Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26,...
-
Upload
timothy-newton -
Category
Documents
-
view
214 -
download
0
Transcript of Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26,...
![Page 1: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/1.jpg)
Henrik JohanssonCERN
March 26, 2013
BUDS workshopINFN Frascati
1201.5366, 1207.6666, 1210.7709
Based on work with: Zvi Bern, John Joseph Carrasco, Lance Dixon, Michael Douglas, Radu Roiban,
Matt von Hippel
Towards Determining the UV Behavior of Maximal Supergravity
![Page 2: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/2.jpg)
H. Johansson, Frascati 2013
SUGRA status in one slide
After 35 years of supergravity, we can only now make
very precise statements about the D=4 ultraviolet
structure.
No D=4 divergence of pure SG has been found to date.
Susy forbids 1,2 loop div., R2, R3 c.t. incompatible with
susy
Pure gravity 1-loop finite, 2-loop divergent Goroff &
Sagnotti
With matter: 1-loop divergent ‘t Hooft & Veltman
Naively susy allows 3-loop div. R4
N=8 SG and N=4 SG 3-loop finite!
N=8 SG: no divergence before 7 loops
7-loop div. in D=4 implies a 5-loop div.
in D=24/5 -- calculation in progress!
UFinite?
N=8 SG
![Page 3: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/3.jpg)
H. Johansson, Frascati 2013
Why is it interesting ?
If N=8 SG is perturbatively finite, why is it interesting ?
It better be finite for a good reason!
Hidden new symmetry, for example
Understanding the mechanism might open a host
of possibilities
Any indication of hidden structures yet?
Gravity is a double copy of gauge theories
Color-Kinematics: kinematics = Lie algebra
Constraints from E-M duality Kallosh,….
Hidden superconformal N=4 SUGRA ? Symmetry?
Gravity
Bern, Carrasco, HJ
Ferrara, Kallosh, Van Proeyen
![Page 4: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/4.jpg)
Henrik Johansson
Gauge Theory Analogy
Gauge theory in D>4 have same problem as D=4
gravity
Non-renormalizable due to dimensionful coupling
However, D=5 SYM has a UV completion: (2,0) theory
in D=6
Is D=5 SYM perturbatively UV finite ? Douglas;
Lambert et al.
If yes, how does it work ?
If no, what do we need to add ?
Solitons, KK modes ? Douglas; Lambert et al.
Understanding D=5 SYM might (or might not)
give clues to how to understand D=4 gravity.
(2,0) theory ?
D=5 SYM
![Page 5: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/5.jpg)
H. Johansson, Frascati 2013
Review UV status N=8 SUGRA and N=4 SYM
4pt amplitudes and UV divergences3,4-loop N=8 SUGRA & N=4 SYM5-loop nonplanar SYM6-loop planar D=5 SYM
5pt amplitudes and UV divergences1,2,3-loop N=8 SUGRA & N=4 SYM
Current 5-loop progress
Conclusion
Outline
![Page 6: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/6.jpg)
H. Johansson, Frascati 2013
UV properties N =8 SG
• N =8 SG: conventional superspace power counting forbids L=1,2
divergences Deser, Kay, Stelle; Howe and Lindström; Green,
Schwarz, Brink; Howe, Stelle; Marcus, Sagnotti
• Three-loop divergence ruled out by calculation: Bern, Carrasco,
Dixon, HJ, Kosower, Roiban, (2007), Bern, Carrasco, Dixon, HJ, Roiban (2008)
• L<7 loop divergences ruled out by counterterm analysis,
using E7(7) symmetry and other methods, but a L=7
divergence is still possibleBeisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove, Kallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, and more….
In D=4 dimensions:
In D>4 dimensions:Through four loops N =8 SG and N =4 SYM diverge in exactly the same dimension:
Marcus and Sagnotti; Bern, Dixon, Dunbar, Perelstein, Rozowsky;Bern, Carrasco, Dixon, HJ, Kosower, Roiban
![Page 7: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/7.jpg)
H. Johansson, Frascati 2013
UV divergence trendPlot of critical dimensions of N = 8 SUGRA and N = 4 SYM
Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle
current trend for N = 8If N = 8 div. at L=7
calculations:
L = 7 lowest loop order for possible D = 4 divergenceBeisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove Kallosh, Ramond, Lindström,
Berkovits, Grisaru, Siegel, Russo, and more….
1-2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel
Finite?
Divergent
![Page 8: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/8.jpg)
H. Johansson, Frascati 2013
UV divergence trendPlot of critical dimensions of N = 8 SUGRA and N = 4 SYM
Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle
current trend for N = 8If N = 8 div. at L=7
calculations:
L = 7 lowest loop order for possible D = 4 divergenceBeisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove Kallosh, Ramond, Lindström,
Berkovits, Grisaru, Siegel, Russo, and more….
1-2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel
26/5 or 24/5 ?
Finite?
Divergent
![Page 9: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/9.jpg)
H. Johansson, Frascati 2013
N =8 Amplitude and Counter Term Structure
4pt amplitude form (any dimension)
divergence occurs in
Counter term
D = 8
D = 6
D = 7
D = 5.5
Looporder
1
2
3
4
If amplitude for L 4 has at least 8 derivatives then by dimensional analysis: no divergence before L = 7 !
D = 24/5 ?5 ??
![Page 10: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/10.jpg)
H. Johansson, Frascati 2013
N =8 Amplitude and Counter Term Structure
4pt amplitude form (any dimension)
divergence occurs in
Counter term
D = 8
D = 6
D = 7
D = 5.5
Looporder
1
2
3
4
If amplitude for L 5 has at least 10 derivatives then by dimensional analysis: no divergence before L = 8 !
D = 26/5 ?5 ??
![Page 11: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/11.jpg)
H. Johansson, Frascati 2013
Earliest appearance of N = 8 Divergence
3 loops Conventional superspace power counting Green, Schwarz, Brink (1982)Howe and Stelle (1989)Marcus and Sagnotti (1985)
5 loops Partial analysis of unitarity cuts; If N = 6 harmonic superspace exists; algebraic renormalisation
Bern, Dixon, Dunbar, Perelstein, Rozowsky (1998)Howe and Stelle (2003,2009)
6 loops If N = 7 harmonic superspace exists Howe and Stelle (2003)
7 loops If N = 8 harmonic superspace exists; string theory U-duality analysis;lightcone gauge locality arguments;E7(7) analysis, unique 1/8 BPS candidate
Grisaru and Siegel (1982);Green, Russo, Vanhove; Kallosh; Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Bossard, Howe, Stelle, Vanhove
8 loops Explicit identification of potential susy invariant counterterm with full non-linear susy
Howe and Lindström; Kallosh (1981)
9 loops Assume Berkovits’ superstring non-renormalization theorems can be carried over to N = 8 supergravity
Green, Russo, Vanhove (2006)
Finite Identified cancellations in multiloop amplitudes; lightcone gauge locality and E7(7),inherited from hidden N=4 SC gravity
Bern, Dixon, Roiban (2006),Kallosh (2009–12), Ferrara, Kallosh, Van Proeyen (2012)
![Page 12: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/12.jpg)
H. Johansson, Frascati 2013
3, 4, 5, 6-Loop Amplitudes
![Page 13: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/13.jpg)
3-loop N =8 SG & N =4 SYMColor-kinematics dual form: Bern, Carrasco, HJ
UV divergent in D=6:Bern, Carrasco, Dixon, HJ, Roiban
![Page 14: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/14.jpg)
4-loops: 85 integral types
![Page 15: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/15.jpg)
H. Johansson, QMUL 2013 15
4-loops N =4 SYM and N =8 SGBern, Carrasco, Dixon, HJ, Roiban 1201.5366
• 85 diagrams • Power counting manifest both N =4 and N =8• Both diverge in D=11/2
up to overall factor, divergence same as for N=4 SYM part
![Page 16: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/16.jpg)
H. Johansson, Frascati 2013
N =4 SYM 5-loop AmplitudeN=4 SYM important stepping stone to N=8 SG1207.6666 [hep-th]
Bern, Carrasco, HJ, Roiban
• 416 integral topologies:
• Used maximal cut method Bern, Carrasco, HJ, Kosower
• Maximal cuts: 410• Next-to-MC: 2473• N2MC: 7917• N3MC: 15156
Unitarity cuts done in D dimensions...integrated UV div. in D=26/5
![Page 17: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/17.jpg)
H. Johansson, Frascati 2013
N =4 SYM 5-loop UV divergence
Non-Planar UV divergence in D=26/5:
Double traces and single-trace NNLC finite in D=26/5,only single-trace LC and NLC are divergent
Vanish!
Bern, Carrasco, HJ, Roiban
![Page 18: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/18.jpg)
H. Johansson, Frascati 2013
Testing D=5 interceptPlot of critical dimensions of N = 8 SUGRA and N = 4 SYM
Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle
current trend for N = 8If N = 8 div. at L=7
calculations:
L = 7 lowest loop order for possible D = 4 divergenceBeisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove Kallosh, Ramond, Lindström,
Berkovits, Grisaru, Siegel, Russo, and more….
1-2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel
![Page 19: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/19.jpg)
H. Johansson, Frascati 2013
6-Loop Planar D=5 SYM
•68 planar diagrams•Given by dual conformal invariance (up to integer 0,1,-1,2,-2... prefactors)
• Independently constructed by: Eden, Heslop, Korchemsky, Sokatchev;
Bourjaily,
DiRe, Shaikh, Spradlin, Volovich
Bern, Carrasco, Dixon, Douglas, HJ, von Hippel
![Page 20: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/20.jpg)
6-Loop Planar D=5 SYM
The Parking Spot Escalation
our secret collaborator…
![Page 21: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/21.jpg)
6-Loop D=5 SYM divergence
•Using integration by parts identities, div. simplifies to 3 integrals
•Numerical integration – modified version of FIESTA 1000 node cluster at Stony Brook
•Result: divergence is nonzero.
•What cancels this divergence ? Solitons/KK modes ? Douglas;
Lambert et al.
Bern, Carrasco, Dixon, Douglas, HJ, von Hippel
![Page 22: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/22.jpg)
Divergence to all loop orders ?Bern, Carrasco, Dixon, Douglas, HJ, von Hippel
Intriguing pattern of UV divergence in critical dimension of maximal susy YM
• Accurate to <1%
• Why do the UV divergences follow this approximate curve?
• Is it asymptotically exact ?
![Page 23: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/23.jpg)
H. Johansson, Frascati 2013
5pt N=8 SUGRA calculations
![Page 24: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/24.jpg)
H. Johansson, Frascati 2013
C-K amplitudes at 1 loop
2
41
3Green, Schwarz,Brink (1982)
Duality-satisfying loop amplitudes:
2
41
3
N=4 SYM:
All-plus QCD:
N=4 SYM andAll-plus QCD:
1106.4711 [hep-th]Carrasco, HJ
![Page 25: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/25.jpg)
SYM UV div in D=8:
Carrasco, HJ 1106.4711 [hep-th]
SU(8) violating SG UV div in D=8:
SG UV div in D=8:
1-loop 5-pts UV divergences
counterterms:
![Page 26: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/26.jpg)
H. Johansson, Frascati 2013
2-loop 5-pts N =4 SYM and N =8 SG
The 2-loop 5-point amplitude with Color-Kin. duality
N = 8 SG obtained from numerator double copies
Carrasco, HJ
1106.4711 [hep-th]
![Page 27: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/27.jpg)
2-loop 5-pts UV divergences
SYM UV div in D=7:
SG UV div in D=7:
Carrasco, HJ 1106.4711 [hep-th]
SU(8) violating SG UV div in D=8:
![Page 28: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/28.jpg)
H. Johansson, Frascati 2013
3-loop 5-point SYM and N =8 SG
(“ladder-like” diagrams)
Carrasco, HJ(to appear)
Non-planar D-dimensional amplitude with manifest color-kinematics duality
(N = 8 SG obtained from squaring the numerators)
![Page 29: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/29.jpg)
H. Johansson, Frascati 2013
3-loop 5-point SYM and N =8 SG
some “Mercedes-like” diagrams… Carrasco, HJ(to appear)
![Page 30: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/30.jpg)
H. Johansson, Frascati 2013
3-loop 5-point SYM and N =8 SGCarrasco, HJ(to appear)
…in total 42 diagrams.
For SYM the UV divergent diagrams (in D=6) are very simple:
(for SG the UV div. comes from the other diagrams as well)
![Page 31: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/31.jpg)
H. Johansson, Frascati 2013
3-loop 5-pts UV divergences
![Page 32: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/32.jpg)
N =8 SG 5-loop StatusWorking on reorganizing 5-loop N=4 SYM
Bern, Carrasco, HJ, Roiban (in progress)
• 416 + 336 = 752 integral topologies• BCJ: 2500 functional Jacobi eqns
Relaxing ansatz:• Non-manifest crossing symmetry• Allow for non-local numerators• Gauge variant numerators• Relax power counting• Allow for triangle diagrams• …
Once we have integrand, integration will take ~ 1 day:• No subdivergences in D=24/5 • No IR divergences since D>4, and absence of bubbles• No more difficult than IBP:s for N=4 SYM < 1 day• If needed, numerical integration ~ few days
![Page 33: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/33.jpg)
SYM SG ?
3 loops, D=6:
4 loops, D=11/2:SYM SG
SYM SG
H. Johansson, Frascati 2013
Fortune-telling from pattern
5 loops, D=26/5:
related to diagrams in thequartic Casimir
![Page 34: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/34.jpg)
Summary
Explicit calculations in N = 8 SUGRA up to four loops show that the power counting exactly follows that of N = 4 SYM -- a finite theory
5 loop calculation in D=24/5 probes the potential 7-loop D=4 counterterm -- will provide critical input to the N = 8 question !
D=5 SYM have a 6-loop UV divergence, showing that the standard perturbative expansion misses some of the (2,0) theory contributions.
Color-Kinematics duality allows for gravity calculations for multiloop multipoint amplitudes -- greatly facilitating UV analysis in gravity.
Numbers in UV divergences of N =8 SUGRA and N =4 SYM coincide, suggesting a deeper connection between the theories
Stay tuned for the 5-loop SUGRA result…
![Page 35: Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666,](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649e7a5503460f94b7abb4/html5/thumbnails/35.jpg)
THANK YOU!