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<ul><li><p>PHYSICAL REVIEW D 70, 115009 (2004)Planck-scale effects on global symmetries: Cosmology of pseudo-Goldstone bosons</p><p>Eduard Masso, Francesc Rota, and Gabriel ZsembinszkiGrup de Fsica Teorica and Institut de Fsica dAltes Energies, Universitat Autonoma de Barcelona</p><p>08193 Bellaterra, Barcelona, Spain(Received 30 April 2004; published 14 December 2004)1550-7998=20We consider a model with a small explicit breaking of a global symmetry, as suggested by gravitationalarguments. Our model has one scalar field transforming under a nonanomalous U1 symmetry, andcoupled to matter and to gauge bosons. The spontaneous breaking of the explicitly broken symmetry givesrise to a massive pseudo-Goldstone boson. We analyze thermal and nonthermal production of this particlein the early universe, and perform a systematic study of astrophysical and cosmological constraints on itsproperties. We find that for very suppressed explicit breaking the pseudo-Goldstone boson is a cold darkmatter candidate.</p><p>DOI: 10.1103/PhysRevD.70.115009 PACS numbers: 14.80.Mz, 11.30.Qc, 95.35.+dI. INTRODUCTION</p><p>It is generally believed that there is new physics beyondthe standard model of particle physics. At higher energies,new structures should become observable. Among them,there will probably be new global symmetries that are notmanifest at low energies. It is usually assumed that sym-metries would be restored at the high temperatures anddensities of the early universe.</p><p>However, the restoration of global symmetries might benot completely exact, since Planck-scale physics is be-lieved to break them explicitly. This feature comes fromthe fact that black holes do not have defined global chargesand, consequently, in a scattering process with black holes,global charges of the symmetry would not be conserved[1]. Wormholes provide explicit mechanisms of such non-conservation [2].</p><p>In the present article, we are concerned with the casethat the high temperature phase is only approximatelysymmetric. The breaking will be explicit, albeit small. Inthe process of spontaneous symmetry breaking (SSB),pseudo-Goldstone bosons (PGBs) with a small mass ap-pear. We explore the cosmological consequences of suchparticle species, in a simple model that exhibits the mainphysical features we would like to study.</p><p>The model has a (complex) scalar field x transform-ing under a global, nonanomalous, U1 symmetry. We donot need to specify which quantum number generates thesymmetry; it might be B-L, or a family U1 symmetry,etc., We assume that the potential energy for has asymmetric term and a symmetry-breaking term</p><p>V Vsym Vnonsym: (1)The symmetric part of the potential is</p><p>Vsym 14jj2 v22; (2)where is a coupling and v is the energy scale of the SSB.This part of the potential, as well as the kinetic termj@j2, are invariant under the U1 global transformation ! ei.04=70(11)=115009(12)$22.50 115009Without any clue about the precise mechanism thatgenerates Vnonsym, we work in an effective theory frame-work, where operators of order higher than four breakexplicitly the global symmetry. The operators would begenerated at the Planck scale MP 1:2 1019 GeV andare to be used at energies below MP. They are multipliedby inverse powers of MP, so that when MP ! 1 the neweffects vanish.</p><p>The U1 global symmetry is preserved by ? jj2but is violated by a single factor . So, the simplest newoperator will contain a factor </p><p>Vnonsym g1</p><p>Mn3Pjjnei ?ei; (3)</p><p>with an integer n 4. The coupling in (3) is in principlecomplex, so that we write it as gei with g real. We willconsider that Vnonsym is small enough so that it may beconsidered as a perturbation of Vsym. Even if (3) is alreadysuppressed by powers of the small factor v=MP, we willassume g small. In fact, after our phenomenological studywe will see that g must be tiny.</p><p>To study the modifications that the small explicitsymmetry-breaking term induces in the SSB process, weuse</p><p> vei=v; (4)with new real fields x and the PGB x. Introducing (4)in (3) we get</p><p>Vnonsym 2gv4vMP</p><p>n3</p><p>cosv </p><p> : (5)</p><p>The dots refer to terms where x is present. We see from(5) that there is a unique vacuum state, with hi v. Tosimplify, we redefine 0 v, and drop the prime, sothat the minimum is now at hi 0. From (5) we easilyobtain the particle mass</p><p>m2 2gvMP</p><p>n1</p><p>M2P: (6)-1 2004 The American Physical Society</p></li><li><p>9 10 11 12 13 14 15</p><p>-3 5</p><p>-3 0</p><p>-2 5</p><p>-2 0</p><p>-1 5</p><p>-1 0</p><p>-5</p><p>log(v/GeV)</p><p>log(</p><p>g)</p><p>FIG. 1. From bottom to the top, the five solid lines are the linesof constant mass, in the g,v plane, for m 2me; 2m; 2mp; 2mb, and 2mt. The dashed line corresponds tothe lifetime equal to the universe lifetime, t0, when ! is the only available mode.</p><p>EDUARD MASSO, FRANCESC ROTA, AND GABRIEL ZSEMBINSZKI PHYSICAL REVIEW D 70, 115009 (2004)Although for the sake of generality we keep then-dependence in (6), when discussing numerics and inthe figures we particularize to the simplest case n 4,with the operator in (3) of dimension five. We will discussin Sec. V what happens for n > 4.</p><p>To fully specify our model, we finally write the cou-plings of the PGB to other particles. We have the usualderivative couplings to fermions [3],</p><p>L f f g0</p><p>2v@ f5f gf f f5f: (7)</p><p>We have no reason to make g0 very different from O1. Tohave less parameters we set g0 1 for all fermions anddiscuss in a final section about this assumption. Then</p><p>gf f mfv</p><p>: (8)</p><p>The PGB couples to two photons through a loop,</p><p>L 18gFF; (9)</p><p>where the effective coupling is g 8v . In the sameway, there is a coupling to two gluons, Lgg 18 ggg</p><p>GaGa with ggg 3sv . Couplings of to</p><p>the weak gauge bosons do not play a relevant role in ourstudy.</p><p>We have now the model defined. It has two free parame-ters: g, which is the strength of the explicit symmetry-breaking term in (3), and v, which is the energy scale of theSSB appearing in (2).</p><p>At this point we would like to comment on the similar-ities and differences between our -particle and the axion[4]. There are similitudes because of their alike origin. Forexample the form of the coupling (7) to matter and (9) tophotons is entirely analogous. A consequence is that wecan borrow the supernova constraints on axions [5] toconstrain our PGB. Another analogy is that axions areproduced in the early universe by the misalignment mecha-nism [6] and by string decay [7] and -particles can alsoarise in both ways.</p><p>However, the fact that the -particle gets its mass justafter the SSB while axions become massive at the QCDscale introduces important differences. While in the axioncase clearly only the angular oscillations matter and alwaysone ends up with axion creation, in our model we need towork in detail the coupled evolution equations of the radialand angular part. We will need to establish when there areand when there are not angular oscillations leading to PGBcreation.</p><p>Also, from the phenomenological point of view, theaxion model is a one-parameter model while our modelhas two parameters v and g. As we will see, this extrafreedom makes the supernova constraints to allow a rela-tively massive -particle, and we will need to investigatethe consequences of the decay in the early universe. There115009is no analogous study for axions, simply because theinvisible axion is stable, in practical terms.</p><p>Let us also mention other previous work on PGBs. In[8], the authors consider the SSB of B-L with explicitgravitational breaking so that they obtain a massive ma-joron. An important difference between their work andours is that we consider nonthermal production, which iscrucial for our conclusions about the PGB being a darkmatter candidate. Also, there is previous work on explicitbreaking of global symmetries [9], and specifically onPlanck-scale breaking [10]. Cosmological consequencesof some classes of PGBs are discussed in [11]. Finally,let us mention a recent paper [12] where a massive majoronis considered in the context of a supersymmetric singletmajoron model [13]. An objective of the authors of [12] isto get hybrid inflation. Compared to our work, they con-sider other phenomenological consequences that the oneswe study.</p><p>The article is organized as follows. We first discuss theparticle properties of . In Sec. III, we work out thecosmological evolution of the fields, focusing in thermaland nonthermal production, and discuss the relic density ofPGBs. The astrophysical and cosmological bounds, and theconsequences of the decay, are presented in Sec. IV. Afinal section discusses the conclusions of our work.</p><p>II. - MASS AND LIFETIME</p><p>We have deduced the expression (6) that gives the massof the -particle as a function of v and g. In terms of thesetwo parameters, we plot in Fig. 1 lines of constant mass,for masses corresponding to the thresholds of electron,muon, proton, bottom, and top final decay, m 2me; 2m; 2mp; 2mb, and 2mt.</p><p>The lifetime of the -particle depends on its mass andon the effective couplings. A channel that is always open is-2</p></li><li><p>PLANCK-SCALE EFFECTS ON GLOBAL SYMMETRIES . . . PHYSICAL REVIEW D 70, 115009 (2004) ! . The corresponding width is</p><p> ! B</p><p> g2m364</p><p>; (10)</p><p>where B is the branching ratio. When m > 2mf, thedecay ! f f is allowed and has a width</p><p> ! f f Bf f</p><p> g2f f</p><p>m8</p><p>%; (11)</p><p>where % 1 4mfm2</p><p>q.</p><p>For masses m < 2me the only available decay is intophotons. When we move to higher masses, as soon asm > 2me, we have the decay ! ee. Actually, thenthe ! ee channel dominates the decay because ! goes through one loop. However, ! / m3whereas ! f f / m. By increasing m we reachthe value m 150me, where ! !ee. For higher m the channel ! dominatesagain. If we continue increasing the mass and cross thethreshold 2m, the decay ! dominates over !ee and ! , because gf f is proportional to thefermion mass mf. This would be true until m 150m,but before, the channel ! p p opens, and so on. Forlarger masses, each time a threshold 2mf opens up, !f f happens to be the dominant decay mode.</p><p>Now we can identify in Fig. 2 the regions with differentlifetimes. We start with the stability region > t0 4</p><p>1017 s [14], the universe lifetime, a crucial region for thedark matter issue. When m < 2me, t0 along thedashed line in Fig. 1 until v 4 1013 GeV and alongthe line m 2me for higher v. Below these lines, relic particles would have survived until now; in practical terms,for values of g and v in that region, we consider as astable particle. Above the lines, < t0 and the particle is9 10 11 12 13 14 15</p><p>-3 5</p><p>-3 0</p><p>-2 5</p><p>-2 0</p><p>-1 5</p><p>-1 0</p><p>-5</p><p> 6</p><p> 7 5 3</p><p> 2</p><p> 4</p><p> 6</p><p> 5</p><p> 4 3</p><p> 4</p><p> 5</p><p> 1</p><p> 3</p><p>stable</p><p>log(v/GeV)</p><p>log(</p><p>g)</p><p>FIG. 2. Zones corresponding to some ranges for the lifetime,. At the bottom, we have the region of the parameter spaceleading to a stable ( > t0), and at the top, the zone 7, where < 1 s. The other regions correspond to: (1) t0 > > 1013 s;(2) 1013 > > 109 s; (3) 109 > > 106 s; (4) 106 > > 104 s;(5) 104 > > 300 s, and (6) 300> > 1 s.</p><p>115009unstable. For future use, we define the ranges:(1) t0 > > 1013 s; (2) 1013 > > 109 s; (3) 109>> 106 s; (4) 106 > > 104 s; (5) 104 > > 300 s; and(6) 300> > 1 s.III. COSMOLOGICAL PRODUCTIONAND PGB DENSITY</p><p>-particles can be produced in the early universe bydifferent mechanisms. There are nonthermal ones, likeproduction associated with the field oscillations, andproduction coming from the decay of cosmic strings pro-duced in the SSB. Also, there could be thermal production.In this section we consider these production mechanismsand the PGB density resulting from them.</p><p>Let us begin with the field oscillation production. Theexpansion of the universe is characterized by the Hubbleexpansion rate H,</p><p>H 12t</p><p>43</p><p>45</p><p>s g?</p><p>p T2MP</p><p>: (12)</p><p>These relations between H and the time and temperature ofthe universe, t and T, are valid in the radiation era. For theperiod of interest, we take the relativistic degrees of free-dom g? 106:75 [15]. In the evolution of the universe,phase transitions occur when a symmetry is broken. Whenthe universe cools down from high temperatures there is amoment when the potential starts changing its shape, andwill have displaced minima. This happens around a criticaltemperature Tcr v and a corresponding critical time tcr.</p><p>In our model, the explicit symmetry breaking is verysmall and the evolution equations of and are approxi-mately decoupled. The temporal development leading and to the minimum occurs in different time scales, sincethe gradient in the radial direction is much greater than thatin the angular one. The evolution happens in a two-stepprocess. First goes very quickly towards the value hi 0, and oscillates around it with decreasing amplitude (par-ticle decay and the expansion of the universe work as afriction). Shortly after, will be practically at its mini-mum. In this first step where evolves, does not practi-cally change, maintaining its initial value 0, which ingeneral is not the minimum, i.e., 0 0. evolves towardsthe minimum once the first step is completely over. This isthe second step where the field oscillates around hi 0.</p><p>We have confirmed these claims with a complete nu-merical treatment of the full evolution equations, using theeffective potential taking into account finite temperatureeffects. We summarize our results in Appendix A.</p><p>Hence, we can write the following -independent evo-lution equation for </p><p>t 3H _t m2 0: (13)Here overdot means d=dt. We do not introduce a decayterm in the motion Eq. (13) and will treat the decay in the-3</p></li><li><p>9 10 11 12 13 14 15</p><p>-3 5</p><p>-3 0</p><p>-2 5</p><p>-2 0</p><p>-1 5</p><p>-1 0</p><p>-5</p><p>I</p><p>II</p><p>III</p><p>IV</p><p>log(</p><p>g)</p><p>log(v/GeV)</p><p>IV</p><p>V</p><p>FIG. 3. We show the various regions according to the pro-duction mechanism in the early universe. With this objective, wedisplay the lines: T v (dotted line), T Tend (solid line),nth nnonth (dashed line) and v vth (dot-dashed line). In (I)and (V) there is only nonthermal production. In (II) and (III)there are nonthermal and thermal processes that generate , in(II) nonthermal dominates while in (III), it is thermal productionthat dominates. In (IV) there is thermally and nonthermallyproduced s, but the last are thermalized. In (V), we have nota general formula for predicting nnonth.</p><p>EDUARD MASSO, FRANCESC ROTA, AND GABRIEL ZSEMBINSZKI PHYSICAL REVIEW D 70, 115009 (2004)usual way. We can do it because has a lifetime that ismuch greater than the period of oscillations.</p><p>Oscillations of the field start at a temperature Tosc suchthat 3HTosc m and this will be much later than theperiod of evolution provided Tosc Tcr v. This isequivalent to</p><p>g1=2 3 1031011 GeV</p><p>v</p><p>1=2: (14)</p><p>We shall see that the phenomenologically viable and in-teresting values of g are very tiny, so that we can assumethat the inequality (14) is fulfilled.</p><p>Since the oscillations are decoupled from oscilla-tions, production is equivalent to the misalignment pro-duction for the axion and we can use those results [6]. Theequation of motion (13) leads to coherent field oscillationsthat correspond to nonrelativistic matter and the coherentfield energy corresponds to a condensate of nonrelativistic particles.</p><p>We define osc and nosc as the energy density, andnumber density, re...</p></li></ul>