Non-Abelian strings and monopoles in supersymmetric gauge theories - - PowerPoint PPT Presentation

Non-Abelian strings and monopoles

in supersymmetric gauge theories Mikhail Shifman and Alexei Yung

1 Introduction

Seiberg and Witten 1994 : Abelian confinement in N = 2 QCD Cascade gauge symmetry breaking:

• SU(N)→ U(1)N−1

VEV’s of adjoint scalars

• U(1)N−1 → 0 (or discrete subgroup)

VEV’s of quarks/monopoles At the last stage Abelian Abrikosov-Nielsen-Olesen flux tubes are formed. π1(U(1)N−1) = ZN−1 (N −1) infinite towers of strings. In particular (N −1) elementary strings

→ Too many degenerative hadron states

In search for non-Abelian confinement non-Abelian strings were suggested in N = 2 U(N) QCD Hanany, Tong 2003 Auzzi, Bolognesi, Evslin, Konishi, Yung 2003 Shifman Yung 2004 Hanany Tong 2004 ZN Abelian string: Flux directed in the Cartan subalgebra, say for SO(3) = SU(2)/Z2 flux ∼ τ3 Non-Abelian string : Orientational zero modes Rotation of color flux inside SU(N).

2 Bulk theory

N = 2 QCD with gauge group U(N) = SU(N) × U(1) and Nf = N flavors of fundamental matter – quarks + Fayet-Iliopoulos term of U(1) factor The bosonic part of the action S =

• d4x

1

4g2

2

• F a

µν

2 + 1

4g2

1

(Fµν)2 + 1 g2

2

|Dµaa|2 + 1 g2

1

|∂µa|2 +

• ∇µqA
• 2 +
• ∇µ¯

˜ q

A

• 2 + V (qA, ˜

qA, aa, a)

• .

Here ∇µ = ∂µ − i 2 Aµ − iAa

µ T a .

The potential is V (qA, ˜ qA, aa, a) = g2

2

2

i

g2

2

f abc¯ abac + ¯ qA T aqA − ˜ qAT a ¯ ˜ q

A

2

+ g2

1

8

• ¯

qAqA − ˜ qA¯ ˜ q

A − Nξ

2

+ 2g2

2

• ˜

qAT aqA

• 2 + g2

1

2

• ˜

qAqA

• 2

+ 1 2

N

• A=1
• (a +

√ 2mA + 2T aaa)qA

• 2

+

• (a +

√ 2mA + 2T aaa)¯ ˜ q

A

• 2

.

Vacuum Φ = 1 2 a + T a aa = − 1 √ 2

     

m1 . . . . . . . . . . . . . . . mN

      ,

For special choice m1 = m2 = ... = mN U(N) gauge group is classically unbroken. qkA =

• ξ

     

1 . . . . . . . . . . . . . . . 1

      ,

¯ ˜ q

kA = 0,

k = 1, ..., N A = 1, ..., N ,

Note

• Color-flavor locking

Both gauge U(N) and flavor SU(N) are broken, however diagonal SU(N)C+F is unbroken q → UqU −1 a → UaU −1

• Two ways to make it valid in quantum regime:
• Nf > 2N The theory is not assymptotically free and stays at

weak coupling (Argyres, Plesser and Seiberg,1996).

• Another way to stay at weak coupling:
• ξ ≫ Λ

8π2 g2

2(ξ) = N log

√ξ Λ ≫ 1

3 Non-Abelian strings

ZN string solution q =

           

φ2(r) ... ... ... ... ... ... φ2(r) ... eiαφ1(r)

           

, ASU(N)

i

= 1 N

           

1 ... ... ... ... ... ... 1 ... −(N − 1)

           

(∂iα) [−1 + fNA(r)] ,

AU(1)

i

≡ 1 2 Ai = 1 N (∂iα) [1 − f(r)] Magnetic U(1) flux of this ZN string is

• d2xF12 = 4π

N

BPS string. First order equations r d dr φ1(r) − 1 N (f(r) + (N − 1)fNA(r)) φ1(r) = 0 , r d dr φ2(r) − 1 N (f(r) − fNA(r)) φ2(r) = 0 , −1 r d drf(r) + g2

1N

4

• (N − 1)φ2(r)2 + φ1(r)2 − Nξ
• = 0 ,

−1 r d drfNA(r) + g2

2

2

• φ1(r)2 − φ2(r)2

= 0 . Tension of the elementary ZN string T = 2π ξ

Profile functions of the string (for N = 2)

f φ φ

3 1

f

2

r 1

Non-Abelian string 1 N

                      

U

           

1 ... ... ... ... ... ... 1 ... −(N − 1)

           

U −1

                      

l p

= −nln∗

p + 1

N δl

p ,

with n∗

l nl = 1

Then q = 1 N [(N − 1)φ2 + φ1] + (φ1 − φ2)

• n · n∗ − 1

N

• ,

ASU(N)

i

=

• n · n∗ − 1

N

• εij

xj r2 fNA(r) , AU(1)

i

= 1 N εij xj r2 f(r) ,

4 CP(N) model on the string

String moduli: x0i, i = 1, 2 and nl, l = 1, ..., N Make them t, z-dependent ZN solution breaks SU(N)C+F down to SU(N − 1) × U(1) Thus the

• rientational moduli space is

SU(N) SU(N − 1) × U(1) ∼ CP(N − 1) S(1+1) = 2β

• dt dz
• (∂k n∗∂k n) + (n∗∂k n)2

where the coupling constant β is given by a normalizing integral β = 2π g2

2

∞

dr

• − d

drfNA +

2

r f 2

NA + d

drfNA

φ2

1

φ2

2

• = 2π

g2

2

Gauge theory formulation of CP(N − 1) model SCP(N−1) = 2β

• d2x|∇knl|2,

where ∇k = ∂k − iAk N complex fields nl, l = 1, ..., N Constraint: |nl|2 = 1. Gauge field can be eliminated: Ak = − i 2 ¯ nl

↔

∂k nl Number of degrees of freedom = 2N − 1 − 1 = 2(N − 1)

5 Confined monopoles

Classical picture ZN Abelian strings ⇐ ⇒ N classical vacua of CP(N − 1) model ASU(N)

i

=

• n · n∗ − 1

N

• εij

xj r2 fNA(r) CP(N − 1) classical vacua: nl = δll0 Higgs phase for quarks = ⇒ confinement of monopoles Elementary monopoles – junctions of two ZN strings monopole flux = 4π × diag 1

2{...0, 1, −1, 0, ...}

In 2D CP(N − 1) model on the string we have N vacua = N ZN strings and kinks interpolating between these vacua

Kinks = confined monopoles

string 1 string 2 vacuum 2 vacuum 1 4D 2D kink monopole

Quantum picture Non-Abelian limit m1 = m2 = ... = mN Φ = 1 2 a + T a aa = − 1 √ 2

     

m1 . . . . . . . . . . . . . . . mN

      ,

t’Hooft-Polyakov unconfined monopoles: Mmonopole = 4π|ml0+1 − ml0| g2

2

→ 0 monopole size ∼ 1/mW ∼ 1/|ml0+1 − ml0| → ∞ Classically monopole disappear

Confined monopoles = kinks are stabilized by quantum (non-perturbative) effects in CP(N − 1) model on the string worldsheet Classically nl develop VEV (|n|2 = 1) There are 2(N − 1) massless Goldstone states. In quantum theory this does not happen SU(N)C+F global symmetry is unbroken Mass gap ∼ ΛCP no massless states (|n|2 = 0) Mmonopole = Mkink ∼ ΛCP monopole size ∼ Λ−1

CP

ξ 1/2

• Λ -1

6 Less supersymmetry

• bulk N = 2 SUSY =

⇒ N = (2, 2) CP(N − 1) on the string Hanany, Tong 2003 Auzzi, Bolognesi, Evslin, Konishi, Yung 2003 Shifman, Yung 2004 Hanany, Tong 2004

• bulk N = 1 SUSY =

⇒ N = (0, 2) hetrotic CP(N − 1) on the string Edalati, Tong 2007 Tong 2007 Shifman, Yung 2008 Shifman, Yung 2008

• bulk non-SUSY =

⇒ non-SUSY CP(N − 1) on the string Gorsky, Shifman, Yung 2004 Gorsky, Shifman, Yung 2005

7 Large N solutions

Witten 1979: solved N = (2, 2) and non-SUSY CP(N − 1) in large N appreoximation Shifman, Yung 2008: generalized Witten’s solution to N = (0, 2) CP(N − 1) N = 2 Bulk = ⇒ N = (2, 2) CP(N − 1) Consider limit e2 → ∞ in S1+1 =

• d2x
• |∇knl|2 + 1

4e2F 2

kl + 1

e2|∂kσ|2 + 1 2e2D2 + 2|σ|2|nl|2 + iD(|nl|2 − 2β) + fermions

• Complex scalar σ, Ak and D form gauge multiplet.

The model has U(1) axial symmetry which is broken by the chiral anomaly down to discrete subgroup Z2N (Witten 1979). The field σ which is related to the fermion bilinear operator transforms under this symmetry as σ → e

2πk N iσ,

k = 1, ..., N − 1. Z2N symmetry is spontaneously broken by the condensation of σ down to Z2, Witten’s solution: Evac = D = 0 = ⇒ N = (2, 2) SUSY is unbroken

√ 2σ = ΛCP e

2πk N i

k = 0, ..., N − 1.

There are N strictly degenerate vacua

σ

σ ∼ ¯ ψLψR

N = 1 Bulk = ⇒ N = (0, 2) CP(N − 1) Bulk: W3+1 = µ 2

• A2 + (Aa)2

, String: Consider limit e2 → ∞ in S1+1 =

• d2x
• |∇knl|2 + 1

4e2F 2

kl + 1

e2|∂kσ|2 + 1 2e2D2 + 2|σ|2|nl|2 + iD(|nl|2 − 2β) + N 2π u |σ|2 + fermions

• u is the deformation parameter N = (2, 2) → N = (0, 2) ,

u =

  

const β g4

2|µ|2

m2

W ,

small µ const β log g2

2|µ|

mW ,

large µ

Solution: 2|σ|2 = Λ2 e−u , iD = Λ2 1 − e−u . Vacuum energy Evac = N 4π iD = N 4π Λ2 1 − e−u .

u Λ

2

2|σ| iD

2

SUSY is broken spontaneously

√ 2σ = Λ e

2πk N i e−u/2

k = 0, ..., N − 1.

There are N strictly degenerate vacua

σ

non-SUSY CP(N − 1) σ = 0

N vacua split

Evac = const N Λ2

CP

  1 + const 2πk

N

2  

8 Kink deconfinement vs confinement

Compare large N solutions of N = (2, 2) , N = (0, 2) and non-SUSY models Common features:

• nl ( and superpartners) acquire mass ∼ Λ
• |nl|2 = 0
• Ak, σ (+ fermions) become dynamical (kinetic energy is generated)

Distinctions:

• N = (2, 2) : SUSY unbroken
• N = (0, 2) : SUSY spontaneously broken
• N = (2, 2) and N = (0, 2) : N degenerate vacua, Z2N → Z2

non-SUSY CP(N − 1): N vacua are split

Kinks = confined monopoles

N = (2, 2) and N = (0, 2) models :

string 1 string 2 vacuum 2 vacuum 1 4D 2D kink monopole

Witten 1979: nl = kink

nl = kink Means that kink acquire global flavor quantum numbers

kinks are in fundamental of SU(N)C+F

Therefore monopole-antimonopole ’meson’ can be singlet or adjoint of flavor SU(N)C+F

N = (2, 2) and N = (0, 2) models: U(1) gauge field Ak is massive → kink deconfinement non-SUSY model: U(1) gauge field Ak is massless → kink confinement

kink anti−kink T T T 1 monopole anti−monopole / /

2D kink confinement = splitting of N vacua T = 2πξ + const N Λ2

CP

  1 + const 2πk

N

2  

9 Conclusions

• Worldsheet internal dynamics of non-Abelian string in U(N) gauge

theory with Nf = N flavors is described by CP(N − 1) model

• N = 2 Bulk =

⇒ N = (2, 2) CP(N − 1) N = 1 Bulk = ⇒ N = (0, 2) heterotic CP(N − 1) non-SUSY Bulk = ⇒ non-SUSY CP(N − 1)

• Non-Abelian confined monopole = CP(N − 1) kink

Stubilized by quantum dynamics on the string at the scale ΛCP

• N = (2, 2) and N = (0, 2) : 2D deconfinement – monopoles can

move freely along the string non-SUSY: 2D confinement – monopoles and antimonopoes form a ’meson’-like configuration on the string