PHYSICAL REVIEW D, VOLUME 63, 054014

Borel resummation of the perturbative free energy of hot Yang-Mills theory

Rajesh R. ParwaniDepartimento di Fisica, Universita’ di Lecce and Istituto Nazionale di Fisica Nucleare, Sezione di Lecce,

Via Arnesano, 73100 Lecce, Italy~Received 25 August 2000; published 5 February 2001!

The divergent perturbative expansion of the free-energy density of thermalSU(3) gauge theory is re-summed into a rapidly convergent series using a variational implementation of the method of conformalmapping of the corresponding Borel series. The resummed result differs significantly from non-perturbativelattice simulations and the discrepancy is attributed to the presence of a pole on the positive axis of the Borelplane. The position of that pole is determined numerically and the difference between the lattice data and theresummed series is related to a phenomenological bag ‘‘constant.’’

DOI: 10.1103/PhysRevD.63.054014 PACS number~s!: 12.38.Mh, 11.15.Bt, 12.38.Cy

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It is generally believed that the collision of heavy ionssufficiently high energies will lead to the formation of a nephase of matter, the quark-gluon plasma, and experimenproduce such a plasma are underway at BrookhavenCERN. As the effective couplinga of quantum chromody-namics~QCD! decreases with an increase in energy, theorhave used perturbative methods to study properties ofplasma at high temperature (T). For example, a completelanalytical calculation of the free-energy density of QCDordera5/2 was performed a few years ago@1,2#. The purelygluonic contribution is given by

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where F0528p2T4/45 is the contribution of non-interacting gluons andm is the renormalization scale in thmodified minimal subtraction (MS) scheme. UnfortunatelyEq. ~1! is an oscillatory, non-convergent, series even fora assmall as 0.2, which is close to the value of physical intere

Padeapproximants~PA’s! were used in Ref.@3# to resumthe series~1!. It was found that the dependence on the artrary scalem was reduced and the convergence of the sesomewhat improved. However PA’s have a number of wknown drawbacks. For those and other reasons, in Ref.@4#the authors abandoned the expansion of the free-energysity with respect to the coupling constant and consideinstead selective resummations of gauge-invariant diagraThough the results of@4# compare favorably with latticesimulations@5#, calculations beyond leading order are coplicated and thus it seems that the issue of convergencleft open. In Ref.@6#, yet another procedure was usedstudy the free-energy of hot QCD: short distance perturtive effects were handled analytically while long-distancefects were described@2# by an effective three-dimensionatheory and studied numerically.

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It appears that a number of separate issues concerning~1! have become confused in the literature. The first issuwhether the given divergent series can be resummed inconvergent series, preferably in a systematic and wmotivated way. The second issue is whether such asummed series accurately represents the physical quanThe third issue is whether one obtains any new physicalsight in the process. It will be the attempt of this papershed some light on these and related questions.

Recall that the divergence of perturbative expansionsquantum field theory is a generic phenomenon@7#. Given aseries

SN~l!511 (n51

N

f nln, ~2!

wherel is the coupling constant, one expects the coefficief n to grow asn! for largen. It is natural then to introduce theBorel transform

B~z!511 (n51

Nf nzn

n!~3!

which has better convergence properties than Eq.~2!. Theseries~2! may be recovered from Eq.~3! through a Laplacetransform

SN~l!51

lE0

`

dze2z/lB~z!. ~4!

In order to proceed non-trivially, one first performs aapproximate summation of the series~3! so that Eq.~4! thengives the resummed version of Eq.~2!. Now, suppose thathe only singularity ofB(z) is at z521/p, with p real andpositive. Then the radius of convergence of the Borel seis 1/p. In order to perform the integral in Eq.~4!, one needsto extend the domain of convergence of the Borel series. Oway to do this is by the method of conformal mapping@7,8#.Define

w~z!5A11pz21

A11pz11~5!

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RAJESH R. PARWANI PHYSICAL REVIEW D63 054014

which maps the Borel plane to a unit circle. The inverseEq. ~5! is given by

z54w

p

1

~12w!2. ~6!

The idea is to rewrite Eq.~3! in terms of the variablew.Therefore, using Eq.~6!, zn is expanded to orderN in w andsubstituted into Eqs.~3!,~4!. The result is

SN~l!5111

l (n51

Nf n

n! S 4

pD n

(k50

N2n~2n1k21!!

k! ~2n21!!

3E0

`

e2z/lw~z!(k1n)dz, ~7!

wherew(z) is given by Eq.~5!. Equation~7! represents aresummation of the original series~2!. This technique hasbeen used in determining critical exponents in statistical stems where the singularity atz521/p is due to instantons@7,8# and in QCD at zero temperature where the singularitdue to renormalons@9#.

Currently no information is available about the exactcation of singularities in the Borel plane of thermal QCthough undoubtedly there is at least one on the negasemi-axis. Therefore in order to apply the resummation~7! toEq. ~1!, a new idea is introduced in this paper: It is firassumed that the only singularity is atz521/p, p.0, withthe value ofp determined by the condition that it be thposition of an extremum of Eq.~7!. That is,p is chosen to bea solution of

S ]SN~l,p!

]p Dl5l0

50. ~8!

SinceSN(l,p) depends on the couplingl, one first fixesl at some reference valuel5l0 ~say, at the mid-point of therange of interest! in order to solve Eq.~8!. Fortunately, itturns out that the solution of Eq.~8!, and hence the convergence ofSN , is not very sensitive to the exact value ofl0.

Let me illustrate the technique by applying it to two caswhere exact results are known. Consider first the integra

I ~l!5E0

`

dze2z

11zl. ~9!

If the right-hand side of Eq.~9! is expanded as a power seriin l, one obtains, atNth order,

I N~l!511 (n51

N

ln~21!nn!. ~10!

Clearly, from Eq.~9!, the exact Borel transform of this serieis B(z)51/(11z) with a singularity atz521. Ignoring thisinformation, let us resum the divergent series~10! using Eq.~7! with f n5(21)nn! and values ofp determined for eachNthrough Eq.~8! at the reference valuel050.5. The resultsfor Eq. ~8! are as follows: There is no extremum forN51.For N52, there is a minimum atp52.65. ForN53 there is

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a local maximum atp51.5 and a minimum atp55.1. ForN54 there is a local minimum atp51.3, a local maximumat p52.3 and a global minimum atp58.4. Jumping aheadto N58, there is a local minimum atp51.075, a local maxi-mum atp51.5 and a global minimum atp53.6.

Thus in general Eq.~8! has more than one solution forgiven N andl0. In Fig. 1, Eq.~7! is plotted forN52, 3, 4,and 8, at the respective minimum. Notice how the curvconverge rapidly to the exact value given by Eq.~9!. Alter-natively, one might choose for eachN the value ofp @fromthe multiple solutions of Eq.~8!# which seems to be part ofconverging sequence. In this case the valuesp52.65(N52), p51.5(N53), p51.3(N54), and p51.075(N58)appear to converge to the exact valuep51. The curves areshown in Fig. 2. Clearly the curves in Fig. 2 converge fasto the exact value than those of Fig. 1, but unless oneinterested in very high numerical accuracy, the differencenot significant. For example, atl50.5, the exact value oEq. ~9! is 0.722657, while the resummed value forN58 isgiven at the global minimump51.075 by S8(l50.5, p51.075)50.722652 and at the local minimump53.6 byS8(l50.5, p53.6)50.722524.

The main points illustrated by this example, which seeto be common to the other cases studied, are~i! the rapid

FIG. 1. Plot of Eq.~7! for the model in Eq.~10!, for N52, 3, 4,and 8 at the respective global minimum of Eq.~8!. The curvesmove upwards with increasingN. The curves forN53,4 practicallycoincide, while the curve for the exact expression~9! is indistin-guishable from that forN58.

FIG. 2. Plot of Eq.~7! for the model in Eq.~10!, for N52, 3, 4,and 8 at the respective values ofp from Eq. ~8! which converge to1. The values of the curves atl50.5 are N52(0.704), N53(0.726),N54(0.7219),N58(0.7227). The curve for the exacexpression~9! is hardly distinguishable from that forN53,4,8.

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BOREL RESUMMATION OF THE PERTURBATIVE FREE . . . PHYSICAL REVIEW D 63 054014

convergence of the resummed series represented by Eq~7!compared to the original wildly oscillating series~10!, ~ii !the relative insensitivity of the convergence and numervalue of the resummed series to the particular extremum csen among the possible multiple solutions of Eq.~8! ~for agiven N and l0), even if the chosen value ofp is quitedifferent from the exact value, and~iii ! the relative insensi-tivity of Eq. ~8!, and hence the convergence of Eq.~7! to theprecise value ofl0.

For another example, consider thermalO(M ) l2f44 field

theory in the limitM→`. The exact free-energy density ithis case has been determined in Ref.@10#. In Eq. ~5.8! ofthat paper1 the perturbative expansion, inl, of the free-energy density is also given up tol6. Defining S5@F(T)2F(0)#/Fideal and choosingm5T for simplicity, the valuesof f n for 2<n<6 can be read off from Eq.~5.8! of Ref. @10#and Eq.~8! solved at some reference value, sayl054. Thesolutions of Eq.~8! in this case areN53, p50.1 ~min!; N54, p50.05~local max!, p50.2~min!; N55, p50.025~lo-cal min!, p50.1 ~local max!, p50.3 ~global min!; N56, p50.1 ~local max!, p50.45 ~global min!. As in the first ex-ample, the convergence of the resummed series is founbe rapid even for large coupling, in contrast to the oscillatbehavior of the ordinary pertubation expansion observed@10#. Figure 3 shows the curves forSN(l), 3<N<6, for thevaluep50.1, which seems to be the valuep converges to asN increases. The exact value of the free-energy densitl58 taken from Fig. 6 of Ref.@10# is about 0.875. Bycomparison the resummed value predicted here is givesixth order byS6(l58,p50.1)50.889. On the other handif one evaluatesS6 not at p50.1 but rather at its globaminimum p50.45, one getsS6(l58,p50.45)50.849, adifference of less than 5%.

Actually, no information is available about the singulaties in the Borel plane for theO(M→`) scalar field studiedin @10#. The good agreement of the results obtained here wthe exact results of Ref.@10# leads one to conjecture that fo

1Note that the definition of the coupling constant used heredifferent from that in Ref.@10#.

FIG. 3. Plot of Eq.~7! for the model in Ref.@10# for N53,4,5and N56 at the valuep50.1. The curves move upwards asNincreases. ForN55,6 they are practically identical. The exact curin Ref. @10# lies slightly below that forN56: At l58 the exactcurve has the value 0.875.

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the free energy of this model, the singularity closest toorigin in the Borel plane might be nearp50.1, that is,z5210.

Note that although the coupling constant in the scalar fitheory model isl2, the perturbative expansion of the freenergy density contains the odd powersl3 andl5 which istypical of thermal field theories with massless particles~or atvery high temperatures! @1,11#. Physically these are due tcollective effects such as Debye screening and it is sotimes suggested in the literature that these terms shoultreated on a different footing. However, as the analysis abshows, from a mathematical point of view these odd poware no different from the other terms in the expansion offree-energy density and can be resummed as part of a sseries.

Finally, the resummation technique of Eqs.~7!,~8! is ap-plied to the free-energy density ofSU(3) gauge theory givenin Eq. ~1!. As in Ref.@4#, I replace (a/p)1/2 by the approxi-mate two-loop running coupling constant defined by

l~c,x!52

A11L~c,x!S 12

51

121

ln@L~c,x!#

L~c,x! D ~11!

where L(c,x)5 ln@(2.28pcx)2#, c5m/2pT and x5T/Tc ,with Tc;270 MeV the critical temperature which separatthe low and high temperature phases@5#. Furthermore, as inRef. @3#, I have absorbed the ln(a) term which appears athree-loop order into the coefficient of thea2 term in Eq.~1!.

Fixing first the reference valuesc051,x053 ~which fixesthe reference value ofl0), the results of Eq.~8! areN52,no extremum;N53, p53.2 ~min!; N54, p57.6~min!; N55, p513.1~min!. Sincec andx appear in Eqs.~1! and~11!only logarithmically, changing these values in the rangesay, 0.5,c,2, 2,x,5 has almost no impact on the solution of Eq. ~8! and hence on the optimal values ofp.

The curves for the resummed series are plotted in Figfor c51, that is at the renormalization scalem52pT. Againthe rapid and monotonic convergence is manifest, the reapproaching the ideal gas value even at moderate temptures ;2Tc . One can estimate the effect of the unknowhigher order,l6, contribution. It turns out to be negligible@12#. Therefore one feels confident that theN55 curve inFig. 4 is numerically close to the~unknown! exact sum of theperturbation series. In Fig. 5 the curve for S5(x,c,p

is

FIG. 4. The resummed~7! free-energy density of hotSU(3)

gauge theory forN53, 4, and 5, at the renormalization pointm52pT. The curves move upwards asN increases.

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RAJESH R. PARWANI PHYSICAL REVIEW D63 054014

513.1) is plotted for three values ofc to indicate its milddependence~less than 1%) to the arbitrary renormalizatioscalem.

Lattice results for the free-energy density of pureSU(3)theory are shown in Fig. 6. The lattice community has incated that their errors are under control~less than5%). Inthat case, I am left with the task of explaining the significadifference~e.g.;15% atT53Tc) between the best analytcally resummed result represented by theN55 curve in Fig.4 and the lattice data. At zero temperature, it is known tnon-Abelian gauge theories are not Borel summable@9#.That is,B(z) contains singularities for positivez, renderingthe integral in Eq.~4! ambiguous. The situation is not expected to be different at non-zero temperature. Usually@9#,the presence of such singularities is taken to indicateexistence of non-perturbative corrections. One can estimthe ambiguity,dS, and hence the non-perturbative corretion, as the residue of the integrand in Eq.~4! at the locationof the singularity@9#. If the singularity ofB(z), on the posi-tive semi-axis, closest to the origin is a pole atz5q, then,from Eq. ~4!,

dS5A

le2q/l ~12!

whereA is a constant. Assuming that the difference betwethe lattice data and Fig. 5 is due to Eq.~12!, the constantsAandq can be determined by rewriting Eq.~12! as

ln~ldS!5 ln~A!2q/l ~13!

FIG. 5. The fifth order resummed free-energy density~7! of hotSU(3) gauge theory atp513.1, for the renormalization scale va

ues m50.5, 1, and 2. The free-energy density increases with

creasingm.

FIG. 6. Mean lattice results for the free-energy density ofSU(3) gauge theory from Ref.@5#. Here Slatt refers to the freeenergy divided by the free energy of an ideal gas of gluons.

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and using fordS the difference between Fig. 6 and the mdian curve in Fig. 5~i.e.,c51). Figure 7 shows the left-hanside of Eq.~13! plotted against 1/l. This givesA5e8.7 andq52.62, and so, withl(x)[l(c51,x),

Slatt5Spert21

l~x!e8.722.62/l(x), ~14!

whereSlatt represents the lattice data for the free energy aSpert the Borel resummed perturbative result, both normized with respect to the ideal gas value.

It is extremely reassuring that both the sign and magtude ofq determined in this way are self-consistent with tassumptions made. In particular, the singularity atz5q52.62 is more than 30 times away from the origin than tsingularity atz521/p521/13.1 and justifiesa posteriorithe resummation procedure~7! which considered only thenearest singularity. Furthermore, sinceSpert is extremelyclose to the ideal gas value, Eq.~14! may be interpreted asgeneralization of phenomenological equations of state@11#for the free energy where the second term on the right-hside of Eq.~14! is called a ‘‘bag constant.’’ In our case th‘‘constant’’ is really temperature dependent and represennon-perturbative contribution to the free energy that vanisat infinite temperature. Note that this interpretation of tsecond term on the right-hand side of Eq.~14! is consistentwith the usual one only because the resummed perturbaresult liesabovethe lattice data.

Let me now summarize the main results of this papernew procedure, a variational version of the well-known coformal mapping of Borel series, was introduced and its prticality illustrated. It was shown that the badly divergent sries for the free-energy density of thermalSU(3) gaugetheory~1! could be resummed in a systematic and relativwell-motivated way into a rapidly convergent series. Hoever, the final result differed significantly from nonperturbative lattice data, suggesting that the discrepancdue to the non Borel summability of the theory. Numericalthe difference was parametrized in terms of two consta@see Eq.~14!#, and it is suggested that the ambiguity in thBorel integral is the bag ‘‘constant’’ used in phenomenolocal models for the free-energy density.

In physical terms, the situation may be described aslows. If the Borel resummed perturbative expansion hagreed with the lattice data, then one would have argued

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BOREL RESUMMATION OF THE PERTURBATIVE FREE . . . PHYSICAL REVIEW D 63 054014

the high-temperature phase ofSU(3) theory is appropriatelydescribed by weakly coupled gluons. However, if the devtion of lattice data from the resummed and convergent pturbation expansion found here is taken at face value, tone is led to conclude that even at temperatures as hightimesTc (;700 MeV), the phase of thermalSU(3) is notaccurately described by weakly coupled gluons but rathemore complicated structures which are responsible fornon-perturbative bag contribution. If one accepts this concsion, then one must also be open to the possibility that wwill be produced at CERN and Brookhaven might be becharacterized as something other than a quark-gluon pla

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For some alternative descriptions of the high-temperatphase of QCD see, for example,@13#.

An extension of the analysis presented here to QCDother thermal gauge theories, a further development ofmethodology itself and its applications to other physicproblems will be presented in an accompanying seriespapers@12#.

I thank Claudio Coriano´ for stimulating discussions anhospitality at Martignano, and the Department of PhysicsLecce for financial support. I also thank the YITP at StoBrook for hospitality during the final stages of this work.

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tt.

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