Table Of ContentPreprinttypesetinJHEPstyle-PAPERVERSION
Numeri
al Study of the Ghost-Gluon Vertex in
5
Landau gauge
0
0
2
n
a
J
7
A. Cu
hieri, T. Mendes and A. Mihara
1
Instituto de Físi
a de São Carlos (cid:21) Universidade de São Paulo
2
C.P. 369, 13560(cid:21)970 São Carlos, SP (cid:21) Brazil
v
4
3
0
Abstra
t: We present a numeri
al study of the ghost-gluon vertex and of the
orrespond-
8 Z (p2) SU(2)
1
0 ing renormalization fun
tion in minimal Landau gauge for latti
e gauge the-
= 44,84,164
4 ory. Data were obtained for tehree di(cid:27)erent latti
e volumes (V ) and for three
0 β = 2.2,2.3,2.4
/ latti
e
ouplings . Gribov-
opy e(cid:27)e
ts have been analyzed using the so-
t
a
alled smeared gauge (cid:28)xing. We also
onsider two di(cid:27)erent sets of momenta (orbits) in
l
-
p order to
he
k for possible e(cid:27)e
ts due to the breaking of rotational symmetry. The vertex
(0;p,−p)
e has been evaluated at the asymmetri
point in momentum-subtra
tion s
heme.
h Z (p2) p > 1
1 ∼
: We (cid:28)nd that is approximately
onstant and equal to 1, at least for momenta
v
i GeV. This
onestitutes a nonperturbative veri(cid:28)
ation of the so-
alled nonrenormalization of
X
the Landau ghost-gluon vertex. Finally, we use our data to evaluate the running
oupling
r α (p2)
s
a
onstant .
SU(2)
Keywords: ghost-gluon vertex, latti
e gauge theory, Landau gauge, running
oupling
onstant.
Contents
1. Introdu
tion 1
2. The ghost-gluon vertex 3
3. Numeri
al Simulations 5
3.1 Ghost-Gluon Vertex on the Latti
e 8
4. Results 10
5. Con
lusions 13
1. Introdu
tion
In the framework of quantum (cid:28)eld theory, Faddeev-Popov ghosts are introdu
ed in order to
quantize non-Abelian gauge theories. Although the ghosts are a mathemati
al artifa
t and
are absent from the physi
al spe
trum, one
an use the ghost-gluon vertex and the ghost
α (p2)
s
propagator to
al
ulate physi
al observables, su
h as the QCD running
oupling ,
using the relation [1℄
Z (p2)Z2(p2)
α (p2) = α 3 3 .
s 0
Z2(p2) (1.1)
1 e
α = g2/4π e Z (p2) Z (p2) Z (p2)
0 0 3 3 1
Here is the bare
oupling
onstant and , and are, respe
-
tively, thegluon, ghostandghost-gluonvertexrenormalizatioen fun
tions.eInLandaugauge,
Z (p2)
1
the vertex renormalization fun
tion is (cid:28)nite and
onstant, i.e. independent of the
p
renormalization s
ale , to all orders eof perturbation theory (see [2℄ and the nonrenormal-
ization theorems in [3, Chapter 6℄). However, a dire
t nonperturbative veri(cid:28)
ation of this
Z (p2)
1
result isstillla
king. Thisisthepurposeof thepresent work. Let usstressthat, if is
(cid:28)nite and
onstant also at the nonperturbative level, the equation above
an be simepli(cid:28)ed,
yielding a nonperturbative de(cid:28)nition of the running
oupling
onstant that requires only
the
al
ulationofgluonandghostpropagators[4,5℄. Anonperturbative investigationofthe
stru
ture of the ghost-gluon vertex is also important for studies of gluon and ghost propa-
gators using Dyson-S
hwinger equations (DSE) [6℄. In fa
t, in these studies, one makes use
of Ansätze for the behavior of the propagators and verti
es in the equations, in order to
obtain solvable trun
ation s
hemes. Of
ourse, a nonperturbative input for these quantities
is important for a truly nonperturbative solution of the DSE.
(cid:21) 1 (cid:21)
Let us re
all that in Landau gauge the gluon and ghost propagators
an be expressed
(in momentum spa
e) as
q q q q F(q2)
Dbc(q,−q) = δbc δ − µ ν D(q2) = δbc δ − µ ν
µν (cid:18) µν q2 (cid:19) (cid:18) µν q2 (cid:19) q2 (1.2)
J(q2)
Gbc(q,−q) = −δbc G(q2) = −δbc
q2 , (1.3)
F(q2) J(q2)
where and are, respe
tively, the gluon and ghost form fa
tor and the
olor
b c 1,2,...,N2 − 1 SU(N )
c c
indi
es and take values in the
ase. Then, in the momentum-
subtra
tion s
heme one has that the gluon and ghost renormalization fun
tions are given
by
F (q2,p2) = Z−1(p2)F(q2)
R 3 (1.4)
J (q2,p2) = Z−1(p2)J(q2)
R 3 (1.5)
e
with the renormalization
onditions
F (p2,p2) = J (p2,p2) = 1.
R R
(1.6)
Z (p2) = 1
1
Thus, if one
an set , the running
oupling (1.1)
an be written as [4, 5℄
e α (p2) = α F(p2)J2(p2) ,
s 0
(1.7)
i.e. one only needs to evaluate the gluon and ghost form fa
tors de(cid:28)ned above. In re
ent
years, the infrared behavior of these form fa
tors has been extensively studied (in Landau
gauge) using di(cid:27)erent analyti
al approa
hes [4, 5, 7, 8, 10, 9, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25℄. Also, numeri
al studies of these form fa
tors and of the running
SU(2)
oupling de(cid:28)ned in eq. (1.7) have been reported in [26, 27, 28, 29℄ for the group and
SU(3)
in [30, 31, 32, 33, 35, 36, 34, 37℄ for the
ase.
An indire
t evaluation of the ghost-gluon vertex renormalization fun
tion has been
Z (p2)
1
re
ently presented in [29℄,
on(cid:28)rming that is (cid:28)nite in the
ontinuum limit. On the
other hand, a dire
t nonperturbative veri(cid:28)
aetion of this result from a numeri
al evaluation
Z (p2)
1
of the ghost-gluon vertex is still missing. Let us stress that a dire
t evaluation of
would allow a study of the running
oupling
onstant [38℄ using eq. (1.1) insteadeof eq.
α (p2)
s
(1.7). Su
h a study may improve the pre
ision of the determination of , sin
e in that
ase one does not need, in prin
iple, to use the so-
alled mat
hing res
aling te
hnique [29℄
β
when
onsidering data obtained at di(cid:27)erent values.
Inthispaperwestudynumeri
ally theghost-gluonvertex andthe
orresponding renor-
Z (p2) SU(2)
1
malization fun
tion for the
asein theminimal Landau gauge. Thede(cid:28)nition
of this renormalizatieon fun
tion (in the
ontinuum) is presented in Se
tion 2. Numeri
al
simulations are explained in Se
tion 3. In parti
ular, in Se
tion 3.1 we de(cid:28)ne the vertex
Z (p2)
1
renormalization fun
tion on the latti
e and
ompare our dire
t determination of to
the indire
t evaluation presented in [29℄. Note that, sin
e the numeri
al evaluatioen of the
ghost-gluon vertex may be a(cid:27)e
ted by Gribov-
opy e(cid:27)e
ts (see dis
ussion in Se
tions 3 and
(cid:21) 2 (cid:21)
q
k
b
µ s
a,
c
Figure 1: Notation (momenta and indi
es) for the ghost-gluon vertex.
4 below), we evalute this quantity using two di(cid:27)erent gauge-(cid:28)xing methods, leading to two
di(cid:27)erent setsof Gribov
opies. A
omparison of the results obtained in the two
ases allows
us to estimate the in(cid:29)uen
e of Gribov
opies (the so-
alled Gribov noise) on the
onsidered
quantity.
Results for the ghost-gluon vertex renormalization fun
tion, the gluon and ghost prop-
α (p2)
s
agators and the running
oupling
onstant are reported in Se
tion 4. Finally, in
Se
tion 5 we draw our
on
lusions. Preliminary results have been reported in [39℄.
2. The ghost-gluon vertex
Aa ηb
µ
Following the notation in Ref. [6℄ (see also Fig. 1) the 3-point fun
tion for (gluon),
η¯c
(ghost) and (anti-ghost) (cid:28)elds (cid:22)
orresponding to the ghost-gluon vertex (cid:22) is given by
Vabc(x,y,z) = hAa(x)ηb(y)η¯c(z)i .
µ µ
(2.1)
Going to momentum spa
e and using translational invarian
e for the 3-point fun
tion, one
gets
Vabc(k;q,s) = d4xd4yd4z e−i(kx+qy−sz) Vabc(x,y,z)
µ µ
Z (2.2)
= d4z e−i(k+q−s)z d4xd4y e−i(kx+qy) hAa(x)ηb(y)η¯c(0)i
µ
Z Z (2.3)
= (2π)4 δ4(k+q−s)Gabc(k,q) ,
µ
(2.4)
δ4(k + q − s)
where implies
onservation of momentum. Then, the ghost-gluon vertex
fun
tion is obtained by (cid:16)amputating(cid:17) the
orresponding 3-point fun
tion (see Fig. 2)
Gabc(k,q)
Γabc(k,q) = µ ,
µ D(k2)G(q2)G(s2) (2.5)
s = k +q D(k2) G(q2)
where and the fun
tions and have been de(cid:28)ned in eqs. (1.2) and
(1.3), respe
tively. At tree level (in the
ontinuum) one obtains [1℄
Γabc(k,q) = ig fabcq ,
µ 0 µ
(2.6)
(cid:21) 3 (cid:21)
Gabc(k,q) Γabc(k,q)
µ µ
Figure2: The3-pointfun
tion anditsrelationwiththefullghost-gluonvertex .
g fabc SU(N )
0 c
where is the bare
oupling and are the stru
ture fun
tions of the Lie
algebra. This implies the well-known result that the ghost-gluon vertex is proportional to
q
the momentum of the outgoing ghost. More generally, we
an write the relation [6℄
Γabc(k,q) = g fabcΓ (k,q) ,
µ 0 µ
(2.7)
Γ (k,q)
µ
where is the so-
alled redu
ed vertex fun
tion. Then, multiplying the previous
fdbcδad a,b,c,d fdbcfabc = N δda
equation by , summing over and using the relation b,c c
P
we get 1
Γ (k,q) = fabcΓabc(k,q) .
µ g N (N2 − 1) µ (2.8)
0 c c aX,b,c
Γ (k,q) = iq
µ µ
Sin
e at tree level one has we
an also write
Γ (k,q) = iq Γ(k2,q2)
µ µ
(2.9)
Γ(k2,q2)
and it is the s
alar fun
tion that gets renormalized, when
onsidering a given
renormalization s
heme.
Clearly, from the above relations we obtain
−i −i 1
Γ(k2,q2) = q Γ (k,q) = q fabcΓabc(k,q) .
q2 µ µ g N (N2 − 1) q2 µ µ (2.10)
Xµ 0 c c Xµ aX,b,c
In our simulations we
onsidered the asymmetri
point with zero momentum for the
k = 0 s = q
gluon ( ), implying . Formula (2.10) then be
omes
−i 1
Ξ(q2) = Γ(0,q2) = q fabcΓabc(0,q) .
g N (N2 − 1) q2 µ µ (2.11)
0 c c Xµ aX,b,c
−i
Let us note that the fa
tor in the equations above disappears, sin
e only the imaginary
Γabc(0,q) Ξ(q2)
µ
part of the vertex fun
tion
ontributes to the quantity , i.e. we
an write
1 1
Ξ(q2) = q fabc Γabc(0,q) .
g N (N2 − 1) q2 µ Im µ (2.12)
0 c c Xµ aX,b,c
(cid:21) 4 (cid:21)
Figure 3: Relation between the full (left) and the bare (right) verti
es.
Finally, inmomentum-subtra
tion s
hemeone(cid:28)xesthevertexrenormalization fun
tion
Z (p2)
1
by requiring (see also Fig. 3)
e Ξ (q2,p2) = Z (p2)Ξ(q2)
R 1
(2.13)
Ξ (p2,p2) = 1e,
R
(2.14)
Γ (0,q)
µ
namely the renormalized redu
ed vertex fun
tion is equal to the tree-level value
iq
µ
at the renormalization s
ale. Then, the vertex renormalization fun
tion is given by
Z−1(p2) = Ξ(p2) .
1 (2.15)
e
We also obtain that the running
oupling
an be written as
g (p2) = g Z−1(p2)Z1/2(p2)Z (p2)
s 0 1 3 3 (2.16)
= g Ξe(p2)F1/2(p2)J(pe2)
0
(2.17)
1 F1/2(p2)J(p2)
= p fabc Gabc(0,p)
N (N2 − 1) p2D(0)G2(p2) µ Im µ (2.18)
c c Xµ aX,b,c
1 F1/2(p2)
= p fabc Gabc(0,p) ,
N (N2 − 1) D(0)G(p2) µ Im µ (2.19)
c c Xµ aX,b,c
where we used eq. (1.3).
3. Numeri
al Simulations
For the simulations (see [40℄ for details) we
onsider the standard Wilson a
tion, thermal-
ized using heat-bath [41℄ a
elerated by hybrid overrelaxation [42, 43, 44℄. Sin
e we are
SU(2) U (x)
µ
onsidering the gauge group we parametrize the link variables as
U (x) = u0(x)⊥1 + i~u (x)~σ ,
µ µ µ
(3.1)
1⊥ 2×2 σb
where is the identity matrix, are the Pauli matri
es and the relation
[u0(x)]2 + [~u (x)]2 = 1
µ µ
(3.2)
(cid:21) 5 (cid:21)
V = N4 44 84 164
Latti
e Volume
No. of Con(cid:28)gurations 1000 400 250
Table 1: Latti
e volumes and total number of
on(cid:28)gurations
onsidered.
is satis(cid:28)ed. We also
onsider the standard de(cid:28)nition of the latti
e gluon (cid:28)eld, i.e.
†
U (x) − U (x)
A (x) = Ab(x)σb = µ µ ,
µ µ 2i (3.3)
Ab(x) ub(x)
µ µ
implying that the latti
e gluon (cid:28)eld
omponent is equal to . The
onne
tion
U (x) A (x)
µ µ
between the latti
e link variables and the
ontinuum gauge (cid:28)eld is given by
U (x) = exp iag Ab(x)tb ,
µ 0 µ
h i (3.4)
tb = σb/2 a g
0
where are the generators of the SU(2) algebra, is the latti
e spa
ing and
2Ab(x)/(ag )
µ 0
is the bare
oupling
onstant. Then, the latti
e quantity approa
hes the
Ab(x) a → 0
µ
ontinuum-gluon-(cid:28)eld
omponent in the naive
ontinuum limit . Let us also
qˆ
re
all that a generi
latti
e momentum has
omponents (in latti
e units)
πq a
µ
qˆ = 2 sin .
µ
(cid:18) L (cid:19) (3.5)
µ
e
L = aN µ q
µ µ µ
Here is the physi
al size of the latti
e in the dire
tion, the quantity takes
⌊−Nµ ⌋+1,...,⌊ Nµ ⌋ N µ
values 2 2 and µ is the number of latti
e points in the dire
etion. If
L q
µ µ
we keep the physi
al size
onstant and indi
ate with the momentum
omponents in
qˆ = aq
µ µ
the
ontinuum, we (cid:28)nd that the latti
e
omponents take values in the interval
(−π, π] a→ 0
when .
In Table 1 we show the latti
e volumes used in the simulations and the
orresponding
numbers of
on(cid:28)gurations. For ea
h volume we have
onsidered three values of the latti
e
β = 2.2,2.3,2.4
oupling, namely . The
orresponding string tensions in latti
e units [29℄
σa2 = 0.220(9),0.136(2) 0.071(1)
are, respe
tively, equal to and .
Theminimal(latti
e) Landau gaugeisimplemented usingthesto
hasti
overrelaxation
(∇·A)2
algorithm [45, 46, 47℄. We stop the gauge (cid:28)xing when the average value of (cid:22) see
10−13
Eq. (6.1) in [47℄ for a de(cid:28)nition (cid:22) is smaller than . The gluon and ghost propagators
in momentum spa
e are evaluated using the relations
1
D(0) = Dbb(0)
12 µµ (3.6)
Xb,µ
1
D(q) = Dbb(q)
9 µµ (3.7)
Xb,µ
e e
1
G(q) = Gbb(q),
3 (3.8)
Xb
e e
where
1
Dbc(q) = exp[iq(x−y)]hAb(x)Ac(y)i ,
µν V µ ν (3.9)
Xx,y
e e
(cid:21) 6 (cid:21)
44 84 164
V = V = V =
β = % % %
2.2 4.3 25.0 97.2
β = % % %
2.3 4.4 16.8 72.8
β = % % %
2.4 4.1 9.8 50.8
Table 2: Per
entage of
on(cid:28)gurations for whi
h the smeared gauge (cid:28)xing has found a (di(cid:27)erent)
β
Gribov
opy. Results are reported for the three latti
e volumes and the three values
onsidered.
Gbc(q) = h M−1 bc(q)i ,
(3.10)
(cid:0) (cid:1)
M−1 bc(qe) = 1 expe[−iq(x−y)] M−1 bc ,
V xy (3.11)
(cid:0) (cid:1) Xx,y (cid:0) (cid:1)
e e
V Mbc
xy
is the latti
e volume and is a latti
e dis
retization of the Faddeev-Popov operator
[(−∂+A)·∂]
. The expression of this matrix in terms of the gauged-(cid:28)xed link variables
SU(N )
c
an be found in [48, eq. (B.18)℄ for the generi
ase and in [49, eq. (11)℄ for the
SU(2)
ase
onsidered in the present work. Let us re
all that this matrix has a trivial null
eigenvalue
orresponding to a
onstant eigenve
tor. Thus, one
an evaluate the inverse of
Mbc
xy
only in the spa
e orthogonal to
onstant ve
tors, i.e. at non-zero momentum. For the
inversion we used a
onjugate gradient method with even/odd pre
onditioning.
Finally, in order to
he
k for possible Gribov-
opy e(cid:27)e
ts we have done (for ea
h ther-
malized
on(cid:28)guration) a se
ond gauge (cid:28)xing using the smearing method introdu
ed in [50℄.
To this end we have applied the APE smearing pro
ess
U (x) → (1 − w)U (x) + wΣ†(x) ,
µ µ µ
(3.12)
Σ (x) U (x)
µ µ
where represents the sumover the
onne
ting staplesforthe link , followed by
w = 10/6
a reunitarization of the link matrix. We have
hosen and stopped the smearing
when the
ondition Tr
W ≥ 0.995
1,1
2 (3.13)
W
1,1
was satis(cid:28)ed. (This is usually a
hieved with a few APE-smearing sweeps.) Here is the
1×1 W
1,1
loop and the average is taken over all loops of a given
on(cid:28)guration. The gauge
(cid:28)xing for the smeared
on(cid:28)guration as well as the (cid:28)nal gauge-(cid:28)xing step have been done
again using the sto
hasti
overrelaxation algorithm. The smeared gauge-(cid:28)xing method is
supposed to (cid:28)nd a unique Gribov
opy, even though this
opy does not always
orrespond
to the absolute minimum of the minimizing fun
tional [50℄. In our simulations we found
44 84 164
V = V = V =
β = % % %
2.2 46.5 52.0 63.8
β = % % %
2.3 38.6 52.2 65.4
β = % % %
2.4 39.0 48.7 65.4
Table 3: Per
entage of (di(cid:27)erent) Gribov
opies for whi
h the smeared gauge (cid:28)xing has found a
smallervalueoftheminimizingfun
tion. Results arereportedforthethreelatti
evolumesandthe
β
three values
onsidered.
(cid:21) 7 (cid:21)
thatthenumber ofdi(cid:27)erent Gribov
opiesobtained withthesmearedgauge(cid:28)xing in
reases
β
with larger latti
e volumes and with smaller values (see Table 2). This is in agreement
with previous studies [51, 52, 53℄ and it should be related to an in
reasing number of lo
al
minima for the minimizing fun
tion when the system is highly disordered. We also found
that the minimum obtained with the smeared gauge (cid:28)xing is not always smaller than the
one obtained without smearing, but at larger latti
e volumes the smearing approa
h seems
to be more e(cid:27)e
tive in getting
loser to the absolute minimum of the minimizing fun
tion
(see Table 3).
3.1 Ghost-Gluon Vertex on the Latti
e
On the latti
e, the de(cid:28)nition of the vertex renormalization fun
tion is obtained as was
done in the
ontinuum (see Se
tion 2). The only di(cid:27)eren
e is that, from the weak-
oupling
expansion (or (cid:16)perturbative(cid:17) expansion) on the latti
e, one obtains that the ghost-gluon
vertex is given at tree level by [54℄
πs a
Γabc(k,q) = ig fabcqˆ cos µ ,
µ 0 µ (cid:18) L (cid:19) (3.14)
µ
e
e e
s = k + q a → 0
where . Clearly, by taking the formal
ontinuum limit of the quantity
Γabc(k,q)/a k = 0
µ e eoneere
overs the
ontinuum tree-level result (2.6). Then, in the
ase ,
s = q
e, eeq. (2.15) is still valid on the latti
e if one sets e
e e −i qˆ
µ
Ξ(q) = Γ (0,q)
e qˆ2 Xµ cos(cid:16)πLqeµµa(cid:17) µ e (3.15)
with
1
Γ (0,q) = fabcΓabc(0,q)
µ g N (N2 − 1) µ (3.16)
0 c c aX,b,c
e e
Gabc(0,q)
Γabc(0,q) = µ
µ D(0)G2(q) (3.17)
e
e
e
and 1/2
4
qˆ = qˆ2 ,
µ
(3.18)
Xµ=1
qˆ q
µ µ
where is de(cid:28)ned in terms of in eq. (3.5). Thus, as in the
ontinuum, we
an write
1 e 1 qˆ
Ξ(q) = µ fabc Γabc(0,q) .
e g0Nc(Nc2 − 1) qˆ2 Xµ cos(cid:16)πLqeµµa(cid:17) aX,b,c Im µ e (3.19)
In order to
he
k for possible e(cid:27)e
ts due to the breaking of rotational symmetry, in our
q = q = q = 0, q = q q =
1 2 3 4 1
simulations we use two types of latti
e momenta, i.e. and
q = q = q = q
2 3 4
. In the following we will indi
ate the (cid:28)erst tyepe ofemomentea as aesymmeetri
e e e e
(cid:21) 8 (cid:21)
L = L = L = L = L
1 2 3 4
and the se
ond one as symmetri
. Clearly, using the relations , we
have πqa
qˆ = qˆ = 2 sin
4
(cid:18) L (cid:19) (3.20)
e
in the asymmetri
ase and
πqa
qˆ = 2 sin
µ
(cid:18) L (cid:19) (3.21)
e
πqa
qˆ = 4 sin
(cid:18) L (cid:19) (3.22)
e
Ξ(q)
in the symmetri
one. Then, the quantity above be
omes
1 1 e 1
Ξ(q) = fabc Γabc(0,q)
g0Nc(Nc2 − 1) qˆ cos πLqa aX,b,c Im 4 (3.23)
e (cid:16) e (cid:17) e
in the asymmetri
ase and
1 1 1
Ξ(q) = fabc Γabc(0,q)
g0Nc(Nc2 − 1) qˆ 2 cos πLqa Xµ aX,b,c Im µ (3.24)
e (cid:16) e (cid:17) e
sin(2θ) = 2 sin(θ) cos(θ)
in the symmetri
one. Using the trigonometri
relation these
formulae
an be rewritten as
1
Ξ(q) = Σ(q) ,
sin 2πqa (3.25)
L
e (cid:16) e (cid:17) e
where 1
Σ(q) = fabc Γabc(0,q)
g N (N2 − 1) Im 4 (3.26)
0 c c aX,b,c
e e
in the asymmetri
ase and
1 1
Σ(q) = fabc Γabc(0,q)
g N (N2 − 1) Im 4 µ (3.27)
0 c c aX,b,c Xµ
e e
in the symmetri
one.
Finally, using eqs. (2.3), (2.4) and (3.11) we obtain
Gabc(0,q) = V hAa(0) M−1 bc(q)i ,
µ µ
(3.28)
(cid:0) (cid:1)
e e
where 1
Aa(0) = Aa(x)
µ V µ (3.29)
Xx
V
and is the latti
e volume. Then, we
an write
V hAa(0) M−1 bc(q)i
Γabc(0,q) = µ .
µ D(0(cid:0))G2(q(cid:1)) (3.30)
e
e
e
(cid:21) 9 (cid:21)
