The strength of crystalline color superconductors
aa r X i v : . [ h e p - ph ] O c t The strength of crystalline color superconductors
Massimo Mannarelli ∗ , Krishna Rajagopal † and Rishi Sharma ∗∗ ∗ Instituto de Ciencias del Espacio (IEEC/CSIC), E-08193 Bellaterra (Barcelona), Spain † Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA ∗∗ Theoretical Division, Los Alamos National Laboratories, Los Alamos, NM 87545, USA
Abstract.
We present a study of the shear modulus of the crystalline color superconducting phaseof quark matter, showing that this phase of dense, but not asymptotically dense, quark matterresponds to shear stress as a very rigid solid. This phase is characterized by a gap parameter D thatis periodically modulated in space and therefore spontaneously breaks translational invariance. Wederive the effective action for the phonon fields that describe space- and time-dependent fluctuationsof the crystal structure formed by D , and obtain the shear modulus from the coefficients of the spatialderivative terms. Within a Ginzburg-Landau approximation, we find shear moduli which are 20 to1000 times larger than those of neutron star crusts. This phase of matter is thus more rigid than anyknown material in the universe, but at the same time the crystalline color superconducting phase isalso superfluid. These properties raise the possibility that the presence of this phase within neutronstars may have distinct implications for their phenomenology. For example, (some) pulsar glitchesmay originate in crystalline superconducting neutron star cores. Keywords:
Quark matter, neutron stars, color superconductor
PACS:
INTRODUCTION
MIT-CTP 3876LA-UR-07-6534At the large baryon number densities found at the cores of neutron stars, quarks may notbe confined into well-defined hadrons. If quarks are deconfined, since the interactionbetween quarks in the color antisymmetric channel is attractive, quarks near their Fermisurfaces will tend to form Cooper pairs. Because the temperature of neutron stars ismuch smaller than the typical pairing energy, if quark matter exists within neutron stars,it must be in some color superconducting phase [1].At asymptotically large densities where the masses of the u , d and s quarks can allbe neglected, quark matter exists in the CFL phase [2] in which quarks of all threecolors and all three flavors form Cooper pairs with zero total momentum. The diquarkcondensate in the CFL phase is antisymmetric in color indices, driven by the attractivecolor interaction, and antisymmetric in the spin indices. Thus it must be antisymmetricin flavor indices, implying that quarks of a given flavor can only pair with quarks of theother two flavors. In the CFL phase, all fermionic excitations are gapped, with a gapparameter D ∼ −
100 MeV.However, at densities relevant for neutron star phenomenology, meaning quark chem-ical potentials at most m ∼
500 MeV, the strange quark mass M s cannot be neglected.In neutral unpaired quark matter in weak equilibrium, M s induces splitting between theFermi surfaces for quarks of different flavor, which can be taken into account to low-est order in M s / m by treating the quarks as if they were massless but with chemicalpotential splittings dm ≡ ( m u − m s ) / dm ≡ ( m d − m u ) / dm = dm ≡ dm = M s / ( m ) . Note that the splitting between unpaired Fermi surfaces increases withecreasing density. In the CFL phase, the Fermi momenta are not given by these optimalvalues for unpaired quark matter; instead, the system pays a free energy price (cid:181) dm m to equalize all Fermi momenta and gains a pairing energy benefit (cid:181) D m . As a functionof decreasing density, there comes a point (at which dm ≈ D / D issmall enough that CFL pairing cannot survive all the way down to the m at which quarkmatter is supplanted by nuclear matter, then the true ground state of intermediate densityquark matter must have a lower free energy than that of the unstable gCFL phase.Crystalline color superconducting quark matter is a possible resolution of the mag-netic instability of the gCFL phase. Crystalline color superconductivity [5, 6, 7, 8] is theQCD analogue of a form of non-BCS pairing first considered by Larkin, Ovchinnikov,Fulde and Ferrell [9] . This phase may be the ground state of matter in the intermedi-ate density regime in which quark matter is favored over nuclear matter but the Fermisurface separations (cid:181) M s / m are large enough to disrupt CFL pairing. In a crystallinecolor superconducting phase, quarks whose Fermi surfaces are separated (as favored inthe absence of pairing) nevertheless pair. Unlike in conventional BCS phases, the quarksin a pair do not have equal and opposite momenta (meaning that the Cooper pairs havenet momentum) allowing both members in a pair to have momenta near their respective,separated, Fermi surfaces. Such phases do not suffer from the magnetic instability [10].A particular crystalline phase is specified by sets of momentum vectors { q I } , meaningthat Cooper pairs with total momentum 2 q aI form for each q aI ∈ { q I } . All the q aI ’s havethe same magnitude q I ≡ | q aI | = hdm I , with h = . dm I the separationbetween the Fermi surfaces of quarks that pair. The directions of the vectors in { q I } must be determined to get structures with the smallest free energy. In position space, thecondensate is h y i a C g y j b i (cid:181) (cid:229) I e I ab e Ii j D I (cid:229) q aI ∈{ q I } exp ( i q aI · r ) . (1)This is antisymmetric in color ( a , b ), spin, and flavor ( i , j ) (where (1, 2, 3) correspondto ( u , d , s ) respectively) indices and is thus a generalization of the CFL condensate tocrystalline color superconductivity. For simplicity, D is set to 0, neglecting h ds i pairingbecause the d and s Fermi surfaces are twice as far apart from each other as each is fromthe intervening u Fermi surface. Hence, I can be taken to run over 2 and 3 only. { q } and { q } define the crystal structures of the h us i and h ud i condensates respectively.We will analyze crystalline color superconductivity in an NJL model, which gives inthe mean field approximation an interaction term L interaction =
12 ¯ y D ( r ) ¯ y T + h . c ., (2)where the proportionality constant in Eq. (1) is conventionally chosen so that, D ( r ) = ( C g ) (cid:229) I e I ab e Ii j D I (cid:229) q aI ∈{ q I } exp ( i q aI · r ) . (3)he authors of [8] calculated the free energy W of several crystalline structures withinthe weak coupling ( dm , D ≪ m ) and Ginzburg-Landau ( D ≪ dm , D ) approximations,and found qualitative features that make a structure favorable. Two particular structuresthat possess these features, called CubeX and 2Cube45z and described below, have alower W than any other crystal patterns that have been analyzed. One or the other orboth of these two is favored over unpaired quark matter and the gapless CFL phase overthe large range of densities given by [8]2 . D < M s m < . D . (4)For D = M s = m at which nuclear matter supersedes quark matter to well abovethe highest m ∼
500 MeV expected in the cores of neutron stars. The robustness ofthese phases is partly due to their having reasonably large gap parameters D , so largethat the Ginzburg-Landau expansion parameter ( D / dm ) can be around a tenth to afourth meaning that this approximation is at the edge of its validity. Nevertheless, theirimpressive robustness over a large range of m relevant for cores of neutron stars makeit worth considering the phenomenological implications of their presence. We also notethat in the case of a simpler crystal pattern, for which results have been obtained withoutmaking the Ginzburg-Landau approximation, this approximation is conservative in thatit always underestimates both D and the condensation energy [7].In the CubeX crystal structure, { q } and { q } each contain four unit vectors,with { ˆq } = { ( / √ )( ±√ , , ± ) } and { ˆq } = { ( √ )( , ±√ , ± ) } . In the2Cube45z crystal structure { q } and { q } each contain eight unit vectors, with { ˆq } = { ( / √ )( ± , ± , ± ) } and { ˆq } = { ( / √ )( ±√ , , ± ) } ∪ { ( / √ )( , ±√ , ± ) } .For these structures, { ˆq } can be exchanged with { ˆq } by rigid rotations, ensuring thatthere are electrically neutral solutions of the gap equation with D = D = D [8], a factthat we will use in the next Section. PHONONS
The crystalline phases of color superconducting quark matter that we have described inthe previous Section are unique among all forms of dense matter that may arise withinneutron star cores in one respect: they are rigid [11]. They are not solids in the usualsense: the quarks are not fixed in place at the vertices of some crystal structure. Instead,these phases are in fact superfluid since the condensates all spontaneously break the U ( ) B symmetry corresponding to quark number. The diquark condensate, althoughspatially inhomogeneous, can carry supercurrents [5, 11]. And yet, we shall see thatcrystalline color superconductors are rigid solids with large shear moduli. It is the spatialmodulation of the gap parameter that breaks translation invariance, and it is this patternof modulation that is rigid. This novel form of rigidity may sound tenuous upon firsthearing, but we shall present the effective Lagrangian that describes the phonons inthe CubeX and 2Cube45z crystalline phases, whose lowest order coefficients have beenalculated in the NJL model that we are employing [11]. We shall then extract the shearmoduli from the phonon effective action, quantifying the rigidity and indicating thepresence of transverse phonons.The shear moduli of a crystal may be extracted from the effective Lagrangian thatdescribes phonons in the crystal, namely space- and time-varying displacements of thecrystalline pattern [12]. In the present context, we introduce displacement fields for the h ud i , h us i and h ds i condensates by making the replacement D I (cid:229) q aI ∈{ q I } e i q aI · r → D I (cid:229) q aI ∈{ q I } e i q aI · ( r − u I ( r )) (5)in (3). One way to obtain the effective action describing the dynamics of the displace-ment fields u I ( r ) , including both its form and the values of its coefficients within theNJL model that we are employing, is to take the mean field NJL interaction given byEq. (2), but with (5), and integrate out the fermion fields. Since the gapless fermions donot contribute to the shear modulus, one can integrate out the fermions completely forthis calculation. As an aside, we mention that this is not true for the calculation of ther-mal or transport properties, where the gapless fermions do contribute and indeed maydominate, as is the case for the heat capacity [13] and neutrino emissivity [14].After integrating out the fermions, we obtain [11] S [ u ] = Z d x (cid:229) I k I × " (cid:229) q aI ∈{ q I } ( ˆ q aI ) m ( ˆ q aI ) n ( ¶ u mI )( ¶ u nI ) − (cid:229) q aI ∈{ q I } ( ˆ q aI ) m ( ˆ q aI ) v ( ˆ q aI ) n ( ˆ q aI ) w ( ¶ v u mI )( ¶ w u nI ) (6)where m , n , v and w are spatial indices running over x , y and z and where we havedefined k I ≡ m | D I | h p ( h − ) . For D = , D = D = D , and h ≃ . k = k ≡ k ≃ . m | D | . S [ u ] is the low energy effective action for phonons in any crystallinecolor superconducting phase, valid to second order in derivatives, to second order inthe gap parameters D I and to second order in the phonon fields u I . Because we areinterested in long wavelength, small amplitude, phonon excitations, expanding to secondorder in derivatives and in the phonon fields is satisfactory. The Ginzburg-Landauapproximation is an expansion in ( D / dm ) , and so is not under quantitative controlfor the most favorable phases, as we have discussed. But, the main requirement fromglitch phenomenology for the shear modulus is that it should be large, and given that wefind much larger values than those obtained for conventional neutron star crusts, thereis no great motivation to go to higher order to improve the precision. (A higher ordercalculation would include couplings between the different u I , meaning they could nolonger be treated independently.)According to the theory of elastic media [15], the shear moduli can be extracted fromthe phonon effective action. Introducing the strain tensor s mvI ≡ (cid:16) ¶ u mI ¶ x v + ¶ u vI ¶ x m (cid:17) , (7)e then wish to compare the action (6) to S [ u ] = Z d x (cid:229) I (cid:229) m r mI ( ¶ u mI )( ¶ u mI ) − (cid:229) I (cid:229) mnvw l mvnwI s mvI s nwI ! , (8)which is the general form of the action quadratic in displacement fields and whichdefines the elastic modulus tensor l mvnwI . In this case, the stress tensor (in general thederivative of the potential energy with respect to s mvI ) is given by s mvI = l mvnwI s nwI . (9)The diagonal components of s are proportional to the compression exerted on the systemand are therefore related to the bulk modulus of the crystalline color superconductingquark matter. Since unpaired quark matter has a pressure ∼ m , it gives a contributionto the bulk modulus that completely overwhelms the contribution from the condensationinto a crystalline phase, which is of order m D . We shall therefore not calculate thebulk modulus. On the other hand, the response to shear stress arises only because of thepresence of the crystalline condensate. The shear modulus is defined as follows. Imagineexerting a static external stress s I having only an off-diagonal component, meaning s mvI = m = v , and all the other components of s are zero.The system will respond with a strain s nwI . The shear modulus in the mv plane is then n mvI ≡ s mvI s mvI = l mvmvI , (10)where the indices m and v are not summed. For a general quadratic potential with s mvI given by (9), n mvI simplifies partially but the full simplification given by the last equalityin (10) only arises for special cases in which the only nonzero entries in l mvnw with m = v are the l mvmv entries, as is the case for all the crystal structures that we consider.For a given crystal structure, upon evaluating the sums in (6) and then using thedefinition (7) to compare (6) to (8), we can extract expressions for the l tensor andthence for the shear moduli. This analysis, described in detail in [11], shows that in theCubeX phase n = k , n = k , (11)while in the 2Cube45z phase n = k , n = k . (12)We shall see in the next Section that it is relevant to check that both these crystals haveenough nonzero entries in their shear moduli n I that if there are rotational vortices areinned within them, a force seeking to move such a vortex is opposed by the rigidity ofthe crystal structure described by one or more of the nonzero entries in the n I . This isdemonstrated in [11].We see that all the nonzero shear moduli of both the CubeX and 2Cube45z crystallinecolor superconducting phases turn out to take on the same value, n CQM = k = . m D = .
47 MeVfm (cid:18) D
10 MeV (cid:19) (cid:16) m
400 MeV (cid:17) , (13)where m is expected to lie between 350 and 500MeV and D may be taken to lie between5 and 25MeV to obtain numerical estimates.From (13) we first of all see that the shear modulus is in no way suppressed relativeto the scale m D that could have been guessed on dimensional grounds. And, second,we discover that a quark matter core in a crystalline color superconducting phase is 20to 1000 times more rigid than the crust of a conventional neutron star [16]. Finally, onecan extract the phonon dispersion relations from the effective action (6). The transversephonons, whose restoring force is provided by the shear modulus turn out to havedirection-dependent velocities that are typically a substantial fraction of the speed oflight, being given by p / p / RIGID QUARK MATTER AND PULSAR GLITCHES
The existence of a rigid crystalline color superconducting core within neutron stars mayhave a variety of observable consequences. For example, if some agency (like magneticfields not aligned with the rotation axis) could maintain the rigid core in a shapethat has a nonzero quadrupole moment, gravity waves would be emitted. The LIGOnon-detection of such gravity waves from nearby neutron stars [17] already limits thepossibility that they have rigid cores that are deformed to the maximum extent allowedby the shear modulus (13), upon assuming a range of breaking strains, and this constraintwill tighten as LIGO continues to run [18]. Perhaps the most exciting implication of arigid core, however, is the possibility that (some) pulsar “glitches” could originate deepwithin a neutron star, in its quark matter core.A spinning neutron star observed as a pulsar gradually spins down as it loses rotationalenergy to electromagnetic radiation. But, every once in a while the angular velocityat the crust of the star is observed to increase suddenly in a dramatic event called aglitch. The standard explanation [19] requires the presence of a superfluid in someregion of the star which also features a rigid structure that can pin the vortices in therotating superfluid and that does not easily deform when the vortices pinned to it areunder tension. As a spinning pulsar slowly loses angular momentum over years, sincethe angular momentum of any superfluid component of the star is proportional to thedensity of vortices, the vortices “want” to move apart. However, if some vortices arepinned to a rigid structure and so do not move, after a time this superfluid componentof the star is spinning faster than the rest of the star. When the “tension” built up in thearray of pinned vortices reaches a critical value, there is a sudden “avalanche” in whichortices unpin, move outwards reducing the angular momentum of the superfluid, andthen re-pin . As this superfluid suddenly loses angular momentum, the rest of the star,including in particular the surface whose angular velocity is observed, speeds up — aglitch. In the standard explanation of pulsar glitches, the necessary conditions are metin the inner crust of a neutron star which features a neutron superfluid coexisting witha rigid array of positively charged nuclei that may serve as vortex pinning sites. In veryrecent work, Link has questioned whether this scenario is viable because once neutronvortices are moving through the inner crust, as must happen during a glitch, they are soresistant to bending that they may never re-pin [20]. Link concludes that we do not havean understanding of any dynamics that could lead to the re-pinning of moving vorticesin the crust, and hence that we do not currently understand the origin of glitches as acrustal phenomenon.By virtue of being simultaneously superfluids and rigid solids, the crystalline phasesof quark matter provide all the necessary conditions to be the locus in which (some)pulsar glitches originate. Their shear moduli (13) make them more than rigid enough forglitches to originate within them. The crystalline phases are at the same time superfluid,and it is reasonable to expect that the superfluid vortices that result when a neutronstar with such a core rotates have lower free energy if they are centered along theintersections of the nodal planes of the underlying crystal structure, i.e. along linesalong which the condensate already vanishes in the absence of a rotational vortex. Acrude estimate of the pinning force on vortices within crystalline color superconductingquark matter indicates that it is sufficient [11]. So, the basic requirements for superfluidvortices pinning to a rigid structure are all present. The central questions that remainto be addressed are the explicit construction of vortices in the crystalline phase andthe calculation of their pinning force, as well as the calculation of the timescale overwhich sudden changes in the angular momentum of the core are communicated to the(observed) surface, presumably via the common electron fluid or via magnetic stresses.Much theoretical work remains before the hypothesis that pulsar glitches originatewithin a crystalline color superconducting neutron star core is developed fully enoughto allow it to confront data on the magnitudes, relaxation timescales, and repeat ratesthat characterize glitches. Nevertheless, this hypothesis offers one immediate advantageover the conventional scenario that relied on vortex pinning in the neutron star crust. Itis impossible for a neutron star anywhere within which rotational vortices are pinned toprecess on ∼ year time scales [21], and yet there is now evidence that several pulsarsare precessing [22]. Since all neutron stars have crusts, the precession of any pulsar isinconsistent with the pinning of vortices within the crust, a requirement in the standardexplanation of glitches. On the other hand, perhaps not all neutron stars have crystallinequark matter cores — for example, perhaps the lightest neutron stars have nuclear mattercores. Then, if vortices are never pinned in the crust but are pinned within a crystallinequark matter core, those neutron stars that do have a crystalline quark matter core canglitch but cannot precess while those that don’t can precess but cannot glitch. CKNOWLEDGMENTS
The work of MM has been supported by the “Bruno Rossi" fellowship program. Thisresearch was supported in part by the Office of Nuclear Physics of the Office of Scienceof the U.S. DOE under grants
REFERENCES
1. K. Rajagopal and F. Wilczek, arXiv:hep-ph/0011333; M. G. Alford, Ann. Rev. Nucl. Part.Sci. , 131 (2001) [arXiv:hep-ph/0102047]; G. Nardulli, Riv. Nuovo Cim. , 1 (2002)[arXiv:hep-ph/0202037]; M. G. Alford, K. Rajagopal, T. Schaefer and A. Schmitt, arXiv:0709.4635[hep-ph].2. M. G. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B , 443 (1999) [arXiv:hep-ph/9804403].3. M. Alford, C. Kouvaris and K. Rajagopal, Phys. Rev. Lett. , 222001 (2004)[arXiv:hep-ph/0311286].4. R. Casalbuoni, R. Gatto, M. Mannarelli, G. Nardulli and M. Ruggieri, Phys. Lett. B , 362 (2005)[Erratum-ibid. B , 297 (2005)] [arXiv:hep-ph/0410401]; K. Fukushima and K. Iida, Phys. Rev. D , 074011 (2005) [arXiv:hep-ph/0501276].5. M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev. D , 074016 (2001)[arXiv:hep-ph/0008208].6. J. A. Bowers and K. Rajagopal, Phys. Rev. D , 065002 (2002) [arXiv:hep-ph/0204079].7. M. Mannarelli, K. Rajagopal and R. Sharma, Phys. Rev. D , 114012 (2006)[arXiv:hep-ph/0603076].8. K. Rajagopal and R. Sharma, Phys. Rev. D , 094019 (2006) [arXiv:hep-ph/0605316]; J. Phys. G , S483 (2006) [arXiv:hep-ph/0606066].9. A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. , 1136 (1964)[Sov. Phys. JETP , 762(1965)]; P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964); S. Takada and T. Izuyama, Prog.Theor. Phys. , 635 (1969).10. M. Ciminale, G. Nardulli, M. Ruggieri and R. Gatto, Phys. Lett. B , 317 (2006)[arXiv:hep-ph/0602180].11. M. Mannarelli, K. Rajagopal and R. Sharma, Phys. Rev. D, to appear, arXiv:hep-ph/0702021.12. R. Casalbuoni, E. Fabiano, R. Gatto, M. Mannarelli and G. Nardulli, Phys. Rev. D , 094006 (2002)[arXiv:hep-ph/0208121].13. R. Casalbuoni, R. Gatto, M. Mannarelli, G. Nardulli, M. Ruggieri and S. Stramaglia, Phys. Lett. B , 181 (2003) [Erratum-ibid. B , 279 (2004)] [arXiv:hep-ph/0307335].14. R. Anglani, G. Nardulli, M. Ruggieri and M. Mannarelli, Phys. Rev. D , 074005 (2006)[arXiv:hep-ph/0607341].15. L. D. Landau and E. M. Lifschitz, Theory of Elasticity , 3rd edition, Oxford, Pergamon (1981).16. T. Strohmayer, H. M. van Horn, S. Ogata, H. Iyetomi and S. Ichimaru, Astrophys. J. , 679 (1991).17. B. Abbott et al. [LIGO Scientific Collaboration], Phys. Rev. D , 042001 (2007)[arXiv:gr-qc/0702039].18. B. Haskell, N. Andersson, D. I. Jones and L. Samuelsson, arXiv:0708.2984 [gr-qc]; L. M. Lin,arXiv:0708.2965 [astro-ph]. The analogous calculation for a star that is rigid throughout was done inB. J. Owen, Phys. Rev. Lett. , 211101 (2005) [arXiv:astro-ph/0503399].19. P. W. Anderson and N. Itoh, Nature , 25 (1975). See [11] for references to subsequent literature.20. B. Link, talk given at INT Workshop on The Neutron Star Crust and Surface, Seattle (2007) andprivate communication.21. A. Sedrakian, I. Wasserman and J. M. Cordes, Astrophys. J. , 341S (1999)[arXiv:astro-ph/9801188]; B. Link, Astron. Astrophys. , 881 (2006) [arXiv:astro-ph/0608319];B. Link, Astrophys. and Space Sci. , 435 (2007).22. I. H. Stairs, A. G. Lyne, S. L. Shemar, Nature, , 484 (2000); T. V. Shabanova, A. G. Lyne,J. O. Urama, Astrophys. J. , 321 (2001); A. E. Chukwude, A. A. Ubachukwu and P. N. Okeke,Astron. Astrophys.399