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String Theory and the LHCTASI 2010. Boulder

Michael Dine

Department of PhysicsUniversity of California, Santa Cruz

March, 2010

Michael Dine String Theory and the LHC

String Theory at the Dawn of the LHC Era

The LHC program is finally underway. Can string theory haveany impact on our understanding of phenomena which we mayobserve?

Supersymmetry?Warping?Technicolor?Just one lonely higgs?


Supersymmetry

In these three lectures, I will focus mainly on supersymmetry,for reasons of time and because here I know how to make themost concrete statements and models.Virtues

1 Hierarchy Problem2 Unification3 Dark matter4 Presence in string theory (often)

The last item is one of the reasons for this focus:supersymmetry is an important crutch in our understanding ofstring theory; most of what we claim to understand is based onstudy of supersymmetric vacua (configurations).


Hierarchy: Two Aspects

1 Cancelation of quadratic divergences2 Non-renormalization theorems (holomorphy of gauge

couplings and superpotential): if supersymmetry unbrokenclassically, unbroken to all orders of perturbation theory,but can be broken beyond: exponentially large hierarchies.


But reasons for skepticism:1 Little hierarchy2 Unification: why generic in string theory?3 Hierarchy: landscape (light higgs anthropic?)


Reasons for (renewed) optimism:1 The study of metastable susy breaking (ISS) has opened

rich possibilities for model building; no longer thecomplexity of earlier models for dynamical supersymmetrybreaking.

2 Supersymmetry, even in a landscape, can account forhierarchies, as in traditional thinking about naturalness

(e− 8π2

g2 )3 Supersymmetry, in a landscape, accounts for stability – i.e.

the very existence of (metastable) states.


Plan of the Lectures

Lecture 1: Low Energy Supersymmetry: Supersymmetryand its (metastable) breaking, supersymmetry and particlephysics - the MSSM.Lecture 2: Microscopic models of supersymmetry breakingand its MediationLecture 3: What might we mean by a stringphenomenology.


Lecture 1. Low Scale Supersymmetry and ItsBreaking. Outline

1 Some features of N = 1 Supersymmetry2 Metastable vs. stable supersymmetry breaking; Simple

Models3 The MSSM4 Gauge Mediated models (continuing into lecture 2)


Supersymmetry Review

I am going to assume some familiarity with supersymmetry, butit is perhaps worth reviewing some basic features of N = 1theories. First, without gravity. Two types of supermultiplets(superfields) describe particles of spins up to 1:

1 Chiral fields: consist of a complex scalar and a Weylfermion

Φ(θ) = φ(θ) +√

2θψ + θ2F . (1)

2 Vector fields: contain a gauge boson and a Weyl fermion(gaugino). Described by a real superfield, V ; for ourdiscussion enough to consider the gauge covariant,spin-1/2 chiral field, Wα

Wα = λα + θβ[δβαD + (σµν)βαFµν . . . (2)

F and D are auxiliary fields; they play an important role as theyare order parameters for supersymmetry breaking.


At the level of terms with two derivatives, a supersymmetriclagrangian is specified by three functions:

1 The superpotential, W (Φi), a holomorphic function of thechiral fields.

2 The Kahler potential, K (Φi ,Φ†i )

3 The gauge coupling functions, fa(Φi), again holomorphicfunctions of the fields (one such function for each gaugegroup.

The lagrangian density has the form, in superspace:∫d4θK (Φ,Φ†) +

∫d2θ

(fa(Φ)W 2

αa + W (Φ))

(3)


Renormalizable Interactions

In this case, K =∑

ΦiΦ†i , fa = − 1

4g2a

and W is at most cubic. Iwill leave the details of the component lagrangian for textbooksand focus here on the scalar potential:

V =∑

F 2i +

12

∑D2

a . (4)

Fi =∂W∂Φi

Da = φ2i T aφi . (5)

Classically, supersymmetry is unbroken if Fi = Da = 0 ∀ i ,a;conversely, it is broken if not. In this case, there is a Goldstonefermion,

G ∝ Fiψi + Daλa. (6)


R Symmetries

In supersymmetry, a class of symmetries known asR-symmetries play a prominent role. Such symmetries can becontinuous or discrete. Their defining property is that theytransform the supercurrents,

Qα → eiαQα Q∗α → e−iαQ∗α. (7)

Necessarily, the superpotential transforms as W → e2iα underthis symmetry.For the question of supersymmetry breaking, these symmetriesplay a crucial role, embodied in a theorem of Nelson andSeiberg:In order that a generic lagrangian (one with all terms allowed bysymmetries) to break supersymmetry, the theory must possessan R symmetry. This theorem is easily proven by examiningthe equations ∂W

∂Φi= 0, and recalling that they are holomorphic.

But, instead, some examples:


Varieties of R symmetric lagrangians

In general, W has R charge 2. Suppose fields, Xi , i = 1, . . .Nwith R = 2, φa, a = 1, . . .M, with R charge 0. Then thesuperpotential has the form:

W =N∑

i=1

Xi fi(φa). (8)

Suppose, first, that N = M. The equations ∂W∂Φi

= 0 are solved if:

fi = 0; Xi = 0. (9)

(R unbroken, 〈W 〉 = 0.) The first set are N holomorphicequations for N unknowns, and generically have solutions.Supersymmetry is unbroken; there are a discrete set ofsupersymmetric ground states; there are generically nomassless states in these vacua.(Again, R unbroken, 〈W 〉 = 0.)


Next suppose that N < M. Then the equations fi = 0 involvemore equations than unknowns; they generally have an M − Ndimensional space of solutions, known as a moduli space. Inperturbation theory, as a consequence of non-renormalizationtheorems, this degeneracy is not lifted. There are masslessparticles associated with these moduli (it costs no energy tochange the values of certain fields).If N > M, the equations Fi = 0 in general do not have solutions;supersymmetry is broken. These are the O’Raifeartaighmodels. Now the equations ∂W

∂φi= 0 do not determine the Xi ’s,

and classically, there are, again, moduli. Quantummechanically, however, this degeneracy is lifted.


Aside 1

The non-renormalization theorems.Quite generally, supersymmetric theories have the propertythat, if supersymmetry is not broken at tree level, then to allorders of perturbation theory, there are no corrections to thesuperpotential and to the gauge coupling functions. Thesetheorems were originally proven by examining detailedproperties of Feynman diagrams, but they can be understoodfar more simply.E.g. consider the N = 1,M = 1 case and restrict attention torenormalizable theories.

W = X(

m2φ2 +

λ

3φ3). (10)

Here I have chosen not to include a linear term in φ; any suchterm can be absorbed in a redefinition of φ.


Now the R symmetry restricts any corrections to W to the formXf (φ). Suppose that λ = 0. Then the theory has a largersymmetry, under which X and φ transform with phases e2iω ande−iω respectively. We can think of λ as itself the expectationvalue of a chiral field with charge 2 under this symmetry. Anycorrection to λ necessarily has the form

δf = λncn (11)

but this does not respect the symmetry. Arguments of this typeapply more generally.


Metastable Supersymmetry Breaking

Intriligator, Shih and Seiberg: example of metastablesupersymmetry breaking in a surprising setting: vectorlikesupersymmetric QCD. At a broader level, brought therealization that metastable supersymmetry breaking is ageneric phenomenon. Consider the Nelson-Seiberg theorem,which asserts that, to be generic, supersymmetry breakingrequires a global, continuous R symmetry. We expect that suchsymmetries are, at best, accidental low energy consequencesof other features of some more microscopic theory.


O’Raifeartaigh Models

This is illustrated by the simplest O’Raifeartaigh model:

W = λX (A2 − µ2) + mYA. (12)

R symmetry with

RZ = RY = 2; RA = 0; X (θ)→ e2iαX (e−iαθ), etc.. (13)

(Also Z2 symmetry, Y → −Y ,A→ −A forbids YA2). SUSY broken;equations:

∂W∂X

=∂W∂Y

= 0 (14)

are not compatible.Features: If m2 > µ2, 〈A〉 = 0 = 〈Y 〉; X undetermined.


Potential for X at one loop (Coleman-Weinberg); 〈X 〉 = 0. Xlighter than other fields (by a loop factor). Scalar components –light pseudomodulus. Spinor is Goldstino.

〈FX 〉 = λµ2 (15)

is the decay constant of the Goldstino.


Aside 2. The Coleman-Weinberg Potential

Basic idea of Coleman Weinberg calculations is simple.Calculate masses of particles as functions of thepseudomodulus. From these, compute the vacuum energy:∑

(−1)F∫

d3k(2π)3

√k2 + m2

i (16)

=∑

(−1)f(

Λ4 + m2i Λ2 +

1(16π2)

m4i ln(mi)

4).

The first two terms vanish because of features ofsupersymmetry. The last must be evaluated, whensupersymmetry is broken.One finds (Shih) that if all fields have R charge 0 or 2, then theR symmetry is unbroken. Shih constructed models for whichthis is not the case. One of the simplest:

W = X2(φ1φ−1 − µ2) + m1φ1φ1 + m2φ3φ−1. (17)


Continuous Symmetry from a Discrete Symmetry

The continuous symmetry of the OR model might arise as anaccidental consequence of a discrete, ZN R symmetry.E.g.

X → e4πiN X ; Y → e

4πiN Y (18)

corresponding to α = 2πN in eqn. 13 above.

Suppose, for example, N = 5. The discrete symmetry nowallows couplings such as

δL =1

M3

(aX 6 + bY 6 + cX 4Y 2 + dX 2Y 4 + . . .

). (19)

Note that W → e4πiN W .

The theory now has N supersymmetric minima, with

X ∼(µ2M3

)1/5αk (20)

where α = e2πi5 , k = 1, . . . ,5.


Metastability

Need a separate lecture to discuss tunneling in quantum fieldtheory (some remarks in Banks’ lectures). Suffice it to say thatin models such as those introduced above, the metastablesupersymmetric state can be extremely long lived. In particular,the system has to tunnel a “long way" (compared withcharacteristic energy scales) to reach the “true" vacuum.Thinking (correctly) by analogy to WKB, the amplitude isexponentially suppressed by a (large) power of the ratio ofthese scales.


The MSSM and Soft Supersymmetry Breaking

MSSM:A supersymmetric generalization of the SM.

1 Gauge group SU(3)× SU(2)× U(1); corresponding (12)vector multiplets.

2 Chiral field for each fermion of the SM: Qf , uf , df ,Lf , ef .3 Two Higgs doublets, HU ,HD.4 Superpotential contains a generalization of the Standard

Model Yukawa couplings:

Wy = yUHUQU + yDHDQD + yLHDE . (21)

yU and yD are 3× 3 matrices in the space of flavors.


Soft Breaking Parameters

Need also breaking of supersymmetry, potential for quarks andleptons. Introduce explicit soft breakings:

1 Soft mass terms for squarks, sleptons, and Higgs fields:

Lscalars = Q∗m2QQ + U∗m2

UU + D∗m2DD (22)

+L∗m2LL + E∗mE E

+m2HU|HU |2 + m2

HU|HU |2 + BµHUHD + c.c.

m2Q, m2

U , etc., are matrices in the space of flavors.2 Cubic couplings of the scalars:

LA = HUQ AU U + HDQ AD D (23)

+HDL AE E + c.c.

The matrices AU , AD, AE are complex matrices3 Mass terms for the U(1) (b), SU(2) (w), and SU(3) (λ)

gauginos:

m1bb + m2ww + m3λλ (24)

So we would seem to have an additional 109 parameters.Michael Dine String Theory and the LHC

Counting the Soft Breaking Parameters

1 φφ∗ mass matrices are 3× 3 Hermitian (45 parameters)2 Cubic terms are described by 3 complex matrices (54

parameters3 The soft Higgs mass terms add an additional 4 parameters.4 The µ term adds two.5 The gaugino masses add 6.

There appear to be 111 new parameters.


But Higgs sector of SM has two parameters.In addition, the supersymmetric part of the MSSM lagrangianhas symmetries which are broken by the general soft breakingterms (including µ among the soft breakings):

1 Two of three separate lepton numbers2 A “Peccei-Quinn" symmetry, under which HU and HD rotate

by the same phase, and the quarks and leptons transformsuitably.

3 A continuous "R" symmetry, which we will explain in moredetail below.

Redefining fields using these four transformations reduces thenumber of parameters to 105.If supersymmetry is discovered, determining these parameters,and hopefully understanding them more microscopically, will bethe main business of particle physics for some time. Thephenomenology of these parameters has been the subject ofextensive study; we will focus on a limited set of issues.


Constraints

Direct searches (LEP, Fermilab) severely constrain thespectrum. E.g. squark, gluino masses > 100’s of GeV,charginos of order 100 GeV. Spectrum must have specialfeatures to explain

1 Absence of Flavor Changing Neutral Currents (suppressionof K ↔ K , D ↔ D mixing; B → s + γ, µ→ e + γ, . . . )

2 Suppression of CP violation (dn; phases in K K mixing).Might be accounted for if spectrum highly degenerate, CPviolation in soft breaking suppressed.


The little hierarchy: perhaps the greatest challenge forSupersymmetry

Higgs mass and little hierarchy:Biggest contribution to the Higgs mass from top quark loops.Two graphs; cancel if supersymmetry is unbroken. Result ofsimple computation is

δm2HU

= −6y2

t16π2 m2

t ln(Λ2/m2t ) (25)

Even for modest values of the coupling, given the limits onsquark masses, this can be substantial.


But another problem: mH > 114 GeV. At tree level mH ≤ mZ .Loop corrections involving top quark: can substantially correctHiggs quartic, and increase mass. But current limits typicallyrequire mt > 800 GeV. Exacerbates tuning. Typically worsethan 1 %.

δλ ∼ 3y4

t16π2 log(m2

t /m2t ). (26)

Possible solution: additional physics, Higgs coupling correctedby dimension five term in superpotential or dimension six inKahler potential.

δW =1M

HUHDHUHD δK = Z †ZH†UHUH†UHU . (27)


Microscopic Supersymmetry

Focus on two general approaches: Gravity mediation andgauge mediation.In both cases, we need to know something about supergravity.We expect supersymmetry to be a local symmetry. The mostgeneral lagrangian with terms up to two derivatives appears inWess and Bagger, elsewhere; a good introduction alsoprovided by Weinberg’s textbook. Specified, again, by Kahlerpotential, superpotential, and gauge coupling functions,For now, some features which will be important for modelbuilding, more general theoretical issues.


Supergravity

Some basic features:Potential (units with Mp = 1):

V = eK[DiWg i iDiW

∗ − 3|W |2]

(28)

Diφ is order parameter for susy breaking:

DiW =∂W∂φi

+∂K∂φi

W . (29)

If unbroken susy, space time is Minkowski (if W = 0), AdS(W 6= 0).If flat space (〈V 〉 = 0), then

m3/2 = 〈eK/2W 〉. (30)


Mediating Supersymmetry Breaking

Generally assumed that supersymmetry is broken by dynamics ofadditional fields, and some weak coupling of these fields to those ofthe MSSM gives rise to soft breakings.The classes of models called "gauge mediated" and “gravitymediated" are distinguished principally by the scale at whichsupersymmetry is broken. If terms in the supergravity lagrangian(more generally, higher dimension operators suppressed by Mp) areimportant at the weak (TeV) scale:

Fi = DiW ≈ (TeV )Mp ≡ M2int (31)

“gravity mediated" (more next lecture). If lower, “gauge mediated";Fi ≈ ∂iW . In the latter case, usually renormalizable operators. Gaugeinteractions simplest; others (= Yukawa couplings) are problematic.In the low scale case, the soft breaking effects at low energies shouldbe calculable, without requiring an ultraviolet completion; theintermediate scale case requires some theory like string theory.


Minimal Gauge Mediation

Main premiss underlying gauge mediation: in the limit that thegauge couplings vanish, the hidden and visible sectorsdecouple. Simple model:

〈X 〉 = x + θ2F . (32)

X coupled to a vector-like set of fields, transforming as 5 and 5of SU(5):

W = X (λ` ¯ + λqqq). (33)

For F < X , `, ¯,q, q are massive, with supersymmetry breakingsplittings of order F . The fermion masses are given by:

mq = λqx m` = λ`x (34)

while the scalar splittings are

∆m2q = λqF ∆m2

` = λ`F . (35)


In such a model, masses for gauginos are generated at oneloop; for scalars at two loops. The gaugino mass computationis quite simple. Even the two loop scalar masses turn out to berather easy, as one is working at zero momentum. The lattercalculation can be done quite efficiently using supergraphtechniques; an elegant alternative uses background fieldarguments. The result for the gaugino masses is:

mλi =αi

πΛ, (36)


For the squark and slepton masses:

m2 = 2Λ2[C3

(α3

4π

)2+ C2

(α2

4π

)2(37)

+53

(Y2

)2 (α1

4π

)2],

where Λ = Fx/x . C3 = 4/3 for color triplets and zero forsinglets, C2 = 3/4 for weak doublets and zero for singlets.


Features of MGM

1 One parameter describes the masses of the threegauginos and the squarks and sleptons

2 Flavor-changing neutral currents are automaticallysuppressed; each of the matrices m2

Q, etc., is automaticallyproportional to the unit matrix; the A terms are highlysuppressed (they receive no one contributions before threeloop order).

3 CP conservation is automatic4 This model cannot generate a µ term; the term is protected

by symmetries. Some further structure is necessary.


General Gauge Mediation

Much work has been devoted to understanding the propertiesof this simple model, but it is natural to ask: just how generalare these features? It turns out that they are peculiar to ourassumption of a single set of messengers and just one singletresponsible for supersymmetry breaking and R symmetrybreaking. Meade, Seiberg and Shih have formulated theproblem of gauge mediation in a general way, and dubbed thisformulation General Gauge Mediation (GGM). They study theproblem in terms of correlation functions of (gauge)supercurrents. Analyzing the restrictions imposed by Lorentzinvariance and supersymmetry on these correlation functions,they find that the general gauge-mediated spectrum isdescribed by three complex parameters and three realparameters. Won’t have time to discuss all of the features here,but the spectrum can be significantly different than that ofMGM. Still, masses functions only of gauge quantum numbersof the particles, flavor problems still mitigated.


Lecture 2. Microscopic Models of SupersymmetryBreaking. Outline

1 MGM and GGM (conclusions)2 Intermediate Scale Supersymmetry Breaking (“Gravity

Mediation")3 Low Energy, Dynamical Supersymmetry Breaking: A

connection to the Cosmological Constant4 The importance of Discrete R Symmetries5 Gaugino condensation and its generalizations6 Building models of Low Energy Dynamical Supersymmetry

Breaking7 Assessment8 If time, a theorem about the superpotential


Intermediate Scale Supersymmetry Breaking (“GravityMediation")

Suppose, e.g., and O’Raifeartaigh like model breakssupersymmetry. W0 chosen so that the cosmological constantvanishes.

FZ ≡ DZ W =∂W∂Z

+∂K∂Z

W 6= 0. (38)

Leads to soft masses for squarks, sleptons;∫

d2θZW 2α coupling

gives gaugino massesProblem:

δK =γ i j

M2 Z †Zφ†iφj (39)

yields flavor-dependent masses for squarks and sleptons.Other potential difficulties include cosmological problems, suchas gravitino overproduction, moduli problems.


Low Energy Supersymmetry Breaking

While I won’t consider string constructions per se, I will focuson an important connection with gravity: the cosmologicalconstant. I will not be attempting to provide a new explanation,but rather simply asking about the features of the low energylagrangian in a world with approximate SUSY and small Λ.


With supersymmetry, an inevitable connection of lowenergy physics and gravity

〈|W |2〉 = 3〈|F |2〉M2p + tiny. (40)

So not just F small, but also W . Why?

1 Some sort of accident? E.g. KKLT assume tuning of Wrelative to F (presumably anthropically).

2 R symmetries can account for small W (Banks). We wewill see, < W > can be correlated naturally with the scaleof supersymmetry breaking.Scale of R symmetry breaking:set by cosmological constant.


Suggests a role for R symmetries. In string theory (gravitytheory): discrete symmetries. Such symmetries are interestingfrom several points of view:

1 Cosmological constant2 Give rise to approximate continuous R symmetries at low

energies which can account for supersymmetry breaking(Nelson-Seiberg).

3 Account for small, dimensionful parameters.4 Suppression of proton decay and other rare processes.


Continuous R Symmetries from Discrete Symmetries

This is a brief recapitulation of points made earlierOR model:

W = X2(A20 − f ) + mA0Y2 (41)

(subscripts denote R charges). If, e.g., |m2| > |f |, FX = f .

Can arise as low energy limit of a model with a discrete Rsymmetry:

X2 → e2πiN X2; Y2 → e

2πiN Y2; A0 → A0. (42)

Allows δW = X N−nY n+1

MN−2p

. N susy vacua far away. Approximate,

accidental R symmetry. SUSY breaking metastable.


〈W 〉: Gaugino Condensation

W transforms under any R symmetry; an order parameter for Rbreaking.

Gaugino condensation: 〈λλ〉 ≡ 〈W 〉 breaks discrete R withoutbreaking supersymmetry.

Readily generalized (J. Kehayias, M.D.) to include orderparameters of dimension one.

E.g. Nf flavors, N colors, Nf < N:

W = ySff ′Qf Q′f + λTrS3 (43)

exhibits a Z2(3N−Nf ) symmetry, spontaneously broken by〈S〉; 〈QQ〉; 〈W 〉.


The dynamics responsible for this breaking can be understoodalong the lines developed in Seiberg’s lectures. Suppose, forexample, that λ� y . Then we might guess that S will acquire alarge vev, giving large masses to the quarks. In this case, onecan integrate out the quarks, leaving a pure SU(N) gaugetheory, and the singlet S. The singlet superpotential follows bynoting that the scale, Λ, of the low energy gauge theorydepends on the masses of the quarks, which in turn depend onS. So

W (S) = λS3 + 〈λλ〉S. (44)

〈λλ〉 = µ3e−3 8π2

bLE g2(µ) (45)

= µ3e−3 8π2

gLE g2(M)+3 b0

bLEln(µ/M)

b0 = 3N − NF ; bLE = 3N (46)


So

〈λλ〉 = M3N−Nf

N e− 8π2

Ng2(M)µNfN . (47)

In our case, µ = yS, so the effective superpotential has the form

W (S) = λS3 + (yS)Nf /NΛ3−Nf /N . (48)

This has roots

S = Λ

(yNf /N

λ

) N3N−NF

(49)

times a Z3N−NF phase.Consistent with our original argument that S large for small λ.Alternative descriptions of the dynamics in other ranges ofcoupling.


Gauge Mediation/Retrofitting

Retrofitted Models (Feng, Silverstein, M.D.): OR parameter ffrom coupling

X (A2 − µ2) + mAY → (50)

XW 2α

Mp+ γSAY .

Need 〈W 〉 = fMp = Λ3, 〈S〉 ∼ Λ, for example.

m2 � µ2

SUSY breaking is metastable (supersymmetric vacuum faraway).


Gauge Mediation and the Cosmological Constant

Gauge mediation: traditional objection: c.c. requires largeconstant in W , unrelated to anything else.Retrofitted models: scales consistent with our requirements forcanceling c.c. Makes retrofitting, or something like it, inevitablein gauge mediation.


Other small mass parameters: m, µ-term, arise from dynamicalbreaking of discrete R symmetry. E.g.

Wµ =S2

MpHUHD. (51)

Readily build realistic models of gauge mediation/dynamicalsupersymmetry breaking with all scales dynamical, no µproblem, and prediction of a large tanβ.


R Symmetry Breaking in Supergravity

Supergravity (moduli): W = f Mp g(X/Mp). X � Mp could giveapproximate R, along lines of Nelson Seiberg. But unclear howone can get a large enough W under these circumstances(suppressed by both R breaking and susy breaking?).Alternatively, again, retrofit scales.These sorts of questions motivate study of W itself as an orderparameter for R-symmetry breaking.


A Bound on W

Theorem: In any theory with spontaneous breaking of acontinuous R-symmetry and SUSY:

|〈W 〉| ≤ 12|F |fa

where F is the Goldstino decay constant and fa is the R-axiondecay constant.Here illustrate in simple models; can prove quite generally, evenfor strong coupling.


The bound in O’Raifeartaigh models

Consider a generic renormalizable OR model with anR-symmetry Φi → eiqiξΦi .

K =∑

i

ΦiΦi , W (Φi) = fiΦi + mijΦiΦj + λijk ΦiΦjΦk

If the theory breaks SUSY at φ(0)i then classically it has a

pseudomoduli space parameterized by the goldstinosuperpartner[Rey; Shih, Komargodski].

G =∑

i

(∂W∂φi

)ψi , Φ =

∑i

(∂W∂φi

)δφi ,

Wherever the R-symmetry is broken there is also a flat directioncorresponding to the R-axion.


Define two complex vectors wi = qiφi and v†i = ∂W∂φi

Since the superpotential has R-charge 2,

2〈W 〉 =∑

j

qjφj∂W (φi)

∂φj= 〈v ,w〉

On the pseudomoduli space we can write

|F |2 =∑

i

(∂W∂φi

)(∂W∂φi

)∗= 〈v , v〉

Parameterizing φi(x) = 〈φi(x)〉eiqi a(x) we obtain for the R-axionkinetic term: (∑

i

|φi(x)|2q2i

)(∂a)2 ⇒ f 2

a = 〈w ,w〉


Then by the Cauchy-Schwarz inequality:

4|〈W 〉|2 = |〈v ,w〉|2 ≤ 〈v , v〉〈w ,w〉 = |F |2f 2a

which is the bound to be established.The bound is saturated if v ∝ w , in which case the R-axionis the Goldstino superpartner.Adding gauge interactions strengthens the bound becausethe D terms contribution to the potential makes |F |2 larger.


Lecture 3. String Theory. Outline

1 What might it mean for string theory to make contact withnature – landscape as our only plausible setting (now).

2 The elephant in the room: The cosmological constant.3 The Banks/Weinberg proposal. BP (string theory fluxes) as

an implementation (details for Denef).4 KKLT as a model. What serves as small parameter? Why

is a small parameter important? Distributions of theories.5 Supersymmetry in string theory and the landscape6 KKLT as a realization of intermediate scale susy breaking.7 A new look at susy breaking in the KKLT framework.8 Landscape perspective on intermediate scale breaking.9 Assessment

10 Discrete symmetries in string theory and the landscape11 Strong CP and axions in string theory and the landscape

(KKLT)12 Assessment: string theory predictions(?)


String Theory and Nature

In Miriam Cvetic’s lectures, you heard how string theory cancome close to reproducing many features of the StandardModel: the gauge group, the number of generations, somefeatures of Yukawa couplings. The constructions she describedtypically come with less desirable features, especially extramassless particles.But these constructions raise obvious questions: why, say,intersecting branes, and not heterotic constructions,non-geometric models, F theory constructions... One answer isthat perhaps we will find the model which describes everything,work out its consequences, and make other predictions.


Another is that we might find some principle which provides theanswer, pointing to a unique string vacuum state. Finally, thereis the point of view advocated by Banks at this school, that thedifferent string “vacua" are actually different theories ofquantum gravity; indeed, there are simply many differenttheoreies, just as there are many possible different fieldtheories.But there is at least one fact which points to a differentpossibility; this is is the cosmological constant.


The cosmological constant problem

Familiar to most of you. Illustrated by our earlier expression forthe vacuum energy. Quartically divergent. Typically, in stringmodels, if no supersymmetry, one obtains a result of thepredicted order of magnitude, with a suitable cutoff (e.g. thestring scale; first done by Rohm). Indeed, since there aretypically moduli (and certainly moduli in any case where weakcoupling computations make sense), this is the case.So no evidence that string theory performs some magic withregards to this problem.


Banks, Weinberg: A proposal

Now I will use words that Tom told you one should never use,but it is to explain a brilliant idea which he put forward andSteven Weinberg fleshed out.A possible way to understand small Λ. Suppose that the underlyingtheory has many, many vacuum states, with a more or less uniformdistribution of c.c.’s (Bousso and Polchinski dubbed a discretuum).Suppose that the system makes transitions between these states, orin some other way samples all of the different states (e.g. they allexist more or less simultaneously). Now imagine a star trek typefigure, traveling around this vast universe. (This requires all sorts ofsuperluminal phenomenon, but we won’t worry about this). Mostuniverses will not be habitable; the c.c. will be of order, say, Planckscale. But sometimes the c.c. will be smaller, the universe will becomparatively flat. Under what conditions might this star trekcharacter find intelligent life. Well, that’s a tough question. As a proxy,Weinberg asked: under what circumstances, assuming all of theother laws of nature are the same, will this observer find galaxies.First, negative c.c. is really bad; if the resulting Hubble parameter isnot much that a billion years or so, the universe will undergo a bigcrunch long before galaxies, much less life, form. Similarly, forpositive c.c., the would be structures will be ripped apart beforegalaxies can form. This sort of argument gives a c.c. about 100 timesas large as observed. . (This was actually a prediction). This isremarkably good (Weinberg was originally rather negative about theresult). On a log scale, it’s excellent. More refined versions of theargument reduce this by an order of magnitude.


Bousso-Polchinski and KKLT

Bousso and Polchinski proposed that such a discretuum mightarise in string theory as a result of fluxes. KKLT: 3-form fluxesin IIB theory. If χ types of fluxes, each taking L values, of orderLχ states. Can easily be enormous, large enough to give thesort of discretuum required. BP just supposed a stable vacuumor state for each flux choice, but not clear this is reasonable.E.g. not clear that moduli are fixed. KKLT provided a moredetailed proposal, and a candidate for a small parameter whichwould allow exploration of some states.Basics of KKLT (builds on much earlier work):

1 IIB Orientifold of CY, or F-theory on CY Four-Fold.2 h2,1 complex structure moduli, h1,1 Kahler moduli, dilaton3 Three form fluxes fix complex structure moduli, dilaton.4 Additional branes required for matter, perhaps susy

breaking.


Fixing the Kahler moduli

At this first stage, the Kahler moduli are undetermined. Lowenergy theory, after integrating out complex structure moduli,and assuming one Kahler modus, described by asupersymmetric effective action with

W = W0 − Ae−ρ/b K = −3 ln(ρ+ ρ†) (52)

Here

ρ = ρ+ ia (53)

where a is an axion, i.e. the theory respects a symmetry ofdiscrete translations of a. Superpotential necessarily e−cρ.Here exponential arises, say, from gaugino condensation onbranes.


KKLT (also Douglas, Denef): W0 distributed uniformly as acomplex variable. If many states, W0 can be small. W0 willserve as the small parameter (*note that if the number of statesis finite, W0 cannot be arbitrarily small, so there is not really asystematic expansion). For W0 small, the superpotential has asupersymmetric stationary point:

DρW ≈ aA/be−ρb − 3ρ+ ρ†

W0 = 0 (54)

ρ = ρ0 ≈ −b ln(|W0|). (55)

This nominally justifies a large ρ expansion, i.e. the α′

expansion (Giddings, Kachru, Polchinski: mechanism toachieve weak string coupling).


Aside 3: How good is this expansion?

Corrections to the metric behave as 1/ρF 2 (compare onepower of curvature in metric with three in flux terms; ρ ∼ 1/R4).Douglas, Denef: number of states of order χh

2,1; sum of fluxes oforder χ. Assuming W0 naturally of order Mp, then smallest W0is

|W0| >1χ

h2,1/2(56)

Flux terms of order χ (Douglas,Denef) If χ ∼ h2,1, then theexpansion is in powers of 1/ log(χ). Probably not extremelysmall (ask Frederick). This estimate is very crude, but it is hardto see how things could be much better than this.


Supersymmetry Breaking in KKLT

So far, susy unbroken. By changing signs in superpotential(shifting a by π) we can find non-susy stationary points, but alsoAdS.KKLT: assumed D3 branes, break susy. Confusing since notclear how to describe in terms of a low energy, supersymmetriclagrangian (perhaps no four d description). In any case, cansimply suppose there is another sector which spontaneouslybreaks susy, perhaps along lines of models we discussed here.What is important is that this extra sector gives a positivecontribution to the cosmological constant (you can fill in Banksobjections at this point).So a plausible (but hardly rigorous) scenario to understand:

1 Large number of states2 Fixing of moduli3 Breaking of supersymmetry4 Distribution of cc’s, other parameters.


General Lessons, and Not

1 Is weak coupling important? At most, only to convinceourselves (to the extent we can) that such a landscapeexists. E.g. with large W0, one might still expect a uniformdistribution of W0 at low energies (Kachru).

2 m2ρ ∼ ρ2m2

3/2; multiplet approximately supersymmetric(axion heavy). How general?

3 We used supersymmetry. Was this important?


What might we extract from the landscape?

Can’t hope to find “the state". (Imagine problem of grad studentgiven a vacuum with a bar code to study. Calculates Λ to thirdorder, it’s very small; get’s very excited; to fourth, still small, tofifth, 40 years later to eighth– it’s still small. Then, at ninthorder; oh well, on to the next one.)Want to ask questions about things like correlations – if suchand such is true, is it often/always true that... Wati Taylor hasstudied such questions in intersecting brane models, looking atcorrelations between numbers of generations and otherquantities, such as gauge groups. No good candidates.


But I would suggest that the most promising questions arethose connected with questions of naturalness. This isprecisely because the phenomena we are trying to explainseem at first sight unlikely.

1 The c.c. (already discussed)2 The strong CP problem3 Fine tuning of the electroweak scale.4 Cosmological issues such as inflation.

There has been much work on the last of these, but it is veryunclear what might be generic. I’ll focus on the second andthird.


WARNING: WE ARE ENTERING A ZONE OF (BIASED?)SPECULATION***Needs a picture ***


Supersymmetry in the Landscape

At first sight, supersymmetry would seem special. Even if it iseasier to explain hierarchies, we are talking about such largenumbers of states that the number of non-susy states exhibitinghuge hierarchies, could well overwhelm those in which thescale arises naturally (Douglas, Susskind).Divide the landscape into branches:

1 States with no supersymmetry2 States with approximate supersymmetry3 States with approximate supersymmetry and discrete R

symmetries.


One possible argument against the first branch and in favor ofbranches two and three: stability (now I will invoke asimple-minded version of Banks’ concerns about “states").In landscape, a typical state with small cc surrounded by vastnumber of states with large negative cc. What prevents decaysto big crunch? In a typical state, no parameter which wouldaccount for smallness of decays. So, even though susyconditions special, this might favor susy states. Can playgames in toy models to try and make quantitative, but probablycan do no more than speculate at this stage.


On supersymmetric branch, why low energy susy favored?Here, conventional naturalness. Douglas/Denef: instances withuniform distribution of gauge couplings. If in significant fractionof typical states, dynamical susy breaking (seems plausiblegiven our earlier discussions) then one might find most stateswith low electroweak scale have low susy breaking scale.Actually, one can do a simple analysis, using

1 Uniformity of W0

2 Requirement of small cc.c.3 Exponential variation of susy scale.4 Fixed electroweak scale (anthropically)

One finds susy scales in fact logarithmically distributed.


R symmetries? Now W0 exponentially distributed; perhapscorrelated with susy breaking scale (as in retrofitting). Now verylow scale susy breaking favored.

P(m3/2) ∝ 1m2

3/2(57)

*** Check this. Consider deriving.***But R symmetry likely to be rare? Must set to zero all fluxeswhich transform; this is, for typical CY’s, a finite fraction of thetotal – exponential suppression of the number of states.But perhaps too naive. Consider tunneling again, now to thedesired state. In toy models, decay to R symmetric statesfavored.


Axions: In KKLT, no light axion. Strong CP problem? Small θnot anthropically relevant. Much pessimism about this (Banks,Dine, Gorbatov; Donoghue).But perhaps KKLT a little too simple as a model. Suppose multipleKahler moduli, ρ1, ρ2, ρ3, say.

W = W0 + Ae−(n1ρ1+n2ρ2+n3ρ3)

b (58)

Now, “heavy" field,

Φ = n1ρ1 + n2ρ2 + n3ρ3 (59)

and two light fields. Φ ∼ b log(|W0|). Low energy effective theory:Kahler potential for light fields, and constant superpotential. Thistheory can break susy, though needs additional fields or large Kahlerpotential modifications (plausible, given our statement aboutapproximations). ρi all fixed, large. Light axions, if non-perturbativeeffects, like instantons,

δW ∼ e−rhii (60)

(could even lead to “axiverse" of Dimopoulos et al; Raby, Acharya,Bobkov...; Dine).


So a coherent picture in which perhaps low energysupersymmetry is about to be found at LHC. But perhaps ahouse of cards. Many assumptions. Perhaps some otherphenomena (warping, large extra dimensions, technicolor?)responsible for ew symmetry breaking. Or worse, just anthropicand we are about to find a single light Higgs. I hope I haveoutlined some questions and some possible approaches, butperhaps your young, fresh minds will come up with better waysof thinking about these questions. And perhaps, within a fewyears, we will have experimental verification of some of theseideas, or clues as to what physics lies Beyond the StandardModel.


Additional Slides Follow Elaborating Topics Above


While we won’t review the analysis Meade et al in detail, it iseasy to see, in simple weakly coupled models, how one canobtain a larger set of parameters. Take, for example, a model,as above, with messengers q, q, `, ¯, but replace the one singletof the earlier model with a set of singlets, Xi . For thesuperpotential, take:

W = λqi Xi qq + λì Xi ¯ . (61)

Now, unlike the case of minimal gauge mediation, the ratio ofthe splittings in the multiplets to the average (i.e. fermion)masses is not the same for q, q and `, ¯. For the fermionmasses:

mq =∑

λqi xi m` =

∑λì xi (62)

while the scalar splittings are

∆m2q =

∑λq

i Fi ∆m2` =

∑λì Fi . (63)


In the case of MGM, the one loop contributions for fieldscarrying color were proportional to ∆m2

q/m2q, while those

contributing to ` were proportional to ∆m2` /m

2` . One now finds,

simply generalizing the previous computation, for the masses ofthe gauginos:

mλ =α3

4πΛq mw =

α2

4πΛ` (64)

mb =α1

4π[2/3Λq + Λ`] .

where

Λq =λi

qFi

λjqxj

Λ` =λi`Fi

λj`xj

(65)

(i and j summed).


Similarly, for the squark and slepton masses we have:

m2 = 2[C3

(α3

4π

)2Λ2

q + C2

(α2

4π

)2Λ2` (66)

+

(Y2

)2 (α1

4π

)2(23

Λ2q + Λ2

` )]


At this point, it is easy to understand the parameter counting ofMeade et al. We can write the general gauge-mediatedspectrum in terms of three independent complex masses forthe gauginos, and parameterize the general sfermion massmatrix as:

m2 = 2[C3

(α3

4π

)2Λ2

qcd + C2

(α2

4π

)2Λ2

w (67)

+

(Y2

)2 (α1

4π

)2Λ2

b],

In the present case, there are two relations among thesemasses, which can be expressed as sum rules. But moregenerally, we have three independent complex parameters, thegaugino masses, and three additional real parameters.


Models with additional fields permit independent values for allof the parameters of GGM. In constructing examples, we willinsist that the messengers fill complete multiplets of SU(5), soas to preserve unification (one can legitimately ask why naturewould be so concerned with achieving unification). Forexample, suppose one has a 10 and 10 of messengers, andmultiple singlets. The messengers can be denoted as Q, Q,U, U, and E , E . One now has three independent parameters,which, by analogy to our previous example, can be denoted asΛQ,ΛU ,ΛE .


The minimal, weak coupling theory which yields the full set ofparameters of GGM consists of a 10 and 10, and two 5, 5 pairs.In this case, however, if the scale of supersymmetry breaking islow, the gauge couplings tend to get strong well below theunification scale. In addition, there is not an automaticmessenger parity, so it is necessary to require additionalstructure in order to suppress the Fayet-Iliopoulos term forhypercharge. Finally, these models don’t actually cover the fullparameter space (though they have the maximal number ofparameters); a strategy for doing this is described by Seiberg etal.


Simple Example of Superpotential Bound

W =

N2∑i=1

Xi fi(φa).

Here there are N2 fields, Xi , of R charge 2, and N0 fields, φa, ofR charge zero. If N2 > N0, supersymmetry is broken, and thereis a classical moduli space (∂W

∂φa= 0 are N0 equations for N2

unknowns). On this moduli space

fa = 2∑|Xi |2 F =

∣∣∣∣∣∑i

Wi

∣∣∣∣∣2

|〈W 〉|2 =∣∣∣∑XiWi

∣∣∣2More intricate R assignments are more interesting (alsoneeded (Shih) to obtain spontaneous R breaking on thepseudomoduli space).


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