Chiral phase structure of QCD with many flavors

Holger Gies; Joerg Jaeckel

REVIEW

Chiral phase structure of QCD with many flavors

Holger Gies, Joerg Jaeckel

High Energy Physics - Phenomenology · Submitted 14 Jul 2005 · 27 references

Pith's verdict: A careful, honest functional-RG determination of Nf^cr; the fermionic side is genuinely tested, but the headline error bar is a hand-set scheme-mismatch judgment rather than a derived systematic.

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arXiv IDhep-ph/0507171 (version 2)

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CitationHolger Gies, Joerg Jaeckel. "Chiral phase structure of QCD with many flavors." arXiv:hep-ph/0507171 (2005). https://pith.science/paper/X75ZEX6I

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Editor's note

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You should read this one for the method, not for the number. Gies and Jaeckel compute the many-flavor chiral transition with the functional RG, and the construction is clean: alpha_cr comes from the four-fermion flow at fixed gauge coupling, alpha*(Nf) from the gauge beta function, and Nf^cr is their crossing. Nothing is fitted to anything, so the reader's no-circularity call is right. The genuinely new bits are the complete basis of pointlike four-fermion operators (Eqs. 4-7, not a subset) and the modified Ward identities (Eq. 13) that fix how four-fermion terms feed the gauge flow. They also check that the lambda-dependent pieces drop out at the fixed point, which is the right check to make. The regulator study is real work: optimized linear regulator versus sharp cutoff, with the strong cancellations between threshold functions noted explicitly. That is a better error estimate than most truncation papers bother with, and it is why the fermionic band in Fig. 3 is narrow. All flow equations and threshold functions are printed, so the calculation is reproducible by hand for its era. Credit where due: this is the most systematic treatment of the operator basis in the pointlike limit that existed at the time. The soft spot is exactly where the stress-test says. The gauge anomalous dimension is not computed in their scheme; they import the four-loop MS result into an RG-scheme flow, footnote 3 concedes possible double counting, and the +1.55/-0.63 band is then set by assumed 30% and 50% uncertainties on the three- and four-loop coefficients, calibrated off a 10% two-loop scheme difference from [22]. That is an honest estimate, clearly labelled, but it is a judgment, and it dominates the headline number. The one-loop relation for eta_A in the threshold functions is a smaller version of the same thing. None of this threatens the qualitative claim, which is the part worth citing: a conformal window well separated from Nf^a.f. survives any plausible scheme shift. The central value 10.0 should be read as 10-ish. Minor point: the argument that classical propagators are self-consistent because the anomalous dimensions vanish at the fixed point uses the truncation to justify the truncation, though it is not load-bearing. Worth a referee's time; the paper is explicit about its own approximations, which is more than most.

Referee report

Significance

The result is a genuinely useful, falsifiable prediction: Nf^cr = 10 for SU(3) is a number that lattice simulations can and do confront, and the paper is careful to state it with an error budget rather than as a point. Several methodological choices deserve explicit credit. The four-fermion basis of Eq. (2) is claimed complete under Fierz transformations, so the truncation captures ladder and non-ladder contributions within the pointlike limit rather than a hand-picked subset. Gauge invariance is not assumed but tracked through the modified Ward-Takahashi identity of Eq. (13), and the observation that the lambda-dependent terms in the gauge flow vanish at any fixed point of the fermionic subsystem is a clean structural result that makes the crossing argument well defined. The use of regulator variation (optimized linear regulator versus sharp cutoff, Eq. 12) as a truncation-error diagnostic is the right instinct, and the smallness of the resulting fermionic band is a meaningful consistency check. The paper also correctly separates what is universal (the existence of the fixed point, the value of Nf^cr) from what is scheme dependent (alpha*, alpha_cr), which is exactly the right way

Major objections (3)

[Sect. II, Eqs. (13)-(14); Sect. III, Fig. 2] The crossing that defines Nf^cr compares two quantities computed in different schemes: alpha_cr from the RG-scheme four-fermion flow with the linear regulator, and alpha*(Nf) from the four-loop MS beta function substituted for the RG-scheme anomalous dimension in Eq. (14). The authors acknowledge this in footnote 3 and correctly note that the lambda-dependent terms in Eq. (13) drop out at the fixed point, but that argument removes the double-counting concern, not the scheme mismatch in the gauge anomalous dimension itself. The residual uncertainty is then assigned by hand (30% on the three-loop and 50% on the four-loop coefficients, calibrated by the 10% two-loop scheme difference quoted from [22]), and the resulting band is asymmetric, +1.55/-0.63, without any stated derivation of the asymmetry. Since this is the dominant contribution to the headline error bar, the manuscript should make the propagation explicit: quote the numerical values of alpha*(Nf) and alpha_cr(Nf) near Nf = 10 and the slope d alpha*/dNf at the crossing, so that a reader can verify that a given delta alpha* maps onto the quoted asymmetric delta Nf^cr. A concrete additional test would be to check the stability of the crossing under a common multiplicative rescaling of both alpha* and alpha_cr, which is the leading effect of a change in the coupling normalization; if Nf^cr is approximately invariant under such a common shift, that would materially strengthen the central value beyond the present one-sided variation of alpha* alone.
[Sect. III, Fig. 3 and the fermionic error band] The fermionic error estimate compares the optimized linear regulator with the sharp cutoff, which changes only the threshold constants l in Eq. (12) while leaving the pointlike vertex and propagator ansatz untouched. The paper argues that strong regulator dependence would signal a failure of the pointlike approximation, which is reasonable, but the inference is one-directional: the absence of regulator dependence bounds the sensitivity of the flow to the momentum localization of the regulator, not the error induced by truncating the momentum dependence of the vertices themselves. The quoted +/-0.29 should therefore be described as a lower bound on the fermionic truncation error rather than as a full error bar, and the sentence in Sect. III stating that the observed cancellations 'provide evidence for the reliability of the pointlike truncation' should be softened accordingly. This does not affect the qualitative conclusion, since the fermionic band is small compared with the gluonic one, but the wording currently overstates what the test establishes.
[Sect. IV, Summary and Conclusions] The concluding sentence states that the result confirms a phase of many-flavor QCD which is asymptotically free 'but has no chiSB and no strict confinement in the IR'. The no-chiSB part follows from the calculation; the no-confinement part does not. The truncation retains only quark and gluon fields with pointlike interactions and, as the authors themselves note, does not facilitate a study of the broken phase or of the infrared dynamics of the gauge sector beyond the fixed-point coupling. The absence of strict confinement is a plausible corollary of an infrared conformal fixed point, but it is not derived here and should either be supported by an explicit argument or removed from the summary sentence, since as written it reads as a result of the present analysis.

Minor objections (8)

[Abstract and Sect. IV] The asymmetric error notation '10.0 +/- 0.29(fermion) [+1.55;-0.63](gluon)' is hard to parse in running text; a form such as Nf^cr = 10.0 +1.55/-0.63 (gluon) +/- 0.29 (fermion), with the asymmetry explained in one clause, would read more cleanly.
[Fig. 2 and Fig. 3] The two figures use different conventions for the same information: Fig. 2 encodes loop order by line colour (red/dark-grey for four loop, green/light-grey for two loop), whereas Fig. 3 encodes it by marker shape (circles, stars, boxes). A consistent convention, or an explicit cross-reference in the Fig. 3 caption, would help the reader connect the crossing in Fig. 2 to the curve in Fig. 3.
[Sect. I] The statement that the transition is 'not conventionally second order' is supported in this paper only by the discontinuity of the lambda_i, which are parameters of the truncated flow rather than observables. It would be worth one sentence making explicit that the evidence is indirect and rests on the identification lambda_i ~ 1/m_i^2 discussed in Sect. III.
[Eq. (2)] There are several typographical problems in the effective action: the field-strength term appears as 'F_mu nu^z F^{z mu nu}' with the colour index misplaced, the gauge-fixing term '(partial_mu A_mu)^2 / 2 alpha' is written without the overall minus sign convention being stated, and the fermion kinetic term shows 'A/' without the slash on the gauge field in the typeset version. These should be cleaned up.
[Eq. (3)] The definition of the renormalized dimensionless couplings uses the same symbol lambda_i on both sides of the equation, which makes the relation ambiguous as printed; a distinct symbol for the bare coupling, or an explicit 'bare' label, would remove the ambiguity.
[Sect. III, footnote 4] The check using the pure one-loop eta_A is described only as lying 'well within the estimated error'. Quoting the numerical shift in Nf^cr that this check produces would make the statement verifiable rather than qualitative.
[Sect. III] The value of Nf^a.f. for SU(3), which sets the upper end of the conformal window, is shown only as a line in Fig. 3 and is never quoted numerically in the text; giving it explicitly (16.5 for SU(3)) would make the statement that the total error is small compared with the distance to Nf^a.f. immediately checkable.
[Throughout] Minor spelling and consistency issues: 'turquois' should be 'turquoise', 'Gaußian' and 'Gaussian' both appear, and the hyphenation of 'Ward-Takahashi' is inconsistent between text and figure captions.

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The paper

The paper claims

Using functional renormalization group flow equations, the authors locate the number of massless quark flavors at which QCD stops breaking chiral symmetry, finding about 10 for SU(3) and confirming a window of asymptotically free but chira

The reading

The paper asks a sharp question about a deformed version of QCD: if you add more and more massless quark flavors, at what point does the theory stop forming a chiral condensate? Light fermions screen the gauge force, so a large flavor number pushes the theory toward weak coupling, where an infrared fixed point can exist and the quarks can stay massless at every scale. The authors want to know whether such a conformal window really exists between ordinary QCD and the point where asymptotic freedom is lost, and where its lower edge sits. They answer by solving flow equations for the effective average action, keeping every pointlike four-fermion interaction allowed by gauge and chiral symmetry, and reading off the crossing between the coupling at which the fermion system goes critical and the coupling the gauge sector actually reaches in the infrared. For SU(3) they quote $N_f^{cr} = 10.0$ with asymmetric errors, and they argue the gap between this and the asymptotic-freedom bound is large enough that the conformal window is not a marginal effect.

What carries the argument

The engine is the Wetterich flow equation for the effective average action, $k$-dependent and exact in principle, truncated to a basis of quark, gluon and four-fermion operators. The four-fermion sector uses a complete basis of pointlike interactions — color and flavor singlets $(V-A)$ and $(V+A)$, the color/flavor-nontrivial $(S-P)$, and the adjoint $(V-A)_{adj}$ — so ladder and non-ladder contributions are both generated by the flow. Gauge invariance is tracked by modified Ward-Takahashi identities, which force the four-fermion contributions to the gauge-coupling flow to be proportional to $c_i (c_i + 2 v_4 l_4^{(F)} c_j)$-type combinations of the $c_i$ coefficients, and these terms vanish at any fixed point of the fermion subsystem, so $N_f^{cr}$ is unaffected by them. Regulator dependence is used deliberately as an error estimator: the difference between the optimized linear regulator and the sharp cutoff bounds the fermionic uncertainty, and the spread across two-, three- and four-loop gauge $\beta$ functions, with 30% and 50% hand-assigned uncertainties on the three- and four-loop coefficients, bounds the gluonic one.

Extended reading notes

Core claim

The central claim is that the zero-temperature chiral phase transition in $SU(N_c)$ gauge theory with many massless flavors happens at a calculable flavor number, and that a genuine window exists above it in which the theory is asymptotically free in the ultraviolet yet never breaks chiral symmetry, flowing instead to a conformal infrared fixed point. For $SU(3)$ the quoted result is $N_f^{cr} = 10.0$, with an error of $0.29$ attributed to the fermionic truncation and $+1.55/-0.63$ to the gluonic sector. The mechanism is a competition of two couplings: the four-fermion self-interactions have an infrared-attractive Gaussian fixed point as long as the gauge coupling stays below a critical value $g_{cr}^2$, and the gauge coupling itself settles at an infrared fixed-point value $g_*^2(N_f)$ that rises as $N_f$ is lowered; the transition is where the two meet. The authors also report a qualitative feature of the transition: the fermionic self-interactions approach finite fixed-point values on the symmetric side and jump discontinuously as $N_f$ crosses $N_f^{cr}$, so the transition is continuous in the chiral order parameter but not conventionally second order, with no light scalar states on the symmetric side.

Load-bearing premise

The gauge sector's running is not computed in the paper's own regularization scheme: the authors substitute the four-loop MS-scheme beta function, concede possible double counting of diagrams, and assign the gluonic error bar by assuming 30 percent and 50 percent uncertainties on the three- and four-loop coefficients, so the headline number inherits whatever scheme mismatch that substitution carries.

Editorial extensions

If this is right

For $SU(3)$, any number of massless flavors between roughly 10 and $16.5$ gives a theory that is asymptotically free in the ultraviolet but has no chiral condensate and no strict confinement in the infrared.
The chiral transition is not a conventional second-order transition: the order parameter moves continuously but the generated four-fermion couplings jump at $N_f^{cr}$, so no Ginzburg-Landau effective theory with light scalars can describe the symmetric side.
The lattice and Dyson-Schwinger estimates that placed the critical flavor number anywhere from about 5 to 12 are now bracketed: values well below 10 would require the gauge sector to behave very differently from the four-loop perturbative running.
Because the estimated total error stays well below the distance to the asymptotic-freedom line, the conformal window is a robust prediction of the truncation rather than a coincidence of the crossing point.

Reading between the lines

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

The same fixed-point-versus-critical-coupling logic should apply to QED in three dimensions and to technicolor models of electroweak symmetry breaking, where the question is likewise whether the gauge coupling can reach a fixed point before the fermion system goes critical.
If the window is real, lattice studies of $SU(3)$ at $N_f = 10, 11, 12$ should show a massless, interacting, non-confining spectrum with a slowly running coupling; the borderline case $N_f = 10$ is the sharpest test because the crossing sits closest to it.
The authors' error budget is dominated by substituting a four-loop perturbative beta function for a flow-equation computation of the gluon anomalous dimension; a genuine scheme-matched calculation would sharpen the central value and could shift it by more than the quoted fermionic error.
The discontinuous jump in the four-fermion couplings suggests that the transition is a boundary in coupling space rather than a critical surface, which would make the symmetric phase's fixed-point values compact and unreachable from the broken side by continuous deformation.

Original abstract

We investigate QCD with a large number of massless flavors with the aid of renormalization group flow equations. We determine the critical number of flavors separating the phases with and without chiral symmetry breaking in SU(Nc) gauge theory with many fermion flavors. Our analysis includes all possible fermionic interaction channels in the pointlike four-fermion limit. Constraints from gauge invariance are resolved explicitly and regulator-scheme dependencies are studied. Our findings confirm the existence of an Nf window where the system is asymptotically free in the ultraviolet, but remains massless and chirally invariant on all scales, approaching a conformal fixed point in the infrared. Our prediction for the critical number of flavors of the zero-temperature chiral phase transition in SU(3) is Nf^{cr}=10.0\pm 0.29(fermion)[+1.55;-0.63](gluon), with the errors arising from approximations in the fermionic and gluonic sectors, respectively.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.88 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We investigate QCD with a large number of massless flavors with the aid of renormalization group flow equations. We determine the critical number of flavors separating the phases with and without chiral symmetry breaking in SU(Nc) gauge theory with many fermion flavors. Our analysis includes all possible fermionic interaction channels in the pointlike four-fermion limit. Constraints from gauge invariance are resolved explicitly and regulator-scheme dependencies are studied. Our findings confirm the existence of an Nf window where the system is asymptotically free in the ultraviolet, but remains massless and chirally invariant on all scales, approaching a conformal fixed point in the infrared. Our prediction for the critical number of flavors of the zero-temperature chiral phase transition in SU(3) is Nf^{cr}=10.0\pm 0.29(fermion)[+1.55;-0.63](gluon), with the errors arising from approximations in the fermionic and gluonic sectors, respectively.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate QCD with a large number of massless flavors with the aid of renormalization group flow equations.	contribution 0.69	0.94	0.03	asserted 0.85
s1	We determine the critical number of flavors separating the phases with and without chiral symmetry breaking in SU(Nc) gauge theory with many fermion flavors.	contribution 0.78	0.95	0.12	asserted 1.00
s2	Our analysis includes all possible fermionic interaction channels in the pointlike four-fermion limit.	method 0.88	0.91	0.03	asserted 0.98
s3	Constraints from gauge invariance are resolved explicitly and regulator-scheme dependencies are studied.	method 0.91	0.93	0.02	asserted 0.98
s4	Our findings confirm the existence of an Nf window where the system is asymptotically free in the ultraviolet, but remains massless and chirally invariant on all scales, approaching a conformal fixed point in the infrared.	result 0.93	0.94	0.74	asserted 1.00
s5	Our prediction for the critical number of flavors of the zero-temperature chiral phase transition in SU(3) is Nf^{cr}=10.0\pm 0.29(fermion)[+1.55;-0.63](gluon), with the errors arising from approximations in the fermionic and gluonic sectors, respectively.	result 1.00	0.95	0.46	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from simulation or computation 0.61.

What was examined, and what was not

sources read

arXiv listing abstract v2

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.** FIG. 1: Sketch of a typical

**Figure 3.** FIG. 3: Critical number of flavors for SU(N

Showing 2 of the paper's 3 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/0507171 by the authors.

References

Works cited (27)

[22]

M. Reuter and C. Wetterich, Phys. Rev. D 56, 7893 (1997) [hep-th/9708051]; H. Gies, Phys. Rev. D 66, 025006 (2002) [hep-th/0202207]

View on Pith Pith review arXiv 1997
[1]

T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982)

View on Pith 1982
[2]

V. A. Miransky and K. Yamawaki, Phys. Rev. D 55, 5051 (1997) [Erratum-ibid. D 56, 3768 (1997)] [hep-th/9611142]

View on Pith Pith review arXiv 1997
[3]

T. Appelquist, J. Terning and L. C. R. Wijewardhana, Phys. Rev. Lett. 77, 1214 (1996) [hep-ph/9602385]; T. Appelquist, A. Ratnaweera, J. Terning and L. C. R. Wijewardhana, Phys. Rev. D 58, 105017 (1998) [hep-ph/9806472]

View on Pith Pith review arXiv 1996
[4]

M. Harada, M. Kurachi and K. Yamawaki, Phys. Rev. D 68, 076001 (2003) [hep-ph/0305018]

View on Pith Pith review arXiv 2003
[5]

Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai and T. Yoshie, Phys. Rev. D 69, 014507 (2004) [hep-lat/0309159]

View on Pith Pith review arXiv 2004
[6]

F. Sannino and J. Schechter, Phys. Rev. D 60, 056004 (1999) [hep-ph/9903359]

View on Pith Pith review arXiv 1999
[7]

T. Appelquist and S. B. Selipsky, Phys. Lett. B 400, 364 (1997) [hep-ph/9702404]; M. Velkovsky and E. V. Shuryak, Phys. Lett. B 437, 398 (1998) [hep-ph/9703345]

View on Pith Pith review arXiv 1997
[8]

M. Harada and K. Yamawaki, Phys. Rev. Lett. 86 (2001) 757 [hep-ph/0010207]

View on Pith Pith review arXiv 2001
[9]

R. S. Chivukula, Phys. Rev. D 55, 5238 (1997) [hep-ph/9612267]

View on Pith Pith review arXiv 1997
[10]

C. Wetterich, Phys. Lett. B 301, 90 (1993); Nucl. Phys. 7 B 352, 529 (1991); Z. Phys. C 48, 693 (1990)

View on Pith 1993
[11]

H. Gies, J. Jaeckel and C. Wetterich, Phys. Rev. D 69 (2004) 105008 [hep-ph/0312034]

View on Pith Pith review arXiv 2004
[12]

U. Ellwanger, M. Hirsch and A. Weber, Z. Phys. C 69, 687 (1996) [hep-th/9506019]; D. F. Litim and J. M. Pawlowski, Phys. Lett. B 435, 181 (1998) [hep-th/9802064]

View on Pith Pith review arXiv 1996
[13]

H. Gies and C. Wetterich, Phys. Rev. D 69, 025001 (2004) [hep-th/0209183]

View on Pith Pith review arXiv 2004
[14]

D. U. Jungnickel and C. Wetterich, Phys. Rev. D 53, 5142 (1996) [hep-ph/9505267]

View on Pith Pith review arXiv 1996
[15]

D. F. Litim, Phys. Lett. B 486 (2000) 92 [hep-th/0005245]; Phys. Rev. D 64 (2001) 105007 [hep-th/0103195]

View on Pith Pith review arXiv 2000
[16]

M. Reuter and C. Wetterich, Nucl. Phys. B 417, 181 (1994)

View on Pith 1994
[17]

U. Ellwanger, Phys. Lett. B 335, 364 (1994) [hep-th/9402077]; F. Freire, D. F. Litim and J. M. Pawlowski, Phys. Lett. B 495, 256 (2000) [hep-th/0009110]

View on Pith Pith review arXiv 1994
[18]

T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390]

View on Pith Pith review arXiv 1997
[19]

M. Czakon, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261]

View on Pith Pith review arXiv 2005
[20]

H. Gies and C. Wetterich, Phys. Rev. D 65 (2002) 065001 [hep-th/0107221]

View on Pith Pith review arXiv 2002
[21]

D. F. Litim, Nucl. Phys. B 631 (2002) 128 [hep-th/0203006]

View on Pith Pith review arXiv 2002
[23]

J. M. Pawlowski, Int. J. Mod. Phys. A 16, 2105 (2001)

View on Pith 2001
[24]

G. ’t Hooft, in Recent Developments in Gauge Theories , NATO Advanced Study Inst., Ser. B, v.59, eds. G. t Hooft et al., Plenum Press, New York (1980)

View on Pith 1980
[25]

H. Gies and J. Jaeckel, Phys. Rev. Lett. 93, 110405 (2004) [hep-ph/0405183]

View on Pith Pith review arXiv 2004
[26]

J. Jaeckel and C. Wetterich, Phys. Rev. D 68 (2003) 025020 [hep-ph/0207094]

View on Pith Pith review arXiv 2003
[27]

J. Jaeckel, hep-ph/0309090

View on Pith Pith review arXiv

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Light fermions in quantum gravity
hep-th 2011-04 conditional novelty 6.0 of 10

Within a Fierz-complete truncation of pointlike four-fermion interactions, metric fluctuations do not drive chiral symmetry breaking, so asymptotically safe gravity remains compatible with light fermions.
Beyond Miransky Scaling
hep-ph 2010-12 accept novelty 6.0 of 10

Miransky scaling near the conformal transition is modified by a power-law factor whose exponent is the gauge coupling's infrared critical exponent.
Conformality or confinement: (IR)relevance of topological excitations
hep-th 2009-06 conditional novelty 6.0 of 10

The sign of the radius dependence of the semiclassically computed gauge mass gap on R3 x S1 is proposed as a marker of the conformal window boundary, yielding estimates such as 2.5N < Nf < 5.5N for fundamental fermions.
Conformality Lost
hep-th 2009-05 accept novelty 6.0 of 10

A conformal phase can be lost when an infrared fixed point annihilates with an ultraviolet fixed point, and the resulting mass gap scales like the BKT correlation length, exp(-c/sqrt(delta)).
The QCD Phase Boundary from Quark-Gluon Dynamics
hep-ph 2008-10 conditional novelty 6.0 of 10

Functional RG flow from the Z mass scale to the chiral regime gives a curvature of the one-flavor QCD phase boundary between 0.4 and 1.3, with the spread set by the choice of running coupling.
Minimal Walking Technicolor: Set Up for Collider Physics
hep-ph 2007-06 conditional novelty 6.0 of 10

The paper assembles a full effective Lagrangian for Minimal Walking Technicolor, including composite scalars, pseudoscalars, vector mesons and technibaryons, and uses modified Weinberg sum rules to relate the spin-one...
Walking in the SU(N)
hep-ph 2006-11 conditional novelty 6.0 of 10

For SU(N) gauge theories with matter in higher-dimensional representations, only the fundamental, adjoint, and two-index representations remain asymptotically free with two or more flavors above nine colors, and the r...
Fierz-complete NJL model study III: Emergence from quark-gluon dynamics
hep-ph 2019-09 conditional novelty 5.0 of 10

Starting from the QCD action with all four-quark couplings set to zero in the ultraviolet, gluon exchange dynamically generates a Fierz-complete set whose dominant channel switches from chiral to diquark as quark chem...
Chiral fermions in asymptotically safe quantum gravity
hep-th 2016-01 conditional novelty 5.0 of 10

Gravity-induced chiral symmetry breaking at the Planck scale is argued to be absent in NJL-type four-fermion models coupled to asymptotically safe gravity, for any number of fermion flavours.
Dynamic AdS/QCD and the Spectrum of Walking Gauge Theories
hep-ph 2013-07 conditional novelty 5.0 of 10

A Dynamic AdS/QCD model with perturbatively input anomalous-dimension running reproduces BKT scaling, an enhanced quark condensate, a light scalar meson, and a vanishing S parameter as the conformal window edge is approached.
Improved Polyakov-loop potential for effective models from functional calculations
hep-ph 2013-02 conditional novelty 5.0 of 10

A temperature rescaling of the Yang-Mills Polyakov-loop potential, extracted from functional QCD calculations, smooths the confinement transition in Polyakov-quark-meson models and improves agreement with lattice QCD.
Conformal Window of Gauge Theories with Four-Fermion Interactions and Ideal Walking
hep-ph 2010-05 conditional novelty 4.0 of 10

Four-fermion interactions shrink the conformal window of SU(N) gauge theories for every matter representation, raising the critical flavor number and pushing the mass anomalous dimension above one at the new boundary.
Conformal Windows of SU(N) Gauge Theories, Higher Dimensional Representations and The Size of The Unparticle World
hep-th 2007-07 conditional novelty 4.0 of 10

Exactly half of asymptotically free supersymmetric SU(N) gauge theories and about a quarter of non-supersymmetric ones develop an infrared fixed point, a fraction the authors use to size the unparticle sector in theory space.
Quantum gravity: a fluctuating point of view
hep-th 2020-07 accept novelty 3.0 of 10

A vertex expansion in graviton fluctuation correlation functions, truncated at the four-point level, reproduces the Reuter ultraviolet fixed point with three attractive directions and shows signs of apparent convergence.
Holography of the Conformal Window
hep-ph 2012-04 conditional novelty 3.0 of 10

Imposing the QCD one-loop anomalous dimension on a D3/D7 brane model yields chiral symmetry breaking at Nf/Nc = 4 via a BKT transition, though the critical value follows from the imposed relation rather than from the ...
Infrared Properties of QCD from Dyson-Schwinger equations
hep-ph 2006-05 accept novelty 3.0 of 10

Landau-gauge Dyson-Schwinger studies indicate an infrared-finite running coupling, a vanishing gluon propagator, and dynamically generated quark mass, though lattice and continuum results still differ at low momentum.
To Break or Not to Break: A Review of a No-Go Theorem on Chiral Symmetry Breaking in QCD-like Theories
hep-ph 2025-09 conditional novelty 2.0 of 10

A review showing that if a QCD-like theory with enough massless flavors is fully color-screened in the infrared, then chiral symmetry must be spontaneously broken.
The QCD phase diagram: Results and challenges
hep-ph 2010-12 conditional novelty 2.0 of 10

Functional renormalisation group calculations reproduce the lattice Yang-Mills deconfinement temperature and place the chiral and deconfinement transitions within roughly 20 MeV of each other in two-flavour QCD.

How the literature uses it

67 arXiv papers cite this in 120 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 citing it · 75 sentences without a confident label

Builds on it 1 paper

Moreover, we improve upon the computation in [ref] by incorporating the gauge and fermion anomalous dimensions derived in this work into the flow of the four-Fermi couplings.

Gauge-Fermion Cartography: from confinement and chiral symmetry breaking to conformality · Convergence of the truncation · 0.97

Uses its data or values 2 papers

For instance, the (non-universal) zero-temperature value of the critical coupling for an optimized RG scheme is $α_cr=g^2_cr/(4π)≃0.8$ for $=3$ and a wide range of $$ [ref] .

Scaling laws near the conformal window of many-flavor QCD · Functional RG results · 0.97
Using the FRG a value of $= 10.0^+1.84_-0.92$ [ref] has been obtained.

On the dependence of the Landau gauge ghost-gluon-vertex on the number of flavors · Introduction · 0.98

Compares with it 6 papers

In particular, our findings are consistent with the results in Refs. [ref] , but different with those in Refs. [ref] . ]

Conformal window from conformal expansion · $SU(N)$ gauge theory with $N_f$ Dirac fermions in various representations · 0.99
This mechanism is similar to the one discovered in 4-dimensional many-flavor QCD [ref] , which gives rise to BKT-type scaling behavior [ref] .

On the Phase Structure of Many-Flavor QED${}_3$ · Conclusions · 0.82
The latter computation displays an $N_f^crit, 4ψ=10$ as shown in <Ref> well in agreement with [ref] and with a

Gauge-Fermion Cartography: from confinement and chiral symmetry breaking to conformality · $γ_m$ in the fRG and perturbation theory · 0.90

Cites it as support 5 papers

Moreover, the results support the merger and annihilation mechanism [ref] , through which the IR fixed point of QED$_3$ merges with the QED$_3$-GNY model and disappears near $N_f^*$.

Searching for gauge theories with the conformal bootstrap · Kinks in the 3D $SO(N)$ vector bootstrap · 0.97
However, provided the flow of the four-quark couplings is governed by the presence of fixed points [ref] , as it is the case in the symmetric regime [ref] , it follows from the analysis of (modified) -identities that the back-reaction of the four-quark couplings on the strong coupling is negligible.

Fierz-complete NJL model study III: Emergence from quark-gluon dynamics · Phase transitions and four-quark interactions · 0.96
This also sustains non-trivially previous functional analyses of the conformal window [ref] where scenario (2) was assumed.

Gauge-Fermion Cartography: from confinement and chiral symmetry breaking to conformality · Pre-condensation at large $N_f$ · 0.97

Cites it as background 26 papers

This region is known as the CBZ window and their properties such as boundaries and magnitude of the fixed point have been widely studied with perturbative [ref] , and non-perturbative methods [ref] as well as lattice simulations [ref] .

On the phase structure of extra-dimensional gauge theories with fermions · UV & IR fixed points · 0.95
Recently several analytic predictions have been provided for the conformal window of nonsupersymmetric gauge theories using different approaches [ref] .

Magnetic Fixed Points and Emergent Supersymmetry · Introduction · 0.85
In particular the critical number of flavors at which chiral symmetry breaking happens has been obtained using many approaches like the gap equation [ref] , [ref] , nonperturbative effective potential [ref] , nonperturbative dynamics of the gauge fields [ref] , lattice calculations [ref] , Nambu Jona Lasinio model [ref] , Bethe Salpeter equation [ref] .

Phase transitions for SU(N) gauge theories with arbitrary number of flavors · Discussion and conclusion · 0.95

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [22]

M. Reuter and C. Wetterich, Phys. Rev. D 56, 7893 (1997) [hep-th/9708051]; H. Gies, Phys. Rev. D 66, 025006 (2002) [hep-th/0202207]

View on Pith Pith review arXiv 1997

[2] [1]

T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982)

View on Pith 1982

[3] [2]

V. A. Miransky and K. Yamawaki, Phys. Rev. D 55, 5051 (1997) [Erratum-ibid. D 56, 3768 (1997)] [hep-th/9611142]

View on Pith Pith review arXiv 1997

[4] [3]

T. Appelquist, J. Terning and L. C. R. Wijewardhana, Phys. Rev. Lett. 77, 1214 (1996) [hep-ph/9602385]; T. Appelquist, A. Ratnaweera, J. Terning and L. C. R. Wijewardhana, Phys. Rev. D 58, 105017 (1998) [hep-ph/9806472]

View on Pith Pith review arXiv 1996

[5] [4]

M. Harada, M. Kurachi and K. Yamawaki, Phys. Rev. D 68, 076001 (2003) [hep-ph/0305018]

View on Pith Pith review arXiv 2003

[6] [5]

Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai and T. Yoshie, Phys. Rev. D 69, 014507 (2004) [hep-lat/0309159]

View on Pith Pith review arXiv 2004

[7] [6]

F. Sannino and J. Schechter, Phys. Rev. D 60, 056004 (1999) [hep-ph/9903359]

View on Pith Pith review arXiv 1999

[8] [7]

T. Appelquist and S. B. Selipsky, Phys. Lett. B 400, 364 (1997) [hep-ph/9702404]; M. Velkovsky and E. V. Shuryak, Phys. Lett. B 437, 398 (1998) [hep-ph/9703345]

View on Pith Pith review arXiv 1997

[9] [8]

M. Harada and K. Yamawaki, Phys. Rev. Lett. 86 (2001) 757 [hep-ph/0010207]

View on Pith Pith review arXiv 2001

[10] [9]

R. S. Chivukula, Phys. Rev. D 55, 5238 (1997) [hep-ph/9612267]

View on Pith Pith review arXiv 1997

[11] [10]

C. Wetterich, Phys. Lett. B 301, 90 (1993); Nucl. Phys. 7 B 352, 529 (1991); Z. Phys. C 48, 693 (1990)

View on Pith 1993

[12] [11]

H. Gies, J. Jaeckel and C. Wetterich, Phys. Rev. D 69 (2004) 105008 [hep-ph/0312034]

View on Pith Pith review arXiv 2004

[13] [12]

U. Ellwanger, M. Hirsch and A. Weber, Z. Phys. C 69, 687 (1996) [hep-th/9506019]; D. F. Litim and J. M. Pawlowski, Phys. Lett. B 435, 181 (1998) [hep-th/9802064]

View on Pith Pith review arXiv 1996

[14] [13]

H. Gies and C. Wetterich, Phys. Rev. D 69, 025001 (2004) [hep-th/0209183]

View on Pith Pith review arXiv 2004

[15] [14]

D. U. Jungnickel and C. Wetterich, Phys. Rev. D 53, 5142 (1996) [hep-ph/9505267]

View on Pith Pith review arXiv 1996

[16] [15]

D. F. Litim, Phys. Lett. B 486 (2000) 92 [hep-th/0005245]; Phys. Rev. D 64 (2001) 105007 [hep-th/0103195]

View on Pith Pith review arXiv 2000

[17] [16]

M. Reuter and C. Wetterich, Nucl. Phys. B 417, 181 (1994)

View on Pith 1994

[18] [17]

U. Ellwanger, Phys. Lett. B 335, 364 (1994) [hep-th/9402077]; F. Freire, D. F. Litim and J. M. Pawlowski, Phys. Lett. B 495, 256 (2000) [hep-th/0009110]

View on Pith Pith review arXiv 1994

[19] [18]

T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390]

View on Pith Pith review arXiv 1997

[20] [19]

M. Czakon, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261]

View on Pith Pith review arXiv 2005

[21] [20]

H. Gies and C. Wetterich, Phys. Rev. D 65 (2002) 065001 [hep-th/0107221]

View on Pith Pith review arXiv 2002

[22] [21]

D. F. Litim, Nucl. Phys. B 631 (2002) 128 [hep-th/0203006]

View on Pith Pith review arXiv 2002

[23] [23]

J. M. Pawlowski, Int. J. Mod. Phys. A 16, 2105 (2001)

View on Pith 2001

[24] [24]

G. ’t Hooft, in Recent Developments in Gauge Theories , NATO Advanced Study Inst., Ser. B, v.59, eds. G. t Hooft et al., Plenum Press, New York (1980)

View on Pith 1980

[25] [25]

H. Gies and J. Jaeckel, Phys. Rev. Lett. 93, 110405 (2004) [hep-ph/0405183]

View on Pith Pith review arXiv 2004

[26] [26]

J. Jaeckel and C. Wetterich, Phys. Rev. D 68 (2003) 025020 [hep-ph/0207094]

View on Pith Pith review arXiv 2003

[27] [27]

J. Jaeckel, hep-ph/0309090

View on Pith Pith review arXiv