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From scattering amplitudes to classical gravity N. Emil J. - - PowerPoint PPT Presentation

Apr 09, 2024 375 likes •751 views

From scattering amplitudes to classical gravity N. Emil J. Bjerrum-Bohr Niels Bohr International Academy, Niels Bohr Institute QCD meets gravity 2019 [Mani Bhaumik Institute] Work together with A. Cristofoli, P . Damgaard, J. Donoghue,

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From scattering amplitudes to classical gravity

  • N. Emil J. Bjerrum-Bohr

Niels Bohr International Academy, Niels Bohr Institute

“QCD meets gravity 2019” [Mani Bhaumik Institute] Work together with

  • A. Cristofoli, P

. Damgaard, J. Donoghue, G. Festuccia,

  • H. Gomez, B. Holstein, L. Plante, P

. Vanhove

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General Relativity as a quantum field theory

— Known for a long time that a particle version of

General Relativity can be derived from the Einstein Hilbert Lagrangian

— Expand Einstein-Hilbert Lagrangian : — Derive vertices as in a particle theory -

computations using Feynman diagrams!

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From scattering amplitudes to classical gravity

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Off-shell computation of amplitudes

— Expand Lagrangian, laborious and tedious process…. — Vertices: 3pt, 4pt, 5pt,..n-pt — Complicated off-shell expressions

(DeWitt;Sannan)

45 terms + sym

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Much more complicated than Yang-Mills theory but still many useful applications..

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From scattering amplitudes to classical gravity

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Gravity as a quantum field theory

— Viewpoint: Gravity as a non-abelian gauge

field theory with self-interactions

— Non-renormalisable theory! (‘t Hooft and Veltman) — Traditional belief : – no known symmetry

can remove all UV-divergences

Dimensionful coupling: GN=1/M2

planck

String theory can by introducing new length scales

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From scattering amplitudes to classical gravity

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Quantum gravity as an effective field theory

— (Weinberg) proposed to view the quantization of

general relativity as that of an effective field theory

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From scattering amplitudes to classical gravity

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Practical quantum gravity at low energies

— Consistent quantum theory:

— Quantum gravity at low energies (Donoghue) — Direct connection to low energy dynamics of string

and super-gravity theories

— Suggest general relativity augmented by higher derivative

  • perators – the most general modified theory

— A somewhat curious application:

Classical physics from quantum theory!

(Iwasaki; Donoghue, Holstein; Kosower, Maybee, O’ Connell…)

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From scattering amplitudes to classical gravity

NB: Contact with General Relativity require some care..! (Many talks..)

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One-loop (off-shell) gravity amplitude computation

Tree Boxes Triangles Bubbles

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From scattering amplitudes to classical gravity

(NEJB, Donoghue, Holstein (2001)

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Tree Boxes Triangles Bubbles

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From scattering amplitudes to classical gravity

(NEJB, Donoghue, Holstein (2001))

One-loop (off-shell) gravity amplitude computation

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Tree Boxes Triangles Bubbles

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From scattering amplitudes to classical gravity

(NEJB, Donoghue, Holstein (2001))

One-loop (off-shell) gravity amplitude computation

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One-loop result for gravity

— Four point amplitude can be deduced to take the form

Focus on deriving these ~> Long-range behavior (no higher derivative contributions) Short range behavior

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From scattering amplitudes to classical gravity

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One-loop and the cut

— It is in fact much simpler to capture the long-range

behavior from unitarity (NEJB, Donoghue, Vanhove)

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From scattering amplitudes to classical gravity

KLT + on-shell 4D input trees recycled from Yang-Mills (Badger et al; Forde Kosower) e.g. D-dimensions (NEJB, Gomez, Cristofoli, Damgaard)

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QCD meets gravity

KLT relationship (Kawai, Lewellen and Tye) (Bern, Dixon, Dunbar, Perelstein, Rozowsky)

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From scattering amplitudes to classical gravity

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All multiplicity S-kernel (NEJB, Damgaard Feng,Søndergaard Vanhove)

(many talks)

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— Will consider scalar-scalar scattering amplitudes

mediated through graviton field theory interaction

—

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Massive scalar-scalar scattering

From scattering amplitudes to classical gravity

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  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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Tree level

Newton’s law through Fourier transform

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Result for the one-loop amplitude

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From scattering amplitudes to classical gravity

1) Expand out traces 2) Reduce to scalar basis of integrals 3) Isolate coefficients (Bern, Dixon, Dunbar, Kosower, NEJB, Donoghue, Vanhove) (See also Cachazo and Guevara)

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  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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One-loop level

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  • From scattering amplitudes to classical gravity

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Classical pieces in loops

From scattering amplitudes to classical gravity

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  • From scattering amplitudes to classical gravity

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Close contour

From scattering amplitudes to classical gravity

Classical pieces in loops

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  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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One-loop level

Ignore quantum pieces Branch (explained by Weinberg)

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  • From scattering amplitudes to classical gravity
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Computational setup

— We use the language of old-fashioned time-ordered

perturbation theory

— In particular we eliminate by hand

— Annihilation channels — Back-tracking diagrams — Anti-particle intermediate states We will also assume (classical) long-distance scattering distances

From scattering amplitudes to classical gravity

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(Cristofoli, Bjerrum-Bohr, Damgaard, Vanhove)

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  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

Relation to a potential

— One-loop amplitude after summing all contributions — How to relate to a classical potential?

— Choice of coordinates — Born subtraction

Super-classical/ singular

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Einstein-Infeld-Hoffman Potential

  • From scattering amplitudes to classical gravity
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— Solve for potential in non-relativistic limit, — Contact with Einstein-Infeld-Hoffmann Hamiltonian

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Post-Newtonian interaction potentials

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(Einstein-Infeld-Hoffman, Iwasaki) Crucial subtraction of Born term to in order to get the correct PN potential (3 – 7/2 -> -1/2 )

  • From scattering amplitudes to classical gravity

From scattering amplitudes to classical gravity

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Relation to a relativistic PM potential

— Amplitude defined via perturbative expansion

around a flat Minkowskian metric

— Now we need to relate the Scattering Amplitude to

the potential for a bound state problem – alternative to matching (Cheung, Solon, Rothstein; Bern, Cheung, Roiban, Shen, Solon, Zeng)

— Starting point: the Hamiltonian of the relativistic

Salpeter equation

  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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Relation to a potential

— Analysis involves solution of the Lippmann-

Schwinger recursive equation:

  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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Tree level

Same result as from matching (Cheung, Solon, Rothstein; Bern, Cheung, Roiban, Shen, Solon, Zeng)

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One-loop

  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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One-loop

  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

Again same result as from matching, no singular term

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Effective potential

In fact we do not have to go through either matching procedure or solving Lippmann-Schwinger to derive

  • bservables such as the scattering angle

Energy relation makes everything simple: (Damour; Bern, Cheung, Roiban, She, Solon, Zeng; Kalin, Porto; NEJB,Damgaard,Cristofoli)

  • From scattering amplitudes to classical gravity
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From scattering amplitudes to classical gravity

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Effective potential

  • From scattering amplitudes to classical gravity
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Thus given the classical amplitude Non-relativistic Hamiltonian with effective potential

From scattering amplitudes to classical gravity

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Scattering angle all orders

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From scattering amplitudes to classical gravity

(Kalin, Porto; NEJB,Damgaard,Cristofoli)

Corrects ‘Bohm’s formula’ + no reference minimal distance

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post-Minkowskian expansion

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Will use similar eikonal setup as for bending of light (extended to massive case): Amplitude computed Eikonal phase b orthogonal and

From scattering amplitudes to classical gravity

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Stationary phase condition (leading order in q)

post-Minkowskian expansion

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From scattering amplitudes to classical gravity

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  • Gravity Amplitudes and General Relativity

post-Minkowskian expansion

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Final result becomes Agrees with (Westpfahl) Light-like limit Extend beyond 2PM…

From scattering amplitudes to classical gravity

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  • From scattering amplitudes to classical gravity
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Any PM order given amplitude…

Confirmation of 3PM & 4PM Bern, Cheung, Roiban, Shen, Solon, Zeng) )

From scattering amplitudes to classical gravity

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— Amplitude toolbox for computations already provided

new efficient methods for computation:

— Double-copy and KLT clearly helps simplify

computations

— Amplitude tools can provide compact trees for

unitarity computations — Very impressive computations by (Bern, Cheung, Roiban,

Shen, Solon, Zeng, and many others) + much more to come…

— Endless tasks ahead / open questions regarding spin,

radiation, quantum terms, high order curvature terms etc

— Clearly much more physics to learn….

Outlook

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From scattering amplitudes to classical gravity