Parameter-Independent Quark Mass Relation in the U(3) × U(3) ′ Model
aa r X i v : . [ h e p - ph ] O c t Parameter-Independent Quark Mass Relationin the U(3) × U(3) ′ Model
Yoshio Koide a and Hiroyuki Nishiura ba Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, JapanE-mail address: [email protected] b Faculty of Information Science and Technology, Osaka Institute of Technology, Hirakata,Osaka 573-0196, JapanE-mail address: [email protected]
Abstract
Recently, we have proposed a quark mass matrix model based on U(3) × U(3) ′ familysymmetry, in which up- and down-quark mass matrices M u and M d are described onlyby complex parameters a u and a d , respectively. When we use charged lepton masses asadditional input values, we can successfully obtain predictions for quark masses and Cabibbo-Kobayashi-Maskawa mixing. Since we have only one complex parameter a q for each massmatrix M q , we can obtain a parameter-independent mass relation by using three equationsfor Tr[ H q ], Tr[ H q H q ] and det H q , where H q ≡ M q M † q ( q = u, d ). In this paper, we investigateits parameter-independent feature of the quark mass relation in the model. PCAC numbers: 11.30.Hv, 12.15.Ff, 12.60.-i,
Recently, we have proposed a quark mass matrix model[1] based on U(3) × U(3) ′ symmetry,in which mass matrices for up-quarks, down-quarks, charged leptons, and neutrinos, M f ( f = u, d, e, ν ) , are described respectively with only one complex parameters a f by( M f ) ji = m f (Φ f ) αi ( S − f ) βα ( ¯Φ f ) jβ . (1 . f and S f are vacuum expectation values (VEVs) matrices. The indexes i, j = 1 , , α, β = 1 , , ′ family. Although Φ f and S f have adimension of ”mass”, we put the factor m f with a dimension of mass in Eq.(1.1), since we treatthose as dimensionless quantities as seen in (1.2), (1.3) and (1.5) later.In (1.1), M ν is a Dirac neutrino mass matrix. Although we consider that the observedneutrinos are Majorana neutrinos and the Majorana neutrino mass matrix is given by a similarmechanism [1, 2] to the so-called neutrino seesaw mechanism [3], we do not discuss the structureof M ν in the present paper because the purpose of the present paper is to discuss the quarkmass relation.We define structure of the matrix Φ f as an dimensionless expressionΦ f = Φ P f , (1 . = diag( z , z , z ) , (1 . P f = diag( e iφ f , e iφ f , e iφ f ) . (1 . ′ is broken into a discrete symmetry S , the matrix ( S − f )is given by ( S f ) − = ( + a f X ) = ( + b f X ) − , (1 . = , X = 13 , (1 . a f is a complex parameter: a f = − b f b f . (1 . M e , the parameter a e is given by a e = 0, so that themass matrix M e is given by M e = m e Φ e Φ † e = m e Φ Φ , (1 . P e = without losing a generality. Namely, we take S e = only for f = e .Therefore, the parameters z i ( i = 1 , ,
3) are given by z i = r m e i m e + m e + m e , (1 . m e , m e , m e ) = ( m e , m µ , m τ ). Here, as the input values ( m e , m µ , m τ ), the running massvalues of the charged leptons at a scale µ = m Z , ( m e , m µ , m τ ) = (0 . , . , . µ = m Z . (The study of the quark mass matrix (1.1) with the form (1.5) has been substantiallydone in Ref.[4] although the model has been based on U(3)-family symmetry, not U(3) × U(3) ′ .)In this model, when we choose suitable values of the complex parameters a q ( q = u, d )together with additional input values, ( m e , m µ , m τ ), we can successfully obtain [1] predictionsfor quark masses and Cabibbo-Kobayashi-Maskawa (CKM)mixing [5]. However, so far, it is notclear whether the successful parameter fitting is unique or not, and that there are another goodparameter solutions or not.In order to settle these questions, parameter-independent mass relations are usefull, whichcan be obtained in this model ; We have three independent equations for the each mass matrix2 q ( q = u, d ), while we have only one complex parameter a q , therefore we can obtain one massrelation. In this paper, we investigate such the parameter-independent mass relation in theU(3) × U(3) ′ model. At present, the observed quark mass values, especially, for the first genera-tion quarks have considerably large error, i.e. m u = 1 . +0 . − . MeV and m d = 2 . +1 . − . MeV at µ = m Z [6]. By obtaining such a parameter-independent quark mass relation, we check whetherthe U(3) × U(3) ′ model is reasonable or not and what values of m u and m d are acceptable to theU(3) × U(3) ′ model. × U(3) ′ model In our model based on U(3) × U(3) ′ symmetry[1, 2], we consider hypothetical fermions F α ( α = 1 , , , ) of U(3) × U(3) ′ , in addition to quarks and leptons f i ( i =1 , ,
3) which belong to ( , ).We assume that the VEV form (1.1) originates from the following 6 × f iL ¯ F αL ) (0) ji (Φ f ) βi ( ¯Φ f ) jα − ( S f ) βα ! f Rj F Rβ ! . (2 . F L ( R ) are heavy fermions with ( , , ) of SU(2) L × U(3) × U(3) ′ . On the other hand, f R are right-handed quarks and leptons, f R = ( u, d, ν, e − ) R , while f L are not physical fields. Theyare given by the following combinations: f L ≡ ( f u , f d , f ν , f e ) L ≡ (cid:18) H H † u q L , H H † d q L , H H † u ℓ L , H H † d ℓ L (cid:19) , (2 . H is a flavon VEV scale, and q L = u L d L ! , ℓ L = ν L e − L ! , H u = H u H − u ! , H d = H + d H d ! . (2 . ′ have been completely broken, the quarks and leptons are describedby the effective Hamiltonian H Y = (¯ ν L ) i ( M ν ) ji ( ν R ) j + (¯ e L ) i ( M e ) ji ( e R ) j + y R (¯ ν R ) i ( Y R ) ij ( ν cR ) j +(¯ u L ) i ( M u ) ji ( u R ) j + ( ¯ d L ) i ( M d ) ji ( d R ) j . (2 . f i are not U(3) family triplet any more in the exact meaning,but they are mixing states between the fermions f and F . However, for convenience, we willstill use the index of U(3) family for these fermion states.3y performing a seesaw-like approximation with Λ = O ( h Φ f i ) ≪ Λ = O ( h S f i ), the massmatrix (2.1) leads to the following Dirac mass matrix of quarks and leptons:( M f ) ji ≃ h H u/d i Λ H h Φ f i αi h ( S f ) − i βα h ¯Φ f i jβ . (2 . q = u or q = d ), the factor h H u/d i / Λ H takes a common value, so that the factor h H u/d i / Λ H does not play any essential role in our study.As seen in this section, Φ f and S f have a dimension of mass. However, for convenience,hereafter, we use a dimensionless expressions (1.2) and (1.5), and define the parameter m f witha mass dimension by Eq.(1.1). When we define an Hermitian mass matrix H q = M q M † q , (3 . H q is given by H q = M q M † q = m q Φ q S − q Φ † q Φ q ( S † q ) − Φ † q = k q P q D / e ( + a q X ) D e ( + a ∗ q X ) D / e P † q , (3 . k q = m q m e . (3 . H q is diagonalized as U q H q U † q = D q ≡ diag( m q , m q , m q ) . (3 . m , m , m ) = 1 k q ( m q , m q , m q ) , (3 . D q ≡ diag( m , m , m ) = 1 k q D q . (3 . H q : c ≡ m + m + m = Tr[ ˜ D q ] = 1 k q Tr[ H q ] , (3 . ≡ m m + m m + m m = 12 (cid:16) (Tr[ ˜ D q ]) − Tr[ ˜ D q ˜ D q ] (cid:17) = 1 k q (cid:8) (Tr[ H q ]) − Tr[ H q H q ] (cid:9) , (3 . c ≡ m m m = det ˜ D q = 1 k q det H q . (3 . A ] as [ A ] simply. By using the explicit form (3.2), weobtain c , c and c as follows: c = (cid:12)(cid:12)(cid:12)(cid:12) a q (cid:12)(cid:12)(cid:12)(cid:12) [ D e ] + 19 | a q | ([ D e ] − [ D e ]) , (3 . c = 12 (cid:12)(cid:12)(cid:12)(cid:12) a q (cid:12)(cid:12)(cid:12)(cid:12) ([ D e ] − [ D e ]) + 29 | a q | [ D e ]det D e , (3 . c = | a q | (det D e ) = | a q | det D e . (3 . Q ≡ [ D q ][ D e ] , Q ≡ [ D q ] − [ D q ][ D e ] − [ D e ] , Q ≡ det D q det D e , (3 . L ≡ [ D e ] − [ D e ][ D e ] , L ≡ [ D e ] det D e [ D e ] − [ D e ] . (3 . Q = (cid:12)(cid:12)(cid:12)(cid:12) a q (cid:12)(cid:12)(cid:12)(cid:12) + 19 | a q | L , (3 . Q = (cid:12)(cid:12)(cid:12)(cid:12) a q (cid:12)(cid:12)(cid:12)(cid:12) + 49 | a q | L , (3 . Q = | a q | . (3 . L and L are given only by the charged lepton masses, and Q , Q and Q areexpressed by quark masses ( m , m , m ) after ( m e , m µ , m τ ) are substituted.5inally, by eliminating the parameter a q from Eqs.(3.15) -(3.17), we obtain the mass relation − b + b ( m + m + m ) − b ( m m + m m + m m ) + b m m m = 0 . (3 . b , b , b , and b are defined by b = (cid:18) L − L (cid:19) , (3 . b = 3(1 − L ) 1[ D e ] , (3 . b = 6 (cid:18) L (cid:19) D e ] − [ D e ] , (3 . b = (1 − L + 2 L ) 1det D e , (3 . m /m versus m /m In order to investigate the behavior of m /m versus m /m , we define parameters x ≡ m m = m q m q , y ≡ m m = m q m q . (4 . x and y are independent of the value of k q defined in (3.3). Then,since ( m , m , m ) are expressed as( m , m , m ) = ( xy, x, m , (4 . − b + b m (1 + x + x y ) − b m ( x + x y + x y ) + b m x y = 0 . (4 . y = f ( x ): y = 1 x s ( b m − b m ) x − ( b − b m )( b m − b m ) x − ( b m − b m ) . (4 . y = f ( x ) has poles at x = 0 , and x = ± s b m − b m b m − b m , (4 . x = ± s b − b m b m − b m . (4 . b , b , b and b are given by b = 1 . ,b = 0 . − ,b = 87 . − ,b = 1 . × (GeV) − . (4 . m e ( µ ) = 0.000486847 GeV, m µ ( µ ) = 0.102751 GeV and m τ ( µ ) = 1.7467 GeV as the charged lepton mass values at µ = m Z [6] .The behavior of y = f ( x ) is illustrated in Fig.1. The behavior depends on the input valueof m . Note that since b b = 1 . , (4 . b − b m ) in (4.4) changes the sign according as m > m or m < m , where m ≡ r b b = 1 . . (4 . m > m as normal type, and the behavior in thecase m < m as non-normal type.Now, let us compare our parameter-independent results with the observed quark massvalues in detail. The observed quark mass values at at µ = m Z [6] are as follows: m u = 0 . +0 . − . GeV , m c = 0 . ± .
084 GeV , m t = 171 . ± . ,m d = 0 . +0 . − . GeV , m s = 0 . +0 . − . GeV , m b = 2 . ± .
09 GeV . (4 . m /m and m /m : m u m c = 0 . +0 . − . , m c m t = 0 . +0 . − . , (4 . m d m s = 0 . +0 . − . , m s m b = 0 . +0 . − . . (4 . m m m ( a ) m = 1( b ) m = 30( c ) m = 100( a )( b )( c )( b )( c ) down quarks region (Xing et. al)up quarks region (Xing et. al) Figure 1: m dependence of the behavior of m /m versus m /m . The curvesof the mass relation y = f ( x ) given in Eq.(4.4) are drown in the ( m /m , m /m )plane for the cases (a) m = 1 GeV, (b) m = 30 GeV, and (c) m = 100 GeV.The shaded square regions are correspond to the observed mass ratios in (4.11) and(4.12) for up-quark sector and down-quark sector respectively obtained by Xing et.al.As seen in Fig.1, the behavior of m /m in the normal type has a maximum whose valueis smaller than ∼ − . On the other hand, as seen in Eq.(4.12), the observe value of m d /m s is m d /m s ≃ .
05. Therefore, the mass ratios for down-quark sector cannot be described bythe behavior of the normal type. Thus we have the solution for Eqs.(4.11) and (4.12) by thebehavior of the normal type for the up-quark sector, and by the behavior of the non-normaltype for the down-quark sector.
Down-quark sector
First, let us see behaviors of the mass ratios ( m /m , m /m ) in the down-quark sector.As seen in Fig.2, we can determine a value m from the observed center values in (4.12) as m = 1 .
03 GeV. If we take m > .
04 GeV, the behavior of the mass ratios becomes the normaltype from non-normal type as seen in the curve (c) of Fig.2. Furthermore, if we take a largervalue m > .
05 GeV, then the curve is out of the error region as seen in the curve (d) of Fig.2.Similarly, there is no solution of m for m < .
91 GeV as seen in the curve (a) of Fig.2. Thus8 m ( a )( b )( c ) ( d ) ( c ) m = 1 (cid:17) a ) m = 0 (cid:17) b ) m = 1 (cid:17) d ) m = 1 (cid:17) m m Figure 2: The m value in the mass relation consistent with the observed values m d /m s and m s /m b in the down-quark sector. The curves of the mass relation y = f ( x ) in Eq.(4.4) are drown in the ( m /m , m /m ) plane for the cases with (a) m = 0 .
91 GeV, (b) m = 1 .
03 GeV, (c) m = 1 .
04 GeV, and (d) m = 1 .
05 GeV.The each curve in the case of (c) and (d) has a left end point, in the left region fromwhich there is no real solutions for m /m . The shaded square region is correspondto the observed mass ratios in (4.12) for down-quark sector obtained by Xing et. al.we obtain m = 1 . +0 . − . GeV , (4 . m = 1 .
03 GeV, we obtain k d = m b m = 2 . , (4 . k d , we have m b = k d m = 2 . +0 . − . GeV , (4 . m b in (4.10). Up-quark sector m m m ( a )( b )( c ) ( b ) m = 114( c ) m = 228( a ) m = 56 (cid:17) Figure 3: The m value in the mass relation consistent with the observed values m u /m c and m c /m t in the up-quark sector. The curves of the mass relation y = f ( x )in Eq.(4.4) are drown in the ( m /m , m /m ) plane for the cases with (a) m = 56 . m = 114 GeV, and (c) m = 228 GeV. The each curve has a left endpoint, in the left region from which there is no real solution for m /m . The shadedsquare region is correspond to the observed mass ratios in (4.11) for up-quark sectorobtained by Xing et. al.In order to get a reasonable value of m in the mass relation, we illustrate curves of themass relation for several values of m in Fig. 3 and Fig 4. We find that there are two solutionsof the m which are consistent with the observed up-quark mass ratios (4.11) as seen in Figs.3and 4: m = 30 . +8 . − . GeV , m = 114 +114 − GeV . (4 . m u /m c , m c /m t ) ≃ (2 . , . × − .However, those two center values m = 30 . m = 114 GeV give k u = 5 .
63 and k u = 1 . , (4 . k d = 2 . k d = 3. It is natural to consider that the relations of quark mass matrices M u and M d to thecharged lepton mass matrix M e take the same weight between the up- and down-quark sectors,10 m m m ( a )( b )( c ) ( c ) m = 39( a ) m = 25 (cid:17) b ) m = 30 (cid:17) Figure 4: Another m value in the mass relation consistent with the observedvalues m u /m c and m c /m t in the up-quark sector. The curves of the mass relation y = f ( x ) in Eq.(4.4) are drown in the ( m /m , m /m ) plane for the cases with (a) m = 25 . m = 30 . m = 39 GeV. The each curve has aleft end point, in the left region from which there is no real solution for m /m . Theshaded square region is correspond to the observed mass ratios in (4.11) for up-quarksector obtained by Xing et. al.i.e. k u = k d , except for the parameters a u and a d . If we want to consider k u = k d = 3, we haveto choose m = 57 GeV from m obst = 172 GeV. Only when we chose the lowest value m = 57GeV in the solution m = 114 +114 − GeV, the value can give k u = 3, so that we can realize therelation k u = k d = 3. (If we require k u = k d = 2 .
98, the case leads to m = 59 . m = 57 GeV.) First generation quark masses
Next, we see the constraints on the first generation quark masses m u and m d . From thecurves (a) m = 0 .
91 GeV, (b) m = 1 .
03 GeV and (c) m = 1 .
04 GeV in Fig.2, we obtain m d = 2 . +3 . − . MeV , (4 . m s ) obs = 55 MeV. Our result (4.18) has wide errorcompared with the observed value [6] ( m d ) obs = 2 . +1 . − . MeV.11imilarly, from Figs.3 and 4, we obtain the following two solutions of m u , m u = 1 . +0 . − . MeV , and m u = 1 . +0 . − . MeV , (4 . m c ) obs = 0 .
619 GeV. We also see that our results (4.19)cannot put any severe constraint on the observed value ( m u ) obs = 1 . +0 . − . MeV.
In conclusion, we have investigate a pameter-independent quark mass relation in the U(3) × U(3) ′ model. Considering our results with the observed quark mass values [6], we conclude that thechoice k u = k d = 3 in the previous work [1] of the explicit parameter fitting of a u and a d wasreasonable. However, we have found that there are two solutions in the up-quark sector as wehave shown in Figs.3 and 4. This is not so serious problem when we take the error range of theobserved quark mass values into consideration.We are convinced that our parameter-independent analysis of the mass relation is usefulfor model checking in future study of the U(3) × U(3) ′ model.We did not investigate a similar parameter-independent study for the CKM mixing. Thesimilar study of the CKM matrix elements V ij cannot been obtained unless the results includequark masses. We are interested in the values V us , V cb , V td and so on, while those values will bedisturbed by existence of large elements V ud , V cs , V tb and so on. It is our future task to obtaina relation without such large contribution terms.One (Y.K.) of the authors was supported by JSPS KAKENHI Grant number JP16K05325. References [1] Y. Koide and H. Nishiura, Phys.Rev. D , 111301(R) (2015).[2] Y. Koide and H. Nishiura, IJMPA , 1750085 (2017).[3] P. Minkowski, Phys. Lett. B , 421 (1977); M. Gell-Mann, P. Ramond, and R. Slan-sky, Proceedings of the Supergravity Stony Brook Workshop, New York, 1979, edited byP. Van Nieuwenhuizen and D. Freedman (North-Holland, Amsterdam, 1979); T. Yanagida,Proceedings of the Workshop on Unified Theories and Baryon Number in the Universe,Tsukuba, Japan 1979, edited by A. Sawada and A. Sugamoto [KEK Report No. 79-18,Tsukuba]; R. Mohapatra and G. Senjanovic, Phys. Rev. Lett. , 912 (1980); J. Schechterand J. W. F. Valle, Phys. Rev. D , 2227 (1980).[4] Y. Koide and H. Fusaoka, Z. Phys. C , 459 (1996).125] N. Cabibbo, Phys. Rev. Lett. , 531 (1963); M. Kobayashi and T. Maskawa,Prog. Theor. Phys. , 652 (1973).[6] Z.-z. Xing, H. Zhang, and S. Zhou, Phys. Rev. D , 113016 (2008). And also see H. Fusaokaand Y. Koide, Phys. Rev. D57