arXiv:hep-ph/0608113 v1 10 Aug 2006
DESY-06-123
hep-ph/0608113
Chiral perturbation theory for
nucleon generalized parton distributions
M. Diehl
1
, A. Manashov
2,3
and A. Sch¨afer
2
1
Theory Group, Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, Germany
2
Institut f¨ur Theoretische Physik, Universit¨at Regensburg, 93040 Regensburg, Germany
3
Department of Theoretical Physics, Sankt-Petersburg State University, St.-Petersburg, Russia
Abstract
We analyze the moments of the isosinglet generalized parton distributions H, E,
˜
H,
˜
E of
the nucleon in one-loop order of heavy-baryon chiral perturbation theory. We discuss in
detail the construction of the operators in the effective theory that are required to obtain all
corrections to a given order in the chiral power counting. The results will serve to improve
the extrapolation of lattice results to the chiral limit.
1 Intro duction
In recent years one has learned that many aspects of hadron structure can be described in the
unifying framework of generalized parton distributions (GPDs). This framework allows one to combine
information which comes from very different sources in an efficient and model-independent manner.
The field was pioneered in [1, 2, 3] and has evolved to considerable complexity, reviewed for instance
in [4, 5, 6, 7]. As GPDs can be analyzed using standard operator product exp ansion techniques
[1, 8], their moments can be and have been calculated in lattice QCD [9]. Lattice calculations of well-
measured quantities can be used to check the accuracy of the method, which may then be employed
to evaluate quantities that are much harder to determine experimentally. This complementarity is
especially valuable in the context of GPDs, because experimental measurements as e.g. in [10] may
not be sufficient to determine these functions of three kinematic variables in a model-independ ent
way. Moreover, several moments of GPDs admit a physically intuitive interpretation in terms of the
spatial and s pin structure of hadrons, see e.g. [2, 11, 12, 13].
A notorious problem of lattice QCD is the need for various extrapolations from the actual sim-
ulations with finite lattice spacing, finite volume and unphysically heavy quarks to the continuum,
infinite volume and physical quark masses. Simple phenomenological fits are often still sufficient in
view of the general size of uncertainties, but w ith increasing numerical precision more reliable meth-
ods have to be applied. C hiral perturbation theory (ChP T) provides such a method [14 ]. Describing
the exact low-energy limit of QCD it predicts the functional form for the dependence of observables
on the finite volume and the pion mass [15] and also the finite lattice spacing [16]. At a given or-
der in th e expansion parameter, ChPT defines a number of low-energy constants which determine
each of these limits. Some of these constants are typically known from independent sources, and the
remaining ones have to be determined from fits to the lattice data. The task of ChPT is thus to
provide the corresponding functional expressions for a sufficient number of observables. In this paper
we contribute to this endeavor by analyzing the moments of the isoscalar nucleon GPDs H, E,
˜
H
and
˜
E in one-loop order.
The analysis of pion GPDs in ChPT has been performed in several papers [17, 18, 19]. In th e case
of the nucleon GPDs, the chiral corrections have been calculated for the lowest moments [20, 17, 21]
in the framework of heavy-baryon ChPT, which performs an expansion in the inverse nucleon m ass
1/M. Due to the kinematic limit taken in this scheme, the sum and difference of the incoming and
outgoing nucleon momenta p
µ
and p
′µ
are of different order in 1/M. As a consequence, the nth
moment of a nucleon GPD contains terms up to nth order in the 1/M expansion. Given the rapidly
growing number of low-energy constants in higher orders of Ch PT, it has been assumed that the
chiral corrections can only be calculated for the terms of lowest order in 1/M, i.e. for the form factors
accompanied by the smallest number of vectors (p
′
− p)
µ
. This would be a serious setback for the
program sketched above. The aim of the present paper is to show that the situation is much better. In
particular, we find that th e corrections of order O(m
π
) and O(m
2
π
) to all form factors p arameterizing
the moments of chiral-even isoscalar nucleon GPDs come from one-loop diagrams in ChPT and the
corresponding higher-order tree-level insertions.
This paper is organized as follows. In Section 2 we recall the relation between moments of nucleon
GPDs and matrix elements of twist-two operators and rewrite it in a form suitable for th e 1/M
expansion. In Section 3 we discuss the construction of twist-two operators in heavy-baryon ChPT
and give a general power-counting sch eme for their contribution to a given nucleon matrix element.
In Sections 4 and 5 we identify the operators that contribute to moments of GPDs at lowest order in
the chiral expansion and give the results of the corresponding loop calculations. We summarize our
findings in Section 6.
1
2 Generalized parton distributions in the nucleon
The nucleon GPDs can be introduced as matrix elements of nonlocal operators. In this paper we
limit ourselves to the chiral-even isoscalar qu ark GPDs, which are defined by
Z
dλ
4π
e
ixλ(aP )
hp
′
| ¯q(−
1
2
λa)
/
a q(
1
2
λa) |pi =
1
2aP
¯u(p
′
)
/
a H(x, ξ, t) +
iσ
µν
a
µ
∆
ν
2M
E(x, ξ, t)
u(p) ,
Z
dλ
4π
e
ixλ(aP )
hp
′
| ¯q(−
1
2
λa)
/
aγ
5
q(
1
2
λa) |pi =
1
2aP
¯u(p
′
)
/
aγ
5
e
H(x, ξ, t) +
a∆
2M
γ
5
e
E(x, ξ, t)
u(p) , (1)
where a sum over u and d quark fields on the l.h.s. is understood, so that H = H
u
+ H
d
etc. Here
a is a light-like auxiliary vector, M is the nucleon mass, and we use the standard notations for the
kinematical variables
P =
1
2
(p + p
′
), ∆ = p
′
− p, t = ∆
2
, ξ = −
∆a
2P a
. (2)
As usual, Wilson lines between the quark fields are to be inserted in (1) if one is not working in the
light-cone gauge a
µ
A
µ
= 0. The x-moments of th e nucleon GPDs are related to the matrix elements
of the lo cal twist-two operators
O
µ
1
µ
2
...µ
n
= S ¯qγ
µ
1
iD
↔
µ
2
. . . iD
↔
µ
n
q ,
e
O
µ
1
µ
2
...µ
n
= S ¯qγ
µ
1
γ
5
iD
↔
µ
2
. . . iD
↔
µ
n
q , (3)
where D
↔
µ
=
1
2
(D
→
µ
− D
←
µ
) and S denotes the symmetrization of all uncontracted Lorentz ind ices and
the s ubtraction of traces, e.g. S t
µν
=
1
2
(t
µν
+ t
νµ
) −
1
4
g
µν
t
λ
λ
for a tensor of r ank two. It is convenient
to contract all open Lorentz indices with the auxiliary vector a,
O
µ
1
...µ
n
→ O
n
(a) = a
µ
1
. . . a
µ
n
O
µ
1
...µ
n
, (4)
and in analogy for
e
O. The matrix elements of the operators (3) can be parameterized as [4, 6]
hp
′
|O
n
(a)|pi =
n−1
X
k=0
even
(aP )
n−k−1
(a∆)
k
¯u(p
′
)
/
a A
n,k
(t) +
iσ
µν
a
µ
∆
ν
2M
B
n,k
(t)
u(p)
+ mod(n + 1, 2) (a∆)
n
1
M
¯u(p
′
)u(p) C
n
(t) ,
hp
′
|
e
O
n
(a)|pi =
n−1
X
k=0
even
(aP )
n−k−1
(a∆)
k
¯u(p
′
)
/
aγ
5
e
A
n,k
(t) +
a∆
2M
γ
5
e
B
n,k
(t)
u(p) . (5)
The moments of the above GPDs are polynomials in ξ
2
,
Z
1
−1
dx x
n−1
H(x, ξ, t) =
n−1
X
k=0
even
(2ξ)
k
A
n,k
(t) + mod(n + 1, 2) (2ξ)
n
C
n
(t) ,
Z
1
−1
dx x
n−1
E(x, ξ, t) =
n−1
X
k=0
even
(2ξ)
k
B
n,k
(t) − mod(n + 1, 2) (2ξ)
n
C
n
(t) ,
2
Z
1
−1
dx x
n−1
e
H(x, ξ, t) =
n−1
X
k=0
even
(2ξ)
k
e
A
n,k
(t) ,
Z
1
−1
dx x
n−1
e
E(x, ξ, t) =
n−1
X
k=0
even
(2ξ)
k
e
B
n,k
(t) . (6)
The restriction to even k in (5) and (6) is a consequence of time reversal invariance.
To calculate the chiral corrections to the nucleons form factors we shall use the formalism of
heavy-baryon chiral perturbation theory, which treats the nucleon as an infinitely heavy particle and
performs a corresponding non-relativistic expansion [22]. The evaluation of nu cleon form factors in
heavy-baryon ChPT is simplified if one works in the Breit frame [23]. It is defined by the condition
~
P = 0, so that the incoming and outgoing nucleons have opposite spatial momenta ~p
′
= −~p =
~
∆/2
and the same energy p
′
0
= p
0
= Mγ, wh ere
γ =
p
1 − ∆
2
/4M
2
. (7)
In the heavy-b aryon formalism the baryon has a additional quantum number, the velocity v, which
in the Breit frame is v = (1, 0, 0, 0). The incoming and outgoing nucleon momenta are thus given by
p = Mγv − ∆/2 and p
′
= Mγv + ∆/2.
The Dirac algebra simplifies considerably in the heavy-baryon formulation. All Dirac bilinears
can be expressed in terms of the velocity v
µ
and the spin operator
S
µ
=
i
2
γ
5
σ
µν
v
ν
. (8)
Using that (v∆) = (vS) = 0, one finds in particular
u(p
′
)u(p) = γ ¯u
v
(p
′
) u
v
(p) ,
u(p
′
)γ
µ
u(p) = v
µ
¯u
v
(p
′
) u
v
(p) +
1
M
¯u
v
(p
′
) [S
µ
, (S∆)] u
v
(p) ,
i
2M
u(p
′
) σ
µν
∆
ν
u(p) = v
µ
∆
2
4M
2
¯u
v
(p
′
) u
v
(p) +
1
M
¯u
v
(p
′
) [S
µ
, (S∆)] u
v
(p) ,
u(p
′
)γ
µ
γ
5
u(p) = 2γ ¯u
v
(p
′
)S
µ
u
v
(p) +
∆
µ
2M
2
(1 + γ)
¯u
v
(p
′
) (S∆) u
v
(p) ,
u(p
′
)γ
5
u(p) =
1
M
¯u
v
(p
′
) (S∆) u
v
(p) , (9)
where the spinors
u
v
(p) = N
−1
1 +
/
v
2
u(p), u
v
(p
′
) = N
−1
1 +
/
v
2
u(p
′
) (10)
with
N =
r
M + vp
2M
=
r
M + vp
′
2M
=
r
1 + γ
2
(11)
3
are normalized as ¯u
v
(p, s
′
) u
v
(p, s) = 2M δ
s
′
s
. With (9) one obtains the following representation for
the matrix elements (5) in the Breit frame:
hp
′
|O
n
(a)|pi =
n
X
k=0
(Mγ)
n−k−1
(av)
n−k
(a∆)
k−1
× ¯u
v
(p
′
)
h
(a∆) E
n,k
(t) + γ [(aS), (S∆)] M
n,k−1
(t)
i
u
v
(p) ,
hp
′
|
e
O
n
(a)|pi =
n
X
k=1
(Mγ)
n−k
(av)
n−k
(a∆)
k−1
× ¯u
v
(p
′
)
2γ (aS)
e
E
n,k−1
(t) +
(a∆)(S∆)
2M
2
f
M
n,k−1
(t)
u
v
(p) , (12)
with
E
n,k
(t) = A
n,k
(t) +
∆
2
4M
2
B
n,k
(t) for k < n , E
n,n
(t) = γ
2
C
n
(t) ,
M
n,k
(t) = A
n,k
(t) + B
n,k
(t) ,
e
E
n,k
(t) =
e
A
n,k
(t) ,
f
M
n,k
(t) =
1
1 + γ
e
A
n,k
(t) +
e
B
n,k
. (13)
The definition of the E
n
and
e
E
n
is conventional but might be confusing as E
n
is not the nth moment
of E(x, ξ, t) etc. We nevertheless use this notation, in order to make it easier to compare our results
with those in the literature. Notice that according to (5) the terms with E
n,k
in (12) are only nonzero
if k is even, whereas those with M
n,k−1
,
e
E
n,k−1
and
f
M
n,k−1
are only nonzero if k is odd. We will
evaluate these form factors in heavy-baryon ChPT. It is straightforward to transform back to the
original form factors using
A
n,k
(t) =
1
γ
2
E
n,k
(t) −
∆
2
4M
2
M
n,k
(t)
, B
n,k
(t) =
1
γ
2
h
M
n,k
(t) − E
n,k
(t)
i
,
e
B
n,k
(t) =
f
M
n,k
(t) −
1
1 + γ
e
E
n,k
(t) . (14)
3 Twist-two matrix elements in heavy-baryon ChPT
Heavy-baryon ChPT combines the techniques of chiral perturbation theory and of heavy-quark ef-
fective field theory [22] (for a detailed review see Ref. [24]). The effective Lagrangian describes the
pion-nucleon interactions in the limit when m
π
, q ≪ M, w here q is a generic momentum. In this
situation the velocity v of the nucleon is preserved in the process. One introduces the nucleon field
with velocity v as [22]
N(x) = e
−iM
0
vx
N
v
(x) + n
v
(x)
, (15)
where M
0
is the nucleon mass in the chiral limit. The fields N
v
(x), n
v
(x) respectively contain the
large and sm all components of the nucleon field and satisfy
/
vN
v
= N
v
,
/
vn
v
= −n
v
. Their Fourier
transform depends on the residual nucleon momentu m, i.e. the original nucleon momentum minus
4
