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The exact mass gap in D=2 asymptotically free field theories and related experimental results
Nuclear Physics B (Proc. Suppl.) 26 (1992) 247-251 North-Holland
THE EXACT MASS GAP IN D=2 ASYMPTOTICALLY FREE FIELD THEORIES AND RELATED EXPERIMENTAL RESULTS Peter Hasenfratz Institute of Theoretical Physics, University of Bern, CH-9012 Bern, Switzerland
Recent results on the exact mass gap of the d = 2,O(N) invariant non-linear a-model and Gross-Neveu model are discussed briefly. The exact mass gap of the 0(3) non-linear o-model can be directly confronted with experiments on the low temperature magnetic properties of La2CU04 compounds. Exact results on the mass gap of d = 2 asymptotically free theories will be considered here together with an application, where these results can be directly compared with experiments. Some of the points have already been discussed by Pierre van Baal a year ago [1] and by Peter Weisz earlier at this conference [2] . I will try to avoid extensive overlap with these contributions .
1 . Introduction and Summary I will consider the O(N) invariant, d = 2 euclidean non-linear (r-model 21 So = d'xô n0 n, fo n'(x)n'(x) = 1, i=1,---,N, N>3, (1) and Gross-Neveu model
xY"ajx - 1/89ó(XX)2 ], d2x[1/ 2j X' Majorana field, i = 1, . . . , N, N>4. (2) SGN
=
Here fo and go are dimensionless couplings . The constraint n2 = 1, and the discrete Z2 symmetry XX --' -XX prevent an explicit mass term in (1) and (2), respectively. Both models are asymptotically free [3], weakly interacting at short dis-
tances while becoming non-perturbative in the infrared . Due to analogies to QCD both models have a long history. It is generally believed that these models, like QCD, have a dynamically generated mass gap m. The spectrum of the nonlinear tr-model is expected to have a massive, Lorentz scalar, O(N) isovector multiplet. The spectrum of the Gross-Neveu model contains a massive, O(N) isovector multiplet corresponding to the fundamental fermions, plus a rich spectrum of bound states [4] . Like in QCD, the mass gap is proportional to the renormalization group invariant A-parameter m=
cAMS-
where, for concreteness, I have chosen the Aparameter defined in the MS scheme of dimensional regularization . The problem of finding the ratio m/AMS attracted many people producing a large number of papers over the years. Strong coupling expansion, small volume expansion, large-N expansion and different numerical simulations are among the techniques applied. A partial list of references can be found in [5]. The exact result for the non-linear or-model has the form [5;6]:
0920--S632N2/$05.00 01992- Elsevier Science Publishers B.V All rights reserved .
248
m/Ams
P . Hasenfratz
IEract mass gap in D= 2 asymptotically freefield theories
(8/e)1 1(N -2 ) I'(1 + 1/(N - 2))
The corresponding results for the mass gap of the fundamental fermion in the Gross-Neveu model was obtained recently [7] (4e) 1/(N-% ) "'/AM-3 r(1- 1 /(N - 2))
These results are exact in the sense that no approximations are involved in deriving them . There were made certain general assumptions, however. It is assumed that a non-zero mass is generated and that the ambiguities in the form of the S-matrix ('CDD-factors') are correctly fixed. Furthermore, the steps leading to the thermodynamic Bethe Ansatz equation - a basic tool In deriving the results - are not completely rigorous. Therefore, as Flyvbjerg and Larsen put it [8], the question arises, whether in addition to being exact, these results are also correct . In the large-N limit (4,5) agree with the 1/Nexpansion up to the order these are available. In the non-linear o,-model it has the form [9] m/AMS = 1 + (3 in 2 -1 +YE)/N +O(1/N2),(6) while for the Gross-Neveu model it reads [10,11] m/AMS = 1 + (21n 2 + 1- YE)/N + 0(1IN2).(7)
Concerning the numerical simulations the situation is rather different in the two models considered . In the fermionic Gross-Neveu model the results, even the most recent ones [12], are not sufficiently quantitative to be compared with (5) . On the other hand, in the non-linear o-model, with the help of the new collective Monte Carlo methods [13], the correlation length is measured very precisely as the function of the bare coupling deep in the continuum limit [14,15] . In spite of that the agreement between the numerical result and (5) is not quite conclusive, especially not for N = 3. The reason is, probably, the very slow
onset of asymptotic (two-loop) scaling . Tricks to avoid this problem did not bring significant improvement [14,16]. The other expansion methods are not sufficiently quantitative for m/A. The only exception is the large-N expansion in Ref. [8,17], where the momentum integrals are evaluated numerically on finite lattices and the expansion is pushed up to 1/N3. These results are comparable in quality with the Monte Carlo results . 2 . An Application: The Magnetic Correlation Length of La2Cu04 Compounds at Low Temperatures The mass gap m = (8/e)A MS of the 0(3) non-linear a-model enters the predictions on the low temperature behaviour of the correlation length f(T) of antiferromagnetic compounds like La2Cu04 . The discovery of superconductivity in doped La2Cu04 [18] has stimulated a large number of experimental studies of this and related materials [19) . The pure undoped La2Cu04 crystal is an insulator . Due to a single unpaired electron on each Cu++ site this materials is an antiferromagnet with unusual properties. It shows a quasi-two-dimensional structure, the in-plane forces being much stronger than the forces between different planes. Additionally, these in-plane forces are an order of magnitude stronger than those observed in other quasi-two-dimensional crystals studied earlier. Doping this material, by Sr say, La2Cu04 --* Lag__.Sr,,Cu04 , x E (0.06, 0.3), the plane becomes metallic and a high temperature superconductor . It is believed that one has to understand the magnetic properties of these precursor insulators in order to understand high-T,; superconductivity after doping . Due to the quasi-two-dimensional structure
P. Hosenfratz /Exact mass gapin D = 2 asymptotically free fieldtheories
the magnetic behaviour of these materials is expected to be described well by the d = d, + 1 = 2 -1- 1 dimensional quantum antiferromagnetic Heisenberg model 71=J
r si si ,
J>0,
where S= is the spin operator at site i, S; = S(S+ 1) with S = 1/2 for the case of La2Cu04. The ground state of this model is antiferromagnetically ordered . This property has been proved rigorously for S > 1 [20] and is almost certainly true for S = 1/2 also [21]. There are two massless excitations - antiferromagnetic magnons - associated with the spontaneously broken rotation symmetry . According to the Mermin-Wagner theorem [22], no massless Goldstone bosons, or magnons can exist at finite temperature in d, = 2. The magnons pick up a tiny mass at low temperatures . The temperature dependence of the correlation length has the exact form [23] t(T)
=
8
(hc/2Rp,) exp(2rp, /T)
[1- 2T/(21rp,) + 0((T/(21rp,»2)] . (9) Here p, is the spin-stiffness (high energy physicists would call it the square the Goldstone boson decay constant, p, = F2 ), while c is the spin-wave velocity . The correct exponential behaviour, including the prefactor he/2ap,, has been predicted correctly earlier in [24]. To get the number e/8 one needs the exact mass gap of the 0(3) non-linear o-model, while the (Tl21rp.) correction requires a detailed calculation in chiral perturbation theory using techniques developed in [25]. This is the point where I have to mention the pioneering work [26], where the first steps have been made towards a systematic treatment of antiferromagnetic magnons using quantum field theoretical methods . Due to the weak between-plane forces the un-
249
0 .03 0 .02 0.01 0
0
100
200
300
400
500
T
600
Fig. 1. The inverse magnetic correlation length of pure crystals of La2Cu04 in 1.728A -1 units. The fit is the theoretical prediction (9) .The data points and the fit is from Ref. [29) .
doped La2Cu04 remains in the broken phase up to TN " 300K* . Eq.(9) applies for TN < T < 27rp, . The spin-wave velocity has been measured to be he = (0.85 f 0.03)e VA [27]. Fig. 1 shows the experimental results [28] compared with a one-parameter fit based on (9) giving 2arp, = 150meV s:ts 1740K° [29]. At T = 500K° the correction Z T/27rp, is - 14% . The fit is rea sonable but not perfect . Small anisotropies in the spin-spin coupling in La2Cu04 might be responsible for the discrepancies [30]. According to Ref. [28] doped samples follow the simple phenomenological rule t!i(z,T) = f-1(x,0) + t-1(O,T), where x is the doping concentration and t(0, T) is given by (9). These problems require further investigations .
3. General Steps and Assumptions Leading to the Mass Gap Predictions The basic idea is the following [6,7]. The free energy of the model is studied as the function of a chemical potential h coupled to one of the
2.50
P Hasenfrata /Exact mass gap in D = 2 asymptotically free fieldtheories
Noether charges . This chemical potential creates a system of finite density. The free energy is calculated in two different ways: a/ The exact free energy fexa,t(h, m) is determined as the function of h and the physical mass to using the exact ;-matrix . b/ For h » m the free energy fpert(h, AWS) can be calculated perturbatively (due to asymptotic freedom) as the function of h and the scale parameter As-, . Comparing the two forms of the free energy one obtains m/AVS . This calculation, therefore takes over all the assumptions which are made in deriving the S-matrix [31] . First, the existence of a non-zero mass gap is assumed. For this reason, considerations [32] which question this common lore are not contradicted by the results here. Second, the factorization equations, O(N) symmetry, crossing, unitarity and analyticity do not fix completely the two-particle S-matrix [31] . The resulting ambiguity (CDD-factor) can be resolved by making qualitative assumptions on the spectrum of the model . The choice made in ef.[31] for the non-linear e-model is certainly correct at large enough N [33] and is consistent with onte Carlo calculations for N = 3 [34] . For the Gross-Neveu model a CDD-factor was chosen [31] corresponding to the bound state spectrum of this model predicted by semiclassical considerations [4] which is believed to be exact . For the thermodynamics a quantization condition for the particles' momenta p = m sink 0 is needed which takes into account the interaction as expressed by the S-matrix . Take a large periodic box of length L and n identical particles . Factorization leads to the quantization condition exp(ip;L) This equation, which occurred earlier in the lit-
erature and is called the thermodynamical Bethe Ansatz [35], is easy to establish in the dilute gas approximation . I am not aware of a rigorous proof valid for arbitrary densities . In ref.[36] arguments are given which leave little doubt, however, that (10) is generally true . The rest of the derivation leading to fesact(h, m) is, although technically somewhat involved, conceptually simple. No assumptions are made in deriving the free energy in perturbation theory fpert . In the GrossNeveu model it is a technically difficult problem since fpe,.t is needed up to ti h2g2 (h) (3-loops!) to fix m/AHS [7]. (In the o-model, where there is a ti h2/ fo classical contribution, a 1-loop calculation is sufficient [5,6] .) The authors of Ref.[7] made a great job in overcoming all the extra conceptual and technical difficulties of the GrossNeveu model producing the closed result in (5).
Acknowledgements I would like to thank F . Niedermayer for explaining some of the details of the work in Ref.[7]. I am indebted to R. J . Birgenau for communicating the results of Ref.[28,29] before publication. I thank P. Weisz and A. Duncan for the valuable discussions during this conference . This work is supported in part by Schweizerischer Nationalfonds.
eferences [1]
P. van Baal, Nud. Phys . B (Proc. Suppl.) 20 (1991) 3. [2] P. Weisz, contribution to this conference . [39 A. TN . Polyakov, Phys . Lett. B59 (1975) 79; E. Brezin and J. Zinn-Justin, Phys . Rev . Lett. 36 (1976) 6-91 ;
P. Hasenfratz /Exact mass gap ii : D =. 2 asymptotically free feldtheories
D. J. Gross and A. Neveu, Phys. Rev . D10 (1974) 3135. [4] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D12 (1975) 2443. [5] P. Hasenfratz, M. Maggiore and F. Niedermayer, Phys. Lett. B245 (1990) 522 . [6] P. Hasenfratz and F . Niedermayer, Phys. Lett. B245 (1990) 529. P. Forgács, F. Niedermayer and P. Weisz, MPI prepr. MPI-Ph/91-37 (1991). [8] H . Flyvbjerg and F. Larsen, Phys. Lett. B266 (1991) 92. P. Biscari, M. Campostrini and P. Rossi, Phys. Lett. B242 (1990) 225 ; H. Flyvbjerg, Nucl. Phys. B348 (1991) 714; H. Lehmann, unpublished. [10] P. Forgács, F. Niedermayer and P. Weisz, MPI prepr. MPI-Ph/91-37 (1991). [11] M. Campostrini and P. Rossi, unpublished . [12] L. Belanger, R. Lacaze, A. Morel, N. Attig, B. Petersson and U. Wolff, Nucl. Phys. B340 (1990) 245 . [13] R. H. Swendsen and J.S. Wang, Phys. Rev. Lett. 58 (1987) 86; F. Niedermayer, Phys. Rev. Lett. 61 (1988) 2026; U. Wolff, Phys. Rev. Let.. 62 (1989) 361 . [14] P. Hasenfratz and F. Niedermayer, Nucl. Phys. B337 (19991% 233; U. Wolff, Phys. Lett. B248 (1990) 335 . [13] J. Apostolakis, C. F. Baillie and G. C. Fox, Boulder prepr., COLO-HEP-231 (1991). [16] I. Dimitrovic, P. Hasenfratz, K . Jansen and Yue Shen, Phys. Lett. B264 (1991) 120. [17] H. Flyvbjerg and S. Varsted, Nucl. Phys. B (Proc. Su=p1.) 17 (1990) 343; Nucl. Phys. B344 (1990) 646 . [18] G. Bednorz and K. A. Müller, Z. Phys. B64 (1986) 189. flgl for an overview, see "Physical Properties of High Temperature Superconductors", D: M . Ginsberg (ed.), Singapore, World Scientific, 1990. [20] J. Neves and F. Peres, Phys. Lett . A114 (1986) 331 ; T. Kennedy, E. H. Lieb and B. S. Shastay, J. Stat. Phys. 53 (1988) 1019. [21] see, for example, T. Barnes, Int . J. of Mod. Phys. C2 (1991) 659. [22] N. D. Mermin and H. Wagner, Phys. Re. Lett. 17 (1966) 1133 . [23] P. Hasenfratz and F. Niedermayer, Phys. Lett . B268
251
(1991) 231 . [24] S. Chakravarty, B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 60 (1988) 1057; Phys. Rev. B39 (1989) 2344. [25] J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142 ; Phys. Lett. B184 (1984) 83; ióid B188 (1987) 477, Nucl. Phys. B307 (1988) 763 ; H. Leutwyler, Phys. Lett. B189 (1987) 197; P. Hasenfratz and H. Leutwyler, Nucl. Phys. B343 (1990) 241 . [26] H. Neuberger and T. Ziinan, Phys. Rev. B39 (1989) 2608. [27] G. APTTL, e t r., . n ~;ö, ácw, Lett. 62 (1989) 2052. [28] B. Keimer, R. J. Birgeneau, A. Cassanho, Y. Endoh, R. W. Erwin, M. A. KLdttner and G. Shirane, unpublished . [29] R. J. Birgeneau, N. Belk, Y. Endoh, R. W. Erwin, M. A. Kastner, B. Keimer and G. Shrane, to appear in the proceedings of ICNS'91 (Oxford, England). [30] R. J. Birgeneau, private communication. [31] A. B. Zamolodchikov and Al. B. Zamolodchikov, Ann. Phys. (NY) 120 (1979) 253 . [32] A. Patrascioiu, Phys. Rev. Lett. 58 (1987) 2285; A . Patrascioiu and E. Seiler, Phys. Rev. Lett. 60 (1988) 875 ; A. Patrascioiu, prepr., AZPH-TH/91-49 (1991) . [33] E. Bruin anri -J. Zinn-Justin, Phys. Rev. B14 (1976) 3110; W. A. Bardeen, B. W. Lee and R. E. Shrock, Phys. Rev. D14 (1976) 3110; B. Berg, M. Karowski, V. Kurak and P. Weisz, Phys. Lett. B76 (1978) 502. [34] M. Lüsher and U. Wolff, Nucl. Phys. B339 (1990) 222 . [35] Al. B. Zamolodchikov, Nucl. Phys. B342 (1990) 695 . [36] P. Hasenfratz and F. Niedermayer, in "Nonperturbative Methods in Low Dimensional Quantum Field Theories", G. Domokos, Z. Horváth and S. K5vesiDomokos, (World Scientific, 1991 ), p.407.
THE EXACT MASS GAP IN D=2 ASYMPTOTICALLY FREE FIELD THEORIES AND RELATED EXPERIMENTAL RESULTS Peter Hasenfratz Institute of Theoretical Physics, University of Bern, CH-9012 Bern, Switzerland
Recent results on the exact mass gap of the d = 2,O(N) invariant non-linear a-model and Gross-Neveu model are discussed briefly. The exact mass gap of the 0(3) non-linear o-model can be directly confronted with experiments on the low temperature magnetic properties of La2CU04 compounds. Exact results on the mass gap of d = 2 asymptotically free theories will be considered here together with an application, where these results can be directly compared with experiments. Some of the points have already been discussed by Pierre van Baal a year ago [1] and by Peter Weisz earlier at this conference [2] . I will try to avoid extensive overlap with these contributions .
1 . Introduction and Summary I will consider the O(N) invariant, d = 2 euclidean non-linear (r-model 21 So = d'xô n0 n, fo n'(x)n'(x) = 1, i=1,---,N, N>3, (1) and Gross-Neveu model
xY"ajx - 1/89ó(XX)2 ], d2x[1/ 2j X' Majorana field, i = 1, . . . , N, N>4. (2) SGN
=
Here fo and go are dimensionless couplings . The constraint n2 = 1, and the discrete Z2 symmetry XX --' -XX prevent an explicit mass term in (1) and (2), respectively. Both models are asymptotically free [3], weakly interacting at short dis-
tances while becoming non-perturbative in the infrared . Due to analogies to QCD both models have a long history. It is generally believed that these models, like QCD, have a dynamically generated mass gap m. The spectrum of the nonlinear tr-model is expected to have a massive, Lorentz scalar, O(N) isovector multiplet. The spectrum of the Gross-Neveu model contains a massive, O(N) isovector multiplet corresponding to the fundamental fermions, plus a rich spectrum of bound states [4] . Like in QCD, the mass gap is proportional to the renormalization group invariant A-parameter m=
cAMS-
where, for concreteness, I have chosen the Aparameter defined in the MS scheme of dimensional regularization . The problem of finding the ratio m/AMS attracted many people producing a large number of papers over the years. Strong coupling expansion, small volume expansion, large-N expansion and different numerical simulations are among the techniques applied. A partial list of references can be found in [5]. The exact result for the non-linear or-model has the form [5;6]:
0920--S632N2/$05.00 01992- Elsevier Science Publishers B.V All rights reserved .
248
m/Ams
P . Hasenfratz
IEract mass gap in D= 2 asymptotically freefield theories
(8/e)1 1(N -2 ) I'(1 + 1/(N - 2))
The corresponding results for the mass gap of the fundamental fermion in the Gross-Neveu model was obtained recently [7] (4e) 1/(N-% ) "'/AM-3 r(1- 1 /(N - 2))
These results are exact in the sense that no approximations are involved in deriving them . There were made certain general assumptions, however. It is assumed that a non-zero mass is generated and that the ambiguities in the form of the S-matrix ('CDD-factors') are correctly fixed. Furthermore, the steps leading to the thermodynamic Bethe Ansatz equation - a basic tool In deriving the results - are not completely rigorous. Therefore, as Flyvbjerg and Larsen put it [8], the question arises, whether in addition to being exact, these results are also correct . In the large-N limit (4,5) agree with the 1/Nexpansion up to the order these are available. In the non-linear o,-model it has the form [9] m/AMS = 1 + (3 in 2 -1 +YE)/N +O(1/N2),(6) while for the Gross-Neveu model it reads [10,11] m/AMS = 1 + (21n 2 + 1- YE)/N + 0(1IN2).(7)
Concerning the numerical simulations the situation is rather different in the two models considered . In the fermionic Gross-Neveu model the results, even the most recent ones [12], are not sufficiently quantitative to be compared with (5) . On the other hand, in the non-linear o-model, with the help of the new collective Monte Carlo methods [13], the correlation length is measured very precisely as the function of the bare coupling deep in the continuum limit [14,15] . In spite of that the agreement between the numerical result and (5) is not quite conclusive, especially not for N = 3. The reason is, probably, the very slow
onset of asymptotic (two-loop) scaling . Tricks to avoid this problem did not bring significant improvement [14,16]. The other expansion methods are not sufficiently quantitative for m/A. The only exception is the large-N expansion in Ref. [8,17], where the momentum integrals are evaluated numerically on finite lattices and the expansion is pushed up to 1/N3. These results are comparable in quality with the Monte Carlo results . 2 . An Application: The Magnetic Correlation Length of La2Cu04 Compounds at Low Temperatures The mass gap m = (8/e)A MS of the 0(3) non-linear a-model enters the predictions on the low temperature behaviour of the correlation length f(T) of antiferromagnetic compounds like La2Cu04 . The discovery of superconductivity in doped La2Cu04 [18] has stimulated a large number of experimental studies of this and related materials [19) . The pure undoped La2Cu04 crystal is an insulator . Due to a single unpaired electron on each Cu++ site this materials is an antiferromagnet with unusual properties. It shows a quasi-two-dimensional structure, the in-plane forces being much stronger than the forces between different planes. Additionally, these in-plane forces are an order of magnitude stronger than those observed in other quasi-two-dimensional crystals studied earlier. Doping this material, by Sr say, La2Cu04 --* Lag__.Sr,,Cu04 , x E (0.06, 0.3), the plane becomes metallic and a high temperature superconductor . It is believed that one has to understand the magnetic properties of these precursor insulators in order to understand high-T,; superconductivity after doping . Due to the quasi-two-dimensional structure
P. Hosenfratz /Exact mass gapin D = 2 asymptotically free fieldtheories
the magnetic behaviour of these materials is expected to be described well by the d = d, + 1 = 2 -1- 1 dimensional quantum antiferromagnetic Heisenberg model 71=J
r si si ,
J>0,
where S= is the spin operator at site i, S; = S(S+ 1) with S = 1/2 for the case of La2Cu04. The ground state of this model is antiferromagnetically ordered . This property has been proved rigorously for S > 1 [20] and is almost certainly true for S = 1/2 also [21]. There are two massless excitations - antiferromagnetic magnons - associated with the spontaneously broken rotation symmetry . According to the Mermin-Wagner theorem [22], no massless Goldstone bosons, or magnons can exist at finite temperature in d, = 2. The magnons pick up a tiny mass at low temperatures . The temperature dependence of the correlation length has the exact form [23] t(T)
=
8
(hc/2Rp,) exp(2rp, /T)
[1- 2T/(21rp,) + 0((T/(21rp,»2)] . (9) Here p, is the spin-stiffness (high energy physicists would call it the square the Goldstone boson decay constant, p, = F2 ), while c is the spin-wave velocity . The correct exponential behaviour, including the prefactor he/2ap,, has been predicted correctly earlier in [24]. To get the number e/8 one needs the exact mass gap of the 0(3) non-linear o-model, while the (Tl21rp.) correction requires a detailed calculation in chiral perturbation theory using techniques developed in [25]. This is the point where I have to mention the pioneering work [26], where the first steps have been made towards a systematic treatment of antiferromagnetic magnons using quantum field theoretical methods . Due to the weak between-plane forces the un-
249
0 .03 0 .02 0.01 0
0
100
200
300
400
500
T
600
Fig. 1. The inverse magnetic correlation length of pure crystals of La2Cu04 in 1.728A -1 units. The fit is the theoretical prediction (9) .The data points and the fit is from Ref. [29) .
doped La2Cu04 remains in the broken phase up to TN " 300K* . Eq.(9) applies for TN < T < 27rp, . The spin-wave velocity has been measured to be he = (0.85 f 0.03)e VA [27]. Fig. 1 shows the experimental results [28] compared with a one-parameter fit based on (9) giving 2arp, = 150meV s:ts 1740K° [29]. At T = 500K° the correction Z T/27rp, is - 14% . The fit is rea sonable but not perfect . Small anisotropies in the spin-spin coupling in La2Cu04 might be responsible for the discrepancies [30]. According to Ref. [28] doped samples follow the simple phenomenological rule t!i(z,T) = f-1(x,0) + t-1(O,T), where x is the doping concentration and t(0, T) is given by (9). These problems require further investigations .
3. General Steps and Assumptions Leading to the Mass Gap Predictions The basic idea is the following [6,7]. The free energy of the model is studied as the function of a chemical potential h coupled to one of the
2.50
P Hasenfrata /Exact mass gap in D = 2 asymptotically free fieldtheories
Noether charges . This chemical potential creates a system of finite density. The free energy is calculated in two different ways: a/ The exact free energy fexa,t(h, m) is determined as the function of h and the physical mass to using the exact ;-matrix . b/ For h » m the free energy fpert(h, AWS) can be calculated perturbatively (due to asymptotic freedom) as the function of h and the scale parameter As-, . Comparing the two forms of the free energy one obtains m/AVS . This calculation, therefore takes over all the assumptions which are made in deriving the S-matrix [31] . First, the existence of a non-zero mass gap is assumed. For this reason, considerations [32] which question this common lore are not contradicted by the results here. Second, the factorization equations, O(N) symmetry, crossing, unitarity and analyticity do not fix completely the two-particle S-matrix [31] . The resulting ambiguity (CDD-factor) can be resolved by making qualitative assumptions on the spectrum of the model . The choice made in ef.[31] for the non-linear e-model is certainly correct at large enough N [33] and is consistent with onte Carlo calculations for N = 3 [34] . For the Gross-Neveu model a CDD-factor was chosen [31] corresponding to the bound state spectrum of this model predicted by semiclassical considerations [4] which is believed to be exact . For the thermodynamics a quantization condition for the particles' momenta p = m sink 0 is needed which takes into account the interaction as expressed by the S-matrix . Take a large periodic box of length L and n identical particles . Factorization leads to the quantization condition exp(ip;L) This equation, which occurred earlier in the lit-
erature and is called the thermodynamical Bethe Ansatz [35], is easy to establish in the dilute gas approximation . I am not aware of a rigorous proof valid for arbitrary densities . In ref.[36] arguments are given which leave little doubt, however, that (10) is generally true . The rest of the derivation leading to fesact(h, m) is, although technically somewhat involved, conceptually simple. No assumptions are made in deriving the free energy in perturbation theory fpert . In the GrossNeveu model it is a technically difficult problem since fpe,.t is needed up to ti h2g2 (h) (3-loops!) to fix m/AHS [7]. (In the o-model, where there is a ti h2/ fo classical contribution, a 1-loop calculation is sufficient [5,6] .) The authors of Ref.[7] made a great job in overcoming all the extra conceptual and technical difficulties of the GrossNeveu model producing the closed result in (5).
Acknowledgements I would like to thank F . Niedermayer for explaining some of the details of the work in Ref.[7]. I am indebted to R. J . Birgenau for communicating the results of Ref.[28,29] before publication. I thank P. Weisz and A. Duncan for the valuable discussions during this conference . This work is supported in part by Schweizerischer Nationalfonds.
eferences [1]
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P. Hasenfratz /Exact mass gap ii : D =. 2 asymptotically free feldtheories
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