Power corrections to TMD factorization for Zboson production
JHE
Power corrections to TMD factorization for Zboson
I. Balitsky 0 1 2 4
A. Tarasov 0 1 2 3
0 Upton , NY 11973 , U.S.A
1 Newport News , VA 23606 , U.S.A
2 Norfolk , VA 23529 , U.S.A
3 Physics Department, Brookhaven National Laboratory
4 Physics Department, Old Dominion University
A typical factorization formula for production of a particle with a small transverse momentum in hadronhadron collisions is given by a convolution of two TMD parton densities with cross section of production of the practical applications at a given transverse momentum, though, one should estimate at what momenta the power corrections to the TMD factorization formula become essential. In this paper we calculate the rst power corrections to TMD factorization formula for Zboson production and DrellYan process in highenergy hadronhadron collisions. At the leading order in Nc power corrections are expressed in terms of leading power TMDs by QCD equations of motion.
NLO Computations; QCD Phenomenology

HJEP05(218)
1 Introduction
2
3
5
6
7
3.1
3.2
3.3
4.1
4.2
4.3
TMD factorization from functional integral
Power corrections and solution of classical YM equations
Power counting for background elds
Approximate solution of classical equations
Power expansion of classical quark elds
4
Leading power corrections at s
Q2
q2
?
4.3.1
4.3.2
4.3.3
4.3.4
4.4.1
4.4.2
Leading contribution and power corrections from JAB(x)JBA (0) terms
Leading power contribution 4.2.1
Parametrization of leading matrix elements
Power corrections from JAB(x)JBA (0) terms
Fifth line in eq. (4.19): the leading term in N1c
Parametrization of matrix elements from section 4.3.1
Sixth line in eq. (4.19) Parametrization of matrix elements from section 4.3.3 4.4
Power corrections from JA (x)JB (0) terms
Last two lines in eq. (4.56)
Parametrization of TMDs from section 4.4.1
Results and estimates
Power corrections for DrellYan process
Conclusions and outlook
A Nexttoleading quark
elds
B Formulas with Dirac matrices
C Subleading power corrections
C.1 Second, third, and fourth lines in eq. (4.19)
C.2 Second to fth lines in eq. (4.56)
C.4 Power corrections from
(1) elds
C.3
Gluon power corrections from JA (x)JA (0) terms
{ i {
Introduction
A typical analysis of di erential cross section of particle production in hadronhadron
collisions at small momentum transfer of the produced particle is performed with the help
of TMD factorization [1{10]. However, the question of how small should be the momentum
transfer in order for leading power TMD analysis to be successful cannot be resolved at
the leadingpower level. The sketch of the factorization formula for the di erential cross
section is [1, 11]
d
d d2q
?
=
XZ
f
d2b?ei(q;b)? Df=A(xA; b?; )Df=B(xB; b?; ) (f f ! H)
+ power corrections + Y
terms;
(1.1)
where is the rapidity, q is the momentum of the produced particle in the hadron frame (see
ref. [1]), Df=A(x; z?; ) is the TMD density of a parton f in hadron A, and
(f f ! H) is
the cross section of production of particle H in the scattering of two partons. The common
wisdom is that when we increase transverse momentum q2 of the produced hadron, at rst
the leading power TMD analysis with (nonperturbative) TMDs applies, then at some point
?
power corrections kick in, and
nally at q2
?
into socalled Yterm making smooth transition to collinear factorization formulas. In this
paper we try to answer the question about the rst transition, namely at what q2 power
Q2, where Q2 = q2, they are transformed
corrections become signi cant.
?
In our recent paper [12] we calculated power corrections
duction by gluongluon fusion. The result was a TMD factorization formula with matrix
elements of threegluon operators divided by an extra power of m2H . In this paper we
calculate power corrections
q
2
Q?2 for Zboson production which are determined by
quarkquarkgluon operators. In the leading order Zboson production was studied in [13{21].
The interesting (and unexpected) result of our paper is that at the leadingNc level matrix
elements of the relevant quarkquarkgluon operators can be expressed in terms of leading
power quarkantiquark TMDs by QCD equations of motion (see ref. [22]). The method of
calculation is very similar to that of ref. [12] so we will streamline the discussion of the
general approach and pay attention to details speci c to quark operators.
The paper is organized as follows. In section 2 we derive the TMD factorization from
the double functional integral for the cross section of particle production. In section 3,
which is central to our approach, we explain the method of calculation of power corrections
based on a solution of classical YangMills equations. In section 4 we
nd the leading
power correction to particle production in the region s
perform the orderofmagnitude estimate of power corrections and in section 6 present our
TMD factorization from functional integral
We consider Zboson production in the DrellYan reaction illustrated in gure 1:
hA(pA) + hB(pA) ! Z(q) + X ! l1(k1) + l2(k2) + X;
(2.1)
where hA;B denote the colliding hadrons, and l1;2 the outgoing lepton pair with total
momentum q = k1 + k2.
The relevant term of the Lagrangian for the fermion elds i describing coupling
between fermions and Zboson is (sW
sin W , cW
cos W )
LZ =
Z d4x J Z (x);
J =
e
X
2sW cW i
i (giV
giA 5) i;
(2.2)
where sum goes over di erent types of fermions, and coupling constants giV = (t3L)i 2qis2W
and giA = (t3L)i are de ned by week isospin (t3L)i of the fermion i, see ref. [23]. In this paper
we take into account only u; d; s; c quarks and e; leptons. We consider all fermions to
The di erential cross section of Zboson production with subsequent decay into e+e
d
dQ2dydq2 =
?
e2Q2
1
192ss2W c2W (m2Z
4s2W + 8s4W
Q2)2 + 2Z m2Z
where we de ned the \hadronic tensor" W (pA; pB; q) as
[ W (pA; pB; q)];
(2.3)
W (pA; pB; q) d=ef
XZ d4x e iqxhpA; pBjJ (x)jXihXjJ (0)jpA; pBi
=
1 Z d4x e iqxhpA; pBjJ (x)J (0)jpA; pBi:
(2.4)
1
Z
DB~ DB
d4x e iqxhpA; pBjJ (x)jXihXjJ (0)jpA; pBi
(2.5)
HJEP05(218)
approximation.
functional integral
(2 )4W (pA; pB; q) = XZ
X denotes the sum over full set of \out" states. It should be mentioned that
there is a power correction coming from the leptonic tensor term
q q . However, if we
consider quarks to be massless, the only e ect of the q q term comes from the (square of)
axial anomaly which has an extra factor s2, and such twoloop factor is beyond our tree
The sum over full set of \out" states in eq. (2.4) can be represented by a double
d4x e iqxZ A~(tf )=A(tf )
DA~ DA
Z ~(tf )= (tf )
D ~
D ~D
D
pA (A~~(ti); ~(ti))
pB (A~~(ti); ~(ti))e iSQCD(A~; ~)eiSQCD(A; )J~ (x)J (0) pA (A~(ti); (ti)) pB (A~(ti); (ti)):
In this double functional integral the amplitude hXjJ (0)jpA; pBi is given by the integral
over ; A
elds whereas the complex conjugate amplitude hpA; pBjJ (x)jXi is represented
by the integral over ~; A~ elds. Also,
p(A~(ti); (ti)) denotes the proton wave function at
the initial time ti and the boundary conditions A~(tf ) = A(tf ) and ~(tf ) =
(tf ) re ect
the sum over all states X, cf. refs. [24{26].
We use Sudakov variables p =
p1 + p2 + p , where p1 and p2 are lightlike vectors
?
close to pA and pB, and the notations x
s
so that p q = ( p q + q p) 2
(p; q)? where (p; q)?
lightcone coordinates (x = p 2s x+ and x = p s x ). Our metric is g
2
= (1; 1; 1; 1)
piqi. Throughout the paper, the
x p
1 and x
x p2 for the dimensionless
sum over the Latin indices i; j; : : : runs over two transverse components while the sum over
Greek indices ; ; : : : runs over four components as usual.
Following ref. [12] we separate quark and gluon elds in the functional integral (2.5)
into three sectors (see
gure 2): \projectile"
elds A ; A with j j <
a, \target"
elds
B ; B with j j < b and \central rapidity" elds C ; C with j j > b and j j > a and get
pA (A~~(ti); ~A(ti)) pA (A~(ti); A(ti))
pB (B~~(ti); ~B(ti)) pB (B~ (ti); B(ti))
1This procedure is obviously gaugedependent. We have in mind factorization in covarianttype gauge,
Our goal is to integrate over central elds and get the amplitude in the factorized form,
i.e. as a product of functional integrals over A
elds representing projectile matrix elements
(TMDs of the projectile) and functional integrals over B
elds representing target matrix
elements (TMDs of the target).
In the spirit of background eld method, we \freeze" projectile and target elds and
j j <
get a sum of diagrams in these external elds. Since j j <
a in the projectile elds and
b in the target elds, at the treelevel one can set with power accuracy
the projectile elds and
= 0 for the target elds  the corrections will be O
and
O
mb2Ns , where mN is the hadron's mass. Beyond the tree level, one should expect that the
integration over C
elds will produce the logarithms of the cuto s a and b which will
cancel with the corresponding logs in gluon TMDs of the projectile and the target. The
m2N
as
result of integration over C elds has the schematic form
and eSe represents a sum of disconnected diagrams (\vacuum bubbles") in external elds.
As usual, since the rapidities of central C
elds and of A, B
elds are very di erent, the
result of integration over C
elds is expressed in terms of Wilsonline operators made form
A and B
elds.
Z
pB (B~~(ti); ~B(ti))
pB (B~ (ti); B(ti))eSe (A; A;A~; ~A;B; B;B~; ~B)O(q; x; A; A; A~; ~A; B; B; B~; ~B):
(2.8)
{ 4 {
From integrals over projectile and target elds in the above equation we see that the
functional integral over C
elds should be done in the background of A and B
elds
satisfying
A~(tf ) = A(tf );
~A(tf ) =
A(tf ) and
B~(tf ) = B(tf );
~B(tf ) =
B(tf ):
Combining this with our approximation that at the tree level
for B, B~
B~ = B~(x ; x?), we see that for the purpose of calculation of the functional integral over
central elds (2.7) we can set
and
A(x ; x?) = A~(x ; x?);
A(x ; x?) = ~A(x ; x?)
B(x ; x?) = B~(x ; x?);
B(x ; x?) = ~B(x ; x?):
(2.10)
In other words, since A,
and A~, ~ do not depend on x , if they coincide at x = 1 they
should coincide everywhere. Similarly, since B, B and B~, ~B do not depend on x , if they
coincide at x = 1 they should be equal.
Next, in ref. [12] it was demonstrated that due to eqs. (2.10) the e ective action
Se (A; A; A~; ~A; B; B; B~; ~B) vanishes for background elds satisfying conditions (2.9).2
Summarizing, we see that at the tree level in our approximation
= O(q; x; A; A; B; B);
where now SC = SQCD(C + A + B; C +
A +
B)
SQCD(A; A)
SQCD(B; B) and
SC = SQCD(C~ + A + B; ~C + A + B) SQCD(A; A) SQCD(B; B). It is well known that
~
in the tree approximation the double functional integral (2.11) is given by a set of retarded
Green functions in the background
elds [27{29] (see also appendix A of ref. [12] for the
proof). Since the double functional integral (2.11) is given by a set of retarded Green
functions (in the background
elds A and B), calculation of the treelevel contribution to
in the r.h.s. of eq. (2.11), is equivalent to solving YM equation for (x) (and A (x))
with boundary conditions such that the solution has the same asymptotics at t !
1 as
the superposition of incoming projectile and target background elds.
The hadronic tensor (2.8) can now be represented as3
1
Z
2It corresponds to cancellation of socalled \Glauber gluons", see discussion in ref. [1].
3As discussed in ref. [12], there is a subtle point in the promotion of background
elds to operators.
When we calculate O as the r.h.s. of eq. (2.11) the
elds
A and
B are cnumbers; on the other hand,
after functional integration in eq. (2.5) they become operators which must be timeordered in the right
separated either by spacelike distances or lightcone distances so all of them (anti) commute and thus can
be treated as cnumbers.
{ 5 {
evaluated between the corresponding (projectile or target) states: if
XZ
m;n
O^(q; x; A^; ^A; B^; ^B) =
dzmdzn0cm;n(q; x) ^ A(zm) ^ B(zn0)
(2.13)
(where cm;n are coe cients and
can be any of A ,
or ) then
m;n
W =
1
Z
As we will demonstrate below, the relevant operators are quark and gluon
elds with
Wilsonline type gauge links collinear to either p2 for A elds or p1 for B
elds.
3
3.1
equations4
Power corrections and solution of classical YM equations
Power counting for background
elds
As we discussed in previous section, to get the hadronic tensor in the form (2.12) we need
to calculate the functional integral (2.11) in the background of the
elds (2.10). Since
we integrate over elds (2.10) afterwards, we may assume that they satisfy YangMills
iD= A A = 0;
iD= B B = 0;
DAA
a = g X
DBBa = g X
f
A
f
B
f
f
t
t
a f
a f
A
B
;
;
where A
ig[A ; ) and similarly for B
elds.
It is convenient to choose a gauge where A
= 0 for projectile elds and B
target elds. The rotation from a general gauge (Feynman gauge in our case, see footnote 1)
to this gauge is performed by the matrix
(x ; x ; x?) satisfying boundary conditions
(x ; x ; x?) x !! 1 [x ;
1 ]xA ;
(x ; x ; x?) x !! 1 [x ;
1 ]xB ;
where A (x ; x?) and B (x ; x?) are projectile and target elds in an arbitrary gauge and
[x ; y ]zA denotes a gauge link constructed from A elds ordered along a lightlike line:
[x ; y ]zA = P e 2sig Ryx dz A (z ;z?)
and similarly for [x ; y ]zB .
The existence of matrix
are \frozen".
(x ; x ; x?) was proved in appendix B of ref. [12] by explicit
construction. The relative strength of Lorentz components of projectile and target elds
4As we mentioned, for the purpose of calculation of integral over C elds the projectile and target elds
{ 6 {
(3.1)
(3.2)
(3.3)
2 Z x
s
1
need to calculate the functional integral (2.5) in the background elds of the strength given
by eqs. (3.4).
3.2
Approximate solution of classical equations
As we discussed in section 2, the calculation of the functional integral (2.11) over C elds
in the tree approximation reduces to
nding elds C and
C as solutions of YangMills
Bf +
f
C
) = 0;
D F a (A + B + C) = g X( Af +
Bf +
f
C
) ta( Af +
Bf +
Cf ):
in this gauge was found in ref. [12]
p
=
1 A(x ; x?)
p
=
1 B(x ; x?)
m5=2;
?
spm ;
?
A (x ; x?)
B (x ; x?)
? is a scale of order of mN or q?. In general, we consider W (pA; pB; q) in the region
q2 ; m2N , while the relation between q2 and m2N and between Q2 and s may
be arbitrary. So, for the purpose of counting of powers of s, we will not distinguish between
?
s and Q2 (although at the
nal step we will be able to tell the di erence since our
nal
expressions for power corrections will have either s or Q2 in denominators). Similarly, for
the purpose of power counting we will not distinguish between mN and q? so we introduce
m
? which may be of order of mN or q? depending on matrix element.
Note also that in our gauge (3.6) (3.7) (3.8)
As we discussed above, the solution of eq. (3.6) which we need corresponds to the sum of
set of diagrams in background eld A + B with retarded Green functions, see gure 3. The
retarded Green functions (in the backgroundFeynman gauge) are de ned as
1
+g2(xj p2 +i p0 O
1
+i p0 jy) (xj p2 +i p0 jy) g(xj p2 +i p0 O
p2 +i p0 jy)
1
1
p2 +i p0 O
p2 +i p0 jy)+: : : ;
(xj P2g
1
+2igF
where
and similarly for quarks.
F
fp ; A + B g + g(A + B)2 g
+ 2iF
ig[A + B ; A + B ];
f
{ 7 {
2 Z x
s
1
1
1
$
tions of the central elds are given by retarded propagators.
Hereafter we use Schwinger's notations for propagators in external elds normalized
according to
where we use spacesaving notation d np
confusion, we will use shorthand notation
(xjF (p)jy)
d 4p e ip(x y)F (p);
1
O
O0(x)
d4z(xj jz)O0(z):
D F
a =
X g f ta
f
P= f = 0;
Z
Z
1
O
f
;
{ 8 {
The solution of eqs. (3.6) in terms of retarded Green functions gives elds C and
that vanish at t !
1. Thus, we are solving the usual classical YM equations5
where
A
P
with boundary conditions6
= C + A + B ;
f =
Cf +
Af +
f
B
;
F
(3.12)
A (x) x != 1 A (x ; x?);
A (x) x != 1 B (x ; x?);
(x) x != 1
(x) x != 1
A(x ; x?);
t! 1 0. These boundary conditions re ect the fact that at t !
!
we have only incoming hadrons with A and B
5We take into account only u; d; s; c quarks and consider them massless.
6It is convenient to
x redundant gauge transformations by requirements Ai( 1 ; x?) = 0 for the
projectile and Bi( 1 ; x?) = 0 for the target, see the discussion in ref. [30].
(2dn)pn . Moreover, when it will not lead to a
(3.9)
(3.10)
C
(3.11)
(3.13)
L
a
L
( 1)a =
D F
2g
a + g [0] t
The explicit form of gluon linear terms L(0)a and L(1)a is presented in eq. (3.26) from our
paper [12]. For our purposes we need only the leading term L
With the linear terms (3.15) and (3.18), a couple of rst terms in our perturbative
series are
for quark elds and
P
are operators in external zeroorder elds (3.14). Here we denote the order of expansion in
the parameter ms2? by (: : :)(n), and the order of perturbative expansion is labeled by (: : :)[n]
as usual. The powercounting estimates for linear term in eq. (3.15) comes from eq. (3.4)
in the form
The gluon linear term is
L(0)
m5=2;
?
As discussed in ref. [12], for our case of particle production with qQ?
to nd the approximate solution of (3.11) as a series in this small parameter. We will solve
eqs. (3.11) iteratively, order by order in perturbation theory, starting from the zeroorder
approximation in the form of the sum of projectile and target elds
and improving it by calculation of Feynman diagrams with retarded propagators in the
background elds (3.14).
The rst step is the calculation of the linear term for the trial con guration (3.14).
The quark part of the linear term has the form
A[1]a(x) =
A[2]a(x) = g
d4z
Z
Z
d4z(xj P2g
"
+ (xj P2g
1
jz)abLb (z);
1
{ 9 {
for gluon
elds (in the backgroundFeynman gauge). Next iterations, like
A[3]a(x), will give us a set of treelevel Feynman diagrams in the background elds A + B
Let us consider the elds in the rst order in perturbative expansion:
Here , , and p? are understood as di erential operators
Now comes the central point of our approach. Let us expand quark and gluon
propagators in powers of background
elds, then we get a set of diagrams shown in gure 3.
The typical bare gluon propagator in gure 3 is
1
Since we do not consider loops of C elds in this paper, the transverse momenta in tree
diagrams are determined by further integration over projectile (\A") and target (\B") elds
in eq. (2.8) which converge on either q
? or mN . On the other hand, the integrals over
converge on either q or
Since q qs = Q2
1 and similarly the characteristic 's are either q or
2
q2 , one can expand gluon and quark propagators in powers of p?
(3.22)
s
( + i )( + i ) jy) =
s( + i )( + i )
( + i )( + i )
k
1
s
(xj
(xj
1
?
1
p
=
1
+ i
1
1
+ i jy) =
+ i jy) =
+
=
=
=
1 +
+
d 2p
d 2p
d 2p
?
?
?
Z
Z
Z
p
=
2
+ i
s
2
s
2
s Z
2
s Z
2
s Z
2
s
2
p2?=s
p
=
?
d
+ i
d
+ i
d
d
+ i
+1i , +1i , and ( +i )( +i ) is
1
d
e i (x y) i (x y) +i(p;x y)?
i (2 )2 (2)(x?
?
y ) (x
y ) (x
y );
i (2 )2 (2)(x?
(2 )2 (2)(x?
?
y ) (x
d
+ i
?
y ) (x
e i (x y) i (x y) +i(p;x y)?
y ) (x
y );
e i (x y) i (x y) +i(p;x y)?
y ) (x
y ):
(3.24)
After the expansion (3.23), the dynamics in the transverse space e ectively becomes trivial:
all background elds stand either at x or at 0. The formula (3.21) turns into expansion
elds in section 2.7 The reason is that in the diagrams
like gure 3 with retarded propagators (3.24) one can shift the contour of integration over
and/or
to the complex plane away to avoid the region of small
or . It should be
mentioned, however, that such shift may not be possible if there is pinching of poles in the
integrals over
1
or . For example, if after the expansion (3.23) we encounter ( +i )( i ) ,
the expansion was not justi ed since actual 's in the integral are
was misidenti ed: we have a propagator of B eld rather than of C eld. Fortunately, at
the tree level all propagators are retarded and the pinching of poles never occurs. In the
higher orders in perturbation theory Feynman propagators in the loops cannot be replaced
1
by retarded propagators so after the expansion (3.23) we can get ( +i )( +i 0) . In such
case the pinching may occur so one needs to formulate a subtraction program to get rid of
p
2
s? and hence the eld
pinched poles and avoid double counting of the elds.
Note that the background
elds are also smaller than typical p2
s. Indeed, from
HJEP05(218)
Also (pi + Ai + Bi)2
eq. (3.4) we see that p = 2s
q
2
?
A
p2.8
k
?
m2 ( because
3.3
Power expansion of classical quark
elds
2
Now we expand the classical quark elds in powers of pp?2
k
q
ms2? ) and similarly p
B .
k
m2
s? (the expansion of classical
gluon elds is presented in eqs. (3.35){(3.38) in ref. [12]). From the previous section it
is clear that the leading power correction comes only from the
rst term displayed in
eq. (3.19). Expanding it in powers of p2?=p2 as explained in the previous section, we obtain
k
(x) =
[0](x) +
[1](x) +
[2](x) +
=
(A0) +
(B0) +
(A1) +
(B1) + : : : ;
(3.26)
where
7Such cuto s for integrals over C
e ective theory (SCET), see review [31].
8The only exception is the
e ectively the expansion in powers of these elds is cut at the second term.
elds are introduced explicitly in the framework of softcollinear
elds B i or A i which are of order of sm? but we saw in ref. [12] that
A
(0) =
A
(0) =
B
(0) =
B
(0) =
A + 2A;
A + 2A;
B + 1B;
B + 1B;
s
s
gp=2 iBi
gp=1 iAi
A
B
1
1
i
i
1
+ i
1
+ i
A;
B;
iBi s
gp=2
iAi s
gp=1
;
:
(3.27)
2A =
2A =
1B =
1B =
+ i
1
i
A(x ; x?)
(x ; x?)
i
i
Z x
Z x 1
dx0
1
dx0
A(x0 ; x?);
A(x0 ; x?)
It is easy to see that power counting of these quark elds has the form
(0)
A
(0)
B
m3=2:
?
As to quark elds
(1), we present their explicit form in appendix A and prove in
appendix C that their contribution is small in the kinematic region s
Q2.
4
Leading power corrections at s
Q2
q2
?
As we mentioned in the introduction, our method is relevant to calculation of power
corrections at any s; Q2
physically interesting case s
?
Q2
q
?2 which we consider in this paper.9
q2 ; m2N . However, the expressions are greatly simpli ed in the
As we noted above, we take into account only u; d; s; c quarks and consider them
massless. The hadronic tensor takes the form
= JA + JB + JAB + JBA;
9We also assume that Zboson is emitted in the central region of rapidity so
qs
qs
Q2.
10We denote the weak coupling constant by e=sW and reserve the notation \g" for QCD coupling constant.
where (cW
cos W , sW
sin W )10
J =
e
where
Z
1
d2x? ei(q;x)? W ( q; q; x?);
dx dx e i qx i qx hpA; pBjJ (x ; x ; x?)J (0)jpA; pBi;
After integration over central elds in the tree approximation we obtain
dx dx e i qx i qx hpAjhpBjJ (x ; x ; x?)J (0)jpAijpBi;
and similarly for JB and JBA. Hereafter we use notation
of au;c = (1
83 s2W ) or ad;s = (1
43 s2W ) depending on quark's avor.
where
The quark elds are given by a series in the parameter ms2? , see eqs. (3.27) and (A.2),
can be any of u; d; s or c quarks.11 Accordingly, the currents (4.4) can be expressed
(a
5) where a is one
as a series in this parameter, e.g.
(0)
JAB
(1)
JAB
=
=
e
e
The leading power contribution comes only from product JAB(x)JBA (0) (or
JBA(x)JAB (0)), while power corrections may come from other terms like JA (x)JB (0).
We will consider all terms in turn.
Leading contribution and power corrections from JAB(x)JBA (0) terms
Power expansion of JAB(x)JBA (0) reads
(4.6)
q
2
In appendix C.4 we demonstrate that terms
q
2
q?s which are much smaller than
(1) lead to power corrections
q
2
Q?2 if Zboson is emitted in the central
region of rapidity. Note that since we want to calculate the leading power corrections,
hereafter we substitute Q2 with Q2. In the limit s
Q2
q2 this change of variables can
only lead to errors of the order of subleading power terms.
?
(A0)(0), they can be decomposed using
First, let us consider the leading power term coming from the rst term in the r.h.s. of this
equation.
11As we mentioned, we will need only rst two terms of the expansion given by eqs. (3.27) and (A.2).
4.2
JB(0A) (x)JA(0B) (0). Using Fierz transformation
As we mentioned, the leadingpower term comes from JA(0B) (x)JB(0A) (0) and
( A
=
[a
where
is implied).
with au;c = (1
83 s2W ) and ad;s = (1
43 s2W ) one obtains
16s2W c2W hpAjhpBjJA(0B) (x)JB(0A) (0) + (x $ 0)jpAijpBi =
o
n
^u(0)jAi;
h Bu(x)
Bu(0)i
hBj ^u(x)
^u(0)jBi
and similarly for other matrix elements (summation over color and Lorentz indices
As usual, after integration over background elds A and B we promote A,
B,
B to operators A^, ^. A subtle point is that our operators are not under Tproduct
ordering so one should be careful while changing the order of operators in formulas like
Fierz transformation. Fortunately, all our operators are separated either by spacelike
intervals or lightlike intervals so they commute with each other.
In a general gauge for projectile and target elds these expressions read (see eq. (3.2))
hAj ^f (x)
hBj ^f (x)
^f (0)jAi = hAj ^f (x ; x?) [x ;
^f (0)jBi = hBj ^f (x ; x?) [x ;
1 ]x[x?; 0?] 1 [ 1 ; 0 ]0 ^f (0)jAi;
1 ]x[x?; 0?] 1 [ 1 ; 0 ]0 ^f (0)jBi (4.11)
and similarly for hAj ^f (0)
^f (x)jAi and hBj ^f (0)
^f (x)jBi.
From parametrization of twoquark operators in section 4.2.1, it is clear that the leading
power contribution to W (q) of eq. (4.1) comes from the product of two f10 s in eq. (4.13)
and (4.15). It has the form [32]
e
2
(4.9)
(4.10)
A and
All other terms in the product of eqs. (4.13) and (4.15) give higher power contributions
q
2
s? W lt(q) (but not
2
q
Q?2 W lt(q))12 so they can be neglected at Q2
s. Similarly, the
contribution of two matrix elements in eq. (4.17) is
can be neglected as well.
Parametrization of leading matrix elements
m2
s? in comparison to W lt(q) so it
Let us rst consider matrix elements of operators without 5. The standard parametrization
of quark TMDs reads
dx d2x? e i x +i(k;x)? hAj ^f (x ; x?)
dx d2x? ei x i(k;x)? hAj ^f (x ; x?)
s
2m2N f f ( ; k2 );
3
?
dx d2x? ei x i(k;x)? hAj ^f (x ; x?) ^f (0)jAi = mN ef ( ; k2 )
?
for quark distributions in the projectile and
for the antiquark distributions.13
$
12The trivial but important point is that any f (x; k?) may have only logarithmic dependence on Bjorken
x but not the power dependence
1 . Indeed, at small x the cuto
x
of corresponding longitudinal integrals
comes from the rapidity cuto
a, see the discussion in section 2. Thus, at small x one can safely put x = 0
and the corresponding logarithmic contributions would be proportional to powers of
s ln a (or, in some
cases, s ln2
a, see e.g. ref. [33]). Also, a more technical version of this argument was presented on page 12.
13In an arbitrary gauge, there are gauge links to
1 as displayed in eq. (4.11).
The corresponding matrix elements for the target are obtained by trivial replacements
dx d2x? e i x +i(k;x)? hBj ^f (x ; x?)
^f (0)jBi
= p2 f1f ( ; k?2) + k f f ( ; k?2) + p1
? ?
s
2m2N f f ( ; k2 );
3
?
dx d2x? e i x +i(k;x)? hBj ^f (x ; x?) ^f (0)jBi = mN ef ( ; k2 );
?
dx d2x? ei x i(k;x)? hBj ^f (x ; x?)
^f (0)jBi
=
s
2m2N f f ( ; k2 );
3
?
dx d2x? ei x i(k;x)? hBj ^f (x ; x?) ^f (0)jBi = mN ef ( ; k2 ):
?
(4.14)
(4.15)
(4.16)
Matrix elements of operators with 5 are parametrized as follows:
1 Z
16 3
1 Z
16 3
The corresponding matrix elements for the target are obtained by trivial replacements
Finally, for future use we present the parametrization of timeodd TMDs
1 Z
16 3
1 Z
16 3
=
=
1
mN
(k p
? 1
+
2mN (k p
? 2
1
mN
(k p
? 1
2mN (k p
? 2
s
s
dx d2x? e i x +i(k;x)? hAj ^f (x ; x?)
$
)h1?f ( ; k?2) +
s
2mN (p1 p2
$
)hf ( ; k?2)
dx d2x? ei x i(k;x)? hAj ^f (x ; x?)
$
$
$
)h1?f ( ; k?2)
)h3?f ( ; k?2)
s
2mN (p1 p2
$
)hf ( ; k?2)
and similarly for the target with usual replacements p1 $ p2, x
$
.
Note that the coe cients in front of f3, gf?, h and h3? in eqs. (4.13), (4.15), (4.17),
and (4.18) contain an extra 1s since p2 enters only through the direction of gauge link so
the result should not depend on rescaling p2 !
not contribute to W (q) in our approximation.
p2. For this reason, these functions do
4.3
Power corrections from JAB(x)JBA (0) terms
The terms in eq. (4.7) proportional to elds are
First, as we demonstrate in appendix C.1, the terms in the second, third, and fourth lines
lead to negligible power corrections
fth and sixth lines.
q?s , so we are left with contribution of the
m
A (x) (a
1 + a
2
=
=
+ (1
1 + a
2
2
a
2
a
5) 1mB(x)
n
B(0) (a
m
A (x)
n
2A(0)
n
B(0)
Next, separating colorsinglet contributions
hA; Bj( A (Bj )nk k
m
A)(
n
B(Ai)
ml Bl)jA; Bi = hA; Bj( A (Ai)
m
ml k
A)(
B(Bj )nk Bl)jA; Bi
n
=
1
Nc
hAj( AmAiml Al)jAihBj( BnBjnk Bk)jBi
Fifth line in eq. (4.19): the leading term in
Let us start with the term
mation (4.8) we obtain
1B(x)
2A(0) . Performing Fierz
transfor1+a2
2
+(1 a2)
g
=
2Nc
1 + a
2
+
a
2
s2
s2
1
2a
s2
(1
(1
1B(x)
2A(0)
B(0)
j 1
j 1
2
j 1
A(x)Ak(x) ip=2
A(0)
B(0)Bj(0) ip=
A(x)Ak(x)p=
A(0)
B(0)Bj(0)p=
a
A(x)Ak(x) i 5p=2
A(0)
B(0)Bj(0) ip=
1
1
1
k 1
k 1
k 1
B(x) +( i
B(x)
( j
B(x) +( i 5
i
k
$
i
$ i 5
j
5
$ i
Using equations (B.3), (B.4), and (B.8) from appendix B we can rewrite eq. (4.22) as
A(x) (a
5) 1B(x)
B(0) (a
5) 2A(0)
A(x)p= [Ai(x)
2
i 5A~i(x)]
1
A(0)
B(0)p= [Bi(0)
1
i 5B~i(0)]
1
B(x)
A(x)Ak(x)p=2 j
A(0)
1
B(0) hBj (0)p=
1
k
2
A(x)p= [ 5Ai(x)
B(0)p= [Bi(0)
1
i 5B~i(0)]
B(x)
+ O
j $ k + gjkBi(0)p=1 i
i 1
B(x)
iA~i(x)]
A(0)
1
1
m8
s
? :
(4.20)
m8
?
s
:
(4.21)
(4.22)
i
k
5)
5)
i
5) :
(4.23)
For forward matrix elements we get
Z
Z
=
=
hAj ^(x ; x?)p=2[A^i(x ; x?)
i 5A~^i(x ; x?)] 1 ^(0)jAi
s
k
:
1+a2
s2 hAj ^(x ; x?)p=2(A^i i 5A~^i)(x ; x?) ^(0)jAi
s2 hAj ^(x ; x?)A^j (x ; x?)p=2 j ^(0)jAi
and similarly for other Lorentz structures in eq. (4.23). The corresponding contribution of
HJEP05(218)
the r.h.s. of eq. (4.23) to W ( q; q; x?) takes the form14
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
Note that for unpolarized hadrons hBj ^(0)(A^j (0)p=1
is easy to see that the last line of eq. (4.23)
j $ k) ^(x ; x?)jBi = 0. Also, it
2a
s2 hAj ^(x)p=2[A^i(x)
^
i 5A~i(x)] ^(0)jAihBj ^(0)p=1( 5A^i(0)
iA~^i(0)] ^(x)jBi
gives zero contribution. Indeed, let us consider the rst term in the r.h.s. of this equation.
hAj ^(x)p=2[A^i(x)
hBj ^(0)p=1( 5A^i(0)
^
i 5A~i(x)] ^(0)jAi
iA~^i(0)] ^(x)jBi
xi;
ij xj ;
this term vanishes (and similarly all other terms in the r.h.s. of eq. (4.26) do vanish too).
Repeating the same steps for the second term in the fth line in eq. (4.19) we get
2A(x) (a
5) B(x)
1B(0) (a
5) A(0)
Since
Ncg 2
=
1+a2
s2
+
1 a
2
s2
2a
s2
redundant.
A
1
A
1
A
1
B
B
1
1
(x)p=2[Ai(0)+i 5A~i(0)] A(0)
B
1
(0)p=1[Bi(x)+i 5B~i(x)] B(x)
(x)Ak(0)p=2 j A(0)
(0)[Bj (x)p=1
k j $ k +gjkBi(x)p=1 i] B(x)
(x)p=2[ 5Ai(0)+iA~i(0)] A(0)
(0)p=1[Bi(x)+i 5B~i(x)] B(x) +O
m8
s
? :
14After specifying the projectile and target matrix elements the \A" and \B" labels of the elds become
g Z
8 3s
g Z
8 3s
+
+
g Z
8 3s
+
Z x
g Z
8 3s
+
Z 0
1
1
example:
hAj
(
! s
2 Z x
1
s
2 F^ i(0)
Z 0
1
s
2 F^ i(x ; x?)
Z x
1
)
Z x
1
dx?dx e i x +i(k;x)? hAj
^f (x ; x?)A^i(x ; x?)
)
k2
dx0 ^f (x0 ; x?) 2s F^ i(x ; x?) p=2 i ^f (0)jAi = i m?N htfw3( ; k?2)
ih~tfw3( ; k?2) ;
dx?dx e i x +i(k;x)? hAj
^f (0)A^i(0)
k2
dx0 ^f (x0 ; 0?) 2s F^ i(0) p=2 i ^f (x ; x?)jAi = i m?N htfw3( ; k?2)
ih~tfw3( ; k?2)
and similarly for the target matrix elements.
For completeness, let us present the structure of gauge links in an arbitrary gauge, for
In this section we present parametrization of matrix elements from section 4.3.3. Similarly
to eqs. (4.40){(4.42) we de ne
dx0 ^f (x0 ; 0?) jAi = i ? htfw3( ; k?2) + ih~tfw3( ; k?2) ;
k
2
mN
dx0 ^f (x0 ; x?) jAi = i ? htfw3( ; k?2) + ih~tfw3( ; k?2)
and similarly for the target matrix elements. Note that unlike twoquark matrix elements,
quarkquarkgluon ones may have imaginary parts which we denote by functions with tildes.
By complex conjugation we get
)
(
)
(
k
2
mN
(A0)(x)
^f (x ; x?)A^j (x ; x?) +
dx0 ^f (x0 ; x?) 2s F^ j (x ; x?) p=2 i ^f (0)jAi
)
dx0 hAj
n ^f (x ; x?)[x ; x0 ]xF^ j (x0 ; x?)[x0 ;
+ ^f (x0 ; x?)[x0 ; x ]xF^ j (x ; x?)[x ;
o
1]x [x?; 0?] 1 [ 1 ; 0 ]0? p=2 i ^f (0)jAi:
4.4
Power corrections from JA (x)JB (0) terms
Power corrections of the second type come from the terms
A(x)
+
+
(A1)(x)
(A0)(x)
In appendix C.4, we will demonstrate that terms
(1) are small in our kinematical region
Q2
Terms
As we prove in appendix C.2, the leading power correction comes from last two lines in
eq. (4.56). We will consider them in turn.
Using eq. (3.27) and separating colorsinglet matrix elements, we rewrite the sixth line in
eq. (4.56) as
A(x)
B(0)
1B(0) +[ A(x)
2A(x)
1B(0)
(4.57)
=
=
+
+
g
2
where we used eqs. (B.4) and (B.5). For the forward matrix elements
dx e i qx hAj
^ 1
(x ; x?)p=2A^i(0) ^(x ; x?)jAi
A(0)
1
Z
Z
Z
Z
=
=
=
=
q
1
1
1
1
dx0 hAj ^(x0 ; x?)
s
2p=2 F^ i(0) ^(x ; x?)jAi;
dx e i qx hAj ^(x ; x?)p=2A^i(0)
1 ^(x ; x?)jAi
dx e i qx hBj ^(0)p=1A^i(x ; x?)
1 ^(0)jBi
s
2p=2 F^ i(0) ^(x0 ; x?)jAi;
s
dx e i qx hBj
^ 1
s
2p=1 F^ i(x ; x?) ^(0)jBi:
(4.58)
The corresponding contribution to W ( q; q; x?) takes the form
2p=1 F^ i(x ; x?) i 5F~^ i(x ; x?) ^(x0 ; 0?)jBi
(Z x
1
dx0
Z 0
1
s
+hAj ^(x ; x?) s
hBj ^(x0 ; 0?) s
2p=2 F^ i(0)+i 5F~^ i(0) ^(x ; x?)jAi
2p=2 F^ i(0) i 5F~^ i(0) ^(x0 ; x?)jAi
2p=1 F^ i(x ; x?)+i 5F~^ i(x ; x?) ^(0)jBi +x $ 0
Similarly, for the seventh line in eq. (4.56) using eqs. (3.27) and (B.6) one obtains
(4.59)
)
1+O
:
m2
?
s
A(x)
1B(0)
A
1
A(x)p=2 hAi(0) i 5A~i(0)i 1
B(0)p=1 hBi(x) i 5B~i(x)i 1
2a
A
1
(x)p=2 hAi(0)+i 5A~i(0)i
A(x)
B
1
(0)p=1 h 5Bi(x)+iB~i(x)i
B(0)
A(x)p=2 hAi(0) i 5A~i(0)i 1
B(0)p=1 h 5Bi(x) iB~i(x)i 1
B (0)
B (0)
Using eq. (4.58) one obtains the contribution to W ( q; q; x?) in the form
g2e2(a2 + 1)
8(2 )4s2W c2W (Nc2
1)Q2s2
Z
(Z x
1
dx0
Z 0
1
2p=2 F^ i(0) + i 5F~^ i(0) ^(x ; x?)jAi
hBj ^(x0 ; 0?) s
+ hAj ^(x ; x?) s
2p=1 F^ i(x ; x?) + i 5F~^ i(x ; x?) ^(0)jBi
2p=2 F^ i(0) i 5F~^ i(0) ^(x0 ; x?)jAi
s
hBj ^(0)
2p=1 F^ i(x ; x?) i 5F~^ i(x ; x?) ^(x0 ; 0?)jBi + x $ 0 :
(4.61)
)
Here we used the fact that the last term in eq. (4.60)
2a
2a hAj ^(x0 ; x?)
2p=2 F^ i(0) + i 5F^~ i(0) ^(x ; x?)jAi
hBj ^(x0 ; 0?)
+ hAj ^(x ; x?)
2p=1
s
2p=1
s
2p=2 F^ i(0)
5F^ i(x ; x?) + iF~^ i(x ; x?) ^(0)jBi
i 5F^~ i(0) ^(x0 ; x?)jAi
5F^ i(x ; x?)
iF~^ i(x ; x?) ^(x0 ; 0?)jBi
(4.62)
(4.63)
(4.64)
(4.65)
gives no contribution since
hAj ^(x0 ; x?)
hBj ^(x0 ; 0?)
2p=1
s
2p=2 F^ i(0)
5F^ i(x ; x?)
i 5F^~ i(0) ^(x ; x?)jAi
iF~^ i(x ; x?) ^(0)jBi
xi;
ij x
j
same as in eq. (4.27).
of the 6th and 7th lines in eq. (4.56) in the form
Next, using parametrizations (4.66) from the next section we obtain the contribution
W 6+7th( q; q; q?) =
e
2
8s2W c2W (Nc2 1)Q2
Z
d2k? (k; q k)?hn2(1+a2u)
j1twu3( q; k?)j2twu3( q; q?
k?) ~j1twu3( q; k?)~j2twu3( q; q?
k )
?
+(1 a2u) j1twu3( q; k?)j1twu3( q; q?
k?)+~j1twu3( q; k?)~j1twu3( q; q?
k )
?
+j2twu3( q; k?)j2twu3( q; q?
k?)+~j2twu3( q; k?)~j2twu3( q; q?
k?) + q $ q
o
+nu $ co+nu $ do+nu $ soi
1+O
m2
?
s
;
where q $
q contribution comes as usually from the (x $ 0) term in eq. (4.59).
4.4.2
Parametrization of TMDs from section 4.4.1
We parametrize TMDs from section 4.4.1 as follows
d2x?dx e i x +i(k;x)? Z x
2p=2 F^ i(0) + i 5F^~ i(0) ^(x ; x?)jAi
g Z
8 3s
g Z
8 3s
= ki j1tw3( ; k?2) + i~j1tw3( ; k?2) ;
d2x?dx e i x +i(k;x)? Z x
= ki j2tw3( ; k?2)
i~j2tw3( ; k?2) :
1
1
s
2p=2 [F^ i(x)
i 5F^~ i(x)] ^(x0 ; 0?)jAi
dx0 hAj ^(x0 ; 0?)
2p=2 F^ i(x) + i 5F^~ i(x) ^(0)jAi
(4.66)
HJEP05(218)
g Z
8 3s
g Z
8 3s
F i $ F^ i.
^
gauge:
= ki j1tw3( ; k?2)
i~j1tw3( ; k?2) ;
d2x?dx e i x +i(k;x)? Z 0
Target matrix elements are obtained by usual substitutions
, p=2 $ p=1, x
$ x , and
For completeness let us present the explicit form of the gauge links in an arbitrary
^(x0 ; x?)F^ i(0) ^(x ; x?) !
Results and estimates
Combining eqs. (4.12), (4.31), (4.51), and (4.64) we get the leading term and rst power
corrections to W (q) in the kinematic region s
Q2
?
q2 in the form
e
2
d2k
?
"(
(1+a2u) 1 2
k2 (q k)2
k?)+2(a2u 1) ?m2N Q2
+
2k?2(q k)2
(Nc2 1)Q2m2N (a2u 1) hhtuw3( z; k?)htuw3( z; q?
?
Nc2 1
Q2
Nc (k; q k)? 2(1+a2u) j1twu3( z; k?)j2twu3( z; q?
? h1?u( z; k?)h1?u( z; q?
k )
?
k?)+h~tuw3( z; k?)h~tuw3( z; q?
k )
?
i
k?) ~j1twu3( z; k?)~j2twu3( z; q?
k )
?
+(1 a2u) j1twu3( z; k?)j1twu3( z; q?
k?)+j2twu3( z; k?)j2twu3( z; q?
k )
?
+~j1twu3( z; k?)~j1twu3( z; q?
k?)+~j2twu3( z; k?)~j2twu3( z; q?
)
n
1+O
o
n
o
n
o
#
k )
?
m2
?
s
;
(5.1)
where the momentum of the produced Zboson is q =
zp1 + zp2 + q .
?
i
(Nc2 1)Q2 2(1+a2u) @i?j1twu3( z; b?)@i?j2twu3( z; b?) @i?~j1twu3( z; b?)@i?~j2twu3( z; b?)
+( z $ z) + u $ c + u $ d + u $ s
1+O
n
n
#
m2
?
;
(5.2)
where f1u( z; b?)
R d2k?e i(k;b)? f1u( z; k?) etc.
Note that in the leading order in Nc power corrections are expressed in terms of leading
power functions f1 and h1?. To estimate the order of magnitude of power corrections, one
can assume that N1c is a good parameter and leave only rst term in the r.h.s. of eq. (5.1):
2
8s2W c2W Nc
Z
"(
d2k
?
n
o
n
o
+
n
u $ s
1) ?
k2 (q
o
#
2
(k; q
k)2
Q2
k)?
? h1?u( z; k?)h1?u( z; q?
k )
?
1 + O
+ O
:
m2
?
s
1
Nc
For completeness, let us present our nal result in the transverse coordinate space
2
"(
(1+a2u) f1u( z; b?)f1u( z; b?)
2m(a2N2u Q12) @?2h1?u( z; b?)@?2h1?u( z; b?)
2
2(a2u 1)
(Nc2 1)Q2m2N
Nc
2
tails of TMD's f1
approximate
Next, eq. (5.3) is a treelevel formula and for an estimate we should specify the rapidity
calculated only tree diagrams made of C elds we have a = z and b =
cuto s for f1's and h1?'s. As we discussed in section 2, the rapidity cuto for f1( z; k?2) is
a and for f1( z; k?2) b, where a and b are rapidity bounds for central elds. Since we
z in eq. (5.1).15
Next, power corrections become sizable at q2
?
m2N where we probe the perturbative
?
k12 and h?
1
?
k14 [34]. So, as long as m2N
Q2 we can
(up to logarithmic corrections). Similarly, for the target we can use the estimate
f1( z; k?2) '
f (k2z) ; h1?( z; k?2) '
m2N h( z) ; f1 '
f (k2z) ; h1? '
f1( z; k?2) '
f (k2z) ; h1?( z; k?2) '
m2N h( z) ; f1 '
f (k2z) ; h1? '
k4
?
k4
?
the doublelog and/or singlelog evolution of TMDs.
15In general, we should integrate over C elds in the leading log approximation and match the logs to
?
?
k
2
?
m2N h( z)
k4
?
m2N h( z)
k4
?
(5.3)
(5.4)
(5.5)
2
2
(k; q
d k
? k2 (q
?
k)2
?
1)[hu( z)hu( z) + hu( z)hu( z)] mQ2N2 )
o
+ u $ c + u $ d + u $ s
n
HJEP05(218)
e
2
k)2
?
1
2
(k; q
k)?
Q2
hn(1 + a2u)[fu( z)fu( z) + fu( z)fu( z)]o + nu $ co + nu $ do + nu $ soi:
Here we used the fact that due to the \positivity constraint" h1?(x; k?2)
we can safely assume that the numbers f (x) and h(x) in eqs. (5.4) and (5.5) are of the same
jmk?Nj f1?(x; k?2) [
35
],
order of magnitude so the last term in the third line in eq. (5.6)
Thus, the relative weight of the leading term and power correction is determined by the
mQ2N2 can be neglected.
factor 1
2 (k;q
Q2k)? . The integrals over k
by m2N and from above by Q2 so we get an estimate
? are logarithmic and should be cut from below
Substituting this to eq. (5.1) we get the following estimate of the strength of power
corrections for Zboson production
(5.6)
o
#
(5.7)
(6.1)
hn(1 + a2u)[fu( z)fu( z) + fu( z)fu( z)]o + nu $ co + nu $ do + nu $ soi;
where we assumed that the
rst term is determined by the logarithmical region
q
2
k
2
m2N and the second by Q2
rection reaches the level of few percent at q?
2
correction becomes bigger, but the validity of the approximation Qq2
?
2
?
q2 . By this estimate, the power
cor20 GeV. Of course, when q2 increases, the
?
1 worsens. Moreover, we have ignored all logarithmic (and doublelog) evolutions which can signi cantly change the relative strength of power corrections.
6
Power corrections for DrellYan process
In this section we consider contribution to the cross section of the DrellYan process which is determined by the hadronic tensor
e
2
4s2W c2W Nc q
1
2 ln
q
2
m?2N +
1
Q2 ln q2
Q2
W (pA; pB; q) =
where J em = eu u
Z 2
1 2 Z
16 4 s
for active avors in our kinematical region.
d x? ei(q;x)?W ( q; q; x?);
dx dx e i qx i qx hpA; pBjJ em(x ; x ; x?)J em(0)jpA; pBi;
u + ed d
d + es s
s + ec c
c is the electromagnetic current
2
u Nc2
1
Nc
k)2
?
?
?
j1twu3( q; k?)j1twu3( q; q?
~j1twu3( q; k?)~j1twu3( q; q?
k )
?
k )
?
j2twu3( q; k?)j2twu3( q; q?
k )
?
~j2twu3( q; k?)~j2twu3( q; q?
i
k?) + ( q $
o
u $ c +
u $ d
n
u $ s
#
o :
Let us present also the largeNc estimate similar to eq. (5.6)
2u 1 2
? k2 (q k)2
(k; q k)? [fu( q)fu( q)+fu( q)fu( q)]
+2e2u[hu( q)hu( q)+hu( q)hu( q)]
'
? k2 (q k)2
1 2
n
o
n
+ u$c + u$d + u$s
o
n
From the results of the present paper it is easy to extract power corrections to W .16
We replace constants au in eq. (5.1) by ef2 and remove factors \1" from expressions like
a
2
1. One can formally set au ! 1 in
and multiply by e2u. After that, we repeat the procedure for other avors and get
(au
5), divide the result (5.1) by a2u,
"(
2u 1
2
(k; q
Q2
k)? f1u( q; k?)f1u( q; q?
k )
?
? h1?u( q; k?)h1?u( q; q?
k )
?
htuw3( q; k?)htuw3( q; q?
k?) + h~tuw3( q; k?)h~tuw3( q; q?
k )
?
(k; q
1
Q2
k)? h2j1twu3( q; k?)j2twu3( q; q?
k )
?
2~j1twu3( q; k?)~j2twu3( q; q?
k )
?
e2u[fu( q)fu( q)+fu( q)fu( q)]+(u $ c)+(u $ d)+(u $ s) :
Obviously, the relative strength of leadingtwist terms and power corrections is the same as
for Zboson production so from our nave estimate (5.7) one should expect power corrections
of order of few percent starting from q
7
Conclusions and outlook
41 Q.
In this paper we have calculated the highertwist power correction to Zboson production
(and DrellYan process) in the kinematical region s
Q2
q2 . Our backoftheenvelope
estimation of importance of power corrections tells that they reach a few percent of the
?
leadingtwist result at q?
4
ref. [21] by comparing leadingorder ts to experimental data.
1 Q which surprisingly agrees with the same estimate made in
16The problem of calculating power corrections for W
with nonconvoluted indices is a separate issue
which we hope to address in a di erent publication.
(6.2)
(6.3)
o
#
Of course, we made our estimate without taking into account the TMD evolution,
notably the most essential doublelog (Sudakov) evolution. One should evolve projectile
TMD from
a = q to ~a = q
2
q
2
q?s = q Q?2 , target TMDs from
b =
q to ~b = q
2
q?s =
2
q Qq?2 ,
and match to the result of leadinglog calculation of integral over central elds in the
rapidity interval between ~a and ~b.
To accurately match these evolutions, we hope to use logic borrowed from the operator
product expansion. We write down a general formula (2.14)
W =
1
(2 )4
Z
where the coe cient functions cm;n(q; x) are determined by integrals over C elds and do
not depend on the form of projectile or target. To
nd these coe cients in the rstloop
order, we integrate over C elds in eq. (2.11) with action SC = SQCD(C + A + B; C +
A +
B)
SQCD(B; B) but without any rapidity restrictions on C elds,
and subtract matrix elements of the operators ^ A(zm) ^ B(zn0) in the background elds A,
A and B,
B multiplied by treelevel coe cients. Both the integrals over C elds in
eq. (2.11) and matrix elements of ^ A(zm) ^ B(zn0) will have rapidity divergencies which will
be canceled in their sum so what remains are the logarithms (or double logs) of the ratio
of kinematical variables (Q2 in our case) to the rapidity cuto s a of the operators ^ A(zm)
and b of ^ B(zn0). Using the above logic we hope to avoid the problem of doublecounting
of elds which arises when integrals over longitudinal momenta of C elds got pinched at
small momenta (see the discussion in the end of secttion 3.2). The work is in progress.
It should be mentioned that, as discussed in ref. [12], our rapidity factorization is
?
di erent from the standard factorization scheme for particle production in hadronhadron
scattering, namely splitting the diagrams in collinear to projectile part, collinear to target
part, hard factor, and soft factor [1]. Here we factorize only in rapidity and the Q2 evolution
arises from k
2 dependence of the rapidity evolution kernels, same as in the BK (and NLO
BK [36]) equations. Also, since matrix elements of TMD operators with our rapidity cuto s
are UV nite [37, 38], the only UV divergencies in our approach are usual UV divergencies
absorbed in the e ective QCD coupling.
It is worth noting that recently the treatment of power corrections was performed
operators will be available.
in the framework of SCET theory (see e.g. refs. [39{41]). However, since our rapidity
factorization is di erent from factorization used by SCET, the detailed comparison of
power corrections to Zboson (or Higgs) production would be possible when SCET result
for TMD corrections in the form of m12Z times matrix elements of quarkantiquarkgluon
Let us note that we obtained power corrections for DrellYan hadronic tensor
convoluted over Lorentz indices. It would be interesting (and we plan) to calculate the
highertwist correction to full DY hadronic tensor. Also, it is well known that for semiinclusive
deep inelastic scattering (SIDIS) and for DY process the leadingorder TMDs have di erent
directions of Wilson lines: one to +1 and another to
1 [42, 43]. We think that the same
directions of Wilson lines will stay on in the case of power corrections and we plan to study
this question in forthcoming publications.
Acknowledgments
The authors are grateful to S. Dawson, A. Prokudin, T. Rogers, R. Venugopalan, and
A. Vladimirov for valuable discussions. This material is based upon work supported by
the U.S. Department of Energy, O
ce of Science, O
ce of Nuclear Physics under
contracts DEAC0298CH10886 and DEAC0506OR23177 and by U.S. DOE grant
DEFG0297ER41028.
A
Nexttoleading quark
elds
In this section we present the explicit expressions for the nexttoleading quark elds (1).
It is convenient to separate these elds in \left" and \right" components:
(1) =
(11) +
1
p=1p=2 (1);
(1)
2
p=2p=1 (1):
The nexttoleading term in the expansion of the elds (3.19) has the form:
(11A) =
(21A) =
(11B) =
(21B) =
(11A) =
(11B) =
(21B) =
1
1
1
1
gp=1 iBi A
2gp=2p=1 B
gp=2 iAi B
2gp=1p=2 A
1
1
A
B
s2 2 p=2B Bj A +
g i
s2 p=1p=2 Pi
1
1 jBj A
1
2g
s2 p=1p=2 (A[1])(0) A;
g i
s2 p=2p=1 Pi
1 jBj A
1
2g
s2 p=2p=1 (A[1])(0) A
g i j
s2 p=
1
g i
s2 p=2p=1 Pi
1
1 kBk A;
1
2g
s2 p=2p=1 (A[1])(0) B;
1 jAj B
1
2g
s2 p=1p=2 (A[1])(0) B
1 kAk B;
s2 2 p=1A Aj B +
A iBi s(
1
i
B
A
B
B iAi s(
1
i
A
2gp=1p=2
s2
2g2 j
2gp=2p=1
s2
2g2 j
ABjB p=2 s2 2 + A kBk
g i
s2 p=1p=2 Pi
g i j
s2 p=
A jBjp=2p=1
A jBjp=1p=2
1
1
i Pi
1
1
i Pj
1
B jAjp=1p=2
B jAjp=2p=1
1
1
i Pi
1
i Pi
1
BAjA p=1 s2 2 + B kAk Pj Pi p=
g j i
1 s2 ;
i Pi
1
i Pi
1 g i 2g
i s2
1
g j i
i p=2 s2 ;
1 g i 2g
i s2
s2 A(A[1])(0) 1
i p=2p=1;
s2 A(A[1])(0) 1
i p=1p=2
1 g i 2g
i s2
s2 B(A[1])(0) 1
i p=1p=2;
1 g i 2g
i s2
s2 B(A[1])(0) 1
i p=2p=1
(A.1)
(A.2)
where Pi = i@i + gAi + gBi, see eq. (3.16). The expressions for
should be read from right
to left, e.g.
A j Bj p=2p=1
1
i Pi
1
(x) s2
Z
dz
A(z) j Bj (z)p=2p=1(zj
1
i Pi
1
i jx) s2
(and 1
1
1
+1i as usual). It is easy to see that the power counting of quark elds
has the form (cf eq. (3.29)):
The gluon elds A(0) and A(0) were calculated in ref. [12]:
(1)
1A
(1)
1B
(1)
2A
(1)
2B
m7=2
? :
HJEP05(218)
and their power counting reads
A(0) = A + (A[1])(0);
A(0) = B + (A[1])(0);
(A[1]a)(0) =
(A[1]a)(0) =
2
AjabBjb
gA
gB
m2 ;
gAi
gBi
m :
j
Ak ip=2
Ak j p=2 i
Ak j p=2 i
Bj ip=
Bj k
j k
B
B
1
k = p=2(Ai
p=1 i = p=2(Ai + iA~i 5)
p=1 i = p=2(Ai
iA~i 5)
iA~i 5)
iB~i 5);
p=1(Bi
p=1(Bi + iB~i 5);
p=1(Bi + iB~i 5);
j ip
=
1
k = p=2(Ai + iA~i 5)
p=1(Bi
iB~i 5):
17We use conventions from Bjorken & Drell where 0123 = 1 and
= g
+ g
g
5. Also, with this convention ~
= i
1
2
(A.3)
(A.4)
(A.5)
(A.6)
(B.1)
(B.2)
(B.4)
B
Formulas with Dirac matrices
In the gauge A = 0 the eld Ai can be represented as
(see eq. (3.5)). It is convenient to de ne a \dual" eld
2 Z x
Ai(x ; x?) =
dx0 A i(x0 ; x?)
2
where F~
= 12
and therefore
2 Z x
F
1
2
A~i(x ; x?) =
dx0 A~ i(x0 ; x?);
B~i(x ; x?) =
dx0 B~ i(x0 ; x?);
as usual.17 With this de nition, we get
2 Z x
ij Aj = A~i;
ij Bj =
Bi ) Ai
B~i =
Ai
i
B ; Ai
~
Bi = Ai
B
(B.3)
and hence
Ak ip=2 j(a
Ak jp=2 i(a
Ak ip=2 j(a
Ak jp=2 i(a
we get
2
s (p^2 ip^1
k ip^2
+ g
5)
5)
5)
5)
Bj ip=1 k(a
Bj kp=1 i(a
Bj kp=1 i(a
5)
5)
Next, using formula ~ ~ =
(g g
g g )
Akp=2 j
Bjp=1 k
Akp=2 j 5
Bjp=1 k + Akp=2 k
j
A p=1 k
Bjp=2 k +
2
s Ai jk
Bjp=1 k 5
Bjp=1 j
Bi jk
For appendix C we also need
5
Bj k
p= i
1 ;
Bj ip=1 k
= (a2 + 1)p=2(Ai iA~i 5) p=1(Bi iB~i 5) 2ap=2(Ai iA~i 5) p=1( 5Bi iB~i);
(B.5)
(B.6)
(B.7)
(B.8)
s j
A
4
2 k
A
+ g
Bj
Bk
2
s Ak ki
2Akp=2 j
B j
j i
Bkp=1 j:
p^1Bi + p^2 ip^1 5
Bi k + k ip^2 5
p=1Bj + p=2 5
p=1Bj + p=2 5
i j
i j
p^1 5Bi) =
p^1(Bi + iB~i 5)
i
5
p^1 5(Bi + iB~i 5);
Bi k 5 = p^2 (Bi + iB~i 5) i + p^2 5 (Bi + iB~i 5) i 5;
p=1 5Bj = p=2
p=1 5Bj = p=2
(Bi + iB~i 5)p=1 + 5p=2
p=1(Bi iB~i 5) + p=2 5
5(Bi + iB~i
5)p=1;
p=1 5(Bi iB~i 5):
Subleading power corrections
Second, third, and fourth lines in eq. (4.19)
In this appendix we show that second, third, and fourth lines in eq. (4.19) yield subleading
power corrections and can be neglected in our approximation.
Let us consider for example the last term in the third line of eq. (4.19). The Fierz
transformation (4.8) yields
1 + a2
1 + a2
2
2
Am(x)
2nA(0)
Bn(0)
a2)
2nA(0)
Bn(0)
Am(x) 2nA(0)
Am(x) i 2nA(0)
Bn(0) i Bm(x) + ( i
Am(x) i 2nA(0)
Bn(0) i
5 Bm(x) + ( i
a2)
Am(x) 2nA(0)
Sorting out the colorsinglet contributions18 we get
(1
(1
$
5 $
5
5
5)
1 $ 5
i
$ i 5
1 $ 5
i 5)
5)
m8
5) + O
? :
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
hA; Bj( Am(Bj )nk Ak)( Bn Bm)jA; Bi = hA; Bj( Am Ak)( Bn(Bj )nk m
B )jA; Bi
1
=
Nc hAj( Al Al)jAihBj( BnBjnk Bk)jBi
B(x)
A(x) ip=2
A(x)p=2
A(x) i 5p=2
B(0)
j 1
j 1
j 1
A(0)
A(0)
A(0)
2A(0)
B(0)gBj (0) i B(x) + ( i
$ i 5
i 5)
B(0)gBj (0) B(x)
(1
1 $ 5
5)
B(0)gBj (0) i B(x) + ( i 5
$ i
i 5) ;
and therefore
2s
1
s
s
2
Z
where 1
+1i , see eq. (3.28).
For the forward matrix elements we get
dx?dx e i qx +i(k;x)? hAj ^(x ; x?)p=2
dx?dx e i qx +i(k;x)? hAj ^(x ; x?)p=2
1
q
f ( q; k?2);
where
is any of matrices with transverse indices. Next, consider
dx?dx e i qx +i(k;x)? hBj ^(0)gB^i(0) ^(x ; x?)jBi
1
dx?dx e i qx +i(k;x)? Z 0
dx0 hBj ^(0)gF^ i(x0 ; 0?) ^(x ; x?)jBi
m
ki f tw3( q; k?2);
18Recall that after the promotion of background elds to operators we can still move those operators
freely since all of them commute, see footnotes 2, 10 and 11.
where f tw3( q; k?2) is some function of order one (by power counting (3.4) this matrix
1). Also, this function may have only logarithmical singularities in q
as q ! 0 but not the power behavior 1 .19 The corresponding contribution to W (q) of
q
eq. (4.1) is proportional to
d 2k?f i ( q; k?)kif tw3( q; k?2)
? W lt
Q?2 W lt
(C.6)
so it can be neglected in comparison to the contributions
that Zboson is produced in a central range of rapidity so q
way one can show that the remaining three terms in the second and third lines of eq. (4.19)
give small contributions to W (q).
Next, it is easy to see that the matrix element of the fourth line of eq. (4.19) vanishes.
Indeed, let us consider the rst term in the fourth line and perform Fierz
transformation (4.8):
2
q
Q?2 W lt (recall that we assume
2
q?s ' Qp
?
q
2
2
Qq?2 ). In a similar
2A(0)
1 + a2
2
2mA(x)
2nA(0)
Bn(0)
(1
1 $ 5
$
5 $
5
5
5) :
5)
)
From the explicit form of 2A and 2A in eq. (3.27) we see that the last term in the r.h.s.
vanishes while the rst two are small. Indeed,
2mA(x)p=1 2nA(0)
Bn(0)p=2 Bm(x) jA; Bi
k 1
A
(x) i p=2 j 1 Al(0)
Bn(0)g2B^ikm(x)B^jnl(0)p=2 Bm(x) jA; Bi
s
p
(x) i =2 j 1 ^(0)jAihBj ^(0)g2A^j (0)A^i(x)p=2 ^(x)jBi
O
? ;
m8
s
2
hA; Bj
= hA; Bj
2
sNc hAj
2
(C.8)
(C.9)
so the contribution to W is of order of ms24? W lt.
C.2
Second to fth lines in eq. (4.56)
Here we show that second to fth lines in eq. (4.56) either vanish or can be neglected.
Obviously, matrix element of the operator in the second line vanishes. Formally,
Z
hAj ^(x ; x?)
^(x ; x?)jAi = ( q)hAj ^(0)
Z
^(0)jBi = ( q)hBj ^(0)
19Large x correspond to lowx domain where matrix elements can be calculated in a shockwave
background of the target particle. The typical propagator in the shockwave external eld has a factor
tum [44, 45]. The integration over large x gives then
restricted from above by a, such terms cannot give 1q (cf. refs [37, 38]).
q + p
2
s
1 and since the integration over
is
and, nonformally, one hadron cannot produce Zboson on his own. For a similar reason,
projectile or target matrix element will be of eq. (C.9) type. In addition, from the explicit
form
's in eq. (3.27) it is easy to see that the fth line in eq. (4.56) can be rewritten
as follows:
2A(x)
B(0)
1B(0)
(C.10)
(x) igBi(x) 2
2 kgBk(x)
A(x)
B(0)
B(0)
A(x)
(0) igAi(0) 1
1 kgAk(0)
1
p
=
A(x)
p
=
5) kgBk(x)
B
1
(0) igAi(0) s21 (a
1
B(0) +x $ 0
B(0)p=2(a
5) B(0)
5) kgAk(0)
1
B(0) +x $ 0:
ms8? , so we are left with the
From the power counting (3.4) we see that this term is
contribution of the last two lines in eq. (4.56).
C.3
Gluon power corrections from JA (x)JA (0) terms
There is one more contribution which should be discussed and neglected:
where we neglected terms which cannot contribute to W due to the reason discussed after
eq. (C.9), i.e. that one hadron (\A" or \B") cannot produce Zboson on its own.
Let us consider the rst term in the r.h.s. of this equation
A(x) + 2A(x)
A(0) + 2A(0)
A(0) + 2A(0)
2A(x)
A(x)
A(x)
2A(0)
A(0)
2A(0) +
A(x)
A(0) +
A(x)
2A(x)
2A(x)
A(0)
2A(0)
A(0)
2A(0) ; (C.11)
= 2
A
1
A
1
p
=
1
B
p
=
p
=
(x) igBi(x) s22 (a
5) A(x)
JA (x)JA (0) =
e
2
16s2W c2W
e
2
16s2W c2W
2A(x)
2A(x)
f:int:
A(x)
2
2Nc(Nc2
2
2Nc(Nc2
1) hAj
1) hAj
A(0)
2A(0)
(x) i 2
p
=
^(x) ^(0)
p
=
s2 j
1 ^(0)jAihBjA^ai(x)A^aj (0)jBi
p= p
(x) i s2 k ^(x) ^(0) k =s2 j
1 ^(0)jAi
hBjA^ai(x)A^aj (0)jBi + O
m8
? ;
(C.12)
(C.13)
?
where f:int: denotes functional integration over A and B
elds in eq. (2.8).
either to 2xixj + x2 gij or to gij . Since the former structure does not contribute due to
The matrix element hBjA^ai(x)A^aj (0)jBi for unpolarized hadrons can be proportional
(2xixj + x2 gij ) ip=2
?
k
kp=2 j = 0
A(x)
2
hA; Bj 2A(x)
A(0)
2A(0) jA; Bi
2Nc(Nc2
1)s2 hAj ^ 1 (x)p=2(a
5) ^(x) ^(0)p=2(a
5)
m8
? :
For the forward target matrix element one obtains
Z
dx e i qx
hBjA^ia(x)A^ai(0)jBi
(C.14)
(C.15)
s2
dx e i qx Z x
1
dx00 hBjF^ai(x0 ; x?)F^ai(x00; 0?)jBi
dx e i qx
hBjF^ai(x ; x?)F^ai(0)jBi =
8 2 sDg( q; x?);
1
(A0)(x)
1 + a2
2
+ (1
a
a
2
2
quark ones.
q?s so they can be neglected.20
either m2
q?s or m2
m2
where we used parametrization (4.6) from ref. [12]. Since the gluon TMD Dg(xB; x?)
behaves only logarithmically as xB ! 0 [38], the contribution of eq. (C.14) to W (q) is of
Q?2 (note that the projectile TMD in the r.h.s. of eq. (C.12) does not have
1 terms for the same reason as in eq. (C.22)). Similarly, all other terms in eq. (C.11) are
C.4
Power corrections from
A(x)
A(x) A(0)
A(x)
elds
give zero contribution since
A(x)
A(x) does not depends on x so
R dx e i qx = 2
( q).
Let us consider now the last two lines in the power expansion (4.6) of JAB(x)JBA (0):
(A0)(x)
(B0)(x) (B0)(0)
After Fierz transformation (4.8) the rst term in the above equation turns to
(B0)(x) (B0)(0)
(A1)(0)
A
A
B
m(0)(x) + (
A
B
5 $
5)
m(0)(x) nA(1)(0)
A
Am(x)p=2 1nA(1)(0)
n(0)(0) Bm(0)(x)
B
(1
1 $ 5
5)
Am(x)p=2 1nA(1)(0)
m8
s
? :
20It is worth mentioning that if Zboson is produced in the region of rapidity close to the projectile, the
contribution (C.15) may be the most important since gluon parton densities at small xB are larger than
gp=2p=1 iBi A +
g i
1
2g
s p=2 (A[1])(0)
1 + a2 ( "
s2
2a ( "
s2
Am(x) p=2p=1 i
1
Bi
+ p= i j 1
2
PiBj
1
1
+ 2p=2 (A[1])(0)
Am(x) p=2p=1 i
1
Bi
+ p= i j 1
2
PiBj
1
1
+ 2p=2 (A[1])(0)
Let us start with the rst term in parentheses in the second line of eq. (C.20). Using
eq. (B.9) the corresponding matrix element can be rewritten as
2sNc
2sNc
ghAj ^(x) i ^(0)jAihBj ^(0)p=1
ghAj ^(x) i
5 ^(0)jAihBj ^(0)p=1
1 (A^i + i 5A~i) (0) ^(x)jBi
^
1
( 5A^i + iA~i) (0) ^(x)jBi:
nk
p= ) :
nk
Ak(0)
#
Ak(0)
#
(C.19)
(C.20)
(C.21)
(C.22)
(C.23)
Let us consider
where we used
2i Z
1
+ i
Z
Z
1
2i Z z
A^k(z ; z?) =
dz0 A^k(z0 ; z?) =
dz0 (z
z0) F^ k(z0 ; z?):
Let us compare this matrix element to that of eq. (4.46):
dx e i qx
(0)A^i(0) ^(x ; x?)jBi
dx e i qx
dx0 ^(x0 ; 0?) 2s F^ i(0)
^(x ; x?)jBi:
We see that 1z in eq. (C.23) is traded for an extra x0
1 in eq. (C.22) (recall that x0 in the
target matrix elements is inversely proportional to characteristic 's in the target which are
of order 1). Consequently, power correction due to matrix element (C.22) can be neglected
in our kinematic region since the matrix element (C.21) is
? rather than
Similarly, the second term in eq. (C.21) does not contribute in our kinematical region.
This is the same reason why we neglected power corrections (C.6). In general, as
we discussed in ref. [12], the way to gure out integrations that give 1 is very simple:
take q ! 0 and check if there is an in nite integration of the type R x q
1
over x0 after one sets q = 0 in the relevant matrix element. Note that to get the terms
Q12 =
q1 qs we need to nd contributions which satisfy both of the above conditions.
Next, consider the second term in parentheses in the second line of eq. (C.20). Using
eq. (4.44) the corresponding matrix element can be rewritten as
q qs
1+a2 "
s2
Am(x)p=2 i j
Am(x)p=2 i j
1
PiBj
1
Bj
+ Am(x)p=2 i j 1
nk 1 Ak(0)
nk 1
nk 1 Ak(0)
Bn(0)p=1 Bm(x)
Bn(0)p=1 Bm(x)
Ai Bj
nk 1 Ak(0)
f:int: 1+a2
Ncs2 ghAj ^(x)p=2 i j 1 i@i ^(0)jAihBj ^(0)p=1
1 ^
Aj (0) ^(x)jBi
1+a2
Nc(Nc2 1)s2 g2hAj ^(x)p=2 i j A^i 1 ^(0)jAihBj ^(0)p=1
1 A^j (0) ^(x)jBi
1+a2
Ncs2 ghAj ^(x)p=2 i j 1
A(0)jAihBj ^(0)p=1
1
(i@iA^j +gA^iA^j )(0) ^(x)jBi: (C.24)
Using eq. (B.9) we get
1+a2 "
s2
am(x)p=2 i j
Ncs2 ghAj ^(x)p=2
1
1
1+a2
Ncs2 ghAj ^(x)p=2
2(1+a2)
Ncs3
ghAj ^(x)p=2
1
+(p=2 p=1 $ p=2 5 p=1 5):
1+a2
Nc(Nc2 1)s2 g2hAj ^(x)p=2A^i 1 ^(0)jAihBj ^(0)p=1
1 (A^i iA~^i 5)(0)
1
PiBj
nk 1 ak(0)
i@i ^(0)jAihBj ^(0)p=1
bn(0)p=1 bm(x) +(p=2 p=1 $ p=2 5 p=1 5)
1 (A^i iA~^i 5)(0)
A(0)jAihBj ^(0)p=1
1
(i@iA^i +gA^iA^i)(0)
A(0)jAihBj ^(0)p=1 5
1 F^~ (0)
It is easy to see that projectile matrix elements lead to terms
x , for example
Z
dx e i qx
hAj ^(x)p=2A^i(0)
1 ^(0)jAi =
dx ei qx Z x
1
dx0
hAj ^(0)p=2[F^ i(x ; x?) ^(x0 ; x?) + F^ i(x0 ; x?) ^(x ; x?)jAi:
1 after integration over
(C.25)
(C.26)
On the other hand, the target matrix elements cannot give a 1 factor. For the rst two
lines in the r.h.s. of eq. (C.25) we proved this in eq. (C.22) above. As to the last lines in
eq. (C.25), the target matrix element can be rewritten as
Z
(i@iA^i +gA^iA^i)(0)
^(x ; x?)jBi =
Z x0
1
2g F^ i(x00; 0?)
Z x00
1
1
dx0
dx000F^ i(x000; 0?)
^(x ; x?)jBi
2i Z
and
Z
dx e i qx
hBj ^(0)p=1 5
1 F^~ (0)
^(x ; x?)jBi
iZ dx e i qx Z 0
1
dx0 hBj ^(0)p=1 5F^~ (x0 ; 0?) ^(x ; x?)jBi:
We see that at q = 0 there are no unrestricted integration over any longitudinal variable
so the r.h.s. of these equations cannot give 1q factor and therefore the contribution to W (q)
is
m2
mQ2?2 .
Finally, we should consider the third term in eq. (C.20)
1 + a2
s2
s2
g
Am(x)p=2 (A[1]nl)(0) Al(0)
Bn(0)p=1 Bm(x)
Am(x)p=2
Am(x)p=2
1
1
Aj (0)
Aj (0)
nk
kl
Al(0)
Al(0)
# "
# "
Bn(0)p=1
Bn(0)p=1
1 Bj (0)
1 Bj (0)
kl
nk
Bm(x)
Bm(x) ;
where we used eq. (A.5) for (A[1])(0). Taking projectile and target matrix elements and
separating colorsinglet contributions using eq. (4.44), we obtain
1)
^(0)jAihBj ^(0)p=1
1 A^j (0)
(C.29)
It has been demonstrated in eq. (C.22) that such matrix elements cannot give 1q and
1 so their contribution to W (q) is small in our kinematical region. Moreover, since the
above arguments do not depend on presence (or absence) of 5, we proved that all terms
in eq. (C.20) give small contributions at q; q
1. In a similar way, one can demonstrate that the other three terms in eq. (C.17) can be neglected. (C.27) (C.28)
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