The ρmeson lightcone distribution amplitudes from lattice QCD
Received: December
meson lightcone distribution amplitudes from lattice QCD
Vladimir M. Braun 0 1 2 4 5 6 7
Peter C. Bruns 0 1 2 4 5 6 7
Sara Collins 0 1 2 4 5 6 7
John A. Gracey 0 1 2 3 5 6 7
Michael Gruber 0 1 2 4 5 6 7
Meinulf Gockeler 0 1 2 4 5 6 7
Fabian Hutzler 0 1 2 4 5 6 7
Paula PerezRubio 0 1 2 4 5 6 7
Andreas Schafer 0 1 2 4 5 6 7
Wolfgang Soldner 0 1 2 4 5 6 7
Andre Sternbeck 0 1 2 5 6 7
Philipp Wein 0 1 2 4 5 6 7
Email: 0 1 2 5 6 7
Open Access 0 1 2 5 6 7
c The Authors. 0 1 2 5 6 7
0 avors of dynamical clover
1 P. O. Box 147, Liverpool, L69 3BX , United Kingdom
2 Universitatsstra e 31 , 93040 Regensburg , Germany
3 Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool
4 Institut fur Theoretische Physik, Universitat Regensburg
5 decay at the smaller pion masses is not taken
6 In this calculation the possibility of
7 mesons , respectively
We present the results of a lattice study of the normalization constants and second moments of the lightcone distribution amplitudes of longitudinally and transversely polarized mesons. The calculation is performed using two fermions at lattice spacings between 0:060 fm and 0:081 fm, di erent lattice volumes up to m L = 6:7 and pion masses down to m = 150 MeV. Bare lattice results are renormalized nonperturbatively using a variant of the RI0MOM scheme and converted to the MS scheme. The necessary conversion coe cients, which are not available in the literature, are calculated. The chiral extrapolation for the relevant decay constants is worked out in detail. We obtain for the ratio of the tensor and vector coupling constants f T =f = 0:629(8) and the values of the second Gegenbauer moments a2 = 0:132(27) and a2 = 0:101(22) at the scale = 2 GeV for the longitudinally and transversely polarized The errors include the statistical uncertainty and estimates of the systematics arising from renormalization. Discretization errors cannot be estimated reliably and are not included.
cTheoretischPhysikalisches Institut; FriedrichSchillerUniversitat Jena

into account.
1 Introduction 2 3 4
General formalism
Continuum formulation
Lattice formulation
Lattice correlation functions
Decay constants
Second moments  the longitudinal case
Second moments  the transverse case
Details of the lattice simulations
Renormalization
Data analysis
Results and conclusion
A Transversity operators in the continuum
B Chiral extrapolation
B.1 E ective eld theory framework
B.2 Chiral Lagrangians for resonances
B.3 Extrapolation formulae
B.4 Contributions to
matrix elements
B.5 Loop functions
Introduction
V =
nal state are attracting increasing attention. Prominent examples
are provided by Bmeson weak decays, B ! V , B ! V ` `, B ! V , B ! V +
Among these processes, the decays B ! K
are of particular
relethat are sensitive to new physics, see, e.g., ref. [3] for a recent review. Another example
high energy, eN ! e N , that, besides deeplyvirtual Compton scattering (DVCS), allows
one to resolve the transverse distribution of partons inside the nucleon. The
corresponding cross sections were measured by the HERA collider experiments H1 and ZEUS and
xed target experiments HERMES (DESY), CLAS (JLAB), and Hall A (JLAB) at
small and moderate values of the Bjorken momentum fraction xBj, respectively. In the
the electronion collider (EIC) [4].
The standard framework for the theoretical description of such processes is based on
the transverse degrees of freedom are integrated out. In general, meson and baryon DAs
the momentum fractions) are given by matrix elements of local operators. From the
pheat the origin) and the rst nontrivial Gegenbauer moment that characterizes the width of
the DA are the most relevant quantities. For example, knowledge of the second moment
of the DA of the longitudinally polarized
meson is crucial for global ts of generalized
parton distributions from the DVMP and DVCS data [5].
The meson coupling to the vector current is known experimentally and the other
parameters were estimated in the past using QCD sum rules [6], see also ref. [7] for an
update. Lattice calculations of the tensor coupling have been reported in refs. [8{12] and
the second moments in ref. [13].
In this work we present new results using two avors of dynamical clover fermions at
lattice spacings between 0:060 fm and 0:081 fm, di erent lattice volumes and pion masses
down to m
= 150 MeV.
Our approach is similar to the strategy used in our paper
on the pion DA [14]. In addition to a much larger set of lattices as compared to the
previous studies, a new element of our analysis is a consistent use of nonperturbative
coe cients for the conversion between our nonperturbative renormalization scheme on
the lattice and the MS scheme are not available in the literature for tensor operators, we
accuracy. The chiral extrapolation for the relevant quantities is worked out in detail.
Although our calculation presents a considerable improvement as compared to earlier
we only consider
mesons and leave the e ects of the SU(3) avor breaking for a future
study. Likewise, we do not consider ! mesons that would require the calculation of
dise ects due to the
decay that becomes possible at the smaller pion masses used
seems unlikely that such e ects are of principal importance. Last but not least,
discretizabe an important problem in such calculations. We expect to be able to improve on some
framework of the CLS initiative [15]. This work, aiming in the long run at smaller lattice
spacings with the help of open boundary conditions, is in progress.
The presentation is organized as follows. Section 2 is introductory, we collect the
necessary de nitions and specify the quantities that will be considered in this work. Section 3
contains a list of the correlation functions that we compute on the lattice. The lattice
ensembles at our disposal and the procedure used to extract the signal are described in
section 4. A nonperturbative calculation of the necessary renormalization factors is
deof the same operators in the continuum and sketch a two loop calculation of the
correcoe cients between our RI0SMOM scheme (de ned as in ref. [14]) and the MS scheme are
presented in the auxiliary le attached to the electronic version of this paper. Section 6 is
devoted to the data analysis and the extrapolation to the physical pion mass using, where
available, chiral e ective eld theory expressions derived in appendix B. The nal section 7
contains a summary of our results and a discussion.
meson has two independent leading twist (twist two) DAs,
[16],
corbreaking and electromagnetic e ects, the DAs of charged
and neutral 0 mesons are
related so that it is su cient to consider one of them, for example, +
. The DAs are de ned
h0jd(z1n)n=[z1n; z2n]u(z2n)j +(p; )
[z1n; z2n]u(z2n)j +(p; )
= m f (e( ) n)
General formalism
Continuum formulation
e( ) p = 0 ;
X e( )e( ) =
e( ; ) = e( )
and we use the notation
= if T e( 0) e( ) (p n)
tion . The straightline pathordered Wilson line connecting the quark elds, [z1n; z2n], is
inserted to ensure gauge invariance. The meson polarization vector e( ) has the following
The variable x has the meaning of the fraction of the
meson's lightcone momentum p n
which is carried by the uquark, whereas 1
x is the momentum fraction carried by the
antiquark d, and
is the renormalization scale (we assume the MS scheme). The scale
dependence will often be suppressed in what follows.
The couplings f and f T appearing in (2.1) are de ned as matrix elements of local
h0jd(0) u(0)j +(p; )i = f m e( ) ;
u(0)j +(p; )i = if T e( )p
in ref. [17] for a detailed discussion. One obtains [17]
f + = (210
4) MeV ;
f (0u) = (221:5
3) MeV ;
f (0d) = (209:7
3) MeV ;
where for the neutral
meson we quote separate values for the uu and dd currents. The
difwhich will be neglected throughout this study.
The tensor coupling f T is scale dependent and is not directly accessible from
experiment. To leading order one obtains
The symmetry property (2.8) implies
and, neglecting isospin breaking e ects, are symmetric under the interchange of the
momentum fractions of the quark and the antiquark,
For convenience we introduce a generic notation h
as weighted integrals of the type
; for the moments of the DAs de ned
f T ( ) = f T ( 0)
CF = 0
where CF = (Nc2
Nf the number of active avors.
The DAs are normalized to unity,
1)=(2Nc), 0 = (11Nc
dx ; (x) = 1 ;
; (x) =
x)li ; =
x)kxli ; = h(1
and in addition we have the (momentum conservation) constraint
x)kxli ; = h(1
= 2x
corresponding to the di erence of the momentum fraction between the quark and the
antiquark and consider
or, alternatively, Gegenbauer moments
n ; = h(2x
n = 2; 4; 6; : : : ;
n; =
2(2n + 3)
3(n + 1)(n + 2) h Cn3=2(2x
relations, e.g.,
2; =
dence at the one loop level
The anomalous dimensions are given by
(0) = 8CF
2(n + 1)(n + 2)
As Gegenbauer polynomials form a complete set of functions, the DAs can be written as
an expansion
Gegenbauer coe cients, e.g.,
an( ) = an( 0)
an( ) = an( 0)
; (x; ) = 6x(1
x) 41 + X an; ( )Cn3=2(2x
; (x; ) = 3 1 + X an; ( ) :
n(0)=(2 0)
(0) =(2 0)
(0) = 8CF
Since the anomalous dimensions increase with n, the higherorder contributions in the
term survives, usually referred to as the asymptotic DA:
! 1) = as(x) = 6x(1
Beyond the leading order, higher Gegenbauer coe cients an mix with the lower ones,
ak; k < n [18, 19]. This implies, in particular, that Gegenbauer coe cients with higher
values of n are generated by the evolution even if they vanish at a low reference scale.
This e ect is numerically small, however, so that it is usually reasonable to employ the
Gegenbauer expansion to some xed order.
Lattice formulation
From now on we work in Euclidean space, using the same conventions as in ref. [14].
The renormalized lightray operators entering the de nition of the DAs are de ned as the
means that moments of the DAs, by construction, are given by matrix elements of local
operators and can be evaluated on the lattice using the Euclidean version of QCD.
Our aim in this work is to calculate the couplings f , f T and the second DA moments.
To this end we de ne bare operators
V (x) = d(x) u(x) ;
T (x) = d(x)
(x) = d(x)
(x) = d(x)
D~D~ + D~ D~
D~D~ + D~ D~
On the lattice the covariant derivatives will be replaced by their discretized versions.
Projection onto the leading twist corresponds to symmetrization over the maximal
possible set of Lorentz indices and subtraction of traces. The operation of
symmetrization and trace subtraction will be indicated by enclosing the involved Lorentz indices in
parentheses, for instance, O( ) = 12 (O
O . Note that for the operators
involving the
matrix also those traces have to be subtracted which correspond to index
pairs where one of the indices equals
Using the shorthand D~~ = D~
)(x) can be rewritten as
)(x) = d(x) ( D~~D~~)u(x)
and its matrix element between the vacuum and the
state is proportional to the bare
value of the second moment h 2i :
+(p; )i = N(
where N(
) is a kinematical prefactor. The operator V(+
to the second derivative of the vector current,
)(x) in the continuum reduces
)(x) = @( @ d(x) )u(x) ;
+(p; )i = N(
with the same prefactor. While in the continuum h
12ibare = 1 by construction, this is no
longer true on the lattice because the Leibniz rule holds for discretized derivatives only
up to lattice artefacts and hence (2.25) is violated. As we will see below, the deviation
from unity for the renormalized h
12i is small. Nevertheless, it still has to be taken into
account and a ects the relation between h 2i and the Gegenbauer moment at nite lattice
spacing [14]:
a2 =
The situation with the tensor operators T (
) and the corresponding matrix elements
The operators V(
) mix under renormalization even in the continuum, as
. Additional mixing could result from the fact that the continuous
Fortunately, in the case at hand it is possible to avoid additional mixing by using suitably
chosen operators, which will be detailed below.
Lattice correlation functions
The basic objects from which moments of the
DAs can be extracted on the lattice are
2point correlation functions. In order to \create" the
meson we use the interpolating
current V (x), which is de ned as V (x) with smeared quark
elds. For details of our
smearing algorithm see section 4. Let O be a local (unsmeared) operator, e.g., one of the
operators de ned in eq. (2.22) above. One then obtains for the 2point function in the
region where the ground state dominates
3 X e ip x
hO(t; x)V y(0)i =
A(O; V j p) =
Here T is the time extent of the lattice, a is the lattice spacing, and E denotes the energy
state. The sign factors
are determined by the Dirac matrices in the creation
operator (which is in our case always
table 1), and nt is the number of time derivatives in O.
O are the analogous factors for O (see
For the decay constants and the second DA moments of the
meson we have to evaluate
the following set of correlation functions:
Decay constants
Cjj (t; 0) = Z
Cjj (t; 0) = m f pZ
C4jj (t; 0) =
In the actual ts we average over the forward and backward running states. As in
in the time direction is reached (see
gure 1 for an example), the \mixing" of these two
contributions is completely negligible. Therefore we work with simple exponential ts,
where the averaging operator t^ is de ned as
1 X3 t^+Cjj (t; 0) =
1 X3 t^+Cjj (t; 0) = f
1 X3 t^ C4jj (t; 0) =
t^ C(t; p) =
The decay constants f and f T can be obtained by simultaneously
tting the
correlation functions (3.5a){(3.5c). The result for the mass is then dominated by the two
from excited states so that the isolation of the ground state is less reliable. Therefore we
rst t the correlator with a smeared operator at the sink, (3.5a), to extract Z and m .
These values are then inserted in eqs. (3.5b) and (3.5c) in order to obtain f and f T as well
an estimation of the statistical error.
Second moments  the longitudinal case
ments consist of the operators
Here and in the following f
g denotes symmetrization of the enclosed n indices with an
form according to the irreducible representation
(4) of the hypercubic group H(4) [21].
other, but mixing with additional operators of the same or lower dimension is forbidden.
The amplitudes (3.2) of the 2point functions (3.1) where O is one member of these
multiplets are related to the amplitudes where O is a component of the vector current V by
A(O1 ; V j p) =
A(O2 ; V j p) =
A(O3 ; V j p) =
A(O4 ; V j p) =
p2p3A(V4; V j p)+ip2EA(V3; V j p)+ip3EA(V2; V j p) ;
p1p3A(V4; V j p)+ip1EA(V3; V j p)+ip3EA(V1; V j p) ;
p1p2A(V4; V j p)+ip2EA(V1; V j p)+ip1EA(V2; V j p) ;
p1p2A(V3; V j p)+p1p3A(V2; V j p)+p2p3A(V1; V j p) :
In order to be able to write these and some of the following formulae in a compact form
we have introduced the notation R , where R+ (R ) is the bare value of h12i (h 2i ).
We will try to increase the signaltonoise ratio by considering only correlation functions
lattice of spatial extent L. Therefore we exclude O4 from our calculation. After averaging
over all suitable combinations as well as over forward and backward running states, the
O1 = Vf234g ;
O2 = Vf134g ;
O3 = Vf124g ;
O4 = Vf123g :
second longitudinal moments can be obtained from the ratio
j=1 k=1
k6=j
1 X3
p^+t^+Ckk t; 2L ej
where momentum averaging is accounted for by the operator p^ :
Second moments  the transverse case
In the transverse case we consider the following multiplets:
p^ C(t; p) =
C(t; p) :
O1;T = T13f32g + T23f31g
O2;T = T12f23g + T32f21g
O3;T = T12f24g + T42f21g
O4;T = T21f13g + T31f12g
O5;T = T21f14g + T41f12g
O6;T = T31f14g + T41f13g
The two multiplets O1+;T ; : : : ; O6;T and O1;T ; : : : ; O6;T both transform according to the
+
irreducible representation 2
(6) of the hypercubic group H(4). As in the case of the
multiplets (3.7), mixing with additional operators of the same or lower dimension is forbidden
by symmetry. The amplitudes (3.2) of the 2point functions (3.1) where O is one member
of the multiplets (3.11) are related to the amplitudes where O is a component of the tensor
A(O1;T ; V j p) =
p2p3 A(T13; V j p) + p1p3 A(T23; V j p)
A(O2;T ; V j p) =
p2p3 A(T12; V j p) + p1p2 A(T32; V j p)
+ ip2E A(T41; V j p) + ip1E A(T42; V j p) ;
A(O3;T ; V j p) =
p1p2 A(T42; V j p)
p1p3 A(T43; V j p)
+ ip3E A(T41; V j p) + ip1E A(T43; V j p) ;
+ ip2E A(T12; V j p) + ip3E A(T31; V j p) ;
A(O4;T ; V j p) =
p1p3 A(T21; V j p) + p1p2 A(T31; V j p)
A(O5;T ; V j p) =
p1p2 A(T41; V j p)
p2p3 A(T43; V j p)
+ ip3E A(T42; V j p) + ip2E A(T43; V j p) ;
+ ip1E A(T21; V j p) + ip3E A(T32; V j p) ;
A(O6;T ; V j p) =
p1p3 A(T41; V j p)
p2p3 A(T42; V j p)
+ ip1E A(T31; V j p) + ip2E A(T23; V j p) :
Here R+ (R ) is the bare value of h12i (h 2i ).
1999( 4)
1998( 2)
1999( 1)
2028( 2)
1237( 2)
1599( 3)
982( 2)
1999( 2)
2178( 2)
m [MeV]
Nconf( Nsrc)
= 5:20, a = 0:081 fm, a 1 = 2400 MeV
= 5:29, a = 0:071 fm, a 1 = 2800 MeV
= 5:40, a = 0:060 fm, a 1 = 3300 MeV
hopping parameter , the pion mass m , the
nite volume corrected pion mass m1 determined
in ref. [22], the lattice size, the value of m L, where L is the spatial lattice extent, the number of
con gurations Nconf and the number of sources Nsrc used on each con guration. Note that the pion
masses have been slightly updated compared to the numbers in ref. [14]. The ensembles marked
earlier within the QCDSF collaboration.
As in the longitudinal case, we only consider correlation functions with the smallest
nonzero momentum in one spatial direction and perform averages similar to those in
eq. (3.9). This leads to the following ratio for the second transverse moments:
1 X3
j=1 l=1 k=1
l6=j k6=j
k6=l
Details of the lattice simulations
For this work we used gauge con gurations which have been generated using the Wilson
fermions. A list of the ensembles used is shown in table 2. We used lattices with three
different inverse couplings
0:06 fm and 0:081 fm. The pion masses vary between 150 MeV and 500 MeV, with spatial
volumes between (1:9 fm)3 and (4:5 fm)3.
In order to increase the overall statistics we performed multiple measurements per
conguration. The source positions of these measurements were selected randomly to reduce
the autocorrelations. To obtain a better overlap with the ground state we applied
Wuppertal smearing [23] in the interpolating current V using APE smeared gauge links [24].
calculated from the time and
momentumaveraged correlation functions according to eq. (3.9) on the
= 5:29,
= 0:13632, L = 32a, T = 64a
ensemble. The cyancolored bar indicates the tted value of R , the error and the tting range.
For the statistical analysis we generate 1000 bootstrap samples per ensemble using
a binsize of 4 to further eliminate autocorrelations. For the purpose of maximizing the
statistics of the second moments, we average for each bootstrap sample over all suitable
and backward running states as pointed out in eqs. (3.9) and (3.13). In order to reduce
contributions from excited states the choice of the starting point of the t range is
important. As an example, gure 1 demonstrates that, with increasing sourcesink distance, the
excited states fall below the noise and plateaus of the correlation functions for R
The starting time tstart is then chosen in such a way that ts with even larger starting
times no longer show any systematic trend in the tted values. Multistate ts (over larger
t ranges) yield consistent results.
Renormalization
Having computed the bare values of the second DA moments, we are left with the task
of renormalizing these bare quantities to obtain results in the standard continuum MS
there is only mixing between the respective + and
operator multiplets, so we have to
determine 2
2 mixing matrices such that
One then obtains for the second moments of the DAs in the MS scheme
= Z11O
= Z22O+ :
= 1;1 R
= 2;2 R +; ;
ij =
ij =
with the renormalization factors ZV and ZT of the vector and the tensor currents,
respectively. Note that one cannot expect 2;2 to be equal to one, since the Leibniz rule holds on
the lattice only up to discretization artefacts.
We want to evaluate the renormalization and mixing coe cients nonperturbatively
on the lattice employing a variant of the RI0MOM scheme, because lattice perturbation
theory is not su ciently reliable. Since forward matrix elements of the + operators vanish
in the continuum limit, we cannot work with the momentum geometry of the original
RI0MOM scheme but must use a kind of RI0SMOM scheme [25]. We follow exactly the
the MS vertex functions of our operators in order to convert the results from our SMOM
scheme to the MS scheme. While these are known to two loops in the longitudinal case,
i.e., for the operators (2.22a), see ref. [26], as well as for the currents (2.21), see ref. [27],
the corresponding results for operators with derivatives involving the matrix
, e.g., the
operators (2.22b), are not yet available in the literature. Therefore we discuss the latter
case, the socalled transversity operators, in appendix A.
In the end, we determine the matrix Z(a; 0) (and analogously (a; 0)) at the reference
the expression
Z(a; )MC = W ( ; 0)Z(a; 0) + A1a
2 2 + A2(a2 2)2 + A3(a2 2)3 ;
where the three matrices Ai parametrize the lattice artefacts and W ( ; 0) describes the
running of Z in the three loop approximation of continuum perturbation theory.
Ignoring the very small statistical errors, we estimate the much more important
systematic uncertainties of Z(a; 0) by performing a number of ts, where exactly one element
of the analysis is varied at a time. More precisely, we choose as representative examples
for t intervals 4 GeV2 <
2 < 100 GeV2 and 2 GeV2 <
2 < 30 GeV2, and we use the
Finally, we consider values for r0 and r0 MS corresponding to the results given in ref. [28].
The various t possibilities are compiled in table 3.
As in the case of the pion DA, the largest e ect comes from the variation of nloops.
In order to obtain our nal numbers for the second moments of the DAs we extract them
from the bare data R
; using each of these sets of values for 11, 12 and 22. So we
have six results for each of our gauge eld ensembles. As our central values we take the
results from Fit 1. De ning i as the di erence between the result obtained with the s
from Fit i and the result determined with the s from Fit 1, we estimate the systematic
uncertainties due to the renormalization factors as p 22 + (0:5
one loop to two loops. This should amount to a rather conservative error estimate. The
(in GeV2)
artefacts (in fm)
A3 6= 0
A3 6= 0
A3 6= 0
A3 = 0
A3 6= 0
A3 6= 0
renormalization factors ZV and ZT needed for the evaluation of f and f T , respectively,
are calculated in the same way.
Data analysis
From the bare values of f etc. we obtain renormalized results in the MS scheme with the
help of our renormalization (and mixing) coe cients on each of our gauge eld ensembles.
With the range of ensembles available (see table 2) we are able to study the pion mass
dependence and, to only a limited extent, volume and discretization e ects. Since our
for our nal numbers.
decay constants
Considering the pion mass dependence, we make use of Chiral Perturbation Theory
(ChPT) for vector mesons [29{31] to obtain the one loop extrapolation formulae for the
Re f = f (0) 1
Re f T = f T (0) 1
= f (0) 1 +
+ f (2)m2 + f (3)m3 + O(m4 ) ;
+ f T (2)m2 + f T (3)m3 + O(m4 ) ;
+ f (2)m2 + f (3)m3 + O(m4 ) :
Details on the ChPT calculation are given in appendix B. For 2m
< m , i.e., below the
neglect instability e ects in our lattice computation, which is necessarily done on
volumes, we use only the real part to t the mass dependence of our data. The pion decay
constant F is kept
xed at its physical value 92:4 MeV, and the chiral renormalization
is chosen to be 775 MeV.
Estimates within ChPT suggest that the thirdorder term / m3 is not negligible for
most of our masses (see appendix B). Our data con rm this expectation  the third order
term is required in order to t over the full range of pion masses. Consistent ts are obtained
including only second order terms for m
< 300 MeV, however, we have, essentially, only
L = 40a
L = 48a
L = 64a
L = 40a
L = 64a
L = 40a
L = 64a
L = 40a
L = 48a
L = 64a
L = 40a
L = 64a
L = 40a
L = 64a
ts using eqs. (6.1) for the decay constants f , f T and their ratio, including
(left) and excluding (right) the data point at m
position of the physical pion mass. The band indicates the one sigma statistical error.
two pion masses in this range. Alternatively, one can ignore the information from ChPT
and perform polynomial ts, i.e., drop the logarithmic term in the t functions (6.1). This
yields very similar results. We expect that a t including the larger pion masses will yield
more reliable numbers than simply taking the values at m
= 150 MeV as our nal results
because, in particular, the lattice used at this pion mass is relatively small.
In order to get at least some idea of the in uence of the instability of the
two kinds of ts, including all masses or excluding the results at m
= 150 MeV, which
constants are shown in
gure 2. Note that the extrapolated values at the physical point
are reasonably consistent with the data at the lowest pion mass.
L = 40a
L = 64a
L = 40a
L = 64a
L = 40a
L = 64a
L = 40a
L = 64a
the data point at m
mass. The band indicates the one sigma statistical error.
For the second moments of vector meson distribution amplitudes (see gure 3) no ChPT
and the nucleon [34] do not contain chiral logarithms in leading oneloop order. The
reasons are rather generic and may apply to vector mesons as well. Therefore we stick to
simple linear ts in m2 depicted in
gure 3. There is no discernible dependence on the
gures 2{3. We perform an extrapolation for every choice given in table 3 and compute
renormalization factors from the di erences of the extrapolated numbers as indicated at
the end of section 5.
Although our data do not allow us to study nitesize and discretization e ects
systematically, we can make some observations. Considering volume e ects, for
= 5:29,
The e ects for the decay constants are sizable, see gures 2 and 4. Unlike the wellknown
forward to compute the leading
nitevolume corrections for this ratio (see appendix B).
= 5:29,
L = 40a
L = 64a
L = 40a
L = 64a
indicates the position of the physical pion mass.
It turns out that the corrections are numerically tiny so that from the ChPT analysis
one expects that
themselves. This expectation is in agreement with our data, as shown in
gure 4: the
vector coupling f itself (left panel). Since in phenomenological studies of hard reactions f
accessible, is a much more interesting quantity. So we do not perform an in nitevolume
extrapolation for f and use this measurement mostly for normalization purposes (e.g.
computing the second moments). On the other hand, the observed volume dependence
would not have any signi cant e ect.
One can see from
gure 3 that the second moments tend to increase with the spatial
no leading chiral logarithms and a very mild
nitevolume dependence. We have checked
noticeable in uence on our results.
f T [MeV]
the main text. The numbers in parentheses denote the statistical error and our estimate of the
uncertainty introduced by the renormalization procedure.
Discretization errors are notoriously di cult to control. A certain insight can be
obtained looking at the quantities h12iMS
and h12iMS, which indicate the violation of the
Leibniz rule at
nite lattice spacing. In the continuum limit they should equal one for
all pion masses. Results for all ensembles are plotted in
gure 5. Again only statistical
smaller. While h12iMS
deviations are noticeably smaller than what we found in the case of the pion [14].
of about 1%, we observe deviations from one of up to 2% for h12iMS. Note that these
equals one within the statistical errors with a maximal deviation
Results and conclusion
In table 4 we compare the results of the two kinds of nal ts that we have performed.
The values in the row labelled \analysis 1" have been obtained by ts to all data points,
while the row labelled \analysis 2" contains the results from
ts where the data with the
t functions (6.1), whereas the second Gegenbauer moments have been tted with linear
functions of m2 . One sees that the results of the two ts are in very good agreement, which
may be an indication that meson decay,
, is not of major importance for the
shortmight be more important, but, unfortunately, cannot be estimated reliably using the set
of lattices at our disposal. We expect to be able to quantify the discretization errors using
the CLS initiative [15].
Comparing to the pion case we observe that for the
meson we are able to access
nents in order to compute a2 in the pion. This helps to reduce the statistical noise and the
corresponding error.
As our nal results we adopt the numbers from analysis 1. Although the systematic
in the case of the decay constants f T and f , the agreement with the experimental value
Sum rule calculations [6, 7] yield f
numbers given in ref. [7] at the renormalization scale
= 1 GeV have been evolved to
lattice [10]
lattice [12]
lattice [13]
sum rules [6, 7] 0:74(5)
renormalization scale is
= 2 GeV.
the systematics are not yet fully controlled, the discrepancies do not look worrying.
of our results have been added in quadrature. Again, the sum rule numbers at
= 2 GeV
have been obtained from the original results at
with Nf = 2.
Some of these quantities have already been investigated on the lattice. The BGR
= 0:742(14)
All existing results are, generally, in good agreement, apart from the ratio of decay
investigations. This ratio depends strongly on the pion mass, cf. gure 2, and the
extrapolation could be a ected by the
decay at this level of accuracy. Clari cation of this
at large recoil (see, e.g., ref. [17]), where, in some cases, there is a tension with predictions
of the Standard Model. Our value for the second Gegenbauer coe cient a2 is signi cantly
more precise compared to previous results. At this level of accuracy, we start to be
sensitive to the di erence between the longitudinally and transversely polarized mesons. Our
results suggest that a2 may be slightly larger than a2, although the di erence is not yet
statistically signi cant. The 20% accuracy for a2 achieved in our work is interesting for
GPD formalism [5]. Such processes will be investigated with high priority at the JLAB
12 GeV upgrade and, in the future, at the EIC.
tions [15]. Apart from the study of discretization errors our goal is to consider DAs of the
whole SU(3)f meson octet, with emphasis on properties of the K
meson, which is of prime
importance for avor physics. This work is in progress.
Acknowledgments
IRG 256594. The ensembles were generated primarily on the QPACE systems of the
thank Benjamin Gla le for software support. Part of the analysis was also performed on
of various institutions which we acknowledge below.
One of the authors (JAG) thanks Dr. P.E.L. Rakow for useful discussions. Helpful
conversations with G.S. Bali are gratefully acknowledged. The work of JAG was carried
out with the support of STFC through the Consolidated Grant ST/L000431/1.
The authors gratefully acknowledge the Gauss Centre for Supercomputing (GCS) for
providing computing time for a GCS LargeScale Project on the GCS share of the
suand NordrheinWestfalen (MIWF).
Transversity operators in the continuum
we refer the interested reader for more background. First, the two classes of operators we
are interested in are the avor nonsinglet operators,
where the operators with a single derivative have been included for completeness. We
= 12 [ ; ] which is related to
OT2 = S &
= S &
@T2 = S@
@T3 = S@
@@T3 = S@ @
= i& :
Our use of &
is to retain the same conventions with earlier renormalization of similar
operators [26, 27, 40] and our use of generalized
matrices which we discuss later. To
de ne the action of the symbol S, which imposes certain symmetrization and tracelessness
conditions, it is best to consider the generalized transversity operators OT 1::: i::: n from
O 1::: i::: n = 0 (i
O 1::: i::: j::: n = 0
for the rst operator of the T2 sector with again a parallel de nition for the total derivative
operator [42]. In our construction for the T3 operators we have taken the convention to
as well as for the nontotal derivative operator of each set. It will be clear from the context
which is meant. The labelling for each derivative of a total derivative operator is one @
symbol applied to the sector label. In de ning the operators we have omitted the explicit
avor indices and note that our perturbative renormalization will be for massless quarks;
in other words we are in the chiral limit. The total derivative operators are required since
there is operator mixing within each separate sector. It would not usually be necessary to
include these but since the Green's functions they are needed for are nonforward matrix
elements, then a momentum will ow through the operator insertion and the mixing will
OiTol = ZiTjl OjTl
ZiTj2 =
ZiTj3 = B 0
We use 1 and 2 to label the elements of the T2 matrix where 1 is the operator T2. Similarly
1, 2 and 3 label the T3 matrix elements which respectively correspond to T3, @T3 and @@T3.
The explicit mixing matrix for the T2 system has been determined in ref. [42] to three loops
in the MS scheme. Prior to the results we present here, the T3 matrix was known only
operator T3 itself and the remaining two diagonal entries are the same as the operator T2
and the tensor current [42{44]. In other words the operators of the T2 system without
because the nonzero entries of the nal two rows of ZiTj3 are the nonzero entries of the ZiTj2
matrix. We have determined the nal two o diagonal elements of ZiTj3 by renormalizing
legs is nulli ed. In other words there is a nonzero momentum
owing through the inserted
operator. This was the method used to determine a similar mixing matrix for the third
moment of the usual twist2 Wilson operators in deep inelastic scattering [26]. However,
in ref. [26] it was noted that such a computational setup was not su cient to determine
piece of information was required. This is achieved here for T3 by the identity
Z1T23 = Z2T23
which is straightforward to establish by integration by parts. Thus to deduce these
reloop renormalization of the operator T3. As the resulting anomalous dimensions for T2 are
given in ref. [42], we record the rst row of the three loop anomalous dimension mixing
matrix for T3 which is
1T13(a) =
CF a + CF [1195CA
+ CF 10368 3CA2 + 126557CA2
31104 3CACF
30197CACF
67392 3CANf TF
38900CANf TF + 20736 3CF2
17434CF2 + 67392 3CF Nf TF
50552CF Nf TF
18557CACF + 10368 3CANf TF
10368 3CF Nf TF + 10696CF Nf TF
1T23(a) =
1T33(a) =
3 CF a + CF [ 125CA + 34CF + 64Nf TF ] 27
+ 694CANf TF
6790CA2 + 15552 3CACF
10368 3CF2 + 16736CF2
+ O(a4) ;
+ O(a4) ;
3 CF a + CF [11CA 109CF + 20Nf TF ] 54
47952 3CA2 + 32969CA2 + 132192 3CACF
138749CACF
+ 25920 3CANf TF
3200CANf TF
72576 3CF2 + 27332CF2
25920 3CF Nf TF + 39040CF Nf TF + 2000Nf2TF2 4860
+ O(a4) ;
gauge parameter cancels as it ought to for gauge invariant operators in the MS scheme.
Having summarized the renormalization of the operators of interest the next stage
is to provide the perturbative corrections to the Green's function where the operator is
Green's function
the fully symmetric point given by
from which we have
inserted in a quark 2point function. As we are considering nonforward matrix elements
there is a momentum
owing through the operator.
More speci cally we consider the
two independent external momenta p and q and we will evaluate the Green's function at
is a mass scale. For this section we will take this scale to be the same mass scale
2 dimensions to ensure the coupling
will not have any logarithms of mass parameter ratios. As each Green's function has free
Lorentz indices we choose to decompose them into a basis of Lorentz tensors denoted by
(p; q) and P(k)
(p; q). Here T2 and T3 indicate the sector as the basis will be
the same for the Green's function with the total derivative operators of each sector too.
The choice of tensors in each basis is not unique. However, each basis is large due to the
number of objects available to build the tensors. These include the momenta p and q as
. In addition there are Lorentz tensors built from the matrices. As in previous
perturbative evaluations [26, 40] we use the generalized matrices of [46{48] denoted by
for integers n
span the spinor space when dimensional regularization is used. As an aside we note that
it is in this context that our choice of &
in the operator de nition
ts naturally. The
algebra and properties of these matrices is wellestablished [49, 50]. We note one speci c
property which is important here which is
p2 = q2 = (p + q)2 =
p q =
where there is no sum over repeated m or n and I 1::: m 1::: n is the generalized unit matrix.
The key point is that this trace partitions the space spanned by the tensors in the basis
into distinct sectors. As we consider the operators in massless QCD, only
will be needed. For T3 it might be expected that
the symmetrization conditions exclude this matrix from the basis. Finally with these
objects we have constructed the tensor basis for each sector. For T2 that involves 30
tensors consistent with the symmetry properties of the inserted operator. A sample set is
presented below. For T3 there are 42 tensors and for space reasons these as well as the full
T2 set are given in the attached data le.
The next step is to compute the coe cients in the decomposition of each Green's
func1::: 6 would be required but
p2=q2=
= Cl
for the di ering dimensionalities of the tensor basis and Green's functions for each sector.
Thus we have C2 =
i and C3 =
2. The algorithm to determine these coe cients has been
given in refs. [26, 40] for instance. Brie y, to apply the multiloop perturbative integration
techniques to
nd these amplitudes we have to extract scalar Feynman integrals which is
achieved by a projection method. The projection matrix, MiTjl , required for each sector is
constructed from the respective tensor basis [26, 40] as it is the inverse of the matrix
NiTjl = tr
(p; q)P(Tjl) 1::: l+1 (p; q) p2=q2=
Due to the size of the matrices, their explicit form is given in the auxiliary data le provided.
shortcut. Hence we have
p2=q2=
Next we brie y note the practical details of actually carrying out the two loop
evaluanotation after all the Lorentz and color indices have been included. There are 3 graphs at
one loop. At two loops there are 32 graphs for OT2 and 37 for OT3
with fewer graphs
for total derivative operators. After this the Feynman rules for either operator together
out to produce a large number of scalar Feynman integrals that need to be calculated. To
products of the momenta in the numerators of the integrals are written in terms of the
propagators. In addition there may be propagator forms which are not present which are
referred to as irreducible. In this format the Laporta algorithm [54] is then applied which
uses integration by parts to systematically construct all the algebraic relations between
expansion is known from direct computation [55{58]. Therefore, we are able to
and two loops. Whilst this is in essence the Laporta method [54] one has to construct the
relations in a practical way. We have chosen to use the Reduze package [59]. Moreover,
topology at one loop and two at two loops. The latter are the ladder and nonplanar
three cases. The nal stage is to carry out the overall renormalization. This is achieved by
and gauge parameter, following the procedure introduced in ref. [60] for automatic
symbeing extracted at the end to leave the nite expressions for each scalar amplitude.
To allow orientation to the full data available in the attached data le we give a selection
each (n)matrix partition for both operators of the T2 sector. For instance, we have
(T22)(p; q) =
1:000000 + [0:271008 + 2:395659]a
(T223)(p; q) = [0:472269 + 1:416806]a
(T229)(p; q) = [ 0:222222
+ [1:329626 2 + 2:430759
6:178403Nf + 55:151461]a2 + O(a3) ;
+ [1:795895 2 + 3:195370
2:817413Nf + 36:018151]a2 + O(a3) ;
+ [ 0:808446 2
4:040708 + 0:886539Nf
14:783322]a2 + O(a3) ;
(@2T)2 (p; q) =
1:000000 + [ 0:062325 + 0:062325]a
+ [0:054445 2 + 0:640942
1:600114Nf + 17:009954]a2 + O(a3) ;
(@2T32)(p; q) = [0:347245 + 1:041736]a
(@2T92)(p; q) = [ 1:041737
+ [1:302171 2 + 3:618039
1:851976Nf + 25:400736]a2 + O(a3) ;
+ [ 3:906512 2
10:854117 + 5:555928Nf
76:202209]a2 + O(a3) ;
is the gauge parameter and the restriction
stands for evaluation at (A.9)
and (A.10). Although we are only interested in the values in the Landau gauge, de ned
in the MS scheme.
Next we summarize some aspects of the tensor basis and projection matrix for the T2
sector. Indeed one purpose of this summary is to provide an aid to the understanding of
the full information given in the attached data le for both T2 and T3. Due to the size of
the bases and matrices we used, a useable electronic format is more appropriate for their
representation. First, we present a selection of tensors in the T2 basis choosing several
representatives from each
(n)matrix partition.
When one of the external momenta is
(p; q) = & q + & q + [2& qq q
(p; q) = & p
+ [2& pq q + d& pq q + & pq q + d& qp q + 3& qp q + d& qp q
+& qp q + d& qq q + 2& qq q + d& pq q + & pq q + d& qp q
(p; q) = & pp p
(p; q) =
MT2 = B
where the subscript on the block matrices corresponds to the label of the analogous
(n)matrix appearing in the projection tensor. Each of these partitions is of di erent size being
respectively 22, 6 and 2 dimensional. Given the size of the rst two submatrices it is again
not feasible to display all entries. Instead we choose to give a few reference entries to
facilitate the extraction of the full matrices from the data le. We have
M(T22) 6 20 =
M(T02) 3 6 =
M(T22) 15 10 =
M(T02) 4 2 =
where indices of M(0) i j range from 1 to 6 and these can be mapped to the labels of the
tensor basis by adding 22. Finally, the remaining sector is compact enough to record it
(p; q) = [p p p
(p; q) =
We have only shown one tensor from the nal partition as the other is given by replacing
the uncontracted vector p by q.
For each of the bases we have explicitly constructed the projection matrix coe cients.
For T2 as there are 30 projectors this would correspond to a 30
30 matrix where the
entries are rational polynomials in d. However, as we are using the generalized basis of
matrices in ddimensions the projector matrix is block diagonal due to the property
of (A.12). In other words
completely as
M(T42) =
Overall the matrix MT2 is symmetric as is MT3 . Finally, this information should be
su cient to connect with the full electronic representation for both sectors.
Chiral extrapolation
E ective
eld theory framework
In the speci c framework of Chiral Perturbation Theory (ChPT, see, e.g., refs. [61{63])
applied here, the generating functional of all QCD correlators is evaluated by means of a
path integral involving an e ective lowenergy Lagrangian Le (U; v; a; s; p; : : :) (compare
with ref. [61], and eqs. (1) and (2) of ref. [30]),
eiZ[v;a;s;p;t] = h0j T exp i
[dU ] exp i
Formally, all QCD Green's functions can be obtained by taking functional derivatives
axialvector and antisymmetric tensor source
elds s; p; v ; a ; t . It should be noted
that the tensor structure with an additional 5 is not independent due to the identity
5 = 2i
. The dots stand for other possible source elds (for example, the
coupling to symmetric tensor elds has been considered in refs. [64, 65]). The tensor source
has been incorporated in ref. [66]. The matrix
eld U collects the pion (Goldstone
elds in a convenient way (see below). The e ective Lagrangian has to be
invariant under local chiral transformations of the Goldstone boson and source elds, and
shares all other symmetries of LQCD. A formal proof that lowenergy QCD can indeed
be analyzed in this way has been given by Leutwyler [62]. Under chiral transformations
(L; R) 2 SU(2)L
SU(2)R, the quark and external source elds transform as
qL :=
l := v
s + ip ! R(s + ip)Ly ;
qR :=
! Ll Ly + iL@ Ly ;
r := v + a
! Rr Ry + iR@ Ry ;
:= P
= P
The e ective Lagrangian and the perturbative series are ordered by a lowenergy power
(or quark masses). For details and further references, we refer to refs. [61{63]. At leading
chiral order, the e ective Lagrangian describing the interaction of the pseudoGoldstone
bosons (pions) with the external source elds and each other is given by (see ref. [61])
L(M2) =
part of s. The brackets h
constant in the chiral limit (F
Here U = exp(ip2 =F ) with
i denote the
avor (or isospin) trace, F is the pion decay
86 MeV), and r U = @ U
i(v + a )U + iU (v
labels the speci c pion, and
are the pertaining channel matrices. We write out
+ =
= p
where the a are the Pauli matrices. The matrix eld U transforms as U ! RU Ly under
chiral transformations.
K(L; R; U ) (which is also unitary). Below we shall set the external elds p; a to zero,
= ta
2 . At fourth order, we have
L(M4) = i `6 F + [u ; u ]i
where we only show the terms needed for our present work (see refs. [61, 66, 67] for the
complete Lagrangian at that order, and eq. (B.7) for the de nition of the operators u ,
and T ).
Chiral Lagrangians for resonances
of ChPT already in refs. [29, 30]. In the following, a \heavy vector meson" framework was
set up [31, 68{71] to deal with problems related to the modi ed power counting in the
mass in the chiral limit). Today, it is better understood how to deal with such problems in
scheme for the e ective eld theory [72{74]. Such methods have been applied to the case
of heavy meson resonances in refs. [75{82]. We refer to these references for details on the
vector meson e ective eld theory outlined below.
Keeping in mind the transformation behavior of the external source elds v , a and
given above, we can write down the following terms describing the interaction of the
vector mesons with the external source elds and the pions (compare also the previous
LiVnVt =
= uF L uy
= uyt uy
LiVnt =
F R;L = @ (v
r U = @ U
i(v + a )U + iU (v
u = iuy (r U ) uy ;
u = p
= D V
D V := @ V
i(v + a )]u + u[@
Extrapolation formulae
For the sake of completeness, we rst discuss the pion matrix elements
see eq. (B.3) for the channel matrices . We have used a largeNc argument here to cast
and ! elds in the matrix form of the last line in eq. (B.7), compare also with eq. (27)
of ref. [31]. The dots indicate terms of higher chiral order, terms involving external source
elds s; p (which are not needed here), or terms involving more derivatives, which result
in contributions of the same form as those resulting from the terms given above, when
using the equations of motion or eld transformations [84]. The vector eld propagator in
momentum space is
(q) = ( i)
f v (k2) = 1
f t (k2) =
+ rmM 2 + rkk2 + O(p4) ;
The standard framework of ChPT yields
p)i = i abc(2p
p)i = abc(k p
I (k2) + O(p4) ; (B.11)
where the loop function I (k2) is given at the end of this appendix, in eq. (B.25) (it
vanishes for k2 ! 0, and is complex for k2 > 4m2 ), and `r6; rm; rk are renormalized
lowenergy constants, which depend on the scale
. M is the leading term in the quarkmass
expansion of the pion mass m , derived from the Lagrangian (B.2) (at the order we are
working, it can be set equal to the pion mass). We note that, up to corrections of two
elastic unitarity,
Im f v (k2) = f v (k2) (k2)t11 (k2) ;
Im f t (k2) = f t (k2) (k2)t11 (k2) ;
4m2 < k2 < 16m2 ;
where (s) :=
t11(s) =
for 4m2 < s < 16m2 :
It easily follows that the form factors f v;t must have the phase 11(s) in the elastic region.
Contributions to
matrix elements
Here we use the de nitions
Here we have to set k2
s equal to the rho pole, k2
function renormalization factor is derived from the
! spole = m2
[85]. The wave
where the contribution of the one loop graphs to the selfenergy is given by (compare
loop(s) :=
qj b(k; )i = i ab(e( )k
e( )k )f T =p2 ;
V M 2 +
A IA!(k2) :
and nd at the one loop level up to O(p4)
The \tadpole" terms can be taken to be energy independent at the order we are working
to. The integral IA can be deduced from eqs. (B.23){(B.26) below, and IA! is given
by eq. (B.29) (with mV ! m!
associated with local operators hF +
m ). The local terms proportional to cV , cTV can be
i etc., and can be used to absorb
(real) terms of O(M 2) from the loop integrals. The loop functions are given at the end of
this appendix (IA! = IAV (mV ! m!
graphs). The leading nonanalytic term in IA! is given by
m )). In the one loop approximation, we evaluate
m2!) =
and the terms of order M 0 and M 2 are absorbed in the corresponding LECs. The chiral
logarithm of this integral is of O(M 4). One nds
where c is the following combination of (renormalized) LECs,
c :=
The coe cient of the leading chiral logarithm is in agreement with ref. [67]. With gV
(see ref. [80], and references therein), the coe cient of the thirdorder term should be
. Inserting this estimate, and
= 770 MeV, the thirdorder term
becomes comparable to the leading chiral logarithm for M & 200 MeV, so it might give a
nonnegligible contribution for most data points.
motivates the extrapolation formula (6.1c), while the formulae (6.1a) and (6.1b) result
from (B.16) and (B.17), respectively, upon inserting the explicit expressions for the loop
functions given below. The cusp e ects and imaginary parts in the chiral behavior of the
on Euclidean lattices with a nite volume. A more thorough analysis is needed to deal with
given above are not a icted by this de ciency. This can be deduced from the fact that
they agree with the corresponding results in the heavy vector meson framework [67, 71],
where the unitarity e ects due to the
loops are either dropped or derived from contact
terms of a nonHermitean part of the e ective Lagrangian (see, e.g., ref. [68]).
In the expression for the ratio given in eq. (B.21), the factors of pZ
and the
nonanalytic terms in the loop function IA containing the imaginary part cancel at oneloop
order. Due to this simpli cation, it is straightforward to compute the
nitevolume
corrections for this ratio. Here, we attempt only an estimate of the leading
correction, related to the O(M 2) `chirallog' term contained in the tadpole loop integral
I (compare eq. (B.27) below). According to the standard formalism of ChPT in a nite
I(L) = I +
06=k2Z3
Here K1(z) is the modi ed Bessel function of the second kind, which decays exponentially
for large positive z, K1(z) !
2z e z . Inserting (B.22) in (B.16) and (B.17) yields the
nitevolume correction to the ratio of eq. (B.21) upon a straightforward chiral
Loop functions
in the formulae above. The loop integral with two pion propagators is given by
I (s) =
k2 =: s :
It diverges when the spacetime dimension d approaches 4,
I (0) = 2 +
however the di erence I (s) := I (s)
I (0) is nite,
I (s) =
where it is understood that real values of s are approached from the upper complex plane
for s 2 [4M 2; 1). Explicitly,
In the chiral limit (M ! 0),
I (s) =
1 + 0(s) artanh
0(s) :=
s > 4M 2. We have also employed the abbreviation
0(s) (s 4M 2)=(16 ) for real
A :=
I :=
M 2 = 2M 2
for d ! 4. The scalar integral including two di erent propagators can be written as
s) =
= I V (m2V )
and we refer to appendix B of ref. [80] for details on the chiral expansion. We also use
where IV is given by the formula for I with M ! mV . Here, the letter V stands for the
vector meson running in the loop ( ; !; : : :).
Open Access.
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