Color-magnetic flux tubes in quark matter cores of neutron stars

# Color-magnetic flux tubes in quark matter cores of neutron stars

Mark G. Alford Physics Department, Washington University, St. Louis, MO 63130-4899, USA    Armen Sedrakian Institute for Theoretical Physics, J. W. Goethe-University, D-60438 Frankfurt am Main, Germany
10 Feb 2010
###### Abstract

We argue that if color-superconducting quark matter exists in the core of a neutron star, it may contain a high density of flux tubes, carrying flux that is mostly color-magnetic, with a small admixture of ordinary magnetic flux. We focus on the two-flavor color-superconducting (“2SC”) phase, and assume that the flux tubes are energetically stable, although this has not yet been demonstrated. The density of flux tubes depends on the nature of the transition to the color-superconducting phase, and could be within an order of magnitude of the density of magnetic flux tubes that would be found if the core were superconducting nuclear matter. We calculate the cross-section for Aharonov-Bohm scattering of gapless fermions off the flux tubes, and the associated collision time and frictional force on a moving flux tube. We discuss the other forces on the flux tube, and find that if we take in to account only the forces that arise within the 2SC core region then the timescale for expulsion of the color flux tubes from the 2SC core is of order years.

###### pacs:
97.60.Jd,12.38.Mh,47.32.C-,03.65.Ta,67.10.Jn

## I Introduction

It has long been conjectured that neutron stars might contain cores of quark matter, and one of the challenges facing nuclear astrophysics is to find signatures by which the presence of such matter could be inferred from observations of the behavior of neutron stars. This requires us to develop a good understanding of the differences between the properties of nuclear matter and quark matter, taking in to account the effects of magnetic fields, which are known to be present in neutron stars. In this paper we study quark matter in magnetic fields  Gauss, which are astrophysically plausible and high enough to affect transport (see for example Huang:2009ue ()) but not so large as to modify the phase structure of the material Ferrer:2006vw (); Fukushima:2007fc (); Menezes:2008qt ().

Nuclear matter at high densities and low temperatures is expected to be a type-II electrical superconductor, with the magnetic field distributed in an Abrikosov lattice of flux tubes Baym:1969 (). In this paper we investigate the possibility that quark matter in the two-flavor color superconducting phase (“2SC”) Alford:2007xm () could be a type-II superconductor with respect to the color gauge fields Iida:2002ev (); Giannakis:2003am (), with color flux tubes that scatter electrons, muons, and ungapped quarks via the Aharonov-Bohm effect. These tubes are not topologically stable, and their energetic stability has not yet been determined; in this paper we investigate the role they might play in transport, and their expulsion time, if they turn out to be stable or to have a lifetime that is sufficiently long. As we explain below, the tubes carry flux that is mostly color-magnetic (hence they can reasonably be called “color-magnetic flux tubes”) with a small admixture of ordinary magnetic flux. We will argue that the density of color-magnetic flux tubes could be high, perhaps only about an order of magnitude less than that of ordinary flux tubes in superconducting nuclear matter. Color magnetic flux tubes may appear in other color superconducting phases, such as the color-flavor-locked (CFL) phase, but the CFL phase has no gapless charged excitations, so in this paper we focus on the 2SC phase. We calculate the Aharonov-Bohm interaction between the flux tubes and unpaired quarks or electrons/muons. We calculate the associated damping time and the forces on the flux tubes. We defer the calculation of other contributions to relaxation and transport in the 2SC phase, such as scattering of the unpaired quarks and electrons off each other, to future work.

The behavior of quark matter phases in magnetic fields is complicated by the intertwined breaking of the strong interaction “color” gauge symmetry and the electromagnetic gauge symmetry. In the 2SC phase, a condensate of Cooper pairs of up () and down () quarks leads to the gauge symmetry breaking pattern Alford:1997zt (); Alford:1999pb (). The unbroken symmetry ensures confinement of particles that carry net red or green color, with a confinement scale around 10 MeV Rischke:2000cn (). The unbroken gauge symmetry is a linear combination of the original electromagnetic and color symmetries, called “rotated electromagnetism”. The associated gauge field, the “ photon”, is a combination of the original photon and one of the gluons. It is massless and propagates freely in 2SC quark matter. The orthogonal combination is a broken gauge generator, and the associated magnetic field has a finite penetration depth.

The situation is closely analogous to the Higgs mechanism in the standard model, where one linear combination of the hypercharge and gauge bosons remains massless (the photon), while the orthogonal combination becomes massive (the ). The flux tubes are therefore analogous to “-strings” Vachaspati:1992fi () which have been found to be stable only in a small region of the standard model parameter space James:1992wb (), although the stable region may be enlarged when bound states are taken in to account Vachaspati:1992mk (). There are differences between the 2SC phase of QCD and the Higgs phase of the standard model: the gluon mass is proportional to the quark chemical potential, not the superconducting order parameter Alford:2007xm (); the non-Abelian gauge group is rather than and is only partly broken, leaving an unbroken confining as well as an unbroken in the low temperature phase. This means that a separate stability calculation will be needed for the 2SC case.

Because electromagnetism is much more weakly coupled than the strong interaction, the massless gauge field is almost identical to the photon, with a small admixture of a color gauge boson. Conversely, the broken gauge field is almost identical to one of the gluons, with a small admixture of the photon Alford:1999pb (). Thus the flux tubes can be described as “color-magnetic flux tubes”. However, because they contain a small admixture of ordinary magnetic flux, they interact with electrons/muons as well as with unpaired (blue) quarks. In summary, the 2SC phase is not a superfluid, but it is a superconductor with respect to the gauge fields, and a conductor with respect to the gauge fields, with current mainly being carried by the gapless electrons and blue quarks (one of which is neutral, the other has charge +1). Strange quarks and muons, if present, will have a lower Fermi momentum because of their higher mass, and hence less phase space near their Fermi surface. Thus their contribution to the processes discussed in this paper will be subleading, and we ignore it.

The picture given above is valid below the critical temperature for 2SC pairing and above an unknown critical temperature at which there will be a transition to a phase in which there is self-pairing of the blue up and down quarks. Such pairing would break the symmetry, so there could be both and flux tubes. Models of the strong interaction between quarks do not give us much idea of the value of . They agree that, because the strong attraction is much weaker in the single-color channel, will be many orders of magnitude lower than the critical temperature for 2SC pairing, perhaps as low as 1 eV ( K) Alford:1997zt (); Schafer:2000tw (); Alford:2002rz (). In this paper we will be concerned with temepratures above , where the gauge symmetry remains unbroken.

Depending on the ratio of the -flux penetration depth to the coherence length of the condensate, the 2SC phase may be type-I or type-II with respect to the magnetic field Iida:2002ev (). In this paper we will be concerned with the possibility of type-II behavior, and the presence of flux tubes containing -flux in the 2SC quark matter core of a compact star. Even if the average magnetic field strength in the core is below the lower critical field, such flux tubes may end up “frozen in” if the quark matter had cooled in to the 2SC state in the presence of the magnetic field. The magnetic field would then be resolved in to a part, which would pass freely through the 2SC quark matter, and an part, which would become trapped in flux tubes (Sec. III).

The paper is structured as follows. In Sec. II we calculate the Ginzburg-Landau parameter for 2SC quark matter, and conclude that it is a type-II superconductor with respect to the broken generator as long as the pairing gap is large enough. We estimate that will suffice, which for typical quark chemical potentials requires MeV. In Sec. III we discuss the nucleation scenario by which the flux tubes can occur in the 2SC superconductor, even when the magnetic field intensities are below the lower critical field. We estimate the density of such flux tubes in the hypothetical 2SC quark matter core of a neutron star. In Sec. IV we calculate the Aharonov-Bohm scattering cross section for electrons or unpaired quarks interacting with color magnetic flux tubes. Sec. V is devoted to the computation of relaxation time of massless electrons and unpaired blue quarks interacting with flux tubes via Aharonov-Bohm cross-section. In Sec. VI we estimate the timescale for expulsion of the flux tubes from the 2SC core, taking in to account the forces on the color-magnetic flux tubes in the 2SC core and at its boundary, but neglecting any forces on the magnetic flux lines outside the core. We summarize our results in Sec. VII.

In our calculations we use “Heaviside-Lorentz” natural units with , where is the Boltzmann constant and is the vacuum permittivity; the electric charge is related to the fine structure constant by .

## Ii Type-II color superconductivity in quark matter

A superconductor is of type II if it obeys the condition

 κ≡λξ>1√2, (1)

where is the Ginzburg-Landau (GL) parameter, is the penetration depth, and is the coherence length for the superconductor. In a system of relativistic fermions with chemical potential and pairing gap , we expect , , so . (In the case of 2SC quark matter the relevant broken gauge symmetry is the “” which is mostly color, so the coupling is approximately the strong coupling constant.) We therefore expect that 2SC quark matter will be a type-II color superconductor if the gap is sufficiently large.

To make a more accurate determination we follow the approach of Bailin and Love Bailin:1983bm () and Iida and Baym Iida:2002ev (). We start with the effective free energy density (Ginzburg-Landau theory) for a relativistic BCS superconductor (Ref. Bailin:1983bm (), (3.12))

 F=Fn+αψ∗ψ+12β(ψ∗ψ)2+γ(∇ψ∗−2ieAψ∗)(∇ψ+2ieAψ)+12μ0(B−μ0H)2 . (2)

(We have followed Ref. Bailin:1983bm () in writing the magnetic field free energy in SI units; in natural units .) Here is the gap parameter; for negative the free energy has a minimum at , with penetration depth , and coherence length given by

 ψ20=−αβ,λ2=12γq2pair|ψ0|2,ξ2=−γα , (3)

where is the charge of the Cooper pair. The GL parameter is then given by

 κ2=λ2ξ2=12q2pairβγ2. (4)

The coefficients in the Ginzburg-Landau functional are Bailin:1983bm ()

 α=ντGL2,β=ν7ζ(3)16(πTc)2,γ=β6p2Fμ2, (5)

where . The fermions have Fermi momentum , so the density of states near the Fermi surface is . The parameter is a degeneracy factor that is for a single-species system, and for the 2SC phase (see Ref. Bailin:1983bm (), Eq. (4.63)). The Ginzburg-Landau theory is most reliable for temperatures close to , however we will use it at . The low-temperature gap parameter is related to the critical temperature by : note that then differs from by a factor of about 1.7. Expressing the coefficients in terms of ,

 κ≈32.74qpair√NΔμ, . (6)

We can check this result by noting that for a relativistic electronic superconductor, and where . Substituting these values into (6) we find , in agreement with Ref. Bailin:1983bm (), (3.24).

In 2SC quark matter, the degeneracy factor is and the charge of the Cooper pair is the charge of the 2SC condensate. From Eqs. (29) and (31) of Sec. IV we find

 qpair=qce(X)=g√3cosφ≈g√3, (7)

where the mixing angle is defined in Eq. (29). We estimate the strong coupling constant by assuming that , so . Substituting these values into (6) we find

 κ2SC≈11Δμq . (8)

We conclude, using (1), that 2SC quark matter will be of type II if the pairing gap is sufficiently large, . In quark matter we expect , so this only requires the 2SC pairing gap to be greater than about 25 MeV, which is well within typical estimates Brown:1999aq (); Brown:1999yd (). Our general conclusion agrees with that of Refs. Iida:2002ev (); Giannakis:2003am () who also noted that a sufficiently large 2SC pairing gap yields a type-II superconductor. Our specific result (8) differs from Eq. (112) of Ref. Iida:2002ev () by a factor of , but given the uncertainty in the strong coupling constant this numerical discrepancy does not affect our conclusion.

## Iii Color-magnetic flux tubes in the 2SC phase

### iii.1 The nucleation and density of flux tubes

When the quark matter core of the star cools below a critical value, a 2SC condensate forms. We expect that this happens before the nuclear mantle becomes superconducting because the gap parameter for quark matter is expected to be an order of magnitude larger than that for proton pairing Alford:2007xm (); Dean:2002zx (); Muther:2005cj (); Sedrakian:2006xm (). The electromagnetic field is then resolved in to a component and an component. The 2SC core is not a superconductor with respect to , so the component is undisturbed Alford:1999pb () (on this we disagree with Ref. hep-ph/0012383 (), which we believe imposes an incorrect boundary condition on the gluon field). However, the core is a superconductor with respect to the component, and we have argued above that it may well be a type-II superconductor. The lower critical field for the -superconductivity is very high,  Gauss Iida:2002ev (), and typical neutron star magnetic fields are expected to be lower than this, but, as we now argue (see also Alford:1999pb () and footnote [8] of Ref. Iida:2002ev ()), it is still quite possible for the -flux to form flux tubes threading the quark matter core. The only way the -flux could be expelled from the core is if the transition from hot quark matter to 2SC happens smoothly from the center of the star outwards. However, it seems more likely that the transition to 2SC matter will proceed by nucleation of 2SC regions (“bubbles”) in the quark matter, which then grow and coalesce. The -flux will be expelled from the 2SC bubbles, but will then be trapped in the non-superconducting regions between the bubbles. As the bubbles grow, these regions become smaller, concentrating the flux there until the local field strength rises above , at which point the bubbles stop growing. At this stage, the core consists of 2SC quark matter with channels of non-superconducting quark matter running though it, carrying the -flux. If the 2SC phase is a type-II superconductor then these channels are unstable and will fragment into flux tubes, each carrying a single quantum of -flux, with a short-range repulsion between the flux tubes. The fact that the average field strength was below the lower critical field for a sphere of 2SC matter in a uniform magnetic field will now manifest itself as an outwardly-directed boundary force on the flux tubes at the point where they meet the edge of the 2SC core. We will study this in Sec. VI.

Because the 2SC phase is a conductor with respect to charge, it supports eddy currents which make it very difficult for the magnetic field in the 2SC core to change. The timescale for expulsion of the magnetic field is estimated to be longer than the age of the universe Alford:1999pb (). Thus we are justified in treating the magnetic field as a fixed background.

If we assume that all the -flux is trapped in the manner described above, then the density of flux tubes is just , the density of magnetic -flux, divided by , the -flux of a single flux tube. is obtained by projecting out the -component of the original electromagnetic flux (see (27)), so . The flux quantum is

 ΦX=2πqpair≈√3παs , (9)

where is the -charge of the 2SC condensate (see (7) and (29)). We can relate it to the flux quantum for an ordinary superconductor where the charge of the condensate is ,

 ΦX=2eqpairΦ0=6sin(φ)Φ0 . (10)

We conclude that

 nv=BXΦX=16BΦ0 . (11)

This is the upper limit on the flux tube density in 2SC matter. Interestingly, as anticipated in Ref. Blaschke:2000gm (), it only differs by a factor of from the density of electromagnetic flux tubes that would result if the core were an electromagnetic superconductor due to electron or proton pairing. Projection on to the component reduces the magnetic flux by a factor , but because the fields are strongly coupled their flux quantum is smaller by a similar factor, so the flux tube density ends up being independent of the mixing angle. The actual density will depend on details of how the transition to 2SC matter was completed. For an internal field  Gauss (), the maximum flux tube density is .

### iii.2 Properties of the flux tube

The thickness of the flux tubes is given by the penetration depth for magnetic -flux in the 2SC phase. This follows from equations (4) to (7). Assuming for relativistic quarks,

 λ=3πgμq|τGL|1/2=(1.3fm)(400 MeVμq)(1−TTc)−1/2 . (12)

The energy per unit length (tension) of the flux tube is given by where is the energy per unit length of the magnetic flux if it were uniformly spread over an circle of radius (Ref. Tinkham (), Sec. (5.1.2)), and is a factor of order 1. In Heaviside-Lorentz natural units , where , so

 εX=Φ2X4πλ2lnκX (13)

(compare Ref. Iida:2002ev (), Eq. (107); see also Ref. Blaschke:2000gm ()). To estimate the tension we work to lowest order in and use (12), (9), and (34). In the low temperature limit we find

 εX=μ2q3πlnκX . (14)

Assuming that in 2SC quark matter is in the 350 to 500 MeV range, and that the logarithmic factor is of order 1, we conclude that the tension will be of order to  MeV/fm.

## Iv Aharonov-Bohm scattering by flux tubes

The Aharonov-Bohm effect provides a remarkably strong interaction between a charged particle and a flux tube containing magnetic flux. For the simple case of a single gauge group (electromagnetism), the differential cross-section per unit length is (see, for example, Ref. Alford:1988sj ())

 dσdϑ=sin2(π~β)2πksin2(ϑ/2), (15)

where

 ~β=qpqc , (16)

where is the charge of the scattering particle. For a flux tube that arises as a topological soliton in an Abelian Higgs model, is the charge of the condensate field whose winding by a phase of characterizes the flux tube; is the momentum in the plane perpendicular to the string, and is the scattering angle. Aharonov-Bohm scattering has several important features:

• The cross-section vanishes if is an integer, but is otherwise non-zero.

• The cross section is independent of the thickness of the flux tube: the scattering is not suppressed in the limit where the symmetry breaking energy scale goes to infinity, and the flux tube thickness goes to zero.

• The cross section diverges both at low energy and for forward scattering.

It is therefore of great interest to determine the values of for scattering of the fermions that are ungapped in the 2SC phase off a flux tube containing magnetic flux associated with the broken gauge symmetry.

### iv.1 The gauge groups and charges

#### iv.1.1 The light fermions

In the 2SC phase we will focus on the gauge group consisting of electromagnetism and the part of the color gauge symmetry that mixes with electromagnetism. The relevant particles are the quarks and the electron:

 ψ=(ru,gd,rd,gu,bu,bd,e−), (17)

where “” means the red up quark, etc. “” is the electron. Muons would have the same interaction as the electron, so we do not include them separately. In this basis, the generators of the two gauge groups are just the diagonal matrices of their electric and color charges,

 Qψ=diag(+23,−13,−13,+23,+23,−13,−1),Tψ=12√3diag(1,1,1,1,−2,−2,0). (18)

The normalization of is fixed by the conventional electric charges of the particles. For we have used the conventional normalization for generators of the color gauge group Donoghue_Golowich_Holstein (). The kinetic term in the lagrangian of the fermions is , where the covariant derivative of the fermion fields is

 Dμψ=∂μψ−ieAQμQψψ−igATμTψψ . (19)

The electromagnetic gauge coupling is , and the QCD gauge coupling is . The photon gauge field is , and the gluon gauge field is . With the normalization of (18), , and .

#### iv.1.2 The 2SC condensate

The 2SC condensate is a diquark condensate,

 ϕij=⟨ψiCγ5ψj⟩ , (20)

where the indices and live in the color-flavor space of (17). The condensate only involves the red and green up and down quarks, so its color-flavor structure is

 ϕ∝⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝01000001000000000−100000−10000000000000000000000000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ . (21)

From (18) we can see how , considered as a matrix in the color-flavor space of (17), transforms under an infinitesimal electromagnetism or color rotation. Each of the quarks in the diquark feels its own color-flavor phase, so each index and is separately transformed:

 Qϕϕ=Qψ⋅ϕ+ϕ⋅Qψ,Tϕϕ=Tψ⋅ϕ+ϕ⋅Tψ, (22)

where the “” on the right hand side signifies ordinary matrix multiplication of the two matrices, and we have used the fact that and are both diagonal, and hence symmetric.

The lagrangian of the 2SC condensate (i.e.  the G-L theory) contains the kinetic term , where the covariant derivative is

 Dμϕ=∂μϕ−ieAQμQϕϕ−igATμTϕϕ . (23)

This determines the coupling of the 2SC condensate to the gauge fields.

### iv.2 The broken/unbroken basis

When the 2SC condensate forms, one linear combination of and is spontaneously broken: we will call it “”. The other remains unbroken: we will call it “”,

 ~Q=Q+η1T ,X=−η2Q+T . (24)

We determine by requiring that the 2SC condensate be invariant under gauge transformations,

 ~Qϕϕ=0 , (25)

which implies that

 η1=−1√3 . (26)

It is natural to work in the basis rather than the basis, so we define new “rotated” gauge fields

 A~Q=cosφAQ−sinφAT,AX=sinφAQ+cosφAT, (27)

where the mixing angle is analogous to the Weinberg angle in the standard model which parametrizes the mixing of the hypercharge and gauge bosons to yield the photon (analogous to here) and the (analogous to here). It is important that the mixing of the gauge fields is expressed in terms of an angle, so it maintains their normalization, so the gauge field kinetic terms for and are still conventionally normalized. In the case of the generators, which we defined in (24), the overall normalization is not important, since it is absorbed in the new gauge couplings.

In the new basis, the covariant derivative of the fermions is

 Dμψ=∂μψ−ie(~Q)A~Qμ~Qψψ−ie(X)AXμXψψ . (28)

We will determine the new gauge couplings and , and the mixing parameters and , by requiring that (28) be equivalent to (19) for all gauge field configurations.

### iv.3 X-charges of the particles and condensate

Flux tubes will contain magnetic -flux, so to determine the Aharonov-Bohm scattering parameter for each particle, we need to find the -charge of each particle, corresponding to in (16). This follows straightforwardly from (28). We will also need to know the -charge of the 2SC condensate, corresponding to in (16).

Requiring that (28) be equivalent to (19) for all gauge field configurations, and using (26), we find

 cosφ=√3g√e2+3g2η2=−e2√3g2=−√3tan2φe(~Q)=√3eg√e2+3g2=ecosφe(X)=√3g2√e2+3g2=gcosφ . (29)

There is a new “rotated” electromagnetism, with coupling , which is slightly smaller than the usual electromagnetic gauge coupling. The charges of the fermions under this gauge group are

 ~Q=diag(12,−12,−12,12,1,0,−1) . (30)

This agrees with the well-known results for the 2SC phase Alford:2007xm ().

The action of the -charge matrix on the 2SC condensate determines the -charge of the condensate, in units of ; from (21), (24), and (29),

 Xϕ+ϕX=qcϕ,whereqc=1√3(1+e23g2)=1√3cos2φ. (31)

The -charge matrix of the fermions is

 X=1√3diag(12+2tan2φ,12−tan2φ,12−tan2φ,12+2tan2φ,−1+2tan2φ,−1−tan2φ,−3tan2φ). (32)

Dividing by (31) we find the Aharonov-Bohm -factors of the fermions, in the basis defined by (17),

 ~βψ=diag(12+32sin2φ,12−32sin2φ,12−32sin2φ,12+32sin2φ,−1+3sin2φ,−1,−3sin2φ) . (33)

Expanding in powers of (since ), we find

 sin2(φ)≈α3αs (34)

so to lowest order in ,

 ~βψ=diag(12+α2αs,12−α2αs,12−α2αs,12+α2αs,−1+ααs,−1,−ααs) . (35)

We conclude that the gapped quarks have close to , which means that they have near-maximal Aharonov-Bohm interactions with an -flux tube. Among the lighter (and hence more phenomenologically relevant) fermions, the -neutral has zero Aharonov-Bohm interaction with the flux tube, while the and electron have the same Aharonov-Bohm factor

 sin(π~βbu)=sin(π~βe)≈−πααs . (36)

## V Relaxation via scattering off flux tubes

### v.1 Relaxation time calculation

In this section we compute the characteristic timescale for a perturbation from equilibrium to relax away due to scattering of the fermions off the color magnetic flux tubes. This relaxation time is a measure of the mean free time between collisions of the fermions with the flux tubes, so we will also refer to it as a collision time. Our calculation applies equally to electrons and the unpaired component of blue colored quarks in the 2SC phase, the key difference being factors in the cross-section. The Boltzmann kinetic equation for blue-quark/electron distribution function is

 ∂f(p,t)∂t = 2πNvV∫d3p′(2π)3{W(p;p′)f(p′,t)[1−f(p,t)] (37) − W(p′;p)f(p,t)[1−f(p′,t)]}δ(ε(p)−ε(p′)),

where is the number of flux tubes, is the volume, and is the transition probability between the states described by momenta and . Time-reversal symmetry implies . In equilibrium the fermion distribution function is given by the Fermi-Dirac distribution function

 f0(p)=11+exp[(p−μi)/T] (38)

where is the temperature and is the chemical potential of blue quarks () and electrons . To solve the Boltzmann equation we shall apply the variational method, where the perturbations from equilibrium are described by variational trial functions whose functional form is dictated by the form of applied perturbation Flowers:1976 (); Flowers:1979 (). The resulting transport coefficients are lower bounds on their exact values. The number of adjustable trial functions, which are used to maximize the entropy production via scattering, could be large. In the following we shall use one linear function , in which case there is no need for variation, since the variational parameter cancels out. It should be kept in mind that the resulting transport coefficients are still lower bounds on their exact values. For small perturbations from equilibrium the Boltzmann equation can be linearized by writing where the (small) perturbation from the Fermi-Dirac form (38) is

 δf(p,t)=−df0(p)dε(p)ϕ(p,t), (39)

where is the trial function. The linearized Boltzmann equation then reads

 −∂ϕ(p,t)∂tf0(p)[1−f0(p)] = 2πNvV∫d3p′(2π)3[ϕ(p,t)−ϕ(p′,t)] (40) × W(p;p′)f0(p′)[1−f0(p)]δ(ε(p)−ε(p′)).

To obtain this form of the kinetic equation we used the detailed balance conditions and It is convenient to work with the Laplace transformed trial function

 ϕ(p,t)=∫dse−stϕ(p,s). (41)

Upon Laplace transforming Eq. (40) we find

 sϕ(p,s)f0(p)[1−f0(p)] = 2πNvV∫d3p′(2π)3[ϕ(p,s)−ϕ(p′,s)] (42) × W(p;p′)f0(p′)[1−f0(p)]δ(ε(p)−ε(p′)).

To define a characteristic relaxation rate we assume that the trial function can be written as

 ϕ(p,s)=ϕ(p)δ(s−s0), (43)

in which case Eq. (42) becomes

 s0ϕ(p)f0(p)[1−f0(p)] = 2πNvV∫d3p′(2π)3[ϕ(p)−ϕ(p′)] (44) × W(p;p′)f0(p′)[1−f0(p)]δ(ε(p)−ε(p′)),

where the perturbation functions are now independent of . We can identify with the relaxation rate (i.e. the inverse of the relaxation time) by comparing the computed kinetic coefficients with standard expressions for transport coefficients, e.g. , the electrical conductivity with the Drude formula.

To formulate the variational principle Ziman () we write Eq. (44) in the compact form

 X(p)=∫[ϕ(p)−ϕ(p′)]P(p,p′)d3p′, (45)

where stands for the left-hand side of Eq. (44); the scattering operator is easily read-off from the kernel on the right-hand side of Eq. (44). Since the factor and the transition probability are positive definite, the operator is positive definite. Further it is linear and self-adjoint (symmetric). Following ref. Ziman () we define an inner product

 ⟨ϕ,ψ⟩≡∫ϕ(p)ψ(p)dp, (46)

in terms of which

 ⟨ϕ,Pψ⟩≡12∫dp∫dp′[ϕ(p)−ϕ(p′)]P(p,p′)[ψ(p)−ψ(p′)]. (47)

The variational principle states that the expression

 ⟨ϕ,X⟩=⟨ϕ,Pϕ⟩ (48)

attains its maximum for the exact value , which satisfies Eq. (45); for any other trial function , that satisfies Eq. (48), . Explicitly, Eq. (48) reads

 s0∫d3p(2π)3ϕ(p,s)2f0(p)[1−f0(p)] = 2πNvV∫d3p(2π)3∫d3p′(2π)312[ϕ(p,s)−ϕ(p′,s)]2 (49) W(p;p′)f0(p′)[1−f0(p)]δ(ε(p)−ε(p′)).

It is also straightforward to check that the variation of Eq. (49) leads us back to “equation of motion” (44). From Eq. (49) we obtain the variational relaxation rate

 s0 = 2πNvVD∫d3p(2π)3∫d3p′(2π)312[ϕ(p)−ϕ(p′)]2W(p;p′)f0(p′)[1−f0(p)]δ(ε(p)−ε(p′)),

where

 D=∫d3p(2π)3ϕ(p)2f0(p)[1−f0(p)]. (51)

The exact relaxation rate . We specify the form of the trial function appropriate to the problem at hand, which is the relaxation of uniform blue-quark/electron velocity on a flux tube, as

 ϕ(p)=p⋅v C(p2), (52)

where is the scalar part of the trial function. In the following we will adopt the simple choice . The differential transition probability can be obtained from the Aharonov-Bohm scattering cross-section, Eq. (15), and is given by (for details see Appendix A)

 dW=2πδ(ε′−ε)2πδ(pz−p′z)4Lsin2(π~β)sin2(ϕ/2)12εVd3p′(2π)32ε′, (53)

where the initial and final state momenta and energies, and , are unprimed and primed respectively (in this section we use for the scattering angle, as opposed to in Sec. IV). Here we have used the cylindrical coordinates coaxial with the flux tube to write . Combining Eqs. (V.1) and (53) and carrying out the phase space integrals (the details are given in Appendix B) we obtain, to lowest order in the low-temperature expansion,

 s0 = p3Fiv2nvT6π2D sin2(π~β), (54)

where is the blue-quark/electron Fermi momentum, is the density of flux tubes. Eq. (51) with the trial function (52) can be computed in the low-temperature limit by approximating to obtain , where is the fermion fluid velocity (52). The relaxation rate for particles of species scattering off flux tubes of area density is then given by

 τ−1if≡s0=nvpFisin2(π~βi) . (55)

It easy to understand the final result (55). It is of the standard form for classical gases , where is the speed of the particles, is the density of scattering centers, and is the cross section for Aharonov-Bohm scattering. Eq. (55) is relevant for thermal relaxation of the gapless fermion species in the 2SC phase. One of these, the blue down quark, has no A-B interaction with the flux tubes (). The other two, the electron and blue up quark, have identical A-B factors (36) although their Fermi momenta are different.

### v.2 Comparison with Coulomb scattering

To find out whether scattering off flux tubes is likely to be an important source of relaxation, and hence a significant contributor to transport properties, it is useful to compare Eq. (55) with the collision time for screened Coulomb scattering via exchange of photons.

The 2SC phase is a -conductor, with two species of gapless charged fermions: the quarks (with -charge +1 and chemical potential ) and the electrons (with -charge +1 and chemical potential ). There may also be muons, but their Fermi momentum will be much smaller. As mentioned in the introduction, the red and green quarks will be confined to bound states whose mass is expected to be of order 10 MeV Rischke:2000cn (), so they play no role in transport at neutron star temperatures. Since , the quarks are more numerous than the electrons and have a larger phase space near their Fermi surface, so they will make the largest contribution to the collision time.

The Coulomb collision time depends on the in-medium photon spectrum, which will be affected by Debye screening and Landau damping arising from the presence of gapless charged excitations, dominantly the quarks because of their larger phase space. A simple estimate can be obtained by assuming that the dispersion relation is dominated by a plasmon pole. The plasma frequency is given by

 ω2p=~αnqμq=43π2~αμ2q (56)

where is the fine structure constant for the “rotated” electromagnetism (29). The collision frequency is given by (see Eq. (10),(12),(18) of Shternin:2006uq ()),

 τ−1qq=8ζ(3)μ2qπ3ω2p~α2T=6ζ(3)π2~αT (57)

where . This result is valid for , which is the relevant regime for neutron stars since is in the range. Eq. (57) is analogous to Ref. Heiselberg:1993cr ()’s Eq. (62) for the thermal conduction timescale, with electromagnetic interactions (so their is replaced by ) and a different number of quark species. The quark-quark Coulomb collision frequency (57) is proportional to temperature whereas the particle-flux-tube collision frequency is independent of temperature. We can therefore define a temperature below which flux tubes dominate the relaxation of deviations from thermal equilibrium. From (57) and (55) we find

 Tf=π26ζ(3)sin2(πβbu)~αnvμq . (58)

To make a numerical estimate we assume that the 2SC core contains the maximum flux tube density given by Eq. (11), and that . Using (36) and the fact that , we find

 (59)

We conclude that for reasonable values of the magnetic field, only at very low temperatures is Aharonov-Bohm scattering off flux tubes likely to be an important source of thermal relaxation. However, it is important to note that the thermal relaxation timescale is not the only one that is relevant to transport. There is also the viscous relaxation rate (Ref. Heiselberg:1993cr (), Eq. (51)) and the momentum relaxation rate (Ref. Heiselberg:1993cr (), Eq. (32)) both of which have a much stronger () suppression at low temperatures. We defer a full discussion of transport in the 2SC phase to later work.

## Vi Forces on the flux tubes

We argued in Sec. III that even if the magnetic field in the core of the star is below the lower critical field, color magnetic flux tubes will still be produced in the transition to the 2SC phase. In this section we study the forces on those flux tubes, and estimate the timescale for their expulsion from the 2SC core. For this initial estimate we take in to account only the forces on the flux tubes within the 2SC core, or at its boundary. Depending on the nature of the material surrounding the core there may be additional forces, and these may modify the expulsion time in a way that would have to be calculated on a case-by-case basis.

The velocity of the flux tube is , the velocity of the normal fluid is , and the velocity of the 2SC condensate is . The forces we consider are mutual friction (“mf”), the non-dissipative (lifting) Magnus-Lorentz force (“ML”), the Iordanskii force (“Iord”), forces arising from zero modes (“zm”), and boundary forces (“bf”) at the quark-hadronic boundary. We assume that local magneto-hydrostatic-gravitational equilibrium is established quickly after the transition to the 2SC phase, so there is no additional buoyancy force 1985SvAL…11…80M (); 2009MNRAS.397.1027J (). We note that there may be additional forces due to density-dependence of the 2SC pairing gap Hsu:1999rf (), but we do not include these since there is as yet no reliable estimate of the density dependence. The equation of motion of a flux tube then has the form

 mVdvLdt=fmf+fML+fIord+fzm+fbf , (60)

where is the effective mass of a flux tube per unit length and is a force per unit length. The boundary forces tend to pull the flux tube in a radial direction, expelling it from the 2SC core. This is resisted by the combination of the other forces.

In our calculations we will assume that the flux tubes are straight. A bent flux tube will feel an additional restoring force determined by its tension.

### vi.1 The background ~Q magnetic field

In our calculations we will neglect the effect of the magnetic field that penetrates the 2SC core. Because of this field, -charged particles, including the quarks and electrons, will feel a Lorentz force. This will have a significant effect on the behavior of the normal fluid of quarks and electrons when the cyclotron frequency becomes larger than the inverse of the characteristic time for equilibration, which as we argued in Sec. V.2, is the quark-quark collision time (57). The dominant component of the fluid is the quarks, with -charge , and , so

 ωc=eBpF , (61)

and we can neglect the effects of the magnetic field on transport when , where

 ωcτqq=2π33ζ(3)1√4παBμqT=0.32(B1012G)(108KT)(400MeVμq) . (62)

We conclude that only for high magnetic fields (above  G) or low temperatures (below  K) might the magnetic field affect thermal relaxation. We defer a discussion of this regime to future work.

### vi.2 Mutual friction

Mutual friction is a frictional force on a flux tube arising from its Aharonov-Bohm interaction with the normal fluid of gapless particles through which it is moving. Consider a vortex moving relative to the normal fluid with velocity . In the relaxation time approximation

 fmf=τ−1ifnv∫d3p(2π)3pf0(p,u), (63)

where is the collision rate between fermions of species and flux tubes (55).

We will assume that the blue up quarks dominate the friction. This is reasonable because the blue down quarks have no charge and hence no Aharonov-Bohm interaction with the flux tube, and the electron Fermi momentum is smaller than that of the blue quarks. The equilibrium Fermi-Dirac thermal distribution of the quarks is

 f0(p,u)={exp[(ε−μi+p⋅u)/T]+1}−1 , (64)

where the term is a correction due to the motion of the vortex relative to the thermal bath with velocity . We will compute the force to linear order in . The leading contribution arises at the first order in velocity

 fmf=τ−1ifnv∫d3p(2π)3p(p⋅u)∂f0(ε)∂ε=ηu. (65)

Carrying out the integral and using (55) we obtain the mutual friction drag coefficient

 η=pFiniτ−1ifnv=nisin2(π~βi) (66)

where is the fermion density and is their Aharonov-Bohm factor (35). As one would expect, the friction coefficient is independent of the magnetic field (i.e. the density of flux tubes). It is proportional to the fermion density, so, as noted above, the quark contribution will dominate the electron contribution.

### vi.3 Boundary forces

Next we wish to calculate the force exerted on the flux tube at the point where it reaches the interface between the 2SC quark matter core and the nuclear mantle of a neutron star (Fig. 1). When the -magnetic flux tube reaches the edge of the 2SC core it combines with the magnetic flux in the core to re-constitute the ordinary magnetic field from which it was originally formed. The form in which the flux continues through the nuclear mantle, and hence the boundary energy, may therefore be influenced by the state of the nuclear matter. In this analysis we will include only the forces arising from the contribution due to the 2SC core itself. We briefly discuss other contributions, but a proper analysis including them would have to be done in the context of a specific model of the whole neutron star and the properties of all regions within it.

The outward force per unit length on the flux tube is (see Fig. 1)

 fb=1ldEdr=rR2−r2εX . (67)

Here should be the total energy of magnetic flux inside and outside the core, but we neglect the outside contribution; is the energy per length of flux tubes. Then from (14),

 fb≈rR2−r2μ2q3πlnκX . (68)

Taking in to account the energy of the magnetic field outside the core will reduce the right hand side of (67), and weaken the outward force on the flux tube. We now discuss the magnitude of such terms in various cases.

If the nuclear mantle is a type-II superconductor, the magnetic field penetrates the nuclear mantle in the form of Abrikosov flux tubes (dashed line in Fig. 1). From (11) we know that each flux tube will spawn 6 Abrikosov flux tubes in the nuclear matter, each of which has energy per unit length

 εnuc=Φ204πλ2nuclnκnuc . (69)

, so if we assume that the logarithmic factor is of order 1 then for in the 50 to 100 fm range, is in the 0.2 to 0.7 MeV/fm range. This means that even when multiplied by a factor of 6, is small in comparison with the tension of the flux tube, which is greater than 10 MeV/fm (14), so (68) is still a good estimate of the boundary force. Of course, in a type-II nuclear mantle there may be other forces, for example if it is also a superfluid there may be entanglement of Abrikosov flux tubes with superfluid vortices, but we neglect those here because they depend on details of the nuclear mantle.

If there is no Cooper pairing of the protons then the nuclear matter is a conductor. In this case the energy gained from shortening the flux tube is counteracted by the field energy of the magnetic field it connects to in the nuclear matter mantle. The criterion for the tension of the flux tube to dominate is the same as the criterion for the magnetic field to be below its lower critical value. Since neutron star magnetic fields are well below the lower critical field for the 2SC phase, we can assume that the 2SC flux tube tension will dominate and we can use (68) again. The only complication is that conducting nuclear matter supports eddy currents which will resist any change in the magnetic field in the nuclear mantle. This may make it much harder to move the -flux tubes in the 2SC core. Again, we do not attempt to include such forces that depend on details of the constitution of the nuclear mantle.

If the nuclear mantle were a type-I proton superconductor Sedrakian:2004yq (); Alford:2005ku (); Alford:2007np (); Charbonneau:2007db () then flux tubes in the 2SC core would connect to non-superconducting domains in the nuclear mantle Sedrakian:2004yq (); Charbonneau:2007db (). In this case we cannot compute the boundary force because the domain structure of the type-I proton superconductor is not known; the possible (layered, cylindrical, etc) structures in type-I superconductors essentially depend on the history of the nucleation of the superconducting phase.

### vi.4 Magnus-Lorentz force

The Magnus-Lorentz force is a non-dissipative force, directed orthogonally to the flux tube velocity, that arises from the superposition of the winding “flow” of the 2SC order parameter around the flux tube and the background flow of the charged superfluid of fermions PhysRev.140.A1197 (); 1991ApJ…380..530M (); PhysRevB.55.485 (). (There is controversy about this in the literature; for example, Jones 1991MNRAS.253..279J (); 2009MNRAS.397.1027J () has suggested that this is cancelled by another contribution from ungapped fermions. Pending a definitive resolution of this disagreement we will use the standard form of the Magnus-Lorentz force.)

The Magnus-Lorentz force per unit length on a flux tube is

 fML=−(jX×^nΦX) , (70)

where is the -flux through the flux tube (9), is a unit vector pointing along the flux tube, and is the current of charge seen by the flux tube, arising from the charge density of the 2SC condensate, moving relative to the flux tube

 jX=ρX(vS−vL) . (71)

We can write where is the density of quarks in the condensate. Since there are 4 quark species in the condensate, and at low temperature all fermions are part of the condensate,

 fML=−ρ(vS−vL)×^n ,ρ≡ρXΦX=πns=4μ33π . (72)

Note that the charge of the Cooper pairs cancels in this expression.

### vi.5 Iordanskii force

The mutual friction force described above is the force on the flux tube in the longitudinal direction (i.e. parallel to its velocity relative to the normal fluid of unpaired quarks), due to Aharonov-Bohm scattering of the unpaired quarks. The Iordanskii force is the transverse component of that same force PhysRevB.55.485 (),

 fIor