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Preview Quantum Bosonic String Energy-Momentum Tensor in Minkowski Space-Time

LPTHE/95-26, SUSX-TH/95-31, December 1995 Quantum bosonic string energy-momentum tensor in Minkowski space-time. E. J. Copeland†, H. J. de Vega†† and A. V´azquez† 6 †School of Mathematical and Physical Sciences, 9 University of Sussex, 9 1 Falmer, Brighton BN1 9QH, U. K. n ††LPTHE,1 Universit´e P. et M. Curie (Paris VI) et Universit´e D. Diderot (Paris VII), a J Tour 16, 1er ´etage, 4, place Jussieu, 75252 Paris Cedex 05, France. 4 1 Abstract v 2 1 The quantum energy-momentum tensor Tˆµν(x) is computed for strings in Minkowski 0 space-time. We compute its expectation value for different physical string states both 1 for open andclosedbosonic strings. The states consideredare describedby normalizable 0 6 wave-packets in the center of mass coordinates. We find in particular that Tˆµν(x) is 9 finite which could imply that the classical divergence that occurs in string theory as we / approach the string position is removed at the quantum level as the string position is h t smeared out by quantum fluctuations. p- FormassivestringstatestheexpectationvalueofTˆµν(x)vanishesatleadingorder(genus e zero). Formasslessstringstatesithasanon-vanishingvaluewhichweexplicitlycompute h and analyze for both spherically and cylindrically symmetric wave packets. The energy- : v momentum tensor components propagate outwards as a massless lump peaked at r=t. i X r a 1Laboratoire Associ´e au CNRS UA 280. 1 Introduction. String theory has emerged as the most promising candidate to reconcile general rela- tivity with quantum mechanics and unify gravity with the other fundamental interactions. It makes sense therefore to investigate the gravitational consequences of strings as we approach the Planck scale. When particles scatter at energies of the order of or larger than the Planck mass, the interaction thatdominatestheircollision isthegravitational one,attheseenergiesthepicture ofparticlefieldsorstringsinflatspace-timeceasestobevalid,thecurvedspace-timegeometry created by the particles has to be taken into account. This has been our motivation to investigate the possible gravitational effects arising from an isolated quantum bosonic string living in a flat space-time background, so we may begin a study of the scattering process of strings merely by the gravitational interaction between them. The systematic study of quantum strings in physically relevant curved space-times was started in [1] and is reviewed in extenso in ref.[2]. Inthispaper,wecalculatetheenergy-momentumtensorofbothclosedandopenquantum bosonicstringsin3+1dimensions. Ourtargetspaceisthedirectproductoffourdimensional Minkowski space-time times a compact manifold taking care of conformal anomalies. It has been shown [3] that the back-reaction for a classical bosonic string in 3 + 1 di- mensions has a logarithmic divergence when the spacetime coordinate x X(σ) that is, → when we approach the core of the string. This divergence is absorbed into a renormalization of the string tension. Copeland et al [3] showed that by demanding that both it and the divergence in the energy-momentum tensor vanish forces the string to have the couplings of compactified N = 1, D = 10 supergravity. In this paper we are able to see that when we take into account the quantum nature of the strings we lose all information regarding the position of the string and therefore any divergences that may appear when one calculates the back-reaction of quantum strings are not related to our position with respect to the position of the string. In the presentwork, the energy-momentum tensor of the string, Tˆµν, is a quantum opera- toranditmayberegardedasthevertexoperatorfortheemission(absorption)ofgravitonsin the case of closed strings or, for the open string case, as the vertex operator for the emission (absorption) of massive spin 2 particles. We compute its expectation value in one-particle string states, choosing for the string center of mass wave function a wave packet centered at the origin. Ourresultsdramatically dependonthemassofthestringstatechosen. Formassivestates the expectation value of the string energy-momentum tensor identically vanishes at leading order (genus zero), thus massive states of closed and open strings give no contribution to Tˆµν. Massless string states (such as gravitons, photons or dilatons) yield a non-zero results which turn out to be spin independent, that is the same expression for gravitons, photons and dilatons. We consider spherically symmetric and cylindrically symmetric configurations. The com- ponents for the string energy density and energy flux behave like massless waves, with the string energy being radiated outwards as a massless lump peaked at r = t (for the spheri- cally symmetric case). We provide integral representations for < Tˆµν(r,t) > [eq.(14)]. After exhibiting the tensor structure of < Tˆµν > [eqs.(16) and (19)], the asymptotic behaviour of 1 < Tˆµν(r,t) > for r and t fixed and for t with r fixed is computed. In the first → ∞ → ∞ regime the energy density and the stress tensor decay as r−1 whereas the energy flux decays as r−2. For t with r fixed, the energy density tends to 0− as t−6. That is, the spherical → ∞ wave leaves behind a rapidly vanishing negative energy density. For cylindrically symmetric configurations, < Tˆµν(ρ,t) > progagates as outgoing (plus ingoing ) cylindrical waves. For large ρ and fixed t the energy density decays as 1/ρ and the energy flux decays as 1/ρ2. For large t with ρ fixed, the same phenomenom of a → ∞ negative energy density vanishing as t−6 appears. We restrict ourselves to bosonic strings in this paper. We expect analogous results from superstrings, since only massless string states contribute to the expectation values of Tˆµν. The structure of this paper is as follows: in section 2, we consider the string energy- momentum tensor as a quantum operator and discuss its quantum ordering problems, ob- servingthatitcanberegardedas avertex operator.Wecalculate theexpectation valueofthe energy-momentum tensor for differentphysical states of thestrings (for both openand closed strings). In section 3, we calculate the string energy-momentum tensor expectation value for massless states in spherically symmetric configurations, whilst in section 4 we calculate it for cylindrically symmetric configurations which are relevant when we study for example cosmic strings which are essentially very long strings. Finally, in section 5 some final remarks about our results are stressed. 2 The string energy-momentum tensor. The energy-momentum tensor for a classical bosonic string with tension (α′)−1 is given by 1 Tµν(x) = dσdτ (X˙µX˙ν X′µX′ν)δ(x X(σ,τ)) (1) 2πα′ − − Z and the string coordinates are given in Minkowski space-time by 1 Xµ(σ,τ) = qµ+2α′pµτ +i√α′ [αµ e−in(τ−σ) +α˜µ e−in(τ+σ)] (2) n n n n6=0 X for closed strings and 1 Xµ(σ,τ) = qµ+2α′pµτ +i√α′ αµ e−inτ cosnσ (3) n n n6=0 X for open strings. For closed strings we can set α′ = 1/2, so inserting eq. (2) in eq. (1) and rewriting the four dimensional delta function in integral form we obtain for closed strings, 1 d4λ pµ Tµν(x) = dσdτ pµpν + [ανe−in(τ−σ) +α˜νe−in(τ+σ)]+ π (2π)4{ √2 n n Z n6=0 X pν + (αµe−in(τ−σ) +α˜µe−in(τ+σ)) + n n √2 n6=0 X + [αµα˜ν e−in(τ−σ)e−im(τ+σ) +α˜µαν e−in(τ+σ)e−im(τ−σ)] eiλ·x e−iλ·X(σ,τ). (4) n m n m } n6=0m6=0 X X 2 We can write X(σ,τ) = X +X +X , cm + − where X = q + pτ is the centre of mass coordinate and X and X refer to the terms cm + − with α and α in X(σ,τ) respectively. In this way, we can see now that our energy- n>0 n<0 momentum tensor has the same form as that of a vertex operator, when we recall that the vertex operator for closed strings has to have conformal dimension 2 whereas for open strings it has to have conformal dimension 1 [10] in order that the string is anomaly free. Eq.(1) is meaningful at the classical level. However, at the quantum level, one must be careful with the order of the operators since X˙µ and X˙ν do not commute with X(σ,τ). We shall define the quantum operator Tˆµν(x) by symmetric ordering. That is, 1 Tˆµν(x) dσdτ ≡ 2πα′ 1 Z X˙µX˙νδ(x X(σ,τ)) + X˙µδ(x X(σ,τ))X˙ν +δ(x X(σ,τ))X˙µX˙ν 3 − − − (cid:26) h i X′µX′νδ(x X(σ,τ)) (5) − − o This definition ensures hermiticity: Tˆµν(x)† = Tˆµν(x) Let us consider a string on a mass and spin eigenstate with a center of mass wave function ϕ(p~)δ(p0 p~2+m2). An on-shell scalar string state is then − p Ψ = d4p ϕ(p~) δ(p0 p~2+m2) p~ . | i − | i Z q Here we assume the extra space-time dimensions (beyond four) to be appropriately compact- ified, and consider string states in the physical (uncompactified) four dimensional Minkowski space-time. Now, normal ordering eq.(4) using eq.(2) and taking the expectation value with respect to the fundamental scalar state (tachyonic), we get Ψ Tˆµν(x)Ψ 1 d4λ h | | i Tˆµν(x) = d4p d4p dσdτ eiλ·x [pµpν +pµpν +pµpν] Ψ Ψ ≡ h i 3π (2π)4 1 2 1 1 2 2 1 2 h | i Z p e−iλ·Xcm p ϕ∗(p~ )ϕ(p~ )δ(p0 p~ 2+m2)δ(p0 p~ 2+m2). (6) h 1| | 2i 1 2 1 − 1 1 2 − 2 2 q q Writing τλ2 p1 e−iλ·Xcm p2 = e−i 2 −iλp2τδ4(λ+p1 p2), h | | i − eq.(6) becomes 2 d4λ Tˆµν(x) = d4p d4p dτ eiλ·x [pµpν +pµpν +pµpν] h i 3 (2π)4 1 2 1 1 2 2 1 2 Z e−iτλ22−iλp2τ δ4(λ+p p )ϕ∗(p~ )ϕ(p~ )δ(p0 p~ 2+m2)δ(p0 p~ 2+m2). 1− 2 1 2 1 − 1 1 2 − 2 2 q q 3 Performing the λ and τ integrals 4 Tˆµν(x) = d4p d4p ei(p2−p1)·x [pµpν +pµpν +pµpν] h i 3(2π)3 1 2 1 1 2 2 1 2 Z δ((3p p ) (p p )) ϕ∗(p~ )ϕ(p~ ) δ(p0 p~ 2+m2)δ(p0 p~ 2+m2). (7) 2 − 1 · 2− 1 1 2 1− 1 1 2 − 2 2 q q The calculation for open strings is obtained by substituing eq.(3) into eq.(5), with the result (setting α′ = 1) 2 Tˆµν(x) = d4p d4p ei(p2−p1)·x [pµpν +pµpν +pµpν] h i 3(2π)3 1 2 1 1 2 2 1 2 Z δ((3p p ) (p p )) ϕ∗(p~ )ϕ(p~ )δ(p0 p~ 2+m2)δ(p0 p~ 2+m2). 2 − 1 · 2− 1 1 2 1 − 1 1 2 − 2 2 q q The extra factor two in eq.(7) comes from the fact that, for closed strings, we have two independent sets of oscillation modes. Now, in general, we want to take the expectation value with respect to particle states with higher (mass)2 and spin than the tachyonic case. Since we are interested in expectation values we must have the same particle in both states, hence m = m = m. As we shall 1 2 show later, only for massless particle states has the energy-momentum tensor a non-zero expectation value. Therefore, let us first consider massless string states. For the closed string there is the graviton p~;s = Pil(n)α˜l αi p~ (8) | i s −1 −1 | i and the dilaton p~ = Pil(n) α˜l αi p~ (9) | i −1 −1 | i where ~p is the momentum (p2 = 0), s = labels the graviton helicity, Pil(n), 1 i,l 3 ± s ≤ ≤ projects into the spin 2 graviton states and Pil(n) into the (scalar) dilaton state, pi ni ≡ p~ | | Pil(n) = Pli(n) , niPil(n) = 0 s s s Psll(n) = 0 , Psil(n)Psi′l(n)= δss′ Pil(n) = Pil( n) , Pil(n) δil ninl . s s − ≡ − The massless vector states (photons) for open strings are given by p~;i = Pil(n)αl p~ . | i −1| i In analogy to the tachyon case eq.(6) we obtain for closed strings 1 d4λ p~ α˜lαiTˆµν(x)α˜j αm p~ = d4p d4p dσdτ Aljim [pµpν +pµpν +pµpν] h 1| 1 1 −1 −1| 2i 12π (2π)4 1 2 1 1 2 2 1 2 Z p eiλ·Xe−iλ·Xcm p ϕ∗(p~ ) ϕ(p~ )δ(p0 p~ 2+m2)δ(p0 p~ 2+m2), (10) h 1| | 2i 1 2 1 − 1 2 − 2 q q 4 where Aljim = 4δljδim 2δljλiλm 2δimλlλj +λlλjλiλm . − − Whereas for the open string case we get, 2 ei(p2−p1)·X p~ αiTˆµν(x)αm p~ = d4p d4p [δim (pµpν +pµpν +pµpν) h 1| 1 −1| 2i 3 1 2 (2π)3 1 1 2 2 1 2 Z 3 ηµν(p p )i(p p )m]δ((3p p ) (p p )) 2 1 2 1 2 1 2 1 −2 − − − · − ϕ∗(p~ )ϕ(p~ ) δ(p0 p~ 2+m2)δ(p0 p~ 2+m2). (11) 1 2 1− 1 2 − 2 q q having performed the σ, τ and λ integrations. Projecting on the massless physical states (8) and (9) and integrating over p0 and p0, 1 2 eqs.(10) and (11) become 1 p ,s Tˆµν(x)p ,s = d3p d3p ei(p2−p1)·x (pµpν +pµpν +pµpν) h 1 | | 2 i 6(2π)3 1 2 1 1 2 2 1 2 Z ϕ∗(p~ )ϕ(p~ ) δ(p p ), (12) 1 2 2 1 · 1 p Tˆµν(x)p = d3p d3p ei(p2−p1)·x (pµpν +pµpν +pµpν) h 1| | 2i 6(2π)3 1 2 1 1 2 2 1 2 Z ϕ∗(p~ )ϕ(p~ )δ(p p ) (13) 1 2 2 1 · respectively. It is easy to check that this energy-momentum tensor is indeed conserved. We will evaluate eq.(12) and eq.(13) shortly. For the massive case (m = 0) we will now 6 show that the equivalent result has to be identically zero. It is convenient to parametrize the momenta p and p for m = 0 in eq.(10) and eq.(11) as follows: 1 2 6 p = m(coshu,sinhuuˆ ) , p = m(coshv,sinhvuˆ ) 1 1 2 2 with uˆ and uˆ unit three-dimensional vectors and u,v 0. Then, 1 2 ≥ m2 p p m2 = [(1 uˆ uˆ )cosh(u+v)+(1+uˆ uˆ )cosh(u v) 2] 2 1 1 2 1 2 · − 2 − · · − − The only real root for m = 0 corresponds to 6 uˆ = uˆ , u v = 0 1 2 − and arbitrary u+v. This means that the delta function in eq.(13) sets p = p and we arrive 1 2 to a constant (x-independent) result for Tˆµν(x) . h i More precisely, in spherical coordinates uˆ = (cosαsinγ,sinαsinγ,cosγ) , uˆ = (cosβsinδ,sinβsinδ,cosδ) 1 2 and we find 8√2π δ(p p m2) = p p m2 δ(α β)δ(γ δ)δ(u v) 2· 1− sinγ(cosht 1)m3 | 2· 1− | − − − − q 5 So, actually the expectation value Tˆµν(x) vanishes for m2 = 0 since the argument of the √ h i 6 vanishes at α = β,γ = δ,u = v. Actually, the only reasonable constant (x-independent) value for Tˆµν(x) describing a h i localized object is precisely zero. From this result we can see that open as well as the closed string massive states do not contribute to the expectation value of Tˆµν(x). It seems that any particle emitted by a freely moving quantum string is massless such as photons, gravitons and dilatons. 3 Tˆµν(x) for massless string states in spherically symmetric configurations. Let us now consider the expectation value of the energy-momentum tensor for the mass- less closed string state given by eq.(12) and eq.(13). Notice that such expectation value is independent of the value of the particle spin (zero, one or two). For such a case, it is convenient to use the parametrization: p = E (1,uˆ ) , p =E (1,uˆ ) 1 1 1 2 2 2 with E ,E 0. 1 2 ≥ Then we see that p p = 0 implies uˆ uˆ = 1 (unless E or E vanishes). Therefore E 2 1 1 2 1 2 1 · · does not need to be equal to E and we find here a non-constant result. 2 More precisely, we find in spherical coordinates 2π δ(p p ) = δ(α β)δ(γ δ) 2 1 · E E sinγ − − 1 2 Inserting this result in eq.(12) and integrating over uˆ yields 2 π ∞ ∞ Tˆµν(x) = E dE E dE ϕ∗(E ,uˆ )ϕ(E ,uˆ ) ei(E2−E1)(t−~x·uˆ1) h i 3(2π)3 1 1 2 2 1 1 2 1 Z0 Z0 [pµpν +pµpν +pµpν] duˆ (14) 1 1 2 2 1 2 1 where duˆ sinγdγdα, p = E (1,uˆ ) and now p = E (1,uˆ ). 1 1 1 1 2 2 1 ≡ Let us consider for simplicity spherically symmetric wave packets ϕ(E,uˆ)= ϕ(E). If we take ~x= (0,0,r) we can then integrate over the angles in eq.(14) with the result 1 ∞ ∞ Tˆ00(t,r) = E dE E dE ϕ∗(E )ϕ(E ) ei(E2−E1)t 1 1 2 2 1 2 h i 6π Z0 Z0 sin(E E )r 2− 1 E2+E2+E E (15) (E E )r 1 2 1 2 2 1 − h i We can relate the result for arbitrary x = (t,~x) with the special case x = (t,0,0,z) using rotational invariance as follows, Tˆ0i(x) = xˆi C(t,r) , i =1,2,3, h i Tˆij(x) = δij A(t,r)+xˆixˆj B(t,r) , i,j = 1,2,3. (16) h i 6 Here C(t,r) = Tˆ03(t,r = z) h i A(t,r) = Tˆ22(t,r = z) , B(t,r)= Tˆ33(t,r = z) Tˆ11(t,r = z) (17) h i h i−h i with xˆi = xi the unit vector, and r 1 ∞ ∞ ei(E2−E1)t Tˆ11(t,r = z) = E dE E dE ϕ∗(E )ϕ(E ) h i −6π 1 1 2 2 1 2 (E E )2r2 Z0 Z0 2− 1 sin(E E )r cos(E E )r 2− 1 E2+E2+E E , 2− 1 − (E E )r 1 2 1 2 (cid:20)1 ∞ ∞ 2− 1 (cid:21)h i Tˆ33(t,r = z) = E dE E dE E2+E2+E E h i 6π 1 1 2 2 1 2 1 2 Z0 Z0 h i ei(E2−E1)t cos(E E )r ϕ∗(E )ϕ(E ) sin(E E )r+2 2− 1 1 2 2 1 (E E )r − (E E )r 2 − 1 (cid:20) 2− 1 sin(E E )r 2 1 2 − , − (E E )2r2 2− 1 (cid:21) i ∞ ∞ Tˆ03(t,r = z) = E dE E dE E2+E2+E E h i 6π 1 1 2 2 1 2 1 2 Z0 Z0 h i ei(E2−E1)t sin(E E )r ϕ∗(E )ϕ(E ) cos(E E )r 2− 1 . (18) 1 2 2 1 (E E )r − − (E E )r 2 − 1 (cid:20) 2− 1 (cid:21) The other components satisfy Tˆ22(t,r = z) = Tˆ11(t,r = z) , Tˆ01(t,r = z) = Tˆ02(t,r = h i h i h i h z) = Tˆ12(t,r = z) = Tˆ13(t,r = z) = Tˆ23(t,r = z) = 0, as they must be from rotational i h i h i h i invariance. As we can see the trace of the expectation value of the string energy-momentum tensor vanishes. In other words the trace of the energy-momentum tensor induced by a quantum string vanishes. Only massless particles are responsible for the field created by the string. Notice that 3A(t,r)+B(t,r)= Tˆ00(t,r) h i due to the tracelessness of the energy-momentum tensor. The r and t dependence in the invariant functions Tˆ00(t,r) ,A(t,r), B(t,r) and C(t,r) h i writing eqs.(15), (17) and (18) as 1 Tˆ00(t,r) = [F(t+r) F(t r)], h i r − − 1 1 A(t,r) = [H(t+r)+H(t r)]+ [E(t+r) E(t r)], −r2 − r3 − − 1 3 3 B(t,r) = [F(t+r) F(t r)]+ [H(t+r)+H(t r)] [E(t+r) E(t r)], r − − r2 − − r3 − − 1 1 C(t,r) = [F(t+r)+F(t r)] [H(t+r) H(t r)] . (19) −r − − r2 − − where 7 1 ∞ ∞ F(x) = E dE E dE ϕ(E )ϕ(E ) 1 1 2 2 1 2 12π Z0 Z0 sin(E E )x 2− 1 E2+E2+E E . (20) (E E ) 1 2 1 2 2 1 − h i Notice that F(x) = H′(x) and H(x) = E′(x). These relations guarantee the conserva- − tion of Tˆµν(x) , h i ∂ ∂ Tˆ0ν(x) + Tˆiν(x) = 0 . ∂th i ∂xih i We choose areal wave-packet ϕ(E) decreasing fast withE andtypically peaked at E = 0. For example a gaussian wave-packet ϕ(E): ϕ(E) = 2α 3/4 e−αE2, (21) π (cid:18) (cid:19) We see from eq.(19) that the energy density Tˆ00(r,t) and the energy flux Tˆ0i(r,t) h i h i behave like spherical waves describing the way a string massless state spreads out starting from the initial wave-packet we choose. In order to compute the asymptotic behaviour of the function F(x) we change in eq.(20) the integration variables E E = vτ/x , E +E = v . 2 1 2 1 − We find 1 ∞ x dτ v τ v τ F(x) = v4dv ϕ( [1+ ])ϕ( [1 ]) 192π τ 2 x 2 − x Z0 Z0 τ2 τ4 sin(vτ) 3 2 . " − x2 − x4# Now, we can let x with the result → ∞ 1 ∞ 2ϕ(0)2 1 F(x) x→=±∞ E4dE ϕ(E)2 + +O( ) ±4 15π x5 x7 Z0 We find through similar calculations, M N ϕ(0)2 1 x→±∞ H(x) = x + + +O( ) , − 4 | | 4π 30π x4 x6 M N x ϕ(0)2 1 E(x) x→=±∞ x2 sign(x)+ +O( ) . − 8 4π − 90π x3 x5 Here, ∞ ∞ M E4dE ϕ(E)2 and N E3dE ϕ(E)2 . ≡ ≡ Z0 Z0 For the gaussian wave packet (21) they take the values 3 1 M = and N = . 16πα (2π)3/2 √α 8 To gain an insight into the behaviour of Tˆµν , we consider the limiting cases: r , t h i → ∞ fixed, r = t and t , r fixed with the following results: → ∞ → ∞ a) r , t fixed → ∞ M 4 ϕ(0)2 1 Tˆ00(r,t) r→=∞ + +O( ) , h i 2 r 15π r6 r7 M t2 4 ϕ(0)2 1 r→∞ A(r,t) = 1 +O( ) , 4 r − r2!− 45π r6 r7 M 3 t2 8 ϕ(0)2 1 r→∞ B(r,t) = 1 + +O( ) −4 r − r2 ! 15π r6 r7 and M t 1 r→∞ C(r,t) = +O( ) . 2 r2 r7 b) r = t and large M ϕ(0)2 1 Tˆ00(r,t) r=t=→∞ + +O( ) h i 4 r 240π r6 r7 5ϕ(0)2 1 r=t→∞ A(r,t) = +O( ) −1440π r6 r7 M 7 ϕ(0)2 1 r=t→∞ B(r,t) = + +O( ) 4 r 480π r6 r7 3M N ϕ(0)2 1 r=t→∞ C(r,t) = + +O( ) 4 r 4π r2 − 480π r6 r7 c) t ,r fixed → ∞ 4ϕ(0)2 1 Tˆ00(r,t) t→=∞ +O(t−8) h i − 3π t6 A(r,t) t→=∞ 0+O(t−6) , B(r,t) t→=∞ 0+O(t−6) , C(r,t) t→=∞ 0+O(t−7) . Thus, as we mentioned earlier the energy density, the energy flux and the components of thestresstensorpropagateassphericaloutgoingwaves. Fortfixed,theenergydensitydecays as r−1 while the energy flux decays as r−2. This corresponds to a r-independent radiated energy for large r. For r fixed and large t, the energy density decays rapidly as O(1/t6). We curiously find a negative energy density in this regime. The spherical wave seems to leave behind a small but negative energy density. Notice that T00 is not a positive definite quantity for strings [see eq.(1)]. 9

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