Table Of ContentPerturbative Renormalization Factors of Baryon Number
Violating Operators for Improved Quark and Gauge Actions in
Lattice QCD
Sinya Aokia, Yoshinobu Kuramashib ∗, Tetsuya Onogic and Naoto Tsutsuic
aInstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
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bDepartment of Physics, Washington University, St. Louis, Missouri 63130, USA
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2 cDepartment of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan
n
a (February 1, 2008)
J
5
1
v
Abstract
5
0
0
1 We calculate one-loop renormalization factors of three-quark operators,
0
0 which appear in the low energy effective Lagrangian of the nucleon decay,
0
/
t for O(a)-improved quark action and gauge action including six-link loops.
a
l
-
p This calculation is required to predict the hadronic nucleon decay matrix ele-
e
h ments in the continuum regularization scheme using lattice QCD. We present
:
v
i detailed numerical results of the one-loop coefficients for general values of
X
r
a the clover coefficients employing the several improved gauge actions in the
SymanzikapproachandintheWilson’s renormalization groupapproach. The
magnitudes of the one-loop coefficients for the improved gauge actions show
sizable reduction compared to those for the plaquette action.
11.15.Ha,12.38.Gc,13.30.-a
Typeset using REVTEX
∗On leave from Institute of Particle and Nuclear Studies, High Energy Accelerator Research
Organization(KEK), Tsukuba, Ibaraki 305-0801, Japan
1
I. INTRODUCTION
While nucleon decay is one of the most exciting prediction from grand unified theo-
ries (GUTs) with and without supersymmetry, none of the decay modes have been exper-
imentally detected up to now. Furthermore, the ongoing Super-Kamiokande experiment is
now pushing the lower limit on the partial lifetimes of the nucleon by an order of magni-
tude from the previous measurements. In principle this would give a strong constraint on
(SUSY-)GUTs, however, the uncertainties in the theoretical prediction of lifetimes due to
poor knowledge of quantum effects at low energy such as hadron or SUSY scales obscure
the direct impact of the experimental lifetime bound on the physics at the GUT scale. In
particular, one of the main uncertainties has been found in the evaluation of the hadron
matrix elements for the nucleon decays, for which various QCD models have given estimate
differing by a factor of ten. Therefore precise determination of the nucleon decay matrix
elements from first principles is required, for which Lattice QCD can play a crucial role.
Recently we carried out a model-independent calculation of the nucleon decay matrix
elements employing the Wilson quark action and the plaquette gauge action in the quenched
approximation [1]. Although one naively expects the error in the discretization and quench-
ing approximation which are the two main systematic errors, it would be desirable to reduce
these unknown systematic errors for high precision calculation. As a step toward this goal
we are required to reduce the scaling violation effects by improving quark and gauge actions.
In this article we present perturbative results for renormalization factors of baryon number
violating operators for improved quark and gauge actions: the O(a)-improved “clover” ac-
tion originally proposed by Sheikholeslami and Wohlert [2] and the gauge action improved
by addition of six link loops to the plaquette term in the Symanzik approach [3] and in
the Wilson’s renormalization group approach [4,5]. Values of the one-loop coefficients of
the renormalization factors are numerically evaluated for combinations of general values of
the clover coefficients in the quark action and some specific values of the coefficients of the
six-link loop terms in the gauge action.
2
This paper is organized as follows. In Sec.II we give the improved quark and gauge
actions on the lattice and their Feynman rules relevant for their calculation. Our calcula-
tional procedure of the renormalization factors for the baryon number violating operators
is described in Sec.III, where we present expressions and numerical values for the one-loop
coefficients of the renormalization factors. Our conclusions are summarized in Sec.IV.
The physical quantities are expressed in lattice units, and the lattice spacing a is sup-
pressed unless necessary. Throughout this paper we use the same notation for quantities
defined on the lattice and their counterparts in the continuum. In case of any possibility of
confusion, however, we shall make a clear distinction between them.
II. ACTIONS AND FEYNMAN RULES
For the gauge action we consider the following general form including the standard
plaquette term and six-link loop terms:
1
S = c TrU +c TrU +c TrU +c TrU (1)
gauge g2 0 pl 1 rtg 2 chr 3 plg
plaXquette recXtangle cXhair paralXlelogram
with the normalization condition
c +8c +16c +8c = 1, (2)
0 1 2 3
where six-link loopsare1×2 rectangle, abent 1×2 rectangle(chair) anda three-dimensional
parallelogram obtained by multiplying the link variables
U = exp ig TaAa(n+µˆ/2) . (3)
n,µ µ
a !
X
The free gluon propagator is obtained in Ref. [3]:
1
D (k) = (1−A )kˆ kˆ +δ kˆ2A (4)
µν (kˆ2)2 " µν µ ν µν σ σ νσ#
X
with
k
ˆ µ
k = 2sin , (5)
µ
2 !
4
kˆ2 = kˆ2. (6)
µ
µ=1
X
3
The matrix A satisfies
µν
(i) A = 0 for all µ, (7)
µµ
(ii)A = A , (8)
µν νµ
(iii)A (k) = A (−k). (9)
µν µν
(iv)A (0) = 1 for µ 6= ν, (10)
µν
and its expression is given by
1
A (k) = (kˆ2 −kˆ2)(q q kˆ2 +q q kˆ2 +q q kˆ2)
µν ∆ ν µρ µτ µ µρ ρτ ρ µτ ρτ τ
4
h
+(kˆ2 −kˆ2)(q q kˆ2 +q q kˆ2 +q q kˆ2)
µ νρ ντ ν νρ ρτ ρ ντ ρτ τ
+q q (kˆ2 +kˆ2)(kˆ2 +kˆ2)+q q (kˆ2 +kˆ2)(kˆ2 +kˆ2)
µρ ντ µ ρ ν τ µτ νρ µ τ ν ρ
−q q (kˆ2 +kˆ2)2 −(q q +q q )kˆ2kˆ2
µν ρτ ρ τ µρ νρ µτ ντ ρ τ
−q (q kˆ2kˆ2 +q kˆ2kˆ2 +q kˆ2kˆ2 +q kˆ2kˆ2) , (11)
µν µρ µ τ µτ µ ρ νρ ν τ ντ ν ρ
i
with µ 6= ν 6= ρ 6= τ the Lorentz indices. q and ∆ are written as
µν 4
q = (1−δ ) 1−(c −c −c )(kˆ2 +kˆ2)−(c +c )kˆ2 , (12)
µν µν 1 2 3 µ ν 2 3
h i
∆ = kˆ4 q + kˆ2kˆ2q (q q +q q ). (13)
4 µ νµ µ ν µν µρ ντ µτ νρ
Xµ νY6=µ µ>ν,ρ>τ,{Xρ,τ}∩{µ,ν}=∅
In the case of the standard plaquette action, the matrix A is simplified as
µν
Aplaquette = 1−δ . (14)
µν µν
For the quark action we consider the O(a)-improved quark action:
1
S = ψ¯ (−r +γ )U ψ +ψ¯ (−r −γ )U† ψ +(m +4r) ψ¯ ψ
quark 2 n µ n,µ n+µˆ n µ n−µˆ,µ n−µˆ 0 n n
Xn Xµ n o Xn
r
¯
−c ig ψ σ P (n)ψ , (15)
SW n µν µν n
4
n µ,ν
XX
where we define the Euclidean gamma matrices in terms of the Minkowski matrices in the
Bjorken-Drell convention: γ = −iγj (j = 1,2,3), γ = γ0 , γ = γ5 and σ = 1[γ ,γ ].
j BD 4 BD 5 BD µν 2 µ ν
The field strength P in the “clover” term is given by
µν
4
1 4 1
P (n) = U (n)−U†(n) , (16)
µν 4 2ig i i
Xi=1 (cid:16) (cid:17)
U (n) = U U U† U† , (17)
1 n,µ n+µˆ,ν n+νˆ,µ n,ν
U (n) = U U† U† U , (18)
2 n,ν n−µˆ+νˆ,µ n−µˆ,ν n−µˆ,µ
U (n) = U† U† U U , (19)
3 n−µˆ,µ n−µˆ−νˆ,ν n−µˆ−νˆ,µ n−νˆ,ν
U (n) = U† U U U† . (20)
4 n−νˆ,ν n−νˆ,µ n+µˆ−νˆ,ν n,µ
From the quark action (15) we obtain the free quark propagator
S−1(p) = i γ sin(k )+m +r (1−cos(p )). (21)
q µ µ 0 µ
µ µ
X X
In order to calculate renormalization factors of the baryon number violating operators up
to one-loop level, we need the following vertices,
p +q p +q
Va(p,q) = −igTa γ cos µ µ −irsin µ µ , (22)
1µ µ 2 2
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
r p −q
Va (p,q) = −gTac σ sin(p −q ) cos µ µ (23)
c1µ SW2 ν µν ν ν ! (cid:18) 2 (cid:19)
X
with p incoming quark momentum and q outgoing quark momentum. The first vertex
µ µ
originates from the Wilson quark action and the second one is the interaction due to the
clover term. In the present calculation of the vertex corrections for the baryon number
violating operators, the two-gluon vertices with quarks give no contribution.
We should note that the baryon number violating operators contain a charge conjugated
field, whose action is obtained from eq.(15) with the replacement of
Ta −→ −(Ta)T, (24)
where the superscript T means the transposed matrix. According to this change, the Feyn-
man rule of the quark-gluon vertices in eqs.(22) and (23) should be modified for the charge
conjugated field.
III. RENORMALIZATION FACTORS FOR BARYON NUMBER VIOLATING
OPERATORS
5
A. Calculational procedure
We consider the following baryon number violating operators in the continuum and on
the lattice:
Ocont = ǫabc (ψ¯c)aΓ (ψ )b [Γ (ψ )c] , (25)
X,Y δ 1 X 2 Y 3 δ
(cid:16) (cid:17) h i
Olatt = ǫabc {1+rm (1−z)}(ψ¯c)aΓ (ψ )b
X,Y 0 1 X 2
δ
(cid:16) (cid:17) rh ← →
+z (ψ¯cD/)aΓ (ψ )b −(ψ¯c)aΓ (D/ψ )b
2 1 X 2 1 X 2
(cid:26) (cid:27)
r2 ← →
−z2 (ψ¯cD/)aΓ (D/ψ )b
4 1 X 2 #
r r →
× 1+ m (1−z) Γ (ψ )c −z Γ (D/ψ )c (26)
0 Y 3 Y 3
2 2
(cid:20)(cid:26) (cid:27) (cid:21)δ
with
→ 1
D/ψ = γ U ψ −U† ψ , (27)
n 2 µ n,µ n+µˆ n−µˆ,µ n−µˆ
Xµ (cid:16) (cid:17)
← 1
ψ¯cD/ = ψ¯c UT −ψ¯c U∗ γ , (28)
n 2 n+µˆ n,µ n−µˆ n−µˆ,µ µ
Xµ (cid:16) (cid:17)
where ψ¯c = ψTC with C = γ γ is a charge conjugated field of ψ. Dirac structures are
4 2
represented by Γ ⊗ Γ = P ⊗ P ,P ⊗ P ,P ⊗ P ,P ⊗ P with the right- and left-
X Y R R R L L R L L
handed projection operators P = (1±γ )/2. The summation over repeated color indices
R,L 5
a,b,c is assumed.
Ultraviolet divergences of composite operators are regularized by the cutoff a−1 in the
lattice regularization scheme, while this is achieved by a reduction of the space-time di-
mension from four in some continuum regularization schemes, where we consider the naive
dimensional regularization (NDR) scheme and the dimensional reduction (DRED) scheme.
Operators defined in different regularization schemes can be related by renormalization fac-
tors:
Ocont(µ) = Z (µa)Olatt (a)+Z O˜latt (a) (29)
X,Y diag X,Y mix X,Y
with µ the continuum renormalization scale. The explicit chiral symmetry breaking due to
6
the Wilson term in the quark action (15) causes the mixing between operators with different
chiral structures, which is denoted by O˜latt .
X,Y
Since QCD is a asymptotically free theory, Z are expected to be perturbatively
diag,mix
calculable in terms of the coupling constant g2/(16π2) at high energy scales, which gives the
following expressions,
g2 3 8
Z (µa) = 1+ −(1−z)rΣ − ln(µa)+∆ +8ln(µa)+∆ , (30)
diag 16π2 2 0 3 ψ V,diag
(cid:20) (cid:18) (cid:19) (cid:21)
g2
Z = ∆ , (31)
mix 16π2 V,mix
where Σ denotes the additive mass renormalization onthe lattice, ∆ is a contribution from
0 ψ
the wavefunction and ∆ is fromthe vertex function. Σ and ∆ areobtained by calculating
V 0 ψ
the continuum and lattice quark self-energies Σcont,latt and ∆ is from the continuum and
V
lattice vertex functions Λcont,latt.
The quark self-energies in the continuum and on the lattice are defined through the
inverse full quark propagators:
g2
(Scont)−1(p) = ip/ +m− Σcont(p), (32)
q 16π2
g2
(Slatt)−1(p) = i γ sin(p )+m +r (1−cos(p ))− Σlatt(p), (33)
q µ µ 0 µ 16π2
µ µ
X X
where we consider massless quark. The difference between Σcont and Σlatt, which originates
from the regularization scheme dependence of the self-energy, gives the quark wavefunction
renormalization factor,
8 ∂Σcont(p) ∂Σlatt(p)
− ln(µa)+∆ = − . (34)
ψ
3 i∂p/ (cid:12) i∂p/ (cid:12)
(cid:12)p=0 (cid:12)p=0
(cid:12) (cid:12)
(cid:12) (cid:12)
The additive quark mass renormalization is expr(cid:12)essed as (cid:12)
g2
m − Σ (35)
0 16π2 0
with
Σ = Σlatt(p = 0). (36)
0
7
Notice that the renormalization factor (31) is given for the case of m = g2/(16π2)Σ , where
0 0
we consider the renormalization for massless quark.
The vertex functions up to one-loop level in the continuum and on the lattice are ex-
pressed in the following way,
g2
Λcont = ǫabcΓ ⊗Γ + ǫabcΓ ⊗Γ Vcont, (37)
X,Y X Y 16π2 X Y X,Y
g2
Λlatt = ǫabcΓ ⊗Γ + ǫabc Γ ⊗Γ Vlatt +zV′latt −z2V′′latt
X,Y X Y 16π2 X Y X,Y X,Y X,Y
h (cid:16) (cid:17)
+Γ˜ ⊗Γ˜ V˜latt +zV˜′latt −z2V˜′′latt , (38)
X Y X,Y X,Y X,Y
(cid:16) (cid:17)i
where the number of prime in the superscript of the lattice vertex corrections denotes the
number of covariant derivative applied to the quark fields at the vertex. Γ˜ ⊗ Γ˜ term
X Y
represents the mixing contribution. The difference between Λcont and Λlatt leads to
X,Y X,Y
8ln(µa)+∆ = Vcont − Vlatt +zV′latt −z2V′′latt , (39)
V,diag X,Y X,Y X,Y X,Y
(cid:16) (cid:17)
∆ = − V˜latt +zV˜′latt −z2V˜′′latt . (40)
V,mix X,Y X,Y X,Y
(cid:16) (cid:17)
We note that the lattice quark-self energy and the lattice vertex corrections are general
function of the clover coefficient c in the quark action and the six-link loop parameters
SW
c inthe gaugeaction. Calculation of ∆ was already carried out inRef. [6]employing the
1,2,3 ψ
general values for c . For c they choose some specific values: c = −1/12,c = c = 0
SW 1,2,3 1 2 3
in the tree-level Symanzik improvement, c = −0.252,c = 0,c = −0.17 suggested by
1 2 3
Wilson based on renormalization group improvement and c = −0.331,c = c = 0 and
1 2 3
c = −0.27,c +c = −0.04 by Iwasaki. According to Ref. [6] we evaluate ∆ for general
1 2 3 V
values of c and for the specific values of c that they employed.
SW 1,2,3
B. Vertex corrections
We calculate the vertex corrections of the operators in eqs.(25) and (26) in the Feynman
gauge employing the massless quarks and and the massless charge conjugated quark with
8
momenta p = p = p = 0 as external states. The infrared singularities are regularized by
1 2 3
a fictitious gluon mass λ introduced in the gluon propagator.
One-loop vertex corrections on the lattice are illustrated in Fig.1. We find that the
lattice vertex corrections is a second polynomial function of the clover coefficients c . The
SW
relevant diagrams for Vlatt and V˜latt are Figs.1(a)−(i), the sum of which gives
X,Y X,Y
Γ ⊗Γ V +Γ˜ ⊗Γ˜ V˜
R/L Y R/L,Y R/L Y R/L,Y
1
= Γ ⊗Γ C 6ln + c (i)v(i)
R/L Y B/ λ2a2 SW diag
(cid:12) (cid:12) i=0,1,2
(cid:12) (cid:12) X
+ Γ ⊗Γ ± 1 (cid:12)(cid:12) γ γ(cid:12)(cid:12) ⊗Γ γ c (i)v(i) (41)
L/R Y 4 µ µ 5 Y µ!i=0,1,2 SW mix
X X
(i) (i)
with C = (N+1)/(2N) in the SU(N) group. The explicit forms of v and v are given
B/ diag mix
by
π d4k 1 6
v(0) = C 48(∆ +r2∆2)2 +8I +16I −θ(π2 −k2)
diag B/ Z−π π2 "F02G0 n 3 1 a bo (k2)2#
+C 6ln|π2|, (42)
B/
π d4k 1
v(1) = C 8r2∆ I , (43)
diag B/ Z−π π2 "F02G0 n 1 ao#
π d4k 1
v(2) = C 4r4∆2I , (44)
diag B/ Z−π π2 "F02G0 n 1 ao#
π d4k 1
v(0) = C 16r2∆2(∆ −4∆µ ) , (45)
mix B/ Z−π π2 "F02G0 n 1 3 1,0 o#
π d4k 1
v(1) = C 8r2∆ I , (46)
mix B/ Z−π π2 "F02G0 n 1 ao#
π d4k 1
v(2) = C −4r2∆ I , (47)
mix B/ Z−π π2 "F02G0 n 3 ao#
where
r2
F = sin2(k )+ (kˆ2)2, (48)
0 µ
4
µ
X
G = (kˆ2)2, (49)
0
I = ∆µ −4∆2 +(16∆ −4sin2(k ))∆µ , (50)
a 1,1 3 3 µ 1,0
I = −∆µ +∆2 +4(−∆ +sin2(k ))∆µ , (51)
b 1,1 3 3 µ 1,0
9
1
∆ = kˆ2, (52)
1
4
1
∆ = sin2(k ), (53)
3 µ
4
µ
X
∆µ = (δ +A )sin2(k )sin2(k ), (54)
1,1 µν µν µ ν
ν
X
k k
∆µ = (δ +A )cos2 µ sin2 ν . (55)
1,0 ν µν µν 2 ! 2 !
X
We do not take the sum over the index µ for ∆µ and ∆µ . In a similar way we obtain the
1,1 1,0
expressions of V′latt and V˜′latt from Figs.1(a)−(i):
X,Y X,Y
Γ ⊗Γ V′ +Γ˜ ⊗Γ˜ V˜′
R/L Y R/L,Y R/L Y R/L,Y
= Γ ⊗Γ c (i)v′(i)
R/L Y SW diag
i=0,1,2
X
1
+ Γ ⊗Γ ± γ γ ⊗Γ γ c (i)v′(i), (56)
L/R Y 4 µ µ 5 Y µ!i=0,1,2 SW mix
X X
where
π d4k 1
v′(0) = C 16r2∆ I +32r2∆ I
diag B/ Z−π π2 "F02G0 n 1 a 1 bo#
π d4k 1
+C 12r2∆ (∆ −4∆µ ) , (57)
F π2 F G 1 3 1,0
Z−π (cid:20) 0 0 n o(cid:21)
π d4k 1
v′(1) = C −8(r2∆ −r4∆2)I
diag B/ Z−π π2 "F02G0 n 3 1 ao#
π d4k 1
+C 3r2I , (58)
F a
π2 F G
Z−π (cid:20) 0 0 n o(cid:21)
π d4k 1
v′(2) = C −8r4∆ ∆ I , (59)
diag B/ Z−π π2 "F02G0 n 1 3 ao#
π d4k 1
v′(0) = C −32r2∆ ∆ (∆ −4∆µ )
mix B/ Z−π π2 "F02G0 n 1 3 3 1,0 o
1
+ 8r2∆ (∆ −4∆µ ) , (60)
F G 1 3 1,0
0 0 n o(cid:21)
π d4k 1 1
v′(1) = C −8(r2∆ −r4∆2)I + 2r2I , (61)
mix B/ Z−π π2 "F02G0 n 3 1 ao F0G0 n ao#
π d4k 1
v′(2) = C −8r4∆ ∆ I (62)
mix B/ Z−π π2 "F02G0 n 1 3 ao#
with C = (N2 −1)/(2N) in the SU(N) group. The sum of Figs.1(a)−(k), which include
F
the tadpole diagrams at the vertex, yields V′′latt and V˜′′latt,
X,Y X,Y
10
