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Stringhe, buchi neri e coerenza quantistica Gabriele Veneziano Has string theory solved the information paradox? Agreement between BH-entropy and counting of states in the case of extremal (BPS) BH (Strominger-Vafa) @ weak coupling Result can be extended to strong coupling thanks to SUSY Decay of quasi extremal BHs follows Hawking iff one traces over initial brane configuration (density matrix) Amati-Russo result for strings and Hagedorn T. Questions: What happens if one starts from a pure state? At weak coupling it does not work, but may work at strong coupling Are there corrections to the pure thermal spectrum? How does all this extend to more ordinary (say Kerr) BHs? Outline 1. Black holes, fundamental strings, “stringholes” 2. Unitary string-hole production in SuperPlanckian collisions (hep-th/0410166) 3. Summary Outline 1. Black holes, fundamental strings, “stringholes” 2. Unitary string-hole production in SuperPlanckian collisions (hep-th/0410166) 3. Summary String vs Black-Hole entropy D= d+1, h = c = numerical factors =1 ls = string length Tree-level string entropy Counting states (FV, BM (‘69), HW (‘70)) Sst = M/Ms = L/ls = No. of string bits in the total string length => Hagedorn temperature in ST: THag ~ Ms NB: no coupling, no G appears! Black-Hole entropy SBH = M RS = (RS/lP)D-2 ~ M(D-2)/(D-3) (GM = RSD-3 , Gh = lPD-2 , 1/TBH = d S/dM = RS /h) to be contrasted with previous Sst = M/Ms = L/ls ***** For any D: Sst /SBH > 1 @ small M, Sst /SBH < 1 @ large M Where do the two entropies meet? Obviously at RS = ls i.e. at TBH = Ms = THag ! “string holes” = states satisfying this entropy matching condition The condition RS = ls looks strongly D-dependent (Cf. dep. of RS on M) However, using string unification @ the string scale, (lP /ls)D-2 = gs2 ~ αGUT entropy matching occurs, at any D, for M = Ms gs-2 = MP gs-2(1-1/(D-2)) and the common value of Sst and SBH is simply Ssh = gs-2 ~ 1/αGUT (~ 102 ) In string theory gs2 is actually a field, the dilaton, and its value is free in PT. Consider the (M, gs2) plane Playing with the correspondence M/Ms Much more difficult to establish except for extremal case R S > ls Black Holes (= Strings? ) R S < ls Strings ≠ BH RS = l s , “string hole” curve many properties match here gs2 Safe conclusion since these strings are larger than RS M D=4 Horowitz & Polchinski, ‘97, ‘98 Damour & GV, ‘00 strong gravity effects weak gravity effects S~M g0s2 gs2 Collapse @ fixed M. Gravitational binding can increase (log of) density of states from linear to quadratic in the physical mass. Turning string entropy into BH entropy S Black hole String (naïve) gs-2 M = gs-2 Ms= Msh M Evaporation at fixed gs or how to turn a BH into a string (Bowick, Smolin,.. 1987) M/Ms trajectory of evaporating BH Black Holes RS = l s Strings string-holes gs2 Is singularity at the end of evaporation avoided thanks to ls? Outline 1. Black holes, fundamental strings, “stringholes” 2. Unitary string-hole production in SuperPlanckian collisions (hep-th/0410166) 3. Summary The problem at hand (Super)-Planckian-energy collisions of light particles within a consistent quantum theory of gravity (superstring theory) in d = D-1 “large” dimensions Since superstring theory is essentially an S-matrix theory, the description will be, naturally, quantum and unitary Three reasons for going about it 1. Weinberg’s: “Because I can!” 2. Theoretical: Information paradox 3. Is the S-matrix description breaking down in some regime? Is quantum coherence lost? And, if not, is the final state (close to) thermal? “Phenomenological”: finding signatures of string/quantum gravity @ future colliders: In KK models with large extra dimensions; In brane-world scenarios; in general: If we can lower the true QG scale down to the TeV NB. Future colliders at best marginal for producing BHs! With very few exceptions, we do not have much of a handle on string theory in extreme regimes (strong coupling, strong curvature) Superplanckian collisions may help as gedanken experiments by providing: A way to find out how QST is able to reproduce expectations from CGR at large distances A way to find out how QST modifies gravity at short distances Two complementary approaches (>1987): 1. 2. Gross-Mende + Mende-Ooguri (1987-1990) ‘t-Hooft; Muzinich & Soldate; Amati, Ciafaloni & GV; Verlinde & V.; FPVV… Arcioni, de Haro, ‘t-Hooft; …(1987-’05) Gross-Mende-Ooguri (GMO) Genus by genus (i.e. loop by loop) calculation (GM, 1987-’88) of elastic scattering at very high energy and fixed sc. angle θ (h = number of loops): (from complex saddle trajectory) All genus resummation (MO, 1990) only justified in an energy window, fails at infinite energy (gs << 1) Small, probably too conservative (result given below) Amati, Ciafaloni, GV (ACV) et al. Work in energy-impact parameter space, A(E,b) Can go to arbitrarily high energy provided b is also increased accordingly, One then goes over to A(E, q~ θ E) by FT and trusts (leading?) contributions coming from the above region of b. In this way one can reach the regime of fixed θ << 1 In gravity, fixed θ scattering at very high E is dominated by large distance physics (opposite of QCD!). Reason explained below.. GMO vs ACV et al. -t UNPHYSICAL le g n a d e x Fi ) O M (G Overlap (small θ) low energy M2P Fixed t (ACV et al) s WITHOUT STRING THEORY b Corr’s to eik. ~ (RS/b)2(D-3) I SCATTERING θ ~ (RS/b)D-3 θ ~ 2π lP II COLLAPSE RS(E) CGR arguments for Collapse @ b < RS Penrose 1974 (unpublished) CTS arguments: 1. Eardley and Giddings, gr-qc/0201034, 2. Giddings and Rychkov, hep-th/0409131 Finite-size effects: 1. Yurtsever, 1988 2. Kohlprath and GV, gr-qc/0203093 In string theory the collapse criterion should be amended! We shall take the string coupling fixed and very small (gs << 1). Cf. our defs. b WITH STRING THEORY I corr’s to eik. ~ (RS/(b+ls))2(D-3) II ls large small BH III lP Eth = Msgs-2 = Msh lP ls RS(E) Three regimes in super-Planckian scattering I) Small angle scattering (relatively easy) II) Large angle and collapse (very hard, all attempts have failed so far) III) Stringy (easy again) This is where GMO and ACV can be compared with amazingly good agreement given the completely different approaches (q~ θ E) Cf. tree level fixed t vs. fixed, small θ Approximate (but exactly unitary) S-matrix in regions I and III Operator formula encoding previous ACV results: (E, p) Xu b+ΔX (E, -p) Xd b Inserting tree-level amplitude (and forgetting ^s!) we get Therefore, for b >> bI (Region I), we can forget about C, C+ . In this region we also find: Inserting this phase shift, and going over to scattering angle θ, we find a saddle point at corresponding precisely to the relation between impact parameter and scattering angle in the (AS) metric of a relativistic particle: clearly, fixed θ , large E probe large b This also explains why Because of eikonal exponentiation, Re δ also gives the average loop-number. Thus the total huge momentum transfer q = θ E is shared among Re δ gravitons to give: meaning that the process is soft at large bs Lesson: while in QCD it is better to get a large transverse momentum via the exchange of as few gluons as possible, in QSG it is better to share it among as many gravitons as possible! We still have to take into account the operators in δ Physically, they describe “diffractive excitation” (DE) via graviton exchange. For lack of time I will not discuss this complication (unless you ask..) Region III Let us forget for a moment that Im δ ≠ 0, C and C+ The saddle point condition now gives the relation: corresponding to deflection from an homogeneous beam of transverse size lsY: b b EL θ <<1 DE θ =gs θ =1 D=4 ls Y fixed θ curves BH max θ @ this E lP ls Y RS(E) Region III: diffractive excitation We will skip it once more Region III: effects of Im δ ≠ 0 The operators C and C+ are now “activated” , recall: The elastic amplitude, <0|S|0>, is now suppressed as exp(-2 Im δ) and therefore: NB: exponent is of order -(s/M*2) => -gs-2 = -Ssh @ E= Eth (M* = MP in D=4, M* > MP for D>4) (see below for a possible origin of M* in QSG) Which final states saturate unitarity? Recall once more: The final state, S|0>, is a coherent state of quanta associated with C, C+. What are they? In order to arrive at above expression for S one had to use the AGK rules of Gribov’s Reggeon Calculus: C (C+) annihilates (creates) a cut gravi-Reggeon*) (CGR) The probability of producing n CGRs obeys a Poisson distribution with an average given by: *) The GR is the stringy graviton; a CGR is whatever is dual to it in the sense of old DHS duality At this point we can compute the average energy of a final state associated with a single CGR: We have thus found that the final-state energies obey a sort of «anti-scaling» law This antiscaling is very unlike what we are familiar with in HEP It is however similar to what we expect in BH physics! In particular: For D=4 Teff ~ THaw even at E < Eth , while for D>4 Teff --> THaw for E --> Eth Ms/gs2 window BH Ms/gs E E-1/(D-3)~TH Ms Ms ~E-1 <E>cgr MD M* =Ms/gs Ms/gs2 E Typical final state via the optical theorem Unitarity cut through 5 CGRs 1. Our results show that, at least (much) below Eth, there is no loss of quantum coherence. However the spectra are not thermal 2. When we go above Eth we can no-longer neglect “classical” corrections. They correspond to interactions among our CGRs: hopefully, they will turn their Poisson distribution into a (approximately) thermal one for their decay products, but there is no reason to expect a breakdown of unitarity 3. The diffractive states may carry out conserved global quantum numbers (if there are any) How does s/M*2 turn into SBH ~ E RS above Eth ? A possible answer is that ls-dependence at b, RS < ls, turns into RS-dependence at b < ls < RS. If so: Finally, what’s the origin of M*? After use of EE’s: stress-energy tensor Summarizing the main points We have been able to recast the main results of ACV in the form of an approximate, but exactly unitary, Smatrix whose range of validity covers a large region of the kinematic energy--angular-momentum plane; We have studied the nature of the dominant final states in a window of energy and impact parameter at whose boundary we expect black-hole formation to begin; We have found a sort of precocious black-hole behaviour, in particular an ``anti-scaling" dependence of the average energy of the final particles from the initial energy, quite reminiscent of the inverse relation between black-hole mass and temperature; This anti-scaling behaviour introduces, through the variable x = ω E/ M*2, a new energy scale M* = Ms /gs, whose physical origin we have tried to trace back These results may have a twofold application: • a conceptual one within the search for an explicit resolution of the information paradox, • a more phenomenological one in the context of the string/quantum-gravity signals expected at colliders in models with large extra dimensions.