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PERSPECTIVES

3. C. M. Grzywacz, Monitoring for Gaseous Pollutants in

Museum Environments (Getty Publications, Los Angeles,

CA, 2006).

4. F. Flieder, C. Capderou, Sauvegarde des Collections du

Patrimoine (CNRS Editions, Paris, 1999).

5. M. Strli , J. Kolar, Eds., Ageing and Stabilisation of Paper

(National and University Library, Ljubljana, 2005).

6. W. Kautek et al., Proc. SPIE 4402, 130 (2001).

7. P. Baglioni, R. Giorgi, Soft Matter 2, 293 (2006).

8. J. G. Neevel, Restaurator 16, 143 (1995).

9. J. Kolar et al., e-PS 4, 19 (2007).

10. J. Male i et al., e-PS 2, 13 (2005).

11. T. Doering, Altes Papier–Neue Techniken, Zerstörgsfreie

Untersuchungen von Papier mit Festphasenmikroextraction

(SPME) (Berliner Wissenschafts-Verlag, Berlin,

2007).

12. M. Strli et al., e-PS 1, 35 (2004).

13. M. Strli et al., in Ageing and Stabilisation of Paper,

M. Strli , J. Kolar, Eds. (National and University Library,

Ljubljana, 2005), pp. 133–148.

14. J. Wouters, Chem. Int. 30, 4 (2008).

15. International Council of Museums–Committee for

Conservation (ICOM-CC), Terminology to Characterize

the Conservation of Tangible Cultural Heritage (2008);

www.icom-cc.icom.museum.

16. J. Levin, J. Druzik, Conservation, the Getty Conserv. Inst.

Newsl. 22(3), 10 (2007).

10.1126/science. 1164991

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cv

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cv

The reason for excitement surrounding

the start-up of the Large Hadron

Collider (LHC) in Geneva, Switzer-

land, has often been conveyed to the general

public as the quest for the origin of mass—

which is true but incomplete. Almost all of

the mass (or weight) of the world we live in

comes from atomic nuclei, which are com-

posed of neutrons and protons (collectively

called “nucleons”). Nucleons, in turn, are

composed of particles called quarks and glu-

ons, and physicists have long believed that

the nucleon’s mass comes from the compli-

cated way in which gluons bind the quarks to

each other, according to the laws of quantum

chromodynamics (QCD). A challenge since

the introduction of QCD (1–3) has been

to carry out an ab initio calculation of the

nucleon’s mass. On page 1224 of this issue,

Dürr et al. (4) report the first such calculation

that incorporates all of the needed physics,

controls the numerical approximations, and

presents a thorough error budget. Because

these accurate calculations agree with labora-

tory measurements, we now know, rather than

just believe, that the source of mass of every-

day matter is QCD.

The key tool enabling this advance is lat-

tice gauge theory (5), a formulation of QCD

and similar quantum field theories that

replaces space-time with a four-dimensional

lattice. To picture the lattice, think of a crystal

with cubic symmetry evolving in discrete time

steps. Lattice gauge theory has theoretical and

computational advantages. The watershed

theoretical result of QCD is the weakening at

short distances of the coupling between

quarks and gluons, called asymptotic freedom

(2, 3). The flip side is the strengthening of the

coupling at large distances, which is responsi-

ble for confining quarks and gluons inside one

of the broad class of particles called hadrons.

Confinement emerges naturally in lattice

gauge theory at strong coupling (5). The lat-

tice also reduces everything we would want to

calculate to integrals that, in principle, can be

evaluated numerically on a computer. Thus,

30 years ago a challenge was set: Use numeri-

cal computations to connect strongly coupled

lattice gauge theory with weakly coupled

asymptotic freedom, thereby recovering the

hadron masses of our world, with continuous

space and time.

The first numerical efforts (6) showed this

approach to be sound, but as the subject

developed, two major obstacles arose, both

connected to physics and to compu-

tation. The first obstacle is describ-

ing the “vacuum.” In classical

physics, the vacuum has nothing in

it (by definition), but in quantum

field theories, such as QCD, the vac-

uum contains “virtual particles” that

flit in and out of existence. In partic-

ular, the QCD vacuum is a jumble of

gluons and quark-antiquark pairs, so

to compute accurately in lattice

QCD, many snapshots of the vac-

uum are needed.

The second obstacle is the

extremely high amount of computa-

tion needed to incorporate the influ-

ence of the quark-antiquark pairs on

the gluon vacuum. The obstacle

heightens for small quark masses,

and the masses of the up and down

quarks are very small. An illustra-

tion of the vacuum and the quark-

antiquark pairs is shown in the fig-

ure. The fluid material is a scientifi-

cally accurate snapshot, sometimes

called the QCD lava lamp, of a typical gluon

field drawn from a lattice-QCD computation

(7). Three of the quarks (up + up + down, or

uud for short) could constitute the proton, but

a strange quark (s) and strange (s) antiquark

have popped out of the vacuum: The proton

has fluctuated into a Λ hyperon (uds) and a

kaon (su).

To make progress despite limited com-

puting power, 20 years’worth of lattice QCD

calculations were carried out omitting the

extra quark-antiquark pairs. The computa-

tion of the nucleon’s mass passed some tech-

nical milestones (8, 9) but was still unsatis-

factory. As well as demonstrating the validity

of strongly coupled QCD, we want to com-

pute properties of hadrons ab initio, to help

Ab initio calculations of the proton and

neutron masses have now been achieved, a

milestone in a 30-year effort of theoretical

and computational physics.

The Weight of the World IsQuantum ChromodynamicsAndreas S. Kronfeld

PHYSICS

Theoretical Physics Group, Fermi National AcceleratorLaboratory, Batavia, IL 60510, USA. E-mail: ask@ fnal.gov

The busy world of the QCD vacuum. The interior of a hadron, suchas the proton or neutron, is not static. Gluons fluctuate in a collec-tive fashion, illustrated by the red-orange-yellow-green-blue fluid.Sometimes the gluon field produces extra quark-antiquark pairs;here, a proton (uud) has fluctuated into a Λ hyperon (uds) and akaon (su). Animations of these phenomena are available at (7).

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interpret experiments in particle and in

nuclear physics. Without the quark-antiquark

pairs, it is impossible to quantify the associ-

ated uncertainty.

A breakthrough came 5 years ago, with the

first wide-ranging calculations incorporating

the back-reaction of up, down, and strange

quark pairs (10, 11). This work used a mathe-

matical representation of quarks that is rela-

tively fast to implement computationally (12),

and these methods enjoyed several notewor-

thy successes, such as predicting some then-

unmeasured hadron properties (13). This for-

mulation is, however, not well suited to the

nucleon, and so a principal task for lattice

QCD remained unfinished.

Dürr et al. use a more transparent formula-

tion of quarks that is well suited to the nucleon

and other baryons (hadrons composed of three

quarks). They compute the masses of eight

baryons and four mesons (hadrons composed

of one quark and one antiquark). Three of

these masses are used to fix the three free

parameters of QCD. The other nine agree

extremely well with measured values, in most

cases with total uncertainty below 4%.

For example, the nucleon mass is com-

puted to be 936 ± 25 ± 22 MeV/c2 compared

with 939 MeV/c2 for the neutron, where c is

the speed of light and the reported errors are

the statistical and systematic uncertainties,

respectively. The final result comes after care-

ful extrapolation to zero lattice spacing and to

quark masses as small as those of up and down

(the two lightest quarks, with masses below 6

MeV/c2). The latter extrapolation may not be

needed in the future. Last July, a Japanese col-

laboration announced a set of lattice-QCD

calculations (14) of the nucleon and other

hadron masses with quark masses as small as

those of up and down.

These developments are serendipitously

connected to the work honored by this year’s

Nobel Prize in physics. The lightest hadron—

the pion—has a mass much smaller than the

others. Before QCD, Nambu (15) proposed

that this feature could be understood as a con-

sequence of the spontaneous breaking of chiral

symmetry (16). In QCD, it has been believed,

the spontaneous breaking of this symmetry by

the vacuum predominates over an explicit

breaking that is small, because the up and

down quarks’ masses are so small. Lattice

QCD (4, 10, 11, 13, 14) simulates and, we see

now, validates these dynamical ideas in the

computer. Moreover, this success puts us in a

position to aid and abet the understanding of

the role of quark flavor (17), including asym-

metries in the laws of matter and antimatter

(18), for which Kobayashi and Maskawa

received their share of the Nobel Prize.

Dürr et al. start with QCD’s defining equa-

tions and present a persuasive, complete, and

direct demonstration that QCD generates the

mass of the nucleon and of several other

hadrons. These calculations teach us that even

if the quark masses vanished, the nucleon

mass would not change much, a phenomenon

sometimes called “mass without mass” (19,

20). It then raises the question of the origin of

the tiny up and down quark masses. The way

nature generates these masses, and the even

tinier electron mass, is the subject of the LHC,

where physicists will explore whether the

responsible mechanism is the Higgs boson or

something more spectacular.

References and Notes

1. H. Fritzsch, M. Gell-Mann, H. Leutwyler, Phys. Lett. B 47,365 (1973).

2. D. J. Gross, F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).

3. H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).4. S. Dürr et al., Science 322, 1224 (2008).5. K. G. Wilson, Phys. Rev. 10, 2445 (1974).6. M. Creutz, Phys. Rev. Lett. 45, 313 (1980).7. D. Leinweber, “Visualizations of Quantum

Chromodynamics (www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel).

8. F. Butler, H. Chen, J. Sexton, A. Vaccarino, D. Weingarten,Nucl. Phys. B 430, 179 (1994).

9. A. Ali Khan et al. (CP-PACS Collaboration), Phys. Rev. D

65, 054505 (2002).10. C. T. H. Davies et al. (HPQCD, MILC, and Fermilab Lattice

Collaborations), Phys. Rev. Lett. 92, 022001 (2004).11. C. Aubin et al. (MILC Collaboration), Phys. Rev. D 70,

094505 (2004).12. C. Bernard et al. (MILC Collaboration), Phys. Rev. D 64,

054506 (2001).13. A. S. Kronfeld, J. Phys. Conf. Ser. 46, 147 (2006).14. S. Aoki et al. (PACS-CS Collaboration), http://arXiv.org/

abs/0807.1661 (2008).15. Y. Nambu, Phys. Rev. Lett. 4, 380 (1960).16. J. Goldstone, Nuovo Cim. 19, 154 (1961).17. N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).18. M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49, 652

(1973).19. F. Wilczek, Phys. Today 52, 11 (November 1999).20. F. Wilczek, Phys. Today 53, 13 (January 2000).21. Fermilab is operated by Fermi Research Alliance LLC,

under Contract DE-AC02-07CH11359 with the U.S.Department of Energy.

10.1126/science.1166844

www.sciencemag.org SCIENCE VOL 322 21 NOVEMBER 2008 1199

PERSPECTIVES

Insects use a variety of strategies to fight

pathogens at different stages of infection,

which may guide antimicrobial development

for human use.Rogue Insect ImmunityDavid S. Schneider and Moria C. Chambers

MICROBIOLOGY

Two recent studies have quietly and sub-

versively broken the models we’ve

used to describe insect immunity.

Impressively, they’ve accomplished this by

using gross observational studies rather than

mechanistic approaches. On page 1257 in this

issue, Haine et al. (1) suggest that what we’ve

considered the central pillar of insect immu-

nity—antimicrobial peptides—may perform

a “mopping up” role in clearing pathogens.

Hedges et al. (2) show that heritable epige-

netic properties can have as large an impact on

insect immunity as any genetically encoded

pathway yet tested. Both studies teach us

important lessons about the way a host organ-

ism interacts with microbes and may have

immediate practical applications.

We often study host-pathogen interactions

in insects to model human infections. A good

illustration of this approach was the discovery

that the immune-activated signaling pathway

mediated by the Toll receptor in the fly

Drosophila melanogaster is partially con-

served by a family of Toll-like receptors in

humans (3, 4). This is now a cornerstone of the

field, and much of the recent work in insects

has involved further dissection of the molecu-

lar details of the Toll pathway and the related

Immune deficiency (Imd) pathway (5). When

activated by microbes, these two pathways

induce the massive production of antimicro-

bial peptides by host cells, and the loss of sig-

naling that is caused by mutations in pathway

components deeply compromises the fly’s

immune response (6).

Haine et al. show that the vast majority of

bacteria are cleared from an insect before

antimicrobial peptides are produced. In the

mealworm beetle, Tenebrio molitor, imme-

diate-acting immune responses, such as

engulfment of bacteria by phagocytes (spe-

cialized immune cells) and melanization (in

which bacteria are killed by reactive oxy-

gen) likely do most of the heavy lifting when

it comes to clearing microbes; more than

99.5% of injected bacteria are cleared from

beetles, in the first hour of infection, before

Department of Microbiology and Immunology, StanfordUniversity, CA 94305–5124, USA. E-mail: dschneider@stanford.edu

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23. R. W. Romani, Astrophys. J. 470, 469 (1996).24. P. Goldreich, W. H. Julian, Astrophys. J. 157, 869 (1969).25. Data provided by EGRET; ftp://legacy.gsfc.nasa.gov/

compton/data/egret/.26. F. Lucarelli et al., Nucl. Instr. Meth. A 589, 415 (2008).27. A. K. Harding, private communication.28. We thank the electronics division at the Max-Planck-Insitut,

Munich, for their work in developing and producing theanalog sum trigger system, especially O. Reimann, R. Maier,S. Tran, and T. Dettlaff. We also thank L. Stodolsky forcomments. We acknowledge the Instituto de Astrofísica forproviding all infrastructure on the Roque de los Muchachosin La Palma. The support of the German Bundesministeriumfür Bildung, Wissenschaft, Forschung und Technologie andMax-Planck-Gesellschaft, the Italian INFN and INAF, theSwiss Schweizerische Nationalfonds, and Spanish Ministeriode Ciencia e Innovación is acknowledged. This work was alsosupported by ETH research grant TH 34/043, by the PolishMinistertwo Nauki i Szkolnictwa Wyższego grant N N203390834, and by the Young Investigators Program of theHelmholtz Gemeinschaft.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/1164718/DC1SOM TextFigs. S1 to S11References

15 August 2008; accepted 8 October 2008Published online 16 October 2008;10.1126/science.1164718Include this information when citing this paper.

List of authors and affiliations:E. Aliu,1 H. Anderhub,2 L. A. Antonelli,3 P. Antoranz,4 M. Backes,5

C. Baixeras,6 J. A. Barrio,4 H. Bartko,7 D. Bastieri,8 J. K. Becker,5

W. Bednarek,9 K. Berger,10 E. Bernardini,11 C. Bigongiari,8†

A. Biland,2 R. K. Bock,7,8 G. Bonnoli,12 P. Bordas,13 V. Bosch-Ramon,13

T. Bretz,10 I. Britvitch,2 M. Camara,4 E. Carmona,7 A. Chilingarian,14

S. Commichau,2 J. L. Contreras,4 J. Cortina,1 M. T. Costado,15,16

S. Covino,3 V. Curtef,5 F. Dazzi,8 A. De Angelis,17 E. De Cea del Pozo,18

R. de los Reyes,4 B. De Lotto,17 M. De Maria,17 F. De Sabata,17

C. Delgado Mendez,15 A. Dominguez,19 D. Dorner,10 M. Doro,8

D. Elsässer,10M. Errando,1M. Fagiolini,12 D. Ferenc,20 E. Fernandez,1

R. Firpo,1 M. V. Fonseca,4 L. Font,6 N. Galante,7 R. J. GarciaLopez,15,16 M. Garczarczyk,7 M. Gaug,15 F. Goebel,7 D. Hadasch,5

M. Hayashida,7 A. Herrero,15,16 D. Höhne,10 J. Hose,7 C. C. Hsu,7

S. Huber,10 T. Jogler,7 D. Kranich,2 A. La Barbera,3 A. Laille,20 E.Leonardo,12 E. Lindfors,21 S. Lombardi,8 F. Longo,17 M. Lopez,8‡E. Lorenz,2,7 P. Majumdar,7 G. Maneva,22 N. Mankuzhiyil,17 K.Mannheim,10 L. Maraschi,3 M. Mariotti,8 M. Martinez,1 D. Mazin,1

M. Meucci,12 M. Meyer,10 J. M. Miranda,4 R. Mirzoyan,7 M. Moles,19

A. Moralejo,1 D. Nieto,4 K. Nilsson,21 J. Ninkovic,7 N. Otte,23,7‡§I. Oya,4 R. Paoletti,12 J. M. Paredes,13 M. Pasanen,21 D. Pascoli,8

F. Pauss,2 R. G. Pegna,12 M. A. Perez-Torres,19 M. Persic,24 L.Peruzzo,8 A. Piccioli,12 F. Prada,19 E. Prandini,8 N. Puchades,1 A.Raymers,14W. Rhode,5 M. Ribó,13 J. Rico,1,25 M. Rissi,2‡ A. Robert,6

S. Rügamer,10 A. Saggion,8 T. Y. Saito,7 M. Salvati,3 M. Sanchez-Conde,19 P. Sartori,8 K. Satalecka,11 V. Scalzotto,8 V. Scapin,17 T.Schweizer,7‡M. Shayduk,7‡ K. Shinozaki,7 S. N. Shore,26 N. Sidro,1

A. Sierpowska-Bartosik,18 A. Sillanpää,21 D. Sobczynska,9 F.Spanier,10 A. Stamerra,12 L. S. Stark,2 L. Takalo,21 F. Tavecchio,3

P. Temnikov,22 D. Tescaro,1 M. Teshima,7 M. Tluczykont,11 D. F.Torres,18,25 N. Turini,12 H. Vankov,22 A. Venturini,8 V. Vitale,17 R. M.Wagner,7 W. Wittek,7 V. Zabalza,13 F. Zandanel,19 R. Zanin,1 J.Zapatero,5 O.C. de Jager,27∥ E. de Ona Wilhelmi1∥¶

1Institut de Física d'Altes Energies, Edifici Cn, CampusUniversitatAutònoma de Barcelona E-08193 Bellaterra, Spain. 2Eidgen-össische Technische Hochschule (ETH), Zürich CH-8093, Switzer-land. 3L'Istituto Nazionale di Astrofisica (INAF), I-00136 Rome,Italy. 4Universidad Complutense, E-28040 Madrid, Spain.5Technische Universität Dortmund, D-44221 Dortmund, Ger-many. 6Universitat Autònoma de Barcelona, E-08193 Bellaterra,

Spain. 7Max-Planck-Institut für Physik, D-80805 München,Germany. 8Università di Padova and Istituto Nazionale di FisicaNucleare (INFN), I-35131 Padova, Italy. 9University of Lodz,PL-90236 Lodz, Poland. 10Universität Würzburg, D-97074Würzburg, Germany. 11Deutsches Elektronen Synchrotron, D-15738 Zeuthen, Germany. 12Università di Siena and INFN Pisa,I-53100 Siena, Italy. 13Universitat de Barcelona, Institut deCiencias del Cosmos (ICC)/Institut d'Estudis Espacials deCatalunya (IEEC), E-08028 Barcelona, Spain. 14Yerevan PhysicsInstitute, AM-375036 Yerevan, Armenia. 15Instituto de Astro-física de Canarias, E-38200, La Laguna,Tenerife, Spain.16Departamento de Astrofisica, Universidad, E-38206 LaLaguna, Tenerife, Spain. 17Università di Udine and INFN Trieste,I-33100 Udine, Italy. 18IEEC–Consejo Superior de Investiga-ciones Cientificas (CSIC), E-08193 Bellaterra, Spain. 19Institutode Astrofisica de Andalucia (CSIC), E-18080 Granada, Spain.20University of California at Davis, Davis, CA 95616–8677, USA.21Tuorla Observatory, Turku University, FI-21500 Piikkiö,Finland. 22Institute for Nuclear Research and Nuclear Energy,BG-1784 Sofia, Bulgaria. 23Humboldt-Universität zu Berlin, D-12489 Berlin, Germany. 24INAF/Osservatorio Astronomico andINFN, I-34143 Trieste, Italy. 25Institució Catalana de Recerca iEstudis Avançats, E-08010 Barcelona, Spain. 26Università di Pisaand INFN Pisa, I-56126 Pisa, Italy. 27Unit for Space Physics,Northwest University, Potchefstroom 2520, South Africa.

†Present address: Instituto de Física Corpuscular, CSIC–Universitatde València, E-46071 Valencia, Spain.‡To whom correspondence should be addressed. E-mail:tschweiz@mppmu.mpg.de (T.S.); nepomake@scipp.ucsc.edu (N.O.);Michael.Rissi@phys.ethz.ch (M.R.); shayduk@mppmu.mpg.de(M.S.); marcos.lopezmoya@pd.infn.it (M.L.M.)§Present address: Santa Cruz Institute for Particle Physics, Universityof California, Santa Cruz, CA 95064, USA.∥These authors are not members of the MAGIC Collaboration.¶Present address: Astroparticule et Cosmologie, CNRS, UniversiteParis, F-75205 Paris Cedex 13, France.

Ab Initio Determination ofLight Hadron MassesS. Dürr,1 Z. Fodor,1,2,3 J. Frison,4 C. Hoelbling,2,3,4 R. Hoffmann,2 S. D. Katz,2,3S. Krieg,2 T. Kurth,2 L. Lellouch,4 T. Lippert,2,5 K. K. Szabo,2 G. Vulvert4

More than 99% of the mass of the visible universe is made up of protons and neutrons. Bothparticles are much heavier than their quark and gluon constituents, and the Standard Model ofparticle physics should explain this difference. We present a full ab initio calculation of themasses of protons, neutrons, and other light hadrons, using lattice quantum chromodynamics.Pion masses down to 190 mega–electron volts are used to extrapolate to the physical point,with lattice sizes of approximately four times the inverse pion mass. Three lattice spacings areused for a continuum extrapolation. Our results completely agree with experimentalobservations and represent a quantitative confirmation of this aspect of the Standard Modelwith fully controlled uncertainties.

The Standard Model of particle physicspredicts a cosmological, quantum chromo-dynamics (QCD)–related smooth transi-

tion between a high-temperature phase dominatedby quarks and gluons and a low-temperature phasedominated by hadrons. The very large energy den-sities at the high temperatures of the early universehave essentially disappeared through expansionand cooling. Nevertheless, a fraction of this energyis carried today by quarks and gluons, which areconfined into protons and neutrons. According tothe mass-energy equivalence E = mc2, we ex-perience this energy as mass. Because more than99% of the mass of ordinary matter comes fromprotons and neutrons, and in turn about 95% of

their mass comes from this confined energy, it isof fundamental interest to perform a controlled abinitio calculation based on QCD to determine thehadron masses.

QCD is a generalized version of quantum elec-trodynamics (QED), which describes the electro-magnetic interactions. The Euclidean Lagrangianwith gauge coupling g and a quark mass of mcan be written as L ¼ −1=ð2g2ÞTrFmuFmuþy½gmð∂m þ AmÞþmy, where Fmn=∂mAn− ∂nAm +[Am,An]. In electrodynamics, the gauge potentialAm is a real valued field, whereas in QCD it is a3 × 3matrix field. Consequently, the commutatorin Fmn vanishes in QED but not in QCD. The yfields also have an additional WcolorW index in

QCD, which runs from 1 to 3. Different WflavorsWof quarks are represented by independentfermionic fields, with possibly different masses.In the work presented here, a full calculation of thelight hadron spectrum in QCD, only three inputparameters are required: the light and strangequark masses and the coupling g.

The action S of QCD is defined as the four-volume integral of L. Green's functions areaverages of products of fields over all field con-figurations, weighted by the Boltzmann factorexp(−S). A remarkable feature of QCD is asymp-totic freedom, which means that for high ener-gies (that is, for energies at least 10 to 100 timeshigher than that of a proton at rest), the interac-tion gets weaker and weaker (1, 2), enabling per-turbative calculations based on a small couplingparameter. Much less is known about the otherside, where the coupling gets large, and the phys-ics describing the interactions becomes nonper-turbative. To explore the predictions of QCD inthis nonperturbative regime, the most systematicapproach is to discretize (3) the above Lagrangian

1John von Neumann–Institut für Computing, DeutschesElektronen-Synchrotron Zeuthen, D-15738 Zeuthen andForschungszentrum Jülich, D-52425 Jülich, Germany. 2BergischeUniversität Wuppertal, Gaussstrasse 20, D-42119 Wuppertal,Germany. 3Institute for Theoretical Physics, Eötvös University,H-1117 Budapest, Hungary. 4Centre de Physique Théorique(UMR 6207 du CNRS et des Universités d'Aix-Marseille I, d'Aix-Marseille II et du Sud Toulon-Var, affiliée à la FRUMAM), Case907, Campus de Luminy, F-13288, Marseille Cedex 9, France.5Jülich Supercomputing Centre, FZ Jülich, D-52425 Jülich,Germany.

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on a hypercubic space-time lattice with spacing a,to evaluate its Green's functions numerically andto extrapolate the resulting observables to the con-tinuum (a→0). A convenient way to carry out thisdiscretization is to place the fermionic variables onthe sites of the lattice, whereas the gauge fieldsare treated as 3 × 3 matrices connecting thesesites. In this sense, lattice QCD is a classical four-dimensional statistical physics system.

Calculations have been performed using thequenched approximation, which assumes thatthe fermion determinant (obtained after integrat-ing over the y fields) is independent of thegauge field. Although this approach omits themost computationally demanding part of a fullQCD calculation, a thorough determination ofthe quenched spectrum took almost 20 years. It

was shown (4) that the quenched theory agreedwith the experimental spectrum to approximately10% for typical hadron masses and demonstratedthat systematic differences were observed be-tween quenched and two-flavor QCD beyondthat level of precision (4, 5).

Including the effects of the light sea quarkshas dramatically improved the agreement be-tween experiment and lattice QCD results. Fiveyears ago, a collaboration of collaborations (6)produced results for many physical quantitiesthat agreedwell with experimental results. Thanksto continuous progress since then, lattice QCDcalculations can now be performed with light seaquarks whose masses are very close to their phys-ical values (7) (though in quite small volumes).Other calculations, which include these sea-quark

effects in the light hadron spectrum, have alsoappeared in the literature (8–16). However, all ofthese studies have neglected one or more of theingredients required for a full and controlled cal-culation. The five most important of those are, inthe order that they will be addressed below:

1) The inclusion of the up (u), down (d), andstrange (s) quarks in the fermion determinantwith an exact algorithm and with an actionwhose universality class is QCD. For the lighthadron spectrum, the effects of the heaviercharm, bottom, and top quarks are included inthe coupling constant and light quark masses.

2) A complete determination of the masses ofthe light ground-state, flavor nonsinglet mesonsand octet and decuplet baryons. Three of theseare used to fix the masses of the isospin-averagedlight (mud) and strange (ms) quark masses and theoverall scale in physical units.

3) Large volumes to guarantee small finite-size effects and at least one data point at asignificantly larger volume to confirm the small-ness of these effects. In large volumes, finite-sizecorrections to the spectrum are exponentiallysmall (17, 18). As a conservative rule of thumb,MpL >

e4, withMp the pionmass and L the lattice

size, guarantees that finite-volume errors in thespectrum are around or below the percent level(19). Resonances require special care. Their finitevolume behavior is more involved. The literatureprovides a conceptually satisfactory frameworkfor these effects (20, 21), which should be in-cluded in the analysis.

4) Controlled interpolations and extrapola-tions of the results to physical mud and ms (oreventually directly simulating at these massvalues). Although interpolations to physical ms,corresponding to MK ≅ 495 MeV, are straight-forward, the extrapolations to the physical value of

Fig. 2. Pion mass dependence of the nucleon (N) andW for all three values ofthe lattice spacing. (A) Masses normalized by MX, evaluated at thecorresponding simulation points. (B) Masses in physical units. The scale inthis case is set byMX at the physical point. Triangles on dotted lines correspondto a ≈ 0.125 fm, squares on dashed lines to a ≈ 0.085 fm, and circles on solidlines to a ≈ 0.065 fm. The points were obtained by interpolating the latticeresults to the physicalms (defined by setting 2MK

2 –Mp2 to its physical value).

The curves are the corresponding fits. The crosses are the continuumextrapolated values in the physical pion mass limit. The lattice-spacingdependence of the results is barely significant statistically despite the factor of3.7 separating the squares of the largest (a ≈ 0.125 fm) and smallest (a ≈0.065 fm) lattice spacings. The c2/degrees of freedom values of the fits in (A)are 9.46/14 (W) and 7.10/14 (N), whereas those of the fits in (B) are 10.6/14(W) and 9.33/14 (N). All data points represent the mean T SEM.

A B

Fig. 1. Effective massesaM = log[C(t/a)/C(t/a +1)], where C(t/a) is thecorrelator at time t, forp, K, N, X, and W at ourlightest simulation pointwithMp ≈ 190 MeV (a ≈0.085 fm with physicalstrange quark mass). Forevery 10th trajectory, thehadron correlators werecomputed with Gaussiansources and sinks whoseradii are approximately0.32 fm. The data pointsrepresent mean T SEM.The horizontal lines indi-cate the masses T SEM,obtained by performingsingle mass-correlated cosh/sinh fits to the individual hadron correlators with a method similar to thatof (29).

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mud, corresponding toMp ≅ 135MeV, are difficult.They need computationally intensive calculations,withMp reaching down to 200 MeVor less.

5) Controlled extrapolations to the contin-uum limit, requiring that the calculations beperformed at no less than three values of thelattice spacing, in order to guarantee that thescaling region is reached.

Our analysis includes all five ingredientslisted above, thus providing a calculation of thelight hadron spectrum with fully controlled sys-tematics as follows.

1) Owing to the key statement from renor-malization group theory that higher-dimension,local operators in the action are irrelevant in thecontinuum limit, there is, in principle, an un-limited freedom in choosing a lattice action.There is no consensus regarding which actionwould offer the most cost-effective approach tothe continuum limit and to physical mud. We usean action that improves both the gauge andfermionic sectors and heavily suppresses non-physical, ultraviolet modes (19). We perform aseries of 2 + 1 flavor calculations; that is, weinclude degenerate u and d sea quarks and anadditional s sea quark. We fix ms to its approxi-mate physical value. To interpolate to the phys-ical value, four of our simulations were repeatedwith a slightly different ms. We vary mud in arange that extends down to Mp ≈ 190 MeV.

2) QCD does not predict hadron masses inphysical units: Only dimensionless combinations(such as mass ratios) can be calculated. To set theoverall physical scale, any dimensionful observ-able can be used. However, practical issues in-fluence this choice. First of all, it should be aquantity that can be calculated precisely andwhose experimental value is well known. Sec-ond, it should have a weak dependence on mud,so that its chiral behavior does not interfere withthat of other observables. Because we are con-sidering spectral quantities here, these two con-ditions should guide our choice of the particlewhose mass will set the scale. Furthermore, theparticle should not decay under the strong in-teraction. On the one hand, the larger the strangecontent of the particle, the more precise the massdetermination and the weaker the dependence onmud. These facts support the use of theW baryon,the particle with the highest strange content. Onthe other hand, the determination of baryon dec-uplet masses is usually less precise than those ofthe octet. This observation would suggest thatthe X baryon is appropriate. Because both theW and X baryon are reasonable choices, wecarry out two analyses, one withMW (theW set)and one withMX (the X set). We find that for allthree gauge couplings, 6/g2 = 3.3, 3.57, and 3.7,both quantities give consistent results, namelya ≈ 0.125, 0.085, and 0.065 fm, respectively. Tofix the bare quark masses, we use the mass ratiopairs Mp/MW,MK/MW or Mp/MX,MK/MX. Wedetermine the masses of the baryon octet (N, S,L, X) and decuplet (D, S*, X*, W) and thosemembers of the light pseudoscalar (p, K) and

vector meson (r, K*) octets that do not requirethe calculation of disconnected propagators.Typical effective masses are shown in Fig. 1.

3) Shifts in hadron masses due to the finitesize of the lattice are systematic effects. Thereare two different effects, and we took both ofthem into account. The first type of volume de-pendence is related to virtual pion exchange be-tween the different copies of our periodic system,and it decreases exponentially with Mp L. UsingMpL >

e4 results in masses which coincide, for

all practical purposes, with the infinite volumeresults [see results, for example, for pions (22)and for baryons (23, 24)]. Nevertheless, for oneof our simulation points, we used several vol-umes and determined the volume dependence,which was included as a (negligible) correction atall points (19). The second type of volume de-pendence exists only for resonances. The cou-pling between the resonance state and its decayproducts leads to a nontrivial-level structure infinite volume. Based on (20, 21), we calculatedthe corrections necessary to reconstruct the reso-nance masses from the finite volume ground-state energy and included them in the analysis(19).

4) Though important algorithmic develop-ments have taken place recently [for example

(25, 26) and for our setup (27)], simulating di-rectly at physical mud in large enough volumes,which would be an obvious choice, is still ex-tremely challenging numerically. Thus, the stan-dard strategy consists of performing calculationsat a number of larger mud and extrapolating theresults to the physical point. To that end, we usechiral perturbation theory and/or a Taylor expan-sion around any of our mass points (19).

5) Our three-flavor scaling study (27) showedthat hadron masses deviate from their continuumvalues by less than approximately 1% for latticespacings up to a ≈ 0.125 fm. Because the sta-tistical errors of the hadron masses calculated inthe present paper are similar in size, we do notexpect significant scaling violations here. This isconfirmed by Fig. 2. Nevertheless, we quantifiedand removed possible discretization errors by acombined analysis using results obtained at threelattice spacings (19).

We performed two separate analyses, settingthe scale with MX and MW. The results of thesetwo sets are summarized in Table 1. The X set isshown in Fig. 3. With both scale-setting proce-dures, we find that the masses agree with thehadron spectrum observed in nature (28).

Thus, our study strongly suggests that QCDis the theory of the strong interaction, at low

Fig. 3. The light hadronspectrum of QCD. Hori-zontal lines and bands arethe experimental valueswith their decay widths.Our results are shown bysolid circles. Vertical errorbars represent our com-bined statistical (SEM) andsystematic error estimates.p, K, and X have no errorbars, because they areused to set the light quarkmass, the strange quarkmass and the overallscale, respectively.

Table 1. Spectrum results in giga–electron volts. The statistical (SEM) and systematic uncertaintieson the last digits are given in the first and second set of parentheses, respectively. Experimentalmasses are isospin-averaged (19). For each of the isospin multiplets considered, this average iswithin at most 3.5 MeV of the masses of all of its members. As expected, the octet masses are moreaccurate than the decuplet masses, and the larger the strange content, the more precise is theresult. As a consequence, the D mass determination is the least precise.

X Experimental (28) MX (X set) MX (W set)

r 0.775 0.775 (29) (13) 0.778 (30) (33)K* 0.894 0.906 (14) (4) 0.907 (15) (8)N 0.939 0.936 (25) (22) 0.953 (29) (19)L 1.116 1.114 (15) (5) 1.103 (23) (10)S 1.191 1.169 (18) (15) 1.157 (25) (15)X 1.318 1.318 1.317 (16) (13)D 1.232 1.248 (97) (61) 1.234 (82) (81)S* 1.385 1.427 (46) (35) 1.404 (38) (27)X* 1.533 1.565 (26) (15) 1.561 (15) (15)W 1.672 1.676 (20) (15) 1.672

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energies as well, and furthermore that latticestudies have reached the stage where all sys-tematic errors can be fully controlled. This willprove important in the forthcoming era in whichlattice calculations will play a vital role inunraveling possible new physics from processesthat are interlaced with QCD effects.

References and Notes1. D. J. Gross, F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).2. H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).3. K. G. Wilson, Phys. Rev. D Part. Fields 10, 2445 (1974).4. S. Aoki et al., Phys. Rev. Lett. 84, 238 (2000).5. S. Aoki et al., Phys. Rev. D Part. Fields 67, 034503 (2003).6. C. T. H. Davies et al., Phys. Rev. Lett. 92, 022001 (2004).7. S. Aoki et al., http://arxiv.org/abs/0807.1661 (2008).8. C. W. Bernard et al., Phys. Rev. D Part. Fields 64,

054506 (2001).9. C. Aubin et al., Phys. Rev. D Part. Fields Gravit. Cosmol.

70, 094505 (2004).

10. N. Ukita et al., Proc. Sci. LAT2007, 138 (2007).11. M. Gockeler et al., Proc. Sci. LAT2007, 129 (2007).12. D. J. Antonio et al., Phys. Rev. D Part. Fields Gravit.

Cosmol. 75, 114501 (2007).13. A. Walker-Loud et al., http://arxiv.org/abs/0806.4549

(2008).14. L. Del Debbio, L. Giusti, M. Luscher, R. Petronzio,

N. Tantalo, J. High Energy Phys. 02, 056 (2007).15. C. Alexandrou et al., http://arxiv.org/abs/0803.3190 (2008).16. J. Noaki et al., Proc. Sci. LAT2007, 126 (2007).17. M. Luscher, Commun. Math. Phys. 104, 177 (1986).18. M. Luscher, Commun. Math. Phys. 105, 153 (1986).19. See supporting material on Science Online.20. M. Luscher, Nucl. Phys. B 354, 531 (1991).21. M. Luscher, Nucl. Phys. B 364, 237 (1991).22. G. Colangelo, S. Durr, Eur. Phys. J. C 33, 543 (2004).23. A. Ali Khan et al., Nucl. Phys. B 689, 175 (2004).24. B. Orth, T. Lippert, K. Schilling, Phys. Rev. D Part. Fields

Gravit. Cosmol. 72, 014503 (2005).25. M. A. Clark, Proc. Sci. LAT2006, 004 (2006).26. W. M. Wilcox, Proc. Sci. LAT2007, 025 (2007).27. S. Durr et al., http://arxiv.org/abs/0802.2706 (2008).

28. W. M. Yao et al., J. Phys. G33, 1 (2006).29. C. Michael, A. McKerrell, Phys. Rev. D Part. Fields 51,

3745 (1995).30. Computations were performed on the Blue Gene

supercomputers at FZ Jülich and at IDRIS and on clustersat Wuppertal and CPT. This work is supported in part byEuropean Union (EU) grant I3HP; Országos TudományosKutátasi Alapprogramok grant AT049652; DeutscheForshungsgemeinschaft grants FO 502/1-2 and SFB-TR55; EU grants RTN contract MRTN-CT-2006-035482(FLAVIAnet) and (FP7/2007-2013)/ERC no. 208740; andthe CNRS's GDR grant 2921. Useful discussions withJ. Charles and M. Knecht are acknowledged.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/322/5905/1224/DC1SOM TextFigs. S1 to S5Tables S1 and S2References

14 July 2008; accepted 1 October 200810.1126/science.1163233

4D Imaging of Transient Structuresand Morphologies in UltrafastElectron MicroscopyBrett Barwick, Hyun Soon Park, Oh-Hoon Kwon, J. Spencer Baskin, Ahmed H. Zewail*

With advances in spatial resolution reaching the atomic scale, two-dimensional (2D) and 3Dimaging in electron microscopy has become an essential methodology in various fields of study.Here, we report 4D imaging, with in situ spatiotemporal resolutions, in ultrafast electronmicroscopy (UEM). The ability to capture selected-area-image dynamics with pixel resolution and tocontrol the time separation between pulses for temporal cooling of the specimen made possiblestudies of fleeting structures and morphologies. We demonstrate the potential for applications withtwo examples, gold and graphite. For gold, after thermally induced stress, we determined theatomic structural expansion, the nonthermal lattice temperature, and the ultrafast transients ofwarping/bulging. In contrast, in graphite, striking coherent transients of the structure wereobserved in both image and diffraction, directly measuring, on the nanoscale, the longitudinalresonance period governed by Young’s elastic modulus. The success of these studies demonstratesthe promise of UEM in real-space imaging of dynamics.

Electrons, because of their wave-particleduality, can be accelerated to have pico-meter wavelength and focused to image

in real space (1). With the impressive advancesmade in transmission electronmicroscopy (TEM),augmented by scanning and aberration-correctionfeatures, it is now possible to image with highresolution (2–7), reaching the sub-angstrom scale.Together with the progress made in electron crys-tallography, tomography, and single-particle im-aging (8–13), today the electron microscope indifferent variants of two-dimensional (2D) and3D recordings has become a central tool in manyfields, frommaterials science to biology (14–16).For all conventional microscopes, the electronsare generated either thermally by heating the

cathode or by field emission, and as such theelectron beam is made of random single-electronbursts with no control over the temporal behav-ior. In these microscopes, time resolution ofmilliseconds or longer, being limited by the videorate of the detector, can be achieved, while main-taining the high spatial resolution, as demon-strated in environmental-TEM studies (17).

Ultrafast imaging, using pulsed photoelectronpackets, provides opportunities for studying, inreal space, the elementary processes of structuraland morphological changes. In electron diffrac-tion, ultrashort time resolution is possible (18),but the data are recorded in reciprocal space.With nanosecond and submicron image resolu-tions (19, 20) limited by space charge, ultrashortprocesses cannot be observed. To achieve ultra-fast resolution in microscopy, the concept ofsingle-electron pulse imaging (18) was realizedas a key to the elimination of the Coulombrepulsion between electrons while maintainingthe high temporal and spatial resolutions. As long

as the number of electrons in each pulse is belowthe space-charge limit, the packet can have a fewor tens of electrons, and the temporal resolution isstill determined by the femtosecond (fs) opticalpulse duration and the energy uncertainty, whichis also on the fs time scale (21), and the spatialresolution is atomic scale (22). However, the goalof full-scale dynamic imaging can be attainedonly when, in the microscope, the problems of insitu high-spatiotemporal resolution for selectedimage areas and of heat dissipation (for reversibleprocesses) are overcome.

Here, we present the methodology of ultra-fast imaging with applications in studies ofstructural and morphological changes in single-crystal gold and graphite films, which exhibitentirely different dynamics. For both, the changeswere initiated by in situ fs impulsive heating,while image frames and diffraction patterns wererecorded in the microscope at well-defined timesafter the temperature jump. The time axis in themicroscope is independent of the response timeof the detector, and it is established using avariable delay-line arrangement; a 1-mm changein optical path of the initiating (clocking) pulsecorresponds to a time step of 3.3 fs.

Shown in Fig. 1 is a picture of the second-generation ultrafast electron microscope (UEM-2)built at the California Institute of Technology(Caltech). The integration of two laser systemsto a modified electron microscope is indicated inthe figure, together with a representative imageshowing the resolution of a 3.4 Å lattice spacingobtained in UEM without the field-emission-gun (FEG) arrangement of conventional TEM.In the figure, the fs laser system is used to gen-erate the single-electron packets, whereas the nslaser system was used for both single-shot andstroboscopic recordings (23). In the single-electronmode of operation, as in UEM-1 (24), the co-herence volume is well defined and appropriatefor image formation in repetitive events (25). Thedynamics are fully reversible, retracing an identi-cal evolution after each initiating laser pulse; eachimage is constructed stroboscopically, in seconds,

Physical Biology Center for Ultrafast Science and Technology,Arthur Amos Noyes Laboratory of Chemical Physics, CaliforniaInstitute of Technology, Pasadena, CA 91125, USA.

*To whom correspondence should be addressed. E-mail:zewail@caltech.edu

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Supplementary Online MaterialAb-initio Determination of Light Hadron Masses

S. Durr1, Z. Fodor1,2,3, J. Frison4, C. Hoelbling2,3,4,R. Hoffmann2, S. D. Katz2,3, S. Krieg2, T. Kurth2,

L. Lellouch4, T. Lippert2,5, K.K. Szabo2, G. Vulvert4

1NIC, DESY Zeuthen, D-15738 Zeuthen and FZ Julich, D-52425 Julich, Germany.2Bergische Universitat Wuppertal, Gaussstr. 20, D-42119 Wuppertal, Germany.

3Institute for Theoretical Physics, Eotvos University, H-1117 Budapest, Hungary.4CPT∗, Case 907, Campus de Luminy, F-13288 Marseille Cedex 9, France.

5Julich Supercomputing Centre, FZ Julich, D-52425 Julich, Germany.

Budapest-Marseille-Wuppertal Collaboration

Details of the simulations

We use a tree-level,O(a2)-improved Symanzik gauge action (S1) and work with tree-level,clover-improved Wilson fermions, coupled to links which have undergone six levels of stoutlink averaging (S2). (The precise form of the action is presented in (S3).)

Simulation parameters, lattice sizes and trajectory lengths after thermalization are summa-rized in Table S1. Note, that we work on spatial volumes as large asL3≃(4 fm)3 and temporalextents up toT≃8 fm. Besides significantly reducing finite-volume corrections, this choice hasa similar effect on the statistical uncertainties of the results as increasing the number of trajecto-ries at fixed volume. For a given pion mass, this increase is proportional to the ratio of volumes.Thus, forT ∝ L, 1,300 trajectories atMπL=4 are approximately equivalent to 4,000 trajec-tories atMπL=3. (A factorL3 comes from the summation over the spatial volume required to

∗CPT is “UMR 6207 du CNRS et des universites d’Aix-MarseilleI, d’Aix-Marseille II et du Sud Toulon-Var,affiliee a la FRUMAM”.

1

project the hadron correlation functions onto the zero-momentum sector and an additionalLcomes from the fact that more timeslices are available for extracting the corresponding hadronmass.)

The integrated autocorrelation times of the smeared plaquette and that of the number ofconjugate gradient iteration steps are less than approximately ten trajectories. Thus every tenthtrajectory is used in the analysis. We calculate the spectrum by using up to eight timeslicesas sources for the correlation functions. For the precise form of the hadronic operators seee.g. (S4). We find that Gaussian sources and sinks of radii≈ 0.32 fm are less contaminated byexcited states than point sources/sinks (see Figure S1). The integrated autocorrelation times forhadron propagators, computed on every tenth trajectory, are compatible with 0.5 and no furthercorrelations were found through binning adjacent configurations. In order to exclude possiblelong-range correlations in our simulations, we performed arun with 10,000 and one with 4,500trajectories. No long-range correlations were observed. Further, we never encountered algo-rithmic instabilities as illustrated by the time history ofthe fermionic force in Figure S2 anddiscussed in more detail in (S3). Note that the fermionic force, which is the derivative of thefermionic action with respect to the gauge field, is directlyrelated to the locality properties ofour action (see Figure S3).

Finite volume corrections and resonances

For fixed bare parameters (gauge coupling, light quark mass and strange quark mass), the ener-gies of the different hadronic states depend on the spatial size of the lattice (in a finite volumethe energy spectrum is discrete and all states are stable). There are two sources of volume de-pendence, which we call type I and type II. These were discussed in a series of papers by M.Luscher (S5–S8). Both effects were quantified in a self-consistent manner in our analysis, usingonly the results of our calculations (i.e. no numerical inputs from experiments were used).

Type I effects result from virtual pion exchanges between the different copies of our peri-odic system. These effects induce corrections in the spectrum which fall off exponentially withMπL for large enough volumes (S5). For one set of parameters (Mπ≈320 MeV ata≈0.125 fm),additional runs have been carried out for several spatial volumes ranging fromMπL≈3.5 to7. The size dependences of the different hadron massesMX are successfully described byMX(L) = MX + cX(Mπ) · exp(−MπL)/(MπL)3/2. Figure S4 shows the volume dependenceatMπ=320 MeV for the two statistically most significant channels: the pion and nucleon chan-nels. The fittedcX coefficients are in good agreement with those suggested by (S9, S10) whichpredicts a behavior ofcX(Mπ) ∝M2

π . Our results for these and other channels confirm the ruleof thumb:MπL>∼4 gives the infinite volume masses within statistical accuracy. Nevertheless,we included these finite volume corrections in our analysis.

The other source of volume dependence (type II) is relevant only to resonant states, inregions of parameter space where they would decay in infinitevolume (five out of the twelveparticles of the present work are resonant states). Since inthis case the lowest energy state with

2

the quantum numbers of the resonance in infinite volume is a two particle scattering state, weneed to take the effects of scattering states into account inour analysis. For illustration we startby considering the hypothetical case where there is no coupling between the resonance (whichwe will refer to as “heavy state” in this paragraph) and the scattering states. In a finite boxof sizeL, the spectrum in the center of mass frame consists of two particle states with energy√

M21 + k

2 +√

M22 + k2, wherek = n2π/L, n ∈ Z3 andM1,M2 are the masses of the lighter

particles (with corrections of type I discussed in the previous paragraph) and, in addition, of thestate of the heavy particleMX (again with type I corrections). As we increaseL, the energy ofof any one of the two particle states decreases and eventually becomes smaller than the energyMX of X. An analogous phenomenon can occur when we fixL but reduce the quark mass (theenergy of the two light particles changes more thanMX). In the presence of interactions, thislevel crossing disappears and, due to the mixing of the heavystate and the scattering state, anavoided level crossing phenomenon is observed. Such mass shifts due to avoided level crossingcan distort the chiral extrapolation of hadron masses to thephysical pion mass.

The literature (S6–S8) provides a conceptually satisfactory basis to study resonances in lat-tice QCD: each measured energy corresponds to a momentum,|k|, which is a solution of acomplicated non-linear equation. Though the necessary formulae can be found in the litera-ture (cf. equations (2.7, 2.10-2.13, 3.4, A3) of (S8)), for completeness the main ingredients aresummarized here. We follow (S8) where theρ-resonance was taken as an example and it waspointed out that other resonances can be treated in the same way without additional difficulties.Theρ-resonance decays almost exclusively into two pions. The absolute value of the pion mo-mentum is denoted byk = |k|. The total energy of the scattered particles isW = 2(M2

π +k2)1/2

in the center of mass frame. Theππ scattering phaseδ11(k) in the isospinI = 1, spinJ = 1channel passes throughπ/2 at the resonance energy, which correspond to a pion momentumkequal tokρ = (M2

ρ/4 −M2π)1/2. In the effective range formula(k3/W ) · cot δ11 = a + bk2,

this behavior impliesa = −bk2ρ = 4k5

ρ/(M2ρ Γρ), whereΓρ is the decay width the resonance

(which can be parametrized by an effective coupling betweenthe pions and theρ). The basicresult of (S7) is that the finite-volume energy spectrum is still given byW = 2(M2

π + k2)1/2 butwith k being a solution of a complicated non-linear equation, which involves theππ scatteringphaseδ11(k) in the isospinI = 1, spinJ = 1 channel and readsnπ − δ11(k) = φ(q). Herekis in the range0 < k <

√3Mπ, n is an integer,q = kL/(2π) andφ(q) is a known kinematical

function which we evaluate numerically for our analysis (φ(q) ∝ q3 for smallq andφ(q) ≈ πq2

for q ≥ 0.1 to a good approximation; more details onφ(q) are given in Appendix A of (S8)).Solving the above equation leads to energy levels for different volumes and pion masses (forplots of these energy levels, see Figure 2 of (S8)).

Thus, the spectrum is determined by the box lengthL, the infinite volume masses of theresonanceMX and the two decay productsM1 andM2 and one parameter,gX , which describesthe effective coupling of the resonance to the two decay products and is thus directly related tothe width of the resonance. In the unstable channels our volumes and masses result in resonancestatesMX which have lower energies than the scattering states (thereare two exceptions, seelater). In these casesMX can be accurately reconstructed fromL, M1, M2 andgX . However,

3

since we do not want to rely on experimental inputs in our calculations of the hadron masses,we choose to use, for each resonance, our set of measurementsfor variousL, M1 andM2 todetermine bothMX andgX . With our choices of quark masses and volumes we find despitelimited sensitivity to the resonances’ widths, that we can accurately determine the resonances’masses. Moreover, the finite volume corrections induced by these effects never exceed a fewpercent. In addition, the widths obtained in the analysis are in agreement with the experimentalvalues, albeit with large errors. (For a precise determination of the width, which is not our goalhere, one would preferably need more than one energy level obtained by cross-correlators. Suchan analysis is beyond the scope of the present paper.)

Out of the 14·12=168 mass determinations (14 sets of lattice parameters/volumes–see Ta-ble S1–and 12 hadrons) there are two cases for whichMX is larger than the energy of thelowest scattering state. These exceptions are theρ and∆ for the lightest pion mass point ata≈0.085 fm. Calculating the energy levels according to (S7, S8) for these two isolated cases,one observes that the energy of the lowest lying state is already dominated by the contributionfrom the neighboring, two particle state. More precisely, this lowest state depends very weaklyon the resonance mass, which therefore cannot be extracted reliably. In fact, an extraction ofMX from the lowest lying state would require precise information on the width of the reso-nance. Since one does not want to include the experimental width as an input in an ab initiocalculation, this point should not be used to determineMρ andM∆. Thus, for, and only for theρ and∆ channels, we left out this point from the analysis.

Approaching the physical mass point and the continuum limit

We consider two different paths, in bare parameter space, tothe physical mass point and contin-uum limit. These correspond to two different ways of normalizing the hadron masses obtainedfor a fixed set of bare parameters. For both methods we follow two strategies for the extrap-olation to the physical mass point and apply three differentcuts on the maximum pion mass.We also consider two different parameterizations for the continuum extrapolation. All residualextrapolation uncertainties are accounted for in the systematic errors. We carry out this analysisboth for theΞ and for theΩ sets separately.

We call the two ways of normalizing the hadron masses: 1. “theratio method”, 2. “massindependent scale setting”.

1. The ratio method is motivated by the fact that in QCD one cancalculate only dimen-sionless combinations of observables, e.g. mass ratios. Furthermore, in such ratios cancella-tions of statistitical uncertainties and systematic effects may occur. The method uses the ratiosrX=MX /MΞ and parametrizes the mass dependence of these ratios in terms of rπ=Mπ/MΞ andrK=MK /MΞ. The continuum extrapolated two-dimensional surfacerX=rX(rπ,rK) is an unam-biguous prediction of QCD for a particle of typeX (a couple of points of this surface have beendetermined in (S3)). One-dimensional slices (2r2

K − r2π was set to 0.27, to its physical value) of

the two-dimensional surfaces forN andΩ are shown on Figure 2 of our paper. (Here we write

4

the formulas relevant forΞ set; analogous expressions hold for theΩ set. The final results arealso given for theΩ set).

A linear term inr2K (or M2

K) is sufficient for the small interpolation needed in the strangequark mass direction. On the other hand, our data is accurateenough that some curvature withrespect tor2

π (or M2π ) is visible in some channels. In order to perform an extrapolation to the

physical pion mass one needs to use an expansion around some pion mass point. This point canberπ=0 (Mπ=0), which corresponds to chiral perturbation theory. Alternatively one can use anon-singular point which is in a range ofr2

π (orM2π) which includes the physical and simulated

pion masses. We follow both strategies (we call them “chiralfit” and “Taylor fit”, respectively).In addition to a linear expression inM2

π , chiral perturbation theory predicts (S11) anM3π

next-to-leading order behavior for masses other than thoseof the pseudo-Goldstone bosons.This provides our first strategy (“chiral fit”). A generic expansion of the ratiorX around areference point reads:rX = rX(ref) + αX [r2

π − r2π(ref)] + βX [r2

K − r2K(ref)] + hoc, where

hoc denotes higher order contributions. In our chiral fit,hoc is of the formr3π, all coefficients are

left free and the reference point is taken to ber2π(ref)=0 andr2

K(ref) is the midpoint betweenour two values ofr2

K , which straddler2K(phys). The second strategy is a Taylor expansion inr2

π

andr2K around a reference point which does not correspond to any sort of singularity (“Taylor

fit”). In this case,r2K(ref) is again at the center of our fit range andr2

π(ref) is the midpointof region defined by the physical value of the pion mass and thelargest simulated pion massconsidered. This choice guarantees that all our points are well within the radius of convergenceof the expansion, since the nearest singularities are atMπ = 0 and/orMK = 0. Higher ordercontributions,hoc, of the formr4

π turned out to be sufficient.We extrapolate to the physical pion mass following both strategies (cubic term of the “chiral

fit” or a quartic contribution of the “Taylor fit”). The variations in our results which follow fromthe use of these different procedures are included in our systematic error analysis.

The range of applicability of these expansions is not precisely known a priori. In caseof the two vector mesons the coefficients of the higher order (r3

π or r4π) contributions were

consistent with zero even when using our full pion mass range. Nevertheless, they are includedin the analysis. For the baryons, however, the higher order contributions are significant. Thedifference between the results obtained with the two approaches gives some indication of thepossible contributions of yet higher order terms not included in our fits. To quantify thesecontributions further, we consider three different rangesof pion mass. In the first one we includeall 14 simulation points, in the second one we keep points upto rπ = 0.38 (thus dropping twopion mass points) and in the third one we apply an even stricter cut at rπ = 0.31 (whichcorresponds to omitting the five heaviest points). The pion masses which correspond to thesecuts will be given shortly. The differences between resultsobtained using these three pion massranges are included in the systematic error analysis.

To summarize, the “ratio method” uses the input datarX , rπ andrK to determinerX(ref),αX andβX and, based on them, we obtainrX at the physical point. The determination of thisvalue is done with the two fit strategies (“chiral” and “Taylor”) for all three pion mass ranges.

2. The second, more conventional method (“mass independentscale setting”) consists of

5

first setting the lattice spacing by extrapolatingMΞ to the physical point, given by the physicalratios ofMπ/MΞ andMK/MΞ. Using the resulting lattice spacings obtained for each bare gaugecoupling, we then proceed to fitMX vs. Mπ andMK applying both extrapolation stratagies(“chiral” and “Taylor”) discussed above. We use the same three pion mass ranges as for the“ratio method”: in the first all simulation points are kept, in the second we cut atMπ=560 MeVand the third case this cut was brought down toMπ=450 MeV.

As shown in the2+1 flavor scaling study of (S3), typical hadron masses, obtained in cal-culations which are performed with ourO(a)-improved action, deviate from their continuumvalues by less than approximately 1% for lattice spacings upto a ≈ 0.125 fm. Moreover, (S3)shows that these cutoff effects are linear ina2 asa2 is scaled froma ∼ 0.065 fm to a ∼ 0.125 fmand even above. Thus, we use the results obtained here, for three values of the lattice spacingdown toa ∼ 0.065 fm, to extrapolate away these small cutoff effects, by allowing rX(ref)(or MX(ref)) to acquire a linear dependence ina2. In addition to the extrapolation ina2, weperform an extrapolation ina and use the difference as an estimate for possible contributions ofhigher order terms not accounted for in our continuum extrapolation.

The physical mass and continuum extrapolations are carriedout simultaneously in a com-bined, correlated analysis.

Statistical and systematic error analysis

Systematic uncertainties are accounted for as described above. In addition, to estimate the possi-ble contributions of excited states to our extraction of hadron masses from the time-dependenceof two-point functions, we consider 18 possible time intervals whose initial time varies fromlow values, where excited states may contribute, to higher values, where the quality of fit clearlyindicate the absence of such contributions.

Since the light hadron spectrum is known experimentally it is of extreme importance tocarry out a blind data analysis. One should avoid any arbitrariness related e.g. to the choiceof some fitting intervals or pre-specified coefficients of thechiral fit. We follow an extendedfrequentist’s method (S12). To this end we combine several possible sets of fitting procedures(without imposing any additional information for the fits) and weight them according to theirfit quality. Thus, we have 2 normalization methods, 2 strategies to extrapolate to the physicalpion mass, 3 pion mass ranges, 2 different continuum extrapolations and 18 time intervals forthe fits of two point functions, which result in 2·2·3·2·18=432 different results for the mass ofeach hadron.

In lattice QCD calculations, electromagnetic interactions are absent and isospin is an exactsymmetry. Electromagnetic and isospin breaking effects are small, typically a fraction of 1% inthe masses of light vector mesons and baryons (S16). Moreover, electromagnetic effects are asmall fraction of the mass difference between the members ofa same isospin multiplet (S16).We account for these effects by isospin averaging the experimental masses to which we compareour results. This eliminates the leading isospin breaking term, leaving behind effects which are

6

only a small fraction of 1%. For the pion and kaon masses, we use isospin averaging andDashen’s theorem (S17), which determines the leading order electromagnetic contributions tothese masses. Higher order corrections, which we neglect inour work, are expected to be belowthe 3 per mil level (see e.g. (S18)). All of these residual effects are very small, and it is safe toneglect them in comparing our results to experiment.

The central value and systematic error bar for each hadron mass is determined from the dis-tribution of the results obtained from our 432 procedures, each weighted by the correspondingfit quality. This distribution for the nucleon is shown in Figure S5. The central value for eachhadron mass is chosen to be the median of the corresponding distribution. The systematic erroris obtained from the central 68% confidence interval. To calculate statistical errors, we repeatthe construction of these distributions for 2000 bootstrapsamples. We then build the bootstrapdistribution of the medians of these 2000 distributions. The statistical error (SEM) on a hadronmass is given by the central 68% confidence interval of the corresponding bootstrap distribu-tion. These systematic and statistical errors are added in quadrature, yielding our final errorbars. The individual components of the total systematic error are given in Table S2.

7

β amud ams L3 · T # traj.

3.3

-0.0960 -0.057 163 · 32 10000-0.1100 -0.057 163 · 32 1450-0.1200 -0.057 163 · 64 4500-0.1233 -0.057 163 · 64 / 243 · 64 / 323 · 64 5000 / 2000 / 1300-0.1265 -0.057 243 · 64 2100

3.57

-0.0318 0.0 / -0.01 243 · 64 1650 / 1650-0.0380 0.0 / -0.01 243 · 64 1350 / 1550-0.0440 0.0 / -0.007 323 · 64 1000 / 1000-0.0483 0.0 / -0.007 483 · 64 500 / 1000

3.7

-0.0070 0.0 323 · 96 1100-0.0130 0.0 323 · 96 1450-0.0200 0.0 323 · 96 2050-0.0220 0.0 323 · 96 1350-0.0250 0.0 403 · 96 1450

Table S1: Bare lagrangian parameters, lattice sizes and statistics. The table summarizes the 14simulation points at three different lattice spacings ordered by the light quark masses. Note thatdue to the additive mass renormalization, the bare mass parameters can be negative. At eachlattice spacing 4-5 light quark masses are studied. The results of all these simulations are usedto perform a combined mass and continuum extrapolation to the physical point. In addition, forone set of Lagrangian parameters, different volumes were studied and four of our simulationsatβ=3.57 were repeated with different strange quark masses.

4 8 12t/a

0

0.5

1

1.5

2

2.5

aMpef

f

Point-PointGaussian-Gaussian

4 8 12t/a

0

0.5

1

1.5

2

2.5

aMNef

f

Point-PointGaussian-Gaussian

Figure S1: Effective masses for different source types in the pion (left panel) and nucleon (rightpanel) channels. Point sources have vanishing extents, whereas Gaussian sources, used onCoulomb gauge fixed configurations have radii of approximately 0.32 fm. Clearly, the extendedsources/sinks result in much smaller excited state contamination.

8

continuum extrapolation chiral fits/normalization excited states finite volume

ρ 0.20 0.55 0.45 0.20K∗ 0.40 0.30 0.65 0.20N 0.15 0.90 0.25 0.05Λ 0.55 0.60 0.40 0.10Σ 0.15 0.85 0.25 0.05Ξ 0.60 0.40 0.60 0.10∆ 0.35 0.65 0.95 0.05Σ∗ 0.20 0.65 0.75 0.10Ξ∗ 0.35 0.75 0.75 0.30Ω 0.45 0.55 0.60 0.05

Table S2: Error budget given as fractions of the total systematic error. Results represent av-erages over theΞ andΩ sets. The columns correspond to the uncertainties related to the con-tinuum extrapolation (O(a) or O(a2) behavior), to the extrapolation to the physical pion mass(obtained from chiral/Taylor extrapolations for each of three possible pion mass intervals usingthe ratio method or the mass independent scale setting), to possible excited state contamination(obtained from different fit ranges in the mass extractions), and to finite volume corrections(obtained by including or not including the leading exponential correction). If combined inquadrature, the individual fractions do not add up to exactly 1. The small (<∼20%) differencesare due to correlations, the non-Gaussian nature of the distributions and the fact that the verysmall finite volume effects are treated like corrections in our analysis, not contributions to thesystematic error (the effect of yet higher order corrections is completely negligible). The finitevolume corrections of the decuplet resonances increase with increasing strange content. This isonly due to the fact that these are fractions of decreasing total systematic errors. The absolutefinite volume corrections of these resonances are on the samelevel.

9

10 20 30 40trajectory

0

5

10

15

pseudofermion 1pseudofermion 2pseudofermion 3s-quark (RHMC)gauge field

Figure S2: Forces in the molecular dynamics time history. Weshow here this history for atypical sample of trajectories after thermalization. Since the algorithm is more stable for largepion masses and spatial sizes, we present –as a worst case scenario– the fermionic force for oursmallest pion mass (Mπ≈190 MeV; MπL≈4). The gauge force is the smoothest curve. Then,from bottom to top there are pseudofermion 1, 2, the strange quark and pseudofermion 3 forces,in order of decreasing mass. No sign of instability is observed.

10

0 1 2 3 4 5 6 7|z|/a

10-6

10-5

10-4

10-3

10-2

10-1

100

||¶D

(x,y

)/¶Um(

x+z)

||

a~~0.125 fma~~0.085 fma~~0.065 fm

Figure S3: Locality properties of the Dirac operator used inour simulations. In the literature,the term locality is used in two different ways (see e.g. (S13–S15)). Our Dirac operator isultralocal in both senses. First of all (type A locality), inthe sum

∑

x,y ψ(x)D(x, y)ψ(y) thenon-diagonal elements of ourD(x, y) are by definition strictly zero for all(x, y) pairs exceptfor nearest neighbors. The figure shows the second aspect of locality (type B), i.e., howD(x, y)depends on the gauge fieldUµ at some distancez: ‖∂D(x, y)/∂Uµ(x + z)‖. In the analyseswe use the Euclidian metric for|z|. We take the Frobenius norm of the resulting antihermitianmatrix and sum over spin, color and Lorentz indices. An overall normalization is performedto ensure unity at|z|=0. The action is by definition ultralocal, thus‖∂D(x, y)/∂Uµ(x + z)‖depends only on gauge field variables residing within a fixed range. Furthermore, within thisultralocality range the decay is, in very good approximation, exponential with an effective massof about 2.2a−1. This is much larger than any of our masses, even on the coarsest lattices.

11

12 16 20 24 28 32 36L/a

0.21

0.215

0.22

0.225

aMp(

L)

c1+ c2 e-Mp L

-3/2 fit

L

Colangelo et. al. 2005

volume dependenceMpL=4

12 16 20 24 28 32 36L/a

0.7

0.75

0.8

aMN(L

)

c1+ c2 e-Mp L

-3/2 fit

L

Colangelo et. al. 2005

MpL=4 volume dependence

Figure S4: Volume dependence of theπ (left panel) andN (right panel) masses for one of oursimulation points corresponding toa ≈ 0.125fm andMπ ≈ 320 MeV. The results of fits to theform c1+c2 exp(−MπL)/(MπL)3/2 are shown as the solid curves, withc1 = aMX(L = ∞) andc2 = acX(Mπ) given in the text (X = π,N for pion/nucleon). The dashed curves correspondto fits with thec2 of refs. (S9,S10).

900 920 940 960 980MN [MeV]

0.05

0.1

0.15

0.2

0.25median

Figure S5: Distribution used to estimate the central value and systematic error on the nucleonmass. The distribution was obtained from 432 different fitting procedures as explained in thetext. The median is shown by the arrow. The experimental value of the nucleon mass is indicatedby the vertical line.

12

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