Calculation of Relativistic Nucleon-Nucleon Potentials in Three-Dimensions

2 MB·

by  M. R. Hadizadeh

EPJ manuscript No.
(will be inserted by the editor)
Calculation of Relativistic Nucleon-Nucleon Potentials in
Three-Dimensions
M. R. Hadizadeh1,2a and M. Radin3b
1 Institute of Nuclear and Particle Physics and Department of Physics and Astronomy, Ohio University, Athens, OH 45701,
USA,
2 College of Science and Engineering, Central State University, Wilberforce, OH 45384, USA,
3 Department of Physics, K. N. Toosi University of Technology, P.O.Box 16315–1618, Tehran, Iran.
7
Received: date / Revised version: date
1
0
2 Abstract. In this paper, we have applied a three-dimensional approach for calculation of the relativistic
nucleon-nucleonpotential.Thequadraticoperatorrelationbetweenthenon-relativisticandtherelativistic
n
nucleon-nucleoninteractionsisformulatedasafunctionofrelativetwo-nucleonmomentumvectors,which
a
J leadstoathree-dimensionalintegralequation.Theintegralequationissolvedbytheiterationmethod,and
thematrixelementsoftherelativisticpotentialarecalculatedfromnon-relativisticones.Spin-independent
9
Malfliet-Tjon potential is employed in the numerical calculations, and the numerical tests indicate that
1
the two-nucleon observables calculated by the relativistic potential are preserved with high accuracy.

]
h PACS. 21.45-v Few-body systems – 21.45.Bc Two-nucleon system – 24.10.Jv Relativistic models
t
-
l
c
u 1 Introduction method,thenonlinearintegralequationissolvedusingthe
n iterationmethodtogetrelativisticandboostedpotentials
[ Theinputsfortherelativisticthree-body(3B)boundand from non-relativistic ones. It is successfully implemented

scattering state calculations [1]-[9] are the fully off–shell in the NN problem, but it has not yet been extended to
1
v relativistic nucleon–nucleon (NN) t−matrices, which can a three-dimensional (3D) approach.
9 beobtainedbysolvingtherelativisticLippmann–Schwinger Anothermethodistomultiplythenon-relativisticpo-
7 (LS) integral equation using relativistic NN interactions. tential by a function that depends on NN relative mo-
2 It is known that there is a nonlinear operator rela- menta, in such a way that both the non-relativistic and
5 tion between the non-relativistic and the relativistic NN the relativistic potentials leads to same phase shifts and
0 interactions. So, the first step toward the calculation of observables[13].Thefunctionisdefinedinsuchawaythat
.
1 relativistic t−matrices is the calculation of the relativis- it changes the non-relativistic kinetic energy to relativis-
0 tic potentials from non-relativistic ones. To this aim, the tic kinetic energy by rescaling the momentum variables,
7 matrix elements of the relativistic NN potential in mo- which leads to the same 2N binding energy for both non-
1 mentum space are traditionally calculated by solving the relativistic and relativistic potentials.
:
v nonlinear equation using the following different methods. In the past decade a 3D approach based on momen-
i Inthespectralexpansionmethod,thequadraticequa- tumvectorvariableswasdevelopedtostudythefew–body
X
tion is solved by inserting a completeness relation of the bound and scattering problems [14]-[40]. In the 3D ap-
ar NN boundandscatteringstatesinto therightsideofthe proachoneworksdirectlywithvectorvariableswhichlead
quadratic equation and by projecting the result into the to 3D integral equations, whereas the partial wave (PW)
momentumspace[10,11].So,byhavingthenon-relativistic representation in the angular momentum basis leads to
potentialonecanfirstcalculatetheNN boundstatewave coupled equations. In the PW representation, depending
function and scattering half-shell t−matrix and used the on the energy scale of the problem, one must sum PWs,
result to solve the nonlinear equation. and consequently at higher energies one needs to consider
Intheiterationmethod,thenonlinearequationissolved a larger number of PWs, however the 3D approach auto-
by iteration. Kamada and Gl¨ockle introduced a powerful matically contains all PWs and the number of equations
numerical technique to calculate the matrix elements of is energy independent.
the relativistic NN potential directly from the matrix el- We would like to point out that as Polyzou and El-
ements of the non-relativistic NN potential [12]. In this ster have shown one can directly calculate the relativistic
t−matrixfromthenon-relativisticone,withoutneedingto
a [email protected] solve the nonlinear equation. Consequently, one does not
b [email protected] needtosolvetheLSequationfortheembeddedNN inter-
2 M. R. Hadizadeh, M. Radin: Calculation of Relativistic Nucleon-Nucleon Potentials in Three-Dimensions
action,andonecancalculatethefullyoff-shellrelativistic (3) as a function of the magnitude of the momentum vec-
t−matrixbyfollowingatwo-stepprocess.Thefirststepis tors and the angle between them. In our calculations we
to obtain the relativistic right–half–shell (RHS) t−matrix have used the spin independent Malfliet–Tjon (MT) po-
from the non-relativistic RHS t−matrix by an analytical tential, which is a superposition of short–range repulsive
relation proposed by Coester et al. [41]. The second step and long–range attractive Yukawa interactions [45]
is to calculate the fully–off–shell t−matrix from the RHS
(cid:18) (cid:19)
t−matrix by solving a first resolvent equation. Keister et V (p,p(cid:48))= 1 VR + VA , (4)
al.[42]proposedthemethodanditisimplementedforthe nr 2π2 q2+µ2 q2+µ2
R A
first time in a 3B scattering calculation [34] in this way.
Using the direct calculation of the relativistic t−matrix where q=p(cid:48)−p. The parameters of the MT–I potential
from the non-relativistic one, recently the relativistic ef- are given in Table 1. In order to obtain the matrix ele-
fects were studied in the 3B binding energy using a 3D ments of therelativistic potential, we have solved Eq. (3)
scheme[14,16].Therelativistic3Bwavefunctionwascal- by the iteration method. A coordinate system is defined
culated for the first time, and it was shown that the rel- by choosing the relative momentum vector p parallel to
ativistic effects lead to a reduction of about 3% in the z−axis and vector p(cid:48) in the x−z plane, so that Eq. (3)
3B binding energy for two models of a spin-independent can be written explicitly as
Malfliet-Tjon type potential. Since the 3D approach au-
tomatically considers all PWs, if it works for the bound
Table 1. Parameters of the Malfliet–Tjon I potential.
state, it can also be extended to the scattering problem,
independent of the range of energy. The next step is to
V (MeV fm) µ (fm−1) V (MeV fm) µ (fm−1)
considerthespinandisospindegreesoffreedomandwork A A R R
with realistic NN interactions. -626.8932 1.550 1438.7228 3.11
Inthiswork,wehaveappliedtheiterationmethodpro-
posed by Kamada and Gl¨ockle to construct the relativis-
tic NN potential from the non-relativistic Malfliet–Tjon
potential in a 3D scheme, without using the PW decom- 1 (cid:90) ∞ (cid:90) 1 (cid:90) 2π
V (p,p(cid:48),x(cid:48))+ dp(cid:48)(cid:48)p(cid:48)(cid:48)2 dx(cid:48)(cid:48) dφ(cid:48)(cid:48)
position. r ω(p)+ω(p(cid:48))
0 −1 0
4mV (p,p(cid:48),x(cid:48))
×V (p,p(cid:48)(cid:48),x(cid:48)(cid:48))V (p(cid:48)(cid:48),p(cid:48),y)= nr , (5)
r r ω(p)+ω(p(cid:48))
2 Three-dimensional formulation of the
quadratic operator relation between the where
relativistic and non-relativistic NN potentials (cid:112) (cid:112)
y =pˆ(cid:48)(cid:48)·pˆ(cid:48) =x(cid:48)x(cid:48)(cid:48)+ 1−x(cid:48)2 1−x(cid:48)(cid:48)2cosφ(cid:48)(cid:48),
According to Bakamjian and Thomas [43] and Fong and x(cid:48) =pˆ(cid:48)·pˆ,
Sucher [44], the relativistic NN dynamics is specified in x(cid:48)(cid:48) =pˆ(cid:48)(cid:48)·pˆ. (6)
terms of the NN mass operator h
We start the iteration with
(cid:104)p|h|p(cid:48)(cid:105)=ω(p)δ(p−p(cid:48))+V (p,p(cid:48)), (1)
r 4mV (p,p(cid:48),x(cid:48))
(cid:112) V(0)(p,p(cid:48),x(cid:48))= nr , (7)
where ω(p) = 2E(p) = 2 m2+p2, m is the mass of r ω(p)+ω(p(cid:48))
the nucleons and p is the relative momentum of two nu-
cleons. The connection between the relativistic and non- and stop it when the calculated relativistic potential sat-
relativistic NN potentials, i.e. V and V , is defined by isfiesEq.(5)witharelativeerrorof10−6 ateachsetpoint
r nr
the quadratic operator equation [12] (p,p(cid:48),x(cid:48)). To speed up the convergence procedure in solv-
ing Eq. (5) we can redefine the relativistic potential in
1 (cid:16) (cid:17)
V = ω(pˆ)V +V ω(pˆ)+V2 . (2) each step of the iteration as a linear combination of the
nr 4m r r r
calculated relativistic potential in the last two successive
The matrix elements of the relativistic potential can be iterations as
obtained from the non-relativistic NN potentials by the
projection of Eq. (2) into the NN basis states |p(cid:105) αV(n)(p,p(cid:48),x(cid:48))+βV(n−1)(p,p(cid:48),x(cid:48))
V(n)(p,p(cid:48),x(cid:48))−→ r r ;
1 (cid:90) r α+β
(cid:104)p|Vr|p(cid:48)(cid:105)+ ω(p)+ω(p(cid:48)) dp(cid:48)(cid:48)(cid:104)p|Vr|p(cid:48)(cid:48)(cid:105)(cid:104)p(cid:48)(cid:48)|Vr|p(cid:48)(cid:105) n=1,2,... (8)
Kamada and Gl¨ockle have used α = β = 1 in their cal-
4m(cid:104)p|V |p(cid:48)(cid:105)
= nr . (3) culations for the AV18 potential. Our numerical analysis
ω(p)+ω(p(cid:48))
showsthatthelargervaluesofαcanleadtofasterconver-
InourstudywehavefollowedKamadaandGl¨ockle’sstrat- genceinthesolutionofEq.(5).InTable2wehaveshown
egy [12] to obtain the matrix elements of the relativis- the number of iterations to reach convergence in Eq. (5)
tic NN potential, i.e. (cid:104)p|V |p(cid:48)(cid:105), directly from the non- fordifferentvaluesofαandβ.Itindicatesthatα=4and
r
relativistic one, i.e. (cid:104)p|V |p(cid:48)(cid:105) without using PW decom- β = 1 leads to faster convergence for the calculation of
nr
position. Here we discuss the numerical solution of Eq. the relativistic potential from the MT–I bare potential.
M. R. Hadizadeh, M. Radin: Calculation of Relativistic Nucleon-Nucleon Potentials in Three-Dimensions 3
Table 2. ThenumberofiterationsN ,toreachtheconver- Table 3. The convergence of the matrix elements of the rela-
iter
gence in the solution of equation (5) for MT–I potential as a tivistic potential V (p,p(cid:48),x(cid:48)) (in units of MeV fm3) as a func-
r
function of averaging parameters α and β. tion of iteration number calculated by MT–I bare potential in
the fixed points (p = 0.87 fm−1, p(cid:48) = 2.09 fm−1, x(cid:48) = 0,±1).
α β Niter The values of the MT–I bare potential Vnr(p,p(cid:48),x(cid:48)) are also
given.
1 0 17
1 1 18 x(cid:48) =−1 x(cid:48) =0 x(cid:48) =+1
2 1 12
3 1 10 V (p,p(cid:48),x(cid:48))
nr
4 1 8 1.1099096 0.7084511 -1.6327025
5 1 10
Iteration # V (p,p(cid:48),x(cid:48))
r
0 1.0525724 0.6718530 -1.5483583
For the discretization of the continuous momentum
1 0.7838006 0.3920976 -1.8512322
and angle variables we used the Gauss-Legendre quadra- 2 0.8895160 0.4979858 -1.7453002
ture.Forthemomentumvariablesahyperbolicpluslinear 3 0.8853336 0.4938142 -1.7495059
mappingisusedtocovertheintegrationdomain[0,∞)by 4 0.8848131 0.4932883 -1.7500527
(cid:83) (cid:83)
the subintervals [0,p1] [p1,p2] [p2,pmax] 5 0.8847858 0.4932573 -1.7500929
6 0.8847910 0.4932608 -1.7500930
1+x
p= , (9) 7 0.8847932 0.4932623 -1.7500928
1 +( 2 − 1 )x 8 0.8847938 0.4932626 -1.7500930
p1 p2 p1
9 0.8847939 0.4932626 -1.7500931
p −p p +p
p= max 2 x+ max 2. (10) 10 0.8847939 0.4932626 -1.7500932
2 2 11 0.8847939 0.4932626 -1.7500933
12 0.8847939 0.4932626 -1.7500933
The typical values for p , p and p are 4, 9 and 60
1 2 max
fm−1 respectively. In our calculations we have used 100
mesh points for the momentum variables, 50 mesh points
for the spherical and 10 mesh points for the azimuthal Table 3 shows an example of the convergence of the
anglevariables.Ineachiterationweneededtointerpolate matrix elements of the relativistic potential by iteration
ontheanglevariableyandtoavoidextrapolationwehave number for the fixed points (p = 0.87 fm−1, p(cid:48) = 2.09
added the extra points ±1 to the angle mesh points x(cid:48). In fm−1, x(cid:48) =0,±1) for α=2 and β =1.
ordertosaveruntimeandmemoryinsolutionofequation
(5) we have used the symmetry property of the kernel to
calculate the integration over azimuthal angle φ(cid:48)(cid:48) on the 3 Numerical tests of the relativistic potential
[0,π/2] domain
3.1 NN bound state
(cid:90) 2π (cid:90) π (cid:20) (cid:21)
dφ(cid:48)(cid:48)f(cosφ(cid:48)(cid:48))=2 2 dφ(cid:48)(cid:48) f(cosφ(cid:48)(cid:48))+f(−cosφ(cid:48)(cid:48)) .
The total Hamiltonian of two interacting nucleons in the
0 0
(11) center of mass system is:
In Figs. 1 and 2 we have shown our numerical results (cid:104)p|H|p(cid:48)(cid:105)=H (p)δ(p−p(cid:48))+V (p,p(cid:48)), (12)
0 nr
fortherelativisticpotentialcalculatedfromtheMT–Ipo-
tential. The bare MT–I potential as well as the difference where H (p) = p2 is the free Hamiltonian, V (p,p(cid:48))
0 m nr
betweenthebareandconstructedrelativisticpotentialsis is the non-relativistic NN interaction and p(p(cid:48)) is the
also shown. The plots of Fig. 1 show the non-relativistic initial (final) relative momentum of two nucleons. The
and relativistic potentials as well as their difference as a Lippmann–Schwingerequationforthetwo–nucleonbound
function of the relative momenta p = p(cid:48) and the angle state is given as
between them x(cid:48). It seems the solution of the quadratic
equation for the relativistic potential completely changes 1
|ψ (cid:105)= V |ψ (cid:105), (13)
the structure of the potential at forward angles for diago- d E −H nr d
d 0
nalmatrixelementsp=p(cid:48),andtherelativisticpotentialis
which can be represented in momentum space as the fol-
almostsmoothincomparisonwiththenon-relativisticpo-
lowing eigenvalue equation
tential.ThecorrespondingplotsinFig.2,showthepartial
wave projection of the non-relativistic and the relativistic 1 (cid:90)
p3DoterenptiraelssenatnadtioanlsobythVe(irp,dpi(cid:48)ff)e=ren2πce(cid:82)s,+c1adlxcu(cid:48)Plat(exd(cid:48))fVro(mp,pt(cid:48)h,ex(cid:48)), ψd(p)= Ed− pm2 dp(cid:48)Vnr(p,p(cid:48))ψd(p(cid:48)). (14)
l −1 l
as a function of the relative momenta p and p(cid:48). As we The relativistic Schr¨odinger equation for the two–nucleon
can see the matrix elements of the relativistic and non- bound state has the form
relativisticpotentialsarelargerforthelowerpartialwaves
and consequently their differences become higher. h|ψ (cid:105)=M |ψ (cid:105), (15)
d d d
4 M. R. Hadizadeh, M. Radin: Calculation of Relativistic Nucleon-Nucleon Potentials in Three-Dimensions
3m) 3m)
Vf 2 Vf 2 3m) 2
(Me 0 (Me 0 MeVf 0
′p,x)−−42 ′p,x)−−42 −V(r−2
Vp,(nr−16 Vp,(r−16 Vnr−14
0.5 60 0.5 60 0.5 60
0 40 0 40 0 40
−0.5 20 −0.5 20 −0.5 20
x′ −1 0 p (fm−1) x′ −1 0 p (fm−1) x′ −1 0 p (fm−1)
Fig.1.Thematrixelementsofthenon-relativistic(leftpanel),therelativistic(middlepanel)NN potentialsandtheirdifferences
(right panel) calculated by MT–I potential as a function of 2B relative momenta p=p(cid:48) and the angle between them x(cid:48).
V l (MeV fm3) V l (MeV fm3) V l − V l (MeV fm3)
nr r nr r

l = 2 l = 2 0 l = 2
8 0.2
−1
6
0
−2
4
−0.2
−3
2
−4 −0.4

) 0.8
1 8 l = 1 l = 1 0 l = 1
0.6
−
0.4
m 6
−5 0.2
f 4
( 0
′ 2 −0.2
p −10
−0.4

8 l = 0 l = 0 0 l = 0 3.5
6 −20
3
4 −40
2.5
2
−60

0 2 4 6 8 2 4 6 8 2 4 6 8
−
p (fm 1)
Fig.2.Thematrixelementsofthepartialwaveprojectionofthenon-relativistic(firstcolumn),therelativistic(secondcolumn)
NN potentials and their differences (third column) calculated by MT–I potential as a function of 2B relative momenta p and
p(cid:48).
M. R. Hadizadeh, M. Radin: Calculation of Relativistic Nucleon-Nucleon Potentials in Three-Dimensions 5
Table 4. Deuteron binding energy calculated for MT–I bare where
and relativistic potentials and their relative difference.
T (p ,p ,x(cid:48))=T (p ,p ,x(cid:48))+T (p ,p ,−x(cid:48)).(19)
sym 0 0 nr 0 0 nr 0 0
Enr (MeV) Er (MeV) (Enr−Er)/Enr%
d d d d d
Consequently, the total cross section can be obtained di-
-2.23100 -2.23229 0.05782
rectly from the differential cross section as
102 σ =(2π)5(cid:16)m2 (cid:17)2(cid:90) +1dx(cid:48)(cid:12)(cid:12)(cid:12)(cid:12)Tsym(p0,p0,x(cid:48))(cid:12)(cid:12)(cid:12)(cid:12)2. (20)
−1
0
10
) The relativistic NN scattering can be described by the
2 V ; E = -2.231 MeV
3/m nr d relativistic form of the Lippmann–Schwinger equation as
10-2 V ; E = -2.232 MeV
(f r d T (p,p(cid:48))=V (p,p(cid:48))
|(p) 10-4 r +(cid:90)rdp(cid:48)(cid:48) ω(p0V)r−(pω,(pp(cid:48)(cid:48)(cid:48)(cid:48)))+i(cid:15)Tr(p(cid:48)(cid:48),p(cid:48)).(21)
d
ψ
| -6 Therelativisticdifferentialandtotalcrosssectionscanbe
10
obtained by Eqs. (18) and (20) and by replacing m with
(cid:112)
m2+p2.
-8 0
10 In Table 5, our numerical results for the total elastic
0 10 20 30 40 50 60
NN scatteringcross sectionsobtainedby the constructed
-1
p (fm ) relativisticpotentialfromtheMT–Ipotentialaregivenas
afunctionoftheon–shellmomentump .Aswecanseethe
0
Fig. 3. The deuteron wave function calculated by the MT–I relativistic total cross sections are in excellent agreement
bare and relativistic potentials. with the corresponding non-relativistic cross sections and
have a percentage relative difference of less than 0.007.
NN phase shifts in the PW scheme are calculated by
where M is the deuteron mass. The relativistic deuteron
d
wave function |ψd(cid:105) satisfies the eigenvalue equation δ (p )=arctan(cid:18)Im Tl(p0)(cid:19), (22)
1 (cid:90) l 0 ReTl(p0)
ψ (p)= dp(cid:48)V (p,p(cid:48))ψ (p(cid:48)). (16)
d Md−ω(p) r d where the partial wave T−matrix, i.e. Tl(p0), can be ob-
tainedfromthe3DformoftheT−matrix,i.e.T(p ,p ,x(cid:48)),
0 0
Ournumericalresultsforthedeuteronbindingenergyand
as
wavefunctioncalculatedbyrelativisticandnon-relativistic
potentials are given in Table 4 and Fig. 3. As we can see (cid:90) +1
theconstructedrelativisticpotentialpreservesthedeuteron Tl(p0)=2π dx(cid:48)Pl(x(cid:48))T(p0,p0,x(cid:48)). (23)
binding energy obtained by the bare MT–I potential with −1
high accuracy and the relative percentage difference of
In Table 6, we have shown our numerical results for the
about 0.06.
s− and p−wave NN phase shifts as a function of the on–
shell momentum p calculated from the projection of the
0
3Dformofthenon-relativisticandrelativisticT−matrices
3.2 NN scattering byEq.(23).Aswecanseetherelativistics−andp−wave
NN phase shifts are in excellent agreement with the cor-
TheinhomogeneousLippmann–Schwingerequationwhich responding non-relativistic ones and have a relative per-
describes two–nucleon scattering can be represented in centagedifferenceoflessthan0.004and0.01respectively.
momentum space as
T (p,p(cid:48);E)=V (p,p(cid:48))
nr nr 4 Discussion and outlook
(cid:90) V (p,p(cid:48)(cid:48))
+ dp(cid:48)(cid:48) nr T (p(cid:48)(cid:48),p(cid:48);E).(17)
pm20 − pm(cid:48)(cid:48)2 +i(cid:15) nr In this paper, we have used a three-dimensional approach
to formulating the relativistic nucleon-nucleon potential
The differential cross section for elastic NN scattering as
asafunctionofthetwo-bodyrelativemomentumvectors.
afunctionofincidentprojectileenergyElab =2Ecm = 2mp20 Thequadraticequationwhichconnectstherelativisticand
is given by non-relativistic nucleon-nucleon interactions is presented
in momentum space as a three-dimensional integral equa-
ddΩσ =(2π)4(cid:16)m2 (cid:17)2(cid:12)(cid:12)(cid:12)(cid:12)Tsym(p0,p0,x(cid:48))(cid:12)(cid:12)(cid:12)(cid:12)2, (18) teiqouna.tFioonr itshesofilvrsetdnbuymtehreicsapliinm-ipnldeempeenntdaetniotnM, tahlefliientt-eTgjroanl
6 M. R. Hadizadeh, M. Radin: Calculation of Relativistic Nucleon-Nucleon Potentials in Three-Dimensions
Table 5. The total elastic NN scattering cross section as a Acknowledgements
functionoftheon–shellmomentump calculatedbytheMT–I
0
bare and relativistic potentials. We thank Professor Hiroyuki Kamada for helpful discus-
sions and also thank Dr. Jeremy Holtgrave for reading
p (MeV) σ (mb) σ (mb) |(σ −σ )/σ |%
0 nr r nr r nr the manuscript in detail and suggesting substantial im-
1 15274.4 15273.3 0.00720 provements. This work is performed under the auspices
10 14526.4 14525.5 0.00620 of the National Science Foundation under Contract No.
25 11485.5 11484.9 0.00522 NSF-HRD-1436702 with Central State University. M. R.
50 6367.46 6367.31 0.00236 H. acknowledges the partial support from the Institute of
75 3432.10 3432.08 0.00058 Nuclear and Particle Physics at Ohio University.
100 1926.16 1926.17 0.00052
200 297.580 297.586 0.00202
300 129.008 129.015 0.00543
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