# On spaces with a $\pi$-base whose elements have an H-closed closure

Davide Giacopello

###### Abstract

We deal with the class of Hausdorff spaces having a $\pi$-base whose elements
have an H-closed closure. Carlson proved that $|X|\leq
2^{wL(X)\psi_{c}(X)t(X)}$ for every quasiregular space $X$ with a $\pi$-base
whose elements have an H-closed closure. We provide an example of a space $X$
having a $\pi$-base whose elements have an H-closed closure which is not
quasiregular (neither Urysohn) such that $|X|>2^{wL(X)\chi(X)}$ (then
$|X|>2^{wL(X)\psi_{c}(X)t(X)}$). Still in the class of spaces with a
$\pi$-base whose elements have an H-closed closure, we establish the bound
$|X|\leq 2^{wL(X)k(X)}$ for Urysohn spaces and we give an example of an
Urysohn space $Z$ such that $k(Z)<\chi(Z)$. Lastly, we present some equivalent
conditions to the Martin’s Axiom involving spaces with a $\pi$-base whose
elements have an H-closed closure and, additionally, we prove that if a
quasiregular space has a $\pi$-base whose elements have an H-closed closure
then such space is Baire.

Keywords: H-closed spaces, $\pi$-bases, quasiregular spaces, Urysohn spaces,
cardinal bounds, cardinal inequalities, Baire spaces.

AMS Subject Classification: Primary 54A25, Secondary 54D10, 54D20.

##  1 Introduction

Throughout the paper we mean by “space”, an Hausdorff topological space. And
for semplicity if a space has a $\pi$-base whose elements have an H-closed
(compact) closure, we say that the space has an H-closed (compact) $\pi$-base.
In 1978, Bell, Ginsburg, and Woods established, in [3], that if a space $X$ is
normal, then $|X|\leq 2^{wL(X)\chi(X)}$, and they constructed an example of a
Hausdorff space in which the aforementioned inequality does not hold. In the
same paper, they posed the question of whether the hypothesis of normality
could be weakened in some way. In [11], Dow and Porter achieved a powerful
result: $|X|\leq 2^{\psi_{c}(X)}$ for every H-closed space $X$. In [5], Bella
and Carlson proved that the inequality $|X|\leq 2^{wL(X)t(X)\psi(X)}$ is true
for every regular space $X$ with a compact $\pi$-base. In [6], Bella, Carlson
and Gotchev proved that almost the same inequality (with the closed
pseudocharacter $\psi_{c}(X)$ instead of the pseudocharacter $\psi(X)$) holds
also for spaces with a compact $\pi$-base. That is,

######  Theorem 1.1.

[6] If $X$ is a space with a compact $\pi$-base. Then $|X|\leq
2^{wL(X)t(X)\psi_{c}(X)}$.

Since H-closedness is a natural generalization of compactness, Bella, Carlson
and Gotchev posed the following question.

######  Question 1.2.

[6] Let $X$ be a space with a $\pi$-base whose elements have H-closed
closures. Is it true that $|X|\leq 2^{wL(X)t(X)\psi_{c}(X)}$?

In [5], some further investigations on spaces having a $\pi$-base with some
properties on the closure of the elements have led to the following result.

######  Theorem 1.3.

[5] Let $X$ be a space and ${\cal B}$ an open $\pi$-base. Suppose for all
$B\in{\cal B}$ that $\overline{B}$ is $H$-closed, (or normal, or Lindelöf, or
has the ccc). Then $d_{\theta}(X)\leq 2^{wL(X)\chi(X)}$ and if $X$ is
quasiregular or Urysohn then $|X|\leq 2^{wL(X)\chi(X)}$.

Then we pose the following questions.

######  Question 1.4.

Does the inequality $|X|\leq 2^{wL(X)t(X)\psi_{c}(X)}$ hold for quasiregular
spaces having an H-closed $\pi$-base?

######  Question 1.5.

Does the inequality $|X|\leq 2^{wL(X)t(X)\psi_{c}(X)}$ hold for Urysohn spaces
having an H-closed $\pi$-base?

Recall that the Banach-Mazur game on the space $X$ is played by two players
Alice and Bob in $\omega$-many innings. At the beginning of the game, Alice
chooses a nonempty open set $U_{0}$ and Bob responds by choosing a nonempty
open set $V_{0}\subset U_{0}$. At the $n$-th inning ($n>0$), Alice chooses a
nonempty $U_{n}\subset V_{n-1}$ and Bob responds by choosing a nonempty open
set $V_{n}\subset U_{n}$, and so on. The player Bob wins if and only if
$\bigcap_{n\in\omega}V_{n}\neq\emptyset$. Banach, Mazur and Oxtoby proved that
the spave $X$ has the Baire property if and only if Alice does not have a
winning strategy in the Banach-Mazur game on $X$. A space $X$ is said Choquet
if Bob has a winning strategy in the Banach-Mazur game on $X$. Choquet spaces
were introduced in 1975 by White who called them weakly $\alpha$-favorable
spaces. Choquet spaces are Baire. A Bernstein subset of reals witnesses that
Baire spaces need not be Choquet.
Bella, Carlson and Gotchev proved the following result.

######  Theorem 1.6.

[7] A space $X$ with a $\pi$-base whose elements have closures that are
compact is Choquet.

It is natural to ask the following question.

######  Question 1.7.

Is any space $X$ with an H-closed $\pi$-base a Choquet space?

In Section 3 we give a negative answer to Question 1.2 (see Example 3.5)
providing an example of a space $X$ having an H-closed $\pi$-base which is not
quasiregular (neither Urysohn) such that $|X|>2^{wL(X)\chi(X)}$ (then
$|X|>2^{wL(X)\psi_{c}(X)t(X)}$). Moreover Carlson answered Question 1.4 (see
Theorem 3.4) proving that $|X|\leq 2^{wL(X)\psi_{c}(X)t(X)}$ for every
quasiregular space with an H-closed $\pi$-base. Example 3.5 shows, inter alia,
that the quasiregular hypotesis is essential. In the following, we present
some further results on spaces with a $\pi$-base whose elements have compact
boundaries.
Additionally, we establish the bound $|X|\leq 2^{wL(X)k(X)}$ for Urysohn
spaces having an H-closed $\pi$-base. The function $k(X)$, defined later on,
was introduced by Alas and Kocinac in [1]. We provide an example of an Urysohn
space $Z$ having an H-closed $\pi$-base such that $k(Z)<\chi(Z)$.
In Section 4 we present some equivalent conditions to the Martin’s Axiom
involving spaces with an H-closed $\pi$-base and, additionally, we give a
partial answer to Question 1.7 (see Theorem 4.5).

##  2 Notation and terminology

For a subset $A$ of a topological space $X$ we will denote by $[A]^{<\lambda}$
$([A]^{\lambda})$ the family of all subsets of $A$ of cardinality $<\lambda$
$(=\lambda)$. We consider cardinal invariants of topological spaces (for
further informations see [12, 14, 15]). In particular, given a topological
space $X$, we will denote with $d(X)$ its density, $\chi(X)$ its character,
$\pi\chi(X)$ its $\pi$-character, $t(X)$ its tightness, $\psi(X)$ its
pseudocharacter, $\psi_{c}(X)$ its closed pseudocharacter, $L(X)$ its Lindelöf
number.
Given a subset $A\subseteq X$ the $\theta$-closure (see [17]) of $A$,
$cl_{\theta}(A)$, is the subset $\\{x\in X:\overline{U}\cap A\not=\emptyset$
for every open neighborhood $U$ of $x\\}$. Clearly, $\overline{A}\subseteq
cl_{\theta}(A)$ for every $A\subseteq X$. Recall that, given a space $X$, the
cardinal invariant $d_{\theta}(X)$, closely related to the density $d(X)$, is
defined as follows. A subspace $D\subseteq X$ is $\theta$-dense in $X$ if
$D\cap\overline{U}\neq\emptyset$ for every non-empty open set $U$ of $X$. The
$\theta$-density $d_{\theta}(X)$ is the least cardinality of a $\theta$-dense
subspace of $X$. Observe that $d_{\theta}(X)\leq d(X)$ for any space $X$ (see
[8] and [10] for more details about $\theta$-density). The $\theta$-tightness,
$t_{\theta}(X)$, is the least cardinal $\kappa$ such that if $x\in
cl_{\theta}(A)$, then there exists $B\subseteq A$ such that $|B|\leq\kappa$
and $x\in cl_{\theta}(B)$. If $X$ is Urysohn is possible to define the
$\theta$-pseudocharacter, $\psi_{\theta}(X)$, that is the least cardinal
$\kappa$ such that for each point $x\in X$ there exists a family ${\cal
V}_{x}$ of open subsets containing $x$ such that $|{\cal V}_{x}|\leq\kappa$
and $\\{x\\}=\bigcap_{V\in{\cal V}_{x}}cl_{\theta}(\overline{V})$. Clearly,
$\psi_{c}(X)\leq\psi_{\theta}(X)$ for every Urysohn space $X$.
Given a space $X$, its weak Lindelöf number, $wL(X)$, is the least cardinal
$\kappa$ such that for every open cover $\cal U$ of $X$ there exists a
subcover ${\cal V}\in[{\cal U}]^{\leq\kappa}$ such that
$\overline{\bigcup_{V\in{\cal V}}V}=X$.
A space is called $H$-closed if it is Hausdorff and it is closed in every
Hausdorff space in which it is embedded. Moreover, a more operative
characterization of H-closedness of a space $X$ is given by the following
statement: “for every open cover $\cal U$ of $X$ there exists a finite
subfamily ${\cal V}\in[{\cal U}]^{<\omega}$ such that $\overline{\bigcup{\cal
V}}=X$”. Two other characterizations of H-closed spaces are that each open
filter (open ultrafilter) on the space has a nonempty adherence (convergers,
respectiely).
A subset $U\subseteq X$ is called regular closed if $U=\overline{int(U)}$ and
regular open if $U=int(\overline{U})$. It is known that if a space is H-closed
every regular closed subset of it inherits the property. Given a space
$(X,\tau)$, the semiregularization of $X$, denoted by $X_{s}$, is the space
$X$ endowed with the topology generated by the basis
$\\{int(\overline{U}):U\in\tau\\}$ of all the regular open sets of $X$.
Let $({\mathbb{P}},\leq,1)$ be a poset. $D\subseteq{\mathbb{P}}$ is a dense
subset (in the sense of posets) if for every $p\in{\mathbb{P}}$ there exists
$d\in D$ such that $d\leq p$. $p,q\in{\mathbb{P}}$ are called compatible
($p\not\perp q$) if there exists $r\in{\mathbb{P}}$ such that $r\leq p$ and
$r\leq q$, otherwise they are called incompatible ($p\perp q$). A poset
$({\mathbb{P}},\leq,1)$ is said to have the countable chain condition (ccc is
the sense of posets) if every antichain (subset of ${\mathbb{P}}$ having
pairwise incompatible elements) is countable. A subset $G\subseteq\mathbb{P}$
is called filter if (a) the maximum $1$ belongs to $G$; (b) for every $p,q\in
G$ there exists $r\in G$ such that $r\leq p$ and $r\leq q$; (c) for each
$p,q\in\mathbb{P}$, if $p\in G$ and $p\leq q$, then $q\in G$.

##  3 Upper bounds on the cardinality of spaces with an H-closed $\pi$-base

It is known that for regular spaces the H-closedness and the compactness are
equivalent properties. In the following we provide an example of a space
having the property we are dealing with and does not have a compact
$\pi$-base.

######  Example 3.1.

An axample of a non quasiregular (hence non regular) space having an H-closed
$\pi$-base, but it does not have a compact $\pi$-base.

Consider the space $(\mathbb{R},\tau_{\mathbb{Q}})$ where $\tau_{\mathbb{Q}}$
is the topology generated by the open set of the form
$\\{x\\}\cup(U\cap{\mathbb{Q}}$), with $x\in{\mathbb{R}}\setminus\mathbb{Q}$
and $U$ an open set in the standard topology on $\mathbb{R}$. Let $X=[0,2]$
with the topology inherited from $(\mathbb{R},\tau_{\mathbb{Q}})$. $X$ is a
Hausdorff non regular nowhere locally compact space. Then it does not have a
compact $\pi$-base. In [13] Herrlich proved that $X$ is H-closed then it
admits a $\pi$-base whose elements have H-closed closures. Now we prove that
there is not a $\pi$-base whose elements have quasiregular closure. Suppose
that ${\cal B}$ is a $\pi$-base of $X$ having elements with quasiregular
closures. Let’s take the open set $(0,2)\cap{\mathbb{Q}}$, then there exists
$U\in{\cal B}$ such that $U\subseteq(0,2)\cap{\mathbb{Q}}$, by hypotesis
$\overline{U}$ is quasiregular. Therefore there is $V\in{\cal B}$ such that
$V\subseteq\overline{V}\subseteq U$ so $\overline{V}\subseteq{\mathbb{Q}}$,
which is a contradiction. By Corollary 3.3 (see below) follows that $X$ is not
quasiregular. $\hfill\triangle$

In [5] it is proved that $|X|\leq 2^{wL(X)\chi(X)}$ holds for quasiregular (or
Urysohn) spaces having an H-closed $\pi$-base. Later on we show that such
bound can be a little improved.

######  Lemma 3.2.

(Carlson) Let $X$ be a space with a $\pi$-base ${\cal B}$ whose elements have
quasiregular closure. Then $X$ is quasiregular.

Let $U$ be a non-empty open subset of $X$. There exists $V\in{\cal B}$ such
that $V\subseteq U$ and $\overline{V}$ is quasiregular. The set
$V=V\cap\overline{V}$ is open in $\overline{V}$. As $\overline{V}$ is
quasiregular, there exists a non-empty open subset $W$ of $X$ such that
$\emptyset\neq
W\cap\overline{V}\subseteq\overline{(W\cap\overline{V})}^{\overline{V}}=\overline{(W\cap\overline{V})}\cap\overline{V}\subseteq
V$.

As $W\cap\overline{V}\not=\emptyset$ then $W\cap V\not=\emptyset$.
Furthermore, we have $\overline{(W\cap
V)}\subseteq\overline{(W\cap\overline{V})}$ and $\overline{(W\cap
V)}\subseteq\overline{V}$. Thus $\overline{(W\cap V)}\subseteq V\subseteq U$.
As $W\cap V$ is open in $X$ and non-empty, we conclude $X$ is quasiregular.

######  Corollary 3.3.

A space $X$ has a $\pi$-base ${\cal B}$ whose elements have quasiregular
closure if, and only if, $X$ is quasiregular.

The previous corollary gives us the motivation to place the hypotesis of
quasiregularity on the space instead of putting it on the closure of the
elements of a $\pi$-base.
Recall that, in [4], Bella and Cammaroto proved that $|X|\leq
d(X)^{t(X)\psi_{c}(X)}$ for every Hausdorff space $X$.

######  Theorem 3.4.

(Carlson) Let $X$ be a quasiregular space with an H-closed $\pi$-base. Then
$|X|\leq 2^{wL(X)t(X)\psi_{c}(X)}$.

Let $\kappa=wL(X)t(X)\psi_{c}(X)$ and let ${\cal B}$ be a $\pi$-base of non-
empty open sets with closures that are H-closed and quasiregular. For each
$B\in{\cal B}$, as $\overline{B}$ is H-closed, by the Dow-Porter’s result (in
[11]) we have that $|\overline{B}|\leq 2^{\psi_{c}(\overline{B})}\leq
2^{\psi_{c}(X)}\leq 2^{\kappa}$. Since $\psi_{c}(X)\leq\kappa$, for each $x\in
X$ we can fix a collection ${\cal V}_{x}$ of open neighborhoods of $x$ such
that $|{\cal V}_{x}|\leq\kappa$ and $\bigcap\\{\overline{V}:V\in{\cal
V}_{x}\\}=\\{x\\}$. Without loss of generality we may assume that each ${\cal
V}_{x}$ is closed under finite intersections.

We will construct by transfinite recursion a non-decreasing chain of open sets
$\\{U_{\alpha}:\alpha<\kappa^{+}\\}$ such that

  * (1)

$|\overline{U_{\alpha}}|\leq 2^{\kappa}$ for every $\alpha<\kappa^{+}$, and

  * (2)

if $X\setminus\overline{\bigcup{\cal M}}\not=\emptyset$ for some ${\cal
M}\in[\bigcup\\{{\cal V}_{x}:x\in\overline{U_{\alpha}}\\}]^{\leq\kappa}$, then
there is $B_{\cal M}\in{\cal B}$ such that $B_{\cal M}\subset
U_{\alpha+1}\setminus\overline{\bigcup{\cal M}}$.

Let $B_{0}\in{\cal B}$ be arbitrary. We set $U_{0}=B_{0}$. Then
$|\overline{U_{0}}|\leq 2^{\kappa}$. If $\beta=\alpha+1$, for some $\alpha$,
then for every ${\cal M}\in[\bigcup\\{{\cal
V}_{x}:x\in\overline{U_{\alpha}}\\}]^{\leq\kappa}$ such that
$X\setminus\overline{\bigcup{\cal M}}\not=\emptyset$, we choose $B_{{\cal
M}}\in{\cal B}$ such that $B_{{\cal M}}\subseteq
X\setminus\overline{\bigcup{\cal M}}$. We define
$U_{\beta}=U_{\alpha}\cup\bigcup\\{B_{\cal M}:{\cal M}\in[\bigcup\\{{\cal
V}_{x}:x\in\overline{U_{\alpha}}\\}]^{\leq\kappa},X\setminus\overline{\bigcup{\cal
M}}\not=\emptyset\\}$. Therefore, by Bella-Cammaroto’s inequality, we have
that $|\overline{U_{\beta}}|\leq 2^{\kappa}$. If $\beta<\kappa^{+}$ is a limit
ordinal we let $U_{\beta}=\bigcup_{\alpha<\beta}U_{\alpha}$. Then clearly
$|U_{\beta}|\leq 2^{\kappa}$, hence $|\overline{U_{\beta}}|\leq 2^{\kappa}$.

Let $F=\bigcup\\{\overline{U_{\alpha}}:\alpha<\kappa^{+}\\}$. Then $|F|\leq
2^{\kappa}$. Since $t(X)\leq\kappa$, $F$ is closed and therefore
$F=\overline{\bigcup\\{U_{\alpha}:\alpha<\kappa^{+}\\}}$. Thus, $F$ is a
regular-closed set.

We will show that $X=F$. Suppose that $X\not=F$. As $X\setminus F$ is open,
$X$ is quasiregular, and the fact that ${\cal B}$ is a $\pi$-base there exists
$B\in{\cal B}$ such that $\overline{B}\subseteq X\setminus F$. Now, fix $x\in
F$. We have $\overline{B}\cap\bigcap\\{\overline{V}:V\in{\cal
V}_{x}\\}=\emptyset$. We will show that there exists $V\in{\cal V}_{x}$ such
that $V\cap\overline{B}=\emptyset$. Suppose by way of contradiction that
$V\cap\overline{B}\not=\emptyset$ for every $V\in{\cal V}_{x}$. The family
${\cal W}=\\{V\cap\overline{B}:V\in{\cal V}_{x}\\}$ is an open filter base on
$\overline{B}$ as it is closed under finite intersections. ${\cal W}$ can then
be extended to an open ultrafilter ${\cal U}$ on $\overline{B}$. As
$\overline{B}$ is H-closed, ${\cal U}$ must converge to a point
$p\in\overline{B}$. Therefore for every $V\in{\cal V}_{x}$ we have
$p\in\overline{(V\cap\overline{B})}^{\overline{B}}=\overline{(V\cap\overline{B})}\cap\overline{B}\subseteq\overline{V}\cap\overline{B}$
and thus $p\in\overline{B}\cap\bigcap\\{\overline{V}:V\in{\cal V}_{x}\\}$. But
this is a contradiction as $\overline{B}\cap\bigcap\\{\overline{V}:V\in{\cal
V}_{x}\\}=\emptyset$.

Therefore for every $x\in F$ there exists $V_{x}\in{\cal V}_{x}$ such that
$V_{x}\cap\overline{B}=\emptyset$. Clearly $\\{V_{x}:x\in F\\}$ is an open
cover of $F$. Since $wL(X)$ is hereditary with respect to regular-closed sets,
there exists ${\cal M}\in\\{V_{x}:x\in F\\}^{\leq\kappa}$ such that
$F\subseteq\overline{\bigcup{\cal M}}$. Then there exists $\alpha<\kappa^{+}$
such that ${\cal M}\in[\bigcup\\{{\cal
V}_{x}:x\in\overline{U_{\alpha}}\\}]^{\leq\kappa}$. As
$\overline{B}\cap\bigcup{\cal M}=\emptyset$ it follows that $B\subset
X\setminus\overline{\bigcup{\cal M}}$, hence $X\setminus\overline{\bigcup{\cal
M}}\not=\emptyset$. Thus, there exists $B_{\cal M}\in{\cal B}$ such that
$\emptyset\not=B_{\cal M}\subseteq U_{\alpha+1}\setminus\overline{\bigcup{\cal
M}}\subseteq F\setminus\overline{\bigcup{\cal M}}=\emptyset$. Since this is a
contradiction, we conclude that $X=F$ and the proof is completed.

We present an example that witnesses the fact that the hypotesis of
quasiregularity is essencial. Such example is a Hausdorff not Urysohn not
quasiregular space.

######  Example 3.5.

A space $X$ having an H-closed $\pi$-base such that $|X|>2^{wL(X)\chi(X)}$
(hence $|X|>2^{wL(X)t(X)\psi_{c}(X)}$).

Let $\kappa$ be a cardinal. Consider the space
$X=({\mathbb{Q}}\times\kappa)\cup({\mathbb{R}}\setminus{\mathbb{Q}})$. If
$(q,\alpha)\in{\mathbb{Q}}\times\kappa$, a basic open neighborhood is
$U_{n}(q,\alpha)=\\{(r,\alpha):r\in{\mathbb{Q}}\hbox{ and
}|r-q|<\frac{1}{n}\\}$ for each $n\in\omega$. If
$x\in{\mathbb{R}}\setminus{\mathbb{Q}}$, a basic open neighborhood is
$U_{n}(x)=\\{x\\}\cup\\{(r,\alpha):r\in{\mathbb{Q}},\alpha<\kappa\hbox{ and
}|r-x|<\frac{1}{n}\\}$. Clearly, $t(X)\psi_{c}(X)\leq\chi(X)\leq\omega$,
moreover $wL(X)\leq{\mathfrak{c}}$. The collection
$\\{U_{n}(q,\alpha):n\in\omega,q\in{\mathbb{Q}}\hbox{ and }\alpha<\kappa\\}$
is a $\pi$-base in $X$. We prove that each $\overline{U_{n}(q,\alpha)}$ is
H-closed for each $n\in\omega,q\in{\mathbb{Q}}\hbox{ and }\alpha<\kappa$. Let
$\cal U$ be a basic open cover of $\overline{U_{n}(q,\alpha)}$. For each
$U\in{\cal U}$ there exists $W_{U}$ which is an open subset of $\mathbb{R}$ in
the standard topology such that
$\overline{U}=\\{(q,\alpha):q\in\overline{W_{U}}\cap{\mathbb{Q}}\\}\cup(({\mathbb{R}}\setminus{\mathbb{Q}})\cap\overline{W_{U}})$.
Since $[q-\frac{1}{n},q+\frac{1}{n}]$ is a compact (hence H-closed) subset of
$\mathbb{R}$ and $\\{W_{U}:U\in{\cal U}\\}$ is an open cover of this interval,
there exists a finite subfamily $\\{U_{1},...,U_{k}\\}$ such that
$[q-\frac{1}{n},q+\frac{1}{n}]\subseteq\bigcup_{i=1}^{k}\overline{W_{U_{i}}}$.
Therefore
$\overline{U_{n}(q,\alpha)}\subseteq\bigcup_{i=1}^{k}\overline{U_{i}}$.
Considering a cardinal $\kappa>2^{\mathfrak{c}}$ we have that
$|X|>2^{wL(X)\chi(X)}$ (hence
$|X|>2^{wL(X)t(X)\psi_{c}(X)}$).$\hfill\triangle$

Notice that for the space $X$ in Example 3.5
$d_{\theta}(X)=\omega<d(X)=2^{\omega}$. Observe also that $d(X)\leq
2^{wL(X)\chi(X)}$ cannot hold in spaces with an H-closed $\pi$-base. Indeed,
the inequality $|X|\leq d(X)^{\chi(X)}$ for Hausdorff spaces, will lead to the
inequality $|X|\leq 2^{wL(X)\chi(X)}$ for spaces with an H-closed $\pi$-base,
which is not true.

We can prove the following result.

######  Lemma 3.6.

Let $X$ be a space with a $\pi$-base $\cal B$ whose elements have compact
boundaries and $F\subseteq X$ a closed set such that $X\backslash
F\neq\varnothing$. Then there is $B\in{\cal B}$ such that $\overline{B}\cap
F=\varnothing$.

There is $B\in{\cal B}$ such that $B\cap F=\varnothing$. We are done if
$\overline{B}\cap F=\varnothing$. Otherwise, $(\overline{B}\backslash B)\cap
F\neq\varnothing$ and $L=(\overline{B}\backslash B)\cap F$ is compact. Let
$y\in B$. As $X$ is Hausdorff, there are open sets $V_{y}$ and $W_{y}$ such
that $y\in V$, $(\overline{B}\backslash B)\cap F\subseteq W_{y}$, and
$V_{y}\cap W_{y}=\varnothing$. There is $C\in{\cal B}$ such that
$\varnothing\neq C\subseteq B\cap V_{y}$ implying
$\overline{C}\subseteq\overline{B\cap V_{y}}\subseteq\overline{V_{y}}$. As
$\overline{C}\cap F\subseteq\overline{B}\cap F=(\overline{B}\backslash B)\cap
F\subseteq W_{y}$ and $W\cap\overline{V_{y}}=\varnothing$, it follows that
$\overline{C}\cap F=\varnothing$.

By Lemma 3.6 and by Theorem 3.4 we obtain the following results.

######  Corollary 3.7.

A space with $\pi$-base whose elements have compact boundaries is
quasiregular.

######  Corollary 3.8.

If a space $X$ has an H-closed $\pi$-base whose elements have compact
boundaries, then $|X|\leq 2^{wL(X)t(X)\psi_{c}(X)}$.

Now we deal with the class of Urysohn spaces. We start with a well known
result that characterizes compactness in the semigularization and we give the
proof for sake of completeness.

######  Theorem 3.9.

(Katětov) A space $X$ is Urysohn and H-closed if, and only if, $X_{s}$ is
compact.

Since a space is compact iff it is H-closed and regular it suffices to show
that $X_{s}$ is regular. Let A be a regular-closed subspace of X and $p$ be a
point not in $A$. For each $q\in A$, there are disjoint regular-open sets
$U_{q}$ and $V_{q}$ such that $p\in U_{q}$, $q\in V_{q}$, and
$\overline{U_{q}}$ and $\overline{V_{q}}$ are disjoint. There is a finite
subset $F$ of $A$ such that $A$ is contained in
$V=\bigcup\\{\overline{V_{q}}:q\in F\\}$. Let $U=\bigcap_{q\in F}U_{q}$. Then
$U\cap V=\emptyset$. We have that $V$ is a regular closed set and that $p\in
X\setminus\overline{V}\subseteq X\setminus A$. This completes the proof. Using
the previous theorem, we prove next result.

######  Lemma 3.10.

Let $X$ be a Urysohn space. If there exists an H-closed $\pi$-base in $X$,
then there exists a compact $\pi$-base in $X_{s}$.

Let ${\cal B}$ be an H-closed $\pi$-base in $X$. Since
$\overline{B}=\overline{int(\overline{B})}=cl_{s}(int(\overline{B}))$, where
$cl_{s}(A)$ denotes the closure in the semiregularization $X_{s}$, by Theorem
3.9, we have that $cl_{s}(int(\overline{B}))$ is compact in $X_{s}$. We have
to prove that $\\{int(\overline{B}):B\in{\cal B}\\}$ is a $\pi$-base in
$X_{s}$. Fix a basic open set $int(\overline{V})$. There exists $B\in{\cal B}$
such that $B\subseteq V$. Then $int(\overline{B})\subseteq int(\overline{V})$.
This completes the proof.

In [1] Alas and Kocinac introduced for Hausdorff spaces, the cardinal function
$k(X)$ that is the least cardinal $\kappa$ such that for every $x\in X$ there
exists a family ${\cal V}_{x}$ of open neighborhoods of $x$ such that $|{\cal
V}_{x}|\leq\kappa$ and for every regular closed subset $\overline{U}$,
containing $x$, there exists $V\in{\cal V}_{x}$ such that
$\overline{V}\subseteq\overline{U}$. It is straightforward that
$k(X)\leq\chi(X)$. The proof of the following lemma is once again direct.

######  Lemma 3.11.

[1] $k(X)=\chi(X_{s})$ for every space $X$.

######  Theorem 3.12.

Let $X$ be a Urysohn space with an H-closed $\pi$-base. Then $|X|\leq
2^{wL(X)k(X)}$.

From Lemma 3.10 it follows that $X_{s}$ has a compact $\pi$-base. Since
$wL(X_{s})=wL(X)$ and by Lemma 3.11, we have that $|X|=|X_{s}|\leq
2^{wL(X_{s})\chi(X_{s})}=2^{wL(X)k(X)}$.

The following result is assumed without proof in [11] however a proof of it
can be found in [9].

######  Lemma 3.13.

[11] Let $X$ be an H-closed space. Then $\chi(X_{s})\leq\psi_{c}(X)$.

Since the relations $\psi_{c}(X)=\psi_{c}(X_{s})\leq\chi(X_{s})$ are true for
every space $X$, then we can say that $k(X)=\chi(X_{s})=\psi_{c}(X)$ holds for
H-closed spaces.

######  Question 3.14.

Is it true that for Urysohn spaces having an H-closed $\pi$-base
$k(X)=\psi_{c}(X)$?

The following example shows that Theorem 3.12 is an actual improvement of the
bound $|X|\leq 2^{wL(X)\chi(X)}$ for Urysohn spaces having an H-closed
$\pi$-base (in [5]).

######  Example 3.15.

There exist an Urysohn H-closed space $Z$ such that $k(Z)<\chi(Z)$.

Consider the space $Z=2^{\omega}$ with the following topology on it: a basic
open neighborhood of a point $\alpha$ is of the form $(U\setminus T)$, where
$U$ is a basic open subset in $2^{\omega}$ containing $\alpha$ and $T$ is a
subset of $2^{\omega}$ such that $|T|\leq\omega$ and $\alpha\not\in T$.
Clearly, $Z$ is Urysohn. We want to show that $Z$ is H-closed. Let $\cal U$ be
an open cover of $Z$ made by basic open sets of the form $B\setminus S$. Since
$2^{\omega}$ is compact there exist $B_{1},...,B_{k}$ such that
$Z=2^{\omega}\subseteq\bigcup_{i=1}^{m}B_{i}\subseteq\bigcup_{i=1}^{m}\overline{B_{i}}^{2^{\omega}}=\bigcup_{i=1}^{m}\overline{B_{i}\setminus
S_{i}}^{Z}$. This proves that $Z$ is H-closed. Then every $\pi$-base is an
H-closed $\pi$-base. As an additional remark, we can say that $wL(Z)<\omega$.
It is easy to see that $(Z)_{s}=2^{\omega}$. Then, by Lemma 3.11,
$k(Z)=\chi(2^{\omega})=\omega$. It is straightforward to see that
$\chi(Z)=|[2^{\omega}]^{\leq\omega}|=2^{\omega}$. $\hfill\triangle$

It can be easily proved that $\psi_{\theta}(X)\leq k(X)$ for every Urysohn
space $X$. Then it is worthwhile to pose the following question.

######  Question 3.16.

Does the inequality $|X|\leq 2^{wL(X)t_{\theta}(X)\psi_{\theta}(X)}$ hold for
every Urysohn space $X$ with an H-closed $\pi$-base?

##  4 Martin’s Axiom and spaces with a quasiregular H-closed $\pi$-base

Recall the topological definition of Martin’s Axiom for a cardinal $\kappa$,
$\aleph_{0}\leq\kappa<2^{\aleph_{0}}$. $MA(\kappa)$ states that for each
compact space $X$ with the ccc, if $(U_{\alpha}:\alpha<\kappa)$ is a family of
open dense subsets of $X$, then
$\bigcap\\{U_{\alpha}:\alpha<\kappa\\}\not=\emptyset$.

######  Theorem 4.1.

[16] Let $\aleph_{0}\leq\kappa<2^{\aleph_{0}}$. The followings are equivalent.

  1. 1.

$MA(\kappa)$.

  2. 2.

For each space $X$ with the ccc such that $\\{x\in X:x\hbox{ has a compact}$
$\hbox{ neighborhood}\\}$ is dense in $X$, if $(U_{\alpha}:\alpha<\kappa)$ is
a family of open dense subsets of $X$, then
$\bigcap\\{U_{\alpha}:\alpha<\kappa\\}$ is a dense subset of $X$.

  3. 3.

For each poset $({\mathbb{P}},\leq,1)$ with the ccc, if $\cal D$ is a family
of dense subsets of $\mathbb{P}$ with cardinality $<\kappa$ then there exists
a filter $G\subseteq\mathbb{P}$ such that for every $D\in\cal D$ one has
$D\cap G\not=\emptyset$.

######  Theorem 4.2.

Let $\aleph_{0}\leq\kappa<2^{\aleph_{0}}$. The following is equivalent to the
conditions in Theorem 4.1.

  * 4.

For each quasiregular H-closed space $X$ with the ccc, if
$(U_{\alpha}:\alpha<\kappa)$ is a family of open dense subsets of $X$, then
$\bigcap\\{U_{\alpha}:\alpha<\kappa\\}\not=\emptyset$.

3.$\implies$4\. Let $(X,\tau)$ be a quasiregular H-closed space having the
ccc. Let $(U_{\alpha}:\alpha<\kappa)$ be a family of open dense subsets of
$X$. Consider the following poset
$({\mathbb{P}},\leq,1)=(\tau\setminus\\{\emptyset\\},\subseteq,X)$. For each
$\alpha<\kappa$ construct a family ${\cal
E}_{\alpha}=\\{U\in{\mathbb{P}}:\overline{U}\subseteq U_{\alpha}\\}$. For the
quasiregularity of $X$, the families ${\cal E}_{\alpha}$, $\alpha<\kappa$, are
non-empty and since $U_{\alpha}$ is an open dense subset of $X$, ${\cal
E}_{\alpha}$ is dense in $\mathbb{P}$ in the sense of posets. Then there exist
a filter ${\cal F}\subseteq{\mathbb{P}}$, which is an open filter on $X$ such
that ${\cal E}_{\alpha}\cap{\cal F}\not=\emptyset$ for every $\alpha<\kappa$.
Then for every $\alpha<\kappa$ choose $F_{\alpha}\in{\cal E}_{\alpha}\cap{\cal
F}$. Since $X$ is H-closed the adherence of $\cal F$ is non-empty, then there
exists $x\in\bigcap\\{\overline{F}:F\in{\cal F}\\}$. Therefore
$x\in\bigcap\\{\overline{F_{\alpha}}:\alpha<\kappa\\}\subseteq\bigcap\\{U_{\alpha}:\alpha<\kappa\\}$.
The other implication is proved in the following theorem.

######  Theorem 4.3.

Let $\aleph_{0}\leq\kappa<2^{\aleph_{0}}$. The followings are equivalent to
the statements of Theorem 4.1 and Theorem 4.2.

  1. 5.

For each space $X$ with the ccc such that $\\{x\in X:x\hbox{ has a
quasiregular}$
$\hbox{H-closed neighborhood}\\}$ is dense in $X$, if
$(U_{\alpha}:\alpha<\kappa)$ is a family of open dense subsets of $X$, then
$\bigcap\\{U_{\alpha}:\alpha<\kappa\\}$ is a dense subset of $X$.

  2. 6.

For each space $X$ with the ccc and a compact $\pi$-base, if
$(U_{\alpha}:\alpha<\kappa)$ is a family of open dense subsets of $X$, then
$\bigcap\\{U_{\alpha}:\alpha<\kappa\\}$ is dense in $X$.

  3. 7.

For each space $X$ with the ccc and a quasiregular H-closed $\pi$-base, if
$(U_{\alpha}:\alpha<\kappa)$ is a family of open dense subsets of $X$, then
$\bigcap\\{U_{\alpha}:\alpha<\kappa\\}$ is dense in $X$.

4.$\implies$5\. Let $X$ be a space with the ccc and let $D=\\{x\in X:x$ has a
quasiregular H-closed neighborhood$\\}$ be a dense subset in $X$. Let
$(U_{\alpha}:\alpha<\kappa)$ be a family of open dense subsets of $X$. Fix a
non-empty open subset $W$ of $X$, then there exist $x\in W\cap D$ and a open
neighborhood $V$ of $x$ such that $\overline{V}$ is quasiregular and H-closed.
Therefore $Y=\overline{V\cap W}$ is a quasiregular H-closed space with the
ccc. Moreover $U_{\alpha}\cap(W\cap V)$, for each $\alpha<\kappa$ is a non-
empty open dense subset of $Y$. Then by hypotesis,
$\emptyset\not=\bigcap\\{U_{\alpha}\cap(W\cap
V):\alpha<\kappa\\}\subseteq\bigcap\\{U_{\alpha}:\alpha<\kappa\\}\cap W$.
5.$\implies$7\. Let $(X,\tau)$ be a space with ccc and let the subset
$D=\\{x\in X:x\hbox{ has a quasiregular H-closed neighborhood}\\}$ dense in
$X$. For each $U\in\tau\setminus\\{\emptyset\\}$ there exists $x\in U\cap D$,
then choose a open neighborhood $V_{U}$ of $x$, such that $\overline{V_{U}}$
is quasiregular and H-closed. Therefore ${\cal B}=\\{U\cap
V_{U}:U\in\tau\setminus\\{\emptyset\\}\\}$ is a quasiregular H-closed
$\pi$-base.
7.$\implies$6\. is trivial.
6.$\implies$2\. Notice that if there exists a compact $\pi$-base $\cal B$ then
$\bigcup{\cal B}$ is a dense subset of $X$ and every point in $\bigcup{\cal
B}$ has a compact neighborhood.

As a corollary one can obtain the following.

######  Corollary 4.4.

If $X$ is a space with the ccc and a quasiregular H-closed $\pi$-base, then
$X$ is Baire.

Actually we can remove the hypotesis ccc and obtain the following.

######  Theorem 4.5.

Let $X$ be a space with a quasiregular H-closed $\pi$-base. Then $X$ is a
Baire space.

Let $\\{U_{n}:n<\omega\\}$ be a family of open dense subsets of $X$ and ${\cal
B}$ be a quasiregular H-closed $\pi$-base. We show $\bigcap_{n<\omega}U_{n}$
is dense in $X$. Let $U$ be a nonempty open set. We show that
$U\cap\bigcap_{n<\omega}U_{n}\not=\emptyset$.

As $U\cap U_{0}\not=\emptyset$, there exists $B\in{\cal B}$ such that
$B\subseteq U\cap U_{1}$ and $\overline{B}$ is a quasiregular and H-closed
subspace. Then there exists an open subset $B_{0}$ of $\overline{B}$ such that
$\overline{B_{0}}^{\overline{B}}\subseteq B\subseteq U\cap U_{0}$. The set
$B_{0}\cap(U\cap U_{1})$ is an open subset of $\overline{B}$, then, by its
quasiregularity, there exists an open subset $B_{1}$ of $\overline{B}$, such
that $\overline{B_{1}}^{\overline{B}}\subseteq B_{0}\cap(U\cap U_{1})$.
Consider the open subset $B_{1}\cap(U\cap U_{2})$ of $\overline{B}$ and
continue by induction. Then we obtain a decreasing sequence
$\\{\overline{B_{n}}^{\overline{B}}:n<\omega\\}$ of closed subsets of
$\overline{B}$ such that $\overline{B_{n}}^{\overline{B}}\subseteq U\cap
U_{n}$. Since $\overline{B}$ is H-closed we have that
$\emptyset\not=\bigcap_{n<\omega}\overline{B_{n}}^{\overline{B}}\subseteq(\bigcap_{n<\omega}U_{n})\cap
U$ .

By Corollary 3.3, we obtain the following corollary.

######  Corollary 4.6.

Let $X$ be a quasiregular space with an H-closed $\pi$-base. Then $X$ is
Baire.

In [2] the authors showed that if $X$ is a Hausdorff Baire space with a rank
2-diagonal, then $|X|\leq wL(X)^{\omega}$ and if $X$ is a Hausdorff Baire
space with a $G_{\delta}$-diagonal, then $d(X)\leq wL(X)^{\omega}$. Therefore,
as corollaries of these results and Theorem 4.5 we have the following:

######  Corollary 4.7.

Let $X$ be a space with a quasiregular H-closed $\pi$-base.

  * (a)

If $X$ has a $G_{\delta}$-diagonal, then $d(X)\leq wL(X)^{\omega}$.

  * (b)

If $X$ has a rank 2-diagonal, then $|X|\leq wL(X)^{\omega}$.

Acknowledgement: The author express his sincere gratitude to N. Carlson and M.
Bonanzinga for their useful suggestions and inspiring discussions. He would
also like to thank the “National Group for Algebric and Geometric Structures,
and their Applications”(GNSAGA-INdAM) for their invaluable support throughout
the course of this research.

## References

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I.R.Boyko, V.V.Bytev, A.S.Zhemchugov

Joint Institute for Nuclear Research, Dubna, Russia

Two-photon physics at future electron-positron colliders

###### Abstract

Two photon collisions offer a variety of physics studies that can be performed
at the future electron-positron colliders. Using the planned CEPC parameters
as a benchmark we consider several topics to be studied in the two-photon
collisions. With the full integrated luminosity the Higgs boson
photoproduction can be reliably observed. A large statistics of various
quarkonium states can be collected. The LEP results on the photon structure
function and the tau lepton anomalous magnetic moment can be improved by 1-2
orders of magnitude.

##  1 Introduction

The flagship of the modern particle physics, the LHC, is expected to continue
running for more than 10 year from now. However, the planning of the next
generation colliders has already started.

It is widely believed that the next major collider project can be a high-
energy, high-luminosity electron-positron collider. Currently four mature
projects of e+e- colliders are under consideration: linear colliders CLIC [1]
and ILC [2] and circular colliders CEPC [3] and FCC-ee [4].

The scientific programs of the future e+e- colliders are well developed. The
main goals are the Higgs boson physics; high-precision measurements at the Z
pole energy; the top quark physics; searches for new physics phenomena. In
this paper we investigate prospects for another branch of the experimental
program: the physics of two photon collisions.

The collisions of (quasi-)virtual photons are responsible for a significant
fraction of the total event rate at electron-positron colliders. They are
especially abundant at the linear colliders due to high acceleration
gradients. Such events represent an unpleasant background for the studies of
other physics processes. On the other hand, the two-photon collisions
themselves provide opportunities for interesting physics studies. In this
paper we analyze several topics which can be investigated using the two-photon
collisions at the future e+e- colliders.

At the linear colliders the rate and the energy spectrum of the two-photon
collisions depend of the particular configuration of beams. Therefore our
study is restricted to the case of circular colliders, where the differential
gamma-gamma luminosity can be predicted from the first principles.

Both CEPC and FCC are expected to take large data sets at 240 GeV center-of-
mass energy (a point close to the maximum yield of Higgs bosons). The CEPC
expected integrated luminosity is 5.6 ab-1. FCC plans to collect at least 5
ab-1 at 240 GeV, followed by 1.7 ab-1 at the energies of top pair production
(340-365 GeV). In this paper we use the CEPC planned energy and luminosity as
a benchmark scenario; however, our qualitative results are also applicable to
the FCC which has a similar expected performance.

##  2 Two-photon collisions at electron-positron colliders

The era of two-photon physics had been started in early 1970th, when a
relatively high cross section of the 4-th order QED process $e^{+}e^{-}\to
e^{+}e^{-}e^{+}e^{-}$ was observed [5] in Budker Institute of Nuclear Physics
(Novosibirsk).

It should be noted that the first theoretical predictions for a
photoproduction of pions [6], [7] in $e^{+}e^{-}$ collision had been made back
around 1960, the calculated cross-sections deemed unmeasurably small and no
further elaboration has been done till the late 60th. At that time a series of
papers covering multiple final states of $e^{+}e^{-}$ colliders had been
published, but only after experimental observation at Novosibirsk the most
remarkable mechanism of cross-section enhancement due to small virtuality of
photons emitted by the electrons was revealed [8].

It is the small virtuality of intermediate photons that features two main
characteristics of two-photon processes: enhancement of the yield with the
increase of colliding beam energies and the possibility to study new type of
processes $\gamma\gamma\to X$ with two variable parameters - the virtualities
of intermediate photons.

The two-photon process can occur at electron-positron or electron-electron
colliders:

$e^{\pm}+e^{-}\to e^{\pm}+e^{-}+\gamma^{*}+\gamma^{*}\to e^{\pm}+e^{-}+X,$

where $X$ describes an arbitrary final state. Not any final state can be
produced in the two-photon processes: since two photons have even C parity,
only states with C=+1 are possible. As it was mentioned above, two-photon
cross-section slowly rises with the energy and becomes greater than the one of
the annihilation channel at $\sqrt{s}/2\gtrsim 1$ GeV.

In a two-photon process the initial particles (electrons and/or positrons)
emit two photons with virtualities $q_{1}$ and $q_{2}$ and the latter merge in
some final system of particles X with invariant mass square
$W^{2}={(q_{1}+q_{2})^{2}}$, see Fig.1.

Figure 1: Kinematics of two-photon process.

The cross-section of two-photon scattering of electron (positron) can be
calculated as a convolution of amplitudes describing the emission of virtual
photons off initial particle and the $\gamma\gamma\to X$ transition. The first
one is calculated within the QED, and the second can be expanded in
independent tensors, the choice of which is quite arbitrary up to the
conservation of the Lorenz invariance, T- and gauge invariance. For the sake
of physical interpretation, it is encoded in five structure functions. Three
of them can be expressed through the cross section $\sigma_{a,b}$ for scalar
($a,b=S$) and transverse photons ($a,b=T$). The other structure functions
$\tau_{TT}$ and $\tau_{TS}$ correspond to transitions with spin-flip for each
of the photons with total helicity conservation [9]:

$\displaystyle\mbox{d}\sigma=\frac{\alpha^{2}}{16\pi^{4}q_{1}^{2}q_{2}^{2}}\sqrt{\frac{(q_{1}q_{2})^{2}-q_{1}^{2}q_{2}^{2}}{(p_{1}p_{2})^{2}-m_{e}^{2}m_{e}^{2}}}\biggl{(}4\rho^{++}_{1}\rho^{++}_{2}\sigma_{TT}+2|\rho_{1}^{+-}\rho_{2}^{+-}|\tau_{TT}\cos(2\tilde{\phi})$
$\displaystyle+2\rho^{++}_{1}\rho^{00}_{2}\sigma_{TS}+2\rho^{00}_{1}\rho^{++}_{2}\sigma_{ST}+\rho^{00}_{1}\rho^{00}_{2}\sigma_{SS}-8|\rho^{+0}_{1}\rho^{+0}_{2}|\tau_{TS}\cos(\tilde{\phi})\biggr{)}\frac{\mbox{d}^{3}p_{1}^{\prime}\mbox{d}^{3}p_{2}^{\prime}}{E_{1}E_{2}},$
(1)

where $\rho_{i}^{ab}$ are the density matrices of the virtual photon in the
$\gamma\gamma$-helicity basis, $p_{i},(p_{i}^{\prime})$ corresponds to the
momentum of initial (scattered) leptons, $\tilde{\phi}$ is the angle between
the scattering planes of colliding particles in c.m.s. of photons, $m_{e}$ is
the mass of the initial lepton.

The exact formula (1) provides an accurate estimation of two-photon process at
any kinematical region and could be used for quantitatively correct
simulations [10, 11]. At the limit $q_{i}^{2}\to 0$ one can see a logarithmic
enhancement of cross-section, with natural kinematic ”regularization” value at
$m_{e}^{2}$.

The small $q^{2}_{i}$ domain gives the main contribution to the cross-section
of the process under consideration. In this limit the expression for (1) can
be simplified under the procedure called Equivalent Photon Approximation
(Weizacher-Williams’ method).

Keeping in mind this approximation and taking into account only leading terms
we can see that the cross-section of the two-photon process factorizes into
two different parts: one part is connected with emission of two real photon by
initial particles and the other is the final state production by the two
photons [13]:

$\displaystyle\frac{\mbox{d}\sigma^{(0)}}{\mbox{d}W^{2}\mbox{d}\Gamma}=2\biggl{(}\frac{\alpha}{\pi}\biggr{)}^{2}\frac{1}{W^{2}}\ln^{2}\frac{E}{m_{e}}f(\frac{W}{2E})\frac{\mbox{d}\sigma_{\gamma\gamma\to
X}(W^{2})}{\mbox{d}\Gamma},$ (2)

where E is the energy of the initial particle and

$\displaystyle
f(\gamma)=-(2+\gamma^{2})^{2}\ln\gamma-(1-\gamma^{2})(3+\gamma^{2})$ (3)

is the factor describing the luminosity dependence over the invariant mass of
the colliding photons.

The cost of this simplicity is the underestimation of cross-section in some
specific kinematics, the loss of the information on the initial particle
scattering angle and missing deep virtual scattering behaviour of cross-
section, when one of the photon has a rather large invariant mass. Of course,
events with big virtuality of an intermediate photon are much more rare, since
there is no $q_{i}^{2}$ pole enhancement.

Small $q^{2}_{i}$ invariant mass of emitted photons means that the scattered
leptons proceed undetected (so called no-tag events), but due to large cross-
section of the process the no-tag events can be selected even without full
final state recovery [14], using the requirements of the small total
transverse momentum of the produced system and small value of its invariant
mass ($W\ll 2E$). The gamma-gamma collisions tend to occur at the c.m.s.
energies much less than the nominal collider energy, as can bee seen in Fig. 2
(left). Nevertheless, the amount of collisions at rather high energies is also
significant and can be used to study different aspects of the two-photon
physics.

Practically, one can tag two photon events by detection of one or both
scattered electrons (single tag or double tag mode). Tagging allows one to
suppress background significantly at the cost of steep reduction of available
statistics. In this case the requirement of tagging the lepton(s) in a given
energy and angular range means non-vanishing virtualities of photon mass and
gives one the possibility to measure the $q_{i}^{2}$ dependence of the two-
photon cross-section (1).

The two-photon processes were always very interesting for physicists due to
the fascinating opportunity to study conversion of pure light to matter. In
this paper we cover rather conservative and well-established topics of the
two-photon physics, namely quarkonium spectroscopy, Higgs production, tau pair
production and photon hadronic structure. All suggested measurements are aimed
to improve significantly the precision of existing experiments and without any
doubt are achievable under the planned characteristics of the future
$e^{+}e^{-}$ colliders. The integrated luminosity for the gamma-gamma
collisions above the $W^{+}W^{-}$ threshold (see Fig. 2, right) will be
slightly less than 1 fb-1 which is comparable to the total $e^{+}e^{-}$
luminosity recorded by the LEP2 experiments.

Figure 2:  Left: number of $\gamma\gamma$ collisions as a function of the
$\gamma\gamma$ invariant mass relative to the $e^{+}e^{-}$ CMS energy. Right:
integrated luminosity of the $\gamma\gamma$ collisions above certain collision
energy, under the assumption of 5.6 ab-1 integrated luminosity of $e^{+}e^{-}$
collisions.

##  3 Quarkonium spectroscopy

Quarkonium is regarded as the simple hadronic systems to explore the QCD
aspects at the low energy regimes through its spectroscopy [15].

Quarkonia, the bound states of a heavy quark c, b and the corresponding
antiquark, can be most effectively studied at $e^{+}e^{-}$ colliders. Since
the discovery of the $J/\psi$ in 1974, a lot of information about bounded
$b\bar{b}$ and $c\bar{c}$ states was elaborated.

Quarkonium can be described as bound quark-antiquark state, with Coulombic
short-distance potential with logarithmic modification of coupling strength
and linear long-distance potential for quark confinement description
[15],[16].

Although there exist a lot of different ways to quantitatively describe
quarkonia, e.g. Lattice QCD [17], NRQCD methods [18], Light front quark
model[19] or exotic like instanton liquid model [20], there are still many
inconsistencies between predicted and measured radial excitation mass spectra
of quarkonium states.

Due to negative charge parity of the photon, only neutral particles with even
charge conjugation $C=1$ can be produced in the two-photon collisions. There
is a lot of interesting tasks concerning light mesons constituted from light
$u,d$ quarks, e.g. $\pi$, $\eta$, $\eta^{\prime}$ and their excited states,
but we expect that CEPC experiments will be insensitive to the hadronic
systems with masses below 3 GeV due to the issues of detector resolution and
experimental environment. Therefore the most straightforward studies can be
done with heavy $c$ and $b$ quarks, which lead us to the charmonia and
bottomonia spectroscopy.

fraction | $\eta_{c}$ | $\chi_{c_{0}}$ | $\chi_{c_{1}}$ | $\chi_{c_{2}}$ | $\eta_{c}(2S)$
---|---|---|---|---|---
$10^{-4}$ | $1.57\pm 0.12$ | $2.04\pm 0.09$ | $<6.3\times 10^{-2}$ | $2.85\pm 0.1$ | $1.9\pm 1.3$
Table 1: Charmonium low state radiative decay branching ratio

Charmonium states with even charge conjugation $C=1$ up to the $\eta_{c}(2S)$
mass of 3637 GeV are well studied, their radiative decays to photons have been
seen and measured (except $\chi_{c_{1}}$, where only upper limit on branching
ratio was established, see Table 1). From numerical estimation of event number
(see Sect. 3.2, double tag set-ups) we can conclude that it is definitely
possible to improve accuracy of $\gamma\gamma$ branching ratio for
$\chi_{c_{1}}$ and $\eta_{c}(2S)$ charmonium states, and to measure at least
the $Q^{2}$ dependence of resonance formation in single-tag set-up, where we
expect approximately $10^{4}$ events for each charmonium state.

The event yield estimation of charmonium production with the mass larger than
that of $\eta_{c}(2S)$ is more tricky. From one point of view the $c\bar{c}$
spectroscopy at that region of mass states is very interesting due to lack of
knowledge about internal structure and branching ratio to two photons of
resonances (see Table 2), so any results on measurement or lower limit
estimation of $\gamma\gamma$ width are highly appreciated. Moreover, the
measurement of $Q^{2}$ dependence of resonance formation could give us hard
restriction over quantum numbers and internal structure of resonances under
consideration.

The charmonium bound states with masses above open charm threshold can not be
described only in the frames of constituent quark model that describes the
observed meson spectrum as $q\bar{q}$ bound states (see Table 2). To explain
unexpected quantum numbers, masses, branching ratios and other properties of
the heavy resonances that form the charmonium spectrum, various possibilities
of new physics states are considered: meson states that are made of bound
gluons (glueballs), qq-pairs with an excited gluon (hybrids), multiquark color
singlet states such as qqqq (tetraquarks) and molecular bound states of
4-quark system, six-quark and ’baryonium’ bound states.

The other side of the problem is that without a knowledge of the internal
structure it is not possible to make a firm estimation of the charmonium state
production. Even a naive equivalent photon approximation with narrow resonance
approximation is not applicable due to unknown branching ratio to two photons.

A rough estimation of production rate we suggest to make under assumption that
all charmonium states mostly consist of $c\bar{c}$ states. We know that
branching ratio of the two photon quarkonium decay at lowest order does not
depend on the state mass and has slight (up to a factor of few) dependence on
its quantum numbers (see [21] and Table 1). So two photon branching ratio
could be estimated at the level of $10^{-4}$. That estimation gives us about
$10^{7}-10^{8}$ events in no-tag mode (although we do not observe scattered
leptons, it is possible to reduce background events by imposing a strict
transverse-momentum balance along the beam axis for the final-state hadronic
system [22]), and about $10^{4}$ events in single (minimal angle of detection
at 6 degrees) and double tag (using the luminosity calorimeter) modes. All
estimations are made with accelerator parameters described in Sect. 3.2.

As a result we can conclude that if the recently discovered charmonium
resonances (Table 2) are similar to the charm quark bound state, we should
definitely discover their quantum numbers and measure the two photon branching
ratios.

name | $J^{PC}$ | Width, MeV | $\Gamma_{\gamma\gamma}$ | nature
---|---|---|---|---
$\chi_{c0}(3860)$ | $0^{++}$ | $201\pm 101$ | seen |  $c\bar{c}$ +possible non-$q\bar{q}$ states
$\chi_{c1}(3872)$ | $1^{++}$ | $<1.2$ | - | candidate for an exotic structure
X(3915) | 0 or $2^{++}$ | $20\pm 5$ | seen |  $c\bar{c}$ +possible non-$q\bar{q}$ states
$\chi_{c2}(3930)$ | $2^{++}$ | $24\pm 6$ | seen |  $c\bar{c}$ +possible non-$q\bar{q}$ states
X(3940) | $?^{??}$ | $37\pm 17$ | - |  $c\bar{c}$ +possible non-$q\bar{q}$ states
X(4050) | $?^{?+}$ | $82\pm 28$ | - | candidate for an exotic structure
X(4100) | $?^{??}$ | $152\pm 70$ | - | candidate for an exotic structure
$\chi_{c}1$(4140) | $1^{++}$ | $22\pm 7$ | not seen | candidate for an exotic structure
X(4160) | $?^{??}$ | $139\pm 60$ | - |  $c\bar{c}$ +possible non-$q\bar{q}$ states
X(4250) | $?^{?+}$ | $177\pm 70$ | - | candidate for an exotic structure
$\chi_{c1}$(4274) | $1^{++}$ | $49\pm 12$ | - | candidate for an exotic structure
X(4350) | $?^{?+}$ | $13\pm 10$ | seen |  $c\bar{c}$ +possible non-$q\bar{q}$ states
$\chi_{c0}$(4500) | $0^{++}$ | $92\pm 29$ | - | candidate for an exotic structure
$\chi_{c0}$(4700) | $0^{++}$ | $120\pm 50$ | - | candidate for an exotic structure
Table 2: Charmonium exotic states that could be seen at $\gamma\gamma$
collision

For the bottomonium states only ground state $\eta_{b}(9399)$ decay to two
photons was seen, and other $C=+1$ states radiative decay were not measured
[23], so any information about them, e.g. upper limit on the two photon
branching or decay event observation, are highly appreciated.

At the Sect. 3.2 we estimate low bottomonium states $1S,1P,2S$. Event number
of positive charge parity states with higher radial quantum number, namely
$\chi_{b0}(2P)$, $\chi_{b1}(2P)$, $\chi_{b2}(2P)$, $\chi_{b1}(3P)$,
$\chi_{b2}(3P)$ are estimated as counterparts with radial quantum number equal
to unity (see Fig. 3).

With the number of bottomonium state events of approximately $10^{5}$ in no-
tag mode and $10^{2}$ with double-tag we could conclude that we definitely can
at least put upper limit on radiative decay branching of bottomonium states.

Figure 3:  The level scheme of the $b\bar{b}$ b states. The figure was taken
from [23].

###  3.1 Theoretical estimation

We can estimate theoretically the full cross-section of two photon meson
production with the help of equivalent photon approximation formula (2), where
cross-section of meson with spin J, mass M photoproduction can be estimated
trough:

$\displaystyle\sigma_{mes}(W^{2})=(2J+1)8\pi^{2}\frac{\Gamma_{\gamma\gamma}}{M}\frac{1}{\pi}\frac{M\Gamma}{(M^{2}-W^{2})^{2}+M^{2}\Gamma^{2}},$
(4)

here $\Gamma_{\gamma\gamma}$ and $\Gamma$ are the two photon and total width
of meson decay.

###  3.2 Number of event estimation

Simulation was performed with the GALUGA [10], [11], [12] two photon
production generator, where virtualities $Q^{2}$ of photons are fully taken
into account through five structure functions describing photon scattering
with different polarizations. Such a consideration gives one the possibility
to make a robust estimation of meson production at large photon virtualities
(large electron or positron scattering angle). Resonance formation in two-
photon scattering are calculated in the framework of constituent-quark model
[11].

The expected event yields are presented for the total beam energy
$\sqrt{s}=240$ GeV and integrated luminosity $5$ ab-1.

name | No Tag | S Tag 10 | S Tag 6 | D Tag 6 | D Tag 6-1.9 | D Tag 1.9-1.9
---|---|---|---|---|---|---
$\eta_{c}$ | $1.\times 10^{9}$ | $5.3\times 10^{5}$ | $1.7\times 10^{5}$ | $7.6\times 10^{2}$ | $1.2\times 10^{4}$ | $8.6\times 10^{4}$
$\chi_{c_{0}}$ | $2.\times 10^{8}$ | $6.5\times 10^{4}$ | $2.2\times 10^{4}$ | $6.3\times 10^{1}$ | $9.\times 10^{2}$ | $8.3\times 10^{3}$
$\chi_{c_{1}}$ | $7.\times 10^{6}$ | $4.4\times 10^{4}$ | $1.1\times 10^{4}$ | $1.7\times 10^{2}$ | $2.4\times 10^{3}$ | $1.7\times 10^{4}$
$\chi_{c_{2}}$ | $9.5\times 10^{7}$ | $3.1\times 10^{4}$ | $9.8\times 10^{3}$ | $5.4\times 10^{1}$ | $7.2\times 10^{2}$ | $6.7\times 10^{3}$
$\eta_{c}(2S)$ | $2.4\times 10^{8}$ | $2.1\times 10^{5}$ | $6.4\times 10^{4}$ | $4.1\times 10^{2}$ | $6.\times 10^{3}$ | $4.1\times 10^{4}$
Table 3: Estimation of event number for charmonium states. Singe tag, no tag
and double tag set-ups are considered. Cuts on electron or positron scattering
angle are 1.9, 6 and 10 degrees.

Quarkonium production was estimated in three different modes, namely no tag,
where no final scattered electron or positron detected, single tag, where
electron or positron are detected and double tag (both electron and positron
are detected). The minimal angles are taken equal to 6 or 10 degrees (various
detector characteristics are considered), and 1.9 degrees for the case when
scattered electron or positron can be detected in the luminosity calorimeter.

name | No Tag | S Tag 10 | S Tag 6 | D Tag 6 | D Tag 6-1.9 | D Tag 1.9-1.9
---|---|---|---|---|---|---
$\eta_{b}(9399)$ | $1.3\times 10^{6}$ | $1.4\times 10^{4}$ | $3.9\times 10^{3}$ | $1.1\times 10^{2}$ | $1.\times 10^{3}$ | $3.6\times 10^{3}$
$\chi_{b_{0}}(1P)$ | $9.6\times 10^{4}$ | $5.2\times 10^{2}$ | $1.4\times 10^{2}$ | $2.7$ | $3.\times 10^{1}$ | $1.5\times 10^{2}$
$\chi_{b_{1}}(1P)$ | $3.9\times 10^{3}$ | $5.1\times 10^{2}$ | $1.3\times 10^{2}$ | $6.6$ | $5.3\times 10^{1}$ | $1.4\times 10^{2}$
$\chi_{b_{2}}(1P)$ | $8.3\times 10^{4}$ | $4.9\times 10^{2}$ | $1.2\times 10^{2}$ | $3.4$ | $3.4\times 10^{1}$ | $1.6\times 10^{2}$
$\eta_{b}(9999)$ | $4.6\times 10^{5}$ | $5.8\times 10^{3}$ | $1.6\times 10^{3}$ | $4.9\times 10^{1}$ | $4.4\times 10^{2}$ | $1.5\times 10^{3}$
Table 4: Estimation of event number for bottomonium states. Singe tag, no tag
and double tag set-ups are considered. Cuts on electron or positron scattering
angle are 1.9, 6 and 10 degrees.

From the Tables (3,4) one can see that the numbers of registered mesons are
drastically dependent on the minimal angle of the scattered lepton detection.
By using the luminosity calorimeter for detection of scattered leptons (the
minimal detection angle of 1.9 degrees) one can enhance the statistics of
registered mesons by two orders of magnitude. This emphasizes the significance
low-angle calorimeter for the two-photon physics.

As one can see, in no-tag mode, where no lepton are registered at final state,
we expect a large number of events, $10^{7}-10^{8}$ for quarkonium states and
$10^{5}$ for bottomonium. But the absence of kinematical constraints in the
no-tag mode will result in huge background contamination. Some special
techniques like strict transverse-momentum balance for final state could be
applied to reduce background. The exact estimation of no-tag quarkonium
sensitivities could be produced only with a detailed detector simulation.

For quarkonium two-photon physics on CEPC we suggest to use a conservative
estimation, namely the single-tag mode (one detected lepton helps to
reconstruct final state kinematics and drastically reduces the background
events) and efficiency of detection about $10\%$. It that case we will
reconstruct about $10^{2}-10^{4}$ events for each charmonium state. For
bottomonium states we could definitely measure low-state $\eta_{b}$ with
number of registered events about $4\times 10^{2}$ and possibly see some
events or make upper bound for higher bottomonium states.

##  4 Higgs boson production

The vertex $H\gamma\gamma$ is forbidden in the SM at tree level. The decay
process $H\rightarrow\gamma\gamma$ as well as the production process
$\gamma\gamma\rightarrow H$ proceed mostly through the top-quark and W loops.
Both processes are strongly suppressed due to the large mass of the top quark.
An observation of an excess in the $\gamma\gamma\rightarrow H$ process would
indicate a new physics phenomena, e.g. a contribution of the anomalous
$H\gamma\gamma$ vertex.

At $e^{+}e^{-}$ colliders the main background to the process
$\gamma\gamma\rightarrow H$ (referred as “signal” hereafter) are the Higgs
boson production via the fusion of virtual Z bosons ($ZZ\rightarrow H$) and
$\gamma\gamma$ collisions with final states identical to those of Higgs decays
but without a formation of the intermediate Higgs boson. The background from
$\gamma\gamma$ collisions can be strongly suppressed by selecting the “single-
tag” events where one of the beam particles is scattered to a significant
angle and is detected in the luminosity calorimeter. The signal reduction due
to this selection is relatively small (by only a factor of 3 to 5) since the
Higgs boson production is characterized by a large $q^{2}$ transfer. In the
following we assume an event selection with a beam particle scattered by at
least 30 milliradians, which is well within the acceptance of the luminosity
monitor [24].

###  4.1 Higgs photoproduction measurement plans

Today the Higgs photoproduction at future lepton colliders attracts
surprisingly low attention.

Nevertheless there was a vivid discussion in frames of Higgs boson production
in ultraperipheral collisions (UPC) at LHC (proton and heavy ion collision
cases was considered, see Silveira talk [27] and references therein ) where
estimations of Higgs photoproduction was made for different set-ups and
energies.

Recently there was published a paper [28] devoted to the estimation of
photoproduction of Higgs boson at the LHeC [29], proposed electron-proton
collider at LHC.

Also there exist a lot of estimations and proposals to measure double Higgs
photoproduction at a photon collider, particularly due to the possibility to
probe trilinear Higgs interactions [25],[26].

###  4.2 Theoretical estimation of event number

For estimation of Higgs production rate due to two-photon mechanism we could
utilize the equivalent photon approximation, elaborated in the papers [6],
[30], and [9]:

$\sigma_{ee\to eeH}\approx
2\biggl{(}\frac{\alpha}{\pi}\biggr{)}^{2}\biggl{(}\ln\frac{E}{m_{e}}\biggr{)}^{2}\int_{0}^{4E^{2}}\frac{\mbox{d}W^{2}}{W^{2}}f(\frac{W}{2E})\times\sigma_{\gamma\gamma\to
H}(W),$ (5)

here $\sigma_{\gamma\gamma\to H}(W)$ means Higgs photoproduction cross-section
estimation and function $f$ was defined in eq. (3). Here we consider only
leading logarithm approximation.

The subprocess cross section for the two photon Higgs production can be
calculated trough narrow resonance estimation [31]:

$\displaystyle\sigma_{\gamma\gamma\to H}(W)\approx
8\pi^{2}\frac{\Gamma_{\gamma\gamma}}{M_{H}}\frac{1}{\pi}\frac{M_{H}\Gamma}{(M_{H}^{2}-W^{2})^{2}+M_{H}^{2}\Gamma^{2}}\approx
8\pi^{2}\frac{\Gamma_{\gamma\gamma}}{M_{H}}\delta(W^{2}-M_{H}^{2}),$ (6)

here $\Gamma_{\gamma\gamma}$ and $\Gamma$ are the two photon and total width
of Higgs decay correspondingly, $M_{H}$ is the Higgs mass.

Given the energy of initial electron beam $E=120$ GeV and
$\Gamma_{\gamma\gamma}=2.27\times 10^{-3}\Gamma$, $\Gamma\approx 4.2$ MeV [23]
we could roughly estimate the Higgs two photon production at CEPC energies at
the level of $0.25$ fb.

It is very interesting that the Higgs production rate through two photon
mechanism at CEPC and LHC are comparable. In the latter case the estimation
gives the cross section about $0.1$ fb, see [32],[33],[34]

###  4.3 Background from the $ZZ$ fusion

The energy dependencies of the signal $\gamma\gamma\rightarrow H$ and the
background $ZZ\rightarrow H$ processes [35] are compared in Fig.4. One can see
that at high energy colliders the $ZZ\rightarrow H$ background rate is much
higher than that of the signal. However, at the $e^{+}e^{-}$ CMS energies near
240 GeV the background drops abruptly and becomes comparable to the signal.
The CEPC and FCC-ee colliders are therefore perfectly suited to study the
$\gamma\gamma\rightarrow H$ process.

Figure 4:  Left: energy dependence of $e^{+}e^{-}\rightarrow He^{+}e^{-}$
cross-section for the signal $\gamma\gamma\rightarrow H$ and background
$ZZ\rightarrow H$ contributions. Right: energy dependence of the signal to
background ratio. Vertical lines indicate 240 GeV CMS energy.

At 240 GeV the total signal cross-section is 0.26 fb [35], to be compared with
0.50 fb for $ZZ\rightarrow H$ background. The background can be significantly
reduced using the fact that the typical $q^{2}$ transfer in the $ZZ\rightarrow
H$ events is much larger than in the $\gamma\gamma\rightarrow H$ process. The
distribution of the scattering angle is presented in Fig.5. Nearly all
background is removed with the requirement that the beam particle is scattered
by less than 24∘.

Figure 5:  Distribution of the beam particle scattering angle for
$\gamma\gamma\rightarrow H$ signal and $ZZ\rightarrow H$ background. The
number of events is normalized to 5.6 ab-1 integrated luminosity.

Fig.6 shows the energy distribution for the scattered beam particles with the
scattering angle between 30 mrad and 24∘. An additional energy cut $E>15$ GeV
is applied to the scattered particle to ensure a reliable identification in
the luminosity calorimeter. After the cuts on the angle and the energy of the
scattered particle the cross-section is 0.049 fb for the signal and 0.027 fb
for the $ZZ\rightarrow H$ background. Assuming 5.6 ab-1 integrated luminosity,
this corresponds to 273 and 154 events, respectively.

Figure 6:  Energy distribution of the beam particle with scattering angle
between 30 mrad and 24∘ for $\gamma\gamma\rightarrow H$ signal and
$ZZ\rightarrow H$ background.

###  4.4 Background from non-resonant $\gamma\gamma$ collisions

The most abundant (58%) Higgs decay channel is $H\rightarrow bb$. In this
channel the most significant background is the non-resonant production of b
quark pairs in the $\gamma\gamma$ collisions, $\gamma\gamma\rightarrow bb$. In
addition, there is a reducible background $\gamma\gamma\rightarrow cc$, where
both jets from c quarks are tagged as b jets. The background from light quarks
can be neglected since it is efficiently suppressed by the b-tagging.

With the “single-tag” selection described above the $\gamma\gamma\rightarrow
bb$ cross-section is 94 fb for the full range of the bb invariant masses. It
is reduced to 0.77 fb for $M_{bb}>$ 100 GeV. According to the CEPC CDR [24],
the invariant mass resolution in $H\rightarrow bb$ decays is about 5 GeV.
Within the $\pm 1\sigma$ window around the Higgs mass ($120<M_{bb}<130$ GeV)
the $\gamma\gamma\rightarrow bb$ background is 0.124 fb, more than 5 times
higher than the signal (taking into account the branching fraction of the
$H\rightarrow bb$ decay).

The total cross-section of the single-tag $\gamma\gamma\rightarrow cc$
production is 2086 fb. It is reduced to 13.6 fb for $M_{cc}>$ 100 GeV and to
1.8 fb for $120<M_{cc}<130$ GeV. Although the cc background is almost 2 orders
of magnitude higher than the signal, it can be efficiently reduced by the
b-tagging. According the the CEPC CDR, the c jet rejection factor is 10 for
the 80% b jet efficiency. Applying the b-tagging to both jets, one gets 64%
efficiency for the signal tagging and a factor of 100 reduction of the charm
background, making the latter much smaller than the $\gamma\gamma\rightarrow
bb$ background.

The non-resonant background can be additionally reduced using a cut on the
direction of the produced b and c jets. Fig. 7 shows the distribution of the
quark polar angle. The background jets are concentrated near the beam axis. We
apply a polar angle cut $\Theta>20^{\circ}$ for both quarks. Within this
acceptance we assume (for the signal and the background) a 75% efficiency to
reconstruct both jets from b or c quarks.

Figure 7:  Distribution of the polar angle of the quark closest to the beam
line. Shaded area is the $\gamma\gamma\rightarrow H$ signal, open histograms
represent the non-resonant background. The events are shown within the
invariant mass window $118<M_{bb}<132$ GeV.

The $M_{bb}$ invariant mass distribution is presented in Fig.8. The Higgs
signal is smeared assuming a 5 GeV mass resolution. A fit to the signal and
background yields results in about $4.1\sigma$ signal significance for the
$H\rightarrow bb$ channel, after the subtraction of the $ZZ\rightarrow H$
background. The signal significance can be further improved by including other
Higgs decay modes. We conclude that the $\gamma\gamma\rightarrow H$ signal can
be reliably observed with the planned CEPC luminosity.

Figure 8:  Distribution of the invariant masses of the Higgs candidates.
Shaded area represents $\gamma\gamma\rightarrow bb/cc$ backgrounds, open
histogram is the signal, with a small contribution from the $ZZ\rightarrow H$
events.

##  5 Tau pair production

At present, the anomalous magnetic moments of electron and muon are measured
with an enormous precision, better than one per billion and one per million,
respectively. These measurements provide an extremely important test of the
Standard Model. At the same time, the anomalous magnetic moment of the tau
lepton, $a_{\tau}$, is known with rather poor accuracy.

The measurement of $a_{\tau}$ is interesting in two respects. First, because
of the large tau mass, $a_{\tau}$ is sensitive to the contributions of the new
physics at higher scales. Second, many theoretical models predict that the new
physics effect manifest themselves only in the properties of the third-
generation fermions. Thus, the overwhelming success of the Standard Model
observed in the sector of anomalous magnetic moment might be just a
consequence of performing the high precision measurements with the leptons of
only first generations.

The most precise determination of $a_{\tau}$ (17 permille) was performed by
the DELPHI experiment at LEP2. The total luminosity was approximately 0.5 fb-1
taken at the c.m.s. energies between 182 and 208 GeV. The tau anomalous
magnetic moment was extracted from the absolute cross-section of the tau pair
production in gamma-gamma collisions. The simplest final state was selected
with one tau decaying to an electron, another to a muon. The DELPHI precision
of the cross-section measurement was about 4%.

At CEPC the integrated luminosity will be increased by 4 orders of magnitude
with respect to LEP2. Thus, an improvement of $a_{\tau}$ precision by a large
factor can be expected.

At the collision energy of 240 GeV the QED cross-section of the
$e^{+}e^{-}\rightarrow e^{+}e^{-}\tau^{+}\tau^{-}$ process is 570 pb [36].
This corresponds to nearly 3 billions events with 5 ab-1 of integrated
luminosity, or 165 millions events with the $e-\mu$ final state.

At LEP2 the selection efficiency for the $e-\mu$ final state was 15-20%. In
the CEPC environment a tighter selection might be necessary to cope with the
high background. We consider the following severe cuts: at least one of the
two tracks must have the transverse momentum $p_{T}>5$ GeV/c, the other track
must have $p_{T}>3$ GeV/c. In addition, we require that directions of the both
tracks are more that 20 degrees from the beam axis, and that the total energy
of the two particles is less than 30 GeV to remove the annihilation events.

Fig.9 shows the distribution of the transverse momentum of the leading and
subleading tracks in the $\gamma\gamma\rightarrow\tau\tau\rightarrow e\mu$
events. One can see that only a very small fraction of the events satisfy the
selection cuts. Fig.10 shows the distribution of the invariant mass of the
electron and the muon after applying the selection cuts.

Figure 9:  Distribution of the transverse momenta of the leading (left) and
subleading (right) tracks in the $\gamma\gamma\rightarrow\tau\tau\rightarrow
e\mu$ events.  Figure 10:  Distribution of the visible invariant masses of the
selected $\gamma\gamma\rightarrow\tau\tau\rightarrow e\mu$ events.

The above selection has the generator-level efficiency of 0.42%, corresponding
to the total statistics of about 700 thousands events. Thus, the statistical
error is expected to be at the permille level and the measurement is likely to
be systematically limited.

Given the very clean and simple final state, one can optimistically expect to
keep the total systematic error at 0.5% level. This will include the absolute
luminosity determination (0.1%), tracking and Particle Identification
efficiency (0.15% per track for the tracking and a similar number for the
PID), trigger efficiency, residual background.

With the above systematic error, the DELPHI precision can be improved by a
factor of 8 for the cross-section measurement and the sensitivity to the
anomalous magnetic moment would be improved by a similar factor. Adding other
final states (e.g. $e-\rho$ and $\mu-\rho$) is not expected to provide a large
improvement of the total error, since the measurement will be systematically
dominated. However, those channels will have partially independent systematic
uncertainties, providing important cross-checks and certain reduction of the
overall systematic errors.

##  6 Photon hadronic structure

The measurement of the photon structure function (PSF) has a long history.
Although the photon is considered as a point-like particle, due to the quantum
effects it can fluctuate to a quark pair or a rho meson. These two processes
are usually described respectively as point-like and hadron contribution to
the photon structure function. The latter could be estimated in the framework
of vector meson dominance model [30] while the former was calculated within
quark model in leading order QCD corrections [37] and later in series of
papers at NLO [38],[39] and NNLO accuracy [40].

The most recent measurements of PSF have been performed at LEP about 20 years
ago, and since then publications are scarce, the most novel review of state-of
the-art could be find at [41],[42],[43].

The photon hadronic structure can be tested via the measurement of inclusive
hadron production in gamma-gamma collisions
$\rm{e^{+}e^{-}}\rightarrow\rm{e^{+}e^{-}}\gamma\gamma^{*}\rightarrow\rm{e^{+}e^{-}+hadrons}$.
The high-virtuality photon $\gamma^{*}$ is radiated off an electron which
scatters at relatively large angle and can be detected in the experimental
setup (“tagged electron”). The second electron is usually scattered at a very
small angle and thus remain undetected (“untagged electron”). The photon
$\gamma$ radiated by the untagged electron can be considered as quasi-real.

The overall reaction can be described as a deep inelastic scattering (DIS)
$\rm{e}\gamma$ of the tagged electron on the real photon. In such a scattering
the hadronic nature of the target photon is effectively revealed.

Pioneering experiment of PSF $F_{2}$ measurement was done at DESY by PLUTO
collaboration in 1981 [44], within the range $Q^{2}\in(1,15)$ GeV2 at an
average beam energy of $15.5$ GeV. Since then many experiments made
contributions so the $Q^{2}$ range was enlarged up to 780GeV2 at PLUTO
[45],[46], ALEPH [47],[48] AMY [49],[50], DELPHI [51],[52], JADE [53], L3
[54],[55] , OPAL [56], TASSO [57], TOPAZ [58], TCP$/2\gamma$ [59].

The most recent experiments have been done at LEP by collaborations ALEPH, L3
and OPAL with similar characteristics: center of mass energy $\approx 200$
GeV, integrated luminosity $\approx$ 600 pb-1 while the minimum angle for
detection in the luminosity calorimeter was $0.024$, $0.03$ and $0.033$ rad
correspondingly. Comparing this data with similar CEPC parameters, namely
c.m.s. energy 240 GeV, integrated luminosity 5 ab-1 and minimal detection
angle of about 1.6 degree ($0.028$ rad) one can naively estimate the number of
events at CEPC to be about $10^{4}$ times bigger than that at LEP.

The hadronic photon structure function $F^{\gamma}_{2}(x,Q^{2})$ can be
extracted from the differential cross-section $d\sigma/dQ^{2}dxdy$, where
$Q^{2}$ is the virtuality of $\gamma^{*}$ and $x,y$ are the Bjorken scaling
variables.

For the case of single-tagged events, when one of the photon is almost real
(photon invariant mass $P\approx 0$), the general process cross-section
$e^{+}e^{-}\to e^{+}e^{-}\gamma\gamma\to e^{+}e^{-}X$

$\displaystyle\frac{d\sigma(ee\to
eeX)}{dxdzdQ^{2}dP^{2}}=f_{\gamma/e}\frac{d\sigma(e\gamma\to eX)}{dxdQ^{2}},$
(7)

are factorized into almost real photon luminosity function $f_{\gamma/e}$

$\displaystyle
f_{\gamma/e}=\frac{\alpha}{2\pi}(\frac{1+(1-z^{2})}{z}\frac{1}{P^{2}}-\frac{2m_{e}^{2}z}{P^{4}}),~{}~{}~{}z={E_{\gamma}}/{E_{beam}}$
(8)

and deep inelastic proton-electron scattering cross-section [30]

$\displaystyle\frac{d\sigma(e\gamma\to
eX)}{dxdQ^{2}}=\frac{2\pi\alpha^{2}}{xQ^{4}}((1+(1-y)^{2})2xF_{2}^{\gamma}-y^{2}F_{L}^{\gamma}).$
(9)

Neglecting the virtuality of the quasi-real photon, one has:
$Q^{2}=2E_{beam}E_{tag}(1-{\rm cos}\theta_{tag})$, $x=Q^{2}/(Q^{2}+W^{2})$,
$y=1-(1+{\rm cos}\theta_{tag})E_{tag}/(2E_{beam})$, where $W$ is the
$\gamma\gamma^{*}$ invariant mass which is experimentally measured as the mass
of the hadronic system, $\theta_{tag}$ and $E_{tag}$ are scattering angle and
energy of the tagged lepton.

Usually the experimentally accessible kinematic region corresponds to small
values of $y$ ($y^{2}\ll 1$), so the contribution of the term proportional to
the longitudinal structure function $F_{L}^{\gamma}$ is negligible, and one
can determine PSF $F_{2}^{\gamma}$ directly from the cross-section (9).

The measurement of the energy $E_{tag}$ and the polar angle $\theta_{tag}$ of
the scattered electron is straightforward. The most difficult part of the
analysis is the reconstruction of the hadronic invariant mass $W$. In addition
to the finite detector resolution and efficiency, the $W$ reconstruction is
also affected by the acceptance issue, since the hadronic system is typically
boosted along the beam axis and part of it remains undetected or poorly
reconstructed. A sophisticated unfolding procedure is required to convert the
measured visible invariant mass into the true one. An additional difficulty
will be the background from overlapping interaction. This background has
little impact on the hard processes, but must be carefully taken into account
in the studies of gamma-gamma collisions.

The scattered electron can be reconstructed either in the luminosity monitor
(probing the small $Q^{2}$ values with high statistics), or in the forward
electromagnetic calorimeter. In the latter case the domain of high $Q^{2}$
values can be accessed. The available range of very high $Q^{2}$ is limited by
the low statistics due to the steeply falling spectrum of scattering angles.
Given the unprecedented luminosity, the future $e^{+}e^{-}$ colliders will be
able to study the photon structure function in the high $Q^{2}$ domain which
was never accessed by other experiments.

The huge statistics expected in the 240 GeV running of the future colliders
allows one to explore the kinematical regions that could not be accessed by
the past experiments due to the limited statistics. Among them, the following
can be mentioned:

  * •

Measurement of the photon structure function at the very high virtualities
$Q^{2}$, corresponding to the very large electron scattering angles.

  * •

Double-tagged events, where both beam particles are detected. In this
situation the whole event is fully reconstructed, which will dramatically
reduce the systematic uncertainty at the expense of the low available
statistics.

This case of deeply virtual target photons $Q^{2}\gg P^{2}\gg\Lambda_{QCD}$ is
very interesting because it is purely perturbative and allows one to compare
the experimental values with absolute QCD predictions.

##  7 Conclusions

In this paper we study several interesting topics of two-photon physics that
could be considered at future electron-positron colliders, CEPC and FCC-ee. As
a reference we take the planned CEPC set-up with c.m.s. energy $\sqrt{s}=240$
GeV, integrated luminosity 5.6 $ab^{-1}$ and assume that a forward
electromagnetic calorimeter will cover the polar angles down to $1.9$ degrees.
Our qualitative results can be applied to FCC-ee project, taking into account
its similar expected performance.

We have considered a rather limited set of topics of two-photon physics,
namely quarkonium spectroscopy, Higgs and tau pair production, photon hadronic
structure. Our estimations are based on a conservative approach and without
any doubt the quantitative characteristics can be surpassed in the real
experiments. New physics problems like searching for MSSM heavy Higgs,
anomalous top quark interactions, new physics state searching will be
considered elsewhere.

In Sect. 3 we consider quarkonium two-photon physics in three possible
detection modes: no-tag, single- and double tag. We show that using low-angle
calorimeter (1.9 degree minimal detection angle) could drastically improve
statistics of tagged events by two orders of magnitude. In no-tag mode we
expect about $10^{7}$-$10^{8}$ quarkonium and $10^{5}$ bottomonium events.
Taking into account the efficiency of detection about $10\%$ and considering
single-tag mode for better reconstruction of final state and reducing
background events, we estimate about $10^{2}-10^{4}$ registered events for
charmonium states. For bottomonium we could expect about $4\times 10^{2}$ of
low-state $\eta_{b}$ and possibly see some events or make upper bound for
higher states.

We find (Sect. 4) that photoproduction of Higgs can be observed with total
cross-section about $0.25$ fb and luminosity of collider 5.6 $ab^{-1}$.
Possible background issues from non-resonant $\gamma\gamma$ collisions and
$ZZ\to H$ were discussed and it was shown that by choosing appropriate
selection cuts one could achieve more than $4\sigma$ signal significance for
the $H\to bb$ channel.

The tau pair production cross-section and anomalous magnetic moment
measurement can be improved up to a factor 8 compared with the precision of
DELPHI experiment at LEP (Sect. 5).

Statistical error of photon structure function measurement (Sect. 6) could be
improved by about two orders of magnitude compared to the LEP experiments.
Also we expect that due to the high luminosity of the future colliders it will
be possible to perform measurements of PSF at very high virtualities $Q^{2}$.

More detailed analysis and thorough simulation of events, background,
systematic and statistical errors of physical quantities to measure can be
estimated after exact characteristics of planned colliders and detectors will
be known.

##  8 Acknowledgement

Authors are grateful to prof. Li Haibo for fruitful discussions which inspired
this work.

The work of V.V.B. was supported in part by the Heisenberg-Landau Program.

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