Three-Loop Results in QCD with Wilson Fermions

3 Pages·0.08 MB·English

by  B. Alles

1
Three-Loop Results in QCD with Wilson Fermions
B. All´esa , C. Christoub , A. Feob,c , H. Panagopoulosb∗ and E. Vicaric,d
aDipartimento di Fisica and I.N.F.N., Milano, Italy
bDepartment of Natural Sciences, University of Cyprus
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9 cDipartimento di Fisica, Universit`a di Pisa, Italy
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1 dI.N.F.N., Sezione di Pisa, Italy

n
a WecalculatethethirdcoefficientofthelatticebetafunctioninQCDwithWilsonfermions,extendingthepure
J gaugeresults of Lu¨scherand Weisz; weshow howthiscoefficient modifiesthescaling function onthelattice. We
2 alsocalculatethethree-loopaverageplaquetteinthepresenceofWilson fermions. Thisallowsustocomputethe
1 lattice scaling function both in thestandard and energy schemes.

2
v
1. INTRODUCTION presentwork,weareextendingourcalculationto
8
include contributions from Wilson fermions.
1
In lattice formulations of field theories the rel-
0 We also present here the calculation of the 3-
evant region for continuum physics is the one
0 loop coefficient w of the average plaquette, in
3
1 wherescalingisverified. Scalinginthecontinuum QCD with Wilson fermions. In the absence of
7 limit requires that all RG-invariant dimension-
fermions,this quantitywascalculatedbysomeof
9 less ratios of physical quantities approach their
the present authors [3].
/
t continuum value with non-universal corrections OurresultsforbL andw allowustostudythe
a which get depressed by integer powers of the in- 2 3
l corrections to asymptotic scaling in Monte Carlo
- versecorrelationlength. Testsofasymptoticscal-
p simulations of dimensionful quantities, for both
inginvolvetherelationshipbetweenthebarecou-
e the standard and energy schemes.
h pling and the cutoff as an essential ingredient.
The involved algebra of lattice perturbation
: This relationship is expressed in the bare lattice
v theory was carried out by making use of an ex-
Xi β function which reads, in standard notation: tensive computer code developed by us in recent
ar βL(g0)≡−addga0 |gr,µ=−bL0g03−bL1g05−bL2g07−...(1) ytieoanr,st[h4e];cfoordethweapsuerxptoesnedseodf tthoeinpcrleusdenetfecramlciuolnas-
and form factors.
The first two coefficients in Eq. (1) are scheme-
independent and well known. Our purpose is to
computethethirdcoefficient,bL,inSU(N)Yang- 2. CALCULATION OF β2L
2
Mills theory with N species of Wilson fermions.
f We performed this calculation using the back-
M. Lu¨scher and P. Weisz [1] have calculated
ground field method. In order facilitate compari-
this coefficient in the pure gluonic case, using a
son we have adopted the notation of Ref.[1].
coordinate space method for evaluating the lat-
We set out to compute d (µa) in:
2
tice integrals. We have verified their result [2],
using a different technique where firstly the inte-
α (µ)=α +d (µa)α2+d (µa)α3+...,
grands are Taylor expanded in powers of the ex- MS 0 1 0 2 0
ternalmomentaandthencomputedtermbyterm
α =g2/4π, α (µ)=g2(µ)/4π (2)
with the introduction of an IR regulator. In the 0 0 MS
∗Presentedthepaper. As we shall see, bL will follow immediately from
2
2
this quantity. d (µa) can be expressed as: (sub)divergences and extractionof logarithms by
2
opportune additions/subtractions (in the case at
d (µa)=(4π)2{(ν(1)(p)−ν(1)(p))2−ν(2)(p)+ν(2)(p) hand, this step gives rise to hundreds of types of
2 R R
expressions,eachcontainingtypicallyhundredsof
−λ(∂ν(1)(p)/∂λ)(ω(1)(p)−ω(1)(p))} (3) terms); automaticgenerationofhighlyoptimized
R R λ=λ0
Fortrancodefortheloopintegrationofeachtype
in terms of the gauge parameter λ, and of ω(p)
of expression.
and ν(p) (the gauge- and background- two point
The integrals are then performed numerically
lattice functions, respectively):
on finite lattices. Our programs perform extrap-
olationsofeachexpressiontoabroadspectrumof
XΓ(2,0,0)(p,−p)aµbµ=−δab3pˆ2[1−ν(p)]/g02 functional forms of the type: e (lnL)j/Li,
Pi,j ij
µ analyze the quality of each extrapolation using a
variety of criteria and assign statistical weights
XΓ(0,2,0)(p,−p)aµbµ=−δabpˆ2[3(1−ω(p))+λ0] to them, and finally produce a quite reliableesti-
µ
mate of the systematic error.
Several consistency checks can be performed
∞
ν(p)=Xg02lν(l)(p) oOn(pt0h)epaserptsaroabteeycsoenvterriablurteiloantsionosf,eaancdhmduiasgtrsaumm:
l=1
∞ up to zero by gauge invariance; Lorentz non-
ω(p)=Xg02lω(l)(p) (4) invariantterms Pµp4µ/p2 mustcancel; the single
and double logarithms are related to known re-
l=1
sults. We have performed all these checks both
ThesubscriptRreferstotheanalogousMSrenor-
analytically and numerically;this was also a use-
malized quantities.
ful verification of the correctness of our error es-
The 1-loop quantitites appearing above,
timates.
ν(1),νR(1),ω(1),ωR(1), are known. For the MS 2- Ourfinalresultforν(2)(p)is(wetaker =1for
loopfunctionν(2)(p)therearefournewdiagrams the Wilson parameter):
R
with respect to the pure gluonic case. We find:
ν(2)(p)=ν(2)(p)| (6)
Nf=0
ν(2)(λ=1, p)(16π2)2 =N2[8ρ+577/18−6ζ(3)]+
R
+N [(−1/N +3N)ln(p2)/(16π2)2
f
N [(−3ρ−401/36)N+(ρ+55/12−4ζ(3))/N]
f
−0.000706(3)/N+0.000544(5)∗N]
with: ρ=ln(µ2/p2). (5)
Also, we have:
The only remaining quantity to calculate (and
by far the most complicated) is ν(2)(p). The in- ν(1)(p)−ν(1)(Nf=0,p)=ω(1)(p)−ω(1)(Nf=0,p)
clusion of fermions brings in 20 additional dia-
=N ∗[ln(p2)/24π2−0.01373219(1)] (7)
grams (of these, two correspond to fermion mass f
renormalization). Atableofpartialresultswillappearinaforth-
The evaluation of these diagrams requires coming publication [5], where we also expect to
tremendous analytical effort. We have developed provide better accuracy.
an extensive computer code for analytic compu- From the relations given above, we can now
tations in lattice perturbation theory [4]. In a easily construct d (µa) and d (µa). Writing:
1 2
nutshell, this computer code performs: Contrac- d (µa)=d +d ln(µa), d (µa)=(d (µa))2+
1 10 11 2 1
tion among the appropriate vertices; simplifica- d +d ln(µa), the coefficient bL follows imme-
20 21 2
tion of color/Dirac matrices; use of trigonome- diately:
try and momentum symmetries for reduction to
a more compact, canonical form; treatment of bL =bMS+(d d −d d )/(128π3) (8)
2 2 20 11 21 10
3
where Now, the scaling violation parameter q = (b2 −
1
bEb )/(2b3) becomes (at N =3,N =3,r=1):
−1 2 0 0 f
bMS = [−5714N3+N2(−224N+66/N)
2 108(4π)6 f q(k =0.1675)=0.2103(4)
+N (3418N2−561−27/N2)] (9) q(k =0.1560)=0.1764(4) (18)
f
In particular: This indeed represents an improvement over the
standard case. We are presently working on im-
bL(N=3,N =3,r=1)=−0.002284(3) (10)
2 f provingtheaccuracyofourresults,forthelonger
(cf. bL(N=3,N =0)=−0.00159983) (11) write-ups [5,6].
2 f
The deviation, q = (b2 − bLb )/(2b3), in the
1 2 0 0
asymptotic scaling formula:
aΛ =exp(−1/2b g2)(b g2)−b1/2b0 1+qg2+... k 103×w2 103×w3
L 0 0 0 0 (cid:2) 0 (cid:3)
0.156 18.2522(2) 9.15(1)
is now found to be: 0.1575 17.7713(2) 8.91(1)
0.16 16.9767(1) 8.51(1)
q(N=3,N =3,r=1)=0.3694(4) (12)
f
0.164 15.7243(2) 7.89(1)
0.1675 14.6480(6) 7.35(1)
3. THE 3-LOOP MEAN PLAQUETTE
Table 1
We have computed the free energy to 3 loops Values of w ×103 and w ×103. N=3, N =3,
2 3 f
r=1, k: hopping parameter.
−lnZ/V =const.+F g2+F g4+... (13)
2 0 3 0
in QCD with Wilson fermions. A total of 24 dia-
grams involving fermions contribute to F .
3
The equality:
h1−2/Ni=−(1/6V)(∂lnZ/∂β) (14) REFERENCES
(valid by virtue of hS i = 0 beyond tree 1. M. Lu¨scher and P. Weisz, Nucl. Phys. B452
fermion
level) relates F and F directly to the quanti- (1995) 234.
2 3
ties w and w in the expansion of the average 2. B. All´es, A. Feo and H. Panagopoulos, Nucl.
2 3
plaquette: Phys. 491 (1997) 498.
3. B. All´es, M. Campostrini, A. Feo and H.
h1−2/Ni=w1g02+w2g04+w3g06+... (15) Panagopoulos,Phys. Lett. B324 (1994) 433.
4. B. All´es, M. Campostrini, A. Feo and H.
We report below the final values for w and w
2 3
Panagopoulos,Nucl. Phys.B413 (1994)553.
atN=3,N =3,r=1. Abreakdownofthe results,
f
5. C. Christou,A. Feo, H. PanagopoulosandE.
along with the explicit N- and N -dependence
f
Vicari, The Three-Loop β Function in QCD
andimprovederrorestimateswillbepresentedin
with Wilson Fermions, Pisa-Cyprus preprint,
a forthcoming publication [6].
in preparation.
From the plaquette coefficients, we may now
6. B.All´es,A.FeoandH.Panagopoulos,Scaling
construct the coupling constant in the “energy”
Corrections in QCD with Wilson Fermions
scheme:
from the 3-Loop Average Plaquette, Pisa-
gE2 =g02+(w2/w1)g04+(w3/w1)g06+... (16) Cyprus preprint, in preparation.
In terms of this improved definition of the cou-
pling constant, the β function reads:
bE =bL+(b w −b w −b w2/w )/w (17)
2 2 0 3 1 2 0 2 1 1