# Approximation of Banzhaf indices and its application to voting games

Acharya  V. Krishna<EMAIL_ADDRESS>Mukherjee  Himadri
<EMAIL_ADDRESS>Sahoo  K. Jajati<EMAIL_ADDRESS>
###### Abstract

In this paper, we propose an improved version of the power index related to
the Banzhaf power index for weighted voting systems. This index now takes into
account the mutual persuasion power matrix(PPM) existing among the voters.
This improved index is calculated for European Union voting by basing the PPM
on immigration data among the EU countries. We also provide better
approximation bounds for the Monte Carlo approximation method for computing
power indices.
keywords: rem.
2010 MSC classification: 91A80. 62P20. 97M40. 91B82. 91A40

## 1 Introduction

In the year 1963 the supreme court of the United States gave the historical
order of one person one vote, several scientists justified the decision using
mathematical analysis of the weighted voting system. The power index for
weighted voting systems developed by Banzhaf in 1965, in the article (Banzhaf
1965), is one of the pioneers in this field. A good introduction of the theory
is the original paper by Banzhaf [8], a relatively modern approach can be
found in [15, 16, 32]. Probabilistic models that construct the index as
probabilities can be found in [33, 35]. There is also an axiomatic approach
that constructs the indices through an axiomatization that shows the
uniqueness of such indices when a set of axioms are satisfied. There are
numerous approaches to streamline the axiomatization as in [15, 18, 20]. Let
$p_{1},p_{2},p_{3},\ldots,p_{m}$ be a number of players with weights
$w_{1},w_{2},w_{3},\ldots,w_{m}$. A coalition C, subset of the set of players,
is called a wining coalition with respect to a preassigned quota $q$ if
$\displaystyle{\sum_{p_{i}\in C}w_{i}\geqslant q}$. Let the set of wining
coalitions be $W_{q}$ or simply $W$ if the quota is clear form the context.

######  Definition 1.1.

A player $p_{i}$ is called critical or swinger for a coalition $C$ if the
following two conditions are met:

$C\in W$ $C-\\{p_{i}\\}\notin W$

Let us call $(p_{i},C)$ a critical pair when $p_{i}\in C$ is critical. Let
$\mathfrak{C}_{i}$ be the set of all pairs where $p_{i}$ is critical. The
Banzhaf power index for the player $p_{i}$ is denoted by $\beta_{i}$ (or
$\tilde{\beta_{i}}$).

$\mathfrak{C}_{i}=\\{(p_{i},C)|p_{i}\in C\mbox{ is critical}\\}$
$\gamma_{i}=\left|{\mathfrak{C}_{i}}\right|$ ${\beta_{i}}=\gamma_{i}/2^{m-1}$
$\tilde{\beta_{i}}=\gamma_{i}/\sum_{j=1}^{m}\gamma_{j}$

$\tilde{\beta_{i}}$, is known as the normalized Banzhaf index. $\beta_{i}$,
the absolute Banzhaf index will, however, be our main focus for this article.

The index, though coming to life as a refutal of weighted voting systems, sees
a lot of applications in real life. The applications are numerous in social
science [29, 14]. In Electoral College voting - members of the parliament
possess weighted votes in proportion to the number of constituencies they
represent. In intergovernmental organization of nations such as the EU the
member countries get their voting weights according to a number of parameters
such as population etc [25, 19]. In the context of EU it is a great debate
among the researchers if the voting power game used to determine the voting
weights is fair [25] and often contains warning against such use [19]. In this
paper, we will try to explain yet again why the voting power method used in
the EU Parliament may not be the right approach to find the voting rights, now
from the perspective of the association between the players. It is evident
that only the absolute quantity of the weights measured in economic terms,
population, area fail to capture the true ‘power of a member’ since such power
should also include the diplomatic prowess of a nation. In the industry,
search engines such as Google use page rank algorithms applying techniques of
weighted voting games among others. From a purely theoretical point of view
the index has generated interests in the line of inverse problems [2] which is
to construct the weights of the players from the indices, or relate the
indices with convex geometry [9] or mathematical extensions [30]. The Banzhaf
index has been a source of great activities recently. The recent study
includes an attack on the computational challenges on the NP complete[31]
problem. There have been multiple algorithms that tackle the computational
difficulties of the index[28, 13] or find theoretical bounds [22]. A good
place to look at different approaches to approximating the index is the survey
[27]. The Monte Carlo method by the [4, 3] uses Hoeffding’s inequality to give
a very strong approximation within a large confidence interval. In [17] the
authors consider heuristics based on density functions. In [26] the authors
employ generating functions.

In this paper, we introduce an improved version of the power index that takes
into account situations where not only the individual weights of the players
are considered but the associations between players as well. To explain the
situation let us consider a set of politicians discussing a motion,
irrespective of their partisan interests they can vote or abstain according to
their personal preferences and equations. One obvious example is the Indian
upper house (Rajya Sabha) debates where the members are frequently persuaded
to vote based on issues related to the constituency they represent not
completely complying with the partisan goals. Thus we can often form a matrix
of associations between the players that does not necessarily represent the
ideology based political decisions. In other fields such as geopolitics, we
often see the actual strength of a nation is not merely the GDP or the might
of its military or population but also the amount of influence that it can
exert on the other nations by several means of soft power. In this paper, we
quantify the soft power of a nation based on the association matrix and the
improved power index developed in this model. To be precise let us say the
players $\\{p_{1},p_{2},p_{3},\ldots,p_{m}\\}$ are engaged in a voting game.
And $a_{ij}\in\mathbb{R}$ are numbers that quantify strength of the
association between the players, this represents the persuasion power of the
player $p_{i}$ has over $p_{j}$. We postulate that the strength of a coalition
is not only the sum of the weights of the players but also the amount of total
persuasive power that it holds to sway the motion in its favour by converting
other players. Thus the definition of a swinger is changed so that the
diplomatic loss of a player leaving the coalition is taken into account. In
the real life, the matrix $A=(a_{ij})$ will almost surely be not symmetric, as
the ’associations’ are not symmetric. Ideas to restrict coalitions or
equivalently the critical pairs can be observed in [9] where the authors
restrict the allowed coalitions to those coalitions forming a convex geometry,
this is claimed to represent the EU scenario better. In our scenario, we will
restrict the critical pairs in a uniform way to get control over the critical
pairs.

We see a lot of applications in the fields of social sciences and computer
science arising out of the improved Banzhaf power index, we list a few such
directions at the end of this article.

## 2 Mathematical Model and Methodology

In the recent studies of power indices in voting games, We see a systematic
study to improve the applicability of the index to the practice of coalition
formation (Burgin, 2001) as well as studies to improve the computational
aspects of the power indices [8, 7, 34, 27]. It has not come to our notice any
attempt to propose a model where a voter’s individual relation (association)
with another voter influences the outcome of the coalition formation. For an
example consider a game with six players $a_{1},a_{2},\cdots,a_{6}$ wherein
$a_{1}$ has very strong positive association with the other players, then any
coalition containing $a_{1}$ will have negative impact on the other coalition
effectively reducing the total weight of the opponent coalition and increasing
the weight of coalition containing $a_{1}$. Such scenarios are clearly visible
in Parliamentary and legislative assembly election, for instance, members of
parliament have strong positive associations with other members having similar
political motives. An attempt using the classical power indices or current
variations of them will not properly predict the actual strength of a
coalition in such realistic scenarios.

### 2.1 Mathematical Formulation

Let $\\{p_{1},p_{2},\cdots,p_{m}\\}$ be a set of players, with weights
$w_{1},w_{2},\cdots,w_{m}$ respectively. Consider association between $p_{i}$
and $p_{j}$ as $a_{ij}\in\mathbb{R}$, with $|a_{ij}|\leq 1$ and $a_{ii}=1$ for
all $i.$ The matrix $\Phi=(a_{ij})$ is called the association matrix for the
players and the goal is to propose a power index $\beta_{\Phi}(p_{i})$ which
takes the association matrix into account. It is interesting to note that if
all off-diagonal elements in $\Phi$ are zero i.e $a_{ij}=0,~{}\forall~{}i\neq
j$, then calculations for this formulation reduces to a $k-$quota banzhaf
index as studied in [17]

#### 2.1.1 Single quota voting game

Consider a voting game with association matrix $\Phi$ and a quota $q>0$. Let S
be the set of all possible coalitions. $C\subseteq S$ is called a winning
coalition if

$\sum_{p_{i}\in C}w_{i}\geq q.$ (1)

Let $W_{q}=\\{C\subseteq S~{}:~{}C~{}\mbox{ is wining}\\}$ be the set of
winning coalition. A tuple $(p_{i},C)$ is called a critical pair if

$\mbox{(i) }p_{i}\in C\mbox{ and }C\in W_{q}~{}~{}~{}\mbox{
(ii)}\sum_{p_{j}\in C}w_{j}-\sum_{k=1}^{m}a_{ik}w_{k}<q.$ (2)

The improved Banzhaf power index for the player $p_{i}$ is defined as:

$\beta_{\Phi,i}=\frac{|\gamma_{i}|}{2^{m-1}}$ (3)

where $\gamma_{i}=\\{(p_{i},C)~{}|~{}(p_{i},C)~{}\mbox{is critical}\\}$

One can extend the definition to multi-weight and multi-quota voting systems
in the following manner. We call a vector $v=(v_{1},v_{2},\cdots,v_{k})$ non
negative if $v_{i}\geq 0$ for all $i$. For any $u,v\in\mathbb{R}^{k}$, $u\geq
v$ if $u-v$ is non-negative. In $k-$dimensional voting game, the weight for
player $p_{i}$ and the quota required for winning coalitions are defined as:

$\vec{w_{i}}=(w_{1},w_{2},\cdots,w_{k}),~{}\vec{q}=(q_{1},q_{2},\cdots,q_{k})$
(4)

If we follow equation 2 and vector comparison we can get critical pairs for
$k-$dimensional voting games. The improved Banzhaf index for multidimensional
voting game is again defined by equation 3. Multidimensional voting games are
used in many real life problems such as the EU election [1].

### 2.2 Calculation of the Banzhaf index and its Approximation

Algorithm 1 $\beta_{\Phi,i}$ for a k-quota game with an association matrix
$\Phi$ of dimension m$\times$ m

procedure Exact-Banzhaf-Asso($\Phi,\vec{w},\vec{q}$)

$\gamma_{i}$ = 0

for all coalitions C containing $p_{i}$ do

if $(p_{i},C)$ is a critical pair as defined in the extension 4

then $\gamma_{i}=\gamma_{i}+1$

end if

end for

$\beta_{\Phi,i}=\frac{\gamma_{i}}{2^{m-1}}$

return $\beta_{\Phi,i}$

Exact calculation of the Banzhaf index for a general case is NP-complete as
shown in [31], so we need to find an alternative to Algorithm 1.

In [4], the authors have used a Monte Carlo method to approximate the power
indices for the players. We extend this idea for multi-quota games with
associations. The approximate banzhaf index for player $p_{i}$,
$\hat{\beta}_{\Phi,i}$ will be estimated by sampling random coalitions
containing player $p_{i}$, then calculating the proportion of these samples
where $p_{i}$ is a critical. We assume that each sample has a probability
$\beta_{\Phi,i}$ of being a coalition where player $p_{i}$ is critical, so we
can approximate $\beta_{\Phi,i}$ by taking into consideration several such
samples. More precisely the estimation procedure is as follows:
Let $C_{1},C_{2},\cdots,C_{n}$ be $n$ randomly sampled coalitions containing
player $p_{i}.$ For $1\leq k\leq n,$ define a Bernoulli random variable

$X_{k}=\left\\{\begin{array}[]{cc}1&~{}~{}\mbox{ If player $p_{i}$ is critical
in $C_{k}.$}\\\ 0&~{}~{}\mbox{otherwise.}\end{array}\right.$

Now, we estimate the index $\beta_{\Phi,i}$ for $p_{i}$ by using the following
estimator

$\hat{\beta}_{\Phi,i}=\frac{\sum_{k=1}^{n}X_{k}}{n}$ (5)

Algorithm 2 is used to compute the improved Banzhaf indices. The number of
samples required, $n$, error bounds and convergence analysis is discussed in
Section 4.

Algorithm 2 $\hat{\beta_{\Phi,i}}$ for a $k-$quota game with an association
matrix $\Phi$

procedure Approx-Banzhaf-Asso($\Phi,\vec{w},\vec{q},n$)

$\gamma_{i}$ = 0, j = 0

while $j\leq n$ do

Choose a coalition C at random containing $p_{i}$

if $(p_{i},C)$ is a critical pair as defined in equation 2

then $\gamma_{i}$ =$\gamma_{i}$ \+ 1

end-if

j = j + 1

end while

$\hat{\beta_{\Phi,i}}$ = $\frac{\gamma_{i}}{j}$

return $\hat{\beta}_{\Phi,i}$

## 3 Some theoretical results

Given a player $p_{i}$ let us define two functions $h(p_{i})$ or simply
$h(i)=\min\\{h\,|\,w_{1}+w_{2}+\cdots+w_{h}>q,\,\forall i,\,w_{i}\neq w\\}$
and $t(i)=\max\\{t\,|\,w_{1}+w_{2}+w_{3}+\cdots+w_{t}<q\,\exists
i,\,w_{i}=w\\}$. Note that in the definitions of $h(i),t(i)$ we will consider
the weights of distinct players as distinct $w_{i}$s even if they are same!
With this notation in place we have the following estimates for the Banzhaf
index of a player $p_{i}$.

######  Proposition 3.1.

$\beta_{i}\leq\,\frac{1}{2^{n}}(2^{n}-2^{t(i)}-2^{n-h(i)})$

###### Proof.

The set of coalitions that are subsets of the set of players
$w_{1},w_{2},\ldots,w_{t(w)}$ such that
$w_{1}+w_{2}+w_{3}+\cdots+w_{t}<q\,\exists\;i,\,w_{i}=w$ is not at all winning
coalitions so any such coalitions containing the player $i$ is not a critical
coalition for the player. Similarly in the other side any superset containing
the player $i$, of the set of players with weights
$w_{1},w_{2},\ldots,w_{h(i)}$ such that $w_{1}+w_{2}+\cdots+w_{h}>q,\,\forall
i,\,w_{i}\neq w$ is not a critical coalition for the player as the coalition
is winning without the player $i$. Now $t$ being maximum and $h$ being the
minimum we have the result. ∎

In the above proposition the greater the variation in the weights of the
players better the estimate from the above inequality. In particular if there
is a dominating weight or a very large quota or equivalently a very tiny quota
the estimate provided by the above becomes close. Other than that it is of
theoretical interest as the estimate provided by the above is not practical.
It will be interesting to study in detail such estimates.

######  Lemma 3.2.

Let $C$ be a coalition, $w(C)$ be the total weight of $C$ and $q$ the quota.
If every player in $C$ is critical then $w(C)<\frac{|C|q}{|C|-1}$.

###### Proof.

Let $C$ be a coalition with $w_{i}\in C$, since $w_{i}$ is a critical player
for the coalition we have the inequality $w(C)-w_{i}<q$ or

$w(C)<q+w_{i}$

If we sum both sides of the inequality we get

$w(C)|C|<q|C|+w(C)$

Or

$w(C)(|C|-1)<q|C|$ $w(C)<\frac{q|C|}{|C|-1}$

∎

Let us define given an association matrix $A$, and a player $p_{i}$ with
weight $w_{i}$ we define $\sum_{j}a_{ij}w_{j}=d_{i}+a_{ii}w_{i}=d_{i}+w_{i}$;
this number in a sense represents the overall relation of the player $p_{i}$
with the other players in the set of players. When the numbers $d_{i}$ are all
positive for all $i$ we call such game an overall positive association game.
Note that for an association voting game if the association matrix is positive
then the game is also overall positive relation game. In the following
proposition we will relate the Banzhaf index of an overall positive voting
game with the Banzhaf index of the related game without association. If
$\beta_{\phi,i}$ denote the Banzhaf index of the player $p_{i}$ with the
association and let $\beta_{i}$ be the index for the same player without the
association. Let us also define $w_{i}+q=q_{i}$ and let the total weight of a
coalition $C$ be denoted by $w(C)$.

######  Proposition 3.3.

With the notation as in the above we have the following statements relating
the actual Banzhaf index with the association based Banzhaf index.

  1. 1.

$\beta_{\phi,i}\leq\beta_{i}$ for all $i$ such that $d_{i}\geq 0$

  2. 2.

$\beta_{\phi,i}\geq\beta_{i}$ for all $i$ such that $d_{i}\leq 0$

  3. 3.

$\beta_{\phi,i}-\beta_{i}$ is precisely the number of coalitions $C$ such that
$d_{i}+q_{i}<w(C)<q_{i}$ minus the number of coalitions such that
$q_{i}<w(C)<d_{i}+q_{i}$

###### Proof.

Note that the first and the second statements follow from the third so we will
prove the third statement. Let us define $v(C)$ for a coalition to be 1 if
$w(C)>q$ and 0 otherwise and let us define ${v^{\prime}}_{i}(C)$ to be $1$ if
$w(C)-d_{i}-w_{i}>q$. With these functions defined it is easy to observe that
$\beta_{i}=\frac{1}{2^{n-1}}\displaystyle{\sum v(C)-v(C\setminus i)}$ and
$\beta_{\Phi,i}=\frac{1}{2^{n-1}}\displaystyle{\sum v(C)-v^{\prime}(C)}$ where
the sums are running over all possible coalitions.

$\beta_{\Phi,i}-\beta_{i}=\frac{1}{2^{n-1}}\Big{(}\displaystyle{\sum
v(C)-{v^{\prime}}_{i}(C)}-\displaystyle{\sum v(C)-v(C\setminus i)}\Big{)}$

Or

$\beta_{\Phi,i}-\beta_{i}=\frac{1}{2^{n-1}}\Big{(}\displaystyle{\sum
v(C\setminus i)-{v^{\prime}}_{i}(C)}\Big{)}$

Let us analyse $v(C\setminus i)-{v^{\prime}}_{i}(C)$ depending on the various
possibilities in the following way. Note that if a coalition satisfies
$w(C)>\max\\{q_{i}+d_{i},q_{i}\\}$, or $w(C)<\min\\{q_{i}+d_{i},q_{i}\\}$ then
both $v(C\setminus i)$ and ${v^{\prime}}_{i}(C)$ are equal to 1, rest 0 making
the difference 0, similarly if $C$ satisfies $q_{i}+d_{i}<w(C)<q_{i}$ then the
difference is 1, and if $q_{i}<w(C)<q_{i}+d_{i}$ then the difference is -1.
This immediately gives the result. ∎

In the following proposition we will find a few general bounds for the power
indices for a general voting game. Let us define for a set of players with
weights $W=\\{w_{1},w_{2},w_{3},\ldots w_{n}\\}$ and a quota $q$ two numbers
$m,M$ in the folllowing way. Let $m$ be the smallest natural number such that
$m\max{W}<q$ and $M\min{W}-max{W}>q$ where $\min{W},\max{W}$ denotes the
minimum and the maximum weights among $W$ respectively.

######  Proposition 3.4.

With the above notations we have the following inequalities for the indices.

  1. 1.

$\beta_{i}\leq\displaystyle{\frac{\displaystyle{\sum_{i=m+1}^{\min\\{M,n\\}}\binom{n}{i}-2^{n-1}}}{2^{n}}}$

  2. 2.

$\beta_{i}\leq\displaystyle\frac{\displaystyle{\sum_{i=m+1}^{\min\\{M,n\\}}i\binom{n}{i}}}{n2^{n}}-\frac{1}{2}$

###### Proof.

Note that due to the definition of the numbers $m,M$ any subset of the set of
players of size less than $m$ cannot be critical for any player since the
total weight does not exceed $q$ and similarly any coalition of size more than
$M$ cannot be critical since removing even the maximum weight player does not
reduce the weight of the coalition below $q$ so the coalitions that are
relevant for the indices are of size between $m+1$ and $\min\\{M,n\\}$. Now
out of these coalitions there are $2^{n-1}$ that do not contain the player
indexed $i$ thus we get the first inequality. For the next inequality notice
if all the players are critical in all possible coalitions we have all indices
equal and maximum possible. In this hypothetical situation we have the total
number of pairs $(p_{i},C)$ where $p_{i}\in C$ critical is the sum
$\displaystyle{\sum_{i=m+1}^{\min\\{M,n\\}}i\binom{n}{i}-n2^{n-1}}$ so now
dividing the number by $n$ and normalizing we get the result. ∎

######  Remark 3.5.

Note that the above proposition 3.4 is a generalization of [15, Dubey ’79] as
the binomial coefficients $\binom{n}{i}$ in the part one of the inequality
above is replaced by the largest binomial coefficient $\binom{n}{m}$ for
$m=\lfloor\frac{n}{2}\rfloor$ and then we sum the index for all $i$.

######  Remark 3.6.

Note that the lemma 3.2 shows that an assumption such as all players are
critical for any coalition is not at all realistic as it puts severe condition
on the size of the coalition. This also further shows there are much more
scope to improve such bounds by scrutinizing the critical coalitions in
detail.

######  Remark 3.7.

For a large set of players with a small variation in the weights the above
bounds in 3.2 give better result in the simulation done in the next section.

In our calculation we have observed that for the relative Banzhaf index
$\beta_{i}$ the following relation is valid for any voting game. We have not
been able to prove the claim yet but we state it as a Conjecture as it is an
interesting and a very narrow bound.

######  Conjecture 3.8.

Let $w_{1},w_{2},w_{3},\ldots w_{m}$ be a set of positive weights, let $w$ be
the maximum of these weights, further let $N=\sum_{i=1}^{m}w_{i}$ then the
relative Banzhaf indices $\beta_{i}\leq\frac{2w}{N}$.

## 4 Error Bounds and Convergence Analysis

It is easy to show that the estimator for the Banzhaf index used in 2.2 is an
unbiased estimator

######  Lemma 4.1.

Let $C_{1},C_{2},\cdots,C_{n}$ be set of randomly sampled coalitions
containing player $p_{i}$ and $X_{1},X_{2},\cdots,X_{n}$ be the random
variables as defined in previous section. Then the estimator
$\hat{\beta}_{\Phi,i}$ is an unbiased estimator for ${\beta}_{\Phi,i}.$

###### Proof.

The random variable $X_{k}$ is a Bernoulli random variable with probability of
success ${\beta}_{\Phi,i}$, so the random variable X = $\sum_{k=1}^{n}X_{k}$
is Binomial with mean $n\beta_{\Phi,i}$. The maximum likelihood estimator for
the parameter ${\beta}_{\Phi,i}$ is $\hat{\beta}_{\Phi,i}$ = $\frac{X}{n}$.
Since $E\left(\frac{X}{n}\right)=\beta_{\Phi,i}$ it implies that the estimator
$\hat{\beta}_{\Phi,i}$ is an unbiased maximum likelihood estimator. ∎

### 4.2 Confidence intervals

Before trying to calculate, the confidence interval by using different
approximation we first state few results which will help to find the
confidence bounds.

######  Definition 4.3.

A function $f:X^{n}\rightarrow\mathbb{R}$ is said to have self-bounding
property if there exist a function $f_{i}:X^{n-1}\rightarrow\mathbb{R}$ such
that for all $x\in X^{n}$ and $1\leq i\leq n$,

  * (a)

$0\leq f(x)-f_{i}(x^{(i)})\leq 1.$

  * (b)

$\sum_{i=1}^{n}\left(f(x)-f_{i}(x^{(i)})\right)\leq f(x),$ where $x^{(i)}$ is
obtained by dropping the $i^{th}$ component of $x$ i.e.,
$x^{(i)}=(x_{1},x_{2}\cdots,x_{i-1},x_{i+1},\cdots,x_{n})\in X^{n-1}.$

Several generalization and weaker version of self- bounding (one can refer
[11, 10]) are defined by the following definitions.

######  Definition 4.4.

A function $f:X^{n}\rightarrow\mathbb{R}$ is called $(a,b)-$ self-bounding if
there exist positive numbers $a$ and $b$ such that for all $x\in X^{n}$ and
$1\leq i\leq n$,

  * (a)

$0\leq f(x)-f_{i}(x^{(i)})\leq 1.$

  * (b)

$\sum_{i=1}^{n}\left(f(x)-f_{i}(x^{(i)})\right)\leq af(x)+b.$

######  Definition 4.5.

A function $f:X^{n}\rightarrow\mathbb{R}$ is called weakly $(a,b)-$ self-
bounding if there exist positive numbers $a$ and $b$ such that for all $x\in
X^{n}$ and $1\leq i\leq n$,

$\sum_{i=1}^{n}\left(f(x)-f_{i}(x^{(i)})\right)^{2}\leq af(x)+b.$

######  Lemma 4.6.

The function $f:\chi^{n}\rightarrow\mathbb{R}$ given by
$f(x_{1},x_{2},\ldots,x_{n})=\frac{1}{n}\sum_{i=1}^{n}x_{i}$ is selfbounding
and $(1/n,0)$ weakly selfbounding.

###### Proof.

Let us take $f_{i}(x^{i})=\frac{1}{n}\sum_{j=1,j\neq i}^{n}x_{j}$ then
$f-f_{i}(x^{i})=\frac{x_{i}}{n}$ thus $0\leq(f-f_{i}(x^{i})\leq 1$ for all
$i$. And further $\sum_{i=1}^{n}(f-f_{i}(x^{i}))=\frac{1}{n}x_{i}=f$ which
proves that the function is selfbounding. And
$\sum_{i=1}^{n}(f-f_{i}(x^{i}))^{2}=\frac{x_{i}^{2}}{n^{2}}=\frac{1}{n^{2}}\sum_{i=1}^{n}x_{i}^{2}\leq\frac{1}{n^{2}}\sum_{i=1}^{n}x_{i}$.
Since
$\sum_{i=1}^{n}(f-f_{i}(x^{i}))^{2}\leq\frac{1}{n^{2}}\sum_{i=1}^{n}x_{i}$ we
get the weakly selfbounding for $a=\frac{1}{n}$ and $b=0$. ∎

######  Remark 4.7.

Note that any positive constant multiple of a selfbounding function ($(a,0)$
weakly selfbounding) is also selfbounding (weakly $(a,0)$ selfbounding ).

To draw best estimates and bounds for any random variable when the variance
information is unknown, Hoeffding’s inequality[21] is most preferable to use.

######  Theorem 4.8 (Hoeffding’s inequality).

Let $X_{1},X_{2},\cdots,X_{n}$ be independent and identically distributed
random variables and $Z=\frac{1}{n}\sum_{i=1}^{n}X_{i}.$ If for each $1\leq
i\leq n,$ $X_{i}$ is bounded by the interval $[a,b],$ then the following
holds.

$\mbox{Prob}\left\\{|Z-E(Z)|<\epsilon\right\\}\geq
1-2\exp\left(-\frac{2n\epsilon^{2}}{(b-a)^{2}}\right).$

Now, we state few basic results in the context of self bounding and weak self
bounding which we will use for finding one-sided confidence intervals for the
indices.

######  Lemma 4.9.

Let $X_{1},X_{2},\cdots,X_{n}$ be independent and identically distributed
random variables. If $Z=f(X_{1},X_{2},\cdots,X_{n})$ and $f$ is weakly $(a,0)$
self bounding then for all $t>0,$

$\mbox{Prob}\left\\{(Z-E(Z))\geq
t\right\\}\leq\exp\left(-\frac{t^{2}}{2aE(Z)+at}\right).$

Further, if $f$ has $(a,0)$ self bounding then

$\mbox{Prob}\left\\{(Z-E(Z))\leq-t\right\\}\leq\exp\left(-\frac{t^{2}}{2\max(a,1)E(Z)}\right).$

In particular, if $f:X^{n}\rightarrow\mathbb{R}^{n}$ is defined as
$f(x)=Z=\sum_{k=1}^{n}X_{k}.$ Since $E(Z)=n\beta_{i}$ and $f$ is $(1,0)$ self
bounding. Then by using Lemma 4.9 we get better estimates as the followings

$\mbox{Prob}\left\\{(Z-E(Z))\geq
n\epsilon\right\\}\leq\exp\left(-\frac{n\epsilon^{2}}{2\beta_{i}+\epsilon}\right)~{}~{}\mbox{
and}$ (6)
$\mbox{Prob}\left\\{(Z-E(Z))\leq-n\epsilon\right\\}\leq\exp\left(-\frac{n\epsilon^{2}}{2}\right).$
(7)

Based on the above results, we arrive a statement to give a precise bound for
the estimator and writing as the following lemma

######  Lemma 4.10 (Confidence intervals).

For a given $n$ randomly samples and accuracy $\epsilon>0$ with confidence
level atleast $1-\delta$, the following confidence intervals holds:

a.
$\displaystyle\left[\hat{\beta}_{\Phi,i}-\sqrt{\frac{1}{2n}\ln\left(\frac{2}{\delta}\right)},~{}\hat{\beta}_{\Phi,i}+\sqrt{\frac{1}{2n}\ln\left(\frac{2}{\delta}\right)}\right]$
b.
$\displaystyle\left[\hat{\beta}_{\Phi,i}-t_{\delta/2}\frac{S}{\sqrt{n}},~{}\hat{\beta}_{\Phi,i}+t_{\delta/2}\frac{S}{\sqrt{n}}\right]$

###### Proof.

We are looking an $\epsilon$ for which
$P(|\hat{\beta}_{\Phi,i}-\beta_{\Phi,i}|<\epsilon)$ is atleast $1-\delta.$
Since $X_{i}$’s are bounded by $[0,1]$, so by using Hoeffding’s inequality,
the $\epsilon$ should satisfy

$\displaystyle 1-2e^{-{2n\epsilon^{2}}}\geq
1-\delta,~{}~{}~{}\Rightarrow\delta\geq 2e^{-{2n\epsilon^{2}}}$
$\displaystyle\Rightarrow$ $\displaystyle\ln\left(\frac{2}{\delta}\right)\leq
2n\epsilon^{2},~{}~{}~{}\Rightarrow\epsilon\geq\sqrt{\frac{\ln\left(\frac{2}{\delta}\right)}{2n}}$

This proves first part of the lemma. Since $X_{1},X_{2},\cdots,X_{n}$ are
independent and Bernoulli trials with mean $\displaystyle\mu=\beta_{\Phi,i}$
and variance $\displaystyle\sigma^{2}=\beta_{\Phi,i}(1-\beta_{\Phi,i}).$ So
for large $n,$ and using central limit theorem, the random variable
$\displaystyle\frac{\overline{X}-\beta_{\Phi,i}}{\sigma/\sqrt{n}}$ is
approximately standard normal. If $\sigma$ is unknown and $n$ is large, then
$\displaystyle\frac{\overline{X}-\beta_{\Phi,i}}{S/\sqrt{n}}$ is approximately
$t-$ distribution with degrees of freedom $(n-1),$ where $S^{2}$ is the sample
variance and defined by
$S^{2}=\displaystyle\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}.$ Let
us denote $P(t>t_{\alpha})=\alpha.$ Using this notation,
$P(|\overline{X}-\beta_{\Phi,i}|\leq\epsilon)\geq 1-\delta$ implies
$\displaystyle\frac{\epsilon\sqrt{n}}{S}\geq t_{\delta/2}.$ Therefor
$\epsilon\geq\displaystyle\frac{St_{\delta/2}}{\sqrt{n}}.$ Which completes the
second part of the theorem. ∎

From the above Lemma, it can be observed that the number samples required to
get the accuracy $\epsilon$ is at least
$\displaystyle\left[\frac{\ln\left(2/\delta\right)}{2\epsilon^{2}}\right]$ or
$\displaystyle\frac{S^{2}t_{\delta/2}^{2}}{\epsilon^{2}}$ in case of part
$(b).$

######  Theorem 4.11.

Let $X_{1},X_{2},\cdots,X_{n}$ be $n$ random samples. Then for given accuracy
$\epsilon>0,$ the confidence interval for $\beta_{\Phi,i}$ (with confidence
level $1-\delta$) is

$\left(\hat{\beta}_{\Phi,i}-\frac{1}{\sqrt{n}}\sqrt{B\ln\left(\frac{2}{\delta}\right)},~{}~{}~{}\hat{\beta}_{\Phi,i}+\frac{1}{\sqrt{n}}\sqrt{B\ln\left(\frac{2}{\delta}\right)}\right).$

###### Proof.

By using equation 6,we obtain

$P\left\\{\hat{\beta}_{\Phi,i}-\beta_{\Phi,i}>\epsilon\right\\}\leq
e^{\left(-\frac{n\epsilon^{2}}{2\beta_{i}+\epsilon}\right)}.$

Using proposition 3.4, and denoting the bound for $2\beta_{i}+\epsilon$ as
$B=B_{w,q,\epsilon},$ we have the following bound:

$P\left\\{\hat{\beta}_{\Phi,i}-\beta_{\Phi,i}>\epsilon\right\\}\leq
e^{\left(-\frac{n\epsilon^{2}}{B}\right)}.$

Since the random variable $\hat{\beta}_{\Phi,i}-\beta_{\Phi,i}$ is
approximately normal distribution with mean zero.

$\displaystyle\Rightarrow e^{\left(-\frac{n\epsilon^{2}}{B}\right)}$
$\displaystyle\leq$ $\displaystyle\left(\frac{\delta}{2}\right)$
$\displaystyle\epsilon$ $\displaystyle\geq$
$\displaystyle\frac{1}{\sqrt{n}}\sqrt{B\ln\left(\frac{2}{\delta}\right)}.$

This completes the proof. ∎

######  Remark 4.12.

The number of samples required within accuracy $\epsilon$ is atleast
$\frac{B\ln\left(\frac{2}{\delta}\right)}{\epsilon^{2}}.$

## 5 Experimental Results

In this section, we will discuss the comparison of $3-$quota banzhaf indices
without association (WTA) and with the association matrix(WA). In the
comparison analysis, the European Union voting system [36] is considered along
with the following quota.

  * (a)

Weight quota ($74\%$ of voting weights).

  * (b)

Population quota ( $62\%$ of population).

  * (c)

Majority of the number of countries ($50\%+1$).

For our analysis we consider the countries with weights more than equal to
7,i.e, $w\geq 7$. The weight-data and population data can be found in
111http://www.consilium.europa.eu/en/council-eu/voting-system/qualified-
majority/
222https://en.wikipedia.org/wiki/List_of_European_Union_member_states_by_population.
The association matrix $\Phi$ is taken based on the immigration between the
respective EU countries. We define the association matrix in the following
manner.
Let us denote $M_{i,j}$ be the number of people migrating from country $i$ to
country $j$ and $M_{j,i}$ be the number of people migrating from country $j$
to country $i.$ Define $M=\displaystyle\max_{i\neq j}|M_{i,j}-M{j,i}|.$ Now
the association matrix $\Phi_{i,j}$ defined as

$\Phi_{i,j}=\left\\{\begin{array}[]{ccc}1&~{}~{}\mbox{ If $i=j.$}\\\
\frac{M_{j,i}-M_{i,j}}{M}&~{}~{}\mbox{ if $1\leq j<i\leq 18.$}\\\
-\Phi_{j,i}&~{}~{}\mbox{if $1\leq i<j\leq 18.$}\end{array}\right.$

Using the above immigration association matrix, improved Banzhaf indices are
computed and given in Table 1.

Table 1: Without Association v/s with association(based on immigration) Country | Weight | Popln | WTA | WA-immgr
---|---|---|---|---
Austria(AUT) | 10 | 8.58 | 0.03549 | 0.05554
Belgium(BEL) | 12 | 11.25 | 0.04403 | 0.06485
Czech Republic(CZE) | 12 | 10.53 | 0.04403 | 0.04681
Germany(DEU) | 29 | 82.30 | 0.09560 | 0.12898
Denmark(DNK) | 7 | 5.66 | 0.02629 | 0.03317
Spain(ESP) | 27 | 46.46 | 0.08853 | 0.10302
Finland(FIN) | 7 | 5.47 | 0.02629 | 0.03097
France(FRA) | 29 | 66.99 | 0.09560 | 0.11263
Britain(GBR) | 29 | 65.11 | 0.09560 | 0.12486
Greece(GRC) | 12 | 10.81 | 0.04403 | 0.02750
Hungary(HUN) | 12 | 9.85 | 0.04403 | 0.00396
Ireland(IRL) | 7 | 4.63 | 0.02629 | 0.03236
Italy(ITA) | 29 | 60.79 | 0.09560 | 0.06765
Netherlands(NLD) | 13 | 17.10 | 0.04418 | 0.06505
Poland(POL) | 27 | 38.56 | 0.08853 | 0.0
Portugal(PRT) | 12 | 10.37 | 0.04403 | 0.03527
Slovakia(SVK) | 7 | 5.42 | 0.02629 | 0.01762
Sweden(SWE) | 10 | 10.01 | 0.03549 | 0.04968
All 18 Countries | 291 | 469.93 | 1.0 | 1.0

Now, we consider a random matrix with entries from $-1$ to $1$ as association
matrix between EU countries. The Banzhaf indices with random association (WA-
Random) are obtained by averaging the value of the banzhaf indices over 100
runs, where we take a new random association matrix in each run and the
Banzhaf indices without association (WTA) are obtained in a single run. The
computed indices are provided in Table 2.

Table 2: Without Association v/s with random association Country | Weight | Popln | WTA | WA-Random
---|---|---|---|---
Austria(AUT) | 10 | 8.58 | 0.03549 | 0.04364
Belgium(BEL) | 12 | 11.25 | 0.04403 | 0.04509
Czech Republic(CZE) | 12 | 10.53 | 0.04403 | 0.04717
Germany(DEU) | 29 | 82.30 | 0.09560 | 0.08706
Denmark(DNK) | 7 | 5.66 | 0.02629 | 0.04134
Spain(ESP) | 27 | 46.46 | 0.08853 | 0.07813
Finland(FIN) | 7 | 5.47 | 0.02629 | 0.04330
France(FRA) | 29 | 66.99 | 0.09560 | 0.08860
Britain(GBR) | 29 | 65.11 | 0.09560 | 0.08013
Greece(GRC) | 12 | 10.81 | 0.04403 | 0.04831
Hungary(HUN) | 12 | 9.85 | 0.04403 | 0.04863
Ireland(IRL) | 7 | 4.63 | 0.02629 | 0.03904
Italy(ITA) | 29 | 60.79 | 0.09560 | 0.07439
Netherlands(NLD) | 13 | 17.10 | 0.04418 | 0.04785
Poland(POL) | 27 | 38.56 | 0.08853 | 0.06762
Portugal(PRT) | 12 | 10.37 | 0.04403 | 0.04858
Slovakia(SVK) | 7 | 5.42 | 0.02629 | 0.03724
Sweden(SWE) | 10 | 10.01 | 0.03549 | 0.03380
All 18 Countries | 291 | 469.93 | 1.0 | 1.0

The approximated Banzhaf indices without association (WTA-Approx) and with
random association matrix (WA-Random-Approx) are computed by using Algorithm 2
which discussed in Section 2. Note that while computing the indices, we use a
single random matrix and obtain the exact value of WA-Random. and for
computing the approximated indices, we run the algorithm $20$ times and then
averaged. Similar computation followed for WTA-Approx. Also we consider
$\epsilon=0.01$ and $\delta=0.01$ for each of the approximation algorithm
runs.

Table 3: Exact v/s Approximation Country | WTA | WTA-Approx | WA-Random | WA-Random-Approx
---|---|---|---|---
AUT | 0.03549 | 0.03540 | 0.06774 | 0.06757
BEL | 0.04403 | 0.04428 | 0.07138 | 0.07150
CZE | 0.04403 | 0.04384 | 0.07138 | 0.07138
DEU | 0.09560 | 0.09557 | 0.09339 | 0.09353
DNK | 0.02629 | 0.02629 | 0.06305 | 0.06277
ESP | 0.08853 | 0.08844 | 0.0 | 0.0
FIN | 0.02629 | 0.02621 | 0.03484 | 0.03474
FRA | 0.09560 | 0.09579 | 0.02633 | 0.02653
GBR | 0.09560 | 0.09588 | 0.06871 | 0.06916
GRC | 0.04403 | 0.04392 | 0.00084 | 0.00083
HUN | 0.04403 | 0.04436 | 0.06967 | 0.06997
IRL | 0.02629 | 0.02624 | 0.06383 | 0.06371
ITA | 0.09560 | 0.09532 | 0.08334 | 0.08319
NLD | 0.04418 | 0.04429 | 0.01747 | 0.01740
POL | 0.08853 | 0.08871 | 0.07458 | 0.07480
PRT | 0.04403 | 0.04395 | 0.06468 | 0.06457
SVK | 0.02629 | 0.02617 | 0.06245 | 0.06228
SWE | 0.03549 | 0.03526 | 0.06623 | 0.06597

The result displayed in Table 3 justifies that the Banzhaf indices
$\hat{\beta_{\Phi,i}}$ are obtained satisfies
$P(|\hat{\beta_{\Phi,i}}-\beta_{\Phi,i}|>\epsilon)\leq\delta$.

## 6 Conclusion

We have an improved and generalized power index that captures the internal
dynamic of players in real life situations that typically arise as examples of
voting games. The few main lines of future research directions that can be
picked up from here are that of showing theoretical results regarding the
index. Better approximations of the index as in [15] could be also interesting
to investigate. In the direction of application one can look into ways of
estimating the association matrix data in real life, and the subsequent
theoretical analysis of the effect of the association matrix on the index. In
the third line we would like mention that with a restricted domain of weights
such as weights satisfying some conditions as in is a very interesting future
direction. A complete development of axiomatics of the improved index as in
[12, 16] is also a theoretical problem.

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# Antikaon-induced $K_{1}(1270)^{+}$ meson production on nuclei near threshold

E. Ya. Paryev1,2
1Institute for Nuclear Research, Russian Academy of Sciences,
Moscow 117312, Russia
2Institute for Theoretical and Experimental Physics,
Moscow 117218, Russia

###### Abstract

We study the inclusive strange axial-vector meson $K_{1}(1270)^{+}$ production
in ${K^{-}}A$ reactions at near-threshold laboratory incident antikaon momenta
within a nuclear spectral function approach, which describes incoherent direct
$K_{1}(1270)^{+}$ meson production in $K^{-}$ meson–proton
${K^{-}}p\to{K_{1}(1270)^{+}}\Xi^{-}$ production processes and accounts for
three different options for its in-medium mass shift (or for its effective
scalar potential) at central density $\rho_{0}$. We calculate the absolute
differential and total cross sections for the production of $K_{1}(1270)^{+}$
mesons on 12C and 184W target nuclei at laboratory angles of 0∘–45∘ by $K^{-}$
mesons with momenta of 2.5, 2.8 and 3.5 GeV/c, which are close to the
threshold momentum ($\approx$ 2.95 GeV/c) for $K_{1}(1270)^{+}$ meson
production off the free target proton at rest. The intrinsic properties of
carbon and tungsten target nuclei have been described in terms of their
spectral functions, which take into account the momenta of target protons and
the energies of their separation from the considered nuclei. We show that the
differential and total (absolute and relative) $K_{1}(1270)^{+}$ antikaon-
induced production cross sections at initial momenta not far from threshold –
at momenta $\sim$ 2.8–3.5 GeV/c, at which there is yet no strong drop in their
strength, reveal a distinct sensitivity to changes in the in-medium shift of
the $K_{1}(1270)^{+}$ mass, studied in the paper, both in the
$K_{1}(1270)^{+}$ meson low-momentum (0.1–1.0 GeV/c) and in its full-momentum
ranges. This would permit evaluating this shift. Experimental data necessary
for this aim can be obtained in a dedicated experiment at the J-PARC Hadron
Experimental Facility.

## 1 Introduction

The study of the in-medium properties (masses and widths) of hadrons at finite
density has received considerable interest in the last two decades due to the
hope to extract valuable information on the partial restoration of chiral
symmetry in a nuclear medium (see, for example, [1–6]). Another prominent case
of medium modification of hadrons is that of the strange vector $K^{*}(892)$
and axial-vector $K_{1}(1270)$ mesons with the same charge states (or with the
same quark structure $q{\bar{s}}$ or ${\bar{q}}s$ with $q=u,d$), which are
chiral partners and whose in-medium mass difference, as is expected [7–10], is
sensitive to the chiral order parameter and, hence, will give the possibility
to identify unambiguously the effect of chiral symmetry breaking in nuclear
medium. Their isospins, spins-parities quantum numbers are
$I(J^{P})=\frac{1}{2}(1^{-})$ for the $K^{*}(892)$ and
$I(J^{P})=\frac{1}{2}(1^{+})$ for the $K_{1}(1270)$. They have a relatively
small vacuum decay widths of 50 and 90 MeV, respectively.

At present, in literature there is a number of publications devoted to the
study of the in-medium properties of hadronic resonances $K^{*}(892)$ and
$K_{1}(1270)$. The properties of ${\bar{K}}^{*}(892)$ and $K^{*}(892)$ mesons
in cold nuclear matter have been studied in Refs. [11–13] and [13, 14],
respectively, basing on chirally motivated model of the meson selfenergies.
Thus, for instance, a moderately attractive and repulsive real low-energy
nuclear potentials (or their in-medium mass shifts) of about -50 and +40 MeV
at density $\rho_{0}$ have been predicted for the ${\bar{K}}^{*}(892)$ and
$K^{*}(892)$ mesons, respectively. These are similar to those for light
strange mesons ${\bar{K}}$ and $K$. On the other hand, a negative mass shift
of about -20 MeV has been predicted for the $K^{*}(892)^{+}$ meson, having the
same quark content $u{\bar{s}}$ as the $K^{+}$ one, at rest at saturation
density $\rho_{0}$ within the quark-meson coupling model [15]. Other recent
calculations [8] based on QCD sum rule predict that the upper limits of the
mass shifts of $K_{1}^{-}$ and $K_{1}^{+}$ mesons in nuclear matter are -249
and -35 MeV, respectively.

The influence of the $K^{*}(892)^{+}$ meson in-medium mass shift on its yield
from ${\pi^{-}}A$ reactions at near-threshold laboratory initial pion momenta
of 1.4–2.0 GeV/c has been recently analyzed within the collision model [16]
using an eikonal approximation. It has been shown that both differential and
total $K^{*}(892)^{+}$ production cross sections possess a high sensitivity to
changes in this shift in the low-momentum region of 0.1–0.6 GeV/c. In addition
to a feasibility study performed in Ref. [16] in order to investigate the
impact of the mass shift of $K^{*}(892)^{+}$ mesons in nuclear matter on their
yield in near-threshold ${\pi^{-}}A$ reactions, we give here the predictions
for the absolute differential and total cross sections for near-threshold
production of $K_{1}(1270)^{+}$ mesons in ${K^{-}}^{12}{\rm
C}\to{K_{1}(1270)^{+}}X$ and ${K^{-}}^{184}{\rm W}\to{K_{1}(1270)^{+}}X$
reactions at laboratory angles of 0∘–45∘ by incident antikaons with momenta of
2.5–3.5 GeV/c as well as for their relative yields from these reactions within
three different scenarios for the $K_{1}(1270)^{+}$ in-medium mass shift.
These nuclear targets were also adopted in calculations [16] of
$K^{*}(892)^{+}$ meson production in ${\pi^{-}}A$ reactions and, therefore,
can be used in studying the ${K^{-}}A\to{K_{1}(1270)^{+}}X$ interactions here.
The present calculations are based on a first-collision model, developed in
Ref. [16] for the description of the inclusive $K^{*}(892)^{+}$ meson
production and extended to account for different scenarios for the
$K_{1}(1270)^{+}$ in-medium mass shift and width change. The medium
modification of the $K_{1}(1270)^{\pm}$ (and $K^{*}(892)^{\pm}$) mesons could
be probed at the J-PARC Hadron Experimental Facility through the $K^{-}$
reaction on various nuclear targets [8] 111)It is appropriate to note that the
measuring of $K_{1}(1270)^{-}$ and $K^{*}(892)^{-}$ mesons would require less
energy to produce compared to that for $K_{1}(1270)^{+}$ and $K^{*}(892)^{+}$
using the $K^{-}$ beam due to more lower their lowest production thresholds in
elementary reactions $K^{-}p\to K_{1}(1270)^{-}p$ and $K^{-}p\to
K^{*}(892)^{-}p$ ($\approx$ 1.94 and 1.08 GeV/c, respectively) compared to
those for $K_{1}(1270)^{+}$ and $K^{*}(892)^{+}$ production in the relevant
elementary channels $K^{-}p\to K_{1}(1270)^{+}\Xi^{-}$ and $K^{-}p\to
K^{*}(892)^{+}\Xi^{-}$ ($\approx$ 2.95 and 1.95 GeV/c, correspondingly).
Therefore, such experimental possibility may be provided for them already now
by the available at J-PARC high-intensity separated secondary antikaon beam
with the momentum up to 2 GeV/c. In this connection, it is interesting to
extend in the future the study of Ref. [16] and of present one also to the
consideration of $K_{1}(1270)^{-}$ and $K^{*}(892)^{-}$ production on nuclei
in near-threshold $K^{-}A$ interactions to get a deeper insight into the
possibility of observing here of their modification in nuclear medium in these
interactions.) . Comparison the results of such measurement for
$K_{1}(1270)^{+}$ and those from $K^{*}(892)^{+}$ measurement in ${\pi^{-}}A$
reactions with our calculations – previous [16] and present – will shed light
on the partial restoration of chiral symmetry in nuclear matter [8].

## 2 Framework: direct process of $K_{1}(1270)^{+}$ meson production in nuclei

Direct production of $K_{1}(1270)^{+}$ mesons in $K^{-}A$ (A=12C and 184W)
interactions at near-threshold incident antikaon beam momenta below 3.5 GeV/c
may occur in the following $K^{-}p$ elementary process, which has the lowest
free production threshold momentum ($\approx$ 2.95 GeV/c):

$K^{-}+p\to K_{1}(1270)^{+}+\Xi^{-}.$ (1)

We can ignore in the momentum domain of interest the contribution to the
$K_{1}(1270)^{+}$ yield from the processes
$K^{-}p\to{K_{1}(1270)^{+}}\Xi^{-}{\pi^{0}}$,
$K^{-}p\to{K_{1}(1270)^{+}}\Xi^{0}{\pi^{-}}$,
$K^{-}n\to{K_{1}(1270)^{+}}\Xi^{-}{\pi^{-}}$ and
$K^{-}p\to{K_{1}(1270)^{+}}\Xi(1530)^{-}$ due to larger their production
thresholds ($\approx$ 3.33 and 3.57 GeV/c, respectively) in $K^{-}N$
collisions.

Following [16], we simplify the ensuing calculations via accounting for the
in-medium modification of the final $K_{1}(1270)^{+}$ meson, involved in the
production process (1), in terms of its average in-medium mass
$<m_{K_{1}}^{*}>$ instead of its local effective mass $m_{K_{1}}^{*}(|{\bf
r}|)$ in the in-medium cross section of this process, with $<m_{K_{1}}^{*}>$
defined as:

$<m^{*}_{K_{1}}>=m_{K_{1}}+V_{0}\frac{<{\rho_{N}}>}{{\rho_{0}}},$ (2)

where $m_{K_{1}}$ is the $K_{1}(1270)^{+}$ free space mass, $V_{0}$ is the
$K_{1}(1270)^{+}$ effective scalar nuclear potential (or its in-medium mass
shift) at normal nuclear matter density ${\rho_{0}}$, and $<{\rho_{N}}>$ is
the average nucleon density. In the present work, for nuclei 12C and 184W of
interest, the ratio $<{\rho_{N}}>/{\rho_{0}}$, was chosen as 0.55 and 0.76,
respectively. No experimental data exist presently on the $K_{1}(1270)^{+}N$
interaction. Recent calculations performed in Ref. [8] within the QCD sum rule
approach show, as was noted above, that the upper limit of the mass shift of
$K_{1}(1270)^{+}$ mesons in nuclear matter is -35 MeV at zero momentum
relative to the surrounding nuclear matter. Therefore, we will adopt for the
quantity $V_{0}$ in this work three following options: i) $V_{0}=-35$ MeV, ii)
$V_{0}=-20$ MeV and iii) $V_{0}=0$ MeV. Since there is no information about
momentum dependence of this mass shift, we will use the above options for it
at finite momenta, accessible in calculation of the $K_{1}(1270)^{+}$
production in $K^{-}A$ reactions at beam momenta of interest, with allowance
for the Fermi motion of intranuclear protons (at momenta $\sim$ 0.1–2.7 GeV/c,
see below). The experimental knowledge on the $\Xi^{-}N$ interaction is poor
currently. The study of $\Xi^{-}$ hypernuclei provides a valuable information
on this interaction at low energies [6, 17]. At present, there are only a few
sets of experimental data on the $\Xi^{-}$ hyperon properties in the nuclear
medium. Following phenomenological information deduced in Refs. [18, 19] from
old emulsion data 222)It is worth noting that the new recent emulsion KEK-PS
E373 [20] and J-PARC E07 [21] experiments, in which the remarkable events
named, respectively, ”KISO” and ”IBUKI” were observed, reported the evidence
of a likely the Coulomb-assisted nuclear 1$p$ single-particle $\Xi^{-}$ state
of the bound $\Xi^{-}$–14N(g.s.) system with binding energy of about 1 MeV,
which also testifies in favor of shallow $\Xi^{-}$–nucleus potential.), from
missing-mass measurements [22–25] in the inclusive ($K^{-}$,$K^{+}$) reaction
on nuclear targets at incident momenta of 1.6–1.8 GeV/c in the $\Xi^{-}$ bound
and quasi-free regions with insufficiently good (10 MeV or worse) [22–24] and
moderate (5.4 MeV) [25] energy resolutions, from the analysis the results of
these measurements yet in Refs. [26, 27] using Green’s function method of the
DWIA, the theoretical predictions [28–31] that the $\Xi^{-}$ cascade hyperon
feels only a weak attractive potential in nuclei, the depth of which is not so
large, $\sim$ -(10–20) MeV at central nuclear density and at rest as well as
the predictions [32, 33] within the SU(6) quark model that the $\Xi^{-}$
single-particle potential in symmetric nuclear matter becomes almost zero at
high $\Xi^{-}$ momenta around 1.3 GeV/c corresponding to the $K_{1}(1270)^{+}$
momentum range of $\sim$ 1–2 GeV/c where the $K_{1}(1270)^{+}$ production
cross sections are the greatest, we will ignore the modification of the free
space mass $m_{\Xi^{-}}$ of the $\Xi^{-}$ hyperons, produced together with the
$K_{1}(1270)^{+}$ mesons in the process (1), in the nuclear medium. To improve
essentially our understanding of the $\Xi^{-}N$ interaction the high-quality
data on the doubly strange $S=-2$ hypernuclei are needed. It is expected that
in the near future high resolution (up to 1.4 MeV) and high statistics data on
the missing-mass spectra for the 12C($K^{-}$,$K^{+}$) reaction around the
$\Xi^{-}$ production threshold and the X-ray data on the level shifts and
width broadening of the $\Xi^{-}$ atomic states in the $\Xi^{-}$ atoms will
become available from the planned dedicated E70 and E03 experiments at the
J-PARC Hadron Experimental Facility [34]. Accounting for that the in-medium
threshold energy 333)Determining mainly the strength of the $K_{1}(1270)^{+}$
production cross sections in near-threshold antikaon–nucleus collisions.)
$\sqrt{s^{*}_{\rm th}}=<m_{K_{1}}^{*}>+m_{\Xi^{-}}$ of the process (1) looks
like that for the final charged particles, influenced also by the respective
Coulomb potentials, due to the cancelation of these potentials, we will
neglect here their impact on these particles as well.

The total energy $E^{\prime}_{K_{1}}$ of the $K_{1}(1270)^{+}$ meson in
nuclear matter is expressed via its average effective mass $<m^{*}_{K_{1}}>$
and its in-medium momentum ${\bf p}^{\prime}_{K_{1}}$ by the expression [16]:

$E^{\prime}_{K_{1}}=\sqrt{({\bf
p}^{\prime}_{K_{1}})^{2}+(<m^{*}_{K_{1}}>)^{2}}.$ (3)

The momentum ${\bf p}^{\prime}_{K_{1}}$ is related to the vacuum
$K_{1}(1270)^{+}$ momentum ${\bf p}_{K_{1}}$ as follows [16]:

$E^{\prime}_{K_{1}}=\sqrt{({\bf
p}^{\prime}_{K_{1}})^{2}+(<m^{*}_{K_{1}}>)^{2}}=\sqrt{{\bf
p}^{2}_{K_{1}}+m^{2}_{K_{1}}}=E_{K_{1}},$ (4)

where $E_{K_{1}}$ is the $K_{1}(1270)^{+}$ total energy in vacuum.

As the $K_{1}(1270)^{+}$–nucleon elastic cross section is expected to be small
similar to the $K^{*}(892)^{+}N$ elastic cross section [35], we will ignore
quasielastic ${K_{1}(1270)^{+}}N$ rescatterings in the present work. Then,
taking into account the attenuation of the incident antikaon and the final
$K_{1}(1270)^{+}$ meson in the nuclear matter in terms, respectively, of the
$K^{-}N$ total cross section $\sigma_{{K^{-}}N}^{\rm tot}$ and the total width
$\Gamma_{K_{1}}(|{\bf r}|)$ in the $K_{1}(1270)^{+}$ rest frame, taken at the
point ${\bf r}$ inside the nucleus, as well as using the results given in
[16], we represent the inclusive differential cross section for the production
of $K_{1}(1270)^{+}$ mesons with vacuum momentum ${\bf p}_{K_{1}}$ on nuclei
in the direct process (1) as follows:

$\frac{d\sigma_{{K^{-}}A\to{K_{1}(1270)^{+}}X}^{({\rm prim})}({\bf
p}_{K^{-}},{\bf p}_{K_{1}})}{d{\bf
p}_{K_{1}}}=I_{V}[A,\theta_{K_{1}}]\left(\frac{Z}{A}\right)\left<\frac{d\sigma_{{K^{-}}p\to
K_{1}(1270)^{+}{{\Xi^{-}}}}({\bf p}_{K^{-}},{\bf p}^{\prime}_{{K_{1}}})}{d{\bf
p}^{\prime}_{{K_{1}}}}\right>_{A}\frac{d{\bf p}^{\prime}_{{K_{1}}}}{d{\bf
p}_{{K_{1}}}},$ (5)

where

$I_{V}[A,\theta_{{K_{1}}}]=A\int\limits_{0}^{R}r_{\bot}dr_{\bot}\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{\sqrt{R^{2}-r_{\bot}^{2}}}dz\rho(\sqrt{r_{\bot}^{2}+z^{2}})\exp{\left[-\sigma_{{K^{-}}N}^{\rm
tot}A\int\limits_{-\sqrt{R^{2}-r_{\bot}^{2}}}^{z}\rho(\sqrt{r_{\bot}^{2}+x^{2}})dx\right]}$
(6)
$\times\int\limits_{0}^{2\pi}d{\varphi}\exp{\left[-\int\limits_{0}^{l(\theta_{{K_{1}}},\varphi)}\frac{dx}{\lambda_{K_{1}}(\sqrt{x^{2}+2a(\theta_{{K_{1}}},\varphi)x+b+R^{2}})}\right]};$
$a(\theta_{K_{1}},\varphi)=z\cos{\theta_{K_{1}}}+r_{\bot}\sin{\theta_{K_{1}}}\cos{\varphi},\,\,\,\,b=r_{\bot}^{2}+z^{2}-R^{2},$
(7)
$l(\theta_{K_{1}},\varphi)=\sqrt{a^{2}(\theta_{K_{1}},\varphi)-b}-a(\theta_{K_{1}},\varphi),$
(8) $\sigma_{{K^{-}}N}^{\rm tot}=\frac{Z}{A}\sigma_{{K^{-}}p}^{\rm
tot}+\frac{N}{A}\sigma_{{K^{-}}n}^{\rm tot},$ (9) $\lambda_{K_{1}}(|{\bf
r}|)=\frac{p^{\prime}_{K_{1}}}{m^{*}_{K_{1}}(|{\bf r}|)\Gamma_{K_{1}}(|{\bf
r}|)},\,\,\,m^{*}_{K_{1}}(|{\bf r}|)=m_{K_{1}}+V_{0}\frac{{\rho_{N}}(|{\bf
r}|)}{{\rho_{0}}}$ (10)

and

$\left<\frac{d\sigma_{{K^{-}}p\to K_{1}(1270)^{+}{\Xi^{-}}}({\bf
p}_{K^{-}},{\bf p}^{\prime}_{K_{1}})}{d{\bf
p}^{\prime}_{K_{1}}}\right>_{A}=\int\int P_{A}({\bf p}_{t},E)d{\bf p}_{t}dE$
(11) $\times\left\\{\frac{d\sigma_{{K^{-}}p\to
K_{1}(1270)^{+}{\Xi^{-}}}[\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},{\bf
p}^{\prime}_{K_{1}}]}{d{\bf p}^{\prime}_{K_{1}}}\right\\},$
$s^{*}=(E_{K^{-}}+E_{t})^{2}-({\bf p}_{K^{-}}+{\bf p}_{t})^{2},$ (12)
$E_{t}=M_{A}-\sqrt{(-{\bf p}_{t})^{2}+(M_{A}-m_{N}+E)^{2}}.$ (13)

Here,
$d\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}[\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},{\bf
p}^{\prime}_{K_{1}}]/d{\bf p}^{\prime}_{K_{1}}$ is the off-shell inclusive
differential cross section for the production of ${K_{1}(1270)^{+}}$ meson and
$\Xi^{-}$ hyperon with modified mass $<m_{K_{1}}^{*}>$ and free mass
$m_{\Xi^{-}}$, respectively. The $K_{1}(1270)^{+}$ meson is produced with in-
medium momentum ${\bf p}^{\prime}_{{K_{1}}}$ in process (1) at the ${K^{-}}p$
center-of-mass energy $\sqrt{s^{*}}$. $E_{K^{-}}$ and ${\bf p}_{K^{-}}$ are
the total energy and momentum of the incident antikaon
($E_{K^{-}}=\sqrt{m^{2}_{K}+{\bf p}^{2}_{K^{-}}}$, $m_{K}$ is the free space
kaon mass); $\rho({\bf r})$ and $P_{A}({\bf p}_{t},E)$ are the local nucleon
density and the spectral function of the target nucleus A normalized to unity
(the concrete information about these quantities, used in the subsequent
calculations, is given in Refs. [36–39]); ${\bf p}_{t}$ and $E$ are the
internal momentum and removal energy of the target proton involved in the
collision process (1); $Z$ and $N$ are the numbers of protons and neutrons in
the target nucleus ($A=Z+N$), $M_{A}$ and $R$ are its mass and radius; $m_{N}$
is the free space nucleon mass; $\theta_{K_{1}}$ is the polar angle of vacuum
momentum ${\bf p}_{{K_{1}}}$ in the laboratory system with z-axis directed
along the momentum ${\bf p}_{{K^{-}}}$ of the incident antikaon beam;
$\sigma_{{K^{-}}p}^{\rm tot}$ and $\sigma_{{K^{-}}n}^{\rm tot}$ are the total
cross sections of the free ${K^{-}}p$ and ${K^{-}}n$ interactions at beam
momenta belonging to the range of 2.5–3.5 GeV/c 444) We ignore the medium
modification of the incoming $K^{-}$ meson mass in this momentum range [40].)
. In this range $\sigma_{{K^{-}}p}^{\rm tot}\approx$ 28 mb and
$\sigma_{{K^{-}}n}^{\rm tot}\approx$ 22 mb [41]. With these, the quantity
$\sigma_{{K^{-}}N}^{\rm tot}$, entering into Eqs. (6) and (9), amounts
approximately to 25 mb for both considered target nuclei 12C and 184W. We will
use this value in our present calculations.

According to [16], we suppose that the off-shell differential cross section
$d\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}[\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},{\bf
p}^{\prime}_{K_{1}}]/d{\bf p}^{\prime}_{K_{1}}$ for $K_{1}(1270)^{+}$
production in process (1) is equivalent to the respective on-shell cross
section calculated for the off-shell kinematics of this process as well as for
the final $K_{1}(1270)^{+}$ meson and $\Xi^{-}$ hyperon in-medium mass
$<m_{K_{1}}^{*}>$ and free mass $m_{\Xi^{-}}$, respectively. Taking into
account the two-body kinematics of the process (1), we get the following
expression for the differential cross section
$d\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}[\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},{\bf
p}^{\prime}_{K_{1}}]/d{\bf p}^{\prime}_{K_{1}}$:

$\frac{d\sigma_{K^{-}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}[\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},{\bf
p}^{\prime}_{K_{1}}]}{d{\bf
p}^{\prime}_{K_{1}}}=\frac{\pi}{I_{2}[s^{*},<m_{K_{1}}^{*}>,m_{\Xi^{-}}]E^{\prime}_{K_{1}}}$
(14)
$\times\frac{d\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},\theta^{*}_{K_{1}})}{d{\bf\Omega}^{*}_{K_{1}}}$
$\times\frac{1}{(\omega+E_{t})}\delta\left[\omega+E_{t}-\sqrt{m_{\Xi^{-}}^{2}+({\bf
Q}+{\bf p}_{t})^{2}}\right],$

where

$I_{2}[s^{*},<m_{K_{1}}^{*}>,m_{\Xi^{-}}]=\frac{\pi}{2}\frac{\lambda[s^{*},(<m_{K_{1}}^{*}>)^{2},m_{\Xi^{-}}^{2}]}{s^{*}},$
(15)
$\lambda(x,y,z)=\sqrt{{\left[x-({\sqrt{y}}+{\sqrt{z}})^{2}\right]}{\left[x-({\sqrt{y}}-{\sqrt{z}})^{2}\right]}},$
(16) $\omega=E_{K^{-}}-E^{\prime}_{K_{1}},\,\,\,\,{\bf Q}={\bf p}_{K^{-}}-{\bf
p}^{\prime}_{K_{1}}.$ (17)

Here,
$d\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},\theta^{*}_{K_{1}})/d{\bf\Omega}^{*}_{K_{1}}$
is the off-shell differential cross section for the production of
$K_{1}(1270)^{+}$ mesons in process (1) under the polar angle
$\theta^{*}_{K_{1}}$ in the ${K^{-}}p$ c.m.s. This cross section is assumed to
be isotropic in our calculations of $K_{1}(1270)^{+}$ meson production in
${K^{-}}A$ interactions:

$\frac{d\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s^{*}},<m_{K_{1}}^{*}>,m_{\Xi^{-}},\theta^{*}_{K_{1}})}{d{\bf\Omega}^{*}_{K_{1}}}=\frac{\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s^{*}},\sqrt{s^{*}_{\rm
th}})}{4\pi}.$ (18)

Here,
$\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s^{*}},\sqrt{s^{*}_{\rm
th}})$ is the ”in-medium” total cross section of reaction (1) having the
threshold energy $\sqrt{s^{*}_{\rm th}}$ defined above. In line with above-
mentioned, it is equivalent to the vacuum cross section
$\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s},\sqrt{s_{\rm th}})$,
in which the vacuum threshold energy $\sqrt{s_{\rm
th}}=m_{K_{1}}+m_{\Xi^{-}}=2.594$ GeV is replaced by the in-medium one
$\sqrt{s^{*}_{\rm th}}$ and the free center-of-mass energy squared $s$,
presented by the formula

$s=(E_{K^{-}}+m_{N})^{2}-{\bf p}_{K^{-}}^{2},$ (19)

is replaced by the in-medium one $s^{*}$, defined by the expression (12).

For the free total cross section
$\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s},\sqrt{s_{\rm th}})$
we have adopted the following estimates, based on the available only two
experimental data points (6.2$\pm$0.6) ${\rm\mu}$b and (1.7$\pm$0.5)
${\rm\mu}$b for the vacuum total cross sections of the processes $K^{-}p\to
K_{1}(1270)^{+}\Xi^{-}\to K{\rho}\Xi^{-}$ and $K^{-}p\to
K_{1}(1270)^{+}\Xi^{-}\to K^{*}(892){\pi}\Xi^{-}$ at 0.4 GeV excess energy
$\sqrt{s}-\sqrt{s_{\rm th}}$ (or at the beam momentum of 4.15 GeV/c) [42] as
well as on the relatively large amount of existing experimental data for the
total cross section of the $K^{-}p\to K^{+}\Xi^{-}$ reaction [41]. Accounting
for the branching ratios of (42$\pm$6)% and (16$\pm$5)% for the channels
$K_{1}(1270)^{+}\to K{\rho}$ and $K_{1}(1270)^{+}\to K^{*}(892){\pi}$ and
these data points, the values of (14.8$\pm$2.6) ${\rm\mu}$b and (10.6$\pm$4.6)
${\rm\mu}$b for the free cross section
$\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s},\sqrt{s_{\rm th}})$
are obtained at $\sqrt{s}-\sqrt{s_{\rm th}}=$0.4 GeV. They are shown in Fig. 1
by full triangles. Here, the vacuum total cross section of the reaction
$K^{-}p\to K^{+}\Xi^{-}$ is presented as well as a function of the respective
excess energy $\sqrt{{\tilde{s}}}-\sqrt{{\tilde{s}}_{\rm th}}$ available in
this reaction with threshold energy $\sqrt{{\tilde{s}}_{\rm
th}}=m_{K}+m_{\Xi^{-}}$. Full squares are experimental data taken from the
compilation of Flaminio et al. [41]. Solid curve is their parametrization by
the formula

$\sigma_{{K}^{-}p\to{K^{+}}{\Xi^{-}}}(\sqrt{{\tilde{s}}},\sqrt{{\tilde{s}}_{\rm
th}})=235.6\left(1-\sqrt{{\tilde{s}}_{\rm
th}}/\sqrt{{\tilde{s}}}\right)^{2.4}\left(\sqrt{{\tilde{s}}_{\rm
th}}/\sqrt{{\tilde{s}}}\right)^{16.6}~{}[{\rm mb}],$ (20)

suggested in Ref. [43].

Figure 1: (Color online) Total cross sections for the reactions
$K^{-}p\to{K^{+}}{\Xi^{-}}$ and $K^{-}p\to{K_{1}(1270)^{+}}{\Xi^{-}}$ as
functions of the available excess energies
$\sqrt{{\tilde{s}}}-\sqrt{{\tilde{s}}_{\rm th}}$ and $\sqrt{s}-\sqrt{s_{\rm
th}}$ above their thresholds $\sqrt{{\tilde{s}}_{\rm th}}$ and $\sqrt{s_{\rm
th}}$, respectively. For notation see the text. Figure 2: (Color online) In-
medium change of the $K_{1}(1270)^{+}$ meson width and its total in-medium
width in its rest frame as functions of the $K_{1}(1270)^{+}$ in-medium mass
shift $V_{0}$ at normal nuclear matter density.

Visual inspection of Fig. 1 yields the following ratio of
$K_{1}(1270)^{+}{\Xi^{-}}$ to $K^{+}{\Xi^{-}}$ production cross sections at
the same excess energies $\sqrt{s}-\sqrt{s_{\rm th}}$ and
$\sqrt{{\tilde{s}}}-\sqrt{{\tilde{s}}_{\rm th}}$ above the
$K_{1}(1270)^{+}{\Xi^{-}}$ and $K^{+}{\Xi^{-}}$ thresholds equal to 0.4 GeV:

$\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s},\sqrt{s_{\rm
th}})/\sigma_{{K}^{-}p\to{K^{+}}{\Xi^{-}}}(\sqrt{{\tilde{s}}},\sqrt{{\tilde{s}}_{\rm
th}})\approx 0.1.$ (21)

Since no data on the total cross section
$\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s},\sqrt{s_{\rm th}})$
exist for other excess energies, it is natural to estimate it at them in the
threshold region assuming that the ratio (21) is also valid at the same excess
energies $\sqrt{s}-\sqrt{s_{\rm th}}$ and
$\sqrt{{\tilde{s}}}-\sqrt{{\tilde{s}}_{\rm th}}$ differing from 0.4 GeV. Under
this assumption, the center-of-mass energies $\sqrt{{\tilde{s}}}$ and
$\sqrt{s}$, at which the cross sections entering into the ratio (21) are
calculated, are linked by the relation

$\sqrt{{\tilde{s}}}-\sqrt{{\tilde{s}}_{\rm th}}=\sqrt{s}-\sqrt{s_{\rm th}}.$
(22)

Thus, we have

$\sqrt{{\tilde{s}}}=\sqrt{s}-\sqrt{s_{\rm th}}+\sqrt{{\tilde{s}}_{\rm
th}}=\sqrt{s}-m_{K_{1}}+m_{K}.$ (23)

When reactions ${K}^{-}p\to{K^{+}}{\Xi^{-}}$ and
${K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}$ go in medium on an off-shell target
protons, instead of the quantity $\sqrt{{\tilde{s}}_{\rm th}}$, appearing in
Eqs. (20)–(23), we should use the in-medium threshold energy
$\sqrt{{\tilde{s}}^{*}_{\rm th}}=<m_{K}^{*}>+m_{\Xi^{-}}$, where $<m_{K}^{*}>$
is the $K^{+}$ meson average in-medium mass defined by the expression
analogous to that given by Eq. (2) and in which $V_{0}=+22$ MeV [38]. And
instead of the quantities $\sqrt{s}$ and $\sqrt{s_{\rm th}}$, entering into
Eqs. (21)–(23), one needs to adopt, respectively, the in-medium expression
(12) and in-medium threshold energy $\sqrt{s^{*}_{\rm th}}$ determined above.
The foregoing leads, as is easy to see, to the following substitutions
$m_{K_{1}}\to$ $<m_{K_{1}}^{*}>$, $m_{K}\to$ $<m_{K}^{*}>$ in the second
relation of Eq. (23).

At incident antikaon momenta of interest $p_{K^{-}}$ $\leq$ 3.5 GeV/c,
$\sqrt{s}\leq 2.785$ GeV and $\sqrt{{\tilde{s}}}\leq 2.007$ GeV. Taking into
account that the relation between the free collision energy squared
${\tilde{s}}$ and the total energy ${\tilde{E}}_{K^{-}}$ and momentum
${\tilde{p}}_{K^{-}}$ of the $K^{-}$ meson inducing the reaction
${K^{-}}p\to{K^{+}}{\Xi^{-}}$ is similar to that given above by Eq. (19), we
easily get that the latter corresponds to its laboratory momenta
${\tilde{p}}_{K^{-}}$ $\leq$ 1.466 GeV/c. They are not far from the threshold
momentum of 1.047 GeV/c of this reaction. For the free total cross section
$\sigma_{{K}^{-}p\to{K^{+}}{\Xi^{-}}}(\sqrt{{\tilde{s}}},\sqrt{{\tilde{s}}_{\rm
th}})$ at these momenta we will employ in our calculations the expression
(20). Using it and the relations (21)–(23), we calculated the vacuum total
cross section
$\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(\sqrt{s},\sqrt{s_{\rm th}})$
of the process (1). It is plotted with a dashed curve in Fig. 1. As can be
seen from this figure, the on-shell cross section
$\sigma_{{K}^{-}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}$ amounts approximately to 16
${\rm\mu}$b for the initial antikaon momentum of 3.5 GeV/c, corresponding to
the excess energy $\sqrt{s}-\sqrt{s_{\rm th}}$ of about 0.2 GeV. This offers
the possibility of measuring the $K_{1}(1270)^{+}$ yield in $K^{-}A$
collisions both at the above-threshold and below-threshold beam momenta at the
J-PARC Hadron Experimental Facility with sizable strength. Here, the
$K_{1}(1270)^{+}$ mesons could be registered via the hadronic decays
$K_{1}(1270)^{+}\to K{\rho}$ and $K_{1}(1270)^{+}\to K^{*}(892){\pi}$.

Figure 3: (Color online) Momentum differential cross sections for the
production of $K_{1}(1270)^{+}$ mesons from the direct
${K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}$ channel in the laboratory polar
angular range of 0∘–45∘ in the interaction of $K^{-}$ mesons of momentum of
3.5 GeV/c with 12C (left) and 184W (right) nuclei, calculated for different
values of the $K_{1}(1270)^{+}$ meson effective scalar potential $V_{0}$ at
density $\rho_{0}$ indicated in the inset. The arrows indicate the boundary
between the low-momentum and high-momentum regions of the $K_{1}(1270)^{+}$
spectra. Figure 4: (Color online) The same as in Fig. 3, but for the incident
antikaon beam momentum of 2.8 GeV/c. Figure 5: (Color online) The same as in
Fig. 3, but for the incident antikaon beam momentum of 2.5 GeV/c. Figure 6:
(Color online) Ratio between the differential cross sections for
$K_{1}(1270)^{+}$ production on 12C (left) and 184W (right) target nuclei in
the angular range of 0∘–45∘ by 3.5 GeV/c $K^{-}$ mesons, calculated with and
without the $K_{1}(1270)^{+}$ meson in-medium mass shift $V_{0}$ at density
$\rho_{0}$ indicated in the inset, as a function of $K_{1}(1270)^{+}$
momentum. The arrows indicate the boundary between the low-momentum and high-
momentum regions of the $K_{1}(1270)^{+}$ spectra. Figure 7: (Color online)
The same as in Fig. 6, but for the incident antikaon beam momentum of 2.8
GeV/c. Figure 8: (Color online) The same as in Fig. 6, but for the incident
antikaon beam momentum of 2.5 GeV/c. Figure 9: (Color online) Total cross
sections for the production of $K_{1}(1270)^{+}$ mesons from the direct
${K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}$ channel on C and W target nuclei with
momenta of 0.1–1.0 GeV/c (upper two panels) and with all allowed momenta
$\geq$ 0.1 GeV/c at given beam momentum (lower two panels) in the laboratory
polar angular range of 0∘–45∘ by 2.5, 2.8 and 3.5 GeV/c $K^{-}$ mesons as
functions of the $K_{1}(1270)^{+}$ in-medium mass shift $V_{0}$ at normal
nuclear density. The lines are visual guides. Figure 10: (Color online) Ratio
between the total cross sections for the production of $K_{1}(1270)^{+}$
mesons from the direct ${K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}$ channel on 12C
and 184W target nuclei at laboratory angles of 0∘–45∘ with momenta of 0.1–1.0
GeV/c (upper two panels) and with all allowed momenta $\geq$ 0.1 GeV/c at
given beam momentum (lower two panels) by 2.5, 2.8 and 3.5 GeV/c $K^{-}$
mesons, calculated with and without the $K_{1}(1270)^{+}$ in-medium mass shift
$V_{0}$ at normal nuclear density, as function of this shift. The lines are
visual guides.

In Eqs. (6)–(8) we suppose that the direction of the $K_{1}(1270)^{+}$ meson
three-momentum is not changed during its propagation inside the nucleus in the
relatively weak nuclear field, considered in the present work, from the
production point here to the vacuum. As a result, the quantities
$\left<d\sigma_{{K^{-}}p\to K_{1}(1270)^{+}{{\Xi^{-}}}}({\bf p}_{K^{-}},{\bf
p}^{\prime}_{{K_{1}}})/d{\bf p}^{\prime}_{{K_{1}}}\right>_{A}$ and $d{\bf
p}^{\prime}_{{K_{1}}}/d{\bf p}_{{K_{1}}}$, entering into Eq. (5), can be
expressed in the following simple forms $\left<d\sigma_{{K^{-}}p\to
K_{1}(1270)^{+}{{\Xi^{-}}}}(p_{K^{-}},p^{\prime}_{{K_{1}}},\theta_{K_{1}})/p^{\prime
2}_{K_{1}}dp^{\prime}_{{K_{1}}}d{\bf\Omega}_{K_{1}}\right>_{A}$ and
$p^{\prime}_{{K_{1}}}/p_{{K_{1}}}$, where
${\bf\Omega}_{K_{1}}(\theta_{K_{1}},\varphi_{K_{1}})={\bf
p}_{{K_{1}}}/p_{{K_{1}}}$. Here, $\varphi_{K_{1}}$ is the azimuthal angle of
the $K_{1}(1270)^{+}$ momentum ${\bf p}_{K_{1}}$ in the laboratory system.
Taking into account the fact that in the considered incident momentum region
$K_{1}(1270)^{+}$ mesons are mainly ejected, due to the kinematics, in a
narrow cone along the beam line 555) Thus, for instance, at a beam momentum of
3.5 GeV/c the $K_{1}(1270)^{+}$ laboratory production polar angles in reaction
(1) proceeding on the target proton being at rest are $\leq$ 18.5∘.) , we will
calculate the $K_{1}(1270)^{+}$ momentum differential and total production
cross sections on 12C and 184W targets for laboratory solid angle
${\Delta}{\bf\Omega}_{K_{1}}$=$0^{\circ}\leq\theta_{K_{1}}\leq 45^{\circ}$,
and $0\leq\varphi_{K_{1}}\leq 2{\pi}$. Integrating the full inclusive
differential cross section (5) over this angular domain, we can represent the
differential cross section for $K_{1}(1270)^{+}$ meson production in
${K^{-}}A$ reactions from the direct process (1), corresponding to this angle,
in the following form:

$\frac{d\sigma_{{K^{-}}A\to{K_{1}(1270)^{+}}X}^{({\rm
prim})}(p_{K^{-}},p_{K_{1}})}{dp_{K_{1}}}=\int\limits_{{\Delta}{\bf\Omega}_{K_{1}}}d{\bf\Omega}_{K_{1}}\frac{d\sigma_{{K^{-}}A\to{K_{1}(1270)^{+}}X}^{({\rm
prim})}({\bf p}_{K^{-}},{\bf p}_{K_{1}})}{d{\bf p}_{K_{1}}}p_{K_{1}}^{2}$ (24)
$=2{\pi}\left(\frac{Z}{A}\right)\left(\frac{p_{K_{1}}}{p^{\prime}_{K_{1}}}\right)\int\limits_{\cos
45^{\circ}}^{1}d\cos{{\theta_{K_{1}}}}I_{V}[A,\theta_{K_{1}}]\left<\frac{d\sigma_{{K^{-}}p\to{K_{1}(1270)^{+}}{\Xi^{-}}}(p_{K^{-}},p^{\prime}_{K_{1}},\theta_{K_{1}})}{dp^{\prime}_{K_{1}}d{\bf\Omega}_{K_{1}}}\right>_{A}.$

We define now the $K_{1}(1270)^{+}$ meson total in-medium width
$\Gamma_{K_{1}}(|{\bf r}|)$ appearing in Eq. (10) and adopted in our
calculations of $K_{1}(1270)^{+}$ production in $K^{-}A$ interactions.
According to Ref. [8], we can represent it in the following form:

$\Gamma_{K_{1}}(|{\bf r}|)=\Gamma_{K_{1}}+\Gamma_{0}\frac{{\rho_{N}}(|{\bf
r}|)}{{\rho_{0}}},$ (25)

where $\Gamma_{K_{1}}=90$ MeV is the vacuum total $K_{1}(1270)^{+}$ decay
width in its rest frame and $\Gamma_{0}$ is the in-medium change of the
$K_{1}(1270)^{+}$ width at saturation density $\rho_{0}$. A work [8] finds a
linear correlation between the width change $\Gamma_{0}$ and the in-medium
mass shift $V_{0}$, shown by dashed curve in Fig. 2. Analytically, a relation
between the quantities $\Gamma_{0}$ and $V_{0}$ can be easily expressed as
follows:

$\Gamma_{0}=38~{}{\rm MeV}+(38~{}{\rm MeV}/35~{}{\rm
MeV}){\cdot}V_{0},\,\,\,-35~{}{\rm MeV}\leq V_{0}\leq 0~{}{\rm MeV}.$ (26)

One can see that the maximum change of the width is +38 MeV when the
$K_{1}(1270)^{+}$ meson mass remains the same in the medium, while for the
upper limit of its mass shift of -35 MeV the width is not changed here at all.
In these cases, the resulting total in-medium width (25) of the
$K_{1}(1270)^{+}$ meson, depicted by the solid curve in Fig. 2, reaches the
values of 128 and 90 MeV, respectively. So, once the change of the
$K_{1}(1270)^{+}$ meson mass is allowed, its total in-medium width gets
smaller. This will lead to a more larger $K_{1}(1270)^{+}$ decay mean ”free”
path $\lambda_{K_{1}}$, which in turn will cause more weaker absorption of
$K_{1}(1270)^{+}$ mesons in the nuclear matter. Thus, for example, Eq. (10)
shows that for typical values $p^{\prime}_{K_{1}}\approx m^{*}_{K_{1}}$ and
total in-medium $K_{1}(1270)^{+}$ meson decay width in its rest frame of 128
MeV, corresponding to no in-medium effects on its mass, this mean ”free” path
is equal to 1.5 fm. Whereas in the case of maximum change of $K_{1}(1270)^{+}$
mass in the medium, when the above total in-medium width equals to the vacuum
width $\Gamma_{K_{1}}=90$ MeV, the $K_{1}(1270)^{+}$ decay mean ”free” path
$\lambda_{K_{1}}$ is larger and is equal to 2.2 fm. These values are
comparable with the radius of 12C of 3 fm and they are much less than that of
184W of 7.4 fm. The above means that the dropping $K_{1}(1270)^{+}$ mass
scenario will lead to an enhancement (see below) of the yield of the
$K_{1}(1270)^{+}$ mesons in $K^{-}A$ reactions at near-threshold incident beam
momenta of interest both due to in-medium shift of the elementary production
threshold to lower energy and due to their weaker absorption in the nuclear
matter.

## 3 Results and discussion

The model described above makes it possible to calculate the absolute
$K_{1}(1270)^{+}$ momentum differential cross sections from the direct
$K_{1}(1270)^{+}$ production process in $K^{-}$12C and $K^{-}$184W collisions.
These cross sections were calculated according to Eq. (24) for three adopted
values of the $K_{1}(1270)^{+}$ in-medium mass shift $V_{0}$ at density
$\rho_{0}$ at laboratory angles of 0∘–45∘ and for initial antikaon momenta of
3.5, 2.8 and 2.5 GeV/c. They are presented, respectively, in Figs. 3, 4 and 5.
One can see from these figures that the $K_{1}(1270)^{+}$ meson differential
cross sections are notably sensitive to its in-medium mass shift, mostly in
the low-momentum region of 0.1–1.0 GeV/c, for both target nuclei and for all
antikaon momenta considered. Here, the differences between all calculations
corresponding to different options for the $K_{1}(1270)^{+}$ in-medium mass
shift are well separated and experimentally distinguishable. They are
practically similar to each other for each target nucleus at initial antikaon
momenta considered. Thus, for example, for incident $K^{-}$ and outgoing
$K_{1}(1270)^{+}$ meson momenta of 3.5 and 0.5 GeV/c, respectively, and in the
case of 12C nucleus the $K_{1}(1270)^{+}$ yield is enhanced at mass shift
$V_{0}=-20$ MeV by about a factor of 2.2 as compared to that obtained for the
shift $V_{0}=0$ MeV. When going from $V_{0}=-20$ MeV to $V_{0}=-35$ MeV the
enhancement factor is about 1.8. In the case of 184W target nucleus these
enhancement factors are about 3.3 and 2.0. At initial antikaon momentum of 2.8
GeV/c and the same outgoing kaon momentum of 0.5 GeV/c the corresponding
enhancement factors are similar and are about 1.9 and 1.7 as well as 1.7 and
1.4 in the cases of 12C as well as 184W target nuclei, respectively. And for
incident beam momentum of 2.5 GeV/c and final kaon momentum of 0.5 GeV/c these
enhancement factors are about 1.9 and 1.5 as well as 1.7 and 1.5 for 12C as
well as 184W nuclei, correspondingly. However, the $K_{1}(1270)^{+}$ low-
momentum (and high-momentum) production differential cross sections at beam
momentum of 2.5 GeV/c are less than those at antikaon momenta of 2.8 and 3.5
GeV/c by about of one to two orders of magnitude. Therefore, the
$K_{1}(1270)^{+}$ meson differential cross sections measurements in the near-
threshold incident $K^{-}$ momentum region (at 2.8–3.5 GeV/c) will open an
opportunity to shed light on its possible mass shift in cold nuclear matter.
Such measurements could be performed in the future at the J-PARC Hadron
Experimental Facility using the high-intensity and high-momentum (up to 10
GeV/c) separated secondary $K^{-}$ beams in the designed now the K10 beamline
[34, 44].

More detailed information about the sensitivity of the differential cross
sections, presented in Figs. 3, 4 and 5, to the $K_{1}(1270)^{+}$ mass shift
is contained in Figs. 6, 7 and 8, where the momentum dependence of the ratios
of these cross sections calculated for the $K_{1}(1270)^{+}$ mass shift
$V_{0}$ at the central density $\rho_{0}$ to the analogous cross section as
determined at $V_{0}=0$ MeV is shown on a linear scale for carbon and tungsten
nuclei at $K^{-}$ momenta of 3.5, 2.8 and 2.5 GeV/c, respectively 666) We
recall that a comparison of a similar model cross section ratios with data on
$\eta^{\prime}$ meson photoproduction off carbon and niobium nuclei was
employed in [45, 46] to extract its in-medium mass shift at a central nuclear
density.) . It should be pointed out that such relative observables are more
favorable compared to those based on the absolute cross sections for the aim
of obtaining the information on particle in-medium modification, since the
theoretical uncertainties associated with the particle elementary production
cross section essentially cancel out in them. It can be seen from these
figures that at the $K_{1}(1270)^{+}$ meson momenta less than 1.0 GeV/c there
are indeed substantial differences between the results obtained by using
considered options for its in-medium mass shift for both target nuclei and all
adopted initial momenta. Moreover, Figs. 7 and 8 clearly show that the
relative $K_{1}(1270)^{+}$ yield experiences rather noticeable variation for
the mass shift range of $V_{0}=0$ MeV to -35 MeV also in the high-momentum
region (at kaon momenta $\geq$ 1.0 GeV/c) for subthreshold $K^{-}$ momenta of
2.8 and 2.5 GeV/c. Thus, for incident antikaon momentum of 2.8 GeV/c and in
the case of a carbon nucleus, the distinctions between the ratios
corresponding to zero mass shift and shift values of -20 and -35 MeV are of
the order of 30% and 60% in the $K_{1}(1270)^{+}$ momentum range $\sim$
1.0–1.6 GeV/c, where the absolute cross sections are the greatest. In a case
of a tungsten target nucleus, these distinctions are about 40% and 90%,
respectively. At an initial beam momentum of 2.5 GeV/c and the same outgoing
kaon momenta of 1.0–1.6 GeV/c, the corresponding distinctions are larger: they
are about 50% and 90% as well as 60% and 220% in the cases of 12C as well as
184W nuclei, respectively. At the same time, as the antikaon beam momentum
increases to 3.5 GeV/c, the sensitivity of the high-momentum parts of the
ratios (and absolute momentum distributions) considered to variations in the
$K_{1}(1270)^{+}$ in-medium mass shift becomes somewhat lower. Thus, at this
incident momentum and the same final kaon momenta as above, the respective
distinctions, as follows from Fig. 6, are smaller (but yet are measurable)
than at beam momenta of 2.8 and 2.5 GeV/c: they are about 15% and 30% as well
as 20% and 40% for 12C as well as 184W nuclei, correspondingly. This means,
accounting for the above considerations, that the measurements of the
$K_{1}(1270)^{+}$ meson momentum distributions (absolute and relative) might
permit to shed light also on the momentum dependence of its in-medium mass
shift (or of its effective scalar potential $V_{0}$) at saturation density
$\rho_{0}$. It should be noticed that the ratios of differential cross
sections for $K_{1}(1270)^{+}$ meson production on 184W nucleus by 3.5 and 2.8
GeV/c antikaons, shown in Figs. 6 and 7, respectively, exhibit dips at momenta
$\sim$ 0.4–0.5 GeV/c for adopted mass shift values of -20 and -35 MeV. This
can be explained by the fact that the above cross sections ”are bent down” at
these momenta (see Figs. 3 and 4) due to the off-shell kinematics of the
direct $K^{-}p$ collision and the role played by the nucleus-related effects
such as the target proton binding and Fermi motion, encoded in the nuclear
spectral function $P_{A}({\bf p}_{t},E)$. The spectral functions for 12C and
184W, used in our calculations, are different [37–39].

The sensitivity of the $K_{1}(1270)^{+}$ meson production differential cross
sections to its in-medium mass shift $V_{0}$, shown in Figs. 3, 4, 5 and 6, 7,
8, can also be studied from the measurements of the total cross sections for
$K_{1}(1270)^{+}$ production in ${K^{-}}^{12}$C and ${K^{-}}^{184}$W
collisions at the threshold incident $K^{-}$ momenta. Such cross sections,
calculated by integrating Eq. (24) over the $K_{1}(1270)^{+}$ momentum
$p_{K_{1}}$ at 2.5, 2.8 and 3.5 GeV/c antikaon momenta in the low-momentum
region (0.1–1.0 GeV/c) and in the full-momentum region allowed for given beam
momentum are shown in Fig. 9 as functions of the mass shift $V_{0}$. It is
seen from this figure that again the low-momentum region of 0.1–1.0 GeV/c
shows the highest sensitivity to this shift. Thus, for example, the ratios
between the total cross sections for $K_{1}(1270)^{+}$ production by 2.5, 2.8,
3.5 GeV/c $K^{-}$ mesons on 12C and 184W nuclei in this momentum region,
calculated with the shift $V_{0}=-35$ MeV, and the same cross sections as
those obtained with $V_{0}=0$ MeV, are about 2.1, 1.9, 1.6 and 2.3, 2.3, 1.8,
respectively. Whereas the same ratios in the full-momentum regions are only
somewhat smaller: they are about 2.0, 1.6, 1.2 for 12C and 2.2, 2.0, 1.4 for
184W, correspondingly. The highest sensitivity of the total cross sections for
$K_{1}(1270)^{+}$ production in the low-momentum and full-momentum ranges
under consideration to the mass shift $V_{0}$ is observed, as one would
expect, at beam momentum of 2.5 GeV/c. However, the cross sections at this
momentum are small and they are less than those at $K^{-}$ momenta of 2.8 and
3.5 GeV/c by about of one and two orders of magnitude, respectively. Since the
latter ones have a measurable strength $\sim$ 0.01–25 ${\rm\mu}$b, the total
cross section measurements of $K_{1}(1270)^{+}$ meson production on nuclei
both in the low-momentum (0.1–1.0 GeV/c) and in the full momentum regions for
incident antikaon momenta not far from threshold (for momenta $\sim$ 2.8–3.5
GeV/c) with the aim of obtaining information on its mass shift in nuclear
medium look quite promising as well.

The above findings of Fig. 9 that both the low-momentum (0.1–1.0 GeV/c) and
the full-momentum regions reveal a definite sensitivity to the
$K_{1}(1270)^{+}$ in-medium mass shift $V_{0}$ at saturation density
$\rho_{0}$ for incident $K^{-}$ momenta of interest are nicely supported also
by the results presented in Fig. 10. Here, the ratios of the $K_{1}(1270)^{+}$
meson production total cross sections calculated for its mass shift $V_{0}$
and given in Fig. 9 to the analogous cross sections determined at $V_{0}=0$
MeV are shown as functions of this mass shift. It should be mentioned that an
analysis of these ratios, as in the case of those shown in Figs. 6–8, has the
advantage that the theoretical uncertainties associated with the elementary
cross section for particle production substantially cancel out in them. It is
clearly seen from this figure that the highest sensitivity of the ratios in
both considered kinematic ranges to the mass shift $V_{0}$ is indeed observed
at far subthreshold antikaon momentum of 2.5 Gev/c, corresponding to the free
space excess energy of -164 MeV. For instance, at this momentum and for these
ranges the cross section ratios for $V_{0}=-20$ MeV are about 1.5 for both
target nuclei. At this mass shift they are similar to those at momentum of 2.8
GeV/c. The latter ones are about 1.4 and 1.5 for both considered kinematic
ranges for 12C and 184W, respectively. As the $K^{-}$ momentum increases to
3.5 GeV/c, the sensitivity of the total cross section ratios to changes in the
$K_{1}(1270)^{+}$ in-medium mass shift $V_{0}$ decreases. Thus, in the case
where $K_{1}(1270)^{+}$ mesons of momenta of 0.1–1.0 GeV/c are produced by 3.5
GeV/c antikaons incident on 12C and 184W targets, the ratios being considered
for $V_{0}=-20$ MeV take smaller but yet measurable values of about 1.3 and
1.4, respectively. The analogous ratios for the production of the
$K_{1}(1270)^{+}$ mesons in the full-momentum ranges by 3.5 GeV/c $K^{-}$
mesons in 12C and 184W nuclei are somewhat yet smaller: they are about 1.1 and
1.2, respectively.

Therefore, one can conclude that a comparison of the low-momentum and full-
momentum ”integral” results shown in Figs. 9, 10 with the respective precise
experimental data, which could be also obtained in future experiments at
J-PARC [34, 44] employing near-threshold $K^{-}$ beams, should allow to
distinguish between possible weak ($V_{0}$ $\sim$ -20 MeV) and relatively weak
($V_{0}$ $\sim$ -35 MeV) $K_{1}(1270)^{+}$ meson mass shifts in nuclear
matter, considered in the present work.

Finally, accounting for the above considerations, we can conclude that the
$K_{1}(1270)^{+}$ differential and total cross section measurements in
${K^{-}}A$ reactions at initial momenta not far from threshold (at momenta
$\sim$ 2.8–3.5 GeV/c) both in the $K_{1}(1270)^{+}$ meson low-momentum
(0.1–1.0 GeV/c) and in its full-momentum ranges will allow us to shed light on
the possible $K_{1}(1270)^{+}$ in-medium mass shift.

## 4 Summary

Our present study was aimed at studying the possibility of extracting
information about the predicted in-medium change in the $K_{1}(1270)^{+}$
meson mass. For this, we calculated the absolute differential and total cross
sections for the production of $K_{1}(1270)^{+}$ mesons on 12C and 184W target
nuclei at laboratory angles of 0∘–45∘ by $K^{-}$ mesons with momenta of 2.5,
2.8 and 3.5 GeV/c, which are close to the threshold momentum (2.95 GeV/c) for
$K_{1}(1270)^{+}$ meson production off the free target proton at rest. These
calculations have been performed within a nuclear spectral function approach,
which describes incoherent direct $K_{1}(1270)^{+}$ meson production in
$K^{-}$ meson–proton ${K^{-}}p\to{K_{1}(1270)^{+}}\Xi^{-}$ production
processes and accounts for three different options for its in-medium mass
shift (or for its effective scalar potential) at central density $\rho_{0}$.
We show that the differential and total (absolute and relative)
$K_{1}(1270)^{+}$ antikaon-induced production cross sections at initial
momenta not far from threshold – at momenta $\sim$ 2.8–3.5 GeV/c, at which
there is yet no strong drop in their strength, reveal a distinct sensitivity
to changes in the in-medium shift of the $K_{1}(1270)^{+}$ mass, studied in
the paper, both in the $K_{1}(1270)^{+}$ meson low-momentum (0.1–1.0 GeV/c)
and in its full-momentum ranges. This would permit evaluating this shift.
Experimental data necessary for this aim can be obtained in a dedicated
experiment at the J-PARC Hadron Experimental Facility.

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