O(a^2) corrections to 1-loop matrix elements of 4-fermion operators with improved fermion/gluon actions

8 Pages·0.12 MB·English

by  Martha Constantinou

O(a2) corrections to 1-loop matrix elements of
4-fermion operators with improved fermion/gluon
actions
0
1
0
2 Martha Constantinoua∗, Vittorio Lubiczb, HaralambosPanagopoulosa,Apostolos

n Skouroupathisa∗,FotosStylianou†a∗
a
a DepartmentofPhysics,UniversityofCyprus
J
P.O.Box20537,NicosiaCY-1678,Cyprus
8
b DipartimentodiFisica,UniversitàdiRomaTreandINFN,SezionediRomaTre,

] ViadellaVascaNavale84,I-00146Roma,Italy
t
a
E-mail: [email protected],[email protected],[email protected],
l
- [email protected],[email protected]
p
e
h We calculatethecorrectionstothe amputatedGreen’sfunctionsof4-fermionoperators,in
[
1-loopLattice Perturbationtheory. The novelaspectof our calculationsis that they are carried

1 outtosecondorderinthelatticespacing,O(a2).
v
We employtheWilson/cloveractionformassless fermions(alsoapplicableforthe twisted
1
massaction in thechirallimit) andthe Symanzikimprovedaction forgluons. Ourcalculations
4
2 havebeencarriedoutinageneralcovariantgauge.Resultshavebeenobtainedforseveralpopular
1
choicesofvaluesfortheSymanzikcoefficients(Plaquette,Tree-levelSymanzik,Iwasaki,TILW
.
1 andDBW2action).
0 WepayparticularattentiontoD F=2operators,bothParityConservingandParityViolating
0
1 (F standsforflavour: S,C,B). We studythemixingpatternoftheseoperators,to O(a2), using
: theappropriateprojectors. Ourresultsforthecorrespondingrenormalizationmatricesaregiven
v
i as a function of a large numberof parameters: couplingconstant, clover parameter, numberof
X
colors,latticespacing,externalmomentumandgaugeparameter.
ar TheO(a2)correctionterms(alongwithourpreviousO(a2)calculationofZY )areessential
ingredientsforminimizingthelatticeartifactswhicharepresentinnon-perturbativeevaluations
ofrenormalizationconstantswiththeRI′-MOMmethod.
Alongerwrite-upofthiswork,includingnon-perturbativeresults,isinpreparationtogetherwith
membersoftheETMCollaboration[1].
PACS: 11.15.Ha, 12.38.Gc, 11.10.Gh, 12.38.Bx
TheXXVIIInternationalSymposiumonLatticeFieldTheory
July26-31,2009
PekingUniversity,Beijing,China
∗Work supported in part by the Research Promotion Foundation of Cyprus (Proposal Nr: TEXN/0308/17,
ENIS X/0506/17).
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
O(a2)correctionsto1-loopmatrixelementsof4-fermionoperators FotosStylianou
1. Introduction
A number of flavour-changing processes are currently under study in Lattice simulations.
Among the most common examples are the decay K → pp and K0–K¯0 oscillations. From ex-
perimental evidence, we know that these weak processes violate the CP symmetry. In theory, the
calculation oftheamountofCPviolation inK0–K¯0 oscillations requirestheknowledgeofB .
K
TheKaonB parameterisobtained fromtheD S=2weakmatrixelement:
K
hK¯0|OˆD S=2|K0i
B = , (1.1)
K 8hK¯0|s¯gm d|0ih0|s¯gm d|K0i
3
where s and d stand for strange and down quarks, and OˆD S=2 is the effective 4-quark interaction
renormalized operator, corresponding tothebareoperator:
OD S=2=(s¯gmLd)(s¯gmLd), gmL=gm ( −g5). (1.2)
1
The above operator splits into parity-even and parity-odd parts; in standard notation: OD S=2 =
OD S=2 −OD S=2 . Since the above weak process is simulated in the framework of Lattice QCD,
VV+AA VA+AV
whereParityisasymmetry,theparity-oddpartgivesnocontribution totheK0–K¯0 matrixelement.
Thus,weconclude thatB canbeextracted fromthecorrelator (x >0,y <0):
K 0 0
CKOK(x,y)=h(d¯g5s)(x)OˆVD SV=+2AA(0)(d¯g5s)(y)i, OVD SV=+2AA=(s¯gm d)(s¯gm d)+(s¯gm g5d)(s¯gm g5d), (1.3)
whereOD S=2 isthebareoperatorandOˆD S=2 istherespective renormalized operator.
VV+AA VV+AA
In place of the operator in Eq. (1.3) it is advantageous to use a four-quark operator with a
different flavourcontent(s,d,s′,d′),andwithD S=D s+D s′=2,namely[2]:
OVD VS=+2AA=(s¯gm d)(s¯′gm d′)+(s¯gm g5d)(s¯′gm g5d′)+(s¯gm d′)(s¯′gm d)+(s¯gm g5d′)(s¯′gm g5d), (1.4)
where now the correlator is given by: CKOK′(x,y)=h(d¯g5s)(x)2OVD VS=+2AA(0)(d¯′g5s′)(y)i. Making
use of Wick’s theorem one checks the equality: CKOK′(x,y)=CKOK(x,y), which means that both
correlators containthesamephysicalinformation.
Theaforementioned matrix elements are very sensitive to various systematic errors. A major
issue facing Lattice Gauge Theory, since its early days, has been the reduction of effects induced
bythefinitenessoflatticespacing a,inordertobetterapproach theelusivecontinuum limit.
Inordertoobtainreliablenon-perturbative estimatesofphysicalquantities(i.e. improvingthe
accuracy of B ) it is essential to keep under control the O(a) systematic errors in simulations or,
K
additionally, reducethelatticeartifactsinnumericalresults. Suchareduction, regardingrenormal-
ization functions, can be achieved by subtracting appropriately the O(a2) perturbative correction
termspresented inthispaper, fromrespective non-perturbative results.
In this paper we calculate the amputated Green’s functions and the renormalization matrices
of the complete basis of 20 four-fermion operators of dimension six which do not need power
subtractions (i.e. mixing occurs only with other operators of equal dimensions). The calculations
are carried out up to1-loop in Lattice Perturbation theory and up toO(a2)in lattice spacing. Our
results are immediately applicable to other D F =2 processes of great phenomenological interest,
suchasD−D¯ orB−B¯mixing. Letusalsomentionthatingenericnewphysicsmodels(i.e. beyond
thestandardmodel),thecompletebasisof4-fermionoperatorscontributestoneutralmesonmixing
amplitudes; thisisthecaseforinstanceofSUSYmodels(seee.g. [3]).
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O(a2)correctionsto1-loopmatrixelementsof4-fermionoperators FotosStylianou
2. Amputated Green’s functions of4-fermion D S=D s+D s′ =2 operators.
Inthisworkweevaluate,uptoO(a2),the1-loopmatrixelementofthe4-fermionoperators1:
O ≡(s¯Xd)(s¯′Yd′)≡(cid:229) (cid:229) (cid:229) s¯c (x)X dc (x) s¯′d (x)Y d′d (x) (2.1)
XY k1 k1k2 k2 k3 k3k4 k4
x c,dk1,k2,k3,k4(cid:16) (cid:17)(cid:16) (cid:17)
OF ≡(s¯Xd′)(s¯′Yd)≡(cid:229) (cid:229) (cid:229) s¯c (x)X d′c (x) s¯′d (x)Y dd (x) (2.2)
XY k1 k1k2 k2 k3 k3k4 k4
x c,dk1,k2,k3,k4(cid:16) (cid:17)(cid:16) (cid:17)
with a generic initial state: d¯′a4(p )s′a3(p )|0i, and a generic final state: h0|d¯a2(p )sa1(p ). Spin
i4 4 i3 3 i2 2 i1 1
indicesaredenotedbyi,k,andcolorindicesbya,c,d,whileX andY correspond tothefollowing
setofproductsoftheDiracmatrices:
1
X,Y ={ ,g 5,gm ,gm g 5,s mn ,g 5s mn }≡{S,P,V,A,T,T˜}; s mn = [gm ,gn ]. (2.3)
1 2
OurcalculationsareperformedusingmasslessfermionsdescribedbytheWilson/cloveraction.
By taking m =0, our results are identical also for the twisted mass action and the Osterwalder-
f
Seiler action in the chiral limit (in the so called twisted mass basis). For gluons we employ a
3-parameter family of Symanzik improved actions, which comprises all common gluon actions
(Plaquette, tree-level Symanzik, Iwasaki, DBW2, Lüscher-Weisz). Conventions and notations for
the actions used, as well as algebraic manipulations involving the evaluation of 1-loop Feynman
diagrams(uptoO(a2)),aredescribed indetailinRef. [4].
Toestablish notation andnormalization, letusfirstwritethetree-level expression fortheam-
putated Green’sfunctions oftheoperators O andOF :
XY XY
L tXrYee(p1,p2,p3,p4,rs,rd,rs′,rd′)ai11ia22i3ai34a4 =Xi1i2Yi3i4d a1a2d a3a4, (2.4)
(L F)tXrYee(p1,p2,p3,p4,rs,rd,rs′,rd′)ai11ia22i3ai34a4 =−Xi1i4Yi3i2d a1a4d a3a2, (2.5)
whereristheWilsonparameter, oneforeachflavour.
We continue with the first quantum corrections. There are twelve 1-loop diagrams that enter
our4-fermioncalculation, sixforeachoperatorO ,OF . Thediagramsd −d corresponding to
XY XY 1 6
theoperatorO areillustratedinFig. 1. Theothersixdiagrams,dF−dF,involvedintheGreen’s
XY 1 6
function of OF are similar to d −d , and may be obtained from d −d by interchanging the
XY 1 6 1 6
fermionic fieldsd and d′ along withtheir momenta, color and spin indices, and respective Wilson
parameters.
Theonly diagrams thatneed tobecalculated from firstprinciples ared ,d andd ,whilethe
1 2 3
rest can be expressed in terms of the first three. In particular, the expressions for the amputated
Green’sfunctions L XY −L XY canbeobtained viathefollowingrelations:
d4 d6
⋆
L Xd4Y(p1,p2,p3,p4,rs,rd,rs′,rd′)ai11ia22i3ai34a4 = L Xd1Y(−p2,−p1,−p4,−p3,rd,rs,rd′,rs′)ai22ia11i4ai43a3 , (2.6)
L Xd5Y(p1,p2,p3,p4,rs,rd,rs′,rd′)ai11ia22i3ai34a4 = (cid:16)L Yd2X(p3,p4,p1,p2,rs′,rd′,rs,rd)ai33ia44i1ai12a2, (cid:17) (2.7)
L Xd6Y(p1,p2,p3,p4,rs,rd,rs′,rd′)ai11ia22i3ai34a4 = L Yd3X(p3,p4,p1,p2,rs′,rd′,rs,rd)ai33ia44i1ai12a2. (2.8)
Oncewehaveconstructed L XY −L XY wecanuserelation:
d4 d6
(L F)XdjY(p1,p2,p3,p4,rs,rd,rs′,rd′)ai11ia22i3ai34a4 =−L XdjY(p1,p4,p3,p2,rs,rd′,rs′,rd)ai11ia44i3ai32a2, (2.9)
1ThesuperscriptletterFstandsforFierz.
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O(a2)correctionsto1-loopmatrixelementsof4-fermionoperators FotosStylianou
s’ s s’ s s’ s s’ s s’ s s’ s
p p p p p p p p p p p p
3 1 3 1 3 1 3 1 3 1 3 1
Y X Y X Y X Y X Y X Y X
p p p p p p p p p p p p
4 d’ d 2 4 d’ d 2 4 d’ d 2 4 d’ d 2 4 d’ d 2 4 d’ d 2
d1 d2 d3 d4 d5 d6
Figure 1: 1-loop diagrams contributing to the amputated Green’s function of the 4-fermi operator OXY.
Wavy(solid)linesrepresentgluons(fermions).
toderivetheexpressionsfor(L F)XY−(L F)XY. FromtheamputatedGreen’sfunctionsforalltwelve
d1 d6
diagramswecanwritedownthetotal1-loopexpressions fortheoperators O andOF :
XY XY
6 6
L XY = (cid:229) L XY, (L F)XY = (cid:229) (L F)XY. (2.10)
1−loop dj 1−loop dj
j=1 j=1
Inouralgebraicexpressionsforthe1-loopamputatedGreen’sfunctionsL XY,L XY andL XY we
d1 d2 d3
kept the Wilson parameters for each quark field distinct, that is: r , r , r , r for the quark fields
s d s′ d′
s, d, s′ and d′ respectively. For the required numerical integration of the algebraic expressions of
theintegrands, corresponding toeachFeynmandiagram,weareforcedtochoosethesquareofthe
valueforeachrparameter. Asinallpresentdaysimulations, weset:
r2=r2 =r2 =r2 ≡1. (2.11)
s d s′ d′
Concerning the external momenta p (shown explicitly in Fig. 1) we have chosen to evaluate the
i
amputated Green’sfunctions attherenormalization point:
p = p = p = p ≡ p. (2.12)
1 2 3 4
Itiseasyandnottimeconsumingtorepeatthecalculations forotherchoicesofWilsonparameters
and for other renormalization prescriptions. The final 1-loop expressions for L XY, L XY and L XY,
d1 d2 d3
uptoO(a2),areobtained asafunction of: thecoupling constant g, cloverparameter c ,number
SW
ofcolorsN ,latticespacing a,externalmomentum pandgaugeparameterl .
c
The crucial point of our calculation is the correct extraction of the full O(a2) dependence
from loop integrands with strong IR divergences (convergent only beyond 6 dimensions). The
singularities areisolated using the procedure explained in Ref. [4]. Inorder toreduce thenumber
ofstrong IRdivergent integrals, appearing indiagram d ,wehaveinserted theidentity below into
1
selected 3-pointfunctions:
1= 1 k\+ap2+k\−ap2−2kˆ2+16(cid:229) sin(ks )2sin(aps )2 , (2.13)
ap2 s
(cid:16) (cid:17)
where qˆ2 =4(cid:229) m sin2(q2m c) and k(p) is the loop (external) momentum. The common factor in Eq.
(2.13)canbetreatedbyTaylorexpansion. Forourcalculations itwasnecessary onlytoO(a0):
1 = 1 +(cid:229) s ps4 +O(a2p2). (2.14)
ap2 a2p2 (p2)2
c
4
O(a2)correctionsto1-loopmatrixelementsof4-fermionoperators FotosStylianou
HerewepresentoneofthefourintegralswithstrongIRdivergences thatenterinthiscalculation:
Z−pp (2dp4k)4kˆ2ski\n+(kam )p2sik\n−(kan )p2 =d mn 0.002457072288−ln6(a42pp22)+a2p2 0.00055270353(6)−ln5(1a22pp22)
h (cid:16) (cid:17)
−a2p2m 0.0001282022(1)+ln7(6a82pp22) +0.000157122310a2(cid:229) sp2ps4 +a2pm pn 0.001870841540a21p2
(cid:16) (cid:17) i h
−0.00029731225(4)+ln(a2p2)−0.000047949674(p2m +p2n )+0.000268598599(cid:229) s p4s +O(a4p4).
768p 2 p2 (p2)2
i
The results for the other three integrals can be found in Ref. [4]. Integrands with simple IR
divergences (convergent beyond4dimensions) canbehandled bywell-knowntechniques.
Duetolackofspacewepresent onlytheresults forL XY andforthespecialchoices: c =0,
l =0(LandauGauge),r =r =r =r =1,andtree-levd1el Symanzikaction: SW
s d s′ d′
g2 d d 1
L XY(p)a1a2a3a4 = d d − a1a2 a3a4 × X Y − ln(a2p2)−0.05294139(2)
d1 i1i2i3i4 16p 2(cid:18) a1a4 a3a2 Nc (cid:19) (cid:26) i1i2 i3i4(cid:20) 2 (cid:21)
+(cid:229) (Xg m ) (Yg m ) [−0.507914049(6)]+(cid:229) (Xg m g n ) (Yg m g n ) 1ln(a2p2)+0.0185984988(9)
i1i2 i3i4 i1i2 i3i4 8
m m ,n (cid:20) (cid:21)
+ (cid:229) (Xg m g r ) (Yg n g r ) 0.3977157268533pm pn +a(L )XY+a2(L )XY , (2.15)
i1i2 i3i4 p2 O(a1) d1 O(a2) d1
m ,n ,r (cid:20) (cid:21) (cid:27)
where:
(L O(a1))Xd1Y =(cid:229)m (Xg m )i1i2Yi3i4+Xi1i2(Yg m )i3i4 × ipm −14ln(a2p2)+0.09460083(1)
(cid:16) (cid:17) (cid:20) (cid:18) (cid:19)(cid:21)
+(cid:229) (Xg m g n )i1i2(Yi3i4g n )+(Xg n )i1i2(Yg m g n )i3i4 × ipm 116ln(a2p2)+0.1692905881(6) , (2.16)
m ,n (cid:16) (cid:17) (cid:20) (cid:18) (cid:19)(cid:21)
and:
(L O(a2))Xd1Y =Xi1i2Yi3i4 p2 −1772 ln(a2p2)+1.32362250(4) +0.06213649(4)(cid:229) sp2p4s
(cid:20) (cid:18) (cid:19) (cid:21)
+(cid:229) (Xg m )i1i2(Yg m )i3i4 p2 −478ln(a2p2)+0.059895142(8) +1.01694823(2)p2m
m
(cid:20) (cid:18) (cid:19) (cid:21)
+(cid:229) (Xg m g n ) Y +X (Yg m g n ) × 0.00592406(2)pn p3m
i1i2 i3i4 i1i2 i3i4 p2
m ,n " #
(cid:16) (cid:17)
+(cid:229) (Xg m )i1i2(Yg n )i3i4 pm pn −16ln(a2p2)−0.19915360(1)
m ,n (cid:20) (cid:18) (cid:19)(cid:21)
+(cid:229) (Xg m g n ) (Yg m g n ) p2 7 ln(a2p2)−0.089628048(6) −0.048180850(cid:229) s ps4
i1i2 i3i4 240 p2
m ,n (cid:20) (cid:18) (cid:19)
29
+p2m − ln(a2p2)+0.16608907(3)
180
(cid:18) (cid:19)(cid:21)
+ (cid:229) (Xg m g r )i1i2(Yg n g r )i3i4 pm pn 34610 ln(a2p2)−0.21865900(2)+0.140961390(cid:229) sp2p4s
m ,n ,r (cid:20) (cid:18) (cid:19)
(p3m pn +pm p3n ) pm pn p2r
−0.110138790 −0.477634781(8) . (2.17)
p2 p2
(cid:21)
Similarexpressions existforL XY andL XY.
d2 d3
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O(a2)correctionsto1-loopmatrixelementsof4-fermionoperators FotosStylianou
3. Mixingand RenormalizationofO and OF onthe lattice.
XY XY
Thematrixelement hK¯0|OD S=2 |K0iisvery sensitive to various systematic errors. Themain
VV+AA
roots of this problem are: a) O(a) systematic errors due to numerical integration, b) the operator
OD S=2 mixeswithother 4-fermion D S=2operators ofdimension six. Mixing withoperators of
VV+AA
lowerdimensionality isimpossible because thereisnocandidate D S=2operator.
Inordertoaddresstheseproblemswehavecalculatedthemixingpattern(renormalizationma-
trices) of the Parity Conserving and Parity Violating 4-fermion D S=2operators (defined below),
byusingtheamputatedGreen’sfunctionsobtainedintheprevioussection. Amoreextensivetheo-
reticalbackground andnon-perturbative results,concerningrenormalization matricesof4-fermion
operators, can befound inRef. [5](see also[2, 6, 7]). Nextwesummarize allimportant relations
fromRef. [5]neededforthepresentcalculation.
One can construct a complete basis of 20 independent operators which have the symmetries
ofthegenericQCDWilsonlattice action(ParityP,ChargeconjugationC,Flavourexchange sym-
metry S≡(d ↔d′),Flavour Switching symmetries S′≡(s↔d,s′ ↔d′)and S′′≡(s↔d′,d ↔s′)),
with4degeneratequarks. Thisbasiscanbedecomposedintosmallerindependent basesaccording
to thediscrete symmetries P,S,CPS′,CPS′′. Followingthe notation of Ref. [5] wehave 10Parity
Conservingoperators,Q,(P=+1,S=±1)and10ParityViolatingoperators,Q,(P=−1,S=±1):
QS=±1≡ 1 O ±OF +1 O ±OF , QS=±1≡ 1 O ±OF +1 O ±OF ,
1 2 VV VV 2 AA AA 1 2 VA VA 2 AV AV
Q2S=±1≡ 12(cid:2)OVV ±OVFV(cid:3)−12(cid:2)OAA±OAFA(cid:3), nQ2S=±1≡ 12(cid:2)OVA±OVFA(cid:3)−21(cid:2)OAV±OAFV(cid:3),

Q3S=±1≡ 12(cid:2)OSS±OSFS (cid:3)−21 O(cid:2)PP±OPFP ,(cid:3) (Q3S=±1≡ 12(cid:2)OPS±OPFS(cid:3)−21 (cid:2)OSP±OSFP (cid:3),

Q4S=±1≡ 12(cid:2)OSS±OSFS(cid:3)+21(cid:2)OPP±OPFP(cid:3), Q4S=±1≡ 12(cid:2)OPS±OPFS(cid:3) +12(cid:2)OSP±OSFP(cid:3) ,
Q5S=±1≡ 12(cid:2)OTT ±OTFT(cid:3) , (cid:2) (cid:3) (Q5S=±1≡ 21(cid:2)OTT˜±OTFT˜(cid:3) . (cid:2) (cid:3)


Summation ov(cid:2)er all indepe(cid:3)ndent Lorentz indices (if any), of the D(cid:2)irac matrices(cid:3), is implied. The
operators shown above are grouped together according to their mixing pattern. This implies that
therenormalizationmatricesZS=±1(ZS=±1),fortheParityConserving(Violating)operators,have
theform:
Z Z Z Z Z S=±1 Z 0 0 0 0 S=±1
11 12 13 14 15 11
Z Z Z Z Z 0 Z Z 0 0
 21 22 23 24 25  22 23 
ZS=±1= Z Z Z Z Z , ZS=±1= 0 Z Z 0 0 . (3.1)
31 32 33 34 35 32 33
Z41 Z42 Z43 Z44 Z45  0 0 0 Z44 Z45
Z51 Z52 Z53 Z54 Z55  0 0 0 Z54 Z55
   
   
Now the renormalized Parity Conserving (Violating) operators, QˆS=±1 (QˆS=±1), are defined
viatheequations:
QˆS=±1=ZS=±1·QS=±1, QˆS=±1=ZS=±1·QS=±1, (3.2)
l lm m l lm m
where l,m=1,...,5 (a sum over m is implied). The renormalized amputated Green’s functions
LˆS=±1 (LˆS=±1) corresponding to QS=±1 (QS=±1), are given in terms of their bare counterparts
LS=±1 (LS=±1)through:
LˆS=±1=Z−2ZS=±1·LS=±1, LˆS=±1=Z−2ZS=±1·LS=±1, (3.3)
l Y lm m l Y lm m
6
O(a2)correctionsto1-loopmatrixelementsof4-fermionoperators FotosStylianou
whereZY isthequarkfieldrenormalization constant.
Inordertocalculatetherenormalization matricesZS=±1(ZS=±1),wemakeuseoftheappro-
priateParityConserving (Violating)ProjectorsPS=±1 (PS=±1):
P +P P +P
PS=±1 ≡ + VV AA , PS=±1 ≡ − VA AV ,
1 64N (N ±1) 1 64N (N ±1)
c c c c
P −P P −P P −P P −P
PS=±1 ≡ + VV AA ± SS PP , PS=±1 ≡ − VA AV ∓ SP PS ,
2 64(N2−1) 32N (N2−1) 2 64(N2−1) 32N (N2−1)
c c c c c c
P −P P −P P −P P −P
PS=±1 ≡ ± VV AA + SS PP, PS=±1 ≡ ∓ VA AV − SP PS,
3 32N (N2−1) 16(N2−1) 3 32N (N2−1) 16(N2−1)
c c c c c c
P +P P P +P P
PS=±1 ≡ + SS PP ∓ TT , PS=±1 ≡ + SP PS ∓ TT˜ ,
4 32Nc(Nc2−1) 32Nc(Nc2−1) 4 32Nc(Nc2−1) 32Nc(Nc2−1)
2Nc±1 2Nc±1
P +P P P +P P
PS=±1 ≡ ∓ SS PP + TT , PS=±1 ≡ ∓ SP PS + TT˜ ,
5 32Nc(Nc2−1) 96Nc(Nc2−1) 5 32Nc(Nc2−1) 96Nc(Nc2−1)
2Nc∓1 2Nc∓1
where P ≡(X ⊗Y )d d . Again, summation is implied over all independent Lorentz
XY i2i1 i4i3 a2a1 a4a3
indices (if any) of the Dirac matrices. The above Projectors are chosen to obey the following
orthogonality conditions:
Tr(PS=±1·LS=±1 )=d , Tr(PS=±1·LS=±1 )=d , (3.4)
l m(tree) lm l m(tree) lm
wherethetraceistakenoverspinandcolorindices,andLS=±1,LS=±1arethetree-levelamputated
(tree) (tree)
Green’sfunctions oftheoperators QS=±1,QS=±1 respectively.
Weimposetherenormalization conditions:
Tr(PS=±1·LˆS=±1)=d , Tr(PS=±1·LˆS=±1)=d . (3.5)
l m lm l m lm
By inserting Eqs. (3.3) in the above relations, we obtain the renormalization matrices ZS=±1,
ZS=±1 intermsofknownquantities:
ZS=±1=ZY2 DS=±1 T −1, ZS=±1=ZY2 DS=±1 T −1, (3.6)
h i h i
(cid:0) (cid:1) (cid:0) (cid:1)
where:
DS=±1≡Tr(PS=±1·LS=±1), DS=±1≡Tr(PS=±1·LS=±1). (3.7)
lm l m lm l m
NotethatDS=±1 andDS=±1 havethesamematrixstructure asZS=±1 andZS=±1 respectively.
Due to lack of space we provide only the matrix DS=+1 (Parity Violating P=−1, Flavour
exchange symmetry S=+1) for the special choices: c =0, l =0 (Landau Gauge), r =r =
SW s d
r =r =1,N =3,andtree-levelSymanzikaction:
s′ d′ c
1+D 0 0 0 0 S=+1
11
0 1+D D 0 0
 22 23 
DS=+1= 0 D 1+D 0 0 (3.8)
32 33
 0 0 0 1+D44 D45 
 0 0 0 D54 1+D55
 
 
7
O(a2)correctionsto1-loopmatrixelementsof4-fermionoperators FotosStylianou
where:
D =+ g2 7.607190(2)+2ln(a2p2)+ 2.642227(3)−19 ln(a2p2) a2p2−2.79899088(3)a2(cid:229) s ps4 ,
11 16p 2 18 p2
(cid:20) (cid:16) (cid:17) (cid:21)
D =+ g2 2.299519(2)+ln(a2p2)+ 1.846794(4)−25 ln(a2p2) a2p2−0.87361421(2)a2(cid:229) s ps4 ,
22 16p 2 36 p2
(cid:20) (cid:21)
(cid:16) (cid:17)
D =− g2 1.1931473(4)+ 1.4685426(7)−1 ln(a2p2) a2p2−0.89270364(3)a2(cid:229) s ps4 ,
23 16p 2 3 p2
(cid:20) (cid:16) (cid:17) (cid:21)
D =− g2 10.970216(2)−6ln(a2p2)+ 6.711307(3)−7 ln(a2p2) a2p2−1.7027590(1)a2(cid:229) s p4s ,
32 16p 2 6 p2
(cid:20) (cid:21)
(cid:16) (cid:17)
D =+ g2 11.595959(2)−8ln(a2p2)+ 3.102499(4)−4 ln(a2p2) a2p2+1.92846914(2)a2(cid:229) s ps4 ,
33 16p 2 9 p2
(cid:20) (cid:21)
(cid:16) (cid:17)
D =+ g2 10.269734(3)−5ln(a2p2)− 0.286209(5)− 1 ln(a2p2) a2p2+2.01567490(4)a2(cid:229) s ps4 ,
44 16p 2 18 p2
(cid:20) (cid:16) (cid:17) (cid:21)
D =+ g2 9.732710(2)−5ln(a2p2)+ 4.602710(4)−7 ln(a2p2) a2p2−1.07855465(9)a2(cid:229) s p4s ,
45 16p 2 9 p2
(cid:20) (cid:16) (cid:17) (cid:21)
D =+ g2 1.1783609(6)+1 ln(a2p2)+ 1.255191(1)−17 ln(a2p2) a2p2−0.98220341(2)a2(cid:229) s ps4 ,
54 16p 2 3 54 p2
(cid:20) (cid:16) (cid:17) (cid:21)
D =+ g2 2.297078(2)+17 ln(a2p2)+ 0.828945(3)−23 ln(a2p2) a2p2−3.06215785(3)a2(cid:229) s ps4 .
55 16p 2 3 27 p2
(cid:20) (cid:16) (cid:17) (cid:21)
In order to obtain ZY for a given renormalization prescription, one must make use of the inverse
fermion propagator, S−1, calculated (up to 1-loop and up to O(a2) for massless Wilson/clover
fermionsandSymanzikgluons)inRef. [4].
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