# A standard convention for particle-level Monte Carlo event-variation weights

MCnet Community

###### Abstract

Streams of event weights in particle-level Monte Carlo event generators are a
convenient and immensely CPU-efficient approach to express systematic
uncertainties in phenomenology calculations, providing systematic variations
on the nominal prediction within a single event sample. But the lack of a
common standard for labelling these variation streams across different tools
has proven to be a major limitation for event-processing tools and analysers
alike. Here we propose a well-defined, extensible community standard for the
naming, ordering, and interpretation of weight streams that will serve as the
basis for semantically correct parsing and combination of such variations in
both theoretical and experimental studies.

A standard convention for particle-level
Monte Carlo event-variation weights

The MCnet Community

Editors: E. Bothmann1, A. Buckley2,*, C. Gütschow3,

S. Prestel4, M. Schönherr5, P. Skands6

Supporters: J.R. Andersen5, S. Bhattacharyaa, J. Butterworth3, G. Singh
Chahal5,

L. Corpeb, L. Gellersen6, M. Gignacc, S. Höched, D. Kare, F. Krauss5,

J. Kretzschmarf, L. Lönnblad4, J. McFaydeng, A. Papaefstathiouh, S. Plätzeri,

S. Schumann1, M.H. Seymourk, F. Siegertk, A. Siódmokl

1 Institut für Theoretische Physik, Georg-August-Universität, Göttingen,
Germany

2 School of Physics & Astronomy, University of Glasgow, Glasgow, UK

3 Department of Physics & Astronomy, University College London, London, UK

4 Department of Astronomy & Theoretical Physics, Lund University, Lund, Sweden

5 Institute for Particle Physics Phenomenology, Durham University, UK

6 School of Physics & Astronomy, Monash University, Clayton VIC 3800,
Australia

a Department of Physics and Astronomy, Northwestern University, Evanston IL,
USA

b CERN, Meyrin, Switzerland

c University of California Santa Cruz, Santa Cruz CA, USA

d Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

e University of Witwatersrand, Johannesburg, South Africa

f University of Liverpool, Liverpool, UK

g Physics and Astronomy Department, University of Sussex, Brighton, UK

h Department of Physics, Kennesaw State University, Kennesaw, GA 30144, U.S.A.

i Institute of Physics, University of Graz, 8010 Graz, Austria

j University of Manchester, Manchester, UK

k Institut für Kern- und Teilchenphysik, TU Dresden, Dresden, Germany

l Jagiellonian University, Cracow, Poland

##  1 Introduction

With the inexorable rise in CPU cost of high-precision Monte Carlo (MC)
collision-event samples [1, 2, 3], and the experimental need for huge volumes
of such samples, reweighting as opposed to explicit re-running of MC event
samples has become the norm. In this approach, a systematic variation to the
_nominal_ MC generator configuration — the one used to perform the original
sampling of event properties — can be obtained without regeneration by
identifying the ratio of probability density between the variation and the
nominal for each event. In MC programs this ratio is often easily calculable
by rapid re-evaluation of a matrix-element or parton-shower kernel formula.
The ratio factor is multiplied on to the nominal-configuration event weight
(resulting from weighted or kinematically biased generation strategies [4]) to
give the event-specific _variation weight_.

Events which are more likely under the variation than the nominal assumptions
have a variation weight greater than the nominal weight, and _vice versa_.
Provided each systematic variation’s phase-space distribution has sufficient
overlap with the nominal distribution from which the events are sampled (i.e.
that the two distributions have common support), using the variation weight in
place of the nominal one for histogramming or other physics applications with
the single event sample produces observable distributions asymptotically equal
to those from explicit generation from the variation model. In practice,
weighting dilutes the statistical power of an MC event sample, but for mild
variations and for uncertainty estimation purposes, the convenience, speed,
and reduced storage requirements make this compromise acceptable.

Including hard-process scale variations, PDF error sets, variations in parton-
shower tunes, electroweak corrections, heavy-flavour schemes, BSM physics, and
more besides, in current usage it is not uncommon for hundreds of physics
variations to be found in each MC event, stored in its _weight vector_ 111An
ordered tuple, rather than a physics vector in the sense of vector spaces. [5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

However, the character-string names used to identify these vectors’ elements —
the weight elements corresponding to one variation through an event dataset
being called a _weight stream_ and having a unique name — are not yet
standardised between generators, leading to complexity, duplicated decoding
attempts, and inevitably errors in analysis codes within experimental
collaborations and in public MC tools such as Rivet [24] and MadAnalysis [25].
Previous attempts [26] have been made to standardise the names and values in
weight streams for the parton-level LHE format [27], and are heavily used.
However, the connected attempt to standardise the propagation of such event
weights to particle-level, i.e. fully exclusive events in the absence of
detector interactions and primarily in the HepMC format [28], have not in
practice seen sufficient uptake and a variety of _de facto_ naming formats are
in circulation via different generator codes. This situation causes problems
and effort-duplication for the many direct and indirect users of particle-
level events.

In this document we address these issues with a new, more complete,
extensible, and formally defined format for particle-level weight-stream
naming, in particular for event generators producing the HepMC format and for
users of such events. We additionally place essential requirements on the
ordering of such variations in generator weight-vectors, for efficient and
reproducible downstream processing, and codify the semantic meaning of the
variation-weight values. This summarises the consensus from a year-long
discussion process among representatives of the main ME+shower+hadronisation
event generator teams, which combines the best features of the various
preceding _ad hoc_ schemes, and defines a new community standard for LHC Run
3.

##  2 Particle-level weight-name structure

The main role of event weights is to indicate the shifts of differential
cross-sections due to variations in the physics model — from alternative hard-
process scales and PDFs, to parton shower options, heavy-flavour treatments,
electroweak corrections, and so on. Simultaneous combinations of such
elementary variations are also commonplace. The component variations hence
need to be clearly identifiable from the names of the weight streams.

In HepMC these stream-names are simple character strings: there is no more
complex data-structure available, and hence particle-level variations need to
be encoded in a “flat” form, with each stream treated as independent, i.e.
there is no opportunity for “grouping” of streams or “inheritance” of common
variations into subsets of streams. In this proposal we hence describe a
scheme for structuring of the available string to specify in a uniform way the
parameters varied in each stream. This notably does not specify exact rules
for combination, although in many standard cases these are well-established:
we leave this possibility, which requires also detailed discussion and perhaps
updating of the LHE event format, as a possible extension to be built on the
foundation of this particle-level standard.

Our agreed general form for particle-level weight names is the following:
either a special nominal weight name, preferably the empty string but with a
few acceptable alternatives, or a variation weight name, in the format

⬇

PREFIX__LEVEL.KEY1=VALUE__LEVEL.KEY2=VALUE...

i.e. a prefix and a non-empty set of key=value blocks separated by double-
underscores. The double-underscore serves the dual purpose of both allowing
single underscores to appear in block components, and of more clearly visually
distinguishing the blocks; both features aid human readability. The keys may
have multiple levels, separated by dots. The prefix and the =VALUE part of
each block are optional, and the prefix and blocks obviously cannot themselves
contain the double-underscore separator. A practical example could be

⬇

MUR=1.0__MUF=2.0__PDF=123456__CKKW.MUQ=20 .

A more formal specification of weight names in general may hence be given in
W3C EBNF [29] notation:

⬇

WEIGHTNAME ::= NOMNAME | VARNAME

NOMNAME ::= "" | "NOMINAL" | "DEFAULT" | "WEIGHT" | "0"

VARNAME ::= (PREFIX "__")? BLOCK ("__" BLOCK)*

PREFIX ::= "USER" | "AUX" | "EXTRA" | "IRREG"

BLOCK ::= KEY ("=" VALUE)?

KEY ::= IDENT ("." IDENT)* - NOMNAME

VALUE ::= IDENT | NUM

IDENT ::= ALPHA (ALPHA | DIGIT | "_")* - "__"

NUM ::= ("+" | "-")? DIGIT+ ("." DIGIT+)? EXPO?

EXPO ::= ("e" | "E") ("+" | "-")? DIGIT+

DIGIT ::= [0-9]

ALPHA ::= [A-Za-z]

where NOMNAME represents the special-case name for the default or nominal
weight stream, to be discussed in Section 2.2. Analysis tools must always
perform a case-insensitive string comparison, as all conceivable case-
conventions already exist in established MC generator outputs.

The context given, we now explain in turn the allowed characters, the nominal
weight name, the role and allowed values of the variation prefix, and the
formatting principles and predefined types of variation key/value block.

###  2.1 Allowed characters

For both nominal and variation weights, there is little argument for use of a
character set beyond the ASCII set, especially as extended “wide” character
sets are problematic in many older or numerically focused languages and it is
simplest to avoid issues around use of extended character encodings. All
weight names are hence restricted to the basic alphanumeric characters and the
following minimal set of punctuation characters:

  * •

equal sign: =

  * •

underscore: _

  * •

full stop: .

  * •

plus: +

  * •

minus: -

where the plus and minus symbols are only allowed as a sign for numerical
values or in exponentials, but not in level or key names, e.g. as a hyphen.

This set of characters has been arrived at by consideration of several factors
that constrain a free-for-all approach:

  * •

Restricting the set to core Unix and bash file-name characters should minimise
the need for escaping, quoting or character translation in shell commands or
file names. This excludes characters such as quote marks, $, >, <, or ;, which
have special meaning in many scripting languages.

  * •

For familiarity and for ease of translation to and from executable-code forms,
the usual programming language restrictions on identifier names to be made
purely of alphanumeric characters apply, including a prohibition on starting
an identifier name with a digit.

  * •

Weight names are often placed in brackets or parentheses of some sort to
indicate that they are variations — including bracketing characters in the
weight names themselves would therefore cause parsing difficulties that are
best avoided.

  * •

White-space characters are not permitted as they can be a source of headaches
for the user. The weight name is a key in a C++ map and so “Name”, “ Name”
(outer padding using a single white-space character on each side) and “ Name”
(outer padding using two white-space characters on each side) would be
interpreted as distinct keys.

  * •

In particular, newline and tab characters in a weight name lead to confusion
when printed, are difficult to distinguish from (also discouraged) spaces, and
can cause parsing difficulties in text-based formats.

  * •

Comma or colon characters are not allowed, as they are natural separators or
indicators of hierarchies, and hence introduce confusion with the double-
underscore and dot syntax used for those concepts. Blocking their use in
weight names means they remain available for use as separator characters in
higher-level applications which need to handle weight names.

We also note that the minus, plus, and equals symbols and potentially other
“operators” can cause issues when trying to save branches in ROOT [30].
However, this is a reflection of one format’s limitation, and when
automatically handling numerical variations minus signs are a natural
occurrence both in the radix and exponent. Translators of event-weighted data
to ROOT-based data formats (and any other base format with such naming
sensitivity) are advised to explicitly perform a translation between literal
plus and minus signs and a ROOT-safe alphanumeric encoding of them, e.g. -1.0
$\to$ _MINUS_1.0, and if necessary similar replacement of the equals-sign
key/value separators with text or a ROOT-safe symbol.

###  2.2 The nominal/default weight

While several MC-generator variation modes are of _a priori_ equal physics
validity, particularly between the central members of several PDF global fits,
a run of simulated events always has a nominal or default configuration: the
one from which the matrix element and parton shower phase-spaces were
rejection-sampled. This is typically unweighted, and hence the events will
have uniform or near-uniform rejection sampling weights in the nominal stream.
The other — variation — weight streams then re-evaluate the kinematics, mostly
without the possibility of rejection, to obtain a modified weight: as this
modification depends on the full event phase-space, each event’s variation
weights will deviate from uniformity in a unique way.

Without making a judgement about the physicality of particular configurations,
although the nominal configuration is almost always a sensible one, it is
extremely useful for an MC sample to have an easily identifiable default
weight. It is most obvious that this should correspond to the nominal
configuration used for the original sampling. Even for setups with multiple
equivalent candidates for the default weight, one of them should be assigned
default status, rather than forcing decoding on the user even for the simplest
visualisation.

The nominal weight name should be easily identifiable even when parsing a list
of jumbled up weight names. Recognising the range of established conventions
in long-public generator codes, a set of acceptable names is required: these
are NOMINAL, DEFAULT, WEIGHT, the digit 0, or the empty string. Exactly one
instance from this set must appear in a given list of weight-vector names.

###  2.3 Variation prefixes

The optional prefix and its delimiting double-underscore are for the purpose
of discriminating physically meaningful weight streams from debug or
bookkeeping ones, and for distinguishing between “official” and user-defined
physical weights.

Much as every event-record graph contains elements intended more for debugging
use by MC authors than as physically-meaningful event history for analysers,
it is natural that event generator authors wish to encode debugging elements
into their events’ weight vectors. These, however, must appear in an
unambiguous form which can be automatically skipped (at least by default) in
analysis tools.

In discussion, it has been identified that there are at least two reasons to
prefix weights with a “class” marker: one is this hiding of truly auxiliary,
unphysical weight streams, and the other is to distinguish between physical
weight streams “blessed” by the generator authors, and those which are
physical but unblessed: “user” weights. The following prefixes may be used to
classify such auxiliary weights:

No prefix:

Canonical cross-section variations: good to process and plot (e.g. standard
scale and PDF variations, for which a standard naming convention is given in
the following section). As many weights as possible should come in this
minimal form.

USER, AUX:

Optional indication of non-standard cross-section variations: good to process
and plot (e.g. user-defined shower variations). This prefix exists only to
indicate the “unblessed” nature of the variations from the perspective of the
generator authors: it should be treated equivalently to the no-prefix form for
observable calculations by downstream tools.

EXTRA:

Supplementary information, exists for every event, but either not a cross-
section variation (e.g. Sherpa’s NTrials) or requiring further manipulation to
obtain a weight variation that can be used equivalently to the nominal weight.
They are hence not required to behave like “physical” weight variations, and
are not to be plotted or otherwise used as a physical variation by default.

IRREG:

Irregular/generator specific: weights of this type are not processed by
default (e.g. fixed-order information). This type of variation weight should
always be appended at the end of the weight vector, such that it’s easier to
skip the IRREG-type group of variation weights (see Section 3).

While the choice between no-prefix and a USER/AUX prefix is largely a matter
of taste, correctly indicating EXTRA and IRREG weight streams is mandatory, to
ensure physically correct processing of events.

###  2.4 Variation blocks

Each block should represent one distinguishing characteristic of the weight
stream, with the full set of blocks fully specifying how the stream deviates
from the nominal configuration. The expectation is that any implied run
settings not listed take the values used by the nominal stream, though null-
variations such as scale multipliers of $\times 1.0$ are allowed to be
included if desired, at the cost of a more lengthy weight name, whose physics
purpose is less immediately obvious to human readers.

It consists of a key, which can contain several dot-delimited levels, and an
optional value appended with a delimiting equal sign to the right-hand side of
the key. Neither the key nor the value can contain a double-underscore as this
is the block delimiter, but single underscores are permitted. If the equals-
sign and RHS value are omitted, the key will be interpreted as equal to a
boolean “true” value.

Several standard key structures are established by existing practice and
common use-cases, with an eye to coherent extension of the scheme as MC
generator mechanisms evolve in sophistication and ability to use variation
weights:

  * •

For PDF variations, the most common scheme will be to specify a single PDF
set, either for both beams or for the single composite-particle beam. In this
case the key should either be LHAPDF, to specify that the following value is
the LHAPDF set ID [31] and can be used for automatic uncertainty-band
construction by standard analysis tools; or a custom key ending in PDF — e.g.
PY8PDF — to specify a PDF identifier specific to the generator or application.
In the latter case, it is recommended that the generator document the custom
key in the event metadata, e.g. in the HepMC GenEvent or GenRun objects’
custom attributes.

  * •

For use-cases where multiple composite beams are in use at the same time,
there may be a need to specify separate PDFs and factorization scales for each
beam. In this case, the *PDF and MUF block names can be followed by an extra
level specifying which beam it refers to, e.g. LHAPDF.BEAM1 or MUF.BEAM2. The
versions described above without these qualifying beam identifiers are to be
interpreted as a compact syntax for simultaneously setting equal values for
both beams.

  * •

For scale variations, similar structures using MUR and MUF keys (without
underscores) followed by scale-modifying multiplicative factors for
renormalization and factorization scale shifts should be used. These are
easily extendable to other types of scale variations, but should always
include MU. In general, the scaling given should be linear in the energy
scale, not the squared form. However, a squared scale (e.g. MUR2) may be
specified to avoid square roots in the token values.

  * •

As well as SM parameters in the calculation, variations to encode changes in
BSM-model parameters are a major use-case of event weights. There are too many
models to pre-specify standard schemes for them all. They should, however,
follow the general syntax described here, and as consistency aids cooperation
we advise that model developers and users coordinate to match naming schemes
as far as possible, e.g. based on the names and parameters in established UFO
packages.

In this context we can see that the variation name

⬇

MUR=0.5__MUF=2.0__PDF=42

represents scalings down and up respectively of the renormalization and
factorization scales by factors of two, and use of the internal PDF number 42.
As there are no distinguishing levels to these keys, they may be assumed to
apply to all parts of the MC generation. Introducing such levels we might
find, e.g.

⬇

ME.MUR=0.5__ME.MUF=2.0__LHAPDF=123456__CKKW.MUQ=20

where the scale variations apply only to the matrix element (ME), a new
merging scale MUQ has been introduced to the CKKW jet-merging component, and
the PDF ID is now the global LHAPDF ID code rather than a run-specific index.
Further distinctions such as application of different scalings to different
parton showers can be applied, e.g.

⬇

ME.MUR=1.0__ME.MUF=2.0__ME.LHAPDF=123456__ISR.MYPDF=14__ISR.MUR=1.0__ISR.MUF=2.0__FSR.MUR2=2__FSR.MUR=1.0__CKKW.MUQ=20

The keys used in variation blocks can be different from the exact forms used
in these examples, although certain weight name components are fairly well
established already, and uniformity is a virtue for users if there is no
compelling reason to deviate. Values corresponding to absolute energy scales
shall be given in GeV.

Ideally, attention should be paid to how floating point values are
represented, e.g. MUR=1.000000E+00__MUF=2.000000E+00 is unnecessarily precise
and impacts legibility. However, this formatting control may be technically
limited by implementation language, and parsers should expect to recognise
numeric tokens in a variety of standard forms. More extensive descriptions of
the functional forms of scales as well as weight combination prescriptions can
be added to the run meta if required.

##  3 Weight-vector selection and ordering

It has long been standard for the first weight in the vector to be the
nominal, before weight names became common. This ordering has been broken in
recent years by some generators using weight names, not helped by a bug in the
HepMC ASCII-format writer which re-sorted output names (but not their values!)
into alphabetical order.222This bug was fixed in HepMC 2.06.11 and should no
longer be considered a viable excuse for not paying attention to ordering.
While not an absolute requirement if using the standardised nominal-stream
names, we strongly recommend that the nominal weight be placed first in the
weight vector.

For analysis tools, the selection of weight subsets is typically very
important. The benefit of weighting is its great speed with respect to
explicit re-generation with a different nominal configuration. But having
$\mathcal{O}(1000)$ weights in a Monte Carlo setup adds often-unwanted
analysis overhead for users who do not always want to analyse all variations
for every iteration of their analysis. For example, the Rivet analysis toolkit
permits both positive- and negative-condition filtering of weight streams
based on numeric-index and regular-expression matching. As string processing
is computationally slow and current generators use a stable ordering of weight
streams in the vector, this matching is performed only once: a list of the
desired weight-subset’s numerical indices is obtained from the first event,
and used directly from then on. The alternative of checking thousands of
filter conditions for every event in case the weight-vector ordering had
changed would be prohibitively slow. This established behaviour and
opportunity for substantial performance optimisations in event processing
means it is important to codify this ordering stability:

  * •

the number of physical weight streams and their weight-vector positions must
not change from event to event.

The existence of auxiliary weights, and of complex setups such as correlated
NLO sub-events (where the set of weight names can differ event-to-event, as
each event type is produced in an independent code block) can lead to
situations where the maximal length of the weight vector is often only known
at the end of the run. However, these latter weight streams are typically also
auxiliary and not intended for analysis. Due to the “first event” inference of
physical weight locations, in such cases it is important to structure the
weight vector in such a way that the physical variation names for physics uses
(and hence which need to be known right from the start) come first:

  * •

Auxiliary weights that might not even exist for every event must be appended
at the end such that only the unphysical tail of the weight vector changes in
length and/or ordering.

##  4 Weight manipulation

Finally, a note on the meaning of weight values:

  * •

Standard weights (i.e. those with empty, USER, or AUX prefixes) must all have
equivalent semantic meaning to the nominal weight.333The careful wording here
reflects that the nominal stream may not represent an entire physical event,
cf. correlated counter-events in higher-order MC simulations. But weight
streams are orthogonal to this: they must not introduce new semantic meanings
to the event over those already in use for the nominal.

This rule forbids use of partially-meaningful physical weight streams which
need to be combined with the nominal or with other variation weights in order
to be used for physics predictions. Any weight-combining logic of this sort
should happen inside the Monte Carlo event generators and be hidden from the
user. For example:

  * •

if a setup contains a default prediction and renormalisation-scale variations,
the scale-variation weight should correspond to the final cross-section
contribution (apart from global normalisation factors);

  * •

if a setup contains a default prediction and a weight that allows to filter
out a subset of the events, satisfying certain selection properties, the
filter weight should be equal to the nominal weight if the event is to be
accepted or zero otherwise.

Of course, this does not mean that sets of variations, e.g. for PDF
uncertainties, should be subjected to a combination attempt prior to the
generator writing out the event, provided each variation in the set
corresponds to a standalone physical variation of the event prediction.

##  5 Summary

Variation weight-streams for MC-generator events are an increasingly important
tool for propagating systematic uncertainties in the phenomenology
calculations into observables for both theoretical and experimental studies. A
major limitation on this usefulness is the lack of a common standard for
labelling — and hence semantically correct processing and combination — of the
variation streams. In this document we have proposed a well-defined base
standard for such naming, including a general syntax and a minimal set of
agreed names for the most common variation concepts. This format has been
agreed by the signed MC-generator authors and users, and will serve as the
basis for more portable event processing, and further community standards.

## Acknowledgements

This work was supported in part by the European Union as part of the Marie
Sklodowska-Curie Innovative Training Network MCnetITN3 (grant agreement no.
722104). Of the editors, AB thanks the Royal Society for University Research
Fellowship grant UF160548; AB and CG thank STFC for funding through the UK
experimental particle physics consolidated grants programme; CG also thanks
STFC for supporting the SWIFT-HEP project (grant ST/V002627/1); SP would like
to acknowledge funding from the Swedish Research Council (Vetenskapsrådet),
contract numbers 2016-05996 and 2020-04303; MS is funded by the Royal Society
through a University Research Fellowship (URF\R1\180549) and an Enhancement
Award (RGF\EA\181033 and CEC19\100349), and supported by the STFC under grant
agreement ST/P001246/1.

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# Nucleon form factors and structure functions from $N_{\mathrm{f}}=2$ clover
fermions

QCDSF/UKQCD Collaboration: S. Collins,a M. Göckeler,a Ph. Hägler,a T.
Hemmert,a R. Horsley,b Y. Nakamura,a,c A. Nobile,a H. Perlt,d ,e P.E.L.
Rakow,f A. Schäfer,a G. Schierholz,e A. Sternbeck,a H. Stüben,g F. Wintera and
J.M. Zanottib
a Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg,
Germany
b School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK
c Center for Computational Sciences, University of Tsukuba, Ibaraki 305-8577,
Japan
d Institut für Theoretische Physik, Universität Leipzig, 04109 Leipzig,
Germany
e Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany
f Theoretical Physics Division, Department of Mathematical Sciences,
University of Liverpool, Liverpool L69 3BX, UK
g Konrad-Zuse-Zentrum für Informationstechnik Berlin, 14195 Berlin, Germany
Email

###### Abstract:

We give an update on our ongoing efforts to compute the nucleon’s form factors
and moments of structure functions using $N_{\mathrm{f}}=2$ flavours of non-
perturbatively improved Clover fermions. We focus on new results obtained on
gauge configurations where the pseudo-scalar meson mass is in the range of
170-270 MeV. We will compare our results with various estimates obtained from
chiral effective theories since we have some overlap with the quark mass
region where results from such theories are believed to be applicable.

##  1 Introduction

Over years significant efforts have been made to use lattice techniques to
investigate the structure of the nucleon. Of particular interest are the
Parton Distribution Functions (PDFs) and form factors. The latter encode
information about charge distribution and magnetization while the PDFs tell us
about the distribution of momentum and spin. While some of the related
observables can be determined with good accuracy by experiments (e.g. the
nucleon’s axial charge $g_{\mathrm{A}}$) other quantities are difficult to
access (like the tensor charge $g_{\mathrm{T}}$).

A precise determination of moments of nucleon PDFs and form factors on the
lattice turned out to be rather challenging. It continues to be difficult to
reach sufficient control on all systematic errors such as finite size effects,
lattice artefacts and the influence of the chiral extrapolation. Simulations
are performed in volumes of a size where some quantities exhibit significant
finite size effects. The available lattice data of the quantities of interest
show no significant discretization effects. But current simulations only probe
a small window of lattice spacings thus providing us with limited control on
the continuum extrapolation. From chiral effective theories (ChEFT) there are
indications that the quark mass dependence close to the physical pion mass is
very strong. Therefore extending lattice simulations into the region where
$m_{\pi}\leq m_{\mathrm{PS}}\lesssim 300~{}\mbox{MeV}$ has become a major goal
for recent calculations.

##  2 Simulation details

For our simulations we use Wilson glue and $N_{\mathrm{f}}=2$ degenerate
flavours of Clover fermions, where the improvement coefficient
$c_{\mathrm{SW}}$ has been determined non-perturbatively. Most of our
configurations have been generated using the BQCD implementation of the HMC
algorithm [1]. Various algorithmic improvements have been applied which
accelerate this algorithm, such as the Hasenbusch preconditioning and the use
of multiple time-scales, or reducing the time spent for matrix inversion. For
instance, chronological guess and the Schwarz Alternating Procedure (SAP) are
used to start the inversion and to precondition the fermion matrix [2].

These algorithmic improvements plus recent increase in computing resources
enabled simulations in the region of small quark masses, i.e. in a region
where the pseudo-scalar mass is smaller than 300 MeV. Fig. 1a shows the
parameter region of our simulations. When approaching physical quark masses
larger lattices are needed to stay in the region $m_{\mathrm{PS}}L\gtrsim 3$
where finite size effects are expected to be sufficiently small (see Fig. 1b).
To investigate such finite size effects we have also performed simulations
with $m_{\mathrm{PS}}L<3$.

(a)

(b)

Figure 1: The left panel shows the simulation points in the
$m_{\mathrm{PS}}^{2}$ vs. $m_{\mathrm{PS}}L$ plane. In the right panel dashed
lines show the lattice spacing and box sizes for which simulations have been
performed. In both figures the continuous lines show where
$m_{\mathrm{PS}}L=3$.

We compute the quark propagators using point sources which we (Jacobi) smear
to improve overlap with the ground state. For the three-point correlation
functions we apply standard sequential source techniques. The distance between
source and sink is about 1 fm. Throughout this paper we will ignore
contributions coming from disconnected terms. While these anyhow cancel in the
iso-vector cases, results for the iso-scalar case maybe affected by an
uncontrolled systematic error.

To set the scale we use the Sommer parameter $r_{0}/a$ which we extrapolated
to the chiral limit at each beta. While on the lattice this quantity can be
determined with small statistical errors, there is no experimental
determination. We therefore computed the dimensionless quantity
$(am_{N})(r_{0}/a)$ on the lattice and use the experimentally well known mass
of the nucleon $m_{N}$ to obtain $r_{0}=0.467\,\mbox{fm}$.

Most of the quantities considered in this paper need to be renormalised. The
renormalisation constants have been determined using the
$\mathrm{RI}^{\prime}\mathrm{-MOM}$ scheme [3], except for the vector current
renormalisation constant $Z_{V}$. Here we applied the condition that the
nucleon’s local vector current at zero momentum must be 1. If necessary, the
results are converted into $\overline{\mathrm{MS}}$ scheme using the 4- and
2-3-loop expressions of the $\beta$ function and corresponding anomalous
dimension $\gamma$, respectively.

##  3 Lowest moments of PDFs

Let us first consider the lowest moment of the polarized nucleon PDF $\langle
1\rangle_{\Delta q}$ (also known as axial coupling constant $g_{\mathrm{A}}$).
This quantity is determined from the renormalised axial vector current
$A_{\mu}^{R}=Z_{A}\,(1+b_{A}\,am_{q})A_{\mu}$, where
$am_{q}=(1/\kappa-1/\kappa^{(S)}_{c})/2$. $Z_{A}$ is known non-perturbatively
[3], for $b_{A}$ we use a tadpole improved one-loop perturbation theory
result.

(a)

(b)

Figure 2: The left panel shows $g_{\mathrm{A}}$ as a function of
$m_{\mathrm{PS}}^{2}$. The open and filled diamonds show the lattice results
before and after correction of finite size effects, respectively. The star
indicates the experimental result. The line shows a fit to the data as
described in the text. The right panel shows the relative finite size effects
determined on the lattice (symbols) and obtained from a fit to an expression
from ChEFT.

We have fitted our lattice results to an expression from ChEFT based on the
SSE formalism. Using this formalism both the quark mass dependence [4] and the
finite volume dependence [5] have been calculated. Since our results for
different lattice spacings do not exhibit clear discretization effects we
combine all our results where $m_{\mathrm{PS}}\leq 450\,\mbox{MeV}$. The fit
range has been chosen such that stable fits are obtained. Our data is not
sufficiently precise to determine all parameters. We therefore fix a few
parameters to their phenomenological value and keep only $g_{\mathrm{A}}$ in
the chiral limit, the leading $\Delta\Delta$-coupling $g_{1}$ and the SSE
coupling term $B_{9}^{r}(\lambda)$ as free fit parameters. The resulting fit
and the lattice data are shown in Fig. 2a.

In our fit we only included results for the largest lattice at a given set of
bare parameters. For some data sets we have results for different volumes. We
thus can compute the relative shift

$\delta_{g_{\mathrm{A}}}(L)=\frac{g_{\mathrm{A}}(L)-g_{\mathrm{A}}(\infty)}{g_{\mathrm{A}}(\infty)}$
(1)

both from the fit as well as from the lattice data taking the results on the
largest lattice as approximation of $g_{\mathrm{A}}(\infty)$. In Fig. 2b we
compare the relative shift for different values of the quark mass with our
lattice results at $m_{\mathrm{PS}}\simeq 270\,\mbox{MeV}$. The shift
predicted from ChEFT only slightly underestimates the relative shift computed
on the lattice.

Also after correcting for finite size effects we observe a significant
difference to the experimental value. It is interesting to notice that a much
better agreement with the experimental value is observed for the ratio
$g_{\mathrm{A}}/f_{\mathrm{PS}}$ (see Fig. 3a). In this ratio the
renormalization constant $Z_{\mathrm{A}}$ drops out.

In Fig. 3b we show our results for the nucleon tensor charge $\langle
1\rangle_{\delta q}=g_{\mathrm{T}}$. We observe only a very mild quark mass
dependence and the data reveals no systematic discretization effects. This
quantity is not well known experimentally. Our values are larger than the
phenomenological results presented in [6].

(a)

(b)

Figure 3: The left panel shows $g_{\mathrm{A}}/f_{\mathrm{PS}}$ as a function
of $m_{\mathrm{PS}}^{2}$. The right panel shows our results for
$g_{\mathrm{T}}^{\overline{\mathrm{MS}}}$ at a scale $\mu=2\,\mbox{GeV}$.

##  4 $n=2$ moments of PDFs

The lowest moment of the unpolarized PDF $\langle x\rangle_{q}=v_{2}$
corresponds to the momentum fraction carried by the quarks in the nucleon.
Lattice results from different collaborations tend to be significantly larger
than the phenomenological value. Fig. 4a and 4b show our most recent results
for the iso-vector and iso-scalar channel. In the latter case disconnected
contributions have been ignored.

Also shown are the results from a fit to results utilizing methods of
covariant Baryon Chiral Perturbation Theory (BChPT) [7]. Fits have been
performed with most parameters fixed to phenomenologically known values. The
iso-vector (iso-scalar) channel data is fitted with only 2 free parameters:
$v_{2}$ in the chiral limit and the coupling $c_{8}$ ($c_{9}$). Near the
physical light quark masses, BChPT predicts $v_{2}$ to become larger when the
quark mass becomes heavier. In our data for $m_{\mathrm{PS}}\lesssim
250\,\mbox{MeV}$ we do not see any indication for a bending down when
approaching the physical pion mass. It thus does not seem that a lack of
results at sufficiently small quark masses could explain the large discrepancy
between the phenomenological value and the lattice results. There are some
indications that part of the discrepancy can be explained by excited state
contamination [8].

(a)

(b)

Figure 4: The left and right panel show results for the second moment of the
iso-vector and iso-scalar unpolarized PDFs, respectively, as a function of
$m_{\mathrm{PS}}^{2}$. The solid lines show the fits to an expression from
ChEFT.

In Fig. 5a the results for the second moment of the polarized PDF $\langle
x\rangle_{\Delta q}=a_{1}$ is shown. Discretization effects again seem to be
absent in data. From a comparison of the results for different volumes it
seems that also finite size effects are small. Results from Heavy Baryon
Chiral Perturbation Theory (HBChPT) [9] lead to the following expression:

$a_{1}^{(u-d)}(m_{\mathrm{PS}})=C\left[1-\frac{4g_{\mathrm{A}}^{2}+1}{2(4\pi
f_{\mathrm{PS}})^{2}}m_{\mathrm{PS}}^{2}\ln\left(\frac{m_{\mathrm{PS}}^{2}}{\mu^{2}}\right)\right]+\cdots$
(2)

In Fig. 5a we plot this expression using $\mu=m_{\mathrm{N}}$ and $C$ chosen
such that it matches the phenomenological value. The bending down which we
observe in our data for $m_{\mathrm{PS}}\lesssim 0.5\,\mbox{MeV}$ is much less
than one would expect from this HBChPT result.

##  5 Electromagnetic form factors

(a)

(b)

Figure 5: The left panel shows the results for the second of the polarized
PDFs as a function of $m_{\mathrm{PS}}^{2}$. In the right panel the results
for the Dirac form factor radius $r_{1}$ are plotted. The dashed lines show
results from ChEFT as described in the text.

To compute the electromagnetic form factors one makes use of the standard
decomposition of the nucleon electromagnetic matrix elements $\langle
p^{\prime},s^{\prime}|V_{\mu}|p,s\rangle=\overline{u}\left[\gamma_{\mu}F_{1}(Q^{2})+\frac{\sigma_{\mu\nu}\,q_{\nu}}{2m_{\mathrm{N}}}F_{2}(Q^{2})\right]u$,
(in Euclidian space) where we use the local vector current $V_{\mu}$. $p$
($s$) and $p^{\prime}$ ($s^{\prime}$) denote initial and final momenta
(spins), $q=p^{\prime}-p$ the momentum transfer (with $Q^{2}=-q^{2}$). To
calculate form factor radii and the anomalous magnetic we have to parametrize
the lattice results. Here we use the ansatz

$F_{i}(Q^{2})=\frac{F_{i}(0)}{\left[1+\frac{Q^{2}}{pm_{i}^{2}}\right]^{p}}$
(3)

with $p=2$ and $p=3$ for the Dirac and Pauli form factors $F_{1}$ and $F_{2}$,
respectively. Our data is not sufficiently precise to favour a particular
parametrization (see [10] for another parametrization).

From fits to Eq. (3) we determine the form factor radii $r_{1}$ and $r_{2}$ as
well as the anomalous magnetic moment $\kappa$. The quark mass dependence of
these quantities has been calculated using the SSE formalism [11]. For $r_{1}$
the parameters are known and we therefore restrict ourselves to a comparison
of the SSE result and the lattice data (see Fig. 5b). While for
$m_{\mathrm{PS}}\gtrsim 300\,\mbox{MeV}$ the lattice results are significantly
smaller than the phenomenological value, towards smaller quark masses we
observe an increase of the radius. This is consistent with predictions from
ChEFT. For $r_{2}$ and $\kappa$ we find a similar behaviour. Since there are
no phenomenological estimates for all parameters of the SSE expressions we
perform a combined fit. The results are plotted in 6a and 6b.

(a)

(b)

Figure 6: The left panel and right panel shows the results for the Pauli
radius $r_{2}$ and the anomalous magnetic moment $\kappa$. The solid lines
show fits to an expression from ChEFT.

##  6 Summary and outlook

We have presented an update of QCDSF results on the lowest moments of
unpolarized, polarized and tensor PDFs as well as the electromagnetic form
factors. Some of our results at light quark masses with
$m_{\mathrm{PS}}\lesssim 300\,\mbox{MeV}$ confirm the expectations from ChEFT
that light quark mass effects are significant. However, this possibly does not
explain all of the observed discrepancies from phenomenological values.

## Acknowledgements

The numerical calculations have been performed on the APEmille and apeNEXT
systems at NIC/DESY (Zeuthen), the BlueGene/P at NIC/JSC (Jülich), the
BlueGene/L at EPCC (Edinburgh), the Dell PC-cluster at DESY (Zeuthen), the
QPACE systems [12] of the SFB TR-55, the SGI Altix and ICE systems at LRZ
(Munich) and HLRN (Berlin/Hannover). This work was supported in part by the
DFG (SFB TR-55) and by the European Union (grants 238353, ITN STRONGnet and
227431, HadronPhysics2, and 256594).

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