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2021 116 Edua do Royo Amonda ain Non-pe u ba i e physics in la ice gauge heo ies Di ec o /es Azcoi i Pé ez, Vicen e Follana Adín, Edua do © Uni e sidad de Za agoza Se icio de Publicaciones ISSN 2254-7606 Edua do Royo Amonda ain NON-PERTURBATIVE PHYSICS IN LATTICE GAUGE THEORIES Di ec o /es Azcoi i Pé ez, Vicen e Follana Adín, Edua do Tesis Doc o al Au o 2021 UNIVERSIDAD DE ZARAGOZA Escuela de Doc o ado P og ama de Doc o ado en Física Reposi o io de la Uni e sidad de Za agoza – Zaguan h p://zaguan.uniza .es Non-pe u ba i e physics in la ice gauge heo ies Doc o al disse a ion o Edua do Royo Amonda ain Supe ised by Vicen e Azcoi i P´ e ez Edua do Follana Ad´ın UNIVERSIDAD DE ZARAGOZA Depa amen o de F´ ısica Te´ o ica & Cen o de As opa ´ ıculas y F´ ısica de Al as Ene g´ ıas 2020 Reposi o io de la Uni e sidad de Za agoza – Zaguan h p://zaguan.uniza .es Aknowledgemen s Ag adecimien os Lle a a buen pue o el abajo que aqu´ı se p esen a, y que ha sido o igen de no pocos des elos, hab ´ıa sido imposible de no con a con el apoyo de muchas pe sonas. Dada la di icul ad de nomb a a odas ellas en an pocas l´ıneas, que ´ıa empeza po da las g acias de o ma gene al a quien, desde su p opio ´ambi o y en mayo o meno medida, haya con ibuido en es a emp esa. Sin pe juicio de lo an e io , y ogando se disculpe cualquie omisi´on que pueda p oduci se a con inuaci´on, en lo que sigue quisie a des aca algunos nomb es que conside o especialmen e ele an es. En lo acad´emico, esul a ineludible econoce la labo del P o eso ado con el que he enido la opo unidad de coincidi en las di e en es e apas del Sis ema Educa i o. Buenos ejemplos se ´ıan mi p o eso a de Lengua en 1ode la E.S.O. o mi u o du an e ese mismo cu so, de Dibujo, sin ol ida la p o eso a que, en 2ode la E.S.O., me in oluc ´o en la Olimpiada Ma em´a ica. Del mismo modo, ya du an e la e apa uni e si a ia, pude dis u a de un g an elenco de docen es que supie on ansmi i su pasi´on po es a disciplina, con menci´on especial pa a el Depa amen o de F´ısica Te´o ica, cuyas asigna u as siemp e dejaban con ganas de segui indagando en la ma e ia. El ag adecimien o al Depa amen o es doble, de hecho, pues su acogida y acompa˜namien o a lo la go de es os a˜nos han sido dignas de elogio. Ha sido pa a m´ı un au ´en ico hono habi a , siquie a empo almen e, los pasillos de Te´o ica, en los que siemp e me han hecho sen i como en casa. A caballo en e lo acad´emico y lo pe sonal, engo que da las g acias a mis di ec o es, Vicen e y Edua do, cuya gu´ıa po los no-siemp e-cla os caminos del doc o ado ha sido encomiable. De ellos he podido ap ende mucho, pe o po si es o ue a poco, su con ibuci´on asciende ampliamen e lo p o esional. Adem´as de se buenos colegas, en el sen ido m´as o mal del ´e mino, ambi´en han demos ado se buenos amigos. En es e mismo sen ido, no pod ´e ag adece lo su icien e la excelen e acogida de Giuseppe du an e mis es ancias en L’Aquila. Muy di ´ıcilmen e se puede da en la ida con una pe sona de mayo calidad humana. No quie o ol ida me ampoco de o os compa˜ne os ocasionales de iaje, como Alex o Ma eo, ni po i ii supues o de mi compa˜ne o de a igas y despacho, Filibe o. Ya en el e eno pe sonal, g acias a odas las amis ades que han es ado ah´ı en an os momen os. A mis compa˜ne os del S adium Casablanca, con los que du an e muchos a˜nos he podido dis u a del balonmano. A los del Ins i u o y a los de la ca e a, con menci´on especial al eje e o mis a-populis a. A las Colonas, eino animal incluido, que sol´ıamos se de Ca ´an—y que cada ez somos m´as. Y, po supues o, g acias a mi amilia. G acias a Menchu, Mus a, No a y A i, que siemp e me han hecho sen i como uno m´as. G acias a A a e, Jes´us, Alejand o y Ga o, po que deci que siemp e me hab´eis apoyado incondicionalmen e se ´ıa queda se co o. Hab´eis sido el mejo apoyo que un hijo o un he mano puede ene . G acias ambi´en, Robin, po saca me a pasea odos los d´ıas, pues no hay mejo mane a de da le una uel a a un p oblema ebelde. Finalmen e y po encima de odo, g acias a Cecilia y a Yasmina, sob e cuyo abajo de cuidados descansa el g ueso de es a esis. Ambas hab´eis sido condici´on sine qua non pa a que es e abajo haya podido sali adelan e, an o como lo sois pa a m´ı. Va po oso as. iii Es e abajo ha sido inanciado po el Minis e io de Econom´ıa y Compe i i idad a a ´es de la ayuda p edoc o al pa a la o maci´on de doc o es BES-2013-063567. De igual modo, cabe menciona los p oyec os del Minis e io de Econom´ıa y Com- pe i i idad/Fondo Eu opeo de Desa ollo Regional FPA2012-35453 y FPA2015- 65745-P, y de la Dipu aci´on Gene al de A ag´on/Fondo Social Eu opeo 2015-E24/2. 4CONTENTS unde lying in e ac ions o qua ks and gluons in he QCD amewo k, cons i u es a e y ac i e ield o esea ch, ha includes a la ge a ie y o app oaches [24]. The in e es ha his ma e gene a es is di ided be ween he beha io o he coupling in he in a ed egion—whe e nowadays he e is no consensus abou αS(q2→0) ending o ze o, eezing o e en di e ging—and a la ge momen a, whe e pe - u ba i e QCD can be applied and bo h expe imen al and heo e ical me hods y o p o ide he mos accu a e app oxima ion. In his con ex , la ice-based s a egies ha e been capable o deli e ing esul s bo h in he in a ed egion and in he high ene gy egime, whe e in ac hey p o ide he mos p ecise de e mina- ion o αS(MZ) [2]. Ou wo k can be amed p ecisely in o hese app oaches ha come om la ice QCD, and i elies upon a ghos -gluon e ex compu a ion, as in [25–27]. This hesis is o gnized as ollows. In Chap e 1 we e iew he undamen als o some o he opics p esen ed in his in oduc ion, including a b ie his o ical e iew o he la ice app oach. A mo i a ion o he inclusion o a θ e m in he QCD ac ion is de eloped in Chap e 2, oge he wi h a b ie e iew o he exis ing app oaches o his p oblem. Chap e 3, based on he wo k p esen ed in [28], is dedica ed o he Ising model wi hin an imagina y magne ic ield. He ea e we add ess he s udy o he massi e Schwinge model wi h a θ e m. The 1- la o esul s, elying on he wo k p esen ed in [29], a e epo ed in Chap e 4, whe eas he 2- la o case is ea ed wi h a pseudo e mions app oach in Chap e 5. Finally, Chap e 6 co e s he compu a ion o αs(q2) ia he ghos -gluon e ex. Las ly, ou conclusions a e summa ized in he homonymous chap e and echnical de ails conce ning he compu a ion o he cumulan expansion pe o med in Chap e 3 a e gi en in Appendix A. In oducci´on Va ias d´ecadas han pasado desde que la c omodin´amica cu´an ica (QCD, po sus siglas en ingl´es) se es ableci´o como la eo ´ıa que desc ibe las in e acciones ue es. Es ampliamen e acep ada como una de las eo ´ıas m´as exi osas de la ´ısica mode na y ha sido pues a a p ueba de o ma exhaus i a, an o desde el pun o de is a e´o ico como del expe imen al. A al as ene g´ıas, QCD es asin ´o icamen e lib e, lo que signi ica que sus cons i- uyen es undamen ales, qua ks ygluones, in e accionan con una in ensidad que dec ece con o me la ene g´ıa alcanza escalas m´as al as. En es a si uaci´on, esul a ac ible aplica la eo ´ıa de pe u baciones pa a esol e las in e acciones a co a dis ancia. Al es udia secciones e icaces en p ocesos de al as ene g´ıas, es necesa io ene en cuen a las in e acciones p oducidas an o a co a como a la ga dis ancia. Sin emba go, con la ayuda de los eo emas de ac o izaci´on [1], es posible com- bina QCD pe u ba i a con cie o inpu no pe u ba i o, p o enien e de uen es emp´ı icas o e´o icas, y ob ene as´ı p edicciones que pueden se con on adas ex- pe imen almen e. De hecho, es e p og ama ha sido lle ado a cabo po una amplia comunidad de cien ´ı icos y cien ´ı icas, abajando en cien os de Uni e sidades, lab- o a o ios y habi ualmen e o ganizados en g andes colabo aciones in e nacionales, incluyendo a ios acele ado es de pa ´ıculas. La impo an e can idad de e idencias ecogidas en es e p oceso ha pe mi ido a QCD con e i se en un componen e muy iable del ac ual Modelo Es ´anda de ´ısica de pa ´ıculas. Adem´as, QCD pe u - ba i a es hoy en d´ıa un campo de in es igaci´on muy ac i o, oda ez que cen os como el G an Colisionado de Had ones p o een de nue os da os expe imen ales cada a˜no [2], y debe in e i se un g an es ue zo e´o ico en el c´alculo de los p ime os ´e minos de las expansiones co espondien es. Po o o lado, pa a escalas de ene g´ıa no an ele adas, la in e acci´on ue e no puede se educida a una se ie con e gen e de diag amas de Feynman. De he- cho, una de sus p opiedades ca ac e ´ıs icas es el llamado con inamien o de colo . Es o signi ica que los qua ks se encuen an siemp e (excluyendo ´egimenes de al a densidad) en es ados ligados, llamados had ones, que son neu ales espec o del colo . En es a si uaci´on pu amen e no pe u ba i a, hay pocas ´ecnicas que puedan analiza la eo ´ıa con ´exi o. P obablemen e la que mejo es ablecida es ´a es QCD 5 6CONTENTS en el e ´ıculo, denominada comunmen e la ice QCD. Desde el abajo undacional de Wilson en 1974 [3], el ´exi o del m´e odo ha ido c eciendo con el iempo. Si bien du an e los p ime os a˜nos ealiza los c´alculos necesa ios pa a ex ae esul ados signi ica i os de QCD pa ec´ıa muy lejano, el p og esi o e inamien o de los algo i - mos jun o con el c ecimien o exponencial de la capacidad compu acional mundial dio la uel a a la si uaci´on. Muchos hi os han sido ya alcanzados: simulaciones p ecisas incluyendo los e ec os de loops de qua ks i uales [4], la de e minaci´on del espec o de had ones lige os con e o es sis em´a icos o almen e con olados [5] o, m´as ecien emen e, la compu aci´on de los spli ings de isosp´ın (es o es, de las di e encias de masa en e neu ´on y p o ´on, u o os canales had ´onicos) con g an acue do con los da os expe imen ales, incluso excediendo su p ecisi´on en algunos casos [6]. Todos ellos son buenos ejemplos del ´exi o de es e en oque. Po los mo i os ci ados, QCD es conside ada como la eo ´ıa que desc ibe co ec- amen e la in e acci´on ue e, an o pa a al as como pa a bajas ene g´ıas, y la ice QCD es econocido po la comunidad como un m´e odo ab ini io iable que iene una in e acci´on ´u il con lo expe imen al, pa a aseando a Wilson [7]. Es en ado pensa que, con la e oluci´on ac ual de la po encia de c´alculo, se ´ıa simplemen e cues i´on de iempo que el en oque de la la ice en en ase y esol iese cada uno de los p oblemas no pe u ba i os que oda ´ıa espe an una soluci´on. Po supues o, las cosas no son an sencillas, e incluso cuando pa a un subconjun o de p oblemas bas- a ´ıa con con a con equipos m´as po en es, exis en emas undamen ales que hoy en d´ıa cons i uyen p egun as abie as. Al menos dos p oblemas compa en es e es- a us: el compo amien o de la ma e ia a densidad ba i´onica ini a—incluyendo su diag ama de ases de empe a u a-densidad—y los es udios que in oluc an e ec os opol´ogicos en QCD. Aunque, como e emos, los in en os han sido nume osos, los a ances en ambos campos han sido escasos. La p incipal di icul ad de ´as de es e modes o p og eso en ambas ´a eas es la misma: la acci´on de la eo ´ıa es compleja, y no exis e e o mulaci´on conocida que pueda e i a la apa ici´on de un p oblema de signo se e o (SSP, po sus siglas en ingl´es). El SSP puede se de inido como el p oblema de e alua num´e icamen e la in- eg al de una unci´on muy oscila o ia, que adem´as depende de un g an n´ume o de a iables. Pe enece a la clase de p oblemas NP-comple os [8], lo que signi ica que encon a un algo i mo que supe ase el SSP en iempo polin´omico equi ald ´ıa a p oba que P=NP, uno de los sie e P oblemas del milenio p opues os po el Clay Ma hema ics Ins i u e. En o as palab as, si un algo i mo as´ı exis iese, e- sol e ´ıa odos los p oblemas de clase NP en iempo polin´omico. Pe o, has a el d´ıa de hoy, el dilema P s NP con in´ua sin espues a, y consecuen emen e no exis e soluci´on gene al pa a el SSP que pe mi a la aplicaci´on de las ´ecnicas de Mon eca lo usuales. As´ı pues, los es ue zos dedicados a in en a supe a es a di- icul ad se concen an en la elabo aci´on de di e en es m´e odos, adap ados a las CONTENTS 7 pa icula idades del p oblema en cues i´on. Es e es el caso de QCD con compo- nen e imagina ia en la acci´on, que apa ece cuando es ´an p esen es los ´e minos co espondien es al po encial qu´ımico (µ > 0) o a e ec os opol´ogicos (θ6= 0). Con el obje i o de a anza en la comp ensi´on de QCD con acci´on compleja, a ias p opues as se han desa ollado a lo la go de las ´ul imas d´ecadas. La din´amica de Lange in compleja [9–11], los m´e odos desa ollados po Azcoi i e al [12,13] y, m´as ecien emen e, los dedales de Le sche z [14–16] y el m´e odo de la densidad de es ados [17–19], son ejemplos ele an es de algunos de es os in en os. En gene al, es as es a egias han pe mi ido es udia un n´ume o conside able de oy models y, en algunos casos, han ob enido esul ados in e esan es pa a QCD con po encial qu´ımico ini o. Sin emba go, en el caso de θQCD no se ha podido ob ene casi ning´un p og eso, debido bien a las di e en es limi aciones undamen- ales de las que los m´e odos ci ados adolecen, bien po las di icul ades p ´ac icas que sus espec i as implemen aciones implican. En es e con ex o, la pa e p incipal de es a esis se ha dedicado al es udio de modelos que su en de un SSP, como el modelo de Ising bidimiensional con campo magn´e ico pu amen e imagina io, o el modelo de Schwinge masi o con un ´unico la o y un ´e mino θ. En el p ime caso, es udiamos el conocido modelo po medio de ´ecnicas anal´ı icas, explo ando una egi´on del espacio de pa ´ame os ( empe a u a eal y campo magn´e ico imagina io) algo desa endida en la li e - a u a, posiblemen e debido a la di icul ad de aplica ´ecnicas an o anal´ı icas como num´e icas. De acue do con los pocos abajos que explo an es e sis ema [20,21], se espe a una es uc u a de ases con cie a iqueza. Nues o abajo p e ende p o- ba la alidez de uno de los m´e odos de Azcoi i e al [13,21] en es e escena io y, dado que puede a oja algo de luz en una egi´on c ´ı ica que su e de un SSP (que ha obs aculizado la consecuci´on de una soluci´on num´e ica) espe amos que pueda se i como e e encia pa a o os m´e odos que aspi en a supe a el p oblema del signo en cualquie eo ´ıa gauge en el e ´ıculo. Con el obje i o de en en a sis emas simila es a QCD con un ´e mino θ, y de es e modo desa olla los m´e odos que lidian con el SSP, hemos es udiado el mod- elo de Schwinge masi o con un la o y ´e mino θ, que se co esponde con QED en dimensi´on 1 + 1. Como e emos, compa e un g an n´ume o de p opiedades con QCD, incluyendo el con inamien o y, en cie o modo, la libe ad asin ´o ica, po lo que de hecho es ampliamen e usado como su oy model. Adem´as, de ini la ca ga opol´ogica en es e modelo es casi i ial, en con as e con cualquie a de las de iniciones usuales pa a es e obse able en QCD, que esul an mucho m´as in incadas. Es e hecho nos ha pe mi ido explo a el modelo con un cos e com- pu acional ac ible, ob eniendo esul ados compa ibles con la p edicci´on anal´ı ica de Coleman [22] y, lo que es m´as impo an e, poninedo a p ueba el m´e odo de- sa ollado en [13] en una eo ´ıa gauge con e miones, lo que cons i uye un paso 8CONTENTS impo an e en el camino a su aplicaci´on en QCD. Como de i ado de la l´ınea de abajo an e io , y empujados po la necesidad de una mayo op imizaci´on de los algo i mos an e io es, ambi´en hemos analizado la e si´on de 2 la o s del modelo de Schwinge . En es e caso, se ha e i ado el c´alculo comple o del de e minan e e mi´onico siguiendo un en oque basado en la ´ecnica de los pseudo e miones [23]. M´as all´a del es udio de sis emas a ec ados po un p oblema de signo, o o ema, incluido den o de la ice QCD, se ha a ado en es a esis: el unning coupling αS. La dependencia de αS(q2) con el momen o ans e ido q, que codi ica las in e acciones subyacen es de qua ks y gluones en el ma co de QCD, cons i uye un campo de in es igaci´on muy ac i o, que incluye una g an a iedad de me odolog´ıas [24]. El in e ´es que es a ma e ia gene a se di ide en e el compo amien o de es e acoplamien o en la egi´on in a oja, donde hoy en d´ıa no exis e consenso sob e si αS(q2→0) iende a ce o, se congela o incluso di e ge, y a momen os al os, donde QCD pe u ba i a puede se aplicada y m´e odos an o e´o icos como expe imen ales in en an p o ee la ap oximaci´on m´as p ecisa. En es e con ex o, las es a egias basadas en el e ´ıculo han sido capaces de o ece esul ados pa a ambos casos, consiguiendo de hecho la de e minaci´on m´as p ecisa pa a αS(MZ) [2]. Nues o abajo puede ubica se p ecisamen e en e los en oques que ienen de la la ice, y se apoya en un c´alculo del ´e ice ghos -gluon, como en [25–27]. Es a esis se o ganiza como sigue. En el Cap´ı ulo 1 epasamos lo undamen al de algunos de los emas p esen ados en es a in oducci´on, incluyendo un b e e esumen his ´o ico sob e el o igen de la ice QCD. Una mo i aci´on pa a la inclusi´on del ´e mino θen la acci´on de QCD se desa olla en el Cap´ı ulo 2, jun o con un b e e epaso de de los en oques exis en es pa a es e p oblema. El Cap´ı ulo 3, basado en el abajo p esen ado en [28], se dedica al modelo de Ising en un campo magn´e ico pu amen e imagina io. A con inuaci´on en en amos el es udio del modelo de Schwinge masi o con ´e mino θ. Los esul ados pa a el caso de un ´unico la o , p esen ados en [29], se o ecen en el Cap´ı ulo 4, mien as que el caso de 2 la o s se a a con un m´e odo basado en pseudo e miones en el Cap´ı ulo 5. Finalmen e, el Cap´ı ulo 6 cub e la compu aci´on de αs(q2) median e el ´e ice ghos -gluon. Pa a acaba , nues as conclusiones se esumen en el cap´ı ulo hom´onomio, y algunos de alles ´ecnicos, que concie nen al c´ompu o de la expansi´on de cumulan es ealizada en el Cap´ı ulo 3, se discu en en el Ap´endice A. Chap e 1 The la ice app oach In his chap e we co e he essen ial poin s o he la ice app oach o QCD, includ- ing a b ie his o ical e iew o i s bi h and e olu ion o e he pas ew decades. The main aspec s o he o malism a e explained, discussing he s enghs and lim- i a ions o Mon e Ca lo me hods when s udying la ice gauge heo ies. Finally, some conside a ions abou he ype o e o s associa ed wi h his me hodology a e discussed, ecalling how we can con ol hem and, e en ually, in which way we can p o ide a p ecise es ima ion o a gi en obse able. 1.1 The dawn o colo The appea ance o qua ks and gluons as he undamen al cons i uen s o ba yons and mesons is ela i ely ecen . In o de o gi e he necessa y con ex , i is desi - able o go back o he middle o he las cen u y. The disco e y o he pion h ough cosmic ay expe imen s in 1947 [30], which was soon ollowed by hose o he i s s ange pa icles, he kaon and he Λ, ma ked he beginning o a endency ha con inued du ing he 50s, by means o di e en expe imen s in ol ing pa icle col- lide s. This expe imen al ac , i.e., he disco e y o a la ge numbe o new pa icles and esonances ha we e somehow ela ed, claimed o an explana ion in e ms o a educed se o deg ees o eedom. I was in 1961 when Mu ay Gell-Mann (who had p e iously in oduced he s angeness as a quan um numbe , conse ed by bo h elec omagne ic and s ong in e ac ions) p o ided a success ul explana ion o he so-called pa icle zoo, in his amous The Eigh old Way [31]. By ex ending he SU(2) isospin symme y, Gell-Mann p oposed a SU(3) la o symme y, b oken by mass di e ences, ha was capable o o ganize all he obse ed had onic s a es. Mo eo e , his symme y p edic ed he exis ence o he Ω−ba yon, a pa icle ha was obse ed h ee yea s la e , wi h a mass ha ma ched accu a ely he alue an icipa ed by he model [32]. P ecisely Gell-Mann in 1964, and independen ly 9 10 CHAPTER 1. THE LATTICE APPROACH Zweig [33], p oposed ha bo h mesons and ba yons we e composed o mo e un- damen al cons i uen s (called qua ks by he men o , and aces by his pupil) which hold ac ional elec ic cha ge. A his poin , some ques ions s ill emained o comple e he qua k puzzle. Pa - icula ly ele an was he ac ha he Ω−pa icle, composed o h ee squa ks in i s g ound s a e, should ha e a symme ic wa e unc ion. This was in con adic- ion wi h Pauli exclusion p inciple, which equi ed an an isymme ic wa e unc ion unde he exchange o wo qua ks, which we e assumed o be e mions o spin 1/2. Ano he conce n in ol ed he decay ampli udes p edic ed by he qua k model, which di e ed om he alues measu ed in elec on-posi on collide s. Bo h issues we e add essed by Gell-Mann, F i zsch and Ba deen, who du ing 1971 and 1972 in oduced a new exac SU(3) symme y o he qua ks, named colo [34–36], which was exac ly conse ed. Colo was soon in e p e ed as a gauge symme y and a ield heo y was cons uc ed by Gell-Mann, F i zsch and Leu wyle [37] and independen ly by G oss and Wilczek [38] in 1973, who emphasized (as Poli ze in [39]) one o he cha ac e is ic p ope ies o he new heo e ical a i ac : asymp- o ic eedom. La e , F i zsch and Gell-Mann would inally gi e he heo y i s mode n name: Quan um Ch omodynamics. In o de o comple e he pic u e o QCD, ano he phenomenon had o be ex- plained: why expe imen s only measu e colo single s, i.e., ba yons and mesons a e ee o any colo cha ge, and indi idual qua ks a e no p esen in na u e as ee pa icles. To explain his con inemen o qua ks and gluons in o had ons, Wilson demons a ed in 1974 how la ice gauge heo ies con ine cha ged s a es in he s ong coupling limi [3]. This wo k se he basis o he quali a i e unde - s anding o colo con inemen —and in his way closes he ea ly pe iod in which QCD was cons uc ed—bu i s in luence o e he heo y would anscend by a his objec i e, since he egula iza ion de eloped by Wilson opened a whole new ield wi hin high ene gy physics. Wi h espec o con inemen i sel , i should be no ed ha , al hough he e exis b oad nume ical e idence o i s alidi y, oday a igo ous ma hema ical p oo is s ill lacking. 1.2 A disc e ized space ime The o mula ion o Quan um Ch omodynamics as he dynamical heo y o he s ong in e ac ion, oge he wi h he unde s anding o asymp o ic eedom and con inemen , is wi hou doub one o he key miles ones eached du ing he pas cen u y. E en so, and due o he s ongly coupled na u e o he heo y, he p e- dic i e powe o QCD was se e ely limi ed du ing i s i s s eps. Specially a low ene gies, pe u ba i e me hods—de eloped wi h g ea success o p ocesses in ol - ing Quan um Elec odynamics—we e o no use, making obse ables such as he 1.2. A DISCRETIZED SPACETIME 11 had on spec um un eachable. In his con ex , Wilson in oduced—in which is conside ed o be he ounda- ional wo k o he la ice app oach [3]—a no el non-pe u ba i e egula iza ion o QCD. The essen ial ideas o he p ocedu e, applicable o any gauge heo y, ha e endu ed up o he p esen day, and a e ela i ely simple. S a ing om Feynman’s pa h in eg al o malism, space ime is disc e ized in a ou -dimensional (Euclidean) hype cube. All objec s a e now de ined in he si es o he hype cube, o he links joining hem. This p ocedu e is done in such a way ha gauge symme y is ex- ac ly p ese ed. Al hough Lo en z (o Euclidean) in a iance is los o any ini e la ice spacing a, Wilson a gued ha his obs acle could be o e come by means o a eno maliza ion-g oup app oach, possible i he e exis s a c i ical poin a some alue o he gauge-coupling o he heo y. A his poin , and e en i la ice QCD had elucida ed he con inemen phe- nomenon, i was unclea how he app oach could be exploi ed in o de o calcula e ele an physical obse ables. The u ning poin came by he end o he decade, when a p ocedu e widely used a he ime in S a is ical Mechanics was in oduced in o he ealm o Field Theo y. In 1978, Wilson [40] p oposed o apply Mon e Ca lo me hods, namely a Me opolis algo i hm, o la ice gauge heo ies, in o de o elucida e nume ically he con inemen phenomenon. Wi h a de ailed p esc ip- ion o how obse ables should be measu ed in such a amewo k, he also poin ed ou ha he i s calcula ions we e al eady ongoing o he SU(2) gauge heo y. The i s nume ical esul s, ha demons a ed he po en ial o he new echnique, came by C eu z, Jacobs and Rebbi in 1979 [41], who pe o med a simula ion o he gauge Z(2) model in ou dimensions, obse ing a i s -o de ansi ion ha suppo ed he con inemen hypo hesis by coun e ing a p e ious conjec u e due o Migdal [42]. In he same yea , Wilson [43] p esen ed ano he eno maliza ion- g oup app oach o he SU(2) gauge heo y, impo ing block-spin echniques om S a is ical Mechanics. Al eady in 1980, C eu z [44] p o ided u he e idence o bo h con inemen and asymp o ic eedom in he SU(2) gauge heo y, ensu ing in his way he possibili y o aking he con inuum limi by holding cons an a physical obse able like he s ing ension. These no el ideas c ys allized in se e al wo ks o e he yea s o ollow, including he pionee ing compu a ions o he ho meson mass o a disc e e app oxima ion o he SU(2) heo y, by Weinga en [45], o he calcula ion o se e al had on masses due o Hambe and Pa isi [46], pe o med al eady wi h SU(3) as he gauge g oup. In bo h cases, he limi s imposed by compu a ional esou ces we e alle ia ed by he use o he so-called quenched app oxima ion, which neglec s en i ely he e ec o i ual qua k loops—wi h he associa ed addi ion o uncon olled e o s. The inclusion o ull dynamical e mions, i.e., aking in o accoun bo h gluon and qua k dynamics, was achie ed i s in 1983 by Azcoi i and Nakamu a [47], who, leaning 12 CHAPTER 1. THE LATTICE APPROACH on he pseudo e mions me hod p oposed by Fuci o e al [23], compu ed he mass spli ing o he ho and omega mesons o he icosahed al app oxima ion o SU(2) as he gauge g oup. A simila compu a ion, also elying in he pseudo e mionic app oach, was made by Hambe in 1985 [48], al eady using SU(3) as he gauge symme y g oup. These ea ly s udies, e en when hey we e made wi h e y modes compu- a ional powe by oday s anda ds, and wi h a numbe o sou ces o uncon- olled sys ema ic e o s—especially signi ican o hose made wi hin he quenched app oxima ion—we e success ul in p o ing he easibili y o he la ice app oach. Du ing he subsequen decades, compu ing esou ces kep g owing s eadily. In pa - allel, new algo i hmic and heo e ical ad ances we e de eloped, which ul ima ely made possible b eak h oughs such as he compu a ion o he ligh had on spec um om i s p inciples [5] o , mo e ecen ly, he de e mina ion o he isospin mass spli ings [6]. A p esen ime, he e exis many esea ch g oups ac i ely wo king in his a ea; s a e-o - he-a esul s can be e iewed annually in he p oceedings o he In e na ional Symposium on La ice Field Theo y. Fo mo e de ails abou he his o ical de elopmen o he la ice app oach, in- cluding a mo e echnical discussion on he issue, we e e he in e es ed eade o he ex ensi e e iew o Fodo and Hoelbling [49]. 1.3 The o malism A his poin , and be o e discussing some o he ca ea s o he la ice app oach, i seems desi able o ske ch a leas he basic p emises o he me hod. In any case, o a mo e exhaus i e in oduc ion o he opic, we ecommend he in e es ed eade he lec u e no es o Da ies [50] o he mo e ecen book by Ga inge and Lang [51], which in ac has se ed as a e e ence o some o he opics p esen ed in his sec ion. 1.3.1 A ini e pa h in eg al The s a ing poin o he app oach is he pa h in eg al o malism [52,53], which, o a gi en quan um ield heo y, allows o w i e i s pa i ion unc ion as a unc ional in eg al Z=ZDΦe−S[Φ],(1.1) whe e Sis he euclidean ac ion o he heo y—e en being conside ed a imagina y ime, physical in o ma ion can be eco e ed as long as he heo y ul ills a se o axioms [54,55]—and he in eg a ion is mean o be pe o med o e all possible con- igu a ions o he ields o he heo y, deno ed gene ically by Φ. I we pa icula ize 1.3. THE FORMALISM 13 o QCD, he ields o be conside ed a e qua ks ψ, an iqua ks ¯ ψand gluons Aµ, and (1.1) can be o mula ed schema ically as ZQCD =ZDψD¯ ψDAµe−SQCD[ψ, ¯ ψ,Aµ],(1.2) wi h acuum expec a ion alues o ope a o s O(ψ, ¯ ψ, Aµ) being gi en by hOi =1 ZQCD ZDψD¯ ψDAµO(ψ, ¯ ψ, Aµ)e−SQCD[ψ, ¯ ψ,Aµ].(1.3) The p ecise meaning o he in eg a ion symbol p esen in (1.2) and (1.3) is a sub le ma hema ical issue; a his poin he heo y would be ill-de ined and a egula iza- ion is manda o y in o de o ex ac any physical in o ma ion and a oid di e - gences. To his end, he e a e se e al possibili ies ha lean on pe u ba i e ex- pansions in he coupling, such as Pauli-Villa s (o dimensional) egula iza ion [56], which ha e been widely used bo h in QED and QCD. The p esc ip ion o Wilson ha was in oduced ea lie [3], is howe e he only ully non-pe u ba i e egu- la iza ion known, allowing o s udy quan um ield heo ies om i s p inciples, especially QCD beyond sho -dis ance in e ac ions, whe e con inemen — oge he wi h he highly non- i ial s uc u e o he QCD acuum—poses an insu moun - able obs acle o pe u ba i e-based app oaches. The eby, going ahead wi h QCD, we disc e ize he con inuum ou -dimensional euclidean space- ime in an hype cubic g id wi h cons an la ice spacing a; his pa ame e will be he egula o o he heo y, in which limi a→0 i s o iginal e sion is eco e ed. Mo eo e , we es ic he ull space ime o a simple 4-d box o ini e ex en , limi ing in his way he spa ial olume and he imagina y- ime e olu ion o he sys em. By doing so, he numbe o a iables o conside be- comes ini e and he o iginal p oblem can begin o be pic u ed as compu a ionally ea able. I we e u n o (1.2), he ields ψ(x) and ¯ ψ(x) now ake alues only o x=a(n1, n2, n3, n4), whe e nia e in ege s sa is ying 0 ≤nia < Li,Libeing he spa ial ( empo al) ex en o he 4-d box in each dimension. Wi h espec o he gluon ield Aµ(x), i is con enien o pos pone b ie ly i s de ini ion on he la ice. Fi s , we should no e ha he ac ion o he heo y, SQCD in (1.2), is he space ime in eg al o he ollowing euclidean1lag angian densi y, LQCD =¯ ψ(iγµDµ+m)ψ+1 2g2T (FµνFµν),(1.4) 1The o dina y eal- ime e sion o LQCD, i.e., wi h Minkowski space ime me ic, is jus ¯ ψ(iγµDµ−m)ψ−T (Fµν Fµν)/2g2.No e ha we a e using a a he compac no a ion, whe e only space ime indexes a e explici , while hose co esponding o colo , spin and la o a e kep implici . A mo e de ailed no a ion could label qua k (and an iqua k) ield componen s by ψ α,c, since ψis a 3-colo ec o , a 4-Di ac spino and a n - la o ec o . In he same way, mis a diagonal ma ix in la o space, con aining each one o he qua k masses. 20 CHAPTER 1. THE LATTICE APPROACH p esc ip ions ha minimize u he he o de o he e o s, he o iginal p oposal o Wilson p o ides a good balance be ween he p ecision achie ed and he complexi y o i s de ini ion. In any case, imp o ed gauge ac ions a e buil by adding highe o de e ms o (1.17), in ol ing loops la ge han he plaque e ha a e supp essed by powe s o a. This ype o gluonic ac ions a e s a e-o - he-a in mode n la ice compu a ions. 1.3.3 In e lude: chi al symme y in he con inuum Jus be o e add essing he cons uc ion o he e mionic ma ix M, we ound nec- essa y o make a leas a shallow e iew o he ole o chi al symme y in QCD, as i s o mula ion on he la ice has deep and di ec implica ions in how dynamical e mions can en e in o he la ice ac ion. In i s con inuum o mula ion, he ac ion o QCD o N la o s is gi en by he space ime in eg al o he lag angian (1.4). Focusing on he e mionic pa and keeping a compac no a ion, whe e ψand ¯ ψ ields a e unde s ood o be N - ec o s in la o space, we ha e SQCD-F =Zd4x¯ ψ(iγµDµ+m)ψ. (1.18) Fo he pa icula case o ze o e mion mass, i.e., i all he componen s o he diagonal ma ix m anish, he e exis wo se s o ans o ma ions ha lea e he abo e ac ion in a ian . The i s amily is composed by he so-called ec o ans- o ma ions, gi en by ψ→eiα1ψ¯ ψ→¯ ψe−iα1,(1.19) ψ→eiαTiψ¯ ψ→¯ ψe−iαTi,(1.20) whe e he Tima ices a e he gene a o s o he la o g oup, SU(N ), and so, he index i uns om 1 o N2 −1. When conside ing al oge he (1.19) and (1.20), he g oup ex ends o U(N ). The o he amily o ans o ma ions can be cons uc ed by including a γ5ma ix in he p e ious ones, and hus hey a e gi en by ψ→eiαγ51ψ¯ ψ→¯ ψe−iαγ51,(1.21) ψ→eiαγ5Tiψ¯ ψ→¯ ψe−iαγ5Ti.(1.22) This las se ecei es he name o axial ans o ma ions. When conside ed oge he wi h ec o ans o ma ions, hey ecei e he name o chi al ans o ma ions; in he same way, he massless limi —in which hey p ese e SQCD—is called chi al limi . To di e en ia e he wo sec o s ha compose chi al symme y, i is cus oma y o label hem wi h a Vo a Asubsc ip . In his way, he ull chi al g oup is gi en by SU(N )V×SU(N )A×U(1)V×U(1)A.(1.23) 1.3. THE FORMALISM 21 I is impo an o no e ha axial ans o ma ions a e symme ies o he massless ac ion, bu his symme y is explici ly b oken o ini e mass qua ks. On he con a y, ec o ans o ma ions (1.20) p ese e he ac ion also in he degene a e mass case—i.e., N species o equal non-ze o mass—con o ming he well-known isospin symme y. Fu he mo e, (1.19) is always a symme y o he ac ion, which implies ba yon numbe conse a ion. Equally ele an is he so-called axial anomaly. Al hough U(1)Ais a symme y o he massless ac ion, i is explici ly b oken in he ully quan ized heo y. This educes he ull exp ession (1.23) o he ollowing symme y g oup: SU(N )V×SU(N )A×U(1)V.(1.24) This g oup is usually exp essed in sligh ly di e en e ms. To his aim, i su ices o ecall ha chi al symme y spli s bo h he qua k ields and he ac ion in o wo sepa a e pieces, commonly named le - and igh -handed. De ining he p ojec o s PLand PRas PL=1−γ5 2PR=1+γ5 2,(1.25) i is possible o pa i ion he ull la o space, since in oducing he p ojec ed qua k and an iqua k ields ψL,R ≡PL,Rψ¯ ψL,R ≡¯ ψPR,L,(1.26) allows o exp ess he qua k (an iqua k) ield as he sum o ψLand ψR, sepa a ing in his way he ac ion in o se e al componen s: SQCD-F =Zd4x¯ ψL(iγµDµ)ψL+¯ ψR(iγµDµ)ψR+¯ ψLmψR+¯ ψRmψL. (1.27) F om he abo e exp ession i i ially ollows ha , in he chi al limi , only he i s wo componen s su i e. Rema kably, hey a e comple ely decoupled in his limi , in e ac ing only h ough he mass e ms in he abo e o mula. Fo his easons, he chi al symme y g oup (1.24) is o en e o mula ed in e ms o i s le - and igh -handed deg ees o eedom, i.e., as SU(N )L×SU(N )R×U(1)V.(1.28) In any case, when conside ing qua ks o ini e bu degene a e mass, he axial sec o symme y is b oken explici ly—jus he SU(N ) pa , since he one co esponding o U(1)Awas al eady b oken by he anomaly—and he abo e g oup ge s educed o i s ec o sec o U(N )V, o ollowing he s uc u e o (1.24) SU(N )V×U(1)V.(1.29) 22 CHAPTER 1. THE LATTICE APPROACH E en ually, i one conside s N la o s wi h di e en masses, he abo e symme y ge s educed o he enso p oduc o N copies o U(1)V. Be o e closing his in e lude, we wan o s ess a couple o issues conce ning chi al symme y. The i s one is ha chi al symme y—o mo e p ecisely, i s axial sec o —is spon aneously b oken in QCD a ze o empe a u e. Le us conside he ac ion wi h jus wo la o s: up and down. Since hese qua ks ha e a e y small mass compa ed o he QCD scale, i s explici symme y b eaking, co esponding o he shi be ween (1.28) and (1.29) o N = 2, should gi e ise o expe imen al e ec s. In pa icula , some pa icles—as, e.g., pa i y pa ne s— should ha e almos -degene a e masses. Howe e , his expec ed symme y does no ma ch he obse a ions; mass di e ences due o he explici b eaking o he QCD ac ion a e signi ican ly smalle han he expe imen al measu es. The o igin o hese disc epancies is, p ecisely, ha he global symme y (1.28) is spon aneously b oken, since he acuum o he heo y is no in a ian unde he co esponding ans o ma ions. In o he wo ds, al hough he ac ion is almos p ese ing chi al symme y, he g ound s a e o he sys em is no . The second poin o discuss is jus a co olla y o he p e ious a gumen : since he e is a con inuous symme y being spon aneously b oken, Golds one’s heo- em p edic he appea ance o se e al massless bosons. This is in ac he case, since i we conside jus qua ks uand d, we should expec h ee ligh bosons— since he symme y is sligh ly explici ly b oken—which can be iden i ied wi h he h ee pions π0,π±. Including also he s ange qua k s, would accoun o eigh bosons, which can also be iden i ied wi h he ou kaons K0,¯ K0, K±and he η meson, which sums up o he h ee pions. No e ha i we o e look he anomaly, in he las case 9 ligh mesons should appea , and he η0pa icle should be also conside ed wi hin his pic u e. I is p ecisely he anomaly wha allows o explain he η0−π0mass di e ence, as was s essed in he p e ious subsec ion. Wi h hese conside a ions, we can now p oceed o alk o e mions in he la ice, wi h a las wo d o cau ion ha summa izes he p e ious a gumen . The ligh es pa icles o he QCD spec um gain hei low masses hanks o he spon aneous b eaking o chi al symme y. I we we e o weak he heo y and emo e he chi al in a iance om he ac ion, he explici b eaking o his symme y would p e en hese pa icles om being pseudo Golds one bosons, and consequen ly hei masses would be expec ed o inc ease. 1.3.4 Dynamical e mions: doubling and chi al symme y In o de o implemen ull QCD in he la ice, we need o choose an ac ion Sla which, as in (1.12), can be decomposed in o wo componen s. Fi s , a pu ely glu- onic e m Sg, o which a iable candida e has al eady been e iewed in subsec ion 1.3.2. In second place, a e m in ol ing qua k and an iqua k ields is needed. Now 1.3. THE FORMALISM 23 ha he basics o chi al symme y in con inuum QCD ha e been discussed, we can ace he ask o cons uc ing his e mionic e m, comple ing in his way he la ice ac ion. The eby, in his subsec ion we will analyze he di icul ies in p oposing a p ope la ice e sion o he e mionic ma ix M, which de ines how qua ks and an iqua ks in e ac . While he cons uc ion o he Wilson gauge ac ion (1.17) was no e y ou- blesome, he si ua ion changes subs an ially when ying o p oceed wi h he e mionic pa o he ac ion in a simila way. The simples app oach o dis- c e ize he con inuum e m consis s in eplacing he i s o de de i a i es in LQCD by a s anda d symme ic di e ence. In his way, he con inuum e mionic e m ¯ ψ(iγµDµ+m)ψbecomes 1 2a¯ ψ(n)iγµUµ(n)ψ(n+ ˆµ)−U† µ(n−ˆµ)ψ(n−ˆµ)+mψ(n),(1.30) which is indeed gauge-in a ian , as bilocal p oduc s o qua k-an iqua k ields a e connec ed by he co esponding link a iables. Un o una ely, his p ocedu e in- oduces he so-called double s: o each o hese nai e e mions included in he la ice ac ion, 16 copies appea in he con inuum limi , 15 o hem being unphys- ical. In o de o ix his si ua ion, an al e na i e is o use Wilson e mions, which add an ex a e m o he nai e o mula ion. I s aim is o assign a di e gen mass, o o de O(a−1), o each o he unwan ed double s, decoupling hem om he he- o y in he con inuum limi . This mechanism is su icien o ix he con inuum limi o he ac ion, howe e , he axial sec o o chi al symme y ge s explici ly b oken in he la ice by he ex a e m—no only he anomalous pa U(1)A, which in ac should be b oken, bu he whole g oup U(N )A. In con inuum QCD, as i was discussed in he p e ious subsec ion, chi al symme y plays a cen al ole, since i s non-anomalous sec o is spon aneously b oken, p oducing app oxima e Golds one bosons. Wi h an explici ly b oken chi al symme y, his mechanism is no possi- ble anymo e: pa icles such as pions, kaons o he η, which con o m he so-called meson oc e , acqui e masses well abo e i s o iginal alues. This beha io hinde s ac ual simula ions om eaching he physical poin . In ac , du ing decades o la ice nume ical wo ks, he achie ed mass o he pion—a e y ele an quan i y, being he ligh es had on in he spec um—has been e y a om i s physical alue, which makes necessa y o pe o m ex apola ions o he physical poin . The p e ious discussion ises a na u al ques ion: is i possible o ind a e mion o mula ion ha , whi hou in oducing unphysical mul iplici ies, p ese es chi al symme y? Fo a long ime, i was belie ed ha , un o una ely, i was no . In ac , a om being a echnical complica ion, he double s issue has i s oo in he chi al anomaly o QCD. In he nai e la ice o mula ion o (1.30) he anomaly dis- appea s, canceled exac ly by he ex a double s. Adding he Wilson e m, which decouples he ex a pa icles, can be in e p e ed as in oducing a la ice e sion o 24 CHAPTER 1. THE LATTICE APPROACH he anomaly by hand—wi h one undesi able e ec : i also b eaks he non-anomalus pa o he symme y, spoiling he spon aneous b eaking mechanism. In ac , as was p o en in 1981 by Ka s en and Smi [67], his is a gene al esul : ei he he axial anomaly is canceled by he p esence o ex a e mions, o i is in oduced in he la ice by a e m ha , necessa ily, explici ly b eaks chi al symme y, in- cluding i s non-anomalous sec o . In he same way, a con empo a y esul due o Nielsen and Ninomiya— he so-called no-go heo em—s a es ha i is no possible o cons uc a la ice o mula ion o QCD ha has a he same ime absence o double s, chi al symme y, and locali y [68, 69]. Fu he mo e, he e also exis s a scheme-independen e sion o his esul , wi h sligh ly di e en condi ions [70], ha makes he impossibili y o egula izing a heo y wi h chi al e mions a a he p o ound ques ion, no exclusi e o he la ice app oach. As a consequence, a choice mus be made be ween p ese ing chi al symme y in i s con inuum o m o a oiding unwan ed deg ees o eedom; e e y e mion o mula ion on he la ice su e s om one pa hology o he o he . Bu , e en hough he abo e esul s a e co ec , a wo ka ound o una ely exis s. Almos 40 yea s ago, Ginspa g and Wilson ealized ha a emnan o chi al symme- y could be iden i ied wi hin he la ice o mula ion, in such a way ha he axial anomaly is s ill p ope ly p ese ed and, a he same ime, no unphysical deg ees o eedom a e in oduced [71]. While he con inuum o m o chi al symme y—i.e., he p ese a ion o he ans o ma ion se s (1.19) o (1.22)— equi es he massles Di ac ope a o o an icommu e wi h γ5, he emnan symme y p oposed in [71] e i ies a b oade es ic ion, namely Dγ5+γ5D=aDγ5D, (1.31) whe e D≡iγµDµs ands o he massless Di ac ope a o . This condi ion is in ac di e en om i s con inuum coun e pa o e e y ini e la ice spacing a, al hough hey con e ge in he a→0 limi . In his sense, (1.31) can be in e p e ed as an ex ension o he con inuum de ini ion, wi h a singula ad an age: i is able o e ade he Nielsen-Ninomiya no-go heo em, since only he con inuum o m o chi al symme y—wi h a anishing igh -hand side in (1.31)—is a ec ed by i [72]. Mo eo e , a modi ied chi al o a ion can be de ined in he la ice, in such a way ha he gauge ields ans o m essen ially as i s con inuum e sions, p ese ing ele an esul s as, i.e., he index heo em [73]. Howe e , he solu ion gi en in [71] was no cons uc i e, in he sense ha a pa icula o m o Dcould no be ound a ha momen ; i would ook almos wo decades o ind a p ac ical implemen a ion o hese ideas. Wi hin he abo e scena io, o he al e na i es would be explo ed o e he yea s o come. In ac , a numbe o hese s a egies we e de eloped and nowadays a e pa o he cu en lo e o he ield. They can be classi ied in g oups acco ding o 1.3. THE FORMALISM 25 how hey deal wi h he doubling p oblem. Some app oaches choose o gi e up chi al symme y; his is he case o Wilson e mions. In his ca ego y a e also included clo e e mions, a Symanzik imp o ed e sion o Wilson e mions ha emo es O(a) disc e iza ion e o s [74], and wis ed mass e mions, which conside pai s o mass degene a e Wilson e mions oge he wi h an isospin mass spli ing e m [75]. An al e na i e app oach consis s in p ese ing chi al symme y, while assuming some o he double s degene acy. He e he p ime example a e Kogu -Susskind o s agge ed e mions [76]. This a ian is cons uc ed om he nai e o mula ion, bu dis ibu es he usual Di ac 4-spino componen s o e neighbo ing si es, in such a way ha he double s degene acy ge s educed om 16 o 4 species. Mo eo e , o some obse ables i is possible o emo e he esidual deg ees o eedom by aking he ou h oo o he e mion de e minan —a p ocedu e commonly e e ed o as oo ing, o iginally p oposed by Ma ina i, Pa isi and Rebbi, in he con ex o he massi e Schwinge model [77]. Al hough i s alidi y was a i s con o e sial, bo h nume ical e idence [78] and heo e ical a gumen s [79] suppo ha his echnique leads o he co ec heo y as long as con inuum and chi al limi s a e aken p ecisely in his o de . As wi h Wilson e mions, he e also exis Symanzik imp o ed e sions o he s agge ed ac ion; wo o hem, widely used by he la ice communi y, a e he Asq ad ac ion [80]— o a-squa ed adpole imp o ed—and he HISQ ac ion [81]— o highly imp o ed s agge ed qua ks. In Chap e 4 we make use o s agge ed qua ks, while on Chap e 6 he analyzed con igu a ions we e gene a ed wi h he HISQ ac ion. Fo he sake o comple eness—e en i i is less ela ed wi h he wo k de el- oped in his hesis— he e is s ill ano he class o e mions ha dese es a leas a men ion: he pa icula solu ions o he Ginspa g-Wilson equa ion (1.31). They p ese e chi al symme y wi hou he double s pa hology, so in his sense hey a e he bes possible e mions, and hey should be p e e ed when compa ed o o he al e na i es. Howe e , all hei implemen a ions su e om he same illness— hey a e by a he mos expensi e e mions in compu a ional e ms. As a consequence, i s use is ese ed o si ua ions whe e chi al symme y is needed wi h conside able p ecision. The e exis wo di e en solu ions ha a e b oadly used: hey a e called o e lap [82–84] and domain-wall [85,86] e mions. In he i s case, he e mionic ma ix is cons uc ed by ope a ing wi h a Wilson-like ma ix— he esul being a cos ly non-spa se ma ix. Fo domain-wall e mions, an ex a i h dimension o in ini e ex en is needed o p ese e chi al symme y. Since in p ac ice his is implemen ed as an addi ional dimension in a ini e la ice, a mild iola ion o he symme y s ill emains, al hough i can be con olled. 26 CHAPTER 1. THE LATTICE APPROACH 1.3.5 Mon e Ca lo: ensembles and obse ables Summing up all he p e ious conside a ions, we ha e now a ully egula ized e - sion o QCD. The eby, he o iginal exp essions o he pa i ion unc ion (1.2) and he acuum expec a ion alue (1.3) o a gi en obse able acqui e now a well-de ined ma hema ical meaning. I we now wan o conside , e.g., a pu ely gluonic obse able O(U), we should compu e he ollowing in eg al: hOi =RDUO(U)e−Sla (U) R la hcalDUe−Sla (U).(1.32) The abo e in eg a ion measu e DUis composed by he p oduc o as many indi- idual Haa measu es as links a e in he gi en la ice, which amoun o V≡QiLi imes he dimension o he la ice. So, o almos any size ha we can hink o , (1.32) con ains a highly mul idimensional in eg al, wi h a as con igu a ion space ha , e en o small la ices, can no be ully explo ed by any compu a ional means. The e exis howe e a class o nume ical app oaches, commonly e e ed o as Mon e Ca lo me hods, ha a e capable o p ope ly sampling hese kind o spaces— o apply hem o he la ice egula iza ion was, in ac , one o he key p oposals o Wilson [40]. In o de o compu e he expec a ion alue o (1.32), a i s Mon e Ca lo ap- p oach could consis in gene a ing a andom sample o link con igu a ions, i.e., a collec ion o ensemble o gauge ields Ui, each one con aining he in o ma ion o each indi idual link Uµ(n) in he la ice. Then, hOi could be compu ed as a weigh ed a e age o e he p e ious ensemble, wi h weigh e−Sla (U). In a simila ashion, s a is ical e o s could be compu ed by s anda d echniques. Howe e , his app oach is oo nai e o he p oblem a hand, since he exponen ial ac o in (1.32) highly supp esses a g ea majo i y o he con igu a ion space elemen s; in o he wo ds, only a small subse o he whole space con ibu es signi ican ly, and a nai e andom sampling o he space would miss he a ea o in e es . In o de o ix his issue, impo ance sampling is o be applied: again, an ensemble o con igu a- ions Uiis gene a ed, bu hey should be selec ed wi h p obabili y e−Sla (Ui). Then, p o ided a well-dis ibu ed ensemble o Nelemen s is a ailable, an es ima o o he expec a ion alue o a gi en obse able can be s aigh o wa dly compu ed as ¯ O=1 N N X iO(Ui).(1.33) The o iginal p oblem is now shi ed o he gene a ion o he Uicollec ion. This can be achie ed by s a ing om a gi en ini ial con igu a ion, say U1, and ollowing a Ma ko chain p ocess, in which he selec ion o he nex con igu a ion is egula ed 1.3. THE FORMALISM 27 by a p obabili y ha depends only on he immedia e p e ious s a e. The idea is ha , e en i one s a s a om he meaning ul con igu a ions, he Ma ko p ocess should d i e he sys em o an equilib ium s a e, in which he dis ibu ion e−Sla (Ui) is ep oduced. To gua an ee ha , he ansi ion p obabili ies om each Ui o any o he Ujneed o be adequa ely de ined. I is su icien (bu no necessa y) o equi e he de ailed balance condi ion o be ul illed, so i we label he ansi ion p obabili y o he chain as p(Ui|Uj), p(Ui|Uj)e−Sla (Uj)=p(Uj|Ui)e−Sla (Ui)(1.34) would be equi ed o e e y iand j. In his case, i he sys em is also e godic— meaning e e y con igu a ion is accessible om any o he in a ini e numbe o s eps—i is gua an eed ha , wi h independence o he ini ial con igu a ion chosen, he Ma ko p ocess will each equilib ium and, consequen ly, a well-dis ibu ed ensemble will be gene a ed. Then, o elabo a e a pa icula algo i hm ha ollows he desc ibed Ma ko p ocess consis s hen in speci ying how he nex elemen o he chain is selec ed, and which ansi ion p obabili ies connec he con igu a ion space. Wi h e- spec o he la e poin , mos o he algo i hms used by he la ice communi y e i y he de ailed balance condi ion. Maybe he mos elemen a y o hem is Me opolis algo i hm, which accep s a change be ween Uiand Ujwi h p obabili y max {1, e−∆Sla }. This is exac ly he app oach ollowed in Chap e 3. Mo e in- ol ed al e na i es include he hea ba h algo i hm [44], which o en includes some o e elaxa ion s eps [87–89]. Fu he mo e, o he mo e gene al case in which e mions a e included in o he ac ion, one o he mos widely used algo i hms is he Hyb id Mon e Ca lo [90,91], which a each upda e combines a mic ocanonical e olu ion wi h a inal me opolis s ep o accep o ejec he p oposed change. In any case, he s ochas ic na u e o Mon e Ca lo me hods in oduce some un- ce ain y in o he compu ed obse ables, in he o m o s a is ical e o s associa ed o he co esponding es ima o s. Ne e heless, hese can be deal wi h wi hou much di icul y, jus by aking in o accoun wo undamen als. Fi s , ha he gene a ed ensemble has o be he malized, meaning ha su icien i e a ions need o be spen o each he equilib ium dis ibu ion o he Ma ko chain. This can be achie ed by moni o ing a se o obse ables ha allow o de e mine when he Ma ko e olu ion is s a iona y. Secondly, and e en mo e impo an , is he inhe - en co ela ion o he con igu a ions gene a ed by he upda e p ocess—gene ally a local algo i hm ha needs a high numbe o i e a ions o p oduce a new con igu- a ion a om he o iginal. In his case, he use o s anda d e o analysis ools, such as jackni e binning, o e en he di ec compu a ion o au oco ela ion imes o each obse able, allows o es ima e in a eliable way he s a is ical e o s o any compu ed obse able. 28 CHAPTER 1. THE LATTICE APPROACH 1.4 Reaching he con inuum and sou ces o e o Up o his poin , we ha e co e ed how con inuum QCD can be egula ized in o a space ime la ice whe e, aking ad an age o Mon e Ca lo me hods, i is possible o es ima e acuum expec a ion alues o gi en obse ables. The compu ed ob- se ables will depend in gene al on he dimensions o he la ice and on he ba e alues o he gauge coupling βand he N masses o he e mion species in ol ed. Al hough knowing hese alues can be enough o some applica ions, in mos cal- cula ions he objec i e is o ob ain a physically measu able quan i y ha can be e en ually con as ed wi h expe imen . This is no he case o he p esen hesis, since he wo k being p esen ed is no di ec ly conce ned abou he las pa o a gene al la ice calcula ion; howe e , in o de o gi e a gene al pe spec i e o wha a comple e compu a ion would equi e, and o he sake o comple eness, he ea e we p oceed wi h a b ie o e iew o he issues in ol ing he las s eps o he la ice app oach. Ob aining a physical p edic ion om ba e la ice esul s equi es a somewha in ol ed limi p ocedu e; no only he la ice spacing ashould anish— his would be he con inuum limi —bu also he ba e couplings should go o hei physical alues. In o he wo ds, he physical poin needs o be eached; i no , he simula ed heo y would be desc ibing an al e na i e scena io, one wi h, e.g., di e en had on masses. In ac , he couplings o he heo y a e no di ec ly obse able (since con inemen p e en s qua k ields o mani es ou side a colo -single , he mass o he qua k is no an obse able and depends on he eno maliza ion scheme). In place, o he obse ables, such as had on masses a ios—which o cou se depend on he a o emen ioned pa ame e s—a e o be compu ed in he la ice, so hey can be compa ed wi h he expe imen al alues. Thus, in o de o compu e a gi en quan i y, i is necessa y o keep ack o se e al addi ional obse ables, one o each o he ee pa ame e s—o la o species— ha a e included in o he calcula ion. Mo o e , i should be no ed ha he la ice spacing ais no a ee pa ame e o he la ice heo y, bu ano he quan i y ha needs o be measu ed wi hin he la ice. The abo e scena io can be summa ized as ollows: we need o ake he con in- uum limi , a→0, while d i ing a se o physically measu able quan i ies o hei expe imen ally de e mined alues. Mo eo e , since doing he compu a ions on a la ice o in ini e ex en is ou o each, he p ocedu e is o be pe o med on a 4D box o ini e physical size.5In o de o ake he a→0 limi , i is equi ed o e- pea he compu a ions a di e en la ice sizes, keeping he physical size o he box cons an . In his way, la ices wi h di e en alues o he la ice spacing a—and consequen ly o he numbe o nodes N—a e compu ed, while he olume o he 5In ac , epea ing he calcula ions in se e al box sizes allows o ex apola e he esul s o he he modynamic limi . 1.4. REACHING THE CONTINUUM AND SOURCES OF ERROR 29 box a4Nis kep cons an . I should be ecalled ha while Nis a ee pa ame e , ais an obse able ha needs o be measu ed (depending no only on he gauge coupling bu also on he inclusion o e mionic species). So, as a i s consequence, keeping he physical olume cons an is no a comple ely i ial issue. The same p oblema ic ha appea s when ying o ix he olume o he 4D box applies o e e y physical obse able ha is in ended o be kep cons an (o d i en o a sui able alue, ob ained om expe imen al sou ces). To achie e his, essen ially wo op ions a e possible: ei he he ee pa ame e s a e uned in an i e a i e p ocess—one la ice a a ime up o he physical poin —o he compu a- ions a e epea ed o se e al alues o hese pa ame e s, and hen in e pola ed o he desi ed poin . A less ideal a ia ion o he la e is o each he physical poin by means o an ex apola ion. This was in ac he ule du ing se e al decades o la ice nume ical compu a ions, since he di e en ypes o e mion disc e iza ions ( e iewed in Sec ion 1.3.4) ha e many di icul ies in eaching a su icien ly low mass o he pion.6 As a inal ema k, i is impo an o s ess ha any ull compu a ion on he la ice should include a ca e ul analysis o all sou ces o e o s. Fi s , s a is ical e o s a e in oduced by he Mon e Ca lo e alua ion p ocess. These a e howe e ela i ely easy o deal wi h, since hey can be es ima ed by s anda d me hods and, e en mo e impo an ly, can be sys ema ically educed by inc easing he compu a- ion ime. A mo e challenging obs acle is in es ima ing he di e en issues leading o sys ema ic e o s, specially when a high p ecision is desi ed—and p e iously unno iced sys ema ic e o s can become big enough o be aken in o accoun . A non-exhaus i e bu common eco d o he di e en sou ces o sys ema ics consid- e ed would include he e o s associa ed wi h he uning o in e pola ion o he physical poin and he con inuum limi p ocess. Usually less se e e a e hose e- la ed wi h he he modynamic limi — ini e olume e ec s—o wi h he exclusion om he ac ion o elec omagne ic e ec s, which, gi en ha he qua ks a e elec- ically cha ged pa icles, should be accoun ed o . In gene al, any app oxima ion assumed should accoun o i s co esponding e o —as, e.g., he case o some la - ice e mion ypes which conside uand dqua k ba e masses as degene a e, and hus a e equi ed o accoun o he co esponding isospin b eaking e ec s. 6Al hough Ginspa g-Wilson e mions p ese e chi al symme y, hey su e om a la ge com- pu a ional o e head ha spoils hei ad an age wi h espec o o he e mion ypes, when each- ing a physical pion mass is conside ed. 36 CHAPTER 2. TOPOLOGY IN QCD pieces, Lθ-QCD =LQCD +iθ g2 64π2µνρσFa µνFa ρσ,(2.8) whe e LQCD s ands o he s anda d exp ession (1.4), which is bo h CP conse ing and eal. In con as , he second piece in he igh hand side o (2.8) b eaks CP— as was s essed in he p e ious sec ion—and, mo e impo an ly o he upcoming discussion, is a pu ely imagina y numbe . The opological na u e o he θ e m has di ec physical e ec s: he acuum o he heo y canno be cons uc ed as a quan um luc ua ion a ound a classical de ini e s a e [97]. The e o e, in o de o p ope ly analyze in which way QCD depends on he θpa ame e , a non-pe u ba i e app oach is equi ed. Unde his ci cums ances, i would seem adequa e o deal wi h he inclusion o he ex a e m Lθwi hin he la ice egula iza ion o malism. In p inciple, i would su ice o ind a p ope disc e iza ion o he opological cha ge Q, and include i s e ec s in he co esponding Mon e Ca lo algo i hm. Incon enien ly, his is a om being enough i non- anishing alues o he acuum angle a e o be explo ed, since he θ e m amoun s o a complex phase in he ac ion ha gi es ise o a se e e sign p oblem. Usual impo ance sampling me hods a e o no use in his si ua ion, and wo ka ounds need o be ound. A p esen , much o he achie ed p og ess in his opic in ol es compu a ions o opological quan i ies a θ= 0, whe e s anda d MC me hods a e s ill applicable. Al hough he de ini ion o he opological cha ge on he la ice is a sub le ques ion, i is possible o compu e quan i ies such as he opological suscep ibili y χ, e en wi h some di icul ies [102]. Howe e , li le p og ess has been achie ed in he las decades in wha conce ns he s udy o he θ > 0 case. I is wo h no ing ha he inclusion o he θ e m is no he only example o a sign p oblem being induced by a complex componen in o he ac ion o QCD. Ano he majo example is gi en by ini e densi y QCD, which akes place when a non-ze o chemical po en ial e m is included in o he e mionic ma ix—a equi ed addi ion when conside ing high ba yonic densi ies. This, in ac , is a ele an scena io which a ec s se e al a eas. I is needed in he s udies o as ophysical objec s such as neu on s a s. Mo eo e , in ea ly Uni e se in es iga ions, dealing wi h ex emely high alues o bo h empe a u e and densi y is equi ed. And, las bu no leas , pa icle collide acili ies ep oduce hese condi ions in hea y ion expe imen s. Al hough, hanks o asymp o ic eedom, pe u ba i e expansions can o e an insigh o some limi ing cases—namely high ene gy, which ansla es in high To high µ—and he µ= 0 case is accessible o he s anda d impo ance sampling echniques, almos he ull µ−Tphase diag am is beyond he each o hese app oaches. In bo h o he cases abo e, conside ing a complex ac ion esul s in he appea - ance o a sign p oblem, which, as i was wa ned a he beginning o his hesis, 2.3. TRYING TO OVERCOME THE SIGN PROBLEM 37 cons i u es one o he Millenium p oblems—in pa icula one ha is no expec ed o be sol ed wi h a posi i e ou come, which o ou in e es s would be P=NP. The e o e, he communi y has di ec ed hei e o s owa ds he de elopmen o di e en al e na i es ha y o e ade, o in some cases amelio a e, he sign p ob- lem in he sys ems o in e es , i.e., in QCD o QCD-like models. In he nex sec ion, we men ion some o he mo e popula s a egies, including a b ie e iew o he me hods de eloped by Azcoi i e al [12, 13], which in ac a e applied in Chap e 4, when econs uc ing he θdependence o he Schwinge model. 2.3 T ying o o e come he Sign P oblem In o de o p og ess in he s udy o complex ac ion sys ems, di e en me hods ha e been de eloped o e he yea s. In he cases whe e he sign p oblem is mild enough, such as in ini e densi y QCD o small alues o he chemical po en ial µ, se e al s a egies can be ollowed wi h success. Pe haps he mo e s aigh o wa d ap- p oach is he echnique known as eweigh ing. I in oduces an auxilia y pa i ion unc ion wi h non-nega i e densi y, which is used o e o mula e he expec a ion alue o a gi en obse able—compu ed in he o iginal ensemble—in e ms o o he expec ed alues ha a e o be de e mined wi hin he auxilia y ensemble. Indeed, his change does no elimina e he p oblem, since he compu a ional cos o his me hod escala es exponen ially wi h he olume o he sys em; howe e , i is use ul when applied o small sys ems, o in ce ain egions o he pa ame e space, such as QCD wi h a small chemical po en ial µ, whe e he e ec s o he sign p oblem a e a om being se e e. In his scena io, al e na i e app oaches include Taylo expansions a ound µ= 0 and analy ic con inua ions om pu ely imagina y chem- ical po en ial, al hough i s scope is hea ily bounded by he p esence o a SSP; o an ex ended discussion on his opic, we ecommend he eade [103,104]. Apa om he abo e me hods, ha p o ide some insigh o mild sign p oblem egions—bu ail o deli e o he wise—o he p ocedu es ha ha e, in p inciple, g ea e scope, ha e been de eloped o e he las decades. In ch onological o de , he i s is Complex Lange in. The o iginal wo ks o Pa isi and Klaude p oposed o gene alize he Lange in equa ion o malism o he case o a complex- alued dis ibu ion [9,10]. E en when igo ous p oo s we e lacking—e.g., he exis ence o a s a iona y solu ion was a conjec u e— he echnique seemed o gi e co ec esul s in some cases [105]. Howe e , in o he sys ems he algo i hm ailed o con e ge, o i did bu o he w ong limi [106]. In e es in he me hod was ebounded when Be ges and S ama escu ealized ha he ins abili ies o he Lange in e olu ion can be deal wi h i he s epsize is educed enough [11]. In ac , using an adap a i e s epsize elimina es his p oblem [107]. No wi hs anding i s ecen successes, he app oach s ill has some ca ea s ha need o be add essed, since in some scena ios 38 CHAPTER 2. TOPOLOGY IN QCD i con inues o con e ge o a w ong esul . Cu en e o s a e de o ed o iden i y unde which condi ions he p ope solu ion can be eached, and wha mechanisms can be used o moni o i he Lange in e olu ion is o be us ed; o a e iew on his opic, we e e o [108]. Wi h a special ocus on sys ems wi h a opological e m in he ac ion, a di e en s a egy was p oposed by Azcoi i e al in 2002 [12]. In summa y, he opological cha ge dependence on θis econs uc ed om he p obabili y dis ibu ion unc ion a θ= 0, which is o be compu ed om simula ions a pu ely imagina y alues o θ, possible since in his case he ac ion becomes eal. These esul s a e o be adjus ed o a sui able analy ical exp ession ha , once in eg a ed, allows o ob ain q(θ) wi h he help o mul ip ecision algo i hms. This app oach p o ed o wo k well in a numbe o sys ems, including he one dimensional Ising model and he U(1) compac model in wo dimensions, gi ing p edic ions o CP3and, in a la e wo k, o CP9—a model ha , as QCD, exhibi s con inemen and asymp o ic eedom [109]. Ne e heless, he scope o he app oach ge s hampe ed by he ac ha i canno econs uc a non-mono one o de pa ame e [21]. In o he wo ds, i a gi en model b eaking CP ge s his symme y es o ed a θ=π, hen q(θ) needs o dec ease a some poin ; in his si ua ion, he me hod ails o ep oduce his beha io and a la ening is obse ed—which, on he o he hand, i is gene ally a oided in models wi h (spon aneously) b oken symme y a θ=π. Ini ially wi h he aim o c osschecking he abo e me hod, an al e na i e ap- p oach, using he same inpu —i.e., Mon e Ca lo simula ions a imagina y alues o he acuum angle θ—was p oposed in 2003 [13]. In his case, an addi ional assump ion needs o be made, namely ha in o de o econs uc he ull θde- pendence, no c i ical poin s a e allowed, excep a mos a θ=π. This poses a se e e obs acle o i s applicabili y in ini e densi y QCD, whe e a ich phase diag am is expec ed, bu is in p inciple well adap ed o θQCD, whe e only one phase ansi ion—i any—is su mised, p ecisely a θ=π[102]. Besides he la e assump ion, a pa icula ex apola ion is equi ed, ha needs ce ain obse ables o a y as slowly as possible. Fo his eason, he me hod is expec ed o wo k well, among o he s, in asymp o ically ee gauge heo ies. This app oach, which has al eady ob ained good esul s in a a ie y o models [109–111], has been applied in Chap e 4 o econs uc he θdependence o he massi e Schwinge model; o a mo e ex ensi e e iew, co e ing he speci ics o he ac ual implemen a ion, we e e he eade o Sec ion 4.3. Finally, o he ecen app oaches o complex ac ion sys ems can be men ioned, such as Le sche z himbles [14–16] o he densi y-o -s a es o LLR me hod [17–19]. Thei de ails a e beyond he scope o his hesis, since hei applica ion o QCD wi h a opological e m seems, o now, emo e. Chap e 3 Ising model wi h a θ e m In his chap e we p esen ou wo k on he wo-dimensional an i e omagne ic Ising model wi h a pu ely imagina y magne ic ield, which can be in e p e ed as a oy model o he usual θphysics, and ha was published in [28]. Ou mo i a ion, as i was an icipa ed a he beginning o his hesis, is wo old. Fi s , we p e end o p o ide a benchma k calcula ion in a sys em which su e s om a s ong sign p oblem, so ha ou esul s can be used o es Mon e Ca lo me hods de eloped o ackle such p oblems. In second place, we wan o es he p edic ions o he me hod de eloped by Azcoi i e al. [13] ega ding his sys em [21], since i s pe o mance wi h he expec ed non- i ial phase diag am could en ail addi ional obs acles o i s eliable applica ion in o he scena ios. In he o ecoming sec ions, we jus i y he choice o model and e iew hei undamen als. Then, we discuss he analy ical echniques applied, including he exac compu a ion o he i s eigh cumulan s o he expansion o he e ec i e Hamil onian in powe s o he in e se empe a u e, which allows o calcula e phys- ical obse ables o a la ge numbe o deg ees o eedom wi h he help o s an- da d mul i-p ecision algo i hms. Finally, we epo accu a e esul s o he ee ene gy densi y, in e nal ene gy, s anda d and s agge ed magne iza ion, and he posi ion and na u e o he c i ical line, which con i m he mean- ield quali a i e pic u e o [21], and which should be quan i a i ely eliable, a leas in he high- empe a u e egime, including he en i e c i ical line. 3.1 Why Ising? As has been a gued along he i s chap e s o his wo k, nume ical simula ion o sys ems wi h a se e e sign p oblem is one o he majo challenges o high-ene gy heo is s—a s a emen which is also alid o hei solid-s a e colleagues. I we deno e he mic oscopic s a es o a gi en physical sys em by s, and he he mody- 39 40 CHAPTER 3. ISING MODEL WITH A θTERM namics o such sys em is desc ibed by a pa i ion unc ion o he o m Z=X s P(s),(3.1) we say ha he sys em in ques ion p esen s a sign p oblem i he “weigh s” P(s) a e no eal and posi i e: This implies ha we canno in e p e P(s) as a p ope p obabili y dis ibu ion, and he s anda d, e icien Mon e Ca lo algo i hms canno be applied. No all sign p oblems a e equally se e e. Le us es ic ou sel es o simplici y o he case whe e he P(s) a e eal bu no posi i e de ini e1. One can easily de ise a eweigh ing algo i hm ha uses he absolu e alue |P(s)|as he weigh o each s a e, and shi s he sign o P(s) in o he obse ables. Now a s anda d Mon e Ca lo me hod is applicable, and in he limi o in ini e s a is ics we should ob ain he co ec esul . Wi h ini e s a is ics, howe e , a key quan i y is he he modynamic a e age o he sign o each con ibu ion o he pa i ion unc ion, ha is, hsign(P(s))i. I his quan i y goes o ze o exponen ially wi h he olume, hsigni ∝ e−αV , hen we would need an exponen ial amoun (in he olume o he sys em V) o s a is ics o ge co ec esul s, which is o cou se impossible in p ac ice. In his case we say ha he sign p oblem is se e e. Beyond QCD a ini e ba yon densi y o QCD wi h a opological e m in he ac- ion, he e exis o he physically ele an sys ems which su e om a SSP. Some o he mos popula examples include chains o quan um spins wi h an i e omagne ic in e ac ions, he wo-dimensional O(3) non linea sigma model wi h a opological e m o he Hubba d model. The exis ence o a SSP is he main eason o he li le p og ess made on he heo e ical unde s anding o hese physical sys ems ou side o phenomenological models. In o de o check no el Mon e Ca lo me hods designed o ackle such p ob- lems, i is highly desi able o ha e a se o benchma k calcula ions as ex ensi e as possible. Fo e y ew sys ems an analy ic solu ion is known, o example, he one-dimensional an i e omagne ic Ising model wi h an imagina y magne ic ield, he wo-dimensional compac U(1) model wi h opological e m, o he wo- dimensional Ising model wi h an imagina y magne ic ield h=iπ/2. In a ew o he cases he sign p oblem can be a oided by e o mula ing he physical sys em wi h new deg ees o eedom, aking ad an age o he ac ha a good choice o hese de- g ees o eedom p o ides an equi alen physical sys em ee om he sign p oblem, which can he e o e be simula ed by s anda d me hods; see e.g. [112] o a ecen discussion on his dualiza ion app oach. Un o una ely his idea wo ks only in a ew cases which, un il now, a e no he mos in e es ing physical sys ems—indeed none o he examples p e iously men ioned ha e been sol ed wi h his idea. Wi hin he abo e scena io, ou in en ion in his wo k is o p o ide a benchma k calcula ion o a sys em o which we do no ha e an analy ic solu ion a ailable, 1The discussion o complex weigh s does no add any undamen al di icul y. 3.1. WHY ISING? 41 no a e o mula ion ha a oids he sign p oblem. We s udy he wo-dimensional an i e omagne ic Ising model wi h a pu ely imagina y magne ic ield, which can be hough o as a oy model o he usual θphysics. Indeed he Euclidean pa i ion unc ion o QCD wi h a non anishing θ e m can be w i en in he o m ZV(θ) = X n pV(n)eiθn (3.2) whe e n, he opological cha ge, is an in ege , and pV(n) is, up o a no maliza ion, he p obabili y o he opological sec o na θ= 0. This has he same s uc u e as he pa i ion unc ion o he an i e omagne ic Ising model in an ex e nal pu ely imagina y magne ic ield, as we will see in de ail la e on, and we expec ha he SSP in bo h sys ems should also be simila . This sys em was s udied in [20] by loca ing he ze os o he pa i ion unc- ion in he complex empe a u e-magne ic ield plane, and hey ound, o pu ely imagina y magne ic ield, a ich phase s uc u e wi h wo phases cha ac e ized by a anishing (non anishing) s agge ed magne iza ion, sepa a ed by a phase ansi- ion line. We s udy his sys em by an exac cumulan expansion o eigh h o de , ollowed by he analy ic compu a ion o he pa i ion unc ion and o he physical quan i ies o a la ge numbe o deg ees o eedom wi h he help o a s anda d mul ip ecision algo i hm. This amoun s essen ially o he compu a ion o he e - ec i e Hamil onian up o o de T−8, and he e o e is expec ed o wo k well in he high- empe a u e egime, and we p o ide s ong e idence ha his is indeed he case. Ou esul s a e consis en wi h [20], and ex end he esul s o [21], ob ained h ough he applica ion o algo i hms de eloped in [12,13], and h ough a mean- ield analysis. We a e able o ob ain a mo e p ecise quan i a i e de e mina ion o he ansi ion line sepa a ing he pa amagne ic and an i e omagne ic phases o he model. Fo some sys ems wi h a SSP, we know a p io i ha he pa i ion unc ion will be posi i e, o example sys ems in he mal equilib ium wi h a (He mi ian) Hamil onian desc ip ion. Such is he case in a quan um ield heo y wi h a θ e m. In he oy model we s udy he e, al hough we do no ha e a igo ous p oo in his case,2we ha e e idence ha , a leas in he egion whe e he app oxima ion we use is alid, he pa i ion unc ion is indeed posi i e (i is i ially always eal). Such e idence is wo old. Fi s , we can p o e igo ously ha up o he i h cumulan , he pa i ion unc ion is indeed posi i e. Un o una ely we ha e no been able o ex end his p oo o highe cumulan s, bu in ou mul ip ecision calcula ions wi h up o eigh cumulan s, we ha e ne e seen an ins ance whe e 2This would imply a non i ial es ic ion on he posi ion o he Lee-Yang ze os o he an i e omagne ic Ising model. To he bes o ou knowledge, e y li le is igo ously known abou such ze os. 42 CHAPTER 3. ISING MODEL WITH A θTERM he pa i ion unc ion is nega i e o anishes. This is highly non i ial: I ins ead o a cons an imagina y magne ic ield we y, o example, o pu a s agge ed imagina y ield in ou la ice ( his is o cou se equi alen o he e omagne ic model wi h a cons an imagina y ield), we immedia ely ge a luc ua ing sign o he pa i ion unc ion. Second, he e ha e been s udies loca ing he Lee-Yang ze os o he an i e o- magne ic wo-dimensional Ising model up o 142la ices [113], and in 12 ×13 la ices [20]. Up o ha size he e is no sign o any ze os cu ing he imagina y axis a any empe a u e. Whe eas his by no means amoun s o a igo ous p oo , we belie e i p o ides a s ong indica ion ha , a leas in he egion o in e es o ou wo k, his model should ha e a posi i e pa i ion unc ion. He ea e , Sec ion 3.2 is de o ed o o mula e he model and o ecall he main ing edien s and esul s o he mean- ield app oxima ion de eloped in [21]. In Sec. 3.3 we in oduce he cumulan expansion, epo he analy ical esul s o he i s eigh cumulan s in he wo-dimensional model, and w i e he analy ical exp essions o he ee ene gy and mean alues o in e es ing physical quan i ies. The esul s o he s agge ed magne iza ion, suscep ibili y, and phase diag am o he model a e epo ed in Sec. 3.4, whe e we also compa e ou esul s a h= 0 and iπ/2 wi h he analy ical solu ions o [114–116]. In Sec. 3.5 we epo ou conclusions. The echnical de ails o he analy ical compu a ion o he cumulan expansion can be ound in Appendix A. 3.2 Two-dimensional Ising model The Ising model [20, 114–119] has been s udied o a long ime now, and i has known analy ical solu ions in he one-dimensional case a any ex e nal magne ic ield h[117], and in wo dimensions only o he case wi hou magne ic ield h[114] and o h=iθ/2 = iπ/2 [115, 116]. The model wi h a pu e imagina y magne ic ield su e s om a SSP in any numbe o dimensions. In addi ion o ha , he expec ed phase diag am o d≥2 is non i ial [21], making he econs uc ion o he θdependence o he obse ables e en mo e challenging. All his makes he model a good heo e ical labo a o y o es new me hods designed o deal wi h he SSP. I is he e o e wo hwhile o ca y ou a de ailed s udy o his model a pu ely imagina y magne ic ield, pa icula ly because li le p og ess has been achie ed on econs uc ing he θdependence o he obse ables, apa om he analysis o [21] and he ecen s udy in [120]. 3.2. TWO-DIMENSIONAL ISING MODEL 43 The pa i ion unc ion o he model, ollowing he con en ions o [21], is: Z=X {si} exp FX <ij> sisj+iθ1 2X i si!.(3.3) The hal magne iza ion M 2≡1 2X i si,(3.4) is an in ege aking any alue be ween −N/2 and N/2, whe e Nis an e en numbe deno ing he o al numbe o spins in he la ice. I is in his sense ha we iden i y M/2 wi h a opological cha ge and ega d he imagina y magne ic ield e m in he ac ion as a θ e m. I is impo an o men ion ha , om now on, we will conside only he an i e omagne ic case F < 0, since he model wi h imagina y ield does no de ine a uni a y heo y o a bi a y alues o he e omagne ic coupling [115,121]. As we shall see in de ail in he nex sec ion, by di iding he ec angula la ice in o wo subla ices, in oducing he espec i e magne iza ions M1and M2, making a cumulan expansion and keeping only he i s cumulan , we a i e a he ollow- ing app oxima ion o he pa i ion unc ion (whe e ddeno es he dimensionali y o he la ice): Z1c(F, θ) = X {si} exp iθM1+M2 2+ 4Fd NM1M2.(3.5) We ecall now he mean- ield analysis ca ied ou in [21]. The esul ing pa i ion unc ion, ZMF (F, θ) = X {si} exp iθM1+M2 2−Fd N(M1−M2)2,(3.6) is di e en om Eq. (3.5). Howe e , i can be seen o gi e he same quali a i e esul s o he obse ables and he phase diag am. In his ega d, we will conside he i s -cumulan expansion Z1cas a mean- ield app oxima ion o Z, and he gene al expansion i sel as an imp o emen o i , a leas o small F, whe e he expansion is expec ed o con e ge. Applying s anda d saddle-poin echniques o he mean- ield pa i ion unc ion [21], one ob ains he F−θphase diag am shown in Fig. 3.1. A second o de c i ical line, dFc=1 2cos2θc 2,(3.7) sepa a es wo di e en phases: a s agge ed one, wi h hmsi 6= 0, o F > Fc(θ), and a pa amagne ic one, wi h hmsi= 0, o F≤Fc(θ). 44 CHAPTER 3. ISING MODEL WITH A θTERM 0 0.1 0.2 0.3 0.4 0.5 0π/2π hmsi= 0 hmsi 6= 0 d|F| θ dFc(θ) Figu e 3.1: Phase diag am o he mean- ield app oach o [21] o he an i e omag- ne ic Ising model in he F−θplane. 3.3 Cumulan expansion and obse ables Ou in e es is ocused on he an i e omagne ic model, whe e he s agge ed mag- ne iza ion is a good o de pa ame e . F om now on we will wo k wi h a ec angula wo-dimensional la ice, al hough he me hod is easily gene alizable o any numbe o dimensions. We di ide he la ice in o wo subla ices Ω1and Ω2in a chess- boa d ashion. In he wo-dimensional la ice his means ha i iand jindex, espec i ely, he ow and he column o a gi en spin, his spin will be in he i s (second) subla ice i he sum i+jis e en (odd). Fo simplici y we will equi e bo h leng hs o he la ice o be e en. Deno ing by N he o al numbe o poin s in he la ice, we de ine he magne iza ion densi ies m1and m2as mj≡Mj N/2≡Pi∈Ωjsi N/2j= 1,2,(3.8) and he densi y o s agge ed magne iza ion is ms≡m1−m2 2.(3.9) Le us deno e by g(m1, m2) he numbe o mic os a es wi h magne iza ion 3.3. CUMULANT EXPANSION AND OBSERVABLES 45 densi ies m1and m2in subla ices Ω1and Ω2, espec i ely, ha is, g(m1, m2) = X {si} δ X i∈Ω1 si−M1!δ X i∈Ω2 si−M2!.(3.10) A i ial compu a ion gi es: g(m1, m2) = N/2 N1+N/2 N2+,(3.11) wi h Nj+≡N(1 + mj)/4 o j= 1,2. Now, by es ic ing ou sel es o he se o con igu a ions wi h gi en magne iza ion densi ies m1and m2, i is s aigh o wa d o de ine he expec a ion alue o a gene al obse able O({si}) wi hin his subse — i.e., a ixed m1, m2—as: hOim1,m2≡1 g(m1, m2)X {si} δ(X i∈Ω1 si−M1)δ(X i∈Ω2 si−M2)O({si}).(3.12) Then, he sum o e all possible spin con igu a ions in he o iginal pa i ion unc ion (3.3) can be pa ially summed up—a leas o mally—g ouping oge he sec o s wi h equal magne iza ion densi ies m1, m2. By doing so, and aking in o accoun he abo e de ini ions, he e o mula ed pa i ion unc ion akes he ollowing o m: Z=X m1,m2 g(m1, m2)*exp iθ 2X i si+FX <ij> sisj!+m1,m2 .(3.13) The θ e m in Eq. (3.13) is jus iθ (m1+m2)N/4, and he e o e cons an a ixed m1and m2; we can ake i ou o he expec a ion alue, a i ing a Z=X m1,m2 g(m1, m2)e1 4Niθ(m1+m2)*exp FX <ij> sisj!+m1,m2 .(3.14) We canno e alua e exac ly he expec a ion alue in Eq. (3.14), as ha would be equi alen o sol ing exac ly he model o a bi a y alues o he ex e nal ield. Ins ead we pe o m a cumulan expansion and unca e a a gi en o de . Le us ecall he de ini ion: e X≡exp ∞ X n=1 κn n n!!,(3.15) whe e he n h cumulan κnis an n h deg ee polynomial in he i s nnoncen al momen s o X, gi en by he ollowing ecu sion o mula: κn=µ0 n− n−1 X m=1 n−1 m−1κmµ0 n−m, µ0 n≡ hXni.(3.16) 52 CHAPTER 3. ISING MODEL WITH A θTERM 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 e|ns (F) |F| k=1 k=4 k=8 Analy ic Figu e 3.6: Nonsingula pa o he in e nal ene gy a θ=π, N = 2000. −0.25 −0.2 −0.15 −0.1 −0.05 0.1 0.2 0.3 0.4 0.5 0.6 c (F) |F| k=1 k=4 k=8 Analy ic Figu e 3.7: Speci ic hea a θ=π, N = 2000, plo ed agains he analy ical exp ession. 3.5. CONCLUSIONS 53 0 0.5 1 1.5 2 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 c (F) |F| k=1, N=2000 k=4, N=2000 k=8, N=2000 k=8, N=6000 Analy ic Figu e 3.8: Speci ic hea a θ= 0, plo ed agains he analy ical solu ion. A θ= 0, Fc= log(1 + √2)/2≈0.4407. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 hm2 si |F| N = 400 N = 800 N = 1600 N = 3200 Figu e 3.9: hm2 sicu es a θ= 2, k = 8. Solid lines a e jus a guide o he eye. 54 CHAPTER 3. ISING MODEL WITH A θTERM 0 0.5 1 1.5 2 2.5 3 3.5 4 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 dhm2 si/dθ |F| N = 100 N = 200 N = 400 N = 800 N = 1600 N = 3200 Figu e 3.10: Scaling o dhm2 si/dθ a θ= 2, k = 8. Solid lines a e a guide o he eye. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 π/4π/2 3π/4π0 |F| θ k = 1 k = 4 k = 8 Ma ee and Sh ock, 2008 Figu e 3.11: The c i ical line Fc(θ), compu ed as he maximum o dhm2 si/dθ a N= 2000. The maximal Fpoin s ob ained in [20] a e also shown. 3.5. CONCLUSIONS 55 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 c (F) |F| N = 400 N = 800 N = 1600 N = 3200 Figu e 3.12: Speci ic hea c wi h k= 8 and θ= 2. Solid lines a e jus a guide o he eye. magne iza ion as an o de pa ame e . The ini e-size scaling sugges s ha he wo phases a e sepa a ed by a con inuous phase ansi ion line. The posi ion o he c i ical poin a θ= 0 is in e y good ag eemen wi h he exac esul Fc= log(1 + √2)/2≈0.4407, and he ee and in e nal ene gy densi ies a θ=π ag ee also well wi h he analy ical p edic ion, a leas in he high- empe a u e egime, hus gi ing eliabili y o ou esul s in his egion. The e o e his model could be a good labo a o y o check p oposals o simula e physical sys ems a lic ed by a SSP. Mo eo e , we ha e con i med ha he sys em unde discussion has a mo e in ol ed phase diag am han in he expec ed θQCD case, which would ha e, a mos , a single c i ical poin in θ=π. In his sense, he econs uc ion me hod o [13], which when applied in [21] was capable o de ec ing he p esence o a c i ical egion, is s ongly suppo ed. 56 CHAPTER 3. ISING MODEL WITH A θTERM Chap e 4 Massi e 1- la o Schwinge model wi h a θ e m We analyze he e he massi e 1- la o Schwinge model wi h a θ e m and a quan- ized opological cha ge. Ou wo k, published in [29], elies on he app oach o Azcoi i e al in [13]. We a e able o calcula e he ull dependence o he o de pa ame e wi h θin a sys em ha includes dynamical e mions. Mo eo e , ou e- sul s a θ=πa e compa ible wi h Coleman’s conjec u e [22] on he phase diag am o his model. This chap e is o ganized as ollows: a e mo i a ing he opic in he i s sec ion, we summa ize some ele an ea u es o he Schwinge model wi h a opo- logical e m in Sec. 4.2. Since he p oposal [13] o analyze physical sys ems wi h a opological e m in he ac ion has been ound o be pa icula ly well sui ed o bypass he sign p oblem in asymp o ically ee gauge heo ies, we decided o apply i , and Sec. 4.3 con ains a b ie e iew o he me hod. In Sec. 4.4 we gi e some echnical de ails conce ning he la ice se up and he compu e simula ions pe - o med. Sec. 4.5 shows ou esul s o he opological cha ge densi y as a unc ion o θa se e al e mion masses and gauge couplings, and inally we end his chap e by epo ing ou conclusions. 4.1 Mo i a ion The na u e and o igin o da k ma e cons i u e one o he mos wide open ques- ions in mode n physics. To gain some insigh in his puzzling p oblem, i is highly desi able o elucida e he exis ence o new low-mass, weakly in e ac ing pa icles om a heo e ical, phenomenological and expe imen al poin o iew. As was ou - lined in Chap e 2, he ligh pa icle ha has ga he ed he mos a en ion is he axion, p edic ed by Weinbe g and Wilczek [99], and Wilczek [100] in he Peccei and 57 58 CHAPTER 4. SCHWINGER MODEL WITH A θTERM Quinn mechanism [98] o explain he absence o pa i y and empo al in a iance iola ions induced by he QCD acuum. The axion is one o he mo e in e es - ing candida es o make he da k ma e o he uni e se, and he axion po en ial, ha de e mines he dynamics o he axion ield, plays a undamen al ole in his con ex . The QCD axion model ela es he opological suscep ibili y χTwi h he axion mass maand decay cons an a h ough he ela ion χT=m2 a 2 a. The axion mass is, on he o he hand, an essen ial ing edien in he calcula ion o he axion abundance in he Uni e se. The e o e, a p ecise compu a ion o he opological p ope ies o QCD and o hei empe a u e dependence becomes o p imo dial in e es in his con ex . Unde s anding he ole o he θpa ame e in QCD and i s connec ion wi h he s ong CP p oblem is one o he majo challenges o high ene gy heo is s [122]. The calcula ion o he opological suscep ibili y in QCD is al eady a challenge, bu calcula ing he comple e po en ial equi es a s a egy o deal wi h he p esence o a highly oscilla ing e m in he pa h in eg al; in o he wo ds, one needs o ci cum en a se e e sign p oblem. In ac euclidean la ice gauge heo y, ou main non-pe u ba i e ool o s udying QCD om i s p inciples, has no been able o help us much because o he imagina y con ibu ion o he ac ion coming om he θ e m, ha p e en s he applicabili y o he impo ance sampling me hod [102]. This is he main eason why he only p og ess in he analysis o he ini e empe a u e θdependence o he acuum ene gy densi y in pu e gauge QCD, ou side o app oxima ions, educes o he compu a ion o he i s ew coe icien s in he expansion o he ee ene gy densi y in powe s o θ[123], and he si ua ion in ull QCD wi h dynamical e mions is, on he o he hand, e en wo se [124–129]. Much expe ience has been de eloped in he las yea s conce ning he s eng hs and weaknesses o he app oaches [12,13], which aspi e o e en ually o e come he sign p oblem in θQCD. As a ma e o ac , i has been applied success ully o he compu a ion o he acuum ene gy densi y and he opological cha ge densi y in a hand ul o in e es ing physical sys ems [21,109–111,130]. Ou pu pose in he p esen chap e is o ake ad an age o his expe ience o pe o m a i s s ep in he ambi ious p og am o compu ing he θdependence o he QCD acuum ene gy densi y. The eby, we analyze he θdependence o a oy model o QCD, he Schwinge model, on he la ice. S ic ly speaking, he Schwinge model in he con inuum is no asymp o ically ee, as QCD, since i is supe - eno malizable and he Callan- Symanzik β- unc ion anishes. Howe e , in he la ice e sion, since he con inuum coupling is dimension ul, he con inuum heo y is eached a in ini e in e se squa e gauge coupling β= 1/e2a2, much in he same way as ou -dimensional asymp o - ically ee gauge heo ies such as QCD. Fu he mo e he model is con ining [131], 4.2. THE MASSIVE SCHWINGER MODEL WITH A θTERM 59 exac ly sol able a ze o e mion mass, has non- i ial opology and shows explici ly he UA(1) axial anomaly [132] h ough a non- anishing alue o he chi al conden- sa e in he chi al limi , in he one- la o case. These a e basically he easons why his model has been ex ensi ely used as a oy model o QCD. I should be no ed ha , o wo dimensional sys ems such as he Schwinge model wi h a θ e m, he e exis nume ical me hods such as Hamil onian me hods [133–135] and he G assmann enso eno maliza ion g oup me hod [136] ha ha e been applied success ully. Howe e , such app oaches a e cu en ly only applicable o wo-dimensional sys ems, whe eas ou aim is o es a me hod ha should, in p inciple, be applicable also o ou -dimensional heo ies such as QCD. 4.2 The massi e Schwinge model wi h a θ e m The Schwinge model is Quan um Elec odynamics in 1+1-dimensions [137]. The euclidean con inuum ac ion eads S=Zd2x¯ ψ(x)γµ(∂µ+ieAµ(x)) ψ(x) + m¯ ψ(x)ψ(x) + 1 4F2 µν(x),(4.1) whe e mis he e mion mass and eis he elec ic cha ge o gauge coupling, which has he same dimension as m. A e a simple escaling o he ields he ac ion can be w i en as S=Zd2x¯ ψ(x)γµ(∂µ+iAµ(x)) ψ(x) + m¯ ψ(x)ψ(x) + 1 4e2F2 µν(x),(4.2) whe e Fµν(x) = ∂µAν(x)−∂νAµ(x) and γµa e 2×2 ma ices sa is ying he algeb a {γµ, γν}= 2gµν.(4.3) whe e gµν s ands o he Euclidean me ic enso . A he classical le el his ac ion is in a ian in he chi al limi unde he UA(1) global ans o ma ions ψ→eiαγ5ψ, (4.4) ¯ ψ→¯ ψeiαγ5,(4.5) leading o he conse a ion o he axial cu en JA µ(x) = ¯ ψ(x)γµγ5ψ(x).(4.6) 60 CHAPTER 4. SCHWINGER MODEL WITH A θTERM Howe e he axial symme y is b oken a he quan um le el because o he axial anomaly, as was discussed in de ail in Sec ion 1.3.3. The di e gence o he axial cu en is ∂µJA µ(x) = 1 2πµνFµν(x),(4.7) wi h µν he an isymme ic enso , and he e o e does no anish. The axial anomaly induces a opological θ e m in he ac ion o he o m S op =iθ 4πZd2xµνFµν(x),(4.8) whe e he opological cha ge Q=1 4πRd2xµνFµν(x) is an in ege . Ou pu pose is hen o analyze he θdependence o he model desc ibed by he ac ion (4.2)+(4.8) S=Zd2x¯ ψγµ(∂µ+iAµ)ψ+m¯ ψψ +1 4e2F2 µν +iθ 4πµνFµν.(4.9) A simple analysis o his model on he la ice sugges s ha i should unde go a phase ansi ion a some in e media e e mion mass mand θ=π, e en a ini e la ice spacing. Indeed he la ice model is analy ically sol able in he in ini e e mion mass limi (pu e gauge wo-dimensional elec odynamics wi h opological e m) [138, 139], and i is well known ha he densi y o opological cha ge ap- p oaches a non- anishing acuum expec a ion alue a θ=π o any alue o he in e se squa e gauge coupling β, exhibi ing spon aneous symme y b eaking. On he o he hand by expanding he acuum ene gy densi y in powe s o m, ea ing he e mion mass as a pe u ba ion [140], one ge s o he acuum expec a ion alue o he densi y o opological cha ge he ollowing θdependence: h−iqi=mΣsinθ +1 2m2sin (2θ) (χP−χS) + ··· ,(4.10) wi h Σ he acuum expec a ion alue o he chi al condensa e in he chi al limi and a θ= 0 (Σ = eγee/2π3/2in he con inuum limi ), and χPand χS he pseu- doscala and scala suscep ibili ies espec i ely. Equa ion (4.10) shows how he Z2 symme y a θ=πis ealized o de by o de in he pe u ba i e expansion o he opological cha ge in powe s o he e mion mass m, and he e o e a c i ical poin sepa a ing he la ge and small e mion mass phases is expec ed. Indeed he model was analyzed in he con inuum by Coleman in [22], whe e he conjec u ed he exis ence o a phase ansi ion a θ=π, and some in e media e e mion mass msepa a ing a ”weak coupling” phase ( e m<< 1), whe e he Z2 symme y o he model a θ=πis spon aneously b oken, om a ”s ong cou- pling” phase ( e m>> 1) whe e he Z2symme y is ealized in he acuum. This 4.3. COMPUTING THE ORDER PARAMETER AS A FUNCTION OF θ61 conjec u e was co obo a ed in [133, 134] using he la ice Hamil onian app oach wi h s agge ed e mions, and mo e ecen ly in [136] using he G assmann enso eno maliza ion g oup and Wilson e mions. 4.3 Compu ing he o de pa ame e as a unc- ion o θ To compu e he θdependence o he densi y o opological cha ge we use he app oach p oposed in e e ence [13]. The only assump ion in his app oach is he absence o phase ansi ions a eal alues o θexcep a mos a θ=π. The me hod is based in ex apola ing a sui ably de ined unc ion o he o igin. This unc ion u ns ou o be e y smoo h in all he cases conside ed up o now [21, 109–111], and his makes us con iden on he whole p ocedu e. He e we summa ize he main s eps. F om nume ical simula ions o ou physical sys em a imagina y alues o θ= −ih ( eal alues o h), which a e ee om he se e e sign p oblem, we compu e he densi y o opological cha ge q(−ih) as a unc ion o h, and in oduce he ollowing unc ions: z= cosh h 2,(4.11) y(z) = q(−ih) anh h 2 .(4.12) The p ocedu e o ind ou he densi y o opological cha ge a eal alues o θ elies on scaling ans o ma ions [13]. We de ine he unc ion yλ(z) as yλ(z) = yeλ 2z.(4.13) Fo nega i e alues o λ, he unc ion yλ(z) allows us o calcula e he o de pa- ame e  anh h 2y(z)below he h eshold z= 1. I y(z) is non- anishing o any posi i e z,1 hen we can plo yλ/y agains y. Fu he mo e, in he case ha yλ/y is a smoo h unc ion o yclose o he o igin, hen we can ely on a simple ex apola ion o y= 0. O cou se, a smoo h beha io o yλ/y canno be aken o g an ed; howe e no iola ions o his ule ha e been ound in he exac ly sol able models. 1E en hough he possibili y o a anishing y(z) o some alue z > 0 canno be comple ely excluded, i does no happen o any o he analy ically sol able models we know. 68 CHAPTER 4. SCHWINGER MODEL WITH A θTERM 0 0.5 1 1.5 2 0 0.01 0.02 0.03 0.04 0.05 0.06 γ yλ β = 2 β = 3 β = 4 m = 0.0 Figu e 4.4: Exponen γ o m= 0.0 and a ious coupling cons an s. The shaded a eas gi e an es ima ion o he ambigui y in he ex apola ion o yλ= 0. The con inuous ed line is he analy ic esul in he pu e gauge heo y, co esponding o in ini e e mion mass. We plo in Fig. 4.6 he esul s o each o he independen analysis o an in e al o θ. As can be seen, he e o s we would ob ain by a e aging he independen poin s a e ully consis en wi h he syn he ic-da a es ima ion. In Fig. 4.6 we p esen q(θ) a β= 3 o wo masses in he symme y es o ed phase, as well as a β= 2 and m= 0.5, in he symme y b oken phase (and also he co esponding analy ic esul s o he pu e gauge case a bo h alues o β o compa ison). In Fig. 4.7 we show he esul s o m= 0 and he h ee di e en alues o he coupling cons an we ha e simula ed. We can clea ly see he es o a ion o he symme y as we app oach θ=π. In Fig. 4.8 we show, o β= 3.0 and m= 0, he o de pa ame e q(θ) in he icini y o θ=π. Fi ing q(θ) nea θ=πin he symme y es o ed phase allows us o ex ac he exponen (π−θ), which is ela ed o γby =γ−1.6We p esen in Table 4.1 ou esul s o . 6The nume ical p ocedu e used o ex ac he wo exponen s is di e en , and he e o e he 4.5. RESULTS 69 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 y yλ β = 3.0 , m = 0.0 Figu e 4.5: Fi o y e sus yλ. Table 4.1: β m  2.0 0.0 0.67(4) 2.0 0.05 0.43(5) 3.0 0.0 0.92(7) 3.0 0.05 0.70(21) 4.0 0.0 0.94(19) To inish his Sec. we wan o discuss a li le bi mo e on he esul s o he massless Schwinge model epo ed in Fig. 4.7. I is well known ha he con inuum o mula ion o he massless Schwinge model shows no θdependence, because he θ e m in he ac ion can be canceled by an anomalous chi al ans o ma ion which esul s, al hough compa ible wi hin e o s, will also be di e en . 70 CHAPTER 4. SCHWINGER MODEL WITH A θTERM 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.5 1 1.5 2 2.5 3 q θ β = 3.0 β = 2.0 m = ∞ m = 0.5 m = 0.05 m = 0.0 Figu e 4.6: O de pa ame e as a unc ion o θ. The da a a m= 0.0 and m= 0.05 co espond o β= 3.0, whe eas he poin s a m= 0.5 co espond o β= 2. Blue poin s, co esponding o he esul s o ou independen uns, a e also shown, o p o ide a di e en es ima e o he e o . The con inuous line labeled m=∞is he pu e gauge analy ic esul o β= 3.0, whe eas he do ed line is he co esponding analy ic esul o β= 2.0. does no change he e mion-gauge ac ion i he e mion mass anishes. Hence he non- i ial θdependence o he densi y o opological cha ge shown in Fig. 4.7 may seem su p ising. Howe e , he massless s agge ed Di ac ope a o does no ha e exac ze o-modes, and he e o e, o a gi en gauge con igu a ion, a nonze o alue o he quan ized opological cha ge Qdoes no imply he exis ence o a co esponding numbe o ze o-modes in he s agge ed Di ac ope a o , as would be he case, o example, wi h he o e lap Di ac ope a o . Wha we should expec ins ead is ha , as we app oach he con inuum limi , he opological cha ge densi y anishes. This is indeed wha seems o happen, as is sugges ed by Fig. 4.9. 4.6. CONCLUSIONS AND OUTLOOK 71 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0 0.5 1 1.5 2 2.5 3 q θ m = 0.0 β = 2.0 β = 4.0 β = 3.0 Figu e 4.7: O de pa ame e as a unc ion o θ, a m= 0.0 and di e en coupling cons an s. 4.6 Conclusions and ou look All ou esul s a e compa ible wi h he s anda d lo e on his model, and in pa ic- ula wi h Coleman’s conjec u e on he exis ence o wo dis inc phases a θ=π, a symme y b eaking phase a la ge mass, and a symme y es o ed phase a small mass. Ou simula ions a e a p oo o concep , and a e no ex ensi e enough o de- e mine p ecisely he posi ion o he c i ical mass a θ=πo i s p ope ies in de ail. Bu he impo an poin is ha we ha e succeeded in calcula ing he ull dependence o he o de pa ame e in θin a gauge heo y wi h e mions and a quan ized opological cha ge, using a me hod ha should, in p inciple, wo k also in highe dimensional heo ies. 72 CHAPTER 4. SCHWINGER MODEL WITH A θTERM 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 2.8 2.85 2.9 2.95 3 3.05 3.1 q θ β = 3.0 , m = 0.0 Figu e 4.8: O de pa ame e as a unc ion o θnea θ=π. 4.6. CONCLUSIONS AND OUTLOOK 73 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.5 1 1.5 2 2.5 3 β2q θ β = 2 β = 3 β = 4 m = 0.0 Figu e 4.9: Rescaled opological cha ge densi y a m= 0.0 and di e en coupling cons an s. 74 CHAPTER 4. SCHWINGER MODEL WITH A θTERM Chap e 5 P elimina y esul s on wo- la o Schwinge —a pseudo e mionic app oach He ea e we p esen ou wo k wi h he massi e Schwinge model wi h a θ e m and wo dis inc e mionic species. Al hough he s a ing poin o he s udy consis s in he applica ion o he same Mon e Ca lo algo i hm ha was de eloped o Chap- e 4—a s anda d implemen a ion o Kogu -Susskind e mions wi h a Me opolis upda e—a sea ch o mo e e icien algo i hms p o es o be necessa y in o de o ully es he capabili ies o he q(θ) econs uc ion app oach o [13]. 5.1 Mo i a ion The s udy o he Schwinge model wi h a single la o o massi e e mions and he inclusion o a θ e m was ca ied ou in he p e ious chap e wi h conside able success, since he dependence o he opological cha ge on θwas de e mined on he la ice o he whole domain o he acuum angle, up o θ=π. This was achie ed hanks o s anda d Mon e Ca lo simula ions pe o med a pu ely imagina y alues o θ, which a e he basic inpu o he econs uc ion me hod o [13]. In hese eal-ac ion compu a ions, s agge ed (Kogu -Susskind) e mions we e used, and he de e minan o he e mionic ma ix was calcula ed a e e y Me opolis s ep by ex ac ing nume ically all i s eigen alues—a echnique as eliable as ine icien . In o de o adjus he compu a ion o he one- la o case, he squa e oo o his de e minan needs o be aken; since he ac ual weigh employed by he algo i hm depends on he loga i hm o he de e minan , his oo ing p ocedu e amoun s o mul iplying by a ac o o 1/2. I ins ead we conside a ac o o N /2, he discussion is alid o he Schwinge model wi h N e mion species—in ac , his 75 76 CHAPTER 5. PRELIMINARY RESULTS ON NF= 2 SCHWINGER is he only change equi ed in he algo i hm. Ha ing de eloped an algo i hm ha can be i ially ex ended o he mul i- la o ed case, i seems na u al o apply i , a leas , o he cases ha a e closely ela ed o one o ou main in e es s du ing his hesis: he s udy o opological objec s on la ice gauge heo ies, and i s implica ions in QCD. In ac , his is he case o he wo- la o ed e sion o he model: i s ac ion holds a U(2) symme y in he chi al limi , o which i s axial U(1) subg oup is b oken by he anomaly, much in he same way as QCD. The emaining SU(2) g oup cons i u es a ue symme y o he heo y ha , con a y o wha occu s in low empe a u e QCD, is exac ly p ese ed—as g an ed by a Theo em due o Coleman,1a con inuous symme y canno be spon aneously b oken in a wo-dimensional sys em, as long as in e ac ions a e kep su icien ly local. This exac ly p ese ed symme y esul s o be an in e es ing p ope y, as long as i is sha ed by he high empe a u e phase o QCD. In o he wo ds, QCD a high empe a u es has in he chi al limi an exac ly p ese ed chi al symme y (con a y o he less exo ic low empe a u e phase, as was discussed in Sec ion 1.3.3). This ac a o s he s udy he wo- la o Schwinge model as a mechanism o gain insigh abou he opological p ope ies o ini e empe a u e QCD. Beyond i s in e es as a oy model o QCD, he N = 2 Schwinge model p esen s a mo e in ol ed θbeha iou han i s single la o ed coun e pa . The la e p esen s, a θ=π, wo dis inc phases depending on he coupling e/m. While Psymme y is spon aneously b oken a weak coupling, in he s ong coupling (o ligh mass) limi he acuum ene gy densi y can be expanded in e ms o he e mion mass m, i s leading con ibu ion being E(θ)∼me cos θ. (5.1) As a consequence, he symme y is exac ly p ese ed and he opological suscep- ibili y emains ini e. By he con a y, he wo- la o e sion o he model, which has a simila beha io in he weak coupling egion, has a mo e in ol ed θdepen- dence on he acuum angle. As i was shown by Coleman [22], a s ong coupling app oxima ion allows o w i e he ene gy densi y θdependence as E(θ)∼m4 3e2 3cos4 3θ 2,(5.2) which e en ually leads o Pexac conse a ion a θ=π, bu wi h a di e gen opo- logical suscep ibili y— he cha ac e is ic o a con inuous phase ansi ion. This ap- p oxima ion, alid in p inciple when e/m >> 1, implies a alue o δ= 1/3 o he 1Al hough his esul is p o en by Coleman in he con ex o quan um ield heo ies [147], i is commonly known as Me min-Wagne heo m, since hey a i ed o he same conclusions in s a is ical physics [148]. 5.2. THE MODEL 77 associa ed c i ical exponen , which desc ibes how he opological cha ge densi y anishes as θapp oaches π. Addi ionally, he ligh es bosons o he spec um a e p edic ed o be an iso iple and an isosingle , he quo ien o i s masses being √3. I is wo h no ing ha p ecisely his mass a io has been ecen ly he subjec o some con o e sy, since a ecen wo k by Azcoi i [149] ound a sub an ial dis- c epancy wi h espec o he o iginal compu a ion o Coleman [22]. Fu he mo e, Geo gi [150] has d own e en mo e a en ion o his model, by p oposing a solu ion o he h ee ques ions posed by Coleman in [22] ha could en ail he exis ence o a no el mechanism, capable o gene a ing he appea ance o ine- uning in low- ene gy e ec i e heo ies and, consequen ly, wi h p omising po en ial conce ning any o he hie a chy p oblems ha a lic he S anda d Model. In any case, since a c i ical poin is expec ed in his model a θ=π, his sys em poses a ele an challenge o he econs uc ion me hod ha was applied du ing Chap e 4; o his e ec , i se es as an addi ional mo i a ion o his wo k—a pa icula ly p agma ic one, a guably. 5.2 The model The ac ion o he one- la o massi e Schwinge model wi h a θ e m can be easily gene alized o i s mul i- la o e sion by adding an index , unning om 1 o N , o he o iginal exp ession (4.9). In his manne , he ac ion o N la o s o equal cha ge eand mass myields SN =Zd2x   N X 1 ¯ ψ [γµ(∂µ+iAµ) + m]ψ +1 4e2F2 µν +iθ 4πµνFµν   .(5.3) Following he easoning o he p e ious chap e , i is possible o disc e ize he abo e con inuum ac ion by using Kogu -Susskind e mions and he s anda d Wil- son ac ion o he gauge pa , as in (4.15). A his poin we ecall ha a p ocedu e commonly known as oo ing was needed o ge he one- la o heo y om he co esponding la ice ac ion, since s agge ed e mions a e no comple ely ee o he doubling p oblem— hey desc ibe wo degene a e species o e mions, in wo dimensions. Bu , as long as we a e in e es ed in he wo- la o e sion, i su ices o conside he exac same ac ion (4.15) and dismiss he oo ing s ep. The nex s ep would imply pe o ming a Mon e Ca lo simula ion, much in he same way as in Chap e 4. Howe e , while in ou one- la o s udy i was enough o pe o m a p oo -o -concep calcula ion, ou aim wi h he N = 2 case is o de e mine mo e in ol ed quan i ies, such as he c i ical exponen o he expec ed θ=πphase ansi ion. E en i he b u e o ce app oach o he p e ious chap e was able o deli e esul s in, oughly speaking, a ew mon hs o compu e 84 CHAPTER 5. PRELIMINARY RESULTS ON NF= 2 SCHWINGER 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 hqi h Me hod [29] Pseudo e mions, n= 10 Figu e 5.5: Resul s o he wo- la o Schwinge model in a 16 ×16 la ice, wi h β= 3 and m = 0.12. δmax = 0.25 and n= 10. The pseudo e mion app oach is con on ed wi h he me hod o Chap e 4, wi h g ea ag eemen o small alues o h, al hough sys ema ic de ia ions a e o he wise obse ed. Chap e 6 Explo a o y ghos -gluon s udy on αswi h HISQ e mions In his las chap e we pu aside he s udy o opological e ec s o ace a non- pe u ba i e quan i y o he u mos ele ance in QCD—i s unning coupling con- s an αs. Ou in en ion is o explo e he dependence o he coupling on he mo- men um ans e q, by means o a pu ely gluonic me hod, ollowing S e nbeck e al [26]. In his way, i is possible o compu e gluon and ghos p opaga o s, G(q2) and D(q2), in he la ice, e en ually leading o he de e mina ion o αs(q2). In he ollowing sec ions we mo i a e he opic and e iew b ie ly he mos ele an ma hema ical ela ions, desc ibing wi h some de ail he echnical issues in ol ed in he p opaga o s de e mina ion. A e ha , ou esul s—ob ained om la ge se s o con igu a ions gene a ed by he MILC collabo a ion—a e p esen ed. The chap e is inished wi h some conside a ions conce ning how he p esen s udy could be ex ended. 6.1 Mo i a ion The de e mina ion o he αscoupling cons an cons i u es a e y ac i e ield o esea ch. In ac , he Pa icle Da a G oup pe iodically p o ide a global a e age o his quan i y, including bo h heo e ical calcula ions o la ice simula ions and expe imen al de e mina ions, such as om had onic τdecays o e+e−annihila ion p ocesses, ha allow o gi e an es ima e—a a gi en scale, usually ha o he Z boson—o he s ong coupling αs[2]. Rema kably enough, he las decade has been ma ked by a se back o he p ecision achie ed when calcula ing he global a e age o his quan i y, in pa due o he exis ence o p e iously unde es ima ed sou ces o sys ema ic e o ha we e p esen in la ice compu a ions. In his si ua ion, he e cohabi se e al independen app oaches wi hin he la - 85 86 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS ice o malism, de eloped by a numbe o in e na ional collabo a ions such as HPQCD [4, 151, 152], PACS-CS [153], ETM [154], and o he esea ch g oups [155, 156]. These app oaches di e in a numbe o echnicali ies, including how e mions a e implemen ed on he la ice. Mo eo e , di e en obse ables can be s udied in o de o ex ac αs, an example being hea y qua k co ela o s in [152], o he de e mina ion o he s a ic po en ial in [156], which in ac gi es a alue o αs(MZ) ha exhibi s some ension wi h he es o la ice p edic ions. Ou in en ion in he cu en chap e is o explo e he po en ial o combining he ghos -gluon e ex echnique o S e nbeck e al [26], which was al eady applied o wis ed mass e mions in [154], wi h he highly imp o ed s agge ed ac ion (HISQ), used by he HPQCD collabo a ion [78]. To his aim, we will analyze ghos and gluon ields in a collec ion o ensembles gene a ed by he MILC collabo a ion wi h he HISQ ac ion [157]. 6.2 αsand he ghos -gluon e ex Be o e going any u he , i should be no ed ha he unning coupling αsis no a physically obse able quan i y. Ins ead, i acqui es a p ecise meaning only in he con ex o pe u ba ion heo y. Then, in o de o compa e wo gi en esul s o he coupling i is necessa y o ake in o accoun he eno maliza ion scheme. Typical choices in he li e a u e a e momen um sub ac ion schemes, he mo e common including MS, MS and MOM. While he s anda d compu ed alue o αs(MZ) is ipically gi en in he li e a u e in he MS scheme [2], he p esen app oach is de ined in a MOM scheme, in which eno maliza ion cons an s a e de ined by equi ing wo- and h ee-poin unc ions o equal hei ee le el exp essions a a gi en ene gy scale µ[158]. The e exis a numbe o ways ha allow o calcula e he alue o he unning coupling αson he la ice. As i has been jus men ioned, he p esen wo k ollows he app oach o S e nbeck e al [26], al hough wi h he ocus se in a di e en egion (since we a e no pa icula ly in e es ed in he in a ed limi o αs). He ea e , we e iew he essen ials o he me hod. The compu a ion o αsin he la ice s a s by ealizing how ghos and gluon p opaga o s can be exploi ed. I s d essing unc ions can be used o de e mine he unning coupling as a eno maliza ion g oup in a ian , in a momen um sub ac ion scheme [25], as αs(q2) = g2 0 4πZD(q2)Z2 G(q2),(6.1) whe e ZDand ZGa e espec i ely he ba e d essing unc ions o gluon and ghos p opaga o s; we discuss how o compu e hese unc ions in he nex sec ions. 6.3. THE GLUON PROPAGATOR 87 Al hough in usual la ice compu a ions, as we e iewed in Chap e 1, ixing he gauge is no necessa y, he ma hema ical exp essions o ghos and gluon p opa- ga o s adop a simple o m in he Landau gauge—which in ac makes he whole compu a ion easible. Consequen ly, in wha ollows all exp essions will be unde - s ood o be alid wi hin he Landau gauge. 6.3 The gluon p opaga o To begin wi h, he s a ing poin is he s anda d 4-dimensional la ice o Vsi es, and 4Vlink a iables Ux,µ ∈SU(Nc= 3). The la ice gluon ields, which li e in he mid-poin o each link, Ax,µ ≡Aµ(x+ ˆµ/2), a e de ined as Ax,µ := 1 2i(Ux,µ −U† x,µ)−1 6iT (Ux,µ −U† x,µ).(6.2) Addi ionally, we ecall ha he colo componen s Aa x,µ o he gluon ield can be compu ed as Aa x,µ := 2 T (TaAx,µ)=2·Im T (TaUx,µ),(6.3) whe e he de ini ion (6.2) has been used. Wi h hese exp essions, we can compu e he ba e gluon p opaga o on he la ice as Dab µν(k) = D˜ Aa µ(k)˜ Ab ν(−k)EU,(6.4) whe e ˜ Aµ=˜ Aa µTaa e he Fou ie ans o med gluon ields. In o he wo ds, Dab µν(q(k)) = 1 V*X x,y Aa x,µAb y,νeik·(x+ˆµ/2)e−ik·(y+ˆν/2)+U ,(6.5) whe e he momen um q(k) is gi en by qµ(kµ) = 2 asin πkµ Lµ.(6.6) I we assume now ha Dab µν(q(k)) has he same enso s uc u e han i s con inuum coun e pa , Dab µν(q) = δab δµν −qµqν q2D(q2),(6.7) i su ices a bi o algeb a o ob ain an exp ession o he scala pa o he p op- aga o D(q2), D(q2) = 1 (D−1)(N2 c−1) X aµ Daa µµ(q),(6.8) 88 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS which is ela ed wi h he d essed p opaga o simply by ZD(q2)≡q2D(q2).(6.9) In p ac ical e ms, he key s ep in his compu a ion—in e ms o compu a ional complexi y—is he Fou ie ans o ma ion o he gauge ields. Fo una ely, Fas Fou ie T ans o m algo i hms allow o compu e e y e icien ly exp ession (6.5), especially aking in o accoun ha a single applica ion o he algo i hm deli e s he p opaga o o e e y la ice alue o he momen a qµa once. 6.4 The ghos p opaga o Following [27], he ghos p opaga o in he la ice is de ined in he Landau gauge as Gab(k) = a2*X xy M−1ab xy eik·(x−y)+=δabG(q),(6.10) whe e he eal symme ic ma ix Mis he Fadee -Popo ope a o , de ined by Mab xy =X µAab x,µδx,y −Bab x,µδx+ˆµ,y −Cab x,µδx−ˆµ,y(6.11) wi h Aab x,µ =Re T {Ta, Tb}(Ux,µ +Ux−ˆµ,µ),(6.12) Bab x,µ = 2 ·Re T TbTaUx,µ,(6.13) Cab x,µ = 2 ·Re T TaTbUx−ˆµ,µ.(6.14) In o de o compu e (6.10), he ollowing sys em o equa ions needs o be sol ed    Max,bycby c=δac cos (k·x), Max,bysby c=δac sin (k·x).(6.15) The 8V-componen ec o s cc,sca e compu ed wi h he conjuga e g adien me hod and can be used o de e mine he in e se o M. Toge he wi h (6.10), and assuming he enso s uc u e o he con inuum, Gab(qµ) = δabG(q2), we ha e G(q2) = 1 (N2 c−1) X ax [cos (k·x)cax a+ sin (k·x)sax a],(6.16) wi h he co esponding d essed p opaga o being gi en by ZG(q2)≡q2G(q2).(6.17) 6.5. RESULTS 89 β m0 l/m0 sm0 sm0 cN3 s×N a( m) # o cn gs 6.00 1/5 0.0509 0.0635 243×64 0.1218(7) 1053 6.30 1/5 0.0370 0.0440 323×96 0.0879(5) 1008 6.72 1/5 0.0240 0.0286 483×144 0.0573(4) 1017 Table 6.1: De ails o he h ee ensembles s udied. In con as wi h he gluon de e mina ion o he p e ious sec ion, he s eps he e desc ibed a e much mo e expensi e in compu a ional e ms. In pa icula , nume - ically sol ing he sys em o equa ions (6.15) is a e y demanding ask which, a la ge la ice sizes—as he ones s udied in his chap e a e— equi es la ge esou ces, bo h in e ms o memo y and p ocessing ime. Fu he mo e, la ice a i ac s a e expec ed o be mo e in ense bo h a la ge and a o -diagonal momen a, due o he lack o o a ional symme y on he la ice in he la e case [159]. Fo his easons, he ghos p opaga o , and as a consequence also αs, ha e been compu ed only o a hand ul o selec ed diagonal momen a. 6.5 Resul s We ha e analyzed h ee se s o con igu a ions, made a ailable by he MILC collab- o a ion [157]. The de ails o hei pa ame e s a e summa ized in Table 6.1. P io o he p opaga o s de e mina ion, we ixed e e y con igu a ion o Landau gauge. To his end, i is necessa y o make use o an i e a i e op imiza ion algo i hm. As is well known, his ype o algo i hms could su e om a se e e c i ical slowing down p oblem. In he case o la ge la ices, as some o he ones analyzed he e, his obs acle can u n insu moun able. Howe e , we ha e e aded his di icul y by ap- plying a Fou ie -accele a ed algo i hm, o iginally p oposed by Da ies e al [160], which allows o alle ia e he compu a ional o e head, making he gauge ixing p ocedu e easible. In his p ocess, he ixing p ocedu e was s opped only when e e y local gauge ield e i ied he ans e sali y condi ion— he la ice e sion o ∂µAµ= 0—up o Θ <10−14, wi h he same de ini ion o [160]. Such le el o p ecision was p o ed o be necessa y, since he p opaga o s a e qui e sensi i e o he gauge condi ion. Once he whole se s we e ixed o Landau gauge, we ha e compu ed bo h p opaga o s, G(q2) and D(q2), o a o al o se en diagonal momen a, k= (n, n, n, n) o n= 1,...,7.(6.18) As we men ioned ea lie , he ghos compu a ion, and in a lesse way he gauge 90 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS # o con igu a ions 1 1053 Landau gauge- ixing ime (co e-h) 7.1 7.1k P opaga o s o 7 momen a (co e-h) 19.7 20.8k To al ime (7 momen a) (co e-h) 26.928.3k Each addi ional momen um (co e-h) +3.0k Table 6.2: Dis ibu ion o compu ing imes in ol ed in he 243×64 ensemble. ixing p ocedu e, a e e y demanding in e ms o compu ing esou ces. This being he case, ac ual calcula ions ha e equi ed o be pe o med in la ge clus e acil- i ies, which p o ide bo h he compu ing powe and he memo y needed o s o e he la ges con igu a ions. To his end, he Uni e si y o Camb idge compu ing se ices ha e been used, including he clus e Da win, and i s 2017 upda e CSD3. Fo he i s se , o olume 243×64, 28k co e-hou s we e used in Da win; o exem- pli y how hese a e dis ibu ed, see Table 6.2. Fo he 323×96 se , a o al o 135k co e-hou s, also in Da win, we e used. Finally, he ensemble co esponding o he bigge la ice size, 483×144, was analyzed wi h a o al cos o 452k co e-hou s o he newe clus e CSD3. Ou esul s o bo h ba e un eno malized p opaga o s a e shown in Figu e 6.1, as a unc ion o he squa ed la ice momen a in physical uni s. By means o Eq. 6.1, he p e ious esul s can be applied o de e mine αs. Taking in o accoun ha , acco ding wi h he pa icula implemen a ion o he HISQ ac ion in he analysed MILC con igu a ions, g2 0≡5 3 2Nc β=10 β,(6.19) ou inal esul s o he un eno malized e sion o he unning coupling a e shown in Fig. 6.2. As a p elimina y conclusion, hese esul s can be checked o be quali a i ely in ag eemen wi h hose o he li e a u e, see e.g. [25,26,154]. In any case, in o de o p o ide an es ima e o he αs(MZ) in he MS eno maliza ion scheme, he co esponding β unc ion should be ca e ully s udied and in eg a ed, oge he wi h a ho ough s udy o p esen olume e ec s and la ice a i ac s, since he knowledge o i s dependence would allow o es ima e he sys ema ic e o s o he me hod. Finally, he ela i ely low alue o he es ima ed s a is ical e o s in he s udied ensembles d i e us o conclude ha comple ing he p esen analysis would be wo hwhile, po en ially leading o a aluable con ibu ion o he wo ld a e age o he s ong unning coupling. 6.5. RESULTS 91 0 1 2 3 4 1 10 0 1 2 3 4 5 6 1 10 q2[GeV 2] G(q)−243×64 323×96 483×144 q2[GeV 2] D(q)−243×64 323×96 483×144 Figu e 6.1: Ghos and gluon ba e p opaga o s, in physical uni s. 0 0.5 1 1.5 2 1 10 q2[GeV2] 243×64 323×96 483×144 Figu e 6.2: The unning coupling αs(q2) as de ined in Eq. 6.1. Shaded a eas connec 1σin e als. 92 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS Conclusions and ou look In he beginning o his hesis we emphasized he key ole o la ice QCD as he main ool capable o dealing wi h non-pe u ba i e phenomena, in QCD and be- yond. Thei successes a e many, as i has been documen ed in Chap e 1. Up o he p esen day, he e exis se e al in e na ional collabo a ions ha wo k in ensi ely o p o ide heo e ical es ima es ha ma ch he e e -inc easing p ecision o he ex- pe imen al measu es. Bu , no wi hs anding he achie emen s in phenomenology g ounds, he e exis open ques ions ha seem o e ade any a emp o esolu ion. Indeed, his is he case o QCD when he opological θ e m is included in o i s ac ion, as i has been e iewed in Chap e 2. Deeply connec ed wi h he s ong CP p oblem, and o g ea impo ance in axion physics, he s udy o he θdependence o QCD has been he long-sigh ed goal o his hesis. Ce ainly, his is an ambi ious objec i e, since he e exis many di icul ies when conside ing θQCD on he la ice, one o he mos undamen al being he p esence o a se e e sign p oblem ha makes he sys em una ainable o s anda d Mon e Ca lo simula ions. In his scena io, ou e o s ha e been i s di ec ed owa ds he es ing o a p omising me hod ha would in p inciple allow o econs uc he ull θdependence o obse ables as he opological cha ge [13]. The eby, in Chap e 3, ollowing [28], we ha e c osschecked a p e ious wo k ha , making use o he econs uc ion me hod o Azcoi i e al, ound signs o a ich phase s uc u e in he an i e omagne ic Ising model wi hin an imagina y magne ic ield—which can be hough o as a θ e m [21]. By making use o a combina ion o analy ical and nume ical—bu exac — echniques, ou esul s ha e suppo ed he p e ious quali a i e pic u e, hus con i ming he po en ial o he econs uc ion me hod in a heo y ha , when compa ed o θQCD, holds a mo e in ica e phase diag am. In he same spi i , Chap e 4 has been de o ed o he s udy o he massi e one- la o Schwinge model wi h a θ e m in he ac ion, which equen ly se es as a oy model o QCD, as p esen ed in [29]. To apply he econs uc ion me hod [13] in his sys em implies a s ingen exam conce ning i s possibili ies o being success ully applied o ull QCD, as i sha es many o i s po en ial obs acles— bu no all, since i holds a e y elemen a y de ini ion o he opological cha ge, in con as wi h he di icul ies ha appea in he SU(3) 4d heo y. Ac ually, we ha e 93 100 CHAPTER 6. EXPLORATORY GHOST-GLUON STUDY ON αS Appendix A Compu a ion o he cumulan s κn In o de o use exp essions (3.17) and (3.20), we need o compu e he cumulan s κn. The n h cumulan can be calcula ed in e ms o he i s nnoncen al momen s µ0 n, µ0 n≡* X <ij> sisj!n+m1,m2 ,(A.1) by means o he ecu sion ela ion (3.16). The summa ion o e < ij > uns o e each couple o neighbo ing spins, o in o he wo ds, o e each link. Two neighbo ing spins always belong o di e en subla ices. Be o e going u he , le us commen on wo in e media e esul s. Fi s , we conside a la ice o Nspins, he magne iza ion o which is he sum m=Pisi, and ask abou he expec ed alue o he p oduc o no hese spins a ixed m(o ixed N+, he numbe o posi i e spins), ha is, hs1s2···snim. One can pe o m his calcula ion by means o he mic ocanonical o malism, a i ing a hs1s2···snim=1 N N+ n X k=0 (−1)kn k N−n N+−n+k.(A.2) In he abo e exp ession, kcan be ead as he numbe o nega i e spins in he p oduc s1s2···sn. In his way, he i s summand, k= 0, coun s he numbe o s a es wi h ze o nega i e spins in he p oduc s1s2···snand mul iplies i by he expec a ion alue o he p oduc in his case, (−1)0= 1. The second one, k= 1, does he same o one nega i e spin in s1···sn, and so on. Di iding he sum by he o al numbe o con igu a ions wi h magne iza ion m= 2N+/N −1, one ob ains he p e ious expec a ion alue a ixed m. Secondly, conside an obse able O(m1, m2) in ou wo subla ice sys em, wi h a dependence on m1and m2such as we can w i e i as O1(m1)O2(m2). In his case, om he de ini ion 101 102 APPENDIX A. COMPUTATION OF THE CUMULANTS κN (3.12) o he expec a ion alue a ixed m1and m2, we ha e hO1(m1)O2(m2)im1,m2=hO1(m1)im1hO2(m2)im2.(A.3) This immedia ely applies o he spin p oduc s1s2···sn. We can always di ide i in o wo p oduc s sa···sband sα···sβ, each one con aining he spins o one o he subla ices, and hen hs1s2···snim1,m2=hsa···sbim1hsα···sβim2.(A.4) Wi h he p e ious couple o esul s, we come back o Eq. (A.1), and apply he linea i y o he expec a ion alue, a i ing a µ0 n=X <ij>,<kl>,··· ,<pq> hsisjsksl···spsqim1,m2,(A.5) which is he sum o he expec a ion alues o he p oduc o nlinks, unning o e all pe mu a ions wi h epe i ions o hese links. Then, in e e y summand we ha e he p oduc o 2nspins, in some cases wi h some o hem iden ical. Taking in o accoun ha s2 i= 1 ∀i, each summand can be educed o he expec a ion alue o he p oduc o n1+n2di e en spins, n1and n2being he numbe o spins in each subla ice. Since by means o Eq. (A.2) we al eady ha e an exp ession ha compu es hs1···snim, he p oblem is educed o coun how many summands in Eq. (A.5) ha e (n1, n2) spins. We call hese numbe s geome ical ac o s, and deno e hem by G(n1, n2). Following his con en ion, we can w i e he n h cen al momen as µ0 n=X {n1,n2}G(n1, n2)hsa···sb | {z } n1spins im1hsα···sβ | {z } n2spins im2,(A.6) whe e he sum uns o e he couples o in ege s (n1, n2) he sum o which is e en and less han o equal o n. The compu a ion o he geome ical ac o s G(n1, n2) can be done by hand o he i s ew cumulan s. As an example, o he second noncen al momen µ0 2we ha e o compu e ou cases: he wo links being he same (sha ing bo h spins), sha ing only one spin belonging o he i s o he second subla ice, and inally no sha ing any spin a all. Tha is, in e ms o he p e ious no a ion, {(n1, n2)}={(0,0),(2,0),(0,2),(2,2)}.(A.7) The ac o s G(n1, n2) can be compu ed easily in his case, e en o an hype cubic la ice o a bi a y dimension d, a i ing a he ollowing exp ession o he second momen µ0 2=Ndh1i+Nd(d−1)(hs1s2im1+hs1s2im2) +Nd(Nd −2(d−1) −1)hs1s2im1hs1s2im2. (A.8) A.1. TRANSLATIONAL SYMMETRY 103 We can use his exp ession o calcula e he second cumulan κ2, κ2=µ0 2−µ02 1 N→∞ −−−→ Nd(m2 1−1)(m2 2−1),(A.9) whe e we ha e aken he he modynamic limi , keeping only he e ms o o de O(N), which is he leading o de o all cumulan s. Subleading o de s can be p ese ed i needed, bu hey a e no ele an o ou pape . The di icul y o he p e ious compu a ion escala es quickly wi h he o de no he cumulan , and i is qui e cumbe some o jus n≥4. In o de o ge beyond his limi a ion, we ha e de eloped a p og am which compu es he geome ical ac o s G(n1, n2) nume ically o a ini e L×Lbidimensional la ice. Since hese ac o s G(n1, n2) a e polynomials in No o de ≤n(and wi h in ege coe icien s), we can un he p og am o la ices o n+ 1 di e en sizes, ob aining a se o (N,G(N)) poin s, which we can use o eco e he exac in ege coe icien s o each geome ical ac o , by means o he Lag ange in e pola ion o mula. The basic idea o he p og am is e y simple. We jus cons uc a pe iodic ec angula L×Mla ice, wi h L, M > n,nbeing he o de o he cumulan we wan o compu e. Wi h his es ic ion we a oid p oduc s o links c ossing he en i e la ice, ha would no appea in he he modynamic limi o any ini e cumulan . Once we ha e his, we s a a loop unning o e all he pe mu a ions wi h epe i ions o nlinks, and pe o m he ollowing s eps, •We ha e a p oduc o nlinks, o equi alen ly 2nspins, s1···s2n. •Recu si ely, we emo e couples o equal spins om his p oduc . •We classi y he emaining p oduc by he numbe o spins in each subla ice, (n1, n2). •We add one o he geome ic ac o G(n1, n2) and p oceed o he nex i e a- ion. When he algo i hm inishes, we ob ain all he G(n1, n2) alues o a gi en N= LM. The compu a ional cos is associa ed o he numbe o i e a ions o he main loop, which g ows as (LM)n, ha is, exponen ially wi h he o de o he cumulan . In p ac ice, we ha e only eached he compu a ion o he ou h cumulan wi h his p og am. Howe e , a numbe o op imiza ions can be implemen ed in o de o each highe o de cumulan s, which we summa ize in wha ollows. A.1 T ansla ional symme y Ou la ice is symme ic unde ansla ions, implying ha all geome ical ac o s a e p opo ional o Nd, he numbe o links. Fixing, e.g., he i s link o he 104 APPENDIX A. COMPUTATION OF THE CUMULANTS κN p oduc , one ob ains he same G(n1, n2), bu di ided by a common ac o Nd. The same ac o is gained in he o e all speed o he p og am. In addi ion o ha , he deg ee o he polynomials G(n1, n2) is also educed by one, and i su ices wi h n(ins ead o n+ 1) di e en sizes in o de o eco e he Ndependence. One can go e en u he by ealizing ha he geome ical ac o co esponding o non-neighbo ing links, G(n, n), is he only one wi h maximum deg ee Nn−1. This allows us o exp ess i in e ms o he emaining ac o s, 1 NdG(n, n) = (Nd)n−1 −1 Nd X {(n1,n2)} (n,n)G(n1, n2),(A.10) which a e only o o de n−2 o less. This means ha i is enough o un he p og am o n−1 la ice sizes, compu e all he geome ical ac o s bu G(n, n) ia he Lag ange in e pola o , and hen wi h he p e ious exp ession ind he N dependence o his las ac o . A.2 F om pe mu a ions o combina ions The p oduc o links commu es, so i s con ibu ion o he geome ical ac o s is he same ega dless o he o de . Then, we can change he main loop o e pe mu a ions wi h epe i ion o a loop o e combina ions wi h epe i ion, by aking in o accoun he mul iplici y o each combina ion. Schema ically, we pe o m Pi,j,...,k con ib(lilj···lk) →X i≤j≤···≤k mul ×con ib(lilj···lk),(A.11) whe e con ib ep esen s a unc ion in ou p og am ha akes a p oduc o links and e u ns he con ibu ion o he geome ical ac o s. I he e a e di e en links, each one appea ing k1, . . . , k imes, he mul iplici y o he combina ion is gi en by mul = n! k1!···k !.(A.12) A.3 Blocks - G ouping links oge he Many o he link p oduc s ha e ew, i any, epea ed spins, and hei con ibu ions o he geome ical ac o s can be coun ed wi hou ha ing o analyze one by one each o hem. This is possible by g ouping hem in se s o links ha we will call in A.4. CLUSTERS OF BLOCKS 105 wha ollows blocks, and eplacing he loop o e link p oduc s by a loop o e block p oduc s. When he blocks in a p oduc a e no neighbo s (i.e., hey do no ha e any common spin), we do no need o pe o m he compu a ion link by link and he con ibu ion can be summed up i ially. Le b1and b3be wo non-neighbo ing blocks, each one composed by Nblinks, and le us deno e he con ibu ions o he geome ical ac o s by λ(n1, n2), whe e λis an in ege coun ing how many p oduc s o links ha e n1(n2) spins in he i s (second) subla ice. Then we ha e con ib(b1b3) = N2 b(2,2),(A.13) o in gene al, o he p oduc o knon-neighbo ing blocks, Nk b(k, k). Following his s a egy, we di ide ou la ice in o unidimensional blocks o 2Mlinks, in a way ha he j h block, bj, con ains all links he i s spin o which belongs o he j h column. As a consequence, bjis a neighbo o blocks j−1 and j+ 1, and, aking in o accoun he bounda y condi ions, b0and bL−1a e neighbo s oo. When we ha e a p oduc o neighbo ing blocks, we p oceed as be o e, analyzing he link p oduc s one by one, and he e is no compu a ional sa ing. Bu when he nblocks a e no neighbo s, we mo e om (Nd)ni e a ions o a single one. A.4 Clus e s o blocks The block me hod, as de ined abo e, ails o sa e any compu a ion ime i wo o mo e blocks a e neighbo s in a gi en block p oduc . Howe e , we can ex end he me hod by di iding each block p oduc in o se e al subp oduc s, which we will deno e as clus e s. In each clus e , one can always connec one block o ano he by he equi alence ela ion o being neighbo s (sha ing spins). And in he same way, in each p oduc di e en clus e s ne e sha e any spin. This allows us o compu e he con ibu ions o each clus e sepa a ely, and hen compose hem wi h he ollowing law, λ1(a, b)⊕λ2(c, d) = λ1λ2(a+c, b +d).(A.14) I he con ibu ions o he clus e s in ol e mo e han one geome ical ac o , lin- ea i y applies, X ab λab(a, b)⊕X cd λcd(c, d) = X ab,cd λabλcd(a+c, b +d).(A.15) P ocessing one clus e wi h kblocks akes a compu ing ime p opo ional o (Nd)k. So di iding he whole block p oduc in smalle clus e s implies o almos e e y 106 APPENDIX A. COMPUTATION OF THE CUMULANTS κN block p oduc a signi ican amoun o ime sa ed. Only when all he blocks a e pa o he same clus e he e is no speed up. Ano he majo op imiza ion can be pe o med by ealizing ha ansla ional in a iance can also be applied he e, since a gi en clus e , say b0b1b1, and any o i s ansla ions, b0+ b1+ b1+ , ha e he same con ibu ion o he geome ical ac o s. Then, when a clus e is going o be compu ed, we can exp ess i in e ms o i s equi alence class, compu e i s con ibu ion, and s o e i in memo y. E e y ime one o i s ansla ions appea s, we jus ake he alue om he memo y, sa ing a lo o compu ing ime. In addi ion o ha , once we ha e compu ed he ac o s G(n1, n2) o he i s size L×M, we know in ad ance all he clus e con ibu ions o any L0×Mla ice ( he blocks keep i s size cons an ). Since almos all he compu ing ime is spen in igu ing ou he clus e con ibu ions, we educe in his way he ull p oblem o compu ing he geome ical ac o s in la ices o n−1 di e en sizes o only one size, he smalles one, M×M. In p ac ice, he ime spen by he es o he sizes needed is ba ely he 1 −2% o ha o he i s size. A.5 Compu a ion o a clus e The las op imiza ion conce ns he compu a ion o he clus e s hemsel es. Un il now i is done simply by pe o ming a loop o e each possible pe mu a ion o links belonging o each o he blocks in he clus e . Howe e , one can go one s ep u he and di ide he blocks composing he clus e in o smalle se s, ha we will call si es. 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