Scalar form factor of the nucleon and nucleon--scalar meson coupling constant in QCD

0.27 MB·English

by  T. M. Aliev

Scalar form factor of the nucleon and nucleon–scalar
meson coupling constant in QCD
7 T. M. Aliev ∗†, M. Savcı ‡
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0 Physics Department, Middle East Technical University, 06531 Ankara, Turkey
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a
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Abstract
5
1
Scalar form factor of the nucleon is calculated in the framework of light cone

2 QCD sum rules, using the most general form of the baryon current. Using the result
v
on scalar form factor of the nucleon, the nucleon–scalar σ and a meson coupling
4 0
4 constants are estimated. Our results on these couplings are in good agreement with
1
the prediction of the external field QCD sum rules method.
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1
6 PACS number(s): 11.55.Hx, 13.40.Em, 14.20.Jn
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∗e-mail: [email protected]
†permanent address:Institute of Physics,Baku,Azerbaijan
‡e-mail: [email protected]
1 Introduction
In spite of the great success of QCD theory of strong interactions, the quark structure
of scalar mesons is still an open question and widely discussed in literature (see [1] for a
review). There are many scenarios for the scalar meson structures like two–quark q¯q, four–
quark q¯qq¯q states, meson–meson bound states and even a scalar glueball. It is very likely
thatthescalarmesonstructureisnotmadeofsuchsimplestructures, butaresuperpositions
of these contents. Therefore different scenarios can give quite different predictions on the
production and decays of scalar mesons which can be tested in experiments. One efficient
way to make progress about the nature scalar bosons is to study their role in the various
two–baryon reactions (NN, ΛN, ΣN, ΛΛ, ΣΣ).
Ouraiminthispaperistocalculatethescalarformfactoroftheprotonintheframework
of light cone sum rules (LCSR) [2, 3] using the most general form of the proton current
and then relate it to the nucleon–scalar σ or a mesons coupling constants. This method
0
is an alternative approach to the traditional sum rule [4]. LCSR is based on the operator
product expansion (OPE) near the light cone, which is an expansion of the time ordered
product over the twist rather than the dimension of the operators. Main contribution of
this approach comes from the operators having lower twist. The main ingredients of the
LCSR are the wave functions of hadrons which define the matrix elements of non–local
operators between the vacuum and one hadron states.
This method applied to study the nucleon electromagnetic form factors [5], and the
weak Λ → pℓν¯ decay [6]. The higher twist wave functions of baryons have been obtained
b ℓ
recently in [7].
The plan of this article is as follows: In section 2 we derive the sum rule for the proton
form factor due to the scalar current. Section 3 is devoted to the numerical analysis and
discussion.
2 Sum rules for the proton form factor due to the
scalar current
In the present section we derive sum rules for the scalar form factor of the proton using the
most general form of nucleon interpolating current. For this aim we start with the following
correlation function:
Π(p,q) = i d4xeiqxh0|T{η(0)J(x)}|N(p)i , (1)
Z
where J(x) = u¯u + (−1)Id¯d is the interpolating current for the scalar σ (I = 0) and a
0
(I = 1)mesons, respectively, andη isasuitableinterpolating current withnucleon quantum
number. Few words about the interpolating current of the nucleon are in order. There exist
the following currents with the isotopic spin 1/2, without derivative terms (see [8]):
2
η (x) = 2εabc uTa(x)CAℓdb(x) Aℓuc(x) , (2)
1 1 2
Xℓ=1(cid:16) (cid:17)
2
η (x) = εabc uTa(x)C/zub(x) γ /zdc(x)− uTa(x)C/zdb(x) γ /zuc(x) , (3)
2 5 5
3
h(cid:16) (cid:17) (cid:16) (cid:17) i
1
where A1 = I, A1 = γ , A2 = γ , A2 = β, and in C is the charge conjugation operator,
1 2 5 1 5 2
a,b,c are the color indices, and z in Eq. (3) is a light–like vector with z2 = 0. In Eq. (2),
β = −1 corresponds to the Ioffe current [9]. Note that in [5] η (x) is modified as
2
η (x) = εabc uTa(x)C/zub(x) γ /zdc(x) .
3 5
h(cid:16) (cid:17) i
The nucleon scalar form factor which is obtained using the current η (x) in the light cone
3
QCD sum rule approach is studied in [10]. However, the current η (x) couples also to
3
the states with isospin 3/2, and these contributions should be eliminated from the final
result, which is not done in [10]. Moreover, terms that are proportional to z//x and x//z in
the coordinate space which give contribution to the considered structure in [11], are also
neglected. In the present work η is chosen as the general form of the interpolating current.
1
Let us firstly calculate the hadronic part of the correlator function. By inserting the
complete set of states between the currents in Eq. (1) with quantum numbers of the
corresponding nucleon, we get the following hadronic representation of the correlator
h0|η|N(p′)ihN(p′)|J|N(p)i h0|η|h(p′)ihh(p′)|J|N(p)i
Π(p,q) = + , (4)
m2 −p′2 m2 −p′2
N h h
X
where p′ = p−q, and q is the momentum carried by the scalar current. The second term in
Eq. (4) takes into account higher states and continuum contributions and hadrons h form
a complete set of baryons having the same quantum number as the ground state baryon B
(here B represents p and n). The matrix elements in Eq. (4) are defined as
h0|η|N(p′)i = λ u (p′) , (5)
N N
hN(p′)|J|N(p)i = g(Q2)u¯(p′)u(p) . (6)
Using Eqs. (4)–(6) and performing summation over nucleon spin, for the phenomenological
part of the correlator we obtain
λ g(Q2)(6p′ +m )u(p)
N N
Π(p,q) = +··· , (7)
m2 −(p−q)2
N
where···represents contributionsfromthehigherstatesandthecontinuum. Itfollowsfrom
Eq. (7) that the correlator function contains three different structures, namely structures
∼6 p, ∼6 q and structure proportional to the unit operator I. In further analysis we will
choose the structure ∼6p.
Here we would like to make the following remark: The nucleon–scalar meson S coupling
constant g can be related to the scalar form factor g(Q2), and for this aim we define
NNS
the following coupling constant
hN(p′)S|N(p)i ≡ g u¯(p′)u(p) . (8)
NNS
It follows then from Eq. (1) that
h0|J|SihN(p′)S|N(p)ih0|η|N(p′)i
Π(p,q) = . (9)
(m2 −p′2)(m2 −q2)
N S
2
The matrix element of the scalar current J between the vacuum and the scalar meson state
is defined as
h0|J|Si = λ . (10)
S
Using Eqs. (8)–(10) and comparing with Eq. (7), we get the following relation between
nucleon scalar form factor and g
NNS
g λ
2 NNS S
g(Q ) = . (11)
m2 +Q2
S
At large Euclidean momenta p′2 = (p−q)2 and q2 = −Q2, the QCD part of the correlator
can be calculated from the explicit formof interpolating currents J(x) and η(0), from which
we obtain
1 2
Π = d4xeiqx CAℓ AℓS(−x) 4ǫabc 0 T ua(0)ub(x)dc(0) N
2 Z Xℓ=1((cid:16) 1(cid:17)αγh 2 iρβ D (cid:12) n α β γ o(cid:12) E
(cid:12) (cid:12)
+ Aℓ CAℓ T S(−x) 4ǫabc 0 T ua(0)ub(x(cid:12) )dc(0) N (cid:12)
2 1 α β γ
ρα γβ
(cid:16) (cid:17) h(cid:16) (cid:17) i D (cid:12) n o(cid:12) E
(cid:12) (cid:12)
+ Aℓ CAℓS(−x) 4ǫabc 0 T u(cid:12)a(0)ub(0) dc(x) N (cid:12) , (12)
2 1 α β γ
ρβ αγ )
(cid:16) (cid:17) h i D (cid:12) n o (cid:12) E
(cid:12) (cid:12)
(cid:12) (cid:12)
where S(x) is the light quark propagator. It follows from Eq. (12) that for calculation of
the correlator from QCD side we need to know the explicit form light quark propagator
and the nucleon distribution amplitudes (DAs). The full propagator of the massless quark
is
i 6x hqq¯i m2x2 1 6x i
S(x) = − 1+ 0 −ig dv G σµν −vxµG γν . (13)
2π2x4 12 16 ! s 0 "16π2x4 µν µν 4π2x2#
Z
The contribution ∼ G can give rise to four and five particle nucleon DAs. These DAs are
µν
usually small [12, 13] and can be neglected.
The second term in Eq. (13) does not give any contribution to the sum rule after the
Borel transformation. So, only the first term is relevant for our calculations.
As has already been noted, in order to calculate the non–perturbative contributions to
the theoretical part of the correlator function one needs to know the matrix element of
ǫabch0|ua(a x)ub(a x)dc(a x)|N(p)i between the proton and vacuum states. This matrix
α 1 β 2 γ 3
element of the non–local operator is parametrized in terms of the nucleon DAs [7, 13, 14,
15, 16, 17]. Their explicit forms are given in the Appendix–A.
Taking intoaccountthethreevalencequarklight-coneDAsuptotwist-6andperforming
integration over x in the coordinate space, we finally obtain the following expression for the
correlator function:
1 1 1−t2 6q− 6pt2
Π = m dt dt 2(1+β)V (t )+2[3∓1+(3±1)β]V (t )
4 N Z0 2Z0 1(q −pt2)2n 2 i 3 i
± 2(1+β)P (t )−2[2±1−(2∓1)β]S (t )e+2(1+β)A (t )
1 i 1 i 2 i
+ 2[3±1+(3∓1)β]A (t )+[2∓1−(2±1)β]T (t )+12(1−β)T (t )
3 i 2 i e 7 i
e
3
+ [6∓1−(6±1)β]T (t )
4 i
m3 1 τ λ o 1−t2 6q− 6pτ
− N dτ dλ e dt dt ±(1−β)V (t )
2 0 1 1 2 0 1(q −pτ)4 6 i
Z Z Z Z n
∓ (1−β)A (t )+3[2∓1−(2±1)β]T (t ) e
6 i 8 i
m 1 λ 1−t2 6p o
− N deλ dt dt e [2∓1+(2±1)β]V (t )
2 0 1 2 0 1(q −pλ)2 2 i
Z Z Z n
+ [2±1+(2∓1)β]A (t )+[2∓1−(2±1)β]T (t ) e
2 i 2 i
± 2(1+β)T (t ) (14)
4 i e e
o
where any functionFe(ti) means F(ti) = F(t1,t2,1−t1,t2), andfunctions with ∼aredefined
as
V (t ) = V (t )−V (t )−V (t ) ,
2 i 1 i 2 i 3 i
A (t ) = −A (t )+A (t )−A (t ) ,
e2 i 1 i 2 i 3 i
A (t ) = A (t )−A (t )+A (t )+A (t )−A (t )+A (t ) ,
e6 i 1 i 2 i 3 i 4 i 5 i 6 i
T (t ) = T (t )+T (t )−2T (t ) ,
e2 i 1 i 2 i 3 i
T (t ) = T (t )−T (t )−2T (t ) ,
e4 i 1 i 2 i 7 i
T (t ) = 2T (t )−2T (t )−2T (t )+2T (t )+2T (t )+2T (t ) ,
e6 i 2 i 3 i 4 i 5 i 7 i 8 i
T (t ) = −T (t )+T (t )+T (t )−T (t )+2T (t )+2T (t ) .
e8 i 1 i 2 i 5 i 6 i 7 i 8 i
Explicit expressions of A (t ), T (t ) and V (t ) can be found in [17]. Note also that in
e α i α i α i
deriving Eq. (14) we omit the terms proportional with the unit operators which do not give
contribution to the structure ∼6p. Upper (lower) sign in Eq. (14) corresponds to the scalar
form factor due to the scalar current with the quantum numbers of σ (a ).
0
InordertosuppressthecontinuumandhigherstatecontributionsweneedtoapplyBorel
transformation to the phenomenological and theoretical parts of the correlation function
with respect to the variable (q −p)2.
For the theoretical part the Borel transformation and the continuum subtraction can
be done by using the following substitution rules
ρ(x) ρ(x)
dx → − dx e−s/M2 ,
(q −xp)2 x
Z Z
ρ(x) 1 ρ(x) ρ(x )
dx → dx e−s/M2 + 0 e−s0/M2 , (15)
(q −xp)4 M2 x2 Q2 +x2M2
Z Z 0
where
1−x
2 2
s = (1−x)M + Q ,
x
(Q2 +s −m2 )2 +4m2 Q2 −(Q2 +s −m2 )
0 N N 0 N
x = ,
0 q 2m 62
N
and Q2 = −q2. Finally, we get the following sum rule for the scalar form factor.
g(Q2) = em2N/M2 1m 1dt 1−t2 e−s/M2 2(1+β)V (t )+2[3∓1+(3±1)β]V (t )
N 2 2 i 3 i
λN "4 Zx0 Z0 n
e
4
± 2(1+β)P (t )−2[2±1−(2∓1)β]S (t )+2(1+β)A (t )
1 i 1 i 2 i
+ 2[3±1+(3∓1)β]A (t )+[2∓1−(2±1)β]T (t )+12(1−β)T (t )
3 i 2 i e 7 i
+ [6∓1−(6±1)β]T (t )
4 i e
+ m3N 1dτ τ dλe λdto 1−t2 dt 1 e−s/M2 ±(1−β)V (t )
2 Zx0 Z1 Z1 2Z0 1τM2 n 6 i
e
∓ (1−β)A (t )+3[2∓1−(2±1)β]T (t )
6 i 8 i
+ x0 e x0dλ λdt 1−t2 dt e1 eo−s/M2 ±(1−β)V (t )
Q2 +x20m2N Z1 Z1 2Z0 1τM2 n 6 i
e
∓ (1−β)A (t )+3[2∓1−(2±1)β]T (t )
6 i 8 i
!
o
+ 1m 1edt2 1−t2 dt e−s/M2 [2∓1e+(2±1)β]V (t )
N 1 2 i
2 Zx0 t2 Z0 n
+ [2±1+(2∓1)β]A (t )+[2∓1−(2±1)β]T (te)
2 i 2 i
± 2(1+β)T (t ) e e (16)
4 i
#
o
e
It follows from Eq. (16) that for determining the scalar form factor, the residue λ of the
N
nucleon needs to be known. The residue λ is obtained from the mass sum rules for the
N
baryons (see for example [18])
M6 hu¯ui
λ2Ne−m2N/M2 = 256π4E2(x)(5+2β +β2)+ 6 −6(1−β2)hd¯di+(−1+β2)hu¯ui
m2 h i
0 2 ¯ 2
− hu¯ui −12(1−β )hddi+(−1+β )hu¯ui , (17)
24M2
h i
where x = s /M2, and
0
n xk
E (x) = 1−ex ,
n
k!
k=0
X
which describes the subtraction of higher states and continuum contributions.
3 Numerical Analysis
In this section we present the numerical analysis for the scalar form factor g(Q2). It
follows from Eq. (16) that the main ingredients of LCSR are the DAs, which involve
non–perturbative parameters f , λ , λ , fu, fd, fd, Au, Vd (see Appendix–B).
N 1 2 1 1 2 1 1
In the numerical calculations we use three different sets of the DAs (for more details,
see [5] and [17]).
1) QCD sum rules based DAs,
2) Asymptotic DAs,
3) A model for the nucleon DAs. The parameters in this case are chosen in such a way
that, the form of DAs describes very well the existing experimental data on nucleon form
factors.
5
The values of the above–mentioned non–perturbative parameters for these cases at µ =
1 GeV scale are given in [5] and [17]. Our numerical calculations show that the values of
the scalar form factors of nucleon due to the scalar current in QCD based DAs case are
larger compared to that of the other two scenarios, and, for customary reasons we present
our numerical results only for this case.
In the problem at hand we have three auxiliary parameters, namely, Borel mass square
M2, continuum threshold s and β in the interpolating current of nucleon. Obviously, any
0
physically measurable must be independent of the auxiliary parameters. So, in the first
hand we have to find the appropriate regions of these parameters, in where the form factor
is independent of them. The mass sum rules analysis shows that optimal value for the
continuum thresholds is in the region s (2÷2.5) GeV2.
0
For this aim we consider the following three–step procedure. In the first step we try
to find the working region of M2 where scalar form factor is independent of M2 at fixed
values of s and β. In Figs. (1), (2), (3) and (4) (Figs. (5), (6), (7) and (8)) we present
0
the dependence of the scalar form factor induced by I = 0 (I = 1)scalar current on M2 at
different fixed values of Q2 and β at s = 2.0 GeV2 and s = 2.5 GeV2, respectively.
0 0
From these figures we see that, in both cases, the scalar form factor is practically inde-
pendent of M2 at different values of Q2, β and s . To decide on an upper bound for M2 we
0
requirethecontributionofcontinuumbelessthanthecontributionofcontinuum–subtracted
sum rules and the lower bound can be determined by requiring that the contribution of the
highest power of 1/M2 to be less than, say, 30% of the higher powers of M2. As a result
of our analysis we find that the region 1 GeV2 ≤ M2 ≤ 2.5 GeV2 satisfies both conditions
simultaneously.
The parameter β is an auxiliary parameter, and hence, the physical quantities should be
independent of it. For this reason we need to find the working region of β where the scalar
form factor is independent of it. In determining this region of β two conditions should be
satisfied: Eq. (17) and its logarithmic derivative with respect to 1/M2 should be positive
and independent of β. From an analysis of Eq. (17) it is found that −0.5 < β < 0.5 is
unphysical [18]. It is further shown in [18] that the working region of β where the second
condition is satisfied is obtained for −0.6 < cosθ < 0.3, where tanθ = β, which yields
β < −1.3 and β > 3.3.
We perform calculations depicting the dependence of the scalar form factors on cosθ at
fixed values of s (s = 2.0 GeV2 and s = 2.5 GeV2), M2 (M2 = 1.0 GeV2 and M2 =
0 0 0
1.5 GeV2) at three fixed values of Q2 (Q2 = 3.0 GeV2, Q2 = 4.0 GeV2 and Q2 = 5 GeV2);
and confirmed that in the regions when β < −1.3 and β > 3.3 ,the form factor is practically
independent of β. So, the common region where mass sum and form factor are independent
of β is the region−0.5 < cosθ < 0.3, which corresponds to β < −1.3 andβ > 3.3 and which
we will use in further discussions. Note that, the Monte Carlo analysis that is performed in
[19] predicts the optimal value of β to be β = −1.2. Our lower bound is slightly different
compared to this value.
We perform the OPE in the light cone with large Euclidean Q2 and (p−q)2 momenta,
where the scalar from factor g(Q2) can reliably be determined at the range Q2 > 2 GeV2.
In the region Q2 < 2 GeV2 OPE becomes questionable and for this reason in determining
g(Q2) we restrict ourselves to the region Q2 > 2 GeV2.
In Figs. (9), (10), (11) and (12) we present the dependence of g(Q2) induced by I =
6
0 (I = 1) current on Q2 at fixed values of β which lies in the above–mentioned region of β;
at M2 = 1 GeV2 and s = 2 GeV2, s = 2.5 GeV2, respectively, for the central values all
0 0
parameters. We observe that the value of g(Q2) seems to be insensitive to the choice of s .
0
We further see from these figures that for positive (negative) values β, the sign of the form
factor is positive (negative). Therefore determination of the sign of g(Q2) can give valuable
information about parameter β.
Having the obtained the Q2 dependence of the form factor, we can determine g . It
NNS
follows from Eq. (10) that
(m2 +Q2)g(Q2)
g = S .
NNS
λ
S
The residues λ and λ of the scalar currents are determined from two–point QCD
σ a0
sum rules analysis in [20, 21], which is predicted to have the value λ = 0.2 and λ = 0.3.
σ a0
Using the results of Figs. (3) and (4), for the scalar form factor g we get
NNσ
6±1 for β = 5 ,
|g | =
NNσ 
 8±2 for β = −5 ; β = −3 ,

5.5±1.5 for β = 5 ,
|g | =
NNa0 
 7±1 for β = −5 ; β = −3 .
For completeness we present the results for gNNS coupling for the Ioffe current case,
despite that β = −1 lies outside the working region of β. Our results are
|g | = 12±2 ,
NNσ
|g | = 11±2 .
NNa0
Note that his coupling constant in framework of the external–field QCD sum rules
methodusing theIoffecurrent isevaluated in[22], andit isobtainedthatg = 14.4±3.7.
NNσ
This same coupling constant is also shown to have the value g = 16.9 [1] from a fit to
NNσ
the NN scattering data.
A comparison of our result for β = −1 shows that, within the error limits, there is a
good agreement between our result and the above–mentioned one.
In conclusion, we have calculated the nucleon scalar form factor in the framework of
light cone QCD sum rules, using the most general form of the baryon current. Using
the obtained result on scalar form factor, we estimate g . Our result in regard to the
NNS
coupling constant g is in a good agreement with the prediction of the external field
NNσ
QCD sum rules method for the Ioffe current. Measurement of the nucleon scalar meson
coupling constant can give valuable information about the quark structure of scalar meson
as well as about the value of parameter β.
Acknowledgments
One of the authors (T. M. A) is grateful to TU¨BI˙TAK for partially support of this work
under the project 105T131.
7
Appendices
A Nucleon distribution amplitudes
In this appendix we present the three–quark distribution DAs from twist–3 to twist–6 (for
more details, see [7] and [17]).
The matrix element of the non–local operator ǫabch0|ua(a x)ub(a x)dc(a x)|N(p)i be-
α 1 β 2 γ 3
tween the proton and vacuum states is parametrized in terms of the nucleon DAs as follows:
4ǫabc 0 ua(a x)ub(a x)dc(a x) P = S m C (γ N) +S m2 C (6xγ N)
α 1 β 2 γ 3 1 N αβ 5 γ 2 N αβ 5 γ
+ P1mND(γ5(cid:12)(cid:12)(cid:12)C)αβNγ +P2m2N(γ5C)α(cid:12)(cid:12)(cid:12)β(6xEN)γ + V1 + x2m4 2NV1M!(6PC)αβ(γ5N)γ
+ V m (6PC) (6xγ N) +V m (γ C) (γµγ N) +V m2 (6xC) (γ N)
2 N αβ 5 γ 3 N µ αβ 5 γ 4 N αβ 5 γ
+ V m2 (γ C) (iσµνx γ N) +V m3 (6xC) (6xγ N)
5 N µ αβ ν 5 γ 6 N αβ 5 γ
x2m2
+ A + NAM (6Pγ C) N +A m (6Pγ C) (6xN) +A m (γ γ C) (γµN)
1 1 5 αβ γ 2 N 5 αβ γ 3 N µ 5 αβ γ
4 !
+ A m2 (6xγ C) N +A m2 (γ γ C) (iσµνx N) +A m3 (6xγ C) (6xN)
4 N 5 αβ γ 5 N µ 5 αβ ν γ 6 N 5 αβ γ
x2m2
+ T + NT M (Pνiσ C) (γµγ N) +T m (xµPνiσ C) (γ N)
1 1 µν αβ 5 γ 2 N µν αβ 5 γ
4 !
+ T m (σ C) (σµνγ N) +T m (Pνσ C) (σµρx γ N)
3 N µν αβ 5 γ 4 N µν αβ ρ 5 γ
+ T m2 (xνiσ C) (γµγ N) +T m2 (xµPνiσ C) (6xγ N)
5 N µν αβ 5 γ 6 N µν αβ 5 γ
+ T m2 (σ C) (σµν 6xγ N) +T m3 (xνσ C) (σµρx γ N) . (A.1)
7 N µν αβ 5 γ 8 N µν αβ ρ 5 γ
The calligraphic DAs do not have definite twist and can be related to the one with definite
twist as:
S = S , (2P·x)S = S −S ,
1 1 2 1 2
P = P , (2P·x)P = P −P ,
1 1 2 2 1
V = V , (2P·x)V = V −V −V ,
1 1 2 1 2 3
2V = V , (4P·x)V = −2V +V +V +2V ,
3 3 4 1 3 4 5
(4P·x)V = V −V , (2P·x)2V = −V +V +V +V +V −V ,
5 4 3 6 1 2 3 4 5 6
A = A , (2P·x)A = −A +A −A ,
1 1 2 1 2 3
2A = A , (4P·x)A = −2A −A −A +2A ,
3 3 4 1 3 4 5
(4P·x)A = A −A , (2P·x)2A = A −A +A +A −A +A ,
5 3 4 6 1 2 3 4 5 6
T = T , (2P·x)T = T +T −2T ,
1 1 2 1 2 3
2T = T , (2P·x)T = T −T −2T ,
3 7 4 1 2 7
(2P·x)T = −T +T +2T , (2P·x)2T = 2T −2T −2T +2T +2T +2T ,
5 1 5 8 6 2 3 4 5 7 8
(4P·x)T = T −T , (2P·x)2T = −T +T +T −T +2T +2T .
7 7 8 8 1 2 5 6 7 8
These DAs can be written as
F(a p·x) = dx dx dx δ(x +x +x −1)e−ip·xΣixiaiF(x ) . (A.2)
i 1 2 3 1 2 3 i
Z
These light cone DAs are scale dependent and can be expanded next–to–leading spin
accuracy with the conformal operators whose explicit expressions are,
0 +
V (x ,µ) = 120x x x [φ (µ)+φ (µ)(1−3x )] ,
1 i 1 2 3 3 3 3
8
0 +
V (x ,µ) = 24x x [φ (µ)+φ (µ)(1−5x )] ,
2 i 1 2 4 4 3
V (x ,µ) = 12x ψ0(µ)(1−x )+ψ−(µ)[x2 +x2 −x (1−x )]
3 i 3 4 3 4 1 2 3 3
+ n
+ ψ (µ)(1−x −10x x ) ,
4 3 1 2
V (x ,µ) = 3 ψ0(µ)(1−x )+ψ−(µo)[2x x −x (1−x )]
4 i 5 3 5 1 2 3 3
n+ 2 2
+ ψ (µ)[1−x −2(x +x )] ,
5 3 1 2
0 + o
V (x ,µ) = 6x [φ (µ)+φ (µ)(1−2x )] ,
5 i 3 5 5 3
0 +
V (x ,µ) = 2[φ (µ)+φ (µ)(1−3x )] ,
6 i 6 6 3
A (x ,µ) = 120x x x φ−(µ)(x −x ) ,
1 i 1 2 3 3 2 1
A (x ,µ) = 24x x φ−(µ)(x −x ) ,
2 i 1 2 4 2 1
A (x ,µ) = 12x (x −x ) [ψ0(µ)+ψ+(µ)]+ψ−(µ)(1−2x ) ,
3 i 3 2 1 4 4 4 3
A (x ,µ) = 3(x −x ) −nψ0(µ)+ψ−(µ)x +ψ+(µ)(1−2x )o ,
4 i 2 1 5 5 3 5 3
A (x ,µ) = 6x (x −xn)φ−(µ) , o
5 i 3 2 1 5
A (x ,µ) = 2(x −x )φ−(µ) ,
6 i 2 1 6
1
T (x ,µ) = 120x x x [φ0(µ)+ (φ− −φ+)(µ)(1−3x )] ,
1 i 1 2 3 3 3 3 3
2
0 +
T (x ,µ) = 24x x [ξ (µ)+ξ (µ)(1−5x )] ,
2 i 1 2 4 4 3
T (x ,µ) = 6x (ξ0 +φ0 +ψ0)(µ)(1−x )+(ξ− +φ− −ψ−)(µ)[x2 +x2 −x (1−x )]
3 i 3 4 4 4 3 4 4 4 1 2 3 3
+n + +
+ (ξ +φ +ψ )(µ)(1−x −10x x ) ,
4 4 4 3 1 2
3 o
T (x ,µ) = (ξ0 +φ0 +ψ0)(µ)(1−x )+(ξ− +φ− −ψ−)(µ)[2x x −x (1−x )]
4 i 5 5 5 3 5 5 5 1 2 3 3
2
n
+ + + 2 2
+ (ξ +φ +ψ )(µ)[1−x −2(x +x )] ,
5 5 5 3 1 2
0 + o
T (x ,µ) = 6x [ξ (µ)+ξ (µ)(1−2x )] ,
5 i 3 5 5 3
1
T (x ,µ) = 2[φ0(µ)+ (φ− −φ+)(µ)(1−3x )] ,
6 i 6 6 6 3
2
T (x ,µ) = 6x (−ξ0 +φ0 +ψ0)(µ)(1−x )+(−ξ− +φ− −ψ−)(µ)[x2 +x2 −x (1−x )]
7 i 3 4 4 4 3 4 4 4 1 2 3 3
n+ + +
+ (−ξ +φ +ψ )(µ)(1−x −10x x ) ,
4 4 4 3 1 2
3 o
T (x ,µ) = (−ξ0 +φ0 +ψ0)(µ)(1−x )+(−ξ− +φ− −ψ−)(µ)[2x x −x (1−x )]
8 i 5 5 5 3 5 5 5 1 2 3 3
2
n
+ + + 2 2
+ (−ξ +φ +ψ )(µ)[1−x −2(x +x )] ,
5 5 5 3 1 2
S (x ,µ) = 6x (x −x ) (ξ0+φ0 +ψ0 +ξ+ +φ+ +ψo+)(µ)+(ξ− +φ− −ψ−)(µ)(1−2x ) ,
1 i 3 2 1 4 4 4 4 4 4 4 4 4 3
3 h i
S (x ,µ) = (x −x )[−(ψ0 +φ0 +ξ0)(µ)+(ξ− +φ− −ψ−)(µ)x
2 i 2 1 5 5 5 5 5 5 3
2
+ + +
+ (ξ +φ +ψ )(µ)(1−2x )] ,
5 5 5 3
P (x ,µ) = 6x (x −x )[(ξ0 −φ0 −ψ0 +ξ+ −φ+ −ψ+)(µ)+(ξ− −φ− +ψ−)(µ)(1−2x )] ,
1 i 3 2 1 4 4 4 4 4 4 4 4 4 3
3
P (x ,µ) = (x −x )[(ψ0 +ψ0 −ξ0)(µ)+(ξ− −φ− +ψ−)(µ)x
2 i 2 1 5 5 5 5 5 5 3
2
9