Debye Screening” in the Monopole-Anti-Monopole Plasma Extends to Locally 4D Theories

Home , Dual photon, Supersplit supersymmetry

P A C I F I C many features of the world are not - as we’ve heard on this workshop - exP lA ined by the C urrent I dea of the F undamental I nteraC tions - the “Standard Model” of particle physics to this end, in the past 30+ years, theorists have come up with a multitude of so-called “Beyond the Standard Model” scenarios, which have raised many questions: for example: is “the Higgs” a fundamental scalar ﬁeld? is it a composite object? is strongly-coupled dynamics involved? is there supersymmetry? is the theory “natural”? ... (dark matter, CP, baryogenesis, inﬂation) ...

time has now come to pay the piper... What will the LHC discover? resonaances.blogspot.com

Blogger Jester says:

Higgs boson. ... Non-SM Higgs boson. ... New Beyond SM Particles. ... Strong Interactions. ... Dark matter. ... Little Higgs and friends. ... Supersymmetry. ... Dragons. ... Black Holes. What will the LHC discover? resonaances.blogspot.com

Blogger Jester says: Here are my expectations. The probabilities were computed using all currently available data and elaborated Bayesian statistics.

Higgs boson. Probability 80% ... Non-SM Higgs boson. Probability 50% ... New Beyond SM Particles. Probability 50% ... Strong Interactions. Probability 20% ... Dark matter. Probability 5% ... Little Higgs and friends. Probability 1% ... Supersymmetry. Probability 0.1% ... -S Dragons. Probability e dragon ... -S Black Holes. Probability 0.1* e dragon ... What will the LHC discover? My purpose here is not to discuss “scenarios” aka “model-building” - a separate and very long subject just offer a few remarks: we have come up with many scenarios Higgs boson. ... Non-SM Higgs boson. it is not clear which (if any) of these ... scenarios are true New Beyond SM Particles. ... weakly-coupled scenarios generally suffer Strong Interactions. from ﬁne-tuning problems ... 2 M __Z 2 Dark matter. e.g., ~ 4,050 = 1,924,050 - 1,920,000 ( GeV ) ... 2 Little Higgs and friends. ... strong-coupling ideas are plagued by our Supersymmetry. inability to calculate ... It is important to understand the signatures of the various scenarios and the discovery potential at the LHC [much work, understandably so!]. What will the LHC discover? My purpose here is not to discuss “scenarios” aka “model-building” - a separate and very long subject just offer a few remarks: we have come up with many scenarios Higgs boson. ... Non-SM Higgs boson. it is not clear which (if any) of these ... scenarios are true New Beyond SM Particles. ... weakly-coupled scenarios generally suffer Strong Interactions. from ﬁne-tuning problems ... 2 M __Z 2 Dark matter. e.g., ~ 4,050 = 1,924,050 - 1,920,000 ( GeV ) ... 2 Little Higgs and friends. ... strong-coupling ideas are plagued by our Supersymmetry. inability to calculate ...

It is also of interest to understand the “theory space” involved [less work...]. My focus will be on this... What will the LHC discover?

Higgs boson. In all “scenarios” for “Beyond the ... Non-SM Higgs boson. Standard Model” physics, new gauge ... New Beyond SM Particles. dynamics is invoked, at some scale.

...... unless “supersplit supersymmetry” turns Strong Interactions. out to be nature’s choice...... Dark matter. ... When the weak force is turned off, these Little Higgs and friends. gauge theories can be “chiral” (L-R ... Supersymmetry. asymmetric) or “vectorlike”. ... What is the “theory space” involved? How well do we understand the dynamics? Is there any progress we can make, without using numerics or assuming extensive amount of supersymmetry? (Is it “useful” progress?...) P A C I F I C 2 0 1 1

Moorea, French Polynesia

Deformations, topological molecules, and (de)conﬁnement

Erich Poppitz oronto . . with Mithat Unsal SLAC/Stanford (2008-present) and Mohamed Anber, Toronto (2011)

also based on works of Unsal w/ Shifman, Simic, Yaffe 2007-2010, and inspired by earlier work of others, to be mentioned as time goes on

What I will talk about applies to the entire “gauge theory space”, as described below... “gauge theory space” conventional wisdom:

pure YM - “formal” but see www.claymath.org/millennium/ “gauge theory space” conventional wisdom: pure YM - “formal” but see www.claymath.org/millennium/

SUSY - very “friendly” to theorists beautiful - exact results gauge theories with “applications”: boson-fermion superpartner masses; supersymmetry breaking degeneracy: in chiral SUSY theories or metastable vacua in new spacetime symmetry vectorlike theories; SUSY compositeness, ﬂavor...

I believe that one of the most important “applications” of supersymmetry is to teach us about the many strange things gauge ﬁeld theories could do - often very much unlike QCD:

- massless monopole/dyon condensation-conﬁnement and chiral symmetry breaking - “magnetic free phases” - dynamically generated gauge ﬁelds and fermions - chiral-nonchiral dualities - last but not least: gauge-string duality made concrete - AdS/CFT! “gauge theory space” conventional wisdom: pure YM - “formal” but see www.claymath.org/millennium/

SUSY - very “friendly” to theorists beautiful - exact results

QCD-like - hard, leave it to lattice folks (vectorlike) but (m, a, V, $) gauge theories with “applications”: varying number of W-, Z-boson masses: “walking” (or “conformal”) technicolor massless vectorlike “unparticles” fermions

- upon increasing the number of fermion “ﬂavors” believed to become conformal - large current lattice effort to determine phase diagram (proves to be not easy!) “gauge theory space” conventional wisdom: pure YM - “formal” but see www.claymath.org/millennium/

SUSY - very “friendly” to theorists beautiful - exact results

QCD-like - hard, leave it to lattice folks (vectorlike) but (m, a, V, $) non-SUSY chiral - poorly understood strong dynamics gauge theories ...almost nobody talks about them anymore “applications”: extended technicolor: fermion mass generation; massless fermions with quark and lepton compositeness; speculations on W, Z, t masses by L/R asymmetric coupling monopole condensation

- non-QCD-like behavior, e.g. “conﬁnement without chiral symmetry breaking”: massless composite fermions (probably true) - ”tumbling” - dynamical generation of different scales (no idea if true... after 30 years!) “gauge theory space” conventional wisdom: pure YM - “formal” but see www.claymath.org/millennium/

SUSY - very “friendly” to theorists beautiful - exact results

QCD-like - hard, leave it to lattice folks (vectorlike) but (m, a, V, $) non-SUSY chiral - poorly understood strong dynamics gauge theories ...almost nobody talks about them anymore

moral:

We don’t know that much about generic non-supersymmetric gauge dynamics. Nature’s analogue computer is not (yet) available and the theory tools are limited: nonperturbative “gauge theory space” tools:

- ‘t Hooft anomaly matching SUSY - “power of holomorphy” - mass and ﬂat direction “deformations” - semiclassical expansions - strings/branes - gauge-string dualities

- lattice “classic”: QCD-like - the others mentioned already - “MAC” (vectorlike) for QCD (EFT...) (most attractive channel) - semiclassical expansions - truncated Schwinger-Dyson equations non-SUSY - ‘t Hooft anomaly matching “postmodern”: chiral - semiclassical expansions - postulated thermal inequality - AdS/QCD theories - postulated beta functions - extrapolating semiclassical results outside region of validity ...

nonperturbative “gauge theory space” tools: tools you don’t really trust - unless conﬁrmed by experiment or the tools - ‘t Hooft anomaly matching on the left SUSY - “power of holomorphy” “voodoo QCD” - mass and ﬂat direction “deformations” [Intriligator] - semiclassical expansions - strings/branes - gauge-string dualities

- relatively recent, having shown some unexpected features - gives another example of ideas borrowed from strings/SUSY, which usefully apply to more general situations! - ﬁnally, all of this turns out to be quite theoretically elegant...

The general theme is about inferring properties of inﬁnite-volume theory by studying (arbitrarily) small-volume dynamics.

The small volume may be

or of characteristic size “L” some history:

Eguchi and Kawai (1982) showed that “loop eqns” (=the inﬁnite set of Schwinger-Dyson eqns) for Wilson loops in pure Yang-Mills theory are identical in small-V and inﬁnite-V theory, to leading order in 1/N, provided:

- “center-symmetry” unbroken - translational symmetry unbroken (see Yaffe, 1982)

= + O(1/N) provided topologically nontrivial (winding) Wilson loops have vanishing expectation value of any expectation value of (folded) expectation value Wilson loop at inﬁnite-L Wilson loop at small-L (= unbroken center) “EK reduction” “large-N reduction” “large-N volume-independence” - one of the few exact results in QFT (albeit not widely known yet, for a reason, see below)

If it can be made to work, potentially exciting, for :

1) simulations may be cheaper (use single-site lattice?) 2) raises theorist’s hopes (that small-L, single-site, easier to solve?) Some intuition of how EK reduction - valid at any coupling - works: in perturbation theory: at strong coupling: from spectra (& Feynman graphs) gravity dual of N=4 SYM - a conformal ﬁeld in appropriate backgrounds theory - Wilson loops, appropriate correlators - insensitive to box if center-symmetric V(r) ~ 1/r : CFT result obtains in center-symmetric vacuum for any r (L) insensitive to box size r . .

L Unsal, EP 2010 effective space size = L N - so even L=1 OK if large N

the “modern” point of view on EK reduction/large-N orbifold equivalence is that of a volume reducing orbifold (e.g., orbifold by translations)

Kovtun, Unsal, Yaffe (2004) Essentially, VEVs and correlators of operators that are center-neutral and carry momenta quantized in units of 1/L (in compact direction) are the same on, say as in inﬁnite-L theory in the large-N limit.

calculating vevs (symmetry breaking, phase structure) - OK, even if all dimensions small calculating spectra (for generic theories/reps no Gross-Kitazawa trick) - need at least one large dimension

“EK reduction” “large-N reduction” “large-N volume-independence” - one of the few exact results in QFT (albeit not widely known yet, for a reason, see below) However, Bhanot, Heller, Neuberger (1982) noticed immediate problem

- center symmetry breaks for L < L c remedies: e.g., Gonzales-Arroyo, Okawa (1982) - TEK... + others later argued to have problems

Recently realized remedy, allowing reduction to arbitrarily small L (single-site) Unsal, Yaffe (2007) if adjoint fermions (more than one Weyl) - no center breaking, so reduction holds at all L double-trace deformations (deform theory to prevent center breaking; deformation “drops out” of loop equations at inﬁnite-N)

Unsal; used for current lattice studies of theoretical studies Unsal-Yaffe; Unsal-Shifman; “minimal walking technicolor” (Sannino) Unsal-EP 2007-9 is 4 (...3,5...) Weyl adjoint theory conformal or not? ﬁx-N, take L-small - semiclassical studies of conﬁnement in center symmetric vacuum- small-L(=1) large-N simulations (2009-) Polyakov’s 3d mechanism works also in a Hietanen-Narayanan; Bringoltz-Sharpe; Catterall et al locally 4d theory - now due to novel strange (nonselfdual) topological small-N large-L simulations (2007-) Catterall et al; del Debbio et al; Hietanen et al... excitations, whose nature depends on fermion content (vectorlike or chiral) (many issues to still be resolved... including who wins!) a complementary regime to that of volume independence - a (calculable!) other uses are in lattice supersymmetry and shadow of the dynamics of the 4 numerical studies of AdS/CFT... dimensional “real thing” one motivational slide with theoretical dreams for small-L large-N not the regime I will discuss further, because: - currently numerically explored on lattice - analytic approaches at large-N, small-L, any g need new ideas... (ambitious task: equivalent to solving!) use large-N “deformation equivalence” to avoid center breaking

now use volume- independence (valid to L=1) with generic fermion representations phase transition, breaks center

by commutativity of diagram, learn about the large-N theory you started with instead, describe developments where controlled theoretical studies are possible and yield quite a pretty picture - study ﬁxed-N, small-L instead:

double-trace deformations (deform theory to prevent center breaking; deformation “drops out” of loop equations at inﬁnite-N) if adjoint fermions (more than one Weyl) Unsal; theoretical studies Unsal-Yaffe; - no center breaking transition at small L Unsal-Shifman; Unsal-EP 2007-9

ﬁx-N, take L-small - semiclassical studies of conﬁnement in center symmetric vacuum- Polyakov’s 3d mechanism works also in a locally 4d theory - now due to novel strange (nonselfdual) topological excitations, whose nature depends on fermion content (vectorlike or chiral)

These studies provides a complementary regime to that of volume independence - give a (calculable!) shadow of the dynamics of the 4 dimensional “real thing”... For this talk enough to consider 4d SU(2) theories

with Nw = multiple adjoints Weyl fermions

“applications”:

N w =1 is ~ Seiberg-Witten theory N=1SUSY YM with soft-breaking mass

N w =4 - “minimal walking technicolor” - happens to be N=4 SYM without the scalars N w =5.5 asymptotic freedom lost MAIN CLAIMS: In 4d theories with periodic adjoint fermions, for small-L, dynamics of confinement, mass gap, chiral symmetry... is semiclassically calculable. Polyakov’s 3d mechanism of confinement by “Debye screening” in the monopole-anti-monopole plasma extends to locally 4d theories. However, the “Debye screening” is now due to composite objects, the “magnetic bions”, which only exist at L>0. In 4d theories with periodic adjoint fermions, for small-L, confining dynamics is semiclassically calculable.

is now an adjoint 3d scalar Higgs ﬁeld but it is a bit unusual - a compact Higgs ﬁeld: such shifts of A vev absorbed into shift of KK number “n” 4 thus, natural scale of “Higgs vev” is leading to

hence, semiclassical if L << inverse strong scale exactly this happens in theories with more than one periodic Weyl adjoints follows from two things, without calculation: 1.) existence of deconﬁnement transition in pure YM and 2.) supersymmetry in pure YM, at small L (high-T), Veff min at A4 =0 & max at pi/L (Gross,Pisarsky,Yaffe 1980s) in SUSY Veff=0, so one Weyl fermion contributes the negative of gauge boson Veff Q.E.D. Polyakov’s 3d mechanism of conﬁnement by “Debye screening” in the monopole-anti-monopole plasma extends to (locally) 4d theories. However, the “Debye screening” is now due to composite objects, the “magnetic bions” of the title. since SU(2) broken to U(1) at scale 1/L

there are monopole-instanton solutions of ﬁnite Euclidean action, (the ‘t Hooft-Polyakov static monopole on R^(3,1) with the time axis removed = 3d instanton) constructed as follows:

monopole-instanton M trivially embedded in 4d monopole-instanton M trivially embedded in 4d new input - ﬁrst seen in D-brane physics - a “KK tower”, whose lowest action member of the tower can be pictured like this (as opposed to the no-twist):

“twisted” or “Kaluza-Klein”: monopole embedded in 4d by a twist by a “gauge transformation” periodic up to center - in 3d limit not there! (inﬁnite action) KK

M K. Lee, P. Yi, 1997 charges: magnetic topological suppression M Euclidean D0-brane

KK Euclidean D0-brane

in SU(2), 1/2 of the ‘t Hooft suppression center-symmetric vev coupling matching in a purely bosonic theory, vacuum would be a dilute M-M* plasma - but interacting, unlike instanton gas in 4d (in say, electroweak theory)

3d Euclidean space-time

physics is that of Debye screening; by analogy: electric ﬁelds are screened in a charged plasma (“Debye mass for photon”) in the monopole-antimonopole plasma, the dual photon (3d photon ~ scalar) obtains mass from screening of magnetic ﬁeld:

also by analogy with Debye mass: “(anti-)monopole operators” 2 aka “disorder operators” - not locally expressed in dual photon mass ~ M-M* plasma density terms of original gauge ﬁelds (Kadanoff-Ceva; ‘t Hooft - 1970s)

(for us, v = pi/L) Polyakov, 1977: dual photon mass ~ conﬁning string tension “Polyakov model” = 3d Georgi-Glashow model or compact U(1) (lattice) but our 4d theory has M and KK which have fermion zero modes index theorem: Nye-Singer (2000), Unsal, EP (2008) symmetries imply: topological excitations generating “Debye” screening must have i. magnetic charge 2 ii. no fermion zero modes M: + ... KK: - ... M + KK* bound state? M*: - ... KK*: + ... - same magnetic charge ~ 1/r-repulsion - fermion exchange ~ log(r)-attraction

2 M + KK* = B - magnetic “bion” size ~ L/g (L) >> L (our “lattice spacing”) 4

B: + ... + B*: - ... -

Unsal 2007: dual photon mass is induced by magnetic “bions” - the leading cause of conﬁnement in SU(N) with adjoints at small L (including SYM) 3d pure gauge theory vacuum monopole plasma Polyakov 1977

circles = M(+)/M*(-) 4d QCD(adj) fermion attraction M-KK* at small-L Unsal 2007, ....

circles = M(+)/M*(-) squares = KK(-)/KK*(+) - require nonzero L, i.e. 4d 4d QCD(adj) bion plasma at small-L Unsal 2007, ....

circles = M(+)/M*(-) squares = KK(-)/KK*(+) - require nonzero L, i.e. 4d blobs = Bions(++)/Bions*(--) - require nonzero L, i.e. 4d 4d QCD(adj) bion plasma at small-L Unsal 2007, .... 1/g 2(L) Le 4

M + KK* = B - magnetic “bions” - - carry 2 units of magnetic charge - no topological charge (non self-dual) L (locally 4d nature crucial: no KK in 4d) bion stability is due to fermion attraction balancing Coulomb L/g2 (L) repulsion - results in scales as indicated 4 - bion/antibion plasma screening generates mass for dual photon

“magnetic bion confinement” operates at small-L in any theory with massless Weyl adjoints, including N=1 SYM (& N=1 from Seiberg-Witten theory) it is “automatic” at small-L: no need to otherwise “deform” theory first time confinement analytically shown in a non-SUSY, continuum, locally 4d theory where we left out further subleading, at small L, contributions. Recalling that 0 = (22 can calculate mass gap, string tension...An additional argument, basedUnsal, on e⇥ective EP field theory,2009, is appropriate Anber, since EP we are2011 work- ing at small L. The BPS and leading twisted (KK) monopole solutions that constitute the 4nf )/3, we find: bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicit solutions (3.25)) of the ± fields and can be e⇥ectively described by a 3D theory that only involves these lowest modes.14 8 2n The1/ coupling2 in this 3D theory is given by g(L)/L (we do not distinguish between the energy Wf 2c˜ log 1 scales ⇥/L or 2⇥/L here, a di⇥erence that will only introduce an inessential correction). Since 3 ( L )the 3D theory is Higgsed at the scale ⇥/L, there is no further running of the 3D coupling M (L) e and all the physics(less should be expressed relevant in terms of g(L contributions)), obeying the usual (unbroken) 4D , (4.40) ⇤ renormalization⇥ group equation. Thus, we argue that the dependence of Zbion on g is given to two-loop order by:

2 8⇥ (1+cg (L)) g2 (L) 2-Loop 1 2-Loop strong scale O(1), positive Zbion (g(L)) 14 8n e (4.37) ⌅ g f (L) where we use the positive numberc ˜ =2⇥c 0/2. 1-Loop where denotes coe⌅cients that play no role in determining the dependence of the dual- ⌅ photon mass on the energy scale. We note that in the limit of asymptotically smallM L 1/, where the perturbative calcu- 8 2n 4.4 Dual photon mass and previous⌅ small-L “estimates” of conformal window f lation is justified, the correction to the leadingIn this subsection, semiclassical we determine the dependence of result the photon mass on the S(1 size toL two-loop) 3 is dominated order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads: 15 ⇤ 42c/g(L) by dependence of the bion action on the nonzero8Z Higgsbion(g) mass, e . As the size L =4⇥ (4.38) M ↵ g2L2 ⇤ 2 is increased, g(L) increases, hence the exponential1 decreases—and4⇥ 2 the corresponding “Higgs exp 2 (1 + cg2-Loop)+(2nf 4) log g1-Loop(L) , ⌅ L ⇧g2-Loop(L) ⌃ contribution” to the dual photon mass increases.and g(L) is the runningFor coupling atn thef energy< scale 14/L. Plugging and then appropriatef > loop 4thise⇥ect does order of (2.2)into (recall that = (22 4n )/3, = (136 64n )/3), we obtain: M 0 f 1 f 1 1 not change the leading behavior dictated by the first factorlog log 2 2 on the r.h.s. of (4.40). However, B 0 1 1 1 M exp log 1 L log L +(4 2n ) log log S 2 2 2 1 f 2 2 ⌅ ⌥ 4 L log L ⇥ P ⇥ ⇤ 0 2L2 ⌅ 1 0 ⌦ for the four Weyl adjoint theory, n = 4, where the1 2 leading dependence of on the S size f 2c log( ) (1+...) e ⇤ L ⇥ 1 1/2 4 2nf M 2 0 1 40 0 2c log 1 (L) 2 e 2 L log , (4.39) vanishes, we find that the next leading contribution⌅ ⇥ to ML ,shownin(4.40) is an increasing ⇥

14Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- function of L. The other terms shown intion ( of4.39 the tower of) twisted and monopole solutions omitted in such a framework. in Since deconstruction (4.40 approximates) do not change this the “extra” dimension only by a finite number of Kaluza-Klein modes, the corresponding tower of twisted monopoles is also finite. conclusion; this... how is most dare easily you seenstudy from non-protected the fact that quantities? their dependence on the gauge coupling is power-law, rather than exponential. – 22 – Thus, the dual photon mass (n = 4) increases with increasing L. Since the bion M f plasma density is proportional to the square of the dual photon mass, this means that the topological excitations do not dilute away in the decompactification limit—at least for su⌅- ciently small L, where this calculation is valid. Thus, according to the conjecture of [13], which ties conformality on R4 to dilution vs. nondilution of the mass gap on R3 S1 at ⇥ increasing L,QCDwithN = 2, and nf = 4 Weyl fermions in the adjoint representation should not exhibit conformal behavior in the large L limit. Taking the “estimate” of [13] at face value means that the conformal window should be 4

5. Summary and discussion

In this paper, we studied in some detail the SU(2) gauge theory with nf massless adjoint Weyl fermions on R3 S1, our main focus being the bion mechanism of confinement of [2]. ⇥ We described in detail the tools and approximations involved and discussed the stability of magnetic bions. The relevant scales in the problem at L are shown on Figure 1.Weused ⌅ methods and approximations familiar from QCD instanton calculations. We also studied the behavior of the mass gap (or string tension) as a function of L at fixed for nf =5, 4, 3, 2. Already the earlier leading-order semiclassical result [13] indicated that the nf = 5 theory is perhaps conformal on R4, with (likely) a weakly-coupled infrared fixed point. The scenario,

15While the analytic expansion of eqn. (4.11) of the non-BPS action is only valid for asymptotically small 3 g mH /mW 10 , see [35], the numerical results for the mH /mW g dependence of the action show that ⇥ ⇤ ⇥ at weak coupling, g 1, the action is a monotonically increasing and approximately linear function, hence our conclusion is valid throughout the weak-coupling regime.

– 23 – An additional argument, basedAn on additional e⇥ective field argument, theory, based is appropriate on e⇥ective since field we theory, are work- is appropriate since we are work- ing at small L. The BPS anding leading at small twistedL. The (KK) BPS monopole and leading solutions twisted that (KK) constitute monopole the solutions that constitute the bions of smallest charge and action,bions of i.e. smallest the one charge most and relevant action, at small i.e. theL, one involve most only relevant the at small L, involve only the lowest Kaluza-Klein modes (withlowest KK Kaluza-Klein numbers 0, modes1, see the (with explicit KK numbers solutions 0, (3.251, see)) of the the explicit solutions (3.25)) of the ± ± 14 fields and can be e⇥ectively describedfields and by can a 3D be theory e⇥ectively that onlydescribed involves by a these 3D theory lowest that modes. only involves these lowest modes.14 The coupling in this 3D theoryThe is given coupling by g( inL) this/L (we 3D theory do not is distinguish given by g between(L)/L (we the do energy not distinguish between the energy scales ⇥/L or 2⇥/L here, a di⇥erencescales ⇥ that/L or will 2⇥ only/L here, introduce a di⇥erence an inessential that will correction). only introduce Since an inessential correction). Since the 3D theory is Higgsed at thethe scale 3D theory⇥/L, there is Higgsed is no atfurther the scale running⇥/L of, there the 3D is no coupling further running of the 3D coupling and all the physics should beand expressed all the in physics terms shouldof g(L), be obeying expressed the in usual terms (unbroken) of g(L), obeying 4D the usual (unbroken) 4D renormalization group equation.renormalization group equation. Thus, we argue that the dependence of Z on g is given to two-loop order by: Thus, we arguebion that the dependence of Zbion on g is given to two-loop order by:

8⇥2 (1+cg (L)) 8⇥2 1 g2 (L) 2-Loop (1+cg (L)) 2-Loop 1 g2 (L) 2-Loop Zbion (g(L)) e 2-Loop (4.37) 14 8nf Zbion (g(L)) e (4.37) ⌅ 14 8nf g (L) ⌅ 1-Loop g1-Loop (L) where denotes coe⌅cients thatwhere playdenotes no role in coe determining⌅cients that the play dependence no role in of determining the dual- the dependence of the dual- ⌅ ⌅ photon mass on the energyphoton scale. mass on the energy scale. M M where4.4 we Dual left out photon further mass subleading, and at previous small L, small- contributions.L “estimates” Recalling that of conformal0 = (22 window can calculate mass gap, string4.4 tension... DualAn additional argument, photon basedUnsal, on e⇥ective EP field mass theory,2009, is appropriate Anber, and since EP we are2011 work- previous small- L “estimates” of conformal window ing at small L. The BPS and leading twisted (KK) monopole solutions that constitute the 4nf )/3, we find: bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicit solutions (3.25)) of the ± 14 1 In this subsection, we determine thefields dependence and can be e⇥ectively described by a 3D of theory the that only involves photon these lowest modes. mass on the S size to two-loop 1 8 2n In thisThe1/ coupling2 subsection, in this 3D theory is given by g(L we)/L (we do determine not distinguish between the energy the dependence of the photon mass on the S size to two-loop Wf 2c˜(log 1 )scales ⇥/L or 2⇥/L here, a di⇥erence that will only introduce an inessential correction). Since (L) 3 e L the 3D theory is(less Higgsed at the relevant scale ⇥/L, there is no contributions) further running of the 3D coupling , (4.40) order. TheM dual photon mass is givenand all the by physics should eqn. be expressed (4.6 in terms of),g(L), which obeying the usual (unbroken) after 4D substituting (4.37), reads: ⇤ order.renormalization The⇥ group dual equation. photon mass is given by eqn. (4.6), which after substituting (4.37), reads: Thus, we argue that the dependence of Zbion on g is given to two-loop order by:

2 8⇥ (1+cg (L)) g2 (L) 2-Loop 1 2-Loop strong scale O(1), positive Zbion (g(L)) 14 8n e (4.37) ⌅ g f (L) where we use the positive numberc ˜ =2⇥c 0/2. 1-Loop 8Zbion(g) where denotes coe⌅cients that play no role in determining the dependence of the dual- ⌅ 8Z (g) photon mass on the energy scale. bion We note that in=4 the limit⇥ of asymptotically smallM L 1/, where the perturbative calcu- (4.38) 2 2 =4⇥ 8 2n (4.38) 4.4 Dual photon mass and previous⌅ small-L “estimates” of conformal2 window2 f M ↵ g L M g L lation is justified, the correction to the leadingIn this subsection, semiclassical we determine the dependence of result the↵ photon mass on the S(1 size toL two-loop) 3 is dominated order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads: 15 _ 2 ⇤ 42c/g(L) by dependence of the bion action on the nonzero8Z Higgsbion(g) mass, e . As the size L 1 4⇥ =4⇥ (4.38) 2 2 2 2 M ↵ g L 1 ⇤ 4⇥ 2 exp volume (1 +1 cg4⇥2-Loop2 )+(2nvolumef 4) log g (L) , is increased, g(L) increases, hence the2 exponentialexp decreases—andsemiclassical(1 + cg )+(2expn 4) log g2 the(L) , corresponding(11-Loop + “Higgscg )+(2n 4) log g (L) , ⌅ L g2 (L) 2-Loop f 1-Loop 2-Loop f 1-Loop L independence ⇧ 2-Loop ⌃ 2 ⌅ ⇧g2-Loop(L) Abelian⌅ L independenceg (L) ⌃ contribution” to thesemiclassical dual photonregime mass increases.and g(L) is the runningFor coupling atn thef energy< scale 14/L. Plugging and⇧ theregimen appropriatef2-Loop> loop 4thise⇥ect does ⌃ order of (2.2)into (recallconfinement that = (22 4n )/3, = (136 64n )/3), we obtain: Abelian M 0 f 1 f 1 1 not change the leading behavior dictated by the first factorlog log 2 2 on the r.h.s. of (4.40). However, 0 1 1 L 1 confinement M exp log 1 log L +(4 2nf ) log log ⌅ ⌥ 4 2L2 2 log 1 2L2 ⇥ and g(L) is the running coupling at the energy⇥ ⇤ 0 2L2 scale⌅ 1/L. Plugging the1 appropriate loop and g(L ) is0 the running coupling⌦ at the energy scale 1/L. Plugging the appropriate loop for the four Weyl adjoint theory, n = 4, where the1 2 leading dependence of on the S size f 2c log( ) (1+...) e ⇤ L ⇥ 1 1/2 4 2nf M 0 2 2c 0 log 1 1 40 order of (2.2)into (recall that = (222 L 4n )/3, = (136 64n )/3), we obtain: 0 (L) 2 e log f , 1 (4.39) f vanishes, wesmall find # thatflavors: the topological next leadingexcitationsorder contribution of⌅ (2.2 large)into⇥ to # flavors:ML ,shownin( dilution(recall of topological that4.40) is0 an= increasing (22 4nf )/3, 1 = (136 64nf )/3), we obtain: M ⇥ become more dense with increasing L - 14 excitationsM with increasing L - Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- function of L. The other terms shown intion ( of4.39 the tower of) twisted and monopole solutions omitted in such a framework. in Since deconstruction (4.40 approximates) do not change this - indicates confinement at infinite L? the “extra” dimension onlyvanishing by a finite number of Kaluza-Klein mass modes, gap? the corresponding tower of twisted monopoles is also finite. 1 conclusion; this is most easily seen from the fact that- indicates their1 dependenceconformal at infinite on theL? gauge coupling 1 0 1 1 log log 2 2 1 1 0 L 1 1 log log 2L2 1 is power-law,M exp rather thanlog exponential. 1 – 22 – log L +(4 2nf ) log log cf continuing2 2 latticeM studies sinceexp2 2007- & recall1 (m,a,V,$)log 1 2 2log L +(4 2nf ) log log ⌅ ⌥ 4 L 0 log 2 2 22 2 1 L ⇥ 2 2 Thus, the dual photon mass ⇥ ⇤(nf =⌅ 4) increases⌥4L with⌅ increasing L ⇤L. Since0 log the bion2 2 ⌅ L ⇥ M ⇥ L plasma density is proportional0 to the square of the dual photon mass, this means that the ⌦ 1 0 ⌦ 2 1 topological excitations2c log( doL not) (1+ dilute...) away in the decompactification2 limit—at least for su⌅- e 2c log( L ) (1+...) ⇤ e ⇤ ciently small⇥ L, where this calculation is valid. Thus, according1 to the conjecture of [13], 1/2 ⇥ 4 2nf 1 2 04 1 40 1/2 3 41 2nf 0 2c log 1 2 0 1 40 which ties conformality2 on R2 to dilutionL vs. nondilution0 2 ofc thelog mass gap on R 1 S at (L) e log 2 2 , L ⇥ (4.39) increasing⌅ L,QCDwithN = 2, and⇥ n =( 4L WeylL) fermionse in the adjointlog representation , (4.39) f⌅ ⇥ ⇥ L should not exhibit conformal behavior in the large L limit. Taking the “estimate” of [13]⇥ at face14 value means that the conformal window should be 4

In this paper, we studied in some detail the SU(2) gauge theory with nf massless adjoint Weyl fermions on R3 S1, our main focus being the bion mechanism of confinement of [2]. ⇥ – 22 – We described in detail the tools and approximations involved and discussed the stability– 22 of – magnetic bions. The relevant scales in the problem at L are shown on Figure 1.Weused ⌅ methods and approximations familiar from QCD instanton calculations. We also studied the behavior of the mass gap (or string tension) as a function of L at fixed for nf =5, 4, 3, 2. Already the earlier leading-order semiclassical result [13] indicated that the nf = 5 theory is perhaps conformal on R4, with (likely) a weakly-coupled infrared fixed point. The scenario,

8⇥2 (1+cg (L)) 8⇥2 1 g2 (L) 2-Loop (1+cg (L)) 2-Loop 1 g2 (L) 2-Loop Zbion (g(L)) e 2-Loop (4.37) 14 8nf Zbion (g(L)) e (4.37) ⌅ 14 8nf g (L) ⌅ 1-Loop g1-Loop (L) where denotes coe⌅cients thatwhere playdenotes no role in coe determining⌅cients that the play dependence no role in of determining the dual- the dependence of the dual- ⌅ ⌅ photon mass on the energyphoton scale. mass on the energy scale. M M 4.4 Dual photon mass and previous small-L “estimates” of conformal window 4.4 DualAn additional argument, photon based on e⇥ective ﬁeld mass theory, is appropriate and since we are work- previous small-L “estimates” of conformal window we knew already that small-L regimeing at small L. The is BPS complementary and leading twisted (KK) monopole solutions thatto constitute the bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicit solutions (3.25)) of the that of volume independence (if it was identical, I’d be talking at± Clay...) ﬁelds and can be e⇥ectively described by a 3D theory that only involves these lowest modes.14 1 In this subsection, we determine theThe dependence coupling in this 3D theory is given by g of(L)/L (we the do not distinguish photon between the energy mass on the S size to two-loop 1 - gives a calculable “shadow”In this of scalesthe⇥ subsection,/L 4dor 2⇥ /L“realhere, a di ⇥thing”erence that will we only introduce determine an inessential correction). Since the dependence of the photon mass on the S size to two-loop the 3D theory is Higgsed at the scale ⇥/L, there is no further running of the 3D coupling order. The- dualcalculability photon not often mass there is in given nonperturbativeand all the by physics should eqn. be expressed problems (4.6 in terms of),g(L), which obeying the usual (unbroken) after 4D substituting (4.37), reads: order.renormalization The group dual equation. photon mass is given by eqn. (4.6), which after substituting (4.37), reads: - so, not to be taken lightly - I think “itThus, pays we argueto calculate” that the dependence of Zbion on g is given to two-loop order by:

2 8⇥ (1+cg (L)) - bound to learng2 something...(L) 2-Loop 1 2-Loop Zbion (g(L)) 14 8n e (4.37) ⌅ f g1-Loop (L) 8Zbion(g) where denotes coe⌅cients that play no role in determining the dependence of the dual- ⌅ 8Z (g) photon mass on the energy scale. bion =4⇥ 2 2 M =4⇥ (4.38) (4.38) M ↵ g L 4.4 Dual photon massM and previous small-L “estimates” of conformalg2L window2 In this subsection, we determine the dependence of the↵ photon mass on the S1 size to two-loop order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads: _ 2 8Z (g) 1 4⇥ =4⇥ bion (4.38) 2 2 2 2 M ↵ g L 1 4⇥ 2 exp volume (1 +1 cg4⇥2-Loop2 )+(2nvolumef 4) log g (L) , 2 exp semiclassical(1 + cg )+(2expn 4) log g2 (L) , (11-Loop + cg )+(2n 4) log g (L) , ⌅ L g2 (L) 2-Loop f 1-Loop 2-Loop f 1-Loop L independence ⇧ 2-Loop ⌃ 2 ⌅ ⇧g2-Loop(L) Abelian⌅ L independenceg (L) ⌃ semiclassical regime and g(L) is the running coupling at the energy scale 1/L. Plugging⇧ theregime appropriate2-Loop loop ⌃ order of (2.2)into (recallconfinement that = (22 4n )/3, = (136 64n )/3), we obtain: Abelian M 0 f 1 f 1 1 0 1 1 log log 2L2 1 confinement M exp log 1 log L +(4 2nf ) log log ⌅ ⌥ 4 2L2 2 log 1 2L2 ⇥ and g(L) is the running coupling at the energy⇥ ⇤ 0 2L2 scale⌅ 1/L. Plugging the appropriate loop and g(L ) is0 the running coupling⌦ at the energy scale 1/L. Plugging the appropriate loop 2c log( 1 ) 2 (1+...) e ⇤ L ⇥ 1 1/2 4 2nf 0 2 2c 0 log 1 1 40 order of (2.2)into (recall that = (222 L 4n )/3, = (136 64n )/3), we obtain: 0 (L) 2 e log f , 1 (4.39) f small # flavors: topological excitationsorder of⌅ (2.2 large)into⇥ # flavors:L dilution(recall of topological that 0 = (22 4nf )/3, 1 = (136 64nf )/3), we obtain: M ⇥ become more dense with increasing L - 14 excitationsM with increasing L - Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- tion of the tower of twisted monopole solutions in such a framework. Since deconstruction approximates - indicates confinement at infinite L? the “extra” dimension onlyvanishing by a finite number of Kaluza-Klein mass modes, gap? the corresponding tower of twisted monopoles is also finite. 1 - indicates1 conformal at infinite L? 1 0 1 1 log log 2 2 1 1 0 L 1 1 log log 2L2 1 M exp log 1 – 22 – log L +(4 2nf ) log log cf continuing2 2 latticeM studies sinceexp2 2007- & recall1 (m,a,V,$)log 1 2 1 2 2log L +(4 2nf ) log log ⌅ ⌥ 4 L ⇤ ⌅0 log⌥ 42 2 ⌅ 2L2 log L ⇥ 2L2 ⇥ ⇥ L ⇥ ⇤ 0 2L2 ⌅ 0 ⌦ 1 0 ⌦ 2 1 2c log( L ) (1+...) 2 e 2c log( L ) (1+...) ⇤ e ⇤ ⇥ 1 1/2 ⇥ 4 2nf 1 2 0 1 40 1/2 4 2nf 0 2c log 1 2 0 1 40 2 2 L 0 2c log 1 (L) e log(L) 2 e 2 , L log , (4.39) (4.39) ⌅ ⇥ ⌅ L ⇥ L ⇥ ⇥

14 Such a theory would be relatively14 straightforwardSuch a theory towould obtain be relativelyvia “deconstruction”—see straightforward to [41] obtain for a via construc- “deconstruction”—see [41] for a construc- tion of the tower of twisted monopoletion solutions of the tower in such of twisted a framework. monopole Since solutions deconstruction in such a approximates framework. Since deconstruction approximates the “extra” dimension only by a finitethe “extra” number dimension of Kaluza-Klein only by modes, a finite the number corresponding of Kaluza-Klein tower of modes, twisted the corresponding tower of twisted monopoles is also finite. monopoles is also finite.

– 22 – – 22 – An additional argument, based on e⇥ective ﬁeld theory, is appropriate since we are work- we knew already that small-L regimeing at small L. The is BPS complementary and leading twisted (KK) monopole solutions thatto constitute the bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicit solutions (3.25)) of the that of volume independence (if it was identical, I’d be talking at± Clay...) ﬁelds and can be e⇥ectively described by a 3D theory that only involves these lowest modes.14 The coupling in this 3D theory is given by g(L)/L (we do not distinguish between the energy - gives a calculable “shadow” of scalesthe⇥/L 4dor 2⇥ /L“realhere, a di ⇥thing”erence that will only introduce an inessential correction). Since the 3D theory is Higgsed at the scale ⇥/L, there is no further running of the 3D coupling - calculability not often there in nonperturbativeand all the physics should be expressed problems in terms of g(L), obeying the usual (unbroken) 4D renormalization group equation.

- so, not to be taken lightly - I think “itThus, pays we argueto calculate” that the dependence of Zbion on g is given to two-loop order by:

2 8⇥ (1+cg (L)) - bound to learng2 something...(L) 2-Loop 1 2-Loop Zbion (g(L)) 14 8n e (4.37) list, so far: ⌅ f g1-Loop (L)

where denotes coe⌅cients that play no role in determining the dependence of the dual- ⌅ photon mass on the energy scale. i. Polyakov’s 3d mechanism of conﬁnement byM “Debye screening” due to monopoles extends inﬁnitesimally into 4d - holds 4.4in Dual all photon theories mass and previous (some small-L “estimates”must be of conformal deformed window to In this subsection, we determine the dependence of the photon mass on the S1 size to two-loop ensure abelianization) order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads:

8Z (g) study shows that massless fermion content plays=4⇥ bioncrucial role in the confinement(4.38) M ↵ g2L2 1 4⇥2 mechanism, contrary to what many thought exp (1 + cg )+(2 n 4) log g 2 ( L ) , ⌅ L g2 (L) 2-Loop f 1-Loop ⇧ 2-LoopUnsal w/ Shifman, Yaffe, EP, Anber,⌃ Argyres.. ’07-’11 and g(L) is the running coupling at the energy scale 1/L. Plugging the appropriate loop order of (2.2)into (recall that = (22 4n )/3, = (136 64n )/3), we obtain: M 0 f 1 f 1 ii. chiral symmetry (discrete/abelian) breaking due to disorder1 operator vevs 0 1 1 log log 2L2 1 M exp log 2 2 1 2 1 log L +(4 2nf ) log log 2 2 ⌅ ⌥ 4 L ⇤ 0 log 2 2 ⌅ L Unsal, ⇥ EP ’09 generic ⇥ L 0 ⌦ 2c log( 1 ) 2 (1+...) e ⇤ L ⇥ 1 1/2 4 2nf 2 0 1 40 0 2c log 1 (L) 2 e 2 L log , (4.39) ⌅ ⇥ L iii. all calculable abelian confinement mechanisms are continuously⇥ related: 14Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- “Seiberg-Witten confinement” “=” “Polyakov-liketion of the tower of twisted monopole solutions bion in such aconfinement” framework. Since deconstruction approximatesUnsal, EP ’11 the “extra” dimension only by a finite number of Kaluza-Klein modes, the corresponding tower of twisted monopoles is also finite. iv. even in SUSY, power of small-L not fully explored yet, e.g. - gave yet another argument for ISS I=3/2 model does not break SUSY – 22 – Unsal, EP ’09 An additional argument, based on e⇥ective field theory, is appropriate since we are work- we knew already that small-L regimeing at small L. The is BPS complementary and leading twisted (KK) monopole solutions thatto constitute the bions of smallest charge and action, i.e. the one most relevant at small L, involve only the lowest Kaluza-Klein modes (with KK numbers 0, 1, see the explicit solutions (3.25)) of the that of volume independence (if it was identical, I’d be talking at± Clay...) fields and can be e⇥ectively described by a 3D theory that only involves these lowest modes.14 The coupling in this 3D theory is given by g(L)/L (we do not distinguish between the energy - gives a calculable “shadow” of scalesthe⇥/L 4dor 2⇥ /L“realhere, a di ⇥thing”erence that will only introduce an inessential correction). Since the 3D theory is Higgsed at the scale ⇥/L, there is no further running of the 3D coupling - calculability not often present inand nonperturbative all the physics should be expressed inproblems terms of g(L), obeying the usual (unbroken) 4D renormalization group equation.

- not to be taken lightly - I think “it paysThus, to wecalculate” argue that the dependence of Zbion on g is given to two-loop order by:

2 8⇥ (1+cg (L)) - bound to learng2 something...(L) 2-Loop 1 2-Loop Zbion (g(L)) 14 8n e (4.37) ⌅ g f (L) list, so far, contd.: 1-Loop where denotes coe⌅cients that play no role in determining the dependence of the dual- ⌅ photon mass on the energy scale. M v. finite-T deconfinement transition at4.4 small-L Dual photon mass due and previous to “competing” small-L “estimates” of conformal electric window and In this subsection, we determine the dependence of the photon mass on the S 1 Simic,size to two-loop Unsal, ’10 magnetic excitations order. The dual photon mass is given by eqn. (4.6), which after substituting (4.37), reads: older speculation: Liao, Shuryak ’06 8Z (g) =4⇥ bion (4.38) M ↵ g2L2 vi. depending on gauge group and fermion matter1 content4⇥2 deconfinement transition at exp (1 + cg )+(2n 4) log g2 (L) , ⌅ L g2 (L) 2-Loop f 1-Loop small-L described by various 2d models ⇧ 2-Loop ⌃ and g(L) is the running coupling at the energy scale 1/L. Plugging the appropriate loop - some solved long ago - Ising, Z_N parafermions,order of (2.2)into (recall that XY-spins = (22 4n )/3, with= (136 64 nZ_K)/3), we perturbation obtain: M 0 f 1 f - some new - “XY”-multi-spin models on root space w/ perturbations1 1 0 1 1 log log 2L2 1 M exp log 1 log L +(4 2nf ) log log - can study rank of transition, various ⌅ critical⌥ 4 2L2 exponents 2 log 1 2L2 ⇥ ⇥ ⇤ 0 2L2 ⌅

via analytic (new 2d cfts?), numerical, cold atom0 (?) methods ⌦ 2c log( 1 ) 2 (1+...) e ⇤ L Anber, Unsal, EP ’11-’12 ⇥ 1 1/2 4 2nf 2 0 1 40 0 2c log 1 (L) 2 e 2 L log , (4.39) ⌅ ⇥ L vii. ⇥ one can even take messages for infinite-L14 deconfinement transition: Such a theory would be relatively straightforward to obtain via “deconstruction”—see [41] for a construc- tion of the tower of twisted monopole solutions in such a framework. Since deconstruction approximates Seiberg-Witten-inspired - role of newthe “extra” types dimension only “topological by a finite number of Kaluza-Klein modes,molecules” the corresponding tower of twisted monopoles is also finite. Unsal, EP ’11-’12 vs older work of Dyakonov ’06

– 22 – many questions remain - feel free to ask your own!