QCD knows new quarks

We find that a big gap between indicators for the breaking strengths of the global chiral SU(2) and U(1) axial symmetries in QCD of the Standard Model (SM) can be interpreted as a new fine-tuning problem. This may thus imply calling for a class of Beyond the SM, which turns out to favor having a new chiral symmetry and the associated massless new quark insensitive to the chiral SU(2) symmetry for the lightest up and down quarks, so that the fine-tuning is relaxed. Our statistical estimate shows that QCD of the SM is by more than 300 standard deviations off the desired parameter space, which is free from the fine-tuning, and the significance will be greater as the lattice measurements on the QCD hadron observables get more accurate. As one viable candidate, we introduce a dark QCD model with massless new quarks, which can survive current experimental, cosmological, and astrophysical limits, and also leave various phenomenological and cosmological consequences, to be probed in the future. This is a new indication from QCD, which gives a new avenue to deeper understand QCD, and provides a new guideline to consider going beyond the SM.


I. INTRODUCTION
QCD possesses the intrinsic scale of O(1) GeV, at which scale the QCD interaction gets strong enough to form hadrons along with the color confinement.All the physical quantities in hadron physics should therefore arise associated with this order of O(1) GeV.However, e.g., the mass difference between proton (p ∼ uud) and neutron (n ∼ udd) is not of this order: individual masses, being mainly fed by the isospin-symmetric dynamical quark mass, are indeed of O(1) GeV, but the mass difference is of O(10 −3 ) GeV.This is thought of as a fine-tuning.
We now know that the tiny mass difference is due to small violation of the isospin symmetry for up and down quarks, including the current quark masses m u and m d arising from the Higgs via the electroweak symmetry breaking and their electromagnetic charge difference.The mass difference in fact goes to zero in the symmetric limit, and, in this sense, the relaxation of the fine-tuning is guaranteed within QCD, or the SM of particle physics alone, without invoking new physics.
The second example is the extraordinary lightness of isotriplet pions composed as the bound states of up and down quarks (e.g.π 0 ∼ ūu+ dd).The pion masses have been observed to be ∼ 140 MeV, which is by one order of magnitude smaller than the 1 GeV scale.Then a question is: why is it so smaller?This question is also actually trivial and self-resolved within QCD.It is explained by a small size of explicit violation of the chiral symmetry for the up and down quarks.In fact, the established chiral perturbation theory [1,2] shows that due to the chiral symmetry, the pion mass keeps small even against quantum corrections arising from the strong coupling scale of O(1) GeV, due to the chiral symmetry.This pion is thus called technical natural [3] a la 't Hooft [4].
Thus the notion of symmetry can make a big gap in scales filled to relax the fine-tuning.Several types of fine-tunings have so far been pointed out, which cannot be resolved by the SM of particle physics or cosmology alone.All those involve an unsatisfactory big cancellation; e.g. the gauge hierarchy problem [5][6][7][8] and the strong CP problem [9][10][11], where the former belongs to the category of technical unnaturalness, while the latter does not because the QCD-theta term does not get renormalization in the continuum theory.The associated fine-tuned small observables have been confirmed: the size of the Higgs mass much smaller than the Planck scale, and the yet unobserved electromagnetic dipole moment of neutron, respectively.This class of fine-tunings can generically be related to existence of a hidden new symmetry which relaxes the big cancellation, so that the theory becomes free from the fine-tuning in the symmetric limit.Taking them seriously into account has so far motivated people to refine or go beyond the standard theories with such a new hidden symmetry, and opened numerous frontiers in research directions along the theoretical particle and cosmological physics.
In this paper, we pose a new and nontrivial fine-tuning problem in QCD of the SM, and propose a new hidden symmetry which relaxes the fine-tuning.
QCD has been well explored and confirmed, but actually we know less precisely how low-energy QCD and the vacuum depends on quark flavors (described like the Columbia plot), in particular, little understand how the relatively heavy strange quark contributes there.This important open issue is thought of as an analogy to the top quark contribution to the electroweak-symmetry broken vacuum in the Higgs potential of the Standard Model, called the electroweak-vacuum stability problem.What the present work focuses on is such a still nontrivial quark-flavor structure of the QCD vacuum, in particular, the essential gap between the breaking strengths of the chiral SU (2) symmetry for the light up and down quarks and U (1) axial symmetry.
A finely tuned big gap is found in three fundamental quantities of QCD: susceptibility functions for the chiral SU (2) L × SU (2) R symmetry, the U (1) A axial symmetry, and the topological susceptibility, which are essential to characterize the vacuum structure of QCD with the lightest three flavors (up, down, and strange quarks).The chiral symmetry is operative only for up and down quarks, while the latter two are correlated by their axial charges.Those susceptibilities are not direct observables in terrestrial experiments nor astrophysical observatories, in contrast to the existing fine-tuning problems as aforementioned.Those can rather be observed in the lattice QCD, though would not have correlation with definite phenomenological observables.
The three susceptibilities are robustly related to each other by the anomalous Ward identities for the chiral SU (3) L × SU (3) R symmetry, to hold the form symbolically like ⟨Chiral SU(2)⟩ = ⟨U(1) Axial⟩ − ⟨Topological⟩ , where brackets stand for vacuum expectation values.(For the precise expressions written in terms of the susceptibilities, see Eq. (8) in the later section, Sec.II.) ⟨Chiral SU(2)⟩ and ⟨U(1) Axial⟩ are indicators of breaking strengths for the chiral SU (2) L × SU (2) R symmetry and U (1) A axial symmetry that go to zero, when those symmetries are restored, respectively.The new fine-tuning reads presence of a big gap in magnitude between ⟨Chiral SU(2)⟩ and ⟨U(1) Axial⟩, because of nonzero and sizable ⟨Topological⟩ in QCD.There a drastic cancellation between two independent infrared singularities is observed, which is responsible for existence of the soft pions and required to yield the finite quark condensate.This thus causes the fine-tuning and yields a gigantic gap between the chiral SU (2) and U (1) axial breaking strengths.
To demonstrate the new fine-tuning in a quantitative way, one needs to work on QCD in the deep infrared region, which is, however, highly nonperturbative because of the strong coupling nature in the low-energy scale.The best method to compute such nonperturbative dynamics is the numerical simulations of QCD on lattices, which have however never measured the three susceptibilities in Eq.( 8) on the same lattice setting at the same time.
Instead of the lattice simulation, we can invoke effective models of low-energy QCD, on the spirit of Weinberg [12], which realize the same breaking structure of the chiral and axial symmetries, and so forth, as that in the low-energy QCD.In this paper, as the low-energy QCD description, we thus adapt a class of the Nambu-Jona Lasinio (NJL) model made of only quarks with several quarkonic interactions.The NJL model has extensively been utilized in the field of hadron physics, and so far provided us with lots of qualitative interpretations for the low-energy QCD features, associated with the chiral and axial symmetry breaking, together with successful phenomenological predictions [13].
We first show statistical good fitness of the model with the lattice simulation data on hadronic observables for QCD with three flavors at physical point.We then regard the model prediction to the three susceptibilities (⟨Chiral SU(2)⟩, ⟨U(1) Axial⟩, and ⟨Topological⟩) as the prediction from full QCD of the SM, with possible theoretical uncertainty.We find more than 300 standard deviation for QCD in the standard model away from the desired parameter space free from the fine-tuning.
The big gap between (⟨Chiral SU(2)⟩ and ⟨U(1) Axial⟩ vanishes only in the limit where the strange quark mass m s is sent to zero, no matter what nonzero small values of m u and m d are taken, as long as the chiral symmetry acts as a good symmetry.The symmetry, which sends m s to zero, has nothing to do with the chiral and isospin symmetries that make QCD fine-tuning free in a view of the existing hadron spectra, as noted above.More remarkably, the new fine-tuning is present even for the two-flavor QCD with quenched strange quark (m s → ∞), and still survives even when the theory approaches the massless two-flavor limit with m u , m d → 0.
Observing that the strange quark acts as a spectator for the chiral SU (2) symmetry for up and down quarks, we deduce that in place of the strange quark, adding a massless new chiral-singlet quark protected by a new symmetry makes QCD free from the new fine tuning.It is also interesting to note that presence of such massless new quarks can potentially solve the strong CP problem as well, in the same way as the massless up quark solution [9,14].
As one viable candidate to make QCD with any m l and m s free from the fine-tuning, we address a dark QCD model with massless new quarks.The model possesses a new chiral symmetry, which protects the new quarks from being massive at the classical level, to be broken by quantum anomalies via QCD gluon and dark QCD gluon interactions.The dark QCD acts like a mirror of QCD, to keep the massless new quark contribution to the anomalous Ward-identity as in Eq.( 1) below the scale of order of 1 GeV.This model is shown to survive current experimental, astrophysical, and cosmological constraints.Several smoking-guns of the presently introduced benchmark model are also discussed.
Several future prospects are finally commented in the section describing our conclusion.

II. CHIRAL WARD IDENTITIES AND TOPOLOGICAL SUSCEPTIBILITY
We begin by introducing the key equation showing a relation between indicators of the breaking strengths for the chiral SU (2) L × SU (2) R symmetry and U (1) A axial symmetry, together with the topological susceptibility.
We first introduce a set of generic anomalous Ward identities for the three-flavor chiral SU (3) L × SU (3) R symmetry in QCD [15][16][17] (see also Appendix B): where the isospin symmetric limit m u = m d ≡ m l has been taken, χ uu p , χ dd P , χ ud P , χ ss P , χ us P , and χ ds P are the pseudoscalar susceptibilities and χ π is the pion susceptibility, that are defined as with ⟨• • •⟩ conn being the connected part of the correlation function.Second, we introduce the topological susceptibility χ top , which is related to the θ vacuum configuration of QCD.It is defined as the curvature of the θ-dependent vacuum energy V (θ) in QCD at θ = 0: Performing the U (1) A rotation for quark fields together with the flavor-singlet condition [18,19], one can transfer the θ dependence coupled to the topological gluon configurations, via the axial anomaly, into current quark mass terms.
where m . Throughout the present paper, we take the signs of quark condensates and quark masses to be negative and positive, respectively, so that χ top < 0 (hence ⟨Topological⟩ in Eq.( 1) contributes as a negative term).Note that χ top → 0, when either of quarks becomes massless (m l or m s → 0), reflecting the flavor-singlet nature of the QCD vacuum.By combining Ward identities in Eq.( 2), χ top in Eq.( 4) is expressed to be where χ η is the η meson susceptibility, which is defined as Equation ( 6) can be written as where χ δ is the susceptibility for the δ meson channel (which is a 0 meson in terms of the Particle Data Group identification), defined in the same way as χ π in Eq.( 2) with the factors of (iγ 5 ) replaced with identity 1. Renaming (χ η − χ δ ) and (χ π − χ δ ) as which corresponds respectively to ⟨Chiral SU(2)⟩ and ⟨U(1) axial⟩ in Eq.( 1), in the Introduction.
In deriving Eq.( 8) one could choose another scalar susceptibility χ σ , which makes the chiral SU (2) and axial partners for χ π and χ η , respectively.[The definition of χ σ is the same as χ η with the (iγ 5 ) factors replaced by the identity (1).]Then Eq.( 8) would be replaced by χ , where χ ′ chiral ≡ χ σ − χ π and χ ′ axial ≡ χ σ − χ η .Even if this alternative identification is taken, the present proposal of the new fine-tuning problem still holds: a big gap between χ ′ axial and the χ ′ chiral is relaxed by a symmetry sending m s → 0 to make χ top /m 2 l vanishing.

III. NEW FINE-TUNING PROBLEM AND HIDDEN SYMMETRY
Now we discuss a new fine-tuning problem based on the key Eq.(8).
FIG. 1: A schematic cartoon on the chiral and axial transformations for susceptibilities First of all, consider In the case with small enough m l and finite strange quark mass m s (m s ≫ m l → 0), as in QCD of the SM at physical point, the topological susceptibility χ top can approximately be evaluated as #1   χ top where m = (2/m l + 1/m s ) −1 .Noting that ⟨ūu⟩, ⟨ dd⟩, and ⟨ss⟩ keep nonzero even when m l = 0, because of the dynamical generation of quark condensates in QCD at the scale of O(1) GeV, we find that for small m l ≪ m s , the ⟨ūu⟩ and ⟨ dd⟩ terms are dominant in Eq. (10), so that the χ top term in Eq.( 8) is well approximated as with the minus sign of the quark-condensate value taken into account.Thus the size of the χ top term gets larger than [O(1) GeV] 2 (with minus sign).Note also that χ chiral > 0, χ axial > 0, and χ chiral < χ axial due to the measured meson spectroscopy (for more details, see Appendix A).Therefore, in the case with small m l and finite m s , we meet a big destructive cancellation in Eq.( 8) between χ axial and the χ top term, both of which are on the order bigger than [O(1) GeV] 2 , to have a highly suppressed χ chiral : Equation (8 2 for m l ≲ MeV.This can be interpreted as a fine-tuning unless some symmetry is present to explain the extraordinary small χ chiral which can relax the big subtraction, as elaborated in Introduction.Note, however, that even the conventional chiral SU (2) symmetry (m l → 0) makes the accidental big cancellation more serious.As it will turn out later, QCD of the SM with light up and down quarks, and relatively heavier strange quark actually suffers from this kind of big subtraction.
This conclusion is unambiguous and would not be altered even if the Ward identity in Eq.( 8) were subtracted by another scalar susceptibility χ σ (which is defined as in the same manner as χ η in Eq. (7) with the (iγ 5 ) factors replaced by identity, and forms the chiral partner of χ π and the axial partner of χ η .) The existence of the fine-tuning is due to the accidental cancellation between two individual infrared singularities responsible for the soft pions in QCD: as discussed in the literature [17], χ axial ∼ 1/m 2 π ∼ 1/m l , χ chiral ∼ constant, and χ top /m 2 l ∼ 1/m l (see also Eq.( 11)) for m l ≪ m s and m l → 0, hence in this limit Eq.( 8) looks like finite = ∞ − ∞.This observation may imply that the chiral limit, on the base of which QCD can be expanded in a way of the chiral perturbation, hence is widely accepted and well established, is faced with an accidental fine-tuning.Thus the proposed fine-tuning is completely separated from the already existing fine-tuning, e.g., on the tiny mass difference between proton and neutron.
Going away from QCD of the SM, we shall consider a counter limit where m s → 0 with keeping finite m l .In Eq.( 8) the χ top term then goes vanishing as m s → 0 [See also Eq.( 10)] reflecting the flavor-singlet nature [17], so that the indicators for the breaking strengths of the chiral and axial symmetries become identically equal each other: In this case the χ top term (χ top /m 2 l ) is adjusted to zero by a big destructive subtraction, i.e., a fine-tuning between χ chiral and χ axial .However, this fine-tuning can be gone in the limit m s → 0 which makes (χ top /m 2 l ) sent to zero, in contrast to QCD of the SM argued above, though the case with m l ≫ m s → 0, where χ chiral ∼ χ axial , is unrealistic.
#1 Throughout the present Letter, we take the signs of quark condensates and quark masses to be negative and positive, respectively, so that χtop < 0.
Therefore, the presently addressed fine-tuning has nothing essentially to do with the smallness of up and down quarks, i.e., the existing chiral symmetry.Hence it is completely separated from the already existing fine-tuning, e.g., on the tiny mass difference between proton and neutron.
Note that the strange quark currently acts as a spectator for the chiral SU (2) symmetry, being singlet.Hence introduction of a new massless quark, protected by own chiral symmetry i.e., hidden new symmetry, should play the same role as the strange quark to solve the fine-tuning problem, with keeping massive enough strange quark in accordance with the observation.We will later introduce an explicit and phenomenologically viable model having massless new quarks (χ) with a new chiral symmetry, which makes real-life QCD free from the fine-tuning: with m l and m s at physical point.
In the next section we will explicitly demonstrate the new fine-tuning of QCD in a more quantitative way.

IV. QUANTIFYING THE FINE-TUNING
We define a ratio [17] which also reads via the Ward identity in Eq.( 8).Thus the deviation from R = 1 dictates a fine-tuning, hence R serves as the estimator of the fine-tuning.
To compute the estimator R, one needs to work on QCD in the deep infrared region, which is highly nonperturbative because of the strong coupling nature in the low-energy scale.The best method to compute such nonperturbative dynamics is the numerical simulations of QCD on the lattice.However, the lattice simulations have never measured the susceptibilities at vacuum with varying m s #2 .Instead of the lattice simulation, in the spirit of Weinberg [12] we can invoke effective models of low-energy QCD, which realize the same breaking structure of the chiral and axial symmetries, and so forth, as that in the low-energy QCD.In this paper, as the low-energy QCD description, we thus adapt a class of the Nambu-Jona Lasinio (NJL) model made of only quarks with several quarkonic interactions.The NJL model has extensively been utilized in the field of hadron physics, and so far provided us with lots of qualitative interpretations for the low-energy QCD features, associated with the chiral and axial symmetry breaking, together with successful phenomenological predictions [13].

V. BEST-FIT MODEL ESTIMATE
We employ an NJL model with three flavors, which takes the form (for a review, see [13]): where the quark field q is represented as the triplet of SU (3) group in the flavor space, q = (u, d, s) T , and λ a (a = 0, 1, • • • , 8) are the Gell-Mann matrices with λ 0 = 2/3 • 1 3×3 .The determinant in L KMT acts on the flavor indices, and M = diag{m l , m l , m s }.L 4f is the standard-scalar four-fermion interaction term with the coupling strength G S .This is the most minimal interaction term involving the smallest number of quark fields for Lorentz scalar and pseudoscalar channels, which #2 This is mainly because firstly it has not been well motivated, and moreover, costs of lattice calculations for small mass are proportional to 1/m, where m is the lightest quark mass.Simulations for light strange quarks can be performed using the same technology as in [20], and employing similar calculations in [21,22]    could be generated at low-energy QCD via the gluon exchange.The a=0 (λ a /2)θ a ] and the chiral phases θ a .The mass term in L explicitly breaks U (3) L × U (3) R symmetry.The determinant term L KMT is called the Kobayashi-Maskawa-'t Hooft [23][24][25][26] term, which is a six-point interaction induced from the QCD instanton configuration coupled to quarks, with the effective coupling constant G D .This interaction gives rise to the mixing between different flavors and also uplift the η ′ mass to be no longer a Nambu-Goldstone boson.The KMT term preserves SU (3) L ×SU (3) R invariance (associated with the chiral phases labeled as a = 1, • • • , 8), but breaks the U (1) A (corresponding to a = 0) symmetry.
The approximate chiral SU (3) L × SU (3) R symmetry is spontaneously broken down to the vectorial symmetry SU (3) V , when the couplings G s and/or G D get strong enough, by nonperturbatively developing nonzero quark condensates ⟨qq⟩ ̸ = 0, to be consistent with the underlying QCD feature.The present NJL model monitors the spontaneous breakdown by the large N c expansion, where N c stands for the number of QCD colors.
The NJL model itself is a (perturbatively) nonrenormalizable field theory because L 4f and L KMT describe the higher dimensional interactions with mass dimension greater than four.Therefore, a momentum cutoff Λ needs to be introduced to make the NJL model regularized.
There are five model parameters that need to be fixed: the light quark mass m l , the strange quark mass m s , the coupling constants G S and G D , and the (three-) momentum cutoff Λ.Since the present NJL model does not incorporate the isospin breaking as well as radiative electromagnetic and weak interactions, it would not be suitable to input experimental values of QCD observables that implicitly include all those corrections.We thus use as inputs observables in lattice QCD with 2 + 1 flavors in the isospin symmetric limit at the physical point available from the literature [27,28], which are exclusive for the gauge interactions external to QCD.We apply the least-χ 2 test to fix the parameters by using five representative observables as in Table I.The resultant values of the best-fit model parameters are given in Table II.The least χ 2 test shows good agreement with the lattice data within the 1σ uncertainties.The best-fit NJL model predicts the susceptibilities relevant to R in Eq.( 14) of the main text as χ chiral = (2.2784± 0.0026) × 10 5 , χ axial = (4.8597± 0.0077) × 10 6 , and With the best fit parameters in Table II, the present NJL model predicts χ top = (0.025 ± 0.002) /fm 4 .For this χ top , comparison with the results from the lattice QCD simulations with 2 + 1 flavors is available, which are χ top = 0.019(9)/fm 4 [29], and χ top = 0.0245 (24) stat (03) flow (12) cont /fm 4 [30].Here, for the latter the first error is statistical, the second one comes from the systematic error, and the third one arises due to changing the upper limit of the lattice spacing range in the fit.Although their central values do not agree each other, we may conservatively say that the difference between them is interpreted as a systematic error from the individual lattice QCD calculation.Thus the present NJL model is, in that sense, in good agreement with the lattice QCD results on χ top .This supports reliability for the present model to estimate R as the QCD prediction, as well as the good fitness of the model with hadronic observable data on the lattice QCD.
We thus compute the estimator R at the best-fit point including the errors associated with the lattice data, and for the present NJL model to match full QCD of the SM has also been reflected there (see also the text), which is drawn by light-blue arrows.The size of deviations is maximally about 340σ, and will be at least over 66σ even when the theoretical uncertainty of 30% is considered. find This clarifies that QCD at physical point is by about 340 standard deviation off the desired theory with R = 1 free from the fine-tuning!This is due to too large m s , as noted above.
We may take into account a possible theoretical uncertainty of about 30%, which could arise from the leading order approximation in the 1/N c expansion, on that the present NJL-model prediction is based.Currently disregarded corrections, associated with the isospin breaking, electromagnetic, and electroweak interactions, would also be small enough to be covered by the 30% uncertainty.Therefore, the estimated value of R in Eq.( 16) with the theoretical uncertainty of 30% would be the one corresponding to the prediction of the SM.Combining this 30% ("theor.")with the error in Eq.( 16) associated with the uncertainties of inputs from the lattice data ("lat."),we would then have R = 0.0469 ± (0.0028) lat.± (0.0141) theor. .It is still about 66 standard deviations.
To make this disfavor visualized, varying the value of m s and m l , with other model parameters fixed at the best-fit values, we plot contours of the estimator R on the (m l , m s ) plane, which is displayed in Figure 2.
The value of R tends to saturate to be ≃ 0.02, even in the massive two-flavor limit with m s → ∞ and m l = 5.75 MeV.This trend is exactly what we have suspected from the m s scaling of the χ top term in Eq.( 11).With m s fixed, say, to the physical point, R tends to get close to 1 as m l becomes larger, and actually reaches 1 before the decoupling limit of the up and down quarks, as clarified in the literature [17].
The deviation in Eq.( 16) can be interpreted as an indication of violation of a new symmetry as conjectured in Eq.( 13), which the SM does not possess.The significance of this new symmetry is subject to the accuracy in the current lattice simulation reflected in the size of the error of R, which is as low as 10%.This significance may therefore be compared to the significance for the discovery of the small isospin breaking in the W and Z boson masses observed at UA1 and UA2 experiments [31][32][33][34], which was the same 10% level in accuracy at the final stage of the discovery era (with data taking til 1985) [35,36].The estimator R can be defined also for the W and Z masses as , which actually forms a big cancellation structure, hence could be thought of as a fine-tuning, in the same way as in χ axial and χ top /m 2 l with R in Eq.( 14).The final UA2 result reads [36] R 1982−1985 WZ = 0.876 ± 0.026, so it is about 4.8σ deviation from the isospin symmetric limit R WZ = 1.This is, however, of course trivial and can be explained by the isospin breaking (related to the so-called custodial symmetry) in the SM, due to the hypercharge gauge interaction and the presence of isospin breaking in the quark masses.Compared to this, the new indication from QCD in Eq.( 16) is by about one order of magnitude more significant.
Current precision measurements on m W and m Z gives R 2022 W Z = 0.88147 ± 0.00013 [37], which corresponds to 890σ deviations from the isospin symmetric limit.Similarly the error of R in Eq.( 16) is also expected to become smaller as the precision in the lattice simulations gets higher in the future, hence the significance of the violation of a new symmetry will be enlarged to be as big as the current one for the isospin breaking.
This prospected significance might also become comparable with the current significance of the isospin breaking in the proton and neutron mass difference with R pn ≡ , which is read as R 2022 pn = 0.998623477(316) [37], leading to ∼ 4 × 10 4 σ.
The gauge hierarchy problem caused by the big destructive loop correction to the Higgs mass , where M p denotes the Planck scale ∼ 10 18 GeV, m h (m h ) ∼ 125 GeV, and we have simply taken into account only the top loop with the top Yukawa coupling ∼ 1.This R H is estimated to be ∼ 6.3 × 10 −32 .A similar estimate can be done also for the strong CP problem, which would yield R θ = θQCD θEW < 10 −10 , where θ QCD and θ EW respectively denote the QCD and electroweak origins for the CP phase of the quark mass term.Thus the statistical significance of R H will keep biggest unless the accuracy in measuring the Higgs mass and top Yukawa coupling gets better than the level of ∼ 10 −32 , the size of R H .
The fine-tuning problem that we presently address is nothing sensory, but is essentially related to existence of a hidden symmetry which makes R = 1 or χ top /m 2 l = 0 or χ chiral = χ axial in terms of our estimator.This is in contrast to the conventional argument: Something delicately fine or not is controlled merely whether the tuning to some extent is necessary to make a big cancellation.The quantification of the present fine-tuning is unambiguously made on the basis of the statistical significance, and the standard deviations are subject to the accuracy in the measurement of Rs, which should therefore be compared with those having the same level of accuracy, as done above.
Note that the strong CP problem is trivially solved in the limit m l → 0, whereas the new fine-tuning problem gets more serious (R → 0).This discrepancy in the two problems can be understood via the Ward identity Eq.( 8), where χ top itself can be sent to zero when m l → 0, which is the massless up quark solution to the strong CP problem, however, χ top /m 2 l then blows up, leaving the new fine-tuning problem.Thus, the two problems are generically separated.This fact also proves that the chiral SU (2) symmetry (with m l → 0) does not make QCD free from the fine-tuning, in sharp contrast to the naive folklore.
We could start with the definition of R, R = χ chiral /χ axial , instead of the Ward identity Eq.( 8), and discuss the difference ∆ axial−chiral = χ axial −χ chiral , so that R = 1−∆ axial−chiral /χ axial .Then the form of Eq.( 8) is unambiguously fixed as it stands, and tells us that the symmetric limit R = 1 is realized when χ top /m 2 l → 0, which cannot be made when m l → 0 (because χ top ∼ m l for small m l ), but can be achieved when m s → 0, which is based on the symmetry argument.This alternative view would also help readers to more definitely see that the m l = 0 limit separates the new fine-tuning problem from the strong CP problem, where R → 0 and R θ = θ QCD /θ EW → 1, respectively.

VI. A CANDIDATE SOLUTION: NEW QUARKS WITH DARK REPLICA OF QCD COLORS
The proposed new fine-tuning problem is present at the scale only around the order of 1 GeV.When the electroweak symmetry becomes manifest at higher scales, the fine-tuning problem will be obscure because it explicitly breaks the global chiral SU (2) and U (1) axial symmetries as well as quark masses (or Yukawa couplings between the Higgs and quarks), so that the key Eq.( 8) will be modified involving the electroweak "topological" susceptibilities.Also at scales ≲ m π , the fine-tuning will be nontransparent due to the decoupling of pions, which is the most dominant source to generate the big gap between χ chiral and χ axial .Thus the new fine-tuning problem needs to be solved by a new physics with the scale Λ NEW in a range of m π ≲ Λ NEW ≲ 1 GeV.
The hint for this avenue is seen in Eq.( 12), which indicates introducing massless new quarks.In fact, the topological susceptibility χ top goes to zero, when a massless new quark couples to the other three quarks, due to the flavor-singlet nature, so that Eq.( 13) is realized.The detailed proof is given in Appendix B. This motivates one to consider an explicit model beyond the SM.
We consider a new chiral quark (χ) to be neutral under the electroweak charges, instead, carries a dark color of SU (N d ) group under the fundamental representation.The group representation table for the χ quark thus goes like Y , where the latter three symmetries correspond to the SM's ones (QCD color, weak, and hypercharge).The dark color symmetry as well as the electroweak neutrality forbids creating undesired light hadrons composed of the ordinary light quarks and the χ quark, such as ūχ and uuχ.
For simplicity, we also assume that the dark QCD coupling g d gets strong almost at the same scale as the ordinary QCD coupling does, (Λ d ∼ Λ QCD = O(1) GeV), namely, g d ∼ g s .
Below the scale ∼ 1 GeV, the dark QCD dynamically breaks the dark chiral U (3) L × U (3) R symmetry for the χ quark, down to the vectorial part, where the extra factor of 3 in the number of flavors comes from the QCD color multiplicity.Only the hadrons singlet under both the dark and ordinary QCD colors survive in the vacuum.Then there emerges only one composite Nambu-Goldstone boson, η d , which becomes pseudo due to the axial anomaly in the dark QCD sector and acquires the mass of O(1) GeV.Besides, at almost the same scale, ordinary QCD breaks the approximate chiral SU (3 + N d ) L × SU (3 + N d ) R symmetry involving the χ quark down to the vectorial one, where again only the color singlets are relevant.The spontaneous breaking of this extended chiral symmetry does not yield excessive meson spectra made of the ordinary quarks, because of the double color symmetries, as aforementioned.Thus the new low-lying spectra consist only of the dark sector: η d ∼ χiγ 5 χ and its dark chiral partner σ d ∼ χχ, as well as spin-1 dark mesons ( χγ µ χ and χγ µ γ 5 χ), and dark baryons (∼ χχχ • • • χ).All those low-lying dark hadrons have the mass on the order of 1 GeV, by feeding the chiral breaking contributions from both the ordinary QCD and dark QCD sectors.
Relaxing the new fine-tuning is tied with vanishing curvature of the QCD vacuum at around the QCD scale: χ top → 0. The new fine-tuning problem is present irrespective to the place of the QCD vacuum, i.e., the value of the (net) QCD θ parameter: even a shifted QCD vacuum with θ gone, say by assumption of a QCD axion, keeps nonzero χ top as the developed axion potential energy including the axion mass, which takes precisely the same flavor-singlet form as in Eq. (10).Thus the new fine-tuning problem is definitely separated from the strong CP problem.
Actually, the dark QCD solution instead implies a nontrivial relation between θ and θ d , the theta parameter in dark QCD, to realize χ top = 0 in the presence of the massless new χ quark: the anomalous axial rotation of the massless χ leaves the θ-dependence into the dark QCD topological sector, ( The required relation might be trivial when QCD itself relaxes θ to 0 at an deep infrared fixed point to be consistent with realization of the confinement, as recently discussed in lattice QCD [38][39][40].This self-relaxation is applicable also to dark QCD, hence in that case one has θ = θ d = 0, and χ top = 0 in the presence of the massless χ quark. In contrast, solving the new fine-tuning problem disfavors QCD axion models as the solution of the strong CP: first of all, the QCD axion needs not to be present until the QCD pions are decoupled from R, otherwise the axion potential energy necessarily yields nonzero χ top even along with massless new quarks.In this sense a composite axion [41,42] with the dynamical/composite scale scaled down to the QCD scale (or lower) might be the candidate, where the composite scale is set to ∼ m π ∼ 4πf a with the axion decay constant f a .However, such a low-scale QCD axion model with both a QCD axion and its small decay constant on the QCD scale has already been ruled out by the LEP search for Z → πγ [43] due to too large axion coupling to diphoton.Thus the QCD axion is incompatible with the solution for the new fine-tuning problem.
In the next couple of subsections, we will discuss several characteristic features of the presently introduced dark QCD model below and above the scale ∼ 1 GeV, in the aspects of phenomenology as well as astrophysical and cosmological observations.
The η d dark meson can mix with the ordinary QCD η ′ by sharing the axial anomaly via QCD.The mixing can be seen through the non-conservation law of the dark axial current J µ η d = χγ µ γ 5 χ: with N c = 3 and (G µν Gµν ) and (G µν Gµν ) being the topological operators of the ordinary QCD and dark QCD, respectively.The size of mixing with η ′ , which couples to the (G µν Gµν ) term, can be evaluated by constructing the two-point , and focusing on the cross-term amplitude ⟨N c (G µν Gµν )•N d (G µν Gµν )⟩.Estimate of the order of magnitude for the mixing amplitude can be made by working on the dual large-N c and -N d expansion with α s ∼ 1/N c and α d ∼ 1/N d .Since the mixing amplitude arises necessarily due to the χ quark loop, it is estimated to be on the order of O For Feynman diagram interpretation, see Fig. 3.This is compared with the non-mixing term amplitudes ⟨N c (G µν Gµν ) ), dominated by the dark gluon loop, and where in the last second equality we have used α s ∼ α d and Λ QCD ∼ Λ d in magnitude, and in the last one a typical size of α s at around 1 GeV [44], α s (1 GeV) ≃ 0.3, has been as a reference value, which is precisely the same order of magnitude as expected from the large N c scaling (i.e., α s ∼ 1/N c ∼ 30%).The mixing angle is of O(10 −2 ) in unit of degree, hence can safely be neglected compared to the mixing between η ′ and η in the ordinary QCD, which is estimated to be about 28 • from the recent lattice simulation at physical point for 2 +1 flavors [28], consistently with the prediction from the chiral perturbation theory, ∼ 20 • [2] and also the current Particle Data Group.Thus the successful η ′ physics in ordinary QCD is not substantially altered.
B. Probing dark mesons via couplings to photon η d ∼ χiγ 5 χ and σ d ∼ χχ can couple to the ordinary quarks through the χ loop with two gluon exchanges.Therefore, they can also couple to diphoton through the ordinary quark loops with the loop-induced η d -digluon and σ d -digluon vertices.See Fig. 4. The large-N c and -N d counting can evaluate the order of magnitude for couplings to diphoton at the nontrivial leading order.The induced photon couplings can be summarized by the following effective Lagrangian: where α em is the fine structure constant for the electromagnetic coupling, which is ≃ 1/137 at the scale of order of 1 GeV, and Q q em denotes the electromagnetic charge for the ordinary quark q in the standard model.The dark-meson decay constant f d can be related to the intrinsic scale Λ d as f d ∼ √ N d /(4π) • Λ d with the large N d scaling taken into account.Then the size of the coupling to diphoton can be estimated as η d and σ d with this size of photon coupling at the mass around 1 GeV are too short-lived (with the lifetime τ ∼ 10 −12 s) to have sensitivity for astrophysical observations, such as cosmic ray telescopes, but can be probed by collider experiments in the same manner as the axionlike particle (a) searches.The currently available observation limit around the target mass and photon coupling comes from the Belle II experiment at the SuperKEKB collider on e − e + → 3γ event with 496 pb −1 data.This experiment has placed the upper bound on the photon coupling, g aγγ ≲ 10 −3 GeV −1 [45].The Belle II future prospects with 20 fb −1 and 50 ab −1 data will reach g aγγ ∼ 10 −4 GeV −1 at the mass around 1 GeV [46], which can probe η d and σ d via the 3γ signal.More precise estimate on the g η d (σ d )γγ coupling is necessary to give more definite prediction to the 3γ event, which will be pursed elsewhere.

C. More on dark meson -ordinary meson mixing
Through loop processes generating the η d -and σ d -photon couplings with external photon legs replaced by the ordinary QCD mesons, η d and σ d can also mix with the ordinary mesons in the isosinglet channel, such as η and σ = f 0 (500).See Fig, 5.This goes like quadratic-mass mixing, the order of which can be evaluated by the large -N c and -N d expansion, to be GeV and m σ ∼ m η ∼ 500 MeV, the size of the mixing angle is estimated as This mixing is, again, safely negligible in comparison with the η − η ′ mixing (by the angle ∼ 20 • − 30 • ), and can be insensitive to highly uncertain and large mixing structure of the isosinglet mesons in QCD.A similar argument is applicable also to other mesons with higher spins, like vector and axialvector mesons.Thus the successful hadron physics is intact.

D. Cosmological abundance of dark baryon
The dark baryons are formed as color-singlet for both ordinary QCD and dark QCD colors.The wavefunction of the ground-state spin-1/2 dark baryon takes the form, e.g., for The dark baryon is completely stable due to the exact dark baryon number conservation in dark QCD, hence can be a dark matter.Since the χ quark can be thermalized with the ordinary QCD gluons in the thermal plasma of the standard model, the dark baryon n d as well as the dark mesons (σ d , η d , • • • ) could be thermally produced as well.As in the case of the ordinary baryons, n d could annihilate into the lightest meson pairs, i.e., via n d nd → η d η d , or more multiple η d states (and also into σ d states), which would determine the freeze out of the number density of n d .
Rough, but, conservative estimate of the thermal relic abundance of n d could be done by assuming the standard freeze-out scenario for this annihilation with the size of the classical cross section ⟨σv⟩ ∼ 4π/m 2 n d .This crude approximation could be justified in the large-N c and −N d limit, where the dark baryon behaves like almost static, nonrelativistic, and a classical rigid body with finite radius (i.e.impact parameter) of O(1/m n d ).The thermal relic abundance is evaluated as [47] , where x FO = m n d /T FO with T FO being the freeze-out temperature; M pl is the Planck scale ∼ 10 18 GeV; g * (T FO ) the effective degree of freedom of relativistic particle at T = T FO .The standard freeze-out scenario gives x FO ∼ 20, and g * (T FO ) = O(50) at T below 1 GeV [37].The relic abundance of n d is then estimated to be of O(10 −9 ) for m n d = O(N d = 3 − 5) GeV, which is compared with the observed abundance of the cold dark matter (CDM) today ∼ 0.1.Thus the dark baryon n d cannot fully account for the presently observed dark matter abundance.The model needs to be improved to be completed, so as to include another dark matter, which yields the main component of the CDM abundance today.

E. Direct detection of dark baryon
Regarding the direct detection experiments by recoils of heavy nuclei with dark matters, the dark matter n d with mass of O(N d ) GeV could contribute to the spin-independent scattering cross section.n d strongly couples to η d and σ d , which can convert into the ordinary η and σ mesons through the mixing with angle θ σ−σ d of O(10 −3 ) in unit of radian, as discussed above.Since the pseudoscalar-mediator contribution vanishes at the leading order at zero momentum transfer relevant to such a nonrelativistic scattering process, the dominant contribution would come from the σ d -σ meson-mixture portal process.Analogously to the Higgs portal scenario, the spin-independent dark matter -nucleon (N ) cross section σ SI (n d N → n d N ) can then be evaluated as where Λ d ; g σN N stands for the σ meson coupling to nucleon, which can be evaluated via so-called the nucleon σ term σ πN = ⟨N |m l ql q l |N ⟩ as g σN N = σ πN fπ .We assume that the n d -dark matter has the same velocity distribution in the dark matter halo as in the case of CDM.Using σ πN ≃ 0.05m N [48][49][50], f π ≃ 0.0924 GeV, m N ≃ 0.940 GeV, and s pole σ ∼ (0.5 − 0.3i) GeV [37], we and Ω n d h 2 = 10 −9 .This signal can be explored by the planning detection experiment searching for sub-GeV dark matters, called ALETHEIA [51].
Incorporation of the η d -portal contribution at loop-level might be crucial at this order of the cross section, and might enhance the cross section as discussed in the literature [52].Furthermore, the gluonic-nucleon matrix element ⟨N | αs π G 2 µν |N ⟩ together with the operator nd n d G 2 µν could also contribute to the cross section, which might be comparable with the σ d − σ portal contribution in Eq. (21).More detailed analysis is to be performed elsewhere.

F. Collider detection prospect of dark hadrons
Exotic hadrons can be created as hybrid bound sates of the ordinary QCD quarks and the dark χ quarks.Those are forced to form like a molecular type, such as qi q j χχ and qi q j χχ • • • χ, because of the QCD and dark QCD color symmetries.The lightest ones would be four-quark bound states made of the ordinary QCD pions (π) and η d or σ d .
In a view of collider phenomenology, the four-quark states (πη d ) and (πσ d ) finally decay to 4γ or µ(e) + 2γ + missing energy with a few GeV scale.Those exotic hadrons could be produced at hadron colliders through rescattering of π and η d or σ d , or n d , in which the dark QCD hadrons could be produced via the gluon fusion process, while the ordinary QCD pion via the initial-gluonic bremsstrahlung almost colinear to the produced dark hadrons.The signal events as above would be hard to detect over the huge backgrounds with hadronic jets with energy on the GeV scale, hence is so challenging at the current status of hadron collider experiments.A photon-photon collider with high accuracy in photon production events with the GeV scale might be more practical to explore those signals.

G. Constraints from QCD running coupling
Extra massless quarks contribute to the running evolution of the ordinary QCD coupling g s , where in the present case N d species of new quarks in the fundamental representation of SU (3) c group come into play.To keep the asymptotic freedom, at least N d has to be < (33/2) − 6 ∼ 10, which is determined by the one-loop perturbative calculation.Collider experiments have confirmed the asymptotic freedom with high accuracy in a wide range of higher energy scales, in particular above 10 GeV, over 1 TeV [37].When α s is evolved up to higher scales using α s (M Z ) measured at the Z boson pole as input, the tail of the asymptotic freedom around O(1) TeV can thus have sensitivity to exclude new quarks.
Current data on α s at the scale around O(1) TeV involve large theoretical uncertainties.This results in uncertainty of determination of α s (M Z ) for various experiments (LHC-ATLAS, -CMS, and Tevatron-CDF, and D0, etc.), which yields α s (M Z ) ≃ 0.110 − 0.130, being consistent with the world average α s (M Z ) ≃ 0.118 within the uncertainties [37].
We work on the two-loop perturbative computation of α s .The dark QCD running coupling (α d ) contributes to the running of α s at two-loop level.This contribution is, however, safely negligible: when α s ∼ α d at low-energy scale as desired, because α d ≪ α s at high energy due to the smaller number of dark QCD quarks (with the net number 3 coming in the beta function of α d ) than that of the ordinary QCD quarks (with the net number 5 or 6 + N d in the beta function of α s ).
Taking into account only the additional N d quark loop contributions to the running of α s , we thus compute the two-loop beta function, and find that as long as N d is moderately large (N d ≤ 5), the measured ultraviolet scaling (for the renormalization scale of µ = 10 GeV − a few TeV) can be consistent with the current data [37] within the range of α s (M z ) above.
Precise measurements in lower scales ≲ 10 GeV have not well been explored so far, due to the deep infrared complexity of QCD.The low-energy running of α s is indeed still uncertain, and can be variant as discussed in a recent review, e.g., [44].The present dark QCD could dramatically alter the infrared running feature of α s , due to new quarks and the running of the dark QCD coupling α d .This will also supply a decisive answer to a possibility of the infrared-near conformality of the real-life QCD, to which point we will come back in the conclusion section.
Other stringent bounds on the extra light quarks or colored scalars come from the ALEPH search for gluino and squark pairs tagged with the multijets at Large Electron Positron (LEP) collider experiment [53].However, this limit has no sensitivity below the mass ∼ 2 GeV, hence is not applicable to the present benchmark model.
Thus a few massless new quarks can still survive constraints on α s at the current status.More precise measurements of α s in the future will clarify how many light or massless new quarks can be hidden in QCD, which will fix the value of N d in terms of the present dark QCD.

VII. OTHER THEORETICAL REMARKS ON FINE-TUNING FREE QCD
Finally, we argue one nontrivial dynamical issue related to successful realization of the fine-tuning free condition with massless new quarks.

A. A nontrivial dynamical issue
It is true that χ top = 0 and χ η = χ π when massless new quarks are assumed to be present, but one might naively think that it implies m π ∼ m η because the size of susceptibility follows the associated meson mass, as noted in Sec.III.This would obviously be contradicted with the observation.However, it should not be the case.
First of all, the susceptibilities correspond to meson-correlation functions at zero momentum transfer, at off-shell of mesons, so do not exactly equal to propagators of mesons.Pions are light enough, and supposed to have the mass close to such soft momenta, and χ π gets affected only from the up and down quark loops [see Eq.( 2) with Eqs.(A7) and (A8) in Appendix A].Therefore, χ π would almost scale with 1/m 2 π .In contrast, the η meson mass is thought to be far off soft, and indeed χ η gets contributions not only from u and d quark loops, but also the strange quark's [see Eq. (7) with Eq.(A6) in Appendix A], where the latter contribution is crucial to yield χ η < χ π and does not simply follow the inverse mass scaling ∼ 1/m 2 η .Now consider the case with the new χ quark, where the loop corrections to χ η involve the χ quark term, as well as the other three quarks terms.[χ η will be constructed from the pseudoscalar susceptibilities labelled as χ 00 P , χ 08 P , χ 015 P , χ 88 P , χ 815 P , and χ 1515 P associated with the generators of U (4).]The χ loop corrections would thus be a key to realize χ η = χ π , while keeping m η > m π , which would make the model parameter space of the dark QCD limited (e.g. the separation of QCD and dark QCD gauge couplings in size would be constrained).
More precise discussion is subject to explicit nonperturbative computation of susceptibilities as well as meson spectra with the χ quark in QCD which also couples to dark QCD.This could be possible by lattice simulations, or also by working on NJL-like models, or applying the chiral perturbation theory.This is, however, beyond the current scope, to be left as the important dynamical issue, in the future.

VIII. CONCLUSION
Toward the deeper understanding of the flavor-dependence of the QCD vacuum, in the present work we have focused on a gap between the breaking strengths of the chiral SU (2) symmetry for the light up and down quarks and U (1) axial symmetry.It might be too premature to conclude that this gap is related merely to the mystery of the quarkgeneration structure in the SM.We have found an alternative interpretation based on the symmetry argument: the gap can be relaxed by a chiral symmetry for the strange quark, and the role of the massless/light strange quark can be replaced by a new massless/light quark (called χ).
The existence of the symmetric limit m s → 0, where the "chiral SU (2) = U (1) axial" is realized as above, is manifest in the flavor dependence of the QCD vacuum, and we have shown how the gap, i.e., the fine-tuning is seriously large, on the basis of the statistical analysis along with comparison with the existing fine-tuned quantities.
Thus, QCD may be yet incomplete and QCD of the SM calls for more quarks to keep the equivalence of the strengths of the chiral SU (2) and U (1) axial breaking, to be free from the fine-tuning.New quarks need protected to be (nearly) massless by a new chiral symmetry, and can be introduced in QCD consistently with existing experiments, as demonstrated in the present work.This symmetry is independent of the existing chiral or isospin symmetry, which ensures the smallness of masses of light quarks and had so far played the role to make QCD free from the fine-tuning, e.g., for the small proton -neutron mass difference compared to individual proton and neutron masses.
The new fine-tuning problem is triggered due to the large strange quark mass, and brings a big gap between the chiral and axial order parameters, detected as a small size of the estimator R in Eq.( 14).This trend actually persists even in a whole temperature, as can be understood by tracing the analysis in the recent literature [17].This small R can be checked on the lattice QCD simulations in the future.
In this paper, a benchmark of the fine-tuned QCD was placed based on the well-known low-energy effective model, NJL.This is a pioneer step, and should be motivated and confirmed by various approaches in the future, such as the lattice calculation and functional-renormalization group analysis.One can also work on the chiral perturbation theory to evaluate R. No work has so far been done properly taking into account the flavor singlet condition for χ top , hence it would also be an interesting issue to be left in the future.
Besides the phenomenological and cosmological consequences discussed in Sec.VI, the notion of the fine-tuning free QCD potentially provides rich perspective toward deeper understanding of the real-life QCD today and past in the thermal history of the universe.Several of those are listed below.
(i) In the thermal history of the universe, massless new quarks should have contributed to the thermal QCD phase transition.As was clarified by the nonperturbative renormalization group analysis [54] and the lattice simulation [55], a large number of light quarks generically decrease the critical temperature of the chiral symmetry restoration, T c ; e.g., T c gets smaller by a factor of about 2/3 and 1/3 [55], when the number of quarks is increased from 3 to 6 (corresponding to N d = 3 in the dark QCD scenario) and 8 (N d = 5), respectively.The lattice QCD simulations with 2 +1 flavors at physical point have reported the pseudo-critical temperature of the chiral crossover, T pc ≃ 155 MeV [56][57][58][59][60].When this T pc is simply scaled by the above flavor dependence, we have T pc ≃ 103 MeV for N d = 3, and T pc ≃ 52 MeV for N d = 5.In particular, this reduction of T pc would give a significant impact on evaluation of the cosmological abundance of dark matters related via the scattering with, or the annihilation into, quarks, when the freeze-out temperature is in a range from O(T pc /10) to O(10T pc ).
Even in this refinement, the dark baryon dark matter would still yield a negligibly small thermal abundance.When a popular Higgs-portal dark matter, which is sensitive to the Higgs invisible decay as well, is for instance additionally considered, this refinement of T pc would suggest substantial re-analysis on a viable parameter space.
(ii) When the massless new quarks have a new color like "N d " of dark QCD in the benchmark model introduced in the present paper, the scenario of the chiral phase transition will be complicated and rich, where the net chiral phase transitions will be categorized into two : the dark-chiral phase transition related to the 3 massless quarks will be first order, which will take place at around the conventional T c , meanwhile the chiral phase transition (crossover) in QCD associated with the three light 3 + N d quarks would follow at around T = T c /3 − 2T c /3, for N d = 5 − 3. The latter crossover might undergo a "jump" at around T c due to the interference with dark QCD colored quarks which undergoes the first order phase transition.This would generate a sharp drop at around T = T c characteristic to the first-order phase transition, to deform the conventional chiral crossover to be like a first-order type.Exploration of the feasibility of such an induced "jump" would be worth pursing by lattice simulations, or the nonperturbative renormalization group analysis, or chiral effective models such as the NJL model.The nonperturbative analysis on an NJL model including QCD as well as dark QCD interactions, which is sort of an extended variant of so-called the gauged NJL model [61][62][63][64][65][66][67][68], would also be intriguing to pursue.
(iii) If the chiral crossover experiences a "jump" in the QCD phase transition epoch of the thermal history of the universe as argued in (ii), nonzero latent heat might also be promptly created during the first-order like phase transition.The associated bubbles might then be nucleated to be expanded and collided each other over the Hubble evolution, and develop gravitational waves, to reach us today, to be detected by gravitational wave interferometers [69][70][71].Thus, the evidence of the chiral phase transition in QCD, namely the origin of nucleon mass in the thermal history, might directly be probed by gravitational wave interferometers in the future.This would be an innovative probe of the first-order QCD phase transition without invoking a flavor-dependent phase transition at the vicinity of the critical endpoint expected to be present with nonzero baryon chemical potential.The dedicated study on the gravitational wave signal would be necessary, to be left in a separated publication.
(iv) If the number of massless (light) new quarks is five, as in the case of one of the presently addressed dark-QCD benchmark models, the real-life QCD of SU (3) group would possess eight flavors at the low-energy.This implies that actually, the real-life QCD might be what is called walking dynamics, having the infrared-near conformality characterized by the Caswell-Banks-Zaks infrared fixed point [72,73].Therefore, it would be worth investigating the fine-tuning free QCD in depth, even in the context of such an infrared conformality of QCD.Actually, the infrared feature would be more intricate, because the five new quarks make the dark QCD communicated with the ordinary QCD in the renormalization group equations.The Caswell-Bank-Zaks infrared fixed point would thus be generalized in the two-coupling space (α s , α d ), which is particularly noteworthy to explore.
(vi) The walking QCD indicated in (iv) and (v), as the fine-tuning free QCD, would also be relevant to a longstanding question on whether the σ meson in QCD could be a composite pseudo Nambu-Goldstone boson, so-called the pseudo dilaton, associated with the spontaneous-scale symmetry breaking in QCD.The lattice simulations on eight flavor QCD with fundamental representation fermions have proven the evidence of the light flavor-singlet qq meson, identified as the composite dilaton, with mass comparable with the mass of the pseudo Nambu-Goldstone boson (like pion in the conventional QCD) associated with the spontaneous breaking of the eight-flavor chiral symmetry [93].To properly match the currently favored walking QCD setup, it would be necessary to work on QCD with 2 + 1 + 5 flavors, where the latter five light fermions are charged also under dark QCD of SU (5) group, and measure the lightest flavor-singlet qq meson signal, which is identified as the σ meson in the real-life QCD, and then confirm the σ meson mass consistent with the experimentally observed value.This study might lead to a heuristic solution to the aforementioned question on the QCD dilaton, and would simultaneously resolve complexity of the scalar meson puzzle with the striking answer of no significant mixing with four-quark states, nor glueballs.
In closing, other than the dark QCD model addressed in the present paper, it would be worth investigating modeling beyond the SM with taking into account making QCD fine-tuning free.
Here is the recipe: first of all, one needs to introduce new Dirac massless quarks, which act as a spectator of the global chiral SU (2) symmetry for the up and down quarks.They are generically allowed to feel the electroweak charge, whichever way it is ganged vectorlikely or chirally.The former case would be phenomenologically viable in light of the electroweak precision tests, and the limit on the number of quark generations placed from the Z boson decays.Second, those new quarks would be preferable not to form the Yukawa interaction with ordinary quarks and Higgs fields (doublets and triplets, and so on) which develop the vacuum expectation values at the weak scale, and yield the mass for the new quarks.If new quarks could be coupled to such Higgses, solving to the strong CP problem (without introducing an axion) as well as keeping the light enough new quarks down until the QCD scale would be hard and challenging.
Given this recipe, one might think that though it would sound somewhat ad hoc, the presumably most minimal setup would be to introduce a electroweak-singlet quark with a negative charge under a new parity, while assign a positive charge for ordinary quarks, so as to avoid spoiling the successful light hadron spectroscopy.In the dark QCD model introduced in the present paper, the role of such a new parity has been played by the dark QCD color charge.Such alternatives are to be addressed in details elsewhere.with m−1 = 2 m l + 1 ms + 1 mχ .Using the relations in Eq.(B3) together with those in Eq.( 2), we find In the case of dark QCD modeling as in the text, actually, the dark QCD coupling to the χ quark explicitly breaks the chiral SU (4) L × SU (4) R symmetry, as well as the mass terms.However, this breaking effect does not modify the anomalous identities associated with the chiral transformations for a = 1, 2, 3, 8, 15, because the dark QCD coupling to the χ quark only breaks the vectorial SU ( where Q d top denotes the dark QCD topological charge.Thereby, one gets χ top = (N c /N d ) 2 χ d top ̸ = 0 with χ d top being the topological susceptibility in dark QCD, unless θ d = −(N c /N d ) θ.This nontrivial relation is required no matter what size θ is, i.e., which is independent of the strong CP problem.

FIG. 4 : 5 σFIG. 5 :
FIG.4:A typical Feynman graph describing generation of σ d -γ -γ and η d -γ -γ vertices.Graphical notations are the same as those in Fig.5 where the symbols ϵ's and f denote the group structure constants for SU (3) c and SU (N d = 5) groups, respectively.The dark baryon mass m n d scales as ∼ N d • N c in the large-N c and -N d expansion, so that it can be larger than the masses of the lightest dark mesons η d and σ d , to be m n d ∼ (N c N d /3)m n normalized to the ordinary QCD baryon mass m n ∼ 1 GeV.

m l m χ χ uχ P + χ dχ P = m s m χ χ sχ P = 1 4 m
l ⟨ūu⟩ + ⟨ dd⟩ + m 2 l χ uu P + χ dd P + 2χ ud P = m s ⟨ss⟩ + m 2 s χ ss P = m χ ⟨ χχ⟩ + m 2 χ χ χχ P .(B5) 4) flavor symmetry down to SU (3) × U (1), which still keeps those chiral symmetries:T a , 0 3×3 1 ∼ [T a , T b=15 ] = 0.The argument in this Appendix can straightforwardly be extended to the case with more extra quarks, like 3 + N d quarks.The presence of the dark QCD theta parameter as well as the QCD's one modifies χ top in Eq.(B5) asχ top → χ top + (N c m/m χ )χ dtop with the dark QCD topological susceptibility χ d top , N c = 3, and m χ in the original χ top as well as m replaced by m χ /N d .

TABLE I :
[27,28]ult on the least χ 2 statistical test of the present NJL model derived by fitting to the lattice QCD data with 2 + 1 flavors in the isospin symmetric limit at the physical point[27,28].The details of the global fit of the NJL model including other available lattice data deserve in another publication.

TABLE II :
The best-fit values of the model parameters.