# Corrigendum to “Comparative Aspects of Spin-Dependent Interaction Potentials for Spin-1/2 and Spin-1 Matter Fields”

Authors: P. C. Malta; L. P. R. Ospedal; K. Veiga; J. A. Helayël-Neto
Journal: Advances in High Energy Physics
 (2017)

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## Body

## 1. Introduction

We present punctual corrections to the paper “Comparative Aspects of Spin-Dependent Interaction Potentials for Spin-1/2 and Spin-1 Matter Fields” [1]. The main conclusions remain unaltered.In [1], we calculated the interparticle potentials for sources with spin-1/2 and spin-1 in the nonrelativistic (NR) regime. We have inadvertently made a few mistakes and here we present the correct version of some of the interparticle potentials obtained.We note further that we have included contact terms coming from Dirac delta contributions of the following Fourier transforms:(1)
∫
d
3
q
→
2
π
3
q
→
i
q
→
j
q
→
2
+
m
2
e
i
q
→
·
r
→
=
δ
i
j
3
δ
3
r
→
+
e
-
m
r
4
π
r
3
δ
i
j
1
+
m
r
-
r
^
i
r
^
j
m
2
r
2
+
3
m
r
+
3
,
(2)
∫
d
3
q
→
2
π
3
q
→
2
q
→
2
+
m
2
e
i
q
→
·
r
→
=
δ
3
r
→
-
m
2
e
-
m
r
4
π
r
,which are the massive generalizations of the “massless” results from [2].In what follows we list the corrections, indicated byΔ
V, that must be added to some of the potentials given in [1]. The mass of the mediator, m, corresponds to either m
ϕ or m
A, as the case may be. We also highlight that not all the potentials from [1] are affected and in this note we only indicate those which are modified (Moreover, we need to use S
i
j
k
=
-
i
ϵ
i
j
k in all potentials. In [1], we had the opposite sign, so one must make S
→
→
-
S
→ throughout.).


## 2. Correcting the Potentials

### 2.1. Potentials (Spin-1/2)

TheS
-
P
S and V
-
P
V potentials from [1] are not affected. The other potentials receive partial corrections. For convenience, we define the function: (3)
I
r
≡
1
4
1
m
1
2
+
1
m
2
2
p
→
2
-
m
2
4
e
-
m
r
4
π
rso that the potentials in equations (25), (30) (As mentioned in equation (30) in [1], the P
S
-
P
S potential for s
=
1
/
2 has the same functional form as the P
S
-
P
S potential for s
=
1.), (44), and (45) in [1] receive the respective corrections:(4)
Δ
V
S
-
S
s
=
1
/
2
=
g
S
1
g
S
2
δ
1
δ
2
I
r
,
(5)
Δ
V
P
S
-
P
S
s
=
1
/
2
=
-
g
P
S
1
g
P
S
2
σ
→
1
·
σ
→
2
12
m
1
m
2
δ
3
r
→
,
(6)
Δ
V
V
-
V
s
=
1
/
2
=
-
e
1
e
2
δ
1
δ
2
I
r
+
δ
1
δ
2
8
1
m
1
2
+
1
m
2
2
+
σ
→
1
·
σ
→
2
6
m
1
m
2
δ
3
r
→
,
(7)
Δ
V
P
V
-
P
V
s
=
1
/
2
=
+
g
P
V
1
g
P
V
2
σ
→
1
·
σ
→
2
I
r
+
1
8
1
m
1
2
+
1
m
2
2
-
1
m
2
δ
3
r
→
3
.Equation (4) modifies the full potential by eliminating a momentum-independent monopole-monopole t
e
r
m
~
m
2
1
/
m
1
2
+
1
/
m
2
2
e
-
m
r
/
4
π
r. A similar modification, but momentum-dependent, is induced by (6). For the other potentials, the modifications involve spin-spin interactions, mostly coupled to contact terms.


### 2.2. Potentials (Spin-1)

#### 2.2.1. Vector Representation

The corrections to equations (28), (30), (47), and (48) in [1] are, respectively:(8)
Δ
V
S
-
S
s
=
1
=
g
S
1
g
S
2
2
δ
1
δ
2
m
1
m
2
I
r
-
1
48
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(9)
Δ
V
P
S
-
P
S
s
=
1
=
-
g
P
S
1
g
P
S
2
S
→
1
·
S
→
2
3
δ
3
r
→
,
(10)
Δ
V
V
-
V
s
=
1
=
-
e
1
e
2
2
δ
1
δ
2
I
r
+
2
3
S
→
1
·
S
→
2
m
1
m
2
+
δ
1
δ
2
6
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(11)
Δ
V
P
V
-
P
V
s
=
1
=
g
P
V
1
g
P
V
2
S
→
1
·
S
→
2
12
m
1
m
2
δ
3
r
→whereby, in (10), I
(
r
) cancels another term in (47) from [1].


#### 2.2.2. Tensor Representation

The correction to theP
S
-
P
S potential in this representation is Δ
V
P
S
-
P
S
s
=
1
/
4
m
1
2
m
2
2 (cf. (9)). Also, the correction to the P
V
-
P
V potential in this representation is the same as in (11).In [1], equations (65) receive the respective corrections:(12)
Δ
V
S
-
S
s
=
1
=
g
S
1
g
S
2
2
δ
1
δ
2
m
1
m
2
I
r
-
5
48
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(13)
Δ
V
V
-
V
s
=
1
=
-
e
1
e
2
2
δ
1
δ
2
I
r
+
2
3
S
→
1
·
S
→
2
m
1
m
2
+
δ
1
δ
2
3
1
m
1
2
+
1
m
2
2
δ
3
r
→
.We note that the correction in (13) excludes one momentum-dependent term associated with p
→
2
1
/
m
1
2
+
1
/
m
2
2
e
-
m
r
/
4
π
r in the V
-
V potential in equation (65) from [1].


## 2.1. Potentials (Spin-1/2)

TheS
-
P
S and V
-
P
V potentials from [1] are not affected. The other potentials receive partial corrections. For convenience, we define the function: (3)
I
r
≡
1
4
1
m
1
2
+
1
m
2
2
p
→
2
-
m
2
4
e
-
m
r
4
π
rso that the potentials in equations (25), (30) (As mentioned in equation (30) in [1], the P
S
-
P
S potential for s
=
1
/
2 has the same functional form as the P
S
-
P
S potential for s
=
1.), (44), and (45) in [1] receive the respective corrections:(4)
Δ
V
S
-
S
s
=
1
/
2
=
g
S
1
g
S
2
δ
1
δ
2
I
r
,
(5)
Δ
V
P
S
-
P
S
s
=
1
/
2
=
-
g
P
S
1
g
P
S
2
σ
→
1
·
σ
→
2
12
m
1
m
2
δ
3
r
→
,
(6)
Δ
V
V
-
V
s
=
1
/
2
=
-
e
1
e
2
δ
1
δ
2
I
r
+
δ
1
δ
2
8
1
m
1
2
+
1
m
2
2
+
σ
→
1
·
σ
→
2
6
m
1
m
2
δ
3
r
→
,
(7)
Δ
V
P
V
-
P
V
s
=
1
/
2
=
+
g
P
V
1
g
P
V
2
σ
→
1
·
σ
→
2
I
r
+
1
8
1
m
1
2
+
1
m
2
2
-
1
m
2
δ
3
r
→
3
.Equation (4) modifies the full potential by eliminating a momentum-independent monopole-monopole t
e
r
m
~
m
2
1
/
m
1
2
+
1
/
m
2
2
e
-
m
r
/
4
π
r. A similar modification, but momentum-dependent, is induced by (6). For the other potentials, the modifications involve spin-spin interactions, mostly coupled to contact terms.


## 2.2. Potentials (Spin-1)

### 2.2.1. Vector Representation

The corrections to equations (28), (30), (47), and (48) in [1] are, respectively:(8)
Δ
V
S
-
S
s
=
1
=
g
S
1
g
S
2
2
δ
1
δ
2
m
1
m
2
I
r
-
1
48
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(9)
Δ
V
P
S
-
P
S
s
=
1
=
-
g
P
S
1
g
P
S
2
S
→
1
·
S
→
2
3
δ
3
r
→
,
(10)
Δ
V
V
-
V
s
=
1
=
-
e
1
e
2
2
δ
1
δ
2
I
r
+
2
3
S
→
1
·
S
→
2
m
1
m
2
+
δ
1
δ
2
6
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(11)
Δ
V
P
V
-
P
V
s
=
1
=
g
P
V
1
g
P
V
2
S
→
1
·
S
→
2
12
m
1
m
2
δ
3
r
→whereby, in (10), I
(
r
) cancels another term in (47) from [1].


### 2.2.2. Tensor Representation

The correction to theP
S
-
P
S potential in this representation is Δ
V
P
S
-
P
S
s
=
1
/
4
m
1
2
m
2
2 (cf. (9)). Also, the correction to the P
V
-
P
V potential in this representation is the same as in (11).In [1], equations (65) receive the respective corrections:(12)
Δ
V
S
-
S
s
=
1
=
g
S
1
g
S
2
2
δ
1
δ
2
m
1
m
2
I
r
-
5
48
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(13)
Δ
V
V
-
V
s
=
1
=
-
e
1
e
2
2
δ
1
δ
2
I
r
+
2
3
S
→
1
·
S
→
2
m
1
m
2
+
δ
1
δ
2
3
1
m
1
2
+
1
m
2
2
δ
3
r
→
.We note that the correction in (13) excludes one momentum-dependent term associated with p
→
2
1
/
m
1
2
+
1
/
m
2
2
e
-
m
r
/
4
π
r in the V
-
V potential in equation (65) from [1].


## 2.2.1. Vector Representation

The corrections to equations (28), (30), (47), and (48) in [1] are, respectively:(8)
Δ
V
S
-
S
s
=
1
=
g
S
1
g
S
2
2
δ
1
δ
2
m
1
m
2
I
r
-
1
48
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(9)
Δ
V
P
S
-
P
S
s
=
1
=
-
g
P
S
1
g
P
S
2
S
→
1
·
S
→
2
3
δ
3
r
→
,
(10)
Δ
V
V
-
V
s
=
1
=
-
e
1
e
2
2
δ
1
δ
2
I
r
+
2
3
S
→
1
·
S
→
2
m
1
m
2
+
δ
1
δ
2
6
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(11)
Δ
V
P
V
-
P
V
s
=
1
=
g
P
V
1
g
P
V
2
S
→
1
·
S
→
2
12
m
1
m
2
δ
3
r
→whereby, in (10), I
(
r
) cancels another term in (47) from [1].


## 2.2.2. Tensor Representation

The correction to theP
S
-
P
S potential in this representation is Δ
V
P
S
-
P
S
s
=
1
/
4
m
1
2
m
2
2 (cf. (9)). Also, the correction to the P
V
-
P
V potential in this representation is the same as in (11).In [1], equations (65) receive the respective corrections:(12)
Δ
V
S
-
S
s
=
1
=
g
S
1
g
S
2
2
δ
1
δ
2
m
1
m
2
I
r
-
5
48
1
m
1
2
+
1
m
2
2
δ
3
r
→
,
(13)
Δ
V
V
-
V
s
=
1
=
-
e
1
e
2
2
δ
1
δ
2
I
r
+
2
3
S
→
1
·
S
→
2
m
1
m
2
+
δ
1
δ
2
3
1
m
1
2
+
1
m
2
2
δ
3
r
→
.We note that the correction in (13) excludes one momentum-dependent term associated with p
→
2
1
/
m
1
2
+
1
/
m
2
2
e
-
m
r
/
4
π
r in the V
-
V potential in equation (65) from [1].


## 3. Concluding Remarks

We have presented the corrections to some of the results given in [1]. The corrections introduced affect mostly the monopole-monopole sectors (e.g., (4), (8), and (12)).The conclusions presented in [1] remain unaltered: in general, the interparticle potentials associated with NR spin-1/2 and spin-1 sources present similarities, especially, in the spin-spin sectors of the S
-
S, S
-
P
S, P
S
-
P
S, and V
-
V interactions. The most important modifications are due to the presence of contact terms, which do not play a role in macroscopic interactions.



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*Source: 9152437-2017-02-21.xml*