CConundrums at Finite Density ∗ Rajiv V. Gavai † Fakult¨at f¨ur PhysikUniversit¨at Bielefeld33615 BielefeldGermanyExtending the successes of lattice quantum chromodynamics(QCD) atzero as well as nonzero temperatures to nonzero density is extremely de-sirable in view of the quest for the QCD phase diagram both theoreticallyand experimentally. It turns out though to give rise to some conundrumswhose resolution may assist progress in this exciting but difficult area, andshould therefore be sought actively.PACS numbers: 12.38.Gc, 11.35.Ha, 05.70.Jk
1. Introduction
The theory of strong interactions, Quantum Chromo Dynamics (QCD),has intriguing properties such as confinement or chiral symmetry breakingwhich have been enigmas for over half a century. A major reason is, ofcourse, the dominance of large coupling in such hadronic properties. Cru-cial clues in building physical pictures to understand them were provided bythe studies of simple models such as the bag model or NJL-model. The dis-covery of instanton solutions and the subsequent investigations of instanton-based models enhanced our understanding further by emphasizing the roleof the zero and near-zero modes of the Dirac equation for interacting quarks.Investigating all these models in extreme environments such at high temper-atures/densities led to a variety of phase diagrams of strongly interactingmatter. It may not come as a surprise that even qualitative features of thesemodel phase diagrams differed substantially, not to mention the quantitativedetails. For instance, the early sketches of the QCD phase diagram display ∗ Presented at the international conference on “Criticality in QCD and the HadronResonance Gas”, 29-31 July 2020, Wroclaw Poland. † Address after 1 January 2021 : Indian Institute of Science Education & Research,Bhopal bypass, Bhauri, Bhopal 462066, India. (1) a r X i v : . [ h e p - l a t ] N ov Rajiv V. Gavai separate deconfinement and chiral transitions for all temperatures and den-sities [1]. Nevertheless, they pointed to an interesting path to fathom chiralsymmetry breaking and/or confinement.QCD formulated on a space-time lattice has yielded a more firm guidancein refining these pictures at finite temperatures to give us a reliable, insome cases even quantitative knowledge of the phase structure. However,extending this to finite densities or equivalently nonzero chemical potential,one encounters conundrums many of which are unrelated to the latticizationand were hitherto still unknown to exist. These pose significant hurdlesin excursions inside the diagram from the temperature axis. There is, ofcourse, the famous fermion sign(phase) problem at nonzero baryon densityor equivalently nonzero baryon chemical potential. The aim of this talk isto draw attention to the other, perhaps equally serious, problems.
2. The µ (cid:54) = 0 problems : I. Divergences Fig. 1. Space-time(inverse temperature) lattice depicting a smallest loop (plaque-tte) with time links and a quark covariant derivative term.
Let us begin by recalling that the (baryonic) chemical potential is aLagrange multiplier to enforce the constraint of (net baryon) number con-servation in the grand canonical ensemble: ∂ µ J Bµ = 0 is the current conser-vation equation and N B = (cid:82) d x J B is the conserved charge. Followingthe same principle on lattice, one obtains [2] a point split version for theconserved number. Thus introducing the chemical potential on the latticeamounts to multiplying the forward [backward] time-like links with f ( µa )[ g ( µa )], with f ( µa ) = 1 + µa [ g ( µa ) = 1 − µa ] as seen in Figure 1 for the onundrums at finite density naıve fermions. This form of f and g , which has been shown to remain thesame for Wilson/Staggered/Improved local fermions as well, leads [2] to thefollowing form of the energy density and quark number density for a gas offree quarks: (cid:15) = c a − + c µ a − + c µ + c µ T + c T (1) n = d a − + d µa − + d µ + d µT + d T . Here c i and d i are constants, a is the lattice spacing, and the subscriptB of µ has been dropped for simplicity as well as to indicate that theseexpressions hold for any conserved charge such as strangeness or electriccharge. In the continuum limit of a →
0, one obtains a leading quarticdivergence and a subleading µ -dependent quadratic one. Subtracting offthe vacuum contribution at T = 0 = µ eliminates the leading divergence ineach case. However, the µ -dependent a − divergences persist in both theenergy density and the quark number density. Note that these divergencesare present for the free theory itself. As a solution to this problem differentforms of f and g have been proposed. The popular exponential choice [3], f ( µa ) = exp( µa ) and g ( µa ) = exp( − µa ) as well as another choice [2], f ( µa ) = (1 + µa ) / (cid:112) (1 − µ a ) along with g ( µa ) = (1 − µa ) / (cid:112) (1 − µ a ),both lead to their corresponding c = 0 = d , which then lead to finiteresults in the continuum limit. Indeed the µ -dependent divergences areeliminated for all f · g = 1 [4]. One anticipates this analytical proof ofthe lack of µ -dependent divergences for free quarks to hold true in anyorder-by-order perturbative inclusion of interactions with gluons. However,numerical simulations are needed, and were employed [5], to extend theproof for the non-perturbative interacting case as well, as shown in Figure2. Both the lack of any diverging behaviour as well as a unique continuumlimit is evident in the data.A natural question arises as to why are there three (or more) latticeQCD actions when the continuum QCD has only one. The usual answeris universality. As long as all these actions reduce to the continuum QCDaction in the limit of a →
0, universality tells us that physics should be thesame for all of them in that limit. Expanding the functions f , g in powersof µa , one finds that the three actions differ by terms of O ( µ a ) and higherwhich vanish in the a → irrelevant terms. Paradox :
These irrelevant terms vanish from action as a → Rajiv V. Gavai / T χ Fig. 2. Continuum limit for quark number susceptibilities with different actions. Alinear behaviour of the data and convergence to a unique continuum limit indicatesthe absence of any divergence. Taken from Ref. [5].
These modified/improved actions have a further problem. One can workout the current conservation equation for the Lagrangian with µ (cid:54) = 0. Itremains unchanged only for the linear µ -case. It acquires µ a and higherorder terms of even powers in all the other cases. Thus integrating overspatial dimensions, one obtains a conserved charge on the lattice only for thelinear µ -form. For all the divergence eliminating form of actions one has no conserved charge on the lattice anymore! Consequently, Z (cid:54) = exp( − β [ ˆ H − µ ˆ N ]) on the lattice for them, and therefore one cannot define an exact canonical partition function on lattice from the Z defined this way. Z = (cid:80) n z n Z Cn on the lattice only for the naıve linear µ -action. Once again, onehas to hope that it is possible at least in the continuum limit of a → µ -dependence for Z arises solely due to loops with time-like links, andhence is ∝ ( f · g ) l , where l is the number of positive timelike links in theloop. This is illustrated for the simplest case of l = 1 in Figure 1. Quarkloops of all sizes and types contribute for the naıve case of f , g = 1 ± µa ,as is indeed the case also in the continuum. However, since f · g = 1 for theother two actions, it is clear that only limited number of loops contribute.Indeed, only quark loops winding around the T -direction contribute to µ B dependence for these cases. Again if all the actions were to lead to the same physics, as they ought to, small quark loops which are topologically trivialmust start also contributing, as a →
0. It is far from clear how this mayhappen since for all non-vanishing a , the f · g = 1 condition applies and onundrums at finite density these loops do not contribute to any µ -dependence. One possible way outmaybe that the small loops sum up to a µ -independent constant, preferablyzero. It is far from clear how this might come about in the interactingtheory. This is yet another conundrum which universality suggests shouldresolve itself in the continuum limit, and needs to be verified by explicitcomputations.
3. Divergences exist in the continuum too
Recall that the conundrums discussed in the section above were relatedto the differences in the f ( µ ) and g ( µ ) : f · g = 1 − µ a for the naive linearcase and f · g = 1 for the other two. This, in turn, arose, as the latter gotrid of the µ -dependent divergences that arise for the former choice. Since inthe continuum limit one finally has the only linear form, one may wonderwhether the µ -dependent divergences exist in the continuum as well, and thelattice as a regulator is merely reproducing them systematically or whetherthe latticization itself introduces the divergences.Indeed, it turns out that contrary to the common belief, the free theorydivergences are not lattice artifacts. They exist in continuum too. Insteadof the lattice regulator, one can employ a momentum cut-off Λ in the contin-uum theory to show [6] the presence of µ Λ terms in number density easily.We summarise below why one ought to expect them in the continuum itself.The quark number density, or equivalently third of the baryon numberdensity for a single flavour, is defined as, n = TV ∂ ln Z ∂µ | T =fixed (2)with Z for free fermions given by Z = (cid:90) D ¯ ψ D ψ e (cid:82) / T0 d τ (cid:82) d x [ − ¯ ψ ( γ µ ∂ µ +m − µγ ) ψ ] , (3)Evaluating the quark number density, n in the momentum space for themassless free quark gas, one has n = 2 iTV (cid:88) m (cid:90) d p (2 π ) ( ω m − iµ ) p + ( ω m − iµ ) ≡ iTV (cid:90) d p (2 π ) (cid:88) ω m F ( ω m , µ, (cid:126)p ) , where p = p + p + p and ω m = (2 m + 1) πT . In the usual contourmethod, the sum over m or ω m gets replaced as an integral in the complex ω -plane. Together with the subtracted vacuum ( µ =0) contribution, one hasin the complex ω -plane line integrals along the directed arms 3 and 1 in Rajiv V. Gavai − Λ + iµ Λ + iµ Λ − Λ Fig. 3. The contour diagram for calculating the number density for free fermionsat zero temperature. P denotes the pole.
Figure 3. Adding and subtracting the side arm line integrals, one obtainsthe canonical answer from the residue of the pole P in Figure 3. One stillhas to evaluate the side arm contributions.Introduce a cut-off Λ for all 4-momenta at T = 0 for a careful evaluationof the divergent arms 2 & 4 contributions in the Figure 3. The µ Λ termscan be seen to arise [6] from the arms 2 & 4. (cid:90) d p (2 π ) (cid:16) (cid:90) + (cid:90) (cid:17) dωπ ωp + ω = − π (cid:90) d p π ln (cid:20) p + (Λ + iµ ) p + (Λ − iµ ) (cid:21) . (4)Utilising the fact that Λ (cid:29) µ , the integrand can be expanded in µ/ Λto discover that while the leading Λ terms do indeed cancel there is anonzero coefficient for the subleading µ Λ term. It may be worth notingthat the arms 2 & 4 make a finite contributions to the µ term as well. Oneusually ignores the subleading contribution from the arms 2 & 4, amountingto a subtraction of the ‘free theory divergence’ in the continuum. Thispractice suggests a prescription of subtracting the free theory divergence byhand on the lattice as well. Such a prescription surely works in includingthe interactions in a perturbation theory. It has been tested in numericalsimulations, and found to work excellently.In order to test whether the divergence is truly absent in simulations, oneneeds to take the continuum limit a → N T → ∞ at fixed T − = aN T . For quenched QCD at T /T c = 1 .
25 & 2 and for quark mass m/T c = 0 .
1, lattices with N T = 4, 6, 8, 10 and 12 were employed [6]. On 50-100 independent configurations quark number susceptibility was computed.Since it is a derivative of the number density with µ , it should have a sim-ple a − divergence. The 1 /a -term for free fermions on the corresponding onundrums at finite density N ×∞ lattice was subtracted from the computed values of the susceptibilityin simulations. The results are displayed in Figure 4 as a function of 1 /N T .If the interactions were to induce additional non-perturbative divergent con-tribution over and above the subtracted free theory ones, the susceptibil-ity should behave as χ /T = c ( T ) N T + c ( T ) + c ( T ) N − T + O ( N − T ).The divergent c -contribution would then lead to a rapid shoot-up near the χ /T -axis. c is the expected continuum result with c governing theapproach to the limit. χ / T T2 c ,m/T c =0.1exp µ continuumlin µ continuum 0 0.5 1 1.5 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 χ / T T2 c ,m/T c =0.1exp µ continuumlin µ continuum Fig. 4. The quark number susceptibility at 1 . T c (left panel) and 2 T c (right panel)for m/T c = 0 .
1. Taken from Ref. [6]
A glance at both the panels of the Figure 4 shows an evident lack of anydivergent rise in both or equivalently c (cid:39) ± aµ ) action [7].
4. The µ (cid:54) = 0 problem : II. Quark Type Placing quark fields on a lattice has the famous doubling problem.Mostly staggered quarks are used in lattice QCD simulations, as they pos-sess some chiral symmetry. Consequently, the chiral condensate, (cid:104) ¯ ψψ (cid:105) canbe employed as an order parameter to investigate the QCD phase diagramas a function of T and µ B . However, flavour and spin symmetry are brokenfor them. Moreover, flavour singlet U A (1) symmetry is broken explicitlyand thus the question of the U A (1) anomaly is mute. On the other hand,the holy grail of phase diagram, namely, the QCD critical point needs twolight flavours and the anomaly to persist [8] for the chiral transition on the µ B = 0 axis to be of second order, and hence it to be a cross over for physicallight quarks. Domain Wall or Overlap quarks are therefore a better choice Rajiv V. Gavai due to their “exact” chiral symmetry on the lattice. Although their nonlo-cality makes them computationally expensive, one can at least in principleemploy them to study the QCD critical point. Defining chemical potentialfor them turns out, however, to be tricky. In particular, introducing chemi-cal potential for either of them faces yet another conundrum related to thedivergence problem discussed above.The usual Wilson Dirac fermion matrix is at the heart of definitionof both these nonlocal quarks. Adapting the exponential prescription for D Wilson , Bloch and Wettig [9] introduced µ . This definition was shown tohave no divergences in the free theory [10, 11]. Unfortunately the BW-prescription breaks the lattice chiral symmetry at any finite density [11],leaving us without any order parameter.Luckily, a lattice action with continuum-like chiral symmetries for quarksat nonzero µ has been proposed already [12]. Since the massless continuumQCD action for nonzero µ can be written explicitly as a sum over rightand left chiral modes of quarks, the key idea was to employ similar chiralprojections for the Overlap quarks to construct the action at nonzero µ . Itwas shown to have exact chiral invariance on the lattice, and thus chiralcondensate works as an order parameter for the entire T - µ B plane [12].Moreover, using the Domain Wall formalism, it was also shown why this isphysically the right thing to do: it counts only the physical (wall) modes asthe baryon number while the BW action includes all the unphysical heavymodes as well.It turns out, however, that this chirally invariant Overlap action withnonzero µ is linear in µ , i.e., comes with divergences. Furthermore, in-venting the f , g in this case which will i) eliminate the divergences and ii)still preserve the exact chiral invariance on the lattice has so far not beenpossible. Recently, it has been shown that SLAC fermions, which too arenonlocal but possess exact chiral symmetry, also need a linear form in µ atfinite density, and it too possesses these divergences [13]. Thus the linearform seems the natural physical choice if chiral symmetries are to be exacton the lattice, although the resultant free theory has divergences. As in theprevious section, these divergences can always be subtracted out especiallyif eliminating them using nonlinear forms for f , g leads to the conundrumsalready discussed.
5. The µ (cid:54) = 0 problems : III. Topology Instanton vacuum provides a nice physical picture of chiral symmetrybreaking and the chiral phase transition [14]. Overlap Dirac operator spec-tra have been used to investigate topology and to understand the nature ofthe high temperature phase. In particular, the number of low quark eigen- onundrums at finite density modes get depleted [15] as T goes up and the number of exact zero modes, ameasure of topological susceptibility, falls exponentially in the quark-gluonplasma phase. Naturally, one can envisage doing a similar study for the highdensity phase. However, it is not easy for QCD due to the sign problem.QCD at nonzero isospin density as well as two colour QCD do not havea sign problem, as the quark determinant is real in each of these cases. Alot of work on both the cases has been done in studying the phase struc-ture [16, 17]. In both these theories, it has been observed that an increasein number density and a drop in the chiral condensate occurs at the same µ c . Interestingly though the spectra of the corresponding low quark modesappear unaffected [18, 19] as a function of the corresponding µ even whenone runs through µ c restoring the chiral symmetry. This observation in twodifferent theories raises an interesting possibility that the chiral symmetryrestoration is decoupled from a change in topology, and thus from perhapsthe deconfinement transition, at finite density/chemical potential in general.Figure 5 displays the eigenvalue distribution [18] on the log scale tohighlight differences in the near-zero modes in the low and high densityphases for the nonzero isospin case: F r equen cy - Log (Λ) Fig. 5. A comparison of the near-zero quark mode distributions below (blue) andabove (red) the finite isospin chemical potential at which chiral symmetry restora-tion occurs. No visible difference is evident. Taken from Ref. [18], where furtherdetails can be found.
Similarly very little or no change is visible in the number of exact zeromodes, or equivalently, the topological susceptibility in both the cases [18,19] across the corresponding chiral symmetry restoring transition. Rajiv V. Gavai
6. Summary
Investigations at finite density using the reliable lattice QCD techniquesface many hurdles, the most famous of which is the sign/phase problemof the quark determinant. We pointed out that the introduction of thechemical potential on the lattice itself is plagued with conundrums. Mostof these, including the µ -dependent divergence, are not due to latticization.Indeed, lattice only reproduces faithfully what exists in the continuum fieldtheory. Elimination of the divergence by modifications of action, as is com-monly done, leads to apparent conflicts with universality which need to beresolved by carrying out continuum limit computations for many differentways of adding chemical potential.Chiral and flavour invariance is crucial for the QCD critical point inves-tigations. Eventually one will have to employ the overlap quarks at finitedensity for reliable simulations. Doing so while retaining the chiral sym-metry seems to lead to a linear µ -dependent action always. Subtraction offree theory divergences was demonstrated to suffice nonperturbatively andshould be tested for the overlap action as well.Numerical simulations suggest that the distribution of the topologicalcharge, Q , changes very little in going from the low T & low density phaseto the low T & high density phase as one goes across the isospin chemicalpotential µ I or µ N c =2 phase transitions, although the chiral condensatedrops and number density picks up at each of these phase transition. Thisis in contrast to the change of low T to high T phase, which exhibits an(exponential) fall-off. This may be a hint towards a possible separation ofthe chiral symmetry restoring transition and the deconfining phase at finitedensity. It will be challenging to check if this is indeed so for the finitedensity QCD.
7. Acknowledgements
It is a pleasure to gratefully acknowledge the financial support by theDeutsche Forschungsgemeinschaft (DFG) through the the Project grant No.315477589-TRR 211 (Strong-interaction matter under extreme conditions).REFERENCES [1] G. Baym, Nucl. Phys.
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