(1+1)-dimensional supersymmetry at finite temperature. A variational approach

Volume 251, number 4

PHYSICS LETTERS B

29 November 1990

( 1 + 1 )-dimensional supersymmetry at finite temperature. A variational approach A. Mishra a, H. Mishra b, S.P. Misra b and S.N. Nayak b a Physics Department, Utkal University, Bhubaneswar 751004, India b Institute o f Physics, Bhubaneswar 751005, India

Received 4 June 1990

We study a ( 1+ 1)-dimensional supersymmetric model with Z(2 ) symmetry at finite temperature with a variational method. For this purpose we use the thermofield dynamics developed by Umezawa et al. and show that at any finite temperature supersymmetry is broken, and that Z(2) symmetry is restored for temperatures greater than a critical temperature T¢.

1. Introduction S u p e r s y m m e t r y ( S U S Y ) [ 1 ] at finite t e m p e r a t u r e has considerable interest, since this s y m m e t r y is believed to be good at high energy scales. The work o f Das and K a k u [ 2 ] was the first to illustrate the spontaneous b r e a k d o w n o f s u p e r s y m m e t r y at finite temperature. In this letter we analyze s y m m e t r y breaking in a ( 1 + 1 )-dimensional supersymmetric model with Z ( 2 ) s y m m e t r y as an illustration o f v a c u u m destabilization through a variational method. The p a p e r is organized as follows. In section 2, we study s y m m e t r y breaking with an explicit construction o f v a c u u m at zero temperature. Here Z ( 2 ) symmetry is broken, but s u p e r s y m m e t r y remains unbroken. In section 3, we consider the above at finite t e m p e r a t u r e with the help o f thermofield d y n a m i c s [ 3 ] with a self-consistent m e t h o d o f calculation for the particle masses and the effective potential. We note that s u p e r s y m m e t r y is broken at any finite temperature, and, at a critical t e m p e r a t u r e To, Z ( 2 ) symmetry is restored. Section 4 contains the results and discussions, along with the identification o f goldstino modes for t e m p e r a t u r e s below and above To.

checked by the action o f the SUSY generators on the vacuum. I f SUSY generators Q,~ annihilate the vacuum, that is Q ~ I 0 ) = 0, then s u p e r s y m m e t r y is not broken. On the other hand, i f Q~ I 0 ) # 0, then SUSY is broken spontaneously. In supersymmetric theories, from the graded algebra we know that Q 2 = H . Thus we can regard, as is well known, that ( 01 HI 0 ) is an order parameter, which when nonzero indicates that s u p e r s y m m e t r y is broken [4,5]. Also with F as the auxiliary field, the supersymmetric transformation a¢~u= ( F ) ~¢: 0

(1)

implies that s u p e r s y m m e t r y is broken [5] when (F) #0. S u p e r s y m m e t r y usually means fermion and boson mass degeneracy. Any departure from this degeneracy implies b r e a k d o w n o f supersymmetry. However, s u p e r s y m m e t r y m a y be b r o k e n without mass degeneracy being lifted [ 6 ], as will be seen later. We shall now study s u p e r s y m m e t r y at zero temperature with a variational ansatz for the construction o f the v a c u u m in a simple model before proceeding to the m o r e c o m p l i c a t e d calculations at finite temperature. We take the lagrangian density as [ 7 ]

2. Supersymmetry at zero temperature W h e t h e r s u p e r s y m m e t r y is broken or not can be 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

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For the real Majorana spinor g/, we take [9] the quantization with constraints as

Le = -~ 1 ((Ou(9)2+iffTuOug/+F2_x/~2FS(q))

@~),

~/z7 dS(¢)

(2)

where ~ is a real scalar field, ~ is a real fermionic Majorana field, F is the auxiliary field and S(0) is any well behaved function of ~. For the y-matrices, we take the representation y0=0" 2 and y~ =ith. The above supersymmetric lagrangian density can be constructed from a single superfield q~(q~, ~, F) by the superfield formalism [ 1,8 ]. On elimination of the auxiliary field by the Euler-Lagrange equation, we get F = x/~-~2S(O ) .

(3)

Then by choosing [ 8 ] S(O)

1(

~G=

- ~ [U~(k)d(k) exp(ikx-iEt)

+ V,~(k)d(k)* e x p ( - i k x + i E t ) ] dk, U,~(k)*g#(k)+V.(-k)*V#(-k)=2EO~,tj,

) "

(5)

along with the quantization conditions [ 9 ] [d(k), d(k' )*] + =4nEa(k-k' ).

(11 )

v(k)=( (E+k),,2

.-i(E-k)~/2 ] ,

with E = (m ~ + k 2) 1/2, but otherwise, E is arbitrary. We shall now consider symmetry breaking as an example of vacuum destabilization. Thus we take [vac' ) = U(~)[vac) ,

We shall now consider symmetry breaking with the lagrangian density as in eq. (5) as an example of vacuum destabilization through a variational method instead of treating the problem in a classical manner. Here we shall deal with a nonperturbative calculation considering only equal time quantum algebra. For this purpose we take the field expansions for the bosonic field q~and its conjugate momentum, 0~at time t = 0 as 1

0 ( z ) = 2 x / ~ ~ [a(z)+a(z) t]

=(~(z)an+O(z) c~, 0(z) = i x/½og~ [-a(z)+a(z)*],

[a(z), a(z' )*] = ~ ( z - z ' ) .

(13)

where we take a coherent state type of construction [ 10] for U(¢) given as

U(~)=exp@ f dzx/½ogz [ a ( z ) * - a ( z ) ] ) .

(14)

We shall determine ~ in U(~) through a variational procedure. The perturbative vacuum [vac) is annihilated by a(z) and d(k). We note that for the coherent state type of construction for [vac' ) as above, we have

(6a)

(vac' I¢lvac' ) =~.

(6b)

The field operator ¢' corresponding to Ivac' ) is now given as

where the operators a(z) and a(z) t satisfy the commutation algebra (7)

Clearly, in eq. (6), o~ is arbitrary. However, if we take a free field basis, then o 9 ~ = [ - ( 0 / 0 z ) 2 +

542

(10)

(12)

2-2--x/~q~¢~'

m~] 1/2

(9)

where

U ( k ) = \i(E_k)l/2 j ,

( 0 u 0 ) 2 a r i ~ u 0 u I//-½j'~4+m2q~2

m" -

(8)

This is consistent with the field operator expansion

(4)

'

with m 2 > 0, the lagrangian density becomes ~= 2

[~,.(x, 0), ~,p(y, 0)1+ = ~ ( x - y ) .

The solutions for the spinors U(k) and V(k) of eq. (9) for the free Dirac equation are given by

m2

=0 2- - 2-

29 November 1990

(b' = U ( ~)(~U(~) - ' = ~ - ~ .

(15)

(16)

We wish to examine whether [vac' ) is energetically more favoured than Ivac). Using eq. (5), the hamiltonian density 3 .o0 becomes y-oo= ~ o + ~jvo + #-~o,

(17)

Volume 25 l, number 4

where

PHYSICS LETTERS B

1(

29 November 1990

some self-consistency requirements [ 11 ] as shown below.

m4)

~-°a°= ~ 0 ( - V 2 ) 0 + ~ 2 + ½ 2 0 4 - m 2 ~ 2+ -~- ,

(18a) ~-°° = - ½iff~,l8, ~u, ~~i~°= V / ~ ~Tqu¢.

Thus the effective potential at zero temperature is given as V(O---- (vac' 19-°° Ivac' ) m4

= ~2~4- ½mZ~2+ ~ - .

(19)

The parameter ~ is then determined by minimizing the energy density as ~ ~"~"~min =

•

(20)

The boson mass is given by m g --

,

3. Supersymmetry at finite temperature

(18b,c) We shall now consider the dynamics at finite temperature with the help of the thermofield dynamics method of Umezawa et al. [ 3 ]. Temperature dependent field theory with thermofield dynamics involves a doubling of Hilbert space together with a definition of thermal vacuum where the expectation value of any operator with respect to the thermal vacuum is the same as the conventional thermal expectation value. The thermal vacuum here is obtained from the zero temperature vacuum by a Bogoliubov transformation in the extended Hilbert space. The ansatz for the thermal vacuum Ivac'; fl) is given as [vac';fl) = (21a)

Ua(fl)UF(fl)Ivac'

) ,

(25)

with

d ' ~ ~ ~min

and by identifying the coefficient of ~ in the effective lagrangian density, the fermion mass is given by mv = ~ / ~

~min ,

(21b)

Us(fl)=exp ( f g( k, fl) × [a'(k)*a'(-k)*-a'(-k)a'(k)]

dk), /

(26a)

such that

ma=.v/2 m=mv .

(22)

We note that in fact the effective lagrangian density is given as ~eff= U(~min)~U(~min) -I ,

(23)

which is equivalent to the replacement of 0 = ~min+ ¢' as is usually taken. Clearly, Z (2) symmetry is broken and symmetry breaking has been associated explicitly with vacuum destabilization. From eqs. (3) and (4) the vacuum expectation value of the auxiliary field is given by (vac' IFIvac' ) =

UF(fl) =exp ( ~ f( k, fl)

(24)

so that supersymmetry is unbroken. We shall next discuss symmetry breaking at finite temperature, with a new "vacuum state", which now becomes temperature dependent. We shall use the thermofield method of Umezawa [3] and utilize

× [d(k)*_d(-k)*-d(-k)d(k) ] dk).

(26b)

In the above a"s are the annihilation operators corresponding to the field 0' parallel to eqs. (6) and a' (k) and _' (k)* are the new annihilation and creation operators for the bosonic sector and _d(k) and _d(k)* are the new annihilation and creation operators for the fermionic sector. Now the bosonic operators that annihilate the thermal vacuum [vac'; fl) are given by

a' (k, fl) = Ua(fl)a' (k) U a ( f l ) - I , a_' (k, fl)= Ua (fl)a_' (k)UB(fl) -1 .

(27a) (27b)

On explicit evaluation, we see that the temperature 543

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independent bosonic operators are related to temperature dependent bosonic operators through the Bogoliubov transformation

a'(k) ~ (coshg(k, fl) a ' ( - k ) * J = ksinhg(k, fl)

sinhg(k, fl)~ coshg(k, fl)J

29 November 1990

From thermofield dynamics method [ 3 ] 1

sinh2g(k' fl) = exp [rico(k, fl) ] - 1 ' where

¢a( k, fl) =x/k2 + ms(fl) 2 . x \ a ' ( - k , fl)tJ"

(28)

Similarly for the fermionic sector the operators annihilating the thermal vacuum are given as

d(k, fl) = U~(fl)d(k) U F ( f l ) - ' ,

(29a)

d(k, fl) = UF(fl)d(k) UF(fl)-'

(29b)

By explicit evaluation, we see that the temperature independent operators are related to temperature dependent operators through the Bogoliubov transformation

d(k) ~ ( cosf(k, fl) d ( - k ) t J = \ - s i n f ( k , fl)

(34)

(35)

Here ms(fl) is the effective mass of the boson at finite temperature to be determined selfconsistently. Similarly for the evaluation of the expectation values of g-OF° and Ti°° given by eqs. (18b) and (32), the following formulae will be useful:

( vac' ; fl[d( k ) *d( k' ) lvac' ; fl) =sinZf( k, fl) 4nE6( k - k ' ) ,

(36a)

(vac';flld(k)d(k') Ivac'; fl) = 0 ,

(36b)

(vac';flld(k)*d(k')*lvac'; fl) = 0 ,

(36c)

which are obtained by using the Bogoliubov transformation given in eq. (30). In the above [3],

sinf(k, fl) cosf(k, fl)J

1

sin2f(k' fi) = exp[fl~(k, fl) ] + 1 '

x kd(-k,P)V

"

(37)

(30) where

Our next job is to evaluate the finite temperature correction to the effective potential given by eq. ( 19 ). For this purpose we need to calculate the expectation value of ~-oo with respect to the thermal vacuum Ivac ' , fl). Using eqs. (16) and ( 18 ) the expressions for yOB0and 9"~m°tbecome ~BO ~. 1 ~ , ( __V2)~, "t- ½~'2"Jt" 1/I,{]}'4"Jl-~¢~2~'3

m 4

9 i ° = x / ~ (O'qJ~+¢q/¢/).

V(~,fl) = ~

-o~ co(k, fl){exp [rico(k, fl) ] - l}

(32)

( v a c ' ; f l l a ' ( k ) a ' ( k ' ) Ivac';fl) = 0 ,

(33a)

( v a c ' ; ilia' (k)*a' (k')* Ivac';fl) = 0 ,

(33b)

( v a c ' ; f l l a ' (k)*a' ( k ' ) Ivac';fl)

544

31¢'~2-m2+k2+co(k'fl)2 co(k, fl){exp[fico(k, fl) ] _ l } dk

(31)

For the evaluation of the expectation value of d-°B° with respect to Ivac', fl) the following formulae will be useful. Using eq. (28), we have

=sinheg( k, fl) ,~(k - k ' ) .

(38)

Here mv(fl) is the effective mass of the fermion at finite temperature to be determined in a self-consistent manner. Thus by using eqs. (18b), (31), (32), (33) and (36), the effective potential at finite temperature is given as

1 i

4- ~ ( 3 ~ 2 - - m2)0'2 q- (,~.~3--m2~)~'+ 12~4 - ½m2¢2+ ~ - ,

e( k, fl) =x/k2 + mF(fl) 2 .

l 4 - ~m2~ 1 + ~2~ 2

m4 l ~ k2dk +-42 + ~ J e(k, fl){exp[fle(k, fl)]+l} +x/~

¢ m~(#)

--or

E(k,/~){exp[Be(k, fi) ] + 1} " (39)

(33c)

We note that we have only two-dimensional param-

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29 November 1990

eters m and 2 in the lagrangian and further the temperature fl- l is the third dimensional parameter. For numerical calculations, let us write the above equation in terms o f dimensionless parameters such as ~ , ' = J . / m 2, / , t B ( f l ) = m B ( f l ) / m , # F ( f l ) = m F ( f l ) / m , y=flm and x=k/m, so that

tive potential as a function o f ( ~ ) and also the auxiliary field VEV at any given temperature. The results are quoted below.

V(~,fl)= ( ½ I i ( y ) + ~ 2 '

We plot V(~, fl) as a function of ~ for different values of temperature for 2 = 0 . 5 in fig. 1. Since V(~, fl) > 0 for all ~, supersymmetry is broken at any finite temperature. We also find that there is a critical temperature Tc at which the Z ( 2 ) symmetry is restored as is clear from the change in the shape o f the potential. In fig. 2, we plot the masses o f the fermion, o f the boson and the vacuum expectation value o f the auxiliary field F as functions of temperature in curves I, II and III, respectively. From curve III, we see that as T ~ 0 , ( F ) ~ 0 , meaning that SUSY is unbroken for T = 0 and that SUSY is broken at any finite temperature. From the curves I and II, which are the same

I 2 ( y ) 2 + ~,~.'~4-- ½~2

1

+ -4~ + I3(y)+h(y)

) m 2,

(4o)

where

1T I1 (Y) = ~ J o

h(y)-- ~

o

1i I3(Y) = ~

32'~2-1+x2+to(x'Y)2 to(x,y){exp[yto(x,y)]_l}dX,

(41a)

to(x,y){exp[yto(x,y)l-1}'

(41b)

x2dx

E(x,y){exp[y~(x,y)]+l}'

4. Discussions

(41c)

0

0

E(x, y){exp[ye(x, y) ] + 1} " (41d)

In the above for bosons and fermions we have substituted

to(x, y) = x / x 2 + / t a ( y ) 2 ,

(42a)

~(x, y) = x / X 2 + # F ( y ) 2 .

(42b)

For a specific coupling 2, we minimize the effective potential at finite temperature through a self-consistent method. For this purpose, we take some initial values f o r / t a ( y ) and ~F(Y) and minimize V(~,fl) with respect to ~. Then corresponding to the value of ~min for which V(~, fl) is m i n i m u m we calculate/~a (y) and fiE(Y) by using eq. (21). We take these as initial values for boson and fermion masses for the next step and repeat the operation. This iteration procedure is repeated for a self-consistent numerical evaluation of the effective masses until the output masses become the input masses. After determining the boson and fermion masses in such a manner we derive the effec-

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

Fig. 1. We plot here the effective potential V(~, fl) in units ofm 2 against ~ for different values of temperature. The change in the shape of the potential may he noted.

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2.0

I.S

1.0

0.5

-fi o, g >*

0.0

-0.5

txl -1.0

-"%o'.~

i

'

o'.~'

,io'

,i~

'

,.~

t

~'.2 ' 2.~

Temperature 3"

Fig. 2. The fermion mass and the boson mass, both in units o f m are plotted as functions of temperature T in units of rn as curves I and II, respectively. In curve III, we plot the vacuum expectation value of auxiliary field F as a function of temperature T, both in units ofm. All the curves indicate that T¢~ 1.8m.

for T~< To, we see that the masses of the fermion and boson are the same although supersymmetry is broken. At T-- To, the masses of the fermion and boson are both zero. Also, from curves I and II, we see that for T> To, the fermion mass remains zero whereas the boson mass continuously increases. We plot the energy density as a function of temperature in fig. 3, which is nonzero, meaning that SUSY is broken at finite temperature. As expected, the energy density increases with the temperature. We note that the supersymmetric charge Q is given

0.0

0.2

0.6

1.0 1.4 Temperature

1.8

2.2

2.6

T

Fig. 3. The effective potential after minimization is plotted as a function of temperature T, both expressed in units of m.

thermal fermion of equal and opposite momenta. We recall that in thermofield dynamics, the hamiltonian for thermal modes has opposite sign to that of normal bosons and fermions [ 3 ]. Hence for T < To, the super pairs as above have zero energy due to mass degeneracy for fermions and bosons, and thus will give rise to a modified version of the Goldstone theorem in the absence of Lorentz invariance [ 6 ]. However, for T > To, the fermion becomes massless, and we have a conventional goldstino particle.

as

Q~ = f dx [ ~ ( 0 o q ~ ) - (T°71~t)~(0~) +iF(T°q/),~].

Acknowledgement

(43) The following observations regarding supersymmetry breaking and the presence ofgoldstino modes may be relevant. During symmetry breaking, Q~ [vac'; fl) 0 yields the goldstino mode. With the Bogoliubov transformations at finite temperature, it is clear that Q,~ Ivac'; fl) consists of super pairs [ 6,12 ] consisting of a fermion and a thermal boson or a boson and a 546

The authors are thankful to N. Barik, Snigdha Mishra, Susama Mishra, P.K. Panda for manY useful discussions. One of the authors (A.M.) would like to acknowledge the Council of Scientific and Industrial Research (CSIR) for a fellowship.

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References [ 1 ] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton U.P., Princeton, 1983). [2] A. Das and M. Kaku, Phys. Rev. D 18 ( 1978 ) 4540. [ 3 ] H. Umezawa, H. Matsumoto and M. Tachiki, Thermofield dynamics and condensed states (North-Holland, Amsterdam, 1982). [4] A. Das and V.S. Mathur, Indian J. Phys. 61 B (1987) 214; Phys. Rev. D 35 (1987) 2053. [5] H.P. Nilles, Phys. Rep. 110 (1984) 1.

29 November 1990

[ 6 ] H. Matsumoto, M. Nakahara, Y. Nakano and H. Umezawa, Phys. Rev. D 29 (1984) 2838. [ 7 ] T. Murphy and L. O'Raifeartaigh, Nucl. Phys. B 218 (1983) 484. [ 8 ] L.J. Boya and J. Casahorran, preprint DFTUZ88-19. [9] M.G. Mitchard, A.C. Davis and A.J. Macfarlane, preprint KUNS 966, HE (TH) 89/02. [10] S.P. Misra, Phys. Rev. D 35 (1987) 2607. [ 11 ] H. Mishra and S.P. Misra, Gross-Neveu model at finite temperature, submitted. [ 12 ] A. Das, Physica A 158 (1989) 1.

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