CP Violation and Baryogenesis in the Presence of Black Holes
aa r X i v : . [ h e p - t h ] M a y UTTG-09-15 TCC-002-15 RUNHETC-2015-7 SCIPP 15/06
CP Violation and Baryogenesis in the Presence of BlackHoles
Tom BanksDepartment of Physics and SCIPPUniversity of California, Santa Cruz, CA 95064 and
Department of Physics and NHETCRutgers University, Piscataway, NJ 08854E-mail: [email protected]. FischlerDepartment of Physics and Texas Cosmology CenterUniversity of Texas, Austin, TX 78712E-mail: fi[email protected]
Abstract
In a recent paper[1] Kundu and one of the present authors showed that there weretransient but observable CP violating effects in the decay of classical currents on thehorizon of a black hole, if the Lagrangian of the Maxwell field contained a CP violatingangle θ . In this paper we demonstrate that a similar effect can be seen in the quantummechanics of QED: a non-trivial Berry phase in the QED wave function is produced byin-falling electric charges. We also investigate whether CP violation, of this or any othertype, might be used to produce the baryon asymmetry of the universe, in models whereprimordial black hole decay contributes to the matter content of the present universe.This can happen both in a variety of hybrid inflation models, and in the HolographicSpace-time (HST) model of inflation[2]. The CP violating term θ QED e π Z F ∧ F, (1)1s a total derivative, R F ∧ F = R B A ∧ F, where B is the boundary of space-time. It iswidely believed that this term has no effect on physics, except in the presence of magneticmonopoles[3]. However in [1], Kundu and WF showed that the θ QED term induces a vorticityin the transient flow of classical current induced on a black hole horizon by a charge fallingthrough it.There is no real contradiction with conventional wisdom, because it is obvious thatfor supported observers outside the black hole, the horizon is a non-trivial boundary. This isthe basis of the Membrane Paradigm[4]. Indeed, the vorticity observed by [1] was proportionalto the Chern-Simons invariant on the horizon.Our aim in the first part of this paper is to show that similar effects can be seen in thequantum mechanics of electrodynamics in the presence of horizons. As first noted by Jackiw[5],in temporal gauge the effect of the θ QED term is to induce a Berry phase[6] into the wavefunctional. The phase is proportional to the Chern-Simons term and can have observableeffects if there is a non-trivial closed loop in the configuration space of fields. We will show thatthe infall of charged particles produces such a closed loop.One may ask whether these phenomena might have some effect in cosmology. More generally,one expects CP violating physics to be important near the horizon of a black hole, if there areCP violating terms of any dimension in the effective Lagrangian. Since black hole decay violatesbaryon number, it is interesting to ask if a population of primordial black holes could contributeto the baryon asymmetry of the universe. In the late 70’s Barrow[7] started to investigatebaryogenesis from primordial black holes. Hook[8] has recently explored this possibility. Heworks in the context of hybrid inflation models, some of which produce a primordial black holepopulation and uses the mechanism of spontaneous baryogenesis [9].The point is that, to leading order, the emission of Hawking radiation is a thermal process,and CPT prevents thermal emission of a baryon excess. Hook uses the violation of CPT by theexpansion of the universe to overcome this objection. In fact however, by the very nature ofthe fact that it is a decay , Hawking radiation is not exactly thermal. The Hawking temperaturechanges with time and the emission spectrum is not exactly thermal. We should expect abaryon excess to be generated at time t , proportional to some positive power of | dT /dt | . Wewill argue, in a very general way that the power is 1.The implications of Hawking induced baryogenesis for hybrid inflation models depends ondetails of the model. The primordial black hole density and the reheat temperature of theuniverse depend on different regions in the inflaton potential. The baryon asymmetry producedin black hole decay will also be model dependent. Thus one must explore a range of modelsto find whether there is a “plausible” regime where the correct baryon asymmetry is producedwithout violating other constraints.By contrast, in the HST model of the early universe, a primordial population of black holesis produced automatically, and dominates the energy density of the universe until the blackholes decay. The radiation dominated Big Bang is produced by black hole decay, and thereheat temperature of the universe is related to the size of primordial curvature fluctuationsby a formula involving a small number of universal parameters[2]. The amount of baryonasymmetry generated depends on one new effective parameter, the ratio of CP violating to CPconserving matrix elements of the Hamiltonian responsible for black hole decay. We show thatif this ratio is o (1) we can reproduce the observed asymmetry within the uncertainties of thecalculation . Given these estimates, it is unlikely that the CP violation induced by θ QED , which The back of the envelope estimates of inflationary parameters given in [2] produce an asymmetry of order α em π could be responsible for the baryon excess, but primordial black holes in theearly universe communicate with much more general sources of CP violation. The addition of the total derivative term to the action does not change the equations of mo-tion, but it changes the canonical momenta as a function of the time derivatives of the vectorpotential. In any coordinate system it shifts the definition of the canonical momentum byΠ i → Π i − θ QED B i . (2)When we choose temporal gauge, in any coordinate system, this shift amounts to δiδA i ( x ) → δiδA i ( x ) − θ QED B i . (3)We can compensate for this shift in the Schrodinger equation by multiplying the wave functionalby a phase Ψ[ A i ] → e iθ QED R A i B i Ψ[ A i ] . (4)The phase is the Chern-Simons action of the three dimensional gauge fields. As pointed outby Berry[6] such an overall phase can lead to observable effects if there is a closed loop inthe configuration space such that the phase does not come back to itself. The fact that θ QED produces a Berry phase in the QED wave function, was pointed out by Jackiw[5].Now let us consider a horizon and study the metric in the near horizon limit. For Schwarzschildblack holes, and the horizon of de Sitter space, the near horizon geometry approaches that ofRindler space . ds → − rdt + dr r + R d Ω . (5)In the above formula, we restrict attention to r >
0. This coordinate system is appropriateto an accelerated detector, which never falls through the horizon. For a supported detector,the horizon is a genuine boundary of space-time and the Membrane Paradigm[4] shows usthat treating it as such captures all of the physics of the horizon, as seen by the detector. Infact, in the Membrane paradigm, the boundary is taken to be the stretched horizon , a timelikehyperboloid whose space-like distance to the horizon is a few Planck lengths.Consider a process in which a charged particle falls through the stretched horizon. Theelectric field E r = ∂ t A r , (6)is, in this gauge, entirely due to a time dependent vector potential A r . We can always write A r = ∂ r W , where W is the open radial Wilson line running from infinity to r . W will betime dependent when we have an infalling charged particle. The initial value of W near r = 0vanishes.Now, for a very large black hole, consider a time ≪ R ln( RM P ) after the particle has passedthrough the stretched horizon [4]. W will have changed by a finite amount. In addition, because − , but the uncertainties in these estimates are large. For genuine Rindler space, the horizon radius R in the formula above goes to infinity and the transversegeometry is flat. F θφ is non-zero there. Notethat so far we have not made a closed loop in configuration space. The phenomena noted in[1], are therefore not connected to the Berry phase. Indeed, the θ QED parameter multiplies atotal time derivative in the action plus a total spatial divergence. The Hall current of [1] isrelated to the integral of the spatial divergence in the bulk (in the temporal gauge this appearsas a modification of Gauss’ law), while the Berry phase is related to the time derivative in theLagrangian. If we now take t larger than the scrambling time, so that F θφ vanishes on thestretched horizon, we have performed a closed loop in configuration space. A r has changedfrom zero to a pure gauge with gauge function equal to the asymptotic value of the Wilson lineon the stretched horizon. The other fields vanish and so does the Berry phase.Now consider the same system in the presence of an external magnetic field, constant intime, with non-zero components F θφ . This field might be generated by a neutron star orbitingthe black hole, or simply a stationary configuration of plasma. In addition, rather than droppingone particle into the hole, we start with no charges, create a pair, separate them and drop eachinto the horizon. The final configuration has only the static magnetic field and A r = ∂ r ( W ) .W is a Wilson line which goes from one point on the stretched horizon to another. Moreprecisely, the r dependent W in the vicinity of the stretched horizon is the Wilson line ona shifted stretched horizon, a distance r from the actual horizon. From the point of view ofthe supported observer, the charged particles take an infinite time to reach the horizon. Theyreach the stretched horizon in a finite time. Using Stokes’ theorem, we can write the Wilsonline describing the creation of the charged particle pair and their subsequent fall through thestretched horizon in terms of a Wilson line connecting them on a fixed surface of constant r .The vector potential A r on the stretched horizon r = xL P is the difference of two such Wilsonlines. After the particles have fallen through the stretched horizon it becomes time independent,and therefore pure gauge. The presence of a boundary at r = xL P and the assumption that thefield F θφ is non-zero on the stretched horizon both before and after the infall tells us that wehave performed a closed loop in the configuration space of fields outside the stretched horizon,with a non-trivial Berry phase.The Berry phase has evolved from 0 to θ QED Z ∂ r W F θφ = θ QED Z dθ d φW F θφ . (7)Here we’ve used the fact that there is no monopole source for the magnetic field. Genericallythis will be non-vanishing because the two points on the horizon can be chosen arbitrarily, ascan the external field. The way in which the horizon provides a non-trivial two sphere in space-time, such that the change in Chern-Simons invariant over this sphere is physical, is reminiscentof the way that black hole horizons force us to include monopole U (1) bundles in the low energyeffective field theory of Maxwell’s equations[10].We have thus demonstrated that quantum processes in the presence of a horizon haveamplitudes which are sensitive to the value of θ QED and thus violate CP. These phases areobviously of order α em . We may pick up contributions far from the horizon, corresponding to the flux of ambient electromagneticfields through the closed loop, but these are irrelevant to the present discussion. Baryogenesis in Hawking Radiation
At first sight, the idea that we could generate a baryon asymmetry in the decay of a blackhole seems wrong even in the presence of CP violation and baryon number violation. To a goodapproximation the radiation is thermal, and CPT guarantees that a thermal distribution hasequal probabilities to be a baryon or an anti-baryon. On the other hand, we are talking abouta decay , and decays violate CPT by their very nature. In a CPT invariant state, every decay isbalanced by a recombination event. Indeed, the quintessential example of a baryogenesis mech-anism is the out of equilibrium decay of an unstable particle. There is indeed no baryogenesisin the unstable equilibrium state (the Hartle-Hawking state) of a black hole in a radiation bath.However, a decaying black hole is only described approximately by this state.The consistency between the two contradictory sentences in the previous paragraph lies inthe phrase “to a good approximation”. Black hole decay is thermal only in the approximationthat its rate is slow, so that the temperature changes adiabatically. Thus, we may contemplateproducing a baryon asymmetry in black hole decay, at a rate which is proportional to somepositive power of the rate of change of temperature dT /dt . The question we need to answer is,“What is the power?”The decay of black holes has a number of features, which distinguish it radically from thedecay of lepto-quarks in old fashioned grand unified models of baryogenesis. There is no sensein which the early universe is an equal mixture of black holes and “anti-black holes” . Wewill first discuss the decay of a single black hole and then argue that we can simply sum thisresult over the gas of black holes produced in the early universe by your favorite mechanism.The eigenstates of the black hole cannot be assigned a baryon number, because baryon numberhas no reason to be even approximately conserved by the quantum gravity dynamics thatdetermines the black hole spectrum. The same is likely to be true for CP . Nonetheless, CPTguarantees that, in the thermal approximation, the expected value of baryon excess vanishes.There are fluctuation corrections to this, but these must be averaged over all the black holes inthe universe, and are typically very tiny .The baryon excess should have a power series expansion in the black hole decay rate d lnM dt ,and CPT indicates that only odd powers contribute . The dominant term will be the firstpower and there is no reason to believe it should be zero. The result for the instantaneouschange in baryon number in a black hole decay should be proportional to the black hole area,the CP violating parameter, and dT /dt the intensive signal of the fact that the decay is out ofequilibrium. We get dBdt = − ǫ CP M M − d ( M/M P ) dt , (8)where ǫ CP is the strength of CP violating processes. Thus, the amount of baryon number Everywhere in what follows, we use B instead of B − L to save writing. Electroweak B violation impliesthat the only interesting primordial asymmetry is that in B − L . Our arguments apply equally well to thisquantum number. Here we consider neutral black holes. We’ll return to the case of charged black holes in the next section. Even in standard model processes, the improbability of CP violation is not a consequence of the intrinsicsmallness of CP violating matrix elements, but rather of the hierarchy of masses and mixings of the quarks. We suggested that these corrections might be sufficiently large in [2] but upon more detailed study we foundthem to be negligible. In the completely unrealistic scenario in which CP violation is small and black hole decay vanishes in thelimit of CP conservation, we would get a square root dependence. B = − ǫ CP ( M/M P ) . (9)This has a universal sign for neutral black holes, and as long as the black hole gas is sufficientlydilute that Hawking particles emitted by one black hole have a negligible probability to beabsorbed by another, it will simply be additive over the whole universe. The initially producedbaryon density is thus ∆ b = ǫ CP n BH ( M/M P ) . (10) n BH is the initial number density of black holes, which is less than M − M P . The black holeswill evolve as a non-relativistic gas until they decay at a time τ ∼ M M P . The inflationary era,whether hybrid or HST, which produced the black holes, will also produce a radiation gas at atemperature T RH . For the HST model, this primordial T RH = 0.At time τ the black hole energy density is M n BH a − ( τ ) , while the radiation energy densityis gT RH a − ( τ ), where g is the effective number of massless particle states into which the blackhole decays. At this point the black holes have all decayed, so we have a radiation gas attemperature gT = gT RH a − ( τ ) + M n BH a − ( τ ) . (11)The baryon to entropy ratio is∆ bσ = ( gT ) − n BH ǫ CP ( M/M P ) a − ( τ ) . (12)There are two simple limits in which we can evaluate these formulae. In the first, the blackhole energy density dominates the primordial radiation at τ and we have∆ bσ = ( g ) − / M − / n / BH ǫ CP ( M/M P ) a − / ( τ ) . (13)If, as in the HST model, the black hole energy dominated the radiation initially, then a − / ( τ ) =( q πMn BH M P τ + 1) − / . In this limit, the initial black hole number density n BH drops out of theformula for the baryon asymmetry. Up to “order 1” numerical factors, we get∆ bσ = ( g ) / ( MM P ) − / ǫ CP . (14)In the HST model, M is about 10 M P . Taking g ∼ we get∆ bσ ∼ × (10) − ǫ CP . (15)For ǫ CP of order 1 this overshoots the observed value a bit, while for ǫ CP ∼ α QED π as one wouldexpect from an order one θ QED it undershoots by a factor ∼ − θ QED in this model, seems a bit problematic. The formulae above are applicableto primordial black holes from hybrid inflation as well, as long as the energy density in blackholes dominates that in radiation until the time of black hole decay. The value of the averageblack hole mass M can take on a variety of values in different hybrid inflation models, and6e have not made a survey to determine which of these models could account for the baryonasymmetry by our mechanism.We note that Hook’s[8] formulae for the baryon yield differ dramatically from our own,particularly in their dependence on the reheat temperature. This is easy to understand. ForHook, the thing driving the departure from equilibrium is the expansion of the universe, and alow reheat temperature means that the expansion is slow when most of the black holes decay.By contrast, in our model, departure from equilibrium is independent of the expansion rate,and the reheat temperature appears only in the denominator of the baryon to entropy ratio.Hook’s baryon yields scale with a positive power of T RH while ours scale as T − RH .Another simple limit to study is one in which the black hole energy density is always smaller than that of the radiation gas. This will not occur in HST, but could occur in some hybridinflation models. In this case we have gT = gT RH a − ( τ ) . (16)The baryon to entropy ratio is∆ bσ = ( gT RH ) − n BH ǫ CP ( M/M P ) a ( τ ) . (17)In addition a ( τ ) = ( 32 π / T RH M P ( M P τ ) / = ( MM P ) / ( 32 π / T RH M P g − / . (18)The final result is ∆ bσ = ( 32 π / g − / ( MM P ) / n BH T RH ǫ CP . (19)Different hybrid inflation models will have different values of M, g and T RH and we are notsufficiently familiar with the hybrid inflation literature on primordial black hole production todo a survey. However, the constraints that the black hole energy density never dominate theradiation density, combined with the inequality M > M P > T RH suggest that the factor n BH T RH will be very small in the regime where the above estimate applies. It seems unlikely thatmodels satisfying these constraints could give a large enough asymmetry. We have not studiedthe intermediate case, where black holes are initially subdominant, but come to dominate theenergy density before they decay. CP violating effects of a non-vanishing θ QED are enhanced by the presence of magneticallycharged black holes, and such black holes also catalyze baryon number violation via the Callan-Rubakov effect. In an ancient iteration of the HST cosmology, we claimed that the universewould be populated by a very dilute gas of very heavy monop-holes with large magnetic chargeand speculated that they could produce the baryon asymmetry.Our current understanding suggests that those claims were incorrect. They were based on atoo-literal reading of the phrase dense black hole fluid , which we used to analyze the properties7f the p = ρ phase of cosmic evolution. We also mixed up effective field theory notions withour new holographic models in a way that makes no sense. The proper way to analyze theprobability of forming a magnetic charge, whether for the horizon filling black hole of the p = ρ era, or the individual post-inflationary black holes of [2] is to use the black hole entropy formulabut only after the p = ρ and inflationary eras have ended. In the model of [2], the universe hasno localized black holes, to which this formula could apply, until the end of inflation.The entropy formula gives the probability that a black hole of fixed mass has integer electricand magnetic charges ( n e , n m ) as P [( n e , n m )] = e − α [ ( n e + θ π n m ) + π α n m ] . (20)Even if we use the value of the fine structure constant at the unification scale, then as long asperturbative unification is valid, the probability of magnetic charges is very small and it wouldseem that only the lowest value n m = 1 could be of any significance. Of course, estimating theprobability from the black hole entropy is really only valid for macroscopic amounts of charge,so what we can really conclude is that only o (1) values of the monopole charge, where the blackholes are far from extremal, will be present with o (1) probability.On the other hand, the decay of such black holes will, if θ QED is o (1) violate both baryonnumber and CP by amounts of order 1 per unit time. Recall[11] that a Planck mass monopoleblack hole will create a distorted region of QCD vacuum around it whose size is much largerthan the Schwarzschild radius. Particles created in the decay of the more numerous neutralblack holes will have interaction cross sections at the QCD scale with this cloud and be suckedinto the region where we can see explicit CP violation from the monopole’s electric field, aswell as explicit baryon violation at the black hole horizon.The decay process of a monop-hole can be modeled like that of an X boson in grand unifiedmodels, except that since the decay amplitudes are not perturbative, there is no analog of theloop suppression of CP violation in those Feynman diagram calculations[12]. We proceed froma thermal soup of magnetically neutral and magnetically charged black holes. Let f m denote thefraction of holes with non-zero positive magnetic charge. There’s an equal number of negativecharges. The basic process that leads to baryogenesis is the transition( M, n m ) → ( M − ∆ M, n m ) + ∆ B, (21)where δB is a collection of particles with total energy ∆ M ≪ M in the monop-hole rest frame,and baryon number ∆ B . Order 1 CP violation means that the process( M, ¯ n m ) → ( M − ∆ M, ¯ n m ) − ∆ B, (22)has a different probability. These probabilities are not computed in a perturbative loop ex-pansion and they are of order 1. As in X boson decay, the very fact that the monop-holes aredecaying means that we are out of thermal equilibrium. The black hole gas is dilute, and inverseprocesses in which particles emitted by one black hole are absorbed by another, are much morerare than the decay processes. In the approximation that we neglect them, the Callan-Rubakoveffect seems unimportant.Also, the fact that the black holes are charged means that we do not require a factor of dT /dt in the decay rate, in order to account for the asymmetry. We can think of the blackhole charge as a chemical potential, shifting the equilibrium of black hole emission to one that8refers particles over anti-particles. The rate of baryon violation in single monop-hole decay isthus dBdt ∼ ( M/M P ) ∆ P, (23)where ∆ dPdt is the difference in probabilities per unit time for the two processes above. Thisdifference should be o (1) if θ QED is o (1). This should be multiplied by f m n BH to calculate thetotal baryon asymmetry of the universe produced by primordial monop-hole decay.We thus have ∆ B neutral ∆ B monop − hole ∼ ǫ CP − neutral ǫ CP − monop − hole M P f m M . (24)If f m > M P /M , then baryogenesis via monop-hole decay dominates over that from the muchmore numerous neutral black holes. This implies M/M P > , which is not compatible withthe HST model, where M/M P ∼ . This scenario is perilously close to violating the Parkerbound on the monopole density. We have ∼ ( M P /M ) monop-holes per baryon. The relevantform of the Parker bound, for the extremal monop-hole mass ∼ M P , and a presumed velocity v relative to the earth of ∼ × − c [13] is n monop − hole < − ( cm ) − × − cv . (25)Given the observed baryon density n b = 10 − ( cm ) − , (26)we get a bound ( M/M P ) > v × − c . (27)The condition that most baryons be produced in monop-hole, rather than neutral black holedecay is M/M P > f − m ∼ , which is much stronger than the observational bound. Thebounds from monopole catalyzed baryon decay in stars and Jovian planets give bounds onthe monopole density 3 to 14 orders of magnitude stronger than the Parker bound[14], whilethe theoretical constraints of our scenario only give us about 7 orders of magnitude leeway.Depending on whether one believes the strongest astrophysical constraints, one might be forcedto conclude that M/M P > . The black hole life-time for such black holes is uncomfortablylong ∼
10 seconds, and the reheat temperature from black hole decay is too low to accommodatenucleosynthesis. It is very unlikely that a model with radiation domination at a temperature ∼
10 MeV, could ever be constructed in the presence of such massive monop-holes. We concludethat the origin of the baryon asymmetry is unlikely to be the decay of primordial magneticallycharged black holes.The probability of finding electrically charged black holes with charge < α unification is of order1, according to the black hole entropy formula. These will not exhibit large CP violation merelyin response to an order 1 value of θ QED , but their decays certainly violate baryon number andwould also violate CP since there is evidence for o (1) CP violation in the CKM matrix. Thus,initial decays of the charged black hole will produce an asymmetry. This is not the mass M of the non-extremal hole before Hawking evaporation has taken place. .Secondly, because electrons are light compared to the initial Hawking temperature, the holewill discharge itself. In the small charge regime this will happen exponentially rapidly[8, 15].Thus, the baryon asymmetry produced by this mechanism will be of order zero in M/M P andis less important than the charge independent asymmetry produced by the shifts in the blackhole equilibrium state. We’ve examined two different questions in this paper, because we initially thought there couldbe a connection between them. The first was evidence for non-trivial effects of the CP violatingangle θ QED , in the presence of black hole horizons, even when no monopoles are present. Wefound a non-trivial Berry phase in processes where neutral systems with separated charges weredropped into the black hole. We then attempted to see whether CP violation via θ QED couldbe the origin of baryogenesis, in models where primordial black holes are produced in the earlyuniverse.We found that the answer to the second question was negative but we also discoveredthat, in the presence of other sources of CP violation, the decay of black holes could producethe asymmetry. This question was previously studied by Hook[8], who considered the case ofHawking radiation in the context of an expanding universe. The expanding universe providesin Hook’s proposal, the violation of equilibrium necessary to produce a baryon excess. Inthe case of HST, the change in temperature in black hole evaporation provides the source ofnon-equilibrium. We showed that it is quite plausible that a baryon excess of the observedmagnitude can be produced in HST and possibly also in a class of hybrid inflation models.
Acknowledgments
This work was begun while TB was a guest of the Physics Dept. at Georgia Tech, and completedwhile he attended the workshop on Quantum Gravity Foundations at KITP-UCSB. He thanksboth of those institutions for their hospitality. The work of T.B. was supported in part by theDepartment of Energy. The work of W.F. was supported in part by the TCC and by the NSFunder Grant PHY-0969020
References [1] W. Fischler and S. Kundu,“Hall Scrambling on Black Hole Horizon,” arXiv:1501.01316[hep-th][2] T. Banks and W. Fischler, “Holographic Inflation Revised,” arXiv:1501.01686 [hep-th][3] E. Witten, “Dyons of Charge eθ π ,” Phys. Lett.B 86, 283 We neglected to include a similar suppression in the calculation of monop-hole decays, because it is somewhatless severe. At any rate, it only worsens the phenomenological problems of that scenario.83