Magnetic catalysis and the chiral condensate in holographic QCD

We investigate the effect of a non-zero magnetic field on the chiral condensate using a holographic QCD approach. We extend the model proposed by Iatrakis, Kiritsis and Paredes in arXiv:1010.1364 that realises chiral symmetry breaking dynamically from 5d tachyon condensation. We calculate the chiral condensate, magnetisation and susceptibilities for the confined and deconfined phases. The model leads, in the probe approximation, to magnetic catalysis of chiral symmetry breaking in both confined and deconfined phases. In the chiral limit, $m_q=0$, we find that in the deconfined phase a sufficiently strong magnetic field leads to a second order phase transition from the chirally restored phase to a chirally broken phase. The transition becomes a crossover as the quark mass increases. Due to a scaling in the temperature, the chiral transition will also be interpreted as a transition in the temperature for fixed magnetic field. We elaborate on the relationship between the chiral condensate, magnetisation and the (magnetic) free energy density. We compare our results at low temperatures with lattice QCD results.


Introduction
Despite our best efforts, many phenomena of strongly coupled field theories remain enigmatic. While we have understood the fundamental building blocks of QCD for some six decades and with the most complex machines in the world at our disposal, confinement, chiral symmetry breaking, phenomenology of the strongly coupled quark gluon plasma, and more remain outside the grasp of a complete mathematical description. To be able to map out the phase diagram of QCD from first principles is a holy grail of quantum field theory. Despite the fact that we have yet to crack these issues, there are tools at our disposal which have given us key insights and allowed us to answer questions about some of these phenomena in interesting ways. Lattice QCD has lead the way for many decades, and with increasing computational tools, both algorithmic and hardware, there are surely interesting times ahead for this approach. Heavy Quark Effective Theory [2], chiral perturbation theory [3] and the Schwinger-Dyson equations [4] are other powerful methods which give a window into certain parameter regions of QCD.
The gauge/gravity duality, based on the AdS/CFT correspondence [5], has been the other major branch in understanding strongly coupled quantum field theories (for a set of pedagogical introductions see [6][7][8][9]). While we are a long way from having a gravity dual of QCD it is clear that certain QCD-like phenomena do show up in simple and elegant gravity duals of less-realistic field theories. Meson spectra, chiral symmetry breaking, confinement/deconfinement phase transitions and more are all accessible in such models.
To create the most realistic QCD gravity dual is clearly of the most important goals of the gauge/gravity duality, and so any step in this direction is worth pursuing. In the topdown approach there have been important advances, such as the breaking of supersymmetry [10][11][12][13], the addition of fundamental matter [14], 1 the addition of chemical potential [22] and external magnetic field [23][24][25][26]. Those advances have provided us with models which mimic QCD-like behaviour.
Regarding chiral symmetry breaking, in QCD we know that there are at least two effects present. In the massless case, the Lagrangian is chirally symmetric at high energies but the vacuum breaks the chiral symmetry spontaneously at low energies. The other effect is the explicit chiral symmetry breaking due to the presence of massive quarks.
Chiral symmetry breaking has been one of the effects that the gauge/gravity duality has been able to model for some time. In the top-down approach, this problem has been considered in the Klebanov-Strassler [10] and Maldacena-Nunez [11,12], the dilaton-flow geometry of Constable-Myers [27,28,32] as well as the D3/D7 [14] and D4/D6 [29] brane models. A model that stands out is the Sakai-Sugimoto model [30,31], that describes the breaking of a U (N f ) × U (N f ) chiral symmetry (through the addition of N f pairs of D8-D8-branes on the non-extremal Witten D4-brane background [13]). This is considered the closest holographic model for QCD and has been studied in great detail and in many regimes over the past years. An alternative geometrical realisation of chiral symmetry breaking was introduced in [33] by Kuperstein and Sonnenschein, through the addition of a pair of D7-D7-branes on the conifold geometry [34] 2 .
In addition to the geometrical realisations described above there is an alternative holographic description of chiral symmetry breaking in terms of open string tachyon condensation developed in [1,37,38]. This is the approach that we will follow in this paper and will be described in detail in the next sections.
Although the top-down approach has given us a wealth of information about what sorts of backgrounds give rise to which phenomena, it is derisable to build bottom up models which are five dimensional models that have a small set of ingredients necessary to describe nonperturbative QCD dynamics. The archetypical bottom-up constructions that incorporate chiral symmetry breaking and mesonic physics are the hard-wall model [39,40], inspired by the Polchinski-Strassler background [41], and the soft-wall model [42] 3 . A more sophisticated model that captures the dynamics of QCD more accurately is Veneziano-QCD (V-QCD) [46,47] (see also [48]). It combines the model of improved holographic QCD (IHQCD) [49][50][51][52] for the gluon sector and a tachyonic Dirac-Born-Infeld (DBI) action proposed by Sen for the quark sector [53,54]. In order for the model to match the predictions of QCD phenomenology one has to adopt a bottom-up approach. The action is generalised to one which contains several freely-defined functions. The form of those functions is chosen in a way that qualitative QCD features are reproduced and are fitted using lattice and experimental data. In the IHQCD case this has been considered in [52], while in the full V-QCD case the detailed comparison was initiated in [47,55].
In QCD, the presence of very strong magnetic fields eB > Λ 2 QCD triggers a plethora of interesting phenomena, among which are the Magnetic Catalysis (MC) (see e.g. [56,57]) and the Inverse Magnetic Catalysis (IMC) of chiral symmetry breaking (see e.g. [58][59][60]). Magnetic fields with a magnitude of eB/Λ 2 QCD ∼ 5 − 10 are realised during non-central heavy ion collisions. Even though the magnetic field strength decays rapidly after the collision, it remains very strong when the quark-gluon plasma (QGP) initially forms. As a consequence it affects the plasma evolution and the subsequent production of charged hadrons [61].
Magnetic catalysis is the phenomenon by which a magnetic field favours chiral symmetry breaking. This phenomenon is characterised by the enhancement of the chiral condensate and in QCD occurs at low temperatures. The physical (perturbative) mechanism behind this is that the strong magnetic field reduces the effective dynamics from (3+1) to (1+1) dimensions, since the motion of the charged particles are restricted to the lowest Landau level. As a consequence of the magnetic field, the lowest Landau level is degenerate and that leads to an enhancement of the Dirac spectral density. This leads, via the Banks-Casher relation [62], to an enhancement of the chiral condensate (magnetic catalysis).
Inverse Magnetic catalysis is the phenomenon by which a magnetic field disfavours chiral symmetry breaking and it is characterised by a quark condensate that decreases in the presence of a strong magnetic field. In QCD this occurs at temperatures approximately higher than 150 MeV. IMC is a non-perturbative effect and the current understanding is that it originates from strong coupling dynamics around the deconfinement temperature. A promising explanation, coming from a lattice perspective, is that IMC is due to a competition between valence and sea quarks in the path integral [63]. The valence contribution is through the quark operators inside the path integral (i.e. the trace of the inverse of the Dirac operator). The magnetic field catalyses the condensate, since it increases the spectral density of the zero energy mode of the Dirac operator. The sea contribution is through the quark determinant, which is responsible for the fluctuations around the gluon path integral. The dependence of the determinant on B and T is intricate and the net result is a suppression of the condensate close to the deconfinement temperature.
For recent reviews on magnetic catalysis and inverse magnetic catalysis see e.g. [64][65][66]. Inverse magnetic catalysis appears also at finite chemical potential where there is a competition between the energy cost of producing quark antiquark pairs and the energy gain due to the chiral condensate. For a nice review of IMC at finite chemical potential see [67].
There have been several attempts to address MC and IMC in a holographic framework, including . Here we single out the approach that was put forward in [80], since it allows for a consistent description of the chiral condensate, based on the VQCD approach [46,47]. They proposed a holographic model that makes manifest the competition between the valence and sea quark contributions to the chiral condensate. In that framework the role of the valence contribution is played by the tachyon (the bulk field dual to the quark bilinear operator), while the role of the sea contribution comes from the backreaction of the magnetic field on the background, the latter relevant for IMC in this scenario. Another interesting proposal was presented in [84,85], suggesting that the cause of IMC is the anisotropy induced by the magnetic field, rather than the charge dynamics that it creates. 4 Lastly, when it comes to distinguishing MC from IMC, besides the chiral condensate, it was realised in [81,82] that the magnetisation plays a very important role.
In this paper we will extend the model of [1,37] to investigate the effect of a nonzero magnetic field on the chiral condensate. Although the model in [1,37] is less realistic than constructions such as VQCD [46,47], it has the privilege of simplicity. There are fewer parameters to fix with the lattice computations and the potentials of the tachyon action are predetermined. We will arrive at a model that includes all of the necessary ingredients for describing the chiral condensate in the presence of a magnetic field. Besides the chiral condensate, the model allows for a consistent description of the magnetisation and provides a very interesting holographic description of magnetic catalysis.
The outline of the paper is as follows. In section 2, we review the holographic approach introduced in [38] to describe the dynamics of chiral symmetry breaking. This includes a specific action for the tachyon field and the implementation of a confinement criterion. Moreover, we describe the specific gravity setup where those ideas were materialised [1,37]. In section 3, we extend the model of [1,37] to describe the effects of the addition of an external magnetic field on the tachyon dynamics. We study in detail the equation of motion for the tachyon in the confined and the deconfined phases and the dynamical breaking of chiral symmetry. In section 4, we calculate and analyse the chiral condensate and the magnetisation. The MC phenomenon is a common feature for both phases. In the deconfined phase at zero quark mass and for a sufficiently strong magnetic field, there is a second order phase transition from the chirally restored phase to a chirally broken phase, signifying the spontaneous breaking of chiral symmetry above a critical value. Going away from the massless limit, the chiral transition becomes a crossover. In addition to remarking on a variety of interesting qualitative properties, we finish the section with a quantitative comparison of the gravity dual predictions with computations from lattice QCD at low temperatures. The main text is supplemented with two appendices. In appendix A we describe the Wess-Zumino (WZ) term of the tachyonic action. In appendix B we perform the detailed IR asymptotic analysis for the equation of motion of the tachyon. While the analysis in the deconfined case is a straightforward generalisation of [1], in the confined case the IR divergence emerges in a systematic and non-trivial way.

The setup
In [38] a holographic picture was proposed which describes the dynamics of chiral symmetry breaking by open string tachyon condensation in the gravity side. In [1,37] a particular setup was developed which allows for a quantitative description of chiral symmetry breaking, and in this section we review these models in detail.
The setup proposed in [38] consists of a system of N f coincident D-brane anti-D-brane pairs in a gravitational background generated by a stack of colour branes. This framework is an extension of the Dirac-Born-Infeld (DBI) plus Wess-Zumino (WZ) actions, which takes into account the effects of open string tachyon condensation [54,94].
The tachyonic mode, τ , is an open string complex scalar, which is in the spectrum of open strings stretching between the brane-antibrane pairs, and transforms in the bifundamental representation of the U (N f ) L × U (N f ) R flavour group. More specifically, τ transforms in the antifundamental of U (N f ) L and in the fundamental of U (N f ) R and vice versa for τ † . Fixing the mass term for τ appropriately, it naturally couples to the 4d quark bilinear operatorqq at the boundary. Then the 4d breaking of the global chiral symmetry is mapped to a 5d Higgs-like breaking of gauge symmetry triggered by τ , as realised in [39,40]. In QCD, spontaneous breaking of chiral symmetry is associated with a nonzero vev for the quark bilinear operator. In the holographic setup of [38], this is realised via a nontrivial IR behaviour for τ generated dynamically. The model also describes the explicit breaking of chiral symmetry associated with having a nonzero mass term m q qq in the 4d theory.
In the case of massless QCD the global chiral symmetry U (N f ) L ×U (N f ) R is preserved in the UV and spontaneously broken at low energies to the diagonal subgroup U (N f ) V . In the holographic setup of [38], this corresponds to a vanishing tachyon at the boundary that grows as it moves away from the boundary and becomes infinite at the end of space. This process can be thought as a recombination of the brane-antibrane pair.
In the next subsections we will describe the tachyon plus DBI and WZ actions of the above model. The physics of the DBI part yields the vacuum configuration and excitations thereon and the WZ part is related to global anomalies and to a holographic realisation of the Coleman-Witten theorem [95].

The Tachyon-DBI action
The general construction consists of a system of N f overlapping pairs of Dq-Dq flavour branes in a fixed curved spacetime generated by a set of N c Dp colour branes. We will be particularly interested in the case p = q = 4 where the colour branes generate the asymptotic AdS 6 cigar geometry [101] and the flavour brane anti-branes are 5d defects associated with quark degrees of freedom [1,37] . For simplicity we focus on the Abelian case N f = 1, corresponding to a single pair of D4-D4 branes. The corresponding DBI action can be written as ( [38], see also [54,94]) and we have defined 5 The symmetric tensor g mn denotes the (five-dimensional) world-volume metric whereas the antisymmetric tensors F L/R mn denote the field strengths associated with the Abelian gauge fields A L/R m . The symmetric tensor h mn describes the dynamics of a complex scalar field τ (the tachyon) The covariant derivative is the one associated with a bifundamental field, i.e.
is the corresponding axial gauge field. The explicit form of h mn is then given by h mn = 2κ ∂ (m τ * ∂ n) τ + j (m a n) + a m a n τ * τ , (2.5) where we have introduced the Abelian current j m = iτ ← → ∂ m τ * and we are using the symmetric tensor notation X (mn) = (X mn + X nm )/2 . For the tachyon potential, we consider the Gaussian form This form was proposed in [38], inspired by the computation in flat space that was derived in boundary string field theory [96,97]. Remarkably, this potential leads to linear Regge trajectories for the mesons [38], something which is otherwise hard to model 6 . The parameters that we use in this paper are related to those defined in [1] by The square roots in (2.1) can be written as The expression for the tachyon-DBI action includes also the transverse scalars that live on the flavour branes. As mentioned in [38], these modes (that appear in a critical string theory setup) are ignored in a holographic QCD analysis, since they do not have an obvious QCD interpretation. 6 The only other approach that leads to linear linear Regge trajectories for the mesons is the soft wall model, based on the IR constraint for the dilaton field [42]. and we have introduced the totally antisymmetric 4-tensor The upper indices in (2.8) are raised using the effective metricg mn 7 . For the analysis of the following subsection (and also in [38]), we consider a five-dimensional metric of the form g mn = diag (g zz (z), g tt (z), g xx (z), g xx (z), g xx (z)) . (2.10) This metric preserves an SO(3) symmetry and the components depend solely on the radial coordinate z, as expected for a holographic QCD background.

Confinement criterion for dynamical chiral symmetry breaking
In this subsection we will describe the connection between confinement and the singular behaviour of the tachyon profile in the IR of the geometry, which was investigated in [38]. Since, at leading order, it is consistent to set the gauge fields to zero, the equation of motion for the tachyon comes from considering solely the DBI part of the action. Following the standard procedure, we set the phase of the complex tachyon to zero and arrive at a differential equation for τ that schematically looks like where # 1 , # 2 and # 3 are combinations of the metric components which can be found in [38]. This is a second order non-linear differential equation and the two integration constants, via the standard AdS/CFT dictionary, can be related to the quark bare mass and condensate. This relationship is found by studying the UV behaviour of the tachyon. Assuming that the space is asymptotically AdS and the tachyon is dual to the quark bilinearqq with conformal dimension ∆ = 3, we arrive at the following expression for the UV limit of the tachyon profile τ = c 1 z + . . . + c 3 z 3 + . . .
(small z) (2.12) where the source coefficient c 1 is proportional to the quark mass m q whereas the vev coefficient c 3 is related to the quark condensate qq . For the IR analysis of the tachyon equation (2.11) we consider the results of [99], according to which a sufficient condition for a gravity background to exhibit confinement is for some value of z = z div . Identifying the point where the divergence appears with the confinement scale, namely z IR = z div , and assuming that the divergence of the metric component g zz is a simple pole near z IR , we conclude that the tachyon diverges near z IR as follows (2.14) The main lesson from this analysis is that the IR consistency condition for the tachyon (2.14) can be used to fix c 3 in terms of c 1 , which is equivalent to fixing the chiral condensate qq in terms of the quark mass m q . As usual, this will be implemented using a shooting technique.
In the seminal paper of Coleman and Witten [95] it was proved that in the limit N c → ∞ and for massless quarks, the chiral symmetry of QCD is spontaneously broken The main message from the analysis of the current subsection is that for a confining theory the tachyon has to diverge in the IR of the geometry while it goes to zero in the UV limit. Since τ transforms in the bifundamental representation of the flavour group, τ = 0 means that the symmetry has been broken down to U (N f ) V . Therefore, the presence of confinement implies spontaneous chiral symmetry breaking, and this is therefore a holographic implementation of the ideas and results of [95].
The analysis of the WZ part of the action is related to the study of anomalies of the chiral symmetry, when there is a coupling between flavour currents and external sources. A gauge transformation of the WZ part of the action produces a boundary term that is matched with the global anomaly of the dual field theory.
In [38], a precise computation of the gauge variation of the 5d WZ action was performed in the case of a real tachyon τ = τ * . The conclusion was that the result is given by a 4d boundary term. This boundary term precisely matches the expected anomaly for the residual U (N f ) V group after imposing the appropriate boundary conditions for τ . In fact, the divergent behaviour for the tachyon (arising from the confinement criterion) in the IR, cf. (2.14), is crucial in the match to the QCD anomaly term. The authors of [38] interpreted this result as a holographic realisation of the Coleman-Witten theorem.
For more details on the WZ term of the tachyon action and the currents see the discussion in appendix A.

The Iatrakis, Kiritsis, Paredes (IKP) model
A simple holographic model of QCD that describes chiral symmetry breaking and the associated mesonic physics was proposed in [1,37], by Iatrakis, Kiritsis and Paredes (IKP). It is a construction that makes explicit the ideas introduced in [38], namely that chiral symmetry breaking and the physics of the flavour sector is encoded in an effective description of a brane-antibrane system with a tachyonic field.
The quarks and antiquarks are introduced through the brane and antibrane, and the physics of interest comes about by condensation of the lowest lying bifundamental scalar on the open strings connecting those branes through a tachyonic instability. The next important step was the choice of the holographic geometry in which these ideas can be realised. The background should be smooth and asymptotically AdS and consistent with confinement in the IR. A simple choice is the AdS 6 soliton geometry [101], which is a solution of the two derivative approximation of subcritical string theory. While the construction in [1,37] is initially top-down, in order to reproduce QCD-like features, one goes beyond the limit in which the two derivative action is a controlled low energy approximation of string theory because we are in a regime where the curvature scale is of the same order as the string length. Thus, we think of this approach as an effective, bottom-up description.
In terms of both complexity and correctly capturing the features of QCD, the IKP model stands somewhere between the hard wall model [39,40] and the VQCD approach [46,47]. The most interesting qualitative features of this approach to QCD physics, as summarized in [1,37], are that • Towers of excitations with J P C = 1 −− , 1 ++ , 0 −+ , 0 ++ are included in the model.
• Dynamical chiral symmetry breaking is realised through tachyon condensation • The excited states have Regge trajectories of the form m 2 n ∼ n.
• The ρ-meson mass increases due to the increase of the pion mass.

The IKP model at finite magnetic field
The fundamental novelty of the current work is to investigate the effects of a finite external magnetic field on the dynamics already described by the IKP model. While an apparently small addition, the extended phase space is rich. In order to do this we need to study the tachyon field along with the gauge potential which leads to the magnetic field. The ansatz for these is Under this ansatz, the field strengths take the form

The Euler-Lagrange equation for τ
We work with the diagonal metric (2.10) and one can show that the ansatz (3.1) leads to a diagonal tensor h mn and that the effective metricg mn takes the form where we have introduced the function which dresses one component of the metric. Note, in particular, that the square root of the determinant of (3.2) can be written as It can be shown that functions Q L(R) of (2.8) are given by Therefore, the DBI Lagrangian given in (2.1) and (2.8) reduces to In appendix A we describe the Chern-Simons (Wess-Zumino) term for a general configuration of gauge fields and a complex tachyon. For the particular case of the ansatz in (3.1), we obtain j = 0, Ω Recalling the definition of Θ z in (3.3), we can split (3.7) into linear and non-linear terms For the case B = 0, we have Q 0 = 1 and (3.8) reduces to eq. (3.6) of [1].

UV asymptotic analysis
The small z limit is asymptotically AdS and therefore in this region (3.8) reduces to where the non-linear terms are sub-leading. In order that the tachyon is dual to the quark mass operatorqq with conformal dimension ∆ = 3 the 5d mass of the scalar field τ must be set such that one has the identification The asymptotic solution for τ takes the form (3.11) The source coefficient c 1 is proportional to the quark mass whereas the vev coefficient c 3 will be related to the chiral condensate.

The confined phase
Until now we have not specified the geometry, but only put constraints on what the UV and IR asymptotic limits must look like. We know that in the confined phased, there must a mass gap corresponding to some point where the geometry stops. An appropriate space-time to consider is thus the 6d cigar-geometry given by The spatial coordinate η is compact, i.e. 0 ≤ η ≤ 2πR. At the tip of the cigar z = z Λ smoothness of the geometry implies that The mass scale M KK plays an important role in the description of confinement and the glueball spectrum. The D4 − D4 pair of flavour branes will be located at η = 0. It is convenient to define the dimensionless radial coordinate u ≡ z/z Λ . Moreover in order to fully eliminate the presence of z Λ and m τ from the Lagrangian, we rescale the tachyon, the magnetic field, the constant V 0 and the field theory coordinates in the following way With these rescalings, the equation of motion for the tachyon becomes where T ≡ ∂ u T and now the functions Q 0 and f Λ are defined as follows The dimensionless coordinate u runs from 0 at the AdS boundary to 1 at the tip of the cigar. Note that the T differential equation (3.15) depends only on the dimensionless parameter B.

IR asymptotic analysis
Near the tip of the cigar, the asymptotic behavior of the tachyon is given by a double series expansion involving a power law behaviour which has both an integer and fractional part which can be separated. This can be parameterised as and g n,m are constant coefficients. Note from (3.17) that the n = 0, m = 0 term means that T is singular at u = 1 so long as g 0,0 does not vanish. Indeed for any non-trivial solution g 0,0 = 0. Plugging (3.17) into (3.15), the latter becomes a double series and the coefficients g n,m are obtained by solving the double series at each order. This is described in appendix B.1. Here we show the first coefficients Note that for small C 0 one has to be careful with the radius of convergence of the series. An important feature of the solution in (3.17) is that all the coefficients depend solely on one parameter, C 0 . This is a nontrivial consequence of the nonlinear terms in the differential equation (3.15) arising from the particular behavior of the tachyon potential, and can be thought as an effective reduction of the original second order differential equation into a first order one 8 .

Numerical analysis of the tachyon equation
In order to solve the equation of motion for the tachyon, (3.15), with the UV and IR behaviours given by (3.11) and (3.17), respectively we fix c 1 and use a shooting technique to numerically integrate, tuning the value of c 3 such that both the UV and IR asymptotics are respected. As happens in the B = 0 case of [1], for a fixed value of c 1 9 there is more than one In order to find which of the two solutions is energetically favoured we must compare their free energies. This is slightly complicated by the fact that the values of c 1 are the same between the solutions that we are comparing, but the values of c 3 are different. In order to compare solution 1, described by (c 1 , c 1 3 ) and solution 2, described by (c 1 , c 2 3 ), we must calculate the difference with i = 1, 2 and where the finite term in (3.19) is coming from the subtraction of the counterterms for each solution (the analysis of the holographic renormalisation is presented in section 4.2). Note that this is not the only counterterm in the free energy but since we perform the calculation in (3.19) at a fixed value of c 1 , this is the only one that survives when we look at their differences.
In for blue, orange, green, red and purple respectively. It is clear that a single value of c 1 may have multiple solutions, however studying the energetics shows that the top branch is always the favoured one, giving the first indication that chiral symmetry is broken at finite magnetic field.

The deconfined phase
Having studied the confined phase, we consider the 6d black-brane for the deconfined phase. The metric in this case is given by The black-brane temperature is given by The deconfinement transition maps to a gravitational Hawking-Page transition between the cigar geometry (3.12) and the black-brane geometry (3.21) 10 . This transition is first order and occurs when z T = z Λ which corresponds to a critical temperature The D4 − D4 pair of flavour branes is again located at η = 0. As in the confined case, it is convenient to define a new radial coordinate v ≡ z/z T and this time we rescale the quantities as With this rescaling the equation of motion for the tachyon becomes where T ≡ ∂ v T and now the functions Q 0 and f T are defined as follows The dimensionless coordinate v runs from 0 to 1 and the T differential equation (3.25) now depends only on the dimensionless parameter B. We will show later in the paper that the dimensionless B will be proportional to the physical value of the magnetic field and inversely proportional to the square of the temperature.

IR asymptotic analysis
Near the horizon (v close to 1), the tachyon field has to be regular. As a consequence, the asymptotic solution takes the form of an ordinary Taylor expansion This time the differential equation (3.25) becomes a simple series and the coefficients C n are obtained by solving the series at each order. This is described in appendix B.2. Here we show the first subleading coefficients We find from (3.27) that the tachyon solution depends solely on one parameter C 0 . Again, this is a consequence of the non-linearity of the differential equation (3.25) that effectively reduces a second order differential equation to a first order one in the near horizon limit. The UV analysis remains the same in both the confined and deconfined cases and thus doesn't need to be treated separately here.

Numerical analysis of the tachyon equation
In this subsection we present the details of the numerical solution of the equation of motion for the tachyon in the deconfined case (3.25) with UV and IR boundary conditions given in (3.11) and (3.27), respectively.
For small values of the magnetic field B the analysis of the tachyon equation is similar to the B = 0 case of [1]. In figure 3 we have plotted tachyon profiles for different values of the magnetic field, at a fixed value of c 1 . Increasing the value of B changes the profile in a continuous way and does not affect its shape. This in turn is reflected in the three other plots of figure 3 that present C 0 and c 3 as functions of c 1 for different values of B. Notice here that the non-monotonic behavior for c 3 that was observed in [1] for the B = 0 case disappears as soon as we increase the value of B. Once we choose the value of c 1 , the values of C 0 and c 3 are determined dynamically by the IR boundary condition, using the shooting technique to numerically solve the equation of motion for the tachyon. Chiral symmetry remains unbroken (spontaneously) for the range of values that we consider in figure 3, since for c 1 = 0 the value of c 3 is also zero, and as a result T = 0 for all v. This observation was put forward also in [1].
For values of B ≥ 10 we see an interesting behaviour appear. In figure 4 we plot C 0 as a function of c 1 for two values of the magnetic that are just below and just above the value B = 10, namely B = 9.99 and B = 10.01. From this plot one can see that when the value of the magnetic field exceeds the (critical) value B ≈ 10, C 0 as a function of c 1 becomes multivalued. Notice that the same behavior appears in the plot of c 3 as a function of c 1 . For B = 10.01 and values of |c 1 | 10 −4 there are three values of C 0 and consequently three different profiles.
In the upper part of figure 5, and for B = 11, we have plotted the three tachyon profiles that correspond to the value c 1 = 1/40. Zooming into the first two plots (red and green) close to the boundary it can be seen that they do not have a monotonic behavior. It is only the last profile, which corresponds to the largest value of C 0 , or equivalently of c 3 that is monotonic.
To distinguish between the three solutions and determine the energetically favored one, we have to compare the free energies of the different tachyon profiles for a fixed value of c 1 . As in the confined case, we have to calculate the difference in free energies, given by (3.19). In the deconfined case W i is given by the following expression where i indexes the different solutions.
In figure 6 we present c 3 as a function for c 1 for B = 11 and B = 12, after the comparison of the free energies of the tachyon profiles has been performed. The energetically favored profile is the one with the highest value of c 3 (or equivalently C 0 ). In this way an

The chiral condensate and magnetisation
Until now we have discussed the parameters which describe the UV asymptotics of the tachyon solution, c 1 and c 3 but not their corresponding field theory quantities. In this section we will connect them with the phenomenological gauge theory parameters of the quark mass and quark bilinear condensate. We then go on to study the magnetisation and the magnetic free energy density.

The renormalised action
In this section we follow the analysis of [1], modified accordingly when there is an external magnetic field. Here we will concentrate on the confined phase, but the deconfined phase follows a very similar analysis. The DBI Lagrangian, after using the redefinitions of (3.14), depends on two constants, namely B & V 0 , and becomes In the deconfined phase we exchange u with v and the term √ 1 − u 5 in the denominator is absent. We regularise the action by introducing a UV cut-off at u = and integrate from to the tip of the cigar at u = 1 In the following we need the appropriate covariant counterterms that will be added to the regularised action in order to cancel the divergences at the boundary. The necessary expression is where γ is the induced metric at u = , namely √ −γ = R 4 4 . The finite counterterms depending on the constants α 1 and α 2 capture the scheme dependence of the renormalised action. The last term in (4.3) cancels the divergence due to the presence of the magnetic field close to the boundary and has the standard form which is known from the probe brane physics analysis [25,72]. The renormalised action is obtained from the following expression (4.4)

The chiral condensate and magnetic catalysis
The quark condensate is defined as usual in the following way q q = δS ren δm q (4.5) To calculate the variation of the regularised action S reg with respect to m q , we need to compute the functional derivative with respect to T , since To calculate the functional derivative of S reg with respect to T , we have to use the equation of motion for the tachyon and we arrive to the following expression [1] δS (4.7) Finally, for the computation of the functional derivative of the tachyon with respect to c 1 we have to take into account that c 3 is a function of c 1 . Putting together all the ingredients and using the UV expansion of the tachyon we arrive to following result for the functional derivative of the renormalised action with respect to c 1 The source coefficient c 1 is proportional to the quark mass m q .
where ζ is a normalisation constant, usually fixed as ζ = √ N c /2π to satisfy large N c counting rules [107]. In this way we obtain The quark mass m q in (4.9) and the chiral condensate in (4.10) are dimensionless because of the redefinitions (3.14). Both m q andqq will be redefined in subsection 4.5 in their dimensionful forms in order to compare with lattice data. Note that the chiral condensate depends implicitly on the magnetic field through the vev coefficient c 3 and on the renormalisation scheme, through the parameter α 1 . In figure 7 we plot the chiral condensate as a function of the quark mass in the confined and deconfined phases. To avoid the presence of the scheme dependent parameter we either fix it to α 1 = −1 (so that the chiral condensate becomes proportional to the vev coefficient c 3 ) or subtract the value of the chiral condensate at zero magnetic field. In this way the subtracted chiral condensate is independent of the renormalisation scheme. In both panels the magnetic field B takes the values 1 (blue), 5 (red), 9 (green) and 13 (orange). We have used the formulae (4.9) and (4.10) with V 0 = 1 and ζ = 1. In both panels the quark mass and chiral condensate are given in units where z Λ = z T = 1. There is actually a nontrivial scaling in M KK and T for the confined and deconfined phases, to be discussed in the next subsection. Note that the chiral condensate for the confined and deconfined phase have the same behaviour in the limit of heavy quark mass, suggesting a universal description. The chiral condensate on the left panel depends on the renormalisation scheme and the chosen scheme was α 1 = −1. The subtracted chiral condensate on the right panel is independent of the renormalisation scheme.
In figure 8 we plot the chiral condensate as a function of the magnetic field in the confined and deconfined phases. The presence of the scheme dependent parameter is fixed/subtracted as in figure 7. Note that for c 1 = 0 the chiral condensate changes drastically as the magnetic field crosses the critical value B = 10. This is a consequence of the spontaneous chiral symmetry breaking that is analysed in figures 4 and 6. A common feature for both figures 7 and 8 is that for large values of the mass or a very strong magnetic field the chiral condensate of the confined the deconfined phases almost coincide. This is related to the fact that in both phases we work in units where z Λ = z T = 1. We have performed a fit in the regime of large B and found that the chiral condensate grows as #B 3/2 in both phases. This asymptotic behaviour will be important in the last subsection, where we compare our results against lattice QCD. In the specific gravity model that we are working on, it seems that the IR boundary conditions do not affect significantly the physics in the limit of very large mass of the quarks or very strong magnetic field. Note that this phenomenon is generally obtained in holographic brane constructions where in the regime of large quark mass (or large magnetic field) the brane is always far from away from the deep IR (see e.g. [6] ). The (subtracted) chiral condensate as a function of the magnetic field in the confined (solid) and deconfined (dashed) phases. In both panels the quark mass parameter c 1 takes the values 0 (blue), 2 (red), 4 (green) and 6 (orange). We have used the formula (4.10) with V 0 = 1 and ζ = 1. In both panels the chiral condensate and magnetic field are given in units where z Λ = z T = 1. There is actually a nontrivial scaling in M KK and T for the confined and deconfined phases, to be discussed in the next subsection. The chiral condensate on the left panel depends on the renormalisation scheme and the chosen scheme was α 1 = −1. The subtracted chiral condensate on the right panel is independent of the renormalisation scheme.
In figure 9 we elaborate on the spontaneous chiral symmetry breaking that occurs when the magnetic field exceeds the critical value B = 10. For that reason we plot the unsubtracted chiral condensate as a function of the magnetic field B, for values of the quark mass that are close to zero. The analysis indicates that at zero quark mass there is a second order phase transition. As the quark mass increases this phase transition degenerates to a crossover. The scaling in T for the deconfined phase implies that the chiral transition when varying B has actually two interpretations. Since B is proportional to B/T 2 , either we fix T and increase the dimensionful magnetic field B or we fix B and decrease the temperature. In the last subsection we will present a plot that describes the chiral transition when varying the temperature, at fixed values of the magnetic field.

The magnetisation
In this section we focus the analysis on the computation of the magnetisation. As we did for the condensate, we restrict the analysis to the confined phase. Magnetisation is defined in the usual way as where F is the free energy and the calculation is at fixed z Λ (the tip of the cigar). In the deconfined phase the calculation will be performed at fixed temperature (z T is inversely proportional to the temperature). Starting from (4.11) we will first compute the part of the magnetisation due to the regularised action in (4.2) where in the last step we have used the equation of motion for the tachyon. That expression can be further simplified. The contribution from the boundary terms reads The second ingredient of (4.11) comes from the contribution of the functional derivative of (4.3) with respect to B. As can be seen by substituting the approximate expression for the tachyon contribution of that term is (4.14) Combining (4.12), (4.13) and (4.14) we arrive to the following expression for the magnetisation It can be checked explicitly that as u → the infinite contribution from the last integral of (4.15) is canceled by the logarithmic counterterm. Note that the magnetisation depends on the renormalisation scheme through the parameter α 2 .
In figure 10 we plot the magnetisation as a function of the quark mass in the confined and deconfined phases. On the left panel of the figure we fix the scheme dependent parameter to α 2 = 0, 11 while in the right panel we plot the difference between the magnetisation from equation (4.15) and the magnetisation for zero quark mass mass. In this way the scheme dependent parameter vanishes and the subtracted magnetisation is independent of the renormalisation scheme.
In figure 11 we plot the magnetisation as a function of the magnetic field in the confined and deconfined phases. The presence of the scheme dependent parameter is fixed/subtracted as in figure 10. In the subtracted magnetisation there is a discontinuity in the first derivative at B = 10 (it will become more evident in the plot of susceptibility), which is a consequence of the spontaneous chiral symmetry breaking in the deconfined case. Note that the calculation is performed in two (complementary) ways: First, we apply the formula (4.15) and in the following we calculate the numerical derivative of the renormalised free energy (4.4) with respect to the magnetic field. The results we obtain from the two calculations are identical; this is a non-trivial confirmation of the formula we derived in (4.15), especially taking into account that the renormalised free energy depends on the scheme dependent parameter α 1 while the magnetisation does not.
Since we do not have an analytic solution for the tachyon profile for either small or large values of the magnetic field, we cannot approximate the magnetisation. However from the numerical analysis, we have verified that for large values of B the magnetisation in the left panel of figure 11 (for α 2 = 0) is approximated by the following expression   The (subtracted) magnetisation as a function of the quark mass in the confined (solid) and deconfined (dashed) phases. We have used the formulas (4.9) and (4.15) with ζ = 1 and V 0 = 1. In both panels the magnetic field B takes the values 1 (blue), 5 (red), 9 (green) and 13 (orange). In both panels the magnetisation and quark mass are given in units where z Λ = z T = 1. There is actually a nontrivial scaling in M KK and T for the confined and deconfined phases, to be discussed in the next subsection The magnetisation on the left panel depends on the renormalisation scheme and the chosen scheme was α 2 = 0. The subtracted magnetisation on the right panel is independent of the renormalisation scheme.
In figure 12 we plot the susceptibility (first derivative of the magnetisation with respect to the magnetic field) as a function of the magnetic field in the confined and deconfined phases. The presence of the scheme dependent parameter α 2 is fixed/subtracted as in figure 10. The jump of the susceptibility of the deconfined phase at the critical value of the magnetic field B = 10 in the right panel of figure 12 is inherited from the right panel of figure 11. The jump in the susceptibility that appears in the left panel of figure 12, is due to the fact that this curve corresponds to zero value for the the quark mass parameter c 1 .

The condensate contribution to the magnetisation
We will extract the contribution to the free energy due to chiral symmetry breaking, which means in our framework having a nonzero tachyon. We start with the dimensionless (bare) free energy density. In the Lorentzian signature the free energy density is identified with the Hamiltonian density, i.e. (4.17) In the confined phase, for instance, L DBI is given in (4.1). We work in units where z Λ = z T = 1 but there is a nontrivial scaling in M 4 KK (vacuum energy) and T 4 (plasma free energy) for the confined and deconfined phases to be addressed in subsection 4.5. When the tachyon is zero the (confined) free energy reduces to and the counterterms contribution becomes The renormalised free energy takes the form F = F bare + F ct . In figure 13 we compare the full renormalised free energy F against the zero tachyon free energy F (T =0) . Notice the black curve (zero tachyon) and the two (solid and dashed) blue curves (i.e. c 1 = 0). In the confined case the black and blue solid curves never coincide, since we always have spontaneous chiral symmetry breaking. However in the deconfined case and for B < 10 the black and blue dashed curves will be on top of each other, while for B > 10 the curves will be different. This last observation is hard to see in figure 13. Similarly, at zero tachyon, the bare magnetisation reduces to (4.20) and the counterterms contribution becomes  that contains the tachyon contribution to the renormalised magnetisation.
In figure 14 we plot the subtracted magnetisation that is defined in (4.22) as a function of the quark mass and as a function of the magnetic field, in the confined and deconfined phases. For the numerical analysis we fix the renormalisation scheme to α 1 = −1 and α 2 = 0. The results can be easily extended to other renormalisation schemes. 12 The main motivation behind this plot is to emphasise the contribution to the magnetisation coming from the chiral condensate.
There is an important thermodynamic identity regarding mixed partial derivatives of the free energy This identity implies the following relation between the magnetisation and condensate In the model at hand, the thermodynamic identity (4.24) holds because the renormalised (magnetic) free energy is smooth in m q and B. We have explicitly checked this identity in the confined and deconfined phases. The curves on the left panel of figure 14 suggest the following approximation where f 1 (B) depends only on the magnetic field. Integrating (4.25) in m q we find that where f 0 (B) depends solely on B and can be identified with the subtracted magnetisation at zero quark mass, i.e. ∆M | mq=0 = f 0 (B). Setting to zero the value of f 0 (B) we obtain the following empirical formula that provides a crude but reasonable approximation for ∆M , as can be seen from the comparison with (4.26) that appears on the right panel of figure 14. The formula (4.27) can be thought as the dominant contribution to the magnetisation arising from the chiral condensate.

Comparison to lattice data
The lattice QCD formalism has been a powerful method for investigating magnetic catalysis (MC) and inverse magnetic catalysis (IMC), see e.g. [59,[103][104][105][106]. In this subsection we will compare some of our results for the condensate and magnetisation at low temperatures with lattice QCD results. First we remind the reader that the dimensionful quark mass m q , chiral condensate qq and magnetic field B can be extracted from the dimensionless variables c 1 , c 3 and B in the confined phase by the relations (4.28) where z Λ = (5/2)M −1 KK is an IR length scale associated with confinement. In the deconfined phase we replace z Λ by z T = 5/(4πT ) which implies nontrivial scalings of the form m q /T , qq /T 3 and B/T 2 . The relevant parameters in our model are V 0 = V 0 R 5 , βR −2 , m τ and ζ. In [1] it was shown that, in order to obtain the appropriate normalisation for the 2-point correlation functions, the model parameters should obey the relations 13 where k is a parameter that controls the meson phenomenology. Without loss of generality, we can fix m τ to 1. We also take N c = 3 and use the phenomenological values (4.30) obtained in [1] from a fit to the meson spectrum. This fixes completely our model parameters:  The physical quark mass and chiral condensate are given by Note in particular that m q qq B 2 = 0.035 For the deconfined phase we find the relations The critical temperature for the first order deconfinement transition, cf. (3.23), becomes T c = 0.22 GeV. In the confined phase the dimensionful free energy and magnetisation are given by where F and M are the dimensionless free energy and magnetisations described in the previous subsections. In the deconfined phase the dimensionful free energy and magnetisation take the form In order to compare our results to lattice QCD, we introduce the subtracted condensate This quantity is scheme independent and therefore free of ambiguities. We evaluate this quantity in the confined and deconfined phases. In figure 15 we compare our results for ∆ qq , as a function of the magnetic field, against lattice QCD results at temperatures T = 113 MeV and T = 142 MeV, obtained in [59]. We have set the quark mass to the physical value m * q = 2.9 MeV. Our results always provide a subtracted condensate increasing with the magnetic field, which is interpreted as MC. The lattice results, on the other hand, show an increasing behaviour at T = 113 MeV and a decreasing behaviour at T = 142 MeV. This is, of course, the well known transition from MC to IMC. A clear description of these results was given in [63] in terms of a valence and sea contribution to the chiral condensate. Since we work in the probe approximation, our model only describes the effect of the magnetic field on the quark mass operator and neglect magnetic effects on the gluonic vacuum (or plasma). Including backreaction it should be possible to describe these effects and therefore the transition from MC to IMC. A nice feature of the IKP model is that we can vary the quark mass m q and go from the regime of light quarks (and mesons) to the heavy quark (heavy meson) regime. Of course, a more realistic description would imply a non-Abelian description that distinguishes the different quark flavours (up, down, strange, etc). However, the Abelian approximation is good enough to explore the transition from light quarks to heavy quarks. In our model we find an interesting behaviour for the RG invariant product of quark mass and (subtracted) chiral condensate, i.e. m q ∆ qq , with ∆ qq defined in (4.38). At small B the quantity m q ∆ qq can be expanded as #B 2 + #B 4 + . . . . Interestingly, the B 2 coefficient grows quickly with the quark mass and reaches a plateau at m q ∼ 1 GeV, suggesting a scaling law in the heavy quark regime. This is shown in Fig. 16 for the confined and deconfined phases. Lattice QCD results for this coefficient were obtained in [104] 14 , represented by the orange curve in Fig. 16. [104] provided a nice weak coupling interpretation for the plateau in the heavy quark regime. In our case, we expect some scaling in the regime of large m q due to an approximate conformal symmetry for the theory at nonzero B after subtracting the (conformal symmetry breaking) B = 0 term. We suspect that the difference between the plateau we found and the plateau found in lattice QCD is associated with the fact that in the holographic QCD model at hand we are always in the strongly coupled regime whereas in real QCD there is a transition between the strongly coupled regime to the weakly coupled regime 15 . Next, we compare our results for the magnetisation against laticce QCD results. For this purpose it is convenient to work the subtracted quantity where T 0 is a reference temperature. This subtracted magnetisation is scheme independent and should therefore be free of ambiguities. On the left panel of Fig. 17 we present our results for the (dimensionful) magnetisation M in the confined and deconfined phases. The black solid curve corresponds to the magnetisation in the confined phase and it is independent of the temperature. The black, blue and red dashed lines represent the results for the deconfined phase at the temperatures T = 114 MeV, T = 130 MeV and T = 142 MeV respectively. At those temperatures the deconfined phase is in a metastable phase 15 Another important difference between our model and real QCD is that at high energies the gluon sector becomes a five dimensional theory. (the confined phase is thermodynamically preferred). On the right panel of Fig. 17 we compare our results for the subtracted magnetisation, defined in (4.39), against the lattice QCD results, obtained in [103,104]. Since the lattice QCD data starts at T = 114 MeV we take that value as our reference temperature T 0 . The black solid line depicts the trivial result ∆M = 0 for the confined phase. The blue and red dashed lines represent the results for ∆M in the deconfined phase at T = 130 MeV and T = 142 MeV. The blue and red dots (and error bars) represent the lattice QCD results obtained in [103,104]. Interestingly, the left panel of Fig. 17 shows that as we go from the confined phase to the deconfined phase there is a transition between a diamagnetic behaviour to a paramagnetic behaviour. This result might be particular to this model, although recent lattice QCD results indicate a similar transition [106]. On the other hand, the right panel of Fig. 17 shows big quantitative differences between our results and the lattice QCD results. Since the lattice QCD data starts at high temperatures, these differences are already expected because, as explained previously, we expect backreaction effects to be important at high temperatures. We note, however, that the lattice QCD data initially shows a competition between an increasing and decreasing behaviour for ∆M. Further analysis of the lattice QCD data reveals that at higher temperatures the subtracted magnetisation ∆M becomes monotonically increasing, where we expect a dominant IMC effect. This suggests that the competition of increasing and decreasing behaviours in ∆M may be related to the competition between IMC and MC at intermediate temperatures. In our model we always find a decreasing behaviour for ∆M in the deconfined phase, consistent with an interpretation in terms of MC, and we suspect that backreaction effects will lead to an increasing behaviour for ∆M, absent in the current model.
Lastly, we display in Fig. 18 our results for the subtracted chiral condensate, defined in (4.38), as a function of the temperature for fixed values of the magnetic field. We compare our results against the lattice QCD results obtained in [103,104]. The magnetic field varies from B = 0.2 GeV (black lines) to B = 1 GeV (orange lines). The solid horizontal lines correspond to the confined phase and the dashed curved lines correspond to the deconfined phase. The dotted lines represent fits to the lattice QCD data [103,104] using the empirical formula obtained in [65]. The subtracted chiral condensate in the confined phase is independent of the temperature because it corresponds to the thermal extension of the QCD vacuum. The deconfined phase, on the other hand, leads to an interesting temperature dependence for the subtracted chiral condensate. The chiral transition described in subsection 4.2 for a varying dimensionless magnetic field B ∼ B/T 2 now is interpreted as a chiral transition for a varying temperature. As a matter of fact, since the tachyon solution only depends on B ∼ B/T 2 , one can obtain the subtracted condensate for any B and T from the solution found at some fixed B.
At very low temperatures and large magnetic fields our results for the subtracted condensate in the confined and deconfined phases agree. This can be explained by the universal scaling c 3 = #B 3/2 , found in subsection 4.2, for the dimensionless condensate c 3 in the regime of large B. Regarding the comparison to lattice QCD, Fig. 18 shows that our results differ significantly from the lattice results in the regime of high temperatures but there is a reasonable agreement at low temperatures. The agreement improves as the magnetic field increases. The disagreement at high temperatures was expected since our model is not able to describe the transition from MC to IMC. We expect that backreaction effects would allow for such a description.

Conclusions
In this paper we have studied the effects of a non-zero magnetic field on the chiral condensate of a QCD-like theory using a holographic QCD model. As emphasised, this model, with chiral symmetry breaking using a tachyon, has been studied in detail before in the absence of a magnetic field, but here we have shown a variety of behaviours in both the confined as well as deconfined phases with a magnetic field present. As expected in the quenched approximation, the addition of the magnetic field has given us catalysis of chiral symmetry breaking, whereby the value of the condensate goes up with increasing magnetic field. There is one caveat to this that in the case of zero quark mass and in the deconfined case, there is a critical value of the magnetic field below which there is no chiral symmetry breaking, and above which it is induced, signifying spontaneous chiral symmetry breaking. This second order phase transition only exists in the chiral limit, and at any non-zero quark mass it becomes a cross-over phase transition.  [103,104], using the empirical formula given in [65].
For large quark masses we have seen that the behaviour of the confined and deconfined phases converge, as expected when the mass scale of the constituents is greater than the dynamical and thermal mass scales of the theory. The universal asymptotic behaviour in the regime of large quark mass suggest some approximate conformal symmetry in the dual field theory, after the subtraction of the (conformal symmetry breaking) mass term and it seems to be related to the AdS asymptotics of the gravity dual. It should be noted however that at large energies the gluon dynamics of these theories are not only conformal but also 4+1 dimensional. As noted in subsection 4.2, this phenomenon is generally obtained in holographic brane constructions.
In addition to the spontaneous symmetry breaking we have been able to study the magnetisation of this theory, where in the deconfined phase the second order phase transition is again apparent in both the magnetisation as well as the susceptibility. Due to the explicit nature of the tachyon in the DBI action we have been able to extract the condensate contribution to the magnetisation. We arrived at a simple empirical formula relating the magnetisation and chiral condensate. Since both quantities are important order parameters for magnetic catalysis and inverse magnetic catalysis, our formula could be useful for unveiling the physical mechanisms behind those phenomena.
As noted, we are here working in the quenched approximation where, in this model, we wouldn't expect to see anything but the magnetic field catalysing chiral symmetry breaking. A clear extension to this work would be to go beyond the probe approximation and allow for back-reaction on the geometry by the tachyon field. This would allow us to also investigate inverse magnetic catalysis, but this calculation will be an order of magnitude more complicated, particularly as the equations of motion would involve a divergent tachyon backreacting on the geometry when confinement is present.
With or without backreaction, several other phenomena could still be investigated in this model. As noted earlier, this model is particularly interesting as it gives rise to realistic Regge trajectories for the mesons, and so the effects of the magnetic field on these trajectories would be extremely interesting to investigate. Given this one could also study the Gellman-Oakes-Renner [116] relation between the quark mass and condensate and the pion mass in the presence of a magnetic field. Investigating the fluctuations on top of this background would also allow for an explicit construction of the chiral effective theory from the 5d flavour action, whereby the Gasser-Leutwyler coefficients [117] could be compared with lattice data. Such calculations have been performed before [28], but this model would likely give results closer to those of QCD.
Extending the model in [1,37] to the non-Abelian case would also be a natural next step. This would allow for a more realistic description of chiral and flavour symmetry breaking as well as the meson phenomenology. Although the original proposal in [38] describes the non-Abelian tachyonic DBI and WZ terms, there are some subtleties when describing spontaneous chiral symmetry breaking and the QCD anomalies in the non-Abelian case.
Another interesting future direction could be the addition of baryons and the further study of the holographic model. First in the probe approximation and then taking into account the backreaction of the baryon in the geometry. That would give access to the low temperature and high density region of the phase diagram. In a top-down framework the baryon vertex corresponds to a D-brane wrapping an internal sphere and connecting to the boundary with N c fundamental strings [108] (see also [109,110]). In the Sakai-Sugimoto model, which is the closest holographic model for QCD, baryons appear as 5d instantons in the non-Abelian flavour sector [111,112]. Interestingly, these 5d instantons are the holographic dual of 4d skyrmions dressed by vector mesons. The interplay between baryon density and magnetic field has also been investigated in [115] for the Sakai-Sugimoto model, in which inverse magnetic catalysis was also observed. In a bottom-up scenario, a baryon solution exists both in AdS/QCD [113] and V-QCD [114].

A The Tachyon-WZ term
It was shown in [38] that the Wess-Zumino (WZ) action can be written as a 5d Chern-Simons action. For the Abelian case it takes the form where γ is a constant proportional to N c and Ω 5 is a 5-form satisfying the equation We have introduced a set of forms , We have used the definition of the 1-form covariant derivative Dτ = dτ + i(A L − A R )τ and the 1-form current j = i (τ dτ * − τ * dτ ). There are two possible solutions for Ω 5 related to each other by a total derivative. The simpler solution is An alternative solution for Ω 5 was given in [38] for the case of a real tachyon. For a complex tachyon it takes the form Ω II 5 ≡ Ω 5 + ∆Ω 5 , where ∆Ω 5 is a total derivative given by From (A.5) and (A.7) we obtain the explicit form This is the form that appears in [38] for the case of a real tachyon (τ = τ * and j = 0). In that case we can easily find that under the residual gauge transformation U (1) V , i.e. A L/R → A L/R + dα, the Chern-Simons form in (A.8) transforms as where we have introduced the 3-form The variation in (A.9) is a boundary term. Imposing the boundary conditions for τ ; namely a vanishing tachyon near the boundary and a divergent tachyon at the end of space we find that the first term in (A.9) vanishes whilst the second term reduces to a 4d anomaly term for the residual U (1) V symmetry, as expected in QCD. The description of the full U (1) L × U (1) R anomaly term is more subtle because it requires a very careful analysis of the variation of (A.8). A first look at the problem suggests that a correction to (A.8) is required in order to describe the full QCD anomaly term.

B.1 IR asymptotic analysis in the confined phase
To find the asymptotic solution near the tip of the brane u = 1, it is convenient to work with the variable x ≡ 1 − u. The ansatz for T will be a series expansion in powers of x. First we write (3.15) as where we have defined the following quantities The advantage of writing the T differential equation in terms of the operators x∂ x is that these operators can act on powers without changing the exponents. The functions P(x), Q(x) and R(x) have nonzero values at x = 0 and can be (Taylor) expanded in powers of x. We consider the series ansatz where α n are real numbers satisfying the inequality α 0 < α 1 < · · · < α n < α n+1 < . . . and g n (x) are analytic functions of x. Plugging the ansatz (B.3) into (B.1), we obtain ∞ n=0 where we have introduced the operator O n ≡ x∂ x +α n . The quantity (O n g n ) is the function obtained when the operator O n has already acted on the function g n . Since the functions g n admit a Taylor expansion around x = 0 we find that (O n g n ) becomes α n when x → 0.
Since the exponents α n are non-integer, at each order in the series we obtain differential equations for the coefficients g n (x). Let us focus on the first exponent α 0 . There are three cases: α 0 = 0, α 0 > 0 and α 0 < 0.
Case I: α 0 = 0 The first term in the series (B.4) is of order x 0 and we find This is a non-linear equation for g 0 and in the limit x → 0 it holds automatically, since O 0 g 0 = α 0 = 0 in that limit.
Case III: α 0 < 0 The physically interesting case is when the solution is singular at x = 0 (which is a good property according to the anomaly story in the IKP framework). Now the first term in the series (B.4) is of order x 3α 0 and we obtain the equation We have two situations: (O 0 g 0 ) = 0 and (O 0 g 0 ) = − g 0 /R(x). Since (O 0 g 0 ) = x∂ x g 0 + α 0 g 0 the equations are first order and can be solved. In the first case we find g 0 ∼ x −α 0 which contradicts the assumption that g 0 is regular. In the second case the equation can be written as dg 0 + g 0 (a 0 dx) = 0 with a 0 (x) = 1 . (B.8) Multiplying (B.8) by an integrating factor u 0 (x) such that du 0 = u 0 (a 0 dx), the l.h.s. of (B.8) becomes an exact differential d(g 0 u 0 ) and we find g 0 (x) = g 0 (0) exp − x 0 dx a 0 (x ) . (B.9) In the limit x → 0 we have that x∂ x g 0 → 0 and g 0 → 1 so we find that where we have also introduced the positive real number r = −α 0 . Note that R(0) = 1/r. The function g 0 (x) can be Taylor expanded as g 0 (x) = ∞ n=0 g 0,n xn where g 0,0 = g 0 (0) ≡ C 0 (B.11) and the following coefficient in the series expansion is g 0,1 = g 0 (0) = − a 0 (0) g 0 (0) = − 3 10 Now we make the following assumption: the exponents α n are integer powers of r, namely α n = (n − 1) r, which is compatible with the case n = 0. The strategy now is to find equations for g n at each different order in the series (B.4). We have already found the first equation (B.7) at order x −3r . At the next order x −2r we find the equation Taking the limit x → 0 in (B.14), we find that The differential equation (B.14) allows us to find g 2 (x) given g 0 (x), the latter found in (B.9). It can be put in the canonical form where a 2 (x) = 1 and we remind the reader that α 2 = r. Following a procedure similar to the one used for g 0 (x) we find g 2 (x) = g 2 (0) e − x 0 dx a 2 (x ) 1 + from considering the equation g 2 = −a 2 g 2 + b 2 . In the limit B → 0 we obtain g 2,1 /g 2,0 = − 1479 3380 which agrees with the result in [1]. The method described above extends in a straightforward manner and we can extract higher order terms in the expansion.
Case IV: α 0 = 0 The first term in (B.24) is of order y −1 and we obtain the equation O 0 g 0 = y∂ y g 0 = 0 with solution g 0 (y) = const. We conclude from this analysis that α 0 = 0. Let us now find the subleading exponent α 1 . We will prove that α 1 is an integer and the series (B.23) actually reduces to an ordinary Taylor expansion. The proof is by contradiction. Assuming that α 1 is not an integer we have 3 possibilities: 0 < α 1 < 1/2, α 1 = 1/2 and α 1 > 1/2.
Case III: α 1 > 1/2 The second term in (B.24) is of order y α 1 and we obtain Taking the limit y → 0 and using P (0) = 1 and Q(0) = 3/5 we find g 1 (0) = 0 which is not a valid solution. From the analysis above we conclude that α 0 = 0 and α 1 is a (positive) integer. Therefore, it is reasonable to assume that the series (B.23) reduces to an ordinary Taylor expansion T (y) = ∞ n=0 y n C n . (B.29) Had we found a noninteger solution for α 1 such that 0 < α 1 < 1 that would have corresponded to the spurious case where T is not singular but has a singular derivative. Plugging the ansatz (B.29) into eq. (B.21) and using the Taylor expansions for P (y), Q(y) and R(y) we obtain ∞ n=0 y n C n n 2 + The terms of order y −1 and y 0 in (B.30) vanish automatically whereas the the term of order y leads to the condition The next term in (B.30) is of order y 2 and vanishes if where we have used the results Q 0 = 3/5, Q 1 /Q 0 = 4, R 0 = 5/6 and P 1 = (1−B 2 )/(1+B 2 ).