Supersymmetry breaking, monopole condensation, …hitoshi.berkeley.edu ›...

Supersymmetry breaking, monopole condensation, and confinement

Brian HenningUC Berkeley

NYU CCPP High Energy SeminarSeptember 24, 2014

with Hitoshi Murayama and Yuji TachikawaarXiv 1410.XXXX

Brian Henning NYU CCPP Seminar 24/Sep/2014

Something exotic*

The tetraquark state Z(4430)

Discovery significance:Belle 2007: 5.2σ

LHCb 2014: 13.9σ

PRL 112, 222002 (2014), arXiv:1404.1903

An exotic meson neither qq →nor qqq

Does not fit into standard quark model.

● Exotic states not present in large N limit

*This talk is not about exotic mesons. This introduction is to serve a point we'll come to shortly.

(see next slide)


Something exoticThe tetraquark state Z(4430)

PRL 112, 222002 (2014)

Conventional wisdom says exotic states are not present in large N limit:

Quark bilinear, i.e. a meson

Tetraquark

Consider two-point fxn of tetraquark

O(N2) (leading order): factorizes to O(N): Tetraquark pole two freely propagating mesons suppressed by 1/N


Something exoticThe tetraquark state Z(4430)

PRL 112, 222002 (2014)

Conventional wisdom says exotic states are not present in large N limit:

Quark bilinear, i.e. a meson

Tetraquark

Consider two-point fxn of tetraquark

O(N2) (leading order): factorizes to O(N): Tetraquark pole two freely propagating mesons suppressed by 1/N

Actually it is possible to have exotics in large N limit(Weinberg 2013, 1303.0342)

What matters is decay width: normally this is broad O(N) but if there aren't — —

Still a pole for tetraquark, even if residue is 1/N suppressed.

light mesons for tetraquark to decay into, decay rate can be suppressed

→ becomes a question of kinematicsNote: Weinberg's points are purely conceptual; relavance to real world exotics requires more analysis


What was the point of this brief discussion on exotics?


We do not understand 4d asymptotically free gauge theories

First and foremost:



More specifically, our tetraquark discussion highlights two interesting points (in my opinion):

First and foremost:

● Lots of mysteries that have experimental data

● Still have conceptual issues to discover/sort out

– Some are even relatively simple, e.g. Weinberg's points (from 2013!) on exotics and 1/N

– Theoretical progress not limited to (daunting) task of solving strong coupling.


So, what is this talk about?


Brief outline

Going to examine new N = 1 theories in the Coulomb phase

Seiberg-Witten N = 2 theoriesmost familar example of the Coulomb phase

In many ways, the solutions to our theories are just more complicated versions of the original SW sol'n (with some obvious N = 1 caveats)

We'll review broader notions of duality and confinement and how they arise in the particular SW sol'n

The points emphasized have a straight-forward generalization to the more complicated looking theories discussed later

Following this review, we'll state our main claims about our N = 1 theories and new phenomena they exhibit, then examine these

claims in detail



The obvious issue is strong coupling the lack of a small ↔

parameter to expand in

One physical manifestation of this is the confinement


Wouldn't it be great if...

physical system

Some local, gauge description

Some other local, gauge description

duality


Perhaps not so crazyDuality is present in

electromagnetism at the classical level!

Two separate U(1) gauge theories with





Spontaneously break U(1) gauge theory and

the Meissner effect

Image credit: Strassler, 2001 Trieste lectures

Magnetic flux is confined!





Spontaneously break U(1) gauge theory and

the Meissner effect

Image credit: Strassler, 2001 Trieste lectures

Magnetic flux is confined!

Notions of duality and confinement at work even in simple electromagnetism...we should take this seriously!


Quick comment on meaning of confinementRest of this talk: confinement means actual confinement

→Wilson loop follows area law

→There are charges in the theory which cannot be screened by matter content of the theory

Stable flux tubes connect those charges which cannot be screened

e.g.

Image credit: Strassler


20 years ago came an explicit realization

N=2 SU(2) gauge theory

generically broken to U(1) on the classical

moduli space → classical EM duality

persists quantum mechanically

SW


Lightning summary of SW sol'n

Solution parameterized by modulus u

Gauge coupling and strong coupling scale

Low energy effective theory of a (supersymmetric) photon

In the quantum theory:

Classical singularity at u = 0 removed

Quantum singularities at u = ±Λ2

Massless monopoles at these points


Goal: calculate the exact form of τ

Finding τ tells you the exact low-energy effective superpotential**

Analogous to finding the quantum moduli space and non-perturbative Weff in N=1 gauge theories such as SQCD

The SW sol'n/Coulomb phase is also a quantum modified moduli space **In N=2 susy you get much

more information since holomorphy also restricts the form of the Kahler potential


How duality transformations act on τDuality transformations act as SL(2,Z)

Inversion of coupling constant

Shift of θ by 2π

SL(2,Z) is the symmetry group of the torus

Natural to interpret τ as the modular parameter

of a torus

Image credit: Terning Modern SUSY“ ”


The torus as an algebraic curve

Structure is encoded in an elliptic curve

Image credit: Argyres SUSY lecture notes

Curve gives value of the

coupling τ at every point in the moduli space


Singularities of curve = new light particles

Curve is singular when two branch points coincide

Torus degenerates as cycle shrinks to zero size

Singular at u = §¤2

τ blows up at these points


Singularities of curve = new light particles

Curve is singular when two branch points coincide

Torus degenerates as cycle shrinks to zero size

Singular at u = §¤2

τ blows up at these points

Means (dual) coupling vanishes, g 0, →near these points

Two major implications:

1) Dual theory is weakly coupled near these points

2) There must be new massless particles to renormalize the dual coupling to zero in the IR

→ These are massless magnetic monopoles/dyons


Takeaway math points from curve


Image credit: Argyres SUSY lecture notes

Sidenote: we will tend to use the curve in the form:


The strong coupling scale smooths out the curve

The moduli determine the shape of the torus


Confinement as a dual Meissner effect!

Add mass for adjoint and integrate out

Left with N = 1 SU(2) theory believed to confine→

What happens in the SW sol'n?



Add mass for adjoint and integrate out

Left with N = 1 SU(2) theory believed to confine→

Only monopole points survive

What happens in the SW sol'n?

Monopoles condense in the weakly coupled, dual description!



This explicit realization of confinement as a dual Meissner effect is obviously satisfying, yet it begs the question:

To what degree does the SW mechanism of confinement reflect physical properties we expect for

N=1 SYM and non-susy YM?



This explicit realization of confinement as a dual Meissner effect is obviously satisfying, yet it begs the question:

To what degree does the SW mechanism of confinement reflect physical properties we expect for

N=1 SYM and non-susy YM?

A simple place to start is to look at the properties long, confining strings

Turns out, in calculable regime of SW sol'n softly broken to N=1 SYM there are some physical shortcomings from

general expectations of pure N=1 SYM


Physical shortcomings in SW confinementToo many quasi-stable states in the system

N=1 and non-susy SU(2) gauge theory expected to have a Z2

confining string

SW confinement is a broken U(1) → strings have integer fluxes

Strassler 1998


Physical shortcomings in SW confinementToo many quasi-stable states in the system

Image credit: Strassler

N=1 and non-susy SU(2) gauge theory expected to have a Z2

confining string

SW confinement is a broken U(1) → strings have integer fluxes

In calculable regime (m ¿ ¤) strings

have to be very long before they break!

Well...full UV theory doesn't allow integer flux strings

Strassler 1998



This simple topological reasoning shows that the confinement mechanism in the calculable regime of SW sol'n does not provide a completely accurate physical

description of confinement in the YM universality class

Nevertheless, there are still interesting questions about the dynamics of the confining vacua in the SW

For example, are confining strings attractive or repulsive, i.e. is the vacuum a Type I or Type II superconductor?


Type I vs. Type II superconductivity

Type I Type II

Vainshtein & Yung 2000

Tension of string carrying n units of flux

→ n-flux string cannot decay

→ exotic strings stable

→ n-flux string decays

→ exotic strings can

Seiberg-Witten theory is a type I superconductor

dynamically relax


Clearly, there are very physical motivations for looking at SW theories.

Yet, there are also some physical shortcomings (from non-susy perspective)

Our questions

To what degree do N = 1 theories in the in the Coulomb phase exhibit the usual

SW structure? Is there any new phenomena in N = 1?

The rest of this talk is dedicated to reporting some findings in these directions


Claims/outline

● We have found all N = 1 gauge theories with no tree level superpotential that are in the Coulomb phase

● A suitable definition of some of these theories can lead to dynamical susy breaking

● Physics of the confining vacua in these susy breaking models might be different from usual SW story

N


Criteria to be in Coulomb phaseHere we simply state the criteria for a gauge theory with no superpotential to be in the Coulomb phase

1) Matter content of each group is such that

Dynkin of ith rep Dynkin of adjoint

Each gauge group has matter content such that N“ F = NC”

Ensures has right symmetries to modify the quantum moduli space

(these criteria should become more clear when we study a specific example)


Criteria to be in Coulomb phaseHere we simply state the criteria for a gauge theory with no superpotential to be in the Coulomb phase

1) Matter content of each group is such that

Dynkin of ith rep Dynkin of adjoint

Each gauge group has matter content such that N“ F = NC”

Ensures has right symmetries to modify the quantum moduli space

2) No field can transform solely in fundamental of a single group

(these criteria should become more clear when we study a specific example)

AllowedNot allowed

Ensures there is a gauge invariant distinction between different phases


Example N = 1 theories in

Coulomb phase

N



Coulomb phase

N

Some of these theories are chiral



Coulomb phase

N

We will focus on this theory for the rest of the talk

(picked for the physics it exhibits and the fact that it is, computationally, among the simplest of these theories)

Some of these theories are chiral


A specific N = 1 SW theory

Matter Content


Symmetries

The non-anomalous R-symmetry under which all fields have zero charge is the condition that allows a quantum moduli space


Can distinguish phasesDiagonal centers of

Sp(4) x SU(2)1,2

cannot

be screened

For example, consider separating test charges T in fundamental of Sp(4)

TT

TQT Q

Sp(4) flux gets screened

SU(2) flux unscreened

Might, e.g., produce Q from vacuum, but there is still unscreened charge

Physical point: there exists gauge invariant distinction between phases 2

!Wilson loop is order parameter


In the Coulomb phase

Easy to see theory is in Coulomb phase by taking

various limits

Below ¤1,2 SU(2) charges confine

Low energy theory:

SO(5) ' Sp(4) gauge theorywith three vectors: S, (QQ), and (QQ)




various limits


Low energy theory:

SO(5) ' Sp(4) gauge theorywith three vectors: S, (QQ), and (QQ)

SO(5) with three vectors generically breaks to SO(2) ' U(1))in the Coulomb phase Intriligator & Seiberg

N=1 SO(N) theories 1995




various limits

Analysis in opposite limit shows theory is also in Coulomb phase(see backup slides for detail)


Counting for the Coulomb phase

Counts fields left after Higgs mechanism

Can parameterize low energy fields by gauge inv.





10 421 = 5 + 8 + 8

16 = 10 + 3 + 3

S Q Q

Sp(4) SU(2) SU(2)





10 421 = 5 + 8 + 8

16 = 10 + 3 + 3

S Q Q

Sp(4) SU(2) SU(2)

Low energy theory has one gauge generator the photon!—


Finding the low energy effective theory● Classically, theory is in Coulomb phase at generic

points on the moduli space

– Expect quantum mechanically that the entire moduli space is in the Coulomb phase

)Low energy U(1) with coupling ()

= moduli fields

● Holomorphic function of moduli● Described by an elliptic curve

● To find elliptic curve need to

– Parameterize moduli space (list gauge invariants + constraints)

– Match elliptic curve to other theories in various limits● Holomorphy guarantees that this approach works

(holomorphic fxn determined by singularities and asymptotics)


The moduli space

Six gauge invariants that we will keep in the analysis

Keep

There are four more gauge invariants

These four can be eliminated with constraints (see next slide)

Moduli space parameterized by gauge invariants

Eliminate with constraints


Relations among gauge invariants

Gauge invariants that are eliminated by constraints




Classical constraint

Quantum modified constraint




Classical constraint

Quantum modified constraint

A quick check is to note symmetries work out:


Parameterize moduli space

Calculate elliptic curve:

✓

)Match to other theories in various limits


Deriving the curve: limit ¤1,2 À ¤4

Elliptic curve for SO(5) with three vectors originally found by Intriligator & Seiberg (1995)

Meson matrix

Need to account for SU(2) quantum constraints (next slide)




Mesons subject to quantum modified constraint




Mesons subject to quantum modified constraint

Meson matrix is then given by


Deriving the curve: limit ¤4 À ¤1,2

The limit ¤4 À ¤1,2 tells us how ¤4 enters the curve

We will skip the details and simply state the result


In the previous limits we found

where


In the previous limits we found

where

Piecing it together we get


Location of singularities

Curve is singular when two roots of cubic coincide

Roots coincide at

On these singular sub-manifolds, a pair of monopoles/dyons

becomes massless

k is function of six gauge invariants this defines two sub-—manifolds of complex dimension five


Summary of the elliptic curve

Dynamical scales smooth out the curve

These gauge invariants parameterize the moduli space

Curve has two singular sub-manifolds. On each sub-manifold point a pair of monopoles becomes massless


Can add mass deformations to the theory

For non-zero masses we can integrate out fields

→ Get locked to singular sub-manifolds

→ Pair of monopoles condenses, signifying confinement in electric theory

After integrating out matter, we find

→ New, exact superpotentials

→ Some previously known exact superpotentials (good consistency check)


Can add mass deformations to the theory

For non-zero masses we can integrate out fields

→ Get locked to singular sub-manifolds

→ Pair of monopoles condenses, signifying confinement in electric theory

After integrating out matter, we find

→ New, exact superpotentials

→ Some previously known exact superpotentials (good consistency check)

These superpotentials have two branches, stemming from the two singular submanifolds

Physically, can trace to fact that these theories can distinguish different types of phases one branch →from confinement, one from oblique confinement


However, for the rest of this talk we will be

interested in other types of deformations

Can we deform the theory to break susy?


Review: susy breaking on quantum modified

Introduce deformation that breaks susy

)Leaves classical flat directions

Outline for susy breaking

moduli spaces

)Using IYIT to build intuition

)Understand in various asymptotic regions

)Look at strongly coupled regime and F-term equations

Comments on physics of the susy breaking vacua


IYIT model of susy breaking

Consider SU(2) gauge theory with 4 doublets

Has 6 gauge invariants, (QQ)ij = -(QQ)ji, subject to quantum constraint

Izawa & Yanagida (1996), Intriligator and Thomas (1996)

Gauge symmetry

Flavor symmetry


IYIT model of susy breaking

Consider SU(2) gauge theory with 4 doublets

Has 6 gauge invariants, (QQ)ij = -(QQ)ji, subject to quantum constraint

Add singlets (in 6 of SU(4) flavor symm) to lift the flat directions

F-term equations inconsistent with quantum constraint SUSY broken!→

Izawa & Yanagida (1996), Intriligator and Thomas (1996)

Gauge symmetry

Flavor symmetry

Inconsistent with quantum constraint which wants


How our model looks like IYIT

Our setup is like IYIT with part of the flavor symmetry gauged...

...perhaps we can break susy?

IYIT superpotential for each SU(2)

In the limit ¤4 0, this → is just two decoupled IYIT sectors and

susy is indeed broken

Introduce two extra singlet fields to lift

flat directions


Dynamical susy breaking

This intuition from IYIT turns out to be correct

Straightforward to see that this leads to no F-term soln's


Will discuss in a few slides


Dynamical susy breaking

This intuition from IYIT turns out to be correct

Straightforward to see that this leads to no F-term soln's


What is less obvious is that

Also breaks susy for either λ and/or λ non-zero

Not all classically flat directions are lifted with this superpotential!

Will discuss in a few slides


susy breaking with flat directionsIt is possible to have susy breaking even when flat directions

are not lifted Intriligator & Thomas (1996), Shirman (1998), Shadmi & Shirman (1999)

“Quantum removal of flat directions”

Basic idea:

Along flat direction matter gets heavy and is

integrated out gauge dynamics get stronger→

In contrast to usual situation where, along the flat

direction, the only effect is to higgs a gauge group,

making the gauge dynamics weaker


susy breaking with flat directions

Classically, sets Q and Q to zero

Flat direction is S

→ Quarks get heavy along this direction → Integrate them out

matching



Classically, sets Q and Q to zero

Flat direction is S

→ Quarks get heavy along this direction → Integrate them out

→ Sp(4) broken to SU(2)L x SU(2)

R

along flat direction

Sp(4) gets higgsed and is therefore weaker“ ”SU(2) groups have matter integrated out and are stronger“ ”

matching

matching



Each SU(2) undergoes gaugino condensation

We've assumed S is large in a classical regime→

Easy to see susy is broken in this regime

(continued from previous slide)


Previous arguments suggestive that susy does break

)IYIT intuition

)Examined asymptotic region

To actually determine if susy breaks

we need to look at strongly coupled regime

)Examine theory near monopole points


Comments on Kahler potential

Quantum moduli space is smoothly described by the meson fields (the gauge invariants)

)Anomaly matching provides evidence for this

)Similar to quantum deformed and s-confining theories

)Anomaly matching provides evidence for this

as well as susy breaking models such as IYIT and ISS

To leading order, expect canonical Kahler potential

N = 1 susy does not control the Kahler potential; however...

Upshot: since Kahler is non-singular, it is sufficient to just look at F-term equations to determine if susy is broken


Reminder of gauge invariants


Near the monopole point

F-term eqns are clearly going to be messy...let's point out a few things ahead of time




is clearly a special point




is clearly a special pointis clearly a special point

Can show that for Wtree = 0, confining branches (where monopoles condense) require




Theory has a Z2 exchange symmetry between the two SU(2) gauge groups

→ Symmetry in F-term eqns



And the F-term eqns are


No solutions


No solutions● Using physical intuition it was straightforward

to see that motivated vacuum candidates don't preserve susy

● More technical math methods, while less physically transparent, provide easier way to see no solutions exist to F-term equations

– F-term eqns define a variety

– They also define an ideal

– variety being the empty set equivalent to 1 being in the ideal (Hilbert's Nullstellensatz)

– Can find a different basis for the ideal (Hilbert's basis thm)

● Convenient basis: Grobner basis algorithms to find, →easy to implement in, e.g., Mathematica

I'm grateful to Alexey Bondal for discussions on these points


Location of susy breaking vacuum

in the susy breaking vacuum, the monopoles condense

It is straightforward to see that


Location of susy breaking vacuum

in the susy breaking vacuum, the monopoles condense

It is straightforward to see that

What is the physics of this susy breaking vacuum?

Is the mechanism for confinement different than usual SW theory?

Type I or type II superconductor?


Type I vs. Type II superconductivity

Type I Type II

Vainshtein & Yung 2000

Tension of string carrying n units of flux




→ exotic strings can

Seiberg-Witten theory is a type I superconductor

dynamically relax


N = 2 SW confinementType I vs. Type II superconductivity

To leading order in adjoint mass deformation, SW theory is right on border b/w type I and II

→ At L.O., flux strings are BPS and non-interacting

→ BPS states not preserved at higher orders

→ Theory is type I superconductor Vainshtein & Yung 2000

Hanany, Strassler, Zaffaroni (1998)

→ Structure dictated by N = 2 susy, which also controls Kahler

Type I Type II




→ exotic strings candynamically relax


More flexibility in our theory?

Superpotential and Kahler not related in our N = 1 theory

We also have more parameters to vary

Perhaps more flexibility in the type of confinement that arises?

Our theory

N = 2 theory


Summary

●New N = 1 SW theories

– We've found all N=1 gauge theories that are in the

Coulomb phase at generic points on the moduli space

● Some of these theories exhibit dynamical susy breaking

– Break susy even with classically flat directions

● SUSY breaking vacua of the theory occur at points where magnetic monopoles condense

– Physics of these vacua are potentially different from usual SW story


Thank you!


New N = 1 SW theory

Easy to see theory is in Coulomb phase by taking various limits

By matching to the curves in these various limits we can determine the curve of the full theory




various limits

Below ¤4 Sp(4) charges confine

Low energy theory:

SU(2) £ SU(2) gauge theorywith two (2,2): (QQ) and (SQQ)

Generic breaking pattern: SU(2) £ SU(2) ! SU(2)D ! U(1)

)in the Coulomb phase Intriligator & SeibergPhases of N=1 (1994)

N=1 SO(N) theories (1995)



The limit ¤4 À ¤1,2 tells us how ¤4 enters the curve

Proceed the same way as previous limit:

—Consider theory below ¤4

—Match onto known result (Intriligator & Seiberg 1994) while subject to the two Sp(4) constraints (one classical + one quantum)

We will skip the details and simply state the result

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