At high temperature, every $(d+1)$-dimensional theory can be reformulated as
an effective theory in $d$ dimensions. We test the numerical accuracy of this
Dimensional Reduction for (3+1)-dimensional SU(2) by comparing perturbatively
determined effective couplings with lattice results as the temperature is
progressively lowered. We observe an increasing disagreement between numerical
and perturbative values from $T=4 T_c$ downwards, which may however be due to
somewhat different implementations of dimensional reduction in the two cases.Comment: Lattice2001(hightemp), AMS-LaTeX v1.2, 3 pages with 3 figure

Hasenbusch

Hasenfratz

Kajantie

LaCock

Laine

Philippe de Forcrand

Philipsen

Slavo Kratochvila

English

arXiv

At high temperature, every $(d+1)$-dimensional theory can be reformulated as an effective theory in $d$ dimensions. We test the numerical accuracy of this Dimensional Reduction for (3+1)-dimensional SU(2) by comparing perturbatively determined effective couplings with lattice results as the temperature is progressively lowered. We observe an increasing disagreement between numerical and perturbative values from $T=4 T_c$ downwards, which may however be due to somewhat different implementations of dimensional reduction in the two cases.At high temperature, every $(d+1)$-dimensional theory can be reformulated as an effective theory in $d$ dimensions. We test the numerical accuracy of this Dimensional Reduction for (3+1)-dimensional SU(2) by comparing perturbatively determined effective couplings with lattice results as the temperature is progressively lowered. We observe an increasing disagreement between numerical and perturbative values from $T=4 T_c$ downwards, which may however be due to somewhat different implementations of dimensional reduction in the two cases

Kratochvila, Slavo

De Forcrand, Philippe

CERN Document Server

1Testing Dimensional Reduction in SU(2) Gauge TheorySlavo Kratochvilaa and Philippe de ForcrandabaInstitute for Theoretical Physics, ETH Zu¨rich, CH-8093 Zu¨rich, SwitzerlandbTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandAt high temperature, every (d + 1)-dimensional theory can be reformulated as an effective theory in d dimen-sions. We test the numerical accuracy of this Dimensional Reduction for (3+1)-dimensional SU(2) by comparingperturbatively determined effective couplings with lattice results as the temperature is progressively lowered. Weobserve an increasing disagreement between numerical and perturbative values from T = 4Tc downwards, whichmay however be due to somewhat different implementations of dimensional reduction in the two cases.1. INTRODUCTIONOne of the major thrusts of nuclear physics inthe next decade will be the eort to study thequark-gluon plasma. At the high temperaturesreached, one can consider applying DimensionalReduction. To test the accuracy of DimensionalReduction, a purely gluonic system provides asimple, but relevant benchmark. Therefore, herewe study SU(2) gauge theory at nite tempera-ture. Up to now, only gauge invariant correla-tion functions of colourless states have been mea-sured. This approach gives no information aboutthe eective action. Here, we measure directlythe eective coupling constants. The analytics ofDimensional Reduction gives us perturbative ex-pressions for these coupling constants. A numer-ical, non-perturbative determination allows us tomonitor the two ways in which the perturbativecalculation can fail: (i) As the temperature is low-ered, the couplings increasingly depart from theirperturbative values. (ii) The eective action be-comes less local - couplings of higher-dimensionoperators, e.g. A60, completely neglected in theperturbative approach, become sizeable.The non-perturbatively determined coupling con-stants allow us, with certainty in principle, to ex-tend the validity of Dimensional Reduction belowtemperatures achieved perturbatively. In thatsense, we attempt a non-perturbative improve-ment of the dimensionally reduced action.∗Talk presented by S. Kratochvila2. DIMENSIONAL REDUCTIONThe leading infrared behavior of 4d-SU(2)gauge theory at high temperatures is governed byits static (zero Matsubara frequency) sector, ob-tained by integrating out its non-static modes toleave behind an eective three-dimensional the-ory. The non-static modes are suppressed in theinfrared. At high temperature, g(T ) becomessmall due to asymptotic freedom. A perturba-tive treatment of the non-static modes is justied.This is called Dimensional Reduction.The Euclidean action isS4d,E =∫d3x∫ β0dτ14F aµνFµν,a. (1)Expanding the Aµ-elds into Fourier series in thetemporal direction, keeping only the static modesand integrating out all the others, we end up witha 3d-SU(2) gauge theory minimally coupled to aself-interacting adjoint scalar eld [1]:S3d,eff =∫d3xf14F aij,0Fij,a0 + Tr[Dj, Aa0,0][Dj, Aa0,0]+ m2DTr(Aa0,0)2 + λA(Tr(Aa0,0)2)2g (2)where the rst term is the 3d-SU(2) gauge theoryand the second term is the kinetic term of thestatic adjoint Higgs Field A0,0. The coecients ofthis 3d eective action, the gauge coupling g3, theDebye mass mD and the quartic coupling λA, canbe determined from the perturbative expansion,or numerically as follows.23. CANONICAL DEMON METHODAny action can be parameterized byS = −∑αβαSα, (3)where Sα’s are interaction terms, and the βα’s areunknown eective coupling constants. The latticeaction corresponding to the continuum theory (2)is after some reformulation [2]:S = βG∑x,i<j[1− 12TrUi(x)Uj(x + i^)Uyi (x + j^)Uyj (x)]+ βA∑x,i12Tr ~A0(x)Ui(x) ~A0(x + i^)Uyi (x)+∑x[−β2 12Tr~A20 + β4(12Tr ~A20)2](4)where a is the lattice spacing, βG = 4/(ag23), β2 =[3 + (mDa)2/2]βA and β4 = aλAβ2A/4. βA isarbitrary under a rescaling of ~A0 / aA0,0kσk. Inaddition we introduce an auxiliary system, calleddemon system, given by the actionSD = −∑α=G,A,2,4βαdα, (5)here the βα’s are the same as in S, and the dα’sare the "demon energies", constrained to lie in theinterval [−dmax, dmax]. The total partition func-tion factorizes into the one of the original systemand that of the single demons. We can computethe distribution of each demon energy dα. Solv-ing this equation with respect to βα allows us todetermine the eective coupling constants:hdαi = 1Z∫dαe−βαdαddα =1βα− dmaxtanh(βαdmax).(6)We follow closely the method of [3]. First, wegenerate a statistically independent congura-tion C with a canonical update of the 4d-SU(2)-System only. On this given conguration C, weperform one microcanonical update of the (4d-SU(2)+Demon)-System, as follows: The changeof a link, under multiplication by a randomly gen-erated matrix close to the identity, causes a shiftof the eective energy of the lattice, Sα, whichmust be compensated with the demon energiesdα. The trial is accepted if all the new demonenergies remain in the allowed region. After visit-ing all links, the demons "move" to another gaugeconguration C’ while keeping their energies. Be-cause our ansatz for the 3d-reduced action is cor-rect in the presence of small fluctuations only, werestrict the microcanonical update to a small stepsize, and perform only one microcanonical sweepon each gauge conguration.4. RESULTSThe βG,A,2,4 are related to the physical cou-pling constants g3(µ),m2D(µ) and λA(µ), as de-ned after (4). For m2D(µ) we must add countert-erms, calculated in [4], to remove UV-divergenciesand compare with continuum MS-scheme the-oretical calculations. We measure the eectivecouplings as a function of temperature T/Tc. Tomonitor eects of the lattice discretization, westudy, at each temperature, several systems ofidentical physical size and increasingly ne lat-tice spacing (Nt = 2, 3, 4 and 5).4.1. g23(µ)-resultsAs shown in Fig. 1, we observe a clear, largedisagreement between the measurements and theperturbative prediction. The measured values ofthe coupling appear to depend only mildly onthe lattice spacing. Unfortunately, the disagree-ment with the perturbative prediction worsens asthe continuum limit is approached. The origin ofthe problem, we believe, lies in our lattice-basedapproach. The perturbative calculation is per-formed in the continuum theory, and the Green’sfunctions are computed for small spatial momentaj~pj  T . In contrast, the spatial momenta on ourlattices are cut-o by the lattice spacing, not bythe temperature. As we reduce the lattice spac-ing to approach the continuum limit, we allow forlarger momenta and depart more and more fromthe infrared, perturbative result.As a rst attempt to mimic the eect of our lat-tice approach, we tried to replace MS ! MSfin the continuum perturbative predictions. Asshown in Fig. 1, a factor f  300 is necessary to3reach reasonable agreement with our data, to becompared with the value 19.82 relating the MSand the lattice renormalization schemes [5]. InFigs.2 and 3, the dashed line shows the eect ofthis rescaling f = 300 on the other couplings.1 1.5 2 2.5 3 3.5 4 4.511.522.533.544.555.56g32(µ)/T for 83x2 and 123x2T/Tcg 32(µ)/TMeasurement 83x2, 750000 configsMeasurement 123x3, 400000 configsMeasurement 163x4, 40000 configsMeasurement 203x5, 5000 configsTheoryTheory with rescaled ΛFigure 1. g23(µ)/T vs T/Tc. Measurements andperturbative predictions (solid line) clearly dis-agree. Rescaling MS by a factor 300 restoresagreement (dashed line).4.2. m2D(µ)-resultsm2D(µ) stays somewhat below the theoreticalpredictions for all temperatures studied (up to4Tc). Because the Debye mass is only obtainedafter subtraction of UV-divergent counterterms,statistical errors quickly increase as the contin-uum limit is approached. There is a tendencyfor better agreement with perturbation theory atsmaller a. As explained above, agreement shouldnot be expected anyway since we keep larger spa-tial momenta than in the continuum approach.Nevertheless, the disagreement for m2D is muchsmaller than for g23, and could be accounted forby a milder rescaling of the  parameter.1 1.5 2 2.5 3 3.5 4 4.50.511.522.533.544.555.5mD2 (µ)/T2 for 83x2 and 123x3T/TcmD2(µ)/T2Measurement 83x2, 750000 configsMeasurement 123x3, 400000 configsTheoryTheory with rescaled ΛFigure 2. m2D(µ)/T2 vs T/Tc.4.3. λA(µ)-resultsLike the other couplings, λA(µ) stays below theperturbative prediction at all temperatures. Scal-ing violations seem reasonably small.1 1.5 2 2.5 3 3.5 4 4.500.10.20.30.40.50.60.70.8λA(µ)/T for 83x2 and 123x3T/Tcλ A(µ)/TMeasurement 83x2, 750000 configsMeasurement 123x3, 400000 configsTheoryTheory with rescaled ΛFigure 3. λA(µ)/T vs T/Tc.5. CONCLUSIONIn perturbative Dimensional Reduction, oneconsiders only two and four-point functions, toone- or at most two-loop level [2], and neglectshigher-dimension operators. Then, DimensionalReduction appears to be valid down to  2Tc[6]. Numerically determined eective couplingconstants allow a non-perturbative improvementof the theory. Higher-dimension operators, withcouplings determined by additional demons, areeasily included. We have added an A60-term inthe eective action, but the eect on the othercouplings is small. Our results still disagree withcontinuum perturbative predictions even at 4Tc.We intend to compare them with the perturba-tive results of [7], obtained directly on the lattice.We are grateful to Mikko Laine for many help-ful discussions and advice.REFERENCES1. K. Kajantie et al., Nucl.Phys.B458(1996) 90.2. K. Kajantie et al., Nucl.Phys.B503(1997)357.3. M. Hasenbusch, K. Pinn and C. Wiecz-erkowski, Phys. Lett. B 338 (1994) 308.4. M. Laine, Nucl. Phys. B 451 (1995) 484.5. A.&P. Hasenfratz, Phys.Lett.B 93 (1980)165.6. O. Philipsen, Nucl.Phys.Proc.Suppl.94(2001)49.7. P. LaCock et al., Nucl.Phys.B 369 (1992)501.

Testing Dimensional Reduction in SU(2) Gauge Theory

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Testing dimensional reduction in SU(2) gauge theory

Testing Dimensional Reduction in SU(2) Gauge Theory

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