David A Kosower Institut de Physique Th é orique CEA Saclay work with Ben Maybee and Donal OConnell Edinburgh arXiv181110950 by Ben Maybee Donal OConnell and Justin Vines 190609260 ID: 814689
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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
Slide2Gravitational 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
Slide3Gravitational 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,…
Slide4Theorists’ Role
Waveform templates
For detection
For measurements Three Phases:Inspiral: a weak-field perturbative approach worksMerge: strong-field, numerics neededRingdown: normal modes
Slide5Approaches
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
Slide6Direct 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
Slide7Our 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
Slide8Set-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
Slide9Wave Packets
Point particles: localized positions and momenta
Wavefunction
Initial state: integral over on-shell phase space
Notation tidies up
s
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
Momentum
Insert a complete set of states and rewrite,
Think of
as final-state momenta: connection to scattering amplitudes
Impulse
aka
time integral of change in momentum
Write
and use unitarity
Expression holds to all orders in perturbation theory
Impulse
Diagrammatically (
is momentum mismatch,
momentum transfer)
First term is linear in amplitude
+
Slide14Radiated 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
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
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
Impulse at LO
O
(
)
Only first term contributes, with tree-level amplitude
Integrate over wavefunctions under
, take
Scalar QED Impulse at LO
2
2 amplitudeImpulse (use
)Evaluate the integral
(
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
Slide20Impulse at NLO
O
(): both terms contributeFirst term, with one-loop amplitudeSecond term, with tree amplitudes
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
Slide22Radiation 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
Spin kick
Maybee, O’Connell, Vines
Spin using Pauli–
Lubanski vector
A few subtleties
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