Higher loop renormalization of fermion bilinear operators

Abstract

We compute the two-loop renormalization functions, in the RI' scheme, of local bilinear quark operators $\bar\psi\Gamma\psi$, where $\Gamma$ denotes the Scalar and Pseudoscalar Dirac matrices, in the lattice formulation of QCD. We consider both the flavor non-singlet and singlet operators; the latter, in the scalar case, leads directly to the two-loop fermion mass renormalization, $Z_m$. As a prerequisite for the above, we also compute the quark field renormalization, $Z_\psi$, up to two loops. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in $c_{SW}$, in terms of both the renormalized and bare coupling constant, in the renormalized Feynman gauge. We also confirm the 1-loop renormalization functions, for generic gauge. A longer write-up of the present work, including the conversion of our results to the MSbar scheme and a generalization to arbitrary fermion representations, can be found in arXiv:0707.2906 .

Full text

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arXiv:0708.4087v1 [hep-lat] 30 Aug 2007
Higher loop renormalization of fermion bilinear
operators
A. Skouroupathis∗†and H. Panagopoulos
University of Cyprus, Department of Physics, Cyprus
E-mail: php4as01@ucy.ac.cy, haris@ucy.ac.cy
We compute the two-loop renormalization functions, in the RI′scheme, of local bilinear quark
operators ¯ ψΓψ, where Γ denotes the Scalar and Pseudoscalar Dirac matrices, in the lattice for-
mulation of QCD. We consider both the flavor non-singlet and singlet operators; the latter, in the
scalar case, leads directly to the two-loop fermion mass renormalization, Zm.
As a prerequisite for the above, we also compute the quark field renormalization, Zψ, up to two
loops.
We use the clover action for fermions and the Wilson action for gluons. Our results are given
as a polynomial in cSW, in terms of both the renormalized and bare coupling constant, in the
renormalized Feynman gauge. We also confirm the 1-loop renormalization functions, for generic
gauge.
A longer write-up of the present work, including the conversion of our results to the MS scheme
and a generalizationto arbitraryfermionrepresentations,can be foundin arXiv:0707.2906.
The XXV International Symposium on Lattice Field Theory
July 30-4 August 2007
Regensburg, Germany
∗Speaker.
†Work supported in part by the Research Promotion Foundation of Cyprus (Proposal Nr: ENIΣX/0506/17).
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/

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Renormalization of fermion bilinears
A. Skouroupathis
1. Introduction
Studies of hadronic properties using the lattice formulation of QCD rely on the computation
of matrix elements and correlation functions of composite operators, made out of quark fields.
A whole variety of such operators has been considered and studied in numerical simulations, in-
cluding local and extended bilinears, and four-fermi operators. A proper renormalization of these
operators is most often indispensable for the extraction of physical results from the lattice.
In this work we study the renormalization of fermion bilinears O = ¯ ψΓψ on the lattice, where
Γ=ˆ1, γ5. Weconsider both flavor singlet and nonsinglet operators. Thecases Γ=γµ, γ5γµ, γ5σµν,
will be presented in a sequel to this work. In order to obtain the renormalization functions of
fermion bilinears we also compute the quark field renormalization, Zψ, as a prerequisite.
We employ the standard Wilson action for gluons and clover-improved Wilson fermions. The
number of quark flavors Nf, the number of colors Ncand the clover coefficient cSWare kept as free
parameters. Our two-loop calculations have been performed in the bare and in the renormalized
Feynman gauge. For 1-loop quantities, the gauge parameter is allowed to take arbitrary values.
The main results presented in this work are the following 2-loop bare Green’s functions (am-
putated, one-particle irreducible (1PI)): • Fermion self-energy: ΣL
scalar ¯ ψψ : ΣL
q : external momentum)
In general, one can use bare Green’s functions to construct ZX,Y
for operator O, computed within a regularization X and renormalized in a scheme Y. We employ
the RI′scheme to compute the various renormalization functions: ZL,RI′
sponding quantities in the MS scheme: ZL,MS
ψ
S
[1].
The flavor singlet scalar renormalization function is equal to the fermion mass multiplicative
renormalization, Zm, which is an essential ingredient in computing quark masses.
Finally, as one of several checks on our results, we construct the 2-loop renormalized Green’s
functions in RI′: ΣRI′
values ofall these functions, computed on thelattice, coincide with values computed in dimensional
regularization (as can be inferred, e.g., from [2]).
The present work is the first two-loop computation of the renormalization of fermion bilinears
on the lattice. There have been madeseveral attempts to estimate ZOnon-perturbatively; see Ref.[1]
for a list of recent references. Some results have also been obtained using stochastic perturbation
theory [3]. A related computation, regarding the fermion mass renormalization Zmwith staggered
fermions can be found in [4].
ψ(q,aL), • 2-pt function of the
P(qaL) (aL: lattice spacing,
S(qaL), • 2-pt function of the pseudoscalar ¯ ψγ5ψ : ΣL
O, the renormalization function
ψ
, ZL,RI′
S
, ZL,RI′
P
. The corre-
, ZL,MS
, ZL,MS
P
, are presented in the longer write-up
O(q, ¯ µ) (O ≡ ψ,S,P), as well as their counterparts in MS: ΣMS
O(q, ¯ µ). The
2. Formulation of the problem
Lattice action: We will make use of the Wilson formulation of the QCD action on the lattice, with
the addition of the clover (SW) term for fermions. The clover coefficient cSWin the Langrangian
is treated here as a free parameter; The fermion mass moin the Langrangian is a free parameter.
However, since we will be using mass independent renormalization schemes, all renormalization
2

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Renormalization of fermion bilinears
A. Skouroupathis
functions which we will be calculating, must be evaluated at vanishing renormalized mass, that is,
when mois set equal to the critical value mcr: mo→ mcr= 0+O(g2
Definition of renormalized quantities: As a prerequisite, we will need the renormalization func-
tions for the gluon, ghost and fermion fields (Aa
parameter α, defined as follows:
µo=√ZAAa
o).
µ, ca, ψ), and for the coupling constant g and gauge
Aa
µ, ca
o=√Zcca, ψo=?Zψψ , go= µεZgg , αo= Z−1
The scale µ enters the relation between goand g only in dimensional regularization (D = 4−2ε
dimensions). We will need ZA, Zc, ZαandZgto 1 loop and Zψto 2 loops.
Definition of the RI′scheme: This renormalization scheme [7, 8, 9]is more immediate for a lattice
regularized theory. It is defined by imposing a set of normalization conditions on matrix elements
at a scale ¯ µ, where (just as in the MS scheme): ¯ µ = µ(4π/eγE)1/2
In Euclidean space, the fermion self energy ΣL
aZAα
(2.1)
(γEis the Euler constant)
o) is renormalized by:
ψ(q,aL) = i/ q+mo+O(g2
lim
aL→0
?
ZL,RI′
ψ
(aL¯ µ)tr?ΣL
ψ(q,aL)/ q?/(4iq2)
?
q2=¯ µ2= 1(2.2)
The trace here is over Dirac indices; a Kronecker delta in color and in flavor indices has been
factored out of the definition of ΣL
self energy) and ZA, Zα(extracted from the gluon propagator). We have checked explicitly that
ZL,RI′
α
= 1 up to one loop, in agreement with the continuum.
For consistency with the Slavnov-Taylor identities, Zgin the RI′scheme is defined as in the
MS scheme. In dimensional regularization (DR) this is achieved by tuning the value of Zgin such
a way as to absorb only the poles in ε which appear in the gluon-fermion-antifermion 1PI vertex
function GA ¯ ψψ(together with matching powers of ln(4π)−γE); this leads to a result for Gfinite
which is finite but not unity. Alternatively, a similar procedure can be performed on the gluon-
ghost-antighost vertex GA¯ cc, leading to exactly the same value for Zg.
The corresponding renormalization condition can be applied on the lattice vertex, GL
quiring that the renormalized expression Gfinite
for GL
g
, using either one of GL
same result is obtained.
Renormalization of fermion bilinears: The renormalization of lattice operators OΓ= ¯ ψΓψ is
defined by: ORI′
Γ
For the scalar (S) and pseudoscalar (P) operators, ZL,RI′
sponding bare 2-point functions ΣL
ψ. Similar conditions hold for Zc(extracted from the ghost
A ¯ ψψ
A ¯ ψψ, re-
A ¯ ψψis the same as that stemming from DR,and similarly
A¯ ccor GL
A¯ cc. We have calculated ZL,RI′
A ¯ ψψ, and have verified that the
Γ= ZL,RI′
(aL¯ µ)OΓo
Γ
can be obtained through the corre-
Γ(qaL) (amputated, 1PI) on the lattice, in the following way:
lim
aL→0
?
ZL,RI′
ψ
ZL,RI′
S
ΣL
S(qaL)
?
q2=¯ µ2=ˆ1 , lim
aL→0
?
ZL,RI′
ψ
ZL,RI′
P
ΣL
P(qaL)
?
q2=¯ µ2= γ5
(2.3)
Once ZL,RI′
Γ
have been calculated, one may convert them to the MS scheme (see Ref.[1]).
3. Computation and Results
There are 28 Feynman diagrams contributing to the fermion self-energy ΣL
loops, and 21 diagrams contributing to ΣL
extra diagrams, shown in Fig.1, in which the operator insertion occurs inside a closed fermion loop.
ψ(q,aL) at 1 and 2
S(qaL), ΣL
P(qaL). For flavor singlet bilinears, there are 4
3

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Renormalization of fermion bilinears
A. Skouroupathis
The evaluation and algebraic manipulation of Feynman diagrams, leading to a code for nu-
merical loop integration, is performed automatically using our software for Lattice Perturbation
Theory, written in Mathematica. The most laborious aspect of the procedure is the extraction of the
dependence on the external momentum q. This is a delicate task at two loops; for this purpose, we
cast algebraic expressions (typically involving thousands of summands) into terms which can be
naively Taylor expanded in q to the required order, plus a smaller set of terms containing superficial
divergences and/or subdivergences. The latter can be evaluated by analytical continuation to D > 4
dimensions, and splitting each expression into a UV-finite part (which can thus be calculated in the
continuum), and a part which is polynomial in q.
Some of the diagrams contributing to ΣL
when considered separately, and thus must be grouped together in order to give finite results, for ex-
ample, diagrams (1-2), (3-4) in Fig.1. As mentioned before, all calculations should be performed at
vanishing renormalized mass; this can be achieved by working with massless fermion propagators,
provided an appropriate fermion mass counterterm is introduced on 1-loop diagrams.
ψ(q,aL), ΣL
S(qaL) and ΣL
P(qaL) are infrared divergent
3
4
1
2
FIG. 1. Extra two-loop diagrams contributing to ZS,singletA cross denotes an insertion of a flavor singlet
operator. Wavy (solid) lines represent gluons (fermions)
All two-loop diagrams have been calculated in the bare Feynman gauge (αo= 1). One-loop
diagrams have been calculated for generic values of αo; this allows us to convert our two-loop
results to the renormalized Feynman gauge (αRI′ = 1 or αMS= 1).
Numerical loop integration was carried out by our “integrator” program, a metacode written
in Mathematica, for converting lengthy integrands into efficient Fortran code. Two-loop integrals
were evaluated for lattices of size up to L = 40; the results were then extrapolated to L → ∞. The
extrapolation error can be estimated quite accurately (see, e.g. Ref. [10]), given that L-dependence
of results can only span a restricted set of functional forms.
Our 1-loop results for Zψ, ZSand ZPconfirm existing results [5] (there is, however, a difference
in ZL,MS
P
, due to an extra finite renormalization factor, Z5, which is required in MS [11]).
Two-loop results: The evaluation of all Feynman diagrams leads directly to the bare Green’s
functions ΣL
functions ZL,Y
S
P
(Y = RI′or MS), via Eqs.(2.2) and (2.3). To this end, we need the
one-loop expression for ZL,RI′
A
. To express our results in terms of the renormalized g, we also need
the one-loop expression for ZL,RI′
g
. Our 1-loop computation of ZL,RI′
with older references (see e.g. [6]). We present below ZL,RI′
the renormalized Feynman gauge αRI′ = 1. Bare Green’s functions can be easily recovered from
the corresponding Z’s. The errors result from the L → ∞ extrapolation. We employ a standard
normalization for the algebra generators: tr(TaTb)=δab/2. (ao≡g2
ψ, ΣL
ψ , ZL,Y
Sand ΣL
P. These, in turn, can be converted to the corresponding renormalization
and ZL,Y
A
and ZL,RI′
and ZL,RI′
g
is in agreement
to two loops in
ψ
, ZL,RI′
S
P
o/16π2, cF≡(N2
c−1)/(2Nc))
4

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Renormalization of fermion bilinears
A. Skouroupathis
ZL,RI′
ψ
= 1+aocF
?
ln(a2
L¯ µ2)+11.852404288(5)−2.248868528(3)cSW−1.397267102(5)c2
2cF+2
−6.36317446(8)Nf+0.79694523(2)NfcSW−4.712691443(4)Nfc2
+49.83082185(5)cF−2.24886861(7)cFcSW−1.39726705(1)cFc2
+Nf
?
+cF
?
+Nc
?
?
?
+ln(a2
?
+66.0780218(9)cF−23.213749(2)cFcSW+4.1411425(3)cFc2
+Nf
?
+cF
?
+Nc
?
?
?
+ln(a2
?
+37.1489292(7)cF+6.746606(1)cFcSW−6.1080465(3)cFc2
+Nf
?
+cF
?
+Nc
?
All expressions reported thus far for ZSand ZPrefer to flavor non singlet operators. In the case
of ZP, all diagrams of Fig.1 vanish, so that singlet and non singlet results coincide. ZSon the other
hand, receives an additional finite contribution (which is the same for the MS scheme as well):
SW
?
+a2
ocF
?
ln2(a2
L¯ µ2)
?1
3Nf−8
3Nc
?
+ln(a2
L¯ µ2)
?
SW
SW+29.03029398(4)Nc
?
−7.838(2)+1.153(1)cSW+3.202(3)c2
505.39(1)−58.210(9)cSW+20.405(5)c2
−20.59(1)−3.190(5)cSW−23.107(6)c2
SW+6.2477(6)c3
SW+4.0232(6)c4
SW
?
SW+18.8431(8)c3
SW+4.2793(2)c4
SW
?
SW−5.7234(5)c3
SW−0.7938(1)c4
SW
??
(3.1)
ZL,RI′
S
= 1+aocF
3 ln(a2
L¯ µ2)−17.9524103(1)−7.7379159(3)cSW+1.38038065(4)c2
2cF+Nf−11
−8.1721694(5)Nf+2.3908354(3)NfcSW−14.13807433(4)Nfc2
SW
?
+a2
ocF
ln2(a2
L¯ µ2)
?9
2Nc
?
L¯ µ2)
SW
SW+55.7975008(9)Nc
?
24.003(3)+11.878(5)cSW+25.59(1)c2
SW+22.078(3)c3
SW−6.1807(8)c4
SW−2.688(1)c4
SW
?
?
−602.35(6)+66.80(7)cSW+75.42(5)c2
−38.16(4)−120.26(5)cSW−16.18(3)c2
SW−27.759(4)c3
SW
SW+12.576(3)c3
SW+1.0175(8)c4
SW
??
(3.2)
ZL,RI′
P
= 1+aocF
3 ln(a2
L¯ µ2)−27.5954414(1)+2.248868528(3)cSW−2.03601561(4)c2
2cF+Nf−11
−8.1721694(4)Nf+2.39083540(6)NfcSW−14.13807433(4)Nfc2
SW
?
+a2
ocF
ln2(a2
L¯ µ2)
?9
2Nc
?
L¯ µ2)
SW
SW+55.7975008(7)Nc
?
38.231(3)−7.672(5)cSW+55.32(1)c2
−876.98(4)+85.80(2)cSW+37.37(4)c2
−104.35(3)−38.70(2)cSW−13.93(3)c2
SW−7.049(3)c3
SW+19.974(3)c3
SW+4.7469(8)c4
SW
?
SW+2.873(1)c4
SW
?
??
SW−4.429(2)c3
SW−1.2898(7)c4
SW
(3.3)
ZL,RI′
S,singlet= ZL,RI′
S
+a2
ocFNf
?− 107.76(1)+82.27(2)cSW−29.727(4)c2
+ 3.4400(7)c3
SW
SW+2.2758(4)c4
SW
?
(3.4)
5