The Massive Phase for the U(N) Chern-Simons Gauge Theory in D=3 at Large N

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William A. Bardeen, Fermilab! 1! ICHEP2014, Valencia, July 5, 2014

The Massive Phase for the

U(N) Chern-Simons Gauge Theory in D=3 at Large N

William A. Bardeen Fermilab

arXiv:1402.4196 [hep-th]! arXiv:1404.7477 [hep-th]!

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William A. Bardeen, Fermilab! 2! ICHEP2014, Valencia, Jul7 5, 2014

Outline

• Introduction

• Conformal theories

• Large N expansions

• Conformal phase of U(N) Chern-Simons gauge theory, dualities

• Massive phase – scalar quarks, pure φ⁶ theory

• Massive phase – scalar quarks in U(N) Chern-Simons theory

• Massive phase for fermions - spinor quarks in U(N) CS theory

• Conclusions

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Conformal Symmetry

• Standard Model is classically conformal

- all gauge and Yukawa couplings are dimensionless

• Higgs Potential:

- conformal symmetry breaking - electroweak symmetry breaking

€

V = λ

(

H⁺H - v²

)

², v ~ 175GeV

• Two possible conformal limits:

- v -> 0: no EWSB, all particles massless - conformal phase - λ -> 0: flat potential, EWSB possible - massive phase

- dynamical symmetry breaking with <H> = v

- all Standard Model particles are massive: top, W, Z, etc - Higgs particle remains massless, the Goldstone mode of the dynamically broken scale symmetry - a dilaton

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Large N Limit

• QCD with a large number of colors, N – ‘t Hooft expansion - U(N) gauge theory with N -> ∞, g²->0, g²N fixed

- Leading order terms are the planar diagrams - systematic 1/N expansion

• Meson effective theory

- Leading order - meson tree diagrams - 1/N expansion is meson loop expansion

- factorization of color singlet composite operators - AdS/CFT duality: meson theory, IR-brane models

• Many applications

- chiral Lagrangians, vector meson dominance

- weak matrix elements, K-meson decay amplitudes

- Buras, Gerard and Bardeen, arXiv:1401.1385[hep-ph], Eur.Phys.J. C

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Chern-Simons Gauge Theory

• U(N) Chern-Simons gauge theories in D=3 - QCD analogy

- dimensionless gauge coupling constant

- quarks as scalar bosons or spinor fermions with color - planar expansion in 1/N

- exact conformal invariance at leading order at large N

• Chern-Simons Action in R³

• Matter Fields in fundamental representation (complex N vector) - Scalar Quarks:

- Spinor Quarks:

€

, λ = N κ

€

S_scalar =

∫

d³^{x D}^⎡_⎣^⎢

(

^µ^ϕ^!⁺

) (

^D^µ^ϕ^!

)

⁺^m²^ϕ^!⁺^{ϕ +}^! _2N¹ ^λ⁴

( )

^ϕ^!⁺^ϕ^!² ⁺ _6N¹ ² ^λ⁶

( )

^ϕ^!⁺^ϕ^!³^⎤_⎦^⎥

€

S

_CS

= i κ

8 π ε

_µνρ

∫ d

³

x ⎧ ⎨ ⎩ A

^µ^a

∂

^ν

A

^ρ^a

+ i 2 3 A

^µ^a

A

^ν^b

A

^ρ^c

f

^abc

⎫ ⎬ ⎭

€

S_fermion =

∫

d³^x

[

^ψ^!⁺

(

^γ^µ^D^µ ⁺ ^m

)

^ψ^!

]

^,^D^µ ⁼^∂^µ ⁺ îA^µâ^Tâ

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Chern-Simons Gauge Theory

• Massless Conformal Phase

- exact solutions at leading order of 1/N expansion

- Giombi, Minwalla, Prakash, Trevedi, Wadia, Yin (2011-2013)

- arXiv:1104.3317, 1105.4011, 1110.4386, 1207.4750

- Aharony, Gur-Ari, Yacoby, Maldecena (2011-2013)

- arXiv:1110.4382, 1207.4593, 1211.1866, 1211.4843

- thermal partition functions, R²xS¹ - conserved higher spin currents

- dualities, AdS₄,Vasiliev’s higher-spin gravity theory (1987)

- Fradkin, Visiliev – Phys.Lett. B189,89(1987)

- novel use of Euclidean space light-cone gauge:

€

A₋^a =

(

A₁^a −iA₂^a

)

^{/ 2} ⁼⁰

€

G₊₃

( )

p ⁼ ^−G³⁺

( )

^p ⁼ ⁴^πⁱ

κ 1

p₋ = 4πi λ N

1

p₋, λ ≡ N

κ , (N,κ) ^→∞

€

∂

∂p₊ 1 p −q

( )₋ ⁼ ²^πδ

2(p −q)^, ^∂

∂p₊ ( )p₋ ⁼ ^0, ^∂

∂p₊ p_s = p₋

p_s , p_s² = 2p₊p₋

(7)

Dynamical Mass Generation

I. Scalar quark theory at λ=0

- Bander, Bardeen, Moshe: Phys.Rev.Lett. 52,1188(1984) - Amit, Rabinovici: Nucl.Phys. B257,371(1985)

- Matter Action:

- Gap Equation:

- Massive phase at critical coupling:

- Induced four-point coupling:

- Scalar Current Correlator:

- Bubble sum:

- Dilaton pole - the spontaneous breaking of scale invariance

€

M_quark² = M² = 1 2

λ₆ N²

ϕ ! ⁺ϕ ^!² = 1

2λ₆ − M 4π

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

= λ₆ 32π² ^M

2

€

S_scalar =

∫

d³^{x D}^⎡_⎣⎢

(

^µ^ϕ^!⁺

) (

^D^µ^ϕ^!

)

⁺_6N¹² ^λ⁶

( )

^ϕ^!⁺^ϕ^!³⎤

⎦ ⎥

€

1= λ₆ 32π²

€

L_eff = 1 2

λ^eff₄ N

ϕ ! ⁺ϕ ^!

( )

²^,^λ^eff⁴ ⁼ ^λ_N⁶ ^ϕ^!⁺^ϕ^! ⁼⁻⁸^π^M

€

J₂^scalar !

( )

k ⁼ ^ϕ^!⁺^ϕ^!

( )

^k^!^ϕ^!⁺^ϕ^!

( )

⁻^k^! ^,

€

J₂^scalar !

( )

k ^/^N ⁼ ^B^S k ! ²

( )

1+λ^eff₄ B_S ! k ²

( )

^→

3 2π

M!

k ² = f_D² N

! 1

k ² , ! k →0

€

B_s !

( )

k ^:

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Dynamical Mass Generation

II. Scalar quark U(N) gauge theory, gauge coupling, λ≠0

- Bardeen, Moshe: arXiv:1402.4196

- Gauge Propagator:

- Currents:

- Gap Equation (single gluon exchange term vanishes):

- Renormalized critical coupling:

€

M_quark² = M² = 1 2

1

N²

(

8π²λ² +λ₆

)

^ϕ^!⁺^ϕ^!² ⁼^⎛_⎝^⎜¹₄ ^λ² ⁺ ₃₂^λ_π⁶ ²^⎞_⎠^⎟^M²

€

1= 1

4 λ² + λ₆ 32π² ^≡

λ^eff₆ 32π²

€

G₊₃

( )

p ⁼ ⁻^G3+

( )

p ⁼ ⁴^πⁱ _N^λ _p¹

−

€

Single gluons: J₃

(

q,q+ p

)

⁼

(

^2q⁺ ^p

)

3, J₋

(

q,q+ p

)

⁼

(

^2q⁺ ^p

)

₋

Seagulls: J₃₃

(

q,q+ p+ p'

)

⁼^δ³³^;^J−−

(

q,q+ p+ p'

)

⁼⁰

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Dynamical Mass Generation

II. Scalar quark U(N) gauge theory, gauge coupling, λ≠0 - Scalar Vertex Functions:

- Ladder Vertex (integral equation):

-

- Seagull exchange (diagram sum is local, renormalizes λ₆)

- Induced four-point coupling (as for λ=0):

- Renormalized bubble:

- Bubble sum:

- Dilaton pole, effective decay constant,

€

λ^eff₄ = λ^eff₆ N

ϕ ! ⁺!

ϕ = −8πM

€

J₂^scalar !

( )

k ^/^N ⁼ ^B^CS k ! ,λ

( )

1+ λ^eff₄ B_CS ! k ,λ

( )

^→

3 2π

M 1−λ²

! 1

k ² = f_D² N

! 1

k ² , ! k →0

€

ϕ ! ⁺!

ϕ

( )

k₃ ^,^k³ ⁻longitudinal momentum

€

V p,k

(

₃

)

⁼¹⁺ⁱ^λ^⎧^⎨_⎩^2arctan

(

^k³^/2 ^p²⁺^M² ⁻^arctan

(

^k³^/2^M

) )

^⎫^⎬_⎭^→1,^k³ ^→0

€

B_CS ! k ,λ

( )

⁼^tan^⎛_⎝^⎜^λ^arctan^⎛_⎝^⎜ ^k^!² ^/2^M^⎞_⎠^⎟^⎞_⎠^⎟^{/ 4}^⎛_⎝^⎜^πλ ^k^!²^⎞_⎠^⎟^,

€

f_D = 3NM/2π

(

1−λ²

)

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Dynamical Mass Generation

III. U(N) gauge theory, spinor quarks, the fermion version

- Bardeen: arXiv:1404.7477

- Matter action:

- two component spinors, ψ:

- Gap equation (noncovariant self-energy functions):

€

S⁻¹

( )

p ⁼

(

ⁱ^γµp_µ +Σ

( )

p

)

^,^Σ

( )

^p ⁼^γ⁻^Σ⁺

( )

^p ⁺^Σ^o

( )

^p

€

γ ! = !

σ , Pauli spin matrices

€

S_fermion =

∫

d³^x

[

^ψ^!⁺

(

^γ^µ^D^µ ⁺ ^m

)

^ψ^!

]

^,^D^µ ⁼^∂^µ ⁺ îA^µâ^Tâ

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III. U(N) gauge theory, spinor quarks, the fermion version - Singlet self-energy function:

- Noncovariant self-energy function:

€

Σ_o

( )

p ⁼ ^m ⁻^4πλi ^d

3q 2π

( )³

1 p−q

( )

₋

iq₋ D q

( )

∫

⁼ ^m ⁻^4πλi ^π₂^d²^q

( )π ³

1 p−q

( )

₋

iq₋ q_s² + M²

∫

= m − λ

2 dq_s²

p_s² Λ²

∫

¹

q_s² + M² = m+λ

(

p_s² + M² − Λ

)

→m+λ p_s² + M², p_s² =2p₊p₋

€

2ip₋Σ₊

( )

p ⁼

(

^Σ^o

( )

^p

)

² ⁻^M²

=λ²^p_s² + 2λ^{m p}_s² + M² +

(

m² −

(

1−λ²

)

^M²

)

=λ²^p_s² + 2λ^m

(

^p_s² + M² −M

)

⁺

(

⁽^m⁺^λ^M⁾² ⁻ ^M²

)

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Dynamical Mass Generation

- Gauge invariant, Lorentz invariant scalar current correlator:

- Conformal limit:

- next to leading term in large momentum expansion,

€

J₀ !

( )

k ^J⁰

( )

⁻^k^! ^/^N ⁼ ⁻₂¹_π _λ¹ ^m

− 1 2π

1 λ ⁻λ

⎛

⎝ ⎜ ⎞

⎠ ⎟ ! k ² /2

( )

¹⁺

k ! ² /2(m+λM)

( )

^tan^⎛⎝ ⎜ ^λ^arctan

(

^k^!² ^/2^M

)

⎞

⎠ ⎟

⎛

⎝ ⎜ ⎞

⎠ ⎟

tan λarctan !

k ² /2M

( )

⎛

⎝ ⎜ ⎞

⎠ ⎟ − !

k ² /2(m+λM)

( )

⎛

⎝ ⎜ ⎞

⎠ ⎟

€

J₀ !

( )

k ^J⁰

( )

⁻^k^! ^/^N ^→₄¹_π^⎛_⎝^⎜¹_λ ⁻^λ^⎞_⎠^⎟ ^k^!² ^tan^⎛_⎝^⎜^λ^π₂^⎞_⎠^⎟^,^λ ^<^2,^k^!^→∞

€

J₀ !

( )

k ^J⁰

( )

⁻^k^! _NLO ⁼ ^m₂^λ_π ^⎧^⎨_⎩⁻¹⁺^⎛_⎝^⎜_λ¹² ⁻¹^⎞_⎠^⎟^tan²^⎛_⎝^⎜^λ^π₂^⎞_⎠^⎟^⎫^⎬_⎭

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III. U(N) gauge theory, spinor quarks, the fermion version - Infrared behavior of massive phase – dilaton pole.

- singularity at zero momentum:

- bare mass parameter tuned to critical point:

- At critical point:

- Near critical behaviour, dilaton mass:

€

J₀

(

k₃⁾^J⁰

(

^−k³⁾ ^/^N ^{→ −}₂¹_π ^⎛_⎝^⎜¹_λ ⁻^λ^⎞_⎠^⎟ ¹

λ/M −1/(m+ λM⁾

( )

⁼

1

2π 1− 1 λ²

⎛

⎝ ⎜ ⎞

⎠ ⎟ M m( + λM⁾ m−(1/λ − λ⁾^M

€

m = M /λ − λM

€

J₀ !

( )

k ^J⁰

( )

⁻^k^! ^/^N ^→₂³_π _λ¹² ⁴_k^!^M² ³ ⁺ ^finite,^k^!^→⁰

€

J₀ !

( )

k ^J⁰

( )

⁻^k^! _k^!_→0 ^→ _k^!² ^f₊^D²_µ² ^, ^f^D ⁼ ⁶^NM_πλ² ³

€

µ² =

(

1/

(

m+ λM

)

⁻ ^λ^/^M

)

¹²^M

3

λ

(

1− λ²

)

⁼¹²^{M M}

( (

^1/^{λ − λ}

)

⁻ ^m

)

λ 1− λ²

(14)

Dynamical Mass Generation

III. U(N) gauge theory, spinor quarks, the fermion version - weak coupling, λ=0.001.

a. Massless dilaton, µ=0 b. Near critical, µ≠0

(15)

III. U(N) gauge theory, spinor quarks, the fermion version - moderate coupling, λ=0.5.

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Dynamical Mass Generation

III. U(N) gauge theory, spinor quarks, the fermion version - near boundary for upper gauge coupling, λ=1.999.

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Dynamical Issues

III. U(N) gauge theory, spinor quarks, the fermion version - Noncovariant gap equation:

- Discrepancy at requires additional zero mode contribution to gap equation:

- Particular artifact of light-cone gauge?, or added constraint?

- Would imply no dynamical symmetry breaking for fermions.

€

Σ₊

( )

p ⁼ ⁻⁴^πλⁱ ^d

3q

( )

2π ³

1 p −q

( )

₋

Σ_o

( )

q D q

( )

∫

⁼ ⁻⁴^πλⁱ ^π^d

2q

( )

2π ³

1 p −q

( )

₋

Σ_o

( )

q q_s² + M²

∫

= − i p₋

λ

2 dq²

0 p_s²

∫

^Σ^o

( )

^q

q_s² + M² = − i p₋

λ²

2 p_s² + λm

(

p_s² + M² − M

)

⎛

⎝ ⎜ ⎞

⎠ ⎟

€

p

_s²

= 0

€

Σ₊

( )

p ⁼ ⁻ _pⁱ

−

R₊ + λ²

2 p_s² +λ^m ^ps

2 + M² −M

( )

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎧ ⎨

⎩

⎫ ⎬

⎭ , 2R₊ = (m+λ^M)² ⁻^M²

€

2R₊ = 0 ⇒ m =

(

1− λ

)

^M^,^critical ^coupling^:^λ ⁼¹

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Dynamical Issues

- if no zero mode, must require m=(1-λ)M; critical point at λ=1.

- no dilaton in massive phase -  hidden explicit breaking

-  or the massless, conformal phase is true ground state

1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09

1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

<Jo(k)Jo(-k)> vs k

€

J₀ !

( )

k ^J⁰

( )

⁻^k^! _{λ →1} ⁼ ₂¹_π

k ! ² arctan !

k ² /2M

( )

^,^m ⁼ ⁰

(19)

Conclusions

• U(N) Chern-Simons gauge theories are an interesting laboratory for studying the mechanisms of the dynamical breaking of

conformal symmetry.

• The theories remain conformal to leading order in the large N expansion where N is the number of colors.

• We obtain exact solutions in the massive phase for both scalar and spinor quarks in the fundamental representation of U(N).

• The scalar current density represents the order parameter of the spontaneous breaking of the conformal symmetry. The explicit computation of the scalar current correlator reveals the

presence of the dilaton pole in the massive quark phase.

• A zero mode pole seems essential to the massive fermion phase but remains a puzzling artifact of the light-cone gauge.

(20)

F William A. Bardeen, Fermilab! 20! University of Granada Seminar, May 13, 2010

Additional Slides

(21)

III. U(N) gauge theory, spinor quarks, the fermion version - Vertex equations at large N:

- Noncovariant vertex functions:

€

V p,k( ₃) ⁼¹⁺^2πλi ^d

3q 2π

( )³

∫

₍_p₋¹_q₎

−

⋅

{

γ₃S_F( )q ^{V q,k}( 3)^S^F(^q⁺^k³)^γ⁻ ⁻^γ⁻^S^F( )^q ^{V q,k}( 3)^S^F(^q⁺^k³)^γ³

}

€

V p

(

,k₃) ⁼^V⁰

(

^p,k³) ⁺^γ⁻^V⁺

(

^p^,k³)

€

V₀

(

p,k₃

)

⁼¹⁺²^πλⁱ ^d

3q 2π

( )³

∫

₍_p₋¹_q₎

−

1

q₃² +q_s² +M²

( )

1 q₃ +k₃

( )

²⁺^q^s²⁺^M²

( )

⋅

{ [

2k₃q₋ +4iq₋Σo( )q

]

^V⁰

(

^q,^k³

)

⁺

[ ]

⁴^q⁻² ^V⁺

(

^q,^k³

) }

€

V₊(p,k₃) ⁼²^πλⁱ ^d

3q 2π

( )³

∫

₍_p₋¹_q₎

−

1 q₃² +q_s² + M²

( )

1 q₃ +k₃

( )² ⁺^q^s²⁺ ^M²

( )

⋅ −2q₃(q₃ +k₃)⁻^4q⁺^q⁻ ⁺^4iq⁻^Σ⁺( )^q ⁺^2Σo 2( )q

[ ]

^V⁰⁽^q,^k³⁾⁺

[

^2q⁻^k³ ⁻^4iq⁻^Σ^o^{( )}^q

]

^V⁺⁽^q,^k³⁾

{ }

(22)

III. U(N) gauge theory, spinor quarks, the fermion version - Solutions for vertex equations:

with

- Boundary conditions:

- Determine the A and B coefficients:

€

V₀

(

p,k₃

)

⁼ ^{A k}

(

³

)

⁺ ^{B k}

(

³

)

^Φ

(

^p^,k³

)

€

ip₋V₊

(

p,k₃

)

⁼

(

^Σ^o

(

^p

)

⁻ ^ik³ ^{/ 2}

)

^{A k}

(

³

)

⁺ ^{B k}

(

³

) (

^Σ^o

(

^p

)

⁺ ^ik³ ^{/ 2}

)

^Φ

(

^p^,k³

)

€

Φ

(

p,k₃

)

⁼^exp

(

⁻²ⁱ^δ

(

^p,^k³

) )

€

δ

(

p,k₃

)

⁼ ¹₂^λ^k³

∫

^dx

[

^x⁽¹⁻ ^x⁾^k³² ⁺ ^p^s²⁺ ^M²

]

^{−1/ 2} ⁼^λ^arctan

(

^k³^/2 ^p^s²⁺ ^M²

)

€

p→ ∞, V₀

(

p,k₃

)

^→^1; ^p ^→ ^0,^ip−V₊

(

p,k₃

)

^→ ⁰

€

1= A k

( )

₃ ⁺^{B k}

( )

³

€

0 =

(

m+λM −ik₃ /2

)

^exp^{( )}ⁱ^δ ^{A k}

( )

³ ⁺^{B k}

( )

³

(

^m⁺^λ^M ⁺^ik³^/2

)

^exp⁽⁻ⁱ^δ⁾

€

δ =δ

( )

k₃ ⁼^δ

(

^0,^k³

)

⁼ ^λ^arctan

(

^k³^/2^M

)

(23)

Dynamical Mass Generation

III. U(N) gauge theory, spinor quarks, the fermion version - Solutions for A(k₃) and B(k₃):

- Color singlet, scalar current correlator:

- from integral equation for vertex functions:

€

A k

( )

₃ ^,^{B k}

( )

³

( )

⁼ ¹₂

(

⁺^,⁻

)

¹₂ⁱ¹⁺

(

^k³^/2

(

^m⁺ ^λ^M

)

^tan^δ

( )

^k³

)

tanδ

( )

k₃ ⁻

(

^k³^/2

(

^m ⁺^λ^M

) )

€

J₀(x) ≡ !

ψ ⁺ψ ^!

( )

x

€

J₀

( )

k₃ ^J⁰

(

^−k³

)

^/^N ⁼ ⁻ ^d

3q 2π

( )

³

∫

^{tr S q}

{ ( )

^{V q,}

(

^k³

)

^{S q}

(

⁺^k³

) }

€

J₀( )k₃ ^J⁰(⁻^k³) ^/^N ⁼ ¹

2π

( )²^λⁱ ^d

∫

2^p^∂^p⁺^⎛_⎝^⎜¹₂^tr

{

^γ⁺^{V p,}⁽ ^k³⁾

}

^⎞_⎠^{⎟ =} _{( )}₂_π¹²_λ_i

∫

^d²^p^∂^p⁺^V⁺⁽^p,^k³⁾

€

J₀

( )

k₃ ^J⁰

(

^−k³

)

^/^N ⁼ ⁻₂¹_π ^Λ⁺ ₂^λ_π ^{i k}

(

³^/2

)

⁻ ₂¹_π ¹_λ ^m

− 1 2π

1 λ ⁻λ

⎛

⎝ ⎜ ⎞

⎠ ⎟

(

k₃/2

)

¹⁺

(

^k³^/2⁽^m⁺^λ^M⁾

)

^tan

(

^λ^arctan

(

^k³^/2M

) )

tan

(

λarctan

(

k₃/2M

) )

⁻

(

^k³^/2⁽^m⁺^λ^M⁾

)