new developments in theory

New Developments in Theory

Shamit KachruLepton-Photon 2009

Saturday, August 15, 2009

Summarizing new developments in a largeendeavor like string theory is a tricky business. In the past few

years, there have been advances in many disparate areas, like:

* Understanding of perturbation theory insupergravity

Could maximal SUGRA be finite?

Is gravity more generally better behaved than we thought in the UV?

c.f. Bern, Dixon, Kosower,...Green, Vanhove,...


* Building new models of inflationary theory where the inflaton potential is calculable, and signatures for future CMB or large-scale structure experiments (gravity waves?

non-gaussian density fluctuations? cosmic strings?) are predicted by the different classes of models:

c.f. CMBPol Mission Concept Study, arXiv:0811.3919;Recent reviews by Baumann; McAllister & Silverstein;...


I will instead focus on a class of developments related to “holography” or “gauge/gravity

duality.” A major tool in all approaches to understanding interactions of the fundamental particles is quantum field theory. Holography

provides the most versatile new tool we’ve had to understand otherwise intractable, strongly-coupled quantum field theories in quite some

time. Such field theories play a crucial role in many proposals for physics beyond the Standard

Model.


Holography: a rapid introduction

The roots of holography lie in the relations of black hole thermodynamics, discovered by Bekenstein, Hawking and others.

Zeroth law: Stationary hole has constant surface gravity

First law:

Second law:

Very clear analogy with laws of thermodynamics and statistical mechanics!


In this analogy, entropy maps to surface area of the event horizon.

Now, a black hole is the natural endpoint of the jeans instability which causes matter to collapse:

This is odd because in 3+1 space-time dimensions, we intuitively expect entropy to scale with volume, not

area!


According to the holographic principle enunciated by ‘t Hooft and Susskind, in quantum gravity, this is not so: the number of degrees of freedom in the exact quantum theory

describing a region of space of volume V, is actually controlled by the surface area A!

As you might imagine, this wide ranging and slightly vague statement is hard to make precise. But it has been made very precise in one large class of gravitational space-times: Anti

de Sitter spaces.


D+1 dimensional Anti de Sitter space can be thought of as the hyperboloid:

A more useful description for our purposes is in terms of the metric in D+1-dimensions itself. One can show that this

takes the form:

It is the homogenous, isotropic, simply-connected space of negative curvature; the simplest solution of Einstein’s

equations with negative cosmological term (whose size is related to the curvature radius L).

( rL )2 (ηµνdxµdxν) + dr2

r2 , µ, ν = 1, .., D − 1

−x20 − x2

1 + x22 + ... + x2

D+1 = −L2


This metric is highly symmetric. In addition to the Poincare symmetry of the Minkowski slices, it enjoys a scale invariance:

x→ λx, t→ λt, r → rλ

In fact, the SO(D-1,1) Poincare symmetries and the non-compact scaling symmetry are combined in a larger group of symmetries

of Anti de Sitter space, SO(D,2).

Peculiar fact: This is the same as the symmetry group of a D dimensional quantum field theory that enjoys conformal

symmetry, i.e. that sits a fixed point of the renormalisation group equations:

βi({gj})|gj=g∗j= 0


An astounding claim, first made somewhat precise by Maldacena, is that holography actually operates to give a precise equivalence between gravity in D+1 dimensional AdS space, and a conformal field theory

in D dimensions, which lives at the boundary of AdS (at infinite r, which light rays reach in finite time from the interior):

Maldacena;Gubser, Klebanov, Polyakov;

Witten


The way that one precisely associates a boundary conformal field theory to a bulk theory of Anti de Sitter gravity is not understood in general. The specific examples we know (which include infinite

classes of conformal theories with various amounts of supersymmetry in 2-6 dimensions) are derived from string theory.The gravity background arises as the solution sourced by string

theoretic “branes,” and the dual field theory is the theory of the worldvolume excitations on those “branes.”

Polchinski


If we take the known examples and try to summarize the relationship between the gravity and the field theory, we come to the following

basic map:

g2Y M ↔ gstring

SU(N) gauge group↔ Curvature scale ( lstringL )4 = 1

g2YMN

global symmetry group G ↔ gauge group G in AdS

Operator O of scaling dimension ∆↔ Field with mass m(∆)

Note the wonderful fact that the string theory is weakly coupled and weakly curved when the field theory has strong ‘t Hooft

coupling! This gives an entirely new approach to formulating and calculating in theories with many colors at strong coupling!


Another feature of the duality which cannot be emphasized strongly enough: intuitively, the extra dimension represented via the AdS radial coordinate “r” is geometrizing the energy

scale in the quantum field theory. (One can see that the natural notion of energy scale varies with r because energy is conjugate

to time, and the tt component of the metric is r dependent).

So if we cut off the AdS geometry at some maximal value corresponding to energy scale EMAXrMAX

E(r < rMAX) ∼ EMAX × rrMAX

For instance in a confining gauge theory which is conformal over a wide range of scales, one expects instead of an exact AdS

spacetime:

ds2 = (r/L)2(ηµνdxµdxν) + dr2

r2 , r > rMIN

with the region at smaller r effectively excised. This is a crude toy model, but real string constructions give qualitatively this behavior.


Holography and Particle Physics Model Building

Several of the most interesting ideas about electroweak symmetry breaking involve strongly coupled quantum gauge

theories.

* A priori, minimal supersymmetry does not require any strong coupling:

The SM gauge group and all forces experienced by quarks and leptons can remain perturbative to a very high scale.

Georgi, Glashow;Dimopoulos,

Raby, Wilczek


In my view this is a bit of a red herring.New techniques may be useful even if we find sparticles at the TeV scale.

While unification is a beautiful idea and its success in the MSSM is intriguing, in most existing “natural”

theories of supersymmetry breaking, and in many ideas about how to explain flavor (e.g. the structure of

yukawa couplings) in SUSY theories, strong dynamics plays a crucial role.

We’ll say more about this momentarily. But first, we’ll discuss applications of holography to a second idea,

whose exemplification has proved difficult in its absence.


* A second beautiful idea for explaining the Planck/Weak hierarchy is “technicolor” (which I’ll group with its close

relative, “composite Higgs models”).

In such theories, the breaking of electroweak symmetry is explained by a condensate of a charged fermion bilinear. This is aesthetically pleasing: the scale can be naturally

low due to asymptotic freedom, and we already know Nature uses similar mechanisms in BCS superconductors

and in chiral symmetry breaking in QCD!

However, there is no commonly accepted working model of technicolor based on 4D gauge theory

computations.

Susskind;Weinberg


There are two basic problems: electroweak precision and explaining the origin of flavor.

I won’t say anything about the first, except that the Peskin-Takeuchi S-parameter:

is highly constrained by experiment, and this constraint is violated by simple 4D technicolor models.

Flavor physics is more interesting. In a model where the Higgs is literally replaced by a fermion bilinear

condensate, one must get Yukawas through dimension 6 operators:

) CMJOCUL

cannot h*

ICMtirmj

at

LYukawa ∼ cΛ2 〈ψψ〉TC Quc + ...


Lq

R

(a) (b)

q

TR

TL

R

TL TR

T

TL

! T ! T

Figure 1: Graphs for ETC generation of masses for (a) quarks and leptonsand (b) technipions. The dashed line is a massive ETC gauge boson. Higher–order technicolor gluon exchanges are not indicated; from Ref. [15].

F 2T M2

πT! 2

g2ETC

M2ETC

〈T̄LTRT̄RTL〉ETC . (4)

Here, mq(METC) is the quark mass renormalized at METC . It is a hardmass in that it scales like one (i.e., logarithmically) for energies below METC .Above that, it falls off more rapidly, like Σ(p). The technipion decay constantFT = Fπ/

√N in TC models containing N identical electroweak doublets of

color–singlet technifermions. The vacuum expectation values 〈T̄LTR〉ETC and〈T̄LTRT̄RTL〉ETC are the bilinear and quadrilinear technifermion condensatesrenormalized at METC . The bilinear condensate is related to the one renor-malized at ΛTC , expected by scaling from QCD to be

〈T̄LTR〉TC = 12〈T̄ T 〉TC ! 2πF 3

T , (5)

by the equation

〈T̄ T 〉ETC = 〈T̄T 〉TC exp

(

∫ METC

ΛTC

dµ

µγm(µ)

)

. (6)

The anomalous dimension γm of the operator T̄T is given in perturbationtheory by

γm(µ) =3C2(R)

2παTC(µ) + O(α2

TC) , (7)

where C2(R) is the quadratic Casimir of the technifermion SU(NTC)–representationR. For the fundamental representation of SU(NTC), it is given by C2(NTC) =

8

Lecture notes by K. Lane, 2002

The simplest idea to generate Yukawas is to add “extended technicolor” interactions which cause quarks and leptons

to communicate with the technifermions.

One then finds masses of order:

y ∼ c× Λ3T C

Λ2ET C

, c ∼ O(1), ΛTC ∼ TeV

FCNC constraints indicate that the ETC scale is 1000 TeV or more; and getting large enough masses (notably the top!) is very

challenging.


Holographic Technicolor

A natural idea for solving the hierarchy problem, given the warped geometries present in AdS/CFT, is the following:

Randall, Sundrum;.....

Agashe, Delgado,May, Sundrum;

.....

This cartoon is pretty involved, but what it represents is simple enough:

ds2 = e−2x5/L (ηµνdxµdxν) + dx25, 0 ≤ x5 ≤ KL

Just the (truncated) AdS metric again! K represents (log of the) hierarchy of scales. K = 30 can solve Planck/Weak hierarchy.

(Extra dim here can be GUT scale or smaller in size here; not like “large extra” scenario, here the mileage comes from warped AdS geometry).


The localized wavefunctions of Higgs, quarks, and leptons in this picture are both natural and important. Key facts:

1. Higgs and Top localized at very warped region in geometry = deep IR in dual field theory.

QFT interpretation: Higgs (and Top) a composite of the strongly interacting, almost conformal dynamics. Like

Technicolor or composite Higgs model.


2. 2nd and 3rd generation particles are localized at the far “ultraviolet” end of the geometry, far from the Higgs.

QFT interpretation: These are “elementary” or point-like states, external to the strongly interacting sector.

3. Yukawa couplings come from wavefunction overlaps with the Higgs. The hierarchy of Yukawas is explained by the

geometry of the wavefunctions!

QFT interpretation: The composite states mix via relevant operators with the CFT and can get O(1) couplings to

technifermion condensate; the elementary states only talk to technifermions through irrelevant operators.

Note that FCNCs can be sufficiently suppressed by the locality of wavefunctions as well!


So, using the 5d gravity duals of strongly coupled conformal field theories (and their more elaborate modifications to give gravity duals of theories which confine after a long period of

RG running, and give rise to composite matter), it seems that one may be able to make working technicolor models using the ideas

of holography.

There are problems remaining, both phenomenological and formal:

* in at least some models, the S parameter scales with N, ruling out a successful model in the regime where gravity

description is applicable

* Unlike SUSY, here no full model exists! The 5d gravity cartoon is attractive. But one needs to show the existence of the QFT with

the desired properties, either directly or by constructing a string model that really gives this 5d gravity solution. This is important:

the properties assumed are quite detailed and are not obviously going to arise in any CFT.


Some steps have been taken towards the construction of full models in string theory, with know gauge duals:

* Construction of stable throat geometries with, however, SUSY at theAdS scale was completed some time ago.

H. Verlinde;Giddings, Kachru, Polchinski

* More recently, string compactifications incorporating fully non-supersymmetric AdS throats (dual to non-supersymmetric gauge theories) with the requisite large hierarchy have also been constructed. The gauge

theories are somewhat elaborate!

Strassler;Kachru, Simic, Trivedi

So at zeroth order, we need to find AdS/CFT

dual pairs where:

1. There is a global symmetry G under which

all relevant operators are charged.

2. We can preserve a sufficiently large

subgroup of G when we “compactify the

throat” that all relevant operators are

still forbidden:

While the precise duality applies to the noncompactCalabi-Yau with flux, one can also construct string solutions

where the extra dimensions are compact, but are well modeled in some neighborhood by the noncompact “warped

conifold” solution. Giddings, Kachru, Polchinski

In such models, the flux-generated potential on themoduli space, allows one to give the Calabi-Yau moduli

a large mass -- roughly l2s/R3

This orbifold maps to a freely-acting

orbifold on the near-horizon geometry (the

fixed point at the origin is no longer

present). The light states on the gravity

side are then just the invariant KK modes in

AdS5 × S5.

Example: k=5

Figure 1: Quiver diagram of the k = 5 case. White arrows denote fermions, and black arrows denotescalars. We thank the authors of [13] for permission to reproduce this figure.

For this purpose it is useful to examine first how the relevant operators in the N = 4

theory transform under the SU(3) × U(1)R symmetry. Let us start with single trace gauge

invariant operators.

The N = 4 theory has three kinds of operators which are bilinears in the scalar:

1) Tr(ZiZj) : These have dimension 2. They transform like a 6 of SU(3) and carry

charge 4/3 under U(1)R. Thus they are not singlets under SU(3) × U(1)R. The operators,

Tr(Z̄iZ̄j), which are complex conjugates transform in the complex conjugate representation

under the global symmetries and are also not singlets.

2) Tr(ZiZ̄j) −13δi

jTr(ZiZ̄i): These also have dimension 2. They are singlets under the

U(1)R but transform like an 8 of SU(3) and are therefore not singlets under the global

symmetry.

3)Tr(ZiZ̄i): This operator is a singlet. However it has an anomalous dimension which

goes like ∆ ∼ (gsN)1/4 and thus is much bigger than unity in the large ’t Hooft coupling

limit. It is therefore not relevant.

In the orbifold theory there are also scalar bilinears which arise from the Qim fields and

their complex conjugates. However these operators inherit their SU(3) × U(1) quantum

numbers and also their anomalous dimensions (to leading order in N) from the N = 4 theory.5 Thus we conclude that there are no scalar bilinears in the orbifold theory which are global

singlet relevant operators (GSROs).

The discussion above brings out one of the central points of the paper, so it is worth

emphasising in more general terms. At strong coupling (large ’t Hooft coupling) in the

supersymmetric parent theory, only protected operators have anomalous dimensions of order

unity; these operators are charged under the global symmetries of the parent theory and thus

are not GSROs. If we can arrange for a sufficiently big subgroup of the global symmetry

group to be preserved by the daughter orbifold theory, it too will not contain any GSROs. In

5This is consistent with the fact that in the sugra approximation the mass of invariant KK modes is left

unchanged by the orbifolding procedure.

– 7 –


Many steps remain to be taken. Certainly, no one has come remotely close to justifying the existence of a full (approximate) CFT with the

structure of composites required to reproduce the Yukawas and flavor physics of the Standard Model. Ideas to do this by adding

branes to the existing stringy throat models are reasonable but will not obviously succeed.

Holography in SUSY model building?

Let me close by discussing why I think there is also a potential role for these new ideas even in supersymmetric model

building. In the standard paradigm:

II. Supersymmetric models with composite

quarks and leptons

The existing paradigm of supersymmetric

model building always starts with a picture

of the form:

A standard paradigm for SUSY model

building involves a “hidden sector” where

SUSY breaks, and then messenger fields

which transmit this breaking to the

Standard Model:

I. Introduction


Standard examples include gauge and gravity mediation. These come with various problems of their own, and do not

shed any light on the quark and lepton mass matrices.

A natural question is: can one build SUSY models with are less modular than this? And that tie the SUSY breaking dynamics to

the origin of flavor?

A simple idea to do this was proposed by Arkani-Hamed, Luty and Terning in the late 1990s. They proposed that:

* SUSY is broken by some strongly interacting sector.

* The same strongly interacting sector generates the first two generations of quarks and leptons as composites!

* The first two generations of sparticles then get large soft masses through their interactions with the SUSY breaking sector. The third generation sparticles only feel soft masses via gauge mediation from

the first two generations; they are much lighter! (And may be the only matter sparticles visible at the LHC). Since the stop is light, no

large tuning is needed for light Higgs.


One could then understand Yukawa hierarchies. For instance, if the first two generations arise as bilinear composites of strongly

interacting “quarks” of the hidden sector at high energies, we’d have:

• In our models, the first two generations arise as composite dimension two operatorsin the UV of the electric theory – these are “meson models,” in the parlance of [2].

Therefore, we can hope to explain some but not all of the structure of the Yukawacouplings that is evident in the Standard Model. Assuming that the Higgs particlesare added as elementary states relative to the strongly coupled dynamics, the Yukawa

couplings of the first two generations (neglecting mixing with the third generation) arisefrom dimension 6 operators in the high-energy theory, and are naturally suppressed.

More precisely, one expects the Yukawa couplings to be generated by a superpotentialof the schematic form

WYuk ∼1

M2flavor

(QQ̃)H(QQ̃) +1

Mflavor(QQ̃)HΨ(3) + Ψ(3)HΨ(3) . (6.2)

This gives rise to a Yukawa matrix whose basic structure is

ε2 ε2 ε

ε2 ε2 εε ε 1

(6.3)

controlled by the small parameter ε ∼ Λe

Mflavor. For ε ∼ 10−2, this is a reasonable starting

point for obtaining the correct Yukawa couplings, though some additional structure

remains to be explained.Clearly, one could do a better job of explaining the flavor structure visible in

the Standard Model with a composite model where the first generation arises from acomposite operator which has (a lowest component in its supermultiplet of) dimension

three in the high energy theory, while the second generation comes from an operator ofdimension two. It would be interesting to construct such models following our generalstrategy. The metastable vacua discussed by [14], which arise in supersymmetric QCD

with additional adjoint matter fields, would likely be a good starting point, since themagnetic duals incorporate additional mesons which are cubic in the electric variables.

• The MSSM gauginos and the third generation elementary states get their soft masses,in these models, from gauge mediation arising from both the composite MSSM gener-

ations, and from the massive messengers (additional vector-like MSSM charges) whichexist at the scale ∼ hµ. For this reason, it is desirable to have the first two generations

of sparticles (and the messengers) at a mass scale somewhat above the TeV scale. Forh ∼ 1, we should choose µ ∼ 5 TeV. The resulting stop mass will be small enough toavoid excessive fine-tuning; the resulting large masses for the squarks in the first two

generations will help with FCNCs, as in the scenarios of [3, 4, 5, 6]. It is interesting toconsider fully solving the supersymmetric flavor problem by raising the masses of the

14

giving rise to a yukawa matrix with the basic pattern:

• In our models, the first two generations arise as composite dimension two operatorsin the UV of the electric theory – these are “meson models,” in the parlance of [2].

Therefore, we can hope to explain some but not all of the structure of the Yukawacouplings that is evident in the Standard Model. Assuming that the Higgs particlesare added as elementary states relative to the strongly coupled dynamics, the Yukawa

couplings of the first two generations (neglecting mixing with the third generation) arisefrom dimension 6 operators in the high-energy theory, and are naturally suppressed.

More precisely, one expects the Yukawa couplings to be generated by a superpotentialof the schematic form

WYuk ∼1

M2flavor

(QQ̃)H(QQ̃) +1

Mflavor(QQ̃)HΨ(3) + Ψ(3)HΨ(3) . (6.2)

This gives rise to a Yukawa matrix whose basic structure is

ε2 ε2 ε

ε2 ε2 εε ε 1

(6.3)

controlled by the small parameter ε ∼ Λe

Mflavor. For ε ∼ 10−2, this is a reasonable starting

point for obtaining the correct Yukawa couplings, though some additional structure

remains to be explained.Clearly, one could do a better job of explaining the flavor structure visible in

the Standard Model with a composite model where the first generation arises from acomposite operator which has (a lowest component in its supermultiplet of) dimension

three in the high energy theory, while the second generation comes from an operator ofdimension two. It would be interesting to construct such models following our generalstrategy. The metastable vacua discussed by [14], which arise in supersymmetric QCD

with additional adjoint matter fields, would likely be a good starting point, since themagnetic duals incorporate additional mesons which are cubic in the electric variables.

• The MSSM gauginos and the third generation elementary states get their soft masses,in these models, from gauge mediation arising from both the composite MSSM gener-

ations, and from the massive messengers (additional vector-like MSSM charges) whichexist at the scale ∼ hµ. For this reason, it is desirable to have the first two generations

of sparticles (and the messengers) at a mass scale somewhat above the TeV scale. Forh ∼ 1, we should choose µ ∼ 5 TeV. The resulting stop mass will be small enough toavoid excessive fine-tuning; the resulting large masses for the squarks in the first two

generations will help with FCNCs, as in the scenarios of [3, 4, 5, 6]. It is interesting toconsider fully solving the supersymmetric flavor problem by raising the masses of the

14

Even better patterns can be easily engineered by having the first and second generation differ in their composite nature (e.g., the first generation can be a “baryon” of the hidden sector theory

that breaks SUSY).


With S. Franco, we were recently able to produce completely calculable examples of such “single sector” models just using

small modifications of supersymmetric QCD as the SUSY breaking sector, with the SUSY-breaking provided by the more

recently developed mechanism of Intriligator, Seiberg and Shih.

While this is an interesting idea, for a long time, no calculable examples which would actually do this were

known. This is not a surprise, since the dynamics involved in both SUSY breaking and formation of

composites can be rather intricate.

A crucial role is played by the “Seiberg duality” of the gauge theory, which allows one to map the strong dynamics to a weakly

coupled “magnetic” description.


But as is clear from this talk, another very promising tool for making such models, is gauge/gravity duality! One can

simply change the details of the 5d cartoon:

2. Composite Models

Here, we spread the matter fields around

in the microscopic 10D theory, by choosing

different flavor-brane embeddings:

zi1 = µ̃i, µ̃3 >> µ̃1,2

In terms of pictures, the two scenarios

then look like:

21

Other scenarios: Compositeness

!! Natural extension allow the position of matter to vary

SM matter emerging as composite of

the SUSY-breaking field theory Arkani-Hamed, Luty and Terning

Gabella, Gherghetta and Giedt

!! Known field theory examples are non-calculable

e.g.: SU(4)!SU(18) ![SU(18)]

!! Compositeness can (partially) explain some issues about flavor physics

composite

SUSY is broken in the IR (where the Higgs was localized in the earlier, non-SUSY models). The Higgs is localized at the

UV; this is the geometric dual of dynamical SUSY breaking with an elementary Higgs.

Depending on where the fermions are, they can be elementary (left picture) or the first two generations can be composite

(right).


The geometry then makes it clear, why in these “single sector” models:

* The Yukawas are correlated with compositeness (more composite generations have less overlap with the Higgs

wavefunction).

* The more composite multiplets get larger soft masses (closer to the source of SUSY breaking at the IR!).

Gauge/gravity duality should provide a powerful tool engineer classes of such models, as alternatives to gauge and

gravity mediation. Gabella, Gherghetta, Giedt;Benini, Dymarsky Franco, Kachru, Simic, Verlinde


Conclusion

One of the most fruitful recent developments in the study of both quantum field theory and quantum gravity, has

been the application of holography to map questions about strongly interacting field theory to geometric questions about 5d spacetimes. I believe this technique may be able

to shed light, at least qualitatively, on fascinating questions of strong dynamics that are also relevant to the

physics of the weak scale in many interesting models.

I have only summarized some developments applying the ideas of holography in particle theory. Similar methods are also being

applied to RHIC physics, to the study of strongly coupled condensed matter systems, and perhaps elsewhere. I do not

have time to discuss all of this!


new developments in theory

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