The pseudo-Goldstone mass spectrum

Physics Letters B 285 (1992) 103-112 North-Holland P t-t YS I C S k E T T E R S B

The pseudo-Goldstone mass spectrum

R. C a s a l b u o n i a, S. D e C u r t i s b, N. D i B a r t o l o m e o c, D. D o m i n i c i d, F. F e r u g l i o e a n d R. G a t t o ¢ a Dipartimento di Fisica, Universit~ di Firenze and 1NFN, Sezione di Firenze, 1-50125 Florence, Italy b INFN, Sezione di Firenze, 1-50125 Florence, Italy c D#partement de Physique Th#orique, Universit# de Genbve, CH-1211 Geneva 4, Switzerland d Dipartimento di Matematica e Fisica, Universita di Camerino, 1-62032 Camerino, Italy e Dipartimento di Fisica, Universitdt di Padova, 1-35131 Padua, Italy

Received 15 April 1992

In schemes of strong electroweak symmetry breaking, independently of the details of the underlying fundamental theory, we develop a framework for the quantitative evaluation of the pseudo-Goldstone masses, including, beside the gauge contributions, also the contributions from those interactions which are responsible for the masses of the ordinary fermions. The effective low- energy Yukawa interactions among pseudo-Goldstone bosons and ordinary fermions appears to play an essential role. All those states which, neglecting such Yukawa terms, would remain massless, tend to acquire mass and we find that these masses, barring special cancellations, are close to those of the heaviest fermions of the theory, the top and the bottom quarks.

1. Introduction

Theories of the electroweak symmetry breaking (SB) which avoid the in t roduct ion of fundamental scalars are usually character ized by a large global invar iance under a chiral symmetry group G. The dy- namical mechanism which is responsible for the elec- t roweak SB triggers, at the same time, a spontaneous breaking of G into a subgroup H. To each generator o f G / H then corresponds a Golds tone boson, which is exactly massless as long as one neglects addi t ional interactions. Such a si tuat ion is, however, not realistic and, typically, gauge interact ions explicit ly breaking the symmetry group G are present from the be- ginning. For instance (but it cannot be the whole s tory) the s tandard model gauge interactions, related to the local group G w = S U ( 3 ) × S U ( 2 ) L × U ( 1 ) r , will in general break the chiral symmetry associated to G. This fact will force an interact ion which will break the degeneracy among the previously equiva- lent vacua Jg2). In the absence of such an induced orientat ion, all the states [.Q) would have the same energy and the small oscil lat ions along the direct ions connecting the different vacua would correspond to

* Partially supported by the Swiss National Foundation.

strictly massless Golds tone bosons. Due to the selec- t ion of a par t icular or ientat ion some of the previous oscil lations will now correspond to massive modes [ 1 ], or, as it is commonly referred to, to pseudo- Golds tone bosons.

Because of the possible lightness o f some of these states, their presence is at the same t ime desirable and embarrassing. Desirable, for the possibi l i ty they offer to be discovered soon at the available or p lanned fa- cilities, embarrassing for the potential conflict they could cause with the amount of data already accu- mulated. It is therefore of p r imary impor tance to try to give a descr ipt ion as accurate as possible o f the propert ies of these particles.

F rom the previous discussion it follows that basic to the predic t ion of the mass spectrum of the pseudo- Golds tones is the precise knowledge o f the possible interact ions which explicit ly break the chiral symmetry G. Among those, the gauge interact ions associated to Gw were first s tudied and their effects on the pseudo-Golds tone boson mass spectrum of simple technicolor models were quant i ta t ively assessed [2,3].

It is however clear that other interact ions explicit ly breaking G must also be present and taken into account. We are here referring in par t icular to the

Elsevier Science Publishers B.V. 103

Volume 285, number 1,2 PHYSICS LETTERS B 2 July 1992

mechanism which is responsible for the generation of the masses of the ordinary fermions.

If, as an illustration of these ideas, we think of an extended technicolor scheme, the gauge interactions associated to the generators connecting ordinary fermions to technifermions will in general break the chiral symmetry G, which in this model is related to the technifermion sector. Since the interactions considered are those responsible for the generation of the fermion masses, it is natural to expect that the induced pseudo-Goldstone masses are somehow related to the fermionic mass spectrum.

Aim of this note is to provide a framework for the explicit and quantitative evaluation of the contribution to the pseudo-Goldstone mass spectrum, coming from the interactions which provide masses to the ordinary fermions. In doing this we will avoid the detailed introduction of a specific fundamental theory. Instead, we will work with the low energy effective theory, essentially characterized by the groups G, H and Gw.

Our strategy will consist in writing the allowed effective Yukawa couplings between the ordinary fermions and the pseudo-Goldstones. In the low-energy theory, such effective Yukawa couplings are sup- posed to carry the relevant information independently of the fundamental mechanism which gives origin to the observed fermion spectrum.

Allowed couplings are those which are invariant under the gauge symmetry Gw, surviving in the low- energy sector. According to the general prescription of ref. [4], such couplings are specific functions of the fermion fields and of the pseudo-Goldstone bosons, which transform nonlinearly both under G and Gw. The expansion of these couplings contains, at lowest order, the mass terms for the ordinary fermions. This allows us to obtain a set of relations among the Yukawa couplings and the fermion masses.

In general, the number of independent Yukawa couplings exceeds the number of masses, so that some free parameters are still present in the analysis. Nevertheless, as explicitly illustrated in the following, this fact does not prevent the possibility of mak- ing certain quantitative statements about the pseudo- Goldstone masses.

Once the Yukawa couplings are given, the natural tool to analyse the problem of vacuum orientation is the effective potential. We will evaluate the one-loop

effective potential including the effects of both ordinary gauge interactions and Yukawa couplings. Fi- nally, from the one-loop effective potential we will derive the pseudo-Goldstones mass spectrum.

For illustrative purposes here we will discuss two commonly used choices for G and H: case A with G = S U ( 4 ) × S U ( 4 ) , H = S U ( 4 ) , and case B with G = SU ( 8 ) X SU (8), H = SU (8). These special pos- siblities correspond to the cases discussed in ref. [ 2 ].

When fermion masses ar taken into account, the chiral symmetry G is so completely broken that, in general, no massless Goldstone boson survives in the theory. Pseudo-Goldstone bosons remaining massless when only the ordinary gauge interactions Gw are considered, receive in general masses. Those masses have a natural range situated around the heaviest among the ordinary fermions, namely rn,op and t~tbouo m. Special choices of the parameters characterizing the fermion masses are nevertheless possible so as to make some of these pseudo-Goldstones light, or even worse, to destabilize the theory.

In the next section we illustrate our formalism in the example A. We then consider the case B. Finally we present our conclusions.

2. Case A: G = SU(4) X SU(4), H = SU(4)

In this section we consider a toy model, just to illustrate in a simple case how the mechanism of generation of pseudo-Goldstone masses works. This model has a chiral symmetry SU(4 )XSU(4 ) spontaneously broken to the diagonal SU (4) subgroup.

Although irrelevant to our effective low-energy formalism, it is suggestive to think of technicolor interactions, associated to some technicolor group GTO with technifermions given by two doublets of techniquarks: (U, D), (C, S). These doublets have the same quantum numbers as the corresponding ordinary quarks, with respect to SU(2)LXU(1)y. This realizes the case d = 0 of ref. [ 2 ]. Such techniquarks are colorless with respect to the ordinary color interactions, so that, in the limit of massless techniquarks, the chiral symmetry is SU(4 )XSU(4 ) (the singlet axial current is expected to be afflicted by anomaly).

The spontaneous breaking of G = SU (4) X SU (4) down to the diagonal subgroup H---SU (4) produces 15 Goldstone bosons, associated to the generators of

104


G/H . For reference a list of these generators is given in table 1. The Goldstone bosons n ~ ( a = 1, 2, 3), associated to the generators T ~, are adsorbed by the electroweak gauge bosons W -+ and Z. The remaining 12 bosons ~ , riD, n~', z~, (kt=0, a) are physical particles in the spectrum and may acquire a mass.

The low-energy effective lagrangian will contain interaction terms among the Goldstones and the ordinary gauge bosons, collected in ~ :

= ~v2Tr(~l, U * ~ U) , (2.1)

where

U = e x p ( ~ ) . (2.2)

We have here denoted, collectively, the generators of table 1 T ~, and the Goldstone bosons nr(r= 1 ..... 15 ). We have chosen the normalization Tr(T~TS)=~rL The covariant derivative ~ U is given by

~,,U=8,,U+ WuU-UB~,, (2.3)

where

W~ = igT a W~ ,

Bu = ig'T 3B,. ( 2.4 )

In the previous equations W~ and B~ are the gauge vector fields associated to S U ( 2 ) L × U ( 1 )r. In this model no term containing the gluon field is present in eq. (2.3)( in a technicolor realization the techniquarks are colorless).

We now consider the Yukawa couplings among the ordinary fermions and the n. To this purpose it is ad-

Table 1 SU(4) generators. Here r ~ are the Pauli matrices, and au--- (~2, r ~) -

1 ff o

rD=~ - - - - \ 0 [ -12 /

,/' o/<, . ' ,, a,, l o /

1 Za 0

_ 1 ( o \ - i a u [ 0 /

vantageous to decompose the matrix U in the following way:

Ud~) u _ gdo g~a Ua~ - Uc. gcd gcc Uc, (2.5)

u,. usd us~ Uss

(The index ud, for instance, refers to the fact that the element Uu~ can be thought of as made of the techniquarks UL and IS)R). Given this representation of the matrix U, it is straightforward to write the Yukawa couplings which are Gw invariant. For the case we are considering, we will not provide a complete clas- sification of all possible Yukawa couplings, which instead we will do for the case presented in the next section. Moreover, only the couplings with the quarks of the third generation will be included, namely those with the quarks top and bottom. The generalization to the whole set of fermions belonging to the three known generations presents no difficulty, without being particularly illuminating. We will assume that the Yukawa couplings are given by the following terms:

o~gy~ 77 t i m3(tRUcctL +{iR t i UscbL +h.c.)

t t i +m4(~iRUcstL +~R t i Ussbe +h.c.)

q_ 7~ t i 77 t i tR Ud, bL + ms(lRUuutL + h.c.)

t i z + m6 (b-~ Uuote +fiR Utodb~L +h.c. ) . (2.6)

The parameters m3, m4, rn5 and m6 are here taken to be real. The index i is the color index. It is immediate to verify that ~ r is indeed invariant under Gw. By expanding the matrix U in powers of ~z, one recovers all the tower of couplings with one, two, n Gold- stones, as needed to properly ensure the required gauge invariance. In particular, from the first term of the expansion, we can read the masses of the top and bottom quarks:

rnt=m3 +ms, mb=m4 +m6. (2.7)

In our approximation, all other fermions are massless. The interaction terms Lfg and ~v are what we need to evaluate the effective potential.

Before doing this, some comments are in order. Since the low energy effective theory is nonrenormal- izable, the one-loop effective potential is divergent. By introducing an ultraviolet cut-offA, roughly situ-

105


ated in the TeN range, such divergences can be class- ified. In general one expects quadratic divergences, logarithmic divergences and finite terms. In the following we will evaluate explicitly only the terms quadratic in the cut-offA, neglecting the other contributions. The situation is here completely analogous to that of ref. [2]. The remaining contributions can of course be evaluated, but they are expected to be less important than the quadratic ones, when these are present.

The introduction of an ultraviolet cut-offA should, ultimately, correspond to the following procedure. Since the terms L#g and Yv in the effective lagrangian produce one-loop divergences which cannot be absorbed into a redefinition of the couplings contained in ~+Sey, the low-energy effective action must be supplemented with appropriate counterterms ~ , such that the results of a one-loop computation are finite and coincide with those of the underlying fundamental theory. The counterterms ~ , are expected to in- clude all those short distance effects which are pro- duced by the integration over the heavy modes of the fundamental theory and which are not contained in ~g+ 5% If the scale characterizing the heavy modes is A, it is natural to assume that the net result from the one-loop amplitude of 5eg+~r and the O(h) counterterms in 5t~t is simply given by the amplitude evaluated with a cut-off at momenta close to A. Al- though it is clear that this guess may contain some oversimplification of the complete physical picture, one hopes to be able in this way to extract the main results of the theory.

Particular care must be paid in analyzing the re- suit. To be sure that the potential obtained is not an artefact of the regularization procedure, one must explicitly check that the terms generated in the effective potential are themselves invariant under transfor- mations of Gw and that they possess the correct chiral behaviour: they must vanish whenever a special choice of parameters in 5#g+5~r restores the chiral symmetry G.

We are now in a position to present the result for this toy model. In the Landau gauge we obtain

3A 2 Neff= 8792 ( m 2 J l -+-m2J2-l-m2J3 q - m 2 j 4

The quantities J i ( I = 1 .... ,6 ) are given by

J, = Uou u L + Udu u L ,

& Uud + + = Uud + Udd Udd,

J~ = U~c u~*~ + u~ u1~,

J4=U.U*cs + U,,U~,,

.15 = Uuu Uc*c + t & u~*~ + h . c . ,

• ]6 = Uud Ucts "l- Udd U~s +h.c. (2 .9)

We observe that in this case there is no contribution from ~g to the one-loop effective potential Neff. Ne- glecting LPy, all the 15 Goldstone bosons remain massless (at the order A 2 ). This result agrees with ref. [2] (recall that we are considering the limit A=0 of the model studied there). On the other hand the Yu- kawa interactions provide for the effective potential given in eq. (2.8). One can easily check that the quantities J, ( i= 1 ..... 6) are invariant under Gw. Moreover, the only choice of parameters which restores the chiral symmetry at the level of Yr is m3= m4 = m5 = m6 = 0. We list below the mass eigenstates and the corresponding eigenvalues of the mass- squared matrix, derived from the potential Neff ofeq. (2.8):

m2(Tr a) -----0, a = 1,2, 3,

rn2(2 + ) = m2(ff - ) = 3A--~-2 (m3m5 +mare6) 7~2U2

rn 2 {;~3- Z~o'~ 6A 2

m 2 {23+nD'~ 6A 2 U U u ) = Tje " , m , ,

{ p o + p 3 ~ = m 2 {/~3o +/53"~

3A 2 - 2rr2v2 ( m 3 + m s ) 2 ,

m 2 {po _p3"~ = m 2 (zOo _f3"~

3A 2 - 2~2v 2 (m4+m6) 2 , (2.10)

+ m3 m5Js -t- m4 m6J6 ) . (2.8)

106


rn2( P; )=m2( fi; )

3A 2 -- 27t2p2 ( m 2 + m 2 + m 3 m 5 + m 4 m 6 ) ,

m2(p¢) = m2(pf )

3A 2 - 2~rzv2 (rn~ +m~ +m3ms +m4m6) ,

(2.10 cont 'd)

where we have introduced the linear combinations

rtg - i r ~ P~'= x /~ (2.11)

a n d / ~ = (P~)*, and we have defined P3 ~ in the standard way.

We note that the would-be Goldstone bosons ga, which are going to be absorbed by the gauge bosons W and Z, are massless. We also observe a curious change of sign: the minus sign of the left-hand side of eq. (2.8), coming from the fermion loop, has been cancelled by a minus sign coming from the expansion of the invariants J~. In this way, for instance, the combinations (po + p 3 ) / x / ~ and (po _p33)/x/~ acquire a positive mass squared. This situation is op- posite to what is usually obtained in computing the dependence of the effective potential from the radial mode. Such a mode, here deliberately omitted from the low-energy spectrum, receives from the fermion loops negative contributions to its squared mass, which tends to destabilize the theory. One can think, as an example, to the Higgs particle of the standard model. Our computation shows that, on the contrary, the angular modes contained in U will not in general obey the same rule as the radial mode, and they can receive from fermion loops positive contributions to their squared mass. From eq. (2.10) we read the conditions for the stability of the one-loop effective potential (remember that the parameters m, are here taken to be real ):

m3m5 > 0 , m4m6 > 0 (2.12)

Such conditions can be easily met by choosing m3 and m5 of the same sign and similarly for m4 and m6.

The structure of the mass spectrum of the pseudo- Goldstones can be understood by observing that in the chiral limit one has a SU (2) c × SU (2) R symmetry for each fermionic doublet. Then the mass matrix

must be a combination of a scalar and of generators of S U ( 2 ) L X S U ( 2 ) R ~ O ( 4 ) which commute with the electric charge. This corresponds to require that these generators commute with the diagonal T3= ½ (T3L+ T3R). In 0 ( 4 ) there is only one such generator:/£3 = ½ ( T3L-- T3R). Therefore the general structure of the mass matrix must be

M2=A + BKB + CT3 . (2.13)

Notice that the ( ½, ½ ) representation of SU (2) X SU(2) , to which the Goldstone bosons belong, de- composes as 1 +_3 with respect to 0 ( 3 ) c 0 ( 4 ) and therefore the following sum rules for the masses are implied for all the quadruplets:

m~(T3 = 0 ) = Z m2(K3 = 0 ) . (2.14)

In the model previously considered, one has for example

mZ f ~t3-- nD" ~ {ff3+nD'~ ~ , - - - ~ j + m 2 ~ - J

=m2(2+)+m2(fr -) . (2.15)

In the following we collect the allowed range for the various masses:

m20r a )=0 , a = 1 , 2 , 3 ,

m2(ft+- )<~ 3A____~ 2 (m2 +m 2) 47~2V 2

m 2 {~3-zto'~ 3A2 2 ~ - ~ ] ~< 2rt2v----5 mb,

m Z / ~ 3 + g D " ~ 3A 2 mt z

m2{P°+P]" ~ 3A 2 m 2 \ - U - ) = 2 2v--m '

{po_p3"~ 3A z m 2 m2t /- ,

3A 2 m2(p~)~< 27r2v---- 5 (m~,+m~) ,

mZ(P~)<~ 3A-----~-2 (m2+m2t). (2.16) 2~2V 2

As we see from the previous list, the pseudo-Gold- stone masses have values which are naturally close to those of the massive fermions of the theory, in this

107


case the top and the bottom. For numerical illustration just take m b = 5 GeV, mt = 150 GeV and A = 1000 GeV. One then finds, in this toy model:

m2(na)=o, m 2 ( ~ ± ) ~ ( 1 6 8 G e V ) 2 ,

m 2 / , ~ - ~ - ] ~ (8 GeV)~,

(7~ 3 --1- 7~D "~ m2 \ ~ 7 - j ~< (238 GeV)2 ,

{eo+p3, mZk~-~f~--,) = ( 2 3 8 GeV) 2 ,

m2 k ~ ) = (8 OeV) 2 ,

rn2(p~ ) ~ (238 OeV) 2 ,

mZ(P~ ) <~ (238 GeV) 2 .

3. Case B. G = SU(8) × SU(8), H = SU(8)

We consider here the case of a chiral symmetry SU (8) × SU (8), spontaneously broken to the diagonal SU (8) subgroup. Such a pattern of global sym- metries is realized for instance in a popular technicolor model, with technifermions given by a generation (U, D, N, E) of techniquarks and techni- leptons, with the same quantum numbers as the corresponding ordinary fermions. The spontaneous breaking of SU (8) × SU (8) to the diagonal SU (8)

subgroup produces 63 Goldstone bosons. The corresponding generators are listed in table 2. The Gold- stone bosons are commonly divided into three classes. One has seven colorless Goldstones ha, ~a and nD associated to the generators T a, T~ and TD ( a = 1, 2, 3). As before, the states n a are those absorbed by W and Z via the Higgs mechanism. A second class contains 32 color octet Goldstones: n~' and n~ ~ (o~= 1, ..., 8). They are related to the generators T~ < and T~ '~. Finally, there are 24 Goldstone bosons which are triplets: Pg' and/~g ~ (/x=0, 1, 2, 3 and i= 1, 2, 3). They are linear combinations of the fields ng ~ and z~3 ~ associated to the generators T~" and 7~g ~ (see eq. (2.11 )). The low energy gauge interactions of the Goldstone bosons are described by the following terms:

~g= ~6 v2 Tr( ~ U* ~, U) , (3.1)

where

The generators T ~ ( s= 1 ..... 63) are normalized according to Tr ( T ~T') = 2d ~s. The covariant derivative ~,U is given by

2,~,U=O,,U+ ~ , U - U . ~ , , (3.3)

where

Table 2 SU ( 8 ) generators. Here )t,~ are the GeU-Mann matrices and (i are three orthonormal vectors in a three-dimensional vector space.

T = 2 \ 0 l e ' i

r o = m / - - f - - ! \ 0 1 -12 /

T ~ = ~ \ 0 l O /

~/'r"®13/3 I 0 \

1 [ z~®2" ] 0"~ /

\ o / o /

o / ~/2\- ia~®~'T| 0 /

108


dp=Wp+&+G,, s?~=&+B;+G,,

W,=igT”W”, , BP=ig’T3BP,

BL =ig’ $$Bp, GP=i%TPGz. (3.4)

In eq. (3.4), W;, B, and G; are the gauge vector

fields related to the gauge group Gw. In order to write the Yukawa couplings between the

Goldstone bosons and the ordinary fermions we de-

compose the matrix U according to

(U& U:d u’;-, vi,\

The notation is similar to that of the previous section. The indices i, j, k and 1, running from 1 to 3, are color indices. Also in this case, we will only consider the coupling of U to the fermions of the third generation. From the previous example it is clear that, since the generated pseudo-Goldstone masses are proportional to the fermion masses, the couplings we are neglecting would provide no significant contribution. The most general Yukawa coupling, invariant with

respect to Gw, is given by

where the A” are the Gell-Mann matrices and all the parameters mi and pL are taken to be real. The appar- ently strange numeration of the couplings in eq. (3.6) is due to the absence of uR. The missing couplings m3, m, and ,u3 would correspond to Yukawa interactions with the vR. From _?Z,, we easily recover the expres- sions for the fermion masses:

m,=m,+3m,+~m,,+m,,

mb=m2+3m6+~m,z$ms,

m,=m,-t3m,o,

m,,=O. (3.7)

All the other fermions remain massless in this

approximation. Starting from the interaction lagrangian ygpg+ 2’r we

now evaluate the one-loop effective potential. As ex- plained in the previous section we will consider only the terms quadratic in the ultraviolet cut-off/i. In the

Landau gauge we obtain

v,, = l%F + v,‘ff 7

where

(3.8)

Tr( Tf UtT; U)

+y Tr (( $) u+$u)], (3.9) T’+

and

A2 v,Y,=- ii;fi [(mf+fmf, -p:)Z,

+(m~+fm~2-p~)Z2+(3m~-p~)Z3

+(3m~+m~)I,+(2m,m,+3m~+~m,m,,)15

+(2m2m,+3m~+~m,m,2+m~,)Z,

+(m,m,+3m5m,+$m7m,,)Z,

+(m2m,+3m,m,+~m,m,2+m,m,o)Z~

+(2m,m,,+am:,)Z9+(2m2m7212+~m:2)1101,

(3.10)

where the quantities 1, (i= 1, . . . . 10) are given by

109


I~ - lI*J rTw + Uau U~d - - ~ u u v u u

6 = u~d u ~ + ua~ u~A,

6 = u,,,,u*,,,, + Ue,, U'e,,,

=U~oU~e+UeeU*eo,

15 - tT" tT*JJ + U*d~ U*d~ - - - - u u ~ u u

I~ -- tT~,i U*u'a + u ~ vy~

17 - - i i + Ud~ U~ +h.c . ,

-- Uud U v e + Udd Uee +h.c . , i8 _ ii "t +i

/9 = T r ( U~,2" Uu,;t" + U*~,2 '~ Uo~2"),

Im=Tr(U*ud2"Uud2~+U~d2"Udd2 '~) . (3.11)

One can show that these quantities are indeed invariant under Gw. The mass eigenstates and eigenvalues are derived expanding at the second order the invariants L in the effective potential. The mass spectrum we obtain is

m2(7~ a) =0, a = 1, 2, 3 ,

2A 2 m2( 7~+ ) = m2(7/- ) = ~ (P7 + P s ) ,

m 2 {:~3- rrD'~ 4A 2

m 2 ( f f J -~nD'~ 4A 2 \ - T T j = ,

m2(zr~+ ) = m 2 ( ; ~ - )

A 2 -4rr2v2 [3(ps +P6 +P9 +P,o) +2(p7 +P8) ] ,

m 2 ( ~ + ~ 3 " ~ A 2 ~ ~ , 1 = 2~r 2v----~ (3ps + 2p7 + 3p9 ) ,

m 2 { 7~c~ -- 7~ c~ 3 " ~ a 2 \ - - - , 7 - ) = 2,~v - - ~ (3p6 + 2p, + 3p,o),

rn 2 {pO, +p3,~ m ~ (/~o,+ff3,'~ = vT --J

a 2

-- ~Pg) , (3.12) 21./ .2/)2 ( P l +P3 +3ps + 4 t 0 7 "3V 4

110

{poi_p3i '~ - o _ -3i

A 2 - ~P to ) , 2nZv2 (P2 +P4 +3p6 +4p. + "

mZ(Py i) = m 2 ( p y i)

A 2 - + ~Pto, 2~2V2 (P2 +P3 + 3p6 + 3p7 +P8 4

m2(p~ -,) = m 2 (/6~ -/)

A 2 = 27~2V2 (191 + P 4 + 3p5 + P 7 + 3p8 + 4p9) ,

(3.12 cont 'd)

where

pl = 2 m 1 2 + 4 2 2 ]v2g,2 ~ m l t - 2 / ~ l +

pz=2mZ2+ 4~m 212 --2fl 2-1gv 2 g t2 ,

P3 =6m27,

p4=2m4 2 + 6 m 2 - 2 a 2+½vzg '2,

P s = 4 m t m s + 6 m 2+~rnSml l ,

P6 =4m2m6 + 6 m 2 + ~rn6ml3 + m~o,

p v = 2 m l m 7 + 6 m s m 7 + ~ m v m l l ,

P8 =2mzm9 +6m6m9+ ~m9ml2 + 2 m 4 m m ,

7 2 3. 2~2 p 9 = 4 m l m l l + 3 m t t +av gs ,

p m = 4 m 2 m , z + 7rn22 + 3v2g 2. (3.13)

As in the SU(4) model the zc ~ remain massless and are absorbed as longitudinal degrees of freedom of W and Z. In general all the other pseudo-Goldstone bosons get a mass, even those colorless and the mass rule ofeq. (2.14) holds for all the multiplets. The stability of the one-loop effective potential requires that the masses given in eq. (3.12 ) are positive. This fact of course puts some constraints on the Yukawa couplings, but we will not pursue a general analysis here. As an example, one can satisfy the stability conditions by taking/~i= 0 ( i= 1, 2, 4) and all the mi of the same sign. We notice that the parameters/G i= 1, 2, 4, give a negative contribution to the squared masses of all color triplets and therefore tend to destabilize the potential. Taking into account only the bosonic contribution to the effective potential one finds that only the colored sector receives mass; putting ~ y = 0 we find the same spectrum as in ref. [ 2 ]:


k - W - ; = k - U - ) 9A 2

- - 8~ 2 g 2 ,

m2{P°'+P3') m2{e°i-lp3i'~

Z 2 _ 2~ 2 (g~ +½g,2) ,

A z - ~ g ) , m 2 ( p ; , ) = ~ (g2 , ,2

A 2 + g g ) (3.14) m 2 ( p f ' ) = ~ 2 (g2 5 , 2 .

To compare with ref. [2 ] one should make the iden- tification 3A 2/27r = M 2.

Coming to the general case, ~ r # 0, it is easy to derive the following allowed ranges for the masses of the colorless pseudo-Goldstone bosons:

ma(Tt a) =0, a = 1,2, 3 ,

2A 2 ~ m ; ) m2(~'±)~ ~ (½m2 +½ m2 + ' , ,

{7~3 -- 7/'D "~ 2A 2 im.~) m 2 k _ _ _ ~ } ~ < _ _ ( m 2 q _ l ~ 2 u 2 2 ,

m 2 [7~ 3 "~" 7[D "~ 2A 2 k ~ - ] ~< n-~Sv2 m~. (3.15)

These ranges are obtained by observing that the corresponding square-masses depend on few combinations of the whole set of original parameters. For instance the mass of the pseudo-Goldstone boson (t~3 + ~rD)/X f2 is proportional to the product rnArnu with mA = m7 and mB = m~ + 3ms + 4 3mj~ and, at the same time, from eq. (3.7), one has mt = mA + roB. We observe that necessary conditions for the presence, in the colorless sector, of terms proport ional to the top and the bot tom masses are, respectively, a nonvan- ishing m7 and m 9. Numerical values from eq. ( 3.15 ) for m ~ = l . 7 GeV, rob=5 GeV, m , = 1 5 0 GeV and A = 1000 GeV are

m 2 ( ~ a) = 0 ,

m2(2 +-)~< (194 GeV) 2 ,

m 2 k ~ ] ~ < ( 9 G e V ) 2,

k ~ , ] ~< (274 GeV) 2 .

We stress that, barring very specific choices for the initial Yukawa couplings, the natural range for the masses of ~ , z~ 2 and (7/3 + Zto)/x/2 is around the top quark mass, whereas for ( ~ 3 - f ro) /x f2 it is close to the bot tom quark mass. The model analyzed so far is probably not realistic and a detailed discussion about its phenomenology would be inappropriate. Never- theless we note that, as far as the colorless sector is concerned, there is no conflict with the present data, for values of the masses close to the upper limit of the range given in eq. (3.15). We recall that the Z may decay into the two charged combinat ions 7/+ and

- of ~ ~ and ~2 but not into a pair o f neutral states [5 ]. The three-body decays requires again the presence of z~ ÷ and z/-. On the other hand the radiative decay of Z into a photon plus the light combinat ion ( ~ 3 - 7to) /x/2 proceeds via an anomalous coupling [ 6 ] and has probably a too low rate [ 7 ] to be seen at LEP I without high luminosity option.

Finally, just to illustrate our results with an example, we give a numerical estimate of the pseudo- Goldstone boson spectrum, obtained by taking/zi = 0, ml=m7=½mtop, m2=mg=½mb, rn i=0 for i ~ 1, 2, 7, 9, A = 1 TeV, cq=0.12 and m t = 150 GeV:

m 2 ( z a)=O, a = 1 , 2 , 3 ,

A 2 m 2 ( ~ + ) = ~ (mr 2 + m 2 ) = (194 GeV) z,

m e [ z~3 -n° '~ 2A2 m 2 = ( 9 G e V ) 2

{~3+~D" ~ 2A 2 m t 2 = ( 2 7 4 G e V ) 2 ,

A 2 "~ 9 2 2

= (426 GeV) 2,

m2 {~r~ + ~ 3 " ~ A 2 = - - + a v gs) k ~ ) 27/'2V2 (m2 9 2 2

= (437 GeV) 2 , (3.16)

111


--2~2V 2 t t t b l 4 ~ ~ s

= (415 GeV) 2 ,

[po, +e3,'~ A 2 rn z ~ , ~ - ~ - - J = 27rzv~ ( 4m2 +vZg~ + ~v~g 'z)

= (392 GeV) 2 ,

{ poi _ p3,'~ A 2 m 2 \ ~ ] = 2~2v---- ~ (4m~ + vZg 2 + ~v2g 'z)

= (280 GeV) 2 ,

A 2 - - mu + v g~ - ~v2g '2) r n 2 ( p ~ i ) = 2rt2v 2 ( 3 m 2 + 2 2 2

= (363 GeV) 2 ,

A 2 - - + 3 m b + v g~ 'b 5vZg '2) m 2 ( p ~ i )= 2zcZv2 (m• 2 2 2

= (316 GeV) 2 . (3.16 con t 'd )

To conclude this section we comment on the parameters et and e2, or T a n d U def ined in refs. [8,9]. We observe that a large isospin splitting, as obta ined in the colorless sector, could in pr inciple give rise to a sizeable cont r ibut ion to Ao=e~ and t2. We have computed the one-loop effect of the pseudo-Gold- stone bosons [9] to el and ~2 using the same cut-off A = 1 TeV. The relevant contr ibut ions come from the color tr iplets and f rom the colorless sector. The result is that they generally give a sizeable, negative contri- but ion to el which is quadra t ic in the pseudo-Gold- stone masses, while the cont r ibut ion to e2 is negligi- ble. For the mass spectrum given in eq, (3.16) we obta in a cont r ibut ion to e~ of about - 0 . 0 0 3 to be compared to the usual s tandard model top contr ibu- t ion equal to 0.007. In general the overall effect of the pseudo-Golds tone loops is to modify the upper l imit on the top mass coming from the SM radia t ive cor- rections. We f ind that depending on the actual value of the cut-off A the negative cont r ibut ion from the pseudo-Golds tone bosons can compensate that of the SM or even become the dominan t one.

4. Conclusions

In theories o f dynamica l electroweak symmet ry breaking without scalar fields, the spontaneous breaking of a large init ial global symmetry G leads to

pseudo-Golds tone bosons in the low-energy spectrum. The masses of such states are in direct relat ions to the possible sources of an explicit breaking of G. Among these sources, the mechanism which gives rise to masses to the known fermions is of p r imary rele- vance, By considering the low-energy effective theory, where the mechanism of fermionic mass gener- a t ion is encoded in the allowed Yukawa couplings, we have provided a f ramework for an explicit and quanti tat ive evaluation of the pseudo-Goldstone mass spectrum. Our approach proceeds through the eval- uat ion of the one-loop effective potent ial in the lead- ing approximat ion , where counter terms of the de- sired symmetry propert ies are generated. The results show that, in general all pseudo-Golds tone bosons receive masses from fermion loops, even those which remain massless when only the gauge bosons correc- t ions come into play. The stabil i ty condi t ions of the effective potent ial have solutions in large por t ions of the pa ramete r space: the contr ibut ions to pseudo- Golds tone masses are quite often posit ive, contrary to what happens, for instance, with the ordinary Higgs o f the s tandard model. Those states which stay massless when Yukawa couplings are neglected, tend to acquire masses close to the values of the heaviest fermions of the low-energy theory, namely the top and the bo t tom quarks. To facilitate the compar ison with the existing l i terature we have developed our formalism only for two specific examples. However the underlying mechanism and the conclusions presented here are expected to be general proper t ies of a whole class of theories.

References

[ 1 ] S. Weinberg, Phys. Rev. D. 13 (1976) 974. [2] M.E. Peskin, Nucl. Phys. B 175 (1980) 197. [3l J.P. Preskill, Nucl. Phys. B 177 (1981) 21. [4] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969)

2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247.

[ 5 ] S. Chadha and M.E. Peskin, Nucl. Phys. B 185 ( 1981 ) 61. [6] J. Ellis, M.K. Gaillard, D.V. Nanopoulos and P. Sikivie, Nucl.

Phys. Phys. B 182 (1981) 529; A, Ali and M.A.B. Beg, Phys. Lett. B 103 ( 1981 ) 376.

[ 7 ] A. Manohar and L. Randall, Phys. Len. B 246 (1990) 537. [8] M.E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990)

964: G. Altarelli and R. Barbieri, Phys. Lett. B 253 ( 1991 ) 161.

[9] R. Casalbuoni, A. Deandrea, S. De Curtis, N. Di Bartolomeo, D. Dominici F. Feruglio and R. Gatto, to be published.

112

Documents

The pseudo-Goldstone mass spectrum