Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise

Michael Dine

Department of Physics University of California, Santa Cruz

Nambu Memorial Symposium, University of Chicago, 2016 Work with P. Draper, L. Stevenson-Haskins, D. Xu, Francesco D’Eramo. Important input from N. Seiberg

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise From my graduate student days, when I first encountered his work on symmetry breaking in the strong interactions, Nambu has been one of my intellectual heroes. This was reinforced during my years at City College, where I regularly heard stories of Nambu from Bunji Sakita, who himself was an admirer. While somewhat younger, Sakita often regaled me with stories of the War years in Japan, and both his experiences and Nambu’s. I finally got to know Nambu in the 1980’s, and all of my interactions with him were intellectually stimulating and enhanced by his charm and wit. I remember many conversations at Chicago, but remarks he made at the 1984 Argonne meeting on String Theory, which were thoughtful but cautionary, particularly stand out. My final interactions came shortly after his Nobel Prize. Like many, I sent him a congratulatory note, only to receive a "mailbox is full" message. A year later, though, I received the most thoughtful note reply.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise In my own work, Nambu’s inﬂuence is perhaps heaviest in the area of string theory and in the appearance of light scalars in the case of continuous global symmetry breaking. His work with Jona-Lasinio is instructive in that it takes a model which in detail cannot be taken too seriously, but extracts important, universal features. Some of what I say today I hope can be viewed as a modest effort in this style. While much of my discussion will center on Nambu-Goldstone bosons, I will also consider some questions in strong dynamics, where fermionic condensates will play important roles.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise P H YSI CAL R EVI EW VOLUME &22, NUMBER AI RII,

Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. P

Y. NAMBU AND G. JONA-LASINIoj' The Enrico terms Institute for Nuclear StuCkes and the Department of Physics, The University of Chicago, Chicago, Illinois (Received October 27, 1960)

It is suggested that the nucleon mass arises largely as a self-energy of some primary fermion field through the same mechanism as the appearance of energy gap in the theory of superconductivity. The idea can be put into a mathematical formulation utilizing a generalized Hartree-Fock approximation which regards real nucleons as quasi-particle excitations. We consider a simplified model of nonlinear four-fermion interaction which allows a p5-gauge group. An interesting consequence of the symmetry is that there arise automatically pseudoscalar zero-mass bound states of nucleon-antinucleon pair which may be regarded as an idealized pion. In addition, massive bound states of nucleon number zero and two are predicted in a simple approximation. The theory contains two parameters which can be explicitly related to observed nucleon mass and the pion-nucleon coupling constant. Some paradoxical aspects of the theory in connection with the p5 trans- formation are discussed in detail.

I. INTRODUCTION equations' 4: " 'N this paper we are going to develop a dynamical E4~= e lto~+40 (1.1 - theory of elementary particles in which nucleons and E0 ~*= eA ~*—+44~, mesons are derived in a unified way from a fundamental near the Fermi surface. is the component of the spinor field. In basic physical ideas, it has thus the 11„+ excitation corresponding to an electron state of mo- characteristic features of a compound-particle model, mentum and corresponding to but unlike most of the existing theories, dynamical P spin +(up), andri ~* a hole state of momentum and spin which means treatment of the interaction makes up an essential part p +, an absence of an electron of momentum — and spin of the theory. Strange particles are not yet considered. p —(down). is the kinetic energy measured from the The scheme is motivated by the observation of an eo Fermi surface; is a constant. There will also be an interesting analogy between the properties of Dirac g equation complex conjugate to describing particles and the quasi-particle excitations that appear Eq. (1), another of excitation. in the theory of superconductivity, which was originated type Equation gives the eigenvalues with great success by Bardeen, Cooper, and Schrieffer, ' (1) and subsequently given an elegant mathematical forlnu- E„=a (e,'+y')-*'. (1.2) ' lation by Bogoliubov. The characteristic feature of the The two states of this quasi-particle are separated in BCS theory is that it produces an energy gap between 2 In the ground state of the system all energy by ~ E„~. the ground state and the excited states of a supercon- the quasi-particles should be in the lower (negative) w'hich ductor, a fact has been confirmed experimentally. energy states of Eq. (2), and it would take a finite The is caused due to the fact that the attractive gap energy 2)E„~ )~2~&~ to excite a particle to the upper phonon-mediated interaction between electrons produces state. The situation bears a remarkable resemblance to correlated pairs of electrons with opposite momenta and the case of a Dirac particle. The four-component Dirac spin near the Fermi surface, and it takes a finite amount equation can be split into two sets to read of energy to break this correlation. can be EP,=o"Pter+ res, Elementary excitations in a superconductor — conveniently described by means of a coherent mixture Egs ——o"Pigs+ nell r, of electrons and holes, which obeys the following E„=W (p'+nt') l,

* Supported by the U. S. Atomic Energy Commission. where tPt and Ps are the two eigenstates of the chirality f' Fulbright Fellow, on leave of absence from Instituto di Fisica operator ys —y jy2y3y4. dell Universita, Roma, Italy and Istituto Nazionale di Fisica According to Dirac's original interpretation, the Nucleare, Sezione di Roma, Italy. world has all the electrons 'A preliminary version of the work was presented at the ground state (vacuum) of the Midwestern Conference on Theoretical Physics, April, 1960 (un- in the negative energy states, and to create excited published). See also Y. Nambu, Phys. Rev. Letters 4, 380 (1960); states (with zero particle number) we have to supply an and Proceedings of the Tenth Annual Rochester Conference on High-Energy Nuclear Physics, 1960 (to be published). energy &~2m. ' J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, In the BCS-Bogoliubov theory, the gap parameter @, 162 (1957). which is absent for free electrons, is determined es- 3 N. N. Bogoliubov, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 58, self-consistent (Hartree-Fock) representa- 73 (1958) Ltranslation: Soviet Phys. -JETP 34, 41, 51 (1958)g; sentially as a N. N. Sogoliubov, V. V. Tolmachev, and D. V. Shirkov, A %em tion of the electron-electron interaction eGect. Methodin the Theory of Supercondlctivity (Academy of Sciences of 4 U, S.S.R., Moscow, 1958). J. G. Valatin, Nuovo cimento 7, 843 (1958). 345

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Y. NAM BU AN 9 G. JONA —LASI NIO tendency for partial cancellation between contributions which lead to q'= 0, and C:D= 1 —2m'I (0):mI (0). from diferent mesons or nucleon pairs. From Eq. (4.8), we have 0(2m'I(0) (-.', . We already remarked before that the model treated here is not realistic enough to be compared with the (b) Put r„5—(iv„vs+2mv, q„/q')Fi(q') was show that actual nucleon problem. Our purpose to + (iv.v~ —iv qvnq. /q')F2(q') (A 3) a new possibility exists for 6eld theory to be richer and more complex than has been hitherto envisaged, even This is seen. to satisfy the integral equation if though the mathematics is marred by the unresolved divergence problem. In the subsequent paper we will attempt to generalize — the model to allow for isospin and finite pion mass, and F2= J~(q')/[I »(q')), (A.4) draw various consequences regarding strong as well as »(q') = 2m'I(q') —J'v(q') weak interactions. where J(q') was defined in Eq. (4.13). APPENDIX On the mass shell, F„5 reduces to We treat here, for completeness, the problem created by the coupling of pseudoscalar and pseudovector terms (iv„vg+ 2mv Sq„/q') F(q'), encountered in the text. As we have seen, such an effect F(q') = 1+F2 (q') = 1/[1 —» (q')). is not essential for the discussion of y~ invariance, but rather adds to complication, which however naturally For q'=0, we have J(q') =0 so that 1&F(0)=1/ appears in the ladder approximation. [1—2m'I(0)) &2. First let us write down the integral equation for a (c) From the structure of the inhomogeneous term, vertex part j. : it is clear that the scattering matrix is given by

I'(p+ :q, p :q)-—— 2go(P )f(v ) +go(P')f(iv v ) 2zgp where I"5 is the pseudoscalar vertex function. =v(p+ ,q, p lq)+--v. Trl v.&(p'+ :q)- (2~)4 Again, from Eq. (A.1), I'6 is determined as

k—)-)d'p'— x(lp'+ 'qp 'q)&-i:(p I'& —VS[1—2m'I(q'))/q'I(q') miV —qV5/q', (A.6)

Sgo v~v. )»[vip(p'+kq) which has an entirely different behavior from the bare (2a-)4 y5 for small q'. The scattering matrix is then

XI (p'+ ', q, p 2q)S, (p-', q—))d p'. (A—.1-) g (v,)f(v,) 2g,[I 2m2I(q2))/q2I(q2) [(~v —qv5)f(vs) * (v~)f—(iv qv~) ~)2.mgo/q' This embraces three special cases depending on the — — inhomogeneous term y: (iv qv )f(iv qv~) go»(q')/q'[I »(q')) + (iv„vs) f (iv„vs);go/[1 —» (q')]. (A.7) (a) v =0 for the Bethe-Salpeter equation for the pseudoscalar meson; The first three terms have a pole at q'=0. The coupling for the pseudovector vertex function I"„~, (b) v=iv„vs constants of the pseudoscalar meson are then (c) v= 2go(vg)f(v~); —go(v„v~)f(v„v~), for the nucleon- antinucleon scattering through these interactions. pseudoscalar coupling: Here i and f refer to initial and final states, and the integral kernel of Eq. (A.1) operates on the f pa, rt. G '= 2go[1 —2m'I(0))/I(0), We will consider them successively. (a) We make the ansatz I'=Cv5+iDv~v q. The inte- pseudovector coupling: grals in Eq. (A.1) then reduce to the standard forms considered in the text. use of Making Eqs. (4.9), (4.17), G„„'=g,»(0)/[1 —» (0)] a,nd (6. we get" 9), Michael Dine Light= go2m'I Scalars(0)/'[1 —2m'I and(0)]. the(A.8) Cosmos: Nambu-Goldstone and Otherwise C=C—(C+2mD)q'I, Their relative sign is such that the equivalent pseudo- D= (C+2mD)mI, scalar coupling on the mass shell is (A.2) l go d~' Jf 4m') 1—2m'I(0): 2m'I(0) I(q')=-- p G„"=4m'gp (A.9) 4a.2 J q2+~2 ( ~2 ) 2m'I (0) 1—2m'I(0) In his famous work on symmetry breaking, the light scalars were the pions of the strong interactions. Today, particularly important roles for light scalars arise in cosmology. Examples include axions as dark matter, but also candidates for the inflaton of slow roll inflation. Indeed, the following statements are often made about inflation:

1 The Planck satellite ruled out hybrid inflation 2 If tensor fluctuations are observed (requiring Planck scale variation of the inflaton) then inflation is necessarily “natural", or “chaotic". 3 In the case of “natural" inflation, this must be understood as “monodromy" inflation or “aligned axions". Today I want to push back (a bit) on these statements.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Part I: Aspects of Natural Inﬂation/Monodromy Inﬂation

1 Chaotic Inflation and Natural Inflation 2 Problems for Natural infation ⇒ Monodromy inflation, Aligned Axions (won’t consider) 3 Aspects of Monodromy in Field Theory 4 Summary of Part I: plausability of chaotic/natural inflation

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Small vs. Large Field Inﬂation

Categories of Inﬂationary models:

1 Large Field: φ Mp. Chaotic inflation, natural inflation, monodromy inflation. Predict potentially observable gravity waves.

2 Small Field: φ Mp. Associated with “hybrid inﬂation". No observable gravity waves. For hybrid, challenges to understand ns as reported by Planck (ns = 0.9603 ± 0.0073.)

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Theoretical Challenges in Each Framework

1 Large field: a. Chaotic inflation: typically monomial potentials. Require φn small coefficients (why?). Suppression of n−4 for many n. Mp Why? b. Natural inflation: requires fa Mp, doesn’t seem to be realized in string theory (failure of explicit constructions; “Weak Gravity Conjecture". Alternative: monodromy inflation, multi-natural inflation, aligned axions... (roughly equivalent to mondromy inflation.) 2 Small field: Planck scale corrections still important. Very δρ tiny couplings to account for ρ . More generally, in any framework: challenging to make predictions which would tie to a detailed microscopic picture. We will propose an alternative view in this talk.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Nambu-Goldstone Bosons as the Actors in Inﬂation: Natural Inﬂation and its Variants

4 V (a) = Λ cos(θ/fa). (1)

Slow roll conditions, e.g.

V 00 η = M2 1 (2) p V

requires fa Mp. Difﬁcult to realize in string theory. (Banks, Fox, Gorbatov, M.D.) Leads to Weak Gravity Conjecture – Arkani-Hamed, Motl, Nicolis, Vafa. Alternatives: Monodromy inﬂation, multiple axions (with alignment).

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Monodromy Inﬂation

Silverstein, Westphal: in string models, argue that axions can “wind". Periodicity greater than 2π. Monomial potentials, e.g. φ2/3 over a broad range in ﬁeld space. The string constructions somewhat complicated. Consider realizations in ﬁeld theory.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise SU(N) supersymmetric gauge theory (Draper, M.D.)

3 2πik iθ/N hλλi = Λ e N e . (3)

If, for example, a susy-breaking term, mλλλ, then θ 2πk V (θ) = m NΛ3 cos( + ) λ N N

So naive periodicity θ → θ + 2π becomes θ → θ + 2πN. Compensated by changing branch.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Elevate θ to a (pseudo) Nambu-Goldstone boson, θ = a(x)/fa. As the axion moves, it crosses over to other branches, but provided mλ is small, the tunneling rate is low (suppression as − C m3 e λ .

For sufﬁciently large N (say N > 100 and suitably small mλ, would be what we seem to require for successful natural inﬂation. Can debate whether this a particularly plausible story (e.g. is such large N in the Swampland – Ooguri’s talk).

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Other Field Theory Realizations

Witten: η0 current algebra at large N: Witten argued that at large N, anomaly could be treated as a perturbation. Argued that one should not think of η0 potential (and θ-dependent effects) as arising from instantons (∼ e−N ), but rather from resuming of perturbation theory. Then from large N counting arguments, a V (U) = TrMU + (log det U + θ)2. N Again, monodromy. θ (axion) can wander some distance. Potential is a monomial. E.g. for small M, very large N, potential has of order N minima. What are these? Puzzling: θ, η0 periodicities? Over what range of ﬁelds might one trust this?

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Large N Pure Gauge Theory

Subsequently, Witten argued for similar behavior in pure gauge theory (monodromy), based in part on AdS/CFT correspondence.

4 2 V (θ) = mink c Λ (θ + 2πk) (4)

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Essentially this branched structure arose in SUSY QCD with a gluino mass above; indeed, for θ + 2πk N,

θ 2πk θ 2πk m NΛ3 cos( + ) ≈ C − m NΛ3( + )2 λ N N λ N N

We should recover ordinary QCD for large mλ. For large mλ, there is no reason to expect stability for large θ + 2πk. It would seem unlikely that there are order N metastable vacua.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise We can consider the η0 case similarly. In particular, study SUSY QCD with 1 ﬂavor (for simplicity), large N. Include a mass term for the quark, and also soft mass terms for scalars and the gaugino. In the regime

mq m˜ mλ Λ

η0 2πk + θ + f V (θ, η0) = NΛ3m cos( η )2. λ N Again, expanding for large N gives Witten’s η0 action, with the correct N scalings. As before, there are, in this limit, N metastable minima associated with the ZN symmetry of the supersymmetric theory. Again, the behavior should change as 2 we allow mλ, m˜ to become large. Metastability not likely to survive.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise How Badly Do Instantons Lie

Witten, in his 1979 papers, noted that the e−N argument for irrelevance of instantons is formal, due to ir divergences, but dismissed this loophole. It is interesting to consider the large N ideology, in light of facts we now understand about the supersymmetric theories. In particularly, in what sense are instanton effects order e−N at large N?

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Pure susy gauge theory: Gauginos condense. The condensate can be computed by studying a theory with Nc − 1 ﬂavors, ﬁrst at small mass, where one can compute a superpotential due to instantons, and then taking the limit of large mass. The gaugino condensate survives at large N. How does this work? In the underlying theory,

2N+1 Λ ¯ W = + mf Qf Qf . det(QQ¯ )

The coefﬁcient Λ2N+1 is of order e−N , and does arise from instantons. But at the stationary points of the superpotential, det(QQ¯ ) is also of order e−N , and hW i ∼ 1. This object is hλλi at large m.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Heuristic argument directly in the low energy theory

Shifman et al: hλλλ . . . λλi = CΛ3Nf Order e−N , because one is calculating a very high dimension operator. Should be hλλiN , but off by a ﬁnite factor. This can be understood in terms of dilute gas corrections (Festuccia, M.D).

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise hλλλ . . . λλi = CΛ3Nf

2 X − 8π (1+2n) Z = e g2 dρρ3N×2nΛ3N+3N×2n n This is highly infrared divergent. If we assume that the ir cutoff is Λ, then we obtain a result proportional to

Λ3N for each term, i.e there is no e−N suppression for each higher order term in the expansion. This is consistent with the known exact results in the theory (esp. Hollowood et al). It is also consistent with the θ dependence we saw earlier for the vacuum energy once one includes a small gaugino mass term.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Cutoff Instantons in Ordinary QCD

Applying this sort of reasoning, in pure gauge QCD, cutting off the ir divergences as above, would suggest

4 X V (θ) = Λ cn cos(nθ) (5) n Similar considerations of instantons in the theory with a single (light) massive ﬂavor would give

4 X 0 V (θ) = Λ dn cos(nθ + η /fπ) (6) n

More generally, compatible with the ZNf symmetry expected in such theories.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Lessons for Inﬂation

1 Chaotic inflation with monimials: still the usual puzzles (suppression of operators at large fields – why?) 2 Monodromy inflation: simple realization in field theory, but requires very large gauge groups. How plausible? How plausible are the various string constructions (long interval of monomial growth not duplicated in field theory examples where one would seem to have greater conrol). So return to other possibilities. Indeed, string theory suggests a simple alternative picture.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Non-Compact String Moduli as an Arena for Inﬂation

In string theory, esp. with supersymmetry, there are other light moduli. By itself not a new idea (e.g. Banks, 1990’s). But will add some new elements. Will assume some degree of low energy supersymmetry, to justify existence pseudomoduli space; scalar partners of compact moduli (axions). Large/small field inflation: close parallels to large/small field solutions of strong CP problem.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Outline of Part II

1 Review of (small field) hybrid inflation – general lessons. 2 Large/small field solutions of the strong CP problem 3 Inflation with non-compact moduli 4 Ingredients for successful modular inflation 5 Large field excursions in the moduli space 6 Outlook for large field hybrid inflation

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Hybrid Inﬂation: Small Field

Often described in terms of fields and potentials with rather detailed, special features, e.g. so-called waterfall field. Can be characterized in a more conceptual way. Inflation occurs in all such models on a pseudomoduli space, in a region where supersymmetry is badly broken (possibly by a larger amount than in the present universe) and the potential is slowly varying. Essentially all hybrid models in the literature are small field models; this allows quite explicit constructions using rules of conventional effective field theory, but it is not clear that small field inflation is selected by any deeper principle.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Simplest (supersymmetric) hybrid model:

W = I(κφ2 − µ2) (7)

φ is known as the waterfall ﬁeld. Classically, for large I, the potential is independent of I;

4 Vcl = µ (φ = 0) independent of I. The quantum mechanical corrections control the dynamics of the inﬂaton:

κ2 V (I) = µ4(1 + log(|I|2/µ2)). (8) 16π2

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise κ is constrained to be extremely small in order that the ﬂuctuation spectrum be of the correct size; κ is proportional, in fact, to VI, the energy during inﬂation. The quantum corrections determine the slow roll parameters.

−8 2 3 VI = 2.5 × 10 Mp (9)

µ 2 µ 2 κ = . × = . × 5 × . 0 17 15 7 1 10 (10) 10 GeV MP

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Kahler Potential Corrections

One expects corrections in powers of Mp. Organize the effective ﬁeld theory in powers of I (Bose,Monteux,Stephenson-Haskins,M.D.). The quartic term in K , α = † † K 2 I II I (11) Mp

gives too large an η unless α ∼ 10−2. Irreducible Fine Tuning.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Superpotential Corrections

Also corrections to the superpotential:

In δ = W n−3 (12) Mp

At least the low n terms must be suppressed. This might occur as a result of discrete symmetries.The leading power of I in the superpotential controls the scale of inﬂation. Higher scales require larger n; this would seem to point to a low scale of inﬂation.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise But pointing in the opposite direction is κ, which gets smaller rapidly with V0,

In addition, achieving ns < 1, consistent with Planck, required a balancing of Kahler and superpotential corrections. Indeed, from the abstract of the Planck theory paper: “the simplest hybrid inﬂationary models, and monomial potential models of degree n > 2 do not provide a good ﬁt to the data."

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise So the theoretical arguments for small field over large field inflation are hardly so persuasive. Even at low scales it is necessary to have control over Planck scale corrections, and tuning of parameters (at least at the part in 10−2 level) is required. One also needs a very small dimensionless parameter, progressively smaller as the scale of inflation becomes smaller.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Generalizing hybrid inflation to large fields: moduli inflation

So it is clearly interesting to explore the possibility of inflation on (non-compact) moduli spaces with Planck scale fields undergoing variations of order Planck scale or larger. Such moduli spaces are quite familiar from string theory. First consider another situation where such a small field/large field dichotomy arises.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Small Field and Large Field Solutions to the Strong CP Problem

To solve the strong CP problem one must account for an accidental global symmetry which is of extremely high quality . Small ﬁeld solutions: Most models designed to obtain a Peccei-Quinn symmetry are constructed with small axion decay constant, fa Mp, with fa = hφi Organize the effective ﬁeld theory in powers of φ/Mp.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Axion Quality

Require 1 ∂V ∂V ≡ = 4 < −11. Qa 2 10 fa 10 fama ∂a ∂a

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise In small ﬁeld models, if φ contains axion ﬁeld, PQ symmetry iα φN φ → e φ need to suppress N−3 up to very high N. E.g. ZN , Mp with N > 11 or more, depending on fa.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Large ﬁeld solutions of the Strong CP Problem

String theory has long suggested a large ﬁeld perspective on the axion problem (Witten). Frequently axions; exhibit continuous shift symmetries in some approximation (e.g. perturbatively in the string coupling). Non-perturbatively broken, but usually a discrete shift symmetry left which is exact.

a → a + 2π is an exact symmetry of the theory

fa depends on the precise form of the axion kinetic term. The (non-compact) moduli which accompany these axions typically have Planck scale vev’s ). Calling the full chiral axion superﬁeld A = s + ia + ... , this periodicity implies that, for large s, in the superpotential the axion appears as e−A. Solving the strong CP problem requires suppressing only a small number of possible terms (Bobkov, Raby; Dine, Festuccia, Wu).

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Typically several moduli which must be stabilized. Not a well understood problem (many speculations). Whatever the mechanism, the axion multiplet is special. If the superpotential plays a signiﬁcant role in stabilization of the saxion, it is difﬁcult to understand why the axion should be light. e−A would badly break the PQ symmetry if responsible for saxion stabilization. Requires an interplay of different moduli.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Non-Compact Moduli as Inﬂatons

So the strong CP problem points to Planck scale regions of field space as the arena for phenomenology. Ingredients for moduli as setting for inflation: 1 In the present epoch, one or more moduli responsible for hierarchical supersymmetry breaking. 2 In the present epoch, a modulus whose superpotential is highly suppressed, whose compact component is the QCD axion. This is not necessary for inflation, but is the essence of a modular (large field) solution to the strong CP problem. 3 At an earlier epoch, a stationary point in the effective action with higher scale supersymmetry breaking and a positive cosmological constant. 4 At an earlier epoch, a field with a particularly flat potential which is a candidate for slow roll. 5 Dynamics such that inflation ends, i.e. one transitions from the region with large susy breaking to a region with smaller.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Fields need not play the same role in the inflationary era that they do now. The Peccei-Quinn symmetry might be badly broken during inflation. Then the axion will be heavy during this period and isocurvature fluctuations may not be an issue.

In such a case the initial axion misalignment angle, θ0, would be fixed rather than being a random variable. 4 We know that the scale of inflation is well below Mp . So plausible that even during inflation moduli have large vev’s, e−A, e−T 1, though much smaller than at present.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Won’t explore particular models here. Simply state that one can construct models, but no claims that they are derived from some deeper framework such as string theory. Some features: 1 Typically tuned. The lower the scale of inﬂation, the greater the level of tuning. 2 Over some range, often resemble chaotic models

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Summary of Our Observations

Explaining inflation from an underlying microscopic theory is an extremely challenging problem, quite possibly inaccessible to our current theoretical technologies. As we have reviewed, even in so-called small field inflation, it requires control over Planck scale phenomena. Within string theory, this requires understanding of supersymmetry breaking (whether large or small) and fixing of moduli in the present universe as well as at much earlier times. It requires an understanding of cosmological singularities, and almost certainly of something like a landscape.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Helpful, then, to understand what might be generic features of the underlying microscopic theory. As we reviewed, the essence of hybrid inflation is motion on a (non-compact) pseudomoduli space. In string theory, at least at the classical level, such moduli spaces are ubiquitous, and the features of these moduli suggest a picture for inflation in which the (canonical) fields have Planck scale motions. We have stressed a parallel between small/large field inflation and small/large field solutions to the strong CP problem. The existence of moduli in string models is strongly suggestive of the large field solutions to both problems. The proposal we have put forward here is similar to the large field solutions of the strong CP problem.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise Several moduli likely play a role in inflation, achieving the needed degree of supersymmetry breaking and slow roll. We have noted that small r is more tuned than large r, giving some weight to the former possibility. We have noted the contrast with small field inflation, where extreme tuning to achieve low scale inflation is replaced by the requirement of an extremely small dimensionless coupling. Returning to the strong CP problem, any would-be Peccei-Quinn symmetry is an accident, and the accident which holds in the current configuration of the universe need not hold during inflation; this would resolve the axion isocurvature problem. It would imply that θ0 is not a random variable.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise The inflationary paradigm is highly successful; the question is whether we can provide some compelling microscopic framework and whether it is testable. In the present proposal, one does not attempt (at least for now) a detailed microscopic understanding, but considers a class of theories. Within those considered here: 1 Higher scales of inflation are preferred 2 High scale axions (even an axiverse). In a more detailed picture, one might hope to connect some lower energy phenomenon, such as supersymmetry breaking, with inflation.

Michael Dine Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise