# The Instanton-Dyon Liquid Model V:

Twisted Light Quarks

###### Abstract

We discuss an extension of the instanton-dyon liquid model that includes twisted light quarks in the fundamental representation with explicit symmetry for the case with equal number of colors and flavors . We map the model on a 3-dimensional quantum effective theory, and analyze it in the mean-field approximation. The effective potential and the vacuum chiral condensates are made explicit for . The low temperature phase is center symmetric but breaks spontaneously flavor symmetry with massless pions. The high temperature phase breaks center symmetry but supports finite and unequal quark condensates.

###### pacs:

11.15.Kc, 11.30.Rd, 12.38.Lg## I Introduction

In the QCD ground state confinement and chiral symmetry breaking are intertwined as lattice simulations have now established LATTICE . The loss of confinement with increasing temperature as described by a jump in the Polyakov line is followed by a rapid cross-over in the chiral condensate for flavors. When the quarks are in the adjoint representation, the cross over occurs much later than the deconfinement transition. There is increasing lattice evidence that the topological nature of the underlying gauge configurations maybe key in understanding some aspects of these results CALO-LATTICE .

This work is a continuation of our earlier studies LIU1 ; LIU2 ; LIU3 ; LIU5 of the gauge topology using the instanton-dyon liquid model. The starting point of the model are the KvBLL instantons threaded by finite holonomies and their splitting into instanton-dyon constituents KVLL , with strong semi-classical interactions DP ; DPX ; LARSEN . At low temperature, the phase preserves center symmetry but breaks spontaneously chiral symmetry. At sufficiently high temperature, the phase restores both symmetries as the constituent instanton-dyons regroup into topologically neutral instanton-anti-instanton molecules. The importance of fractional topological constituents for confinement was initially suggested through instanton-quarks in ARIEL , and more recently using bions in UNSAL .

The instanton-dyons carry fractional topological charge and are able to localize chiral quarks into zero modes. For quarks in the fundamental representation, as the KvBLL fractionate, the zero-mode migrates to the heavier instanton-dyon constituent KRAAN . The random hopping of these zero modes in the instanton-dyon liquid is at the origin of the spontaneous breaking of chiral symmetry as has been shown both numerically SHURYAK1 ; SHURYAK2 and using mean-field methods LIU2 . In supersymmetric QCD some arguments were presented in TIN .

At finite temperature the light quarks are subject to anti-periodic boundary conditions on to develop the correct occupation statistics in bulk. General twisted fermionic boundary conditions on amounts to thermal QCD with Bohm-Aharanov phases that alter fundamentally the nature of the light quarks RW ; RMT . A particularly interesting proposal consists of a class of twisted QCD boundary conditions with resulting in a manifestly symmetric QCD dubbed -QCD JAP . The confined phase is both center and chiral symmetric eventhough the boundary conditions are flavor breaking. The deconfined phase is center and chiral symmetry broken JAP ; TAKUMI .

The purpose of this paper is to address some aspects of twisted fermionic boundary conditions in the context of the instanton-dyon liquid model. Since the localization of the zero-modes on a given instanton species is very sensitive to the nature of the the twist on , this deformation offers an insightful tool for the possible understanding of the fundamental aspects of the spontaneous breaking of chiral symmetry through the underlying topological constituents. Similar issues were addressed using PNJL models JAP and more recently monopole-dyons and without anti-monopole-dyons for small THOMAS . A numerical analysis in the the instanton-dyon liquid model with was recently presented in LARS3 .

In section 2 we briefly review the model and discuss the general case of twisted boundary conditions. The special cases of are given and the corresponding normalizable zero-modes around the center symmetric point constructed. We derive explicitly the pertinent hopping matrices between the instanton-dyons and the instanton-anti-dyons for the case of which are central to the quantitative study of the spontaneous breaking of chiral symmetry. In section 3 we use a series of fermionization and bosonization transformations to map the instanton-dyon partition function on a 3-dimensional effective theory. For , additional discrete symmetries combining charge conjugation and exchange between conjugate flavor pairs are identified, with the same chiral condensates at high temperature. In section 4 we derive the effective potential for the ground state of the 3-dimensional effective theory. We explicitly show that it supports a center symmetric state with spontaneously broken chiral symmetry. The center asymmetric phase at high temperature supports unequal chiral condensates. Our conclusions are in section 5.

## Ii Effective action with twisted fermions

### ii.1 General setting

For simplicity we detail here the general setting for . The pertinent changes for any will be quoted when appropriate. For a fixed holonomy with and , the SU(2) KvBLL instanton KVLL is composed of a pair of instanton-dyons labeled by L, M (instanton-anti-dyons by ). In general, there are are BPS instanton-dyons and only one twisted instanton-dyon. As a result the global gauge symmetry is reduced through .

For example, the grand-partition function for dissociated KvBLL instantons and anti-instantons and massless flavors is

Here and are the 3-dimensional coordinate of the i-dyon of m-kind and j-anti-dyon of n-kind. Here a matrix and a matrix whose explicit form are given in DP ; DPX . is the streamline interaction between dyons and antidyons as numerically discussed in LARSEN . For the SU(2) case it is Coulombic asymptotically with a core at short distances LIU1 . We will follow our original discussion with light quarks in LIU2 , with the determinantal interactions in (LABEL:SU2) providing for an effective core repulsion on average. We omit the explicit repulsion between the cores as in LIU5 , for simplicity. The fugacities are related to the overall instanton-dyon density, and can be estimated using lattice simulations CALO-LATTICE . Here they are external parameters, with a dimensionless density

(2) |

For definiteness, the KvBLL instanton action to one-loop is

(3) |

The fermionic determinant with twisted quarks will be detailed below. In many ways (LABEL:SU2) resembles the partition function for the instanton-anti-instanton ensemble ALL .

### ii.2 Twisted boundary conditions and normalizable zero modes

Consider QCD on with the following anti-periodic boundary conditions modulo a flavor twist in the center of

(4) |

with and respectively. Under a twisted gauge transformation of the type

(5) |

(4) is symmetric following the flavour relabeling . As a result the theory is usually referred to as -QCD JAP . In contrast, (4) breaks explicitly chiral flavor symmetry through

(6) |

To construct explicitly the fermionic zero modes in a BPS or KK dyon with the twisted boundary conditions (4), we consider the generic boundary condition

(7) |

and redefine the quark field through . The latter satisfies a modified Dirac equation with an imaginary chemical potential RW ,

(8) |

In a BPS dyon with periodic boundary conditions, the solution to (8) asymptote

(9) |

which is normalizable for . For anti-periodic boundary condition, the requirement for the existence of a normalizable zero mode in a BPS dyon is .

### ii.3 Case:

For , the flavor twisted boundary condition (4) takes the explicit form

(10) |

The d,s boundary conditions in (II.3) admit a discrete symmetry under the combined charge conjugation and the flavor exchange .

The normalizability condition for the quark zero modes following from the flavor twisted boundary conditions in (8-9) shows that always support a normalizable KK zero mode, while support BPS zero modes that are at the edge of the normalizability domain in the symmetric phase with . The BPS modes carry a time dependence of the form as , while the KK mode carries a time dependence of the form . In both cases, we are restricting the modes to the lowest frequencies in Euclidean -time, for simplicity. This means a moderatly large temperature ranging from the center symmetric to asymmetric phase.

The explicit form of the twisted zero modes in a BPS dyon and satisfying the twisted boundary condition (7) can be obtained in closed form in the hedgehog gauge,

(11) |

in color-spin, with and

Here and refers to respectively. Asymptotically, the BPS zero modes take the compact form in the hedgehog gauge

(13) |

For the KK instanton-dyon, we recall the additional time-dependent gauge transformation from the BPS instanton-dyon. The explicit form of the zero modes are also similar (11-II.3) with now . We note that for the flavor twisted boundary condition (4), corresponds to (mod ) in (II.3) which are not normalizable BPS zero modes at exactly . Following our analysis in LIU5 , we choose to regulate the zero modes by approaching the holonomies in the center symmetric phase as follows ()

(14) |

As a result, the M1-instanton-dyon carries 2 zero modes (d,s), the M2-instanton-dyon carries none, and the L-dyon carries 1 zero mode (u). This regularization enforces the Nye-Singer index theorem for fundamental quarks NS and the discrete symmetry noted earlier.

### ii.4 Case:

For the case of , a more general set of twisted boundary conditions will be analyzed with

(15) |

which is (4) for . (II.4) is seen to have the additional discrete symmetry when and at . Thus, only the range will be considered. In this case, the M-instanton-dyon carries 1 zero-mode (d), while the L-instanton-dyon carries 1 zero-mode (u). For (II.4) the normalizable zero modes are asymptotically of the form (II.3) with .

For completeness we note the Roberge-Weiss boundary condition RW

(16) |

In the range , the M-instanton-dyon carries 2 zero modes with none on the L-instanton-dyon. In the range they jump back on the M-instanton-dyon. We note that for with , the M-zero mode moves to be an L-zero mode with the asymptotic , the 2 zero modes jump onto the L-instanton-dyon. In the range

(17) |

This is to be compared to the case with with the asymptotic

(18) |

### ii.5 Twisted fermionic determinant

The fermionic determinant can be viewed as a sum of closed fermionic loops connecting all instanton-dyons and instanton-antidyons. Each link – or hopping – between an instanton-dyon and -anti-instanton-dyon is described by the hopping chiral matrix

(19) |

Each of the entries in is a “hopping amplitude” of a fermionic zero-mode from an instanton-dyon to a zero-mode (of opposite chirality) of an instanton-anti-dyon

with , and similarly for the other components. In the hedgehog gauge, these matrix elements can be made explicit in momentum space. Their Fourier transform is

(21) |

with the contribution from the lowest Matubara mode retained. We recall that the use of the zero-modes in the string gauge to assess the hopping matrix elements, introduces only minor changes in the overall estimates as we discussed in LIU2 (see Appendix A).

#### ii.5.1 Case

The key physics in the Fourier transforms is captured by retaining only the flux-induced mass-like in the otherwise massless asymptotics, i.e.

(23) |

The i-assignments are respectively given by

(24) |

In the center symmetric phase with , (II.5.1) are long-ranged for the M-instanton-dyons,

(25) |

Here is a normalization constant fixed by the regularization detailed in (II.3).

#### ii.5.2 Case

(26) |

The correponding hopping matrix is ()

(27) |

with the assignments

(28) |

and

(29) |

It follows that

(30) |

Using (17-18) we note that the hopping matrix element (30) satisfies the anti-periodicity condition

(31) |

with the -argument exhibited for clarity.

## Iii Su() ensemble

Following DP ; LIU1 ; LIU2 the moduli determinants in (LABEL:SU2) can be fermionized using pairs of ghost fields for the instanton-dyons and for the instanton-anti-dyons. The ensuing Coulomb factors from the determinants are then bosonized using boson fields for the instanton-dyons and similarly for the instanton-anti-dyons. The result is

(32) |

For the streamline interaction part , we note that as a pair interaction in (LABEL:SU2) between the instanton-dyons and instanton-anti-dyons, it can be bosonized using standard methods POLYAKOV ; KACIR in terms of and fields. As a result each dyon species acquire additional fugacity factors of the form

(33) |

with and the ith root of the Lie generator, and its affine root due to its compacteness. Therefore, there is an additional contribution to the free part (32)

(34) |

where for simplicity we approximated the streamline by a Coulomb interaction, and the interaction part is now

without the fermions. We now show the minimal modifications to (LABEL:FREE3) when the fermionic determinantal interaction is included.

### iii.1 Fermionic fields

To fermionize the determinant in (LABEL:SU2) and for simplicity, consider first the case of fermionic zero-modes attached to the kth instanton-dyon, and define the additional Grassmanians with so that

(36) |

We can re-arrange the exponent in (36) by defining a Grassmanian source with

(37) |

and by introducing 2 additional fermionic fields . Thus

(38) |

with a chiral block matrix

(39) |

with entries . The Grassmanian source contributions in (38) generates a string of independent exponents for the L-instanton-dyons and -instanton-anti-dyons

(40) |

The Grassmanian integration over the in each factor in (40) is now readily done to yield

(41) |

for the k-instanton-dyon and -instanton-anti-dyon. The net effect of the additional fermionic determinant in (LABEL:SU2) is to shift the k-instanton-dyon and -instanton-anti-dyon fugacities in (LABEL:FREE3) as follows

(42) |

where we have now identified the chiralities with . Note that for the instanton-dyons and instanton-anti-dyons with no zero-mode attached, the fugacities remain unchanged.

### iii.2 Resolving the constraints

In terms of (32-LABEL:FREE3) and the substitution (42), the instanton-dyon partition function (LABEL:SU2) for finite can be exactly re-written as an interacting effective field theory in 3-dimensions,

(43) | |||||

with the additional chiral fermionic contribution . Since the effective action in (43) is linear in the , the latters integrate to give the following constraints

and similarly for the anti-dyons.

To proceed further the formal classical solutions to the constraint equations or should be inserted back into the 3-dimensional effective action. The result is

(45) |

with the 3-dimensional effective action

Here is in (34) plus additional contributions resulting from the solutions to the constraint equations (III.2) after their insertion back. This procedure for the linearized approximation of the constraint was discussed in LIU1 ; LIU2 .

For the general case with

(47) |

these contributions in (LABEL:NEWS) are only symmetric, which is commensurate with (6). The determinantal interactions preserve the individual vector flavor symmetries, but upset the individual axial flavor symmetries. However, the latters induce the shifts

(48) |

which can be re-absorbed by shifting back the constant magnetic contributions

(49) |

thanks to the free form in (34). This observation is unaffected by the screening of the magnetic-like field, since a constant shift can always be reset by a field redefinition. This hidden symmetry was noted recently in THOMAS . We note that this observation holds for the general form of the streamline interaction used in LIU2 as well, due to its vanishing form in momentum space. From (49) it follows that , so that only the axial flavor singlet is explicitly broken by the determinantal contributions in (LABEL:NEWS) as expected in the instanton-dyon-anti-dyon ensemble. As a result, (LABEL:NEWS) is explicitly symmetric.

### iii.3 Special cases:

For the case with the twisted boundary condition (II.3), the fermionic terms in the effective action (LABEL:NEWS) are explicitly

following the regulartization (II.3) around the center symmetric point. As noted earlier, (III.3) is explicitly symmetric under the combined charge conjugation and the flavor exchange since . With this in mind, (III.3) is symmetric under .

For the case with the twisted boundary condition (II.4), the fermionic terms in the effective action (LABEL:NEWS) are now

while for the Roberge-Weiss boundary condition (II.4) they are

## Iv Equilibrium state

To analyze the ground state and the fermionic fluctuations we bosonize the fermions in (45-LABEL:NEWS) by introducing the identities

(53) |

and by re-exponentiating them to obtain

with

(55) |

The ground state is parity even so that . By translational invariance, the ground state corresponds to constant . We will seek the extrema of (LABEL:ZDDEFF2) with finite condensates in the mean-field approximation, i.e.

(56) |

With this in mind, the classical solutions to the constraint equations (III.2) are also constant

(57) |

with

(58) |

and similarly for the anti-dyons. The expectation values in (IV-58) are carried in (LABEL:ZDDEFF2) in the mean-field approximation through Wick contractions.

### iv.1 in symmetric phase

In the center-symmetric phase, with all holonomies being equal , the pressure simplifies to

with the individual fermionic terms being

(60) |

Here and are dimensionless. From (24), we recall the assignment of quark phases , for respectively. The center symmetric phase breaks spontaneously chiral symmetry, as the gap equation have nonzero solution. Each of the flavor chiral condensate is found to be

(61) |

We now note that at asymptotically low temperatures, the contribution in the hopping matrix element (25) is dominant.

### iv.2 = in general asymmetric phase

In general asymmetric phase the holonomies have values away from the center

(62) |

Note that in general, the parameters are not small. With these choices for the holonomies (IV.2) , the u-flavor rides the L-instanton-dyon, and the ds-flavors ride the -instanton-dyons. For the ds-flavors, the hopping matrix elements between the instanton-dyon and anti-instanton-dyon are given by

(63) |

with