Les Houches Lectures on Effective Field Theories and Gravitational Radiation

0.35 MB·English

by  Walter D. Goldberger

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LES HOUCHESLECTURES ON EFFECTIVE FIELD

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THEORIESAND GRAVITATIONALRADIATION
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1 Walter D.Goldberger

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DepartmentofPhysics,YaleUniversity,NewHaven,CT06520
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Contents
1. LectureI 3
1.1. Introductionandmotivation 3
1.2. Effectivefieldtheories:areview 6
1.2.1. Matching 10
1.2.2. Correctionstothematchingcoefficients 11
1.2.3. Powercounting 16
2. LectureII 18
2.1. ThebinaryinspiralasanEFTcalculation 18
2.2. TheEFTforisolatedcompactobjects 20
2.3. Calculatingobservables 23
2.4. Integratingouttheorbitalscale 29
2.5. Radiation 34
2.6. Finitesizeeffects 38
3. Conclusions 40
4. Acknowledgments 41
AppendixA. Redundantoperators 41
References 43
2
Abstract
These lectures give an overview of the uses of effective field theories in de-
scribinggravitationalradiationsourcesforLIGOorLISA.Thefirstlecturereviews
someofthestandardideasofeffectivefieldtheory(decoupling, matching,power
counting)mostlyinthecontextofasimpletoymodel. Thesecondlecturesetsup
theproblemofcalculatinggravitationalwaveemissionfromnon-relativisticbinary
starsbyconstructingatowerofeffectivetheoriesthatseparatelydescribeeachscale
intheproblem: theinternalsizeofeachbinaryconstituent,theorbitalseparation,
andthewavelengthofradiatedgravitons.
1. LectureI
1.1. Introductionandmotivation
Theselecturesdescribetheusesofeffectivefieldtheory(EFT)methodstosolve
problemsingravitationalwavephysics. Manyofthesignalsrelevanttogravita-
tionalwaveexperimentssuchasLIGO[1],VIRGO[2]andtheplannedLISA[3]
correspond to astrophysical sources whose evolution involve a number of dis-
tinctlyseparatedlengthscales. Inordertocomputesignaltemplatesthatcapture
thephysicsaccurately,itisimportanttosystematicallyaccountforeffectsarising
atallthesedifferentscales. Itisexactlythissortofproblemthatisbesttreated
by EFT methods, analogous to the EFTs constructed to unravel multiple scale
problemsinhighenergyphysicsorcondensedmatter.
Forthesakeofconcreteness,IwillfocusintheselecturesontheEFTformu-
lationoftheslow“inspiral”phaseofcompactbinarystars(thatis,withneutron
star (NS) or black hole (BH) constituents). A more comprehensive review of
gravitationalwave sourcesandphenomenologycan be foundin the lecturesby
A.Buonnanoatthisschool(alsosee, e.g.,ref.[4]). Theinspiralphaseplaysan
importantroleingravitationalwavephysics,andcorrespondstotheperiodinthe
evolution of the binary in which the system is non-relativistic, the bound orbit
slowlydecayingduetotheemissionofgravitationalradiation.
An appealingfeatureofthe binaryinspiralphaseis that, theoretically,itisa
verytractableproblem. Inprinciple,thedynamicsiscalculableasaperturbative
3
4 W.D.Goldberger
expansion of the Einstein equations in the parameter v 1, a typical three-
≪
velocity.1 To get a handle on the scales involved, it is worthwhile to calculate
someofthefeaturesofthebinaryinspiralattheorderofmagnitudelevel. Let’s
consider inspiral events seen by LIGO, which for illustration we will assume
operatesinthefrequencybandbetween10Hzand1kHz. Atsuchfrequencies,
the relevant inspiral sources correspond to neutron star binaries, with m
NS
∼
1.4m ,(m isthesolarmass)orperhapsblackholeswithtypicalmassm
BH
10m⊙. Sinc⊙ethetimeevolutionisnon-relativistic,itistoagoodapproximatio∼n
treate⊙d in terms of Newtonian gravity. In particular, the orbital parameters are
relatedby
G m r
v2 N s, (1.1)
∼ r ≡ 2r
where r is the orbitalradius, and we have introducedthe Schwarzschild radius
r = 2G m. The frequency of the radiation emitted during this phase is of
s N
order the orbital frequency 2πν v/r, so for a binary inspiral that scans the
∼
LIGOband,thetypicalorbitalradiusismeasuredinkilometers
1/3 1/3
m m
r(10Hz) 300km r(1kHz) 14km , (1.2)
∼ m → ∼ m
(cid:18) ⊙(cid:19) (cid:18) ⊙(cid:19)
which is therefore well separated from the size of the compact objects, r
s
∼
1kmm/m . Theexpansionparameterv alsoevolvesasthebinarysweepsthe
detectorfre⊙quencyband:
1/3 1/3
m m
v(10Hz) 0.06 v(1kHz) 0.3 , (1.3)
∼ m → ∼ m
(cid:18) ⊙(cid:19) (cid:18) ⊙(cid:19)
indicatingthatthevelocityisagoodexpansionparameteruptothelastfewcycles
oforbitseenbythedetector.Asv 1,theinspiralphaseends,andtheevolution
→
mustbetreatedbynumericalsimulations.
Duringtheinspiralphase,theboundorbitisunstabletotheemissionofgravi-
tationalradiation.Thepoweremittedingravitationalwavesiswellapproximated
bythe quadrupoleradiationformula. Takingthe orbitto be circular, the power
radiatedis
dE 32
dt = 5 G−N1v10. (1.4)
1Eventhoughwewillbedoingclassicalphysicsformostoftheselectures,Iuseparticlephysics
unitsc=¯h=1throughout.
LesHouchesLecturesonEffectiveFieldTheoriesandGravitationalRadiation 5
ThemechanicalenergyofthebinaryisjustE = 1mv2 toleadingorderinv,
−2
sobyconservationofenergy
d 1 32
dt −2mv2 =− 5 G−N1v10, (1.5)
(cid:18) (cid:19)
weobtainanestimateforthedurationoftheinspiraleventseenbyLIGO
8/3
5 1 1 m −
∆t= r 5min. , (1.6)
512 s"vi8 − vf8#∼ (cid:18)m⊙(cid:19)
andthenumberoforbitalcyclesasthesignalsweepsthedetectorband
N tf ω(t)dt= 1 1 1 4 104 m −5/3 radians, (1.7)
∼Zti 32"vi5 − vf5#∼ × (cid:18)m⊙(cid:19)
withωtheorbitalangularfrequency.
Eqs. (1.2), (1.3), (1.6) give an estimate for the typical length, velocity, and
duration of an inspiral event that sweeps the LIGO frequency band. Inspiral
events seen by LISA follow the same dynamics. However, because LISA will
operate in a frequencyrange complementaryto LIGO, 10 5Hz < ν < 1Hz,
−
thesourcescorrespondtoobjectsofmuchlargemass(e.g.,BH/BHbinarieswith
m 105 8m orBH/NSsystemswithm 105 7).
BH − BH −
∼ ⊙ ∼
For either LIGO or LISA, the numberof orbitalcyclesspent in the detector
frequencyband is quite large, see e.g. Eq. (1.7). Thus even a slight deviation
betweentheoreticalcalculationsof the gravitationalwave phase (the waveform
“templates”) and the data will become amplified over the large number of cy-
clesofevolution.Consequently,inspiralwavesignalscarrydetailedinformation
aboutgravitationaldynamics. Infact,ithasbeendeterminedthatLIGOwillbe
sensitive to corrections that are order v6 in the velocity expansion beyond the
leadingorderquadrupoleradiationpredictions[5].
Theprocedureforcomputingcorrectionsto themotionofthebinarysystem
inthenon-relativisticlimitbyaniteratedexpansionoftheEinsteinequationsis
called the post-Newtonianexpansionof generalrelativity [6]. What makes the
calculationsdifficult(andinteresting)isthatthereisaproliferationofphysically
relevanteffectsoccurringatdifferentlengthscales,
r = sizeofcompactobjects,
s
r = orbitalradius,
λ = wavelengthofemittedradiation,
6 W.D.Goldberger
allcontrolledbythesameexpansionparameterv:
r
s v2 r v, (1.8)
r ∼ λ ∼
wherethefirstestimatefollowsfromKepler’slawandthesecondfromthemulti-
poleexpansionoftheradiationfieldcoupledtonon-relativisticsources. Thusto
dosystematiccalculationsathighordersinv,itisnecessarytodealwithphysics
atmanydifferentscales. Aconvenientwayofdoingthisistoconstructachain
ofEFTsthatcapturetherelevantphysicsateachscaleseparately.
Thegoaloftheselecturesistorecasttheproblemofcomputinggravitational
waveobservablesfornon-relativisticsourcesintermsofEFTs. Inthenextsec-
tionwebrieflyreviewtheEFTlogic,usingastandardparticlephysicsexample,
the low energy dynamics of Goldstone bosons in a theory with spontaneously
brokenglobalsymmetry,toillustratethemainideas.Insec.2,weturntotheEFT
formulationofthebinaryinspiralproblem. Thepresentationfollows[7],which
isbasedonsimilarEFTsfornon-relativisticboundstatesinQEDandQCD[8].
However,thefocushereisnotondetailedcalculations,butindescribinghowto
“integrateout”physicsateachofthescalesr ,andrtoobtainatheorywithwell
s
definedrulesforcalculatinggravitationalwaveobservables(atthescaleλ)asan
expansioninv. Possibleextensionsoftheideaspresentedheretootherproblems
ingravitationalwavephysicsarediscussedintheconclusions,sec.3.
1.2. Effectivefieldtheories: areview
EFTsareindispensablefortreatingproblemsthatsimultaneouslyinvolvetwoor
morewidelyseparatedscales. Asatypicalapplication,supposeoneisinterested
incalculatingtheeffectsofsomesortofshortdistancephysics,characterizedby
ascaleΛ,onthedynamicsatalowenergyscaleω Λ. IntheEFTdescription
≪
of such a problem, the effects of Λ on the low energy physics become simple,
making it possible to construct a systematic expansion in the ratio ω/Λ 1.
≪
Herewebrieflyreviewthebasicideasthatgointotheconstructionandapplica-
tionsofEFTs. Moredetailedreviewscanbefoundinrefs.[9,10,11,12].
The key insight that makes EFTs possible is the following: consider for in-
stanceafieldtheorywithlightdegreesoffreedomcollectivelydenotedbyφ(e.g,
asetofmasslessfields)andsomeheavyfieldsΦwithmassneartheUVscaleΛ.
TheinteractionsofthesefieldsaredescribedbysomeactionfunctionalS[φ,Φ].
Supposealsothatweareonlyinterestedindescribingexperimentalobservables
involvingonlythelightmodesφ. Thenitmakessensetointegrateoutthemodes
Φ to obtain an effectiveaction that describesthe interactionsof the light fields
amongeachother
eiSeff[φ] = DΦ(x)eiS[φ,Φ]. (1.9)
Z
LesHouchesLecturesonEffectiveFieldTheoriesandGravitationalRadiation 7
Itturnsoutthatingeneral,S [φ]canbeexpressedasalocalfunctionalofthe
eff
fieldsφ(x),
S [φ]= c d4x (x), (1.10)
eff i i
O
i Z
X
for(ingeneralaninfinitenumber)localoperators (x). Thecoefficientsc are
i i
O
usually referred to as “Wilson coefficients”. If (x) has mass dimension ∆ ,
i i
O
thenthe Wilson coefficients, evaluatedata renormalizationpointµof orderΛ,
scaleaspowersofΛ
α
c (µ=Λ)= i , (1.11)
i Λ∆i−4
with α (1). From this observation we conclude that the short distance
i
∼ O
physicscanhavetwotypesofeffectsonthedynamicsatenergiesω Λ:
≪
Renormalizationofthecoefficientsofoperatorswithmassdimension∆ 4.
• ≤
Generation of an infinite tower of irrelevant (i.e. ∆ > 4) operators with
•
coefficientsscalingasinEq.(1.11).
This result regarding the structure of the low energy dynamics encoded in
S [φ]isusuallyreferredtoasdecoupling.Thisconcepthasitsrootsinthework
eff
ofK.Wilsonontherenormalizationgroup[13]. Decouplingasusedbypractic-
ingfieldtheoristswas firstmaderigorousin [14]. Theideaisusefulbecauseit
statesthatthedependenceofthelowenergyphysicsonthescaleΛisextremely
simple. All UV dependenceappearsdirectlyin the coefficientsof the effective
Lagrangian, and determining the Λ dependence of an observable follows from
powercounting(essentiallyageneralizedformofdimensionalanalysis). Effec-
tiveLagrangiansaretypicallyusedinoneoftwoways:
1. The “full theory” S[φ,Φ] is known: In this case, integrating out the heavy
modesgivesasimplewayofsystematicallyanalyzingtheeffectsoftheheavy
physics on low energy observables. Because only the low energy scale ap-
pearsexplicitlyintheFeynmandiagramsoftheEFT,amplitudesareeasierto
calculateandtopowercountthaninthefulltheory.
Examples: IntegratingouttheW,Z bosonsfromtheSU(2) U(1) Elec-
L Y
×
troweak Lagrangian at energies E m results in the Fermi theory of
W,Z
≪
weakdecayspluscorrectionssuppressedbypowersofE2/m2 . Itiseasier
W,Z
to analyze electromagnetic or QCD corrections to weak decays in the four-
FermitheorythaninthefullElectroweakLagrangian,asthegraphsofFig.1
indicate. See,e.g.,ref.[15]. AnotherexampleistheuseofEFTstocalculate
heavyparticlethresholdcorrectionstolowenergygaugecouplings[16].This
hasapplications,forinstance,inGrandUnifiedTheoriesandinQCD.
8 W.D.Goldberger
2. The full theory is unknown (or known but strongly coupled): Whatever the
physics at the scale Λ is, by decoupling it must manifest itself at low ener-
giesasaneffectiveLagrangianoftheformEq.(1.10). Ifthesymmetries(eg.
Poincare,gauge,global)thatsurviveatlowenergiesareknown,thentheop-
erators (x) appearinginS [φ]mustrespectthosesymmetries. Thusby
i eff
O
writingdownaneffectiveLagrangiancontainingthe mostgeneralsetofop-
eratorsconsistentwiththesymmetries,wearenecessarilyaccountingforthe
UVphysicsinacompletelymodelindependentway.
Examples: TheQCDchiralLagrangianbelowthescaleΛ ofSU(3)
χSB L
×
SU(3) SU(3) chiral symmetry breaking [17, 18]. Here the full the-
R V
→
ory,QCD,isknown,butbecauseΛ isoforderthescaleΛ 1GeV
χSB QCD
∼
wheretheQCDcouplingisstrong,itisimpossibletoperformthefunctional
integralin Eq. (2.16) analytically. Another exampleis generalrelativity be-
low the scale m 1019GeV. This theory can be used to calculate, e.g.,
Pl
∼
graviton-graviton scattering at energies E m . Above those energies,
Pl
≪
however,scatteringamplitudescalculatedin generalrelativitystartviolating
unitaritybounds,andtheeffectivefieldtheorynecessarilybreaksdown.Thus
general relativity is an effective Lagrangian for quantum gravity below the
strong coupling scale m . The EFT interpretation of general relativity is
Pl
reviewedinmoredetailinrefs.[19,20]
Finally,itisbelievedthattheStandardModelitselfisaneffectivefieldtheory
belowscalesoforderΛ = 1TeVorso(seelecturesbyH.Murayamaatthis
school).Thisscalemanifestsitselfindirectly,intheformofSU(2) U(1)
L Y
×
gauge invariant operators of dimension ∆ > 4 constructed from Standard
Modelfields[21],certainlinearcombinationsofwhichhavebeenconstrained
experimentally using collider data from the LEP experiments at CERN and
from SLD at SLAC (see [22] for a recent analysis of precision electroweak
constraintsusingeffectiveLagrangians). Ifthereisindeednewphysicsatthe
TeVscale,itwillbeseendirectly,attheCERNLHCwhichisduetocomeon
lineinthenextfewyears.
In eitherofthese two classes ofexamples, integratingouttheheavyphysics
asinEq.(2.16)resultsinaaneffectiveLagrangianthatcontains,ingeneral,an
infinitenumberofoperators (x). However,because
i
O
1. Anoperator (x)with[ ]=∆ >4contributestoanobservableatrelative
order O O O
ω ∆O 4
− 1, (1.12)
Λ ≪
(cid:16) (cid:17)
2. Agivenobservablecanonlybedetermineduptoafiniteexperimentalresolu-
tionǫ 1,
≪
LesHouchesLecturesonEffectiveFieldTheoriesandGravitationalRadiation 9
onemaytypicallytruncatetheseriesinEq.(1.10)afterafinitenumberofopera-
tors,thosewithmassdimension∆ N +4,where
≤
ω N
ǫ=expt.error= . (1.13)
Λ
(cid:16) (cid:17)
ThereforeS [φ]ispredictiveaslongastherearemoreobservablesthanoper-
eff
atorswithmassdimension∆ N +4.
≤
Tomakethesegeneralremarksmoreconcrete,itisworthstudyingatoyexam-
pleinsomedetail. We’llconsiderthequantumfieldtheoryofasinglecomplex
scalarfieldφ(x)withLagrangian(the“fulltheory”)
= ∂ φ2 V(φ), (1.14)
µ
L | | −
where
V(φ)= λ φ2 v2/2 2. (1.15)
2 | | −
(cid:0) (cid:1)
ThistheoryisinvariantundertheglobalU(1)symmetry
U(1):φ(x) eiαφ(x), (1.16)
→
forsomeconstantphaseα,aswellasadiscretechargeconjugationsymmetry
C :φ(x) φ∗(x). (1.17)
→
Classically, the ground state of this theory is determined by the minimum of
V(φ). Because of the U(1) symmetry the ground state is degenerate, and the
vacuum manifold is the circle φ2 = v2/2 in the complex φ plane. To study
| |
fluctuationsaboutthevacuum,weexpandthefieldsaboutanyofthese(equiva-
lent)vacuua,forinstancethepoint
v
φ = . (1.18)
h i √2
Expandingaboutthis pointspontaneouslybreaksthe U(1)symmetry,resulting
inoneGoldstoneboson.Itisconvenienttowritetheoriginalfieldφ(x)as
1
φ(x)= (v+ρ(x))eiπ(x)/v. (1.19)
√2
Undertheoriginalsymmetries,thenewfieldstransformas
ρ(x) ρ(x),
U(1): → (1.20)
π(x)/v π(x)/v+α,
(cid:26) →
10 W.D.Goldberger
and
ρ(x) ρ(x),
C : → (1.21)
π(x) π(x).
(cid:26) →−
TheLagrangianintermsofthenewfieldsisgivenby
1 1 ρ 2 λ
= (∂ ρ)2+ 1+ (∂ π)2 (vρ+ρ2/2)2, (1.22)
µ µ
L 2 2 v − 2
(cid:16) (cid:17)
so the spectrum of excitations about the vacuum consists of a “modulus” field
ρ(x) with tree level mass given by m2 = λv2 and the Goldstone boson π(x)
ρ
withm =0. (Wetakeλ 1sothataperturbativetreatmentisvalid).
π
≪
Supposethatweareinterestedinworkingoutthepredictionsofthistheoryat
energiesω m . Atsuchscales,onlythedynamicsofthemasslessfieldπ(x)
ρ
≪
is non-trivial, and following our general discussion it is convenientto write an
effectiveLagrangian. The generalstructure of this effective Lagrangianis dic-
tated by the symmetries of the originaltheory. In particular the U(1), realized
non-linearlyastheshiftsymmetryπ(x)/v π(x)/v+α,restrictstheeffective
→
Lagrangiantobeafunctionof∂ πonly.Thereforethesymmetriesofthelowen-
µ
ergytheoryaloneexplainanumberofconsequencesofthelowenergydynamics,
forinstance
The field π(x) must be massless, since a mass term would break the shift
•
symmetry.ThisisjustthestatementofGoldstone’stheoreminthecontextofthe
lowenergyEFT.
The field π(x) is derivatively self-coupled. This implies in particular that
•
scatteringamplitudesare“soft”, vanishingaspowersofthetypicalenergyω in
thelimitω 0.
→
Inadditiontotheconstraintsfromthenon-linearlyrealizedU(1),thereisalsoa
constraintfromthe chargeconjugationsymmetryπ π whichsays thatthe
→ −
EFT must be even in π. Thus the effective Lagrangian must be of the general
form
1 c
= (∂ π)2+ 8 (∂ π∂µπ)2+ , (1.23)
LEFT 2 µ 4Λ4 µ ···
whereonlytheleadingoperatorswithtwoandfourπ’shavebeendisplayed. In
thisequationΛ m andc issomedimensionlessconstantoforderλ.
ρ 8
∼
1.2.1. Matching
Sinceweknowthefulltheory,Eq.(1.14),itispossibletoexplicitlyperformthe
functionalintegralofEq.(1.9)inordertoobtaintheEFTparameterslikeΛand
c intermsofthecouplingsλ,vappearinginthefulltheoryLagrangian. Rather
8