N_c and m_pi dependence of rho and sigma mesons from unitarized Chiral Perturbation Theory

0.06 MB·English

by  G. Rios

r s
N and m dependence of and mesons from
p
c
unitarized Chiral Perturbation Theory
G. Ríos , C. Hanhart† and J. R. Peláez
∗ ∗
1 DepartamentodeFísicaTeóricaII.UniversidadComplutensedeMadrid.
∗
1 †InstitutfürKernphysikandJülichCenterforHadronPhysics,ForschungzentrumJülichGmbH.
0
2
Abstract. We review our work on the r and s resonances derived from the Inverse Amplitude
n
Method.Inparticular,westudytheleading1/N behavioroftheresonancesmassesandwidthsand
a c
J theirevolutionwith changingmp . The1/Nc expansiongivesa cleardefinitionofq¯qstates, which
is neatly satisfied by the r butnotby the s , showingthatits dominantcomponentis notq¯q. The
0
1 mp dependenceoftheresonancepropertiesisrelevanttoconnectwithlatticestudies.Weshowthat
ourpredictionscomparewellwithsomelatticeresultsandwefindthattherpp couplingconstant
] ismp independent,incontrastwiththespp coupling,thatshowsastrongmp dependence.
h
p Keywords: Scalarmesons,chirallagrangians,1/Ncexpansion
- PACS: 14.40.Cs,12.39.Fe,13.75.Lb,11.15.Pg
p
e
h
Light hadron spectroscopy lies beyond the realm of perturbative QCD. At low ener-
[
gies,however,onecanusetheQCDlowenergyeffectivetheory,namedChiralPerturba-

1
tionTheory(ChPT)[1],todescribethedynamicsofthelightestmesons.ChPTdescribes
v
6 theinteractionsoftheGoldstonebosonsoftheQCDchiralsymmetrybreaking,namely,
6 the pions, by means of a effective lagrangian compatible with all QCD symmetries in-
7
volving only the pion field. The infinite tower of terms in this lagrangian is organized
1
. as a low energy expansion in powers of p2/L 2c , where p stands either for derivatives,
1
0 momenta or masses, and L c 4p fp , where fp denotes the pion decay constant. ChPT
≃
1 is renormalized order by order by absorbing loop divergences in the renormalization of
1
higher order parameters, known as low energy constants (LECs), that parametrize the
:
v high energy QCD dynamics and carry no energy or mass dependence. They depend on
Xi aregularizationscalem butafterrenormalizationtheobservablesareindependentofthis
r scale. The value of the LECs depend on the underlying QCD dynamics and are deter-
a
mined from experiment. Up to the desired order, the ChPT expansion provides a sys-
tematic and model independent description of how observables depend on some QCD
parameterslikethelightquark massmˆ =(m +m /2)orthenumberofcolors,N [2].
u d c
The use of ChPT is limited to low energies and masses, nevertheless, combined
with dispersion relations and elastic unitarity it leads to a successful description of
meson dynamics up to energies around 1 GeV, generating resonant states not originally
present in the lagrangian, without any a priori assumption on their existence or nature.
In particular, we find the r and s resonances as poles on the second Riemann sheet
of pp elastic scattering amplitudes. With this approach we can then study some of
theseresonancesproperties,liketheirspectroscopicnaturethroughtheirmassandwidth
dependenceonN ,ortheirdependenceonthepionmassinordertoconnectwithlattice
c
studies. In the following sections we review this “unitarized ChPT” approach, named
the Inverse Amplitude Method (IAM) [3, 4, 5], and then apply it to study the leading
1/N behaviorand thechiral extrapolationofther and s mesons.
C
The r and s resonances appear as poles on the second Riemann sheet of the(I,J)=
(1,1) and (I,J) = (0,0) pp scattering partial waves of definite isospin, I and angular
momentum J, respectively. Elastic unitarity implies for these partial waves, t(s), and
physicalvaluesofs belowinelasticthresholds,that
Imt(s)=s (s) t(s) 2 Im1/t(s)= s (s), with s (s)=2p/√s, (1)
| | ⇒ −
where s is the Mandelstam variable and p is the center of mass momentum. Conse-
quently, the imaginary part of 1/t is known exactly. However, ChPT amplitudes, being
anexpansiont t +t + , witht =O(pk),can onlysatisfyEq.(1)perturbatively
2 4 k
≃ ···
Imt (s)=0, Imt (s)=s (s)t2(s) ... (2)
2 4 2
Theresonanceregionliesbeyondthereach ofstandardChPT.Thisregionhowever,can
bereached combiningChPT withdispersiontheory throughtheIAM [3,4, 5].
The analytic structure of the pp scattering amplitude t(s), consisting on a right cut
extendingfroms =4m2 to¥ ,andaleftcutfrom ¥ to0,allowstowriteadispersion
th p
relationfortheauxiliaryfunctionG(s) t2(s)/t(s)−
≡ 2
s3 ¥ ImG(s)
G(s)=G(0)+G(0)s+1G (0)s2+ ds ′ +LC(G)+PC, (3)
′ 2 ′′ p Z ′s3(s s ie )
sth ′ ′− −
where the integral over the left cut has been abbreviated as LC(G) and PC stands from
the pole contributions in the scalar wave corresponding to the Adler zero. The terms in
Eq.(3) are evaluated using unitarity and ChPT as follows: The right cut (RC) is exactly
evaluatedconsideringtheelasticunitarityconditionsEqs.(1),(2):ImG(s)= Imt (s).
′ 4 ′
−
The subtraction constants only involve the amplitude and its derivatives evaluated at
s = 0, so they can be safely approximated with ChPT: G(0) t (0) t (0), G(0)
2 4 ′
t (0) t (0),G (0) t (0).TheLC,beingsuppressedby1/≃s3(s −s),isweighted≃at
2′ − 4′ ′′ ≃− 4′′ ′ ′−
low energies, so it is appropriate to approximate it with ChPT: LC(G) LC(t ). The
4
PC counts O(p6), it has been calculated explicitly [6] and it is numer≃ica−lly negligible
exceptneartheAdlerzero, away fromthephysicalregion.
Neglecting PC for the moment, taking into account that t (s) is just a first order
2
polynomial in s, and that a dispersion relation can be also written for t , we can write
4
Eq.(3)as
G(s) t (0) t (0)s t (0) t (0)s 1t (0)s2 RC(t ) LC(t )=t (s) t (s), (4)
≃ 2 − 2′ − 4 − 4′ −2 4′′ − 4 − 4 2 − 4
which immediately leads to the IAM formula tIAM(s) = t22(s) . The IAM formula
t2(s) t4(s)
satisfies exact elastic unitarity and, when reexpanded at low−energies, reproduces the
ChPT expansion up to the order used to approximate the subtraction constants and the
left cut. Here we have presented an O(p4) IAM but it can be generalized to higher
chiral orders. Note that in the IAM derivation ChPT has been always used at low
energies, to evaluate parts of a dispersion relation whose elastic unitarity cut has been
3 3
q-q scaling q-q scaling
r
2.5 G s / G sphys s 2.5 Ms / Msphys
2 2
Ms / Msphys
1.5 1.5
Mr / Mrphys
1 1
G s / G sphys
0.5 G r / G rphys 0.5
0 0
4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20
Nc Nc
FIGURE1. Left:r ands 1/N scalingO(p4).Right:s 1/N scalingO(p6).
c c
taken into account exactly. Thus, there are no additional model dependencies in the
approach,whichisreliableuptoenergieswhereinelasticitiesbecomeimportant.Taking
thepole contributioninto account leads to a modified IAM formula[6] which is almost
indistinguishable from the ordinary one except in the Adler zero region, where the
modifiedformulashouldbeused.Actually,weusethemodifiedIAMinthisworksince,
asit willbeshownbelow,oneamplitudepolegets neartheAdlerzero region.
ThissimpleIAMformulaisabletoreproducepp scatteringdatauptoroughly1GeV
andgeneratesther ands poleswithvaluesoftheLECscompatiblewithstandardChPT
[5].The1/N expansionisimplementedinChPTthroughtheLECs,whoseleading1/N
c c
scaling is known from QCD. Also, the mp dependence of IAM agrees with ChPT up to
the order used. Hence, it is straightforward to study the leading 1/N behavior and the
C
mˆ dependence oftheresonances generated withtheIAM.
The QCD 1/N expansion [2] provides a clear definition of q¯q bound states: their
c
masses and widths scale as O(1) and O(1/N ) respectively. The QCD leading 1/N
c c
behavior of the ChPT parameters (fp , mp and the LECs) is well known. Hence, by
scaling with N the ChPT parameters in the IAM, the N dependence of the r and s
c c
mesons mass and width has been determined [7, 8]. They are defined from the pole
positionsas√spole=M iG .NotethatweshouldnottaketoolargeNC values,sincethe
N ¥ isaweaklyinter−actinglimit,wheretheIAMapproachislessreliable[9].Also,
c
→
for very large N , a tiny admixture of q¯q in the physical state could become dominant,
c
butthisdoesnotgiveanyinformationabout thephysicalstatedominantcomponent.
Fig. 1 (left) shows the r and s mass and width N scaling. It can be seen that the r
c
followsremarkablywelltheexpectedbehaviorofaq¯qstate,confirmingthatthemethod
obtainsthecorrectN behaviorofwellknownq¯qstates.Incontrast,thes doesnotfollow
c
thatq¯qpattern, allowingusto concludethat itsdominantcomponentis notq¯q.
Loop contributions play an important role in determining the s pole position. Since
they are 1/N suppressed compared to tree level terms, it may happen that for larger
c
N they become comparable to tree level O(p6) terms, which are subdominant in the
c
ChPT series, but not N suppressed. Thus we checked the O(p4) results with an O(p6)
c
IAM calculation [8]. We defined a c 2-like function to measure how close a resonance
is from a q¯q behavior. First, we used it at O(p4) to show that it is not possible to find a
set of LECs that makes the s to behave predominantlyas a q¯q state. Next, we obtained
an O(p6) data fit where the r q¯q behavior was imposed. Figure 1 (right) shows the Ms
and G s Nc scaling obtained from that fit. Note that both Ms and G s grow near Nc =3,
1.1
1.5
1 Adler 1
) zero
mp0.5 V)
(cid:190)(cid:214)m (s / -0.05 M (Ger 00..89 EMTIIMALMCC
I-1-.51 threshold 0.7 CP-PARCBSC+-AUQdKCeQPlDaDCiSdGDFe
0 1 pp2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6
(cid:190)
Re ((cid:214) s / mp ) mp (GeV)
2
2 1,5 rpp
Mass of Width of 1.8 spp (upper)
s vs. r s vs. r 1.6 spp (lower)
1.5 1 phys|g / g| 11..24
1 0.5 1
0.8
0.6
0.5 0
0 1 2 3 0 1 2 3 0 50 100 150 200 250 300 350 400 450
mp / mpphys mp / mpphys mp (MeV)
FIGURE 2. Top Left: Movement of the s (dashed lines) and r (dotted lines) poles for increasing
mp on the second sheet. The filled (open) boxes denote the s (r ) pole positions at mp = 1, 2, and
phys
3 mp ,respectively.TopRight:ComparisontheIAMMr dependenceonmp withsomerecentlattice
re×sults[14]. Bottom Left: Comparison of the r (light) and s (dark) mass dependence on mp . Bottom
Center:Comparisonof ther (light)ands (dark)widthdependenceonmp . Thedotted(r )dot-dashed
(s )linesshowthedecreaseduetoonlyphasespaceassumingaconstantcouplingtopp .BottomRight:
r ands couplingscalculatedfromthepoleresidue.Inallpanels,thebandscovertheLECsuncertainty.
confirmingtheO(p4)resultofanonq¯qdominantcomponent.However,forN between
c
8 and 15, where we still trust the IAM, Ms becomes constant and G s starts decreasing.
This may hint to a subdominant q¯q component, arising as loops become suppressed as
N grows.Finally,byforcingthes tobehaveas aq¯q,wefoundthatinthebestcasethis
c
subdominantcomponentcouldbecomedominantaroundN >6 8,butalwayswithan
c
N ¥ mass above 1 GeV instead of its physical 450 MeV v−alue. This supports the
c
→ ∼
emergingpictureoftwolowenergyscalarnonets,oneofexoticnaturebelow1GeVand
anotherofordinary q¯q natureabove1 GeV.
ChPT also provides an expansion of mp in terms of mˆ (at leading order mp2 mˆ).
∼
Thus, by changing mp in the amplitudes we see how the IAM poles depend on mˆ. We
reporthere ouranalysisofther ands properties dependenceonmp [10].
Thevaluesofmp consideredshouldfallwithintheChPTapplicabilityrangeandallow
for some elastic regime below KK¯, that would almost disappear if mp > 500, which
would be the mostoptimisticapplicabilityrange. We expect higher order corrections to
bemorerelevantas mp increases. Thus,ourresultsbecomeless reliableas mp grows.
Fig.2(topleft)showstheevolutionofthes andr polepositionsasmp isincreased.In
ordertoseethepolemovementsrelativetothepp threshold,whichisalsoincreasing,we
useunitsofmp ,sothethresholdisfixedat√s=2.Bothpolesmoveclosertothreshold
andtheyapproachtherealaxis.Ther polesreachtherealaxisatthesametimethatthey
crossthreshold.Oneofthemjumpsintothefirstsheetandbecomesaboundstate,while
its conjugate partner remains on the second sheet practically at the very same position
as that in the first. In contrast, the s poles go below threshold with a finite imaginary
part before they meet in the real axis, still on the second sheet, becoming virtual states.
As mp increases, one pole moves toward threshold and jumps through the branch point
to the first sheet stayingin the real axis below threshold.The other s polemoves down
in energies away from threshold and remains on the second sheet. Similar movements
werefoundwithinquark models[12]and afinitedensityanalysis[13].
Fig. 2 (top right) shows our results for Mr dependence on mp compared with some
lattice results [14] and the Mr PDG value. In view of the incompatibilities between
differentlatticecollaborations,wefindaqualitativegoodagreementwithlatticeresults.
The Mr dependence on mp agrees also with estimations for the two first coefficients of
itschiralexpansion[15].
In Fig. 2 (bottomleft) we compare the mp dependence of Mr and Ms , normalized to
their physical values. The bands cover the LECs uncertainties. Both masses grow with
mp ,butMs growsfasterthanMr .Above2.4mpphys,weshowtwobandssincethetwos
poleslieon thereal axiswithtwodifferent masses.
In the bottom center panel of Fig. 2 we compare the mp dependence of G r and G s
normalizedto theirphysicalvalues:note that bothwidths becomesmaller. Wecompare
this decrease with the expected phase space reduction as resonances approach the pp
threshold. We find that G r follows very well this expected behavior, which implies that
the rpp coupling is almost mp independent. In contrast, G s deviates from the phase
space reduction expectation. This suggests a strong mp dependence of the s coupling
to two pions, which we confirm with a explicit calculation of the resonances couplings
fromthepoleresidues asshowninthebottomleft panel.
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