# Drawing Prolog Search Trees:
A Manual for Teachers and Students of Logic Programming

Johan Bos
University of Groningen
<EMAIL_ADDRESS>
###### Abstract

Programming in Prolog is hard for programmers that are used to procedural
coding. In this manual the method of drawing search trees is introduced with
the aim to get a better understanding of how Prolog works. After giving a
first example of a Prolog database, query and search tree, the art of drawing
search trees is systematically introduced giving guidelines for queries with
variables, conjunction, disjunction, and negation. Further examples are
provided by giving the complete search trees that are shown in Learn Prolog
Now!

###### Contents

  1. 1 Introduction
  2. 2 Drawing Search Trees
    1. 2.1 The general idea
    2. 2.2 A first example
    3. 2.3 Next steps
  3. 3 The Search Tree Expansion Rules
    1. 3.1 A simple query without variables
    2. 3.2 Simple queries with variables
    3. 3.3 More than one query: conjunctive queries
    4. 3.4 A disjunctive query
    5. 3.5 Lists and variable renaming
  4. 4 Search Trees with Negation and the Cut
    1. 4.1 The built-in control predicates
    2. 4.2 The cut
    3. 4.3 An example using cut
    4. 4.4 Negation as failure
  5. 5 Search Trees in Learn Prolog Now!
    1. 5.1 Proof Search, Chapter 2 from Learn Prolog Now!
    2. 5.2 Recursion, Chapter 3 from Learn Prolog Now!
    3. 5.3 Recursion, Chapter 3 from Learn Prolog Now!
    4. 5.4 More Lists, the predicate append/3
    5. 5.5 Search tree for the last query of Exercise 2.2 from Learn Prolog Now!
    6. 5.6 Complete answers to Exercise 4.7 from Learn Prolog Now!

###### Listings

  1. 1 Prolog database for a complicated romance.
  2. 2 Prolog database for a small animal kingdom.
  3. 3 Definition of member/2
  4. 4 Prolog database to illustrate the cut
  5. 5 A Prolog database and definition for youngest/1.
  6. 6 Prolog database to illustrate proof search.
  7. 7 Prolog definition for add/3.
  8. 8 Definition of append/3.

###### List of Figures

  1. 1 Search tree expansion rules for queries without variables.
  2. 2 Search tree expansion rules for queries with variables.
  3. 3 Search tree expansion rules for conjunctive queries.
  4. 4 Search tree expansion rules for disjunctive queries.
  5. 5 Search tree expansion rules for the logical predicates.
  6. 6 Search tree expansion rules for the cut.
  7. 7 Search tree expansion rules for negation.

##  1 Introduction

Prolog, a declarative programming language, requires a different mindset for
programmers who are used to code in procedural programming languages like C,
Python, or Java. It is astonishingly hard to achieve this different attitude
to coding. One approach that sheds light on the way Prolog executes programs
is to apply the brutal method of drawing complete search trees. It is well
worth the effort: mastering Prolog will enrich the life of a programmer
considerably, because it adds a completely new (and beautiful) perspective on
solving computer science problems.

Prolog’s backbone is based on a depth-first search strategy that operates on
facts and rules declared in a database (Clocksin and Mellish, 1987; Bratko,
2001; Blackburn et al., 2006). This sounds harmless, but it is not always
straightforward to predict how a Prolog program behaves in response to a
certain query (sometimes also called question or goal) given to it. The best
way to find out how Prolog reacts to a query, given a certain knowledge base,
is to draw a search tree. In the standard logic programming course books there
is too little space devoted to drawing seach trees.

Clocksin and Mellish (1987), one of the first complete introductions to
programming in Prolog, look at the way Prolog solves queries by providing a
flow of satisfaction, which is based on the tracing model that Prolog offers
for debugging programs. Bratko (2001) probably was the first to introduce
search trees, but he calls them execution traces. The search trees in this
manual are inspired by those of Bratko.

Even though drawing search trees requires some effort, students of logic
programming will get a much better understanding of how their Prolog programs
work by doing so, instead of peeking at the output of overrated tools such as
the trace facility. This manual is therefore meant to serve as an aid in the
classroom, both for students and teachers of logic programming. Trees can be
drawn in a free fashion by paper and pencil, with the aid of sophisticated
drawing software, or just by a simple straightforward standard text
characters. The latter method will be used in this manual.

This manual is not an introduction to Prolog, and assumes the reader has some
basic experience with logic programming, and is familiar with the concepts
clause, fact, rule, predicate, backtracking, and unification. We will first
show the basic idea of drawing search trees in Section 2, where we give an
example for a simple query, a Prolog database, and the corresponding search
tree. This will give the reader a first feeling of what search trees are and
what their potential is. In Section 3, we show the search tree expansion
rules, for simple queries (with and without variables), conjunctive queries,
queries with disjunction, and queries for some built-in predicates In Section
4 we show how to draw search trees for queries with negation and the cut.
Finally, in Section 5 we give our version of all search trees that occur in
Blackburn et al. (2006).

##  2 Drawing Search Trees

###  2.1 The general idea

The programming language Prolog, one of the oldest (and one of the oddest)
programming languages in the world, has its roots in (classical) logic. It
therefore divides everything in true or false. Like classical logic, Prolog’s
ingredients are based on conjunction, disjunction, implication, and negation,
although the use of these operators is restricted and Prolog’s negation is
based on the closed world assumption (see Section 4).

Prolog works by answering queries with respect to a knowledge base. Queries
are like yes/no-questions in natural language: they are answered by Prolog as
yes (following the information in the knowledge base), or no (the knowledge
base does not support the query). A Prolog database comprises clauses that are
either rules or facts. Rules introduce new queries, facts lead to positive
answers, and if a piece of information cannot be deduced, it will be
considered false. Together, the facts and rules define one or more predicates
in the knowledge base. Queries can also contain variables (recall that
variables are denoted by uppercase letters in Prolog). A query with variables
for which Prolog returns a positive answer, will also instantiate the variable
with values that make the query true. Such queries are the natural language
equivalent of wh-questions (questions starting with who, what, where, and
when).

A search tree shows the way Prolog finds an answer to a query. It shows all
the steps of reasoning to come to an answer, so you may also call it a proof
tree, using more logical terminology. In this manual, we use the following
strategy for drawing search trees:

  1. 1.

Write down the query in a box on the top of the page—the box acts as a sort of
to-do list;

  2. 2.

Draw the branches based on the number of clauses defined for the predicate
that the query addresses;

  3. 3.

Proceed with the branches in a left-to-right fashion. If a branch corresponds
to a fact, go to Step 4. If a branch corresponds to a rule, go to Step 5.

  4. 4.

If the query does not unify with the fact draw a cross (x, denoting a dead
branch); otherwise (the query unifies with the fact) draw an empty box ([],
signalling that there is nothing left to do);

  5. 5.

If the query does not unify with the head of the rule draw a cross; otherwise
draw a new box with the body of the rule as new query. Go to Step 2.

Of course, search trees can be drawn in many different ways. It doesn’t matter
which style you’re using, as long as you’re consistent (for instance, instead
of the cross you can also write fail or no, and instead of the empty box you
can also write yes or success). You can draw them with pencil and paper, or
use a handy graphical visualisation tool.

In this manual we follow, by and large, the graphical conventions introduced
in Learn Prolog Now! by Blackburn et al. (2006). It is a very simple way to
draw trees in a normal text format. However, note that not all search trees in
Learn Prolog Now! are complete (see Section 5).

Let’s have look at a first example.

###  2.2 A first example

The example we consider is taken from Learn Prolog Now! (Blackburn et al.,
2006, p.38), and describes a situation with rather complicated love
relationships between three people and a (rather naive) definition of
jealousy, as shown in Listing 1.

⬇

1loves(vincent,mia).

2loves(marsellus,mia).

3

4jealous(A,B):- loves(A,C), loves(B,C).

Listing 1: Prolog database for a complicated romance.

Suppose we are interested in the answers to the query jealous(X,Y). Following
Step 1 of the recipe given above, we start our first search tree by putting
the query in an empty box (we use square brackets to visualise boxes):

⬇

1 [jealous(X,Y)]

This query addresses the predicate jealous/2. There is only one clause
defining this predicate in the database. Hence, we only draw one branch:

⬇

1 [jealous(X,Y)]

2 |

3 |

4 |

Because the clause in the database is a rule, we do two things: first we check
whether the head of the rule unifies with the query. It does, because we have
for both arguments variables in the query, and variables always unify. What
happens next is that the body of the rule in line 4 of Listing 1 will form the
new query, taking into account the variable bindings that occurred while
matching the query with the head of the rule. So this gives us:

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

We now proceed at looking at this new query. It is a compound query, and it
contains two sub-queries combined with a conjunction. In this case we first
look at the left conjunct: loves(A,C). The loves/2 predicate is defined by two
clauses in the database as Listing 1 shows in line 1–2. So we will draw two
new branches—one for each clause:

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

7 / \

8 / \

9 / \

Note, at this point, that the order of the clauses in the database—from top to
bottom—corresponds to a left-to-right order in the search tree that we draw.
This is just a conventional habit that we adopt (in a culture where one would
write from right to left, we probably would have picked a different order).
But this order is important. As Prolog searches answers to queries by
examining clauses from top to bottom, the order of answers (in case there is
more than one answer) should be reflected in the search tree. Hence, the order
of the answers in a search tree should be read from left to right. And this
explains why we proceed with the first branch from the left, and do not look
at the second, open, branch (because recall, Prolog’s search strategy is
depth-first!). The left-most branch targets the first clause of loves/2, shown
in line 1 of Listing 1. This is a fact. So the answer is either true or false.
It is true if the fact matches with the query, which it does, because once
again, the query contains only variables: variable A unifies with atom
vincent, and variable C unifies with atom mia.

Does this mean we have our first answer? No! We were dealing with a compound,
conjunctive, query. We did the first part of this compound query, but still
have to solve the remaining sub-query. So our new query becomes loves(B,mia)
and the search tree is expanded as follows:

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

7 / \

8 A=vincent / \

9 C=mia / \

10 / \

11[loves(B,mia)]

Once more, we deal with the predicate loves/2, and there are two possibilities
because it is defined by two clauses, so again we draw two new branches:

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

7 / \

8 A=vincent / \

9 C=mia / \

10 / \

11[loves(B,mia)]

12 / \

13 / \

14 / \

We proceed by examining the left-most branch. The corresponding fact in the
database matches with the query, and variable B unifies with atom vincent.
Because a fact does not introduce new queries, we have our first answer, and
we signify this with an empty box in the search tree, showing that we
satisfied all queries: there is nothing more left to do on our to-do list
(instead of an empty box in line 15 you could also label the node with “yes”
or “true”):

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

7 / \

8 A=vincent / \

9 C=mia / \

10 / \

11 [loves(B,mia)]

12 / \

13B=vincent / \

14 / \

15 []

The answer that Prolog will give to the original query is: true, with the
variable X unified with the atom vincent (because X unifies with A), and
variable Y unified with atom vincent (because Y unifies with B). (So the
answer to our original query, “Who is jealous of whom?”, is “Vincent is
jealous of himself”. This isn’t perhaps the answer you would expect, but as we
have remarked before, the definition if jealousy in Listing 1 is a rather
naive one and incomplete.)

So far so good—we got our first search tree, but it is not complete yet. There
are still open branches that might give us alternative (positive) answers. To
make the search tree exhaustive, we perform, like Prolog does if you ask more
for solutions, backtracking. We go up in the tree, and find the first open
branch. We continue to expand this branch. This will give us the following
search tree and a second solution for the query:

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

7 / \

8 A=vincent / \

9 C=mia / \

10 / \

11 [loves(B,mia)]

12 / \

13B=vincent / \ B=marsellus

14 / \

15 [] []

Is this search tree exhaustive? No, it isn’t: there is yet another open
branch. When we turn to that, we arrive at the following search tree:

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

7 / \

8 A=vincent / \ A=marsellus

9 C=mia / \ C=mia

10 / \

11 [loves(B,mia)] [loves(B,mia)]

12 / \

13B=vincent / \ B=marsellus

14 / \

15 [] []

The current query that we work on the search tree, loves(B,mia), is a query we
have already seen before, and is already part of the larger search tree. So we
know how it will expand, as the queries in search trees are independent from
each other. This means the complete, exhaustive search tree can be graphically
realised as follows:

⬇

1 [jealous(X,Y)]

2 |

3 | X=A

4 | Y=B

5 |

6 [loves(A,C),loves(B,C)]

7 / \

8 A=vincent / \ A=marsellus

9 C=mia / \ C=mia

10 / \

11 [loves(B,mia)] [loves(B,mia)]

12 / \ / \

13B=vincent / \ B=marsellus B=vincent / \ B=marsellus

14 / \ / \

15 [] [] [] []

Note that variable bindings, when they appear, are decorated along the
branches of the search tree. Each Prolog answer to a query corresponds to a
path from the top query to an empty box. All the variable bindings along this
path are the instantiations used in answering the query. For this particular
query we get four different answers: Vincent is jealous of Vincent, Vincent is
jealous of Marsellus, Marsellus is jealous of Vincent, and Marsellus is
jealous of Marsellus.

###  2.3 Next steps

In the previous example we got acquainted with a first search tree with a
couple of exciting developments. But there are a lot of other things that can
happen when constructing search trees. Not all paths may lead to an answer:
they die, instead, leading to dead branches. There are compound queries as we
have seen, but sometimes the queries are disjunctive instead of conjunctive,
and there are cases where we have to rename the variables. The next section
gives a schematic overview of all cases that may appear (excluding negation
and the cut, to which we define a separate section).

##  3 The Search Tree Expansion Rules

###  3.1 A simple query without variables

First we place the query in a box or list. Next we check how many clauses are
defined in the Prolog database for the predicate matching the query. Then we
draw a branch for each clause that is defined by the matching predicate. For
instance, say you get a query p(a), and the predicate p/1 happens to be
defined by two clauses in the knowledge base. Then we can draw a tree with two
branches, and we concentrate on the left-most branch. There are three
possibilities: (i) there is no fact p(a) in the database: in this case we draw
a cross to indicate that this possibility is “dead”; (ii) there is a fact p(a)
in the database: we draw an empty box to indicate we found an answer; (iii)
there is a rule whose head matches the query, e.g. p(a):- Q., in the database:
draw a new box for the query Q. All three options are shown here:

⬇

NO FACT: p(a). FACT: p(a). RULE: p(X):- Q.

[p(a)] [p(a)] [p(a)]

/ \ / \ / \

/ \ / \ / \

/ \ / \ / \

x ... [] ... [Q] ...

Figure 1: Search tree expansion rules for queries without variables.

Let’s have a look at a concrete example where we can apply the rules shown in
Figure 1. Consider the following knowledge base about a small animal kingdom
with various mammals, reptiles, and birds (Listing 2).

⬇

1mammal(fox).

2mammal(whale).

3mammal(bat).

4

5bird(pelican).

6bird(swan).

7

8reptile(snake).

9

10animal(X):- mammal(X).

11animal(X):- reptile(X).

12animal(X):- bird(X).

Listing 2: Prolog database for a small animal kingdom.

This knowledge base contains the definition for four predicates using six
facts and three rules. Suppose we want to know whether a bat is an animal,
according to this database. The corresponding search tree for animal(bat) is
drawn as follows:

⬇

1 [animal(bat)]

2 / | \

3 / | \

4 / | \

5 [mammal(bat)] [reptile(bat)] [bird(bat)]

6 / | \

7 / | \

8 [] x x

And indeed, a bat is an animal, because a bat is a mammal, and every mammal is
an animal. The search tree shows this by the branch with the empty box in line
9.

What about queries that contain variables? How do the expansion rules look for
such queries? Very similar, but we need to keep track of the variable
instantiations. Let’s have a look.

###  3.2 Simple queries with variables

Here we consider search trees for queries with variables. As with any query,
we place it in a box. Then we check how many clauses are defined in the Prolog
database for the predicate matching the query. We draw a branch for each
clause that is defined by the matching predicate. But now, because there are
variables in the query, each branch also needs to be annotated in case
variable bindings occur. Like before, we need to concentrate on the left-most
branch first. There are three possibilities: (i) there is a fact in the
database that matches the query: in this case we draw an empty box and
decorate the branch with variable bindings; (ii) there is a rule whose head
matches the query, for instance p(Y):- q(Y)., and there no variables in the
rule with the same name as in the query: add the body of the rule as a new
query to the tree; (iii) there is a rule whose head matches the query that
contains variables with the same name as in the query (e.g. p(X):- q(X): in
this case you need to rename the variables of the rule to avoid confusion. All
three options are shown in Figure 2.

⬇

FACT: p(a). RULE: p(Y):- q(Y). RULE: p(X):- q(X).

[p(X)] [p(X)] [p(X)]

/ \ / \ / \

X=a / \ X=Y / \ X=X’/ \

/ \ / \ / \

[] ... [q(Y)] ... [q(X’)]

Figure 2: Search tree expansion rules for queries with variables.

Consider again the knowledge base about the animal kingdom (Listing 2). The
search tree for animal(Animals) (what is an animal?) is drawn as follows.

⬇

1 [animal(Animal)]

2 / | \

3 / | \

4 Animal=X / | Animal=X \ Animal=X

5 / | \

6 / | \

7 [mammal(X)] [reptile(X)] [bird(X)]

8 / / \ | / \

9 / / \ | X=snake / \

10 X=fox / / \ X=bat | / X=pelican \ X=swan

11 / / X=whale \ | / \

12 / / \ | / \

13 [] [] [] [] [] []

###  3.3 More than one query: conjunctive queries

In case there is a conjunctive query (two or more queries separated by a
comma), we first consider the left-most query. The remaining queries are
carried over if the first one succeeds.

⬇

FACT: p(a) RULE: p(A):- q(A).

[p(X),Q] [p(X),Q] [p(X),q(X),Q]

/ \ / \ / \

X=a / \ X=A / \ X=A / \

/ \ / \ / \

[Q] [Q] [q(A),Q]

Figure 3: Search tree expansion rules for conjunctive queries.

###  3.4 A disjunctive query

The semi-colon denotes a disjunctive query. Here it is important to look at
the scope of the disjuncts (it is good practive to use brackets to
disambiguate scope ambiguities). Disjunction always creates two new branches,
one for each disjunct. Nothing happens to the variables, and further queries
in the box are duplicated (Figure 4).

⬇

[p(X);q(X)] [(p(X);q(X)),Q]

/ \ / \

/ \ / \

/ \ / \

[p(X)] [q(X)] [p(X),Q] [q(X),Q]

Figure 4: Search tree expansion rules for disjunctive queries.

###  3.5 Lists and variable renaming

Sometimes variables need to be renamed when drawing a search tree. This is
required when there is a variable occuring in a query that is also used in the
rule. We illustrate this procedure of renaming with the help of the standard
definition for member/2:

⬇

1member(X,[X|_]).

2member(X,[_|L]):- member(X,L).

Listing 3: Definition of member/2

Say we follow the example in Learn Prolog Now! and have a query
member(X,a,b,c]) (Blackburn et al., 2006, p.244). The query happens to contain
the variable X. This variable is also used by the rule in the definition of
member/2. To avoid confusion we need to rename the variables in the rule each
time we use it. It is safe to do this: the scope of Prolog variables is
restricted to the rule they occur in. So the rule above can be replaced by
`member(Y,[_|L]):- member(Y,L).`, for instance. You can pick any (unused so
far in the search tree) variable name that you like. In the tree below we have
chosen X’, X’’, and so on, but we could have also chosen X1 and X2 or other
variables.

⬇

1 [member(X,[a,b,c])]

2 / \

3 X=a / \X=X’

4 / \

5 [] [member(X’,[b,c])]

6 / \

7 X’= b / \X’=X’’

8 / \

9 [] [member(X’’,[c])]

10 / \

11 X’’=c / \X’’=X’’’

12 / \

13 [] [member(X’’’,[])]

14 / \

15 / \

16 x x

The result of this query are three solutions: `X=a`, `X=b` (because `X=X’` and
`X’=b`), and `X=c` (because `X=X’`, `X’=X’’` and `X’’=c`).

##  4 Search Trees with Negation and the Cut

###  4.1 The built-in control predicates

There are a couple of built-in predicates that affect the way search trees are
constructed. One important one is the cut. In order to get good understanding
of the cut, we first need to look at the logical predicates. The predicate
true/0 always succeeds; the predicates fail/0 (or the equivalent false/0)
always fail. Hence, their corresponding search trees are as follows:

⬇

[fail] [fail,Q] [false] [false,Q] [true] [true,Q]

| | | | | |

| | | | | |

x x x x [] [Q]

Figure 5: Search tree expansion rules for the logical predicates.

###  4.2 The cut

Without doubts, one of the most interesting (and one of the most trickiest)
tools of Prolog is the cut predicate. It is a predicate that always succeeds.
But it has an extremely useful side-effect: it cuts open branches from the
search tree, thereby giving direct control on memory management. Cuts are used
to make Prolog programs efficient. This often goes at the cost of making
Prolog programs less declarative and readable. The fact or the matter is that
cuts are needed in practical applications of logic programming.

The cut is not an easy concept to understand. Some newcomers to Prolog confuse
cut with a halt or a stop. But it has little to do with that, because it
always succeeds. What cut does is pruning open branches (choice points) in the
search tree. But, and here is where it gets tricky, not necessarily all open
branches in the search tree. The effect of the cut predicate is to discard all
choice points (open branches) created since entering the predicate in which
the cut appears. In other words, commit to the clause (rule) in which the cut
appears and discard choice points that have been created by goals to the left
of the cut in the current clause. So the general schemata look like:

⬇

[!] [!,Q]

| |

| |

[] [Q]

Figure 6: Search tree expansion rules for the cut.

But these rules don’t show anything about cutting branches. So let’s have a
look at an example using cut.

###  4.3 An example using cut

Consider the following Prolog database:

⬇

1a(X):- b(X).

2a(X):- c(X).

3

4b(X):- d(X), !.

5

6b(X):- e(X).

7

8c(1).

9c(2).

10

11d(3).

12d(4).

13

14e(5).

Listing 4: Prolog database to illustrate the cut

This Prolog database defines four predicates: a/1, b/1, c/1, and d/1, using
three rules and four facts. It doesn’t have any practical usage, except from
explaining how the cut works. Now consider the query a(A). As there are two
rules for a/1, we can draw two branches:

⬇

1 [a(A)]

2 / \

3 / \

4 / \

The first branch corresponds to a rule, introducing a new query:

⬇

1 [a(A)]

2 / \

3 A=X / \

4 / \

5 [b(X)]

This query can be satisfied by two clauses, so we create two new branches, and
concentrate on the first one. This is a query with a cut:

⬇

1 [a(A)]

2 / \

3 A=X / \

4 / \

5 [b(X)]

6 / \

7 / \

8 / \

9 [d(X),!]

As this is a compound, conjunctive query (the second part just happens to be a
cut), we first look at the left conjunct, the query d(X). There are two
possibilities to satisfy this query, so we draw two branches. The first branch
succeeds, and the variable X is unified with atom 3. And now we have to deal
with the cut:

⬇

1 [a(A)]

2 / \

3 A=X / \

4 / \

5 [b(X)]

6 / \

7 / \

8 / \

9 [d(X),!]

10 / \

11X=3 / \

12 / \

13 [!]

This cut, as all cuts do, succeeds. But it cuts down all open branches that
were created since it was introduced. There are three open branches in the
tree above. The first open branch was introduced before the cut came in to
play, so it is not affected (just check the Prolog database to verify this).
What about the second branch? This branch was triggered by the predicate b/1
that also introduced the cut. As a consequence, this branch is pruned away
from the search tree, as is the third open branch. This gives the following
tree, with the two branches removed (signalled by X):

⬇

1 [a(A)]

2 / \

3 A=X / \

4 / \

5 [b(X)]

6 / \

7 / X

8 / \

9 [d(X),!]

10 / \

11X=3 / X

12 / \

13 [!]

14 |

15 |

16 []

At this point, backtracking will jump to the one open branch that is left,
corresponding to the second possibility for satisfying the query a(A). The
final search tree, then, looks like this:

⬇

1 [a(A)]

2 / \

3 A=X / \ A=X

4 / \

5 [b(X)] [c(X)]

6 / \ | \

7 / X X=1| \ X=2

8 / \ | \

9 [d(X),!] [] []

10 / \

11X=3 / X

12 / \

13 [!]

14 |

15 |

16 []

In sum, there are three solutions for this query. Without the cut in the
program there would have been five answers.

###  4.4 Negation as failure

In Prolog, a negated query Q is written as not(Q) or `\+ Q`. A negated guery
succeeds if an attempt to falsify it fails (Clocksin and Mellish, 1987). This
is called negation as failure (Bratko, 2001). Prolog’s negation makes use of
the cut and the built-in predicate fail/0. When used, it creates two new
branches in the tree. In the first branch the query in the scope is placed
followed by the cut and fail/0. The second branch succeeds (but it will be cut
from the tree in case the query of the first branch succeeds). This is how the
search tree expansion rule for negation looks like:

⬇

[\\+ p(X)] [\\+ p(X),Q]

/ \ / \

/ \ / \

/ \ / \

[p(X),!,fail] [] [p(X),!,fail] [Q]

Figure 7: Search tree expansion rules for negation.

The expansion rules in Figure 7 also show why variables in the scope of a
Prolog negation are never bound. This is because Prolog negation introduces
two branches, and variables on different branches are independent of each
other. So even if the query Q in in Figure 7 would contain an occurrence of
the variable X, it would be effected by any bindings that were made while
trying to proof the query p(X).

Now let’s look at an example. We have three children (Ann, Ben and Cai) with
their ages stored in a Prolog database. And we have defined a predicate that
computes who is the youngest child (Listing 5).

⬇

1age(ann,5).

2age(ben,4).

3age(cai,6).

4

5youngest(X):- age(X,Y), \\+ (age(_,Z), Z < Y).

Listing 5: A Prolog database and definition for youngest/1.

Here, in line 5 of Listing 5, “the youngest” is defined as a child with a
certain age Y, such that there is no child that has an age that is smaller
than Y. Well, who is the youngest? We can find out by drawing the search tree
for youngest(Who). There is only one rule for youngest/1, so we draw just one
branch with a complex query that is formed by the body of the rule:

⬇

1 [youngest(Who)]

2 |

3 | Who=X

4 |

5 [age(X,Y), \\+ (age(_,Z), Z < Y)]

This is a compound query, with the first conjunct a simply query with
variables, and the second conjunct a negated query of another compound query.
We look at the left conjunct first. There are three facts for the predicate
age/2 in the database (line 1–3 in Listing 5), so we draw three branches. We
look at the first branch first: perhaps Ann with the age of 5 is the youngest.

⬇

1 [youngest(Who)]

2 |

3 | Who=X

4 |

5 [age(X,Y), \\+ (age(_,Z), Z < Y)]

6 / | \

7 X=ann / | \

8 Y=5 / | \

9 /

10 [\\+ (age(_,Z), Z < 5)]

Now we deal with the negation, using the search tree expansion rule for
negation, introducing the cut-fail combination:

⬇

1 [youngest(Who)]

2 |

3 | Who=X

4 |

5 [age(X,Y), \\+ (age(_,Z), Z < Y)]

6 / | \

7 X=ann / | \

8 Y=5 / | \

9 /

10 [\\+ (age(_,Z), Z < 5)]

11 / \

12 / \

13 / \

14 / []

15 /

16 [age(_,Z), Z < 5, !, fail]

Once again, for age/3 we have three possibilities, so we draw three branches.
The first branch fails (because Ann is not younger than Ann), the second
intially succeeds (Ben is younger than Ann), but then the cut removes the two
open branches:

⬇

1 [youngest(Who)]

2 |

3 | Who=X

4 |

5 [age(X,Y), \\+ (age(_,Z), Z < Y)]

6 / | \

7 X=ann / | \

8 Y=5 / | \

9 /

10 [\\+ (age(_,Z), Z < 5)]

11 / \

12 / X

13 / \

14 / []

15 /

16 [age(_,Z), Z < 5, !, fail]

17 / | \

18 Z=5 / Z=4 | X

19 / |

20 [5 < 5,!,fail] [4 < 5,!,fail]

21 / |

22 / [!,fail]

23 x |

24 [fail]

25 |

26 x

No, Ann is not the youngest. The cut was activated and removed two of the open
branches. The fail/0 predicates ensures that we look at the two other
possibilities. Perhaps Ben or Cai is (we don’t redraw the part of the search
tree for reasons of limited space):

⬇

1 [youngest(Who)]

2 |

3 | Who=X

4 |

5 [age(X,Y), \\+ (age(_,Z), Z < Y)]

6 / | \

7X=ann / X=ben | \

8Y=5 / Y=4 | \

9 / |

10 x [\\+ (age(_,Z), Z < 4)]

11 / \

12 / \

13 / []

14 /

15 [age(_,Z), Z < 4, !, fail]

16 / | \

17 Z=5 / Z=4 | \ Z=6

18 / | \

19[5<4,!,fail] [4<4,!,fail] [6<4,!,fail]

20 / | |

21 / | |

22x x x

Yes, we have a positive answer. The cut is not activated, and the top query
succeeds with variable Who unified with atom ben. There is still one open
branch, but backtracking will not lead to more solutions here.

##  5 Search Trees in Learn Prolog Now!

Some of the search trees that you find in Learn Prolog Now! (Blackburn et al.,
2006, 2007) are incomplete, or are not drawn in the way shown in this manual.
We will present them here in the order they appear in the book.

###  5.1 Proof Search, Chapter 2 from Learn Prolog Now!

In this chapter of Learn Prolog Now, proof search is illustrated with the
following knowledge base (Blackburn et al., 2006, p.34):

⬇

1f(a).

2f(b).

3

4g(a).

5g(b).

6

7h(b).

8

9k(X):- f(X), g(X), h(X).

Listing 6: Prolog database to illustrate proof search.

The query posed is k(Y). There is only one answer. The way Prolog works this
out is illustrated by the following search tree:

⬇

1 [k(Y)]

2 |

3 X=Y |

4 |

5 [f(Y),g(Y),h(Y)]

6 / \

7 Y=a / \ Y=b

8 / \

9 [g(a),h(a)] [g(b),h(b)]

10 / \ / \

11 / \ / \

12 [h(a)] x x [h(b)]

13 / \

14 / \

15 x []

Even though in Learn Prolog Now! the tree is shown with variable renaming,
this is not required for this example. Another difference with Learn Prolog
Now! are the two branches that we draw for the query g(a). Here it is correct
to draw two branches following our rules for drawing search trees, as the
predicate g/1 is defined by two clauses. Having said that, it is also clear
that the second branch fails immediately, and in cases like these we could
consider some flexibility and be less strict in showing all search
possibilities in the tree, leaving out obvious pre-mature dead branches.

###  5.2 Recursion, Chapter 3 from Learn Prolog Now!

The search tree for the query descent(anne,donna) in Learn Prolog Now! is
incomplete (Blackburn et al., 2006, p.54). The search tree requires variable
renaming for the variable Z, and we do so by introducing Z1, Z2, and so on.
The complete search tree is enormous, but we managed to get it on one page:

⬇

1 [descend(anne,donna)]

2 / \

3 / \

4[child(anne,donna)] [child(anne,Z),descend(Z,donna)]

5 / / \ \ / / \ \

6 / / \ \ Z=bridget / / \ \

7x x x x / x x x

8 /

9 /

10 [descend(bridget,donna)]

11 / \

12 / \

13 / \

14 [child(bridget,donna)] [child(bridget,Z1),descend(Z1,donna)]

15 / / \ \ / / \ \

16 / / \ \ / / \ \

17 x x x x x Z1=caroline / x x

18 /

19 /

20 [descend(caroline,donna)]

21 / \

22 / \

23 / \

24 [child(caroline,donna)] [child(caroline,Z2),descend(Z2,donna)]

25 / / \ \ / / \ \

26 / / \ \ / / \ \

27 x x [] x x x \ Z2=donna x

28 \

29 \

30 [descend(donna,donna)]

31 / \

32 / \

33 / \

34 [child(donna,donna)] [child(donna,Z3),descend(Z3,donna)]

35 / / \ \ / / \ \

36 / / \ \ / / \ \

37 x x x x x x x \ Z3=emily

38 \

39 \

40 [descend(emily,donna)]

41 / \

42 / \

43 / \

44 [child(emily,donna)] [child(emily,Z4),descend(Z4,donna)]

45 / / \ \ / / \ \

46 / / \ \ / / \ \

47 x x x x x x x x

Note that there is only one successful branch, shown in line 27 with the empty
box.

###  5.3 Recursion, Chapter 3 from Learn Prolog Now!

Example 4 of Chapter 3 in Learn Prolog Now! is about the recursive predicate
add/3 (Blackburn et al., 2006, p.57), which we give here again in Listing 7.

⬇

1add(0,Y,Y).

2add(succ((X),Y,succ(Z)):- add(X,Y,Z).

Listing 7: Prolog definition for add/3.

The search tree given two pages later is incomplete (Blackburn et al., 2006,
p.59). Here is the complete search tree:

⬇

1[add(succ(succ(succ(0))),succ(succ(0)),R)]

2 / \

3 x \ R=succ(Z)

4 \

5 [add(succ(succ(0)),succ(succ(0)),Z)]

6 / \

7 x \ Z=succ(Z’)

8 \

9 [add(succ(0),succ(succ(0)),Z’)]

10 / \

11 x \ Z’=succ(Z’’)

12 \

13 [add(0,succ(succ(0)),Z’’)]

14 / \

15 Z’’=succ(succ(0)) / x

16 /

17 []

The query succeeds, and the variable R unifies with
succ(succ(succ(succ(succ(0))))).

###  5.4 More Lists, the predicate append/3

In Chapter 6 of Learn Prolog Now!, the idea behind the standard Prolog
predicate append/3 is explained (Blackburn et al., 2006, p.106). The recursive
definition provided contains two clauses (a fact and a rule):

⬇

1append([],L,L).

2append([H|T],L2,[H|L3]):- append(T,L2,L3).

Listing 8: Definition of append/3.

This is done with the help of a search tree for the query
append([a,b,c],[1,2,3],X) a couple of pages later. The search tree is
incomplete: it only shows the succesful branches (Blackburn et al., 2006,
p.108). The complete search tree is realised as follows:

⬇

1[append([a,b,c],[1,2,3],X)]

2 / \

3 / \ X = [a|L3]

4 / \

5 x [append([b,c],[1,2,3],L3)]

6 / \

7 / \ L3 = [b|L3’]

8 / \

9 x [append([c],[1,2,3],L3’)]

10 / \

11 / \ L3’ = [c|L3’’]

12 / \

13 x [append([],[1,2,3],L3’’)]

14 / \

15 L3’’ = [1,2,3] / \

16 / \

17 [] x

There is one closed branch, so Prolog will answer in the affirmative way and
unify the variable X with the list [a,b,c,1,2,3].

###  5.5 Search tree for the last query of Exercise 2.2 from Learn Prolog Now!

In Exercise 2.2 we are asked to draw the search tree for magic(Hermione). The
answer given is correct (Blackburn et al., 2006, p.239) but it consists of
unnecessary variable renaming. A simplified search tree is:

⬇

1 [magic(Hermione)]

2 / | \

3 / | \

4 Hermione=X / Hermione=X | \ Hermione=X

5 / | \

6 / | \

7 [house_elf(X)] [wizard(X)] [witch(X)]

8 / | / | \

9X=dobby / x / | \ X=rita_skeeter

10 / / | \

11 [] X=hermione / | []

12 / |

13 / | X=’McConagall’

14 [] |

15 []

###  5.6 Complete answers to Exercise 4.7 from Learn Prolog Now!

The answers to Exercise 4.7, drawing three search trees for the predicate
member/3, are all incomplete (Blackburn et al., 2006, p.244). Here are the
complete search trees:

⬇

1 [member(a,[c,b,a,y])]

2 / \

3 / \

4 / \

5 x [member(a,[b,a,y])]

6 / \

7 / \

8 / \

9 x [member(a,[a,y])]

10 / \

11 / \

12 / \

13 [] [member(a,[y])]

14 / \

15 / \

16 x [member(a,[])]

17 / \

18 / \

19 x x

⬇

1 [member(x,[a,b,c])]

2 / \

3 / \

4 / \

5 x [member(x,[b,c])]

6 / \

7 / \

8 / \

9 x [member(x,[c])]

10 / \

11 / \

12 / \

13 x [member(x,[])]

14 / \

15 / \

16 x x

⬇

1 [member(X,[a,b,c])]

2 / \

3 X=a / \X=X’

4 / \

5 [] [member(X’,[b,c])]

6 / \

7 X’= b / \X’=X’’

8 / \

9 [] [member(X’’,[c])]

10 / \

11 X’’=c / \X’’=X’’’

12 / \

13 [] [member(X’’’,[])]

14 / \

15 / \

16 x x

## References

  * Blackburn et al. (2006) Blackburn, P., J. Bos, and K. Striegnitz (2006).  Learn Prolog Now! College Publications.
  * Blackburn et al. (2007) Blackburn, P., J. Bos, and K. Striegnitz (2007).  Prolog, tout de suite ! College Publications.
  * Bratko (2001) Bratko, I. (2001).  Prolog Programming for Artificial Intelligence.  Addison-Wesley.
  * Clocksin and Mellish (1987) Clocksin, W. and C. Mellish (1987).  Programming in Prolog (Third, Revised and Extended Edition ed.).  Springer Verlag.


# A Precision Relation between $\Gamma(K\to\mu^{+}\mu^{-})(t)$ and ${\cal
B}(K_{L}\to\mu^{+}\mu^{-})/{\cal B}(K_{L}\to\gamma\gamma)$

Avital Dery<EMAIL_ADDRESS>Department of Physics, LEPP, Cornell
University, Ithaca, NY 14853, USA  Mitrajyoti Ghosh<EMAIL_ADDRESS>Department of Physics, LEPP, Cornell University, Ithaca, NY 14853, USA  Yuval
Grossman<EMAIL_ADDRESS>Department of Physics, LEPP, Cornell University,
Ithaca, NY 14853, USA  Teppei Kitahara<EMAIL_ADDRESS>Institute
for Advanced Research, Nagoya University, Nagoya 464–8601, Japan  Kobayashi-
Maskawa Institute for the Origin of Particles and the Universe, Nagoya
University, Nagoya 464–8602, Japan  KEK Theory Center, IPNS, KEK, Tsukuba
305–0801, Japan  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
Stefan Schacht<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Manchester, Manchester, M13 9PL, United Kingdom

###### Abstract

We find that the phase appearing in the unitarity relation between
$\mathcal{B}(K_{L}\rightarrow\mu^{+}\mu^{-})$ and
$\mathcal{B}(K_{L}\rightarrow\gamma\gamma)$ is equal to the phase shift in the
interference term of the time-dependent $K\rightarrow\mu^{+}\mu^{-}$ decay. A
probe of this relation at future kaon facilities constitutes a Standard Model
test with a theory precision of about $2\%$. The phase has further importance
for sensitivity studies regarding the measurement of the time-dependent
$K\rightarrow\mu^{+}\mu^{-}$ decay rate to extract the CKM matrix element
combination $|V_{ts}V_{td}\sin(\beta+\beta_{s})|\approx
A^{2}\lambda^{5}\bar{\eta}$. We find a model-independent theoretically clean
prediction, $\cos^{2}\varphi_{0}=0.96\pm 0.03$. The quoted error is a
combination of the theoretical and experimental errors, and both of them are
expected to shrink in the future. Using input from the large-$N_{C}$ limit
within chiral perturbation theory, we find a theory preference towards
solutions with negative $\cos\varphi_{0}$, reducing a four-fold ambiguity in
the angle $\varphi_{0}$ to a two-fold one.

††preprint: KEK-TH-2469

##  I Introduction

A recent proposal has shown that short-distance parameters of the decay
$K\to\mu^{+}\mu^{-}$ can be cleanly extracted from a measurement of the
$K_{L}\text{--}K_{S}$ time-dependent rate DAmbrosio:2017klp ; Dery:2021mct ;
Brod:2022khx . The time-dependent rate for a beam of initial $K^{0}$ particles
can be written as

$\displaystyle\frac{1}{{\cal
N}}\frac{d\Gamma(K^{0}\to\mu^{+}\mu^{-})}{dt}\,=f(t)\equiv\,C_{L}\,e^{-\Gamma_{L}t}+C_{S}\,e^{-\Gamma_{S}t}+2\,C_{\text{Int.}}\cos(\Delta
M_{K}t-\varphi_{0})e^{-\frac{\Gamma_{L}+\Gamma_{S}}{2}t}\,,$ (1)

where ${\cal N}$ is a normalization factor, $\Gamma_{L}$ ($\Gamma_{S}$) is the
$K_{L}$ ($K_{S}$) decay width, and $\Delta M_{K}$ is the $K_{L}\text{--}K_{S}$
mass difference. Then, the four experimental parameters characterizing the
time dependence,

$\left\\{C_{L},\,C_{S},\,C_{\text{Int}},\,\varphi_{0}\right\\}\,,$ (2)

are directly related to the four theory parameters describing the system
Dery:2021mct ,

$\left\\{|A(K_{S})_{\ell=0}|,\,\,|A(K_{L})_{\ell=0}|,\,\,|A(K_{S})_{\ell=1}|,\,\,\arg\big{[}A(K_{S})_{\ell=0}^{*}\,A(K_{L})_{\ell=0}\big{]}\right\\}\,,$
(3)

where the subscripts $\ell=0$ ($s$-wave symmetric wave function) and $\ell=1$
($p$-wave anti-symmetric wave function) correspond to the CP-odd and -even
$(\mu^{+}\mu^{-})$ final states, respectively. The relations between the
experimental and theory parameters are given by

$\displaystyle\begin{aligned} C_{L}\,\,\,\,&=\,|A(K_{L})_{\ell=0}|^{2}\,,\\\
C_{S}\,\,\,\,&=\,|A(K_{S})_{\ell=0}|^{2}+\beta_{\mu}^{2}|A(K_{S})_{\ell=1}|^{2}\,,\\\
C_{\text{Int.}}\,&=\,|A(K_{S})_{\ell=0}^{*}\,A(K_{L})_{\ell=0}|=|A(K_{S})_{\ell=0}||A(K_{L})_{\ell=0}|\,,\\\
\varphi_{0}\,\,\,\,\,&=\,\arg\big{[}A(K_{S})_{\ell=0}^{*}\,A(K_{L})_{\ell=0}\big{]}\,,\end{aligned}$
(4)

with

$\displaystyle\beta_{\mu}=\sqrt{1-\frac{4m_{\mu}^{2}}{m_{K^{0}}^{2}}}\,.$ (5)

The experimental parameter $\varphi_{0}$, which is the phase shift of the
oscillating rate in Eq. (1), is a combination of the relative weak and strong
phases between the $K_{S}$ and $K_{L}$ amplitudes to the CP-odd final state.
In this paper, we demonstrate that this phase shift is closely related to the
proportionality coefficient in the ratio between the rates of
$K_{L}\to\mu^{+}\mu^{-}$ and $K_{L}\to\gamma\gamma$.

The ratio between the rates of $K_{L}\to\mu^{+}\mu^{-}$ and
$K_{L}\to\gamma\gamma$ is of historical significance. Using CPT invariance,
unitarity, and the well-motivated assumption that the absorptive part of the
$K_{L}\to\mu^{+}\mu^{-}$ amplitude is dominated by the two-photon intermediate
state, the ratio between the rates of $K_{L}\to\mu^{+}\mu^{-}$ and
$K_{L}\to\gamma\gamma$ is bounded by the lower limit Christ:1971hr ;
Stern:1973xs ; Sehgal:1966wr ; Sehgal:1969zok

$\displaystyle
R_{K_{L}}\equiv\frac{\Gamma(K_{L}\to\mu^{+}\mu^{-})}{\Gamma(K_{L}\to\gamma\gamma)}\geq
1.195\times 10^{-5}\,.\ $ (6)

However, back in the 1970’s, this conflicted with the contemporary
experimental upper bound of $0.4\times 10^{-5}$ Clark:1971kj ; Arnold:1968zza
, leading to the so-called “$K_{L}\to\mu^{+}\mu^{-}$ puzzle” Stern:1973xs ;
Chen:1971kfc  that gained much attention. But today, this ratio is measured in
the experiment as Workman:2022ynf

$\displaystyle R^{\mathrm{exp}}_{K_{L}}=(1.250\pm 0.022)\times 10^{-5}\,.$ (7)

This means that the observed branching ratio ${\cal
B}(K_{L}\to\mu^{+}\mu^{-})=(6.84\pm 0.11)\times 10^{-9}$ Workman:2022ynf ,
along with ${\cal B}(K_{L}\to\gamma\gamma)=(5.47\pm 0.04)\times 10^{-4}$
Workman:2022ynf  known today obey the lower limit prescribed by CPT invariance
and unitarity.

In this paper, we show that the phase $\varphi_{0}$ is cleanly predicted in
the Standard Model (SM), up to a four-fold discrete ambiguity, making its
measurement a potent test of the SM. The discrete ambiguity can be partially
resolved by using further theory input from the literature in the
large-$N_{C}$ limit of chiral perturbation theory (ChPT). This result is
additionally significant for sensitivity estimations of a future measurement
of the short-distance parameters.

Leptonic kaon decays have been a field that received a lot of attention in the
literature recently. Effects from CPV in kaon mixing on
$K\rightarrow\mu^{+}\mu^{-}$ have been taken into account in Refs.
DAmbrosio:2017klp ; Brod:2022khx , and implications for physics beyond the SM
have been studied in Refs. Chobanova:2017rkj ; Endo:2017ums ; Dery:2021vql .
Another future high precision test of the SM employing the ratio
$\mathcal{B}(K_{S}\rightarrow\mu^{+}\mu^{-})_{\ell=0}/\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})$
has been identified in Ref. Buras:2021nns . Advances in calculating
$K_{L}\rightarrow\mu^{+}\mu^{-}$ and $K_{L}\rightarrow\gamma\gamma$ on the
lattice can be found in Refs. Christ:2020bzb ; Zhao:2022pbs ; Christ:2022rho .
On the experimental side, the LHCb collaboration recently found an improved
bound on $K_{S}\rightarrow\mu^{+}\mu^{-}$ LHCb:2020ycd  and
$K_{S,L}\rightarrow 2(\mu^{+}\mu^{-})$ Gomez:2022 .

In Sec. II we introduce our notation and summarize key results from the
literature. In Sec. III we determine $\cos^{2}\varphi_{0}$, which predicts
$\varphi_{0}$ up to a four-fold ambiguity, in a model-independent way, only
assuming that the long-distance contributions are SM-like. In Sec. IV we
reduce this ambiguity to a two-fold one by using the large-$N_{C}$ limit and
assuming that the short-distance physics is known. We demonstrate that the
remaining ambiguity cannot be resolved using current knowledge in Sec. V. We
conclude in Sec. VI.

##  II Setup and notation

Figure 1: Leading order Feynman diagram of the long-distance two-photon
contribution to $K_{L}\rightarrow\mu^{+}\mu^{-}$.

We work within a framework defined by the following approximations, as
detailed in Ref. Dery:2021mct :

  1. (i)

We neglect CPV in mixing, which is a sub-dominant effect for our purposes.
Note that when considering higher order corrections, this effect can be taken
into account consistently Brod:2022khx .

  2. (ii)

We neglect CPV in the long-distance contribution.

  3. (iii)

We assume that the leptonic current is (axial-)vectorial, i.e., given by
$\bar{\mu}\gamma^{\mu}(a+b\gamma^{5})\mu$.

In the SM, all three approximations are fulfilled within the precision
relevant for our findings. In addition, in the following we assume that

  1. (iv)

The long-distance contribution to the $K_{L}\to\mu^{+}\mu^{-}$ amplitude is
SM-like. That is, the only non-negligible intermediate state is the di-photon
state.

Regarding the short-distance contribution, we explicitly state whenever our
results are relevant regardless of any assumption on the nature of the short-
distance physics, and when SM input is used.

Adopting this setup, it has been shown that the decay of $K_{S}$ to the CP-odd
final state, $(\mu^{+}\mu^{-})_{\ell=0}$ involves only short-distance physics.
Additionally, the $K_{L}\to\mu^{+}\mu^{-}$ decay proceeds only to the CP-odd
final state, $(\mu^{+}\mu^{-})_{\ell=0}$. However, the latter involves two
contributions of different underlying physics:

  1. 1.

A short-distance (SD) contribution, arising to leading order from box and
electroweak penguin diagrams, for which the ingredients for a precise SM
prediction are straight-forward to derive.

  2. 2.

A long-distance (LD) contribution, strongly dominated by the on-shell two
photon intermediate state, see Fig. 1.

The on-shell two photon contribution is known to be significantly larger in
magnitude than both the SD contribution and the off-shell part of the LD
contribution.

The $K^{0}\to(\mu^{+}\mu^{-})_{\ell=0}$ and
$\overline{K}^{0}\to(\mu^{+}\mu^{-})_{\ell=0}$ amplitudes can be written as a
sum of two general contributions, with corresponding weak and strong phases,

$\displaystyle A_{\ell=0}$ $\displaystyle=$
$\displaystyle|A_{SD}|e^{i\theta_{SD}}e^{i\delta_{SD}}+|A_{LD}|e^{i\theta_{LD}}e^{i\delta_{LD}}\,,$
(8) $\displaystyle\overline{A}_{\ell=0}$ $\displaystyle=$
$\displaystyle-\left(|A_{SD}|e^{-i\theta_{SD}}e^{i\delta_{SD}}+|A_{LD}|e^{-i\theta_{LD}}e^{i\delta_{LD}}\right)\,,$

where the overall minus sign for $\overline{A}_{\ell=0}$ is due to the CP
nature of the final state. Using the convention

$\displaystyle|{K_{S}}\rangle=p|{K^{0}}\rangle+q|{\overline{K}^{0}}\rangle\,,\qquad|{K_{L}}\rangle=p|{K^{0}}\rangle-q|{\overline{K}^{0}}\rangle\,,$
(9)

the mass eigenstate amplitudes are related to $A_{\ell=0}$ and
$\overline{A}_{\ell=0}$ by Dery:2021mct

$\displaystyle A(K_{S})_{\ell=0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[A_{SD}(1+\lambda_{SD})+A_{LD}(1+\lambda_{LD})\right]\,,$
(10) $\displaystyle A(K_{L})_{\ell=0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[A_{SD}(1-\lambda_{SD})+A_{LD}(1-\lambda_{LD})\right]\,,$

where

$\displaystyle A_{SD}$
$\displaystyle\equiv|A_{SD}|e^{i\theta_{SD}}e^{i\delta_{SD}}\,,$
$\displaystyle A_{LD}$
$\displaystyle\equiv|A_{LD}|e^{i\theta_{LD}}e^{i\delta_{LD}}\,,$ (11)
$\displaystyle\overline{A}_{SD}$
$\displaystyle\equiv-|A_{SD}|e^{-i\theta_{SD}}e^{i\delta_{SD}}\,,$
$\displaystyle\overline{A}_{LD}$
$\displaystyle\equiv-|A_{LD}|e^{-i\theta_{LD}}e^{i\delta_{LD}}\,,$ (12)

and

$\lambda_{SD}\equiv\frac{q}{p}\frac{\overline{A}_{SD}}{A_{SD}}\,,\qquad\lambda_{LD}\equiv\frac{q}{p}\frac{\overline{A}_{LD}}{A_{LD}}\,.$
(13)

As we focus here mainly on amplitudes with $\ell=0$ final states, compared to
the notation of Ref. Dery:2021mct  we use for brevity the notation

$\displaystyle\lambda_{SD}$ $\displaystyle\equiv\lambda_{0}^{SD}\,,$
$\displaystyle\lambda_{LD}$ $\displaystyle\equiv\lambda_{0}^{LD}\,,$ (14)
$\displaystyle A(K_{S})$ $\displaystyle\equiv A(K_{S})_{\ell=0}\,,$
$\displaystyle A(K_{L})$ $\displaystyle\equiv A(K_{L})_{\ell=0}\,.$ (15)

The amplitudes are normalized such that

$\displaystyle\mathcal{B}(K_{S,L}\rightarrow\mu^{+}\mu^{-})_{\ell=0}$
$\displaystyle=\frac{\tau_{K}\beta_{\mu}}{16\pi m_{K}}|A(K_{S,L})|^{2}\,.$
(16)

We also use

$\displaystyle\mathcal{B}(K_{L}\rightarrow\gamma\gamma)$
$\displaystyle=\frac{\tau_{K}}{32\pi
m_{K}}|A(K_{L}\rightarrow\gamma\gamma)|^{2}\,.$ (17)

Up to this point, the labels LD and SD are just naming. However, in the
following we will treat them as corresponding to what we think of as long-
distance and short-distance amplitudes. It is important to note that the
separation into LD and SD contributions is not well-defined. We think of
short-distance physics as having no sources for a strong phase, while long-
distance physics can go on-shell. However, any on-shell intermediate state can
also be considered to contribute off-shell. Therefore there is no way to
unambiguously define the separation. In the following we keep the strong
phases general, while we insert knowledge of the SM weak phase of the long-
distance contribution.

We then have Dery:2021mct

$\displaystyle\lambda_{LD}$
$\displaystyle=-\left(\frac{V_{cd}V_{cs}^{*}}{V_{cd}^{*}V_{cs}}\right)\left(\frac{V_{ud}^{*}V_{us}}{V_{ud}V_{us}^{*}}\right)=-e^{-2i\theta_{uc}}\,.$
(18)

Equation (18) corrects a typo in Eq. (42) of Ref. Dery:2021mct . Here,

$\displaystyle\theta_{uc}\equiv\mathrm{arg}\left(-\frac{V_{cd}V_{cs}^{*}}{V_{ud}V_{us}^{*}}\right)=\mathcal{O}(\lambda^{4})\,,$
(19)

such that, to $\mathcal{O}(\lambda^{4})$,

$\displaystyle\arg\lambda_{LD}=-2\theta_{LD}+\pi=\pi\,,$ (20)

that is

$\displaystyle\theta_{LD}$ $\displaystyle=0\,,\qquad\lambda_{LD}=-1\,.$ (21)

No assumptions are made for the short-distance phase,

$\displaystyle\arg\lambda_{SD}=-2\theta_{SD}+\pi\,.$ (22)

We have therefore

$\displaystyle A(K_{S})$
$\displaystyle=\frac{1}{\sqrt{2}}A_{SD}(1+\lambda_{SD})=i\sqrt{2}|A_{SD}|\sin\theta_{SD}e^{i\delta_{SD}}\,,$
(23) $\displaystyle A(K_{L})$
$\displaystyle=\frac{1}{\sqrt{2}}\left[A_{SD}(1-\lambda_{SD})+2A_{LD}\right]=\sqrt{2}\Big{(}|A_{LD}|e^{i\delta_{LD}}+|A_{SD}|\cos\theta_{SD}e^{i\delta_{SD}}\Big{)}\,.$
(24)

Note that $A(K_{S})$ is pure short-distance and is manifestly CP-odd. The
oscillation term in the rate is then controlled by the interference term:

$\displaystyle A(K_{S})^{*}A(K_{L})$ $\displaystyle=$
$\displaystyle-2i|A_{SD}|\sin\theta_{SD}\left(|A_{LD}|e^{i\Delta\delta}+|A_{SD}|\cos\theta_{SD}\right)\,,$
(25)

where $\Delta\delta\equiv\delta_{LD}-\delta_{SD}$.

##  III Determination of $\cos^{2}\varphi_{0}$: model-independent

From Eqs. (4) and (25), we have

$\displaystyle\cos^{2}\varphi_{0}\,=\,\frac{{\rm
Re}\big{[}A(K_{S})^{*}A(K_{L})\big{]}^{2}}{|A(K_{S})A(K_{L})|^{2}}=\frac{(\sqrt{2}|A_{LD}|\sin\Delta\delta)^{2}}{|A(K_{L})|^{2}}\
=\frac{\big{[}A(K_{L})_{\text{absorptive}}\big{]}^{2}}{|A(K_{L})|^{2}}\,,$
(26)

where

$A(K_{L})_{\text{absorptive}}\equiv{\rm Im}\big{[}A(K_{L})\big{]}\,,$ (27)

and we define the imaginary part relative to the strong phase of the SD
amplitude (or in the basis in which the SD amplitude carries no strong phase,
hence $\delta_{LD}=\Delta\delta$). This convention corresponds to choosing
$C_{\text{had.}}$, introduced in Sec. IV below, to be real without loss of
generality.

The numerator is simply the on-shell long-distance contribution to the $K_{L}$
amplitude. Under the assumption that the only non-negligible intermediate
state is the di-photon state Martin:1970ai , this absorptive part is equal to
the discontinuity of the three-point diagram (see Fig. 1),

$\displaystyle\left|\sqrt{2}|A_{LD}|\sin\Delta\delta\right|=\big{|}A(K_{L})_{\text{absorptive}}\big{|}=\left|\frac{1}{2i}{\rm
Disc}(K_{L}\to\gamma\gamma\to\mu^{+}\mu^{-})\right|\,.$ (28)

This discontinutiy can be computed in a straightforward way using Cutkosky
rules, which entail considering the cut intersecting both photon propagators.
In this way the left part of the cut diagram is directly related to the
measured rate of $K_{L}\to\gamma\gamma$, and the right
$\gamma\gamma\to\mu^{+}\mu^{-}$ part of the diagram requires QED only,
allowing to extract the magnitude of the discontinuity completely model
independently. This leads to

$\displaystyle\cos^{2}\varphi_{0}$
$\displaystyle=\frac{\big{[}A(K_{L})_{\text{absorptive}}\big{]}^{2}}{|A(K_{L})|^{2}}=C_{\mathrm{QED}}^{2}\frac{\Gamma(K_{L}\rightarrow\gamma\gamma)}{\Gamma(K_{L}\rightarrow\mu^{+}\mu^{-})}\,,$
(29)

_i.e._ ,

$\displaystyle\cos^{2}\varphi_{0}\,{\cal
B}(K_{L}\to\mu^{+}\mu^{-})\,=\,C_{\text{QED}}^{2}\,{\cal
B}(K_{L}\to\gamma\gamma)\,,$ (30)

where $C_{\mathrm{QED}}$ describes the $\gamma\gamma\rightarrow\mu^{+}\mu^{-}$
transition and is given as Martin:1970ai

$C_{\text{QED}}=\frac{\alpha_{em}m_{\mu}}{\sqrt{2\beta_{\mu}}m_{K}}\,\log\left(\frac{1-\beta_{\mu}}{1+\beta_{\mu}}\right)+{\cal
O}(\alpha_{em}^{2})\,.$ (31)

Equation (30) demonstrates that $\cos^{2}\varphi_{0}$ is simply the
proportionality factor parameterizing to what extent is the
$K_{L}\to\mu^{+}\mu^{-}$ rate saturated by the absorptive contribution from
the intermediate $\gamma\gamma$ state. Using the measured ratio of rates
$R_{K_{L}}$, see Eq. (6), we have a clean SM prediction for the phase
$\varphi_{0}$, up to a four-fold discrete ambiguity,

$\displaystyle\cos^{2}\varphi_{0}\,$ $\displaystyle=$
$\displaystyle\,\frac{C_{\mathrm{QED}}^{2}}{R_{K_{L}}^{\mathrm{exp}}}\,.$ (32)

This SM prediction is dependent only on

  1. 1.

The measurement of $R_{K_{L}}$, currently with an uncertainty of ${\cal
O}(2\%)$;

  2. 2.

A QED calculation, here taken up to relative corrections of ${\cal
O}(\alpha_{em})$;

  3. 3.

The assumption that other intermediate on-shell contributions ($3\pi$,
$\pi\pi\gamma$) are negligible Martin:1970ai .

Inserting

$\displaystyle C_{\rm QED}^{2}=1.195\times 10^{-5}\big{[}1+{\cal
O}(\alpha_{em})\big{]},$ (33)

and

$\displaystyle R_{K_{L}}^{\rm exp}=(1.250\pm 0.022)\times 10^{-5}\,,$ (34)

which we obtain, not taking into account any correlations, from
Workman:2022ynf

$\displaystyle\mathcal{B}(K_{L}\rightarrow\mu^{+}\mu^{-})$
$\displaystyle=(6.84\pm 0.11)\times 10^{-9}\,,$ (35)
$\displaystyle\mathcal{B}(K_{L}\rightarrow\gamma\gamma)$
$\displaystyle=(5.47\pm 0.04)\times 10^{-4}\,,$ (36)

using Gaussian error propagation, we arrive at $\cos^{2}\varphi_{0}=0.96\pm
0.02$ , where the quoted error reflects only the experimental error on
$R_{K_{L}}^{\rm exp}$.

There are two sources of theoretical errors. One of the them is the higher
order QED calculation, resulting in an error of order $\alpha_{em}\sim 1\%$.
This error is reducible, that is, if needed the calculation of the higher-
order corrections can be done. The other source of uncertainty are the
intermediate states that we neglected, such as $3\pi$ and $\pi\pi\gamma$
Martin:1970ai . These contributions are estimated to be at most $1\%$ of the
two-photon state that we considered. Each of the two effects results in about
$1\%$ error, and thus we conservatively add them linearly resulting in a total
theory error of $0.02$ to arrive at our final estimate,

$\cos^{2}\varphi_{0}=0.96\pm 0.02_{\mathrm{exp}}\pm 0.02_{\mathrm{th}}\,.$
(37)

Combining the experimental and theoretical errors in quadrature we then have

$\cos^{2}\varphi_{0}=0.96\pm 0.03.$ (38)

Note that the error quoted in Eq. (38) is therefore to be interpreted as an
estimate of the total uncertainty, rather than as a statistical error only.

Given the value of $\cos^{2}\varphi_{0}$ there are four possible values for
the phase shift, $\varphi_{0}$. Two of them correspond to overall constructive
interference in the time-dependent rate and two correspond to destructive
interference, depending on the sign of $\cos\varphi_{0}$. We plot the time-
dependent rate for the four possibilities in Fig. 2 for both a $K^{0}$ or a
$\overline{K}^{0}$ beam. Note that, for a $K^{0}$ beam, a positive (negative)
$\cos\varphi_{0}$ results in constructive (destructive) interference. As
pointed out in Ref. DAmbrosio:2017klp , the situation is reversed for a
$\overline{K}^{0}$ beam where a negative (positive) $\cos\varphi_{0}$ results
in constructive (destructive) interference.

Figure 2: (a) Illustration of the four-fold ambiguity arising from
$\cos^{2}\varphi_{0}=0.96$. (b) The time dependence of the
$K^{0}(t)\to\mu^{+}\mu^{-}$ rate, $f(t)$, as defined in Eq. (1), for the four
values of $\varphi_{0}$. We have used a normalization in which $C_{L}$ of Eq.
(1) is set to unity. (c) The same for an initial pure $\overline{K}^{0}$ beam.

##  IV Determination of $\cos\varphi_{0}$: model dependent

The four-fold ambiguity cannot be resolved in a model-independent way. Thus,
in this section we use input from theory in order to try to reduce it. We
first consider the sign of $\cos\varphi_{0}$. We discuss below how chiral
perturbation theory (ChPT) and lattice QCD can help in this regard.

###  IV.1 Chiral perturbation theory

Within chiral perturbation theory and using the large-$N_{C}$ limit, it has
been shown that the sign of the absorptive amplitude relative to the short-
distance contribution, which determines the sign of $\cos\varphi_{0}$, can be
determined GomezDumm:1998gw ; Isidori:2003ts .

In the following, we relate our notation for the long-distance contribution to
that appearing in the literature in order to apply existing results, primarily
of Refs. Isidori:2003ts ; GomezDumm:1998gw ; Knecht:1999gb ; DAmbrosio:1997eof
. We rewrite the long-distance amplitude for $K_{L}\to\mu^{+}\mu^{-}$ as

$\displaystyle\sqrt{2}|A_{LD}|e^{i\Delta\delta}\,$
$\displaystyle=\,\big{[}A_{LD}\big{]}_{\rm
dispersive}+i\big{[}A_{LD}\big{]}_{\rm absorptive}\,$
$\displaystyle\equiv\,C_{\rm had.}\Big{[}\big{(}A_{LD}^{\rm local}+{\rm
Re}A_{LD}^{\gamma\gamma}\big{)}+i\,{\rm Im}A_{LD}^{\gamma\gamma}\Big{]}\,,$
(39)

where $C_{\rm had.}$ encodes the hadronic behavior of the effective
$K_{L}\gamma\gamma$ vertex. Without loss of generality, we take $C_{\rm had.}$
to be real, consistent with Eq. (27). The dispersive part is split in two
parts to be consistent with the literature,

$\displaystyle\big{[}A_{LD}\big{]}_{\rm dispersive}$ $\displaystyle=C_{\rm
had.}\left(A_{LD}^{\rm local}+{\rm Re}A_{LD}^{\gamma\gamma}\right)\,,$ (40)
$\displaystyle\big{[}A_{LD}\big{]}_{\rm absorptive}$ $\displaystyle=C_{\rm
had.}\,{\rm Im}A_{LD}^{\gamma\gamma}\,,$ (41)

where $A_{LD}^{\rm local}$ denotes the local counterterm, which is real
Isidori:2003ts . Using the fact that the decay $K_{L}\to\gamma\gamma$ has the
same hadronic behavior, we define

$A(K_{L}\to\gamma\gamma)\equiv C_{\rm had.}\,A_{\gamma\gamma}\,.$ (42)

Note that $A_{\gamma\gamma}\neq A^{\gamma\gamma}_{LD}$. The former is part of
the $K_{L}\rightarrow\gamma\gamma$ amplitude, while the latter is the two-
photon contribution to $K_{L}\rightarrow\mu^{+}\mu^{-}$. Then we have

$\displaystyle\frac{\Gamma(K_{L}\to\mu^{+}\mu^{-})}{\Gamma(K_{L}\to\gamma\gamma)}\,$
$\displaystyle=$
$\displaystyle\,2\beta_{\mu}\left|\frac{\sqrt{2}|A_{LD}|e^{i\Delta\delta}+\sqrt{2}|A_{SD}|\cos\theta_{SD}}{A(K_{L}\to\gamma\gamma)}\right|^{2}$
$\displaystyle=$ $\displaystyle 2\beta_{\mu}\frac{\left(A_{LD}^{\rm
local}+{\rm Re}A_{LD}^{\gamma\gamma}+\sqrt{2}|A_{SD}|\cos\theta_{SD}/C_{\rm
had.}\right)^{2}+({\rm
Im}A_{LD}^{\gamma\gamma})^{2}}{|A_{\gamma\gamma}|^{2}}\,.$

Note that $C_{\rm had.}$ now appears as a factor accompanying the short-
distance contribution. We can now easily relate to the notations of Ref.
Isidori:2003ts , with (the superscript “IU” denotes the initials of the
authors of Ref. Isidori:2003ts )

$\displaystyle\frac{A_{LD}^{\rm local}}{|A_{\gamma\gamma}|}$
$\displaystyle=\frac{\alpha_{em}m_{\mu}}{\pi
m_{K}}\big{[}\chi_{\gamma\gamma}(\mu)\big{]}^{\rm IU}\,,$ (44)
$\displaystyle\frac{{\rm Re}A_{LD}^{\gamma\gamma}}{|A_{\gamma\gamma}|}$
$\displaystyle=\frac{\alpha_{em}m_{\mu}}{\pi m_{K}}\left[{\rm
Re}C_{\gamma\gamma}-\frac{5}{2}+\frac{3}{2}\log\left(\frac{m_{\mu}^{2}}{\mu^{2}}\right)\right]^{\rm
IU}\,,$ (45) $\displaystyle\frac{\sqrt{2}|A_{SD}|\cos\theta_{SD}}{C_{\rm
had.}|A_{\gamma\gamma}|}\,$ $\displaystyle=\,\frac{\alpha_{em}m_{\mu}}{\pi
m_{K}}\big{[}\chi_{\rm short}\big{]}^{\rm IU}\,,$ (46)
$\displaystyle\frac{{\rm Im}A_{LD}^{\gamma\gamma}}{|A_{\gamma\gamma}|}$
$\displaystyle=\frac{\alpha_{em}m_{\mu}}{\pi m_{K}}\big{[}{\rm
Im}C_{\gamma\gamma}\big{]}^{\rm IU}=\frac{1}{\sqrt{2\beta_{\mu}}}C_{\rm
QED}\,.$ (47)

The assumptions and findings of the literature, as conveyed in Refs.
Isidori:2003ts ; GomezDumm:1998gw , can now be summarized as the following:

  1. 1.

Using results in the large-$N_{C}$ limit, Refs. Isidori:2003ts ;
GomezDumm:1998gw  find destructive interference between the short-distance and
the local long-distance contributions,

$\frac{\cos\theta_{SD}}{C_{\rm had.}A_{LD}^{\rm local}}\,<\,0\,.$ (48)

  2. 2.

Ref. Isidori:2003ts  uses phenomenological analyses of the form factor in
$K_{L}\to\gamma e^{+}e^{-}$, $K_{L}\to\gamma\mu^{+}\mu^{-}$ and $K_{L}\to
e^{+}e^{-}\mu^{+}\mu^{-}$ from data, together with theory considerations, to
estimate the local counter term. Using up-to-date inputs, we update their
estimation (see App. A) and find

$\big{[}\chi_{\gamma\gamma}(m_{\rho})\big{]}^{\rm IU}\,=\,{\frac{\pi
m_{K}}{\alpha_{em}m_{\mu}}}\frac{A_{LD}^{\rm
local}}{|A_{\gamma\gamma}|}=(6.10\pm 1.01)\,>\,0\,,$ (49)

where we have set $\mu^{2}=m_{\rho}^{2}$ here and in the following whenever we
make use of specific numerical estimates. Hence, using Eq. (48),

${\rm sgn}\big{[}C_{\rm had.}\big{]}\,=\,-{\rm
sgn}\big{[}\cos\theta_{SD}\big{]}\,.$ (50)

  3. 3.

This, in turn, determines the sign of the absorptive long-distance amplitude
relative to the short-distance contribution,

$\displaystyle{\rm sgn}\left(\big{[}A_{LD}\big{]}_{\rm absorptive}\right)\,$
$\displaystyle=$ $\displaystyle\,{\rm sgn}\big{[}\cos\theta_{SD}\big{]}\,,$
(51)

where we used the fact that ${\rm sgn}\left[{\rm
Im}A_{LD}^{\gamma\gamma}\right]={\rm sgn}\left[C_{\rm QED}\right]=-1$.

We conclude that within a model for the short-distance contribution, and
adopting the assumptions in the literature regarding the long-distance
physics, i.e., the large-$N_{C}$ limit, the sign of $\cos\varphi_{0}$ is
determined,

$\displaystyle{\rm sgn}\Big{[}\cos\varphi_{0}\Big{]}\,$ $\displaystyle=$
$\displaystyle\,{\rm sgn}\left[\frac{{\rm
Re}\big{[}A(K_{S})^{*}A(K_{L})]}{|A(K_{S})A(K_{L})|}\right]$ $\displaystyle=$
$\displaystyle\,{\rm sgn}\left[\frac{\big{[}A(K_{L})\big{]}_{\rm
absorptive}\,{\rm sgn}[\sin\theta_{SD}]}{|A(K_{L})|}\right]$ $\displaystyle=$
$\displaystyle\,{\rm sgn}\Big{[}\tan\theta_{SD}\Big{]}\,.$

###  IV.2 Detailed assumptions within ChPT

The considerations leading to the assumption of destructive interference
between the short-distance and the local long-distance contributions, as
conveyed in Eq. (48), involve some details of the structure of the
$K_{L}\to\gamma\gamma$ amplitude within the ChPT. According to Ref.
Cirigliano:2011ny , the on-shell tensor amplitude for $K_{L}\to\gamma\gamma$
starts from at $\mathcal{O}(p^{6})$ in the ChPT as

$\displaystyle-
iA(K_{L}\to\gamma\gamma)=\varepsilon^{\mu\nu\rho\sigma}\epsilon_{1\mu}(q_{1})\epsilon_{2\nu}(q_{2})q_{1\rho}q_{2\sigma}\,c^{(6)}(0,0)\,,$
(53)

where $c^{(6)}(q_{1}^{2},q_{2}^{2})$ is an amplitude for
$K_{L}\to\gamma^{\ast}(q_{1})\gamma^{\ast}(q_{2})$. We factorize the $\pi^{0}$
exchange contribution in $c^{(6)}(0,0)$ and define a dimensionless reduced
amplitude as $c_{\rm red}^{(6)}$,

$\displaystyle
c^{(6)}(0,0)\,=\,-\frac{2}{\pi}\alpha_{em}F_{0}\left(G_{8}-G_{27}\right)\frac{1}{1-r_{\pi}^{2}}c_{\rm
red}^{(6)}\,,$ (54)

and $c_{\rm red}^{(6)}$ can be expanded in the ChPT as

$\displaystyle\begin{aligned} c_{\rm
red}^{(6)}\,=\,&1+\frac{1-r_{\pi}^{2}}{3\left(1-r_{\eta}^{2}\right)}\left[\left(1+\xi\right)c_{\theta}+2\sqrt{2}\,\hat{\rho}\,s_{\theta}\right]\left(\frac{F_{\pi}}{F_{\eta_{8}}}c_{\theta}-2\sqrt{2}\frac{F_{\pi}}{F_{\eta_{1}}}s_{\theta}\right)\\\
&-\frac{1-r_{\pi}^{2}}{3\left(1-r_{\eta^{\prime}}^{2}\right)}\left[2\sqrt{2}\,\hat{\rho}\,c_{\theta}-\left(1+\xi\right)s_{\theta}\right]\left(\frac{F_{\pi}}{F_{\eta_{8}}}s_{\theta}+2\sqrt{2}\frac{F_{\pi}}{F_{\eta_{1}}}c_{\theta}\right)\,,\end{aligned}$
(55)

with $\sqrt{2}F_{0}=\sqrt{2}F_{\pi}=f_{\pi}=(130.2\pm 0.8)$ MeV
FlavourLatticeAveragingGroupFLAG:2021npn , $F_{\eta_{8}}=(1.27\pm
0.02)\,F_{\pi}$, $F_{\eta_{1}}=(1.14\pm 0.05)\,F_{\pi}$ Escribano:2015yup ;
Gan:2020aco , and $r_{P}\equiv m_{P}/m_{K}$. Combining the above formulae
gives

$\displaystyle-
iA(K_{L}\to\gamma\gamma)=-\frac{\sqrt{2}\alpha_{em}f_{\pi}\left(G_{8}-G_{27}\right)}{\pi\left(1-r_{\pi}^{2}\right)}c_{\rm
red}^{(6)}\varepsilon^{\mu\nu\rho\sigma}\epsilon_{1\mu}(q_{1})\epsilon_{2\nu}(q_{2})q_{1\rho}q_{2\sigma}\,.$
(56)

Here, the point is that $\mathcal{O}(p^{4})$ contributions vanish within the
ChPT, which are proportional to $c^{(4)}_{\rm red}$ and

$\displaystyle c_{\rm
red}^{(4)}\,=\,1+\frac{1-r_{\pi}^{2}}{3\left(1-r_{\eta_{8}}^{2}\right)}\,=\,\frac{4-3r_{\eta_{8}}^{2}-r_{\pi}^{2}}{3\left(1-r_{\eta_{8}}^{2}\right)}\,=\,0\,,$
(57)

where the Gell-Mann–Okubo mass formula,
$4m_{K}^{2}=3m_{\eta_{8}}^{2}+m_{\pi}^{2}$, is used. Therefore, $c_{\rm
red}^{(6)}$ amplitudes correspond to the violation of the Gell-Mann–Okubo
formula implying that the sign of $c_{\rm red}^{(6)}$ is sensitive to the
$\eta$–$\eta^{\prime}$ mixing angle $\theta$ in the octet–singlet basis, the
$SU(3)_{F}$ breaking $\xi$ Donoghue:1986ti ; He:2002as , the nonet symmetry
breaking $\hat{\rho}$ Donoghue:1983hi ; Cheng:1990mw ; DAmbrosio:1997hlp , and
their higher-order corrections. $(1+\xi)$ is proportional to the
$K_{L}\to\eta_{8}$ form factor, while $\hat{\rho}$ is proportional to the
$K_{L}\to\eta_{1}$ one. However, by considering the typical parameter regions;
$\theta\approx-20^{\circ}$, $\xi\sim 0.0\text{--}0.2$, and $\hat{\rho}\approx
0.8$ Cirigliano:2011ny , one can predict $\text{sgn}[c_{\rm red}^{(6)}]>0$,
which leads to

$\displaystyle{\rm sgn}\Big{[}A(K_{L}\to\gamma\gamma)\Big{]}={\rm
sgn}\Big{[}A(K_{L}\to\pi^{0}\to\gamma\gamma)\Big{]}\,.$ (58)

Combining this relation with $\text{sgn}[G_{8}-G_{27}]$ which can be extracted
from the $\Delta S=1$ effective Lagrangian in the large-$N_{C}$ limit
Pich:1995qp ; Isidori:2003ts ; Gerard:2005yk , $\text{sgn}[C_{\rm had.}]>0$
can be predicted, see Eq. (50), where, in the SM we have
$\cos\theta_{SD}^{\mathrm{SM}}<0$.

###  IV.3 Lattice QCD

In the last decade, lattice QCD made paramount progress in the treatment of
$K\rightarrow\pi\pi$ RBC:2020kdj ; RBC:2015gro ; Blum:2015ywa ; Blum:2011ng .
Moreover, recently, lattice QCD made advances in the calculation of
$K_{L}\rightarrow\mu^{+}\mu^{-}$ and $K_{L}\rightarrow\gamma\gamma$
Christ:2020bzb ; Zhao:2022pbs ; Christ:2022rho . ChPT parameters like $G_{8}$
can now also be extracted from fits to lattice results, as shown in Ref.
Pich:2021yll .

While it seems to us that the data is available to extract the sign, we were
unable to find it from the available publications. It would be interesting to
use the available lattice data to obtain it. Such an extraction would be
interesting to confront the ChPT results.

###  IV.4 SM prediction for the short-distance physics

Within the SM, the short distance contribution arises from the following
effective Hamiltonian Buchalla:1995vs

$\displaystyle{\cal H}_{\rm
eff}=-\frac{G_{F}}{\sqrt{2}}\frac{\alpha_{em}}{2\pi\sin^{2}\theta_{W}}\left[V_{ts}^{*}V_{td}Y(x_{t})+V_{cs}^{*}V_{cd}Y_{NL}\right]\left[(\bar{s}d)_{V-A}(\bar{\mu}\mu)_{V-A}\right]+\text{h.c.}\,.$
(59)

We can then write (in the basis where $\delta_{SD}=0$)

$\displaystyle A(K_{L})_{SD}^{\rm SM}\,$ $\displaystyle=$
$\displaystyle\,\big{(}\sqrt{2}|A_{SD}|\cos\theta_{SD}\big{)}^{\text{SM}}$
(60) $\displaystyle=$
$\displaystyle\,\frac{\sqrt{2}G_{F}\alpha_{em}}{\pi\sin^{2}\theta_{W}}\big{|}V_{ts}^{*}V_{td}Y(x_{t})+V_{cs}^{*}V_{cd}Y_{NL}\big{|}f_{K}m_{\mu}m_{K}\,\cos\theta_{SD}^{\rm
SM}\,,$

where we identify

$\theta_{SD}^{\text{SM}}\,=\,\arg\left(-\frac{V_{ts}^{*}V_{td}+V_{cs}^{*}V_{cd}Y_{NL}/Y(x_{t})}{V_{cs}^{*}V_{cd}}\right)\,.$
(61)

We therefore find,

$\displaystyle\tan\theta_{SD}^{\rm SM}\,$ $\displaystyle=$
$\displaystyle\,-\frac{\eta}{(1-\rho)+\frac{1}{A^{2}\lambda^{4}}\frac{Y_{NL}}{Y(x_{t})}}+{\cal
O}(\lambda^{6})\,<\,0\,.$ (62)

Hence, from Eq. (IV.1), within the SM and under the aforementioned model-
dependent assumptions, we have

$\displaystyle\big{[}\cos\varphi_{0}\big{]}^{\rm
SM}_{\text{large}\,N_{C}}\,<\,0\,.$ (63)

Together with the result of section III, $\cos^{2}\varphi_{0}=0.96\pm 0.03$,
we have

$\displaystyle\big{[}\cos\varphi_{0}\big{]}^{\rm
SM}_{\text{large}\,N_{C}}\,=\,-0.98\pm 0.02\,.$ (64)

Note that the error combines both a statistical error from experiment as well
as a theory component, _i.e._ is to be interpreted as an estimate of the total
uncertainty.

##  V Going beyond the two-fold ambiguity

In order to determine the sign of $\sin\varphi_{0}$ and get rid of the
remaining ambiguity, we would need to determine the signs and magnitudes of
the competing short-distance and long-distance dispersive contributions,

$\displaystyle\sin\varphi_{0}$ $\displaystyle=\frac{{\rm
Im}\big{[}A(K_{S})^{*}A(K_{L})\big{]}}{|A(K_{S})A(K_{L})|}$
$\displaystyle=-\frac{\left(\big{[}A_{LD}\big{]}_{\rm
dispersive}+\sqrt{2}|A_{SD}|\cos\theta_{SD}]\right)\mathrm{sgn}[\sin(\theta_{SD})]}{|A(K_{L})|}\,.$
(65)

We recall that $\big{[}A_{LD}\big{]}_{\rm dispersive}$ can be written as

$\displaystyle\big{[}A_{LD}\big{]}_{\rm dispersive}$
$\displaystyle=\sqrt{2}|A_{LD}|\cos\Delta\delta=C_{\rm had.}\left(A_{LD}^{\rm
local}+{\rm Re}A_{LD}^{\gamma\gamma}\right)\,,$ (66)

see Eq. (IV.1). Existing semi-phenomenological theory estimations of
$\big{[}A_{LD}\big{]}_{\rm dispersive}$ come with large theory uncertainties.
Using the estimate of Ref. Isidori:2003ts  as in Eq. (49), updated with
existing data (see App. A), we have for the long-distance contribution,

$\displaystyle\frac{\big{[}A_{LD}\big{]}_{\rm
dispersive}}{|A(K_{L}\to\mu^{+}\mu^{-})|}\,$ $\displaystyle=$
$\displaystyle\,\frac{C_{\rm
had.}}{|A(K_{L}\to\mu^{+}\mu^{-})|}\left(A_{LD}^{\rm local}+{\rm
Re}A_{LD}^{\gamma\gamma}\right)$ $\displaystyle\,=\frac{C_{\rm
had.}|A_{\gamma\gamma}|}{|A(K_{L}\to\mu^{+}\mu^{-})|}\frac{\alpha_{em}m_{\mu}}{\pi
m_{K}}\left(\big{[}\chi_{\gamma\gamma}(m_{\rho})\big{]}^{\rm IU}+\left[{\rm
Re}C_{\gamma\gamma}-\frac{5}{2}+\frac{3}{2}\log\left(\frac{m_{\mu}^{2}}{m_{\rho}^{2}}\right)\right]^{\rm
IU}\right)$ $\displaystyle=\,\text{sgn}\left[C_{\rm
had.}\right]\sqrt{\frac{2\beta_{\mu}}{R_{K_{L}}}}\frac{\alpha_{em}m_{\mu}}{\pi
m_{K}}\left[(6.10\pm 1.01)-5.14\right]\in\,\big{[}-0.009,\,0.37\big{]}\,,$

where $\text{sgn}[C_{\rm had.}]>0$ derived in the previous section is used.

For the short-distance SM contribution, we have (see Eq. (60))

$\displaystyle\frac{A(K_{L})_{SD}^{\rm SM}}{|A(K_{L}\to\mu^{+}\mu^{-})|}\,$
$\displaystyle=$
$\displaystyle\,\frac{\sqrt{2}G_{F}\alpha_{em}(m_{Z})}{\pi\sin^{2}\theta_{W}}\frac{\big{|}V_{ts}^{*}V_{td}Y(x_{t})+V_{cs}^{*}V_{cd}Y_{NL}\big{|}f_{K}m_{\mu}m_{K}\,\cos\theta_{SD}^{\rm
SM}}{|A(K_{L}\to\mu^{+}\mu^{-})|}$ $\displaystyle=$ $\displaystyle\,-0.331\pm
0.008\,,$

where we use the following inputs,

$\displaystyle Y(x_{t})=0.931\pm 0.005~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Brod:2022khx}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,\qquad\qquad
Y_{\rm NL}=(2.95\pm 0.46)\times 10^{-4}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Gorbahn:2006bm}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
$\displaystyle
A=0.790^{+0.017}_{-0.012}\,,\qquad\qquad\qquad\quad\qquad\lambda=0.22650\pm
0.00048\,,$
$\displaystyle\bar{\rho}=0.141^{+0.016}_{-0.017}\,,\qquad\qquad\qquad\quad\,\,\,\,\qquad\bar{\eta}=0.357\pm
0.011\,,$ $\displaystyle m_{K}=497.61\,{\rm
MeV}\,,\qquad\qquad\qquad\,\,\,\quad m_{\mu}=105.658\,{\rm MeV}\,,$
$\displaystyle G_{F}=1.166378\times 10^{-5}\,{\rm
GeV}^{-2}\,,\quad\quad\,\,\,f_{K}=155.7\,{\rm MeV}\,,$
$\displaystyle\alpha_{em}=1/137\,,\qquad\qquad\qquad\qquad\qquad\alpha_{em}(m_{Z})=1/129\,,$
$\displaystyle\sin^{2}\theta_{W}=0.23\,,\qquad\qquad\qquad\,\,\quad\qquad
m_{\rho}=775.26\,\text{MeV}\,.$

In particular, the CKM input results in $\cos\theta_{SD}^{\mathrm{SM}}=-0.94$.

Hence, the large theory uncertainty on the dispersive long-distance
contribution, as reflected in the range given in Eq. (V), does not allow to
determine if it is larger or smaller than the short-distance SM contribution,
Eq. (V). We conclude that with current knowledge on the dispersive long-
distance contribution the sign of $\sin\varphi_{0}$ cannot be determined.

Therefore, we have, using Gaussian error propagation,

$\displaystyle\big{[}\cos\varphi_{0}\big{]}^{\rm
SM}_{\text{large}\,N_{C}}\,=\,-0.98\pm
0.02\,,\qquad\big{[}\sin\varphi_{0}\big{]}^{\rm
SM}_{\text{large}\,N_{C},\,\text{theory}}\,=\,\pm\big{(}0.21\pm
0.07\big{)}\,.$ (70)

We note that due to the nature of the theoretical error, the error should not
be interpreted as a statistical error but rather as an estimate of the
uncertainty.

##  VI Discussion and Conclusions

In this work we have related the phase shift, $\varphi_{0}$, appearing in the
time-dependent decay rate of a neutral kaon to a dimuon pair to the ratio of
integrated rates of $K_{L}\to\mu^{+}\mu^{-}$ and $K_{L}\to\gamma\gamma$. This
relation holds to an excellent approximation under the well-motivated
assumption that the two-photon intermediate state dominates the absorptive
contribution, and within any model in which the short-distance leptonic
current is axial or vectorial, as in the SM. The only input required other
than the ratio of integrated rates is a coefficient calculated within QED. We
find that $\cos^{2}\varphi_{0}$ is precisely predicted model independently,
and given by

$\cos^{2}\varphi_{0}\,=\,0.96\pm 0.02_{\mathrm{exp}}\pm 0.02_{\mathrm{th}}\,.$
(71)

The experimental error comes from the error of $R_{K_{L}}$ and the theory
error is our estimate of the size of higher order QED corrections and the
contribution from other intermediate states beside the di-photon state. The
result leaves a four-fold ambiguity in $\varphi_{0}$.

The phase shift, $\varphi_{0}$, is also of experimental significance since it
controls the integrated number of interference events. For a $K^{0}$ beam a
positive value of $\cos\varphi_{0}$ is preferred as it enhances the
interference and improves the feasibility of extracting clean short-distance
information from the interference term. For a $\overline{K}^{0}$ beam the
situation is reversed, and a negative value of $\cos\varphi_{0}$ is preferred.
Thus, for experiments employing a proton beam on target, where the number of
$K^{0}$ particles is expected to exceed that of $\overline{K}^{0}$ particles,
a positive $\cos\varphi_{0}$ would be preferred.

We were unable to determine the sign of $\cos\varphi_{0}$ completely model
independently. Thus, with the use of several assumptions, that is, within the
framework of ChPT and using a typical parameter region (motivated by the
large-$N_{C}$ limit, as detailed in Sec. IV.2), the sign of $\cos\varphi_{0}$,
relative to the short-distance contribution, can be predicted. We find that
within this framework, and assuming that the short-distance contribution is
SM-like, there is a theory preference towards $\cos\varphi_{0}<0$. New Physics
can potentially yield a different sign for $\cos\varphi_{0}$. A measurement of
the angle $\varphi_{0}$ is therefore a test of the validity of several
assumptions, pertaining to both the short-distance physics and the ChPT
description of the dispersive long-distance physics.

Given the assumptions that were made to arrive to the conclusion about the
sign of $\cos\varphi_{0}$, and the fact that the prediction can be modified
for models beyond the SM, we conclude that neither solution is unequivocally
theoretically favored. That is, we cannot conclude that we know the sign of
$\cos\varphi_{0}$ to high confidence. Thus, when planing to perform the
experiment we encourage the experimental collaborations to consider both
possible signs for $\cos\varphi_{0}$ for purposes of sensitivity estimations.

We therefore conclude by emphasizing that the time dependence of the kaon
decay rate to two muons provides two very clean SM predictions:

  1. 1.

The coefficient of the interference term allows the extraction of the
theoretically clean decay rate ${\cal B}(K_{S}\to\mu^{+}\mu^{-})_{\ell=0}$. In
the SM, this observable is proportional to the CKM combination
$|V_{ts}V_{td}\sin(\beta+\beta_{s})|$.

  2. 2.

Although it includes long-distance as well as short-distance physics, the
phase shift in the interference term, $\varphi_{0}$, is predicted cleanly up
to a four-fold ambiguity.

Thus, an experiment that performs the time-dependence studies of the
$K\to\mu^{+}\mu^{-}$ decay rate provides two independent tests of the SM from
the same measurement. While the phase $\varphi_{0}$ is not determined model
independently, it directly impacts the measurement of the $K\to\mu^{+}\mu^{-}$
time dependent rate as it affects the interference between the $K_{S}$ and
$K_{L}$ amplitudes in the total rate. The phase shift $\varphi_{0}$ is
therefore a quantity of critical importance in kaon physics, as a way to test
the SM, and in extension, as a probe of new physics beyond the SM because of
its sensitivity to short distance effects.

###### Acknowledgements.

We thank Joachim Brod, Hector Gisbert, Antonio Pich, and Emmanuel Stamou for
useful discussions. Y.G. is supported in part by the NSF grant PHY1316222.
S.S. is supported by a Stephen Hawking Fellowship from UKRI under reference
EP/T01623X/1 and the Lancaster-Manchester-Sheffield Consortium for Fundamental
Physics, under STFC research grant ST/T001038/1. The work of T.K. is supported
by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for
Early-Career Scientists (Grant No. 19K14706) and the JSPS Core-to-Core Program
(Grant No. JPJSCCA20200002). For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising. This work uses existing data which is available at
locations cited in the bibliography.

##  Appendix A Update of the theory estimate for $\chi_{\gamma\gamma}(\mu)$

In this section we update the theory estimate of Ref. Isidori:2003ts . The
experimental input that goes into this estimate comes from analyses of the
form factor in $K_{L}\to\gamma e^{+}e^{-}$, $K_{L}\to\gamma\mu^{+}\mu^{-}$ and
$K_{L}\to e^{+}e^{-}\mu^{+}\mu^{-}$, and can be summarized by the parameter
$\alpha_{\rm exp.}$, defined by

$\alpha_{\rm
exp.}=-m_{\rho}^{2}\frac{d}{dq^{2}}f(q^{2},0)\biggr{\rvert}_{q^{2}=0}\,,$ (72)

where $f(q_{1}^{2},q_{2}^{2})$ is the $K_{L}\rightarrow\gamma\gamma$ form
factor. The experimental value derived from $K_{L}\to\gamma e^{+}e^{-}$ has
been updated by the KTeV collaboration after Ref. Isidori:2003ts  was
published. We take KTeV:2007ksh

$\alpha_{\rm exp.}|_{ee}=-1.73\pm 0.05\,,$ (73)

combined with KTeV:2001sfq ; KTeV:2002kut

$\displaystyle\alpha_{\rm exp.}|_{\mu\mu}$ $\displaystyle=$
$\displaystyle-1.54\pm 0.10\,,$ (74) $\displaystyle\alpha_{\rm
exp.}|_{ee\mu\mu}$ $\displaystyle=$ $\displaystyle-1.59\pm 0.37\,,$

and arrive at the weighted average

$\alpha_{\rm exp.}=-1.691\pm 0.044\,.$ (75)

Comparing with Ref. Isidori:2003ts  (who quote $\alpha_{\rm exp.}=-1.611\pm
0.044$), the central value has gone up by $\sim 5\%$ while the relative error
remains the same.

Inserting this into Eq. (22) of Ref. Isidori:2003ts , we have

$\displaystyle\chi_{\gamma\gamma}(m_{\rho})=(6.10\pm 0.16_{\rm
exp.})-\Delta_{\Lambda}\,,$ (76)

(updated from $(5.83\pm 0.15_{\rm exp.})-\Delta_{\Lambda}$). Following Ref.
Isidori:2003ts  we take

$|\Delta_{\Lambda}|\leq 1.0\,,$ (77)

and reach the result

$\Big{[}\chi_{\gamma\gamma}(m_{\rho})\Big{]}_{\rm IU}=6.10\,\pm\,0.16_{\rm
exp.}\,\pm\,1.0_{\rm th.}\,.$ (78)

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