U(1)_A Topological Susceptibility and its Slope, Pseudoscalar Gluonium and the Spin of the Proton

0.12 MB·English

by  Stephan Narison

U(1)ATOPOLOGICALSUSCEPTIBILITYANDITSSLOPE,
a
PSEUDOSCALARGLUONIUMANDTHESPINOFTHEPROTON .
StephanNARISON
LaboratoiredePhysiqueThe´oriqueetAstrophysiques,
UM2,PlaceEuge`neBataillon,34095-MontpellierCedex05-FR.
6 E-mail:[email protected]
0
0
We review the determinations of the pseudoscalar glueball and eventual radial excitation of the η
2 masses and decay constants from QCD spectral sum rules (QSSR).The glueball mass is(2.05±
n 0.19)whichonecancomparewiththeeventualexperimentalcandidateX(1835),whiletheη(1400)
a is likely a radial excitation of the η′-meson. Their effects on the estimates of U(1)A topological
J susceptibilityanditsslopeaswellastheimpactofthelatterintheestimateofthespinoftheproton
isdiscussed. Wepredictthesingletpolarizedpartondistributionstobea0(Q2 = 4GeV2)=0.32±
0
0.02, whichisaboutafactortwosmallerthantheOZIvalue, butcomparablewiththeCOMPASS
1 measurementof0.24±0.02.

1
v
6 Prologue
6
0 It is a great honour and pleasure for me to write this review for Professor Adriano Di Gi-
1 acomo for celebrating his 70th birthday. My scientific connection with Adriano started
0 when I studied at CERNin 82, the SVZ-expansion of QSSR1,2, and where the Pisa group
6
3 has presented the first quenched lattice results values of the gluon condensates α G2
0 s
h i
/ and g3f GaGbGc at the 1st Montpellier Conference on Non-pertubative methods in
h abch i
85. About the same time, my previous interest4 on the topological susceptibility χ(0) of
p
- the U(1) anomaly5 and its slope χ′(0) continued after several discussions with Gabriele
A
p
Veneziano during my stay at CERN . A program on the estimate of these quantities using
e
QSSR has started7. In parallel, the Pisa group has runned their lattice simulations, and,
h
: for his 2nd participation at the Montpellier conference (QCD 90), Adriano has presented
v thelatticePisagroup8 results forχ(0)andχ′(0)whichconfirmedourprevious resultsin7.
i
X Since then, there was a common interest between the Pisa group and Montpellier on these
non-perturbativeaspectsofQCD.Morerecently,thesecommoninterestsconcerntheproton
r
a spin problem9,10,11,theestimate ofthegluon condensate using QSSR12 andlattice13,and
theestimateofthequark-gluon mixedcondensates usingQSSR14andfieldcorrelators15.
In 94, Adriano joined the international organizing committee of the QCD-Montpellier Se-
ries of Conferences and, in 2001, the one of the Madagascar High-Energy Physics (HEP-
MAD)SeriesofConferences. Heisamongthefewcommitteememberswhosendregularly
speakers andparticipate continuously totheseconferences. Theorganization teamofthese
meetingshasalwaysappreciated hishumankindness andfriendship.
1 Introduction
The U(1) anomaly is one the most fundamental and fascinating problem of QCD,which
A
has been solved in the large N limit by ’t Hooft-Witten-Veneziano5,6, where Veneziano
c
aThisreviewiswritteninhonourofProfesssorAdrianoDiGiacomoforhis70thbirthday
(Adrianofest26-27thJanuary2006,Pisa,Italy)
found a solution without the uses of instantons. The U(1) topological susceptibility is
A
definedas:
χ(0) ψ (0) i d4x Q(x)Q†(0) , (1)
gg
≡ ≡ hT i
Z
where:
1
Q(x) = α GµνG˜a , (2)
8π s a µν
istheU(1) anomalyofthesingletaxial-current:
A
∂µA (x) = J 2m ψ¯ (iγ )ψ +2n Q(x). (3)
µ q q q 5 q f
≡
q=Xu,d,sh i
where n is the number of light quark flavours, ψ the quark fields, m the current quark
f q q
massand: G˜a (1/2)ǫ Gρσ .Thetopological chargeisdefinedas:
µν µνρσ a
≡
Q = d4xQ(x), (4)
Z
whichisanintegerforclassicalfieldconfigurations (2ndChernnumber). Argumentsbased
onlargeN forSU(N ) SU(n )leadto6 :
c c f
×
f2
χ(0) = 2nπ Mη2′ +Mη2−2MK2 . (5)
f
(cid:16) (cid:17)
where,f = (92.42 0.26)MeVb .Forn = 3:
π f
±
χ(0) (180MeV)4 . (6)
≃
GabrieleVenezianoand,lateron,GiancarloRossihavechallengedustoanalyzethevalidity
ofthisapproximation atfiniteN = 3,bycomputing χ(0)andχ′(0)usingQSSR,because
c
the previous result is obtained at q2 = M2 but not at q2 = 0, where, they have used the
η′
expansion:
χ(q2) χ(0)+q2χ′(0)+..., (7)
≃
2 Thepseudoscalargluonium/glueballmassfromQSSR
QSSRhasbeenusedtocalculatethetopological chargeinpureYang-MillsQCD.Insodo-
ing,onehasfirstextractedthemassanddecayconstantofthepseudoscalarglueball/gluonium
usingeitheranumericalfittingprocedure4,7 orsumruleoptimization criteria17,18 fromthe
unsubtracted sumrules(USR):
tc d
(τ) = dte−tτImχ(t) and (τ) = log (τ), (8)
L R −dτ L
Z0
bAgeneralizationoftheresultincludingSU(3)breakingsforthedecayconstantscanbefoundin16.
wheretheQCDmodelcomingfromthediscontinuity oftheQCDgraphshasbeenusedfor
thecontinuum. Theanalysis gives:
M = (2.05 0.19)GeV and f = (8 17)MeV, (9)
G G
± −
corresponding tot =(6 7)GeV2,andwheref isnormalized as:
c G
−
0Q(x)G = √2f M2 , (10)
G G
h | | i
i.e. like f /(2n ) with f = (92.42 0.26)(92.42 0.26) MeV. The positivity of the
π f π
± ±
spectral function gives:
M (2.34 0.42)GeV. (11)
G
≤ ±
Onecancomparetheprevious valueswiththequenched latticeresults19:
M = (2.1 2.5)GeV. (12)
G
−
Recent sum rule analysis using instanton liquid model leads to similar results: M 2.2
G
GeV and f 17 MeV in our normalization20, though the same approach leads to≃some
G
inconsistencie≃s in the scalar channel21. One should note that in the previous analysis17,18
and20, the value of the gluonium decay constant is smaller than the corresponding value
of the η′ one which is about 24 MeV in the chiral limit4,7,9 (see section 5.2). In a recent
paper22,themassoftheeventual experimental pseudoscalar gluonium candidate X(1835)
has been used asinput and the Laplace sum rule has been exploited forfixing the corre-
L
sponding decayconstant. Though(apriori)self-consistent, thisanalysis islessconstrained
than the previous sum rules used in21,20, where the two sum rules and have been si-
L R
multaneously used for constraining the gluonium mass and decay constant. The resulting
value of the gluonium decay constant is about (8-12) MeV22 and agrees with the previ-
ous values. On the other, a G-η mixing due to direct instanton has been considered in24
c
for pushing the unmixed gluonium mass of about 2.1 GeV down to 1.8 GeV with a mix-
ing angle of about 170.This value is comparable with the early value of about θ = 120
P
of the meson-gluonium mixing angle using the OPE of the off-diagonal light quark-gluon
correlator:
i
ψ (q2) d4xeiqx Q(x)J†(0) , (13)
gq ≡ 2n hT q i
f Z
which is proportional to m 25. Indeed, it is also plausible that the X(1845) comes from a
s
complicatedmixingoftheglueballwiththeη(1440)andη . Aconfirmationofthegluonium
c
natureoftheX(1835)requires somemoreindependent tests.
3 Onthenaturesoftheη(1295) andη(1400) fromQSSR
There are twoother experimental candidates which arethe η(1295) and η(1400)23, which
one can intuitively interpret as the first radial excitations of the η(547) and η′(958). Here,
weshalltestiftheη(1400)satisfiesthisinterpretation. Usingthesameapproach,thenature
of the η(1400)23 has been tested by measuring its coupling to the U(1) singlet current
A
Q(x)inthechirallimit. Inthiscase,wecanworkwiththeSSR:
tc dt 1
e−tτ Imχ(t), (14)
t π
Z0
asoneexpectsfromlargeN argumentsthatthetopologicalchargeχ(0)vanishesduetothe
c
θ-independence oftheQCDLagrangian, thankstothepresenceofthesingletη . Including
1
in the sum rule, the contributions of the η′, η(1400), G and the QCD continuum with t
c
fixed previously, one cannot find any room to put the η(1400), i.e. f 0. Relaxing the
η
≈
constraint byreplacing theQCDcontinuum withtheη(1400), onecandeduce:
fη(1400) 16MeV fη′ (24.1 3.5)MeV, (15)
≤ ≪ ≃ ±
which should be a weak upper bound within this assumption. This feature indicates that
the η(1400) is likely the first radial excitation of the η′, as intuitively expected, while the
glueball hasahighermass.
4 TheU(1)Atopological susceptibilityfromQSSR
4.1 ThepureYang-Millsresult
Applying theBorel/Laplace operator tothesubtracted quantity:
χ(q2) χ(0)
noquarks
− | , (16)
q2
one can derive acombination of the unsubtracted sum rule (USR)and subtracted sum rule
(SSR)forthetopological suceptibility inpureYang-Millswithoutquarks4,7:
tc dt tτ 1
χ(0) = e−tτ 1 Imχ(t)
noquarks
| t − 2 π −
Z0 (cid:18) (cid:19)
α 2 2 1
s τ−2 +2π2 G2 +6π2g G3 τ3 . (17)
8π π2 logτΛ2 h i h i
(cid:18) (cid:19) (cid:26) (cid:27)
The value of the gluon α G2 = (0.07 0.01) GeV2 from e+e− data and heavy-quark
s
h i ±
mass-splitting12,2 has been confirmed by lattice calculations with dynamical fermions13,
while the value of the triple gluon condensate g3f GaGbGc is about 1.5 GeV2 α G2
abc s
h i h i
from instanton model26 and lattice calculations3. The previous combination of sum rules
is more interesting than the alone SSR, as the effect of the QCD continuum is minimized
here. Atthesumruleoptimization scaleofτ about0.5GeV−2,itleadstothevalue4,7:
α 2
χ(0) [(106 122)MeV]4 4 s G2 , (18)
noquarks
| ≃ − − ≈ − 8π h i
(cid:18) (cid:19)
indicating the role of the gluon condensate in the determination of χ(0). The sign and the
size of this result, though inaccurate are in agreement with the large N results obtained
c
previously. Inpuregauge,latticegivesthevalue8,27,28:
χ(0) [(167 25)MeV]4 , χ(0) [(191 5)MeV]4 , (19)
SU(2) SU(3)
| ≃ − ± | ≃− ±
whichonecancomparewiththetwoformeronesfromlargeN andfromthesumrules.
c
4.2 Resultinthepresence ofquarks
In this case the analysis is more involved as one also has to consider the diagonal quark-
quarkcorrelatorc:
i
ψ (q2) d4xeiqx J (x)J†(0) , (20)
qq ≡ (2n )2 hT q q i
f Z
andoff-diagonal quark-gluon correlator inEq. (13). Then,thefullcorrelator reads:
ψ (q2)= ψ (q2)+2ψ (q2)+ψ (q2). (21)
5 gg gq qq
Inthiscase,ψ (0)isnotvanishingandcanbededucedfromchiralWardidentitiestobe6,10:
5
4
ψ (0) = m q¯q , (22)
5 −(2n )2 qh i
f
q=u,d,s
X
Therefore,thetopologicalchargeχ(0)vanishesinthechirallimitduetotheθindependence
of the QCD Lagrangian. Lattice calculations in full QCD for two degenerate dynamical
fermionsfind29:
ψ (0) = [(163 6)MeV]4 , (23)
5
±
in agreement with the large N result5,6 but does not have enough accuracy for checking
c
thelinearm -dependence expectedfromcurrentalgebra.
q
Inthis case ofmassivequarks, weinclude SU(3) breakings tothesum ruleinorder tosee
theeffectofthephysicalη′. IncludingboththegluoniumandQCDcontinuumcontribution
totheappropriatesumrule,onecanseefrom4,7 thattheircontributions tendtocompensate,
andthengivetheapproximate numerical formulatoleadingorder:
2Mη2′fη2′e−Mη2′τ 1− M2η2′τ! ≃ ψ5(0)+ 4π32 2mnsf!2τ−1 . (24)
Usingthequarkmodelprediction:
1
fη′ √3fπ 27MeV, (25)
≃ 2n ≃
f
the physical η′ mass and the value m (τ) 100 MeV30, we can deduce at the sum rule
s
≃
optimization scaleτ 0.5GeV−2:
≃
ψ (0) [157MeV]4 , (26)
5
≃
in good agreement with the large N and lattice results, then showing the consistency of
c
the sum rule approach. Armed with these consistency checks, we shall now evaluate with
confidence theslopeofthetopological susceptibility χ′(0).
cThese correlatorsinclude intheir definitions the quark mass through J , whilein10, this quark mass is
q
factorizedout.
′
5 Theslopeχ (0)ofthetopological susceptibilityfromQSSR
Using the q2 expansion in Eq. (7), one can derive a sum rule for χ′(0) by applying, the
Laplacesumruleoperator, tothetwicesubtracted quantity:
χ(q2) χ(0) q2χ′(0) dt 1 1
− − = Imχ(t), (27)
(q2)2 t t q2 iǫπ
Z − −
5.1 ThepureYang-Millscase
Oneobtainsinthiscase:
∞ dt 1
e−tτ Imχ(t) = (τ) χ(0)τ +χ′(0), (28)
t2 π F1 −
Z0
where (τ) comes from the OPE expression of the correlator χ(q2)4. Eliminating χ(0)
1
F
byusingEq.(17),onecandeducethesumrule7:
∞ dt tτ 1 α 2 2
e−tτ 1 tτ 1 Imχ(t) χ′(0)+ s τ−1
t2 − − 2 π ≃ 8π π2 ×
Z0 h (cid:18) (cid:19) i (cid:18) (cid:19)
α β
1+ s 10 2γ β 2 2 log logτΛ2 +4π2g G3 τ3 , (29)
E 1
π − − β − h i
(cid:26) (cid:18) (cid:19)h 1 (cid:16) (cid:17)i (cid:27)
whereβ = 11/2andβ = 51/4forpuregaugeSU(3) . Usingτ-andt -stability cri-
1 2 c c
− −
teria,andsaturatingthespectralfunctionbythepseudoscalarglueballandQCDcontinuum,
onecandeduce atτ 0.5GeV−27:
≃
χ′(0) [7 3)MeV]2 , (30)
noquarks
| ≃ − ±
indicating that: χ′(0)M2 χ(0),whichjustifiestheaccuracyofthelargeN result. This
η′ ≪ c
valueandthesignhasbeenconfirmedbylatticecalculations8:
χ′(0) [(9.8 0.9)MeV]2 . (31)
quenched
| ≃ − ±
5.2 Resultinthepresence ofmasslessquarks
Inthiscase,quarkloopsenterintothevalueoftheβ functionandoftheQCDscaleΛ. The
η′ entersnowintothespectralfunction. Itscouplingtothegluoniccurrentcanbeestimated
fromtheSSR:
tc dt tτ 1
e−tτ 1 Imχ(t), (32)
t − 2 π
Z0 (cid:18) (cid:19)
takingintoaccountthefactthatinthechirallimitχ(0) = 0. Therefore, oneobtains9:
fη′ τ 0.5GeV−2 (24.1 3.5)MeV, (33)
≃ ≃ ±
(cid:16) (cid:17)
wherethedecayconstant hasaweakτ-dependence because ofrenormalization:
8
fη′(τ) ≃ fˆη′exp β2logτΛ2 , (34)
(cid:18)− 1 (cid:19)
where fˆη′ is RG invariant. This result can be compared with the quark model prediction
in Eq. (25). Using the Laplace and FESR versions of the twice subtracted sum rule, one
finds9:
χ′(0) (τ) [(26.5 3.1)MeV]2 . (35)
massless
| ≃ ±
The result has changedqsign compared to the one from pure Yang-Mills, while its absolute
value is about 12 times higher. The main effect is due to the η′ which gives the dominant
contribution tothe spectral function compared tothe gluonium and QCDcontinuum. That
canbeunderstoodbecauseofitslowmassandofthefactthatitsdecayconstantfη′ islarger
thanthegluonium decayconstantf .
G
Similarestimateofψ (0)inthechirallimitforthesingletchannelhasbeendonein31,where
5
thevalueofχ(0)′ issignificantlylargerbyaboutafactor2.5thanours. However,wefound
some inconsistencies in the two sides of the SR:in the experimental side the contributions
of the singlet and octet mesons have been considered and the phenomenological value of
the decays constants, masses and mixing angles have been used; while in the QCD side,
only the correlator associated to the singlet current Q(x), or to massless quark has been
accounted for.
Ontheother,somecriticismsraisedinaseriesofpaper32,duetodirectinstantonbreakingof
theOPE,donotapplyinourcase: ourdifferentsumrulesoptimizeatalargescaleτ 0.5
GeV−2, where this contribution become irrelevant like other high-dimension conden≤sates.
In20,ithasbeenarguedthatscreeningcorrections cancelthedirectinstanton contributions,
indicating that these effects are not yet well understood. However, in your approach, the
introduction of thenew 1/q2-term in theOPE,which isalso negligible, isanalternative to
thedirectinstanton contributions33,34. Someanswerstothesecriticisms havebeenalready
explicitly givenin10.
5.3 Resultinthepresence ofmassivequarks
As can be seen from the expression of the two-point correlator ψ (q2) in Eq. (21), the
5
analysis is more involved. In this case, its slope has been estimated from the substracted
Laplacesumrules. Atthestability pointτ (0.2 0.4)GeV−2 ,onefinds10:
≃ −
ψ′(0) = (33.5 3.9)MeV, (36)
5 ±
q
whiletheη′-decayconstantbecomes:
fη′(τ) (27.4 3.7)MeV, (37)
≃ ±
compared with the results for massless quarks in Eqs. (35) and (33) and agrees quite well
withthequarkmodelprediction inEq. (25). OnecannoticethattheSU(3)breakingeffect
is about 10% and 20% respectively showing a smooth dependence on the strange quark
mass as expected. Similar analysis can be done in the flavour non-singlet case by working
withthecorrelator associated totheη-mesoncurrent:
0∂µJ8 η = f M2 , (38)
h | µ5| i η η
where in the SU(3) limit and in this normalization without 1/2n , f = f = (92.42
f η π
±
0.26)MeV.Therefore, oneobtains10:
ψ′88(0) = (43.8 5.0)MeV, and f /f = 1.37 0.16. (39)
5 ± η π ±
q
Instead of the value offexp = (92.42 0.26) MeV,we have used the sum rule prediction
π
±
10:
f = (107 12)MeV, (40)
π
±
foraself-consistency ofthewholeresults. ThevalueoftheSU(3)breaking ratio f /f is
η π
inlinewithf /f =1.2,whereweexpectbiggereffectsfortheη thanfortheK.
K π
6 ApplicationsoftheQSSRresultstotheprotonspin
The previous results have been applied to the proton spin problem, where one expects that
the gluon content of the proton is due to the U(1) anomaly. This property can be made
A
explicitintheapproachofDISwherethematrixelementsfromtheOPEarefactorised into
composite operators and proper vertex functions35. In the case of polarised µp scattering,
the composite operator can be identified with the slope χ′(0) of the topological suscepti-
bility, which is an universal quantity and then target independent, while the corresponding
propervertexisrenormalisationgroupinvariant. Thefirstmomentofthepolarisedstructure
function reads,intermsoftheaxialcharges oftheproton:
1 1 1 1
Γp(Q2) dxgp(x,Q2)= CNS(α (Q2)) a3+ a8 + CS(α (Q2))a0(Q2),
1 ≡ 1 12 1 s 3 9 1 s
Z0 (cid:18) (cid:19)
(41)
wheretheWilsoncoefficientsarisefromtheOPEofthetwoelectromagnetic currents:
1
CNS = 1 a 3.583a2 20.215a3 , CS = 1 a 0.550a2 , (42)
1 − s− s − s 1 − 3 s− s
wherea α /π. Theaxialcharge aredefinedfromtheforwardmatrixelementsas:
s s
≡
1 1
p,s J3 p,s = a3s , p,s J8 p,s = a8s , p,s J0 p,s = a0(Q2)s ,
h | µ5| i 2 µ h | µ5| i 2√3 µ h | µ5| i µ
(43)
where Ja are the axial currents and s the proton polarisation vector. Using QCD parton
µ5 µ
model,theaxialcharges read,intermsofmomentsofpartondistributions36:
n
a3 = ∆u ∆d, a8 = ∆u+∆d 2∆s, a0 = [∆Σ ∆u+∆d+∆s] fa ∆g(Q2).
s
− − ≡ − 2
(44)
a3 anda8 areknownintermsoftheF andDcoefficients frombetaandhyperon decays:
a3 = F +D and a8 = 3F D, (45)
−
where37:
F +D = 1.257 0.008 and F/D = 0.575 0.016, (46)
± ±
so that an experimental determination of the first moment of gp in polarised deep inelastic
1
scattering (DIS) allows a determination of the singlet axial charge a0(Q2). In the na¨ıve
quark or valence quark, parton model, one expects that ∆s = ∆g = 0, and then a0 = a8,
which is the OZI prediction. The proton spin problem is that the experimental value of
a (Q2) is much smaller than a8, which would be its expected value if the OZI rule were
0
exactinthischannel. ThefirstSMCdatafound39,40 :
a0(Q2 = 10GeV2) = 0.19 0.17 and a0(Q2 = 5GeV2)= 0.19 0.06, (47)
± ±
whichareconfirmedandimprovedbytherecentCOMPASSdataatQ2 = 4GeV241:
a0(Q2 = 4GeV2)= 0.24 0.02. (48)
±
TheseresultsaremuchsmallerthantheOZIprediction (∆g = 0):
∆Σ = 3F D = 0.579 0.021. (49)
OZI
| − ±
Inserted into Eq. (41), the OZI results lead to the Ellis-Jaffe sum rule38. It has been
conjecturedin36thatthesuppressionofa0withrespecttoa8isduetothegluondistribution.
However,theSMCmeasurement of∆Σgivesavalue40:
∆Σ 0.38 0.04, (50)
SMC
| ≃ ±
indicatingthatindependentlyof∆g,thereisalsoalargedifferencebetweentheOZIpredic-
tionandthedatad. Inthefollowing, wewillnottrytosolvethisdiscrepancy butwillshow
ourprediction fora0,wherethesumofthequarkandgluoncomponents areconcerned.
6.1 Predictioninthechirallimit
Using the composite operator proper vertex functions approach9, one can write the de-
⊕
composition, inthechirallimit:
1 1
Γp1singlet = 92M 2nfC1S(αs(Q2)) χ′(0)|Q2Γˆη0NN . (51)
N
q
Thekeyassumption isthatthevertexiswellapproximated byitsOZIvalue:
Γˆη0NN = √2Γˆη8NN , (52)
while all OZIviolation in Γp iscontained inside χ′(0). Comparing the result with
1singlet
theOZIprediction ofa8,onecandeduce:
p
a0(Q2) √6
a8 = f χ′(0)|Q2 = 0.60±0.12, (53)
π
q
whichgivesouroriginal protonspinsumrule9:
a0(Q2 = 10GeV2)= 0.35 0.05 = Γp(Q2 = 10GeV2)= 0.143 0.005, (54)
± ⇒ 1 ±
wherewehaveused:
χ′(0)|Q2=10GeV2 = (23.2±2.4)MeV, (55)
q
afterrunningtheresultinEq. (35)from2to10GeV2. Theseresultscanbecomparedwith
theOZIvalueinEq. (49)andagreewiththelastSMCdataatQ2 = 10GeV240:
Γp(Q2)= 0.145 0.008 0.011 = a0(Q2)= 0.37 0.07 0.10, (56)
1 ± ± ⇒ ± ±
dIn42,anestimateofthescalarseaquarkcontentofthenucleonindicatesthatitissuppressedcomparedto
thevalenceone.Similarresultsmayapplyhere.
6.2 Thecaseofmassivequarks
Inthiscase,therelationinEq. (53)isreplaced by10:
a0(Q2) 1 ψ5′(0)
= = 0.55 0.02, (57)
a8 √2 qψ′88(0) ±
5
q
where ψ (q2) has been defined in Eq. (21) and ψ88(q2) is the two-point correlator for the
5 5
octetcurrent. Therunning ofthesubtraction constant ψ′(0)isverysmoothfromτ−1=3to
5
10GeV2. Usingψ′88(0) a8 = 0.58,theprevious relationleadsto:
≡
a0(Q2 = 4GeV2) a0(Q2 =10GeV2) 0.32 0.02, (58)
≃ ≃ ±
whichisalmostequaltoitsvalueinthechirallimit. Thisprediction iscomparable withthe
COMPASSresult (0.24 0.02)41 giveninEq. (48) butisabout afactor 2smaller than its
±
OZIvalueinEq. (49). Usingthisresult,wepredict:
Γp(Q2 = 4GeV2)= 0.145 0.002, (59)
1 ±
where wehave used the value of α (M ) 0.347 0.0343,44,23, and the previous values
s τ
≃ ±
of F and D from37. Improvements of these results at COMPASS energy require a good
controlofthehighertwistcorrections inthePTexpressions ofΓp whichareexpectedtobe
1
biggerherethanatSMC.Usingthesumruleestimatein45 ofabout 0.03ofthecoefficient
of the 1st power 1/Q2 corrections, one obtains a correction of abo−ut -(0.008 0.008) to
Γp at 4 GeV2, where the error is an educated guess taking into account larg±er absolute
1
value obtained from the fitof theBjorken sum rule in9. Acontrol ofthe corrections inthe
assumption ofthevalidity ofOZIforthesinglet andnon-singlet vertexfunctions isnotyet
testablee. Including thepowercorrection, weconsider asafinalestimate:
Γp(Q2 = 4GeV2)= 0.137 0.008. (60)
1 ±
Several tests of the approach have been also proposed in the literature47. Another crucial
test ofour result willbealattice measurement ofψ′(0) withmassive dynamical fermions,
5
whichwewishthatthePisagroupputsinitsagenda. Unfortunately, anattempttomeasure
a0(Q2)onthelattice11 hasnotbeenconclusive, asitgivesavalue:
a0(Q2) = 0.04 0.04 0.20, (61)
± ±
wherethelasterrorisaneducated guessoftheeffectiveerrorexpected bytheauthors.
7 Conclusions
WehaveusedQSSRforpredicting thegluonium decayconstant andmass,andforarguing
that theη(1440) islikely theradial excitation oftheη′(958). Theseresults havebeen used
for predicting the topological susceptibility and its slope, which are useful inputs in the
resolution ofthe“proton spinproblem”.
eAQSSRevaluationoftheηNN couplinghasnotbeenconclusive46.