# Ward identities, $\bm{B\to V}$ transition form factors and applications

M. Ali Paracha  Laboratório de Física Teórica e Computacional, Universidade
Cruzeiro do Sul, 01506-000 São Paulo SP, Brazil; Department of Physics, School
of Natural Sciences, National University of Science and Technology, Islamabad,
Pakistan<EMAIL_ADDRESS>Bruno El-Bennich  Laboratório de Física
Teórica e Computacional, Universidade Cruzeiro do Sul, 01506-000 São Paulo,
SP, Brazil; Instituto de Física Teórica, Universidade Estadual Paulista,
01140-070 São Paulo, SP, Brazil<EMAIL_ADDRESS>M. Jamil
Aslam  Department of Physics, Quaid-i-Azam University, Islamabad 45320,
Pakistan<EMAIL_ADDRESS>Ishtiaq Ahmed  Laboratório de Física Teórica e
Computacional, Universidade Cruzeiro do Sul, 01506-000 São Paulo, SP, Brazil;
National Centre for Physics, Quaid-i-Azam University Campus, Islamabad 45320,
Pakistan<EMAIL_ADDRESS>
###### Abstract

Long distance effects are studied in the rare exclusive semileptonic
$B_{(d,s)}\to V\ell^{+}\ell^{-}$ decays, where $V$ denotes a $K^{*}$ or $\phi$
meson. The form factors, which describe the meson transition amplitudes in the
effective Hamiltonian approach, are calculated by means of Ward identities,
experimental constraints and extrapolated within a general vector meson
dominance framework. These form factors are then compared to the ones obtained
in Lattice QCD simulations, with Light Cone Sum Rules and a Dyson-Schwinger
equation approach. Additionally, the $B_{d}\to K^{*}\ell^{+}\ell^{-}$ and
$B_{s}\to\phi\ell^{+}\ell^{-}$ branching ratios are computed and the
differential branching fractions are given as a function of the squared-
momentum transfer.

##  1 Introduction

Rare $B$ meson decays have been a subject of great interest for the past two
decades. These decays not only provide a stringent tests of the Standard Model
(SM), but also serve as a tool to extract physics beyond the SM in the flavor
sector. We remind that in the SM rare decays do not occur at tree order but
proceed at loop level through the Glashow-Iliopoulos-Maiani (GIM) mechanism
[1]. Attempts to unravel the imprints of physics in and beyond the SM in the
flavor sector imply the study of inclusive [2] and exclusive [3] $B$-meson
decays. From an experimental point of view, exclusive decays are easier to
observe than inclusive decays. However, theoretically, exclusive decays are
more challenging due to the uncertainties in the calculation of hadronic
transition form factors which, so far, are model-dependent quantities when
computed for the entire range of squared momentum transfer $q^{2}$. Different
frameworks have been used to compute transition form factors, namely within
the constituent quark model (CQM), light cone sum rules (LCSR), QCD sum rules
(QCDSR), Dyson-Schwinger equation (DSE) approaches and lattice QCD (LQCD)
amongst others. These form factors are the ingredients of physical
observables, such as branching ratios and different asymmetries related to
final particle states, especially in $B\to K(K^{\ast})\ell^{+}\ell^{-}$ [4, 5,
6], $B\to\gamma\ell^{+}\ell^{-}$ [7] decays.

Nowadays, rare decays, and in particular the decays $B_{d}\to
K^{\ast}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$, are of particular
interest as they are the object of intense experimental scrutiny by the LHCb
Collaboration [8]. Updated values of experimental branching ratios for these
decays have been published [9] and their numerical values are given by,

$\displaystyle\mathrm{Br}(B_{d}\to K^{\ast}\mu^{+}\mu^{-})$ $\displaystyle=$
$\displaystyle(1.06\pm 0.09)\times 10^{-6},$ (1)
$\displaystyle\mathrm{Br}(B_{s}\to\phi\mu^{+}\mu^{-})$ $\displaystyle=$
$\displaystyle(7.6\pm 1.5)\times 10^{-7}\ .$ (2)

Here, we study the transition form factors, $B_{d}\to K^{\ast}$ and
$B_{s}\to\phi$, using Ward identities to compute their values at $q^{2}=0$ and
then extrapolate them in a general vector meson dominance model to larger
$q^{2}$ values. We compare these form factors with those obtained in Refs. [4,
10, 11]. In addition, for the decay $B_{d}\to K^{\ast}\mu^{+}\mu^{-}$, we also
compare the transition form factors with the results of a DSE model of QCD
[12].

The vector dominance approach has been successfully applied to different
heavy-to-light semileptonic decays, such as $B\to\rho$ [13], $B\to\gamma$
[14], $B\to K_{1}$[15] and $B_{c}\to D_{s}^{\ast}$ [16]. The main purpose here
is to apply the same technique to $B\to K^{*}(\phi)$ decays, where we relate
the form various factors of the transition in a model-independent way via Ward
identities. This allows us to make a clear separation between pole and non-
pole type contributions [17, 18]. The residue of the pole is then determined
in a self-consistent manner in terms of $\xi_{\perp}(0)$ or $g_{+}(0)$ which
in turn defines the couplings of vector and axialvector $B$ mesons in the
$B\to V$ channel. The form factors in the $q^{2}\to 0$ limit are expressed in
terms of a universal function, $\xi_{\perp}(0)\equiv g_{+}(0)$, that is
introduced in Large Energy Effective Theory (LEET) [17, 18] of heavy-to-light
form factors, and the masses of the particles. The value of $g_{+}(0)$ will be
extracted from the experimental values of the branching ratios of
corresponding $B_{d,s}\to V\gamma$ decays. Eventually, using the above form
factors, we compute branching ratios and compare them with their experimental
values in Eqs. (1) and (2) as well as with the predictions of the LCSR,
Lattice QCD and DSE approaches.

##  2 Theoretical Framework

###  2.1 Effective Hamiltonian

At the quark level, the decays $B\to V\ell^{+}\ell^{-}$ ($V=K^{\ast},\phi$)
are governed by the transition $b\to s\ell^{+}\ell^{-}$. The phenomenology of
the rare decays is commonly described by the heavy-quark effective
Hamiltonian,

$\displaystyle
H_{\mathrm{eff}}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V^{\ast}_{ts}\,\sum\limits_{i=1}^{10}C_{i}({\mu})O_{i}({\mu})\
,$ (3)

where the Wilson coefficients $C_{i}(\mu)$ encode contributions from energy
scales above the renormalization point $\mu$; due to asymptotic freedom
property in QCD, these can be computed in perturbation theory, for instance in
the naive dimensional regularization scheme [19]. The local quark operators,
$O_{i}$, describe via their matrix elements the strong and electromagnetic
contributions from all scales below $\mu$ and thus require a nonperturbative
approach to their determination. Additional SM/NP operators can be inserted in
the above effective Hamiltonian. The operators which describe the decay $B\to
V\ell^{+}\ell^{-}$ in the SM are given by,

$\displaystyle O_{7}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}m_{b}\left(\bar{s}\sigma_{\mu\nu}P_{R}b\right)F^{\mu\nu},\,$
$\displaystyle O_{9}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{l}\gamma^{\mu}l),\,$
(4) $\displaystyle O_{10}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma_{\mu}P_{L}b)(\bar{l}\gamma^{\mu}\gamma_{5}l),$

where $P_{L,R}=\left(1\mp\gamma_{5}\right)/2$ are left- and right-handed
projectors. The effective Hamiltonian given in Eq. (3) yields the following
matrix element for the decay $B\to V\ell^{+}\ell^{-}$,

$\displaystyle\mathcal{M}(B\rightarrow V\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}G_{F}}{2\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}\bigg{[}\langle
V(k,\varepsilon)|\overline{s}\gamma^{\mu}(1-\gamma^{5})b|B(p)\rangle\left\\{C_{9}^{\mathrm{eff}}(\overline{l}\gamma^{\mu}l)+C_{10}(\overline{l}\gamma^{\mu}\gamma^{5}l)\right\\}$
(5) $\displaystyle-2\,C_{7}^{\mathrm{eff}}m_{b}\langle
V(k,\varepsilon)|\overline{s}\,i\sigma_{\mu\nu}\tfrac{q^{\nu}}{q^{2}}(1+\gamma^{5})b|B(p)\rangle(\overline{l}\gamma^{\mu}l)\bigg{]},$

where $\alpha_{\mathrm{em}}$ is the electromagnetic coupling constant while
$p$ and $k$ are the momenta of $B$ and $V$ mesons, respectively. Moreover,
$\varepsilon$ is the polarization vector of final state vector meson and
$q^{2}=(p-k)^{2}$ is the squared momentum transfer. The explicit expressions
of Wilson Coefficients $C_{7,9}^{\mathrm{eff}}$ and $C_{10}$ are given in Ref.
[20] and we do not reproduce them here.

###  2.2 Form Factors and Ward Identities

Both decays, $B_{d}\to K^{\ast}\ell^{+}\ell^{-}$ and
$B_{s}\to\phi\ell^{+}\ell^{-}$, involve the hadronic matrix elements of the
quark operators introduced in Eq. (4) between the $B$ and $V$
($V=K^{\ast},\phi$) meson states. These matrix elements can be parameterized
in terms of transition form factors which are functions of the square of
momentum transfer, $q^{2}$. The different matrix elements can be written as

$\displaystyle\left\langle
V(k,\varepsilon)\left|\bar{s}\gamma_{\mu}b\right|B(p)\right\rangle$
$\displaystyle=$
$\displaystyle\frac{2\epsilon_{\mu\nu\alpha\beta}}{M_{B}+M_{V}}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}V(q^{2}),$
(6) $\displaystyle\left\langle
V(k,\varepsilon)\left|\bar{s}\gamma_{\mu}\gamma_{5}b\right|B(p)\right\rangle$
$\displaystyle=$ $\displaystyle
i\left(M_{B}+M_{V}\right)\varepsilon^{\ast\mu}A_{1}(q^{2})$ (7)
$\displaystyle-i\,\frac{(\varepsilon^{\ast}\\!\cdot
q)}{M_{B}+M_{V}}\left(p+k\right)^{\mu}A_{2}(q^{2})$
$\displaystyle-2iM_{V}\,\tfrac{q^{\mu}}{q^{2}}\,(\varepsilon^{\ast}\\!\cdot
q)\left[A_{3}(q^{2})-A_{0}(q^{2})\right],$

where

$\displaystyle A_{3}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B}+M_{V}}{2M_{V}}A_{1}(q^{2})-\frac{M_{B}-M_{V}}{2M_{V}}A_{2}(q^{2}),$
(8)

and where the relation $A_{3}(0)=A_{0}(0)$ holds. Likewise,

$\displaystyle\left\langle
V(k,\varepsilon)\left|\bar{s}\sigma_{\mu\nu}q^{\nu}b\right|B(p)\right\rangle$
$\displaystyle=$ $\displaystyle
2\,F_{1}(q^{2})\epsilon_{\mu\nu\alpha\beta}\,\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}$
(9) $\displaystyle\left\langle
V(k,\varepsilon)\left|\bar{s}\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B(p)\right\rangle$
$\displaystyle=$ $\displaystyle
i\left[\\!\\!\begin{array}[]{c}\left(M_{B}^{2}-M_{V}^{2}\right)\varepsilon_{\mu}^{\ast}-(\varepsilon^{\ast}\cdot
q)(p+k)_{\mu}\end{array}\\!\\!\right]F_{2}(q^{2})$ (11) $\displaystyle+$
$\displaystyle i\,(\varepsilon^{\ast}\cdot
q)\left[\\!\begin{array}[]{c}q_{\mu}-\frac{q^{2}}{M_{B}^{2}-M_{V}^{2}}(p+k)_{\mu}\end{array}\\!\right]F_{3}(q^{2})\
,$ (13)

where $F_{1}(0)=F_{2}(0)$. Now, the form factors appearing in Eqs. (6) to (13)
can be related to each other by means of Ward identities [13, 26], i.e.,

$\displaystyle\left\langle
V(k,\varepsilon)\left|\bar{s}\sigma_{\mu\nu}q^{\nu}b\right|B(p)\right\rangle=-(m_{b}+m_{s})\left\langle
V(k,\varepsilon)\left|\bar{s}\gamma_{\mu}b\right|B(p)\right\rangle$ (14)
$\displaystyle\left\langle
V(k,\varepsilon)\left|\bar{s}\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B(p)\right\rangle=(m_{b}-m_{s})\left\langle
V(k,\varepsilon)\left|\bar{s}\gamma_{\mu}\gamma_{5}b\right|B(p)\right\rangle\
.$ (15)

Making use of Eqs. (6) to (13) in Eqs. (14) and (15), one can relate the
transition form factors as follows:

$\displaystyle F_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{m_{b}+m_{s}}{M_{B}+M_{V}}\,V(q^{2}),$ (16) $\displaystyle
F_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{m_{b}-m_{s}}{M_{B}-M_{V}}\,A_{1}(q^{2}),$ (17)
$\displaystyle F_{3}(q^{2})$ $\displaystyle=$
$\displaystyle-(m_{b}-m_{s})\tfrac{2M_{V}}{q^{2}}[A_{3}(q^{2})-A_{0}(q^{2})]\
.$ (18)

Note that the form factor relations as stated in Eqs. (16), (17) and (18) are
model independent. Yet, in deriving the above Ward identities [13], one
assumes that the light-quark degrees of freedom can be neglected and hence
takes $p=p_{b}$ as suggested by heavy quark effective theory in the limit
$m_{b}\to\infty$ with $p_{b}/m_{b}$ constant. This is a sensible approach and
justified in the case of $B$ mesons, though nonperturbative
$\Lambda_{\mathrm{QCD}}/m_{c}$ corrections due to the light quarks may be
important in charmed mesons. As known from a series of nonperturbative
studies, heavy quark effective theory is not generally a good guide to charm
physics [21, 22, 23, 24, 25] since the charm quark is neither light nor really
heavy. Eventually, a reliable calculation of heavy-light form factors,
couplings and decay constants requires the correct description of dynamical
chiral symmetry breaking, that is the effect of the light degrees of freedom.

Figure 1: Comparison of the $B_{d}\to K^{\ast}$ transition form factors, Eqs.
(33), (34) and (35) (black solid curve) with predictions from LQCD [10]
(points with error bars), LCSR [4] (red dashed curve) and DSE [12] (green
dashed curve). Figure 2: Comparison of form factors for $B_{s}\to\phi$; curve
description as in Fig. 1 though no DSE results available in this transition.

The normalization of the form factors at $q^{2}=0$ can be expressed in terms
of $g_{+}(q^{2}),g_{-}(q^{2})$, $h(q^{2})$ and $h_{1}(q^{2})$ which are
discussed in detail in Refs. [13, 16]. Namely, we me make use of

$\displaystyle\langle
V(k,\varepsilon^{\ast})|\bar{s}i\sigma_{\alpha\beta}b|B(p)\rangle$
$\displaystyle=$
$\displaystyle-i\,\epsilon_{\alpha\beta\rho\sigma}\varepsilon^{\ast\rho}[(p+k)^{\sigma}g_{+}+q^{\sigma}g_{-}]-(\varepsilon^{\ast}\\!\cdot
q)\,\epsilon_{\alpha\beta\rho\sigma}(p+k)^{\rho}q^{\sigma}h$ (19)
$\displaystyle-i[(p+k)^{\alpha}\epsilon_{\beta\rho\sigma\tau}\varepsilon^{\ast\rho}(p+k)^{\sigma}q^{\tau}-\alpha\leftrightarrow\beta]\,h_{1}\
,$

so the form factors $V(q^{2}),A_{1,2}(q^{2}),F_{1,2,3}(q^{2})$ become,

$\displaystyle F_{1}(q^{2})$ $\displaystyle=$ $\displaystyle
g_{+}(q^{2})-q^{2}h(q^{2})\ ,$ (20) $\displaystyle F_{2}(q^{2})$
$\displaystyle=$ $\displaystyle
g_{+}(q^{2})+\frac{q^{2}}{M^{2}_{B}-M^{2}_{V}}\,g_{-}(q^{2})\ ,$ (21)
$\displaystyle F_{3}(q^{2})$ $\displaystyle=$ $\displaystyle-
g_{-}(q^{2})-(M^{2}_{B}-M^{2}_{V})h(q^{2})\ ,$ (22)

and

$\displaystyle V(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B}+M_{V}}{m_{b}+m_{s}}\left[\,g_{+}(q^{2})-q^{2}h_{1}(q^{2})\right]\
,$ (23) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B}-M_{V}}{m_{b}-m_{s}}\left[g_{+}(q^{2})+\frac{q^{2}}{M^{2}_{B}-M^{2}_{V}}\,g_{-}(q^{2})\right]\
,$ (24) $\displaystyle A_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{M_{B}-M_{V}}{m_{b}-m_{s}}\,\left[g_{+}(q^{2})-q^{2}h(q^{2})\right]-\frac{2M_{V}}{M_{B}-M_{V}}\,A_{0}(q^{2})\
.$ (25)

At $q^{2}=0$, the form factors $F_{1}$, $F_{2}$, $V$ and $A_{1}$ in Eqs. (20),
(21), (23) and (24) are thus parameterized by a single constant $g_{+}(0)$
whereas $A_{2}$ and $A_{0}$ in Eq. (25) can be expressed in terms of
$g_{+}(0)$ and $A_{0}(0)$. It is well known that in case of real photon
emission in $B_{d,s}\to V\gamma$ decays, the decay rate is a function of
$g_{+}(0)$ [18]:

$\displaystyle\Gamma(B\to
V\gamma)=\frac{G_{F}^{2}\,\alpha_{\mathrm{em}}}{32\pi^{4}}|V_{tb}V_{ts}^{\ast}|^{2}m^{2}_{b}M^{3}_{B}\left(1-\tfrac{M^{2}_{V}}{M^{2}_{B}}\right)^{\\!3}|C_{7}^{\mathrm{eff}}|^{2}|g_{+}(0)|^{2}.$
(26)

Using the experimental values of branching ratios of $B_{d}\to K^{\ast}\gamma$
and $B_{s}\to\phi\gamma$ decays [9],

$\displaystyle\mathrm{Br}(B_{d}\to K^{\ast}\gamma)$ $\displaystyle=$
$\displaystyle(4.33\pm 0.15)\times 10^{-5},$ (27)
$\displaystyle\mathrm{Br}(B_{s}\to\phi\gamma)$ $\displaystyle=$
$\displaystyle(3.6\pm 0.4)\times 10^{-5}\ ,$ (28)

the extracted $g_{+}(0)$ for these decays are,

$\displaystyle g_{+}(0)^{B_{d}\to K^{\ast}}$ $\displaystyle=$ $\displaystyle
0.365^{+0.025}_{-0.025},$ (29) $\displaystyle g_{+}(0)^{B_{s}\to\phi}$
$\displaystyle=$ $\displaystyle 0.335^{+0.02}_{-0.02}\ .$ (30)

With $g_{+}(0)$ in hand, the other unknown, i.e. $A_{0}(0)$, can be expressed
in terms of it as [13],

$\displaystyle A_{0}(0)$ $\displaystyle=$
$\displaystyle\left(\frac{1-M_{V}^{2}/M_{B}^{2}}{1+M_{V}^{2}/M_{B}^{2}}+\frac{M_{B}}{M_{V}}\right)g_{+}(0)\
.$ (31)

It is worth mentioning that the form factor relations derived from Ward
identities do not hold for the entire physical momentum, $q^{2}$, region.
Therefore, we employ the following parametrization between $q^{2}=0$ and near
the poles:

$\displaystyle F(q^{2})$ $\displaystyle=$
$\displaystyle\frac{F(0)}{(1-q^{2}/M^{2})(1-q^{2}/M^{\prime 2})}\ ,$ (32)

where $M$ is $M_{B^{\ast}}(1^{-})$ or $M_{B^{\ast}_{A}}(1^{+})$ and
$M^{\prime}$ is the radial excitation of $M$. The parametrization given in Eq.
(32) not only takes into account the correction to the single pole dominance,
as suggested by dispersion relations [26], but also help us to determine the
couplings of $B^{\ast}$ or $B^{\ast}_{A}$ to the $BV$ channel [13, 14], which
is not the aim of this work. Finally, using the above parameterization, the
form factors can be expressed as:

$\displaystyle V(q^{2})$ $\displaystyle=$
$\displaystyle\frac{V(0)}{(1-q^{2}/M^{2}_{B^{\ast}})(1-q^{2}/M^{\prime
2}_{B^{\ast}})},$ (33) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{A_{1}(0)}{(1-q^{2}/M^{2}_{B^{\ast}})(1-q^{2}/M^{\prime
2}_{B^{\ast}})}\left(1-\frac{q^{2}}{M^{2}_{B}-M^{2}_{V}}\right),$ (34)
$\displaystyle A_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{{\tilde{A}_{2}}(0)}{(1-q^{2}/M^{2}_{B^{\ast}})(1-q^{2}/M^{\prime
2}_{B^{\ast}})}-\frac{2M_{V}}{M_{B}-M_{V}}\frac{A_{0}(0)}{(1-q^{2}/M^{2}_{B})(1-q^{2}/M^{\prime
2}_{B})},$ (35)

where $\tilde{A}_{2}(0)$ is defined as [13]:

$\displaystyle\tilde{A}_{2}(0)$ $\displaystyle=$
$\displaystyle\left(\frac{M_{B}-M_{V}}{m_{b}-m_{s}}\right)\,g_{+}(0)\ .$ (36)

The transition form factors in Eqs. (33), (34) and (35) are plotted as a
function of $q^{2}$ in Figs. 1 and 2 for $B_{d}\to K^{\ast}$ and
$B_{s}\to\phi$, respectively, where we use the Particle Data Group values for
the masses of $B$, $B^{*}$, $B_{A}^{*}$ and their excited states [9]. In the
same figures, we also compare our form factors with those obtained in the
Lattice QCD and LCSR approaches and in case of the $B_{d}\to K^{\ast}$
transition the DSE form factors [12] are also included. The trend of the
$V(q^{2})$ evolution we obtain parallels that of the DSE form factor, whereas
our $A_{1}(q^{2})$ and $A_{2}(q^{2})$ compare more favorably with the LCSR
results, as becomes clear from Fig. 1. Similarly, for the
$B_{s}\to\phi\mu^{+}\mu^{-}$ decay, our form factors are mostly in agreement
with LCSR predictions except for larger momenta, $q^{2}\gtrsim 10$ GeV2; see
Fig. 2. It is also apparent that lattice QCD simulations produce a softer
slope of the $A_{1}(q^{2})$ and $V(q^{2})$ form factors than the other
calculations. For reference, we also list the form factor values at $q^{2}=0$
and $q^{2}=q^{2}_{\mathrm{max}}$ as well as their ratios in Tables 1 and 2.

| $V$ | $A_{1}$ | $A_{2}$ | $\frac{V(q^{2}_{\mathrm{max}})}{V(0)}$ | $\frac{A_{1}(q^{2}_{\mathrm{max}})}{A_{1}(0)}$ | $\frac{A_{2}(q^{2}_{\mathrm{max}})}{A_{2}(0)}$
---|---|---|---|---|---|---
Present Work | 0.456 (2.858) | 0.365 (0.4476) | 0.316 (1.007) | 6.267 | 1.226 | 3.186
LCSR | 0.457 (2.064) | 0.337 (0.577) | 0.282 (0.830) | 4.516 | 1.712 | 2.943
LQCD | 0.30 (1.93) | 0.30 (0.62) | 0.25 (0.44) | 6.433 | 2.06 | 1.76
DSE | 0.37 (2.98) | 0.29 (1.28) | 0.30 (2.177) | 8.054 | 4.431 | 7.256
Table 1: Variation of form factors for the transition $B_{d}\to K^{\ast}$ at $q^{2}=0$ and $q^{2}=q^{2}_{\mathrm{max}}$ in different approaches. The first value represents the form factors at $q^{2}=0$ whereas the value at maximum $q^{2}$ is in parentheses. The ratios of the form factors at $q^{2}_{\mathrm{max}}$ and $q^{2}=0$ are listed in the last three columns. | $V$ | $A_{1}$ | $A_{2}$ | $\frac{V(q^{2}_{\mathrm{max}})}{V(0)}$ | $\frac{A_{1}(q^{2}_{\mathrm{max}})}{A_{1}(0)}$ | $\frac{A_{2}(q^{2}_{max})}{A_{2}(0)}$
---|---|---|---|---|---|---
Present Work | 0.433 (2.566) | 0.335 (0.436) | 0.282 (0.828) | 5.926 | 1.301 | 2.936
LCSR | 0.434 (2.455) | 0.311 (0.637) | 0.234 (0.676) | 5.656 | 2.048 | 2.888
Lattice | 0.24 (1.74) | 0.29 (0.62) | 0.25 (0.41) | 7.25 | 2.137 | 1.64
Table 2: Variation of form factors for the transition $B_{s}\to\phi$ at
$q^{2}=0$ and $q^{2}=q^{2}_{\mathrm{max}}$ in different approaches. Table
entries are as in Table 1.

##  3 Applications of Transition Form Factors: Branching Fractions

To conclude, we present the calculated branching ratios of the $B\to
K^{*}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$ decays, where we remind
that the transition form factors are the major hadronic input as well as
source of uncertainties. Here we use the form factors introduced in Section
2.2 and extrapolated via Eqs. (33), (34) and (35) to compute the differential
branching fractions and compare them to those obtained with form factors of
the corresponding LCSR, Lattice QCD and DSE approaches.

The formula for the differential decay rate is given by,

$\displaystyle\frac{d^{2}\Gamma(B\to V\ell^{+}\ell^{-})}{d\cos\theta\,dq^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2M^{3}_{V}}\frac{2\beta\sqrt{\lambda}}{(8\pi)^{3}}|\,\mathcal{M}|^{2}$
(37)

with $\mathcal{M}$ from Eq. (5) and where
$\beta\equiv\sqrt{1-4m^{2}_{\ell}/s}$,
$\lambda\equiv\lambda(M_{B},M_{V},s)=M^{4}_{B}+M^{4}_{V}+q^{4}-2M^{2}_{{B}}M^{2}_{V}-2q^{2}M^{2}_{B}-2q^{2}M^{2}_{V}$.
The explicit form of differential decay rates for the decay $B_{d}\to
K^{\ast}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$ can be found in Ref.
[27]. The plot of the differential branching fractions, $d\Gamma(B\to
V\ell^{+}\ell^{-})/dq^{2}$, for the above mentioned decays in different
approaches are depicted in Figs. 3a and 3b. The functional form of the
differential branching fractions we obtain is comparable with the one using
other form factor calculations, albeit with variations in magnitude. Note,
however, that the DSE approach leads to a stronger variation of
$d\Gamma/dq^{2}$ with $q^{2}$ in the $B\to K^{*}$ transition.

Furthermore, numerical values of branching ratios of $B_{d}\to
K^{\ast}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$ decays are listed in
Table 3. The $B_{d}\to K^{\ast}\mu^{+}\mu^{-}$ branching ratio obtained with
transition form factors of the DSE based model lies close to the experimental
value. Within theoretical and experimental errors, both of which may be
underestimated, our numerical results for both decays do agree with the PDG
average. We cannot make a statement in case of the LQCD and LCSR results as no
error estimate is given, yet the LQCD branching ratios are considerably larger
than experimental values. This could be due to the fact that LQCD does not
produce as strongly rising form factors, hence might not describe a VMD
behavior. In the case of the $B_{s}\to\phi\mu^{+}\mu^{-}$ decay, the
discrepancy between the experimental average and LCSR and LQCD values is
equally large. It is worthwhile to remind that the values of $g_{+}(0)$, Eqs.
(29) and (30), are very similar. Given that the functional form of the
$V(q^{2})$, $A_{1}(q^{2})$ and $A_{2}(q^{2})$ form factors of the $B_{d}\to
K^{\ast}$ transition is not noticeable different from the one in
$B_{s}\to\phi$ and that the phase space difference is minor, we do not expect
very different branching ratio values.

Figure 3: Differential branching fractions, $d\Gamma/dq^{2}$, for the decays $B_{d}\to K^{\ast}\mu^{+}\mu^{-}$ (left panel) and $B_{s}\to\phi\mu^{+}\mu^{-}$ (right panel). The gray-shaded band represents the estimated error of the differential branching fraction in the present approach; the same branching fractions computed with the LCSR, Lattice QCD and DSE form factors are plotted with red-dashed, green-dashed and orange-dashed curves, respectively.  | Br$(B_{d}\to K^{\ast}\mu^{+}\mu^{-})$ | Br$(B_{s}\to\phi\mu^{+}\mu^{-})$
---|---|---
Present Work | $(1.37\pm 0.2)\times 10^{-6}$ | $(1.21\pm 0.15)\times 10^{-6}$
LCSR | $1.31\times 10^{-6}$ | $1.70\times 10^{-6}$
Lattice | $1.97\times 10^{-6}$ | $1.67\times 10^{-6}$
DSE | $1.18\times 10^{-6}$ | —
PDG | $(1.06\pm 0.09)\times 10^{-6}$ | $(7.6\pm 1.5)\times 10^{-7}$
Table 3: Numerical values of branching ratios for the semileptonic decays
$B_{d}\to K^{\ast}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$ in
different form factors approaches.

## Acknowledgments

M. A. P., B. E. and I. A. are grateful to the organizers of the XXXVII Reunião
de Trabalho sobre Física Nuclear no Brasil in Maresias, São Paulo, for their
kind invitation and acknowledge support by the São Paulo Research Foundation
(FAPESP) under grant nos. 2012/13047-2, 2013/23177-3 and 2013/16088-4. M. A.
P. and I. A. would also like to thank the Physics Department at Quaid-i-Azam
University for the kind hospitality during the last stages of the work. M. J.
A would like to thank the support by Quaid-i-Azam University through the
University Research Fund. B. E. is partially supported by a CNPq fellowship
no. 301190/2014-3.

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# Deterministic Quantum Field Trajectories and Macroscopic Effects

M. Akbari-Moghanjoughi  Faculty of Sciences, Department of Physics, Azarbaijan
Shahid Madani University, 51745-406 Tabriz, Iran
<EMAIL_ADDRESS>
###### Abstract

In this work the root to macroscopic quantum effects is revealed based on the
quasiparticle model of collective excitations in an arbitrary degenerate
electron gas. The $N$-electron quantum system is considered as $N$ streams
coupled, through the Poisson’s relation, which are localized in momentum space
rather than electron localization in real space, assumed in ordinary many body
theories. Using a new wavefunction representation, the $N+1$-coupled system is
reduced to simple pseudoforce equations via quasiparticle (collective quantum)
model leading to a generalized matter wave dispersion relation. It is shown
that the resulting dual lengthscale de Broglie’s matter wave theory predicts
macroscopic quantum effects and deterministic field trajectories for charges
moving in the electron gas due to the coupling of the electrostatic field to
the local electron number density. It is remarked that any quantum many body
system composed of large number of interacting particles acts as a dual arm
device controlling the microscopic single particle effects with one hand and
the macroscopic phenomena with the other. Current analysis can be further
extended to include the magnetic potential and spin exchange effects. Present
model can also be used to confirm macroscopic entanglement of charged
particles embedded in a quantum electron fluid.

###### pacs:

52.30.-q,71.10.Ca, 05.30.-d

##  I Introduction

It is more than a decade that walking droplets on the surface of a
periodically vibrating fluid have fascinated scientists with their many
pseudoquantum features [1, 2]. Floating of a droplet on a vibrating bath of
fluid for the first time was described by Jearl Walker in 1978 in a Scientific
American amateur scientist article. The recent pioneering experiments have [3,
4] revealed that the macroscopic millimeter-sized walking droplets on a
suitably driven fluid surface can mimic variety of quantum-like effects which
were already assumed to emerge at atomic and molecular scales [5]. The most
prominent interference effect has been experimentally confirmed for droplets
passing through single and double slit barriers [6, 7] which are quite
reminiscent to their quantum counterparts. However, at the complete contrast
to the Copenhagen interpretation, the mobile droplets are observed to pass
through only one slit at a time. The later effect seems to give a credit to
the initially ignored deterministic pilot wave quantum theory of de Broglie
[8, 9] which was then re-established by Bohm in 1952 [10, 11]. Subsequent
experiments on walking droplets have shown yet more interesting hydrodynamic
quantum analog aspects such as quantum tunneling [12], bound state orbital
quantization [13, 14, 15] and Landau levels [16]. Therefore, classical
experiments with tiny magic droplets seem to fundamentally alter our
conceptions of wave-particle duality and consequently foundations of quantum
mechanics itself in near future [17].

On the other hand, while the novel classical analogy between quantum particles
and bouncing droplets may have shed light on the understanding of quantum
weirdness, the underlying physical and mathematical connections between de
Broglie-Bohm pilot-wave and the hydrodynamic theories is still a matter of
debate [18]. It is however apparent that the akin similarities between quantum
wave-particle phenomenon and the interacting droplet with its own hydrodynamic
wave is pointing at a much deeper physical description of the quantum
mechanics. The later suggestion has motivated an intensified research [19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] towards detailed theoretical and
experimental investigation of the dynamics of magic bouncing droplet and its
interaction with the surface waves produced by the vibrating platform.
Unfortunately, not much of a progress has been made towards a unified
theoretical description of the phenomenon due to the very complex nature of
collective interactions in fluids and the rapid temporal evolution of the
droplet specific parameters. Recent studies based on the Schrödinger-Poisson
system has shown interesting features of quantum electron fluids due to
electrostatic coupling of single electron excitations and electron
correlations [32, 33, 34, 35]. The collective quantum excitations of electron
gas has been shown to behave as if an electron possesses two distinct de
broglie’s wavelengths, simultaneously [36]. Due to such effect hot electrons
in semiconductor manage to tunnel out of the plasmonic device in resonant
processes [37, 38, 39].

In this research we show that quantum mechanics is a double arm device acting
at different scales at the same time. We employ a newly developed [40]
quasiparticle approach to study collective excitations in the electron gas
without the need for Madelung hydrodynamic formalism in which each electron is
assumed to be a quantum stream localized in the momentum space thereby being
couples to other streams through the Poisson’s relation in a multistream
model. We provide the quasiparticle formalism in Sec. II. In Sec. III the
generalized matter wave dispersion relation is obtained. The examples of
deterministic quantum field trajectories is given in Sec. IV based on dual-
tone quantum theory. The macroscopic quantum effects are introduced in the
electron gas in Sec. V and conclusions are drawn in Sec. VI.

##  II Quasiparticle Representation of Collective Modes

Collective excitations in electron gas can be effectively modeled using the
quasiparticle concept. Starting from $N$ single-electron time-dependent
Schrödinger equations coupled through Poisson’s relation, we have a
generalized multistream electron system as follows [40]

$\displaystyle i\hbar\frac{{\partial{{\cal{N}}_{j}}({\bf{r}},t)}}{{\partial
t}}={\cal{H}}{{\cal{N}}_{j}}({\bf{r}},t),$ (1a)
$\displaystyle\Delta\phi({\bf{r}})=4\pi
e\left[{\sum\limits_{j=1}^{N}{{{{\cal{N}}_{j}}({\bf{r}},t){\cal{N}}_{j}^{*}({\bf{r}},t)}-{n_{0}}}}\right].$
(1b)

in which the subscript $j$ denotes the $j$-th stream, $n_{0}$ denotes the
neutralizing charge number density and ${\cal{N}}_{j}$ represents the
corresponding single-electron wavefunction. The Hamiltonian operator acting on
each electron ${\cal{H}}={\cal{K}}+{\cal{U}}$ is the addition of kinetic
energy operator and the potential energy in the form of
${\cal{U}}=-e\phi+\mu$, in which $\phi$ is the self consistent electrostatic
potential and $\mu$ the chemical potential of the electron gas. In the
quasiparticle representation the collective modes in a quantum electron gas
can be studied using the the following effective Schrödinger-Poisson system

$\displaystyle i\hbar\frac{{\partial{\cal{N}}({\bf{r}},t)}}{{\partial
t}}=-\frac{{{\hbar^{2}}}}{{2m}}\Delta{\cal{N}}({\bf{r}},t)-e\phi({\bf{r}}){\cal{N}}({\bf{r}},t)+\mu{\cal{N}}({\bf{r}},t),$
(2a) $\displaystyle\Delta\phi({\bf{r}})=4\pi
e\left[{|{\cal{N}}({\bf{r}},t){|^{2}}-{n_{0}}}\right].$ (2b)

where $n={{{{\cal{N}}}({\bf{r}},t){\cal{N}}^{*}({\bf{r}},t)}}$ is the local
electron number density defined through the quasiparticle statefunction
${\cal{N}}({\bf{r}},t)=\sum\limits_{j=1}^{N}{{{\cal{N}}_{j}}({\bf{r}},t)}$.
Note that in the quasiparticle representation electrons do not follow
independent particle or mean-field assumptions but instead are coupled via the
electrostatic and chemical potential in the electron gas to the Poisson’s
relation and appropriate equation of state (EoS), respectively. In the
classical limit the chemical potential vanishes, whereas, in the fully
degenerate electron gas the chemical potential becomes identical with the
constant Fermi energy of the system. In the following analysis we assume that
the chemical potential of the system does not vary significantly over the
characteristic quantum lengthscale, for simplicity. The quasiparticle
Hamiltonian can be further generalized to include the magnetic potential,
electron exchange and other effects.

##  III Generalized de Broglie’s Wavelengths and Matter Wave Dispersion

The quasiparticle model of electron gas in Eq. (2) has been shown to lead to
dual-lengthscale theory giving rise to two distinct de Broglie wavelengths
with the ordinary wavelength characterising the single-electron excitations
and the other representing the collective excitations. The linearized system
($\psi^{0}=1,\phi^{0}=0,\mu^{0}=\mu_{0}$) after the variable separation leads
to the following coupled pseudoforce system

$\displaystyle i\hbar\frac{{d\varphi(t)}}{{dt}}=\epsilon\varphi(t),$ (3a)
$\displaystyle\Delta\Psi({\bf{r}})+\Phi({\bf{r}})+2E\Psi({\bf{r}})=0,$ (3b)
$\displaystyle\Delta\Phi({\bf{r}})-\Psi({\bf{r}})=0,$ (3c)

in which $\Psi({\bf{r}})=\psi({\bf{r}})/\sqrt{n_{0}}$ with
${{\cal{N}}}({\bf{r}},t)=\psi({\bf{r}})\varphi(t)$ is the normalized
quasiparticle wavefunction and $\Phi({\bf{r}})=e\phi({\bf{r}})/E_{p}$ is the
normalized electrostatic potential energy with $E_{p}=\sqrt{4\pi
e^{2}n_{0}/m}$ is the plasmon energy and the normalized energy parameter
$E=(\epsilon-\mu_{0})/E_{p}$ is the total kinetic energy of electron gas with
$\epsilon=\sum\limits_{j=1}^{N}{{\epsilon_{j}}}$ being the energy eigenvalue
of quasiparticle excitations and $\epsilon_{j}$ the energy of $j$-th electron
in the system. In this normalization scheme the space and time are normalized
with respect to the plasmon length $l_{p}=1/k_{p}$ with
$k_{p}=\sqrt{2mE_{p}}/\hbar$ being the plasmon wavenumber and the inverse
plasmon frequency $\omega_{p}=E_{p}/\hbar$. The first equation in (3) leads to
the trivial time-dependence solution, $\varphi(t)=\exp(i\epsilon t/\hbar)$.
Linearizing the system leads to the generalized energy dispersion relation
$E=k^{2}/2+1/2k^{2}$ in which the wavenumber is normalized to the plasmon
wavenumber. The dispersion relation can be regarded as composed of a term
charcterizing the free electron parabolic dispersion ($k^{2}/2$) and an extra
term ($1/2k^{2}$) due to long range electrostatic correlations. Therefore, in
the quasiparticle model, the collective excitations are characterized by two
de Broglie wavelength which are actually reciprocal of each other [36].
Therefore, the quasiparticle model of collective electron excitation is a
dual-lengthscale quantum theory.

Figure 1: (a) The variation of characteristic plasmon length in electron gas
in terms of the equilibriumm electron number density. (b) The variation of
characteristic plasmon energy in electron gas in terms of the equilibriumm
electron number density. (c) Variation of the chemical potential of isothermal
arbitrary degenerate electron gas with respect to the equilibrium electron
number density. The generalized matter wave (energy) dispersion in which
energy anf wavenumber are normalized to corresponding plasmon parameters. The
dashed curve in Fig. 1(d) denotes the free electron matter wave dispersion.

Figure 1 depicts the variations in different plasmon parameters and the
generalized energy dispersion relation of collective excitations in
interacting electron gas. Figure 1(a) shows the variations in the plasmon
length in nanometer unit. It is seen that this length decreases sharply with
increase in the electron gas number density dropping to few tenth of a
nanometer in elemental metals. In Fig. 1(b) the plasmon energy is shown in
electron Volt unit. It is remarked that this quantity increases rapidly with
increase on the electron number density reaching few $eV$ in typical metals.
The chemical potential of the electron gas in free electron model is related
to the electron number density and quantum statistical pressure via the well-
known isothermal equation of state (EoS)

$\displaystyle{n_{e}(\mu,T)}=\frac{{{2^{1/2}}m{{}^{3/2}}}}{{{\pi^{2}}{\hbar^{3}}}}\int_{0}^{+\infty}{\frac{{\sqrt{{\varepsilon}}d{\varepsilon}}}{{{e^{\beta({\varepsilon}-\mu)}}+1}}},$
(4a)
$\displaystyle{P_{e}(\mu,T)}=\frac{{{2^{3/2}}m{{}^{3/2}}}}{{3{\pi^{2}}{\hbar^{3}}}}\int_{0}^{+\infty}{\frac{{{{\varepsilon}^{3/2}}d{\varepsilon}}}{{{e^{\beta({\varepsilon}-{\mu})}}+1}}.}$
(4b)

where $\beta=1/k_{B}T$ with $T$ being the equilibrium electron temperature and
$P_{e}$ being the quantum statistical electron gas pressure. Variation of the
chemical potential with electron number density at room temperature is
depicted in Fig. 1(c). This parameter is vanishingly small for classical
electron gas and is saturated in complete degeneracy limit ($n_{0}\simeq
10^{22}$cm-3). The dispersion relation is shown in Fig. 1(d) in which the
single-electron and collective branches connect at the quantum beating point
$k=1$ [36]. For a given value of the quasiparticle energy (orbital) $E$
exceeding the band gap value, $E=2E_{p}$, the energy level intersects the
dispersion curve in two de Broglie wavenumbers with the smaller one
characterizing the long range electrostatic interactions and the larger one
due to single electron excitations. The single-electron excitation dispersion
branch is asymptotically approaches the free electron dispersion curve for
higher quasiparticle energy values. It is remarked that, as the quasiparticle
energy increases, the single electron and collective mode scalelengths differ
significantly.

##  IV Deterministic Quantum Field Trajectory

The pseudoforce system (3) in one dimension has the following simple solution
[36]

$\left[{\begin{array}[]{*{20}{c}}{{\Phi}(x)}\\\
{{\Psi}(x)}\end{array}}\right]=\frac{1}{{2\alpha}}\left[{\begin{array}[]{*{20}{c}}{{\Psi_{0}}+k_{2}^{2}{\Phi_{0}}}&{-\left({{\Psi_{0}}+k_{1}^{2}{\Phi_{0}}}\right)}\\\
{-\left({{\Phi_{0}}+k_{1}^{2}{\Psi_{0}}}\right)}&{{\Phi_{0}}+k_{2}^{2}{\Psi_{0}}}\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{\cos({k_{1}}x)}\\\
{\cos({k_{2}}x)}\end{array}}\right],$ (5)

in which $\Phi_{0}$ and $\Psi_{0}$ are the constants assuming
$\Phi^{\prime}(0)=\Psi^{\prime}(0)$=0, for simplicity. The characteristic
wavenumbers are given as

${k_{1}}=\sqrt{E-\alpha},\hskip 8.53581pt{k_{2}}=\sqrt{E+\alpha},\hskip
8.53581pt\alpha=\sqrt{{E^{2}}-1},$ (6)

satisfying the complementarity-like relation $k_{1}k_{2}=1$. Note that the
coupling of the electrostatic field to the wavefunction $\Psi(x)$ (the local
electron density, $n$) leads to deterministic quantum trajectories for charge
particles moving in the electron gas excited to given quasiparticle orbital.
The normalized classical guiding equation is

$\Gamma\frac{{{d^{2}}x}}{{d{t^{2}}}}+Q\frac{{d\Phi}}{{dx}}=0,$ (7)

where $\Gamma$ and $Q$ are the particle mass and charge ratios with respect to
that of electron. This is obviously different from the guiding equation in de
Broglie-Bohm pilot wave theory defined through the probability density as

$\frac{{dx}}{{dt}}={\mathop{\rm\Im}\nolimits}{\mathop{\rm}\nolimits}\left[{\nabla\ln\left(\Psi\right)}\right]=\frac{{J(x)}}{{n(x)}},$
(8)

where $\Im$ denotes the imaginary part and

$J(x)=\frac{i}{2}\left({\Psi\frac{{\partial{\Psi^{*}}}}{{\partial
x}}-{\Psi^{*}}\frac{{\partial\Psi}}{{\partial x}}}\right).$ (9)

The former determinism is defined through the particle-field coupling in
quasiparticle model, however, the later is defined based on current density
representation in Madelung fluid formalism.

Figure 2: (a) Variation of local electron number density in quasiparticle
orbital ($E=4E_{p}$). (b) Variation of electrostatic potential energy in
quasiparticle orbital ($E=4E_{p}$). (c) Deterministic one-dimensional field
trajectories of a test particle at orbital $E=2$ and different initial speeds.
(d) Deterministic one-dimensional field trajectories of a test particle at
different orbital and initial speed $v_{0}=v_{p}$. $v_{p}=(\hbar/m)k_{p}$ is
the corresponding plasmon speed of the particle.

Figure 2(a) and 2(b) show the local electron number density and electrostatic
potential energy profiles at given quasiparticle orbital ($E=2$). The electron
density $n=\Psi^{*}\Psi$ and electrostatic potential energy variations are
dual tone in nature. The later is due to two distinct de Broglie’s wavenumbers
governing the quasiparticle excitations in the electron gas. Figure 2(c) shows
the classical charged particle trajectory in quantum-coupled electrostatic
field assuming $\Gamma=Q=1$, for simplicity. It is remarked that for initial
particle speed $v_{0}=0.82,0.85$ in a given quasiparticle orbital $E=2$ the
oscillations are stationary with the particle being localized in the electron
gas. However, for higher initial particle speeds the particle moves within the
gas in an oscillatory path. Figure 2(d) depicts the particle trajectory for
$v_{0}=1$ (the speed is normalized to plasmon speed $v_{p}=(\hbar/m)k_{p}$) at
different quasiparticle orbital. The particle is apparently localized at low
energy orbital, whereas, moving at higher quasiparticle orbital.

##  V Macroscopic Quantum Effects

The system of equations (3) has a polar solution of the form [34]

$\left[{\begin{array}[]{*{20}{c}}{\Phi(r)}\\\
{\Psi(r)}\end{array}}\right]=\frac{Q}{{2\alpha}r}\left[{\begin{array}[]{*{20}{c}}{{\Psi_{0}}+k_{2}^{2}{\Phi_{0}}}&{-\left({{\Psi_{0}}+k_{1}^{2}{\Phi_{0}}}\right)}\\\
{-\left({{\Phi_{0}}+k_{1}^{2}{\Psi_{0}}}\right)}&{{\Phi_{0}}+k_{2}^{2}{\Psi_{0}}}\end{array}}\right]\left({\begin{array}[]{*{20}{c}}{{{\rm{e}}^{\pm
i{k_{1}}r}}}\\\ {{{\rm{e}}^{\pm i{k_{2}}r}}}\end{array}}\right),$ (10)

where $Q$ is the corresponding pole charge. On the other hand, in the
cartesian form we have

$\displaystyle\frac{{{\partial^{2}}\Psi(x,y,z)}}{{\partial{x^{2}}}}+\frac{{{\partial^{2}}\Psi(x,y,z)}}{{\partial{y^{2}}}}+\frac{{{\partial^{2}}\Psi(x,y,z)}}{{\partial{z^{2}}}}+\Phi(x,y,z)=-2E\Psi(x,y,z),$
(11a)
$\displaystyle\frac{{{\partial^{2}}\Phi(x,y,z)}}{{\partial{x^{2}}}}+\frac{{{\partial^{2}}\Phi(x,y,z)}}{{\partial{y^{2}}}}+\frac{{{\partial^{2}}\Phi(x,y,z)}}{{\partial{z^{2}}}}-\Psi(x,y,z)=0,$
(11b)

where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and the dipole solution has the following
form [34]

$\displaystyle\Phi=\frac{{\left({1+k_{2}^{2}}\right)\exp\left[{i{k_{1}}\sqrt{{{(x-a)}^{2}}+{y^{2}}+{z^{2}}}}\right]-\left({1+k_{1}^{2}}\right)\exp\left[{i{k_{2}}\sqrt{{{(x-a)}^{2}}+{y^{2}}+{z^{2}}}}\right]}}{{2\alpha\sqrt{{{(x-a)}^{2}}+{y^{2}}+{z^{2}}}}}$
(12a)
$\displaystyle+\frac{{\left({1+k_{2}^{2}}\right)\exp\left[{i{k_{1}}\sqrt{{{(x+a)}^{2}}+{y^{2}}+{z^{2}}}}\right]-\left({1+k_{1}^{2}}\right)\exp\left[{i{k_{2}}\sqrt{{{(x+a)}^{2}}+{y^{2}}+{z^{2}}}}\right]}}{{2\alpha\sqrt{{{(x+a)}^{2}}+{y^{2}}+{z^{2}}}}},$
(12b)
$\displaystyle\Psi=\frac{{\left({1+k_{1}^{2}}\right)\exp\left[{i{k_{2}}\sqrt{{{(x-a)}^{2}}+{y^{2}}+{z^{2}}}}\right]-\left({1+k_{2}^{2}}\right)\exp\left[{i{k_{1}}\sqrt{{{(x-a)}^{2}}+{y^{2}}+{z^{2}}}}\right]}}{{2\alpha\sqrt{{{(x-a)}^{2}}+{y^{2}}+{z^{2}}}}}$
(12c)
$\displaystyle+\frac{{\left({1+k_{1}^{2}}\right)\exp\left[{i{k_{2}}\sqrt{{{(x+a)}^{2}}+{y^{2}}+{z^{2}}}}\right]-\left({1+k_{2}^{2}}\right)\exp\left[{i{k_{1}}\sqrt{{{(x+a)}^{2}}+{y^{2}}+{z^{2}}}}\right]}}{{2\alpha\sqrt{{{(x+a)}^{2}}+{y^{2}}+{z^{2}}}}},$
(12d)

in which we have taken $Q=1$ and $\Phi_{0}=\Psi_{0}=1$ and
$\Phi^{\prime}_{0}=\Psi^{\prime}_{0}=0$, for simplicity. Also, $a$ is the pole
spacing. The solution (8) represents double quantum interference effect for
wavefunction (electron number density) as well as for electrostatic potential
field. Using (10) and the standard proceedure it can be shown [34] that the
total probability current is $J_{t}=J+J_{d}$

$\displaystyle
J=\frac{i}{2}\left({{\Psi}\nabla\Psi^{*}-\Psi^{*}\nabla{\Psi}}\right),$ (13a)
$\displaystyle
J_{d}=-\frac{i}{2}\left({{\Phi}\nabla\Phi^{*}-\Phi^{*}\nabla{\Phi}}\right),$
(13b)

where $J$ is the conventional probability current (density) governing all
charged and uncharged particles and the new term $J_{d}$ only rules the
charged species.

Figure 3: (a) The dipole interference pattern for quasiparticle wavefunction
at orbital $E=20$ with dipole spacing $a=3$. (b) The dipole interference
pattern for quasiparticle potential energy disptribution at orbital $E=20$
with dipole spacing $a=3$. (c) The dipole interference probability density for
quasiparticle wavefunction at orbital $E=20$ with dipole spacing $a=3$. (d)
The deterministic microscopic interference trajectories for quasiparticle
wavefunction at orbital $E=20$ with dipole spacing $a=3$.

Figure 3 shows the quantum dipole interference which is quite analogous to the
well known double-slit phenomenon. We have used $z=0$ in order to study the
interference in $x$-$y$ plane. The absolute value of wavefunction is shown at
Fig. 3(a) for $E=20$ and $a=3$. The interference pattern shows fringes due to
constructive and destructive quantum interferences. In Fig. 3(b) we show the
potential field interference for the same parameters. However, while the weak
interference pattern is detectable, interference fringes are missing. Figure
3(c) depicts the normal probability current indicating similar pattern as in
Fig. 3(a). This figure shows the quantum particle flow intensity in the fluid
representation. Figure 3(d) shows the streamplot of the probability current
showing the similar pattern as in 3(a) and 3(c). It shows the virtual paths
occurring in quantum flow due to dipole interference. This feature is the
quite analogous as the pilot-wave theory interpretation of quantum
interference effect [8, 9].

Figure 4: (a) The dipole interference pattern for quasiparticle wavefunction
at orbital $E=80$ with dipole spacing $a=1$ (microscopic level). (b) The
dipole interference pattern for quasiparticle potential energy disptribution
at orbital $E=80$ with dipole spacing $a=160$ (macroscopic level). (c) The
dipole interference pattern for quasiparticle wavefunction at orbital $E=180$
with dipole spacing $a=1$ (microscopic level). (d) The dipole interference
pattern for quasiparticle potential energy disptribution at orbital $E=100$
with dipole spacing $a=160$ (macroscopic level).

Figure 4 shows the interference patterns for higher quasiparticle energies.
Figure 4(a) shows the interference pattern for $|\Psi(x,y)|$ for $E=80$ and
$a=1$. Figure 4(b) reveals that fringes appear for $|\Phi(x,y)|$ at this
orbital and larger scale ($a=160$). This is expected due to larger de
Broglie’s wavelength for collective branch compared to that of single-electron
one. In Figs. 4(c) and 4(d) we have further increased the quasiparticle energy
to see the interference pattern shift. It is remarked that for $E=180$ the
spacing between fringes decrease/increase for wavefunction/potential pattern.
This feature is also obvious from the energy dispersion curve in Fig. 1(d). It
is revealed that, while with increase of quasiparticle orbital energy the
wavefunction interference goes microscopic the potential field interference
leads to macroscopic effects.

Figure 5: (a) The dipole interference pattern for quasiparticle potential
energy disptribution at orbital $E=12$ with dipole spacing $a=100$
(macroscopic level). (b) The dipole interference pattern for quasiparticle
potential energy disptribution at orbital $E=50$ with dipole spacing $a=200$
(macroscopic level). (c) The dipole interference pattern for quasiparticle
potential energy disptribution at orbital $E=80$ with dipole spacing $a=300$
(macroscopic level). (d) The dipole interference pattern for quasiparticle
potential energy disptribution at orbital $E=800$ with dipole spacing $a=800$
(macroscopic level).

The later prediction is confirmed in Fig. 5. In this figure we have
consecutively increased the energy and dipole spacings from 5(a) to 5(d) to
reach a macroscopic level. It is evident that by proportional increase of
these parameters one can arbitrarily increase the macroscopic behavior of the
quantum system. This is obviously due to coupling of the macroscopic
electrostatic (or other) field to the microscopic quantum effects. As depicted
in Fig. 1(a) the characteristic plasmon length varies widely over the electron
number density changes. For a typical value of $l_{p}=10$nm ($n_{0}\simeq
10^{17}$cm-3) the interference lengthscale of Fig. 5(d) falls into the
millimeter range.

Figure 6: (a) The dipole interference pattern for quasiparticle wavefunction
at orbital $E=12$ with dipole spacing $a=100$ (macroscopic level). (b) The
dipole interference pattern for quasiparticle wavefunction at orbital $E=50$
with dipole spacing $a=200$ (macroscopic level). (c) The dipole interference
pattern for quasiparticle wavefunction at orbital $E=80$ with dipole spacing
$a=300$ (macroscopic level). (d) The dipole interference pattern for
quasiparticle wavefunction at orbital $E=800$ with dipole spacing $a=800$
(macroscopic level).

In figure 6 we have shown the effects used in Fig. 5 in the case of
wavefunction interference. Figure 6(a) reveals an interesting complex
wavefunction quantum interference pattern at macroscopic level. Increase of
the interference scale parameter in Figs. 6(b) and 6(d) is shown to lead to
emergence of fractal-like fine structure in the interference pattern until in
Fig. 6(d) the interference pattern fades away at macroscopic level. Therefore,
while the potential interference becomes clearer at macroscopic level the
wavefunction interference clears away. The later effect is due to the
complementarity-like relation $\lambda_{1}\lambda_{2}=1$ between the
microscopic and macroscopic de Broglie’s wavelengths.

Figure 7: (a) The dipole interference pattern for electric field distribution
at orbital $E=50$ with dipole spacing $a=200$ (macroscopic level). (b) The
dipole interference pattern for charge probability density at orbital $E=50$
with dipole spacing $a=200$ (macroscopic level). (c) The streamplot of
electric field curves at orbital $E=20$ and $a=2$ (microscopic level). (d) The
streamplot of charge current trajectories at orbital $E=20$ and $a=2$
(microscopic level).

Figure 7(a) depicts the absolute value of the electric field showing similar
interference pattern as in Fig. 5(b). However, there are fine structure
modulated over the interference pattern due to the local quantum electron
density variations. The presence of such microscopic density pattern makes
harder to detect the macroscopic quantum effects. Figure 7(b) shows the
probability current part due to the charge transfer as defined in (13) for
similar interference parameters as in Fig. 7(a). The pattern is quite similar
to the electric field interference pattern and predicts a macroscopic effec
However, the modulated quantum electron density pattern somehow vagues the
vivid interference pattern present in microscopic level. Figures 7(c) and 7(d)
show streamplots of electric field and charge probability density revealing
the similar structures present in Fig. 3(d) at the microscopic level.

##  VI Conclusion

Using the quasiparticle model of excitations in arbitrary degenerate electron
gas we have shown that macroscopic quantum effects are observable due to dual-
tone nature of collective excitations. The generalized energy dispersion of
quasiparticle model revealed that many body quantum systems act like a dual
arm device acting simultaneously at the microscopic and macroscopic level due
to the coupling of quantum field particle interactions to the local particle
number density. Current analysis may also have fundamental implications for
macroscopic level particle quantum entanglement. This model may be further
generalized to include the effects such as electromagnetic interactions and
electron spin effects.

##  VII Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  VIII Conflict of Interest

The authors have no conflicts to disclose.

##  IX References

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