Large N phase transitions in Supersymmetric gauge theories with massive matter

Transcription

1 Lar phas trastos Suprsytrc au thors wth assv attr Mul Trz Uvrsdad Copluts d Madrd Basd o: J. Russo ad K. arbo arv: A. Barraco ad J. Russo arv: J. Russo G. Slva ad M.T. arv:

2 Pla Study o act rsults au thors both at lar ad t Eact = all ordr coupl clud both prturbatv ad o-prturbatv cotrbutos. Four-dsoal = SQCD ad = * SYM at lar. = U CS-attr thory at lar 3. = U CS-attr thory wth t

3 Localzato I. Eact partto ucto or = suprsytrc YM thors o S 4 wth arbtrary attr cott. [Pstu 07.84] II. Eact partto ucto or = suprsytrc CS attr thors o S 3 [Kapust Wlltt ad Yaaov ] = SYM thors our dsos: Partto ucto localzs to a atr tral ovr Coulob odul Scl a d a a a z S cl 4 S 4 d 4 R tr 8 a loop a z st a; da a a VEV o scalar o vctor ultplt = Eact dpdc Ths a coplcatd tral whch w stll d to coput ordr to udrstad th udrly physcs.

4 = 4 Supr Ya-Mlls thory o S 4 Istatos do ot cotrbut. -loop corrctos cacl Gaussa atr odl: a a a a d 8 At lar th tral s doatd by a saddl-pot. Itroduc th valu dsty: a th saddl-pot quato rads y y dy Evalus ar dstrbutd a scrcl Wr s law r _

5 = 4 SYM : Wlso loop [Ercso So ad arbo ] [Drur Gross 00074] W C d I 3/ 4 Rproducd by holoraphy Epad th Bssl ucto at sall l o ts th prturbatv srs that ust rproduc th Fya dara calculatos W C I 8 9 Thr s a sooth dpdc wth l all th way ro 0 to ty. o phas trasto btw th prturbatv l << r ad th stro coupl l >> r dscrbd by AdS/CFT dualty. What about = SYM thors? Is th trpolato btw wa ad stro coupl stll sooth? Could thr b a quatu phas trasto at so valu o l?

6 . Suprsytrc Chr-Sos wth assv attr Do Chr-Sos-attr thors udro quatu phas trastos l th aaloous our-dsoal cas? Cosdr th = suprsytrc U Chr-Sos thory wth lvl coupld to a attr cott v by udatals ad atudatals chral ultplts o ass. Th partto ucto localzs to [Kapust Wlltt Yaaov ] U d 4sh [4cosh cosh ] Ca ths tral b coputd plctly? It dpds o our paratrs Cosdr th t plaar lt. Th th partto ucto ca b dtrd by a saddl-pot calculato. Vzao lt t d d Th saddl-pot quatos ar th d coth tah tah t Ths s a actly solvabl odl [Barraco J.R ]

7 Lar soluto th dcopactcato lt d coth tah tah t Rpulso attracto to or attracto to = +- cobd ct: attracto to = 0 Rstor R dpdc ad ta dcopactcato lt R R I t s d th ths lt ust dcoupls attr ultplts whch t a t ass. Th thory rducs to pur = Chr-Sos thory. A spcto o th saddl-pot quato shows that th lt R = ty s rular at th sa t t os to ty wth t d R Th th dpdc o R copltly cacls out ad o obtas d s s s

8 Pottal V - -

9 Cosdr th tral quato: A d A s s s As th coupl l s crasd ro 0 th syst os throuh drt phass I II III whr: I. A < II. A = III. A > Phas I l < : arss wh A < ply that μ <. Th th s uctos cacl out. [ ] Th dstrbuto pads as l s crasd utl th dpots ht + /-. Phas II < l < /-z: A = [ ] Phas III l > /-z: arss wh A >. [ ] W assud: 0 0 Wh oly phas I II ra

10 Evalu dsty t : r approachs th Wr dstrbuto t=0. =50 z = 0.5 Phas I : r or = 50 z = 0.5 t = 47. Phas II : r or = 50 z = 0.5 t = 60. Phas II : Cas >. r or = 50 z = t = 50. Phas III : r or = 50 z =0.5 t = 50.

11 Fr ry: F lo F F F I II III R 6 6 R R Phas I Phas II Phas III Ths pls a dscotuty th thrd drvatv at both crtcal pots λ = ad λ = ζ : FI FII R F F R II III / Thror both phas trastos ar thrd ordr.

12 3. = U Chr-Sos-attr wth t Cosdr U d [J. Russo G. Slva ad M.T. arv: ] 4sh [4 cosh cosh ] Th prvous saddl-pot thod oly coputs th spcal corr o th paratr spac volv th plaar =ty lt Ft : O ca us th thod o orthooal polyoals. Th coput rducs to th probl o coput th tral: I at ct Sc 9th ctury ths tral appard ay trst cotts. I partcular studs o Ra zta uctos by Sl ad Moc thta uctos by Raaua. I 933 t was studd rat dtal a classcal papr by Mordll who av th rsult or all possbl valus o paratrs a b c d. bt d dt

13 U partto ucto ro orthooal polyoals Us th thod o orthooal polyoals th partto ucto ca b wrtt as ollows: U d dt! Cas = d I I I sh Th tral I s a partcular cas o th Mordll tral. It ca ral b valuatd trs o prssos volv t sus. Howvr two cass t rducs to a t su o trs. Lucly o o ths cass s wh = p / s a tr whch CS thory s rqurd by au varac.

14 Us Mordll s orula w d I Eapls: 0 4 U Th orula cotas prturbatv as wll as o-prturbatv trs Prturbatv:.. p / o-prturbatv:.. pp/ Masslss cas U Hhr ra U Ft dsoal Hlbrt spac?

15 Lar coupl ad phas trastos Th partto ucto s proportoal to th dtrat o J dd by th basc tral J d 4cosh cosh Wh s lar th a cotrbuto o th tral cos ro th saddl-pot at = l D = p ad cosdr lar wth d p. Ths s quvalt to th dcopactcato lt cosdrd bor but ow w ta t at t v low.. = Ths pls cosh p p p J d p d p p p p d p d Whch tr s doat dpds o whthr th saddl-pot ls wth th trval -pp or outsd. Ths lads to thr drt cass whch ar o-to-o corrspodc to th thr phass coutrd th plaar lt.

16 I - <. I ths cas w sply hav Lar : th r ry bcos II -- < -. I ths cas s v by a t product. At lar III -- W thus rproduc th r rs oud th plaar lt phass ad 3 start wth act t prssos.! 3 / 6 S CS U F U F U U / l F U

17 Utary atr odl orulato ad lar A trst lt s th lar lt at d Chr-Sos lvl. A covt approach s to orulat th utary vrso o th atr odl whr th valus l o S. I pur CS thory o ca show that ths approach vs quvalt rsults [Roo Trz 03.4] Th utary atr odl ca b vwd as a dorato o th cotour o trato to aary valus. Th partto ucto ow has trootrc uctos U d ] cos [4cos 4s ~ Ths ca b wrtt as a trato o th copact trval U q q d d d 4s ] cos [4cos ] cos [4cos 4s ] cos [4cos 4s ~ 3 ] 0 [ ] 0 [ ] 0 [ whr w usd Posso rsuato orula. Ths or prts to coput th lar lt by wrt as a Topltz dtrat ad us th Sző thor.

18 Topltz dtrats ad Sző thor Lt l l d T T z z 0... dt ] [ [l ] ar th cocts o th Fourr paso o l ad G=p[lo ] 0.. th Topltz dtrat s th partto ucto o a U utary atr odl Th th stro Sző lt thor stats that z z whr G T ] [l l ] [l ] [l p dt a h b a c b a d c b a Topltz atr H-Sző orula

19 I our cas ths vs ~ U q q I w urthr ta th lt o to ty at d / th ths rproducs th prsso o or phas I. Th othr phass II ad III caot b rcovrd bcaus th utary odl ± ca vr b zro.

20 Coclusos Massv suprsytrc au thors hbt lar phas trastos at crtcal coupls. Trastos occur wh th valu dstrbuto pads ad th larst valu hts th ass. Th tra asslss stats cotrbut to th plaar r ry. Eapls: Four dsoal = SYM ad = SQCD. = thory has a ravty dual. It prdcts soth spcal occurr at = U Chr-Sos wth assv lavors: Th plaar thory prsts thr phass wh 0 < < ad two phass wh. Ft : Th atr tral d ca b coputd plctly us Mordll trals. Lar wth d : ca b coputd us Sző lt thor. Mass dord ABJM Modl Rch structur o phas trastos at aary valus o th coupls aaloous to = SYM [Adrso arbo ]. Gravty dual phoo trs o M bras? Phas trastos also 5d = SYM+CS tr ad adot attr [Maha dl ] Phas trasto at so crtcal rlatv valu o th two coupls. Pur CS cas wth adot attr: vdc or t ubr o phas trastos.