Pion Chiral Symmetry Breaking in the Quark-Level Linear Sigma Model and Chiral Perturbation Theory

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by  Michael D. Scadron

Pion Chiral Symmetry Breaking in the
Quark-Level Linear Sigma Model and Chiral Perturbation Theory
Michael D. Scadron∗
Physics Department, University of Arizona, Tucson, AZ 85721, USA
Frieder Kleefeld† and George Rupp‡
Centro de F´ısica das Interacc¸˜oes Fundamentais (CFIF),
Instituto Superior T´ecnico, Av. Rovisco Pais, P-1049-001 LISBOA, Portugal
(Dated: February 2, 2008)
Chiralsymmetrybreaking(ChSB)isreviewedtosomeextentwithinthequark-level-linear-sigma-
model(QLLσM)theoryandstandardchiralperturbationtheory(ChPT).Itisshown,onthebasisof
severalexamplesrelatedtothepion,asawell-knownGoldstonebosonofchiralsymmetrybreaking,
6 that even the non-unitarized QLLσM approach accounts, to a good approximation, for a rather
0 simple, self-consistent, linear, andverypredictivedescription ofNature. On theotherhand,ChPT
0 — even when unitarized — provides a highly distorted, nonlinear, hardly predictive picture of
2 Nature, which fits experiment only at the price of a lot of parameters, and requires a great deal
of unnecessary theoretical effort. As the origin of this distortion, we identify the fact that ChPT,
n
reflecting only direct ChSB by nonvanishing, current-quark-mass values, does not — contrary to
a
QuantumChromodynamics(QCD)andtheQLLσM—containanymechanismforthespontaneous
J
generationofthedynamicalcomponentoftheconstituentquarkmass. Thisleadstoaverypeculiar
4
picture of Nature, since the strange current quark mass has to compensate for the absence of
2
nonstrangedynamicalquarkmasses. WethusconcludethatstandardChPT—contrarytocommon

wisdom — is unlikely to be the low-energy limit of QCD. On the contrary, a chiral perturbation
1
theory derived from the QLLσM, presumably being the true low-energy limit of QCD, is expected
v
instead to provide a distortion-free description of Nature, which is based on the heavy standard-
6
9 modelHiggsbosonaswellaslightscalarmesons,asthesourceofspontaneousgenerationofcurrent
1 and dynamical quark masses, respectively.
1
0 PACSnumbers: 14.40.Aq,11.30.Rd,11.30.Qc,14.40.Cs
6
0
/ I. INTRODUCTION quantity to discuss and measure the amount of ChSB in
h
the pionisnotonlyits massm ,but alsothe underlying
p π
(nonstrange) constituent quark mass m . The latter
- Thefundamentallawsofphysicsallrespectsomeexact con
p canbe additivelydecomposedasm =m +m [3]
symmetries. In Nature, however, we are mostly — and con dyn cur
e into a bulk part called dynamical quark mass m as-
fortunately – facing symmetries which are broken. In dyn
h
sociated with chiral-symmetric strong interactions, and
: fact, if the electromagnetic, weak, strong, and gravita-
v a smallcorrectioncalled current quark mass m , being
tional interactions obeyed exactly the same symmetries, cur
i nonzero only in the presence of ChSB electroweak inter-
X the observable world would be a dull place. It is pre-
actions. Hence, in the CL we have m 0, m 0,
r cisely the mismatch between the symmetries respected π → cur →
a individually by the different interactions that gives rise mdyn → mCL, and mcon → mCL, where mCL may be
called chiral-limiting quark mass.
to the amazing diversity and beauty of Nature at high,
Thus, it becomes clear that a self-consistent picture
intermediate, and particularly low energies [1].
of ChSB will only be achieved through a complete un-
Inthispaperweshallfocusonthechiralsymmetryun-
derstandingofelectroweakandstronginteractions,espe-
derlying the theory of strong interactions, as they inter-
cially their interplay. Unfortunately, and maybe some-
fere with chiral-symmetry-breaking (ChSB) electroweak
whatsurprisingly,neithertheexperimentalnorthetheo-
interactions. For convenience, our attention will be con-
reticalsituationisevenclosetosettled,despitetheseem-
centratedon observablesassociatedwith the gold-plated
ingly overwhelming success of the present-day standard
test particle of ChSB, i.e., the pion [2], which, being a
model of particle physics (SMPP), as we explain next.
Goldstone boson, would be massless in the so-called chi-
In the first place, despite the consensus on the mech-
ral limit (CL), that is, the limit of strong interactions
anism for generating the tiny up/down current quark
without electroweak interactions.
masses ( 101 MeV) in the Lagrangian density of the
Aswillbediscussedinmoredetailbelow,aconvenient SMPP,th∼ecorrespondingheavyscalar(m >105MeV),
H
the Higgs boson, remains undetected.
Secondly,thereissubstantialexperimentalevidence[1]
for the existence of a complete nonet of light (m < 1
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] GeV) scalar mesons, which can be considered respon-
‡Correspondingauthor;Electronicaddress: [email protected] sible for the dynamical generation of the surprisingly
2
large m ( 300 MeV). However, our present master- isreproduced,i.e.,inthechirallimitthemassofthelight-
dyn
∼
ing of QCD [4], the commonly accepted theory under- estscalarmesonequalstwice the dynamicalquarkmass,
lying strong interactions, does not allow to predict the withthepionbeingmassles,ofcourse,inthe samelimit.
existence of light scalar mesons, not to speak of their Clearly, the simplicity and linearity of the QLLσM is
multiplet structure and pole positions. But at least one owing to the inclusion of the σ meson, now experimen-
can gain, e.g. on the basis of QCD-inspired variational tallyconfirmed[1],asanelementaryfieldbeingthechiral
chiral quark models [5], Bethe-Salpeter equations (BSE) partner of the pion [24, 25] whereby in loop order both
[6–9], or Dyson-Schwingerequations (DSE) [9, 10], some areself-consistently recoveredas 0−+ and 0++ q¯q states,
understanding about the dynamical generation of m , respectively. In contrast, ChPT requires unitarization,
dyn
once small ChSB values for m are given. Unfortu- i.e., infinite order, to resuscitate the σ [14, 15]. As we
cur
nately, most DSE approaches are Euclidean, whereas it shalldemonstratethroughseveralexamples,theQLLσM
is not clear whether a Wick rotation from Minkowski is doing a much better job, even without unitarization,
[11, 12] to Euclidean space will yield identical results. in reproducing a large variety of low-energy observables
thanChPT,withonlyatinyfractionoftheeffortneeded.
Finally, chiral perturbation theory (ChPT) (see e.g.
Ref. [13]), the mainstreamlow-energy effective approach Finally,weshallpresentargumentswhytheQLLσMmay
even constitute a full-fledged, asymptotically free theory
to strong interactions, is only capable of recovering the
of strong interactions.
lightscalarmesonswhenitisunitarized[14,15]byhand.
Inthepresentpaper,weshallpassinreviewthemech-
However,itsexactrelationtoQCDremainsunexplained,
anism of ChSB for a variegation of low-energy observ-
despitewell-knownclaimsmadebythelargeChPTcom-
ables and relations, employing the QLLσM as compared
munity. Moreover,ChPTreliesuponastronglydistorted
to standard ChPT, namely: pion and nucleon sigma
pictureofNature,namelybasedoncurrentquarks,whose
terms(Sec.II),mesonchargeradii(Sec.III),Goldberger-
massesareatleasttwoordersofmagnitudesmallerthan
Treiman relations (Sec. IV), Goldberger-Treiman dis-
those of most observed hadrons. As ChPT does not
crepancies (Sec. V), constituent quark mass via baryon
provide any mechanism to generate m , contrary to
dyn
magneticmoments(Sec.VI),effective currentquarkand
QCD as well as chiral quark models and BSE/DSE ap-
ChSB pion, kaon, η masses (Sec. VII), ground-state
proaches, it is the strange current quark mass, taken at 8
scalar q¯q nonet (Sec. VIII), I = 0 ππ scattering length
roughly 25 times the nonstrange current mass, which
(Sec. IX), and the process γγ π0π0 (Sec. X). A sum-
partly has to take over the role of m . This, we be-
dyn →
mary with discussion and conclusions is presented in
lieve, is the origin, of, for example, the excessively large
Sec. XI.
strangeness content of the proton predicted by ChPT,
and, as discussed below, the unacceptably large value
for σs¯s ( 10 MeV), which has been measured to be
2 MπNeV≃[16]. Furthermore, an additional and very siz- II. σππ AND σπN CHIRAL BREAKING TERMS
≤
able amount of glue, which logically should already be
accountedforinChPTasthesupposed“low-energylimit The pion σ term is considered a c-number (t-
of QCD”,is still needed e.g.to explain the nucleonmass independent). It is defined at q2 = t = m2π via PCAC
[17]. as (with fπ 93 MeV for π0, the Particle Data Group
≃
(PDG) 2004 [1] takes f (92.42 0.26) MeV; ETC
The aim of the present paper is to show that there π ≃ ±
stands for “Equal-Time Commutator”)
exists an alternative approach to strong interactions,
much simpler than ChPT and free from its distortions σ δ = π [Q5,i∂Aπ] π
ππ ij h i| π ETC| ji
and shortcomings, yet highly predictive and also intu-
1
itively appealing, namely the quark-level linear σ model ≃ f h0|∂Aπi(0)|πji=m2πδij . (1)
(QLLσM). The original LσM, with nucleons instead of π
quarks as the fermionic fields, had been suggested by The analogue πN σ term is defined as [26]
Schwinger [18], and was then proposed by Gell-Mann
σ u¯ u = N [Q5 ,i∂Aπ] N , (2)
and L´evy [19] as the preferable model for strong interac- πN N N h | π3 3 ETC| i
tions featuring spontaneousChSB, in particularrespect- also taken as a c-number in the 1960’s.
ingpartialconservationofaxial-vectorcurrents(PCAC). Boththe nonvanishingvalues ofEqs.(1)and(2)char-
Onthe otherhand, inthe same epochNambuandJona- acterize SU(2) SU(2) ChSB.
Lasinio (NJL) [20] presented a nonrenormalizablemodel Inthelangua×geoftheLσM,Eqs.(1)and(2)arerepre-
for the pion, originally encompassing (anti)nucleons but sentedby“nonquenched”(NQ)σ tadpolegraphs,shown
straightforwardly extensible to quarks, which was based in Fig. 1, with LσM couplings 2g =(m2 m2)/f
on dynamical ChSB. Then, the LσM was generalized to m2/f , g m /f . Both taσdπpπole “heσa−ds” πare gπen≃-
quarksbyL´evy[21]andCabibbo&Maiani[22]. Finally, eraσtedπbyσtNheNC≃hSBNHamπiltonian(H meanssemi-strong
SS
Delbourgo and Scadron [23] managed to dynamically Hamiltonian) implying
generate the QLLσM self-consistently in one-loop order,
2
whichconsiderablyincreaseditspredictivepower. Forin- σNQ = mπ m 40MeV, (3)
stance,thefamouschiral-limitingNJLresultmσ =2mCL πN (cid:18)mσ(cid:19) N ≃
3
H H
SS SS
σ σ
π π
N N
FIG. 1: Tadpoles generated by theChSB “semi-strong” (SS) Hamiltonian HSS.
for an averaged nucleon mass m = 938.9 MeV and ChPTschemeof1990[39]useddoubly-subtracted disper-
N
chiral-limiting σ mass mCL of 650.8 MeV (see Sec. VIII sion relations (adding 6 new paramters) via a low-mass-
σ
and Ref. [23]), predicting an “on-shell” broad σ mass of HiggsdecayH ππ,yettobemeasured. Then,in1991,
[27–29] m 665 MeV, for (mCL)2 m2 m2. ChPT obtained→[40] a non-c-number πN σ term with
σ ≃ σ ≃ σ− π
The perturbative “quenched” πN σ term found by
σ (t=0) 45MeV, (9)
GMOR [30], or via the APE collaboration [31], is about πN ≃
and a larger πN σ term at t = 2m2 of 60 MeV
π
σGMOR = mΞ+mΣ−2mN m2π as (HOChPT stands for “higher-order ChPT”) [41] (in
πN 2 m2 m2 MeV)
(cid:18) K − π(cid:19)
26MeV, (4) σ (2m2) = σGMOR(25)+σHOChPT(10)
≃ πN π πN πN
+ σs¯s (10)+σt-dep.(15)
so the net πN σ term in the LσM is the sum πN πN
60MeV, (10)
σLσM =σGMOR+σNQ (26+40)MeV=66MeV. (5) ≃
πN πN πN ≃ becausethe latter “threepieces happen to havethe same
sign as σGMOR” [41]. Note that the σs¯s term has been
WhiletheAdler–Weinberg[32,33]πN low-energythe- πN πN
measuredas 2MeV[16]. Sucha“happening”suggests
orems have large background terms, the Cheng–Dashen ≤
that ChPT in Eq. (10) is physically less significant and
(CD) [34] theorem at t = 2m2 has a very small back-
π much more complicated than Eqs. (5), (7), or (8) above.
ground term (for the isospin-even πN amplitude), viz.
F+(ν =0,t=2m2)= σπN+ (m4) 1.020m−1 , (6)
π f2 O π ≃ π III. MESON CHARGE RADII
π
as measured by Koch and Ho¨hler [35], corresponding to Using vector and axial-vector form factors, the PDG
(for f 93 MeV) tablesonpages499and621ofRef.[1]statethatthepion
π
≃
and kaon charge radii are measured as
f2
σπN =(1.018) π 63MeV, (7) rπ+ = (0.672 0.008)fm, (11)
mπ+ ≃ ±
r = (0.560 0.031)fm. (12)
K+
±
wherethesmallO(m4π)CDterminEq.(6)is0.002m−π1. In fact, the vector-meson-dominance (VMD) and LσM
Or instead one can work in the infinite-momentum schemes give nearby values [42] (with mˆ (m +m )/2
u d
frame(IMF),whichsuppressesthetadpolesandpredicts and h¯c 197.3 MeVfm) ≡
[36] (requiring squared baryon masses with cross term ≃
2mNσπN) rπVMD = √6/mρ ≃ 0.623fm, (13)
rLσM = 1/mˆ 0.63fm, (14)
m2 +m2 2m2 m2 π CL ≃
σπN = Ξ 2mΣN− N (cid:18)m2K −πm2π(cid:19)≃63MeV. rrKVLMσMD == √2/6(/mˆm+K∗m ) ≃ 00..5541ffmm,. ((1156))
(8) K s CL ≃
NotethatthedatainEq.(7)ortheIMFvalueinEq.(8) Note that mˆ 313 MeV as we shallpoint out later in
CL
of63MeVarecompatiblewiththe LσMvalueinEq.(5) Sec. VI. Then t≃he q¯q pion is very tightly bound, almost
of 66 MeV. a fused q¯q meson.
However, ChPT leads to a more complicated scheme. Moreover,on page 498 of Ref. [1], the pion vector and
By 1982, the review by Gasser and Leutwyler preferred axial-vector form factors are observed at q2 =0 as
σ 24 to 35 MeV [37]. In 1984, their revised ChPT
[3π8N] r≃uled out some LσM in Appendix B “... as a realis- FVπ(0) = 0.017±0.008, (17)
tic alternative to QCD ...”, andthe“scalarformfactor” Fπ(0) = 0.0116 0.0016. (18)
A ±
4
Using f 93 MeV, the theoretical CVC estimates of This would imply, in combination with Eq. (26) and
π
these form≃factors are [43] (see also Ref. [42]) r2 1/mˆ2 =1/(gf )2 N /(4π2f2), that
π ≃ π ≃ c π
FVπ(0) = 8mπ2πf+π ≃ 0.0190, (19) (cid:10) r(cid:11)S2 = m62 8πm22πf2 − 19123π2mf22π
m π (cid:18) π π (cid:19)
Fπ(0) = π+ 0.0127, (20) (cid:10) (cid:11) 13 3 11
A 12π2fπ ≃ = 1− 24 4π2f2 ≃ 24 rπ2 ≃ 0.19fm2 ,
(cid:18) (cid:19) π
reasonably near the data in Eqs. (17) and (18). (cid:10) (cid:11) (28)
Sakurai’s [44] vector-meson-coupling universality
(VMU) [45] suggests g g 6, whereas the LσM which is clearly in strong conflict with the ChPT results
ρ ρππ
predicts [46] that VMU≈receives∼a correction of 1g Eqs. (24) and (25).
6 ρππ
via the mesonic πσπ loop, giving g =g + 1g or InordertoextendtheLσMtoSU(3),theK+ e+νγ
ρππ ρ 6 ρππ →
form factor sum is measured, according to page 621 in
gρππ 6 Ref. [1], as
= , (21)
g 5
ρ FK(0)+FK(0) = 0.148 0.010. (29)
V A ±
close to the data [1, 42]
The SU(3(cid:12)) LσM pole term(cid:12)s add up to [42, 52]
(cid:12) (cid:12)
gρππ ≃5.95, gρ ≃4.96. (22) FVK(0)+FAK(0) LσM = 0.109+0.044 = 0.153, (30)
These couplings in (22) follow from the newly measured
co(cid:12)mpatible with d(cid:12)ata in Eq. (29). The quark-loop pre-
decay rates [1] Γ(ρππ) = 150.3 MeV and Γ(ρee) = (cid:12) (cid:12)
diction for the pion charge radius was initially found in
7.02 keV (neglecting errors), as in Eqs. (11) and (12)
Refs. [53] via quark (and meson) loops in a LσM frame-
of Ref. [42].
work,butnot viaChPT.Theabove-measuredvectorand
Although VMD and the LσM reasonably match
axial-vectorformfactorsinEqs.(17),(18),and(29)have
charge-radii data, the one-loop-order ChPT prediction
not been extended to the mentioned ChPT scalar form
gives [47]
factors,as was suggestedin deriving Eq.(9), and will be
r2 = 12L /f2, (23) suggested in Eqs. (43) and the nonlinear a00 in Secs. V,
π 9 π
IX, respectively.
whichdoesnot uniq(cid:10)uel(cid:11)ypredictdata,sinceL isaChPT
9
low-energy-constant(LEC) parameter.
Whereas ChPT does not explain the pion or kaon IV. GOLDBERGER-TREIMAN RELATIONS
charge radii, the LσM continues to match data.
Furthermore,two-loopChPT[48,49]suggeststhatthe First we note that the pion-quark coupling in the
pion scalar-form-factorradius obeys QLLσM is [23] g = 2π/√3 3.6276, also following
πqq
≃
from infrared QCD [54] and the Z = 0 compositeness
r2 = (0.61 0.04)fm2. (24)
S ± condition [55]. Then the pion Goldberger-Treiman rela-
This value is d(cid:10)eba(cid:11)ted [50, 51] as tion(GTR) givesthe nonstrangeconstituentquarkmass
as
r2 = (0.75 0.07)fm2. (25)
S ± f g = mˆ = 93MeV 2π 337.4MeV. (31)
π
Weinturnclaim(cid:10) th(cid:11)attheaverageofEqs.(24)and(25)is × √3 ≃
75% greater than the square of the VMD and LσM pion
The nonrelativistic quark model predicts [1], via mag-
chargeradiusvaluesinEqs.(13)and(14),the lattertwo
netic dipole moments (see Sec. VI, Eq. (57)), practically
being compatible with data in Eq. (11). Moreover, the
the same value, viz.
scalar-form-factorbasisofEqs.(24)and(25)hasneither
been verified experimentally, nor theoretically via VMD 1
mˆ = (m +m ) 337.5MeV, (32)
u d
or the LσM, in contrast with the vector & axial-vector 2 ≃
form factors in Eqs. (17–20), or Eqs. (29,30) to follow. with m m 4 MeV following from the kaon [1, 56]
d u
If we followed the discussion of Gasser and Leutwyler mass diffe−rence≃or the Σ− Σ+ baryon mass difference,
in Ref. [38], the identity yielding the constituent ma−sses
ffπ(m(02π)) −1 = 61m2π rS2 + 19123π2mf22π + O(m4π) (26) mu ≃ 335.5MeV , md ≃ 339.5MeV. (33)
π π Asforthestrangequark,thedataratiofor[1]f /f
(cid:10) (cid:11) K π ≃
would hold. From Eq. (40) in the upcoming Sec. V, we 1.22 gives
know that, to a good approximation,
m f
s K
2 1 1.44 (34)
fπ(m2π) 1 = m2π + (m4). (27) mˆ ≃ fπ − ≃
fπ(0) − 8π2fπ2 O π ⇒ ms ≃ 1.44mˆ ≃ 486.0MeV. (35)
5
Then the kaon GTR However,the ChPT scalar form factors predict
1
fKg = 2(ms+mˆ) = 411.75MeV (36) fπ(m2π) 1 67..72%% vviiaa otwnoe lloooopps [[6409]],, (43)
f (0) − ∼ 
π 8.2% via two loops [51].
predicts, via g =2π/√3, 
Even though thereis a two-loop debate as to how
1
fK = (ms+mˆ)/(2π/√3) = 113.50MeV, (37) ChPT should proceed, none of these 6.7%, 7.2%, 8.2%
2
predictions (based on scalar form factors) are near
giving the ratio the 3%chiral-breakingmodel-independentpredictionsof
Eqs. (40), (41), or (42).
f 113.50
K
= 1.22, (38)
f 93 ≃
π
VI. CONSTITUENT QUARK MASS VIA
a second test of g = 2π/√3 (Eq. (31) being the first BARYON MAGNETIC MOMENTS
test). Alternatively, from Eq. (32) with f 93 MeV,
π
≃
fK/fπ 1.22, Thenonrelativisticvalence(V)quarkmodelhasaxial-
≃
vector spin components of the nucleon [61]
2f g mˆ = m 485.7MeV, (39)
K s
− ≃
4 1
very close to Eq. (35), a third test of g =2π/√3. ∆uV = , ∆dV = , ∆sV =0, (44)
3 −3
ChPT appears not to alter these GTRs, nor does so
PCAC, even though both were first obtained theoret- with total spin ΣV =∆uV +∆dV +∆sV =1. Although
ically via the LσM (not with quarks, but nucleons as ∆u ∆d 1.27 [1] is about 30% lower than the valence
− ≈
fermions [19]). Recall that ChPT rules out some kind value from Eq. (44), good valence predictions from the
of LσM in Appendix B of Ref. [38], which is an indi- nucleon magnetic moments (m.m.) stem from [61]
rect mark against ChPT, because our GTR-LσM chiral
scheme in Eqs. (31)–(39) above is a close match with µp(uud) = µu∆uV +µd∆dV +µs∆sV , (45)
data. µ (ddu) = µ ∆u +µ ∆d +µ ∆s , (46)
n d V u V s V
predicting the ratio
V. GOLDBERGER-TREIMAN DISCREPANCIES
µ ∆u
u V
+1+0
A once-subtracted dispersion relation (containing no µp = (cid:18)µd ∆dV (cid:19) = 9 = 1.5 (47)
a0d.2d6i)tiMoneaVl [p1a])ratmheetqe2rsv)arpiarteidoinctosf f(forasf[π57]= (92.42 ± µn ∆∆udV + µµu +0 −6 −
π V d
fπ(m2π) 1 = m2π 1+ m2π 2.84%. (40) (using quark m.m. values µu = e/3mu, µd = −e/6md),
f (0) − 8π2f2 10mˆ2 ≃ close to the observed [1] ratio 2.793/( 1.913) 1.46.
π π (cid:18) (cid:19) Also, the nucleon m.m. difference [1, 6−1] is ≈ −
The dominant 2.79% (m2) term in Eq. (40) was first
obtained in Refs. [58].OTheπ smaller 0.05% (m4) term e
O π µp µn = (µu µd)(∆uV ∆dV) = 4.706 ,
was found in Ref. [29], having been anticipated numeri- − − − (cid:18)2mN(cid:19)
cally in Ref. [57]. In any case, Eq. (40) is a Taylor series (48)
in m2, m4, needing only two terms. Alternatively, we predicting the average constituent quark mass (for mˆ
π π ≈
invoke data to explain the GT discrepancy as [27–29] mu,md)
mNgA 5/3
∆ = 1 = (2.07 0.57)%, (41) mˆ = m 332.5MeV, (49)
− fπgπNN ± N4.706 ≈
using mN =938.92MeV, gA/gV =1.2695 0.0029[1], for∆uV ∆dV =5/3,nearmˆ mp/µp 336MeV,and
| | ± − ≈ ≈
fπ = (92.42 0.26) MeV [1], and gπNN = 13.17 0.06 near the GT mass in Eq. (31), or the anticipated m.m.
± ±
[59]. Prior 1971 measurements gave gπNN 13.40 and quark mass in Eq. (32).
≃
∆ 3.8%. The analysis of Sec. II yields Asamatteroffact,assuming∆s=0,thesumµ +µ
p n
≃
predictsalargervaluemˆ 355.6MeV,whichreducesto
σ 63 ≈
πN = 3.35%. (42) Eqs. (31,32) only if there is a slight strangeness compo-
2mN 2 938.92 ≃ nent in the nucleon, viz.
×
All of the above analyses suggest a 3% chiral-breaking 337.5 355.6
effect. ∆s − 5.4%. (50)
≈ 337.5 ≈ −
6
This result is compatible with recent data [16] finding Recall that m was first estimated [6], via quark-
dyn
∆s 6%. dressingandbindingequations,asm 300–320MeV,
dyn
− ≤ ∼
In a similar manner, the Σ to N m.m. valence- or ideally
difference ratio is predicted as
m
N
m = 313MeV. (59)
µΣ+(uus)−µΣ−(dds) = ∆uV −∆sV = 4/3 =0.800 dyn 3 ≈
µ (uud) µ (ddu) ∆u ∆d 5/3
p − n V − V (51) This scale matches infrared QCD [3, 6, 66], for αs ≈0.5
at a 1 GeV cutoff,
only 4% higher than the observed [1] difference ratio
0.769. Moreover,the present β-decay value [1] 1
4π 3
m = α q¯q 313MeV, (60)
∆u ∆d = gA = 1.2695 0.0029, (52) dyn 3 sh− i ≈
− ± (cid:20) (cid:21)
and the λ component found from various hyperon
8 for q¯q (245 MeV)3. Also, from Sec. III, recall that
semileptonic weak decays give h− i≈
m = r−1 313MeV, (61)
∆u+∆d 2∆s = g 3f −d 0.584 (53) dyn π ≈
A
− f +d ≈
(cid:20) (cid:21)A for VMD pion charge radius rπ 0.63 fm, near data [1].
≈
(fortheempiricallydeterminedratio[62](d/f) 1.74). As for the ChSB current-quark-mass scale, the ChPT
A
As we shall now see, Eq. (53) requires a slight ∆≈s = 0, review of 1982 [37] takes
6
i.e., ∆s= 5.7%.
− (f m )2
Lastly, adding and subtracting Eqs. (52,53) leads to mˆ π π 5.5MeV, (62)
cur
[63] ≈ 2 q¯q ≈
h− i
∆u ∆s 0.927 , ∆d ∆s 0.343. (54) for f 93 MeV and q¯q (245 MeV)3. Refer-
π
− ≈ − ≈ − ≈ h− i ≈
ence [37] implies this relation is due to GMOR [30], but
Then the latter four equations uniquely predict
Eq. (62) and the f scale are hard to find in GMOR,
π
∆u 0.87, ∆d 0.40, ∆s 0.057, ∆Σ 0.41, which focuses instead on the good-bad structure of q¯λiq
≈ ≈− ≈− ≈ (55) andq¯γ5λiqmatrixelements[67]. Infact,Ref.[68]stresses
together with the good valence results Eqs. (47,48,50) on page 462 that this GMOR SU(3) assumption (lead-
and the constituent quarks mass Eq. (49). This quark- ing directly to our Eq. (62)) may not be correct. More
spinpattern(55)iscompatiblewiththeQCDpredictions recently,Ref. [66]notedthat, ifEq.(62)weretrueandif
[64] N s¯sN = 0 (in fact, ∆s 0.057 is compatible with
h | | i ≈ −
data [16]), then it is easy to show that the πN σ term
∆u = 0.85 0.03 , ∆d = 0.41 0.03, should be in theory
± − ± (56)
∆s = 0.08 0.03 , ∆Σ = 0.37 0.07.
− ± ± 3(m m ) 606MeV
Also, dynamical tadpole leakage is similar in spirit σπthN = msΞ− Λ ≈ ms . (63)
to the QLLσM, but now with axial-vector f (1285), 1 1
1 mˆ cur− mˆ cur−
f1(1420)mixing[65]. Thisgeneratesquarkspinscloseto (cid:16) (cid:17) (cid:16) (cid:17)
Eqs. (55,56), with ∆s = 6.0%, near the values 5.7% If we follow recent ChPT with [40] σChPT = 45 MeV,
− − πN
in Eq. (53) and 5.4% in Eq. (50). then the latter equation requires (m /mˆ) 14.47,
s cur
− ≈
Given this consistent pattern of quark spins with ap- and so m 80 MeV for mˆ 5.5 MeV. “If
s,cur cur
≈ ≈
proximate average quark mass in Eq. (49), for mˆ = this were true then a significant fraction of the nucleon’s
(m +m )/2, µ = e/3m , µ = e/6m and leading- mass would be due to strange quarks — in contradic-
u d u u d d
−
order mˆ = mp/µp 336.0 MeV, and further folding in tion with the quark model” [66]. Moreover, such a re-
≈
md mu 4 MeV, we obtain (in MeV) a quadratic sult would be in conflict with the usual ChPT ratio [37]
− ≈
equation for mˆcon, viz. (ms/mˆ)cur 25. We shall return to current quarks in
≈
the next section.
m 14
mˆ2 p mˆ + = 0, (57)
con− 2.792847 con 9
(cid:18) (cid:19)
VII. EFFECTIVE CURRENT QUARK AND
whoseonlypositivesolutionismˆ 337.5MeV,asan-
con
≈ CHIRAL-SYMMETRY-BREAKING PION, KAON,
ticipated in Eq. (32). The latter is near Eq. (49), nearer
η8 MASSES
stilltom /µ 336.0MeV,andevenclosertothe GTR
p p
≈
quark mass in Eq. (31). As mentioned in the Introduc-
Following Ref. [69], we use QCD to express a
tion, this bulk constituent quark mass can be decom-
momentum-dependent dynamical quark mass as
posed into its CL dynamically generated part m and
dyn
its chiral-brokencurrent-quark-masscomponent m :
cur m3
m (p2) = dyn , (64)
mcon = mdyn+mcur . (58) dyn p2
7
sothatm atp2 =m2 is,asrequiredbyconsistency, orroughly,fromEq.(69)givenEq.(58)andm 313
dyn dyn dyn ≈
MeV,
m (p2 =m2 ) m . (65)
dyn dyn ≡ dyn m2 mˆ (m +mˆ ) mˆ 53MeV,
π ∼ cur dyn cur ⇒ cur ∼
Furthermore, we take the nonstrange constituent quark
(70)
mass mˆ as 337.5 MeV from Secs. IV,VI and compute
con
the effective current quark mass as (avoiding any good-
taking m 313 MeV as in Sec. VI.
dyn
bad assumptions besides Eq. (62), but keeping the de- ≃
Alternatively, we can take the baryon d/f ratio or
composition Eq. (58), and also Eqs. (59–61))
structure-function integrals as [67]
m3dyn d f¯ 2f¯ +f¯
δmˆ = mˆ 68.3MeV, (66) u d s
con− mˆ2con ≃ f = −f¯u f¯s
−
with δmˆ → 0 when mˆcon → mdyn in the CL, as one = 3 3m2Σ−m2N −m2Λ−m2Ξ 0.28,
expects for the current quark mass. Note that δmˆ −5 m2 m2 ≈ −
≃ Ξ− N
6668.3MeMVeVofiSsecv.erIyI, nalesaoratsheexcpheicrtaeld-b!reaking σπN ≃ 63– ff¯¯d ≈ 0.64 , f¯s = 0 (near f¯s ≈ −0.057).(71)
Then for the q¯q pion, its mass is predicted from u
Eq. (66) as
In fact, the spectator-helicity rule [67] is slightly altered
to (using squared baryon masss as in Eq. (71), rather
m = δmˆ +δmˆ 136.6MeV, (67)
π ≃ than the f¯ =7.9, ... as found in Eq. 39 of Ref. [67])
u
midwaybetweentheobserved[1]m =134.98MeVand
π0 5
mπ± =139.57 MeV. h¯ = , f¯u 7.64, f¯d 4.85,
Instead,theChPTprediction(62)suggestingm 2 ≈ ≈
u,cur
o5fMReeVf.a[7n0d]mhadv,ciunrg∼fi9niMtee-vVo,luomrteakeffinegcttsh,edCohnPoTt spcrheedmic∼et ff¯¯d ≈ 0.635 , f¯u+f¯d ≈ 12.49 for f¯s = 0.(72)
u
a pion mass anywhere in the region of 140 MeV. The
QLLσM assumes standard pion q¯q states π+ = d¯u , Note that the latter ratio is compatible with f¯d/f¯u =
| i
π− = u¯d , π0 = u¯u d¯d /√2, and takes the 0.64 in Eq. (71), and the latter sum implies the Jaffe–
p|ioni as a| tiightly bound| q¯qi−Nambu-Goldstone “fu(cid:12)(cid:12)sed(cid:11)” Llewellyn-Smith form [74] for the current quark mass
(cid:12) (cid:11) (cid:0) | i (cid:12) (cid:11)(cid:1) (squared) is
nonstrange me(cid:12)son (verified via(cid:12)the QLLσM, VMD, and
measuredmesonchargeradiusinSec.III). Theobserved
m σ
protonchargeradius[1]R =(0.870 0.008)fmsuggests mˆ2 = N πN
the proton is a “touching”pquark-pyr±amiduud state [25] cur f¯u+f¯d
and there is minimal s¯s , either from data [16], phe- 938.9MeV 63MeV
h iN = × = (68.8MeV)2,(73)
nomenology [71], or from the magnetic-moment scheme 12.49
of Sec. VI.
very near the predicted effective current quark mass of
Instead, we return to studying the nonstrange quark
68.3 MeV in Eq. (66). Thus, we deduce in several in-
mass mˆ , recalling that mˆ = δmˆ 68.3 MeV in
cur cur
≈ dependent ways (Eqs. (68), (70), (73)) that mˆ
Eq.(66)(alsoseeRef.[72])successfullypredictsthepion cur ≃
62.4 MeV, 53 MeV, 68.8 MeV, all near δmˆ
mass at mπ =2δmˆ =136.6 MeV in Eq. (67). ∼ ∼ ≈
68.3 MeV from Eq. (66), but not near the ChPT value
Moreover, mˆ is sometimes called the current mass
cur
mˆ 5.5 MeV in Eq. (62).
via neutral PCAC (for a review, see Ref. [58]). Using cur ≈
Also, following Ref. [68], we have
quark structure functions, one predicts the pion mass
(squared) as
m2 m2 = f¯ (m2 mˆ2) = 0.5417GeV2 ,
Σ− N d s− cur (74)
m2π = 2mˆ2curh¯ , h¯ = 52 m2Ξ−m2N = f¯u(m2s−mˆ2)cur = 0.8556GeV2 .
⇒ mˆcur = m√5π ≈ 62.4MeV, (68) Tthheis(mle2adsmˆt2o) thescraaleti)o, nf¯eda/rf¯uthe≈va0l.u63e30.(6i4ndfoepunenddaebnotvoef.
s− cur
Moreover, if we sum the two equations in Eq. (74) and
awfrounhrldeemrFe[6πua7±rs]l.aG≈nMN[17Oo33t9R]e.s5[uf73rgo0gMm]eaesntVEed,qdsCit.nhhv(Pao6tkT8it)nhtgtaehkrtaeh.thme.hs2πs.epro∝eefcEtmmˆaqct2πsuo.rr∝(.-6hF8emˆu)libc2ccuiiatnrny,i u(t1hs.ee39fv¯u7a3l+u/e1f¯2dmˆ.4≈c9u)1r2G=.4e9V0f.20ro6≈m830E.1Gq1e.1V(972Gfr)oe,Vmw2e.EgqSe.utb((6smt6i2s)t−uytmiˆien2lgd)cstuhrthe≈ne
consistent ratios
behave as either mˆ or mˆ2 , depending on the renor-
cur cur
malization scale. At a 1 GeV scale we can write
m 0.1119
s
+1 = 5.00 (75)
m2π = hπ|HChSB|πi ∼ mˆcur hπ|u¯u+d¯d|πi , (69) (cid:16) mˆ (cid:17)cur ≈ s(0.0683)2
8
from (f¯ +f¯) above, pseudoscalar mesons (π(137), K(496), η(548), η′(958))
u d
that play the role of Goldstone bosons associated with
m 0.1125 ChSB. What is not so well-known and accepted is that
s
+1 = 5.01 (76)
mˆ cur ≈ s(0.0683)2 thereisalsoexperimentalevidence[1]foracorresponding
(cid:16) (cid:17) lightscalar-mesonnonet[7,77]f (600)(σ),K∗(800)(κ),
0 0
from (f¯u f¯d) above, f0(980),a0(985). Themembersofthelatternonet,being
− muchtoolighttobeaccomodatedasnaive(unquenched)
m 2m2 quark-model states, are rather to be interpreted as the
s = K 1 = 5.00 (77) chiral partners (see e.g. Ref. [24]) of the former pseu-
mˆ cur s m2π − doscalar mesons. In this spirit, it is useful to define the
(cid:16) (cid:17)
from the PCAC K-to-π ratio [72], for m2K =13m2π, ffloalvloowrinnognfieetsldomf sactarliacresanSd(xp)saeunddoPsc(axla),rfmoresUon(3s),×resUp(e3c)-
m 2m tively:
s K
= 1 = 6.21 (78)
(cid:16) mˆ (cid:17)cur mπ − σuu¯ a+0 κ+ ηuu¯ π+ K+
from the light cone [75], and S = a−0 σdd¯ κ0  , P = π− ηdd¯ K0  .
ms = 3∆m2N +1 = 4.93 (79)  κ− κ¯0 σss¯ K− K¯0 ηss¯ (82)
mˆ cur smNσπN For later convenience, we also define σ (σ +
(cid:16) (cid:17) n ≡ uu¯
sfrhoimft ([7sq2]ua∆remd2N) f=rom0.4t6heGbeVar2y,obne-oincgtetthSeUn(u3c)leovanlumeaossf ησdddd¯¯))//√√22,,aσn3d η≡3 ≡(σ(uηu¯uu¯−−σηddd¯d)¯/)/√√22≃≃πa000., ηn ≡ (ηuu¯ +
(1.158 GeV)2. The average ratio from Eqs. (75–79) The still somewhat controversialstatus of the lightest
membersoftheground-statescalar-mesonnonet,namely
is 5.23, which when combined with mˆ = 68.3 MeV
cur
the σ(600) and the κ(800), is due to both experimen-
implies m = 357 MeV, very near the analogue of
s,cur
Eq.(66),i.e.,m =m m3 /m2 =356MeV, tal and theoretical difficulties. On the one hand, the
s,cur s,con− dyn s,con amplitudes in elastic ππ and Kπ scattering rise very
for m = 486 MeV and m = 313 MeV. Note that
s,con dyn
slowly from threshold upwards, due to nearby Adler ze-
the ChPT ratio (m /mˆ) 25 scaled to mˆ 5.5
s cur cur
≈ ≈ ros[78,79]just belowthreshold. Onthe otherhand, the
MeV predicts m 137.5 MeV, leading to an incon-
s,cur
≈ theoretical description of light scalar-meson resonances
sistent picture of Nature.
Finally, we extend the ππ σ term σ m2 from turns out to be quite cumbersome, owing to the large
ππ ≈ π unitarizationeffects,whichdemandamanifestlynonper-
Eq. (1) to the three ChSB meson σ terms
turbative treatment.
1 1 In a field-theoretic framework, the most efficient and
σ , σ , σ =m2 1, , (80)
ππ KK η8η8 π 2 3 obvious approach to the light scalars is indubitably the
(cid:18) (cid:19)
QLLσM [23, 25, 27, 42], which contains from the outset
as found in Refs. [76]. Note that all three vanish in the — contrary to e.g. QCD — all the relevant (experimen-
CL, with m2π →0. To obtain the chiral-broken η8 mass, tally observable) degrees of freedom, i.e., quarks as well
we invoke the Gell-Mann–Okubo (GMO) value (see also asscalarandpseudoscalarmesons. Moreover,alreadyat
Eqs. (A23)) tree level important nonperturbative features of strong
interactions are included. The interaction Lagrangianof
mGMO = 4m2K −m2π √17m 563MeV, (anti)quarksand(pseudo)scalarmesonsintheQLLσMis
η8 r 3 ≈ π ≈ given by int(x) =√2gq¯(x)(S(x)+iP(x)γ5)q(x), with
(81) L
g the strong coupling constant given by g 2π/√3,
using m2 13m2, and the average ChSB pion mass | | ≃
K ≈ π as follows from one-loop dynamical generation [23, 80].
136.6 MeV from Eq. (67). Since the observed [1] m is
η Byintegratingout(anti)quarks,theQLLσMLagrangian
547.75 MeV, mGMO 563 MeV from Eq. (81) is about
η8 ≈ can be converted into an effective U(3) U(3) LσM
2.8%greaterthanm ,similartothe3%GTRdiscrepan- ×
η meson Lagrangian displaying strong similarity with the
cies in Sec. V, but much lower than the 6.7–8.2% ChPT
well-known “traditional” U(3) U(3) LσM Lagrangian
predictions in Eq.(43). Moreoever,folding the GMO re- ×
[21, 22, 27, 81, 82], briefly summarized in Appendix A.
lationintothethreemesonσ termsofEq.(80),onefinds
TheQLLσM(without(axial-)vectormesons)intheCL
(38σ1η)8,η8g−iv4inσgKaKc+onσsπiπst=en0t,paasttaenrtnic(i3p/a3te)din(4E/2q)s.+(810=)a0n.d (m2π = m2K = 0) predicts [6, 7, 27] the following NJL-
− like [20] mass relations and chiral-limiting scalar-meson
masses:
VIII. GROUND-STATE SCALAR q¯q NONET (mCL)2 = (2mˆ )2 (626MeV)2 , (83)
σn CL ≃
(mCL)2 = (2mCL)2 (901MeV)2 , (84)
It is well-known and commonly accepted that there σs s ≃
exists (within a U(3) U(3) flavor scheme) a nonet of (mCL)2 = (2mˆ )(2mCL) (751MeV)2 , (85)
× κ CL s ≃
9
withmˆ =313MeV,mCL =1.44 mˆ =450.72MeV. As for the I =1/2 scalar κ meson, seemingly doomed
CL s × CL
Beyond the CL, the leading contributions to the scalar- not so long ago [85], despite our old theoretical predic-
mesonmassesdue toquarkloopswillleadtoexpressions tions [7, 77], it was first experimentally rehabilitated by
equivalent to Eqs. (83–85), but now with non-CL quark the E791 collaboration [86], providing a mass value of
masses, i.e. (see e.g. Ref. [25]),
m = (797 19)MeV. (91)
κ
m2 (2mˆ)2 = (675MeV)2 , (86) ±
σn ≃ This is to be compared to 810 MeV from Eq. (88),
m2 (2m )2 = (972MeV)2 , (87)
σs ≃ s 800 MeV from Refs. [7, 87], as well as the pole posi-
m2 (2mˆ)(2m ) = (810MeV)2 , (88) tions (727 i 263) MeV [77] and (714 i 228) MeV
κ ≃ s − × − ×
[88]. Notethatthelattertwocoupled-channelmodelcal-
withmˆ =337.5MeV,m =1.44 mˆ =486.0MeV,which
s culationsbothgiveapeakcrosssectionaround800MeV.
×
predictionslieimpressivelyclosetotheexperimentalval-
Ontheotherhand,averyrecentanalysis[89]ofKπdata
ues.1
from different high-statistics experiments yields the pole
A realistic nonstrange-strange mixing angle of φ
s position
18◦ [84]implies,withthehelpofEqs.(A16)and(A17≃),
±
the following values for mσ and mf0: mκ = (760±20±40)−i×(420±45±60)MeV. (92)
1 m2 m2 As a final remark on the κ, we should stress the impor-
m2 = m2 +m2 σs − σn
σ 2 σs σn − cos(2φ ) tanceofunitarization[82,90]effects,whicharecapableof
(cid:18) S (cid:19) e.g.shiftingtheLagrangianmassparameterinEq.(A13)
= (630.8MeV)2 , (89)
from1128MeVtoapolevalueof 700–800MeV,besides
∼
1 m2 m2 dynamically generating extra states [77, 88, 91]. How-
m2 = m2 +m2 + σs − σn ever, in the latter reference, the authors introduce by
f0 2 σs σn cos(2φ )
(cid:18) S (cid:19) hand a rather unphysical Adler zero, which apparently
= (1001.3MeV)2 . (90) moves the dynamically generated κ pole far away from
the physical region [79].
Of course, the latter mass predictions are still subject,
Next we discuss the f (980). The mass of (980 10)
in principle, to further corrections stemming from non- 0 ±
MeV listed by the PDG [1] is entirely compatible with
zero pseudoscalar-meson masses, as can be seen from
theabovevaluesof972MeVand1001.3inEqs.(87)and
Eqs. (A9–A12) predicted by the “traditional” LσM ap-
(90), respectively. So are the f (980) pole positions of
proach discussed in the Appendix. However, as noticed 0
(998 4) i (17 4) MeV and (994 i 14) MeV
there, several uncertainties persist within such a formal- { ± − × ± } − ×
given in the very recent analysis of Ref. [89].
ism, which make it hard to establish a one-to-one corre-
Concerningthea (985),thecoupled-channelapproach
spondence with the more straightforwardQLLσM. ofRef.[77],whichin0cludesthe crucialηπ andKK¯ chan-
Comparing now with the experimental scalar-meson
nels, predicts a reasonable pole mass of (968 i 28)
masses,weseethatthenon-CLQLLσMresultsdoavery − ×
MeV, to be compared to (1036 i 84) MeV in the
goodjob. Startingwiththeσ meson,the PDGtables [1] − ×
experimental analysis in Ref. [89]. As already noted in
very cautiously list the f (600) with an extremely wide
0 Ref. [25], we consider the approximate mass degeneracy
mass range of 400–1200MeV, but giving a large number
ofthea (985)andf (980)purelyaccidental,eventhough
of recent experimental findings or analyses tending to 0 0
thetheoreticalsituationinaQLLσM/LσMframeworkis
convergearounda typicalmassvalue ofabout600MeV.
still unsettled, due to the apparent relevance of meson
Alsotheoreticalcoupled-channelmodelsreproducingthe
loops [25]. Namely, were we to neglect meson contri-
correspondingS-waveππ phaseshiftsandaccountingfor
butions to the a (985) mass in Eq. (A9), then it would
a low-lying σ pole lead to similar predictions. For in- 0
be approximately degenerate with the σ . On the other
stance, the unitarized model of Ref. [77] found a pole at n
hand, if we simply evaluate Eq. (A9) with mˆ = 337.5
(470 i 208)MeV,withapeak massaround600MeV.
More−re×cently, the ππ ππ,KK¯ analysis in Ref. [28] MeV, ξ =1, andφP =(41.2±1.1)◦ [82, 92], we obtaina
→ surprisingly accurate mass prediction of 1012 MeV. The
found an even broader σ pole, with a real part of 570–
more realistic coupling ratio ξ = 0.9064 (see our discus-
600 MeV, referring to the σ as the “f (665)”. Although
0 sionintheAppendix)wouldevenimplym =990MeV.
theprecisemass tobeattributedtoabroadresonanceis a0
Analternative,moreempiricalwayto estimatescalar-
alwaysmodel-dependent,thelatterresultsarefullycom-
mesonmasses is by usingquadraticmass differences mo-
patible with the value 2mˆ 675MeV ofEq.(86), orthe
≃ tivated by the IMF. The resulting equal-splitting laws
more realistic value 630.8 MeV of Eq. (89).
have a remarkable predictive power (see e.g. Ref. [25],
Appendix C), being fully compatible with the foregoing
QLLσM predictions.
While these σ (675), κ(810), σ (972), a (985) q¯q
1 The PDG [1] further states that the nonstrange process n s 0
scalars are in agreement, to a very good approxima-
a1(1260) → σπ is the dominant mode, and that a1(1260) →
f0(980)π is not seen, presumably because the f0(980) is (ap- tion, with the QLLσM theory, even without unitariza-
proximately)ascalars¯sstate[83,84]. tion, one-loop (non-unitarized) ChPT is not compatible
10
withaspecificLσM,asshowninAppendixBofRef.[38]. for s = 4m2, t = u = 0, ε = m2/m2 0.045. Equa-
π π σ ≃
Therefore, non-unitarized ChPT is at odds with the en- tion(97)iscompatiblewiththerecentlymeasured(K )
e4
tireground-statescalarq¯q nonet2,assummarizedinthis E865 data [94]
section, if indeed the LσM of Ref. [38] possesses similar
properties as the QLLσM discussed above. a00 = 0.216±0.013m−π1 . (98)
However two-loop ChPT in Refs. [49, 51] predicts
a = 0.22m−1 or a = 0.23m−1, using Roy equations
IX. I =0 ππ SCATTERING LENGTH 00 π 00 π
insteadoftheLσM.ButwesuggestEqs.(93–97)tellthe
wholeLσMstory(andChPTwithverymanyparameters
UsingtheLσM,PCAC,andcrossingsymmetry,Wein- adds nothing new). We would like to stress that — even
berg originally predicted [33] for the I = 0, J = 0 ππ thoughEq.(97)hasbeenderivedwithouttakingintoac-
scattering length count t-channel vector-meson-exchange contributions —
the result will change at most very slightly when includ-
7 m
a = π 0.159m−1 , (93) ing such contributions, as they are “chirally shielded” in
00 32π f2 ≃ π I = 0 S-wave ππ scattering close to the ππ threshold,
π
due to the presence of an extra contact term [95].
for mπ+ =139.57 MeV, fπ 92.42 MeV [1]. It is worth pointing out that Weinberg’s result
≃
Stated slightly differently, and following V. de Alfaro Eq. (93), amounting to leading-order non-unitarized
et al. (see Ref. [19]), when working at the soft point s= ChPT,is obtainedfromthe LσM resultEq.(97)by tak-
m2, one sees that the net ππ amplitude “miraculously ing the limit m . This limit essentially implies a
π σ
→ ∞
vanishes” (via chiral symmetry), i.e., nonlinear-σ-model framework, which unfortunately has
beendominatingtheoreticaldescriptionsofππscattering
Mnet = Mcontact+Mσ-pole during the past three decades, despite its technical com-
ππ ππ ππ
= λ+2g2 (m2 m2) = 0, (94) plications, deviations from experimental data, and the
σππ π − σ mounting experimental evidence for a light scalar-meson
since g = (m2 m2)/(2f ) = λf from the LσM nonet.
σππ σ − π π π
Lagrangian. Thus, the contact term “chirally eats” the
σ pole at s=m2, yielding the famous amplitude “Adler
π X. γγ → π0π0
zero”slightlybelowtheππ threshold[78]. Crossingsym-
metry then extends Eq. (94) to [93]
Assuming that the S-wave process γγ π0π0 pro-
Mabcd(s,t,u) = A(s,t,u)δabδcd ceedsviaanintermediateσ resonance(γγ →σ π0π0),
ππ → →
+ B(s,t,u)δacδbd with mσ ≃700 MeV, Ref. [96] anticipated that the cross
section would be very small (<10 nb). In fact, this was
+ C(s,t,u)δadδbc , (95)
later confirmed [97] with Crystal-Ball data. Then, in
1991, Ref. [98] studied a π(ππ) , and used the
1 S-wave
with →
Dirac identity
ALσM(s,t,u) = 2λ 1 2λfπ2 1 2mγ 1 = γ 1 1 γ (99)
− (cid:18) − m2σ−s(cid:19) 6p−m 5 6p−m − 5 6p−m − 6p−m 5
m2 m2 s m2
= σ− π − π , (96) to verify that the sum of the quark-box and -triangle
m2 s f2
σ− π diagrams vanishes. The same holds for γγ σ π0π0
→ →
[99]. The amplitude for the latter process would even
sothattheI =0 S-waveamplitude3A+B+C generates
vanish exactly in the CL. In the QLLσM, this “chiral
a 23% enhancement of Eq. (93):
shielding”,alreadymentionedabove,hasbeenmanifestly
built in.
7+ε m
a00|LσM = 1 4ε32ππf2 +O(ε2) However,alreadyintheabstractofthe1993ChPTpa-
− π per Ref. [100], it was stated that “the one-loop [ChPT]
7 m
1.23 π 0.20m−1 , (97) computation does not fit data, even close to threshold,
≃ 32π fπ2 ≃ π because unitarity effects areimportant, evenat verylow
energies”. In effect, the latter authors were arguing to
unitarize ChPT, which later turned out [14, 15] to rein-
statethebroadσpole,andpresumablyevenrecoversthe
2 Very recently [15], Roy equations have been used to extract a QLLσM theory. But then the nonperturbative effects of
σ polefromlow-energyππ scattering data, intheframeworkof
the σ meson with a finite mass would have to be taken
ChPT. However, the given pole position, especially the claimed
into account, too, in contrastwith non-unitarized ChPT
verysmallerroronit,ishighlyquestionableinviewoftheneglect
oftheKK¯ channel. briefly discussed at the end of Sec. IX.