The diagonal and off-diagonal quark number susceptibility of high temperature and finite density QCD
aa r X i v : . [ h e p - l a t ] A p r HIP-2008-07/THNSF-KITP-08-10
The diagonal and off-diagonal quark number susceptibility ofhigh temperature and finite density QCD
A. Hietanen a , K. Rummukainen b , a Theoretical Physics Division, Department of Physicsand Helsinki Institute of Physics,P.O.Box 64, FI-00014 University of Helsinki, Finland b Department of Physics,University of Oulu P.O.Box 3000, FI-90014 Oulu, Finland
Abstract
We study the quark number susceptibility of the hot quark-gluon plasma at zero andnon-zero quark number density, using lattice Monte Carlo simulations of an effectivetheory of QCD, electrostatic QCD (EQCD). Analytic continuation is used to obtainresults at non-zero quark chemical potential µ . We measure both flavor singlet (diagonal)and non-singlet (off-diagonal) quark number susceptibilities. The diagonal susceptibilityapproaches the perturbative result above ∼ T c , but below that temperature we observesignificant deviations. The results agree well with 4d lattice data down to temperatures ∼ T c . The off-diagonal susceptibility is more prone to statistical and systematic errors,but the results are consistent with perturbation theory already at 10 T c . . Introduction The quark (baryon) number susceptibility of hot QCD matter characterizes the “softness”of the equation of state. It is directly related to the event-by-event fluctuations observedin heavy ion collision experiments [1], probing the phase diagram and the properties of thehot QCD plasma. Thus, it is of significant interest to calculate it theoretically as accuratelyas possible. Hence, several calculations of susceptibility have been published using latticesimulations [2, 3, 4, 5, 6, 7, 8] or perturbation theory [9, 10, 11].In this work we use lattice Monte Carlo simulations in order to measure the diagonal (flavorsinglet) and off-diagonal (non-singlet) quark number susceptibilities at high temperatures andat non-zero densities. Instead of full 4-dimensional QCD, the theory we study on the latticeis a dimensionally reduced effective theory of the hot quark-gluon plasma phase of QCD,electrostatic QCD (EQCD) [12, 13, 14, 15, 16]. It is by now well established that EQCD canaccurately describe many properties of the hot QCD plasma, and it provides a very convenientstarting point for studying high-temperature QCD using perturbative analysis [17, 18, 19] ornon-perturbative lattice simulations.The validity of the effective theory approach is based on the fact that at high enoughtemperatures the gauge coupling constant g becomes small, giving rise to three relevantmomentum scales (neglecting quark masses): hard scale p ∼ πT , corresponding to non-zeroMatsubara frequencies, soft electric scale ∼ gT and supersoft magnetic scale ∼ g T . EQCDis obtained by (formally) integrating over the hard scales perturbatively, leaving an effectivetheory for soft and supersoft scales. All infrared divergences inherent in finite temperaturefield theories are correctly contained in the effective theory. A crucial feature of EQCD isthat all of the fermionic modes are integrated over, leaving a purely bosonic theory.EQCD offers an interesting alternative to standard high-temperature lattice simulations.Above all, the theory is three-dimensional and purely bosonic, making it much cheaper tosimulate. The standard QCD lattice simulations work well at temperatures up to 5–10 T c , but due to the sheer cost of the simulations with light quarks it can be very difficult toobtain accurate results. In contrast the perturbative analysis works at temperatures T > ∼ T c (albeit with slow convergence), but since the infrared singularities in the magnetic sectorcannot be treated perturbatively the accuracy is limited to some order (depending on theobservable) in the coupling constant expansion. The lattice simulations of EQCD fully includethe effects of the infrared singularities, thus offering a clear way to improve on the perturbativeresults. While EQCD cannot describe the QCD phase transition, it has been observed to bequantitatively accurate down to temperatures of order 2–4 T c , depending on the quantity ofinterest. On the other hand, it is relatively easy to do EQCD simulations at arbitrarily hightemperatures, enabling one to quantify the convergence to the perturbation theory and therole of the infrared singularities. Lattice simulations of EQCD have been used to calculateQCD pressure at high temperature [20, 21], spatial string tension [22], and spatial screeninglengths [23, 24, 25]. 1n this paper we present the lattice calculations using EQCD to measure the diagonaland off-diagonal quark number (baryon number) susceptibilities at zero and non-zero baryonchemical potential. At non-zero chemical potential EQCD suffers from a sign problem, al-beit this is milder than in full QCD. The finite chemical potential results are obtained byperforming simulations with imaginary values of the chemical potential and then analyticallycontinuing to real chemical potential. We observe that the deviations from the perturba-tion theory are significant up to temperatures of order 20 T c . On the other hand, EQCD isobserved to work at surprisingly low temperatures: our results agree well with existing 4dlattice simulations even slightly below 2 T c . The method also is well suited for simulations atnon-zero chemical potential, because our observations agree those of [5] and extend to evenhigher values of chemical potential. The results have been partly published in [26, 27].The paper is organised as follows. In Sec. 2 we give the theoretical background and specifythe considered observables. In Sec. 3 we present the numerical results of lattice Monte CarloSimulations. Conclusions are given in Sec. 4.
2. Effective theory
The electrostatic QCD with finite chemical potential µ is defined by the action S E = Z d x L E L E = 12 Tr[ F ij ] + Tr[ D i , A ] + m Tr[ A ] + iγ Tr[ A ] + λ (Tr[ A ]) , (2.1)where F ij = ∂ i A j − ∂ j A i + ig [ A i , A j ] and D i = ∂ i + ig A i . F ij , A i and A are traceless3 × A = A a T a , etc). The theory has 4 parameters: g (3-dimensionalgauge coupling), m , λ and γ , with dimensions [ g ] = [ λ ] = GeV, [ γ ] = GeV / and[ m ] = GeV . Non-zero value of the parameter γ , caused by non-zero quark chemicalpotential, renders the action complex. Thus, this theory is not free from the sign problem offinite density QCD.It is convenient to define three dimensionless ratios y = m g , x = λ g , z = γ g , (2.2)leaving only g dimensionful. Through the dimensional reduction process (perturbativematching of suitable observables in EQCD and real QCD), the parameters of EQCD be-come functions of physical 4d parameters: the temperature T and the chemical potential µ (the quark masses are set to zero). The parameters are also functions of the renormalizationscale Λ MS used in the derivation of the effective theory. If we denote the number of quark2avors by N f , for N c = 3 the relations are [16, 24]: g = 24 π − N f T Λ g / Λ MS − N f X i =1 − N f D (¯ µ i ) x + O ( x ) ! (2.3) x = 9 − N f − N f x / Λ MS − N f X i =1 − N f D (¯ µ i ) x + O ( x ) ! (2.4) y = (9 − N f )(6 + N f )144 π x N f X i =1
36 + N f ¯ µ i ! +486 − N f − N − N π (9 − N f ) N f X i =1 N f )(9 − N f )486 − N f − N − N ¯ µ i ! + O ( x ) (2.5) z = N f X i =1 ¯ µ i π (cid:18) N f − N f x (cid:19) + O ( x ) , (2.6)where ¯ µ = µ/ ( πT ), and, for small ¯ µ , D (¯ µ ) ≈ − ζ (3)¯ µ /
2, andΛ g = 4 πT exp (cid:18) − N f log 466 − N f − γ E (cid:19) , (2.7)Λ x = 4 πT exp (cid:18) −
162 + 102 N f − N + (36 N f − N ) log(4)594 − N f + N − γ E (cid:19) . (2.8)The dimensional reduction scheme is expected to be valid temperatures down to ∼ T c andchemical potential up to µ ∼ πT or ¯ µ ∼
1. For N f = 2 these values correspond to x ∼ . z ∼ .
1. At higher temperatures x becomes rapidly smaller. Hence, the higher ordercorrections in x in above formulas become in practice very small, and we ignore corrections O ( x ) in above expressions. We shall further restrict ourselves to 2 massless quarks, N f = 2: g = g | µ =0 x = x | µ =0 y = y | µ =0 N f X i =1 ¯ µ i ! ≡ y N f X i =1 ¯ µ i ! z = N f X i =1 ¯ µ i π . (2.9)See [24] for more discussion about the effect of this approximation. We define the quark number susceptibility in EQCD as: χ ,ij = 1 V ∂ ∂ ¯ µ i ∂ ¯ µ j ln Z = 1 V ∂ ∂ ¯ µ i ∂ ¯ µ j ln Z D A k D A exp ( − S E ) , (2.10)3here i, j stands for quark flavors u and d, and label 3 indicates that this is a result from3-dimensional effective theory. Thus, there are two independent components of the suscep-tibility: diagonal ( i = j ) and off-diagonal ( i = j ). Using the shorthand notation for thedimensionless volume averages ˆ A n ≡ g n V Z d x Tr A n ( x ) , (2.11)and defining the condensates C = (cid:10) ˆ A (cid:11) C = V g (cid:16)(cid:10) ( ˆ A ) (cid:11) − (cid:10) ˆ A (cid:11) (cid:17) C = V g (cid:16)(cid:10) ( ˆ A ) (cid:11) − (cid:10) ˆ A (cid:11) (cid:17) (2.12) C = V g (cid:16)(cid:10) ˆ A ˆ A (cid:11) − (cid:10) ˆ A (cid:11)(cid:10) ˆ A (cid:11)(cid:17) , we can write the susceptibility as χ ,ij g = − δ ij y C − π C + 916 ¯ µ i ¯ µ j y C + i π (¯ µ i + ¯ µ j ) y C (2.13)We note here the rather striking fact that the expectation value in C is purely imaginary for real ¯ µ , rendering the full expression real. The imaginary expectation value comes fromthe complex measure; ˆ A and ˆ A itself are always real-valued. The sign problem of finite density QCD is manifested here as an imaginary term in the EQCDaction, Eq. (2.1). This makes the standard Monte Carlo importance sampling impractical,except for very small chemical potentials and/or small volumes. One option to circumventthis problem is to use analytic continuation to complex values of ¯ µ : the sign problem vanishesfor purely imaginary ¯ µ .However, we emphasize that the direct analytic continuation in ¯ µ is clearly suboptimal andunnecessary in this case: of the terms appearing in EQCD action Eq. (2.1), only iγ Tr[ A ]is responsible for the sign problem. Thus, it is sufficient to analytically continue γ (or z ) toimaginary values and leave the other parameters to the values determined by the desired valueof ¯ µ . By far the dominant effect of non-zero ¯ µ is due to the ¯ µ -dependence of the parameter y in Eq. (2.9), we can take into account almost all of the effects of the chemical potential byjust using the correct y (¯ µ ). The remaining small corrections are then taken into account byanalytic continuation z → iz . In [26] the susceptibility was evaluated by ignoring this correction; the improved statistics here make thesmall correction non-negligible. z → − z and A → − A ,the partition function must be an even function of z (and µ ). From this follows that theexpectation values (cid:10) ˆ A n (cid:11) are even (odd) functions of z for even (odd) n . Therefore, we canTaylor expand the condensates C i appearing in the expression for the susceptibility (2.13) inpowers of z as appropriate: C i ( z ) = X n c i,n z n = X n i n c i,n ( − iz ) n . (2.14)The analytic continuation now proceeds as follows: we perform simulations with imaginaryvalue of z and determine the Taylor series coefficients c i,n for each of the condensates up tothe desired order. Using Eq. (2.14) we obtain the the condensates C i at real values of z ,which can be inserted in Eq. (2.13) in order to obtain the susceptibility.The dependence of the condensates on z is very mild, as expected, and it turns out to besufficient to expand the condensates to very low order: C = a + a z C = a C = a C = ∂C ∂ ( iz ) = − ia z. (2.15)Note that we assume that C and C are independent of z . This is indeed the case to thestatistical accuracy we can reach.If we now denote with C i ( z I ) the condensates measured from simulations with imaginary z = (0 , z I ), the susceptibility at real z = ( z R ,
0) becomes χ ,ij ( z R ) g = − δ ij y (cid:18) C ( z I ) + z z I C ( z I ) (cid:19) − π C ( z I )+ 916 ¯ µ i ¯ µ j y C ( z I ) + 14 π (¯ µ i + ¯ µ j ) y z R z I C ( z I ) . (2.16)We note here that one simulation at some z I is sufficient to obtain the condensates and thesusceptibility at all (small enough) z R . However, because both y and z depend on ¯ µ , onlythe value of z R which corresponds to ¯ µ used in evaluating y is physical. Thus, for each valueof the chemical potential we need to do a new simulation. We also choose to use z I = z R inour simulations, eliminating the ratios z I /z R in Eq. (2.16). In what follows we shall use thenotation z = z R = z I to refer to both quantities.The phase diagram of EQCD has 3 distinct phases: a symmetric phase with (cid:10) ˆ A (cid:11) = 0and 2 broken phases with non-zero (cid:10) ˆ A (cid:11) , related by reflection (cid:10) ˆ A (cid:11) ↔ − (cid:10) ˆ A (cid:11) [28]. In orderto properly represent 4d QCD, the effective theory must remain in the symmetric phase.In the absence of the chemical potential the symmetric phase is at most metastable, whenthe parameters x and y are fixed to values which correspond to 4d QCD. This is normallynot a problem, because the metastability is very strong and for all practical purposes thesymmetric phase remains stable. 5pplying imaginary chemical potential to the full action would decrease the value of theparameter y ( µ ), Eq. (2.9). Hence, the metastability would be reduced and finally completelylost at some value of imaginary µ . However, for our method of analytic continuation thisproblem is completely avoided: because we calculate y ( µ ) with real µ , the value of y increasesas µ increases. Thus, the physical symmetric phase remains stable at all values of µ . The relation between χ ,ij and the physical 4d susceptibility is given by χ ij T = g π T χ ,ij + ∂ ∂µ i ∂µ j ∆ p, (2.17)where ∆ p = p QCD − p EQCD is the perturbative 3d ↔
4d matching coefficient for pressure.This is perturbatively computable order-by-order in coupling constant expansion, because allperturbatively problematic infrared singularities of high temperature QCD are fully containedin EQCD. The matching coefficient is currently known to order O ( g ) [17]. The simulation results in Sec. 3 indicate that the O ( g ) and higher order contributionsto the matching coefficient are very small; indeed, if we compare our results with the 4dsimulation results, we obtain an excellent fit when we assume that these contributions vanish.Thus, the O ( g ) and above contributions to the susceptibility are strongly dominated by thecontributions coming from EQCD.Because EQCD is derived using perturbation theory, the final results depend on the per-turbative scale Λ MS . We shall use here the value Λ MS = 245 MeV, which has been obtainedfrom lattice simulations with 2 light Wilson quarks [30]. For the critical temperature we use T c = 170 MeV, yielding the ratio T c / Λ MS = 0 . The comparison between EQCD and 4dQCD simulation results is somewhat sensitive to the precise value of this ratio, but it canvary ±
10% without significantly affecting the quality of the match. The value 0.7 turns outto be close to the optimal one for the matching.Due to the perturbative nature of the matching equations it turns out to be convenient todo the matching by subtracting the known 3d perturbative susceptibility and adding the 4done: χT = g π T (cid:16) χ latt3 − χ pert3 (cid:17) + χ pert T . (2.18)Here χ pert3 and χ pert are 3d and 4d perturbative results. We also note that the quantities χ uu = χ dd and χ ud = χ du are related to those used in [5] by χ q = 2( χ uu + χ ud ) (2.19) χ I = 12 ( χ uu − χ ud ) (2.20) χ C = 59 χ uu − χ ud . (2.21) For the pressure the matching coefficient has been calculated to O ( g ) in a much simpler theory in Ref. [29]. We obtain the same value by using the results r T c = 0 .
438 [31] and r Λ MS = 0 .
62 [30]. .5. On the lattice The theory in Eq. (2.1) is discretized in a standard way, as described in [16]. Due to thesuperrenormalizability of the 3d theory the couplings λ and g do not run, and m haswell-known linear and logarithmic divergences as the lattice spacing a →
0. When thesedivergences are subtracted the continuum limit is straightforward.The evaluation of the quark number susceptibility requires the measurement of the con-densates in Eq. (2.13) on the lattice. Due to the superrenormalizable nature of the theory,measurements can be rigorously converted to MS scheme in the lattice continuum limit; be-cause MS was used in in the perturbative matching to 4d QCD, this also allows us to compareto 4d results.The relations between the condensates on the lattice and in continuum can be written inthe limit the lattice spacing a → β ≡ / ( g a ) → ∞ ) as [16, 32] C , MS = C ,a − ˜ c β − ˜ c (cid:0) ln β + ˜ c ′ (cid:1) + O (1 /β ) ,C , MS = C ,a − (cid:2) ¯ c (cid:0) ln β + ¯ c ′ (cid:1)(cid:3) + O (1 /β ) ,C , MS = C ,a + O (1 /β ) ,C , MS = C ,a + O (1 /β ) . (2.22)Here labels MS and a indicate that the quantity is calculated in MS or lattice regularization,respectively. The numerical coefficients are˜ c ≈ . , ˜ c = 3 d A (4 π ) ≈ . , ˜ c ′ ≈ . , ¯ c = 516 π ≈ . , ¯ c ′ ≈ . . (2.23)
3. Lattice simulations
The lattice simulations were carried out using two massless quark flavors ( N f = 2). We usednine different values of temperature T , varying from T ≈ . MS up to ∼ × Λ MS . Thetemperature values are shown in Table 3. While the largest temperature is huge in physicalunits, in 3d parameters the variation is much milder; this is related to the fact that QCDapproaches weakly coupled theory at high T extremely slowly. Thus, an extreme range ofhigh temperatures is required in order to reliably assess the convergence to the perturbationtheory.At each temperature we use 6 values for z = ( µ u + µ d ) / (3 π T ), µ u = µ d , up to z = 0 .
15 or µ u /T ≈ .
22. This amounts to 54 different (
T, µ ) pairs. For each physical point simulations7 /T c y ( T ) x ( T )1 .
32 0.357 0.132 .
31 0.448 0.1011 . . × . × . × . × y , x -values used in the simulations. For eachtemperature quark chemical potential has 6 values, parametrized by z ≡ µ q / (3 π T ) =0 , . , . , . , . , .
15, and for non-zero z y is modified according to Eq. (2.9). Ateach (
T, µ )-pair the simulations are done using 6 different lattice spacings, parametrized by β ≡ / ( g a ) = 32 , , , , , β = 6 / ( g a ) = 32 . . . β = 120), the largest lattice size varies between 256 to 320 .In addition to the simulations at physical parameter values, we also did a several series ofruns at fixed x , y and varying z . While these simulations do not correspond to any physicalparameter set, they enable us to look at the z -dependence of the condensates separately. Allin all, our dataset contains 693 individual runs. It turns out that the accuracy requirement are so high that the continuum limit extrapolationof the condensates have to be taken with great care. Especially the continuum extrapolation of (cid:10) ˆ A (cid:11) is critical, because it strongly dominates the susceptibility. While we know the divergent(as a →
0) and constant contributions appearing in the continuum limit, Eq. (2.22), O ( a ), O ( a ln a ) and higher order terms are not yet known. Thus, we use an ansatz (cid:10) ˆ A (cid:11) a − C.T. = c + c β + c ′ β log( β ) + c β , (3.1)where C.T. indicate the known counterterms in Eq. (2.22) and c i are fit parameters. Theexistence of the logarithmic term in the ansatz increases the errors of the extrapolation anorder of magnitude compared to the case without the logarithmic term. However, c ′ isexpected to be a constant independent of y : by dimensional grounds the expansion of (cid:10) ˆ A (cid:11)
8n powers of the lattice spacing can be written as (cid:10) ˆ A (cid:11) a = D a + D g + a (cid:2) D g + D m + D g λ + D λ (cid:3) + O ( a ) . (3.2)The form of the O ( a ) coefficients D , D and D is known and they do not contain a termlogarithmic in a [16, 32], whereas the coefficient D , which is constant in y (or m ), mightinclude one. Thus, the possible a ln a -contribution should indeed be independent of y .The existence of the logarithmic term can be seen in Fig. 1, where we show the parameter c ′ obtained from continuum fits using the ansatz (3.1). Note that here c ′ is fitted independentlyfor each physical parameter set, allowing arbitrary y (and z ) dependence. As expected, theresult is fairly well consistent with constant c ′ ≈ .
69; the remaining systematic discrepanciesin the fit can be caused by contributions which are of higher order than O ( a ), includingterms of type a ln a . Thus, we shall fix c ′ to this value in Eq. (3.1) for all continuum limitextrapolations which follow.We note that the value of c ′ has negligible effect on the results at small (physically relevant)temperatures; c ′ could be set to zero without affecting the continuum limit. It is significantonly at very large T , where it potentially has a role when we compare simulations with theperturbation theory. We observe deviations from perturbative results even at very high T if c ′ < ∼ . ∼
15% around 0.69 do not affect thefinal results.Nevertheless, it is clear that an analytic calculation of O ( a ) effects in EQCD would behighly desirable. There is an ongoing calculation using stochastic perturbation theory [33],which will hopefully confirm our results.The contributions of the other condensates are numerically much smaller and we were notable to see any sign of logarithmic a -dependence in those. It turns out that it is advantageousto make the continuum extrapolation using the full expression of the susceptibility (2.16),instead of extrapolating individual condensates. (Naturally, after the subtraction of theknown counterterms in Eq. (2.22).) This extrapolation is shown in Fig. 2.The lattice volumes are chosen large enough so that finite volume effects become negligible.We have tested this by doing simulations at selected parameter values using different volumes;at the smallest lattice spacing ( β = 120) the volume varies from V = 144 up to V = 320 .No systematic finite volume effects inside two sigma errors. For more discussion of finite sizeeffects on a related model see Ref. [21].The finite chemical potential dependence is studied using the method described in Sec. 2.3.The condensates C = V g ( (cid:10) ( ˆ A ) (cid:11) − (cid:10) ˆ A (cid:11) ) and C = V g ( (cid:10) ( ˆ A ) (cid:11) − (cid:10) ˆ A (cid:11) ) should belargely independent of z (for fixed x, y ) for the equation (2.15) to be valid. This indeedturns out to be the case, within the statistical errors, and any remnant z -dependence iscompletely drowned out by the contributions from z -independent parts in Eq. (2.15). Indeed,the overall z -dependence of each of the condensates in Eq. (2.15) turns out to be statisticallyalmost invisible, with the exception of C = V g ( (cid:10) ˆ A ˆ A (cid:11) − (cid:10) ˆ A (cid:11)(cid:10) ˆ A (cid:11) ), which has a linear9 c ’ z = 0z = 0.025z = 0.05z = 0.075z = 0.1z = 0.15Fit: c’ = 0.69(2) Figure 1: Fitting of the logarithmic coefficient c ′ in continuum extrapolation. The data isconsistent with the assumption that c ′ is a constant χ / d . o . f ≈ / β β χ , uu y = 6.6214y = 5.3080y = 3.9945y = 3.0853y = 2.0244y = 1.1801y = 0.7110y = 0.4483y = 0.3574 z=0 z=0.1 Figure 2: Continuum extrapolation of the diagonal susceptibility χ , uu at chemical potential z = 0 and z = 0 .
1. The statistical errors are too small to be visible.10 -dependence. In practice the µ -dependence of the susceptibility is almost completely due tothe µ -dependence of the parameter y and the µ C -term Eq. (2.16) . Nevertheless, here wedo take into account the small z -dependence of C = (cid:10) ˆ A (cid:11) and C , although it will affect thefinal results by only about 1 sigma. Now we are in position to compare the continuum limit results with the perturbation the-ory. First we shall look at the diagonal susceptibility χ , uu = χ , dd . The susceptibility hasbeen calculated in perturbation theory up to order g ln 1 /g [9]. In 3-dimensional units theperturbative result can be written as a power series in 1 / √ y , with the following result: χ pert3 , uu g = 8 + 9¯ µ p µ y / π − (9 − x )¯ µ − x ) + 6(4 + 3¯ µ ) ln(4 + 3¯ µ ) + 6(4 + 3¯ µ ) ln( y )2(4 + 3¯ µ ) 3 y (4 π ) − (8 + 3¯ µ )(89 + 4 π −
44 ln(2))8(4 + 3¯ µ ) / y / (4 π ) + n − µ )¯ µ ] + 119313 π + 640(4 + 3¯ µ ) ln(4 + 3¯ µ )+ 640(4 + 3¯ µ ) ln( y ) o µ ) (4 π ) + 803(4 π ) β M2 + O ( y − / ) . (3.3)We have set here ¯ µ u = ¯ µ d = ¯ µ . As can be observed in Fig. 3 the overall agreement betweenthe lattice result and the perturbation theory is very good, especially at large y (large tem-perature). The result contains an unknown ¯ µ -independent order O ( y ) -term denoted by β M2 in [18]. The same term appears also in the off-diagonal susceptibility, Eq. (3.5), and it turnsout that it gives much tighter constraints for the value of β M2 than the diagonal one. The fitto the off-diagonal susceptibility gives β M2 = − . ± .
3. This value is small enough that itseffect is negligible for the diagonal susceptibility, nonetheless we set here β M2 = − . χ latt3 − χ pert3 and fit a function of form b /y − / + b /y to the result. The fitresults are shown in Fig. 4 and Table 2. We note that for small z the 1 /y / -term is muchsmaller than the 1 /y -term, indicating that the O ( g ) -contribution arising from EQCD issmaller in magnitude to the O ( g ) term, at least for all physically relevant temperatures. Atlarge z the statistical errors grow rapidly; this is due to the term ∝ z V g ( (cid:10) ( ˆ A ) (cid:11) − (cid:10) ˆ A (cid:11) )in Eq. (2.16).Finally, we obtain the physical 4d result for the diagonal susceptibility from Eq. (2.18).As described in Sec. 2.4, the 3d ↔
4d mapping remains sensitive to the unknown O ( g ) andhigher order perturbative contributions to the matching coefficient. In Fig. 5 we show the This fact was used in the preliminary results published in ref. [27] y χ T/ Λ MS __ z = 0z = 0.025z = 0.5z = 0.075z = 0.1z = 0.15 y -0.05-0.04-0.03-0.02-0.010 χ - χ , p T/ Λ MS __ z = 0z = 0.025z = 0.05 Figure 3: Left: the diagonal quark number susceptibility χ , uu /g at different values ofchemical potential. The symbols indicate the lattice measurements, and the solid lines arethe perturbative result. Right: The difference between the lattice and perturbation theory.z fit χ / dof0 0 . / √ y − . /y . / √ y − . /y . / √ y − . /y − . / √ y − . /y − . / √ y − . /y − . / √ y + 0 . /y b / √ y + b /y to ( χ lat3 , uu − χ pert3 , uu ) /g . The smallest y (lowest temperature) points are left out of the fit.12 -1/2 -0.03-0.025-0.02-0.015-0.01-0.0050 0 0.4 0.8 1.2 1.6y -1/2 -0.03-0.025-0.02-0.015-0.01-0.0050 0 0.4 0.8 1.2 1.6y -1/2 -0.03-0.025-0.02-0.015-0.01-0.00500 0.4 0.8 1.2 1.6y -1/2 -0.03-0.02-0.010 0 0.4 0.8 1.2 1.6y -1/2 -0.03-0.02-0.0100.01 0 0.4 0.8 1.2 1.6y -1/2 -0.04-0.03-0.02-0.010z = 0 z = 0.025 z = 0.05z = 0.075 z = 0.1 z = 0.15 Figure 4: The diagonal susceptibility ( χ lat3 , uu − χ pert3 , uu ) √ y/g as a function of 1 / √ y with differentvalues of the chemical potential. Solid line is a 1st order polynomial fit. The data at y − / ≈ . µ = 0 with these unknown contributions set to zero. We observe that theresult fits the 4d lattice simulations very well, clearly indicating that the magnitude of thesecontributions must be small, and in what follows we shall set them to zero. On the otherhand, it should be noted that the difference between the purely perturbative result and EQCDsimulation result is substantial at T < ∼ T c , as indicated by the two lines in Fig. 5. This isa clear indication that the contributions beyond the currently known perturbative ones havenon-negligible effect at experimentally accessible temperatures.The µ -dependence of the diagonal susceptibility is shown in Fig. 6, normalized to theStefan-Boltzmann value χ SB ( µ ) = T + 3 π µ . (3.4)We note that at temperatures above 100 T c the deviation from the Stefan-Boltzmann lawis independent of µ , but at lower T there is significant µ -dependence. The µ -dependencematches very well the 4d lattice results by Allton et al. [5], also shown in Fig. 6.13
10 100T/T c χ uu / T our resultGavai et al.Karsch et al.perturbation theory Figure 5: The diagonal susceptibility χ uu in 4d units at µ = 0. The data points indicate thefull EQCD result. The continuous line is the result of the fit in Table 2. The dashed line showsthe perturbative result alone, Eq. (3.3), using the same matching as in the EQCD result.The difference between these two curves indicates the magnitude of the non-perturbativecontributions. The agreement with the 4d-lattice results of Gavai et al. [2] and Karsch et al.[4] is good. c χ uu / χ S B ( µ ) µ /T = 0 µ /T = 0.37 µ /T = 0.74 µ /T = 1.11 µ /T = 1.48 µ /T = 2.22 µ q /T0.7511.251.51.7522.252.5 χ uu / T our result, T = 1.32T c our result, T = 2.0T c Allton et al. T = 1.36T c Allton et al. T = 1.98T c Figure 6: Left: The diagonal susceptibility at different µ , normalized to Stefan-Boltzmannlaw. Right: µ -dependence of the susceptibility compared with the 4d lattice results of Alltonet al. [5]. 14 .3. Off-diagonal susceptibility The perturbative result for the off-diagonal susceptibility in 3d units is χ pert3 , ud g = 9¯ µ p µ y / π + 27¯ µ µ y (4 π ) + 27¯ µ (89 + 4 π −
44 log(2))8(4 + 3¯ µ ) / y / (4 π ) + n − π + 15360¯ µ )¯ µ + 2560(4 + 3¯ µ ) ln(4 + 3¯ µ )+2560(4 + 3¯ µ ) ln( y ) o µ ) (4 π ) + 803(4 π ) β M2 + O ( y − / ) (3.5)where β M2 is the same unknown coefficient which appears in the diagonal susceptibility,Eq. (3.3). In this case we can fit the value at z = 0, obtaining β M2 = − . ± . . (3.6)This value is small enough to have in practice negligible effect on the final results. Again thesimulation data is very well described by the perturbation theory, Fig. 7; only at z = 0 or atlowest temperatures can we observe deviations from perturbation theory.After matching to 4d, we obtain the result for off-diagonal susceptibility χ ud , shown inFig. 8. At T > ∼ T c the results match the perturbation theory very well, but at lower tem-peratures there are deviations: most significantly, at T = 1 . T c and µ = 0 the simulationresults clearly undershoot the perturbation theory. On the other hand, the 4d lattice resultsin [5] at µ = 0 indicate small but non-zero value, which agrees well with perturbation the-ory. [9, 10]. This can be an indication that this point is already outside the validity rangeof EQCD; however, we also note that by increasing T c / Λ MS the EQCD results are broughtcloser to 4d lattice results [5]. The agreement with the perturbation theory and 4d latticeresults is rather good already at T = 2 . T c .We also note that the physical value of the off-diagonal susceptibility is obtained in EQCDby a subtraction of two divergent as a → T = 1 . T c .Nevertheless, the overall µ -dependence of χ ud is in rough accordance with the 4d latticeresults [5] already at T = 1 . T c , as shown on the right panel in Fig. 8, and at T = 2 . T c theagreement is already very good. The non-diagonal susceptibility is seen to behave quite wellup to large values of µ/T ∼
2. 15 y χ , ud T/ Λ MS __ z = 0z = 0.025z = 0.05z = 0.075z = 0.1z = 0.15 y -0.004-0.003-0.002-0.00100.001 χ , ud - χ , ud , p T/ Λ MS __ z = 0z = 0.025 Figure 7: Left: the off-diagonal susceptibility χ , ud /g in 3d units. Right: the differencebetween lattice and perturbative susceptibilities ( χ latt3 , ud − χ pert3 , ud ) /g , shown at 2 smallest ¯ µ .The statistical errors grow rapidly as ¯ µ increases. c χ ud / T µ /T = 0 µ /T = 0.37 µ /T = 0.74 µ /T = 1.11 µ /T = 1.48 µ /T = 2.22 µ q /T-0.02500.0250.050.0750.1 χ ud our data, T = 1.32T c Allton et al. T = 1.36T c our data, T = 2.3T c Allton et al. T = 1.98T c Figure 8: Left: The off-diagonal susceptibility in 4d. At low temperatures we obtain signif-icantly different values from the perturbation theory (solid lines), but there is no deviationanymore at T = 10 T c . Right: µ -dependence of off-diagonal susceptibility compared with All-ton et al. [5]. The precision of results from [5] are probably not accurate enough to predictthe behaviour at region µ/T >
1. 16 . Conclusions
We have measured the quark number susceptibility of high temperature finite density QCDusing lattice simulations of EQCD, an effective 3-dimensional theory of full 4d QCD. Thevery good match to the 4d lattice results with 2 light quark flavors at low temperatures andwith the perturbation theory at high temperatures shows the wide range of applicability ofthe method. The diagonal susceptibility is seen to agree with 4d simulations by Allton etal. [5] even below 2 T c , including the dependence on µ . On the other hand, we observe asubstantial deviation from the known perturbative result up to temperatures ∼ T c . Theoff-diagonal susceptibility is compatible with perturbation theory already at T > ∼ T c . Theresults also agree with the 4d simulations [5] except perhaps at lowest temperatures, T < T c .The results clearly indicate that EQCD is a viable method to obtain quantitatively signif-icant results of the hot QCD plasma down to T ∼ T c . Equally significant is the observationthat the currently known perturbative result alone deviates significantly from the correctresult: while the perturbative result can be made to match the 4d lattice data by adjust-ing the still unknown (high perturbative order) matching coefficients, EQCD allows us todirectly measure the differences between simulations and perturbative calculations withoutany scale or matching ambiguities. Thus, simulations of EQCD are exceptionally well suitedfor observing the convergence of the perturbation theory. It is worth noting that while theEQCD susceptibility also suffers from matching ambiguity, we obtain an excellent fit to 4dsimulations by assuming these matching coefficients vanish, indicating that the contributionfrom these is necessarily very small.
5. Acknowledgements
We acknowledge useful discussion with K. Kajantie, M. Laine and A. Vuorinen. This workhas been partly supported by the Magnus Ehrnrooth Foundation, a Marie Curie Fellowshipfor Early Stage Researchers Training, and the Academy of Finland, contract number 114371.KR also acknowledges partial support by the National Science Foundation under Grant No.PHY05-51164. Simulations have been carried out at the Finnish IT Center for Science (CSC).
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