# How Infrared Singularities Affect Nambu-Goldstone Bosons
at Finite Temperatures

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

###### Abstract

Ordered phases realized through broken continuous symmetries embrace long-
range order-parameter fluctuations as manifest in the power-law decays of both
the transverse and longitudinal correlations, which are similar to those at
the second-order transition point. We calculate the transverse one-loop
correction to the dispersion relation of nonrelativistic Nambu-Goldstone (NG)
bosons at finite temperatures, assuming that they have well-defined dispersion
relations with some integer exponents. It is found that the transverse
correlations make the lifetime of the NG bosons vanish right on the assumed
dispersion curve below three dimensions at finite temperatures. Combined with
the vanishing of the longitudinal “mass”, the result indicates that the
correlations generally bring both an intrinsic lifetime to the excitations and
a qualitative change in the dispersion relation at long wavelengths below
three dimensions at finite temperatures, unless it is protected by some
conservation law. A more detailed calculation performed specifically on a
single-component Bose-Einstein condensate reveals that the gapless Bogoliubov
mode predicted by the perturbative treatment is modified substantially after
incorporating the correlations to have (i) an intrinsic lifetime and (ii) a
peak shift at long wavelengths from the linear dispersion relation. Thus, the
NG bosons may be fluctuating intrinsically into and out of the ordered
“vacuum” to maintain the order temporally.

##  I Introduction

Goldstone’s theorem Nambu61 ; Goldstone61 ; GSW62 ; Weinberg95  states that
broken continuous symmetries necessarily accompany gapless excitations called
Nambu-Goldstone (NG) bosons in the ordered phases. Examples include spin waves
of isotropic ferromagnets Bloch30 ; HP40 ; Dyson56  and
antiferromagnets,Anderson52 ; Kubo52  and the Bogoliubov mode of single-
component Bose-Einstein condensates (BECs).Bogoliubov47 ; LHY57 ; Beliaev58 ;
HP59 ; GN64  Their existence has been elucidated theoretically prier to the
proof of the theorem GSW62  based on perturbative treatments around zero
temperature.Bloch30 ; HP40 ; Anderson52 ; Kubo52 ; Dyson56 ; Bogoliubov47 ;
LHY57 ; Beliaev58 ; HP59 ; GN64  The connection among the number of broken
continuous symmetries, that of emergent NG bosons, and their dispersion
relations has been clarified for non-relativistic systems NC76 ; Schafer01 ;
WB11 ; WM12 ; Hidaka13  by implicitly assuming existence of well-defined
(long-lived) excitations and analyticity in their dispersion relations.

On the other hand, the gapless excitations have been known to cause infrared-
singular behaviors in the ordered phases similar to those at the second-order
transition point.Ma76 ; Amit84 ; Justin96  Most notably, the transverse order-
parameter correlation function diverges as $k^{-2}$ in the wave vector space
for ${\bm{k}}\\!\rightarrow\\!{\bm{0}}$, as can be shown rigorously based on
the Bogoliubov inequality,Hohenberg67 ; MW66  to make any long-range order
impossible below two dimensions at finite temperatures; the exponent $-2$
without the anomalous dimension is crucial in reaching the
conclusion.Hohenberg67  Moreover, the longitudinal correlation function also
diverges as $k^{d-4}$ in $d$ dimensions for $2\\!\leq d\\!<\\!4$ due to the
transverse correlations, as can be concluded based on various
techniques.ABDS99  Hence, the longitudinal order-parameter susceptibility,
which is predicted to recover a finite value in the ordered phases by the
mean-field analysis,Ma76 ; Amit84  remains divergent from the second-order
transition point down to zero temperature.Amit84 ; ABDS99  Indeed, this fact
has been found separately and independently for specific systems such as
isotropic ferromagnets HP40 ; Larkin67 ; SM70 ; BWW73 ; PP73  and single-
component BECs.NN75  Thus, the transverse correlations bring a qualitative
change in those of the longitudinal branch from massive to massless in the
terminology of relativistic quantum field theory.Weinberg95 ; ABDS99

The purpose of the present paper is to study how the infrared singularities
affect the NG bosons themselves, which have been regarded as well-defined
quasiparticles in the transverse channel having definite dispersion relations
with some integer exponents.NC76 ; Schafer01 ; WB11 ; WM12 ; Hidaka13  For
example, the Bogoliubov theory on the single-component BECs with a single
broken symmetry in the phase angle predicts that there should be a gapless
excitation with a linear dispersion relation at long wavelengths,Bogoliubov47
; OKSD05  in agreement with the general argument of assuming analyticity.WM12
However, the predicted linearity results from the assumption inherent in the
perturbative treatment that the longitudinal susceptibility is finite, which
no longer holds after the correlations through the transverse channel are
incorporated, as mentioned above. Hence, one may expect a substantial
modification of the long-wavelength dispersion relation from the linear one
with an infinite lifetime.

It should be noted that the topic on the BECs has already been studied at zero
temperature NN75 ; CCPS97  to conclude that the Bogoliubov mode no longer
exists because of the infrared singularities to be replaced by another gapless
branch with a linear dispersion relation. An implicit and crucial assumption
in reaching the result is again the analyticity of the inverse Green’s
function $\hat{G}^{-1}(\vec{k})$ at $\vec{k}\\!=\\!\vec{0}$, on which it is
expanded in $\vec{k}\\!=\\!({\bm{k}},\varepsilon)$ up to the second order,
especially up to $\varepsilon^{2}$ in terms of the frequency $\varepsilon$
beyond the standard linear order to find the new mode. However, the presumed
analyticity is not self-evident at all in view of the divergence of the
longitudinal susceptibility; hence, existence of the distinct phonon-like mode
at $T\\!=\\!0$ in the single-particle branch remains unestablished yet. This
is more so at finite temperatures where the Bose distribution function
$f(\varepsilon)$ with a pole at $\varepsilon\\!=\\!0$ is relevant, so that the
singular $\varepsilon\\!=\\!0$ branch should be treated separately from those
of $\varepsilon\\!\neq\\!0$, similarly as in the treatment of the critical
phenomena.Ma76 ; Amit84 ; Justin96

This paper is organized as follows. In Sec. 2, we give a general estimate on
the lifetime of a Nambu-Goldstone boson assuming a well-defined dispersion
relation initially and incorporating the transverse correlations
perturbatively. In Sec. 3, we present a more detailed calculation on the
excitation spectrum of a single-component Bose-Einstein condensate. In Sec. 4,
we provide concluding remarks. We use the units of $\hbar=k_{\rm B}=1$.

##  II Analytic Estimation of Lifetime

In this section, we present an analytic one-loop estimation of the lifetime of
an NG boson at finite temperatures, which is assumed to have a well-defined
dispersion relation $E_{k}\propto k^{n}$ with $n\geq 1$. Unlike previous
studies,HM65 ; Pitaevskii97 ; Giorgini98  we carry it out by retaining the
full spectral resolution in terms of the wave vector; see the last paragraph
of this section on this point.

The corresponding transverse Green’s function $G_{t}$ can be expressed in the
Lehmann representationAGD63 ; FW71  as

$\displaystyle
G_{t}(k,z)\\!=\\!\int_{-\infty}^{\infty}\frac{d\varepsilon}{2\pi}\frac{\rho_{t}(k,\varepsilon)}{z-\varepsilon}.$
(1)

where $\rho_{t}$ is the spectral function

$\displaystyle\rho_{t}(k,\varepsilon)=2\pi[w_{k}^{+}\delta(\varepsilon-
E_{k})-w_{k}^{-}\delta(\varepsilon+E_{k})],$ (2)

given in terms of the excitation energy $E_{k}>0$ and its spectral weight
$w_{k}^{\pm}\geq 0$. The weight $w_{k}^{-}$ becomes finite when the relevant
broken symmetry yields a connection between the particle and hole channels, as
in the cases of BEC and antiferromagnetism. It has been shown
generallyHohenberg67 ; MW66  that $G_{t}(k,z=0)\propto k^{-2}$ holds in the
limit of $k\rightarrow 0$, which for Eq. (1) with Eq. (2) can be expressed
alternatively as

$\displaystyle\frac{w_{k}^{\pm}}{E_{k}}\propto k^{-2}.$ (3)

We now consider the transverse one-loop process that is responsible for the
infrared singularities of the continuous-symmetry-breaking phases. It can be
expressed analytically in terms of $G_{t}(k,z)$ by

$\displaystyle\chi_{tt}(q,i\omega_{\ell})\equiv-\frac{T}{V}\\!\sum_{\bm{k}}\\!\sum_{n}G_{t}(|{\bm{k}}+{\bm{q}}|,i\varepsilon_{n}+i\omega_{\ell})G_{t}(k,i\varepsilon_{n}),$
(4)

where $T$ and $V$ are the temperature and volume, respectively, and
$\varepsilon_{n}\\!\equiv\\!2n\pi T$ and $\omega_{\ell}\\!\equiv\\!2\ell\pi T$
($n,\ell\\!=\\!0,\pm 1,\cdots$) are boson Matsubara frequencies
$(\hbar\\!=\\!k_{\rm B}\\!=\\!1)$. Substituting Eq. (1) and using the Bose
distribution function $f(\varepsilon)\equiv(e^{\varepsilon/T}-1)^{-1}$, we can
transform the sum over $n$ in Eq. (4) into a contour integral, collect
residues on the real axis, and perform the analytic continuation
$i\omega_{\ell}\rightarrow\omega+i0_{+}$.AGD63 ; FW71  We thereby obtain the
imaginary part of the retarded response function as

$\displaystyle{\rm Im}\chi_{tt}^{\rm R}(q,\omega)=$
$\displaystyle\,\frac{1}{2V}\sum_{\bm{k}}\int_{-\infty}^{\infty}\frac{d\varepsilon}{2\pi}\rho_{t}(k,\varepsilon)\rho_{t}(|{\bm{k}}+{\bm{q}}|,\varepsilon+\omega)$
$\displaystyle\,\times[f(\varepsilon+\omega)-f(\varepsilon)].$ (5)

Substitution of Eq. (2) into Eq. (5) yields

$\displaystyle\,{\rm Im}\chi_{tt}^{\rm R}(q,\omega)$ $\displaystyle=$
$\displaystyle\,-\frac{\pi}{V}\sum_{\bm{k}}\sum_{\sigma=\pm}\biggl{\\{}f(E_{k})w_{k}^{\sigma}w_{|{\bm{k}}+{\bm{q}}|}^{\sigma}\delta(\omega-
E_{|{\bm{k}}+{\bm{q}}|}+E_{k})$
$\displaystyle\,+\left[f(E_{k})+\frac{1}{2}\right]w_{k}^{\sigma}w_{|{\bm{k}}+{\bm{q}}|}^{-\sigma}\delta(\omega-
E_{|{\bm{k}}+{\bm{q}}|}-E_{k})$
$\displaystyle\,\,-\,(\omega\rightarrow-\omega)\biggr{\\}},$ (6)

where we have transformed $f(-E)\\!=\\!-1-f(E)$ for $E>0$ and subsequently
made a change of variables ${\bm{k}}+{\bm{q}}\\!=\\!-{\bm{k}}^{\prime}$ for
terms with the factor $f(E_{|{\bm{k}}+{\bm{q}}|})$ or
$f(E_{|{\bm{k}}+{\bm{q}}|})+\frac{1}{2}$.

Let us focus on the first term in the curly brackets of Eq. (6), where we can
estimate

$\displaystyle f(E_{k})w_{k}^{\sigma}\propto(T/E_{k})w_{k}^{\sigma}\propto
k^{-2}.$ (7a) for $k\rightarrow 0$ at $T>0$ based on Eq. (3). Note that it is
the pole $\varepsilon\\!=\\!0$ of $f(\varepsilon)$ that brings the
singularity. Moreover, integration of
$w_{|{\bm{k}}+{\bm{q}}|}^{\sigma}\delta(\omega-E_{|{\bm{k}}+{\bm{q}}|}+E_{k})$
over the angle $\theta\equiv\arccos({\bm{k}}\cdot{\bm{q}}/kq)$ yields another
factor $k^{-1}$ at $\omega=E_{q}$,
$\displaystyle\,\int_{0}^{\pi}d\theta\sin^{d-2}\theta\,w_{|{\bm{k}}+{\bm{q}}|}^{\sigma}\delta(\omega-
E_{|{\bm{k}}+{\bm{q}}|}+E_{k})$ $\displaystyle=$
$\displaystyle\,\frac{1}{kq}\int_{E_{|k-q|}}^{E_{k+q}}dE_{k_{1}}\frac{k_{1}\sin^{d-3}\theta}{dE_{k_{1}}/dk_{1}}w_{k_{1}}^{\sigma}\delta(\omega-
E_{k_{1}}+E_{k})$
$\displaystyle\stackrel{{\scriptstyle\omega=E_{q}}}{{\longrightarrow}}$
$\displaystyle\,\frac{1}{kq}\left[\frac{k_{1}\sin^{d-3}\theta}{dE_{k_{1}}/dk_{1}}w_{k_{1}}^{\sigma}\right]_{E_{k_{1}}=E_{q}+E_{k}}\propto\frac{1}{k},$
(7b)

where we have made a successive change of variables $\theta\rightarrow
k_{1}\equiv|{\bm{k}}+{\bm{q}}|\rightarrow E_{k_{1}}$. Since
$E_{|k-q|}<E_{q}+E_{k}\leq E_{k+q}$ holds for $E_{k}\propto k^{n}$ with $n\geq
1$, the factor in the square brackets in the third line is finite and
positive; see the original expression of the first line on the latter point.

The same transformation can be performed for the other terms with $f(E_{k})$
in the curly brackets of Eq. (6). They yield additional contributions for
which $E_{k_{1}}\\!=\\!E_{q}+E_{k}$ in the third line of Eq. (7b) is replaced
by $E_{k_{1}}\\!=\\!E_{q}-E_{k}$, $E_{k_{1}}\\!=\\!-E_{q}+E_{k}$, and
$E_{k_{1}}\\!=\\!-E_{q}-E_{k}$, respectively. However, the latter two
conditions cannot be satisfied in the limit of $k\rightarrow 0$, as seen from
the fact that they reduce to $E_{q}=-E_{q}$. Hence, the latter two terms
vanish for $k\rightarrow 0$, leaving the first two finite contributions that
are additive.

It follows from Eq. (7) that the integrand of Eq. (6) at $\omega=E_{q}$ with
$q>0$ behaves as $k^{(d-1)-3}$ for $k\rightarrow 0$. Hence, ${\rm
Im}\chi_{tt}^{\rm R}(q,\omega)$ at $\omega\\!=\\!E_{q}$ is concluded to
diverge for $d\leq 3$ at finite temperatures. We also note that, with the
emergence of three-point vertices in the ordered phases, $\chi_{tt}^{\rm
R}(q,\omega)$ contributes directly to the self-energy; see also Fig. 1(b) on
this point. Hence, the divergence of ${\rm Im}\chi_{tt}^{\rm R}(q,\omega)$ at
$\omega=E_{q}$ implies that the lifetime of the quasiparticles with the
dispersion relation $\omega=E_{q}$ is zero right on the dispersion curve, in
contradiction to the original assumption of Eq. (2). This fact implies that
the long-range transverse correlations in the continuous-symmetry-breaking
phases generally prevent any gapless excitation from having a well-defined
dispersion relation with an integer exponent below three dimensions at finite
temperatures. There can be exceptional cases where well-defined dispersion
relations are protected by some conservation laws. Otherwise, cancellation of
the infrared singularities to recover a sharp dispersion relation is unlikely
to happen; no such mechanism at finite temperatures has been known to date.

It should be noted that the first and second terms in the curly brackets of
Eq. (6), known as the Landau and Beliaev dampings, respectively, have
certainly been investigated theoretically.HM65 ; Pitaevskii97 ; Giorgini98
However, the damping rate in those finite-temperature studies is defined by a
sum of Eq. (6) over ${\bm{q}}$ for a given $\omega$ with no spectral
resolution in terms of ${\bm{q}}$. Hence, the result obtained above has been
overlooked. Whereas the results of Ref. HM65, ; Pitaevskii97, ; Giorgini98,
show the stability of the entire system against the thermal activation of
quasiparticle excitations, the present one reveals the presence of an
intrinsic lifetime in each excitation which is not describable perturbatively.
Combined with the vanishing of the longitudinal mass, it tells us that the
dispersion relation of the NG bosons may also be modified qualitatively at
long wavelengths from that of the mean-field analysis.

##  III Detailed consideration on BEC

In this section, we study in more detail how the $k^{-2}$ correlations modify
the spectral function of Eq. (2) based on a formalism where the longitudinal
susceptibility vanishes manifestly. Specifically, we consider an isotropic
homogenous Bose-Einstein condensate composed of identical particles with mass
$m$ and spin $0$ at $T>0$ interacting via a two-body potential $U({\bm{r}})$,
which can be described in terms of the scalar fields
$(\psi,\psi^{*})\\!\equiv\\!(\psi_{1},\psi_{2})$.

###  III.1 Vanishing of transverse and longitudinal masses

Let us introduce Green’s functions by
$G_{jj^{\prime}}(\vec{r},\vec{r}^{\,\prime})\\!\equiv\\!-\langle
T_{\tau}\psi_{j}(\vec{r})\psi_{j^{\prime}}(\vec{r}^{\,\prime})\rangle+\langle\psi_{j}(\vec{r})\rangle\langle\psi_{j^{\prime}}(\vec{r}^{\,\prime})\rangle$
with $j=1,2$ and $\vec{r}\\!\equiv\\!({\bm{r}},\tau)$, where $\tau$ lies in
$[0,T^{-1}]$ with the periodic boundary conditions.AGD63 ; FW71  The Fourier
transform of $\hat{G}\\!\equiv\\!(G_{jj^{\prime}})$ can be parametrized in
units of $m\\!=\\!1/2$ as

$\displaystyle\hat{G}(\vec{k})\\!=\\!\begin{bmatrix}-\Delta(\vec{k})&-i\varepsilon_{n}\\!-\\!k^{2}\\!-\\!\Sigma(-\vec{k})\\!+\\!\mu\\\
i\varepsilon_{n}\\!-\\!k^{2}\\!-\\!\Sigma(\vec{k})\\!+\\!\mu&-\Delta(\vec{k})\end{bmatrix}^{-1}$
(8)

where $\mu$ is the chemical potential and
$\vec{k}\equiv({\bm{k}},i\varepsilon_{n})$ in this expression.Kita19  The
anomalous self-energy satisfies $\Delta(-\vec{k})\\!=\\!\Delta(\vec{k})$, and
the Hugenholtz-Pines relationHP59  $\Sigma(\vec{0})-\mu=\Delta(\vec{0})$
ensures existence of a gapless excitation branch in the system. Indeed,
choosing the condensate wave function
$\Psi_{j}\\!\equiv\\!\langle\psi_{j}\rangle$ as real
($\Psi_{1}\\!=\\!\Psi_{2}\\!\equiv\\!\Psi$) and unitary-transforming

$\displaystyle\psi_{j}\\!=\\!\frac{1}{\sqrt{2}}\\{\phi_{l}+i(-1)^{j-1}\phi_{t}\\},$
(9)

we can decompose the field $\psi_{j}$ into the longitudinal and transverse
ones $(\phi_{l},\phi_{t})$ with $\langle\phi_{l}\rangle\\!=\\!\sqrt{2}\Psi$
and $\langle\phi_{t}\rangle\\!=\\!0$. Elements of the transformed inverse
Green’s function are given by

$\displaystyle G^{-1}_{ll}(\vec{k})=$ $\displaystyle\,-\xi_{ll}(\vec{k}),$
(10a) $\displaystyle G^{-1}_{tt}(\vec{k})=$
$\displaystyle\,-\xi_{tt}(\vec{k}),$ (10b) $\displaystyle
G^{-1}_{lt}(\vec{k})=$
$\displaystyle\,-G^{-1}_{tl}(\vec{k})=i\left\\{i\varepsilon_{n}-\frac{\Sigma(\vec{k})-\Sigma(-\vec{k})}{2}\right\\},$
(10c)

with

$\displaystyle\begin{array}[]{l}\xi_{ll}(\vec{k})\\\
\xi_{tt}(\vec{k})\end{array}\biggr{\\}}\equiv
k^{2}+\frac{\Sigma(\vec{k})+\Sigma(-\vec{k})}{2}-\mu\pm\Delta(\vec{k}),$ (13)

so that $\xi_{ll}(\vec{0})=2\Delta(\vec{0})$ and $\xi_{tt}(\vec{0})=0$
according to the Hugenholtz-Pines relation. Thus, the transverse branch is
gapless and should behave as $\xi_{tt}({\bm{k}},0)\propto k^{2}$ for
$k\rightarrow 0$.Hohenberg67  The transformation (9) also clarifies that the
Bogoliubov mode with the linear dispersion relation, which is non-analytic in
$k^{2}={\bm{k}}\cdot{\bm{k}}$, is realized by the mixing of the gapless
transverse branch $\xi_{tt}\propto k^{2}$ with the apparently gapped
longitudinal one $\xi_{ll}(\vec{0})\propto 2\Delta(\vec{0})$ through a finite
frequency $i\varepsilon_{n}$. However, the longitudinal branch also becomes
gapless as $\Delta({\bm{k}},0)\propto k^{d-4}$ for
${\bm{k}}\rightarrow{\bm{0}}$ due to the transverse correlations.ABDS99 ; NN75

Both the Hugenholtz-Pines relation and vanishing of the longitudinal mass
$\Delta(\vec{0})=0$ can be derived based on Goldstone’s theorem.GSW62 ;
Weinberg95  Specifically, the grand potential (or effective action) $\Gamma$
as a functional of the condensate wave function is invariant under the global
gauge transformation

$\displaystyle\delta\Psi_{j}(\vec{r})\\!\equiv\\!i\,\delta\theta\sum_{j^{\prime}}(\hat{\sigma}_{3})_{jj^{\prime}}\Psi_{j^{\prime}}(\vec{r})$

by the phase angle $\delta\theta$, where $\hat{\sigma}_{3}$ denotes the third
Pauli matrix and the $\vec{r}$ dependence originates from a coupling of
$\psi_{j}$ to an external source field $J_{j}(\vec{r})$.Weinberg95 ; Amit84 ;
ABDS99  This invariance of $\Gamma$ reads

$\displaystyle\sum_{jj^{\prime}}\int
d^{4}r\frac{\delta\Gamma}{\delta\Psi_{j^{\prime}}(\vec{r})}(\hat{\sigma}_{3})_{j^{\prime}j}\Psi_{j}(\vec{r})\\!=\\!0.$

Differentiating the equality $n$ times in terms of
$\Psi_{j_{i}}(\vec{r}_{j_{i}})$ ($i\\!=\\!1,2,\cdots,n$), switching off the
source field, and adopting the $\vec{k}$ representation, we obtain Amit84 ;
ABDS99 ; Kita19

$\displaystyle\,\sum_{i=1}^{n}\sum_{j_{i}^{\prime}}\Gamma^{(n)}_{j_{1}\cdots
j_{i}^{\prime}\cdots
j_{n}}(\vec{k}_{1},\cdots,\vec{k}_{n})(\hat{\sigma}_{3})_{j_{i}^{\prime}j_{i}}$
$\displaystyle\,+\sum_{j^{\prime}j}\Gamma^{(n+1)}_{j_{1}\cdots
j_{n}j^{\prime}}(\vec{k}_{1},\cdots,\vec{k}_{n},\vec{0})(\hat{\sigma}_{3})_{j^{\prime}j}\Psi_{j}=0,$
(14)

where $\Gamma^{(n)}_{j_{1}\cdots
j_{n}}(\vec{k}_{1},\cdots,\vec{k}_{n})\\!\propto\\!\delta_{\vec{k}_{1}+\cdots+\vec{k}_{n},\vec{0}}$
is the $n$-point vertex defined by the Fourier transform of
$\delta^{n}\Gamma/\delta\Psi_{j_{1}}(\vec{r}_{1})\cdots\delta\Psi_{j_{n}}(\vec{r}_{n})$.
The $n\\!=\\!1$ case of Eq. (14) is the identity known standardly as
Goldstone’s theorem Weinberg95  and also equivalent to the Hugenholtz-Pines
relation.HP59  The latter fact can be seen by substituting the stationarity
condition $\Gamma^{(1)}_{j}(\vec{0})=0$ of the grand potential and also the
connection $\hat{\Gamma}^{(2)}(-\vec{k},\vec{k})=-\hat{G}^{-1}(\vec{k})$
between $\hat{\Gamma}^{(2)}\equiv(\Gamma^{(2)}_{jj^{\prime}})$ and Eq.
(8).Amit84 ; Weinberg95  On the other hand, the case of $n=2$ in Eq. (14)
yields the vanishing of $\Delta(\vec{0})$. To see this, it is convenient to
transform Eq. (14) into the $(\phi_{l},\phi_{t})$ representation through
$\hat{\sigma}_{3}\rightarrow\hat{\sigma}_{2}$ and
$\Psi_{j}\rightarrow\delta_{jl}\sqrt{2}\Psi$, where the Hugenholtz-Pines
relation is given by

$\displaystyle\Gamma^{(2)}_{tt}(\vec{0},\vec{0})=0.$ (15)

The choice $(j_{1},j_{2})=(l,t)$ then yields

$\displaystyle\Gamma^{(2)}_{ll}(-\vec{k},\vec{k})-\Gamma^{(2)}_{tt}(-\vec{k},\vec{k})=\sqrt{2}\Gamma^{(3)}_{ltt}(-\vec{k},\vec{k},\vec{0})\Psi$

so that

$\displaystyle\Gamma^{(2)}_{ll}(\vec{0},\vec{0})=\sqrt{2}\Gamma^{(3)}_{ltt}(\vec{0},\vec{0},\vec{0})\Psi$
(16)

holds according to Eq. (15). On the other hand, the self-energies can be
expressed graphically as Fig. 1.Kita21  The graph (b) contains the transverse
contribution
$c_{1}\chi_{tt}(\vec{0})\Gamma^{(3)}_{ltt}(\vec{0},\vec{0},\vec{0})$ to
$\Gamma^{(2)}_{ll}(\vec{0},\vec{0})$ given in terms of some constant $c_{1}$,
whereas the other processes yield a finite value $c_{0}$. Hence,
$\Delta(\vec{0})=\Gamma^{(2)}_{ll}(\vec{0},\vec{0})/2$ can be written as

$\displaystyle\Delta(\vec{0})=\frac{c_{0}/2}{1-c_{1}\chi_{tt}(\vec{0})/\sqrt{2}\Psi},$
(17)

where $\chi_{tt}(\vec{0})$ diverges for $d<4$ at finite temperatures as seen
by inspecting Eq. (4) with $G_{tt}(k,0)\propto k^{-2}$ for $k\rightarrow 0$.
Thus, we conclude that $\Delta(\vec{0})=0$ holds for $d<4$ at $T>0$.Kita19
This argument is the finite-temperature extension of the one given by
Nepomnyashchiĭ and Nepomnyashchiĭ NN75  that found $\Delta(\vec{0})=0$ for
$d\leq 3$ at $T=0$.

Figure 1: (Color online) Self-energy diagrams. Small and medium circles
represent symmetrized bare and renormalized vertices, respectively.

###  III.2 Numerical study

The qualitative argument given above clarifies the crucial importance of the
identities of Eq. (14) for studying the excitations of BECs. We now perform a
more detailed calculation of its spectral function by adopting a contact
potential $U({\bm{r}})=U\delta({\bm{r}})$. The formalism developed previously
Kita21  so as to satisfy Eq. (14) yields the following expressions for the
self-energies in the weak-coupling regime

$\displaystyle\Sigma(\vec{k})-\mu=\Delta(\vec{k})=\Psi^{2}\tilde{U}(\vec{k})$
(18)

given in terms of the effective potential

$\displaystyle\tilde{U}(\vec{k})=\frac{U}{1-U[2\chi_{GG}(\vec{k})\\!+\\!2\chi_{FF}(\vec{k})\\!-\\!\chi_{GF}(\vec{k})\\!-\\!\chi_{FG}(\vec{k})]},$
(19)

where $\chi_{GF}(\vec{k})$, for example, is obtained from Eq. (4) by the
replacement $(G_{t},G_{t})\rightarrow(G,F)\equiv(G_{12},-G_{11})$ given in
terms of the upper elements of Eq. (8). Equation (19) is compatible with all
the qualitative results mentioned in the previous paragraph, as seen by noting
$G(k,0)\approx F(k,0)\approx\frac{1}{2}G_{tt}(k,0)$ for $k\rightarrow 0$. The
Bogoliubov theory is reproduced from Eq. (18) by omitting the correlation
effects as $\tilde{U}(\vec{k})\rightarrow U$. Indeed, substituting the
corresponding self-energy into Eq. (8), we obtain the Bogoliubov spectrum

$\displaystyle E_{\rm B}(k)=k\sqrt{k^{2}+2\Delta_{\rm B}},\hskip
28.45274pt\Delta_{\rm B}\equiv\Psi^{2}U,$ (20)

which has a linear dispersion relation for $k\xi\\!\lesssim\\!0.5$ where
$\xi\equiv\Delta_{\rm B}^{-\frac{1}{2}}$. Below we present results of the
first iteration based on Eqs. (8), (18) and (19) starting from the Bogoliubov
theory, in which $\chi_{GG}(\vec{k})$ etc. are estimated by the one-loop
approximation given by Eq. (4). We have adopted a weak-coupling interaction of
$\bar{n}U/T_{0}=0.1$ ($\bar{n}$: particle density), where we can also
approximate $\Psi^{2}\approx[1-(T/T_{0})^{3/2}]\bar{n}$ using the non-
interacting condensation temperature $T_{0}$.

Figure 2: (Color online) Normalized effective potential
$\tilde{U}(q,\omega)/U$ at $T/T_{0}=0.3$. Three-dimensional plots of (a) the
real part and (b) the imaginary part. Detailed plots of (c) the real part and
(d) the imaginary part as functions of $\omega/E_{\rm B}(q)$ at $q\xi=0.005$.

Figure 2 shows the normalized effective potential $\tilde{U}(q,\omega)/U$
calculated at $T/T_{0}=0.3$. In Fig. 2(a), a vertical fault with a deep steep
valley is seen to develop in the plateau of ${\rm Re}\,\tilde{U}$ along the
Bogoliubov dispersion $\omega\\!=\\!E_{\rm B}(q)$. The imaginary part ${\rm
Im}\,\tilde{U}$ of Fig. 2(b), on the other hand, is characterized by sharp
mountainous peaks along $\omega\\!=\\!E_{\rm B}(q)$ that rise and broaden
increasingly towards the origin. A closer look at the region of
$\omega\\!=\\!E_{\rm B}(q)$ reveals that both ${\rm Re}\,\tilde{U}$ and ${\rm
Im}\,\tilde{U}$ vanish along the Bogoliubov dispersion due to the divergence
of the denominator in Eq. (19), as seen in Fig. 2(c) and (d) calculated for
$q\xi=0.005$. The approaches to $0$ are logarithmic and non-analytic in three
dimensions. Thus, the first iteration predicts that the vanishing of the
longitudinal mass $\Delta(0,0)\\!=\\!0$ extends to the finite $(q,\omega)$
region along the Bogoliubov dispersion curve. Figure 3 shows the corresponding
spectral function $\rho(k,\varepsilon)\equiv-2{\rm Im}G^{\rm
R}(k,\varepsilon)$. The spectral weight vanishes along the Bogoliubov
dispersion to produce a double-peak structure for each $k$ as a function of
$\varepsilon$. Moreover, the higher peak shifts towards zero frequency as
$k\rightarrow 0$ due to $\tilde{U}(0,0)=0$. The fault in Fig. 2(a) decreases
in magnitude as $T\rightarrow 0$ to disappear completely at zero temperature,
but $\tilde{U}(0,0)$ still vanishes logarithmically (i.e., non-analytically)
even in the limit.NN75

Figure 3: (Color online) (a) Spectral function
$\rho(k,\varepsilon)\\!\equiv\\!-2{\rm Im}G^{\rm R}(k,\varepsilon)$ at
$T/T_{0}=0.3$. (b) Detailed plots of the normalized spectral function as a
function of $\varepsilon/E_{\rm B}(k)$ for three different values of
$k\xi=5.0\times 10^{-2}$, $5.0\times 10^{-3}$, $5.0\times 10^{-4}$.

The above results of the first iteration elucidate essential modifications to
the mean-field dispersion relation by the transverse correlations. Most
notably, the spectral weight has disappeared completely along the mean-field
dispersion curve with the $\delta$-function spectrum. They indicate presence
of an intrinsic lifetime in the excitations and a qualitative change in the
dispersion relation from the mean-field analytic one at long wavelengths due
to the vanishing of the longitudinal mass, as seen by setting $\Delta_{\rm
B}\rightarrow 0$ for $k\rightarrow 0$ in Eq. (20). Note that the longitudinal
mass vanishes even at zero temperature logarithmically, so that the latter
statement holds true even at zero temperature. An intrinsic lifetime in the
excitations may also persist down to $T\\!=\\!0$, as for the case of some
sorts of magnon systems.CZ09

Experiments on the Bogoliubov excitations of Eq. (20) have been performed for
trapped Bose-Einstein condensates.Vogels02 ; Ozeri02 ; OKSD05  The particle-
hole mixing of Eq. (4) predicted by the Bogoliubov theory has been observed
clearly at $k\xi=0.38$,Vogels02 ; OKSD05  and a reasonable agreement between
the Bogoliubov theory and measured excitation energies has been reported for
$k\xi\gtrsim 0.5$.Ozeri02 ; OKSD05  However, the observed excitation spectra
are accompanied by substantial broadenings of the order of the excitation
energies themselves,Vogels02 ; OKSD05  which may be caused partly by the
inhomogeneity of the trap potential. Hence, they cannot tell whether the
dispersion relation is really a sharp distinct one or accompanied by some
essential broadening and deviation from Eq. (20), especially at the low
excitation energies of $k\xi\lesssim 0.1$. It remains to be performed both
theoretically and experimentally to develop methods to detect them in high
resolutions and clarify the excitation spectrum definitely.

##  IV Concluding Remarks

We have studied how the long-range transverse correlations affect the
dispersion relation of Nambu-Goldstone bosons at finite temperatures by
incorporating the correlations perturbatively at the one-loop level. It is
found that the divergence of the longitudinal susceptibility (i.e., vanishing
of the longitudinal mass), which is characteristic of systems with broken
continuous symmetries, extends to the finite energy-momentum region as
vanishing of the spectral weight right on the assumed dispersion curve below
three dimensions at finite temperatures.

We finally point out the necessity of distinguishing various correlation
functions that have been focused in respective studies of Goldstone’s theorem.
Specifically, its first proof,GSW62 ; Weinberg95  given around Eq. (14), is
relevant to the two-point function of the field $\psi_{j}(\vec{r})$, whereas
the second proof GSW62 ; Weinberg95  concerns the correlations of
$\psi_{j}(\vec{r})$ with the field $j_{\alpha}^{0}(\vec{r})$ that constitutes
the conserved charge $Q_{\alpha}\equiv\int j_{\alpha}^{0}(\vec{r})\,d^{3}r$.
Finally, the dispersion relations of the NG bosons have been discussed in
terms of the two-point functions of $j_{\alpha}^{0}(\vec{r})$.NC76 ; Schafer01
; WB11 ; WM12 ; Hidaka13  The three kinds of correlation functions are
generally different and can have different poles. This may be realized clearly
in terms of the single-component BECs where $j_{\alpha}^{0}(\vec{r})$ is the
density operator $n(\vec{r})=\psi_{2}(\vec{r})\psi_{1}(\vec{r})$. The fact
that its four-point vertices have different $q$\- and $\omega$-limits,Kita21
contrary to the assumption made by Gavoret and Nozières,GN64  implies that the
system can embrace collective modes in the correlation functions of $(n,\psi)$
and $(n,n)$ that are distinct from the single-particle excitations in
$(\psi,\psi)$, similarly as in the case of Fermi liquids and superfluid Fermi
liquids.AGD63 ; SR83  The topic here is relevant to the poles of $(\psi,\psi)$
concerning the first proof.

## Acknowledgment

This work was supported by JSPS KAKENHI Grant Number JP20K03848.

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# Measurement of $B^{+}_{c}$ properties at CDF

T.S. Nigmanova111Speaker, on behalf of the CDF Collaboration, K.R. Gibsonb,
M.P. Hartzc, P.F. Shepardb aUniversity of Michigan, Ann Arbor, MI 48109, USA
bUniversity of Pittsburgh, Pittsburgh, PA 15260, USA  cUniversity of Toronto,
Toronto M5S, Canada

###### Abstract

The $B^{+}_{c}$ meson is composed of two heavy quarks of distinct flavor.
Measurements of its lifetime and production properties have been made based on
semileptonic $B^{+}_{c}\to J/\psi+l^{+}+X$ decays using data collected with
the CDF II detector corresponding to an integrated luminosity of 1 fb-1. The
$B^{+}_{c}$ average lifetime $c\tau$ is measured to be
142.5${}^{+15.8}_{-14.8}$(stat)$\pm 5.5(syst)$ $\mu m$. The measurements of
the ratio of the production cross section times branching ratio of
$B^{+}_{c}\to J/\psi\mu^{+}\nu$ relative to $B^{+}\to J/\psi K^{+}$ were done
for two $p_{T}(B)$ thresholds: for $p_{T}(B)>$ 4 GeV/$c$ as $0.295\pm
0.040~{}\mbox{(stat.)}^{+0.033}_{-0.026}~{}\mbox{(syst.)}\pm
0.036~{}(p_{T}~{}\mbox{spectrum})$ and for $p_{T}(B)>$ 6 GeV/$c$ as $0.227\pm
0.033~{}\mbox{(stat.)}^{+0.024}_{-0.017}~{}\mbox{(syst.)}\pm
0.014~{}(p_{T}~{}\mbox{spectrum})$.

##  I Introduction

The $B^{+}_{c}$ meson charge_conjugate  is composed of an anti-bottom quark
$\bar{b}$ and a charm quark $c$. The presence of two relatively heavy quarks
with different flavors is unique to the $B^{+}_{c}$ system and affects the
decay and production properties. The theoretically predicted lifetime kiselev
is about a factor of three times smaller than that of other B mesons. The
expected $B^{+}_{c}$ production cross section chang  is about 3 orders of
magnitude lower than the production cross section of the $B^{+}$ CDF-bplus-
Xsect . The first observation of the $B^{+}_{c}$ was made using data taken
with the CDF detector at the Fermilab Tevatron during run I bc-observation .
Precise mass measurements have been made by the CDF Collaboration using fully
reconstructed $B^{+}_{c}\rightarrow J/\psi\pi^{+}$ decays, where $J/\psi$
decays through $J/\psi\rightarrow\mu^{+}\mu^{-}$ bc-mass .

In this work we report preliminary measurements of the $B^{+}_{c}$ lifetime in
the semileptonic decay modes $J/\psi\mu^{+}X$ and $J/\psi eX$, and the
production cross section times branching ratio of the decay mode
$B^{+}_{c}\rightarrow J/\psi\mu^{+}\nu$ relative to the $B^{+}\rightarrow
J/\psi K^{+}$ decay. The results presented here are based on a data sample
with an integrated luminosity of 1 fb-1 at $\sqrt{s}$=1.96 TeV collected by
the CDF II detector.

##  II  The $B^{+}_{c}$ lifetime measurement concept

To measure the lifetime of the $B^{+}_{c}$, we construct a per event lifetime
that is defined using variables measured in the transverse plane. If all of
the decay products of the $B^{+}_{c}$ decay are identified, the lifetime $ct$
is the lifetime of the $B^{+}_{c}$ meson in its rest frame measured in units
of microns of light travel time. It is expressed as

$ct=\frac{mL_{xy}}{p_{T}}$ (1)

where $m$ is the mass of the $B^{+}_{c}$, $p_{T}$ is the momentum of the
$B^{+}_{c}$ in the plane transverse to the direction of the proton beam, and
$L_{xy}$ is the decay length of the $B^{+}_{c}$ projected along the transverse
momentum. The mass of the $B^{+}_{c}$ used in this measurement is $m=6.286$
GeV/c2 bc-mass . However, we do not measure all of the particles in the
semileptonic $B^{+}_{c}$ final state. Instead we must define a pseudo lifetime

${ct^{*}=\frac{mL_{xy}(J/\psi l^{+})}{p_{T}(J/\psi l^{+})}}$ (2)

where $L_{xy}$ and $p_{T}$ are evaluated using the $J/\psi+l^{+}$ system. We
can obtain the true $B^{+}_{c}$ lifetime by defining a factor $K$, where
$ct=Kct^{*}$. We evaluate the $K$ factor distribution, $H(K)$, for $B^{+}_{c}$
events using Monte Carlo simulation. We are then able to express the
distribution of $ct^{*}$ for $B^{+}_{c}\rightarrow J/\psi l^{+}X$ as

${F_{B_{c}}(ct^{*},\sigma)=\int
dKH(K)\frac{K}{c\tau}\theta(ct^{*})exp(-\frac{Kct^{*}}{c\tau})\otimes
G(\sigma)}$ (3)

where $c\tau$ is the average $B^{+}_{c}$ lifetime, $\sigma$ represents the
estimated error on the measurement of $ct^{*}$ for each event, and $G(\sigma)$
is defined as

${G(\sigma)=\frac{1}{\sqrt{2\pi}s\sigma}e^{-\frac{1}{2}(\frac{ct^{*}}{s\sigma})^{2}}}$
(4)

The measurement of the $B^{+}_{c}$ average lifetime is be carried out by
minimizing $-2Log(L)$, which is evaluated for the candidate $J/\psi+l$ events.
$L$ is the likelihood function for the $ct^{*}$ and $\sigma$ measured in
candidate events and includes $c\tau$ as a free parameter.

##  III  The $B^{+}_{c}$ cross section measurement concept

We measure

${\frac{\sigma(B^{+}_{c})BF(B^{+}_{c}\to
J/\psi\mu^{+}\nu)}{\sigma(B^{+})BF(B^{+}\to J/\psi
K^{+})}=\frac{N(B^{+}_{c})}{N(B^{+})}\times\epsilon_{rel}}$ (5)

The basic strategy of the $B^{+}_{c}$ cross section measurement is to
reconstruct the number of $B^{+}_{c}\to J/\psi\mu^{+}\nu$ relative to the
number of $B^{+}\to J/\psi K^{+}$ candidates, $N(B^{+}_{c})/N(B^{+})$,
determine the relative detector and reconstruction efficiency,
$\epsilon_{rel}$ = $\epsilon(B^{+})/\epsilon(B^{+}_{c})$, and use these to
determine the ratio of the final production cross section times branching
ratio.

##  IV  The event selection

The analysis presented here is based on the events recorded with a di-muon
trigger that is dedicated to $J/\psi\to\mu^{+}\mu^{-}$ decays. Both analysis
use the same $J/\psi\to\mu^{+}\mu^{-}$ sample. The muon pair is reconstructed
within a pseudo-rapidity range $|\eta|<$ 1.0. We select about 6.9$\times$106
$J/\psi$ candidates, measured with a mass resolution of approximately 12
MeV/$c^{2}$. The di-muon invariant mass distribution is shown in Fig. 1.

Figure 1: The $J/\psi\rightarrow\mu^{+}\mu^{-}$ invariant mass distribution.
Events within a mass range of $\pm$50 MeV/$c^{2}$ around the central $J/\psi$
mass value were used for both the lifetime and cross section analysis.

In addition to the di-muons from the $J/\psi$ decay, we require a third track
that is matched to the same vertex as the $J/\psi$. This third track can be
from any of three samples of interest: $B^{+}_{c}\to J/\psi l^{+}$X decays,
$B^{+}\to J/\psi K^{+}$ decays, or just $J/\psi+track$ decays. The last sample
represents sources of backgrounds to the $B^{+}_{c}$ semileptonic decay.

##  V  The $B^{+}_{c}$ backgrounds overview

###  V.1  Common for both analysis

Both the lifetime and cross section analysis have some common sources of
background: misidentified $J/\psi$, misidentified third muons, and $b\bar{b}$
backgrounds. The misidentified $J/\psi$ background occurs when one of the
muons is actually a mis-reconstructed hadron or muon from other sources that
produce a mass consistent with that of the $J/\psi$. The misidentified third
muon background can arise from the following sources. The $J/\psi$ in the
$J/\psi+track$ system is highly populated by non-$B^{+}_{c}$ sources. The
third track associated with the $J/\psi$ could be a $\pi^{+}$ or $K^{+}$ that
can either decay-in-flight or punch-through the calorimeter and the steel
absorber and produce the muon signature. The $b\bar{b}$ events represent cases
when a $J/\psi$ is produced from one $b$ jet and the third muon originates
from the other $b$ in same event.

Figure 2 shows the misidentified muon rates for $\pi^{\pm}$, $K^{\pm}$, and
p($\bar{p}$) and the pseudo-proper decay length ct∗ for misidentified third
muons.

Figure 2: The misidentified muon rates from $\pi^{\pm}$, $K^{\pm}$, and
p($\bar{p}$) as a function of hadron $p_{T}$ (left), and the pseudo-proper
decay length ct∗ for misidentified third muons (right).

###  V.2  Specific for each analysis

There is an additional background for the cross section measurement. The
selected $B^{+}_{c}\to J/\psi\mu^{+}X$ sample contains contributions from
other $B^{+}_{c}$ decays with a tri-muon in the final state. For example, a
$B^{+}_{c}$ can decay into $\psi(2S)\mu^{+}\nu$ followed by
$\psi(2S)\rightarrow J/\psi X$.

The following backgrounds are specific to the lifetime analysis: misidentified
e±, residual conversions, and prompt $J/\psi$. The misidentified e± can arise
from cases when a $\pi^{\pm}$, K±, or $\bar{p}$ from the $J/\psi+track$ system
satisfies the e± likelihood function based on the calorimeter responses. The
residual conversions are e± from $\gamma$-conversion or $\pi^{o}$ Dalitz
decays. The prompt $J/\psi$ are additional $J/\psi l^{\pm}$ candidates where
the $J/\psi$ originates from prompt non-$B^{+}_{c}$ sources. Figure 3
illustrates the $e^{+}e^{-}$ veto efficiencies and the pseudo-proper decay
length ct∗ distribution for the $J/\psi$+Conversion e background sample.

Figure 3: The $e^{+}e^{-}$ veto efficiencies as a function of electron $p_{T}$
(left), and the pseudo-proper decay length ct∗ distribution for the
$J/\psi$+Conversion e background sample (right).

##  VI  The $B^{+}_{c}$ lifetime results

###  VI.1  Lifetime systematic uncertainties

The systematic uncertainties in the $B^{+}_{c}$ lifetime measurement originate
from uncertainties in our models for background and signal events. Some of the
largest systematic uncertainties are summarized below:

  * •

Resolution function - choice of model for detector resolution: $\pm$3.8 $\mu
m$

  * •

Pythia model for $b\bar{b}$ background - relative contribution of QCD
processes: $\pm$2.4 $\mu m$

  * •

Vertex detector alignment - uncertainties in the positions of silicon
detectors: $\pm$2.0 $\mu m$

  * •

$e^{+}e^{-}$ veto efficiency - uncertainties related to modeling $e^{+}e^{-}$
veto efficiencies: $\pm$1.5 $\mu m$

  * •

$B_{c}$ spectrum - variations of the K factor distribution due to variations
in the $B_{c}$ production spectrum: $\pm$1.3 $\mu m$

We add the individual uncertainties in quadrature to obtain a total
uncertainty of $\pm$5.5 $\mu m$.

###  VI.2  Lifetime results

We fit the $ct^{*}$ distributions for signal candidates in the $J/\psi\mu^{+}$
and $J/\psi e^{+}$ channels separately using likelihood functions based on our
models for signal and background events. The fitted data is shown in Fig. 4
for $B^{+}_{c}\to J/\psi\mu^{+}X$ and for $B^{+}_{c}\to J/\psi e^{+}X$ decays.

Figure 4: The pseudo-proper decay length ct∗ distributions for $B^{+}_{c}\to
J/\psi\mu^{+}X$ decays (left) and for $B^{+}_{c}\to J/\psi e^{+}X$ decays
(right) with their the background models superimposed.

The $B^{+}_{c}$ lifetime is found to be 179.1${}^{+32.6}_{-27.2}$(stat) $\mu
m$ for the $J/\psi\mu^{+}$ final state and 121.7${}^{+18.0}_{-16.3}$ (stat)
$\mu m$ for the $J/\psi e^{+}$ decay mode, respectively. We performed the
simultaneous fit of both samples and found an average $B^{+}_{c}$ lifetime of
142.5${}^{+15.8}_{-14.8}(stat)\pm 5.5(syst)$ $\mu m$. Figure 5 shows our
$B^{+}_{c}$ average lifetime comparison with other measurements.

Figure 5: World average of $B^{+}_{c}$ lifetime, which includes the CDF Run I
$B_{c}$ lifetime, the most recent D0 Run II result, and the result presented
in this paper. The lifetimes are weighted by the total variance of the
individual measurements in the average.

##  VII  The $B^{+}_{c}$ relative cross section

In order to measure the $B^{+}_{c}$ relative cross section we need to find the
numbers of $B_{c}^{+}$ and $B^{+}$, $N(B^{+}_{c}$) and $N(B^{+}$), and
determine the relative efficiency, $\epsilon_{rel}$ =
$\epsilon(B^{+})/\epsilon(B^{+}_{c})$. We select 229 (214) $B^{+}_{c}$
candidates with the requirement $p_{T}(B^{+}_{c})>$ 4 (6) GeV/$c$,
respectively. The number of $B^{+}_{c}$ signal events after backgrounds
subtraction is presented in the following subsection. The number of $B^{+}\to
J/\psi K^{+}$ signal events is found to be 2333 $\pm$ 55 (2299 $\pm$ 53) for
$p_{T}(B^{+})>$ 4 (6) GeV/$c$, respectively. The combinatoric and $B^{+}\to
J/\psi\pi^{+}$ contributions are subtracted.

###  VII.1  The $B^{+}_{c}$ backgrounds and excess

The backgrounds and the resulting number of signal events for the
$B^{+}_{c}\to J/\psi\mu^{+}\nu$ decays are summarized in Table 1.

Table 1: Observed $N(B^{+}_{c}\to J/\psi\mu^{+}\nu$) for the $p_{T}(J/\psi\mu)>$ 4 GeV/c (6 GeV/c) threshold. |  $p_{T}(B)>$ 4 GeV/$c$ |  $p_{T}(B)>$ 6 GeV/$c$
---|---|---
$N(B^{+}_{c}$) observed | 229$\pm$15.1(stat) | 214$\pm$14.6(stat)
Misidentified $J/\psi$ | 21.5$\pm$3.3(stat) | 20.5$\pm$3.2(stat)
Misid. third muon | 55.8$\pm$2.0(stat) | 53.6$\pm$1.9(stat)
Doubly misid. | -8.8$\pm$0.4(stat) | -7.5$\pm$0.3(stat)
$b\bar{b}$ background | 37.7$\pm$7.3(st+sys) | 35.4$\pm$7.0(st+sys)
Other $B^{+}_{c}$ modes | 5.2$\pm$0.5(stat) | 4.8$\pm$0.4(stat)
Total background | 111.4$\pm$8.3(stat) | 106.9$\pm$8.0(stat)
$B^{+}_{c}$ signal | 117.6$\pm$17.2(stat) | 107.1$\pm$16.7(stat)

The background identified as “Doubly misidentified” in Table 1 represents the
subsample of misidentified $J/\psi$ and misidentified third muons that needs
to be subtracted only once to avoid double counting.

Figure 6 shows the invariant mass distribution of the $B^{+}_{c}\to
J/\psi\mu^{+}X$ data events with the experimental backgrounds and a Monte
Carlo simulation of the signal sample superimposed (left), and with the
background subtracted (right).

Figure 6: The invariant mass distribution of the $B^{+}_{c}\to J/\psi\mu^{+}X$
data events with the experimental backgrounds and a Monte Carlo simulation of
the signal sample superimposed (left), and with the background subtracted
applied (right).

###  VII.2  The relative efficiency $\epsilon_{rel}$

In order to determine the relative efficiency, we simulate $B^{+}\to J/\psi
K^{+}$, $B^{+}_{c}\to J/\psi\mu^{+}\nu$, and $B^{*+}_{c}\to B^{+}_{c}\gamma$
decays. As a description of the $\eta-p_{T}$ spectrum for $B^{+}_{c}$ we use
the most recent theoretical work by Chang et al. chang . For the $B^{+}$ we
used the spectrum from Ref. FONLL , which is found to be in good agreement
with CDF measurements. All Monte Carlo simulation events are passed through
the full detector and trigger simulation. The Monte Carlo simulation samples
were processed in the same way as for the data. The efficiencies
$\epsilon_{B^{+}_{c}}$ and $\epsilon_{B^{+}}$ for $p_{T}(B)>$ 4 (6) GeV/$c$,
along with the relative efficiency, are presented in Table 2.

Table 2: Efficiencies for $B^{+}_{c}$ and $B^{+}$ for $p_{T}(B)>$ 4 (6) GeV/$c$. Efficiency |  $p_{T}(B)>$ 4 GeV/$c$ |  $p_{T}(B)>$ 6 GeV/$c$
---|---|---
$\epsilon_{B^{+}_{c}}$ (%) | $0.0551\pm 0.0010$ | $0.1232\pm 0.0024$
$\epsilon_{B^{+}}$ (%) | $0.3231\pm 0.0022$ | $0.6005\pm 0.0042$
$\epsilon_{rel}$ |  $5.867\pm 0.068$ (stat) |  $4.873\pm 0.060$ (stat)

The $p_{T}$ spectra for data and Monte Carlo simulation are shown in Fig. 7.

Figure 7: The comparison of the $p_{T}$ spectra of data versus Monte Carlo
simulation for $B^{+}_{c}\to J/\psi\mu^{+}\nu$ (left), and for $B^{+}\to
J/\psi K^{+}$ decays (right).

###  VII.3  Cross section systematic uncertainties

We divide the systematic uncertainties into two categories: uncertainties on
the number of $B^{+}_{c}$ signal events and uncertainties in the determination
of the relative efficiency.

####  VII.3.1  $B^{+}_{c}$ background uncertainty

The systematic uncertainty considered in the determination of $N(B^{+}_{c})$
arise from events which do not originate from $B^{+}_{c}$ decays. All
backgrounds are assigned a systematic uncertainty except the misidentified
$J/\psi$ background. Since the estimation of this background is determined
directly from the sidebands of the $J/\psi$, we do not know of any source of
systematic error that should be included.

The largest source of uncertainty in the misidentified third muon calculation
is due to the poor knowledge of the proton fraction in the $J/\psi+track$
sample. The particle identification method, dE/dx, does not allow us to
separate protons from kaons in our kinematic region. Consequently, we measure
the proton fraction in other momentum ranges using the time-of-flight (TOF)
particle identification combined with dE/dx information and then extrapolate
the fraction to our momentum range according to measured trends in Monte Carlo
simulation.

The other $N(B^{+}_{c})$ systematic uncertainty arises from a poor knowledge
of non-exclusive $B^{+}_{c}\to J/\psi\mu^{+}X$ branching ratios and is
estimated by varying the branching ratios of eleven $B^{+}_{c}$ decay modes
that may contribute to the sample of tri-muon events. In order to assign a
systematic uncertainty we double and halve the branching ratios of the non-
exclusive decays with respect to the rate of $B^{+}_{c}\to J/\psi\mu^{+}\nu$.
We choose the larger variation in either direction for the systematic
uncertainty.

Table 3 summarizes all of the $B^{+}_{c}$ background systematic uncertainties
assigned.

Table 3: Systematic uncertainties on the number of $B^{+}_{c}\to J/\psi\mu^{+}\nu$ events for different $p_{T}(B)$ thresholds. The total uncertainty is calculated by adding all the individual uncertainties in quadrature. $N(B^{+}_{c})$ uncertainties |  $p_{T}(B)>$ 4 GeV/$c$ |  $p_{T}(B)>$ 6 GeV/$c$
---|---|---
Misid. third muon |  $\pm 5.7$ (sys) |  $\pm 5.5$ (sys)
Doubly misid. |  $\pm 0.9$ (sys) |  $\pm 0.8$ (sys)
Other $B^{+}_{c}$ decays |  ${}^{+6.0}_{-2.8}$ (sys) |  ${}^{+5.6}_{-2.5}$ (sys)
Total |  ${}^{+8.3}_{-6.4}$ (sys) |  ${}^{+7.9}_{-6.1}$ (sys)

####  VII.3.2  Relative efficiency systematic uncertainty

We consider the systematic uncertainty in the prediction of the relative
efficiency due to the measured statistical uncertainty of the $B^{+}_{c}$
lifetime, knowledge of the production spectra for $B^{+}_{c}$ and $B^{+}$, and
differences between $K$ and $\mu$ triggering rates at the first level of the
CDF trigger system, the extremely fast trigger, XFT.

The relative efficiency systematic uncertainty due to $B^{+}_{c}$ lifetime
uncertainty is estimated by varying the lifetime by $\pm$14 $\mu$m of its
default value. This variation represents approximately one standard deviation
in the average lifetime result.

The relative efficiency systematic uncertainty due to the knowledge of
$B^{+}_{c}$ $p_{T}$ spectrum is fully based on the theoretically predicted
$p_{T}$ spectra from work in Ref. chang . We consider the variations between:

  * •

doubling the $q\bar{q}$ contribution relative to the nominal approach;

  * •

the pure gluon fusion model, called the fixed flavor number ( FFN) model, and
the more complete $gg+g\bar{b}+gc$ model, known as the general-mass variable-
flavor-number (GMVFN) model;

  * •

the combined $B^{+}_{c}+B^{*+}_{c}$ spectrum and a pure $B^{+}_{c}$ spectrum.

The systematic uncertainty due to knowledge of the $B^{+}$ $p_{T}$ spectrum is
estimated by varying the Monte Carlo simulated spectrum below 10 GeV/$c$ to
bring it into agreement with the data (see the right plot in Fig. 7. The
difference between the nominal and recalculated relative efficiency is
assigned as the uncertainty due to $B^{+}$ $p_{T}$ spectrum.

Another source of systematic uncertainty that we consider is the different XFT
efficiencies of kaons and muons that exist in the data and are not modeled in
the simulation.

The total $\epsilon_{rel}$ systematic uncertainty is summarized in Table 4.

Table 4: Systematic uncertainty assigned to $\epsilon_{rel}$ for different $p_{T}(B)$ thresholds. The numbers 0.720 and 0.298 in the “Total” represent the uncertainties due to of $B^{+}_{c}$ spectrum. $\epsilon_{rel}$ uncertainties |  $p_{T}(B)>$ 4 GeV/$c$ |  $p_{T}(B)>$ 6 GeV/$c$
---|---|---
$B^{+}_{c}$ lifetime | ${}^{+0.393}_{-0.223}$ | ${}^{+0.354}_{-0.160}$
$B^{+}_{c}$ spectrum |  $\pm$ 0.720 |  $\pm$ 0.298
$B^{+}$ spectrum |  $\pm$ 0.340 |  $\pm$ 0.161
XFT systematics |  $\pm$ 0.192 |  $\pm$ 0.160
Total |  ${}^{+0.554}_{-0.450}\pm$0.720 |  ${}^{+0.420}_{-0.278}\pm$0.298

##  VIII  The $B^{+}_{c}$ relative cross section results

We have performed a measurement of the relative production cross section of
$B^{+}_{c}\to J/\psi\mu^{+}\nu$ in inclusive $J/\psi$ data with an integrated
luminosity of 1 fb-1. We have identified a sample of 229 (214) events with an
estimated background from all sources of 111$\pm$8 (107$\pm$8) events for
$p_{T}(J/\psi\mu^{+})>$ 4 (6) GeV/$c$, respectively. The final numbers used in
the cross section measurement, including systematic uncertainties, are given
in Table 5.

Table 5: Final numbers used in the calculation of the relative $B^{+}_{c}\to J/\psi\mu^{+}\nu$ production cross section times the branching ratio for two different $p_{T}(B)$ thresholds. Final values |  $p_{T}(B)>$ 4 GeV/$c$
---|---
$N(B^{+}_{c}$) | 117.6$\pm$17.2 (stat) ${}^{+8.3}_{-6.4}$(sys)
$N(B^{+}$) | 2333$\pm$ 55 (stat)
$\epsilon_{rel}$ | 5.867$\pm$0.068 (stat) ${}^{+0.554}_{-0.450}$(sys)
|  $\pm$0.720 ($B^{+}_{c}$ spectrum)
Final values |  $p_{T}(B)>$ 6 GeV/$c$
$N(B^{+}_{c}$) | 107.2$\pm$16.7 (stat) ${}^{+7.9}_{-6.1}$ (sys)
$N(B^{+}$) | 2299$\pm$53 (stat)
$\epsilon_{rel}$ | 4.872$\pm$0.060 (stat) ${}^{+0.420}_{-0.278}$ (sys)
|  $\pm$0.298 ($B^{+}_{c}$ spectrum)

We give the result for the ratio $\frac{\sigma(B^{+}_{c})BF(B^{+}_{c}\to
J/\psi\mu^{+}\nu)}{\sigma(B^{+})BF(B^{+}\to J/\psi K^{+})}$ with $p_{T}(B)>$ 4
GeV/$c$ thresholds as

$0.295\pm 0.040~{}\mbox{(stat.)}^{+0.033}_{-0.026}~{}\mbox{(syst.)}\pm
0.036~{}(p_{T}~{}\mbox{spec})$

and for $p_{T}(B)>$ 6 GeV/$c$ as

$0.227\pm 0.033~{}\mbox{(stat.)}^{+0.024}_{-0.017}~{}\mbox{(syst.)}\pm
0.014~{}(p_{T}~{}\mbox{spec})$.

Of the two results, the measurement with the $p_{T}(B)>$ 6 GeV/$c$ threshold
has the lower systematic error. Below 6 GeV/$c$ there is uncertainty in the
$B^{+}$ efficiency that appears to introduce a significant systematic
discrepancy between the simulated spectrum and the spectrum as determined from
the data.

Using theoretical assumptions and independent measurements, we are then able
to calculate the total $B^{+}_{c}$ cross section. Using the measured
quantities $BR(B^{+}\to J/\psi K^{+})$ = $(1.007\pm 0.035)\times 10^{-3}$
Ref:PDG  and $\sigma(B^{+})=2.78\pm 0.24~{}\mu$b for $p_{T}(B^{+})>$ 6 GeV/$c$
CDF-bplus-Xsect , we calculate

$\displaystyle\sigma(B^{+}_{c})*BR(B^{+}_{c}\to J/\psi\mu^{+}\nu)=0.64\pm
0.20~{}\mbox{nb}$

for $p_{T}(B^{+}_{c})>$ 6 GeV/$c$. Assuming that the branching ratio
$BR(B^{+}_{c}\to J/\psi\mu^{+}\nu)=2.07\times 10^{-2}$ Ref:BcIvanov , we find
the total $B^{+}_{c}$ cross section to be

$\displaystyle\sigma(B^{+}_{c})=31\pm 10~{}\mbox{nb}$

##  IX  Conclusions

We have performed measurements of the $B^{+}_{c}$ lifetime and production
properties based on semileptonic $B^{+}_{c}\to J/\psi+l^{+}+X$ decays using
data from $p\bar{p}$ collisions collected with the CDF II detector
corresponding to an integrated luminosity of 1 fb-1 at $\sqrt{s}$=1.96 TeV.

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