- Email: [email protected]
Supersymmetry at finite temperatures
Volume 123B, number 3,4
PHYSICS LETTERS
31 March 1983
SUPERSYMMETRY AT FINITE TEMPERATURES Klusau TESHIMA l The Enrtco Fermi Instltute, The University o f Chtcago, Chzcago, IL 60637, USA Received 21 November 1982 The effects of finite temperature on supersymmetrlc models are examined Associated with the breaking of the supersymmetry, a zero-mass pole (hke a Nambu-Goldstone fermion) is found to contribute to correlation functions
In usual cases, thermal fluctuation has the effect o f restoring various symmetries. In the case o f supersymmetry, however, finite temperature breaks the symmetry [ 1 - 3 ] . This IS because fermlons and bosons obey different statistics. This breaking of supersymmetry appears in the non-vanishing expectation value of the hamiltonlan (square of the supercharge), while the lagranglan and the field equations remain unchanged Thus it has something in common with the spontaneous symmetry-breaking; therefore, it is natural to ask whether a massless N a m b u - G o l d s t o n e (NG) fermion appears associated with the symmetry-breaking at finite temperatures. Glrardello et al. [2] investigated the effects o f fimte temperature on supersymmetrlc models. They examined the Ward-Takahashi (WT) identity associated with the supersymmetry At zero temperature, we need a zero-mass NG fermion in order to saturate the WT identity if the hamiltoman has a non-zero expectation value. At finite temperatures, however, they found that an excess (symmetry-breaking) term appears in the WT identity because different boundary conditions are imposed on the bosomc and fermIonlc fields in the imaginary-time formalism [4]. Because o f this symmetry-breaking term they claimed that the non-vanishing expectation value o f the hamdtoman does not necessarily Imply the appearance of a NG fermlon In fact, the global supersymmetry transformation cannot be defined at finite temperatures in the imaginarytime formalism because the anticommutmg parameter of the transformation should be antlperIodlc m time. There seems, therefore, to be no reason for a NG fermlon to appear. Glrardello et al. [2] examined a simple W e s s - Z u m i n o model [5] in the leading order o f the high-temperature expansion (at the one-loop level) [4]. In this model supersymmetry is not broken spontaneously at zero-temperature. They found that the supersymmetry is broken (the mass-spectrum splits) at finite temperature The fermion in this model is found to be massive at low temperatures. They concluded, therefore, that the supersymmetry is broken without a NG fermion (in accordance with the above reasoning). They then examined a modified O'Raxfeartalgh model [6], m which supersymmetry Is broken spontaneously at zero-temperature (with a NG fermlon). They found, however, that the NG fermion remains massless at fimte temperatures at the one-loop level (although the discrete chiral symmetry of this model is stall broken spontaneously). How can we understand the pecuhar remnant of the supersymmetry at finite temperatures in the latter case? Does supersymmetry actually make sense at fimte temperatures (as was suggested by van Hove [7]) 9 Or is the oneloop approximation leading us astray9 It is desirable ff we can go beyond the lowest orders. One of the ways to sum up contributions in higher orders systematically is the 1IN expansion. Unfortunately, some difficulties are found In the 1IN expansion m four dimensions [8]. We must analyze in lower dimensions HIgashijlma and Uematsu [9] examined an O(n)-symmetric W e s s - Z u m i n o model m two dimensions In the hmit 1 Work supported m part by Soryushx-Shogakkai, and the Nishma Memorial Foundation, Japan, and the NSF, Grant No PHY-7923669. 226
0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
Volume 123B, number 3,4
PHYSICS LETTERS
31 March 1983
N-~ co (at zero-temperature). They found a phase transition from the supersymmetric phase to the spontaneously broken phase [with an O(N)-smglet NG fermion]. As we shall see below, there is no such phase transition at fimte temperatures, because fimte temperature always breaks the supersymmetry. We can refer from analytlcity that at low temperatures the smglet fermlon is either always massless or always masstve lrrespectwe of whether it ~s massive or massless at zero-temperature. It seems unnatural that the massive fermmn (at zero-temperature) could suddenly become massless at fimte temperatures. One might conclude, therefore, that the smglet fermion would be always massive at low temperatures. We shall find, however, that this is not the case. A zero-mass pole is always found to contribute to the correlatmn functmn of the smglet fermion at fimte temperatures because of the (broken) supersymmetry 0rrespective of whether the supersymmetry is spontaneously broken or not at zero-temperature). Let us first describe the model we shall examine The lagrangian is [9]
.t2=fd20(¼D'~V,O#),+½Ug(~30 +UX~b0 +gqSo~,~bt), where summation over i = 1,. eters 0~ (a = 1,2).
¢,(x, o) = A,(x) +
(1)
, N is understood. Ct (i = O, 1 ..... N ) are the superfields with antlcommuting param-
O~,~(x) +7' OaO,~F,(x)
(2)
D ~ xs the covarlant derwatwe m the superspace. In terms of the component fields, .C =½ (3uA/)2 +½1 ¢i7 u 0u t~t +~ F 2 +Ng(FoA 2 _ A 0 t~Ot~O) +NXFo +g(2AoFtA, - A o ~Ot¢, +FoA 2 - 2~0 etA,) • (3) After integrating overA t, fit, and Fi, we obtain the effective lagranglan "~eff =N{Fo(gA20 + ~) -gZot~o~o - ½1 t r I n (l')'/'t 0/.t -- 2 g h 0 ) +-~l tr In [ - O2 - 4g2A 2 + 2gF 0 - 4gZff0(iTuOu - 2gA0) - I ¢0]}-
(4)
In the leading order of the I/N expansion, the effective potential is *x
Veff(A,F, r ) : V(eO)(A,F) + V(el~(A,F; V), V(eO~(A,F) = (N/8g 2) p2(m2 - / a 2) + (N/81r) [p2 ln(,2 /eM2) _ m 2 ln(m2/eM2)], V(D( eft'~A , F , T ) = - ~N ? d k { l n ( 1 - e x p [ - [ 3 ( k 2 + m 2 ) l / 2 ] ) - l n ( l +
e x p [ - f l ( k 2 +,2)1/2]}},
(5)
where 1/3 = T is the temperature, and M the mass-scale introduced an renormahzlng ~. A = (A 0 ), and F = (F0).
m 2 = 4g2A 2 _ 2gF is the squared mass of A i, and , 2 = 4g2A 2 is that of ~z. In order to find the minimum point, we differentiate Veff b y A and F. 0=
O-
0Veff(A ,F; T) _N _ , 2 +g_2 In OF =4g rr M 2 0 Veff (A ,F, T)
OA
4-g2
dk
,
(6)
7r 0 (k2+m2)l/2{exp~3(k2+m2)l/2] - 1
~2 /~2 [ - N A L m 2 - p 2 + = - ln=-rr m 2
oo
rr
(k 2 + m2) 1/2 {exp[fl(k 2 + m2) 1/2] - 1} (k 2 + . 2 ) 1 / 2 {exp[fl(k2+.2)l/2] + 1}
.1 Note that at T ~ 0 the effective potentml is not identical to the expectatmn value of the hamiltoman, and is not positive definite 227
Volume 123B, number 3,4
PHYSICS LETTERS
31 March 1983 --4 ---4 -"~ -.'~ --4 ,,., . ,, ,, OO .O .OO "--- N~ ¢..~ -~' L "
r--0
'~
/"
,\
0
/
_
<£,[5~'~"
05 I
I
Jl , 05
I
,I I0 m/(grr -I/2)
~,"
,~o
I 15
Ftg, 1 Solutions of eq (7) (solid lines) and eq (6) (dashed lines) at T = 0 The solution/a = rn of eq (7) Is supersymmetric. Crossing points give the extrema of Veff
0
05
m/lg
IO_tn)
15
Ftg 2 Solutions of eq. (7) (independent of M) and eq (6) (for M = 0.7 M c) at T = 0 1-0.5 (m umts ofg/x/~)
The curves of the solutions of eqs. (6) and (7) at T = 0 are shown m fig. 1. The points at which the curves meet give the e x t r e m a o f Veff. F o r m 2 < M c2 =g2/(rre), there are two m i m m u m points on the line/a = m (supersymmetric hne), while for M > M c the m l m m u m point is on another b r a n c h of the curve o f eq. (7) on which # 4: m. Thus analytic, ty is b r o k e n at M = M c (phase transiUon) At T v~ 0, we find s m o o t h curves o f eq. (7) w i t h o u t branch points (fig. 2). No transition takes place from the supersymmetric phase to the b r o k e n phase. We can write the effective potential in terms of A only, e l i n u n a t m g F b y the use of eq. (6). Veff(A , T) with M = 0.7 M c is depicted in fig 3 for several temperatures. The discrete chlral s y m m e t r y , which is b r o k e n spontaneously (i.e.,A :# (3) at low temperatures, xs restored above critical temperatures (fig. 4), It may be interesting to note that although at T = 0 two supersymmetric vacua are degenerate (for M < Mc) the
06
M:O7M c
04
,~,~ 02
T,
06, 04
:>-02 -04
,f 05 I0 ffllgrr-I/2): 2
15
Fig 3 The effectwe potentml Veff(A, T) for M = 0 7 M c, 228
Ch,ral-symmetrhc
02 0
I
I
L
I0
20
30
MIMc
Fig 4 The phase diagram, which shows that the chtral symmetry is restored above critical temperatures
Volume 123B, number 3,4
PHYSICS LETTERS
31 March 1983
vacuum with smaller A is always chosen when the system is cooled from high temperatures (see fig. 3). It is because the finite-temperature part V(I~ is monotonically decreasing as A decreases; this result does not depend on the details o f models and on the dimensionality of s p a c e - t i m e . Now we turn to the problem o f the self-energy 2;(p, T) of the singlet fermion 40 with momentum pU We can evaluate g (p, T) in the imaginary-time formalism. Note, however, that the "imaginary-time" itself is not a physical coordinate. Only the "equal-time functions" in this formalism are related to physical correlation functions. Therefore, we should perform a Fourier transformation after calculating Feynman graphs in momentum space. The transformation, however, includes infinite summation over the discrete energies , 2 , and the behavior of the transform as not easy to see analytically. Alternatively, we can make use of the real-time formalism, in which one analytically continues the time to the real ax~s, so that the Fourier transformation is the integral over the continuous variable. Then the position of the poles in momentum space gives information on the behavior o f the correlation functions at large distances. Making use of the usual manipulation [4], we find 0=
a Veff (A , F , T) aA
=_gNAF_8Ng2A(d2k( lgF + r r f ( k 2 - / ~ 2) + r r f ( k 2 - m 2) a(27r)z \(k2-112+le)(k2-m2+ie) e x p ( 3 1 k ° l ) + 1 e x p ( 3 l k ° l ) - 1 ] = F Z (0, T).
(8)
We thus find ~ ( 0 , T ~ 0) = 0, because F 4= 0 at finite temperatures. This result may be surprising because
Y-,(O,O) = 2gA -g/(2rrA2)=/= hm 1~(0, T) = 0, T~ 0
for 3 t < M c Note that the temperature dependent terms in the propagators vanish very rapidly as T -+ 0 like
exp(-m/T) However, we also find a factor 2gF =/12 - m 2 in the denominator of the temperature-dependent term in Z (0, T), which IS as small as exp(-m/T) (for M < Me) This factor enhances the net temperature effect to be O(1) (Note that the above result does not depend on the dlmenslonallty o f the s p a c e - t i m e . ) Of course, the zero-temperature limit should coincide with the value at zero-temperature physically. Evahiatmg Z(p, T) at pU ~ 0 (for M < Mc) , we find that at low temperatures E ( p , T) increases rapidly (proportionally to p 2 ) until it reaches the zero-temperature value at p2 ~ m 2 e x p ( - m 3 ) . We then find three poles (on the real axis) in the two-point function atp u = 0, a t p 2 ~ m 2 e x p ( - f l m ) , and a t p 2 ~ [4gA(1-4rrA2)] 2. In the limit T--> 0, the contributions of the first two poles cancel each other, and the third pole gives the same result as at T = 0 Physically speaking, the correlation function damps as e x p ( - T r ) [not as e x p ( - m r ) ] at very remote distances [r >> m - 1 exp(3m)] because o f the contribution of the pole at pU = 0. In the limit t -+ 0, however, it has already damped as e x p ( - m r ) in the region r ~< m - 1 e x p ( 3 m ) , so that the behavior like e x p ( - T r ) cannot be observed (for M < M c) We could make use o f the WT identity in the real-time formalism (not In the imaginary time formalism o f ref. [2] ) to show the necessity o f a zero-mass pole in the fermlonic correlation function. The WT identity would also show that the contribution o f the zero-mass pole vanishes rapidly (exponentially) as T ~ 0 in cases where the supersymmetry is not broken at T = 0 (i e., the expectation value of the hamiltonian vanishes exponentially as T-+ 0). On the other hand, in cases where the supersymmetry IS broken spontaneously at T = 0, the contribution o f the zero-mass pole o f the NG fermlon can be observed at any temperature. Supersymmetry thus manifests itself at finite temperatures. *2 The "energy" here is m fact the Fourmr-conjugate variable to the imaginary tune, and not the energy of the physical system 229
Volume 123B, number 3,4
PHYSICS LETTERS
31 Match 1983
I would hke to thank all the members of the theoretical group at KEK for their warm hospitality. I am indebted to S. Watamura for collaboration at the early stage of this investigation. I also thank N. Sakai and D. Friedan for critical comments, and M. Gross for reading the manuscript carefully.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
A Das and M Kaku, Phys. Rev. D18 (1978) 4540. L GlrardeUo, M.T. Gnsaru and P Salomonson, Nucl Phys. B178 (1981) 331 K. Fujlkawa, INS-report-454 (1982). See, for example, L Dolan and R Jackiw, Phys Rev D9 (1974) 3320 J Wess and B Zummo, Nucl. Phys. B78 (1974) 1. L. O'Ralfeartalgh, Nucl Phys. B96 (1975) 331. L. Van Hove, CERN preprmt TH. 3280 (1982) A. Hlguchi and Y. Kazama, Nucl. Phys. B206 (1982) 152 K Hlgashillma and T. Uematsu, Tokyo preprmt UT-385 (1982)
230
PHYSICS LETTERS
31 March 1983
SUPERSYMMETRY AT FINITE TEMPERATURES Klusau TESHIMA l The Enrtco Fermi Instltute, The University o f Chtcago, Chzcago, IL 60637, USA Received 21 November 1982 The effects of finite temperature on supersymmetrlc models are examined Associated with the breaking of the supersymmetry, a zero-mass pole (hke a Nambu-Goldstone fermion) is found to contribute to correlation functions
In usual cases, thermal fluctuation has the effect o f restoring various symmetries. In the case o f supersymmetry, however, finite temperature breaks the symmetry [ 1 - 3 ] . This IS because fermlons and bosons obey different statistics. This breaking of supersymmetry appears in the non-vanishing expectation value of the hamiltonlan (square of the supercharge), while the lagranglan and the field equations remain unchanged Thus it has something in common with the spontaneous symmetry-breaking; therefore, it is natural to ask whether a massless N a m b u - G o l d s t o n e (NG) fermion appears associated with the symmetry-breaking at finite temperatures. Glrardello et al. [2] investigated the effects o f fimte temperature on supersymmetrlc models. They examined the Ward-Takahashi (WT) identity associated with the supersymmetry At zero temperature, we need a zero-mass NG fermion in order to saturate the WT identity if the hamiltoman has a non-zero expectation value. At finite temperatures, however, they found that an excess (symmetry-breaking) term appears in the WT identity because different boundary conditions are imposed on the bosomc and fermIonlc fields in the imaginary-time formalism [4]. Because o f this symmetry-breaking term they claimed that the non-vanishing expectation value o f the hamdtoman does not necessarily Imply the appearance of a NG fermlon In fact, the global supersymmetry transformation cannot be defined at finite temperatures in the imaginarytime formalism because the anticommutmg parameter of the transformation should be antlperIodlc m time. There seems, therefore, to be no reason for a NG fermlon to appear. Glrardello et al. [2] examined a simple W e s s - Z u m i n o model [5] in the leading order o f the high-temperature expansion (at the one-loop level) [4]. In this model supersymmetry is not broken spontaneously at zero-temperature. They found that the supersymmetry is broken (the mass-spectrum splits) at finite temperature The fermion in this model is found to be massive at low temperatures. They concluded, therefore, that the supersymmetry is broken without a NG fermion (in accordance with the above reasoning). They then examined a modified O'Raxfeartalgh model [6], m which supersymmetry Is broken spontaneously at zero-temperature (with a NG fermlon). They found, however, that the NG fermion remains massless at fimte temperatures at the one-loop level (although the discrete chiral symmetry of this model is stall broken spontaneously). How can we understand the pecuhar remnant of the supersymmetry at finite temperatures in the latter case? Does supersymmetry actually make sense at fimte temperatures (as was suggested by van Hove [7]) 9 Or is the oneloop approximation leading us astray9 It is desirable ff we can go beyond the lowest orders. One of the ways to sum up contributions in higher orders systematically is the 1IN expansion. Unfortunately, some difficulties are found In the 1IN expansion m four dimensions [8]. We must analyze in lower dimensions HIgashijlma and Uematsu [9] examined an O(n)-symmetric W e s s - Z u m i n o model m two dimensions In the hmit 1 Work supported m part by Soryushx-Shogakkai, and the Nishma Memorial Foundation, Japan, and the NSF, Grant No PHY-7923669. 226
0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
Volume 123B, number 3,4
PHYSICS LETTERS
31 March 1983
N-~ co (at zero-temperature). They found a phase transition from the supersymmetric phase to the spontaneously broken phase [with an O(N)-smglet NG fermion]. As we shall see below, there is no such phase transition at fimte temperatures, because fimte temperature always breaks the supersymmetry. We can refer from analytlcity that at low temperatures the smglet fermlon is either always massless or always masstve lrrespectwe of whether it ~s massive or massless at zero-temperature. It seems unnatural that the massive fermmn (at zero-temperature) could suddenly become massless at fimte temperatures. One might conclude, therefore, that the smglet fermion would be always massive at low temperatures. We shall find, however, that this is not the case. A zero-mass pole is always found to contribute to the correlatmn functmn of the smglet fermion at fimte temperatures because of the (broken) supersymmetry 0rrespective of whether the supersymmetry is spontaneously broken or not at zero-temperature). Let us first describe the model we shall examine The lagrangian is [9]
.t2=fd20(¼D'~V,O#),+½Ug(~30 +UX~b0 +gqSo~,~bt), where summation over i = 1,. eters 0~ (a = 1,2).
¢,(x, o) = A,(x) +
(1)
, N is understood. Ct (i = O, 1 ..... N ) are the superfields with antlcommuting param-
O~,~(x) +7' OaO,~F,(x)
(2)
D ~ xs the covarlant derwatwe m the superspace. In terms of the component fields, .C =½ (3uA/)2 +½1 ¢i7 u 0u t~t +~ F 2 +Ng(FoA 2 _ A 0 t~Ot~O) +NXFo +g(2AoFtA, - A o ~Ot¢, +FoA 2 - 2~0 etA,) • (3) After integrating overA t, fit, and Fi, we obtain the effective lagranglan "~eff =N{Fo(gA20 + ~) -gZot~o~o - ½1 t r I n (l')'/'t 0/.t -- 2 g h 0 ) +-~l tr In [ - O2 - 4g2A 2 + 2gF 0 - 4gZff0(iTuOu - 2gA0) - I ¢0]}-
(4)
In the leading order of the I/N expansion, the effective potential is *x
Veff(A,F, r ) : V(eO)(A,F) + V(el~(A,F; V), V(eO~(A,F) = (N/8g 2) p2(m2 - / a 2) + (N/81r) [p2 ln(,2 /eM2) _ m 2 ln(m2/eM2)], V(D( eft'~A , F , T ) = - ~N ? d k { l n ( 1 - e x p [ - [ 3 ( k 2 + m 2 ) l / 2 ] ) - l n ( l +
e x p [ - f l ( k 2 +,2)1/2]}},
(5)
where 1/3 = T is the temperature, and M the mass-scale introduced an renormahzlng ~. A = (A 0 ), and F = (F0).
m 2 = 4g2A 2 _ 2gF is the squared mass of A i, and , 2 = 4g2A 2 is that of ~z. In order to find the minimum point, we differentiate Veff b y A and F. 0=
O-
0Veff(A ,F; T) _N _ , 2 +g_2 In OF =4g rr M 2 0 Veff (A ,F, T)
OA
4-g2
dk
,
(6)
7r 0 (k2+m2)l/2{exp~3(k2+m2)l/2] - 1
~2 /~2 [ - N A L m 2 - p 2 + = - ln=-rr m 2
oo
rr
(k 2 + m2) 1/2 {exp[fl(k 2 + m2) 1/2] - 1} (k 2 + . 2 ) 1 / 2 {exp[fl(k2+.2)l/2] + 1}
.1 Note that at T ~ 0 the effective potentml is not identical to the expectatmn value of the hamiltoman, and is not positive definite 227
Volume 123B, number 3,4
PHYSICS LETTERS
31 March 1983 --4 ---4 -"~ -.'~ --4 ,,., . ,, ,, OO .O .OO "--- N~ ¢..~ -~' L "
r--0
'~
/"
,\
0
/
_
<£,[5~'~"
05 I
I
Jl , 05
I
,I I0 m/(grr -I/2)
~,"
,~o
I 15
Ftg, 1 Solutions of eq (7) (solid lines) and eq (6) (dashed lines) at T = 0 The solution/a = rn of eq (7) Is supersymmetric. Crossing points give the extrema of Veff
0
05
m/lg
IO_tn)
15
Ftg 2 Solutions of eq. (7) (independent of M) and eq (6) (for M = 0.7 M c) at T = 0 1-0.5 (m umts ofg/x/~)
The curves of the solutions of eqs. (6) and (7) at T = 0 are shown m fig. 1. The points at which the curves meet give the e x t r e m a o f Veff. F o r m 2 < M c2 =g2/(rre), there are two m i m m u m points on the line/a = m (supersymmetric hne), while for M > M c the m l m m u m point is on another b r a n c h of the curve o f eq. (7) on which # 4: m. Thus analytic, ty is b r o k e n at M = M c (phase transiUon) At T v~ 0, we find s m o o t h curves o f eq. (7) w i t h o u t branch points (fig. 2). No transition takes place from the supersymmetric phase to the b r o k e n phase. We can write the effective potential in terms of A only, e l i n u n a t m g F b y the use of eq. (6). Veff(A , T) with M = 0.7 M c is depicted in fig 3 for several temperatures. The discrete chlral s y m m e t r y , which is b r o k e n spontaneously (i.e.,A :# (3) at low temperatures, xs restored above critical temperatures (fig. 4), It may be interesting to note that although at T = 0 two supersymmetric vacua are degenerate (for M < Mc) the
06
M:O7M c
04
,~,~ 02
T,
06, 04
:>-02 -04
,f 05 I0 ffllgrr-I/2): 2
15
Fig 3 The effectwe potentml Veff(A, T) for M = 0 7 M c, 228
Ch,ral-symmetrhc
02 0
I
I
L
I0
20
30
MIMc
Fig 4 The phase diagram, which shows that the chtral symmetry is restored above critical temperatures
Volume 123B, number 3,4
PHYSICS LETTERS
31 March 1983
vacuum with smaller A is always chosen when the system is cooled from high temperatures (see fig. 3). It is because the finite-temperature part V(I~ is monotonically decreasing as A decreases; this result does not depend on the details o f models and on the dimensionality of s p a c e - t i m e . Now we turn to the problem o f the self-energy 2;(p, T) of the singlet fermion 40 with momentum pU We can evaluate g (p, T) in the imaginary-time formalism. Note, however, that the "imaginary-time" itself is not a physical coordinate. Only the "equal-time functions" in this formalism are related to physical correlation functions. Therefore, we should perform a Fourier transformation after calculating Feynman graphs in momentum space. The transformation, however, includes infinite summation over the discrete energies , 2 , and the behavior of the transform as not easy to see analytically. Alternatively, we can make use of the real-time formalism, in which one analytically continues the time to the real ax~s, so that the Fourier transformation is the integral over the continuous variable. Then the position of the poles in momentum space gives information on the behavior o f the correlation functions at large distances. Making use of the usual manipulation [4], we find 0=
a Veff (A , F , T) aA
=_gNAF_8Ng2A(d2k( lgF + r r f ( k 2 - / ~ 2) + r r f ( k 2 - m 2) a(27r)z \(k2-112+le)(k2-m2+ie) e x p ( 3 1 k ° l ) + 1 e x p ( 3 l k ° l ) - 1 ] = F Z (0, T).
(8)
We thus find ~ ( 0 , T ~ 0) = 0, because F 4= 0 at finite temperatures. This result may be surprising because
Y-,(O,O) = 2gA -g/(2rrA2)=/= hm 1~(0, T) = 0, T~ 0
for 3 t < M c Note that the temperature dependent terms in the propagators vanish very rapidly as T -+ 0 like
exp(-m/T) However, we also find a factor 2gF =/12 - m 2 in the denominator of the temperature-dependent term in Z (0, T), which IS as small as exp(-m/T) (for M < Me) This factor enhances the net temperature effect to be O(1) (Note that the above result does not depend on the dlmenslonallty o f the s p a c e - t i m e . ) Of course, the zero-temperature limit should coincide with the value at zero-temperature physically. Evahiatmg Z(p, T) at pU ~ 0 (for M < Mc) , we find that at low temperatures E ( p , T) increases rapidly (proportionally to p 2 ) until it reaches the zero-temperature value at p2 ~ m 2 e x p ( - m 3 ) . We then find three poles (on the real axis) in the two-point function atp u = 0, a t p 2 ~ m 2 e x p ( - f l m ) , and a t p 2 ~ [4gA(1-4rrA2)] 2. In the limit T--> 0, the contributions of the first two poles cancel each other, and the third pole gives the same result as at T = 0 Physically speaking, the correlation function damps as e x p ( - T r ) [not as e x p ( - m r ) ] at very remote distances [r >> m - 1 exp(3m)] because o f the contribution of the pole at pU = 0. In the limit t -+ 0, however, it has already damped as e x p ( - m r ) in the region r ~< m - 1 e x p ( 3 m ) , so that the behavior like e x p ( - T r ) cannot be observed (for M < M c) We could make use o f the WT identity in the real-time formalism (not In the imaginary time formalism o f ref. [2] ) to show the necessity o f a zero-mass pole in the fermlonic correlation function. The WT identity would also show that the contribution o f the zero-mass pole vanishes rapidly (exponentially) as T ~ 0 in cases where the supersymmetry is not broken at T = 0 (i e., the expectation value of the hamiltonian vanishes exponentially as T-+ 0). On the other hand, in cases where the supersymmetry IS broken spontaneously at T = 0, the contribution o f the zero-mass pole o f the NG fermlon can be observed at any temperature. Supersymmetry thus manifests itself at finite temperatures. *2 The "energy" here is m fact the Fourmr-conjugate variable to the imaginary tune, and not the energy of the physical system 229
Volume 123B, number 3,4
PHYSICS LETTERS
31 Match 1983
I would hke to thank all the members of the theoretical group at KEK for their warm hospitality. I am indebted to S. Watamura for collaboration at the early stage of this investigation. I also thank N. Sakai and D. Friedan for critical comments, and M. Gross for reading the manuscript carefully.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
A Das and M Kaku, Phys. Rev. D18 (1978) 4540. L GlrardeUo, M.T. Gnsaru and P Salomonson, Nucl Phys. B178 (1981) 331 K. Fujlkawa, INS-report-454 (1982). See, for example, L Dolan and R Jackiw, Phys Rev D9 (1974) 3320 J Wess and B Zummo, Nucl. Phys. B78 (1974) 1. L. O'Ralfeartalgh, Nucl Phys. B96 (1975) 331. L. Van Hove, CERN preprmt TH. 3280 (1982) A. Hlguchi and Y. Kazama, Nucl. Phys. B206 (1982) 152 K Hlgashillma and T. Uematsu, Tokyo preprmt UT-385 (1982)
230