A mechanism for fermion mass generation

Volume 142B, number 4

PHYSICS LETTERS

26 July 1984

A MECHANISM FOR FERMION MASS GENERATION ~r

James G. McCARTHY and Burt A. OVRUT Department of Physics, The Rockefeller University, New York, N Y 10021, USA

Received 23 January 1983 Revised manuscript received 17 April 1984

We present a mechanism for fermion mass generation, and apply it to the lepton sector of the Weinberg-Salam model. Extra fields, necessary to produce the large mass splittings of the observed fermions, form bound states more massive than the electroweak scale.

Despite the success o f SU(3)C X SU(2)L X U ( l ) y gauge theories in describing strong and electroweak interactions, theoretical understanding of fermion masses and mixing angles remains incomplete. There have been many different approaches to this question [ 1]. One is to generate these parameters at tree level, using group theoretical constraints [2]. Another is based upon the proposition that "all fermions are created equal" at tree level. Symmetry breaking and radiative corrections then generate the mass spectrum and mixing angles through the renormalization group. This approach is due to Georgi et al. [3] who use it to derive quark masses and mixing angles. In this paper we consider the lepton mass spectrum only. We follow the spirit of ref. [3], but make some significant modifications. Specifically, we do the following. (a) Lepton families are constrained to have identical Yukawa couplings by a horizontal SU(M) symmetry. (b) SU(M) is spontaneously broken to SU(M - 1) at scale M G. The leptons naturally divide into three families which renormalize differently under SU(M 1). (c) (Almost) exponential, large splitting o f the running Yukawa couplings is achieved by constraining the theory to have (approximately) vanishing S U ( M - 1) Work supported in part by the Department of Energy under Contract Grant Number DE-AC02-81ER40033B.000. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

beta-function, and by taking M sufficiently big. (d) S U ( M - 1) is spontaneously broken to S U ( M - 2) at scale M I. The three observed lepton families are singlets under S U ( M - 2), and their mass ratios are determined. (e) For M ~> 5, the gauge parameter of S U ( M - 2 ) is asymptotically free, and becomes strong coupling at scale AM_ 2. Therefore, since all the unobserved leptonic families transform non-trivially under S U ( M - 2), they form into bound states at low energy. For Meven, these bound states have mass o f O(AM_2). (f) SU(2)L X U(1)y is spontaneously broken to U(1)E M by a combination o f the usual Higgs field and bound state scalars at scale 250/x/~GeV. (g) Inputting the known values of aEM and sin20 w at 102 GeV, as well asM I and M, we can predict the values OfMG, AM_ 2 , mu/m e , and m r / m e. We want to emphasize step (e), which is new to our approach. This step greatly simplifies the theory, since it obviates the introduction o f new Higgs fields to give the unobserved fermions large mass. Also, we stress that our approach has very little arbitrariness from the group theoretical point o f view. The constraints o f anomaly cancellations, hypercharge assignments, and so on lead to an almost unique theory. The values of mu/m e and m r / m e depend on a detailed analysis of the model. We present typical results at the end o f this paper. The gauge group is G = SU(3)C X SU(2)L X U(1)X1 X SU(M)H 1 . U(1)X~ combines with U(1) subgroups of 281

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Table 1 Fields under SU(M)H 1. LL-BiR are fermions, and H - S are scalars. Indices a = 1, 3 and i = 1, N B.

SU(3) C

SU(2)L

U(1)X1

SU(M)H1

1 1 1

XL XE XN 1/6 -1/3 XB XB

M(M+I)/2 M(M+I)/2 M(M+I)/2 1 1 M ( M - 1)/2 M ( M - 1)/2

XH XK

M M

LL ER NR 4 d~, BiL BlR

3

3 1 1

2 1 1 2 1 1 1

H K

1 1

1 1

0

1

2

1/2

1

S

1

1

XS

M

SU(M)H 1 to form the hypercharge group, U ( 1 ) y . SU(M)H 1 is a horizontal gauge group that acts nontrivially on leptonic fields, and some Higgs fields, only. It constrains all leptonic families to have the same Yukawa coupling. The field content is given in table 1. L L, and ER, contain M ( M + 1)/2 families o f left chiral, and electron4ike right chiral, leptons respectively. N R contains M(M+ 1)/2 families o f right chiral neutrinos. N R is added to allow the theory to be anomaly free. The SU(M)H1 representation, M(M+ 1)/2, is chosen because it is the one o f lowest dimension that breaks into three families under subgroup S U ( M - 1)H2 o f SU(M)H 1 . Fields qaL, uaR , and daR, where a --- 1 , 3 , are the usual quark families. BiL, BiR, and S, where i = 1 ,NB, are added to allow the 3-function o f gauge group S U ( M - 1)H2 to vanish. Scalar H develops a vacuum expectation value (VEV) that breaks SU(M)H, to S U ( M - 1 ) H 2 at scale M G. Scalar K develops a ~ E V that breaks SU(M - 1)t-12 to S U ( M - 2)H; at scale M I. Finally, ¢ is the usual Higgs field. The theory is anomaly free if the following constraints are satisfied.

X L : - 3 / M ( M + 1),

xE=xL+_,2,

xN=xL

+~-1 .

(1) 1. SU(M)H1 breaking to S U ( M - 1)Hz . The potential energy can be arranged so that H develops a VEV, (H), that spontaneously breaks SU(M)H i to 282

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Table 2 Fields under SU(M - 1)H2, ~eL-6i R are fermions, and I~, are scalars. Coefficients A = a (M - 2)[ 2M(M - 1) ] -1/'2, and B =A/(M - 2).

SU(3)C

SU(2) L

U(1)X2

SU(M- I)H2

~.

1

2

-1/2

1

L[ eR E~, E~ veR N~ N~ .

1 1 1 1 1 1 1

2 1 1 1 1 1 l

X L + 2B

( M - 1)M/2

-1

1

X1E - A X E + 2B 0 xN-A x N + 2B

M- l (M - 1)M/2 1 M- I (m-1)m/2

1

x

b~,bZR

1

1

X~ + 2B

( M - 1) (M-2)/2

1 1

1 1

xKI+B X~+B

M-1 M- 1

-A

i

S U ( M - 1 ) H 2 at scale M G. Note that (H)also breaks U(1)X 1 . There is only one broken generator in SU(M)til that commutes with all generators o f S U ( M - 1)1t2. Denote it by T 1 . If we define

X 2 = X 1 + aT 1

(2)

then X 2 is unbroken by (H) if

X~ = a[(M- 1)/2MI 1/2.

(3)

It follows that the unbroken gauge group at energies below M G is G 2 = SU(3)C X SU(2)L X U(1)X2 X S U ( M - 1)H2- The fields with mass smaffer than M G are given in table 2. Note that LL, ER, and N R each decompose as [M(M + 1)/21 = [ l l + [ M - II + [ ( M - 1)M/2],

(4)

under S U ( M - 1)I-/2. Fields~eL, eR, and ve R represent the left chiral electron doublet, right chiral electron, and electron neutrino respectively. Since they are singlets under S U ( M - 1)H2, X 2 is identical to the usual hypercharge, Y/2, on these fields. Setting X2(~e L) --- --1/2, X2(eR) = 0, and X2(veR) = 0, we find that X

=X

-i,

X

=X

+i,

(s) M ( M + I) - 6 (~M__~_I)I/2 a = 4M(M+ 1)

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The fields in LL, ER, and N R that transform as [M 1 ] and [ ( M - 1)M/2] under SU(M 1)H2 represent the muonic and tauonic families respectively. Henceforth, we ignore neutrino masses. At M G all families have the same Yukawa coupling ~E. However, these couplings are renormalized. The dominant graphs are due to the horizontal gauge interactions. Since the electron, muonic, and tauonic families are each in different representations of S U ( M - 1)H2, they all renorrealize differently. The renormalization group equations can be solved to give

26 July 1984

under SU(M - 2)H 3 . Lr L , Er R , and Nr R each decompose as

1)/21,

[ ( M - 1)M/2] = [1] + [ 3 / - 2 ] + [ ( M - 2 ) ( M -

(13) under S U ( M - 2)H3. The singlet fields in (12) and (13) represent the muon and tauon families respectively. Setting Y/2( [ 1]) to the usual Weinberg-Salam model values we find that

O~M_1(0) ai_l(t ) = l_(bi_l/27r)C~M_l(O) t ,

(6)

b-

xR(t ) = ~.E [e~M_I(O)/aM_I(t)] 3CM_1 (R)/bM-l ,

(7)

Below scale MI, the electron, muon, and tauon families, being singlets under SU(M - 2)H3, no longer receive radiative corrections to their Yukawa couplings from horizontal gauge interactions. Therefore, it follows from eq. (9) that the mass ratios of the leptons

where t = In Q / M G, R is the representation [ 1 ], [ M - 1], or [(M - 1)M/2], CM_I(R) is the quadratic Casimir coefficient of R, and bM_ 1 is the coefficient of the 3-function of O~M_1 . We want ~R(t) to be as large as possible for arbitrary t < 0 . This will be the case if bM_ 1 = ½(12N B - 7 M + 2 0 ) = 0 .

(S)

It follows that eq. (7) becomes kR(t) = k E e x p [ - ( 3 / 2 7 r ) C M _ l ( R ) a M _ 1 ( O ) t ] .

(9)

2. S U ( M - 1)H2 breaking to S U ( M - 2}H3. The potential energy can be arranged so that K develops a VEV, (I~), that spontaneously breaks S U ( M - 1)H2 to SU(M 2)H3 at scale M I. Note that (K) also breaks U(1)X2 . Denote the unique broken generator in SU(M 1)H2 that commutes with all generators of S U ( M - 2)H3 by T 2. If we define

(10)

Y = X 2 + bT 2 ,

Y/2 is unbroken by (I() if X K = -a[2M(M-

1)] -1/2 + b [ ( M - Z ) / 2 ( M -

1)] 1/2. (ll)

It follows that the unbroken gauge group at energies below M I is G 3 = SU(3)C × SU(2)L × U(1)y × S U ( M - 2 ) H 3. LUL, EUR, NU R each decompose as [M

i] = [1] + [ 3 /

2],

(14)

are

mu _ ( M G ] (3/2.)[M(M-2)/2(M-1)IaM_ 1 (0)

me

\-~-i !

,

m

[Mr._ ~(3/2.) [ (M- 2)(M+I)/(M-1)] aM_ , (0)

m-~r=/~\--MII-I!/

This renormalization continues until S U ( M - 1)Hz is spontaneously broken.

1

M ( M + 1) - 6 - 2)]-1/2 2 ( M + 1) [ 2 ( M - 1)(3/ .

(15)

.

(16)

The fields with mass smaller than M I are given in table 3. None o f these fields are observed at low energy so they must acquire a mass greater than the electroweak scale. How is this possible? The coefficient o f the 3-function for the S U ( M - 2)H 3 gauge parameter Table 3 Fields under SU(M - 2)Ha. All fields are fermions. Coefficients C = b [2(M - 1)(M - 2)] -rf2, and D = C(M - 3). SU(3)C

SU(2)L U(1)y

SU(M- 2)H3

L~ (~[1 (fL2 E ~(

1 1 1 1

2 2 2 1

X~ - A + C x L + 2B - D X~+ 2B+ 2C xE1 - A + C

M-2 M- 2 ( M - 2)(M-1)/2 M- 2

ER1 E'~(2

1 1

1 1

xEI + 2 B - D x E + 2B + 2C

M- 2 (M-2)(M-1)/2

~ NR1 NR2

1 1 1

1 1 1

XN1-A+C M- 2 XN + 2 B - D M-2 x N + ZB + 2C ( M - 2 ) ( M - 1 ) / 2

(12) 283

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is given by

Tile ratios of the lepton masses are given by

bM_2 = - ~ M + 10.

(17)

It follows that this parameter is asymptotically free w h e n M ~> 5. Henceforth,we assume this to be the case. Let AM_ 2 be the scale at which O~M_2 becomes strong coupling (OtM_ 2 -~ 0.3). Therefore, below this energy, all fields in table 3 form bound states. F o r M even, it can be shown that these bound states all have mass of O(AM_2). Henceforth, we assume that M is even, and that AM_ 2 is not smaller than the electroweak scale.

3. SU(2)L × U(1)y breaking to U(1)EM. The assumed decoupling o f scalars S at M I necessitates the breakdown of SU(2)L X U(1)y, since bound states involving S must also be heavy. Hence, AM_ 2 must be of O (1 TeV), the usual technicolo r value. Field q~ mixes with bound state, composite scalars in the (,mass) 2 matrix. The largest mixings are of o(XrA~/_2), where Xr is evaluated at AM_ 2 . For arbitrary Xr, the parameters o f the theory can be arranged so that one o f the mass eigenstates, ~', developes a VEV which breaks SU(2)L X U(1)y to U(1)E M at mass scale 250/X/~ GeV. Parameter k g can be chosen to give the correct electron mass of 0.511 MeV. We consider the renormalization group scaling proproperties o f the gauge couplings a2, a y , and aX2 of SU(2)L, U(1)y, and U(1)X2 respectively. At 102 GeV, aEM = 1/127 and sin~2'0W = 0.225. Starting with these boundary conditions, we scale a 2 and ~ y (which changes into aX2 at MI) up to M G . We must constrain all parameters so that a2, a y , and aX2 are weak couplings in this domain. One constraint, which slows the growth o f CZx2 at large momentum, is X S = - ( M + 3) (M - 2)/4M(M 2 - 1).

(18)

Also, with eventual grand unification in mind, we constrain the parameters so that ~2 = ~X2 = °~M-1 at M G. These constraints are rather difficult to satisfy. One solution, which satisfies all the requirements discussed in this paper, is M =8,

N B=3,

XB =-0.3,

M G = 3.52 X 1011 GeV,

(19)

M I = 4.75 X 104 GeV.

For these parameters we find that

aM_ 1 = 0.14, 284

26 July 1984

AM_ 2 = 3.00 × 103 G e V .

(20)

mu/m e = 37.6,

mr/m e = 3 4 9 0 .

(21)

The mr/m e ratio is exactly correct, whereas mu/m e is too small by a factor o f 5.5. It is clear from (15) and (16) that the mass ratios are determined by the Casimir coefficients. In this model these imply that m S m e ~-(mr~me) 1/2, instead o f the observed ratio (mr/me)2~ 3 . Hence, the above values depend on our choice o f SU(M) as the horizontal group, and on the representation content of the theory. It is likely that a more detailed study o f the group theoretical structure will yield a better prediction for m f f m e. Now consider the same theory, but without the scalars S. Parameter aM_ 1 now has a small, nonvanishing/3-function. Since/3 is small the conclusions of sections 1 and 2 are essentially unchanged. The decoupling theorem no longer implies that SU(2)L X U(L)y must be dynamically broken when the bound states form. Although a most-attractive-channel analysis suggests that the symmetry breaks, it is not conclusive. We will assume that it does not. The parameters of the theory can be arranged so that the electroweak breaking scale is 250/x/~ GeV and the electron has mass of 0.511 MeV. Constraining the gauge parameters as above, we find that one solution, which satisfies all the requirements in this paper, is M =8,

N B =3,

S B=-0.145,

MG =2.23X1017GeV,

(22)

MI=2.65X 109GeV.

For these parameters aM_I(MG)=0.118,

A M _ z = 9 X 107 G e V .

(23)

The ratios o f lepton masses are the same as in (21). We want to acknowledge H. Georgi for stimulating our interest in this problem, and H. Pagels for helpful discussion s.

References [ 1 ] See, e.g., S. Weinberg, Phys. Rev. Lett. 29 (1972) 388 ; H. Georgi and S.L. Glashow, Phys. Rev. D7 (1973) 2457, S.M. Barr and A. Zee, Phys. Rev. D17 (1978) 1854; F. Wilczek and A. Zee, Phys. Rev. Lett. 42 (1979) 421 ; C.D. Frogatt and H.B. Nielsen, Nucl. Phys. B147 (1979) 277 ; H. Fritzch, Nucl. Phys. B155 (1979) 189;

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S.M. Barr, Phys. Rev. D21 (1980) 1424; D24 (1981) 1895; M.J. Bowick and P. Ramond, Phys. Lett. 103B (1981) 338; R. Barbieri, D.V. Nanopoulos and A. Masiero, Phys. Lett. 104B (1981) 194; R. Barbieri, D.V. Nanopoulos and D. Wyler, Phys. Lett. 106B (1981) 303.

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[2] S. Dimopoulos, Phys. Lett. 129B (1983) 417. [3] H. Georgi, A. Nelson and A. Manohar, Phys. Lett. 126B (1983) 169; see also: D.B. Kaplan, H. Georgi and S. Dimopoulos, HUTP-83/A079; D.B. Kaplan and H. Georgi, HUTP-83/A069.

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