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in recent papers it has been proposed a mechanism based on the r - mode instability for the evolution of the internal magnetic field in millisecond accreting compact stars @xcite .
r - modes represent a class of oscillation modes in rotating neutron stars , which are unstable with respect to the emission of gravitational waves @xcite .
only the existence of efficient damping mechanisms can suppress this instability and allows neutron stars to rotate at high frequencies .
an important mechanism for damping is dissipation which can be provided by shear and bulk viscosities converting the rotational energy of the star into heat .
another very efficient source of damping is given by the coupling of the r - modes with the pre - existing poloidal magnetic field of the star : it has been shown that r - modes are responsible for the formation of huge internal toroidal magnetic fields which could reach values of @xmath1  g or larger in the case of newly born neutron stars .
smaller values are obtained in the case of old and accreting neutron stars . a large fraction of the rotational energy of the star is thus stored in the magnetic field  @xcite .
this configuration becomes unstable due to the so - called tayler instability @xcite : a new poloidal component is generated which can then be wound - up itself closing the dynamo loop .
the internal fields evolve into a stable configuration in which the toroidal component can be significantly stronger than the poloidal @xcite .    here
we analyze the evolution of the external magnetic field during / after accretion due to the diffusion outside the core of the magnetic field generated by r - modes .
the increase of the external magnetic field implies a growth of the spin down rate @xmath2 of the millisecond nss .
finally we show that this mechanism can account for the large @xmath0 observed for the two pulsars j1823 - 3021a and j1824 - 2452a .
the toroidal magnetic fields generated in the core of accreting millisecond neutron stars are in the range @xmath3 $ ]  g  @xcite .
these strong toroidal fields can be stabilized with respect to the tayler instability by a much smaller poloidal component  @xcite that we assume in the range @xmath4 $ ]  g. we indicate with @xmath5 the moment at which the new internal poloidal component is fully developed .    in order for the poloidal field to affect the spin frequency evolution of the star it is necessary that the it diffuses outside of the core .
a crucial issue concerns the time - scale @xmath6 on which the internal poloidal component can diffuse .
unfortunately there are no precise estimates about the value of @xmath6 in the presence of a layer of superconducting material in the inner crust .
a detailed description of the diffusion of the internal magnetic field is beyond the scope of this paper .
we limit ourselves to note that if @xmath6 is shorter than a few ten million years the star enters a propeller regime too quickly and it is impossible to accelerate the ns to the higher observed spin frequencies . on the other hand
if @xmath6 is much larger than @xmath7  yr the expulsion of the magnetic field would be phenomenologically irrelevant .
it is thus necessary that the diffusion of the internal poloidal field takes place on a time - scale @xmath8 $ ]  yr .
these estimates are compatible with the theoretical predictions given by @xcite and @xcite .
we begin our analysis at @xmath9 .
we follow the evolution of the external magnetic field @xmath10 while the internal field @xmath11 diffuses outside of the core . for simplicity
we assume an exponential growth of the external magnetic field : @xmath12 \label{evb}\ ] ] where @xmath13 and the initial _ external _ magnetic field is assumed to be @xmath14  g. the latter is a rather typical strength for neutron stars in low mass x - ray binaries ( lmxbs ) .    in figure  [ fig.1 ]
we show the evolution of the _ external _ magnetic field obtained by using eq .
( [ evb ] ) .    in the following
we consider the diffusion of the new generated internal magnetic field separately in the case in which mass accretion is still active and in the case it has terminated .
$ ]  yr . the blue area indicates the value of the internal poloidal component @xmath15 $ ]  g which is about two orders of magnitude smaller than the toroidal component @xmath16 . here @xmath17  g.,width=340,height=264 ]      here
we consider the evolution of a _ recycled _ neutron star in a scenario in which the mass accretion phase ends within a few million years after @xmath18 . in this case
the evolution of the spin period of the star is quite simple because it is not necessary to take into account the interaction between the accretion disk and the magnetic field during the diffusion of @xmath19 .
the total angular momentum of the star satisfies the equation  @xcite : @xmath20 where @xmath21 is the speed of light , @xmath22 is the angular momentum of the star and @xmath10 is obtained from eq .  [ evb ] .    in the following we indicate with @xmath23 the spin period at the time @xmath24 and we treat @xmath23 as a free parameter .
notice that the internal magnetic field can develop only if the star enters the r - mode instability region @xcite .
the upper limit of this region is characterized by a period @xmath25 whose precise value depends on the viscous properties of the star and on the possible existence of the ekman layer  @xcite .
therefore also the parameter @xmath23 is bound by the condition @xmath26  ms .
we stress that @xmath27 and @xmath28 are not the same for every star .
they actually depend on the very complicated evolution of the internal magnetic field which in turn is related to the values of @xmath29 and @xmath30 ( see for instance figure  3 of @xcite ) .
results for the evolutionary paths in the @xmath31 plane are shown in figure  [ fig.2 ] .
the shaded strip includes the possible trajectories of the stars ( the red arrows are included as guidelines for the temporal evolution ) .
the shape of this strip is regulated mainly by the three parameters @xmath6 , @xmath27 and @xmath25 ( @xmath30 plays a marginal role ) .
the width of the strip at the beginning is fixed by @xmath25 and @xmath32 .
the latter parameter corresponds to the minimal period reachable by a recycled msp .
observationally its value is of the order of @xmath33  ms  @xcite .
@xmath27 fixes the width of the strip at late times and @xmath6 regulates the height of the elbow : the smaller @xmath6 the higher the elbow .
stellar objects under study are labeled .
the dashed line corresponds to stars accreting at the eddington limit  @xcite .
the shaded area includes possible paths ( indicated by the red arrows for the case without mass accretion ) of msps after @xmath18 ( see text ) .
notice that the two @xmath34 stars lie well inside the shaded area.,width=340,height=264 ]    remarkably , we can populate the region above the line corresponding to the eddington limit ( dashed line ) .
this is exactly the region where j1823 - 3021a and j1824 - 2452a lie .      if the mass accretion phase lasts tens or hundreds of million years after @xmath18 , the evolutive scenario is more complicated .
the plasma flowing onto the neutron star forms an accretion disk whose inner edge is given by the coupling radius of the magnetosphere ( of the same order of magnitude of the alfvn radius ) @xmath35
the interaction between the magnetic field of the compact object and the conducting material flowing from its companion can provide the necessary torque @xmath36 to spin up the pulsar  @xcite . however , under certain conditions , it is possible that this interaction leads to a torque reversal .
the growth of the external field @xmath10 causes the magnetic boundary @xmath37 to move outward relative to the corotation radius @xmath38 defined as the distance at which the spin frequency of the star is equal to the keplerian frequency .
the star enters a propeller phase when @xmath39 : a centrifugal barrier prevents the material to flow onto the star and a new braking torque @xmath40 acts to slow down the pulsar  @xcite .
( green line ) , @xmath41 ( red line ) and period @xmath42 of the star ( blue line ) .
the dashed black line indicates the moment at which the spin equilibrium phase takes place .
here we assume @xmath43  ms , @xmath44  g and @xmath45 .
_ panel ( b ) _ : frequencies at which the spin equilibrium phase takes place as a function of the mass accretion rate .
we show two examples , corresponding to @xmath46  ms ( gray area ) and @xmath47  ms ( cyan area ) .
the width of the strips corresponds to the uncertainty on @xmath48 ( see yellow area in figure  [ fig.1 ] ) .
the red dot refers to the case shown in panel ( a ) .
, width=340,height=264 ]    in figure  [ fig.3]a we show a typical evolutionary path for an accreting neutron star if the diffusion of the internal poloidal component @xmath11 is taken into account . the growth of the external magnetic field @xmath49 leads to the end of the spin - up phase .
the star evolves into an equilibrium spin phase , it alternates between spin - up and spin - down ( @xmath50 ) and finally it appears as a radio pulsar at the end of the mass accretion phase .    finally ,
from the analysis presented above we can notice that in order to reach high spin frequencies it is necessary a mass accretion rate @xmath51 ( see figure  [ fig.3]b ) .
in the previous section we have presented a general scenario for the evolution of msps when the production and the diffusion of the internal magnetic field is taken into account . in this section
we concentrate on the cases of the two isolated pulsars j1823 - 3021a and j1824 - 2452a .    in the most widely discussed model ,
msps have been re - accelerated through mass accretion , in particular by having spent a fraction of their lives as part of a lmxb .
the way by which they end up as isolated pulsars is still not fully understood .
a possibility is that the companions are destroyed through ablation caused by energetic radiation from the pulsar @xcite .
another possibility requires mass loss from the companion which can take place in very close binary systems due to tidal dissipation @xcite . a totally different scheme to produce an isolated msp is the one in which the pulsar had an interaction with another star or binary in the globular cluster , and was spun - up in the process @xcite . in the following
we will concentrate on the scenario in which msps are produced in lmxbs .
+ the pulsar j1823 - 3021a , located in the globular cluster ngc 6624 , was discovered in 1994  @xcite .
the fermi large area telescope has recently detected the @xmath52 ray counterpart of its pulsations  @xcite with a luminosity of @xmath53 ergs s@xmath54 , the highest observed @xmath52 ray luminosity for any msp .
the pulsation period @xmath42 is @xmath55 ms and the observed spinning down rate is extremely large @xmath56 s s@xmath54 , larger by far than the typical value for other msps .
these properties make j1823 - 3021a an extremely interesting stellar object .
the total observed @xmath52 ray emission implies that a significant fraction of @xmath57 is due to the intrinsic spinning down rate @xmath0  @xcite . within the standard magnetic dipole model for pulsar emission ,
such a large @xmath0 is generated by a large surface magnetic field @xmath58 , larger than @xmath59 g. this fact represents a puzzle for standard msp theory  @xcite : if j1823 - 3021a was spun - up by mass accretion from a companion , due to its large magnetic field @xmath58 , even assuming an accretion rate at the eddington limit @xmath60 , the star would have reached a period @xmath61 ms , which is larger than its present period .
+ the isolated millisecond pulsar j1824 - 2452a is located in the globular cluster m28 at @xmath62  kpc from the sun @xcite .
the pulsation period is @xmath63  ms and the observed spinning down rate is @xmath64 s s@xmath54 .
indirect evidence exists that this large @xmath0 observed is indicative of the true value , e.g. the detection of glitches @xcite , giant pulses @xcite , strong x - ray pulses @xcite and a large second period derivative @xmath65 @xcite . as for j1823 - 3021a ,
the large @xmath0 implies a surface magnetic field @xmath66  g and the combination of values ( @xmath31 ) is difficult to explain in the standard msps theory .
+ note that these anomalous pulsars can be explained in the standard model if the accretion occurred at super - eddington rate .
about such a possibility , we note that all the accreting compact stars detected up to now in lmxbs have steady mass accretion rates @xmath67  @xcite presented in @xcite in parentheses refer to these active periods and not to steady accretion . ] .
moreover , taking into account also the interaction between the external magnetic field and the accretion disk , a minimal value for the mass accretion rate @xmath68 is necessary to accelerate the stars up to the present spin frequency . for lower values of @xmath29
the accreting ns enters a propeller regime that prevents a further acceleration .
in our analysis we have presented a scheme in which the msps can populate the region occupied by j1823 - 3021a and j1824 - 2452a .
it is important to remark that the existence of a propeller phase does not invalidate the possibility to interpret these two special msps in our scheme .
indeed , in figure  [ fig.3]a we show that the spin frequency of the star does not change significantly during the spin equilibrium phase : even if a fast spin - down of the star takes place during the propeller regime @xcite , the star nevertheless will continue to appear as a radio pulsar in the shaded area of figure  [ fig.2 ] . concerning the two anomalous stellar objects they are no more accreting and if a propeller phase took place during accretion it is clearly already over .
the exact position in the @xmath42-@xmath2 plane before a possible propeller phase can not be determined precisely , but it was located at higher frequencies : the existence of this phase would make therefore even more difficult to interpret these two objects within the standard pulsar model ( because they would lie further from the eddington limit ) but it is easy to accommodate in our model .
we have discussed the evolution of the external magnetic field of recycled msps due to the diffusion of strong internal magnetic fields generated by r - modes during the mass accretion phase .
our analysis is based on the previous work of @xcite and @xcite where it has been shown that r - modes generate strong internal magnetic fields in accreting compact stars .
we have shown that the growth of the external magnetic field affects the evolution of the spin - down rate @xmath0 of the msps and that this mechanism can explain the high @xmath0 of j1823 - 3021a and of j1824 - 2452a .
our analysis open new evolutionary paths for the _ recycled _ msps in the @xmath31 plane .
g.p . acknowledges financial support from the italian ministry of research through the program rita levi montalcini .
r - modes can generate strong magnetic fields in the core of accreting millisecond neutron stars ( nss ) . the diffusion of these fields outside the core causes the growth of the external magnetic field and thus it affects the evolution of the spin down rates @xmath0 of the millisecond pulsars ( msps ) . 
 the diffusion of the internal magnetic field provides a new evolutionary path for the msps . 
 this scenario could explain the large @xmath0 of the pulsars j1823 - 3021a and j1824 - 2452a .    
 [ firstpage ]    millisecond pulsars  magnetic fields  r - modes .