You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We examine the transverse and longitudinal components of the drag force upon a straight vortex line due to the scattering of liquid $^4$He excitations. For this purpose, we consider a recently proposed Hamiltonian that describes the dissipative motion of a vortex, giving an explicit expression for the vortex-quasiparticle interaction. The involved dissipative coefficients are obtained in terms of the reservoir correlation function. Most of our explicit calculations are concerned to the range of temperatures below 0.4 K, at which the reservoir is composed by phonon quasiparticle excitations. We also discuss some important implications in the determination of possible scattering processes leading to dissipation, according to the values of vortex mass found in the literature.'
address: |
     Departamento de Física, Facultad de Ciencias Exactas y Naturales,\
    Universidad de Buenos Aires, RA-1428 Buenos Aires, Argentina\
    and Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina
author:
- 'H M Cataldo, M A Despósito, and D M Jezek'
title: Longitudinal and transverse forces on a vortex in superfluid $^4$He
---

Although it is well established that vortex dynamics plays an important role in the behavior of superfluids, there are many controversial results in the literature about its dissipative motion. At zero temperature the motion of the vortex is provoked by the transversal Magnus force, but at finite temperatures the scattering of collective excitations by the vortex provides a dissipation mechanism that damps its cyclotron motion. It is widely accepted that for low enough temperatures, and only taking into account phonon-vortex scattering, the longitudinal component of this dissipative force behaves as fifth power in temperature [@ior]. Nevertheless, the form of the transverse component is still controversial [@so] since very different results were reported so far and it does not appear that any of them can be regarded as definitive.

In this context, we have recently presented [@ca1] a model for the dissipation of a straight vortex line in superfluid $^4$He in which the vortex is regarded as a macroscopic quantum particle whose irreversible dynamics can be described in the frame of Generalized Master Equations. This procedure enables us to cast the effect of the coupling between vortex and heat bath as a drag force with one reactive and one dissipative component, in agreement with phenomenological theories. In this paper we shall investigate the components of the drag force considering that the reservoir is composed by the quasiparticle (qp) excitations of the superfluid. The dissipative vortex dynamics arises then from the scattering processes ocurring between the vortex and the qp’s. Most of our explicit calculations will be concerned to the range of temperatures below 0.4 K, at which the qp excitations constitute a phonon reservoir[@do].

Let us begin by performing a description of our model. Choosing a coordinate system fixed to the superfluid, the Hamiltonian for a straight vortex line parallel to the $z$-axis, may be written as [@ca1; @do] $$H_v = \frac{1}{2 m_v} \left [{\bf p} - q_v {\bf A}({\bf r})\right]^2 
\,\, ,
\label{Hfree}$$ where $ {\bf r} $ and $ {\bf p} $ denote respectively the vortex position and momentum operators, $${\bf A}({\bf r}) = \frac{h\,\rho_s\,l}{2} (y,-x)$$ is the vector potential for the Magnus force, $m_v$ is the inertial mass of the vortex, $\rho_s$ the number density of the superfluid, $h$ Planck’s constant, $l$ the system length along the $z-$ axis and $q_v = \pm 1$ the sign of the vorticity according to the right handed convention.

In what follows we shall consider the excitations to be at rest, so that the velocity of the normal fluid vanishes. In a previous work [@ca2] we have considered an interaction Hamiltonian of the form $$H_{int} = - {\bf B}\cdot \,\, {\bf v}
\label{hin}$$ where $ {\bf B} $ and $ {\bf v} $ are vectors that depend on the reservoir and vortex operators respectively. We have proven that the only linear combination of the vortex observables that leads to a dynamics in accordance with phenomenological descriptions is, $${\bf v} = \left(\frac{p_x}{m_v}- \frac{\Omega}{2} \,y, 
\frac{ p_y}{m_v} +
\frac{\Omega}{2}\,x\right)\,\,,
\label{vfree}$$ being $\Omega$ the “cyclotron frequency", $$\Omega= \frac{q_v \,h\,\rho_s\,l}{m_v}.
\label{Ome}$$

In order to obtain an equation of motion for the mean value of the complex vortex position operator $R = x + i y $, in previous works [@ca1; @ca2] we have derived by means of a standard reduction-projection procedure, a generalized master equation for the density operator of the vortex. We have employed a usual weak-coupling approximation, in which the vortex dynamics is affected by the reservoir degrees of freedom only through the second order time correlation tensor $\langle {\bf B}(t) {\bf B}\rangle$, where the angular brackets indicate an average over the reservoir equilibrium ensemble and ${\bf B}(t)$ denotes a free time evolution for the reservoir operators. In addition, such tensor is naturally assumed to be isotropic in the $x-y$ plane. We have also made use of the Markovian approximation which assumes that such correlations are short lived within observational times. From such master equation, we then extracted equations of motion for the expectation values of vortex position and momentum operators. Finally, after some algebra and by elimination of the momentum, we obtained the desired equation for $\langle R(t)\rangle$ [@ca1; @ca2]: $$m_v\langle\ddot{R}\rangle
=i (m_v\Omega + \gamma) \,\langle\dot{R}\rangle \,\, .
\label{queda}$$ The complex coefficient $\gamma$ is defined as $$\gamma = \frac{2\Omega}{\hbar}\int_{0}^{\infty}\,d\tau\, 
\psi(\tau)\,e^{\displaystyle - i\Omega \tau } \,\, ,
\label{integ}$$ where $\psi(\tau)$ is the imaginary part of the time correlation isotropic tensor element, $$\psi(\tau)={\cal I}m\,(<B_j(\tau)\,B_j>) \,\, .
\label{corr}$$

In the right hand side of equation (\[queda\]) one can identify two forces, namely the Magnus force [@so] and the drag force. In particular, the drag force $$F_D = i \gamma
 \,\,\langle\dot{R}\rangle$$ has two components. One is parallel to the Magnus Force which we shall call the Transverse Force (TF) :

$$F_T = i\, {\cal R}e (\gamma)
 \,\,\langle\dot{R}\rangle,$$

and the other one is parallel to the velocity and we shall refer to it as the Longitudinal Force (LF) :

$$F_L = - {\cal I}m ( \gamma)
 \,\,\langle\dot{R}\rangle .
\label{flo}$$

We want to recall that it is well known that the LF must be opposite to the velocity (in our notation this means ${\cal I}m (\gamma)
>0 $). With respect to the TF great controversy still exists upon its direction [@ior; @so; @pita] or even more, upon its proper existence [@tho; @vol].

In the present work, the reservoir vector operator $\bf B$ of Eq. (\[hin\]) is chosen as to describe qp scattering by the vortex. In terms of creation $a^+_{\bf{q}}$ and annihilation $a_{\bf{q}} $ operators for a qp of momentum $\bf{q}$ it reads, $${\bf B} =  \sum_{ {\bf k} , {\bf q} }  \, \,   ( {\bf k} -  {\bf q} ) 
  \,\,  \Lambda_{k,q}    \,\,
a_{{\bf k}}^+ \,  a_{ {\bf q}} \,\, ,
\label{bintn}$$ being $ \bf{k} -\bf{q} $ the transferred momentum and $ \Lambda_{k,q}$ a vortex-qp coupling constant which, in order to preserve isotropy, is assumed to depend only on the modulus of qp momentum ($q=|{\bf q}|$).

To calculate the drag coefficients we must first compute the correlation function in (\[corr\]). Taking into account the time evolution $a_{ \bf{q}}(t)=  a_{ \bf{q}}(0)
e^{-i w_q t }$ and after some calculations, we have $$< B_j(t)\, B_j> =\frac{1}{2} \sum_{ \bf{k} ,  \bf{q} } 
|\Lambda_{k,q}|^2 
(k^2+q^2) e^{i(w_k-w_q) t}  n(w_k) [1 + n(w_q)]    \,\, ,
\label{bx}$$ where $n(w_q)$ denotes the mean number of qp’s of energy $\hbar w_q$, $$n (w_q) = <a_{\bf{q}}^+ \,  a_{ \bf{q}}>  = 
\frac{1}{e^{\beta \hbar w_q} -1 }  \, \, .
\label{eneq}$$

Note that at temperature $T=0$ the correlation function (\[bx\]) vanishes due to the factor $n(w_k)$, yielding a vanishing drag force as expected. This behavior would not be reproduced by a linear interaction in the qp operators, as for example the one proposed in the Hamiltonian of Ref. [@niu].

The parameter $\gamma$ (Eq. (\[integ\])) can be written in terms of the Fourier transform $\psi[w]$ of the imaginary part of the correlation function (\[bx\]) as $$\begin{aligned}
{\cal R}e (\gamma)  & = & \frac{4}{i\hbar}\Omega\,\,\,
 {\cal P} \int_{0}^{\infty}\,dw 
\frac{w}{ \Omega^2-w^2 } \psi[w]
\label{rein} \\
{\cal I}m (\gamma) & = & \frac{ 2\pi}{i\hbar}\Omega \psi[\Omega]
\label{imin}\end{aligned}$$ where ${\cal P}$ refers to the principal part and $$\psi[w] = 
\frac{i}{4}  \sum_{ {\bf k},  {\bf q} } (k^2+q^2)
 |\Lambda_{k,q}|^2 [n(w_q)-n(w_k)]\delta (w_k-w_q-w)\,\, .
\label{delsin}$$ The above expressions (\[imin\]) and (\[delsin\]) for the dissipative coefficient in (\[flo\]) can be interpreted from the scattering processes embodied in our interaction Hamiltonian. In fact, we first note that the vortex Hamiltonian (\[Hfree\]) has an equally spaced level spectrum of separation $\hbar\Omega$ (the so-called Landau levels[@landau]) and, in addition, the operator ${\bf v}$ which couples the vortex to the reservoir qp’s, can be expressed from (\[vfree\]) as a linear combination of creation and destruction operators of a vortex energy quantum[@landau]. Thus, according to (\[hin\]) and (\[bintn\]), we may identify the scattering processes embodied in our model as those involving one vortex energy quantum $\hbar\Omega$ jointly with the creation and destruction of one qp. Such processes can also be identified from (\[delsin\]) as follows. Each term proportional to $n(w_q)$ represents the process by which a qp of energy $\hbar w_q$ combines with a vortex quantum $\hbar\Omega$ to form a qp of energy $\hbar w_k=\hbar w_q+\hbar\Omega$ (this may be seen from (\[imin\]) and the Dirac delta in (\[delsin\])). The weight $n(w_q)$ of this process can be easily understood by taking into account the [*thermal*]{} origin of the “incoming” qp of energy $\hbar w_q$ in contrast to the [*interaction*]{} origin of the “outgoing” qp of energy $\hbar w_k$. Similarly, the terms weighted by $n(w_k)$ in (\[delsin\]) represent the annihilation of a qp of energy $\hbar w_k$ jointly with the creation of a vortex quantum $\hbar\Omega$ and a qp of energy $\hbar w_q$. Finally, we note that the positive sign of the dissipative coefficient ${\cal I}m (\gamma)$ arises from the factor $n(w_q)-n(w_k)>0$ in (\[delsin\]) i. e., the weight of processes involving vortex energy loss must be greater than those causing vortex energy gain.

At low enough temperatures it is to be expected that only scattering processes involving phonons should be relevant. Therefore, we shall use the phonon dispersion relation $w_q=c_sq$ (being $c_s$ the sound velocity) and impose a cutoff momentum to the qp’s. This amounts to neglect all the scattering processes which yield not a phonon as the “outgoing” qp. Note that the “incoming” [*thermal*]{} qp will be surely a phonon. Under such an approximation Eq. (\[delsin\]) can be written as,

$$\begin{aligned}
\psi[w] & =  & \frac{i}{4c_s^2}
\int_{0}^{\infty}\,dw'\, S(w',w'+w) [w'^2 +(w'+w)^2]\nonumber\\
 & \times & [n(w') - n(w'+w)] 
\label{psiwm}\end{aligned}$$

where we introduce the so-called scattering function [@mar], defined as $$S(w',w'') = \sum_{{\bf k},{\bf q} } |\Lambda_{k,q}|^2 
\delta (w'-w_q)\delta(w''-w_k)  \,\,
\label{scat}$$ and being related to the scattering of the environmental excitations between states of frequencies $w'$ and $w''$.

The integral in (\[psiwm\]) has to be solved numerically except in the limit $
T \rightarrow 0 $. In fact, for $\hbar w/k_BT\rightarrow\infty$ the gain term $n(w'+w)$ can be neglected and the loss one $n(w')$ “filters” all but the lowest frequency “incoming” phonons. This means that the factors accompanying $n(w')$ in (\[psiwm\]) can be approximated to lowest order in $w'$. In particular, the scattering function (\[scat\]) is assumed to be a continuous symmetric function of both variables [@mar] satisfying $S(w',w)\simeq S(w)\,w'^p$ for $w'\rightarrow 0$. Thus equation (\[psiwm\]) can be approximated for $T\rightarrow 0$ as follows, $$\begin{aligned}
\psi[w] & \simeq & \frac{i}{4c_s^2}w^2\,S(w)\,\int_0^{\infty}
dw'\,w'^p\,n(w')=\nonumber\\
 & = & \frac{i}{4 c_s^2}
\, p!\,\zeta(p+1)\,w^2\,S(w) 
\left(\frac{k_BT}{\hbar}\right)^{p+1} 
\,\, ,
\label{Ttend0}\end{aligned}$$ where $\zeta(n)\,\,\,(n\geq 2)$ denotes the Riemann zeta function.

It is convenient to notice that our model is unable to provide an [*a priori*]{} explicit form for the scattering function, because we treat vortex and qp excitations as separate entities which are assumed to interact by means of a generic Hamiltonian. Any additional information should be based upon experimental results or more fundamental theories. In fact, for the lowest temperature domain, only theoretical results are at present available and they predict a $T^5$ dependence for the LF [@ior]. Hence, we set $p=4$ in (\[Ttend0\]) and accordingly a low frequency $\sim
w^4$ behavior for the scattering function. To perform numerical calculations which illustrate our results we shall assume a simple form for the scattering function (\[scat\]), as a generalization of the usual super-Ohmic dissipation with an exponential cutoff [@weiss]: $$S(w',w'') \propto
 w'^4 e^{- w'/w_o} w''^4 e^{- w''/w_o} \,\, ,
\label{sca2}$$ being $w_o$ a frequency cutoff parameter which allows us to select only the phonon part of the $^4$He qp excitations spectrum i. e., the frequencies below $w_c\simeq$1.2 ps$^{-1}$. This is evident from Fig. 1 where we plot the one variable scattering function $S(w,w)$ for $w_o$=0.06 ps$^{-1}$. In addition, we plot the frequency spectrum $f(w)$ of the normal fluid density (i. e., $\rho_n=\int_0^\infty
f(w)dw$) for T=0.4 K, which shows that at most the first half of the phonon spectrum makes a relevant contribution to the qp excitations according to a similar behavior for the one variable scattering function.

From Eqs. (\[rein\]) and (\[imin\]) we note that the value of cyclotron frequency, which in turn depends on the vortex mass $m_v$, (see Eq. (\[Ome\])) is necessary for the determination of the drag force. Unfortunately, there is also a controversy regarding the calculation of the vortex mass[@niu; @duan2; @duan; @niu2] that makes the possible value of $\Omega$ to range from a lower bound[@duan2] $\Omega_{min}\simeq 0.1$ ps$^{-1}$ to the upper bound[@do] $ \Omega_{max}\simeq
3$ ps$^{-1}$. This suggests that a study of the drag force dependence on $\Omega$ could be illustrative.

In Fig. 2 we show the coefficients of both, the transverse and the longitudinal components of the force, that is ${\cal R}e
(\gamma)$ and ${\cal I}m (\gamma)$ respectively, as functions of $
\Omega $ for different temperatures. We see that, apart from a change of scale, the shape of curves does not change appreciably along the temperature range $0<T\leq 0.4K$ that we are considering. The only effect seems to be a shift to low frequencies with increasing temperatures (note that for a fixed vortex mass, the cyclotron frequency (\[Ome\]) decrease with increasing temperature due to the factor $\rho_s$, however this variation is negligible for $T<0.4K$).

Regarding ${\cal I}m
(\gamma)$ we first indicate that according to (\[imin\]), the plots of this function in Fig. 2 turn out to be proportional to the Fourier transform of $\dot{\psi}(t)$. In addition, we notice that ${\cal I}m(\gamma)$ vanishes for $\Omega> w_c $. Such absence of dissipation is simply understood as a direct consequence of energy conservation in the scattering processes. This is imposed by the Dirac delta in (\[delsin\]), since it reflects the fact that phonon-to-phonon scattering events can only take place if the vortex energy quantum does not exceed the most energetic phonon of the spectrum. Therefore, we may conclude that dissipation due to phonon$\rightarrow$phonon scattering should be negligible unless the vortex mass yields a value of cyclotron frequency less than $ w_c \simeq$1.2 ps$^{-1}$, which is an intermediate value between the quoted frequencies $\Omega_{min}$ and $\Omega_{max}$. Moreover, the value $\hbar\Omega_{max}$, arising from a usual hydrodynamical prescription for the calculation of the vortex mass[@do], exceeds the energy of any undamped qp excitation[@pist]. So, we may extend our previous conclusion by saying that any dissipation via qp$\rightarrow$qp scattering events should be negligible for such value of vortex mass, suggesting thus a possible scenario of dissipation via multi-qp scattering processes[@marko].

Regarding the TF, we remark that if it were possible to draw some experimental information about at least the direction of this component, this would also shed more light on the value of the vortex mass. In fact, from Fig. 2 we note that at intermediate values between $\Omega_{min}$ and $w_c$, the TF changes its direction. In particular, a positive (negative) value of this component implies $ \Omega > 0.45 $ ps$^{-1}$ ($\Omega < 
0.3$ ps$^{-1}$), while a change of sign yields some intermediate value of $\Omega$. Unfortunately no experimental data are available for temperatures below $ 0.4 $ K.

It is important to note, in concluding this report, that our study has been restricted to a strictly rectilinear vortex, neglecting thus any possible contribution to the forces arising from vortex line curvatures. Clasically, vortex lines can be deformed as helical waves known as Kelvin waves, and there is experimental evidence of similar modes for quantized vortices in Helium II [@do]. From the theoretical viewpoint, such waves could be regarded as a particular form of elementary excitation bound to the vortex. In fact, it has been shown that the elementary excitation spectrum of an imperfect Bose gas in presence of a straight vortex line consists of both, phonons and Kelvin-like waves [@pife]. Viewed from this perspective, being two different kinds of independent elementary excitations, the consequence of scattering processes involving phonons and Kelvin modes appears eventually as a possible secondary effect upon the values of longitudinal and transverse forces. This issue has been scarcely addressed in the literature. In particular, Sonin has long ago performed a simplified study [@sofi] neglecting the vortex velocity due to Kelvin modes, but taking into account the effect of such modes on the scattering of phonons by means of an average over a classical Rayleigh-Jeans distribution for the oscillations of the vortex filament. He concluded that such effect could at most modify the calculated forces for a strictly rectilinear vortex in 1.7 % at $T=0.5K$. Unfortunately, within our formalism a quantitative study of the dynamics of a true three-dimensional curved vortex filament, exhibits a high degree of difficulty.

In summary, starting from a microscopic model we have performed a calculation of the drag force on a moving vortex due to the scattering of qp excitations at temperatures below 0.4 K. We have also discussed, by analyzing both the longitudinal and transverse forces as functions of the cyclotron frequency, some important implications in the determination of possible scattering processes leading to dissipation, according to the values of vortex mass found in the literature.

This work was supported by grant PEI 0067/97 from Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina.

Iordanskii S V 1966 [*Sov. Phys. JETP*]{} [**22**]{} 160 Sonin E B 1997 [*Phys. Rev.*]{} B [**55**]{} 485 Cataldo H M, Despósito M A, Hernández E S and Jezek D M 1997 [*Phys. Rev*]{} B [**55**]{} 3792 Donnelly R J 1991 [*Quantized Vortices in Helium II*]{} (Cambridge: Cambridge University Press) Cataldo H M, Despósito M A, Hernández E S and Jezek D M 1997 [*Phys. Rev*]{} B [**56**]{} 8282 Pitaevskii L P 1959 [*Sov. Phys. JETP*]{} [**8**]{} 888 Thouless D J, Ao P and Niu Q 1996 [*Phys. Rev. Lett.*]{} [**76**]{} 3758 Volovik G E 1996 [*Phys. Rev. Lett.*]{} [**77**]{} 4687 Niu Q, Ao P and Thouless D J 1994 [*Phys. Rev. Lett.*]{} [**72**]{} 1706 Landau L D and Lifshitz E M 1965 [*Quantum Mechanics, Nonrelativistic Theory*]{} (Oxford: Pergamon Press) chap. XVI; Cohen-Tannoudji C, Diu B and Laloë F 1977 [*Quantum Mechanics*]{} (New York: Wiley) vol. I, chap. VI Castro Neto A H and Caldeira A O 1992 [*Phys. Rev.*]{} B [**46**]{} 8858 Weiss U 1993 [*Quantum Dissipative Systems*]{} (Singapore: World Scientific) Duan J M 1994 [*Phys. Rev.*]{} B [**49**]{} 12381 Duan J M 1995 [*Phys. Rev. Lett.*]{} [**75**]{} 974 Niu Q, Ao P and Thouless D J 1995 [*Phys. Rev. Lett.*]{} [**75**]{} 975 Excitations above two times the roton energy should be unstable towards decay into two rotons. See, Pistolesi F 1998 [*Phys. Rev. Lett.*]{} [**81**]{} 397, and references therein. Notice that such conclusions are based upon the Dirac delta energy conserving in (\[delsin\]) that derives from the Markov approximation. The lifetime for heat bath correlations ($\psi(t)$) can be estimated from the inverse of the frequency cutoff in (\[delsin\]) and it is determined by the highest undamped qp energy which is two times the roton energy. This yields a lifetime of order 0.4 ps which should be less than any observation time in the time scale of dissipation $m_v/{\cal I}m(\gamma)$. Such condition should be met for a sufficiently weak vortex-qp coupling, or equivalently for low enough temperatures. From Donnelly’s book[@do] we may identify $\frac{{\cal I}m(\gamma)}{l}=\gamma_0$, where the parameter $\gamma_0$ is a well decreasing function for temperatures below $2K$. The lowest temperature measure of $\gamma_0$ ($1.3K$) yields $\frac{m_v}{{\cal
I}m(\gamma)}=\frac{q_vh\rho_s}{\Omega\gamma_0}=26.8
\Omega^{-1}$. Taking into account a $\Omega^{-1}$ value of at least $\frac{1}{3}$ ps, it is clear that the Markov approximation is very likely to be valid for temperatures below $1.3K$. Pitaevskii L P 1961 [*Sov. Phys. JETP*]{} [**13**]{} 451; Fetter A L 1965 [*Phys. Rev.*]{} [**138**]{} A709. Sonin E B 1976 [*Sov. Phys. JETP*]{} [**42**]{} 469.