color-kinematics duality for pure yang-mills and gravity...

Physics|Amplitudes

Color-Kinematics Duality for Pure Yang-Mills and Gravity at

One and Two Loops Josh Nohle

[Bern, Davies, Dennen, Huang, JN - arXiv: 1303.6605] [JN - arXiv:1309.7416]

[Bern, Davies, JN – unpublished]

21 November, 2014 HKUST

Outline • Introduction and Motivation • One-slide review of color-kinematics

duality (BCJ) and double-copy • Non-SUSY BCJ at one loop

• Punchline: Everything works

• Non-SUSY BCJ at two loops • Punchline: Interesting yet tractable obstacles to gravity construction

• UV divergences in the corresponding non-SUSY theory of gravity

• Conclusion

2/36 Physics|Amplitudes HKUST 21 November, 2014

Physics|Amplitudes

Introduction and Motivation

Introduction • Both string theory and field theory

frameworks suggest an intimate connection between Yang-Mills theory and gravity

• Gravity ~ (Yang-Mills)2

• The observed duality between color and kinematics in certain amplitudes of (super-)Yang-Mills theory allows us to make this connection between Yang-Mills and gravity explicit in perturbation theory [Bern, Carrasco, Johansson]

• Proved for trees, conjectured for loops


• Duality-satisfying numerators already found for: • n pts. at tree level (L=0) [Bjerrum-Bohr, Damgaard, Sondergaard, Vanhove]

• N=4 sYM: • Up to 4 loops at 4 pt. [Bern, Carrasco, Johansson; Bern, Carrasco, Dixon, Johansson, Roiban]

• Up to 7 pts. at 1 loop [Bjerrum-Bohr, Dennen, Monteiro, O’Connell]

• 5 pts. up to 3 loop [Carrasco, Johansson]

• N=1 sYM (first reduced SUSY numerators) [Carrasco, Chiodaroli, Gunaydin, Roiban]

• 4-point, 1-loop, D=4

• A lot of work in self-dual YM [Boels, Isermann, Monteiro, O’Connell]

• n-pt., 1-loop, all-plus- or single-minus-helicity pure YM

• Different matter content and other theories beyond (super-)YM • See talks by Johansson and Huang

Color-Kinematics Duality (BCJ)

Physics|Amplitudes HKUST 21 November, 2014 5/36

Motivation • Beyond theoretical implications [eg. Monteiro

and O’Connell], color-kinematics duality is a spectacular tool for constructing (super)gravity integrands

• Allows us to probe UV structure [see Davies talk]

• Ultimate goal is to investigate D8R4 counterterm in N=8 SUGRA [see Davies talk]

• 7 loops in D=4 •  [Bossard,Howe,Stelle; Elvang,Freedman,Kiermaier; Green,Russo,

Vanhove; Green,Bjornsson; Bossard,Hillmann,Nicolai; Ramond,Kallosh; Broedel,Dixon; Elvang,Kiermaier; Beisert,Elvang,Freedman,Kiermaier,Morales,Stieberger]

• 5 loops in D=24/5


Motivation • Complications with BCJ arise at 5 loops

in N=4 super-Yang-Mills theory, which is used to construct N=8 SUGRA

• Goal: Find a more tractable example where similar

obstructions occur

• Examine one- and two-loop amplitudes in non-supersymmetric Yang-Mills


Physics|Amplitudes

Review

• BCJ Conjecture : For m points and L loops, we can regroup terms between kinematic numerators to make the numerators obey the same Jacobi relations as the color factors [Bern, Carrasco,

Johansson]

• Double Copy : [Bern, Dennen, Huang, Kiermaier]

Color-Kinematics Duality (BCJ)


Physics|Amplitudes

One Loop

• Most BCJ representations found are for SUSY theories • Common question: “Does BCJ work for less supersymmetric theories?” Answer: “Yes. It works rather seamlessly, at least for one loop.”

• Found color-kinematics duality-satisfying numerators at four points, one loop for non-SUSY Yang-Mills theory

• Valid in D dimensions, with formal polarization vectors

• Actually, found a rep. for “general” adjoint field content in loop • Special restrictions can be implemented when field content is supersymmetric

NonSUSY One-Loop BCJ


• Method: • The seven diagrams needed to construct the color-ordered amplitude

• Ordering (1,2,3,4)

• Other diagrams will not survive the unitarity cut

• Need to find BCJ-satisfying numerators for these diagrams



• Method (cont.): • No constructive approach exists yet, so start with ansatz for box diagram • Want general dimensions, so need to use formal polarization vectors

• Write down all possible terms

• Need 4 polarizations and 4 powers of momenta in each term. Also,

• 468 possible terms with undetermined coefficients

External Polarization

Loop Momentum

External Momenta



• Method (cont.): • Demand that box ansatz obeys rotation and reflection symmetries

• 387/468 coefficients fixed

• Generate other diagram numerators using Jacobi identities



• Method (cont.): • Match to the two 2-particle unitarity cuts of the amplitude

• We actually computed amplitude using Feynman rules and cut that

• (We let arbitrary adjoint matter flow in the loop)




• Method (cont.) – The “wishlist”:

• Other optional constraints involving bubble-on-external-leg and tadpole diagrams

• Gives correction UV divergence using vacuum integrals

• Gives good power counting for SUSY theories




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Introduction Why Amplitudes?

• Renormalization: The Dark Secret of Infinities • Many theories need to absorb infinities into parameters of the theory • Simple example: scalar field theory

• Absorb divergences into field, mass, and coupling parameters (the Z’s).

• Harder example: Yang-Mills theory

Physics|Amplitudes PhD Qualifying Exam May 15, 2013

Ng: # of gluons | Nf: # of fermions | Ns: # of scalars

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Dg = D - 2

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Df : # of on-shell fermionic degrees of freedom

Physics|Amplitudes

Two Loops

• Motivation (win-win): • If it works simply: More evidence for BCJ for no SUSY • If it does not work: Tractable example of bad behavior

• Eye on BCJ for five-loop N=4 sYM N=8 SUGRA

• Approach: ①  Ansätze for master diagram numerators

②  Use kinematic Jacobi relations to build other diagram numerators ③  Demand relabeling symmetries and other Jacobi restrictions

④  Force the ansatz to obey spanning set of cuts

• Results and Resolution

NonSUSY Two-Loop BCJ


• Approach: ①  Ansätze for master diagram numerators — propagators are Feynman


Max powers: p4 | q4 # of terms: 9814

Max powers: p3 | q5 # of terms: 9452


• Approach: ②  Use kinematic Jacobi relations to build other diagram numerators • 14 topologies that we care about

• Define other diagram numerators by kinematic Jacobi identities • This enforces BCJ

• Example: Non-planar double-box numerator



• Approach: ③  Demand relabeling symmetries and other Jacobi restrictions • Example: Planar double-box

• Need to impose symmetries on

• All other diagrams inherit symmetry through Jacobi relations


3 2

4 1 q p


• Approach: ③  Demand relabeling symmetries and other Jacobi restrictions

• Example: Non-planar double-box numerator



• Approach: ④  Force the ansatz to obey spanning set of cuts (color-ordered) • Demands correct YM answer

• Here, we simply cut (non-supersymmetric) Feynman rules



• Results: ①  Ansätze for master diagrams ②  Use kinematic Jacobi relations to build other diagrams

③  Demand relabeling symmetries and other Jacobi restrictions ④  Force the ansatz to obey spanning set of cuts


Looks like a tractable example to study


• Resolution: • We have most of the gravity answer from utilizing double-copy

• Checked double-copy gravity cuts numerically on the two solved cuts

• YM Feynman rules, in fact, can give the final vertical-vertical cut, where BCJ is satisfied on that cut (which is really all that we need)

• Then, that gravity cut is trivial using double-copy

• Assimilate the vertical-vertical cut into the two solved cuts using known method (“cut merging”) [arXiv: hep-ph/0404293, Bern, Dixon, Kosower]



• Merging • Need to merge numerators of shared diagrams • Horizontal-vertical and 3-particle cuts fix all but the middle contact term

• Terms proportional to (p+q)^2

• Write numerators in inverse propagator momentum basis

• Merging procedure example in double-copy theory of gravity:


:= - +

2 2 2 2 “ ”


Physics|Amplitudes

Gravity

• Square to Get Gravity:

(YM)2 Gravity

Gluon Gluon Graviton Anti-Symmetric Tensor

Scalar

Gravity


• Lagrangian comes from low-energy string theory limit:

• Square BCJ numerator to obtain gravity amplitude

• Now, extract UV-divergent piece of integrals

Gravity


UV Divergences

• Divergences (or lack thereof) in D = 4: • Everything finite except φφφφ, AAAA, and φφAA amplitudes

  Example: φφφφ

corresponding to the operator

  Matches ‘t Hooft and Veltman’s 1974 result up to a factor of 2/3 because we also have the anti-symmetric tensor in the loop

(dim. reg.)


• Divergences also calculated in D = 6 and D = 8

• Two-loop counterterm in D = 4 (from all-plus helicity)

UV Divergences

  Example: 4 External Graviton Counterterm in D = 6

  Example: 4 External Graviton Counterterm in D = 4 at two loops


Conclusion • Main Results:

• 1) Color-kinematics and double-copy works for four points at one loop with no SUSY

• 2) Uncovered an example where color-kinematics duality is not so simple at two loops

• Finished two-loop calculation via the “cut-merging” procedure

• Continue investigating what constraints need to be loosened to get BCJ to work globally

• BCJ only on cuts? Locality? Higher power-counting? Asymmetric representation? [See Carrasco talk]

• Connect findings to N=4 sYM at five loops


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