Generalized Gross-Neveu models and chiral symmetry breaking from string theory

Anirban Basu1,* and Anshuman Maharana2,†

1Institute for Advanced Study, Princeton, New Jersey 08540, USA2Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA

(Received 31 October 2006; published 8 March 2007)

We consider intersecting D-brane models which have two dimensional chiral fermions localized at theintersections. At weak coupling, the interactions of these fermions are described by generalized Gross-Neveu models. At strong coupling, these configurations are described by the dynamics of probe D-branesin a curved background spacetime. We study patterns of dynamical chiral symmetry breaking in thesemodels at weak and strong coupling, and also discuss relationships between these two descriptions.

DOI: 10.1103/PhysRevD.75.065005 PACS numbers: 11.30.Rd, 11.15.Pg, 11.25.Uv

I. INTRODUCTION

Four dimensional quantum chromodynamics (QCD) de-scribes the force of strong interactions in nature. Thistheory is asymptotically free and is strongly coupled atlow energies, thus making it difficult to analyze. In par-ticular, it has proven difficult to solve the theory even in thelimit of large number of colors. In the limit in which thequark masses in the QCD Lagrangian can be neglected, thistheory has a chiral flavor symmetry which is broken in thepresence of the mass terms. Understanding the dynamics ofchiral symmetry breaking in strongly coupled gauge theo-ries continues to be a challenge.

An extremely successful phenomenological model tounderstand chiral symmetry breaking in QCD4 was devel-oped originally by Nambu and Jona-Lasinio (NJL) [1]where they constructed a field theory of chiral fermionswith a four-fermion interaction. In this model, the chiralflavor symmetry gets dynamically broken to the diagonalsubgroup and the mesons are identified with the Nambu-Goldstone bosons of the broken symmetry generators.However, this four-fermion interaction is nonrenormaliz-able and so the predictions depend on the UV cutoff of thetheory.

In two dimensions, an asymptotically free quantum fieldtheory was constructed by Gross and Neveu (GN) [2]which has a renormalizable four-fermion interaction. Sothe coupling undergoes dimensional transmutation [3],leaving the number of colors as the only free parameterin the theory. This model can be exactly solved in the limitof large number of colors and exhibits dynamical chiralsymmetry breaking. Thus the GN model has proven to bean extremely interesting toy model in understanding chiralsymmetry breaking in QCD. It should be noted that thechiral symmetry is broken only when the number of colorsNc is strictly infinite, and is restored for any finite value ofNc [4,5], however large. As described in [4], the 1=Ncexpansion is reliable in studying the spectrum of this modelas well as related ones like the Thirring model. So though

we shall continue to use the term ‘‘symmetry breaking’’ asused in the literature, it is important to remember that thefermions are massive, no continuous symmetries are bro-ken, and the massless bosons are not Nambu-Goldstonebosons.

It is believed that string theory techniques will play animportant role in understanding strong coupling issues likeconfinement and chiral symmetry breaking in QCD. Aconfining pure Yang-Mills U�Nc� gauge theory in fourdimensions was constructed in [6] by compactifying oneof the world-volume dimensions of Nc D4-branes on acircle of radius R with supersymmetry breaking boundaryconditions. The fermions and the scalars of the world-volume theory of the D4-branes get masses at tree leveland one-loop level, respectively, and so the infra-red theoryis pure Yang-Mills which exhibits confinement. In order tomodel more realistic theories involving chiral fermions, theauthors of [7,8] considered aD-brane configuration involv-ing ‘‘flavor’’ D8-branes alongwith the ‘‘color’’ D4-branes,where the flavor branes intersect the color branes alongthree spatial dimensions, such that there are no directionstransverse to both the flavor and the color branes. Theflavor branes are separated by a distance L along the D4world-volume. One gets chiral fermions in this modelwhich are given by the 4–8 strings stretching betweenthe flavor and color branes. These chiral fermions arelocalized at the intersections of the color and flavor branes.Though classically this configuration preserves the chiralflavor symmetry, this model exhibits dynamical chiralsymmetry breaking (which is a global symmetry from thepoint of view of the color brane theory). This was demon-strated at large values of a classically dimensionless cou-pling constant �=L (where � is essentially the ’t Hooftcoupling of five-dimensional Yang-Mills theory) in [7], byconsidering the D8-branes as probes [9]1 in the near-horizon geometry of the Nc D4-branes. Chiral symmetrybreaking manifests itself as a wormhole solution of the D8-brane action connecting the D8-D8 pair.

Now in these models when one takes the transverseseparation L to be of the same order of magnitude as the

*Electronic address: abasu@ias.edu†Electronic address: anshuman@physics.ucsb.edu 1Also see [10–12] for related discussions.

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circle radius R, the scales of chiral symmetry breaking andconfinement are comparable and it becomes harder toanalyze the dynamics of chiral symmetry breaking withouttaking into account the effects of confinement, and viceversa. Thus to look at the dynamics when chiral symmetryis broken but the theory is not confining, one can considerthe limit when R! 1. This has been done in [13]2 and theconfiguration has been analyzed in the limits �=L! 0 and�=L! 1, and it has been found that the chiral symmetryis broken in both the regimes. The �=L! 0 regime isdescribed by a nonlocal version of the NJL model (see[15] for a review of phenomenological applications of thenonlocal NJL model), while the opposite regime is de-scribed by a wormhole configuration similar to that men-tioned above. It has also been conjectured that thesesolutions lie in the same universality class as QCD4 whichis obtained by taking R to be finite.

A similar analysis to study chiral symmetry breaking intwo dimensions has been done in [16]. The D-brane con-figuration is given by the color D4-branes which intersectthe flavor D6-branes along one spatial dimension such thatthere are no directions transverse to both the color and theflavor branes. Dynamical breaking of chiral symmetryoccurs both in the �=L! 0 and �=L! 1 limits. The�=L! 0 regime is described by the GN model, whilethe opposite regime is described by a wormhole configu-ration. This setup is expected to be in the same universalityclass as QCD2, which is obtained by wrapping three of theworld-volume directions of the D4-branes on T3.

In this paper, we generalize the construction of [16] byconsidering multiple stacks of flavor D6 (and D6)-braneswhich intersect the color D4-branes along one spatialdirection. The flavor branes are placed at various pointsin R3 which parametrizes the world-volume directions ofthe D4-branes that are transverse to the intersection. Ourmotivation is two-fold: to understand the pattern of chiralsymmetry breaking in multibrane constructions, and tostudy the relationship between the weak and strong cou-pling descriptions. In the limit �=Li ! 0 (where Li refersto the distances between the various D6 and D6 pairs in R3)theD-brane configurations reduce to generalized GN mod-els,3 where we exhibit various patterns of chiral symmetrybreaking. In the opposite regime, we describe these con-figurations in terms of probe D6-branes in the near-horizongeometry of the color branes. We find a close relationshipbetween the two descriptions.

The plan of the paper is as follows. We begin by describ-ing the D-brane setup in the next section, followed by theanalysis of various setups where all the flavor branes arecollinear in the transverse R3. These configurations havesome interesting features, and we discuss the relationship

between chiral symmetry breaking in the strong and weakcoupling limits. We next generalize this construction toflavor branes spanning an R2 in R3, and we finally considergeneral flavor brane configurations in R3.

The pattern of chiral symmetry breaking has distinctfeatures. In all our examples we find that at weak coupling,not all the possible condensates are nonvanishing. Thestrong coupling analogue of condensates is the presenceof wormholes connecting the brane and antibrane pairs.Just like at weak coupling, the wormhole configuration isalso determined by the energetics, and leads to patterns ofchiral symmetry breaking similar to weak coupling.

Finally we describe patterns of restoration of the chiralsymmetries as the temperature of the system is raised. At asufficiently high temperature, all the symmetries arerestored.

II. THE D-BRANE SETUP

The GN model can be realized in string theory [16] byconsidering Nc color D4-branes extending along the(01234) directions and two stacks of flavor D6 andD6-branes which extend in the directions (0156789). Todiscuss generalizations of the GN model, we shall considermultiple stacks of D6 and D6-branes given by

0 1 2 3 4 5 6 7 8 9D4: x x x x x

D6s;D6s: x x x x x x x

All the flavor branes extend in the (0156789) directions,and are located at different points in R3 spanned by thedirections (234). We shall take every stack of flavor branesto be composed of Nf branes, our results can be easilygeneralized to the case where the stacks consist of differentnumbers of branes.

It is straightforward to deduce the massless spectrumcorresponding to theseD-brane configurations. Apart fromthe obvious U�Nc� gauge theory on the D4-brane world-volume, and the U�Nf�i gauge theory on the world-volumeof the i-th stack of Nf D6 (or D6)-branes, there are extranormalizable massless modes coming from the 4–6 stringsthat stretch between the D6 (D6)-branes and the D4-branes. These massless modes are spacetime fermionscoming from the Ramond sector, the lowest modes in theNeveu-Schwarz sector are massive. These 4–6 strings arelocalized in the directions (01) and give rise to chiralfermions in two dimensions. Thus every stack of D6-branes gives rise to left-moving fermions qL which trans-form in the �Nf; Nc� of U�Nf�L �U�Nc�. Similarly, everystack of Nf D6-branes produces right-moving fermions qRwhich transform in the �Nf;Nc� of U�Nf�R �U�Nc�. Thusthe spectrum of a configuration consisting of m stacks ofD6 and n stacks of D6-branes has m left-movers qL and nright-movers qR. There is a U�Nf� symmetry associatedwith each stack of flavor branes. The fermions are charged

2The issue of separation of scales was discussed in a differentcontext in [14].

3GN models with more than one coupling have been consid-ered, for example, in [17].

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under the flavor group of the stack they are localized on,and transform trivially under the flavor group of otherstacks.

We will be always working in the limit where �0 ! 0,gs ! 0, Nc ! 1, with gsNc and Nf fixed.4 In this limit,the gauge coupling of the flavor branes vanish compared tothat of the color branes, so the only relevant coupling is the’t Hooft coupling

� �g2

5Nc4�2 ; (1)

where g25 � 4�2gs�

01=2 is the dimensionful coupling of thefive-dimensional Yang-Mills theory. As in [16], we shalldiscuss the physics of the system in two tractable regimesgiven by the classically dimensionless couplings �=Li ! 0and �=Li ! 1, where Li are the distances between theD6-D6 pairs.

In the former limit, the system is well described by aninteracting theory of the chiral fermions with the D4-branegauge field. These fermions are derivatively coupled to theD4-brane scalars and so these couplings vanish in the �0 !0 limit. As we shall demonstrate below, one can integrateout the gauge field along the lines of [16] to obtain gener-alized GN models. The symmetries associated with theflavor stacks U�Nf�L for the left-moving fermions, andU�Nf�R for the right-moving ones appear as globalsymmetries.

In the later limit �=Li ! 1, the ’t Hooft coupling � islarge and/or the flavor branes are close to each other. Thenthe five-dimensional gauge theory effects are large, and wecannot use the above description. However we now have analternate weakly coupled description [18] which involvesanalyzing the D6-brane dynamics in the near-horizon ge-ometry of the color D4-branes, which we will also discussbelow.

III. THREE COLLINEAR FLAVOR BRANES

We begin with simplest nontrivial examples of chiralsymmetry breaking which involve three stacks of flavorbranes placed along a straight line in the R3 spanned by thedirections (234). These will illustrate the general featuresof symmetry breaking and vacuum structure in our models.We shall rely heavily on these configurations while discus-sing later the general flavor brane setups in R3. With threecollinear branes, there are two physically distinct order-ings,5 given by D6� D6� D6 and D6� D6� D6 whichwe discuss below. In what follows, it will be useful tointroduce indices on the stacks of flavor branes. We labelthe D6-brane stack by 1 and the D6-brane stacks by labels�1 and �2.

A. D6�D6�D6

Consider a stack of D6-branes and two stacks ofD6-branes placed along the 4 direction in the order D6�D6� D6 as shown in Fig. 1. The stack of D6-branes islocated at the origin while the D6-brane stacks have coor-dinates L1 and �L2 in the 4 direction.

In the regime where the separation between the stack ofD6-branes and the stacks of D6-branes is much largercompared to the five-dimensional ’t Hooft coupling �,the effective action is given by

S �Zd5x

��

1

4g25

F2MN � �

3� ~x�qy1 �i@� � A��q1

� �3� ~x� L1x̂4�qy�1�i@� � A��q�1

� �3� ~x� L2x̂4�qy�2�i@� � A��q�2

�; (2)

where q1 is the left-handed fermion localized at stack 1,and q�1 and q�2 are the right-handed fermions localized atstacks �1 and �2 respectively. Also FMN ,M,N � 0, 1, 2, 3, 4,is the field strength for the five-dimensional gauge fieldAM, and A� � A0 � A1 are its light-cone components.

To leading order in �=Li, the dynamics of the fermionscan be studied by integrating out the five-dimensionalgauge field in the single gluon exchange approximation[16]. This can be easily done in Feynman gauge, and oneobtains an action with a four-fermion interaction

S �Zd2x�qy1 i@�q1 � q

y�1i@�q�1 � q

y�2i@�q�2

�g2

5

4�2

Zd2xd2y�G�x� y; L1��q

y1 �x� q�1�y���q

y�1�y�

q1�x�� �G�x� y; L2��qy1 �x� q�2�y���q

y�2�y� q1�x��;

(3)

where G�x; L� is the Euclidean propagator

G�x; L� �1

�x2 � L2�3=2: (4)

D6 D6 D6

0 L1−L2

2 1 1

FIG. 1. Collinear D6� D6� D6.

4Thus, we will not be considering processes involving theannihilation of the branes and the antibranes.

5Other configurations are related to the ones we discuss bycharge conjugation.

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In the four-fermion interactions, the dot represents colorindex contractions, while the flavor indices are contractedin the obvious way. This action is a generalization of thenonlocal GN model. Now we can further consider the locallimit of this model where the fields do not fluctuate overdistances of order Li, and we consider the theory at dis-tance scales much larger then Li.

6 In this limit the propa-gator essentially behaves as a delta function smeared overdistances Li, and using

Zd2xG�x; L� �

2�L

(5)

in (3), we get a generalized local GN model with action

S �Zd2x�qy1 i@�q1 � q

y�1i@�q�1 � q

y�2i@�q�2

�1

Nc

Zd2x

�2��L1�qy1 q�1��q

y�1 q1�

�2��L2�qy1 q�2��q

y�2 q1�

�: (6)

Note that the couplings of the four-fermion interactions aregiven by the ratios of the five-dimensional ’t Hooft cou-pling and the separations between the branes and theantibranes.

This model can be exactly solved in the large Nc limit.We briefly outline the steps involved in analyzing thevacuum structure. We begin by introducing auxiliary bo-sonic fields and writing the action as

S �Zd2x�qy1 i@�q1 � q

y�1i@�q�1 � q

y�2i@�q�2

�Zd2x

��

1

Nc

L1

2��j�1j

2 �1

Nc

L2

2��j�2j

2

� ��1qy1 q�1 ��2q

y1 q�2 � h:c:�

�: (7)

In the large Nc limit, the effective potential can be explic-itly evaluated by integrating out the fermions.7 Usingdimensional regularization, we get

Veff

Nc�

L1

2��j�1j

2 �L2

2��j�2j

2 �j�1j

2 � j�2j2

4�

�

�ln�j�1j

2 � j�2j2

�2

�� 1

�; (8)

where � is the renormalization scale.

The effective potential (13) has three extrema

j�1j j�2j Veff

A: �e�L1=� 0 � �2Nc4� e�2L1=�

B: 0 �e�L2=� � �2Nc4� e�2L2=�

C: 0 0 0

(9)

While C is the global maximum, the global minimum isdetermined by the relative magnitudes of L1 and L2.Without loss of generality, we take L1 < L2 for our dis-cussion,8 and thus the extremum A corresponds to thevacuum. The field �1 which corresponds to the fermionbilinear qy1 q�1, acquires a nonvanishing vacuum expecta-tion value, while �2 does not condense. As a result theclassical chiral symmetry U�Nf�1 �U�Nf��1 �U�Nf��2 isdynamically broken to U�Nf�diag�1;�1� �U�Nf��2. The fermi-ons q1 and q�1 acquire mass much smaller than the energyscale � [2], while q�2 continues to be massless. Note thatthe extremum B is tachyonic (along the �1 direction) withmass

m2 �@2Veff

@�1@ ��1

�2���L1 � L2�< 0: (10)

Many of these statements shall have interesting counter-parts in the strong coupling regime, which we discuss next.

When �=Li is large, the configuration admits a weaklycoupled dual description [16] which we briefly review.Consider a stack of D6-branes and a stack of D6-braneslocated at� L

2 along the 4 direction, respectively, as shownin Fig. 2. To study the behavior of the system at strongcoupling, consider a probe D6-brane in the near-horizongeometry of Nc D4-branes, which is given by the metricand the dilaton

D6

L/2−L/2

D6

0

FIG. 2. D6-D6 pair.

6As in [16], one can continue to work with the nonlocal model,however in the limit we work in, it gives the same result.

7We quote results for Nf � 1, these generalize in a straight-forward manner for arbitrary Nf. 8We shall treat the case L1 � L2 separately.

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ds2 �

�UR

�3=2����dx�dx� � �dx4�2 �

�UR

��3=2�dU2

�U2d�24�;

e� � gs

�UR

�3=4;

where

R3 � ��; (11)

U is the radial coordinate, and �4 labels the angulardirections in (56789). The analysis in [16] revealed asolution where the D6-brane extends in the (01) directions,wraps the four sphere labeled by �4 and has a wormholelike profile U�x4� which asymptotes to the undeformed D6and D6-branes at infinity as shown in Fig. 3. The wormholeconnecting the branes manifestly exhibits chiral symmetrybreaking. In fact, one can show that this configuration hasless energy than the undeformed D6-D6 pair with

�E�L� � ��2

L4 : (12)

Our configuration admits two wormhole solutions, be-cause the D6-branes in stack 1 can connect onto either ofthe two stacks of D6-branes at �1 and �2 (see Figs. 4 and 5).

From (12) we see that energetics requires that the D6-branes in stack 1 connect to the closer D6-brane stack at �1,while �2 remains disconnected, as depicted in Fig. 4. Thusas in the weak coupling regime, the classical chiral sym-metry U�Nf�1 �U�Nf��1 �U�Nf��2 is broken toU�Nf�diag�1;�1� �U�Nf��2.

Note that the strong coupling counterpart of vanishing�2 condensate is the absence of a wormhole connectingstacks 1 and �2. This illustrates an interesting point aboutgeneral D-brane configurations at strong coupling. Atstrong coupling, all ‘‘condensates’’ except those corre-sponding to the pairs of branes and antibranes which areconnected by wormholes vanish. In all examples we willconsider later in the paper, we find that the weak coupling

vacuum structure has behavior which seems to be reminis-cent of this property.

The energetically disfavored configuration in Fig. 5 hasthe same symmetries as the extremum B in (9) and it isnatural to think of it as the strong coupling continuation ofB. We note that the corresponding wormhole configurationis metastable, while B in (9) has a tachyon given by (10).

Finally, we consider the particular case when L1 �L2 � L. In the weak coupling regime, from (3) we seethat the nonlocal GN model (and consequently the localGN model) has an enhanced U�Nf�L �U�2Nf�R chiralsymmetry under which

q�1

q�2

� �

transforms in the 2Nf dimensional representation ofU�2Nf�R. Thus

�1

�2

� �

also transforms in the same way. In this case, the vacuumconfiguration of the GN model is given by

j�1j2 � j�2j

2 � �2e�2L=�; (13)

which leads to the vacuum energy

Veff � ��2Nc

4�e�2L=�: (14)

−L/2 L/2

D6 D6

FIG. 3. The wormhole.

112

FIG. 5. Metastable configuration.

1 12

FIG. 4. Vacuum configuration.

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Thus the U�Nf�L �U�2Nf�R chiral symmetry is dy-namically broken to U�Nf�diag�L;R� �U�Nf�R. At strongcoupling, this symmetry breaking manifests itself in thefact that the wormhole can connect the D6-brane witheither of the two D6-branes as they both have the sameenergy.

B. D6�D6�D6

The other configuration involves placing the branes andantibranes in the order D6� D6� D6 as shown in Fig. 6.

We place the stack of D6-branes at the origin and the twostacks of D6-branes at L1 and L2, where L2 > L1. Theanalysis can be easily carried out making use of the resultsof the previous section. At weak coupling, the couplingsbetween the left and right-handed fermions depend only ofthe distances between the brane and antibrane pairs. Hencethe vacuum energy density is identical to the previous caseand is given by (8). From (9) we see that �1 has a non-vanishing vacuum expectation value, and so the classicalU�Nf�1 �U�Nf��1 �U�Nf��2 chiral symmetry is dynami-cally broken to U�Nf�diag�1;�1� �U�Nf��2. At strong cou-pling, the D6-brane is connected to the anti D6-branecloser to it through a wormhole as shown in Fig. 7.

IV. MORE PATTERNS OF CHIRAL SYMMETRYBREAKING

Having analyzed chiral symmetry breaking in three col-linear stacks of flavor branes, we now consider moregeneral patterns involving D-brane configurations whichhave equal numbers of D6 and D6-branes. First, we shallconsider four collinear stacks of flavor branes. Then weshall consider two simpleD-brane configurations spanningan R2 in R3. Apart from being useful models to studypatterns of chiral symmetry breaking, these constructionswill be helpful in analyzing the general pattern of symme-try breaking involving arbitrary configurations of flavorbranes in R3. As before we find that the patterns of chiralsymmetry breaking are similar at weak and strongcoupling.

A. Four collinear flavor branes

In this section we shall consider configurations with twostacks of D6-branes, and two stacks of D6-branes placedalong the 4 direction. We shall label the D6-brane stacks byindices 1, 2, and denote their coordinates by L1, L2 re-spectively. Similarly, we denote the D6-brane stacks by �1,�2, and denote their coordinates by L�1, L�2 respectively. Wealso use Li �j to denote the distance between the i-thD6-brane and the �j-th D6-brane.

We first consider the weak coupling limit and obtain theeffective potential. As in the three stack case, to leadingorder in �=Li �j, the dynamics of the fermions at lengthscales much greater than the D-brane separations can bedescribed by a local GN model obtained by integrating outthe five-dimensional gauge field in the single gluon ex-change approximation. Once again, the effective field the-ory is a generalized GN model with two left-moving andtwo right-moving fermions, and is given by

S �Zd2x

�Xi

qyi i@�qi �X

�j

qy�j i@�q �j

�1

Nc

Xi; �j

2��Li �j�qyi q �j��q

y�j qi�

�: (15)

After introducing auxiliary fields �i �j corresponding tofermion bilinears qyi q �j and integrating out the fermions,one obtains the effective potential in the large Nc limit

Veff

Nc�Xi; �j

Li �j2��

j�i �jj2 �

������p

8�

�ln��

�����p

2�2

�� 1

�

��

�����p

8�

�ln��

�����p

2�2

�� 1

�; (16)

where

D6 D6 D6

1 1 2

0 L1 L2

FIG. 6. Collinear D6� D6� D6.

1 1 2

FIG. 7. -

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�Xi; �j

j�i �jj2;

� ��Xi; �j

j�i �jj2

�2� 4j�1�1�2�2 ��1�2�2�1j

2:(17)

The properties of the vacuum configuration obtained byminimizing (16) are closely related to the ordering of thestacks (i.e., the relative magnitudes of Li �j). Upto chargeconjugation, there are three distinct orderings D6� D6�D6� D6, D6� D6� D6� D6 and D6� D6� D6�D6. We discuss each case separately, also illustrating therelationship to wormhole solutions at strong coupling.

1. D6�D6�D6�D6

For this ordering given by Fig. 8, the minimum of theeffective potential (16) is given by

j�1�2j � �e�L1�2=�; j�2�1j � �e�L2�1=�;

�1�1 � 0; �2�2 � 0:(18)

Thus the classical U�Nf�1 �U�Nf�2 �U�Nf��1 �U�Nf��2chiral symmetry is broken toU�Nf�diag�1;�2� �U�Nf�diag�2;�1�.At strong coupling the wormhole configuration with thelowest energy is shown in Fig. 9, which exhibits the samesymmetry breaking pattern.9

2. D6�D6�D6�D6

In this case given by Fig. 10, the minimum of theeffective potential (16) is given by

j�1�1j � �e�L1�1=�; j�2�2j � �e�L2�2=�;

�1�2 � 0; �2�1 � 0;(19)

and the chiral symmetry is dynamically broken to

U�Nf�diag�1;�1� �U�Nf�diag�2;�2�. Again, the energetically fa-vorable wormhole configuration is shown in Fig. 11, whichexhibits the same symmetry breaking pattern.

3. D6�D6�D6�D6

This case as shown in Fig. 12 exhibits more interestingstructure of chiral symmetry breaking. The nature of thevacuum structure explicitly depends on the separations ofthe stacks. At weak coupling, simple energetics using theeffective potential (16) shows that for

D6 D6 D6 D6

2 1 2 1

FIG. 8. Collinear D6� D6� D6� D6.

2 1 2 1

FIG. 9. -

D6 D6 D6 D6

2 2 1 1

FIG. 10. Collinear D6� D6� D6� D6.

2 2 1 1

FIG. 11. -

9The difference between the radii of the two wormholes at thepoint of closest approach is much larger than

������0p

for sufficientlylarge values of �=Li �j, thus the gravity construction can betrusted.

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e�2L1�1=� � e�2L2�2=� > e�2L1�2=� � e�2L2�1=�; (20)

the vacuum has condensates

X : j�1�1j � �e�L1�1=�; j�2�2j � �e�L2�2=�;

�1�2 � 0; �2�1 � 0:(21)

On the other hand, when the inequality in (20) is reversedthe condensates are

Y : j�1�2j � �e�L1�2=�; j�2�1j � �e�L2�1=�;

�1�1 � 0; �2�2 � 0:(22)

Note that X and Y represent two distinct phases, withdifferent symmetries of the vacuum.

Similarly in the strong coupling regime, there are twopatterns of chiral symmetry breaking determined by theenergetics of the wormhole configurations. For

1

L41�1

�1

L42�2

>1

L41�2

�1

L42�1

; (23)

the energetically favored configuration is shown in Fig. 13.On the other hand, the configuration in Fig. 14 is energeti-cally favored when the inequality in (23) is reversed. Theconfigurations in Figs. 13 and 14 preserve the same sym-metries as the phases X and Y respectively. They should bethought of as the strong coupling continuations of thesephases.

From the structure of the inequalities (20) and (23), weexpect that at an arbitrary coupling the vacuum is in aphase determined by an inequality of the form

f��L1�1

�� f

��L2�2

�> f

��L1�2

�� f

��L2�1

�; (24)

where f�x� is a monotonically increasing function of x, and

f�x� !�e�2=x; as x! 0;x4; as x! 1:

B. FOUR FLAVOR BRANES SPANNING TWODIMENSIONS

So far we have considered stacks of flavor branes that liealong a straight line in R3. In order to understand thegeneral patterns of chiral symmetry breaking, it is usefulto look at flavor brane configurations that are not collinear.With this in mind, we analyze chiral symmetry breaking inthe two simplest D-brane configurations spanning an R2 inR3, both at weak and strong coupling. These two configu-rations are given by two stacks of D6-branes and two stacksof D6-branes lying in a plane in R3, such that they form arectangle. There are two distinct orderings of the D-branes(upto charge conjugation), as we discuss below.

1. D6�D6�D6�D6 rectangle

We consider the D6 and D6-branes as depicted in Fig. 15forming a rectangle in R2. In the diagram, each stack of D6(D6)-branes is represented by a point, given by a vertex ofthe rectangle.

Proceeding as before, at weak coupling, the nonvanish-ing condensates are given by

j�1�2j � �e�L1�2=� � �e�L2�1=� � j�2�1j;

�1�1 � 0; �2�2 � 0:(25)

Similarly in the strong coupling limit, there are two worm-holes connecting the corresponding vertices of therectangle.

D6 D6 D6 D6

2 2 1 1

FIG. 12. Collinear D6� D6� D6� D6.

2 2 1 1

FIG. 13. -

2 12 1

FIG. 14. -

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2. D6�D6�D6�D6 rectangle

Finally, we consider the other possible configuration ofD6 and D6-branes which form a rectangle in R2 as de-picted in Fig. 16. At weak coupling, if L1�1 > L1�2, thenonvanishing condensates are given by

j�1�2j � �e�L1�2=� � �e�L2�1=� � j�2�1j;

�1�1 � 0; �2�2 � 0:(26)

On the other hand, if L1�2 > L1�1, the nonvanishing conden-sates are given by

j�1�1j � �e�L1�1=� � �e�L2�2=� � j�2�2j;

�1�2 � 0; �2�1 � 0:(27)

At strong coupling, there are corresponding wormhole

solutions along the corresponding edges of the rectanglein an obvious way.

Now when L1�1 � L1�2, the D-brane configuration has anenhanced U�2Nf�L �U�2Nf�R chiral symmetry classi-cally, which gets dynamically broken to U�Nf�diag�L;R� �

U�Nf�diag�L;R� both at weak and strong coupling.

V. GENERAL PATTERNS OF CHIRAL SYMMETRYBREAKING AND RESTORATION AT HIGH

TEMPERATURE

From the above examples, one can see the generalpatterns of chiral symmetry breaking both at weak cou-pling as well as at strong coupling. In the weak couplinglimit, in the single gluon exchange approximation, wealways get a generalized nonlocal GN model.Furthermore restricting to length scales much larger thatthe separations between the flavor branes, we end up get-ting local GN models. Interactions in these models dependonly on the distances Li �j between a stack of D6-branes andanother stack of D6-branes placed at Li and L �j respec-tively, through the corresponding GN coupling �=Li �j. Inparticular, the coupling between a D6-D6 pair does notdepend on the angular orientation of the two stacks in R3.Thus in general, we will have generalized GN models withall possible couplings, and the vacuum configuration willbe determined by the energetics;, namely, the vacuumconfiguration will have nonvanishing condensates of onlythose fermion bilinears connecting the D6-D6 pairs suchthat the energy is at its global minimum. On the basis of theconstructions we have described, we expect that in anyD-brane configuration with equal number of stacks of D6and D6-branes, the vacuum configuration will always havenonvanishing condensates such that all the fermions getmass dynamically. We also expect that if there are P stacksof D6-branes and Q stacks of D6-branes such that P � Q,the vacuum configuration will have jP�Qj stacks ofunpaired D6 (D6� branes, depending on whether P>Q�P<Q�. For certain D-brane configurations, there canbe enhanced chiral symmetries. For example, if a stack ofD6-branes is equidistant from K stacks of D6-branes in R3,then the configuration has an enhanced U�Nf�L �U�KNf�R chiral symmetry, which is dynamically brokento U�Nf�diag�L;R� �U��K � 1�Nf�R. At weak coupling, onehas a condensate given by a fermion bilinear involving theleft-moving fermions from the D6-brane stack and theright-moving fermions from any one of the K D6-branestacks. In the strong coupling limit, the analogues of con-densates are wormholes. The vacuum configuration is de-termined by the the energetics of wormholes connectingpairs of D6-branes and D6-branes.

We have so far analyzed the chiral symmetry breakingwhich occurs at infinite Nc at zero temperature. If we nowstart heating the system to higher and higher temperatures,

D6 D6

D6D6

1 1

22

FIG. 16. D6� D6� D6� D6 rectangle.

D6 D6

D6D6

1 2

12

FIG. 15. D6� D6� D6� D6 rectangle.

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the chiral symmetry gets restored as we now explain. In theGN model, it is well known that the chiral symmetry isrestored at a temperature Tc given by [19–21]

Tc � 0:57mf; (28)

where mf is the dynamically generated mass for the fer-mions at zero temperature. In fact, this phase transition issecond order in nature. Thus in the weak coupling limitwhere we obtain the generalized GN models, there is chiralsymmetry restoration at temperatures given by

Tc�Li �j� � 0:57�e�Li �j=����: (29)

So as we heat up the system from zero temperature, thecondensates start evaporating as soon as the correspondingphase transition temperatures are attained. Thus at a suffi-ciently high temperature, all the symmetries are restored.The strong coupling analysis is similar, with the phasetransition temperature given by [16]10

Tc�Li �j� �0:205

Li �j: (30)

However, now the phase transition is first order in nature.Thus we see that the D-brane configurations we have

discussed above exhibit interesting patterns of chiral sym-

metry breaking. The construction of generalized GN mod-els from string theory opens up several directions whichmight be worth looking at. It would be nice to understandthe spectrum of bound states, solitons and their interactionsin these theories both at weak and strong coupling, andpossibly also at finite values of the coupling. Also onemight be interested in analyzing the integrability of thesetheories to compute the S-matrix. It might also be useful toanalyze configurations where there are directions trans-verse to both the color and the flavor branes, which enableus to give tunable masses to the fermions. These techniquesare generalizable to other D-brane configurations andmight be useful in understanding generalized models ofchiral symmetry breaking in other dimensions, thus gen-eralizing the results in [24]. Finally, one can also analyzegeneralized models of QCD2 by wrapping the D4-braneson T3 [16]11 as mentioned before.

ACKNOWLEDGMENTS

We would like to thank S. Giddings, J. Maldacena, P.Ouyang, and J. Polchinski for useful discussions. The workof A. B. and A. M. is supported by NSF Grant No. PHY-0503584, and DOE Grant No. DE-FG02-91ER40618respectively.

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11A construction of QCD2 using a different D-brane configu-ration has been done in [25].

10This has been analyzed for the D4�D8�D8 case in[22,23].

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