Understanding the Ultraviolet Properties of Supergravity Theories

N=4 supergravity at three loops: Z. Bern, SD, T. Dennen, Y.-t. Huang [arXiv: 1202.3423 [hep-th]] N=4 supergravity at four loops: Z. Bern, SD, T. Dennen, in progress

Scott Davies, UCLA

International School of Subnuclear Physics Ettore Majorana Foundation and Centre for Scientific Culture July 2, 2013 Why supersymmetry?

 Supersymmetry can potentially explain  Hierarchy problem  Dark matter  High-energy unification of forces

 Supersymmetry also results in better ultraviolet behavior in gauge theories.  N=4 super-Yang-Mills is ultraviolet finite in four dimensions.  How much can supersymmetry improve the ultraviolet behavior of gravity? 2/15 Gravity in the Ultraviolet  In D=4, pure Einstein gravity is  Finite at one loop,  Divergent at two loops. (Goroff, Sagnotti (1985))

 Supersymmetry helps gauge theory. Look at supergravity. (Freedman, van Niewenhuizen, Ferrara (1976))  One- and two-loop divergences ruled out in all pure theories. (Grisaru (1977); Tomboulis (1977); Deser, Kay, Stelle (1977))  No pure (no matter) supergravity divergence in D=4 in any theory has ever been found explicitly.

3/15 Three-loop N=4 SUGRA is finite!

 N=4 pure supergravity (Cremmer, Scherk, Ferrara (1978)) was expected by

many to diverge at three loops. (Bossard, Howe, Stelle; Bossard, Howe, Stelle, Vanhove (2011))

(Bern, SD, Dennen, Huang (2012)) 4/15 What about four loops?

5/15 What goes into the calculation?

Construct the integrand

Integrate to get ultraviolet divergence Reduce to log divergent Tensor reduce Reduce integrals to basis Subtract subdivergences Add it up

6/15

Constructing the Integrand  Use the duality between color and kinematics in Yang-Mills. Sum over diagrams Kinematic numerator

Color factor

Propagators Integration over loop momenta  BCJ representation for Yang-Mills where color and kinematics obey same Jacobi identities:

(Bern, Carrasco, Johansson (2008,2010))

7/15 Constructing the Integrand  Use the duality between color and kinematics in Yang-Mills. Sum over diagrams Kinematic numerator

Color factor

Propagators Integration over loop momenta  BCJ representation for Yang-Mills where color and kinematics obey same Jacobi identities:

(Bern, Carrasco, Johansson (2008,2010))

 Double-copy construct gravity: Kinematic numerators

(Bern, Carrasco, Johansson (2008,2010)) 7/15 Constructing the Integrand

 Only one copy needs to satisfy the duality relation,

Use BCJ representation; known to four loops. Use Feynman diagrams.

N=4 N=4 SUSY pure non-SUSY ~ x SUGRA Yang-Mills Yang-Mills

8/15 Constructing the Integrand  N=4 super-Yang-Mills BCJ numerators for three loops:

Loop momentum

External momentum

(Bern, Carrasco, Johansson (2010)) Loop momenta enters Loop momenta enters

N=4 super-Yang-Mills Pure Yang-Mills  Degree of divergence for Yang-Mills is logarithmic.  Gravity diagrams are worse behaved in the ultraviolet.

9/15 Reduction to Log Divergent  Infinite parts of integrals are polynomial in external momenta and mass (after subtractions – more on this later). (Caswell, Kennedy (1982))  Power-count degree of divergence.

Quadratically divergent

Logarithmically divergent

10/15 Reduction to Log Divergent  Infinite parts of integrals are polynomial in external momenta and mass (after subtractions – more on this later). (Caswell, Kennedy (1982))  Power-count degree of divergence.

Quadratically divergent

Logarithmically divergent

 Can set external momenta to zero in log divergent integrals without affecting divergence.

10/15 Reduction to Log Divergent  In N=4 supergravity,  Three loops: linearly divergent.  Four loops: quadratically divergent.  Dimension operator extracts dependence on external momenta. Quadratically divergent Dimension operator

11/15

Reduction to Log Divergent  In N=4 supergravity,  Three loops: linearly divergent.  Four loops: quadratically divergent.  Dimension operator extracts dependence on external momenta. Quadratically divergent Dimension operator

Quadratically divergent Integral is log divergent

Set external momenta In propagators to zero

11/15

Quadratically divergent Integral is log divergent

Set external momenta In propagators to zero  Regulate infrared divergence with mass: 11/15

Tensor Reduction  Complicated tensor structure in numerators.  We want scalar (tensor-free) integrals.  Solution in Appendix A of Peskin and Schroeder.

d=4-2є

12/15 Tensor Reduction  Complicated tensor structure in numerators.  We want scalar (tensor-free) integrals.  Solution in Appendix A of Peskin and Schroeder.

d=4-2є  With no dependence on external momenta, all we can write down are metric tensors.

 Contract both sides with metric tensors and solve.

Scalar integrals

 Four-loop computation has up to 10-tensor integrals. 12/15

Basis of Scalar Integrals

 Use FIRE (Smirnov (2008)) to reduce to a basis.

 Evaluate integrals using MB (Czakon (2006)) which implements a Mellin- Barnes representation (Smirnov (1999)).  Three-loop basis:

13/15 What goes into the calculation?

Construct the integrand

Integrate to get ultraviolet divergence Reduce to log divergent Tensor reduce Reduce integrals to basis Subtract subdivergences Add it up

Subdivergences  Subtracting subdivergences is an alternative to using lower-loop counterterms. One-loop counterterm vertex

One-loop subdivergences

14/15 Subdivergences  Subtracting subdivergences is an alternative to using lower-loop counterterms. One-loop counterterm vertex

One-loop subdivergences

 There are no lower-loop divergences, so why subtract?  Non-leading pieces are regulator dependent.

Regulator-dependent divergent integrals

Regulator-independent subtracted integrals 14/15 Status  Three-loop pure N=4 SUGRA is finite.

15/15 Status  Three-loop pure N=4 SUGRA is finite.

 Leading ultraviolet divergence in an

L-loop integral is .

 First divergences in a theory will be order .

15/15 Status  Three-loop pure N=4 SUGRA is finite.

 Leading ultraviolet divergence in an

L-loop integral is .

 First divergences in a theory will be order .

 At four loops, we have cancellation of and .

 Still working on and .

 109 terms.

 Subdivergences are a complication.

15/15

Status  Three-loop pure N=4 SUGRA is finite.

 Leading ultraviolet divergence in an

L-loop integral is .

 First divergences in a theory will be order .

 At four loops, we have cancellation of and .

 Still working on and .

 109 terms.

 Subdivergences are a complication.

 We will have four loops soon.

 If finite, will surprise the supergravity community.  If divergent, first ever pure supergravity divergence in D=4. 15/15