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Monodromy in QCD: Insights from Supersymmetric Theories with Soft Breakings

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Monodromy in QCD: Insights from Supersymmetric Theories with Soft Breakings Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings Michael Dine Department of Physics University of California, Santa Cruz NATIFEST, Institute for Advanced Study, 2016 Work with P. Draper, L. Stevenson-Haskins, D. Xu Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings I’ll save most of my personal remarks for the banquet. I will say that Nati and I have had a great friendship, both personal and scientific, starting in about 1982. Lots of things happened to us in 1983 [Photograph: Miri Seiberg, courtesy of N.Seiberg]. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings Nuclear Phy51cs B241 (1984) 493-534 North-Holland Pubhshlng Compan) DYNAMICAL SUPERSYMMETRY BREAKING IN SUPERSYMMETRIC QCD tan AFFLECK Joveph Hentq Laboratorte~, Princeton Utm,ersat~, Princeton. ¥ew Jer~e~ 08544, l_'S 4 Michael DINE and Nathan SEIBERG The ln~trtute for 4d~'an~ed Studl, Prtn~eton, Ne. Jet~e~ 08540, L'S 4 Received 12 December 1983 (Revised 1 March 1984) The massless lmut of supers~mmctrlc QCD ~ath ~) flavors and ,'~ colors is analyzed m detail For ~% < N there is a unique superpotentlal which anaght be generated by non-perturbatl~e effects We show that it indeed appears, thus v~olatlng the non-renormallzataon theorems For "vI = ~r 1 mstantons produce the ~uperpotent~al For ~/j < N - 1 it ~ again generated, provided that a m~ld assumption about the d?namac~ of pure supersymmetrlc gauge theories is correct For ~t >~ ~ no mvanant superpotent~al e,~asts, the classical vacuum degeneracy is a propert5 of the full quantum theo~ When a small quark mass term as added to the theory (for A/t < N), ~v supersymmetrlc ground states, identifiedMichael with those Dine found b~ W~ttenMonodromy exist As m ~ in 0 QCD:the~e ,'v Insights ~a~ua wander From to Supersymmetric Theories with Soft Breakings mfimt 5, lea~mg the massless theor~ wnhout a ground state 1. Introduction Supersymmetry offers hope for a sohitton to the hierarchy problem for two related reasons [1,2] First, because of the non-renormahzatlon theorems of perturbation theory [3], scalar fields are protected from developing large masses This fact allows the construction of particle physics models with stable gauge hierarchies Second, because the non-renormallzat~on theorems have been proven only within perturba- tton theory, one may hope that small, non-perturbatlve effects break supersymmetry and explain the origin of the large hierarchy [1] In lower dimensions, examples of this phenomenon are known In supersymrnetrlc quantum mechamcs, instantons have been shown to spontaneously break supersymmetry [1.41 In 2 + 1 dimensions. Affleck, Harvey and Witten have exhibited a non-perturbatlve breakdown of the non-renormahzation theorems due to mstantons [5] However, In four dimensions, dynamical supersymmetry breaking has remained elusive. Wltten has in fact proven that supersymmetry is unbroken in many interesting theories [6] In particular, he has computed the index, A = Tr(-1) g in pure SU(N) gauge theory, and shown that this theory has at least N supersymmetrlc 493 Nati has gone on to greater and greater glory: Just latest of many honors reflecting an extraordinary career in science: About ICTPVisit ICTPSupport ICTPICTP for Women Dirac Medallists 2016 • Nathan Seiberg (Institute for Advanced Study, Princeton) • Mikhail Shifman (University of Minnesota) • Arkady Vainshtein (University of Minnesota) ICTP's 2016 Dirac Medal and Prize are awarded to Nathan Seiberg (Institute for Advanced Study, Princeton), Mikhail Shifman (University of Minnesota) and Arkad Vainshtein. Professor Seiberg has made major contributions to supersymmetric field theories elucidating the power of holomorphy to establish the non renormalisation theorems, deciphering the different phases of N=1 supersymmetric theories and uncovering a strong-weak coupling duality known as Seiberg duality. He (in collaboration with Edward Witten) also made major contributions towards a full non-perturbative understanding of N=2 theories that has led to many further developments in theoretical physics and mathematics. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings Two of the others in the picture have also gone on to greatness: Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings The fouth person in the picture has not advanced so much. Today – Instantons in Supersymmetric QCD Again – and other . things. Still stuck on equationI 3.33 dffle~k etof al/ thatD~namwalsupersvmrnetr~ paper: breahmg 509 where S v is the fern-non propagator Thus, gathering factors of v and Including factors of O required on dimensional grounds, we obtain - 16,<' ef d4x°dPvSF(X - xo)Sv(y- Xo) P ( 8¢72 , ~t'2) ×exp g2(p) 4~r ~p- ~g_ - (3 33) Extracting the external legs, we obtain the desired fermlon mass term m x - ASu -4 (3 34) just as in eq (3.15) Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings We can also extract the scalar potential by a simple but somewhat indirect argument Consider the gauge-lnvarlant dural field ~= 1 2~Qc eueZgQ'/Q~ (3 35) In the vacua we study, q~ = )/~ v + qv + (terms quadratic in the fields), (3 36) where cp is the massless field of eq (3 14) Writing =A~ + v~O¢~ + 02F~, (3 37) we have (vl { Q,~, +,~ }Iv) = ~-(vl F~lv) (3 38) Here It,) denotes the perturbatlve vacuum under study Now, if the lnvanant superpotentlal of eq (2 18) is generated, the fern-non X IS a "Goldstone ferrmon" m the sense that the supercurrent creates this field m this "vacuum" More precisely, if a source is introduced wtuch puts the system in this state, then (v[l~lx( p )) - ( Fq,~ o~/~ (3 39) ~/2p ° and the energy of the state is E= I(F~)I 2 (3 40) Two related issues in QCD: θ dependence and the U(1) problem. Instantons provide an understanding of both, but in ordinary QCD, plagued by infrared issues. These are strong coupling questions. Qualitative expectations from instantons 4 P Pure gauge theory: V (θ) = Λ n cn cos(nθ) with cn’s not calculable systematically (ir problems) 0 0 4 0 0 With matter, η potential: V (θ; η ) ∼ Λ cos(θ + η =fη) Singling out the η0 is heuristic, at best; η0 is not particularly light. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings Witten long ago suggested that instantons are not a reliable guide, and proposed an alternative: large N. Large N approximation: N ! 1; g2N fixed: consistent with many qualitative features of QCD (existence of resonances, Zweig’s rule...) At large N, instantons effects should behave as 2 − 8π e g2 ∼ e−c N . U(1) problem reappears. Instead, θ, η0 potentials from resumming of perturbation theory. N dependence as from Feynman diagrams. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings . Implications of this viewpoint: Correlation functions at zero momentum of FF~ behave as (Nf = 0): d n Z n E(θ) / h d 4x FF~ i / N2−n dθn This is not compatible with a simple cos(θ), or more P generally n cn cos(nθ) behavior for E. If Nf N, quarks as a perturbation – anomaly as a perturbation. η0 a pseudogoldstone boson, with mass of order 1=N. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings Branched structure of QCD: multiple (metastable) ground states at fixed θ How then to account for the 2π periodicity expected of θ? Witten suggested that QCD should be branched. For pure QCD: 4 2 E(θ; k) = ΛQCD(θ + 2πk) : Then θ ! θ + 2π compensated by k ! k − 1. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings 0 Including quarks, θ ! θ + η =fπ: Now one can write an expression in terms of the Goldstone boson matrix, p πAλA=2 iη0= 2N i U = e f e fπ : 0 0 2 2 η 2 E(θ; η ) = fπ Trµi U + (θ + 2πk + ) : fπ Focus on mq Λ=N: p 1 0 0 2 2 Mass of the η : mη / 1=N (Fπ / N). η0 2 Interactions of the η0 suppressed as: 1 ( )n: Nn−2 fπ Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings Supersymmetric Gauge Theories: A Testing Grounds In QCD these problems are hard. In Supersymmetric gauge theories with small soft breakings, we can address all of these questions. Today we will see: In pure gauge case, with small mλ, branches, N-dependence, N-dependence exactly as anticipated. Branches arise from spontaneous breaking of approximate, discrete symmetries. Small numbers of matter fields can be treated systematically as a perturbation η0 lagrangian as anticipated. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings But: While N dependence as anticipated, interpretation in terms of Feynman graphs obscure When instanton computations are possible, they are unsuppressed at large N – indeed, same N counting as anticipated from perturbation theory. Critical role of spontaneously broken, approximate discrete symmetries; these symmetries badly broken as mλ ! ΛQCD. Do these branches survive? Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings We will see that even in the regime of weak coupling, phase transitions as function of mλ=mq between N and Nf ground states. Whether the N branches survive, or disappear, as the soft breakings increase and we approach real QCD is a question for lattice experiments. Michael Dine Monodromy in QCD: Insights From Supersymmetric Theories with Soft Breakings Supersymmetric Gauge Theory Without Matter We have understood much about strongly interacting supersymmetric gauge theories, exploiting, particularly, holomorphy of quantities like the superpotential as functions of couplings. In the supersymmetric limit, the gauge theory without matter possesses a ZN symmetry, spontaneously broken by a gaugino condensate: 2 3 2πik hλλi = 32π Λhol e N First question: is this consistent with expectations from large N.
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