The Low soft-photon theorem again

V. S. Fadina and V. A. Khozeb

a Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia

b Institute for Particle Physics Phenomenology, Durham University, DH1 3LE, UK

###### Abstract

It is shown that contrary to claims of Ref. [1] the formulated in the proper
physical variables Low theorem [2] for soft photon emission does not require
any modification. We also reject the criticism in Ref. [1] of the papers [3,
4]. At the same time, we identify some inaccuracies in Ref. [3] in the
presentation of the soft-photon theorem for the case of spin-one-half
particles. We also point out shortcomings in consideration of the Low theorem
in the classic textbooks [5, 6].

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Nearly 65 years ago in his celebrated paper [2] Francis Low proved that the
first two terms in the series expansion of the differential radiative cross-
section in powers of photon energy $k$ can be expressed via the corresponding
non-radiative amplitude.

This result was extended to the cross-sections of unpolarized multiparticle
processes involving charged spin 1/2 particles by Burnett and Kroll in [3] and
generalized subsequently by Bell and Van Royen [7] to particles of arbitrary
spin. The studies of soft photon radiation have a very long history and has
continued to attract attention until now, see [1], [8] \- [12] and references
therein. Recently the authors of [1](see also [8, 9]) have questioned the Low
theorem, in what concerns the validity of the second term in the expansion
over photon momentum. They also disputed some results of the basic works [3,
4].

This paper aims to demonstrate that the critical comments of [1] concerning
the consistency of the Low theorem and some results of [3, 4] are unjustified.
While addressing this issue we found some inaccuracies in [3] and in the
presentation of the Low theorem in the popular textbooks [5, 6].

In the next section, we consider the soft-photon theorem for the case of two
spin-zero particles exemplified similarly to [1] by the radiation in the
$\pi^{-}\pi^{0}$ scattering. In section 3 we address the Low theorem for the
case of the process with spin-one-half emitter, and in section 4 we consider
the Burnett-Kroll extension of the soft photon theorem. Some inconsistencies
in the presentation of the soft theorem in [5, 6] are discussed in section 5
and we conclude in section 6.

##  2 The Low theorem for scalar particles

Let us use for particle momenta the same notation as in Ref. [1] and consider
the processes

$\pi^{-}(p_{a})+\pi^{0}(p_{b})\rightarrow\pi^{-}(p_{1})+\pi^{0}(p_{2})$ (1)

with

$p_{a}+p_{b}=p_{1}+p_{2},\;\;p_{a}^{2}=p_{1}^{2}=m_{a}^{2},\;\;p_{b}^{2}=p_{2}^{2}=m_{b}^{2},$
(2)

and

$\pi^{-}(p_{a})+\pi^{0}(p_{b})\rightarrow\pi^{-}(p^{\prime}_{1})+\pi^{0}(p^{\prime}_{2})+\gamma(k,e)$
(3)

with

$p_{a}+p_{b}=p^{\prime}_{1}+p^{\prime}_{2}+k,\;\;p_{a}^{2}=p_{1}^{{}^{\prime}2}=m_{a}^{2},\;\;p_{b}^{2}=p_{2}^{{}^{\prime}2}=m_{b}^{2},\;\;k^{2}=0~{}.$
(4)

It is convenient to introduce notation

$j_{\mu}=\frac{p^{\prime}_{1\mu}}{(p^{\prime}_{1}k)}-\frac{p_{a\mu}}{(p_{a}k)}~{},\;\;{\cal
P}_{f\mu}=t_{f}^{\mu\rho}p^{\prime}_{2\rho}~{},\;\;{\cal
P}_{i\mu}=t_{i}^{\mu\rho}p_{b\rho}~{},$ (5)

where

$t_{i}^{\mu\rho}=\frac{p_{a}^{\mu}}{(p_{a}k)}k^{\rho}-g^{\mu\rho}~{},\;\;t_{f}^{\mu\rho}=\frac{p^{\prime\mu}_{1}}{(p^{\prime}_{1}k)}k^{\rho}-g^{\mu\rho}~{}.$
(6)

Then the original Low theorem [2] reads (see Eq. (2.16) in [2])

$M_{\mu}=e\Bigg{[}j_{\mu}+\Bigg{(}{\cal P}_{i\mu}+{\cal
P}_{f\mu}\Bigg{)}\frac{\partial}{\partial\nu}\Bigg{]}T(\nu,\Delta)~{},$ (7)

where

$\nu=(p_{a}p_{b})+(p_{1}^{\prime}p_{2}^{\prime})~{},\;\;\Delta=(p_{b}-p_{2}^{\prime})^{2},$
(8)

and
$T(\nu,\Delta)=T(p_{1}^{{}^{\prime}2},p_{a}^{2},;\nu,\Delta)|_{p_{1}^{{}^{\prime}2}=p_{a}^{2}=m_{a}^{2}}$,
where $T(p_{1}^{{}^{\prime}2},p_{a}^{2},;\nu,\Delta)$ is the off-mass-shell
amplitude (amputated Green function) of elastic process. It is clearly stated
in [2] (see Eq.(2.9) and a phrase below it) that this amplitude conserves
momentum and energy but not mass. Let us emphasize that the amplitude of the
process (1) is a function of two invariant variables, $T(\nu_{i},t_{b})$, with
$\nu_{i}=2(p_{a}p_{b})$ and $t_{b}=(p_{b}-p_{2})^{2}$.
The important point here is that the Low theorem is formulated in terms of the
physical momenta of the radiative process (3). The authors of [1] claim that
such consideration is incorrect because these momenta cannot be the momenta of
the non-radiative (elastic) process. Below Eq.(3.40) in [1] it is written ”The
term of order $\omega^{0}$ 111Note that in the notation of Ref. [1] $\omega$
is the photon energy. in the expansion of the amplitude given in [2]
corresponds to the fictitious process (3.31) which does not respect energy-
momentum conservation.” Here (3.31) means

$\pi^{-}(p_{a})+\pi^{0}(p_{b})\rightarrow\pi^{-}(p_{1})+\pi^{0}(p_{2})+\gamma(k,e),$
(9)

i.e. the inelastic process (3), but with momenta of the final pions
corresponding to the non-radiative case (3). However, such an equation has
never been presented in [2]. Actually, everywhere in [2] the physical momenta
of the process (3) satisfying the energy-momentum conservation (4) are used.
In contrast, the momenta used in [1] correspond to the elastic process. It is
true, that the momenta $p_{a},p_{b},p^{\prime}_{1},p^{\prime}_{2}$ used in [2]
cannot be momenta of the elastic process, because they do not satisfy energy-
momentum conservation for this process, $p_{a}+p_{b}\neq
p^{\prime}_{1}+p^{\prime}_{2}$. But the arguments $\nu_{i}$ and $t_{b}$ of the
elastic scattering amplitude $T(\nu_{i},t_{b})$ are independent variables, so,
certainly, they can be taken equal to
$\nu=(p_{a}p_{b})+(p_{1}^{\prime}p_{2}^{\prime})$ and
$\Delta=t_{b}=(p_{b}-p^{\prime}_{2})^{2}$. Instead of using these variables
the authors of [1] introduced artificial momenta $p_{1}$ and $p_{2}$
satisfying the elastic conservation law $p_{a}+p_{b}=p_{1}+p_{2}$ and
reformulate the theorem in terms of these momenta. The momenta $p_{1}$ and
$p_{2}$ in [1] are written as

$p_{1}=p_{1}^{\prime}+l_{1}~{},\;\;p_{2}=p_{2}^{\prime}+l_{2}~{},$ (10)

where the momenta $l_{1}$ and $l_{2}$ are small (${\cal O}$ (k)) and

$l_{1}+l_{2}=k~{}.$ (11)

Since both $p_{1}$ and $p_{1}^{\prime}$, as well as $p_{2}$ and
$p_{2}^{\prime}$ are on-shell, we get

$2(p^{\prime}_{1}l_{1})=-l_{1}^{2},\;\;2(p^{\prime}_{2}l_{2})=-l_{2}^{2}~{}.$
(12)

In Ref.[1] some particular representation for the momenta $l_{i}$ is used,
which is not essential for the discussion below. The amplitude of the
radiative process is presented there (see Eqs. (A1) or (3.27) and (3.28) in
[1]) at $k^{2}=0,(\epsilon^{*}k)=0$, where $\epsilon^{*}$ is the photon
polarization vector, as

${\cal M}_{\lambda}=e{\cal
M}^{(0)}(s_{L},t,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})\Bigg{[}\frac{p_{a\lambda}}{(p_{a}k)}-\frac{p_{1\lambda}}{(p_{1}k)}-\frac{p_{1\lambda}(l_{1}k)-l_{1\lambda}(p_{1}k)}{(p_{1}k)^{2}}\Bigg{]}$
$+2e\frac{\partial}{\partial s_{L}}{\cal
M}^{(0)}(s_{L},t,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})\Big{[}p_{b\lambda}-p_{a\lambda}\frac{(p_{b}k)}{(p_{a}k)}\Bigg{]}$
$-2e\frac{\partial}{\partial t}{\cal
M}^{(0)}(s_{L},t,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})\Bigg{[}\frac{p_{a\lambda}}{(p_{a}k)}-\frac{p_{1\lambda}}{(p_{1}k)}\Bigg{]}\Bigg{[}\left((p_{a}-p_{1})k\right)-(p_{a}l_{1})\Bigg{]}~{}.$
(13)

Here
$s_{L}=(p_{a}p_{b})+(p_{1}p_{2}),\;\;t=(p_{a}-p_{1})^{2}=(p_{b}-p_{2})^{2},\;\;{\cal
M}^{(0)}(s_{L},t,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})$ is the
elastic scattering amplitude.

Note that $s_{L}\neq\nu$ and $t\neq\Delta$. With the accuracy to order $k$
accounting for Eq. (12), we have

$s_{L}=\nu+k(p^{\prime}_{2}+p^{\prime}_{1})=\nu+k(p_{2}+p_{1})~{},\;\;t=\Delta-2(l_{2}p_{b})~{}.$

Therefore, taking the first term in (13) the expansion in $k$

${\cal M}^{(0)}(s_{L},t,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})={\cal
M}^{(0)}(\nu,\Delta,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})$
$+\left(k(p_{a}+p_{b})\right))\frac{\partial}{\partial\nu}{\cal
M}^{(0)}(\nu,\Delta,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})$
$-2(l_{2}p_{b})\frac{\partial}{\partial\Delta}{\cal
M}^{(0)}(\nu,\Delta,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})$ (14)

and setting in the second and the third terms $s_{L}=\nu,\;\;t=\Delta$ (that
is allowed because these terms are ${\cal O}$ (k)), we arrive at the original
formulation of the Low theorem (apart from the difference in common sign of
the amplitudes in [1] and [2]). Thus, the formulation of the Low theorem
proposed in [1] fully agrees with the original one [2]. However, it uses
artificially constructed variables that differ from the physical variables of
the radiative process used in Ref. [2]). It is worth emphasizing here that
such an agreement holds just for the first two terms in the expansion in the
photon energy. Only these two terms obey the theorems for soft-photon
radiation based on the gauge invariance. Let us note here that the choice in
[2] of $\nu$ and $\Delta$ as two independent variables of the non-radiative
scattering amplitude is a convenient, but obviously not the only option (see,
e.g. footnote 3 in [3]) Therefore, different formulations of the theorem are
possible. For example, taking as the two independent variables
$\nu_{i}=2(p_{a}p_{b})$ and $\Delta$ and using the relation

$\nu=\nu_{i}-(k(p_{a}+p_{b}))~{},$ (15)

we obtain with the ${\cal O}$ (k) accuracy

$T(\nu,\Delta)=-(k(p_{a}+p_{b}))\frac{\partial}{\partial{\nu_{i}}}T(\nu_{i},\Delta)~{}.$
(16)

Inserting (16) in (7) with an account for momentum conservation and neglecting
in the $M_{\mu}$ terms proportional to $k_{\mu}$ we get

$M_{\mu}=e\Bigg{[}j_{\mu}+2P_{i\mu}\frac{\partial}{\partial{\nu_{i}}}\Bigg{]}T(\nu_{i},\Delta)~{}.$
(17)

Similarly, we can obtain

$M_{\mu}=e\Bigg{[}j_{\mu}+2P_{f\mu}\frac{\partial}{\partial{\nu_{f}}}\Bigg{]}T(\nu_{f},\Delta)~{},$
(18)

where $\nu_{f}=2(p^{\prime}_{1}p^{\prime}_{2})$. In the expressions (7), (17),
(18) the first arguments of the amplitude of the non-radiative process differ
by the ${\cal O}$ (k) terms. It is always possible to set the second argument
also being different by the terms of the same order. For example, we can write
the singular term as $ej_{\mu}T(\nu_{i},\Delta_{p})$, where
$\Delta_{p}=(p_{a}-p^{\prime}_{1})^{2}=\Delta+2k(p_{a}-p^{\prime}_{1})$. In
this case, with the required accuracy

$T(\nu_{i},\Delta)=T(\nu_{i},\Delta_{p})-2(k(p_{a}-p^{\prime}_{1}))\frac{\partial}{\partial{\Delta_{p}}}T(\nu_{i},\Delta_{p})$
(19)

and we obtain from (17)

$M_{\mu}=e\Bigg{[}j_{\mu}+2{\cal
P}_{i\mu}\frac{\partial}{\partial\nu}-2\big{(}{\cal R}_{i\mu}+{\cal
R}_{f\mu}\big{)}\frac{\partial}{\partial{\Delta_{p}}}\Bigg{]}T(\nu_{i},\Delta_{p})~{},$
(20)

with

${\cal R}_{f}^{\mu}=t_{f}^{\mu\rho}p_{a\rho}~{},\;\;{\cal
R}_{i}^{\mu}=t_{i}^{\mu\rho}p^{\prime}_{1\rho}~{},$ (21)

where $t_{i,f}^{\mu\rho}$ are defined in (6).

Very often the soft radiation theorem is formulated using the partial
derivatives with respect to momenta. Such an expression can be derived from
the original one with the help of relations

${\cal
P}_{i}^{\mu}\frac{\partial}{\partial\nu}T(\nu,\Delta)=t_{i}^{\mu\rho}\frac{\partial}{\partial{p^{\rho}_{a}}}T(\nu,\Delta)~{},$
${\cal
P}_{f}^{\mu}\frac{\partial}{\partial\nu}T(\nu,\Delta)=t_{f}^{\mu\rho}\frac{\partial}{\partial
p^{{}^{\prime}\rho}_{1}}T(\nu,\Delta)~{},$ (22)

where $\nu$ and $\Delta$ are defined in (8). Let us note that the amplitude
$T(\nu,\Delta)$ is defined on the mass shell of all particle momenta,
therefore, generally speaking, the partial derivatives over the components of
the momenta are ill-defined, because their definition requires exit from the
mass shell. But due to the properties

$t_{f}^{\mu\rho}p^{\prime}_{1\rho}=0,\;\;t_{a}^{\mu\rho}p_{a\rho}=0,$ (23)

which follow from (6), the amplitudes in Eq. (22) remain on the mass shell, so
that such a problem does not arise. Using the relations (22) we obtain from
(7)

$M_{\mu}=e\Bigg{[}j_{\mu}(k)+\Bigg{(}D^{\mu}_{i}+D^{\mu}_{f}\Bigg{)}\Bigg{]}T(\nu,\Delta)~{},$
(24)

where

$D^{\mu}_{f}=t_{f}^{\mu\rho}\frac{\partial}{\partial{p^{{}^{\prime}\rho}_{1}}}~{},\;\;D^{\mu}_{i}=t_{i}^{\mu\rho}\frac{\partial}{\partial{p^{\rho}_{a}}}~{},$
(25)

Using the definitions of
$\nu_{i}=2(p_{a}p_{b}),\nu_{f}=2(p^{\prime}_{1}p^{\prime}_{2}),\nu=\frac{1}{2}(\nu_{i}+\nu_{f}),\Delta=(p_{b}-p^{\prime}_{2})^{2},\Delta_{p}=(p_{a}-p^{\prime}_{1})^{2}$
one can easily check that

$\Bigg{(}D^{\mu}_{i}+D^{\mu}_{f}\Bigg{)}T(\nu_{i},\Delta)={\cal
P}_{i}^{\mu}\frac{\partial}{\partial{\nu_{i}}}T(\nu_{i},\Delta)~{},\;\;\Bigg{(}D^{\mu}_{i}+D^{\mu}_{f}\Bigg{)}T(\nu_{f},\Delta)={\cal
P}_{f}^{\mu}\frac{\partial}{\partial{\nu_{f}}}T(\nu_{f},\Delta)~{},$ (26)
$\Bigg{(}D^{\mu}_{i}+D^{\mu}_{f}\Bigg{)}T(\nu,\Delta_{p})=\Bigg{(}{\cal
P}_{i}^{\mu}+{\cal
P}_{f}^{\mu}\Bigg{)}\frac{\partial}{\partial\nu}-2\Bigg{(}{\cal
R}_{i}^{\mu}+{\cal
R}_{f}^{\mu}\Bigg{)}\frac{\partial}{\partial{\Delta_{p}}}\Bigg{]}T(\nu,\Delta_{p})~{}.$
(27)

This means that all relations (7), (17),(18),(20) have the same differential
in momenta form (24) with the corresponding arguments of amplitude $T$.

It is easy to see that the differential form (24) does not change if we take
instead of $\nu$ and $\Delta$ any scalar variables, which are equal to them at
$k=0$, which means $a\nu_{i}+(1-a)\nu_{i}$ and $b\Delta+(1-b)\Delta_{p}$ with
the coefficients $0\leq a,b\leq 1$. Indeed, using relation
$\nu_{i}-\nu_{f}=2\big{(}k((p^{\prime}_{a}+p^{\prime}_{b}))\big{)}=2\big{(}k((p_{a}+p_{b}))\big{)}$
we get

$j^{\mu}(\nu_{i}-\nu_{f})=\frac{p^{{}^{\prime}\mu}_{a}}{(kp^{\prime}_{a})}2\big{(}k((p^{\prime}_{a}+p^{\prime}_{b}))\big{)}-\frac{p_{a}^{\mu}}{(kp_{a})}2\big{(}k((p_{a}+p_{b}))\big{)}=2{\cal
P}^{\mu}_{f}-2{\cal P}^{\mu}_{i}~{},$ (28)

while

$D^{\mu}(\nu_{i}-\nu_{f})=\big{(}D^{\mu}_{i}+D^{\mu}_{f}\big{)}\big{(}2(p_{a}p_{b})-2(p^{\prime}_{a}p^{\prime}_{b})\big{)}=2{\cal
P}^{\mu}_{i}-2{\cal P}^{\mu}_{f}~{},$ (29)

so that

$\Big{(}j^{\mu}+D^{\mu}\Big{)}(\nu_{i}-\nu_{f})=0~{}.$ (30)

At first sight, the result (29) may look confusing, since
$D^{\mu}\sim{\cal}k^{0}$ and $(\nu_{i}-\nu_{f})\sim{\cal}k^{1}$, so the right-
hand side of (29) should also be $\sim{\cal}k^{1}$. The point is that when
calculating derivatives, the vectors
$p_{a},p_{b},p^{\prime}_{a},p^{\prime}_{b}$ are considered as being
independent (not related to $k$ by momentum conservation). Similarly, we
obtain

$\big{(}j^{\mu}+D^{\mu}\big{)}(\Delta-\Delta_{p})=j^{\mu}2\big{(}k((p^{\prime}_{a}-p_{a}))\big{)}+\big{(}D^{\mu}_{i}+D^{\mu}_{f}\big{)}\big{(}(p_{b}-p^{\prime}_{b})^{2}-(p_{a}-p^{\prime}_{a})^{2}\big{)}=0~{}.$
(31)

Accounting for these remarks, we get for squared modulus of the matrix element
$\epsilon^{\mu}M_{\mu}$ with the real photon polarisation vector
$\epsilon^{\mu}$

$|\epsilon^{\mu}M_{\mu}|^{2}=e^{2}(\epsilon^{\mu}j_{\mu}(k))\Bigg{[}(\epsilon^{\mu}j_{\mu}(k))+\epsilon^{\mu}D_{\mu}\Bigg{]}|T(a\nu_{i}+(1-a)\nu_{f},b\Delta+(1-b)\Delta_{p})|^{2}~{}.$
(32)

##  3 The Low theorem for spin-one-half particles

In addition to Eq.(7) for the scalar particles, the original paper [2]
contains also the derivation of a similar equation (Eq.(3.16) in [2]) for the
case when the emitter is a one-half fermion. For consideration of this case,
it is convenient to use the same notation as in [2]). Then, for the process

$f(p_{i})+\pi^{0}(q_{i})\rightarrow f(p_{f})+\pi^{0}(q_{f})+\gamma(e,k)~{},$
(33)

where $f$ is a spin-one-half fermion of charge $e$, anomalous magnetic moment
$\lambda$ and mass $m$, the result of [2] reads

$M_{\mu}=\bar{u}(p_{f})\Big{[}(e\gamma_{\mu}+\frac{\lambda}{2}[\gamma_{\mu},\hat{k}])\frac{1}{\hat{p}_{f}+\hat{k}-m}T+T\frac{1}{\hat{p}_{i}-\hat{k}-m}(e\gamma_{\mu}+\frac{\lambda}{2}[\gamma_{\mu},\hat{k}])$
$+e\Big{(}{\cal P}_{f\mu}+{\cal
P}_{i\mu}\Big{)}\left(\frac{\partial}{\partial\nu}T\right)\Big{]}u(p_{a})~{},$
(34)

where ${\cal P}_{f\mu}$ and ${\cal P}_{i\mu}$ are given by (5) and (6) with
the replacement $p_{a}\rightarrow p_{i},p^{\prime}_{1}\rightarrow p_{f}$, and

$T=A(\nu,\Delta)+\frac{1}{2}(\hat{q}_{i}+\hat{q}_{f})B(\nu,\Delta),\;\nu=(p_{i}q_{i})+(p_{f}q_{f}),\;\Delta=(q_{i}-q_{f})^{2}~{}.$
(35)

We recall that the Low theorem is written via the physical momenta of the
process (33), and therefore, the criticism of [1] repeated in [9] is not
justified.

The representation (34) can be rewritten with the ${\cal O}$ (k) accuracy as

$M_{\mu}=e\bar{u}(p_{f})\Bigg{[}j_{\mu}T+\Bigg{(}{\cal P}_{f\mu}+{\cal
P}_{i\mu}\Bigg{)}\left(\frac{\partial}{\partial\nu}T\right)$
$+\frac{[\gamma_{\mu},\hat{k}])}{4(kp_{f})}\Bigg{(}1+\frac{\lambda}{e}(\hat{p}_{f}+m)\Bigg{)}T-T\Bigg{(}1+\frac{\lambda}{e}(\hat{p}_{i}+m)\Bigg{)}\frac{[\gamma_{\mu},\hat{k}]}{4(kp_{i})}\Bigg{]}u(p_{i})~{},$
(36)

where $j_{\mu}$ is given by (5) with the replacement $p_{a}\rightarrow
p_{i},p^{\prime}_{1}\rightarrow p_{f}$. unpolarized, non-radiative In Ref.[2]
the soft-photon theorem was considered on the amplitude level, and Eq. (34) is
the final result of this paper. The extension of the Low theorem developed in
[3] concerns its application to the radiative cross sections for unpolarized
particles, and the main result is proof that ”the first two terms of an
expansion in the photon energy depend on the unpolarized, non-radiative cross-
section only”.

Let us obtain the square of the matrix element summed over fermion
polarizations taking the original Low theorem in its differential in momenta
form, that is using

$j_{\mu}T+\big{(}{\cal P}_{f\mu}+{\cal
P}_{i\mu}\big{)}\left(\frac{\partial}{\partial\nu}T\right)=\Big{(}j_{\mu}+D_{\mu}\big{)}{\cal
T}~{}.$ (37)

Recall here that, as shown in the previous section, in the differential form
(37) we can set in $T$ (35) instead of $\nu$ and $\Delta$ any scalar
variables, which are to equal them at $k=0$. Using(36) and (37), we obtain for
real photon polarisation vectors $\epsilon^{\mu}$

$\sum_{spin}|\epsilon^{\mu}M_{\mu}|^{2}=e^{2}\Bigg{[}(\epsilon
j)\Big{(}(\epsilon j)Tr\Big{(}(\hat{p}_{i}+m))T(\hat{p}_{f}+m)\overline{T}$
$+Tr\Big{(}(\hat{p}_{i}+m)\big{(}(\epsilon
D)T\big{)}(\hat{p}_{f}+m)\overline{T}\Big{)}+\Big{(}(\hat{p}_{i}+m)T\big{(}\hat{p}_{f}+m)\big{(}(\epsilon
D)\overline{T}\big{)}\Big{)}$ $+\frac{(\epsilon
j)}{4(kp_{f})}Tr\Bigg{(}(\hat{p}_{f}+m)T(\hat{p}_{i}+m)\overline{T}\Big{(}1+\frac{\lambda}{e}(\hat{p}_{f}+m)\Big{)}[\hat{k},\hat{e}]$
$+(\hat{p}_{f}+m)[\hat{e},\hat{k}]\Big{(}1+\frac{\lambda}{e}(\hat{p}_{f}+m)\Big{)}T_{0}(\hat{p}_{i}+m)\overline{T}\Bigg{)}$
$-\frac{(\epsilon
j)}{4(kp_{i})}Tr\Bigg{(}(\hat{p}_{f}+m)T_{0}(\hat{p}_{i}+m)[\hat{k},\hat{\epsilon}]\Big{(}1+\frac{\lambda}{e}(\hat{p}_{i}+m)\Big{)}\overline{T}$
$+(\hat{p}_{f}+m)T\Big{(}1+\frac{\lambda}{e}(\hat{p}_{i}+m)\Big{)}[\hat{\epsilon},\hat{k}](\hat{p}_{i}+m)\overline{T}_{0}\Bigg{)}~{},$
(38)

where $\sum$ means summation over the fermion polarizations. It is easy to see
from (38) that (as already observed in [3]) the terms with the anomalous
magnetic moment have canceled. Then using formulas

$\frac{1}{4(kp_{f})}\Big{(}[\hat{k},\hat{\epsilon}](\hat{p}_{f}+m)+(\hat{p}_{f}+m)[\hat{k},\hat{\epsilon}]\Big{)}=\epsilon_{\mu}t_{f}^{\mu\rho}\gamma_{\rho}=(\epsilon
D)(\hat{p}_{f}+m),$
$\frac{1}{4(kp_{i})}\Big{(}[\hat{k},\hat{\epsilon}](\hat{p}_{i}+m)+(\hat{p}_{i}+m)[\hat{k},\hat{\epsilon}]\Big{)}=\epsilon_{\mu}t_{i}^{\mu\rho}\gamma_{\rho}=(\epsilon
D)(\hat{p}_{i}+m),$ (39)

we get from (38)

$\sum_{spin}|\epsilon^{\mu}M_{\mu}|^{2}=e^{2}(\epsilon j)\Big{(}(\epsilon
j)+(\epsilon D)\Big{)}\sum_{spin}|M|^{2}~{},$ (40)

where

$\sum_{spin}|M|^{2}=Tr\big{(}(\hat{p}_{i}+m)T(\hat{p}_{f}+m)\overline{T}\big{)}=\sum_{spin}|\bar{u}(p_{f})Tu(p_{i})|^{2}~{}.$
(41)

Recall that $T$ here is expressed in terms of the momenta of the radiative
process (33), and that we can take in $T$ (35) instead of $\nu$ and $\Delta$
any scalar variables, that are equal to them at $k=0$.
Note that in Refs. [8, 9] the soft-photon theorem for the pion-proton
scattering was formulated following the same approach as in [1] for the pion-
pion scattering. Derivation of the result presented in [9] ”involved a lengthy
and complex analysis” given in [8]. The result is also complicated, so that
the proof of its equivalence to the original formulation is not presented here
and will be considered elsewhere.

##  4 On the Burnett-Kroll extension of Low theorem

In Ref. [2] the soft-photon theorem was considered on the amplitude level, and
Eqs. (7) and (34) are the final results of paper [2] for the amplitudes of
soft photon emission by spin-zero and spin-one-half particles respectively in
scattering on a neutral spin-zero particle. Our Eqs. (32) and (40) were
obtained by a direct application of the results of [2] to the amplitudes
squared, summed over the fermion polarizations in the second case. Note that
our results seem to look like a confirmation (or repetition) of the
conclusions of [3]. However, there is an the essential difference between
these results.

For illustration, let us reproduce the final result (Eq. (11)) of the paper
[3]

$\sum_{spins}|T_{\gamma}(\epsilon,k,p)|^{2}=\sum_{a}Q_{a}\frac{\epsilon\cdot
p_{a}}{k\cdot p_{a}}\sum_{b}Q_{b}\Big{[}\frac{\epsilon\cdot p_{b}}{k\cdot
p_{b}}-\epsilon\cdot D_{b}(k)\Big{]}\sum_{spins}|T(p^{\prime})|^{2},$ (42)

where $T_{\gamma}(\epsilon,k,p)$ and $T(p^{\prime})$ are the radiative and
non-radiative amplitudes, $p_{a}$ and $Q_{a}$ are the momentum and charge of
the particle $a$, and $k$ and $\epsilon$ are the momentum and polarisation
vector of the photon. All particles except the photon are considered as
incoming, so that $\sum_{a}p_{a}=k$,
$p^{\prime}_{a}=p_{a}-\xi_{a}(k),\;\;\sum_{a}\xi_{a}(k)=k,\;\;p_{a}\cdot\xi_{a}=0$,

$D_{a}(k)=\frac{p_{a}}{k\cdot
p_{a}}k\cdot\frac{\partial}{\partial{p_{a}}}-\frac{\partial}{\partial{p_{a}}}~{}.$
(43)

Note, that Eqs. (32) and (40) (as well as Eqs. (7) , (34) ) are expressed
through momenta of the radiative processes, while in Ref. [3] (see Eq.(42)
above), the momenta of non-radiative processes are used. There is a large
uncertainty in the choice of these momenta at the given momenta of the
radiative process. An important restriction on this choice imposed in [3] is
that ”scalar invariants are the same to first order in $k$ whether expressed
in terms of $p$ or $p^{\prime}$ ”, i.e. momenta of the radiative or non-
radiative processes. Of course, only the independent invariants are
considered, because it is impossible to have simultaneously, for example,
$(p^{\prime}_{i}q^{\prime}_{i})=(p_{i}q_{i})$ and
$(p^{\prime}_{f}q^{\prime}_{f})=(p_{f}q_{f})$ since
$(p^{\prime}_{i}q^{\prime}_{i})=(p^{\prime}_{f}q^{\prime}_{f})$, but
$(p_{i}q_{i})=(p_{f}q_{f})+(k((p_{i}+q_{i}))$. It was assumed in (42) that
$\sum_{spins}|T(p^{\prime})|^{2}$ is expressed in invariants constructed from
the momenta of the radiative process. Note that the imposed restriction leaves
a choice of momenta of the non-radiative process far from being unique.

For the case of radiation by the scalar particles, formulations of the soft-
photon theorem via momenta of the radiative and non-radiative processes
coincide. Indeed, the form (32) is in a perfect agreement with Eq. (11) of [3]
(see Eq.(42) above), because $|T(p^{\prime})|^{2}$ expressed in invariants
constructed from the momenta of the radiative process is nothing more than
$|T(a\nu_{i}+(1-a)\nu_{f},b\Delta+(1-b)\Delta_{p})|^{2}$. The choice of
different coefficients $a$ and $b$ corresponds to the choice of different
scalar invariants which are the same to first order in $k$, whether expressed
via $p$ or $p^{\prime}$.

The paper of Barnett and Kroll [3] was criticized in Ref.[1] (see the end of
Appendix B) on the grounds that ”their results contain derivatives of the non-
radiative amplitudes with respect to one momentum keeping the other ones
fixed”. This is incorrect, as it follows from the sentence in [3] after Eq.
(4) there: ”The prescription then is that $T$ is expressed in terms of the
momenta $p_{a}$ which are differentiated as independent variables, then
evaluated with momenta $p^{\prime}_{a}$”. Note, that In Eq. (42) it is assumed
that before differentiation $\sum_{spins}|T(p^{\prime})|^{2})$ is expressed in
terms of the invariants constructed from the momenta of the radiative process.

We could see a weak point in the paper [1] regarding the partial derivatives
$\frac{\partial}{\partial{p_{a}}}$, which are generally poorly defined, as it
was already noted, because their definition requires exit from the mass shell.
But these derivatives enter only in the combinations
$\xi_{a}\cdot\frac{\partial}{\partial{p_{a}}}$ and $D_{a}(k)$, which do not
require a shift from the mass shell,
so the only reproach is that the authors of [1] did not explain this.
Therefore, in the case of radiation by scalar particles, the result of [1] is
completely consistent with Eq. (32), and we have only minor comments regarding
its derivation. For radiation by the spin-one-half particles, the situation is
somewhat more complicated. In this case, the result of [1] written in the
notation of Eq.(33) takes the form

$\sum|\epsilon^{\mu}M_{\mu}|^{2}=e^{2}(\epsilon j)\Big{(}(\epsilon
j)+(\epsilon D)\Big{)}\sum_{spin}|M^{\prime}|^{2}~{},$ (44)

where prime labels momenta of the non-radiative process, and

$j^{\mu}=\frac{p^{\mu}_{f}}{(p_{f}k)}-\frac{p_{i}^{\mu}}{(p_{i}k)}~{},\;\;D^{\mu}=t_{i}^{\mu\rho}\frac{\partial}{\partial{p_{i\rho}}}+t_{f}^{\mu\rho}\frac{\partial}{\partial{p_{f\rho}}}~{},$
$t_{i}^{\mu\rho}=\frac{p_{i}^{\mu}}{(p_{i}k)}k^{\rho}-g^{\mu\rho}~{},\;\;t_{f}^{\mu\rho}=\frac{p^{\mu}_{f}}{(p_{f}k)}k^{\rho}-g^{\mu\rho}~{}.$
(45)
$\sum_{spin}|M^{\prime}|^{2}=Tr\big{(}(\hat{p}^{\prime}_{i}+m)T(\hat{p}^{\prime}_{f}+m)\overline{T}^{\prime}\big{)}=\sum_{spin}|\bar{u}(p^{\prime}_{f})T^{\prime}u(p^{\prime}_{i})|^{2}~{},$
(46)
$T^{\prime}=A(\nu^{\prime},\Delta^{\prime})+\frac{1}{2}(\hat{q}^{\prime}_{i}+\hat{q}^{\prime}_{2})B(\nu^{\prime},\Delta^{\prime})~{},\nu^{\prime}=2(p^{\prime}_{i}q^{\prime}_{i})=2(p^{\prime}_{f}q^{\prime}_{f})~{},\Delta^{\prime}=(q^{\prime}_{i}-q^{\prime}_{f})^{2}=(p^{\prime}_{i}-p^{\prime}_{f})^{2}~{}.$
(47)

The restriction that scalar invariants are the same to first order in $k$,
whether expressed in terms of non-primed or primed momenta (i.e. momenta of
radiative or non-radiative processes) is very important because it needed is
for the definition of the derivatives in (44). Indeed, (44) contains the
derivatives in non-primed momenta acting on a scalar function of the primed
momenta, therefore, some prescription for the transformation of the scalar
products of the primed momenta into the scalar products of the non-primed
momenta is required.
Let us take $\nu^{\prime}_{i}=2(p^{\prime}_{i}q^{\prime}_{i})$ and
$\Delta^{\prime}_{p}=(p^{\prime}_{i}-p^{\prime}_{f})^{2}$ as the independent
variables, so that

$\nu^{\prime}_{i}=\nu_{i},\;\;2(p^{\prime}_{i}q^{\prime}_{i})=2(p_{i}q_{i})~{},\;\;\Delta^{\prime}_{p}=\Delta_{p},\;\;(p^{\prime}_{i}-p^{\prime}_{f})^{2}=(p_{i}-p_{f})^{2}~{},$
(48)

and take

$p^{\prime}_{i}=p_{i},\;p^{\prime}_{f}=p_{f},\;\;q_{i}=q^{\prime}_{i}+\eta_{i},\;\;q_{f}=q^{\prime}_{f}-\eta_{f}~{}.$
(49)

Therefore, $T^{\prime}$ becomes

$T^{\prime}=A(\nu_{i},\Delta_{p})+\frac{1}{2}(\hat{q}^{\prime}_{i}+\hat{q}^{\prime}_{2})B(\nu_{i},\Delta_{p})~{},\;\;\nu_{i}=2(p_{i}q_{i})~{},\;\;\Delta_{p}=(p_{i}-p_{f})^{2}~{}.$
(50)

The equalities (48) and the on-mass shell conditions for $q_{i},q_{f}$ lead to
the relation

$(\eta_{i}q_{i})=0,\;\;(\eta_{f}q_{f})=0,\;\;(\eta_{i}p_{i})=0~{}.$ (51)

Using that from momentum conservation law it follows $\eta_{i}+\eta_{f}=k$,
these conditions can be rewritten as

$(\eta_{i}q_{i})=0,\;\;(\eta_{i}q_{f})=(kq_{f}),\;\;(\eta_{i}p_{i})=0~{}.$
(52)

It is easy to see that this system of equations has an infinite number of
solutions except for the very special choice of kinematics.

Recall that in Eq. (40), in the expression for $T$ (35) instead of $\nu$ and
$\Delta$ we can take any variables, which are equal to $\nu$ and $\Delta$ at
$k=0$. Let us set these as $\nu_{i}$ and $\Delta_{p}$, i.e. substitute in (41)
$T$ with ${\cal T}$,

${\cal
T}=A(\nu_{i},\Delta_{p})+\frac{1}{2}(\hat{q}_{i}+\hat{q}_{f})B(\nu_{i},\Delta_{p})~{},$
(53)

so that

${\cal
T}=T^{\prime}+\frac{1}{2}(\hat{\eta}_{i}-\hat{\eta}_{f})B(\nu_{i},\Delta_{p})~{}.$
(54)

Therefore, the difference between the factors in front of $e^{2}(\epsilon
j)^{2}$ in (40) and (44) is (for compactness we drop the arguments of $A$ and
$B$)

$\sum_{spin}|M|^{2}-\sum_{spin}|M^{\prime}|^{2}=\frac{1}{2}Tr\Big{(}(\hat{p}_{f}+m)(\hat{\eta}_{i}-\hat{\eta}_{f})B(\hat{p}_{i}+m)\big{(}A^{*}+\frac{1}{2}(\hat{q}^{\prime}_{i}-\hat{q}^{\prime}_{f})B^{*}\big{)}$
$+\frac{1}{2}Tr\Big{(}(\hat{p}_{f}+m)\big{(}A+\frac{1}{2}(\hat{q}^{\prime}_{i}-\hat{q}^{\prime}_{f})B\big{)}(\hat{p}_{i}+m)(\hat{\eta}_{i}-\hat{\eta}_{f})B^{*}\Big{)}=(AB^{*}+BA^{*})2m\big{(}(\eta_{i}-\eta_{f})(p_{i}+p_{f})\big{)}$
$+2BB^{*}\Big{[}\big{(}p_{i}(\eta_{i}-\eta_{f})\big{)}\big{(}p_{f}(q^{\prime}_{i}+q^{\prime}_{f})\big{)}+\big{(}p_{f}(\eta_{i}-\eta_{f})\big{)}\big{(}p_{i}(q^{\prime}_{i}+q^{\prime}_{f})\big{)}-\big{(}(\eta_{i}-\eta_{f})(q^{\prime}_{i}+q^{\prime}_{f})\big{)}\big{(}(p_{i}p_{f})-m^{2}\big{)}\Big{]}~{},$
(55)

which is not zero. Indeed, using $\eta_{f}=k-\eta_{i}$ and the relations (52)
we obtain

$\sum_{spin}|M|^{2}-\sum_{spin}|M^{\prime}|^{2}=(AB^{*}+BA^{*})2m\big{(}-(k(2q_{f}+p_{f}+p_{i})\big{)}$
$+BB^{*}\big{[}2\nu_{i}\big{(}-(k(2q_{f}+p_{f}+p_{i}))\big{)}-2\Delta_{p}\big{(}(k(p_{i}+q_{i}))\big{)}\big{]}~{}.$
(56)

The difference $\sum_{spin}|M|^{2}-\sum_{spin}|M^{\prime}|^{2}$ (56) was
presumed to be zero in [3] because it was written as

$\frac{1}{2}\Big{(}\big{(}\eta_{i}\frac{\partial}{\partial{q_{i}}}-\eta_{f}\frac{\partial}{\partial{q_{i}}}\big{)}Tr\big{(}(\hat{p}_{f}+m)T(\hat{p}_{i}++m)\overline{T}\big{)}\Big{)}\big{|}_{q_{i}=q^{\prime}_{i},q_{f}=q^{\prime}_{f}}~{}.$
(57)

Indeed, the latter expression nullifies if we take the derivatives after the
calculation of traces with $q_{i}=q^{\prime}_{i},q_{f}=q^{\prime}_{f}$ and
present the result in terms of $\nu_{i}$ and $\Delta_{p}$. It was supposed in
[3] that these operations are commutative, but this is untrue. Such an
assumption was used in [3] where the term
$\xi_{b}\frac{\partial}{\partial{p_{b}}}\sum_{spin}|T|^{2}$ in Eq. (10) was
set to zero. This circumstance on its own would be sufficient to argue that
the final result of [3], Eq. (11), is incorrect. But the term with derivatives
in this equation is also erroneous because the same wrong assumption
(commutability of taking derivatives and summation over the spins followed by
the presentation of the result in terms of scalar variables expressed in
momenta of the radiative process) was used in its derivation. Therefore, it is
necessary to calculate the difference

$D^{\mu}\sum_{spin}|M|^{2}-D^{\mu}\sum_{spin}|M^{\prime}|^{2}~{},$ (58)

where

$\sum_{spin}|M|^{2}=Tr\big{[}(\hat{p}_{f}+m)\big{(}A+\frac{1}{2}(\hat{q_{i}}+\hat{q_{f}})B\big{)}(\hat{p}_{i}+m)\big{(}A^{*}+\frac{1}{2}(\hat{q_{i}}+\hat{q_{f}})B^{*}\big{)}\big{]}~{},$
(59)

and $\sum_{spin}|M^{\prime}|^{2}$ is obtained from (59) by the replacement
$q_{i}\rightarrow q^{\prime}_{i},q_{f}\rightarrow q^{\prime}_{f}$. Note that
since in the difference (58) only ${\cal O}$ $(k^{0})$ terms must be kept, and
invariant amplitudes $A$ and $B$ are the same in both terms, their derivatives
do not contribute to the difference so that they will be considered in what
follows as constants.

In the first term, the derivatives are taken before the calculation of the
trace, so that

$D^{\mu}\sum_{spin}|M|^{2}=t_{f}^{\mu\rho}Tr\big{[}\gamma_{\rho}\big{(}A+\frac{1}{2}(\hat{q^{\prime}_{i}}+\hat{q^{\prime}_{f}})B\big{)}(\hat{p}_{i}+m)\big{(}A^{*}+\frac{1}{2}(\hat{q^{\prime}_{i}}+\hat{q^{\prime}_{f}})B^{*}\big{)}\big{]}$
$+t_{i}^{\mu\rho}Tr\big{[}(\hat{p}_{f}+m)\big{(}A+\frac{1}{2}(\hat{q^{\prime}_{i}}+\hat{q^{\prime}_{f}})B\big{)}\gamma_{\rho}\big{(}A^{*}+\frac{1}{2}(\hat{q^{\prime}_{i}}+\hat{q^{\prime}_{f}})B^{*}\big{)}\big{]}~{}.$
(60)

Direct calculation gives

$D^{\mu}\sum_{spin}|M|^{2}=4AA^{*}\big{(}R_{i}^{\mu}+R_{f}^{\mu}\big{)}+2m(AB^{*}+BA^{*})\big{(}2{\cal
P}_{i}^{\mu}+2{\cal P}_{f}^{\mu}-R_{i}^{\mu}-R_{f}^{\mu}\big{)}$
$+BB^{*}\Big{[}2\nu_{i}\big{(}2{\cal P}_{i}^{\mu}+2{\cal
P}_{f}^{\mu}-R_{i}^{\mu}-R_{f}^{\mu}\big{)}+2\Delta_{p}\big{(}{\cal
P}_{i}^{\mu}+{\cal
P}_{f}^{\mu}\big{)}-4\mu^{2}\big{(}R_{i}^{\mu}+R_{f}^{\mu}\big{)}\Big{]}~{}.$
(61)

where here $\mu$ is the mass of the scalar particle.

In the second term in (58) the derivatives $D^{\mu}$ act after calculation of
the trace and the presentation of the result in terms of $\nu_{i}$ and
$\Delta_{p}=(p_{i}-p_{f})^{2}$, which gives

$D^{\mu}\sum_{spin}|M^{\prime}|^{2}=D^{\mu}\Big{[}2AA^{*}(4m^{2}-\Delta_{p})+(AB^{*}+BA^{*})2m(2\nu_{i}+\Delta_{p})+BB^{*}(2\nu^{2}_{i}+2\nu_{i}\Delta_{p}+2\mu^{2}\Delta_{p})\Big{]}$
$=4AA^{*}\big{(}R_{i}^{\mu}+R_{f}^{\mu}\big{)}+2m(AB^{*}+BA^{*})\big{(}4{\cal
P}_{i}^{\mu}-2R_{f}^{\mu}-2R_{i}^{\mu}\big{)}$
$+BB^{*}\Big{[}4\nu_{i}\big{(}2{\cal
P}_{i}^{\mu}-R_{i}^{\mu}-R_{f}^{\mu}\big{)}+4\Delta_{p}{\cal
P}_{i}^{\mu}-4\mu^{2}\big{(}R_{i}^{\mu}+R_{f}^{\mu}\big{)}\Big{]}~{}.$ (62)

Using (61) and (62) we arrive at

$D^{\mu}\sum_{spin}|M|^{2}-D^{\mu}\sum_{spin}|M^{\prime}|^{2}=2m(AB^{*}+BA^{*})\big{(}-2{\cal
P}_{i}^{\mu}+2{\cal P}_{f}^{\mu}+R_{f}^{\mu}+2R_{i}^{\mu}\big{)}$
$+BB^{*}\Big{[}2\nu_{i}\big{(}2{\cal P}_{f}^{\mu}-2{\cal
P}_{i}^{\mu}+R_{i}^{\mu}+R_{f}^{\mu}\big{)}+2\Delta_{p}\big{(}{\cal
P}_{f}^{\mu}-{\cal P}_{i}^{\mu}\big{)}\Big{]}~{}.$ (63)

It is quite straightforward to see using (63), (56) and explicit expressions
for $j^{\mu},\;\;{\cal P}_{i,f}^{\mu},\;\;R_{i,f}^{\mu}$ that

$\Big{(}j^{\mu}+D^{\mu}\Big{)}\sum_{spin}|M|^{2}-\Big{(}j^{\mu}+D^{\mu}\Big{)}\sum_{spin}|M^{\prime}|^{2}=0,$
(64)

which means, two errors made in [3] during the derivation of Eq. (42) as a
result of the incorrect assumption of commutability of the two operations
(taking derivatives and summation over the spins followed by the presentation
of the result in terms of scalar variables written in momenta of the radiative
process) cancel each other, as was shown above by the direct calculations. In
reality, these calculations were not needed at all, because the cancellation
can be proved using Eqs. (30) and (31).

The net outcome of our consideration is the proof of the validity of the
result (44), obtained in [3]. However, in our opinion, it is much more
convenient to use the form (40), which does not require an introduction of the
artificial and ambiguously defined momenta of the non-radiative processes.

And finally, about the criticism in [1] concerning Ref.[4] (see Appendix A in
[1]). It was based on the fact that the formula

${\cal
M}_{\lambda}\Bigg{|}_{Lipatov}=e\Bigg{[}\frac{p_{a\lambda}}{(p_{a}k)}-\frac{p^{\prime}_{1\lambda}}{(p^{\prime}_{1}k)}\Bigg{]}{\cal
M}^{(0)}(s_{L},t,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})$
$-e(p_{a}-p_{1},k)\Bigg{[}\frac{p_{a\lambda}}{(p_{a}k)}-\frac{p^{\prime}_{1\lambda}}{(p^{\prime}_{1}k)}\Bigg{]}\frac{\partial}{\partial
t}{\cal M}^{(0)}(s_{L},t,m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi},m^{2}_{\pi})~{},$
(65)

obtained from the equations of [4] for the case of the process (3), is
different from Eq. (13), due to the absence of the term with
$\frac{\partial}{\partial{s_{L}}}{\cal M}^{(0)}$ in (65), and that the term
proportional to $\frac{\partial}{\partial t}{\cal M}^{(0)}$ is different from
the result of [1].

It is easy to see that this criticism has no basis. Ref.[4] considered the
high energy process in multi-Regge kinematics, assuming the Regge behaviour of
the inelastic amplitude. Therefore, the term with
$\frac{\partial}{\partial{s_{L}}}{\cal M}^{(0)}$ was dropped in [4] because at
large energies it falls down. As for the term proportional to
$\frac{\partial}{\partial t}{\cal M}^{(0)}$, it has to be different from the
result of [1], since in [4] it was used the original form (7) of the Low
theorem, which is formulated in terms of momenta of the radiative process.
There is no term with $\frac{\partial}{\partial t}{\cal M}^{(0)}$ in the
original formula (7). In (65) such a term appeared because in this formula
$t=\frac{1}{2}((p_{a}-p^{\prime}_{1})^{2}+(p_{b}-p^{\prime}_{2})^{2})$ was
used instead of $\Delta=(p_{b}-p^{\prime}_{2})^{2}$ in (7). It is easy to see
that the second term in (65) is the result of the transition in (7) from
$\Delta$ to $t$.

##  5 The Low theorem in the textbooks

As discussed above, a conclusion in Ref.[1] that the representation of soft
photon emission amplitudes, given in [2] is incomplete, is based on the
assumption that it is written in terms of momenta of the non-radiative
processes that do not satisfy energy-momentum conservation. Unfortunately,
such an erroneous assumption is promoted by some popular textbooks. In
particular, this applies to the textbook [5](and its originals in Russian
[6]). To clarify this issue, let us first note that the initial particle
momenta are labeled in [5] as $p_{1}$ and $p_{2}$, whereas in [1] as $p_{a}$
and $p_{b}$. Here for convenience, we continue using the same notation (1) and
(3). Then we obtain from Eqs. (140.1), (140.9) and (140.10) of [5]

$M=(\epsilon^{*})^{\mu}\left(M^{(-)}_{\mu}+M^{(0)}_{\mu}\right),\;M^{(-)}_{\mu}=ej_{\mu}M^{(el)},\;M^{(0)}_{\mu}=2e\Bigg{(}{\cal
P}_{i\mu}+{\cal P}_{f\mu}\Bigg{)}\frac{\partial}{\partial s}M^{(el)}~{},$ (66)

so that

$M_{\mu}=M^{(-)}_{\mu}+M^{(0)}_{\mu}=e\Bigg{[}j_{\mu}+2\Bigg{(}{\cal
P}_{i\mu}+{\cal P}_{f\mu}\Bigg{)}\frac{\partial}{\partial
s}\Bigg{]}M^{(el)})~{},$ (67)

where $j_{\mu}$ and ${\cal P}_{i,f\mu}$ are defined in (5) and (6) and
$M^{(el)}$ is defined in [5] as elastic scattering amplitudeite. Eq.(67) is
analogous to the original Low equation (7), but it does not coincide with it.
In particular,(67) contains an extra factor 2 in front of $({\cal
P}_{i\mu}+{\cal P}_{f\mu})$. The appearance of this factor is caused by the
contradictory definitions of $M^{(el)}$. As it follows from the unnumbered
equation after Eq. (140.7) (as well as from Eq. (140.8), Eq. (140.10) and the
unnumbered equation after it), $M^{(el)}$ is considered a function of $s$ and
$t$. There is no problem with $t$ defined in Eq. (140.6) as

$t=(p_{b}-p_{2}^{\prime})^{2}~{},$ (68)

since both $t$ (Eq.(68) in [5]) and $\Delta$ (Eq. (8) in [2]) are expressed in
terms of the same momenta, so that they just coincide, $\Delta\equiv t$. It is
not so neither for the amplitudes $M^{(el)}(s,t)$ and $T(\nu,\Delta)$, nor for
their arguments $s$ and $\nu$. The point is that the amplitude $M^{(el)}(s,t)$
is not defined in [5] so clearly, as $T(\nu,\Delta)$ in [2]. Indeed,
$M^{(el)}(s,t)$ is defined in [5] by the diagram (140.5) as the amplitude of
the elastic scattering process, but with the same particle momenta, as for the
inelastic process (3). Moreover, there is the equality in Eq. (140.6) in [5]
which in the accepted notation [1] reads as

$s=(p_{a}+p_{b})^{2}=(p_{1}^{\prime}+p_{2}^{\prime})^{2}~{}.$ (69)

This is correct for the momenta of the non-radiative process, but not for the
momenta of the process (3). Thus, the momenta of radiative and non-radiative
processes are confused in [5], and this leads to an error in the presentation
of the theorem.

The book [5] also contains a differential form of the equation (67), which is
claimed to be derived from (67), but this derivation is questionable. It is
based on the equation following (140.10), which reads as

$2p_{b\nu}\frac{\partial}{\partial s}{\Bigg{|}}_{t}=\frac{\partial}{\partial
p^{\nu}_{a}}{\Bigg{|}}_{p_{b},p^{\prime}_{1},p^{\prime}_{2}}$ (70)

and an analogous equation for $\frac{\partial}{\partial
p^{\prime}_{1^{\prime}}}$. We stress again that these equations themselves are
incorrect, since the partial derivatives over components of the momenta are
ill-defined, as was already discussed in Section 2, because their definition
requires exit from the mass shell. It should be noted that this circumstance
is not essential, since in the final formula for the differential form in [5],
which can be written using expressions (25) as

$M_{\mu}=e\big{(}j_{\mu}+D_{i\mu}+D_{f\mu}\big{)}M^{(el)},~{}$ (71)

the derivatives appear in the combinations $D_{(i,f)\mu}$ which keep momenta
on the mass shell. But the disputed equations also have another, much more
serious drawback. These equations are confusing because the elastic scattering
momenta are related by the conservation law, and if three of them are fixed,
the fourth one is also fixed, so taking a derivative with respect to it makes
no sense. Actually, it is always possible to set a partial derivative with
respect to a certain momentum to zero expressing the amplitude via other
momenta.

Due to these inaccuracies in the derivation, the equation (71) turns out to be
unequivalent to the original one (67). Indeed, if we assume that $M^{(el)}$ in
Eq. (71) is $M^{(el)}((p_{a}+p_{b})^{2},t)$ then we obtain (67) without the
term $2{\cal P}_{f\mu}$, and if we assume that $M^{(el)}$ in Eq. (71) is
$M^{(el)}((p^{\prime}_{1}+p^{\prime}_{2})^{2},t)$ then we obtain (67) without
the term $2{\cal P}_{i\mu}$. Note that in the first case, we actually get the
formulation (17), and in the second (18). This means that the formulation of
the Low theorem in the differential form (71) can be considered correct if we
assume that $M^{(el)}$ in it is $M^{(el)}(s,t)$, where $s$ and $t$ are
expressed in terms of the momenta of the radiative process. But, in our
opinion, the formulation (67) could not be justified.

##  6 Conclusion

Recently there has been a renewal of interest in the physics of soft photon
emission and the status of the celebrated Low theorem [2], see for instance
[1], [8] \- [12]. In particular, the authors of [1] (see also [8, 9])
questioned the validity of the Low theorem and claimed that the term of the
order of $k^{0}$ in the expansion of the radiative amplitude in photon energy
$k$ needs modification. In this paper, we demonstrated that contrary to this
claim, the Low theorem when formulated in terms of the final state momenta,
does not require revision. It is shown here that the formulation of the soft-
photon theorem proposed in [1] is in complete agreement with the original
result of Low [2]. Note that though here we considered explicitly only the
processes with spin zero and spin-one-half emitters, similar arguments should
hold for the soft photon emission in other scattering processes with an
arbitrary number of external charged particles (as already mentioned in
Refs.[2, 3]). We also address the criticism in [1] of the papers [3] and [4]
and explain why it is unsubstantiated.

It is worth emphasizing that the agreement of the formulations does not mean
their equivalence, because different forms of presentation could differ by the
values of the omitted terms. In some sense, the situation here reminds the
expansion in terms of running the coupling constant in perturbation theory. It
is well known there that when a finite number of terms in the perturbative
expansion is taken, the definition of the coupling constant at different
scales leads to different results. A lot of work has been devoted to this
problem in QCD, where running is really important, and many recipes have been
proposed for the choice of the best scale (that is, how to get the finite
number of terms closer to the exact result), see for instance, the book [13].
However, it seems to be impossible to suggest a prescription that is
universally applicable to all cases. In the case of the soft-photon theorem,
the situation is even worse, since we are talking not about the choice of one
parameter, but about the choice of the amplitude in front of the expression
which is singular in photon energy (classical current), depending on many
parameters. Finally, in this paper, we address some inaccuracies in the
discussion of the Low theorem in Ref. [3] and in the classic textbooks [5, 6].

## References

  * [1] P. Lebiedowicz, O. Nachtmann and A. Szczurek, “High-energy $\pi$$\pi$ scattering without and with photon radiation,” Phys. Rev. D 105 (2022) no.1, 014022, [arXiv:2107.10829 [hep-ph]].
  * [2] F.E. Low, “Bremsstrahlung of Very Low-Energy Quanta in Elementary Particle Collisions,” Phys. Rev. 110 (1958) no. 4, 974-977.
  * [3] T. H. Burnett and N. M. Kroll, “Extension of the low soft photon theorem,” Phys. Rev. Lett. 20 (1968), 86.
  * [4] L. N. Lipatov, “Massless Particle Bremsstrahlung Theorems for High-energy Hadron Interactions,” Nucl. Phys. B 307, 705-720 (1988).
  * [5] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii. QUANTUM ELECTRODYNAMICS, volume 4 of Course of Theoretical Physics. Pergamon Press, Oxford, 1982.
  * [6] E.M. Lifshitz and L.P. Pitaevsky, “Relativistic quantum theory, part 2,” Fizmatlit, Nauka, Moscow,1971; V.B. Berestetsky, E.M. Lifshitz and L.P. Pitaevsky, “Quantum Electrodynamics,” Fizmatlit, 2002.
  * [7] J. S. Bell and R. Van Royen, “On the low-burnett-Kroll theorem for soft-photon emission,” Nuovo Cim. A 60 (1969), 62-68.
  * [8] P. Lebiedowicz, O. Nachtmann and A. Szczurek, “Soft-photon theorem for pion-proton elastic scattering revisited,” [arXiv:2307.12673 [hep-ph]].
  * [9] P. Lebiedowicz, O. Nachtmann and A. Szczurek, “Soft-Photon Theorem for Pion-Proton Scattering: Next to Leading Term,” [arXiv:2307.13291 [hep-ph]].
  * [10] T. Engel, A. Signer and Y. Ulrich, “Universal structure of radiative QED amplitudes at one loop,” JHEP 04, 097 (2022), [arXiv:2112.07570 [hep-ph]]
  * [11] R. Balsach, D. Bonocore and A. Kulesza, “Soft-photon spectra and the LBK theorem,” [arXiv:2312.11386 [hep-ph]].
  * [12] R. Balsach, D. Bonocore and A. Kulesza, “The emission of soft-photons and the LBK theorem, revisited,” [arXiv:2401.01820 [hep-ph]].
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ISBN 978-1-107-42475-3, 978-0-521-63148-8, 978-0-511-71744-4


$\displaystyle\cup~{}\\{(f^{*}(u),f^{*}(u\cdot\mathrm{R.B}))\mid
u,u\cdot\mathrm{R.B}\in\Delta^{\circ}\\}$

The preceding definition is a simplified version of Definition 20 from (?) as
we are not considering role inclusions nor inverse roles. We also make a
slight modification – using $\mathsf{Ind}(\mathcal{K}^{\dagger})$ rather than
only individuals from the ABox – to make it compatible with nominals.

We have phrased the definition directly in terms of our desired function
$f^{*}$, but other functions $f^{\prime}$ with domain $\Delta^{\circ}$ can be
used instead. Depending on which $f^{\prime}$ is used to define the
interlacing, the resulting interpretation $\mathcal{I}^{\prime}$ may or may
not be a model of the considered KB. It was shown in (?) that if $f^{\prime}$
is pseudo-injective, then the $f^{\prime}$-interleaving is a model and
moreover maps homomorphically into the starting interpretation. This property
(stated in Lemma 7 below, phrased for our KB $\mathcal{K}^{\dagger}$) was
proven for $\mathcal{ALCHI}$ KBs in normal form, but is easily shown to also
hold for $\mathcal{ALCO}$ KBs in normal form.

######  Definition 9.

A function $f^{\prime}:\Delta^{\circ}\rightarrow E$ is _pseudo-injective_ if:
for all $u,v\in\Delta^{\circ}$, if $f^{\prime}(u)=f^{\prime}(v)$, then
$f(u)=f(v)$.

######  Lemma 7.

If $f^{\prime}:\Delta^{\circ}\rightarrow E$ is pseudo-injective, then the
$f^{\prime}$-interlacing ${\mathcal{I}}^{\prime}$ is a model of
$\mathcal{K}^{\dagger}$ and the following mapping is a homomorphism from
${\mathcal{I}}^{\prime}$ to $\mathcal{I}^{\dagger}$:

$\begin{array}[]{cccl}\sigma:&\Delta^{{\mathcal{I}}^{\prime}}&\rightarrow&\Delta^{\mathcal{I}^{\dagger}}\\\
&f^{\prime}(u)&\mapsto&f(u)\end{array}$

As $f^{\prime}$ is pseudo-injective, $\sigma$ is well defined.

It was proven in (?) that $f^{*}$ is pseudo-injective, and the same arguments
apply to our modified $f^{*}$.

######  Lemma 8.

$f^{*}$ is pseudo-injective.

It follows from the preceding lemmas that ${\mathcal{I}}^{*}$ is a model of
$\mathcal{K}^{\dagger}$ which maps homomorphically into
$\mathcal{I}^{\dagger}$ via the mapping $\sigma$. In fact, due to the way
$f^{*}$ is defined, we can be more precise about the homomorphism $\sigma$:

######  Lemma 9.

The homomorphism $\sigma$ from ${\mathcal{I}}^{*}$ to $\mathcal{I}^{\dagger}$
is such that $\sigma(a)=a$ for all $a\in\mathsf{Ind}(\mathcal{K}^{\dagger})$.

We are now ready to show that ${\mathcal{I}}^{*}$ retains the desired
properties of $\mathcal{I}^{\dagger}$ (and $\mathcal{I}$).

######  Lemma 10.

${\mathcal{I}}^{*}\not\models q$.

###### Proof.

We could obtain the result by examining the proof of Lemma 5 in (?), which
shows that the query matches in ${\mathcal{I}}^{*}$ (there denoted
$\mathcal{I}^{\prime}$ and defined w.r.t. the original definition of
$\Delta^{*}$) are injectively mapped into $\mathcal{I}^{\dagger}$ (and hence
there cannot be any additional matches in ${\mathcal{I}}^{*}$). However, it
turns out that the argument is much simpler in our case, since we are start
from an interpretation $\mathcal{I}^{\dagger}$ with
$\mathcal{I}^{\dagger}\not\models q$. The existence of the homomorphism
$\sigma$ from ${\mathcal{I}}^{*}$ to $\mathcal{I}^{\dagger}$ that is the
identify on $\mathsf{Ind}(\mathcal{K}^{\dagger})$ (see Lemmas 7 and 9), hence
also on $\mathsf{Ind}(q)$, means that any match of $q$ in ${\mathcal{I}}^{*}$
can be reproduced in $\mathcal{I}^{\dagger}$. Therefore, from
$\mathcal{I}^{\dagger}\not\models q$, we immediately obtain
${\mathcal{I}}^{*}\not\models q$. ∎

######  Lemma 11.

The cost of ${\mathcal{I}}^{*}$ does not exceed that of
$\mathcal{I}^{\dagger}$:
$\mathit{cost}_{\mathcal{K}_{\omega}}({\mathcal{I}}^{*})\leq\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}^{\dagger})$.
In particular, we have that $vio_{\tau}({\mathcal{I}}^{*})\subseteq
vio_{\tau}(\mathcal{I}^{\dagger})\subseteq\Delta^{*}$ for every
$\tau\in\mathcal{T}\setminus\mathcal{T}_{\infty}$.

###### Proof.

First note that since ${\mathcal{I}}^{*}$ is a model of
$\mathcal{K}^{\dagger}$, it is a model of $\mathcal{A}^{\dagger}$ and hence
satisfies all assertions in
$\\{\alpha\in\mathcal{A}\mid\mathcal{I}\models\alpha\\}=\\{\alpha\in\mathcal{A}\mid\mathcal{I}^{\dagger}\models\alpha\\}$.
Since $\mathcal{I}^{\dagger}\not\models\alpha$ for
$\alpha\in\mathcal{A}\setminus\mathcal{A}^{\dagger}$, it follows that

$vio_{\mathcal{A}}({\mathcal{I}}^{*})\subseteq
vio_{\mathcal{A}}(\mathcal{I}^{\dagger})$

Let us now consider violations of TBox axioms. First note that since
${\mathcal{I}}^{*}$ is a model of $\mathcal{K}^{\dagger}$, it is a model of
$\mathcal{T}_{\infty}^{vio}$ (Lemma 5, item (iii)). It follows that
${\mathcal{I}}^{*}\models\tau$ for every $\tau\in\mathcal{T}_{\infty}$. Now
consider some $\tau\in\mathcal{T}\setminus\mathcal{T}_{\infty}$ and suppose
that $e\in vio_{\tau}({\mathcal{I}}^{*})$. By Lemma 5, item (v),
$vio_{\tau}(\mathcal{I}^{*})=A_{\tau}^{\mathcal{I}^{*}}$, so $e\in
A_{\tau}^{\mathcal{I}^{*}}$. From the definition of ${\mathcal{I}}^{*}$, we
know that there exists $w\in\Delta^{\circ}$ such that $e=f^{*}(w)$. As
$\sigma$ is a homomorphism from ${\mathcal{I}}^{*}$ to $\mathcal{I}^{\dagger}$
(Lemma 7) and $e=f^{*}(w)\in A_{\tau}^{\mathcal{I}^{*}}$, we must also have
$f(w)\in A_{\tau}^{\mathcal{I}^{\dagger}}$. As $\mathcal{I}^{\dagger}$ is a
model of $\mathcal{K}^{\dagger}$, we can again apply Lemma 5 item (v) to get
$vio_{\tau}(\mathcal{I}^{\dagger})=A_{\tau}^{\mathcal{I}^{\dagger}}$. We thus
obtain $f(w)\in vio_{\tau}(\mathcal{I}^{\dagger})$, and thus
$f(w)\in\Delta^{*}$ (due to our definition of $\Delta^{*}$). But
$f(w)\in\Delta^{*}$ implies that $e=f^{*}(w)=f(w)$, and hence that $e\in
A_{\tau}^{\mathcal{I}^{\dagger}}$. We have thus shown that
$vio_{\tau}({\mathcal{I}}^{*})\subseteq vio_{\tau}(\mathcal{I}^{\dagger})$.

As every violation of an axiom or assertion from $\mathcal{K}_{\omega}$ in the
interpretation ${\mathcal{I}}^{*}$ is also present in $\mathcal{I}^{\dagger}$,
we can conclude that
$\mathit{cost}_{\mathcal{K}_{\omega}}({\mathcal{I}}^{*})\leq\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}^{\dagger})$.
∎

##### Quotient construction to get bounded-size interpretation

It now remains to ‘shrink’ the interpretation ${\mathcal{I}}^{*}$ to obtain an
interpretation with the same properties but having the required size. To do
so, we can proceed exactly as in Chapter 3.4 of (?), by defining a suitable
equivalence relation, and then considering the quotient interpretation
obtained by merging elements from the same equivalence class.

We will now describe how the equivalence relation $\sim_{n}$ is defined, but
without giving every detail as the definition is rather involved and we will
not be modifying it except by using our own definition of $\Delta^{*}$. The
definition of $\sim_{n}$ involves the notion of the $n$-neighbourhood of an
element $d$ in $\mathcal{I}^{*}$ relative to $\Delta^{*}$, denoted
$\mathcal{N}^{\mathcal{I}^{*},\Delta^{*}}_{n}(d)$, consisting of the elements
in $\Delta^{\mathcal{I}^{*}}$ that can be reached by taking at most $n$
‘steps’ along role edges, starting at $d$ and stopping whenever an element of
$\Delta^{*}$ is reached. If $d\in\Delta^{*}$, then $d$ itself is the only
element in its $n$-neighbourhood (for any $n$). However, when
$d\in\Delta^{\mathcal{I}^{*}}\setminus\Delta^{*}$, we know that $d=aw$ for
some $a\in\mathsf{Ind}(\mathcal{K}^{\dagger})$ and $w\in\Omega^{*}$. Using the
tree-shaped structure of the domain $\Delta^{\circ}$, we can identify a unique
‘root’ prefix $r_{n,d}$ of $d=aw$ such that
$f^{*}(r_{n,d})\in\mathcal{N}^{\mathcal{I}^{*},\Delta^{*}}_{n}(d)$ and for
every $d^{\prime}\in\mathcal{N}^{\mathcal{I}^{*},\Delta^{*}}_{n}(d)$, there is
a unique word $w^{d^{\prime}}_{n,d}\in\Omega^{*}$ such that
$d^{\prime}=f^{*}(r_{n,d}\cdot w_{n,d}^{d^{\prime}})$ and
$|w^{d^{\prime}}_{n,d}|\leq 2n$. With this uniform way of refer to the
elements in neighbourhood of a considered element $d$, we can define a
function $\chi_{n,d}$ whose output tells us for each word $w\in\Omega^{*}$
with $|w|\leq 2n$ whether there is an element in the neighbourhood whose word
is $w$, and if so, whether that element belongs to $\Delta^{*}$, and if not,
which concept names it satisfies in $\mathcal{I}^{*}$. The equivalence
relation $\sim_{n}$ then groups together those elements
$d,e\in\Delta^{\mathcal{I}^{*}}\setminus\Delta^{*}$ which have the same
associated word (i.e. $w^{d}_{n,d}=w^{e}_{n,e}$) and same associated function
($\chi_{n,d}=\chi_{n,e}$), plus an additional condition on the length of $d$
and $e$ (namely, $|d|=|e|\textrm{ mod }2|q|+3$).

With the equivalence relation $\sim_{n}$ at hand, it now suffices to merge
elements which are equivalent w.r.t. $\sim_{|q|+1}$. Formally, we consider the
quotient interpretation $\mathcal{J}:=\mathcal{I}^{*}/\sim_{|q|+1}$ whose
domain is $\Delta^{\mathcal{J}}=\\{[e]_{\sim_{|q|+1}}\mid
e\in\Delta^{\mathcal{I}^{*}}\\}$ and whose interpretation function
$\cdot^{\mathcal{J}}$ is as follows:

  * •

$A^{\mathcal{J}}=\\{[e]_{\sim_{|q|+1}}\mid e\in A^{\mathcal{I}^{*}}\\}$

  * •

$R^{\mathcal{J}}=\\{([d]_{\sim_{|q|+1}},[e]_{\sim_{|q|+1}})\mid(d,e)\in
R^{\mathcal{I}^{*}}\\}$

  * •

$a^{\mathcal{J}}=[a^{\mathcal{I}^{*}}]_{\sim_{|q|+1}}$

where $[e]_{\sim_{|q|+1}}$ is the equivalence class of $e$ w.r.t.
$\sim_{|q|+1}$.

It has been shown that $\mathcal{J}$ remains a model of the considered KB and
does not contain any additional query matches. Given that the interpretation
$\mathcal{I}^{*}$ we consider here is built in exactly the same way as the
interleaving $\mathcal{I}^{\prime}$ considered in (?), except that we work
with a larger domain of interest $\Delta^{*}$ (which includes the one used to
build $\mathcal{I}^{\prime}$), exactly the same arguments can be used to show
Lemma 12 below. We point out that the presence of nominals in
$\mathcal{K}^{\dagger}$ is not problematic as all of its individuals (whether
present in the ABox or TBox) are included in $\Delta^{*}$ and are therefore
left untouched by the construction.

######  Lemma 12.

The interpretation $\mathcal{J}$ is a model of $\mathcal{K}^{\dagger}$ such
that $\mathcal{J}\not\models q$.

It remains to show that the quotient operation does not increase the cost.

######  Lemma 13.

The cost of $\mathcal{J}$ is no higher than the cost of $\mathcal{I}^{*}$:
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{J})\leq\mathit{cost}_{\mathcal{K}_{\omega}}({\mathcal{I}}^{*})$.

###### Proof.

First note that since $\mathcal{J}$ is a model of $\mathcal{K}^{\dagger}$, it
satisfies all assertions in $\mathcal{A}^{\dagger}$. It follows that
$vio_{\mathcal{A}}(\mathcal{J})\subseteq
vio_{\mathcal{A}}({\mathcal{I}}^{*})$. The fact that $\mathcal{J}$ is a model
of $\mathcal{T}^{\dagger}$ means that it satisfies every
$\tau\in\mathcal{T}_{\infty}$. Consider now an axiom
$\tau\in\mathcal{T}\setminus\mathcal{T}_{\infty}$, and suppose that $e\in
vio_{\tau}(\mathcal{J})$. Since $\mathcal{J}$ is a model of
$\mathcal{K}^{\dagger}$, we have $e\in A_{\tau}^{\mathcal{J}}$ (recall Lemma
5, item (v)). It follows from the definition of $\mathcal{J}$ that there
exists $e^{*}$ such that $e=[e^{*}]_{\sim_{|q|+1}}$ and $e^{*}\in
A_{\tau}^{\mathcal{I}^{*}}$. However, since $\mathcal{I}^{*}$ is also a model
of $\mathcal{K}^{\dagger}$, we can infer from $e^{*}\in
A_{\tau}^{\mathcal{I}^{*}}$ that $e^{*}\in vio_{\tau}({\mathcal{I}}^{*})$. By
Lemma 11, we know that $vio_{\tau}({\mathcal{I}}^{*})\subseteq\Delta^{*}$. As
$e^{*}\in\Delta^{*}$, we have $[e^{*}]_{\sim_{|q|+1}}=\\{e^{*}\\}$. It follows
that $vio_{\tau}(\mathcal{J})\subseteq\\{\\{e^{*}\\}\mid e^{*}\in
vio_{\tau}({\mathcal{I}}^{*})\\}$, and hence
$|vio_{\tau}(\mathcal{J})|\leq|vio_{\tau}({\mathcal{I}}^{*})|$. Putting
everything together, we can therefore conclude that
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{J})\leq\mathit{cost}_{\mathcal{K}_{\omega}}({\mathcal{I}}^{*})$.
∎

To complete the proof of Proposition 8, we now examine the steps in the
construction in order to place a bound on the size of the interpretation
$\mathcal{J}$. By analyzing the number of equivalence classes, the following
upper bound on $|\Delta^{\mathcal{J}}|$ was shown in (?):

$\displaystyle(2|q|+3)\times|\mathcal{T}|^{|q|+2}\times(|\Delta^{*}|+2^{\mathsf{sig}(\mathcal{T})}+1)^{|\mathcal{T}|^{2|q|+2}}$

It therefore remains to bound the set $|\Delta^{*}|$ (using our definition of
$\Delta^{*}$):

$\displaystyle|\Delta^{*}|$
$\displaystyle\leq|\mathsf{Ind}(\mathcal{K}^{\dagger})|+|\bigcup_{\tau\in\mathcal{T}\setminus\mathcal{T}_{\infty}}vio_{\tau}(\mathcal{I}^{\dagger})|$
$\displaystyle\leq|\mathsf{Ind}(\mathcal{K}^{\dagger})|+\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}^{\dagger})$
$\displaystyle\leq|\mathsf{Ind}(\mathcal{A})|+|\mathsf{Ind}(\mathcal{T})|+\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})$
$\displaystyle\leq|\mathsf{Ind}(\mathcal{A})|+|\mathsf{Ind}(\mathcal{T})|+k$

By substituting the latter quantity for $|\Delta^{*}|$ in the bound above, we
obtain the desired upper bound on $|\Delta^{\mathcal{J}}|$ that is polynomial
w.r.t. $k$ and $|\mathcal{A}|$, assuming $|\mathcal{T}|$ and $|q|$ are treated
as constants.

###  C.3 Lower Bounds

See 14

###### Proof.

We use a reduction from the $\Theta^{p}_{2}$-complete problem of deciding
whether a given vertex belongs to all the independent sets of maximum size of
a graph (?; ?, Lemma 6). Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a
graph and $w\in\mathcal{V}$ a vertex. Define an $\mathcal{EL}_{\bot}$ WKB
$\mathcal{K}_{\omega}=(\langle\mathcal{T},\mathcal{A}\rangle,\omega)$ as
follows.

$\displaystyle\mathcal{A}$ $\displaystyle=\\{In_{i}(v)\mid
v\in\mathcal{V},i\in\\{1,2\\}\\}$
$\displaystyle\cup\\{Edge(x,y)\mid(x,y)\in\mathcal{E}\\}$
$\displaystyle\cup\\{Distinguish(w)\\}$ $\displaystyle\cup\\{NoGoal(w)\\}$
$\displaystyle\mathcal{T}$ $\displaystyle=\\{In_{i}\sqcap\exists
Edge.In_{i}\sqsubseteq\bot\mid i\in\\{1,2\\}\\}$
$\displaystyle\cup\\{In_{i}\sqcap Distinguish\sqsubseteq Goal\mid
i\in\\{1,2\\}\\}$ $\displaystyle\cup\\{Goal\sqcap NoGoal\sqsubseteq\bot\\}$
$\displaystyle\omega$ $\displaystyle(\tau)=\infty\text{ for
}\tau\in\mathcal{T}$ $\displaystyle\omega$ $\displaystyle(In_{i}(v))=1\text{
for }v\in V,i\in\\{1,2\\}$ $\displaystyle\omega$
$\displaystyle(Edge(x,y))=\infty\text{ for }(x,y)\in\mathcal{E}$
$\displaystyle\omega$ $\displaystyle(Distinguish(w))=\infty$
$\displaystyle\omega$ $\displaystyle(NoGoal(w))=1$

We show that $w$ does not belong to all the independent sets of maximum size
of $\mathcal{G}$ iff $\mathcal{K}_{\omega}\models_{p}^{opt}NoGoal(w)$.

For every independent set $I\subseteq\mathcal{V}$ of $\mathcal{G}$, define the
interpretation $\mathcal{I}_{I}=(\mathcal{V},\cdot^{\mathcal{I}_{I}})$ by

  * •

$In_{i}^{\mathcal{I}_{I}}=I$, $i\in\\{1,2\\}$

  * •

$Edge^{\mathcal{I}_{I}}=\mathcal{E}$

  * •

$Distinguish^{\mathcal{I}_{I}}=\\{w\\}$

  * •

$Goal^{\mathcal{I}_{I}}=\\{w\\}$ if $w\in I$ and
$Goal^{\mathcal{I}_{I}}=\emptyset$ otherwise

  * •

$NoGoal^{\mathcal{I}_{I}}=\emptyset$ if $w\in I$ and
$NoGoal^{\mathcal{I}_{I}}=\\{w\\}$ otherwise

It is easy to check that
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}_{I})=2|\mathcal{V}\setminus
I|+\delta_{w}^{I}$ where $\delta_{w}^{I}=1$ if $w\in I$ and $\delta_{w}^{I}=0$
otherwise.

Moreover, for every interpretation $\mathcal{I}$ of finite cost, note that for
$i\in\\{1,2\\}$, $In_{i}^{\mathcal{I}}\cap\mathcal{V}$ is an independent set
of $\mathcal{G}$ (since $\mathcal{I}\models In_{i}\sqcap\exists
Edge.In_{i}\sqsubseteq\bot$).

($\Rightarrow$) Suppose there is an independent set $M$ of $\mathcal{G}$ of
maximal size that does not contain $w$. By construction,
$\mathcal{I}_{M}\models NoGoal(w)$. We show that $\mathcal{I}_{M}$ has minimal
cost. Suppose for a contradiction that there exists $\mathcal{I}$ such that
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})<\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}_{M})$.
Then $|\mathcal{V}\setminus In_{1}^{\mathcal{I}}|+|\mathcal{V}\setminus
In_{2}^{\mathcal{I}}|\leq\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})<2|\mathcal{V}\setminus
M|$ (since $\delta_{w}^{M}=0$). We then have that
$|In_{1}^{\mathcal{I}}\cap\mathcal{V}|>|M|$ or
$|In_{2}^{\mathcal{I}}\cap\mathcal{V}|>|M|$ which contradicts the fact that
$M$ is of maximum size. Therefore $\mathcal{I}_{M}$ has optimal cost and
$\mathcal{K}_{\omega}\models_{p}^{opt}NoGoal(w)$.

($\Leftarrow$) Suppose that $\mathcal{K}_{\omega}\models_{p}^{opt}NoGoal(w)$,
i.e., there is $\mathcal{I}$ with optimal cost such that $\mathcal{I}\models
NoGoal(w)$. Since $\mathcal{I}$ has finite cost,
$In_{1}^{\mathcal{I}}\cap\mathcal{V}$ and
$In_{2}^{\mathcal{I}}\cap\mathcal{V}$ are independent sets of $\mathcal{G}$.
Suppose for a contradiction that there is an independent set $M$ of
$\mathcal{G}$ such that
$|M|>\max(|In_{1}^{\mathcal{I}}\cap\mathcal{V}|,|In_{2}^{\mathcal{I}}\cap\mathcal{V}|)$.
Then $\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}_{M})\leq
2|\mathcal{V}\setminus M|+1<|\mathcal{V}\setminus
In_{1}^{\mathcal{I}}|+|\mathcal{V}\setminus In_{2}^{\mathcal{I}}|$ i.e.
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}_{M})<\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})$
(since $\mathcal{I}\models NoGoal(w)$), which contradicts our assumption.
Hence $In_{1}^{\mathcal{I}}\cap\mathcal{V}$ or
$In_{2}^{\mathcal{I}}\cap\mathcal{V}$ is an independent set of $\mathcal{G}$
of maximum size that does not contain $w$.

We may therefore conclude that IQA${}^{opt}_{p}$ for $\mathcal{EL}_{\bot}$ is
$\Theta^{p}_{2}$-hard in data complexity (even with weights bounded
independently from $|\mathcal{A}|$ and encoded in unary).

We now consider IQA${}^{opt}_{c}$ and show that $w$ belongs to all the
independent sets of maximum size of $\mathcal{G}$ iff
$\mathcal{K}_{\omega}\models_{c}^{opt}Goal(w)$.

($\Rightarrow$) Suppose that $w$ belongs to all independent sets of
$\mathcal{G}$ of maximum size. Let $\mathcal{I}$ be an interpretation with
optimal cost w.r.t. $\mathcal{K}_{\omega}$. Since $\mathcal{I}$ has finite
cost, then for $i\in\\{1,2\\}$, $In_{i}^{\mathcal{I}}\cap\mathcal{V}$ is an
independent set of $\mathcal{G}$. Assume for a contradiction that neither
$In_{1}^{\mathcal{I}}\cap\mathcal{V}$ nor
$In_{2}^{\mathcal{I}}\cap\mathcal{V}$ is of maximal size and let
$M\subseteq\mathcal{V}$ be an independent set of $\mathcal{G}$ of maximum
size. Then $w\in M$ and
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}_{M})=2|\mathcal{V}\setminus
M|+1<\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})$, contradicting the
optimal cost of $\mathcal{I}$. Therefore there is $i\in\\{1,2\\}$ such that
$In_{i}^{\mathcal{I}}\cap\mathcal{V}$ is an independent set of maximum size,
which implies that $w\in In_{i}^{\mathcal{I}}$. Finally, as
$\mathcal{I}\models In_{i}\sqcap Distinguish\sqsubseteq Goal$ and
$\mathcal{I}\models Distinguish(w)$, we have that $\mathcal{I}\models
Goal(w)$. We have thus shown that
$\mathcal{K}_{\omega}\models_{c}^{opt}Goal(w)$.

($\Leftarrow$) Let $\mathcal{K}_{\omega}\models_{c}^{opt}Goal(w)$, and suppose
for a contradiction that there exists an independent set
$M\subseteq\mathcal{V}$ of $\mathcal{G}$ of maximal size such that $w\not\in
M$. We show that $\mathcal{I}_{M}$ has optimal cost. Indeed, suppose
$\mathit{optc}(\mathcal{K}_{\omega})<\mathit{cost}_{\mathcal{K}}(\mathcal{I}_{M})$
and let $\mathcal{I}$ be an interpretation of optimal cost. As
$\mathcal{K}_{\omega}\models_{c}^{opt}Goal(w)$ we have that $w\in
Goal^{\mathcal{I}}$. Thus there must exist $i\in\\{1,2\\}$ such that $w\in
In_{i}^{\mathcal{I}}$ (otherwise the interpretation $\mathcal{I}^{\prime}$
obtained from $\mathcal{I}$ by removing $w$ from $Goal^{\mathcal{I}}$ and
adding $w$ to $NoGoal^{\mathcal{I}}$ would have a smaller cost than
$\mathcal{I}$). Therefore
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})=|\mathcal{V}\setminus
In_{1}^{\mathcal{I}}|+|\mathcal{V}\setminus
In_{2}^{\mathcal{I}}|+1<\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I}_{M})=2|\mathcal{V}\setminus
M|$ since $\delta_{w}^{M}=0$ as $w\notin M$. Thus $\min(|\mathcal{V}\setminus
In_{1}^{\mathcal{I}}|,|\mathcal{V}\setminus
In_{2}^{\mathcal{I}}|)<|\mathcal{V}\setminus M|$ and we deduce
$|In_{i}^{\mathcal{I}}\cap\mathcal{V}|>|M|$ for some $i\in\\{1,2\\}$ which is
a contradiction since $In_{i}^{\mathcal{I}}\cap\mathcal{V}$ is an independent
set. Hence $\mathcal{I}_{M}$ is an interpretation of optimal cost w.r.t
$\mathcal{K}_{\omega}$ and $w\in Goal^{\mathcal{I}_{M}}$. It follows that
$w\in In_{i}^{\mathcal{I}_{M}}$ for $i\in\\{1,2\\}$ (with the same argument as
before), i.e., $w\in M$. Thus $w$ belongs to all independent sets of
$\mathcal{G}$ of maximum size.

We have thus shown that IQA${}^{opt}_{c}$ for $\mathcal{EL}_{\bot}$ is
$\Theta^{p}_{2}$-hard in data complexity (even with weights are bounded
independently from $|\mathcal{A}|$ and encoded in unary). ∎

See 15

###### Proof.

The proof is by reduction from the following $\Delta^{P}_{2}$-hard problem:
given a satisfiable boolean formula $\varphi$ over variables
$x_{1},\dots,x_{n}$ such that $\varphi$ is a set of clauses
$c_{1},\dots,c_{m}$ where each clause has exactly two positive and two
negative literals and any of the four positions in a clause can be filled
instead by one of the truth constants true and false, and given
$k\in\\{1,\dots,n\\}$, decide whether the lexicographically maximum truth
assignment $\nu_{max}$ satisfying $\varphi$ with respect to
$(x_{1},\dots,x_{n})$ fulfills $\nu_{max}(x_{k})=1$. It follows from (?) and
from the reductions from SAT to 3SAT and from 3SAT to 2+2SAT (?) that this
problem is $\Delta^{P}_{2}$-hard.

Let $\varphi$ be an instance of the problem as previously defined. We define
$\mathcal{K}_{\omega}=(\langle\mathcal{T},\mathcal{A}\rangle,\omega)$ an
$\mathcal{EL}_{\bot}$WKB as follows:

$\displaystyle\mathcal{A}=$ $\displaystyle\\{S(\varphi,c_{j})\mid 1\leq j\leq
m\\}\cup$
$\displaystyle\\{P_{\ell}(c_{k},x_{j})\mid\ell\in\\{1,2\\},x_{j}\text{ is the
}\ell^{th}\text{ pos. lit. of }c_{k}\\}\cup$
$\displaystyle\\{N_{\ell}(c_{k},x_{j})\mid\ell\in\\{1,2\\},\neg x_{j}\text{ is
the }\ell^{th}\text{ neg. lit. of }c_{k}\\}\cup$
$\displaystyle\\{F(x_{i}),T(x_{i}),T^{\prime}(x_{i})\mid 1\leq i\leq
n\\}\cup\\{F(f),T(t)\\}$ $\displaystyle\mathcal{T}=$ $\displaystyle\\{F\sqcap
T\sqsubseteq\bot,F\sqcap T^{\prime}\sqsubseteq\bot\\}\cup$
$\displaystyle\\{\exists S.(\exists P_{1}.F\sqcap\exists P_{2}.F\sqcap\exists
N_{1}.T\sqcap\exists N_{2}.T)\sqsubseteq\bot\\}$

We set $\omega(\tau)=\infty$ for every $\tau\in\mathcal{T}$, and
$\omega(\alpha)=\infty$ for every assertion $\alpha$ built on $P_{\ell}$ and
$N_{\ell}$ as well as for $T(t)$ and $F(f)$. For the remaining assertions, we
define the weights through the following prioritization (see Lemma 6.2.5 in
(?)):

  * •

$L_{1}=\\{T(x_{i}),F(x_{i})\mid 1\leq i\leq n\\}$

  * •

$L_{2}=\\{S(\varphi,c_{j})\mid 1\leq j\leq m\\}$

  * •

$L_{3}=\\{T^{\prime}(x_{1})\\}$, $L_{4}=\\{T^{\prime}(x_{2})\\}$, $\dots$,
$L_{n+2}=\\{T^{\prime}(x_{n})\\}$

Let $u=max(2n,m)+1$ and define $\omega(\alpha)=u^{n+2-i}$ for every $\alpha\in
L_{i}$.

We show that $\mathcal{K}_{\omega}\models_{c}^{opt}T^{\prime}(x_{k})$ iff
$\mathcal{K}_{\omega}\models_{p}^{opt}T^{\prime}(x_{k})$ iff
$\nu_{max}(x_{k})=1$.

Let $\mathcal{I}$ be an interpretation with optimal cost. By definition of
$\omega$, $\mathcal{I}\models\mathcal{T}$ and $\mathcal{I}$ satisfies all
$P_{\ell}$/$N_{\ell}$ assertions, $T(t)$ and $F(f)$. First note that for every
$1\leq i\leq n$, $\mathcal{I}$ satisfies exactly one of the assertions
$T(x_{i}),F(x_{i})$. Indeed, it cannot contain both since $\mathcal{I}\models
F\sqcap T\sqsubseteq\bot$ and if there was $i$ such that
$\mathcal{I}\not\models T(x_{i})$ and $\mathcal{I}\not\models F(x_{i})$, then
$\mathcal{J}$ obtained from $\mathcal{I}$ by setting
$T^{\mathcal{J}}=T^{\mathcal{I}}\cup\\{x_{i}^{\mathcal{I}}\\}$,
$S^{\mathcal{J}}=\emptyset$ and $X^{\mathcal{J}}=X^{\mathcal{I}}$ for every
other concept or role name $X$ would be such that
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{J})<\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})$
since $m\times u^{n}<u^{n+1}$ because $m<u$. Moreover, it is easy to see that
$\mathcal{I}\models T^{\prime}(x_{i})$ exactly when $\mathcal{I}\not\models
F(x_{i})$, i.e. when $\mathcal{I}\models T(x_{i})$.

Let $\nu$ be the valuation of $x_{1},\dots,x_{n}$ such that $\nu(x_{i})=1$ iff
$\mathcal{I}\models T(x_{i})$ (iff $\mathcal{I}\models T^{\prime}(x_{i})$ iff
$\mathcal{I}\not\models F(x_{i})$). Assume for a contradiction that $\nu$ does
not satisfy $\varphi$. There exists a clause $c_{j}$ of $\varphi$ such that
for every positive literal $x_{i}$ of $c_{j}$, $\nu(x_{i})=0$, i.e.
$\mathcal{I}\models F(x_{i})$, and for every negative literal $\neg x_{i}$ of
$c_{j}$, $\nu(x_{i})=1$, i.e. $\mathcal{I}\models T(x_{i})$. It follows that
$\mathcal{I}\not\models S(\varphi,c_{j})$ (otherwise $\varphi^{\mathcal{I}}$
would be in $(\exists S.(\exists P_{1}.F\sqcap\exists P_{2}.F\sqcap\exists
N_{1}.T\sqcap\exists N_{2}.T))^{\mathcal{I}}$ and the cost of $\mathcal{I}$
would be infinite. Since $\varphi$ is satisfiable, there exists $\nu^{\prime}$
that satisfies $\varphi$. Let $\mathcal{J}$ be the interpretation such that
$P_{\ell}^{\mathcal{J}}=P_{\ell}^{\mathcal{I}}$ and
$N_{\ell}^{\mathcal{J}}=N_{\ell}^{\mathcal{I}}$ for $1\leq\ell\leq 2$,
$T^{\mathcal{J}}=\\{t^{\mathcal{I}}\\}\cup\\{x_{i}^{\mathcal{I}}\mid\nu^{\prime}(x_{i})=1\\}$,
$F^{\mathcal{J}}=\\{f^{\mathcal{I}}\\}\cup\\{x_{i}^{\mathcal{I}}\mid\nu^{\prime}(x_{i})=0\\}$,
and $S^{\mathcal{J}}=\\{(\varphi,c_{j})\mid 1\leq j\leq m\\}$. Then
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{J})<\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})$.
Indeed, $\mathcal{I}$ and $\mathcal{J}$ both violate the same number of
$T$/$F$-assertions and the cost $u^{n}$ of violating the single assertion
$S(\varphi,c_{j})$ is greater than the cost of violating all
$T^{\prime}$-assertions
$\Sigma_{i=3}^{n+2}u^{n+2-i}=\Sigma_{i=0}^{n-1}u^{i}=\frac{u^{n}-1}{u-1}$.
Hence, $\nu$ satisfies $\varphi$.

Finally, we show that $\nu=\nu_{max}$. Define $\mathcal{J}$ by
$P_{\ell}^{\mathcal{J}}=P_{\ell}^{\mathcal{I}}$ and
$N_{\ell}^{\mathcal{J}}=N_{\ell}^{\mathcal{I}}$ for $1\leq\ell\leq 2$,
$T^{\mathcal{J}}={T^{\prime}}^{\mathcal{J}}=\\{t^{\mathcal{I}}\\}\cup\\{x_{i}^{\mathcal{I}}\mid\nu_{max}(x_{i})=1\\}$,
$F^{\mathcal{J}}=\\{f^{\mathcal{I}}\\}\cup\\{x_{i}^{\mathcal{I}}\mid\nu_{max}(x_{i})=0\\}$,
$S^{\mathcal{J}}=\\{(\varphi,c_{j})\mid 1\leq j\leq m\\}$. If there exists $l$
such that $\nu$ and $\nu_{max}$ coincide on $x_{i}$ for $i<l$ but not on
$x_{l}$, i.e. $\nu(x_{l})=0$ and $\nu_{max}(x_{l})=1$ (since $\nu_{max}$ is
the lexicographically maximum truth assignment satisfying $\varphi$), then
$\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{J})<\mathit{cost}_{\mathcal{K}_{\omega}}(\mathcal{I})$.
Indeed, the cost $u^{n-l}$ of violating $T^{\prime}(x_{l})$ is greater than
the cost
$\Sigma_{i=l+3}^{n+2}u^{n+2-i}=\Sigma_{i=0}^{n-l-1}u^{i}=\frac{u^{n-l}-1}{u-1}$
of violating all $T^{\prime}(x_{i})$ for $i>l$. Hence $\nu$ and $\nu_{max}$
coincide on every $x_{i}$, $1\leq i\leq n$.

We have shown that for every interpretation with optimal cost $\mathcal{I}$,
$\mathcal{I}\models T^{\prime}(x_{k})$ iff $\nu_{max}(x_{k})=1$. Hence
$\mathcal{K}_{\omega}\models_{c}^{opt}T^{\prime}(x_{k})$ iff
$\mathcal{K}_{\omega}\models_{p}^{opt}T^{\prime}(x_{k})$ iff
$\nu_{max}(x_{k})=1$. ∎