From Scattering Amplitudes to Classical Observables - PowerPoint Presentation

Slide1

From Scattering Amplitudes to Classical Observables

David A. Kosower

Institut

de Physique

Th

é

orique

, CEA–

Saclay

work with

Ben

Maybee

and

Donal

O’Connell (Edinburgh)

[arXiv:1811.10950];

by

Ben

Maybee

,

Donal

O’Connell, and Justin Vines [1906.09260]

@ Amplitudes 2019, Trinity College, Dublin— July 2, 2019

Slide2

Gravitational Waves

The dawn of gravitational-wave astronomy: LIGO & VIRGO

25 events since Sept 2015, 15 in 2019 (to June 30)

gracedb.ligo.org/latest/2+ neutron-star pair mergers, 18 black hole pair mergers, rest ?Dramatic direct confirmation starting at the centennial of GR

This past May also marked the 100

th

anniversary of the solar eclipse which yielded the first observational confirmation of General Relativity

Slide3

Gravitational Waves

Can expect dramatic discoveries in

Cosmology: presence and distribution of

heavy objects, limits [1906.08000]Astrophysics: neutron-star equation of state—GW170817; S190425z; S190510g?Need theoretical input for waveformsHigher-order terms in the Post-Newtonian expansion still await

General relativists have worked hard for many resultsUsing an Effective One-Body formalismBuonanno, Pan,

Taracchini

,

Barausse, Bohé, Cotesta, Shao, Hinderer, Steinhoff, Vines; Damour, Nagar, Bernuzzi, Bini, Balmelli, Messina; Iyer, Sathyaprakash; Jaranowski, SchaeferNumerical Relativity Pretorius; Campanelli et al.; Baker et al.Joined by Effective Field TheoristsGoldberger, Rothstein; Goldberger, Li, Prabhu, Thompson; Chester; Porto,… Kol; Levi,…

Slide4

Theorists’ Role

Waveform templates

For detection

For measurements Three Phases:Inspiral: a weak-field perturbative approach worksMerge: strong-field, numerics neededRingdown: normal modes

Slide5

Approaches

Traditional: solve General Relativity perturbatively

Effective Field Theory: use separation of scales to compute better in General Relativity

An idea: use scattering amplitudesCalculate only what’s needed for physical quantitiesDouble copy: Gravity ~ (Yang–Mills)2Kawai, Lewellen, Tye; Bern, Carrasco, Johansson

Slide6

Direct Route to Predictions

Compute Potential

Compute effective-field theory scattering amplitude from parametrized & match amplitude to EFT

Cheung, Rothstein, Solon; Bern, Cheung, Roiban, Shen, Solon, Zeng  Zvi

Bern’s talkExtract potential from form of terms in scattering amplitudeBjerrum-Bohr, Damgaard

,

Festuccia

, Planté, Vanhove; Foffa, Mastrolia, Sturani, SturmChung, Huang, Kim, Lee  Sangmin Lee’s talkGuevara, Ochirov, Vines  Alexander Ochirov’s talkFeed potential into Effective-One-Body formalismCompute Effective ActionPlefka, Shi, Steinhoff, Wang  Jan Plefka’s talkOther connections to classical scattering

Goldberger, Ridgway; Goldberger, Ridgway, Li, Prabhu; Shen

Slide7

Our Strategy: Take the Scenic Route

Pick well-defined observables in the quantum theory

That are also relevant classically

Express them in terms of scattering amplitudes in the quantum theoryUnderstand how to take the classical limit efficiently

Slide8

Set-up

Scatter two massive particles

Look at three observables:

Change in momentum (‘impulse’) of one of themMomentum radiated during the scatteringSpin kickWe all love scattering amplitudesBut they aren’t the final goal or physically meaningful on their own

Slide9

Wave Packets

Point particles: localized positions and momenta

Wavefunction

Initial state: integral over on-shell phase space

Notation tidies up

s

 

Slide10

Classical Limit, part 1

Classical limit requires

: restore

We’re still relativistic field theorists: keep

Dimensional analysis

[

Ampln]

in couplings:

;

Distinguish wavenumber from momentum:

Net:

n

-point,

L

-loop amplitude in scalar QED scales as

Not the whole story, of course

 

Slide11

Momentum

Insert a complete set of states and rewrite,

Think of

as final-state momenta: connection to scattering amplitudes

 

Slide12

Impulse

aka

time integral of change in momentum

Write

and use unitarity

Expression holds to all orders in perturbation theory

 

Slide13

Impulse

Diagrammatically (

is momentum mismatch,

momentum transfer)

First term is linear in amplitude

 

+

Slide14

Radiated Momentum

Expectation of messenger (photon or graviton) momentum

Insert complete set of states

Expressible as scattering amplitude squared or cut of amplitude

Valid to all orders

 

Slide15

Classical Limit, part 2

Three scales

: Compton wavelength

: wavefunction spread

: impact parameter

Particles localized:

Well-separated wave packets:

More careful analysis confirms this ‘Goldilocks’ condition

 

 

Slide16

Classical Limit, part 2

In-state wavefunctions

and

both represent particle

Should be sharply peaked

Overlap should be

Angular-averaged on-shell

functions transmit constraint to Shrinking on :

Fixed on

:

Natural integration variables for messenger (massless) momenta are wavenumbers, not momenta:

mismatch

,

transfer

(from analysis of outgoing states),

loop

(from unitarity)

More factors of

to extract

 

Slide17

Impulse at LO

O

(

)

Only first term contributes, with tree-level amplitude

 

Integrate over wavefunctions under

, take

 

Slide18

Scalar QED Impulse at LO

2



2 amplitudeImpulse (use

)Evaluate the integral

 

Slide19

(

Dilaton

) Gravity Impulse at LO

Squaring pure Yang–Mills gives gravity + dilatonImpulseEvaluate integral (same as QED)Remove the dilaton to obtain the classic GR result

Portilla; Westpfahl,

Goller

;

Ledvinka, Schäfer, Bičák; Damour

Slide20

Impulse at NLO

O

(): both terms contributeFirst term, with one-loop amplitudeSecond term, with tree amplitudes

 

Slide21

Impulse at NLO

Massless loops are purely quantum

as are vertex,

wavefn, propagator corrections—after renormalizationLoops with masses are not purely quantumLeft with triangles, boxes, and cut boxesTechnical note: summing over

gives functionsExample: triangle contribution

Boxes and cut boxes individually

singular

as Cancel between contributionsNeed to Laurent expandTechnical note: need to retain terms in s until after expansionGeneral proof of cancellation lackingResult agrees with direct classical calculation 

Slide22

Radiation at LO

Leading contribution is

O

()Five-point tree amplitude

Scalar QED matches classical resultMomentum conservation automatic, no need to add Abraham–Lorentz–Dirac force as in the classical theory

 

Slide23

Spin kick

Maybee, O’Connell, Vines

Spin using Pauli–

Lubanski vector

A few subtleties

 

Slide24

Summary

Gravitational-wave astronomy makes new demands on theoretical calculations in GR

Opportunity for scattering amplitudes, as the double copy offers a simpler avenue to such calculations

Study observables valid in both quantum and classical theoriesOrganize formalism to take limit in a simple wayCount s

Momenta for massive particles, wavenumbers for masslessExamples: impulse, radiated momentum, spin kick