Axion-Assisted Electroweak Baryogenesis
PPreprint typeset in JHEP style - HYPER VERSION
SU-ITP-10-18OUTP-10-10P
Axion-Assisted Electroweak Baryogenesis
Nathaniel Craig
Institute for Theoretical Physics, Stanford University, Stanford, CA 94306, USAE-mail: [email protected]
John March-Russell
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road,Oxford, OX1 3NP, UKE-mail: [email protected]
Abstract:
We consider a hidden-valley gauge sector, G , with strong coupling scale Λ ∼ TeV and CP-violating topological parameter, θ , as well as a new axion degree of freedomwhich adjusts θ to near zero in the current universe. If the G -sector couples to the StandardModel via weak-scale states charged under both, then in the early universe it is possiblefor the CP-violation due to θ (which has not yet been adjusted to zero by the hiddenaxion) to feed in to the SM and drive efficient baryogenesis during the electroweak (EW)phase transition, independent of the effectively small amount of CP violation present in theSM itself. While current constraints on both the new axion and charged states are easilysatisfied, we argue that the LHC can investigate the vast majority of parameter space whereEW-baryogenesis is efficiently assisted, while the hidden axion should comprise a significantfraction of the dark matter density. In the supersymmetric version, the “messenger” mattercommunicating between the SM- and G -sectors naturally solves the little hierarchy problemof the MSSM. The connection of the hidden scale and masses of the “quirk”-like messengersto the EW scale via the assisted electroweak baryogenesis mechanism provides a reason forsuch new hidden valley physics to lie at the weak scale. Keywords:
Beyond Standard Model. a r X i v : . [ h e p - ph ] J un ontents
1. Introduction 22. Generating the Baryon Asymmetry 33. A Toy Model 54. Experimental Constraints 11 G -color cosmology 144.3.2 G -axion cosmology 15
5. Phenomenology 15
6. Supersymmetric Axion-assisted Electroweak Baryogenesis 17
7. Conclusions 18A. Axion mass and evolution 19 – 1 – . Introduction
The string landscape of vacua allows a rich variety of sectors hidden, or partially hidden,from the Standard Model (SM) which can lead to experimentally or observationally ac-cessible new phenomena. Recent discussions have focussed particularly on the superlightaxion-like states ubiquitous in string compactifications—the “axiverse” [1]—or weak-scalesupersymmetric states, such as photini from hidden U(1)’s [2], or goldstini from indepen-dent supersymmetry breaking sectors [3], both of which can strongly change LHC phe-nomenology, astrophysical observations, and early universe cosmology.In this paper we turn to another way in which a mildly-sequestered hidden sector,now possessing a heavy axion, may impact the physics and phenomenology of the SM. Theessential idea is simple to state: Consider a hidden sector gauge theory analogous to QCDwith its own additional CP-violating topological θ -parameter, as well as a (heavier thanQCD-) axion degree of freedom which automatically adjusts this theta angle to near zero inthe current universe. We argue that if this extra sector couples to the SM, say via TeV-scalebi-fundamental states, then, in the early universe, it is possible for the CP-violation due to θ (which has not yet been adjusted to zero by the hidden axion) to feed in to the SM anddrive efficient baryogenesis during the electroweak phase transition. In contrast to theCP violation present in the SM itself, which even for large CKM phase is very small due tosuppression by powers of the squared differences of quark Yukawa couplings, the physicalCP violation induced at the time of the EWPT can be large in our mechanism. Moreover,the stringent present experimental constraints on CP violation in extensions of the SM aresimply and easily met by this construction as the hidden-sector θ -angle is naturally relaxedto near zero at current temperatures.One may ask why a similar mechanism is not operable in an even simpler extension ofthe SM, the SM with QCD θ -term now relaxed to very near zero by the conventional Peccei-Quinn-Weinberg-Wilczek QCD axion [4, 5, 6]: Similar to our model the QCD axion hasnot yet relaxed to its θ QCD -cancelling minimum at the time of the EWPT, so it seems thatlarge CP violation is available in this situation as well. This is not the case, however, as the θ -term is topological and only leads to physical effects via topologically non-trivial QCDfield configurations, which at temperatures T (cid:29) Λ QCD have large action and are suppressed[7]. This leads to an effective size of CP-violation suppressed by ∼ (Λ QCD /T EW ) (cid:28) θ QCD ∼ Assuming, as usual, that the EWPT is sufficiently out-of-equilibrium. Here our focus is on the alter-ations to standard EW-baryogenesis due to the hidden sector θ -term and axion, and we take the view thatthe new sector and couplings to the SM are “retrofitted” to one of the many TeV-completions of the SMwith sufficiently out-of-equlibrium dynamics (not necessarily only a strong 1st order EW phase transition).We will return to the detailed investigation of EWPT out-of-equlibrium dynamics with axion-assistance ina later work. – 2 –ouplings to the SM have attracted a considerable amount of study on account of theirnovel phenomenology (see, e.g., [8, 9, 10, 11, 12, 13, 14]); our viewpoint is that the con-nection to electroweak baryogenesis provides a reason for such new physics to lie at theweak scale. These features lead in turn to experimental signatures of our scenario at theLHC, and associated constraints from precision electroweak tests and direct searches atLEP/Tevatron. The constraints are easily satisfied, and leave a large range of parameterspace which is accessible to investigation at the LHC. An amusing feature of a supersym-metric version of our mechanism is that the “messenger” matter communicating betweenthe SM and hidden sectors naturally solves the little hierarchy problem of the MSSM forthe range of parameters for which EW-baryogenesis is effectively assisted. Even thoughthe hidden sector axion is heavier than the QCD axion by a factor ∼ ( v EW / Λ QCD ) ∼ ,the prospects for detecting the new axion itself are similar to that of the standard QCDaxion for equivalent axion decay constant f G . Intriguingly, as we show in detail below, effi-cient axion-assisted electroweak baryogenesis (AAEB) favours, f G ∼ GeV, the naturalstring value of the axion decay constant[15, 1], and disfavours the traditional axion win-dow, 10 GeV < ∼ f G < ∼ GeV. Thus, as f G is in the anthropic range requiring a mildlyfine-tuned initial misalignment angle, a further natural consequence of our mechanism isthat the hidden axion comprises a significant fraction of the dark matter density.The setup of our paper is as follows: In Section 2 we review the relation between theobserved baryon asymmetry and a class of CP-odd effective operators. In Section 3 wepresent a simple model, based on a confining hidden sector with an axion-like couplingand matter fields charged under the SM, and show that confinement near the electroweakscale may naturally give rise to the required degree of CP violation during electroweakbaryogenesis. Such additional degrees of freedom are naturally subject to a variety ofexperimental and cosmological constraints, which we consider in Section 4. The presenceat the weak scale of new strongly-interacting fields charged under the SM raises the prospectof various LHC signatures and other phenomenological consequences, to which we turn inSection 5. In Section 6 we briefly consider a supersymmetrized version of the model, forwhich the hidden sector matter content naturally solves the little hierarchy problem. Weconclude in Section 7, and reserve for the Appendix a detailed discussion of the axion massand evolution.
2. Generating the Baryon Asymmetry
A particularly simple way of generating a sufficient amount of CP violation is to incorporatenew physics giving rise, at low energies, to an effective (and CP-odd) irrelevant operatorof the form [16] L CP = g π W µν ˜ W µν Φ( T, H ) (2.1)where W is the SU (2) gauge field strength and Φ is some temperature-dependent functionalof the Higgs field(s). The functional Φ acts like a type of T -dependent, and thus, in the In this Section we follow closely the discussion of the QCD case presented in Ref.[7]. – 3 –arly universe, time-dependent, effective “axion” for W ˜ W , driving CP violation into theSM sector during the EW transition and thus assisting EW baryogenesis.Using the anomaly equation, g π W µν ˜ W µν = ∂ µ j µCS , an interaction of the above formmay be expressed, by integrating by parts and focusing on the time component, as achemical potential for the Chern-Simons number density for SU (2), which in turn biasesthe change in baryon number L CP = j CS ∂ Φ = n CS d Φ /dt . Hence the effective Lagrangianarising from irrelevant operators of the form Eq.(2.1) looks like a chemical potential forChern-Simons number density, where the effective chemical potential is given by µ CS =˙Φ. It is then relatively straightforward to evaluate the contribution to CP violation inbaryogenesis for various forms of the functional Φ. The size of CP violation ultimatelydepends on both the size of the coefficient in Φ and the degree of time variation duringbaryogenesis. In general, if the only time-variation in Φ comes from the time-variation of T , then µ CS is suppressed by a factor T /M P encoding the rate of Hubble expansion. It’sonly during a first-order phase transition that we expect more rapid time variation of thequantity Φ, and hence an unsuppressed chemical potential for n CS .Although there are a variety of nonequilibrium phenomena that may lead to baryo-genesis, we will focus here on baryogenesis during the electroweak phase transition [17],under the assumption that the phase transition is first-order and sufficiently strong. Inprinciple, the phase transition proceeds through bubble nucleation, and the generationof baryon number asymmetry occurs only in the bubble wall when ˙Φ is significant. Inpractice, however, it is difficult to make any precise quantitative statements about a nu-cleation phase transition without extensive numerical simulation. One must understandnot only the details of the phase transition and bubble wall propagation, but also particletransport at the phase boundary in the presence of CP violation. Qualitative estimatesmay be made if we consider a spinodal decomposition phase transition, in which the scalarfield rolls uniformly to the true vacuum; such a transition gives a spatially uniform, buttime-varying, phase transition. The assumption is that the results should be similar tothose of a nucleation phase transition, as Lorentz invariance in principle relates processeswith time-varying fields to those with space-varying fields. The resulting expression forbaryon asymmetry is expected to reflect the correct parametric dependence on microscopicparameters, up to O (1) coefficients. We emphasize, however, that this simplifying assump-tion is made strictly for the purpose of understanding the correct parametric dependence ofbaryon asymmetry upon the microphysical parameters of our model; it is reasonable insofaras the magnitude of CP violation is independent from the details of the phase transition.Assuming we have some free energy difference ∆ F between neighbouring minima, anda rate Γ a for fluctuations between neighbouring minima (in the absence of bias, as weassume the bias is a small perturbation), the master equation for baryon number is [22] dn B dt = − a T ∆ F (2.2)We can write this in a somewhat more useful form via ∆ F = n f ∂F∂B (where n f = 3) to For excellent reviews, see [18, 19, 20]. For a recent discussion see, e.g., [21] and references therein. – 4 –btain dn B dt = − a µ B T = − Γ a µ CS T (2.3)The number density of baryons created during the phase transition is then given by n B = n f T (cid:90) ∞ dt Γ a ( t ) µ CS ( t ) . (2.4)The crucial part, of course, rests in accurately evaluating the chemical potentials biasingbaryon number production, as well as the time evolution of rates.The transition rate for fluctuations between neighbouring minima depends entirelywhether electroweak symmetry is broken. In the unbroken phase, sphaleron transitions areunsuppressed, while in the broken phase there arises the usual exponential suppression.That the transition rates in the symmetric and broken phases are, respectively,[18]Γ a (cid:39) α w ) T (symmetric) (2.5)Γ a (cid:39) ( α w T ) − m W e − E sph /T (broken) (2.6)Clearly the contribution from the symmetric phase dominates. In the spinodal decompo-sition transition, we may assume that these values interpolate smoothly. Treating Γ a as astep function and T as essentially constant during the phase transition, we can estimatethe integral to find a total baryon asymmetry∆ ≡ n B s (cid:39) πg ∗ n f α w T i T f δ Φ (2.7)where T i , T f are the temperatures before and after the phase transition; s = (2 π g ∗ / T ,and δ Φ ≡ Φ( T i , H = g W T i ) − Φ( T i , H = 0). This estimate should be compared to theobserved value, n B /s (cid:39) − . Although this expression has been obtained for a spinodaldecomposition transition with a number of simplifying assumptions, it is parametricallysimilar to the analogous expression for bubble nucleation; see, e.g., [23, 24]. In the case ofbubble nucleation, the expression ( T i /T f ) is typically replaced by ∼ ( m t /T c ) ( m h /T c ) , butthe parametric dependence on δ Φ is unaltered. It is then a fairly straightforward procedureto compute the size of CP violation δ Φ given a particular functional Φ, to which we willnow turn.
3. A Toy Model
Consider now an additional nonabelian gauge sector G with fundamental and antifunda-mental fermions carrying vector-like charges under SM SU (2) L × U (1) Y . We will assumethat these fields are uncharged under SU (3) C ; this assumption may be relaxed, whichleads to more stringent constraints on vector masses and significantly altered collider phe-nomenology, among other things. The matter content is such that the group G is asymp-totically free and confines at some scale Λ G ; such additional confining sectors have beenthe subject of considerable study in recent years [8, 9, 11, 12, 13]. We will henceforth refer– 5 –o the gauge sector G as G -color, and the fermions charged under G as G -quarks or G -fermions. While there are a variety of candidate models that exhibit the same qualitativefeatures, among the simplest such candidates is G = SU ( N ) G with the matter contentshown in Table 1.The added G -fermion matter content is SU ( N ) G SU (2) L U (1) Y Q (cid:3) Y Q Q (cid:3) − Y Q U (cid:3) / Y Q U (cid:3) − / − Y Q Table 1:
Model matter content. Here Y Q isan arbitrary hypercharge assignment. free of gauge and gravitational anomalies. Wemay introduce vector masses for the G -fermions,as well as appropriate couplings to the SMHiggs field H : L G ⊃ − µ Q QQ − µ U U U − λH † QU − λ (cid:48) HQU + h.c.(3.1)The G -fermions acquire mass from both theirvector masses and from electroweak symmetry breaking (EWSB) via their coupling to theHiggs. The mass eigenstates consist of two Dirac G -fermion-antifermion pairs of electriccharge ± ( Y Q + 1 /
2) and one Dirac G -fermion-antifermion pair of charge ± ( Y Q − ). Themass of the latter G -fermion pair is simply µ Q , while the mass matrix for the charge ± ( Y Q + 1 / G -fermions is M = (cid:32) µ Q √ λv ( T ) √ λ (cid:48) v ( T ) µ U (cid:33) (3.2)where have emphasized the temperature-dependence of the EWSB Higgs vev which will beimportant to our mechanism. In the simplest case with µ Q = µ U = µ , the mass eigenvaluesfor the three Dirac fermions are then m Q ( T ) = µ, µ ± (cid:114) λλ (cid:48) v ( T ) . (3.3)In Section 6, in the context of a supersymmetric version of our toy model, we will arguethat the vector-like mass terms, µ Q , µ U , arise in the same fashion as the SUSY-Higgs µ -term, thus justifying both their link to the EW scale, and our notation. At present we willjust take them to be free parameters of the toy model.Of course, this matter content spoils na¨ıve unification of SM gauge couplings. Althoughconventional gauge coupling unification is by no means a necessary ingredient, it is possibleto restore gauge coupling unification by adding additional G -quarks charged under SU (3) C ,with suitable SM charge assignments filling out complete + multiplets under SU (5) GUT .We will assume that such additional G -quarks, if present, have vector masses (cid:38) G -color gauge sector is asymptotically free, and confines in the IR at a scaleΛ G = µ e − π /b G g G, (3.4)given an initial value of the G gauge coupling g G, at a scale µ . In this case b G = T ( G ) − N f T ( R ) = N −
2. We will be interested in a range of possible values of Λ G – 6 –elative to m Q . In any case, since the G -fermions are vectorlike under the SM, a potentialcondensate of G -fermions leaves electroweak symmetry unbroken.It is completely consistent with the symmetries of the theory to include also a CP-odd θ -term for the G -color gauge group, L G ⊃ − α G θ G π G µν ˜ G µν . (3.5)Such θ terms are generically present in nonabelian hidden sectors, some consequences ofwhich were considered in [25]. Of course, the bare angle θ G may be shifted by rotationsof the G -quark mass matrix, θ G → θ G + arg det M ≡ θ G ; for clarity we will leave thisredefinition implicit. While the analogous CP angle in the SM is constrained to satisfy | θ QCD | < − , there are no direct constraints on the value of θ G . The CP angle for the G -color sector becomes particularly interesting in the event thatthere is also a pseudo-Goldstone axion a G coupling to G -color via the usual operator L G ⊃ α G π a G f G G ˜ G (3.6)where f G is the scale of PQ symmetry breaking for the G -axion. Although such axionsare not necessary ingredients of a nonabelian hidden sector (unlike the SM axion, requiredto explain the smallness of θ QCD ), they arise quite naturally in many string compactifica-tions [15]. The axion acquires a zero-temperature mass from confinement of G of order m a ∼ m Q Λ G f G , which may also accumulate finite-temperature corrections. There also maybe contributions to the G -axion mass coming from string instantons, but these are sub-dominant to the mass coming from G -color confinement if the string-compactification issuch that the QCD axion solves the usual strong-CP-problem, as discussed in [1].In general, the G -axion obtains a nonzero vacuum expectation value; as with the QCDaxion, its expectation value may naturally be of order (cid:104) a G (cid:105) ∼ f G when its PQ symmetry isbroken, leading to a nonzero angle θ G . When the amplitude of the G -axion is nonvanishing,there is a condensate of G ˜ G . We can parameterize this as usual via α G π (cid:104) G ˜ G (cid:105) = m a ( T ) f G sin θ G (3.7)where m a ( T ) is appropriately the temperature-dependent axion mass.Below the scale m Q , integrating out the massive G -fermions gives rise to a variety ofeffective operators, the most important of which (for our purposes) is the coupling to the SU (2) L W ˜ W term: L eff ∼ α W α G π m Q W µν ˜ W µν G µν ˜ G µν (3.8) Notice that this may change if the G -quarks are charged under QCD, in which case there arises aneffective contribution to θ QCD of order ∼ α G π (cid:104) G ˜ G (cid:105) m Q θ G . This mechanism for enhancing CP violation during baryogenesis may also work in a hidden sectorwithout an axion, provided a nonzero vacuum expectation value for the G -gluon condensate (cid:104) G ˜ G (cid:105) , butthis lacks the tidy relaxation of θ G at late times. Moreover, additional axions are a natural consequence ofstring compactification, and therefore not unexpected in this context. – 7 –henever a nonzero amplitude for the G -axion gives a nonvanishing expectation value to G ˜ G , this produces an effective θ angle for SU (2) L . The result is a temperature-dependent(and thus time-dependent) effective operator L eff ∼ g π W µν ˜ W µν (cid:34)(cid:88) i m Q,i ( T ) m a ( T ) f G sin θ G (cid:35) . (3.9)This is precisely of the form Eq.(2.1). Thus in this case we obtain a chemical potential forCS-number of order µ CS (cid:39) sin θ G f G ddt (cid:32)(cid:88) i m a ( T ) m Q,i ( T ) (cid:33) , (3.10)and can identify the change δ Φ appearing in the expression for the final baryon asymmetry,Eq.(2.7) to be δ Φ( T, H ) ∼ δ (cid:32)(cid:88) i m a ( T ) m Q,i ( T ) (cid:33) f G sin θ G . (3.11)If the time dependence of the G -color fermion and axion masses is significant aroundthe time of the electroweak phase transition, the G -color sector will feed a large amountof CP violation to the SM. The resulting contribution to CP violation during electroweakbaryogenesis will, in general, be unsuppressed by factors of H/T ; both m Q and m a dependon the Higgs vev, which is rapidly changing during the phase transition. In order toestimate the parametric size of CP violation, we must first parameterize the G -axion mass m a as a function of temperature and Higgs vev. Assuming the dominant contribution tothe axion mass comes from G -color confinement, this depends sensitively on the relativevalues of m Q , Λ G , and the critical temperature T c of the electroweak phase transition.In general, experimental constraints will restrict us to the case m Q > T c , leaving threepossibilities:1. T c < Λ G < m Q , i.e., G -quarks above confinement, confinement before EWSB;2. T c < m Q < Λ G , i.e., G -quarks below confinement, confinement before EWSB;3. Λ G < T c < m Q , i.e., G -quarks above confinement, EWSB before confinement.The parametric dependence of the effective chemical potential, µ CS , or equivalentlythe change δ Φ in the equation for the final baryon asymmetry, Eq.(2.7), is fairly sensitiveto the hierarchy of scales. Let us consider each in turn. • T c < Λ G < m Q Since in this case there are no light G -quarks at the confining scale, the axion massmay merely be estimated from instanton effects as m a f G ∼ Λ G . (3.12)– 8 –pon integrating out the massive G -quarks, however, the IR scale Λ G picks up adependence on the masses m Q (for fixed value of the G -coupling-constant in theUV), and thus the Higgs vev v ( T ), which at 1-loop is given byΛ G = Λ G,UV (cid:18) Λ G,UV m Q (cid:19) ( b ,UV /b ,IR − (3.13)where b ,UV and b ,IR are the 1-loop beta-function coefficients of the G -color theoryabove and below the G -quark mass threshold, and Λ G,UV is the G -color scale assuming m Q = 0. In our case we desire the change in IR scale due to the electroweak vevturning on, which arises due to the originally degenerate G -quark mass eigenstatesof mass µ splitting, as in Eq.(3.3). For the toy model, the confinement scale belowthe mass of all the G -quarks is given in terms of the U V scale byΛ G = Λ G,UV (cid:32) µ ( µ − λλ (cid:48) v )Λ G,UV (cid:33) / N . (3.14)In the case T c < Λ G < m Q , we expect the ratio v ( T ) /µ to be small, so expandingthe expression for Λ G , we find that, for our toy model, the leading change at theelectroweak phase transition is given by δ Λ G = − λλ (cid:48) N vδvµ Λ (4 − / N ) G,UV µ / N . (3.15)Similarly the leading change in the denominator m Q in Eq.(3.11) is δ (cid:88) i m Q,i ( T ) = 10 λλ (cid:48) vδvµ . (3.16)Putting these together we find that δ Φ = sin θ G (cid:18) − N (cid:19) (cid:18) λλ (cid:48) vδvµ (cid:19) (cid:18) Λ G,UV µ (cid:19) (4 − / N ) , (3.17)Since δv ≤ v , we see from Eq.(3.17) that it is not possible to raise the G -quark massscale arbitrarily above the EW-scale v by raising the vector-like mass µ , withoutstrongly suppressing the final baryon asymmetry. • T c < m Q < Λ G In this case the mass comes from confinement effects, with somenumber of light species of G -quark. The axion mass may thus be estimated usingchiral perturbation theory, in analogy with the QCD axion, giving m a f G ∼ m Q Λ G . (3.18)Dependence on the Higgs vev arises again through dependence on m Q ; in this casethe result is simply δ Φ (cid:39)
10 sin θ G λλ (cid:48) vδv Λ G µ . (3.19)Ultimately, this will prove to be an uninteresting region of parameter space fromthe perspective of electroweak baryogenesis due to the rapid relaxation of θ G whenΛ G > m Q . – 9 – Λ G < T c < m Q Of course, if Λ G < T c , the electroweak phase transition occurs before confinement.In this case, the axion still obtains a mass through instanton effects that may beestimated for large T using the dilute-instanton-gas approximation [26]. In this case,the axion mass scales as m a f G ≈ Λ G (cid:18) Λ G T (cid:19) (11 N − . (3.20)The full calculation is fairly tedious; see the Appendix for details. Once again, the G -quark mass enters into Λ G via scale-matching at the scale m Q , and in the case ofthe toy model the axion mass may be expressed in terms of UV parameters as m a f G ≈ Λ G,UV (cid:32) µ ( µ − λλ (cid:48) v )Λ G,UV (cid:33) / (cid:18) Λ G,UV T (cid:19) N − . (3.21)Then for our toy model we have parametrically in this case δ Φ (cid:39)
283 sin θ G (cid:18) λλ (cid:48) vδvµ (cid:19) (cid:18) Λ G,UV µ (cid:19) (cid:18) Λ G,UV T (cid:19) N − . (3.22)In all of these cases it is then a relatively straightforward matter to compute the baryonasymmetry ∆ using Eq.(2.7). There is, of course, an important caveat, which is that the G -axion evolution has not relaxed θ G to zero by the time of the electroweak phase transition.Approximately speaking, θ G begins to relax when m a ( T ) (cid:38) H ( T ); the equation of motionfor θ G is that of an underdamped oscillator with temperature-dependent angular frequency m a ( T ). In practice, the situation is somewhat more delicate, and requiring m a ( T ) = 3 H ( T )at a temperature T < T c is overly conservative. Rather, we may study the time evolutionof θ G with a temperature-dependent axion mass and impose the reasonable constraint that θ G not pass through zero before T = T c ; a detailed discussion is reserved for the Appendix. This results in a significant constraint on the G -color confinement scale even for an axiondecay constant approaching the Planck scale. Indeed, for f G ∼ GeV, requiring theaxion to not be relaxing implies the scale Λ G < ∼ GeV, for which the final baryon asymmetryis insufficient for baryogenesis, thereby excluding our mechanism. The situation is muchmore favorable for GUT-scale axions with f G ∼ GeV, for which confinement scalesas high as a few hundred GeV are allowed without relaxing θ G before the EWPT. Theimplications of this are twofold: First, GUT-scale G -axions are strongly favored if thismechanism for baryogenesis is to be effective. Second, the G -color confinement scale mustbe near the weak scale – and certainly not much higher – in order to prevent the CPviolating angle from relaxing too soon. This is extremely fortuitous from the perspectiveof collider phenomenology, as it forces the scales of the G -color sector to lie within reachof the LHC. It might be possible that significant evolution towards a small effective axion mis-alignment angle isacceptable, but there are concerns about the oscillating behavior of the axion substantially washing out theproduced baryon asymmetry even in this case. – 10 – (cid:76) G (cid:72) GeV (cid:76) Μ (cid:72) G e V (cid:76) N (cid:61)
3; sin Θ G (cid:61) f G (cid:61) GeV; T c (cid:61)
100 GeV; Λ v (cid:61)
100 GeV
50 100 150 200 250 300300400500600700 (cid:76) G (cid:72) GeV (cid:76) Μ (cid:72) G e V (cid:76) N (cid:61)
3; sin Θ G (cid:61) .01; f G (cid:61) GeV; T c (cid:61)
100 GeV; Λ v (cid:61)
100 GeV
Figure 1:
Baryon asymmetry as a function of Λ G , µ for representative choices N = 3, T c = 100GeV, (cid:113) λλ (cid:48) v = 100 GeV, and sin θ G = 1, f G = 10 GeV (left), sin θ G = 10 − , f G = 10 GeV(right). We have taken a smooth interpolation of the various limiting values of δ Φ. The red contoursindicate 10 − < ∆ < − from dark to light. The region shaded in blue is excluded by relaxationof θ G before the EWPT (see Appendix for details; this represents a fairly conservative constraint,and it may be possible for baryogenesis to be efficiently assisted even in this region). The regionshaded in purple ( µ <
300 GeV) is excluded by collider limits on µ (see Section 4.1). We allowa large range of ∆ to take account of the uncertainties introduced by our approximate treatmentof the EWPT dynamics; the correlation between Λ G and µ could be made more precise given anexplicit theory of the first-order EWPT dynamics and associated baryon number creation at thebubble-wall. For hidden-axion decay constant, f a < ∼ GeV, the initial axion misalignment angle θ G can be large, while for the string-preferred value, f a ∼ GeV, the misalignment must satisfy θ G < ∼ − , as discussed in Section 4.3.2. Representative values of ∆ as a function of the vector mass µ and strong coupling scaleΛ G are shown in Fig. 1 along with constraints coming from the relaxation of θ G . As notedabove, the constraints from relaxation of θ G entirely exclude axions with low scales of PQsymmetry breaking, f G (cid:46) GeV; in this case θ G always begins oscillating before theelectroweak phase transitions (though, again, this is a conservative criterion, and efficientbaryogenesis may still be possible once θ G has begun to evolve). The situation for GUT-scale PQ breaking is more favorable; θ G relaxes after the EWPT as long as Λ G (cid:46) G , the G-quark masses cannot be much more than 600 GeV inorder to produce sufficient baryon asymmetry. Thus the favored parameter space for bothΛ G and µ is tightly constrained to lie around the weak scale. Although the precise valueof the baryon asymmetry depends on the details of the electroweak phase transition andtransport across bubble walls, there is significant room to reproduce a sufficient amount ofCP violation if the G -quarks and confinement scale lie in the range GeV–TeV.
4. Experimental Constraints
The allowed parameter range of G -color confinement scale Λ G and G -quark masses m Q – 11 –s constrained by a variety of considerations, including limits from nonobservation at theTevatron; constraints from precision electroweak measurements; and cosmological limits onboth G -color fields and the G -axion. Ultimately these considerations place lower boundson m Q and Λ G , as well as a relation between f G and θ G , but do not significantly impactthe parameter space for axion-assisted baryogenesis. The constraints on fermions of the G -color sector coming from nonobservation at the Teva-tron and LEP are not tremendously stringent, owing both to the lack of specific searches forhidden sector G -fermions and, in the models considered here, the absence of light G -quarkscharged under SU (3) C . G -fermions may be pair produced at colliders through an off-shell photon, Z , W , orHiggs. The production rates for the first three processes are fixed entirely by the gaugecouplings, while production through the Higgs depends on the size of the λ, λ (cid:48) Yukawacouplings as well as the Higgs mass. Their production at colliders is essentially that ofthe quirk scenario at large Λ [11, 13, 14]: pair-produced G -fermions form a bound stateconnected by a G flux tube, radiating kinetic energy and angular momentum throughemission of photons, hadrons, and G glueballs before recombining and annihilating intolighter SM states. For G -fermions produced via an off-shell photon or Z , the primaryannihilation channel is into G glueballs. G -fermions produced via an off-shell W cannotannihilate into an SM charge-neutral state, and so instead annihilate into leptons or quarksvia an off-shell W .Current collider limits are fairly weak. There are a variety of potential bounds comingfrom different channels at the Tevatron. Two primary channels giving rise to representativelimits are • l + γ + E / T ; CDF Run II limits on anomalous events involving a high- p T chargedlepton and photon with missing transverse energy based on 929 pb − of data [27]exclude masses for color-singlet G -fermions below 200 GeV; the bound for colored G -fermions is ∼
250 GeV. • γ + γ and 2 γ + τ ; CDF Run II limits on the inclusive production of diphoton eventswith a third photon based on 1155 pb − [28] place a similar limit on G -fermionmasses, excluding (cid:46)
200 GeV. Limits from a diphoton plus tau search on 2014 pb − place limits on G -quark masses below 250 GeV assuming λ ∼
1, which may be relaxedsomewhat for smaller Yukawa couplings.Taken together, these searches collectively limit m Q (cid:38)
200 GeV for the lightest G -quark(or m Q (cid:38)
250 GeV for colored G -quarks). Given that the lightest G -quark mass is m Q = µ − (cid:113) λλ (cid:48) v , for (cid:113) λλ (cid:48) v ∼
100 GeV this suggests a lower limit of µ (cid:38)
300 GeV on the vectormasses. It is important to emphasize that bounds on a SM fourth generation from Higgssearches at the Tevatron are vitiated in this case, since the G -fermions do not acquire theirentire mass from electroweak symmetry breaking.– 12 – (cid:45) (cid:45) (cid:45) (cid:68) S (cid:68) T (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) S (cid:68) T Figure 2:
Corrections to electroweak precision observables
S, T for µ = 300 , , , , , (cid:112) λλ (cid:48) / v = 100 GeV assuming G = SU (3) with hypercharge assignments Y Q = 0 (left)and Y Q = 1 / ,
95% CL ellipses are shown; best fit is achieved by (∆ S, ∆ T ) =(0 . , . As with any theory involving additional states carrying electroweak quantum numbersand interactions violating the custodial symmetry of the Higgs sector, the G -color sectorcontributes to precision electroweak observables. If the only source of mass for G -colorfermions arose from electroweak symmetry breaking, this would pose a stringent constrainton the matter content of the G -color sector. However, the addition of vector masses for G -color fermions allows them to be safely decoupled; vector masses above a few hundredGeV are more than sufficient to satisfy precision electroweak constraints.As usual, the contributions to precision electroweak observables may be quantifiedin terms of the S and T parameters [29] (contributions to U are negligible and weaklyconstrained). For Λ G > M Z , the appropriate degrees of freedom are composites of thestrong G -color interactions, and a precise calculation of contributions to the S and T parameters is not possible. However, a reasonable estimate may be made on the basis ofthe one-loop contribution due to the fundamental fermions of the G -color sector.For example, among the effective operators induced below the scale m Q there arises acontribution to the S parameter of the form L eff ∼ N gg (cid:48) π λλ (cid:48) µµ (cid:48) m Q H † HW µν B µν (4.1)It is clear from the level of the effective operator that this contribution to S is suppressedby at least ( λv/µ ) , which rapidly diminishes as a function of µ . For µ = µ (cid:48) with anexpansion in small m u ≡ λv , m d ≡ λ (cid:48) v , and M W , the new fermion contributions to S and– 13 – may be computed more precisely using [30]:∆ T = N πs W M W µ (cid:2) m u + m d ) + 2( m u m d + m d m u ) + 18 m u m d (cid:3) (4.2)∆ S = N πµ (cid:2) m u + m d ) + m u m d (3 + 20 Y Q ) (cid:3) (4.3)where Y Q is the weak hypercharge of Q and ∆ S, ∆ T indicate the deviations from the valuesof S, T predicted by the SM alone for m t = 173 . m h = 115 GeV. Of course,more exact estimates would need to take into consideration the effects of confinement, butthe ( λv/µ ) suppression will still persist as an effective form factor, and confinement is notexpected to significantly alter the na¨ıve prediction and effects of decoupling. In general,precision electroweak constraints are readily satisfied for modest values of the vector mass µ ; representative points in the parameter space of (∆ S, ∆ T ) are shown in Fig. 2. Thelower bound on µ coming from direct detection limits already renders safe the G -quarkcontributions to precision electroweak variables, even when the SU (2) doublet G -quarkscarry nonzero hypercharge. G -color cosmology Naturally, one might be concerned that the addition of a strongly coupled gauge sector withcouplings to the SM would pose a host of challenges to conventional cosmology. Althoughthe safest cosmology might simply entail reheating temperatures T RH (cid:46) MeV so that the G -color sector is never thermalized, such a low reheating temperature is fairly unattractivefrom the perspective of electroweak baryogenesis. Higher reheating temperatures lead toa cosmological abundance of G -quarks and G -color glueballs, whose evolution must beconsidered in detail. However, for the confinement scales of interest here, the G -colorsector is surprisingly safe from a cosmological perspective. As the G -quarks carry conserved G -color charges, the lightest G -quark is absolutelystable. However, below the confinement scale Λ G , G -color interactions ensure that all G -color bound states annihilate efficiently into the lightest G -color mesons. The lightest G -color mesons then decay rapidly into SM states via their coupling to electroweak gaugebosons or the Higgs; for Λ G > G < m Q the lightest states of the G -color sector tend to be G -color glueballs.These bound states decay into SM states via loops of G -color quarks, which then decay intoSM states via the Higgs or electroweak gauge bosons. The leading dimension-six operatorcomes from Higgs boson exchange via the operator L ⊃ α G λ πm Q H † HG µν G µν (4.4)which leads to a lifetime for the scalar 0 ++ G -color glueball of order [14] τ ∼ − s × (cid:16) m Q
300 GeV (cid:17) (cid:18)
100 GeVΛ G (cid:19) . (4.5) Surprisingly, G -color sectors with much smaller Λ G – as low as O (10 eV) – are viable as well, thoughthis often requires the lightest G -quarks to be charged under SU (3) C [31, 32]. – 14 –eedless to say, such decays occur well before BBN and pose no particular challenges toconventional cosmology. G -axion cosmology Although the G -quarks and G -color glueballs are cosmologically safe due to the relativelylarge values of m Q and Λ G , the cosmology of the light G -axion – as with the cosmology ofthe QCD axion [33, 34, 35] – is rather less trivial. As with the QCD axion, the G -axion isinitially a random field on superhorizon scales with vacuum expectation value a G ∼ f G θ G,i ;the fluctuations of θ G,i are naturally expected to be of order one. Not far above the G -color confinement scale Λ G , G -color instantons (and perhaps also string instantons)induce a potential for the G -axion. When m a ∼ H , the axion begins to oscillate aroundthe resulting minimum as an underdamped harmonic oscillator of frequency m a . Thesecoherent axion oscillations contribute as cold dark matter to the critical energy density anamount Ω a h ∼ (cid:18) f G M P (cid:19) (cid:18) Λ G T i (cid:19) (cid:18) a G,i f G (cid:19) (4.6)In this case T i is set by m a ( T i ) ∼ H. Consequently, the observed energy density constrains f G (cid:16) a G,i f G (cid:17) (cid:46) GeV. If a G,i f G = θ G,i ∼
1, then necessarily f G (cid:46) GeV. However, thisshould not be construed as ruling out higher values of f G ; indeed, it is entirely possiblefor larger values of f G that the initial angle θ G,i is anthropically constrained to satisfy thecritical density bound [36]. For example, for a G -color PQ scale of order f G ∼ GeV,the critical density bound implies θ G,i (cid:46) − .
5. Phenomenology
Ultimately, generating the observed baryon asymmetry from the strong CP angle of aconfining hidden sector is most interesting if the hidden sector degrees of freedom areexperimentally accessible. As we have seen in previous sections, the observed value ofbaryon asymmetry is most readily produced if the G -quark masses lie in the range 200 –600 GeV and the confinement scale lies in the range 50 – 250 GeV , an ideal scenario forproduction at the LHC. If the G -axion accounts for the bulk of CP violation during baryogenesis, it is necessarily thecase that the G -color confinement scale Λ G and G -quark masses m Q are in a range amenableto extensive production at the LHC. Indeed, effective axion-assisted baryogenesis providesa “reason” for the hidden sector to lie near the weak scale. The LHC phenomenology of anadditional strong sector is quite striking, and has been studied extensively in [37, 13, 14];here we will confine ourselves to a brief review of the most compelling results.For m Q , Λ G in the range 100 GeV − G -color fieldsare prompt, occur within the detector, and may lead to distinctive signatures. As discussedearlier, CDF limits constrain the lightest G -quark to satisfy m Q (cid:38)
200 GeV. G -quark pairsin the range 200 GeV < m Q < TeV may be directly produced at the LHC via off-shell– 15 –lectroweak gauge bosons or the Higgs. For m Q > Λ G , the primary decays of G -quarkbound states are into G -color glueballs, whose subsequent decays into SM states are visiblefor Λ G > G -quark bound states carrying electroweak quantum numbers decayinto G -color glueballs and SM states via an off-shell W , which may provide a particularlypromising signal at the LHC [37].The decays of G -color glueballs are fairly spectacular, occuring primarily into elec-troweak bosons as well as the Higgs. The final states of these decays are rich in jets,leptons, and photons, with the most likely discovery channels coming from two- or four-photon resonances. In the parameter range of interest, Λ G (cid:38)
50 GeV, the distinctivelystringy signatures of the original quirk scenario [11] are less apparent due to the high scaleof confinement, but instead the decays occur promptly in the detector and are by no meansuninteresting.Of course, definitively connecting the detection of a confining hidden sector at the LHCto axion-assisted electroweak baryogenesis is a much more delicate issue. Although the ob-servation of G -quarks and G -color glueballs would not be an unequivocal indicator, theirpresence near the weak scale would be a compelling suggestion that hidden sector physicsmay be relevant to electroweak baryogenesis. Further evidence would be provided if theparameters of new physics indicated that the electroweak phase transition may have beensufficiently strongly first-order. For example, in the case of the Minimal SupersymmetricSM (MSSM), a Higgs mass below 127 GeV and stop mass below 120 GeV would be sug-gestive of a first-order electroweak phase transition sufficient for electroweak baryogenesis[38]. As f G < ∼ M P l , the G -axion can be no lighter than m a ∼ − eV. This puts it outside therange of the interesting gravitational signatures of ultra-light axions considered in [1]. The G -axion does develop a coupling to (cid:126)E · (cid:126)B at two loops, raising the prospect of interestingsignatures from decays into photon pairs, but in general this rate is far too small to be ofobservable interest. For the most part, the prospects for detecting the G -axion throughdirect or indirect means are not terribly better or worse than those for the QCD axion.It should be noted that the effective θ angle for SU (2) induced by Eq.(3.9) is invisibleto measurements in the current era irrespective of the action of the G -axion, owing to thefact that a time-independent weak θ angle may be eliminated by ( B + L ) transformations;even in the presence of ( B + L )-violating irrelevant operators, the effective θ angle is highlysuppressed. Only the time-dependent angle during baryogenesis is of any relevance tophenomenology in this case. It is amusing to note that the addition of weak-scale fermions coupled to the Higgs maysignificantly increase the strength of the electroweak phase transition [39]. In general, theeffects are greatest for λv (cid:29) µ . Making the SM phase transition truly first order requires(for a moderate number of G -color degrees of freedom) strong coupling to the Higgs oforder λ (cid:38) .
5; such strong couplings raise the unpleasant prospect of Yukawa coupling– 16 –andau poles several decades above the weak scale. It is clear that, given the range of λ and µ considered here, the G -color sector cannot render the phase transition strongly first-order on its own. Nonetheless, the presence of G -fermions coupled to the Higgs certainlyincreases the strength of the phase transition. Such effects may be even more significant inthe supersymmetric case, where the MSSM phase transition may already be weakly firstorder in some regions of parameter space. Indeed, they may even raise the bounds on Higgsand stop masses required for a first-order electroweak phase transition above ∼
120 GeV.
6. Supersymmetric Axion-assisted Electroweak Baryogenesis
Although we have been working thus far in a nonsupersymmetric context, it is fairlystraightforward to generalize our results from the SM to the MSSM. In this case thefermions
Q, Q, U, U may be promoted to superfields with superpotential terms W G = µ Q QQ + µ U U U + λH u QU + λ (cid:48) H d QU (6.1)Assuming the squarks and sleptons are heavier than a few hundred GeV, the existingconstraints from colliders, cosmology, and precision electroweak considered in previoussections are essentially unaltered.It is important to emphasize that the vector masses µ Q , µ U may be generated by thesame physics that gives rise to the supersymmetric Higgs sector µ -term µ H H u H d , andhence naturally lie around the weak scale. This provides a natural explanation for thecoincidence problem between the scale of G -color and SM sectors.Assuming unbroken R parity, the lightest supersymmetric particle (LSP) will be stableand may contribute to dark matter relic abundance. The LSP in a supersymmetric theorydepends on the scale of supersymmetry breaking and its communication to both the G -color sector and the MSSM. One potential candidate for a G -color-sector LSP is the G -colorgluino, leading to the formation of G -color gluino-gluon hadrons. Another candidate is the G -color axino, the fermionic superpartner of the G -axion. In both cases, the relic abundanceis in general incalculable owing to the strong G -color interactions, but both hidden sectorLSP candidates may lead to consistent cosmologies and dark matter abundances. In theevent that the LSP arises in the G -color sector, MSSM LOSP decays into the hidden sectorLSP are expected to give rise to the usual host of interesting hidden valley signals. It has been observed [30, 40] that a vector-like fourth generation carrying SM quantumnumbers and coupling to the Higgs may ameliorate the little hierarchy problem of theMSSM by virtue of their additional radiative contributions to the lightest Higgs mass.Strictly speaking, nothing requires this generation to carry SM quantum numbers alone; G -quarks carrying electroweak quantum numbers and coupling to the Higgs may play thesame role as a vector-like fourth generation. Crucially, G -color quark loop contributionsto the lightest Higgs mass will be N enhanced, akin to the color enhancement of fourth-generation vector quarks. For moderate values of tan β and λ ∼ O (1), vector masses inthe range 300 GeV < µ <
550 GeV may suffice to naturally push the Higgs mass above– 17 –14 GeV. Conveniently, this mass range is ideally suited to generating the observed baryonasymmetry and distinctive G -color phenomenology at the LHC. We find this amusing.
7. Conclusions
In this work we have explored a mechanism to generate the additional CP violation requiredfor successful EW baryogenesis. If there exists a hidden gauge sector with a strong couplingscale close to the electroweak scale, then the CP-violating strong dynamics of the hiddensector θ -term may feed into the SM sector and induce an effective chemical potential forbaryon number in the early universe, allowing non-perturbative (“sphaleron”) processes inthe SM to generate the observed baryon asymmetry. Communication of CP violation to theSM requires messenger states charged under both the confining hidden sector gauge groupand, at least, SU (2) × U (1) Y . The mechanism is efficient when the hidden strong scale isat or slightly above the electroweak scale, and the messenger states are also parametricallyclose to the electroweak scale. These scales are bounded from below by collider searchesand bounded from above by the relaxation of the hidden sector θ term. In this case theeffective CP-violating chemical potential in the SM is large precisely due to the combinationof the strong dynamics of the hidden sector and the fast (compared to the Hubble time atelectroweak temperatures) dynamics of the EW phase transition. Thus, for a sufficientlyout-of-equilibrium EW phase transition, large CP violation can easily be induced by ourmechanism, independent of the CP violation in the SM, or MSSM, sector. At temperaturesparametrically below the electroweak scale the CP violation turns off due to the relaxationof the hidden θ -angle by the hidden sector axion (which is heavier than the QCD axionassuming equal decay constants).Simple supersymmetric and non-supersymmetric toy realisations of this mechanism in-volving SU (2) L × U (1) Y -charged vector-like states also charged under the hidden group werepresented. The hidden sector, including the matter messengers in particular, easily evadecurrent experimental and observational constraints, but the messengers are experimentallyaccessible at the LHC over the preferred parameter space. The collider phenomenology ofthe messengers and hidden sector gauge fields has much in common with “hidden valley” or“quirk” scenarios, while the connection of the hidden sector and messenger mass scales withthe EW scale via successful axion-assisted EW baryogenesis provides a reason why suchnew, hidden-valley, physics sits parametrically close to the weak scale. Intriguingly, in thesupersymmetric case the messenger matter naturally solves the little hierarchy problem ofthe MSSM when the messengers are at a mass scale that allows efficient axion-assisted EWbaryogenesis. Finally, our axion assistence mechanism favours the natural string value ofthe axion decay constant, f G ∼ GeV, implying that the hidden axion should comprisea significant fraction of the observed dark matter density.
Acknowledgements
We would like to thank Savas Dimopoulos, Sergei Dubovsky, Peter Graham, Juan Garcia-Bellido, Lawrence Hall, and Jay Wacker for useful discussions. JMR thanks the StanfordInstitute for Theoretical Physics and the Berkeley Center for Theoretical Physics for their– 18 –ind hospitality during a Visiting Professorship at Stanford during the initiation of thiswork; NC thanks the Dalitz Institute for Fundamental Physics and the Department ofTheoretical Physics, Oxford University for hospitality during the completion of this work.NC is supported by the NSF GRFP and the Stanford Institute for Theoretical Physics underNSF Grant 0756174. JMR is partially supported by the EU FP6 Marie Curie Researchand Training Network UniverseNet (MPRN-CT-2006-035863), by the STFC (UK), and bya Royal Society Wolfson Merit Award.
A. Axion mass and evolution
At finite temperature, the axion mass may be determined (in the limit T (cid:29) Λ G ) byintegrating over instantons of all sizes, m a f G = (cid:82) dρ n ( ρ ), where ρ is the instanton size and n ( ρ ) the number density. Using the dilute-instanton-gas approximation [26], one finds thezero-temperature result n ( ρ, T = 0) = C N ρ (cid:18) πg (cid:19) N N f (cid:89) i ξρm i exp (cid:0) − π g (cid:1) (A.1)where where ξ = 1 . C N = (0 . ξ − ( N − ( N − N − , and g is given by the renormalizationgroup improved running coupling at the scale ρ ,4 π g ( ρ ) = 16 (11 N − N f ) ln(1 /ρ Λ) + 12 [17 N − N f (13 N − / N ](11 N − N f ) ln ln(1 /ρ Λ) + O (1 / ln ρ Λ)(A.2)The finite-temperature density is related by n ( ρ, T ) = n ( ρ, × exp (cid:18) − (cid:26) λ (2 N + N f ) + 12 A ( λ )[1 + 16 ( N − N f )] (cid:27)(cid:19) (A.3)where λ = πρT , A ( λ ) (cid:39) − ln(1+ λ / α (1+ γλ − / ) − , α = 0 . γ = 0 . T (cid:29) Λ G : m a ( T ) = Λ G f G (cid:18) m Q Λ G (cid:19) N f (cid:18) Λ G T (cid:19) (11 N + N f − I (A.4)where I ≡ . N − N − (cid:18) N − N f (cid:19) N ξ N f − N +2 (A.5) × (cid:90) ∞ x (11 N + N f − ln ( T /x Λ G ) N − a exp[ f ( x )] dx Here f ( x ) ≡ − π x (2 N + N f ) + 16 (6 + N − N f ) (cid:104) ln(1 + π x / − α (1 + γ/π / x / ) − (cid:105) (A.6)– 19 – (cid:45) (cid:45) (cid:72) t (cid:144) t G (cid:76) log (cid:64) H (cid:144) m a (cid:68) Θ G (cid:145) Θ G , i Figure 3:
The evolution of θ G as a function of time (where t G = t ( T = Λ G )) for µ = 300 GeV,Λ G = 400 GeV, f G = 10 GeV, where we have taken for m a ( T ) a smooth interpolation between m a ( T (cid:29) Λ G ) and m a ( T = 0). The straight line denotes log [3 H ( T ) /m a ( T )]. Note that θ G beginsto relax well after 3 H ( T ) = m a ( T ). and a = [17 N − N f (13 N − / N ](11 N − N f ) . This expression gives the correct finite-temperature axionmass in the limit m Q (cid:29) Λ G by taking N f → G must then berelated to Λ G,UV via the usual scale-matching). The instanton density n ( ρ, T = 0) mayalso be used to obtain an estimate of the zero-temperature axion mass, but here the issue ismore delicate. While at finite temperature it is possible to integrate over instantons of allsizes because T provides the appropriate large-distance cutoff, at zero temperature thereis no such cutoff. At best, one may estimate the zero-temperature mass by assuming thatinstantons of size ρ ∼ G dominate, which gives the parametrically correct result.Having determined the axion mass for (at least for T = 0 and T (cid:29) Λ G ), we may nowturn to the evolution of θ G . This is of critical importance in determining the scale at whichCP violation from the G -color sector begins to decrease. The equation of motion for θ G issimply ¨ θ G + 3 H ˙ θ G + m a ( T ) θ G = 0 (A.7)(where we have disregarded the negligible Γ a ˙ θ G term). This evolution becomes interestingthanks to the temperature dependence of the axion mass, which becomes important when m a ( T ) (cid:38) H , at which point coherent oscillations of θ G commence. We may solve theequations of motion numerically, assuming a functional form for m a ( T ) that smoothlyinterpolates between m a ( T (cid:29) Λ G ) and m a ( T = 0) . A representative evolutionary historyfor θ G is shown in Figure 3. There are two details worth emphasizing: The first is that θ G begins to evolve significantly after the point 3 H ( T ) = m a ( T ) is reached, by as much as anorder of magnitude. The second is that this evolution commences after t G ≡ t ( T = Λ G ),i.e., after the confining phase transition, despite the fact that 3 H ( T ) = m a ( T ) occurs before t G . Both considerations become significant when determining constraints on axion-assistedelectroweak baryogenesis coming from the relaxation of θ G .– 20 – eferences [1] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, StringAxiverse , arXiv:0905.4720 .[2] A. Arvanitaki, N. Craig, S. Dimopoulos, S. Dubovsky, and J. March-Russell, String Photiniat the LHC , Phys. Rev.
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