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M. STRIKMAN AND C. WEISS PHYSICAL REVIEW C 97, 035209 (2018)
The LF components of the longitudinal auxiliary four-
vectors L and T ,Eqs.(2.20) and (2.21), are obtained as
L
+
=
1 −
ξ
d
2x
d
p
+
d
,L
−
=
M
2
d
+
Q
2
ξ
d
1
p
+
d
, L
T
= 0
T
+
= 0,T
−
= 0, T
T
= p
pT
.
(2.38)
The vector L has only collinear components, while T has only
transverse components and coincides with the recoil hadron
transverse momentum in the collinear frame. Because the
momentum transfer q is along the (negative) z direction, the
initial and final electron momenta have the same transverse
components,
p
eT
= p
e
T
(collinear frame), (2.39)
such that they define a plane together with the z axis (electron
scattering plane). The recoil angle φ
p
, defined in terms of
invariants in Eq. (2.28), then is the azimuthal angle of p
p
,
measured relative to the electron scattering plane. It is con-
ventional to choose the electron transverse momenta in the x
direction, such that the electron scattering plane is the xz plane.
In this case the angle φ
p
becomes the conventional azimuthal
angle of p
p
in the xy plane, cos φ
p
= p
x
p
/|p
pT
| (see Fig. 3).
The deuteron plus momentum p
+
d
> 0 in the above formulas
remains arbitrary and defines a particular member of the
class of collinear frames. Longitudinal boosts (along the z
axis) can be performed simply by changing the value of
p
+
d
in the above formulas. Note that the class of collinear
frames contains several special cases of interest: (a) the target
rest frame, p
+
d
= M
d
; (b) the Breit frame, p
+
d
=
Q
2
/ξ
d
,in
which q
0
= (q
+
+ q
−
)/2 = 0; (c) the CM frame of the virtual
photon and the deuteron, p
+
d
=
√
Q
2
+ ξ
d
M
2
d
/
√
ξ
d
(1 − ξ
d
), in
which q
z
= (q
+
− q
−
)/2 =−p
z
d
=−(p
+
d
− p
−
d
)/2 and thus
q =−p
d
. For reference we note that the collinear frames used
here are equivalent to the covariant formulation of the collinear
expansion in terms of lightlike vectors of Ref. [49].
D. Recoil momentum variables
The tagged structure functions of the deuteron in Eq. (2.22)
depend on the usual DIS variables (e.g., W
d
and Q
2
)aswellas
the recoil nucleon momentum. The latter dependence involves
two independent variables formed from p
p
, related to the
two invariants (p
p
p
d
) and (p
p
q); the dependence on the t hird
invariant (p
p
p
e
) is encoded in the explicit φ
p
dependence
of the cross section. Here we describe several physically
interesting choices of recoil momentum variables that are used
in the subsequent calculations. We present their relation to
the rest-frame recoil momentum, their kinematic limits, and
the corresponding phase space elements. We assume isospin
symmetry and define the nucleon mass as the average of the
proton and neutron masses
M
N
= (M
p
+ M
n
)/2 = 0.9389 GeV. (2.40)
The deuteron binding energy and mass are taken at their exact
values
d
= 2.2MeV,
M
d
= M
p
+ M
n
−
d
= 2M
N
−
d
= 1.8756 GeV. (2.41)
Note that the relation between the deuteron binding energy and
mass is not affected when replacing the proton and neutron
masses by their average.
In a collinear frame defined by Eqs. (2.33) the tagged
structure functions can be regarded as functions of the LF
plus momentum fraction of the recoil proton and the transverse
momentum modulus of the recoil momentum,
α
p
≡
2p
+
p
p
+
d
=
2
p
0
p
+ p
z
p
p
0
d
+ p
z
d
, (2.42)
|p
pT
|≡
p
x
p
2
+
p
y
p
2
. (2.43)
The definition of α
p
in Eq. (2.42), as the fraction relative to
p
+
d
/2, is natural and leads to simple expressions in the nuclear
structure calculations below. The kinematic limits of α
p
are
dictated by LF plus momentum conservation in the scattering
process and given by
α
p
/2 < 1 − ξ
d
. (2.44)
The invariant phase space element in terms of α
p
and |p
pT
|
takes the form
d
3
p
p
E
p
=
dα
p
α
p
d
2
p
pT
. (2.45)
An important variable is the invariant momentum transfer
between the initial-state deuteron and the final-state nucleon
(see Fig. 2),
t ≡ (p
p
− p
d
)
2
, (2.46)
or the reduced variable
t
≡ t − M
2
N
. (2.47)
The theoretical analysis of tagged DIS, Eq. (2.1), relies
essentially on the analytic properties of the cross section in
t
; see Secs. IV and VI below. The invariant t
is related in a
simple manner to the energy of the recoiling nucleon in the
deuteron rest frame (we use “RF” to denote rest-frame energy
and momentum),
t
= M
2
d
− 2 M
d
E
p
(RF),
E
p
(RF) ≡
|p
p
(RF)|
2
+ M
2
N
. (2.48)
The kinematic limit of t
is attained at p
p
(RF) = 0,
t
<t
min
≡ M
2
d
− 2M
d
M
N
=−M
d
d
=−0.0041 GeV
2
.
(2.49)
Inside the physical region the rest-frame momentum is ob-
tained from t
as
|p
p
(RF)|
2
=−
t
2
1 −
t
2M
2
d
+
M
2
d
4
− M
2
N
. (2.50)
A s impler relation is obtained if we neglect the t
/(2M
2
d
)termin
the parentheses; this approximation is well justified for t ypical
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