Chiral-Scale Perturbation Theory About an Infrared Fixed Point

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by  R. J. Crewther

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Chiral-Scale Perturbation Theory About an Infrared Fixed Point
n
a
J
1 R.J.Crewther1,a and LewisC.Tunstall1,2,b,c
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1CSSMandARCCentreofExcellenceforParticlePhysicsattheTera-scale,DepartmentofPhysics,Uni-

versityofAdelaide,AdelaideSA5005Australia
]
h 2AlbertEinsteinCentreforFundamentalPhysics,InstituteforTheoreticalPhysics,UniversityofBern,Sidler-
p strasse5,CH-3012Bern,Switzerland
-
p
e Abstract. We review the failure of lowest order chiral SU(3) ×SU(3) perturbation
L R
h
theory χPT toaccount foramplitudes involving the f (500) resonance and O(m )ex-
[ 3 0 K
trapolations inmomenta. We summarize our proposal toreplace χPT3 with anew ef-

1 fective theory χPTσ based on a low-energy expansion about an infrared fixed point in
v 3-flavourQCD.Atthefixedpoint,thequarkcondensatehq¯qi ,0inducesnineNambu-
vac
8 Goldstonebosons: π,K,ηandaQCDdilatonσwhichweidentifywiththe f (500)res-
0
3 onance. We discuss the construction of the χPT Lagrangian and its implications for
σ
2 mesonphenomenology atlow-energies. Ourmainresultsincludeasimpleexplanation
8 forthe∆I =1/2ruleinK-decaysandanestimatefortheDrell-Yanratiointheinfrared
.
1 limit.
0
4
1
: 1 Three-flavorchiralexpansions: problems in the scalar-isoscalarchannel
v
i
X Chiral SU(3)L × SU(3)R perturbation theory χPT3 is nowadays well established as the framework
to systematically analyze the low-energy interactions of π,K,η mesons — the pseudo Nambu-
r
a Goldstone (NG) bosons of approximate chiral symmetry. The method relies on expansions about
aNG-symmetry,viz., low-energyscatteringamplitudesandmatrixelementscanbedescribedbyan
asymptoticseries
A={A +A +A +...} (1)
LO NLO NNLO χPT3
inpowersandlogarithmsofO(m )momentumandquarkmassesm = O(m2),withm /m held
K u,d,s K u,d s
fixed. The scheme works provided that contributions from the NG sector {π,K,η} dominate those
fromthenon-NGsector{ρ,ω,...};anassumptionknownasthepartialconservationofaxialcurrent
(PCAC)hypothesis.
Ithasbeenobserved[1],however,thattheχPT expansion(1)isafflictedwithapeculiarmalady:
3
it typically diverges for amplitudes which involve both a 0++ channel and O(m ) extrapolationsin
K
momenta. The originof thisphenomenoncanbe tracedto the f (500)resonance,a broad0++ state
0
whosecomplexpolemassandresidue[2]
m =441−i272MeV and |g |=3.31GeV (2)
f0 f0ππ
ae-mail:[email protected]
be-mail:[email protected]
cSpeaker.
EPJWebofConferences
NGbosonsp·p′=O(m2K)
0 (mass)2 β
π K η Nf=3proposal
χPT
3 O α
f0 ρ αIR s
(mass)2
0
NotNGbosons
Nf=0lattice
χPTσ0NGbosonsp·p′=O(m2K) sepsacraaletion NbootsoNnGs (mass)2 (flba)vPorroQpoCsDedwβi-tfhunincftriaorned(sfiolxieddlipnoei)nftoαrINR.f T=h3e
π f0 K η ρ dashedlineshowstheYang-Mills(Nf =0)lat-
ticeresult[6] forcontinued growthinα with
s
(a)ScaleseparationsbetweenNambu-Goldstone(NG)sec- decreasingscaleµ. Despiteextensiveliterature
tors and other hadrons for each type of chiral perturba- [7]concerningtheexistenceofα ,thereiscur-
IR
tion theory χPT discussed in this proceeding. In con- rently no consensus which of the above two,
ventional three-flavortheoryχPT (topdiagram), thereis physically distinct, scenarios is actually real-
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no scale separation: the non-NG boson f (500) sits in ized in QCD. In particular, it is unclear how
0
the middle of the NG sector {π,K,η}. Our three-flavor sensitiveexistingresultsaretovariationsinN .
f
proposal χPT (bottom diagram) for O(m ) extrapola- Thisisperhapsunsurprising,sincemoderncal-
σ K
tionsinmomentaimpliesaclearscaleseparationbetween culationsutilizedifferent,nonperturbativedefi-
the NG sector {π,K,η,σ = f } and the non-NG sector nitionsofα ,therebymakingcomparisonsbe-
0 s
{ρ,ω,K∗,N,η′,...}. tweenvariousanalysesdifficult.
Figure1
havebeendeterminedto remarkableprecision. Since χPT classes f pole termsas next-to-leading
3 0
order(NLO),figure1ashowswhythelow-energyexpansion(1)fails:thelocationof f anditsstrong
0
couplingtoπ,K,ηmesonsinvalidatestherequirementsofPCAC.
2 Three-flavorchiral-scaleexpansionsabout an infraredfixed point
Inthisproceeding,wesummarizeourproposal[3]tosolvetheconvergenceproblemofχPT expan-
3
sions(1)bymodifyingtheleadingorder(LO)ofthe3-flavortheory. Inshort,oursolutioninvolves
extendingthestandardNGsector{π,K,η}toinclude f (500)asaQCDdilatonσassociatedwiththe
0
spontaneousbreakingofscaleinvariance. ThescalesymmetriccounterpartofPCAC—partialcon-
servationofdilatationcurrent(PCDC)—thenimpliesthatamplitudeswithσ/f poletermsdominate,
0
comparedwithcontributionsfromthenon-NGsector{ρ,ω,K∗,N,η′,...}.1
ThisscenariocanoccurinQCDifatlowenergyscalesµ ≪m ,thestrongcouplingα forthe
t,b,c s
3-flavortheoryrunsnonperturbativelyto an infraredfixed pointα (figure1b). Atthe fixed point,
IR
thegluonicterminthestrongtraceanomaly[9]
β(α )
θµµ = 4αs GaµνGaµν+(cid:0)1+γm(αs)(cid:1) X mqq¯q (3)
s q=u,d,s
vanishes,whichimpliesthatinthechirallimit
θµ = 1+γ (α ) (m u¯u+m d¯d+m s¯s)→0, (4)
µ(cid:12)(cid:12)αs=αIR (cid:0) m IR (cid:1) u d s
(cid:12)
1AdiscussiononviolationsofPCDCandWeinberg’spowercountingscheme[8]inγγchannelsiscontainedin[3].
MENU2013
andthushq¯qi actsasacondensateforbothscaleandchiralSU(3) ×SU(3) transformations.2 By
vac L R
consideringinfraredexpansionsaboutthecombinedlimit
m ∼0 and α .α , (5)
u,d,s s IR
our proposal is to replace χPT by chiral-scale perturbation theory χPT , where the strange quark
3 σ
massm in (4) sets thescale ofm2 aswell asm2 and m2 (figure1a, bottomdiagram). Asa result,
s f0 K η
the rules forcountingpowersof m are changed: f pole amplitudes(NLO in χPT ) are promoted
K 0 3
toLO.ThatfixestheLOproblemforamplitudesinvolving0++channelsandO(m )extrapolationsin
K
momenta.NotethatweachievethiswithoutupsettingsuccessfulLOχPT predictionsforamplitudes
3
whichdonotinvolvethe f ;thatisbecausetheχPT Lagrangianequalstheσ→0limitoftheχPT
0 3 σ
Lagrangian.
In the physicalregion0 < α < α , the effectivetheoryconsists ofoperatorsconstructedfrom
s IR
the SU(3) field U=U(π,K,η) and chiral invariant dilaton σ, with terms classified by their scaling
dimensiond:
L =L σ,U,U† =:Ld=4+Ld>4 +Ld<4 : . (6)
χPTσ (cid:2) (cid:3) inv anom mass
Explicitformulasforthestrong,weak,andelectromagneticinteractionsareobtainedbyscalingLa-
grangianoperatorssuchasK U,U† = 1F2Tr(∂ U∂µU†)andK = 1∂ σ∂µσbyappropriatepowers
(cid:2) (cid:3) 4 π µ σ 2 µ
ofthed =1fieldeσ/Fσ. Forexample,theLOstrongLagrangianreads
Lidn=v4,LO =(cid:8)c1K +c2Kσ+c3e2σ/Fσ(cid:9)e2σ/Fσ,
Ladn>o4m,LO =(cid:8)(1−c1)K +(1−c2)Kσ+c4e2σ/Fσ(cid:9)e(2+β′)σ/Fσ,
Ld<4 =Tr(MU†+UM†)e(3−γm)σ/Fσ, (7)
mass,LO
whereF ≈100MeVisthedilatondecayconstant,whosevalueisestimatedbyapplyingananalogue
σ
oftheGoldberger-TreimanrelationtoanalysesofNN-scattering[10].Heretheanomalousdimensions
γ =γ (α )andβ′ =β(α )areevaluatedatthefixedpointbecauseweexpandinα aboutα . The
m m IR IR s IR
low-energyconstantsc andc arenotfixedbysymmetryargumentsalone,whilevacuumstabilityin
1 2
theσdirectionimpliesthatbothc andc areO(M). From(7),oneobtainsformulasforthedilaton
3 4
massm
σ
m2F2 = F2 m2 + 1m2 (3−γ )(1+γ ),−β′(4+β′)c (8)
σ σ π(cid:0) K 2 π(cid:1) m m 4
andσππcoupling
L = 2+(1−c )β′ |∂π|2−(3−γ )m2|π|2 σ/(2F ). (9)
σππ (cid:8)(cid:0) 1 (cid:1) m π (cid:9) σ
Note that (9) is derivative, so an on-shell dilaton is O(m2) and consistent with σ being the broad
σ
resonance f (500).
0
OurproposedreplacementforχPT possessessomedesirablefeatures,theforemostbeing:
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1. The∆I =1/2ruleforK-decaysemergesasaconsequenceofχPT ,withadilatonpolediagram
σ
(figure2a)accountingforthelargeI =0amplitudeinK →ππ. Here,vacuumalignment[13]
S
oftheeffectivepotentialinducesaninteractionL =g K σwhichmixesK andσinLO.
KSσ KSσ S S
The effective couplingg is fixed by data on γγ → π0π0 and K → γγ, with ourestimate
KSσ S
|g | ≈ 4.4× 103keV2 accurate to a precision . 30% expected from a 3-flavor expansion.
KSσ
Combinedwithdataforthe f width(Eq.(2)),wefindanamplitude A ≈0.34keVwhich
0 (cid:12) σ-pole(cid:12)
accountsforthelargemagnitude|A | = 0.33keV. Consequently,(cid:12)(cid:12)theLO(cid:12)(cid:12) ofχPT explains
0expt. σ
the∆I =1/2ruleforkaondecays.
2Theformerpropertyisasimpleconsequenceofthefacttheq¯qisnotasingletunderdilatations. Thedualroleofhq¯qivac
wasexplored[4,5]insomedetailpriortotheadventofQCD.
EPJWebofConferences
γ π
π
σ
σ
g g
KS g8,27 + gKSσ gσππ γ σγγ σππ π
π (b) Dilaton pole in γγ → ππ. Low-
(a)TreediagramsintheeffectivetheoryχPT forthedecayK → est order χPT includes other tree di-
σ S σ
ππ.Thevertexamplitudesdueto8and27contactcouplingsg and agrams(forπ+π− production) andalso
8
g aredominated bytheσ/f -poleamplitude. Themagnitude of π±,K± loopdiagrams(suppressedbya
27 0
g isfoundbyapplyingχPT toK →γγandγγ→ππ. factor1/N )coupledtobothphotons.
KSσ σ S c
Figure2
2. Our analysis of γγ channelsand the electromagnetictrace anomaly [11, 12] yields a relation
betweentheeffectiveσγγcouplingandthenonperturbativeDrell-YanratioR atα :
IR IR
2α
g = R − 1 . (10)
σγγ 3πF (cid:16) IR 2(cid:17)
σ
AphenomenologicalvalueforR isdeducedbyconsideringγγ → π0π0 inthelarge-N limit
IR c
(figure2b). Dispersiveanalyses[14]ofthisprocessesareabletodeterminetheradiativewidth
of f (500),whichinturnconstrainsg andyieldstheestimateR ≈5.
0 σγγ IR
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