Z. Phys. C - Particles and Fields 36, 489~-94 (1987) zo,so.n. P a r t i c l e s for Physik C

and Fiek:ts �9 Springer-Verlag 1987

Dynamical breakdown of supersymmetry and chiral symmetry in 1 + 1 dimensions*

Y.-P. Yi 1 and Y.-P. Kuang 2 1 Department of Physics, Tsinghua University, Beijing, China 2 Center of Theoretical Physics, CCAST (World Laboratory), Beijing, and, Department of Physics, Tsinghua University, and Institute of Theoretical Physics, Academia Sinica, Beijing, China

Received 22 June 1987

Abstract. Dynamical breakdown of supersymmetry and chiral symmetry is studied through a 1 + 1 dimen- sional field theoretical model in the large-N limit in 1IN expansion. The study is based on the calculation of bilinear field condensates in an effective potential approach. It is shown that the condition for super- symmetry breaking is related to the values of the parameters in the model and chiral symmetry breaking exists in most cases.

Supersymmetry and chiral symmetry are two beautiful ideas related to most of the new developments in modern particle theory. The real world does not actually respect supersymmetry, so that the study of spontaneous supersymmetry breaking mechanisms is of special importance. In perturbation theory, radia- tive correction does not induce supersymmetry break- ing due to the non-renormalization theorem. Therefore more attention has been paid to the investigation of whether supersymmetry can be broken non-perturba- tively, for example, through dynamical symmetry breaking. Dynamical breakdown of chiral symmetry is also a very interesting problem since the hadron world seems to reflect this broken symmetry while in the preon world (composite models of quarks and leptons) certain unbroken chiral symmetry is needed for a natural explanation of the light masses [1]. Various investigations show that dynamical chiral symmetry breaking seems to take place in QCD-like theories [2]. Whether this also happens in chiral gauge theories and supersymmetric gauge theories is still under investigation [3]. Dynamical symmetry break- ing is a long standing difficult problem due to the mathematical complexity. For the sake of simplicity,

* Work supported by the Science Fund of the Chinese Academy of Science

it is often studied in 1 + 1 dimensional field theoretical models [4-8]. Although such models are not realistic themselves, the study has proved to be useful since such models have all the interesting features as re- normalizability, asymptotic freedom and the desired symmetries analogous to the four dimensional realistic theory. Also 1 + 1 dimensional field theory is closely related to the currently interesting superstring theory which is regarded as a promising theory for the unification of fundamental interactions.

Let us consider a 1 + 1 dimensional model with an O(N) symmetry and study non-perturbative spontaneous symmetry breakings in the large-N limit in 1/N expansion. The simplest supersymmetric model consists of only an O(N) vector scalar superfield

q~i=Ai+ffT,+�89 i= 1,.. . ,N. (1)

and an O(N) singlet scalar superfield

~o = Ao + 0~o + lOOFo, (2)

where 0 is the anticommuting coordinate, A~Fi, Ao, F o are real scalars and 0, ~ , To are Majorana spinors. The Lagrangian can be written as

~a = [ � 8 9 + �89

+ W(r ~3]./~o (3)

where

0 0 D~ - Off~ i(7~'O)~ox" (4)

and the superpotential W(~o, ~i) is taken to be

W(~o, ~i) = - �89 ~o ~i ~i - ~h ~o ~o ~o (5)

with dimensionful coupling constants g and h. When h = 0 this model reduces to the first model of [7]. In

490

terms of the component fields, (3) can be written as �9 i r 1 6 2

= �89 o + {O,A f f 'A , + ~ o r ~Po +

1 2 1 !-'2 1; ~ 2 + hAZ)Fo gAoAiFi + ~Fo + ~ri -- ~ g n i

+ �89189 (6)

Equation (3) is manifestly supersymmetric and O(N) invariant. Furthermore, we can see from (6) that A ~ has the following discrete chiral symmetry

A i ---~ - - A i , tI'ti --* 7 5 t~i , F i ---~ F i (7)

A o ~ - A o, ~ o - ~ 7 ~ u o, F o ~ F o.

Supersymmetry will be broken if some of the F-terms

( CI)0 )(1/2)00 = V o

(q~o 4'o)~v~o = 2AoFo - q'o %

(~i ~i)(1/2)oo = 2AiFi - ~'i ~i (8)

develop nonvanishing vacuum expectation values (VEV). Nonvanishing VEV of ~ ~ will break the chiral symmetry (7) dynamically�9 Here we only consi- der VEV's of O(N) singlet operators since it has been pointed out by Coleman [9] that in 1 + 1 dimensions the continuous symmetry O(N) cannot be broken.

Davis, Salomonson and van Holten [7] studied the effective potential as function of the classical fields of A o and F o in the h -- 0 model and got the conclusion that supersymmetry is unbroken from the vanishing of the VEV of F o. As we have seen, a complete discussion should also include the calculation of VEV's of other F-terms in (8). Furthermore, Fo can be seen as the auxiliary field related to bilinear scalar field operators in the sense of dynamical symmetry breaking as can be seen from the Euler-Lagrange equation of Fo:

F o = �89 + hA2). (9)

Hence the calculation in [7] concerns only composite operators of scalar fields and thus gives no information about chiral symmetry breaking. In this paper, we will give an improved discussion by calculating also the VEV's of (q~o q~o)(1/2)~0 and ((1)i(I)i)(1/2)~ O. Since this contains the knowledge of the VEV ( ~ ~u~) 0, chiral symmetry breaking can also be discussed. The conventional auxiliary field method [4, 10] for studying composite field condensates is not sufficient here since it is inconvenient to introduce auxiliary fields for such composite field operators. In [11] an improved ef- fective potential approach for calculating composite field condensates has been developed which is more effective than the conventional auxiliary field method. We will use the method of [11] to do our calculation. The main steps of the method can be summarized as follows:

i. Introduce external sources coupling to the corn-

posite field operators at hand and write down the external source terms properly to guarantee that no composite field condensate effect appears in the tree level generating functional of connected Green's func- tions We.

ii. Calculate the whole generating functional Win large-N limit of 1/N expansion.

iii. Do renormalization and get the finite W. iv. Use the definition of classical fields (includ-

ing composite ones) to derive the relations between external sources and classical fields�9

v. Use the above relations to make Legendre trans- formation and thereby effective potential as function of all classical fields is obtained�9 Its minimum deter- mines the relevant VEV's.

As has been emphasized by Murphy and O'Raifeartaigh [12] that the functional integration over the auxiliary field F~ is mathematically ill-defined due to the positive sign of �89 in (6). The well-defined way is to eliminate Fi by its Euler-Lagrange equation which is now

F i = gAoA i. (10)

In general, F o should also be understood in the sense of (9). However, in large-N limit, effect o fF o integration is negligible. So that there is no problem to keep F o uneliminated as an auxiliary field in our calculation and this will simplify the calculation.

In this paper, we only consider composite operators up to bilinear operators of O(N) non-singlet fields. To this extent, the VEV's of the first three F-terms in (8) are

<F)o = � 89 o + �89

((q~o q~o)r )o = 2 ( A o )o ( Fo )o (11)

<(~i~i)(,/z)~o)o = 2 g ( A o ) o < A i A i ) o -- < ~ %)o.

Note that (~Uo) o must vanish. Since ((q~oq~o)(~/2)0o>o is expressed in terms of ( Ao )o and ( Fo )o, the only extra composite operator we should consider is ( ~ q~i)(1/2)~o = 2gAoAiA i -- ~ i~i .

Now consider the generating functional

Z [ I ~, J o, J i, H, K ] = e iW[ Ii'J~

= S ~ A i ~ i e x p { i ~ d 2 x [ ~ + JoAo + JiAi + ~ i

+ H F o + �89 , - ~P~ tFi) + P(K)] } (12)

where Ii, Jo, Ji, H, K are external sources and P(K) is a polynomial of K necessary for getting consistent result that no condensate effect occurs in the tree level We. We shall determine P(K) later. In (12) we only write explicitly the functional integration over the O(N) non-singlet fields which are relevant to the loop expansion calculation. Also, since < ~o)o = 0 and it does not have to be considered in the effective potential approach, we have not introduced the external sources coupling to ~u o in (12).

Let the classical fields of A o, Fo, Ai, 7~, etc. be

ao, fo, ai, Oi, etc. For example

6W 6W 5W a o - 3 j o, f o - sH , a i - f J i'

3W ~ = 6~-i' etc.

(13)

composite operator A classical field (p of the 2gAoAiAi _ Vpi ~ is defined as

6W �89176 d~i~i + (P) = f-K" (14)

When external sources are switched off, the classical fields give the VEV's of corresponding fields, especially (p_gives the VEV of the composite operator 2gAoA~A ~- tPi~i, i.e. (((~)i(l)i)(1/2)O0>O, since VEV's of O(N) non- singlet fields all vanish due to Coleman's theorem [9].

It is easy to see that 6" W/6K"[ . . . . . . . . o contains only contributions of various kinds of condensates. Thus the consistency condition that the tree level We con- tains no condensates can be written as [11]

6"Wc 6K" s . . . . . ~=o =0 . (15)

This condition is sufficient to determine P(K). Let Aoo Ale, ~goc, ~r be a nontrivial (source

dependent) solution of the classical field equations. Then Wc is [13]

W c = Id2x{~A'[Aoc,Aic, tffOc, tffic, Foc ]

+ doAo~ + JiAi~ + ~ ~i~ + HFoc

+ �89 - ~ ) + P(K)}. (16)

To determine P(K), we can, for simplicity, switch off all external sources except K and solve the classical field equations

Fo = �89 + hA 2)

A~(F o + gA~ - 2KAo) - ~'o ~ = 0

Ao(hr o + g2A2) - gKA~ -- �89 ~o ~o - 3g ~ ~ -= 0

(gA o -- K)~ i + gAi ~ o = 0

hA o tP o + gA~ ~ = 0. (17)

The nontrivial solution of these equations (with all fields real) has complicated forms depending on the range of h/g. Lengthy but elementary calculation shows that the so determined P(K) always has the form

P(K) = C(g, h)K 4. (18)

The simplest case is the range h/g => 2 in which

h 2 C(g, h) - (19) 8g4"

Our next step is to do loop expansion round the classical solution. In large-N limit, only l-loop cal- culation is necessary. Using the standard technique for evaluating the l-loop contribution to the generat-

491

ing functional W1[13], we easily get

Wl = ~-~[. d2x ((g2A~c + gFoc- 2gKAoc)

.(In gaAzc+gF~176 l )

where A is a momentum cut-off and the fields Aoc, Fo~ are solutions of the Euler-Lagrange equations

~w~ ~Wc - - = 0, = 0, etc. (21) 6Fo~ 6Aoc

with nonvanishing external sources. W1 is divergent as A--* oc. With a supersymmetric renormalization [5], we get the finite WI:

N - 2 f 2 2 W a = - ~ j d x~(g Ao~ + gFoc -- 2gKAoc )

�9 ( ln g2A2~+gF~176 ) M 2 1

where M is a dimensionful parameter related to the renormalization and can be regarded as the overall scale parameter in the theory. The form of (22) is equivalent to the one obtained from minimal sub- straction in dimensional regularization [7].

The exact classical fields ao,fo defined in (13) are related to the classical solutions Aoo Foe by

a o = Ao~ + #o 1)

fo = Fo~ +f~o x) (23)

where #01) and f~o 1) are contributions of quantum fluctuations and are of the order of l-loop level. Since W 1 itself is a l-loop quantity, up to l-loop all Aoc and Fo~ in (22) can be replaced by ao andfo. Furthermore, by means of (21), we can see that up to l-loop

Wr fo ... . ] = Wc[Aoc, Fo~,...]. (24)

In large-N limit we assume that Ng 2 and Nh 2 a r e

finite as N ~ ~ . Thus making the replacement

g h

and rescaling the fields by [7]

a o ~ x / ~ a o, f o ~ v / N f o , r etc. (26)

we express the total generating functional as

w=w~+wl

N r d 2 f l ~'2 = J x~gJo- �89 2 +ha~))fo-g2a~)aZi (

492

+ �89 + Joao + Jia, + LiP, + Hfo + �89 - tffCpi) + P(K)

+ , r 2,Kao,

.(lngZa~\ + gfoM 2- 2gKao - 1)

,gao 0]} (27)

Here we have ignored ~o which are irrelevant in the following calculations.

To do Legendre transformation, we need to derive the relation between the external source K and clas- sical fields from the definition of the classical fields (13), (14). Using (21) and (24), we can write the l-loop form of (14) as

dP(K) �89162 = d--K-- + ,4 (28)

where A is of order of l-loop level,

i ~ W 1 g_( (~dl a2t~Wl 2aoai6W1 - 3W 1 ) a = t72-- +

(29)

Solving (28) we can get K as a function of the classical fields. We shall do this later in specific discussions.

The Legendre transformation is

F[ao, fo, ai, Oi, q)]

= W[J o, H, Ji, Hi, K] -- N~d2x { Joao + Hfo

+ J,a, + LO, + �89 - ~,~, + ~o)}. (30)

Since Ai and ~v i cannot have nonvanishing VEV's we shall ignore them in the following calculations. The effective potential is then

Veff(ao, fo, ~o)

= - F / [ . d 2 x = N { 1 2 - x f o + lha2 fo + �89 - P(K)

1 [ - 2 2 ~ | ( g ao + gfo - 2gKao)

\ / g2a2 ) �9 ( l n o + gfo - 2gKao

M2 1

-(gao-K)2(ln(ga~ K)2 1 ) 1 }. (31)

Here K is to be expressed in terms of the classical fields. The VEV's are determined by the stationary condition (corresponding to vanishing sources):

0 Vef f O Vef f = 0, {~ Veff 0fo = 0 ' 0a ~ ~ - = 0 . (32)

We know that in theories with global supersym- metry, the vaccum energy is semi-positively definite. A supersymmetric vacuum has zero vacuum energy while broken supersymmetry corresponds to positive vacuum energy. Thus if (32) has a supersymmetric solution fo = 0, r = 0(((~o ~o)(1/21o0)o also vanishes as can be seen from (11)), this solution, having lowest vacuum energy, must represent the true physical vacuum and hence supersymmetry is not spontane- ously broken. Spontaneous supersymmetry breaking occurs only if fo = 0, 9 = 0 is not a solution of (32). Therefore our first thing to do is to examine whether fo = 0, q~ = 0 satisfies (32).

Let us study the effeotive potential in a small vicinity of the minimum fo = 0, r = 0 (External sources are in the small vicinity of Jo = H = K = 0). First we have to solve (28) and express K in terms of classical fields. Take Jo = H = 0 and K # 0 but small. We see from (18) that dP(K)/dK = 4C(g, h)K 3. Next we evaluate A ((29)). Since VEV's of all O(N) non-singlet and Lorentz spinor operators vanish, we only have to deal with (1/N)(dW/dK) in (29). W 1 depends on K both explicitly and implicitly through ao(K ) and fo(K). Now

(6a~ K + . . . ao(K)=(Ao)o+ ~K- o

We know that (3ao/(~K)= (62W/3Jo3K), (6fo/3K) = (62W/3H6K). In this paper, we only take into account condensates of composite operators bilinear in O(N) non-singlet fields. To this extent, (6ao/SK)o , (6fo / 3K)0 vanish as ( F o ) o = 0. Thus in this small vicinity W1 only depends on K explicitly and so it is easy to see that (28) reads

�89162 = - l l n g2(A~176 K M2 + O(K 2) (33)

and we get

ye(Ao)~ K = - 2n~p/ln M2 (34)

With this K in Veff, (32) reads

2 f o - h a 2

+ / 4 n a "In 92(A~ '~ +gfo g J +4~ -In M 2

/

2ncp/lng2 ( A~2 ~ ~ hao o- ( ao+( )) I 4ngao~o/lng2(Ao) 2

. g24+gJ~ M ) In M2

= 0

(35a)

4 9 3

( g a o + ( . , ., g~<AoYo'~'?-[

- I n \ M 2 ] = 0 (35b)

, ( . , �9 a2(Ao)2"~ -1 )

g 4 rcoa~ q~ /ln g2 (~2 ) 2~

�9 a o In M2

_(gao+(2rcq~/ln92(Ao)2"~'~

( . , . , o~<AoYohV -] (ga~176 ~ ) ) I = �9 In Ms _j O. (35c)

Evidently fo = 0, ~o = 0 satisfies (35b) and (35c) while with fo = 0, 4o = 0, (35a) becomes

hag g In g2 a2~ = ~ - ~ (36)

which is a nontrivial equation for ao 2. It is easy to see that a 2 = 0 is not a solution and for a given scale parameter M, (36) can have a 2 > 0 solution only if

92 h/9 < �9 (37) = 47rM2e

Furthermore, we see from (11) that ( F o ) o = 0 , (Ao)o 2 >= 0 and (AiAi)o > 0 can simultaneously hold only if hi9 < O. Thus the vacuum is supersymmetric only if hi9 < O. Therefore whether supersymmetry is spontaneously broken depends on the value of hi9:

h/9 < 0 ~ unbroken supersymmetry

h/9 > 0 =~ spontaneously broken supersymmetry.

(38)

Since ( A o ) o # 0 in the h/9<O case, (Fo)o = ((1~i~i)(112)00) 0 ~-- 0 in (11) means that ( ~i Ti)o #0. Hence chiral symmetry (7) is dynamically broken in this case. The only possibility of obtaining both unbroken supersymmetry and chiral symmetry is h = 0 .

Next we consider the broken supersymmetry case, h/g > 0. The above way of solving (28) is inconvenient now. For nonvanishing ~o, we can use the method of [11] to do this calculation. With dP(K)/dK= 4C(g, h)K 3, up to l-loop, we can formally write (28) as

K = ( � 8 9 1/3 I f ~o'~ 1/3 ~ 7 t 7 ) -(C~o)-UaA. (39) \4C(h.h) )

Substituting (39) into (31), we get, up to l-loop,

g, ff(ao,fo,q))= - fg+haofo+~-~,~)

1 [ ( /~o \U3\ 4,< o'a +oS~176 )

�9 ,) - (gao - �89 2 (In (gao - gqVc)'3) 2

(40) The stationary condition, (32), now reads

2fo - ha z + 9 1 n 92az + ofo gao(q~/C) 1/3

m 2 - - 0 (41a)

haofo - ~-~(gao - �89 '/3)

.[ln9ZaZ + ofo~Mgz ao(cp/C)'/3

_ln(gao-~_/C)l")~J= o (41b)

�89 i3 (q9/C)-2/3[ 2 2

+ 12rcC 0% In 9 ao + gfoM 2 - Oao(tp/C) U3

- ~(qUcy-) ~ ] -(#ao-�89 (ga~ M2 =0 . (41c)

Equation (41) can be solved numerically. As an example, we take M = 1, g = 1, h = 4, then C(g,h) is given by (19). Numerical calculation gives the following solution

(Ao)o = 0.0480, (Fo)o = 0.0935,

( ((~i (~i)(1/2)00)0 = - - 0.0259, (42)

and hence

((q)o r = 2 ( A o ) o ( F o ) o = 0.0090. (43)

It is easy to check that this is really a minimum of Veff with Veer > 0. From (11) and (42) we see that

(AiA,) o = 0.178, (~ i ~ ) o = 0.0430. (44)

So that chiral symmetry (7) is dynamically broken in this case.

We see that even in a model requiring renormaliza- tion supersymmetry may be spontaneously broken.

In summary, we have calculated (((~0)(1/2)00)0' ((~o ~o)(1/2)~0)o, and ((~i ~i),~2)g0)o up to the contribu- tions of bilinear operators of O(N) non-singlet fields in effective potential approach in large-N limit to examine the possibility of dynamical breakdown of supersymmetry and chiral symmetry (7). Whether supersymmetry is broken depends on the values of t h e coupling constants g and h. When h/g <= 0 super- symmetry is unbroken, otherwise supersymmetry is spontaneously broken. In most cases, chiral symmetry (7) is broken dynamically by the condensate ( ~ ~ ) o .

494

References

1. G. 't Hooft, in: Recent developments in Gauge Theories. G. 't Hooft et al. (eds.) New York: Plenum 1980

2. For instance, S. Coleman, E. Witten: Phys. Rev. Lett. 45 (1980) 100; D. Weingarten: Phys. Rev. Lett. 51 (1983) 1830; C. Vafa, E. Witten, Nucl. Phys. B234 (1984) 173; C. Vafa, E. Witten: Commun. Math. Phys. 95 (1984) 257

3. Soine examples of recent investigations are: M.E. Peskin: 1982 Les Houches Lectures, SLAC-PUB-3021; J.M. Gerard, N.P. Nills: Phys. Lett. 129B (1983) 243; E. Eichten, R.D. Peccei, J. Preskill, Z. Zeppenfeld: Nucl. Phys. B268 (1986) 161; E. Eichten, J. Preskill; Nucl. Phys. B268 (1986) 179

4. D.J. Gross, A. Neveu: Phys. Rev. D10 (1974) 3235

5. O. Alvarez: Phys. Rev. D17 (1978) 1123 6. A.C. Davis, P. Salomonson, J.W. van Holten: Phys. Lett. l13B

(1982) 472 7. A.C. Davis, P. Salomonson, J.W. van Holten: Nucl. Phys. B208

(1982) 484 8. I.K. Affleck: Phys. Lett. 121B (1983) 245 9. S. Coleman: Commun. Math. Phys. 31 (1973) 259

10. K. Higashijima, T. Uematsu, Y.-Z. Yu: Phys. Lett. 135B (1984) 302

11. Y.-P. Kuang, X.-Y. Li, Z.-Y. Zhao: Commun. Theor. Phys. 3 (1984) 251

12. T. Murphy, L. O'Raifeartaigh: Nucl. Phys. B218 (1983) 484 13. R. Jackiw: Phys. Rev. D9 (1974) 1686