Ultraviolet Behavior of Supergravity

1
Ultraviolet Behavior of
Supergravity

• Lance Dixon (SLAC)
• International School of Subnuclear Physics
• Erice, Sicily
• Based on work with Z. Bern, J.J. Carrasco, D.
  Dunbar, H. Johansson, D. Kosower, M. Perelstein,
  R. Roiban, J. Rozowsky
• Lecture II 1 September 2009

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Compare spectra
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KLT copying
Bern, LD, Dunbar, Perelstein, Rozowsky (1998)

• KLT relations give the N8 SUGRA cuts
• products of N8 SUGRA trees, summed over all
  internal states very simply in terms of
• sums of products of two copies of N4 SYM cuts
• Need both planar (large Nc) and non-planar terms
• in corresponding multi-loop N4 SYM amplitude

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KLT copying at 3 loops
Using
it is easy to see that
N8 SUGRA
N4 SYM
N4 SYM
rational function of Lorentz products of external
and cut momenta all state sums already performed
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Extreme simplicity at 1 and 2 loops

• 1 loop

Green, Schwarz, Brink Grisaru, Siegel (1981)
color dresses kinematics
N8 supergravity just remove color, square
prefactors! Dc automatically same as for N4
SYM for L 1,2.
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Ladder diagrams (Regge-like)
In N4 SYM
In N8 supergravity
Schnitzer, hep-th/0701217
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For Lgt2, UV behavior of generic graphs looks
worse in N8 than N4
N4 SYM
N8 supergravity
2 from HE behavior of gravity
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But Cancellations expected beginning at L3
L-particle cut of same integral reveals a
one-loop (L2)-point amplitude but with a
numerator factor which has too many loop
momenta to be consistent (strongly violates
no-triangle property)

• Implies that there have to be additional
  cancellations
• (but how strong?), from other types of integrals,
  
• not detectable in 2-particle cuts
• Inspired computation of full 4-graviton
  amplitude at 3 4 loops

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3 loop amplitude
Bern, Carrasco, LD, Johansson, Kosower, Roiban,
th/0702112 Bern, Carrasco, LD, Johansson, Roiban,
0808.4112
Nine basic integral topologies
Seven (a)-(g) long known (2-particle cuts ?
easily determine using rung rule)
BDDPR (1998)
Two new ones (h), (i) have no 2-particle cuts
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N4 numerators at 3 loops
manifestly quadratic in loop momentum
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N8 numerators at 3 loops
Omit overall
also manifestly quadratic in loop momentum
BCDJR (2008)
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N8 no worse than N4 SYM in UV
Manifest quadratic representation at 3 loops
same behavior as N4 SYM implies same critical
dimension still for L 3
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4 loops

• Begin with the 4-loop cubic vacuum graphs
• Decorate them with 4 external legs to generate
• 50 nonvanishing cubic 4-point graphs
• Determine the 50 numerator factors, first for
  N4 SYM,
• then, using KLT, for N8 supergravity

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Simplest (rung rule) graphsN4 SYM numerators
shown
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Most complex graphsN4 SYM numerators shownN8
SUGRA numerators much larger
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Checks on the final N4 result

• Lots of different products of MHV tree
  amplitudes.
• NMHV7 anti-NMHV7 and MHV5 NMHV6
  anti-MHV5
• evaluated by Elvang, Freedman, Kiermaier,
  0808.1720

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All the N8 SUGRA detailsyou could ever want
and more

• In two locations we provide all 50 numerator
  factors for the 4-loop N8 SUGRA amplitude, in
  Mathematica readable files
• aux/ in the source of the arXiv version of
  0905.2326 hep-th
• EPAPS Document No. E-PRLTAO-103-025932
  (Windows-compatible!)

• Plus you can
• get rotatable 3D views of all 50 graphs
• extract all possible information about the 50
  numerators,etc.

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UV behavior of N8 at 4 loops

• All 50 cubic graphs have numerator factors
  composed of terms
• 
  loop momenta l
• 
  external momenta k
• Maximum value of m turns out to be 8 in every
  integral

• Manifestly finite in D4 4 x 4 8
  26 - 2 lt 0

• Not manifestly finite in D5 4 x 5 8 26
  2 gt 0

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Cancellations between integrals

• Cancellation of k4 l8 terms vanishing of
  coefficient of
• is fairly simple just set external momenta ki ?
  0,
• collect coefficients of the 2 resulting vacuum
  diagrams,
• observe that the coefficients cancel.

• Cancellation of k5 l7 and k7 l5 terms is
  trivial
• Lorentz invariance does not allow an odd-power
  divergence.

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Identities among 4-loop integrals

• At order k6 l6 , relevant for UV behavior in
  D5-2e,
• after collecting all contributions, 30 vacuum
  integrals remain.
• There are multiple ways to take the limit ki ?
  0
• because one can first shift the loop momenta, by
• Comparing the different limits (which must be
  the same)
• provides consistency relations between the
  integrals, such as
• Checked by evaluated all UV poles all 30
  integrals
• Using these relations

UV pole cancels in D5-2e N8 SUGRA
still no worse than N4 SYM in UV at 4 loops!
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Beyond four loops
higher-loop cancellations inferred from 1-loop
n-point (no triangle hypothesis)
inferred from 2-, 3-, 4-loop 4-point, via cuts
Same UV behavior as N4 super-Yang-Mills
UV behavior unknown
of loops

explicit 1, 2, 3 and 4 loop computations
of contributions
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E7(7)
Cremmer, Julia (1978,1979)

• N8 SUGRA has a continuous symmetry group,
  a noncompact form of E7. The 70 scalars
  parametrize the coset space E7(7)/SU(8), so
  non-SU(8) part is realized nonlinearly.
• We didnt use this symmetry at all in our
  explicit computations. KLT relations make
  manifest only an SU(4) x SU(4) subgroup of SU(8).
• However, it is quite possible that
  understanding E7(7) better will be a key to
  further progress.
• Recently
• - Used to construct light-cone superspace
  Hamiltonian
• Explicit action on covariant fields described
• - Implications for tree-level S-matrix elements
  discussed

Brink, Kim, Ramond, 0801.2993
Kallosh, Soroush, 0802.4106

Bianchi, Elvang, Freedman, 0805.0757
Arkani-Hamed, Cachazo, Kaplan, 0808.1446
Kallosh, Kugo, 0811.3414
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What might it all mean?

• Lets suppose N8 SUGRA is finite to all loop
  orders.
• Would this prove that it is a nonperturbatively
  consistent theory of quantum gravity?
• No!
• There are at least two reasons to think it might
  need a nonperturbative completion
• Likely L! or worse growth of the order L
  coefficients,
• L! (s/MPl2)L
• Different E7(7) behavior of the perturbative
  series (invariant!) compared with the E7(7)
  behavior of the mass spectrum of black holes
  (non-invariant!)

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Is N8 SUGRA only as good as QED?

• QED is renormalizable, but its perturbation
  series has zero radius of convergence in a
  L! aL
• UV renormalons associated with UV Landau pole
• But for small a it works pretty well
  ge - 2 agrees with
  experiment to 10 digits, etc.
• We know of many pointlike nonperturbative UV
  completions for QED asymptotically free GUTs
• What is/are the nonperturbative UV completion(s)
  for N8 SUGRA? Could some be pointlike
  too?
• Some say N8 SUGRA is in the Swampland not
  connected to string theory
  Green, Ooguri, Schwarz
• If so, then maybe the completion has to be
  pointlike?!

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Conclusions

• Through 4 loops, 4-graviton scattering amplitude
  of N8 supergravity has UV behavior no worse
  than the corresponding 4-gluon amplitude of N4
  SYM.
• Not only is N8 SUGRA finite at 3 and 4 loops,
  but
• Dc 6 at L3 Dc
  5.5 at L4
• same as for N4 SYM!
• Will the same continue to happen at higher
  loops? Partial evidence from generalized
  unitarity supports this.
• If so, N8 supergravity would be a
  perturbatively finite, pointlike theory of
  quantum gravity
• Although it may not be of direct
  phenomenological relevance, could it point the
  way to other, more realistic, finite theories?
  (with less than maximal SUSY, sadly)
• We may be engaged with, not a swampland, but a
  wetlands restoration project!

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Extra slides
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Numerator factors for N8 SUGRA
old numerator factors
BCDJKR (2007)
quartic in
new
BCDJR (2008)
(e-i) reshuffled so all terms are quadratic in
loop momenta
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4 loop graphs
Scope of computational problem illustrated by
number of cubic 4-point graphs with nonvanishing
coefficients and various topological properties
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Box cut
Bern, Carrasco, Johansson, Kosower, 0705.1864

• If the diagram contains a box subdiagram, can use
  the simplicity of the 1-loop 4-point amplitude to
  compute the numerator very simply
• Planar example

• Only five 4-loop cubic topologies
• do not have box subdiagrams.
• But there are also contact terms
• to determine.

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Twist identity

• If the diagram contains a four-point tree
  subdiagram, can use a Jacobi-like identity to
  relate it to other diagrams.
• 
  Bern, Carrasco,
  Johansson, 0805.3993

• Relate non-planar topologies to planar, etc.
• For example, at 3 loops, (i) (e) (e)T
  contact terms

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