# 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

In the article titled “Comparative Aspects of Spin-Dependent Interaction Potentials for Spin-1/2 and Spin-1 Matter Fields” [1], calculations for some interparticle potentials for sources with spin-1/2 and spin-1 in the nonrelativistic (NR) regime contained inadvertent errors which should be corrected as follows. The main conclusions are unaltered.We note further that we have included contact terms coming from Dirac delta contributions of the following Fourier transforms:(1)∫d3q→2π3q→iq→jq→2+m2eiq→·r→=δij3δ3r→+e-mr4πr3δij1+mr-r^ir^jm2r2+3mr+3,(2)∫d3q→2π3q→2q→2+m2eiq→·r→=δ3r→-m2e-mr4πr,which are the massive generalizations of the “massless” results from [2].In what follows we list the corrections, indicated byΔV, that must be added to some of the potentials given in [1]. The mass of the mediator, m, corresponds to either mϕ or mA, 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 Sijk=-iϵijk 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-PS and V-PV potentials from [1] are not affected. The other potentials receive partial corrections. For convenience, we define the function: (3)Ir≡141m12+1m22p→2-m24e-mr4πrso that the potentials in equations (25), (30) (As mentioned in equation (30) in [1], the PS-PS potential for s=1/2 has the same functional form as the PS-PS potential for s=1.), (44), and (45) in [1] receive the respective corrections:(4)ΔVS-Ss=1/2=gS1gS2δ1δ2Ir,(5)ΔVPS-PSs=1/2=-gPS1gPS2σ→1·σ→212m1m2δ3r→,(6)ΔVV-Vs=1/2=-e1e2δ1δ2Ir+δ1δ281m12+1m22+σ→1·σ→26m1m2δ3r→,(7)ΔVPV-PVs=1/2=+gPV1gPV2σ→1·σ→2Ir+181m12+1m22-1m2δ3r→3.Equation (4) modifies the full potential by eliminating a momentum-independent monopole-monopole term~m21/m12+1/m22e-mr/4πr. 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)ΔVS-Ss=1=gS1gS22δ1δ2m1m2Ir-1481m12+1m22δ3r→,(9)ΔVPS-PSs=1=-gPS1gPS2S→1·S→23δ3r→,(10)ΔVV-Vs=1=-e1e22δ1δ2Ir+23S→1·S→2m1m2+δ1δ261m12+1m22δ3r→,(11)ΔVPV-PVs=1=gPV1gPV2S→1·S→212m1m2δ3r→whereby, in (10), I(r) cancels another term in (47) from [1].


#### 2.2.2. Tensor Representation

The correction to thePS-PS potential in this representation is ΔVPS-PSs=1/4m12m22 (cf. (9)). Also, the correction to the PV-PV potential in this representation is the same as in (11).In [1], equations (65) receive the respective corrections:(12)ΔVS-Ss=1=gS1gS22δ1δ2m1m2Ir-5481m12+1m22δ3r→,(13)ΔVV-Vs=1=-e1e22δ1δ2Ir+23S→1·S→2m1m2+δ1δ231m12+1m22δ3r→.We note that the correction in (13) excludes one momentum-dependent term associated with p→21/m12+1/m22e-mr/4πr in the V-V potential in equation (65) from [1].


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

TheS-PS and V-PV potentials from [1] are not affected. The other potentials receive partial corrections. For convenience, we define the function: (3)Ir≡141m12+1m22p→2-m24e-mr4πrso that the potentials in equations (25), (30) (As mentioned in equation (30) in [1], the PS-PS potential for s=1/2 has the same functional form as the PS-PS potential for s=1.), (44), and (45) in [1] receive the respective corrections:(4)ΔVS-Ss=1/2=gS1gS2δ1δ2Ir,(5)ΔVPS-PSs=1/2=-gPS1gPS2σ→1·σ→212m1m2δ3r→,(6)ΔVV-Vs=1/2=-e1e2δ1δ2Ir+δ1δ281m12+1m22+σ→1·σ→26m1m2δ3r→,(7)ΔVPV-PVs=1/2=+gPV1gPV2σ→1·σ→2Ir+181m12+1m22-1m2δ3r→3.Equation (4) modifies the full potential by eliminating a momentum-independent monopole-monopole term~m21/m12+1/m22e-mr/4πr. 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)ΔVS-Ss=1=gS1gS22δ1δ2m1m2Ir-1481m12+1m22δ3r→,(9)ΔVPS-PSs=1=-gPS1gPS2S→1·S→23δ3r→,(10)ΔVV-Vs=1=-e1e22δ1δ2Ir+23S→1·S→2m1m2+δ1δ261m12+1m22δ3r→,(11)ΔVPV-PVs=1=gPV1gPV2S→1·S→212m1m2δ3r→whereby, in (10), I(r) cancels another term in (47) from [1].


### 2.2.2. Tensor Representation

The correction to thePS-PS potential in this representation is ΔVPS-PSs=1/4m12m22 (cf. (9)). Also, the correction to the PV-PV potential in this representation is the same as in (11).In [1], equations (65) receive the respective corrections:(12)ΔVS-Ss=1=gS1gS22δ1δ2m1m2Ir-5481m12+1m22δ3r→,(13)ΔVV-Vs=1=-e1e22δ1δ2Ir+23S→1·S→2m1m2+δ1δ231m12+1m22δ3r→.We note that the correction in (13) excludes one momentum-dependent term associated with p→21/m12+1/m22e-mr/4πr 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)ΔVS-Ss=1=gS1gS22δ1δ2m1m2Ir-1481m12+1m22δ3r→,(9)ΔVPS-PSs=1=-gPS1gPS2S→1·S→23δ3r→,(10)ΔVV-Vs=1=-e1e22δ1δ2Ir+23S→1·S→2m1m2+δ1δ261m12+1m22δ3r→,(11)ΔVPV-PVs=1=gPV1gPV2S→1·S→212m1m2δ3r→whereby, in (10), I(r) cancels another term in (47) from [1].


## 2.2.2. Tensor Representation

The correction to thePS-PS potential in this representation is ΔVPS-PSs=1/4m12m22 (cf. (9)). Also, the correction to the PV-PV potential in this representation is the same as in (11).In [1], equations (65) receive the respective corrections:(12)ΔVS-Ss=1=gS1gS22δ1δ2m1m2Ir-5481m12+1m22δ3r→,(13)ΔVV-Vs=1=-e1e22δ1δ2Ir+23S→1·S→2m1m2+δ1δ231m12+1m22δ3r→.We note that the correction in (13) excludes one momentum-dependent term associated with p→21/m12+1/m22e-mr/4πr 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-PS, PS-PS, 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: 9152437-2017-04-19.xml*