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arXiv:1002.0925v1 [hep-lat] 4 Feb 2010
DESY 10-019
SFB/CPP-10-21
Status and prospects for the calculation of
hadron structure from lattice QCD
Dru B. Renner∗
NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
E-mail: dru.renner@desy.de
Lattice QCD calculations of hadron structure are a valuable complement to many experimental
programsas well as an indispensabletool to understandthe dynamicsof QCD. I present a focused
review of a few representative topics chosen to illustrate both the challenges and advances of our
community: the momentumfraction, axial charge and charge radius of the nucleon. I will discuss
the current status of these calculations and speculate on the prospects for accurate calculations of
hadron structure from lattice QCD.
The XXVII International Symposium on Lattice Field Theory
July 26-31, 2009
Peking University, Beijing, China
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
1. Introduction
Our most stringent constraints on the structure of any hadron follow from the underlying sym-
metries of QCD. Translational invariance dictates that the momentum of a hadron is the sum of
the momenta of all its constituents, giving rise to a powerful sum rule for the nucleon. Rotational
invariance demands that the spins of hadrons have the values n/2 for an integer n, hence the nu-
cleon spin is precisely 1/2 and not 0.5 with some experimental error. Similarly, we know that the
electric charge of the nucleon is exactly 1 and the net strangeness is 0. These statements are so
commonplace that they may even appear trivial, but each of these exact results serves as an entry
to different aspects of the dynamics of QCD.
The momentum of the nucleon is built up from that of the quarks and gluons. While trans-
lational symmetry constrains the entire contribution, the individual contributions of each quark
flavor and the gluon depend on the details of QCD. Similarly, the spin of the nucleon arises from
the quark spin, quark orbital motion and the gluon angular momentum, each of which are individ-
ually unconstrained by symmetries but must sum to 1/2. The charge of the nucleon arises from the
quark charges that yield a total charge of 1, but the distribution of this charge in the spatial degrees
of freedom probes the nonperturbative structure of the nucleon. Finally, the strange quarks in the
nucleon occur in precisely matched pairs of quarks and anti-quarks. Despite the net absence of
strangeness, the consequences of this hidden flavor are felt in many observables. In the following,
I will elaborate on a few of these points and discuss what we currently know from lattice calcu-
lations, phenomenology and experimental measurements. Along the way, I’ll offer my opinion
regarding where our calculations may have to go to provide accurate results for hadron structure.
The past several years have seen extensive reviews of the calculations presented each year
at the annual Lattice conference. You can find the few most recent reviews in [1, 2, 3], and a
very recent and exhaustive collection of results can be found in [4]. Rather than duplicating these
efforts, I will instead present some of the key examples mentioned above.1This will unfortunately
prohibit me from discussing all the hadron structure efforts presented at Lattice 2009, but I hope
these proceedings will still provide a useful overview, nonetheless.
2. Nucleon Momentum
The momentum of a nucleon with energy E and mass m is precisely constrained by Lorentz
symmetry as p2=E2−m2. This is such a common statement in particle physics that it seems nearly
trivial to even mention it. However, here we want to question how this momentum arises from the
underlying QCD degrees of freedom. In other words, what is the distribution of this momentum
among the nucleon’s constituents? This question concerns the details of the dynamics of QCD and
presents a challenge to lattice calculations of nucleon structure.
2.1 Parton Distribution Functions
The proper field theoretic response to the question above requires constructing the momen-
tum distributions of the nucleon’s constituents, the parton distribution functions (PDFs). I’ll pro-
1At the lattice conference I discussed strangeness in the nucleon. A recent review [5] covers much of this, so I will
not include these results in this review.
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Status and prospects for the calculation of hadron structure from lattice QCD
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0 0.2 0.4
0.6
0.81
x
0
0.2
0.4
0.6
x f(x,µ=2GeV)
up
down
anti-up
anti-down
gluon
Figure 1: Momentum distribution of quarks and gluons in the nucleon at µ = 2 GeV. For each parton
xf(x,µ) is plotted where f(x,µ) is the correspondingPDF. At largex, the up anddownquarksare the largest
componentsof the nucleon momentum. At low x, the gluon dominates and the anti-quarkdistributions grow.
The curves were generated using the LHAPDF library [6] and the MSTW2008 NNLO [7] dataset at a
renormalization scale of µ = 2 GeV.
vide a proper definition of the PDFs shortly, but for the moment we want to focus on a more
intuitive understanding of the quark and gluon distributions, q(x) and g(x). In the parton model
q(x) or g(x) would be the probability to find a quark, of some flavor, or a gluon with momentum
pparton=xpnucleon. This interpretation is retained in QCD, however, q(x,µ) and g(x,µ) now carry a
renormalization scale µ that loosely gives the energy resolution at which the distribution is probed.
While this muddies the picture a bit, the PDFs, nonetheless, are well-defined universal properties
of the nucleon that are probed in many different experiments.
2.1.1 Phenomenological Results for PDFs
There is now a well-established industry dedicated to extracting the PDFs from global anal-
yses. The analyses differ in many details, including the order of the perturbative expansion used,
the treatment of the quark masses and the handling of the statistical and systematic errors from the
many experimental inputs. These variations have an impact on the precision ultimately obtained,
however, these details do not impact the discussion at hand. Hence, as merely one example among
several, I show the PDFs from the MSTW collaboration in Fig. 1.
Field theory effects ultimately handicap any simple interpretation of the PDFs, however, we
can still see that the PDFs at a low scale, µ = 2 GeV in Fig. 1, retain features that one might expect
for the nucleon. Theupand downquark distributions are thelargest component for high momentum
(x ? 1). These distributions in fact have peaks at reasonable values of x, as one might expect for a
3
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
0 0.2 0.4
0.6
0.81
x
0
0.2
0.4
0.6
x f(x,µ=100GeV)
up
down
anti-up
anti-down
gluon
Figure 2: Momentum distribution of quarks and gluons in the nucleon at µ = 100 GeV. The details are the
same as in Fig. 1 but now µ = 100 GeV. At high scales, the PDFs for the quarks and gluons mix and the
simple non-relativistic nucleon structure, still visible at µ = 2 GeV, is suppressed at µ = 100 GeV.
hadron dominantly composed of two up quarks and one down quark. However, the most prominent
feature in the plot for low x is clearly the gluon distribution. This is a clear indication of QCD
physics at work. This has a counterpart in the simple observation that the quark masses directly
contribute only about 1% of the nucleon’s mass. Beyond the dynamics of QCD, the presence of
the anti-quarks is a striking field theory effect. This statement may seem mundane, but we must
remember the role played by the anti-quarks in nucleon structure to appreciate why we go through
all the difficulty to calculate hadronic matrix elements from fully dynamical lattice QCD rather
than being content with models or quenched calculations.
As mentioned, field theory complicates the interpretation of the PDFs. To illustrate this, I
show the same distributions again in Fig. 2 but now evolved to a scale of µ = 100 GeV. Any hint
of nucleon physics is now well hidden and all the constituents of the nucleon appear to play nearly
equally important roles. We will return to this issue again when discussing the evolution of the
momentum fraction.
2.1.2 Operator Definition of PDFs
Theparton distribution functions can be defined asnucleon matrix elements ofquark and gluon
fields separated by light-like distances. As an example, we record here the operator definition of
the unpolarized quark distribution q(x,µ2) [8],
q(x,µ2) =1
2
?dλ
2πeixp·λn?p,s| q(−λ/2 n) / nWn(−λ/2 n,λ/2 n) q(λ/2 n) |p,s?|µ2 .
(2.1)
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
For each flavor there are two other twist-two quark distributions, the helicity and transversity dis-
tributions, denoted as ∆q(x,µ2) and δq(x,µ2) respectively. The factor Wn(−λ/2 n,λ/2 n) is the
Wilson line extending along the arbitrary light-cone direction n from λ/2 n to −λ/2 n,
Wn(−λ/2 n,λ/2 n) = Pexp
?
ig
?λ/2
−λ/2dα A(αn)·n
?
.
The scale, µ, and scheme dependence of q(x,µ2) comes from the renormalization of the operator
in Eq. 2.1.
The expression in Eq. 2.1 highlights the difficulty of lattice calculations of PDFs. Lattice
QCD calculations are performed in Euclidean space, however, the PDFs involve quark and gluon
fields separated along the light-cone. These inherently Minkowski-space observables are difficult
to construct explicitly in Euclidean space. But, as we will see in the next section, the moments in x
of these distributions can be calculated directly in Euclidean space.
2.2 Moments of Parton Distributions
As just discussed, the PDFs as a function of x are essentially Minkowski-space objects. But
the moments in x are related to local operators that can be calculated in Euclidean space on the
lattice.
2.2.1 Mellin Transform: from x to n
The moments in x are defined as follows,
?xn?q,µ2 =
?1
−1dxxnq(x,µ2) =
?1
0dxxn?q(x,µ2)−(−1)nq(x,µ2)?.
The sign in the above equation is determined by the identification of q(x) for negative x with the
anti-quark distribution as q(x,µ2)=−q(−x,µ2).2These moments can be related to forward matrix
elements of twist-two operators. First we present the complete result but then sketch the argument
in the following.
?p,s| q(0) γ{µ1iDµ2···iDµn}q(0) |p,s?|µ2 = 2?xn?q,µ2 p{µ1...pµn}
The brackets in T{µ1···µn}denote symmetrization of the indices of the tensor T and subtraction of
the traces. The precise meaning of this operation will be clarified shortly.
The derivation of Eq. 2.2 is almost elementary, but it is so essential to the method underlying
the lattice calculations that wesketch the arguments using theunpolarized PDFsasanexample. The
first step isto introduce light-cone coordinates and gauge. Thisisauseful firststep inunderstanding
several aspects of the PDFs. In defining the light-cone coordinates, you introduce two light-cone
directions nµ
for a four-vector v. (Similarly, / n±= γ ·n±= γ∓.) Light-cone gauge is the choice of A(x)·n = 0,
2Thisisafrequent source of confusion when comparing tophenomenological determinations of thePDFs. Alsonote
that thesign isdifferent for thetransversity distributionbut thesame for thehelicity distribution: δq(x,µ2)=δq(−x,µ2)
and ∆q(x,µ2) = −∆q(−x,µ2).
(2.2)
±= (1,0,0,±1)/√2. The two new light-cone coordinates are given by v·n±= v∓
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
which sets the Wilson line Wn(−λ/2 n,λ/2 n) to 1. Then choosing n = n−in Eq. 2.1, imposing
the light-cone gauge for n−and relabeling λ as y−gives
q(x,µ2) =1
2
?dy−
2πeixp+y−?p,s| q(−y−/2) γ+q(y−/2) |p,s?|µ2 .
(2.3)
The next step in relating the moments to local operators is to use the known limited support of
q(x) to the region −1 ≤ x ≤ 1 to expand the integration range used to define the moments,
?1
?xn?q,µ2 =
−1dxxnq(x,µ2) =
?∞
−∞dxxnq(x,µ2).
The remaining steps are relatively elementary. Thekey sequence follows from combining the above
with Eq. 2.3,
?∞
?∞
where ∂+= ∂/∂y−. The above manipulations must be understood as acting under the
Eq. 2.3. The final result is
−∞dxxneixp+y−= (ip+)−n
= (−1)n(ip+)−n
?∞
−∞dx(∂+)neixp+y−=
−∞dxeixp+y−(∂+)n= (p+)−nδ(p+y−)(i∂+)n
?dy−in
?xn?q,µ2 = 2−1(p+)−(n+1)?p,s| q(0) γ+(i∂+)nq(0) |p,s?|µ2 .
By inspection, this expression can be seen as the light-cone coordinate, light-cone gauge form of
the following
?p,s| q(0) / n (in·D)nq(0) |p,s?|µ2 = 2?xn?q,µ2(n· p)n+1.
This expression relates ?xn?q,µ2 to diagonal matrix elements of local operators. The more familiar
form follows from writing the above as
nµ1nµ2...nµn?p,s| q(0) γµ1iDµ2···iDµnq(0) |p,s?|µ2 = 2?xn?q,µ2 nµ1...nµnpµ1...pµn.
This form makes it clear precisely what the symmetrization and trace removal in Eq. 2.2 means.
2.2.2 Phenomenology of ?x?u−d
The global analysis illustrated in Figs. 1 and 2 can also be used to examine the moments of the
PDFs. This time taking as an example the results from the CTEQ collaboration, I plot the results
for ?x?qand ?x?gin Fig. 3. These results follow from numerically integrating curves similar to
those in Figs. 1 and 2. Furthermore, the state-of-the-art results for the particular quantity of interest
to us, ?x?u−d, are collected in Tab. 1. The phenomenological analyses often present results in terms
of the so-called valence distribution qv(x) = q(x)−q(x) and the anti-quark distribution q(x), but
Tab. 1 gives the correct combination that is comparable to lattice results.
What is particularly interesting about the momentum fraction ?x? is that it obeys a sum rule,
1 =∑
q
?x?q,µ2 +?x?g,µ2
(2.4)
3The error given in [14] is asymmetric, but here we take only the upper error to compare to the lattice results that
are all higher than the phenomenological values.
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101
102
103
104
µ2 [GeV2]
0
0.2
0.4
0.6
<x>(µ)
gluon
up
down
strange
charm
Figure 3: Running of the momentum fraction of the quarks and gluons in the nucleon. As µ increases, the
gluon momentumfraction ?x?gincreases towards the asymptotic value of 4/7 and ?x?q, for each of the quark
flavors, approaches the common value of 3/28. (These values hold for the 4 flavor theory.) Despite the
unique limiting values for x, the non-perturbativeinput at low µ clearly dictates the values of the momentum
fraction for each parton over many orders of magnitude. The results come from the LHAPDF library [6]
using the CTEQ6.6C [9] dataset.
where ∑qis the sum over all relevant flavors. This sum rule imposes constraints on the scale
evolution of ?x?µ2 such that ?x?µ2 asymptotically approaches a perturbatively calculable limit for
large µ.
lim
µ→∞?x?q,µ2 =
3
16+3NF
16
16+3NF
lim
µ→∞?x?g,µ2 =
For NF= 4, we have lim?x?q= 3/28 ≈ 10% and lim?x?g= 4/7 ≈ 60%. However, we can clearly
see in Fig. 3 that the asymptotic results for ?x? bear little resemblance to the results at any reason-
able value of µ. In fact, the hierarchy shown in Fig. 3 is quite clear. The gluons carry about 40% of
the nucleon momentum, the up and down quarks carry about 35% and 20%, and the strange takes
most of the remaining 5%. It is precisely this pattern that we eventually hope to understand from
lattice calculations.
2.3 Lattice Calculation of ?x?u−d
The lowest non-trivial moments of the unpolarized quark PDFs are ?x?q.4These are the quark
contributions that enter the momentum sum rule in Eq. 2.4. Here I use the particular combination
4The lowest moments ?1?qare known in terms of the quark valence structure of the nucleon.
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[x]uv−dv
0.1790±0.0023
0.1747±0.0039
0.1640±0.0060
0.1645±0.00463
0.1820±0.0056
0.1754±0.0041
?x?u−d
0.1646±0.0027
0.1603±0.0041
0.1496±0.0062
0.1501±0.0048
0.1676±0.0058
0.1610±0.0043
ABMK
BBG
JR
MSTW
AMP06
BBG
Table 1: Phenomenologicalvalues for ?x?u−dat µ =2 GeV. For each calculationwe give the momentof the
non-singlet valence distribution, denoted by [x]uv−dv. (The square brackets denote simply the integral over
the region 0 ≤ x ≤ 1 as opposed to the region −1 ≤ x ≤ 1 used in the angular brackets.) These values were
collected in [10]. The original references are [10] (ABMK), [11, 12] (BBG), [13] (JR), [14] (MSTW), [15]
(AMP06) and [11, 12] (BBG N3LO). This is combined with the result [x]u−d= −0.0072±0.0007 from
[10] to produce a result for ?x?u−d, which can be compared to lattice calculations.
?x?u−das a benchmark observable to evaluate the lattice calculation of moments of PDFs.
?p,s|qγ{µiDν}τ3q|p,s?
???µ2= 2?x?u−d,µ2p{µpν}
Here q = (u,d) is the doublet of the light quarks. The flavor combination u−d eliminates discon-
nected diagrams, which would otherwise require a substantial computational investment. Addition-
ally, this combination also eliminates any mixing with gluonic operators, thus greatly simplifying
the renormalization of this quantity.5Without the mixing, this observable then only requires a
multiplicative renormalization. Finally, you can calculate the bare matrix elements using nucleons
with ? p = 0. Hence, ?x?u−dis basically the most accurate scale-dependent observable in nucleon
structure that we calculate on the lattice.
Figure 4 illustrates all the dynamical lattice QCD results for ?x?u−dfor pion masses less than
700 MeV. The values for ?x?u−d, mπand a come from a variety of already published sources and
numerous private communications. (The references are provided in the caption to Fig. 4.) The
most striking feature that you observe in Fig. 4 is that, despite the community’s efforts to calculate
with many actions, several lattice spacings and volumes and a broad range of pion masses now
approaching 250 MeV, the calculations still overestimate the experimental measurement by at least
30% and maybe as much as a factor of 2. The spread amongst the groups obviously suggests some
systematic variations, and I will examine two possible explanations shortly, but it is important to
note that the individual calculations are visibly much more consistent with themselves than with
the other calculations. Additionally, notice that some groups are beginning to perform calculations
“at the physical point,” as illustrated by the QCDSF calculation in Fig. 4. This phrase quite often
means simply mπ< 200 MeV. Nonetheless, the next generation of lattice calculations will likely
shed some much needed light on the chiral behavior. However, the discrepancy amongst the groups
at pion masses where we should be able to reliably calculate today is an issue that needs to be
5Strictly speaking, the disconnected diagrams and the mixing with gluons only vanish for lattice actions with an
exact flavor symmetry.
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Figure 4: World’s dynamical lattice QCD results for ?x?u−d. The lattice calculations all overestimate
the phenomenologically determined value. They also show a fair amount of scatter amongst themselves.
The possible role of finite size and renormalization effects are discussed in the text. The lattice results
are from [16, 17, 18] (RBC NF= 2+1), [19, 20] (RBC NF= 2), [21] (LHPC), [22, 23] (ETMC) and
[24] (QCDSF). The experimental result is generated using the LHAPDF library [6] and the CTEQ6.6C
dataset [9].
addressed. Without resolving these discrepancies, it will be hard to confidently establish physical
results even with calculations approaching the physical pion mass.
Before discussing finite size and renormalization effects, I want to make a quick comment on
the lattice spacing dependence of ?x?u−d. This is currently poorly studied. Of the five calculations
shown in Fig. 4, only QCDSF has calculated beyond a single lattice spacing. However, even in that
case, the range in a that is used to establish scaling is not large. Their results are an encouraging
hint that lattice artifacts are not a substantial part of the discrepancy in Fig. 4, but there is nothing
universal about such effects and all the groups must make a stronger effort to calculate at multiple
lattice spacings.
A persistent concern in nucleon structure calculations is the role of finite size effects. In
fact, the results at Lattice 2009 have added much to this issue even if they haven’t resolved it.
In Fig. 5, I examine several finite size studies by various collaborations. Excluding the lightest
calculations at mπ= 260 MeV, one observes no statistically significant finite size effects for any
of the remaining calculations. These results are consistent with the common rule-of-thumb that
mπL ≈ 4 is sufficient.6However, the recent results of QCDSF at mπ= 260 MeV potentially stand
in contrast to the finite size dependence observed at higher pion masses. This calculation suggests
that mπL = 4 is at best just barely sufficient to capture the large volume limit of ?x?u−d. It is
6My use of mπL to gauge finite size effects is, of course, not strictly correct. I am loosely assuming a discussion in
or near the chiral limit in which 1/mπwill be the dominant length scale.
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Figure 5: Finite size studies for ?x?u−d. The dependenceof ?x?u−don mπL is shown for several calculations.
Notice the essentially flat behavior for mπL >4 for all but the lightest pion mass. The QCDSF calculation at
the lightest pion mass may suggest the emergence of a more substantial finite size effect as mπis decreased.
The curve is a very simple scaling to the mπ= 260 MeV results and is meant solely to guide the eye. The
results are from the same references given in Fig. 4.
possible that this observation would vanish with higher statistics. In fact, the three calculations at
mπ=260 MeV do essentially agree statistically, but the trend in the results is suggestive, especially
in comparison to all the other results at larger mπ. Of course, it is also possible that ?x?u−dwould
drop even further with still larger L.
In a review of this nature it is difficult to perform a detailed infinite volume limit of all the
results presented at the conference. However, we can illustrate the impact of finite size effects
by simply making the crude restriction to mπL > 4. This is illustrated in Fig. 6. The picture
is certainly clearer and one may even be left with the impression of some downward curvature for
some of the lattice calculations, but we must guard against wishful thinking. Excluding the QCDSF
results, each of the groups is statistically consistent with a linear dependence on m2
0.75. The one exception is QCDSF. A linear fit gives χ2/dof = 2.1. This is not overwhelming
evidence of non-linearity and all the apparent curvature comes solely from the lightest point at
mπ= 260 MeV. This point is 2.8σ less than the second lightest QCDSF point. Increased statistics
in these calculations and confirmation of this new finite size behavior by other groups will be
necessary to understand what is happening at mπ= 260 MeV.
Given the results in Figs. 5 and 6, there is no concrete conclusion that we can draw yet regard-
ing finite size effects, but I hope that Fig. 5 will stand as a warning that the finite size effects can,
and likely do, change substantially as we decrease the pion mass and that collaborations pushing
towards the physical pion mass must keep these effects in mind. At this point one may argue that
we should investigate the finite size effects as indicated from chiral perturbation theory. It is my
πwith χ2/dof <
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Status and prospects for the calculation of hadron structure from lattice QCD
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Figure 6: Large volume results for ?x?u−d. The results are fromthe same references givenin Fig. 4, but only
those calculations with mπL > 4 are shown. The results from each calculation are consistent with a smooth,
nearly flat mπdependence. However there is a systematic variation between the calculations.
personal opinion that, given an absence of any real evidence for curvature in ?x?u−dand given that
such non-analytic behavior is the hallmark of chiral perturbation theory, it is theoretically question-
able to force a fit to chiral perturbation theory. While this is my opinion, it was clear at the lattice
conference that many presenters also simply chose to not show chiral fits. Of course, calculations at
still lighter pion masses may show the expected chiral behavior and systematic errors at the current
pion masses may obscure this behavior, but we should keep in mind that in the end calculations
down to the physical point may be necessary to convincingly establish results for ?x?u−d.
One obvious feature of Fig. 6 is that the results from each collaboration have a flat behavior
but there are clear shifts between the groups. One natural concern in this respect is the renormal-
ization of the operator used to determine ?x?u−d. This is multiplicative and, importantly, quark
mass independent. However it does depend on the lattice action and hence will vary between the
different calculations. By taking the ratio of ?x?u−dto the value ?x?ref
ence mass, which I arbitrarily choose to be mπ= 500 MeV, we can eliminate the renormalization
factor consistently for each calculation. The result of this is shown in Fig. 7, and we find that the
various calculations appear to collapse onto a universal curve. As with the finite size effects, we
can not make a specific conclusion, but this result is suggestive of potential problems in the renor-
malization of ?x?u−d. There are other possible explanations of the systematic shifts between the
groups, such as the lattice spacing effects mentioned earlier or systematics due to plateaus that are
too short in the bare matrix element calculations [19, 25]. Independent of the ultimate explanation
for these problems, I hope that the difference between Fig. 6 and Fig. 7 encourages us to examine
the systematics of our calculations.
u−dat some canonical refer-
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Figure 7: Renormalization free ratio: ?x?u−d/?x?ref
form the ratio ?x?u−d/?x?ref
the renormalization factors from each calculation. The calculations are now all consistent with each other.
This demonstrates that a systematic error, possibly in the renormalization, could possibly account for the
discrepancy among the collaborations.
u−d. The calculations with mπL > 4 from Fig. 6 are used to
u−d, where the reference scale is chosen as mref
π= 500 MeV. This ratio eliminates
3. Nucleon Spin
As discussed in the introduction, the nucleon spin is exactly 1/2 due to rotational symmetry,
and similar to the momentum fraction, it obeys a sum rule [26],
1
2=1
2∑
q
?1?∆q,µ2 +∑
q
Lq,µ2 +Jg,µ2 ,
(3.1)
which relates the nucleon spin to the contributions from quark helicity ?1?qand orbital angular
momentum Lqand a net contribution from the gluons Jg. The asymptotic evolution of the total
quark contribution Jq=1
2∑q?1?∆q+∑qLqand the total gluon contribution Jgis given by
µ→∞Jq,µ2 =1
16
16+3NF.
For NF= 4, we find lim Jq= 2/7 ≈ 0.60·1/2 and lim Jg= 3/14 ≈ 0.40·1/2. Again we find the
gluons playing a substantial role, now in the nucleon spin and earlier in the nucleon momentum.
This is a particularly surprising conclusion in light of the naive quark model result ∑q?1?∆q= 1
and Jg= 0. However, similar to the asymptotic results for the momentum fraction, we should be
suspicious that non-perturbative QCD dynamics can alter the picture at low scales.
lim
2
3NF
16+3NF
lim
µ→∞Jg,µ2 =
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Unfortunately, much less is known experimentally regarding the decomposition of the nucleon
spin. The one quantity that is known well is the axial coupling, gA= ?1?∆u−?1?∆d, that is mea-
sured accurately in neutron beta decay. A recent review [27] gives gA= 1.2750(9) and the most
recent PDG [28] world average is gA= 1.2694(28). The remaining quark contributions are es-
sentially the only other known pieces. The experimental result from HERMES in 2007 [29] gives
?1?∆u= 0.842(12), ?1?∆d= −0.427(12) and ?1?∆s= −0.085(17). QCD sum rules [30] or model
estimates [31] indicate that Jg≈ 0.25 at low scales. Taking the experimental results for ?1?∆qand
this estimate for Jg, we can estimate that 50% of the nucleon spin is given by the gluons with the
remaining divided into 33% from quark spin and 18% from quark orbital motion. This picture is
much less certain than the momentum sum rule, but again, it provides a challenge to the lattice
QCD effort.
3.1 Lattice Calculation of gA
The nucleon axial charge can be defined by
?p,s|qγµγ5τ3q|p,s? = 2gAsµ
or gA= ?1?∆u−∆d. The moments ?1?∆qare the lowest moments of the polarized PDFs. For the
unpolarized distributions, the moments ?1?qare fixed by the valence structure of the nucleon, but
this is not the case for the polarized distribution. This difference actually makes the moments ?1?∆q
even simpler than the momentum fractions, ?x?q, discussed in the previous section. In fact, using
a lattice action with chiral symmetry would eliminate the need to renormalize the lattice operator
used to determine ?1?∆qaltogether.7However, many lattice actions in use today do require this
renormalization, but it is not scale dependent and is in many ways simpler than the renormaliza-
tion required for ?x?q. Additionally, since the power of x in ?1?∆qis one lower than in ?x?q, the
lattice operator does not contain a derivative and is a simple quark bilinear. Consequently, the
bare matrix elements can be calculated more accurately for ?1?∆qthan for ?x?q. As in the case
of ?x?u−d, we focus on the u−d combination again to eliminate the computationally demanding
disconnected diagrams. Furthermore, this combination gives the axial coupling gA= ?1?∆u−∆dthat
is very accurately measured in neutron beta decay.
In Fig. 8, I collect all the full QCD lattice results for gA, again with mπ< 700 MeV. Similar
to ?x?u−d, gAserves as a benchmark observable for lattice calculations of nucleon structure. As
in Fig. 4 for ?x?u−d, we find that the lattice results show a degree of scatter and all consistently
underestimate the experimental measurement. However, the discrepancy, between 10% and 20%,
is more mild and the scatter in the lattice results is less severe. In fact, as I try to argue next, it
appears that this scatter may be almost entirely accounted for by finite size effects, at least to the
current level of statistical precision.
In Fig. 9, I examine several finite size studies that are available for gA. The results for the
heaviest values of the pion mass show a strong finite size effect. This has led to the current view
that mπL > 6 may in fact be needed to reliably determine gA. However, we find that, as quantified
by mπL, the finite size effects are diminishing as mπis lowered. This can be seen quite clearly in
7This is only true for the non-singlet moments, such as ?1?∆u−∆d. The axial anomaly generates mixing for the
singlet moments.
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
Figure 8: World’s dynamical results for gA. The lattice results are from [32, 33] (BGR), [25] (RBC NF=
2+1), [19, 20] (RBC NF=2), [34] (LHPC), [35] (ETMC) and [24] (QCDSF). The experimentalresult is the
PDG 2008 value [28]. The discrepancy between the lattice calculations and the experimental measurement
and the scatter among the lattice calculations are both smaller than for ?x?u−dshown in Fig. 4.
Fig. 9 by examining the lighter pion mass calculations of both QCDSF and ETMC. It is possible
that the volumes for these calculations at mπ= 310 MeV or 260 MeV may still be too small to
see the asymptotic volume dependence. We can also consult chiral perturbation theory, which does
allow for this sort of behavior; however, given that gAis essentially flat for the largest volumes, one
might reasonably question the use of chiral perturbation theory at these pion masses.
In order to attempt to estimate the infinite volume limits for the lattice calculations presented
here, I simply require mπL > 6. This may turn out to be an excessive requirement for low mπ,
which would actually be good news regarding finite size effects, but it is clearly required for heavier
pion masses. The lattice results satisfying this are shown in Fig. 10. As is clear from the figure,
this restriction is very severe, however, the resulting lattice calculations show a strong level of
agreement amongst themselves. The results are still lower than the experimental measurement, but,
unlike for ?x?u−d, only a mild curvature is required to reconcile the current calculations with the
physical limit. In particular, notice that, with one exception, there is no systematic shift between the
various collaborations. As mentioned earlier, the renormalization of gAis generically easier than
that of ?x?u−d, and this lends a bit more support to the hypothesis that differences in renormalization
may be driving part of the variation of ?x?u−din Fig. 6. The one small exception is the result of
QCDSF which is just slightly low compared to all the other calculations. However, this may be
consistent with a small discrepancy in fπ, which renormalizes with the same factor as gA, that is
also present for QCDSF [36].
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Status and prospects for the calculation of hadron structure from lattice QCD
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Figure 9: Finite size effects in gA. The mπL dependence is shown for several calculations. The results at the
heaviest pionmasses indicatea significantfinite size effect in gA, suggestingthat mπL of 6 may be necessary.
But the light mπresults of QCDSF and ETMC may in fact be showing a weakening of the finite size effects
in gAas measured in terms of mπL. The lattice results and experimental measurement are the same as in
Fig. 8, and the curves are simple scaling fits to guide the eye.
4. Nucleon Charge
The charge of the nucleon, or any hadron, again appears to be an essentially trivial topic.
There is no doubt that the proton has one unit of charge and that the neutron is, well, neutral.
The certainty in our understanding of the nucleon’s charge arises from the flavor symmetries of
QCD, however, the distribution of this charge in the spatial degrees of freedom of the nucleon’s
constituents is not prescribed by symmetry and provides another probe of the dynamics of QCD
within the nucleon. This idea leads to a broad range of calculations on the lattice and quickly
involves the generalized parton distributions, but to avoid excessive complications and to explain
the concepts in the simplest setting, I will discuss just the neutron charge [37].
4.1 Neutron Transverse Charge Distribution
Much of the interest in form factors originates with the interpretations that we can assign to
them. There is now a rich field theoretic discussion dedicated to this issue, but in order to maintain
our footing and retain a strong connection to the experimental measurements, it is valuable to
remember that ultimately, the form factors parametrize the QCD contributions to cross sections,
independent of any interpretation we attach to them.
dσ
dΩ=
?dσ
dΩ
?
Mott
?F2
1+τF2
2+2τ(F1+F2)2tan2(θ/2)?
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
Figure 10: Results for gAwith mπL > 6. The lattice results from Fig. 8 that satisfy mπL > 6 are shown. All
the lattice calculations, but one, are in statistical agreement. The results still underestimate the experimental
value, but the discrepancy is much less severe than in ?x?u−d. The QCDSF results are just slightly low.
This may indicate some small systematic error in the renormalization or it may simply illustrate the lattice
artifacts that are present in all the calculations.
Here F1(q2) and F2(q2) are the nucleon form factors and τ = q2/(4m2). For some in our commu-
nity, this is where the discussion ends. That is, of course, a perfectly valid point of view. However,
the understanding provided to us by the generalized parton distributions and the relationship to
transverse quark distributions [38] allows us to go further.
Historically the form factors were interpreted as charge and magnetization densities. In par-
ticular the charge density was constructed from GE= F1−τF2as
?
ρ3D(? r) =
d3? q
(2π)3e−i? r·? q
m
?m2+? q2GE(? q2).
This is plotted for the neutron in Fig. 11. There we see that the charge distribution is negative for
large r. This is usually understood in terms of the pion cloud generated by n → pπ−fluctuations.
Overall neutrality of the neutron forces the core to be positive leading to a negative charge radius.
But this picture is spoiled by relativistic corrections that enter as (m?r?)−1, where ?r? is some
relevant length scale for the charge distribution. These corrections would vanish in the non-
relativistic limit m → ∞ but are substantial for the nucleon. The introduction of the transverse
distributions in the infinite-momentum frame (IMF) eliminates these corrections, which enter now
as (pz?r?)−1with pz→ ∞ defining the IMF. The transverse distributions [38] are given by
?d2? q⊥
ρ2D(?b) =
(2π)2e−i?b·? q⊥F1(? q2
⊥).
This is shown for the neutron in Fig. 12. The distribution still has a negative tail for large r, but
16
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
0
0.5
1
1.5
2
r [fm]
0
0.1
4πr2 ρ3D(r) [fm-1]
Kelly 2004
Figure 11: Naive 3d neutron charge distribution. The 3d Fourier transform of the GEform factor gives the
rest frame charge distribution of the nucleon up to relativistic corrections. The negative distribution at large
r is interpreted as the pion cloud arising from the n → pπ−fluctuations. This gives rise to a negative charge
radius for the neutron. The curveswere generatedby numericallyintegratingthe Kelly 2004parametrization
of the nucleon form factors [39].
now we find a negative core. Furthermore, the charge radius is positive for the transverse charge
distribution. Understanding this new view of the neutron is a challenge onto itself. Additionally,
the introduction of the transverse coordinates in a field-theoretically clean manner opens many new
avenues for investigation of hadron structure from lattice QCD.
4.2 Lattice Calculation of ?r2?u−d
As for the nucleon momentum and spin structure, I again introduce a benchmark observable
that allows us to summarize the status of lattice calculations of form factors. Here I choose the
isovector charge radius, ?r2?u−d. As before, the flavor structure is u−d, so disconnected diagrams
again vanish. Furthermore, this observable enters matrix elements of the vector current. These two
facts combine to make ?r2?u−dan accurate quantity to calculate on the lattice. Additionally, all the
lattice calculations can now use an exactly conserved vector current. This eliminates the need for
any renormalization and simplifies the calculation significantly.
The form of the nucleon’s electromagnetic interaction is given by
γµF1(q2)+iσµνqν
?p′,s′|Jµ|p,s? = u(p′,s′)
?
2mF2(q2)
?
u(p,s)
and the charge radius is then given by
?r2?u−d= −6dFu−d
1
dq2
?????q2=0
.
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
0
0.5
1
1.5
2
b [fm]
-0.2
-0.1
0
0.1
2πb ρ2D(b) [fm-1]
Kelly 2004
Figure 12: Transverse2d neutronchargedistribution. The 2d Fourier transformof the F1form factor results
in the transverse charge distribution of the nucleon. Relativistic corrections are absent but the distribution
must be understood in the infinite momentum frame. Notice the negative core, in contrast to Fig. 11. Also
the charge radius is now positive in this frame. The distribution was constructed from the same form factor
parametrization in Fig. 11.
The factor of 6 is the conventional choice, but a factor of 4 would instead give the correct transverse
charge radius in the IMF. To avoid confusion, we stick to the conventional choice, but we should
keep in mind the theoretically clean interpretation that is available for the transverse distribution.
In Fig. 13 I gather all the full QCD results for ?r2?u−d. It is immediately evident that all
the lattice calculations show a remarkable agreement for this observable. The strong scatter that
is present for ?x?u−dand to a lesser extent gAis clearly absent for ?r2?u−d. This is particularly
surprising given the fact that this quantity is expected to diverge in the chiral limit as indicated by
the curve in Fig. 13. We might even expect large finite size effects in such a quantity. However,
we must keep in mind that ?r2?u−dis calculated from the slope of F1(q2) at q2= 0 and that the
momentum quantization imposed by a finite volume forces us to extract this slope using the first
non-zero value of q2, which satisfies qL ≈ 2π. We may, in fact, be extrapolating from a region
with weaker L dependence through the region with stronger L dependence. Ultimately, whatever
the explanation, we find a near absence of volume dependence for ?r2?u−d.
Lacking any real guidance on the necessary volumes for ?r2?u−d, I simply follow the conven-
tional wisdom and take mπL > 4. These lattice results are shown in Fig. 14. An optimist may find
a mild increase, but this would be hard to defend strongly. Additionally, this observable requires
much more than a mild increase to agree with the experimental value at the physical point. But it
does appear that all the calculations seem to agree. Given that ?r2?u−drequires no renormalization,
this is consistent with the speculation suggesting that renormalization may be partially culpable for
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Status and prospects for the calculation of hadron structure from lattice QCD
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Figure 13: World’s dynamical results for ?r2?1/2
[19, 20] (RBC NF= 2), [40] (LHPC/MILC), [41] (LHPC/RBC), [35] (ETMC) and [24] (QCDSF). The
experimental result comes from the Kelly 2004 parametrization of the form factors [39]. The leading order
heavy baryonchiral perturbationtheory prediction (LO HBχPT) contains no unknownlow energyconstants
other than?r2?1/2
size effects in this quantity, but this does not appear to be the case.
u−d. The lattice calculations are [25] (RBC NF= 2+1),
u−dandis logarithmicallydivergentas mπ→0. Onemight, therefore,expectsubstantialfinite
the systematic variation in ?x?u−ddiscussed earlier. Despite the fact that this observable still shows
a significant discrepancy with the experimental result, the agreement amongst all the lattice groups
is encouraging.
5. Conclusions and Outlook
In order to illustrate the status and promise of lattice calculations of hadron structure, I chose
to review three benchmark observables that represent the broad range of hadronic physics that we
can address on the lattice: the momentum fraction, axial charge and charge radius of the nucleon.
Nearly all the lattice groups have pushed the pion mass towards 300 MeV or lower and some are
now breaking through 200 MeV. This is an enormous accomplishment, but unfortunately most
hadronic observables are failing to show the long-sought chiral curvature. Independent of the
expectations of chiral perturbation theory, the three benchmark observables still do not show a
convincing approach to the experimental results.
The obvious conclusion is that yet lighter, and maybe even physical, pion masses will be
required. This may be the case, but we should not do this at the expense of other systematic errors.
Finite size effects must remain a concern as we lower mπ. Both ?x?u−dand gAdemonstrate that the
volume dependence can have a non-trivial dependence on mπand our conventional rule-of-thumb
that mπL ≈ 4 may not necessarily be sufficient for all quantities.
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
Figure 14: Large volume results for ?r2?1/2
not easily calculate the form factors at the low Q2needed to determine ?r2?1/2
volume. Lacking any detailed knowledge of the finite size effects in ?r2?1/2
with mπL > 4 are shown. There is general agreement among all the lattice groups and a mild increase as
mπis lowered. However ?r2?1/2
measurement.
u−d. Due to momentum quantization at finite volume, one can
u−dwhile also working at small
u−d, only the results from Fig. 13
u−dis not yet rising nearly as fast as it needs to match onto the experimental
Lattice artifacts are still a concern, if only because so few groups have seriously examined
them. The agreement of the groups for ?r2?u−dmight indicate weak artifacts for some quantities,
but the systematic shifts between the groups for ?x?u−dis a warning that renormalization, an essen-
tial part of the continuum limit for most quantities of interest, may be partially responsible for these
problems. The role of renormalization is further implicated by the general agreement of the groups
for the renormalization free ratio of ?x?u−d/?x?ref
ing a scale-dependent renormalization) to gA(requiring only a finite renormalization) to ?r2?u−d
(requiring no renormalization) will be a strong test of our control of renormalization issues.
u−d. In fact, the progression from ?x?u−d(requir-
These systematic effects will be a part of any critical review of lattice calculations, but we
should remember the rich QCD physics that provides the motivation for these calculations. Each of
the benchmark observables is a stand-in for a broad range of experimentally relevant and intellec-
tually interesting observables: the decomposition of momentum among the nucleon’s constituents,
the nature of the nucleon’s spin and the distribution of the nucleon’s charge. Apart from these
observables, Lattice 2009 saw a sizeable effort to study strangeness in the nucleon, results for the
pion, delta and other hadrons, and a range of exploratory calculations. Clean control of systematics
and a continued push toward the physical pion mass will ultimately allow detailed calculations that
will complement the experimental programs and advance our insight into hadron structure.
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Status and prospects for the calculation of hadron structure from lattice QCD
Dru B. Renner
Acknowledgments
I gratefully appreciate the many conversations and email exchanges with colleagues while
preparing for the Lattice 2009 conference. I would like to thank D. Alexandrou, G. Bali, W. Bi-
etenholz, T. Blum, T. Doi, M. Engelhardt, W. Freeman, Ph. Hägler, P.-A. Harraud, T. Hemmert,
K. Jansen, T. Kaneko, T. Korzec, M. Lin, Z. Liu, D. Mohler, B. Musch, J. Negele, H. Ohki, S. Ohta,
J. Osborn, G. Schierholz, W. Schroers, T. Streuer, K. Takeda, N. Ukita, A. Walker-Loud and
J. Zanotti. Additionally, this work was supported by the DFG Sonderforschungsbereich/Transregio
SFB/TR9-03
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