Euclidean to Minkowski Bethe-Salpeter amplitude and observables

1.2 MB·

by  J. Carbonell

Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
Euclidean to Minkowski Bethe-Salpeter amplitude and
observables
J. Carbonella,1, T. Fredericob,2, V.A. Karmanovc,3
1InstitutdePhysiqueNucleaire,Universit´eParis-Sud, IN2P3-CNRS,91406OrsayCedex, France
2InstitutoTecnolo´gico deAerona´utica, DCTA,12228-900,S.Jos´edosCampos,Brazil
3LebedevPhysicalInstitute,LeninskyProspekt53, 119991 Moscow,Russia
7
1
Received:date/Accepted:date
0
2

n Abstract We propose a method to reconstruct the formfactors)oneneedstheMinkowskispaceamplitude
a Bethe-SalpeteramplitudeinMinkowskispacegiventhe or, equivalently, the Euclidean one with complex ar-
J
Euclidean Bethe-Salpeter amplitude – or alternatively guments [2]. During the recent years, new methods to
0
the Light-Front wave function – as input. The method find the Minkowski solution for the two-body bound
1
isbasedonthenumericalinversionoftheNakanishiin- stateweredevelopedandprovedtheirefficiency,atleast

] tegralrepresentationandcomputingthecorresponding for simple kernels like the one-boson exchange (OBE).
h
weight function. This inversion procedure is, in gen- Someofthem[3,4,5,6]arebasedontheNakanishiinte-
p
- eral, rather unstable, and we propose several ways to gralrepresentation[7]oftheBSamplitudeandprovide,
p
considerably reduce the instabilities. In terms of the at first, the Nakanishi weight function, which then al-
e
h Nakanishi weight function, one can easily compute the lows to restore the BS amplitude and the light front
[ BS amplitude, the LF wave function and the electro- (LF) wave function.

1 magnetic form factor. The latter ones are very stable Other methods based on the appropriate treatment
v in spite of residual instabilities in the weight function. of the singularities in the BS equation, provide the
9 This procedure allows both, to continue the Euclidean Minkowski solution directly, both for bound and scat-
7
BSsolutionintheMinkowskispaceandtoobtainaBS tering states [8].
4
2 amplitude from a LF wave function. The straightforward numerical extrapolation of the
0 solutionfromEuclideantoMinkowskispaceisveryun-
Keywords Bethe-Salpeter equation, Nakanishi
1. stable and has not achieved any significant progress of
Representation, Light-Front
0 practicalinterest.However,theabovementionedNakan-
7 ishiintegralrepresentationprovidesamorereliablemethod.
1
ItisvalidfortheEuclideanΦ aswellasfortheMinkowski
: 1 Introduction E
v Φ solutions and both solutions are expressed via one
M
i
X and the same Nakanishi weight function g. This inte-
Amongthemethodsinquantumfieldtheoryandquan-
gral representation, given in detail in the next section,
r tummechanicalapproachestorelativisticfew-bodysys-
a can be symbolically written in the form:
tems , the Bethe-Salpeter (BS) equation [1], based on
firstprinciplesandexistingalreadyformorethansixty
Φ =K g (1)
years, remains rather popular. E E
The solution of the BS equation in Euclidean space
and
ismuchmoresimplethaninMinkowskione.Ratherof-
ten the Euclidean solution is enough, mainly when one
Φ =K g, (2)
is interested in finding the binding energy. However, in M M
many cases (e.g. for calculating electromagnetic (EM)
where K and K denote respectively the Nakanishi
E M
two-dimensionalintegralkernelsinEuclideanandMinkowski
ae-mail:[email protected]
bCorresponding author; e-mail:[email protected] spacesandgtheweightfunctionforthetwo-bodybound
ce-mail:[email protected] state.
2
Solving the integral equation (1) relative to g and hand, this equation has a unique solution. The chal-
substitutingtheresultintorelation(2),onecaninprin- lenge here is to use an appropriate method allowing
ciple determine the Minkowski amplitude Φ starting us to find this solution. These mathematical methods
M
with the Euclidean one Φ . This strategy seems ap- are developed and well known. Using them and know-
E
plicable since, in contrast to the direct extrapolation ing either Φ or ψ we aim to extract g and find
E LF
Φ → Φ , it uses the analytical properties of the BS from it Φ and the corresponding observables. This
E M M
amplitudewhichareimplementedintheNakanishirep- will demonstrate that this procedure is feasible to get
resentation. the solution.
A similar integral relation exist expressing the LF The appearance of mathematically well defined but
wave function ψ in terms of the Nakanishi weight func- numerically ill-posed problems is not a rare exception.
tion g [4]: It is, for instance, manifested when using the Stieltjes
and Lorentz integral transforms [15,16,17] to solve the
ψLF =Lg (3) scatteringfew-bodyproblemswithboundstatebound-
ary conditions or when extracting the general parton
One of the most remarkable interests of the Nakan- distributions (GPD) from the experimental data [18].
ishirepresentationisthatitconstitutesacommonroot
In this work we will proceed according to the fol-
forthethreedifferentformalobjects(Φ ,Φ ,ψ )al-
E M LF lowing steps.
lowingtodetermineanyofthemonceknownanyother,
In a first step, we will solve numerically equations
and thus providing a link between two theoretical ap-
(1) and (3) in a toy model where Φ , ψ and g are
proaches–Bethe-SalpeterandLight-FrontDynamics– E LF
known analytically. The comparison of the numerical
which are in principle and in practice quite different.
solutions for g with the analytical one will tell us the
Since the LF wave function can be also found inde-
reliabilityofourmethodinsolvingtheonedimensional
pendentlyeitherbysolvingthecorrespondingequation
and two-dimensional Fredholm first kind equations.
[9](seeforinstance[10])orbymeansofQuantumField
Inasecondstep,wewillgooveramorerealisticdy-
Theory (QFT) inspired approaches like the Discrete
namical case. There, Φ is obtained by solving the Eu-
Light Cone Quantization [11] or the Basis Light Front E
clidean BS equation. The LF wave function is given by
Quantization approach [12], one can pose the problem
equation(3),wheregiscalculatedbyusingthemethods
of finding the Nakanishi weight function from equation
introduced in [4], to solve the Minkowski BS equation,
(3). This equation is only one-dimensional. Its solution
and based on the LF projection and Nakanishi repre-
is more simple, it requires matrices of smaller dimen-
sentation. In order to keep trace of its origin we will
sionandthereforeitismorestablethanthesolutionof
denote it by g .
the 2D equation (1). M
It is worth noticing that the same method can be By solving equations (1) and (3), we find g in two
applied, far beyond the OBE dynamics, to the com- independent ways – denoted respectively gE and gLF –
pleteQFTdynamicalcontentofthelatticecalculations. tobecomparedwitheachotheraswellaswithgM.Our
Therein, the full (not restricted to the OBE kernel) methods should be consistent as far as all these weight
Euclidean BS amplitude can be obtained and, via the functions are reasonably close to each other.
Nakanishi integral, the corresponding BS amplitude in Thiswavefunctionψ ,foundbyprojectingheBS
LF
Minkowski space can be calculated as well as the ob- amplitude in the light-front, and the ψ one obtained
LF
servables.TheNakanishirepresentationwasusedin[13] from solving the LF equation, are practically indistin-
to calculate the parton distribution amplitude for the guishable from each other. We find also the Minkowski
pion. BS amplitude Φ and using it, we calculate observ-
M
The first results of our research in this field were ables, namely, EM form factor and momentum distri-
published in [14]. We have found that the solution of bution, represented by the LF wave function. We also
the integral equation (1) relative to g was rather un- calculate the form factor independently, expressing it
stable.Inordertofindnumericalsolution,thisintegral via the LF wave function ψ . It turned out that the
LF
equationwasdiscretizedusingtwoquitedifferentmeth- formfactorscalculatedbythesetwowaysareveryclose
ods(Gaussquadraturesandsplines)anditresultedinto to each other and practically insensitive to instabilities
a linear system. The instability of its the solution dra- of g’s remaining after their suppression by the method
matically increases with the matrix dimension. we use. That demonstrates that indeed, knowing the
It turned out that the equation (1) is a Fredholm Euclidean BS amplitude and using the methods devel-
equationofthefirstkind,whichmathematicallyaclas- opedinthepresentpaper,onecancalculateelectroweak
sical example of an ill-posed problem. On the other observables.
3
In Sec. 2 we present the formulas for the Nakan- kernel (OBE, for instance). As mentioned, it can be
ishi integral representation, both for the BS amplitude also found in lattice calculations [19], which are much
andtheLFwavefunction.Achangeofvariables(map- difficultnumericallybutincludealltheQuantumField
ping) introduced to simplify the integration domain is Theory dynamics.
described in Sec. 3. The numerical method for solving Weremind,thatalternativelytoeq.(3)theLFwave
the discretized equation is presented in Sec. 4. The an- function ψ canbefoundby solvinga2Dequationin
LF
alytically solvable model, on which we will test the nu- the LF dynamics [9] or by using other QFT inspired
merical solutions, comparing them with the analytical methods like DLCQ [11] or BLFQ [12].
ones, is presented in Sec. 5. In Sec. 6 we study the sta- We will therefore assume that one of the functions
bility of these solutions. In Sec. 7 we find numerically Φ or ψ is known and can be used as input for solv-
E LF
the Nakanishi weight function using as input the Eu- ing equations (5) or (6) relative to Nakanishi weight
clidean BS amplitude and the LF wave function found functiong.Onceobtainedinthisway,g canbeusedto
from the OBE interaction kernel. These results are ap- calculate any of the remaining quantities in the triplet
plied to calculate EM form factors in Sec. 8. Finally, (ψ,Φ ,Φ ) and corresponding observables.
E M
Sec.9containsagenraldiscussionandtheconclusions.
3 Mapping
2 Nakanishi representation
Though,thetime-likemomentumvariablevariesinthe
TheBSamplitudeinMinkowskispaceΦ (k,p),foran range −∞ < k < ∞, we can reduce the problem to
M 4
S-wavetwo-bodyboundstatewithconstituentsmasses the half-interval 0 < k < ∞ and real arithmetic, by
4
m ant total mass M, depends in the rest frame p = assuming Φ (k ,−k ) = Φ (k ,k ). Furthermore, we
E v 4 E v 4
(M,0) on two variables k and k . We represent the will make in Eq. (5) the following mapping
v 0
four-vector k as k =(k ,k) and denote k =|k|.
0 v 0<γ(cid:48),k ,k <∞ → 0<x(cid:48),x,z <1
TheNakanishirepresentationforthisamplitudereads v 4
[7]: by:
Φ (k ,k )= (4) x(cid:48) x z
M v 0 γ(cid:48) = , k = , k = . (7)
(cid:90) 1 (cid:90) ∞ g(γ(cid:48),z(cid:48)) 1−x(cid:48) v 1−x 4 1−z
dz(cid:48) dγ(cid:48) ,
−1 0 (γ(cid:48)+κ2−k02+kv2−Mk0z(cid:48)−i(cid:15))3 Eq. (5) takes then the form:
where κ2 = m2 − M2 and g is the Nakanishi weight Φ (x,z)=2(cid:90) 1dx(cid:48)(cid:90) 1dz(cid:48) g(x(cid:48),z(cid:48)) (8)
4 E (1−x(cid:48))2
function. The power of the denominator is in fact an 0 0
arbitraryintegerandwehavechosen3forconvenience. ×Re(cid:20) x2 + z2 + x(cid:48) +κ2−iM zz(cid:48) (cid:21)−3.
This is just the equation which is symbolically written (1−x)2 (1−z)2 1−x(cid:48) 1−z
in (2). to be solved in the compact domain [0,1]×[0,1]. The
In the Euclidean space, after the replacement k = factor 1 istheJacobian.Werewritethisequation
0 (1−x(cid:48))2
ik , this formula is rewritten as: as:
4
(cid:90) 1 (cid:90) 1
ΦE(kv,k4)= (5) Φ (x,z)= dx(cid:48) dz(cid:48)K(x,z;x(cid:48),z(cid:48))g(x(cid:48),z(cid:48)) (9)
E
(cid:90) 1 (cid:90) ∞ g(γ(cid:48),z(cid:48)) 0 0
dz(cid:48) dγ(cid:48) ,
−1 0 (γ(cid:48)+k42+kv2+κ2−iMk4z(cid:48))3 where
2
which was represented symbolically in (1). K(x,z;x(cid:48),z(cid:48))= (10)
(1−x(cid:48))2
One can also express through g the LF wave func-
tion as [4]: (cid:20) x2 z2 x(cid:48) zz(cid:48) (cid:21)−3
×Re + + +κ2−iM .
(1−x)2 (1−z)2 1−x(cid:48) 1−z
1(cid:90) ∞ (1−z2)g(γ(cid:48),z)dγ(cid:48)
ψLF(γ,z)= 4 (cid:104) (cid:105)2. (6) For the normal solutions g(x(cid:48),−z(cid:48)) = g(x(cid:48),z(cid:48)) that
0 γ(cid:48)+γ+z2m2+κ2(1−z2) comes from the symmetry of Φ with respect to k ,
E 4
which separates out the abnormal solutions for the two
This equation corresponds to (3). As usually found in
identical boson case.
the literature, the LF wave function ψ is considered
LF After introducing in Eq. (6) for the variables γ,γ(cid:48)
as depending on variables k2,x. They are related to
⊥ the mapping defined in (7), we get that:
γ,z by γ =k2, z =2x−1 (see e.g. [9]).
⊥
TheEuclideanBSamplitudeΦ (k ,k )canbeeas- (cid:90) 1
E v 4 ψ (x,z)= dx(cid:48)L(x,x(cid:48);z)g(x(cid:48),z) (11)
ilyfoundfromthecorrespondingequationwithagiven LF
0
4
where We substitute g in Eq. (11), calculate the integral nu-
(1−z2) merically and validate the equation in the N Gaussian
L(x,x(cid:48);z)= (12)
4(1−x(cid:48))2 quadraturepoints{xj}intheinterval0<x<1.Equa-
tion(11)transformsintotheinhomogeneouslinearsys-
(cid:18) x x(cid:48) (cid:19)−2
× + +z2m2+κ2(1−z2) . tem (the parameter z is omitted):
1−x 1−x(cid:48)
N
Here z plays the role of a parameter. (cid:88)
ψ (x )= L c , (18)
LF j ji i
i=1
4 Solving equations (9) and (11) where
(cid:90) 1
L = dx(cid:48)L(x ,x(cid:48);z)G (x(cid:48)), (19)
Wewill lookfor the solutionof Eq. (9)byexpanding it ji j i−1
0
on Gegenbauer polynomials in both variables:
with L(x,x(cid:48);z) defined in (12). For a given z, the sys-
N(cid:88)x,Nz tem(18)ofN linearequationsissolved,andoncedeter-
g(x,z)= c G (x)G (z) (13)
ij i−1 j−1 minedthecoefficientsc ,Eq.(17)providesthesolution
i
i,j=1 g(x,z) everywhere.
where Concerning the number of points N used in the
√ (1) discretization of the integral equations there exists a
Gn(x)= 2n+1Cn2 (2x−1), (14) ”plateauofstability”,correspondingtoanoptimalvalue
of N. For a small values of N, the accuracy is not
(1)
Cn2 (2x − 1) is a standard (non-normalized) Gegen- enough but, as we will see below, by increasing N the
bauer polynomial. Whereas, the polynomial Gn(x) are solution becomes oscillatory and unstable. This is just
orthonormalized: a manifestation of above mentioned fact that the equa-
(cid:90) 1 tions (9) and (11) represent both a Fredholm integral
dxG (x)G (x)=δ
n n(cid:48) nn(cid:48) equation of the first kind which is a classical example
0
ofanill-posedproblem.Theirkernelsarequadratically
We substitute g(z,x) from (13) in Eq. (9), calculate
integrablewhatensurestheexistenceanduniquenessof
the integrals numerically and validate the equation in
thesolution.However,directnumericalmethodsdonot
N =N ×N discretepoints(x ,z ),withi=1,...,N
x z i j x allowtofindthesolutioninpractice,sincetheyleadto
andj =1,...,N .Wechoseasvalidationpointsx (z )
z i j unstableresults.Toavoidinstability,onecan,ofcourse,
the N (N ) Gauss points in the interval 0 < x < 1.
x z keepN smallenough.However,forsmallN theexpan-
Eq. (9) transforms into the following linear system:
sions (13) and (17) give a very crude reproduction of
the unknown g. Therefore, to find solution, we will use
Φ =N(cid:88)z,NxKi(cid:48)j(cid:48)c (15) a special mathematical method – the Tikhonov regu-
ij ij i(cid:48)j(cid:48)
larization (TRM) [21].
i(cid:48)j(cid:48)
Notethatanothermethod(MaximumEntropyMethod)
where
wasrecentlyproposed[20]tosolveEq.(5)andsuccess-
Φij =ΦE(xi,zj), (16) fully applied, at least in the case of monotonic g’s.
(cid:90) 1 (cid:90) 1 We will follow here the standard and straightfor-
Ki(cid:48)j(cid:48) = dz(cid:48) dxK(x ,z ;z(cid:48),x(cid:48))G (x(cid:48))G (z(cid:48))
ij i j i(cid:48)−1 j(cid:48)−1 ward way explained above: by discretization of the in-
0 0
tegral equation we turn it into a matrix equation and
Eq.(15)isaN×N systemoflinearequationsofthe
solve it by inverting the matrix, however, regularizing
type B = AC with the inhomogeneous term given by
the inversion problem. The TRM [21] allows to find a
the euclidean BS amplitude B ≡Φ and as unknowns
ij stable solution for sufficiently large N. Namely, follow-
the array C ≡ c of coefficients of the expansion (13).
ij ing to [21], we will solve an approximate minimization
The solution of this system C = A−1B, will provide
problem,i.e.,wewillfindC providingtheminimumof:
the coefficients c and by this the Nakanishi weight
ij
function g(x,z) at any point. (cid:107) AC − B (cid:107) .
ThesolutionofEq.(11)isfoundsimilarly.However,
In the normal form, that corresponds to the replace-
since z is a parameter, the solution is found for a fixed
ment of the equation AC = B by the regularized one
z and its decomposition is one-dimensional:
A†AC +(cid:15)C = A†B with (cid:15) (cid:28) 1. The solution of this
N equation reads:
(cid:88)
g(x,z)= c G (x) (17)
i i−1 C =(A†A+(cid:15)I)−1A†B, (20)
i=1 (cid:15)
5
where I is the identity matrix. For small (cid:15) (but not for wherethekernelL ,theinhomogeneoustermψ and
1 LF
infinitesimal) the solution C of this equation is very the solution g read:
(cid:15)
close to the solution of the original equation AC = B. (cid:104) (cid:105)
ψ (x)= (2−x)x+2(1−x)log(1−x)
However, the solution of Eq. (20) is much more stable LF
than the solution of AC =B. (1−x)2
× , (26)
In our previous work [14] we also used a regulariza- x3
tionprocedure,butinamorenaiveform.Theequation 1 1
L (x,x(cid:48))=
AC = B was replaced not by A†AC +(cid:15)C = A†B, but 1 (cid:104) (cid:105)2(1−x(cid:48))2
x + x(cid:48) +1
by (A+(cid:15)I)C = B. Then, instead of the solution (20) 1−x 1−x(cid:48)
we got (1−x)2
= , (27)
(1−xx(cid:48))2
C =(A+(cid:15)I)−1B. (21) g(x)=(1−x)2. (28)
(cid:15)
We will consider ψ , given by Eq. (26) as input, solve
LF
Thisincreasesthestabilitybutnotsostronglylike(20), numerically, Eq. (25) in the form of the decomposition
as we are going to illustrate in what follows. (17) and compare the solution with the analytical g
given in (28).
5 Analytically solvable model
As the first step, we will check the methods and study
theappearanceofnumericalinstabilitiesbysolvingthe
simplestone-dimensionalequation(6)inamodelwhere
thefunctionsψ andgareknownanalytically.Wewill
LF
comparethenumericalsolutionwiththeanalyticalone.
Inequation(6)thevaluesz,mandκaretheparameters
which can be chosen arbitrary (keeping the kernel non-
singular). We can also put an arbitrary factor at the
frontoftheintegralinr.h.-side,whichonlychangesthe
normalization of ψ . Using this freedom, we choose
LF
z2m2+κ2(1−z2)=1 and replace the factor 1(1−z2)
4
also by 1. Then the equation (6) obtains the form:
Fig. 1 The numerical solution of Eq. (25) for g(x) (dashed
curve), found in the form of Eq. (17), for the discretization
(cid:90) ∞ g(γ(cid:48))dγ(cid:48) rank N = 11 with (cid:15) = 0 in (21) in comparison to the exact
ψ (γ)= . (22) solutiong(x)=(1−x)2(not-distinguishablefromthedashed
LF 0 (γ+γ(cid:48)+1)2 curve).
As a solution we take:
1
6 Studying stability
g(γ)= (23)
(1+γ)2
In our precedent paper [14] we have found that the in-
Its substitution it into (22) provides the l.h.-side: version of the Nakanishi integral is rather unstable rel-
ative to the increase of the number of points N, using
(cid:20) (cid:21) theregularizationmethodgivenbyEq.(21).Therefore,
1 γ(2+γ)
ψ (γ)= −2log(1+γ) (24) solvingEq.(18),wewillfirststudytheonsetofinstabil-
LF γ3 (1+γ)
ity and its suppression by the Tikhonov regularization
(20).
Inthemappingvariablestheequation(22)isrewrit- The solution of the non-regularized equation (18),
ten as: by using Eq. (21) with (cid:15) = 0 and N = 11, is shown
in Fig. 1. It coincides, within the thickness of the lines,
(cid:90) 1 with the exact one, given by Eq. (28). However, when
ψ (x)= L (x,x(cid:48))g(x(cid:48))dx(cid:48), (25)
LF 1 increasing N up to N = 14, we can observe (Fig. 2)
0
6
Fig. 2 Thesameasin Fig.1butforN =14, (cid:15)=0. Fig. 4 The sameas in Figs. 1,2, 3, but withTikhonovreg-
ularization,for N =16and(cid:15)=10−18.
the onset of instability: the numerical solution g(x) be-
comes oscillating around the exact solution. Note also
that the determinant of A is very small and it quickly
decreases when N increases. For N = 11, det(A) ∼
10−35 while for N =14 one has det(A)∼10−59.
Fig. 5 ThesameasinFig.4butwiththeregularizationby
Eq.(21), N =16and(cid:15)=10−10.
with N = 16, the solution found by TRM, Eq. (20),
is shown in Fig. 4. It has some oscillations, which are
strongly enhanced in the case with regularization (21)
and (cid:15)=10−10. The same happens with the regularized
Fig. 3 Thesameasin Fig.1butforN =16, (cid:15)=0.
solution for N = 14, (cid:15) = 10−10 (not shown), when we
replace the Tikhonov regularization (20) by (21).
FurtherincreaseofN resultsinextremelystrongos- OurstudyshowsthatusingtheTRMmethodthere
cillations. The solution for N =16 and (cid:15)=0, strongly is almost no (cid:15) dependence of the results in a rather
oscillates and dramatically differs from the exact one, wide limits. However, when (cid:15) is taken relatively large
as it can be seen in Fig. 3 . For N =32 and (cid:15)=0, the (e.g. N = 16, (cid:15) = 10−4), the numerical solution can
solution oscillates even more strongly and is wrong by sensibly differs from the exact one. On the other hand,
orders of magnitude; we find for instance g(0) ≈ −103 asitwasmentioned,whenusingtoosmall(cid:15)onerecovers
instead of g(0)=1 according to the analytic result. the problem of oscillations, as it is seen in Fig. 4 (N =
Now let us solve the corresponding equation by us- 16 and (cid:15) = 10−18). For N = 16,(cid:15) = 10−17 one can
ing TRM, Eq. (20). For N = 16 and (cid:15) = 10−10, we observe the first, yet weak signs of oscillations. Hence,
found that the oscillations completely disappear. The for N = 16, the stability (absence of oscillations) and
numerical solution coincides with the exact one within insensitivity to the value of (cid:15) (cid:54)= 0 are valid in rather
precision better than 1%, similarly to what we observe large interval (cid:15)=10−4÷10−17.
in the figure 1. To avoid repetition, we do not show The regularized solution for N = 32,(cid:15) = 10−10,
the corresponding figure. For much smaller (cid:15) = 10−18 found by Eq. (20) and TRM, again coincides with the
7
exact one within thickness of lines. We do not show it cialmethods,inparticularthosebasedontheTikhonov
since the curves are the same as in Fig. 1. For N =32, regularization (20), allows to solve it.
the oscillations appear at (cid:15)≤10−16. In the next section we will show that the Nakanishi
For higher N the situation is similar. Like in the representation (5) for the Euclidean BS amplitude can
case N = 32, the solution for N = 64,(cid:15) = 0, found bealsoinverted.Thiswillbeshownnotinatoymodel,
by C =A−1B strongly oscillates and it strongly differs but for the BS solution with a OBE kernel.
from the exact one, even stronger than in Fig. 3. The
calculated value g(0) ≈ −107 instead of g(0) = 1 is
again completely wrong. The regularized solution for 7 OBE interaction
N = 64,(cid:15) = 10−10, found by Eq. (20) also coincides
with theexact one withinthicknessof lines, like itis in Let us now consider the dynamical case of two spinless
Fig. 1. However, the strong oscillations appear earlier, particle of unit mass (m = 1), interacting via OBE
at (cid:15)≤10−13 (in contrast to (cid:15)≤10−16 for N =32). kernelwithexchangedbosonmassµ=0.5,andforming
We compare now the two ways of regularization a bound state of total mass M =1.0.
givenbyEqs.(21)and(20).InFig.5thesolutiongiven All the necessary solutions in this model – LF wave
byEq.(21)forN =16,(cid:15)=10−10isshown.Itoscillates, function ψ , Euclidean BS amplitude Φ and Nakan-
LF E
while, the solution found by using TRM, Eq.(20), with ishiweightfunctiong –havebeencomputedbysolving
sameNand(cid:15)(notshown)coincideswiththeanalytical thecorrespondingequations.Theweightfunctiongwas
one. As mentioned above, the solution for N =16 and foundindependentlyfromtheLFwavefunctionandBS
(cid:15)=10−18 (Fig. 4) reveals moderate oscillations, which solutions by solving the equation derived in [4,6] with
disappear at (cid:15)>10−17. the same OBE kernel. We will use for ψ and g the
LF
Thereasonwhichmakesthedeterminantverysmall results [6,22] and for Φ – our own calculations.
E
and turns the solution of the linear system in an ill- We will take profit from the simplicity of the Eu-
conditionedproblemisthepresenceofverysmalleigen- clideansolutionandextracttheNakanishiweightfunc-
values of the kernel of the Fredholm integral equation tion g from Φ by inverting the Nakanishi represen-
E E
(25). As the dimension of the matrix increases, more tationinEuclideanspace(5)viatheTikhonovregular-
small eigenvalues are present, and the eigenstates are ization method and compare it with g found from an
oscillating functions, that mixes with the solution ob- equationderivedfromtheMinkowskispaceBSequation
tained within a numerical accuracy. These contribu- [6,22].Thequalityofthesolutiong willbecheckedby
E
tions to the solution are oscillatory, building the pat- computing the LF wave function ψ and comparing
LF
tern seen in Figs. 3 and 5, by increasing dimension of it with ψ found via g provided by [22]. Schemati-
LF
the matrix equation. The amplitude of the eigenvec- cally the above extraction procedure is represented as
tor contribution to the solution increases as the eigen- Φ → g → ψ . Another possibility to extract g is
E E LF
value decreases, making the oscillations of the solution ψ →g →Φ . Namely, starting with the LF wave
LF LF E
divergent. The regularization by (cid:15) cuts the contribu- function, get g and calculate with it the Euclidean
LF
tion of the small eigenvalues, and to some extend the Φ and compare this result with the initial Φ . We
E E
numerical stability can be found, at the expense of will also compare the EM form factors calculated ini-
a finite (cid:15). Both regularization methods from Eq. (20) tially via ψ and, independently, via Φ and finally
LF M
(Tikhonovmethod)and(21)improvethenumericalso- expressedviag.Wewillseethattheobservablesareal-
lution.TheTikhonovregularizationwiththereduction most insensitive to the residual uncertainties in either
to the normal form provides larger eigenvalues as com- g org ,whichsurviveaftersuppressingtheinstabil-
E LF
paredto(21),asitisevidentfromthestabilityanalysis ities by the Tikhonov regularization (20). This ensures
that in practice allows much smaller (cid:15)’s before consid- reliable results for the observables, when one uses the
erable oscillations of the solution appear. See for ex- EuclideanBSamplitudeasinput,calculatesgbyinvert-
ample Fig. 5, where the method (21) provides huge os- ing the Nakanishi integral and then with the extracted
cillations. Whereas the TRM allows (cid:15) small as 10−18 g goes to the observables.
compared with the solution with (cid:15)=10−10. The Nakanishi weight function g obtained by in-
E
The calculations presented in this section demon- verting eq. (9) with N = 4,N = 2 and the TRM
x z
strate that the Nakanishi representation, at least, for with (cid:15) = 10−10, is shown in Fig. 6 (dashed line). Solid
the LF wave function, Eq. (6), can be indeed inverted line represents the direct solution g – denoted below as
numerically,thatis,besolvedrelativetotheNakanishi g – computed in [6,22] by Frederico-Salm`e-Viviani
FSV
weight function g. Though this representation, consid- bysolvingadynamicalequationforgandnormalizedin
ered as an equation for g, is an ill-posed problem, spe- adifferentwaythanΦ .Tocomparebothsolutionsfor
E
8
g, we normalize g so that the two solutions coincide ThecorrespondingLFwavefunctionscalculatedfrom
E
(and equal to −1) at x=0, z =0.3. g andg byusingEq.(11)areshowninFig.8.We
E FSV
multiplyψ calculatedbyEq.(11)byanormalization
LF
factor.Bothψ ’swellcoincidewitheachotherdespite
LF
the difference between g and g seen in Fig. 6.
E FSV
Fig. 6 The solution g (x,z = 0.3) (dashed) found solving
E
Eq. (9) with N = 2,N = 4 and (cid:15) = 10−10, with Φ ,
z x E
using the Tikhonov regularization method, compared to the
solutiongFSV(x,z=0.3)[6,22] (solid). Fig. 8 LF wave function calculated by Eq. (11) (multiplied
by a normalization factor) for both extracted g and g
LF E
comparedtotheactualresultsobtainedwithg in[6,22]
FSV
To check the quality of g , solution of Eq. (9), we from the Minkowski space solution of the BS equation. All
E
substitute it in r.h.-side of Eq. (9), calculate Φ and the curves overlap with each other within the width of the
E
lines.
compare it with the input Φ . The result is shown in
E
Fig. 7. We display by the solid curve the input Eu-
clidean BS amplitude c Φ as a function of x for a
2 E
fixed value z = 0.3. The dashed curve corresponds to
the BS amplitude Φ (x,z = 0.3) calculated via g by
E E
Eq. (9). Up to a factor c =0.88 they are very close to
2
each other.

Fig. 9 The Nakanishi weight function g (x,z = 0.3) ex-
E
tracted from Φ , with N = 2,N = 4 (dashed curve) and
E z x
N =5,N =3(solidcurve)with(cid:15)=10−10 usingTRM.
x z
Thesolutiong forN =5,N =3,(cid:15)=10−10 mul-
E x z
tipliedbyanormalizationfactor,isshowninFig.9.For
Fig. 7 ThesolidcurvecorrespondstotheEuclideanBSam-
comparison, we add the solution for N = 4, N = 2,
plitude c Φ (z = 0.3,x) (c = 0.88) calculated via the BS x z
2 E 2 (cid:15)=10−10,shownbydashedlineinFig.6.Inspiteofthe
equationwiththeOBEkernel.Thedashedcurvecorresponds
totheBSamplitudeΦ (z=0.3,x)calculatedviag byEq. visible difference between the dashed and solid curves,
E E
(9). these two solutions give coinciding LF wave functions
9
andtheEuclideanBSamplitudes(afterequivalentnor-
malizations), similarly to ones shown in Figs. 7 and 8.
8 Calculating EM form factor
Once we have the Nakanishi weight function g by Eqs.
(4) and (6), we can find both the BS amplitude in
Minkowski space and the LF wave function. The EM
form factor can be expressed via both of them. In this
way, we obtain two expressions in terms of g. We will
use both to calculate the form factor and we will com-
pare the results.
The electromagnetic vertex is expressed in terms of
the BS amplitude by:
(p+p(cid:48))νFBSM(Q2)=i(cid:90) d4k (p+p(cid:48)−2k)ν Fig. 10 EM form factor calculated via LF wave function
(2π)4 by Eq. (32), for Nz = 2, Nx = 4 (dashed curve) and for
(cid:18)1 (cid:19) (cid:18)1 (cid:19) Nz = 3,Nx = 5 (solid curve), i.e., via solutions gE shown
×(k2−m2)Φ p−k;p Φ p(cid:48)−k;p(cid:48) , (29) Fig.9.
M 2 M 2
whereQ2 =−q2,q isthefour-momentumtransfer,and
TheEMformfactorcalculatedviaLFwavefunctionby
p(cid:48) =p+q. Substituting Φ in the form (4) and calcu-
M
Eq. (32), for N = 4, N = 2 (dashed curve) and for
latingtheintegraloverd4k,onegets(seeEq.(13)from x z
N =5,N =3 (solid curve), i.e., via the solutions for
[23]): x z
g given in Fig. 8, is shown in Fig. 10. We see that the
E
1 (cid:90) ∞ (cid:90) 1 apparently distinct solutions shown in Fig. 6 does not
FBSM(Q2)= dγ dzg(γ,z)
27π3N result in a considerable difference of the corresponding
BSM 0 −1
(cid:90) ∞ (cid:90) 1 (cid:90) 1 f form factors. Notice that the differences represent only
× dγ(cid:48) dz(cid:48)g(γ(cid:48),z(cid:48)) duu2(1−u)2 fn4um (30) a 5% deviation at Q2 ∼10m2
0 −1 0 den
with
f =(6ξ−5)m2+[γ(cid:48)(1−u)+γu](3ξ−2)
num
1
+2M2ξ(1−ξ)+ Q2(1−u)u(1+z)(1+z(cid:48))
4
f =m2+γ(cid:48)(1−u)+γu−M2(1−ξ)ξ
den
1
+ Q2(1−u)u(1+z)(1+z(cid:48)),
4
where ξ = 1(1+z)u+ 1(1+z(cid:48))(1−u).
2 2
ThenormalizationfactorN isdeterminedfrom
BSM
the condition FBSM(0)=1.
The form factor is expressed via LF wave function
as follows (see e.g. Eq. (6.14) from [9]):
1 (cid:90) d2k dx
FLF(Q2)= ⊥
(2π)3 2x(1−x)
×ψ (k ,x)ψ (k −xQ ,x), (31)
LF ⊥ LF ⊥ ⊥
Fig. 11 EM form factor calculated via Minkowski BS am-
whereQ2 =Q2.Substitutingin(31)theLFwavefunc-
⊥ plitude, by Eq. (30) with the same gE used to calculate the
tion ψ (k ,x) determined by Eq. (6), one finds (see
LF ⊥ curvesinFig.10;thecurvesareindicatedas inFig.10.
Eq. (26) from [23]):
1 (cid:90) ∞ (cid:90) ∞ (cid:90) 1 (cid:90) 1
FLF(Q2)= dγ(cid:48) dγ dx du
25π3N Thesameformfactor,calculatedviathesameNakan-
LF 0 0 0 0
x(1−x)u(1−u)g(γ,2x−1)g(γ(cid:48),2x−1) ishi weight functions, but in the Minkowski BS frame-
× .
[uγ+(1−u)γ(cid:48)+u(1−u)x2Q2+m2−x(1−x)M2]3 work by Eq. (30) is shown in Fig. 11. The difference
(32) between the form factors corresponding to the two so-
10
wave function ψ . It is the same g that enters in the
LF
Minkowski BS amplitude (4) and in the form factor
(30), including the higher sector contributions.
As it was just mentioned, the form factor (30) (cal-
culatedwithg extractedfromthetwo-bodyLFwave
LF
function)includesimplicitlynotonlythetwo-bodycom-
ponent contribution but also the higher Fock compo-
nents. So, this higher Fock sector contribution is found
from the two-body component which was taken as in-
put.Thoughitseemsalittlebitparadoxical(thepossi-
bility to get information about higher Fock states from
thetwo-bodyone),thisis,probably,amanifestationof
self-consistency of the Fock decomposition embedded
in the Nakanishi integral representation. In the field-
theoreticalframework(inwhichonlytheBSamplitude
Fig. 12 The EM form factor for the four calculations given can be defined via the Heisenberg field operators [1]),
in Figs. 10 and 11 with g shown in Fig. 9. Solid (upper)
E the number of particles is not conserved and the exis-
curve:resultsobtainedviaLFwavefunctionbyEq.(32),for
tence of one Fock component requires the existence of
N = 5, N = 3 and via Minkowski BS amplitude, by Eq.
x z
(30),forN =4, N =2,whichareindistinguishablewithin other ones. The set of them, corresponding to differ-
x z
thicknessoftheline.Dotted (middle) curve:resultsobtained entnumbersofparticles,ensuresalsothecorrecttrans-
via LF wave function by Eq. (32) for N = 4, N = 2.
x z formation properties of the full state vector |p(cid:105) since
Dashed (bottom) curve: results obtained via Minkowski BS
theFockcomponentsaretransformedbydynamicalLF
amplitude,byEq.(30),for N =5, N =3.
x z
boosts in each other.
The fact that the difference between form factors
lutions for g shown in Fig. 6 is larger than in Fig. 10, determined by the two-body LF wave function and the
E
though it is still not so significant. form factor including higher components is small was
foundalsoinWick-Cutkoskymodel[24].Thecontribu-
All four versions from Figs. 10 and 11 are shown
tionofmany-bodycomponentswithn≥3reaches36%
forcomparisontogetherinFig.12.Onecandistinguish
inthefullnormalizationintegralF(0)onlyforthecou-
only three curves of the four since two of then coin-
plingconstantsohugethatthetotalmassofthebound
cide with each other. We compare form factors calcu-
system tends to zero.
lated via LF wave function and Minkowski BS ampli-
A final remark: the stability of the form factors ob-
tude using g . The form factor calculated via LF wave
E
tained from g , though quite unexpected in view of
function by Eq. (32), for N = 5, N = 3 and cal- E
x z
the sizeable difference of the Nakanishi weight func-
culated via Minkowski BS amplitude, by Eq. (30), for
tions(seee.g.Fig.8),isinfactappropriate.Aswehave
N = 4, N = 2 are shown by the solid line (upper
x z
discussed in the analytical example, the reason for the
curve). They are indistinguishable within thickness of
instabilityofthesolutionofthelinearsystemisthecon-
lines.
tributionfromtheverysmalleigenvalues.Thereforethe
It is worth to make the two following remarks. (i)
differencebetweentheg ’scomesfromthecorrespond-
ThoughtheformfactorsFLF(Q2)andFBSM(Q2)turned E
ingeigenvectorswhichinducetheobservedoscillations.
outtobeveryclosetoeachother(asitisseenfromthe
However, when using the Minkowski BS amplitude to
presentcalculationsandfrom[23]),theyshouldnotco-
compute the form factor, the contributions from the
incideexactly.TheirdeviationseeninFig.12iscaused
eigenvectors with small eigenvalues are damped in the
not only by numerical uncertainties, but also by phys-
same way they were enhanced in the inversion. It is
ical reasons. This deviation cannot be completely re-
very likely that this result follows from the fact that
movedbymoreprecisecalculations.Frompointofview
the spectra of the Nakanishi kernels in Minkowski and
of the Fock decomposition of the state vector, the form
Euclidean spaces are the same.
factor FLF(Q2) is determined only by the two-body
contribution in the state vector, whereas FBSM(Q2),
determinedbythetwo-bodyBSamplitude,includesim- 9 Discussion and conclusion
plicitlytheeffectofhigherLFFockstates.(ii)Thecon-
tribution of the higher Fock components can be found Wehavedemonstratedbyexplicitcalculations,thatthe
knowing only the two-body component. Indeed, invert- Nakanishi representations of the Bethe-Salpeter ampli-
ing Eq. (6), we extract g from the two-body LF tude in Euclidean space (Φ ) and the light-front wave
LF E