ELSEVIER Nuclear Physics B (Proc. Suppl.) 106 (2002) 941-943

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PROCEEDINGS SUPPLEMENTS www.elsevier.com/locate/npe

SUSY Ward identities in 1-1oop perturbation theory Federico Farchioni t a, Alessandra Feo b *, Tobias G alla ¢, Claus Gebert a, Robert Kirchner a, Istvän Montvay a, Gernot Münster b, Roland Peetz b, Anastassios Vladikas d.

DESY-Münster-Roma Collaboration

aDeutsches Elektronen-Synchrotron, DESY, Notkestr. 85, D-22603 Hamburg, Germany

bInstitut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany

¢Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK

dINFN, Sezione di Roma 2, Universitä di Roma "Tor Vergata", 1-00133 Rome, Italy

We present preliminary results of a study of the supersymmetric (SUSY) Ward identities (WIs) for the N = 1 SU(2) SUSY Yang-Mills theory in the context of one-loop lattice perturbation theory. The supersymmetry on the lattice is explicitly broken by the gluino mass and the lattice artifacts. However, the renormalization of the supercurrent can be carried out in a scheine that restores the nominal continuum WIs. The perturbative calculation of the renormalization constants and mixing coefficients for the local supercurrent is presented.

1. I N T R O D U C T I O N

SUSY gauge theories present different non- perturbative aspects which are the object of cur- rent research, for example the possible mech- anisms for dynamical supersymmetry breaking. The simplest SUSY gauge theory is the N = 1 SUSY Yang-Mills theory. For SU(Nc) it has (Nc 2 - 1) gluons and the same number of mass- less Majorana fermions (gluinos) in the adjoint representation of the color group.

To formulate supersymmetry on the lattice we follow the ideas of Curci and Veneziano [1]. They adopt the Wilson formulation for the N = 1 SUSY Yang-Mills theory. Supersymmetry is bro- ken explicitly by the Wilson term and the finite lattice spacing. In addition, a soft breaking due to the introduction of the gluino mass is present. It is proposed that supersymmetry can be recov- ered in the continuum limit by tuning the bare gauge coupling g and the gluino mass m~ to the SUSY point, at m~ -- 0, which also coincides with

*Talk given by Alessandra Feo. ? Present Address: Institut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany

the chiral point. In previous publications [2], and references therein, we have investigated these is- sues.

Another independent way to study the SUSY (chiral) limit is by means of the SUSY Wls. On the lattice they contain explicit SUSY breaking terms. In this framework, the SUSY limit is de- fined to be the point in parameter space where these breaking terms vanish and the SUSY Wls take their continuum form.

Our collaboration is currently focussing on the study of the SUSY WIs on the lattice, either with Monte Carlo methods [3] or by a perturba- tive calculation of the renormalization constants and mixing coefficients of the lattice supercur- rent. For a different perturbative approach to SUSY WIs see [4].

2. S U S Y W A R D I D E N T I T I E S ON T H E L A T T I C E

In the Wilson formulation of the N = 1 SUSY Yang-Mills theory [1] the gluonic part of the ac- tion is the standard plaquette action while the fermionic part reads

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942 E Farchioni et al./Nuclear Physics B (Proc. Suppl.) 106 (2002) 941-943

-~(z + a/i)(~ù + r)uù (x)~(x)Utù (z)) /

Supersymmetry is not realized on the lattiee. One can still define transformations that reduce to the continuum SUSY ones in the limit a ~ 0. A possible choice is

6U u(x) = - a g U u(x )e%A(x)

- a g g % A(z + a/i)Uù (z) i

6A(X) -- f fpröpr(X)Z g

where Gpr is the clover plaquette operator.

2.1. Ward ident i t ies and opera to r mixing Compared to the numerical simulations [3] a

lattice perturbative calculation of the SUSY WIs introduces new aspects. In order to do perturba- tion theory we have to fix the gauge, which im- plies that new terms appear in the SUSY WIs: the gauge fixing term (GF) and the Faddeev- Popov term (FP) while contact terms (CT) ap- pear off-shell [5]• Taking into account all contri- butions coming from the action, the bare SUSY WIs reads

6O < o ~ ù s ù ( ~ ) - 2~oOx(~) + $ ~ 0

- 6SGF - - ( ~ / ~ - - - ~ e=O - - 6 S E R - v ~~--T~ ~=o> = (OXs(~)).

X s is the symmetry breaking term, whose specific form depends on the choice of the lattice super- current. We define the lattice supercurrent to be

Sù (x) = - 2--iT~ {~ù, (x)aù, 7ùA(x)} . g

We choose a non-gauge invariant operator inser- tion O := A~(y)Äb(z) . Gauge dependence im- plies that operator mixing with non-gauge invari- ant terms has to be taken into account for the operator renormalization. X s mixes with opera- tors of equal or lower dimension [6]

X s ( x ) = f ( s ( x ) - (Zs - 1)AùSù(x) -

2~x(x) - ZTaùTù(~) - ~ , Za, A,. i

The additional operators As do not appear in the on-shell gange invariant numerical approach of [3]. They are either BRS-exact, A = 6BRSÄ, or vanish using the equation of motion. Moreover, because the Ai do not appear in the SUSY WIs at tree level, the ZA~ are O(g2).

We are forced to go off-shell, contrary to the numerical simulations that are in the on-shell regime, otherwise it is not possible to separate the contributions of T, and Sù. Finally, infrared di- vergences are treated using the Kawai procedure [7], which gives the general recipe to renormalize a Feynman diagra.m at one-loop order.

2.2. Pe r tu rba t ive calculat ion We calculate the matrix elements of the terms

in the SUSY WIs for general external momenta p (for the gauge field) and q (for the fermion field) and the projections over % and %75 matrices.

The lattice SUSY transformations of the gauge field Au are not identical to the continuum ones: the transformation of the gange link U~ deter- mines the transformation properties of A~. Up to O(g2),

6A~ = i (g(x )T t tAb(x) -F @(x+ a/ i)TBAb(x-F a/i))q-

i - c ~agAb«(C(z)~ù~ (x)- @(~ + a/i)~ù~Œ(~ + a/i) )A~ù

6 c d ;4 a292(26aböcd -- 6acöbd -- aäöbc)AttA" {

(e(x)%A«(z) + e(x + a/i)%A"(x + a/i)) } ,

which reduces to the continuum SUSY transfor- mation, 6Aa~, = 2ig%,A% in the limit a --+ 0.

At one-loop order, three propagator integrals on the lattice are tabulated in [8] in terms of lat- tice constants plus the continuum counterparts:

1 f • 1;ku;kuk~;kuk~k p Co;u;u~;uvp (P, q) __ l j~ , k k2( k + p)2( k+q)2 i

With the help of [9,10] one can express Co(p,q), which for arbitrary external momenta p and q is a complicated expression in terms of Spence func- tions, as

Co(p,q) = ~-(Li2(P'q2Æ)--T'm2¢P'q+Æq---~ )

E Farchioni et aL/Nuclear Physics B (Proc. Suppl.) 106 (2002) 941-943 943

1 log(~, q - A lo "(q -- p)2 ,, +3 _q¥~) g~---U -))' where A 2 = (p . q ) 2 _ ~q2. The Cu(p,q) , Cu~ (19, q), Cu~, p (p, q) can be written recursively in terms of scalar functions p2 ,q2 ,p .q and Co(p,q) multiplying Lorentz components of the external momenta p and q [10]. The Ageneral results for ar- bitrary external momentä p and q are very long (sometimes up to 1000 terms). Therefore a small momenta expansion is reqüired.

2.3. R e n o r m a l i z a t i o n C o n s t a n t s One can write the matrix elements in the form

( O A u S v ) = SF(q)" AAs" D(p) . Job,

where SF(q) and D(p) are the full gluino and gluon propagators, 5ob is the color factor and Aas is the matrix element with amputated external propagators. For small momenta, Aas yields at one-loop order

Aas = 2(19- q)up~a~«%,(1 + Tff)

+i(p - q)u(abJu« - puTa)TT s + . . . ,

where Ts s is the coefficient of the one-loop con- tribution which is proportional to the tree level expression of Aas, TT s is the coefficient of the contribution proportional to the tree level AaT- The tree level matrix element of (OAu Tu ) reads

AaT = i(p -- q)u(AbS~,« - P~T«) •

Collecting the various contributions in the WIs and expanding them in terms of a basis of Lorentz-Dirac structures, the renormalization and mixing constants can be obtained from the coefficients as

- ( Z s - 1 ) = Tff + Tf fT + T~ + TGF + Tff P C T x TTF P - z T = T¢ + T~ + T~ + TGF +

We consider an appropriate choice of projections, which for small momenta and on tree level read,

Tr(%/~LXT) = 4i(pcp« -- p26«a -- Pcq«) +p . qö««)

'I~(7c75AAT) "= 0 Tr(%Aas) '= 8i(pcpa -p26«« - pcq«

+p " qö«c,)

Tr(%75Aas) = 8ipuqÆeup«c,

in order to extract the coefficients. Typically the T(T'),T(s') are constants plus logarithms depend- ing on the external momenta. Our preliminary results for Zs and ZT show a good agreement with the numerical data of [3].

3. O U T L O O K

It is possible to study the SUSY WIs by means of lattice perturbation theory and to determine the coefficients ZT and Zs in the off-shell regime.

The contributions for all diagrams have been written down explicitly for small external mo- menta. Preliminary results are in accordance with Monte Carlo data, but still we have to in- crease the precision of the numerical integrations and to perform several checks before presenting the final results in a forthcoming publication.

Acknowledgements : We thank G. Rossi, M. Stingl and M. Testa for stimulating discussions.

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