PHYSICAL REVIEW D, VOLUME 59, 075009 Higgs- and Goldstone-boson-mediated long range forces F. Ferrer Grup de Fı´sica Teo`rica and Institut de Fı´sica d’Altes Energies, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Barcelona, Spain M. Nowakowski Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil ~Received 9 November 1998; published 4 March 1999! In certain mild extensions of the standard model, spin-independent long range forces can arise by exchange of two very light pseudoscalar spin-0 bosons. In particular, we have in mind models in which these bosons do not have direct tree level couplings to ordinary fermions. Using the dispersion theoretical method, we find a 1/r 3 behavior of the potential for the exchange of very light pseudoscalars and a 1/r 7 dependence if the pseudoscalars are true massless Goldstone bosons. @S0556-2821~99!00509-3# PACS number~s!: 14.80.Mz, 11.10.Wx, 11.55.Fv, 11.80.Fv av a ith g - ot f F ro e lt gh n s er o o in ur - a al ve e s ar- her ce, ge r- ight s of ars. en tan- le esult o the the - elds le to the e- w o I. INTRODUCTION Most studies and investigations on long range forces h always centered, for obvious reasons, around the electrom netic and gravitational interaction. However, starting w the very early example of the Casimir-Polder long ran force @1#, over the Feinberg-Sucher force@2# mediated by two neutrinos~see Fig. 1! and finally going to recent devel opments in supersymmetry and superstrings@3#, there has been continuous interest in effects and detection of ex long range forces@4#. The actual applicability or relevance o these forces is, of course, different from case to case. instance, the Casimir-Polder force is, in principle, of elect magnetic origin. It arises as a consequence of photon change between polarizable neutral systems and the resu potential has a 1/r 7 dependence at long distances. Althou the Casimir-Polder force has been recently detected i laboratory experiment@5#, the neutrino mediated~i.e., in- volving weak interaction couplings! Feinberg-Sucher force i too weak to be of any significance in Earth-based exp ments. If at all, a suitable arena for this force would be astrophysical and/or cosmological dimension~see for in- stance in this respect@6# and references in@7#!. The result of Feinberg and Sucher has been recently extended to als count for the exchange of very light Dirac@8# and Majorana @9# neutrinos. Temperature-dependent corrections includ the exchange of thermalized neutrinos at finite temperat such as the relic cosmic neutrinos atT21;1 mm, have been calculated in@10,7#. Finally let us mention that exten sions of the standard model can allow, in principle, for variety of different long range forces@4#, mediated, for in- stance by very light or massless scalars or pseudosc @11#. The former force acting between neutrinos themsel has been discussed, e.g., in@12#. The potential due to the exchange of two pseudoscalar particles~box diagrams! was computed in@13,14#. Furthermore, new exotic long rang forces can appear also in the context of gauge mediated persymmetry breaking and in superstring theories@3#. The implications of a new long range force due to an extraU(1) gauge group have been discussed recently by Fayet in@15#. 0556-2821/99/59~7!/075009~7!/$15.00 59 0750 e g- e ic or - x- ing a i- f ac- g e, ars s u- Should neutrinos be the only massless or very light p ticles in spectrum of the theory, then the Feinberg-Suc result would be the only possible exotic long range for regardless of the model. This is clear since all long ran forces~including the electromagnetic and gravitational inte action! arise as a consequence of an exchange of very l quanta. However, as mentioned above, many extension the standard model also predict very light pseudoscal Usually diagrams involving two such pseudoscalars will th result in a spin-independent long range force between s dard fermions@13,14#. Recall that an exchange of a sing pseudoscalar between fermions gives a spin-dependent r for the potential@4#. Indeed, a covariant calculation with tw pseudoscalars exchange has been recently performed in@14# in the context of a generic theory where the coupling of pseudoscalarf to fermions is taken either asfc̄g5c or, alternatively, as the derivative version (]mf)c̄g5gmc. In the latter casef can represent a generic Goldstone boson. In first case the authors obtain a 1/r 3 dependence of the poten tial whereas the double exchange of Goldstone bosons yi a more drastic fall-off, viz. 1/r 5. However, very often, i.e., in a wide class of models, these pseudoscalars do not coup standard fermions~often not even to gauge bosons! on ac- count of some symmetry arguments~see the Appendix where one such model is briefly sketched!. However they do always have a tree level coupling to Higgs-scalar particles of theory. Indeed, it is difficult to imagine a reasonable symm try argument which would forbid such couplings. We no FIG. 1. One of the diagrams in the S.M. giving rise to the tw neutrino force in four-Fermi effective theory. ©1999 The American Physical Society09-1 er ra on Fi t e ll r n re en th er ar r on so e te th em s o r ar n l in e tio fo i te er al rm end l- fer e e y rity tion r s ou- is ct two the the f ve - e- re F. FERRER AND M. NOWAKOWSKI PHYSICAL REVIEW D59 075009 assume that the scalars themselves couple to standard f ons, which is the case in most models. If so, then the diag in Fig. 2 displays a very nice analogy to the diagram resp sible for the Feinberg-Sucher force~see Fig. 1!. Indeed, we have replaced only fermions by bosons when comparing 1 with Fig. 2. Of course, one expects a differentr depen- dence of the potential arising from the two diagrams due different dimensionality of the coupling constants. If th pseudoscalars have both couplings, to the fermions as we to the Higgs scalars, the result of@14# and our paper should then be added. Since the coupling of the Higgs scala fermions is usually proportional to the mass of the fermio one may suspect that the box-diagrams using the di pseudoscalar-fermion coupling are more important. In g eral this is model dependent, but we can safely state here the pseudoscalar fermion coupling constant is also ‘‘exp mentally’’ restricted by arguments of energy loss in st where one assumes that the bulk of energy of the sta carried away by the standard mechanism in form of phot and neutrinos@16#. If we assume that the pseudoscalar is a Goldstone bo a connection to theU(1) forces considered in@15# can be possibly made as the latter display a ‘‘Goldstone-like’’ b havior as theU(1) coupling approaches zero@15#. The paper is organized as follows. In Sec. II we calcula using dispersion theoretical methods@17#, the long range force due to the diagram in Fig. 2 where we assume that coupling between the Higgs scalar~H! and the very light pseudoscalar~a! is linear and of the formHaa. We also briefly touch upon some issues concerning a possible t perature dependence of the potential. In the subsequent tion we change the linear coupling to a derivative version the formH(]ma)(]ma). In Sec. IV we discuss the particula case of Goldstone bosons exchange. In Sec. V we summ our results. II. LONG RANGE FORCES DUE TO PSEUDOSCALAR-PSEUDOSCALAR-SCALAR NONDERIVATIVE COUPLINGS The dispersion theoretical technique of calculating lo range forces in quantum field theory is reviewed in detai @17#. This method is especially suitable to cope with high order diagrams and relativistic effects and its implementa to compute the neutrino pair exchange force is straight ward @2#. The results agree with the computations done @18# by performing the Fourier transform of the associa FIG. 2. Pseudoscalar mediated long range force without di fermion coupling. 07500 mi- m - g. o as to , ct - at i- s is s n, - , e - ec- f ize g r n r- n d Feynman amplitude in momentum space~this latter strategy is only applicable in general when there is no lower ord long range force and relativistic corrections are negligible!. According to the rules of the dispersion theoretic method we must compute the following Laplace transfo ~we restrict here ourselves to central forces which dep only on the distancer[ur u between the two particles!: V~r !5 2 i 8p2r E4ma 2 ` dt@M# t exp~2At r !, ~2.1! where the integration variablet stands for the usual Mande stam variable which equals the four-momentum trans squared,q2. Here, @M# t denotes the discontinuity of th Feynman amplitude~i.e., the absorptive part of the sam! across the cut in the realt axis and is best computed b taking advantage of the analyticity and generalized unita properties leading to the Cutkosky rules@17#. Let us now consider the case of some generic interac terms of the form Lint5g H f f f̄ f H, L int8 5g Haa aaH, ~2.2! where f are standard fermions,H is the heavy Higgs scala with massmH anda is the very light pseudoscalar with mas ma . We can essentially neglect here possible quartic c plings of the formH2a2 as self-energy corrections due to th quartic coupling would only eventually give rise to conta interactions. It is convenient to define global coupling constants as G[ g H f f g Haa mH 2 , G8[ g H f 8 f 8 g Haa mH 2 , ~2.3! which capture the constants of the four vertices and the Higgs propagators in Fig. 2. For future reference we draw reader’s attention to the fact that we have expanded Higgs propagators inq2 and kept only the zeroth order o this expansion; this then gives themH 2 in the denominators of G and G8 in Eq. ~2.3!. The full matrix element of the dia- gram in Fig. 2 is given by M522 i GG8G@ ū~p18!u~p1!ū~p28!u~p2!#. ~2.4! The one-loop integral is represented above byG, i.e., G[E d4k ~2p!4 i k22ma 21 i e i k̄22ma 21 i e , k̄5k2q, q5p12p185p282p2 , q25t. ~2.5! We assume also the nonrelativistic limit in which we ha ū(p18)u(p1)5ū(p28)u(p2).1. Using the prescriptions aris ing from generalized unitarity, which amount to the replac ment, ct 9-2 e q. e t su n io e ur in - e um d rd e hod nly h to HIGGS- AND GOLDSTONE-BOSON-MEDIATED LONG . . . PHYSICAL REVIEW D 59 075009 1 k22ma 21 i e →22p id~k22ma 2!u~k0!, ~2.6! we obtain for the discontinuity @G# t5 1 ~2p!2E d4k ~2p!4d~k22ma 2!d~ k̄22ma 2!u~k0!u~ k̄0! 5 1 8p A12 4ma 2 t . ~2.7! Obviously we have@M# t522 i GG8@G# t which has to be inserted into Eq.~2.1! to compute the final expression of th potential: V~r !52 GG8 32p3r E 4ma 2 ` dtA12 4ma 2 t exp~2At r ! 52 GG8ma 8p3r 2 K1~2mar !, ~2.8! whereK1 is a modified Bessel function. To show that E ~2.8!, for a very small massma , yields indeed a long rang potential, let us take the limitma→0 in Eq.~2.8! ~equivalent to rma!1). For the leading order of the expansion we ge V~r !.2 GG8 16p3r 3 . ~2.9! For comparison we quote below the Feinberg-Sucher re for massless neutrinos@2# VFS~r !5 GF 2gvgv8 4p3r 5 , ~2.10! whereGF is the Fermi andgv andgv8 weak vector coupling constants. Note that, in contrast to Eq.~2.9!, the Feinberg- Sucher force~2.10! is repulsive. This difference betwee these two forces is due to an extra minus sign for the ferm loop in Eq.~2.10!. We would like to touch at this point briefly upon finit temperature corrections to Eqs.~2.8! and ~2.9!. In doing so we will follow mainly @10# and @7# to which we refer the reader for more details on this subject. At finite temperat T the spin-0 boson propagatorST(k) takes the form ST~k!5 1 k22ma 21 i e 22ipd~k22ma 2!n~T!, ~2.11! 07500 lt n e where n(T) is the particle distribution function with the chemical potential already set to zero. As noted explicitly @10#, the propagator~2.11! is sufficient to calculate the prob lem at hand.1 We will restrict ourselves to Boltzmann distributions n~T!5exp@~2Ek!/T#, ~2.12! in which Ek is the energy. To calculate the potential itself w use now the method of Fourier transforming the moment amplitude, i.e., VT~r !5E d3Q ~2p!3 exp~ iQr !MT~Q! 5 1 2p2r E0 ` dQ QMT~Q!sinQr, ~2.13! where in the static limit we haveq.(0,Q) and in the second equality we have definedQ5uQu and r 5ur u. The second expression in Eq.~2.13! holds for potentials which depen only on r. As before, we can write effectivelyMT. 22iGG8GT such thatGT is the one loop integral involving two ‘‘cross’’ products of two propagators, one the standa vacuum part and the other thermal part, viz., GT5E d4k ~2p!42ipd~k22ma 2!n~T! 3S 1 ~k1q!22ma 2 1 1 ~k2q!22ma 2D . ~2.14! GT can be further evaluated to be GT5 4i ~2p!2E 0 ` dk k2 Ak21ma 2 exp~2Ak21ma 2/T! 3E 21 1 dz 1 4k2z22Q2 , ~2.15! where nowk5uku. Recalling thatMT522iGG8GT and in- serting this into Eq.~2.13! and subsequently performing th integration first overQ and then overz we get 1We depart for a moment from the dispersion theoretical met and use, following@10# and@7#, the traditional Fourier transform to compute theT-dependent effects. In such a situation we need o the real part of the amplitude correctly given by using Eq.~2.11! ~see Ref. @10#!, which is the 121 component of the full 2-dimensional matrix propagator used in the real time approac finite temperature field theory@19#. 9-3 q. its le e e e u pa e he rc th tu en t lar a s ce ng se ak - he- to t is he ue tion t re- w- ad- the a eu- fer- F. FERRER AND M. NOWAKOWSKI PHYSICAL REVIEW D59 075009 VT~r !52 GG8 4p3 1 r 2E 0 ` k dk Ak21ma 2 exp~Ak21ma 2/T!sin~2kr ! 52 GG8 2p2 1 r Tma A11~2rT !2 K1S ma T A11~2rT !2D . ~2.16! Equation ~2.16! is the finite temperature correction to E ~2.8!. It is instructive at this stage to examine different lim of Eq. ~2.16!. First, let us consider the casema→0 as done in Eq. ~2.9! for the vacuum contribution. We get the simp result VT~r !.2 GG8 2p3 1 r T2 11~2rT !2 . ~2.17! Using the last limit~i.e., ma→0) we can also investigate th ranger @T21. In this range~2.17! can be expanded to give VT~r !.2 GG8 8p3r 3 . ~2.18! At these distances, long compared to the inverse temp ture, we can add now to the vacuum part~2.9! Eq. ~2.18! to arrive at the complete answer for the potential Vtot~r !5VT~r !1V~r !.2 3 16 GG8 p3r 3 . ~2.19! This last result is particularly interesting when we compar with the corresponding result in the context of the two ne trino force, calculated at zero and finite temperature@7#. In the neutrino case the total sum consisting of the vacuum and the finite temperature contribution@i.e., an equation cor- responding to Eq.~2.19!# switches the sign of the force in th range r @T21, a repulsive force becomes attractive in t presence of relic neutrinos@7#. This is a quite interesting result which sheds new light on the Feinberg-Sucher fo The reason why a similar reversal does not take place in two boson force@cf. Eq. ~2.19!# ~i.e., why this attractive force does not become repulsive when we add tempera corrections! is due to the fact that the relative sign betwe the vacuum part of the propagator and the thermal par plus in the boson propagator@cf. Eq. ~2.11!# whereas it is minus for fermions@19#. Although the temperature of the very light pseudosca at the present epoch, provided of course these pseudosc exist, is model dependent, it should be comparable~at least in the order of magnitude! to the temperature of relic axion @20# or Majorons@21#. III. THE CASE OF DERIVATIVE COUPLINGS In this section we will also compute the dispersion for arising from Fig. 2, considering however a different coupli scheme between the heavy Higgs scalars and the light p doscalars. For the relevant Lagrangian interaction we t now @22# 07500 ra- it - rt e. e re is s lars u- e L int9 5g̃ Haa H~]ma!~]ma!. ~3.1! To simplify things, we will also start right from the begin ning considering massless pseudoscalars~instead of taking the limit ma→0 at the end of the calculation!. We define also overall couplings in analogy to Eq.~2.3!: G̃[ g H f f g̃ Haa mH 2 , G̃8[ g H f 8 f 8 g̃ Haa mH 2 . ~3.2! As in the preceding section we start with the dispersion t oretical definition of the potential, i.e., Eq.~2.1! where we denote now the matrix element byM̃ given by M̃.22iG̃G̃8•G̃ , G̃5E d4k ~2p!4 i k2 i k̄2 ~k• k̄!2, ~3.3! where as beforek̄5q2k. The rest of the calculation follows essentially on the same lines as in Sec. II. First we have calculate the discontinuity@M̃# t}@G̃# t and insert the resul into Eq. ~2.1!. For the discontinuity we obtain @G̃# t5 qmqn ~2p!2E d4kd~k2!d~ k̄2!kmkn 5 qmqn ~2p!2 p 2 F1 3S qmqn2 1 4 gmnq2D G5 t2 32p ~3.4! with q25t as usual. Calculating the integral transform of th discontinuity remains. To distinguish the potential from t results in the preceding section we will call the potential d to two pseudoscalar exchange arising from the interac ~3.1!, Ṽ. For the latter we get Ṽ~r !52 G̃G̃8 128p3r E 0 ` dt exp~2At r !t252 15G̃G̃8 8p3r 7 . ~3.5! If we compare this expression with the potential~2.9! it be- comes clear that it is theq45t2 dependence of@G# t which gives here the steep fall-off proportional to 1/r 7. In Eq. ~2.9! the corresponding integrand, i.e.,@G# t was simply a constan ~for ma50) giving rise to a milder 1/r 3 dependence. In principle, one could now also calculate temperatu dependent effects as we have done in Sec. II. We will, ho ever, not dwell further on this subject here and instead dress in the next section the interesting question of potential due to the exchange of two Goldstone bosons. IV. LONG RANGE FORCES DUE TO PHYSICAL GOLDSTONE BOSONS In the two preceding sections we have calculated in rather model-independent way the potential due to two ps doscalar exchange according to Fig. 2 and using two dif 9-4 tr fo g o M - pt av of th um u m ng m II. e th tly f cl e e th t tia ad d th n- tive c. q. ible t x- rem cho- ion u- mes q. n- ow o- n HIGGS- AND GOLDSTONE-BOSON-MEDIATED LONG . . . PHYSICAL REVIEW D 59 075009 ent interaction Lagrangians,~2.2! and ~3.1!. Here we would like to address the situation when the pseudoscalar is a ~i.e., strictly massless! Goldstone boson. In the literature one can find numerous papers where Goldstone bosons either the linear scheme~2.2! is used or the derivative one as in Eq.~3.1!, very often with the insis- tence that, for Goldstone bosons, the derivative couplin the correct one. We will examine the two Goldstone bosons potential n in a general model, but using as an example the singlet joron model@23#, briefly sketched in the Appendix. The Ma joron J ~we change the notation here,a→J) is a true Gold- stone boson due to spontaneous breaking of the le number. The two different couplings discussed above h been derived explicitly in the appendix. Equation~A6! cor- responds to the linear scheme whereas Eq.~A8! to the de- rivative one. Also note that, apart from the explicit form the couplings, we can use from now on the results from two preceding sections. Since in the singlet Majoron model the physical spectr consists of twoheavyscalarsH andS and themasslessMa- joron J, instead of one diagram as in Fig. 2, we have fo distinct amplitudes corresponding to the four possible co binations of the heavy scalars, i.e., to the excha HH,SS,HS, andSH. Let us first investigate in detail the linear coupling sche ~A6! which would then fall in the general domain of Sec. All we have to do now is to use the result~2.8! and replace the general couplingGG8 by the concrete example from th Appendix. As mentioned before, we have to sum over different possibilities of heavy scalar exchanges, i.e., ~GG8! Majoron 5 ( P,P85H,S g P f f g P8 f f g PJJ g P8JJ mP 2mP8 2 . ~4.1! Although the coupling of Higgs scalars is not always stric proportional to the fermion mass~for instance, in case o nucleons it also depends on the gluon content of the nu ons! we will use here, as an example, the coupling ofH and S to fundamental fermions. In the singlet Majoron mod they are given byg H f f 52 i (A2GF)1/2mf cosu and g S f f 5 2 i (A2GF)1/2mf sinu. The coupling constants among th spin-0 bosons can be read off from Eq.~A6!. Taking all this into account we obtain ~GG8! Majoron 50. ~4.2! This, of course, does not imply that the potential due to exchange of two Majorons is zero. It means, however, tha is not of the simple 1/r 3 dependence as indicated in Eq.~2.9!. In order to get a meaningful nonzero result for the poten ~due to Majorons!, we have to go one step more in theq2 expansion of the heavy Higgs propagators. We alre stressed in Sec. II that the results presented there are vali the zeroth order expansion, i.e., fully neglecting theq2 in the heavy Higgs propagators. In other words, this means (GG8) Majoron 5(GG8) Majoron (q250)50. The next term in the expansion is 07500 ue r is t a- on e e r - e e e e- l e it l y for at ~GG8! Majoron ~q2![ ( P,P85H,S g P f f g P8 f f g PJJ g P8JJ ~q22mP 2 !~q22mP8 2 ! . GF 2mf mf 8 2 sin2u cos2u tan2 b 3S 1 mH 2 2 1 mS 2D 2 q4. ~4.3! Since the relevant integrand in the form of@G# tuma50 @cf. Eq. ~2.8!# does not give any furtherq2 dependence~it is a con- stant!, the q45t2 term from Eq.~4.3! is the only one to be integrated over. This, of course, resembles theq4 depen- dence in Eq.~3.4!. Indeed, the final expression for the pote tial reads VJJ~r !52 15Gf 2mfmf 8 16p3r 7 sin2~2u!tan2bS 1 mH 2 2 1 mS 2D 2 ~4.4! and has remarkably the samer dependence as Eq.~3.5!. Let us now repeat the steps from above for the deriva coupling scheme~3.1! discussed in the general setting in Se III and given specifically for the singlet Majoron case in E ~A8!. The equation corresponding to Eq.~4.3! reads in this scenario as follows: ~G̃G̃8! Majoron ~q2![ ( P,P85H,S g P f f g P8 f f g̃ PJJ g̃ P8JJ ~q22mP 2 !~q22mP8 2 ! . GF 2mfmf8 2 sin22u tan2bS 1 mH 2 2 1 mS 2D 2 1••• .~G̃G̃8! Majoron ~q250!, ~4.5! i.e., a nonzero result of the expansion here is already poss at the lowest order. Inserting this into Eq.~3.5! we confirm, however, the result~4.4!. This is mainly due to the fact tha (GG8) Majoron (q2) has the sameq2 dependence as@G̃# t . The equivalence of the two coupling schemes, Eqs.~A6! and ~A8!, in calculating the potential due to Majorons e change is a particular example of a more general theo which states that physical results cannot depend on the sen parametrization of the fields@24#. Recall that Eq.~A6! follows directly from choosing the representation~A2! whereas Eq.~A8! is a consequence of the representat ~A7!. Although we have used a particular model in our comp tations, we expect the 1/r 7 behavior to hold for a generic Goldstone boson. From the equivalence of the two sche which allows us to employ nonlinear representations like E ~A7! where they can simplify the calculations and the ge eral properties of decoupling of Goldstone bosons at l energies@25# we conclude that for Goldstone bosons the p tential will always behave as 1/r 7 and the vanishing of the coefficient in front of the 1/r 3 term that only appears whe 9-5 hi , a tiv n th d te igh ta at fo e m s a tt on e ac w s ld / do e- I on u to - s- h a he er io - t the ten m, lep- e ro- e e sted ep- F. FERRER AND M. NOWAKOWSKI PHYSICAL REVIEW D59 075009 using non-derivative couplings is not a coincidence of t particular model. As the calculations in this section show low energies it is more direct and advisable to use deriva couplings whereas the alternative~A2! requires more com- plicated calculations involving cancellations of consta terms to render the same results. V. CONCLUSIONS We have calculated the long range potentials due to exchange of very light or massless pseudoscalars using persion theoretical methods. In particular, we investiga these long range potentials in models where the very l pseudoscalars do not have a tree-level coupling to the s dard fermion. The only possible diagram which in coordin space can then result in long range potentials displays a mal resemblance to the diagram responsible for the two n trino Feinberg-Sucher force. Indeed, the formal difference of fermions versus bosons in the loop. In Sec. II we co puted the long range potential for very light pseudoscalar the linear coupling scheme and also examined some an gies and differences to the Feinberg-Sucher force. The la included some investigation on finite temperature correcti to the potentials. The potential in this case falls off as 1/r 3. In the following section we performed a very similar exercis but considering a derivative coupling scheme for the inter tion between heavy scalars and pseudoscalars. Finally presented a nice equivalence of both coupling scheme calculating the potential due to the exchange of true Go stone bosons. Here the fall-off is much steeper, namely 1r 7. As far as the latter is concerned we add that a 1/r 5 depen- dence is possible, via box diagrams, provided the pseu calars have tree level couplings to fermions. ACKNOWLEDGMENTS This work was partially supported by the CICYT R search Project AEN98-1093. F.F. acknowledges the CIR for financial support. M.N. would like to thank Fundac¸ão de Amparo àPesquisa de Sa˜o Paulo~FAPESP! and Programa de Apoio a Nu´cleos de Exceleˆncia ~PRONEX!. APPENDIX We present below the simplest version of a Major model which is a physical Goldstone boson in the spectr of the theory associated with spontaneous breakdown of lepton numberL @23#. This model, known as a singlet Ma joron model, became well known in connection with invi ible Higgs boson decays@26#. We emphasize that althoug the details will be given here for this particular model, variety of similar models exist. The usual motivation behind a Majoron model lies in t choice of the Majorana mass term. The latter can be eith bare mass term,mMnR TCnR , violating explicitly the lepton number or an interaction term of the formhwnR TCnR which conservesL. The fieldw is aSU(2)^ U(1) complex singlet with L522 which acquires a nonzero vacuum expectat 07500 s t e t e is- d t n- e r- u- is - in lo- er s , - e in - s- T m tal a n value^w&5w/A2 giving rise to a Majorana mass term~h is a dimensionless parameter!. The scalar potentialV(F,w) contains besides the stan dard Higgs doubletF the singletw. The potential is of the form V~F,w!5m1 2~F†F!1m2 2~w* w!1l1~F†F!2 1l2~w* w!21l12~F†F!~w* w! ~A1! such that it conserves the lepton number. We choose firs linear representation for the fields F5S G1 v A2 1 f1 iG0 A2 D , w5 w A2 1 s1 iJ A2 , ~A2! whereG1 and G0 are nonphysical Goldstone bosons ea up by the gauge bosons according to the Higgs mechanisJ is the physical one~Majoron! and v and w are the corre- sponding vacuum expectation values triggering e.w. and ton number S.S.B. After minimization of the potential th mass matrix of the two scalar particles reads ~f s!S l1v2 l12 2 vw l12 2 vw l2w2 D S f s D 5 1 2 mH 2 HH1 1 2 mS 2SS, ~A3! whereH andS are the mass eigenstates obtained by the tation S H SD 5S cosu 2sinu sinu cosu D S f s D . ~A4! Equations~A3! and ~A4! can be combined to deduce th following set of equations: 2l1v25cos2umH 2 1sin2umS 2 , 2l2w25sin2umH 2 1cos2umS 2 , 2l12vw5sin 2u~mS 22mH 2 !. ~A5! Equation~A5! is useful to extract the vertices in terms of th angleu and the scalar masses. We are especially intere here in the trilinear verticesHJ2 andSJ2. They are given by the interaction Lagrangian L int ~1!5 ~A2GF!1/2 2 tanb@mS 2 cosuS2mH 2 sinuH#J21••• ~A6! whereGF is the Fermi coupling constant and tanb5v/w. For comparison, let us also make use of a nonlinear r resentation for the singlet fieldw, viz., 9-6 ve th le - ent the rs also ip- ded el the by nts. HIGGS- AND GOLDSTONE-BOSON-MEDIATED LONG . . . PHYSICAL REVIEW D 59 075009 w5 1 A2 ~w1s8!exp~ iJ/w!. ~A7! The componentsf ands8 will now mix to give the physical scalarsH andS@as in Eqs.~A3! and~A4!#. So far, there is no difference with respect to the linear representation. Howe in the nonlinear representation the interaction terms of Majoron J with the scalars will get generated in the sing kinetic term (]mw* )(]mw) which after rotation to the physi cal scalars gives L int ~2!5~A2GF!1/2 tanb@cosuS2sinuH#~]mJ!~]mJ!1•••. ~A8! S E B s 07500 r, e t As mentioned before, there exist a wide class of differ Majoron models invoking slightly differentU(1) symme- tries to be spontaneously broken. The latter can be either lepton number, a combination of individual lepton numbe or a family symmetry. We refer the reader to@27# for a short account of these models and references. We mention that some, previously popular Majoron models , like the tr let model or the doublet model have been, by now, exclu in their simplest versions through LEP data~through the ab- sence of the decay channelZ→J1Higgs). However, more complicated version~mostly in conjunction with a singlet! can be still viable. Also note that Majoron models which predict a tree lev coupling to ordinary matter are severely constrained by argument of energy loss in stars possibly carried away Majorons. A singlet Majoron model evades these constrai cs ms iss, ett. , k nd @1# H. B. G. Casimir and P. Polder, Phys. Rev.73, 360 ~1948!; E. M. Lifschitz, Zh. Éksp. 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