Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops

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and Two Loops Josh Nohle [Bern, Davies, Dennen, Huang, JN - arxiv: 1303.6605] [JN - arxiv:1309.7416] [Bern, Davies, JN unpublished] 21 November, 2014 HKUST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 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 4/36

Color-Kinematics Duality (BCJ) 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 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 D 8 R 4 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 6/36

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 7/36

Physics Amplitudes Review

Color-Kinematics Duality (BCJ) 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] 9/36

Physics Amplitudes One Loop

NonSUSY One-Loop BCJ 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 11/36

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 BCJsatisfying numerators for these diagrams 12/36

NonSUSY One-Loop BCJ 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 External Polarization Loop Momentum External Momenta Need 4 polarizations and 4 powers of momenta in each term. Also, 468 possible terms with undetermined coefficients 13/36

NonSUSY One-Loop BCJ Method (cont.): Demand that box ansatz obeys rotation and reflection symmetries 387/468 coefficients fixed Generate other diagram numerators using Jacobi identities 14/36

NonSUSY One-Loop BCJ 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) 447/468 coefficients fixed 15/36

NonSUSY One-Loop BCJ 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 468/468 coefficients fixed 16/36

Introduction Why Amplitudes? N g : # of gluons N f : # of fermions N s : # of scalars 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 /30

Introduction Why Amplitudes? Renormalization: The Dark Secret of Infinities D g = D - 2 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 /30

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). D f : # of on-shell fermionic degrees of freedom Harder example: Yang-Mills theory Physics Amplitudes PhD Qualifying Exam May 15, 2013 /30

Physics Amplitudes Two Loops

NonSUSY Two-Loop BCJ 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: Ansätze for master diagram numerators Use kinematic Jacobi relations to build other diagram numerators Demand relabeling symmetries and other Jacobi restrictions Force the ansatz to obey spanning set of cuts Results and Resolution 21/36

NonSUSY Two-Loop BCJ Approach: Ansätze for master diagram numerators propagators are Feynman Max powers: p 4 q 4 # of terms: 9814 Max powers: p 3 q 5 # of terms: 9452 22/36

NonSUSY Two-Loop BCJ Approach: 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 23/36

NonSUSY Two-Loop BCJ Approach: Demand relabeling symmetries and other Jacobi restrictions Example: Planar double-box 3 2 q p 4 1 Need to impose symmetries on All other diagrams inherit symmetry through Jacobi relations 24/36

NonSUSY Two-Loop BCJ Approach: Demand relabeling symmetries and other Jacobi restrictions Example: Non-planar double-box numerator 25/36

NonSUSY Two-Loop BCJ Approach: Force the ansatz to obey spanning set of cuts (color-ordered) Demands correct YM answer Here, we simply cut (non-supersymmetric) Feynman rules 26/36

NonSUSY Two-Loop BCJ Results: Ansätze for master diagrams Use kinematic Jacobi relations to build other diagrams Demand relabeling symmetries and other Jacobi restrictions Force the ansatz to obey spanning set of cuts Looks like a tractable example to study 27/36

NonSUSY Two-Loop BCJ 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] 28/36

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NonSUSY Two-Loop BCJ 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 := + - 30/36

Physics Amplitudes Gravity

Gravity Square to Get Gravity: (YM) 2 Gravity Gluon Gluon Graviton Anti-Symmetric Tensor Scalar 32/36

Gravity Lagrangian comes from low-energy string theory limit: Square BCJ numerator to obtain gravity amplitude Now, extract UV-divergent piece of integrals 33/36

UV Divergences Divergences (or lack thereof) in D = 4: Everything finite except φφφφ, AAAA, and φφaa amplitudes Example: φφφφ (dim. reg.) 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 34/36

UV Divergences Divergences also calculated in D = 6 and D = 8 Example: 4 External Graviton Counterterm in D = 6 Two-loop counterterm in D = 4 (from all-plus helicity) Example: 4 External Graviton Counterterm in D = 4 at two loops 35/36

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 cutmerging 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 Physics Amplitudes HKUST 21 November, 2014 36/36