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# A bounded rationality account of dependency length minimization in Hindi

Sidharth Ranjan<EMAIL_ADDRESS>
University of Stuttgart, 70174 Stuttgart, Germany
Titus von der Malsburg<EMAIL_ADDRESS>
University of Stuttgart, 70174 Stuttgart, Germany

###### Abstract

The principle of dependency length minimization, which seeks to keep
syntactically related words close in a sentence, is thought to universally
shape the structure of human languages for effective communication. However,
the extent to which dependency length minimization is applied in human
language systems is not yet fully understood. Preverbally, the placement of
long-before-short constituents and postverbally, short-before-long
constituents are known to minimize overall dependency length of a sentence. In
this study, we test the hypothesis that placing only the shortest preverbal
constituent next to the main-verb explains word order preferences in Hindi (a
SOV language) as opposed to the global minimization of dependency length. We
characterize this approach as a least-effort strategy because it is a cost-
effective way to shorten all dependencies between the verb and its preverbal
dependencies. As such, this approach is consistent with the bounded-
rationality perspective according to which decision making is governed by
‘fast but frugal’ heuristics rather than by a search for optimal solutions.
Consistent with this idea, our results indicate that actual corpus sentences
in the Hindi-Urdu Treebank corpus are better explained by the least effort
strategy than by global minimization of dependency lengths. Additionally, for
the task of distinguishing corpus sentences from counterfactual variants, we
find that the dependency length and constituent length of the constituent
closest to the main verb are much better predictors of whether a sentence
appeared in the corpus than total dependency length. Overall, our findings
suggest that cognitive resource constraints play a crucial role in shaping
natural languages.

Keywords: Hindi; Word order; Syntactic choice; Locality; Production; Cognitive
modeling

## Introduction

Human working memory is viewed as a limited capacity system, where there is
pressure to minimize the cognitive load by not retaining information longer
than necessary. ? (?, ?, ?) proposed the idea of bounded rationality in
decision making which incorporates the aforementioned cognitive limitation and
proposed that human’s ability to make rational decisions has adapted to making
choices that are satisfactory rather than optimal. In language, speakers’
decisions are influenced by the context in which the decision is
made—availability of information, cognitive resources, and the limited
response time. This tension naturally exerts a challenge for speakers and
constrains them to adopt certain strategies to formulate sentences that are
most efficient for communication. For example, choosing word-order patterns
that are most common in daily discourse or placing the words/phrases early in
the sentence that are more accessible or salient in working memory. In this
work, we examine whether word order variation in Hindi can be seen as a result
of the pressures for efficient communication within the constraints of bounded
rationality.

Prior work has investigated the role of phrase length on the constituent
ordering of sentences (?, ?, ?, ?, ?, ?, ?). The findings suggest that SVO
languages (e.g., English) have a preference for short-before-long linguistic
structure such that shorter phrases are more accessible and get realized
before relatively less accessible or longer ones (?, ?). In contrast, for SOV
languages (e.g., Japanese), long-before-short ordering is prevalent, i.e.,
longer phrases tend to be shifted ahead of shorter ones with a view that long
phrases are semantically rich and conceptually salient, and thus realized
before than the shorter ones (?, ?). In this vein, Gibson’s Dependency
Locality Theory provides a unified explanation for constituent ordering
patterns in both SOV and SVO languages (?, ?). The theory prefers constituent
order that minimizes the overall dependency length of the sentence
irrespective of language. Preverbally, long-before-short and postverbally,
short-before-long constituent ordering minimize the overall dependency length
in the sentence (?, ?).

The principle of dependency length minimization (DLM), is believed to reduce
the cognitive load on working memory by reducing the distance between
syntactically related words in the sentence for both speakers and hearers (?,
?, ?, ?, ?, ?, ?). This has been shown to be the case in natural languages,
which tend to have shorter overall dependency lengths than those produced
randomly (?, ?, ?, ?, ?). The DLM hypothesis has therefore been demonstrated
to be an essential characteristic of efficient word order by virtue of
communicative efficiency in human language (?, ?). However, the extent to
which the minimization of dependency length is applied to human language
systems in not yet fully understood. This serves as our primary motivation,
for which we apply the idea of bounded rationality to DLM and investigate how
well it explains word-order preferences in Hindi.

  * (1)

    * a.

      *

di toffee rote hue bacche=ko baajaar jaate samaye maa=ne di-pfv.f.sg toffee
cry=prog child=acc market going while mother=erg

The mother gave the crying child a candy while going to market.

    * b.

      * maa=ne rote hue bacche=ko baajaar jaate samaye toffee di

    * c.

      * rote hue bacche=ko baajaar jaate samaye maa=ne toffee di

Hindi, a Indo-Aryan language from the Indo-European language family, has
subject-object-verb (sov) as the basic word order and has a rich case-marking
system (?, ?). In contrast to English, Hindi exhibits greater flexibility in
the arrangement of words within a sentence, as illustrated in Example 1.
Example 1a has 4 preverbal constituents and is considered the most preferred
syntactic choice since it originally appeared in the corpus as opposed to all
variants (4! = 24) that are possible with this sentence.

In this work, we test our hypothesis that speakers adopt a least-effort
strategy to determine the word order choices in Hindi as opposed to global
minimization of dependency length. We define least-effort strategy as the
placement of shortest preverbal constituent closest to the main verb as this
is the most economical way of reducing all the dependencies between the verb
and its preverbal dependents (see Figure 3 for an illustration). We generated
different counterfactual variants by randomly rearranging their preverbal
constituents (see Examples 1b-1c to list a few) for each reference sentence
(1a) in Hindi-Urdu Treebank corpus of written text. Next, we conducted a
quantitative analysis of those variants and their corresponding corpus
reference sentences. We then deployed a logistic regression classifier to
distinguish the corpus reference sentence from the artificially generated
variants based on the dependency length and constituent length of preverbal
constituents, and overall dependency length of the sentence. The primary
motivation behind choosing constituent-level predictors was to test the
efficacy of the least-effort strategy (see Figure 3d) against global
minimization constraint (see Figure 3b) under the purview of bounded
rationality. As the number of preverbal constituents increases (say from 3
$\rightarrow$ 5), the number of possible variants also increases significantly
(3! = 6 $\rightarrow$ 5! = 120); overburdening the speakers to make choices. A
boundedly rational speaker would then apply a heuristic strategy that
identifies a satisfactory solution that does not necessarily minimize total
dependency length (?, ?, ?).

Our main contribution through this work is that we show the impact of
dependency length minimization on word order choices in Hindi using large-
scale naturalistic data rather than the constructed experimental stimuli and
offer cross-linguistic evidence essential for the advancement of cognitive
theories (?, ?).

The remainder of this paper is structured as follows. We first briefly
describe the data and methods used to investigate the hypothesis, followed by
the results in this work. Towards the end, we discuss the broader implications
of our findings with a conclusion and future directions.

## Method

(a) Reference sentences (7586 data points)  (b) Variant sentences (184818 data
points)

Figure 1: Percentages of reference and variant sentences with varying
preverbal constituents

Our data set consists of 7,586 declarative sentences from the Hindi-Urdu
Treebank (HUTB) corpus of written text belonging to the newswire domain (?,
?). For each reference sentence in the corpus, we created artificial variants
by randomly permuting the preverbal constituents whose head was immediate
child to the root verb in their projective dependency tree (see Figure 3). In
instances where there were more than 100 variants (an arbitrary
cutoff111Higher or lower cutoffs do not impact our results. to keep our
computation tractable), we randomly selected 99 of them. Our variant
generation approach resulted in 184818 competing variants in total, for the
reference sentences in our dataset. Refer to Example 1 for an illustration.
Figure 1 depicts the percentages of reference-variant sentences for each
number of preverbal constituents. Due to insufficient data points for
sentences with 7 or more preverbal constituents, we only present the corpus
analysis results for sentences with up to 6 preverbal phrases. However, for
the binary classification task presented in the later section of this work, we
used the entire dataset to compute our results 222We also carried out a
similar analysis with only grammatical variants and found that our findings
were consistent with those reported below..

Figure 2: Average constituent length of preverbal constituents for corpus
sentences with 2 to 6 constituents separately

## Corpus analysis

Previous corpus studies in Hindi have revealed a predominance of long-before-
short constituent orders preverbally, and smaller head-dependent distances (?,
?, ?). To test our hypothesis of least-effort strategy, we investigated the
role of all preverbal constituents in minimizing the overall dependency length
of the sentence. Consistent with previous studies, we also found that long-
before-short constituent orders are dominant in Hindi (?, ?). In addition, we
also observed a wide distribution of long-before-short and short-before-long
constituents in corpus sentences with varying preverbal constituents (see
Figure 2). Crucially, the plot in Figure 2 suggests the constituent closest to
the main verb has a strong tendency to be the shortest among all the preverbal
constituents in the sentence. This is consistent with the predictions of the
least effort principle, because moving the shortest constituent to the main
verb minimizes not just that constituent’s dependency length but also the
dependency lengths of all other preverbal dependents of the main verb,
therefore striking a favorable cost-benefit ratio.

Interestingly, Figure 2 also shows that the pressure to move shorter
constituent to the main verb increases with the number of preverbal
constituents, thus suggesting that speakers may dynamically balance production
cost and comprehenders’ cognitive constraints.333Pearson’s correlation
coefficient between sentence length and number of preverbal constituents was
0.45. To further test support the idea that speakers adopt a least-effort
strategy (i.e. place shortest constituent next to the main verb), we compared
the total dependency length of reference sentences (attested in the corpus) to
four alternatives with different constituent orders illustrated in Figure 3
for example sentence 1a:

  1. 1.

Ascending order: Arrange the preverbal constituents in their increasing order
of constituent lengths. This arrangement leads to maximal dependency length of
the sentence (See Figure 3a).

  2. 2.

Descending order: Arrange the preverbal constituents in their decreasing order
of constituent lengths. This arrangement globally minimizes the dependency
length of the sentence (See Figure 3b).

  3. 3.

Random order: Arrange the preverbal constituents randomly (See Figure 3c).

  4. 4.

Least-effort order: Start with a random order of preverbal constituents and
then simply move the shortest constituent next to the main verb (See Figure
3d).

[arc edge, arc angle=80, text only label, label style=above] [column
sep=-0.05cm] toffee & maa & ne & baajaar & jaate & samaye & rote & hue &
bacche & ko & di
& & & & & & & & & &
shortest& & & & & & & longest & & & verb
& & & & & & & & & &
candy & mother & erg & market & going & while & cry & aux & child & acc & gave
‘The & mother & gave & the & crying & child & a candy & while & going & to the
& market.’
1119 (DL=23) 1128 1155 1191 111subj 123obj1 146pp1 1710verb 11111verb

(a) Subfigure 3 list of tables text

[arc edge, arc angle=80, text only label, label style=above] [column
sep=0.05cm] rote & hue & bacche & ko & baajaar & jaate & samaye & maa & ne &
toffee & di
& & & & & & & & & &
& longest & & & & & & & & shortest & verb
& & & & & & & & & &
‘The & mother & gave & the & crying & child & a candy & while & going & to the
& market.’
1137 (DL=13) 1164 1182 11100 114subj 157obj1 189pp1 11010verb 11111verb

(b) Subfigure 4 list of tables text

[arc edge, arc angle=80, text only label, label style=above] [column sep=0cm]
maa & ne & baajaar & jaate & samaye & toffee & rote & hue & bacche & ko & di
& & & & & & & & & &
& & & & & shortest & & & & & verb
& & & & & & & & & &
‘The & mother & gave & the & crying & child & a candy & while & going & to the
& market.’
1119 (DL=20) 1146 1164 1191 112subj 135obj1 166pp1 1710verb 11111verb

(c) Subfigure 1 list of tables text

[arc edge, arc angle=80, text only label, label style=above] [column
sep=0.03cm] maa & ne & baajaar & jaate & samaye & rote & hue & bacche & ko &
toffee & di
& & & & & & & & & &
& & & & & & & & & shortest & verb
& & & & & & & & & &
‘The & mother & gave & the & crying & child & a candy & while & going & to the
& market.’
1119 (DL=17) 1146 1182 11100 112subj 135obj1 169pp1 11010verb 11111verb

(d) Subfigure 2 list of tables text

Figure 3: Dependency length and constituent ordering for Hindi head-final
structure; Only main verb dependencies are depicted in the figures; Total
dependency length (DL) of the structure indicated above each sub-figure;
Constituent dependency length is mentioned above each dependency arc

Our results in Figure 4 show that the dependency length of sentences in the
corpus generally tracks the dependency length of the least-effort solution for
sentences with three or more preverbal constituents. Only sentences with two
preverbal constituents deviate from the least-effort order. This may be
because the memory pressure is so low in these cases that the system does not
optimize dependency length at all, not even with the least effort heuristic.
Consistent with previous studies, the plot also confirms that corpus reference
sentences on average have lower dependency length than the sentences with
random word orders (?, ?, ?, ?, ?).

These results further corroborate the idea that dependency length minimization
in Hindi, and thus constituent order, are governed by a least-effort strategy.
This is consistent with the bounded rationality view of decision making, based
on which we would expect that speakers consider only a limited search space
when planning sentences (?, ?).

In the next section, we further test and validate the least-effort hypothesis
using a decision task with a goal to distinguish reference sentences from
random variants (?, ?, ?).

Figure 4: Average total dependency length (i.e., total dependency length
normalized by the number of words in a sentence) for different constituent
orderings and for different numbers of preverbal constituents (7586 data
points for each ordering)

## Computational simulation

We set up a binary classification task to distinguish the reference sentence
amidst the artificially generated variants. Our original dataset had a
significant class imbalance, with 184818 variants compared to only 7586
reference sentences. To address this imbalance, we applied the technique
suggested by ? (?) which transforms the task of classifying a sentence as
either reference or variant into a task of ranking reference sentence against
each of its variants (pairwise ranking) by training the classifier on the
difference between the feature vectors of variant sentence and corresponding
paired reference sentence (see Equations 1 and 2).

$w~{}\cdot~{}\phi(reference)>w~{}\cdot~{}\phi(variant)$ (1)

$w~{}\cdot~{}(\phi(reference)~{}-~{}\phi(variant))>0$ (2)

Equation 1 represents the objective of a standard binary classifier where
corpus reference sentence is preferred against its corresponding variant
sentence. The classifier must determine a feature weight ($w$) such that the
dot product of $w$ and the reference feature vector ($\phi(reference)$) is
greater than the dot product of $w$ and the variant feature vector
($\phi(variant)$). This objective can be rephrased as Equation 2 where the dot
product of $w$ and the difference between the feature vectors must be greater
than zero. The dataset was structured such that each variant sentence was
paired with its corresponding reference sentence, with the order being
balanced across these pairs. For instance, Example 1 would give (1a-1b) and
(1c-1a). Feature vectors were subtracted and binary labels were assigned to
each transformed data point. The “Reference-Variant” pairs were labeled as “1”
and “Variant-Reference” pairs were labeled as “0”, thus balancing the
previously imbalanced classification task. Post ?’s transformation, we had in
total 184818 data points for our classification task. See ? (?) for more
details.

We used the glm function in R to test the stated hypothesis at the outset
(dependent variable $\sim$ independent variables):
$choice\sim\delta~{}dependency~{}length$ and
$choice\sim\delta~{}constituent~{}length$. Here, choice is a binary choice
dependent variable (1 stands for reference sentence preference, and 0 denotes
the variant sentence preference). The delta ($\delta$) refers to the
difference between the feature vectors of reference sentence and its paired
variant. The classifier predicts label “1” if corpus reference sentence
outranks a variant paired with it and “0” if variant outranks its paired
reference sentence. All the independent variables were normalised to
$z$-scores, i.e., the predictor’s value (centered around its mean) was divided
by its standard deviation. We estimated independent variables as below (also
see Figure 3 for illustration):

  1. 1.

total dependency length: Summation of head-dependent distances in a dependency
tree of a sentence. And the head-dependent distance was estimated by counting
the number of intervening words between them.

  2. 2.

constituent dependency length: Number of words spanned between the head of the
preverbal constituent and the main root on which the constituent is dependent.

  3. 3.

constituent length: Total word counts in the constituent.

### Regression analysis

In this section, we test our main hypothesis that speakers optimize word order
choices in Hindi by adopting least-effort strategy (i.e., shifting the
shortest constituent closest to the main verb) under the constraints of
dependency length minimization. Figure 1b depicts the distribution of variants
with varying preverbal constituents. Since our classification dataset is
dominated by reference-variant pairs with 5 preverbal constituents, we test
the least-effort hypothesis on the transformed version of our dataset with 5
preverbal constituents only (87143 data points) using a logistic regression
model. We estimated the dependency length and constituent length of all 5
preverbal constituents and introduced them as predictors in our regression
models ($choice~{}\sim~{}\delta const1+\delta const2+\delta const3+\delta
const4+\delta const5$). However, before doing so, we first deployed the
Recursive Feature Elimination Cross-validation algorithm (?, ?, RFECV) to find
the best predictors among five preverbal constituents and then reported the
regression results using the obtained optimal features.

Tables 1 and 2 display the results of our regression analysis. Negative
regression coefficients in Tables 1 and 2 suggest that reference sentences
tend to have lower constituent-level dependency length and the length of
constituent, respectively, than their paired variant. The larger magnitude of
the regression coefficient of the last two preverbal constituents closest to
the verb ($const4~{}\&~{}const5$) as opposed to other preverbal constituents
in the model suggests their greater role in minimizing overall dependency
length of reference sentence than the paired variants. Notably, our results
were consistent across reference-variant sentences with 2 to 6 preverbal
constituents, indicating that our findings are not limited to rare
constructions, but are applicable to frequent ones as well in the Hindi
natural corpus. In sum, these findings suggest that speakers indeed, apply the
least-effort strategy to optimize for word orders such that the preverbal
constituents closest to the verb are shorter and also have lower dependency
length.

Predictor | Estimate | Std-Error | z-value
---|---|---|---
Intercept | 0.018 | 0.008 | 1.98
$1^{st}$ constituent’s deplen | -0.004 | 0.012 | -0.35
$2^{nd}$ constituent’s deplen | 0.088 | 0.014 | 6.20
$3^{rd}$ constituent’s deplen | -0.147 | 0.015 | -10.12
$4^{th}$ constituent’s deplen | -0.529 | 0.016 | -32.90
$5^{th}$ constituent’s deplen | -2.720 | 0.023 | -118.03

Table 1: Regression model containing dependency lengths (deplen) of preverbal
constituents as predictors (87143 data points); significant predictors denoted
in bold with p $<$ 0.001

Predictor | Estimate | Std-Error | z-value
---|---|---|---
Intercept | -0.003 | 0.008 | -0.41
$1^{st}$ constituent’s length | -0.083 | 0.009 | -8.44
$3^{rd}$ constituent’s length | 0.058 | 0.010 | 5.72
$4^{th}$ constituent’s length | -0.148 | 0.009 | -15.26
$5^{th}$ constituent’s length | -1.549 | 0.016 | -97.82

Table 2: Regression model containing preverbal constituent lengths as
predictors (87143 data points); significant predictors denoted in bold with p
$<$ 0.001; constituent length = word counts in the constituent

Predictor(s) | Accuracy
---|---
total dependency length | 62.69
2nd-last preverbal constituent’s deplen | 68.48***
last preverbal constituent’s deplen | 72.70***
last + 2nd last preverbal constituent’s deplen | 77.17***

Table 3: Prediction accuracy of distinct models with dependency length as
predictor on full dataset (184818 data points; deplen = dependency length;
McNemar’s two-tailed significance test compared to previous row: ***
$p<0.001$)

### Classification analysis

Based on the insights from the previous section, we deployed the last two
preverbal constituent’s dependency length and constituent length (closest to
the main verb: 2nd last & last constituents) as predictors into the
classification model aimed at predicting reference sentences amidst the
counterfactual variants. We make use of our entire dataset (184818 reference-
variant pairs) and conduct 10-fold cross-validation to evaluate model’s
classification accuracy, _i.e.,_ the percentage of data points where a model
correctly predicted the referent sentence over a paired variant. Tables 3 and
4 present the prediction performance of our models. As illustrated in Table 3,
in terms of individual classification accuracy, the dependency length of the
last constituent turned out to be the best predictor (72.70%), thus validating
the efficacy of the proposed least-effort principle. Interestingly, this
measure supersedes the performance of the overall dependency length measure
(62.69%) of the sentence as well.

Over a baseline model containing the last constituent’s dependency length,
adding the 2nd last constituent’s dependency length induced a significant
increase of 4.47% in the classification accuracy (p $<$ 0.001 using McNemar’s
two-tailed test). We find similar results when the constituent lengths of the
last two constituents were deployed as predictors in the classifier, as shown
in Table 4. These findings lend credence to the idea that speakers adopt
least-effort strategy, as measured by dependency length minimization, in
determining Hindi word-order preferences.

Predictor(s) | Accuracy
---|---
2nd-last preverbal constituent length | 54.35
last preverbal constituent length | 69.62***
last + 2nd last preverbal constituent length | 70.28***

Table 4: Prediction accuracy of distinct models with constituent length as
predictor (184818 data points; McNemar’s two-tailed test compared to previous
row: *** $p<0.001$)

## Discussion and conclusion

Our main findings demonstrate that speakers in Hindi choose optimal word
orders by placing the shortest phrase closest to the main verb, consistent
with the central ideas of the bounded rationality view of decision making.
This least effort approach not only reduces the overall dependency length but
also enables a selective search procedure in a vast search space where finding
the optimal solution is difficult. Therefore the sentence planning system uses
a simple yet powerful heuristic. Additionally, as depicted in Figure 2, the
pressure to shorten the constituent closest to the main verb increases as the
length of the sentence increases. We also observed that sentences that
originally appeared in the corpus had dependency lengths distribution similar
to the proposed least-effort solution evincing the efficacy of bounded
rationality in choosing the appropriate word order in Hindi (See Figure 4). In
addition, the HUTB corpus sentences had lower overall dependency length than
the random word orders consistent with the earlier studies (?, ?, ?, ?).

The prediction performance of two distinct classification models aimed at
distinguishing reference sentences amidst artificially generated variants
using just the constituent lengths (70.28%) and just the dependency lengths of
the last two preverbal constituents that are nearest to the main verb (77.17%)
emphasizes the significance of the proposed least-effort strategy in shaping
the constituent order (?, ?, ?, ?). It’s also interesting to note that the
overall dependency length (62.69%) performs poorly in predicting corpus
reference sentences (amidst variants) than the least-effort strategy (last
preverbal constituent’s dependency length = 72.70%; last preverbal constituent
length = 69.62%). As a part of future work, we plan to examine the
effectiveness of our proposed least-effort measure across various languages
from different language families. Studying how languages may differ in the way
they implement bounded rationality strategies can provide further insight into
the relationship between language and cognition.

The broader implication of our work is that word order choices can be seen as
a result of the pressures for efficient communication within the constraints
of bounded rationality (?, ?, ?). The speaker’s attempt to minimize dependency
length is one such frugal strategy to deal with limited cognitive resources
and the complexity of the task. Given that we often lack the cognitive and
computational resources to solve problems exactly, we tend to use
approximation methods as a more viable strategy, leading to bounded
rationality (?, ?, ?). According to ?, the following mechanisms are used by
people to deal with complex problems and make decisions under the constraints
of bounded rationality – pattern recognition, heuristic search, and
extrapolation. Along similar lines, in the case of language, communicative
efficiency can be achieved by using most common word-order patterns, doing
selective search among possible word orders that place the syntactically
related words close together or salient/predictable words early in the
sentence to overcome the memory load. In addition, speakers can make use to
certain properties of languages, such as word-order flexibility, which
provides helpful cues to the listener for improved understanding. From the
listener’s perspective, flexibility in word order can be challenging, and
relying on various language-specific cues could be a rational strategy for
them to navigate this challenge within the constraints of bounded rationality.
Therefore, our results and interpretations of bounded rationality hold from
both speaker as well as hearer perspectives. A thorough investigation from
these perspectives will pave the way for better understanding. To make more
tangible claims about language production in general, it will also be
necessary to investigate the efficacy of the least-effort principle on spoken
data.

Although bounded rationality may not fully account for all aspects of
language, it may provide an explanation for certain aspects, such as the
extent of dependency length minimization, word-order patterns, individual
differences in decision-making processes, and the cognitive mechanisms
involved in language use. Future work needs to investigate how different
factors, such as context, task demands, and individual cognitive differences,
may affect how speakers and listeners use boundedly rational strategies.

Overall, our results suggest that within the constraints of bounded
rationality, dependency length minimization is a significant predictor of
Hindi word order patterns.

## Acknowledgments

We would like to thank the anonymous CogSci-2023 reviewers for their helpful
suggestions and feedback.

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#  $\bar{B}^{0}\to\bar{K}^{(*)\,0}X$, $B^{-}\to K^{(*)\,-}X$,
$\bar{B}_{s}^{0}\to\eta(\eta^{\prime},\phi)X$ from the $X(3872)$ molecular
perspective

Wei-Hong Liang<EMAIL_ADDRESS>Department of Physics, Guangxi Normal
University, Guilin 541004, China  Guangxi Key Laboratory of Nuclear Physics
and Technology, Guangxi Normal University, Guilin 541004, China  Ting Ban
Department of Physics, Guangxi Normal University, Guilin 541004, China
Eulogio Oset<EMAIL_ADDRESS>Department of Physics, Guangxi Normal
University, Guilin 541004, China  Departamento de Física Teórica and IFIC,
Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de
Paterna, Aptdo.22085, 46071 Valencia, Spain

###### Abstract

We study the decays $\bar{B}^{0}\to\bar{K}^{0}\,X$, $B^{-}\to K^{-}\,X$,
$\bar{B}^{0}_{s}\to\eta(\eta^{\prime})\,X$, $\bar{B}^{0}\to\bar{K}^{*0}\,X$,
$B^{-}\to K^{*-}\,X$ , $\bar{B}_{s}^{0}\to\phi\,X$, with $X\equiv X(3872)$,
from the perspective of the $X(3872)$ being a molecular state made from the
interaction of the $D^{*+}D^{-},D^{*0}\bar{D}^{0}$ and $c.c.$ components. We
consider both the external and internal emission decay mechanisms and find an
explanation for the $\bar{K}^{0}\,X$ and $K^{-}\,X$ production rates, based on
the mass difference of the charged and neutral $D^{*}\bar{D}$ components. We
also find that the internal and external emission mechanisms add
constructively in the $\bar{B}^{0}\to\bar{K}^{0}\,X$, $B^{-}\to K^{-}\,X$
reactions, while they add destructively in the case of
$\bar{B}^{0}\to\bar{K}^{*0}\,X$, $B^{-}\to K^{*-}\,X$ reactions. This feature
explains the decay widths of the present measurements and allows us to make
predictions for the unmeasured modes of
$\bar{B}^{0}_{s}\to\eta(\eta^{\prime})\,X(3872)$ and $B^{-}\to
K^{*-}\,X(3872)$. The future measurement of these decay modes will help us get
a better perspective on the nature of the $X(3872)$ and the mechanisms present
in production reactions of that state.

##  I Introduction

The $X(3872)$, as the first state found demanding an exotic, non $q\bar{q}$,
nature, has been the subject of intense debate concerning its structure. Some
groups claim a molecular nature with the $D^{*+}D^{-}$, $D^{*0}\bar{D}^{0}$
and c.c. components Braaten:2020nmc ; Liu:2020tqy ; Meng:2020cbk ;
Dong:2021juy ; Wu:2021udi ; Gordillo:2021bra ; Dong:2021bvy ; Meng:2021kmi ;
Kamiya:2022thy ; Lin:2022wmj ; Ji:2022uie ; Wang:2022qxe ; Wang:2022xga ;
Kinugawa:2023fbf ; Yang:2023mov ; Wu:2023rrp ; Peng:2023lfw ;
Terashima:2023tun , other groups claim a compact tetraquark state Shi:2021jyr
; Esposito:2021vhu ; Huang:2021poj ; Chen:2022ddj ; Sharma:2022ena , or some
mixture of the two structures Lebed:2022vks ; Wang:2023ovj  (see Refs.
Kalashnikova:2018vkv ; Yamaguchi:2019vea ; Brambilla:2019esw ; Guo:2019twa ;
Chen:2022asf ; Mai:2022eur  for earlier related work). The extreme proximity
of the state to the $D^{*0}\bar{D}^{0}$ threshold has been a main factor to
support the molecular nature Guo:2017jvc ; Dong:2021juy , but, as discussed in
detail in Ref. Song:2023pdq , the proximity of the state to a threshold is not
sufficient to guarantee its molecular nature, although certainly makes it far
more likely. It is found in Ref. Song:2023pdq  that it is possible to have a
state of non-molecular nature very close to a threshold, but one pays a huge
price: the scattering length $a$ for the hadron-hadron components, to which
the state inevitably couples, goes to zero, and the effective range $r_{0}$
goes to infinity in the limit of zero energy binding. With the small but
finite binging of the $X(3872)$, the resulting $a,r_{0}$ magnitudes if one
demands the state to be of non-molecular nature are very small and very large,
respectively, compared with experimental data, to the point that one can
exclude the non-molecular nature with uncertainties smaller than about $5\%$.

Very recently, in the work of Ref. ShenWang , the
$\bar{B}^{0}\to\bar{K}^{0}\,X(3872)$ and $B^{-}\to K^{-}\,X(3872)$ reactions
were proposed as a means to learn about the nature of the $X(3872)$. Indeed,
with a compact tetraquark nature for the $X(3872)$, one expects the branching
ratio of the two decay modes to be equal, but experimentally
$\mathcal{B}(B^{-}\to
K^{-}\,X(3872))/\mathcal{B}(\bar{B}^{0}\to\bar{K}^{0}\,X(3872))\simeq 2$.
Instead, from the molecular perspective this ratio is tied to the loop
functions $G_{i}$ for the neutral and charged components $D^{*0}\bar{D}^{0}$,
$D^{*+}D^{-}$, respectively, which are different at the pole of the state
because of the mass difference between the charged $D^{*+}D^{-}$ and neutral
$D^{*0}\bar{D}^{0}$ components.

In Ref. ShenWang , the decay from the molecular perspective is studied
considering the dominant external emission decay mode. In the present work we
follow closely the idea of Ref. ShenWang  but include in the study also the
internal emission. Then extend the idea to study the
$\bar{B}^{0}\to\bar{K}^{*0}\,X(3872)$, $B^{-}\to K^{*-}\,X(3872)$ and
$\bar{B}^{0}_{s}\to\phi\,X(3872)$ decays. The latter reaction proceeds only
via internal emission, and we would expect a small rate compared to that of
the $\bar{B}^{0}\to\bar{K}^{*0}\,X(3872)$ decay, but, surprisingly, the rates
are similar. Its explanation is found, because in the case of the
$\bar{K}^{0},K^{-}$ pseudoscalar production the external and internal emission
mechanisms add constructively, while in the case of $\bar{K}^{*0},K^{*-}$
production they add destructively. We compare ratios of branching ratios for
presently measured decay modes and make predictions for the unmeasured modes,
$\bar{B}^{0}_{s}\to\eta(\eta^{\prime})\,X(3872)$ and $B^{-}\to
K^{*-}\,X(3872)$. It will be interesting to compare these predictions with
future measurements, which will certainly help to go deeper in our
understanding of the nature of the $X(3872)$ and the mechanisms of its
production.

##  II Formalism

###  II.1 The molecular $X(3872)$ state

We follow here the work of Ref. Song:2023pdq , where the $X(3872)$ is obtained
from the interaction of the $D^{*0}\bar{D}^{0},D^{*+}D^{-}$ ($+c.c.$)
components. The extended local hidden gauge approach is used, with the coupled
channels

$\begin{split}X_{\rm
c}=&\dfrac{1}{\sqrt{2}}(D^{*+}D^{-}-D^{*-}D^{+}),\\\\[5.69054pt] X_{\rm
n}=&\dfrac{1}{\sqrt{2}}(D^{*0}\bar{D}^{0}-\bar{D}^{*0}D^{0}).\end{split}$ (1)

The interaction between these components stems from the exchange of $\rho$ and
$\omega$ vector mesons, and with the label $X_{\rm c}\equiv 1,X_{\rm n}\equiv
2$, one finds the interaction matrix

$\displaystyle\centering V=\left(\begin{array}[]{cc}\tilde{v}&\tilde{v}\\\
\tilde{v}&\tilde{v}\\\ \end{array}\right),\@add@centering$ (4)

with

$\displaystyle\tilde{v}=-g^{2}\;\dfrac{4\,m_{D^{*0}}\,m_{D^{0}}}{M^{2}_{V}};~{}~{}~{}~{}g=\dfrac{M_{V}}{2\,f};~{}~{}~{}M_{V}=800\,\textrm{
MeV},~{}~{}~{}f=93\,\textrm{ MeV},$ (5)

which has been calculated at the $D^{*0}\bar{D}^{0}$ threshold. The isospin
states, with the isospin multiplets phase convention $(D^{+},-D^{0})$,
$(\bar{D}^{0},D^{-})$ (and same for $D^{*}$), are given by

$\begin{split}|X,I=0\rangle=\dfrac{1}{\sqrt{2}}\,(X_{\rm c}+X_{\rm
n}),\\\\[5.69054pt] |X,I=1,I_{3}=0\rangle=\dfrac{1}{\sqrt{2}}\,(X_{\rm
c}-X_{\rm n}),\end{split}$ (6)

and we see that for $I=0$ the interaction is attractive and can produce a
bound state, but for $I=1$ the interaction is zero and we do not expect a
bound state. We note in passing that the strength of the interaction here is
double than for $I=0$ in the $T_{cc}$ state FeijooLiang . This means we should
expect a bigger binding for the $X(3872)$ than for the $T_{cc}$. With the
centroid of mass data and the $X(3872)$ mass of the PDG pdg , which we write
below,

$\displaystyle m_{{D}^{0}}=(1864.84\pm 0.05)\,\textrm{ MeV},$ $\displaystyle
m_{{D}^{+}}=(1869.66\pm 0.05)\,\textrm{ MeV},$ $\displaystyle
m_{D^{*0}}=(2006.85\pm 0.05)\,\textrm{ MeV},$ $\displaystyle
m_{D^{*+}}=(2010.26\pm 0.05)\,\textrm{ MeV},$ (7) $\displaystyle
m_{D^{*0}}+m_{{D}^{0}}=3871.69\,\textrm{ MeV},$ $\displaystyle
m_{D^{*+}}+m_{{D}^{-}}=3879.92\,\textrm{ MeV},$ $\displaystyle
M_{X(3872)}=(3871.65\pm 0.06)\,\textrm{ MeV},$

the binding of the $X(3872)$ with respect to the $D^{*0}\bar{D}^{0}$ threshold
is of $0.04\,\textrm{ MeV}$, but can be larger considering the uncertainties
of the masses.

The scattering matrix with this potential is easily obtained as

$\displaystyle\centering T=[1-VG]^{-1}\,V=\dfrac{1}{\rm
det}\;\left(\begin{array}[]{cc}\tilde{v}&\tilde{v}\\\ \tilde{v}&\tilde{v}\\\
\end{array}\right),\@add@centering$ (10)

with det being the determinant of $[1-VG]$, given by

${\rm det}=1-\tilde{v}\,G_{1}-\tilde{v}\,G_{2},$ (11)

and the diagonal $D^{*}\bar{D}$ loop function $G={\rm diag}[G_{1},G_{2}]$
given with the cutoff regularization by

$G_{i}=\int_{|\vec{q}\,|<q_{\rm max}}\dfrac{{\rm
d}^{3}q}{(2\pi)^{3}}\;\dfrac{\omega_{1}^{(i)}(\vec{q}\,)+\omega_{2}^{(i)}(\vec{q}\,)}{2\,\omega_{1}^{(i)}(\vec{q}\,)\,\omega_{2}^{(i)}(\vec{q}\,)}\;\dfrac{1}{s-[\omega_{1}^{(i)}(\vec{q}\,)+\omega_{2}^{(i)}(\vec{q}\,)]^{2}+i\epsilon},$
(12)

with $\omega_{j}^{(i)}(\vec{q}\,)=\sqrt{m_{j}^{2}+\vec{q}^{\;2}}$ the energy
of particle $j$ in the $i$-th channel. We use $q_{\rm max}=420\,\textrm{
MeV}$, as demanded in the study of the $T_{cc}$, with $D^{*}D$ components
FeijooLiang , but to get the nominal binging of $40\,\rm keV$ we have to
multiply $\tilde{v}$ by a factor $\beta=0.537$. Then we need to know the
couplings of the $X(3872)$ to any of the charged or neutral components and we
find

$g_{1}^{2}=\lim_{s\to
s_{0}}(s-s_{0})\,T_{11},~{}~{}~{}~{}~{}g_{1}g_{2}=\lim_{s\to
s_{0}}(s-s_{0})\,T_{12},$ (13)

with $s_{0}$ the squared of the energy of the $X(3872)$ state. From Eqs. (10)
and (11) using l’Hospital rule for the limit, we find

$g_{1}^{2}=\dfrac{\tilde{v}}{-\tilde{v}\frac{\partial}{\partial
s}\,(G_{1}+G_{2})}=g_{1}\,g_{2}.$ (14)

This means that $g_{1}=g_{2}$ (or $g_{\rm c}=g_{\rm n}$ in the nomencluture
for charged or neutral components). In the formalism below we just need
$G_{1},G_{2}$ at $s_{0}$ and the ratio $g_{\rm n}/g_{\rm c}=1$, which
indicates that we have an $I=0$ state.

The formalism that we use here follows closely the one of Ref. ShenWang  to
study the $\bar{B}^{0}\to\bar{K}^{0}\,X(3872)$ and $B^{-}\to K^{-}\,X(3872)$
decays. In addition we include the hadronization from the internal emission
mode and extend the formalism to the production of vector mesons.

###  II.2 Internal emission for pseudoscalar production

From the molecular point of view, the $X(3872)$ is given by

$X=\dfrac{1}{\sqrt{2}}(X_{\rm c}+X_{\rm n}),$ (15)

where $X_{\rm c}$ and $X_{\rm n}$ are the charged and neutral components of
$D\bar{D}^{*},\bar{D}D^{*}$ respectively, shown in Eq. (1). Altogether

$X=\dfrac{1}{2}(D^{*+}D^{-}-D^{*-}D^{+}+D^{*0}\bar{D}^{0}-\bar{D}^{*0}D^{0}),$
(16)

which has isospin $I=0$ and $C$-parity $C=+$. In $s$-wave this state
corresponds to $J^{P}=1^{+}$, completing the quantum numbers of the state.

At the quark level, the weak decay for $\bar{B}^{0}\to\bar{K}^{0}\,X(3872)$
and $B^{-}\to K^{-}\,X(3872)$ proceeds as shown in Fig. 1.

Figure 1: Internal emission mechanism at the quark level for
$\bar{B}^{0}\to\bar{K}^{0}\,c\bar{c}$ decay (a) and $B^{-}\to K^{-}\,c\bar{c}$
decay (b).

The $X(3872)$ production from these mechanism is accomplished by means of
hadronization of the $c\bar{c}$ component. Let us see how this proceeds. We
start by writing the pseudoscalar ($P$) and vector ($V$) matrices for
$q_{i}\bar{q}_{j}$ in terms of the physical mesons. With the
$\eta,\eta^{\prime}$ mixing of Ref. Bramon , we have

$P=\left(\begin{array}[]{cccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{3}}\eta+\frac{1}{\sqrt{6}}\eta^{\prime}&\pi^{+}&K^{+}&\bar{D}^{0}\\\\[5.69054pt]
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{3}}\eta+\frac{1}{\sqrt{6}}\eta^{\prime}&K^{0}&D^{-}\\\\[5.69054pt]
K^{-}&\bar{K}^{0}&~{}-\frac{1}{\sqrt{3}}\eta+\sqrt{\frac{2}{3}}\eta^{\prime}{}&D_{s}^{-}\\\\[5.69054pt]
D^{0}&D^{+}&D_{s}^{+}&\eta_{c}\\\ \end{array}\right),$ (17)

$V=\left(\begin{array}[]{cccc}\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&\rho^{+}&K^{*+}&\bar{D}^{*0}\\\\[5.69054pt]
\rho^{-}&-\frac{1}{\sqrt{2}}\rho^{0}+\frac{1}{\sqrt{2}}\omega&~{}K^{*0}{}&D^{*-}\\\\[5.69054pt]
K^{*-}&\bar{K}^{*0}&\phi&D_{s}^{*-}\\\\[5.69054pt]
D^{*0}&D^{*+}&D_{s}^{*+}&J/\psi\\\ \end{array}\right).$ (18)

Then, upon hadronization of the $c\bar{c}$, we need to create a vector and a
pseudoscalar, but the order matters and we can have $PV$ or $VP$ combinations.
Hence

${\rm
a)}~{}PV:~{}~{}c\bar{c}\to\sum_{i}c\,\bar{q}_{i}q_{i}\,\bar{c}=\sum_{i}P_{4i}\,V_{i4}=(PV)_{44}=D^{0}\bar{D}^{*0}+D^{+}D^{*-}\,+\cdots,$
(19) ${\rm
b)}~{}VP:~{}~{}c\bar{c}\to\sum_{i}c\,\bar{q}_{i}q_{i}\,\bar{c}=\sum_{i}V_{4i}\,P_{i4}=(VP)_{44}=D^{*0}\bar{D}^{0}+D^{*+}D^{-}\,+\cdots,$
(20)

where the notation “$\cdots$” indicates other terms that have no overlap with
the $X(3872)$ components of Eq. (16). We see that the combination
$(VP)_{44}-(PV)_{44}$,

$(VP)_{44}-(PV)_{44}=D^{*+}D^{-}+D^{*0}\bar{D}^{0}-D^{*-}D^{+}-\bar{D}^{*0}D^{0},$
(21)

has a perfect match with Eq. (16). This hadronization is common to
$\bar{B}^{0}\to\bar{K}^{0}\,X(3872)$ and $B^{-}\to K^{-}\,X(3872)$.

Let us see with the same mechanism, how we can have the decay
$\bar{B}_{s}^{0}\to\eta(\eta^{\prime})\,X(3872)$. The mechanism is shown in
Fig. 2.

Figure 2: Internal emission mechanism at the quark level for
$\bar{B}_{s}^{0}\to\eta(\eta^{\prime})\,c\bar{c}$ decay.

The hadronization in this case gives us the same combination as in Eq. (21),
and the $s\bar{s}$ splits into $\eta$ and $\eta^{\prime}$ according to the
matrix element $P_{33}$ of Eq. (17), as

$s\bar{s}\to-\frac{1}{\sqrt{3}}\,\eta+\sqrt{\frac{2}{3}}\,\eta^{\prime}.$ (22)

###  II.3 External emission for pseudoscalar production

External emission is the dominant mode of weak decay Chau  since it is color
favoured. One expects something like a factor $3$ times bigger amplitude than
for internal emission. The mechanism for external emission is depicted in Fig.
3.

Figure 3: External emission mechanism for $\bar{B}^{0}\to\bar{c}s\,c\bar{d}$
decay (a) and $B^{-}\to\bar{c}s\,c\bar{u}$ decay (b).

Upon hadronization, we have three possibilities,

  * a)

$\bar{c}s\to PV,~{}c\bar{d}\to P\;(D^{+})$:

$s\bar{c}\to\sum_{i}s\,\bar{q}_{i}q_{i}\,\bar{c}=\sum_{i}P_{3i}\,V_{i4}=(PV)_{34}.$
(23)

Hence,

$\bar{c}s\,c\bar{d}\to(K^{-}D^{*0}+\bar{K}^{0}D^{*-}+\cdots)\,D^{+}.$ (24)

  * b)

$\bar{c}s\to VP,~{}c\bar{d}\to P\;(D^{+})$:

$s\bar{c}\to\sum_{i}s\,\bar{q}_{i}q_{i}\,\bar{c}=\sum_{i}V_{3i}\,P_{i4}=(VP)_{34}.$
(25)

Hence,

$\bar{c}s\,c\bar{d}\to(K^{*-}\bar{D}^{0}+\bar{K}^{*0}D^{-}+\cdots)\,D^{+}.$
(26)

  * c)

$\bar{c}s\to PP,~{}c\bar{d}\to V\;(D^{*+})$:

$s\bar{c}\to\sum_{i}s\,\bar{q}_{i}q_{i}\,\bar{c}=\sum_{i}P_{3i}\,P_{i4}=(PP)_{34}.$
(27)

Hence,

$\bar{c}s\,c\bar{d}\to(K^{-}\bar{D}^{0}+\bar{K}^{0}D^{-}+\cdots)\,D^{*+}.$
(28)

We can see that the combination $(PP)_{34}\,D^{*+}-(PV)_{34}\,D^{+}$,

$(PP)_{34}\,D^{*+}-(PV)_{34}\,D^{+}=(D^{*+}D^{-}-D^{*-}D^{+})\,\bar{K}^{0},$
(29)

has again perfect overlap with the wave function of Eq. (16).

We can repeat the procedure for $K^{-}$ production of Fig. 3(b) and we find
again

$(PP)_{34}\,D^{*0}-(PV)_{34}\,D^{0}=(D^{*0}\bar{D}^{0}-\bar{D}^{*0}D^{0})\,\bar{K}^{-},$
(30)

which overlaps with Eq. (16) in the same way.

We should also note that with this decay mode we find no terms that go to
$\eta(\eta^{\prime})\,X(3872)$. Hence, $\eta,\eta^{\prime}$ production
proceeds only via internal emission.

So far we have done the direct production of the components of the $X(3872)$.
In order to produce the resonance, these components must propagate as depicted
in Fig. 4,

Figure 4: Propagation of $D\bar{D}^{*},\bar{D}D^{*}$ components to create the
$X(3872)$: (a) external emission; (b) internal emission.

where the loops stand for the $G$ function of $D\bar{D}^{*}+c.c.$ propagation.
Then we obtain the amplitude for $\bar{B}^{0}\to\bar{K}^{0}\,X(3982)$ as

$\displaystyle t(\bar{K}^{0})$ $\displaystyle=$ $\displaystyle
C\,G_{D^{*+}D^{-}}(M_{X})\,g_{\rm c}+A\,\left[G_{D^{*+}D^{-}}(M_{X})\,g_{\rm
c}+G_{D^{*0}\bar{D}^{0}}(M_{X})\,g_{\rm n}\right]$ (31) $\displaystyle=$
$\displaystyle C\,g_{\rm
c}\,\left\\{G_{D^{*+}D^{-}}(M_{X})+A^{\prime}\left[G_{D^{*+}D^{-}}(M_{X})+\frac{g_{\rm
n}}{g_{\rm c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})\right]\right\\},$

where $A^{\prime}\equiv A/C$ and $g_{\rm c},g_{\rm n}$ are the couplings of
$(D^{*+}D^{-}-D^{*-}D^{+})$ and $(D^{*0}\bar{D}^{0}-\bar{D}^{*0}D^{0})$ to
$X(3872)$ of Eq. (16), respectively.

Similarly, for $K^{-}(\eta,\eta^{\prime})$ production we find

$\displaystyle t(K^{-})$ $\displaystyle=$ $\displaystyle C\,g_{\rm
c}\,\left\\{\frac{g_{\rm n}}{g_{\rm
c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})+A^{\prime}\left[G_{D^{*+}D^{-}}(M_{X})+\frac{g_{\rm
n}}{g_{\rm c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})\right]\right\\},$ (32)
$\displaystyle t(\eta)$ $\displaystyle=$ $\displaystyle C\,g_{\rm
c}\,A^{\prime}\left[G_{D^{*+}D^{-}}(M_{X})+\frac{g_{\rm n}}{g_{\rm
c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})\right]\,(-\frac{1}{\sqrt{3}}),$ (33)
$\displaystyle t(\eta^{\prime})$ $\displaystyle=$ $\displaystyle C\,g_{\rm
c}\,A^{\prime}\left[G_{D^{*+}D^{-}}(M_{X})+\frac{g_{\rm n}}{g_{\rm
c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})\right]\,(\sqrt{\frac{2}{3}}).$ (34)

Experimentally $\Gamma(K^{-})/\Gamma(K^{0})\sim 2$ pdg  and now we can see
qualitatively the reason for this factor, because at the pole
$(G_{D^{*0}\bar{D}^{0}}/G_{D^{*+}D^{-}})^{2}\sim 2$, as can be seen in the
work of Ref. Gamermann , due to the mass difference between the $D^{*+}D^{-}$
and $D^{*0}\bar{D}^{0}$ components. Assuming the contribution of internal
emission small, this could give a qualitative explanation of the experimental
ratio, which is the idea of Ref. ShenWang . We aim at being more quantitative
here and also extend the formalism to the production of vector mesons that we
address below.

###  II.4 Internal emission for vector production

We now look at the reactions $\bar{B}^{0}\to\bar{K}^{*0}\,X(3872)$, $B^{-}\to
K^{*-}\,X(3872)$, $\bar{B}_{s}^{0}\to\phi\,X(3872)$. The internal emission for
vector production is identical to the one exposed before in Figs. 1(a) and
1(b), replacing $\bar{K}^{0}$ by $\bar{K}^{*0}$ and $K^{-}$ by $K^{*-}$
respectively. For $\phi$ production, it is also easy since the mechanism is
the one of Fig. 2, where now $s\bar{s}\to\phi$. The weight $A$ will now be
$\tilde{A}$, since the coupling of the $q\bar{q}$ component to pseudoscalars
or vectors have different weights LiangOset .

###  II.5 External emission for vector production

Following the steps of the former subsection for external emission we find now

  * a)

$\bar{K}^{*0}$ production:

$(VV)_{34}\,D^{+}-(VP)_{34}\,D^{*+}~{}\to~{}(-1)(D^{*+}D^{-}-D^{*-}D^{+})\,\bar{K}^{*0};$
(35)

  * b)

$K^{*-}$ production:

$(VV)_{34}\,D^{0}-(VP)_{34}\,D^{*0}~{}\to~{}(-1)(D^{*0}\bar{D}^{0}-\bar{D}^{*0}D^{0})\,K^{*-}.$
(36)

Note that we obtain the same expression as in Eqs. (29) and (30), simply
replacing $\bar{K}^{0}$ by $\bar{K}^{*0}$ and $K^{-}$ by $K^{*-}$
respectively. There is only one difference which is the sign. This different
sign will be of relevance for the different rates obtained for pseudoscalar
and vector production, as we shall see.

All this said, the amplitudes for vector production are now

$\displaystyle t(\bar{K}^{*0})$ $\displaystyle=$
$\displaystyle\tilde{C}\,g_{\rm
c}\,\left\\{-G_{D^{*+}D^{-}}(M_{X})+\tilde{A}^{\prime}\left[G_{D^{*+}D^{-}}(M_{X})+\frac{g_{\rm
n}}{g_{\rm c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})\right]\right\\},$ (37)
$\displaystyle t(K^{*-})$ $\displaystyle=$ $\displaystyle\tilde{C}\,g_{\rm
c}\,\left\\{-\frac{g_{\rm n}}{g_{\rm
c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})+\tilde{A}^{\prime}\left[G_{D^{*+}D^{-}}(M_{X})+\frac{g_{\rm
n}}{g_{\rm c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})\right]\right\\},$ (38)
$\displaystyle t(\phi)$ $\displaystyle=$ $\displaystyle\tilde{C}\,g_{\rm
c}\,\tilde{A}^{\prime}\left[G_{D^{*+}D^{-}}(M_{X})+\frac{g_{\rm n}}{g_{\rm
c}}\,G_{D^{*0}\bar{D}^{0}}(M_{X})\right],$ (39)

where $\tilde{A}^{\prime}\equiv\tilde{A}/\tilde{C}$.

Note that in Eqs. (31) to (34) and here in Eqs. (37) to (39) the constants
$\tilde{C}$ and $C$ are different, hence we cannot relate pseudoscalar
production and vector production, but we can compare different rates for
pseudoscalar production and different rates for vector production. There is
one more thing, while $C$ and $\tilde{C}$ and $A,\tilde{A}$ are different, the
ratio $A^{\prime}\equiv A/C$ and $\tilde{A}^{\prime}\equiv\tilde{A}/\tilde{C}$
indicates the fraction of internal to external emission, and this is basically
related to the color factor. Hence we shall make the reasonable assumption
that $A^{\prime}\equiv\tilde{A}^{\prime}$.

In addition, for angular momentum conservation we have the extra factor
$p_{B_{(s)}}^{\mu}\,\epsilon_{\mu}(X)$ for pseudoscalar decay ShenWang  and
$\epsilon_{\mu}(V)\,\epsilon^{\mu}(X)$ for vector decay. Upon squaring the
$t_{i}$ amplitudes, $|t_{i}|^{2}$, and summing over polarizations, the first
term for pseudoscalar production gives
$\vec{p}_{X}^{\;2}\,M_{B_{(s)}}^{2}/M_{X}^{2}$ and the second term for vector
production a constant. Since $|\vec{p}_{X}|\equiv|\vec{p}_{P}|$ in the rest
frame of $B_{(s)}$, with $\vec{p}_{P}$ the pseudoscalar momentum, for
calculating ratios we just include the $\vec{p}_{P}^{\;2}$ factor in the
formula of the width.

##  III Results

In Eqs. (31) to (39) we have the amplitudes for each one of the transitions
considered. The decay width corresponding to these amplitudes are

$\Gamma_{i\to
j}=\dfrac{1}{8\pi}\;\dfrac{1}{M^{2}_{B_{i}}}\;|t_{i,j}|^{2}\,p_{j}\,F(p_{j}),$
(40)

with

$p_{j}=\dfrac{\lambda^{1/2}(M^{2}_{B_{i}},m^{2}_{j},M^{2}_{X})}{2\,M_{B_{i}}},$
(41)

where $B_{i}$ is the decaying $B_{(s)}$ meson and $j$ the pseudoscalar or
vector produced in addition to the $X(3872)$. The factor $F(p_{j})$, where a
constant factor, inoperative in ratios, is omitted is

$\displaystyle F(p_{j})=\left\\{\begin{array}[]{ll}p_{j}^{2},&{\rm
for~{}pseudoscalar~{}production};\\\\[2.84526pt] 1,&{\rm
for~{}vector~{}production}.\\\ \end{array}\right.$ (44)

When calculating ratios between pseudoscalar or vector production rates, the
factors $C\,g_{\rm c}$ and $\tilde{C}\,g_{\rm c}$ cancel and the ratios depend
on the only parameter $A^{\prime}\equiv\tilde{A}^{\prime}$. The present
experimental situation, according to the PDG pdg , is the following:

$\displaystyle\mathcal{B}(\bar{B}^{0}\to\bar{K}^{0}\,X(3872))$
$\displaystyle=$ $\displaystyle(1.1\pm 0.4)\times 10^{-4},$
$\displaystyle\mathcal{B}(B^{-}\to K^{-}\,X(3872))$ $\displaystyle=$
$\displaystyle(2.1\pm 0.7)\times 10^{-4},$
$\displaystyle\mathcal{B}(\bar{B}^{0}\to\bar{K}^{*0}\,X(3872))$
$\displaystyle=$ $\displaystyle(1.0\pm 0.5)\times 10^{-4},$ (45)
$\displaystyle\mathcal{B}(B^{-}\to K^{*-}\,X(3872))$ $\displaystyle<$
$\displaystyle 6\times 10^{-4},$
$\displaystyle\mathcal{B}(\bar{B}_{s}^{0}\to\phi\,X(3872))$ $\displaystyle=$
$\displaystyle(1.1\pm 0.4)\times 10^{-4}.$

We determine the value of $A^{\prime}$ to get the ratio between
$\mathcal{B}(\bar{B}^{0}_{s}\to\phi\,X)/\mathcal{B}(\bar{B}^{0}\to\bar{K}^{*0}\,X)$
and then the rest of the ratios are predictions. We find

$A^{\prime}=0.2,$ (46)

which makes sense as the color suppressed internal emission versus external
emission modes. Then we find the ratios of Tables 1 and 2.

Table 1: Ratios of $R\equiv\mathcal{B}(B_{i}\to j)/\mathcal{B}(\bar{B}^{0}\to\bar{K}^{0}\,X(3872))$. The different life times $\tau_{B_{i}}$ are considered. In brackets experimental ratio with errors summed in quadrature. | $B^{-}\to K^{-}\,X$ | $\bar{B}_{s}^{0}\to\eta\,X$ | $\bar{B}_{s}^{0}\to\eta^{\prime}\,X$
---|---|---|---
$R$ | $2.07~{}~{}(1.91\pm 0.94)$ | $0.043$ | $0.048$
Table 2: Ratios of $R^{\prime}\equiv\mathcal{B}(B_{i}\to j)/\mathcal{B}(\bar{B}^{0}\to\bar{K}^{*0}\,X(3872))$. The different life times $\tau_{B_{i}}$ are considered. In brackets experimental ratio with errors summed in quadrature. | $B^{-}\to K^{*-}\,X$ | $\bar{B}_{s}^{0}\to\phi\,X$
---|---|---
$R^{\prime}$ | $5.2~{}~{}(<6)$ | $1.10~{}~{}(1.10\pm 0.68)$

The essential ingredient in understanding these results is the fact that
$|G_{D^{*0}\bar{D}^{0}}|$ at the pole is bigger than $|G_{D^{*+}D^{-}}|$.
$G_{D^{*+}D^{-}}$ looks like $G_{D^{*0}\bar{D}^{0}}$ but displaced about
$7\;\textrm{ MeV}$ to the right, and both are negative. This has a consequence
that $|G_{D^{*0}\bar{D}^{0}}|>|G_{D^{*+}D^{-}}|$ at the pole as can be seen in
Fig. 5.

Figure 5: $G_{D^{*+}D^{-}}$ and $G_{D^{*0}\bar{D}^{0}}$ as a function of the
energy $\sqrt{s}$. The vertical line indicates the mass of the $X(3872)$.

It is interesting to remark that the contribution of the internal emission,
even if small, has been important to get these results. Indeed, if we neglect
the internal emission then $\mathcal{B}(B^{-}\to
K^{-}X)/\mathcal{B}(\bar{B}^{0}\to\bar{K}^{0}\,X)=2.5$, which is bigger than
experiment. On the other hand, in this extreme case
$\mathcal{B}(\bar{B}_{s}^{0}\to\phi X)=0$. The introduction of the internal
emission term, adding constructively to the external emission in Eqs. (32) to
(34), softens the former ratio. On the other hand, for vector production in
Eqs. (37) and (38), the external and internal emissions interfere
destructively for $\bar{K}^{*0}$ and $K^{*-}$ production, reducing their
widths, and has a consequence that the rates of
$\bar{B}^{0}\to\bar{K}^{*0}\,X(3872)$ and $\bar{B}_{s}^{0}\to\phi\,X(3872)$
are similar.

##  IV Conclusions

In this work we have addressed the issue of $B$ decays to a pseudoscalar and
the $X(3872)$ and a vector and $X(3872)$, from the perspective that the
$X(3872)$ is a dynamically generated state from the $D\bar{D}^{*}+c.c.$
interaction in coupled channels. In particular we have studied the decays
$\bar{B}^{0}\to\bar{K}^{0}\,X$, $B^{-}\to K^{-}\,X$,
$\bar{B}^{0}_{s}\to\eta(\eta^{\prime})\,X$, $\bar{B}^{0}\to\bar{K}^{*0}\,X$,
$B^{-}\to K^{*-}\,X$ , $\bar{B}_{s}^{0}\to\phi\,X$, with $X\equiv X(3872)$. We
have seen that the production of the final states requires both the mechanism
of external and internal emission. Although the internal emission mechanism is
suppressed with respect to external emission by a color factor, we have seen
that for the vector production the two terms interfere destructively, while
they add constructively for pseudoscalar production. This has as a consequence
that while $\eta$ and $\eta^{\prime}$ production, which proceed via internal
emission, are small compared to $\bar{K}^{0}$ of $K^{-}$ production, which
proceeds via the sum of the two mechanisms, $\phi$ production, which also
proceeds via internal emission, has comparable strength to $K^{*}$ production,
where both external and internal emission are at work but add destructively.
The reason for an unexpected factor $2$ between $K^{-}$ and $\bar{K}^{0}$
production is traced back to the composite nature of the $X(3972)$ from the
charged and neutral $X_{\rm c},X_{\rm n}$ of Eq. (1) and the mass difference
between $D^{*+}\,D^{-}$ and $D^{*0}\bar{D}^{0}$. Indeed, the production of the
$X(3872)$ requires the propagation of the $D^{*}\bar{D}$ components, which is
accomplished by means of the loop $G$ functions. The one of the charged
components is displaced about $7\,\textrm{ MeV}$ to higher energies, due to
the mass difference between the charged and neutral components, and this has
as a consequence that at the pole energy the ratio $|G_{\rm n}/G_{\rm c}|^{2}$
is about $2.5$ (see Fig. 5). This would be the ratio between $K^{-}$ and
$\bar{K}^{0}$ production rates in the absence of internal emission. The
contribution of internal emission reduces this factor to about $2$. This
provides a natural explanation for this apparently strange ratio that naively
one would expect to be of the order of unity. The fact that the rates of
vector production also appear fine with the same framework adds extra support
to this picture. In addition we have made predictions for $\eta$ and
$\eta^{\prime}$ production in the pseudoscalar sector, and $K^{*-}$ in the
vector sector. Future measurements of these decays widths should provide extra
support for this picture.

###### Acknowledgements.

This work is partly supported by the National Natural Science Foundation of
China (NSFC) under Grants No. 11975083 and No. 12365019, and by the Central
Government Guidance Funds for Local Scientific and Technological Development,
China (No. Guike ZY22096024). R. M. acknowledges support from the CIDEGENT
program with Ref. CIDEGENT/2019/015, the Spanish Ministerio de Economia y
Competitividad and European Union (NextGenerationEU/PRTR) by the grant with
Ref. CNS2022-13614. This work is also partly supported by the Spanish
Ministerio de Economia y Competitividad (MINECO) and European FEDER funds
under Contracts No. FIS2017-84038-C2-1-P B, PID2020-112777GB-I00, and by
Generalitat Valenciana under contract PROMETEO/2020/023. This project has
received funding from the European Union Horizon 2020 research and innovation
programme under the program H2020-INFRAIA-2018-1, grant agreement No. 824093
of the STRONG-2020 project.

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