Nonsplit Domination Cover Pebbling Number for Some Class of Middle Graphs

A. Lourdusamy1, I. Dhivviyanandam2, Lian Mathew3

1Department of Mathematics,

St. Xavier’s College (Autonomous), Palayamkottai

Email<EMAIL_ADDRESS>https://orcid.org/0000-0001-5961-358X.

2 Reg. No : 20211282091003, Department of Mathematics,

St. Xavier’s College (Autonomous), Palayamkottai

Affiliated to Manonmaniam Sundaranar University, Abisekapatti-627012, Tamil
Nadu, India,

Email<EMAIL_ADDRESS>https://orcid.org/0000-0002-3805-6638.

3Department of Mathematics

CHRIST(Deemed to be university), Pune- Lavasa Campus, India.

Email<EMAIL_ADDRESS>https://orcid.org/0000-0002-4926-7756.

###### Abstract

Let $G$ be a connected graph. A pebbling move is defined as taking two pebbles
from one vertex and placing one pebble to an adjacent vertex and throwing away
the other pebble. The non-split domination cover pebbling number,
$\psi_{ns}(G)$, of a graph $G$ is the minimum of pebbles that must be placed
on $V(G)$ such that after a sequence of pebbling moves, the set of vertices
with a pebble forms a non-split dominating set of $G$, regardless of the
initial configuration of pebbles. We discuss some basic results, NP-
completeness of non-split domination number, and determine $\psi_{ns}$ for
some families of Middle graphs.

Keywords: non-split dominating set, non-split domination cover pebbling
number, cover pebbling number, Middle graphs.

2020 AMS Subject Classification: : 05C12, 05C25, 05C38, 05C76.

##  1 Introduction

Lagarias and Saks were the first ones to introduce the concept of pebbling and
F.R.K. Chung[1] used the concept in pebbling to solve a number theoretic
conjecture. Then many others followed suit including Glenn Hulbert who
published a survey of pebbling variants[2]. The subject of graph pebbling has
seen massive growth after Hulbert’s survey. In the past 30 years so many new
variants in graph pebbling have been developed which can be applied to the
field of transportation, computer memory allocation, game theory, and the
installation of mobile towers.

Let us denote $G^{\prime}$s vertex and edge sets as $V(G)$and $E(G)$,
respectively. Consider a graph with a fixed number of pebbles at each vertex.
One pebble is thrown away and the other is placed on an adjacent vertex when
two pebbles are removed from a vertex. This process is known as a pebble move.
The pebbling number of a vertex $v$ in a graph $G$ is the smallest number
$\pi(G,v)$, allowing us to shift a pebble to $v$ using a sequence of pebbling
move, regardless of how these pebbles are located on G’s vertices. The
pebbling number, $\pi(G)$, of a graph $G$ is the maximum of $\pi(G,v)$ over
all the vertices $v$ of a graph. Considering the concept of cover pebbling [6]
and non-split domination [5] we develop a new concept, called the non-split
domination cover pebbling number of a graph, denoted by $\psi_{ns}(G)$. In
paper[6] “The cover pebbling number, $\lambda(G)$ is defined as the minimum
number of pebbles required such that given any initial configuration of at
least $\lambda(G)$ pebbles, it is possible to make a series of pebbling moves
to place at least one pebble on every vertex of $G$” and in [5] The domination
cover pebbling number is defined as “ the minimum number of pebbles required
so that any initial configuration of pebbles can be shifted by a sequence of
pebbling moves so that the set of vertices that contain pebbles form a
dominating set $S$ of $G$”. Kulli, V.R et al. introduced the non-split
domination number in[5]. A dominating set $D$ of a graph $G=(V,E)$ is a non-
split dominating set if the induced $<V-D>$ is connected. The non-split
domination number $\gamma_{ns}(G)$ of $G$ is the minimum cardinality of a non-
split dominating set. We developed the concept of non-split domination cover
pebbling deriving from the concept of cover pebbling and non-split domination
in graphs. Thus, we arrived the definition of the non-split domination cover
pebbling number,$\psi_{ns}(G)$, of a graph $G$ as the minimum of pebbles that
must be placed on $V(G)$ such that after a sequence of pebbling moves, the set
of vertices with a pebble forms a non-split dominating set of $G$, regardless
of the initial configuration of pebbles. We discuss the basic results and
determine $\psi_{ns}$ for path graphs, wheel graphs, cycle graphs, and fan
graph

##  2 Preliminaries

For graph-theoretic terminologies, the reader can refer to[3, 4].

######  Result 1.

[5]

  * 1

. The non-split domination number of a complete graph $K_{n}$ is
$\gamma_{ns}(K_{n})=1$.

  * 2

. The non-split domination number of a Wheel graph is $\gamma_{ns}(W_{n})=1$.

  * 3

. The non-split domination number of a path is $\gamma_{ns}(P_{n})=n-2$.

  * 4

. The non-split domination number of Cycle is $\gamma_{ns}(C_{n})=n-2$.

######  Result 2.

[7] The domination cover pebbling number of the wheel is $\psi(W_{n})=n-2$.

######  Result 3.

[6] The cover pebbling number of path $P_{n}$ is $\gamma(P_{n})=2^{n}-1$.

##  3 Results

######  Theorem 1.

For a simple connected graph $G$, $\psi(G)\leq\psi_{ns}(G)\leq\sigma(G)$.

######  Theorem 2.

For a complete graph $K_{n}$ on $n$ vertices, the nonsplit domination cover
pebbling number is, $\psi_{ns}(K_{n})=1$

######  Theorem 3.

The non-split domination cover pebbling number of a wheel graph $W_{n}$ is,
$\psi_{ns}(W_{n})=\psi(W_{n}).$

###  3.1 NP- Completeness of Nonsplit Domination Problem

The proof is by reduction from the known NP-complete problem ’domination’. The
domination problem asks ‘for a given graph $G$ and an integer $k$, does the
graph $G$ contains a domination set of cardinality at most $k$?’.

######  Theorem 4.

The nonsplit domination problem is NP-complete.

###### Proof.

Let $G$ be any graph. Construct $G^{*}$ as follows: Let the graph $G$ be on
the first level and let the Path $P_{2}$ be on the second level. Join each
vertex of graph $G$ to both the vertices of the path $P_{2}$ (see Figure 1).
It is clear that the construction can be done in polynomial time.
Now, let $G$ be a graph with domination number $k$. Then, $k+1$ vertices will
form the non-split domination set for the graph $G^{*}$ iff $G$ has a
dominating set with $k$ vertices.
Suppose that the graph $G$ has a dominating set with cardinality $k$. Choose
any one vertex from the path $P_{2}$. Then, it is straightforward to see that
these $k+1$ vertices form a dominating set for the graph $G^{*}$. Moreover, if
we remove the $k+1$ vertices and the edges incident to it, we get a connected
graph. Thus, the non-split domination number of $G^{*}$ is $k+1$ whenever $G$
has a domination set with cardinality $k$.
Conversely, assume that $k+1$ vertices form the non-split dominating set for
the graph $G^{*}$. Then, by the definition $k+1$ vertices form a dominating
set for $G^{*}$, and hence the domination set for graph $G$ has cardinality
$k$. Thus, $G$ has a dominating set with $k$ vertices whenever $G^{*}$ has a
non-split domination set with order $k+1$. ∎

Figure 1: Construction of $G$ from $G^{*}$

###  3.2 Nonsplit Domination Cover Pebbling Number for the Middle Graph of
Path

######  Theorem 5.

The non-split domination cover pebbling number of the middle graph of the path
is, $\psi_{ns}(M(P_{n}))=2^{n+1}-3$.

###### Proof.

Let $V(P_{n})=\\{x_{1},x_{2},x_{3},\cdots,x_{n}\\}$ be the vertices of path
$P_{n}$ and $y_{1},y_{2},\cdot,y_{n-1}$ be the added vertices corresponding to
the edges $e_{1},e_{2},\cdots,e_{n-1}$ of $P_{n}$ to obtain $M(P_{n})$.Then
the total number of vertices is $2n-1$ and the edges are $3n-4$. Consider the
non-split domination set $D=\\{y_{i}|1\leq i\leq n-1\\}$ with containing $n$
vertices. The vertices of $D$ dominate all the vertices in $M(P_{n})$ and the
induced subgraph $<V-D>$ is a connected graph. Placing $2^{n+1}-4$ pebbles on
any one of the end vertices we could cover the non-split dominating set’s
vertices $x_{n}$ to $x_{2}$, if we place all the pebbles at $x_{n}$. And also
we don’t get the non-split domination set. Hence, the non-split domination
cover pebbling number of $M(P_{n})$ is $\psi_{ns}(M(P_{n}))\geq 2^{n+1}-3$.

Considering the configuration of $C$ with $2^{n+1}-3$ on the vertices of
$M(P_{n})$, we prove the sufficient condition to cover the non-split
dominating set.

Case 1:  Let the source vertex be $x_{1}$.
If we place all the pebbles on $x_{1}$, to cover $x_{n}$ we need $2^{n}$
pebbles. Thus, to cover the next furthest non-split dominating set is
$x_{n-1}$ which requires $2^{n-1}$. Likewise, we get the series of Pebble
distribution $2^{n}+2^{n-1}+2^{n-2}+\cdots+2^{2}+2^{0}$ to cover all the
vertices of the non-split dominating set. Thus, we used
$\sum_{k=2}^{n}2^{k}+1=2^{n+1}-3$ pebbles.

Case 2:  Let the source vertex be $x_{l},\ 1<l\leq n-1$.
Let us place all the pebbles at $x_{l}$. If $l<\lfloor\frac{n}{2}\rfloor$,
then we need $2^{(n-l)+1}-3$ pebbles to cover $x_{l}$ to $x_{n}$ of the non-
split dominating set. Now to cover the remaining non-split dominating set we
need $2^{l+1}-4$. Thus, we used $2^{(n-l)+1}+2^{l+1}-7<2^{n+1}-3$ pebbles.

Case 3:  Let the source vertex be $y_{1}$.
Let us consider the source vertex either any one of the non-split dominating
vertices of $y_{1}$ or $y_{n}-1$. Consider all the pebbles placed on $y_{1}$.
Then we need 4 pebbles to cover the adjacent vertices $x_{1}$ and $x_{2}$.
Then to cover the remaining non-split dominating set of vertices we need
$2^{n}-4$ pebbles. Thus, we used $2^{n}<2^{n+1}-3$ pebbles. Hence,
$\psi_{ns}(M(P_{n}))=2^{n+1}-3$. ∎

###  3.3 Nonsplit Domination Cover Pebbling Number for The Middle Graph of
Cycle Graphs

######  Theorem 6.

The non-split domination cover pebbling number of the middle graph of the
cycle is,
$\psi_{ns}(M(C_{n}))=\begin{cases}2\sum_{k=0}^{\lceil\frac{n}{2}\rceil}2^{k}-8&\
\ n\ {is\ odd}\\\ \\\
\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor+1}2^{k}-8+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}2^{k}&\
\ n\ {is\ even}.\end{cases}$

###### Proof.

Let $V(C_{n})=\\{x_{1},x_{2},x_{3},\cdots,x_{n}\\}$ and
$y_{1},y_{2},\cdots,y_{n}$ be the inserted vertices corresponding to edges
$e_{1},e_{2},\cdots,e_{n}$ of $C_{n}.$ to construct the middle graph of Cycle
$M(C_{n})$. Then the total number of vertices is $2n$ and the edges are $3n$.
Let the non-split dominating set
$D=\\{{y_{i}}\cup{x_{1},x_{2},x_{3},\cdots,x_{j}}\\}$ where $1\leq i,j\leq n$
and $x_{j}\neq N[y_{i}]$. Thus, $D$ dominates all the vertices of $M(C_{n})$,
and $<V-D>$ is connected. The total number of vertices in $D$ is $n-1$.

Case 1: When $n$ is odd.
Without loss of generality, Let $D=\\{y_{n},x_{2},x_{3},\cdots,x_{n-1}\\}.$
Placing $2\sum_{k=0}^{\lceil\frac{n}{2}\rceil}2^{k}-7$ pebbles on the source
vertex $x_{1}$ we can not put one pebble each on all the vertices of $D$.
Hence, $\psi_{ns}(M(C_{n}))\geq 2\sum_{k=0}^{\lceil\frac{n}{2}\rceil}2^{k}-8,$
when n is odd.

Distribution of $2\sum_{k=0}^{\lceil\frac{n}{2}\rceil}2^{k}-8$ pebbles on the
configuration of $C$, we cover all the vertices of $D$. Now we prove the
sufficient condition for $M(C_{n})$, when $n$ is odd.

Case 1.1: Let the source vertex be $x_{1}$.
Using the Theorem 5 we can cover the non-split dominating set of $D$ from
$x_{2}$ to $x_{\lceil\frac{n}{2}\rceil}$. The total number of pebbles used are
$2^{\lceil\frac{n}{2}\rceil+1}-4$. Similarly, to cover the remaining non-split
dominating set {$y_{n},x_{n-1},\cdots,x_{\lceil\frac{n}{2}\rceil+1}$ we use
$2^{\lceil\frac{n}{2}\rceil+1}-6$ pebbles. Hence we have spent total
$2^{\lceil\frac{n}{2}\rceil+2}-10=2\sum_{k=0}^{\lceil\frac{n}{2}\rceil}2^{k}-8$
pebbles. Case 1.2: Let the source vertex be any one of the vertices of
$y_{i}$, $1\leq i\leq n$. Without loss of generality, let the source vertex be
$y_{n}$. Using the Theorem 5 we can cover the non-split dominating set of $D$
from $x_{2}$ to $x_{\lceil\frac{n}{2}\rceil}$. The total number of pebbles
used are $2^{\lceil\frac{n}{2}\rceil+1}-4$. Similarly, to cover the remaining
non-split dominating set {$y_{n},x_{n-1},\cdots,x_{\lceil\frac{n}{2}\rceil+1}$
we use $2^{\lceil\frac{n}{2}\rceil}-3$ pebbles. Hence we have spent total
$3(2^{\lceil\frac{n}{2}\rceil})-7<2\sum_{k=0}^{\lceil\frac{n}{2}\rceil}2^{k}-8$
pebbles. Hence, $\psi_{ns}(M(C_{n}))\leq
2\sum_{k=0}^{\lceil\frac{n}{2}\rceil}2^{k}-8$, When n is odd.

Case 2: When $n$ is even.
Without loss of generality, Let $D=\\{y_{n},x_{2},x_{3},\cdots,x_{n-1}\\}.$
Placing
$\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor+1}2^{k}-7+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}2^{k}$
pebbles on the source vertex $x_{1}$ we can not put one pebble each on all the
vertices of $D$. Hence,
$\psi_{ns}(M(C_{n}))\geq\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor+1}2^{k}-8+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}2^{k}$,
When n is even.

Distribution of
$\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor+1}2^{k}-8+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}2^{k}$
pebbles on the configuration of $C$, we cover all the vertices of $D$. Now we
prove the sufficient condition for $M(C_{n})$, when $n$ is even.

Case 2.1: Let the source vertex be $x_{1}$.
Using the Theorem 5 we can cover the non-split dominating set of $D$ from
$x_{2}$ to $x_{\lceil\frac{n}{2}\rceil+1}$. The total number of pebbles used
are $2^{\lceil\frac{n}{2}\rceil+2}-4$. Similarly, to cover the remaining non-
split dominating set {$y_{n},x_{n-1},\cdots,x_{\lceil\frac{n}{2}\rceil+2}$ we
use $2^{\lceil\frac{n}{2}\rceil+1}-6$ pebbles. Hence we have spent total
$3(2^{\lceil\frac{n}{2}\rceil+1})-10=\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor+1}2^{k}-8+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}2^{k}$
pebbles. Case 2.2: Let the source vertex be any one of the vertices of
$y_{i}$, $1\leq i\leq n$. Without loss of generality, let the source vertex be
$y_{n}$. Using the Theorem 5 we can cover the non-split dominating set of $D$
from $x_{2}$ to $x_{\lceil\frac{n}{2}\rceil+1}$. The total number of pebbles
used are $2^{\lceil\frac{n}{2}\rceil+2}-4$. Similarly, to cover the remaining
non-split dominating set {$y_{n},x_{n-1},\cdots,x_{\lceil\frac{n}{2}\rceil+2}$
we use $2^{\lceil\frac{n}{2}\rceil}-3$ pebbles. Hence we have spent total
$5(2^{\lceil\frac{n}{2}\rceil})-7<\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor+1}2^{k}-8+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}2^{k}$
pebbles. Hence,
$\psi_{ns}(M(C_{n}))\leq\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor+1}2^{k}-8+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}2^{k}$,
when n is even. Hence proved.

∎

###  3.4 Non-split Domination Cover Pebbling Number for The Middle Graph of
Wheel Graphs

######  Theorem 7.

The non-split domination cover pebbling number of the middle graph of the
wheel is,
$\psi_{ns}(M(W_{n}))=\begin{cases}(\lfloor\frac{n}{2}\rfloor)8+6&\ \ n\ {is\
odd}\\\ \\\ (\lfloor\frac{n}{2}\rfloor)8+10&\ \ n\ {is\ even}.\end{cases}$

###### Proof.

Let $V(W_{n})=\\{x_{0},x_{1},x_{2},x_{3},\cdots,x_{n-1}\\}$ be the vertices of
$W_{n}$ and $x_{0}$ be the center vertices of $W_{n}$. Let
$y_{1},y_{2},\cdots,y_{n-1}$ be the inserted vertices corresponding to edges
$v_{0}v_{i}$ where $1\leq i\leq n-1$ and $a_{1},a_{2},\cdots,a_{n-2}$ be the
inserted vertices on the edges $x_{j}x_{j+1}$ where $1\leq j\leq n-2$ and
$a_{n-1}$ lies in $x_{n-1}x_{1}$. Also the total number of edges in the
$M(W_{n})$ is $3(n-1)+1$.

Case 1: When $n$ is even. Here $d(y_{i})=n+3\ (1\leq i\leq n-1)$
Sub case 1.1: Suppose $i$ is odd.
Consider the set
$D=\\{y_{i},a_{i+1},a_{i+3},\cdots,a_{n-2}a_{i-2},a_{i-5}\cdots,a_{1}\\}$
where $a_{j}\neq N(y_{i})$ and $j=1,3,\cdots,i-2,i+1,i+3,\cdots,n-1$ be a non-
split dominating set and $<V_{D}>$ is connected which is the minimum Non-split
dominating set of $M(W_{n})$. If we place $(\lfloor\frac{n}{2}\rfloor)8+9$
pebbles on $x_{1}$, we can not cover all the vertices of $D$. Hence we require
$(\lfloor\frac{n}{2}\rfloor)8+10$ pebbles to cover $D$. Hence
$\psi_{ns}(M(W_{n}))\geq(\lfloor\frac{n}{2}\rfloor)8+10$, When $n$ is even.

Distribution of $(\lfloor\frac{n}{2}\rfloor)8+10$ pebbles on the configuration
of $C$, we cover all the vertices of $D$. Now we prove the sufficient
condition for $M(W_{n})$, when $n$ is even.

Subcase 1.2: Let the source vertex be $a_{k}$, where $k$ is not in $D$.
Without loss of generality, let the source vertex be $a_{1}$. Since there will
be $\lfloor\frac{n}{2}\rfloor-3$ vertices having the distance 3 and 2 vertices
having the distance 2 from the source vertex then we need
$(\lfloor\frac{n}{2}\rfloor)8+6<(\lfloor\frac{n}{2}\rfloor)8+10$ Pebbles.
Subcase 1.3: Let the source vertex be $x_{0}$.
Let us place all the pebbles on $x_{0}$. Since all the dominating vertices are
at the distance of 2 from the center except $y_{i}$, we need
$2+\lfloor\frac{n}{2}\rfloor)4$ pebbles to cover the non-split dominating set.
Thus, $\psi_{ns}(M(W_{n}))\geq(\lfloor\frac{n}{2}\rfloor)8+10$, When $n$ is
even and $i$ is odd.

Case 1.2: When $i$ is even.
Consider the dominating set
$D=\\{y_{i},a_{i+1},a_{i+3},\cdots,a_{n-1},a_{i-2},a_{i-4},\cdots,a_{2}\\}$
where $a_{j}\neq N[y_{i}]$ and $j=2,4,\cdots,i-2,i+1,i+3,\cdots,n-1$ be the
minimum dominating set and $<V_{D}>$ is connected. Now to prove the Non-split
domination pebbling number of $M(W_{n})$, when n is even and $i$ is even, we
can follow the same method of Case 1, Subcase 1.1, 1.2 and 1.3..

Case 2: When $n$ is odd.
Here $d(y_{i})=n+3\ (1\leq i\leq n-1)$
Subcase 2.1: Suppose $i$ is odd.
Consider the set
$D=\\{y_{i},a_{i+1},a_{i+3},\cdots,a_{n-1}a_{i-2},a_{i-4}\cdots,a_{1}\\}$
where $a_{j}\neq N(y_{i})$ and $j=1,3,\cdots,i-2,i+1,i+3,\cdots,n-1$ be a non-
split dominating set and $<V_{D}>$ is connected which is the minimum Non-split
dominating set of $M(W_{n})$. If we place $(\lfloor\frac{n}{2}\rfloor)8+5$
pebbles on $x_{1}$, we can not cover all the vertices of $D$. Hence we require
$(\lfloor\frac{n}{2}\rfloor)8+6$ pebbles to cover $D$. Hence
$\psi_{ns}(M(W_{n}))\geq(\lfloor\frac{n}{2}\rfloor)8+6$, When $n$ is odd.

Distribution of $(\lfloor\frac{n}{2}\rfloor)8+6$ pebbles on the configuration
of $C$, we cover all the vertices of $D$. Now we prove the sufficient
condition for $M(W_{n})$, when $n$ is odd.

Subcase 1.2: Let the source vertex be $a_{k}$, where $k$ is not in $D$.
Without loss of generality, let the source vertex be $x_{1}$. Since there will
be $\lfloor\frac{n}{2}\rfloor-2$ vertices having the distance 3, one vertex at
the distance of two and one vertex is adjacent to the source vertex, then we
require $(\lfloor\frac{n}{2}\rfloor)8+6$ pebbles.

Subcase 1.3: Let the source vertex be $x_{0}$.
Let us place all the pebbles on $x_{0}$. Since all the dominating vertices are
at the distance of 2 from the center except $y_{i}$, we need
$2+\lfloor\frac{n}{2}\rfloor)4$ pebbles to cover the non-split dominating set.
Subcase 1.4: Let the source vertex be any one of the vertices of $y_{i}$ where
$i$ is odd.
Let us place all the pebbles on $y_{1}$. Since all the dominating vertices are
at the distance of 2 from $y_{1}$ then we need $1+\lfloor\frac{n}{2}\rfloor)4$
pebbles to cover the non-split dominating set. Thus,
$\psi_{ns}(M(W_{n}))=(\lfloor\frac{n}{2}\rfloor)8+6$, When $n$ is even and $i$
is odd.

Case 1.2: When $i$ is even.
Consider the dominating set
$D=\\{y_{i},a_{i+1},a_{i+3},\cdots,a_{n-1},a_{i-2},a_{i-4},\cdots,a_{2}\\}$
where $a_{j}\neq N[y_{i}]$ and $j=2,4,\cdots,i-2,i+1,i+3,\cdots,n-1$ be the
minimum dominating set and $<V_{D}>$ is connected. Now to prove the Non-split
domination pebbling number of $M(W_{n})$, when n is odd and $i$ is even, we
can follow the same method of Case 2, Subcase 2.1, 2.2, 2.3 and 1.4.. Thus
proved. Hence, the non-split domination cover pebbling number of
$M(W_{n})=(\lfloor\frac{n}{2}\rfloor)8+6$.

###  3.5 Non-split Domination Cover Pebbling Number for The Middle Graph of
Fan Graphs

######  Theorem 8.

The non-split domination cover pebbling number of the middle graph of the fan
is,
$\psi_{ns}(M(F_{n}))=\begin{cases}(\lceil\frac{n}{2}\rceil-1)8+6&\ \ n\ {is\
odd}\\\ \\\ (\lfloor\frac{n}{2}\rfloor-2)8+6&\ \ n\ {is\ even}.\end{cases}$

###### Proof.

It follows from Theorem 7. ∎

∎

## References

  * [1] F.R.K. Chung, Pebbling in hypercubes, SIAMJ. Disc. Math., 2(4) (1989), pp. 467-472.
  * [2] G. Hurlbert, A survey of graph pebbling, Congressus Numerantium 139 (1999), pp. 41-64.
  * [3] J.A. Bondy and U.S.R. Murty, Graph theory with applications, (1977).
  * [4] Gary Chartrand, Introductory Graph theory, Dover books on Mathematics, (1985).
  * [5] Kulli, V. R., and B. Janakiram. ”The non-split domination number of a graph.” Indian Journal of Pure and Applied Mathematics 31.4 (2000): 441-448.
  * [6] Crull, B., Cundiff, T., Feltman, P., Hurlbert, G. H., Pudwell, L., Szaniszlo, Z., and Tuza, Z. (2005). The cover pebbling number of graphs. Discrete Mathematics, 296(1), 15-23. https://doi.org/10.1016/j.disc.2005.03.009
  * [7] Cockayne, E. J. ”Domination of undirected graphs—a survey.” Theory and Applications of Graphs. Springer, Berlin, Heidelberg, 1978. 141-147.


On the Discrepancy between the FOPT and CIPT Approaches for Hadronic Tau
Spectral Function Moments

André H. Hoang1,2$\star$ and Christoph Regner1

1 University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Wien,
Austria

2 Erwin Schrödinger International Institute for Mathematics and Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, Austria

*<EMAIL_ADDRESS>
15th International Symposium on Radiative Corrections:
Applications of Quantum Field Theory to Phenomenology,
FSU, Tallahasse, FL, USA, 17-21 May 2021 10.21468/SciPostPhysProc.?

## Abstract

The discrepancy between the FOPT and CIPT approaches for hadronic $\tau$
spectral function moments constitutes the major theoretical uncertainty for
strong coupling determinations from tau decay data. We show the discrepancy
can be analytically understood since the Borel representations – which have
been assumed to be identical for both approaches previously – differ in the
presence of IR renormalons. This implies that the OPE condensate corrections
are different for both approaches and that the discrepancy may eventually be
reconciled. In the talk we explain the difference and some mathematical
aspects of of the FOPT and CIPT Borel representations and show numerical
results.

###### Contents

  1. 1 Introduction
  2. 2 Essence of this talk
  3. 3 Brief Primer on Renormalon Calculus
  4. 4 The FOPT and CIPT Borel Representations
  5. 5 Essential Comments and the Asymptotic Separation
    1. 5.1 Form of the Contour Integration and CIPT OPE Corrections
    2. 5.2 FOPT and CIPT OPE Corrections are indeed different
    3. 5.3 Asymptotic Separation
    4. 5.4 Brief numerical Analysis
  6. 6 Conclusions

##  1 Introduction

Moments of the $\tau$ hadronic spectral functions obtained from LEP [1, 2]
provide an important tool for precise determinations of the strong coupling
$\alpha_{s}$. Predictions for the spectral function moments in the massless
quark limit are based on the vacuum polarization function $\Pi(p^{2})$, which
is known perturbatively to 5 loops (i.e. ${\cal O}(\alpha_{s}^{4})$) in full
QCD [3, 4, 5, 6]. Considering only for first generation quarks and QCD
corrections, the theoretical moments can be written as [7, 8, 9, 10]

$A_{W}(s_{0})\,=\,\textstyle\frac{N_{c}}{2}\,S_{\rm
ew}\,|V_{ud}|^{2}\Big{[}\,\delta^{\rm
tree}_{W}+\delta^{(0)}_{W}(s_{0})+\sum_{d\geq
4}\delta^{(d)}_{W}(s_{0})\Big{]}\,,$ (1)

where $N_{c}=3$, $S_{\rm ew}$ stands for electroweak corrections (which we do
not consider further), $V_{ud}$ is a CKM matrix element and $s_{0}$ is the
upper bound of the spectral function integration. The term $\delta^{\rm
tree}_{W}$ is the tree-level contribution and $\delta^{(0)}_{W}(s_{0})$ stands
for the higher order perturbative QCD corrections. The terms
$\delta^{(d)}_{W}(s_{0})$ represent condensate corrections in the framework of
the operator product expansion (OPE) [11]. They involve vacuum matrix elements
of low-energy QCD operators of increasing dimension resulting from an
expansion in inverse powers of $s_{0}$. The leading dimension $d=4$ term is
related to the well-known gluon condensate
$\langle\alpha_{s}G^{\mu\nu}G_{\mu\nu}\rangle$. There are also so-called
duality-violation corrections which can be important phenomenologically, but
which are not relevant for the subsequent discussion and therefore suppressed
in Eq.(1). Using the 5-loop results [3, 4, 5, 6] an impressive precision of
about $5\%$ has been achieved for $\alpha_{s}(m_{\tau}^{2})$ (corresponding to
an uncertainty of $1.5\%$ for $\alpha_{s}(m_{Z}^{2})$), where the uncertainty
is dominated by the perturbative error in $\delta^{(0)}_{W}(s_{0})$ [9, 10,
12, 13].

The QCD corrections $\delta^{(0)}_{W}(s_{0})$ are obtained from the expression
($x\equiv s/s_{0}$)

$\textstyle\delta^{(0)}_{W}(s_{0})\,=\,\frac{1}{2\pi
i}\,\,\ointctrclockwise_{{\cal C}_{s}}\\!\\!\frac{{\rm
d}s}{s}\,W({\textstyle\frac{s}{s_{0}}})\,\hat{D}(s)\,=\,\frac{1}{2\pi
i}\,\,\ointctrclockwise_{{\cal C}_{x}}\\!\\!\frac{{\rm
d}x}{x}\,W(x)\,\hat{D}(xs_{0})\,.$ (2)

where $\hat{D}(s)$ is the partonic Adler function,
$\frac{1}{4\pi^{2}}(1+\hat{D}(s))\,\equiv\,-\,s\,\frac{{\rm
d}\hat{\Pi}(s)}{{\rm d}s}$ and the weight function $W(x)$ is a polynomial in
$x$ which (together with the choice of $s_{0}$) specifies the type of moment
considered. The contour path ${\cal C}_{s}$ (${\cal C}_{x}$) starts/ends at
$s=s_{0}\pm i0$ ($x=1\pm i0$) and traverses the complex $s$-plane, crossing
the Euclidean axis half way through, with sufficient distance from the origin
such that the strong coupling stays in the perturbative regime. Through
analyticity this path is related to an associated integration along the real
positive $s$-axis over the experimental spectral function data [10].
Frequently a circular path with $|s|=s_{0}$ ($|x|=1$) is considered, but it
may be deformed arbitrarily as long as it stays in the region where the strong
coupling remains perturbative. For
$W_{\tau}(x)=(1-x)^{3}(1+x)=1-2x+2x^{3}-x^{4}$ and $s_{0}=m_{\tau}^{2}$ the
moment $A_{W_{\tau}}(m_{\tau}^{2})$ agrees with the normalized total hadronic
$\tau$ decay rate
$R_{\tau}=\Gamma(\tau^{-}\to\mbox{hadrons}\,\nu_{\tau}(\gamma))/\Gamma(\tau^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\tau}(\gamma))$.

The two widely employed methods to calculate $\delta^{(0)}_{W}(s_{0})$ are
Fixed Order Perturbation Theory (FOPT) and Contour Improved Perturbation
Theory (CIPT). The CIPT approach is based on the perturbative series for the
Adler function of the form111Here we use conventions, where 1-loop
$\beta$-function coefficient has the form $\beta_{0}=11-2n_{f}/3$ and we
furthermore define
$a(-x)\equiv\frac{\beta_{0}\,\alpha_{s}(-s)}{4\pi}=\frac{\beta_{0}\,\alpha_{s}(-xs_{0})}{4\pi}$
and $a_{0}\equiv\frac{\beta_{0}\,\alpha_{s}(s_{0})}{4\pi}=a(1)$. We also take
$n_{f}=3$.

$\displaystyle\hat{D}(s)$ $\displaystyle\,=\,$
$\displaystyle\textstyle\sum_{n=1}^{\infty}\,\bar{c}_{n}\,a^{n}(-x)\,,$ (3)

with real-valued coefficients $\bar{c}_{n}$ (which agree with those of the
real-valued Euclidean Adler function for $s=-s_{0}$) and complex-valued powers
of the strong coupling. One carries out the contour integration over powers of
the complex-valued strong coupling $\alpha_{s}(-s)$. The CIPT series arises
from truncating the sum in Eq. (3). The FOPT approach consists of expanding
the series (3) in powers of $\alpha_{s}(s_{0})$, so that the complex phases
appear exclusively in powers of $\ln(-s/s_{0})$ within the integrands of the
coefficients. For FOPT, the series arises from truncating the sum in powers of
$\alpha_{s}(s_{0})$, so that the powers of the strong coupling can be factored
out of the contour integration for each series term. The CIPT approach differs
from FOPT in that it resums the powers of $\ln(-s/s_{0})$ to all orders along
the integration path [8, 14].

It is an important fact that, after the contour integration (2) is carried out
for the moment series, it is not possible anymore to switch between the FOPT
and CIPT expansions by a scheme change of $\alpha_{s}$. So the difference of
the truncated FOPT and CIPT series for the spectral function moments, is of a
quite different character as the renormalization scale variations usually
carried out for perturbative QCD predictions. A major limitation of
$\alpha_{s}$ determinations from the moments $A_{W}(s_{0})$ is that FOPT and
CIPT calculations of $\delta^{(0)}_{W}(s_{0})$ for moments with good
perturbative convergence yield to systematic numerical differences that do not
seem to be covered by the conventional perturbative uncertainty estimates
related to renormalization scale variations. Since CIPT in general leads to
smaller values for $\delta^{(0)}_{W}(s_{0})$ than FOPT, extractions of
$\alpha_{s}(m_{\tau}^{2})$ based on CIPT generally arrive at larger values
than those based on FOPT.

##  2 Essence of this talk

In this talk we report on the results given in Refs. [15, 16], which
demonstrated that the different character of the FOPT and CIPT spectral
function moment series together with the fact the coefficients $\bar{c}_{n}$
in Eq. (3) contain asymptotic (i.e. non-convergent) contributions due to
infrared (IR) renormalons [17, 18] leads to a systematic disparity in the
high-order behavior of the two types of moment series. It is the above
mentioned property of the FOPT and CIPT methods – that one cannot switch
between them through a change of renormalization scheme – that is a crucial
ingredient in the discussions that follow. The disparity – which we call the
asymptotic separation – can be sufficiently sizeable and manifest itself
already at very low orders to explain the observed discrepancy between 5-loop
FOPT and CIPT moments mentioned above. However, the disparity provides a
resolution to the FOPT-CIPT discrepancy problem only if the asymptotic
character is already manifest in the known perturbative coefficients up to
5-loops, which means that the known 5-loop coefficient of $\hat{D}(s)$ is
already dominated in a sizeable way by the asymptotic behavior of infrared
(IR) renormalons. In practice, the IR renormalon dominance assumption implies
that the dominant gluon condensate IR renormalon governs the behavior of the
Adler function series at 5-loops [19] in a sizeable way and that – within some
uncertainties – one can make relatively rigid predictions for the Adler
function’s perturbative coefficients beyond 5 loops using the renormalon
calculus. Some evidence has been provided supporting the IR renormalon
dominance assumption for $\hat{D}(s)$ [20], but we stress that it cannot be
strictly proven. Thus, even though the disparity between the FOPT and CIPT
series exists as a matter of principle (because of the existence of IR
renormalons in perturbative QCD [17, 18]) our results provide an explanation
of the observed FOPT-CIPT discrepancy at the 5 loop level only in the context
of the IR renormalon dominance assumption. We stress that this talk is not
intended to provide arguments on the validity of the IR renormalon dominance
assumption, but to discuss the principal aspects of the asymptotic separation.

In the following we provide a brief primer to the renormalon calculus (Sec.
3), explain how the character of the FOPT and CIPT series leads to principal
differences in their Borel representation in the presence of IR renormalons
(Sec. 4), we address some mathematical subtleties of the CIPT Borel
representation, and we show numerical results (Sec. 5). We emphasize that the
study of the involved analytic expressions is a complicated matter,
particularly in full QCD, so that in this talk we can primarily state the
outcome without going into technical details. Many analytic results will for
simplicity be written down in the large-$\beta_{0}$ approximation (see Ref.
[16]). We refer to Ref. [15] for all details and the analytic results in full
QCD.

##  3 Brief Primer on Renormalon Calculus

The renormalon calculus provides a convenient way to quantify the large-order
behavior of the coefficients of asymptotic series, which for any perturbative
series in QCD is tied to the IR and UV properties of the $\beta$-function [21,
22, 23, 17, 18]. Furthermore, there is a one-to-one correspondence of the
asymptotic contributions in the series coefficients with IR origin to power
corrections in the context of the OPE. In the following we briefly outline the
basics of renormalon calculus to the extend needed for the understanding of
the following parts of this talk.

Starting from the perturbation series
$\hat{\sigma}=\sum_{n=1}^{\infty}d_{n}(\mu)(\alpha_{s}(\mu)/\pi)^{n}$ for a
quantity $\sigma$ in powers of (the real-valued) $\alpha_{s}(\mu)$, the so-
called Borel function (or Borel transform) of $\hat{\sigma}$, is defined by
$B[\hat{\sigma}](u)=\sum_{n=1}^{\infty}(4^{n}d_{n}(\mu))/(\beta_{0}^{n}\Gamma(n))u^{n-1}$.
In the Borel function the asymptotic $n$-factorial growth of the $d_{n}$
coefficients with $n$ is compensated by the inverse powers of $\Gamma(n)$ such
that the Taylor series for $B[\hat{\sigma}](u)$ in powers of $u$ is absolute
convergent in a circle around the origin of the complex Borel $u$-plane. The
resummed function $B[\hat{\sigma}](u)$ in this circle can be analytically
continued into the entire Borel plane (at least as far as information
accessible to perturbation theory is concerned). The original series (in
powers of $\alpha_{s}$) can be recovered from the $B[\hat{\sigma}](u)$ Taylor
series from the relation
$\hat{\sigma}=\int_{0}^{\infty}du\,B[\hat{\sigma}](u)\,e^{-4\pi
u/(\alpha_{s}(\mu)\beta_{0})}$. The so-called Borel sum is the result of the
same integral using the full function $B[\hat{\sigma}](u)$ in the entire
complex $u$ plane. We call the integral over the full $B[\hat{\sigma}](u)$
function also the Borel representation of $\sigma$. Asymptotic contributions
in the original series are related to non-analytic structures (cuts and poles)
in $B[\hat{\sigma}](u)$ in the complex $u$ plane, where the previously
mentioned radius of convergence is related to the non-analytic structure
located closest to the origin. The closer the non-analytic structure is to the
origin, the larger its impact (i.e. its dominance) in the original series. One
calls these non-analytic structures renormalons, and one furthermore
distinguishes between IR and UV renormalons. The character of these
renormalons is determined from the UV and IR properties of QCD which are
directly tied to the perturbative $\beta$-function and, as far as IR
renormalons are concerned, to the form of the OPE corrections. One can
consider the Borel sum as the “all-order resummed” result of the original
series for $\sigma$. However, if there are non-analytic structures along the
positive real $u$-axis, which usually happens for IR renormalons, the Borel
sum requires some path deformation prescription, such as the principle value
(PV) prescription, to be well-defined. Carrying out a convergent scheme change
for $\alpha_{s}$ (e.g. related to a reexpansion of the series for a different
renormalization scale or when using a different renormalization condition for
the strong coupling) leaves the Borel sum invariant. We also mention that the
Borel representation (and its value within any prescription) is strictly
invariant under a rescaling of the coupling constant,
$\alpha_{s}(\mu)\to\eta(\mu)\equiv\lambda\alpha_{s}(\mu)$ for any positive
real number $\lambda$. The latter invariance will be important in this talk.

The Borel function $B[\hat{D}](u)$ for the perturbative Euclidean Adler
function series with the form
$\hat{D}(-s_{0})=\sum_{n=1}^{\infty}\bar{c}_{n}a^{n}_{0}$ has been studied
intensely in the past [18, 24, 19, 25, 26, 27, 20]. The exact form of
$B[\hat{D}](u)$ is unknown, but each OPE term implies the existence to an
additive contribution in $B[\hat{D}](u)$ with a specific non-analytic
structure that is uniquely tied to the dimension of the non-perturbative
matrix element (condensate), its anomalous dimension, its Wilson coefficient
and the coefficients of the $\beta$-function. The leading OPE term is the
dimension-4 gluon condensate correction, which for the Euclidean Adler
function has the form

$\displaystyle\hat{D}^{\rm OPE}_{d=4}(-s_{0})$ $\displaystyle\,=\,$
$\displaystyle\textstyle
C_{G^{2}}(\alpha_{s}(s_{0}))\frac{\langle\alpha_{s}G^{\mu\nu}G_{\mu\nu}\rangle}{s_{0}^{2}}\,.$
(4)

The gluon condensate correction implies the existence in $B[\hat{D}](u)$ of a
certain linear combination of non-analytic terms $\frac{1}{(2-u)^{\gamma}}$
for different rational values of $\gamma$. In the large-$\beta_{0}$
approximation, where $C_{G^{2}}=1$, this linear combination collapses to the
single term $\frac{1}{2-u}$. In general, OPE condensate corrections with
dimension $d$ are associated to certain linear combinations of non-analytic
terms $\frac{1}{(d/2-u)^{\gamma}}$, each of which then implies contributions
in the coefficients $d_{n}$ of the form
$(\frac{\beta_{0}}{2d})^{n}(\frac{2}{d})^{\gamma-1}\frac{\Gamma(\gamma+n-1)}{\Gamma(\gamma)}$.
This reduces to $(\frac{\beta_{0}}{2d})^{n}\Gamma(n)$ for $\gamma=1$ and makes
the asymptotic character of the series expansion manifest. The smaller the
dimension $d$, the stronger is the increase with $n$. Since OPE corrections
are known to exist for all integer values of $d\geq 4$, $B[\hat{D}](u)$
contain non-analytic renormalon terms of the form $\frac{1}{(p-u)^{\gamma}}$
for $p=2,3,4\ldots$. The practical limitation of the association of an OPE
correction and a specific linear combination of non-analytic renormalon
contributions is that the normalization of this linear combination within
$B[\hat{D}](u)$ is a priori unknown (up to the fact that it is non-zero) and
can only be fixed with additional assumptions. This is the origin of the issue
concerning the renormalon dominance assumption mentioned in Sec. 2. Only for
the large-$\beta_{0}$ approximation, which can be calculated from massless
fermion self-energy insertions into the ${\cal O}(\alpha_{s})$ gluon exchange
diagrams, the Borel function is known exactly [28]. The large-$\beta_{0}$
approximation is believed to exhibit at least the qualitative features of the
Borel function in full QCD.

We also mention that the gluon condensate OPE corrections almost completely
cancel (up to contributions coming to the higher order corrections to its
Wilson coefficient $C_{G^{2}}$) from the contour integral of Eq. (2), if the
weight function $W(x)$ does not contain a quadratic term $x^{2}$. At the same
time, the associated perturbative behavior of $\delta^{(0)}_{W}(s_{0})$ is
much better than for weight functions with a quadratic term [20]. This is the
reason why for most recent phenomenological analyses (aiming for strong
coupling determinations) only moments with weight functions without a
quadratic term have been employed. This observation is consistent with the
norm of the gluon condensate renormalon being quite sizeable, so that it
already governs the size of the known 5-loop corrections.

##  4 The FOPT and CIPT Borel Representations

The central aspect of our work is that the Borel representations of the FOPT
and CIPT spectral function moment series are not identical. To see this we
construct the two Borel representations directly from the series terms using
the form of the Borel function of the Euclidean Adler function $B[\hat{D}]$ as
an input, but making no further assumption about their form.

We first consider CIPT and start from the observation that the contour
integrals over $\frac{1}{x}W(x)(\frac{\alpha_{s}(-xs_{0})}{\pi})^{n}$, which
arise for each CIPT moment series term, do a priori not allow to cleanly
identify the expansion parameter of the series – simply because the
renormalization scale of $\alpha_{s}$ is integration parameter dependent. This
is the very special characteristics of the CIPT approach. It implies that we
should consider the whole integral to be part of the series coefficients and
reintroduce an expansion parameter by hand so that we can apply the principles
in the construction of the Borel function explained in Sec. 3. Applying the
strict invariance mentioned at the very end of the 2nd paragraph, an
appropriate choice of the expansion parameter is $\alpha_{s}(s_{0})$, which
one can conveniently pull out of the series coefficients with the appropriate
power,222Any multiple of $\alpha_{s}(s_{0})$ can be picked as the expansion
parameter without changing the series terms, but $\alpha_{s}(s_{0})$ is
convenient since it also used for the FOPT series and allows for easy
comparison.

$\textstyle\delta^{(0),{\rm CIPT}}_{W}(s_{0})\,=\,\,\frac{1}{2\pi
i}\,\sum_{n=1}^{\infty}\bar{c}_{n}\,\Big{[}\,\ointctrclockwise_{{\cal
C}_{x}}\\!\\!\frac{{\rm
d}x}{x}\,W(x)\,\Big{(}{\textstyle\frac{a(-x)}{a_{0}}}\Big{)}^{n}\,\Big{]}\,a_{0}^{n}\,.$
(5)

Now we can proceed and obtain the Borel function for the CIPT series
$\delta_{W}^{(0),{\rm CIPT}}(s_{0})$,

$\displaystyle B[\delta_{W}^{(0),{\rm CIPT}}(s_{0})](u)$ $\displaystyle=$
$\displaystyle\textstyle\sum_{n=1}^{\infty}\,\Big{[}\,\frac{1}{2\pi
i}\,\ointctrclockwise_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,W(x)\,\Big{(}{\textstyle\frac{a(-x)}{a_{0}}}\Big{)}^{n}\,\Big{]}\,\frac{\bar{c}_{n}}{\Gamma(n)}\,\bar{u}^{n-1}$
$\displaystyle=$ $\displaystyle\textstyle\frac{1}{2\pi
i}\,\ointctrclockwise_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,W(x)\,\Big{(}{\textstyle\frac{a(-x)}{a_{0}}}\Big{)}\,B[\hat{D}]\Big{(}\textstyle\frac{a(-x)}{a_{0}}\bar{u}\Big{)}\,,$

where $B[\hat{D}](u)$ is the Borel function of the Euclidean perturbative
Adler function already mentioned above, defined through the series
$B[\hat{D}](u)=\sum_{n=1}^{\infty}\frac{\bar{c}_{n}}{\Gamma(n)}u^{n-1}$ in its
region of convergence around the origin. The non-analytic structures in the
analytically continued expression for $B[\hat{D}](u)$ in the entire complex
$u$-plane are inherited directly to $B[\delta_{W}^{(0),{\rm
CIPT}}(s_{0})](u)$. The Borel representation of the spectral function moments
in the CIPT approach thus has the form

$\textstyle\delta_{W,{\rm Borel}}^{(0),{\rm
CIPT}}(s_{0})=\int_{0}^{\infty}\\!\\!{\rm d}\bar{u}\,\,\frac{1}{2\pi
i}\,\ointctrclockwise_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,W(x)\,\big{(}{\textstyle\frac{a(-x)}{a_{0}}}\big{)}\,B[\hat{D}]\Big{(}{\textstyle\frac{a(-x)}{a_{0}}}\bar{u}\Big{)}\,e^{-\frac{\bar{u}}{a_{0}}}\,.$
(7)

This derivation does not depend on a particular form of $B[\hat{D}](u)$. It
only assumes that the Taylor series for $B[\hat{D}](u)$ in the complex $u$
plane around the origin specifies the function unambiguously in the entire
complex $u$-plane. This is an assumption that has been made in any past study
of the Borel function of the Euclidean perturbative Adler function, even if
not explicitly stated. Furthermore, it is assumed that swapping the
$x$-integration and the sum over $n$ in Eq. (4), which is correct within the
radius of convergence, is also allowed for the analytically continued
function. We also note that it is allowed to swap the $\bar{u}$ and $x$
integrations when evaluating Eq. (7).

Let us now consider the Borel representation for the perturbative moments in
the FOPT approach. To derive the Borel representation from the prescription in
the 2nd paragraph of Sec. 3 is not an easy task due to the appearance of the
powers of logarithms $\ln(-s/s_{0})$ in the integrals for the series
coefficients, which depend on the $\beta$-function coefficients. The
derivation is, however, straightforward in the large-$\beta_{0}$
approximation, where the series for the Adler function in the complex plane
for the expansion in powers of $\alpha_{s}(s_{0})$ can be written down in
closed form (using $a(-x)=\frac{a_{0}}{1+a_{0}\ln(-x)}$)

$\displaystyle\hat{D}(s)$ $\displaystyle\,=\,$
$\displaystyle\,\textstyle\sum_{n=1}^{\infty}\,a_{0}^{n}\,\sum_{i=0}^{n-1}\frac{(n-1)!}{i!(n-i-1)!}\,\bar{c}_{n-i}\,(-\ln(-x))^{i}\,.$
(8)

It is straightforward to show through algebraic manipulation that the Borel
function of the resulting FOPT moment series has the form (see Ref. [16])

$\textstyle B[\delta_{W}^{(0),{\rm FOPT}}(s_{0})](u)=\frac{1}{2\pi
i}\,\ointctrclockwise_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,W(x)\,B[\hat{D}](u)\,e^{-u\ln(-x)}\,,$ (9)

where we again assume that swapping the contour integration and the sum over
$n$, which is correct within the radius of convergence, is also allowed for
the analytically continued function.333Note that the radius of convergence of
the $\bar{u}$-series for the CIPT moment Borel function in Eq. (4) is by a
factor of $\alpha_{s}(s_{0})/\alpha_{s}(-s_{0})$ larger than the one for the
$u$-series for the FOPT moment Borel function in Eq. (9). Using the relation
$e^{-u\ln(-x)}=e^{-\frac{u}{a(-x)}+\frac{u}{a_{0}}}$ we then obtain the
expression

$\textstyle\delta_{W,{\rm Borel}}^{(0),{\rm FOPT}}(s_{0})={\rm
PV}\int_{0}^{\infty}\\!\\!{\rm d}u\,\,\frac{1}{2\pi
i}\,\ointctrclockwise_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,W(x)\,B[\hat{D}](u)\,e^{-\frac{u}{a(-x)}}\,,$ (10)

for the Borel representation of the spectral function moments in the FOPT
approach. As for Eq. (7), it is allowed to swap the $u$ and $x$ integrations
when evaluating Eq. (10). It has been shown in Ref. [15] that this expression
also applies in full QCD. Prior to our work, the expression in Eq. (9) has
been adopted as the Borel representation for the FOPT moments and the CIPT
moments, where the apparent relation to the CIPT expansion was taken for
granted using the argumentation that its $u$ expansion immediately leads to
the CIPT series of Eq. (5). However, this view did not properly account for
the fact that $\alpha_{s}(-s)$ cannot be used as the expansion parameter of
the CIPT moments series from the mathematical perspective.

Figure 1:  Left: Borel integration paths in the $u$-plane involved for the
FOPT and CIPT Borel representations for an IR renormalon with $p>0$. The red
zig-zag line represents the renormalon cut. Right: FOPT and CIPT spectral
function moment series associated to the weight function $W(x)=1$ and the
Borel function $\frac{1}{2-u}$ in the large-$\beta_{0}$ approximation for
$\alpha_{s}(m_{\tau}^{2})=0.34$ and $s_{0}=\mu^{2}=m_{\tau}^{2}$.

When considering the purely perturbative interpretation of Eqs. (7) and (10)
(i.e. the truncated Taylor series in u or $\bar{u}$, which is always an
analytic function) and expanding either in $\alpha_{s}(s_{0})$ or
$\alpha_{s}(-s)$ prior to the contour integration, both expressions lead to
both the FOPT and CIPT moment series and are equivalent. Furthermore, both
Borel representations are formally related through the complex-valued change
of variables
$u=\bar{u}\,\alpha_{s}(-s)/\alpha_{s}(s_{0})=\bar{u}\,a(-x)/a_{0}$. If the
full Borel function $B[\hat{D}](u)$ were also an analytic function, this
change of variables would be sufficient to prove that both Borel
representations are equivalent. However, Eqs. (7) and (10) are not equivalent
and lead to different Borel sums due to the presence of non-analytic IR
renormalons in the Euclidean Borel function $B[\hat{D}](u)$. Consider a
generic IR renormalon contribution of the form $\frac{1}{(p-u)^{\gamma}}$,
which leads to a cut along the positive real $u$-axis starting at $u=p$, see
the left panel in Fig. 1 showing the complex $u$-plane. This makes the FOPT
Borel representation (10) only well-defined with an additional prescription on
the $u$-integration is imposed. The most common prescription used in the
literature, the principal value (PV) prescription, is to take the average of
the deformations above and below the cut (paths 1a and 1b). We have indicated
it already in Eq. (10). In contrast, the CIPT Borel representation does not
require a prescription because $\alpha_{s}(-s)/\alpha_{s}(s_{0})$ is complex
along the contour integration over $s$ as long as ${\rm Im}[s]\neq 0$. From
the perspective of the $u$-integration in the FOPT Borel representation, the
$\bar{u}$ integration in the CIPT Borel representation never touches the cut
and proceeds either entirely above (shown as path 2 for ${\rm Im}[x]>0$) or
below it. Figure 1 also illustrates that the difference in the FOPT and CIPT
Borel sums, called the asymptotic separation, arises from closing path 2 with
either paths 1a or paths 1b at positive real infinity. For ${\rm Im}[x]>0$,
the situation displayed in the figure, it arises from closing path 2 with path
1a.

##  5 Essential Comments and the Asymptotic Separation

The form of the Borel representation of the CIPT spectral moments in Eq. (7)
is imperative when deriving the Borel function explicitly from the CIPT series
terms. The analytic form of the CIPT Borel representation bears a number of
novel and quite subtle properties which we briefly discuss in the following
and which are important for the numerical computation of the asymptotic
separation. In the following we consider a generic IR renormalon term in the
Euclidan Adler functions Borel function of the form $B^{\rm
IR}_{\hat{D},p,\gamma}(u)=\frac{1}{(p-u)^{\gamma}}$.

###  5.1 Form of the Contour Integration and CIPT OPE Corrections

For the FOPT Borel representation the choice of the complex $x$-integration
path ${{\cal C}_{x}}$ is arbitrary as long as it is ensured that the coupling
$a(-x)$ stays in the perturbative region. For the CIPT Borel representation an
additional restriction arises because the coupling affects the analytic
properties of the CIPT Borel function (4). Let us consider the contribution to
the CIPT Borel representation due to the generic IR renormalon $B^{\rm
IR}_{\hat{D},p,\gamma}(u)$ and for $W(x)=(-x)^{m}$:

$\displaystyle\delta_{\\{(-x)^{m},p,\gamma\\},{\rm Borel}}^{(0),{\rm
CIPT}}(s_{0})$ $\displaystyle=$
$\displaystyle\textstyle\int_{0}^{\infty}\\!\\!{\rm d}\bar{u}\,\frac{1}{2\pi
i}\,\ointctrclockwise_{{\cal C}_{x}}\frac{{\rm
d}x}{x}\,(-x)^{m}\,\big{(}{\textstyle\frac{a(-x)}{a_{0}}}\big{)}\,\frac{e^{-\frac{\bar{u}}{a_{0}}}}{\big{(}p-\frac{a(-x)}{a_{0}}\bar{u}\big{)}^{\gamma}}\,.$
(11)

Apart from the Landau pole and the cut along the positive real $x$ axis
contained in the coupling $a(-x)$, there is an additional cut in the $x$-plane
for real values of $x$ with $\alpha_{s}(-xs_{0})\geq
p\alpha_{s}(s_{0})/\bar{u}$. In the large-$\beta_{0}$ approximation this is
equivalent to
$x\geq\tilde{x}(\bar{u})\equiv-e^{(\bar{u}-p)/pa_{0}}=-(\frac{\Lambda_{\rm
QCD}^{2}}{s_{0}})^{(p-\bar{u})/p}$. The value of $\bar{x}(u)$ is negative so
that it affects the possible choices for the path ${{\cal C}_{x}}$. For
$\bar{u}<p$ the cut is still within the circular path $|x|=1$. For $\bar{u}>p$
it is not, so that we have to deform the integration path ${\cal C}_{x}$
further into the negative real $x$-plane such that it crosses the real
negative axis at a value below $\tilde{x}(\bar{u})$. This property entails
that, when $\bar{u}\to\infty$, the allowed region where the path can cross the
negative real $x$-axis is shifted towards negative infinity. Furthermore, when
the $\bar{u}$ integration is carried out first, this cut stretches to minus
real infinity, so that the contour ${\cal C}_{x}$ must be deformed to minus
negative real infinity as well. This additional cut is an essential issue when
one attempts to apply the analytic structure of Eq. (7) for a calculation of
the Borel sum to the expansion of complex-valued (non-Euclidean) Adler
function of Eq. (3), i.e. when discussing its form without the contour
integration (see Refs. [15, 16] for such an analysis). The analytic form of
the CIPT Borel representation implies that this Borel sum has a cut along the
Euclidean axis that is power-suppressed by a factor
$e^{-\frac{p}{a(-x)}}\sim(\frac{\Lambda_{\rm QCD}^{2}}{-s})^{p}$. Since this
cut is unphysical, one must conclude that the associated OPE corrections to
the Adler function (which does not have such a cut at the hadron level) cannot
have the standard form discussed in Sec. 3. This implies that the OPE
corrections that need to be added to the CIPT spectral function moments differ
from those of the FOPT moments and, furthermore, cannot be computed from the
standard form of the Adler function’s OPE corrections. We note that this
conclusion is not imperative at this point since the contour integration is an
integral part in the derivation of the form of Eq. (7), but we believe that it
is the correct one.444The unphysical cut may be taken as a formal reason to
dismiss the form of Eq. (7) and all its implications. However, because the cut
is power suppressed, it can be compensated by OPE corrections that do not have
standard form or maybe even have a connection to duality violating effect. So
there is no contradiction. We believe that there is sufficient evidence that
supports the view that the Borel representation of Eq. (7) should be taken
seriously.

###  5.2 FOPT and CIPT OPE Corrections are indeed different

The statement that the OPE corrections to the CIPT spectral function moments
differ from those of the moments computed with FOPT and, furthermore, do not
have standard form is is quite intriguing and not easy to accept. It implies
that phenomenological analyses within the CIPT approach may be subject to a
yet unquantified additional uncertainty concerning the treatment of the OPE
corrections.555The size of this uncertainty is only sizeable if the
normalization of the gluon condensate renormalon in full QCD is sizeable as
well. It is therefore worth to spend some time to discuss it further, having
in mind the statement we made earlier on the suppression of the gluon
condensate renormalon for spectral function moments $W(x)$ without a quadratic
term $x^{2}$. In the large-$\beta_{0}$ approximation, where the gluon
condensate renormalon structure in $B[\hat{D}](u)$ has the form
$\frac{1}{2-u}$, it is straightforward to see this suppression when carrying
out the $x$ contour integration for the FOPT Borel representation for this
renormalon structure [29]:

$\displaystyle\delta_{\\{(-x)^{m},2,1\\},{\rm Borel}}^{(0),{\rm FOPT}}(s_{0})$
$\displaystyle=$ $\displaystyle\textstyle{\rm PV}\int_{0}^{\infty}\\!\\!{\rm
d}u\,\frac{1}{2-u}\,\frac{1}{2\pi i}\,\ointctrclockwise_{|x|=1}\frac{{\rm
d}x}{x}\,(-x)^{m}\,e^{-u\ln(-x)}\,e^{-\frac{u}{a_{0}}}$ $\displaystyle=$
$\displaystyle\textstyle{\rm PV}\int_{0}^{\infty}\\!\\!{\rm
d}u\,\,\frac{(-1)^{m}\sin(u\pi)}{\pi(u-m)}\,\frac{1}{2-u}\,e^{-\frac{u}{a_{0}}}\,.$

For $m\neq 2$ the renormalon pole at $u=2$ is completely eliminated, the Borel
function for the FOPT moment becomes analytic in the entire $u$-plane and the
PV prescription can be dropped. This cancellation is accompanied by two more
facts, namely that (i) the associated FOPT series is convergent (see the red
dots in the right panel of Fig. 1 for $W(x)=1$) and that (ii) the gluon
condensate correction (4) vanishes in the $x$-contour integration since the
residue is zero. The associated CIPT series (blue dots), however, is not
convergent.666It is intriguing that this fact has apparently never been
noticed in the literature prior to our work. The CIPT Borel representation for
$W(x)=1$ that arises from carrying out the $x$ contour integration has the
form

$\displaystyle\delta_{\\{1,2,1\\},{\rm Borel}}^{(0),{\rm CIPT}}(s_{0})$
$\displaystyle=$ $\displaystyle\textstyle\int_{0}^{\infty}\\!\\!{\rm
d}\bar{u}\,\left(\frac{-1}{2a_{0}}\right)Q\left(1,0,\frac{2-\bar{u}}{2a_{0}}\right)\,e^{-\frac{\bar{u}}{a_{0}}}\,,$
(13)

with $Q(1,0,\rho)=\frac{i}{2\pi}\,[\ln(\rho+i\pi)-\ln(\rho-i\pi)]$. The
$\bar{u}$-integral along the positive real axis does again not need a
prescription, but the Borel function has cuts located parallel to the real
$\bar{u}$ axis starting at distance $2|1+ia_{0}\pi|$ to the origin. These cuts
signal that the underlying series is not convergent as can be clearly seen in
the figure. For a renormalon cut $\frac{1}{(p-u)^{\gamma}}$ the distance is
$p|1+ia_{0}\pi|$, see Refs. [15, 16] for details and formulae for all cases.
The non-convergence of the CIPT series and the uncancelled cuts, imply that
(in the large-$\beta_{0}$ approximation) that the CIPT series requires a
finite OPE correction. Since the standard gluon condensate OPE corrections
vanishes, the required OPE corrections cannot have the standard form of Eq.
(4).

###  5.3 Asymptotic Separation

In the right panel of Fig. 1 also the FOPT and CIPT Borel sums for the series
associated to the Borel function contribution $\frac{1}{2-u}$ for $W(x)=1$ are
shown as the colored horizontal lines. They can be computed directly from Eqs.
(5.2) and (13). The difference between the two is called asymptotic separation
and clearly visible. The FOPT series clearly converges to its Borel sum, while
the CIPT series approaches its Borel sum at intermediate order prior to
divergence. In general, it is more convenient to calculate the asymptotic
separation by doing first the Borel integration, applying the argumentation
concerning closing the paths 1a and 1b with path 2 with respect to the IR
renormalon cut we have mentioned at the end of Sec. 4. This leads to

$\displaystyle\Delta(m,p,\gamma,s_{0})\,\equiv\,\delta_{\\{(-x)^{m},p,\gamma\\},{\rm
Borel}}^{(0),{\rm CIPT}}(s_{0})\,-\,\delta_{\\{(-x)^{m},p,\gamma\\},{\rm
Borel}}^{(0),{\rm FOPT}}(s_{0})$
$\displaystyle\textstyle=\,\frac{1}{2\Gamma(\gamma)}\,\ointctrclockwise_{{\cal
C}_{x}}\frac{{\rm d}x}{x}\,(-x)^{m}\,{\rm sig}[{\rm
Im}[x]]\,(a(-x))^{1-\gamma}\,e^{-\frac{p}{a(-x)}}\,.$ (14)

for the asymptotic separation for the generic renormalon structure $B^{\rm
IR}_{\hat{D},p,\gamma}(u)$. For $m<p$, which covers all linear weight
functions,777We recall that $B[\hat{D}](u)$ only contain non-analytic
renormalon terms of the form $\frac{1}{(p-u)^{\gamma}}$ for $p=2,3,4\ldots$.
the asymptotic separation $\Delta$ can be computed in the prescribed way, but
for $m>p$, the exponentially suppressed term $e^{-\frac{p}{a(-x)}}$ is beaten
by the divergent $(-x)^{m}$ term when $x$ approaches $-\infty$ in the
remaining contour integral. For this case one needs to determine $\Delta$
through analytic continuation, which boils down to the analytic formula
determined for $m<p$. Details of this analytic continuation are given in Ref.
[15]. For $m=p$, we define by hand $\Delta(p,p,\gamma,s_{0})=0$, because in
this case the renormalon behavior is not suppressed in the moment series and
the FOPT and CIPT series both exhibit an unstable and divergent character [20,
15] such that the discussion of a discrepancy between them is irrelevant from
the purely practical point of view. In the large-$\beta_{0}$ approximation the
analytic formulae are quite simple and read
($e^{-\frac{p}{a_{0}}}=(\frac{\Lambda_{\rm QCD}^{2}}{s_{0}})^{p}$)

$\displaystyle\textstyle\Delta_{\beta_{0}}(m\neq
p,p,1,s_{0})=\frac{(-1)^{p-m}}{p-m}\,e^{-\frac{p}{a_{0}}}\,,$ (15)
$\displaystyle\textstyle\Delta_{\beta_{0}}(m\neq
p,p,2,s_{0})=(-1)^{p-m}\,\bigg{[}\frac{1}{(p-m)^{2}}+\frac{1}{(p-m)a_{0}}\bigg{]}\,e^{-\frac{p}{a_{0}}}\,.$
(16)

The analytic expressions in full QCD are more complicated and written down in
Ref. [15].

Figure 2:  Left: FOPT and CIPT spectral function moment series associated to
the weight function $W(x)=(-x)^{4}$ and the Borel function $\frac{1}{2-u}$ in
the large-$\beta_{0}$ approximation for $s_{0}=m_{\tau}^{2}$. Right: FOPT and
CIPT Moment series $\delta^{(0),{\rm FOPT}}_{W_{\tau}}(m_{\tau}^{2})$ for the
total hadronic $\tau$ decay rate $R_{\tau}$ in the large-$\beta_{0}$
approximation. Horizontal lines represent the FOPT and CIPT Borel sums and the
orange band in the right panel shows the FOPT Borel sum ambiguity. We used
$\alpha_{s}(m_{\tau}^{2})=0.34$ and $\mu^{2}=m_{\tau}^{2}$.

###  5.4 Brief numerical Analysis

We have already shown in the right panel of Fig. 1 that the asymptotic
separation describes the disparity in the behavior of FOPT and CIPT spectral
function moments series very accurately for $W(x)=1$ and a $p=2$ simple pole
IR renormalon in the large-$\beta_{0}$ approximation. The left panel of Fig. 2
shows the corresponding case for $W(x)=(-x)^{4}$, where the analytic
continuation is mandatory to obtain the result for the asymptotic separation.
Here the FOPT series is again convergent (in contrast to the CIPT series), and
the description of the disparity in the behavior of both series is again very
accurately described by the asymptotic separation. This excellent description
can be easily checked for any IR renormalon and any monomial weight function
$W(x)=(-x)^{m}$ also in full QCD and we refer to Ref. [15] for details.

We conclude with showing in the right panel of Fig. 2 the FOPT and CIPT series
for the normalized total hadronic $\tau$ decay rate $R_{\tau}$, where the
weight function $W_{\tau}(x)$ is a linear combination of several monomials
(see text below Eq. (2)) using the full Borel function $B[\hat{D}](u)$ in the
large-$\beta_{0}$ approximation [28], see e.g. Eq. (12) in Ref. [16] for the
expression. Since in the full Borel function IR and UV renormalon poles are
located at all integer values along the real $u$-axis (except for $u=0,1$),
the FOPT and the CIPT series are both asymptotic (and non-convergent). Both
series are shown for the same $\alpha_{s}$ value. The oscillating structures
visible in both series arise from the influence of UV renormalons which are
associated to a sign-alternating increase of the series coefficients. The
impact of these UV renormalons is, however, very small at intermediate orders
below 9 so that we can observe the impact of the IR renormalons.888The leading
UV renormalon pole is located at $p=-1$ and should in principle dominate over
the impact of the gluon condensate renormalon pole at $p=2$ already at very
low orders. However, it happens that in the $\overline{\rm MS}$ scheme the
normalization of these UV renormalons is strongly suppressed compared to the
IR renormalons. We can clearly see the disparity between the FOPT and CIPT
series (around orders 5 to 8), which is the reason why $\alpha_{s}$ values
based on CIPT analyses tend to be larger than for FOPT-based analyses (once
the same OPE corrections are used in both approaches). The FOPT and CIPT Borel
sums are again indicated by the colored horizontal lines and we have also
displayed as the light orange band the standard estimate for the ambiguity of
the FOPT Borel sum in the PV prescription, which is defined as the difference
from using paths 1a and 1b in Fig. 1 (left panel) times a factor
$\frac{1}{\pi}$. We again see that the asymptotic separation describes the
disparity between the FOPT and CIPT series very well, and we also observe that
the asymptotic separation is substantially larger than the FOPT Borel sum
ambiguity (even if we would not include the ad hoc suppression factor
$\frac{1}{\pi}$). Interestingly, $99.8\%$ of the numerical value of the
asymptotic separation comes from the gluon condensate renormalon. This happens
because the Borel function $B[\hat{D}](u)$ of the Euclidean Adler function in
the large-$\beta_{0}$ approximation contains a gluon condensate IR renormalon
cut with a sizeable normalization and because the contribution of IR
renormalons with $p\geq 3$ is strongly power-suppressed by additional factors
of $\Lambda_{\rm QCD}^{2}/s_{0}$, see Eq. (15). From a practical point of
view, only the $p=2$ gluon condensate renormalon is relevant when considering
the implications of the asymptotic separation.

##  6 Conclusions

In this talk we have shown that the Borel representations of the FOPT and CIPT
$\tau$ hadronic spectral function moments have different Borel
representations. The CIPT Borel representation is new and has novel and subtle
features. In the presence of IR renormalons the different analytic properties
lead to a difference in their Borel sums, called the asymptotic separation.
While the FOPT Borel representation has been known before, the CIPT Borel
representation is new and its structure provides the implication that the OPE
corrections that need to be added to the CIPT moments differ from those of the
FOPT approach and do furthermore not have standard form. From a numerical
point of view, the asymptotic separation and its implications are practially
relevant only if the gluon condensate IR renormalon has a substantial
normalization. This is so in the large-$\beta_{0}$ approximation, where the
asymptotic separation nicely describes the disparity between the FOPT and CIPT
spectral function moment series.

## Acknowledgements

We thank Matthias Jamin and Diogo Boito for discussions. This work was
supported in part by the FWF Austrian Science Fund under the Project No.
P32383-N27. We also acknowledge partial support by the FWF Austrian Science
Fund under the Doctoral Program “Particles and Interactions” No. W1252-N27.

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