f0mu y0mu p0mu l0mu j

# Does personality affect the allocation of resources within households?

[Version: ]

(Gastón P. Fernández111Ph.D. student at the University of Leuven (KU Leuven),
Department of Economics, Naamsestraat 69, box 3565, 3000 Leuven (e-mail:
gfernandez@kuleuven.be). I deeply appreciate the invaluable guidance of my
advisors Laurens Cherchye and Frederic Vermeulen. I would also like to thank
Wietse Leleu and all participants at the Conference of the European Society
for Population Economics (ESPE) in Belgrade, the Trans-Atlantic Doctoral
Conference (TADC) in London, and the Public-Labor-Health Seminar, the
Household Economics Gathering, and the ECORES Summer School in Leuven for
their helpful comments. All errors are on my own.
University of Leuven (KU Leuven))

###### Abstract

This paper examines whether personality influences the allocation of resources
within households. To do so, I model households as couples who make Pareto-
efficient allocations and divide resources according to a distribution
function. Using a sample of Dutch couples from the LISS survey with detailed
information on consumption, labor supply, and personality traits at the
individual level, I find that personality affects intrahousehold allocations
through two channels. Firstly, the level of these traits act as preference
factors that shape individual tastes for consumed goods and leisure time.
Secondly, by testing distribution factor proportionality and the exclusion
restriction of a conditional demand system, I observe that differences in
personality between spouses act as distribution factors. Specifically, these
differences in personality impact the allocation of resources by affecting the
bargaining process within households. For example, women who are relatively
more conscientious, have higher self-esteem, and engage more cognitively than
their male partners receive a larger share of intrafamily resources.

JEL Classification Numbers: D1, J12, J22, J24

Keywords: Collective Household Model, Distribution Factors, Personality
Traits.

##  Introduction

There is increasing evidence that personality traits matter for relevant life
outcomes Heckman . (2021). For instance, personality is associated with the
formation of future cognitive skills Cunha . (2010) or with the educational
and occupational choices over the life cycle Todd Zhang (2020). Personality is
also correlated with the probability of marriage and divorce Lundberg (2012)
and is a relevant attribute on which individuals sort into the marriage market
Dupuy Galichon (2014). Nevertheless, much less is currently known about
personality’s impact on intrahousehold consumption patterns. For example, do
personality traits affect the allocation of resources through their impact on
individual preferences over goods? Or are there other mechanisms by which
personality might shape the way couples decide over total resources? Is
personality related to the distribution of power within households?

In this paper, I aim to empirically investigate the questions mentioned above
by structurally testing the role of personality traits in resource allocation
within households. I show that differences in personality traits between
spouses play a significant role in shaping the distribution of resources
within established households. Families are modeled as couples who make static
decisions regarding private and public consumption and also allocate their
time to the labor market. As a starting point, I assume that each adult
household member has their own rational preferences. Additionally, I assume
that couples make Pareto-efficient allocations and distribute resources among
household members through an intrahousehold decision process Chiappori (1988,
1992). By adopting this framework, I can test the concept of collective
rationality, which refers to the collective model, using observed household
allocations. This approach allows me to uncover relevant information
underlying the consumption process. The main focus of this paper is to explore
the hypothesis that personality traits may partially determine how couples
divide resources. To investigate this, I test various theoretical restrictions
of the collective model as formalized by bourguignon2009efficient. The
collective framework not only enables the characterization of couples in terms
of rational decisions but also allows for the integration of individual
personality into a model of household consumption and labor supply.

This article contributes theory-based evidence about new channels that may
explain consumption inequality within households. In the collective model,
couples maximize a weighted sum of individual utilities, where the weights are
referred to as Pareto weights. In the collective literature, when examining
the impact of a specific variable on household behavior, a distinction is made
between two channels: preference and distribution factors. Preference factors
typically influence individual preferences for consumed commodities, while
distribution factors specifically affect the decision-making process within
the household through changes in the Pareto weights. In this sense, the level
of a specific variable (e.g., years of schooling) is often considered as a
preference factor and the relative amount of it (e.g., differences in
education between partners) as a distribution factor. I leverage this notion
and investigate both distribution factor proportionality and the exclusion
restriction of a conditional demand system by utilizing differences in
personality between spouses. I test the testable restrictions derived from
collectively rational behavior and find no evidence to reject that differences
in personality influence the bargaining process. Furthermore, I demonstrate
that certain personality factors, such as differences in conscientiousness or
self-esteem between spouses, are strongly associated with consumption
inequality within the household. These findings provide valuable insights into
the role of personality traits in shaping intrahousehold resource allocation
dynamics.

Distribution factors, which influence household decisions without directly
impacting preferences, have been extensively studied in the collective
literature. These factors encompass a wide range of variables, including
relative wages among spouses and the presence of divorce laws in relevant
matching markets. For instance, browning1994income demonstrate that the
intrahousehold allocation of resources is related to factors such as relative
ages and relative incomes in consumption models. chiappori2002marriage extend
earlier versions of the collective model and test their implications by
introducing the local sex ratio and divorce laws as distribution factors in a
labor supply model. In a nonparametric setting, cherchye2011revealed examine
the relationship between the intrahousehold share of income and differences in
age and educational level between spouses. Furthermore, exploiting exogenous
variation from a randomized cash transfer program in Mexico, several studies
have constructed distribution factors and tested the theoretical restrictions
of the collective model (see bobonis2009allocation, attanasio2014efficient,
de2022household).222See browning2014economics for a comprehensive review.

Building upon the collective framework and the existing applied research on
the impact of personality, this paper contributes novel evidence suggesting
that both intrahousehold rational behavior and consumption inequality are
linked to the personality types of household members. While recent
advancements in personality research have been extensively reviewed (see
john2010handbook), the detailed examination of this particular issue is still
relatively unexplored. In a related study, flinn2018personality develop a
model of household behavior and apply it to Australian data to investigate how
personality traits influence cooperative and non-cooperative interactions
within households, as well as members’ labor supply and wage rates. Their
findings demonstrate that personality directly affects intrahousehold behavior
and also indirectly impacts individual wages. The approach taken in the
present paper differs from flinn2018personality. Instead of applying a
behavioral model to the data, this study leverages a set of testable
restrictions derived from bourguignon2009efficient, which serve as necessary
and sufficient conditions for the collective model. By adopting this approach,
the current study can structurally test the extent to which personality traits
determine the allocation of resources between partners by influencing their
respective bargaining positions within the household.

The rest of the paper unfolds as follows. Section 2 provides an introduction
to the notation used and presents a collective model of household consumption
and labor supply. This section also outlines the testable restrictions of the
model based on observed household behavior, specifically focusing on
distribution factor proportionality and the exclusion restriction of a
conditional demand system. In Section 3, the sample used in the analysis is
described, along with the available measures of personality traits. Section 4
outlines the empirical strategy employed in the study. It discusses how
potential issues of multicollinearity in personality traits are addressed.
Furthermore, it presents the functional form for the household demand
functions and explains how tests of the collective model are derived from
these functions. Section 5 presents the results obtained from testing the
restrictions of the collective model. In Section 6, the relationship between
intrahousehold consumption inequality and personality traits is discussed.
Finally, Section 7 concludes the paper.

##  Theory

The analysis considers households consisting of two adult members: the wife
($f$) and the husband ($m$). These individuals jointly make consumption
decisions involving a Hicksian public good ($C\in\mathbb{R}_{+}$), private
Hicksian assignable goods for each member ($c^{i}\in\mathbb{R}_{+}$), and
individual leisure time ($\ell^{i}=T-L^{i}$), where
$\ell^{i}\in\mathbb{R}_{+}$ represents the amount of leisure time, $T$ is the
time endowment for each individual, and $L$ is the time supplied to labor
($i=m,f$). It is assumed that children, if present, do not participate in the
allocation of the household budget. The prices of all Hicksian goods are
normalized to one and wages ($w^{i}\in\mathbb{R}_{++}$) represent the prices
of individual leisure. The preferences of household members are captured by
well-behaved utility functions. Each individual has an egoistic utility
function denoted as $u^{i}(c^{i},\ell^{i},C;\bm{\xi})$. The utility function
also depends on the vector $\bm{\xi}$, which represents observed heterogeneity
(i.e., taste shifters).

In the collective model of chiappori1988rational, chiappori1992collective, any
Pareto-efficient intrahousehold allocation can be characterized as the
solution of the following optimization program:

$\begin{split}\max_{c^{m},c^{f},\ell^{m},\ell^{f},C}&\biggr{[}u^{m}(c^{m},\ell^{m},C;\bm{\xi})+\mu(w^{m},w^{f},y,\textbf{z})u^{f}(c^{f},\ell^{f},C;\bm{\xi})\biggr{]}\\\
\textrm{s.t.}&\quad c^{m}+c^{f}+C+w^{m}\ell^{m}+w^{f}\ell^{f}\leq y,\\\
&\quad\hskip 119.50148ptc^{i}\geq 0,\\\ &\quad\hskip 119.50148ptC\geq 0,\\\
&\quad\hskip 96.73918ptT\geq\ell^{i}\geq 0,\\\ \end{split}$ (P1)

where $y$ is household full income defined by $y=w^{m}T+w^{f}T+x$ with
$x\in{\rm I\\!R}_{+}$ the household nonlabor income, and $\mu\in$ $]0,1[$ in
the objective function is the Pareto weight that depends on (exogenous) wages,
income, and distribution factors (z). A variation on elements of z could
impact outside options of household members and thus their intrahousehold
bargaining power (see vermeulen2002collective).333In axiomatic bargaining
models, variables that are only applicable for threat points of the bargaining
process can be potential distribution factors. See the discussion about
extrahousehold environmetal parameters in mcelroy1990empirical and about
bargaining models in browning2014economics. I take both household composition
and intrafamily allocation of power as exogenously given. The solution to (P1)
implies a set of differentiable household demand functions for goods and
leisure that depend on prices, full income, observed heterogeneity, and the
distribution function:

$\textbf{g}=\textbf{g}\Bigr{[}w^{m},w^{f},y,\mu(w^{m},w^{f},y,\textbf{z});\bm{\xi}\Bigr{]}\quad\quad\forall\hskip
5.69054pt\textbf{g}\in\\{\textbf{c},\bm{\ell},C\\}.$ (1)

Distribution factor proportionality. As explained by bourguignon2009efficient,
in a setting with no price variation distribution factor proportionality is
necessary and sufficient for the collective model.444The first notions of the
proportionality condition with only private consumption are introduced in
bourguignon1993intra and browning1994income. bourguignon2009efficient extend
these results for public goods and externalities in consumption. This entails
testing a set of cross-equation restrictions based on the estimation of the
household demand system (1):

$\frac{\partial c^{m}/\partial z_{1}}{\partial c^{m}/\partial
z_{k}}=\frac{\partial c^{f}/\partial z_{1}}{\partial c^{f}/\partial
z_{k}}=\frac{\partial\ell^{m}/\partial z_{1}}{\partial\ell^{m}/\partial
z_{k}}=\frac{\partial\ell^{f}/\partial z_{1}}{\partial\ell^{f}/\partial
z_{k}}=\frac{\partial C/\partial z_{1}}{\partial C/\partial
z_{k}}\quad\quad\forall\hskip 5.69054ptk=2,\dots,K.$ (2)

The intuition of equation (2) is that distribution factors (z) only affect the
intrahousehold allocation of consumption and leisure through their impact on
the distribution function ($\mu$). To see this, take the marginal change in
distribution factor $z_{k}$ on the household demand for commodity $j$:

$\begin{split}\frac{\partial g_{j}}{\partial z_{k}}&=\frac{\partial
g_{j}}{\partial\mu}\frac{\partial\mu}{\partial z_{k}}.\end{split}$ (3)

Comparing the effect of two distribution factors, $z_{k}$ and $z_{l}$, we get:

$\begin{split}\frac{\partial g_{j}/\partial z_{k}}{\partial g_{j}/\partial
z_{l}}&=\frac{\partial\mu/\partial z_{k}}{\partial\mu/\partial
z_{l}},\end{split}$ (4)

where the right-hand-side term in equation (4) is independent of the demand
for good $j$.

$z$-conditional demand system. An alternative demand system is the
$z$-conditional system coined by bourguignon2009efficient. Under the
assumption that distribution factor $z_{1}$, say, is strictly monotonic on
commodity $c^{m}$, say, it is possible to invert the demand function for such
good on this (continuous) factor:

$z_{1}=v(w^{m},w^{f},y,c^{m},\textbf{z}_{-1};\bm{\xi}),$ (5)

where $\textbf{z}_{-1}$ is equal to z but excluding the first
element.555Appendix B provides evidence that supports monotonicity between
male private consumption and one of the distribution factors presented in
Section 4. Substituting (5) into the demand for the remaining goods
$\Phi(\cdot)$, we get the $z$-conditional demand system for
$\tilde{\textbf{g}}$ with $\tilde{\textbf{g}}\in\\{c^{f},\bm{\ell},C\\}$:

$\begin{split}\tilde{\textbf{g}}&=\Phi(w^{m},w^{f},y,\textbf{z};\bm{\xi}),\\\
&=\Phi\Bigr{[}w^{m},w^{f},y,v(w^{m},w^{f},y,c^{m},\textbf{z}_{-1};\bm{\xi}),\textbf{z}_{-1};\bm{\xi}\Bigr{]},\\\
&=\tilde{\textbf{g}}(w^{m},w^{f},y,c^{m},\textbf{z}_{-1};\bm{\xi}).\\\
\end{split}$ (6)

The restriction of the collective model based on the estimation of the
(conditional) demand system in equation (6) states that subject to the
conditioning good ($c^{m}$), the demand for the remaining goods should be
independent of all other distribution factors. This translates into the
following testable implication:

$\frac{\partial\tilde{\textbf{g}}(w^{m},w^{f},y,c^{m},\textbf{z}_{-1};\bm{\xi})}{\partial
z_{k}}=0\quad\quad\forall\hskip 5.69054ptk=2,\dots,K.$ (7)

The restriction described in equation (7) implies that, conditional on the
commodity used to invert $z_{1}$, additional distribution factors should not
provide any meaningful additional information about the intrahousehold
behavior. It is important to note that for this restriction to have empirical
significance, it requires at least two distribution factors and at least two
demand functions.

Although the testable implication in equation (7) is empirically more powerful
than implication (2), which is used as a robustness check in the empirical
application, both restrictions capture the same underlying mechanism.666See
Proposition 2 in bourguignon2009efficient and the discussion thereof. The
intuition behind this is illustrated in Figure 1. Suppose we observe an
optimal household demand function that is relatively more representative of
$m$’s preferences, such as $\textbf{g}^{0}$. Now, assume that we want to
reallocate intrahousehold resources in a manner that is more favorable to the
wife’s ($f$) preferences, resulting in household decisions represented by
$\textbf{g}^{1}$. The testable restrictions of the collective model inform us
that variations in the distribution factors z would only impact such a
reallocation of resources by shifting the individual bargaining weights
($\mu$). In other words, distribution factors do not alter the Pareto frontier
since they do not directly affect preferences or the budget constraint.

Figure 1: The collective effect

$u^{m}$$u^{f}$Utility Pareto
frontier$\textbf{g}^{0}=(\tilde{\textbf{c}},\tilde{\bm{\ell}},\tilde{C})$$\textbf{g}^{1}=(\hat{\textbf{c}},\hat{\bm{\ell}},\hat{C})$
$\frac{\partial\mu(\textbf{w},y,\textbf{z})}{\partial z_{k}}$

Source: Based on browning2014economics.

##  Data

I use a sample of Dutch households obtained from the Dutch Longitudinal
Internet Studies for the Social sciences (LISS) panel gathered by CentERdata.
This dataset provides rich information on economic and sociodemographic
variables. Crucially, it also collects detailed data on individual consumption
and a set of member-specific personality scales.

The sample selection criteria for this study are as follows, similar to those
used in other studies such as cherchye2017household and cherchye2012married.
Couples included in the sample must have both adults between the ages of 25
and 65. Both adults in the couple must participate in the labor market for at
least 10 hours per week, as wage information is required. Couples with at
least one self-employed adult are excluded from the sample. This is because
obtaining wage information for self-employed individuals is more complex
compared to salaried workers. The sample includes only couples with no
additional household members apart from children residing in the household.
For example, couples living with friends or parents are excluded. Due to
significant imbalance issues in the panel structure of the data, the study
does not make use of the panel structure and treats the data as a pooled
cross-section. Overall, the sample consists of 1130 couples pooled from five
different years, ranging from 2009 to 2015.

Table 1 provides summary statistics for the main variables used in the
analysis. All economic variables are in weekly real terms. Full income is
defined as the sum of spouses’ wages multiplied by the total time available
(i.e., 112) plus any non-labor income of the household. Leisure for each
partner is derived by subtracting the hours worked by each individual from the
total available time. The dataset includes information on assignable
consumption for each household member. This refers to individual expenditures
on various goods such as food, tobacco, or clothing. In the empirical
analysis, these individual expenditures are treated as a Hicksian aggregate
commodity. Total household private consumption represents the sum of both
spouses’ total private consumption, including their individual assignable
consumption. Household consumption is calculated as the sum of public
consumption and assignable private consumption. Public expenses, such as
mortgage payments, are considered as a Hicksian aggregate commodity. As shown
in Table 1, females work fewer hours and have lower wages compared to males.
In terms of assignable consumption, females spend slightly more per week than
males. The majority of total household consumption comes from public expenses.
Females allocate more time to leisure activities than males, although a
detailed breakdown of non-labor time is not available.777Data about the
individual time allocated to household chores is only available in three
waves. Demographically, males are slightly older and have a higher educational
level compared to females.

The spouses’ personality traits in this study are measured using three
different sources. The first source is Rosenberg’s Self-Esteem Scale Rosenberg
(1965), which assesses individuals’ perceptions of their self-worth. The
second source is the Need For Cognition Scale Cacioppo Petty (1982), which
serves as a proxy for an individual’s inclination to engage in intellectual
activities. The third source is the Big Five Personality Traits questionnaire
Goldberg (1990, 1992), which captures personalities based on five overarching
dimensions.888To construct each personality measure, I consider items with
high loading values from exploratory factor analysis as in
flinn2018personality and todd2020dynamic. These personality measures
demonstrate high internal consistency, as indicated by Cronbach’s alphas
exceeding 0.7. Out of the total 1130 couples in the sample, valid information
on personality traits is available for 583 couples. For households with
missing personality information, the values are imputed by averaging observed
individual personality scores from other waves. This imputation approach takes
into account the stability of personality traits over time, which has been
suggested by previous studies.999See, e.g., cobb2012stability, todd2020dynamic
or fitzenberger2022personality. See appendix A for the stability of
personality traits in the current sample. I test various imputation methods,
such as using the median value, but the main results remain robust. Looking at
the bottom of Table 1, on average, males tend to have higher values than
females in measures of self-esteem, extraversion, and cognitive engagement. In
contrast, females tend to score higher than males in conscientiousness,
neuroticism, and agreeableness. Both males and females exhibit similar levels
of openness. These gender differences in personality traits align with
findings from previous studies conducted on Dutch samples (see, e.g.,
nyhus2005effects or dupuy2014personality). Importantly, the gender differences
in personality traits observed in the sample remain virtually unchanged even
after the imputation of missing personality traits.

Table 1: Summary statistics.

| Mean | Std. dev. | Min | Max
---|---|---|---|---
A. Economic variables: |  |  |  |
Male wage rate | 13.63 | 3.71 | 6.88 | 29.90
Female wage rate | 12.05 | 3.16 | 4.03 | 21.80
Male weekly hours worked | 37.43 | 4.74 | 12 | 60
Female weekly hours worked | 25.98 | 7.99 | 10 | 48
Full income | 2820.69 | 576.79 | 1357.20 | 4770.11
Household private consumption | 2241.59 | 472.04 | 1142.50 | 4089.12
Assig. male private consumption | 89.97 | 51.78 | 15 | 453.72
Assig. female private consumption | 95.25 | 54.11 | 19.38 | 507.66
Public consumption | 579.10 | 229.75 | 102.96 | 1898.35
Total household consumption | 764.32 | 256.07 | 173.21 | 2284.98
Male weekly leisure | 74.56 | 4.74 | 52 | 100
Female weekly leisure | 86.01 | 7.99 | 64 | 102
B. Demographic variables: |  |  |  |
Male age | 47.39 | 9.76 | 25 | 65
Female age | 45.46 | 9.90 | 25 | 65
Number of children | 1.16 | 1.11 | 0 | 5
Male dummy low education | .20 | .40 | 0 | 1
Female dummy low education | .43 | .49 | 0 | 1
Male dummy middle education | .36 | .48 | 0 | 1
Female dummy middle education | .23 | .42 | 0 | 1
Male dummy high education | .43 | .49 | 0 | 1
Female dummy high education | .32 | .47 | 0 | 1
C. Personality traits: |  |  |  |
Male Openness | 3.07 | .26 | 1.37 | 3.87
Female Openness | 3.07 | .28 | 1.87 | 3.87
Male Extraversion | 3.18 | .51 | 1.33 | 4.50
Female Extraversion | 3.12 | .51 | 1.33 | 4.50
Male Agreeableness | 3.07 | .25 | 2.00 | 3.75
Female Agreeableness | 3.16 | .20 | 2.37 | 3.75
Male Neuroticism | 2.29 | .57 | 1.00 | 4.22
Female Neuroticism | 2.59 | .59 | 1.00 | 4.33
Male Conscientiousness | 2.78 | .27 | 1.88 | 3.66
Female Conscientiousness | 2.85 | .24 | 1.77 | 3.55
Male Self-esteem | 5.98 | .65 | 3.80 | 7.00
Female Self-esteem | 5.85 | .72 | 3.70 | 7.00
Male Cognitive engagement | 4.78 | .86 | 2.66 | 7.00
Female Cognitive engagement | 4.39 | .84 | 2.25 | 6.75

Notes: Sample size of 1130 couples. LISS waves 2009, 2010, 2012, 2015, and
2017 pooled up. All economic variables are in weekly 2015 euros.

##  Empirical strategy

In this section, I discuss the measures of relative personality traits that
are employed to examine the restrictions of the collective model outlined in
Section 2. These relative personality traits capture the differences in
personality between spouses within a household. The functional form for the
household demand functions is also introduced. From these demand functions,
several testable implications can be derived to assess the validity of the
collective model.

Multicollinearity in personality traits. To address the issue of
multicollinearity arising from the seven measures of personality traits, I
employ principal component (PC) analysis. This analysis is applied to the
entire sample, which includes both women and men. The goal is to identify the
principal components that explain the majority of the variance in the observed
personality measures. By extracting these principal components, which are
linearly uncorrelated factors, the paper aims to reduce the dimensionality of
the personality traits and mitigate the multicollinearity problem. This
approach allows for a more precise estimation of the effects of personality
traits on intrahousehold consumption behavior Jolliffe (2002).

Table 2 presents the correlations between the principal components (PCs) and
the individual personality measures, as well as the eigenvalues and the share
of observed variance explained by each PC. The results indicate that the two
principal components capture distinct aspects of personality traits. PC1 is
associated with traits such as introversion, lower self-esteem, and lower
cognitive engagement. On the other hand, PC2 is characterized by higher levels
of neuroticism and conscientiousness. The eigenvalues and the proportion of
observed variance explained by each PC reflect their relative importance in
explaining the variability in the original personality measures.

For each couple in the sample, the relative endowment of personality traits
between partners is calculated by constructing the ratio of spouses’ principal
components (PCs). These ratios, representing the relative distribution of
personality traits, are treated as continuous measures and tested as
distribution factors in the collective consumption model presented in Section
2. To facilitate comparison and analysis, the PCs are scaled from 1 to 100,
considering that they can take negative values. Figure 2 displays the
distribution of these ratios. On average, women tend to have higher values for
the common personality factor represented by PC1 compared to men. In contrast,
men exhibit higher values for PC2 relative to women.

Table 2. Principal components

Personality: | PC1 | PC2
---|---|---
1\. Extraversion | $-$ |
2\. Agreeableness |  |
3\. Openness |  |
4\. Conscientiousness |  | $+$
5\. Neuroticism |  | $+$
6\. Self-esteem | $-$ |
7\. Cognitive engagement | $-$ |
Eigenvalue | 1.41 | 1.23
Variance share | 28.58% | 21.75%

Notes: Explained share of the observed variance: 50.33%. The table indicates
the sign of those loadings that are larger than a cut-off of .8 with respect
to the largest coefficient in each component (similar procedure as in
jolliffe2002principal). The largest coefficient in PC1 is self-esteem; in PC2
is conscientiousness.

Figure 2. Within-couple differences in personality traits

Parametrization of unconditional demand functions. To test the restrictions of
the collective model, a functional form for the household demand functions
needs to be specified. I follow bobonis2009allocation and parametrize the
unconditional demand functions $\textbf{g}\in\\{\textbf{c},\bm{\ell},C\\}$ in
budget share form as:

$\omega_{j}=\alpha_{j}+\ln(\textbf{z}^{\prime})\bm{\beta}+a_{j}(y)+b_{j}(y^{2})+\ln(\textbf{w}^{\prime})\bm{\lambda}+\textbf{e}^{\prime}\bm{\delta}+\textbf{m}^{\prime}\bm{\psi}+\bm{\tau}_{j}+\varepsilon_{j},$
(8)

where for each couple in the sample, $\omega$ is the budget share on good $j$,
$a$ and $b$ are functions of full income and its square, w is a vector of
partners’ wages, $\bm{\tau}$ are time dummies capturing heterogeneity over
time, and $\varepsilon$ is unobserved heterogeneity.101010Potential sources of
endogeneity for full income are measurement error in nonlabor income, taste
shocks to total consumption that could be correlated to unobserved
heterogeneity in the budget shares equations, or saving decisions that may be
driving changes in nonlabor income. Prices of composite goods, which are
normalized to one, are assumed to enter through $\bm{\tau}$. The vector z
includes the relative endowment of personality traits, i.e., ratios of PCs
between partners of a household. The additional controls e and m are detailed
below.111111The assumption of a linear-log functional form allows for a
straightforward interpretation of the coefficient estimates in the empirical
model. Additionally, the empirical results remain consistent regardless of the
specific functional form assumption chosen (results can be provided upon
request).

One potential source of endogeneity in equation (8) is the endogenous
selection of couples in the marriage market, wherein individuals may form
couples based on their respective personality traits. Despite the limitations
of the current dataset, I address this potential issue in two ways.121212Fully
addressing selection in personality traits, such as through the estimation of
a structural matching model, is beyond the scope of this paper. First, the
vector of taste shifters (e) includes, among other explanatory variables, the
logarithm of the level of principal components (PCs) of each spouse and their
squares. I include the squares of the PCs to accommodate for potential
nonlinearity in the influence of personality on preferences over commodities,
as suggested in the analysis of borghans2008economics. Second, in all
specifications, I incorporate the vector m to account for marriage market
conditions with respect to personality, as discussed in dupuy2014personality.
This vector incorporates the weighted ratios of the number of husbands and
wives who are of similar age and educational level and who have the same score
in a given personality trait as the husband or wife of each household, divided
by the corresponding number of husbands or wives. These ratios, referred to as
personality ratios, are akin to the sex ratio concept in chiappori2002marriage
and serve to control for the underlying structure of the marriage market in
the sample with respect to personality traits.

The proportionality restriction imposed by collective rationality (as
expressed in equation (2)) on the system of unconditional demand functions can
be formulated as follows:

$\begin{split}\frac{\partial\omega_{j}/\partial\ln(z_{1})}{\partial\omega_{j}/\partial\ln(z_{2})}=&\frac{\partial\omega_{s}/\partial\ln(z_{1})}{\partial\omega_{s}/\partial\ln(z_{2})},\\\
\frac{\beta_{j1}}{\beta_{j2}}=&\frac{\beta_{s1}}{\beta_{s2}}\end{split}$ (9)

for all goods $j$, $s$, with $j\neq s$. If condition (9) is satisfied, it
implies that there is no evidence to reject the hypothesis that the effects of
differences in personality traits between partners on resource allocation
occur solely through their influence on the household’s distribution function.

To test the nonlinear cross-equation restrictions presented in equation (9),
the model is estimated as a system, allowing for correlation between the error
terms across the budget shares equations. The cross-equation hypotheses are
then examined using Wald test formulations. It is important to note that these
formulations may be subject to statistical issues. For instance, in OLS
systems, Wald tests tend to over reject the null hypothesis, and they are not
invariant to the definition of the null hypothesis (see
greene2003econometric). To address these concerns, this study adopts a similar
approach to that of bobonis2009allocation. Firstly, the Wald tests are
conducted using the bootstrap distribution with 1000 replications. Secondly,
as a robustness check of the main results, linear Wald tests are computed
based on the estimation of the $z$-conditional demand system proposed by
bourguignon2009efficient.

Parametrization of the $z$-conditional demand system. Under the additional
assumption that one distribution factor is strictly monotone in one good, we
can derive the demand for that good as a function of the distribution factor.
In my analysis, I find suggestive evidence indicating the presence of a
monotonic correlation between factor $z_{2}=PC_{2}^{f}/PC_{2}^{m}$ and male
private consumption ($c^{m}$).131313Refer to appendix B for detailed evidence
on the monotonicity assumption. It is important to note that for the
collective test based on the conditional demand system presented in this
section, $z_{2}$ needs to be both continuous and statistically significant.
For further discussion on this topic, see de2022household.

In budget share form, the demand for male private consumption ($c^{m}$)
inverted on $z_{2}$ is given by:

$\begin{split}\ln(z_{2})=&\frac{1}{\beta_{c^{m}2}}[\omega_{c^{m}}-\alpha_{c^{m}}-\beta_{c^{m}1}\ln(z_{1})-a_{c^{m}}(y)-b_{c^{m}}(y^{2})\\\
&-\ln(\textbf{w}^{\prime})\bm{\lambda}_{c^{m}}-\textbf{e}^{\prime}\bm{\delta}_{c^{m}}-\textbf{m}^{\prime}\bm{\psi}_{c^{m}}-\bm{\tau}_{c^{m}}-\varepsilon_{c^{m}}].\end{split}$
(10)

Substituting equation (10) in
$\tilde{\textbf{g}}(w^{m},w^{f},y,c^{m},\textbf{z}_{-2};\bm{\xi})$, the demand
for the remaining goods, we obtain the $z$-conditional demand system:

$\begin{split}\omega_{s}&=\varphi_{s}+\theta_{s}\ln(z_{1})+a_{s}(y)+b_{s}(y^{2})+\frac{\beta_{s2}}{\beta_{{c^{m}}2}}\omega_{c^{m}}\\\
-&\frac{\beta_{s2}}{\beta_{{c^{m}}2}}[a_{c^{m}}(y)+a_{c^{m}}(y^{2})+\ln(\textbf{w}^{\prime})\bm{\lambda}_{c^{m}}+\textbf{e}^{\prime}\bm{\delta}_{c^{m}}+\textbf{m}^{\prime}\bm{\psi}_{c^{m}}+\bm{\tau}_{c^{m}}]+\zeta_{s},\end{split}$
(11)

where

$\begin{split}\varphi_{s}&=\alpha_{s}-\frac{\alpha_{c^{m}}\beta_{s2}}{\beta_{{c^{m}}2}},\\\
\theta_{s}&=\beta_{s1}-\dfrac{\beta_{{c^{m}}1}\beta_{s2}}{\beta_{{c^{m}}2}},\\\
\zeta_{s}&=\frac{\beta_{s2}}{\beta_{{c^{m}}2}}\varepsilon_{c^{m}}+\varepsilon_{s}\end{split}$

for all goods $s\neq c^{m}$. One important source of endogeneity that arises
from the estimation of (11), is the fact that the share of male private
consumption is not independent of the new compound error term $\zeta_{s}$. A
natural instrument for men’s consumption is $z_{2}$ which satisfies the
standard requirements for being a relevant and valid instrumental variable. It
is worth noting that equation (10) demonstrates the correlation between
$\omega_{c^{m}}$ and $z_{2}$, while the latter is excluded from equation (11).
To mitigate this endogeneity problem, I employ a control function approach by
incorporating the residuals from the first stage of the conditioning good into
the estimation of equation (11).141414Control functions for testing collective
rationality are also used by bobonis2009allocation, attanasio2014efficient,
de2022household.

The exclusion restriction imposed by the collective model, as inferred from
the estimation of the $z$-conditional demand system in equation (11), can be
stated as follows:

$\frac{\partial\omega_{s}}{\partial\ln(z_{1})}=\theta_{s}=0\quad\quad\forall\hskip
5.69054pts\neq{c^{m}}.$ (12)

For each budget share equation in the system (11), a linear test is conducted
to assess the significance of the parameter estimate of the relative
personality factor. Restriction (12) indicates that once we condition the
demand for the remaining goods on the demand for $c^{m}$, which is
monotonically related to $z_{2}$, the additional variation provided by $z_{1}$
does not play a significant role in determining the household equilibrium.
This condition is equivalent to the requirement of distribution factor
proportionality, as discussed in bourguignon2009efficient. The exclusion
restriction stated in equation (12) carries greater empirical power compared
to the cross-equation restrictions presented in (9). This observation further
strengthens the robustness of the estimation results obtained for the
unconditional demand system.

##  Empirical results

I estimate the unconditional demand system in equation (8) using ordinary
least squares (OLS), while the $z$-conditional demand system in equation (11)
is estimated using a control function approach. In the control function
approach, I incorporate the residuals obtained from a first-stage regression
of male private consumption into the demand for the other commodities. To
account for heteroskedasticity, I use robust standard errors, and I cluster
the standard errors at the household level in all specifications.

Table 3 presents the estimates of the system of unconditional demand
equations. The specifications include several control variables: a linear
control function for household full income and its square, instrumented with
household potential income; the logarithm of spouses’ wages and the
interaction between them; the square of the husband’s wage; husband’s age and
its square; husband’s educational level; the number of children in the couple;
a dummy variable indicating whether the couple is married or cohabiting; and
time dummies.151515Wife’s age and educational level are not included in the
specifications due to multicollinearity issues, as there is a significant
positive assortative mating in age and education in the sample. However, the
results remain robust when using the wife’s characteristics as controls
instead. Additionally, I include the logarithm of the principal components in
levels and their squares as additional taste shifters in the specifications.
The marriage market personality ratios are also included in the analysis.

Firstly, it is observed that the relative endowments of personality between
spouses have a significant impact on male private consumption, leisure time,
and public expenditures. Both personality factors positively affect private
and public consumption, but negatively influence the allocation of leisure.
Secondly, the second distribution factor, which is associated with
conscientiousness and neuroticism, has a larger average effect compared to the
first distribution factor. Thirdly, the ratios of the estimated coefficients
of the distribution factors across commodities, as indicated in equation (9),
are 0.37 for $c^{m}$, 0.17 for $c^{f}$, 0.16 for $l^{m}$, 0.37 for $l^{f}$,
and 0.31 for $C$. These proportional average effects across commodities are
supported by the results of the (bootstrapped) proportionality test presented
at the bottom of Table 3. This evidence suggests that relative personality
influences an individual’s consumption within a partnership, but solely
through its impact on the distribution of power within the household. Finally,
personality also directly affects the allocation of resources through its
impact on preferences, as evidenced by the significant effect of the principal
components in levels on commodities. Due to multicollinearity, the estimated
coefficients of the wife’s principal components are not included in the
analysis.

Table 3. OLS estimates of the effect of relative personality on household
consumption. System of unconditional demand functions.

| Dependent variable: budget share
---|---
| $\omega_{c^{m}}$ | $\omega_{c^{f}}$ | $\omega_{\ell^{m}}$ | $\omega_{\ell^{f}}$ | $\omega_{C}$
$\ln(\frac{PC1^{f}}{PC1^{m}})$ | .033+ | .014 | -.022 | -.082+ | .058
(.016) | (.014) | (.024) | (.035) | (.044)
$\ln(\frac{PC2^{f}}{PC2^{m}})$ | .088+ | .078 | -.136+ | -.218+ | .186+
(.043) | (.054) | (.073) | (.097) | (.110)
$\ln(PC1^{m})$ | .036+ | .005 | -.037 | -.123+ | .119+
(.019) | (.020) | (.031) | (.045) | (.056)
$\ln(PC2^{m})$ | .112+ | .057 | -.231+ | -.243 | .305+
(.068) | (.087) | (.105) | (.152) | (.188)
Additional covariates | Yes | Yes | Yes | Yes | Yes
Time dummies | Yes | Yes | Yes | Yes | Yes
Proportionality test |  $\chi^{2}$(4) = 0.892 ($p-$value = .911)

Notes: Sample size of 1130 couples. Robust standard errors clustered at the
household level are in parentheses. $PC$: principal component. I estimate the
proportionality test’s $p-$value on its bootstrap distribution over 1000
replications. Additional covariates: linear control function for full income
and its square instrumented with household potential income; the log of
spouses’ wages and the interaction between them; the square of husband’s wage;
husband’s age and its square; husband’s educational level; the number of
children the couple has; marital status; the squares of the log of spouses’
PCs in levels; and personality ratios. ${}^{\bm{+}}:$ Significant with at
least 90% of confidence.

Table 4 presents the estimates of the $z$-conditional demand functions based
on equation (11). The same control variables are used as in the unconditional
demand equations. It should be noted that the conditioning good is $c^{m}$,
and the relative level of PC2 is employed to invert the demand for this good.
Importantly, both personality factors have a significant impact on the budget
share equation of $c^{m}$. The most compelling evidence is obtained from
estimations where the budget share equation is responsive to both factors De
Rock . (2022). Additionally, the relative level of PC2 is statistically
significant in four out of five budget share equations. In the unconditional
demand system (Table 3), the relative amount of PC1 is significant for two
commodities (male private consumption and female leisure). However, in the
$z$-conditional demand system (Table 4), it is not significant in any case.
This evidence suggests that the impact of relative personality is indeed one-
dimensional, meaning that relevant information regarding the intrahousehold
allocation of resources is completely summarized by the share of male private
consumption. Crucially, this finding is confirmed by the result of the
collective test at the bottom of Table 4.

Table 4. OLS estimates of the effect of relative personality on household
consumption. System of $z$-conditional demand functions.

Dependent variable: budget share
---
| $\omega_{c^{f}}$ | $\omega_{\ell^{m}}$ | $\omega_{\ell^{f}}$ | $\omega_{C}$
$\ln(\frac{PC1^{f}}{PC1^{m}})$ | -.031 | .055 | .043 | -.067
(.034) | (.047) | (.064) | (.073)
Additional covariates | Yes | Yes | Yes | Yes
Time dummies | Yes | Yes | Yes | Yes
Collective test |  $\chi^{2}$(4) = 3.476 ($p-$value = .502)

Notes: Sample size of 1130 couples. Robust standard errors clustered at the
household level are in parentheses. $PC$: principal component. The
conditioning good is $c^{m}$. I estimate the collective test’s $p-$value on
its bootstrap distribution over 1000 replications. Additional covariates:
linear control function for full income and its square instrumented with
household potential income; the log of spouses’ wages and the interaction
between them; the square of husband’s wage; husband’s age and its square;
husband’s educational level; the number of children the couple has; marital
status; the square of the log of spouses’ PCs in levels; and personality
ratios.

##  Personality and intrahousehold consumption inequality

After providing theory-based evidence that (relative) personality affects the
bargaining weights of household members, it is important to explore the
relationship between personality and within-family inequality. Following the
approach of cherchye2020marital, I analyze intrahousehold consumption
inequality using the women and men relative individual cost of equivalent
bundle (RICEB). For a given couple, these bundles are defined as follows:

$\begin{split}\textrm{RICEB}^{i}&=\frac{c^{i}+w^{i}\ell^{i}+C}{y}\quad\quad\textrm{with}\hskip
5.69054pti\in\\{m,f\\}.\end{split}$ (13)

Member-specific RICEBs describe how household members allocate consumption
relative to the household’s full income, taking into account both scale
economies and the intrahousehold division of resources, thus providing an
assessment of individual welfare.161616It is worth noting that while the
concept of RICEBs is related to the sharing rule concept in the collective
literature, the RICEBs evaluate public expenditures at market prices instead
of Lindahl prices. bostyn2022time utilize RICEBs to analyze individual welfare
in a collective model that incorporates marriage market restrictions. In this
study, intrahousehold consumption inequality is proxied by the difference
between partners’ RICEBs, specifically RICEBf minus RICEBm.

Next, I examine the distribution of intrahousehold consumption inequality for
three categories of couples: (a) households where the female fraction of a
specific personality trait is above the 80th percentile of the distribution of
all female fractions; (b) households where the female fraction of a specific
personality trait is between the 45th and 55th percentiles of the distribution
of all female fractions; and (c) households where the female fraction of a
specific personality trait is below the 20th percentile of the distribution of
all female fractions. This categorization allows for a comparison between
households where the within-household female personality fraction is either
high, moderate, or relatively low. It is important to note that the female
personality fraction ($r_{p}$) for personality trait $p$ is constructed as
$r_{p}=p^{f}/(p^{f}+p^{m})$.171717Appendix C provides a detailed overview of
the distribution of these female personality fractions as well as the RICEB
measures.

Figure 2 illustrates how intrahousehold consumption inequality varies with the
relative amount of personality within couples for each personality measure,
comparing the three types of households mentioned above. First, it can be
observed that couples with a moderate within-family difference in personality
tend to exhibit, on average, a smaller degree of intrahousehold consumption
inequality (indicated by the red dashed lines, which are more concentrated
around zero on the horizontal axis). Second, for almost all personality
measures (with the exceptions of openness and neuroticism), the black solid
line is consistently positioned to the right of the blue dash-dotted line.
This implies that a larger fraction of a woman’s personality relative to her
partner is associated with a greater allocation of intrahousehold resources
towards her. This pattern is particularly pronounced for conscientiousness,
self-esteem, and cognitive engagement.181818Note that self-esteem and
conscientiousness exhibited the highest loadings among the personality traits
in the principal components analysis (see Table 2). Indeed, in those three
cases, as demonstrated in Panel A of Table 5, I strongly reject the null
hypothesis of equal means between couples with a large and small female
personality fraction (referring to the black and blue distributions in Figure
2). In Panel B of Table 5, I present the difference in average intrahousehold
consumption inequality between households with large and small personality
fractions in the sample. For instance, in couples where women exhibit higher
levels of cognitive engagement than their male partners, there is an average
of 4.48% more intrahousehold resources allocated to them compared to couples
where men are more engaged in intellectual activities.

Table 5. Panel A: Test of equal mean in intrahousehold inequality between
couples with large and small female personality fractions. Panel B: Difference
in average intrahousehold inequality between couples with large and small
female personality fractions.

| Panel A: | Panel B:
---|---|---
| Bootstrap statistics | Difference in inequality
|  $t$-statistic |  $p$-value |
Agreeableness | -.411 | .468 | 0.428%
Openness | -1.609 | .225 | 1.851%
Extraversion | -1.097 | .349 | 1.213%
Conscientiousness | -3.506+ | .014 | 3.949%
Neuroticism | .400 | .484 | 0.476%
Self-esteem | -4.022+ | .005 | 3.441%
Cognitive engagement | -3.776+ | .009 | 4.486%

Notes: Panel A shows the results of a bootstrapped $t$-test of equal mean
between the black and blue distributions shown in Figure 2. I estimate both
the $t$-statistic and $p$-value on their bootstrap distribution over 1000
replications. Panel B shows the difference in the average intrahousehold
inequality between black and blue distributions shown in Figure 2.
+ Significant with at least 90% of confidence.

Figure 2. Intrahousehold consumption inequality and relative personality.

1919footnotemark: 19

Notes: This figure shows kernel density plots of intrahousehold inequality
(i.e., RICEBf minus RICEBm) and within-couple female personalities ratios.

##  Conclusion

This paper presents compelling evidence, based on theoretical foundations,
regarding the role of personality in resource allocation within households
when assuming Pareto-efficient decision-making. By examining variations in
personality traits among Dutch couples, this study tests for distribution
factor proportionality and the exclusion restriction utilizing a conditional
demand system estimation. The findings do not allow for the rejection of the
hypothesis that (relative) personality influences the bargaining process
within households. Notably, women who exhibit higher levels of
conscientiousness, self-esteem, and cognitive engagement relative to their
male partners tend to receive a larger proportion of intrafamily resources. To
address potential selection bias in personality, the budget share equations
are conditioned on the level of personality and additional explanatory
variables that capture the structure of the marriage market in relation to
personality traits within the sample.

The findings presented in this paper provide strong support for conducting a
more comprehensive and structural analysis to explore the significance of
personality traits within the family context, as well as the underlying
mechanisms through which these traits exert their influence. Firstly,
employing a model with a more robust parametric structure for preferences and
the sharing rule, similar to approaches utilized by browning2013estimating or
cherchye2017household, would offer deeper insights into the welfare
implications of personality traits. Such an approach could enhance our
understanding of how these traits affect individual well-being. Secondly, it
is worth noting that several studies have demonstrated the importance of
personality traits in assortative mating within the marriage market
(lundberg2012personality or dupuy2014personality). Therefore, it would be
valuable to estimate a matching model and examine the complete structure of
the marriage market as a potential driver of power dynamics. This would allow
for a comprehensive assessment of how personality traits shape partner
selection and subsequent resource allocation within households. Lastly, the
current paper’s framework overlooks intertemporal aspects that are relevant to
household consumption, such as the influence of personality on occupational or
educational choices (todd2020dynamic). Considering these factors in future
research would enhance the richness and applicability of the analysis.

## Appendix

##  Appendix A Stability of personality traits

This section illustrates the evolution of personality over time for women and
men in our sample. Figures A1 and A2 show the average score by age for each
personality measure. I consider all waves together.

Figure A1. Female average personality scores by age. Figure A2. Male average
personality scores by age.

##  Appendix B Monotonic relationship between $z_{2}$ and male private
consumption ($\omega_{c^{m}}$)

Following attanasio2014efficient, I study the relationship between the second
distribution factor ($z_{2}=\ln(\frac{PC2^{f}}{PC2^{m}})$) and the share of
male private consumption ($\omega_{c^{m}}$) by looking at the point estimates
of different polynomials. The direction of the point estimates implies an
increasing relationship between the share of men’s private expenditures and
the second measure of relative personality within households. This
information, together with the fact that both distribution factor influence
significantly $\omega_{c^{m}}$ (see Table 3), support the choice of men’s
private expenditures as the conditioning good.

Table A1. Effect of distributions on the consumption shares.

| Dependent variable: budget share
---|---
| $\omega_{c^{m}}$ | $\omega_{c^{f}}$ | $\omega_{\ell^{m}}$ | $\omega_{\ell^{f}}$ | $\omega_{C}$
$\ln(\frac{PC1^{f}}{PC1^{m}})$ | .022 | .033+ | -.019 | -.104+ | .068
(.021) | (.017) | (.029) | (.039) | (.052)
$[\ln(\frac{PC1^{f}}{PC1^{m}})]^{2}$ | -.001 | .005+ | -.001 | -.004 | .002
(.001) | (.001) | (.002) | (.003) | (.005)
$[\ln(\frac{PC1^{f}}{PC1^{m}})]^{3}$ | .0002 | .001 | -.003 | .001 | .0002
(.001) | (.001) | (.002) | (.003) | (.003)
$\ln(\frac{PC2^{f}}{PC2^{m}})$ | .010 | .029 | -.158+ | -.106 | .225+
(.051) | (.072) | (.084) | (.115) | (.139)
$[\ln(\frac{PC2^{f}}{PC2^{m}})]^{2}$ | .002 | -.005 | -.006 | -.0004 | .010
(.004) | (.005) | (.007) | (.013) | (.013)
$[\ln(\frac{PC2^{f}}{PC2^{m}})]^{3}$ | .014+ | .002 | -.002 | -.018 | .004
(.005) | (.005) | (.007) | (.014) | (.015)
Additional covariates | Yes | Yes | Yes | Yes | Yes
Time dummies | Yes | Yes | Yes | Yes | Yes

Notes: Sample size of 1130 couples. Robust standard errors clustered at the
household level are in parentheses. $PC$: principal component. Additional
covariates: linear control function for full income and its square
instrumented with household potential income; the log of spouses’ wages and
the interaction between them; the square of husband’s wage; husband’s age and
its square; husband’s educational level; the number of children the couple
has; marital status; the log of spouses’ PCs in levels and their squares; and
personality ratios. ${}^{\bm{+}}:$ Significant with at least 90% of
confidence.

##  Appendix C Distribution of female personality fractions and RICEBs

Table C1. Summary statistics for female personality ratios and RICEBs measures ($N$ = 1130 couples) | Mean | Std. Dev. | Min | p25 | Median | p75 | Max
---|---|---|---|---|---|---|---
RICEBf | 0.606 | 0.070 | 0.273 | 0.563 | 0.609 | 0.653 | 0.802
RICEBm | 0.598 | 0.072 | 0.323 | 0.551 | 0.595 | 0.644 | 0.839
Female fractions ($r_{p}$): |  |  |  |  |  |  |
Self-Esteem | 0.494 | 0.037 | 0.391 | 0.474 | 0.496 | 0.515 | 0.631
Extraversion | 0.495 | 0.062 | 0.286 | 0.457 | 0.488 | 0.535 | 0.700
Openness | 0.500 | 0.031 | 0.357 | 0.480 | 0.500 | 0.520 | 0.694
Neuroticism | 0.531 | 0.076 | 0.309 | 0.475 | 0.532 | 0.580 | 0.773
Agreeableness | 0.508 | 0.027 | 0.413 | 0.490 | 0.509 | 0.521 | 0.636
Conscientiousness | 0.507 | 0.033 | 0.356 | 0.489 | 0.505 | 0.529 | 0.612
Cognitive Engagement | 0.479 | 0.064 | 0.299 | 0.435 | 0.478 | 0.518 | 0.679

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# Bound states and electromagnetic radiation of relativistically rotating
cylindrical wells

Matteo Buzzegoli  Kirill Tuchin  Department of Physics and Astronomy, Iowa
State University, Ames, Iowa, 50011, USA

###### Abstract

We compute the effect of rigid rotation on the non-relativistic bound states.
The energy levels of the bound states increase with the angular velocity of
rotation until at certain value of the angular velocity they are completely
pushed out into the continuum which corresponds to dissociation of the bound
states. When the angular velocity exceeds the critical value at which the
ground state disappears into the continuum, no bound state is possible. This
effect should have important consequences for the phenomenology of the quark-
gluon plasma. One of the ways to study it experimentally is to observe the
electromagnetic radiation emitted by a rotating bound state. We compute the
corresponding intensity of electromagnetic radiation and show that it strongly
depends on the angular velocity of rotation.

##  I Introduction

The study of the properties of rotating systems has a long history. Relatively
recent studies focused on thermodynamics and kinetic of rotating quantum
systems Vilenkin:1980zv ; Becattini:2007nd ; Ambrus:2014uqa ; McInnes:2014haa
; McInnes:2015kec ; Ambrus:2015lfr ; Chen:2015hfc ; McInnes:2016dwk ;
Jiang:2016wvv ; Chernodub:2016kxh ; Ebihara:2016fwa ; Chernodub:2017ref ;
Liu:2017spl ; Buzzegoli:2017cqy ; Zhang:2018ome ; Buzzegoli:2020ycf ;
Chen:2019tcp ; Wang:2018zrn ; Wang:2019nhd ; Palermo:2021hlf ;
Sadooghi:2021upd . A great interest in relativistic rotating systems was
recently spurred by the phenomenology of the relativistic heavy-ion collisions
that indicates that the quark-gluon plasma has large local vorticity
STAR:2017ckg ; Csernai:2013bqa ; Deng:2016gyh ; Jiang:2016woz ; Xia:2018tes .
In fact, the vorticity $\Omega$ found in numerical simulations approaches the
relativistic limit $\Omega r=1$, where $r$ is the radial distance from the
rotation axis. The importance of the causal boundary was emphasized by a
number of authors Duffy:2002ss ; Ebihara:2016fwa ; Chernodub:2017ref ;
Chernodub:2017mvp .

One of the plasma signatures is dissociation of the bound states immersed into
it through the mechanism of the Debye screening. The ability of plasma to
break up the bound states depends on the relationship between the Debye radius
which is a decreasing function of temperature and the linear size of the bound
state. On the other hand, the rotating plasma drags the bound state along. As
a result, the rotating bound state possesses extra centrifugal energy that
makes it more fragile. Thus, both the Debye screening and rotation induce the
break up of bound states. The main goal of this paper is to investigate the
effect of rotation on the bound state spectrum.

To this end we consider a model consisting of a cylindrical well of radius $R$
and constant depth $U_{0}$, rotating with constant angular velocity $\Omega$
about an axis passing through its center-of-mass. We neglect all plasma
effects apart from being the source of rotation. Furthermore, we treat the
bound state non-relativistically which allows us to retain essential
qualitative features while keeping algebra simple. Our results thus apply
literally to a heavy quarkonium immersed into and coaxial with a vortex of the
rotating plasma. The critical ingredient of our calculation is the causal
boundary condition at $r=1/\Omega$. The results strongly depend on whether
$\Omega R$ is larger or smaller than unity. We refer to this two cases as
rapid and slow rotation. Our main observation is that the energy levels of the
rotating quarkonium increase with $\Omega$ and get pushed out of well at a
certain $\Omega$ that is larger for deeper wells. We study these matters in
Sec. III.

The dependence of the spectrum on $\Omega$ affects the electromagnetic
radiation emitted by the quarkonium. In particular, we found that when
$\Omega>1/R$ the intensity increases by $(\Omega R)^{6}$ in comparison with
the non-rotating well. This is studied it Sec V. This means that the life-time
of the excited quarkonium states are sharply decrease with $\Omega$. Similar
conclusion can be made with regard to the gluon strong and electroweak
interactions as well. This observation may be instrumental for the
experimental investigation of the rotation effects on the bound states.

##  II Bound states of stationary cylindrical well

First, for the future reference, consider a non-rotating cylindrical well as
shown in Fig. 1. We are interested in the solutions with $E<0$. The
Schrödonger equation inside the well $r<R$, reads

$\displaystyle
H_{0}\psi_{0}=-\frac{1}{2M}\nabla^{2}\psi_{0}-U_{0}\psi_{0}=E_{0}\psi_{0}\,.$
(1)

The corresponding set of eigenfunctions, regular at the origin and bound in
the radial direction, is

$\displaystyle\psi_{0}({\bm{x}})=AJ_{m}(kr)e^{im\phi}e^{ik_{z}z}\,,$ (2)

where $J_{m}(z)$ is the Bessel function of the first kind, $A$ is the
normalization constant and

$\displaystyle k=\sqrt{2M(U_{0}-k_{z}^{2}/2M-|E_{0}|)}\,.$ (3)

Clearly the bound states exist only if $|E_{0}|\leq U_{0}-k_{z}^{2}/2M$.

Outside the well the functions that vanish at $r\to\infty$ are

$\displaystyle\psi_{0}({\bm{x}})=BK_{m}(\kappa r)e^{im\phi}e^{ik_{z}z}\,,$ (4)

where $K_{m}(z)$ is the modified Bessel function of the second kind and

$\displaystyle\kappa=\sqrt{2M(k_{z}^{2}/2M+|E_{0}|)}\,.$ (5)

Eq. (4) satisfy the boundary condition $\psi\to 0$ as $r\to\infty$.

The boundary conditions at $r=R$ are the requirements of continuity of the
wave function and its first derivative:

$\displaystyle AJ_{m}(kR)$ $\displaystyle=BK_{m}(\kappa R)\,,$ (6)
$\displaystyle AJ_{m}^{\prime}(kR)k$ $\displaystyle=BK_{m}^{\prime}(\kappa
R)\kappa\,.$ (7)

The solution exists only if the determinant vanishes:

$\displaystyle J_{m}(kR)K_{m}^{\prime}(\kappa R)\kappa=K_{m}(\kappa
R)J_{m}^{\prime}(kR)k\,.$ (8)

This equation along with

$\displaystyle k=\sqrt{2MU_{0}-\kappa^{2}}$ (9)

gives the spectrum.

##  III Bound states of rotating cylindrical well

The Hamiltonian of the rotating system is $H=H_{0}-\Omega J_{z}$. Thus, in
cylindrical coordinates, $H\psi=E\psi$ is equivalent to
$H_{0}\psi=(E+m\Omega)\psi$, where $m$ is an integer eigenvalue of the
operator $J_{z}=-i\partial_{\phi}$. The general solution of the latter
equation was found in the previous section, we just need to substitute
$E_{0}=E+m\Omega$ and apply the new boundary conditions. We proceed by
considering separately the slowly and rapidly rotating well.

|
---|---
Figure 1:  Potential cylindrical well rotating with angular velocity $\Omega$.
Because of the causality, the wave function must vanish at $r>1/\Omega$. Left
panel: slow rotation, right panel: fast rotation. In the stationary case
$\Omega^{-1}\to\infty$.

###  III.1 Slow rotation $R<1/\Omega$

The wave functions inside the well have the same functional form as (2).
However, $k$ is now given by

$\displaystyle k=\sqrt{2M(U_{0}-k_{z}^{2}/2M-|E+m\Omega|)}\,,\qquad
E+m\Omega<0\,.$ (10)

The solution outside the well reads

$\displaystyle\psi({\bm{x}})=\left[BK_{m}(\kappa r)+CI_{m}(\kappa
r)\right]e^{im\phi}e^{ik_{z}z}\,,$ (11)

where $I_{m}(z)$ is the modified Bessel function of the first kind and

$\displaystyle\kappa=\sqrt{2M(k_{z}^{2}/2M+|E+m\Omega|)}\,.$ (12)

Unlike $\psi_{0}$ in (4), eigenfucntions (11) cannot extend to infinity.
Instead, causality requires that $\psi$ vanishes at $r=1/\Omega$:

$\displaystyle BK_{m}(\kappa/\Omega)+CI_{m}(\kappa/\Omega)=0\,.$ (13)

Additionally, the boundary conditions at $r=R$ yield

$\displaystyle AJ_{m}(kR)$ $\displaystyle=BK_{m}(\kappa R)+CI_{m}(\kappa
R)\,,$ (14) $\displaystyle AJ_{m}^{\prime}(kR)k$
$\displaystyle=BK_{m}^{\prime}(\kappa R)+CI_{m}^{\prime}(\kappa R)\kappa\,.$
(15)

The determinant of the set of linear equations (13), (14), (15) must vanish:

$\displaystyle I_{m}$
$\displaystyle(\kappa/\Omega)\left[-J_{m}(kR)K_{m}^{\prime}(\kappa
R)\kappa+K_{m}(\kappa R)J_{m}^{\prime}(kR)k\right]$ $\displaystyle-
K_{m}(\kappa/\Omega)\left[-J_{m}(kR)I_{m}^{\prime}(\kappa
R)\kappa+I_{m}(\kappa R)J_{m}^{\prime}(kR)k\right]=0.$ (16)

As $\Omega\to 0$ this equation reduces to (8).

Using the convenient notation: $x=kR$, $y=\kappa R$, $\lambda=R\Omega$,
$\beta=\sqrt{2MU_{0}R^{2}}$, we can write the boundary condition (III.1) as:

$\displaystyle I_{m}$
$\displaystyle(y/\lambda)\left[-J_{m}(x)K_{m}^{\prime}(y)y+K_{m}(y)J_{m}^{\prime}(x)x\right]$
$\displaystyle-
K_{m}(y/\lambda)\left[-J_{m}(x)I_{m}^{\prime}(y)y+I_{m}(y)J_{m}^{\prime}(x)x\right]=0\,,$
(17)

where $x$ and $y$ are related to each other as

$\displaystyle x=\sqrt{\beta^{2}-y^{2}}\,.$ (18)

Denoting the roots of (17) as $y_{mn}$, $n=1,2,\ldots$ we obtain the energy
spectrum

$\displaystyle
E_{mnk_{z}}=-\frac{1}{2MR^{2}}y_{mn}^{2}-\frac{m\lambda}{R}+\frac{k_{z}^{2}}{2M}\,.$
(19)

Figure 2: Energy levels (19) and (22) with $m=0$, $k_{z}=0$ in wells of
different depth: $\beta=1$ (dotted line), $\beta=1.5$ (short-dashed line),
$\beta=2$ (long-dashed line), $\beta=2.5$ (dashed-dotted line) and $\beta=3$
(solid line) as a function of $\lambda=R\Omega$. Notice the discontinuity at
$\lambda=1$.

###  III.2 Rapid rotation

If the rotation is fast $R>1/\Omega$, all the region $r>1/\Omega$ is outside
of the rotating spacetime. Thus there is only solution inside the well; its
functional form is (2) and the boundary condition is

$\displaystyle J_{m}(k/\Omega)=0\,.$ (20)

Denoting zeros of the $m$’th Bessel function as $x_{ma}$, $a=1,2,\ldots$ we
obtain $k=\Omega x_{ma}$. The spectrum of a particle in the rapidly rotating
cylindrical well is

$\displaystyle
E_{mak_{z}}=\frac{1}{2M}(\Omega^{2}x^{2}_{ma}+k_{z}^{2})-U_{0}-m\Omega\,,$
(21)

or, equivalently,

$\displaystyle
E_{mak_{z}}=\frac{1}{2MR^{2}}\left(\lambda^{2}x^{2}_{ma}-\beta^{2}\right)+\frac{k_{z}^{2}}{2M}-\frac{m\lambda}{R}\,.$
(22)

Fig. 2 displays the energy level (19), for $\lambda<1$, and (22), for
$\lambda>1$, with $m=0$, $k_{z}=0$ in the potential wells of different depths.
As the angular velocity of rotation increases, the level increases (i.e. its
absolute value decreases) until it reaches $E=0$. The discontinuity at
$R\Omega=1$ is an artifact of our model and disappears when the boundary is
treated more carefully, see Sec. IV. For the range of $\beta$ shown in the
figure, there is only one level, but in a deeper well there are more energy
levels. All of them exhibit similar dependence on $\Omega$.

###  III.3 Shallow and slowly rotating cylindrical well

Consider states with $m=0$, $k_{z}=0$ in the shallow well limit $\beta^{2}\ll
1$, we can expand the Bessel functions in (17) at $x,y\ll 1$ which yields

$\displaystyle
1-\frac{1}{2}(\beta^{2}-y^{2})\ln\frac{2}{\tilde{\gamma}y}=-\frac{\pi\beta^{2}}{2}\exp\left(-\frac{2y}{\lambda}\right)\,,$
(23)

where $\tilde{\gamma}=\exp\gamma\approx 1.78$, with $\gamma$ being Euler-
Mascheroni constant. Considering the right-hand-side of (23) a perturbation we
can write at the leading “non-rotating” approximation

$\displaystyle
1-\frac{1}{2}(\beta^{2}-y_{0}^{2})\ln\frac{2}{\tilde{\gamma}y_{0}}=0\,.$ (24)

The only solution is obtained by neglecting $y_{0}$ compared to $\beta$:

$\displaystyle
y_{0}=\frac{2}{\tilde{\gamma}}\exp\left(-\frac{2}{\beta^{2}}\right)\,.$ (25)

Clearly, $y_{0}$ is much smaller than any positive power of (small) $\beta$.
In usual units

$\displaystyle
E_{0}=-\frac{2}{\tilde{\gamma}^{2}MR^{2}}\exp\left(-\frac{2}{MU_{0}R^{2}}\right)\,.$
(26)

The second iteration of (23) is obtained by expanding $y=y_{0}+y_{1}$ at small
$y_{1}\ll y_{0}$:

$\displaystyle y_{1}=-\pi
y_{0}\exp\left(-\frac{4}{\lambda\tilde{\gamma}}e^{-\frac{2}{\beta^{2}}}\right)\,.$
(27)

Converting this back to the dimensional quantities gives the correction due to
rotation

$\displaystyle\Delta E=-2\pi
E_{0}\exp\left(-\frac{4}{R\Omega\tilde{\gamma}}e^{-\frac{2}{MU_{0}R^{2}}}\right)\,.$
(28)

As $\Omega$ increases, $\Delta E$ also increases with the result that the
energy level moves up.

The level crosses into the continuum when $E_{0}+\Delta E=0$. This happens at
the value $\Omega_{c}$ given by

$\displaystyle\Omega_{c}=\frac{2\sqrt{2M|E_{0}|}}{\ln(2\pi)}\,.$ (29)

##  IV Bound states of rotating 2D $1/r$ potential

Another instructive model of a rotating well with cylindrical symmetry is an
attractive $U=-\alpha/r$ potential in two dimensions. Unlike the previous
example, this potential does not have a sharp boundary and hence no
discontinuity of the energy levels observed in Fig. 2.

As in the previous section we first consider the stationary problem

$\displaystyle\left({\bm{\nabla}}^{2}+\frac{2M\alpha}{r}-2M|E_{0}|\right)\psi_{0}=0\,.$
(30)

Its solution, finite at the origin, is

$\displaystyle\psi_{0}=A\,_{1}F_{1}(-n+|m|+1/2,2|m|+1,\rho)\rho^{|m|}e^{-\rho/2}e^{im\phi}\,,$
(31)

where $\,{}_{1}F_{1}$ is the confluent hypergeometric function of the first
kind and we defined

$\displaystyle\rho=\frac{2\alpha M}{n}r\,,\qquad n=\frac{1}{\sqrt{2|E|}}\,.$
(32)

Requiring that at $r\to\infty$ the wave function (31) is finite gives the
quantization condition $n=n^{\prime}-1/2$ with $n^{\prime}=1,2,\ldots$ and
$|m|\leq n^{\prime}-1$.

The boundary condition of the rotating potential requires vanishing of the
wave function at $r=1/\Omega$, i.e.

$\displaystyle\psi\big{|}_{\rho=2\ell/n}=0\,,\qquad\ell=\frac{\alpha
M}{\Omega}\,.$ (33)

As a result $n^{\prime}$ depends on $\Omega$ and is not integer anymore.
Solving (33) for the given values of $m$ and $\ell$ we find the spectrum of
$n^{\prime}$. The energy levels are then given by

$\displaystyle
E_{n^{\prime},m}=-\frac{\alpha^{2}M}{2(n^{\prime}(\Omega)-1/2)^{2}}-m\Omega\,.$
(34)

Fig. 3 (left panel) shows $E_{n^{\prime},0}$ as a function of $\ell$. The
qualitative behavior is similar to Fig. 2. As the angular velocity of rotation
$\Omega$ increases the energy levels get pushed out of the well until at
$\ell=\ell_{1,0}=0.72$ the last azimuthally symmetric level
($m=0,n^{\prime}=1$)***More precisely, the level that has $m=0$,
$n^{\prime}=1$ at $\Omega=0$. disappears indicating absence of such bound
states. The next azimuthally symmetric level ($m=0,n^{\prime}=2$) disappears
at $\ell=\ell_{2,0}=3.84$.

|
---|---
Figure 3:  Left panel: Energy levels $m=0$, $n^{\prime}=1,2,3,4$ (dotted,
dashed, dashed-dotted and solid lines respectively) in the two-dimensional
potential $U=-\alpha/r$ at different values of $1/\ell$. Right panel: energy
levels with given $n^{\prime}$ (first number) and $m$ (second number) at
$\ell=10$ and $\alpha\ell=1$.

##  V Electromagnetic radiation by rotating cylindrical well

A rotating well binding oppositely charged particles emits electromagnetic
radiation. The intensity of radiation can be used as an effective diagnostic
tool.

The $S$-matrix element describing photon radiation when the system transitions
from the initial state $i$ to the final state $f$ is

$\displaystyle S_{fi}=-ie\int dt\int
d^{3}x\,{\bm{j}}_{fi}(x)\cdot{\bm{A}}^{*}(x)\,,$ (35)

where the transition current is

$\displaystyle{\bm{j}}_{fi}=\frac{i}{2M}\left(\psi_{i}{\bm{\nabla}}\psi^{*}_{f}-\psi^{*}_{f}{\bm{\nabla}}\psi_{i}\right)\,.$
(36)

We will distinguish the quantum numbers of the final state with the prime,
i.e. $i=\\{m,k_{z},\ldots\\}$, $f=\\{m^{\prime},k_{z}^{\prime},\ldots\\}$. The
well wave functions, including their time dependence, read

$\displaystyle\psi(x)=\frac{1}{2\pi}X(r)e^{im\phi}e^{ik_{z}z}e^{-iEt}\,,$ (37)

where $X(r)$ is an appropriate radial function derived in the previous
section. The wave functions are normalized by $\int\psi^{*}\psi d^{3}x=1$
which implies

$\displaystyle\int_{0}^{1/\Omega}drrX^{2}=1\,.$ (38)

The photon wave function is

$\displaystyle{\bm{A}}^{*}(x)=\frac{1}{\sqrt{2\omega
V}}{\bm{\epsilon}}^{*}_{{\bm{p}},\lambda}e^{-i{\bm{p}}\cdot{\bm{x}}+i\omega
t}\,,$ (39)

where ${\bm{\epsilon}}^{*}_{{\bm{p}},\lambda}$ is the polarization vector,
$\lambda=\pm 1$.

Since we treat the well in non-relativistic approximation, the leading
contribution to the radiation intensity is the electric dipole radiation
corresponding to wave lengths much larger than the system size, i.e. $pR\ll
1$. Thus neglecting the phase ${\bm{p}}\cdot{\bm{r}}$ in the exponent in (39)
and substituting it along with (37) into (36) and (35) produces

$\displaystyle S_{fi}=\frac{e(2\pi)^{2}}{2M\sqrt{2\omega
V}}{\bm{\epsilon}}^{*}_{{\bm{p}},\lambda}\cdot{\bm{I}}_{fi}\delta(E-E^{\prime}-\omega)\delta_{mm^{\prime}}\delta_{k_{z}k_{z}^{\prime}}\,,$
(40)

where we defined

$\displaystyle{\bm{I}}_{fi}$
$\displaystyle=I_{r,fi}\hat{{\bm{r}}}+I_{\phi,fi}\hat{{\bm{\phi}}}+I_{z,fi}\hat{{\bm{z}}}\,$
$\displaystyle=\hat{{\bm{r}}}\int_{0}^{1/\Omega}(X_{f}^{\prime}X_{i}-X_{f}X_{i}^{\prime})rdr+2m\hat{{\bm{\phi}}}\int_{0}^{1/\Omega}X_{f}X_{i}dr+2k_{z}\hat{{\bm{z}}}\int_{0}^{1/\Omega}X_{f}X_{i}rdr\,.$
(41)

The total radiation intensity is given by dividing the emission probability
$|S|^{2}$ by the observation time $\Delta t$, multiplying by the photon energy
$\omega$ and summing and integrating over the photon phase space:

$\displaystyle W_{fi}=\sum_{\lambda}\int\frac{|S|^{2}}{\Delta
t}\omega\frac{do_{\hat{{\bm{p}}}}\omega^{2}d\omega V}{(2\pi)^{3}}\,,$ (42)

where $do_{\hat{{\bm{p}}}}$ is the solid angle element in the direction of the
photon momentum. Substituting (42) into (40) and using

$\displaystyle\sum_{\lambda}|{\bm{\epsilon}}^{*}_{{\bm{p}},\lambda}\cdot{\bm{I}}_{fi}|^{2}=\frac{|{\bm{I}}_{fi}\times{\bm{p}}|^{2}}{\omega^{2}}\,,$
(43)

we obtain

$\displaystyle
W_{fi}=\frac{1}{24\pi}\frac{e^{2}\omega^{2}}{M^{2}}|{\bm{I}}_{fi}|^{2}\delta_{mm^{\prime}}\delta_{k_{z}k_{z}^{\prime}}\,.$
(44)

Eqs. (44) and (V) give the intensity of the electromagnetic radiation in the
dipole approximation. The photon energy $\omega$ is fixed by the energy and
momentum conservation and depends on the angular velocity $\Omega$.

###  V.1 Radiation by rapidly rotating cylindrical well

Using the model we developed in the previous section we can compute the
intensity of electromagnetic radiation of the rapidly rotating cylindrical
well $R>1/\Omega$. For simplicity we consider a reference frame where
$k_{z}=0$. The radial wave functions normalized by (38) and satisfying the
boundary condition (20) are

$\displaystyle X(r)=\frac{\sqrt{2}\Omega}{J_{m+1}(x_{ma})}J_{m}(x_{ma}\Omega
r)\,.$ (45)

Using these in (V) and plugging into (44) furnishes

$\displaystyle
W_{fi}=\frac{1}{96\pi}\frac{e^{2}\Omega^{6}}{M^{4}}(x_{ma}^{2}-x_{ma^{\prime}}^{2})^{2}\Omega^{2}(\tilde{I}_{r,fi}^{2}+\tilde{I}_{\phi,fi}^{2})\delta_{mm^{\prime}}\,,$
(46)

where the photon energy is, using (22),

$\displaystyle\omega=E_{ma0}-E_{ma^{\prime}0}=\frac{\Omega^{2}}{2M}(x_{ma}^{2}-x_{ma^{\prime}}^{2})\,,$
(47)

and the functions

$\displaystyle\tilde{I}_{r,fi}$ $\displaystyle=\int_{0}^{1}\frac{2\rho
d\rho}{J_{m+1}(x_{ma})J_{m+1}(x_{ma^{\prime}})}\left[J_{m}^{\prime}(x_{ma^{\prime}}\rho)J_{m}(x_{ma}\rho)x_{ma^{\prime}}-J_{m}(x_{ma^{\prime}}\rho)J_{m}^{\prime}(x_{ma}\rho)x_{ma}\right],$
(48) $\displaystyle\tilde{I}_{\phi,fi}$
$\displaystyle=\int_{0}^{1}\frac{4md\rho}{J_{m+1}(x_{ma})J_{m+1}(x_{ma^{\prime}})}J_{m}(x_{ma^{\prime}}\rho)J_{m}(x_{ma}\rho)\,$
(49)

are independent of $\Omega$. They can be expressed in terms of the
hypergeometric function.

###  V.2 Radiation by rotating infinite cylindrical well

The calculation of the radiation intensity for slowly rotating cylindrical
well $R<1/\Omega$ can also be done analytically. However it leads to bulky
expressions involving Bessel functions. A more instructive model is an
_infinite_ cylindrical well, i.e. the potential $U=0$ if $r\leq R$ and
$U=\infty$ otherwise. The corresponding wave functions are similar to (45)

$\displaystyle X(r)=\frac{\sqrt{2}}{\ell
J_{m+1}(x_{ma})}J_{m}(x_{ma}r/\ell)\,,$ (50)

where $\ell=R$ if $R<1/\Omega$ (slow rotation) and $\ell=1/\Omega$ otherwise.
The intensity is now given by (see (46)):

$\displaystyle
W_{fi}=\frac{1}{96\pi}\frac{e^{2}}{M^{4}\ell^{6}}(x_{ma}^{2}-x_{ma^{\prime}}^{2})^{2}(\tilde{I}_{r,fi}^{2}+\tilde{I}_{\phi,fi}^{2})\delta_{mm^{\prime}}\,.$
(51)

Thus, in intensity is $\Omega$ independent for slow rotation. However, once
$\Omega>1/R$ the intensity increases rapidly in proportion to $\Omega^{6}$:

$\displaystyle
W_{fi}=(R\Omega)^{6}W_{fi}^{\Omega=0}\,,\qquad\text{if}\quad\Omega R>1\,.$
(52)

While the particular functional form of $W_{fi}(\Omega)$ depends on the form
of the potential $U$, the observation of the steep growth of the radiation
intensity with $\Omega$ appears to be quite general.

##  VI Discussion

We computed the energy spectrum of a particle in the non-relativistic
cylindrical potential well rotating with constant angular velocity $\Omega$.
The energy levels increase with the angular velocity $\Omega$ until they are
pushed out of the well at some critical value $\Omega_{c}$ as can be seen in
Fig. 2. The particular form of the functional dependence $E(\Omega)$ is model-
dependent. The discontinuity at $R\Omega=1$ is an artifact of the cylindrical
well approximation; it disappears for a smoother potential such as the
attractive $1/r$ potential discussed in Sec. IV.

To investigate the effect of rotation in a phenomenologically interesting case
of charmonium one would have to consider rotation of the spherical symmetric
potential, which has to be dealt with numerically. Nevertheless, we can make a
ballpark estimate using (34). The binding energy of
$J\mskip-2.0mu/\mskip-0.5mu\psi$ is
$M(\psi^{\prime})-M(J\mskip-2.0mu/\mskip-0.5mu\psi)=\frac{16}{9}\alpha^{2}M=0.59$
GeV. The reduced mass is $M=m_{c}/2=0.63$ GeV which implies that
$\alpha=0.72$. Using the values $\ell_{1,0}$ and $\ell_{2,0}$ listed at the
end of Sec. IV we derive that $J\mskip-2.0mu/\mskip-0.5mu\psi$ and
$\psi^{\prime}$ dissociate into the continuum when they get swirled by a
vortex rotating with angular velocity $\Omega=3$ fm-1 and $\Omega=0.6$ fm-1
respectively. These estimates of $\Omega$ are somewhat larger than the
vorticity achievable at a (relatively) low energy relativistic heavy ion
collisions Deng:2016gyh . We stress that at the above values of vorticity, the
bound states completely dissolve, however the effect of rotation — the
decrease of the binding energy with $\Omega$ — is essential even at sub-
critical angular velocities. Needless to say that these estimates should be
taken only as a motivation for further investigation with more accurate
quarkonium models.

We also computed the intensity of radiation emitted by a rotating bound state
and found that when $R\Omega<1$ it increases by a factor $(R\Omega)^{6}$. This
indicates an increase in transition probability from the excited to the ground
state.

Our results indicate that plasma rotation has significant effect on the hadron
spectra in relativistic heavy-ion which deserves a dedicated quantitative
study.

###### Acknowledgements.

This work was supported in part by the U.S. Department of Energy under Grant
No. DE-FG02-87ER40371.

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