Enhanced mass spectrum of pseudo-Goldstone bosons?

Physics Letters B 277 (1992) 141-147 North-HoUand PHYSICS LETTERS B

Enhanced mass spectrum of pseudo-Goldstone bosons?

Soo- Jong R e y Center for Theoretwal Physws, Sloane Physws Laboratory, Yale Umverstty, New Haven, CT 06511, USA

Received 9 December 1991

In an attempt to clarify recent claims by Lus~gnoh, Maslero and Roncadelh, we study poss~bflmes of enhanced pseudo-Goldstone boson masses through three examples m which a global symmetry Is mtnnslcally as well as spontaneously broken lepton number m a variant of the triplet Majoron model, Pecce~-Qumn symmetry m low-energy supersymmetnc standard models or two Hlggs doublet standard models and chlral symmetries m extended walking technicolor models We obvaate the claims by Lus~gnoh el al by pointing out that the s~ze of mtnnslc global symmetry breaking depends on hierarchies of the vacuum expectaUon values The notions of approximate global symmetry and pseudo-Goldstone bosons hold only when the hierarchy is not too large

1. Introduction

The Goldstone theorem states that whenever a continuous symmetry is spontaneously broken, there arises a massless, spin zero field carrying the same quantum number as the broken symmetry generator In the real world, we hardly observe exactly conserved global symmetries either exphclt symmetry breaking interactions or nonperturbatlve effects af- flict the otherwise exactly conserved global symmetries In such situations, the would-be Goldstone bosons no longer remain massless Rather, they acqmre masses typically of order o f the intrinsic symmetry breaking scale, and are called pseudo-Goldstone bosons [1] Examples include plons in QCD chlral symmetries, mws~ble axlons to solve strong CP prob- lems and many others such as famdons and Majorons

Recently, Luslgnoh, Maslero and Roncadelh [2] examined a variant of the triplet Majoron model [ 3 ] m which the lepton number conservation ~s mtnns~- cally but shghtly as well as spontaneously broken and pointed out that the mass of the Majoron (pseudo- Goldstone boson of spontaneously broken lepton number) can be hierarchically enhanced over a naive estimate In an attempt to clarify their conclusion, we reinvestigate the triplet Majoron model and point out the underlying reasons for such a phenomenon size of the mtrmstc global symmetry breaking ts dynamt- cally determmed by ratzos of vacuum expectatton val-

ues We call them mfrared and ultravtolet mter-lock- mg since the higher and the lower local symmetry breakxng scales provide the ultraviolet and the infrared physics scales of the global symmetry breaklngs

We also point out that s~mllar phenomena anse m various sxtuatxons involving &fferent symmetry breaking scales, and study in detad two addmonal examples o f interest Pecce~-Qulnn symmetry in weak-scale supersymmetnc standard models [4] or two-Hlggs doublet standard models and global technlfermlon chlral symmetries m extended [ 5 ] walking techmcolor models [ 6 ] In particular, m the first example, we observe that the hierarchy of symmetry breaking scales cannot be smaller than 10- s in order for the Peccel-Qulnn mechamsm to solve the strong CP problem

2. Triplet majoron model

One example o f the infrared-ultraviolet interlocking phenomenon is the triplet Majoron model [ 3 ] m extensions of the standard electroweak model 's Hlggs sector The scalar sector o f the triplet Majoron model consists o f two Htggs fields ~ = (2, l, 0) and A= (3, 1, - 2 ) o f [SUL(2) XUy( 1 ) ]local and [ut( 1 ) ]global-lepton quantum numbers

The most general form of renormallzable Htggs po-

0370-2693/92/$0500© 1992ElsevlerSoencePubhshersBV All nghts reserved 141

Volume 277, number 1,2 PHYSICS LETTERS B 27 February 1992

tentxal lnvarlant under the gauge and global symmetries can be written as [ 7 ]

V[(~, Z~] = 2 l (~[~ t ~ _ _ V2)2+22(TrAtA_ VaZ)2

+23 ((P * ~ - V 2) (Yr AtA- V 2)

+a( cb t~ Tr d t , 4 - A t ~ t3)

+fl[ (TrA*A)Z-Tr(A*AA*A) ] (2 1 )

In addition, there exists a unique soft term that breaks the global u/( 1 ) lepton number [ 8 ]

W[q~, A] = ~(2V2q~*~+ V2w Trzf*A

- V a ~ T d * ~ + h c ) (2 2)

By the assumption of a small intrinsic breaking of the lepton number conservation, ~ is necessarily much smaller than the other coupling constants The poten- tial 1$ minimized at nonzero vacuum expectation values ( ~ ) = Vw and (A) = Va whenever the coupling constants satisfy

2 1 , 2 2 , a , f l > O ,

42122 --22 > 0 ,

E>0 ( 2 3 )

The particle spectrum in the broken phase consists of isotriplet Goldstone bosons absorbed into the longitudinal components of W and Z gauge bosons, two charged scalars, two scalar Hlggs and one pseudo- Goldstone boson The last one reduces to the original Majoron [ 3 ] in the limit e--,0

A large hterarchy of the symmetry breaking scales, Vn << Vw, is most easily obtained by decoupllng the two symmetry breaklngs a, f l=0 This simplifica- tion turns out not to modify our subsequent conclu- sions, however Let us denote a ratio of the symmetry breaking scales as 0= Vn/Vw << 1 In the original triplet Majoron model, this was motivated m order to keep the

P - M ~ - ~ 0 w - v ~ + 4 v ~

very close to unity to be compatible with experiments #1

~ Recent LEP data of Z-decay ruhng out the triplet Majoron model, we study the model only for theoretmal Interest and allow the symmetry breaking scale hierarchy to vary as much as we wish

Interested only in the Goldstone boson sector, we focus on neutral components of these pseudoscalar fields The electroweak symmetry breaking scale VW= V2( 1 +02) ~ V~ serves as a high-energy, ultraviolet cutofffor the spontaneous lepton number symmetry breaking Keeping 0 small implies a fine-tuning of the actual lepton number symmetry breaking physics to a low-energy, infrared scale Assuming the Nambu-Goldstone realization, we write ¢)= Vwexp(1Ol/Vw) and A= Va exp(102/V~) Therefore, from eq (2 2), we obtain a mass matrix M z of the pseudoscalars

M~,=4eV 2, M22VwVa, M22=eV 2 (24 )

The physical pseudo-scalar spectrum is obtained by dlagonahzlng the mass matrix One of them is a neutral, exact Goldstone boson S = (1 +402)-1/2× ( 0 1 + 2 0 ~ 0 2 ) , which IS absorbed to the longitudinal component of the massive Z~ gauge boson upon electroweak symmetry breaking The other is a pseudo- Goldstone boson O= ( 1 + 4 0 2 ) - 1/2 ( 0 2 - - 2001 ) with a mass

The mass formula of the pseudo-Goldstone boson, the Majoron, appears to Indicate a mass enhancement compared to a naive estimate M~GB ~ E V 2 unless Vw~ Va, the pseudo-Goldstone bosons are always heavier than a naive estimate based on the spontaneous global symmetry breaking scale Va and the ex- pilot symmetry breakmg scale cA It is in this sense that Lusignoh et al [2] concluded that there is an anomalously large enhancement of PGB masses

However, this behavtor is because the lntnnsic breaking Wbecomes more important than the poten- tial V whenever a fine-tuning exists between the in- terlocked symmetry brealong scales Vw >> Va In other words, the pseudo-Goldstone boson notion makes sense only if W[Vw, Va] ~< V[Vw, VA] This implies that 4 2 V~ i> e V~ Vz~, or equivalently

(Va)2=O z ( 2 6 ) ~ Vww

Alternatively, for a fixed c, one needs to have 0>/x/~ for the approximate symmetry to make sense As one

142


decreases the symmetry breaking scale hierarchy 0 to be smaller than x/Q, the lntnnsxc breaking begins to dominate and the Nambu-Golds tone mode notion breaks down In that case, a proper description is in terms of Wlgner modes even though the lsotnplet A develops a nonzero vacuum expectation value This transition between the Nambu-Golds tone mode and the Wlgner mode is also signalled by strongly Inter- acting Majorons as 0 is decreased to the cntlcal value x/Q Therefore, an "'effectwe'" mtrmstc breakmg stze ts dynamically de termmed to be E (Vw/Va) 2

Dynamics among the pseudo-Goldstone bosons is summarized by a low-energy effective lagranglan below the scale of the lepton number symmetry break- lng From the soft breaking term eq (2 2), we find the quartlc coupling among 0 PGBs

1 24o~ ~i ~ (2 7)

Therefore, we do find that the Majoron PGBs begin to interact strongly in the S-wave channel once their mass is increased through a hierarchical symmetry breakmg, 0 << 1 The bound that the PGBs interact weakly is derived as 0>_-x/Q, which is precisely the bound eq (2 6) that the Nambu-Golds tone mode description o f Majorons holds valid Therefore, the strong nonderlvatlve interaction among PGBs indicates a transition from Nambu-Golds tone mode to Wlgner mode After all, this is consistent with the prediction of the chtral dynamics an increase o f PGB mass is accompanied by an increase o f PGB nonderlvatlve Interaction strength Similarly, one finds also an enhanced n e u t n n o - n e u t n n o - M a j o r o n interaction [8 ] Relationship between Majoron mass, non- derivative coupling, and neutrino coupling is

MEon 2 2 2 = 4 V , t A 4 o = 2 ~ o V a , (2 8)

In complete agreement with chlral dynamics relations In the Wlgner mode description, only W contrib-

utes to the quadratic masses of scalar fields Dlagon- allzmg the mass matrix, one finds the mass o f the pseudo-scalar field to be

M2vs=E(V 2 + V 2) (2 9)

When the Intrinsic symmetry brealang parameter E satisfies the bound eq (2 6), one finds the naive es- timates MZs ~< ~ V 2 Moreover, the fact that the original lsotnplet field d has a nonzero vacuum expecta-

txon value is not so significant The UV scale is set by Vw, which is much higher than the lsotrlplet symmetry breaking scale V,~ Indeed, in the limit that Vao0, the mass o f lsotnplet pseudoscalar field excitation is

V L Therefore, one can say that the Wlgner mode and the Nambu-Golds tone mode coexist, and both descriptions are complementary to each other

3. Weak-scale supersymmetric standard models

Another situation we would hke to exemphfy IS the Hlggs sector o f the weak-scale supersymmetnc standard models [4,9] The model consists of two lsodoublet Hlggs scalar supermultlplets H1 and HE The scale o f the electroweak symmetry breaking is char- acterlzed by the two vacuum expectation values ( H 1 ) = Vl and ( H 2 ) = V 2 such that V~ + V~ ~ (247 GeV) 2, b a m n g a possibility o f nonzero sneutrlno VEV Solution to the hierarchy problem requires the supersymmetry breaking induced gravltlno mass m3/

2 ~ G~ 1/2 The most general gauge mvarlant superpotentlal involving the Hxggs superfields reads

W=f~( Q n l ) U c + fa( QH2) o c

+ f t (LH2)E~+aH1H2 (3 1 )

For concreteness, let us consider the model of Hall and Randall [ 11 ] in which the generalized matter parity [10] is UR(1) ~2 The UR(1) charges are de- fined as follows all standard model particles, 0, hlggsmos, - 1, gaugmos, + 1, squarks and sleptons, + 1 One interesting feature of this model is that the last term in the generic superpotentlal eq (3 1 ) is not allowed by UR ( 1 ) conservation as well as all possible baryon and lepton number violating dimension four operators Therefore, the/z-problem [ 11 ] is naturally solved in their models ~3 In this case, It is known that the superpotentlal possesses an extra, global upQ ( 1 ) Pecce l -Qumn (PQ) symmetry The PQ charges are

a2 This model is already ruled out at its minimal form from the recent LEP Z-data It may sUll be made consistent by relaxing some of the mlmmahty reqmrements given by Hall and Randall

a3 There are other suggested solutions to the #-problem [ 12,13 ] based on nonrenormahzable terms of 0 (M2/MpI) ,~ Gf t/2 reduced by a h~dden sector in many superstnng/supergrawty models

143


assigned as H,, + 1, H2, + 1, U ¢, - 1, D ¢, - 1, E ~, - 1 When the Hlggs field get VEVs, the PQ symmetry is spontaneously broken, and generate the well- known Peccel-Qulnn-Wemberg-Wflczek (PQWW) axmn [ 14 ] as an exact Goldstone boson

In the model of Hall and Randall, below the weak scale, there are also several soft supersymmetry breaking terms One of the soft terms allowed revolv- ing the Hlggs fields Is

L~oft =e ( V 2 [h212+ V 2 [hi 12- II1Vzhlh2 + h c ) ,

( 3 2 )

in whmh h,, 2 are the lowest components of Hlggs superfields H1,2 At the same t~me, this term also breaks exphcltly the aforementioned PQ symmetry Therefore, the soft breakmg term eq (3 2 ) generates a mass of pseudo-Goldstone boson, the PQWW axlon

After the spontaneous symmetry breaking, the physmal spectrum consists of two neutral and one charged Hlggs scalars, and one neutral pseudoscalar Assuming the Nambu-Goldstone mode is reahzed, the pseudo-scalar spectrum is found by setting hi = Vlexp(l(/~l/V1) and h2= V2exp(lO2/V2) in eq (3 2 ) Again, we are Interested in a large hmrarchy of VEVs, 0= V2/II1 << 1, while satisfying V 2 + V 2 (247GeV)2 ~4 One of the pseudoscalars is .~= ( 1 - 02)- i/2 (¢1 - 002), and is absorbed into the longitudinal component of massive Z~ gauge boson The other one is the PQWW axlon O= (I + 02) - 1/2 (¢2+0¢1) of mass

M~ =E( V21 + V~)-~EV2~ ( 3 3 )

The contnbutmn of gluons to the axlon mass through axial anomaly is neglected compared to eq (3 3 ) for the moment

In the hmlt 0--,0, the PQ charge of h: vamshes lin- early to 0 as well Therefore, the PQ symmetry break- lng is predominantly triggered by the h2 field whose VEV is much smaller than that of hi, the Hlggs field that tnggers the electroweak symmetry breaking Therefore, we may suspect that Ma 2 ~ V i 2 in this hmlt The exact mass formula eq (3 3) seems to sug- gest an enhancement of the PGB mass over the naive

We recall that the allowed hmrarchy range is rather hmlted, for example, from the recent LEP data on neutral partmle decay wlths of Z

esUmate by a factor of 1//0 2 Likewise, the nondenvatlve self-interaction of PQWW axlons is of order 240=E(0+1/0)2,~0 -2 Unless ~ 0 2, the PQWW axlons interact very strongly in the S-wave channel, overwhelming P-wave interactions being dictated by the chlral dynamics

This indicates that the physmal spectrum in the 0--,0 hmlt is reahzed by the Wlgner mode Indeed, m terms of the linear reahzatlon of both H~ggs fields Hi,2, we do find two degenerate, panty doublet scalar particles of mass e V~ This agrees with the upper mass bound of the PQWW axton, eq (3 3) As 0 is fine- tuned away unity, the strength of exphclt PQ symmetry breaking by the soft term eq (3 2) grows so large that the notion of approximate PQ symmetry does not hold any more We again emphasize that this happens only in the spontaneously broken phase of the Hlggs sector, and importance of the interlocking between the symmetry breaking scales

Since the coexistence of Nambu-Goldstone and Wlgner mode realizations of a global symmetry is due to a fine-tuned symmetry breaking scale hierarchy, one may mqmre whether the very same situation arises m nonsupersymmetnc standard models with two Hlggs doublets as well In such models, in order to prevent dangerously large tree-level flavor chang- mg neutral currents, one has to impose, for example, a discrete 7/2 symmetry [ 15] hi, uC--,-hl, - u ¢ or, eqmvalently, h2, d ¢, e ¢--, -h2, - d ¢, - e ¢ This symmetry forbids 7/2 odd quadratic Hlggs terms such as the soft breaking one in eq (3 2) In addmon, if we do not allow (hTh2)2 type quartlc couphngs of Hlggs fields, ~t is known that PQ symmetry emerges as an exact symmetry Of course, the PQ symmetry becomes anomalous due to QCD interactions between the PQWW axlons and quarks In this case as well, we do find a similar dependence of the PQWW axlon mass and couplings to matter and photon on the symmetry breaking scale hmrarchms The standard cur- rent algebra estimate of the PQWW axlon mass [ 16 ] gives

M2 = ( 0 + 0 ) 2 ( Z + 1 + 2 ) -l(\247f~m~oGeVJ~2

~ ( 0 + 0)2 (74 keV) 2 ( 3 4 )

in whmh Z=- (mu/ma) . . . . . t It is seen that the QCD

144


anomaly-induced PQWW axlon mass is enhanced by ( 8 + 1 / 8 ) By examining the Yukawa coupling of Hlggs fields, one also finds similar enhancements of axlon-quark (or axlon-hadron), axlon-lepton and ax~on-photon couplings [ 17 ], consistent w~th the chxral dynamics predictions In turn, this lmphes a fine-tuned, large symmetry breaking scale hierarchy is ruled our purely from the point of v~ew of attempt- ing to solve the strong CP problem through the PQWW axlon, apart from all other existing phenom- enologlcal constraints as the h~erarchy is increased too much, the intrinsic PQ symmetry breaking becomes overwhelming and the axlon solution to the strong CP problem does not work any more Indeed, imposing the S-wave channel Interaction be less than order unity, we find that

4

l >~47r(0+ 0 ) (f~m~°)2 8-410-12 (247 GeV) 4 "~ (3 5)

Therefore, the symmetry breaking scale hierarchy purely from consideration of the ax~on solution to the strong CP problem is hmlted to 0>i 10 -3 This is m- tmtlvely consistent, since axlons of heavier mass than the QCD scale can be integrated out of the low-energy effective lagrangmn of hadrons, thus cannot re- lax 0 to zero It ~s amusing to compare th~s with the ratio of the second generation or third generation quark mass ratio ~ ~ , inferred from the recent top quark search constraint of CDF

The invisible axlon models of Dine, Fxschler and Sredmckl and of Zhlnltsky [ 18 ] consist of two lSO- doublet Hlggs fields and one smglet field, all of which carry nonzero PQ charges In this case, however, the PQ scale is predominantly determined by the VEV of the single field Being a stagier field, it is stable against any hierarchies of local symmetry breaking scales from the lsodoublet Hlggs fields Therefore, there is no hierarchy dependence of the invisible axlon mass On the other hand, its coupling to fermlons or pho- tons may still depend on the hierarchy since these are determined by the electroweak scale physics

4. Extended walking technicolor theories

As the last example, we consider extended [ 5 ] walking technicolor models [ 8 ] The basic idea be- hind the walking technicolor models ~s to arrange the

technicolor gauge group and the technlferm~on rep- resentaUon content so that the techmcolor beta func- tion is small enough This enables one to achieve Fxo the techmplon decay constant, kept near infrared scale -~247 GeV, while the technlfermlon condensate ( Q Q ) is enhanced toward an ultraviolet scale Aero the extended technicolor scale The sideway Interac- tions of ETC generate four-fermlon (and hlgher-fer- m~on as well) interactions Among them, we are par- tlcularly interested in

g2 L4 = A2-~Tc Tr JLJ~ + h c

g2 - AzT c daba#(aaQbO_.~Q'#) + h c ( 4 1 )

These four-fermlon operators would respect the [GTc×SUc(3) XSUL(2) XUr(1) ]1o~ gauge xnvan- ance; otherwise, they are completely general In par- tlcular, it may not respect the separate chlral flavor symmetries GF and GI: of techmfermlons Qa and Q~,, but be mvanant under a simultaneous rotation of GF~GF®G~ Therefore, at breaks generically the full technicolor flavor chtral symmetries Let us draw an analogy with the earlier case The spontaneous chlral symmetry brealong of technlfermlons is driven at two competing scales AETC and A-rc the first due to four-fermxon interactions Induced by the extended technicolor interactions and the second due to walking techmcolor gauge interactions On the other hand, a subgroup of the full technlfermlon chlral symmetry is also broken at the higher scale AETC Therefore, the size of the Intrinsic (subgroup of) chlral symmetry breaking is not too small (even though not too large if we fine tune the ETC interactions to be weak enough) In the low-energy effective lagranglan of technxpxons, the ETC reduced four-fermlon operators Induce nondenvatlve terms of the form

( Q Q ) ( Q ' Q ' ) Tr(2J*X') + h c ( 4 2 ) Lo ~g2 A2ETC

This term is mvarxant only to the subgroup of the otherwise chlral symmetry SUL(NF)×SUR(NF), In which only a diagonal choral rotation ,F,~La,SRtd and ,S' ~Ld, F,'Rtd survives as an exact chlral symmetry In other words, the Goldstone boson manifold ~s curved and the vacuum ahgnment should be studied within the unbroken chlral symmetry submanffold

145


In the walking technicolor theories, typically (QQ)~,A2TcFTc and ( Q ' Q ' ) 2 , ~AETcFTc These two technlplon decay constants need not be the same, and, in fact, have a soft hierarchy in many situations There are several sources to produce different technlpion decay constants First, If the technicolor representations of Q and Q' are different, the technicolor gauge theory gives different chtral symmetry breaking scales For example, in the renormalization- group-improved most attractive channel (MAC) analysis,

2n F~rCFxc ~ e x p [OI~TC-(--FTc) \ C2(R)(CE(R [_) - 1 ) ] ,

whtch can lead to a sizable hierarchy of chtral symmetry breaking scales if the second Casimtr lnvar- iants of the two different representations are quite dtfferent Of course, this hierarchy cannot be too large, smce (1) otherwise, the asymptotic freedom of the technicolor gauge theory ts lost and (2) there is not enough room for the coupling to run (or rather walk) between ATC and AET C

In such situations that there are two hierarchically different technlpion decay constants, the mass matrix of the pseudo-Goldstone boson derived from eq (4 2) depends upon the ration O--FTc/F~rc The pattern that pseudo-Goldstone boson masses depend on 0 hierarchically is stmtlar to the previous two examples, and leads to mass and self-couphng enhancements as 0 is increased

Yet another situation ts when the ultraviolet scale is provided by the ETC-lnteractton induced chlral symmetry breaking This will be the case when the diagonal ~¢'s are sizably larger than the off-diagonal ~¢'s, close to the Nambu-Jona-Laslno critical value Then, the chlral symmetry breaking lS driven largely by the ETC mduced four-fermton operators In thts case, F~rc ts driven up toward the ETC scale Of course, this situation is not desirable from the pomt of view of walking technicolor models Nevertheless, tt provides another mstance of theoretical situations that sizable hierarchical symmetry breaking patterns can arise dynamically

5. Conclusion

In thts paper we mvestlgated how an hlerarchl-

cally enhanced mass spectrum of pseudo-Goldstone bosons is possible We examined three examples in all of which there exists a hierarchy of symmetry breaking scales The pseudo-Goldstone boson mass, S-wave Interaction strength and fermton-Goldstone boson interaction strength are shown to depend on the hierarchy All of these quantities increase their size as the hierarchy becomes larger, and eventually loses the notion of Nambu-Goldstone mode since size of the intrinsic symmetry breaking begms to dominate In most cases we are famlltar with [ 1 ], intrinsic breakmgs of global symmetries are controlled by some constant input constant parameters, and are assumed to be small to render a notion of pseudo-Goldstone bosons What is novel in the examples we discuss ts that the parameters of mtnnsic breaking are not constants but determined by details of the vacuum altgn- ments and their hterarchtes

Acknowledgement

The author is grateful to T Appelqulst, R Barbl- erl, M Dine, L J Hall, L Krauss, W Marctano, M Soldate and M Suzuki for many helpful discussions He also thanks to L J Hall for making it possible to visit LBL where this work was initiated This work was supported in part by US Department of Energy under Contract DE-AC03-76SF00098, by the Na- tional Science Foundation under Grant PHY90- 21 139 and by the Texas National Research Labora- tory Commission

References

[I]S Wember$, Phys Rev Lett 29 (1972) 1698, Phys Rev D7 (1973) 2887, see also H Georgl and A Pals, Phys Rev D 12 (1975) 508, T Haglwaraand B W Lee, Phys Rev D 7 (1973)459

[2] M Luslgnoh, A Maslero and M Roncadelh, Phys Lett B 252 (1990) 247

[ 3 ] Y Chlkaslge, R N Mohapatra and R D Pecce:, Phys Lett B98 (1981) 26

[4] See, e g, M B Wise, Low-energy supersymmetry, lecture at TASI-87 (Santa Fe, NM, USA, 1987), L J Hall, Weak scale supersymmetry, lecture at Intern Schol of Subnuclear Physics (Enee, Italy 1990)

[5 ] E Elchten and K Lane, Phys Lett B 90 (1980) 125

146


[6]B Holdom, Phys Lett B 150 (1985) 301, T Appelqmst, D Karabah and L C R Wijewardlmna, Phys Rev Lett 57 (1986) 957, T Appelqulst and L C R WIjewardhana, Phys Rev D 35 (1987) 774, D 36 (1987) 568

[7] H Georgl, S L Glashow and S Nussmov, Nucl Phys B 193 (1981) 297, G B Gelmml and M Roncadelh, Plays Lett B 99 (1981) 411

[8] L -F LI, Y Lm and L Wolfenstem, Phys Lett B 158 (1985) 45

[9] L J Hall and L Randall, Phys Rev Lett 65 (1990) 2939, Nucl Phys B 352 (1991) 289

[10] S Dlmopoulos and H Georgl, Nucl Phys B 193 (1981) 150

[ 11 ] J E Kam and H P Nflles, Phys Lett B 138 (1984) 150 [ 12] J E Kam and H P Nllles, prepnnt MPI-PAE/PTh 16/91,

E J Chun, J E Kam and H P Nllles, prepnnt TUM-TH- 124/91, SNUTP 91/25 ( 1991 )

[ 13] M Dine, private commumcatlon ( 1991 ) [14] R D PeccelandH Qulnn, Phys Rex, Lett 38 (1977) 1440,

Phys Rev D 16 (1977) 1791, S Wemberg, Phys Rev Lett 40 (1978)223, F Wdezek, Phys Rev Lett 40 (1978) 279

[15 ]SL Glashow and S Wemberg, Plays Rev D 15 (1977) 1958

[ 16] W Bardeen and S -H H Tye, Phys Lett B 74 (1978) 229 [17]See, e g , J E Kam, Phys Rep 150(1987) 1 [ 1 8 ] J E Kam, Phys Rev Lett 43 (1979) 103,

M Shlfman, V I Vamstem and V I Zakharov, Nucl Phys B 166 (1980) 4933, M Dine, W Flschler and M Sredmckl, Plays Lett B 104 (1981) 199, A P Zhmltsky, Soy J Nuel Phys 31 (1980)260

147

Documents

Enhanced mass spectrum of pseudo-Goldstone bosons?