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understanding the phase structure of quantum chromodynamics ( qcd ) is one of the key issues in current high - energy physics . thorough phenomenological knowledge of properties of the hadron spectrum as well as nuclear matter is now being complemented by increasingly precise first - principle numerical studies of qcd at nonzero temperature . however , the application of lattice techniques to matter at high baryon chemical potential @xmath1 remains a major challenge due to the infamous sign problem  @xcite . the difficulties encountered in simulations of qcd triggered interest in similar theories which are free of the sign problem . these include simulations at imaginary chemical potential  @xcite , qcd at nonzero isospin density  @xcite , a qcd - like theory with adjoint quarks  @xcite , and two - color qcd  @xcite . the latter , two - color qcd , will be the subject of the present article . two - color qcd differs in several aspects from the world we live in . the most notable difference perhaps is that the colorless baryons are formed from two quarks , and hence are bosons . dense matter is then not realized as an interacting fermi sea of nucleons , but rather as a bose gas of diquarks which undergoes bose einstein condensation ( bec ) at sufficiently low temperature . therefore the ground state of cold and dense two - color quark matter forms a superfluid  @xcite . another noteworthy feature of two - color qcd , stemming from the fact that the @xmath2 gauge group has only pseudo - real representations , is the pauli  grsey symmetry  @xcite connecting quarks with antiquarks . as a consequence , the spectrum of nambu goldstone ( ng ) bosons of the spontaneously broken chiral symmetry contains diquark states in addition to the pseudoscalar mesons . the presence of light particles carrying baryon number ( i.e.  baryonic pions ) is quite peculiar to two - color qcd ( and qcd with adjoint quarks ) and crucial for understanding the phase structure at @xmath3 and @xmath4 by means of the chiral lagrangian approach  @xcite . the model - independent chiral lagrangian arguments have been complemented by investigations in various models such as the linear sigma model  @xcite , the random matrix theory  @xcite , the strong - coupling expansion  @xcite . the nambu  jona - lasinio ( njl ) model was first applied to two - color qcd in ref . @xcite . in fact there has been a tight communication between a number of effective model studies and the monte - carlo simulations on the lattice . recent progress in this direction led to first attempts to probe bec of diquarks and the region of moderate baryon density  @xcite . thus , based on the knowledge achieved in these preceding papers , the present paper aims to make a proposal to use the two - color qcd model as a controllable test setting to clarify the following controversial issues on the real - qcd phase diagram . * in - medium @xmath0 symmetry restoration *  + the qcd lagrangian has global @xmath0 symmetry at the classical level . the quantum anomaly , however , leads to a nonconserving contribution to the axial current , which breaks @xmath0 symmetry explicitly . from the point of view of quantum field theory the anomaly comes from highly ultraviolet modes , and thus , the anomaly should be insensitive to any infrared scales such as @xmath5 , @xmath1 , and the quark mass @xmath6 . in this sense the @xmath0 anomaly is never restored at any @xmath5 nor @xmath1 . in the effective model description the @xmath0 anomaly manifests itself in the form of a @xmath0-breaking interaction which is microscopically induced by instantons  @xcite . because instantons are suppressed at high @xmath5 or @xmath1  @xcite , the @xmath0-breaking interaction is anticipated to weaken in a medium , leading to `` effective restoration '' of the @xmath0 symmetry  @xcite . in fact , a quantitative estimate of the reduction of the @xmath0 effect is crucial for locating the qcd critical point in the @xmath7 phase diagram  @xcite . the numerical study by the lattice monte - carlo simulation is in principle possible in two - color qcd . the clear signal for @xmath0 restoration is degeneracy in the spectra of mesons connected by a @xmath0 rotation  @xcite . that is , in the two - flavor case , the @xmath0 partners are @xmath8 the masses of these multiplets become degenerate when the chiral symmetry is also restored . since the @xmath9 meson involves the so - called disconnected diagrams , the lattice simulation is too noisy to see the degeneracy with @xmath10 ( flavor - singlet @xmath11 ) which is also in a noisy channel . it is , however , feasible to check the degeneracy between @xmath12 and @xmath13 in the finite-@xmath5 and finite-@xmath1 lattice simulation of two - color qcd .  in this paper we will give a quantitative guide from an effective model study .     * superfluidity and quarkyonic matter *  + a new state of matter at high density has been recognized and it is now referred to as _ quarkyonic matter _  @xcite . because the definition of quarkyonic matter is clear only in the large @xmath14 limit , in which color non - singlet interactions are subleading , there seem to be some confusing arguments like that quarkyonic matter overwhelms color superconductivity . this is not quite true especially when @xmath14 is finite . intuitively , quarkyonic matter is characterized by the properties that the thermodynamic quantities ( pressure , baryon density , etc . ) should be almost saturated by the degenerated fermi liquid of quarks and the collective excitations on top of the fermi surface should be colorless ( i.e.  color confined ) . this is actually one of the known properties fulfilled by a certain color - superconducting phase , that is , the color - flavor locked ( cfl ) phase  @xcite . to form a colorless object in the cfl phase , however , we need to treat a meson composed of four quarks , which is technically complicated . instead of real qcd , here , we shall make use of the two - color system ; this exhibits superfluidity at high density which is reminiscent of color superconductivity in qcd , and besides , we need not treat four quarks , since diquarks ( that is , baryons in the two - color world ) can make a color singlet . we will demonstrate that the realization of quarkyonic matter is not so exclusive to disfavor superfluidity and diquark condensation . the goal of our discussions in this part is to make convincing the phase diagram as drawn in fig . [ fig : phase ] . normalized by the `` stefan  boltzmann '' ( see sec . [ sec : phase ] for detailed explanations ) value ranges from 0.4 to 0.6 . ] let us now specify the model that we will use . physics of cooper pairing of quarks ( dense fermionic matter in general ) is well described by njl - type effective models . however , the njl - type models suffer from a serious drawback : the lack of confinement , which may bring about artificial model predictions . in order to capture the essential features of confinement physics and yet maintain the technical simplicity of the njl model , it was augmented with the polyakov loop , equipped with a phenomenological potential designed to reproduce selected lattice data at finite @xmath5 and zero @xmath1  @xcite . ( see also , ref . @xcite for example , for a related approach of the polyakov loop extended quark - meson model . ) this extended model is now called the pnjl model . in spite of its simplicity , the pnjl model has shown remarkable agreement with thermodynamic quantities measured in the lattice qcd simulations . so far , among available finite-@xmath5 and finite-@xmath1 model options at least , the pnjl model is one of the best tools to unveil the qcd phase structure . in this paper , in a sense , we _ downgrade _ the pnjl model into the two - color setting , hoping that our model predictions would guide the future two - color simulations . figure  [ fig : phase ] is actually the phase diagram of two - color two - flavor qcd predicted by means of the pnjl model . we note that the phase diagram is divided into three regimes : one at high @xmath5 consists of deconfined and chiral symmetric particles , which is to be identified as a quark - gluon plasma . the other one at low @xmath5 and low @xmath1 is of course the hadronic phase . the last one at low @xmath5 and high @xmath1 is commonly referred to as a superfluid state . we will later discuss that this state can be regarded as quarkyonic matter so to speak , quarkyonic superfluid of two - color quark matter . keeping in mind this phase structure , we will look at the pole and screening masses of mesons to extract the information on @xmath0 symmetry . the meson spectrum in the pnjl model was first analyzed in ref . @xcite . along the same line , we will perform the calculations and examine the dependence of meson spectrum on the @xmath0-breaking interaction strength . the paper is organized as follows . in sec . [ sec : model ] we introduce the model lagrangian and derive some basic analytic formulas . before going into the numerical study , we describe in sec . [ sec : parfix ] in detail the way we fix the parameters of our model . most of the results have been obtained numerically and are presented in sec . [ sec : results ] . finally , in sec . [ sec : conclusions ] we summarize and conclude . two - color qcd with @xmath15 massless quark flavors has a global @xmath16 flavor invariance at the classical level owing to the pauli  grsey symmetry  @xcite . the axial anomaly explicitly breaks @xmath16 to @xmath17 . in the vacuum , the @xmath17 symmetry is spontaneously broken by the standard chiral condensate down to its @xmath18 subgroup . in the @xmath19 case , which is the subject of the present paper , the symmetry - breaking pattern can be equivalently cast as @xmath20  @xcite . the spectrum of ng modes therefore consists of a single 5-plet , including three pions and two diquarks ( a diquark and an antidiquark  a baryonic and an antibaryonic pion ) . within the njl model , this degeneracy is reflected by the equality of couplings in the meson and diquark channels  @xcite . let us begin the njl analysis with the @xmath0-invariant interaction lagrangian , @xmath21 \,,\end{gathered}\ ] ] where @xmath22 and @xmath23 denote pauli matrices in color and flavor spaces , and @xmath24 the charge conjugation of the dirac spinor @xmath25 with @xmath26 . the interaction @xmath27 is minimal in the sense that it only involves the scalar and pseudoscalar channels with isospin zero and unity . while the above @xmath28 is invariant under a @xmath0 rotation , we further need an interaction which breaks the @xmath0 symmetry . to that end we consider the analogous interaction as follows : @xmath29 \,.\end{gathered}\ ] ] the general interaction lagrangian is thus a sum of these two ; @xmath30 . at @xmath31 only @xmath28 remains and the interaction preserves @xmath0 , whereas at @xmath32 the remaining piece @xmath33 breaks @xmath0 maximally , being equivalent to the two - flavor instanton - induced interaction  @xcite . indeed @xmath34 is a usually used @xmath0-violating parameter  @xcite but we will also use @xmath35 for notation simplicity . in previous works , @xmath36 ( @xmath37 ) was used to discuss the phase structure  @xcite .    performing the hubbard stratonovich transformation in all six channels , we arrive at the total lagrangian as @xmath38 the covariant derivative @xmath39 involves coupling of the quarks to the background gauge field @xmath40 which translates to the polyakov loop in the end . in addition , we note that @xmath6 and @xmath41 denote the current quark mass and the quark chemical potential . later we will introduce @xmath1 to denote the baryon chemical potential ; @xmath42 where 2 comes from the number of colors . the collective fields @xmath43 represent in order the mesons in the scalar - isoscalar , pseudoscalar - isovector , pseudoscalar - isoscalar , and scalar - isovector channels , and the scalar and pseudoscalar diquarks . ( we hereafter omit the subscript `` 0 '' out of @xmath10 and @xmath12 for simplicity . )    in the absence of isospin chemical potential the isovector modes do not develop a vacuum expectation value . moreover , the vafa  witten theorem  @xcite guarantees that the chiral condensate in the vacuum has positive parity . in our model approach , this requires @xmath44 , that is , @xmath45 . ( at the same time , @xmath46 , i.e. , @xmath47 is needed for our mean - field treatment using the hubbard stratonovich transformation @xcite . ) consequently , with the exception of the @xmath0-conserving limit @xmath31 , the scalar chiral and diquark condensates will always be preferred to the pseudoscalar ones . we will therefore take into account only the @xmath9 and @xmath48 condensates . the mean - field thermodynamics of the system is then independent of the parameter @xmath34 , which will only affect the propagation of collective modes , to be discussed in sec .  [ sec : collective ] .      in the pnjl model , one introduces a constant temporal gauge field which couples to the quarks via the covariant derivative . in the polyakov gauge this gauge field is diagonal in the color space , and for the color @xmath2 group it has a form , @xmath49 , where @xmath50 is a real `` phase '' . the order parameter for deconfinement is then the traced polyakov loop given by @xmath51 where @xmath52 is the inverse temperature . in the mean - field approximation , the thermodynamic potential is given by a sum of the gauge and quark parts , @xmath53 in the following , we will refer to the two quark colors for simplicity as the `` red '' and `` green '' . combining the red quark and the green antiquark into the nambu  gorkov spinor , @xmath54 , the background gauge field becomes proportional to the unit matrix in this doubled space and the quark thermodynamic potential can be expressed as @xmath55 where the nambu gorkov hamiltonian reads @xmath56 here @xmath57 is the constituent quark mass and the @xmath9 and @xmath48 now stand for the condensates . the four eigenvalues of the hamiltonian are easily found as @xmath58 and @xmath59 corresponding to the gapped quasiparticle dispersion relations , where @xmath60 the total thermodynamic potential thus becomes @xmath61    + \frac{\sigma^2+\delta^2}{4 g } \\   & - 4\sum_{i=\pm}\int\!\frac{d^3{{\boldsymbol{k}}}}{(2\pi)^3 } \biggl [    e^i_{{{\boldsymbol{k } } } } \ ! + \ ! t\log\left ( 1 \ ! + \ ! 2\phi    e^{-\beta e^i_{{{\boldsymbol{k } } } } } \ ! + \ ! e^{-2\beta e^i_{{{\boldsymbol{k } } } } }    \right)\biggr ] \ , . \end{split } \label{tdpot}\ ] ] the first term is the gauge part @xmath62 having two model parameters @xmath63 and @xmath64 . we assume the simple form motivated by lattice strong - coupling expansion  @xcite . it differs from the three - color expression by a simpler logarithmic term due to the @xmath2 haar measure , and by the rescaled prefactor of the exponential term , which is in general proportional to @xmath65 . note that , as usual in the pnjl model literature , we simulate the effects of gauge dynamics by a constant background temporal gauge field . we then adopt a phenomenological ansatz for the gauge contribution to the mean - field thermodynamic potential , chosen to reproduce selected features of the pure gauge theory . therefore , the parameters @xmath66 only enter the thermodynamic potential since there are no dynamical gauge degrees of freedom in our model lagrangian . it is interesting to recall that in the three - color case the thermodynamic potential in general can not be written in terms of the polyakov loop variable @xmath67 ( and the conjugate @xmath68 ) only , and one has to use two phases analogous to our @xmath50 to parameterize it . on the contrary , in the two - color pnjl model the thermodynamic potential depends just on @xmath67 even in the presence of a diquark condensate . this is because diquarks are colorless . this considerably simplifies the discussion and also avoids technical ambiguities stemming from generally complex effective actions involving the diquark condensate  @xcite . the values of the condensates in thermodynamic equilibrium are determined by minimizing the thermodynamic potential with respect to the variables @xmath9 , @xmath48 , and @xmath67 .    in the quark sector , the effect of the polyakov loop as compared to the simple njl model is to make the replacement @xmath69 in the quasiparticle contribution to the thermodynamic potential . similarly , in the gap equations as well as collective mode propagators , one generalizes @xmath70 , where @xmath71 is the fermi dirac distribution , to @xmath72 \label{eq : mod_f}\\   & = \left(1-\frac{\phi}{\cosh(\beta e)+\phi}\right )   \bigl [ 1 - 2f(2e ) \bigr ] . \label{eq : mod_f2}\end{aligned}\ ] ] the latter two forms ( [ eq : mod_f ] ) and ( [ eq : mod_f2 ] ) of the function @xmath73 illustrate that for @xmath74 thermal excitations are dominated by quark modes with baryon number @xmath75 ( i.e.  deconfinement ) , while for @xmath76 by baryon modes with baryon number @xmath77 ( i.e.  confinement ) . with these replacements , one can readily generalize the results of the pure njl model to the polyakov loop - extended one , as observed in ref . @xcite for the three - color case .    at zero temperature the polyakov loop expectation value is zero for all values of the chemical potential . this led to the suggestion that the pnjl model can naturally describe quarkyonic matter  @xcite at high chemical potential , that is , a phase where chiral symmetry is restored but confinement persists  @xcite . a simple glance at eq .   shows that when @xmath78 , thermal excitations are indeed governed by the term @xmath79 , i.e. , they correspond to colorless baryons . let us take a closer look at how @xmath80 arises as @xmath81 . we consider the gap equation for @xmath67 following from eq . , that is , @xmath82 at high enough @xmath41 and low @xmath5 the system is in the bardeen  cooper  schrieffer ( bcs ) regime where cooper pairing of quarks occurs close to the fermi sea  @xcite . the right - hand side of this equation is then dominated by the particle ( @xmath83 ) part . we can further simplify the calculation by using the high - density approximation , in which we expand the dispersion relation around the fermi surface , @xmath84 , and replace the measure @xmath85 by @xmath86 , where @xmath87 is the density of states at the fermi surface . the integral thus becomes gaussian near @xmath88 and we arrive at the asymptotic result as @xmath89 the polyakov loop itself is hence suppressed exponentially at low temperature . the propagators of the collective modes are easily obtained by a second variation of the effective action that follows from eq .   after integrating the quarks fields out . denoting a set of collective fields symbolically as @xmath90 , the inverse propagator at imaginary ( bosonic matsubara ) frequency @xmath91 is given by @xmath92 , \end{gathered}\ ] ] where @xmath93 and @xmath94 stands for the fermionic matsubara frequencies . the constant @xmath95 is equal to @xmath96 for @xmath97 , to @xmath98 for @xmath99 , to @xmath100 for @xmath101 , and to @xmath102 for @xmath103 . the trace is taken in dirac , flavor , as well as nambu gorkov space . when calculating the partial derivatives of the hamiltonian , that determine the yukawa couplings of quarks to the bosonic modes , we demand that all fields have to be kept as in eq .  . in the diquark condensation phase , the baryon number is spontaneously broken . as a result some of the modes mix and we have to find their dispersion relations by diagonalizing a matrix propagator . this applies to the scalar - isoscalar modes , @xmath104 , as well as pseudoscalar - isoscalar modes , @xmath105 . the only modes that do not mix are @xmath106 and @xmath107 thanks to conservation of isospin and parity . their inverse propagators are trivial in isospin space , namely , @xmath108 . after analytic continuation to real frequencies they acquire the following forms :    @xmath109,\\ d^{-1}_a(\omega,{{\boldsymbol{p}}})=&\frac{\zeta}{2g}-\zeta^2\sum_{i , j=\pm } \sum_{k , l=\pm } \int\!\frac{d^3{{\boldsymbol{q}}}}{(2\pi)^3 }   \frac1{2e^i_{{{\boldsymbol{q}}}+\frac{{{\boldsymbol{p}}}}2 } e^{-j}_{{{\boldsymbol{q}}}-\frac{{{\boldsymbol{p}}}}2 } }   \left(1-ij\frac{m^2-{{\boldsymbol{q}}}^2+\frac{{{\boldsymbol{p}}}^2}{4 } }   { \epsilon_{{{\boldsymbol{q}}}+ \frac{{{\boldsymbol{p}}}}{2 } } \ ,    \epsilon_{{{\boldsymbol{q}}}-\frac{{{\boldsymbol{p}}}}{2}}}\right)\\ & \times\frac{(e^i_{{{\boldsymbol{q}}}+\frac{{{\boldsymbol{p}}}}2}+ik\ ,   \xi^i_{{{\boldsymbol{q}}}+\frac{{{\boldsymbol{p}}}}2})(e^{-j}_{{{\boldsymbol{q}}}-\frac{{{\boldsymbol{p}}}}2 }   + jl\,\xi^{-j}_{{{\boldsymbol{q}}}-\frac{{{\boldsymbol{p}}}}2})-kl\delta^2 }   { \omega+k e^i_{{{\boldsymbol{q}}}+\frac{{{\boldsymbol{p}}}}2 }   + l e^{-j}_{{{\boldsymbol{q}}}-\frac{{{\boldsymbol{p}}}}2 } }   \bigl[\varphi(k e^i_{{{\boldsymbol{q}}}+\frac{{{\boldsymbol{p}}}}2 } )   + \varphi(l e^{-j}_{{{\boldsymbol{q}}}-\frac{{{\boldsymbol{p}}}}2})\bigr ] \ , . \end{split } \label{medium_props}\ ] ]    here , we have changed the momentum notation from @xmath110 to @xmath111 to reserve @xmath112 to take a summation over @xmath113 . it should be noted that in the limit @xmath114 , only the @xmath115 terms survive and the expressions somewhat simplify . the first of the formulas can be used together with the gap equation for @xmath48 to show that in the diquark condensation phase the pion ( pole ) mass is exactly equal to @xmath116 at zero temperature . the proof is straightforward , hence we omit the details .    in the normal phase ( where @xmath117 ) all propagators can be evaluated easily . we provide here the list of expressions that we later in sec . [ sec : num_collective ] use to calculate the masses numerically . to recall that these are the propagators in normal matter , let us write a superscript @xmath118 : @xmath119    propagators of the antidiquarks , @xmath120 and @xmath121 , are obtained from the diquark ones by charge conjugation , i.e.  changing the sign of the chemical potential . armed with these analytic expressions we are now ready to proceed in the numerical calculations . first we will explain our choice of the model parameters . next we will investigate the phase diagram , in which we particularly look into the nature of the superfluid phase . finally we will come to the discussion of the meson spectrum dependence on the @xmath0-breaking parameter @xmath34 . our model has five parameters whose values have to be chosen appropriately : @xmath63 and @xmath64 in the polyakov loop sector , and @xmath122 , @xmath6 , and the sharp three - momentum cutoff @xmath123 in the njl sector . ( the parameter @xmath34 or @xmath124 will be treated as an undetermined free parameter . ) let us first concentrate on the latter . in the three - color njl model one normally fixes the values of @xmath122 , @xmath6 , and @xmath123 from the physical pion mass @xmath125 , decay constant @xmath126 , which are both measured experimentally , and the chiral condensate , which is calculated on the lattice or estimated from the qcd sum rules  @xcite . the authors of ref . @xcite chose to set up the two - color njl model using roughly the same input values , while in ref . @xcite the pion mass , decay constant and the ( three - color ) constituent quark mass were used , resulting in a rather different parameter set . we shall argue here that both these fits may be actually overdetermined and one should be very careful especially when comparing the model outputs with lattice data . the chiral limit qcd , as a yang mills theory coupled to massless quarks in the color fundamental representation , has a single mass scale , namely @xmath127 . all physical quantities should be expressed as an appropriate power of @xmath127 times a dimensionless number . in the njl model one can therefore choose freely a single input quantity , basically just to set the energy unit . dimensionless combinations of observables are a pure prediction of qcd , and shall thus not be tuned to arbitrary values . in particular , either the pion decay constant , the chiral condensate , or the constituent quark mass may serve for this purpose , but not two of them independently . with the fact in mind that the current quark mass is a free parameter in the lattice simulation , we note that the pion mass can in principle acquire any value and represents a new physical scale in addition to @xmath127 .    . the thick lines show the pnjl result ( black solid : chiral condensate in units of the vacuum value @xmath128 ; red dashed : polyakov loop @xmath67 ) , while the thin lines indicate the pure chiral ( njl ) and pure gauge transitions . as expected , the deconfinement transition in the absence of quarks is of second order . ] unfortunately , we are not aware of any lattice data which would provide us with the input needed to fix the njl model parameters unambiguously . in order to make at least an educated guess , we use an argument based on the @xmath14 scaling of physical quantities . using the fact that @xmath126 is proportional to @xmath129 and the chiral condensate to @xmath14 , we rescale the three - color values by factors @xmath130 and @xmath131 , respectively . regarding the polyakov loop sector parameters , the constant @xmath63 is related to the critical temperature @xmath132 for deconfinement in the pure gauge theory by @xmath133 . since @xmath132 in the first approximation does not depend on @xmath14  @xcite , we use the value @xmath134 . the parameter @xmath64 can in principle be adjusted in order to make the chiral and deconfinement crossovers happen at about the same temperature  @xcite . here we use the estimate based on lattice strong - coupling expansion , @xmath135 , where @xmath136 is the physical string tension . the input values as well as the fitted parameter set are summarized in table  [ tab : parameters ] .    in order to check that our parameter set is reasonable , we plot in fig . [ fig : mu0xovers ] the expectation values of @xmath9 and @xmath67 as a function of @xmath5 at @xmath137 . the positions of the two crossovers move very close to each other when the coupling between the quark and polyakov loop sectors is switched on . we note that this observation of simultaneous crossovers is less clear if we use the unrescaled input parameters . 1em 1em       in the vacuum the diquark is degenerate with the pions and its mass is therefore @xmath125 . thus , when @xmath138 exceeds @xmath125 , the diquark condenses and one enters the bec phase which forms a superfluid component . at this moment the constituent quark mass @xmath139 is still rather large . however , as @xmath1 further increases , @xmath139 drops . once @xmath140 , a fermi sea of quarks appears . here we note that the baryon number density , @xmath141 , acquires a contribution from the diquark condensate and becomes nonzero as soon as @xmath142 or @xmath143 . because of the binding energy , naturally , @xmath144 is smaller than @xmath139 , and thus the onset of @xmath145 emerges first and then a quark fermi sea shows up with increasing @xmath41 . in the presence of the fermi sea the diquark condensation is closer to the bcs pairing of quarks sitting near the fermi surface rather than to bec of bound bosonic molecules . there is no phase transition associated with this qualitative change of behavior , so one speaks of a bcs  bec crossover . even though it is not particularly sharp , it can be conveniently defined by the condition @xmath146  @xcite . we will draw the phase diagram of our model in a `` conventional '' way . in the phase diagram the deconfinement crossover is conveniently defined by the condition @xmath147 . the deconfinement temperature is then almost independent of the chemical potential . in fact , the value of the polyakov loop at all temperatures depends on @xmath1 very weakly , as can be seen from fig . [ fig:3dcondensates ] , where all condensates are plotted as a function of @xmath148 and @xmath149 where @xmath125 is fixed to be @xmath150 . this behavior of the polyakov loop can be traced back to the fact that the two - color diquark is a color singlet , and therefore does not break center symmetry to induce nonzero polyakov loop directly .    even at nonzero quark mass the diquark condensate exhibits a clear second - order phase transition as shown in the plot for @xmath151 ( where @xmath128 is the vacuum value of the chiral condensate ) in fig . [ fig:3dcondensates ] . unlike the deconfinement crossover , we can draw a well - defined phase boundary in the phase diagram separating the normal and superfluid phases . as we have already mentioned above , we can confirm that nonzero @xmath48 certainly appears at @xmath142 at @xmath152 .     plane . we indicate the chiral - symmetry breaking ( @xmath153sb ) , bec , and bcs phases . solid line : critical onset of diquark condensation where @xmath48 starts to be nonzero . dashed line : deconfinement crossover ( defined by @xmath147 ) . dash - dotted line : bcs  bec crossover ( defined by @xmath146 ) . ] a compilation of these data leads to the phase diagram we present in fig . [ fig : phase_diagram ] . the solid line represents a second - order phase transition between the normal and superfluid phases . in the superfluid region we have added a dash - dotted line which indicates @xmath146 and can be interpreted as a bcs  bec - type crossover . the dashed line is the deconfinement crossover defined by @xmath147 . figure  [ fig : phase_diagram ] is a basis to understand a more `` advocative '' way of presenting the phase diagram as shown in fig .  [ fig : phase ] . now we are well prepared to discuss the physical meaning of each phase labeled in fig . [ fig : phase ] . the red band which spreads almost straight along the horizontal axis represents the deconfinement crossover . because a crossover has a width and does not have a unique definition , it should be much more reasonable to express the transition region not by a line but a band . the band width in fact tells us how rapid or slow the crossover is . we drew the deconfinement band by the condition that @xmath67 ranges from 0.4 to 0.6 . the blue band showing a sudden decrease around @xmath154 is the chiral crossover defined similarly by the condition that @xmath155 ranges from 0.4 to 0.6 . we see that the two crossovers of deconfinement and chiral restoration take place simultaneously at zero density , and this coincidence persists until around @xmath154 . the pink dotted line represents the superfluid onset , which is a well - defined phase transition . finally the green band which shows behavior similar to the chiral crossover is drawn by the baryon number density @xmath141 . we can compute @xmath141 in the pnjl model and normalize it by the `` stefan  boltzmann '' value . since the pnjl model is a cutoff theory , even the free ( non - interacting ) limit suffers from the cutoff artifact and deviates from the standard formula , which is also the case in the lattice simulation  @xcite . therefore we evaluate the baryon number density in the stefan  boltzmann limit @xmath156 using the pnjl model with @xmath157 and @xmath158 imposed by hand . in this way , we indicate by the green band in fig .  [ fig : phase ] the region in which @xmath159 ranges from 0.4 to 0.6 . because the chiral phase transition controls the dynamical quark mass , it is conceivable that the bcs  bec crossover in fig . [ fig : phase_diagram ] is associated with the chiral crossover . this is indeed the case ; the dash - dotted line in fig . [ fig : phase_diagram ] is covered by the chiral crossover band in fig . [ fig : phase ] . hence , as labeled in fig . [ fig : phase ] , the right - bottom region is characterized by small @xmath67 ( confinement ) and small @xmath155 ( chiral symmetric ) . this is in fact in accord with the identification of quarkyonic matter in refs . @xcite . in this case the blue band is interpreted as the quarkyonic transition . if we use @xmath159 to define the quarkyonic transition according to refs . @xcite , the green band , instead of the blue band , plays the role of the quarkyonic boundary . we would emphasize here that the former criterion makes more physical sense at least in the present case of @xmath160 . actually , our consideration of the bcs  bec crossover provides us with a clear view point on this issue . as we have already discussed , in the chemical potential window @xmath161 , finite baryon number density grows . the carriers of the baryon number are , however , not quarks but baryons ( baryonic pions ) . this is so because @xmath162 in this region . therefore , in the phase region we called bec in fig .  [ fig : phase_diagram ] , the more appropriate physical interpretation should be `` superfluid nuclear matter '' rather than a quarkyonic state in which the pressure is mostly given by fermi - degenerated quarks . this difference in the interpretation makes a contrast to the large-@xmath14 arguments  @xcite , and it still remains a question to which the real world with @xmath163 is closer , infinite @xmath14 or @xmath160 ? answering this question goes beyond our current scope . if the quark - hadron continuity scenario driven by the cfl state is realistic , we can say that the situation at @xmath160 is more relevant . before closing this section we mention the previous studies . in preceding works there was some controversy regarding the order of the phase transition from the diquark condensation phase to the normal phase . in @xcite it was concluded that there is a tricritical point at @xmath149 somewhere in the range @xmath164 , and for higher @xmath1 the transition becomes first order . on the other hand , the authors of  @xcite used the thouless criterion to calculate the critical temperature , which _ assumes _ that the transition is second order . our numerical results for the diquark condensate as a function of @xmath148 and @xmath149 ( see fig .  [ fig:3dcondensates ] ) suggest that the transition is second order everywhere . we have further confirmed the statement that there is no first - order phase transition by looking at the quartic ginzburg landau coefficient in our numerical calculation . detailed computations and arguments are given in the appendix . -breaking parameter @xmath34 . the black solid is @xmath125 , the red dashed @xmath165 , the blue dotted @xmath166 , and the green dash - dotted @xmath167 . we see that @xmath168 and @xmath169 in the @xmath0 symmetric ( @xmath31 ) case . ] we are going to investigate the mass spectrum of collective modes as a function of @xmath5 and @xmath1 for different values of the @xmath0-breaking parameter @xmath34 . a useful starting point therefore is the @xmath34-dependence of the masses in the vacuum . this is shown in fig . [ fig : vacmass ] . it is worth noting that we only display the meson spectrum ; thanks to the unbroken @xmath170 symmetry at @xmath137 , the scalar diquark @xmath48 is degenerate with @xmath171 mesons and the pseudoscalar diquark @xmath172 is degenerate with the @xmath173 mesons . we should perhaps emphasize that whenever we speak of a ( pole ) mass , we have in mind the zero of the real part of the inverse propagator  @xcite . we will therefore sometimes refer to it as the real - part mass . this coincides with the position of the pole if the zero appears below the threshold for decay into quark pairs . another prescription for the mass would be to take the real part of the complex pole of the meson propagator , yielding a somewhat different result , or to compute the spectral function whose peak position and broadness indicate the physical mass and decay width .    in the @xmath0-symmetric limit ( @xmath31 ) , in our calculations , the masses of @xmath10 and @xmath173 are equal to those of @xmath171 and @xmath9 , respectively . the degeneracy of @xmath10 and @xmath171 is generally exact only when the quark mass is strictly zero , so that they are both massless ng bosons . in the ( p)njl model in the mean - field approximation , this exact degeneracy holds regardless of finite quark masses , which is an artifact of the approximation . in fact , @xmath10 and @xmath171 belong to different irreducible representations of the unbroken @xmath170 ; @xmath10 is a singlet and @xmath171 ( with @xmath48 ) form a quintet . the degeneracy is only approximate for small quark mass once higher - order meson loops are taken into account .    off the limit of @xmath31 , both @xmath166 and @xmath167 increase steeply . the typical value of @xmath34 in the three - color njl model with two light quark flavors , that one can obtain either by fitting the physical @xmath174 mass or by a reduction of the three - flavor model , is in the range @xmath175  @xcite . in order to be able to investigate the convergence to the @xmath0-symmetric limit , we consider in the following two particular values , @xmath176 and @xmath177 . that is , if we assume that the vacuum value is @xmath178 , we consider two examples of no reduction at all and @xmath179 reduction of @xmath0 effects . it is quite unlikely that @xmath34 goes to zero in the @xmath5 and @xmath1 range of our interest .     as a function of @xmath148 . the notation for the lines is the same as in fig . [ fig : vacmass ] : the black solid is @xmath125 , the red dashed @xmath165 , the blue dotted @xmath166 , and the green dash - dotted @xmath167 . the thick lines correspond to @xmath166 and @xmath167 at @xmath176 , while the thin lines to @xmath177 . ]    in fig . [ fig : mu0mass ] the meson real - part masses are plotted as a function of @xmath148 while keeping @xmath137 . the reason why some curves exhibit a cusp structure is that the corresponding modes cross the quark antiquark ( or quark  quark in the case of diquarks ) threshold as @xmath5 increases ( and @xmath139 decreases accordingly ) . of course , such a decay into quarks is unphysical and is just an artifact , following from the lack of confinement in the njl model . even in the pnjl model these decay processes are not sufficiently suppressed @xcite , since the coupling to the polyakov loop only imitates confinement in a statistical sense .     as a function of @xmath148 . the notation for the lines is the same as in fig . [ fig : vacmass ] . the black solid is @xmath125 , the red dashed @xmath165 , the blue dotted @xmath166 , and the green dash - dotted @xmath167 . the thick lines correspond to @xmath166 and @xmath167 at @xmath176 , while the thin lines to @xmath177 . ] the real - part masses are difficult to measure on the lattice because the number of lattice sites in the temporal direction is severely limited . instead , one can straightforwardly determine the damping of correlators of the fermion - bilinear interpolating fields for the mesons at large _ spatial _ separations . the exponential decay of the correlations is related to the screening mass of the lightest mode in the selected channel . in the ( p)njl model , this can be found as the pole of the static propagator , @xmath180 , in the complex - momentum plane . the results are shown in fig . [ fig : mu0screen ] . at @xmath152 the real - part and screening masses should coincide thanks to the lorentz invariance . in the model calculation , however , this nice feature is slightly breached by the three - momentum cutoff , but the difference of the two masses in the vacuum turns out to be about a few percent at most . hence the apparent cutoff artifacts are reasonably small . the masses of the chiral partners become degenerate at high temperature , signaling the restoration of chiral symmetry . on the other hand , at any ( fixed ) @xmath181 the masses of the parity partners , connected by a @xmath0 rotation , do not converge even at the highest temperatures considered . from the technical point of view , this is a consequence of the simple structure of the inverse propagators  : when chiral symmetry is restored , the self - energies of the parity partners become equal up to a simple rescaling by @xmath182 . physically , in reality , one should expect the coupling @xmath34 to vary with @xmath5 since it is induced by instantons whose density is exponentially suppressed at high @xmath5  @xcite . if we consider that the @xmath0-breaking interaction strength is proportional to the ( pure ) topological susceptibility , we can infer the @xmath5-dependence from the lattice data in the pure gauge simulation . instead of doing so , in this work , we pick up several values of @xmath34 .    a proper way to understand fig . [ fig : mu0screen ] is thus as follows . at @xmath152 naturally @xmath34 is nonzero , and if precise two - color simulation data is available for @xmath166 and @xmath167 , in principle , @xmath34 can be fixed by the data . we can perform the pnjl model calculations using the determined @xmath34 to go to the higher temperature . if we see a reduction of @xmath166 and @xmath167 toward degenerated @xmath125 and @xmath165 , it is a signal for the effective @xmath0 restoration . we can deduce how far @xmath34 decreases by adjusting @xmath34 to fit @xmath166 and @xmath167 at each temperature . hence , fig . [ fig : mu0screen ] is a demonstration for all these possible investigations once the two - color simulation successfully measures the screening masses in good precision .     and @xmath171 as a function of @xmath149 and @xmath148 in the extreme case of @xmath31 . this quantity indicates the progressive @xmath0 restoration in the hot and/or dense medium . the masses are denoted by @xmath183 with asterisk to make clear that these are in - medium quantities and different from the vacuum @xmath125 which is a fixed parameter of the model ; see table  [ tab : parameters ] . ]    finally , in fig . [ fig:3dmass ] , we plot the difference of screening masses of @xmath173 and @xmath171 as a function of @xmath149 and @xmath148 . since these two modes do not mix even in presence of the diquark condensate , the masses can be calculated straightforwardly also in the bec phase using the in - medium propagators  . the choice of the screening masses instead of the real - part ones here is motivated by the lattice measurement , and also technically favored : while the real - part masses are obscured by landau damping in the diquark condensation phase at @xmath3 , the screening masses remain well defined . the results indicate effective restoration of @xmath0 symmetry at high temperature and/or chemical potential . while in the @xmath5-direction the degeneracy more - or - less copies the restoration of chiral symmetry which is another source of @xmath0 breaking , in the @xmath1-direction the convergence of @xmath184 and @xmath185 is slower as a result of additional @xmath0 breaking by the diquark condensate . we note that fig . [ fig:3dmass ] is the extreme example of @xmath31 , which is not likely near the phase boundaries , but could be the case in the quark - gluon plasma region in view of the lattice data of the topological susceptibility @xcite . in the future , by combining the two - color lattice outputs and the pnjl model analysis , it would be possible to make a 3d plot of @xmath34 which should approach zero at high @xmath5 and/or high @xmath1 . we have adopted the pnjl model as an effective approach to two - color qcd and applied it to the case of two light quark flavors . this is the simplest case which exhibits nontrivial low - energy spectrum due to spontaneous chiral symmetry breaking , and at the same time can be simulated by lattice monte - carlo techniques . we argued that one can fit the parameters of the njl part of the model using physical ( three - color ) observables , but their values have to be rescaled appropriately . once this is done and the quark sector is coupled to the polyakov loop , the model yields locking behavior of chiral and deconfinement crossovers as long as @xmath1 is zero . we checked older results on the phase structure of the two - color njl model in the plane of @xmath148 and @xmath149 , and analyzed its modification induced by the coupling to the polyakov loop . the phase transition between the normal and superfluid phases is second order for all values of the chemical potential considered . in a large part of the diquark condensation phase the expectation value of the polyakov loop is small , which resembles quarkyonic matter predicted using large-@xmath14 arguments . we carefully clarified the realization of quarkyonic matter in the two - color system . the baryon number density @xmath141 appears finite as soon as @xmath1 exceeds the mass of the baryonic pion , but still the quark contribution to @xmath141 is not substantial until @xmath1 surpasses the twice of the dynamical quark mass . after then quark degrees of freedom supersede baryons , meaning a transition from `` superfluid nuclear matter '' into `` quarkyonic superfluid . '' our model analysis of the phase diagram is based on two simplifying assumptions . the first one is the mean - field approximation which treats the system as a gas of noninteracting fermionic quasiparticles . this may not be quantitatively accurate in some regions of the phase diagram such as for @xmath186 at nonzero temperature where the system behaves rather as a dilute bose gas . on the other hand , in the theory of strongly - interacting fermi gases the mean - field approximation is known to be reliable at zero temperature . moreover , the structure of the phase diagram concerning diquark condensation and chiral symmetry restoration is robust , being a direct consequence of the symmetry of two - color qcd . the second assumption is the extrapolation of the gauge part of the thermodynamic potential to nonzero baryon chemical potential . this has been justified for three - color qcd and low baryon chemical potential by a comparison with available lattice data and , in fact , is the source of the predictive power of the pnjl model . therefore , we are confident about the existence of the quarkyonic superfluid phase as depicted in fig .  [ fig : phase ] . on the other hand , the most recent lattice data @xcite suggest that the polyakov loop at a fixed low value of temperature starts to rise at @xmath187 , signalling possible deconfinement . this certainly presents a challenge to the pnjl model and determines the direction of our future research efforts .    finally , we studied the dependence of the spectrum of collective excitations ( scalar and pseudoscalar mesons and diquarks ) on the strength of the axial anomaly . for that sake we introduced a njl - type interaction with a tunable @xmath0-breaking parameter . for all modes we calculated both the real - part mass and the screening mass , which governs the decay of spatial correlators , in order to facilitate a direct comparison with lattice simulations . above the chiral restoration / deconfinement temperature the masses of the chiral partners become degenerate , as expected . at the same time , our results indicate that the restoration of @xmath0 symmetry in terms of the masses of parity partners can not be hidden by chiral restoration unlike the full topological susceptibility , which is good . this would naturally incorporate in the model study the suppression of instanton effects in a hot and/or dense matter . to make this whole argument into a quantitative predictive framework , further physical input from the lattice simulation would be indispensable , e.g.  precise measurement of @xmath167 as a function of @xmath5 and @xmath1 . it would be also interesting that , since the topological susceptibility in the two - color pure gauge theory does not cost much , we can compare the inferred @xmath34 behavior and the suppression of the pure topological susceptibility . the authors are grateful to h.  abuki , j.  o.  andersen , x.  huang , l.  kyllingstad , and w.  weise for useful discussions , and to l.  he for pointing out an error in an earlier version of the manuscript . the work of t.  b.  was supported in part by the alexander von humboldt foundation , and by the extreme matter institute emmi in the framework of the helmholtz alliance program of the helmholtz association ( ha216/emmi ) . k.  f.  is supported by japanese mext grant no.20740134 and also supported in part by yukawa international program for quark hadron sciences . y.  h.  was supported by the grant - in - aid for the global coe program `` the next generation of physics , spun from universality and emergence '' from the mext of japan .    . the solid and dashed lines are the pnjl and njl model results , respectively , with the same input parameters . ] in order to make clear whether the phase transition of diquark superfluidity becomes ( weakly ) first order at high @xmath1 , we performed the ginzburg landau expansion of the thermodynamic potential near the second - order transition line and calculated the coefficient of the quartic term with respect to @xmath48 . in the presence of a tricritical point and the onset of a first - order phase transition , this coefficient would go to zero and then change its sign to negative . the thermodynamic potential @xmath188 depends only on the square of the diquark condensate . in general , @xmath188 depends on other condensates , symbolically denoted by @xmath189 , as well . in the pnjl model , we have @xmath190 , while in the standard njl model the only other condensate would be @xmath9 . to study the behavior of the diquark condensate near the critical temperature , it is most convenient to solve the gap equations for the other condensates , i.e. , @xmath191 . these define @xmath189 implicitly as a function of @xmath192 , and the thermodynamic potential is then a function of @xmath192 solely , that is , @xmath193 . the coefficients of the quadratic and quartic terms in the ginzburg landau functional are now determined by the first and second _ total _ derivatives with respect to @xmath192 , evaluated at @xmath117 . the first derivative vanishes at the transition point by means of the gap equation . the second derivative defines the effective ginzburg landau quartic coupling and is in general expressed as @xmath194 the inverse in the second term is assumed in the matrix sense . the sign of @xmath195 decides whether the transition is of first or second order . in particular in the njl model , this expression acquires a simple form ( with an obvious notation for the partial derivatives ) @xmath196 in the pnjl model , one has to calculate the @xmath197 matrix of second partial derivatives . given the formula for the thermodynamic potential  , these are easily evaluated explicitly at @xmath117 as , @xmath198 \notag\\ & + 4t\sum_{i=\pm}\int\frac{d^3{{\boldsymbol{k}}}}{(2\pi)^3 } \frac1{(\cosh\beta\xi^i_{{{\boldsymbol{k}}}}+\phi)^2 } \,.\end{aligned}\ ] ] the njl limit is recovered by setting @xmath199 . we plot the numerical results in fig . [ fig : glcoupling ] , from which we conclude that the ginzburg landau coupling is always positive , and that the phase transition hence is always second order , within a reasonable chemical potential range ( below the cutoff @xmath123 ) , and in both njl and pnjl models .
we discuss the phase structure of quantum chromodynamics ( qcd ) with two colors and two flavors of light quarks . 
 this is motivated by the increasing interest in the qcd phase diagram as follows : ( 1 ) the qcd critical point search has been under intensive dispute and its location and existence suffer from uncertainty of effective @xmath0 symmetry restoration . 
 ( 2 ) a new phase called quarkyonic matter is drawing theoretical and experimental attention but it is not clear whether it can coexist with diquark condensation . we point out that two - color qcd is nontrivial enough to contain essential ingredients for ( 1 ) and ( 2 ) both , and most importantly , is a system without the sign problem in numerical simulations on the lattice . 
 we adopt the two - flavor nambu  jona - lasinio model extended with the two - color polyakov loop and make quantitative predictions which can be tested by lattice simulations .