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Physics|Amplitudes
Subtleties in Nonsupersymmetric Gravity at
Two Loops
Josh Nohle, UCLA
[Zvi Bern, Clifford Cheung, Huan-Hang Chi, Scott Davies, Lance Dixon, JN – to appear]
[Zvi Bern, Scott Davies, JN – to appear] 18 June, 2015
Radcor/Loopfest
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Physics|Amplitudes
Introduction
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Introduction
• Initial goal: Reproduce and expand on (eg. adding different
fields) the two-loop pure gravity calculation of Goroff & Sagnotti (1986) using modern methods
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 1/17
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Introduction
• Initial goal: Reproduce and expand on (eg. adding different
fields) the two-loop pure gravity calculation of Goroff & Sagnotti (1986) using modern methods
• We found many surprises along the way • Contribution of evanescent operators – vanish in D=4
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 1/17
![Page 5: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/5.jpg)
Introduction
• Initial goal: Reproduce and expand on (eg. adding different
fields) the two-loop pure gravity calculation of Goroff & Sagnotti (1986) using modern methods
• We found many surprises along the way • Contribution of evanescent operators – vanish in D=4
• Quantum (in)equivalence of dilaton and anti-symmetric tensor fields
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 1/17
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Introduction
• Initial goal: Reproduce and expand on (eg. adding different
fields) the two-loop pure gravity calculation of Goroff & Sagnotti (1986) using modern methods
• We found many surprises along the way • Contribution of evanescent operators – vanish in D=4
• Quantum (in)equivalence of dilaton and anti-symmetric tensor fields
• Contribution of three-form field – zero degrees of freedom in D=4
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 1/
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Introduction
• Initial goal: Reproduce and expand on (eg. adding different
fields) the two-loop pure gravity calculation of Goroff & Sagnotti (1986) using modern methods
• We found many surprises along the way • Contribution of evanescent operators – vanish in D=4
• Quantum (in)equivalence of dilaton and anti-symmetric tensor fields
• Contribution of three-form field – zero degrees of freedom in D=4
• [Color-kinematics duality constraints only on spanning set of cuts]
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 1/17
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Physics|Amplitudes
Outline
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Outline ① Layout of Pertinent NonSUSY Gravity Theories
• Graviton, dilaton, antisymmetric tensor, three-form
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 2/17
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Outline ① Layout of Pertinent NonSUSY Gravity Theories
• Graviton, dilaton, antisymmetric tensor, three-form
② Methods • Unitarity, spinor-helicity, [color-kinematics duality]
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 2/17
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Outline ① Layout of Pertinent NonSUSY Gravity Theories
• Graviton, dilaton, antisymmetric tensor, three-form
② Methods • Unitarity, spinor-helicity, [color-kinematics duality]
③ One-Loop Gravity à Gauss-Bonnet • Review of finiteness argument / trace anomaly
• Quantum inequivalence?
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 2/17
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Outline ① Layout of Pertinent NonSUSY Gravity Theories
• Graviton, dilaton, antisymmetric tensor, three-form
② Methods • Unitarity, spinor-helicity, [color-kinematics duality]
③ One-Loop Gravity à Gauss-Bonnet • Review of finiteness argument / trace anomaly
• Quantum inequivalence?
④ Two-Loop Gravity à R3 divergence • Reproduce Goroff & Sagnotti, computing identical-helicity amplitude
• Contribution of evanescent operators
• Quantum equivalence?
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 2/17
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Outline ① Layout of Pertinent NonSUSY Gravity Theories
• Graviton, dilaton, antisymmetric tensor, three-form
② Methods • Unitarity, spinor-helicity, [color-kinematics duality]
③ One-Loop Gravity à Gauss-Bonnet • Review of finiteness argument / trace anomaly
• Quantum inequivalence?
④ Two-Loop Gravity à R3 divergence • Reproduce Goroff & Sagnotti, computing identical-helicity amplitude
• Contribution of evanescent operators
• Quantum equivalence?
⑤ [Color-Kinematics Duality at Two-Loops] UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 2/17
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Physics|Amplitudes
Nonsupersymmetric Gravity Theories
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NonSUSY Gravity Theories
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 3/17
• Pure Gravity Lagrangian:
• Graviton, Three-Form Lagrangian:
• Note the D=4 duality transformation
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NonSUSY Gravity Theories
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 4/17
• Graviton, Antisymmetric Tensor Lagrangian:
• Note the D=4 duality transformation between GD and GA
• Graviton, Dilaton Lagrangian:
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NonSUSY Gravity Theories
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 5/17
• Graviton, Dilaton, Antisymmetric Tensor Lagrangian:
• Low-energy limit of bosonic string theory • This theory emerges from KLT or as a double-copy of
nonsupersymmetric Yang-Mills theory
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Physics|Amplitudes
Methods
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Methods
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 6/17
• Generalized Unitarity [Bern, Dixon, Dunbar, Kosower]
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Methods
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Generalized Unitarity • Compute tree-level amplitudes using
Feynman rules or KLT
[Bern, Dixon, Dunbar, Kosower]
6/17
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Methods
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Generalized Unitarity • Compute tree-level amplitudes using
Feynman rules or KLT
• Get cuts of loop amplitude by sewing together trees
[Bern, Dixon, Dunbar, Kosower]
6/17
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Methods
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Generalized Unitarity • Compute tree-level amplitudes using
Feynman rules or KLT
• Get cuts of loop amplitude by sewing together trees
• Make cuts into loop integrands
[Bern, Dixon, Dunbar, Kosower]
6/17
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Methods
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Generalized Unitarity • Compute tree-level amplitudes using
Feynman rules or KLT
• Get cuts of loop amplitude by sewing together trees
• Make cuts into loop integrands
• Spinor-Helicity Formalism • Used for two-loop “bare” integrands
[Bern, Dixon, Dunbar, Kosower]
6/17
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Methods
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Generalized Unitarity • Compute tree-level amplitudes using
Feynman rules or KLT
• Get cuts of loop amplitude by sewing together trees
• Make cuts into loop integrands
• Spinor-Helicity Formalism • Used for two-loop “bare” integrands
• Color-Kinematics Duality • Useful for GDA theory [See talk by Scott Davies]
[Bern, Dixon, Dunbar, Kosower]
[Bern, Carrasco, Johansson]
6/17
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Physics|Amplitudes
One-Loop Gravity
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 7/17
• Finiteness Argument [‘t Hooft and Veltman (1974)]
• Possible external graviton counterterms in D=4:
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Finiteness Argument [‘t Hooft and Veltman (1974)]
• Possible external graviton counterterms in D=4:
• Ricci scalar and Ricci curvature vanish by equations of motion and can be eliminated by field redefinitions
7/17
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Finiteness Argument [‘t Hooft and Veltman (1974)]
• Possible external graviton counterterms in D=4:
• Ricci scalar and Ricci curvature vanish by equations of motion and can be eliminated by field redefinitions
• In D=4, Gauss-Bonnet Theorem states
• In asymptotically flat space, χ=0 à Riemann curvature vanishes
7/17
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Finiteness Argument [‘t Hooft and Veltman (1974)]
• Possible external graviton counterterms in D=4:
• Ricci scalar and Ricci curvature vanish by equations of motion and can be eliminated by field redefinitions
• In D=4, Gauss-Bonnet Theorem states
• In asymptotically flat space, χ=0 à Riemann curvature vanishes
• With nontrivial topology, pure gravity counterterm is [Capper and Duff (1974);
Tsao (1977); Critchley (1978); Gibbons, Hawking, Perry (1978); Goroff and SagnoF (1986); Bornsen and van de Ven (2009)]
7/17
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 8/17
• The one-loop divergence also manifests itself in the trace/conformal/Weyl anomaly
• Computed long ago for many different fields in the loop (ext. gravitons)
[Capper and Duff (1974); Tsao (1977); Critchley (1978); Gibbons, Hawking, Perry (1978); Duff and van Nieuwenhuizen (1980); Siegel (1980); Grisaru, Nielsen, Siegel, Zanon (1984); Goroff and Sagnotti (1986); Bornsen and van de Ven (2009); Bern, Cheung, Chi, Davies, Dixon, JN; Etc.]
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• The one-loop divergence also manifests itself in the trace/conformal/Weyl anomaly
• Computed long ago for many different fields in the loop (ext. gravitons)
• They show that the effective energy-momentum tensor is proportional to the one-loop counterterm
where
[Capper and Duff (1974); Tsao (1977); Critchley (1978); Gibbons, Hawking, Perry (1978); Duff and van Nieuwenhuizen (1980); Siegel (1980); Grisaru, Nielsen, Siegel, Zanon (1984); Goroff and Sagnotti (1986); Bornsen and van de Ven (2009); Bern, Cheung, Chi, Davies, Dixon, JN; Etc.]
8/17
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• The one-loop divergence also manifests itself in the trace/conformal/Weyl anomaly
• Computed long ago for many different fields in the loop (ext. gravitons)
• They show that the effective energy-momentum tensor is proportional to the one-loop counterterm
where
• For the theories we discussed,
graviton dilaton antisym. 3-form
[Capper and Duff (1974); Tsao (1977); Critchley (1978); Gibbons, Hawking, Perry (1978); Duff and van Nieuwenhuizen (1980); Siegel (1980); Grisaru, Nielsen, Siegel, Zanon (1984); Goroff and Sagnotti (1986); Bornsen and van de Ven (2009); Bern, Cheung, Chi, Davies, Dixon, JN; Etc.]
8/17
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 9/17
• Notice that the dilaton and antisymmetric tensor contribute differently in spite of the D=4 duality transformation,
• Quantum (in)equivalence under duality transformations?
graviton dilaton antisym. 3-form
[Duff and van Nieuwenhuizen (1980); Siegel (1980); Fradkin and Tseytlin (1984); Grisaru, Nielsen, Siegel, Zanon (1984)]
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Notice that the dilaton and antisymmetric tensor contribute differently in spite of the D=4 duality transformation,
• Quantum (in)equivalence under duality transformations?
• Notice that the three-form contributes despite having zero degrees of freedom in D=4.
graviton dilaton antisym. 3-form
[Duff and van Nieuwenhuizen (1980); Siegel (1980); Fradkin and Tseytlin (1984); Grisaru, Nielsen, Siegel, Zanon (1984)]
9/17
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Notice that the dilaton and antisymmetric tensor contribute differently in spite of the D=4 duality transformation,
• Quantum (in)equivalence under duality transformations?
• Notice that the three-form contributes despite having zero degrees of freedom in D=4.
• Dimensional regularization introduces subtleties
graviton dilaton antisym. 3-form
[Duff and van Nieuwenhuizen (1980); Siegel (1980); Fradkin and Tseytlin (1984); Grisaru, Nielsen, Siegel, Zanon (1984)]
[Capper and Kimber (1980)]
9/17
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One-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
• Notice that the dilaton and antisymmetric tensor contribute differently in spite of the D=4 duality transformation,
• Quantum (in)equivalence under duality transformations?
• Notice that the three-form contributes despite having zero degrees of freedom in D=4.
• Dimensional regularization introduces subtleties • Gauss-Bonnet operator is evanescent, but could affect 2 loops
graviton dilaton antisym. 3-form
[Duff and van Nieuwenhuizen (1980); Siegel (1980); Fradkin and Tseytlin (1984); Grisaru, Nielsen, Siegel, Zanon (1984)]
[Capper and Kimber (1980)]
9/17
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Physics|Amplitudes
Two-Loop Gravity
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• There is a valid R3 counterterm and corresponding divergence
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 10/17
[Goroff and Sagnotti (1986); van de Ven (1992)]
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• There is a valid R3 counterterm and corresponding divergence
• For four plus-helicity gravitons, this corresponds to where
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Goroff and Sagnotti (1986); van de Ven (1992)]
10/17
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• There is a valid R3 counterterm and corresponding divergence
• For four plus-helicity gravitons, this corresponds to where
• They performed the bare integration and subtracted any
subdivergences, which can appear even if artificial
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Goroff and Sagnotti (1986); van de Ven (1992)]
10/17
![Page 41: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/41.jpg)
• There is a valid R3 counterterm and corresponding divergence
• For four plus-helicity gravitons, this corresponds to where
• They performed the bare integration and subtracted any
subdivergences, which can appear even if artificial • Are there meaningful subdivergences even though there is no one-
loop divergence in D=4? I.e., does the evanescent Gauss-Bonnet operator (trace anomaly) feed into the two-loop divergence?
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Goroff and Sagnotti (1986); van de Ven (1992)]
10/17
![Page 42: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/42.jpg)
• We aimed to reproduce this result using unitarity and spinor-helicity
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 11/17
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
![Page 43: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/43.jpg)
• We aimed to reproduce this result using unitarity and spinor-helicity
• We constructed the four-graviton all-plus integrands
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
11/17
![Page 44: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/44.jpg)
• We aimed to reproduce this result using unitarity and spinor-helicity
• We constructed the four-graviton all-plus integrands • e.g., the planar double-box has the simple integrand numerator
where the λ’s represent the -2ε-components of the loop momenta
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
11/17
![Page 45: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/45.jpg)
• We aimed to reproduce this result using unitarity and spinor-helicity
• We constructed the four-graviton all-plus integrands • e.g., the planar double-box has the simple integrand numerator
where the λ’s represent the -2ε-components of the loop momenta
• Integrating, we find the “bare” UV divergence
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
11/17
![Page 46: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/46.jpg)
• We aimed to reproduce this result using unitarity and spinor-helicity
• We constructed the four-graviton all-plus integrands • e.g., the planar double-box has the simple integrand numerator
where the λ’s represent the -2ε-components of the loop momenta
• Integrating, we find the “bare” UV divergence
• Does not match Goroff & Sagnotti, but if we look closely, we will see meaningful subdivergences!
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
11/17
![Page 47: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/47.jpg)
• We aimed to reproduce this result using unitarity and spinor-helicity
• We constructed the four-graviton all-plus integrands • e.g., the planar double-box has the simple integrand numerator
where the λ’s represent the -2ε-components of the loop momenta
• Integrating, we find the “bare” UV divergence
• Does not match Goroff & Sagnotti, but if we look closely, we will see meaningful subdivergences!
• We will deal with the subtractions using counterterms
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
11/17
![Page 48: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/48.jpg)
• We now consider the bare result with the single- and double- Gauss-Bonnet counterterm insertions in D=4-2ε
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 12/17
Bare Single GB Double GB
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
![Page 49: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/49.jpg)
• We now consider the bare result with the single- and double- Gauss-Bonnet counterterm insertions in D=4-2ε
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
Bare Single GB Double GB
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
12/17
![Page 50: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/50.jpg)
• We now consider the bare result with the single- and double- Gauss-Bonnet counterterm insertions in D=4-2ε
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
Bare Single GB Double GB
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
12/17
![Page 51: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/51.jpg)
• We now consider the bare result with the single- and double- Gauss-Bonnet counterterm insertions in D=4-2ε
• Matches Goroff & Sagnotti after meaningful subtractions!
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
Bare Single GB Double GB
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
12/17
![Page 52: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/52.jpg)
• For other theories, the coefficient of 1/ε is given by
• [Not shown is graviton + n3 3-forms: total = 209/24 – n3 (15/2)]
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 13/17
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
![Page 53: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/53.jpg)
• For other theories, the coefficient of 1/ε is given by
• [Not shown is graviton + n3 3-forms: total = 209/24 – n3 (15/2)]
• Again, it appears that dilatons and antisymmetric tensors are quantum-mechanically inequivalent
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
13/17
![Page 54: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/54.jpg)
• For other theories, the coefficient of 1/ε is given by
• [Not shown is graviton + n3 3-forms: total = 209/24 – n3 (15/2)]
• Again, it appears that dilatons and antisymmetric tensors are quantum-mechanically inequivalent
• and that the 3-form plays a nontrivial role
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
13/17
![Page 55: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/55.jpg)
• For other theories, the coefficient of 1/ε is given by
• [Not shown is graviton + n3 3-forms: total = 209/24 – n3 (15/2)]
• Again, it appears that dilatons and antisymmetric tensors are quantum-mechanically inequivalent
• and that the 3-form plays a nontrivial role • but…
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
13/17
![Page 56: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/56.jpg)
• …look at the finite log pieces as well • Pure Gravity:
• Gravity + 3-Form
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 14/17
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
![Page 57: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/57.jpg)
• …look at the finite log pieces as well • Pure Gravity:
• Gravity + 3-Form
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
14/17
![Page 58: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/58.jpg)
• …look at the finite log pieces as well • Pure Gravity:
• Gravity + 3-Form
• Can alter divergence with 3-form, but preserve logarithms!
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
14/17
![Page 59: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/59.jpg)
• …look at the finite log pieces as well • Pure Gravity:
• Gravity + 3-Form
• Can alter divergence with 3-form, but preserve logarithms! • No physical effects — “Cheshire Cat” field
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
14/17
![Page 60: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/60.jpg)
• …look at the finite log pieces as well • Pure Gravity:
• Gravity + 3-Form
• Can alter divergence with 3-form, but preserve logarithms! • No physical effects — “Cheshire Cat” field • Similar story for GD vs. GA theories – same logarithms
• Quantum equivalence under duality
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
14/17
![Page 61: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/61.jpg)
• How should we interpret these results?
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 15/17
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
![Page 62: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/62.jpg)
• How should we interpret these results?
• The UV divergences depend of the field representations (UV completion)!
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
15/17
![Page 63: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/63.jpg)
• How should we interpret these results?
• The UV divergences depend of the field representations (UV completion)!
• Better to look at renormalization scale: log(μ2)
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
15/17
![Page 64: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/64.jpg)
• How should we interpret these results?
• The UV divergences depend of the field representations (UV completion)!
• Better to look at renormalization scale: log(μ2)
• For all of the theories that we considered, we find
where Ns = # of states in the theory
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
15/17
![Page 65: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/65.jpg)
• How should we interpret these results?
• The UV divergences depend of the field representations (UV completion)!
• Better to look at renormalization scale: log(μ2)
• For all of the theories that we considered, we find
where Ns = # of states in the theory
• Another angle of attack: • Tw0-loop integrands in D-dimensions with formal polarization
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Cheung, Chi, Davies, Dixon, JN (to appear)]
15/17
![Page 66: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/66.jpg)
Physics|Amplitudes
BCJ at Two Loops
![Page 67: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/67.jpg)
• Constructed Yang-Mills numerators at two loops that obey kinematic Jacobi identities on the spanning set of cuts
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 16/17
[Bern, Davies, JN (to appear)]
![Page 68: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/68.jpg)
• Constructed Yang-Mills numerators at two loops that obey kinematic Jacobi identities on the spanning set of cuts
• Also, demanded symmetry constraints of the diagrams
• D-dimensional, formal polarization
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Davies, JN (to appear)]
16/17
![Page 69: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/69.jpg)
• Constructed Yang-Mills numerators at two loops that obey kinematic Jacobi identities on the spanning set of cuts
• Also, demanded symmetry constraints of the diagrams
• D-dimensional, formal polarization
• Construction did not work when we demanded kinematic Jacobi identities to hold “globally” — off of the cuts
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Davies, JN (to appear)]
16/17
![Page 70: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/70.jpg)
• Constructed Yang-Mills numerators at two loops that obey kinematic Jacobi identities on the spanning set of cuts
• Also, demanded symmetry constraints of the diagrams
• D-dimensional, formal polarization
• Construction did not work when we demanded kinematic Jacobi identities to hold “globally” — off of the cuts
• Square numerators to get GDA theory numerators
Two-Loop Gravity
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015
[Bern, Davies, JN (to appear)]
16/17
![Page 71: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/71.jpg)
• Reproduced Goroff & Sagnotti two-loop divergence
Conclusions
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 17/17
![Page 72: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/72.jpg)
• Reproduced Goroff & Sagnotti two-loop divergence
• Evanescent operators play a nontrivial role
Conclusions
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 17/17
![Page 73: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/73.jpg)
• Reproduced Goroff & Sagnotti two-loop divergence
• Evanescent operators play a nontrivial role
• Field representation choices alter UV divergence, but better to look at log(μ2) terms
Conclusions
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 17/17
![Page 74: Subtleties in Nonsupersymmetric Gravity at Two Loops · ① Layout of Pertinent NonSUSY Gravity Theories • Graviton, dilaton, antisymmetric tensor, three-form ② Methods • Unitarity,](https://reader033.vdocuments.site/reader033/viewer/2022060611/60617bb67b4b6b7853033880/html5/thumbnails/74.jpg)
• Reproduced Goroff & Sagnotti two-loop divergence
• Evanescent operators play a nontrivial role
• Field representation choices alter UV divergence, but better to look at log(μ2) terms
• Future work: • Perform two-loop calculation using formal polarization vectors
• Closer to Goroff & Sagnotti, avoids FDH scheme and more helicity configurations
• Already found BCJ numerators for GDA theory
Conclusions
UCLA Physics|Amplitudes Radcor/Loopfest 18 June, 2015 17/17