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Holographic chiral phase transition with chemical potential

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N. Horigome, Y. Tanii

High Energy Physics - Theory · Submitted 28 Aug 2006 · 24 references

Pith's verdict: A competent, honest finite-density extension of Sakai-Sugimoto; the first-order transition is real, the μ identification is standard, and the low-T argument is the one place that needs tightening.

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arXiv IDhep-th/0608198 (version 3)
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CitationN. Horigome, Y. Tanii. "Holographic chiral phase transition with chemical potential." arXiv:hep-th/0608198 (2006). https://pith.science/paper/6HQHOFEK
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The paper

The paper claims

Adding a baryon-number chemical potential to a holographic model of QCD turns the chiral symmetry transition first order, and above a critical density removes it altogether.

The reading

The paper asks what happens to the chiral symmetry of quarks in a holographic model of QCD once a baryon number density is switched on. The setting is the Sakai-Sugimoto construction: $N_c$ D4-branes provide a gravitational background dual to a confining gauge theory, while $N_f$ probe D8-$overline{mathrm{D8}}$ pairs provide massless quarks whose chiral symmetry is realised geometrically by how the probe branes sit in that geometry. The authors propose that the asymptotic value of the time component of a $U(1)$ gauge field living on the probe branes is the baryon number chemical potential $mu$, and then solve the probe-brane equations at finite temperature $T$ and finite $mu$. What they find is a first-order transition between a connected, curved brane configuration (chiral symmetry broken) and a straight, disconnected one (chiral symmetry unbroken), with the transition temperature falling as $mu$ grows and the transition disappearing entirely above a critical density. The version under review is a revision; the authors state that an error in the first version was pointed out by a later paper.

What carries the argument

The machinery is the DBI action on the probe D8-$overline{mathrm{D8}}$ world-volume, evaluated on an ansatz where only $tau(U)$ and $A_0(U)$ vary, so that the Chern-Simons term drops out. Each equation of motion integrates once to two constants, $U_0$ and $C$. A regularity condition at the turning point, $A_0(U) sim A_0(U_0) + text{const} times C,|U-U_0|^{1/2}$, forces $C=0$ in the low-temperature phase, which is why the chemical potential there is just a constant $mu$ and cannot change the brane shape. In the high-temperature phase the same constant $C$ survives in the straight configuration and is fixed by requiring $A_0$ to vanish at the horizon $U_T$, a condition derived by going to Cartesian coordinates near the tip of the cigar; integrating the resulting expression to infinity gives $mu$ as a function of $u_T$ and $c$, which is what converts the abstract $c$–$u_T$ diagram into a physical $mu$–$T$ diagram.

Extended reading notes

Core claim

The claim is that in the high-temperature phase of the Sakai-Sugimoto model there are exactly two admissible probe-brane configurations once a background $U(1)$ gauge field is turned on, and that which one dominates is decided by comparing their Dirac-Born-Infeld actions. The curved, connected configuration has the D8 and $overline{mathrm{D8}}$ branes joining at a turning point $U_0$ and breaks the chiral symmetry to the diagonal subgroup; the straight configuration has them running separately to the horizon and leaves the symmetry intact. The difference $Delta S$ of the two actions is finite and depends on the horizon parameter $u_T$ and on the integration constant $c$ that measures the gauge-field charge. For $c=0$ the transition sits at a definite $u_{T0}$, reproducing the earlier zero-density result; as $c$ increases the transition moves to smaller $u_T$, a new region of positive $Delta S$ appears near $u_T=0$ once $c>0.2158$, and for $c>0.2252$ the straight configuration wins everywhere, so the chiral symmetry is never broken. Translated through the horizon relation and the asymptotic expansion $A_0 sim mu + v/U^{3/2}$, this gives a phase diagram in the chemical potential–temperature plane whose upper-right region is unbroken and whose lower-left region is broken, with a first-order line between them that terminates at a critical chemical potential.

Load-bearing premise

The load-bearing premise is that the asymptotic value of the $U(1)$ gauge field on the probe branes really is the baryon number chemical potential; the paper offers this as a proposal supported by the AdS/CFT dictionary rather than deriving it, and the entire physical reading of the phase diagram rests on it.

Editorial extensions

If this is right

  • Below the confinement/deconfinement temperature the model admits only the connected configuration, so chiral symmetry stays broken at every chemical potential, in contrast with the expectation from QCD that large density restores it even at zero temperature.
  • There is a critical chemical potential, corresponding to $c approx 0.2252$ in the rescaled variables, above which no chiral transition occurs at any temperature.
  • The transition is first order, since two distinct brane configurations coexist at the transition point rather than one merging continuously into the other.
  • In physical units the transition temperature is of order $L^{-1}$ and the transition chemical potential of order $g^2_{YM} N_c L^{-2} M_{KK}^{-1}$, which is much larger than $M_{KK}$ because $g^2_{YM} N_c gg 1$.
  • The resulting phase diagram can be compared directly against the expected QCD phase diagram, and the comparison singles out the low-temperature region as where the model departs from QCD.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The whole phase diagram is computed as a function of the gauge-field parameter $c$; if the identification of the asymptotic $A_0$ with a baryon-number source is off by a rescaling or refers to a different operator, the shape of the diagram survives unchanged while its horizontal axis is mislabelled, so an independent check of that dictionary entry would be the decisive test.
  • The authors' own remark that back-reaction is neglected suggests the natural next calculation: letting the D8-brane charge density deform the $tau$-$U$ geometry would plausibly generate a low-temperature chiral-restoring branch, which is precisely the branch QCD is expected to have.
  • Because the constant term of $A_0$ is fixed by the horizon regularity condition $A_0(U_T)=0$, a different regularity prescription would shift the map $mu(c,u_T)$ without touching the existence or order of the transition, so the qualitative result is more robust than the numerical critical values.
  • The model has exactly massless quarks by construction, so the first-order character claimed here is a sharp statement about the ideal chiral limit; a comparison with a lattice study at small but nonzero quark mass would test whether the order survives once the symmetry is explicitly broken.
Original abstract

We discuss the Sakai-Sugimoto model at finite temperature and finite chemical potential. It is a holographic model of large N_c QCD with N_f massless quarks based on a D4/D8-\bar{D8} brane system. The near horizon limit of the D4-branes and the probe approximation of the D8-\bar{D8} pairs allow us to treat the D4-branes as a gravitational background and the D8-\bar{D8} pairs as a probe which does not affect the background. We propose that the asymptotic value of a U(1) gauge field on the D8-\bar{D8}-branes is identified with the chemical potential for the baryon number. Using this chemical potential we analyze the phase structure of this model and find a chiral symmetry phase transition of the first order.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,343 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.05 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We discuss the Sakai-Sugimoto model at finite temperature and finite chemical potential. It is a holographic model of large N_c QCD with N_f massless quarks based on a D4/D8-\bar{D8} brane system. The near horizon limit of the D4-branes and the probe approximation of the D8-\bar{D8} pairs allow us to treat the D4-branes as a gravitational background and the D8-\bar{D8} pairs as a probe which does not affect the background. We propose that the asymptotic value of a U(1) gauge field on the D8-\bar{D8}-branes is identified with the chemical potential for the baryon number. Using this chemical potential we analyze the phase structure of this model and find a chiral symmetry phase transition of the first order.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We discuss the Sakai-Sugimoto model at finite temperature and finite chemical potential. contribution 0.74 0.71 0.04 asserted 0.73
s1 It is a holographic model of large N_c QCD with N_f massless quarks based on a D4/D8-\bar{D8} brane system. background 0.71 0.10 0.03 asserted 1.00
s2 The near horizon limit of the D4-branes and the probe approximation of the D8-\bar{D8} pairs allow us to treat the D4-branes as a gravitational background and the D8-\bar{D8} pairs as a probe which does not affect the background. method 0.87 0.23 0.03 asserted 0.99
s3 We propose that the asymptotic value of a U(1) gauge field on the D8-\bar{D8}-branes is identified with the chemical potential for the baryon number. method 0.59 0.77 0.08 speculative 0.50
s4 Using this chemical potential we analyze the phase structure of this model and find a chiral symmetry phase transition of the first order. result 0.99 0.96 0.85 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.71.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The phase diagram of the dual gauge theory.
Figure 2. A D8-D8-brane configuration in the low temperature phase. where C and U0 are integration constants. As in the zero temperature case [11] and the low temperature phase in ref. [13] we have imposed a condition τ ′ (U0) = ∞. A typical configuration of τ (U) is shown in
Figure 3. D8-D8-brane configurations in the high temperature phase. where U0 is an integration constant. As before we have imposed the boundary conditions τ ′ (U0) = ∞ and A0(∞) = µ. A typical configuration of τ (U) is shown in
Figure 4. ∆S as a function of uT for various values of c. From the bottom to the top each line represents the case for c = 0, 0.2158, 0.2252, 0.3 respectively. ∆S > 0. When c > 0.2252, ∆S is positive for all values of uT and the chiral symmetry is always unbroken. From these results we can draw a phase diagram in the c-uT space as shown in
Figure 5. The phase diagram in the c-uT space. for a massless vector field in a six-dimensional bulk, µ is a source coupled to an operator of dimension four O4 on a five-dimensional boundary. The U(1) gauge field A0 defined on the whole of the D8-D8 world-volume contains the gauge fields for both of U(1)V and U(1)A in the flavor symmetry (2.6). The part of A0 which is symmetric for an interchange of D8 and D8 corresponds to U(1)V , while the part which is antisymmetric corresponds to U(1)A [11, 22]. Since the constant term µ is symmetric, it is a background value of the U(1)V gauge field…
Figure 6. The phase diagram in the ˜µ-T˜ space. 5. Discussions We analyzed the Sakai-Sugimoto model at finite temperature and finite baryon number chemical potential. The chemical potential is introduced as an asymptotic value of the U(1) gauge field on the probe D8-D8-branes. Using this model we studied the phase structure of the chiral symmetry breaking and obtained the phase diagram in

Figures reproduced from arXiv:hep-th/0608198 by the authors.

References
How later papers used it

70 later papers cite it

Evidence: substantial. 69 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 118 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 5 papers with this label

  • Followed this paper in assuming a dependence of the additional gauge field component on U, to be fixed by the equations of motion from the probe action.

    A Comparative Study of NJL and Sakai-Sugimoto Models (2009) · 0.89 · paraphrased: the licence does not allow quoting

Compares with it 1 of 7 papers with this label
How the literature uses it

68 arXiv papers cite this in 117 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Papers citing it · 60 sentences without a confident label

Builds on it 1 paper
  • Our calculation builds upon previous work within the Sakai-Sugimoto model in the presence of a magnetic field [ref] , and generalizes the results for the chiral phase transition in the $T$-$μ$ plane at $B=0$ [ref] and the $T$-$B$ plane at $μ=0$ [ref] to the entire $T$-$μ$-$B$ space.

    Inverse magnetic catalysis in dense holographic matter · Introduction · 1.00

Uses its method or tool 5 papers
  • Thus, in addition to the components ([ref]), we include [equation] where following Reference [ref] , we assume the dependence of this component on $U$ and this function must be determined by solving the equations of motion that follow from the probe action.

    A Comparative Study of NJL and Sakai-Sugimoto Models · External Parameters · 0.89

  • This is done by relating the boundary values at $z=±∞$ for the gauge fields $Â_0(z)$ and $A_0(z)$ with the baryon and isospin chemical potentials $μ_B$ and $μ_I$ [ref] .

    Meson supercurrents and the Meissner effect in the Sakai-Sugimoto model · Equations of motion and ansatz including magnetic field, chemical potentials, and supercurrents · 0.81

  • Similar to [ref] - [ref] , in order to investigate this holographic model with a chemical potential, we only need open the zero component of the gauge field on the worldvolume of the flavor D4 branes.

    Holographic phase transition in a non-critical holographic model · Brane configuration · 0.97

Compares with it 7 papers
Cites it as background 37 papers
  • Among these works, the D4-D8 model i.e. the Witten-Sakai-Sugimoto model [ref] , as a top-down approach based on the underlying string theory, becomes famous and successful in the process of time because this model includes mostly all the fundamental elements of QCD in a very simple way so that it could reproduce various elementary features of QCD e.g. deconfinement and chiral phase transition [ref] , baryon spectrum [ref] , glueball spectrum [ref] , the interaction of glueball and meson or baryon [ref] , the theta term [ref] .

    Corrections to the instanton configuration as baryon in holographic QCD · Introduction · 0.93

  • Finite baryon density in the Sakai-Sugimoto model has been studied in [ref] .

    Holographic Nuclear Physics · Introduction · 0.95

  • However attempts to introduce a baryon chemical potential to AdS/CFT have been just started recently [ref] , and there still exists a point which is under debate [ref] .

    Baryon-charge Chemical Potential in AdS/CFT · Introduction · 0.85

Discussion

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