# New form of kernel in equation for Nakanishi function

V.A. Karmanov<EMAIL_ADDRESS>Lebedev Physical Institute, Leninsky
Prospekt 53, 119991 Moscow, Russia

###### Abstract

The Bethe-Salpeter amplitude $\Phi(k,p)$ is expressed, by means of the
Nakanishi integral representation, via a smooth function $g(\gamma,z)$. This
function satisfies a canonical equation $g=Ng$. However, calculations of the
kernel $N$ in this equation, presented previously, were restricted to one-
boson exchange and, depending on method, dealt with complex multivalued
functions. Although these difficulties are surmountable, but in practice, they
complicate finding the unambiguous result.

In the present work, an unambiguous expression for the kernel $N$ in terms of
real functions is derived. For the one-boson scalar exchange, the explicit
formula for $N$ is found. With this equation and kernel, the binding energies,
calculated previously, are reproduced. Their finding, as well as calculation
of the Bethe-Salpeter amplitude in the Minkowski space, become not more
difficult than in the Euclidean one. The method can be generalized to any
kernel given by irreducible Feynman graph. This generalization is illustrated
by example of the cross-ladder kernel.

###### pacs:

03.65.Pm, 03.65.Ge, 11.10.St

##  I Introduction

One of the efficient methods of solving the Bethe-Salpeter (BS) equation [1]

$\displaystyle\Phi(k,p)$ $\displaystyle=$
$\displaystyle\frac{i^{2}}{\left[(\frac{p}{2}+k)^{2}-m^{2}+i\epsilon\right]\left[(\frac{p}{2}-k)^{2}-m^{2}+i\epsilon\right]}$
(1) $\displaystyle\times$
$\displaystyle\int\frac{d^{4}k^{\prime}}{(2\pi)^{4}}iK(k,k^{\prime},p)\Phi(k^{\prime},p),$

directly in Minkowsky space, is based on using the Nakanishi representation
[2] for the BS amplitude:

$\displaystyle\Phi(k,p)$ $\displaystyle=$
$\displaystyle-i\int_{-1}^{1}dz^{\prime}\int_{0}^{\infty}d\gamma^{\prime}$
$\displaystyle\times$
$\displaystyle\frac{g(\gamma^{\prime},z^{\prime})}{\left[\gamma^{\prime}+m^{2}-\frac{1}{4}M^{2}-k^{2}-p\cdot
k\;z^{\prime}-i\epsilon\right]^{3}}.$

(here $m$ is the constituent mass, $M$ is the bound state mass) and finding
the function $g(\gamma,z)$ from corresponding equation. The latter equation is
derived from the BS one after substituting (I) into (1). An advantage of this
method is in the fact that the function $g(\gamma,z)$ satisfies an equation
which is relatively easy solvable and its solution is smooth. Having found it
numerically, one can restore by Eq. (I) the BS amplitude in Minkowski space.
Then, after substituting $\Phi(k,p)$, e.g., in the expression for the
electromagnetic form factor one can integrate analytically over the relative
momentum $k$ and express form factor in terms of the solution $g(\gamma,z)$
[3, 4]. In this way, the Minkowski BS amplitude $\Phi$ appears as an auxiliary
quantity which in the final result is replaced by $g$.

For the massless ladder kernel (Wick-Cutkosky model [5, 6]) the
$\gamma$-dependence of the function $g(\gamma,z)$ is reduced to the delta-
function $\delta(\gamma)$ and to finite sum of its derivatives so the integral
(I) turns into sum of the one-dimensional ones with the functions $g_{n}(z)$,
satisfying a system of equations. In non-relativistic limit, the Coulombien
spectrum and wave functions are reproduced.

For massive ladder exchange kernel:

$K(k,k^{\prime})=-\frac{16\pi
m^{2}\alpha}{(k-k^{\prime})^{2}-\mu^{2}+i\epsilon}$ (3)

($\alpha$ is the coupling constant which appears in the corresponding Yukawa
potential $V(r)=-\frac{\alpha}{r}e^{-\mu r}$) this method was proposed and an
equation for $g(\gamma,z)$ was firstly obtained in [7]. Subsequent researches
[8, 9, 10, 11], including the present one, were aimed to deriving, step by
step, most simple forms of this equation and its kernel, convenient for
practical use, further simplifying the numerical solutions.

In Ref. [8], by substituting $\Phi(k,p)$ in the form (I) into Eq. (1) and
making the light-front projection, the following equation for the function
$g(\gamma,z)$ was derived:

$\displaystyle\int_{0}^{\infty}\frac{g(\gamma^{\prime},z)d\gamma^{\prime}}{\Bigl{[}\gamma^{\prime}+\gamma+z^{2}m^{2}+(1-z^{2})\kappa^{2}\Bigr{]}^{2}}=$
$\displaystyle\int_{0}^{\infty}d\gamma^{\prime}\int_{-1}^{1}dz^{\prime}\;V(\gamma,z,\gamma^{\prime},z^{\prime})g(\gamma^{\prime},z^{\prime}),$
(4)

where $\kappa^{2}=m^{2}-\frac{1}{4}M^{2}$. The kernel
$V(\gamma,z,\gamma^{\prime},z^{\prime})$ was expressed via the kernel
$K(k,k^{\prime})$, eq. (3), of the BS equation (1) in [8].

The equation in the form (I), although solvable numerically, was not yet
convenient enough, since it contained the integrals in both parts. This
resulted in instability of the numerical procedure which though was overcome
in Ref. [8] but required extra methods and studies.

Further efforts in this field were then directed to derivation of equation in
the ”normal” form:

$g(\gamma,z)=\int_{0}^{\infty}d\gamma^{\prime}\int_{-1}^{1}dz^{\prime}\;N(\gamma,z,\gamma^{\prime},z^{\prime})g(\gamma^{\prime},z^{\prime}),$
(5)

which does not contain the integral in l.h.-side. It was firstly derived in
[10], using uniqueness of the Nakanishi representation. This derivation was
real ”matter of art”, however restricted to the ladder kernel only. The
binding energies found in [10] by numerical solution of Eq. (5) were in very
good agreement with the solutions of Eq. (I) found in [8] and with the
Euclidean space solutions. Besides, the stability of results was much better.

Then in Ref. [11] it was noticed that the integral in l.h.-side of Eq. (I) had
a form of the Stieltjes transform which could be inverted analytically.
Inverting it, an equation, again in the form (5), was found. The operator,
inverse to the operator in the l.h.-side of Eq. (I), applied to the r.h.-side
of (I), creates the kernel $N$ of the equation (5). $N$ is expressed through
$V$ as follows [11]:

$\displaystyle
N(\gamma,z,\gamma^{\prime},z^{\prime})=\frac{\gamma}{2\pi}\int_{-\pi+\epsilon}^{\pi-\epsilon}\;d\phi\;e^{i\phi}$
$\displaystyle\phantom{\frac{\gamma}{2\pi}\int_{-\pi}^{\pi}}\times
V\Bigl{(}\gamma
e^{i\phi}-z^{2}m^{2}-(1-z^{2})\kappa^{2},z,\gamma^{\prime},z^{\prime}\Bigr{)}$
(6)

with $\epsilon\to 0$. The equations in the form (5), found in [10] and [11] by
different methods, were equivalent since their kernels coincided with each
other. However, these kernels were obtained in rather different forms, so it
was uneasy to compare them analytically. Their equivalence was demonstrated in
[11] numerically.

For the first glance, the way from the kernel $K$ in the BS equation (1) to
the kernel $N$ in the equation (5) was paved. However, as it was noticed and
demonstrated in examples already in [11], the direct calculation of the kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$ by Eq. (I) was, to some extent,
ambiguous. This problem arised since the kernel
$V(\gamma,z,\gamma^{\prime},z^{\prime})$ was expressed via the functions sqrt
and $\log$, see Eqs. (9) and (10) below, which are multivalued in the complex
plane (where we go after the substitution $\gamma\to\gamma
e^{i\phi}-z^{2}m^{2}-(1-z^{2})\kappa^{2}$). Though this ambiguity can be
eliminated, this delicate problem requires some extra efforts and very
attentive and careful definitions of branches of the multivalued complex
functions. This complicates the direct practical applications of the approach
[11] to solving the BS equation via finding the function $g(\gamma,z)$ from
Eq. (5).

The aim of the present work is to represent the kernel $N$ in Eq. (5) in
another (third) form which does not contain this ambiguity. We emphasize that
we are only talking about a new, more convenient _form_ of one and the same
kernel. The kernel $N$ in new form, derived in the present article, is
identical to ones found in Refs. [10, 11]. However, its form is completely
unambiguous, restricted to the real functions only, does not require delicate
choosing branches of multivalued functions and therefore it makes direct
numerical solving Eq. (5) not more complicated than solving the Euclidean BS
equation. Though the kernel $N$, derived below, might look a little bit
lengthy and cumbersome, it is given by the explicit formulas, does not contain
any integrations (for one-boson exchange) and therefore it is easy,
unambiguously and rapidly computated.

The advantages of this new representation of the kernel $N$ in the equation
(5) make solving the BS equation a simple and routine work.

Further content of the present paper is the following. In Sec. II we derive
this new form of the kernel $N$. Sec. III summarizes the results of Sec. II in
the form of recipes ready to use. In Sec. IV we take the limit of zero
exchange mass $\mu\to 0$ and show analytically that in this limit the equation
(5) with the kernel found in Secs. II and III turns into the Wick-Cutkosky
equation and kernel [5, 6]. This serves as another test of our kernel and
shows that the equation (5) is a direct generalization of the Wick-Cutkosky
one for massive exchange. In Sec. V the equation (5) defined in the interval
$-1\leq z\leq 1$ is reduced to the half interval $0\leq z\leq 1$ and the
corresponding kernel is derived. This allows to increase the precision of
numerical calculations. In Sec. VI we explain, how to derive the kernel in the
canonical equation (5) from the Feynman cross-ladder one. The method can be
applied to any kernel $K$ in the BS equation (1). Sec. VII contains the
concluding remarks.

##  II Calculating the ladder kernel $N$ in the equation (5)

The relation between the original kernel $K$, Eq. (3), appearing in the BS
equation (1), and the kernel $V$ in Eqs. (I), (I) was derived in Ref. [8]. In
the case of the ladder kernel (3), the kernel $V$ takes the form:

$V(\gamma,z,\gamma^{\prime},z^{\prime})=\left\\{\begin{array}[]{ll}W(\gamma,z,\gamma^{\prime},z^{\prime}),&\mbox{if
$-1\leq z^{\prime}\leq z\leq 1$}\\\
W(\gamma,-z,\gamma^{\prime},-z^{\prime}),&\mbox{if $-1\leq z\leq
z^{\prime}\leq 1$}\end{array}\right.$ (7)

where:

$W(\gamma,z,\gamma^{\prime},z^{\prime})=\frac{\alpha
m^{2}(1-z)^{2}}{2\pi[\gamma+z^{2}m^{2}+(1-z^{2})\kappa^{2}]}\int_{0}^{1}dv\left(\frac{v}{D}\right)^{2},$
(8) $\displaystyle D$ $\displaystyle=$ $\displaystyle
v(1-v)(1-z^{\prime})\gamma+v(1-z)\gamma^{\prime}$ $\displaystyle+$
$\displaystyle
v(1-z)(1-z^{\prime})\Bigl{[}1+z(1-v)+vz^{\prime}\Bigr{]}\kappa^{2}$
$\displaystyle+$ $\displaystyle
v\Bigl{[}(1-v)(1-z^{\prime})z^{2}+v{z^{\prime}}^{2}(1-z)\Bigr{]}m^{2}$
$\displaystyle+$ $\displaystyle(1-v)(1-z)\mu^{2}.$

###  II.1 Integrating over $v,\phi$

The integral (8) over $v$ was calculated analytically [8]. It reads:

$W(\gamma,z,\gamma^{\prime},z^{\prime})=\frac{\alpha
m^{2}}{2\pi}\frac{(1-z)^{2}}{[\gamma+z^{2}m^{2}+(1-z^{2})\kappa^{2}]}\frac{1}{b_{2}^{2}(b_{+}-b_{-})^{3}}\left[\frac{(b_{+}-b_{-})(2b_{+}b_{-}-b_{+}-b_{-})}{(1-b_{+})(1-b_{-})}\right.+\left.2b_{+}b_{-}\log\frac{b_{+}(1-b_{-})}{b_{-}(1-b_{+})}\right],$
(9)

where

$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle(1-z)\mu^{2},$
$\displaystyle b_{\pm}$ $\displaystyle=$
$\displaystyle-\frac{1}{2b_{2}}\;\left(b_{1}\pm\sqrt{b_{1}^{2}-4b_{0}b_{2}}\right),$
(10) $\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle\gamma+\gamma^{\prime}-(1-z)\mu^{2}-\gamma^{\prime}z-\gamma
z^{\prime}$ $\displaystyle+$
$\displaystyle(1-z^{\prime})\left[z^{2}m^{2}+(1-z^{2})\kappa^{2}\right],$
$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle-\gamma(1-z^{\prime})-(z-z^{\prime})[(1-z)(1-z^{\prime})\kappa^{2}$
$\displaystyle\phantom{z-\gamma(1)-(z-z^{\prime})}+(z+z^{\prime}-zz^{\prime})m^{2}].$

The kernel $N(\gamma,z,\gamma^{\prime},z^{\prime})$ is now determined by the
integral over $\phi$, Eq. (I).

As mentioned, Eqs. (9) and (10) and, hence, the kernel
$V(\gamma,z,\gamma^{\prime},z^{\prime})$ contains $\log$ and sqrt – the
multivalued functions in the complex plane. This requires, after substituting
$\gamma\to\gamma e^{i\phi}-z^{2}m^{2}-(1-z^{2})\kappa^{2}$, very careful
definition of their branches. This is the reason of difficulty and limitation
in practical use of this method. In the next section we will avoid this
difficulty.

###  II.2 Integrating over $\phi,v$

The method of calculation of the kernel $N$ used in the present work differs
from one used in [11] only by the order of integrations: first over $\phi$,
then over $v$. This minimal, for the first glance, difference turns out to be
crucial. The integrand in Eq. (8), before integrating over $v$, does not
contain any multi-valued functions. Therefore the $\phi$-integration is
straightforward and unambiguous. It results in real functions. The subsequent
$v$-integration is analytical. This calculation of the double integral (I),
(8) will be carried out below.

We take the kernel $V(\gamma,z,\gamma^{\prime},z^{\prime})$ in the form Eqs.
(7), (8), do not integrate over $v$ in (8), but make the substitution
$\gamma\to\gamma e^{i\phi}-z^{2}m^{2}-(1-z^{2})\kappa^{2}$ in the argument
$\gamma$. In this way, the kernel (I) obtains the form:

$N(\gamma,z,\gamma^{\prime},z^{\prime})=\int_{0}^{1}n(\gamma,z,\gamma^{\prime},z^{\prime};v)\,dv,$
(11)

where

$n(\gamma,z,\gamma^{\prime},z^{\prime};v)=c(v)\int_{-\pi+\epsilon}^{\pi-\epsilon}d\phi\frac{1}{(a(v)\exp(i\phi)+b(v))^{2}}$
(12)

and

$\displaystyle a(v)$ $\displaystyle=$ $\displaystyle\gamma
v(1-v)(1-z^{\prime}),$ $\displaystyle b(v)$ $\displaystyle=$
$\displaystyle(1-z)\Bigl{[}(1-v)\mu^{2}$ $\displaystyle+$ $\displaystyle
v\Bigl{(}\gamma^{\prime}+v({z^{\prime}}^{2}m^{2}+(1-{z^{\prime}}^{2})\kappa^{2})\Bigr{)}\Bigr{]},$
$\displaystyle c(v)$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{2}}\alpha
m^{2}v^{2}(1-z)^{2}.$ (13)

Note that $a(v)$, $b(v)$ and $c(v)$ are positive. Besides the argument $v$,
$a(v)$ depends also on $\gamma,z^{\prime}$, $b(v)$ depends on
$z,\gamma^{\prime},z^{\prime}$ and $c(v)$ depends on $z$. This creates
dependence of $N(\gamma,z,\gamma^{\prime},z^{\prime})$ on all four variables
$\gamma,z,\gamma^{\prime},z^{\prime}$. We omit these extra arguments in
$a(v),b(v),c(v)$ for shortness. For some values of parameters and variables it
may happen that $a(v)=b(v)$. Then the denominator in (12) crosses zero when
$\phi\to\pm\pi$ ($\epsilon\to 0$) and the integrand of
$n(\gamma,z,\gamma^{\prime},z^{\prime};v)$ vs. $\phi$ becomes singular. It
remains to be singular after $\phi$-integration. We will show that the
singularity, which survives after the $\phi$-integration, is represented by
the delta-function $\delta[b(v)-a(v)]$ and after integration over $v$ in (11)
it gives a finite, non-singular contribution to
$N(\gamma,z,\gamma^{\prime},z^{\prime})$. However, due to its origin, we will
call it ”singular part contribution”. Another finite contribution results from
the non-singular part of $n(\gamma,z,\gamma^{\prime},z^{\prime};v)$ vs. $v$.
So, we get two contributions, $N_{reg}$ and $N_{sing}$, which we will call
non-singular (or regular) and singular part contributions correspondingly. The
kernel $N(\gamma,z,\gamma^{\prime},z^{\prime})$ is the sum of these two terms:

$N(\gamma,z,\gamma^{\prime},z^{\prime})=N_{reg}+N_{sing},$ (14)

where

$\displaystyle N_{reg}$ $\displaystyle=$
$\displaystyle\int_{v_{1}}^{v_{2}}n_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v)\,dv,$
(15) $\displaystyle N_{sing}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}n_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)\,dv.$
(16)

The functions $n_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v)$ and
$n_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)$ will be found below and are
given by eqs. (18) and (33) correspondingly. The arguments of the functions
$N_{reg}$ and $N_{sing}$ are omitted here. We precise and restore them below,
calculating these functions. Note that, depending on the argument values, the
contribution of $N_{sing}$ may be zero. Or, on the contrary, $N_{reg}$ and
$N_{sing}$ may enter in $N(\gamma,z,\gamma^{\prime},z^{\prime})$ twice, for
different sets of arguments (see Sec. III). In the integral (15) for $N_{reg}$
the integration interval $0\leq v\leq 1$ was replaced by the interval $0\leq
v_{1}\leq v\leq v_{2}\leq 1$ since, as it will be shown below,
$n_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v)$ contains the theta-function
which can make the integration interval more narrow than $0\leq v\leq 1$ and
reduce the integration limits (though, not always).

We start with calculation of the non-singular part $N_{reg}$. Calculating it,
we will also determine the integration limits $v_{1,2}$.

Non-singular part contribution

To calculate the integral (12) for $a(v)\neq b(v)$, we can put in (12)
$\epsilon=0$. We introduce the complex variable $y=\exp(i\phi)$. Then this
integral obtains the form:

$\displaystyle n_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v)$
$\displaystyle=$
$\displaystyle\frac{1}{i}\int_{\mathcal{C}}dy\frac{c(v)}{y[a(v)y+b(v)]^{2}},$
(17)

where integration is carried out over the unit closed circle ${\mathcal{C}}$
around origin in the complex plane $y$. The integrand vs. $y$ has two poles:

$y=0\quad\mbox{and}\quad y=-\frac{b(v)}{a(v)}$

with the corresponding residues:

$Res(y=0)=-\frac{ic(v)}{b^{2}(v)}\quad\mbox{and}\quad
Res\left(y=-\frac{b(v)}{a(v)}\right)=\frac{ic(v)}{b^{2}(v)}$

If both poles are within the unit circle ${\mathcal{C}}$, the residues cancel
each other and the integral is zero. The result is not zero if the pole
$y=-\frac{b(v)}{a(v)}$ is outside the unite circle ${\cal C}$. It happens when
$-\frac{b(v)}{a(v)}<-1\to b(v)>a(v)$. Therefore, the result is:

$\displaystyle n_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v)$
$\displaystyle=$ $\displaystyle\left.2\pi iRes(y=0)\right|_{b(v)>a(v)}$ (18)
$\displaystyle=$ $\displaystyle\displaystyle{\frac{2\pi
c(v)}{b^{2}(v)}}\theta[b(v)-a(v)].$

Its contribution to the kernel $N(\gamma,z,\gamma^{\prime},z^{\prime})$ is
given by the integral (15) over $v$. This integral can be calculated
analytically, in terms of the primitives, since $b(v)$ vs. $v$ is a quadratic
polynomial.

Some tedious analysis is related to the theta-function in (18). Its argument
$b(v)-a(v)$ is also a quadratic function of $v$:

$b(v)-a(v)=A_{0}v^{2}+B_{0}v+C_{0}=A_{0}(v-v_{-})(v-v_{+}),$ (19)

where

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\gamma(1-z^{\prime})+(1-z)[m^{2}{z^{\prime}}^{2}+\kappa^{2}(1-{z^{\prime}}^{2})]>0,\;\;$
(20) $\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle(\gamma^{\prime}-\mu^{2})(1-z)-\gamma(1-z^{\prime}),\;$ (21)
$\displaystyle C_{0}$ $\displaystyle=$ $\displaystyle\mu^{2}(1-z)>0$ (22)

and $v_{\mp}$ are the roots of the quadratic equation $b(v)-a(v)=0$:

$v_{\mp}=\frac{1}{2A_{0}}(-B_{0}\mp\sqrt{D_{0}}),$ (23)

where

$D_{0}=B_{0}^{2}-4A_{0}C_{0}.$ (24)

If $D_{0}<0$, the roots are complex and the polynomial (19) vs. $v$ does not
change the sign. Since $A_{0}>0$, it is always positive. In this case, the
theta-function in (18) does not give any constraint and the integration for
$N_{reg}$ in (15) is carried out in the interval $0\leq v\leq 1$, that is
$v_{1}=0,v_{2}=1$.

If $D_{0}>0$, the equation $b(v)-a(v)=0$ vs. $v$ has two real roots $v_{\mp}$.
These roots, depending on values of the variables
$\gamma,z,\gamma^{\prime},z^{\prime}$, can be inside the interval $0\leq v\leq
1$, one inside, other outside, etc. When $v$ crosses a root, the argument
$b(v)-a(v)$ of the theta-function changes the sign and the theta-function
changes its value from 0 to 1 or back. Since the coefficient $A_{0}$ at
$v^{2}$ in Eq. (19), is positive, the value of polynomial between the roots
$v_{\mp}$ is negative. If the interval $v_{-}\leq v\leq v_{+}$ is inside of
$0\leq v\leq 1$ or partially overlaps with it, their overlapping part (where
$\theta[b(v)-a(v)]=0$) is excluded from integration. In this way, the initial
integration interval $0\leq v\leq 1$ is reduced by the theta-function to one
or two smaller intervals.

Note that according to Eq. (23), if $D_{0}>0$ and since the product
$A_{0}C_{0}$ is positive, then
$\sqrt{D_{0}}=\sqrt{B_{0}^{2}-4A_{0}C_{0}}<|B_{0}|$ and therefore the roots
$v_{\mp}$ cannot have opposite signs: either both roots are negative (if
$B_{0}>0$), or both are positive (if $B_{0}<0$). Therefore we will consider
the following cases only:

  1. 1.

$v_{-}<\;v_{+}<0$. Since the interval between the roots $v_{-}<v<v_{+}$ (where
$b(v)-a(v)<0\to\theta[b(v)-a(v)]=0$) does not overlap with $0\leq v\leq 1$,
whereas for $v>v_{+}$ (and certainly, for $v>0$),
$b(v)-a(v)>0\to\theta[b(v)-a(v)]=1$, in this case, the theta-function in (18)
does not give any constraint and the integration limits in (15) are $v_{1}=0$,
$v_{2}=1$.

  2. 2.

$0<v_{-}<v_{+}<1$. The theta-function is zero if $v_{-}<v<v_{+}$. The
integration limits are $v_{1}=0$, $v_{2}=v_{-}$ and $v_{1}=v_{+}$, $v_{2}=1$.
One should take sum of the integrals over these two intervals.

  3. 3.

$0<v_{-}<1<v_{+}$. The integration limits are $v_{1}=0$, $v_{2}=v_{-}$.

  4. 4.

$1<v_{-}<v_{+}$. The integration limits are $v_{1}=0$, $v_{2}=1$, like in the
case 1.

The integral (15) with $n_{reg}$ defined in (18) is represented as:

$N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1},v_{2})=F(v_{2})-F(v_{1}),$
(25)

where $F(v)$ is the primitive:

$F(v)=\int\frac{2\pi c(v)dv}{b^{2}(v)}=\frac{\alpha
m^{2}}{2\pi}\int\frac{v^{2}dv}{(A_{1}v^{2}+B_{1}v+C_{1})^{2}}$ (26)

and $v_{1,2}$ are the integration limits indicated above for the cases 1-4.
The denominator in r.h.-side of (26) contains

$\displaystyle\frac{b(v)}{1-z}$ $\displaystyle=$
$\displaystyle(1-v)\mu^{2}+v\Bigl{[}\gamma^{\prime}+v\Bigl{(}{z^{\prime}}^{2}m^{2}+(1-{z^{\prime}}^{2})\kappa^{2}\Bigr{)}\Bigr{]}$
(27) $\displaystyle=$ $\displaystyle A_{1}v^{2}+B_{1}v+C_{1},$

where

$\displaystyle A_{1}=m^{2}{z^{\prime}}^{2}+\kappa^{2}(1-{z^{\prime}}^{2})>0,$
$\displaystyle B_{1}=\gamma^{\prime}-\mu^{2},$ $\displaystyle
C_{1}=\mu^{2}>0.$

From the expression (27) we see that since $\gamma^{\prime}>0$, in the
interval $0\leq v\leq 1$ (which is larger than the integration interval
$v_{1}\leq v\leq v_{2}$ in (15)), the value $A_{1}v^{2}+B_{1}v+C_{1}$ is
always positive. This means that if the denominator in (26) has zeros, these
zeros are outside the interval $0\leq v\leq 1$. Therefore, the integrand in
(26) is not singular. However, the form of $F(v)$ depends on the sign of

$D_{1}=B_{1}^{2}-4A_{1}C_{1}.$ (28)

Namely, calculating the integral (26), we find:

$F(v)=\left\\{\begin{array}[]{ll}F^{-}(v),&\mbox{if $D_{1}<0$}\\\
F^{+}(v),&\mbox{if $D_{1}>0$}\end{array}\right.,$ (29)

where

$\displaystyle F^{-}(v)$ $\displaystyle=$ $\displaystyle\frac{\alpha
m^{2}}{2\pi}\left[\frac{B_{1}C_{1}+(B_{1}^{2}-2A_{1}C_{1})v}{A_{1}|D_{1}|[C_{1}+v(B_{1}+vA_{1})]}\right.$
(30) $\displaystyle+$
$\displaystyle\left.\frac{4C_{1}}{|D_{1}|^{3/2}}\arctan\frac{B_{1}+2vA_{1}}{\sqrt{|D_{1}|}}\right]$

and

$\displaystyle F^{+}(v)$ $\displaystyle=$ $\displaystyle-\frac{\alpha
m^{2}}{2\pi}\left[\frac{B_{1}C_{1}+(B1^{2}-2A_{1}C_{1})v}{A_{1}D_{1}[C_{1}+v(B_{1}+vA_{1})]}\right.$
(31) $\displaystyle+$
$\displaystyle\left.\frac{2C_{1}}{D_{1}^{3/2}}\log\left(1-\frac{2\sqrt{D_{1}}}{B_{1}+\sqrt{D_{1}}+2A_{1}v}\right)\right].$

The equation (25) together with Eqs. (29-31) determine the regular part
contribution $N_{reg}$.

Singular part contribution

The singular part contribution results from the integral (12) where
$b(v)=a(v)$ for some value of $v$. To extract it from (12), we calculate this
integral analytically and then take the limit $b(v)\to a(v)$. Calculating the
integral via primitive, we find:

$\displaystyle n(\gamma,z,\gamma^{\prime},z^{\prime};v)$ $\displaystyle=$
$\displaystyle\frac{2c(v)(\pi-\epsilon)}{b^{2}(v)}+\frac{c(v)Arg\Bigl{[}b+a(v)\exp[i(\epsilon-\pi)]\Bigr{]}}{b^{2}(v)}-\frac{c(v)Arg\Big{[}(b(v)+a(v)\exp[i(-\epsilon+\pi)]\Bigr{]}}{b^{2}(v)}$
(32) $\displaystyle-$
$\displaystyle\frac{2a(v)c(v)\sin\epsilon}{b(v)[(b(v)-a(v)\cos\epsilon)^{2}+a^{2}(v)\sin^{2}\epsilon]}.$

At $\epsilon\to 0$ and $b(v)\to a(v)$, the last term in (32) only is singular
and dominates. We keep it, take the limit $\epsilon\to 0$ and make the
replacement $a(v)\epsilon=\epsilon^{\prime}$. In this way we find:

$n_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)=-\frac{2c(v)\epsilon^{\prime}}{b(v)[(b(v)-a(v))^{2}+{\epsilon^{\prime}}^{2}]}.$

Using the formula:

$\frac{1}{\pi}\frac{\epsilon}{(x^{2}+\epsilon^{2})}\stackrel{{\scriptstyle\epsilon\to
0}}{{=}}\delta(x),$

we obtain

$n_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)=-\frac{2\pi
c(v)}{a(v)}\delta[b(v)-a(v)].$ (33)

The difference $b(v)-a(v)$ (a second order polynomial in $v$) is represented
in the form of product, Eq. (19). If roots $v_{\mp}$ of the equation
$b(v)-a(v)=0$ vs. $v$ are real and both are in the interval $0<v<1$, then the
integral over $v$ (the second term in (15)) is reduced to the sum over roots:

$\displaystyle\int_{0}^{1}n_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)dv$
$\displaystyle=$ $\displaystyle
N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v_{-})$ (34) $\displaystyle+$
$\displaystyle N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v_{+})$

where we denoted

$N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)=-\frac{2\pi
c(v)}{A_{0}|v_{+}-v_{-}|\,a(v)},$ (35)

$a(v),\,c(v)$ are defined in (II.2), $A_{0}$ is defined in (20). If only one
root $v_{-}$ is in the interval $0<v<1$, then instead of the sum (34) we
should take the contribution of this root only:

$\int_{0}^{1}n_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)dv=N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v_{-}).$
(36)

As explained above, two real roots $v_{\mp}$ have the same sign. Therefore the
situation when one root $v_{+}$ is in the interval $0<v<1$, whereas the root
$v_{-}$ is outside (and, hence, $v_{-}<0$) is impossible.

If both roots are outside of the interval $0<v<1$, the singular part does not
contribute. This analysis is similar to one given in the points 1-4 above Eq.
(25).

In this way, the equations (34-36) determine the singular part contribution
$N_{sing}$.

##  III Summary

In this section, summarizing the above results, we give the recipe of
calculating the ladder massive exchange kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$ in the equation (5). It takes a few
following steps. Depending on the relation between $z,z^{\prime}$
($z<z^{\prime}$ or $z>z^{\prime}$), the kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$ is expressed, by the formula similar
to eq. (7), via the auxiliary kernel
$\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ which will be constructed in
this section. In its turn, the kernel
$\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ is expressed via
$N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1},v_{2})$, eq. (25), and via
$N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)$, eq. (35). The function
$N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1},v_{2})$ is constructed via
another function $F(v)$, eq. (29). The contributions $N_{reg}$ and $N_{sing}$
to $\tilde{N}$ depend on the sign of $D_{0}$, Eq. (24), determining the
existence of the real roots $v_{\mp}$ of the equation $b(v)-a(v)=0$. If
$D_{0}>0$ and the real roots exist, the result depends on their positions
relative to the interval $0\leq v\leq 1$. For all these cases the kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$ in eq. (5) is defined below. The
construction is a little bit lengthy, but it is coherent and absolutely
unambiguous.

We do not stop at the contruction of the kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$, but we reduce below the equation (5)
to the equation (52) defined in the half-interval $0\leq z\leq 1$. The kernel
$N_{half}(\gamma,z,\gamma^{\prime},z^{\prime})$ in this reduced equation will
be also expressed through $\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$.

We consider the following cases.

  1. 1.

$D_{0}<0$. According to eq. (23), the roots $v_{\mp}$ are complex. Therefore,
the singular contribution $N_{sing}$ is absent. Besides, $b(v)-a(v)>0$. Hence,
always $\theta[b(v)-a(v)]=1$. Therefore

$\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})=N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1}=0,v_{2}=1)$
(37)

with $N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1},v_{2})$ defined in
(25).

  2. 2.

$D_{0}>0$. The roots $v_{\mp}$ given by eq. (23) are real. The form of the
kernel $\tilde{N}$ is different in three following sub-cases, depending on the
positions of the roots.

(a) The roots are out of the integration domain $0\leq v\leq 1$ over $v$:
either $v_{-}<v_{+}<0$, or $1<v_{-}<v_{+}$. Then the kernel
$\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ is still defined by Eq. (37).

(b) Both roots are in the integration interval: $0<v_{-}<v_{+}<1$. Then the
kernel $\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ reads

$\displaystyle\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ $\displaystyle=$
$\displaystyle
N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1}=0,v_{2}=v_{-})+N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1}=v_{+},v_{2}=1)$
(38) $\displaystyle+$ $\displaystyle
N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v_{-})+N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v_{+}).$

with $N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};v_{1},v_{2})$ defined in
(25) and $N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v)$ defined in (35).

(c) One root is in the integration interval $0\leq v\leq 1$, whereas the
second one is at $v>1$. That is: $0<v_{-}<1<v_{+}$. Then the kernel
$\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ takes the form

$\displaystyle\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ $\displaystyle=$
$\displaystyle N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime};0,v_{-})$ (39)
$\displaystyle+$ $\displaystyle
N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime};v_{-}).$

We remind that the roots cannot have opposite signs (see remark below eq.
(24)), therefore the case $v_{-}<0<v_{+}$ is excluded.

The kernel $N(\gamma,z,\gamma^{\prime},z^{\prime})$ is constructed via
$\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ similarly to eq. (7):

$N(\gamma,z,\gamma^{\prime},z^{\prime})=\left\\{\begin{array}[]{ll}\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime}),&\mbox{if
$-1\leq z^{\prime}\leq z\leq 1$}\\\
\tilde{N}(\gamma,-z,\gamma^{\prime},-z^{\prime}),&\mbox{if $-1\leq z\leq
z^{\prime}\leq 1$}\end{array}\right.$ (40)

We emphasize that in the second case (if $-1\leq z\leq z^{\prime}\leq 1$) we
should make the replacement $z\to-z,\;z^{\prime}\to-z^{\prime}$ everywhere,
including Eqs. (20-22) determining the positions of the roots (23).

This completes the calculations. Just this kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$, with
$\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ defined above, enters in the
equation (5) for $g(\gamma,z)$.

Test 1. For test, we take the same parameters which were used in the test
carried out in [11], namely:

$\alpha=1,m=1,\mu=0.15,M=1.9.$

We also take the same values of variables:

$\gamma=0.1,\gamma^{\prime}=1,z=0.2,z^{\prime}=0.35.$

Since $z<z^{\prime}$, according to Eq. (40), the kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$ is determined by
$\tilde{N}(\gamma,-z,\gamma^{\prime},-z^{\prime})$. I.e. in further
calculations, we should change the signs of the variables $z,z^{\prime}$:
$z\to-z$, $z^{\prime}\to-z^{\prime}$. For these new values of variables, we
find $D_{0}=1.036>0$. By Eq. (23), we find the roots of the equation
$b(v)-a(v)=0$: $v_{-}=-2.672$, $v_{+}=-0.0263$. Hence, we deal with the case
2(a). The value of $\tilde{N}$ is determined through $N_{reg}$ by Eq. (37).
The contribution $N_{sing}$ of the singular part is absent. In its turn,
$N_{reg}$ is determined by Eqs. (25), (29) and, since $D_{1}=0.937>0$, finally
by $F_{+}(v)$, Eq. (31). In this way, we find the value of the kernel:
$N=0.1153980591510$, that coincides within all digits with the value found in
Eq. (31) of Ref. [11].

Test 2. For the same set of the parameters $\alpha,m,\mu,M$ we take the
following set of variables:

$\gamma=0.5,\gamma^{\prime}=0.1,z=0.2,z^{\prime}=0.35.$

Since $z<z^{\prime}$, according to Eq. (40), the kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$ is still determined by
$\tilde{N}(\gamma,-z,\gamma^{\prime},-z^{\prime})$. I.e. in further
calculations, we should take the negative values of $z,z^{\prime}$: $z\to-z$,
$z^{\prime}\to-z^{\prime}$. For these new values of variables, we find
$D_{0}=0.239>0$. By Eq. (23), we find the roots of the equation $b(v)-a(v)=0$:
$v_{-}=0.050$, $v_{+}=0.579$. Hence, we now deal with the case 2(b). The value
of $\tilde{N}$ is determined through $N_{reg}$ and $N_{sing}$ by Eq. (38). The
singular part now gives a non-zero contribution. In its turn, $N_{reg}$ is
determined by Eqs. (25), (29) and, since $D_{1}=-0.0127<0$, finally by
$F_{-}(v)$, Eq. (30). In this way, we find the value of the kernel:
$N=-0.0531890858160$, that coincides within all digits with the value found in
Eq. (33) of Ref. [11].

Note that these two tests cover all three functions used to calculate $N$: the
function $F^{-}(v)$, Eq. (30), the function $F^{+}(v)$, Eq. (31), and
$N_{sing}(\gamma.z,\gamma^{\prime},z^{\prime},v)$, Eq. (35).

##  IV Massless exchange

The case $\mu=0$ corresponds to the Wick-Cutkosky model [5, 6]. For
simplicity, we will consider the ground state only. In this model, the ground
state function $g(\gamma,z)$ turns into

$g(\gamma,z)=\delta(\gamma)g(z)$ (41)

and the equation for $g(z)$ was found in [5, 6]. Below we will show that the
equation (5) with the kernel found in the previous sections turns, in the
limit $\mu\to 0$, into the Wick-Cutkosky equation.

According to Sec. III, the formulas which should be used to calculate the
kernel $N$ in the equation (5) depend on the position of the roots, Eq. (23),
relative to the interval $0<v<1$. Decomposing $v_{\mp}$, Eq. (23), in series
of $\mu^{2}$ and denoting $B_{00}=B_{0}(\mu=0)$, we should consider two cases:
$B_{00}<0$ and $B_{00}>0$.

The case $B_{00}<0$
$\left(\gamma^{\prime}<\frac{1-z^{\prime}}{1-z}\gamma\right)$:

$v_{-}=-\frac{1}{B_{00}}(1-z)\mu^{2}>0,\;v_{+}=-\frac{B_{00}}{A_{0}}>0.$

From Eqs. (20), (21) it follows: $A_{0}>0,\;A_{0}+B_{00}>0.$ Therefore
$-\frac{B_{00}}{A_{0}}<1$, $0<v_{-}<v_{+}<1$. We should calculate the kernel
according to the case 2(b) from the Summary, Sec. III, i.e., by Eq. (38).

The case $B_{00}>0$
$\left(\gamma^{\prime}>\frac{1-z^{\prime}}{1-z}\gamma\right)$:

$v_{-}=-\frac{B_{00}}{A_{0}}<0,\;v_{+}=-\frac{1}{B_{00}}(1-z)\mu^{2}<0.$

In this case: $v_{-}<v_{+}<0$. We should calculate the kernel according to the
case 2(a) from Sec. III, i.e., by Eq. (37).

In this way, taking the limit $\mu\to 0$ in Eq. (37) and in the first line of
Eq. (38), we find the regular part contribution:

$N_{reg}(\gamma,z,\gamma^{\prime},z^{\prime})=\left\\{\begin{array}[]{ll}\frac{\alpha
m^{2}(1-z)}{2\pi\gamma(1-z^{\prime})}\frac{1}{\gamma^{\prime}+m^{2}{z^{\prime}}^{2}+\kappa^{2}(1-{z^{\prime}}^{2})},&\mbox{if
$\gamma^{\prime}<\frac{1-z^{\prime}}{1-z}\gamma$}\\\ \frac{\alpha
m^{2}}{2\pi\gamma^{\prime}}\frac{1}{\gamma^{\prime}+m^{2}{z^{\prime}}^{2}+\kappa^{2}(1-{z^{\prime}}^{2})},&\mbox{if
$\gamma^{\prime}>\frac{1-z^{\prime}}{1-z}\gamma$}\end{array}\right.$ (42)

The singular part contribution is calculated similarly. For $B_{0}<0$
$\left(\gamma^{\prime}<\frac{1-z^{\prime}}{1-z}\gamma\right)$ this
contribution is given by the second line of eq. (38). For $B_{0}>0$
$\left(\gamma^{\prime}>\frac{1-z^{\prime}}{1-z}\gamma\right)$ it is absent,
see eq. (37). That is:

$N_{sing}(\gamma,z,\gamma^{\prime},z^{\prime})=\left\\{\begin{array}[]{ll}-\frac{\alpha
m^{2}(1-z)}{2\pi\gamma(1-z^{\prime})}\frac{1}{\gamma^{\prime}+m^{2}{z^{\prime}}^{2}+\kappa^{2}(1-{z^{\prime}}^{2})},&\mbox{if
$\gamma^{\prime}<\frac{1-z^{\prime}}{1-z}\gamma$}\\\ 0&\mbox{if
$\gamma^{\prime}>\frac{1-z^{\prime}}{1-z}\gamma$}\end{array}\right.$ (43)

The full kernel $\tilde{N}$ is the sum of (42) and (43):

$\displaystyle\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}0,&\mbox{\small if
$\gamma^{\prime}<\frac{1-z^{\prime}}{1-z}\gamma$}\\\ \frac{\alpha
m^{2}}{2\pi\gamma^{\prime}}\frac{1}{\gamma^{\prime}+m^{2}{z^{\prime}}^{2}+\kappa^{2}(1-{z^{\prime}}^{2})},&\mbox{\small
if $\gamma^{\prime}>\frac{1-z^{\prime}}{1-z}\gamma$}\end{array}\right.$ (46)
$\displaystyle=$ $\displaystyle\frac{\alpha
m^{2}}{2\pi\gamma^{\prime}}\;\frac{\theta[\gamma^{\prime}(1-z)-\gamma(1-z^{\prime})]}{\gamma^{\prime}+m^{2}{z^{\prime}}^{2}+\kappa^{2}(1-{z^{\prime}}^{2})}$
(47)

The first lines in Eqs. (42) and (43) are the same, up to the opposite signs,
they cancel each other in Eq. (46).

We will show that the solution in the Wick-Cutkosky form (41) indeed satisfies
the equation (5) with the kernel (46). Since the support of the function
$g(\gamma,z)$ is $\gamma>0$, the delta-function $\delta(\gamma)$ is obtained
in the limit $\mu\to 0$ from a function having a peak at $\gamma>0$. To keep
this property, we replace $\delta(\gamma)$ by $\delta(\gamma-\epsilon)$ with
$\epsilon>0$ and at the end of calculation take the limit $\epsilon\to 0$.
With the kernel $\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})$, Eq. (46),
and with $g(\gamma,z)=\delta(\gamma-\epsilon)g(z)$ the equation (5), after
integration in r.h.-side over $\gamma^{\prime}$ by means of the delta-function
$\delta(\gamma^{\prime}-\epsilon)$, obtains the form:

$\delta(\gamma-\epsilon)g(z)=\int_{-1}^{1}dz^{\prime}\frac{\alpha
m^{2}}{2\pi\epsilon}\;\frac{\theta[\epsilon(1-z)-\gamma(1-z^{\prime})]}{\epsilon+m^{2}Q(z^{\prime})}g(z^{\prime}),$

where $Q(z^{\prime})=1-\eta^{2}(1-{z^{\prime}}^{2})$, $\eta=\frac{M}{2m}$.
Note that the function

$h(\gamma)=\frac{1}{\epsilon}\theta[\epsilon(1-z)-\gamma(1-z^{\prime})]$

vs. $\gamma$ differs from zero in the narrow interval
$0<\gamma<\epsilon\frac{1-z}{1-z^{\prime}}$. In this interval it is constant
(equal to $\frac{1}{\epsilon}$) and it is zero outside. The integral:

$\int_{-\infty}^{\infty}h(\gamma)d\gamma=\int_{0}^{\epsilon\frac{1-z}{1-z^{\prime}}}h(\gamma)d\gamma=\frac{1-z}{1-z^{\prime}}$

Hence, the integral over $\gamma$ from the function
$\frac{1-z^{\prime}}{1-z}h(\gamma)$ is 1, this function approximates the
delta-function $\delta(\gamma-\gamma_{0})$, where $\gamma_{0}$ is a value
within the interval $0<\gamma_{0}<\epsilon\frac{1-z}{1-z^{\prime}}$.
Therefore, replacing, in the limit $\epsilon\to 0$,
$h(\gamma)\to\frac{1-z}{1-z^{\prime}}\delta(\gamma)$ and omitting
$\delta(\gamma)$ in both parts of equation, we obtain the following equation
for $g(z)$:

$g(z)=\frac{\alpha}{2\pi}\int_{-1}^{1}dz^{\prime}\;\frac{1-z}{1-z^{\prime}}\;\frac{g(z^{\prime})}{Q(z^{\prime})}.$

This equation is valid when $z^{\prime}<z$. For $z^{\prime}>z$ one should make
the replacement $z\to-z$, $z^{\prime}\to-z^{\prime}$. Therefore we obtain:

$g(z)=\frac{\alpha}{2\pi}\int_{-1}^{1}dz^{\prime}\;R(z,z^{\prime})\frac{g(z^{\prime})}{Q(z^{\prime})},$
(48)

where

$R(z,z^{\prime})=\left\\{\begin{array}[]{ll}\frac{1-z}{1-z^{\prime}},&\mbox{if
$z^{\prime}<z$}\\\ \frac{1+z}{1+z^{\prime}},&\mbox{if
$z^{\prime}>z$}\end{array}\right.$

The equation (48) exactly coincides with Eq. (15) from [6], derived in the
case $\mu=0$, and with Eq. (12) from [4] (for the quantum number $n=1$). This
coincidence can be considered as another test of the kernel presented in Sec.
III.

##  V Equation on the half-interval ${\bf 0\leq z\leq 1}$

It is easy to show that the solution of Eq. (5) with the kernel constructed by
Eq. (40), even with arbitrary function $\tilde{N}$, is either symmetric, or
antisymmetric. For finding its solution numerically, it is useful, using this
symmetry, to reduce it, instead of the interval $-1\leq z\leq 1$, to the half-
interval $0\leq z\leq 1$.

Using the definition (40), we rewrite Eq. (5), in terms of the kernel
$\tilde{N}$ as follows:

$\displaystyle g(\gamma,z)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\gamma^{\prime}\int_{-1}^{0}dz^{\prime}\;\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})g(\gamma^{\prime},z^{\prime})$
$\displaystyle+$
$\displaystyle\int_{0}^{\infty}d\gamma^{\prime}\int_{0}^{z}dz^{\prime}\;\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})g(\gamma^{\prime},z^{\prime})+\int_{0}^{\infty}d\gamma^{\prime}\int_{z}^{1}dz^{\prime}\;\tilde{N}(\gamma,-z,\gamma^{\prime},-z^{\prime})g(\gamma^{\prime},z^{\prime})$

For symmetric $g_{sym}(\gamma,z)=g_{sym}(\gamma,-z)$, making the replacement
of variable $z^{\prime}\to-z^{\prime}$ in the first line of r.h.-side of Eq.
(V), we get:

$\displaystyle g_{sym}(\gamma,z)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\gamma^{\prime}\int_{0}^{1}dz^{\prime}\;\tilde{N}(\gamma,z,\gamma^{\prime},-z^{\prime})g_{sym}(\gamma^{\prime},z^{\prime})$
$\displaystyle+$
$\displaystyle\int_{0}^{\infty}d\gamma^{\prime}\int_{0}^{z}dz^{\prime}\;\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime})g_{sym}(\gamma^{\prime},z^{\prime})+\int_{0}^{\infty}d\gamma^{\prime}\int_{z}^{1}dz^{\prime}\;\tilde{N}(\gamma,-z,\gamma^{\prime},-z^{\prime})g_{sym}(\gamma^{\prime},z^{\prime}).$

Introducing the kernel defined on the half-intervals $0\leq z\leq 1$, $0\leq
z^{\prime}\leq 1$

$N_{half}(\gamma,z,\gamma^{\prime},z^{\prime})=\left\\{\begin{array}[]{ll}\tilde{N}(\gamma,z,\gamma^{\prime},-z^{\prime})+\tilde{N}(\gamma,z,\gamma^{\prime},z^{\prime}),&\mbox{if
$z^{\prime}<z$}\\\
\tilde{N}(\gamma,z,\gamma^{\prime},-z^{\prime})+\tilde{N}(\gamma,-z,\gamma^{\prime},-z^{\prime}),&\mbox{if
$z^{\prime}>z$}\end{array}\right.,$ (51)

where $\tilde{N}$ is defined in Eqs. (37), (38) or (39), depending on the
positions of the roots $v_{\mp}$ (for given arguments $\gamma,\pm
z,\gamma^{\prime},\pm z^{\prime}$), we rewrite the equation (V) as

$g_{sym}(\gamma,z)=\int_{0}^{\infty}d\gamma^{\prime}\int_{0}^{1}dz^{\prime}\;N_{half}(\gamma,z,\gamma^{\prime},z^{\prime})g_{sym}(\gamma^{\prime},z^{\prime}).$
(52)

The antisymmetric solution $g_{asym}(\gamma,z)$ is determined by the equation
(52) with the kernel $N_{half}$ which differs from (51) by the opposite sign
at the first term in r.h.-side in (51), i.e., by the replacement
$\tilde{N}(\gamma,z,\gamma^{\prime},-z^{\prime})\to-\tilde{N}(\gamma,z,\gamma^{\prime},-z^{\prime})$.

The equation (52) is most convenient for the numerical solution, since its
interval in $z$ is twice more narrow than in (5). That allows to twice
increase the density of the discretization points and, in this way, to
increase the precision. Solution $g(\gamma,z)$ in full interval $-1\leq z\leq
1$ is trivially obtained from $g_{sym}(\gamma,z)$ (or from
$g_{asym}(\gamma,z)$) using the symmetry (antisymmetry) of this solution.

Solving Eq. (52) for $\mu=0.15$ and $\mu=0.5$ numerically, by spline
techniques, with number of intervals $N_{\gamma}=N_{z}=24$, we reproduced,
within three digits, the results shown in the Table I in Ref. [10]. Increase
of number of intervals improves the comparison.

##  VI Cross-ladder kernel

As mentioned, the above calculation of the kernel $N$ in the equation (5) was
based on the inversion of the kernel $L$ contained in the l.h.-side of the
equation (I) written symbolically in the form $Lg=Vg$. The recipe of this
inversion is universal for any kernel $V$ contained in the r.h.-side of Eq.
(I). The result of application of the inverse l.h.-side kernel $L$ to $V$:
$N=L^{-1}V$ is given by Eq. (I). Therefore, similar calculation of the kernel
$N$ can be carried out for any kernel $K$ in the BS equation (1), given by an
irreducible Feynman graph (and, correspondingly, for any kernel $V$ in
r.h.-side of Eq. (I)). Though, of course, for more complicated $K$ the
calculations become more cumbersome. We illustrate it (more schematically) for
the cross-ladder kernel.

Figure 1: Cross-ladder kernel

Feynman graph for the cross-ladder kernel is shown in Fig. 1. Corresponding
kernel $V^{(CL)}(\gamma,z,\gamma^{\prime},z^{\prime})$ in the non-canonical
equation (I) was calculated in [9]. We rewrite $V^{(CL)}$ as

$V^{(CL)}(\gamma,z,\gamma^{\prime},z^{\prime})=-\frac{1}{\pi^{2}}\alpha^{2}m^{4}(1-z^{2})^{3}\int_{0}^{1}y_{4}(1-y_{4})^{2}dy_{4}\int_{0}^{1}dy_{3}\int_{0}^{1-y_{3}}dy_{2}\int_{0}^{1-y_{2}-y_{3}}G\,\eta\,dy_{1},$
(53)

where $\eta=1-y_{4}[1-(1-y_{1}-y_{3})(y_{1}+y_{3})]$,

$G=\frac{1}{[\gamma+m^{2}z^{2}+\kappa^{2}(1-z^{2})]D^{3}},$ (54)

and $D$ reads:

$\displaystyle D$ $\displaystyle=$ $\displaystyle
c_{\gamma}\gamma+c_{\gamma^{\prime}}\gamma^{\prime}+c_{\kappa}\kappa^{2}+c_{m}m^{2}+c_{\mu}\mu^{2}$
(55) $\displaystyle-$ $\displaystyle
y_{4}|\tilde{c}|\Bigl{[}\gamma+m^{2}z^{2}+\kappa^{2}(1-z^{2})\Bigr{]}.$

The coefficients $c_{\cdots}$ determining $D$ are given in [9].

Making in (53) the substitution (I), we obtain for the kernel
$N^{CL}(\gamma,z,\gamma^{\prime},z^{\prime})$ the expression

$N^{(CL)}(\gamma,z,\gamma^{\prime},z^{\prime})=-\frac{1}{\pi^{2}}\alpha^{2}m^{4}(1-z^{2})^{3}\int_{0}^{1}y_{4}(1-y_{4})^{2}dy_{4}\int_{0}^{1}dy_{3}\int_{0}^{1-y_{3}}dy_{2}\int_{0}^{1-y_{2}-y_{3}}n\,\eta\,dy_{1},$
(56)

where

$n=\frac{1}{2\pi}\int_{-\pi+\epsilon}^{\pi-\epsilon}\frac{d\phi}{(a_{1}\exp{i\phi}+b_{1})^{3}},$
(57)

and

$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle(c_{\gamma}-y_{4}|\tilde{c}|)\gamma,$ $\displaystyle b_{1}$
$\displaystyle=$ $\displaystyle
c_{\gamma^{\prime}}\gamma^{\prime}-c_{\gamma}(\kappa^{2}(1-z^{2})+m^{2}z^{2})+c_{\kappa}\kappa^{2}$
$\displaystyle+$ $\displaystyle c_{m}m^{2}+c_{\mu}\mu^{2}.$

Like for the ladder kernel, Eq. (14), the result is again given by sum of two
integrals (now - 4D integrals over $y_{1},y_{2},y_{3},y_{4}$) from regular and
singular part contributions: $n=n_{reg}+n_{sing}$. They are calculated
similarly to the ladder case. However, in the ladder case, the values $a$ and
$b$ are positive. The condition that the pole $y=-\frac{b(v)}{a(v)}$ is
outside the unit cycle means $-\frac{b(v)}{a(v)}<-1$, that is $b(v)-a(v)>0$.
In the cross-ladder case, $a_{1}$, depending on the values of variable, can be
positive or negative, that now means: $\left|\frac{b_{1}}{a_{1}}\right|>1$,
that is $|b_{1}|-|a_{1}|>0$. Hence:

$\displaystyle n_{reg}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{C}}dy\frac{1}{y(a_{1}y+b_{1})^{3}}=\displaystyle{\frac{1}{b_{1}^{3}}}\theta(|b_{1}|-|a_{1}|),$
$\displaystyle n_{sing}$ $\displaystyle=$
$\displaystyle-\frac{1}{b_{1}^{2}}\delta(b_{1}-a_{1}).$ (58)

The kernel is obtained by substituting in Eqs. (VI) the values $a_{1},b_{1}$
and integrating according to Eq. (56).

##  VII Conclusion

We have found the new form of the kernel
$N(\gamma,z,\gamma^{\prime},z^{\prime})$ of the canonical equation (5):
$g=Ng$, for the Nakanishi function $g$ for the ladder kernel. This form
differs from two forms found previously in Refs. [11, 10], though all three
kernels are equivalent to each other. The kernel in the form found in the
present paper is expressed via the real functions, does not contain any
ambiguities and the method of its calculation can be applied to any kernel
given by irreducible Feynman graph. We outlined this generalization for the
cross-ladder BS kernel. For the ladder, the integration is fulfilled
analytically and the result for $N$ is given by Eq. (40) in terms of
$\tilde{N}$ determined in Sec. III. The solutions $g(\gamma,z)$ of Eq. (5),
defined in the interval $-1\leq z\leq 1$, are either symmetric relative to
$z\to-z$, or antisymmetric. For both solutions the equation (5) is reduced to
Eq. (52), defined on the half-interval $0\leq z\leq 1$. For symmetric
solution, the kernel is defined in Eq. (51). For antisymmetric one, the kernel
is constructed as explained in Sec. V. For the cross-ladder, the kernel is
given by Eq. (56).

The canonical equation (5), in contrast to Eq. (I), does not contain the
integral in l.h.-side and therefore it is solved easier, its solution is
stable relative to the numerical procedure. The equation (5) is an equation in
two variables, with smooth easy calculated kernel, and therefore finding its
solution is not more difficult than for the Euclidean BS equation. However, it
provides, via Nakanishi function, the BS amplitude in Minkowski space.
Therefore, applications of these methods to solving the BS equation can give
considerable advantages [12, 13].

## Acknowledgement

I am sincerely grateful to Jaume Carbonell kindly provided me his fortran code
which I used to solve the equation (52).

## References

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