In this paper we apply the finite-temperature renormalization group from the point of view of “environmentally friendly” renormalization. We study both λϕ⁴ theory and the magnetic sector of QCD. At one loop level the complete temperature range of λϕ⁴ is successfully described in terms of the parameters of the zero temperature theory. We show also how the critical temperature can be calculated in terms of the latter. For the magnetic sector of QCD, in distinction to λϕ⁴, a one-loop finite temperature renormalization group improvement is not sufficient to 

describe the high-temperature regime

Stephens, C. R.

van Eijck, M. A.

English

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DIAS-STP-92-04THE FINITE-TEMPERATURE RENORMALIZATION GROUPAPPLIED TO )6 THEORY AND QCDM.A. VAN EIJCKInstitute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65,1018 XE Amsterdam, The NetherlandsandC.R. STEPHENSDIAS, 10 Burlington Road, Dublin 4, IrelandABSTRACTIn this paper we apply the finite-temperature renormalization group from the pointof view of “environmentally friendly” renormalization. We study both )çb theoryand the magnetic sector of QCD. At one loop level the complete temperature rangeof A is successfully described in terms of the parameters of the zero temperaturetheory. We show also how the critical temperature can be calculated in terms ofthe latter. For the magnetic sector of QCD, in distinction to a one-loop finitetemperature renormalization group improvement is not sufficient to describe thehigh-temperature regime.1. IntroductionSince its introduction the finite-temperature Renormaiization Group’ has beenrepeatedly studied and applied. Here we consider it as an example of “environmentally friendly” renormalization2.The latter is based on the notion that the effective degrees of freedom (fluctuations) of a system are sensitive to the environment.With environmentally friendly renormalization one renormalizes (reparameterizes)the theory in an environment dependent way so as to enable the renormalized parameters to track the evolving nature of the effective degrees of freedom as a functionof scale in a perturbatively controllable manner.Temperature is an interesting example of a relevant environmental parameter,as a field theory at very high temperatures exhibits qualitatively different effectivedegrees of freedom than those of the zero-temperature theory. In earlier work2’3thiscrossover of the effective degrees of freedom was accessed completely for .\q theory,both in the symmetric and broken phases, using an environmentally friendly renormaiization group. The running parameter used was the finite-temperature mass(inverse screening length) m(T), the limit m(T) —* 0 corresponding to a secondorder phase transition. Although the latter is a natural parameter it is often thecase that only the zero-temperature parameters are experimentally known. To relate these to the finite-temperature parameters one must run a parameter otherthan m(T). The temperature itself is the obvious candidate, or rather one runsan arbitrary renormalized temperature scale r, and after the renormalization groupequations are solved, r is set equal to the physical temperature T. This is necessaryin any case in order that perturbation theory in terms of the renormalized parameters be well behaved. As we will show, by running the environment itself in thisway, one may answer questions such as: what is the critical temperature?2. Running the Environment for \q TheoryIn this section we consider the renormaiization of .Acb4 theory at an arbitraryfiducial temperature scale r. We apply the normalization conditions—0 m \ T — — m2 -‘-r, 7, F’T ) — T 2 i”n1= = r) =F(po = 0,p,mT,AT,T,T = = 1 (1)pwhere O represents the minimum of the effective potential, i.e. it satisfies theequation of stateI’’(p = = r) = 0. (2)The beta functions, obtained by differentiating these renormalization conditionswith respect to r for fixed bare parameters, depend on derivatives d/dT. Thesecan be eliminated with the equation of state (2). The equations relevant for thesymmetric phase areT-2= .AT(N+2)Tf0s_fS_(1m )d.) )2(NH-8) CO 2 dm2=—622 f dsee(2m_r_-2) (4)In fig. 1 we present a plot of the solutions of these equations for the case where thesystem is in a state of broken symmetry at T = 0. We have chosen the arbitrarytemperature r = T. This is not simply because T is the relevant physical scale butalso because without this choice perturbation theory would break down in the vicinity of a phase transition. The place where the mass curve is zero determines thecritical temperature T in terms of the initial conditions for the flow equations (3)and (4). For temperatures T >> T the mass increases linearly with temperature anda “mean field” regime is reached where infrared (IR) fluctuations are suppressed dueto the large thermally induced mass. The solution of these equations can be perturbatively controlled at higher order (though a resummation technique should beused in the vicinity of the phase transition) thereby providing a technique by whichthe critical temperature can be calculated to all orders. Notice that it is in no wayassumed that the behaviour near the critical point is three dimensional. One startsoff at zero temperature, heats the theory up, then examines what occurs withoutany prejudice as to the expected behaviour. We believe this to be an importantadvantage of environmentally friendly methods. In fig. 1 the coupling constant goesto zero as the critical point is approached. As has previously been emphasized2thisdoesn’t imply that the interactions there vanish. The appropriate coupling constantin the vicinity of the phase transition is .\T/m(T) and this approaches a non-zerofixed point.The results here are completely consistent with those found by environmentally friendly methods wherein the finite-temperature mass is the running parameter. It is important to realize that in the above equations we have evaluated the appropriate diagrams with propagators of mass m(r). Hence if we writed/dT = 8/8r + (rdm?,/dT)O/8m, the latter acts non-trivially. Diagramatically onecan think of this as being equivalent to summing up all “daisy” insertions intothe internal lines of the B function diagrams. Such a maneuver is crucial in thatwithout it the tadpole insertions, which constitute the largest temperature effectsin the daisy diagrams, would ensure a breakdown in perturbation theory.3. The QCD caseWe now turn our attention to QCD. We have used as a renormalization conditionthat the static (i.e. zero energy), spatial three-gluon vertex equals the tree-levelvertex in the symmetric momentum configuration= 0,,g,, T== gfabc [gij(pi— p2)k + cycl.]. (5)In contradistiction to the previously discussed case this chosen renormalization condition depends on two parameters, the momentum scale ,, and the temperaturescale r. Therefore we can perform a renormalization group analysis with respect toboth parameters, i.e. we can run more than one environmental parameter at thesame time.For the calculation we have used the Landau gauge Background Field Feynmanrules5 resulting from the Vilkovisky-de Witt effective action in order to get ridof ambiguities arising from gauge. dependence6 Due to the corresponding WardIdentities the calculation is simplified in that we only have to calculate the transversegluon self energy function 11Tr in the static limit. In terms of the coupling a,- :=g,/47r2 the functions are thenda dllTr dakT dll= a,,7- P .. , T = a,,. T . (6)d,c d p‘L dT dTThe ‘r renormalization group is needed to draw conclusions about the temperaturedependence of the coupling. This can not be done using the ic-scheme alone withoutassuming something about the temperature dependence of the initial value of thecoupling used in solving the differential equation.More details of the ingredients of the calculation can be found in reference ‘.The result isda,c,r da,,r= /3vac + /3th, T = /3th, (7)drwhere the vacuum contribution is, as usual,—2 1 11r irN/—‘vac—‘2ic,6C +‘1f)and where, in terms of the JR and TJV convergent integralscx, 2 2k+1F=I dx log —2 (9)Jo e/2T — x — l k=o 2k + 1and[00 1 xG=I dx p , (10)Jo e1/2T — (x —the thermal contribution is given by13th =(F-1 + F’ - - G1) Nf]. (11)Small differences with the thermal gluon contribution obtained by Antikainen et.al.8(which contains + F2’ — — G) are probably caused by the use of theBackground Field Feynman gauge, as may possibly be checked with Gauge Dependence Identities9.Because the two beta functions (7) are not exactly each other’s opposite therenormaJization group improved coupling is not just a function of the ratio i’/’r.There is another dimensionful scale (such as AQCD) that comes from an initial condition for these differential equations. The solution of the set of coupled differentialequations can be written in the form111 (12)(TNC — Nf)‘AqcD — f()where the function f satisfies /3th = ct,icdf/di with the initial condition urn40 f = 0so that we can identify AQCD with the usual zero-temperature QCD scale. Actuallythis function f can be found in terms of the functions F and G:f = (F + + G) N0 + (F + F1) Nf. (13)Fig. 2 is a contour plot of the effective coupling as a function of both momentum-and temperature-scale. The results are best trusted in places where the coupling issmall. For physical reasons we have to restrict ourselves in any case to the regionwhere the coupling is positive, which is below the uppermost line in the graph.The high-temperature behaviour (i.e. for -r >> c) is determined byf —÷ Nfi2+ (N - N) in + 0(1). (14)The coefficient of the dominant contribution is the same as in Landsman’s result’°,but others found different coefficients8”and even different signs’2 with a strong dependence on the gauge parameter and the details of the renormalization condition6.The sign of this coefficient is of crucial significance for the behaviour of the couplingin this limit. For increasing temperatures at fixed momentum scale, our sign makesthe coupling grow to a pole, an indication that we are entering a strong-couplingregime, whereas the opposite sign would lead to asymptotic freedom in this limit.Stimulated by the original belief’3 that high-temperature QCD would be asymptotically free as in the high-momentum situation, Landsman suggested that this signwould be an artifact of the one-loop calculation and that a higher-loop calculationor a resummation could change it. We however believe that this will not happen,as the sign appears quite naturally if one realises that this limit r/ic — 00 is anJR limit where confinement takes place. Unless at higher ioop order the magneticmass increases quickly enough with temperature in order to act as an sufficient JRcutoff, we cannot get around this problem without actually solving confinement. Webelieve this to be an important consideration when considering phase transitionswhich involve non-abelian gauge fields.In the regime r >> i the beta functions behave as in a three-dimensional theoryso that we designate this as the region where dimensional reduction occurs. Here itis natural, as for ?, to use a different dimensionless coupling u = since thenfixed points may turn up more clearly. However in this case such a reparametrizationcannot remove the pole and will not give a different behaviour.If we allow the momentum-scale to change with temperature simultaneously, thehigh-temperature limit can be taken in many ways. In the region r>> i the shape ofthe contours is given by r i in A• This characterizes exactly along which pathsin the (‘r, ic)-plane the coupling increases or decreases. For example at a fixed ratior/c (no matter what this ratio is) we eventually find a coupling that decreases like1/in i, much in the same way as at zero temperature. This is a natural contour toconsider for a weak-coupling regime’4where one could treat the quark-gluon plasmaas a perfect gas, as then the thermal average of the momentum of massless quantaat temperature T is proportional to the temperature. However at low momentathe assumption of weak coupling breaks down. Furthermore, instead of consideringquantities at the average momentum it is more appropriate to use thermal averagesof the quantities themselves as a weighted integral over all momenta’2. But onceagain one runs into problems at the low-momentum end as long as we cannot treatthe strong-coupling regime.4. AcknowledgementsWe thank R. Kobes and G. Kunstatter for organizing the 3rd Workshop onThermal Theories and Their Applications in Banif. CR5 is grateful for the hospitality of the Theoretical Physics group at Imperial College where part of this workwas carried out.5. References1. H. Matsumoto, Y. Nakano and H. Umezawa, Phys. Rev. D29 (1984) 1116.2. Denjoe O’Connor and C.R. Stephens,Nuc. Phys. B360 (1991) 237; “Environmentally Friendly Renormalization”; Preprin THU-93/14, DIAS-STP-93-19; (to be published in mt. Jou. Mod. Phys. A).3. Denjoe O’Connor, C.R. Stephens and F. Freire, Mod. Phys. Lett A25 (1993)1779; Class. Quam. Gray. 23 (1993) S243; F. Freire and C.R. Stephens, Z.Phys. C60 (1993) 127.4. P. Elmfors, NORDITA preprint 92/38P (1992).5. A. Rebhan, Z. Phys. C30 (1986) 309.6. N. Ashida, A. Niegawa, H. Nakkagawa and H. Yokota, Phys. Rev. D44(1991) 473.7. M.A. van Eijck, Can. J. Phys. 71 (1993) 237; M.A. van Eijck, C.R. Stephensand C.W. van Weert, Mod. Phys. Lett. A9 (1994) 309.8. J. Antikainen, M. Chaichian, N.R. Pantoja and J.J. Salazar, Phys. Lett.B242 (1990) 412.9. R. Kobes, G. Kunstatter and A. Rebhan, Nuci. Phys. B355 (1991) 1.10. N.P. Landsman, Phys. Lett. B232 (1989) 240.11. R. Baier, B. Pire and D. Schiff, Z. Phys. C51 (1991) 581.12. K. Enquist and K. Kainulainen, Z. Phys. C53 (1992) 87.13. J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 (1975) 1353.14. K. Enquist and K. Kajantie, Mod. Phys. Lett. A2 (1987) 479.Figure CaptionsFig. 1: Solutions of the coupled differential equations (3,4) for the running mass m and coupling \ ofthe one-component model as a function of temperature T. At the critical temperaturethe coupling goes to zero simultaneously with the mass.Fig. 2: Contour plot of the running coupling a,-(i, r) for QCD with three colours and six fermionflavours. The fermions have been taken massless. Only below the curve aK,T = c (close to= 1000) the coupling is positive and finite.Figal0.100.050.000.5 2.01.0 1.5TFig.2II300-0.1000o a F0.50.210.0-0.10.0IIII0.010.020.030.040.050,0iC/AQCD

The Finite-Temperature Renormalization Group Applied to λϕ⁴ Theory and QCD

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The Finite-Temperature Renormalization Group Applied to λϕ⁴ Theory and QCD

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