extending improvement to fermions

Volume 136B, number 5,6 PHYSICS LETTERS 15 March 1984

EXTENDING IMPROVEMENT TO FERMIONS

Werner WETZEL Institut fiir Theoretische Physik, Universitiit Heidelberg, Heidelberg, Fed. Rep. Germany

Received 18 October 1983

Tree-level improvement of the lattice action for SU(N) gauge theory is extended to Wilson fermions. The cancellation of lattice artefacts is verified for quark-quark and quark-gluon scattering to order a 2 in the lattice spacing a.

Symanzik's improvement program for lattice theories [1 ] has recently been applied to the non-linear o-model [2] and to SU(N) Yang-Mills theory [3,4]. The idea underlying improvement is that the effects of a non-zero lattice spacing a can be reduced to corrections of higher order in a by appropriately choosing the lattice action. For SU(N) Yang-Mills theory it has been shown [4] that the addition of planar plaquettes with 6 links to the original Wilson action [5] allows removal from the heavy quark potential of the O(a 2) terms on the tree level anct the O(a 2 In a) terms on the one-loop level.

In the following we will extend tree level improvement from gluons to quarks and demonstrate explicitly the absence of O(a 2) corrections in quark-quark and quark-gluon scattering.

We start by generalizing Wilson's Ansatz [6] for lattice fermions to include next-to-nearest neighbour terms, i.e.

L l=a3 ~n ( -ma~n@n+l ~d[Cl~n(~l-'Yv')Un't~bn+f~+C2~n(~'2-'~t~)Un'vUn+fL't~n+2f~-~n@n+h'~;~

Here @n represents the quark variables at the lattice site n; Un,~, is the link variable between site n and site n + t~ which is related to the gauge field Aan,u according to

Un, =exp(iga ~ A b T b) (2) b n,lz "

The constants h i which remove the degeneracy in the fermion spectrum, and the weights c i will be conveniently chosen below. The parameter m denotes the quark mass (for simplicity one quark species only). Eq. (1) is to be completed by the Yang-Mills part constructed from elementary plaquettes and planar plaquettes with six links (see for instance refs. [4,7] ).

From (1) we obtain the following expression for the free quark propagator S I (P):

+! SI (p)-1 = m ~ {i7~ [c 1 sin(p~a) + c 2 sin(2p~a)] + ~ - ClX 1 cos(p~a) - c2X 2 cos(2pua)) , (3) a /./

The normalization and improvement condition

SI (p)-I = m + ~ i ' r ,p , + O(a 3) (4) /z

requires that .1

,1 This result has already been obtained in ref. [9] in the context of the Gross-Neveu model.

0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

407


4 1 C 1 - '~, c 2 = - ~ , Xl/X= 1, ~k2/~k = 2. (5)

This leaves ~ as a free parameter. Using (5) we obtain for (3)

SI --1 = m + 1 ~ [iTv(1 + 2ca2b 2) sin(pua) + 2cXa4ib4], (6) a /.t

where ibv = (2/a) sin(pva/2) and c = 1/12. For comparison the propagator corresponding to only nearest-neighbour terms is

s - l = m + l ~ 1 2^2 - X a p v ] . ( 7 ) a [i7~ sin(pua) +

Here the parameter X contributes to order a. We now derive from (1), (5) improved quark-gluon vertices by expanding the link variables in terms of the cou.

piing constant g. With the momentum assignments in fig. la and Q = p + p' we get to order g

2 -2 p2,1 = _igT b ( _2 ikca 3 Qu (P2 + ib~ 2) + 7~ [(1 + ca (p~ +/3~2)) cos(½ Qua) 2 "2 1 + ca Qu cos (~qva ) ] ) -/.t

l 2 2 = -igTbTu(1 +'~ ca qt~) + O(a3). (8)

This vertex satisfies the Slavnov-Taylor identity

l u qu =gTb [ S I ( p ) - I - S I ( P ' ) - I ] " (9) u

In the next order we get referring to fig. lb with Q = p + p '

i 2 "2 _ 2i~aQu cos({ Qua)]+ 2a(T °, T c} F2~ =g26u~ca2{fbcdTCl(s-r)u [7~(1 - -2 a Qu)

xt-xO cos( ' ' + ( X c o s ( { - ' " - - c o s ( { 1 1~ a%Q~) (rus u Qu a) - Pu p~a) cos(~ pua) Qua) -

= g2~vca2 fbcd TdTu(s_r)u + O(a3). (10)

As to the unimproved form of these vertices the reader is referred to ref. [8]. The improved three-gluon vertex and gluon propagator can be found in ref. [7]. In the subsequent application

we only need their expansion up to O(a2). The result for the three-gluon vertex with the momentum assignments of fig. lc is

,0, 3 - . -bat 1 ca2(r 2 + 2 + q2)) + irrel.] 6uv6urca2q 2 (s--r)u} + cycl. perm. + O(a4). (1 1) #vr - tg l {BUy [(S- r)r (1 + ~ s v --

We have omitted from (11) irrelevant terms like 8~vqTr 2 which vanish upon contracting the index ~" with a polarization vector.

For the inverse gluon propagator D~lv we get with the usual covariant gauge ftxing

P~ f

I / " I

I ~. x I IJlb u,b ~,c v,d

a) b) c)

Fig. 1. Momentum assignments for the vertices given in the text, (a): £2,1 ; (b): £2,2 ; (c): £ °,3.

408


-i DI uv = A~v + ° t - lquqv [1 - ~ ca2(q 2 + q2)] + O(a4),

1 2 2 Ag.v = 6uvq2(1 + caZq 2) - quq~ [1 +2 ca (qu +q2)] + O(a4). (12)

It is to be noted that Auv deviates from its continuum limit by terms of order a 2, whereas such terms are absent in S I [see eq. (4)]. In fact since gauge invariance requires ~v Auv qv = 0, improvement cannot be achieved in a gauge invariant manner for all components of A. We expect it however to hold for the physical degrees of polarization.

To investigate this issue, we identify/a = 4 with the time direction and/a = 1,... , 3 with the spatial directions. Consider first the case where q points into the 3 direction, i.e. q l = q2 = 0. The physical polarizations are e = 1 or e = 2, for which A indeed reduces to its continuum limit form. In fact the expression for A given in eq. (12), which was originally derived in ref. [4] by imposing improvement on the heavy-quark potential can be understood as a consequence of the requirement that A is gauge invariant, has cubic symmetry, and shows improvement for the above configuration.

Consider now the more general case that q lies in the (1, 3) plane. As before e = 2 qualifies as a physical polarization, but for the polarization vector in the (1, 3) plane, which is orthogonal to q improvement does not hold. Instead we find from the on-mass-shell condition

Au~,e v Iq2 =0 = O(a4) (13) p

that

1 ca2q2 + O ( a 4 ) ] , (14) 0[ 1 - 2 e u = e/~

where e 0 are the continuum polarizations, i.e. e 0 -q = 0. Note that the order a 2 term in (14) is not effective for e = 2 .

In conclusion, when constructing scattering amplitudes involving gluons, we have to take into account possible order a 2 terms from the polarization vectors. Only then we can expect a complete cancellation of the lattice artefacts to order a 2.

We are now ready to investigate whether the above lattice Feynman rules yield improved tree level quark-quark and quark-gluon scattering amplitudes.

For quark-quark scattering we have to consider the diagram of fig. 2. Due to the Slavnov-Taylor identity (9), 2 2 1 only the 6uv part from the gluon propagator contributes for on-mass-shell quarks yielding the factor (1 + ca q u ) - "

According to (8) this factor is cancelled by two factors (I 1 2 2 + 2 ca qu) from the vertices. The Feynman graphs relevant to quark-gluon scattering are depicted in fig. 3. Diagrams 3a, 3b receive O(a 2)

terms from the vertices and the polarization vectors. According to (8) and (14) these lattice artefacts cancel each other.

Diagrams 3c is more complicated to analyze because O(a 2) terms arise from the quark-gluon vertex, the gluon propagator, the three-gluon vertex and the polarization vectors. The different sources are schematically indicated in fig. 4.

Due to the on-mass-shell condition for the external legs (meant to include the transversality of physical polarizations) only the 6uv part of the gluon propagator contributes. Adding figs. 4a and 4b we have effectively one diagram where the internal gluon is contracted with a polarization vector of the form (14) with e 0 oq = 0, i.e. the complement to figs. 4c, 4d except for the fact that the gluon is virtual. It is now immediately evident from (11)

Fig. 2. Quark-quark scattering.

409


)::: ) > >. + ".,.'" + ----<</ ." + k.. "% ,%

a) b) e) d)

Fig. 3. Quark-gluon scattering.

. > " k . > ." > < ÷ , , "-<, ÷ + - - , , . + -

a) b) c) d~ e)

Fig. 4. The various sources of O(a 2) terms for diagram 3c indicated by a dot.

that the O(a 2) contributions from the 3 polarization vectors cancel the O(a 2) terms from fig. 4e which are propor. t ional to the continuum limit form of the three-gluon vertex.

Since the two external gluon legs are on the mass shell there remains from the yet uncancelled 5u~6ur structure of the three-gluon vertex a term proport ional to q2(r u - s#), where q refers to the internal gluon. This contribu- t ion is cancelled by diagram 3d whose O(a 2) contr ibut ion was given in eq. (10).

To summarize, we have shown form SU(N) lattice gauge theory that improvement on the tree level can be extended to the fermion sector by including besides the planar plaquettes o f size 2, quark couplings which extend along straight lines across 2 lattice units.

References

[ 1] K. Symanzik, in: Mathematical problems in theoretical physics, Lecture notes in physics, Vol. 153 (Springer, Berlin, 1982); DESY preprints DESY 83-016, 83-026.

[2] G. Martinelli, G. Parisi and R. Petronzio, Phys. Lett. 100B (1981) 485; B. Berg, S. Meyer, I. Montvay and K. Symanzik, Phys. Lett. 126B (1983) 467; S. Meyer, Universit~it Kaiserslautern preprint; M. Falcioni et al., Frascati preprint LNF-83/7 (P); R. Musto, F. Nicodemi and R. Pettorino, Phys. Lett. 129B (1983) 95.

[3] S. Belforte, G. Curci, P. Menotti and G. Paffuti, Phys. Lett. 131B (1983) 423; B. Berg, A. Billoire, S. Meyer and C. Panagiotakopoulos, Phys. Lctt. 133B (1983) 359.

[4] P. Weisz, Nucl. Phys. B212 (1983) 1; G. Curci, P. Menotti and G. Paffuti, Phys. Lett. 130B (1983) 205.

[5] K.G. Wilson, Phys. Rev. D10 (1974) 2445. [6] K.G. Wilson, Erice Lectures (1975). [7] W. Bernreuther and W. Wetzel, Phys. Lett. 132B (1983) 382. [8] H. Kawai, R. Nakayama and K. Seo, Nucl. Phys. B189 (1981) 40. [9] T. Eguchi and R. Nakayama, Phys. Lett. 126B (1983) 89.

410

extending improvement to fermions

Documents