Scattering in large N matter Chern SimonsTheory

Shiraz Minwalla

Department of Theoretical PhysicsTata Institute of Fundamental Research, Mumbai.

Developments in M-theory, Korea Jan 15

Shiraz Minwalla

References

S. Jain, M. Manglik, S. M, S. Wadia, S. Yokoyama : ArXiv1404.6373Y. Dandekar, M. Manglik, S.M. ArXiv 1407.1322K. Inbasekar, S. Jain, S. Mazumdar, S.M. V. Umesh, Y.Yokoyama: in progressY. Dandekar, M. Manglik, S.M. : in progress

Shiraz Minwalla

Introduction

The last 20 years has seen the discovery of many strongweak coupling dualities in supersymmetric quantum fieldtheories in 2, 3 and 4 dimensions.It thus seems clear that field theory-field theory dualitiesare common in the space of susy quantum field theories.

Shiraz Minwalla

Introduction

Why have all dualities discovered over the last 20 yearsbeen supersymmetric?Exp 1: Susy is an essential ingredient in field theory fieldtheory dualities, atleast in dimensions greater that two.Exp 2: Field theory - field theory dualities are common inthe space of all quauntum field theories. Dualities havebeen discovered only in susy qfts as these are the onlytheories in which we can do exact computations.How to distinguish 1 from 2? One strategy : find a contextin which exact computations are possible at strongcoupling without susy. If you find a lot of duality, suggests 2is the correct explanation. One example : this talk

Shiraz Minwalla

Chern Simons Theory

CS theory. Gauge theory in 3 dimensions.

S =κ

4πTr∫ (

AdA +23

A3)

+ Smatter

Note that:Equation of motion κF = J. No local degrees of freedom inthe gauge field.Action is gauge invariant even though Lagrangian is not.This requires κ = integer.Effective coupling constant λ = N

κ . Dimensionless.Discrete. Cannot be continuously renormalized.Consequence no running of the gauge coupling. Easy toconstruct superconformal field theories without susy.Effective fixed lines in the large N t’ Hooft limit.

Shiraz Minwalla

Chern Simons Theory: Notation

We pause to define notation. The integer behind the ChernSimons action, κ depends on regulation scheme andneeds a clear definition.Throughout this talk we will use the symbol k for the levelof the WZW theory obtained in studying Chern Simonstheory on a manifold with a boundary.

κ = k + sgn(k)N

Under level rank duality duality of pure Chern Simonstheory, k ′ = −sgn(k)N, N‘ = |k |This implies κ′ = −κ, N ′ = |κ| − N. In other words

λ′ = λ− sgn(λ)

Shiraz Minwalla

Chern Simons theory coupled to fermionic matter

Simple example of conformal theories.

Smatter =

∫ψ̄ (Dµγ

µ + mF )ψ

where ψ is a fermion in any representation of the gaugegroup.Atleast at small enough λF this theory is conformal atmF = 0 in dim reg. Proof. Gauge coupling cant run. Noother relevant or marginal operator.This talk. Fermion taken to be in the fundamentalrepresentation. Will be able to solve theory at all λF atlarge N because effectively vector model (no dof in gaugefield)

Shiraz Minwalla

Chern Simons - Bosonic matter theories

The second theory we study in this talk is the ChernSimons theory coupled to bosonic matter∫ (

Dµφ̄Dµφ+ m2Bφ̄φ+

12NB

b4(φ̄φ)2)

We will be particularly interested in a scaling limit of thebosonic theory. To explain this we rewrite the theory as∫ (

Dµφ̄Dµφ+ m2Bφ̄φ+

b4

2NB(φ̄φ)2 − NB

2b4

(σ − b4

NBφ̄φ−m2

B

)2)

∫ (Dµφ̄Dµφ+ σφ̄φ+ NB

m2B

b4σ − NB

σ2

2b4

)(1)

Shiraz Minwalla

Chern Simons coupled to critical bosons

The so called critical limit of the bosonic theory is definedby the limit

b4 →∞, mB →∞,4πm2

Bb4

= mcriB = fixed. (2)

In this limit

S =

∫ (Dµφ̄Dµφ+ σφ̄φ+ NB

mcriB

4πσ

)

Atleast in the large N limit, this theory is also conformal atmcri

B = 0. mcriB = 0 is a relevant deformation of this critical

theory.

Shiraz Minwalla

Conjectured Duality

It has been conjectured that the Fermionic Chern Simonstheory is dual in the large N limti to the critical bosonChern Simons theory under the duality map

κF = −κB,

NF = |κB| − NB,

λB = λF − sgn(λF ),

mF = −mcriB λB.

(3)

The duality map is stated in the t’ Hooft large N limit andcould receive subleading corrections at finite N and κ.Note that the map for the t’ Hooft coupling is precisely thatof level rank duality.Prior to this talk: three kinds of concrete evidence.

Shiraz Minwalla

Evidence for duality

Specialize to mF = mcri = 0. Has been shown that thespectrum of single trace operators and their three pointfunctions match between the two sides.At arbitrary mF the thermal partition function of boththeories has been computed on an S2 at temperatures ororder

√N. Highly nontrivial. Undergo two phase transitions

as function of temperature. Match exactly between the twosides.Duality conjecture generalized to general theory of bosonsand fermions interaction with CS. 7 continuousparameters. Duality map determined. Leaves N = ∈ andN = 1 submanifolds fixed. Giveon Kutasov type (reallyBenini et al) duality N = 2. Continuous deformation fromknown susy duality to purely bose-fermi.

Shiraz Minwalla

Scattering amplitudes and bosonization

Bose-Fermi or bosonization duality. Sounds interesting.Would like to understand in more detail. Best case:bosonisation formula like ψ = eφ. Too much to ask for incurrent context as ψ and φ are not gauge invariant.While arbitrary insertions of ψ and φ are not meaningful,insertions that are taken to infinity along lines of equationsof motion - i.e. S matrices - are well defined. For thisreason the map between S matrices is a ‘poor mansbosonization’ map between these two theories.This talk: We indpependently compute the S matrix for thebosons and fermions, and examine their interrelationshipunder duality. Also uncover interesting structural featuresalong the way.

Shiraz Minwalla

Scattering Amplitudes: Colour Kinematics

In either the bosonic or fermionic theory we refer to aquantum the transforms in the fundamental of U(N) as aparticle and a quantum that tranforms in theantifundamental of U(N) as an antiparticle.We study the most general 2× 2 scattering process. Thereare three such processes

Pi + Pm → Pj + Pn

Pi + Aj → Pn + An

Aj + An → Ai + Aj

(4)

All these S matrices are contained in appropriate on shelllimits of the single correlator. In e.g. the bosonic theorythis is

< φiφmφ̄j φ̄n >

Shiraz Minwalla

Color Kinematics

It follows from U(N) invariance that

< φiφmφ̄j φ̄n >= aδj

i δnm + bδn

i δjm

Consequently, in each channel we need to compute 2distinct scattering amplitudes. For particle-particlescattering it is convenient to work in the following basis forU(N) singlets

< φiφmφ̄j φ̄n >= Tsym

(δj

i δnm + δn

i δjm

)+ Tas

(δj

i δnm − δn

i δjm

)TSym = a+b

2 is the S matrix in the channel of symmetricexchange, while Tas = a−b

2 is the S matrix in the channel ofantisymmetric exchange.

Shiraz Minwalla

Colour Kinematics, 1N

On the other hand for particle - antiparticle exchange thefollowing basis is most useful

< φiφmφ̄j φ̄n >= TSing

δji δ

nm

N+ TAdj

(δn

i δjm −

δji δ

nm

N

)TSing = Na + b is the scattering matrix in the singletchannel, while TAdj = b is the scattering matrix in theadjoint channel.It is not difficult to convince oneself that, at leading order inthe 1

N expansion

a ∼ b ∼ Tsym ∼ Tas ∼ TAdj ∼1N

On the other handSSing ∼ O(1)

Shiraz Minwalla

Unitarity and 1N

We have chosen the colour structures multiplying TSing etcto be simply projectors onto the exchange representations.As projectors square to unity it follows from unitarity that

Tc − T †c = TcT †c

Tc is any one of the four T matrices described above. Wehave used the fact that e.g. 2→ 4 production contributesto unitarity only at subleading order in 1

N .The unitarity equation is rather trivial for Tsym, Tas, Tadj , asthe RHS is subleading in 1

N . However it imposes a highlynontrivial nonlinear constraint on Tsing , the most nontrivialof the four scattering matrices.

Shiraz Minwalla

Non Relativistic Limit

The S matrix for non relativistic particles interacting viaChern Simons exchange was worked out in the early 90s,most notably by Bak, Jackiw and collaborators.The main result is strikingly simple. Consider the scatteringof two particles in representations R1 and R2, in exchangechannel R. It was demonstrated that the S matrix equalsthe scattering matrix of a U(1) charged particle of unitcharge scattering off a point like flux tube of magnetic fieldstrength ν = c2(R1)+c2(R2)−c2(R)

κ .This quantum mechanical S matrix was computedoriginally by Aharonov and Bohm and generalized by Bakand Camillio to take account of possible point likeinteractions between the scattering particles.

Shiraz Minwalla

Non Relativistic Scattering Amplitude

TNR = −16πicB (cos (πν)− 1) δ(θ) + 8icB sin(πν)Pv(

cotθ

2

)+ 8cB| sinπν|

1 + eiπ|ν| ANRk2|ν|

1− eiπ|ν| ANRk2|ν|

,

ANR =−1w

(2R

)2|ν| Γ(1 + |ν|)Γ(1− |ν|)

.

(5)

Here wR|2ν| is a measure of the strength of theBak-Camillio contact interaction between the scatteringparticles. In the limit w(kR)|2ν| → 0 ANR →∞ and thesecond line of the scattering amplitude simplifies to

8cB| sinπν|,the original Aharonov Bohm result

Shiraz Minwalla

The δ function and its physical interpretation

The non relativistic amplitude above has a very unusualfeature, a piece in the scattering amplitude proportional tothe δ(θ).This term in the scattering amplitude was missed in theoriginal paper of Aharonov and Bohm. The amplitude wascorrected with the addition of this piece in the early 80s. Inthe early 90s Jackiw and collabortors emphasized that isterm is necessary to unitarize Aharonov Bohm scattering.Infact this term has a simple physical interpretation (whiteboard). The physical interpretation makes no reference tothe non relativistic limit, so we assume that the δ functionpiece is present unmodified even in relativistic scattering.

Shiraz Minwalla

Effective value of ν

In the large N limit there is a simple formula for thequadratic Casimir of representations with a finite number ofboxes plus a finite number of antiboxes.

c2(R) =N(nb + na)

2

using this result in the formula for ν we find

νsym ∼ νas ∼ νAdj ∼ O(1/N)

On the other handνSing | = λ

Now the non relativistic limit is obtained by taking k → 0 atfixed ν. If ν ∼ O(1/N) and if we work to leading order in 1

Nwe effectively take ν → 0 first. In other words the results ofAharonov Bohm Bak Camillio yield a sharp prediction forthe non relativistic limit only of Tsing .

Shiraz Minwalla

Exact Propagators: method

Work in Lorentzian space. Set A− = 0. Main advantage:∫A3 = 0. No gauge boson self interactions. Also no IR

divergences (contrast with A0 = 0).If we want to understand scattering we first haveunderstand free propagation.No gauge self interactions plus planarity gives a simpleSchwinger Dyson equation (like t’ Hooft model). Nonlinearintegral equation. Quite remarkably admits exact solution

Shiraz Minwalla

Exact Bosonic Propagator

In the case of the bosons we have

〈φj(p)φ̄i(−q)〉 =

(2π)3δij δ

3(−p + q)

p2 + c2B

(6)

where the pole mass, cB is a function of mB,b4 and λB,given by

c2B =

λ2B

4c2

B −b4

4π|cB|+ m2

B. (7)

Shiraz Minwalla

Exact Fermionic Propagator

⟨ψj(p)ψ̄i(−q)

⟩=δi

j (2π)3δ3(−p + q)

iγµpµ + ΣF (p), (8)

where

ΣF (p) =iγµΣµ(p) + ΣI(p)I,

ΣI(p) =mF + λF

√c2

F + p2s ,

Σµ(p) =δ+µp+

p2s

(c2

F − Σ2I (p)

),

c2F =

(mF

sgn(mF )− λF

)2

. (9)

Nontrivial (and necessary) that pole in propagatorp2 − c2

F = 0 is Lorentz invariant.

Shiraz Minwalla

Restrictions on soln of 4 point function

Now turn to evaluating the 4 point function needed forcomputing scattering. Linear integral equation. Shamefullyunable to solve in general. Exact solution for the specialcase q± = 0.Defect has different implications in different channelsIn the singlet exchange channel it turns out that qµ is thecentre of mass energy. If q± = 0 then momentum centre ofmass momentum is spacelike. Incompatable with puttingonshell. Solution not useful for directly computing Tsing .In the other three channels qµ is the momentum transfer.In the scattering of two particles of equal mass, themomentum exchange is always spacelike. Setting q± = 0is simply a choice of Lorentz frame. Assuming the answeris covariant, there is no loss of information. Can read offfull results for TAdj , Tsym, Tas.

Shiraz Minwalla

Onshell 4 point function for bosons

Evaluating the onffshell 4 pt function subject to thisrestriction and taking the onshell limit we find

T =4πiκB

εµνρqµ(p − k)ν(p + k)ρ

(p − k)2 + j(√

q2). (10)

Where

j(q) =4πiqκB

(

4πiqλB + b̃4

)+(−4πiqλB + b̃4

)( 12+

cBiq

− 12+

cBiq

)λB

(4πiqλB + b̃4

)−(−4πiqλB + b̃4

)( 12+

cBiq

− 12+

cBiq

)λB

.

(11)

Shiraz Minwalla

Limits of j(q)

In the limit λB → 0 j(q) reduces to

j =−b4

1 + b4H(q)(12)

where

H(q) =

∫d3r

(2π)31(

r2 + c2B

) ((r + q)2 + c2

B

) (13)

Matches well known exact answer for large N φ4 scattering.If we take b4 →∞ and then take the limit λB → 1 we getthe even simpler answer

j(q) = −8πcB. (14)

Matches tree Fermionic scattering amplitude!

Shiraz Minwalla

Fermionic Results and duality

We can repeat the computation for the fermions at allvalues of λF . Can show that the final result exactlymatches with the Bosonic onshell amplitude once wesubstitute

λF = λB − sgnλB

This implies T BAdj = T F

Adj

MoreoverT B

sym = T Fas, T B

as = T Fsym

This is the result we should have expected both from levelrank duality as well as from basic statisitics.

Shiraz Minwalla

Scattering in the singlet channel (bosons)

According to standard lore we can obtain Tsing byanalytically continuing the above results.Performing the naive analytic continuation gives a resultthat cannot be right. To start with it does not have the deltafunction piece that we know must be there on physicalgrounds. Indeed no analytic continuation can give thispiece as it is not analytic.Clearly the rules for crossing symmetry must be modified.By playing around we have come up with the followingconjecture

Tsing =sin(πλB)

πλBT ac

sing − i(cos(πλB)− 1)I(p1,p2,p3,p4)

where T trialS is the singlet amplitude one obtaines from

naive analytic continuation

Shiraz Minwalla

Properties of conjectured S channel formula

Our conjecture for the exact scattering matrix has the followingfeatures

It agrees with the well known result for φ4 scattering asλ→ 0 (trivial check).Agrees with tree level scattering for Fermions at λB → 1.More generally it maps under duality to a similarconjectured formula on the Fermionic side.Exactly obeys the nonlinear unitarity equation

T − T † = TT †

in a highly nontrivial mannerAgrees at one loop with explicit computation in LorentzgaugeHas the correct nonrelativisitic limit. More below.

Shiraz Minwalla

Explicit Conjecture

Our conjecture is

Tsing =

8πi√

s(1− cos(πλB))δ(θ) + 4i√

s sin(πλB)Pv(

cot(θ

2

))+

4√

s sin(π|λB|)F ;

F =

(

4π|λB|√

s + b̃4

)+ eiπ|λB |

(−4π|λB|

√s + b̃4

)( 12+

cB√s

12−

cB√s

)|λB |

(4π|λB|

√s + b̃4

)− eiπ|λB |

(−4π|λB|

√s + b̃4

)( 12+

cB√s

12−

cB√s

)|λB |

,

(15)

Shiraz Minwalla

Straight forward non relativistic limit

The straight forward non relativistic limit is taken by taking√s → 2cB. In this limit F = sgn(λB) and

Tsing = 8πi√

s(1− cos(πλB))δ(θ) + 4i√

s sin(πλB)Pv(

cot(θ

2

))+ 4√

s| sin(π|λB|)|(16)

This is in perfect agreement with the Aharonov Bohmresult. Natural quesiton: is there any way to get the Bak-Camillio modification out of our formula? Ans: yes! Thereis a second, more sophisiticated non relativistic limit onecan take.

Shiraz Minwalla

Tuned Non Relativistic limit

We first note that our conjectured S matrix has a pole forb̃4 ≥ b̃crit

4 = 8πcB|λB| indicating the existence of a particle -antiparticle bound state in the singlet channel at thesevalues of parametersAs b̃4 approaches b̃crit

4 from above, the mass of the boundstate approaches 2cB. Setting b̃4 = b̃crit

4 + δb4, it may beshown that EB ∼ (δb4)1/|λB |) at small δb4

This observation motivates study of the scaled nonrelativistic limit

δb4

cB→ 0,

kcB→ 0,

kcB

(cB

δb4

) 12|λB |

= fixed. (17)

Shiraz Minwalla

Reduction in tuned non relativistic limit

In the tuned non relativistic limit the nontrivial term in ourconjectured S matrix simplifies to

8cB| sin(πλB)|1 + eiπ|λB |

[δb4

(2cB

k

)2|λB |

16π|λB |cB

]

1− eiπ|λB |

[δb4

(2cB

k

)2|λB |

16π|λB |cB

] ,

This agrees exactly with the Bak -Camillio generalization ofthe Aharonov Bohm scattering provided we identify

− w (cBR)2|λB | =cB

δb4

(16π|λB|

Γ(1 + |λB|)Γ(1− |λB|)

). (18)

I view this nontrivial match as an extremely nontrivial checkof our conjecture

Shiraz Minwalla

Possible explanation of sinπλπλ

Interesting observation: sinπλπλ equals Witten’s result for the

expectation value of a circular Wilson loop on S3 in pureChern Simons theory in the large N limit.This observation suggests a possible explanation of themodified crossing symmetry rules (details on white board)If this is right it is the tip of the iceberg. Finite N and κ.

Shiraz Minwalla

Bosons-Fermions?

Let us accept the conjecture for the moment. How didfermions turn into bosons? Different answers in differentchannels.Adjoint channel. Boring. No phase, no statistics. Boring.Sym and as channel. No phase but statistics. Level rankduality on Young Tableax of exchange representationsSinglet channel. Most interesting. Anyonic phase. Neitherbosons nor fermions. eiπλB = eiπλF e−iπsgnλF impliesλB = λF − sgnλF . Equating anyonic phases in the singletchannel gives a derivation of the anyonic phase!

Shiraz Minwalla

Brief Summary of ongoing work

Idea: generalize computation to theories one boson andone fermion that preserves atleast N = 1 sypersymmetry.Motivation. To check duality for these theories. To checkthat the conjecture for crossing symmetry continues towork for these theories. Make contact with extensiveliterature on scattering in N = 6 theories.Method. Have found an offshell N = 1 generalization oflightcone gauge. Removes gauge boson self interactions.Can repeat all calculations presented here in superspace.Have done this. Have verified that, in the on shell limit, ourresults are in perfect agreement with the duality map onthis manifold that I conjectured with Jain and Yokoyamafrom the study of partition functionsHave also verified that our results dramatically simpilfywhen the N = 1 theories become N = 2 susy.In process of verifying that the same conjecture for Schannel S matrix unitarizes scattering.

Shiraz Minwalla

Conclusions

Presented computations and conjectures for all ordersscattering matrices in Chern Simons matter theories.Results in perfect agreement with dualitySuggest novel transformations of S matrices undercrossing symmetry. Current conjectures are large Nresults. Study of crossing at finite N and k a veryinteresting problem.Generalizations to susy theories underway. Shouldevenually make contact with scattering in ABJM theories.

Shiraz Minwalla