††Address: My Ismail University of Meknes, Department of Mathematics B. P. 11
201 Zitoune, Meknès, Morocco††Emails<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
# On the rationality conjecture of some finite CW-complexes

A. Boudjaj and Y. Rami

###### Abstract.

In this paper, we establish the rationality conjecture raised in [7] for any
$(r-1)$-connected ($r\geq 2$) $kr$-dimensional CW-complex $X$ ($k\geq 2$)
having a unique spherical cohomology class $u\in\tilde{H}^{r}(X,\mathbb{Z})$
such that $u^{k}\not=0$. Next, we illustrate (topologically) our result by
giving the minimal cell structure of such a CW-complex whose cohomology is a
truncated polynomial algebra.

###### Key words and phrases:

LS-category, (Higher) Topological complexity, James generalized Hopf
invariants, Zero-divisor-cup-length.

###### 1991 Mathematics Subject Classification:

Primary 55P10; Secondary 55M30

##  1\. Introduction and statement of the results

Throughout this paper, unless stated otherwise, $X$ will denote an
$(r-1)$-connected finite CW-complex $X$ ($r\geq 2$) and $\mathbb{K}$ an
arbitrary field.

Very recently, M. Farber et al. have posed in [7], the following rationality
conjecture

######  Conjecture 1.0.1.

The $TC$-generating function
$\mathcal{F}_{X}(x)=\sum_{n=1}^{n=\infty}TC_{n+1}(X)x^{n}$ is a rational
function with a single pole of order 2 at $x=1$. More explicitly, there exists
$P\in\mathbb{Z}[X]$ such that:

$\mathcal{F}_{X}(x):=\sum_{n=1}^{n=\infty}TC_{n+1}(X)x^{n}=\frac{P(x)}{(1-x)^{2}}.$

To make precise terms of $\mathcal{F}_{X}(x)$, recall that the sectional
category (or Schwartz’s genus [15]) of a fibration $F\rightarrow
E\stackrel{{\scriptstyle p}}{{\rightarrow}}B$ is the minimal integer $k\geq 0$
denoted $secat(p)$ such that there exists an open cover $B=U_{0}\cup
U_{1}\cup\ldots\cup U_{k}$ with the property that over each set $U_{i}$ the
fibration admits a continuous section. $secat(-)$ is indeed an homotopy
invariant. Special cases useful for us are the (higher) topological
complexities $TC_{n}(X)=secat(\pi_{n})$ ($n\geq 2$) introduced firstly by M.
Farber in [5] for $n=2$ and generalized for any $n\geq 2$ by Y. Rudyak in [14]
(see also [2]). Here

$\begin{array}[]{cccc}\pi_{n}:&P(X)&\rightarrow&X^{n}\\\
&\gamma&\mapsto&(\gamma(0),\gamma(\frac{1}{n}),\ldots,\gamma(\frac{n-1}{n}),\gamma(1))\end{array}$

is the fibrational substitute for the iterated diagonal map
$\Delta_{n}:X\rightarrow X^{n}$. In fact, $\pi_{n}$ describes a motion
planning algorithm of a physical system moving in its configuration space $X$
between any two positions (the input and the output) but having to reach $n-2$
intermediate stats. Referring to [7, Lemma 1], if $X$ satisfies (1.0.1) then
for all $n$ large enough $TC_{n+1}(X)-TC_{n}(X)$ becomes constant. The
invariants $TC_{n}(X)$ are usually difficult to determine and are often
approximated by the algebraic invariants cup-length and zero-divisors-cup-
length defined respectively as follows:

$cup_{\mathbb{K}}(X)=sup\\{j\geq 1,\;\exists
a_{1},a_{2},\ldots,a_{j}\in\tilde{H}^{*}(X,\mathbb{K})\;\hbox{such
that}\;a_{1}a_{2}\ldots a_{j}\not=0\\}$

and

$zcl_{n}(X,\mathbb{K})=sup\\{j\geq 1,\;\exists a_{1},\;\ldots,\;a_{j}\in
Ker(\smile_{n,\mathbb{K}})\;\hbox{such that}\;a_{1}\ldots a_{j}\not=0\\}$

where $\smile_{n,\mathbb{K}}:H^{*}(X,\mathbb{K})^{n\otimes}\rightarrow
H^{*}(X,\mathbb{K})$ denotes the graded algebra multiplication generalizing
the standard cup-product $u\smile 1_{\mathbb{K}}=\smile_{2,\mathbb{K}}$. These
will be subsequently used.

Our goal in this paper is to study the conjecture (1.0.1) for any
$(r-1)$-connected finite CW-complex ($r\geq 2$) satisfying the following
condition:

(1) ${\it\dim
X=kr,\;\hbox{and}\;\tilde{H}^{r}(X,\mathbb{Z})\cong\mathbb{Z}u,\;\hbox{with}\;u^{k}\not=0\;\hbox{some}\;k\geq
2.}$

Using the Universal Coefficients Theorem and the Hurewicz Theorem we obtain
$\tilde{H}^{r}(X,\mathbb{Z})\cong
Hom_{\mathbb{Z}}(\tilde{H}_{r}(X,\mathbb{Z}),\mathbb{Z})\stackrel{{\scriptstyle{hur^{\vee}_{X}}{\cong}}}{{\longleftarrow}}Hom_{\mathbb{Z}}(\pi_{r}(X),\mathbb{Z}).$
It results that, if $a\in\tilde{H}_{r}(\mathbb{S}^{r},\mathbb{Z})$ and
$[f]\in\pi_{r}(X)\cong\mathbb{Z}$ are generating homology and homotopy classes
respectively, then, $<f^{*}(u),a>=<u,f_{*}(a)>\not=0$ and consequently,
$f^{*}(u)\not=0$. Thus $u\in\tilde{H}^{r}(X,\mathbb{Z})$ is indeed a spherical
cohomology class [10, Lemma 3.1].

To establish our main result, we have to determine the characteristics of the
fields $\mathbb{K}$ for which $u^{k}_{\mathbb{K}}$ is non-zero. Then, we
specify among the latter, those that realize equalities
$TC_{n}(X)=zcl_{n}(X,\mathbb{K})$ ($\forall n\geq 2$). From this last
condition, there arises the integers $\lambda_{(3,k)}=\sum_{0\leq i\leq
k}(-1)^{i}(C_{k}^{i})^{3}$ which depend only on $k$.

Based on [7, Theorem 1], we establish the following

######  Proposition 1.0.2.

Let $X$ be an $(r-1)$-connected finite CW-complex satisfying (1). Then $X$
verifies (1.0.1) provided there exists a field $\mathbb{K}$ with
characteristic zero or a prime number not dividing $\lambda_{(3,k)}$ (resp.
$2\lambda_{(3,k-1)}$) when $k$ is even (resp. when $k$ is odd)

As a immediate consequence of Dirichly’s Theorem, we state our main result as
follows:

######  Theorem 1.0.3.

Any $(r-1)$-connected finite CW-complex satisfying (1) verifies the conjecture
(1.0.1).

Using similar arguments we obtain the following (cf. Remark 3.2.1 below for
the notion of $r$-admissible fields):

######  Theorem 1.0.4.

If $X$ is an $(r-1)$-connected $kr$-dimensional CW-complex whose cohomology
over any choice of $r$-admissible field $\mathbb{K}$ is a truncated polynomial
algebra, i.e. $H^{*}(X,\mathbb{K})\cong\mathbb{K}[v]/(v^{k+1})$, then $X$
verifies the conjecture (1.0.1).

Examples of spaces verifying the above theorem are those satisfying in
addition to (1) the condition: $\dim H^{*}(X,\mathbb{K})=k$ for every
$\mathbb{K}$ with $char(\mathbb{K})\in\mathcal{P}_{(i)}$ (resp.
$char(\mathbb{K})\in\mathcal{P}_{(ii)}$) (cf. §2 for the definitions of
$\mathcal{P}_{(i)}$ and $\mathcal{P}_{(ii)}$).

In §2, we determine fields $\mathbb{K}$ for which $u^{k}_{\mathbb{K}}\not=0$.
§3 is devoted to proofs of our main results and in §4 we describe minimal
structures of $(r-1)$-connected $kr$-dimensional CW-complexes such that
$H^{*}(X,\mathbb{K})\cong\mathbb{K}[u_{\mathbb{K}}]/(u^{k+1}_{\mathbb{K}})$
for any choice of $r$-admissible field $\mathbb{K}$. We end the paper by §5 by
giving a range of examples of $kr$-dimensional CW-complexes of the form
$K=\mathbb{S}^{r}\cup_{\beta_{1}}e^{2r}\cup\ldots\cup_{\beta_{r}}e^{kr}$ which
satisfy (1.0.1).

##  2\. Fields satisfying $u^{k}_{\mathbb{K}}\not=0$

Throughout this section, $X$ will denote an $(r-1)$-connected finite CW
complex satisfying condition (1).

Recall that the cup-product can be defined in terms of the Alexander-Whitney
diagonal approximation:

(2) $\smile:H^{p}(X,{G_{1}})\otimes H^{q}(X,{G_{2}})\rightarrow
H^{p+q}(X,{G_{1}\otimes G_{2}})$

given, in general, for any groups $G_{1}$ and $G_{2}$ by:

$(a\smile b)(\sigma)=a(\sigma_{\mid[v_{0},v_{1},\ldots v_{p}]})\otimes
b(\sigma_{\mid[v_{p},v_{p+1},\ldots v_{p+q}]}).$

Therefore, by putting in (2) $G_{1}=\mathbb{Z}$ and $G_{2}=\mathbb{K}$; any
field and $u_{\mathbb{K}}:=u\smile 1_{\mathbb{K}}$, we obtain
$u_{\mathbb{K}}^{k}=u^{k}\smile 1_{\mathbb{K}}$. Thereafter, we will make use
of the identification:

$H^{kr}(X,\mathbb{Z})=\mathbb{Z}^{m_{k}}\oplus(\mathbb{Z}/p_{1}^{\alpha_{1}}\mathbb{Z})\oplus(\mathbb{Z}/p_{2}^{\alpha_{2}}\mathbb{Z})\oplus\ldots\oplus(\mathbb{Z}/p_{{\theta(k)}}^{\alpha_{\theta(k)}}\mathbb{Z}).$

and will consider the family:

$\mathcal{P}_{u^{k}}:=\\{p_{1},p_{2},\ldots,p_{{\theta(k)}}\\}.$

Next, although $u\in\tilde{H}^{r}(X,\mathbb{Z})$ has infinite additive order,
it may exist $m\in\mathbb{N}^{*}$ such that $mu^{l+1}=0$ for some $\;2\leq
l\leq k-1$ (i.e. in the chain level, $mu^{l+1}=\delta v$ for some $v\in
C^{{(l+1)}r+1}(X)$). Thus, we should discus two cases depending on the
additive order $o(u^{k})$ of $u^{k}$. In all what follows, for any group $G$,
its free part will be denoted by $Free(G)$.

  1. (i)

If $o(u^{k})$ is infinite so that $Free(H^{lr}(X,\mathbb{Z}))\not=0$ for each
$\;1\leq l\leq k$, condition (1) implies $u^{k}_{\mathbb{K}}\not=0$ for all
fields $\mathbb{K}\in\mathcal{P}\backslash\mathcal{Q}_{u^{k}}$, where
$\mathcal{Q}_{u^{k}}:=\\{q_{1},q_{2},\ldots,q_{{\gamma(k)}}\\}$ is such that
$u^{k}=q_{1}^{f_{1}}q_{2}^{f_{2}}\ldots
q_{{\gamma(k)}}^{f_{\gamma(k)}}w;\;w\in{\mathbb{Z}^{m_{k}}}$.

  2. (ii):

If $o(u^{k})$ is finite, so that $mu^{l+1}=0$ for some $m\in\mathbb{N}^{*}$
and $\;2\leq l\leq k-1$. Hence, $mu^{k}=0$ and therefore
$o(u^{k})=:p_{1}^{\beta_{1}}p_{2}^{\beta_{2}}\ldots
p_{{\beta(k)}}^{\beta_{\theta(k)}}/m$ such that for any $1\leq
i\leq\theta(k)$, we have $0\leq\beta_{i}\leq\alpha_{i}$ and at least one
$\beta_{i}>0$. Thus, on one hand, for $\mathcal{Q}_{u^{l}}$ being defined
similarly as $\mathcal{Q}_{u^{k}}$, condition (1) implies that
$u^{l}_{\mathbb{K}}\not=0$ for all $\mathbb{K}^{{}^{\prime}s}$ such that
$char(\mathbb{K})\in\mathcal{P}\backslash\mathcal{Q}_{u^{l}}$ where
$\mathcal{P}$ stands for the set of all prime numbers. On the other one, for
any field whose characteristic lies to
$\mathcal{P}\backslash\mathcal{P}_{u^{k}}$;
$u^{l+1}_{\mathbb{K}}\not=0,\ldots,u^{k}_{\mathbb{K}}\not=0$.

In all what follows we put

$\mathcal{P}(i)=\\{0\\}\cup\mathcal{P}\backslash\mathcal{Q}_{u^{l}}\;\hbox{and}\;\mathcal{P}(ii)=\mathcal{P}\backslash(\mathcal{P}_{u^{k}}\cup\mathcal{Q}_{u^{l}}).$

As a Consequence, we have established the following

######  Lemma 2.0.1.

$\dim Span\\{u_{\mathbb{K}},u^{2}_{\mathbb{K}},\ldots,u^{k}_{\mathbb{K}}\\}=k$
if and only if $char(\mathbb{K})\in\mathcal{P}(i)$ or else
$char(\mathbb{K})\in\mathcal{P}(ii)$.

##  3\. Proofs of our main results

The following lemma will clarify the behavior of $zcl_{n}(X,\mathbb{K})$
towards $char(\mathbb{K})$ and the parity of the nil-potency order $k$. For
any $n\geq 3$ we put:

$A_{1}=u_{\mathbb{K}}\otimes 1\otimes\dots\otimes 1,\;A_{2}=1\otimes
u_{\mathbb{K}}\otimes 1\otimes\dots\otimes
1,\;\dots,\;A_{n}=1\otimes\dots\otimes 1\otimes u_{\mathbb{K}}.$
$\xi_{(n,k)}=[(\prod_{i=2}^{i=n}(A_{1}-A_{i}))(A_{2}-A_{3})]^{k}$

and

$\mu_{(n,k)}=\xi_{(n,k-1)}[(A_{1}-A_{2})(A_{1}-A_{3})\ldots(A_{1}-A_{n})]=\xi_{(n,k)}/(A_{2}-A_{3}).$

######  Lemma 3.0.1.

Let $\mathbb{K}\in\mathcal{P}(i)$ (resp. $\mathcal{P}(ii)$). Then, for any
$n\geq 3$:

  1. (i)

$\;\;\;\xi_{(n,2k)}=\lambda_{(3,2k)}u^{k}_{\mathbb{K}}\otimes\dots\otimes
u^{k}_{\mathbb{K}}\;\;\;$ and $\;\;\;\xi_{(n,2k+1)}=0$.

  2. (ii)

$\;\;\;\mu_{(n,2k+1)}(A_{1}-A_{n})=2(-1)^{n-1}\lambda_{(3,2k)}u^{k}_{\mathbb{K}}\otimes\dots\otimes
u^{k}_{\mathbb{K}}.$ where:

$\lambda_{(3,2k)}=2[\sum_{0\leq i\leq
k}(-1)^{i}(C_{2k}^{i})^{3}]+(C_{2k}^{k})^{3}.$

###### Proof.

Notice first the following inductive formulas for any $n\geq 3$, $k\geq 2$ and
$l\geq 2$:

$\xi_{(n+1,k)}=\xi_{(n,k)}(A_{1}-A_{n+1})^{k}\;\;\;\;\hbox{and}\;\;\;\xi_{(n,k+l)}=\xi_{(n,k)}\xi_{(n,l)}.$

Now, since $u^{j}_{\mathbb{K}}=0,\forall j>k$, a straightforward calculus
gives

$\xi_{(n,k)}=\lambda_{(n,k)}u^{k}_{\mathbb{K}}\otimes\dots\otimes
u^{k}_{\mathbb{K}}$

(some constant $\lambda_{(n,k)}\in\mathbb{Z}$). Thus, using again
$u^{j}_{\mathbb{K}}=0,\forall j>k$ we deduce that in the relation
$\xi_{(n+1,k)}=\xi_{(n,k)}(A_{1}-A_{n+1})^{k}$, the term $(-1)^{k}A_{n+1}^{k}$
is the only one to be retained from teh equality
$(A_{1}-A_{n+1})^{k}=\sum_{i=0}^{i=k}(-1)^{k-i}C_{k}^{i}A_{1}^{i}A_{n+1}^{k-i}$.
Hence:

$\xi_{(n+1,k)}=(-1)^{k}\xi_{(n,k)}A_{n+1}^{k}=(-1)^{k}\lambda_{(n,k)}u^{k}_{\mathbb{K}}\otimes\dots\otimes
u^{k}_{\mathbb{K}}$

(with $n+1$ factors $u^{k}_{\mathbb{K}}$) so
$\lambda_{(n+1,k)}=(-1)^{k}\lambda_{(n,k)}=\ldots=(-1)^{(n-2)k}\lambda_{(3,k)}.$
Consequently, we vave:

(3) $\lambda_{(n,k)}=(-1)^{(n-1)k}\lambda_{(3,k)},\forall n\geq 3.$

Now, by using Newton’s formula for each term in

$\xi_{(3,k)}=(-1)^{k}(A_{1}-A_{2})^{k}(A_{3}-A_{2})^{k}(A_{2}-A_{3})^{k},$

we get the deterministic coefficient:

$\lambda_{(3,k)}=\sum_{0\leq i\leq k}(-1)^{i}(C_{k}^{i})^{3}.$

Thus,

$\lambda_{(n,2k)}=2[\sum_{0\leq i\leq
k}(-1)^{i}(C_{2k}^{i})^{3}]+(C_{2k}^{k})^{3}\;\;\;\hbox{and}\;\;\;\lambda_{(n,2k+1)}=0.$

Next, consider
$\mu_{(n,k)}=\xi_{(n,k-1)}[(A_{1}-A_{2})(A_{1}-A_{3})\ldots(A_{1}-A_{n})]$.
Formulas lying coefficients and roots in
$Q(A_{1})=(A_{1}-A_{2})(A_{1}-A_{3})\ldots(A_{1}-A_{n})$ and the constraints
$u^{j}_{\mathbb{K}}=0,\forall j>k$ lead us to take only into account the
constant coefficient $q_{0}=(-1)^{n-1}A_{2}A_{3}\ldots A_{n}$ as well as that
of $A_{1}$, i.e. that of $q_{1}=(-1)^{n-2}\sum_{2\leq i_{1}<\ldots<i_{n-2}\leq
n}\limits A_{i_{1}}\ldots A_{i_{n-2}}$. Therefore,

$\mu_{(n,k)}=(-1)^{n-1}\lambda_{(n,k-1)}[\widehat{u^{k-1}_{\mathbb{K}}}\otimes
u^{k}_{\mathbb{K}}\ldots\otimes
u^{k}_{\mathbb{K}}-\sum_{j=2}^{j=n}u^{k}_{\mathbb{K}}\otimes\ldots\otimes
u^{k}_{\mathbb{K}}\otimes\widehat{u^{k-1}_{\mathbb{K}}}\otimes
u^{k}_{\mathbb{K}}\dots\otimes u^{k}_{\mathbb{K}}]$

where $\widehat{u^{k-1}_{\mathbb{K}}}$ means that this factor is in the
$j^{-th}$ place. It is then immediate that

$\;\;\;\mu_{(n,2k+1)}(A_{1}-A_{n})=2(-1)^{n-1}\lambda_{(3,2k)}u^{k}_{\mathbb{K}}\otimes\dots\otimes
u^{k}_{\mathbb{K}}.$

∎

###  3.1. Proof of Proposition 1.0.2

The following theorem will be subsequently used:

######  Theorem 3.1.1.

[4, 2]: For any $s$-connected CW-complex $X$, any field $\mathbb{K}$ and any
integer $n\geq 2$ we have

$cup_{\mathbb{K}}(X)\leq cat_{LS}(X)\leq\frac{\dim
X}{s+1}\;\hbox{and}\;zcl_{n}(X,{\mathbb{K}})\leq TC_{n}(X)\leq\frac{n\dim
X}{s+1}.$

###### Proof.

By the above theorem, $(r-1)$-connectedness with $r\geq 2$ and dimension $kr$
of $X$ with $k\geq 2$ imply that $cat_{LS}(X)\leq k$ and $TC_{n}(X)\leq nk$
for every $n\geq 2$. We continue considering the following steps (we fix
$\mathbb{K}$ a field satisfying hypothesis of Proposition 1.0.2):

  1. (1)

By Lemma 2.0.1 we have $u^{k}_{\mathbb{K}}\not=0$, hence, $k\leq
cup_{\mathbb{K}}(X)\leq cat_{LS}(X)$ [4, Proposition 1.5] and so
$cup_{\mathbb{K}}(X)=k=cat_{LS}(X)$.

  2. (2)

If $n=2$ one shows easily (for any $\mathbb{K}$) that
$\xi_{(2,k)}=(A_{1}-A_{2})^{2k}\not=0$, thus, $2k\leq zcl_{2}(X,{\mathbb{K}})$
and therefore $zcl_{2}(X,{\mathbb{K}})=2k=TC_{2}(X)$.

  3. (3)

Assume for the rest that $n\geq 3$:

(i) if $k$ is even, $k\geq 2$, using Lemma 3.0.1, we see that

$\xi_{(n,k)}\not=0\Leftrightarrow\lambda_{(n,k)}\not=0\Leftrightarrow\lambda_{(n,k)}\wedge
char(\mathbb{K})=1$

in which case we have $nk\leq zcl_{n}(X,{\mathbb{K}})$. Therefore,
$zcl_{n}(X,{\mathbb{K}})=nk=TC_{n}(X)$.

(ii) if $k$ is odd, the same argument using $\mu_{(n,k)}(A_{1}-A_{n})$ gives
us the same equality $zcl_{n}(X,{\mathbb{K}})=nk=TC_{n}(X)$.

To conclude, it remain to use [7, Theorem 1]. ∎

###  3.2. Proof of Theorem 1.0.3

###### Proof.

Proposition 1.0.2 is satisfied for any field whose characteristic lies to
$\mathcal{P}_{(i)}$ or else $\mathcal{P}_{(ii)}$ and does not divide the
integer $\lambda_{(3,k)}$ or $\lambda_{(3,k-1)}$ depending on the parity of
$k$. It suffice then to make use of Dirichely’s theorem by which we may fix
such a field. ∎

Next, to establish Theorem 1.0.4 recall that a field $\mathbb{K}$ is said
$r$-admissible if there exists an $(r-1)$-connected CW-complex $X$ such that
$H^{*}(X,\mathbb{K})\cong\mathbb{K}[v]/(v^{k+1})$. Referring to [18, Theorem
2] and [11, Theorem 4L.9 and the remark just after] we note on one hand that
if $char(\mathbb{K})=p$, a prim number, then the nil-potency order $k$ of the
indeterminate $v$ should be greater than $p-1$ and on the other we resume in
the following remark complete results concerning this notion:

######  Remark 3.2.1.

(a): Let $\mathbb{K}$ be a field of characteristic zero or a prim $p$. Then,
$\mathbb{K}$ is $r$-admissible

  1. (1)

for every even $r\geq 2$ if characteristic is zero.

  2. (2)

for $r\in\\{2,\;4\\}$ if $p\geq 2$.

  3. (3)

for $r=8$ if $p=2$ or an arbitrary odd prim of the form $p=4s+1$ some $s\geq
1$.

  4. (4)

for every even $r\notin\\{2,4,8\\}$ if $p$ is an arbitrary odd prim of the
form $p=\frac{r}{2}s+1$ some $s\geq 1$.

(b): According to Dirichly’s theorem, there are infinitely many primes in any
arithmetic progression $p=\frac{r}{2}s+1$, that is, to each integer $r\geq 2$
it is associated infinitely many $r$-admissible fields.

####  3.2.1. Proof of Theorem 1.0.4

Notice first that with hypothesis of Theorem 1.0.4 and Remark 3.2.1 above,
Lemma 2.0.1 and Lemma 3.0.1 are satisfied for any choice of $r$-admissible
field $\mathbb{K}$ when we replace $v$ by $u_{\mathbb{K}}$.

###### Proof.

It suffice to use successively similar arguments as in proves of Proposition
1.0.2 and Theorem 1.0.3. ∎

##  4\. Minimal cell structure of CW-complexes with truncated polynomial
cohomology

Our goal in this section is to give a " simultaneously " example of spaces $X$
satisfying (1) and having a truncated polynomial algebra cohomology over any
choice of $r$-admissible field $\mathbb{K}$ i.e.
$H^{*}(X,\mathbb{K})\cong\mathbb{K}[u_{\mathbb{K}}]/(u^{k+1}_{\mathbb{K}})$.
These (as it is mentioned in the introduction) form a particular class among
those given by Theorem 1.0.4. Simple examples of such spaces are
$\mathbb{C}P^{k}$ and $\mathbb{H}P^{k}$ [11, Corollary 4L. 10]. Notice however
that real projective spaces $\mathbb{R}P^{2k}$ and $\mathbb{R}P^{2k+1}$ do not
satisfy (1) since
$H^{*}(\mathbb{R}P^{2k},\mathbb{Z})\cong\mathbb{Z}[\alpha]/(2\alpha,\alpha^{k+1})$;
$|\alpha|=2$ and
$H^{*}(\mathbb{R}P^{2k+1},\mathbb{Z})\cong\mathbb{Z}[\alpha,\beta]/(2\alpha,\alpha^{k+1},\beta^{2},\alpha\beta)$;
$|\alpha|=2$ and $|\beta|=2k+1$ [11].

More examples are among CW-complexes of the form
$L=\mathbb{S}^{n_{1}}\cup_{\beta_{2}}e^{n_{2}}\cup\ldots\cup_{\beta_{k-1}}e^{n_{k-1}}$,
with $n_{i+1}-1\geq n_{i}\geq 2$. Indeed, I. M. James associated to any
$\beta_{k}\in\pi_{n_{k}-1}(L)$ such that $n_{k}=n_{k-1}+n_{1}$ its generalized
Hopf invariant $h_{k}([\beta_{k}])$ to be the integer $m_{k}$ satisfying
$x_{1}x_{{k-1}}=m_{k}x_{k}$. Here $x_{i}$ stands for the generating element in
$H^{n_{i}}(L\cup_{\beta_{k}}e^{n_{k}},\mathbb{Z})$. He then showed that this
correspondence defines an homomorphism
$h_{k}:\pi_{n_{k}-1}(L)\rightarrow\mathbb{Z}$. As a particular case, if
$n_{1}=r\geq 2$ and $n_{i+1}=n_{i}+n_{1}$ for each $1\leq i\leq k-1$, we get
$K^{\prime}=\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup\ldots\cup_{\beta_{k-1}}e^{(k-1)r}$
and $h_{k}:\pi_{rk-1}(K^{\prime})\rightarrow\mathbb{Z}$ is an homomorphism
extending the usual Hopf invariant
$H=:h_{2}:\pi_{2r-1}(\mathbb{S}^{r})\rightarrow\mathbb{Z}$. Thereafter, we
denote
$K=\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup\ldots\cup_{\beta_{r}}e^{kr}$
obtained from $K^{\prime}$ and $\beta_{k}$ via the relation
$x_{1}x_{k-1}=h_{k}([\beta_{k}])x_{k}=:m_{k}x_{k}$ ($k\geq 3$). Now, by using
the long cohomology exact sequences of pairs

$(\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup\ldots\cup_{\beta_{i}}e^{ir}\;,\;\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup\ldots\cup_{\beta_{i+1}}e^{(i+1)r})$

we exhibit a family of isomorphisms

$H^{ir}(\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup\ldots\cup_{\beta_{i+1}}e^{(i+1)r})\cong
H^{ir}(\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup\ldots\cup_{\beta_{i}}e^{ir})$

($2\leq i\leq k-1$). Knowing that
$H^{r}(\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r})\cong
H^{r}(\mathbb{S}^{r})\cong\mathbb{Z}x_{1}$ we may identify each $x_{i}$ with
its antecedent, hence, if we put $H([\beta_{2}])=h_{2}([\beta_{2}])=:m_{2}$,
we see by induction that :

$m_{2}\ldots m_{k-1}m_{k}x_{k}=x_{1}^{k}.$

Consequently, every CW-complex
$K=\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup\ldots\cup_{\beta_{r}}e^{kr}$ whose
attaching maps are such that $m_{2}\ldots m_{k-1}m_{k}\not=0$ satisfies (1)
and $o(x_{1}^{k})$ is infinite since the one on $x_{k}$ is (due to the
hypothesis $r\geq 2$). Moreover, (cf. §2),
$\mathcal{Q}_{x_{1}^{k}}=\\{q_{i}\in\mathcal{P},\;:\;q_{i}/m_{2}\ldots
m_{k-1}m_{k}\\}\;\hbox{and}\;\mathcal{P}_{(i)}=\mathcal{P}\backslash\mathcal{Q}_{x_{1}^{k}}.$

Next, put $L=K\cup_{\beta^{\prime}_{l+1}}e^{(l+1)r+1}$ (some $l<k$) such that
(in cellular homology) the cell $e^{(l+1)r+1}$ satisfies

$\delta(e^{(l+1)r+1})=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots
p_{{\theta(k)}}^{\alpha_{\theta(k)}}e^{(l+1)r}.$

It is clear that $L$ satisfies (1) with $x_{1}^{l+1}$ and consequently
$x_{1}^{k}$ of finite order, and
$\mathcal{Q}_{x_{1}^{l}}=\\{q_{i}\in\mathcal{P},\;:\;q_{i}/m_{2}\ldots
m_{l-1}m_{l}\\}\;\hbox{and}\;\mathcal{P}_{(ii)}=\mathcal{P}\backslash(\mathcal{P}_{x_{1}^{k}}\cup\mathcal{Q}_{x_{1}^{l}}).$

Our main result in this section is given with the following terms already used
in [11, Proposition 4C.1] :
$\tilde{e}^{n}$ will denote the pair of cells
$(e^{n+1},e^{n},\delta(e^{n+1})=q^{t}e^{n})$ where $q^{t}$ is the additive
order of the homology (hence the cohomology) class induced by a summand
$\mathbb{Z}/q\mathbb{Z}$.

######  Theorem 4.0.1.

Let $X$ be a finite $kr$-dimensional CW-complex satisfying (1) with cohomology
a truncated polynomial algebra
$H^{*}(X,\mathbb{K})\cong\mathbb{K}[u_{\mathbb{K}}]/(u^{k+1}_{\mathbb{K}})$
for any choice of $r$-admissible field $\mathbb{K}$.

  * (a)

If $o(u^{k})$ is infinite, then $X$ is homotopy equivalent to a CW-complex
$K=\mathbb{S}^{r}\cup_{\eta_{2}}e^{2r}\cup\ldots\cup_{\eta_{k}}e^{kr}$ such
that $h_{2}(\eta_{2})h_{3}(\eta_{3})\ldots
h_{k}(\eta_{k})=q_{1}^{f_{1}}q_{2}^{f_{2}}\ldots
q_{{\gamma(k)}}^{f_{\gamma(k)}}$ where $q_{i}\in\mathcal{Q}_{u^{k}}$.

  * (b)

If $o(u^{k})$ is finite and $l\geq 2$ is the greatest power of $u$ with
infinite order, then $X$ is homotopy equivalent to a CW-complex
$K=L\cup_{\varrho}L^{\prime}$ where
\- $L=\mathbb{S}^{r}\cup_{\eta_{2}}e^{2r}\cup\ldots\cup_{\eta_{l}}e^{kl}$ such
that $h_{2}(\eta_{2})h_{3}(\eta_{3})\ldots
h_{l}(\eta_{l})=q_{i_{1}}^{g_{i_{1}}}q_{i_{2}}^{g_{i_{2}}}\ldots
q_{i_{\gamma(l)}}^{g_{i_{\gamma(l)}}}$ with $q_{i_{j}}\in\mathcal{Q}_{u^{l}}$
and $0\leq g_{i_{j}}\leq f_{i_{j}}$ and
\- $L^{\prime}$ is formed, for each $p_{i}\in\mathcal{P}_{u^{k}}$, by $k-l$
cells: an $\tilde{e}_{i}^{jr}$ or else an $\tilde{e}_{i}^{jr-1}$ for each
$l+1\leq j\leq k-1$, each of these cells is of finite additive order
${p_{i}}^{\alpha^{\prime}_{i}}$ (${\alpha^{\prime}_{i}}\geq{\alpha_{i}}$) and
an $\tilde{e}_{i}^{kr-1}$ of finite additive order ${p_{i}}^{\alpha_{i}}$.

###### Proof.

We note at the beginning that, by Lemma 2.0.1, every $r$-admissible field is
indeed in $\mathcal{P}_{(i)}$ or else in $\mathcal{P}_{(ii)}$. This explains
notations in the statement of the theorem.

  * (a)

Assume that $o(u^{k})$ is infinite. By hypothesis on the cohomology of $X$ the
UCT with $\mathbb{Q}$-coefficients, condition (1) and Remark 3.2.1 imply that
$H_{ir}(X,\mathbb{Z})\cong\mathbb{Z}$ for each $0\leq i\leq k$. Therefor $X$
is homotopy equivalent to the CW-complex
$K=\mathbb{S}^{r}\cup_{\eta_{2}}e^{2r}\cup\ldots\cup_{\eta_{k}}e^{kr}$ whose
attaching maps satisfy the condition $h_{2}(\eta_{2})h_{3}(\eta_{3})\ldots
h_{k}(\eta_{k})=q_{1}^{f_{1}}q_{2}^{f_{2}}\ldots
q_{{\gamma(k)}}^{f_{\gamma(k)}}$ where $q_{i}\in\mathcal{Q}_{u^{k}}$.

  * (b)

Assume that $o(u^{k})$ is finite and let $l\geq 2$ be the greatest power of
$u$ with infinite order. First of all, since the powers $u^{i},\;1\leq i\leq
l$ are in the free part of $\tilde{H}^{*}(X,\mathbb{Z})$, we should introduce
cells $e^{0},\;e^{r},\;\ldots,\;e^{lr}$ in the minimal structure of $X$. These
form $L=\mathbb{S}^{r}\cup_{\eta_{2}}e^{2r}\cup\ldots\cup_{\eta_{l}}e^{kl}$
such that $h_{2}(\eta_{2})h_{3}(\eta_{3})\ldots
h_{l}(\eta_{l})=q_{i_{1}}^{g_{i_{1}}}q_{i_{2}}^{g_{i_{2}}}\ldots
q_{i_{\gamma(l)}}^{g_{i_{\gamma(l)}}}$ with $q_{i_{j}}\in\mathcal{Q}_{u^{l}}$
and $0\leq g_{i_{j}}\leq f_{i_{j}}$.

Now, by Lemma 2.0.1, in order to realize hypothesis on the cohomology of $X$,
the cells $e^{{(l+1)}r},\;\ldots,\;e^{kr}$ should be introduced for every
$p=char(\mathbb{K})\in\mathcal{P}_{(ii)}=\mathcal{P}\backslash(\mathcal{P}_{u^{k}}\cup\mathcal{Q}_{u^{l}})$.
Hence, in order to rich the (full dimension)
$\dim\tilde{H}^{*}(X,\mathbb{K})=k$, the UCT:

$H^{ir}(X,\mathbb{K})\cong Hom(H_{ir}(X,\mathbb{Z}),\mathbb{K})\oplus
Ext(H_{ir-1}(X,\mathbb{Z}),\mathbb{K})$

imposes to add (for each $p_{i}\in\mathcal{P}_{(ii)}$):
\- $k-l-1$-generator cells $e^{{jr}}_{i}$ or else $e^{{jr-1}}_{i}$ ($l+1\leq
j\leq k$) and its corresponding relater-cells $e^{{jr+1}}_{i}$ or else
$e^{{jr}}_{i}$ satisfying
$\delta(e^{{jr+1}}_{i})={p_{i}}^{\alpha^{\prime}_{i}}e^{jr}_{i}$ or else
$\delta(e^{{jr}}_{i})={p_{i}}^{\alpha^{\prime}_{i}}e^{jr-1}_{i}$ with
$\alpha^{\prime}\geq\alpha$.
\- a cell of degree $rk$ which should came from the Ext term, that is from a
pair ($e^{rk},\;e^{rk-1}$) such that
$\delta(e^{{jr}}_{i})={p_{i}}^{\alpha_{i}}e^{jr-1}_{i}$. We then obtain
$L^{\prime}$ as it is stated in the theorem.

∎

##  5\. Examples and a final remark

To have an idea on sequences of prime numbers introduced in Proposition 1.0.2,
we give in the table below (Table 1) all prime factors of $\lambda_{(3,k)}$
for $2\leq k\leq 40$. This in possible thanks to the Scientific Workplace 5.5
software:

Table 1. List of prime factors of $\lambda_{(3,k)}$ up to $40$ k | prime factors of $\lambda_{(3,k)}$
---|---
2 | 2, 3
4 | 2, 3, 5
6 | 2, 3, 5, 7
8 | 2, 3, 5, 7, 11
10 | 2, 3, 7, 11, 13
12 | 2, 3, 7, 11, 13, 17
14 | 2, 3, 5, 11, 13, 17, 19
16 | 2, 3, 5, 11, 13, 17, 19, 23
18 | 2, 3, 5, 11, 13, 17, 19, 23
20 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
22 | 2, 3, 5, 7, 13, 17, 19, 23, 29, 31
24 | 2, 3, 5, 7, 13, 17, 19, 23, 29, 31
26 | 2, 3, 5, 7, 17, 19, 23, 29, 31, 37
28 | 2, 3, 5, 17, 19, 23, 29, 31, 37, 41
30 | 2, 3, 5, 11, 17, 19, 23, 29, 31, 37, 41, 43
32 | 2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47
34 | 2, 3, 5, 7, 11, 19, 23, 29, 31, 37, 41, 43, 47
36 | 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53
38 | 2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 53
40 | 2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 53, 59

###  5.1. Examples

The following theorem that we recall in our notations ($h_{i}^{r}$ is denoted
above by $h_{i}$!) is paramount to get examples of CW-complexes of the form
$K=\mathbb{S}^{r}\cup_{\beta_{2}}e^{2r}\cup_{\beta_{3}}\ldots\cup_{\beta_{k}}e^{kr}$
satisfying the conjecture (1.0.1):

######  Theorem 5.1.1.

[3, Thorem 1]

$Im(h_{i}^{r})=\Bigg{\\{}\begin{array}[]{lr}\mathbb{Z},&\hbox{if}\;r=2,\;4,\,8\;\hbox{and}\;i=2\\\
\mathbb{Z},&\hbox{if}\;r=2\;\hbox{and i a prime number}\\\
i\mathbb{Z},&\hbox{otherwise}\end{array}\Bigg{.}$

######  Example 5.1.2.

  1. (1)

For $r=2$ and $k=2$, the space $K=\mathbb{S}^{2}\cup_{\beta_{2}}e^{4}$, where
$\beta_{2}:\mathbb{S}^{3}\rightarrow\mathbb{S}^{2}$ is the Hopf map, that is
such that $h_{2}(\beta_{2})=1$, clearly satisfies Proposition 1.0.2 for any
field of characteristic $p\notin\\{2,3\\}$.

  2. (2)

For $r=2$ and $k=3$, let
$\beta_{3}=[i_{2},[i_{2},i_{2}]]:\mathbb{S}^{5}\rightarrow\mathbb{S}^{2}\cup_{\beta_{2}}e^{4}$
satisfying $h_{3}(\beta_{3})=3$. The space

$K=\mathbb{S}^{2}\cup_{\beta_{2}}e^{4}\cup_{\beta_{3}}e^{6}$

satisfies hypothesis of Proposition 1.0.2 for any field of characteristic
$p\notin\\{2,3\\}$.

  3. (3)

For $r=4$ and $k=3$, let $\beta_{2}:\mathbb{S}^{7}\rightarrow\mathbb{S}^{4}$
be the Hopf map, that is such that $h_{2}(\beta_{2})=1$, and
$\beta_{3}=[i_{4},[i_{4},i_{4}]]:\mathbb{S}^{11}\rightarrow\mathbb{S}^{4}\cup_{\beta_{2}}e^{8}$
satisfying $h_{3}(\beta_{3})=3$. The space

$K=\mathbb{S}^{4}\cup_{\beta_{2}}e^{8}\cup_{\beta_{3}}e^{12}$

satisfies hypothesis of Proposition 1.0.2 for any field of characteristic
$p\notin\\{2,3\\}$.

  4. (4)

For $r=4$ and $k=4$, in addition to $\beta_{2}$ and $\beta_{3}$ of the above
item, consider
$\beta_{4}:\mathbb{S}^{15}\rightarrow\mathbb{S}^{4}\cup_{\beta_{2}}e^{8}\cup_{\beta_{3}}e^{12}$,
provided by Theorem 5.1.1, such that $h_{4}(\beta_{4})=4$. Hence, the space

$K=\mathbb{S}^{4}\cup_{\beta_{2}}e^{8}\cup_{\beta_{3}}e^{12}\cup_{\beta_{4}}e^{16}$

satisfies hypothesis of Proposition 1.0.2. for any field of characteristic
$p\notin\\{2,3,5\\}$.

  5. (5)

For $r=8$ and $k=7$, let $\beta_{2}:\mathbb{S}^{15}\rightarrow S^{8}$ be the
Hopf map, that is with $h_{2}(\beta_{2})=1$ and consider the maps
$\beta_{3}:\mathbb{S}^{23}\longrightarrow S^{8}\cup_{2}e^{16}$,
$\beta_{4}:\mathbb{S}^{31}\rightarrow\mathbb{S}^{8}\cup_{2}e^{16}\cup_{3}e^{24}$,
$\beta_{5}:\mathbb{S}^{39}\rightarrow\mathbb{S}^{8}\cup_{2}e^{16}\cup_{3}e^{24}\cup_{4}e^{32}$,
$\beta_{6}:\mathbb{S}^{47}\rightarrow\mathbb{S}^{8}\cup_{2}e^{16}\cup_{3}e^{24}\cup_{4}e^{32}\cup_{5}e^{40}$
and
$\beta_{7}:\mathbb{S}^{55}\rightarrow\mathbb{S}^{8}\cup_{2}e^{16}\cup_{3}e^{24}\cup_{4}e^{32}\cup_{5}e^{40}\cup_{6}e^{48}$,
provided by Theorem 5.1.1, such that $h_{3}(\beta_{3})=3$,
$h_{4}(\beta_{4})=4$, $h_{5}(\beta_{5})=5$, $h_{6}(\beta_{6})=6$ and
$h_{7}(\beta_{7})=7$. Hence, the space

$K=\mathbb{S}^{8}\cup_{\beta_{2}}e^{16}\cup_{\beta_{3}}e^{24}\cup_{\beta_{4}}e^{32}\cup_{\beta_{5}}e^{40}\cup_{\beta_{6}}e^{48}\cup_{\beta_{7}}e^{56}$

satisfies hypothesis of Proposition 1.0.2 for any field of characteristic
$p\notin\\{2,3,5,7\\}$.

  6. (6)

For $r=8$ and $k=8$ we consider $\beta_{2}$, $\beta_{3}$, $\beta_{4}$,
$\beta_{5}$, $\beta_{6}$, $\beta_{7}$ given in the previous item and
$\beta_{8}:\mathbb{S}^{63}\rightarrow\mathbb{S}^{8}\cup_{\beta_{2}}e^{16}\cup_{\beta_{3}}e^{24}\cup_{\beta_{4}}e^{32}\cup_{\beta_{5}}e^{40}\cup_{\beta_{6}}e^{48}\cup_{\beta_{7}}e^{56}$
such that $h_{8}(\beta_{8})=8$. Hence, the space

$K=\mathbb{S}^{8}\cup_{\beta_{2}}e^{16}\cup_{\beta_{3}}e^{24}\cup_{\beta_{4}}e^{32}\cup_{\beta_{5}}e^{40}\cup_{\beta_{6}}e^{48}\cup_{\beta_{7}}e^{56}\cup_{\beta_{8}}e^{64}$

satisfies hypothesis of Proposition 1.0.2 for any field of characteristic
$p\notin\\{2,3,5,7,11\\}$.

  7. (7)

The above examples correspond to cases where $r=2,4,8$. Now, by Theorem 5.1.1
when $r\not=2,\;4,\;8$ we obtain a CW-complex $K$ such that the product of its
generalized Hopf invariants is $h_{2}(\beta_{2})\ldots h_{k}(\beta_{k})=k!$.
In order to illustrates the impact of $\lambda_{(3,k)}$ in the process, we
finish by fixing $r=6$ and taking some arbitrary values of $k$. The table
below dresses fields characteristics to exclude when

$K=\mathbb{S}^{6}\cup_{\beta_{2}}e^{12}\cup_{\beta_{3}}\ldots\cup_{\beta_{k}}e^{6k}:$

Table 2. Finite characteristic $p$ not allowed, for some values of $k$ ($r=6$) k |  finite characteristic $p$ not allowed
---|---
4 | 2, 3, 5
5 | 2, 3, 5
16 | 2, 3, 5, 7, 11, 13, 17, 19, 23
18 | 2, 3, 5, 7, 11, 13, 17, 19, 23
20 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
22 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

###  5.2. Final remark

In this paper, we have treated the case where
$u\in\tilde{H}^{r}(X,\mathbb{Z})$ in condition (1) is the unique spherical
cohomology class (thus of infinite order). To achieve our approach in studying
the rationality conjecture for finite CW-complexes $X$ satisfying the
relations $zcl_{n}(X)=TC_{n}(X)\;\forall n\geq 2$, it remains, on the one
hand, to deal with the case where $u\in\tilde{H}^{r}(X,\mathbb{Z})$ is of
finite order (e.g. $\mathbb{R}P^{n}$) and, on the other hand, to consider more
than one generating cohomology class in $\tilde{H}^{r}(X,\mathbb{Z})$
spherical or not. This is our main project for the future.

## References

  * [1]
  * [2] I. Basabe, J. González, Y. B. Rudyak, and D. Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topol. Volume 14, Number 4 (2014), 2103-2124.
  * [3] J. H. Baues, Hopf Invariants For Reduced Products of Spheres, Pro. of The Ame. Math. Soc Volume S9, Number 1, August 1976.
  * [4] O. Cornea, G. Lupton, J. Oprea, D. Tanré, Lusternik-Scnirelmann Category, Mathematical Surveys and Monographs, Vol. 103 (2003).
  * [5] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29, 211–221 (2003) (MR1957228).
  * [6] M. Farber, Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zurich (2008) (MR2455573)
  * [7] M. Farber, D. Kishimoto, and D. Stanley, Generating functions and topological complexity, Topology Appl. 278 (2020), 107235.
  * [8] M. Farber, S. Tabachnikov, and S. Yuzvinsky, Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 34, 1853 – 1870 (2003).
  * [9] J. González , M. Grant, and L. Vandembroucq, Hopf Invariants for sectional category with applications to topological robotics, Quarterly Journal of Mathematics, 70(4) (2019), 1209-1252. https://doi.org/10.1093/qmath/haz019. Homology Homotopy Appl. 15(2) (2013), 73-81.
  * [10] L. Guijarro, T. Schick, and G. Walschap, Bundles with spherical Euler class, Pacific J. of Math., Vol. 27, No. 2, (2002), 377-392. 19\. 974.
  * [11] A. Hatcher, Algebraic Topology. Cambridge University Press, 2002.
  * [12] I. M. James, Note on cup-products, Proc. Amer. Math. Soc. 8 (1957), 374-383. MR 19\. 974.
  * [13] D. Kishimoto and A. Yamaguchi On The Growth of Topological Complexity, arXiv 205.09847v1 [math.AT] 20 May 2020.
  * [14] Y. Rudyak, On higher analogous of topological complexity, Topology and its Applications Volume 157, Issue 5, 1 April 2010, Pages 916-920.
  * [15] A. Schwarz, The genus of a fiber space, Am. Math. Soc. Trans. 55, 49–140 (1966) (MR0154284).
  * [16] A. Shar, The Homotopy Groups of Spaces whose Cohomology is a Truncated Polynomial Algebra, Proceedings of the American Mathematical Society Volume 38, Number 1, March 1973.
  * [17] H. Toda, p-primary components of homotopy groups IV. Compositions and toric constructionS, Memories of the College of Science, Univ. of Kyoto, A Vol. XXXII, Mathematics No. 2, 1959.
  * [18] H. Toda, Note on Cohomology Ring of Certain Spaces, Proceedings of the American Mathematical Society Vol. 14, No. 1 (Feb., 1963), pp. 89-95.
  * [19] J.H.C Whitehead, A certain exact sequence, Ann. Math. 52 (1950), 51-110, 2012, pp. 15-21.


# Collision-induced radiative quenching and other disintegration modes of the
$2s$ state of muonic hydrogen and deuterium atoms

V.P. Popov<EMAIL_ADDRESS>V.N. Pomerantsev  pomeran@nucl-
th.sinp.msu.ru Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, 119991 Moscow, Russia

###### Abstract

The formation and various disintegration modes of $2s$ states for muonic
hydrogen and deuterium atoms at kinetic energies both above and below the
$2s-2p$ threshold are studied. The cross sections of the collision-induced
radiative quenching, elastic scattering, and Coulomb deexcitation of
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms in collisions with ordinary
hydrogen and deuterium atoms at collision energies below the $2s-2p$ threshold
are calculated in the framework of the close-coupling approach. The basis set
includes all open and closed channels corresponding to exotic-atom states with
principal quantum number $n$ from 1 up to 30. Results of these numerical
quantum-mechanical calculations of cross sections are used as input data for
detailed cascade calculations. The kinetics of atomic cascade for $\mu^{-}p$
and $\mu^{-}d$ atoms is studied in the wide range of the relative target
densities, $\varphi=10^{-8}-1$ applying the improved version of the cascade
model. It is shown that the collision-induced radiative quenching gives a
significant contribution to the total disintegration of the $2s$ state of
muonic atoms at energies below the $2s-2p$ threshold. The initial population
of the $2s$ state for both muonic hydrogen and deuterium atoms, absolute
intensities and probabilities of the different disintegration modes as well as
their lifetimes are calculated for the fractions with kinetic energies above
and below the $2s-2p$ threshold. The obtained results are compared with the
known experimental data.

##  I Introduction

The negatively charged muons stopped in $H_{2}$ or $D_{2}$ gaseous target form
$\mu^{-}p$ or $\mu^{-}d$ atoms in highly excited states KP ; KPF ; JC . The
formation of muonic atoms of hydrogen isotopes is followed by a number of the
subsequent radiative and collisional processes. In the process of
deexcitation, an $\mu^{-}p$ or $\mu^{-}d$ atom passes through many
intermediate states until it reaches the ground state or weak decay of the
meson occurs ($\mu^{-}\to e^{-}+\bar{\nu_{e}}+\nu_{\ mu}$) in the excited
state of the atom. During this deexcitation cascade, a fraction of $\mu^{-}p$
or $\mu^{-}d$ atoms reaches the $2s$ state. The number of atoms reached $2s$
state is determined by several factors: the initial
$(n,\,l,\,E)$-distributions (e.g., see PPForm ; PPIsot ), the competition of
collisional processes and radiative transitions during the deexcitation
cascade, as well as the target temperature and density.

Muonic hydrogen and muonic deuterium atoms are of special interest among
exotic atoms due to their simplest structure and possibility to investigate a
number of problems both the exotic atom physics and the bound-state QED. The
$2s$ state plays a particular role in these atoms due to $(2s-2p)$ Lamb shift,
$E_{L}=202.0$ meV, and has no analog in hadronic atoms. A high-energy
component of $(\mu p)_{1s}$ with kinetic energy $\sim 0.9$ keV was discovered
RPD ; PPK  from the analysis of the time-of-flight spectra (at low gas
pressures $p_{H_{2}}$ = 16 and 64 hPa) and attributed to non-radiative
quenching of the $2s$ state due to the formation of the muonic molecule in a
resonant $(\mu p)_{2s}+{\rm H}_{2}$ collision and subsequent Coulomb
deexcitation of the $(pp\mu)^{+}$ complex (see WJKF  and references therein).
Theoretical estimation of the quenching rate in the framework of this model
gives $\sim 5\times 10^{10}$ $s^{-1}$ at liquid-hydrogen density (LHD=4.25
$\times 10^{22}$ atoms/cm3) that is about an order of magnitude smaller than
the value $4.4^{+2.1}_{-1.8}\times 10^{11}s^{-1}$ extracted from experiments
PPK . In our paper PPForm  the observed collisional quenching and high-energy
$(\mu p)_{1s}$ component was explained by the direct Coulomb deexcitation (CD)
process,

$(\mu^{-}p)_{2s}+{\rm H}\to(\mu^{-}p)_{1s}{\rm(0.9keV)}+{\rm H},$ (1)

The problem of the $(\mu^{-}p)_{2s}$ collisional quenching in the hydrogen
target has been studied previously both experimentally Eg81 ; HA2 ; Breg ;
FKott ; RPohl ; RPD  and theoretically KL71 ; Muel75 ; CF77 ; CohB81 ;
CascalII ; PPForm . Experimental investigations with muonic hydrogen and
deuterium atoms at Paul Scherrer Institute PDH ; Ludh ; Natur ; Mud  performed
at a low gas pressure have revived interest in the subject of the formation
and disintegration modes of $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms at
kinetic energies both above and below the $2s-2p$ threshold.

The theoretical description of the collisional processes and kinetics of the
atomic cascade was improved significantly in our works (e.g., see PPForm ;
PPKin ; KPP ; diff ; pion ; PPInHyp ; PPInd ; PPIsot  and refs. therein).

In the collision of muonic hydrogen or deuterium atoms in the $2s$ state with
the target atom or molecule, a strong coupling between this state and the
unstable with respect to the radiative $2p\to 1s$ transition $2p$ state leads
to the collision-induced radiative quenching of the $2s$ state at kinetic
energies below the $2s-2p$ threshold. The consequences of this mechanism can
be observed in both the absolute x-ray yield of the $K\alpha$ line and the
lifetime of the muonic hydrogen and deuterium atoms in the $2s$ state.
According to our knowledge, the process of the collision-induced radiative
quenching of the $(\mu p)_{2s}$ state has been treated in the framework of the
semiclassical approaches Muel75 ; CohB81  and was never taken into account in
cascade calculations, since the delayed $K_{\alpha}$ x-rays induced during the
collisions have not been experimentally observed HA2 .

In this paper, we investigate the further history of $(\mu^{-}p)$ and
$(\mu^{-}d)$ atoms formed in the $2s$ state with kinetic energies both above
and below the $2s-2p$ threshold. Their fate is determined by the interplay of
a few processes: the elastic $2s-2s$ scattering, the $2s\to 2p$ Stark
transition followed by the fast radiative $2p\to 1s$ deexcitation, the
collision-induced radiative quenching, the Coulomb deexcitation $2s\to 1s$,
and $\mu$-decay.

The paper is organized as follows. Section II presents the results of
calculations of the cross sections for elastic $2s-2s$ scattering, $2s-2p$
Stark transitions, collision-induced radiation quenching, and $2s-1s$ Coulomb
deexcitation for muonic hydrogen and deuterium atoms performed in the
framework of the close-coupling approach. Results of cascade calculations are
presented in Sec. III. Initial populations and kinetic energy distributions of
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms are presented in Sec. III A.
Collisional rates for $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms and
intensities of the different modes of their disintegration are given in Secs.
III B and III C, correspondingly. The contributions of different processes,
forming the absolute x-ray yield of $K\alpha$ line are discussed in Sec. III
D. Calculated probabilities of different disintegration modes and a lifetime
of $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms with energies both above and
below the $2s-2p$ threshold are presented in Sec. III E. The results are
summarized in Sec. IV.

Atomic units are used throughout unless otherwise stated.

##  II Cross sections of collisional processes for muonic hydrogen and
deuterium atoms

In our paper PPIsot , the cross sections of all processes at collisions of the
excited $\mu^{-}p$ and $\mu^{-}d$ atoms with hydrogen isotope atoms in the
ground state have been presented. In these calculations, performed in the
framework of the close-coupling approach, the radiative widths or instability
of the excited states of muonic atoms and the possibility of the collision-
induced radiative transitions were not taken into account. This approximation
is completely justified for all excited $nl$ states except the $2p$ state.
Indeed, the radiative widths of highly excited states are very small and the
rates of the collision-induced radiative transitions are negligible compared
to the rate of the direct radiative dipole transitions $nl\to n^{\prime}l\pm
1$.

The situation is quite different in the case of the $2s$ state. At collision
energies below the $2s-2p$ threshold, the Stark transition $2s\to 2p$ is
energetically forbidden and the collision-induced radiative quenching

$(\mu^{-}a)_{2s}+A\to(\mu^{-}a)_{1s}+A+\gamma(1.9\,\rm{keV})$ (2)

($a=p$ or $a=d$, and $A=H,D$) may prove to be an important mechanism of the
disintegration of the $2s$ state. Therefore, in this paper devoted to the fate
of the $2s$ state in muonic hydrogen and deuterium atoms, the cross sections
of the elastic scattering

$(\mu^{-}a)_{2s}+A\to(\mu^{-}a)_{2s}+A$ (3)

and Coulomb deexcitation (CD)

$(\mu^{-}a)_{2s}+A\to(\mu^{-}a)_{1s}(0.9\,\rm{keV})+A(1.0\,\rm{keV})$ (4)

in $(\mu^{-}p)_{2s}+H$ and $(\mu^{-}d)_{2s}+D$ collisions calculated earlier
PPForm ; PPIsot  at energies below the $2s-2p$ threshold were recalculated
taking into account the radiation width or the instability of the $2p$ state.

Calculations were performed in the framework of the closed coupling approach
by using the propagation matrix method described in details in our paper
PPForm . The main advantage of this method is that it allows to describe both
open and closed unstable-state channels as well as to avoid numerical errors
caused by the exponentially increasing linearly-independent solutions of the
system of the close-coupling equations. This method has been developed and
applied earlier by the authors, in particular, for the description of the
collision-induced absorption or annihilation in the case of hadronic atoms
PPInd ; PPInHyp .

The present calculations were performed with the extended basis set including
all states of the muonic hydrogen (deuterium) atom with the principal quantum
number values from $n=1$ up to $n_{max}=30$. The value $\Gamma_{2p}=80$
$\mu$eV of the radiative width of the $2p$ state was used for both muonic
hydrogen and muonic deuterium atoms. All other states were considered as
stable ones. In the case of the $2s$ state at kinetic energies below the
$2s-2p$ threshold, the $S$ matrix in the subspace of the open channels is not
unitary due to the instability of the $2p$ state. The unitary defect allows to
determine the cross section of the collision-induced radiative transition from
the $2s$ state as follows:

$\sigma^{\rm
ind}_{2s}=\frac{\pi}{k^{2}}\sum_{J,n^{\prime}=1,2}(2J+1)\left(\delta_{2sJ,n^{\prime}sJ}-\left|S^{J}_{2s\to
n^{\prime}s}\right|^{2}\right).$ (5)

Here, $k^{2}=2M_{r}E_{\rm cm}$, $E_{\rm cm}$ is the relative motion energy in
the entrance channel, and $M_{r}$ is the reduced mass of the scattering
problem.

Cross sections for $(\mu^{-}p)_{2s}+H$ collisions calculated with
$\Gamma_{2p}=80$ $\mu$eV and $\Gamma_{2p}=0$ are shown in Fig. 1.

Figure 1:  The energy dependence of cross sections for $(\mu^{-}p)_{2s}+H$
collisions: elastic scattering $2s-2s$, Coulomb deexcitation $2s\to 1s$,
collision-induced radiative quenching, and Stark $2s\to 2p$ transition. Cross
sections of the elastic scattering and Coulomb deexcitation calculated with
$\Gamma_{2p}=0$ PPForm  are shown by dashed lines for comparison.

As can be seen from Fig.1, the cross sections of the elastic scattering and CD
practically do not change when the radiative width $\Gamma_{2p}$ of the $2p$
state is taken into account except for the resonance region. This means that
the width can be considered as a small perturbation, and the cross section of
the collision-induced process is proportional to the width. The dependence of
the cross section of the collision-induced process on the value $\Gamma$ has
been studied in detail in our paper PPInd . At the same time, in the resonance
region, the cross sections of the elastic scattering and the CD are strongly
suppressed (by several times in comparison with results presented in PPForm )
due to the process (2) of the collision-induced radiative quenching. In the
resonance region, the cross section of the process (2) even exceeds the cross
section of the elastic scattering.

Figure 2:  Cross sections of the collision-induced radiative quenching for
$(\mu^{-}p)_{2s}+H$ collisions vs the laboratory kinetic energy below the
$2s-2p$ threshold: the solid line and dashed-dotted line are the results of
the present calculations with the basis sets $n_{max}=30$ and $n_{max}=2$,
respectively, and the dashed line is obtained from Fig. 4 of CohB81 .

The comparison of the collision-induced radiative cross sections calculated in
the present fully quantum-mechanical approach with the cross sections
calculated in CohB81  within a simple model of the potential scattering with
the complex potential is shown in Fig.2. Here the cross section of the
collision-induced radiative quenching calculated with the simplest basis set
in which only wave functions of the $1s$, $2s$, and $2p$ states of the muonic
atom were included is also presented. In the paper CohB81 , the Born-
Oppenheimer approximation was applied using the four-term basis set with the
lowest $1s$ and $2s$ orbitals in variational calculations. In such
approximation, the scattering problem is reduced to a simple model of the
potential scattering with the complex potential.

Figure 2 shows that the cross section of the collision-induced radiative
quenching calculated in CohB81  is too small even in comparison with our
close-coupling result obtained with the minimal basis set in which only the
$1s$, $2s$, and $2p$ muonic-atom wave functions were taken into account.
Besides, the cross section of collision quenching calculated in Muel75  is
somewhat less in comparison with the result of CohB81 . Therefore, the
adiabatic description and correspondingly the Born-Oppenheimer approximation
is not justified for the realistic treatment of the collision-induced
radiative quenching of the $2s$ state in $(\mu^{-}p)_{2s}+H$ collisions at
kinetic energies below the $2s-2p$ threshold.

Figure 3 shows the energy dependence of the cross sections for
$(\mu^{-}d)_{2s}+D$ collisions. In contrast to the case of the
$(\mu^{-}p)_{2s}$, there are no pronounced sub-threshold resonances, and,
therefore, taking into account the 2p-state width does not at all affect the
elastic scattering and CD cross sections.

Figure 3:  The same as in Fig. 1 for $(\mu^{-}d)_{2s}+D$ collisions. Cross
sections of the elastic scattering and Coulomb deexcitation calculated with
$\Gamma_{2p}=0$ from PPIsot  are shown by dashed lines.

##  III Results of cascade calculations

The cascade calculations for muonic hydrogen and muonic deuterium atoms were
performed in the framework of the improved version of the cascade model
developed in our recent papers PPForm ; PPIsot . In particular, the
probability densities of $n$ and $l$ initial (at the instant of the exotic
atom formation) distributions as well as of the initial kinetic energy
distribution (see Eqs. 22-24 in PPIsot ) were used in the present cascade
calculations. Besides, the exact kinematic of the binary collision taking into
account the target motion was applied in the present cascade calculations (for
detail see PPIsot ). Finally, the present cascade model includes new results
given in Sec. II for cross sections of the elastic scattering, collision-
induced radiative quenching, and the CD for $(\mu^{-}p)_{2s}$ and
$(\mu^{-}d)_{2s}$ at kinetic energies below the $2s-2p$ threshold.

To obtain good statistics, the fate of the $10^{8}$ muonic atoms has been
analyzed in the present cascade calculations for each value of the target
density in the relative density range, covering nine orders of magnitude
$\varphi=10^{-9}-1$, where $\varphi$ is the target density in units of the
liquid hydrogen density, $N_{lhd}=4.25\cdot 10^{22}$ atoms/cm3.

The various characteristics of muonic hydrogen and deuterium atoms in the $2s$
state were calculated: the arrival population and kinetic energy distribution,
the intensities and probabilities of the different disintegration modes, and
lifetimes. Some of present results are compared with the available
experimental data HA2 ; PDH ; RPD ; Mud .

###  III.1 Initial population and kinetic energy distribution of
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$

We define the total initial or arrival population of the $2s$ state,
$\varepsilon^{\rm tot}_{2s}$, as the probability that the formed muonic
hydrogen or deuterium atom will reach this state during the deexcitation
cascade regardless of whether its kinetic energy is above or below of the
$2s-2p$ threshold.

The density dependence of the total arrival population for $(\mu^{-}p)_{2s}$
and $(\mu^{-}d)_{2s}$ atoms calculated using the present version of the atomic
cascade is shown in Fig. 4. Calculations were performed at relative hydrogen
density $\varphi=(10^{-9}-1)$ and at target temperature $T=300$ K.

Figure 4:  The total arrival population $\varepsilon^{\rm tot}_{2s}$ of
$\mu^{-}p$ (solid line) and $\mu^{-}d$ (dashed line) atoms vs the density of
the target calculated at a target temperature $T=300$ K. The experimental data
for are from HA2 .

The total arrival populations for the $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$
have similar dependences on the target density. They are about 2% in the
density range $\varphi=(10^{-9}-10^{-7})$ and grow up to 88% and 57% at a
liquid hydrogen density for $\mu^{-}p$ and $\mu^{-}d$ atoms, respectively. In
the case of muonic hydrogen the density dependence of the $\varepsilon^{\rm
tot}_{2s}$ has been explained in detail in the paper PPForm . The same
explanation is also valid for the muonic deuterium.

At densities below $10^{-7}$, the role of collisional processes in the
deexcitation cascade is extremely weak and the arrival population of the $2s$
state is mainly determined by $(n,l)$ distributions at the instant of the
exotic atom formation and the rates of radiative transitions from the higher
$n\,p$ states ($n\geqslant 3$). So, in this density range, the arrival
population $\varepsilon^{\rm tot}_{2s}$ is equal to about 2% and is very
weakly dependent on the density for both muonic hydrogen and muonic deuterium
atoms.

In the density range $\varphi=10^{-7}-2.5\cdot 10^{-4}$, the arrival
population $\varepsilon^{\rm tot}_{2s}$ increases from about 2% up to 6.77%
and 6.34% for $\mu^{-}p$ and $\mu^{-}d$ atoms, respectively. In this density
range, the total population of the $2s$ state grows due to an increase of the
intensity of all collisional processes, which in turn lead to a larger
population of $np$ states ($n\geqslant 5$) and accordingly to an increase of
the intensity of the radiative $np\to 2s$ transitions.

This explanation is in accordance with the density dependence of the relative
x-ray yields, $Y_{i}=I(K_{i}/I(K_{tot})$, for $K_{i}$ lines
($i=\alpha,\beta,\gamma$, etc) (see also Figs. 17 and 18 in PPIsot ) and a
strong correlation between the relative yields of $K_{i}$ lines and the total
population $\varepsilon^{\rm tot}_{2s}$. At densities below $2.5\cdot
10^{-4}$, the calculated arrival population $\varepsilon^{\rm tot}_{2s}$ is in
perfect agreement with the estimation based on formula (see Eq. (21) in PPForm
)

$\varepsilon^{\rm rad}_{2s}=0.134(1-Y_{\alpha})+0.01Y_{>\beta}.$ (6)

At these densities, the calculated values of $\varepsilon^{\rm tot}_{2s}$ are
in very good agreement with the experimental data obtained from the measured
relative $X$-ray yields HA2  at $H_{2}$ gas pressures between 0.25 and 150
Torr ($T=300$ K) that corresponds to the range of the relative densities from
$3.8\cdot 10^{-7}$ up to $2.3\cdot 10^{-4}$.

At densities above $2.5\cdot 10^{-4}$, the populations $\varepsilon^{\rm
tot}_{2s}$ and $\varepsilon^{\rm rad}_{2s}$ have quite different dependencies
on the density. The population $\varepsilon^{\rm rad}_{2s}$ reaches its
maximum value 7.7% and 7.5% at $\varphi=2\cdot 10^{-3}$ and then rapidly
decreases to about 2.4% and 1.2% at $\varphi=1$, respectively for $\mu^{-}p$
and $\mu^{-}d$ atoms. Contrary to this, the arrival population
$\varepsilon^{\rm tot}_{2s}$ in the density range $\varphi=2\cdot 10^{-3}-1$
rapidly grows from 10.4% (for $\mu^{-}p$) and 8.6% (for $\mu^{-}d$) up to 88%
and 57% at a liquid hydrogen density for $\mu^{-}p$ and $\mu^{-}d$ atoms,
respectively. It is important to note, that these values of the total arrival
population significantly exceed the statistical weight of the $2s$ state (25%)
especially in the case of the muonic hydrogen.

In the density range $10^{-4}\lesssim\varphi\lesssim 3\cdot 10^{-3}$, Stark
transitions increase the population of $np$ states ($3\leqslant n\leqslant 5$)
from which the radiative $np\to 2s$ transitions populate the $2s$ state. This
explanation is supported by the density dependence of the absolute x-ray
yields of the $K\beta,K\gamma$ and $K\delta$ lines (see Fig. 15 in PPIsot ).

At a density above $\approx 3\cdot 10^{-3}$, the CD $5\to 4$, $4\to 3$, and
$3\to 2$ together with Auger and radiative transitions result in very fast
increasing of the population $\varepsilon^{\rm tot}_{2s}$. Besides, at the
density $\varphi\gtrsim 10^{-2}$ the rate of the Stark $2p\to 2s$ transition
is comparable with the rate of the radiative transition $2p\to 1s$ and becomes
even about two orders of the magnitude more at a liquid hydrogen density
($\varphi~{}=~{}1$). Therefore, at the density above $10^{-2}$ the initial
population of the $2s$ state is mainly formed by the Auger transition $3p\to
2s$, the CD $3l\to 2s\,(l=0,1,2)$, and the always open Stark $2p\to 2s$
transition.

The kinetic energy of the muonic atom changes during the atomic cascade as a
result of the complex interplay of various cascade processes preceding its
arrival in the $2s$ state. In cascade calculations, the kinetic energy
distribution of exotic atoms at the time of their transition to the $2s$ state
is described by the probability density, $\omega_{2s}(E;\varphi,T)$, which is
determined by the preceding cascade processes as well as by the target density
and temperature.

Figure 5:  Kinetic energy distributions of muonic hydrogen (solid lines) and
deuterium (dashed lines) atoms on arrival in the $2s$ state vs. the laboratory
kinetic energy calculated at a target temperature $T=300$ K for three values
of the relative target density: $7\cdot 10^{-7}$, $7\cdot 10^{-5}$ and
$10^{-2}$. The dotted line corresponds to the initial energy distribution of
muonic atoms at the instant of their formation (see Eq. (8)).

In Fig. 5 we present the distribution function of the probability,

$W_{2s}(E;\varphi,T)=\int_{0}^{E}\omega_{2s}(E^{\prime};\varphi,T)dE^{\prime},$
(7)

which determines the probability that the muonic atom has kinetic energy in
the laboratory frame less than $E$ at the instant of the $2s$ state formation.
Further, this probability will be called the kinetic energy distribution of
muonic hydrogen or muonic deuterium atoms on arrival in the $2s$ state.
Calculations were performed at a room target temperature $T=300$ K for three
values of the relative target density: $\varphi=7\cdot 10^{-7}$, $7\cdot
10^{-5}$, and $10^{-2}$. The dotted line in the figure shows the kinetic
energy distribution of muonic atoms at the instant of their formation, given
by the formula

$W_{\rm{in}}(E)=1-\varkappa\exp(-E/E_{1})-(1-\varkappa)\exp(-E/E_{2}),$ (8)

where parameters $\varkappa=0.805$, $E_{1}=0.469$ eV, and $E_{2}=4.822$ eV
were determined in PPForm  from the fit of experimental data RPD  at the
density corresponding to the target pressure $p_{\rm{H}_{2}}=0.0625$ hPa and
temperature $T=300$ K.

As it is seen from Fig. 5, the kinetic energy distributions of the $2s$ state
for both muonic hydrogen and deuterium atoms are practically indistinguishable
at the density $\varphi=7\cdot 10^{-7}$ and have a small differences from the
initial energy distribution given by Eq. (8). At very low target densities
$\varphi\lesssim 10^{-6}$ the deexcitation cascade is practically purely
radiative and the initial kinetic energy distributions of muonic atoms at the
instant of their formation are conserved up to the end of the cascade. In this
density range the kinetic energy distributions of both muonic hydrogen and
deuterium atoms in the $2s$ state show a very small acceleration due to the
Coulomb deexcitation of highly-excited states.

At densities $\varphi\gtrsim 10^{-6}$, the effect of the CD in the highly-
excited states (preceding the population of the $2s$ state) becomes more
pronounced providing a high-energy component in the kinetic energy
distribution of the $2s$ state. In particular, at a density $\varphi=7\cdot
10^{-5}$ (see Fig. 5), the contributions of the CD $8\to 7$, $7\to 6$, $6\to
5$, and $5\to 4$ are explicitly seen in kinetic energy distributions of both
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms. The observed here small
differences between kinetic energy distributions of $(\mu^{-}p)_{2s}$ and
$(\mu^{-}d)_{2s}$ atoms are explained by the weak isotopic effect in cross
sections of the CD for highly-excited states (for details see PPIsot ). With
the further density increasing the role of the CD becomes much more prominent.
In particular, Figure 5 illustrates the significant isotopic effect of the CD
in the kinetic energy distributions of the $2s$-state of muonic hydrogen and
deuterium atoms at a density $\varphi=10^{-2}$. Here the strong isotopic
effect in the cross sections and accordingly in rates of the CD $5\to 4$,
$4\to 3$, and especially $3\to 2$ (e.g., see Figs. 8, 9,11, and 12 in PPIsot )
leads to significant differences in the kinetic energy distributions of atoms
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ at the instant of their formation.

In accordance with the kinetic energy distribution of the $2s$ state (see Fig.
5) the total arrival population $\varepsilon^{\rm tot}_{2s}$ can be divided
into two fractions

$\varepsilon^{\rm tot}_{2s}(\varphi)=\varepsilon^{\rm
a}_{2s}(\varphi)+\varepsilon^{\rm b}_{2s}(\varphi),$ (9)

where $\varepsilon^{\rm a}_{2s}(\varphi)$ and $\varepsilon^{\rm
b}_{2s}(\varphi)$ denote arrival populations of fractions with the kinetic
energy above and below the $2s-2p$ threshold, respectively. Absolute values of
$\varepsilon^{\rm a}_{2s}(\varphi)$ and $\varepsilon^{\rm b}_{2s}(\varphi)$
calculated for $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms at target
densities from $\varphi=10^{-8}$ up to 1 and a target temperature $T=300$ K
are shown in Table 1.

Table 1: The density dependence of arrival populations $\varepsilon^{\rm a}_{2s}$ and $\varepsilon^{\rm b}_{2s}$ (per formed muonic atom) of $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ fractions above and below of the $2s-2p$ threshold, respectively. , $\varphi$ |  $(\mu^{-}p)_{2s}$ |  |  $(\mu^{-}d)_{2s}$ |
---|---|---|---|---
(lhd) |  $\varepsilon^{\rm a}_{2s}$(%) |  $\varepsilon^{\rm b}_{2s}$(%) |  $\varepsilon^{\rm a}_{2s}$(%) |  $\varepsilon^{\rm b}_{2s}$(%)
$10^{-8}$ | 1.06 | 0.87 | 1.05 | 0.88
$10^{-7}$ | 1.11 | 0.90 | 1.12 | 1.02
$10^{-6}$ | 1.48 | 0.98 | 1.48 | 1.03
$10^{-5}$ | 2.82 | 1.10 | 2.57 | 1.14
$10^{-4}$ | 4.64 | 1.28 | 4.10 | 1.41
$10^{-3}$ | 6.13 | 2.21 | 4.76 | 2.69
$10^{-2}$ | 15.74 | 3.35 | 7.89 | 4.95
$10^{-1}$ | 54.67 | 5.37 | 29.05 | 7.07
$10^{0}$ | 76.02 | 11.93 | 46.55 | 11.04

It is important to notice some peculiarities in the formation of these
fractions in muonic hydrogen and muonic deuterium atoms. First, the arrival
population $\varepsilon^{\rm a}_{2s}$ for the muonic hydrogen and deuterium
atoms with kinetic energies above the $2s-2p$ threshold is larger than the
arrival population $\varepsilon^{\rm b}_{2s}$ with kinetic energies below the
$2s-2p$ threshold at all values of the target density. Second, the arrival
population $\varepsilon^{\rm a}_{2s}$ grows with the density much faster than
the arrival population $\varepsilon^{\rm b}_{2s}$ for both $(\mu^{-}p)_{2s}$
and $(\mu^{-}d)_{2s}$ atoms. Finally, the arrival population $\varepsilon^{\rm
b}_{2s}$ of the muonic deuterium is greater than the arrival population of the
same fraction of the muonic hydrogen at all densities of the target except for
the region very close to the liquid hydrogen density. On the contrary, at
$\varphi>10^{-6}$, the arrival population $\varepsilon^{\rm a}_{2s}$ for the
muonic hydrogen is more (and even significantly more at $\varphi\geqslant
10^{-3}$) than the arrival population of the same fraction in the case of the
muonic deuterium.

###  III.2 Collisional rates of $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms

After the formation of the hydrogen exotic atoms in the $2s$ state, their
further fate is determined by the rate of the muon decay
($\lambda_{\mu}=4.54\cdot 10^{5}$ ps-1) and by the rates of the collisional
processes: the elastic scattering $2s-2s$, the collision-induced radiative
quenching, the Stark transition $2s\to 2p$ followed by the fast radiative
transition $2p\to 1s$, and the CD $2s\to 1s$. The rate of the collisional
process in the laboratory frame is defined as follows

$\lambda_{nl\to n^{\prime}l^{\prime}}(E_{\rm{lab}})=\varphi
N_{\rm{LHD}}\sigma_{nl\to
n^{\prime}l^{\prime}}(E_{\rm{lab}})\sqrt{\frac{2E_{\rm{lab}}}{M}},$ (10)

where $E_{\rm{lab}}$ and $M$ are the kinetic energy in the laboratory frame
and the mass of the muonic atom, respectively.

Figure 6 shows the energy dependence of collisional rates for the elastic
scattering $2s-2s$, CD $2s\to 1s$, and collision-induced radiative quenching
of the $2s$ state for muonic hydrogen and deuterium atoms at the liquid
hydrogen density ($\varphi=1$). The rate of the radiative transition
$\lambda^{rad}(2p\to 1s)=0.12$ ps-1 is shown for comparison. A small
difference of about 5% between radiative rates of the $2p\to 1s$ transition in
muonic hydrogen and deuterium due to the difference of their reduced masses is
not visible at the scale of the figure.

Figure 6:  Collisional rates of the elastic $2s-2s$ scattering, CD $2s\to 1s$,
and collision-induced radiative quenching of the $2s$ state as a function of
the laboratory kinetic energy for $(\mu^{-}p)$ (solid lines) and $(\mu^{-}d)$
(dashed lines) atoms calculated at the liquid hydrogen density.

The isotope effect observed in cross sections for scattering of muonic atoms
in states with values of the principal quantum number $n<8$ (see PPIsot ) is
practically absent for highly excited states with $n>8$. The only source of
the isotopic effect in the kinetics of the atomic cascade for states with
$n>8$ is the trivial kinematic factor $\sqrt{2E_{\rm{lab}}/M}$ in Eq. (10). In
the case of the $2s$ state, the isotopic effect revealed in cross sections
(see Figs. 1 and 3) is additionally enhanced by this factor associated with
the difference of the exotic atom velocities at the same energy in the
laboratory system. It is seen in Fig. 6, that the isotopic effect in rates of
all collisional processes is very strong.

###  III.3 Intensities of the different disintegration modes of
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$

The intensities of the decay modes, i.e. the number of atoms (per one formed
muonic atom) disintegrated through a given mode were calculated within the
present cascade model for muonic hydrogen and deuterium atoms in $2s$ state.
The intensities of the destruction of the $2s$ state for $(\mu^{-}p)$ and
$(\mu^{-}d)$ atoms are shown as functions of the target density in Figs. 7 and
8, respectively, and are also presented in Tables 2 and 3 for several density
values.

Figure 7: Intensities (per one formed $\mu^{-}p$ atom) for all disintegration
modes of $(\mu^{-}p)_{2s}$ vs the relative target density: collision-induced
radiative quenching (solid line), Stark $2s\to 2p$ transition (line with open
circles), Coulomb de-excitation (line with asterisks), and $\mu$-decay (dashed
and dashed-dotted lines for fractions below and above the $2s-2p$ threshold,
respectively).

The figures and tables show the intensities of the following destruction
modes:
$I_{\rm rqu}$ – collision-induced radiative quenching,
$I_{\rm Str}$ – Stark transition $2s\to 2p$ followed by the fast radiative
$2p\to 1s$ transition,
$I_{\rm CD}$ – Coulomb deexcitation $2s\to 1s$,
$I^{\rm a}_{\mu}$ – $\mu^{-}$-decay at kinetic energies above the $2s-2p$
threshold,
$I^{\rm b}_{\mu}$ – $\mu^{-}$-decay at kinetic energies below the $2s-2p$
threshold.

Figure 8:  The same as in Fig. 7 for $(\mu^{-}d)_{2s}+D$ collisions.

At the lowest densities, the muon decay is the main process resulting in the
disintegration of the $2s$ state of both muonic hydrogen and deuterium atoms
at kinetic energies above and below the $2s-2p$ threshold (see also Table II).

At kinetic energy above the $2s-2p$ threshold, the intensity of the
$\mu$-decay decreases rather rapidly with the density increasing. On the
contrary, the intensity of $\mu$-decay at kinetic energies below the $2s-2p$
threshold at first even increases with increasing density (especially in the
case of $(\mu^{-}d)_{2s}$) and then decreases rapidly with further increase in
density at $\varphi>10^{-5}$.

Table 2: The density dependence of intensities (per one formed muonic atom, in percent) of the $\mu$-decay calculated for $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ with kinetic energies both above and below the $2s-2p$ threshold ($I^{\rm a}_{\rm w}$ and $I^{\rm b}_{\rm w}$, respectively). $\varphi$ |  $(\mu^{-}p)_{2s}$ |  |  $(\mu^{-}d)_{2s}$ |
---|---|---|---|---
(lhd) |  $I^{\rm a}_{\rm w}$ |  $I^{\rm b}_{\rm w}$ |  $I^{\rm a}_{\rm w}$ |  $I^{\rm b}_{\rm w}$
$10^{-8}$ | 0.86 | 0.95 | 0.91 | 0.95
$10^{-7}$ | 0.41 | 1.12 | 0.57 | 1.02
$10^{-6}$ | 0.08 | 1.05 | 0.16 | 1.43
$10^{-5}$ | 0.02 | 0.33 | 0.03 | 0.79
$10^{-4}$ | 0.00 | 0.04 | 0.00 | 0.14
$10^{-3}$ | 0.00 | 0.01 | 0.00 | 0.02
Table 3: The density dependence of intensities $I_{\rm CD},I_{\rm rqu},$ and $I_{\rm Str}$ ( in percent) calculated for $(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$. $\varphi$ |  |  $(\mu^{-}p)_{2s}$ |  |  | $(\mu^{-}d)_{2s}$ |
---|---|---|---|---|---|---
(lhd) |  $I_{\rm CD}\,\,$ |  $I_{\rm rqu}$ |  $I_{\rm Str}$ |  $I_{\rm CD}$ |  $I_{\rm rqu}$ |  $I_{\rm Str}$
$10^{-8}$ | 0.00 | 0.00 | 0.11 | 0.00 | 0.004 | 0.07
$10^{-7}$ | 0.004 | 0.04 | 0.43 | 0.00 | 0.04 | 0.27
$10^{-6}$ | 0.04 | 0.40 | 0.90 | 0.01 | 0.31 | 0.58
$10^{-5}$ | 0.13 | 1.32 | 2.13 | 0.06 | 1.36 | 1.48
$10^{-4}$ | 0.17 | 1.74 | 3.97 | 0.11 | 2.25 | 3.01
$10^{-3}$ | 0.26 | 2.66 | 5.41 | 0.18 | 3.70 | 3.55
$10^{-2}$ | 0.39 | 3.91 | 14.76 | 0.28 | 5.69 | 6.85
$10^{-1}$ | 0.94 | 9.33 | 49.52 | 0.45 | 9.09 | 26.45
$10^{0}$ | 4.93 | 48.98 | 32.38 | 1.51 | 30.59 | 24.33

The intensities of Stark $2s\to 2p$ transitions in muonic hydrogen and
deuterium atoms are in accordance with the primary populations of these atoms
at kinetic energies above the $2s-2p$ threshold but turn out to be less than
the values of the corresponding primary populations. The intensity of the
Coulomb deexcitation for $(\mu^{-}p)_{2s}$ are about 1.5 - 2 times higher than
in the case of $(\mu^{-}d)_{2s}$, while the values of the radiation quenching
intensity for these two isotopes are very close at the target density less
than $10^{-5}$ and with increasing density targets up to $10^{-2}$ this ratio
changes in favor of the muonic deuterium atom by about 1.5 times.

The fastest collisional process – the elastic $2s-2s$ scattering – leads to a
deceleration of muonic atoms in the $2s$ state. As a result, some part of the
muonic $2s$-state atoms initially formed with kinetic energies above the
$2s-2p$ threshold turns out to be below the threshold. Thus, the population of
the fraction with kinetic energy above the $2s-2p$ threshold decreases, and
the population of the fraction with kinetic energy below the $2s-2p$ threshold
increases accordingly compared to their values at the instant of formation.
Thus, the total population of the fraction with kinetic energy below the
$2s-2p$ threshold is always greater than its initial population
$\varepsilon^{\rm b}_{2s}$. This influence of elastic scattering on
redistribution of fractions with kinetic energies above and below the $3s-2p$
threshold (in favor of increasing the sub-threshold fraction) increases with
increasing target density.

In Table 4 we present the initial population $\varepsilon^{\rm b}_{2s}$ and
the total number of the disintegrated $(\mu^{-}p)_{2s}$ atoms with the energy
below the $2s-2p$ threshold $I^{\rm b}_{2s}$ (per one formed $\mu^{-}p$ atom):

$I^{\rm b}_{2s}=I^{\rm b}_{\mu}+I_{\rm CD}+I_{\rm rqu}$ (11)

in comparison with the experimental data RPD ; PDH .

Table 4: The initial population $\varepsilon^{\rm b}_{2s}$ and the total number of the disintegrated $(\mu^{-}p)_{2s}$ atoms with the energy below the $2s-2p$ threshold $I^{\rm b}_{2s}$ (per one formed $\mu p$ atom). $\varphi$ |  $\varepsilon^{\rm b}_{2s}$(%) |  |  $I^{\rm b}_{2s}$ (%) |
---|---|---|---|---
(lhd) |  Theory |  Theory |  Exp. |  Ref.
$7.35\cdot 10^{-8}$ | 0.90 | 1.12 | $0.85^{+0.14}_{-0.14}$ | RPD
$2.94\cdot 10^{-7}$ | 0.95 | 1.27 | $0.92^{+0.08}_{-0.08}$ | RPD
$1.18\cdot 10^{-6}$ | 1.05 | 1.51 | $1.10^{+0.08}_{-0.08}$ | RPD
$4.71\cdot 10^{-6}$ | 1.13 | 1.72 | $1.25^{+0.98}_{-0.48}$ | PDH
$1.88\cdot 10^{-5}$ | 1.15 | 1.84 | $1.07^{+0.57}_{-0.30}$ | PDH
$7.53\cdot 10^{-5}$ | 1.28 | 1.93 | $1.63^{+1.08}_{-0.35}$ | PDH

On the whole, there is a satisfactory agreement between the theoretical
results and experimental data. An exception is the experimental value obtained
at a target density of $1.88\cdot 10^{-6}$ PDH , which is much less the
corresponding theoretical value. It should be noted that this experimental
value falls out of the general trend of population growth with increasing
target density and, moreover, there are no arguments to explain it.

###  III.4 Absolute x-ray yield of $K\alpha$ line

The absolute x-ray yield of the $K\alpha$ line is determined as the intensity
of this line per muonic atom formed. The total intensity, $I_{K\alpha}$, of
the $K\alpha$ line can be represented as a sum of three different
contributions corresponding to the preceding history of the atoms involved in
its formation:

$I_{K\alpha}=I_{2p\to 1s}+I_{\rm Str}+I_{\rm rqu}.$ (12)

Here $I_{2p\to 1s}$ denotes the intensity of the direct $2p\to 1s$ radiative
transition occurring after the transition of a muonic atom to the $2p$ state
from states with the principal quantum number $n\geqslant 3$. Remind, that
$I_{\rm Str}$ and $I_{\rm rqu}$ are the intensities of the $2s\to 2p$ Stark
transition (at kinetic energies above the $2s-2p$ threshold) followed by the
$2p\to 1s$ radiative transition and the collision-induced radiative quenching
of the $2s$ state (at kinetic energies below the $2s-2p$ threshold),
respectively.

Figures 9 and 10 show the density dependence of the different modes forming
the total absolute yield of the $K\alpha$ line as well as their sum calculated
at a target temperature $T=300$ K for $\mu^{-}p$ and $\mu^{-}d$ atoms,
respectively.

Figure 9:  The density dependence of the different contributions to the
absolute $K\alpha$-line yield calculated for muonic hydrogen at the target
temperature $T=300$ K.: $I_{2p\to 1s}$ (solid line with triangles), $I_{\rm
Str}$ (dashed line), $I_{\rm rqu}$ (solid line with filled circles), and their
sum (solid line).

As can be seen from Figures 9 and 10, the contribution of the direct radiation
$2p-1s$ transition $I_{2p\to 1s}$ to the absolute yield of the $K\alpha$-line
decreases with increasing target density $\varphi>10^{-6}$. In the case of a
muonic hydrogen atom, this decrease becomes rather rapid at $\varphi>10^{-3}$
and the contribution reaches a value of 0.1 at the density of liquid hydrogen.
In the case of a muonic deuterium atom, we observe a similar dependence on the
density at $\varphi<10^{-3}$, however a further increase in the target density
does not lead to a sharp decrease of the contribution $I_{2p\to 1s}$ (at the
density of liquid hydrogen, this contribution is about 0.4). This difference
in the dependence of $I_{2p\to 1s}$ on the density in muonic hydrogen and
muonic deuterium is explained by isotope effect in the rates of collisional
processes of these atoms (see PPIsot ) in the upper part of the cascade of
deexcitation ($n\geqslant 3$). In particular, this is reflected in the primary
populations of the $2s$ state at kinetic energies above the $2s-2p$ threshold
(see Table 1).

Figure 10:  The same as in Fig. 9 for $\mu^{-}d$ atom.

###  III.5 Probabilities of different disintegration modes and lifetime of
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$

The probability that the muonic atom formed in the $2s$ state disintegrates
due to one of decay modes, $P_{i}(\varphi)$, is given by

$P_{i}(\varphi)=\frac{1}{p_{0}}\Big{\\{}1-\exp\left[-\int_{0}^{\infty}\frac{\lambda_{i}(E;\varphi)}{\lambda_{\rm
tot}(E;\varphi)}F_{i}(E;\varphi)dE\right]\Big{\\}}.$ (13)

Here, the $\lambda_{i}(E;\varphi)$ is the rate of the process $i$ where index
$i$ denotes the following disintegration channels: $i=\mu$ ($\mu$-decay),
$i=\rm CD$ (Coulomb deexcitation $2s\to 1s$), $i=\rm rqu$ (collision-induced
radiative quenching), and $i=\rm Str$ (Stark transition $2s\to 2p$ resulting
in the radiative transition $2p\to 1s$). $\lambda_{\rm tot}(E;\varphi)$ is a
summary rate of all processes, which lead to the disintegration of the $2s$
state:

$\lambda_{\rm tot}(E;\varphi)=\lambda_{\mu}+\lambda_{\rm CD}+\lambda_{\rm
rqu}+\lambda_{\rm Str}.$ (14)

The factor $p_{0}=1-e^{-1}$ provides the correct behavior of the probability
$P_{\mu}$ in the limit $\varphi\to 0$. The kinetic energy distribution
$F_{i}(E;\varphi)$ is defined at the instant of the disintegration of the $2s$
state through the i-th decay mode and normalized by the condition:

$\int_{0}^{\infty}F_{i}(E;\varphi)dE=1.$ (15)

The probability of the different disintegration modes for $(\mu^{-}p)_{2s}$
and $(\mu^{-}d)_{2s}$ atoms as functions on target density is shown in Figures
11 and 12, correspondingly. As can be seen from the figures, probabilities of
$\mu$-decay have a similar density dependence for muonic hydrogen and
deuterium atoms. This is explained by the fact that the $\mu$-decay rate does
not depend on either the energy or the target density, and the kinetic energy
distributions of muonic hydrogen and deuterium are very close in the density
region where the probability of muon decay is not negligible (see Fig. 5).

On the contrary, for the probabilities of the destruction of the $2s$ state in
collisional processes, we observe significant differences between muonic
hydrogen and deuterium in both the values of these probabilities and their
dependence on the target density. These differences are especially pronounced
in the ratio of the probabilities of Stark transitions and collision-induced
radiation quenching. The observed differences are explained by two factors:
significant differences in the ratio of velocities, as well as in the
distributions over the kinetic energies of muonic hydrogen and deuterium
atoms.

Figure 11:  The density dependence of decay probabilities of $(\mu^{-}p)_{2s}$
for various disintegration modes: collision-induced radiative quenching (solid
line), Stark $2s\to 2p$ transition (line with open circles), Coulomb de-
excitation (line with asterisks), and $\mu$-decay (dashed and dashed-dotted,
and dotted lines for fractions below and above the $2s-2p$ threshold and their
sum, respectively).

Figure 12:  The same as in Fig. 11 for $(\mu^{-}d)_{2s}$ atom.

The mean time interval between the formation of the muonic atom in the $2s$
state and its decay we define as the lifetime of the $2s$ state. In this work,
to determine the lifetime of the $2s$ state, we use the $\mu$\- decay
probability

$P_{\mu}(\varphi)=\frac{1}{p_{0}}\left[1-\exp\left(-\lambda_{\mu}\tau_{2s}(\varphi)\right)\right],$
(16)

where a lifetime $\tau_{2s}(\varphi)$ of the $2s$ state is defined by

$\tau_{2s}(\varphi)=\int_{0}^{\infty}\frac{F_{\mu}(E;\varphi)}{\lambda_{\rm
tot}(E;\varphi)}dE$ (17)

From Eqs. (16) and (17) we obtain

$\tau_{2s}(\varphi)=-\tau_{\mu}\ln\left[1-p_{0}\cdot P_{\mu}(\varphi)\right].$
(18)

Formulas similar to Eqs. (16)-(18) allow to define and to calculate the
lifetime $\tau_{2s}^{\rm a}$ for the fraction of atoms in the $2s$ state with
the kinetic energy above the $2s-2p$ threshold and also the lifetime
$\tau_{2s}^{\rm b}$ for the fraction of atoms in the $2s$ state with the
kinetic energy below the $2s-2p$ threshold.

$P_{\mu}^{\alpha}(\varphi)=\frac{1}{p_{0}}\left[1-\exp\left(-\lambda_{\mu}\tau_{2s}^{\alpha}(\varphi)\right)\right],$
(19)

where

$\tau_{2s}^{\alpha}(\varphi)=\int_{0}^{\infty}\frac{F_{\mu}^{\alpha}(E;\varphi)}{\lambda^{\alpha}(E;\varphi)}dE.$
(20)

is the lifetime of the fraction of the $2s$ state with the kinetic energy
above ($\alpha=\rm a$) or below ($\alpha=\rm b$) the $2s-2p$ threshold,
respectively. The corresponding total rates of these fractions are given by:

$\lambda^{\rm a}(E;\varphi)=\lambda_{\mu}+\lambda_{\rm
CD}(E;\varphi)+\lambda_{\rm Str}(E;\varphi)$ (21)

and

$\lambda^{\rm b}(E;\varphi)=\lambda_{\mu}+\lambda_{\rm
CD}(E;\varphi)+\lambda_{\rm rqu}(E;\varphi).$ (22)

Infinite limits of integration in Eq. (20) are of a formal nature, since the
probability densities $F_{\mu}^{\rm a}(E;\varphi)$ and $F_{\mu}^{\rm
b}(E;\varphi)$are different from zero only in the intervals $E\geqslant E_{\rm
thr}$ and $E\leqslant E_{\rm thr}$, respectively. From Eqs. (19) and (20) we
obtain

$\tau_{2s}^{\alpha}(\varphi)=-\tau_{\mu}\ln\left[1-p_{0}\cdot
P_{\mu}^{\alpha}(\varphi)\right].$ (23)

In the present study three variants of the $(\mu^{-}p)_{2s}$ and
$(\mu^{-}d)_{2s}$ lifetime were calculated: the lifetime, $\tau_{2s}^{\rm a}$,
of the fraction with the kinetic energy above the $2s-2p$ threshold, the
lifetime, $\tau_{2s}^{\rm b}$, of the fraction with the kinetic energy below
the $2s-2p$ threshold, and the lifetime, $\tau_{2s}$, of the $2s$ state
independently whether the kinetic energy of muonic atom is above or below the
$2s-2p$ threshold.

Figure 13:  The density dependence of the $(\mu^{-}p)_{2s}$ lifetime
calculated for various fractions: below the $2s-2p$ threshold - dashed line,
above the $2s-2p$ threshold - dashed-dotted line. The lifetime, $\tau_{2s}$,
of the $2s$ state independently whether the kinetic energy of muonic atom is
above or below the $2s-2p$ threshold is shown by the solid line. The
experimental values from PDH

.

Figure 14:  The same as in Fig. 13 for $(\mu^{-}d)_{2s}$. The experimental
values from Mud

.

Results of calculations are presented in Figs. 13 and 14 for the
$(\mu^{-}p)_{2s}$ and $(\mu^{-}d)_{2s}$ atoms, respectively. As it is seen
from the figures, the lifetime of the $2s$ state for both muonic hydrogen and
muonic deuterium has a similar dependence on density for both fractions. At
the lowest densities, the lifetimes are determined by the muon decay rate.
With increasing density, the role of collisional processes increases, which
leads to a decrease in the lifetime of both fractions. The lifetime for the
fraction with kinetic energy below the $2s-2p$ threshold, $\tau_{2s}^{\rm b}$,
decreases much more slowly with increasing target density than for the
fraction with kinetic energy above the $2s-2p$ threshold. At a densities
$\varphi>10^{-5}$, the lifetime of a $2s$ state behaves as $\varphi^{-1}$, and
the ratio $\tau_{2s}^{\rm b}$/$\tau_{2s}^{\rm a}$ is equal to about 15.

In the case of muonic hydrogen, the theoretical values of the lifetime of the
fractions with kinetic energy both above and below the $2s-2p$ threshold are
in excellent agreement with the available experimental data. In the case of
muonic deuterium, the agreement between the theoretical and available at one
density experimental values of the lifetimes of the $2s$ state can be
considered as good for $\tau_{2s}^{\rm b}$ and satisfactory for
$\tau_{2s}^{\rm a}$.

##  IV Conclusion

The fully quantum-mechanical study of the collisional processes — elastic
scattering, collision-induced radiative quenching, and Coulomb deexcitation —
has been performed in the framework of the close-coupling approach for
$(\mu^{-}p)_{2s}+H$ and $(\mu^{-}d)_{2s}+D$ collisions at kinetic energies
below the $2s-2p$ threshold. The cross sections have been calculated with the
basis set including all open and closed channels associated with the exotic
atom states with the principal quantum number up to $n_{max}=30$ and then have
been used as input data in the kinetics of the atomic cascade.

An important result of this work is the elucidation of the role of the
collision-induced radiative quenching in the destruction of a muonic atom in
$2s$-state. As far as the authors know, this problem has not been previously
investigated. Our calculations showed that this process makes a significant
contribution to the total disintegration of the $2s$-state of muonic atoms at
energies below the $2s-2p$ threshold.

The kinetics of the atomic cascade has been investigated for both muonic
hydrogen and muonic deuterium atoms within the updated version of the cascade
model, which has a number of essential improvements over the previous cascade
calculations. The cascade calculations have been performed for target
densities covering eight orders of magnitude from $10^{-8}$ up to 1 (in the
units of the liquid hydrogen density).

As a result of ab initio cascade calculations, a number of characteristics of
the $2s$ state of muonic hydrogen and deuterium atoms were obtained: primary
populations and kinetic energy distributions, absolute values of intensities
different modes of disintegration, the probability of decay of the $2s$ state
through different channels and lifetimes of the $2s$ state at kinetic energy
both above and below the $2s-2p$ threshold. The obtained theoretical results
are compared with the available experimental data. The results of the work can
be used for the planning experiments and analysis of experimental data.

## References

  * (1) G. Ya. Korenman and V. P. Popov, Muon Catalyzed Fusion 4, 145 (1989).
  * (2) G. Ya. Korenman, V. P. Popov, and G. A. Fesenko, Muon Catalyzed Fusion 7, 179 (1992).
  * (3) J. S. Cohen, Rep. Prog. Phys. 67, 1769 (2004).
  * (4) V. P. Popov and V. N. Pomerantsev, Phys. Rev. A 83, 032516 (2011).
  * (5) V. P. Popov and V. N. Pomerantsev , Phys. Rev. A 95, 022506 (2017).
  * (6) R. Pohl, Ph.D. thesis 14096, ETH Zurich, 2001.
  * (7) R. Pohl, H. Daniel, F.J. Hartmann et al., PRL 97, 193402 (2006).
  * (8) J. Wallenius, S. Jonsell, Y. Kino and P. Froelich, Hyperfine Interact. 138, 285 (2001).
  * (9) R. Pohl et al., Hyperfine Interact. 138, 35 (2001).
  * (10) P. O. Egan et al., Phys. Rev. A 23, 1152 (1981).
  * (11) H. Anderhub et al., Phys. Lett. B 143, 65 (1984).
  * (12) N. Bregant et al., Phys. Lett. A 241, 344 (1998).
  * (13) F. Kottman et al., Hyperfine Interact. 119, 3 (1999); 138, 55 (2001).
  * (14) G. Kodosky and M. Leon, Nuovo Cimento B1, 41 (1971).
  * (15) R. O. Mueller, V. W. Hughes, H. Rosenthal, and C. S. Wu, Phys. Rev. A 11, 1175 (1975).
  * (16) G. Carboni and G. Fiorentini, Nuovo Cimento B39, 281 (1977).
  * (17) James S. Cohen and J. N. Bardsley, Phys. Rev. A 23, 46 (1981).
  * (18) T. S. Jensen and V. E. Markushin, Eur. Phys. J. D 21, 271 (2002).
  * (19) Randolf Pohl et al., Phys. Rev. Lett. 97, 193402 (2006).
  * (20) L. Ludhova et al., Phys. Rev. A 75, 040501(R) (2007).
  * (21) R. Pohl et al., Nature (London) 466, 213 (2010).
  * (22) Marc Diepold et al., Phys. Rev. A 88, 042520 (2013).
  * (23) Pomerantsev Vladimir N., Popov Vladimir P., Hyperfine Interact. 209, 75 (2012).
  * (24) G. Ya. Korenman, V. N. Pomerantsev, and V. P. Popov, JETP Lett. 81, 543 (2005).
  * (25) V. N. Pomerantsev and V. P. Popov, JETP Lett. 83, 331 (2006).
  * (26) V. N. Pomerantsev and V. P. Popov, Phys. Rev. A 73, 040501(R) (2006).
  * (27) Pomerantsev Vladimir N., Popov Vladimir P., Hyperfine Interact. 209, 69 (2012).
  * (28) V. P. Popov and V. N. Pomerantsev , Phys. Rev. A 86, 052520 (2012).