Transcript Topological Charge Membranes, the Chiral Condensate, and Goldstone Boson Propagation Chiral Dynamics Workshop, JLAB, Aug.

Topological Charge Membranes, the Chiral
Condensate, and Goldstone Boson Propagation
Chiral Dynamics Workshop, JLAB, Aug. 6-10, 2012
Hank Thacker
University of Virginia
Numerical and theoretical evidence for topological
charge membranes in the QCD vacuum:
Lattice QCD results (Horvath, et al; Ilgenfritz,et al,2003):
 Studies of local TC distribution have revealed a “lasagna
vacuum” of extended, thin, codimension 1 membranes of coherent
TC in a laminated, alternating sign array. Analogous codimension
1 structures are observed in 2D CPN models.
Holographic QCD (Luscher, 1978; Witten, Sakai, Sugimoto,1998):
 Topological charge membranes have a natural holographic
interpretation as D6 branes of IIA string theory = “domain walls”
between k-vacua with loc    2k
Large-Nc Chiral Dynamics (Witten, Veneziano,1979):
 Chiral Lagrangian form of axial U1 anomaly is an eta-prime mass
2
term  log DetU  : ( ')2 . Periodic under  '   ' 2 , but only by a
change of branch for log  multiple discrete k-vacua separated by
domain walls. (Here  / f    is the chiral U(1) phase.)
Theta dependence, phase transitions:
Witten (1979): Large-Nc arguments require a phase
transition (cusp) at    Contradicts instanton
expansion, which gives smooth  - dependence.
E( ) = (free energy)
.
2
large Nc  


instantons  (1  cos )
This large Nc behavior conjectured from chiral
Lagrangian arguments (Witten, 1979)
Clarified by AdS/CFT duality (Witten, 1998)

 ( ) from fractionally charged Wilson loops
(Keith-Hynes and HT, PRD 2008)
CP1
CP5
 ( )
 ( )
 / 2
CP9
 / 2
large N
 ( )
Instanton gas
 / 2
--Multiple discrete k-vacua characterized by an effective
value of  which differs from the  in the action by integer
multiples of 2 = number of units of background RamondRamond flux in the holographic framework.
-- Interpretation of effective similar to Coleman’s discussion
of 2D massive Schwinger model (Coleman, 1976), where
  background E field
In 2D U(1) models (CP(N-1) or Schwinger model): Domain
walls between k-vacua are world lines of charged particles:
  0 vac
q
q
  2 vac
The emerging picture -- A “laminated” vacuum:
 Vacuum is filled with alternating sign dipole layers of topological charge.
Confinement scale set by correlation length of surface orientation vector   .
+-
+-+
+-
-+
+-
+-
+-+
+-+
-+
+++-+
-+
-+
-+
-+
Codimension-one
+++++TC dipole layers
-+
-+
-+
-+
-+
2 steps in 
The nature of the chiral condensate in this picture follows from the
observation that, in the presence of quarks,  becomes the U1
chiral field, so the axial vector current is given by J 5   . This
gives a delta-function on the brane surface.
Conclusion: condensate consists of surface quark
modes on the membranes, with left and right chiral
densities  1   5  living on topological charge sheets
on opposite sides of each membrane.

Membranes are sheets of magnetic RR charge:
A fields on opposite sides of brane must be matched by a
topologically nontrivial gauge transformation g defined on the
brane surface.
The gauge transf on the brane surface can be separated into a 3D
“in brane” transformation and the transf of the A component
transverse to the brane.
CL
CR
The 3D in-brane g.t. represents currents
on the brane surface.
g action on transverse A component
represents transverse fluctuations of
brane surface.

Wu-Yang matching
defines gauge transf.
g on brane
Currents on surface are related to
transverse motion of surface by 4D
gauge invariance.
Brane dynamics, the descent eqns. and anomaly inflow:
Topological charge density is a total divergence Q   K 
where K  is the Chern-Simons current
    2   
K     Tr  A  A  A A A     K 3


3
Integral over 4D k-vacuum bubble reduces to 3D
integral of K 3 over membrane surface.
Effective action for membrane given by integral of
gauge variation of Chern-Simons 3-form  K 3 over
(2+1)D brane surface.

Note that K3 is not gauge invariant. But (Faddeev, 1984)
 Q   dK 3   d  K 3   0
So locally we can write  K 3 as a total deriv. of a 2-form
 K 3  dK 2
But K2 is a WZW 2-form Tr  g 1 g g 1 g K where
g  ei is the gauge transformation that defines  K 3 .
Integral of K2 over a closed 2-surface is ambiguous
mod 2 . Depends on winding number of g in enclosed
3-volume = units of boundary current on 2-surface.
Anomaly inflow idea: Chiral anomaly  J 5  0 is
interpreted as current disappearing from the bulk
theory but reappearing as a subdimensional boundary
current, so that
  J 5     J 5 
0
bulk
boundary
In QCD the bulk AV current is the chiral current   .
The boundary current on the brane is also chiral current,
but it mixes with the Chern-Simons current K  via qq
annihilation.
The anomaly inflow constraint at the brane surface is
implemented by requiring   to transform under a
color gauge transformation in a way that cancels the
variation of K  :
     K      
This defines the chiral field  , with source  K  = CS
fluctuations on brane surface.
CS brane
 K
 K
chiral q-qbar current
 t  0  massless pole in K  K . This is cancelled by
massless Goldstone pole in    . (4D Kogut-Susskind)
Pion propagation and membrane surface states:
In Nf = 1 QCD, quarks occupy 2D surface states and 
becomes the chiral U(1) field  ' . Because of the
repeated qq annihilations induced by  K  , the  
acquires a mass.
Without annihilation,  is a massless excitation :


       bulk     boundary  0
The boundary term represents the flow of the condensate.
The anomaly inflow constraint    K  still applies, but
now  g on brane does not change winding number, so
 K     K2 is globally true:   K   0
  0
Chiral eigenmodes are pairs of Landau orbitals on
opposite sides of brane surface, described by 2+1 D
Chern-Simons theory:
qL
qR
Surface modes can be interpreted as approximate
topological zero modes associated with TC dipole layer.
Summary:
 The QCD vacuum is a condensate of codimension one
membranes where the value of  jumps by 2 across the brane.
A brane is similar to a sheet of magnetic monopoles, in that the A
fields on the two sides of the brane must be matched together by a
topologically nontrivial gauge transformation on the brane surface.
The matching gauge transformation describes surface color
currents g 1 g .
 Variations of the matching gauge transformation correspond to
transverse fluctuations of the brane. The effective action of the
brane is given by the gauge variation of the Chern-Simons 3-form
on the surface (the “world sheet” action).
 In pure YM without quarks, the bulk field  is sourced by the
Chern-Simons current,    K  . For Nf = 1,  becomes the
U(1) chiral field  . The axial current   includes delta-function
surface currents on the branes. The chiral condensate is formed
from surface eigenmodes similar to 2D Landau levels in a
transverse magnetic field.
 In the absence of qq annihilation, gauge invariance and the
anomaly inflow definition of  constrain the combination of surface
and bulk contributions to   to be conserved, giving a massless
Goldstone boson as a propagating surface wave along the brane.
Future:
The description of the chiral condensate in terms of
surface modes of a Chern-Simons/WZW theory defines
a direct connection between QCD and the EFT that
describes the chiral condensate. This should provide
relations between fundamental low energy constants
which will test the picture.
e.g. relation between f and  t .