Page 1

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/

Page 2

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.

2

Page 3

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

00.20.4

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

Page 4

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

00.20.4

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)

4

Page 5

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∓

5

Page 6

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.

6

Page 7

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

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.

7

Page 8

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

[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.

8

Page 9

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

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.

9

Page 10

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

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 <

10

Page 11

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

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-

11

Page 12

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

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 =

12

Page 13

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

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.

13

Page 14

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].

14

Page 15

Status and prospects for the calculation of hadron structure from lattice QCD

Dru B. Renner

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)?

15