Gauge-fixing with Compact Lattice Gauge Fields

ASIT K DE with Mugdha Sarkar (SINP, Kolkata, India) 24 July 2018

36th International Symposium on Lattice Field Theory (LATTICE 2018) MSU, Lansing, MI, USA (22-28 July 2018) Intro & Background • Gauge fixing not necessary for Wilson’s standard lattice gauge theory with compact algebra-valued gauge fields

• Problems in Chiral gauge theories with manifestly local lattice fermions

• These lattice fermions break chiral symmetry explicitly on lattice Lattice chiral gauge theories break gauge invariance

• Functional integration takes place also along the gauge orbit (i.e. in the longitudinal direction or the direction of gauge transformation)

Longitudinal gauge degrees of freedom (lgdof) couple to the physical degrees of freedom in the terms that break gauge invariance

2 For example A  U + U †  † U + † U † µ ⇠ xµ x+µ ! x xµ x+µ x+µ xµ xµ ⇣ ⌘ ⇣ ⌘ where G are radially frozen (random) scalar fields x 2 Hence a theory with gauge-non-invariant terms

Physical fields + lgdof (scalar fields)

The new action S ( x† Uxµx+µ) is now gauge-invariant under the extended local transformations:

U h U h† , h ,h G xµ ! x xµ x+µ x ! x x x 2

S (U ) S (U , ) V xµ ⌘ H xµ x

The reduced model is defined by the action on the trivial orbit: Uxµ † I x+µ ! x Reduced model has a global symmetry: h with h G x ! x 2 Early proposals of Lattice Chiral Gauge Theory

• Smit-Swift / Wilson-Yukawa model • Domain-wall waveguide model all failed because of interaction of the lgdofs with fermions.

Remedy: Gauge fix the lgdofs to control their dynamics

Gauge invariant observables remains intact provided the following integral C C B exp S [U ,C,C,B] D D D D GF µ Z over an orbit is a non-zero constant. A no-go theorem for compact gauge fields Neuberger 1987

Start with a manifestly gauge invariant lattice gauge theory, the Wilson way: Z = U exp( S [U]) D GI Z One then inserts, a la Fadeev-Popov: Z = C C B exp S [U ,C,C,B] GF D D D D GF µ Z S = 2t i tr (BF(U)) + tr C F (U) + ⇠g2tr(B2) GF BRST x x X X ⇥ 2 2⇤ = 2t BRSTtr CF(U) + ⇠g tr(B ) x x X ⇥ ⇤ X ZGF is required to be independent of U , so that only a constant was inserted in Z

If this requirement is fulfilled, gauge-invariant correlation functions of the gauge-fixed theory are identical to those of the unfixed (manifestly gauge invariant) theory. dZ GF = C C B 2tr(CF(U)) dt D D D D BRST " x # Z X =0

Indeed ZGF t=1 = ZGF t=0 independent of U | |

But Z =0 GF|t=0 because the integrand is then devoid of any Grassmann variables

Z = Z =0 ) GF|t=1 GF Expectation value of any gauge-invariant operator has the indeterminate form

If pure Yang Mills on lattice cannot be non-perturbatively gauge fixed, there would be no hope for lattice Chiral Gauge Theories in the gauge-fixing approach. Gauge-fixing with compact gauge fields

Cannot maintain BRST

Need to evade Neuberger’s no-go theorem

For U(1) lattice gauge theory: Break BRST explicitly, restore it by tuning counter-terms in the continuum limit and in the process decouple the lgdofs

For non-Abelian lattice gauge theory (e.g. SU(2)): Modify BRST to gauge-fix only the coset space leaving a subgroup (U(1)) gauge invariant —> eBRST formalism U(1) Lattice gauge theory with HD gauge fixing term Golterman & Shamir 1997

1 S = SW + SGS + Sct S = (1 Re U (x)) W g2 Pµ⌫ x, µ<⌫ X 2 S =˜ (U) (U) B Sct =  Uxµ + U † GS ⇤xy ⇤yz x xµ xyz x ! xµ X X X

2 ⇤xy(U)= (y,x+µUxµ + y,x µUx† µ,µ 2xy) Bx = ( x µ,µ + xµ) /4, with xµ =ImUxµ A A A µ µ X X

The action has an unique absolute minimum at Uxµ =1

In the naive continuum limit, the HD gauge-fixing term

2 4 2 4 2 2 ˜g d x(@µAµ) =(1/2⇠) d x(@µAµ) where ⇠ =1/(2˜g ) Z Z Weak coupling perturbation theory near g =0, ˜ with ⇠ !11 ⇠ What happens near the WCPT point?

Expand action in powers of g , using constant field approximation:

g6 V (A )= g2 A2 + ... + ˜ A2 A4 + ... cl µ µ 2 µ µ µ ! ( µ ! µ ! ) X X X

Critical surface:  FM FMD(g, ˜)=0 ⌘ where the gauge boson (photon) is rendered massless

1/4  FM FMD gAµ = | | , µ for  < FM FMD h i ± 6˜ 8 ✓ ◆ gAµ =0, µ for  FM FMD h i 8

The phase with the vector condensate is called the FMD phase

Continuum limit is to be taken at FM-FMD transition from within the FM phase where gauge symmetry is recovered and as a result the lgdofs decouple

The above picture is validated very well for weak gauge couplings Observables for numerical study

2 1 1 1 1 EP = Re UPµ⌫ (x) E = Re Uxµ V = Im U 6L4 4L4 v4 L4 xµ *x,µ<⌫ + * x,µ + *u µ x ! + X u X X X t

In addition, photon and scalar propagators in momentum space

κ~ 0 0.1 0.2 0.3 0.4 0.5 0 FM

-0.5 PM For weak gauge couplings g 1 -1 FMD for large enough ˜  -1.5 κ WCPT line -2 g = 1.0 FM-FMD approaching FM-FMD from FM side 4 FM-PM -2.5 AM 16 lattice PM-AM reveals free massless photons and FMD-PM -3 FMD-AM the lgdofs appear to be decoupled -3.5

Investigations of U(1) Wilson-Yukawa and domain-wall waveguide models in the reduced limit exhibit free chiral fermions What happens at large gauge coupling? AKD & Mugdha Sarkar 2016, 2017 3.5 50

0.8 ~ 3 κ = 0.6 40 ~ 0.6 g = 1.3 ; κ = 0.7 κ = -1.050 0.4 2.5 30 -0.940 0.2 -1 κ ↓-0.980 2 -1.040 20 Gµµ 0 0 0.2 0.4 0.6 0.8 -1 Hµµ 1.5 10

∆κ = κ - κc = 0.03 1 0 g = 1.3 κ~ = 0.7 κ~ = 0.6 3 0.5 8324 lattice -10 8 24 lattice κ~ = 0.5 0 -20 0 1 2 3 4 5 0 1 2 2 3 4 5 p^2 p^

0.3 -12.5 ~ 4 κ = 0.40, κ = -0.995 κ~ 8 MM -12.6 g = 0.6, = 0.45 4 0.25 10 MM g = 1.3 4 κ = -0.210 10 HMC -12.7 0.2

> -12.8

χ 0.15 – χ ~ < κ = 0.50, κ = -1.010 -12.9 0.1 -13 4 0.05 16 Average action density -13.1 124 0 -13.2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1000 2000 3000 4000 5000 6000 7000 8000 m0 sweep Main Conclusions of the HD Abelian Gauge-fixing at large gauge couplings

g 0.0 0.5 g =0, 1.0 ˜ = 1 1.5 0 I III The physics at the large bare -1 • II gauge couplings (free massless -2 k IV V photons with lgdofs decoupled) -3 appears to be the same as in -4 the weak gauge couplings and 0.0 0.5 is controlled by the perturbative 1.0 é 1.5 point g =0, ˜ k !1

• The tricritical line appears to be the only candidate for non-trivial physics • Multihit Metropolis (MM) fails to produce faithful field configurations at larger values of ˜ , and HMC appears to do much better Gauge fixed Yang Mills theory on lattice Schaden 1999; Golterman & Shamir 2004, 2006, 2013, 2014

One way to evade the no-go theorem: Introduce equivariant BRST (eBRST) Gauge fix only the coset space leaving minimally the Cartan subgroup unfixed

For example, for the SU(2) Yang Mills theory, gauge fix SU(2)/U(1) and the U(1) gauge invariance is left intact

Nilpotency of BRST is modified and a 4-ghost term appears

In addition to its importance in regard to

• lattice chiral gauge theory • an alternate formulation of gauge theories it is interesting to ask: if non-perturbatively the dynamics of the longitudinal sector can affect the physics of the transverse degrees of freedom, given that g andg ˜ are both asymptotically free (˜g2 = ⇠g2) eBRST: SU(2)/U(1) case Uµ =exp(iVµ)

BRST eBRST

Vµ = Wµ1⌧1 + Wµ2⌧2 + Aµ⌧3 Vµ = Wµ1⌧1 + Wµ2⌧2 + Aµ⌧3

C = C1⌧1 + C2⌧2 + C3⌧3 C = C1⌧1 + C2⌧2

s = iC s = iC

sWµ = µ(A)C sVµ = µ(V )C D D sAµ = i[Wµ,C] 2 2 sC = iC sC = iC SU(2)/U(1) =0 | sC = iB sC = iB sB =0 sB =[iC2, C]

2 2 s =0 s = U(1) The eBRST-invariant gauge-fixing action

1 1 ˜ F µ(A)Wµ ˜ = = = ⇠ D 2⇠g2 2˜g2 2

The gauge fixing partition function, a theory on the gauge orbit, does not depend on U andg ˜2 = ⇠g2 Now, because of the presence of the 4-ghost term, it is also not zero, thus evading the no-go For any U, Z (U, ⇠g2) =0 defines a TFT GF 6

INVARIANCE THEOREM (U) = (U) hO iunfixed hO ieBRST Restricting to expectation values of gauge-invariant operators, the eBRST gauge-fixed theory is rigorously equivalent to the unfixed theory Now go to the trivial orbit ( U =1 ) of the eBRST theory

That is the Reduced Model discussed at the beginning, consisting of the lgdofs and the ghost fields, still symmetric under eBRST, local U(1) and now a global SU(2) symmetry (more on this in text talk by Mugdha Sarkar)

Can there be SSB in a TFT? SU(2) U(1) ? global ! global If yes, what is its effect on the full eBRST theory? Is eBRST broken too?

We have implemented the program on the lattice for numerical simulation - a very hard problem

At the very preliminary level, we find evidence for the breaking of the global SU(2) to U(1) in the reduced theory.

What this means for the full eBRST theory is still unclear, but our very preliminary results are so far consistent with the eBRST theory going into a Higgs-like phase with the W and ghost fields appearing to acquire mass. eBRST is probably left unbroken since this symmetry can allow equal mass terms for coset gauge fields and ghosts