One-Loop Self Energy and Renormalization of the Speed of Light for some Anisotropic Improved Quark Actions

45 Pages·0.34 MB·English

by  Stefan Groote

OHSTPY-HEP-T-99-022
MZ-TH/99-64
One-Loop Self Energy and Renormalization of the Speed of Light
for some Anisotropic Improved Quark Actions
Stefan Groote
Newman Laboratory, Cornell University, Ithaca, NY 14853, USA
and
0 Institut fu¨r Physik der Johannes-Gutenberg-Universita¨t,
0
Staudingerweg 7, D-55099 Mainz, Germany
0
2

n
a Junko Shigemitsu
J
Physics Department, The Ohio State University, Columbus, OH 43210, USA
9
1

1
v Abstract
1
2
0
1 One-loopcorrectionstothefermionrestmassM1,wavefunctionrenormaliza-
0
tion Z and speed of light renormalization C are presented for lattice actions
0 2 0
0 that combine improved glue with clover or D234 quark actions and keep the
/ temporal and spatial lattice spacings, a and a , distinct. We explore a range
t t s
a
of values for the anisotropy parameter χ a /a and treat both massive and
l s t
- ≡
p massless fermions.
e
h PACS number(s): 11.10.Gh, 11.15.Ha, 12.38.Bx, 12.38.Gc
:
v
i
X
r
a
Typeset using REVTEX
1
I. INTRODUCTION
Examples of successful employment of anisotropic lattices in lattice QCD simulations
have been increasing lately. They include extensive studies of the glueball spectrum [1],
investigations of heavy hybrid states [2,3] and calculations of quarkonium fine structure [4].
In most cases one is dealing with large states requiring large spatial volumes and also signals
that can only be extracted from high statistics data. Working with highly improved actions
on coarse lattices helps with the large volume and statistics problems, however, a coarse
temporal lattice spacing means that correlation functions fall off very rapidly. This last
problem can be circumvented by going to an anisotropic lattice which allows for a much
finer temporal grid. The correlation functions can be sampled much more frequently in a
given physical time region where the signal is still good.
Another potential use of anisotropic lattices would be in simulations of matrix elements
in hadronic states with large momenta. These typically occur in semileptonic decays of
heavy hadrons. Once one goes beyond spectrum calculations to matrix elements, one is
faced with the matching problem between operators in the coarse highly improved lattice
theory and continuum QCD. In this article we take a first step in accumulating necessary
renormalization factors based on perturbation theory. We carry out the one-loop self energy
calculation in several anisotropic improved quark actions. This gives us renormalization of
the rest mass M , the wave function renormalization Z and the “speed of light” renor-
1 2
malization, C , a quantity which will be defined more precisely in the following sections.
0
We treat both massive and massless quarks. Perturbation theory can be used not only in
operator matchings, but also to fix parameters in the lattice actions. C is an example of
0
one such parameter.
In the next section we introduce the gauge and quark actions considered in this article.
Section 3 describes the general formalism that we employ for the self energy calculation
with massive fermions. We follow closely the work of the Fermilab group [5] which we
could straightforwardly extend to anisotropic actions. Section 4 discusses specific one-loop
contributions for mass, wavefunction and speed of light renormalizations. Our results are
tabulated insection 5 for variouschoices ofactions, fermion masses anddegree of anisotropy.
Some calculational details are left for appendices, where we describe Feynman rules and IR
subtractions in our calculations.
II. GAUGE AND QUARK ACTIONS
We work with two classes of gauge actions denoted I and II [6,7], with
SG SG
I = β 1 cGPss′ +cGRss′ +cGRs′s
SG − x,s>s′ χ ( 0 u4s 1 u6s 1 u6s )
X
P R R
β χ cG st +cG st +cG ts (1)
− x,s ( 0 u2su2t 1 u4su2t 1 u4tu2s)
X
and
2
II = β 1 5Pss′ 1 Rss′ 1 Rs′s
SG − x,s>s′ χ (3 u4s − 12 u6s − 12 u6s )
X
4 P 1 R
st st
β χ . (2)
− x,s (3u2su2t − 12u4su2t)
X
The x sum is over lattice sites and the variable s runs over spatial directions. β 2N /g2,
c
≡
χ is the anisotropy parameter
χ = a /a (3)
s t
and
1
P = Real Tr U (x)U (x+a )U†(x+a )U†(x) , (4)
µν N { µ ν µ µ ν ν }
c
(cid:16) (cid:17)
1
R = Real Tr U (x)U (x+a )U (x+2a )U†(x+a +a )U†(x+a )U†(x) . (5)
µν N { µ µ µ ν µ µ µ ν µ ν ν }
c
(cid:16) (cid:17)
u and u are the tadpole improvement parameters u for spatial and temporal link variables
s t 0
respectively [8].
The parameters cG and cG in action I are constrained to satisfy cG + 8cG = 1. The
0 1 SG 0 1
Symanzik improved gauge action, in which O(a2) errors are removed, corresponds to cG =
0
5/3 and cG = 1/12 [9], whereas cG = 3.648 and cG = 0.331, for χ = 1, leads to one of the
1 − 0 1 −
RGimprovedIwasakiactions[10]. In II parametershavebeenfixedtotheSymanzikvalues.
SG
WewillbeworkingmainlywithSymanzikimprovedactionsandpresentRGimprovedresults
only for a few cases. We note that the action II is intrinsically asymmetric even for the
SG
isotropic limit χ = 1.
The most highly improved quark action that we have analysed is the D234 action [6].
1 1 C 1
I = a3a Ψ γ ( C (3))+ 0~γ (~ C ~(3))+m
SD234 s t c ta ∇t − 6 3t∇t a · ∇− 6 3∇ 0
x (cid:26) t s
X
ra 1 1 1 1
s (2) (4) (2) (4)
( C )+ ( C )
− 2 a2 ∇t − 12 4t∇t a2 ∇j − 12 4∇j 
t s j
X
C iσ F˜µν 
F µν
ra Ψ (6)
s c
− 4 aµaν )
1 C 1
= Ψ γ ( C (3))+ 0~γ (~ C ~(3))+a m
x L( t ∇t − 6 3t∇t χ · ∇− 6 3∇ t 0
X
r 1 1 1
(2) (4) (2) (4)
χ( C )+ ( C )
−2  ∇t − 12 4t∇t χ ∇j − 12 4∇j 
j
X
C a a 
r Fiσ F˜µν s t Ψ . (7)
µν L
− 4 aµaν)
The quark fields Ψ and the dimensionless lattice fields Ψ are related through
c L
Ψ = a3/2Ψ . (8)
L s c
3
The dimensionless derivatives (n) and field strength tensors F˜µν are tadpole improved [8]
∇
and defined in the Appendix. We use the convention σ = 1[γ ,γ ] and set r = 1 in all
µν 2 µ ν
our calculations. At tree-level the coefficients C , C , C , C , C and C are equal to
0 3 3t 4 4t F
one. The quark action is then tree-level accurate through O(a3) and O(a3). C is what
s t 0
we call the “speed of light”. This parameter is adjusted, in general either perturbatively
or nonperturbatively, to ensure correct dispersion relations for particles. In anticipation of
working on anisotropic lattices with a much finer than a , one can drop the higher order
t s
improvement terms in the temporal derivatives by setting C = C = 0, without loosing
3t 4t
accuracy. We call this action II .
SD234
C 1
II = Ψ γ + 0~γ (~ C ~ (3))+a m
SD234 L( t∇t χ · ∇− 6 3∇ t 0
x
X
r 1 1
(2) (2) (4)
χ + ( C )
−2  ∇t χ ∇j − 12 4∇j 
j
X
C a a 
r Fiσ F˜µν s t Ψ (9)
µν L
− 4 aµaν)
The familiar O(a) accurate clover quark [11] action corresponds to setting C = C = 0 in
3 4
the above and using a less improved field strength tensor Fµν (also defined in the Appendix)
rather than F˜µν.
C
= Ψ γ + 0~γ ~ +a m
clover L t t t 0
S ( ∇ χ ·∇
x
X
r 1 C a a
χ (2) + (2) r Fiσ Fµν s t Ψ (10)
−2  ∇t χ j ∇j − 4 µν aµaν) L
X
 
We have carried out one-loop self energy calculations for several combinations of the
above gauge and quark actions, for both massless and massive quarks. We list the specific
actions considered in Table I. For actions A and A′ massless results have already appeared
S S
in [12]. We agree with their results and we include these cases here for completeness. With
action C we treat only the massless case, since our formalism for massive quarks, following
[5], reqSuires that the only time derivatives be in the and (2) terms. Both II and
∇t ∇t SD234
satisfy this condition, but I does not.
Sclover SD234
III. GENERAL FORMALISM FOR SELF ENERGY CALCULATIONS
In this section we summarize the formalism for self energy calculations, along the lines
of reference [5]. Perturbative calculations for massive Wilson quarks are also described in
reference[13]. Weconcentrateonthemassivecase, sincemasslesslatticeperturbationtheory
has been in the literature for decades.
For massive fermions we use either II or . The fermion self energy Σ(p) is
SD234 Sclover
defined in terms of the momentum space propagators G(p) and G (p) for the full and free
0
theories respectively, as
4
−1 −1
G (p) = G (p) Σ(p). (11)
0 −
Carrying out the Fourier transform in p one defines
0
G(t,p~) = π/at dp0eip0tG(p ,p~)
0
2π
Z−π/at
(p~)e−E(p~)tΓ + ... . (12)
2 proj
≡ Z
Γ is a projection operator in Dirac space. The ellipses refer to lattice artifacts and
proj
additional multi-particle states that could be created by the lattice fermion field operator Ψ
beyond the single quark state. The rest mass, M , is defined as
1
M = E(p~ =~0). (13)
1
We do not consider the kinetic mass, M [5] in this article. We will renormalize at the point
2
(p ,p~) = (iM ,~0) and define the wave function renormalization constant
0 1
Z = (p~ =~0). (14)
2 2
Z
For a zero spatial momentum quark propagating forward in time one expects (t > 0)
G(t,0) = π/at dp0eip0tG(p ,0)
0
2π
Z−π/at
1+γ
Z e−M1t 0 + ... . (15)
2
≡ 2
Our goal in this section is to relate Z and M to parameters in the action and to Σ(p). In
2 1
order to orient ourselves, however, it is useful to first consider the free case with Σ(p) = 0.
A. Free Anisotropic Propagator
The free propagator G (p ,~p = 0) for both actions II and becomes (for r = 1)
0 0 SD234 Sclover
1 1
G (p ,~p = 0) =
0 0
a iγ sin(a p )+a m +χ χcos(a p )
t 0 t 0 t 0 t 0
−
iγ sin(a p )+a m +χ χcos(a p )
0 t 0 t 0 t 0
= − − . (16)
(sin(a p ))2 +[(a m +χ) χcos(a p )]2
t 0 t 0 t 0
−
In terms of the variable
z eiatp0 = e−atE (17)
≡
(p = iE), onefinds two zeros ofthedenominator corresponding topositive energy solutions.
0
(a m +χ) (a m +χ)2 +1 χ2
t 0 t 0
z = − − (18)
1 qχ 1
−
5
and
(a m +χ) (a m +χ)2 +1 χ2
t 0 t 0
z˜ = − − . (19)
1 qχ+1
The other two zeros, z and z˜ correspond to negative energy solutions, z = 1/z and
2 2 2 1
z˜ = 1/z˜ . The integral over p in (15) can be done as a contour integral around the unit
2 1 0
circle in the variable z. One picks up contributions from both positive energy solutions (for
t > 0).
(1+γ0) e−M1(0)t
pole at z : , (20)
1
2 (a m +χ)2 +1 χ2
t 0
−
(1 γ0)q e−M˜1(0)t
pole at z˜ : − , (21)
1
2 (a m +χ)2 +1 χ2
t 0
−
q
with
a M(0) = ln(z ), a M˜(0) = ln(z˜ ). (22)
t 1 − 1 t 1 − 1
Clearly, z isthephysicalpositiveenergysolution. Thesecondsolutionz˜ isalatticeartifact,
1 1
similar to the time doubler for r = 1 in isotropic actions. The solution z˜ disappears in the
1
isotropic limit, χ 1, where a M˜6 (0) . In the same limit the physical solution z goes
→ t 1 → ∞ 1
over into the well known result
1
z . (23)
1
→ 1+a m
t 0
The gap between M˜(0) and M(0), measured in units of 1/a is
1 1 s
(χ+1)
˜(0) (0)
a (M M ) = χ ln , (24)
s 1 − 1 (χ 1)
−
independent of m . This becomes at χ = 1 and approaches 2 as χ . The size
0
∞ → ∞
of this gap, a δE 2, is hence equal to or larger than the amount by which conventional
s
≥
spatial doublers are raised through the Wilson mechanism. We will henceforth ignore z˜ and
1
concentrate on the physical pole at z = z . Comparing (20) with (15) one sees that there is
1
nontrivial mass dependent wave function renormalization even at tree-level with
1 1
(0)
Z = = . (25)
2 (0) (0)
(a m +χ)2 +1 χ2 χ sinh(a M )+cosh(a M )
t 0 t 1 t 1
−
q
This has been pointed out several times in the literature [14,15].
A useful way to rewrite (18) is
(0) (0)
a m +χ = χ cosh(a M )+sinh(a M ). (26)
t 0 t 1 t 1
6
B. Mass Renormalization
In the interacting case one has a nontrivial Σ(p) which we write as
1
a Σ(p) = iγ B (p,m )sin(a p )+i [γ B (p,m )sin(a p )]+C(p,m ). (27)
t 0 0 0 t 0 j j 0 s j 0
χ
j
X
The ~p = 0 propagator becomes
1 iγ (1 B )sin(a p )+a m +χ C χcos(a p )
0 0 t 0 t 0 t 0
G(p ,~p = 0) = − − − − , (28)
a 0 (1 B )2[sin(a p )]2 +[(a m +χ) C χcos(a p )]2
t 0 t 0 t 0 t 0
− − −
where, B = B (p ,m ) and C = C(p ,m ) are evaluated at ~p = 0. If p = iE is the location
0 0 0 0 0 0 0
of a pole in (28), the following implicit equation must be satisfied.
(1 B (iE,m ))sinh(a E) = [(a m +χ) C(iE,m ) χcosh(a E)]. (29)
0 0 t t 0 0 t
− ± − −
(0)
One can check that the “+” sign leads to the pole M in the free limit. Hence, the implicit
1
equation for M is given by
1
χ cosh(a M )+sinh(a M ) = a m +χ+B (iM ,m )sinh(a M ) C(iM ,m ). (30)
t 1 t 1 t 0 0 1 0 t 1 1 0
−
In a perturbative calculation of M one expands
1
M = M(0) +α M(1) +O(α2). (31)
1 1 s 1 s
B and C in (30) start out O(α ), so through one-loop their argument can be replaced by
0 s
(0)
the tree-level M . Expanding the LHS also through O(α ) and taking (26) into account,
1 s
one finds
(0) (0) (0)
B (iM ,m )sinh(a M ) C(iM ,m )
α a M(1) = 0 1 0 t 1 − 1 0
s t 1 (0) (0)
χsinh(a M )+cosh(a M )
t 1 t 1
(γ +1)
(0) 0 (0)
= Z tr a Σ(p = iM ,~p = 0) , (32)
− 2 ( 4 t 0 1 )
(0)
where the trace is taken over Dirac space. We note that in the M = 0,m = 0 limit, the
1 0
γ part of the trace tr (γ +1)Σ does not contribute and one has
0 0
{ }
(1)
α a M (0) = tr a Σ(0) /4 = C(0,0). (33)
s t 1 − { t } −
In order to have massless quarks remain massless under renormalization, one needs to carry
(1)
out additive mass renormalization and M in (32) requires a subtraction. This subtraction
1
must be done without jeopardizing the pole condition (30). There is a standard way to
accomplish this. Let m be the value of the bare quark mass parameter m for which the
c 0
physical quark rest mass vanishes (M = 0). Eqn.(30) then tells us that m is implicitly
1 c
defined through
7
a m C(0,m ) = 0. (34)
t c c
−
In equations such as (28) or (30) one always has the combination a m C. Using (34) one
t 0
−
can add and subtract a m so that
t c
a m C a (m m ) (C C(0,m )) = a m C˜. (35)
t 0 t 0 c c t
− → − − − −
Previous derivations go through with m replaced by
0
m m m (36)
0 c
≡ −
and C(iM ,m ) by
1 0
C˜(iM ,m ) = C(iM ,m ) C(0,m ). (37)
1 0 1 0 c
−
In most lattice simulations, m and hence also m are determined nonperturbatively from
c
the simulations themselves. For C˜, however, one still often uses the one-loop result
C˜(iM(0),m ) = C˜(iM(0),m) = C(iM(0),m) C(0,0). (38)
1 0 1 1 −
(0)
M is now given in terms of m rather than m . We will be presenting our results as
1 0
(0)
functions of a M , with the understanding that the shift m m m has been carried
s 1 0 → 0 − c
(0)
out and that, for instance, M is given by
1
(0) (0)
a m+χ = χ cosh(a M )+sinh(a M ) (39)
t t 1 t 1
rather than by (26). In (32) one needs to replace C by C˜. Our final formula for the one-loop
mass correction, measured in units of 1/a then becomes
s
B (iM(0),m)sinh(a M(0)) C˜(iM(0),m)
α a M(1) = χ 0 1 t 1 − 1
s s 1,sub (0) (0)
χsinh(a M )+cosh(a M )
t 1 t 1
(γ +1)
= Z(0) tr 0 a Σ(p = iM(0),p~ = 0,m) a Σ(0,~0,0) . (40)
− 2 ( 4 s 0 1 − s )
h i
(0)
This expression vanishes automatically for M = m = 0. We prefer to measure dimension-
1
ful quantities in terms of 1/a rather than 1/a . When exploring χ 1 it makes more sense
s t
≥
to fix a and let a be arbitrarily fine, rather than to fix a and let a become arbitrarily
s t t s
coarse. In the isotropic limit (40) agrees with formulas in the literature [5,13].
C. Wave Function Renormalization
In order to extract a general formula for the wave function renormalization Z we need
2
to find the residue of G(p ,~p = 0) at the pole p = iM . In terms of the variable z the
0 0 1
Fourier transform in (15) has the form
dz g(z)
(z)t/at , (41)
(2πi)z f(z)
I|z|=1
8
where the integral is taken over the unit circle. To find the residue we expand the denomi-
nator around z = e−atM1
1
df
f(z) = (z z ) + ... . (42)
1
− dz !
z=z1
The contribution from the physical pole to G(t,0) is then
g(z)
e−M1t . (43)
zf′(z)!
z=z1
One finds for the numerator
g(z = z ) = (γ +1)(1 B (iM ,m))sinh(a M ) (44)
1 0 0 1 t 1
−
and for the denominator
2(1 B (iM ,m))sinh(a M )
0 1 t 1
− ×
d
χsinh(a M )+cosh(a M )+ i [iB (p ,m)sin(a p )+C(p ,m)] (45)
 t 1 t 1 d(atp0) 0 0 t 0 0 !p0=iM1
using 
df df
z = i . (46)
dz !z=z1 − d(atp0)!p0=iM1
One can now read off Z and after recognizing the last term in (45) as derivatives acting on
2
different parts of a Σ(p ,p~ = 0,m), one obtains
t 0
(γ +1) d
Z−1 = χsinh(a M )+cosh(a M )+itr 0 Σ(p ,~p = 0,m) . (47)
2 t 1 t 1 4 dp0 0 !p0=iM1
The one-loop approximation to Z is obtained by expanding M once again in α .
2 1 s
Z−1 = χsinh(a M(0))+cosh(a M(0))+α a M(1) (χcosh(a M(0))+sinh(a M(0)))
2 t 1 t 1 s t 1,sub t 1 t 1
(γ +1) d
0
+itr Σ(p ,p~ = 0,m)
0
4 dp0 !p0=iM1(0)
α
(0)−1 s (1) (0) (0) (0)
= Z [ 1+ a M (χcosh(a M )+sinh(a M ))Z
2 χ s 1,sub t 1 t 1 2
(γ +1) d
+itr 0 Σ(p ,p~ = 0,m) Z(0)] + O(α2). (48)
4 dp0 0 !p0=iM1(0) 2 s
(0)−1
In the last expression we have found it convenient to factor out the tree-level Z . Equa-
2
tions (47) and (48) go over into the formulas of [5] in the isotropic limit.
9
D. Speed of Light Renormalization
In order to discuss renormalization of the speed of light one needs to look at the inverse
momentum space propagator at small but nonzero spatial momentum.
−1 −1
a G (p) = a G (p) a Σ(p)
t t 0 − t
1
= iγ (1 B )sin(a p )+i [γ (C K B )sin(a p )]
0 0 t 0 j 0 j j s j
− χ −
j
X
+a m+χ χcos(a p ) C, (49)
t t 0
− −
with K = 1 for and K = (4 cos(a p ))/3 for I,II . One can rewrite G−1(p) as
j Sclover j − s j SD234
1 (C K B )
−1 0 j j
a G (p) = (1 B ) iγ sin(a p )+i [γ − sin(a p )]
t − 0  0 t 0 χ j (1 B ) s j 
 Xj − 0 
+a m+χ χcos(a p ) C. (50)
t − t 0 − 
C is adjusted so that for small a p the relative coefficient of the γ sin(a p ) and the
0 s j 0 t 0
γ a p /χ terms remains equal to unity. (C K B )sin(a p ) = (C B )a p for all quark
j s j 0 j j s j 0 j s j
− −
actions in this limit (of course, K sin(a p ) is a better approximation to the continuum a p
j s j s j
in the D234 action than in the clover action), and one has
C = 1+B (C ) B (C ) 1+B (C = 1) B (C = 1) + O(α2). (51)
0 j 0 − 0 0 ≈ j 0 − 0 0 s
Just as with Z we will define the speed of light renormalization at the zero spatial momen-
2
tum mass shell point p = (iM ,~0). From (27) the two terms B and B needed for C at
1 j 0 0
one-loop can be extracted through
χ ∂
B = i tr γ a Σ(p) , (52)
j j t
− 4 ∂(aspj) !p=(iM(0),~0)
1
1tr(γ a Σ(iM(0),~0))
B = 0 t 1 m > 0 (53)
0 −4 sinh(a M(0))
t 1
or
i ∂
= tr γ Σ(p) m = 0. (54)
0
−4 ∂p0 !p=(0,~0)
WenotethatinthemassivecasethereisnontrivialrenormalizationofC evenintheisotropic
0
limit χ = 1, due to our noncovariant mass shell condition, p = (iM ,~0). Nevertheless, we
1
believe the above definition of the renormalization of the speed of light is a sensible and
physical one.
IV. ONE-LOOP CONTRIBUTIONS TO Σ(P)
In the previous section one-loop corrections for M , Z and C were determined in terms
1 2 0
of traces over Σ(p) or over derivatives acting on Σ(p). In this section we describe the lattice
10