EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-294 LHCb-PAPER-2016-050 December 7, 2016

Observation of the decay ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$

The LHCb collaboration†††Authors are listed at the end of this Letter.

Decays of the ${{\mathchar 28932\relax}^{-}_{b}}$ and ${{\mathchar
28938\relax}^{-}_{b}}$ baryons to the charmless final states
${p}h^{-}h^{\prime-}$, where $h^{(\prime)}$ denotes a kaon or pion, are
searched for with the LHCb detector. The analysis is based on a sample of
proton-proton collision data collected at centre-of-mass energies $\sqrt{s}=7$
and $8\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated
luminosity of $3\mbox{\,fb}^{-1}$. The decay ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ is observed with a
significance of $8.7$ standard deviations, and evidence at the level of $3.4$
standard deviations is found for the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ decay. Results are
reported, relative to the ${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$
normalisation channel, for the products of branching fractions and $b$-hadron
production fractions. The branching fractions of ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ and ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{\pi}^{-}}{{\pi}^{-}}$ relative to
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays are
also measured.

Published in Phys. Rev. Lett.

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

Decays of $b$ hadrons to final states that do not contain charm quarks provide
fertile ground for studies of $C\\!P$ violation, i.e. the breaking of symmetry
under the combined charge conjugation and parity operations. Significant
asymmetries have been observed between $B$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ partial widths in ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$
[1, 2, 3, 4] and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
[3, 4] decays. Even larger $C\\!P$-violation effects have been observed in
regions of the phase space of
${{{B}^{-}}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$,
${{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, ${{K}^{+}}{{K}^{-}}{{K}^{-}}$ and
${{K}^{+}}{{K}^{-}}{{\pi}^{-}}$ decays [5, 6, 7]. A number of theoretical
approaches [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] have been proposed to
determine whether the observed effects are consistent with being solely due to
the non-zero phase in the quark mixing matrix [19, 20] of the Standard Model,
or whether additional sources of asymmetry are contributing.

Breaking of the symmetry between matter and antimatter has not yet been
observed with a significance of more than five standard deviations ($\sigma$)
in the properties of any baryon. Recently, however, the first evidence of
$C\\!P$ violation in the $b$-baryon sector has been reported from an analysis
of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays
[21]. Other $C\\!P$-asymmetry parameters measured in ${{\mathchar
28931\relax}^{0}_{b}}$ baryon decays to ${p}{{\pi}^{-}}$, ${p}{{K}^{-}}$ [3],
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{p}{{\pi}^{-}}$ [22], ${\mathchar
28931\relax}{{K}^{+}}{{K}^{-}}$ and ${\mathchar
28931\relax}{{K}^{+}}{{\pi}^{-}}$ [23] final states are consistent with zero
within the current experimental precision; these comprise the only charmless
hadronic $b$-baryon decays that have been observed to date. It is therefore of
great interest to search for additional charmless $b$-baryon decays that may
be used in future to investigate $C\\!P$-violation effects.

In this Letter, the first search is presented for decays of ${{\mathchar
28932\relax}^{-}_{b}}$ and ${{\mathchar 28938\relax}^{-}_{b}}$ baryons, with
constituent quark contents of ${b}{s}{d}$ and ${b}{s}{s}$, to the charmless
hadronic final states ${p}h^{-}h^{\prime-}$, where $h^{(\prime)}$ is a kaon or
pion. The inclusion of charge-conjugate processes is implied throughout.
Example decay diagrams for the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ mode are shown in Fig.
1. Interference between Cabibbo-suppressed tree and loop diagrams may lead to
$C\\!P$-violation effects. The ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ and ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays proceed by tree-
level diagrams similar to that of Fig. 1 (left). Diagrams for ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ and both ${{\mathchar
28932\relax}^{-}_{b}}$ and ${{\mathchar
28938\relax}^{-}_{b}}\rightarrow{p}{{\pi}^{-}}{{\pi}^{-}}$ require additional
weak interaction vertices. The rates of these decays are therefore expected to
be further suppressed.

Figure 1: (Left) tree and (right) loop diagrams for the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay channel.

The analysis is based on a sample of proton-proton collision data, recorded by
the LHCb experiment at centre-of-mass energies $\sqrt{s}=7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$, corresponding to $3\mbox{\,fb}^{-1}$ of
integrated luminosity. Since the fragmentation fractions, $f_{{\mathchar
28932\relax}^{-}_{b}}$ and $f_{{\mathchar 28938\relax}^{-}_{b}}$, which
quantify the probabilities for a $b$ quark to hadronise into these particular
states, have not been determined, it is not possible to measure absolute
branching fractions. Instead, the product of each branching fraction and the
relevant fragmentation fraction is determined relative to the corresponding
values for the topologically similar normalisation channel
${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ (the ${{B}^{-}}$
fragmentation fraction is denoted $f_{u}$). Once one significant signal yield
is observed, it becomes possible to determine ratios of branching fractions
for decays of the same baryon to different final states, thus cancelling the
dependence on the fragmentation fraction.

The LHCb detector [24, 25] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The pseudorapidity is defined as
$-\ln\left[\tan(\theta/2)\right]$ where $\theta$ is the polar angle relative
to the beam axis. The detector elements that are particularly relevant to this
analysis are: a silicon-strip vertex detector surrounding the $pp$ interaction
region that allows $b$ hadrons to be identified from their characteristically
long flight distance; a tracking system that provides a measurement of the
momentum ($p$) of charged particles; two ring-imaging Cherenkov detectors that
enable different species of charged hadrons to be distinguished; and
calorimeter and muon systems that provide information used for online event
selection. Simulated data samples, produced with software described in Refs.
[26, *Sjostrand:2007gs, 28, 29, 30, 31, *Agostinelli:2002hh, 33], are used to
evaluate the response of the detector to signal decays and to characterise the
properties of certain types of background. These samples are generated
separately for centre-of-mass energies of $7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$, simulating the corresponding data-taking
conditions, and combined in appropriate quantities.

Online event selection is performed by a trigger [34] that consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. At
the hardware trigger stage, events are required to contain either a muon with
high transverse momentum ($p_{\mathrm{T}}$) or a particle that deposits high
transverse energy in the calorimeters. For hadrons, the transverse energy
threshold is typically $3.5\mathrm{\,Ge\kern-1.00006ptV}$. The software
trigger for this analysis requires a two- or three-track secondary vertex with
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one charged particle must have $p_{\mathrm{T}}$ above a threshold of
$1.7\,(1.6){\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
$\sqrt{s}=7\,(8)\mathrm{\,Te\kern-1.00006ptV}$ data. This particle must also
be inconsistent with originating from any PV as quantified through the
difference in the vertex-fit $\chi^{2}$ of a given PV reconstructed with and
without the considered particle ($\chi^{2}_{\text{IP}}$). A multivariate
algorithm [35] is used for the identification of secondary vertices consistent
with the decay of a $b$ hadron.

The offline selection of $b$-hadron candidates formed from three tracks is
carried out with an initial prefiltering stage, a requirement on the output of
a neural network [36], and particle identification criteria. To avoid
potential bias, the properties of candidates with invariant masses in windows
around the ${{\mathchar 28932\relax}^{-}_{b}}$ and ${{\mathchar
28938\relax}^{-}_{b}}$ masses were not inspected until after the analysis
procedures were finalised. The prefiltering includes requirements on the
quality, $p$, $p_{\mathrm{T}}$ and $\chi^{2}_{\text{IP}}$ of the tracks. Each
$b$ candidate must have a good quality vertex that is displaced from the
closest PV (i.e. that with which it forms the smallest
$\chi^{2}_{\text{IP}}$), must satisfy $p$ and $p_{\mathrm{T}}$ requirements,
and must have reconstructed invariant mass loosely consistent with those of
the $b$ hadrons. A requirement is also imposed on the angle $\theta_{\rm dir}$
between the $b$-candidate momentum vector and the line between the PV and the
$b$-candidate decay vertex. In the offline selection, trigger signals are
associated with reconstructed particles. Selection requirements can therefore
be made not only on which trigger caused the event to be recorded, but also on
whether the decision was due to the signal candidate or other particles
produced in the $pp$ collision [34]. Only candidates from events with a
hardware trigger caused by deposits of the signal in the calorimeter, or
caused by other particles in the event, are retained. It is also required that
the software trigger decision must have been caused by the signal candidate.

The inputs to the neural network for the final selection are the scalar sum of
the $p_{\mathrm{T}}$ of all final-state tracks, the values of $p_{\mathrm{T}}$
and $\chi^{2}_{\text{IP}}$ for the highest $p_{\mathrm{T}}$ final-state track,
the $b$-candidate $\cos\left(\theta_{\rm dir}\right)$, vertex $\chi^{2}$ and
$\chi^{2}_{\text{IP}}$, together with a combination of momentum information
and $\theta_{\rm dir}$ that characterises how closely the momentum vector of
the $b$ candidate points back to the PV. The $p_{\mathrm{T}}$ asymmetry
between the $b$ candidate and other tracks within a circle, centred on the $b$
candidate, with a radius $R=\sqrt{\delta\eta^{2}+\delta\phi^{2}}<1.5$ in the
space of pseudorapidity and azimuthal angle $\phi$ (in radians) around the
beam direction [37] is also used in the network. The distributions of these
variables are consistent between simulated samples of signal decays and the
${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ normalisation channel, and
between background-subtracted
${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ data and simulation. The
neural network input variables are also found to be not strongly correlated
with either the $b$-candidate mass or the position in the phase space of the
decay. The neural network is trained to distinguish signal from combinatorial
background in the ${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ channel,
using a data-driven approach in which the two components are separated
statistically using the sPlot method [38] with the $b$-candidate mass as
discriminating variable. The requirement on the neural network output is
optimised using a figure of merit [39] intended to give the best chance to
observe the signal decays. The same neural network output requirement is made
for all signal final states, and has an efficiency of about $60\,\%$.

Using information from the ring-imaging Cherenkov detectors [40], criteria
that identify uniquely the final-state tracks as either protons, pions or
kaons are imposed, ensuring that no candidate appears in more than one of the
final states considered. For pions and kaons these criteria are optimised
simultaneously with that on the neural network output, using the same figure
of merit. The desire to reject possible background from
${{{B}^{-}}}\rightarrow{{K}^{+}}h^{-}h^{\prime-}$ in the signal modes
justifies independent treatment of the proton identification requirement. In
the simultaneous optimisation, the efficiency is taken from control samples
while the expected background level is extrapolated from sidebands in the
$b$-candidate mass distribution. The combined efficiency of the particle
identification requirements is about $30\,\%$ for the ${p}{{K}^{-}}{{K}^{-}}$,
$40\,\%$ for the ${p}{{K}^{-}}{{\pi}^{-}}$ and $50\,\%$ for the
${p}{{\pi}^{-}}{{\pi}^{-}}$ final state.

In order to ensure that any signal seen is due to charmless decays, candidates
with ${p}{{K}^{-}}$ invariant mass consistent with the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}h^{-}\rightarrow{p}{{K}^{-}}h^{-}$ or ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{c}}h^{-}\rightarrow{p}{{\pi}^{-}}h^{-}$ decay chain are
vetoed. Similarly, candidates for the normalisation channel with
${{K}^{+}}{{K}^{-}}$ invariant mass consistent with the
${{{B}^{-}}}\rightarrow{{D}^{0}}{{K}^{-}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$
decay chain are removed. After all selection requirements are imposed, the
fraction of selected events that contain more than one candidate is much less
than $1\,\%$; all such candidates are retained.

The yields of the signal decays are obtained from a simultaneous unbinned
extended maximum likelihood fit to the $b$-candidate mass distributions in the
three ${p}h^{-}h^{\prime-}$ final states. This approach allows potential
cross-feed from one channel to another, due to particle misidentification, to
be constrained according to the expected rates. The yield of the normalisation
channel is determined from a separate fit to the ${{K}^{+}}{{K}^{-}}{{K}^{-}}$
mass distribution.

Each signal component is modelled with the sum of two Crystal Ball (CB)
functions [41] with shared parameters describing the core width and peak
position and with non-Gaussian tails to both sides. The tail parameters and
the relative normalisation of the CB functions are determined from simulation.
A scale factor relating the width in data to that in simulation is determined
from the fit to the normalisation channel. In the fit to the signal modes the
peak positions are fixed to the known ${\mathchar 28932\relax}^{-}_{b}$ and
${\mathchar 28938\relax}^{-}_{b}$ masses [42, 43, 44]; the only free
parameters associated with the signal components are the yields.

Cross-feed backgrounds from other decays to ${p}h^{-}h^{\prime-}$ final states
are also modelled with the sum of two CB functions, with all shape parameters
fixed according to simulation but the width scaled in the same way as signal
components. Cross-feed backgrounds from
${{{B}^{-}}}\rightarrow{{K}^{+}}h^{-}h^{\prime-}$ decays are modelled, in the
mass interval of the fit, by exponential functions with shape fixed according
to simulation. The yields of all cross-feed backgrounds are constrained
according to expectations based on the yield in the correctly reconstructed
channel and the (mis-)identification probabilities determined from control
samples.

In addition to signal and cross-feed backgrounds, components for partially
reconstructed and combinatorial backgrounds are included in each final state.
Partially reconstructed backgrounds arise due to $b$-hadron decays into final
states similar to the signal, but with additional soft particles that are not
reconstructed. Possible examples include ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow
N^{+}h^{-}h^{\prime-}\rightarrow{p}{{\pi}^{0}}h^{-}h^{\prime-}$ and
${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{*-}}h^{-}\rightarrow{p}{{K}^{-}}{{\pi}^{0}}h^{-}$.
Such decays are investigated with simulation and it is found that many of them
have similar $b$-candidate mass distributions. The shapes of these backgrounds
are therefore taken from ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow
N^{+}h^{-}h^{\prime-}\rightarrow{p}{{\pi}^{0}}h^{-}h^{\prime-}$ simulation,
with possible additional contributions considered as a source of systematic
uncertainty. The shapes are modelled with an ARGUS function [45] convolved
with a Gaussian function. The parameters of these functions are taken from
simulation, except for the threshold of the ARGUS function, which is fixed to
the known mass difference $m_{{{\mathchar
28932\relax}^{-}_{b}}}-m_{{{\pi}^{0}}}$ [46, 42]. The combinatorial background
is modelled by an exponential function with the shape parameter shared between
the three final states. Possible differences in the shape between the
different final states are considered as a source of systematic uncertainty.
The free parameters of the fit are the signal and background yields, and the
combinatorial background shape parameter. The stability of the fit is
confirmed using ensembles of pseudoexperiments with different values of signal
yields.

Figure 2:  Mass distributions for $b$-hadron candidates in the (top left)
${p}{{K}^{-}}{{K}^{-}}$, (top right) ${p}{{K}^{-}}{{\pi}^{-}}$, (bottom left)
${p}{{\pi}^{-}}{{\pi}^{-}}$ and (bottom right) ${{K}^{+}}{{K}^{-}}{{K}^{-}}$
final states. Results of the fits are shown with dark blue solid lines.
Signals for ${{\mathchar 28932\relax}^{-}_{b}}$ and ${{{B}^{-}}}$
(${{\mathchar 28938\relax}^{-}_{b}}$) decays are shown with pink (light green)
dashed lines, combinatorial backgrounds are shown with grey long-dashed lines,
cross-feed backgrounds are shown with red dot-dashed lines, and partially
reconstructed backgrounds are shown with dark blue double-dot-dashed lines.

The results of the fits are shown in Fig. 2. The significance of each of the
signals is determined from the change in likelihood when the corresponding
yield is fixed to zero, with relevant sources of systematic uncertainty taken
into account. The signals for ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ and
${p}{{K}^{-}}{{\pi}^{-}}$ decays are found to have significance of
$8.7\,\sigma$ and $3.4\,\sigma$, respectively; each of the other signal modes
has significance less than $2\,\sigma$. The relative branching fractions
multiplied by fragmentation fractions are determined as

$R_{{p}h^{-}h^{\prime-}}\equiv\frac{f_{{{\mathchar
28932\relax}^{-}_{b}}}}{f_{u}}\frac{{\mathcal{B}}({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}h^{-}h^{\prime-})}{{\mathcal{B}}({{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}})}=\frac{\mathcal{N}({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}h^{-}h^{\prime-})}{\mathcal{N}({{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}})}\frac{\epsilon({{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}})}{\epsilon({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}h^{-}h^{\prime-})}\,,$ (1)

where the yields $\mathcal{N}$ are obtained from the fits. A similar
expression is used for the ${{\mathchar 28938\relax}^{-}_{b}}$ decay modes.
The efficiencies $\epsilon$ are determined from simulation, weighted according
to the most recent ${{\mathchar 28932\relax}^{-}_{b}}$ and ${{\mathchar
28938\relax}^{-}_{b}}$ lifetime measurements [42, 43, 44], taking into account
contributions from the detector geometry, reconstruction and both online and
offline selection criteria. These are determined as a function of the position
in phase space in each of the three-body final states. The phase space for
each of the ${\mathchar 28932\relax}^{-}_{b}$ and ${\mathchar
28938\relax}^{-}_{b}$ decays to ${p}h^{-}h^{\prime-}$ is five-dimensional, but
significant variations in efficiency occur only in the variables that describe
the Dalitz plot. Simulation is used to evaluate each contribution to the
efficiency except for the effect of the particle identification criteria,
which is determined from data control samples weighted according to the
expected kinematics of the signal tracks [40, 47]. The description of
reconstruction and selection efficiencies in the simulation has been validated
with large control samples; the impact on the results of possible residual
differences between data and simulation is negligible.

For the ${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$,
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ channels, efficiency
corrections for each candidate are applied using the method of Ref. [48] to
take the variation over the phase space into account. Using this procedure,
the efficiency-corrected and background-subtracted $m({p}{{K}^{-}})_{\rm min}$
distribution shown in Fig. 3 is obtained from ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ candidates. Here
$m({p}{{K}^{-}})_{\rm min}$ indicates the smaller of the two $m({p}{{K}^{-}})$
values for each signal candidate, evaluated with the ${{\mathchar
28932\relax}^{-}_{b}}$ and the final-state particle masses fixed to their
known values [46, 42]. The distribution contains a clear peak from the
${\mathchar 28931\relax}(1520)$ resonance, a structure that is consistent with
being a combination of the ${\mathchar 28931\relax}(1670)$ and ${\mathchar
28931\relax}(1690)$ states, and possible additional contributions at higher
mass. Compared to the ${p}{{K}^{-}}$ structures seen in the amplitude analysis
of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}{{K}^{-}}$ [49], the contributions from the broad ${\mathchar
28931\relax}(1600)$ and ${\mathchar 28931\relax}(1810)$ states appear to be
smaller. A detailed amplitude analysis will be of interest when larger samples
are available.

Figure 3:  Efficiency-corrected and background-subtracted [38]
$m({p}{{K}^{-}})_{\rm min}$ distribution from ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ candidates.

For channels without significant signal yields the efficiency averaged over
phase space is used in Eq. (1). A corresponding systematic uncertainty is
assigned from the variation of the efficiency over the phase space; this is
the dominant source of systematic uncertainty for those channels. The
quantities entering Eq. (1), and the results for $R_{{p}h^{-}h^{\prime-}}$,
are reported in Table 1. When the signal significance is less than
$3\,\sigma$, upper limits are set by integrating the likelihood after
multiplying by a prior probability distribution that is uniform in the region
of positive branching fraction.

Table 1:  Fitted yields, efficiencies and relative branching fractions
multiplied by fragmentation fractions ($R_{{p}h^{-}h^{\prime-}}$). The two
uncertainties quoted on $R_{{p}h^{-}h^{\prime-}}$ are statistical and
systematic. Upper limits are quoted at 90 (95) % confidence level for modes
with signal significance less than $3\,\sigma$. Uncertainties on the
efficiencies are not given as only the relative uncertainties affect the
branching fraction measurements.

Mode | Yield ${\cal N}$ | Efficiency $\epsilon$ (%) | $R_{{p}h^{-}h^{\prime-}}\ (10^{-5})$
---|---|---|---
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ | 82.9$\;\pm\;$ | 10.4 | 0.398 |  $\phantom{\\!-}265$$\;\pm\;$ | 35 $\;\pm\;$ | 47 |
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ | 59.6$\;\pm\;$ | 16.0 | 0.293 |  $\phantom{\\!-}259$$\;\pm\;$ | 64 $\;\pm\;$ | 49 |
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{\pi}^{-}}{{\pi}^{-}}$ | 33.2$\;\pm\;$ | 17.9 | 0.573 |  $\phantom{\\!-1}74$$\;\pm\;$ | 40 $\;\pm\;$ | 36 |  $<$ 147 (166)
${{\mathchar 28938\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ |  $-2.8$$\;\pm\;$ | 2.5 | 0.375 |  $\phantom{\\!1}-9$$\;\pm\;$ | 9 $\;\pm\;$ | 6 |  $<$ 118 1(22)
${{\mathchar 28938\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ |  $-7.6$$\;\pm\;$ | 9.2 | 0.418 |  $\phantom{\\!1}{-23}$$\;\pm\;$ | 28 $\;\pm\;$ | 23 |  $<$ 151 1(62)
${{\mathchar 28938\relax}^{-}_{b}}\rightarrow{p}{{\pi}^{-}}{{\pi}^{-}}$ | 20.1$\;\pm\;$ | 13.8 | 0.536 |  $\phantom{\\!-0}48$$\;\pm\;$ | 33 $\;\pm\;$ | 28 |  $<$ 109 (124)
${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ | 50 490$\;\pm\;$ | 250 | 0.643 | — |

The sources of systematic uncertainty arise from the fit model and the
knowledge of the efficiency. The fit model is changed by varying the fixed
parameters of the model, using alternative shapes for the components, and by
including components that are omitted in the baseline fit. Intrinsic biases in
the fitted yields are investigated with simulated pseudoexperiments, and are
found to be negligible. Uncertainties in the efficiency arise due to the
limited size of the simulation samples and possible residual differences
between data and simulation in the trigger and particle identification
efficiencies [50]. Possible biases in the results due to the vetoes of charm
hadrons are also accounted for. The efficiency depends on the signal decay-
time distribution, and therefore the precision of the ${\mathchar
28932\relax}^{-}_{b}$ and ${\mathchar 28938\relax}^{-}_{b}$ lifetime
measurements [42, 43, 44] is a source of uncertainty. Similarly, the
$p_{\mathrm{T}}$ distribution assumed for signal decays in the simulation
affects the efficiency. Since the $p_{\mathrm{T}}$ spectra for ${\mathchar
28932\relax}^{-}_{b}$ and ${\mathchar 28938\relax}^{-}_{b}$ baryons produced
in LHC collisions have not been measured, the effect is estimated by weighting
simulation to the background-subtracted [38] $p_{\mathrm{T}}$ distribution for
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays
obtained from the data. The difference in the average efficiency between the
weighted and unweighted simulation is assigned as the associated systematic
uncertainty. This is the dominant source of systematic uncertainty for the
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ and
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ modes.

The yield of ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays is sufficient to
use as normalisation for the relative branching fractions of the other
${{\mathchar 28932\relax}^{-}_{b}}$ decays. The results are

$\displaystyle\frac{{\cal B}({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}})}{{\cal B}({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}})}$ $\displaystyle=$
$\displaystyle 0.98\pm 0.27\mathrm{\,(stat)}\pm 0.09\mathrm{\,(syst)}\,,$
$\displaystyle\frac{{\cal B}({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{\pi}^{-}}{{\pi}^{-}})}{{\cal
B}({{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}})}$
$\displaystyle=$ $\displaystyle 0.28\pm 0.16\mathrm{\,(stat)}\pm
0.13\mathrm{\,(syst)}\ \ \ <\ 0.56~{}(0.63)\,,$

where the upper limit is quoted at 90 (95) % confidence level. The same
sources of systematic uncertainty as discussed above are considered. Since the
effects due to the $p_{\mathrm{T}}$ distribution largely cancel, the dominant
contributions are due to the trigger efficiency, fit model and (for the
${{\mathchar 28932\relax}^{-}_{b}}\rightarrow{p}{{\pi}^{-}}{{\pi}^{-}}$ mode)
efficiency variation across the phase space.

In summary, a search for decays of ${{\mathchar 28932\relax}^{-}_{b}}$ and
${{\mathchar 28938\relax}^{-}_{b}}$ baryons to ${p}h^{-}h^{\prime-}$ final
states has been carried out with a sample of proton-proton collision data
corresponding to an integrated luminosity of $3\mbox{\,fb}^{-1}$. The first
observation of the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay, and first
evidence for the ${{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ decay, have been
obtained; there is no significant signal for the other modes. This is the
first observation of a ${{\mathchar 28932\relax}_{b}}$ decay to a charmless
final state. These modes may be used in future to search for $C\\!P$
asymmetries in the $b$-baryon sector.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

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LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, S. Ali43, G. Alkhazov31, P. Alvarez Cartelle55,
A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L. An3, L. Anderlini18, G.
Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B. Appleby56, F. Archilli43,
P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M. Artuso61, E. Aslanides6, G.
Auriemma26, M. Baalouch5, I. Babuschkin56, S. Bachmann12, J.J. Back50, A.
Badalov38, C. Baesso62, S. Baker55, V. Balagura7,c, W. Baldini17, R.J.
Barlow56, C. Barschel40, S. Barsuk7, W. Barter40, M. Baszczyk27, V.
Batozskaya29, B. Batsukh61, V. Battista41, A. Bay41, L. Beaucourt4, J.
Beddow53, F. Bedeschi24, I. Bediaga1, L.J. Bel43, V. Bellee41, N. Belloli21,i,
K. Belous37, I. Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson43, A.
Berezhnoy33, R. Bernet42, A. Bertolin23, C. Betancourt42, F. Betti15, M.-O.
Bettler40, M. van Beuzekom43, Ia. Bezshyiko42, S. Bifani47, P. Billoir8, T.
Bird56, A. Birnkraut10, A. Bitadze56, A. Bizzeti18,u, T. Blake50, F. Blanc41,
J. Blouw11,†, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N.
Bondar31,40, W. Bonivento16, I. Bordyuzhin32, A. Borgheresi21,i, S. Borghi56,
M. Borisyak35, M. Borsato39, F. Bossu7, M. Boubdir9, T.J.V. Bowcock54, E.
Bowen42, C. Bozzi17,40, S. Braun12, M. Britsch12, T. Britton61, J.
Brodzicka56, E. Buchanan48, C. Burr56, A. Bursche2, J. Buytaert40, S.
Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M. Calvo Gomez38,m, A. Camboni38,
P. Campana19, D.H. Campora Perez40, L. Capriotti56, A. Carbone15,e, G.
Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i, L. Carson52, K.
Carvalho Akiba2, G. Casse54, L. Cassina21,i, L. Castillo Garcia41, M.
Cattaneo40, G. Cavallero20, R. Cenci24,t, D. Chamont7, M. Charles8, Ph.
Charpentier40, G. Chatzikonstantinidis47, M. Chefdeville4, S. Chen56, S.-F.
Cheung57, V. Chobanova39, M. Chrzaszcz42,27, X. Cid Vidal39, G. Ciezarek43,
P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J. Closier40, V. Coco59, J.
Cogan6, E. Cogneras5, V. Cogoni16,40,f, L. Cojocariu30, G. Collazuol23,o, P.
Collins40, A. Comerma-Montells12, A. Contu40, A. Cook48, G. Coombs40, S.
Coquereau38, G. Corti40, M. Corvo17,g, C.M. Costa Sobral50, B. Couturier40,
G.A. Cowan52, D.C. Craik52, A. Crocombe50, M. Cruz Torres62, S. Cunliffe55, R.
Currie55, C. D’Ambrosio40, F. Da Cunha Marinho2, E. Dall’Occo43, J. Dalseno48,
P.N.Y. David43, A. Davis3, K. De Bruyn6, S. De Capua56, M. De Cian12, J.M. De
Miranda1, L. De Paula2, M. De Serio14,d, P. De Simone19, C.-T. Dean53, D.
Decamp4, M. Deckenhoff10, L. Del Buono8, M. Demmer10, A. Dendek28, D.
Derkach35, O. Deschamps5, F. Dettori40, B. Dey22, A. Di Canto40, H.
Dijkstra40, F. Dordei40, M. Dorigo41, A. Dosil Suárez39, A. Dovbnya45, K.
Dreimanis54, L. Dufour43, G. Dujany56, K. Dungs40, P. Durante40, R.
Dzhelyadin37, A. Dziurda40, A. Dzyuba31, N. Déléage4, S. Easo51, M. Ebert52,
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Luppi17,g, O. Lupton40, A. Lusiani24, X. Lyu63, F. Machefert7, F. Maciuc30, O.
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M. Martin6, M. Martinelli41, D. Martinez Santos39, F. Martinez Vidal69, D.
Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R. Matev40, A. Mathad50,
Z. Mathe40, C. Matteuzzi21, A. Mauri42, E. Maurice7,b, B. Maurin41, A.
Mazurov47, M. McCann55,40, A. McNab56, R. McNulty13, B. Meadows59, F. Meier10,
M. Meissner12, D. Melnychuk29, M. Merk43, A. Merli22,q, E. Michielin23, D.A.
Milanes66, M.-N. Minard4, D.S. Mitzel12, A. Mogini8, J. Molina Rodriguez1,
I.A. Monroy66, S. Monteil5, M. Morandin23, P. Morawski28, A. Mordà6, M.J.
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C. Nguyen-Mau41,n, S. Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A.
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Goicochea2, A. Otto40, P. Owen42, A. Oyanguren69, P.R. Pais41, A. Palano14,d,
F. Palombo22,q, M. Palutan19, A. Papanestis51, M. Pappagallo14,d, L.L.
Pappalardo17,g, W. Parker60, C. Parkes56, G. Passaleva18, A. Pastore14,d, G.D.
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V. Placinta30, S. Playfer52, M. Plo Casasus39, T. Poikela40, F. Polci8, A.
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A. Rollings57, V. Romanovskiy37, A. Romero Vidal39, J.W. Ronayne13, M.
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Schiller53, H. Schindler40, M. Schlupp10, M. Schmelling11, T. Schmelzer10, B.
Schmidt40, O. Schneider41, A. Schopper40, K. Schubert10, M. Schubiger41, M.-H.
Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A. Semennikov32, A.
Sergi47, N. Serra42, J. Serrano6, L. Sestini23, P. Seyfert21, M. Shapkin37, I.
Shapoval45, Y. Shcheglov31, T. Shears54, L. Shekhtman36,w, V. Shevchenko68,
B.G. Siddi17,40, R. Silva Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S.
Simone14,d, M. Sirendi49, N. Skidmore48, T. Skwarnicki61, E. Smith55, I.T.
Smith52, J. Smith49, M. Smith55, H. Snoek43, l. Soares Lavra1, M.D.
Sokoloff59, F.J.P. Soler53, B. Souza De Paula2, B. Spaan10, P. Spradlin53, S.
Sridharan40, F. Stagni40, M. Stahl12, S. Stahl40, P. Stefko41, S. Stefkova55,
O. Steinkamp42, S. Stemmle12, O. Stenyakin37, H. Stevens10, S. Stevenson57, S.
Stoica30, S. Stone61, B. Storaci42, S. Stracka24,p, M. Straticiuc30, U.
Straumann42, L. Sun64, W. Sutcliffe55, K. Swientek28, V. Syropoulos44, M.
Szczekowski29, T. Szumlak28, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G.
Tellarini17,g, F. Teubert40, E. Thomas40, J. van Tilburg43, M.J. Tilley55, V.
Tisserand4, M. Tobin41, S. Tolk49, L. Tomassetti17,g, D. Tonelli40, S. Topp-
Joergensen57, F. Toriello61, E. Tournefier4, S. Tourneur41, K. Trabelsi41, M.
Traill53, M.T. Tran41, M. Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P.
Tsopelas43, A. Tully49, N. Tuning43, A. Ukleja29, A. Ustyuzhanin35, U. Uwer12,
C. Vacca16,f, V. Vagnoni15,40, A. Valassi40, S. Valat40, G. Valenti15, R.
Vazquez Gomez19, P. Vazquez Regueiro39, S. Vecchi17, M. van Veghel43, J.J.
Velthuis48, M. Veltri18,r, G. Veneziano57, A. Venkateswaran61, M. Vernet5, M.
Vesterinen12, J.V. Viana Barbosa40, B. Viaud7, D. Vieira63, M. Vieites Diaz39,
H. Viemann67, X. Vilasis-Cardona38,m, M. Vitti49, V. Volkov33, A. Vollhardt42,
B. Voneki40, A. Vorobyev31, V. Vorobyev36,w, C. Voß9, J.A. de Vries43, C.
Vázquez Sierra39, R. Waldi67, C. Wallace50, R. Wallace13, J. Walsh24, J.
Wang61, D.R. Ward49, H.M. Wark54, N.K. Watson47, D. Websdale55, A. Weiden42,
M. Whitehead40, J. Wicht50, G. Wilkinson57,40, M. Wilkinson61, M. Williams40,
M.P. Williams47, M. Williams58, T. Williams47, F.F. Wilson51, J. Wimberley60,
J. Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K.
Wraight53, K. Wyllie40, Y. Xie65, Z. Xing61, Z. Xu41, Z. Yang3, Y. Yao61, H.
Yin65, J. Yu65, X. Yuan36,w, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, Y. Zhang63, A. Zhelezov12, Y. Zheng63,
X. Zhu3, V. Zhukov33, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 38
70Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased


# Legendre Transformation of the Luttinger–Ward Functional
from the Bare Interaction Vertex to the Renormalized One

Takafumi Kita Department of Physics, Hokkaido University, Sapporo 060-0810,
Japan

###### Abstract

On the basis of the Luttinger–Ward functional for interacting many-body
systems given in terms of full Green’s function $G$ and the bare interaction
vertex $\Gamma^{(0)}$, we develop a novel Legendre transformation to express
the grand thermodynamic potential $\Omega$ as a functional of $G$ and the
renormalized interaction vertex $\Gamma$ so that (i) $G$ and $\Gamma$ obey the
stationarity conditions $\delta\Omega/\delta G=0$ and
$\delta\Omega/\delta\Gamma=0$ and (ii) $\Gamma$ reduces to $\Gamma^{(0)}$ in
the weak-coupling limit. The formalism enables us to perform microscopic
studies of thermodynamic, single-particle, and two-particle properties in a
unified self-consistent conserving framework.

##  I Introduction

The purpose of the present study is to express the grand thermodynamic
potential $\Omega$ as a functional of full Green’s function $G$ and the
renormalized interaction vertex $\Gamma$ to incorporate single-particle
excitations and collective fluctuations microscopically into thermodynamics.
So far, the vertex renormalization has been performed mostly by procedures
with no explicit connection to $\Omega$, such as the $T$-matrix or shielded-
potential approximation BK61 ; Baym62 ; BS89 , but we then encounter a
difficulty in taking the resulting vertex into $\Omega$. Expressing $\Omega$
in terms of $\Gamma$ is prerequisite to quantitative calculations of
thermodynamics with two-particle or collective excitations, especially around
emerging ordered phases where those fluctuations become dominant.

The topic was pioneered by De Dominicis and Martin DDM64-1 ; DDM64-2  who
introduced the Legendre transformation of $\Omega$ as a key factor in
performing the renormalizations at various levels. Using this transformation,
they presented a diagrammatic vertex renormalization procedure DDM64-2  that
relies on the topological structure of skeleton-type Feynman diagrams for
$\Omega$. However, no explicit proof has been given on the validity of
assigning the same sign and weight as those of the bare perturbation expansion
to each of the renormalized skeleton-type diagrams. Moreover, the resulting
$\Gamma$ has an unfavorable feature that it reduces to $-\Gamma^{(0)}$ instead
of the bare interaction vertex $\Gamma^{(0)}$ itself in the weak-coupling
limit; see Eq. (52) of Ref. DDM64-2,  on this point with
$\Gamma^{(0)}\rightarrow v_{2}$ and $\Gamma\rightarrow C_{2}$. Among other
approaches to incorporating collective excitations into thermodynamics is that
based on the Hubbard–Stratonovich transformation Stratotovich57 ; Hubbard59 .
However, it generally focuses on a single kind of fluctuation such as the
density or spin upon performing the transformation so that the other
fluctuations may be neglected completely in any approximate treatment
afterwards.

With these as a background, in this study we develop an alternative vertex
renormalization procedure based on the Luttinger–Ward functional LW60 , where
the renormalization at the one-particle level, called “mass renormalization”,
has been completed rigorously through expressing $\Omega$ as a functional of
$G$. Since it is written in terms of $\Gamma^{(0)}$, however, the functional
still has room for a further renormalization. A key quantity in it is the
functional $\Phi[G,\Gamma^{(0)}]$ composed of skeleton diagrams for $\Omega$,
which forms a basis for systematic approximations called
$\varPhi$-derivableBaym62  or conservingBS89  approximation that enable us to
incorporate correlation effects progressively beyond the Hartree–Fock theory
to describe equilibrium and nonequilibrium many-body phenomenaLW60 ; BK61 ;
Baym62 ; BS89 ; Kita10 . Using the correlation part of $\Phi[G,\Gamma^{(0)}]$,
we will perform the remaining renormalization to express $\Omega$ as a
functional of $(G,\Gamma)$ in such a way that the functional obeys the
stationarity conditions $\delta\Omega/\delta G=0$ and
$\delta\Omega/\delta\Gamma=0$. The latter condition yields a self-consistent
equation for $\Gamma$ whose solution appropriately reduces to $\Gamma^{(0)}$
in the weak-coupling limit. Once $G$ and $\Gamma$ are known by solving the
coupled equations, we can also obtain the two-particle Green’s function
$G^{\rm II}$, as will be shown below. Thus, the present formalism provides a
unified framework to study $(\Omega,G,G^{\rm II})$ consistently and
simultaneously. For example, it enables us to calculate free energy,
magnetization, and susceptibility through magnetic transitions, incorporating
single-particle and collective excitations on an equal footing.

This paper is organized as follows. In Sect. 2, we present the formulation. In
Sect. 3, we provide several examples of approximate $\Phi$ functionals for
fermions. In Sect. 4, we provide concluding remarks. We use the units of
$\hbar=k_{\rm B}=1$.

##  II Formulation

###  II.1 Luttinger–Ward functional

We consider a system of identical bosons or fermions interacting via a two-
body potential $U({\bm{r}}_{1}-{\bm{r}}_{2})$ in the coordinate-space
representation. We focus on normal states for clarity, for which the
Luttinger–Ward functional is given by LW60 ; Kita10

$\displaystyle\Omega=\frac{\sigma}{\beta}{\rm
Tr}\bigl{[}\ln(-\underline{G}_{0}^{-1}+\underline{\Sigma})+\underline{\Sigma}\,\underline{G}\bigr{]}+\Phi.$
(1)

Here, $\sigma$ takes $+1$ for bosons and $-1$ for fermions, $\beta$ is the
inverse temperature, and $\underline{G}_{0}$ and $\underline{\Sigma}$
respectively are noninteracting matrix Green’s function and self-energy whose
rows and columns are specified by the index $1\equiv(\xi_{1},\tau_{1})$, where
$\xi_{1}$ denotes the space-spin coordinatesKita15  and $\tau_{1}$ is the
imaginary time that lies in $[0,\beta]$. This functional satisfies the
stationarity condition $\delta\Omega/\delta\underline{G}=0$, which yields
Dyson’s equationKita10

$\displaystyle\underline{G}^{-1}=\underline{G}_{0}^{-1}-\underline{\Sigma},$
(2)

where the self-energy is defined by

$\displaystyle\Sigma(1^{\prime},1)=-\sigma\beta\frac{\delta\Phi}{\delta
G(1,1^{\prime})}.$ (3)

Figure 1: (Color online) Diagrammatic expression of $\Phi$ up to the fourth
order in $\Gamma^{(0)}$. A filled circle and a line with an arrow denote
$\Gamma^{(0)}$ and $G$, respectively. The area enclosed by the circle in the
fourth diagram represents an extended vertex $\bar{\Gamma}^{(0)}$ defined by
Eq. (7).

A key quantity in Eq. (1) is the functional $\Phi$. To write it concisely, we
introduce the bare symmetrized vertex,

$\displaystyle\Gamma^{(0)}(1^{\prime}2^{\prime},12)\equiv$
$\displaystyle\,U({\bm{r}}_{1}-{\bm{r}}_{2})\delta(\tau_{1}-\tau_{2})\bigl{[}\delta(1^{\prime},1)\delta(2^{\prime},2)$
$\displaystyle\,+\sigma\delta(1^{\prime},2)\delta(2^{\prime},1)\bigr{]},$ (4)

having the symmetry
$\Gamma^{(0)}(1^{\prime}2^{\prime},12)\\!=\\!\sigma\Gamma^{(0)}(2^{\prime}1^{\prime},12)\\!=\\!\sigma\Gamma^{(0)}(1^{\prime}2^{\prime},21)\\!=\\!\Gamma^{(0)}(12,1^{\prime}2^{\prime})$.
The interaction Hamiltonian can be expressed in terms of $\Gamma^{(0)}$ and
the field operators $(\hat{\psi},\hat{\psi}^{\dagger})$ in the Heisenberg
representation as

$\displaystyle\hat{H}_{\rm int}=$
$\displaystyle\,\frac{1}{4\beta}\hat{\psi}^{\dagger}(1^{\prime})\hat{\psi}^{\dagger}(2^{\prime})\Gamma^{(0)}(1^{\prime}2^{\prime},12)\hat{\psi}(2)\hat{\psi}(1),$
(5)

where integrations over repeated arguments are implied. As given in Fig. 1
graphically, the functional $\Phi$ consists of skeleton diagrams in the bare
perturbation expansion of $\Omega$ in $\hat{H}_{\rm int}$ with the replacement
$\underline{G}_{0}\rightarrow\underline{G}$. The first-order diagram is the
Hartree–Fock or mean-field contribution given analytically by

$\displaystyle\Phi_{\rm MF}\equiv$
$\displaystyle\,\frac{1}{2\beta}{\Gamma}^{(0)}(1^{\prime}2^{\prime},12)G(1,1_{+}^{\prime})G(2,2_{+}^{\prime}),$
(6)

where the subscript + in the equal-time averages of the first order places the
creation operator to the left.LW60 ; Kita10

The remaining diagrams in Fig. 1 constitute the correlation contribution
$\Phi_{\rm c}\equiv\Phi-\Phi_{\rm MF}$. To express $\Phi_{\rm c}$ concisely,
we follow the procedure of De Dominicis and Martin DDM64-1  to decompose
matrix Green’s function formally as
$\underline{G}=\underline{G}^{\frac{1}{2}}\underline{G}^{\frac{1}{2}}$ and
introduce the extended vertex,

$\displaystyle\bar{\Gamma}^{(0)}(1^{\prime}2^{\prime},12)\equiv$
$\displaystyle\,G^{\frac{1}{2}}(1^{\prime},3^{\prime})G^{\frac{1}{2}}(2^{\prime},4^{\prime})\Gamma^{(0)}(3^{\prime}4^{\prime},34)$
$\displaystyle\,\times G^{\frac{1}{2}}(3,1)G^{\frac{1}{2}}(4,2),$ (7)

which represents the structure enclosed by the circle in Fig. 1 with the first
two (latter two) arguments corresponding to outgoing (incoming) lines. Indeed,
we can write $\Phi_{\rm c}$ as a functional of $\bar{\Gamma}^{(0)}$ alone as

$\displaystyle\Phi_{\rm c}\bigl{[}\bar{\Gamma}^{(0)}\bigr{]}=$
$\displaystyle\,-\frac{1}{8\beta}\bar{\Gamma}^{(0)}(1^{\prime}2^{\prime},12)\bar{\Gamma}^{(0)}(12,1^{\prime}2^{\prime})$
$\displaystyle\,+\frac{\sigma^{2}}{2^{3}\cdot
3\beta}\bar{\Gamma}^{(0)}(11^{\prime},22^{\prime})\bar{\Gamma}^{(0)}(22^{\prime},33^{\prime})\bar{\Gamma}^{(0)}(33^{\prime},11^{\prime})$
$\displaystyle\,+\frac{\sigma^{3}}{2\cdot
3\beta}\bar{\Gamma}^{(0)}(1^{\prime}2,12^{\prime})\bar{\Gamma}^{(0)}(2^{\prime}3,23^{\prime})\bar{\Gamma}^{(0)}(3^{\prime}1,31^{\prime})$
$\displaystyle\,-\cdots.$ (8)

The prefactor of the second term on the right-hand side originates from the
product of (i) $-\beta^{-1}$ in the definition of $\Omega$, (ii)
$(-\frac{1}{4})^{3}\frac{1}{3!}$ from the expansion of $e^{-\beta\hat{H}_{\rm
int}}$, (iii) $2!2^{3}$ from the number of ways of connecting the creation and
annihilation operators for the third diagram in Fig. 1, and (iv) $\sigma^{2}$
from the two closed particle lines. That of the third term has been obtained
by replacing the latter two factors above by (iii) $2!2^{5}$ and (iv)
$\sigma^{3}$, respectively, as appropriate for the fourth diagram in Fig. 1.
Specifically on (iii), every factor $2$ originates from choosing one of the
two identical creation or annihilation operators in the interaction
Hamiltonian to connect.

###  II.2 Renormalized vertex and Legendre transformation

Now, we perform the Legendre transformation for changing the vertex variable
in $\Omega$ from $\Gamma^{(0)}$ to $\Gamma$. First of all, we define the
renormalized vertex in terms of $\Phi_{\rm c}$ in Eq. (8) by

$\displaystyle\bar{\Gamma}(1^{\prime}2^{\prime},12)\equiv$
$\displaystyle\,-4\beta\frac{\delta\Phi_{\rm
c}}{\delta\bar{\Gamma}^{(0)}(12,1^{\prime}2^{\prime})}.$ (9)

Practically, we should replace $\bar{\Gamma}^{(0)}(12,1^{\prime}2^{\prime})$
in the denominator by
$\frac{1}{2}[\bar{\Gamma}^{(0)}(12,1^{\prime}2^{\prime})+\sigma\bar{\Gamma}^{(0)}(12,2^{\prime}1^{\prime})]$
to incorporate the symmetry of $\bar{\Gamma}^{(0)}$ manifestly in the
differentiation. Substitution of Eq. (8) into the right-hand side yields

$\displaystyle\bar{\Gamma}(1^{\prime}2^{\prime},12)=$
$\displaystyle\,\bar{\Gamma}^{(0)}(1^{\prime}2^{\prime},12)-\frac{1}{2}\bar{\Gamma}^{(0)}(1^{\prime}2^{\prime},33^{\prime})\bar{\Gamma}^{(0)}(33^{\prime},12)$
$\displaystyle\,-\sigma\bigl{[}\bar{\Gamma}^{(0)}(2^{\prime}3,23^{\prime})\bar{\Gamma}^{(0)}(3^{\prime}1^{\prime},31)$
$\displaystyle\,+\sigma\bar{\Gamma}^{(0)}(1^{\prime}3,23^{\prime})\bar{\Gamma}^{(0)}(3^{\prime}2^{\prime},31)\bigr{]}+\cdots,$
(10)

which certainly reduces to $\bar{\Gamma}^{(0)}$ in the weak-coupling limit.
Equation (10) enables us to express $\bar{\Gamma}^{(0)}$ as a functional of
$\bar{\Gamma}$. We then perform the Legendre transformation,

$\displaystyle\Phi_{\rm
L}\bigl{[}\bar{\Gamma}\bigr{]}\equiv\frac{1}{4\beta}\bar{\Gamma}^{(0)}(1^{\prime}2^{\prime},12)\bar{\Gamma}(12,1^{\prime}2^{\prime})+\Phi_{\rm
c}\bigl{[}\bar{\Gamma}^{(0)}\bigr{]},$ (11)

which satisfies

$\displaystyle 4\beta\frac{\delta\Phi_{\rm
L}\bigl{[}\bar{\Gamma}\bigr{]}}{\delta\bar{\Gamma}(12,1^{\prime}2^{\prime})}=\bar{\Gamma}^{(0)}(1^{\prime}2^{\prime},12),$
(12)

since the implicit dependence on $\bar{\Gamma}$ through $\bar{\Gamma}^{(0)}$
cancels out owing to Eq. (9). Equation (12) is expressible as a stationarity
condition in terms of $\bar{\Gamma}$ by introducing another functional of
$(\bar{\Gamma}^{(0)},\bar{\Gamma})$,

$\displaystyle\bar{\Phi}_{\rm
c}\bigl{[}\bar{\Gamma}^{(0)},\bar{\Gamma}\bigr{]}\equiv-\frac{1}{4\beta}\bar{\Gamma}^{(0)}(1^{\prime}2^{\prime},12)\bar{\Gamma}(12,1^{\prime}2^{\prime})+\Phi_{\rm
L}\bigl{[}\bar{\Gamma}\bigr{]}.$ (13)

Indeed, one can show by using Eq. (12) that

$\displaystyle\frac{\delta\bar{\Phi}_{\rm
c}\bigl{[}\bar{\Gamma}^{(0)},\bar{\Gamma}\bigr{]}}{\delta\bar{\Gamma}(12,1^{\prime}2^{\prime})}=0$
(14)

holds. Note also that $\bar{\Phi}_{\rm c}=\Phi_{\rm c}$ is satisfied, as seen
from Eqs. (11) and (13).

In summary, the grand thermodynamic potential is now given as a functional of
$(G,\Gamma)$ by

$\displaystyle\Omega[G,\Gamma]=$ $\displaystyle\,\frac{\sigma}{\beta}{\rm
Tr}\bigl{[}\ln(-\underline{G}_{0}^{-1}+\underline{\Sigma})+\underline{\Sigma}\,\underline{G}\bigr{]}+\Phi_{\rm
MF}[G,\Gamma^{(0)}]$ $\displaystyle\,+\bar{\Phi}_{\rm
c}\bigl{[}\bar{\Gamma}^{(0)},\bar{\Gamma}\bigr{]},$ (15)

which is stationary with respect to the variations of both $G$ and $\Gamma$,
as given explicitly by Eqs. (2) and (14). The corresponding self-energy is
defined by

$\displaystyle\Sigma(1^{\prime},1)\equiv-\sigma\beta\frac{\delta\bigl{(}\Phi_{\rm
MF}+\bar{\Phi}_{\rm c}\bigr{)}}{\delta G(1,1^{\prime})}$ (16)

instead of Eq. (3). Noting Eq. (14), we can perform the differentiation of
$\bar{\Phi}_{\rm c}$ with respect to $G$ by considering only the dependence
through $\bar{\Gamma}^{(0)}$ given by Eq. (7). We thereby obtain

$\displaystyle\Sigma(1^{\prime},1)=$ $\displaystyle\,\Sigma_{\rm
MF}(1^{\prime},1)$
$\displaystyle\,+\frac{\sigma}{4}\Gamma^{(0)}(3^{\prime}4^{\prime},12)G(2,2^{\prime})\Gamma(1^{\prime}2^{\prime},34)G(3,3^{\prime})G(4,4^{\prime})$
$\displaystyle\,+\frac{\sigma}{4}\Gamma^{(0)}(1^{\prime}2^{\prime},34)G(3,3^{\prime})G(4,4^{\prime})\Gamma(3^{\prime}4^{\prime},12)G(2,2^{\prime}),$
(17)

where $\Sigma_{\rm
MF}(1^{\prime},1)\\!\equiv\\!-\sigma\Gamma^{(0)}(1^{\prime}2^{\prime},12)G(2,2_{+}^{\prime})$
is the mean-field self-energy and $\Gamma$ is defined in terms of
$\bar{\Gamma}$ in the same way as Eq. (7).

This completes our basic formulation. Specifically, Dyson’s equation (2) with
Eq. (17) and the stationarity condition $\delta\Omega/\delta\bar{\Gamma}=0$
constitute a set of self-consistent (i.e., nonlinear) equations for $G$ and
$\Gamma$. Substituting the solution into Eq. (15) yields the grand potential
$\Omega$.

###  II.3 Two-particle Green’s function

The formalism also enables us to obtain the two-particle Green’s function,

$\displaystyle G^{\rm
II}(12,1^{\prime}2^{\prime})\equiv\langle\hat{T}_{\tau}\hat{\psi}(1)\hat{\psi}(2)\hat{\psi}^{\dagger}(2^{\prime})\hat{\psi}^{\dagger}(1^{\prime})\rangle,$
(18a) where $\hat{T}_{\tau}$ is the time-ordering operator.LW60 ; Kita10
Indeed, it is expressible as a sum of the disconnected and cumulant
parts,AGD63  $\displaystyle G^{\rm II}(12,1^{\prime}2^{\prime})=$
$\displaystyle\,G(1,1^{\prime})G(2,2^{\prime})+\sigma
G(1,2^{\prime})G(2,1^{\prime})$
$\displaystyle\,-G(1,3^{\prime})G(2,4^{\prime})\Gamma(3^{\prime}4^{\prime},34)G(3,1^{\prime})G(4,2^{\prime}),$
(18b)

with
$G(1,1^{\prime})\equiv-\langle\hat{T}_{\tau}\hat{\psi}(1)\hat{\psi}^{\dagger}(1^{\prime})\rangle$.
Equation (18b) can be regarded as defining the renormalized interaction vertex
$\Gamma$ in such a way that replacing $(G,\Gamma)$ by $(G_{0},\Gamma^{(0)})$
yields the first-order perturbation result for $G^{\rm II}$. It tells us that,
once $G$ and $\Gamma$ are known, we can also calculate the two-particle
Green’s function. Thus, the present formalism enables us to study two-particle
or collective excitations simultaneously with the free energy and single-
particle excitations in a unified framework.

The validity of using the same $\Gamma$ in Eq. (18b) as that in Eq. (17) can
be confirmed in terms of the interaction energy. Specifically, it follows from
Eqs. (5) and (18a) that the interaction energy is expressible as

$\displaystyle\langle\hat{H}_{\rm int}\rangle=$
$\displaystyle\,\frac{1}{4\beta}\Gamma^{(0)}(1^{\prime}2^{\prime},12)G^{\rm
II}(12,1_{+}^{\prime}2_{+}^{\prime}).$ (19a) On the other hand, by replacing
$\hat{H}_{\rm int}$ by $\lambda\hat{H}_{\rm int}$, differentiating the
resulting $\Omega(\lambda)$ in terms of $\lambda$, and setting $\lambda=1$, we
also obtainLW60 ; Kita10  $\displaystyle\langle\hat{H}_{\rm
int}\rangle=-\frac{\sigma}{2\beta}\Sigma(1^{\prime},1)G(1,1_{+}^{\prime}).$
(19b)

By substituting Eqs. (18b) and (17) into Eqs. (19a) and (19b), respectively,
we obtain an identical expression for $\langle\hat{H}_{\rm int}\rangle$. This
agreement shows the consistency of the present formulation, which generally
cannot be reached by vertices obtained from the Bethe–Salpeter equation using
$\Gamma^{({\rm i})}\propto\delta^{2}\Phi/\delta G\delta G$ as the input called
irreducible vertex.Baym62 ; Kita10 ; Kontani13

###  II.4 Comments

Four comments are in order concerning the formalism. First, the present
definition of the renormalized vertex by Eq. (9) is advantageous over that of
De Dominicis and MartinDDM64-2  in terms of the following: (i) it is simple
and algebraic without recourse to any topological structure of skeleton-type
diagrams; (ii) it reduces to $\Gamma^{(0)}$ instead of $-\Gamma^{(0)}$ in the
weak-coupling limit. The subsequent Legendre transformation from
$\Gamma^{(0)}$ to $\Gamma$ can be performed straightforwardly from Eq. (11)
down to Eq. (13).

Second, the formalism is conserving, i.e., it satisfies the particle,
momentum, and energy conservation laws automatically. The proofs proceed in
exactly the same way as those given for the Luttinger–Ward functionalBaym62 ;
Kita10  in terms of the self-energy in Eq. (16).

Third, Eq. (15) can be extended straightforwardly to ordered phases such as
superconductivity and ferromagnetism, except for Bose–Einstein condensation;
the case of superconductivity will be presented below. Note in this context
that, for fermions, the stationarity condition of $\Omega$ with respect to the
order parameter is naturally included in the stationarity condition of Green’s
function in the static limit, which incorporates the possible emergence of
spontaneous anisotropy or order.

Fourth, the formalism enables us to calculate thermodynamic, single-particle,
and two-particle properties in a single consistent approximation scheme, in
contrast to previous treatments where they have been studied by adopting
different approximations, e.g., the mean-field approximation for $(\Omega,G)$
and the random-phase approximation for $G^{\rm II}$.Anderson58 ; Moriya84
Moreover, it is applicable to both normal and ordered phases on an equal
footing because of the self-consistency procedure for calculating
$(G,\Gamma)$, as already mentioned. These two features may be regarded as
definite advantages of the present formalism, especially for describing
collective fluctuations near second-order transitions. Note also that
practical calculations of $(\Omega,G,G^{\rm II})$ can be performed in terms of
$G$ and $\Gamma^{(0)}$ alone without recourse to the Legendre transformation
at all, by using Eqs. (9) and (18) additionally to study two particle
properties such as transport coefficients.

##  III Examples for Fermions

We focus on the case of fermions with $\sigma=-1$ to present some examples of
approximations, including one for superconductivity.

###  III.1 Particle–particle scattering approximation

First, we consider the $T$-matrix BK61 ; Baym62  or particle–particle
scatteringBS89  approximation suitably generalized to incorporate the exchange
process at each order. This is one of the exceptional cases where Eq. (9) can
be inverted analytically to express $\Gamma^{(0)}$ explicitly in terms of
$\Gamma$ satisfying the antisymmetry requirement:
$\Gamma(1^{\prime}2^{\prime},12)=-\Gamma(2^{\prime}1^{\prime},12)$.
Specifically, let us collect the series of the third and fifth contributions
in Fig. 1 up to infinite order besides that of the second-order diagram.
Accordingly, Eq. (8) is approximated by

$\displaystyle\Phi_{\rm c}\bigl{[}\bar{\Gamma}^{(0)}\bigr{]}=$
$\displaystyle\,\frac{1}{\beta}{\rm
Tr}\biggl{[}\ln\biggl{(}\underline{1}+\frac{1}{2}\underline{\bar{\Gamma}}^{(0)}\biggr{)}-\frac{1}{2}\underline{\bar{\Gamma}}^{(0)}\biggr{]},$
(20)

where $\underline{1}$ and $\underline{\bar{\Gamma}}^{(0)}$ are matrices whose
elements are given by
$(\underline{1})_{11^{\prime},22^{\prime}}\\!\equiv\\!\delta(1,2)\delta(1^{\prime},2^{\prime})$
and
$(\underline{\bar{\Gamma}}^{(0)})_{11^{\prime},22^{\prime}}\\!\equiv\\!\bar{\Gamma}^{(0)}(11^{\prime},22^{\prime})$,
and the logarithm is defined by the power series of
$\underline{\bar{\Gamma}}^{(0)}$. The differentiation of Eq. (9) yields

$\displaystyle\underline{\bar{\Gamma}}=\underline{\bar{\Gamma}}^{(0)}\biggl{(}\underline{1}+\frac{1}{2}\underline{\bar{\Gamma}}^{(0)}\biggr{)}^{\\!\\!-1}.$
(21)

The relation can be inverted as

$\displaystyle\underline{\bar{\Gamma}}^{(0)}=\underline{\bar{\Gamma}}\,\biggl{(}\underline{1}-\frac{1}{2}\underline{\bar{\Gamma}}\biggr{)}^{\\!\\!-1}.$
(22)

Substitution of Eq. (22) into Eq. (11) with Eq. (20) yields

$\displaystyle\Phi_{\rm
L}\bigl{[}\bar{\Gamma}\bigr{]}\equiv\frac{1}{\beta}{\rm
Tr}\biggl{[}-\ln\biggl{(}\underline{1}-\frac{1}{2}\underline{\bar{\Gamma}}\biggr{)}-\frac{1}{2}\underline{\bar{\Gamma}}\biggr{]}.$
(23)

By comparing Eq. (23) with Eq. (20), one realizes that the vertex
renormalization cannot be carried out by assigning the same sign and weight to
each skeleton diagram as those in the bare perturbation expansion, contrary to
the assumption in Ref. DDM64-2, . The transformation from Eq. (8) to Eq. (13)
may be regarded as an algebraic Legendre transformation introduced without
assuming the convexity of the original functional.JKM17

The above results can be put into more familiar expressions by replacing
$\underline{\bar{\Gamma}}\rightarrow\underline{\Gamma}\,\underline{GG}$ with
$(\underline{{\Gamma}})_{11^{\prime},22^{\prime}}\\!\equiv\\!{\Gamma}(11^{\prime},22^{\prime})$
and $\bigl{(}\underline{GG}\bigr{)}_{11^{\prime},22^{\prime}}\equiv
G(1,2)G(1^{\prime},2^{\prime})$.

###  III.2 Particle–hole scattering approximation

Second, we consider the shielded potentialBK61 ; Baym62  or particle–hole
scattering approximationBS89  suitably generalized to incorporate the exchange
process at each order. Specifically, let us collect the fourth and sixth
diagrams in Fig. 1 up to infinite order besides that of the second-order
diagram. Accordingly, Eq. (8) is approximated by

$\displaystyle\Phi_{\rm c}\bigl{[}\bar{\Gamma}^{(0)}\bigr{]}=$
$\displaystyle\,\frac{1}{2\beta}{\rm
Tr}\biggl{[}\ln\bigl{(}\underline{1}-\underline{\bar{\Gamma}}^{(0)}\bigr{)}+\underline{\bar{\Gamma}}^{(0)}+\frac{1}{4}\bigl{(}\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{2}\biggr{]},$
(24)

where the matrix $\underline{\bar{\Gamma}}^{(0)}$ is now defined by $\langle
11^{\prime}|\underline{\bar{\Gamma}}^{(0)}|22^{\prime}\rangle\equiv\bar{\Gamma}^{(0)}(12^{\prime},1^{\prime}2)$.
Performing the differentiation of Eq. (9) yields

$\displaystyle\langle
11^{\prime}|\underline{\bar{\Gamma}}|22^{\prime}\rangle=$
$\displaystyle\,\langle
11^{\prime}|\underline{\bar{\Gamma}}^{(0)}|22^{\prime}\rangle$
$\displaystyle\,+\langle
11^{\prime}|\bigl{(}\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{2}\bigl{(}\underline{1}-\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{\\!\\!-1}|22^{\prime}\rangle$
$\displaystyle\,-\langle
12^{\prime}|\bigl{(}\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{2}\bigl{(}\underline{1}-\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{\\!\\!-1}|21^{\prime}\rangle.$
(25)

In general, the equation can only be inverted numerically to write
$\Gamma^{(0)}$ in terms of $\Gamma$ owing to the presence of the exchange
contribution given by the third term on the right-hand side.

On the other hand, we can express $\Omega$ analytically in terms of a
renormalized vertex $\Gamma^{({\rm ph})}$ without the antisymmetry.
Specifically, we introduce $\Gamma^{({\rm
ph})}(11^{\prime},22^{\prime})\equiv\langle
11^{\prime}|\underline{\Gamma}^{({\rm ph})}|22^{\prime}\rangle$ by performing
the differentiation of Eq. (9) without taking the antisymmetry of
${\bar{\Gamma}}^{(0)}$ into account to obtain

$\displaystyle\underline{\bar{\Gamma}}^{({\rm
ph})}=\underline{\bar{\Gamma}}^{(0)}+2\bigl{(}\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{2}\bigl{(}\underline{1}-\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{\\!\\!-1}.$
(26)

The relation can be inverted algebraically as

$\displaystyle\underline{\bar{\Gamma}}^{(0)}=-\frac{1}{2}\bigl{(}\underline{1}+\underline{\Gamma}^{({\rm
ph})}\bigr{)}+\frac{1}{2}\Bigl{[}\underline{1}+6\underline{\Gamma}^{({\rm
ph})}+\bigl{(}\underline{\Gamma}^{({\rm
ph})}\bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}.$ (27)

The functional $\Phi_{\rm L}[\bar{\Gamma}^{({\rm ph})}]$ can be constructed by
substituting Eq. (27) into Eq. (11) with Eq. (24). The results can be put into
more familiar expressions by the replacement $\underline{\bar{\Gamma}}^{({\rm
ph})}\rightarrow\underline{\Gamma}^{({\rm ph})}\underline{GG}^{\rm T}$ with
$\bigl{(}\underline{GG}^{\rm T}\bigr{)}_{11^{\prime},22^{\prime}}\equiv
G(1,2)G(2^{\prime},1^{\prime})$. The resulting functional
$\bar{\Phi}[G,\Gamma^{({\rm ph})}]$ corresponds to the Luttinger–Ward $\Phi$
functional given in terms of $U$ instead of $\Gamma^{(0)}$. It remains to be
clarified how neglecting the antisymmetry requirement affects various results.
In this context, it is definitely necessary for calculating Eq. (18b) to
substitute $\frac{1}{2}[\Gamma^{({\rm
ph})}(3^{\prime}4^{\prime},34)-\Gamma^{({\rm ph})}(4^{\prime}3^{\prime},34)]$
into $\Gamma(3^{\prime}4^{\prime},34)$ so as to reproduce the antisymmetry of
$G^{\rm II}$ appropriately.

###  III.3 Fluctuation exchange approximation

Third, we consider the fluctuation exchange (FLEX) approximationBS89  of
collecting the third to sixth diagrams in Fig. 1 up to infinite order besides
that of the second-order diagram. There are two ways of performing the
renormalization.

The first one is to solve Eq. (10) in the FLEX approximation to obtain a
single renormalized $\bar{\Gamma}$. This approach is advantageous in that the
mixing between the particle–particle and particle–hole processes is naturally
incorporated, but the approach will also be numerically much more demanding.

The second one neglects the antisymmetry requirement and introduces two kinds
of renormalized vertices corresponding the particle–particle and particle–hole
channels. Specifically, we introduce matrices $\underline{1}$,
$\underline{\bar{\Gamma}}^{(0)}$, and $\underline{C}$ through

$\displaystyle(\underline{1})_{11^{\prime},22^{\prime}}\equiv$
$\displaystyle\,\begin{bmatrix}\delta(1,2)\delta(1^{\prime},2^{\prime})&0\\\
0&\delta(1,2)\delta(1^{\prime},2^{\prime})\end{bmatrix},$ (28a)
$\displaystyle(\underline{\bar{\Gamma}}^{(0)})_{11^{\prime},22^{\prime}}\equiv$
$\displaystyle\,\begin{bmatrix}\frac{1}{2}\bar{\Gamma}^{(0)}(11^{\prime},22^{\prime})&0\\\
0&-\bar{\Gamma}^{(0)}(12^{\prime},1^{\prime}2)\end{bmatrix},$ (28b)
$\displaystyle(\underline{C})_{11^{\prime},22^{\prime}}\equiv$
$\displaystyle\,\begin{bmatrix}2\delta(1,2)\delta(1^{\prime},2^{\prime})&0\\\
0&\delta(1,2)\delta(1^{\prime},2^{\prime})\end{bmatrix},$ (28c)

to incorporate the two channels independently. Equation (8) is then
approximated by

$\displaystyle\Phi_{\rm c}\bigl{[}{\bar{\Gamma}}^{(0)}\bigr{]}=$
$\displaystyle\,\frac{1}{2\beta}{\rm
Tr}\,\underline{C}\biggl{[}\ln\bigl{(}\underline{1}+\underline{\bar{\Gamma}}^{(0)}\bigr{)}-\underline{\bar{\Gamma}}^{(0)}+\frac{1}{3}\bigl{(}\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{2}\biggr{]}.$
(29)

The factor $\frac{1}{3}$ originates from our specific choice of dividing the
second-order process into the particle–particle and particle–hole channels
with the ratio 1:2, but note that there is arbitrariness in how the ratio is
chosen. Since Eq. (29) is diagonal in the particle–particle and particle–hole
indices, we can perform the differentiation of Eq. (10) easily to obtain

$\displaystyle\underline{\bar{\Gamma}}=$
$\displaystyle\,\begin{bmatrix}\underline{\bar{\Gamma}}^{({\rm
pp})}&\underline{0}\\\ \underline{0}&\underline{\bar{\Gamma}}^{({\rm
ph})}\end{bmatrix},$ (30)

where $\underline{\bar{\Gamma}}^{({\rm pp})}$ and
$\underline{\bar{\Gamma}}^{({\rm ph})}$ are defined in terms of Eqs. (21) and
(26), respectively, by subtracting $\frac{2}{3}\underline{\bar{\Gamma}}^{(0)}$
and $\frac{1}{3}\underline{\bar{\Gamma}}^{(0)}$; note the difference in the
definitions of $\underline{\bar{\Gamma}}^{(0)}$ between them. Equation (30)
can be used to express $\underline{\bar{\Gamma}}^{(0)}$ as a functional of
$\underline{\bar{\Gamma}}$. Substituting it into Eq. (11) with Eq. (29), we
obtain $\Phi_{\rm L}[\bar{\Gamma}]$.

###  III.4 FLEX-S approximation for superconductivity

The FLEX approximationBS89  can be generalized concisely into the FLEX-S
approximation for describing superconductivity Kita11 . We will discuss the
vertex renormalization in the FLEX-S approximation without requiring
antisymmetry.

Let us express Eq. (5) in a symmetric form with respect to
$(\hat{\psi},\hat{\psi}^{\dagger})\\!\equiv\\!(\hat{\psi}_{1},\hat{\psi}_{2})$
asKita11

$\displaystyle\hat{H}_{\rm int}=$
$\displaystyle\,\frac{1}{4!\beta}\Gamma^{(0)}_{i^{\prime}i,j^{\prime}j}(1^{\prime}1,2^{\prime}2)$
$\displaystyle\,\times\hat{\cal
N}\hat{\psi}_{3-i^{\prime}}(1^{\prime})\hat{\psi}_{i}(1)\hat{\psi}_{3-j^{\prime}}(2^{\prime})\hat{\psi}_{j}(2),$
(31)

where $\hat{\cal N}$ is the normal-ordering operator of placing creation
operators to the left, and
$\Gamma^{(0)}_{i^{\prime}i,j^{\prime}j}(1^{\prime}1,2^{\prime}2)$ with
$i,j=1,2$ is defined in terms of $\Gamma^{(0)}(1^{\prime}2^{\prime},12)$ in
Eq. (4) by

Figure 2: (Color online) Diagrammatic expression of $\Phi$ up to the fourth
order in terms of the vertex of Eq. (32), which is denoted by a small circle.

$\displaystyle\Gamma^{(0)}_{i^{\prime}i,j^{\prime}j}(1^{\prime}1,2^{\prime}2)\equiv$
$\displaystyle\,\delta_{ij}\delta_{i^{\prime}i}\delta_{j^{\prime}j}\Gamma^{(0)}(1^{\prime}2^{\prime},12)$
$\displaystyle\,-\delta_{i,3-j}\delta_{i^{\prime}i}\delta_{j^{\prime}j}\Gamma^{(0)}(1^{\prime}2,12^{\prime})$
$\displaystyle\,-\delta_{i,3-j}\delta_{i^{\prime}j}\delta_{j^{\prime}i}\Gamma^{(0)}(1^{\prime}1,2^{\prime}2),$
(32)

having the symmetry
$\Gamma^{(0)}_{i^{\prime}i,j^{\prime}j}(1^{\prime}1,2^{\prime}2)\\!=\\!\Gamma^{(0)}_{j^{\prime}j,i^{\prime}i}(2^{\prime}2,1^{\prime}1)\\!=\\!-\Gamma^{(0)}_{i^{\prime}j,j^{\prime}i}(1^{\prime}2,2^{\prime}1)\\!=\\!-\Gamma^{(0)}_{i^{\prime}i,3-j,3-j^{\prime}}(1^{\prime}1,22^{\prime})$.
The equivalence of Eqs. (5) and (31) can be seen easily by substituting Eq.
(32) into the latter and rearranging the resulting expression in the normal
order to remove $\hat{\cal N}$. The advantage of Eq. (31) lies in the
equivalence of the four field operators. Indeed, by using the symmetry of
$\Gamma^{(0)}_{i^{\prime}i,j^{\prime}j}$ and the anti-commutation relations of
the field operators under $\hat{\cal N}$, we can place each of
$\hat{\psi}_{3-j^{\prime}}(2^{\prime})$ and $\hat{\psi}_{j}(2)$ right next to
$\hat{\psi}_{3-i^{\prime}}(1^{\prime})$ and transform the resulting expression
into the same form as Eq. (31) through a change of variables. This equivalence
enables us to perform the perturbation expansion of $\Phi$ in $\hat{H}_{\rm
int}$ concisely in terms of the Feynman diagrams of Fig. 2 without
arrows,Kita11  where an additional summation at each vertex over every
internal index $i=1,2$ is implied. To be more specific, we can associate
$(\hat{\psi}_{i},\hat{\psi}_{3-i})$ in Eq. (31) with
$(\hat{\psi},\hat{\psi}^{\dagger})$ of the normal state to write down a single
contribution analytically for each Feynman diagram in terms of Green’s
function,

$G_{ii^{\prime}}(1,1^{\prime})\equiv-\bigl{<}\hat{T}_{\tau}\hat{\psi}_{i}(1)\hat{\psi}_{3-i^{\prime}}(1^{\prime})\bigr{>}.$
(33)

and subsequently multiply it by a combinatorial factor originating from the
equivalence of the four field operators. By introducing matrices
$\underline{1}$, $\underline{\Gamma}^{(0)}$, and $\underline{GG}^{\rm T}$ as

$\displaystyle\langle
1^{\prime}1_{i^{\prime}i}|\underline{1}|2^{\prime}2_{j^{\prime}j}\rangle\equiv$
$\displaystyle\,\delta_{ij}\delta_{i^{\prime}j^{\prime}}\delta(1,2)\delta(1^{\prime},2^{\prime}),$
(34a) $\displaystyle\langle
1^{\prime}1_{i^{\prime}i}|\underline{\Gamma}^{(0)}|2^{\prime}2_{j^{\prime}j}\rangle\equiv$
$\displaystyle\,\Gamma^{(0)}_{i^{\prime}i,jj^{\prime}}(1^{\prime}1,22^{\prime}),$
(34b) $\displaystyle\langle 1^{\prime}1_{i^{\prime}i}|\underline{GG}^{\rm
T}|2^{\prime}2_{j^{\prime}j}\rangle\equiv$
$\displaystyle\,G_{i^{\prime}j^{\prime}}(1^{\prime},2^{\prime})G_{ji}(2,1),$
(34c)

we can thereby express the $\Phi$ functional in the FLEX-S approximation as
$\Phi=\Phi_{\rm MF}+\Phi_{\rm c}$ withKita11

$\displaystyle\Phi_{\rm MF}=$ $\displaystyle\,-\frac{1}{8\beta}{\rm
Tr}\,\underline{\Gamma}^{(0)}\underline{GG}^{\rm T},$ (35a)
$\displaystyle\Phi_{\rm c}=$ $\displaystyle\,\frac{1}{2\beta}{\rm
Tr}\biggl{[}\ln\biggl{(}\underline{1}-\frac{1}{2}\underline{\Gamma}^{(0)}\underline{GG}^{\rm
T}\biggr{)}+\frac{1}{2}\underline{\Gamma}^{(0)}\underline{GG}^{\rm T}$
$\displaystyle\,+\frac{1}{12}\bigl{(}\underline{\Gamma}^{(0)}\underline{GG}^{\rm
T}\bigr{)}^{2}\biggr{]}.$ (35b)

See Eq. (28) of Ref. Kita11,  and note the differences of factor 2 in the
definition of Eq. (32) and the arrangement of arguments in Eq. (4) from Eqs.
(20b) and (16) in Ref. Kita11, , respectively. Note that Eq. (35b)
appropriately reduces to Eq. (29) in the normal-state limit of
$G_{12}=G_{21}=0$.

By decomposing
$\underline{G}=\underline{G}^{\frac{1}{2}}\underline{G}^{\frac{1}{2}}$ with
$\langle 1_{i}|\underline{G}|1^{\prime}_{i^{\prime}}\rangle\equiv
G_{ii^{\prime}}(1,1^{\prime})$, we can express Eq. (35b) as a functional of

$\displaystyle\underline{\bar{\Gamma}}^{(0)}\equiv\underline{G}^{\frac{1}{2}}\bigl{(}\underline{G}^{\rm
T}\bigr{)}^{\frac{1}{2}}\underline{\Gamma}^{(0)}\underline{G}^{\frac{1}{2}}\bigl{(}\underline{G}^{\rm
T}\bigr{)}^{\frac{1}{2}}$ (36)

alone by the replacement $\underline{\Gamma}^{(0)}\underline{GG}^{\rm
T}\rightarrow\underline{\bar{\Gamma}}^{(0)}$. Equation (9) is now changed into
$\bar{\Gamma}=-4!\beta\delta\Phi_{\rm c}/\delta\bar{\Gamma}^{(0)}$, and the
substitution of Eq. (35b) into its right-hand side yields

$\displaystyle\underline{\bar{\Gamma}}=\underline{\bar{\Gamma}}^{(0)}+\frac{3}{2}\bigl{(}\underline{\bar{\Gamma}}^{(0)}\bigr{)}^{2}\biggl{(}\underline{1}-\frac{1}{2}\underline{\bar{\Gamma}}^{(0)}\biggr{)}^{\\!\\!-1}.$
(37)

The relation can be inverted algebraically as

$\displaystyle\underline{\bar{\Gamma}}^{(0)}=-\frac{1}{2}\biggl{(}\underline{1}+\frac{1}{2}\underline{\Gamma}\biggr{)}+\frac{1}{2}\biggl{[}\underline{1}+5\underline{\Gamma}+\frac{1}{4}\bigl{(}\underline{\Gamma}\bigr{)}^{2}\biggr{]}^{\frac{1}{2}}.$
(38)

We then perform the Legendre transformation

$\displaystyle\Phi_{\rm L}[\bar{\Gamma}]\equiv\frac{1}{4!\beta}{\rm
Tr}\,\underline{\bar{\Gamma}}^{(0)}\underline{\bar{\Gamma}}+\Phi_{\rm
c}[\bar{\Gamma}^{(0)}],$ (39)

which satisfies $\bar{\Gamma}^{(0)}=4!\beta\delta\Phi_{\rm
L}/\delta\bar{\Gamma}$, and introduce

$\displaystyle\bar{\Phi}_{\rm
c}[\underline{\bar{\Gamma}}^{(0)},\underline{\bar{\Gamma}}]\equiv-\frac{1}{4!\beta}{\rm
Tr}\,\underline{\bar{\Gamma}}^{(0)}\underline{\bar{\Gamma}}+\Phi_{\rm
L}[\bar{\Gamma}].$ (40)

The grand thermodynamic potential
$\Omega=\Omega[\underline{G},\underline{\Gamma}]$ is given in terms of Eq.
(40) by

$\displaystyle\Omega=-\frac{1}{2\beta}{\rm
Tr}\,\bigl{[}\ln\bigl{(}-\underline{G}_{0}^{-1}+\underline{\Sigma}\bigr{)}+\underline{G}\,\underline{\Sigma}\bigr{]}+\Phi_{\rm
MF}+\bar{\Phi}_{\rm c}.$ (41)

See Eq. (11) of Ref. Kita11, . It satisfies
$\delta\Omega/\delta\underline{G}\\!=\\!\underline{0}$ and
$\delta\Omega/\delta\underline{\Gamma}\\!=\\!\underline{0}$.

##  IV Concluding Remarks

We have developed a formalism of calculating the grand thermodynamic potential
$\Omega$, one-particle Green’s function $G$, and two-particle Green’s function
$G^{\rm II}$ in a unified consistent framework by expressing $\Omega$ as a
functional of $G$ and the renormalized interaction vertex $\Gamma$. The key
result is Eq. (15) for $\Omega$, which satisfies the stationarity conditions
$\delta\Omega/\delta G=0$ and $\delta\Omega/\delta\Gamma=0$. Using the
solution of the coupled equations, we obtain $G$ by Eq. (2) in terms of the
self-energy given by Eq. (17), $G^{\rm II}$ by Eq. (18b), and $\Omega$ by Eq.
(15); this $\Omega$ naturally contains contributions of both single-particle
and two-particle (i.e., independent and collective) excitations. The
differences from and the advantages of the present formalism over previous
ones are summarized in Sect. II.4. Functional (15) will also be useful in
phenomenological studies of adopting some model form for $\Gamma$ to clarify
collective-mode contributions to thermodynamic observables. Generalizing the
formalism to describe Bose–Einstein condensates with a finite average
$\langle\hat{\psi}\rangle$ has yet to be performed.

A way to incorporate single-particle and collective excitations simultaneously
into calculations of thermodynamic observables has been sought over many
decades with no definite answer reached yet apparently. Back in the 1960,
Doniach and EngelsbergDE67  expressed the interaction energy of nearly
ferromagnetic Fermi liquids in terms of the magnetic susceptibility in the
random-phase approximation obtained earlier by Izuyama et al.,IKK63
integrated it in terms of the coupling constant to extract the extra free
energy due to spin fluctuations, and obtained a qualitative fit to the low-
temperature specific heat of liquid 3He that exhibits a logarithmic non-Fermi-
liquid behavior (see Ref. BP91,  for more references on the subject). The
approach has also been adopted by MoriyaMoriya84  in constructing the theory
of itinerant electron magnetism, where susceptibility is expressed in terms of
several phenomenological parameters whose values can be extracted from
experiments. On the other hand, the logarithmic corrections to Landau’s Fermi-
liquid theory observed in various physical quantities have been described
alternatively by introducing the concept of statistical quasiparticle energy
in the single-particle channel.BP91  The extensive references on the subject
given in Ref. BP91,  indicates the absence of any established microscopic
framework for treating single-particle and collective excitations
simultaneously and consistently for calculating thermodynamic observables such
as specific heat. The situation was unchanged until today, as may be seen in a
recent attempt to incorporate collective Cooper-pair fluctuations into the
Fermi liquid theory near superfluid transitions.LS22  The present formalism,
which can handle $(\Omega,G,G^{\rm II})$ microscopically and simultaneously in
a unified consistent manner, is expected to provide a firm basis for studying
the two kinds of contributions to thermodynamic observables quantitatively.

## Acknowledgment

This work was supported by JSPS KAKENHI Grant Number JP20K03848.

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