[a]Kay Schönwald

#  $\mathcal{O}(\alpha_{s}^{3})$ calculations for the inclusive determination
of $|V_{cb}|$

###### Abstract

For the determination of the Cabbibo-Kobayashi-Maskawa matrix element
$|V_{cb}|$ from inclusive data a precise knowledge of the semileptonic $b\to
c$ decay rate is necessary. Since this observable has a bad convergence
behavior when the heavy quark masses are expressed in the on-shell or
$\overline{\text{MS}}$ scheme the latest determinations have been obtained in
the so called kinetic mass scheme. The relation between the different schemes
needs to be known to high precision as well. In this proceedings we present
our recent calculations which push the precision of both ingredients to
$\mathcal{O}(\alpha_{s}^{3})$. The results can be used to improve the
inclusive determination of $|V_{cb}|$.

##  1 Introduction

Inclusively the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements $|V_{ub}|$ and
$|V_{cb}|$ are extracted from global fits to experimental data on the
semileptonic $B\to X_{c(u)}\ell\overline{\nu}$ decay width and moments of
several kinematic distributions like the ones for the hadronic invariant mass
or the lepton energy [1, 2, 3, 4, 5]. Theoretically these decays can be
described in the heavy quark effective theory (HQET) as a double expansion in
the strong coupling constant $\alpha_{s}$ and the inverse heavy (bottom) quark
mass $1/m_{b}$. The leading term in the expansion in the heavy quark mass is
given by the free quark decay $b\to c(u)\ell\overline{\nu}$. The convergence
of this double series depends crucially on the scheme used to express the
heavy quark mass. Here, the pole mass suffers from renormalon ambiguities [6,
7], which can be avoided by going to, for example, the $\overline{\text{MS}}$
mass scheme, which is often employed in LHC analyses. However, at low energies
it is advantageous to switch to so called threshold masses like the $1S$ [8,
9, 10] or the kinetic [11, 12] mass scheme.

In these proceedings we summarize the recent calculation of the
$\mathcal{O}(\alpha_{s}^{3})$ relation between the pole and kinetic heavy
quark mass (c.f. Refs. [13, 14]) and of the semileptonic decay rate (c.f. Ref.
[15]) as well as their phenomenological implications.

##  2 The Kinetic Heavy Quark Mass to $\mathcal{O}(\alpha_{s}^{3})$

The kinetic heavy quark mass is defined in strong analogy to the relation
between the mass of a heavy meson $M_{H}$ and the respective heavy quark mass
$m_{Q}$:

$\displaystyle M_{H}$
$\displaystyle=m_{Q}+\overline{\Lambda}+\frac{\mu_{\pi}^{2}}{2m_{Q}}+\mathcal{O}\left(\frac{1}{m_{Q}^{2}}\right)~{},$
(1)

where the parameter $\mu_{\pi}^{2}$ is a non-perturbative matrix elements of
local HQET operators and $\overline{\Lambda}$ is the binding energy of the
meson in the heavy quark limit. The relation between the kinetic heavy quark
mass and the pole (or equivalently on-shell) mass is obtained from Eq. (1) by
identifying $M_{H}\to m_{Q}^{\rm OS}$, $m_{Q}\to m_{Q}^{\rm kin}$ and
evaluating the operator matrix elements in perturbation theory [12]. The
explicit relation up to $\mathcal{O}(1/m_{Q})$ reads:

$\displaystyle m_{Q}^{\rm OS}$ $\displaystyle=m_{Q}^{\rm
kin}(\mu)+[\overline{\Lambda}]_{\rm pert}+\frac{[\mu_{\pi}^{2}(\mu)]_{\rm
pert}}{2m_{Q}^{\rm kin}(\mu)}+\mathcal{O}\left(\frac{1}{m_{Q}^{2}}\right)~{}.$
(2)

The relation has been previously calculated up to
$\mathcal{O}(\alpha_{s}^{2})$ [12, 16].

Figure 1: Sample Feynman diagrams for the scattering of an external current
(wavy line) on a heavy quark (solid line). Taken from Ref. [14].

A constructive way to compute the HQET parameters in perturbation is given by
the Small Velocity (SM) sum rules [11]. Here, one considers the scattering of
a heavy quark $Q$ on a current $J$. The current transfers energy to the quark
$Q$ and excites it, causing possibly emissions of further gluons or quarks. We
denote the inclusive final state as $X_{Q}$. Working in the rest frame of the
initial heavy quark we can define the excitation energy $\omega$ by

$\displaystyle\omega$ $\displaystyle=q_{0}-q_{0}^{\rm
min}=q_{0}-\left(\sqrt{\vec{q}^{2}+m_{Q}^{2}}-m_{Q}\right)~{},$ (3)

where $q=(q_{0},\vec{q})$ is the 4-momentum of the current. The velocity of
the system $X_{Q}$ after the scattering is given by $\vec{v}=\vec{q}/m_{Q}$.
The perturbative versions of the operator matrix elements can then be given by

$\displaystyle[\overline{\Lambda}]_{\rm pert}$
$\displaystyle=\lim\limits_{\vec{v}\to
0}\lim\limits_{m_{Q}\to\infty}\frac{2}{\vec{v}^{2}}\frac{\int_{0}^{\mu}\omega\,W(\omega,\vec{v})d\omega}{\int_{0}^{\mu}W(\omega,\vec{v})d\omega}~{},$
(4) $\displaystyle[\mu_{\pi}^{2}]_{\rm pert}$
$\displaystyle=\lim\limits_{\vec{v}\to
0}\lim\limits_{m_{Q}\to\infty}\frac{3}{\vec{v}^{2}}\frac{\int_{0}^{\mu}\omega^{2}\,W(\omega,\vec{v})d\omega}{\int_{0}^{\mu}W(\omega,\vec{v})d\omega}~{},$
(5)

where $W(\omega,\vec{v})$ is the structure function corresponding to the
scattering and the parameter $\mu$ is introduced as a Wilsonian cut-off in
order to separate low and high energy effects. The perturbative versions of
the operator matrix elements are therefore given by moments of the scattering
cross section. However, the non-relativistic description in terms of
excitation energy and velocity given in Eqs. (4) and (5) do not allow a
straight forward expansion on the level of Feynman diagrams. For the
calculation we followed the following strategy (see Ref. [14] for a more
detailed discussion):

  * •

We utilize the optical theorem and consider the discontinuity of the $bJ\to
bJ$ forward scattering diagrams (see Figure 1 for example diagrams).

  * •

We express the non-relativistic quantities $\omega$ and $\vec{v}^{2}$ in terms
of the Lorentz invariants

$\displaystyle
y=m_{Q}-s=-m_{Q}\omega(2+\vec{v}^{2})+\mathcal{O}(\omega^{2},\vec{v}^{4})~{},$
(6) $\displaystyle
q^{2}=-m_{Q}\vec{v}^{2}(m_{Q}-\omega)+\mathcal{O}(\omega^{2},\vec{v}^{4})~{}.$
(7)

  * •

The limit $m_{Q}\to\infty$ can now be realized as the asymptotic expansion
around the threshold $s=m_{Q}^{2}$ (or equivalently $y=0$) for which we use
the strategy of expansion by region [17, 18]. The limit $\vec{v}\to 0$ can
subsequently be realized by a naive Taylor expansion in $q$. When the leading
terms in $y$ and $q^{2}$ of the structure function have been extracted we use
Eqs. (6) and (7) to go back to the non-relativistic quantities and re-expand.

This strategy now allows to use the full machinery of multi-loop calculations,
i.e. we generate one-, two- and three-loop forward scattering diagrams with
qgraf [19] and use FORM [20] to insert the Feynman rules, perform the Dirac
and color algebra and expand all loop-momenta according to the rules of
asymptotic expansion. In the present case the momenta can either scale hard
($k_{i}\sim m_{b}$) or ultrasoft ($k_{i}\sim y/m_{b}$). The corresponding
regions have been cross-checked with the program Asy.m [21]. After the
expansion the denominators become linearly dependent, so a partial fraction
decomposition becomes necessary. For this we used the program LIMIT [22],
which automatizes this step. This program also maps each scalar integral to a
unique integral family, so that we were able to reduce all integrals to a
small set of master integrals using the programs FIRE [23] and LiteRed [24].

If the loop momenta in the asymptotic expansion scale hard ($k_{i}\sim
m_{b}$), the master integrals are given by on-shell propagator integrals,
which are well studied in the literature [25, 26, 27]. For ultrasoft momenta
($k_{i}\sim y/m_{b}$) new types of master integrals appear which were
evaluated using Mellin-Barnes techniques and differential equations in
auxillary parameters.

The final result is given by

$\displaystyle\frac{m^{\text{kin}}}{m^{\text{OS}}}$
$\displaystyle=1-\frac{\alpha_{s}^{(n_{l})}}{\pi}C_{F}\biggl{(}\frac{4}{3}\frac{\mu}{m^{\text{OS}}}+\frac{1}{2}\frac{\mu^{2}}{\left({m^{\text{OS}}}\right)^{2}}\biggr{)}+\left(\frac{\alpha_{s}^{(n_{l})}}{\pi}\right)^{2}C_{F}\Biggl{\\{}\frac{\mu}{{m^{\text{OS}}}}\Biggl{[}C_{A}\biggl{(}-\frac{215}{27}+\frac{2\pi^{2}}{9}+\frac{22}{9}l_{\mu}\biggr{)}$
$\displaystyle\mbox{}+n_{l}T_{F}\biggl{(}\frac{64}{27}-\frac{8}{9}l_{\mu}\biggr{)}\Biggr{]}+\frac{\mu^{2}}{\left({m^{\text{OS}}}\right)^{2}}\Biggl{[}C_{A}\biggl{(}-\frac{91}{36}+\frac{\pi^{2}}{12}+\frac{11}{12}l_{\mu}\biggr{)}+n_{l}T_{F}\biggl{(}\frac{13}{18}-\frac{1}{3}l_{\mu}\biggr{)}\Biggr{]}\Biggr{\\}}$
$\displaystyle\mbox{}+\left(\frac{\alpha_{s}^{(n_{l})}}{\pi}\right)^{3}C_{F}\Biggl{\\{}\frac{\mu}{{m^{\text{OS}}}}\Biggl{[}C_{A}^{2}\biggl{(}-\frac{130867}{1944}+\frac{511\pi^{2}}{162}+\frac{19\zeta_{3}}{2}-\frac{\pi^{4}}{18}+\biggl{(}\frac{2518}{81}-\frac{22\pi^{2}}{27}\biggr{)}l_{\mu}-\frac{121}{27}l_{\mu}^{2}\biggr{)}$
$\displaystyle\mbox{}+C_{A}n_{l}T_{F}\biggl{(}\frac{19453}{486}-\frac{104\pi^{2}}{81}-2\zeta_{3}+\biggl{(}-\frac{1654}{81}+\frac{8\pi^{2}}{27}\biggr{)}l_{\mu}+\frac{88}{27}l_{\mu}^{2}\biggr{)}+C_{F}n_{l}T_{F}\biggl{(}\frac{11}{4}-\frac{4\zeta_{3}}{3}-\frac{2}{3}l_{\mu}\biggr{)}$
$\displaystyle\mbox{}+n_{l}^{2}T_{F}^{2}\biggl{(}-\frac{1292}{243}+\frac{8\pi^{2}}{81}+\frac{256}{81}l_{\mu}-\frac{16}{27}l_{\mu}^{2}\biggr{)}\Biggr{]}+\frac{\mu^{2}}{\left({m^{\text{OS}}}\right)^{2}}\Biggl{[}C_{A}^{2}\biggl{(}-\frac{96295}{5184}+\frac{445\pi^{2}}{432}+\frac{57\zeta_{3}}{16}-\frac{\pi^{4}}{48}$
$\displaystyle\mbox{}+\biggl{(}\frac{2155}{216}-\frac{11\pi^{2}}{36}\biggr{)}l_{\mu}-\frac{121}{72}l_{\mu}^{2}\biggr{)}+C_{A}n_{l}T_{F}\biggl{(}\frac{13699}{1296}-\frac{23\pi^{2}}{54}-\frac{3\zeta_{3}}{4}+\biggl{(}-\frac{695}{108}+\frac{\pi^{2}}{9}\biggr{)}l_{\mu}+\frac{11}{9}l_{\mu}^{2}\biggr{)}$
$\displaystyle\mbox{}+C_{F}n_{l}T_{F}\biggl{(}\frac{29}{32}-\frac{\zeta_{3}}{2}-\frac{1}{4}l_{\mu}\biggr{)}+n_{l}^{2}T_{F}^{2}\biggl{(}-\frac{209}{162}+\frac{\pi^{2}}{27}+\frac{26}{27}l_{\mu}-\frac{2}{9}l_{\mu}^{2}\biggr{)}\Biggr{]}\Biggr{\\}}~{},$
(8)

with $l_{\mu}=\ln\frac{2\mu}{\mu_{s}}$ ($\mu$ denotes the Wilsonian cutoff and
$\mu_{s}$ the renormalization scale of the strong coupling constant) and the
$SU(N_{c})$ color factors are given by $C_{F}=(N_{c}^{2}-1)/2N_{c}$,
$C_{A}=N_{c}$ and $T_{F}=1/2$. Note that this relation takes into account
finite charm quark mass effects. These effects are given by decoupling effects
only, which we showed by explicit calculation. The conversion between the
kinetic mass and other mass schemes has been included in the public programs
RunDec [28] and REvolver [29].

##  3 The Semileptonic Decay Width to $\mathcal{O}(\alpha_{s}^{3})$

For the computation of the semileptonic decay width, we need to calculate the
process

$\displaystyle b(q)\to X_{c}(p_{x})\ell(p_{\ell})\overline{\nu}(p_{\nu}),$ (9)

where $X_{c}$ is an inclusive state containing at least one charm quark and
potentially other light quarks and gluons. We can again use the optical
theorem and consider the imaginary parts of 5-loop forward scattering diagrams
(see Figure 2).

Figure 2: Sample Feynman diagrams for calculation of the semileptonic decay
rate at $\mathcal{O}(\alpha_{s}^{3})$. Straight, curly and dashed lines
represent quarks, gluons and leptons, respectively. The weak interaction
mediated by the $W$ boson is shown as a blob. Taken from Ref. [15].

Since a calculation with complete analytical dependence on the charm and
bottom mass seems out of reach, we consider the diagrams in an asymptotic
expansion around

$\displaystyle\delta$ $\displaystyle=1-\frac{m_{c}}{m_{b}}~{}.$ (10)

Although the expansion parameter is large for physical values of $m_{c}$ and
$m_{b}$ ($\delta\sim 0.7$), it has been shown in Ref.[30] at
$\mathcal{O}(\alpha_{s}^{2})$ that this expansion converges well at the
physical point and can even be extended down to $m_{c}\to 0$ ($\delta\to 1$)
with reasonable precision. Furthermore, it turns out that in this limit the
calculation simplifies:

  * •

For the asymptotic expansion in the limit $\delta\to 0$ we can use expansion
by regions. Here, the loop momenta can be either hard $k_{i}\sim m_{b}$ or
ultrasoft $k_{i}\sim\delta m_{b}$ again.

  * •

The leptonic momenta have to be ultrasoft in order to generate an imaginary
part. This reduces the number of regions to be considered.

  * •

In the $\delta$-expansion one can completely factorize the leptonic system and
integrate it out without IBP reduction. We are therefore left with 3-loop
integrals, although we started from 5-loop diagrams.

For the remaining 3-loop diagrams the scaling of the loop momenta can again
either be hard or ultrasoft and the calculation can be performed in close
analogy to the one of the kinetic mass relation discussed before. However,
since we are not only interested in the leading term of the expansion in
$\delta$ but aim for 8 terms in the expansion, we encounter huge intermediate
expressions of $\mathcal{O}(100\,\text{GB})$ for individual diagrams and
$\mathcal{O}(10^{7})$ scalar integrals with positive and negative indices up
to 12 which needed to be reduced to master integrals. 111We thank A. Smirnov
for providing a private version of FIRE which was essential for the reduction.
Furthermore, the wave function and mass renormalization constants at
$\mathcal{O}(\alpha_{s}^{3})$ allowing for two massive quarks, where only a
few term in the expansion $m_{c}\to 0$ had been known analytically before (see
Ref. [31]), needed to be extended in order to renormalize the present
calculation (see Ref. [32]).

Parametrizing the total decay rate as

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\Gamma_{0}\left(X_{0}+C_{F}\sum\limits_{i=1}^{\infty}\left(\frac{\alpha_{s}}{\pi}\right)^{i}X_{i}\right)$
(11)

we obtain the following contributions at $\mathcal{O}(\alpha_{s}^{3})$

$\displaystyle X_{3}$
$\displaystyle=\delta^{5}\biggl{(}\frac{266929}{810}-\frac{5248a_{4}}{27}+\frac{2186\pi^{2}\zeta_{3}}{45}-\frac{4094\zeta_{3}}{45}-\frac{1544\zeta_{5}}{9}-\frac{656l_{2}^{4}}{81}+\frac{1336}{405}\pi^{2}l_{2}^{2}+\frac{44888\pi^{2}l_{2}}{135}$
$\displaystyle\mbox{}-\frac{9944\pi^{4}}{2025}-\frac{608201\pi^{2}}{2430}\biggr{)}+\mathcal{O}(\delta^{6}).$
(12)

We used the notations $l_{2}=\ln(2)$, $a_{4}=\text{Li}_{4}(1/2)$ and
$\zeta_{i}$ is Riemanns zeta function. In the result above the color factors
are specified to QCD and the renormalization scale $\mu_{s}=m_{b}$ has been
chosen. The full result with general color factors and expanded up to
$\mathcal{O}(\delta^{12})$ can be found in the ancillary file to Ref. [15].
Recently the results of a subset of color factors has been confirmed up to
$\mathcal{O}(\delta^{9})$ in Ref. [33].

##  4 Phenomenological Results

Using the values $\alpha_{s}^{(5)}(M_{Z})=0.1179$ [34],
$\overline{m}_{c}(3\,\text{GeV})=993~{}\text{MeV}$ [35] and
$\overline{m}_{b}(\overline{m}_{b})=4163~{}\text{MeV}$ [36], we obtain

$\displaystyle m_{b}^{\rm kin}(\mu=1\,\text{GeV})$
$\displaystyle=(4163+259+78+26\pm 13)\,\text{MeV}=(4526\pm
13)\,\text{MeV}~{}.$ (13)

We estimate the error as half the $\mathcal{O}(\alpha_{s}^{3})$ correction,
which is also consistent with the residual scale uncertainty and known
contributions in the large $\beta_{0}$ approximation at 4-loop. The same
approach at $\mathcal{O}(\alpha_{s}^{2})$ leads to an uncertainty of
$39\,\text{MeV}$, the three-loop results therefore reduce the perturbative
uncertainty by about a factor of two.

For the semileptonic decay rate in the on-shell scheme with $m_{b}^{\rm
OS}=m_{b}=4.7\,\text{GeV}$ and $m_{c}^{\rm OS}=m_{c}=1.3\,\text{GeV}$ we
obtain

$\displaystyle\Gamma(m_{b},m_{c})$
$\displaystyle=\Gamma_{0}X_{0}\left[1-1.72\frac{\alpha_{s}^{(5)}}{\pi}-13.09\left(\frac{\alpha_{s}^{(5)}}{\pi}\right)^{2}-162.82\left(\frac{\alpha_{s}^{(5)}}{\pi}\right)^{3}\right]$
(14)

One observes the expected bad convergence of the perturbative series. Using
the kinetic scheme for the bottom quark and the $\overline{\text{MS}}$ scheme
for the charm quark mass we obtain

$\displaystyle\Gamma(m_{b}^{\rm kin},\overline{m}_{c}(3\,\text{GeV}))$
$\displaystyle=\Gamma_{0}X_{0}\left[1-1.67\frac{\alpha_{s}^{(4)}}{\pi}-7.25\left(\frac{\alpha_{s}^{(4)}}{\pi}\right)^{2}-28.6\left(\frac{\alpha_{s}^{(4)}}{\pi}\right)^{3}\right]$
(15)

Similar improvements in der perturbative behavior are also observed using
other threshold mass schemes for the bottom quark mass. For a more detailed
discussion see Ref. [15].

Both results have already been used to update the inclusive determination of
$|V_{cb}|$ [5]. The inclusion of the presented $\mathcal{O}(\alpha_{s}^{3})$
corrections resulted in a small shift of the central value but reduced the
uncertainty due to the semileptonic width $\Gamma$ by a factor of two.

## Acknowledgments

This research was supported by the Deutsche Forschungsgemeinschaft (DFG,
German ResearchFoundation) under grant 396021762 — TRR 257 “Particle Physics
Phenomenology after theHiggs Discovery”.

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# GPT Deciphering Fedspeak: Quantifying Dissent Among Hawks and Doves

Denis Peskoff
Office of Population Research
Princeton University
<EMAIL_ADDRESS>
Adam Visokay
Sociology
University of Washington
<EMAIL_ADDRESS>
Sander Schulhoff
Computer Science
University of Maryland
<EMAIL_ADDRESS>
Benjamin Wachspress
Computer Science
Princeton University
<EMAIL_ADDRESS>
Alan Blinder
Economics
Princeton University
<EMAIL_ADDRESS>Brandon M. Stewart
Sociology and OPR
Princeton University
<EMAIL_ADDRESS>
###### Abstract

Markets and policymakers around the world hang on the consequential monetary
policy decisions made by the Federal Open Market Committee (FOMC). Publicly
available textual documentation of their meetings provides insight into
members’ attitudes about the economy. We use GPT-4 to quantify dissent among
members on the topic of inflation. We find that transcripts and minutes
reflect the diversity of member views about the macroeconomic outlook in a way
that is lost or omitted from the public statements. In fact, diverging
opinions that shed light upon the committee’s “true” attitudes are almost
entirely omitted from the final statements. Hence, we argue that forecasting
FOMC sentiment based solely on statements will not sufficiently reflect
dissent among the hawks and doves.

##  1 The Road to FOMC Transparency

The Federal Open Market Committee (FOMC) is responsible for controlling
inflation in the United States, using instruments which dramatically affect
the housing and financial markets, among others. For most of the 20th century,
conventional wisdom held that monetary policy is most effective when decision-
making was shrouded in secrecy; the tight-lipped Alan Greenspan, a past
chairman of the Fed, quipped about “learning to mumble with great
incoherence.” But times change.

Blinder et al. (2008) show how the emergence of greater transparency and
strategic communication became an important feature of 21st century central
banking. Fed communication is now an integral component of monetary policy,
and “Fed watchers” dote on every word. The FOMC first started releasing public
statements following their meetings in February 1994. This meager
documentation grew and now consists of three types for each official meeting:
carefully produced and highly stylized one page statements are released
immediately after each FOMC meeting, followed about three weeks later by
lengthier minutes, and finally five years later by full, verbatim transcripts.
Subsequently this triplet is referred to as documents. We find minutes closely
reflect the content of transcripts, so to avoid redundancy, we focus our
analysis on transcripts and statements.

Figure 1: The transcripts (left) contains opinions and disagreements but
statements (right) are concise. We analyze both datasets using GPT-4
prompting.

Increased FOMC communication has prompted social science research spanning the
disciplines of economics, sociology, finance and political science (Section
2.1). Financial market participants are also keenly interested. Billions, if
not trillions of dollars are traded on the Fed’s words. The
interpretations—right or wrong—of what the FOMC “really means” move markets
and affect the economy. However, relying upon documents as data in the social
sciences is a challenge due to the lack of structure and the cost of
annotation (Grimmer and Stewart, 2013; Gentzkow et al., 2019; Ash and Hansen,
2023).

Figure 2: Text taken from an FOMC meeting on December 11, 2001. A dovish
statement does not reflect the hawkish sentiment of Speaker 1. GPT-4 can
quantify dissent lost from transcripts to statements.

Hansen and Kazinnik (2023) show that Generative Pre-training Transformer (GPT)
models outperform a suite of commonly used NLP methods on text quantification.
Motivated by these results, we set out to quantify the language of the FOMC
using GPT-4 OpenAI (2023) by preparing a combined data set of FOMC documents
from 1994-2016 (Section 2).111https://github.com/DenisPeskoff/FedNLP We
conclude that transcripts contain more dissent than statements (Section 3).

##  2 FOMC Data: Transcript to Statement

The FOMC normally meets eight times per year in order to assess current
economic conditions, ultimately deciding upon the path for monetary policy. We
aggregate and release the official publicly available text documenting these
deliberations by the Fed as an aligned corpora of documents from 1994 to
2016.222https://www.federalreserve.gov/monetarypolicy/fomc_historical.htm
These text documents are similar in content, but transcripts and statements
are dramatically different in style and detail (Figure 1).333Transcripts only
became being released in 1994 and due to the 5 year lag the latest transcripts
are not yet available. For our purposes, the statements required no pre-
processing. For the transcripts, we use regular expressions to partition and
then re-aggregate the text by each unique speaker. See Appendix D for an
example of each document type.

###  2.1 Lessons from Social Science

Past work has used at most one form of FOMC meeting documentation, but rarely
multiple in conjunction. For example, in the finance literature, Mazis and
Tsekrekos (2017) apply Latent Semantic Analysis to FOMC statements to identify
the main "themes" used by the committee and how well they explain variation in
treasury yields. Gu et al. (2022) use minutes to investigate how the tonality
of committee deliberations impacts subsequent stock market valuations.
Political scientists use transcripts to estimate committee members’
preferences on inflation and unemployment Baerg and Lowe (2020). Economists
have assessed the role of communication in achieving monetary policy
objectives by looking at similar documents (Romer and Romer, 2004; Handlan,
2020). Hüpper and Kempa (2023) investigate the extent to which shifting
inflation focus is reflected in full transcripts. Edison and Carcel (2021)
apply Latent Dirichlet Allocation (LDA) to transcripts to detect the evolution
of prominent topics. Hansen et al. (2017) use LDA to quantify transcripts and
identify how transparency affects the committee’s deliberations.

###  2.2 Hawks and Doves

We take the transcripts to best represent FOMC members’ underlying attitudes
and think of the statements as stylized representations of what they wish to
communicate publicly. To identify disagreement, we need to go beyond the
statements and look closely at the language employed by members in their
remarks throughout meeting transcripts. Dissenting votes are rare because of a
strong historical norm: members dissent only if they feel very strongly that
the committee’s decision is wrong. Modest disagreements do not merit
dissent.444Notably, this is not true of the monetary policy committees of
other nations; on some of them, dissent is common—even expected (Tillmann,
2021). That said, members of the committee do frequently voice detectable
disagreements with one another at meetings. Disagreements frequently concern
the state of the economy, the outlook for inflation, and many other things,
including where the range for the federal funds rate should be set that day.
Such debate and deliberation among members is a routine and productive element
of the meetings. These disagreements are more clearly expressed in the
transcripts. Daniel Tarullo’s comments in a 2016 transcript illustrate the
point:

> “it is institutionally important for us to project an ability to agree, even
> if only at a fairly high level, and that is why I abstained rather than
> dissented over each of the past several years […] I have gone out of my way
> in the past four years not to highlight publicly my points of difference
> with the statement.”

Figure 3: Our analysis of statements finds that averaging at the sentence-
level (gray) loses information since the average sentence is Neutral.
Ingesting the overall statement (blue) better mirrors the manual gold label
(gold).

###  2.3 Manual Analysis to Create a Gold Label

Figure 4: Our analysis of transcripts with GPT-4 at the speaker level shows
that meetings consistently have dissenting opinions. Similarly to the previous
figure, 2001 to 2004 is dovish (blue), while 2004 to 2006 is hawkish (orange).

Dissent amongst speakers is normally concentrated on the discussion of the
economic and financial situation of the U.S, specifically inflation targeting.
For example, in the January 2016 meeting transcript, the committee discusses
their 2 percent inflation projection in the context of factors such as oil
prices, the job market, and the Chinese economy. Many of the members argue
that the inflation projection of 2 percent will not be accurate, while others
who do support the 2 percent projection qualify their support with varying
degrees of uncertainty. In this meeting, President Mester concludes,

> “My reasonable confidence that inflation will gradually return to our
> objective over time recognizes there is and has always been large
> uncertainty regarding inflation forecasts.”

While Mr. Tarullo argues in opposition,

> “…I didn’t have reasonable confidence that inflation would rise to 2
> percent. Nothing since then has increased my confidence. To the contrary, a
> few more doubts have crept in.”

If the meeting statement was an accurate representation of what transpired at
the meeting, it would follow that the uncertainty of the committee regarding
their inflation forecast would be communicated. Instead, the diverging
individual opinions are omitted in the final statement, where:

> “The Committee currently expects that, with gradual adjustments in the
> stance of monetary policy, economic activity will expand at a moderate pace
> and labor market indicators will continue to strengthen. Inflation is
> expected to remain low in the near term, in part because of the further
> declines in energy prices, but to rise to 2 percent over the medium term as
> the transitory effects of declines in energy and import prices dissipate and
> the labor market strengthens”

We apply this level of granular analysis to create a gold label for
statements, classifying each one according to Hawk and Dove definitions
proposed by Hansen and Kazinnik (2023): Dovish (-1.0), Mostly Dovish (-0.5),
Neutral (0), Mostly Hawkish (.5), Hawkish (1.0). A trained undergraduate does
a first pass and escalates any borderline cases to a FOMC expert for
adjudication. We manually review the January 2016 transcript and pair it with
a simple computational analysis, which finds that most dissent at that time
surrounded topics relating to inflation (Appendix C).

###  2.4 GPT-4 “Reads” Terse Documents

GPT-4, and Large Language Models more broadly, are a suitable tool for rapid
linguistic processing at scale. We produce three different measurements of
hawk/dove sentiment using statements and two using transcripts. Hansen and
Kazinnik (2023) use GPT-3 to quantify 500 sentences selected uniformly at
random from FOMC statements between 2010 and 2020. We extend this by using
GPT-4 to quantify all 3728 sentences in statements from 1994 to 2016 (Appendix
B). A limitation of this approach is that the holistic sentiment of the
meeting is not captured because each sentence is scored independently—without
context.

The first statement measurement we propose is to simply take an unweighted
mean of all individual sentence scores for each meeting. Because most
sentences have nothing to do with inflation (62% of sentences scored as
Neutral), we hypothesize that this method has further limitations. We resolve
this by ingesting each entire statement into our GPT-4 prompt. Both of these
measurements, along with manual gold labels, can be seen in Figure 3.

For the final statement measurement, we construct a logit-scaled score (Lowe
et al., 2011),

$\theta^{(L)}=\log\left(\frac{Hawk+0.5}{Dove+0.5}\right)$

where Hawk and Dove are the sums of the hawkish and dovish scores,
respectively. In this approach $\theta^{(L)}$ ignores sentences scored as
Neutral, placing more emphasis on the relative rather than absolute
differences between hawkish and dovish sentiment. Furthermore, since
$\theta^{(L)}$ has no predefined end points, this allows us to generate
positions at any level of extremity, which more appropriately reflects the
outlier meetings.

When measuring hawk/dove sentiment using transcripts, the vast amount of text
adds an additional challenge. Rather than evaluating a 100+ page transcript at
the sentence level, we instead evaluate transcripts at the speaker level. That
is to say, we use GPT-4 to quantify hawk/dove sentiment for each distinct
speaker within each transcript, and then aggregate those by meeting date. In
contrast to the 3728 individual sentence observations across all statements,
we evaluate 5691 speaker observations across all transcripts. We then follow
the same steps outlined above to calculate unweighted average and logit-scaled
scores.

The logit-scaled measurements for transcripts and statements track one another
quite closely over most of the years, demonstrating that GPT-4 is effective at
identifying similar content across document types of dramatically different
length and style. It is worth noting, however, that the logit-scaled
transcript scores have larger extremes than statements, especially the upper
bound: (-3.66, 3.37) and (-3.40, 2.40), respectively. This underscores how the
FOMC devotes considerable attention to curating their communication strategy
to convey confidence and unity despite dissent among the hawks and doves. The
logit-scaled scores begin to diverge in 2012, with transcripts trending more
neutral/hawkish while statements remain mostly dovish. Hence, we propose a
direct comparison.

##  3 LLMs Quantify Economic Text

###  3.1 Measuring Dissent

We can use the sentence-level statement and speaker-level transcript scores
from GPT-4 to compute a measure of dissent for each meeting using the
following algorithm:

  1. 1.

From the list of scores for each meeting, count the number of hawkish/mostly
hawkish and the number of dovish/mostly dovish scores.

  2. 2.

If there is at least one hawkish score and at least one dovish score within
the same meeting, assign Dissent = 1. Else, Dissent = 0.

We find that 47% of statements and 82% of transcripts contain dissent. We also
compute the conditional probability of a transcript containing dissent given
the associated statement binary, $P(T=1|S=1)$ and $P(T=1|S=0)$. We find that
when a statement contains dissent, $P(T=1|S=1)$, the transcript agrees more
than 97% of the time. However, for statements scored as having no dissent,
$P(T=1|S=0)$ we find that more than 69% of associated transcripts are scored
as containing dissent. This means that for the 53% of statements that don’t
show signs of dissenting opinions, there is likely dissent in the transcript
as evidenced by the speaker-level hawk/dove scores.

###  3.2 Conclusion and Next Steps

Our method of ingesting the entire statement for an aggregate prediction
better captures the extremes, which more closely mirror the gold label human
annotation and suggests that Large Language Models can avoid the noise in this
nuanced context. The $F_{1}$ score for this comparison is 0.57. While this is
rather low as a measure of model “fit”, it is important to note that the
results rarely flip sentiment (from hawkish to dovish, or mostly hawkish to
mostly dovish), rather, it just seems to mostly disagree on adjacent
categories. See Figure 3 for a visual comparison of the sentence-level, entire
text, and manual scores. Of note, the inconsistent provision of statements and
relatively high volatility in hawk/dove sentiment before 2000 is consistent
with Meade and Stasavage (2008) and Hansen et al. (2017) who have also studied
the 1993 change in FOMC communication strategy. We demonstrate that GPT-4 can
identify the extremes in dissenting hawk and dove perspectives despite the
indications of a clear consensus in the statements. This empirical finding
supports our manual analysis.

While we focus on transcripts and statements, future work may consider an even
more fine grained analysis, incorporating minutes as well. We found the
content of the minutes to more closely resemble the transcripts than the
statements, but differences do exist and remain underexplored.

Additionally, we note that GPT-4 scores made more neutral predictions than the
gold standard manual labels. To improve upon this, we created a balanced few-
shot example using sentences from FOMC statements not included in our sample —
meetings since 2020. This marginally improved the prediction “fit” ($F_{1}$ of
57% to 58%), but we expect that this could be improved much further with
additional prompt engineering.

GPT-4 is able to quickly quantify stylized economic text. Our results from
quantifying dissent support the hypothesis that dissenting opinions on the
topic of inflation omitted from FOMC statements can be found in the associated
transcripts. As LLMs continue to improve, we expect that it will be possible
to study even more nuanced questions than the ones we answer here.

## Limitations

Substantively, strategic signaling in the FOMC is a challenging topic and this
is only an initial investigation. Dissent does not have clear ground truth
labels and thus we are reliant on human judgment and our team’s substantive
expertise on monetary policy. Finally, as with much current research, our work
relies on OpenAI’s GPT API, which poses challenges to computational
reproducibility, as it relies on the stability of an external system that we
cannot control.

## Ethics Statement

The FOMC is a high-stakes body whose activities are already subject to
substantial scrutiny. There is some ethical risk in exploring linguistic
signals of hidden information. For example, based on the substantive
literature we believe that dissent is intentionally signalled in meetings in
order to set up future discussions or lay claims to particular positions.
Nevertheless, attribution of intention (as implied by ’dissent’) always
involves some level of error that could be uncomfortable to meeting
participants who feel mischaracterized. We also emphasize that our approach to
capturing dissent would not be appropriate to use outside this specific
context without careful validation. Finally, by making the FOMC data more
easily available to the NLP community, we also assume some ethical
responsibility for the potential uses of that data (see e.g. Peng et al.,
2021). We spend under $1500 on computation and under $1000 on annotation and
believe our results to be reasonably reproducible. We feel that these concerns
are ultimately minor given that all participants are public officials who knew
their transcripts would ultimately be released.

## Acknowledgements

This material is based upon work supported by the National Science Foundation
under Grant #2127309 to the Computing Research Association for the CIFellows
2021 Project. We thank the Initiative for Data-Driven Social Science at
Princeton University for grant support and OpenAI for technical support.
Visokay is supported by the National Institute of Mental Health of the NIH
under Award Number #DP2MH122405 and by the Center for Statistics and the
Social Sciences at the University of Washington.

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##  Appendix A Appendix

Our appendix contains the GPT prompt we used, a section on our computational
analysis, and an example of the statement (Figure 6), transcript (Figure 7),
and minutes (Figure 8) for the January 29-30, 2008 meetings.

##  Appendix B GPT-4

We used GPT-4 heavily for our experiments and analysis. Recent work OpenAI
(2023); Liu et al. (2023); Guan et al. (2023) that successfully uses GPT-4 for
classification gave us confidence in its quality. Additionally, a human expert
on our team examined GPT-4 generated labels, and found that in a sample of 25,
our expert agreed with 19 labels with high confidence, and 22 labels with at
least moderate confidence.

###  B.1 Methodology

We used variants of a single prompt template for all of our tasks. It contains
the relevant labels (Hawkish, Dovish, etc.) as well as their definitions. It
includes space for some INPUT and asks which label best applies to the input.
When processing different documents, such as transcripts, we would switch out
the word statements in the prompt for the appropriate document word.

For statements, we 0-shot prompted GPT-4 to label each statement as one of the
five labels. We also ran an experiment in which we classified each sentence of
each statements as one of the labels, then averaged the sentence scores to get
to statement score. Finally, we reran this sentence classification with a
10-shot prompt. The prompt was similar to below, except with 10 examples of
sentences and their classifications at the beginning.

For minutes, we 0-shot prompted GPT-4-32K to label each statement as one of
the five labels.

For transcripts, we 0-shot prompted GPT-4-32K to examine all of each speakers
speech, and provide each speaker a single label for each transcript.

For all API calls to OpenAI, we only modified the model (either GPT-4 or
GPT-4-32K). We did not change any other settings.

⬇

prompt_template = """

<statement>

INPUT

</statement>

<labels>

Dovish: Strongly expresses a belief that the economy may be

growing too slowly and may need stimulus through mon\-

etary policy.

Mostly dovish: Overall message expresses a belief that the economy may

be growing too slowly and may need stimulus through

monetary policy.

Neutral: Expresses neither a hawkish nor dovish view and is

mostly objective.

Mostly hawkish: Overall message expresses a belief that the economy is

growing too quickly and may need to be slowed down

through monetary policy.

Hawkish: Strongly expresses a belief that the economy is growing

too quickly and may need to be slowed down through monetary policy.

</labels>

Which label best applies applies to the statement (Dovish, Mostly Dovish,
Neutral, Mostly Hawkish, Hawkish)?

"""

##  Appendix C Computational Analysis

Figure 5: Discussion of inflation (Red and Blue topics) is more contentious
than other topics and the average of all data (dotted line).

We paired our manual review of the January 26-27, 2016 transcript with a
computational analysis of dissent in the meeting. We stratified the meeting
into nine topics, each corresponding to a portion of the transcript content.
As a baseline, we counted the number of speakers in each section to see if
this metric could reflect dissent. This technique, however, seemed to reflect
the length of the conversation as opposed to the degree to which members
disagreed with one another.

Our next approach was to do a sentiment analysis of each topic to see if the
prevalence of negativity could indicate dissent. We supposed that negative
sentiment would be high if the speakers opposed the stance of either other
individuals or the committee as a whole. Using the VADER lexicon (Hutto and
Gilbert, 2014), we calculated the sentiment of each sentence within the nine
topics. Since VADER is trained on web-based social media content, which is
typically more abrupt than the formal language appearing in the FOMC
transcript, we conducted the sentiment analysis by sentence to optimize the
method’s performance.

To analyze dissent more specifically, we computed the fraction of negative
sentences in each topic. For this analysis, we set the threshold negativity
score to be 0.1. That is, sentences with a negativity score of 0.1 or higher
were classified as negative while all others were not. This number determined
by manually reviewing what sentences were captured by varying thresholds and
evaluating whether or not they conveyed dissent. When the threshold was set
too low (0.05), four out of ten randomly selected sentences conveyed dissent.
When set too high (0.15), seven out of ten randomly selected sentences
conveyed dissent, but many sentences that indicated dissent were omitted. At
the threshold of 0.1, still seven out of ten randomly selected sentences
conveyed dissent, and more sentences that conveyed dissent were captured.

##  Appendix D Document Examples

Figure 6: Released statement for the January 26-27, 2016 meeting.

See Figures 6, 7, and 8 for examples of the documents.

Figure 7: A page of the January 26-27, 2016 meeting transcript that shows
dissent. Figure 8: A page of the January 26-27, 2016 meeting minutes that
discusses inflation targeting.