# Cluster expansions and chiral symmetry at large density in 2

## Transcript Of Cluster expansions and chiral symmetry at large density in 2

PoS(LATTICE 2015)187

Cluster expansions and chiral symmetry at large density in 2-color QCD

E. T. Tomboulis∗ Dept. of Physics and Astronomy, University of California, Los Angeles Los Angeles, CA 90095, USA E-mail: [email protected]

SU(Nc) lattice gauge theories with Nf ﬂavors of massless staggered fermions are considered at high quark chemical potential µ and any temperature T . In the strong coupling regime (sufﬁciently small β ) they have been shown to possess a chiral phase of intact global U(Nf ) ×U(Nf ) symmetry. The proof is by cluster expansions which converge in the inﬁnite volume limit. Extension to weaker coupling does not appear feasible in the presence of complex fermion determinant. For theories with real determinant, however, such as 2-color QCD with fundamental fermions, or any Nc with even Nf and adjoint fermions, such large µ cluster expansions can be used to show chiral behavior of fermionic lattice observables at any gauge coupling. Unfortunately, this absence of color superﬂuidity/superconductivity at high µ appears to be a lattice artifact due to lattice saturation, a serious problem plaguing the standard ﬁnite density formalism on the lattice. Some possible ways of circumventing saturation are discussed.

The 33rd International Symposium on Lattice Field Theory 14 -18 July 2015 Kobe International Conference Center, Kobe, Japan*

∗Speaker.

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

http://pos.sissa.it/

Cluster expansions and chiral symmetry

E. T. Tomboulis

PoS(LATTICE 2015)187

1. Introduction

A lot of effort has been devoted in recent years toward elucidating the expected rich structure of the QCD phase diagram. Still, away from a strip along the temperature axis at small density this phase diagram remains largely conjectural. This is due to our inability to perform simulations in Lattice Gauge Theory (LGT) due to the sign (complex fermion determinant) problem. With presently available simulation techniques we are basically restricted to µ/T 1. Even in cases with real determinant simulations at large µ with light fermions appear at least an order of magnitude more demanding than at zero density. It is this regime of high density and low temperature that is physically particularly interesting as it has been argued to engender, depending on the color and ﬂavor content, a variety of color superconductivity/superﬂuidity phases. In light of this state of affairs there have been many studies of ﬁnite density LGT at strong coupling which is amenable to a variety of techniques. Integrating out the gauge ﬁeld in the strong coupling limit with staggered fermions results in a representation of the partition function in terms of monomers, dimers and baryon loops [1], or monomers, dimers and polymers [2]. The sign problem is partly evaded within this representation, thus allowing simulations [1]-[4]. Another approach is based on mean ﬁeld investigations of effective actions obtained by retaining the leading terms in 1/d expansion in the spatial directions while leaving the timelike directions intact [5], [6]. In all such investigations a transition to a chirally symmetric phase is found at some critical µ. The existence of this phase for general SU(Nc) at strong coupling was proven in [7], [8] by means of a cluster expansion shown to converge for large µ in the inﬁnite volume limit. Such large µ strong coupling cluster expansions are reviewed in section 2 below. We then proceed to show how they can be used to extract information for all couplings in the case of LGT with real fermion determinant. We discuss the meaning of such lattice results in the last section.

2. Large µ, strong coupling cluster expansion in SU(Nc) LGT

The lattice action is S = Sg + SF where Sg is the usual gauge ﬁeld plaquette action and SF = ∑x,y ψ¯ (x)Mx,y(U)ψ(y) is the action for massless staggered fermions in the presence of quark chemical potential µ. We take Nf staggered fermions ﬂavors (which corresponds to 4Nf continuum ﬂavors). SF is then invariant under a U(Nf ) × U(Nf ) global chiral symmetry corresponding to independent rotations of fermions on even and odd sublattices.

The basic idea behind the cluster expansion [7] is that the presence of a nonvanishing chemical potential in SF introduces an anisotropy between the spacelike and timelike directions. This can be exploited to set up a cluster expansion for large µ. (There is an analogous anisotropy in the case of large T that was used for the convergent expansion in [9] showing chiral symmetry restoration at high temperature.) The expansion is generated by simply expanding the exponential of the space-like part of the action, exp ∑x,y ψ¯ (x)M(xs,y)(U)ψ(y) and carrying out the fermion integrations in the measure provided by the unexpanded exponential of the timelike part of the action, exp ∑x,y ψ¯ (x)M(xt,)y(U)ψ(y), which depends on µ. In other words, one performs a fermion spacelike hopping expansion with fermions connected in the time direction by propagators given by the µ-dependent timelike part of the action. In the strong coupling limit, i.e., β = 0, where the gauge action Sg is absent, this constitutes the entire expansion [7]. It may be extended to ﬁnite strong

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coupling, i.e. small β , by combining this fermionic expansion with the usual strong coupling plaquette expansion [8]. The latter is obtained by expanding the exponential of the plaquette gauge ﬁeld action in characters:

∏ ∏ ∑ ∏ exp{ β Re trUp} = a0(β )|Λ| 1 + d jc j(β )χ j(Up) ≡ a0(β )|Λ| 1 + fp(Up)

p

Nc

p

j=0

p

(2.1)

and expanding in powers of fp’s. The diagrammatics of these expansions are explained in [7], [8]. In the Polyakov gauge where all bond variables U0(τ, x) are chosen to be independent of τ, i.e.

U0(τ, x) = diag(eiθ1(x)/L, eiθ2(x)/L, · · · , eiθNc (x)/L) ≡ exp(iΘ(x)/L) ,

(2.2)

explicit evaluation of the timelike fermion propagator for propagation from τ to τ gives [7]:

C(τ − τ , Θ(x)) = δ δ [1 − (−1)(τ − τ )] e−iθa(x)(τ − τ )/L e−µ(τ − τ ) ,

ai,b j

ab i j

1 + e−iθa(x)e−µL

for (τ − τ ) > 0, µ > 0

(2.3)

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C(τ − τ , Θ(x)) = −δ δ [1 − (−1)|τ − τ |] e−iθa(x)[1 − |τ − τ |/L] e−|µ|[L − |τ − τ |] ,

ai,b j

ab i j

1 + e−iθa(x)e−µL

for (τ − τ ) < 0, µ > 0

(2.4)

Note that, for τ > τ, propagating backward in time from τ to τ is equivalent to propagating forward from τ winding around the periodic time direction to τ. For µ < 0, i.e., for nonvanishing antiquark chemical potential, a physically distinct situation, replace θa(x) by −θa(x), and reverse the sign condition on (τ − τ ) in (2.3) and (2.4).

As seen from (2.3) - (2.4), C(τ, Θ(x)) vanishes for even τ. This is a consequence of the chiral invariance of the action. The other salient property of C(τ, Θ(x)) is its exponential decay for nonvanishing µ. These are the crucial properties for the convergence of the expansion. Some typical diagrams are shown in Fig. 1. The expansion can be shown to converge in the large volume limit for sufﬁciently large µ and small β [7], [8].

A consequence of such convergence is that the expectation of any local chirally non-invariant fermion operator O(x), e.g., ψ¯ (x)ψ(x) or diquark operators, vanishes identically term by term in the expansion by the invariance of the measure. Correlation functions O(x)O(y) then can receive non-vanishing contributions only from diagrams intersecting both sites x and y. A straightforward consequence of this fact is that:

O(x)O(y) < C0C−|x−y| ,

where C0,C are space-dimension-dependent constants and |x − y| is the minimum number of bonds connecting the two sites. In other words, there is clustering of 2-point (and all higher) correlations: the global U(Nf ) ×U(Nf ) symmetry is intact for sufﬁciently large µ, and small β .

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Cluster expansions and chiral symmetry

=

(a)

(b)

E. T. Tomboulis (c)

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=

(d)

(e)

(f)

Figure 1: Some diagrams in the expansion: (a) - (d) diagrams involving only fermions (as in β = 0); (e) - (f) diagrams including gauge ﬁeld plaquettes. Directed lines represent fermion hopping spacelike links, broken lines represent timelike propagators (their direction shown for clarity only in (a) and (d)), gauge ﬁeld plaquettes shown in solid blue lines.

3. Theories with real fermion determinant

Can we extend this expansion setup to larger regimes of the gauge coupling? Write the partition function in the form of a gauge ﬁeld integration over a fermion partition function ZF (U):

Z = DU eSg(U)ZF (U ) with ZF (U ) = Dψ¯ Dψ eSF(U) = Det M(U ) . (3.1)

The expectation of a general fermionic operator O[ψ¯ , ψ] may then be expressed in the form:

O = dν(U) O F(U) ,

(3.2)

where

1 O F(U) = ZF (U)

Dψ¯ Dψ eSF (U)O[ψ¯ , ψ]

(3.3)

is its expectation in the fermionic measure in the background of the gauge ﬁeld and

dν(U ) ≡ dU eSg(U) DetM(U ) Z

(3.4)

is the (normalized) full effective gauge ﬁeld measure at coupling β .

Now, one may expand the fermion expectation O F (U) given by (3.3) in the same type of expansion as in the previous section. This expansion for O F (U), in generic gauge ﬁeld background

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U and for operators O of bounded support and spatial dimension d ≥ 1, converges absolutely, and uniformly in the spatial lattice size, at any temperature T for sufﬁciently large µ. The result holds for any choice of Nc, Nf . The proof, and associated estimates, proceed as in the strong coupling case above except that it is actually simpler since no integration over the gauge ﬁeld is involved engendering additional connectivity among diagrams.

Expansion of O F (U) leads to expansion of

O = dν(U) O F(U) .

(3.5)

Does this expansion converge? Convergence of the O F (U) expansion implies

| O F (U)| < CO ,

(3.6)

where CO is a constant, which is observable- and spacetime-dimension-dependent, but independent of the background U and the spatial lattice volume. Absolute convergence of the expansion (3.5)

for O now follows from the absolute convergence of the expansion for O F(U) provided the measure dν is real and positive:

| O | = dν(U) | O F(U)| < CO .

(3.7)

dν(U) is real positive if the fermion determinant DetM(U) is real positive. This is the case for Nc = 2 and fundamental rep. fermions; or any Nc, even Nf and adjoint fermions. In the case of Nc = 2 and fundamental fermions one has a pseudo-real representation with gauge ﬁeld matrices satisfying τ2Uτ2 = U∗. In the case of general SU(Nc) and adjoint fermions one has a real representation and the U’s represented by real orthogonal matrices. In both cases then DetM is real: in the two-color and fundamental fermion case DetM = Detτ2Mτ2 = DetM∗; in the general Nc and adjoint fermion case DetM is manifestly real. Furthermore, at µ = 0 the U(Nf ) × U(Nf ) symmetry of staggered fermions is enlarged to U(2Nf ), which has interesting consequences for spontaneous symmetry breaking, cf. [10]. At low T , a sequence of a chiral condensate phase, followed by a diquark

condensate phase, followed by a chiral symmetry restored phase is expected with increasing µ,

as found in mean ﬁeld computations, cf. [1], [5]. Extensive simulations of the 2-color theory at

ﬁnite µ with Dirac fermions have been carried out in [11]. Unless close to the continuum limit,

however, Dirac fermions do not possess any well-deﬁned chiral properties and cannot be compared

to staggered fermions in a meaningfully way.

Operators O of interest here would be the usual chiral condensate order parameter Oq¯q = ψ¯ (x)ψ(x), as well as the Nc = 2 fundamental fermions diquark condensate:

1 Oqq =

ψT (x)τ2ψ(x) + ψ¯ (x)τ2ψ¯ T (x)

.

2

(3.8)

For SU(Nc) an adjoint fermions diquark condensate

Oaqdqi j... = 21 εi j...kl ψk T (x)ψl(x) + ψ¯ k(x)ψ¯ l T (x)

(3.9)

breaks U(Nf ) ×U(Nf ) −→ SU(2) isospin; in particular, the condensate breaks UB(1). Such breaking of the global symmetries engenders superﬂuidity. An operator for color symmetry breaking

condensate would be

Oaqq = 21 ψT (x)taψ(x) + ψ¯ (x)taψ¯ T (x) .

(3.10)

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(3.10) transforms in the adjoint representation of SU(Nc), i.e. as a composite adjoint Higgs ﬁeld, and its condensation would break the color symmetry, as well as the chiral symmetries, resulting into color superconductivity. Note that because it is an adjoint composite, any such phase would be separated from the unbroken conﬁning phase by a true phase boundary.

An immediate consequence of the convergence implied by (3.7), however, is that, just as before (section 2), within the convergence radius the U(Nf ) × U(Nf ) symmetry is preserved. Indeed, the expectation of any local fermion operator non-invariant under this symmetry, such as Oq¯q, Oaqdqi j... or Oaqq above, vanishes identically term by term in our expansion by the invariance of the measure. Equivalently, all 2-point and higher correlation functions of such operators are seen to cluster exponentially, i.e., there is no spontaneous breaking of the global U(Nf )×U(Nf ) symmetry. The result holds for all gauge couplings β and all temperatures T at sufﬁciently large µ. As a consequence no superﬂuidity and/or color superconductivity phase involving breaking of (any part of) these global symmetries occurs at high µ. Rather what may be called a “quarkyonic" phase [12] with intact chiral symmetry obtains at low T .

4. Lattice saturation - Discussion

We saw that at sufﬁciently large quark chemical potential and sufﬁciently large gauge coupling the U(Nf ) × U(Nf ) global symmetry of the SU(Nc) LGT with Nf ﬂavors of massless staggered fermions is intact. This is an exact lattice result obtained by cluster expansions converging in the large volume limit. It accords with a large number of previous simulation and mean-ﬁeld/analytical studies, mostly for Nc = 3, 2, [1]-[6] which ﬁnd a transition to such a phase. Furthermore, in the case of LGT with real fermion determinant we saw how information obtained from these fermionic cluster expansions can be used to extend this result to all couplings. The crucial question, of course, is what relation these lattice results bear to the continuum massless theory.

Unfortunately, the immediate answer appears to be that they are of little direct relevance. This is because they seem to be largely determined by the onset of lattice saturation. Lattice saturation, i.e., every lattice site being occupied by the maximum number of fermions allowed by the Pauli principle, can be a real effect on a physical lattice, as observed in certain condensed matter systems; but it is a regularization artifact in the LGT context. Once saturation sets in no condensates can form. Computation of the quark number density within our expansion indeed shows that saturation is present; the density is at its maximum per site at T = 0 deviating only by small exponential corrections at low T . The saturation effect at strong coupling sets in immediately upon the transition to the chirally symmetric phase. An earlier discussion of this was given in [13]. This raises the question of whether the T = 0 chiral transition at strong coupling seen in simulations, most recently in [4], reﬂects the eminent set-in of lattice saturation rather than the true location of the (expected) transition. To explore such questions would require having some control over the onset of saturation.

In the usual ﬁnite density lattice formalism, employed here and previous cited studies, a chemical potential µ is introduced on the timelike links uniformly throughout the lattice. This inexorably leads to saturation once µ becomes large enough. A possible way of avoiding this is to introduce spatially variable µ, and, in particular, a “thinned-out" distribution obtained by setting µ to a lower or negative value on a subset of the timelike bonds. One may, e.g., partition the spatial lattice into

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Cluster expansions and chiral symmetry

E. T. Tomboulis

cubes of some ﬁxed size and introduce µ¯ < 0 (antiquark chemical potential) on the bonds of the timelike ﬁber(s) extending from one (or more) site(s) in each cube, while having quark chemical potential µ > 0 on the rest of the timelike links. One may thus achieve any mean density (net particle number per unit volume) by adjusting µ versus µ¯ , while allowing local particle number ﬂuctuations which thwart complete saturation. It is not hard to see that such schemes will generally upset the convergence of the expansions above, and may necessitate some series repackaging or resummation leading to a different physical picture.1 Such extensions are currently under investigation.

References

[1] E. Daggoto, A. Moreo and U. Wolff, Phys. Lett. B 186, 395 (1987).

[2] F. Karsch and K. H. Mütter, Nucl. Phys. B 313, 541 (1989); J. U. Klaetke and K. H. Mütter, Nucl. Phys. B 342, 764 (1990).

[3] S. Chandrasekharan and F-J. Jiang, Phys. Rev. D 74, 014506 (2006) [arXiv:hep-lat/0602031]; D. H. Adams and S. Chandrasekharan, Nucl. Phys. B 662, 220 (2003) [arXiv:hep-lat/0303003].

[4] P. de Forcrand, J. Langelage, O. Philipsen and W. Unger, Phys. Rev. Lett. 113, 152002 (12014); P. de Forcrand and M. Fromm, Phys. Rev. Lett. 104, 112005 (2010) [arXiv:0907.1915].

[5] Y. Nishida, K. Fukushima and T. Hatsuda, Phys. Rept. 398, 281 (2004) [arXiv:hep-ph/0306066]; Y. Nishida, Phys. Rev. D 69, 094501 (2004) [arXiv:hep-ph/031231].

[6] N. Kawamoto, K. Miura, A. Ohnishi and T. Ohmura, Phys. Rev. D 75, 014502 (2007) [arXiv:hep-lat/0512023]; K. Miura, T. Nakano and A. Ohnishi, Prog. Theor. Phys. 122, 1045 (2009) [arXiv:0806.3357].

[7] E. T. Tomboulis, J. Math. Phys. 54, 122301 (2013) [arXiv:1304.4678].

[8] E. T. Tomboulis, Int. J. Mod. Ph. A 29, 1445004 (2014) [arXiv:1404.0664].

[9] E. T. Tomboulis and L. G. Yaffe, Phys. Rev. Lett. 52, 2115 (1984); Comm. Math. Phys. 100, 313 (1985).

[10] S. Hands, J. B. Kogut, M-P. Lombardo, and S. Morrison, Nucl. Phys. B 558, 327 (1999); S. Hands et al, Eur. Phys. J. C17, 285 (2000). J. B. Kogut, D. Toublan and D. K. Sinclair, Phys. Rev. D 68, 054507 (2003).

[11] S. Cotter, P. Giudice, S. Hands and J-I. Skullerud, Phys. Rev. D 87, 034507 (2013) [arXiv:1210.4496]; arXiv:1210.6559; T. Boz et al, Eur. Phys. J. A49, 87 (2013) [arXiv:1303.3223].

[12] L. McLerran and R. D. Pisarski, Nucl. Phys. A 796, 83 (2007); Y. Hidaka, L. McLerran and R. Pisarski, Nucl. Phys. A 808,117 (2008) [arXiv:0803.0279].

[13] V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, JHEP 0309, 014 (2003) [arXiv:hep-lat/0307019].

[14] E. T. Tomboulis, Phys. Rev. D 87, 034513 (2013) [arXiv:1211.4842].

1The chiral symmetry transition in going from small Nf /Nc to large Nf /Nc at strong coupling may provide an analogous example of such resummations necessitated by a change in physical parameters and completely altering the physical outcome [14].

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Cluster expansions and chiral symmetry at large density in 2-color QCD

E. T. Tomboulis∗ Dept. of Physics and Astronomy, University of California, Los Angeles Los Angeles, CA 90095, USA E-mail: [email protected]

SU(Nc) lattice gauge theories with Nf ﬂavors of massless staggered fermions are considered at high quark chemical potential µ and any temperature T . In the strong coupling regime (sufﬁciently small β ) they have been shown to possess a chiral phase of intact global U(Nf ) ×U(Nf ) symmetry. The proof is by cluster expansions which converge in the inﬁnite volume limit. Extension to weaker coupling does not appear feasible in the presence of complex fermion determinant. For theories with real determinant, however, such as 2-color QCD with fundamental fermions, or any Nc with even Nf and adjoint fermions, such large µ cluster expansions can be used to show chiral behavior of fermionic lattice observables at any gauge coupling. Unfortunately, this absence of color superﬂuidity/superconductivity at high µ appears to be a lattice artifact due to lattice saturation, a serious problem plaguing the standard ﬁnite density formalism on the lattice. Some possible ways of circumventing saturation are discussed.

The 33rd International Symposium on Lattice Field Theory 14 -18 July 2015 Kobe International Conference Center, Kobe, Japan*

∗Speaker.

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

http://pos.sissa.it/

Cluster expansions and chiral symmetry

E. T. Tomboulis

PoS(LATTICE 2015)187

1. Introduction

A lot of effort has been devoted in recent years toward elucidating the expected rich structure of the QCD phase diagram. Still, away from a strip along the temperature axis at small density this phase diagram remains largely conjectural. This is due to our inability to perform simulations in Lattice Gauge Theory (LGT) due to the sign (complex fermion determinant) problem. With presently available simulation techniques we are basically restricted to µ/T 1. Even in cases with real determinant simulations at large µ with light fermions appear at least an order of magnitude more demanding than at zero density. It is this regime of high density and low temperature that is physically particularly interesting as it has been argued to engender, depending on the color and ﬂavor content, a variety of color superconductivity/superﬂuidity phases. In light of this state of affairs there have been many studies of ﬁnite density LGT at strong coupling which is amenable to a variety of techniques. Integrating out the gauge ﬁeld in the strong coupling limit with staggered fermions results in a representation of the partition function in terms of monomers, dimers and baryon loops [1], or monomers, dimers and polymers [2]. The sign problem is partly evaded within this representation, thus allowing simulations [1]-[4]. Another approach is based on mean ﬁeld investigations of effective actions obtained by retaining the leading terms in 1/d expansion in the spatial directions while leaving the timelike directions intact [5], [6]. In all such investigations a transition to a chirally symmetric phase is found at some critical µ. The existence of this phase for general SU(Nc) at strong coupling was proven in [7], [8] by means of a cluster expansion shown to converge for large µ in the inﬁnite volume limit. Such large µ strong coupling cluster expansions are reviewed in section 2 below. We then proceed to show how they can be used to extract information for all couplings in the case of LGT with real fermion determinant. We discuss the meaning of such lattice results in the last section.

2. Large µ, strong coupling cluster expansion in SU(Nc) LGT

The lattice action is S = Sg + SF where Sg is the usual gauge ﬁeld plaquette action and SF = ∑x,y ψ¯ (x)Mx,y(U)ψ(y) is the action for massless staggered fermions in the presence of quark chemical potential µ. We take Nf staggered fermions ﬂavors (which corresponds to 4Nf continuum ﬂavors). SF is then invariant under a U(Nf ) × U(Nf ) global chiral symmetry corresponding to independent rotations of fermions on even and odd sublattices.

The basic idea behind the cluster expansion [7] is that the presence of a nonvanishing chemical potential in SF introduces an anisotropy between the spacelike and timelike directions. This can be exploited to set up a cluster expansion for large µ. (There is an analogous anisotropy in the case of large T that was used for the convergent expansion in [9] showing chiral symmetry restoration at high temperature.) The expansion is generated by simply expanding the exponential of the space-like part of the action, exp ∑x,y ψ¯ (x)M(xs,y)(U)ψ(y) and carrying out the fermion integrations in the measure provided by the unexpanded exponential of the timelike part of the action, exp ∑x,y ψ¯ (x)M(xt,)y(U)ψ(y), which depends on µ. In other words, one performs a fermion spacelike hopping expansion with fermions connected in the time direction by propagators given by the µ-dependent timelike part of the action. In the strong coupling limit, i.e., β = 0, where the gauge action Sg is absent, this constitutes the entire expansion [7]. It may be extended to ﬁnite strong

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coupling, i.e. small β , by combining this fermionic expansion with the usual strong coupling plaquette expansion [8]. The latter is obtained by expanding the exponential of the plaquette gauge ﬁeld action in characters:

∏ ∏ ∑ ∏ exp{ β Re trUp} = a0(β )|Λ| 1 + d jc j(β )χ j(Up) ≡ a0(β )|Λ| 1 + fp(Up)

p

Nc

p

j=0

p

(2.1)

and expanding in powers of fp’s. The diagrammatics of these expansions are explained in [7], [8]. In the Polyakov gauge where all bond variables U0(τ, x) are chosen to be independent of τ, i.e.

U0(τ, x) = diag(eiθ1(x)/L, eiθ2(x)/L, · · · , eiθNc (x)/L) ≡ exp(iΘ(x)/L) ,

(2.2)

explicit evaluation of the timelike fermion propagator for propagation from τ to τ gives [7]:

C(τ − τ , Θ(x)) = δ δ [1 − (−1)(τ − τ )] e−iθa(x)(τ − τ )/L e−µ(τ − τ ) ,

ai,b j

ab i j

1 + e−iθa(x)e−µL

for (τ − τ ) > 0, µ > 0

(2.3)

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C(τ − τ , Θ(x)) = −δ δ [1 − (−1)|τ − τ |] e−iθa(x)[1 − |τ − τ |/L] e−|µ|[L − |τ − τ |] ,

ai,b j

ab i j

1 + e−iθa(x)e−µL

for (τ − τ ) < 0, µ > 0

(2.4)

Note that, for τ > τ, propagating backward in time from τ to τ is equivalent to propagating forward from τ winding around the periodic time direction to τ. For µ < 0, i.e., for nonvanishing antiquark chemical potential, a physically distinct situation, replace θa(x) by −θa(x), and reverse the sign condition on (τ − τ ) in (2.3) and (2.4).

As seen from (2.3) - (2.4), C(τ, Θ(x)) vanishes for even τ. This is a consequence of the chiral invariance of the action. The other salient property of C(τ, Θ(x)) is its exponential decay for nonvanishing µ. These are the crucial properties for the convergence of the expansion. Some typical diagrams are shown in Fig. 1. The expansion can be shown to converge in the large volume limit for sufﬁciently large µ and small β [7], [8].

A consequence of such convergence is that the expectation of any local chirally non-invariant fermion operator O(x), e.g., ψ¯ (x)ψ(x) or diquark operators, vanishes identically term by term in the expansion by the invariance of the measure. Correlation functions O(x)O(y) then can receive non-vanishing contributions only from diagrams intersecting both sites x and y. A straightforward consequence of this fact is that:

O(x)O(y) < C0C−|x−y| ,

where C0,C are space-dimension-dependent constants and |x − y| is the minimum number of bonds connecting the two sites. In other words, there is clustering of 2-point (and all higher) correlations: the global U(Nf ) ×U(Nf ) symmetry is intact for sufﬁciently large µ, and small β .

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Cluster expansions and chiral symmetry

=

(a)

(b)

E. T. Tomboulis (c)

PoS(LATTICE 2015)187

=

(d)

(e)

(f)

Figure 1: Some diagrams in the expansion: (a) - (d) diagrams involving only fermions (as in β = 0); (e) - (f) diagrams including gauge ﬁeld plaquettes. Directed lines represent fermion hopping spacelike links, broken lines represent timelike propagators (their direction shown for clarity only in (a) and (d)), gauge ﬁeld plaquettes shown in solid blue lines.

3. Theories with real fermion determinant

Can we extend this expansion setup to larger regimes of the gauge coupling? Write the partition function in the form of a gauge ﬁeld integration over a fermion partition function ZF (U):

Z = DU eSg(U)ZF (U ) with ZF (U ) = Dψ¯ Dψ eSF(U) = Det M(U ) . (3.1)

The expectation of a general fermionic operator O[ψ¯ , ψ] may then be expressed in the form:

O = dν(U) O F(U) ,

(3.2)

where

1 O F(U) = ZF (U)

Dψ¯ Dψ eSF (U)O[ψ¯ , ψ]

(3.3)

is its expectation in the fermionic measure in the background of the gauge ﬁeld and

dν(U ) ≡ dU eSg(U) DetM(U ) Z

(3.4)

is the (normalized) full effective gauge ﬁeld measure at coupling β .

Now, one may expand the fermion expectation O F (U) given by (3.3) in the same type of expansion as in the previous section. This expansion for O F (U), in generic gauge ﬁeld background

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U and for operators O of bounded support and spatial dimension d ≥ 1, converges absolutely, and uniformly in the spatial lattice size, at any temperature T for sufﬁciently large µ. The result holds for any choice of Nc, Nf . The proof, and associated estimates, proceed as in the strong coupling case above except that it is actually simpler since no integration over the gauge ﬁeld is involved engendering additional connectivity among diagrams.

Expansion of O F (U) leads to expansion of

O = dν(U) O F(U) .

(3.5)

Does this expansion converge? Convergence of the O F (U) expansion implies

| O F (U)| < CO ,

(3.6)

where CO is a constant, which is observable- and spacetime-dimension-dependent, but independent of the background U and the spatial lattice volume. Absolute convergence of the expansion (3.5)

for O now follows from the absolute convergence of the expansion for O F(U) provided the measure dν is real and positive:

| O | = dν(U) | O F(U)| < CO .

(3.7)

dν(U) is real positive if the fermion determinant DetM(U) is real positive. This is the case for Nc = 2 and fundamental rep. fermions; or any Nc, even Nf and adjoint fermions. In the case of Nc = 2 and fundamental fermions one has a pseudo-real representation with gauge ﬁeld matrices satisfying τ2Uτ2 = U∗. In the case of general SU(Nc) and adjoint fermions one has a real representation and the U’s represented by real orthogonal matrices. In both cases then DetM is real: in the two-color and fundamental fermion case DetM = Detτ2Mτ2 = DetM∗; in the general Nc and adjoint fermion case DetM is manifestly real. Furthermore, at µ = 0 the U(Nf ) × U(Nf ) symmetry of staggered fermions is enlarged to U(2Nf ), which has interesting consequences for spontaneous symmetry breaking, cf. [10]. At low T , a sequence of a chiral condensate phase, followed by a diquark

condensate phase, followed by a chiral symmetry restored phase is expected with increasing µ,

as found in mean ﬁeld computations, cf. [1], [5]. Extensive simulations of the 2-color theory at

ﬁnite µ with Dirac fermions have been carried out in [11]. Unless close to the continuum limit,

however, Dirac fermions do not possess any well-deﬁned chiral properties and cannot be compared

to staggered fermions in a meaningfully way.

Operators O of interest here would be the usual chiral condensate order parameter Oq¯q = ψ¯ (x)ψ(x), as well as the Nc = 2 fundamental fermions diquark condensate:

1 Oqq =

ψT (x)τ2ψ(x) + ψ¯ (x)τ2ψ¯ T (x)

.

2

(3.8)

For SU(Nc) an adjoint fermions diquark condensate

Oaqdqi j... = 21 εi j...kl ψk T (x)ψl(x) + ψ¯ k(x)ψ¯ l T (x)

(3.9)

breaks U(Nf ) ×U(Nf ) −→ SU(2) isospin; in particular, the condensate breaks UB(1). Such breaking of the global symmetries engenders superﬂuidity. An operator for color symmetry breaking

condensate would be

Oaqq = 21 ψT (x)taψ(x) + ψ¯ (x)taψ¯ T (x) .

(3.10)

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PoS(LATTICE 2015)187

(3.10) transforms in the adjoint representation of SU(Nc), i.e. as a composite adjoint Higgs ﬁeld, and its condensation would break the color symmetry, as well as the chiral symmetries, resulting into color superconductivity. Note that because it is an adjoint composite, any such phase would be separated from the unbroken conﬁning phase by a true phase boundary.

An immediate consequence of the convergence implied by (3.7), however, is that, just as before (section 2), within the convergence radius the U(Nf ) × U(Nf ) symmetry is preserved. Indeed, the expectation of any local fermion operator non-invariant under this symmetry, such as Oq¯q, Oaqdqi j... or Oaqq above, vanishes identically term by term in our expansion by the invariance of the measure. Equivalently, all 2-point and higher correlation functions of such operators are seen to cluster exponentially, i.e., there is no spontaneous breaking of the global U(Nf )×U(Nf ) symmetry. The result holds for all gauge couplings β and all temperatures T at sufﬁciently large µ. As a consequence no superﬂuidity and/or color superconductivity phase involving breaking of (any part of) these global symmetries occurs at high µ. Rather what may be called a “quarkyonic" phase [12] with intact chiral symmetry obtains at low T .

4. Lattice saturation - Discussion

We saw that at sufﬁciently large quark chemical potential and sufﬁciently large gauge coupling the U(Nf ) × U(Nf ) global symmetry of the SU(Nc) LGT with Nf ﬂavors of massless staggered fermions is intact. This is an exact lattice result obtained by cluster expansions converging in the large volume limit. It accords with a large number of previous simulation and mean-ﬁeld/analytical studies, mostly for Nc = 3, 2, [1]-[6] which ﬁnd a transition to such a phase. Furthermore, in the case of LGT with real fermion determinant we saw how information obtained from these fermionic cluster expansions can be used to extend this result to all couplings. The crucial question, of course, is what relation these lattice results bear to the continuum massless theory.

Unfortunately, the immediate answer appears to be that they are of little direct relevance. This is because they seem to be largely determined by the onset of lattice saturation. Lattice saturation, i.e., every lattice site being occupied by the maximum number of fermions allowed by the Pauli principle, can be a real effect on a physical lattice, as observed in certain condensed matter systems; but it is a regularization artifact in the LGT context. Once saturation sets in no condensates can form. Computation of the quark number density within our expansion indeed shows that saturation is present; the density is at its maximum per site at T = 0 deviating only by small exponential corrections at low T . The saturation effect at strong coupling sets in immediately upon the transition to the chirally symmetric phase. An earlier discussion of this was given in [13]. This raises the question of whether the T = 0 chiral transition at strong coupling seen in simulations, most recently in [4], reﬂects the eminent set-in of lattice saturation rather than the true location of the (expected) transition. To explore such questions would require having some control over the onset of saturation.

In the usual ﬁnite density lattice formalism, employed here and previous cited studies, a chemical potential µ is introduced on the timelike links uniformly throughout the lattice. This inexorably leads to saturation once µ becomes large enough. A possible way of avoiding this is to introduce spatially variable µ, and, in particular, a “thinned-out" distribution obtained by setting µ to a lower or negative value on a subset of the timelike bonds. One may, e.g., partition the spatial lattice into

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cubes of some ﬁxed size and introduce µ¯ < 0 (antiquark chemical potential) on the bonds of the timelike ﬁber(s) extending from one (or more) site(s) in each cube, while having quark chemical potential µ > 0 on the rest of the timelike links. One may thus achieve any mean density (net particle number per unit volume) by adjusting µ versus µ¯ , while allowing local particle number ﬂuctuations which thwart complete saturation. It is not hard to see that such schemes will generally upset the convergence of the expansions above, and may necessitate some series repackaging or resummation leading to a different physical picture.1 Such extensions are currently under investigation.

References

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1The chiral symmetry transition in going from small Nf /Nc to large Nf /Nc at strong coupling may provide an analogous example of such resummations necessitated by a change in physical parameters and completely altering the physical outcome [14].

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