb | | ± × V = (35.7 3.1) 10−4 ub | | ± × V ub = 0.086 0.008 V ± (cid:12) cb(cid:12) s(cid:12)in2β(cid:12) = 0.739 0.048 (cid:12) (cid:12) ± (cid:12) (cid:12) The errors on sin2β are mostly statistical while those on Vub are largely theoretical. Vcb As more data is taken, sin2β will become more precisely (cid:12)know(cid:12) n, but the precision in (cid:12) (cid:12) Vub will likely remain at the 10% level for some time. (cid:12) (cid:12) Vcb (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)2.2(cid:12)Masses Dueto strong interactions, the masses of thelight quarks arenot known well. The most precise estimates of m /m and m /m come from chiral perturbation theory. There u d s d is some disagreement in the literature on the sizes of the errors of the light quark mass ratios [14]. For this reason, we take m /m and m /m from [15] with doubled errors. u d s d Recent lattice QCD calculations with dynamical quarks have improved our knowl- edge of m , previously the least known of the light quarks. We use the unquenched s lattice QCD result with n = 2 for m [6] and double the error to account for the f s discrepancy with the sum rule result. The central value (Eqn. 2.2) is near the low end of the range given by the PDG [14]. The preliminary result with n = 2+1 indicates f that the strange quark might be even lighter. For the charm quark mass we use a quenched lattice QCD result since an unquenched calculation is not yet available. Quenching errors are known to be about 25% for the strange quark mass and 1 to 2% for the bottom quark mass. Because of the mass hierarchy, it is expected that the quenching error on m will lie somewhere c between these two bounds. Thus we take the lattice QCD result with a (probably conservative) 10 percent systematic (quenching) error as in [4] and double it. – 3 – We use the bottom quark mass from [16] and the top quark pole mass from the Particle Data Group (PDG) 2003 [14]. m u = 0.553 0.043 2 (2.2) m ± × d m s = 18.9 0.8 2 m ± × d m (2GeV) = 89 11 2MeV s ± × m (m ) = 1.30 0.15 2GeV c c ± × m (m ) = 4.22 0.09GeV. b b ± M (pole) = 174.3 5.1GeV t ± m (m ) = 165 5GeV t t ± All running mass parameters are defined in the MS scheme. We note here that the doubled errors we are using almost incorporate the bounds found in the PDG [14]. For the purposes of further analysis we compare to fermion masses evaluated at M . We define the renormalization factor Z mi(MZ) for i = c, b, t mi(mi) η (2.3) i  ≡  mi(MZ) for i = u, d, s. mi(2 GeV) At two loops in QCD we find   η = 0.56, η = 0.69, η = 1.06 (2.4) c b t η = η = η = 0.65. u d s 3. Setup This section introduces our notation for the Yukawa and CKM matrices. Quark masses are expressed in terms of Weyl spinors as = Q YU H uc +Q YD H dc, (3.1) −L i ij h ui j i ij h di j u where i,j = 1,2,3 are the family indices, Q = is the left-handed quark doublet, d (cid:18) (cid:19) uc and dc are the left-handed anti-up and -down quarks, and H and H are the up u d and down Higgs fields. To keep track of phases in YU and YD, which in general are complex, we define φUij ≡ argYiUj and φDij ≡ argYiDj . Two unitary matrices VU and UUc diagonalize the up quark Yukawa matrix by VUYUUUc = YDUiag. – 4 – Similarly, VD and UDc diagonalize YD. We assume that the mass matrices are hierarchical [7] which means Y(U,D) Y˜(U,D) (23,32) 1, 22 1, (3.2) (cid:12)Y(U,D)(cid:12) ≪ (cid:12)Y(U,D)(cid:12) ≪ (cid:12) 33 (cid:12) (cid:12) 33 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12)Y(U,D) (cid:12)(cid:12) (cid:12)(cid:12)Y(U,D)(cid:12)(cid:12) (12,21) 1, 11 1, (cid:12)Y˜(U,D)(cid:12) ≪ (cid:12)Y˜(U,D)(cid:12) ≪ (cid:12) 22 (cid:12) (cid:12) 22 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (U,D)(cid:12) (U,D)(cid:12) (U,D(cid:12)) (U,D(cid:12)) Y (cid:12)Y (cid:12) Y (cid:12) Y (cid:12) 13 31 12 21 , (cid:12) Y(U,D) (cid:12) ≪ (cid:12) Y˜(U,D) (cid:12) (cid:12) 33 (cid:12) (cid:12) 22 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) where (cid:12) (cid:12) (cid:12) (cid:12) Y Y Y˜ Y 23 32 Y˜ eiφ˜22 (3.3) 22 22 22 ≡ − Y ≡ 33 (cid:12) (cid:12) (cid:12) (cid:12) is used since we will consider both of the following(cid:12) ca(cid:12)ses: Y Y Y Y 0 = Y < 23 32 and 0 = Y & 23 32 . (3.4) 22 22 | | Y 6 | | Y (cid:12) 33 (cid:12) (cid:12) 33 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Note, an approximately symmetric matrix ansatz will be hierarchical unless the known (cid:12) (cid:12) (cid:12) (cid:12) CKM mixing angles come as a surprising cancellation of two large angles for up and down quarks. We also note that there are “lopsided” textures in the literature with Y Y which do not fulfill the conditions described above. We do not consider 32 33 | | ∼ | | such asymmetric textures in this paper. For an analysis of these models, see [8]. With no order one off-diagonal terms in YU or YD, we can neglect higher powers of the mixing angles and keep cosθ 1. This makes the mixing matrices quite simple. ≃ 1 sU 0 1 0 sU 1 0 0 − 12 − 13 V = sU∗ 1 0 0 1 0 0 1 sU . (3.5) U  12   − 23 0 0 1 sU∗ 0 1 0 sU∗ 1 13 23     V has the same form. The sU can be considered as generalized mixing angles carrying D ij both a real mixing angle and a phase. The CKM matrix V = V∗VT is then given by CKM U D 1 s∗ +sU∗s sU∗s∗ +s∗ 12 13 23 − 12 23 13 V = s sDs∗ 1 s∗ +sU s∗ , (3.6) CKM − 12 − 13 23 23 12 13  sDs s s sD∗s 1 12 23 − 13 − 23 − 12 13   – 5 – where YU YD sU 12, sD 12, (3.7) 12 ≃ Y˜U 12 ≃ Y˜D 22 22 YU YD sU 13, sD 13, 13 ≃ YU 13 ≃ YD 33 33 YU YD sU 23, sD 23, 23 ≃ YU 23 ≃ YD 33 33 s sD sU , s sD sU , s sD sU . 23 ≡ 23 − 23 12 ≡ 12 − 12 13 ≡ 13 − 13 The above approximations work well, except in the case of first and second generation mixing. For YD = 0, YD = YD , which we will address in this paper, | 11| | 12| | 21| sD = Y1D2 md 1+ md . To capture the rather large md 5% cor- | 12| Y˜D ≃ ms O ms O ms ∼ 22 rection (cid:12)prop(cid:12)erly,qwe m(cid:16)ust inc(cid:16)lude(cid:17)a(cid:17)t least through ( sD 2) in cD (cid:16) 1(cid:17) sD 2. We (cid:12) (cid:12) O | 12| 12 ≡ −| 12| (cid:12) (cid:12) need do this only for the first and second generation mixing in the down sector, for p all other mixing angles are small enough that the errors from our approximations are below 1%. (e.g., mu 2 10−3). mc ∼ × An exact diagonalization of the 2 2 submatrix for down quarks gives × m sD = d (3.8) | 12| m +m r s d and thus s ( V ) is given by,2 12 us | | ⊂ | | m m m s = s d eiφ u) (3.9) 12 | | m +m m − m r s d (cid:12)r s r c (cid:12) (cid:12) (cid:12) where φ (φU φ˜U ) (φD φ˜D). For Y(cid:12)(cid:12)U = YU = 0 we ha(cid:12)(cid:12)ve ≡ 12 − 22 − 12 − 22 12 21 m d s = . (3.10) 12 | | m +m r s d Furthermore, s 0.04 s and we have V = s∗ + sU∗s s∗ within a 1% | 23| ∼ ≪ | 12| us 12 13 23 ≃ 12 error. Therefore, V = s∗ is used. us 12 Wewillalsoneedβ,themostpreciselymeasuredanglewithintheunitaritytriangle. In terms of CKM matrix elements: V V∗ β = arg cd cb . (3.11) −V V∗ (cid:18) td tb(cid:19) 2 U U D InEqn. 3.9thesecondfactorincludesthecosineofthedownquarkmixingangle,i.e. s s c . 12 12 12 → – 6 – 4. Analysis In this section we confront the data. We first consider the 2 2 subsector of light × quarks and show that good fits to the data are obtained. In section 4.2 we extend the analysis to the full 3 3 case. × 4.1 2 2 light quark matrices × The Cabbibo angle, in the original texture by Weinberg [1], is generated solely by the down quark Yukawa matrix. The Yukawa matrices are given by A 0 0 C YU = YD = (4.1) 0 B C D (cid:18) (cid:19) (cid:18) (cid:19) where the parameters A, B, C, D can be taken to be real without loss of generality. Hence m d V = (4.2) us | | m +m r s d which works surprisingly well since the observed V = 0.2240 0.0036 is just the us | | ± same as md = 0.224 0.004. ms+md ± On tqhe other hand, if YU is taken to have the same form as YD then a non- removable phase enters in the determination of V . We then find us | | m m m V = s d eiφ u . (4.3) us | | m +m m − m r s d (cid:12)r s r c(cid:12) (cid:12) (cid:12) Note, using the central value of the quark(cid:12)masses, V has th(cid:12)e right value for φ π. (cid:12) us (cid:12) ∼ 2 However, due to the large uncertainties in the light quark masses, the value of φ is not significantly constrained. In what follows, we always assume YD = YD = 0 and Y(U,D) = 0. These | 12| | 21| 6 11 conditions imply that V is given by either equation (4.2) or (4.3). We will be using us | | these expressions for V throughout the rest of the analysis. us | | 4.2 3 3 quark matrices × Over the next two subsections we address two categories of 3 generation quark textures. In section 4.2.1, we consider textures with zero 11 and 13 elements and symmetric 12 and 21 elements. We show that all such textures, provided they are hierarchical, are ruled out. These include the 5-zero models I, II, and IV, nominally consistent with previous data, classified by Ramond et. al. in [13] and listed in Table 1.3 Section 4.2.2 36-zero symmetric texture models are already ruled out [13]. – 7 – Table 1: The symmetric 5-zero textures classified in [13]. The number of zeros (here, 5) refers to the total number of zero elements in the upper-right and diagonal portions of the up and down matrices. YU YD 0 X 0 0 X 0 I X X 0 X X X     0 0 X 0 X X  0 X 0   0 X 0  II X 0 X X X X     0 X X 0 X X  0 0 X  0 X 0  III 0 X 0 X X X     X 0 X 0 X X  0 X 0   0 X 0  IV X X X X X 0     0 X X 0 0 X  0 0 X  0 X 0  V 0 X X X X 0     X X X 0 0 X     covers textures with non-zero 13 elements. We choose to study models III and V of Table 1, the most constrained of these models, to illustrate that texture models of this type with 5 or fewer zeros are consistent with the data. 4.2.1 Models with 11 = 13 = 0 and 12 = 21 = 0 6 We consider here hierarchical texture models with Y(U,D) = Y(U,D) = 0, YU = YU = 11 13 | 12| | 21| 6 0, and YD = YD = 0 which include type I, II and IV of [13]. By the method | 12| | 21| 6 presented in the Appendix, we see that there are 2 non-removable phases in the mass matrices for I and IV and 3 phases are non-removable for II.4 The location of these phases can be chosen as follows: I φD φU 22 22 II φD φU φU 22 23 32 IV φD φU 22 32 4Note, inour analysis we specifically use the phases fromI, II andIV 5-zeromodels. Nevertheless, sinceweareinterestedin Vub/Vcb andinthemaximumvalueforsin2β ourresultsholdforallmodels | | with 13=31=0 and 12=21=0. 6 – 8 – YU = YD = 0 implies s = 0, and the CKM matrix becomes extremely simple: 13 13 13 1 s∗ sU∗ s∗ 12 − 12 23 V = s 1 s∗ (4.4) CKM  − 12 23  sD s s 1 12 23 − 23   YU m sU = 12 ue−iφ˜U22 (4.5) 12 Y˜U ≃ m 22 r c m sD d e−iφ˜D22 12 ≃ m +m r s d φ˜ φ˜D φ˜U . 22 ≡ 22 − 22 We now use these angles to find the CKM elements and β in terms of mass ratios and phases. In the case of V , we use the expression found in equation (4.3) (also see us | | footnote 2 with regards to β). m m m V = s d e−iφ˜22 u (4.6) us | | m +m m − m r s d (cid:12)r s r c(cid:12) (cid:12) (cid:12) V m(cid:12) (cid:12) ub = sU u(cid:12) (cid:12) V | 12| ≃ m (cid:12) cb(cid:12) r c (cid:12) (cid:12) (cid:12) (cid:12) s sU cD (cid:12) β(cid:12) = arg 12 = arg 1 12 12 arg 1 reiφ˜22 sD − sD ≃ − (cid:18) 12(cid:19) (cid:18) 12 (cid:19) (cid:16) (cid:17) where m m u s r . (4.7) ≡ m m r c d Since r 1/4 1, β is restricted to be a small angle. The maximum value for β, i.e. ∼ ≪ β , is given by max sinβ = r, (4.8) max sin2β = 2r√1 r2. max − Using the data for the quark masses, we get r = 0.22 0.04. Thus this texture model ± results in sin2β = 0.43 0.08. (4.9) max ± – 9 –