The extra vector abeyance (EVA) mechanism as the origin of the KM matrix

Volume 261, number 4

PHYSICS LETTERS B

6 June 1991

The extra vector abeyance (EVA) mechanism as the origin of the KM matrix S.

Kelley, Jorge L. Lopez

i and D.V. N a n o p o u l o s 2

Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77381, USA

Received l 0 December 1990

We examine a general mechanism whereby an extra vector-like quark of mass Mevagenerates realistic quark mixing angles from otherwise flavor diagonal quark Yukawa matrices. For scales above Mevathe vector-like quark is manifest, while the Yukawa matrices are diagonal, i.e., the KM matrix is held in abeyance. Conversely, for scales below Mo,a the vector-like quark is in an abeyant state, while the KM matrix is manifest. This mechanism is naturally realized in many superstring models. In fact, knowledge of the Yukawa matrices at the string scale, together with suitable renormalization group scaling, unambiguously determines the quark mass ratios and the KM angles at low energies. Conversely, these low-energy observables may be used to determine asyet-unknown parameters in the string model. We explicitly apply the EVA mechanism to a toy model inspired by the structure of the flipped SU (5) string model, to illustrate how this mechanism may be used to predict some of the KM angles. We also outline the procedure to be followed in a realistic calculation in the flipped SU ( 5 ) string model.

1. Introduction The fermion masses and the K o b a y a s h i - M a s k a w a ( K M ) matrix are fundamental p a r a m e t e r s o f the s t a n d a r d model. Most o f these parameters are tightly constrained by experiment, and the ones which are not, e.g., the topquark mass a n d the CP-violating phase, are the subject o f intense experimental and theoretical research. One o f the p r i m e m o t i v a t i o n s to study physics b e y o n d the standard model is to try to explain the origin and magnitude of these p a r a m e t e r s which the s t a n d a r d m o d e l neither explains nor predicts. A model which could successfully predict the nine fermion masses and the four angles determining the K M matrix from a few f u n d a m e n t a l p a r a m eters would be an amazing success. The first attempts at understanding the K M regularities in terms o f unified models date back to the E 6 G U T models [ 1,2]. It was realized that small perturbations away from p r o p o r t i o n a l up- and down-quark mass matrices could provide the observed small K M angles. These perturbations were assumed to be due to radiative corrections to the quark mass matrices i n d u c e d by couplings to superheavy vector-like fermions. Later on the possible existence o f vector-like fermions was m a d e m o r e " a p p e a l i n g " by taking them to be light [ 3 - 6 ] and hence have significant mixing with the quarks and leptons o f the standard model, leading to non-unitary K M matrices, tree-level flavor changing neutral currents in the quark [ 3,4,5,7 ] a n d lepton [ 3,4 ] sectors, new models o f C P violation [ 8,4, 5,9 ], a n d constrained K M matrices [ 10 ]. In the context o f unified theories o f all particle interactions, some superstring models [ 1 l - 17 ] have recently progressed significantly in their ability to connect physics at the string scale with low-energy physics. N o t a b l y among these is the superstring derived flipped SU ( 5 ) model [ 1 l - 14 ]. We would like to focus on the fermion mass spectrum aspect o f its newly acquired predictive power. In the flipped SU (5) model, third generation Supported by an ICSC-World Laboratory Scholarship. 2 Supported in part by DOE grant DE-ASOS-81ER40039. 424

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fermion masses have their origin in cubic superpotential couplings [ 11,12 ], while second and first generation fermion masses arise from string-induced nonrenormalizable superpotential couplings [12 ], typically suppressed by powers of ( ~ ) / M ~ 1/ 10, where the ~'s are singlet fields and M E 1018 GeV is the scale of nonrenormalizable interactions [ 18,12 ]. In principle, nonrenormalizable terms can give rise to flavor diagonal as well as flavor off-diagonal entries in the Yukawa matrices. However, in the particular case of the flipped SU (5) model off-diagonal terms are highly suppressed [ 13 ], yielding much too small mixing angles. Fortunately this is not the end of the story. In addition to the three generations of the standard model, many string models have extra vector-like representations in their spectrum [ 6,15,19 ] with nonvanishing standard model quantum numbers. In E 6 superstring models [ 15 ], these fields are usually assumed to receive large enough masses so that they, among other things, do not induce too rapid proton decay. When the extra vector-like representations decouple at a mass scale Meva, the remaining light fields with the same standard model quantum numbers become in general linear combinations of the original, pre-decoupling fields, and are determined by the SU (2) L invariant ( i.e. M = 0 ) mass terms in the superpotential. This amounts to both a rescaling and a rotation of the 3 × 3 Yukawa matrices at the scale Meva where the heavy fields decouple. For scales above Meva, the Yukawa matrices are very simple and hence the KM matrix is held in abeyance, while the vector-like quark is manifest. For scales below Meva, the vectorlike quark decouples, i.e., becomes abeyant, while the KM matrix is now manifest. Hence, we refer to this origin of the KM matrix as the extra vector abeyance (EVA) mechanism. Ignoring phases for the moment, the three independent A / = 0 scaled masses ~ ~ defining the heavy vector-like quark pair ~ / ] 2 3 4 m , U4 (taken here to be an up-type quark), 4

4 u c -u OtiUiU4--[Ot I ~ Oli~UuiCU4 =-- iotUlu~234,nu4 ,

i=1

i=1

4

~. ( a ~ ) 2 = 1 ,

laUl =M~va ,

(1)

i=1

give the same number of degrees of freedom as the three real KM angles. For mass scales M ~ >> Mz, the exotic effects of the heavy vector-like quark are completely negligible at low energies. Below we show that this mechanism is realized naturally in the string flipped SU (5) model. In a unified model, Yukawa couplings and masses at the unification scale are related to those at all energies by renormalization group scaling. If all these parameters are actually known at the unification scale, this yields specific predictions for the A / = 0 masses o~ at the scale M~v~, and for the quark masses and the KM angles at low energies. Conversely, experimental knowledge of these low-energy observables may be used to constrain the Yukawa couplings and masses in the high-energy theory. This is particularly useful in the string example we will consider, since some effective Yukawa couplings originate from string-induced nonrenormalizable terms which depend on some quantities which are in principle calculable, but in practice only approximately known. In the second section we present the simplest realization of the EVA mechanism and discuss some of its generalizations. In the third section we study a toy model inspired by the structure of the flipped SU (5) string model and calculate some of the KM angles as a function of the top-quark mass. We also outline the procedure to be followed in a realistic calculation of the KM angles in the actual string model. Finally, we summarize our results and present our conclusions.

2. The extra vector abeyance (EVA) mechanism

We begin with one of the simplest realizations of the EVA mechanism. Consider a model with the field content of the minimal supersymmetric extension, the standard model plus extra u4 and u] isosinglet quarks. Unitary rotations on the Qi, d c, Li, and l~ ( i = 1, 2, 3 ) superfields can be made to diagonalize the down-quark and lepton mass matrices. We write the 4 × 4 up-quark mass matrix as UT~uUc, where these mass terms arise from two different sources: (a) Yukawa couplings after electroweak symmetry breaking (i.e., A / = 1 masses) 2~Qiu~H--, 2~(H°)uiu~,i=l, 2, 3 , j = 1, 2, 3, 4; and (b) A / = 0 mass terms otju u 4u j,c j = 1, 2, 3, 4. Schematically we have 425

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u~

u~

u~

../l. =u2 U3 U4 a?

;t~



a~

a~

6 June 1991

u~

(2) ~

We now perform a unitary rotation on the u~ fields on the EVA basis of ( 1 ), that is

/U~234m~

/0l? ~1~ ~1~ ~Ul~/U,~

Ueva-~uC1234°'l If'? f'~ f'~ ~'411u31 \u~2~4o,/ \8? ,~ ,~ ,~/\u~/ where t ~ = / ~ / l a " l . The fl~, ~ , ~ are constrained by the unitarity of the rotation matrix but are otherwise arbitrary. This arbitrariness corresponds to rotations among the three light states and hence we can choose the above parameters at our convenience, such that they satisfy unitarity. Once the light-state mass matrix is known, a particularly convenient choice results from requiring that the light states be the mass eigenstates. In the EVA basis the up-quark mass matrix becomes _,,x~e~.~.~va,': with

j[eva=jtu(OUeva)T=(J3ol×ul G3×3) \1

I

0,×3,,'

(4)

where G and J are matrices with O ( 2 ~ ( H °) ) entries. Since the ot~ masses are not related to the electroweak scale, it is usually assumed that c~" >> 2 U ( H ° ) . In this limit one can diagonalize the up-quark mass matrix jl~va to second order in (2U(H ° ) / I o : 1 ) by solving ,~rrrL~,~u~¢va/v, oz~'~¢T ~Jr r '~L . = diag(m 2, m 2, m~, m 2 ). The result is [5] _fA3X4~_( K UL--\B~.×4]--k,--JTK/Iot"[

J/lo:l)+o(,~,/la,i)2 ' 1

(5)

where K is obtained from KTGGTK= diag(m~, m¢2, mt2 ). As is well known, the physical consequences of the charged current interaction are modified in the presence of isosinglet left-handed quarks (u4 here) which do not feel the SU (2)L interactions but still participate in the mass matrix along with the usual left-handed doublet fields. We write

~w=~:v

W.+h.c.,

where ~ ( u , c, t, T) r. Here Vis the non-unitary 4X 3 KM matrix which follows from the substitution u3) =An ~ , that is

V~.AT=(jT/K~ul) ,

(6) (u~, u2, (7'

where K x contains the usual 3 × 3 KM matrix elements. Since UL is unitary one obtains the following constraints on V: VV T ='g-B.B. T and vTv='~, that is Vis unitary in its columns. The physical consequences of the neutral current interaction are also affected by EVA, leading to naturally suppressed tree-level FCNC in the up-quark sector as follows:

g

LP~ - 2 cos 0w 426

[tP~y~'(gLPL+gRPR) ~Jm--~/mT~'PL(BXuB,)~"]Zu,

(8)

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with gL = 1 _4 sinEOw, gR= _4 sin2Ow, and (see ( 5 ) )

B "r~

{(KrJ)(KTj)T/I°tuI2 --(KTj)T/IotU I

u/~u =~,

--(KTJ)/lotUl) 1 "

(9)

Hence, the tree-level Z FCNC among the u, c, t quarks are suppressed by factors of (2 u < H o > / Ia u I ) 2, in accordance to a general result [ 3 ]. The EVA mechanism, even in its simplest form, has many enticing aspects such as: tree-level flavor changing Z currents, supersymmetric flavor changing mass insertions, several possible sources for CP violation, and the extra vectorlike representations themselves. The crucial point is that even if the Yukawa matrix 2~ is very simple, the rotation to the EVA basis will induce a nontrivial structure in the 3 × 3 quark mass matrix G and hence K # ~ in general. We will restrict our present discussion to the generation of a realistic K_M matrix from initially diagonal Yukawa matrices, ignoring any complex phases. It should be clear that the three independent ct ? defining the EVA basis, plus the given Yukawa matrix 2g, determine the quark masses and the three KM angles. Also, we would like to emphasize the most basic property of the EVA mechanism: the low-energy effects of the exotic representations become completely negligible as the EVA scale is raised, while the EVA imprint on the KM matrix remains indelible. There are many generalizations of the EVA mechanism beyond the simple addition of a vector-like Q = isosinglet quark. Extra chiral representations are highly disfavored [20]. More than one extra vector-like u, u c pair could be added. Extra vector-like representations other than u, u c could also be added. However, gauge coupling unification at high energies places severe constraints on the exotic spectrum of a specific unified model [ 21 ], so these extra representations cannot be added randomly. The phenomenological consequences of these various possibilities have been considered to the same extent in generic models before [ 3-10], and more recently also in the context of realistic unified models [ 16,22 ]. For our present purposes, the EVA generalization we consider is to add an extra vector-like lepton doublet pair, L4 and L~, and an extra vector-like lepton singlet pair, l~ and 15, to the three chiral lepton doublets L~, and singlets l? of the standard model ~t. We now have a general 5 × 5 lepton mass matrix lTjg~l c, where the various e ~ ' i,j= 1, 2, 3, 4; (b) aLLeLe, i= 1, 2, 3, 4; (c) ailils, entries in the mass matrix arise from: (a) 2ijLil~H, ~ ~ i=l, 2, 3, 4; (d) 255LCslsH '. Schematically we have

l,(

•ff¢~=/3

aL

/4

ls ot~

o/L

a~

a~

\ 1.

(10)

/

a~4 2 5 5 < H ' ° > ]

We now rotate the L~, l? ( i = 1, 2, 3, 4) fields to the EVA basis defined in analogy with (3), but with OLv~, O~v~ instead. The resulting mass matrix jg~w is in block diagonal form and can be handled in analogy with the previous case. The neutral current interactions ~ generate small tree-level FCNC suppressed by O (2e< H ' ° > / aL,~) 2. These lepton number violating processes have been explored before in the context of general models [ 3,4 ], and recently in the string flipped SU (5) model [ 22 ].

3. A toy model To illustrate the predictive power of the EVA mechanism, in this section we present a toy model inspired by ~l This type of leptonic EVA scenario also arises in the context of E6superstring models [ 16 ]. 427

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the structure of the flipped SU (5) string model [ 12,14]. Our toy model has the field content of the models discussed in section 2, and interacts through the superpotential Wtoy = 2 t Q 3 u ~ I ~ + ~ . b Q 3 d ~ H + ) t x L 3 1 ~ H + 2 c Q 2 u ~ I t + 2 s Q 2 d ~ H + u 4 ( o ~ u +L~(otLLI

+

~ +o~u~ +ot~u~)

a 2LL 2 + 0~3L L3 ) +15 (~x~l l~~ + cz~l$ + o~l~) .

( 11 )

Two important aspect of this toy model are shared by the flipped SU (5) string model: (i) the first generation of quarks and leptons does not appear in W~oy,and (ii) we have taken 6d - 0 , z = u, L, £. Because of the first fact, in the fermionic sector the first generation remains massless and uncoupled to the other two, irrespectively of the value of 6d. On the other hand, in the scalar sector, the supersymmetric mass insertions Arn~e and Am 2 are proportional to 8 ] m 2, where m is the supersymmetry breaking mass, irrespectively of the vanishing of the first generation Yukawa couplings [ 22 ]. Unless c~~ were very small, these mass insertions would generate much too large contributions to K - K and D - D mixing, and the lepton flavor violating decay ~t~ e'/[ 22 ], among other things. The couplings in Wtoy are assumed to evolve down to low energies from the following high-energy unified initial conditions

,~t =,~b =)],x = X/~ g , oLu=o/L =O/~ = a i , i = 1 , 2 , 3.

(12)

When the scale drops below that of the aZ's, the EVA rotation takes place, the last three terms in Wtoydrop out, and the remaining light fields are reexpressed using the EVA transformations O~.,L,~. In the notation of section 2, 2 ~ has only two nonzero entries 222 =2c and ,~-~3=At. Using eqs. (2) and (4), J and G can then be easily calculated, and from G G T also the/£3 x 3 portion of the KM matrix V (7). A nice calculational check comes from verifying that G G "r depends only on the &~, indeed

/000

G = 2¢/~2 2¢#~ 2c/~ (/t°)=~GG T=

(i

0

)2[l_(t~)x ]

0 )

_,~,c,~tdt~ot~ (/~0}2.

(13)

The top- and charm-quark masses are the square roots of the eigenvalues of GG T, and the KM matrix elements I Vcbl = I Vts I = sin 0, I VcsI = I V,b l = cos 0 are the entries of the matrix that diagonalizes G G T (see eqs. (5) and (7)). A similar calculation can be performed to determine the tau lepton mass, starting from the lepton mass matrix J/~ in ( 10 ) with 2e= diag (0, 0, 2,, 0), 255 = 0, and a ~ = a L = 0; see ref. [22] for details. We then obtain m, = x / l _ (~L)2 ~

(a~)2 2, ( H O ) , 2 ~u 2

mb=2b(H0),

m2=22[1--(d~)2](H°}2tan2fl,

sin0~ 2cdt]x/1-(~Y)2-(dt~)2 2t

1 -- ( ~ ) 2

'

c o s f l = g2 ( n 0 ) X/2 Mw

mE-- l2c(o~i) _ ( d t ~ ) 2 (HO)EtanEfl (14)

In evaluating sin 0 it is reasonable to define the KM matrix at the weak scale where the off-diagonal charged current couplings due to the rotation to the quark mass eigenstate basis are introduced. Since the heavy weak bosons giving off-diagonal couplings decouple from the renormalization group equations at this scale, the quark mass matrices will remain diagonal at lower energies, and hence the KM matrix does not renormalize below the weak scale. We will consider renormalization effects using one-loop renormalization group equations for the gauge and Yukawa couplings, and the vector-like masses ot~. In our toy model, the unification scale Mx and EVA scale Meva are found by requiring o~3(Mx)= a z (Mx) = c~r(Mx), starting from the observed low-energy values of sin20 and a3. Taking central values of 1/ C~¢m= 127.9, o/3 (Mz)=0.107, and sinZ0w= 0.233 [21,14] gives M x = 8 . 2 X 1015 GeV and M~va=2.7 × 1015 GeV. 428

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Taking ot3(Mz)=0.102, and sin20w=0.232 gives the biggest gap between the two scales with M x = 3 . 0 X 10 is GeV and Mcva = 4.6 X 10 ~3 GeV. The & ~'s and Yukawa couplings renormalize from Mx to Mcva, where the heavy vector-like representations are isolated by rotating to the EVA basis and are subsequently dropped from the renormalization group evolution. This also entails a rescaling of the running Yukawa couplings to account for the decoupled states. The Yukawa couplings are further renormalized down to the weak scale. One-loop QCD corrections are then used to obtain the quark masses at the weak scale from their physical values. With the renormalization-group scaled low-energy Yukawa couplings 2t, 2b, ;t~, and the evolved dt z values (which depend on the two independent high-energy parameters, e.g., d~1.3) we can use (see (14)) the known mass of the bottom quark to solve for < H ° > and tan r, and the known mass of the tau lepton to fix &l. The topquark mass then fixes &3, and since 0 ~/~3 ~ X/1 -- (til)2, we obtain both an upper and a lower bound on mr. This then gives a prediction for the KM angles in terms of mr. We obtain a function sin 0(mr) in the allowed range 15 5 < mt< 170 GeV. This function vanishes at both ends of the mt range and peaks at the unrealistically small value of sin 0~ 8 X 10 -4. However, there are many important features of the string model which are not reflected in this toy model and which may greatly modify this result. The unification relations between the ot Z,s (12) renormalize so that a u ~ ot L ~ a~ at the EVA scale. In the toy model, the small splitting between the EVA scale and the unification scale results in this effect being negligible. However, in the string model, the EVA scale is expected to be of the order of 10 TeV [ 14 ], and we then expect the ensuing large splittings between the c~z's to produce substantial differences between the toy model and the string model. In addition, the renormalization group evolution in the string flipped SU (5) model involves a rich structure of Yukawa couplings and intermediate scales [ 14 ] which have not been included in the toy model. We now outline how one would actually calculate the KM angles including radiative effects in the flipped SU(5) string model. In this model, the dtf originate from nonrenormalizable terms at the string scale, e.g., cif4f~HH/M, where HH is a specific pair of hidden sector matter fields [ 12 ]. The coupling ci then evolves down to the scale A where the hidden sector gauge group becomes strong and the condensate forms. At this point the vector-like masses, e.g., ot L - c i ( A ) < H H > / M ~ A , are formed. These masses then renormalize down to the scale of the EVA masses O (c~L ), where the vector-like representations are dropped from the renormalization group equations and the rotation to the EVA basis takes place. We should point out that this program is hindered by technical difficulties associated with the computation of hidden sector matter field condensates

[23]. Although this procedure in principle gives a prediction for the KM angles as illustrated by the toy model calculation, and some of the elements of the calculation have been carried out in detail [ 14 ], we still lack the neccessary understanding of the pattern of singlet vacuum expectation values [ 12 ] and hidden sector condensates to do a realistic calculation in the string model. We are confident that these technical difficulties will soon be overcome and that the KM angles will provide yet another test of the flipped string.

4. Conclusion We have presented a new mechanism that explains the origin of the KM angles from otherwise flavor diagonal quark Yukawa matrices. Extra vector-like representations are introduced that couple through A / = 0 mass terms to the chiral representations of the supersymmetric standard model. When these heavy states decouple at a scale Meva, the surviving light states are specific linear combinations of the original states and in general originate a non-trivial KM matrix. The new vector-like states could have observable phenomenological consequences, such as tree-level FCNC. However, these exotic effects are naturally suppressed by powers of (mq,~t/Meva) 2. When the EVA scale is increased, the supersymmetric standard model is recovered at low energies. However, EVA still leaves its imprint behind, as the KM matrix is fully determined in terms of the relative ratios of the heavy vectorlike masses. The extra vector abeyance (EVA) mechanism presented here is naturally realized in many string models. In 429

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p a r t i c u l a r , we h a v e p r e s e n t e d a t o y m o d e l i n s p i r e d b y t h e s t r u c t u r e o f t h e f l i p p e d S U ( 5 ) s t r i n g m o d e l , a n d s h o w n h o w it c a n b e u s e d to p r e d i c t s o m e o f t h e K M a n g l e s as a f u n c t i o n o f t h e t o p - q u a r k m a s s . W e h a v e also o u t l i n e d t h e m e t h o d t o b e f o l l o w e d i n a r e a l i s t i c c a l c u l a t i o n i n t h e s t r i n g m o d e l , as well as t h e o u t s t a n d i n g technical obstacles that need to be overcome before an explicit and decisive calculation can be undertaken. Our work demonstrates the amazing possibility of using the experimental knowledge of fermion masses and KM a n g l e s t o d i r e c t l y t e s t s t r i n g m o d e l s a n d e n c o u r a g e s t h e d e v e l o p m e n t o f t h e n e c e s s a r y c a l c u l a t i o n a l tools.

Acknowledgement W e w o u l d like t o t h a n k H. P o i s f o r u s e f u l d i s c u s s i o n s .

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