# Inverse magnetic catalysis in holographic models of QCD

###### Abstract

We study the effect of magnetic field on the critical temperature of the confinement-deconfinement phase transition in hard-wall AdS/QCD, and holographic duals of flavored and unflavored super-Yang Mills theories on . For all of the holographic models, we find that decreases with increasing magnetic field , consistent with the inverse magnetic catalysis recently observed in lattice QCD for . We also predict that, for large magnetic field , the critical temperature , eventually, starts to increase with increasing magnetic field and asymptotes to a constant value.

^{†}

^{†}institutetext: Department of Physics, University of Illinois, Chicago, Illinois 60607, USA\arxivnumber

## 1 Introduction

Recently, the study of the QCD phase diagram for magnetic field has attracted considerable attention Gusynin:1994re (); Miransky:2002rp (); Mizher:2010zb (); Fraga:2008um (); Gatto:2010pt (); Gatto:2010qs (); Osipov:2007je (); Kashiwa:2011js (); Klimenko:1992ch (); Alexandre:2000yf (); Filev:2007gb (); Albash:2007bk (); Alam:2012fw (); Johnson:2008vna (); Bergman:2008sg (); Evans:2010xs (); Preis:2010cq (); Ballon-Bayona:2013cta (); Bali:2011qj (), see Kharzeev:2012ph () for a review. The main motivation for studying the QCD phase diagram under external magnetic field stems from the fact that strong magnetic field is produced in heavy ion collisions experiments at RHIC and LHC Skokov:2009qp (), due to the charged spectator particles, which has interesting effects on the quark-gluon plasma created during these heavy ion collision experiments Kharzeev:2007jp (); Fukushima:2008xe (); Kharzeev:2004ey (); Voloshin:2008jx (); Abelev:2009uh (); Selyuzhenkov:2011xq (), see Kharzeev:2012ph () for a review. A strong magnetic field is also produced during the electroweak phase transition of the early Universe Vachaspati:1991nm (), and relatively weaker magnetic field is produced in the interior of dense neutron stars Duncan:1992hi ().

Another motivation comes from the fact that the study of the QCD phase diagram with magnetic field is amenable to numerical simulations of QCD on the lattice, without facing the sign problem of lattice QCD that exist in the case of non-zero baryon chemical potential , creating an opportunity to compare the holographic and low energy effective models of QCD directly with QCD itself.

Regarding the study of the QCD phase diagram for magnetic field , most of the models for QCD Gusynin:1994re (); Miransky:2002rp (); Mizher:2010zb (); Fraga:2008um (); Gatto:2010pt (); Gatto:2010qs (); Osipov:2007je (); Kashiwa:2011js (); Klimenko:1992ch (); Alexandre:2000yf (), including the holographic ones Filev:2007gb (); Albash:2007bk (); Alam:2012fw (); Johnson:2008vna (); Bergman:2008sg (); Evans:2010xs (), have studied chiral-symmetry-restoration transition and have predicted that the critical temperature of the transition increases with increasing magnetic field at zero chemical potential . This enhancing effect of the magnetic field on the critical temperature has been termed magnetic catalysis. However, recent lattice QCD result Bali:2011qj () has indicated the opposite effect, that is, the critical temperature decreases with increasing magnetic field , for and zero chemical potential . This inhibiting effect of the magnetic field on the critical temperature has been termed inverse magnetic catalysis.

Even though, the recent lattice QCD result Bali:2011qj () has also indicated that the confinement-deconfinement and chiral symmetry breaking phase transitions occur at the same critical temperature at least for , most holographic calculations so far Filev:2007gb (); Albash:2007bk (); Alam:2012fw (); Johnson:2008vna (); Bergman:2008sg (); Evans:2010xs (); Preis:2010cq () have been concerned only with of the chiral symmetry breaking phase transition.

However, recently, reference Ballon-Bayona:2013cta (), inspired by the recent lattice QCD result Bali:2011qj (), has a priori assumed confinement and chiral symmetry breaking transitions to occur at the same critical temperature in Sakai-Sugimoto model, and has argued that, in this case, must be a decreasing function of , consistent with the recent lattice QCD result Bali:2011qj (), but has not provided a direct computation of .

In this paper, we give a direct computation of the critical temperature of the confinement-deconfinement phase transition in hard-wall AdS/QCD, and holographic duals of flavored and unflavored SYM on where is a circle of length in one of the spatial directions. (Note that, at finite temperature , is really where is the thermal circle with length .) Also, note that, since the fermions of both the flavored and unflavored SYM on obey antiperiodic boundary conditions around the circle , they acquire a tree-level mass . The scalars are periodic around the circle, hence they acquire masses only at the quantum level through their couplings to the fermions CasalderreySolana:2011us (). The gluons, however, do not acquire masses, therefore, at low-energy, both flavored and unflavored SYM on reduce to pure 3D Yang-Mills theory.

It is well known that both flavored and unflavored super-Yang Mills theories (SYM) on flat spacetime are not confining gauge theories. However, they can be made confining in the large- limit by placing them on a compact space with length , and the confinement-deconfinement phase transition occurs at critical temperature Witten:1998zw (); Surya:2001vj (); Sonnenschein:2000qm (), see CasalderreySolana:2011us (); Natsuume:2014sfa () for a review. In our case, the compact space is , that is, we compactify one of the spatial dimensions into a circle of length .

The confinement-deconfinement phase transition both in flavored and unflavored SYM on is holographically modeled by a phase transition between a black hole solution with radius of horizon , and -soliton solution which smoothly ends at . However, to study the confinement-deconfinement phase transition in QCD on at strong coupling, we use the hard-wall AdS/QCD model where the confinement-deconfinement phase transition, of QCD on , is holographically modeled by a phase transition between a black hole solution with radius of horizon , and thermal- solution with hard-wall IR cut-off .

We derive the corresponding thermal- solution which is the holographic dual to the confined phase of QCD on by starting from a black hole solution, which corresponds to the deconfined phase of strongly coupled QCD on , by setting the mass of the black hole to zero Lee:2009bya (). And, we derive the corresponding -soliton solution, which is the holographic dual to the confined phase of flavored and unflavored SYM on , by ”double Wick rotating” a black hole solution CasalderreySolana:2011us (); Natsuume:2014sfa ().

In this paper, we use two black hole solutions in the presence of constant magnetic field . First, we use the black hole solution in the presence of constant magnetic field found in D'Hoker:2009bc () to study the confinement-deconfinement phase transition in strongly coupled QCD on and unflavored SYM on . Then, we use the black hole solution in the presence of constant magnetic field , including the backreaction of flavor or D7-branes for , found in Ammon:2012qs () to study the confinement-deconfinement phase transition in flavored SYM on .

The effect of magnetic field on different observables has also been studied in Basar:2012gh (); Mamo:2013efa (); Arciniega:2013dqa (); Critelli:2014kra (); Rougemont:2014efa () using the backreacted black hole solution of D'Hoker:2009bc () without flavor D7-branes.

Depending on the specific holographic models to QCD, various length and energy scales appear throughout this paper. Some of the relevant length and energy scales are: the radius of the spacetime which we set to , the radius of the black hole horizon which is related to the Hawking temperature of the black hole (which is dual to the field theory temperature ), the radial position of the canonical singularity of the -soliton for flavored and unflavored SYM on , the radial position of the hard-wall in the thermal- solution for the hard-wall AdS/QCD, and an external magnetic field in the range of for the hard-wall AdS/QCD model and for the flavored SYM on .

The outline of this paper is as follows: In section 2, we write down the 5-dimensional Einstein-Maxwell action (which will be used to study confinement-deconfinement phase transition in hard-wall AdS/QCD and holographic dual of unflavored SYM on ) including the Gibbons-Hawking surface term and the appropriate counter terms. We then review the black hole solution in the presence of constant magnetic field found in D'Hoker:2009bc (). Then, starting from the black hole solution, which corresponds to the deconfined phase of strongly coupled QCD on flat spacetime and unflavored SYM on , we derive the corresponding thermal- and -soliton solutions, which correspond to the confined phases of strongly coupled QCD on flat spacetime and unflavored SYM on , respectively. We also determine the on-shell Euclidean actions (free energies) for the black hole, thermal-, and -soliton solutions.

In section 3, we compute the critical temperature of the confinement-deconfinement phase transition in hard-wall AdS/QCD, and holographic duals of flavored and unflavored SYM on . We first compute the critical temperature of the confinement-deconfinement phase transition in hard-wall AdS/QCD by requiring the difference between the black hole and thermal- on-shell Euclidean actions vanish at . Then, we compute the critical temperature of the confinement-deconfinement phase transition in holographic dual of unflavored SYM on by requiring the difference between the black hole and -soliton on-shell Euclidean actions vanish at . Finally, using the insight we gained, in computing the critical temperature of the confinement-deconfinement phase transition in holographic dual of unflavored SYM on , we compute the critical temperature of the confinement-deconfinement phase transition in holographic dual of flavored SYM on by constructing the backreacted -soliton solution from the backreacted black hole metric of model, with magnetic field , found in Ammon:2012qs ().

In Appendix A, we compute the critical temperature of the confinement-deconfinement phase transition in hard-wall AdS/QCD for large magnetic field .

## 2 Einstein-Maxwell theory in 5D

In this section, we review elements of Einstein-Maxwell theory in 5D which will, subsequently, be used to study confinement-deconfinement phase transitions in hard-wall AdS/QCD and holographic dual of unflavored SYM on .

The action of five-dimensional Einstein-Maxwell theory with a negative cosmological constant is D'Hoker:2009bc ()^{*}^{*}*Our conventions here are such that the Ricci scalar here is related to the Ricci scalar there (given in D'Hoker:2009bc ()) by .

(1) |

where the bulk action is

(2) |

and the boundary action is

(3) |

In the boundary action (3), the first term is the Gibbons-Hawking surface term, and the other terms are the counter terms needed to cancel the UV() divergences in the bulk action in accordance with the holographic renormalization procedure Skenderis:2002wp (). Note that the counter terms are entirely constructed from the induced metric on the boundary surface at , that is,

(4) |

And, is the trace, with respect to , of the extrinsic curvature of the boundary given by . Using the matrix formula Natsuume:2014sfa (), we can write D'Hoker:2009bc (); Natsuume:2014sfa ().

In addition to the Bianchi identity, the field equations are D'Hoker:2009bc ()

(5) |

(6) |

From now on we set the AdS radius to unity, that is, .

Turning on a constant bulk magnetic field, in the -direction, , where the bulk gauge potential ,^{†}^{†}†Note that the bulk gauge potential and the corresponding bulk magnetic field are dual to the boundary gauge potential and the corresponding boundary magnetic field which couple to the charged particles of the field theory living at the boundary. Later on, when we start discussing specific holographic models to QCD, we will specify the type of global gauge group and the associated boundary current . which solves Maxwell’s equation (6), and contracting Einstein’s field equation (5), one can find the Ricci scalar to be

(7) |

So, the on-shell Euclidean action (which can be found from the Lorentzian action (1) by analytic continuation in the imaginary time direction, i.e., ) takes the form

(8) |

where the on-shell Euclidean bulk action is

(9) |

and, the on-shell Euclidean boundary action is

(10) |

and, is the UV cut-off while is the radius of the horizon for a black hole solution, and IR cut-off for a thermal- or -soliton solutions. From now on we set . Also, note that the on-shell Euclidean action is related to the free energy by .

### 2.1 Background solutions with

In this subsection, we review the black hole solution in the presence of constant magnetic field found in D'Hoker:2009bc () which corresponds to the deconfined phase of strongly coupled QCD on (flat spacetime) and unflavored SYM on . Then, starting from the black hole solution, by setting the mass of the black hole to zero Lee:2009bya (), we derive the corresponding thermal- solution which is the holographic dual to the confined phase of strongly coupled QCD on flat spacetime . And, by ”double Wick rotating” the black hole solution CasalderreySolana:2011us (); Natsuume:2014sfa (), we derive the corresponding -soliton solution which is the holographic dual to the confined phase of unflavored and strongly coupled SYM on .

#### 2.1.1 Black hole

For and electric charge density , the perturbative black hole solution in powers of , up to an integration constant is given in Eq. 6.1 and 6.16 of Ref. D'Hoker:2009bc (). Here, we set the electric charge density and fix the integration constant so that the perturbative solution in powers of matches the near boundary solution which is also given in Eq. 4.4, 4.5 and 6.16 of D'Hoker:2009bc (). Therefore, the black hole solution in Eq. 6.1 and 6.16 of Ref. D'Hoker:2009bc (), for vanishing electric charge density and , takes the form

(11) | |||||

and, the Hawking temperature becomes

(12) |

where is the mass of the black hole, , the radius of the horizon is defined by requiring , is the Hawking temperature of the black hole, and is the length of the thermal circle which acquired a fixed value as a function of in order to avoid the canonical singularity at the horizon . One can also check that (11) indeed satisfies the Einstein field equation (5) or its contracted version (7).

#### 2.1.2 Thermal-

The thermal- solution can be found from a black hole solution by setting the mass of the black hole to zero, see Lee:2009bya () for the electrically charged black hole case. Therefore, from the black hole solution for (11), we can determine the thermal-AdS solution for by setting the mass of the black hole ,

(13) | |||||

#### 2.1.3 -soliton

The -soliton solution Horowitz:1998ha (); Surya:2001vj () can be determined from the black hole solution (11) by ”double Wick rotation” and CasalderreySolana:2011us (); Natsuume:2014sfa (). Therefore, for the -soliton solution is,

(14) | |||||

where is the length of the circle in the compactified direction which is arbitrary for the black hole solution but in order to avoid the canonical singularity at (where is defined by requiring ), it acquires a finite value which is given in terms of for the -soliton solution (14).

### 2.2 On-shell Euclidean actions with

In this subsection, we determine the on-shell Euclidean actions (free energies) for the black hole, thermal-, and -soliton solutions. And, we compute the difference between the on-shell Euclidean actions of the deconfining geometry (which is the black hole geometry for both hard-wall AdS/QCD and holographic dual of unflavored SYM on ) and the confining geometry (which is the thermal- geometry for hard-wall AdS/QCD, and the -soliton geometry for holographic dual of unflavored SYM on ).

#### 2.2.1 Black hole

The on-shell Euclidean action (8) for the black hole solution with (11) is

(15) |

where the on-shell Euclidean bulk action of the black hole for is

(16) |

and the on-shell Euclidean boundary action of the black hole for is

(17) |

The bulk action (16) (after using the black hole metric for (11), using the fact that , evaluating the integrals, and simplifying) become

(18) |

which diverges when , and the boundary action (17) becomes

(19) |

where we ignored terms which goes to zero in the limit. Also note that (19) diverges when , but the sum of (18) and (19) is finite. Hence, the black hole on-shell Euclidean action (15) is

(20) |

#### 2.2.2 Thermal-

The on-shell Euclidean action (8) for the thermal- solution with (13) is

(21) |

where the on-shell Euclidean bulk action of the thermal- for is

(22) |

and the on-shell Euclidean boundary action of the thermal- for is

(23) |

The thermal- bulk action (22) (after using the thermal- metric for (13), using the fact that , evaluating the integrals, and simplifying) becomes

(24) |

which diverges when , and the thermal- boundary action (23) becomes

(25) |

which diverges as well when . But, the sum of (24) and (25) is finite. Hence, the thermal on-shell Euclidean action (21) becomes

(26) |

where we used at the boundary .

#### 2.2.3 -soliton

Since, black hole (11) and -soliton (14) are equivalent Euclidean geometries, their on-shell Euclidean actions take similar form. In fact, the on-shell Euclidean action of -soliton can be found by merely replacing by in the on-shell Euclidean action for the black hole Natsuume:2014sfa (). Therefore, the difference between the on-shell actions of the black hole (20) and of -soliton geometries is simply

(28) |

## 3 Confinement-deconfinement phase transition in holographic models of QCD for

In this section, we compute the critical temperature of the confinement-deconfinement phase transition in hard-wall AdS/QCD, and holographic duals of flavored and unflavored SYM on .

### 3.1 Confinement-deconfinement phase transition in hard-wall AdS/QCD

For hard-wall AdS/QCDD'Hoker:2009bc ()^{‡}^{‡}‡For the hard-wall AdS/QCD model, the bulk magnetic field and the corresponding bulk gauge potential are dual to the boundary magnetic field and the corresponding boundary gauge potential of the subgroup of the global flavor group of QCD. And, the boundary vector gauge potential couples to the boundary conserved vector current ., we determine the critical temperature of the confinement-deconfinement phase transition by first determining the critical radius of the horizon from the condition that the difference between the Euclidean actions for the black hole and thermal- solutions vanish at , i.e., . For , requiring in (27), we find the constraint equation for the critical radius of the horizon to be

(29) |

which can be solved numerically for . Note that, we have fixed in (27), so that (29) reduces to the constraint equation found in Herzog:2006ra (); BallonBayona:2007vp () at , which is . Once we find the solution for from the constraint equation (29), we can use (12) to find . The plot of the numerical solution for for is given in Fig. 1, and the numerical plot clearly shows that decreases with increasing in agreement with the inverse magnetic catalysis recently found in lattice QCD for Bali:2011qj ().

### 3.2 Confinement-deconfinement phase transition in holographic dual of unflavored SYM on

For the holographic dual of unflavored SYM on ^{§}^{§}§For the unflavored SYM case, the bulk magnetic field and the corresponding bulk gauge potential are dual to the boundary magnetic field and the boundary gauge potential of the subgroup of the global R-symmetry group of SYM which couples to the boundary R-current ., we study the confinement-deconfinement phase transition by using the same Einstein-Maxwell action in as we used for the hard-wall AdS/QCD, and the analysis will be similar to the hard-wall AdS/QCD case but, for the unflavored SYM on case, we compactify the black hole solution in the direction into a circle of length , and compare its free energy with the free energy of -soliton solution (14) instead of the thermal- solution (13) that we used for the hard-wall AdS/QCD.

It is easy to see from (28) that the critical radius of the horizon at which is given by . Therefore, using (12), the critical temperature becomes,

(30) |

Fixing so that we reproduce the correct result , and fixing from the value of at , which we denote as , we can write (30) in terms of as

(31) |

where we defined the critical magnetic field and is the radius of the AdS spacetime. From (31), it is easy to see that is a decreasing function with increasing in qualitative agreement with the recent lattice QCD result Bali:2011qj ().

### 3.3 Confinement-deconfinement phase transition in holographic dual of flavored SYM on

In the previous subsection, we have studied the confinement-deconfinement phase transition in the holographic dual of unflavored SYM on using the backreacted black hole and -soliton geometries, from which, we can infer a simple prescription of finding in any backreacted black hole and -soliton based models.

The prescription is, first find the backreacted metric and the Hawking temperature of the black hole, then the critical temperature is simply given by where can be fixed by the value of .

Therefore, using this prescription, we can determine of the holographic dual of flavored SYM on . To this end we will use the backreacted metric of model given in Ammon:2012qs () where the authors have also found the Hawking temperature including the backreaction of -branes and magnetic field ^{¶}^{¶}¶For the model, we use a bulk magnetic field which corresponds to the Kalb-Ramond two form field . And, also note that for the DBI action of the probe brane, the gauge invariant and physically significant field strength is given by which is the sum of Maxwell’s field strength and the Kalb-Ramond two form field . And, the bulk magnetic field and the corresponding bulk gauge potential are dual to the boundary magnetic field and the boundary gauge potential of the subgroup of the global flavor group of the supersymmetric field theory which couples to the boundary vector current , see, for example, Ref. Hoyos:2011us (). For our case, but . to be, see Eq. 3.1 of Ammon:2012qs ()^{∥}^{∥}∥In Eq. 3.1 of Ammon:2012qs (), is written in terms of and . Here, we have used Eq. 2.35 and 3.4 of Ammon:2012qs () (which relates and , respectively) in order to write explicitly in terms of , , and .,

(32) |

Since the on-shell Euclidean action of the black hole solution (including the backreaction of -branes and magnetic field ) has also been given in Eq. 3.14 of Ammon:2012qs (), in order to find the corresponding Euclidean action of the -soliton, all we need to do is replace by in Eq. 3.14 of Ammon:2012qs (). Hence, the difference between the two on-shell Euclidean actions vanishes at the critical radius of the horizon . And, using in (32), the critical temperature of the confinement-deconfinement phase transition in flavored SYM on becomes

(33) |

which can be written in terms of as

(34) |

where is the value of the ’t Hooft coupling fixed at the horizon , that is, where is the string coupling constant and is the dilaton scalar field.

Note that, for , (34) reduces to

(35) |

which is in a qualitative agreement with the hard-wall AdS/QCD Kim:2007em (), functional renormalization group study of QCD Braun:2006jd (), and lattice QCD Karsch:2000kv () results which show that decreases with increasing number of flavors at zero magnetic field and chemical potential .

We have plotted (34) in Fig. 2 which clearly shows that decreases with increasing in agreement with the inverse magnetic catalysis recently found in lattice QCD for Bali:2011qj ().