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contact interactions have been studied thoroughly in non - relativistic quantum mechanics .
different studies have illustrated some of the analogies between quantum mechanics and quantum field theory @xcite .
nevertheless , the analogies are still to a large extent ambiguous . in our opinion , studying contact interactions relativistically give a better chance for understanding the relation between quantum mechanics and quantum field theory , especially quantum field theory is basically a theory for describing the interaction of relativistic particles .
the problem of relativistic @xmath0-function potential has been studied in the mathematical literature in the context of the theory of self - adjoint extensions of pseudo - differential operators @xcite . lately , we studied the problem by directly solving the relativistic version of the schrdinger equation of the hamiltonian @xmath1 in 1-d @xcite . unlike the textbook case of the non - relativistic @xmath0-function potential , the relativistic problem gives rise to ultra - violet divergences that required regularization and renormalization .
previously , we used dimensional regularization to show that the system has remarkable features .
for example , the relatively simple system shares many features with some complex quantum field theories , like asymptotic freedom , dimensional transmutation in the massless limit , and it also possesses an infra - red conformal fixed point .    in quantum field theory
it is well known that in some cases , different regularization methods give different results for the same problem @xcite .
this is also the case for non - local quantum field theories @xcite . in @xcite
, it was shown that for the non - relativistic @xmath0- function potential in odd @xmath2 , the dimensional regularization gives different results than the cutoff regularization , provided that the calculation is done in a non - perturbative setting .    in the case of the relativistic schrdinger equation ,
the hamiltonian is nonlocal unlike the non - relativistic case .
a naive guess would tell us that for this case , and for a certain problem , different regularization methods give different results .
however , in this paper we prove that this is not the case for the solution of the relativistic schrdinger equation for a point particle in an external 1-dimensional @xmath0-function potential .
we study the solution in a non - perturbative setting using cutoff regularization .
it gives exactly the same results from our previous calculations using dimensional regularization .
this important because it is an evidence of universality .
the rest of this paper is organized as the following : section 2 is for studying the bound state problem non - relativistically and relativistically using contour integrals .
section 3 is for discussing the gap equation and cutoff regularization , and using the bound state energy to define a renormalization condition .
section 4 for studying the scattering state problem non - relativistically and relativistically with the use of contour integrals , and defining the energy - dependence of the running coupling constant .
section 5 is for the conclusions .
the non - relativistic solution of the schrdinger equation for the @xmath0-function potential is a textbook problem .
nevertheless , we discuss it briefly to highlight the similarities and differences between the non - relativistic and relativistic case , especially regarding the contour integration approach used to calculate the wave function in both cases .
the schrdinger equation for this case is @xmath3 in momentum space , the above equation takes the following form @xmath4 where @xmath5 accordingly , in coordinate space the wave function for the bound state is @xmath6 where @xmath7 is the binding energy , @xmath8 , and @xmath9 is the bound state energy . the value of @xmath10 in the above equation can be obtained using contour integral .
the contour in this case has one pole inside the upper half circle at @xmath11 , as it is illustrated in figure.1 top panel . for the non - relativistic @xmath0-function potential ,
the wave function of the bound state is finite at the origin , and it is given by the following equation @xmath12    1.6 cm    the relativistic time - independent schrdinger equation is @xmath13 where @xmath14 is the bare coupling constant . in momentum space , the above equation is @xmath15 solving for @xmath16 , we get @xmath17 from the above equation , we express the bound state wave function in coordinate space as @xmath18 again here , the value of @xmath19 in the above equation can be obtained using contour integral . for a bound state @xmath20 , the contour in this case has one pole inside the upper half circle at @xmath21 .
in addition , it has a branch cut along the positive imaginary axis starting at @xmath22 as it is illustrated in figure.1 ( bottom panel ) . for this case @xmath23 is @xmath24.\ ] ] for a strong bound state when @xmath25 , and for an ultra - strong bound state
when @xmath26 , the pole does not have a residue .
for this case @xmath19 is @xmath27 both wave functions are logarithmically divergent at the origin .    in the non - relativistic limit , @xmath28 , the relativistic wave function of eq.([boundstate ] ) reduces to @xmath29,\ ] ] where @xmath30 .
the wave function indeed reduces to the non - relativistic wave function of eq.([nonrelbound ] ) .
however , the divergence of the relativistic wave function persists for any non - zero value of @xmath31 . on the other hand , the strong bound and the ultra strong bound states
do not have a non - relativistic limit .
they are purely relativistic states .
from eq.([psip ] ) and eq.([fourier ] ) , we get @xmath32 the above equation gives @xmath33 where @xmath34 depends on the value of @xmath9 .
the above equation determines the relation between @xmath14 and the bound state energy @xmath9 .
the integral in eq.([gap ] ) is logarithmically ultra - violet divergent , therefore it needs to be regularized .
we do this by using the momentum cutoff regularization . to proceed with this ,
first we must isolate the term that is causing the divergency .
we do that by expanding the integrand in eq.([gap ] ) in powers of @xmath35 , and then we get @xmath36 where @xmath37 is the momentum cutoff . by integrating term by term , and then taking the limit @xmath38
, we find that all terms under the summation sign are finite , while the first term is logarithmically ultra - violet divergent .
the resultant series from the second term converges for @xmath39 , and can be summed .
accordingly , we can write @xmath34 as @xmath40 or @xmath41 where @xmath42 is the finite part of @xmath34 . in this case , @xmath43 for an ultra - strong bound state with energy @xmath44 the series diverges .
still , the result can be obtained by directly integrating the convergent expression , and taking the limit @xmath38 @xmath45 as a renormalization condition , we fix the binding energy @xmath9 in units of the mass @xmath46 , such that the running bare coupling is given by @xmath47 at the non - relativistic limit , the binding energy @xmath48 is small compared to the rest mass . in this case , the running bare coupling is given by @xcite @xmath49 here it is very useful to compare the above equation with the non - relativistic contact interaction @xmath50 , which does not require renormalization , for the attractive delta @xmath51 , the bound state energy is given by @xmath52 , or @xmath53 .
this suggests to define a renormalized coupling @xmath54 as @xmath55
for the non - relativistic scattering states , we assume that the solution in momentum space has the following ansatz @xcite @xmath56 where @xmath57 . after substituting eq.([generalansatznon ] ) into eq.([schrpnon ] ) , we get @xmath58 @xmath59 in the above equation , the value of @xmath60 can be obtained using contour integral . in this case
, the contour has two poles on the real axis at @xmath61 as it is illustrated in figure.2 top panel . after calculating @xmath60 ,
it is straightforward to calculate @xmath62 , which is @xmath63.\ ] ] for the non - relativistic @xmath0-function potential , the wave function of the scattering state is finite at the origin .    1.6 cm    for the relativistic case , the ansatz takes the following form @xcite @xmath64 by substituting eq.([generalansatz ] ) into eq.([relschrp ] ) , and after solving for @xmath65 , we get @xmath66 where @xmath67 . from eq.([phirel ] ) , and by using momentum cutoff regularization , we can write @xmath68 the integral in the above equation is similar to the one in eq.([gap ] ) , with the exception that @xmath69 . therefore , we can write @xmath70 as in the case of @xmath34 , the term causing the divergency is @xmath71 .
therefore , we can write @xmath72 where @xmath73 is the convergent part of @xmath74 for @xmath75 . it can be obtained from eq.([iebu ] ) after replacing @xmath9 by @xmath76 , and we get @xmath77 now , we substitute for @xmath78 from eq.([phizerodelta ] ) into eq.([phirel ] ) , and also substituting for @xmath14 from eq.([gap ] ) , we get @xmath79 from eq.([iebg ] ) and eq.([ie0 ] ) , it is obvious that ultra - violet divergences of @xmath74 and @xmath34 is canceled .
therefore , we can say in this case that the problem is renormalizable .
accordingly , eq.([phipdelta1final ] ) can be written as @xmath80 certainly @xmath81 is a finite quantity .
the value of @xmath60 can be obtained from eq.([phipdelta1final2 ] ) using contour integral .
the contour in this case has two poles on the real axis at @xmath82 , and it has a branch cut along the positive imaginary axis starting at @xmath22 as it is illustrated in figure.2 bottom panel . after calculating @xmath83 , it is straightforward to calculate @xmath62 , which is @xmath84 , \
e = \sqrt{k^2 + m^2}.\nonumber\\\end{aligned}\ ] ] again , the wave function is logarithmically divergent at the origin . to understand the meaning of the expression @xmath81 , we take the non - relativistic limit when @xmath57 , and @xmath85 , then @xmath86 and the non - relativistic limit of eq.([psiscat ] ) is @xmath87 , \
\delta e = \frac{k^2}{2m}.\end{aligned}\ ] ] in the non - relativistic limit , @xmath88 reduces to the non - relativistic wave function of eq.([psiscatnon ] )
. however , the divergence of the relativistic wave function persists . nevertheless , comparing eq.([psiscatlimit ] ) with eq.([psiscatnon ] ) together with eq.([i - ieq ] ) is telling us that @xmath89 is nothing but the energy - dependent running coupling constant @xmath90 which is renormalized at the scale @xmath9 , and finite for @xmath38 .
therefore , we have @xmath91^{-1}.\end{aligned}\ ] ]
we proved that cutoff regularization gives the same results as dimensional regularization for the problem of the schrdinger equation for a relativistic point particle in an external 1-dimensional @xmath0-function potential . in both cases ,
the ultra - violet divergences have been canceled , and in both cases we get the same energy - dependent running coupling constant @xmath90 , which is renormalized at the scale @xmath9 , and finite .
this means that the features that was obtained using dimensional regularization in @xcite are preserved when using cutoff regularization .
this can be considered as an evidence of universality for a nonlocal hamiltonian .
this publication was made possible by the nprp grant # nprp 5 - 261 - 1 - 054 from the qatar national research fund ( a member of the qatar foundation ) .
the statements made herein are solely the responsibility of the authors .
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we study the solution of the relativistic schrdinger equation for a point particle in 1-d under @xmath0-function potential by using cutoff regularization . 
 we show that the problem is renormalizable , and the results are exactly the same as the ones obtained using dimensional regularization .