Volume 127B, number 1,2 PHYSICS LETTERS 21 July 1983

EXPLICIT EVALUATION OF PHYSICAL QUANTITIES AND SUPERSYMMETRY PROPERTIES

OF THE LATTICE O(N) a MODEL AT LARGE N

P. Di VECCHIA Physics Department, University of Wuppertal, D-5600 Wuppertal 1, West Germany

R. MUSTO, F. NICODEMI, R. PETTORINO Istituto di Fisiea Teoriea dell'Universit~ di Napoli, Naples, Italy and INFN, Sezione di NapoIi, Naples, Italy

P. ROSSI CERN, Geneva, Switzerland

and

P. SALOMONSON Institute of Theoretical Physics, GSteborg, Sweden

Received 4 March 1983

We explicitly show how to extract physical quantities of the continuum theory from two lattice versions of the non- linear o model, that are exactly solvable for N ~ o~. We also build a lattice action which is supersymmetric in the large N limit and suggest how to recognize supersymmetry restoration in the continuum even when starting from a non-supersym- metric lattice formulation.

A very convenient way to study non-perturbative phenomena in gauge theories is to introduce a lattice structure in the euclidean spacetime [1]. The quanti- ties of the continuum theory can then be determined by sending the lattice distance a to zero (this corre- sponds, in an asymptotic free theory, to the limit of small coupling constant) and by extracting, in this limit, a behaviour in agreement with the prediction of the renormalization group. This procedure has been used to compute the string tension [2] and the glue- ball mass [3] in lattice Yang-Mil ls theory by means of the Monte Carlo technique. Other quantities like the topological charge fluctuation [4] and the gluon condensate [5] have also been computed with the same technique; However, in this case the subtraction of a perturbative tail was necessary before one could isolate the renormalization group behaviour. Of course, such a tail depends on the particular definition used

for the quantities on a lattice, while their non-per- turbative behaviour in the scaling region is universal.

In order to show explicitly how to extract physical quantities of the continuum theory from a lattice theory, we consider in this letter the asymptotically free O(N) non-linear o model, that can be explicitly solved for large N both in the lattice and in the conti- nuum theory. In particular, we show that physical quantities as the magnetic susceptibility, the correla-

1 tion length and (g auS3uS) can uniquely be extracted from the lattice theory and that they are independent from the particular lattice version used.

In the second part of this letter we study the prob- lem of supersymmetry on a lattice. In particular, we consider two lattice versions of the O(N) non-linear o model with fermions, that are supersymmetric in the naive continuum limit, but not on the lattice. We show that one of these versions has no phase transi-

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Volume 127B, number 1,2 PHYSICS LETTERS 21 July 1983

tion and its leading order in N is supersymmetric, while the other has a phase transition and stays not supersymmetric also for large N. However, both of them in the scaling region show the existence of a supersymmetric continuum theory, whose physical quantities can be computed using any of the two lat- tice versions.

We start from two different lattice formulations of the O(N) non-linear o model, described by the follow- ing actions:

--8 2 = l ~Sn+ta'S n -S4= I.-~(Sn+v'Sn) 2 , g n ,u ' 2gn,u (1)

together with the constraint S 2 = 1. In order to perform the 1]N expansion of these

theories we introduce in S 4 an auxiliary real field XTr,u, that makes it quadratic in the fields Sn, we use for the 6 function of the constraint the usual exponential representation and, after a rescaling of the fields S n

x / g S n , a n ~ (an]g) we can perform the gaussian functional integration over S n getting the following effective actions:

S~ ff

N - i / 3 ~ a n n

1 ( ) - ~ t r l n ~ c o s p u ~ ( p + q ) + i a ( p + q ) , (2) /.L

S~ ff

N = - i / 3 ~ a n --½/~ ~2`n2,# n n ,~

" (3) where we have introduced the Fourier transforms of the fields a and 2`:

for the quadratic action S 2 and

2a/3 =f0(2`/a) , a - 22, 2 = 1/2/3, (7,8)

for the quartic action S 4 . f0(3') is given by

/r (1 -1

In two dimensions f0(7) = (2/rr)K(27) where K is the complete elliptic function of the first kind. The saddle point condition (6) describes a unique phase, while the conditions (7) and (8) imply a first- order phase transition occurring for 2/3 ~ 0.956 for the quartic action. In the strong coupling phase we have

a = 1/2/3, k = 0 , (10)

while in the weak coupling phase a and 2 ̀are both different from zero. The details of this phase transi- tion are described in a future publication [6], here we are only interested in studying the transition to the continuum theory, which is obtained by staying in the weak coupling phase, since the o model is asymp- totically free.

The spontaneously generated mass of the conti- nuum theory can be obtained from (6) and (7) for large/3 (or equivalently for 3,-+½) by using the asymp- totic formula

f0(7) > l l n 1 6 + O(1 - 23'), (11) ,~1/2 7r 1 - 43 '2

Inserting (11) into (6) and (7) we get the following asymptotic behaviours:

1 ~ ~ - 4 exp (-4rr/3) ,

a n = ~ e x p ( i p n ) a ( p ) , ~n,u = ~ exp( ipn)2`u(P) , P P (4)

and 13 = 1]gN. For the sake of simplicity we use the same symbols g,/3 for the coupling in both actions.

At large N and fixed 13 we can restrict ourselves to constant fields:

a (p ) = i a6(p) , 2`u(p) = X 6(p) , (5)

and we get the following saddle point condition:

2a/3 = fO(1/a) , (6)

2 ̀ i - 4 e x p { - 2 r r / 3 [ 1 +(1-1 / /3 )1 /2 ]} a

1 ~ - 4 exp Or) exp (-47r/3), (12)

for the two theories in (1), respectively. The mass (i.e., the inverse correlation length) is de-

fined in the two theories by the relations:

a m = c h - l ( a - 1), am = ch-l(a/2` - 1) , (13)

and in the continuum theory this reduces to

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Volume 127B, number 1.2 PHYSICS LETTERS 21 July 1983

a2m2 = 2 ( a - 2 ) , aZm2=2(oe/X-2). (13')

Inserting finally the asymptotic formulas (12) in (13) we get:

a2m 2 , 32 exp (-41r/3), /3--+00

a2m 2 ' 32 exp0r ) exp (--41r/3) . (14) fl--+,~

We find therefore the right renormalization group be- haviour, but the constant in front of the exponential is different. This is, however, due to the fact that A~ lattice) 4: A~ lattice) 4: A (c°nfinuum). A relation between

the various A's can be obtained from the requirement that we renormalize the theory keeping a physical parameter fixed as, for instance, the mass. In this case, if we define in the different regularizations the corre- sponding A parameters as follows:

A~ atfice) = a -1 exp (-2rr/3) , A~ lattice) = a -1 exp(-27r/3),

A (c°nfinuum) = A exp (-27r/3), (15)

we get from (14) that the mass spontaneously gener- ated is given by the following formulas in terms of the various A's:

m = r ~ A~latfice) = ~ exp (rr/2) A~ lattice)

= A (c°ntinuum) . (16)

Remember that the saddle point condition corre- sponding to (6) and (7) in the continuum theory, after a Pauli-Villars regularization, is given by

/3 = ( 1/4rr) In (A~cont.)/m 2) . (17)

We now compute the two-point function for the field q~i for N large in the two theories defined by the ac- tions in (1). An easy calculation gives the following result:

i / _ 1 ; e x p [ i p ( n - m ) ] (S nS m) j d2p 8iJ (18) 2c~/3N -,r (21r) 2 1 - 72; u cospu '

where 3' = ( I /c 0 X/a in the case of the quadratic (quar- tic) action. The magnetic susceptibility can easily be obtained from (18) by summing over i and m:

N

(S / m~X/m)_ 1 1 (19) i=1 23/3 1 -- 27 "

By using the asymptotic formulas (12), we find that (19) behaves in the limit/3 -+ '~ as required by the re- normalization group. Then, remembering the rela- tions (15) and (16) we get in this limit the following value for the magnetic susceptibility

N ~ ( S i ~ m s i ) _ 1 (20) i=1 a2m2/3 '

in complete agreement with the result of the conti- nuum theory.

Finally, we want to show how to extract from the lattice theory the non-perturbative renormalization group invariant quantity (~ S[~S) of the continuum theory, which corresponds to the gtuon condensate in Yang-MiUs theory. It can easily be computed in the continuum theory for large N and one gets:

(½SIS]S) = 1 2 ~m . (21)

The continuum theory is defined by the following lagrangian

L = (1/2g) ~uS" a~ S, (22)

with the constraint S • S = 1. In the lattice theory, the quantity (21) can be ex-

tracted by computing the internal energy which is given by:

(~u(Sm+u',..gm- 1)) = a - 2 - 1/2t3, (23)

( l ~ [ ( S m + v ' S m ) 2 - 1 ] ) = X 2 - 1 , (24)

for the quadratic and the quartic actions, respectively. On the right-hand side, we have written the expres- sions obtained for large N. By using the asymptotic formulas (12) together with the identity:

X 2 = (1/83 ,2) [1 + (1 -- 472//3) 1/2 -- 23,2//3],

= x/c~, ( 25 )

we obtain the following behaviour for large/3 for (23) and (24):

(~(Sm+v'Srn-1)) 1

- ' - ' + - g5 + 16 exp (-4~r/3) + ..., (26) fl ---~ oo Lb~

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Volume 127B, number 1,2 PHYSICS LETTERS 21 July 1983

( l~u [(Sm+u'Sm)2-1] )

, [ - ½ + ½(1 - 1//3)1/2 - 1/4/3] fl--~oo

+ 16 exp (rr) exp (-4rr/3) + .... (27)

Therefore we get a perturbative tail which is depen- dent on the particular regularization procedure and which has no meaning in the continuum theory. In addition we get an exponential behaviour which is in agreement with the renormalization group prediction. The use of eqs. (16) tells us that the non-perturbative part of (26) and (27) is universal and is indeed equal to the result (21) of the continuum theory.

We have also computed (26) and (27) by using a mean field approximation.We have found that a nafve [7] and a more sophisticated (1/d expandable) ver- sion [8] of the mean field theory reproduces exactly the perturbative tails of (26) and (27). In addition, the one-loop correction in the expansion around the mean field gives an additional contribution to the free energy thus matching its large N perturbative value, and leaves the internal energy unchanged. It is clear from this example that a mean field theory does not give any information on the quantities of the conti- nuum theory. Our example suggest also that a mean field approximation can be used in order to compute the perturbative tail * a.

The analysis we have performed of the transition from the lattice formulation to the continuum theory is particulary interesting when fermions are introduc- ed in the action. Indeed we will now show that, in the case of the quadratic action, one can introduce fer- mions in such a way to obtain a model supersymme- tric on the lattice in the N-~ oo limit. One can recover a supersymmetric continuum limit even starting with an action which is not such on the lattice.

The supersymmetric O(N) o model in the conti- nuum theory is described by the following free action [101:

1 1 S=fd2x [½(0uS)2 - - $ i ~ , " a u ~ t ~(F)21, (28)

with the addition of the constraints

,1 Such an idea is being applied to compute the gluon con- densate in Yang-MiUs theory in ref. [9].

1 • S 2 = l , S ' ~ t = 0 , F ' S = i 1 ~ T 5 ~ t . (29)

Both the action (28) and the constraints (29) are left invariant by the following supersymmetry transfor- mations:

6S = i e ~ , 8 ~ = (7"0uS+ 3,5F)e,

6F = ieTsT"0u ~ . (30)

It is straightforward to write a lattice verion of the free action (28)

1,

\

+ iOn~'U(*n+p -- gln_u)l + 2 ~_J F2n ). (31) n

where the bosonic part contains fields located at points separated by two links like the fermions in order to have the same number of bosonic and fer- mionic degrees of freedom as is necessary in a super- symmetric theory ,2. In eq. (31), we have added a field independent term which is essential if we are in- terested in computing the vacuum energy. We have omitted this term in the purely bosonic lagrangian, since its presence was inessential for the considera- tions of the first part of this letter. The supersymme- try transformations on the lattice written for fields in momentum space are:

6S(p) = ie~(p) ,

6 ~(p) = (i7 ~ sinpu S(p) + 75F(p))e, (32)

fiE(p) = -e3 ,57u sinpu ~k(p).

It is also interesting to notice that the commutator of two such transformation yields

[f 1 ,62]S(p) = -2e2"rUel sinp~ S ( p ) ,

and corresponding expressions for if(P) and F(P). This amounts in configuration space to a typical lattice translation:

1 [61,6 2] Sn = i 2e2~/Ue I ~ (Sn+ u - Sn_u) , It is easy to check that (31) is left invariant by these transformations. The constraints

S~n=l, SnOrt=O, (33)

,2 We follow here the same procedure as in ref. [12].

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are, however, not preserved by the transformations (32) and therefore the theory described by (31) and (33) is not supersymmetric. It becomes, however, naively supersymmetric in the continuum limit.

Using the constraint (33) on F we can eliminate the field F from the action (31) and get a quartic term for the fermions:

1 ( ~ [(Sn+u'Sn_ u - 1) - $ 2 = ~gg n,/~

+ i*nV~(*n+** -- * n-u)] -- ½ ~ (*nT5 Iltn)2} • n /(34)

S 2 should be supplemented with the constraints:

S 2 = 1, Sn* n = 0 . ( 3 5 )

As in the case of the purely bosonic theory we can write another version of the lattice O(N) cr model de- scribed by the following action:

-S4 = ~ (n~,u {½ [(Sn+u " Sn-u)2 -1]

+ il~nT/a(*n+~ -- ~n-ta)} 1 n~ (~nT5~tn)2)(~6)

together with the constraints (35). The 1/N expansion of the two previous theories

can be performed by introducing the auxiliary fields q~n and Xn, u which make the actions (34) and (36) quadratic in ~ and S and by exponentiating the two constraints (35). We can then integrate over the fields S and ff and we get the following effective lagrangians (with the fermionic multiplier C = 0):

1 2 -S~ ff= -13N ~ (ia n + ~ Cn) n

- ½ N t r l n 2 ( i a ( p + q) + ~ cospuS(o +q))

+ ½Ntr In ( i75¢(p+ q)+ ~ 2 7 u sinpuS(p + q ) ) , /.t

(37)

- S g ff - f iN ian + ~ <b n +

-½Ntrln2(ia(p+q)+ ~ ku(p +q) cos 2pu ) #

(38)

+ ½Ntrln ( i75¢(p + q)+ ~27UsinpuS(p+q)),

respectively, for the actions (34) and (36), where/3 = 1/4gN.

For large N it is sufficient to consider constant fields

a(p) = ia6(p),

and we get the following saddle point conditions

813~ =f0(1/a), 8fi½ (¢2 + 4) =f0(2/(¢2 + 4)) ,

813 c~ = f0(X/a), 813 ½ (q~2 + 4) = f0(2/(q} 2 + 4)) ,

c~- 2X 2 = 1/213,

¢(p) = q~5(p), Xu(p) = 6(p)Xu, (39)

(4O)

(41)

respectively, for the two theories (34) and (36). The two conditions (40) imply a relation between

a and q~

a = ½ (¢2 + 4) , (42)

which, when it is inserted in (37) with the constant fields (39), gibes a vanishing vacuum energy. This fact and the equality between the fermionic and bosonic correlation lengths implied by (42) for any 13 suggests that the large N limit of the quadratic theory is super- symmetric. This is true in fact because the action (37) rewritten before integrating over the fields S and for C = 0 and constant cz and ¢, is invariant under the transformations (32) as it has already been proved.

In the case of the quartic action there is no rela- tion between the two conditions in (41): therefore in general the bosonic and fermionic correlation lengths are different even forN-* oo and the theory is not supersymmetric. This symmetry is, however, restored in the limit/3 -+ oo. In this limit, in fact, the bosonic and fermionic correlation lengths become equal as can be seen by using the asymptotic formulas (11) and (12). Their value is also equal to the common correlation length which is extracted from the theory with the quadratic action reproducing also the result of the continuum theory.

In conclusion, it is nice to have a supersymmetric lattice theory, but if we want to use the lattice only as a way to regularize the quantum theory, it does not matter whether supersymmetry is lost together with Lorentz and translation invariance. If it is not spontaneously broken, its existence will anyway be recognized for sufficiently large/3.

As a consequence of our analysis, the restoration of supersymmetry in the continuum limit for more

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Volume 127B, number 1,2 PHYSICS LETTERS 21 July 1983

compl ica ted theories like four-dimensional lat t ice

gauge theories can be checked, for example by means

o f the Monte Carlo technique or o f strong coupling series, by showing that in the scaling region the fer-

mionic and the bosonic glueballs have the same mass.

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[3] For a recent review, see: B. Berg, lecture notes John Hopkins Workshop (Florence, 1982), CERN preprint TH. 3327 (1982); see also: K.H. Mtltter and K. Schilling, Phys. Lett. I17B (1982) 75; 121B (1983) 267.

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