# Gain-Loss Hedging and Cumulative Prospect Theory

Lorenzo Bastianello, Alain Chateauneuf, Bernard Cornet111Bastianello
(corresponding author): Università Ca’ Foscari Venezia, (email:
lorenzo.bastianello@unive.it); Chateauneuf: IPAG Business School, Université
Paris 1 Panthéon-Sorbonne and Paris School of Economics, (email:
<EMAIL_ADDRESS>); Cornet: Université Paris 1 Panthéon-Sorbonne and
Kansas University (email: cornet@ku.edu). First version: February 2020

###### Abstract

Two acts are comonotonic if they yield high payoffs in the same states of
nature. The main purpose of this paper is to derive a new characterization of
Cumulative Prospect Theory (CPT) through simple properties involving
comonotonicity. The main novelty is a concept dubbed gain-loss hedging: mixing
positive and negative acts creates hedging possibilities even when acts are
comonotonic. This allows us to clarify in which sense CPT differs from Choquet
expected utility. Our analysis is performed under the simpler case of (piece-
wise) constant marginal utility which allows us to clearly separate the
perception of uncertainty from the evaluation of outcomes.

Keywords: Cumulative Prospect Theory, Comonotonicity, Gain-loss hedging, Šipoš
integral, Choquet integral.

JEL Classification Number: D81.

##  1 Introduction

When making everyday decisions, economic agents are often confronted with
uncertainty. For instance, one can think of a decision maker (DM) who needs to
choose how to allocate her wealth between two different portfolios of assets,
or a firm that has to decide whether to invest in an innovative technology or
in a traditional one. The most popular model used under risk and uncertainty
is the expected utility model. This model, proposed first by Bernoulli at the
beginning of the XVIII century, has been axiomatized by de Finetti [6], Savage
[13] and Von Neumann and Morgenstern [27]. However, empirical evidence has
shown that expected utility does not provide a good description of DMs’ actual
choices. Early examples are the famous paradoxes of Allais [1] and Ellsberg
[9].

One of the most prominent and most successful alternative to expected utility
theory is cumulative prospect theory (CPT) of Tversky and Kahneman [21]. The
aim of this paper is twofold: $(i)$ we provide a new mathematical
characterization of the CPT functional under the simplifying assumption of
(piece-wise) constant marginal utility à la Yaari [26]; $(ii)$ we use the
characterization of the previous point to obtain a novel preference
axiomatization of CPT.

Consider acts as functions from a state space $S$ to the set of real numbers.
Thus, given an act $f:S\rightarrow\mathbb{R}$, $f(s)$ can be interpreted as
the amount of money or consumption good that a DM obtains if the state turns
out to be $s$. A central role is played by comonotonic acts. Loosely speaking,
two acts are comonotonic if they are positively correlated. Mixing two
comonotonic acts does not provide a possible hedge against uncertainty. This
idea was exploited in the seminal papers of Schmeidler [14], [15] to extend
expected utility to Choquet expected utility.

One advantage of CPT over the Choquet model is that it allows to disentangle
the behavior of DMs in the domain of gains from the one in the domain of
losses, i.e. when outcomes are respectively above or below a certain reference
point (in our case the reference point is naturally taken equal to 0). This
difference in behavior can be decomposed into two components. The first one is
called loss-aversion and says that “losses loom larger than gains” (Tversky
and Kahneman [21]). Mathematically, it means that losses are multiplied by a
constant $\lambda>1$. The second one is usually called sign dependence and
says that the attitude toward uncertainty (mathematically represented by a
capacity) is different for gains and for losses. We take this behavior as a
starting point for both the mathematical characterization of CPT and its
axiomatization. The intuition behind our properties is that adding comonotonic
acts can still provide some hedge if those acts are of opposite signs and have
non-disjoint supports. We call this property gain-loss hedging.

We describe here the two main properties that we use in Section 3.1 to
characterize mathematically the CPT functional. The first property is well-
known and postulates comonotonic independence (separately) for gains and for
losses. Comonotonic acts do not provide a possible hedge against uncertainty
and therefore adding them should not change the preferences of the DM. Take
three acts $f,g,$ and $h$, all in the domain of gains or all in the domain of
losses, such that $h$ is comonotonic with $f$ and $g$. Then our condition
require that if $f$ and $g$ are indifferent, then adding $h$ to both of them
does not change a DM’s preferences since in both situations $h$ does not
increase nor reduce uncertainty.

The second property, that we call gain-loss hedging, represents the main
behavioral novelty of the paper. The key idea is that adding an act above the
reference point to an act below the reference point may provide an hedge
against uncertainty unless these acts have disjoint supports. To exemplify
suppose that there are two states of the world $S=\\{s_{1},s_{2}\\}$ and that
a DM with a linear utility function over outcomes is indifferent between the
assets $f=(20,0)$ ($f$ is the act that pays 20 if $s_{1}$ is realized and 0
otherwise) and $g=(10,10)$. Consider now the act $h=(0,-10)$ which has
disjoint support with $f$ but not with $g$. When the DM evaluates
$f+h=(20,-10)$ and $g+h=(10,0)$, she may feel $f+h$ more uncertain than $g+h$
and therefore she may prefer $g+h$. Note that indifference between $f$ and $g$
and then a strict preference for $g+h$ is precluded by the expected utility
model (with the utility function being the identity). More interestingly, this
preference pattern would be a paradox even for the more general Choquet
expected utility model of Schmeidler [15] (with the utility function being the
identity). The Choquet model excludes any possible hedging through mixing of
comonotonic acts. In this example however act $h$ is comonotonic with both
acts $f$ and $g$ and therefore no possible hedging would be envisioned by the
Choquet model. Therefore $h$ is a possible hedge to uncertainty when added to
$g$ because gains and losses balance out one another, and not because of
comonotonicity. We elaborate more on this idea in Example 1.

In Section 3.2, we give a preference axiomatization of the CPT model with
piece-wise linear utility. We do not assume the Anscombe and Aumann [2]
framework, and our axioms only appeals to simple properties related to
comonotonicity. Moreover, we propose a new and simple axiom that can be used
to elicit the coefficient of loss-aversion $\lambda$. In order to derive a CPT
representation of preferences, we use the mathematical characterization of
Section 3.1. In a sense, our paper parallels, in the context of prospect
theory, the work of Schmeidler [14], [15] on the Choquet integral.

Empirical evidence not only supports sign-dependence, but it suggests further
that agents are uncertainty averse for gains and uncertainty seeking for
losses, see for instance Wakker [23], Section 12.7 for a review. Section 3.3
provides testable axioms that characterize those opposite behaviors. Finally
we investigate when uncertainty aversion for gains is symmetric to uncertainty
seeking for losses. Behaviorally, this happens if a DM who is indifferent
between an act $f$ and a monetary outcome $\alpha$ is also indifferent between
$-f$ and $-\alpha$. In this case we prove that weights for gains and losses
are dual with respect to each other and that CPT reduces to a Šipoš integral,
see Šipoš [18]. This result clarifies the relation of CPT with the Šipoš
integral that was first noticed by Starmer and Sudgen [19] (see also Wakker
[23] and Kothiyal et al. [12]).

Of course, there are several axiomatizations of CPT available in the
literature. The concept of comonotonicity and the fact that acts are rank-
ordered are crucial, see Diecidue and Wakker [8]. The very first
axiomatization is provided in the seminal paper of Tversky and Kahneman [21]
and relies on comonotonic independence and a property called double matching.
See also Trautmann and Wakker [20] for a recent characterization using these
axioms in a (reduced) Anscombe and Aumann [2] framework. Wakker and Tversky
[24] pair comonotonicity with trade-off consistency (see also Chateauneuf and
Wakker [4] for the case of risk). The conomotonic sure thing principle
approach (or a weakening of it called tail independence) is developed in Chew
and Wakker [5], Zank [28] and Wakker and Zank [25]. The paper closest to our
is the one of Schmidt and Zank [16]. The authors characterize CPT through an
axiom called independence of common increments for comonotonic acts.
Interestingly, they obtain a piecewise linear utility function (with a kink
about the reference point), as in our axiomatization. We refer the reader to
the introductory section of Schmidt and Zank [16] for a detailed discussion
about the advantages of adopting piece-wise linear utility.

The rest of the paper is organized as follows. Section 2 introduces the
framework, the mathematical notations and the behavioral models that we will
consider. Section 3 is divided in three subsections and it contains our main
results. Section 3.1 presents the mathematical characterization of the CPT and
Šipoš functionals, Section 3.2 provides a behavioral characterization of CPT
and Section 3.3 discusses DM’s attitude towards uncertainty. Section 4
concludes. All proofs are gathered in the Appendix.

##  2 Framework

Let $S$ be a set of states of the world endowed with a $\sigma$-algebra ${\cal
A}$. Elements of ${\cal A}$ are called events. We denote $\mathcal{F}$ the set
of all bounded, real-valued, ${\cal A}$-measurable functions over $S$, i.e.
$\mathcal{F}~{}=~{}\\{f:S\rightarrow\mathbb{R}|f\text{ is bounded and }{\cal
A}\text{-mesurable}\\}$. A function $f\in\mathcal{F}$ is called act. An act
can be interpreted as an asset that pays a monetary outcome in $\mathbb{R}$
that depends on the realization of the state of the world. We denote the
positive part of an act $f\in\mathcal{F}$ by $f^{+}=f\vee 0$ and the negative
part by $f^{-}=(-f)\vee 0$. Note that both positive and negative parts are
greater than 0.222Note that several papers studying prospect theory use the
symbol $f^{-}$ to denote $f\wedge 0$. The set
$\mathcal{F}^{+}=\\{f\in\mathcal{F}|f(s)\geq 0,\,\forall s\in S\\}$ is the set
of positive acts, the set of negative acts $\mathcal{F}^{-}$ is defined
analogously. Two acts $f,g\in\mathcal{F}$ have the same sign if either
$f,g\in\mathcal{F}^{+}$ or $f,g\in\mathcal{F}^{-}$. We say that two acts are
of opposite sign if one of them is positive and the other is negative. The
support of an act of $f\in\mathcal{F}$ is the set $supp(f)=\\{s\in S|f(s)\neq
0\\}$. Two acts $f,g\in\mathcal{F}$ are comonotonic if for all $s,t\in S$,
$(f(s)-f(t))(g(s)-g(t))\geq 0$. Let $A\subseteq S$, $1_{A}$ is the indicator
function of the set $A$, i.e. $1_{A}(s):=\begin{cases}1&\text{ if }s\in A\\\
0&\text{ if }s\in A^{c}\end{cases}$. If $\alpha\in\mathbb{R}$, then $\alpha
1_{A}$ denotes the constant act which pays $\alpha$ in every state $s\in A$.

A (normalized) capacity $v$ on the measurable space $(S,{\cal A})$ is a set
function $v:{\cal A}\mapsto\mathbb{R}$ such that $v(\emptyset)=0,\,v(S)=1$ and
for all $A,B\in{\cal A},\,A\subseteq B\Rightarrow v(A)\leq v(B)$. If $v$ is a
capacity, we define its conjugate by $\hat{v}(A)=1-v(A^{c})$ for all
$A\in{\cal A}$. A capacity $v:{\cal A}\mapsto\mathbb{R}$ is convex (concave)
if, for all $A,B\in{\cal A}$, $v(A\cup B)+v(A\cap B)\geq(\leq)v(A)+v(B)$.
Given a capacity $v$ on $(S,{\cal A})$, the Choquet integral of
$f\in\mathcal{F}$ with respect to $v$ is a functional
$C:\mathcal{F}\rightarrow\mathbb{R}$ defined by

$C(f)=\int_{S}{f\,dv}:=\int_{-\infty}^{0}{(v(\\{f\geq
t\\})-1)\,dt}+\int_{0}^{+\infty}{v(\\{f\geq t\\})\,dt},$

In the following we will remove the subscript $S$ from the integral sign
whenever the domain of integration is clear. Given a capacity $v$ on $(S,{\cal
A})$, the Šipoš integral (see Sipoš [18]) of $f\in\mathcal{F}$ with respect to
$v$ is a functional $\check{S}:\mathcal{F}\rightarrow\mathbb{R}$ defined as

$\check{S}(f)=\int f^{+}dv-\int f^{-}dv$

where the two integrals are Choquet integrals. The following Lemma gives an
alternative formulation of the Šipoš integral when the conjugate capacity is
used when evaluating the negative part of a function. Moreover it clarifies
the relation between the Choquet integral and the Šipoš integral.

######  Lemma 1.

Let $v$ be a capacity and $\hat{v}$ its conjugate. Then the following holds:

  * •

$\check{S}(f)=\int f^{+}dv+\int-f^{-}d\hat{v}$;

  * •

$C(f)=\int f^{+}dv+\int-f^{-}dv=\int f^{+}dv-\int f^{-}d\hat{v}$.

The main object of this paper is the (piecewise linear) Cumulative Prospect
Theory (CPT) functional $CPT:\mathcal{F}\rightarrow\mathbb{R}$. It is a
generalization of both Choquet and Šipoš integrals. Consider two capacities
$v^{+}$, $v^{-}$ and a real number $\lambda>0$, then the (piecewise linear)
CPT functional $CPT:\mathcal{F}\rightarrow\mathbb{R}$ is defined by

$CPT(f)=\int f^{+}dv^{+}-\int\lambda f^{-}dv^{-}.$

A preference relation $\succsim$ over $\mathcal{F}$ is a complete and
transitive binary relation with non-empty strict part. As usual, $f\succsim g$
means “$f$ is preferred to $g$”. We denote $\succ$ and $\sim$ the strict and
weak part of $\succsim$. A functional $I:\mathcal{F}\rightarrow\mathbb{R}$
represents $\succsim$ if for all $f,g\in\mathcal{F}$, $f\succsim g$ if and
only if $I(f)\geq I(g)$.

##  3 Main results

This section contains our two main results. The first one, Theorem 2,
characterizes mathematically the CPT functional. The second result, Theorem 5,
studies which behavioral axioms a preference relation should satisfy in order
to be represented by a CPT functional.

###  3.1 The CPT functional

We start with a seminal theorem of Schmeidler [14] who provided a
characterization of the Choquet functional. Before presenting the result we
recall that a functional $I:\mathcal{F}\rightarrow\mathbb{R}$ is monotonic if
$f\geq g\Rightarrow I(f)\geq I(g)$, where $f\geq g$ means $f(s)\geq g(s)$ for
all $s\in S$. Moreover $I$ satisfies comonotonic additivity if, whenever $f$
and $g$ are comonotonic, then $I(f+g)=I(f)+I(g)$.

######  Theorem 1.

(Schmeidler [14]) Let $I:\mathcal{F}\rightarrow\mathbb{R}$ be a given
functional with $I(1_{S})=1$. Then the following are equivalent.

  * (i)

$(a)$ $I$ is monotonic; $(b)$ $I$ satisfies comonotonic additivity.

  * (ii)

$I$ is a Choquet integral.

The CPT functional generalizes the Choquet functional by relaxing comonotonic
additivity. More specifically, comonotonic additivity will be retained only
for comonotonic acts of the same sign and for (comonotonic) acts of opposite
sign with disjoint supports. The following is our first main result.

######  Theorem 2.

Let $I:\mathcal{F}\rightarrow\mathbb{R}$ be a given functional satisfying
$I(1_{S})=1$. Then the following are equivalent.

  * (i)

$(a)$ $I$ is monotonic; $(b)$ $I$ satisfies comonotonic additivity on
$\mathcal{F}^{+}$ and $\mathcal{F}^{-}$ and for acts $f,g$ of opposite sign
such that $supp(f)\cap supp(g)=\emptyset$.

  * (ii)

$I$ is a CPT functional.

Consider item $(i)$ of both Theorem 1 and Theorem 2. Note that part (b) of
Theorem 1 implies (b) of Theorem 2, as acts with opposite sign and disjoint
supports are comonotonic. This relaxation not only characterize a functional
that is more general than the Choquet integral, but it also gives some
important insights from a behavioral point of view.

Recall that comonotonic additivity is a weakening of full-fledged additivity,
a property that would force the functional $I$ to be linear, and hence an
expectation. The behavioral intuition behind comonotonic additivity is that
adding two comonotonic acts does not permit possible hedging against choices
of nature.

Relaxing comonotonic additivity allows us to uncover more sophisticated
attitudes towards uncertainty and more subtle forms of hedging. The fist
remarkable property of the CPT functional is that it differentiates agents’
behavior in the domain of gains (i.e. $\mathcal{F}^{+}$) from the one in the
domain of losses (i.e. $\mathcal{F}^{-}$). The outcome for which behavior
changes, namely the monetary outcome 0, is called the reference point.333In
this paper, the reference point is exogenously given and it is normalized to 0
for convenience (we could have chosen any other reference point
$r\in\mathbb{R}$). Schmidt and Zank [17] provide axioms to make the reference
point endogenous. Comonotonic additivity is preserved whenever acts under
considerations are both above or both below the reference point. Comonotonic
additivity over $\mathcal{F}^{+}$ and $\mathcal{F}^{-}$ weakens a condition
already well known in the literature called cosigned independence. Two acts
$f,g\in\mathcal{F}$ are sign-comonotonic or simply cosigned if they are
comonotonic and there exists no $s\in S$ such that $f(s)>0$ and $g(s)<0$, see
Wakker and Tversky [24] and Trautmann and Wakker [20].

One of the main contribution of the present paper lies in the second
comonotonic additivity requirement that characterizes the $CPT$ functional,
namely $f,g$ of opposite sign such that $supp(f)\cap supp(g)=\emptyset$
implies $CPT(f+g)=CPT(f)+CPT(g)$. This means that comonotonic additivity can
fail if we have $f,g$ of opposite sign and $supp(f)\cap supp(g)\neq\emptyset$
(we underline again that such acts are comonotonic). The behavioral intuition
behind this requirement is that adding the positive and negative parts of two
acts can provide a hedge against possible choices of nature even when acts
under consideration are comonotonic. We call this property gain-loss
hedging.444We thank Peter Wakker for suggesting this terminology. This hedging
possibility it is not considered for instance in the Choquet model, where the
only way to hedge is to add two non-comonotonic acts. The following example
provides more details for the particular case in which CPT reduces to a Šipoš
integral, i.e. $\lambda=1$ and $v^{+}=v^{-}$.

######  Example 1.

Let $S=\\{s_{1},s_{2},s_{3}\\}$ and consider a CPT functional with $\lambda=1$
and $v=v^{+}=v^{-}$ (i.e. a Šipoš integral). Let $v$ be defined as

$A$ | $S$ | $\emptyset$ | $s_{1}$ | $s_{2}$ | $s_{3}$ | $s_{1}\cup s_{2}$ | $s_{2}\cup s_{3}$ | $s_{1}\cup s_{3}$
---|---|---|---|---|---|---|---|---
$v$ | 1 | 0 | ${2\over 3}$ | ${1\over 3}$ | 0 | ${2\over 3}$ | ${2\over 3}$ | 1

Consider now the following acts on $S$.

| $s_{1}$ | $s_{2}$ | $s_{3}$
---|---|---|---
$f$ | 3 | 4 | 4
$g$ | 0 | 11 | 0
$h$ | -3 | 0 | -1
$-h$ | 3 | 0 | 1
$f+h$ | 0 | 4 | 3
$g+h$ | -3 | 11 | -1

Acts $f,g,h$ are comonotonic, but $supp(g)\cap supp(h)=\emptyset$ while
$supp(f)\cap supp(h)\neq\emptyset$. Let $\succsim_{\check{S}}$ be the
preference relation induced by the $\check{S}$ functional, i.e.
$f\succsim_{\check{S}}g\Leftrightarrow\check{S}(f)\geq\check{S}(g)$ and
$\succsim_{C}$ the one induced by the $C$ functional (both functionals
$\check{S}$ and $C$ are defined in Section 2). We have

$\displaystyle\check{S}(f)=C(f)=$ $\displaystyle 3+(4-3){2\over 3}={11\over
3}$ $\displaystyle\check{S}(g)=C(g)=$ $\displaystyle 0+(11-0){1\over
3}={11\over 3}$

and therefore $f\sim_{\check{S}}g$ and $f\sim_{C}g$. Moreover since $h$ is
comonotonic with $f$ and $g$, by comonotonic additivity $f+h\sim_{C}g+h$ (one
can actually verify that $C(f+h)=C(f)+C(h)={7\over 3}=C(g)+C(h)=C(g+h)$).
However we can notice that the act $f+h$ looks much “smoother” than $g+h$ and
moreover $f+h\geq 0$ since gains balance losses. This intuition is captured by
the preference relation induced by the Šipoš integral as

$\displaystyle\check{S}(f+h)=$ $\displaystyle 0+(3-0){2\over 3}+(4-3){1\over
3}={7\over 3}$ $\displaystyle\check{S}(g+h)=$ $\displaystyle
C(g)-C(-h)={11\over 3}-(0+(1-0)1+(3-1){2\over 3})={4\over 3}$

and therefore $f+h\succ_{\check{S}}g+h$.

Example 1 shows that gain-loss hedging is an interesting behavioral feature of
CPT and of Šipoš integrals. Adding positive and negative acts with supports
that are not disjoint, can provide an hedge even when the acts involved are
comonotonic. This happens because gains compensate losses. In the next section
we provide a new behavioral foundation of CPT taking this observation as a
starting point.

Example 1 shows that preferences represented by Šipoš integrals are rich
enough to entail gain-loss hedging behaviors. It is therefore interesting to
mathematically characterize Sipoš integrals. Theorem 3 shows that a symmetric
condition pins down a CPT functional as a Šipoš integral.

######  Theorem 3.

A CPT functional is a Šipoš integral if and only if $CPT(-f)=-CPT(f)$ for all
$f\in\mathcal{F}$.

Theorem 3 says that CPT reduces to a Šipoš integral if and only if the
condition $CPT(-f)=-CPT(f)$ for all $f\in\mathcal{F}$ is satisfied. This is an
interesting result as such condition is a strong one. As an example, if $C$ is
a Choquet functional then $C(-f)=-C(f)$ for all $f\in\mathcal{F}$ if and only
if the capacity $v$ equals its conjugate $\hat{v}$, and therefore it is
additive on events $\\{A,A^{c}\\}$.

###  3.2 A behavioral characterization of CPT

In this section we provide a preference axiomatization of CPT. We recall that
a preference relation $\succsim$ over $\mathcal{F}$ is a complete and
transitive binary relation with non-empty strict part.

The first axiom is a continuity axiom.

A.1 Continuity. The sets $\\{\alpha\in\mathbb{R}|\alpha 1_{S}\succsim f\\}$
and $\\{\alpha\in\mathbb{R}|f\succsim\alpha 1_{S}\\}$ are closed for all
$f\in\mathcal{F}$.

Note that the axiom requires only to compare acts with constants. This
dispenses us to formulate topological assumptions on the set of acts
$\mathcal{F}$.

The second axiom is a monotonicity property.

A.2 Monotonicity. Let $f,g\in\mathcal{F}$ be such that $f\geq g$. Then
$f\succsim g$.

Consider now the well known comonotonic independence axiom (Chateauneuf [3],
Schmeidler [15]). It says that if two acts $f$ and $g$ are indifferent to each
other, then adding a comonotonic act $h$ to both of them does not change the
DM’s preferences. The idea behind this condition is that adding comonotonic
acts does not provide any possible hedge against uncertainty.

A.C Comonotonic Independence. Let $f,g,h\in\mathcal{F}$ such that $h$ is
comonotonic with $f$ and with $g$. Then $f\sim g$ implies $f+h\sim g+h$

Preferences satisfying A.1, A.2 and A.C are represented by a Choquet integral.
We present this result in the next proposition

######  Theorem 4.

(Chateauneuf [3], Schmeidler [15]) Let $\succsim$ be a preference relation
over $\mathcal{F}$. Then the following are equivalent.

  * (i)

$\succsim$ satisfies A.1, A.2 and A.C.

  * (ii)

There exists a (unique) capacity $v$ such that $\succsim$ is represented by a
Choquet functional.

However, as Example 1 shows, Comonotonic Independence may be too strong as it
doesn’t take into account (gain-loss) hedging possibilities that arise adding
positive and negative acts with non-disjoint supports. The following two
axioms, axiom A.3 and A.4, are both implied by Comonotonic Independence. They
are at the heart of our behavioral characterization. They generalizes A.C in
two directions. First, axiom A.3 allows for different attitudes towards
uncertainty in the domain of gain and in the domain of losses. Second, axiom
A.4 takes into account possible gain-loss hedging opportunities that arises in
situations like the one of Example 1.

A.3 Comonotonic Independence for Gain and Losses. Let
$f,g,h\in\mathcal{F}^{+(-)}$ be such that $h$ is comonotonic with $f$ and $g$.
Then $f\sim g$ implies $f+h\sim g+h$.

A.4 $\lambda$-Disjoint Independence. There exists $\lambda>0$ such that for
all $f\in\mathcal{F}^{+}$ and $g\in\mathcal{F}^{-}$ such that $supp(f)\cap
supp(g)=\emptyset$ and such that $f\sim\alpha 1_{S}$ and $g\sim\beta 1_{S}$

  1. 1.

if $\alpha+\lambda\beta\geq 0$ then $f+g\sim(\alpha+\lambda\beta)1_{S}$;

  2. 2.

if $\alpha+\lambda\beta<0$ then
$f+g\sim\left({\alpha+\lambda\beta\over\lambda}\right)1_{S}$.

Axiom A.4 represents the main behavioral novelty. To better understand it,
note that it is implied by the following (stronger) axiom.

A.4∗ Disjoint Independence. For all $f\in\mathcal{F}^{+}$ and
$g\in\mathcal{F}^{-}$ such that $supp(f)\cap supp(g)=\emptyset$ and such that
$f\sim\alpha 1_{S}$ and $g\sim\beta 1_{S}$, one has
$f+g\sim(\alpha+\beta)1_{S}$.

It is easy to see that A.4∗ follows from A.4 imposing $\lambda=1$. A.4∗
requires that the act $f+g$ is evaluated as the sum of its constant
equivalent. In the general case, we can have $\lambda\neq 1$ and in this case
A.4 says that the constant equivalent of $f+g$ depends on the sign of
$\alpha+\lambda\beta$. The interpretation for the case of loss-aversion,
$\lambda>1$, is the following. The DM outweighs losses by a factor of
$\lambda$ and considers $\lambda\beta$ instead of $\beta$ tout-court. If
$\alpha+\lambda\beta>0$ then the DM feels “overall in the domain of gains” and
the certainty equivalent of $f+g$ is positive and such that $\alpha>0$ is
balanced by $\lambda\beta<\beta<0$, i.e. the certainty equivalent $\beta$ of
losses is outweighed by a factor of $\lambda$. If $\alpha+\lambda\beta<0$ then
the DM feels “overall in the domain of losses” and in this case the certainty
equivalent of $f+g$ is negative and equal to $\beta<0$ plus
${\alpha\over\lambda}>0$, i.e. the certainty equivalent $\alpha$ of the
positive part decreased by a factor of $\lambda$ (since
${\alpha\over\lambda}<\alpha$).

Importantly, $\lambda$ can be determined in the lab: take
$f\in\mathcal{F}^{+}$ and $g\in\mathcal{F}^{-}$ such that $supp(f)\cap
supp(g)=\emptyset$, ask the certainty equivalents $\alpha$, $\beta$ and
$\gamma$ of $f$, $g$ and $f+g$ respectively. If $\gamma=\alpha+\beta$, there
is no loss-aversion or seeking. If $\gamma\neq\alpha+\beta$, then if
$\gamma>0$ we have $\lambda={\gamma-\alpha\over\beta}$ and if $\gamma<0$ we
have $\lambda={\alpha\over\gamma-\beta}$. There is lively debate on whether
loss-aversion is a real phenomena or not, with results on both sides. See for
instance Gal and Rucker [10] and Gächter, Johnson and Herrmann [11]. We hope
therefore that A.4 could be helpful to elicit loss-aversion in a setting in
which individuals’ preferences are represented by the CPT functional with
piece-wise constant marginal utility.

When A.C is replaced by A.3 and A.4, we obtain a characterization of the CPT
functional. The following is our second main result.

######  Theorem 5.

Let $\succsim$ be a preference relation over $\mathcal{F}$. Then the following
are equivalent.

  * (i)

$\succsim$ satisfies A.1, A.2, A.3 and A.4.

  * (ii)

There exist two (unique) capacities $v^{+}$, $v^{-}$ and a real number
$\lambda>0$ such that $\succsim$ is represented by a CPT functional.

Note that if we replace A.4 with A.4∗ in Theorem 5, we obtain a CPT functional
with $\lambda=1$, i.e. loss-neutrality.

###  3.3 Attitude towards uncertainty

As we already said above, a remarkable property of CPT is that (unlike the
Choquet functional) it allows to disentangle DMs’ attitude towards uncertainty
in the domain of gains from the one in the domain of losses. This is made
possible since an act is evaluated through the sum of two Choquet integrals
with respect to a capacity $v^{+}$ for gains, and a different one $v^{-}$ for
losses.

Experimental evidence shows that DMs are uncertainty averse for gains and
uncertainty seeking for losses. Loosely speaking, uncertainty aversion
(seeking) means that agents prefer situations in which objective probabilities
of events are (not) available. In our framework, objective probabilities are
not there at all. Therefore an act is not uncertain only if it is a constant
act. Intuitively, in our purely subjective setting, an uncertainty averse
(seeking) DM would prefer acts that are “as close (far) as possible” to
constant acts. We capture this idea with the two following axioms.

A.3’ Let $f,g,h\in\mathcal{F}^{+}$ such that $h$ is comonotonic with $g$. Then
$f\sim g\Rightarrow f+h\succsim g+h$.

A.3” Let $f,g,h\in\mathcal{F}^{-}$ such that $h$ is comonotonic with $f$. Then
$f\sim g\Rightarrow f+h\succsim g+h$.

Axiom A.3’ captures the intuition that DMs are uncertainty averse in the
domain of gains $\mathcal{F}^{+}$. Consider three acts
$f,g,h\in\mathcal{F}^{+}$ such that $f\sim g$ and $h$ is comonotonic with $f$
and $g$. Then adding (the potentially non-comonotone act) $h$ to $f$ increases
the appreciation of $f$ since $h$ may by an hedge against $f$, while at the
same time it decreases the appreciation of $g$ since uncertainty may be
higher. To exemplify, let $A\in{\cal A}$ and consider $f=10\cdot 1_{A}+5\cdot
1_{A^{c}}$, $g=5\cdot 1_{A}+10\cdot 1_{A^{c}}$ and $h=0\cdot 1_{A}+5\cdot
1_{A^{c}}$. Then $f+h=10\cdot 1_{S}$ is a constant act while $g+h=5\cdot
1_{A}+15\cdot 1_{A^{c}}$ is even more uncertain than $g$. A DM who dislikes
uncertainty would clearly prefer $f+h$ to $g+h$. Axiom A.3” can be interpreted
similarly, but in this case the DM is willing to increase the perceived
uncertainty. Notice that similar conditions were proposed by Chateuneuf [3],
see also Wakker [22].

The following theorem shows that if a DM is uncertainty averse for gains and
uncertainty seeking for losses then the capacities appearing in the CPT
functional are both convex.

######  Theorem 6.

Let $\succsim$ be a preference relation over $\mathcal{F}$. Then the following
are equivalent.

  * (i)

$\succsim$ satisfies A.1, A.2, A.3’, A.3”, and A.4.

  * (ii)

There exist two (unique) convex capacities $v^{+}$, $v^{-}$ and $\lambda>0$,
such that $\succsim$ is represented by a CPT functional.

Note that the CPT functional can be rewritten (using Lemma 2 in the Appendix)
as

$CPT(f)=\int f^{+}dv^{+}+\int-\lambda f^{-}d\hat{v}^{-}.$ (3.1)

If one is using this formulation then Theorem 6 implies that the conjugate
capacity $\hat{v}^{-}$ is concave.

We conclude this section providing a testable axiom that characterizes
symmetric attitudes around the reference point with respect to uncertainty. An
interesting question is in fact to understand when one has $v^{-}=v^{+}$ in
the CPT functional.555Or, equivalently if one is using formulation (3.1), when
one gets $v^{-}=\hat{v}^{+}$. Note that if $\lambda=1$ Theorem 3 applies and
one gets a Šipoš integral. Consider the following axiom.

A.5 Gain-Loss Symmetry. Let $f\in\mathcal{F}$ and $\alpha\in\mathbb{R}$. Then
$f\sim\alpha 1_{S}$ if and only if $-f\sim-\alpha 1_{S}$.

Axiom A.5 says that if a DM is indifferent between an (uncertain) act $f$ and
a sure amount $\alpha$, then she should stay indifferent between $-f$ and
$-\alpha$. The intuition is that the DM sees $f$ and $-f$ as symmetric with
respect to the reference point 0, and therefore evaluates them through the
symmetric sure amounts $\alpha$ and $-\alpha$. The following theorem offers a
behavioral characterization of the Šipoš integral.

######  Theorem 7.

Let $\succsim$ be a preference relation over $\mathcal{F}$. Then the following
are equivalent.

  * (i)

$\succsim$ satisfies A.1, A.2, A.3, A.4 and A.5.

  * (ii)

There exists a (unique) capacity $v$ such that $\succsim$ is represented by
the Šipoš integral.

##  4 Conclusion

We provided an axiomatic analysis of CPT with piece-wise linear utility. This
allowed us to focus on (sign-dependent) attitudes towards uncertainty. First,
we mathematically characterized the CPT functional by weakening the
comonotonic additivity property of the Choquet integral. We also gave
conditions to reduce CPT to a Šipos integral. Then we gave an axiomatic
characterization of CPT. The main novelty is given by a gain-loss hedging
property: gains and losses balance each other out and provide an hedge against
uncertainty. Moreover, we introduced an axiom that offers a way to easily
elicit the coefficient of loss-aversion, in case of piece-wise linear utility.
Finally, we characterized uncertainty aversion for losses and uncertainty
loving for gains. Moreover we showed that these attitudes are symmetric with
respect to the reference point if and only if CPT is a Šipos integral.

##  Appendix A Appendix

We begin with an elementary Lemma. The proof if given for sake of
completeness.

######  Lemma 2.

Let $\hat{v}(A)=1-v(A^{c})$ and $f\in\mathcal{F}^{+}$ or $f\in\mathcal{F}-$.
Then $-\int fdv=\int-fd\hat{v}$.

###### Proof.

Let $f\in\mathcal{F}^{+}$, then one has

$\displaystyle-\int fdv$ $\displaystyle=-\int_{0}^{\infty}v(s\in S|f(s)\geq
t)dt$ $\displaystyle=-\int_{0}^{\infty}[1-\hat{v}(s\in S|f(s)>t)]dt$
$\displaystyle=-\int_{0}^{-\infty}[1-\hat{v}(s\in S|f(s)>u)](-du)$
$\displaystyle=\int^{0}_{-\infty}[\hat{v}(s\in S|f(s)>u)-1]du$
$\displaystyle=\int-fd\hat{v}$

The case $f\in\mathcal{F}^{-}$ can be treated similarly. ∎

###### Proof of Lemma 1.

To prove the first point we just need to apply Lemma 2. In fact noticing that
$f^{-}\in\mathcal{F}^{+}$ we have $\check{S}(f)=\int f^{+}dv-\int f^{-}dv=\int
f^{+}dv+\int-f^{-}d\hat{v}.$
The second point, note that $f=f^{+}+(-f^{-})$ and that $f^{+}$ and $-f^{-}$
are comonotonic. Then by the comonotonic additivity of the Choquet integral
proved in Theorem 1, we have

$\int fdv=\int f^{+}+(-f^{-})dv=\int f^{+}dv+\int-f^{-}dv.$

Note that by Lemma 2 one can also write $\int fdv=\int f^{+}dv-\int
f^{-}d\hat{v}$. ∎

###### Proof of Theorem 2.

$(i)\Rightarrow(ii)$. We start with an auxiliary Lemma.

######  Lemma 3.

For all $\alpha>0$, for all $f\in\mathcal{F}^{+}\cup\mathcal{F}^{-}$,
$I(\alpha f)=\alpha I(f)$. Moreover for $\alpha>0$ and $f\in\mathcal{F}^{+}$,
$I(f+\alpha 1_{S})=I(f)+\alpha$.

###### Proof of Lemma 3.

The proof is standard. ∎

Let $v^{+}(A)=I(1_{A})$, then doing the same proof as Schmeidler [14] one can
show that for all $f\in\mathcal{F}^{+}$, $I(f)=\int fdv^{+}$. Now let
$\lambda:=-I(-1_{S})$. By comonotonic additivity of $I$, $I(0)=0$. By
monotonicity of $I$, $I(-1_{S})\leq I(0)=0$. Then $\lambda\geq 0$. Define for
all $A\in{\cal A}$,

$v(A)=-{I(-1_{A})\over\lambda}.$

We have $v(\emptyset)=0$ and $v(S)=1$. Take $A\subseteq B$ so that
$-1_{A}\geq-1_{B}$. Since $I$ is monotonic, $I(-1_{A})\geq I(-1_{B})$ and
therefore $v(A)\leq v(B)$. This show that $v$ is a capacity. Define $v^{-}$ as
the conjugate capacity of $v$, meaning that for all $A\in{\cal A}$,

$v^{-}(A)=1-v(A^{c}).$

We will show that for all $f\in\mathcal{F}^{-}$, $f$ simple, $I(f)=\int\lambda
fdv^{-}$. Let $f\in\mathcal{F}^{-}$ be defined as

$f=x_{1}1_{A_{1}}+\dots+x_{n}1_{A_{n}}$

where $\\{A_{1},\dots,A_{n}\\}$ is a partition of $S$ and $x_{1}\leq\dots\leq
x_{n}\leq 0$. Note that we can rewrite $f$ as

$f=(0-x_{n})(-1_{S})+(x_{n}-x_{n-1})(-1_{A_{n-1}\cup\dots\cup
A_{1}})+\dots+(x_{3}-x_{2})(-1_{A_{2}\cup A_{1}})+(x_{2}-x_{1})(-1_{A_{1}}).$

Define

$h_{i}=(x_{i+1}-x_{i})(-1_{A_{i}\cup\dots\cup A_{1}})$

with the convention that $x_{n+1}=0$. We have that

$f=\sum_{i=0}^{n}h_{i}$

We show now that $h_{i}$ is comonotone with $\sum_{k=i+1}^{n}h_{k}$. Consider
$s,t\in S$ be such that $s\in A_{i}\cup\dots\cup A_{1}$ and
$t\in(A_{i}\cup\dots\cup A_{1})^{c}$, suppose $t\in A_{l}$ for $l>i$. Then
$h_{i}(s)-h_{i}(t)=x_{i}-x_{i+1}\leq 0$ and
$\sum_{k=i+1}^{n}h_{k}(s)-\sum_{k=i+1}^{n}h_{k}(t)=x_{i}-x_{l}\leq 0$, hence
$(h_{i}(s)-h_{i}(t))\left(\sum_{k=i+1}^{n}h_{k}(s)-\sum_{k=i+1}^{n}h_{k}(t)\right)\geq
0$. If $s,t\in A_{i}\cup\dots\cup A_{1}$ or $s,t\in(A_{i}\cup\dots\cup
A_{1})^{c}$ the previous product is 0. This shows that the functions $h_{i}$
and $\sum_{k=i+1}^{n}h_{k}$ are conomotone.

Since $h_{i}$ and $\sum_{k=i+1}^{n}h_{k}$ are negative, by comonotonic
additivity on $\mathcal{F}^{-}$ we have

$I(f)=I(h_{1}+\sum_{i=2}^{n}h_{i})=I(h_{1})+I(h_{2}+\sum_{i=3}^{n}h_{i})=\dots=\sum_{i=1}^{n}I(h_{i}).$

Note that by Lemma 3 and by definition of $v^{-}$ we have

$I(h_{i})=(x_{i+1}-x_{i})\lambda(v^{-}(A_{i+1}\cup\dots\cup A_{n})-1).$

Therefore

$\displaystyle I(f)$ $\displaystyle=\sum_{i=1}^{n-1}I(h_{i})+I(h_{n})$
$\displaystyle=\lambda\left[\sum_{i=1}^{n-1}(x_{i+1}-x_{i})v^{-}(A_{i+1}\cup\dots\cup
A_{n})+\sum_{i=1}^{n-1}(x_{i+1}-x_{i})\right]+(x_{n+1}-x_{n})(-\lambda)$
$\displaystyle=\lambda\left[\sum_{i=1}^{n-1}(x_{i+1}-x_{i})v^{-}(A_{i+1}\cup\dots\cup
A_{n})-x_{n}+x_{1}\right]+\lambda x_{n}$
$\displaystyle=\lambda\left[x_{1}+\sum_{i=1}^{n-1}(x_{i+1}-x_{i})v^{-}(A_{i+1}\cup\dots\cup
A_{n})\right]$ $\displaystyle=\lambda\int fdv^{-}$ $\displaystyle=\int\lambda
fdv^{-}$

Notice that every bounded function can be approximated by a sequence of step
functions as in Schmeidler [14]. This shows that for all
$f\in\mathcal{F}^{-}$, $I(f)=\int\lambda fdv^{+}$. Let now $f\in\mathcal{F}$
and notice that $f=f^{+}+(-f^{-})$ and moreover $supp(f^{+})\cap
supp(f^{-})=\emptyset$. Hence

$I(f)=I(f^{+}+(-f^{-}))=I(f^{+})+I(-f^{-})=\int
f^{+}dv^{+}+\int\lambda(-f^{-})d\bar{v}^{-}$

Let $\hat{v}^{-}$ be the conjugate capacity of $\bar{v}^{-}$, i.e.
$\hat{v}^{-}(A)=1-\bar{v}^{-}(A^{c})$ for all $A\in{\cal A}$. Then by Lemma 2
one has

$\int-\lambda f^{-}d\bar{v}^{-}=-\int\lambda f^{-}d\hat{v}^{-}$

Defining $v^{-}=\hat{v}^{-}$ concludes the “$\Rightarrow$” part of the proof.

$(ii)\Rightarrow(i)$. We prove (a). Suppose $f\geq g$. Then $f^{+}\geq g^{+}$
and $g^{-}\geq f^{-}$. It is well known that the Choquet integral is
monotonic. Hence $I(f)=\int f^{+}dv^{+}-\int\lambda f^{-}dv^{-}\geq
g^{+}dv^{+}-\int\lambda g^{-}dv^{-}=I(g)$ .
We prove (b). Let $f,g$ comonotonic and such that $f,g\geq 0$ (the case
$f,g\leq 0$ is similar). Then $(f+g)^{+}=f+g=f^{+}+g^{+}$ and
$(f+g)^{-}=0=f^{-}=g^{-}$. Therefore

$I(f+g)=\int(f+g)^{+}dv^{+}=\int f^{+}dv^{+}+\int g^{+}dv^{+}=\\\ \int
f^{+}dv^{+}-\int\lambda f^{-}dv^{-}+\int g^{+}dv^{+}-\int\lambda
g^{-}dv^{-}=I(f)+I(g).$

We prove part (b) of $(ii)$. Let $f$, $g$ be of opposite sign (for instance
$f\geq 0$ and $g\leq 0$) and such that $supp(f)\cap supp(g)=\emptyset$. Notice
that $(f+g)^{+}=f=f^{+}$, $(f+g)^{-}=-g=g^{-}$, and $f^{-}=0$, $g^{+}=0$.
Therefore

$I(f+g)=\int(f+g)^{+}dv^{+}-\int\lambda(f+g)^{-}dv^{-}=\int
f^{+}dv^{+}-\int\lambda g^{-}dv^{-}=\\\ \int f^{+}dv^{+}-\int\lambda
f^{-}dv^{-}+\int g^{+}dv^{+}-\int\lambda g^{-}dv^{-}=I(f)+I(g)$

which complete the proof of part (b). ∎

###### Proof of Theorem 3.

$(i)\Rightarrow(ii)$ Let $CPT$ be a Šipoš integral. Then $\lambda=1$ and
$v^{+}=v^{-}$ and hence

$CPT(-f)=\int(-f)^{+}dv-\int(-f)^{-}dv=\int f^{-}dv-\int f^{+}dv=-CPT(f)$

$(ii)\Rightarrow(i)$ Note that $\lambda=1$ since
$-\lambda=CPT(-1_{S})=-CPT(1_{S})=-1$. Let $A\in{\cal A}$ and consider
$f=1_{A}$. Then

$CPT(-f)=0-\int 1_{A}dv^{-}=-v^{-}(A)\text{ and }-CPT(f)=-\int
1_{A}dv^{+}=-v^{+}(A)$

Therefore

$CPT(-f)=-CPT(f)\Leftrightarrow v^{-}(A)=v^{+}(A).$

Since this must be true for all $A\in{\cal A}$, $v^{-}=v^{+}$ and the CPT
functional is a Šipoš integral. ∎

###### Proof of Theorem 5.

$(ii)\Leftarrow(i)$ We only prove A.4. Take $\lambda>0$ of the CPT functional.
Fix $f$ and $g$ s.t. $f\in\mathcal{F}^{+}$ and $g\in\mathcal{F}^{-}$ and s.t.
$supp(f)\cap supp(g)=\emptyset$. Suppose $f\sim\alpha 1_{S}$ and $g\sim\beta
1_{S}$. Note that $\alpha\geq 0$ and $\beta\leq 0$. Therefore

$\displaystyle CPT(f)=CPT(\alpha 1_{S})$ $\displaystyle\Leftrightarrow
CPT(f)=\alpha$ $\displaystyle CPT(g)=CPT(\beta 1_{S})$
$\displaystyle\Leftrightarrow
CPT(g)=-\lambda\int\beta^{-}dv^{-}=-\lambda(-\beta)=\lambda\beta.$

Moreover since $f$ and $g$ have opposite signs and have disjoints supports we
have

$CPT(f+g)=CPT(f)+CPT(g)=\alpha+\lambda\beta.$

Now, if $\alpha+\lambda\beta>0$,
$CPT((\alpha+\lambda\beta)1_{S})=\alpha+\lambda\beta$, and since CPT
represents $\succsim$, $f+g\sim(\alpha+\lambda\beta)1_{S}$. If
$\alpha+\lambda\beta<0$,
$CPT\left({\alpha+\lambda\beta\over\lambda}1_{S}\right)=-\lambda\int\left({\alpha+\lambda\beta\over\lambda}\right)^{-}dv^{-}=-\lambda{-\alpha-\lambda\beta\over\lambda}=\alpha+\lambda\beta$.
Therefore $f+g\sim{\alpha+\lambda\beta\over\lambda}1_{S}$.

$(i)\Rightarrow(ii)$ First, note that for all $f\in\mathcal{F}^{+}$, $f=f^{+}$
and for all $f\in\mathcal{F}^{-}$, $f=-f^{-}$. (One can prove that) By A.1 and
A.2 for all $f\in\mathcal{F}^{+}$ there exists a unique $\alpha_{f^{+}}\geq 0$
s.t.

$f^{+}\sim\alpha_{f^{+}}1_{S}.$

Let $\lambda>0$ be the one of Axiom A.4. Then again by A.1 and A.2 for all
$f\in\mathcal{F}^{-}$ there exists a unique $\alpha_{-f^{-}}\leq 0$ s.t.

$-f^{-}\sim\left({\alpha_{-f^{-}}\over\lambda}\right)1_{S}.$

Define $I:\mathcal{F}\rightarrow\mathbb{R}$ as

$I(f)=I(f^{+})+I(-f^{-})$

where $I(f^{+})=\alpha_{f^{+}}$ and $I(-f^{-})=\alpha_{-f^{-}}$. Note that
$f^{+}\sim I(f^{+})1_{S}$ and
$-f^{-}\sim\left({I(-f^{-})\over\lambda}\right)1_{S}$. Moreover $I(1_{S})=1$
by Monotonicity.

We will prove that $I$ satisfies the conditions of Theorem 2 and it is
therefore a CPT functional.

######  Step 1.

Fix $f\in\mathcal{F}$, then $I(f)\geq 0$ implies $f\sim I(f)1_{S}$ and
$I(f)<0$ implies $f\sim{I(f)\over\lambda}1_{S}$.

###### Proof.

Let $f\in\mathcal{F}$.

  * •

Case 1: $I(f)\geq 0$. Note that $f=f^{+}+(-f^{-})$ and by definition
$f^{+}\sim I(f^{+})1_{S}$ and $-f^{-}\sim{I(-f^{-})\over\lambda}1_{S}$.
Moreover $I(f^{+})+\lambda{I(-f^{-})\over\lambda}=I(f)\geq 0$, hence by A.4
and by the definition of $I(f)$

$f=f^{+}+(-f^{-})\sim\left(I(f^{+})+\lambda{I(-f^{-})\over\lambda}\right)1_{S}=I(f)1_{S}.$

  * •

Case 2: $I(f)<0$. Then reasoning as before and applying A.4 we get

$f=f^{+}+(-f^{-})\sim\left({I(f^{+})+\lambda{I(-f^{-})\over\lambda}\over\lambda}\right)1_{S}=\left({I(f^{+})+I(-f^{-})\over\lambda}\right)1_{S}={I(f)\over\lambda}1_{S}.$

∎

######  Step 2.

$I$ is monotone.

###### Proof.

Let $f,g\in\mathcal{F}$ be such that $f\geq g$. Then $f^{+}\geq g^{+}$ and
$-f^{-}\geq-g^{-}$. Then by Monotonicity $f^{+}\succsim g^{+}$ and
$-f^{-}\succsim-g^{-}$. Then by Step 1 $I(f^{+})1_{S}\sim f^{+}\succsim
g^{+}\sim I(g^{+})1_{S}$ and
${I(-f^{-})\over\lambda}1_{S}\sim-f^{-}\succsim-g^{-}\sim{I(-g^{-})\over\lambda}1_{S}$.
Monotonicity implies $I(f^{+})\geq I(g^{+})$ and $I(-f^{-})\geq I(-g^{-})$.
Summing up we obtain $I(f)\geq I(g)$. ∎

######  Step 3.

$I$ satisfies comonotonic additivity over $\mathcal{F}^{+}$ and
$\mathcal{F}^{-}$.

###### Proof.

We prove comonotonic additivity over $\mathcal{F}^{-}$, the proof for
$\mathcal{F}^{+}$ can be done in a similar way.
Take $f,g\in\mathcal{F}^{-}$ s.t. $f$ and $g$ are comonotone. By Step 1,
$f\sim{I(f)\over\lambda}1_{S}$ and $g\sim{I(g)\over\lambda}1_{S}$. Since
constant acts are comonotone with all other acts and
${I(f)\over\lambda},{I(g)\over\lambda}\leq 0$, by A.3 one gets
$f+g\sim{I(f)\over\lambda}1_{S}+g$ and
$g+{I(f)\over\lambda}1_{S}\sim{I(g)\over\lambda}1_{S}+{I(f)\over\lambda}1_{S}$.
Since $f+g\in\mathcal{F}^{-}$, by Step 1 $f+g\sim{I(f+g)\over\lambda}1_{S}$.
Therefore
${I(f+g)\over\lambda}1_{S}\sim\left({I(f)\over\lambda}+{I(g)\over\lambda}\right)1_{S}$,
and Monotonicity implies $I(f+g)=I(f)+I(g)$. ∎

######  Step 4.

For all $f\in\mathcal{F}^{+(-)}$ and $g\in\mathcal{F}^{-(+)}$ s.t.
$supp(f)\cap supp(g)=\emptyset$, $I(f+g)=I(f)+I(g)$.

###### Proof.

Fix $f\in\mathcal{F}^{+}$ and $g\in\mathcal{F}^{-}$ s.t. $supp(f)\cap
supp(g)=\emptyset$. Define $h=f+g$ and note that $h^{+}=f$ and $-h^{-}=g$.
Therefore by definition of $I$, $I(f+g)=I(h)=I(h^{+})+I(-h^{-})=I(f)+I(g)$. ∎

######  Step 5.

$I$ represents $\succsim$ over $\mathcal{F}$ (i.e. $f\succsim g\Leftrightarrow
I(f)\geq I(g)$).

###### Proof.

Fix $f,g\in\mathcal{F}$. Then we have to consider 4 cases.

  * •

Case 1: $I(f),I(g)\geq 0$. Using Step 1 and Monotonicity $I(f)1_{S}\sim
f\succsim g\sim I(g)1_{S}\Leftrightarrow I(f)\geq I(g)$.

  * •

Case 2: $I(f),I(g)\leq 0$. Using Step 1 and Monotonicity
${I(f)\over\lambda}1_{S}\sim f\succsim
g\sim{I(g)\over\lambda}1_{S}\Leftrightarrow I(f)\geq I(g)$, since $\lambda>0$.

  * •

Case 3: $I(f)\geq 0>I(g)$. Using Step 1 and Monotonicity $I(f)1_{S}\sim
f\succsim g\sim{I(g)\over\lambda}1_{S}\Leftrightarrow I(f)\geq I(g)$. Note
that in this case we cannot have $g\succsim f$.

  * •

Case 4: $I(g)\geq 0>I(f)$. This is the same as Case 3.

∎

Since $I(1_{S})=1$, Steps 2, 3 and 4 prove that $I$ satisfies condition $(i)$
of Theorem 2 and therefore $I$ is a CPT functional. Moreover Step 5 shows that
$I$ represents $\succsim$. Therefore the proof is complete. ∎

###### Proof of Theorem 6.

$(i)\Rightarrow(ii)$ Note that A.3’ and A.3” imply A.3. Hence Theorem 5
applies and $I$ is represented by a CPT functional. It is left to show that
$v^{+}$ and $v^{-}$ are convex. We only show convexity of $v^{-}$. Fix
$A,B\in{\cal A}$ and note that $CPT(-1_{A})=-\lambda
v^{-}(A)=CPT(-v^{-}(A)1_{S})$ and a similar statement holds for $B\in{\cal
A}$. Therefore $-1_{A}\sim-v^{-}(A)1_{S}$ and $-1_{B}\sim-v^{-}(B)1_{S}$.
Since $-1_{B}$ is comonotonic with $-v^{-}(A)1_{S}$, by A.3”
$-v^{-}(A)1_{S}-1_{B}\succsim-1_{A}-1_{B}$. Moreover since $-v^{-}(A)1_{S}$ is
comonotonic with both $-1_{B}$ and $-v^{-}(B)1_{S}$ by A.3” we get
$-1_{B}-v^{-}(A)1_{S}\sim-v^{-}(B)1_{S}-v^{-}(A)1_{S}$. Therefore

$-v^{-}(B)1_{S}-v^{-}(A)1_{S}\sim-1_{B}-v^{-}(A)1_{S}\succsim-1_{A}-1_{B}$

Note that $-1_{A}-1_{B}=-1_{A\cup B}-1_{A\cap B}$ and since $1_{A\cup B}$ and
$1_{A\cap B}$ are comonotonic,

$\displaystyle CPT(-1_{A\cup B}-1_{A\cap B})$
$\displaystyle=-\int\lambda(-1_{A\cup B}-1_{A\cap B})^{-}dv^{-}$
$\displaystyle=-\lambda\left(\int 1_{A\cup B}dv^{-}+\int 1_{A\cap
B}dv^{-}\right)$ $\displaystyle=-\lambda[v^{-}(A\cup B)+v^{-}(A\cap B)].$

Therefore $-\lambda[v^{-}(A)+v^{-}(B)]=CPT(-v^{-}(A)1_{S}-v^{-}(B)1_{S})\geq
CPT(-1_{A\cup B}-1_{A\cap B})=-\lambda[v^{-}(A\cup B)+v^{-}(A\cap B)]$ which
implies $v^{-}(A)+v^{-}(B)\leq v^{-}(A\cup B)+v^{-}(A\cap B)$, i.e. $v^{-}$ is
convex.

$(ii)\Rightarrow(i)$ Left to the reader. ∎

###### Proof of Theorem 7.

$(i)\Rightarrow(ii)$ Since $\succsim$ satisfies A.1, A.2, A.3 and A.4, it can
be represented by a CPT functional $I$ by Theorem 5. Hence for all
$f\in\mathcal{F}$, $f\sim I(f)1_{S}$ and $-f\sim I(-f)1_{S}$. Notice that by
A.5 one has also $-f\sim-I(f)1_{S}$ and hence A.2 implies $I(-f)=-I(f)$. By
Theorem 3, $I$ is a Šipoš integral.

$(ii)\Rightarrow(i)$ Left to the reader. ∎

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# Measurement of the Sequential $3\alpha$ Process in the Photodissociation of
${}^{12}\mathrm{C}$

Resmi K.Bharathan1,2<EMAIL_ADDRESS>Midhun C.V1 M.M Musthafa1
<EMAIL_ADDRESS>Sreena M1,3 Silpa Ajaykumar1,2 Farhana Thesni M.P1 Swapna B1
Vafiya Thaslim T.T1 Shaima A1 Nived K1 Akhil R1 Anagha P.K1 Arunima Dev T.V1
Keerthi E.S1 Akshay K. S1 Arun P.V1,3 S. Ghugre4 1 Nuclear and Radiation
Physics Group, Department of Physics, University of Calicut-673635, India
2Medical Physics Group, Radiation Oncology, Malabar Cancer Centre,
Thalassery-670103, India  3Department of Medical Physics, Indo-American Cancer
Hospital & Research Institute, Hyderabad-500034, India  4UGC-DAE-Consortium
for Scientific Research, Kolkata, West Bengal 700098, India

###### Abstract

The cross sections for the
${}^{12}\mathrm{C}(\gamma,\alpha)^{8}\mathrm{Be}\rightarrow 3\alpha$ reaction
have been successfully measured using exclusive coincidence between three
$\alpha$ particles, minimizing Compton background. Sequential breakup
kinematics are evident, and the cross sections are presented as locally
averaged histogram values. Theoretical Fresco CDCC-CRC calculations reproduce
the experimental data, showing that the process involves electromagnetic
coupling to both ${}^{8}\mathrm{Be}^{0^{+}}$ and ${}^{8}\mathrm{Be}^{2^{+}}$
states. This study confirms that the
${}^{12}\mathrm{C}(\gamma,\alpha)^{8}\mathrm{Be}\rightarrow 3\alpha$ reaction
proceeds via a sequential mechanism, crucial for understanding its
significance in radiotherapy dosimetry.

##  I  Introduction

High-energy photon beams, reaching up to 18 MeV, find widespread application
in radiotherapy, where they deposit energy primarily through secondary
electron interactions, quantified by absorbed dose. To effectively treat deep-
seated tumors, higher energy bremsstrahlung beams are employed, but concerns
arise from nuclear reactions initiated by these beams, particularly
12C$(\gamma,3\alpha)$, which generates high-energy $\alpha$ particles.[1].
These particles, when deposited within a few cubic micrometers, create micro
hotspots, causing complete tissue destruction particularly in the target.
However, current dosimetry techniques struggle to measure these events due to
their small volume and the volume averaging effect of detector.[2] This issue
is critical given the abundance of 12C atoms in biological tissue and the high
photon flux used in radiotherapy. The clinical significance of micro hotspots
is determined by their mean separation and behavior, which are influenced by
the 12C($\gamma$, 3$\alpha$) cross section and the alpha spectrum,
respectively. Understanding these parameters is crucial for optimizing
radiotherapy treatments.

Radiation transport simulations are increasingly favoured over traditional
dosimetric measurements for the prediction of micro hotspots in radiotherapy,
primarily due to inherent limitations in the latter approach. These
simulations necessitate a meticulous examination and analysis of the
12C($\gamma$, 3$\alpha$) reaction cross sections, as they play a fundamental
role in the accurate depiction of radiation interaction dynamics. However,
despite advancements, the full comprehension of the 12C($\gamma$, 3$\alpha$)
reaction kinematics remains elusive, particularly in scenarios where the
excitation energy of 12C surpasses the critical threshold of the $\alpha$
separation energy of 7.36 MeV. Above this energy threshold, the behavior of
the reaction and the ensuing particle spectrum become increasingly complex
under three body kinematics, with higher excitation levels of 12C resulting in
a proportional increase in the kinetic energy exhibited by the resultant
$\alpha$ particles upon their breakup. Moreover, the presence of resonant
couplings within the exit channel of the reaction further complicates the
determination of the excitation function and particle spectrum, introducing
significant variability and unpredictability into the overall process.

The 12C($\gamma$,3$\alpha$) reaction is believed to follow a sequential
process, beginning with the capture of a photon by 12C, which then leads to
the population of the 8Be+$\alpha$ state via electromagnetic coupling.
Subsequently, due to the unbound nature of 8Be, it undergoes instantaneous
breakup into two $\alpha$ particles. Additionally, resonant states of 8Be
contribute to the reaction, alongside overlaps from the
8Be$\rightarrow\alpha+\alpha$ breakup continuum. These various couplings and
overlaps are expected to alter both the
12C($\gamma$,$\alpha$)8Be$\rightarrow$3$\alpha$ excitation function and the
resulting $\alpha$ spectrum. Hence, besides the excitation function, the
resonant couplings has to be well understood for generating evaluated data
sets to be applied in simulations.

There are numerous measurements on 12C, break up into $3\alpha$ reaction.
However, most of them are limited to the resonant breakups of 12C, which are
different from the electromagnetic coupling. Kirsebom et al.[3, 4] has studied
the sequential breakup of 12C$\rightarrow^{8}$Be${}^{*}+\alpha\rightarrow
3\alpha$ via the 10Be$(^{3}\mathrm{He},p)^{12}$C reaction and identified an
electromagnetic coupling to 8Be${}^{*}+\alpha$ states. However the measurement
is with a limited statistics. Similar kind of electromagnetic coupling has
been observed by Laursen et al.[5] in the ${}^{11}\mathrm{B}+p$ reaction,
decaying to 8Be${}^{*}+\alpha$ states. Kirsebom et al.[6] has identified the
sequential breakups of unbound resonant states
$(2^{-},1^{+}\>\mathrm{and}\>4^{-})$ of 12C via the resonant states of 8Be.
Similarly, weak electromagnetic couplings in the transition from Hoyle state
has been observed by T.K Rana et al [7, 8]. These works are based on the
population of 12C states via proton capture, transfer induced or $\beta$
delayed pathways. These reactions are populating the resonant states of 12C
than the 8Be$*+\alpha$ continuum, hence these results cannot directly adapted
for 12C($\gamma$,$\alpha$)8Be$\rightarrow$3$\alpha$ reaction phase.

There are several challenges existing to execute the measurement of
12C($\gamma$,$\alpha$)8Be$\rightarrow$3$\alpha$ reaction such as the
availability of monoenergetic $\gamma$ for sufficient intensity, isolating
Compton electrons and other scattered photons eventing on the detectors. Since
the bremsstrahlung beams of higher intensities are available, reconstructing
true events from the higher background level is the major concern.

Hence to address the real situation of populating $\alpha+^{8}$Be (ground and
$0^{+}$) through electromagnetic coupling, an experiment on
12C($\gamma$,$\alpha$)8Be$\rightarrow$3$\alpha$ has been performed, and
successfully detected the 3$\alpha$ particles, with exclusive event
identification, by minimizing the Compton background. The measured cross
sections has been reproduced with fresco CDCC-CRC calculations.

Figure 1: Schematic diagram of the experimental set up showing the Silicon
detectors positions and target orientations

##  II Materials and Methods

The reaction, 12C($\gamma$,$\alpha$)8Be$\rightarrow$3$\alpha$, has been
performed with 14.6 MeV end point energy bremsstrahlung beam, as there is no
mono energetic source exists in the current scenario. The experimental setup
has been designed in such a way to detect the $3\>\alpha$s in coincidence, for
a limited solid angle, which made the capability of reconstruction of the
photon energy associated to each individual event. The detector angles are
chosen as per the three body sequential kinematics given by R. J. de Meijer
and R. Kamermans [9]. The schematic diagram of experimental setup is given in
Figure 1. The illustration of kinematically correlated three body events are
shown in Figure 2.

The primary $\alpha$ angle is set at $+60^{\circ}$ to mitigate the significant
influence of Compton electrons and photons at extreme forward angles. To
achieve exclusive coincidences, other detector angles are meticulously chosen
at $-80^{\circ}$ and $-110^{\circ}$, as determined by relativistic kinematics
calculations. The experimental setup, depicted in Figure3, illustrates this
configuration. Conducted at the Varian Clinac-iX accelerator facility at
Malabar Cancer Centre, Thalassery, the experiment utilized a bremsstrahlung
beam with an endpoint energy of 14.6 MeV. This beam, meticulously collimated,
was directed towards a 300 $\mu$m thick carbon target (natC). The real
experimental setup with Varian Clinac-iX and the scattering chamber mounted on
the treatment couch is shown in Figure 3.

Figure 2: Calculated event kinematics for achieving the exclusive coincidence

Three silicon surface barrier detectors, each 1500 $\mu$m thick, were employed
to detect the $\alpha$ particles. The detector collimators are chosen as
aluminum, to minimize the electrons generated from the collimator, reaching
the detectors. These detector signals were preamplified using Ortec-IH142
preamplifiers, and the data was acquired via a CAEN- DT5743 digitizer. These
detectors were biased at an optimal voltage to ensure a well-measurable pulse
rise time. To minimize pulse height aberrations at reduced voltage, the energy
was mapped into charge rather than peak voltage. The detectors were calibrated
using 5.48MeV $\alpha$ particles from ${}^{241}\mathrm{Am}$ source. A
precision pulser has been configured with 100Hz, to ensure the calibration, by
identifying the linearity. Particle identification was achieved through a 2D
correlation between charge and pulse rise time, similar to the method
described in [10]. However with pulse rise time instead of zero crossover
timing. Additionally, proper CFD backings were configured for the detectors to
prevent pileups caused by Compton electrons. In addition to recording
individual detector energies, a multiplicity trigger and energy sum generated
from CFD were also acquired to enable fast identification and scaling of
exclusive three-body coincidences. The Compton events were figured with PSD
and rejected from event registration in the offline analysis of the waveform.
The measurement has been performed with the external trigger mode, with the
multiplicity $\geq\>2$.

To determine the flux of the photon beam, accelerator-built transmission-type
beam monitor chambers were employed, maintaining a consistent flux of
approximately $10^{13}\>\mathrm{photons}/cm^{2}$ throughout the experiment.
The built-in monitor chambers were calibrated prior to the experiment using
secondary standard ionization chamber, to a calibration of 1 monitor unit (MU)
equals 1 cGy. This is converted into flux by Kerma formalism used for
dosimetry[11].

A 2D plot between charge and rise time has been constructed for each detector.
The typical particle discrimination plot is shown in Figure 4. The proton and
$\alpha$ bands are well evident without overlaps in the present configuration.
The $\alpha$ events of each detector were banana gated for event localization.
A total energy parameter, gated for triple coincidence alpha events has been
generated. The $\alpha$ spectrum identified in the $60^{\circ}$, without
triple coincidence gate, is given in Figure 5. This has been summed with the
breakup Q value for obtaining the photon energy subjected for the break up. It
is assumed that the 92 keV added by the 8Be decay is lesser than the binning
resolution, hence it is excluded.

Figure 3: Experimental setup showing Clinac-iX Medical Linear Accelerator,
with scattering chamber on the treatment couch Figure 4: The Charge Pulse
shape correlation showing the particle discrimination

The events in the photon energy domain has been binned into locally averaged
histogram values. The event yields in each energy bin has been converted into
cross sections by accounting the photon flux corresponding to each
bremsstrahlung energy bin and the three body kinematic efficiencies. The
intensity scaling factors were accounted from the segmental integration of the
bremsstrahlung spectrum. The spectum for the 14.6 MeV bremsstrahlung has been
adapted through the method by Midhun et al.[12]. Further the kinematic
efficiencies for the experimental setup has been estimated through Monte-Carlo
approach on 3 body sequential emission kinematics. The calculated kinematic
efficiencies and bremsstrahlung weights are given in Figure 6. The obtained
cross sections were compared with fresco  CDCC-CRC calculations[13].

Figure 5: The measured row $\alpha$ spectrum at 60∘ detector

##  III fresco Calculations

The concept adapted for theoretical calculations is structured such that the
$\gamma+^{12}\mathrm{C}$ entrance channel state couples to the final
$\alpha+^{8}\mathrm{Be}$ states through electromagnetic modes. For this
current analysis, the $E_{1}$ mode is considered. As the final state forms,
the ejectile $\alpha$ wave function couples to the ${}^{8}\mathrm{Be}$
resonant or breakup continuum, resulting in a sequential breakup. This concept
has been implemented through Fresco CDCC-CRC calculations.

The entrance channel mass partition 12C+$\gamma$ has been defined with a
pseudo potential for photons. $\alpha$+8Be has been defined as the exit
channel. The 8Be is binned into $\alpha$ core and $\alpha$ valance structure.
Potential having Gaussian shape, with $V_{0}=106\>MeV$ and $R_{0}=2.236\>fm$,
is used to define the $\alpha$+$\alpha$ breakup continuum state of 8Be. The
Guassian potential parameters has been retrieved from Marquez et al.[14] and
Rusek et al.[15]. The $0^{+}\>\mathrm{(ground)}$ and
$2^{+}\>\mathrm{(3.03\>MeV)}$ states of 8Be has been considered. These states
has been defined with $p+^{7}Li$ binning potential, having Woods-Saxon volume
form with parameters $V_{0}=50.6\>MeV$, $r_{0}=1.15\>fm$ , $a_{0}=0.57\>fm$,
$V_{so}=5.5\>MeV$, $r_{so}=1.15\>fm$ and $a_{so}=0.57\>fm$. The potential
shape and parameters were adapted from Chattopadhyay et al. [16].

Along with the CRC states, the break up continuum generated by
$\braket{\alpha+\alpha}{{}^{8}\mathrm{Be}}$ is also considered. The breakup
continuum generated through the relative movement of $\alpha-\alpha$ cluster
in 8Be, has been discritized based on the wave number and angular momentum,
based on,

$k=\frac{\sqrt{2\mu E_{x}}}{\hbar^{2}}$ (1)

Where $\mu$ is the reduced mass of $\alpha-\alpha$ system, and $E_{x}$ being
the excitation energy. The discritized states of continuum having bin width of
0.175 fm has been generated upto $k^{-1}=0.7\>fm$. The spin of each bin has
been taken as the relative angular momentum of the $\alpha-\alpha$ system at
that angular momentum where as the spin of $\alpha$ is zero by default.
$3\alpha$ folding potentials with Gaussian form factor, with parameters
described above, has been used for defining these states. An overlap of these
CDCC states over the $0^{+}$ and $2^{+}$ CRC states has been defined, for
accounting the width broadening of the resonant states.

All the states in exit channel mass partition has been made coupled to the
entrance channel mass partition electromagnetic coupling of $E_{1}$ mode
(electric dipole mode). The spectroscopy factors for $0^{+}$ and $2^{+}$ CRC
states has been used as unity, adapting the value from D.Chattopadhyay et
al.[16]. An average spectroscopy factor of 0.75 has been used for CDCC states.
However it couldn’t be verified in the current scenario as due to the
systematic uncertainties and poor discrimination in the relative energy
determination. The fresco has been run for CRC only for the first instance and
the CDCC+CRC for the second instance. The cross section for each state has
been taken and 60∘ cross sections has been selected. Each bin energy has been
converted to $\gamma$ beam incident energy using relativistic two body
kinematics. The contributions from individual states has been added to get the
final cross sections. The final cross sections has been compared with
experimental cross sections.

##  IV Result and Discussion

Figure 6: Kinematic efficiencies and Bremsstrauhlung yield for computing each
excitation energy  Figure 7: Experimental cross section adding with FRESCO
calculation for $0^{+}$ and $2^{+}$ contributions

The 12C($\gamma$,$\alpha$)8Be$\rightarrow$3$\alpha$ cross sections has been
successfully measured by constructing exclusive coincidence between 3$\alpha$
particle events, by establishing multiplicity trigger condition. The events
were recorded with minimum Compton background, unlike the previous attempts.
The events under sequential kinematics is illustrated in Figure 4. The cross
sections are presented as locally averaged histogram values, in the fraction
of photon energy, in Figure 7. The present cross sections belongs to inclusive
cross section of all states of 8Be, populated in the present scenario, as the
relative energy resolution provided by the present method is not enough to
identify the individual breakup contributions. Since the geometrical
efficiency of the present set up by the 3 body sequential emission kinematics
is similar to the total efficiency. Hence the same has been used. This adds up
a systematic uncertainty of 1.3% to the cross section data.

The fresco CDCC – CRC calculations are shown as solid lines in the Figure 7.
The blue line show the contributions from
12C($\gamma$,$\alpha$)8Be$\rightarrow 3\alpha$, corresponding to the $\alpha$
production from the ground state of 8Be, and Green curve shows the $\alpha$
production from the $2^{+}$ resonance state of 8Be (3.03 MeV). The coherent
sum of the contributions from $0^{+}$ and $2^{+}$ state of 8Be explains the
experimental cross section. Based on the experimental measurement, with the
support of the theoretical calculations, it can elucidate that the
12C($\gamma$,$\alpha$)8Be$\rightarrow 3\alpha$ process is a sequential
process.

The $\gamma+^{12}$C initial capture state populates $\alpha+^{8}$Be state
through electromagnetic coupling in direct mode to
${}^{8}\mathrm{Be}^{0^{+}}+\alpha$ or ${}^{8}\mathrm{Be}^{2^{+}}+\alpha$
states. The present measurement followed by the CDCC-CRC identifies the
presence of the coupling of $2^{+}$ state in the excitation function, above
the $E_{\gamma}$ of 10.8 MeV, where as the Q-value for populating this state
is 10.39 MeV. The $2^{+}$ component is showing a broad resonant width due to
the overlap of $\braket{\alpha+\alpha}{{}^{8}\mathrm{Be}}$ breakup continuum.

Identification of the electromagnetic couplings to
${}^{8}\mathrm{Be}^{0^{+}}+\alpha$ and ${}^{8}\mathrm{Be}^{2^{+}}+\alpha$
states can be coincided along with the observation by Laursen et al.[5] and
Kirsebom et al. [4, 6]. There Kirsebom et al. [4] discusses the population of
the $2^{+}$ state of 8Be through the decay of 12.71 MeV $(1^{+})$ state of
12C. However it was holding an ambiguity about the sequential breakup
mechanism as the life time of the $2^{+}$ state of 8Be is about $1.5\>MeV$.
The present study based on the measurement with direct photons, followed by
CDCC-CRC analysis shows that the population of 8Be states are due to the
direct electromagnetic couplings. However, the continuum states are existing
as a equal counter part. By considering the ${}^{7}\mathrm{Li}+p$ binning
potential for these states, the the population has been well explained and the
ambiguity is thereby clarified.

The exclusive cross sections are typically in fractions of millibarns, and the
average energy of the $\alpha$ particles observed is around 3 MeV. Considering
these cross sections, photon flux used in radiotherapy, and the number of 12C
nuclei present in the unit tissue, the
${}^{12}\mathrm{C}(\gamma,\alpha)^{8}\mathrm{Be}\rightarrow 3\alpha$ events
are very significant, as each event deposit almost 170 Gy dose in a 10$\mu
m^{3}$ volume. As considering serial organs like spine and other neutral
tissue, this makes a permanent damage, and may leads to a paralysis of the
serial organ. Further extensive measurements with greater solid angle coverage
and higher statistics are required to improve our understanding of this
process.

##  V Conclusion

The sequential breakup mechanism of
${}^{12}\mathrm{C}(\gamma,\alpha)^{8}\mathrm{Be}\rightarrow 3\alpha$ has been
experimentally observed in the $\gamma$ energy range of 9.5 MeV to 13.5 MeV.
The experimental cross sections were successfully reproduced using fresco
CDCC-CRC calculations. These calculations indicate that the process is
sequential, the electromagnetic coupling directly populates
${}^{8}\mathrm{Be}^{0^{+}}+\alpha$ and ${}^{8}\mathrm{Be}^{2^{+}}+\alpha$
states. The overlap of breakup continuum on the resonant states makes the
resonant widths broader.

##  VI Acknowledgements

The authors gratefully acknowledge the support of Dr. Geetha Satheeshan, Head
of the Department of Radiation Oncology at Malabar Cancer Center, Thalassery,
as well as the invaluable assistance provided by the radiotherapy technical
staff.

The authors extend their sincere thanks to UGC-DAE-CSR-KC for providing
financial support for this project.

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