Superfield perturbation theory at non-zero temperatures

Volume 200, number 1,2

PHYSICS LETTERS B

7 January 1988

SUPERFIELD PERTURBATION THEORY AT N O N - Z E R O TEMPERATURES R. ANISHETTY, Rahul BASU and H.S. SHARATCHANDRA

The Institute of Mathematical Sciences, Madras 600 113, India Received 21 September 1987

Supergraph Feynman rules are developed for non-zero temperatures. These are used to study the implications on the effective potential and fermion mass at T¢ 0.

Superfield perturbation theory makes transparent many of the features of supersymmetric (SUSY) theories (at T = 0). For instance, in the effective action, there are no loop contributions to the F-type (i.e. 00 component of the chiral superfields) terms while contributions to the D-type (02 62 component) are at least bilinear ~t in F a n d / e [ 1 ]. This directly implies the no-renormalization theorem i.e. if there exists a SUSY minimum of the lagrangian with (A,) = ~ ( F i ) = 0 then these continue to be the minimum of the full effective potential. However, this feature is no longer true at non-zero temperatures, because both ( F i ) and (Ai) shift when loop corrections are included. In this paper, we would like to highlight these new features by developing a superfield perturbation theory for T ¢ 0. These differences and their implications are summarized below. (a) The superfield propagator is no longer local in 0. While T = 0 propagators have 6 4 ( 0 - 0 ' ) , we now get, in addition, a 6 4 ( 0 + 0 ') piece. (b) At T = 0 all superfields in the effective action have the same argument (0, 0). However, at T ¢ 0 , the superfields with arguments (0, 0) and ( - 0 , 0) appear together. (c) No-renormalization of F-terms persists in the effective action. However, at T ¢ 0, A-type (i.e. 0, 6 independent part) and V-type (i.e. Oa'nO component) terms are also generated in addition to D-terms. (d) At T = 0 , we have the Ward identiy

m,j(~)o=0,

(I)

where m o is the fermion mass matrix. Consequently if (F~)o ~ 0 then there exists a goldstino. At T ~ 0 we find mu(T) ( F j ) T ~ 0. Therefore the nonvanishing of ( F j ) v does not imply the existence of a massless fermion. We now proceed to obtain the superfield Feynman rules. The lagrangian density is given by

L=fd4Oi~=cb, cb,+(~d2OP(Cbl .... ,tbn) + h . c . ) ,

(2)

where P ( ~ l ..... ~n) is the superpotential. The partition function at temperature T= 1/fl is given by the functional integral ~ We restrict ourselvesto theories with only chiral superfields, • ,(x, 0) -={A,(x), ~'/(x), F,(x)}, i= 1,...,n.

85

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PHYSICS LETTERS B

7 January 1988

#

o

where all bosonic (fermionic) components of q~i, ~5i obey periodic (anti-periodic) boundary conditions w.r.t. Xo. These boundary conditions can be summarized as q) +i(Xo, x, 0, O)= _+~+_i(Xo +fl, x, 0, 0) ,

(4)

where

q~±i(x, 0, 0) -- ½[ q~,(x, 0, 0) + q~,(x, -0, - 6 ) ] .

(5)

We now obtain the free superfield propagator. For this we first consider the free lagrangian density for one superfield q~:

Lo=fd40~+(fd20½mCb2

+ h.c.).

(6)

Replacing q~ by (qb + + q)_ ) in eq. (6) and noting that there are no cross terms between q~+ and q)_ due to d20 integrations, we get

Lo=fdaO(,f+q~++ct,_q,_)+(fd20(½m,~2++½m, h.c.).~£)+

(7)

Since for chiral superfields [Sq~ ± =0 we may rewrite all terms in Lo as an integral over d40, viz. L o = J d 4 0 { ~ + ~ + +qS_qb_ +½m[qD+(_~D2[-]+ 1) q~+ +~O_(_ID2[Z-1) ~ _ + h.c.]}.

(8)

Here we have used the following identities [ 2]

fd4xfd40=-lfd4xd20[) I7)2D2 q) + =

2,

(9)

16E3q~+

(10)

or ~I)2DZ([-]±)-lqb+

=aiD± ,

(11)

where D , = 0/00"+ icrmc~OaO,,, and ([] ± )- 1 is the Green function for the d'alembertian operator, defined appropriately over periodic (anti-periodic) functions. The superfield Green functions

(12) for the lagrangian (8) obey

PM± G± = P 6 4 ( x - x ') 6 4 ( 0 - 0') - P 6 8 ( z - z ') ,

(12a)

where M+ = 86

1

- ¼m2I~)2 [7 + 1

"

(12b)

Volume 200, number 1,2

PHYSICS LETTERSB

7 January 1988

In eq. (12a) the chiral projection operator [2]

o)

P~a

D2

(12c)

'

and 64+ ( O - - 0 ' ) ~ 1 [ 6 4 ( 0 _ _ 0 , ) -}-84(0.1_0,)] .

(12d)

Solving (12a), a+-

,

,

[2+-m 2

Im2132/[2+

)

88+(z-z')"

(13)

From (12) and (13), and noting that ( qb+ q~_ ) = 0, we write down the ( q~ qb ) Green function as

(00)=(0+0+)+(0

0_),

(14)

and other such analogues ( 0 replaced by 05 etc.). Summarizing the various propagators, we have

,(,[~+-m 2 + D-

= ~ 1 (1U]+ + ~ _m 2

[]_ -ml 2) 84(x-x')64(O+O')

ZI+(X--X') 8 4 ( 0 - - 0 (q~(z) ~ ( z ' ) ) = l m D

m 2 64(x-x')84(O-O')

,) -t- Z~(X--X') 84(0"~-0 ') ,

1 2 ~1 ( rT+ -1m 2 [2+

1( 1 1 +¼ m D 2 ~ [ - ] + _ m 2 []+

~--m2

=- ImD2[Ag(x-x ') 8 4 ( 0 - 0

') +

1

(15)

+ []_ ~ 1 --m __1 gq_ )84(x_x,)84(O_O, ) 1)84(X_X,)64(Oq_O,)

El_

A'(x-x')

84(Ont-O')] .

(16)

The Feynman rules for the vertices [ 1,2] are identical to those at T=0. The major qualitative difference in the Feynman rules compared to T = 0 is that the superfield propagators (15), (16) have a 84(0 + 0') piece. Therefore the thermal propagators are no longer local in superspace. We now consider the structure of the effective action. To this end, we first study the 0 dependence of an arbitrary 1PI vertex which arises from propagator factors 8i~ -=8(0, + 0j), D 2 factors from vertices or propagators, external superfield wave functions and d40i integrations. Consider any propagator joining two particular vertices Vj and V2. We may integrate by parts to remove all the D's acting on the corresponding 6 function [ 1 ] just as at T = 0. If there is more than one propagator joining the same vertices we may simplify the contributions using identities such as [2] 6if, D2D26~ =8~-1 , 8~,D28f,

= - 4 6 ; 1 e x p ( - - 2 i 0 t ~ 0 ~ O,),

8~ D 2I) 265 = 168~, exp ( - 4i0, cr~0 1 0 , ) ,

(17a) 8~117)28fl =

--48~ exp(2i0~a"01O,),

8~1 I) 2D 282~ = 168~ exp (4i0 ~~" ~q,O~).

(17b, c) (17d, e)

We perform the 02 integral using the 6(0 ~-T-02) factor so that 02 is everywhere replaced by + 01. Continue this procedure by considering all propagators joining V2 to a new vertex V3, V3 to V4 and so on till all 8 functions 87

Volume 200, number 1,2

PHYSICS LETTERS B

7 January 1988

have been removed and all 0 integrals, except 01, have been performed. All this is exactly as in the T = 0 case, except for the explicit 0 ~ dependence like exp ( - 2i0 1a" 61 0n) because of (1 7). We have now ended up with a d401 integral with sometimes exponential factors exp(4i0~ an0~ 0n) and external fields ( ~ , 45, Dq~, I)45, D2q), I)245) at various different space-time points with, however, the 0 arguments being either 0 1, or - 0 1. We may now perform the d40 ~ integration. If we consider those contributions wherein every propagator has either only a fi+ or only a fi- factor then there is no explicit exp(4i0~ an0~ 0n) factor, and so we get the D~omponent of the product o f superfields ( ~ , 45, Dtb, D~,...) as in the T = 0 theory. On the other hand, if both + and ~ - factors are present, then we have a exp (4i0a n00n) - 1 +_4i0a n 00n + 1 0 2 ~2 [] factor in general, multiplying superfield wave functions. Therefore, in addition to the D-term, we now get V (i.e. 0a 'n 6 component) or A (i.e. independent of 0, 0)-type components of superfields. This is the qualitative difference with the situation at zero temperature. We are now in a position to analyze the consequences. The effective action has the general form

F= f d 4 x ( f d 2 O p + ; d 2 0 " + fd40 ~= q)~45~+~d40D+ fd400crmOvm+ fd400202~),

(18)

where D, V .... N are temperature dependent functionals of superfields @ ( x , +_0, + 6 ) , 4 5 . D ~ @ , Da q~ , D : q~s, 0 2 45~, all with the same 0 but different space-time coordinates. The first three terms on the RHS are the tree contributions. As we are interested in the effective potential and fermion mass term ~2, we may ignore all derivatives o f fields and therefore make the replacement

q~--,A+x/2Oq/+O2V,

D~x/2N~+20~F,

D2~--*2F,

(19)

with corresponding expressions for conjugate fields. We need terms at most quadratic in either g¢ or g). It is useful to make a Taylor expansion in the arguments D~q~, I) a 45. D(tZ~, 45, D2 ti~, ['-)245, Da@, Da@)

.._, K(D) + ( q//x/2 + OV)( ~U/x/2+ OF) L (D) + (q)/xf2 + 0 f f ) ( ~ / x / 2

+ 0F)

M (D)

+ (~t/V/2+OF)(qz/x/2+OF)(¢t/x/2+OI?)(gt/x/2+OF) N (D) + .... V,~(...)--,( q//xf2 +OF) am( ¢Z/xf2 +OF) R 'v) + ....

(20)

Here we have suppressed the integrations over space-time arguments x and the internal indices. K¢m, L (D), M (D) and N ~D) are functions of ~ , 45 ,D 2 t]5, I) 2 45 only (and not of D~tb and I)a 45). There are similar expressions for ~ with the (D) in the functions K(D),...,N (D) replaced by superscript (A). First we shall look at the effective scalar potential V~ff. On performing the 0 integration, we get

V~ff(A,A,F,F)=FF+FP'+Ffi'+FFKIDo)o+FZFL~Do)o+[~FM~t~o+~FZN~Ddo+ZFFR~Vdo+K~Ado,

(21)

where K~00 stands for the first derivative of K(A, A, F, F) w.r.t, the first two arguments A and ,4, with no differentiation involving the last two arguments. We have used a similar notation for other objects involving subscripts. The four three-terms on the R H S are the tree contributions. The next four terms arise out of the loop contribution of the D-type which are present in the T = 0 case also. These terms are all at least bilinear in F and ~2In the real-time formalism the fermion masses would correspond to poles in E of the propagator St(E, p) for momentum p-~0. By the fermion mass matrix here we mean S ~ t ( E = 0, p = 0). This is sufficient to check for massless particles. Extracting fermion masses from the imaginary-time propagator is more delicate because pO is discrete. However, the propagator has a unique continuation into the real-time propagator. We may expect non-derivative terms of the type q/(xo, x) qZ(Xo,x) in the imaginary-time formalism to correspond to S~-~(E=O, p=O). 88

Volume 200, number 1,2

PHYSICS LETTERS B

7 January 1988

F. However, the last two terms coming from A- and V-type are new to the T # 0 case. It is non-zero (in general) even when F = F = 0 . Therefore F = 0 is no longer a minimum of the full effective potential, even if it is for T=0. To obtain the fermion mass matrix, we collect terms quadratic in ~u and independent of ~. These have the form (D) -- L 6Ado . mr(A, A, F, F) = P" + FK~~do+ FFL ~Ddo+ F2N~Do~o+ Fff2 N.ooo

(22)

The fermion mass matrix is then mr(A-r, Ax, FT, Fv) where the arguments of mr denote thermal expectation values. At T = 0 , if SUSY is unbroken then F = 0 and hence mf=P" to all orders. In addition, at T = 0 , K, R and L terms with superscripts (V) and (A) are absent in eqs. (21) and (22), respectively. Notice that we then get the well-known Ward identity

mr(A,A,F,F, T = 0 ) F=OVefj-(A,.4, F,F, T=O)/OA ,

(23)

which implies a massless goldstino once o#0. At T # 0 , eq. (23) is no longer valid. Hence at non-zero temperatures, FT # 0 does not necessarily imply the existence of a massless fermion [ 3]. Adopting the view that any SUSY breaking order parameter at T ¢ 0 must signal the existence or non-existence of the goldstino, we find that Fv cannot be an order parameter. We now demonstrate with an explicit example, that the/~-term is non-zero at T ¢ 0. Consider a term in the one-loop contribution to the effective potential shown in fig. 1. Using our supergraph Feynman rules and noting that only terms which contain both 6 + and 6 - give an A-term contribution, we find a term of the form

fd4xi d4x2 d40, d402t~(x,, 0,)[A+~i-2I]~A_t~+2U 2 -]'zJ+t~+2[)2A_t~-2D2]~(x2, 02).

(24)

With the help of the delta function identities in eq. (17), this reduces to

fd4x,

d4x2 d 4 0

fl)(xl, O) ,J+ exp(4iOa"O0,,) A_[~(x2, O)+c~(x2,

-0)] .

(25)

There is clearly a non-vanishing contribution (const.) AA to K in Verf, from the above expression. Consequently 0 Vef~/OAhas a term proportional to A and not F. This implies it can never satisfy a Ward identity like (23) since the LHS of (23) is proportional to F. In the literature [4], it has been erroneously claimed that SUSY transformations in the partition function implies eq. (23) even at T # 0. It is not possible to make the usual SUSY transformation becauseA and ~u satisfy different boundary conditions so that A +x/2e~/ has no meaning. However, by considering transformations with the parameter e as x-dependent and with antiperiodic boundary conditions, we obtain

01,]"~-JA~ j + x/2 j~, j--ije.~
(26)

wherejA,j~,,jv are the sources forA, ~, and Frespectively and < >j represents expectation values in the presence of these sources. S~, is the SUSY current. This equation may be converted into a functional identity for F(A, ~,, F), the generating functional for 1PI vertices viz.

O,,
(27)

Fig. 1.

89

Volume 200, number 1,2

PHYSICS LETTERS B

7 January 1988

We m a y try to get a n i d e n t i t y for mf b y m a k i n g a f u n c t i o n a l d i f f e r e n t i a t i o n w.r.t, q/(~0) a n d switching off the f e r m i o n i c sources (i.e. ~u, ~ = 0 ) Using

0 - _ ~ f-4 0Jo(c°) Og/(~a) (Su(x) ) J - - L J a w o--~'~ (¢(co) S,,(x) )j,

(28)

where the s u m m a t i o n Z is over all fields 0, we get

f

02F ov/(o2)

o, a ~ , ( x )

v,=o 0 ~ u

021" + F(X) 0~,(~o) O~t(x)

(S~,(x)

~u=O

~OF v/=o fi(x-~o) ~(a)) >J~=°+ ' / 2 v..,~,

-- i"/2~mOmA(X) O~(~O) Oy2(X) (~2/~

~'=0 = 0

"

(29)

B o y a n o v s k y [4 ] argues that the first t e r m o n the L H S o f ( 2 9 ) b e i n g a total derivative, drops out, o n i n t e g r a t i n g over x a n d he thus recovers eq. ( 2 3 ) o n restricting h i m s e l f to c o n s t a n t A a n d F fields. H o w e v e r , this t e r m cannot be d r o p p e d b e c a u s e (a) there w o u l d be a massless f e r m i o n c o n t r i b u t i n g to (Sv(x) v/(co)} a n d ( b ) e v e n is there is n o massless f e r m i o n , (O/Ox~) (Su(x) ~,(o)) } is a n t i p e r i o d i c in x a n d therefore there are b o u n d a r y cont r i b u t i o n s . In fact, k e e p i n g this t e r m gives us a t a u t o l o g y b e c a u s e (O/Ox,)(S,,(x) ~ , ( ~ 0 ) ) = F ( x ) a ( x - ~ o ) , so that n o useful c o n c l u s i o n can be o b t a i n e d ~3 ~3 Strictly speaking, the fermions in the functional integral should have anti-periodic boundary conditions even at T=0 and then, any attempt at reaching any implications on C would suffer from tautologies, as here. However, correlation functions are now expected to be insensitive to the boundary conditions so that we may use periodic boundary conditions for fermions and obtain useful identities forF.

References [ 1] See e.g., S.J. Gates, M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin Cummings, Menlo Park, CA, 1983). [2] J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton, U.P., Princeton, NJ, 1983). [3] L. Girardello, M.T. Grisaru and P. Salomonson, Nucl. Phys. B 178 (1981) 33l. [4] D. Boyanovsky, Physica D 15 (1985) 152; Phys. Rev. D 29 (1984) 743. [5] H. Matsumoto, M. Nakahara, Y. Nakano and H. Umezawa, Physics D 15 (1985) 163; Phys. D 29 (1984) 2838. [ 6 ] R. Kubo, M. Toda and N. Hashitsume, Statistical physics II (Springer, Berlin, 1985 ) p. 233.

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