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QCD phenomenology from the lattice: Renormalisation of local operators
Nuclear Physics B (Proc. Suppl.) 9 (1989) 121-133 North-Holland, Amsterdam
121
QCD PHENOMENOLOGY FROM THE LATTICE: Renormalisation of Local Operators*
Christopher T. Sachrajda Department of Physics, The University,
Southampton,
S09 5NH, United Kingdom
The problem of extracting phenomenologically useful results from lattice measurements of operator matrix elements is reviewed. We start with a brief discussion of "lattice renormalisation schemes" and lattice perturbation theory. Then, by using the vector current as an example, it is demonstrated that the mixing of operators under renormalisation with other operators of higher mass dimension, an effect which disappears as the lattice spacing goes to zero, is nevertheless numerically significant at present values of p. It is also demonstrated that the power divergences (i.e. terms which diverge as inverse powers of the lattice spacing) which arise when the operators of interest can mix with others of lower dimension, must be subtracted non-perturbatively.
h
(in this talk I will use the MS scheme) whereas
I. INTRODUCTION In many phenomenological
applications of QCD
the non-perturbative physics is contained in
the lattice computations of the matrix elements are performed using some "lattice
matrix elements of local operators taken
renormalisation scheme",
between hadronic states.
depends on the lattice action being used).
During this confer-
ence we will hear about the latest (and in many
(the lattice scheme
cases the first) lattice computations of some
difference between the two schemes; this in-
of these matrix elements 1'2
volves performing a perturbative calculation
However, before
the results of these computations can be compared with experimental measurements,
It
is therefore necessary to bridge the
using Feynman rules derived from the lattice they
action which was used to compute the matrix
have to be combined with perturbative calcula-
elements.
tions of the contributions
which use lattice perturbation theory, in some
from the
short-distance or light cone regions of phase space,
(i.e. with the corresponding Wilson
coefficient function).
In general both the
perturbative calculations and the matrix
I will discuss these calculations,
detail in section 2. To illustrate the content of the opening paragraph let us consider deep inelastic structure functions and weak decay amplitudes.
elements depend on the way the local operators
In each case the result can be expressed as
have been renormalised
the matrix element of a bilocal operator
(i.e. on the
renormalisation scheme and on the
H(x,0) between hadronic states.
renormalisation scale).
inelastic electroproduction H is the product
The perturbative
In deep
calculations have generally been performed in
of two electromagnetic currents, one at x and
some standard continuum renormalisation scheme
the other at 0, whereas for weak decays H is the product of two weak (V-A) currents, convoluted with the W or Z propagator.
Plenary talk presented at the annual lattice conference, LATT88, Fermilab, September 1988
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Using
the operator product expansion H is expanded as a series of local operators:
C.T. Sachrajda / QCD phenomenology from the lattice
122
other choices of the action, H(x,O)
= ~
C i (x;~) 0i(O; ~)
(I.I)
where ~ islthe renormalisation scale and the
For the structure functions in the
Bjorken limit we are interested in light-like separations
(x2~0) whereas for the weak decays
we are interested in the short distance limit, where each component of x is 0(I/Mw).
In each
(the operators
of lowest twist in the case of the structure functions and the operators of lowest dimension in the case of the weak decays).
The matrix
elements of H(x,0) are independent of ~ and of the renormalisation scheme used to define the renormalised operators 0i(O;~).
However,
it
is necessary to calculate the coefficient function C i in the same renormalisation scheme as that used to compute the matrix elements of Oi, a calculation which involves the use of
The plan for the remainder of this talk is In the next section I will
briefly review the use of lattice perturbation theory.
which are 0(a) in numerical calculations, this may lead to significant errors at present values of the lattice spacing a. In section 3, by using the renormalisation constant of the local vector current
example,
(which
I demonstrate that this is indeed
the case.
For the purposes of illustration I
will use the Wilson action for lattice QCD:
Generalisations
to other
operators are discussed. In section 4 I consider power divergences,
i.e. terms in the
renormalisation constants of operators which diverge as an inverse power of the lattice spacing (a), as a goes to 0.
These power
divergences are more severe on the lattice, since Euclidean symmetry is broken (as is chiral symmetry in the case of Wilson fermions).
"lattice perturbation theory".
as follows.
section 5).
is not conserved on the lattice) as an
case we need only keep a small number of operators in the sum in (I,I),
(see
Although it is usual to neglect terms
operators {0 i} form a complete set of local operators.
in particular
for that containing staggered quarks,
It is pointed out that the
power divergences must be subtracted non-perturbatively.
Section 5 contains a
brief review of the calculations which have been performed to date, which use lattice perturbation theory to extract phenomenologically meaningful results from
s : -
lattice computations. x
Section 6 contains a
summary and our conclusions.
^
+
+Ki(x+~)(l-y~lU
(x)¥(x)]-i(x),(x)) 2. LATTICE PERTURBATION THEORY
I
Tr[Up+U;?
(1.2)
Plaquettes
AS an example consider the local vector current V l°c = ~y~¥.
This current is not
where all colour,flavour and spin indices have
conserved in the lattice theory defined by
been omitted for simplicity.
the action (1.2).
parameter,
K is the hopping
~ m 6/g2o where go is the bare
We start by calculating
its renormalisation constant to one loop
coupling constant and Up is the product of
order, using perturbation theory.
links belonging to the elementary plaquette P.
Feynman rules required for this calculation
The discussion however, applies equally well to
are shown in fig,l (rules for other vertices and propagators can be found in 3.
The
Because
the theory is written in terms of links,
C.T. Sachrajda / QCD pheaomenology from the lattice
p,a
k Gluon Propagator
~2
123
~ab
~r
l~J ~'~sinka +½a~z 8ij
k Quark Propagator
a,p a
--21ia(p+q)~ " } T?. ,j •
j
J a,p
b,-J
-½g2os,. {co,½(p÷q), o P
-ija~(P~q2 )~ }{TQ, T b}jj
j
Quark- Gluon Vertices FIGURE I Feynman Rules required for the calculation of the one loop contribution to the renormalisation constant of the local vector current. ~ ~ 2/a sin (k#/2).
C.T. Sachrajda / QCD phenomenology from the lattice
124
(a)
(b)
(c)
FIGURE 2 One-loop diagrams which contribute to the renormalisation constant of the local vector current. rather than potentials,
there are additional
contribution comes from the "tadpole graph"
vertices to those in the continuum theory; e.g.
(fig.2(c)), which is a general feature of
the four point vertex in fig.l.
such computations.
The diagrams
which have to be evaluated are shown in fig.2.
In the next section I will report on the
These are evaluated numerically with the
results of lattice computations of Zvl°C,
results (in the Feynman gauge):
which will be compared to the expected result, i.e. that given in eqn. (2.3).
s CF
fig 2(a) -
[log
1
+ 7.26]
(2.1a)
In
general we are interested in computing the matrix element of some local operator 0
fig 2(b)
~ s CF
I-log
I
+
1.12] (21b)
between an initial state state
fig 2(c) - ~
S
If>.
li> and a final
Renormalising the operator by
using the lattice scheme we have: CF
12.24
(2.1c)
where I is an infrared cut-off introduced by
•
II>LATT =
0(o)[i +
modifying the denominator of the gluon propagator from ~2 t o ~2+ ~2,
factor
4 CF ~ g .
The u l t r a v i o l e t
and t h e c o l o u r
_ss
4, C-yIogC~2a2]
+
+ CLATT)
"'']
(2.4)
(and infrared)
whereas
by using the MS scheme we have
divergence cancels and we find ks zlOC 1 - s V = ~ C F 20.6
+
(2.2)
Taking p = 6.0 (i.e. g = I) as a typical value
2 ylog ~-+ ...]
(2.5)
of the coupling in current lattice where CLATT, c~-~ and the anomalous dimensions calculations gives, y are constants which can be determined from zlOC = 0.83 V
(2.3) one-loop perturbation theory.
The calculations are straightforward, although in general the loop integrals have to be performed numerically.
The largest
bare operator.
0 (°) is the
Thus the matrix elements in
the two renormalisation schemes are related by
C.T. Sachrajda / Q C D phenomenology from the lattice
125
momentum tensor.
Fi>~-~ =LATT (2.6)
states are proportional
S
( c ~ - CLATT?+...]
El + ~ In this
way t h e m a t r i x e l e m e n t d e f i n e d
continuum renormaIisation from that
in the hadron. Among the diagrams contribut-
in a
ing to the renormalisation
scheme i s o b t a i n e d
The corresponding
Of course if we calculate all the
higher order terms on the right hand side then
As the lattice and MS definitions
of ~s
Indeed the contributions
Thus in this example c ~
correction
In practice
to the
the lattice coupling is
This is natural for the example of the
tends to be
For other operators
CLATT > c~[~
for n=O
(2.8a)
CLATT ~ c~-~
for n=l
(2.8b)
CLATT < CM-~
for n~2
(2.8c)
derivatives
are due to lattice
in the operator.
this is not the
case, however the large contributions c~
represents a negligibly small
to the measured value of.
where n is the number of covariant
local vector current given above since c ~ = 0 , and all the corrections effects.
- CLATT is small,
the dominant one it is usually the case that
the one loop term in (2.6) is sensitive
used.
from the two
Since the tadpole contribution
(2.7)
~LATT(~ )
choice.
loop integral is identical
to that for the tadpole graph in fig (2.c).
and eqn.(2.6)
differ c o n s i d e r a b l y 3 ' 4
at ~ ~ 2GeV,
(3).
tadpole graphs are equal and opposite.
the result would not depend on the choice of
2.7
of this operator
is the one loop graph drawn in figure
Which definition of ~s should we use in
~s"
to, the average
fraction of momentum carried by the quarks
m e a s u r e d on t h e l a t t i c e .
eqn.(2.6)?
Forward matrix elements
of this operator between single hadron
to
- CLATT still come from lattice effects.
Thus although
it is not possible
to say that
the use of the lattice coupling is a better approximation
than a continuum definition,
without calculating
higher order corrections,
this is indeed the expectation above observation
that the one loop contribu-
tion comes predominantly I have mentioned contributions
from lattice effects.
above that the tadpole FIGURE 3
to the wave function
renormalisation contribution
based on the
constant
is the largest
to the one loop term in eqn.(2,2),
A "tadpole" diagram which contributes to the renormalisation constant of the energy momentum tensor, and hence.
and generally gives the dominant contribution. Tadpole graphs also arise from the expansion of the link variables
in covariant derivatives
in
is the quark contribution
to the energy
In lattice computations
it is usual to
neglect terms which vanish as the lattice
operators which contain such derivatives. Consider for example the operator ~ygDV~,
3. 0(a) CORRECTIONS
which
spacing goes to zero. attempts
to "improve"
There have been some the calculations
including some of these terms, notably
by
C.T. Sachrajda/ QCD phenomenology from the lattice
126
Symanzik's
program 5,6, but in practice
terms are not kept in phenomenological tions.
these
Here I would like to demonstrate,
means of a numerical neglecting
computation,
where
applicaby
that
four-momentum p.
this matrix
lattice at ~=6.0 we
when the particle is a pion8:
errors.
The example which I will use is again that of the local vector current VB l°c = ~y~¥.
is not conserved
= 0.99 ± 0.05
for K=0.1515
~v = 1.00 ± 0.05
for K=O.1530
= I.II ± 0,06
for K=0.1545
In
the previous section we have seen that this operator
Computing
element on a 20xlO2x40
obtain the following very precise results
terms which are O(a) can lead to
significant
I p > is a meson or baryon state with
in the theory defined
(3.4)
by the action (1.2) and that we should expect its renormalisation 6.0.
constant
The computation
The chiral limit corresponds
to be ~ 0.83 at
to a value of
the hopping parameter K=0.1564(2) 8'9.
of Zvl°C is made
simpler by the existence of a conserved vector For the proton the results are still
current in the theory defined by (1.2)7: 1
~= ~ ~(x) (y-1)u (x)v(x+~) + ~ (x+~) (v+l) U+(x),(x)] The renormalisation ~
consistent with one but are a little
(3.1)
precise.
constant of the operator
(3.1) separately
_loc ZV can be readily
is precisely one.
for the two cases when the
quark, q, is an up quark or a down quark. We find I0
determined by taking ratios of matrix elements, _loc
In this case we compute the matrix
elements of the conserved vector current
~V = 0.87 ± 0.15
for K=0.1515
and q=up
~V = 0.92 ± 0.10
for K=0.1515
and q=down
=
ZV
(3.2)
and = ratio of corresponding functions. zlOC has been computed using both two and V three point correlation
(2.5)
correlation
~V = 0.89 ± 0.29
for K=0.1530 and q=up
~V = 1.08 ± 0.16
for K=0.1530
functions and I will For both the pion and proton,
now discuss these results.
had three-momentum a) 3-Point Correlation Before presenting
Functions.
the results for Zvl°C,
constant of the conserved vector current,
I
a
In this way we check the precision of
any information
(without of course gaining
about the physics content).
Defining this renormalisation < p I ~
[ p >
=l_ 2pp ~v
and the matrix
index ~ chosen to be I.
The results are
consistent with ZV = 1 and we also have some
quantity for which we know the exact result to
the numerical work,
(I/10,0,0)
the hadron
elements were computed with the Lorentz
will present those for the renormalisation
be I.
and q=down
constant by (3.3)
indication of the precision of lattice computations
involving three-point
tion functions.
correla-
The results in (3.4) and
(3.5) were obtained using 15 gauge field configurations.
C. T. Sachrajda / QCD phenomenology
from
the lattice
127
We have confirmed these results on our K 0.1515 0.1530 0.1545
K 0.1515 0.1530
larger
PION = 4 0.76 0.73 0.71
~= 1 0.73 0.71 0.71
lattice 12.
These results are
clearly inconsistent,
the results obtained using the three-point correlation function,
PROTON q = up ~ = 4 0.76 0.73
not only with the
perturbative result of 0.83, but also with
~ = 1 0.72(3) 0.74(10)
(see table I).
What is the reason for the discrepancy between the results obtained using two-point and three-point correlation functions?
K 0.1515 0.1530
It
is not due to the quenched approximation,
PROTON q = dow~ ~ = 4 0.76 0.73
~ = 1 0.73 0.73(4)
since this would affect both computations equally. Moreover,
the one-loop contribution
to Zvl°C does not contain any quark loops Table I. Values of ZvL°C obtained by
anyway.
computing three point correlation functions
being due to the fact that V l°c mixes with
containlng the operator qy~q.
operators of higher dimension with coeffi-
The error is
We interpret the discrepancies as
less than one on the last digit unless
cients which vanish as the lattice spacing
otherwise stated.
goes to zero,
This effect is neglected in
all calculations, In table one we present the results of our
but the numerical results
presented above seem to indicate that it can
calculations of ZV l°c, obtained by taking the
lead to significant errors.
This interpre-
ratio of the three-point correlation functions
tation is supported by the fact that at
with the local and conserved currents
~=6.2 (for which the inverse lattice spacing
(eqn.(3.2)).
is about 3 GeV), Zvl°C increases to a value
The correlation functions were
evaluated at zero momentum transfer. that the statistical
0.6513 .
Notice
Here I have demonstrated,
errors on the ratios are
by using V l°c
tiny, they are generally less than 1 in the
as
second digit.
of higher dimension can lead to significant
The results do depend on the
quark mass, albeit very mildly.
They differ a
an example, that mixing with operators
errors,
(~ 40%?) on the numerical evaluation
little from the one-loop perturbative result of
of matrix elements.
0.83, but in itself this could be accounted for
particularly simple because of the knowledge
by higher order corrections to the perturbative
that the renormalisation constant of the
prediction.
conserved current is precisely I.
What is more significant however
is that the results differ from those obtained
Our example was
For other
operators we do not know the exact value of the renormalisation constant, however, by
using two-point correlation functions. b) Two-Point Correlation Functions
computing the ratio of the matrix elements
Using two-point correlation functions, on a
of two different lattice definitions of the
I03x20 lattice at ~=6.0, Maiani and
same continuum operator, and comparing the
Martinelli II obtained the results
result to the perturbative prediction for
Zvl°C = 0.57 ± 0.01
for K=0.1515
ZvI°C = 0.57 ± 0.01
for K=0.1530
Zvl°C = 0.58 ± 0.02
for K=0 1545
this ratio, one can get some idea of the (3.6)
systematic problems.
The important
question which now arises is whether it is
C.T. Sachrajda/ QCD phenomenology from the lattice
128
possible to include some of the O(a) correc-
the 4 operators 0 3 D ~
transform as the
tions by hand, and hence to reduce the
vector representation
((~,~) in the notation
systematic
of ref (16) where the character table for
error?
This problem is currently
the hypercubic
being studied.
group can be found).
vector current ~ y ~ 4.
POWER DIVERGENCES
transforms
The
in the same
way and the operator 03~D~ mixes with the vector current under renormalisation.
In the previous section we discussed the problem of operators mixing under renormalisation
mixing coefficient
diverges
Hence in the computations
with other operators
of higher
elements of 0 3 ~ ,
The
like I/a 2.
of the matrix
the leading term is this
dimension.
This effect was found to be
mixing term of O(I/a2), which is a lattice
numerically
significant
artefact of no phenomenological
at current values of
the lattice spacing a, but becomes important as a gets smaller.
interest.
This power divergence must be subtracted
less
In this section
before we can extract useful results from
we discuss mixing with operators of lower
lattice calculations.
dimension;
case this is possible by choosing the
in this case the mixing coeffi-
In this particular
cients diverge as inverse powers of a. This
Lorentz indices appropriately.
problem is more severe on the lattice because
the combination
Lorentz
(or Euclidean)
symmetry
invariance
and chiral
(in the case of Wilson fermions)
not symmetries
of the lattice.
renormalisation
schemes,
prevent the existence
transform as different of the Lorentz group.
these symmetries
often
operators may
tensor representations As will be demonstrated
below, the power divergences non-perturbatively
are
must be subtracted
before meaningful
results
can be obtained from the lattice computations. To illustrate divergences 03~vP m ~ Lorentz
indices.
operators
the existence of power
Matrix elements of these
are proportional
tions of hadrons
(4.1)
like the eight dimensional
representation
of the hypercubic
group and
does not mix with operators of lower dimension.
Thus it is possible to evaluate
the matrix elements of 03~vP with the power divergence
removed but at a two-fold price.
First of all the subtraction removes the component
in eqn.(4.1)
in 03411 which
like the (~,~) representation
i.e. it removes the power divergence.
D p ~, symanetrised over the
to the second
moment of the deep inelastic structure
transforms
transforms
consider the operators
y~ D v
03411 - ~ (03422 + 03433 )
In continuum
of power divergences,
since e.g. the corresponding
For example
leads to a considerable
tion with the corresponding precision.
func-
(when the forward matrix
numerical
Secondly,
This
cancella-
loss of
when we take the
matrix element of 0~ vp we are interested the "leading twist" component,
which is the
element between the single hadron states is
term proportional
computed 8'I0,14)
momentum of the hadron. Thus the matrix
or the quark distribution
to p~pVpp, where p is the
amplitude of the pion (when the matrix element
element of the operator in eqn,(4.1)
between the pion and the vacuum is computedlS).
vanishes
Under transformations
non-zero three-momentum.
of the hypercubic
group
possible
in
trivially unless the hadron has a
to evaluate
It is not
the matrix elements of
C.T. Sachrajda / QCD phenomenology from the lattice 0~ vp with the hadron at rest.
129
Power divergences must be subtracted
Note that it would not be possible to avoid
non-perturbatively.
the large numerical cancellations by choosing all 3 Lorentz indices to be different (e.g. by
For the field strength tensor F ~v on the lattice, the definition 19
using the operator 0~ 12 which transforms like the ( ~ )
representation and again does not mix
with operators of lower dimension).
Fgv~ !
4a 2
However
~
4 plaquettes in ~,v plane
1 U + 2i~UpP ~ (4.3)
the symmetrisation over the Lorentz indices which is implicit in the definition of 0~ 12,
has the advantage that it transforms like
again involves the cancellation of power
the ( I , 0 ) ~
divergences.
representation of the hypercubic group.
Moreover using 0~ 12 we would
need to give the hadron two non-zero components
F ~v is defined using a single plaquette,
the ~,v plane as in eqn. (4.3), then it
To illustrate the large numerical cancellations consider Figure 4, where the three-point correlation functions of the operators 0~ 11 and 0~ II - ~ (0~22 + 0~ 33) between pion states with momentum (~/I0,0,0) The subtraction of the power
divergences reduces the result by an order of magnitude.
If
rather than summing over all 4 plaquettes in
of three momentum.
are presented 14.
(0,I) 6 dimensional, reducible
Fortunately the correlations in the
results for 0~ II and ~ (0~22 + 0~ 33) are strong and even after the subtraction there is still a significant signal,
(the situation is
somewhat worse in the case of the quark distribution amplitudel5). For the structure function of the pion the results in fig. 4 give
transforms like a 24 dimensional reducible representation and there is more scope for operators containing F ~v to be able to mix with operators of lower dimension.
For
example, using the single plaquette definition of F ~ ,
the gluonic contribution
to the energy-momentum tensor, F~VF ~p, contains a hypercubic group singlet and mixes with the unit operator under renormalisaiton,
(we have verified numeri-
cally that the vacuum expectation value of F~VF ~p does not vanish when the single plaquette definition of F ~v is usedl2). Hence care has to be taken to choose an
=
0.18 ± 0.05
(4.2)
appropriate lattice definition of F ~V to avoid power divergences, and eqn. (4.3) is
at a scale of about 7 GeV, where experimental data is available.
Unfortunately not all power divergences
In this example even after
the subtraction of power divergences the error is reasonable, but this is not so in some other
can
be removed by using the transformation
properties of operators under the hypercubic group, and choosing the Lorentz indices
cases. It has been suggested that it may be reasonable to subtract the power divergences perturbatively 17.
frequently found to be the optimal choice.
Perturbation theory at ~ = 6 . 0
appropriately.
An important example appears
in the computation of weak matrix elements in the study of the ~I = ~ rule (in
generally gives a small contribution for these
particular when evaluating matrix elements
divergences 17'18, whereas we have
between single meson states).
seen that the full non-perturbative subtraction changes the result drastically.
~±
The operators
=
[SLY d L ULY~UL ± SLY u L ULY~dL ] - u ~ (4.4)
c
C.T. Sachrajda / QCD phenomenology from the lattice
130
I
I
I
C(tx= 15,fy)x 10"l
I
1
[
K = 0.1530
Operator n" 3~11(y)
Operator O~'~ (y)-½(O~ ,,(y)+ O~"(y)) 100 90 80 70 60 50 -4,0--
3020100 0
5
6
7
8
I
I
9
10
FIGURE 4: Three-point correlation functions of the operators 0~ II and 0~ II(0~ 22 + 0~aa). The difference is due to the subtraction of power divergences.
C.T. Sachrajda/ QCD phenomenology from the lattice
131 +
Thus in order to determine combination
which have mass dimension six, mix under renormalisation
of matrix elements
with the scalar density sd, +
which has dimension
3; specifically
A+ = Z± ^± + + O~E N [0LATT + S60- + S5 O- +
C±
+
+
(4.5)
operators with which O± mix.
do not contain any power divergences,
These
(the GIM
The power divergence
of
the
in c-.
In the SU(3) limit, using
chiral perturbation
of the AI = ~ amplitude, configurations 1,20,21.
even with 30
For this reason the amplitudes
(which
do not suffer from these power divergences) directly may seem to be the natural way to
is
+
contained
for the 100%
errors which still exist in the evaluation
evaluation of the K + ~
that the extra mass
factor in 850± is the mass
charm quark).
of power divergences
in eqn. (4.10) is responsible
renormalisation
S60± and S50± are other six and five
dependent
I sd I K+(g) > (4.10)
The subtraction
mechanism guarantees
IO~ERTI K+(O)>
- <~+(O)
where Z is the diagonal
dimensional
+
IO~ERTIK (R)>
sd]
+ ± Z- [0pERT + c ± sd]
constant,
y- we need the
theory to reduce out a pion
study the AI = ~ rule, however in this case one first has to understand
from the K + I~ matrix element, we find that
scalar particle
<~+(k)
coefficients
the role of the
pole 20'21.
We note in passing that in this case, the 10;m N IK+(H)> = y~k.q + gem 2
(4.6) +
where +the K + I~ amplitude Thus y- is given by
c ± obtained non-
perturbatively,
is given by y-.
using the procedure
described above, agree approximately with those obtained in perturbation
[E(q) - m] my ± = <~+(0)
I~;ENI
- <~+(O)
IO~ENI
theory.
K+(R)>
K+(O)>
(4.7)
5. PERTURBATIVE
CALCULATIONS
In this section I briefly review the The factor in square brackets on the left hand side of eqn.(4.7) fluctuations
is small and subject to large
hence reducing
the evaluation
of y±.
cient c ± is determined condition
the precision
The divergent
in
coeffi-
once a renormalisation
is imposed on the operators 0 ±, for
example
calculations
(or are nearing completion),
from those computed on the lattice: i)
Bilinear Operators
of
the sixteen matrices of Dirac theory).
For these operators ± I~RENI K+(O)> = 0
(4.8)
+
± c
<~ (9) = -
condition c- is
the perturbative
have been performed both for
The calculations
include the
cases in which the lattice operators are
+
extended over two lattice sites (connected
10pERTI K+(9 )> (4.9)
+
(~r~ where F is one
Wilson fermioms 22,23 and staggered fermions 24.
+
With this renormalisation given by:
which enable
one to determine continuum matrix elements
calculations <~+(9)
which have been done to date
I sd I
K+(O) >
by the appropriate
link variable so as to
preserve gauge invariance), operators of the form ~ F D ~ ,
so that where D is the
C.T. Sachrajda / QCD phenomenology from the lattice
132
covariant
derivative
ii) Four-fermion
are also included.
operators
and ~I = 3/2 weak decays. cients the calculation whereas for staggered
relevant
2)
for AI =
lattice spacing,
For Wilson coeffi-
is complete 25'26,27, fermions
The corrections
of O(a), where a is the
can be significant.
In
section 3 we presented an explicit example in which this was the case.
it is almost
3)
Power divergences,
i.e. terms which
complete 28,29, but the mixing with the dimen-
diverge as inverse powers of a as a ~ O,
sion five operator
must be subtracted non-perturbatively,
(SsO± in eqn. 4.5)) has
still not been worked out. fermions the corrections
For staggered
are frequently very
large (in some cases the one loop perturbative correction can change the measured
ACKNOWLEDGEMENTS
result by
It is a pleasure to thank my colleagues
over 50%28).
of the European Lattice Collaboration
iii) Lowest twist operators inelastic
structure
relevant for deep
functions.
For Wilson
fermions these have been computed 17. the coefficients
see
section 4.
Phenomenology.
However
tabulated in tables VI and
Particular
thanks goes to
Guido Martinelli with whom many of the ideas and calculations
VII 17 are wrong 3 0 ' 2 3 , 1 4 .
for an
education in so many aspects of Lattice
described
in this talk were
developed.
iv) Lowest dimension and lowest twist baryonic operators.
These operators are relevant for
REFERENCES
proton decay matrix elements and the quark distribution
amplitudes
of nucleons.
Wilson fermions the coefficients
For
v) Renormalisation
of fB' (the decay constant
of the B-mesons).
Eichten 32 has proposed an
approach to the computation
and the corresponding heavy quark propagator
B-parameter,
these proceedings.
2.
G. Martinelli,
3.
H. Kawai, R. Nakayama and K. Seo, Nucl.Phys. B189 (1981) 40.
4.
A. Hasenfratz and P. Hasenfratz, Phys.Lett. 93B (1980) 165.
5.
K. Symanzik, ibid, 205.
6.
J-P. Ma and W. Wetzel, (1986) 441.
7.
L.H. Karseten and J.H. Smit, Nucl.Phys. B183 (1981) 103.
8.
G. Martinelli and C.T. Sachrajda, Phys. B306 (1988) 865.
9.
M.B. Gavela et al, Nucl. Phys. B306 (1988) 677.
these proceedings.
of fB
in which the
Nucl.Phys.
B226 (1983);
is expanded in inverse
powers of the heavy quark mass. sponding perturbative
C, Bernard,
have been
c o m p u t e d 31 .
interesting
I.
The corre-
calculation
Phys.Lett.
176B
is currently
in p r o g r e s s 33,
6. SUMMARY AND CONCLUSIONS
Nucl.
The three main points raised in this talk are : 1)
Matrix
be r e l a t e d
elements
m e a s u r e d on t h e
to
defined
those
lattice
by using
renormalisation
schemes in perturbation
see section 2.
The corrections
significant
see section 5.
I0. G. Martinelli and C.T. Sachrajda, preprint TH.5042/88.
Cern
theory,
are often
and sometimes very large.
such calculations
can
continuum
II. L. Maiani and G. Martinelli, B178 (1986) 265.
Phys.Lett.
Many
have been or are being done,
12. G. Martinelli unpublished.
and C.T. Sachrajda,
C.T. Sachrajda / QGD phenomenology from the lattice 13. M.B. Gavela, L. Maiani, G. Martinelli, S.Petrarca, O. Pene and F. Rapuano, unpublished. 14. G. Martinelli and C.T. Sachrajda, Phys.Lett. 196B (1987) 184. 15. G. Martinelli and C.T. Sachrajda, Phys.Lett. 190B (1987) 151. 16. J. Mandula, G. Zweig and J. Govaerts, Nucl. Phys. B228 (1983) 91 17. A.S. Kronfeld and D.M. Photiadis, Phys.Rev. D31 (1985) 2939. 18. S. Gottileb and A.S. Kronfeld, Phys.Rev. D33 (1985) 227. 19. J. Mandula, G. Zweig and J. Govaerts, Nucl.Phys. B228 (1983) 109. 20. L. Maiani, these proceedings. 21. M.B. Gavela et al., Cern Preprint TH.5032/88. 22. G. Martinelli and Y.C. Zhang, Phys.Lett. 123B (1983) 77. 23. G. Martinelli and Y.C. Zhang, Phys.Lett. 125B (1983) 7. 24. D. Daniel and S.N. Sheard, Nucl.Phys. B302 (1988) 471. 25. G. Martinelli, Phys. Lett. 141B (1984) 395. 26. C. Bernard, T. Draper and A. Soni, Phys.Rev. D36 (1987) 3224 27. G. Curci, E. Franco, L. Maiani and G. Martinelli, Phys.Lett. 202B (1988) 363. 28. S.N. Sheard, Edinburgh Preprint 87/439 (1988). 29. G. Kilcup, A. Patel and S. Sharpe, in preparation. 30. A. Kronfeld, private communcation. 31. D.G. Richards, C.T. Sachrajda and C.J. Scott, Nucl.Phys. B286 (1987) 683. 32. E. Eichten, proceedings of the International Symposium on Field Theory on the lattice, Seillac, France 1987. 33. P. Boucaud, L. Liu and O. Pene, in preparation.
133
121
QCD PHENOMENOLOGY FROM THE LATTICE: Renormalisation of Local Operators*
Christopher T. Sachrajda Department of Physics, The University,
Southampton,
S09 5NH, United Kingdom
The problem of extracting phenomenologically useful results from lattice measurements of operator matrix elements is reviewed. We start with a brief discussion of "lattice renormalisation schemes" and lattice perturbation theory. Then, by using the vector current as an example, it is demonstrated that the mixing of operators under renormalisation with other operators of higher mass dimension, an effect which disappears as the lattice spacing goes to zero, is nevertheless numerically significant at present values of p. It is also demonstrated that the power divergences (i.e. terms which diverge as inverse powers of the lattice spacing) which arise when the operators of interest can mix with others of lower dimension, must be subtracted non-perturbatively.
h
(in this talk I will use the MS scheme) whereas
I. INTRODUCTION In many phenomenological
applications of QCD
the non-perturbative physics is contained in
the lattice computations of the matrix elements are performed using some "lattice
matrix elements of local operators taken
renormalisation scheme",
between hadronic states.
depends on the lattice action being used).
During this confer-
ence we will hear about the latest (and in many
(the lattice scheme
cases the first) lattice computations of some
difference between the two schemes; this in-
of these matrix elements 1'2
volves performing a perturbative calculation
However, before
the results of these computations can be compared with experimental measurements,
It
is therefore necessary to bridge the
using Feynman rules derived from the lattice they
action which was used to compute the matrix
have to be combined with perturbative calcula-
elements.
tions of the contributions
which use lattice perturbation theory, in some
from the
short-distance or light cone regions of phase space,
(i.e. with the corresponding Wilson
coefficient function).
In general both the
perturbative calculations and the matrix
I will discuss these calculations,
detail in section 2. To illustrate the content of the opening paragraph let us consider deep inelastic structure functions and weak decay amplitudes.
elements depend on the way the local operators
In each case the result can be expressed as
have been renormalised
the matrix element of a bilocal operator
(i.e. on the
renormalisation scheme and on the
H(x,0) between hadronic states.
renormalisation scale).
inelastic electroproduction H is the product
The perturbative
In deep
calculations have generally been performed in
of two electromagnetic currents, one at x and
some standard continuum renormalisation scheme
the other at 0, whereas for weak decays H is the product of two weak (V-A) currents, convoluted with the W or Z propagator.
Plenary talk presented at the annual lattice conference, LATT88, Fermilab, September 1988
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Using
the operator product expansion H is expanded as a series of local operators:
C.T. Sachrajda / QCD phenomenology from the lattice
122
other choices of the action, H(x,O)
= ~
C i (x;~) 0i(O; ~)
(I.I)
where ~ islthe renormalisation scale and the
For the structure functions in the
Bjorken limit we are interested in light-like separations
(x2~0) whereas for the weak decays
we are interested in the short distance limit, where each component of x is 0(I/Mw).
In each
(the operators
of lowest twist in the case of the structure functions and the operators of lowest dimension in the case of the weak decays).
The matrix
elements of H(x,0) are independent of ~ and of the renormalisation scheme used to define the renormalised operators 0i(O;~).
However,
it
is necessary to calculate the coefficient function C i in the same renormalisation scheme as that used to compute the matrix elements of Oi, a calculation which involves the use of
The plan for the remainder of this talk is In the next section I will
briefly review the use of lattice perturbation theory.
which are 0(a) in numerical calculations, this may lead to significant errors at present values of the lattice spacing a. In section 3, by using the renormalisation constant of the local vector current
example,
(which
I demonstrate that this is indeed
the case.
For the purposes of illustration I
will use the Wilson action for lattice QCD:
Generalisations
to other
operators are discussed. In section 4 I consider power divergences,
i.e. terms in the
renormalisation constants of operators which diverge as an inverse power of the lattice spacing (a), as a goes to 0.
These power
divergences are more severe on the lattice, since Euclidean symmetry is broken (as is chiral symmetry in the case of Wilson fermions).
"lattice perturbation theory".
as follows.
section 5).
is not conserved on the lattice) as an
case we need only keep a small number of operators in the sum in (I,I),
(see
Although it is usual to neglect terms
operators {0 i} form a complete set of local operators.
in particular
for that containing staggered quarks,
It is pointed out that the
power divergences must be subtracted non-perturbatively.
Section 5 contains a
brief review of the calculations which have been performed to date, which use lattice perturbation theory to extract phenomenologically meaningful results from
s : -
lattice computations. x
Section 6 contains a
summary and our conclusions.
^
+
+Ki(x+~)(l-y~lU
(x)¥(x)]-i(x),(x)) 2. LATTICE PERTURBATION THEORY
I
Tr[Up+U;?
(1.2)
Plaquettes
AS an example consider the local vector current V l°c = ~y~¥.
This current is not
where all colour,flavour and spin indices have
conserved in the lattice theory defined by
been omitted for simplicity.
the action (1.2).
parameter,
K is the hopping
~ m 6/g2o where go is the bare
We start by calculating
its renormalisation constant to one loop
coupling constant and Up is the product of
order, using perturbation theory.
links belonging to the elementary plaquette P.
Feynman rules required for this calculation
The discussion however, applies equally well to
are shown in fig,l (rules for other vertices and propagators can be found in 3.
The
Because
the theory is written in terms of links,
C.T. Sachrajda / QCD pheaomenology from the lattice
p,a
k Gluon Propagator
~2
123
~ab
~r
l~J ~'~sinka +½a~z 8ij
k Quark Propagator
a,p a
--21ia(p+q)~ " } T?. ,j •
j
J a,p
b,-J
-½g2os,. {co,½(p÷q), o P
-ija~(P~q2 )~ }{TQ, T b}jj
j
Quark- Gluon Vertices FIGURE I Feynman Rules required for the calculation of the one loop contribution to the renormalisation constant of the local vector current. ~ ~ 2/a sin (k#/2).
C.T. Sachrajda / QCD phenomenology from the lattice
124
(a)
(b)
(c)
FIGURE 2 One-loop diagrams which contribute to the renormalisation constant of the local vector current. rather than potentials,
there are additional
contribution comes from the "tadpole graph"
vertices to those in the continuum theory; e.g.
(fig.2(c)), which is a general feature of
the four point vertex in fig.l.
such computations.
The diagrams
which have to be evaluated are shown in fig.2.
In the next section I will report on the
These are evaluated numerically with the
results of lattice computations of Zvl°C,
results (in the Feynman gauge):
which will be compared to the expected result, i.e. that given in eqn. (2.3).
s CF
fig 2(a) -
[log
1
+ 7.26]
(2.1a)
In
general we are interested in computing the matrix element of some local operator 0
fig 2(b)
~ s CF
I-log
I
+
1.12] (21b)
between an initial state state
fig 2(c) - ~
S
If>.
li> and a final
Renormalising the operator by
using the lattice scheme we have: CF
12.24
(2.1c)
where I is an infrared cut-off introduced by
•
II>LATT =
0(o)[i +
modifying the denominator of the gluon propagator from ~2 t o ~2+ ~2,
factor
4 CF ~ g .
The u l t r a v i o l e t
and t h e c o l o u r
_ss
4, C-yIogC~2a2]
+
+ CLATT)
"'']
(2.4)
(and infrared)
whereas
by using the MS scheme we have
divergence cancels and we find ks zlOC 1 - s V = ~ C F 20.6
+
(2.2)
Taking p = 6.0 (i.e. g = I) as a typical value
2 ylog ~-+ ...]
(2.5)
of the coupling in current lattice where CLATT, c~-~ and the anomalous dimensions calculations gives, y are constants which can be determined from zlOC = 0.83 V
(2.3) one-loop perturbation theory.
The calculations are straightforward, although in general the loop integrals have to be performed numerically.
The largest
bare operator.
0 (°) is the
Thus the matrix elements in
the two renormalisation schemes are related by
C.T. Sachrajda / Q C D phenomenology from the lattice
125
momentum tensor.
Fi>~-~ =
states are proportional
S
( c ~ - CLATT?+...]
El + ~ In this
way t h e m a t r i x e l e m e n t d e f i n e d
continuum renormaIisation from that
in the hadron. Among the diagrams contribut-
in a
ing to the renormalisation
scheme i s o b t a i n e d
The corresponding
Of course if we calculate all the
higher order terms on the right hand side then
As the lattice and MS definitions
of ~s
Indeed the contributions
Thus in this example c ~
correction
In practice
to the
the lattice coupling is
This is natural for the example of the
tends to be
For other operators
CLATT > c~[~
for n=O
(2.8a)
CLATT ~ c~-~
for n=l
(2.8b)
CLATT < CM-~
for n~2
(2.8c)
derivatives
are due to lattice
in the operator.
this is not the
case, however the large contributions c~
represents a negligibly small
to the measured value of
where n is the number of covariant
local vector current given above since c ~ = 0 , and all the corrections effects.
- CLATT is small,
the dominant one it is usually the case that
the one loop term in (2.6) is sensitive
used.
from the two
Since the tadpole contribution
(2.7)
~LATT(~ )
choice.
loop integral is identical
to that for the tadpole graph in fig (2.c).
and eqn.(2.6)
differ c o n s i d e r a b l y 3 ' 4
at ~ ~ 2GeV,
(3).
tadpole graphs are equal and opposite.
the result would not depend on the choice of
2.7
of this operator
is the one loop graph drawn in figure
Which definition of ~s should we use in
~s"
to
fraction of momentum carried by the quarks
m e a s u r e d on t h e l a t t i c e .
eqn.(2.6)?
Forward matrix elements
of this operator between single hadron
to
- CLATT still come from lattice effects.
Thus although
it is not possible
to say that
the use of the lattice coupling is a better approximation
than a continuum definition,
without calculating
higher order corrections,
this is indeed the expectation above observation
that the one loop contribu-
tion comes predominantly I have mentioned contributions
from lattice effects.
above that the tadpole FIGURE 3
to the wave function
renormalisation contribution
based on the
constant
is the largest
to the one loop term in eqn.(2,2),
A "tadpole" diagram which contributes to the renormalisation constant of the energy momentum tensor, and hence
and generally gives the dominant contribution. Tadpole graphs also arise from the expansion of the link variables
in covariant derivatives
in
is the quark contribution
to the energy
In lattice computations
it is usual to
neglect terms which vanish as the lattice
operators which contain such derivatives. Consider for example the operator ~ygDV~,
3. 0(a) CORRECTIONS
which
spacing goes to zero. attempts
to "improve"
There have been some the calculations
including some of these terms, notably
by
C.T. Sachrajda/ QCD phenomenology from the lattice
126
Symanzik's
program 5,6, but in practice
terms are not kept in phenomenological tions.
these
Here I would like to demonstrate,
means of a numerical neglecting
computation,
where
applicaby
that
four-momentum p.
this matrix
lattice at ~=6.0 we
when the particle is a pion8:
errors.
The example which I will use is again that of the local vector current VB l°c = ~y~¥.
is not conserved
= 0.99 ± 0.05
for K=0.1515
~v = 1.00 ± 0.05
for K=O.1530
= I.II ± 0,06
for K=0.1545
In
the previous section we have seen that this operator
Computing
element on a 20xlO2x40
obtain the following very precise results
terms which are O(a) can lead to
significant
I p > is a meson or baryon state with
in the theory defined
(3.4)
by the action (1.2) and that we should expect its renormalisation 6.0.
constant
The computation
The chiral limit corresponds
to be ~ 0.83 at
to a value of
the hopping parameter K=0.1564(2) 8'9.
of Zvl°C is made
simpler by the existence of a conserved vector For the proton the results are still
current in the theory defined by (1.2)7: 1
~= ~ ~(x) (y-1)u (x)v(x+~) + ~ (x+~) (v+l) U+(x),(x)] The renormalisation ~
consistent with one but are a little
(3.1)
precise.
constant of the operator
(3.1) separately
_loc ZV can be readily
is precisely one.
for the two cases when the
quark, q, is an up quark or a down quark. We find I0
determined by taking ratios of matrix elements, _loc
In this case we compute the matrix
elements of the conserved vector current
~V = 0.87 ± 0.15
for K=0.1515
and q=up
~V = 0.92 ± 0.10
for K=0.1515
and q=down
ZV
(3.2)
and = ratio of corresponding functions. zlOC has been computed using both two and V three point correlation
(2.5)
correlation
~V = 0.89 ± 0.29
for K=0.1530 and q=up
~V = 1.08 ± 0.16
for K=0.1530
functions and I will For both the pion and proton,
now discuss these results.
had three-momentum a) 3-Point Correlation Before presenting
Functions.
the results for Zvl°C,
constant of the conserved vector current,
I
a
In this way we check the precision of
any information
(without of course gaining
about the physics content).
Defining this renormalisation < p I ~
[ p >
=l_ 2pp ~v
and the matrix
index ~ chosen to be I.
The results are
consistent with ZV = 1 and we also have some
quantity for which we know the exact result to
the numerical work,
(I/10,0,0)
the hadron
elements were computed with the Lorentz
will present those for the renormalisation
be I.
and q=down
constant by (3.3)
indication of the precision of lattice computations
involving three-point
tion functions.
correla-
The results in (3.4) and
(3.5) were obtained using 15 gauge field configurations.
C. T. Sachrajda / QCD phenomenology
from
the lattice
127
We have confirmed these results on our K 0.1515 0.1530 0.1545
K 0.1515 0.1530
larger
PION = 4 0.76 0.73 0.71
~= 1 0.73 0.71 0.71
lattice 12.
These results are
clearly inconsistent,
the results obtained using the three-point correlation function,
PROTON q = up ~ = 4 0.76 0.73
not only with the
perturbative result of 0.83, but also with
~ = 1 0.72(3) 0.74(10)
(see table I).
What is the reason for the discrepancy between the results obtained using two-point and three-point correlation functions?
K 0.1515 0.1530
It
is not due to the quenched approximation,
PROTON q = dow~ ~ = 4 0.76 0.73
~ = 1 0.73 0.73(4)
since this would affect both computations equally. Moreover,
the one-loop contribution
to Zvl°C does not contain any quark loops Table I. Values of ZvL°C obtained by
anyway.
computing three point correlation functions
being due to the fact that V l°c mixes with
containlng the operator qy~q.
operators of higher dimension with coeffi-
The error is
We interpret the discrepancies as
less than one on the last digit unless
cients which vanish as the lattice spacing
otherwise stated.
goes to zero,
This effect is neglected in
all calculations, In table one we present the results of our
but the numerical results
presented above seem to indicate that it can
calculations of ZV l°c, obtained by taking the
lead to significant errors.
This interpre-
ratio of the three-point correlation functions
tation is supported by the fact that at
with the local and conserved currents
~=6.2 (for which the inverse lattice spacing
(eqn.(3.2)).
is about 3 GeV), Zvl°C increases to a value
The correlation functions were
evaluated at zero momentum transfer. that the statistical
0.6513 .
Notice
Here I have demonstrated,
errors on the ratios are
by using V l°c
tiny, they are generally less than 1 in the
as
second digit.
of higher dimension can lead to significant
The results do depend on the
quark mass, albeit very mildly.
They differ a
an example, that mixing with operators
errors,
(~ 40%?) on the numerical evaluation
little from the one-loop perturbative result of
of matrix elements.
0.83, but in itself this could be accounted for
particularly simple because of the knowledge
by higher order corrections to the perturbative
that the renormalisation constant of the
prediction.
conserved current is precisely I.
What is more significant however
is that the results differ from those obtained
Our example was
For other
operators we do not know the exact value of the renormalisation constant, however, by
using two-point correlation functions. b) Two-Point Correlation Functions
computing the ratio of the matrix elements
Using two-point correlation functions, on a
of two different lattice definitions of the
I03x20 lattice at ~=6.0, Maiani and
same continuum operator, and comparing the
Martinelli II obtained the results
result to the perturbative prediction for
Zvl°C = 0.57 ± 0.01
for K=0.1515
ZvI°C = 0.57 ± 0.01
for K=0.1530
Zvl°C = 0.58 ± 0.02
for K=0 1545
this ratio, one can get some idea of the (3.6)
systematic problems.
The important
question which now arises is whether it is
C.T. Sachrajda/ QCD phenomenology from the lattice
128
possible to include some of the O(a) correc-
the 4 operators 0 3 D ~
transform as the
tions by hand, and hence to reduce the
vector representation
((~,~) in the notation
systematic
of ref (16) where the character table for
error?
This problem is currently
the hypercubic
being studied.
group can be found).
vector current ~ y ~ 4.
POWER DIVERGENCES
transforms
The
in the same
way and the operator 03~D~ mixes with the vector current under renormalisation.
In the previous section we discussed the problem of operators mixing under renormalisation
mixing coefficient
diverges
Hence in the computations
with other operators
of higher
elements of 0 3 ~ ,
The
like I/a 2.
of the matrix
the leading term is this
dimension.
This effect was found to be
mixing term of O(I/a2), which is a lattice
numerically
significant
artefact of no phenomenological
at current values of
the lattice spacing a, but becomes important as a gets smaller.
interest.
This power divergence must be subtracted
less
In this section
before we can extract useful results from
we discuss mixing with operators of lower
lattice calculations.
dimension;
case this is possible by choosing the
in this case the mixing coeffi-
In this particular
cients diverge as inverse powers of a. This
Lorentz indices appropriately.
problem is more severe on the lattice because
the combination
Lorentz
(or Euclidean)
symmetry
invariance
and chiral
(in the case of Wilson fermions)
not symmetries
of the lattice.
renormalisation
schemes,
prevent the existence
transform as different of the Lorentz group.
these symmetries
often
operators may
tensor representations As will be demonstrated
below, the power divergences non-perturbatively
are
must be subtracted
before meaningful
results
can be obtained from the lattice computations. To illustrate divergences 03~vP m ~ Lorentz
indices.
operators
the existence of power
Matrix elements of these
are proportional
tions of hadrons
(4.1)
like the eight dimensional
representation
of the hypercubic
group and
does not mix with operators of lower dimension.
Thus it is possible to evaluate
the matrix elements of 03~vP with the power divergence
removed but at a two-fold price.
First of all the subtraction removes the component
in eqn.(4.1)
in 03411 which
like the (~,~) representation
i.e. it removes the power divergence.
D p ~, symanetrised over the
to the second
moment of the deep inelastic structure
transforms
transforms
consider the operators
y~ D v
03411 - ~ (03422 + 03433 )
In continuum
of power divergences,
since e.g. the corresponding
For example
leads to a considerable
tion with the corresponding precision.
func-
(when the forward matrix
numerical
Secondly,
This
cancella-
loss of
when we take the
matrix element of 0~ vp we are interested the "leading twist" component,
which is the
element between the single hadron states is
term proportional
computed 8'I0,14)
momentum of the hadron. Thus the matrix
or the quark distribution
to p~pVpp, where p is the
amplitude of the pion (when the matrix element
element of the operator in eqn,(4.1)
between the pion and the vacuum is computedlS).
vanishes
Under transformations
non-zero three-momentum.
of the hypercubic
group
possible
in
trivially unless the hadron has a
to evaluate
It is not
the matrix elements of
C.T. Sachrajda / QCD phenomenology from the lattice 0~ vp with the hadron at rest.
129
Power divergences must be subtracted
Note that it would not be possible to avoid
non-perturbatively.
the large numerical cancellations by choosing all 3 Lorentz indices to be different (e.g. by
For the field strength tensor F ~v on the lattice, the definition 19
using the operator 0~ 12 which transforms like the ( ~ )
representation and again does not mix
with operators of lower dimension).
Fgv~ !
4a 2
However
~
4 plaquettes in ~,v plane
1 U + 2i~UpP ~ (4.3)
the symmetrisation over the Lorentz indices which is implicit in the definition of 0~ 12,
has the advantage that it transforms like
again involves the cancellation of power
the ( I , 0 ) ~
divergences.
representation of the hypercubic group.
Moreover using 0~ 12 we would
need to give the hadron two non-zero components
F ~v is defined using a single plaquette,
the ~,v plane as in eqn. (4.3), then it
To illustrate the large numerical cancellations consider Figure 4, where the three-point correlation functions of the operators 0~ 11 and 0~ II - ~ (0~22 + 0~ 33) between pion states with momentum (~/I0,0,0) The subtraction of the power
divergences reduces the result by an order of magnitude.
If
rather than summing over all 4 plaquettes in
of three momentum.
are presented 14.
(0,I) 6 dimensional, reducible
Fortunately the correlations in the
results for 0~ II and ~ (0~22 + 0~ 33) are strong and even after the subtraction there is still a significant signal,
(the situation is
somewhat worse in the case of the quark distribution amplitudel5). For the structure function of the pion the results in fig. 4 give
transforms like a 24 dimensional reducible representation and there is more scope for operators containing F ~v to be able to mix with operators of lower dimension.
For
example, using the single plaquette definition of F ~ ,
the gluonic contribution
to the energy-momentum tensor, F~VF ~p, contains a hypercubic group singlet and mixes with the unit operator under renormalisaiton,
(we have verified numeri-
cally that the vacuum expectation value of F~VF ~p does not vanish when the single plaquette definition of F ~v is usedl2). Hence care has to be taken to choose an
=
0.18 ± 0.05
(4.2)
appropriate lattice definition of F ~V to avoid power divergences, and eqn. (4.3) is
at a scale of about 7 GeV, where experimental data is available.
Unfortunately not all power divergences
In this example even after
the subtraction of power divergences the error is reasonable, but this is not so in some other
can
be removed by using the transformation
properties of operators under the hypercubic group, and choosing the Lorentz indices
cases. It has been suggested that it may be reasonable to subtract the power divergences perturbatively 17.
frequently found to be the optimal choice.
Perturbation theory at ~ = 6 . 0
appropriately.
An important example appears
in the computation of weak matrix elements in the study of the ~I = ~ rule (in
generally gives a small contribution for these
particular when evaluating matrix elements
divergences 17'18, whereas we have
between single meson states).
seen that the full non-perturbative subtraction changes the result drastically.
~±
The operators
=
[SLY d L ULY~UL ± SLY u L ULY~dL ] - u ~ (4.4)
c
C.T. Sachrajda / QCD phenomenology from the lattice
130
I
I
I
C(tx= 15,fy)x 10"l
I
1
[
K = 0.1530
Operator n" 3~11(y)
Operator O~'~ (y)-½(O~ ,,(y)+ O~"(y)) 100 90 80 70 60 50 -4,0--
3020100 0
5
6
7
8
I
I
9
10
FIGURE 4: Three-point correlation functions of the operators 0~ II and 0~ II(0~ 22 + 0~aa). The difference is due to the subtraction of power divergences.
C.T. Sachrajda/ QCD phenomenology from the lattice
131 +
Thus in order to determine combination
which have mass dimension six, mix under renormalisation
of matrix elements
with the scalar density sd, +
which has dimension
3; specifically
A+ = Z± ^± + + O~E N [0LATT + S60- + S5 O- +
C±
+
+
(4.5)
operators with which O± mix.
do not contain any power divergences,
These
(the GIM
The power divergence
of
the
in c-.
In the SU(3) limit, using
chiral perturbation
of the AI = ~ amplitude, configurations 1,20,21.
even with 30
For this reason the amplitudes
(which
do not suffer from these power divergences) directly may seem to be the natural way to
is
+
contained
for the 100%
errors which still exist in the evaluation
evaluation of the K + ~
that the extra mass
factor in 850± is the mass
charm quark).
of power divergences
in eqn. (4.10) is responsible
renormalisation
S60± and S50± are other six and five
dependent
I sd I K+(g) > (4.10)
The subtraction
mechanism guarantees
IO~ERTI K+(O)>
- <~+(O)
where Z is the diagonal
dimensional
+
IO~ERTIK (R)>
sd]
+ ± Z- [0pERT + c ± sd]
constant,
y- we need the
theory to reduce out a pion
study the AI = ~ rule, however in this case one first has to understand
from the K + I~ matrix element, we find that
scalar particle
<~+(k)
coefficients
the role of the
pole 20'21.
We note in passing that in this case, the 10;m N IK+(H)> = y~k.q + gem 2
(4.6) +
where +the K + I~ amplitude Thus y- is given by
c ± obtained non-
perturbatively,
is given by y-.
using the procedure
described above, agree approximately with those obtained in perturbation
[E(q) - m] my ± = <~+(0)
I~;ENI
- <~+(O)
IO~ENI
theory.
K+(R)>
K+(O)>
(4.7)
5. PERTURBATIVE
CALCULATIONS
In this section I briefly review the The factor in square brackets on the left hand side of eqn.(4.7) fluctuations
is small and subject to large
hence reducing
the evaluation
of y±.
cient c ± is determined condition
the precision
The divergent
in
coeffi-
once a renormalisation
is imposed on the operators 0 ±, for
example
calculations
(or are nearing completion),
from those computed on the lattice: i)
Bilinear Operators
of
the sixteen matrices of Dirac theory).
For these operators ± I~RENI K+(O)> = 0
(4.8)
+
± c
<~ (9) = -
condition c- is
the perturbative
have been performed both for
The calculations
include the
cases in which the lattice operators are
+
extended over two lattice sites (connected
10pERTI K+(9 )> (4.9)
+
(~r~ where F is one
Wilson fermioms 22,23 and staggered fermions 24.
+
With this renormalisation given by:
which enable
one to determine continuum matrix elements
calculations <~+(9)
which have been done to date
I sd I
K+(O) >
by the appropriate
link variable so as to
preserve gauge invariance), operators of the form ~ F D ~ ,
so that where D is the
C.T. Sachrajda / QCD phenomenology from the lattice
132
covariant
derivative
ii) Four-fermion
are also included.
operators
and ~I = 3/2 weak decays. cients the calculation whereas for staggered
relevant
2)
for AI =
lattice spacing,
For Wilson coeffi-
is complete 25'26,27, fermions
The corrections
of O(a), where a is the
can be significant.
In
section 3 we presented an explicit example in which this was the case.
it is almost
3)
Power divergences,
i.e. terms which
complete 28,29, but the mixing with the dimen-
diverge as inverse powers of a as a ~ O,
sion five operator
must be subtracted non-perturbatively,
(SsO± in eqn. 4.5)) has
still not been worked out. fermions the corrections
For staggered
are frequently very
large (in some cases the one loop perturbative correction can change the measured
ACKNOWLEDGEMENTS
result by
It is a pleasure to thank my colleagues
over 50%28).
of the European Lattice Collaboration
iii) Lowest twist operators inelastic
structure
relevant for deep
functions.
For Wilson
fermions these have been computed 17. the coefficients
see
section 4.
Phenomenology.
However
tabulated in tables VI and
Particular
thanks goes to
Guido Martinelli with whom many of the ideas and calculations
VII 17 are wrong 3 0 ' 2 3 , 1 4 .
for an
education in so many aspects of Lattice
described
in this talk were
developed.
iv) Lowest dimension and lowest twist baryonic operators.
These operators are relevant for
REFERENCES
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amplitudes
of nucleons.
Wilson fermions the coefficients
For
v) Renormalisation
of fB' (the decay constant
of the B-mesons).
Eichten 32 has proposed an
approach to the computation
and the corresponding heavy quark propagator
B-parameter,
these proceedings.
2.
G. Martinelli,
3.
H. Kawai, R. Nakayama and K. Seo, Nucl.Phys. B189 (1981) 40.
4.
A. Hasenfratz and P. Hasenfratz, Phys.Lett. 93B (1980) 165.
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K. Symanzik, ibid, 205.
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J-P. Ma and W. Wetzel, (1986) 441.
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L.H. Karseten and J.H. Smit, Nucl.Phys. B183 (1981) 103.
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G. Martinelli and C.T. Sachrajda, Phys. B306 (1988) 865.
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M.B. Gavela et al, Nucl. Phys. B306 (1988) 677.
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C, Bernard,
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I.
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6. SUMMARY AND CONCLUSIONS
Nucl.
The three main points raised in this talk are : 1)
Matrix
be r e l a t e d
elements
m e a s u r e d on t h e
to
defined
those
lattice
by using
renormalisation
schemes in perturbation
see section 2.
The corrections
significant
see section 5.
I0. G. Martinelli and C.T. Sachrajda, preprint TH.5042/88.
Cern
theory,
are often
and sometimes very large.
such calculations
can
continuum
II. L. Maiani and G. Martinelli, B178 (1986) 265.
Phys.Lett.
Many
have been or are being done,
12. G. Martinelli unpublished.
and C.T. Sachrajda,
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133