aa r X i v : . [ h e p - t h ] O c t USTC-ICTS-07-02
Probing α -Vacua of Black Holes in LHC Tower Wang
Institute of Theoretical Physics, Chinese Academy of Sciences,P. O. Box 2735 Beijing 100080, ChinaInterdisciplinary Center for Theoretical Study,University of Science and Technology of China,Hefei, Anhui 230026, China [email protected]
Abstract
Motivated by the idea of α -vacua in Schwarzschild spacetime, we studied thedeformed spectrum of Hawking radiation. Such a deformation would leave signatureson the small black hole evaporation in LHC because their vacuum deviates from theUnruh state. 1or massive scalar fields there is a family of de Sitter-invariant states, includingthe Hartle-Hawking vacuum state as a special case [1, 2]. These states are well knownas the so-called α -vacua because they are related to Hartle-Hawking vacuum by a newparameter α [2]. The choice of physical vacuum from these states might be imprintedon the spectrum of CMBR (see e.g. [3]). The application of α -vacua to cosmologicaldark energy can be found in [4] as a recent review.Starting with Hartle-Hawking vacuum, Chamblin and Michelson in [6] constructedthe α -vacua of scalar field in Schwarzschild, Schwarzschild-dS and Schwarzschild-AdSblack hole spacetimes. However, it is generally conceived that at late times, a blackhole formed from gravitational collapse is well-approximated by an eternal black holewith the scalar field in the Unruh state [9], rather than in the Hartle-Hawking state[8]. On the other hand, assuming large extra dimensions and TeV Plank scale, smallblack holes would be produced and decay in LHC [11, 12, 13], thus provide the firstexperimental test of Hawking’s radiation hypothesis [7]. Their sudden evaporationtakes place not very long after the formation, so naturally their vacuum deviates fromthe Unruh state. Fortunately, since the discussion in [6] does not depend heavily onthe details of vacuum with which one starts, the α -vacua from the Unruh state canalso be constructed, in hopes of characterizing such a deviation with new parameters.In this paper, we explore the modified spectrum of Hawking radiation for scalars,considering effects of the α -vacua of Schwarzschild spacetime. If small black holesare indeed produced in LHC abundantly, such a modification may leave observablefingerprints on their evaporation spectrum.Before getting the spectrum, we would like to work over some details about the α -vacua. We first introduce a set of global coordinates of Schwarzschild spacetime,by embedding it into M , . The global coordinates are a byproduct of this paper,in which the symmetry under antipodal map is more manifest than in other coordi-nates. After partly solving the Klein-Gordon equation, we then deduce a more exactexpression of α -vacua, which relates to the usual Hartle-hawking vacuum modes via atrivial (nonmixing) Bogoliubov transformation and a Mottola-Allen transformation.Subsequently, in order to search some signatures of α -vacua in LHC phenomeno-logically, we turn to the Unruh state and the corresponding α -vacua, and derive theevaporation spectrum following the standard procedure, i.e., by picking up the Bo-goliubov coefficients. The spectrum depends on two new parameters α and γ . When α = 0, it deviates from ordinary greybody spectrum of Hawking radiation. So at the Actually the α -vacua in [2] are defined with two parameters α and β . We will use γ instead of β following the notations of [6]. ds = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r ( dθ + sin θdϕ ) . (1)It can be embedded into M , ds = − dX + dX + dX + dX + dX + dX + dX (2)by setting [5, 6] X = r sin θ cos ϕ, X = r sin θ sin ϕ, X = r cos θ,X = − M r Mr + 4 M r r M , X = 2 √ M r Mr ,X = 4 M r − Mr cosh (cid:18) t M (cid:19) , X = 4 M r − Mr sinh (cid:18) t M (cid:19) . (3)The spacetime outside the horizon of Schwarzschild black hole is given as thealgebraic variety determined by the three polynomials [5, 6]43 X + X − X = 16 M ,X ( X + X + X ) = 576 M , √ X X + X = 24 M . (4)The global coordinates are related to static coordinates via (cid:16) e t M , r, θ, ϕ (cid:17) static → (cid:18) cos σ + tanh τ cos σ − tanh τ , M cosh τ sin σ , θ, ϕ (cid:19) global , (5)which is resulted from the definition X = 2 M sin θ cos ϕ cosh τ sin σ , X = 2 M sin θ sin ϕ cosh τ sin σ , X = 2 M cos θ cosh τ sin σ ,X = − M cosh τ sin σ + 4 M cosh τ sin σ , X = 2 √ M cosh τ sin σ,X = 4 M cosh τ cos σ, X = 4 M sinh τ, (6)and gives the form of metric ds = 16 M cosh τ sin σ { sin σ [sinh τ (1 + cosh τ sin σ ) − cosh τ sin σ ] dτ + cosh τ [cos σ (1 + cosh τ sin σ + cosh τ sin σ ) + cosh τ sin σ ] dσ +2 sinh τ cosh τ sin σ cos σ (1 + cosh τ sin σ + cosh τ sin σ ) dτ dσ } + 4 M cosh τ sin σ ( dθ + sin θdϕ ) . (7)3learly the global coordinates presented above are quite similar to but more compli-cated than those for de Sitter spacetime [16].Correspondingly the measure is √− g = (cid:12)(cid:12)(cid:12)(cid:12) M sin θ cosh τ sin σ (cid:12)(cid:12)(cid:12)(cid:12) , (8)while the non-vanishing components of contravariant metric tensor are g ττ = − M [cos σ (1 + cosh τ sin σ + cosh τ sin σ ) + cosh τ sin σ ] ,g τσ = g στ = sinh τ sin σ cos σ M cosh τ (1 + cosh τ sin σ + cosh τ sin σ ) ,g σσ = − sin σ M cosh τ [sinh τ (1 + cosh τ sin σ ) − cosh τ sin σ ] ,g θθ = cosh τ sin σ M , g ϕϕ = cosh τ sin σ M sin θ . (9)The covariant Klein-Gordon equation( ✷ x − µ )Φ( x ) = 1 √− g ∂ µ ( √− gg µν ∂ ν Φ) − µ Φ = 0 (10)in global coordinates is too complicated to be fully solved by brute force. However,we can still get some details from it as follows.In terms of global coordinates, the antipodal map [2, 6] x → x A reads( τ, σ, θ, ϕ ) → ( − τ, π − σ, π − θ, ϕ ± π ) . (11)It is easy to see the metric (7) is manifestly invariant under this antipodal transfor-mation and further ( ✷ x − µ ) G (1)0 ( x, y ) = 0 ⇒ ( ✷ x A − µ ) G (1)0 ( x A , y ) = ( ✷ x − µ ) G (1)0 ( x A , y ) = 0 (12)for the Hadamard function G (1)0 ( x, y ). That is to say, the Hadamard functions G (1)0 ( x, y ) and G (1)0 ( x A , y ) satisfy the same equation. This property is most mani-fest in global coordinates. We emphasize the property here and confirm the existenceof a complete set of orthonormal modes obeying φ ω,l,m ( x A ) = φ ∗ ω,l,m ( x ) (13)in the following, because they play important roles in proving that the α -vacua re-spect symmetries of the spacetime [2, 6]. For de Sitter spacetime, the symmetry is4he O (1 ,
4) group. While for Schwarzchild spacetime, it is the “
CP T ” invarianceintroduced in [5] and recounted in [6].In [6], it was argued that one can choose the modes satisfying (13) because theequation of motion is invariant under complex conjugation and under the antipodalmap. It seems to me this argument guarantees only the existence of certain specialsolution of differential equation (10) which satisfies Φ( x A ) = Φ ∗ ( x ), rather than theexistence of a set of Hartle-Hawking vacuum modes φ ω,l,m satisfying (13). We wouldlike to fill this gap and make the argument in [6] more solid, by precisely constructingthese modes of Hartle-Hawking vacuum.Our construction will be based on some results presented in [17]. In the book[17], a complete set of Hartle-Hawking modes ˆ φ (1) ω,l,m , ˆ φ (2) ω,l,m , ˆ φ (3) ω,l,m and ˆ φ (4) ω,l,m has beenconstructed. These modes are orthonormal and meet conditions [17]ˆ φ (1) ω,l,m ( − U, − V, θ, ϕ ) = ˆ φ (2) ∗ ω,l,m ( U, V, θ, ϕ ) , ˆ φ (3) ω,l,m ( − U, − V, θ, ϕ ) = ˆ φ (4) ∗ ω,l,m ( U, V, θ, ϕ ) . (14)Here U , V are Kruskal coordinates. In our global coordinates, the conditions aretranslated into ˆ φ (1) ∗ ω,l,m ( − τ, π − σ, θ, ϕ ) = ˆ φ (2) ω,l,m ( τ, σ, θ, ϕ ) , ˆ φ (3) ∗ ω,l,m ( − τ, π − σ, θ, ϕ ) = ˆ φ (4) ω,l,m ( τ, σ, θ, ϕ ) . (15)At the same time, it is trivial to show the factorizationˆ φ ( i ) ω,l,m ( x ) = ˆ f ( i ) ω,l,m ( τ, σ ) Y l,m ( θ, ϕ ) , i = 1 , , , H ± especially, the book [17] gave analytic expressions ofˆ φ ( i ) ω,l,m , and one can show they are factorized as (16) indeed. Combining (15), (16)and the following formula for spherical harmonic functions Y l,m ( π − θ, ϕ ± π ) = ( − l Y l,m ( θ, ϕ ) (17)together, we can check the relationsˆ φ (1) ∗ ω,l,m ( x A ) = ( − l ˆ φ (2) ω,l,m ( x ) , ˆ φ (3) ∗ ω,l,m ( x A ) = ( − l ˆ φ (4) ω,l,m ( x ) , (18) Different from chapter 11.2 of book [17], we will use notations ˆ φ (1) ω,l,m , ˆ φ (2) ω,l,m , ˆ φ (3) ω,l,m and ˆ φ (4) ω,l,m instead of ϕ d ′ J , ϕ p ′ J , ϕ u ′ J and ϕ n ′ J respectively for self-consistent of this paper. φ ω,l,m by the trivial Bogoliubov trans-formation [2] φ (1) ω,l,m ( x ) = 1 √ e iπl/ [ ˆ φ (1) ω,l,m ( x ) + ˆ φ (2) ω,l,m ( x )] ,φ (2) ω,l,m ( x ) = 1 √ e iπ ( l +1) / [ ˆ φ (1) ω,l,m ( x ) − ˆ φ (2) ω,l,m ( x )] ,φ (3) ω,l,m ( x ) = 1 √ e iπl/ [ ˆ φ (3) ω,l,m ( x ) + ˆ φ (4) ω,l,m ( x )] ,φ (4) ω,l,m ( x ) = 1 √ e iπ ( l +1) / [ ˆ φ (3) ω,l,m ( x ) − ˆ φ (4) ω,l,m ( x )] . (19)The modes (21) form a complete set of orthonormal modes. In particular,1. φ ( i ) ω,l,m ( x A ) = φ ( i ) ∗ ω,l,m ( x ).2. ( φ ( i ) ω,l,m , φ ( i ′ ) ω ′ ,l ′ ,m ′ ) = δ i,i ′ δ ω,ω ′ δ l,l ′ δ m,m ′ and ( φ ( i ) ω,l,m , φ ( i ′ ) ∗ ω ′ ,l ′ ,m ′ ) = 0.3. The set of φ ω,l,m is complete and spans the space of ˆ φ ω,l,m .The α -vacua are constructed from this set of modes by a Mottola-Allen transfor-mation [2, 6]˜ φ ( i ) ω,l,m ( x ) = cosh αφ ( i ) ω,l,m ( x ) + e iγ sinh αφ ( i ) ∗ ω,l,m ( x ) , α ≥ , − π < γ ≤ π. (20)One should notice that ˆ φ ( i ) ω,l,m ( x ) and φ ( i ) ω,l,m ( x ) are of positive frequencies with re-spect to the affine parameters on H ± . However, this is not true for ˜ φ ( i ) ω,l,m ( x ) sincethe transformation (20) mixes modes of the same frequency but with different sign.Or equivalently, from another point of view based on (13), it mixes modes on theantipodal points x and x A . The modes (20) are taken as a new “vacuum” state [1, 2]because this transformation is (the Bogoliubov coefficients are) frequency independentand preserves orthonormality.The scalar field in (10) may be decomposed in different bases if we consider dif-ferent vacua, Φ( x ) = X i,ω,l,m h a ( i ) ω,l,m φ ( i ) ω,l,m ( x ) + a ( i ) † ω,l,m φ ( i ) ∗ ω,l,m ( x ) i = X i,ω,l,m h ˜ a ( i ) ω,l,m ˜ φ ( i ) ω,l,m ( x ) + ˜ a ( i ) † ω,l,m ˜ φ ( i ) ∗ ω,l,m ( x ) i . (21)The properties (12) and (13) are the major tricks to study properties of two-pointfunctions, and to prove that α -vacua respect the symmetries of the spacetime [2, 6].6or example, equation (13) leads to a relation between Hadamard function G (1) αγ ( x, y )for α -vacua and G (1)0 ( x, y ), D ( x, y ) for Hartle-Hawking vacuum, G (1) αγ ( x, y ) = cosh(2 α ) G (1)0 ( x, y ) + cos γ sinh(2 α ) G (1)0 ( x A , y ) − sin γ sinh(2 α ) D ( x A , y ) . (22)From (13) it is clear that G (1)0 ( x, y ) and G (1)0 ( x A , y ) obey the same equation of motionand hence respect symmetries of the spacetime. Therefore, G (1) αγ ( x, y ) is CP T [5, 6]invariant for α -vacua with γ = 0 , π . As explained in [2] and reiterated in [6], the α -vacua with sin γ = 0 break the time-reversal symmetry, to which we will come backlater when discussing small black holes in LHC.In the previous part we focused on two tasks:1. making it more manifest that G (1)0 ( x A , y ) obeys the same equation as that of G (1)0 ( x, y ), thus preserves the symmetries of the Schwarzschild spacetime; and2. establishing a complete set of orthonormal modes satisfying (13).Global coordinates (7) of Scharzschild spacetime were a byproduct during our re-search. The coordinates may be not necessary here, but facilitate the first task in away. Their physical implications and applications in various aspects of Schwarzschildblack hole remain unclear, which we would like to study elsewhere in the future.In the following, to be brief, we will work in the matrix formalism. That is, wewill write a basis of modes in a column matrix and multiply it by a square matrixto represent the Bogoliubov transformation. For example, the trivial Bogoliubovtransformation (21) will be written concisely, φ i = P ij ˆ φ j , (23)where the collective subscript i or j denotes the complete set of quantum numbers ω , l , m and superscripts ( i ) that must be specified to describe a mode. Or even morecompactly, φ = P ˆ φ. (24)Likewise, the Mottola-Allen transformation (20) will be denoted in the form˜ φ = cosh αφ + e iγ sinh αφ ∗ , α ≥ , − π < γ ≤ π. (25)In the above, we have been dealing with the Hartle-Hawking type α -vacua, andfocusing on some theoretical problems. In the following, we will turn to a relatively in-dependent issue – a phenomenological problem: calculating the evaporation spectrumfor the Unruh vacuum and those for its α -vacua correspondingly.7or Unruh vacuum, the procedure is standard [9, 18]. If we use ψ ′ i to label thepositive frequency modes on ℑ + and ψ i to label the positive frequency modes on ℑ − ,then the Bogoliubov transformation ψ i → ψ ′ i , i.e., a i → a ′ i can be written as ψ ′ = Aψ + Bψ ∗ , (26)or namely a = a ′ A + a ′† B ∗ , (27)in which A and B are Bogoliubov transformation matrices, while a and a ′ are rowmatrices with elements a i and a ′ i respectively. A solution of the field equation (10)can be expanded asΦ( x ) = X i ( a i ψ i + a † i ψ ∗ i ) = X i ( a ′ i ψ ′ i + a ′† i ψ ′∗ i ) . (28)It has been proved in [18] that there is a relation between the Bogoliubov coefficients B ij = − e − ω i / (2 T H ) A ij , (29)and the late time particle flux through ℑ + given a vacuum on ℑ − is determined by[18] (cid:0) BB † (cid:1) ∗ ii = 1 e ω i /T H − . (30)In (4+n)-dimensions [10], the Hawking temperature can be traded for the black holeradius [13], T H = n + 14 πr H . (31)Taking into consideration of greybody factor, the spectrum of energy flux has theform dE ( ω ) dt = X l σ l,n ( ω ) ωe ω/T H − dk n +3 (2 π ) n +3 . (32)At low energy ωr H ≪
1, an analytic expression for greybody factor σ l,n has beenderived in [14, 15]. In large extra dimensional scenarios, small black holes may emitscalar fields in the bulk as well as on the brane. Our attention in this paper will be“localized” on the brane with the help of analytic formulas given in [14, 15], althoughthe calculation in the bulk can be accomplished in a similar way. In the masslessparticle approximation, corresponding to Unruh vacuum (30), the low energy scalarspectrum on the brane is [14, 15] d E ( ω ) dωdt = r − H X l (2 l + 1)Γ( l +1 n +1 ) Γ(1 + ln +1 ) ( n + 1) Γ( + l ) Γ(1 + l +1 n +1 ) (cid:16) ωr H (cid:17) l ωr H ) e πωr H / ( n +1) − . (33)8or the α -vacua constructed from the Unruh state, we should consider the follow-ing series of transformations ˜ ψ i → ψ i → ψ ′ i , ˜ a i → a i → a ′ i . (34)In other words, we take the α state˜ ψ = cosh αψ + e iγ sinh αψ ∗ , α ≥ , − π < γ ≤ π (35)as the physical vacuum on ℑ − . Inversion of (35) leads to ψ = cosh α ˜ ψ − e iγ sinh α ˜ ψ ∗ . (36)At the same time, one can also formally write ψ ′ = ˜ A ˜ ψ + ˜ B ˜ ψ ∗ , ˜ ψ = ˜ A ′ ψ ′ + ˜ B ′ ψ ′∗ . (37)The expected number of particles in the i th mode is related to (cid:16) ˜ B ˜ B † (cid:17) ∗ ii . Fromthe relations (26), (36) and (37), one immediately gets ψ ′ = ( A cosh α − Be − iγ sinh α ) ˜ ψ + ( B cosh α − Ae iγ sinh α ) ˜ ψ ∗ , (38)thus ˜ B = B cosh α − Ae iγ sinh α. (39)Multiplied by ˜ B † = B † cosh α − A † e − iγ sinh α, (40)it gives˜ B ˜ B † = cosh α (cid:0) BB † (cid:1) + sinh α (cid:0) AA † (cid:1) − sinh α cosh α [ e iγ (cid:0) AB † (cid:1) + e − iγ (cid:0) BA † (cid:1) ] . (41)As a consistency check, when α = 0, apparently it reduces to the expected form˜ B ˜ B † | α =0 = BB † . In virtue of (29), (30) and (31), one can simplify the diagonalentries in (41) and write down (cid:16) ˜ B ˜ B † (cid:17) ∗ ii = sinh αe πω i r H / ( n +1) + cos γ sinh(2 α ) e πω i r H / ( n +1) + cosh αe πω i r H / ( n +1) − . (42)The absorption amplitude and greybody factor are caused by the traversal ofemitted particles in the gravitational background. They are independent of the initial9onditions, i.e., independent of the vacuum we choose on ℑ − . As a result, given an α state as the vacuum on ℑ − , at the late time, the low energy flux through ℑ + is d E ( ω ) dωdt = r − H X l (2 l + 1)Γ( l +1 n +1 ) Γ(1 + ln +1 ) ( n + 1) Γ( + l ) Γ(1 + l +1 n +1 ) (cid:16) ωr H (cid:17) l ωr H ) × sinh αe πωr H / ( n +1) + cos γ sinh(2 α ) e πωr H / ( n +1) + cosh αe πωr H / ( n +1) − . (43)For high energy emissions ωr H ≫
1, equation (43) predicts a divergent energyflux. This suggests the breakdown of (42) and (43) at very high energy. Indeed,during derivation of the spectrum, we have neglected backreactions to the black hole,which are supposed to be small at low energy. In the high energy region, especiallyfor small black holes, the backreaction effect of the particle emission will be large, sothe spectrum (42) together with (43) cannot be trusted there any more.Assuming large extra dimensions and TeV Plank scale, in [11, 12, 13], it has beenproposed that small black holes of TeV scale masses would be produced in LHC andprovide the first experimental test of Hawking’s radiation hypothesis. On the onehand, the lifetime of the small black holes in this scenario is of order τ ∼ M ∗− p (cid:18) MM ∗ p (cid:19) n +3 n +1 ∼ r H (cid:18) MM ∗ p (cid:19) n +2 n +1 (44)or typically 10 − second [12, 15], much shorter than that of ordinary black holes inastrophysics. On the other hand, for a black hole formed by collapse, at a sufficientlylong time after its formation, the Unruh state serves as a good boundary condition ofGreen’s function. While not long after its formation, the boundary condition dependson details of the collapse. If we introduce the α -vacua (35) as a new boundarycondition, there are two additional parameters α and γ , which would capture someuniversal features of the black hole formation in LHC and characterize the deviationof the vacuum from the Unruh vacuum. In this sense, the physical values of α and γ cannot be determined theoretically. However, we observe that for non-eternal blackholes, possibly the value of α depends on their lifetime. Specifically, we guess that α increases with respect to the ratio M ∗ p M , and vanishes in the limit M ∗ p M →
0. Inlarge extra dimensional scenarios, for small black holes produced in LHC, the ratiois of order 1 < M ∗ p M .
10, thus we can take α as a parameter of constant. If α islarge enough, in the low energy region ωr H ≪
1, the greybody profile of Hawking Of course, there might be some arguments favoring vanishing values of α and γ . But that is amatter subject to debate. Say, typically we have M ∗ p ∼ M ∼ −3 ω r H d E / d ω d t [ r H − ] n=0, α =0, γ =0n=0, α =0.02, γ =0n=0, α =0.04, γ =0n=0, α =0.06, γ =0 −4 ω r H d E / d ω d t [ r H − ] n=0, α =0, γ = π n=0, α =0.02, γ = π n=0, α =0.04, γ = π n=0, α =0.06, γ = π Figure 1:
The scalar energy flux d E ( ω ) dωdt on the brane as a function of ωr H with n = 0 , α = 0 (black solid lines), . (red dash-dotted lines), . (bluedashed lines), . (magenta dotted lines) and γ = 0 (the left graph), π (theright graph). radiation will be deformed according to (43). Schwarzschild phase is an importantstage during the evaporation of small black holes. If a non-vanishing value of α isindeed physical for these black holes, signatures of (43) must be imprinted on theevaporation spectrum, thus can be found out by a detailed study of small black holedecay.To give a dramatic impression, we show the low energy spectra of scalar fluxeson the brane for various values of n , α and γ in figures 1 and 2. When plotting thespectrum according to (43), we summed over l up to the third partial wave, sincethe contribution from higher partial waves is negligible. In all of the figures, α =0,0.02, 0.04, 0.06 are indicated by black solid lines, red dash-dotted lines, blue dashedlines, and magenta dotted lines respectively. For comparison, we show the low energyspectra without a large extra-dimension ( n = 0) in figure 1. For scenarios with largeextra-dimensions ( n =2, 3), the spectra are depicted in figure 2. When α = 0, withany value of γ , we are always back to the same spectrum: the greybody spectrum(33) for the Unruh vacuum. So one can compare the other three lines with the blacksolid line in each figure to search some features of α -vacua. A remarkable feature ofthe spectrum is its dependence on γ . For 0 < | γ | ≤ π , the energy flux is enhancedas α is tuned up. Between π < | γ | ≤ π , the flux is depressed with respect to α atlow energy. Particularly, near | γ | = π , for small α , the enhancement or depression isinvisible in the low energy region. Remember that as we have mentioned previously, anon-vanishing γ means the breaking of time-reversal symmetry [2, 6]. Unlike eternal11 ω r H d E / d ω d t [ r H − ] n=2, α =0, γ =0n=2, α =0.02, γ =0n=2, α =0.04, γ =0n=2, α =0.06, γ =0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.0050.010.0150.020.0250.030.035 ω r H d E / d ω d t [ r H − ] n=3, α =0, γ =0n=3, α =0.02, γ =0n=3, α =0.04, γ =0n=3, α =0.06, γ =00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.0020.0040.0060.0080.010.0120.014 ω r H d E / d ω d t [ r H − ] n=2, α =0, γ = π /2n=2, α =0.02, γ = π /2n=2, α =0.04, γ = π /2n=2, α =0.06, γ = π /2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.0050.010.0150.020.025 ω r H d E / d ω d t [ r H − ] n=3, α =0, γ = π /2n=3, α =0.02, γ = π /2n=3, α =0.04, γ = π /2n=3, α =0.06, γ = π /20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.0020.0040.0060.0080.010.0120.014 ω r H d E / d ω d t [ r H − ] n=2, α =0, γ = π n=2, α =0.02, γ = π n=2, α =0.04, γ = π n=2, α =0.06, γ = π ω r H d E / d ω d t [ r H − ] n=3, α =0, γ = π n=3, α =0.02, γ = π n=3, α =0.04, γ = π n=3, α =0.06, γ = π Figure 2:
The scalar energy flux d E ( ω ) dωdt on the brane as a function of ωr H with n = 2 (the left graphs), n = 3 (the right graphs), α = 0 (black solidlines), . (red dash-dotted lines), . (blue dashed lines), . (magentadotted lines) and γ = 0 (the upper graphs), π/ (the middle graphs), π (thelower graphs). α -vacua with γ = 0.Up to now, we have only discussed the spin-0 filed. It is well known that blackholes radiate fields with various spins. α -vacua in de Sitter spacetime for scalar fieldhave been previously extended to other fields, see [19, 20]. If such extensions gothrough in Schwarzschild spacetime, their effects should also be studied to probe α -vacua of small black holes in LHC. Acknowledgement : We would like to thank Miao Li and Wei Song for usefuldiscussions. We are also grateful to the referees for valuable comments which haveenabled us to improve the manuscript substantially.
References [1] E. Mottola, “Particle Creation In De Sitter Space,” Phys. Rev. D , 754 (1985).[2] B. Allen, “Vacuum States In De Sitter Space,” Phys. Rev. D , 3136 (1985).[3] U. H. Danielsson, “Inflation, holography and the choice of vacuum in de Sitterspace,” JHEP , 040 (2002) [arXiv:hep-th/0205227].[4] I. Antoniadis, P. O. Mazur and E. Mottola, “Cosmological dark energy: Prospectsfor a dynamical theory,” New J. Phys. , 11 (2007) [arXiv:gr-qc/0612068].[5] G. W. Gibbons, “The Elliptic Interpretation Of Black Holes And Quantum Me-chanics,” Nucl. Phys. B , 497 (1986).[6] A. Chamblin and J. Michelson, “Alpha-vacua, black holes, and AdS/CFT,”Class. Quant. Grav. , 1569 (2007) [arXiv:hep-th/0610133].[7] S. W. Hawking, “Particle Creation By Black Holes,” Commun. Math. Phys. ,199 (1975) [Erratum-ibid. , 206 (1976)].[8] J. B. Hartle and S. W. Hawking, “Path Integral Derivation Of Black Hole Radi-ance,” Phys. Rev. D , 2188 (1976).[9] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D , 870 (1976).[10] R. C. Myers and M. J. Perry, “Black Holes In Higher Dimensional Space-Times,”Annals Phys. , 304 (1986). 1311] T. Banks and W. Fischler, “A model for high energy scattering in quantumgravity,” arXiv:hep-th/9906038.[12] S. B. Giddings and S. D. Thomas, “High energy colliders as black hole fac-tories: The end of short distance physics,” Phys. Rev. D , 056010 (2002)[arXiv:hep-ph/0106219].[13] S. Dimopoulos and G. L. Landsberg, “Black holes at the LHC,” Phys. Rev. Lett. , 161602 (2001) [arXiv:hep-ph/0106295].[14] P. Kanti and J. March-Russell, “Calculable corrections to brane black hole decay.I: The scalar case,” Phys. Rev. D , 024023 (2002) [arXiv:hep-ph/0203223].[15] P. Kanti, “Black holes in theories with large extra dimensions: A review,” Int.J. Mod. Phys. A , 4899 (2004) [arXiv:hep-ph/0402168].[16] M. Spradlin, A. Strominger and A. Volovich, “Les Houches lectures on de Sitterspace,” arXiv:hep-th/0110007.[17] Valeri P. Frolov and Igor. D. Novikov, Black Hole Physics , Kluwer AcademicPublishers (1998).[18] P. K. Townsend, “Black holes,” arXiv:gr-qc/9707012.[19] J. de Boer, V. Jejjala and D. Minic, “Alpha-states in de Sitter space,” Phys.Rev. D , 044013 (2005) [arXiv:hep-th/0406217].[20] H. Collins, “Fermionic alpha-vacua,” Phys. Rev. D71