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more than thirty years after the first discussions about a phase transition in qcd @xcite , only small portions of the phase diagram have been explored . although the complete phase diagram of qcd is of high dimensionality , experiments can at best explore a three dimensional section of the full phase diagram . further , collisions of heavy - ions have only a single control parameter , the cm energy , @xmath0 . as a result they explore a single line in this three - dimensional phase diagram . by varying the nuclei being collided , one could , perhaps , extend the search to a small patch around the line . the field is wide open for new ideas on experimental coverage of the qcd phase diagram . theoretical work is no less constrained by the tools of the trade . in regions of high symmetry ( for example , when the quark masses vanish ) , universality arguments @xcite have been used to put constraints on the phase diagram . such arguments are realized in models , for example , effective meson models , four - fermi models or random matrix models , which have the same symmetries as qcd . the resulting predictions of universal properties , , the order of the transition and critical indices , are expected to coincide with qcd . since the locations of phase transitions are not universal , models should be used to constrain the topology of the phase diagram rather than quantitative predictions of the location of phase boundaries or critical points . when the symmetries are broken , as in the real world , the usefulness of these models is curtailed further . weak coupling methods for qcd give precise quantitative predictions , but for high temperatures and densities , when the qcd coupling is small enough . lattice computations were long confined to the region with vanishing chemical potentials , extensions to finite chemical potential being constrained by the fermion - sign problem . the first systematic non - perturbative treatments of qcd at non - vanishing chemical potential using lattice methods have now begun , and the first results are now available @xcite . in spite of these limitations , tremendous progress has been made . as we demonstrate in this paper , known results can now be uniquely exended , using only thermodynamic arguments , to yield the topology of the full phase diagram of two flavour qcd and strongly constrain it for three flavours . the topology of the phase diagram of qcd is constrained by its symmetries . it is well - known that qcd possesses a set of approximate global symmetries , called flavour symmetries , related to the phases of quark wavefunctions . qcd with two flavours of massless quarks would possesses a chiral symmetry @xmath1 ( @xmath2 and @xmath3 are transformations on left and right handed quarks respectively ) . the up and down quark masses ( @xmath4 , @xmath5 respectively ) break this symmetry . since @xmath6 is non - zero the chiral symmetry is broken to the diagonal vector symmetry , @xmath7 , called isospin . since @xmath8 is of the order of a few mev , and much smaller than the scale , @xmath9 , chiral symmetry is approximately valid , being broken at the level of 510% . the mass difference @xmath10 is non - zero but small , thus violating isospin symmetry by a small amount . there is also a semi - light quark flavour , the strange , which has mass , @xmath11 , comparable to @xmath9 . including this extends the chiral group to @xmath12 @xcite . according to data , this symmetry is broken at the 25% level down to the two - flavour symmetry @xmath1 @xcite . in addition to these approximate flavour symmetries there is also the exact phase symmetry @xmath13 whose charge is the baryon number . the axial phase symmetry , @xmath14 , is broken at @xmath15 by instantons . there is mounting evidence that this symmetry is not restored through a phase transition @xcite . in this paper we shall assume that there is no @xmath14 restoring phase transition in qcd . the large global symmetries of qcd can be broken in many ways , thus giving rise to a complicated phase diagram . in this paper we confine ourselves to the phase diagram at small chemical potentials . by this we mean that the order parameters involve pairing of quarks and antiquarks . at larger chemical potentials there are other interesting phases where the pairing could be between two quarks @xcite . we do not examine these phases in this work . a range of intermediate chemical potentials may exist where both quark - antiquark and quark - quark pairings need to be taken into account @xcite . if this is so , then some details of the phase diagrams presented here would have to be extended .    since the phase diagram is structured around the breaking of chiral and flavour symmetry , one might expect that it is independent of the number of colours , @xmath16 . this is correct for @xmath17 . for the specific case of @xmath18 , however , the fact that quark representations are real means that the chiral symmetry is enhanced . as a result , the considerations below do not apply to @xmath18 . there is a large body of literature on this particular case , and we refer the interested reader to a recent review @xcite . interestingly enough , as @xmath19 the fact that a baryon contains @xmath16 quarks implies that the region of small chemical potentials , in the technical sense adopted here , increases to @xmath20 . as a result , several interesting new phases open up in the hadronic regime and may be studied using different order parameters @xcite . we argue here that thermodynamic considerations allow us to extend presently available knowledge to large parts of the parameter space of qcd and enable us to build a qualitative picture of the complete phase diagram of qcd for small chemical potential . we deduce the topology of the three dimensional slice of the phase diagram of two - flavour qcd which may be accessible to experimental tests . we also indicate how these arguments allow us to constrain the phase diagram of qcd with up , down and strange quarks . the plan of this paper is the following : in the next section we briefly review the gibbs phase rule in the form that we will use it . the two sections following that deal with @xmath21 and 3 respectively . the final section contains a summary of our results . in this paper we investigate the phase diagram of qcd using an essential tool of thermodynamics : the gibbs phase rule . before stating the rule we recall that a system in thermodynamic equilibrium is fully described by a certain number of extensive thermodynamic quantities . this set of extensive quantities , among which we must always count the entropy @xmath22 and the energy @xmath23 , serve as coordinates in the so - called gibbs space . the thermodynamically stable states of a system are in one - to - one correspondence to a convex surface @xmath24 in gibbs space . the thermodynamic intensive quantities are derivatives of @xmath23 with respect to one of the other extensive quantities , the derivative with respect to @xmath22 being @xmath25 . by the process of taking legendre transforms of @xmath23 with respect to each of the intensive variables , one reaches a description of thermodynamics in terms of the intensive quantities and a free energy , @xmath26 , which is extensive . a phase diagram is obtained by projecting out information on @xmath26 , and describes the regions in which different phases of a system are thermodynamically stable .    by construction almost all points in the phase diagram correspond to one pure phase . this is the one with the lowest free energy among all possible phases . if the system has more than one phase , then at some points in the phase diagram two phases may coexist . if we label the phases by @xmath27 and @xmath28 , then the condition for coexistence is the equality of free energy densities in the two phases , g_a(t,_u , m_u , ) = g_b(t,_u , m_u , ) , [ coexist]if the dimension of the phase diagram is @xmath29 , then two phases coexist along solutions of the above equation , , generically along hypersurfaces of @xmath30 dimensions . three - phase coexistence requires simultaneous equality of three free energies , and hence occurs along hypersurfaces of @xmath31 dimensions . this surface of 3-phase coexistence is clearly the intersection of three surfaces of 2-phase coexistence , obtained by taking the phases pairwise . @xmath32 phases generically coexist along hypersurfaces of @xmath33 dimensions , which are the intersection of @xmath32 surfaces of two - phase coexistence . this is one form of the statement of gibbs phase rule .    a two - phase coexistence surface either has no boundary or ends in a surface of one lower dimension , @xmath31 , called a critical surface . similarly , a 3-phase coexistence surface may have a boundary . such a boundary is a @xmath34 dimensional surface called a tricritical surface . since the 3-phase coexistence surface is the intersection of three 2-phase coexistence surfaces , the tricritical surface is the intersection of three critical surfaces @xcite . boundaries of @xmath32-phase coexistence surfaces , when they exist , are called @xmath32-critical surfaces , and can be viewed as intersections of @xmath32-critical surfaces . clearly a @xmath32-critical surface is one at which @xmath32 different phases simultaneously become indistinguishable . thus another form of gibbs phase rule states that when the dimension of the phase diagram is @xmath29 , then there may be @xmath30 dimensional surfaces of two - phase coexistence ( first order phase transitions ) , @xmath31 dimensional critical surfaces , @xmath34 dimensional tricritical surfaces , . the most well - known application of the rule is to phase diagrams of chemically pure substances ( for example , water ) , characterised by two intensive parameters @xmath25 and the pressure @xmath35 . since @xmath36 , one has lines of two - phase coexistence . such lines can either end in critical points , or two such lines can meet at isolated triple points ( , points of three phase coexistence ) . both these possibilities are realized in the well - known phase diagram of water . a different example is in mixtures of two chemically pure substances ( water - alcohol , binary alloys , qcd with quarks , ) , where an additional intensive parameter , the chemical potential @xmath37 , makes the phase diagram three dimensional . in this case one has surfaces of two - phase coexistence , bounded by critical lines . three surfaces can intersect along lines of three - phase coexistence . the end point of such a line is a tricritical point , and one can view this point also as the crossing point of three critical lines . the realization of these possibilities in @xmath38-@xmath39 mixtures was treated in @xcite . if the phase diagram is known only in some part of the space of intensive variables where calculations are tractable , then the gibbs phase rule allows us to constrain the possibilities that arise from extrapolations to larger parts of the space . as we show in the coming sections , the constraints are enormous , and sometimes they determine the topology of the phase diagram completely .    in qcd a choice of the extensive quantities can be the order parameters for the breaking of chiral symmetry ( one condensate for each flavour of quarks and the net number of quarks of each flavour ) . conjugate to these are the couplings the quark masses and the chemical potentials respectively . adding to this set the total energy , one sees that for @xmath21 the phase diagram has 5 dimensions . for @xmath40 the strange quark mass and chemical potential are added , as a result of which the phase diagram is 7 dimensional . experiments are constrained to work with given quark masses , and hence explore a phase diagram of dimension 3 for @xmath21 and dimension 4 for three flavours . this is true if one examines a strongly interacting system on a time scale much larger than the slowest strong interaction related relaxation time , but much smaller than the time scale of the flavour changing weak interactions . this is the case in heavy - ion collisions , where the expanding fireball cools and freezes out on time scales of the order of a few fermis , whereas the typical strangeness changing time scale is many orders of magnitude larger . this might seem to indicate that the full 4 dimensional phase diagram can be explored . however , in the initial state the only conserved quantum numbers are the baryon number , @xmath41 , and the electrical charge , @xmath42 . since these two , and the total energy , are the only quantities which can be manipulated in experiments , the three dimensional slice of the phase diagram conjugate to these variables is the only part of the full phase diagram which is accessible to any possible experiment . as is well - known , other parts of the @xmath40 phase diagram may become accessible in situations at high densities and relatively low temperatures , where fermi - blocking of the light quark states may increase the lifetime of strange quarks . this situation may well be realized in compact stellar objects . in the early universe , around the time of the qcd phase transition , the baryon density is much higher than now , but still small enough that @xmath43 . also , the inverse hubble time is much smaller than it is now , but much larger than weak the weak interaction time scale . hence , in the early universe , both strong and weak interactions should be considered to be in equilibrium , so that the phase diagram reverts to being 3 dimensional . unfortunately , the chemical potentials in the early universe are so small that essentially only the temperature axis is explored . for @xmath21 qcd the five intensive quantities can be chosen to be @xmath25 , @xmath44 , @xmath45 , @xmath4 and @xmath5 . a common alternative choice is to use @xmath46 , conjugate to the baryon number , @xmath41 , and @xmath47 , conjugate to the third component of isospin , @xmath48 . moving between the ensembles corresponding to these is straightforward ; detailed formulae are given in @xcite . it is useful to note that these intensive quantities are conjugate to the extensive thermodynamic variables n_u = , _ u = , [ extens]and similar relations for down quarks , where @xmath49 is the net number of up quarks and @xmath50 is an extensive quantity whose density is the up quark condensate @xmath51 . the best - known part of the phase diagram for @xmath21 qcd is the temperature axis for @xmath52 @xcite . the action has the chiral symmetry @xmath1 . at @xmath15 this is spontaneously broken to the isospin part , @xmath7 , resulting in three massless pions arising as the goldstone bosons of the broken symmetry . in the high temperature limit the symmetry is restored . the symmetry of the theory is isomorphic to o(4 ) , which is known to have a second order phase transition . universality then indicates that there is a critical temperature , @xmath53 , at which a second order phase transition occurs from the chiral symmetry broken phase to the symmetric phase with critical exponents in the o(4 ) symmetry class . the order parameter is the isoscalar chiral condensate , @xmath54 , which changes from a non - zero value at low temperatures to zero at @xmath53 . for @xmath55 the order parameter is negative for any positive @xmath8 , and flips sign when @xmath8 in negative . hence there is a line of first order phase transitions for @xmath56 and @xmath55 , which ends in the critical point at @xmath53 . there is no finite temperature phase transition at finite @xmath8 . lattice computations are performed at finite quark mass , and are consistent with this picture since they observe only a cross over . chiral extrapolations have given mixed results : there is no agreement on the critical exponents which are seen . this could be a finite lattice spacing artifact . several studies have used staggered quarks which , for @xmath56 , have only an @xmath57 chiral symmetry at finite lattice spacing . the chiral transition in this case should lie in the @xmath58 universality class . it turns out to be hard to distinguish o(4 ) and o(2 ) exponents @xcite . lattice studies are consistent with both @xcite . studies with wilson quarks show results consistent with critical exponents in the @xmath59 universality class @xcite . the situation can be clinched in simulations with dynamical overlap quarks , which realize the full chiral symmetry . however , these simulations are currently in their infancy . recently attention has been focused on the three dimensional section of the phase diagram with varying @xmath25 , @xmath46 and @xmath8 and fixed @xmath60 . using the order parameter @xmath61 one finds the phase diagram shown in figure [ fg.phsdgt0m ] . the plane @xmath62 was described in the previous subsection . the first order line and its o(4 ) critical end point described there extends naturally to three dimensions , becoming a surface of 2-phase coexistence ( corresponding to a sign flip in @xmath61 ) with a critical line as boundary in the @xmath56 plane . for positive @xmath8 there is a 2-phase coexistence surface across which a large negative value of @xmath61 changes discontinuously to a smaller value . for @xmath63 there is a symmetrically placed coexistence surface with the sign of @xmath61 reversed . these three 2-phase coexistence surfaces intersect in a triple line in the @xmath56 plane . since there is a critical line bounding one of these surfaces , it must meet the triple line at a tricritical point . we are led to the conclusion that all three 2-phase coexistence surfaces are bounded by critical lines , and the tricritical point , at @xmath64 , is their common intersection point . one fact about this phase diagram is worth emphasizing . the quantity @xmath61 is an order parameter related to symmetry only on the @xmath56 plane in figure [ fg.phsdgt0m ] , since it is exactly zero everywhere above the triple - phase and critical lines . at finite @xmath8 , above the coexistence surface it is not exactly zero , but has a value proportional to @xmath8 . below this surface it has a much larger value . across the coexistence surface it has a discontinuity , and hence can be used to flag a first order phase transition even at finite @xmath8 . however , along the wing critical lines , @xmath61 is not an order parameter in the sense of being zero in one phase and non - zero in another . this is related to the fact that , for @xmath65 , the phase transition is not related to a symmetry . however , along the critical line there are long range correlations between local fluctuations in @xmath61 , or , equivalently , through a fluctuation - dissipation theorem , there are divergent susceptibilities of @xmath61 . this kind of critical behaviour is said to be a liquid - gas transition , and expected to be in the ising universality class . the slopes of the coexistence surface can be determined by a generalization of the clapeyron - clausius equations . for the case at hand we can write dg_i = s_i dt + b_i d_b + s_i dm , [ cceq]where the subscript @xmath66 refers to the two phases ( say , a and b ) which are at equilibrium along the coexistence surface . as one moves along the surface , the changes @xmath67 , @xmath68 and @xmath69 are related by the fact that the free energy density is the same in the two phases , eq.([coexist ] ) . in order to define a density in a relativistic theory , one notes that all four extensive quantities scale similarly , and hence any one of them can be used as a normalization . in determining the slope of the coexistence surface , one has to hold one of the three intensive quantities fixed , so one can choose to use the extensive quantity conjugate to it to perform the normalization . along lines of constant @xmath8 , one can then normalize all extensive quantities by @xmath61 to get the relation .|_m [ mutslope]since the right hand side is independent of the sign of the chiral condensate , one can use the magnitude @xmath70 in the expression . choose a to be the low temperature phase and b to be the high temperature phase . then , as is shown by lattice computations , @xmath71 and @xmath72 . as a result , the expression on the right is negative , implying that with increasing @xmath25 the surface of coexistence moves to smaller @xmath37 , as shown in figure [ fg.phsdgt0m ] . since the ordering of the entropy and chiral condensate determine the slope , the figure is consistent with the idea that the low temperature phase contains hadrons , whereas the high temperature phase contains quarks @xcite . at finite @xmath8 one then has a line of first order transitions starting at @xmath15 and a large @xmath46 and ending in a critical end point at @xmath73 which is in the ising universality class . these arguments , based on the gibbs phase rule and the clapeyron - clausius equation , reproduce the results of earlier model studies @xcite , which spurred recent developments in lattice qcd @xcite and gave rise to experimental interest in the search for the qcd critical point @xcite . one can extend such arguments to the other two slopes . along lines of constant @xmath46 one finds .|__b = . [ mtslope]given the relative magnitudes of the entropies and chiral condensates , one finds that this slope is negative , , at constant @xmath46 , the temperature of coexistence decreases with increasing quark mass . along lines of constant @xmath25 one finds the slope .|_t = -1b , [ mumslope]which is positive . to the best of our knowledge , arguments about these slopes based on the clapeyron - clausius equations are new . they imply that with increasing quark mass the critical end point , @xmath73 , moves to larger @xmath25 and smaller @xmath46 , as indicated by current lattice data @xcite . the three - dimensional slice through the @xmath21 phase diagram which is obtained for a fixed non - vanishing value of @xmath8 and @xmath74 ( including the special slice which also has @xmath62 ) has been extensively examined in the literature @xcite . this case has one very attractive feature : the quark determinant is positive definite , and hence lattice simulations are possible , and they can be used to check arguments using global symmetries . the drawback is that the symmetry which makes the quark determinant positive definite is responsible for making the phase diagram non - generic .    the plane of @xmath25-@xmath47 , for arbitrary @xmath8 and vanishing @xmath46 and @xmath75,was first discussed in @xcite . using an effective theory , it was argued that along the line @xmath15 , there should be a phase transition which results in the development of a charged pion condensate , @xmath76 . it was argued that the phase transition should be critical and in the @xmath59 universality class with critical @xmath77 , and that an @xmath58 critical line emanate from it . now , the gibbs phase rule does not generically allow a critical line in a two - dimensional phase diagram . however , subsequent lattice computations @xcite saw instead a first order line at finite @xmath25 connecting to a critical or tricritical point with @xmath78 . the phase diagram was extended to the three dimensional slice @xmath25-@xmath46-@xmath47 for finite @xmath8 and @xmath74 by three different methods in @xcite . in all these works , critical lines were obtained in the @xmath47-@xmath46 surfaces for generic @xmath25 . thus there are critical surfaces in this three dimensional phase diagram . such a violation of the gibbs phase rule may occur in subspaces of high symmetry . for example , in the plane @xmath79 , one found the o(4 ) critical line shown in figure [ fg.phsdgt0m ] . however , these exceptions are non - generic ; when extending the phase diagram they do not develop into higher dimensional surfaces . this is clear in the above example . in the same way , one is forced to the conclusion that the slice @xmath74 is not generic . we consider extensions to @xmath80 next . once isospin is broken by a mass difference between the up and down quarks , an isospin chemical potential no longer matches the up quark and down antiquark fermi surfaces exactly . this breaks the symmetry that kept the full quark determinant non - negative , and hence presents the usual problems for lattice simulations . however , it gives a generic phase diagram in @xmath25-@xmath47 . consider the section of the phase diagram for generic non - zero @xmath8 and @xmath62 . changing the sign of @xmath75 is equivalent to toggling the definition of @xmath81 and @xmath82 flavours ( in the absence of electroweak interactions ) and hence to flipping the sign of @xmath47 . thus , if there is charged pion condensation , , @xmath83 , for some value of @xmath47 , then there is a first order transition across @xmath74 , in which the sign of @xmath84 distinguishes the two phases . thus , along some line in the @xmath74 plane one expects triple - phase coexistence , the phases corresponding to @xmath85 and two non - zero values of @xmath86 of opposite sign . as one goes from small to large @xmath47 along a line of fixed @xmath75 and @xmath25 , the order parameter @xmath87 increases . hence , by adapting the argument based on the clausius - clapeyron equations for the @xmath25-@xmath46-@xmath8 phase diagram , one obtains the slope drawn in figure [ fg.conj2 ] . such a 3-phase coexistence line can be accommodated by the three possible phase diagrams shown in figure [ fg.conj2 ] . the first possibility shown in the figure is generic . the triple - phase line , @xmath88 , starts at @xmath15 and continues up . there is no critical boundary to the three surfaces of first order transitions . this phase diagram is ruled out by the results of @xcite which point to some kind of criticality at @xmath89 . however , the @xmath7 isospin symmetry is broken to a vector u(1 ) when @xmath90 . as a result , an o(4 ) critical point is not stable under this perturbation . the second generic possibility shown in figure [ fg.conj2 ] takes this fact into account . the triple - phase line , @xmath88 , ends at a tricritical point . three critical lines emerge from this point . one lies in the symmetry plane @xmath74 and at @xmath15 it is seen to be the o(4 ) critical point identified in @xcite . along the two wing critical lines the remnant vector @xmath91 symmetry is broken by charged pion condensation . this is closely analogous to the phase diagram in figure [ fg.phsdgt0m ] . in this scenario , as one traverses a line of fixed @xmath90 for @xmath15 by varying @xmath47 , one should go continuously from a phase with small @xmath87 to one with a large value of this condensate . however , at @xmath15 and small @xmath47 it is not energetically feasible for pions to condense , hence there must be a range of @xmath47 over which @xmath87 vanishes . in that case a phase transition must occur at @xmath15 for any @xmath47 . this brings us to the final possibility : the triple - phase line ends at @xmath15 in a tricritical point . two wing critical lines emerge from this in the @xmath15 plane . the third critical line , which is the o(4 ) line of the second scenario , is squashed down to a point . in this scenario there is always a phase transition between the charged pion condensed and uncondensed phases . this is the favoured phase diagram , since the lattice computation of @xcite , which sees a critical or tricritical point at @xmath15 and @xmath74 , prefers this over the first scenario . it would be good to check this through another computation at smaller lattice spacing . one might expect that when the isospin symmetry is broken , the flavour singlet order parameter should be replaced by the pair @xmath50 and @xmath92 @xcite . if so , the @xmath25-@xmath46 phase diagram would be extended to the first topology shown in figure [ fg.imposs ] . this contains a line of 4-phase coexistence , labeled q in the diagram with an end point which is a tetracritical point . in a three dimensional phase diagram one does not generically expect either a 4-phase coexistence line or a tetracritical point . we have argued before that non - generic situations might be obtained on sections of high symmetry . however , having two different non - vanishing masses breaks the flavour symmetry maximally , so it is impossible to find a more generic situation . this no - go argument can be evaded by noting that @xmath47 and @xmath93 break the flavour symmetry to the same subgroup . as a result , it is possible that the topology is the same in the two sections @xmath25-@xmath46-@xmath75 for @xmath94 and @xmath25-@xmath46-@xmath47 for @xmath95 separately . in other words , the 4-phase coexistence line and the tetracritical point lie in the intersection of these two sections in this extended four dimensional phase diagram @xmath25-@xmath46-@xmath47-@xmath75 . by the same token , such a topology is ruled out as non - generic in the @xmath25-@xmath46-@xmath47 section for @xmath74 unless it is also seen in this section . arguing differently , one might expect that for @xmath96 the effect of isospin symmetry breaking through the quark masses should not change the phase diagram qualitatively . in that case the two order parameters @xmath50 and @xmath92 would be redundant , and the two sheets of first order transitions merge into a single sheet of first order transitions found with @xmath61 . this possibility is shown as the second topology in figure [ fg.imposs ] . if this second argument is correct , then one must face the question of what happens at large @xmath97 . exactly this question was answered at @xmath15 using an effective theory in @xcite . at finite @xmath93 , the explicit breaking of isospin symmetry to vector u(1 ) allows the @xmath98 to mix with isoscalars ( this allows a change in its mass without changing the masses of @xmath99 ) . when @xmath97 is large , a neutral pion condensate can form , and , at @xmath15 , there is a second order transition between phases with and without this condensate . in the real world a mass difference between the charged and neutral pions is observed , so a mixing of this sort could occur . however , a neutral - pion condensate is not observed at @xmath15 , so one may conclude that the physical value of @xmath97 is not large enough for isospin breaking effects to qualitatively change the phase diagram shown in the second part of figure [ fg.imposs ] .    the mixing of neutral pions with isoscalars is a physical effect which does not exist in the models used in @xcite . this leads us to prefer the second phase diagram shown in figure [ fg.imposs ] . the only lattice study to date of thermodynamics at finite @xmath75 @xcite saw no evidence of two separate crossovers . a computation with a different lattice formulation of quarks will be an interesting and useful additional piece of evidence . the physical phase diagram of qcd is the one in which all possible experimentally tunable parameters are shown for fixed physical values of the remaining parameters . this is the phase diagram in the space of @xmath25-@xmath46-@xmath47 for generic small @xmath8 and @xmath100 . after the analysis of the previous subsections , deducing the topology of this phase diagram is straightforward . the various phases are distinguished by the order parameters @xmath61 and @xmath87 . we discuss the topology of the phase diagram ( shown in figure [ fg.conj3 ] ) only in the quadrant with @xmath101 . the remainder can be constructed by symmetry .    in the @xmath94 plane there is a first order line ending in the qcd critical point . across this line @xmath61 changes discontinuously . following the discussion of the previous subsection , we conclude that this line develops into a surface of 2-phase coexistence . the qcd critical end point develops into a critical line ( in the liquid - gas universality class ) which is the boundary of this surface . in the @xmath62 plane there is a first order line separating phases with vanishing and finite values of @xmath87 with a liquid - gas critical point at @xmath15 . this develops into a 2-phase coexistence surface . the two first order surfaces must meet along some line , @xmath102 . hence @xmath102 is a line of 3-phase coexistence , and another surface of 2-phase coexistence must emanate from it , separating the phase in which both the condensates vanish from that in which neither does . the two ends of @xmath102 , labeled @xmath103 and @xmath104 , must both be tricritical points . at @xmath103 the first order surface of discontinuity in @xmath87 must have a boundary . since the system has no particular symmetry at @xmath103 , all three critical lines meeting at this point must be in the universality class of a liquid - gas transition . at @xmath104 there seem to be two critical lines , both in the liquid - gas universality class . the third critical line is degenerate , as in the third topology shown in figure [ fg.conj2 ] . the derivation of the clapeyron - clausius equations for the physical phase diagram needs little further comment . along planes of constant @xmath46 one can use eq . ( [ mutslope ] ) , since @xmath8 and @xmath75 are also constant along this line . along the planes where @xmath47 are constant , the modification of eq . ( [ mutslope ] ) was already discussed in section [ sc.tmuidm ] , and leads to the opposite slope of the coexistence surface for pion condensation . the remaining part involves scaling the expression @xmath105 by @xmath106 in the phase @xmath66 , and equating the two free energy densities so defined at each coexistence surface . the resulting clapeyron - clausius equation is .|_t = -b . [ mubmutcceq]the first order surface of chiral symmetry restoration , which is in the phase without a pion condensate , then has only weak dependence on @xmath47 . similarly , the first order surface of pion condensation has weak dependence on @xmath46 because the baryon density is low in the phase with chiral symmetry breaking . the surface separating the phase with both condensates vanishing from the phase where both condensates are present has more complicated behaviour . one expects that in the phase without condensates the number densities scale with chemical potentials as @xmath107 and @xmath108 when @xmath25 is small . in that case this first order surface is roughly linear in @xmath46 and @xmath47 . lattice computations at @xmath74 can substantially improve our knowledge of the functional form of @xmath109 and @xmath110 , and therefore the shape of the physical phase diagram through continuation to @xmath90 . the position of @xmath103 and @xmath104 are constrained by these considerations . it is estimated that the qcd critical end point at @xmath111 is at @xmath112 with @xmath113@xmath114 mev . since the effect of isospin breaking is small , one expects that at realistic values of @xmath75 the critical end point does not shift drastically . one also knows that at @xmath115 , the critical point in @xmath116 . the corrections at realistic @xmath75 are not large . one expects that @xmath117200 mev . also , from this topology one could deduce that in heavy - ion collisions where @xmath118 one should expect to see the qcd critical line close to the estimates of @xmath119 and @xmath120 for @xmath40 qcd , the intensive quantities at the phase transition are the temperature @xmath25 , the three quark masses , @xmath4 , @xmath5 and @xmath11 , and the three chemical potentials @xmath44 , @xmath45 and @xmath121 . as before , we can take linear combinations of these variables to parametrize the seven - dimensional phase diagram . we will use the mean light quark mass @xmath6 and the mass splitting between these two , @xmath122 . it is useful to construct the baryon chemical potential , @xmath46 and two other combinations , as outlined in @xcite . the three flavour phase diagram has barely been explored , and we shall not be able to do justice to the many kinds of pairings that it may possess . in this first work we restrict attention to a question which has been asked recently : is the two - flavour phase diagram for chiral symmetry restoration ( figure [ fg.phsdgt0m ] ) a good guide to the corresponding phase diagram in qcd with additional strange quarks ? universality arguments @xcite lead us to expect that the chiral phase transition in three flavour qcd at zero chemical potentials is of first order . the order parameter used is the three - flavour condensate . in the three - flavour chiral limit , since the action is blind to flavour , any linear combination of three flavoured condensates is as good an order parameter as any other , so one could as well continue to use the two - flavour order parameter @xmath61 to study the phase structure . when the strange quark becomes massive with the light quarks remaining massless , one does not expect symmetry restoration in the strange sector while the symmetry remains broken in the light quark sector . similar expectations hold whenever the strange quark mass is heavier than the up and down quark masses . hence , in this whole region one expects that the same phase structure is seen whether one uses the three flavour condensate or @xmath61 . the phase diagram of qcd with three quarks has been explored in several lattice computations @xcite where the two light quarks are degenerate , , @xmath74 . in this case the phase diagram in the section @xmath8-@xmath11-@xmath25 is of the kind presented in figure [ fg.flag ] . in these studies there is only one chiral phase transition observed , and not separate ones for strange and light flavours , in conformity with the above discussion . for infinite @xmath11 and @xmath56 , one has two flavour chiral symmetry , and hence a second order transition in the o(4 ) universality class . when @xmath123 , , three flavour symmetry is exact , one has a line of first order transitions up to a critical end point , which lies in the ising universality class @xcite . moving off the exact @xmath40 line ( termed going to @xmath124 in the literature ) , this develops into a surface of first order transitions bounded by a critical line . since this line is in the liquid - gas universality class @xcite , it must meet the o(4 ) line in a tricritical point . the nature of this critical line is crucial in establishing a phase diagram . if it is in the liquid - gas class , as expected , then this would be visible with any formulation of lattice quarks . the phase diagram shown in figure [ fg.flag ] can be probed entirely with the light quark condensate . there are , of course , other phase transitions in this phase diagram for example the deconfining transition in the corner which contains the quenched theory . this is observed with a different order parameter , the polyakov loop , and hence does not appear in the phase diagram shown . these results are often presented in a figure ( the second of figure [ fg.flag ] ) which is the phase diagram projected down to the plane @xmath8-@xmath11 . this is not a phase diagram since a point on it does not represent a stable thermodynamic phase . instead , each point on this diagram represents whether or not there is a phase transition at some @xmath25 `` above '' it . unfortunately there seems to be no name for such an useful diagram , and we are forced to invent a name for it . in the rest of this paper we call it the `` flag diagram '' , because each point flags whether there is a phase transition above it , and if so , the order of the transition . one of the main open questions about the flag diagram is the location of the physical point . all computations indicate that this lies deep in the crossover region @xcite . a second important question is about the location of the tricritical point . what is the value of @xmath125 ? if the physical strange quark mass is less than @xmath125 then the @xmath21 phase diagram will be a good guide to the phase diagram of real qcd . if , on the other hand , the physical strange quark mass is larger , then the phase diagram in the real world could be significantly different . one naively expects that @xmath126 ; since the physical value of @xmath11 is also similar , this is an interesting and wholly non - perturbative question . one estimate of @xmath125 that we are aware of was made using a linear sigma model with parameters fixed by hadronic data @xcite . this finds that the k meson mass at the tricritical point is at least 1700 mev , far in excess of the physical value . while this would seem to indicate that the physical strange quark mass is lighter than @xmath125 , there are indications that further work may be needed in order to pin down @xmath125 . for example , in @xcite it is estimated that the critical end point along the @xmath40 line occurs at @xmath127 mev . the lattice computations of @xcite indicate that the critical end point may be at @xmath128 mev . whether the discrepancy is due to an insufficiency in the model of @xcite or the multiple extrapolations performed in @xcite , or a combination of the two , remains to be tested in future . in @xcite the estimate @xmath129 mev is presented . indications that the cutoff effects at comparable cutoffs are large come from the results of @xcite . another slice that is interesting from the point of view of forthcoming planned experiments is the four - dimensional @xmath25-@xmath46-@xmath8-@xmath11 slice obtained with @xmath130 . this was investigated recently on the lattice @xcite , and a flag diagram obtained by projecting out the @xmath25 axis was used in the presentation . these two studies differ in the conclusions reached : one @xcite favours the first flag diagram shown in figure [ fg.possible ] ; the other @xcite , we argue below , is consistent with the second possibility shown in the same figure . these are computations which present many technical challenges . we remark later on one way to resolve the issue . two sections of this phase diagram have been discussed the section with @xmath62 was discussed in the previous subsection , and the section with @xmath131 was discussed in the section on the @xmath21 phase diagram in @xmath25-@xmath46-@xmath8 . both contain a line of o(4 ) critical points , and a line of ising critical points , meeting at a tricritical point . the two lines of o(4 ) critical points are in fact the same , since they both originate in the finite-@xmath25 transition for the @xmath21 chiral symmetry restoration . in the 4-dimensional phase diagram one should , therefore , see a single critical surface of o(4 ) transitions . the boundary of this critical surface is the tricritical line , the two ends of which show up as the two tricritical points we have already discussed . at this tricritical line the o(4 ) critical surface is glued on to surfaces of ising criticality . the tricritical line can also be viewed as the boundary of a surface of triple phase coexistence . this much is a direct consequence of the gibbs phase rule . the gibbs phase rule also allows a tetracritical point in a generic four dimensional phase diagram . a tetracritical point is the end point of a line of four - phase coexistence . with a single scalar order parameter , such as @xmath61 , it is not possible to distinguish four phases . therefore , we conclude that a tetracritical point should not be seen in the four dimensional phase diagram observed with @xmath61 . there remain two generic possibilities for the shape of the phase diagram and they are shown in terms of the corresponding flag diagram in figure [ fg.possible ] . the first possibility is that the tricritical line lies entirely on the side @xmath132 . the second is that the tricritical line continues to some @xmath133 and then bends back towards @xmath125 . in the first scenario , the critical line moves to larger @xmath8 and @xmath11 as @xmath46 is increased by a small amount from @xmath62 . in the second , the opposite movement occurs until a large value of @xmath46 . this is precisely the observation which distinguishes the two lattice studies in @xcite . it would be interesting to push lattice studies to smaller cutoffs and decide which scenario is actually obtained in qcd .    in the first case the phase diagram for @xmath134 is qualitatively like the @xmath21 phase diagram , and for @xmath135 , similar to @xmath40 . in the second case for @xmath136 there are two sheets to the phase diagram at fixed @xmath11 : one similar to the @xmath21 phase diagram , the other like the @xmath40 phase diagram , with the two tricritical points joined by an o(4 ) critical line , as shown in figure [ fg.unusual ] . the inset in the figure shows the unusual phase diagram for fixed non - zero value of @xmath8 . it would be very interesting if qcd is eventually seen to yield such a phase diagram . the phase diagram of qcd is complicated . the large global symmetries of qcd imply that there are many thermodynamical intensive parameters , , free parameters which enter the lagrangian . as a result , there are many different kinds of pairings of quarks that can arise as these parameters are changed . in this paper we have considered only small chemical potentials , defined to be the region where the relevant pairings are between a quark and an antiquark . the topology of the phase diagram for qcd with two flavours of quarks has been investigated earlier by many authors , mainly in sections of partial symmetry . in this paper we have extended these considerations to the case of physical quark masses , which break both the chiral symmetry and isospin symmetry . explicit breaking of isospin symmetry by quark masses leads to mixing of isovectors with isoscalars and hence influences the phase diagram ( see section [ sc.tmubdm ] ) . this piece of physics is incorporated into the phase diagram that we construct . the topology of this physical phase diagram is strongly constrained by thermodynamics essentially the gibbs phase rule ( see section [ sc.gibbs ] ) . our results are given in section [ sc.physical ] and summarized in the phase diagram shown in ( see figure [ fg.conj3 ] ) . the physical phase diagram contains a triple line at which phases with and without the chiral and pion condensates coexist . deducing actual values of the critical and tricritical points in this diagram is a much harder task . however , appropriate clapeyron - clausius equations along with the results of current lattice computations @xcite place various constraints on them . improving these constraints through future lattice computations is possible without extreme effort . the phase diagram of three flavour qcd could be more complicated , since there are even larger global symmetries . strange quarks can decay into the light quarks through the weak interactions , so one may ask whether this phase diagram is relevant to any physical system . in the context of heavy - ion collisions , where the system is expanding so fast that weak interactions do not come into play in the thermodynamics , strange quarks are indeed relevant . further , in the physics of compact stars ( involving large chemical potentials ) , where weak decays of strange quarks are suppressed due to fermi - blocking of the light quark states into which they could have decayed , strange quarks also become relevant to thermodynamics . thus the study of the three flavour phase diagram is interesting . comparatively little is known at present about this phase diagram . here we restrict attention to the breaking and restoration of chiral symmetry . the gibbs phase rule places strong constraints on the topology of the phase diagram . as the strange quark mass is increased keeping the light quarks massless , there is a tricritical value , @xmath125 , at which the finite temperature chiral phase transition changes from being first order to second order ( for @xmath134 ) . presently available observations on the lattice @xcite can be patched together into two possible phase diagrams which we have discussed in section [ sc.phasd ] . the three - 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we investigate the phase diagram of qcd at small chemical potentials , , when chiral and flavour symmetry breaking involves the pairing of a quark and an antiquark . 
 the phase diagram of two - flavour qcd at small chemical potentials involves chiral symmetry restoration and charged pion condensation . 
 we extend previous studies of the topology of the phase diagram , in sections with high degree of symmetry , to the physical case of fully broken flavour symmetry , using generic thermodynamic arguments . 
 we argue that the extension is unique and present the result . in three flavour qcd 
 the phase diagram for chiral symmetry restoration is less well constrained . 
 however , we argue that present lattice data allows just two different phase diagrams , which we discuss .