aa r X i v : . [ h e p - ph ] O c t Screened perturbation theory at four loops
Lars Kyllingstad
Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim,Norway
Abstract
We study the thermodynamics of massless φ -theory using screened perturbation theory, whichis a way to systematically reorganise the perturbative series. The free energy and pressure arecalculated through four loops in a double expansion in powers of g and m/T , where m is athermal mass of order gT . The result is truncated at order g . We find that the convergenceproperties are significantly improved compared to the weak-coupling expansion. Key words:
Screened perturbation theory
PACS:
1. Introduction
Recently, Gynther et al . calculated the pressure of massless φ -theory to order g inthe weak-coupling expansion [1]. The weak-coupling pressure for various orders of g isshown in Fig. 1b. Note that it does not seem to converge as higher and higher orders areincluded. This is a well-known problem, not only in scalar field theory, but also in gaugetheories.Many methods have been devised to improve upon the convergence of this expansion.Among them is screened perturbation theory (SPT), which was first introduced in thermalfield theory by Karsch, Patk´os and Petreczky [2]. SPT constitutes a reorganisation of theperturbative series so that one resums selected diagrams from all orders of perturbationtheory. In the following, we will indeed see that using SPT improves the convergencesignificantly.This talk is a brief overview of the calculations and results of Ref. [3].
2. Screened perturbation theory
The Lagrangian density for a massless φ -theory is Preprint submitted to Elsevier 4 November 2018 = 12 ( ∂ µ φ )( ∂ µ φ ) − g φ , (1)where g is the coupling constant. The SPT Lagrangian of this theory is defined as L SPT = L free + L int , (2)where L free = 12 ( ∂ µ φ )( ∂ µ φ ) − m φ and L int = 12 m φ − g φ . (3)If we set m = m , it is clear that L SPT = L . We now take m to be of order g andexpand systematically in powers of g . This defines a reorganisation of the perturbativeseries, in which the expansion is about an ideal gas of massive particles. The mass m hasthe simple interpretation of a thermal mass. A prescription for m is discussed later; fornow we take it formally to be of order gT .
3. Free energy
We calculate the free energy in a double power expansion. That is, first we do a loopexpansion in powers of g , and thereafter we expand each diagram in powers of m/T .The inclusion of the mass term in the interaction yields an additional Feynman rulenot present in the original theory, namely = m . This is called a mass insertion .The free energy can then be written as a series of vacuum diagrams, F = + + + + + + · · · , (4)which we truncate at four loops. Note that mass insertions count as loops, since they areof order g .As an example of an m/T expansion, take the one-loop diagram with a single massinsertion: F ≡ = − m T X p =2 πnT Z p P + m , (5)There are two momentum scales in this sum-integral; the hard scale, which is of order T ,and the soft scale, of order gT . The former arises from the nonzero Matsubara frequencies,whereas the latter comes from the thermal mass m . We isolate the contribution from thezeroth Matsubara mode, as it only contains the soft scale. This yields F = − m T Z p p + m + X p =0 Z p P + m . (6)Since m ≪ P in the second term, we can expand it in a geometric series: F = − m T Z p p + m + X p =0 Z p P (cid:18) m P + m P + · · · (cid:19) . (7)2he mass can now be taken outside the sum-integral in each term, and the result is aseries of easily-evaluable massless sum-integrals. Finally, the results are truncated at g .
4. The tadpole mass
The pressure of the original theory is obtained in the limit where the two masses areequal, and is defined as P = −F| m = m . (8)The parameter m in screened perturbation theory is completely arbitrary, and if we wereable to include all loop orders, the result would indeed be independent of m . To completethe calculation we must instead find a prescription for m which is physically meaningful.The simplest choice is the tadpole, m = = 12 g T X p Z p P + m . (9)In the weak-coupling limit the propagator in the loop is massless, and Eq. (9) reduces to m = g T . (10)Using this value for the mass one obtains the weak-coupling pressure, shown in Fig. 1b.Our result through order g agrees with the N = 1 result in Ref. [1].We can generalise this to higher loop orders by taking m to be the tadpole mass , m = + + + · · · = g ∂ F ∂ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) m = m . (11)With this choice, m is well-defined at all loop orders. Since the propagators in Eq. (11)are massive as well, it means that in calculating the pressure we are doing a selectiveresummation of diagrams from all orders of perturbation theory.
5. Results
Fig. 1a shows the SPT pressure truncated at various loop orders. The two- and three-loop results are indistinguishable from the exact numerical results found in Ref. [4].Convergence is rapid—in the two-loop case terms of order g – g are negligible, while atthree loops one can neglect terms of order g .There are no exact numerical data available for comparison with our result at fourloops, but experience with lower loop orders indicates that this is indeed a good ap-proximation. This can, however, only be confirmed by calculating the pressure through g . 3 (a) (b) PP ideal g (2 πT )0 1.0 2.0 3.0 4.0 0 1.0 2.0 3.0 4.01.041.021.000.980.960.940.920.90 2 loops3 loops4 loops g g g g g Fig. 1. (a) Pressure normalised to P ideal through g for various loop orders in SPT. (b) Weak-couplingpressure through various orders of g [1,5,6,7].
6. Summary and outlook
We have calculated the pressure of a massless φ theory using screened perturbationtheory. As Fig. 1 shows, the successive approximations in SPT seem a lot more stablethan in the weak-coupling expansion. The apparent improved convergence seems to belinked to the fact that SPT is basically an expansion about an ideal gas of massiveparticles, instead of an expansion about an ideal gas of massless particles, which is thecase for the weak-coupling expansion.Note that in Fig. 1b, only terms through order g in the weak-coupling expansion havebeen included. This is because part of the g -contribution arises from five-loop vacuumdiagrams which aren’t considered in Ref. [3]. Evaluation of the free energy to order g iswork currently in progress [8]. Acknowledgements
This work was done in collaboration with Jens O. Andersen. The author would like tothank the organising commitee of SEWM08 for an interesting and stimulating conference.References [1] A. Gynther et al., JHEP , (2007) 094.[2] F. Karsch, A. Patk´os and P. Petresczky, Phys. Lett. B , (1997) 69.[3] J. O. Andersen and L. Kyllingstad, arXiv:0805.4478, (2008). To appear in Phys. Rev. D.[4] J. O. Andersen, E. Braaten and M. Strickland, Phys. Rev. D , (2001) 105008.[5] P. Arnold and C. Zhai, Phys. Rev. D , (1994) 7603.[6] R. R. Parwani and H. Singh, Phys. Rev. D , (1995) 4518.[7] E. Braaten and A. Nieto, Phys. Rev. D , (1995) 6990.[8] J. O. Andersen and L. Kyllingstad. In preparation., (1995) 6990.[8] J. O. Andersen and L. Kyllingstad. In preparation.