Correlator of topological charge densities at low $Q^2$ in QCD: connection with proton spin problem

20 Pages·0.25 MB·English

by  B. L. Ioffe

APH N.S., Heavy Ion Physics 7 (1998) 000–000
HEAVY ION
PHYSICS
(cid:13)cAkad´emiaiKiado´
Correlatorof topologicalcharge densities at low Q2 in QCD:
connection with proton spin problem.
9
9
9 B. L. Ioffe
1 Institute of Theoretical and Experimental Physics,
n 117218,Moscow, B.Cheremushkinskaya 25, Russia,
a e-mail: ioff[email protected]
J

5 Received

1
Abstract. Vacuum correlator of topological charge densities χ(q2) at low q2
v
3 inQCDisdiscussed: itsvalueχ(0)andfirstderivativeχ′(0)atq2 =0andthe
2 contributions of pseudoscalar quasi-Goldstone bosons to χ(q2) at low q2. The
2 QCD sum rule [1], giving the connection of χ′(0) (for massless quarks) with
1
the partofthe protonspinΣ,carriedbyu,d,squarks,is presented. Fromthe
0
requirement of selfconsistancy of the sum rule the values of χ′(0) and Σ were
9
9 found. The same value of χ′(0) follows also from the experimental data on
/ Σ. The contributions of π and η to χ(q2) are calculated basing on low energy
h
theorems. In such calculation the π η mixing, expressed in terms of quark
p
−
- mass ratios is of importance.
p
e
h
: 1.Introduction.
v
i
X
The existence of topological quantum number is a very specific feature of non-
r abelian quantum field theories and, particularly, QCD. Therefore, the study of
a
properties of the topological charge density operator in QCD
α
Q (x)= sGn (x)G˜n (x) (1)
5 8π µν µν
and of the corresponding vacuum correlator
χ(q2)=i d4xeiqx 0 T Q (x), Q (0) 0 (2)
5 5
h | { }| i
Z
is of a great theoretical interest. (Here Gn is gluonic field strength tensor, G˜ =
µν µν
(1/2)ε G is its dual, n are the colour indeces, n = 1,2,...N2 1, N is the
µνλσ λσ c − c
numberofcolours,N =3inQCD).Theexistenceoftopologicalquantumnumbers
c
innon-abelianfieldtheorieswasfirstdiscoveredbyBelavinetal.[2],theirconnection
with non-conservation of U(1) chirality was established by t’Hooft [3]. Gribov [4]
0231-4428/97/ $ 5.00
(cid:13)c1997 Akad´emiaiKiad´o, Budapest
2
was the first, who understood, that instanton configurations in Minkowski space
realize the tunneling transitions between states with different topological numbers.
Crewther [5] derived Ward identities related to χ(0), which allowed him to prove
the theorem, that χ(0) = 0 in any theory where it is at least one massless quark.
An important step in the investigation of the properties of χ(q2) was achieved by
Veneziano[6]andDiVecchiaandVeneziano[7]. Theseauthorsconsideredthelimit
N . Assuming that in the theory there are N light quarks with the masses
c f
→ ∞
m M, where M is the characteristic scale of strong interaction, Di Vecchia and
i
≪
Veneziano found that
Nf 1 −1
χ(0)= 0 q¯q 0 , (3)
h | | i mi!
i
X
where 0 q¯q 0 is the common value of quark condensate for all light quarks
h | | i
and the terms of the order m /M are neglected. 1 The concept of θ-term in
i
the Lagrangian was succesfully exploited in [6] in deriving of (3). Using the same
conceptandstudyingthepropertiesoftheDiracoperatorLeutwylerandSmilga[8]
succeeded in provingeq.3 at any N for the case of two light quarks,u and d. How
c
this method can be generalized to the case of N = 3 is explained in the recent
f
paper [9].
In this paper I discuss χ(q2) in the domain of low q2 , i.e. I suppose that
| |
q2/M2 is a small parameter and restrict myself to the terms linear in this ratio. In
this domain must be accounted terms χ(0), χ′(0)q2 as well as the contributions of
low mass pseudoscalar quasi–Goldstone bosons π and η, resulting to nonlinear q2
dependence.
Strictlyspeaking,onlynonperturbativepartofχ(q2)hasdefinitemeaning. The
perturbative part is divergent and its contribution depends on the renormalization
procedure. Forthisreasononlynonperturbativepartofχ(q2)(2)withperturbative
part subtracted will be considered here. In ref.1 on the basis of QCD sum rules in
theexternalfieldstheconnectionofχ′(0)(itsnonperturbativepart)withthepartof
the protonspin Σ, carriedbyu,d,s quarks,wasestablished. Fromthe requirement
of the selfconsistancy of the sum rule it was obtained
χ′(0)=(2.4 0.6) 10−3 GeV2 (4)
± ×
in the limit of massless u,d and s quarks. The close to (4) value follows also
by the use of experimental data on Σ. At low q2 the terms, proportional to quark
masses,arerelatedtothecontributionsoflightpseudoscalarmesonsasintermediate
states in the the correlator (2). These contributions are calculated below. In such
calculationforthe caseofthreequarksthe mixingofπ0 andη is ofimportanceand
it is accounted.
Thepresentationofthematerialinthepaperisthefollowing. InSec.2theQCD
sum rule for Σ is derived. The important contribution to the sum rule comes from
1Thedefinitionofχ(q2)usedaboveineq.(2),differsbysignfromthedefinitionusedin[4]-[6].
3
χ′(0). Thisallowstodetermineχ′(0)intwoways: usingtheexperimentaldataonΣ
andfromtherequirementoftheselfconsistancyofthesumrule. Bothmethodsgive
the same value of χ′(0), given by eq.4. In Sec.3 the low energy theorems related to
χ(0) are rederived with the account of possible anomalous equal-time commutator
terms. (In [5]-[7]it was implicitly assumed that these terms are zero). In Sec.4 the
case of one and two light quarks are considered. It is proved, that the mentioned
above commutator terms are zero indeed and for the case of two quarks eq.3 is
reproduced without using N limit and the concepts of θ–terms. In Sec.5 the
c
→∞
case of three u,d,s light quarks is considered in the approximation m ,m m .
u d s
≪
The problem of mixing of π0 and η states [10,11] is formulated and corresponding
formulae are presented. The account of π η mixing allows one to get eq.(3) from
−
low energy theorems, formulated in Sec.2 for the case of three light quarks. (At
m ,m m it coincides with the two quark case). In Sec.5 the q2–dependence of
u d s
≪
χ(q2) was found at low q2 in the leading nonvanishing order in q2/M2 as well as
| |
in m /M.
q
2.QCD sum rule for Σ. Connection of Σ and χ′(0).
As well known, the parts of the nucleon spin carried by u,d and s-quarks are
determined from the measurements of the first moment of spin dependent nucleon
structure function g (x,Q2)
1
1
Γ (Q2)= dxg (x,Q2) (5)
p,n 1;p,n
Z
0
The data allows one to find the value of Σ – the part of nucleon spin carried by
three flavours of light quarks Σ =∆u+∆d+∆s, where ∆u,∆d,∆s are the parts
of nucleon spin carried by u,d,s quarks. On the basis of the operator product
expansion (OPE) Σ is related to the proton matrix element of the flavour singlet
axial current j0
µ5
2ms Σ= p,sj0 p,s , (6)
µ h | µ5| i
where s is the proton spin 4-vector, m is the proton mass. The renormalization
µ
scheme in the calculationof perturbative QCD correctionsto Γ can be arranged
p,n
in such a way that Σ is scale independent.
An attempt to calculate Σ using QCD sum rules in external fields was done in
ref.[12]. Let us shortly recall the idea. The polarization operator
Π(p)=i d4xeipx 0T η(x),η¯(0) 0 (7)
h | { }| i
Z
4
was considered, where
η(x)=εabc ua(x)Cγ ub(x) γ γ dc(x) (8)
µ µ 5
!
is the current with proton quantum numbers [13], ua,db are quark fields, a,b,c are
colour indeces. It is assumed that the term
∆L=j0 A (9)
µ5 µ
whereA isaconstantsingletaxialfield,isaddedtoQCDLagrangian. Intheweak
µ
axial field approximationΠ(p) has the form
Π(p)=Π(0)(p)+Π(1)(p)A . (10)
µ µ
Π(1)(p)iscalculatedinQCDbyOPEatp2 <0, p2 R−2,whereR istheconfine-
µ | |≫ c c
mentradius. Ontheotherhand,usingdispersionrelation,Π(1)(p)isrepresentedby
µ
the contributionofthe physicalstates, the lowestofwhichis the protonstate. The
contribution of excited states is approximated as a continuum and suppressed by
the Boreltransformation. The desiredanswer is obtained by equalling of these two
representations. This procedurecanbe appliedto anyLorenzstructureofΠ(1)(p),
µ
but as was arguedin [14],[15]the best accuracycanbe obtainedby considering the
chirality conserving structure 2p pˆγ .
µ 5
An essential ingredient of the method is the appearance of induced by the
external field vacuum expectation values (v.e.v). The most important of them in
the problem at hand is
0j0 0 3f2A (11)
h | µ5| iA ≡ 0 µ
ofdimension3. Theconstantf2 isrelatedtoQCDtopologicalsusceptibility. Using
0
(9), we can write
0j0 0 =lim i d4xeiqx 0T j0 (x),j0 (0) 0 A
h | µ5| iA q→0 h | { ν5 µ5 }| i ν ≡
Z
lim P (q)A (12)
q→0 µν ν
≡
The general structure of P (q) is
µν
P (q)= P (q2)δ +P (q2)( δ q2+q q ) (13)
µν L µν T µν µ ν
− −
Because of anomaly there are no massless states in the spectrum of the singlet po-
larizationoperatorP evenformasslessquarks. P (q2)alsohavenokinematical
µν T,L
singularities at q2 = 0 . Therefore, the nonvanishing value P (0) comes entirely
µν
from P (q2). Multiplying P (q) by q q , in the limit of massless u,d,s quarks we
L µν µ ν
get
5
q q P (q)= P (q2)q2 =N2(α /4π)2i d4xeiqx
µ ν µν − L f s ×
Z
0TGn (x)G˜n (x),Gm (0)G˜m (0)0 , (14)
×h | µν µν λσ λσ | i
where Gn is the gluonic field strength, G˜ = (1/2)ε G .(The anomaly con-
µν µν µνλσ λσ
dition was used, N =3.). Going to the limit q2 0, we have
f
→
4
f2 = (1/3)P (0)= N2χ′(0), (15)
0 − L 3 f
where χ(q2) is defined by (2).
According to Crewther theorem [5]), χ(0) = 0 if there is at least one massless
quark. The attempt to find χ′(0) itself by QCD sum rules failed: it was found [12]
thatOPEdoes notconvergeinthe domainofcharacteristicscalesfor this problem.
However, it was possible to derive the sum rule, expressing Σ in terms of f2 (11)
0
or χ′(0). The OPE up to dimension d = 7 was performed in ref.[12]. Among the
induced by the external field v.e.v.’s besides (11), the v.e.v. of the dimension 5
operator
g 0 q¯γ (1/2)λnG˜n q 0 3h A , q =u,d,s (16)
h | α αβ | iA ≡ 0 β
q
X
was accounted and the constant h was estimated using a special sum rule,
0
h 3 10−4GeV4 . There were also accounted the gluonic condensate d=4 and
0
≈ ×
the square of quark condensate d = 6 (both times the external A field operator,
µ
d = 1). However, the accuracy of the calculation was not good enough for reliable
calculationofΣintermsoff2: thenecessaryrequirementofthemethod–theweak
0
dependence of the result on the Borel parameter was not well satisfied.
In [1] the accuracy of the calculation was improved by going to higher order
terms in OPE up to dimension 9 operators. Under the assumption of factorization
– the saturation of the product of four-quark operators by the contribution of an
intermediate vacuum state – the dimension 8 v.e.v.’s are accounted (times A ):
µ
g 0q¯σ (1/2)λnGn q q¯q 0 =m2 0q¯q 0 2, (17)
− h | αβ αβ · | i 0h | | i
where m2 =0.8 0.2 GeV2 was determined in [16]. In the framework of the same
0 ±
factorization hypothesis the induced by the external field v.e.v. of dimension 9
α 0j(0) 0 0q¯q 0 2 (18)
sh | µ5| iAh | | i
is also accounted. In the calculation the following expression for the quark Green
function in the constant external axial field was used [15]:
0T qa(x), q¯b(0) 0 =iδabxˆ /2π2x4+
h | { α β }| iA αβ
+(1/2π2)δab(Ax)(γ xˆ) /x4 (1/12)δabδ 0q¯q 0 +
5 αβ αβ
− h | | i
6
+(1/72)iδab 0q¯q 0 (xˆAˆγ Aˆxˆγ ) +
5 5 αβ
h | | i −
+(1/12)f2δab(Aˆγ ) +(1/216)δabh (5/2)x2Aˆγ (Ax)xˆγ (19)
0 5 αβ 0" 5− 5#
αβ
The terms of the third power in x-expansion of quark propagator proportional to
A are omitted in (19), because they do not contribute to the tensor structure of
µ
Π of interest. Quarks are considered to be in the constant external gluonic field
µ
and quark and gluon QCD equations of motion are exploited (the related formulae
are given in [17]). There is also an another source of v.e.v. h to appear besides
0
the x-expansionofquarkpropagatorgivenineq.(19): the quarksinthe condensate
absorbthe softgluonicfieldemittedbyotherquark. Asimilarsituationtakesplace
also in the calculation of the v.e.v. (18) contribution. The accounted diagrams
with dimension 9 operators have no loop integrations. There are others v.e.v. of
dimensions d 9 particularly containing gluonic fields. All of them, however,
≤
correspond to at least one loop integration and are suppressed by the numerical
factor (2π)−2. For this reason they are disregarded.
The sum rule for Σ is given by
Σ+C M2 = 1+ 8 em2/M2 a2L4/9+6π2f2M4E W2 L−4/9+
0 − 9λ˜2 ( 0 1 M2!
N
W2 1 a2m2 1 a2
+14π2h M2E L−8/9 0 πα f2 (20)
0 0 M2! − 4 M2 − 9 s 0 M2)
Here M2 is the Borel parameter, λ˜ is defined as λ˜2 = 32π4λ2 = 2.1 GeV6,
N N N
0η p = λ v , where v is proton spinor, W2 is the continuum threshold, W2 =
N p p
h | | i
2.5 GeV2,
a= (2π)2 0q¯q 0 =0.55 GeV3 (21)
− h | | i
E (x)=1 e−x, E (x)=1 (1+x)e−x
0 1
− −
L = ln(M/Λ)/ln(µ/Λ), Λ = Λ = 200 MeV and the normalization point µ
QCD
was chosen µ=1 GeV. When deriving (20) the sum rule for the nucleon mass was
exploitedwhatresultsinappearanceofthefirstterm,-1,intherighthandside(rhs)
of (20). This term absorbs the contributions of the bare loop, gluonic condensate
as well as α corrections to them and essential part of terms, proportional to a2
s
and m2a2. The values of the parameters, a,λ˜2 ,W2 taken above were chosen by
0 N
the best fit ofthe sum rules for the nucleonmass (see [18],Appendix B) performed
at Λ = 200 MeV. It can be shown, using the value of the ratio 2m /(m +
s u
m ) = 24.4 1.5 [19] that a(1 GeV) = 0.55 GeV3 corresponds to m (1 GeV) =
d s
±
153 MeV. α corrections are accounted in the leading order (LO) what results in
s
appearance of anomalous dimensions. Therefore Λ has the meaning of effective Λ
in LO. The unknown constant C in the left-hand side (lhs) of (20) corresponds to
0
7
the contribution of inelastic transitions p N∗ interaction withA p (and
µ
→ → →
ininverseorder). ItcannotbedeterminedtheoreticallyandmaybefoundfromM2
dependence of the rhs of (20) (for details see [18,20]). The necessary condition of
the validity of the sum rule is Σ C M2 exp[( W2+m2)/M2] atcharacteristic
0
| |≫| | −
valuesofM2 [20]. The contributionofthe lasttermintherhsof(20))is negligible.
The sum rule (20) as well as the sum rule for the nucleon mass is reliable in the
interval of the Borel parameter M2 where the last term of OPE is small, less than
10 15%ofthe totaland the contributionof continuumdoes not exceed40 50%.
− −
This fixes the interval 0.85 < M2 < 1.4 GeV2.The M2-dependence of the rhs of
(20) at f2 =3 10−2 GeV2 is plotted in fig.1a. The complicated expression in rhs
0 ×
of (20) is indeed an almost linear function of M2 in the given interval! This fact
strongly supports the reliability of the approach. The best values of Σ = Σfit and
C =Cfit are found from the χ2 fitting procedure
0 0
n
1
χ2 = [Σfit CfitM2 R(M2)]2 =min, (22)
n − 0 i − i
i=1
X
where R(2) is the rhs of (20).
Fig. 1. a) The M2-dependence of Σ+C M2 at f2 =3 10−2 GeV2; b) Σ (solid
0 0 ×
line, left ordinate axis) and χ2, (dashed line, right ordinate axis). as a functions
of f2.
0 p
The values of Σ as a function of f2 are plotted in fig.1b together with χ2.
0
In our approach the gluonic contribution cannot be separated and is included in
p
Σ. The experimental value of Σ can be estimated [21,22] (for discussion see [23])
as Σ = 0.3 0.1. Then from fig.1b we have f2 = (2.8 0.7) 10−2 GeV2 and
± 0 ± ×
χ′(0)=(2.3 0.6) 10−3 GeV2. The error in f2 and χ′ besides the experimentall
± × 0
8
errorincludestheuncertaintyinthesumruleestimatedasequaltothecontribution
of the last term in OPE (two last terms in Eq.(20) and a possible role of NLO α
s
corrections. Allowing the deviation of χ2 by a factor 1.5 from the minimum
we get χ′ = (24 0.6) 10−3GeV2 and Σ = 0.32 0.17 from the requirement
± × p ±
of selfconsistency of the sum rule. At f2 = 2.8 10−2 GeV2 the value of the
0 ×
constant C found from the fit is C = 0.19 GeV−2. Therefore, the mentioned
0 0
above necessary condition of the sum rule validity is well satisfied.
LetusdiscusstheroleofvarioustermsofOPEinthesumrules(20). Toanalyze
it the sum rule (20) was considered for 4 different cases, i.e. when it is taken into
consideration: a) only contribution of the operators up to d=3 (the term –1 and
the term, proportional to f2 in (20)); b) contribution of the operators up to d=5
0
(the term h is added); c) contribution of the operators up to d=7 (three first
0
∼
terms in(20)), d)the result(20),i.e. alloperatorsup to d=9. For this analysisthe
value of f2 = 0.03 GeV2 was chosen, but the conclusion appears to be the same
0
for all more or less reasonable choice of f2. Results of the fit of the sum rules are
0
shown in Table 1 for all four cases. The fit is done in the region of Borel masses
0.9<M2 <1.3GeV2. InthefirstcolumnthevaluesofΣareshown,inthesecond
- values of the parameter C, and in the third - the ratio γ = χ2/Σ, which is
| |
the real parameter, describing reliability of the fit. From the table one can see,
p
that reliability of the fit monotonously improves with increasing of the number of
accounted terms of OPE and is quite satisfactory in the case d
Table 1.
case Σ C(GeV−2) γ
a) -0.019 0.31 10−1
b) 0.031 0.3 5.10−2
c) 0.54 0.094 9.10−3
d) 0.36 0.21 1.3 10−3
·
3.Low energy theorems.
Consider QCD with N light quarks, m M 1 GeV, i = 1,...N . Define the
f i f
≪ ∼
singlet (in flavour)axial current by
Nf
j (x)= q¯(x)γ γ q(x) (23)
µ5 i µ 5
i
X
and the polarization operator
P (q)=i d4xeiqx 0 T j (x),j (0) 0 . (24)
µν µ5 µ5
h | { }| i
Z
9
The general form of the polarization operator is:
P (q)= P (q2)δ +P (q2)( δ q2+q q ) (25)
µν L µν T µν µ ν
− −
Because of anomaly the singlet axial current is nonconserving:
∂ j (x)=2N Q (x)+D(x), (26)
µ µ5 f 5
where Q (x) is given by (1) and
5
Nf
D(x)=2i m q¯(x)γ q (x) (27)
i i 5 i
i
X
It is well known, that even if some light quarks are massless, the corresponding
Goldstone bosons, arising from spontaneous violation of chiral symmetry do not
contribute to singlet axial channel (it is the solution of U(1) problem), i.e. to
polarizationoperatorP (q). P (q2) also have no kinematical singularities at q2 =
µν L
0. Therefore
P (q)q q = P (q2)q2 (28)
µν µ ν L
−
vanishes in the limit q2 0. Calculate the left-hand side (lhs) of (28) in the
→
standard way – put q q inside the integral in (24) and integrate by parts. (For
µ ν
this it is convenient to represent the polarization operator in the coordinate space
as a function of two coordinates x and y.) Going to the limit q2 0 we have
→
lim P (q)q q =i d4x 0 T 2N Q (x), 2N Q (0)
µν µ ν f 5 f 5
q2→0 h | {
Z
+2N Q (x), D(0)+D(x), 2N Q (0)+D(x), D(0) 0
f 5 f 5
}| i
Nf
+4 m q¯(0)q (0) 0
i i i
h| | i
i
X
+ d4x 0 [j (x), 2N Q (0) ] 0 δ(x )=0 (29)
05 f 5 0
h | | i
Z
Inthe calculationof(29)the anomalycondition(26)wasused. The terms, propor-
tional to quark condensates arise from equal time commutator [j (x), D(0)] ,
05 x0=0
calculated by standard commutation relations. Relation (29) up to the last term
was first obtained by Crewther [5]. The last term, equal to zero according to stan-
dardcommutationrelationsandomittedin[5]-[7],iskeeped. Thereasonis,thatwe
deal with very subtle situation, related to anomaly, where nonstandard Schwinger
terms in commutationrelations may appear. (It can be shown, that, in generalthe
only Schwinger term in this problem is givenby the last term in the lhs of (29): no
others can arise.) Consider also the correlator:
10
P (q)=i d4x eiqx 0 T j (x), Q (0) 0 (30)
µ µ5 5
h | { }| i
Z
and the product P (q)q in the limit q2 0 (or q2 of order of the m2, where
µ µ → π
m is the mass of Goldstone boson). The general form of P (q) is P (q) = Aq .
π µ µ µ
ThereforenonvanishingvaluesofP q inthelimitq2 0(oroforderofquarkmass
µ µ
→
m, if q2 m2 – this limit will be also intersting for us later) can arise only from
| |∼ π
Goldstonebosonsintermediatestates in(30). Letus usestimate the corresponding
matrix elements
0 j π =Fq (31)
µ5 µ
h | | i
0 Q π =F′ (32)
5
h | | i
F is of order of m, since in the limit of massless quarks Goldstone bosons are
coupled only to nonsinglet axial current. F′ is of order of m2f m, where f is
π π ∼ π
the pion decayconstant(not consideredto be small), since inmassless quarklimit,
the Goldstone boson is decoupled from Q . These estimations give
5
q2
P q m2 (33)
µ µ ∼ q2 m2
− π
and it is zero at q2 0, m2 =0 and of order of m2 at q2 m2. In what follows I
→ π 6 ∼ π
will restrict myself by the terms linear in quark masses. So, I can put P (q)q =0
µ µ
at q 0. The integration by parts, in the right-hand side (rhs) of (30) gives:
→
lim P (q)q = d4x 0 T 2N Q (x), Q (0)+D(x), Q (0) 0
µ µ f 5 5 5
q2→0 − h | { }| i
Z
d4x 0 [ j (x), Q (0) ] 0 δ(x )=0 (34)
05 5 0
− h | | i
Z
After the substitution of)(3i.n (29) arise the low energy theorem:
i d4x 0 T 2N Q (x), 2N Q (0) 0
f 5 f 5
h | { }| i
Z
Nf
i d4x 0 T D(x), D(0) 0 4 m 0 q¯(0)q (0) 0
i i i
− h | { }| i− h | | i
Z i
X
+i d4x 0 [ j0 (x), 2N Q (0) ] 0 δ(x )=0 (35)
h | 05 f 5 | i 0
Z
The low energy theorem (35), with the last term in the lhs omitted, was found by
Crewther [5].