Geometric Baryogenesis from Shift Symmetry
SSISSA 64/2016/FISI
Geometric Baryogenesis from Shift Symmetry
Andrea De Simone, ∗ Takeshi Kobayashi, † and Stefano Liberati ‡ SISSA and INFN Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy
We present a new scenario for generating the baryon asymmetry of the universe that is inducedby a Nambu–Goldstone (NG) boson. The shift symmetry naturally controls the operators in thetheory, while allowing the NG boson to couple to the spacetime geometry as well as to the baryons.The cosmological background thus sources a coherent motion of the NG boson, which leads tobaryogenesis. Good candidates of the baryon-generating NG boson are the QCD axion and axion-like fields. In these cases the axion induces baryogenesis in the early universe, and can also serve asdark matter in the late universe.
Introduction. — The excess of matter over antimatterin our universe is crucial for our very existence, and is wellsupported by various observations. In particular, mea-surements of the cosmic microwave background (CMB)give the ratio between the baryons and the entropy of theuniverse as n B /s ≈ . × − [1]. However the originof this baryon asymmetry still remains unexplained.In this letter we present a natural framework for creat-ing the baryon asymmetry by a Nambu–Goldstone (NG)boson of a spontaneously broken symmetry which weneed not specify. The guiding principle here is the shiftsymmetry of the NG boson, or an approximate one fora pseudo Nambu–Goldstone (pNG) boson. We arguethat a NG boson coupled to various forces through shift-symmetric operators naturally comes equipped with thebasic ingredients for a successful baryogenesis.From the point of view of shift symmetry, linear cou-plings of a NG boson to total derivatives, such as to topo-logical terms, are not forbidden. Thus with gauge fields,a NG boson can acquire dimension-five operators of theform φF ˜ F . In particular with SU(2) gauge fields, such aterm gives rise, through the anomaly equation, to a cou-pling to the divergence of the baryon current, i.e. φ ∇ µ j µB .On the other hand, gravity also provides a shift-symmetric mass-dimension-five operator φ G , with G = R − R µν R µν + R µνρσ R µνρσ being the topological Gauss–Bonnet term. In an expanding universe, the Gauss–Bonnet coupling yields an effectively linear potential forthe massless NG boson and sources a coherent time-derivative of the NG condensate. This, through itscoupling to the baryon current, shifts the spectrum ofbaryons relative to that of antibaryons, and thereforeallows baryogenesis even in thermal equilibrium whenbaryon number nonconserving processes occur rapidly.In other words, the NG boson mediates the effect of thespontaneous breaking of Lorentz invariance in an expand-ing universe to a shift in the baryon/antibaryon spectra.We will also show that this scenario can be realizedwith the QCD axion, in which case the axion providesthe baryon asymmetry and dark matter in our universe,as well as solve the strong CP problem.Although the mechanism of generating the baryons bythe spontaneous breaking of Lorentz invariance (or CP T symmetry [2]) has been investigated in the past, our sce-nario is quite distinct from the previous studies. “Spon-taneous baryogenesis” [3] is driven by a massive scalarderivatively coupled to the baryon current, with a masstypically as large as m (cid:38) GeV [4, 5]. However sucha scalar condensate can ruin the subsequent cosmologi-cal expansion history. Moreover, the spatial fluctuationof the scalar seeded during inflation produces baryonisocurvature perturbations [6], which are tightly con-strained from CMB measurements. These observationsconstrain the model parameters to lie within a rathernarrow window [5]. On the other hand, in our scenariothe (p)NG boson is (nearly) massless. The small massmakes the boson long-lived, and even allows the baryon-generating pNG boson to play the role of dark matter.The shift symmetry further suppresses the baryon isocur-vature much below the observational bounds.We should also remark that the gravitational back-ground playing an important role in our scenario is remi-niscent of “gravitational baryogenesis” [7], which invokesa derivative coupling between the Ricci scalar and thecurrent, ( ∂ µ R ) j µB . Such a term seems somewhat ad hoc in the sense that gravity is assumed to distinguish be-tween matter and antimatter, however it might arisewith the aid of mediators. Phenomenologically, gravita-tional baryogenesis typically requires a quite high cosmictemperature, and also a trace anomaly for the energy-momentum tensor in order to have a non-vanishing ∂ t R in a radiation-dominated universe. In contrast, the cos-mic temperature in our scenario can be lowered due tothe direct coupling between the NG boson and the baryoncurrent. Furthermore, since G does not vanish during ra-diation domination, our scenario need not rely on traceanomalies.Let us also note the crucial difference with the modelof [8] which considered a coupling ( ∂ µ G ) j µB . Such a termintroduces higher derivative terms in the equations of mo-tion which can lead to ghost instabilities. On the otherhand, the φ G coupling of the NG boson does not yieldhigher derivatives, and thus does not introduce extra de-grees of freedom except for φ itself. Baryogenesis with a NG Boson. — Following the abovearguments, we consider a theory of a shift-symmetric NG a r X i v : . [ h e p - ph ] M a r scalar φ linearly coupled to the divergence of the baryoncurrent, as well as to the Gauss–Bonnet term, describedby the Lagrangian L√− g = M p R − ∂ µ φ ∂ µ φ + φf ∇ µ j µB + φM G + · · · . (1)Here f and M are mass scales suppressing the dimension-five operators, and ∇ µ is a covariant derivative. Wehave specified the relative sign of the two coupling termsfor simplicity; this sign at the end determines whetherbaryons or antibaryons are created.The derivative coupling to the baryon current can orig-inate from the anomalous couplings to the SU(2) gaugefields (in such a case the coupling term is effective whensphalerons are in equilibrium [9]); alternatively, the termcould directly be generated upon spontaneous symmetrybreaking, as in the example of [10]. The gravitationalcoupling may also arise from the symmetry breaking, asin this case M would be naturally associated to the co-herence length of the NG condensate.The NG boson may further couple to the lepton cur-rent, then the produced lepton asymmetry can later beconverted to the baryons; for the purpose of our discus-sion it suffices to just display the baryon current. Regard-ing gravity, a mass-dimension-five Chern–Simons cou- pling φR ˜ R also preserves the shift symmetry of φ [11],however we omit this term since R ˜ R vanishes in a FRWuniverse. Purely from the point of view of shift symme-try, there can also be φ ∇ R , or terms equivalent to thisup to total derivatives. However such terms introduceghostly extra degrees of freedom, and thus we do not ex-pect them to result from a symmetry breaking of a stabletheory [12].Shift-symmetric operators other than those shown arecontained in the dots in (1). We consider them to havesmaller effects on the φ dynamics compared to φ G /M ,either because the coupled non-gravitational fields arenot expected to have large vacuum expectation val-ues, or the operators have mass-dimensions higher thanfive. A pNG φ can also obtain a (possibly temperature-dependent) potential from some nonperturbative effects.For the moment we assume such a potential to be neg-ligible during baryogenesis, until later when we discussthe possibility of φ being an axion. The Lagrangian ofmatter fields other than φ is also included in the dots.Varying the Lagrangian (1) in terms of g µν and drop-ping total derivatives gives the Einstein’s equation (if j µB is a fermion current one should instead use vierbeins,however this actually does not affect the results [5]), M p G µν = T φ ( µν ) + T G ( µν ) + T dots( µν ) , T φµν = g µν (cid:18) − ∂ ρ φ ∂ ρ φ − ∂ ρ φf j ρB (cid:19) + ∂ µ φ ∂ ν φ + 2 ∂ µ φf j Bν ,T G µν = 4 M (cid:0) R ∇ µ ∇ ν φ − g µν R ∇ ρ ∇ ρ φ + 2 R µν ∇ ρ ∇ ρ φ − R ρµ ∇ ρ ∇ ν φ + 2 g µν R ρσ ∇ ρ ∇ σ φ − R ρ σµ ν ∇ ρ ∇ σ φ (cid:1) . (2)Here T ( µν ) = ( T µν + T νµ ), and T dots( µν ) represents thecontributions from the dots in (1). We also used that G g µν − RR µν + 4 R ρµ R νρ + 4 R ρσ R ρµσν − R ρστµ R ρστν vanishes in four spacetime dimensions as a consequenceof the generalized Gauss–Bonnet theorem.Considering a flat FRW universe, ds = − dt + a ( t ) d x , the Gauss–Bonnet term is expressed in termsof the Hubble rate H = ˙ a/a (an overdot denotes a deriva-tive in terms of the cosmological time t ) as G = 24( H + H ˙ H ) . (3)Focusing on the homogeneous mode of the NG scalar, φ = φ ( t ), and ignoring the spatial components of thebaryon current, the Friedmann equation (i.e. (0 ,
0) com-ponent of the Einstein’s equation (2)) reads3 M p H = ˙ φ − ˙ φj B f −
24 ˙ φH M + T dots00 . (4)We suppose the right hand side to be dominated by T dots00 and that φ has a negligible effect on the cosmologicalexpansion; we will evaluate this condition later on. The equation of motion of φ that follows from theterms shown in (1) is0 = ∇ µ ∇ µ φ + ∇ µ j µB f + G M (5)= − a ddt (cid:26) a (cid:18) ˙ φ − j B f − H M (cid:19)(cid:27) . (6)Neglecting for the moment the term with the baryon cur-rent, the velocity of the scalar is obtained as˙ φ = 8 H M + const . × a − . (7)On the right hand side, during inflation when H is nearlyconstant, the second term is expected to become neg-ligibly tiny compared to the first one. After inflation, H redshifts as a − during radiation domination, and as a − / during matter domination, hence the second termgrows relative to the first. Which term dominates duringbaryogenesis is set by the initial condition of ˙ φ , whichin turn is determined by the details of spontaneous sym-metry breaking. Here for simplicity, we assume that thetwo terms are comparable in magnitude at the begin-ning of inflation; then one can easily check that, even ifthe duration of inflation is just enough to solve the hori-zon problem, the first term dominates over the secondthroughout the post-inflationary era until today. Hencehereafter we ignore the a − term in (7).Since the time component of the baryon current de-notes the baryon number density, i.e. j B = n B , one seesfrom the energy-momentum tensor (2) that a nonzero ˙ φ gives a contribution to the energy density as ∆ T φ = − n B ˙ φ/f , hence shifts the energy level of baryons rela-tive to that of antibaryons. When the particles are inthermal equilibrium, this can be interpreted as a particleof type i with baryon number B i obtaining an effectivechemical potential of µ i = B i ˙ φf = 8 B i H f M , (8)and likewise for its antiparticle but with an oppositesign. Thus if some baryon number violating process isin equilibrium during a radiation-dominated epoch, abaryon asymmetry is produced. Supposing the particlesto be relativistic fermions and ignoring their masses, thebaryon density is obtained from the Fermi–Dirac distri-bution as n B = (cid:88) i B i g i µ i T (cid:26) O (cid:16) µ i T (cid:17) (cid:27) , (9)where the sum runs over all particle/antiparticle pairs i coupled to φ , and g i counts the internal degrees of free-dom of the (anti)particle i [13]. Using the expressionsfor the Hubble rate 3 M p H = ( π / g ∗ T and entropydensity s = (2 π / g s ∗ T during radiation domination,the baryon-to-entropy ratio is obtained as n B s = π (cid:80) i B i g i √ g / ∗ g s ∗ T f M M p . (10)This ratio freezes out when the baryon violating interac-tions fall out of equilibrium. Using a subscript “dec” todenote evaluation at the decoupling of the baryon violat-ing interactions (and in particular T dec for the decouplingtemperature), the ratio ( n B /s ) dec should coincide withthe current value of 8 . × − .We have only considered the homogeneous mode of φ inthe above discussions, however the φ field can also pos-sess spatial fluctuations seeded during inflation. Here,note that the baryon asymmetry (10) is independent ofthe field value of φ as a consequence of the shift symme-try; therefore the φ fluctuations do not directly propa-gate into baryon isocurvature perturbations (see also [14]where a related idea was investigated). Still the baryonisocurvature is not strictly zero since the φ fluctuationsare not completely frozen outside the horizon and thusyields fluctuations in ˙ φ . However this effect is suppressed by powers of ( k/aH ) for a comoving wave number k ,which can easily be checked by solving the full equationof motion (5) starting from a Bunch–Davies initial con-dition. Hence the resulting baryon isocurvature is ex-tremely small on CMB scales which are far outside thehorizon at decoupling, being compatible with the non-observation of isocurvature. Backreaction and Consistency. — We now analyze theconditions under which the above calculations can betrusted.In φ ’s equation of motion (6), the term j B /f whichwe have neglected represents the backreaction of the pro-duced baryons on φ . Comparing the last two terms in (6)and substituting for j B from the above calculations, onefinds that the baryon backreaction can be neglected upondecoupling if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) H M (cid:19) − j B f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dec = (cid:80) i B i g i T f (cid:28) . (11)This is basically a requirement that the decoupling tem-perature should be lower than the cutoff f . Violation ofthis condition would signal the breakdown of the effectivefield theory.The effect of the φ condensate on the cosmological ex-pansion can be neglected if its contribution to the Fried-mann equation (4) is much smaller than the total densityof the universe. This imposes, at the time of decoupling, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M p H (cid:32) ˙ φ −
24 ˙ φH M (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dec = 1603 H M M p (cid:28) . (12)Here we substituted the solution for ˙ φ , and also omitted( ˙ φ/f ) j B as it is guaranteed to be smaller than the otherterms under (11).One can also carry out a power counting estimate ofthe cutoff scale from φ G /M along the lines discussedin [15]. Requiring the cutoff to be higher than the rel-evant energy scales gives a condition somewhat similarto (12), although a naive power counting may be mis-leading for a Gauss–Bonnet term. In the following dis-cussions, we adopt (12) as the bound on M . Let us alsoremark that even when H > M , the condition (12) is notnecessarily violated; however, if higher dimensional grav-itational couplings are universally suppressed by M (e.g.( R/M )( ∂φ ) ), then their contributions may become im-portant.The decoupling scale is also bounded from above by theinflation scale H inf , which is constrained by observationallimits on primordial gravitational waves; the Planck con-straint [16] yields H dec < H inf (cid:46) × GeV . (13)The viable parameter space in the f – T dec plane isshown in Figure 1. Here we have chosen (cid:80) i B i g i = 1
11 12 13 14 15 16 17 181112131415161718 log ( f [ GeV ]) l og ( T d ec [ G e V ] ) M = G e V M = G e V M = G e V exceeds Planck boundon inflation scale s i g n i fi c a n t b a r y o n b a c k r e a c t i o n FIG. 1. Parameter space in the f – T dec plane. The coloredregions are excluded due to significant backreaction from thebaryons (red), and by the Planck upper bound on the inflationscale (green). The allowed parameter space is shown in white.The black lines indicate where the right amount of baryonasymmetry is produced, for the choice of M = 10 GeV(solid), 10 GeV (dashed), 10 GeV (dotted). and g ∗ ( T dec ) = g s ∗ ( T dec ) = 106 .
75, and the colored re-gions denote where the conditions are violated; the redregion is excluded due to significant baryon backreac-tion (cf. (11)), and the green region is excluded by the
Planck bound on the inflation scale (cf. (13)). The blacklines indicate where the correct amount of baryon asym-metry ( n B /s ) dec ≈ . × − is achieved (cf. (10)),for M = 10 GeV (solid), 10 GeV (dashed), 10 GeV(dotted). For these choices of M , the condition (12) fromthe gravitational backreaction is comparable to or weakerthan the inflation bound (13), and thus not shown in thefigure. For smaller M , the line of ( n B /s ) dec ≈ . × − moves towards smaller T dec ; the condition (12) does notcut off the line within the ranges of f and T dec shown inthe figure, however for M (cid:46) GeV, the allowed valuesfor H dec exceed M and thus higher dimensional gravita-tional operators may become relevant.Further constraints can be imposed on the parameterspace depending on the nature of the NG boson. Let ussee this directly in the following examples. QCD Axion and Axion-Like Fields. — Here we discussthe possibility that φ is the QCD axion [17] which pro-vides a solution to the strong CP problem. Then inaddition to the linear potential sourced by the Gauss–Bonnet coupling, the axion obtains a periodic potentialfrom non-perturbative QCD effects as V QCD ( φ, T ) = m ( T ) f a (cid:26) − cos (cid:18) φf a (cid:19)(cid:27) . (14)Here f a is the axion decay constant, and the temperature- dependent mass is m ( T ) ≈ . × m a (cid:18) Λ QCD T (cid:19) for T (cid:29) Λ QCD , m a for T (cid:28) Λ QCD , (15)with m a ≈ × − eV (cid:0) GeV /f a (cid:1) , and Λ QCD ≈
200 MeV. Focusing on the field range | φ | (cid:46) f a , then com-parison of V QCD (cid:39) m ( T ) φ with the Gauss–Bonnetcoupling φ G /M in a radiation-dominated universe showsthat the latter dominates over the former at temperatures T (cid:38) GeV (cid:18) | θ ( T ) | Mf a (cid:19) / , (16)where we used θ ≡ φ/f a . As the right hand side dependsweakly on θM/f a , we see that as long as T dec (cid:38) GeVthe QCD effect is negligible during baryogenesis. On theother hand, the Gauss–Bonnet coupling has become neg-ligible by the time the axion starts oscillating along itsQCD potential, which typically occurs at T osc ∼ θ = G f a m a M ∼ − f a M , (17)which (unless for an extremely tiny M ) is much smallerthan the observational bound | θ | (cid:46) − from limitson the neutron electric dipole moment [18]. Thus thebaryon-generating axion solves the strong CP problem.However the QCD axion φ may overclose the universe,as its abundance relative to cold dark matter (CDM) isgiven as [19] Ω φ Ω CDM ∼ θ (cid:18) f a GeV (cid:19) / , (18)where θ osc is the field value at the onset of the axion os-cillations. If f a = f , and taking for instance the allowedvalues on the black lines in Figure 1, then the axion islong-lived and Ω φ can exceed unity. One way to avoidthis is by fine-tuning the misalignment θ osc to a tiny value(perhaps from anthropic reasoning). However the neces-sary fine-tuning is actually more severe when taking intoaccount the axion isocurvature perturbations [20]; in or-der for the total CDM isocurvature to be below the CMBlimit [16], the axion can constitute only a small fraction ofthe entire CDM. Moreover, since the axion field evolves inthe early times due to the Gauss–Bonnet coupling, thisfield excursion should also be taken into account upontuning the initial field value.Alternatively, M could take a low value, providedthat higher dimensional gravitational couplings are some-how suppressed. Then, for instance, M (cid:46) GeV al-lows baryogenesis without significant backreaction with f a ∼ f ∼ GeV; and without fine-tuning the align-ment, i.e. θ osc ∼
1, the QCD axion can generate thebaryon asymmetry as well as constitute the entire CDM.For these parameters, the CDM isocurvature can also beconsistent with observational limits.We also comment on the possibility of φ being one ofthe axion-like fields arising from string theory compactifi-cations [21]. In the simplest case, such a field is describedby a periodic potential (14) with a constant mass m ; thenits abundance is computed asΩ φ Ω CDM ∼ θ (cid:18) f a GeV (cid:19) (cid:16) m − eV (cid:17) / . (19)For example, with θ osc ∼ f a ∼ f ∼ GeV, and m ∼ − eV, the axion-like φ can serve as CDM andgenerate the baryons, cf. Figure 1. One can also checkthat if further M (cid:46) GeV, the corresponding decou-pling temperature allows inflation scales that give CDMisocurvature below the current limit. Such an ultralightaxion CDM is also interesting from the point of view thatit can produce distinct signatures on small-scale struc-tures [22].
Discussion. — Without some extra symmetries, thereis no a priori reason to forbid a NG boson from acquir-ing shift-symmetric couplings to other fields. While mostcoupled fields do not induce coherent effects, the back-ground gravitational field of an expanding universe givesrise to a coherent motion of the NG boson. We haveshown that this leads to the creation of a net baryonasymmetry of the universe. Good candidates for thebaryon-generating NG boson are the axion(-like) fields.This raises the intriguing possibility that an axion couldinduce baryogenesis in the early universe, then serve ascold dark matter in the later universe (and further solvethe strong CP problem if it is the QCD axion).Let us comment on the observable consequences of ourscenario. Theories of a scalar coupled to the Gauss–Bonnet term are known to evade no-hair theorems forblack holes [23], which may be tested by gravitationalwave observations. We also note that if M is not farfrom M p , the corresponding high decoupling tempera-ture implies a high inflation scale, yielding primordialgravitational waves that could be observed by upcom-ing experiments. Furthermore, couplings between thetime-dependent NG boson and parity violating termssuch as F ˜ F may leave signatures in cosmological obser-vations [24, 25]. It would also be interesting to study theexperimental implications of the required baryon violat-ing interactions.We would like to thank Stefano Bertolini, LathamBoyle, Shunichiro Kinoshita, Marco Letizia, ShuntaroMizuno, and Thomas Sotiriou for helpful conversations.T.K. acknowledges support from the INFN INDARKPD51 grant. S.L. wishes to acknowledge the John Tem-pleton Foundation for the supporting grant ∗ [email protected] † [email protected] ‡ [email protected][1] P. A. R. Ade et al. [Planck Collaboration], Astron. Astro-phys. , A13 (2016) [arXiv:1502.01589 [astro-ph.CO]].[2] In local effective field theories the violation of CP T sym-metries implies that of local Lorentz invariance, whileLorentz breaking operators can be either
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