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Perspectives on the double copy: Outlook and Summary Henrik - - PowerPoint PPT Presentation
Perspectives on the double copy: Outlook and Summary Henrik - - PowerPoint PPT Presentation
Perspectives on the double copy: Outlook and Summary Henrik Johansson Uppsala U. & Nordita Dec 13, 2019 UCLA QCD Meets Gravity Double copy Gauge theory Gravity Gauge symmetry Diffeomorphism symmetry Driven by technical simplicity
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Op Opportunities of the double copy
Hidden structures in gauge theory
à BCJ relations Casali, Tourkine, Rodina à kinematic Lie algebra (CK duality) Mason
Unified framework
à web of theories (theory classification) Chiodaroli à recycling of calculations
Mogull, Dixon, Teng
Generality of the DC
à amplitudes (& form factors in gauge theory) à BH solutions Monteiro à classical scattering/radiation Solon, Luna, Ochirov à curved backgrounds, off shell ? Adamo, Nagy
Deeper physical understanding of DC ?
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Textbook perturbative gravity is complicated
After symmetrization 100 terms !
= =
de Donder gauge higher order vertices…
103 terms
complicated diagrams:
104 terms 107 terms 1021 terms 1031 terms
Bjerrum-Bohr
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On On-she shell si simplifications ns
Graviton plane wave: = Gravity scattering amplitude:
Yang-Mills polarization Yang-Mills vertex Yang-Mills amplitude
On-shell 3-graviton vertex: Gravity processes = “squares” of gauge theory ones - entire S-matrix
M GR
tree(1, 2, 3, 4) = st
u AYM
tree(1, 2, 3, 4) ⊗ AYM tree(1, 2, 3, 4)
Bern, Carrasco, HJ Kawai, Lewellen, Tye
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Squ Squaring ng of YM the heory
Gravity processes = squares of gauge theory ones - entire S-matrix
Gravity Yang-Mills pure Yang-Mills → Einstein gravity + dilaton + axion
N =4 super-YM
→
N =8 supergravity → →
E.g.
squared numerators
(BCJ double copy)
Dixon, Parra-Martinez, Teng, Herrmann
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Wh Why/ y/How double copy y works ?
Interactions (3pt) Spectrum (2pt) Gauge symmetry (4pt)
+ +
consistent amplitudes
à
Additionally, for consistency one should check that properties such as factorization, crossing symmetry and unitarity are preserved by double copy
3 ingredients are often sufficient:
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Spectrum
States are obtained by tensoring Lorentz reps. (εh)ij
µν
= ε((i
µ εj)) ν
(graviton) (εB)ij
µν
= ε[i
µεj] ν
(B-field) , (εφ)µν = εi
µεj νδij
D − 2 (dilaton) .
More precisely in D dim: (dilaton) (axion) Chiodaroli, Nagy
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Matter spectrum
Matter spin 1
local global
Matter spin 3/2
Tensors: Spin 1/2: Spin 0: Chiodaroli
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Sp Spectrum – br broken symmetry
Spontaneous symmetry breaking:
broken local broken global
spin 1: Spin 3/2:
(massive) (massive)
Chiodaroli, Gunaydin, HJ, Roiban
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In Interactions
- n-shell:
a ca c a c
a i
a c
a i
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In Interactions - ma matter
- n-shell:
a c a c a c a c
many more examples….
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Hi Higher-de deriv ivativ ive in interactio ions
a c
Gauss-Bonnet:
a c a c
cubic Riemann:
a c
Dixon, Brödel; Bern, Edison, Kosower, Parra-Martinez; Nohle, HJ; …
Rodina
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Gauge & diffeo invariance
BCJ Jacobi identity
− =
cubic diagram form:
gauge invariance: diffeo invariance:
kinematic Jacobi Id
Chiodaroli, Nagy CK duality à kinematic Lie algebra ?
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Example: axion-dilaton gravity
Consider double copy of D-dimensional pure YM:
(εh)ij
µν
= ε((i
µ εj)) ν
(graviton) (εB)ij
µν
= ε[i
µεj] ν
(B-field) , (εφ)µν = εi
µεj νδij
D − 2 (dilaton) . States:
S =
⁄
dDx√−g
C
−1 2R + 1 2(D − 2)∂µφ∂µφ + 1 6e−4φ/(D−2)HλµνHλµν
D
Amplitudes consistent with the theory: In 4D this is axion-dilaton gravity: Symmetry allows for consistent truncation of scalars
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Ex Example: pure GR
Pure 4D Einstein gravity:
HJ, Ochirov
Does not match YM2 spectrum: Deform YM theories with massless fundamental quarks Anti-align the spins of the quarks à gives scalars in GR e.g. become ghosts if Nagy
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Ex Examp mple: YM-Ei Einstein th theory
GR + YM = YM ⊗ (YM + φ3)
Chiodaroli, Gunaydin, HJ, Roiban
N = 0,1,2,4 YM-E
supergravity
N = 0,1,2,4 SYM
YM +
GR+YM amplitudes are “heterotic” double copies
φ3
hµν ∼ Aµ ⊗ Aν
Aµa ∼ Aµ ⊗ φa
Note: N =0,1,2 YM-E are contaminated by axion-dilaton states
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Ex Exception th that t proves th the ru rule…
Not all gauge theories obey color-kinematics dualiy
Imagine the double copy: ? According to conventional wisdom must be a gravitino and is the number of supersymmetries What goes wrong? The theory
- nly obeys color-kinematics duality if supersymmetric à
Kinematic Jacobi Id. à Fierz Id. that enforces SUSY
Chiodaroli, Jin, Roiban
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We Web of double-co copy co construct ctible theories
See review 1909.01358 – Bern, Carrasco, Chiodaroli, HJ, Roiban Chiodaroli
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So Some open problems
20
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Gr Gravitational radiation
LIGO/VIRGO observations à motivates high-order PN, PM calcs.
BH gravitational scattering Non-abelian gauge-theory process
Using double copy for GW, potentials and observables :
Key problems: à scalability
Solon, Zeng, Maier
à higher-spin extensions
Goldberger, Ridgway; G, R, Prabhu, Thompson; G, Li, P Luna, Monteiro, Nicholson, O'Connell, White; Shen; Plefka, Steinhoff, Wormsbecher, Shi, Steinhoff, Wang; Maybee, O'Connell, Vines; Bern, Cheung, Roiban, Shen, Solon, Zeng; […]
many talks: Buonanno, Bjerrum-Bohr, Brustein, Damgaard, Di Vecchia, Guevara, Maia, Maier,
Levi, Luna, Kosower, Kälin, Parra-Martinez, Shen, Solon, Steinhoff, Vines, Veneziano, Yang, ….
Levi, Steinhoff, Vines, Luna, Ochirov, Yang
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Re Removing the dila dilaton ?
For massive processes the dilaton couples to mass
see e.g. Luna, Nicholson, O’Connell, White; Plefka, Shi, Wang Luna, Nicholson, O’Connell, White
Can be removed by compensating diagrams
- r projectors applied to on-shell states Bern, Cheung, Roiban, Shen, Solon, Zeng
However, methods not completely satisfactory: à What it the most efficient approach? à Is removal complete for all physical processes? à General framework for different theories? (cf. HJ, Ochirov for pure GR) Solon
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La Lagrangian fo for massive (QCD)x(QCD) ?
Gluon: Massive quarks: What is the square of QCD ?
HJ, Ochirov [1906.12292]
Lagrangian determined by matching to double copy of QCD ampl.
L(QCD)2 = − 2 κ2 R + ∂µ ¯ Z ∂µZ
1 − 2
4 ¯
ZZ
2 +
Nf
X
ı=1
⇢
−1 2V ∗
ıµ⌫V µ⌫ ı
+ m2
ı V ∗ ıµV µ ı
1− κ 2(Z + ¯ Z)+ κ2 2 ¯ ZZ
- + ∂µϕ∗
ı ∂µϕı − m2 ı ϕ∗ ı ϕı
1 − κ 2(Z + ¯ Z) + κ2 16(Z2+ ¯ Z2+ 8 ¯ ZZ)
- + iκ
4 mı
h
(ϕ∗
ı Vıµ+ V ∗ ıµϕı)∂µ(Z − ¯
Z) − (Z − ¯ Z)(∂µϕ∗
ı Vıµ+ V ∗ ıµ∂µϕı)
i
− iκ2 4 mı(ϕ∗
ı Vıµ+ V ∗ ıµϕı)( ¯
Z∂µZ − Z∂µ ¯ Z) + κ2 8
h
ϕ∗
ı ϕı∂µ ¯
Z ∂µZ + ¯ ZZ∂µϕ∗
ı ∂µϕı
i
+
Nf
X
ı,|=1
⇢κ2
8 ϕ∗
ı ϕı
h
∂µϕ∗
| ∂µϕ| − 3m2 |ϕ∗ |ϕ| + 2m2 |V ∗ |µV µ |
i
+ κ2 4 mım|
h
ϕ∗
ı ϕ∗ |VıµV µ | + ϕıϕ|V ∗ ıµV ∗µ |
+ 2ϕ∗
ı ϕ|V ∗ |µV µ ı
i
+ O(κ3). (1.2)
How do we write down all-orders Lagrangians in general DC theories? Nagy
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Wh What i is t the Kinematic A Algebra ?
Monteiro, O’Connell (’11)
Self dual YM in light-cone gauge: Lie Algebra:
YM vertex
Generators of diffeomorphism invariance:
- - If YM numerators obey Jacobi Id. à kinematic algebra should exist!
- - Algebra may dramatically simplify GR integrand construction.
What is known? Beyond the simplest helicity sectors (NMHV)
Chen, HJ, Teng, Wang [1906.10683]
tensor generator vector generator Adamo
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Th Thanks for a very interesting conference!
Man Many Than anks to: Zv Zvi, , Clifford, Do Donal, , Jo John Jo Joseph, Ra Radu an and d Ira
- A better understanding of gravity perturbation theory and
- A bright future for applications to GW calculations.