DIETER LÜST (LMU, MPI) - Stony Brook University

Large N Graviton Scattering,
Black Hole & String Production and
Some Comments on String Geometry
DIETER LÜST (LMU, MPI)
Simons Center, 6. May 2015
1
Mittwoch, 6. Mai 15
I) Introduction
Outline:
II) Large N graviton scattering amplitudes at
high energies in Gravity and String Theory
(Black Hole Production)
Work in collaboration with Gia Dvali, Cesar Gomez,
Reinke Isermann and Stephan Stieberger,
arXiv: 1409.7405
III) T-duality and Non-commutative &
Non-associative Geometry
(volume uncertainty)
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Mittwoch, 6. Mai 15
I) Introduction
Quantum mechanics:
• Complementary picture:
Wave
N particles
• Heisenberg‘s uncertainty: phase space quantization:
[x, p] = i~ )
Semiclassical limit:
x
p
~ = const. ,
Quantization of area in phase space.
~
N !1
Distances can be still arbitrarily short.
In quantum gravity and in string theory some
of these statements have
to be refined.
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Mittwoch, 6. Mai 15
High energy string scattering:
[Amati, Ciafaloni,Veneziano (1987); Gross, Mende (1987)]
Refinement of Heisenberg relation:
x
~
0
+↵ p
p
x
Smallest possible distance:
)
x
p
~↵0 =
xmin
What about smallest possible volume?
4
Mittwoch, 6. Mai 15
p
In particular two questions and puzzles:
• What is the quantum nature of Black Holes microscopic understanding of black hole entropy?
Two (interconnected ?) claims:
Solve these problems (partially) within Einstein gravity!
Classicalization & the black hole N-portrait
• What is the high energy behavior of graviton
scattering amplitudes ?
Unitarity at tree level ?
Mittwoch, 6. Mai 15
5
Classicalization & the black hole N-portrait:
• Are described by IR physics,
[G. Dvali, C. Gomez, .....]
.. where there is no need to modify gravity in the IR
Quantization of gravity in IR
semiclassical regime.
However there remain still some UV problems:
• Precise coefficient coefficient in black hole entropy:
S =
1
4
A
L2P
• Renormalization, UV finiteness of loop amplitudes
New UV degrees of freedom ➮
6
Mittwoch, 6. Mai 15
String theory !
Unitarity and classicalization:
hµ⌫
hµ⌫
⇠ sL2P
hµ⌫
hµ⌫
It is known that tree level graviton scattering
amplitudes grow like s (center of mass energy).
2
Violation of unitarity at s = MP
One possible solution: Wilsonian approach:
Amplitude is unitarized by integrating in new weakly
coupled degrees of freedom of shorter and shorter
wave lengths (at higher and higher energies).
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Mittwoch, 6. Mai 15
However it is expected that black holes will be
produced in particle scattering processes with high
energies of the order
p
s
1
< Rs ⌘
p
sL2P
[´t Hooft (1987); Antoniadis, Arakani-Hamed,Dimopoulos, Dvali (1998); Banks, Fischler (1999);
Dimopoulos, Landsberg (2001);Yoshino, Nambu (2002); Giddings, Thomas (2002);
Eardley, Giddings (2002); Giddings, Rychkov (2004); ...]
Classicalization: Amplitudes get unitarized by classical
black hole formation.
[G. Dvali, C. Gomez (2010); G. Dvali, G. Giudice, C. Gomez, A. Kehagias (2010)]
(Gravity protects itself at high energies by black hole
formation.)
So we need a better understanding of how black holes
are formed in graviton scattering amplitudes.
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Mittwoch, 6. Mai 15
2 parts of this talk:
(i) New look at graviton scattering at trans-Planckian
energy
• New kinematical regime, relevant for black hole
production:
2
! N
with N ! 1
• The results are in agreement with with
unitarization via classicalization
and the corpuscular picture of black holes.
• The scattering process is still IR in the sense that the
gravitons in the final state are soft and possess
sub-Planckian energies.
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Mittwoch, 6. Mai 15
Crossing the UV barrier:
The 2 → N string amplitude exhibits an interesting
transition property:
• Soft final gravitons:
Unitarization by black holes.
• Hard final gravitons: Unitarization by string Regge states.
New trans-Planckian cross-over energy scale:
E IR/UV = N Mstring
(ii) Some remarks about possible string geometry in the UV:
Use stringy symmetry: T-duality & fluxes
Non-commutative
Non-associative
10
Mittwoch, 6. Mai 15
}
geometry for closed
strings
II) Large N Graviton Scattering Amplitudes
Black hole corpuscular N-portrait:
Quantum black hole
=
(talk by C. Gomez)
Bound state of N gravitons
(Bose-Einstein condensate)
[G. Dvali, C. Gomez (2011 - 2014); G. Dvali, C. Gomez, D.L. (2012)]
Relevant properties:
• N is large and the gravitons are soft.
• Interaction strength among individual gravitons is small:
L2P
↵ = 2 << 1
R
( R ... graviton wave length)
• Collective (`t Hooft like) coupling:
= ↵N
• Black holes are formed at the quantum critical point:
=1
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Mittwoch, 6. Mai 15
(R =
p
N LP )
[G. Dvali, C. Gomez, arXiv:1207.4059;
Flassig, Pritzel, Wintergerst, arXiv:1212.3344]
Black hole bound state (at
• Mass and size: MBH
= 1 ):
p
= N MP ,
RBH =
p
N LP
• Exponential degeneracy, entropy: S ⇠ N
• Semiclassical behavior:
positive
Bogoliubov
energy levels
frequencies,
system of N free
gravitons
|
{z
black hole
}
weakly coupled graviton
Bose-Einstein condensate
1
z
N !1
Bogoliubov
modes become
gapless,
degeneracy of
states
excluded region
}|
{
= ↵N
complex
Bogoliubov
frequencies, no viable
S-matrix state
Can we reconcile this picture in graviton scattering
processes (expressed 12in terms of N and )?
Mittwoch, 6. Mai 15
2
! N graviton amplitude with high center of mass s:
We want to have soft gravitons in the final state.
k2
k1
sij = (ki + kj )2 ⇠
kN
kN
1
Classicalization limit: pin ⇠
s ! 1,
Double scaling limit:
N ! 1,
✏=
1
(
>
>
>
>
:
i, j 2 {1, N } ,
s
N 2
s
(N 2)2
,
i 2 {1, N } , j 2
/ {1, N } ,
,
i, j 2
/ {1, N } .
s and pout ⇠
2
N
p
N
s
2
!0
small impact parameter)
s ! 1 (s >> MP ) with
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Mittwoch, 6. Mai 15
p
8
>
s,
>
>
>
<
=
s
MP2 N
6= 0
(i) Field theory
To compute the graviton scattering amplitudes we use
on-shell methods, KLT techniques and scattering
equation.
[Kawai, Lewellen, Tye (1986)],
N-point gravity tree level scattering:
MN
=
✓

2
◆N
2
S[ (2, . . . , N
X
, 2S(N
AN (1, (2, . . . , N
2), N
1, N )
3)
2), (2, . . . , N
2)]N
1 AN (1, N
• S[. . . , . . . ] is called momentum kernel,
1, (2, . . . , N
S⇠
N
sij
2), N )
3
• AN (. . . ) color ordered MHV Yang-Mills amplitude:
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Mittwoch, 6. Mai 15
The amplitude including its combinatorics can be
[Cachazo, He,Yuan (2013)]
obtained via the scattering equations:
MN ⇠
Z
dN
Vol SL(2, C)
✓
Y X
a
b6=a
sab
a
b
◆
(N-3)! solutions determine positions of N points on sphere.
These equations are hard to solve for arbitrary kinematics.
Simplifies in classicalization regime: [see also Kalousios (2013)]
MN
=
⇥
N
3
2
2
2 8
+
a N
2
b N
2
1+
N s
! 
N!
2
N
N !1
Mittwoch, 6. Mai 15
N s
[(N
2
N
b
2
a 1
2
3)!!]2
a+b 2
2
a+b N
2
15
a
2
1+
3
b N
1 N +a+b
+
2
2
2
a N
b 1
a+b 3
2
2
2
HN (a, b)2
To obtain the physical probability, i.e. the S-matrix element,
we have to consider
d|h2|S|N
2i|2 =
1
(N
2)!
N
Y1
i=2
dp4i |MN |2 4 (Ptotal )
p
Full cross section by integrating over momenta
and
p
s
summing over helicities:
( pin ⇠ s , pout ⇠ N 2 )
Physical 2 ! N 2 perturbative, scattering
probability in classicalization regime:
|h2|S|N
2i| =
2
✓
L2P s
N2
Collective coupling
◆N
N
⌘ ↵N =
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Mittwoch, 6. Mai 15
N! =
✓
◆N
N! ⇠ e
2
s/MP N
N
N
Connection to non-perturbative black hole bound state:
The perturbative amplitude is suppressed by e
N
.
This is just the inverse of the degeneracy of states of a
black hole with entropy S ⇠ N .
Therefore this suppression factor is compensated at the
N
critical point = 1 by e
from the degeneracy of
black hole states:
ABH ⇠
X
j
|h2|S|N i|2p |hN |BHij |2np ⇠
N
e
N
|p ⇥ eN |np
pert.
amplitude
projection on black
hole bound
state
(This was cross checked by a semiclassical
calculation.)
So, black hole is exactly dominating at
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Mittwoch, 6. Mai 15
= 1.
For large s unitarization occurs if N increases appropriately:
This bound implies that
N & Ncrit =
2
sLP
This is the core of the idea of classicalization!
N should be larger than the corresponding entropy of a
black hole with mass equal to the center of mass energy.
Increase energy s
2 −→ N − 2
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Mittwoch, 6. Mai 15
Region of
System
of N
perturbative
essentially
free
unitarity,
energy levels
gravitons
black hole
|
{z
}
weakly coupled graviton
Bose-Einstein condensate
1
excluded region
z
}|
19
Mittwoch, 6. Mai 15
Unitarity
Formationblack
of
threshold,
black
hole,
hole
production
exponential
with
extra nondegeneray
of
perturbative
statesfactorr
entropy
{
Forbidden
Region thatregion,
violates
no viable
S-matrix
unitarity
state
↵N
=
However there remain still some UV problems:
What is happening in the regime where
N < Ncrit = sL2P
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Mittwoch, 6. Mai 15
>1 ?
(ii) Closed string theory
21
Mittwoch, 6. Mai 15
High energy behavior of open/closed string amplitudes
shows exponential fall off due to Regge modes.
[Veneziano (1968); Amati, Ciafaloni,Veneziano (1987); Gross, Mende (1987), Gross, Manes (1989]
Example: 4-point graviton amplitude
(
↵0
4 s)
↵0
( 4 s)
↵0
( 4 t)
↵0
( 4 t)
↵0
4 u)
(
M4 ⇠ K
↵0
( 4 u)
⇢ 0
↵
2
2
0 st
!↵0 !1  |A4 | ⇥ 4⇡↵
exp
(s ln |s| + t ln |t| + u ln |u|)
u
2
Square of
YM-amplitude
Momentum
kernel
String
form factor
(Note: this was basically the state of the art before our paper.)
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Mittwoch, 6. Mai 15
Generalization to arbitrary (large) N:
High energy limit:
2
2
L
L
s >> L2P , gs 2 = 2s >> 1 ) ↵0 s >> 2s >> 1
LP
LP
N 2
2
MN = 
|AY M (1, . . . , N )| FN
String form factor, comprises all stringy physics.
MN can be computed again via scattering equations
and zeroes of Jacobi polynomials:
MN
= (4⇡↵0 )N
3
N
Y3✓
⌫=1
0
FT
⇥ MN
+ O (↵ s)
⌫ ⌫ (↵ + ⌫)↵+⌫ ( + ⌫) +⌫
(↵ + + N 3 + ⌫)↵+ +N 3+⌫
1
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Mittwoch, 6. Mai 15
◆ ↵40 s
Two different
energy
regimes:
p
(i)
s
< Ms : ()
N
< N gs2
„infrared“, field theory regime
Field and ST theory amplitudes agree.
This was already conjectured for the MHV case up to 5 points by [Cheung, O´Connell, Wecht (2010)]
MN =
Fp
N =1
s
2
(ii)
> Ms : ()
> N gs
N
FT
MN
„ultraviolet“, string theory regime
MN ⇠ 
N 2
↵
0N 3
se
↵0
2
(N 3) s ln(↵0 s)
String states dominate.
Amplitude gets tamed by string states (Regge modes).
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Mittwoch, 6. Mai 15
black holes
z
weakly-coupled
gravitons
}|
excluded region
{ z
1
field theory
z
}|
}|
{
string theory
{
z
N gs2
N Ms
}|
{
p
s
Transition occurs at E IR/UV = N Mstring
Gravitons in final state become hard: Ef inal > Ms
25
Mittwoch, 6. Mai 15
black holes
excluded
region
field theory
z
}|
{
z
1
field theory
z
}|
}|
{
z
N gs2
}|
{
string theory
{
z
N gs2
Consistency for all
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Mittwoch, 6. Mai 15
string theory
}|
1
{
What is happening at the point
= N gs2 = 1 ?
Here the F.T. amplitude agrees with the string
amplitude at the critical point = 1 .
This the point where the string effects match the
amplitude from the F.T. black hole formation.
1
gs = p
N
String - black hole correspondence:
black hole can be described by a state of strings.
[Horowitz, Polchinski (1996); Dvali, D.L. (2009); Dvali, Gomez (2010)]
Here the IR is meeting the UV.
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Mittwoch, 6. Mai 15
What about loop corrections or higher order gravity (UV)
corrections?
They should correspond to 1/N corrections to what we
✓ ◆g
✓ ◆g
computed:
1
1
Ag loop ⇠
, AR g ⇠
N
N
g
There is some recent interest in higher order R gravity:
Z
p
4
2
Scale invariant gravity: S ⇠
dx
gR
[Alvarez-Gaume, Kehagias, Kounnas, D.L., Riotto, Toumbas]
- propagating, ghostfree spin-2 only on curved backgrounds
(de Sitter or anti-De Sitter).
- flat backgrounds: only scalar mode, no gravitational
interaction.
Non-trivial 28interplay between UV/IR !
Mittwoch, 6. Mai 15
As we have seen, the gravity amplitudes can be
expressed as sums over Yang-Mills amplitudes.
But we never used the information about the
number of colors Nc .
• Relation between open and closed string coupling:
gs =
2
gopen
p
• At point of string-bh correspondence: gs = 1/ N
• Planar limit of gauge theory:
2
gopen
= 1/Nc
N=
So naively we get:
2
Nc
What is the interpretation of this relation?
29
Mittwoch, 6. Mai 15
III) T-duality and Non-commutative &
Non-associative Geometry
Closed strings on a circle of radius r:
T-duality:
r
↵0
$ r˜ =
r
p $ p˜
˜
(Dual) space coordinates: X $ X
(Dual) momenta:
˜ :
(X, X)
doubled geometry
Shortest possible radius:
r
rc =
p
↵0
At short (substringy) distances it is „better“ to use the
dual description of the geometry (string geometry).
Mittwoch, 6. Mai 15
Double field theory:
W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)
• O(D,D) invariant effective string action containing
momentum and winding coordinates at the same time:
Z
2D
2 0
SDFT = d X e
R
X M = (˜
xm , xm )
Strong constraint
Standard SUGRA
Geometric
backgrounds
Mittwoch, 6. Mai 15
Violation of
strong constraint
SUGRA
D. Andriot, O. Hohm, M. Larfors, D.L., P. Patalong (2011/12);
R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke (2012);
D. Andriot, A. Betz (2013); F. Hassler, D.L. (2014))
Non-geometric backgrounds
(L-R asymmetric) consistent
string
backgrounds
31
Non-geometric backgrounds & non-geometric fluxes:
- Non-geometric Q-fluxes: spaces that are locally still
Riemannian manifolds but not anymore globally.
(Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht (2005); Dabholkar, Hull, 2005)
Transition functions between two coordinate patches are
given in terms of O(D,D) T-duality transformations:
C. Hull (2004)
Q-space will become non-commutative:
- Non-geometric R-fluxes: spaces that are even
locally not anymore manifolds.
R-space will become non-associative:
[X i , X j , X k ] := [[X i , X j ], X k ] + cycl. perm. =
=
(X i · X j ) · X k
X i · (X j · X k ) + · · · 6= 0
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Mittwoch, 6. Mai 15
Non geometric torus, metric is patched
together by a T-duality transformation.
33
Mittwoch, 6. Mai 15
Non geometric torus, metric is patched
together by a T-duality transformation.
34
Mittwoch, 6. Mai 15
Closed string boundary conditions in doubled space:
M
X M ( + 2⇡) = UN
(f lux)X N ( ) + 2⇡P I
X M (⌧ + 2⇡) = VNM (f lux)X N (⌧ ) + 2⇡ P˜ M
X M = (˜
xm , xm ) , P M = (pm , p˜m ) , P˜ M = (˜
pm , pm )
Closed strings move on a „D-brane“ in doubled space.
Non-geometric backgrounds
conditions
Mixed boundary
Non-commutativity/Non-Associativity
(In analogy to open strings with D-N boundary
conditions due to F-flux
Non-commutativity.)
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Mittwoch, 6. Mai 15
Consider deformations of toroidal backgrounds by
(non)-geometric fluxes:
UV
IR
Flat torus
with H-flux
Tx1
Twisted
torus with
f-flux
j
i
[XH,f , XH,f ]
=0
Tx2
Nongeometric
space with
Q-flux
1
2
[XQ , XQ ]
Nongeometric
space with
R-flux
Tx3
' Q p˜
3
1
2
3
They can be computed by
[[XR
, XR
], XR
]'R
- world-sheet quantization of the closed string
- CFT & canonical T-duality
(Blumenhagen, Plauschinn; Lüst (2010); Blumenhagen, Deser, Lüst, Rennecke, Plauschin (2011);
Condeescu, Florakis, Lüst (2012), Andriot, Larfors, Lüst, Patalong (2012); C. Blair (2014); I. Bakas, D.L. to appear soon)
- Membrane action
Mittwoch, 6. Mai 15
(Mylonas, Schupp, Szabo (2012))
17
Q-flux:
1
[XQ (⌧,
2
), XQ (⌧,
)] =
⇡2
3
i Q p˜
6
winding
number
The non-commutativity of the torus (fibre) coordinates is
determined by the winding in the circle (base) direction.
Corresponding uncertainty relation:
1
XQ
2
XQ
3
Ls
3
Q h˜
p i
The spatial uncertainty in the X1 , X2 - directions
grows with the dual momentum in the third direction:
non-local strings with winding in third direction.
Mittwoch, 6. Mai 15
37
R-flux background:
Via T-duality in all three directions, it describes the high
energy (short distance phase) of the H-flux
background):
2
1
2
[XR , XR ]
=) Non-associative algebra:
2
⇡
1
2
3
[[XR
(⌧, ), XR
(⌧, )], XR
(⌧, )] + perm. =
R
6
Use
3
[XR , p3 ]
=
⇡
i R p3
6
=i
Corresponding classical „uncertainty relations“:
1
XR
2
XR
Volume uncertainty:
3
Ls R
1
XR
3
hp i
2
XR
3
XR
L3s R
(see also: D. Mylonas, P. Schupp, R.Szabo, arXiv:1312.1621)
Mittwoch, 6. Mai 15
38
Non-associative algebra:
[x , x ] = ✏
i
j
ijk
pk , [x , p ] = i~
i
j
ij
,
[p , p ] = 0
i
j
[x , x , x ] = [[x , x ], x ] + cycl. perm. = R
i
j
k
3-product:
i
j
k
ijk
R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arXiv:1106.0316
D. Mylonas, P. Schupp, R.Szabo, arXiv:1207.0926, arXiv:1312.162, arXiv:1402.7306.
I. Bakas, D.Lüst, arXiv:1309.3172
(f1 43 f2 43 f3 )(~x) = ((f1 ?p f2 ) ?p f3 ) (~x)
x
x
x
iRijk @i 1 @j 2 @k 3
(f1 43 f2 43 f3 )(~x) = e
This 3-product is non-associative.
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Mittwoch, 6. Mai 15
f1 (~x1 ) f2 (~x2 ) f3 (~x3 )|~x
Summary:
• Exact computation of N-point gravity (string)
amplitudes in trans-planckian energy region in closed form
• We found evidence for classicalization and
black hole production (black hole N-portrait)
in field theory.
• We found an interesting trans-Planckian transition
between field theory and string theory: string black hole correspondence.
Next steps:
[Stieberger (2013/14); Cachazo, He,Yuan (2014)]
• Mixed gauge boson (open)/gravity (closed) amplitudes:
Bh N-portrait with matter
[Dvali, Gomez, D.L. (2013)]
• Bh N-portrait beyond tree level
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Mittwoch, 6. Mai 15
First steps in [Kuhnel, Sundborg (2014)]
• Closed string geometry at very short distances is
possibly non-commutative or even non-associative.
Several question:
What is the correct interpretation of this non-associativity?
(On-shell fields in CFT still behave associative!)
What about non-commutative/non-associative gravity?
Do we have to extend quantum mechanics?
How to represent this algebra?
Similarity with Nambu mechanics (2 Hamiltonians)?
Thank you very much!
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Mittwoch, 6. Mai 15