Gravitational Bremsstrahlung from Transplanckian-Energy Collisions: - - PowerPoint PPT Presentation

Gabriele Veneziano

Gravitational Bremsstrahlung from Transplanckian-Energy Collisions: a progress report

Black holes, quantum information, entanglement, and all that

IHES, 29.05-01.06 2017

Why is this topic of any relevance for this workshop? My excuse is that, by looking at these gedanken experiments at E >> MP in a regime where collapse is not expected, we arrive at an S-matrix which is:

• 1. Unitary (i.e. information preserving),
• 2. Shows the emergence of the Hawking-

temperature scale TH ~ h/GE << MP

Introduction

Outline

• PART I
• An unsolved “textbook” exercise
• A classical GR approach
• A quantum S-matrix approach, comparison with CGR
• GW energy spectrum and a logarithmic divergence
• PART II (if time allows)
• A claim by DGILS and its reinterpretation by ABV.
• A final question

The problem of computing the GWs emitted by a binary system is (almost) as old as GR. It has become gradually relevant for testing GR, for searches of GWs at bars and interferometers and, more recently, after LIGO’s observation of GWs emitted by the coalescence of BH-BH binaries: EOB (Damour, Buonanno), numerical rel. Most of the time this process is in the NR regime, with the exception of the merging itself when high speeds (v/c ~ 0.3-0.6) are reached.

The textbook exercise

Much less attention has been devoted in the past to a more academic problem: Consider the collision of two massless or highly relativistic (γ = E/m >> 1) gravitationally interacting particles in the regime R << b in which they deflect each other’s trajectory by a small angle θE (with γ-1 << θE << 1)

Problem: compute the GW spectrum associated

with this collision to lowest order in θE. How can it possibly be an unsolved problem? (A. Gruzinov, private conversation, early 2014)

θs ≡ θE = 8GE b ≡ 2R b ; c = 1

What we do know

• 1. The zero frequency limit (ZFL).

(see Smarr, prl 1977) We have a solid prediction for dEGW/dω as ω-> 0. It can be obtained either by a classical or by a quantum argument. Latter uses the well- known soft graviton limit (Weinberg 1965, …) The result (after integrating over angles)

dEGW dω → Gs π θ2

E log(θ−2 E ) ;

ω → 0

An interesting problem: Can we get the 1st correction to the ZFL? (Addazi, Bianchi & GV, in progress)

• 2. Work in the seventies:
• P. D’Eath; D’Eath and Payne ~ 1978

• S. Kovacs and K.Thorne 1977

• 3. Numerical Relativity

(F. Pretorius, U. Sperhake, private comm. ~ 04.14)

The calculation in NR is challenging because the deflected particles carry with them two shock waves that travel (almost) as fast as the emitted GWs (and roughly in the same direction) Disentangling the two becomes very tricky for γ’s >~ 3 and θE a bit > γ-1 Maybe some hope for the future?

dEgr d2k dω = Gs R2 exp

• −|k||b| − ω R3

b2 ⇥ ; Gs

• R2

b2 >> 1

A 1st attempt (2007) by D. Amati,

• M. Ciafaloni & GV and an “energy crisis”

Within ACVs approximations one finds:

=> the energy fraction in GWs is O(1) already for b = b* >> R (Gs/h (R/b*)2 =O(1)).

Smells bad (S. Rychkov priv. comm).

Need GR’s answer to:

Q1: What’s the cutoff in ω for the GWs emitted in an ultra-relativistic small angle (b >> R) 2-body collision? Related Qs:

• Q2 Is the massless limit singular?
• Q3: Is the classical limit singular?
• My (tentative) answers to Q2 & Q3: No-No!

What’s GR’s answer for θE > 1/γ?

(in particular for massless particles)

• Recent progress.

Classical: Gruzinov & GV (1409.4555), Spirin &Tomaras (1503.02016);

• Quantum: Ciafaloni, Colferai & GV (1505.06619),

C C Coraldeschi & GV, (1512.00281).

A simple Classical Treatment (A. Gruzinov & GV, 1409.4555)

The calculation is done directly in the c.o.m. system for massless particles at small θs.

Obtained via Huygens principle in Fraunhofer approximation.

b

x

x’

θs θ

θs

z− = 0 z+ = 0 z = 0 z → +∞

1 2

Schematic illustration of Huygens-Fraunhofer

z− = −2R log b

z− = −R log(b − x)2

z− = −2R ✓ log b − b · x b2 ◆ 2R x · b b2 x · θ

Frequency and angular distribution of GW spectrum:

Re ζ2 and Im ζ2 correspond to the two GW polarizations.

• Subtracting the deflected shock wave (cf. P. D’Eath) is crucial!

dEGW dω d2˜ θ = GE2 π4 |c|2 ; ˜ θ = θ − θs ; θs = 2R b b2

c(ω, ˜ θ) = Z d2x ζ2 |ζ|4 e−iωx·˜

θ h

e−2iRωΦ(x) − 1 i

Φ(x) = 1 2 ln (x − b)2 b2 + b · x b2

ζ = x + iy

c(ω, θ) = Z d2x ζ2 |ζ|4 e−iωx·θ  e−iRω ln (x−b)2

b2

− e+2iERω b·x

b2

A quantum treatment of same problem

(Ciafaloni, Colferai & GV, 1505.06619), CC&Coraldeschi & GV, 1512.00281)

Postponing momentarily the properties of this GW spectrum let me jump to:

One observation is that the usual soft-graviton recipe (emission from external legs) has to be amended since the internal exchanged gravitons are almost on shell (“fractionation” of the exchanged transverse momentum).

• Emission from such internal lines is important for

not-so-soft gravitons (hence for the energy loss). Q: is it taken care of by NL correction? (ABV)

In CC(C)V (1505.06619 & 1512.00281) the same problem has been addressed at the quantum level improving on the earlier (ACV07) treatment.

Another point is that, for gravitons with ω > R-1, there are decoherence effects.

• At fixed graviton helicity and momentum production

amplitudes depend in a precise way upon the incidence angle, which changes along the fast-particle trajectory.

• This decoherence causes a break from the flat

spectrum at ω ~ R-1.

Similar -but not identical- to the classical result of

G+V.

If this effect is kept into account when summing

• ver diagrams in which the graviton can be emitted

by any rung in the ladder diagram, the result for c(ω, θ) is:

ΦR(z) ! Φ(z) ⌘ ⇣ ˆ b · z + log

• ˆ

b z

• ⌘

,

hs(z) ⌘ 1 π2z⇤2 ✓E ω log

• ˆ

b ω E z

• log
• ˆ

b z

• ◆

⌘ ΦR(z) π2z⇤2

Z Z

θ

Z d2z Z 1 dξ hs(z)eiωbz·(θ−ξΘs(b))

However, as argued by CCCV, one should also take into account the difference between the eikonal phase of the final 3-particle state and that of an elastic 2-particle state.

• When this is done, the classical result of G+V is

exactly recovered in the limit hω/E << 1!

We have analyzed (mostly numerically) the properties

• f the spectrum in the classical limit.

To be illustrated below in a few pictures. But first some more qualitative remarks. For b-1 < ω < R-1 the E-spectrum is almost flat in ω

• Below ω = b-1 it freezes reproducing the ZFL

dEGW dω → 4G π θ2

sE2 log(θ−2 s )

dEGW dω ∼ −8G π θ2

s log(ωR)

For ω > ω* G+V argued for an ω-2 spectrum (TBC): it

turns out (extrapolating to θs ~1) to be that of a time- integrated BH evaporation!

Above ω = R-1 the energy spectrum becomes scale-invariant

• This gives a log ω* in the “efficiency” for a cutoff at ω*

Using ω* ~ R-1 θs-2 (where our approximations break down

and the “Dyson bound” dE/dt < 1/G is saturated) we find (to leading-log accuracy) the suggestive result:

dEGW dω ∼ θ2

s

E ω EGW √s = 1 2π θ2

s log(θ−2 s )

ωR

θ θs θs 1 1 ZFL Weinberg W Lipatov L log-log plot

1 Gs dEGW dω = θ2

s log(θ−1 s )

1 Gs dEGW dω = θ2

s

Z θs/ωR

θs

dθ θ = θ2

s log 1

ωR

1 Gs dEGW dω = Z θs/

√ ωR

d2(θ − θs) = θ2

s

ωR

(θ − θs) = θs √ ωR

ωR = θs θ

θs-2

ω*?

ω-θ distribution (azimuth integrated)

SOME NUMERICAL RESULTS ON THE SPECTRA

θs = 10-3

• M. Ciafaloni, D. Colferai & GV, 1505.06619

Frequency spectrum

1/(ω R)

ωR = 0.125 ωR = 10-3

• M. Ciafaloni, D. Colferai, F. Coraldeschi & GV, 1512.00281

Angular (polar and azimuthal) distribution

ωR = 8.0 ωR = 1.0 Angular (polar and azimuthal) distribution

If that behavior persists as b -> bc ~ R, the GW/graviton

distribution becomes more and more “isotropic” with <n> ~ Gs/h and (again!) characteristic energy O(h/R ~TH).

The emerging picture is quite appealing: transverse momenta are limited by 1/b while longitudinal ones (and energies) are controlled by the larger scale 1/R (with some leakage at higher frequencies)

We now want to understand what, if any, provides a large-frequency cutoff and extend the reasoning towards the large-angle/collapse regime. First steps (involving some educated guesses) already made by Ciafaloni & Colferai (1612.06923).

END PART I (check time!)

A claim by

Dvali, Gomez, Iserman, Luest, Stieberger (1409.7405) NB: D=4, no attempt to project on fixed impact parameter.

R(E) b ls ls lP

string gravity strong gravity weak gravity Collapse

lP

Critical line?

E= Eth ~ Ms/gs2 >> MP E = MP

Expected phase diagram from collapse criteria

ACV87+GV04 DGILS14

In 1409.7405 Dvali et al. have considered the process 2 UHE gravitons —> N ULE gravitons

They claim (in D=4 & both QFT & QST) that they can estimate the tree-level x-section @ large N

s >> MP2 E ~ MP2 s-1/2 << MP

σ(2 → N) ∼ N! ✓e2c Gs N 2 ◆N ∼ 1 N! (c Gs)N

where c is a constant O(1) they cannot compute. This formula can be roughly understood…

Then, with an argument I cannot follow, they pick up a precise value for N (= e2cGs) and find: NB: This N does NOT correspond to dominant N (Nmax = c G s where x-section is exp.lly large) DGILS add a “final state entropy factor” exp(+SBH) and claim they are able to saturate unitarity.

σ(2 → N) ∼ N! ✓e2c Gs N 2 ◆N ∼ 1 N! (c Gs)N

If correct looks like a nice way to extend the ACV approach above threshold for BH formation…

σ(N = e2c Gs) ∼ e−N ∼ e−SBH(M∼√s)

Exclusive x-sections have IR singularities:

• 1. At tree-level they blow up.
• 2. At fixed multiplicity w/ virtual corrections resummed

they vanish;

• 3. Only suitably defined inclusive enough xsections are

free from IR problems.

In a recent paper (A. Addazi, M. Bianchi & GV, 1611.03643) we gave a reinterpretation of DGILS that seems to resolve its tension w/ ACV and to justify qualitatively their basic picture!

however…

Define an IR-safe gravitational “jet-x section” (NB: no coll. sing. in gravity!)

Ei > ¯ E

Ej < ¯ E , X

j

Ej ≤ ∆

Virtual

and then add virtual and real soft corrections with above “cuts” to correct it. For tree-level exclusive diff. x-section use DGILS: dσ(tree) d log ω1 . . . d log ωN ∼ 1 N! (c Gs)N

Virtual soft corrections to a generic massless (2->N) process (ABV- 1611.03643)

Ιn mi ->0 limit:

1 2

B > 0 except if initial process exactly collinear. Maximal for f.s. in plane orthogonal to collision axis remains O(Gs) independently of N. Using energy-momentum cons. can be rewritten as Adding real soft emission cancels as usual the log λ dependence in favor of one upon log Δ

• 1. The scale TH ~ h/R appears naturally.
• 2. For \bar{E} > TH we find that the total multi-jet x-

section is exponentially suppressed:

Applying this to our case gives following results

• 3. For \bar{E} < TH we can see the possibility of

saturating unitarity, with a single-jet inclusive x-section going like (NB: NOT a Planck spectrum!):

σt( ¯ E TH) < exp ✓ Gs log ¯ E TH ◆

1 σt dσ( ¯ E < TH) dω = Gs ω e−

ω TH

One final question

Is there a “pre-collapse” analog of QCD’s pre-confinement (Amati & GV,’79)?

• The perturbative evolution of a QCD jet

produces a partonic state that consists of many color singlets of limited mass and resembles, energetically, the hadronic final state.

• But some non-perturbative physics is still

necessary to get down to hadrons.

• Present codes (e.g. HERWIG) use very much this

property of QCD jets (plus gluon interference).

• A general pattern seems to emerge where, at the

quantum level, the transition between the dispersive and the collapse phase is smoothed out.

• As some critical value b = bc ~ R is approached,

the nature of the final state appears to change smoothly from one characteristic of a dispersive state to one reminiscent of Hawking’s radiation (high multiplicity & <Ef> = ~ h/R).

• But, even then, NP physics is still needed for

thermalization!

In gravity

R(E) b ls ls

BH

Critical line?

2 = string gravity 3 = strong gravity 1=weak gravity

The classical phase diagram

R(E) b ls ls

BH

No sharp boundary?

Small-angle (in)elastic scattering Large-angle inelastic scattering, collapse

The quantum phase diagram?

2 = string gravity (not this talk) 3 = strong gravity 1=weak gravity

THANK YOU

Further material

The weak-gravity QFT regime: t-channel “fractionation”

Power counting for connected trees: Classical corrections related to “tree diagrams”

Summing tree diagrams => solving a classical field theory. Q: Which is the effective field theory for TP-scattering?

Acl(E, b) ∼ G2n−1sn ∼ Gs R2(n−1) → Gs (R/b)2(n−1)

The string-gravity regime: s-channel “fractionation”

R(E) b ls ls

BH

Critical line?

2 = string gravity 3 = strong gravity 1=weak gravity

In string theory even single gravi-reggeon exchange gives a complex scattering amplitude. Imaginary part is due to

formation of closed-strings in the s-channel (DHS duality)

Exponentially small at b >> ls, important at b < ls .Y1/2

Fast growth of <n>, i.e. s-channel fractionation. As a result:

hni ⇠ ImA(E, b) ⇠ Gs ~Y exp ✓ b2 l2

sY

◆ ; Y = log(α0s)

hEi = ps hni ⇠ ~Y RS > Ms ; b < ls p Y

Softening of final state: a precursor of BH evaporation (GV hep-th/0410166, Gribov’s mem. vol. 05 )?

“Classical corrections” screened, string-corrected leading eikonal can be trusted even for b < R.

Gravi-reggeon exchanged in t-channel giving a complex scattering amplitude Heavy closed string produced in s-channel DHS duality (1967)

many heavy strings in s-channel AGK cutting rules apply Turning the previous diagram by 90o