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Anomalous Fermion Production in Gravitational Collapse

arXiv:gr-qc/9305018v1 24 May 1993DAMTP93/R14Anomalous Fermion Production in Gravitational CollapseG.W. GibbonsAlan R. SteifD.A.M.T.P.Silver St.Cambridge UniversityCambridge, U.K. CB3 9EWgwg1@moria.amtp.cam.ac.ukars1001@moria.amtp.cam.ac.ukABSTRACT: The Dirac equation is solved in the Einstein-Yang-Mills background foundby Bartnik and McKinnon.

We find a normalizable zero-energy fermion mode in the s-wavesector. As shown recently, their solution corresponds to a gravitational sphaleron whichmediates transitions between topologically distinct vacua.

Since the Bartnik-McKinnonsolution is unstable, it will either collapse to form a black hole or radiate away its energy.In either case, as the Chern-Simons number of the configuration changes, there will be anaccompanying anomalous change in fermion number.May 21, 1993

1. IntroductionNeither the Yang-Mills nor Einstein field equations admit static finite-energy, non-singular solutions.

However, such particle-like solutions have been found by Bartnik andMcKinnon [1] in the combined Einstein-Yang-Mills theory.These solutions were laterinterpreted as sphalerons, that is, static saddlepoint solutions lying at the top of an energybarrier in field configuration space separating vacua with different Chern-Simons number[2] [3]. This, in particular, accounts for their instability.

If the sphaleron is perturbed, it willeither radiate its energy to infinity or collapse to form a black hole. Since either processinvolves a change in Chern-Simons number, one expects an equal anomalous change inchiral fermion number.

In this paper we initiate a study of this problem. In Section 2, theBartnik-McKinnon solution and its interpretation as a sphaleron is reviewed.

In Sections3-5, the Dirac equation in the background of the static Bartnik-McKinnon sphaleron isanalyzed. We find a zero energy bound state in the s-wave sector.

In Section 6, a Higgsfield is included. In Section 7, the conformal invariance of the Dirac equation is exploitedto prove a general no-hair theorem for fermions.2.

Bartnik-McKinnon SolutionsIn [1] a class of spherically symmetric particle-like solutions to the Einstein-Yang-Millsequations with SU(2) gauge group were found numerically. It was shown that there arean infinite sequence of solutions (An(r), Bn(r), Kn(r)) for the metric and gauge fieldds2 = −A2dt2 + B2dr2 + r2(dθ2 + sin2 θdφ2)(2.1)Ai = 1 −K2gr ǫijknjτ k∗(2.2)where g is the gauge coupling constant,τ i are the generators of SU(2),and⃗n = (sin θ cos φ, sin θ sin φ, cos θ).

Kn has n nodes and satisfies the boundary con-dition Kn(0) = 1 and Kn(∞) = (−1)n.At large distances, the metric approachesSchwarschild and the gauge field strength decays as 1/r3 with Ai approaching the puregauge Ai = −ig∂iUU −1 where U = 1 for n even and U = −i⃗n · ⃗τ for n odd. Near anode r = r0 (K(r0) = 0), the gauge field corresponds to a P = 1/g Dirac monopole,and the metric to extremal Reissner-Nordstrom.

The masses of the Bartnik-McKinnon* This form of the gauge field is gauge equivalent to that used in [1].1

solutions are proportional to the only mass scale in the problem, g−1G−1/2, where G is thegravitational constant. The masses increase with n approaching the mass of the extremalReissner-Nordstrom black hole with magnetic charge, P = 1/g as n →∞.

Heuristically,in this limit, Kn fluctuates rapidly approaching its mean zero value corresponding to theDirac monopole. The existence of the Bartnik-McKinnon solutions was subsequently es-tablished rigourously [4], and generalized to include horizons [5].Soon after their discovery, it was shown that both particle-like and black hole solutionsare unstable [6].

In fact, the Bartnik-McKinnon solutions correspond to a gravitationalanalog of the sphaleron [2][3]. We recall that sphalerons [7] are static saddle-point solu-tions that lie at the top of an energy barrier separating topologically distinct vacua.

Forasymptotically flat spacetimes, the ADM energy provides a positive definite mass func-tional on field configuration space. The zero energy “vacua” are flat spacetime metricsand pure gauge Yang-Mills configurations Ai = −ig ∂iUU −1.

Each pure gauge yields amap from space into the group manifold of SU(2). Demanding U →1, this becomes amap U : S3 →S3.

Since maps with different winding, or Chern-Simons, numbers cannotbe continuously deformed into one another, paths in configuration space connecting thesevacua must enter non-vacuum regions, and therefore by the positive definiteness of the massfunctional must traverse an energy barrier. * Each such path has a maximal energy config-uration.

Among these configurations the one with minimum energy will be an extremumof the mass functional and therefore correspond to an unstable saddlepoint solution. Byconsidering paths connecting the vacuum sector with Chern-Simons number zero and thesector with Chern-Simons number one, one obtains the Bartnik-McKinnon solution K1.One can see by symmetry that its Chern-Simons number is 1/2.

Repeating this procedure,with paths connecting K1 and its gauge equivalent solution with Chern-Simons number−1/2, the K2 solution with zero Chern-Simons number is obtained. Continuing in thisway, the entire Bartnik-McKinnon sequence is produced.

This procedure also yields theblack hole solutions if one considers asymptotically flat metrics with fixed horizon area.As the fields evolve dynamically along a path in configuration space connectingvacua with different Chern-Simons numbers and passing through the sphaleron, anoma-lous fermion production will occur [8]. The total number of chiral fermions created equals* We shall regard vacua which are related by so called large gauge transformations as distinct.One can, if one wishes, identify them, in which case the paths we refer to below become closednon-contractible loops in this identified configuration space.2

the change in the Chern-Simons number of the field configuration. As one passes betweentopologically distinct vacua, a fermion state will emerge from the negative energy Diracsea, enter the discrete spectrum, and finally merge with the positive energy continuum.By symmetry, therefore, there should be a zero energy bound state at the midpoint of thispath where the sphaleron configuration is located.

If perturbed, the sphaleron will fallin one of two directions. The Yang-Mills field will dominate, in which case the sphaleronwill radiate away its energy to infinity.

The fermion zero mode would then also escapeto infinity perhaps after being reflected at the origin. Alternatively, gravity may inducethe object to collapse to a black hole swallowing the Yang-Mills field, and presumably thefermion mode as well.3.

Dirac EquationWe now proceed to test this picture by solving the Dirac equation in the sphaleronbackground. As expected from the above discussion, we find a zero-energy bound statein the s-wave sector.

We also derive the full set of time dependent radial equations. Todetermine more precisely the fate of the zero mode if the sphaleron collapses to a blackhole would require numerically solving the time dependent equations.

Such an analysiswas done for the Yang-Mills-Higgs sphaleron in [9][10].The massless Dirac equation in an Einstein-Yang-Mills background for an isodoubletfermion is given byiγµ(∇µ −igAµ)Ψ = 0(3.1)where ∇µ is the covariant derivative ∇µ = ∂µ −14ωabµ γaγb. µ and a are tangent andspacetime indices respectively and are related by eaµ ≡ea, a basis of orthonormal one-forms.

ωabµ ≡ωab are the associated connection one-forms obeying dea + ωab ∧eb = 0, andγa are Dirac matrices satisfying {γa, γb} = −2ηab with η00 = −1. The metric and gaugefield are given in (2.1) and (2.2) where τ i are now the Pauli spin matrices.

In terms of theorthonormal basise0 = Adter = Bdreθ = rdθeφ = rsin θdφ(3.2)the connection one-forms areω0r = B−1A′dtωθr = B−1dθωφr = B−1sin θdφωφθ = cos θdφ(3.3)3

where ′ ≡∂∂r. In the chiral representation, the gamma matrices areγ0 =0110γi =0σi−σi0γ5 =−1001(3.4)where σi are Pauli spin matrices.Ψ can be decomposed into its left and right chiralcomponents Ψ =ψLψRwhere ψL(R) carry two-component Lorentz and isospin indices.Since the Dirac equation is massless, the two chiralities decouple.

In the following, weconsider just the right-handed component ψ ≡ψR. The Dirac equation then becomes∂ψ∂t + ⃗σ · ⃗nrA1/2B∂∂r(rA1/2ψ) + Ar DT/ψ −igA⃗σ · ⃗Aψ = 0(3.5)where DT/is the Dirac operator on the unit two sphere and ⃗a·⃗b ≡aibi.

Eqn. (2.2) implies⃗σ · ⃗A = (K−1)2gr (⃗n · ⃗σ × ⃗τ).

The conserved inner product is given by< ψ1|ψ2 >=Zψ1†ψ2 Br2drdΩ. (3.6)We now discuss the effect of charge conjugation, C, and parity, P on the sphaleron.The action for a Dirac fermion coupled to a Yang-Mills field with arbitrary gauge groupis invariant under C and P separately.

Under C, the fields transform asΨ →ΨC = γ2Ψ∗Aµ →ACµ = −A∗µ(3.7)and under P asΨ(xi, t) →ΨP = γ0Ψ(−xi, t)Ai(xi, t) →APi = −Ai(−xi, t)A0(xi, t) →AP0 = A0(−xi, t). (3.8)The metric being real and spherically symmetric is invariant under C and P. BecauseC and P interchange chirality, the action for chiral fermions is only invariant under thecombined CP transformation.

Using (3.7), (3.8), and −τ i∗= τ 2τ iτ 2, the gauge field (2.2)transforms as Ai →ACPi= τ 2Aiτ 2. The CP invariance of the theory then implies thatgiven a solution Ψ to the Dirac equation in the sphaleron background with energy E,τ 2ΨCP is a solution in the same background, but with energy −E.4

4. S-wave Sector and Zero ModeWe now solve the Dirac equation (3.5) by separating variables.

Since the total angularmomentum ⃗K = ⃗L + ⃗S + ⃗T commutes with the Hamiltonian, ψ can be expanded ineigenstates of K2 and Kz with eigenvalues k and m. ⃗L, ⃗S = ⃗σ/2, and ⃗T = ⃗τ/2 arethe orbital angular momentum, spin, and isospin. The s-wave sector corresponding tok = m = 0 is spanned by the two states χ1 and χ2 ≡⃗σ · ⃗nχ1 whereχ1 =1√210S01T−01S10T(4.1)is the hedgehog spinor satisfying (⃗σ + ⃗τ)χ1 = 0.

The action of the various operatorsin the Dirac equation on the two states can be determined from the hedgehog propertyand the spin commmutation relations. The transverse Dirac operator can be written asDT/= (2⃗S·⃗L+1)(⃗σ·⃗n) = (J2−L2−S2+1)(⃗σ·⃗n) where ⃗J ≡⃗L+ ⃗S.

Since [J2,⃗σ·⃗n] = 0 and⃗σ·⃗n changes the value of the orbital angular momentum by one, we find that DT/χ1 = −χ2and DT/χ2 = χ1. The operator ⃗n · ⃗σ × ⃗τ appearing in the gauge field term in the Diracequation also interchanges χ1 and χ2: (⃗n · ⃗σ × ⃗τ)χ1 = −2iχ2 and (⃗n · ⃗σ × ⃗τ)χ2 = 2iχ1.Thus, (3.5) becomes∂f∂t + ∂g∂r∗+ AKr g = 0∂g∂t + ∂f∂r∗−AKr f = 0(4.2)where ψ = r−1A−1/2fχ1 + r−1A−1/2gχ2 and r∗is the “tortoise” coordinate satisfyingdr∗dr = A−1B.Eqn.

(4.2) admits a zero energy bound state of the formf = expZ rr0B Kr dr ,g = 0(4.3)for K = Kn with n odd. For n even, K →1 asymptotically, and therefore f diverges.The zero mode (4.3) vanishes at r = 0 and falls offas 1/r asymptotically.

The chargedensity of the wave function from (3.6) is given by 4πA−1B|f|2. The density of the zeromode (4.3) is peaked in the monopole region, K = 0, which is effectively acting as apotential well.

For a spacetime with horizon at r = rH, the metric functions behave as A ∼(r−rH)1/2(1+O(r−rH)) and B ∼(r−rH)−1/2(1+O(r−rH)). From (4.3) we observe thatthe wavefunction diverges as (r−rH)−1/4, and its norm logarithmically.

This is perhaps tobe anticipated in light of the no-hair theorem for fermions. The effective potential within5

the s-wave sector is found by rewriting the two coupled first order equations as decoupledsecond order equations∂2f∂t2 −∂2f∂r∗2 + V (r)f = 0,˙g = −∂f∂r∗+ Hf(4.4)whereV (r) = ∂H∂r∗+ H2,H ≡AKr . (4.5)Near the horizon, the effective potential vanishes as V ∼(r −rH)1/2.5.

k > 0 ModesThe eigenspaces with eigenvalues k and m are four-dimensional for k ≥1. Since L2and Ki commute, they can be simultaneously diagonalized.

Within the four dimensionaleigenspace, there are two states denoted |k, m, ± > with L2 eigenvalues l = k ± 1, and anorthogonal two-dimensional space with eigenvalue l = k. In addition, either J2 = (L +S)2or R2 ≡(S + T)2 can be diagonalized, but not simultaneously since they do not commute.The eigenstates of J2 are |k, m, ± > with eigenvalues j = k ± 1/2 and |k, m, j± >, lyingin the l = k subspace, with eigenvalues j = k ± 1/2. The eigenstates of R2 are |k, m, ± >both with eigenvalues r = 1 and |k, m, 0 > and |k, m, 1 >, lying in the l = k subspace,with eigenvalues r = 0 and 1.

The J2 and R2 eigenstates in the l = k subspace are relatedby a rotation matrix of angle ξ with tan ξ =qkk+1 :|k, m, j+ > =1√2k + 1(√k + 1|k, m, 0 > +√k|k, m, 1 >)|k, m, j−> =1√2k + 1(−√k|k, m, 0 > +√k + 1|k, m, 1 >). (5.1)6

The R2 eigenstates when expressed in terms of spherical harmonics take the form:|k, m, + >=1p(2k + 2)(2k + 3)p(k + m + 1)(k + m + 2)Y m+1k+1 |1, −1 >−p2(k + m + 1)(k −m + 1)Y mk+1|1, 0 > +p(k −m + 1)(k −m + 2)Y m−1k+1 |1, 1 >|k, m, −>=1p2k(2k −1)p(k −m)(k −m −1)Y m+1k−1 |1, −1 >+p2(k + m)(k −m)Y mk−1|1, 0 > +p(k + m −1)(k + m)Y m−1k−1 |1, 1 >|k, m, 0 >=Y mk |0, 0 >|k, m, 1 >=1p2k(k + 1)p(k −m)(k + m + 1)Y m+1k|1, −1 >+m√2Y mk |1, 0 > −p(k + m)(k −m + 1)Y m−1k|1, 1 >(5.2)where |1, 1 >, |1, 0 >, |1, −1 > are the spin-one triplet of R2.We now proceed to express the various operators appearing in the Dirac equation (3.5)as matrices in the basis|k, m, + >, |k, m, j+ >, |k, m, j−>, |k, m, −>in which K2,Kz, L2, and J2 are diagonalized. Since ⃗σ ·⃗n commutes with J2 and squares to unity, afterappropriate normalization of states, one finds that ⃗σ ·⃗n|k, m, ± >= |k, m, j± >.

Using thisresult, one can show that the transverse Dirac operator DT/= (J2 −L2 −S2 + 1)⃗σ · ⃗ntakes the formDT/=0−k −100k + 1000000−k00k0. (5.3)Finally, one can determine the matrix form of the operator ⃗n · ⃗σ × ⃗τ appearing in thegauge field term in the Dirac equation (3.5).

Since it is antisymmetric in ⃗σ and ⃗τ, it mustinterchange the r = 0, 1 states, and because of the factor ⃗n, it changes the orbital angularmomentum by one. Further calculation shows⃗n · ⃗σ × ⃗τ =2i2k + 10k + 1−pk(k + 1)0−k −100pk(k + 1)pk(k + 1)00−k0−pk(k + 1)k0.

(5.4)Substituting (5.3) and (5.4) in (3.5), the Dirac equation reduces to four coupled linearfirst-order equations. It appears that higher mode zero energy bound states are forbidden7

since the wave function diverges at r = 0. Near the horizon, r = rH, A = α(r −rH)1/2(1+O(r −rH)) and B = β(r −rH)−1/2(1 + O(r −rH)) while K is constant.

Therefore, toleading order in r −rH, the Dirac equation becomes˙ψ + αβ ⃗σ · ⃗n(r −rH)ψ′ + 14ψ= 0(5.5)with solutions ψ ∼e−iEt(r −rh)(±iEβ/α−1/4) which as before diverge at the horizon.6. Higgs FieldIn this section, we consider the effect of a Higgs field.

The Dirac equation then becomesiγµ(∇µ −igAµ)Ψ −ΦΨ = 0(6.1)where Φ = φiτ i is the Higgs field in the adjoint representation. We have absorbed theYukawa coupling constant in Φ.

Consider the following spherically symmetric ansatz forΦφi = F(r)ni. (6.2)Since the two chiralities of the fermion, ψL and ψR, no longer decouple, the Dirac equationnow reduces to four coupled first order equations in the s-wave sector and to eight in thehigher mode sector.

Using the fact that ⃗τ · ⃗nχ1 = −χ2 and ⃗τ · ⃗nχ2 = −χ1, we find thatthe s-wave equations become∂fR(L)∂t±∂gR(L)∂r∗+ AKr gR(L)−iAFgL(R) = 0∂gR(L)∂t±∂fR(L)∂r∗−AKr fR(L)−iAFfL(R) = 0(6.3)where ΨR(L) = r−1A−1/2fR(L)χ1 + r−1A−1/2gR(L)χ2. There is in fact still a zero-energybound state solution to these equations:fR = −ifL = expZ rr0B(Kr −F) dr,gR = gL = 0.

(6.4)Moreover, since the Higgs field causes the wave-function to decay exponentially at infinity,Ψ ∼e−F (∞)r, there is a bound state for both odd and even n. As before, for a space-time with horizon the zero mode diverges there. The s-wave equations can be written asdecoupled second order equations∂2fR∂t2 −∂2fR∂r∗2 + V (r)fR = 0,fL = ifR,˙gR = i˙gL = −∂fR∂r∗+ HfR(6.5)8

whereV (r) = ∂H∂r∗+ H2,H ≡A(Kr −F). (6.6)The equations for the higher modes can be found as well.

The matrix associated with theoperator ⃗τ ·⃗n appearing in the Higgs field term is obtained from ⃗σ ·⃗n by exchanging σ andτ and using the fact that the r = 0, 1 states are antisymmetric and symmetric in σ and τrespectively:⃗τ · ⃗n =0−cos 2ξsin 2ξ0−cos 2ξ00sin 2ξsin 2ξ00cos 2ξ0sin 2ξcos 2ξ0(6.7)where tan ξ =qkk+1.7. Fermion No-Hair TheoremAssuming that the endpoint of gravitational collapse is a stationary black hole, thenthe uniqueness theorems imply it must be Kerr-Newman.

In general, the no-hair theo-rems assert that the only external fields that a black hole can generate are those yieldingconserved charges in the form of a surface integral at infinity. If one attempts to find aneighbouring solution with hair, the solution for the perturbation will necessarily divergeat the horizon of the black hole (assuming it vanishes at spatial infinity.) We should pointout that a Yang-Mills black hole such as the one discussed earlier does not unambiguouslyviolate the no-hair theorems since the non-linear nature of the field allows one to view itand not the black hole as being the source of the external Yang-Mills field.

* After all, onewould certainly ascribe the complicated gravitational field of an accretion disk surroundinga black hole to the disk and not to the black hole. The higher multipoles moments of thegravitational field can be unambiguously identified with the higher moments of the matterdensity distribution of the disk.

The Yang-Mills field, however, does not have well definedmultipole moments. In particular, attempts to construct a total charge by integrating theYang-Mills magnetic field does not yield a gauge invariant quantity since there is a freeuncontracted group index.The fact that the zero mode (4.3) ((6.4)) diverges on the horizon is expected fromthese no-hair theorems.

In this section, we prove a general no-hair theorem for fermions. * In fact, Einstein-Skyrme black holes with Yang-Mills hair have been constructed recently[11],and are stable under linear perturbations[12].9

This has already been done for Schwarschild in [13] and [14] By employing the conformalinvariance of the Dirac equation, we show that all static solutions in a spherically symmetricspacetime diverge at the horizon. Any static, spherically symmetric spacetime may bewritten in isotropic coordinatesds2 = −V 2dt2 + W 2dx · dx(7.1)where V and W are functions of |x|.

(This ansatz is, in fact, more general than beingspherically symmetric applying to multi-black hole metrics as well.) Consider the gaugedDirac equationiγµ(∇µ −igAµ)Ψ −mΨ = 0(7.2)where m might depend on a Higgs field and hence on position.In d spacetime di-mensions we have the following result:if (Ψ, gµν, Aµ, m) is a solution of (7.2), then(Ωd−12 Ψ, Ω−2gµν, Aµ, Ωm) is also a solution.

For the metric (7.1) with A0 = 0, this impliesthat V 3/2Ψ solves (7.2) with mass term V m in the ultrastatic optical metricds2 = −dt2 + W 2V 2 dx · dx. (7.3)As is the case for no-hair theorems, we are interested in time-independent solutions to theDirac equation.

Applying conformal invariance again, but now to the spatial part of themetric implies that χ ≡V 1/2WΨ solves the flat three-dimensional gauged Dirac equationwith mass term mW.For m = Ai = 0 the flat space Dirac equation may be solved in a variety of ways, butperhaps the most illuminating from the present conformal viewpoint is to note that in flatEuclidean n-space,χα = T αi1i2...ilxi1xi2 . .

. xil(7.4)clearly solves the flat space Dirac equation with m = Ai = 0 provided T αij...k satisfiesγiαβT βij...k = 0.

(7.5)Solutions which vanish at spatial infinity can be obtained from (7.4) by an inversion. Toeach static solution χ(xi) to the flat space Dirac equation in n spatial dimensions, there isan inverted solution γixirn χ(xi/r2).

Applying this to (7.4), we obtain the general multipolesolutionχ = γinirl+2 Ti1i2...ilni1ni2 . .

.nil(7.6)10

where ni = xi/r. Thus, if m = Ai = 0, the general solution in the spacetime (7.1) whichdecays at infinity is given byΨ = V −1/2W −1 γinirl+2 Ti1...ilni1 .

. .

nil. (7.7)It is now clear that Ψ will blow up at a non-extreme horizon for which V = 0 for r > 0.In the extreme case for a regular horizon where W →r−1 and V →r, we see that Ψ stillblows up for any permitted value of l.AcknowledgementsWe thank N. Manton for many useful discussions.A.S.

would like to acknowledge the support of the SERC.11

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