Chemical potential on the lattice: Universal or Unique?
aa r X i v : . [ h e p - l a t ] D ec Chemical potential on the lattice: Universal or Unique?
Rajiv V. Gavai a a Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, Germany
Abstract
Lattice techniques are the most reliable ones to investigate non-perturbativeaspects of quantum chromodynamics (QCD) such as its phase diagram inthe temperature-baryon density plane. They are, however, well-known to bebeset with a variety of problems as one increases the density. We addresshere the old question of placing the baryonic (quark) chemical potential onthe lattice. We point out that it may have important consequences for thecurrent and future experimental searches of QCD critical point.
Email address: [email protected] (Rajiv V. Gavai)Address after 1 January 2021 : Indian Institute of Science Education & Research,Bhopal bypass, Bhauri, Bhopal 462066, India.
Preprint submitted to Elsevier December 15, 2020 . Introduction
The behaviour of strongly interacting matter, described by QuantumChromodynamics(QCD), at nonzero temperatures or baryon densities hascontinued attracting attention both theoretically and experimentally for morethan three decades [1, 2, 3]. Since QCD coupling is known to be large at ornear the scale of QCD, Λ
QCD , investigating the QCD phase diagram neces-sitates strong coupling techniques. Lattice QCD is the most successful non-perturbative technique which has provided us with key interesting resultspertaining to the phase diagram. For instance, it is known from indepen-dent lattice studies that the transition from the hadron phase to the quarkgluon plasma phase at zero baryon density is a crossover [4, 5, 6]. Extend-ing these results to non-zero baryon density, or equivalently nonzero quarkchemical potential µ , one encounters the famous sign problem : the quarkdeterminant becomes a complex number, inhibiting the use of the trustedimportance sampling based Monte Carlo methods.Several ways have been proposed to confront the sign problem in QCD[7, 8, 9, 10]. Based on an analysis of model quantum field theories withthe same symmetries as two light flavour QCD [11, 12], a critical end-point is expected to exist in the QCD phase diagram. One expects thebaryon number susceptibility to diverge [13] there. Consequently, its Taylorseries expansion at finite baryon density would have a finite radius of conver-gence, leading to an estimate of the location of the critical end-point [13, 14].First such estimates of the radius of convergence of the Taylor series sug-gested the critical end-point to be at T E /T c = 0 .
94 and µ B /T E = 1 . T E /T c = 0 . , µ B /T E = 1 . σ level disagree [16] with the results of [15]. In heavy-ion experiments atRHIC, the fluctuations of the net proton number are employed as a proxyfor the net baryon number. The STAR experiment at Brookhaven NationalLaboratory has measured the fluctuations of the net proton number up tothe fourth order for a wide range of center of mass energy √ s . At √ s = 19 . th -order fluctuationsmay shed light on whether the crossover at zero baryon density is a shadowof the O (4) criticality in the chiral limit [20]. Clearly, still higher orders willeventually need to be computed for better control over the radius of conver-gence. Thus higher order susceptibilities are, and will continue to remain, ofimmense interest.In this paper, we compare and contrast the different ways of introducingthe chemical potential on the lattice, and assess their impact on these higherorder susceptibilities which also govern the coefficients of the Taylor series.Astonishingly, we find that the results depend on the way chemical potentialis introduced. The differences appear to persist in the continuum limit. Thisobservation also has consequences for all other methods to tackle the signproblem. We argue for a choice closest to the continuum QCD as the best.In section 2, we recall the existing methods to place chemical potential onthe lattice and demonstrate their failure with universality. The next section3 is devoted to a discussion of their other attributes. We finally summariseour results.
2. Universality and Chemical Potential
The lattice QCD partition function in the path integral formalism is givenby Z = Z D U µ D ¯ ψ D ψ e − S G − S F (ma ,µ a) , (1)where ψ ( x ), ¯ ψ ( x ) and U µ ( x ) represent the quark, anti-quark at site x andthe gluon field on the link ( x, ˆ µ ) respectively. S G denotes a suitable choicefor the gluonic action and S F is the quark action. We shall consider belowthe naıve quark action but our considerations are easily generalized to otherlocal actions such as the Wilson action, the staggered action or their improvedversions. Similarly, we will consider only a single flavour with the baryonicchemical potential µ B = 3 µ for simplicity, generalization to more flavoursagain being straightforward. Denoting by ma the quark mass and by µa itschemical potential, the fermionic action is given by S F = ¯ ψM ( ma, µa ) ψ with3 defined as below: S F ( ma, µa ) = X x,µ =1 ¯ ψ ( x ) γ µ (cid:2) U µ ( x ) ψ ( x + ˆ µ ) − U † µ ( x − ˆ µ ) ψ ( x − ˆ µ ) (cid:3) (2)+ X x ¯ ψ ( x ) γ (cid:2) f ( µa ) · U ( x ) ψ ( x + ˆ4) − g ( µa ) · U † ( x − ˆ4) ψ ( x − ˆ4) (cid:3) + ma ¯ ψ ( x ) ψ ( x ) , Three possible choices have so far been used in the literature [21, 22, 23] forthe functions f and g , denoted below by subscripts L (linear), E (exponential)and S (square root): f L ( µa ) = 1 + µa , g L ( µa ) = 1 − µa (3) f E ( µa ) = exp( µa ) , g E ( µa ) = exp( − µa ) f S ( µa ) = (1 + µa ) / p − µ a , g S ( µa ) = (1 − µa ) / p − µ a . Following the natural route of obtaining the conserved charge from the cor-responding current conservation equation on the lattice leads to the naıvelinear choice [23] above. However, it has µ -dependent quadratic divergencesin the number density and the energy density even for the free quark gas.These can be eliminated by the other two options for f and g . Indeed allfunctions satisfying f ( µa ) · g ( µa ) = 1 eliminate [24] those divergences. It isa straigthforward exercise to check that all these actions lead to the same continuum action in the limit of vanishing lattice spacing a → a . Integrating theGrassmannian quark and antiquark fields, one has Z = Z D U µ e − S G DetM(ma , µ a) . (4)A derivative of ln Z with µ leads to the quark number density, or equiv-alently (1/3) the baryon number density, defined by, n = TV ∂ ln Z ∂µ | T =fixed (5)= 1 N t N s a h Tr M − · M ′ i , where M ′ is the derivative of the fermionic matrix M with respect to µa , T= ( N t a ) − is the temperature and V = N s a is the volume. In the process4f obtaining predictions for the signals of either the critical end point or thetwo-flavour chiral transition, one evaluates higher order derivatives of n toobtain various fluctuations such as the variance, skewness or kurtosis etc.In fact, coefficients of µ a have been computed in attempts to locate theQCD critical point [14], and those of µ a terms are expected to assist [20] inpinning down the hints of a critical point in the chiral limit of the two-flavourtheory in the heavy ion collision data.In general, a O ( µ k a k ) will clearly involve up to k -th derivative of thefermion matrix M , and thus of f and g . Using the condition f ( µa ) · g ( µa ) = 1along with the obvious f (0) = g (0) = 1 and f ′ (0) = − g ′ (0) = 1 (to ensure the µN form in the a → f ′′ (0)+ g ′′ (0) = 2. Using thefact that particle-antiparticle symmetry implies f ( µa ) = g ( − µa ), one findsthat the f k (0) = ( − k g k (0), and thus f ′′ (0) = g ′′ (0) = 1. Both f E and f S satisfy this. Unfortunately they differ in all the higher derivatives. There areno more conditions to fix the higher derivatives. Indeed, f ′′′′ (0) = 4 f ′′′ (0) − f ′′′ E (0) = 1 with f ′′′′ E (0) = 1 and f ′′′ S (0) = 3 with f ′′′′ S (0) = 9 do satisfy thisrelation. Thus only the first derivative is identical for all the f ’s in eq.(3).Already the second derivative f ′′ L (0) = 0 but the second derivative is identicalfor f E and f S and is unity. All further higher derivatives are different. Notethese are all pure numbers, i. e. , an approach to continuum limit will notchange these derivatives themselves. This has consequences for the varioushigher order fluctuations of the conserved charge. They too will be differentdepending upon the choice of f from eq.(3) with no hope of their convergingin the continuum limit. A priori all f are on the same footing. This thereforeappears to be then a serious violation of universality, as f ′′′ and f ′′′′ enterexperimentally measurably quantities such as kurtosis or the χ B .One ought to have seen this coming after all since f L has quadratic µ -dependent divergences but the others do not. An easy way to see this is tolook at the expression for quark number susceptibility. It is given by χ = TV hD(cid:0) Tr M − M ′ (cid:1) E + (cid:10) Tr (cid:0) M − M ′′ − M − M ′ M − M ′ (cid:1)(cid:11)i . (6)Since f ′′ L = g ′′ L = 0 for the naıve linear choice for all µa , the first term inthe second expectation value vanishes whereas f ′′ (0) = g ′′ (0) = 1 ensureselimination of the divergence for the other two, and indeed for all such f and g which satisfy f · g = 1. It is important to note that the second derivativecomes from O ( µ a ) terms in f E , f S and g E , g S in eq.(3). What appears5rrelevant at the action level is not so at the susceptibility level. Indeed,the divergence is eliminated precisely due to this fact. It should not comeas a surprise that this phenomena recurs for higher order susceptibilities aswell. One encounters even more prescription dependence at higher orders.Consider for example the fourth order susceptibility [13] : χ = TV "(cid:28) O + 6 O + 4 O + 3 O + O (cid:29) − (cid:28) O + O (cid:29) . (7)Here the notation O ij ··· l stands for the product, O i O j · · · O l . The relevant O i for eq.(7) are [13] O = Tr M − M ′ , (8) O = − Tr M − M ′ M − M ′ + Tr M − M ′′ , O = 2 Tr ( M − M ′ ) − M − M ′ M − M ′′ + Tr M − M ′′′ , O = − M − M ′ ) + 12 Tr ( M − M ′ ) M − M ′′ − M − M ′′ ) − M − M ′ M − M ′′′ + Tr M − M ′′′′ . Since O and O have terms with M ′′′ and M ′′′′ , which in turn contain the f ′′′ , g ′′′ , f ′′′′ and g ′′′′ it is clear that for each choice out of the three in eq.(3),one will obtain a different value on the same set of dynamical gauge con-figurations. Again none of the terms on the RHS of eq.(8) vanishes in thecontinuum limit. Thus χ onwards for all higher order susceptibilities one ob-tains results which depend on the choice of f and g and are thus not universal.This loss of universality is not limited only to the higher order fluctuations ofthe conserved charges computed using lattice QCD simulations. Recall thatthe pressure P can be constructed as a series in µ B with these susceptibilitiesas the coefficients. Hence, the pressure, and consequently all thermodynamicquantities derived from it, are also similarly prescription dependent from thefourth order onwards.In short, the quest to get rid of the µ -dependent divergences lead tomodification of the action in the Euclidean representation of the partitionfunction, ostensibly by adding terms which are irrelevant in the continuumlimit a →
0. The presence of the dimensional parameter µ in these terms,however, spoils this naıve expectation of universality. Employing the f L and g L prescription has the advantage of being faithful to the continuum theoryin reproducing the higher order fluctuations, but also has the disadvantageof a µ -dependent divergence, again as in the continuum theory.6 . Conservation of Charge The linear chemical potential forms f L and g L also have few more continuum-like attributes which the other forms lack. These too appear to suggest addi-tional violations of universality. Since the quark determinant Det M ( ma, µa )is gauge invariant and contains only the gauge link fields U µ ( x ), expandingthe determinant in its various terms, one obtains its representation as a sumof closed Wilson loops of the link fields. One can classify them into spatialand temporal loops. Only the latter can contribute to µ -dependence. Fur-thermore if there are l -timelike positive links in a Wilson loop, it also has l negative timelike links. Its contribution then is proportional to ( f L · g L ) l =(1 − µ a ) l . On the other hand, ( f E · g E ) l = 1 and ( f S · g S ) l = 1. Thereforenone of these loops contributes to µ -dependence of Z if one opted for eitherof the two forms to introduce µ . There is a topological distinct class of loopswhich does contribute to all of them. The simplest amongst them is a loopwinding around the temperature axis once, and contributes f N t or g N t foreach f and g depending on the winding direction. One can, of course havemore windings. Only these topologically nontrivial Wilson lines lead to any µ -dependence in the case of E and S -forms. Note that small topologicallytrivial loops do contribute in the continuum just as in the linear case. Inview of the topological distinction in the classes of loops, it is hard to seehow even in the a → E and S -forms will somehow agree with the L -case and the continuum, although strange cancellations can not be ruledout until a full actual computation is performed.Using fugacity z = exp( µ/T ), one relates the grand canonical partitionfunction to the canonical ones : Z GC = P n z n Z Cn . Since z lat = exp ( N t µa ),such a relation is feasible only for the linear prescription of adding chemicalpotential. Alternatively, one sees this in the conserved number which oughtto remain the same for all µ . Recall that invariance of the action of under aglobal U (1) symmetry leads to a current conservation equation, ∂ µ j µ ( x ) = 0,and hence the conserved charge N = P ~x j ( ~x ). It is worth noting thataddition of µ ¯ ψ ( x ) γ ψ ( x ) term in the Lagrangian does not alter the currentconservation equation in the continuum, with the conserved charge remainingthe same as it should.For the lattice theory one can similarly demand invariance of eq.(2) underthe global U (1) symmetry: For ψ ′ = ψ + δψ and ¯ ψ ′ = ¯ ψ + δ ¯ ψ , δS F = 0,where δψ = iǫψ , and δ ¯ ψ = − iǫ ¯ ψ and ǫ is small. The resultant current7onservation equation is easily worked out as P µ [ j µ ( x − ˆ µ ) − j µ ( x )] = 0for the case µa = 0 when f and g are unity in general. Here j µ ( x ) = (cid:2) ¯ ψ ( x ) γ µ U µ ( x ) ψ ( x + ˆ µ ) + ¯ ψ ( x + ˆ µ ) γ µ U † µ ( x ) ψ ( x ) (cid:3) is the point split version ofthe usual current one obtains in the continuum theory. For the case of µ = 0,one can write the generic f and g as (cid:2) ( f + g ) / ± ( f − g ) / (cid:3) respectively.The δS F = 0 equation can then be simplified similarly with two differences. δS F has an additional term proportional to [ f ( µa ) − g ( µa )] /
2, which is givenby δS addF ( ma, µa ) = [ f ( µa ) − g ( µa )] / X x (cid:2) ¯ ψ ( x ) γ U ( x ) ψ ( x + ˆ4) (9)+ ¯ ψ ( x ) γ U † µ ( x − ˆ4) ψ ( x − ˆ4) − ¯ ψ ( x − ˆ4) γ U ( x − ˆ4) ψ ( x ) − ¯ ψ ( x + ˆ4) γ U † ( x ) ψ ( x ) (cid:3) . Noting that x is a dummy sum variable, and substituting y = x ± ˆ4 inthe two terms on the third line of the eq. (9), it is easy to show that δS addF ( ma, µa ) = 0, bringing the full δS F = 0 to a current conservationform but with a difference. The expression for only j ( x ) is replaced by j mod ( x ) = [ f ( µa ) + g ( µa )] / (cid:2) ¯ ψ ( x ) γ U ( x ) ψ ( x + ˆ4) + ¯ ψ ( x + ˆ4) γ U † ( x ) ψ ( x ) (cid:3) ,resulting in the modified conserved charge being N mod = P ~x j mod ( ~x ). Sub-stituting the f and g from eq. (3), one can work out the consequences ineach case. For the linear case, N mod = N and thus remains unchanged.For the other two cases, namely the exponential and the square root forms, N mod itself is µ -dependent for nonzero a , being cosh( µa ) N ( µa = 0) and N ( µa = 0) / p − µ a respectively. For small µa , these functions can be ex-panded to obtain a quadratic a -approach to the standard conserved chargein the a →
4. Summary
Current and future experimental programs on heavy ion collisions aimto measure fluctuations of conserved charges precisely. The STAR resultsalready exhibit intriguing structure in higher order proton number fluctua-tions such as kurtosis. Still higher order fluctuations ( χ B ) are anticipated toshed light on the nature of the chiral phase transition. Reliable theoreticalpredictions are needed for these for a trustworthy comparison. Lattice QCDat finite density is the best tool one currently has.8efining a conserved charge, for instance the baryon number, from thecorresponding conserved current defined on the lattice and adding it usingthe canonical Lagrange multiplier type linear chemical potential term in thefermion actions on the lattice is most natural. Its µ -dependent divergenceslead in the past to the proposals of other action, including the popular ex-ponential action. We showed that these actions lead to different results forthe same physical quantities, namely the higher order fluctuations startingfrom kurtosis. These differences in the same physical quantity persist in thecontinuum limit of a →
0, and therefore the actions designed to eliminatefree theory µ -dependent divergences violate universality. We also providedtwo other arguments to demonstrate the lack of universality. Only the ac-tion linear in µ has continuum-like attributes of contribution from temporalWilson loops of all sizes, as well as an unchanged current conservation equa-tion and hence the same conserved charge for µ = 0 as in the continuum.Other actions, including the popular exponential form, do not share theseproperties: Only topologically nontrivial Wilson lines contribute to the µ -dependence and the conserved charge itself becomes function of µ . It maybe worth noting that preservation of exact chiral invariance on the latticeseems feasible only for a linear form [25] for the continuum-like overlap andthe domain wall fermions. Since a µ -dependent divergence exists alreadyin the continuum for a gas of free fermions, and is subtracted there, it cansimilarly be subtracted out in simulations [26]. Action with linear chemicalpotential term is thus unique in that it mimics the continuum behaviourfaithfully for both local and nonlocal fermion actions. Modifying the localaction to eliminate the divergence leads to a loss of universality for higherorder susceptibilities, and indeed the full partition function.
5. Acknowledgements
It is a pleasure to gratefully acknowledge the support by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) through thethe CRC-TR 211 ’Strong-interaction matter under extreme conditions’– projectnumber 315477589 – TRR 211. The author is also very happy to acknowledgethe kind hospitality of the Physics Department of the Universit¨at Bielefeld,Germany, in particular that of Profs. Frithjof Karsch and Olaf Kaczmarek.9 eferences [1] J. Cleymans, R. V. Gavai and E. Suhonen, Phys. Rept. et al. (MILC Collaboration), Phys. Rev.
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