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Matthew Reece Harvard University Exploring the Weak Gravity Conjecture Matthew Reece Harvard University Based on 1506.03447, 1509.06374, 1605.05311, 1606.08437 with Ben Heidenreich and Tom Rudelius. & 161n.nnnnn with Grant Remmen, Thomas Roxlo, and Tom Rudelius Ooguri, Vafa 2005 Concretely: How good can approximate symmetries be? Can an approx. global symmetry be “too good to be true” and put a theory in the swampland? Approximate Symmetries For large-field, natural inflation we might like to have a good approximate shift symmetry φ φ + f, f > M ! Pl In effective field theory, nothing is wrong with this. In quantum gravity, it is dangerous. Quantum gravity theories have no continuous global symmetries. Basic reason: throw charged stuff into a black hole. No hair, so it continues to evaporate down to the smallest sizes we trust GR for. True of arbitrarily large charge ⇒ violate entropy bounds. (see Banks, Seiberg 1011.5120 and references therein) The Power of Shift Symmetries Scalar fields with good approximate shift symmetries can play a role in: • driving expansion of the universe (now or in the past) • solving the strong CP problem • making up the dark matter • breaking supersymmetry • solving cosmological gravitino problems These are serious, real-world phenomenological questions! Example: QCD Axion Weinberg/Wilczek/…. taught us to promote theta to field: a ↵s µ⌫ Gµ⌫ G˜ instantons: shift-symmetric potential fa 8⇡ di Cortona et al. 1511.02867 Important point: large field range, small spurion 2 2 2 2 2 1 m f (md/mu) 1 m f (md/mu)((md/mu) (md/mu) + 1) 2 ⇡ ⇡ λ = ⇡ ⇡ − ma = 2 2 4 4 f (1 + (md/mu)) −f (1 + md/mu) Example: QCD Axion 9 • Stellar cooling constraints: fa >~ 10 GeV • This means the quartic coupling <~ 10-41. Any axion is very weakly coupled! • Whole potential has an overall exponentially small spurion -S e in front; relative terms fa suppressed. • Upper bound on fa? Black hole super-radiance (Arvanitaki et al.) rules out a window near the Planck scale • Weak Gravity Conjecture: fa <~ MPl/S. Links how exponentially small the spurion is with relative suppression between terms. What is the Weak Gravity Conjecture? The Weak Gravity Conjecture In 2006, Arkani-Hamed, Motl, Nicolis, and Vafa (AMNV) conjectured that in any consistent gravitational theory with a U(1) gauge field—like electromagnetism—there is at least one particle with m p2eQM Pl This is the opposite of the black hole extremality bound. For such particles, gravity is weaker than E&M. What is the Conjecture? Arkani-Hamed, Motl, M > Q Nicolis, Vafa (“AMNV”) M = Q hep-th/0601001 M < Q Particle exists with M<Q (superextremal). FigureWhy? 2. An extremal Postulate black hole canthat decay extremal only if there black exist particle holess should decay. whose charge exceeds their mass. The difficultiesWhy? involving remnantsElse: arenearly-extremal avoided if macroscopic black black holes canholes evaporate spontaneously all their charge away,evolve and so thesetoward states wouldextremality. not be stable.Sinceextremalblackholes Sequester information and have M = QMPl,inorderforthemtobeabletodecayintoelementaryparticlentropy forever. Seems strange. es, these particles should have m<qMPl.OurconjecturealsonaturallyfollowsfromGell-Mann’s totalitarian principle (“everything that is not forbidden is compulsory”) because there should not exist a large number of exactly stable objects (extremal black holes) whose stability is not protected by any symmetries. Another heuristic argument leading to same limit on Λ is the following. Consider the minimally charged monopole solution in the theory. With a cutoffΛ,itsmassisoforder M Λ/g2 and its size is of order R 1/Λ.Itwouldbesurprisingfortheminimally mon ∼ mon ∼ charged monopole to already be a black hole because the valuesofallchargescarriedby ablackholeshouldbemacroscopic(andeffectively continuous); after all, a black hole is a classical concept. Demanding that this monopole is not blackyields Mmon < < 2 1 Λ gMPl (5) MPlRmon ∼ ⇒ ∼ 2.3 Simple parametric checks It is easy to check the conjecture in a few familiar examples. For U(1)’s coming from closed heterotic strings compactified to four dimensions, for instance, we have gM M , (6) Pl ∼ s 6 Axion Inflation The Weak Gravity Conjecture WGC Constraints on Axion Inflation Loopholes Conclusions and Directions for Future Research The N-Species (Mild)Convex Weak Gravity Hull Conjecture Condition Suppose we have not 1, but N 1-form gauge fields. Multiple U(1) gauge groups: black hole extremality The N-species weak gravity conjecture holds that the bound generically becomes q~i convex hull of the charge-to-mass vectors ~zi = Mp M > (e Q )2 +(e Q )2 +±...+(±e mQi )2M must contain the NBH-dimensional1 1 unit2 ball.2 N N Pl p Stricter WGC than ~z = !eQ/m each U(1) individually! Cheung, Remmen 1402.2287 CHC: convex hull contains the unit ball Figure 6 : The convex hull condition [Cheung, Remmen ’14]. Tom Rudelius October 21, 2015 Super-Planckian Axions? AxionAxion Inflation Inflation TheThe Weak Weak GravityAxion Gravity Conjecture Inflation Conjecture Axion fieldsWGCWGCThe Constraintsin Weak Constraints string Gravity on on Axion Conjecture Axion theory Inflation Inflation are generally found to have WGC Constraints on AxionLoopholes InflationLoopholes LoopholesConclusionsConclusions f < MPl. (Banks, Dine,Conclusions Fox, Gorbatov hep-th/0303252) So can’t immediately use one for large-field inflation. ThreeThreeThree Popular Popular Popular Solutions: Solutions: Solutions: People have tried clever model-building to improve: V!øV" !ø" V!ø" Figure 2 : Decay ø FigureFigure 2 : 2Decay : Decay ø ø Constant Alignment N-flation ConstantConstantAlignment Alignment Alignment Monodromy Figure 1 : N-flation [Kim, Nilles, Peloso Figure 3 : Axion FigureFigure(Dimopoulos, 1 : 1N : -flationN -flationKachru, [Kim,(Kim,[Kim, Nilles, Nilles, Nilles, PelosoPeloso Peloso ’04) Figure(Silverstein,Figure 3 : 3Axion : WestphalAxion ’08; [Dimopoulos et al. ’04] Monodromy [Dimopoulos[DimopoulosMcGreevy, Wacker et al. et al. ’05) ’04]’04] MonodromyMonodromyMcAllister, Flauger, ….) ’05] [McAllister et al. ’08, ’05]’05] [McAllister[McAllister et al. et ’08, al. ’08, Silverstein, Silverstein,Silverstein, WGC provides a handle in the case Westphala = A ’08,ydy WestphalWestphal ’08, ’08, Flauger et al. ’09] FlaugerFlaugerI et al. et ’09] al. ’09] Tom Rudelius October 21, 2015 Tom RudeliusTom Rudelius OctoberOctober 21, 2015 21, 2015 Inflation vs. WGC: Magnetic Also see electric arguments: AMNV; Rudelius; Brown, Cottrell, Shiu, Soler AMNV pointed out that magnetic monopoles have a Weak Gravity bound that looks like mmag < (qmag/e)MPl 1 ~ 2 If we interpret the classical monopole radius 2 B = mmag rcl e as a cutoff length scale for EFT, we have Z 1 (also follows from Lattice WGC ⇤ = rcl− <eMPl because tower of states) If our 5D radius should be bigger than this cutoff, we have 1 R>rcl > eMPl implying again that f<MPl (de la Fuente, Saraswat, Sundrum 1412.3457) 3.1 Warmup: diagonal N-flation Suppose that we have N U(1) gauge fields Ai with couplings ei and that the kinetic mixing among them is negligible. We will also assume that in this basis the charge lattice simply consists of integer electric or magnetic charges under each U(1). These are strong simplifying assumptions,Inflation but provide a usefulvs starting WGC: point. The WeakMulti-Field Gravity Conjecture applied to each gauge field separately implies the existence of electricallyandmagneticallychargedparticles satisfyingWGC, then, certain bounds.tells us But decay the constraint constants for the set ofareN filesselds is than actually the stronger than for any individual field: if we marginally saturate the bound for each U(1) by postulating a magnetic monopolePlanck withscale charge. Butqm =1 N-flation,and mass √ alignment,2MPl/ei,forexample,thenanearly-extremalblack and monodromy holeare withexplicitly large and trying equal charges to get under super-Planckian every U(1) will not be fieldable torange decay. out This is because theof sub-Planckian extremality bound for decay a black hole constants. charged under Does multipl WGCegroupsdependsnotonthesum help here? of the charges but on the charges added in quadrature: for a magnetically charged black hole 2 2 2 2 in four dimensions, the bound is Qeff Q1/e1 + ...+ QN /eN <MBH/(√2MPl). StartConsider with extranaturalN-flationN. -Nflation axions that≡ moves from along N U(1) the diagonal gauge in groups. each axion direction, ! attemptingTry to move to obtain along an eff ectivediagonal decay constant to get 1 2 1 1 f 2 = f 2 + f 2 + ...+ f 2 = + ...+ . (12) eff 1 2 N 2πR e2 e2 " # " 1 N # NoticeMagnetically that this is the charged appropriate black expression holes under obey the assumption of no kinetic mixing and the further assumption that the dominant instanton effects give rise to a potential of the form ci cos(ai/fi),aswouldbegeneratedforexamplefromwrappedworldlineso2 2 2 felectrically Q1 Q2 QN charged particles ofQ chargee↵ 1 under+ each+ gauge...+ group.<M MoreBH/M gePlneral instantons can lead to ⌘ e2 e2 e2 alignment$ phenomena in whichs 1 inflation2 winds aroundN one direction in axion space multiple times. We will return to such a possibility later. 2 The linear combination of 1/ei factors appearing on the right-hand side of (12) is precisely what appears in the extremality bound for a magnetically charged black hole with equal charge Q under all N gauge groups. Let us build some intuition by considering waysthatsucha diagonally magnetically charged black hole could decay: •Itcouldemitamonopoleofdiagonalcharge(q, q, . .q),sothatitscharge-to-massvector points in the
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