The Myriad Uses of Instantons

In quantum chromodynamics (QCD), the role which topologically non-trivial configurations play in splitting the singlet pseudo-Goldstone meson, the $\eta^\prime$, from the octet is familiar. In addition, such configurations contribute to other processes which violate the axial $U(1)_A$ symmetry. While the nature of topological fluctuations in the confined phase is still unsettled, at temperatures well above that for the chiral phase transition, they can be described by a dilute gas of instantons. We show that instantons of arbitrary topological charge $Q$ generate anomalous interactions between $2 N_f |Q|$ quarks, which for $Q = 1$ make the $\eta^\prime$ heavy. For two flavors we compute an anomalous quartic meson coupling and discuss its implications for the phenomenology of the chiral phase transition. A dilute instanton gas suggests that for cold, dense quarks, instantons do not evaporate until very high densities, when the baryon chemical potential is $\gtrsim 2$ GeV.

In quantum chromodynamics (QCD), the up, down and strange quarks are relatively light, and there is an approximate global flavor symmetry of SU (3) L ×SU (3) R ×U (1) A . When the hadronic vacuum spontaneously breaks chiral symmetry, a flavor octet of light pseudo-Goldstone bosons is generated, which are the π, K, and η mesons of broken SU (3) L × SU (3) R . When QCD first emerged, it was a puzzle why there isn't an associated ninth pseudo-Goldstone boson in the flavor singlet channel, the η , from the breaking of the axial U (1) A symmetry.
This occurs because while classically there is an axial U (1) A symmetry, it is not valid quantum mechanically because of an anomaly [1]. There are topologically nontrivial fluctuations which violate the U (1) A symmetry [2] and make the η heavy [3]. Classically these configurations are instantons: these have a topological winding number equal to an integer Q, and an (Euclidean) action equal to 8π 2 |Q|/g 2 , where g is the coupling constant of QCD . Instantons split the singlet η from the octet of pseudo-Goldstone bosons, and also generate the θ parameter of QCD [5].
There are several open questions regarding the nature of topological fluctuations in the QCD vacuum. In absence of a large energy scale to cut-off the size of the instantons, their fluctuations on any length scale become relevant and the integration over their contribution blows up. This is cured non-perturbatively through confinement, where dense topologically non-trivial fluctuations may form an instanton liquid [15][16][17]. Furthermore, it is expected that QCD behaves smoothly as the number of colors, N c , goes to infinity [38,39]. In this limit, the contribution of a single instanton vanishes exponentially, while current algebra can be used to show that the η is still split from the octet of pseudo-Goldstone bosons [38]. This could occur if there are topologically non-trivial fluctuations whose topological charge is not an integer, but an integer times 1/N c ; in certain limits, such as for adjoint QCD on a femto-torus, this can be shown semi-classically [29,30]. * pisarski@bnl.gov † frennecke@bnl.gov However, if the effective coupling is small, e.g. at high temperature or quark density, then a semi-classical analysis is valid, and topologically non-trivial fluctuations can be approximated as a dilute instanton gas [11,12]. Numerical simulations of lattice QCD provide insight into how the topological structure changes with temperature [31][32][33][34][35][36][37]. Remarkably, these demonstrate that the overall power of the topological susceptibility with respect to the temperature T is given by a dilute instanton gas above temperatures as low as a few hundred MeV [33][34][35].
In this Letter we address a modest problem and consider quantities which are nonzero only because of topologically nontrivial configurations, using a dilute instanton gas as an illustrative example. Studies of the phenomenological implications of the axial anomaly, including the effects mentioned above, have been based on effective quark interactions that are generated in a dilute gas of instantons of unit topological charge [4]. Here we generalize this by demonstrating that effective 2N f |Q|-quark interactions are generated in a dilute gas of instantons of arbitrary topological charge Q [7][8][9]. Even though semi-classically such topological field configurations are suppressed exponentially, these interactions can give rise to novel anomalous effects related uniquely to fluctuations of higher topological charge. We explicitly work out the local effective interaction for Q = ±2 for the case where the color orientation of instantons is aligned. At low energies and for two quark flavors this is a quartic meson interaction. We study its qualitative impact on the mass spectrum within a simple mean-field picture. An appendix includes technical details of the computation.
Multi-instanton-induced interactions. We start with an analysis for arbitrary topological charge, generalizing that of 't Hooft [4]. We consider the generating functional of QCD for Gaussian fluctuations around a background of instantons with topological charge Q, which we term Q-instantons. For a Q-(anti-) instanton background, massless quarks have N f |Q| (right-) left-handed zero modes [6]. We show that the functional zero mode determinant of quarks has the structure of a 2N f |Q|-quark correlation function and compute its coupling constant in arXiv:1910.14052v1 [hep-ph] 30 Oct 2019 a dilute gas of Q-instantons.
The zero modes of gauge fields arise from symmetries, such as translations, that yield inequivalent instanton solutions. This defines a moduli space which is parametrized by the collective coordinates of the instantons. The general Q-instanton has been constructed by Atiyah, Drinfeld, Hitchin and Manin (ADHM) [8,9]. It can be viewed as a superposition of Q instantons with unit charge, where each constituent is described by a location z i , a size ρ i and an orientation in the gauge group U i . There are then 4N c collective coordinates for each constituent-instanton, so the moduli space of the Q-instanton has dimension 4N c |Q|. Schematically, the generating functional is where χ = (A µ , c,c, ψ,ψ) contains the fluctuating gluon, ghost and quark fields and χ (Q) = (A (Q) µ , 0, 0, 0, 0) contains the Q-instanton background field A (Q) µ . S[χ] is the gauge-fixed action of QCD in Euclidean spacetime. In the second line we integrate the path integral over the nonzero modes to leading order in the saddle-point approximation, leaving only the integration over the collective coordinates C Q . The instanton density n Q contains the functional determinants of the zero and non-zero modes of gluons and ghosts, the non-zero mode determinant of the quarks and the Jacobian from changing the integration over zero modes to collective coordinates [40].
Our main ingredient is the zero modes of massless quarks [10]. Due to the axial anomaly, the Dirac operator in the presence of the Q-instanton, / D (Q) = γ µ ∂ µ +A  1, . . . , N f is an index for flavor and i = 1, . . . , |Q| is a topological charge index. Because of the zero modes, the generating functional is only nonzero in the presence of a source J, which generates the quark zero mode determinant, det 0 (J), in Eq. (1).
The generating functional in Eq. (1) has first been computed for Q = 1 and N c = 2 [4] and arbitrary N c [41]. For |Q| > 1, the generating functional to one loop order is only known in certain limits [19].
One limit where one can compute is when the distance between the locations of the constituent-instantons are much larger than their sizes; i.e. |R ij | ≡ |z i − z j | ρ i for all i = j. In this case, at leading order, the Q-instanton can be viewed as |Q| instantons of unit charge which are well separated. Expanding the general ADHM-solution in this dilute limit [9], the path integral factorizes into into a product of constituent-instanton contributions, For ease of notation, we assume Q > 0 as anti-instantons with Q < 0 can be treated similarly. The factor of Q! arises because the single instantons can be treated as identical particles. The collective coordinate measure for the i-th constituent-instanton is dC i = dρ i d 4 z i dU i . dU i is the Haar measure of the coset space SU (N c )/I Nc , where the stability group of the instanton I Nc is given by all SU (N c )-transformations that leave the instanton unchanged. We emphasize that in the dilute limit the instanton density only depends upon the sizes ρ i . Deriving the quark zero modes at leading order for the dilute Q-instanton using the methods of [10], one finds that they are simply given by the corresponding zero modes for Q = 1, so that the quark zero mode determinant factorizes and Z (Q) [J] = (Z (1) [J]) Q /Q!. Thus, for a dilute gas of Q-instantons, the effective Lagrangian which results is the Q th power of the 't Hooft determinant, where each determinant is integrated over space-time, What we require, however, is a local interaction, given by a single integral over space-time for the Q th power of Q . To find this, one needs to account for the overlap between the constituent-instantons. To order ρ 4 /(R 2 ) 2 the only change we need to account for is the difference in the quark zero modes [9]. The zero mode for the Q = 1 instanton is where ϕ R is a right-handed spinor so that the zero mode is left-handed. It will be useful later to note that far from the instanton the quark zero mode is proportional to the free quark propagator ∆(x) = γ µ x µ /2π 2 (x 2 ) 2 . For simplicity we consider instantons with charge two, assuming that the constituent-instantons are aligned in color space. Using the zero modes of Ref. [10], the 2N f zero modes for Q = 2 can be expressed in terms of the Q = 1 zero modes as: where So for dilute instantons the Q = 2 zero modes decompose into separate Q = 1 zero modes, connected by the overlap term X i . In general, the determinant depends on the locations of both constituent-instantons, z 1 and z 2 , which can be rewritten as an average position z = (z 1 + z 2 )/2 and their separation R 12 . Integrating over R 12 the zero mode determinant becomes where I N f measures the overlap of the zero modes, For one flavor the overlap integral is infrared-divergent, requiring a cutoff for large distances |x − z i |. Presumably, this cutoff is set by the average separation between an instanton and an anti-instanton. For two or more flavors, a local interaction is generated when all quark zero modes are close to the same constituentinstanton [42]. In this case we find Because zero modes approach free quark propagators at large distances (3), the zero mode determinant (6) has the form of a 2N f Q-quark correlation function. Hence, in direct generalization of [4], the generating functional in the presence of dilute 2-instantons gives rise to an effective interaction between 4N f quarks. Assuming that the topological fluctuations are described by a dilute gas of instantons, the contribution from dilute Q = 2 instantons and antiinstantons generates an anomalous contribution to the local effective Lagrangian in the color-singlet channel [43]: where P R/L = (1±γ 5 )/2 are the right-/left-handed projection operators and K Q, The effective coupling in this semi-classical analysis is, This result generalizes the instanton-induced local interaction to topological charge Q = 2. We note that, while the effective action induced by a single instanton breaks U (1) A down to the cyclic group Z N f , the Q = 2 contribution has a larger residual Z 2N f -symmetry. The computation outlined here can be generalized to arbitrary topological charge and will be discussed in a future publication [44].
A low energy model. To illustrate the physical effect of interactions induced by higher topological charge, we consider a linear sigma model for N f = 2 that includes all anomalous interactions up to quartic order. These are generated in a dilute gas of instantons and anti-instantons with Q = 1 and 2. Classically, the global chiral sym- Effective mesons are given by Φ = (σ + iη) + ( a 0 + i π) τ , with the Pauli-matrices τ . The resulting Lagrangian then is (cf. e.g. [23]) x P a E J n x e l j L W 9 0 R 0 W 2 U F E o g n b 3 r F h y y o 5 Z 9 m + n k j s l 5 G s n K r 7 h G O e I 4 K G F E B I K m n 4 A g Z T f E S p w E B M 7 Q Y d Y Q s 8 3 c Y k u R p n b I k u S I Y g 2 + W / w d J S j i u d M M z X Z H m 8 J u B N m 2 l j g 3 j S K L t n Z r Z J + S v v O f W u w x p 8 3 d I x y V m G b 1 q X i i F H c J q 5 x S c Z / m W H O 7 N X y f 2 b W l c Y F q q Y b n / X F B s n 6 9 L 5 0 1 h l J i D V N x M a G Y T a o 4 Z r z F V 9 A 0 d Z Z Q f b K P Q X b d H x O K 4 y V R k X l i o J 6 C W 3 2 + q z H j L n a m + 5 P p z f m / a V y Z b m 8 t L t S q l X z g R c w h 3 k s c q q r q G E L O 6 z D w z W e 8 I w X 6 9 S 6 s + 6 t h 0 + q 1 Z f n z O L b s h 4 / A P j G l n g = < / l a t e x i t > 2 = 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " B 3 M B m + r a w 3 9 M 9 8 y v 1 r r c r Z X P A F I = " > A A A C 0 3 i c h V F L S 8 N A E J 7 G V 1 t f V Y 9 e g k X w V N I q 2 I t Q 8 I E X o Y J p i 2 0 p S b p N l + Z F s i 3 U 4 k W 8 e v O q / 0 t / i w e / r I m g I t 2 w m d l v Z r 5 5 m Y H D I 6 F p b x l l Y X F p e S W b y 6 + u r W 9 s F r a 2 G 5 E / D i 2 m W 7 7 j h y 3 T i J j D P a Y L L h z W C k J m u K b D m u b o N L Y 3 J y y M u O / d i G n A u q 5 h e 3 z A L U M A u u 1 Y Q 9 6 r q C e q 1 i s U t Z I m j / p X K S d K k Z J T 9 w v v 1 K E + + W T R m F x i 5 J G A 7 p B B E b 4 2 l U m j A F i X Z s B C a F z a G d 1 T H r F j e D F 4 G E B H + N t 4 t R P U w z v m j G S 0 h S w O b o h I l f Z x L y S j C e 8 4 K 4 M e Q X 7 g 3 k n M / j f D T D L H F U 4 h T T D m J O M V c E F D e M y L d B P P t J b 5 k X F X g g Z U l d 1 w 1 B d I J O 7 T + u Y 5 g y U E N p I W l c 6 l p w 0 O U 7 4 n m I A H q a O C e M o p g y o 7 7 k M a U j L J 4 i W M B v h C y H j 6 q E e u u Z p u 9 7 e S r r l R K Z U P S 5 X r o 2 K t m i w 8 S 7 u 0 R w f Y 6 j H V 6 J L q q M N C l m d 6 o V d F V 2 b K g / L 4 5 a p k k p g d + n G U p 0 8 S o 5 H I < / l a t e x i t > 2 = 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " a D 0 O 0 F Q A H i J h l j Q Q 1 B B 9 5 R m E B 1 Y = " > A A A C 1 H i c h V F L S 8 N A E J 7 G V 1 t f V Y 9 e g k X w V J I q 2 I t Q 8 I E X o Y J 9 Q F t K k m 7 T p X m R p I V a P Y l X b 1 7 1 d + l v 8 e C 3 a y K o S D d s Z v a b b 7 6 d 2 T E D h 0 e x p r 1 l l I X F p e W V b C 6 / u r a + s V n Y 2 m 5 E / j i 0 W N 3 y H T 9 s m U b E H O 6 x e s x j h 7 W C k B m u 6 b C m O T o V 8 e a E h R H 3 v Z t 4 G r C u a 9 g e H 3 D L i A G 1 O 9 a Q 9 8 r q i a p r v U J R K 2 l y q X 8 d P X G K l K y a X 3 i n D v X J J 4 with We emphasize that taking into account the contributions from instantons and anti-instantons is necessary to ensure CP-invariance. The term ∼ χ 1 arises from bosonizing the usual 't Hooft determinant from instantons with Q = ±1, while the term ∼ χ 2 is generated by bosonizing interactions with Q = ±2 in Eq. (8) [45]. We focus on the mass spectrum of mesons in the meanfield approximation. We use the σ-, η-, a 0 -meson masses and f π to fix four of the five parameters of L in the vacuum. Chiral symmetry breaking is controlled by the mass parameter m 2 . By varying m 2 relative to its vacuum value, in Eq. (H12) we define a reduced temperature t = t(m 2 ), where t = 0 corresponds to the vacuum and t = 1 to the chiral phase transition. By choosing χ 2 as a free parameter, we can study the impact of the topological charge-two term on the masses in the phases with broken and restored chiral symmetry. The resulting mass spectrum is shown in Fig. 1.
The splitting between the pion and eta mass is due exclusively to the axial anomaly in the chiral limit. Since χ 2 is a quartic coupling, its contribution to the masses is proportional to the chiral condensate. As the condensate melts, this contributions vanishes so that χ 1 is the only anomalous contribution to the masses in the symmetric phase. The larger we choose χ 2 , the smaller χ 1 has to be to reproduce the correct vacuum masses. In the chirally symmetric phase m σ = m π and m η = m a0 , but m σ = m η when χ 1 = 0. Even when χ 1 is small, however, we stress that there are still anomalous effects in the chirally symmetric phase from nonzero χ 2 . These manifest themselves in correlation functions of quartic and higher order.
Needless to say, the effects generated by anomalous coupling from instanton with Q = ±2 depend upon how large it is in vacuum and how rapidly it decreases with temperature T and quark chemical potential µ. In vacuum, the nature of the dominant fluctuations in topological charge is certainly a formidable problem in non-perturbative physics. To estimate these effects, we use a simple gas of dilute instantons. To this end, we adopt a crude bosonization scheme, which yields simple relations between the anomalous meson couplings in (11), and the corresponding quark couplings in the dilute instanton gas: with and κ 2 is given in (9). We introduce a mass scale, M , which is a fundamental parameter of our effective theory. Motivated by the complete computation at one loop order [4], and the partial computation at two loop order [18], for three colors and two massless flavors we take the density of a single instanton in the vacuum to be where g 2 = g 2 (ρΛ M S ) is the running coupling constant at two loop order and d M S is a renormalization-scheme dependent constant. The apparent simplicity of our form for the instanton density belies a major assumption that everywhere the coupling g 2 appears that we can replace it with g 2 (ρΛ M S ). This assumption, while admittedly extreme, is both simple and useful. Owing to the interplay between the running coupling from the classical action in the exponential and the factor ∼ g −12 from the collective coordinate Jacobian, n 1 (ρ) develops a pronounced maximum at ρΛ M S ≈ 1/2. For typical values of Λ M S ≈ 300 MeV [46], this implies typical instanton sizes of ρ ≈ 1/3 fm, which is consistent with the value in an instanton liquid [15][16][17]. Of course we cannot compute reliably at large ρ, since inevitably the instanton size is comparable to the confinement scale, and semiclassical approximations break down.
Since the two anomalous couplings χ 1,2 now only depend on a single free parameter M , we can redo the meanfield analysis of the meson masses, and find a unique value for M in the vacuum. From the dilute instanton gas with Λ M S = 300 MeV we find χ 1 = 0.33 GeV 2 and χ 2 = 0.57 in vacuum. However, χ 2 and all other anomalous effects are very sensitive to the value chosen for Λ M S .
We conclude by discussing how the dilute instanton gas evaporates as T and µ increase. For a single instanton we approximate the change to the instanton density for three colors and two flavors as where m 2 D (T, µ) is the Debye mass at leading order, and A(x) has been determined in [11,12]. Owing to the screening of the color-electric field in the medium, the instanton density decreases both with increasing T and µ. We find that instanton effects are decreased to 10% of their strength in vacuum at about T ≈ 0.7 Λ M S at µ = 0 and µ ≈ 2.4 Λ M S at T = 0. Using realistic values for the critical temperature T c [47] and Λ M S [46], we find that instanton effects are significantly suppressed at temperatures T 1.5T c for µ = 0, consistent with lattice results [31][32][33][34][35][36][37]. As discussed in App. I, at zero temperature in a dilute instanton gas, instantons evaporate only at extremely high densities of µ 1.5 πT c . Using T c = 156 MeV, this corresponds to baryon chemical potentials of µ B 2 GeV.
Summary & outlook. We demonstrated that novel effective interactions are generated by instantons of higher topological charge. In general, instantons of topological charge Q give rise to 2N f |Q|-quark interactions. This opens up the possibility to study the effects of the axial anomaly directly for higher correlation function of quarks or hadrons. Besides the example studied here, it is especially interesting to study QCD with one light flavor, where instantons with Q = ±2 generate a mass for the η meson. These methods can also be used to compute anomalous couplings for heterochiral mesons with J ≥ 1 [28] and tetraquark mesons [27].

Appendix A: Conventions
We use the chiral representation for the Euclidean gamma matrices, i.e. with the Pauli matrices σ i we define and The fifth gamma matrix is then given by Left-(right-) handed fields are defined by having eigenvalue −1 (+1) with respect to γ 5 . Thus, the projection operators on left-and right-handed fields are given by We also define the matrices which are selfdual and antiselfdual respectively, They are related to the 't Hooft symbols η aµν through the SU (2) color generators T a via In terms of the Pauli matrices τ a , the generators are T a = −iτ a /2. For N c > 2 these generator are given by an appropriate embedding of SU (2) into SU (N c ). For instance, one may use the first three Gell-Mann matrices instead of the Pauli matrices for SU (3). Note that we use σ a for the Pauli matrices in spinor space and τ a in color. The 't Hooft symbols are given by They apparently inherit the (anti-) selfduality from the σ's.

Appendix B: Dilute instantons
We start with the gauge field configurations with topological charge Q in the dilute limit. The most general form of the Q-instanton can be obtained by means of the ADHM construction [8,9]. In general, the Q-instanton solution is described by a superposition of 1-instantons, where each of these constituent-instantons is parametrized by a position z i , a size ρ i and an orientation in the gauge group U i . For the special case where all these constituent-instantons are aligned in the gauge group, a Q-instanton solution has first been discovered by 't Hooft [7]. It is based on a superpotential Π(x) satisfying = ∂ µ ∂ µ is the d'Alembertian in Euclidean space. With this, the Q-instanton can be written as withσ µν defined in Eq. (A5). 't Hooft's solution for the superpotential is given by This solution only depends on the location and sizes of the constituent instantons. There is no relative orientation in the gauge group. A global orientation U is indirectly contained in the corresponding gauge field, A where the collective coordinates, z i , ρ i and U i , have a physical interpretation in terms of positions, sizes and gauge group orientations of the Q constituent-instantons that make up the Q-instanton.
Here, we are interested in dilute Q-instantons. This means that the separation between the constituent-instantons is large against their sizes, |z i − z j | ρ i for all i = j. A key feature of this limit is that, at leading-order, the Q-instanton (B4) is zero everywhere except for x close to one of the constituent-instanton locations z i . Hence, in the vicinity of each z i , (B4) looks like a Q = 1 BPST-instanton in singular gauge, To leading order, the dilute Q-instanton is given by a chain of independent 1-instantons. As a result, the generating functional in the saddle-point approximation about this instanton configuration factorizes into a product of generating functionals in a 1-instanton background, as shown in Eq. (2). We refer to [10,19] for a more detailed discussion of this factorization. Following [9], one can derive the general Q-instanton solution to leading order in the dilute/small-instanton limit. This facilitates the generalization of the present discussion to arbitrary orientation in the gauge group and will be discussed in a forthcoming publication [44].

Appendix C: Quark zero modes for the dilute Q-instanton
In the presence of Q-instantons, quarks have zero modes, where is the Dirac operator in the Q-instanton background. It follows from the Atiyah-Singer index theorem that gauge field configurations with topological charge Q give rise to N f |Q| left-handed (for Q > 0) or right-handed (for Q < 0) quark zero modes [6,48]. For 't Hooft's solution for the aligned Q-instanton (B4), they are obtained from the superpotential via [10] ψ where C is a normalization constant, and ϕ R is a right-handed spinor, where α is a spinor index and c is a SU (2) color index. Note that, owing to the gamma-matrix in the zero mode in singular gauge, the zero mode ψ is left-handed for Q > 0, as required by the index theorem. For an anti-instanton, Q < 0, one simply has to replace ϕ R by We first discuss the explicit form for Q = 2: For a dilute 2-instanton the separation of the two constituent-instantons is always far larger than their respective size, Furthermore, in order to be insensitive to the extended nature of the instanton, we consider the zero modes to be far away from the constituent-instanton locations, i.e.
The reason for this limit is that the resulting effective interaction is generated by quarks scattering off the instanton. Due to the extended nature of the instanton, this interaction is in general non-local. In the limit (C8), however, the size of the instanton can be neglected. This allows us to rewrite the zero mode (C6) in a suggestive way: and analogously for the second zero mode ψ (2) f 2 (x). In the first step, we dropped the term ρ 2 1 ρ 2 2 in the denominator in the second line since it is subleading. In the second step we used We define the Q = 1 zero modes as Comparison between the form of the exact Q = 2 quark zero mode ψ (2) f 1 (x) and our approximation in (C9). We used the parameters z1 = 3, z2 = 6 and ρ1 = ρ2 = 0.21 for this plot. Note that we added a small offset to facilitate a logarithmic plot. The baseline of the zero mode is close to zero. The scalar function we plot here is defined in Eq. (C12). and Hence, the Q = 2 quark zero mode looks like the sum of the zero modes corresponding to the two constituent-instantons and the term X i , which quantifies their overlap. We emphasize that ψ f 2 , while being located at z 2 and of size ρ 2 , still has the same gauge group orientation, U 1 , as the first Q = 1 zero mode. This follows from 't Hooft's solution, where the gauge group orientations of the constituent-instantons are aligned. Also note that the leading contribution to ψ (2) We show a comparison between the form of the exact quark zero mode (C6) and our approximation (C9) in Fig. 2. For this figure, we project onto the scalar part of the zero mode ψ for the configuration (x − z 1 ) · (x − z 2 ) = |x − z 1 ||x − z 2 | in order to have a function that only depends on the relative distances. We find that our approximation is very good even close to z 1 and z 2 for ρ/|R| 0.3. The normalization constant C is determined at leading order via which yields C = −1/ √ 2π. To make the following computations more transparent, we use a graphical representation of the zero modes (C9): where the left peak is located at z 1 and the right peak at z 2 .
In the limit of small constituent-instantons, Eq. (C8), we can further simplify the zero modes by taking only the leading terms in |x − z i |/ρ i . Then, (C10) becomes where ∆(x − z) is the free propagator of a massless quark, Hence, the Q = 2 quark zero modes can be represented in terms of free quark propagators and the overlap term, This property facilitates the identification of the quark zero mode determinant with an effective correlation function of quarks, which we will do explicitly below. For arbitrary topological charge, we only mention the leading order in the dilute limit. The corresponding gauge field configuration is given by (B4). With this, the leading contribution to the quark zero modes in the dilute limit for any topological charge is given by: Thus, the N f Q quark zero modes of the dilute Q-instanton are, to leading order, given by a collection of 1-instantoninduced zero modes. This is consistent with our solution for Q = 2 in Eq. (C9), since the term ∼ X i is a sub-leading correction in the dilute limit.
Appendix D: Generating functional in a Q-instanton background: leading order in the dilute limit We first briefly discuss the generating functional of QCD in a Q-instanton background with a focus on the quark zero mode determinant to leading order in the dilute-instanton limit. This will set the stage for the subsequent detailed analysis for Q = 2. The general strategy is to to evaluate the QCD generating functional in the saddle point approximation to leading order, where the stationary point is given by an instanton of topological charge Q. This is, to some extent, natural, since (anti-) self-dual, topological gauge field configurations indeed minimize the classical Yang-Mills action under the assumption that it is finite [2]. Because of this, such an analysis is called semi-classical. The general form of the generating functional is, where χ = (A µ , c,c, ψ,ψ) is the fluctuating multi-field containing gluons, ghosts and quarks. S[χ] is the gauge-fixed action of QCD in Euclidean spacetime.
µ , 0, 0, 0, 0) is the Q-instanton background field. We only introduced a source J for quark-antiquark pairs, since this is the only relevant case for the present purposes. To leading order in the saddle point approximation, one expands the action S χ + χ (Q) about vanishing fluctuating field to quadratic order. The linear terms vanish on the equations of motion. The leading term is given by the action of the Q-instanton, The quadratic terms give rise to the well-known functional determinants. Denoting them as 1 2 A µ M µν A A ν +c M c c + ψ M ψ ψ, the generating functional becomes The exact form of the other terms is irrelevant here. To renormalize the contributions of large eigenvalues, it is understood that all nonzero-mode determinants are normalized with the determinant at vanishing gluon background field. In the presence of the Q-instanton, all fields have zero modes related to the invariance of the action under certain translations, dilatations and global gauge rotations that lead to inequivalent instanton configurations. These symmetries give rise to the 4N c |Q| instanton collective coordinates describing their position (z i ), size (ρ i ) and orientation in the gauge group (U i ). Fluctuations in the directions of zero modes cannot assumed to be small, so strictly speaking the saddle point approximation is only valid for the non-zero modes, while the zero modes have to be treated exactly. To this end, one changes the integration over zero modes to an integration over collective coordinates. We define the Q-instanton density, where C q is the set of all collective coordinates of the Q-instanton. J is the Jacobian from the coordinate change from zero modes to collective coordinates. det / 0 / D is the determinant over the non-zero modes of the quarks. We assume that the quark source J is only a small perturbation and can be neglected in the non-zero mode determinant. With this, the generating functional becomes det 0 (J) is the determinant of the source J in the space of quark zero modes. All this has been discussed in detail in [4], where the generating functional in Eq. (1) has first been computed for Q = 1 and N c = 2. The generalization to SU (N c ) is discussed in [41]. For |Q| > 1, solutions are only known in certain limits, see e.g. [10,19].
For Q-instantons with aligned gauge group orientation at leading order in the dilute limit, the gauge field configuration is given by (B4) and the zero modes are given by (C18). The zero mode determinant then is where we do not sum over the flavor and zero mode indices. The source J is a (N f Q × N f Q)-matrix in the space of zero modes. It is sufficient to only consider the contribution from the diagonal of J. We will match the zero mode determinant to an effective multi-quark interaction, so the different contributions to the determinant can be obtained by permutation of the quark fields (cf. Eq. (G2)). We denote the diagonal elements as J f f ii ≡ J f i and find: where we used that for |x f i − z i | ρ i , this can be expressed in terms of free quark propagators (C16). The quark zero modes have mass-dimension one, so that J has dimension 2. It is convenient to introduce a source with the canonical mass-dimension one, hence we introducedJ f i = ρ i J f i . Since the collective coordinates z i , ρ i and U i are integrated over in the generating functional, we can write this as i.e. to leading order in the dilute limit, the quark zero mode determinant factorizes into independent Q = 1 contributions. We call this determinant non-local, as it depends on all independent instanton locations. As we will show below, if we go beyond leading order, there is also a local contribution, where the determinant only depends on a single location. As discussed e.g. in [9,10,19], the functional determinants of the gluons and ghosts also factorize, even to order ρ 4 /|R| 4 in the dilute limit. Hence, the generating functional factorizes completely, with the Q = 1 collective coordinate integration measure Note that the zero mode determinant depends on all collective coordinates, while the instanton density only depends on the instanton size. We used that the instanton density and the collective coordinate integration measure for arbitrary Q to leading order in the dilute limit also factorize [10,19], i.e. and where Q! is a combinatorial factor related to the permutation-symmetry of the 1-instanton contributions.
Appendix E: Generating functional in a 2-instanton background: N f = 1 Next, we use the form of the quark zero modes in Eq. (C9) to compute the zero mode determinant of the quarks. We start with N f = 1, so we can drop the flavor index. The (diagonal part of the) quark zero mode determinant then is: Using that the zero mode in the dilute case decomposes into a sum of one contribution centered at z 1 and one at z 2 , (C9) and (C14), Eq. (E1) contains various contributions. We drop the integrations over the source locations, but include the integrations over the instanton location here for convenience. One term contains the dominant pieces of each zero mode, z1,z2 1 Since this term has no overlap between the contributions at z 1 and z 2 , it completely decomposes into two separate 1-instanton contributions. This is the leading-order, non-local contribution in the dilute limit. Hence, the corresponding generating functional is given by Eq. (D9). Beyond leading order, there are corrections to this non-local contribution given, e.g., by z1,z2 1 But there are also two local terms in the sense that they can be written as a contributions solely from terms centered around the same point times an overlap-term that can be integrated out. They are given by z1,z2 1 and z1,z2 These terms are indeed given by four Q = 1 zero modes centered around a single z i . The remaining overlap integral , integrates-out the 'leakage' from the contributions of ψ f i (x) around z j to z i (for i = j). This integral can be carried out analytically, For N f = 1, the overlap integral is dominated by large distances |x − z i |. We therefore introduced an infrared cutoff R 0 . Presumably, this is generated by repulsive instanton-anti-instanton interactions [16]. For the generating functional, we use that the gauge contribution factorizes also at next-to-leading order in the dilute limit [9]. Only corrections for the quark zero mode determinant have to be taken into account. The local part of the Q = 2 partition function Z (2) [J] (2) for N f = 1 therefore is We defined the ρ i -independent function, For |x − z i | ρ i this function can be expressed in terms of free quark propagators (C16): Since the expression in (E8) is symmetric under the exchange of the topological charge indices 1 and 2 , we finally arrive at where the Q = 1 instanton density n 1 is given by (14).
Appendix F: Generating functional in a 2-instanton background: N f ≥ 2 The discussion for N f ≥ 2 is a straightforward generalization of the N f = 1 case. Again, it is sufficient to take only the contribution from the diagonal of the (2N f × 2N f )-matrix J into account. The determinant is As for N f = 1. There are numerous non-local contributions. Focussing on the integration over the instanton locations, the leading non-local contribution is given by, i.e. it is a product of two independent terms, each involving 2N f Q = 1 quark zero modes. This is discussed in App. D.
Most of the terms of the determinant give corrections to this non-local term. However, there are again two local terms, which read and The overlap integral for any N f is now given by: In general, this integral depends on the different, arbitrary source locations. However, for a dilute 2-instanton, where we assume that the constituent-instantons are far apart, there naturally arises a contribution that is independent of the source locations. It is precisely given by the limit where the non-local contributions are suppressed. To this end, we note that the zero modes ψ (2) f 2 (x) are responsible for the overlap term in Eq. (F3). This overlap stems from configurations where ψ This limit is consistent with our initial assumption for the dilute 2-instantion in Eqs. (C7) and (C8) as long as |x f 2 − z 1 | ρ 1 , ρ 2 . Furthermore, the non-local terms, which are dominated by configurations where at least one the zero modes ψ The analogous statement is true for the overlap from ψ (2) f 1 (x f 1 ) in Eq. (F4) and the corresponding non-local corrections. Hence, in this case the overlap term only depends on the instanton size and the distance between the instantons, and the quark zero mode determinant is dominated by the local contribution. The overlap integral for N f ≥ 2 then becomes and the Q = 2 partition function for any N f is with F i defined in (E9). We emphasize that since at large distances F i contains two free quark propagators (E10), the generating functional has the form of a 2N f Q-quark correlation function.

Appendix G: The effective interaction
We now discuss the details of the derivation of the effective action from the quark zero mode determinant computed in the previous sections. The main trick is to exploit that far away from the instanton the quark zero mode determinant can be expressed in terms of quark propagators, cf. Eq. (E10). With this it is possible to find a quark correlation function that mimics the zero mode determinant without a topological background field. The location of the effective vertex then coincides with the instanton location.

Any topological charge at leading order
Before we discuss the local interaction for Q = 2, we start with the leading order dilute Q-instanton. We make the following ansatz for the effective generating functional for arbitrary topological charge and flavor: where we note again that, without loss of generality, we assume Q > 0. The index LO indicates that this ansatz is specifically for the leading order in the dilute limit. ω i are constant tensors carrying spin and color which will be determined explicitly below. K Q,N f = (Q!) N f /(N f Q)! is a combinatorial factor. The pre-exponential factor V (Q)+ eff,LO generates a non-local 2N f Q-correlation function with coupling strength κ Q . The superscript + indicates that this is contribution from instantons. We will use − for anti-instantons with Q < 0. We note that, due to Fermi statistics, this term can be rewritten as a determinant, This justifies why we only took the diagonal contribution of the zero mode determinant into account. All other contributions are given by permutations of the quark fields. The correlation function generated by V (Q)+ eff,LO can be computed by expressing the exponential as a power series inJ, and using Wick's theorem to contract the quarks from the sources with the ones in V (Q)+ eff,LO . Note our suggestive notation for the vertex locations in (G1) and the source locations in (G3). The dilute Q-instanton limit corresponds to the assumption that the generating functional is dominated by configurations where the z i in (G1) are widely separated. As a result, all contractions of quark fields are suppressed except for the ones where all quarks sourced at x f i are contracted with all quarks at z j in V (Q)+ eff,LO . All other contraction involve at least one propagator ∆(z i , z j ), with i = j, which is highly suppressed in the dilute limit. Hence, only the term of order N f Q inJ in (G3) can contribute to the correlation function. Then, for fixed f , there are Q! equivalent ways to contract the quarks at x f i to the ones at z j . Since this can be done for each f , there are (Q!) N f equivalent contributions. All other contractions are suppressed since they contain at least one ∆(z i , z j ). Hence, by expanding the exponential in powers of the the sources and using Wick's theorem in the dilute limit, the generating functional Z To compensate for this combinatorial factor, we introduced the factor 1/K Q,N f for the effective coupling in (G1). Taking all of this into account, we find for the 2N f Q-quark correlation function: We can now compare this to the generating functional in the dilute Q-instanton background in (D9), where we use the representation of the quark zero mode determinant in (D7), i.e. we demand that the effective generating functional (G4) and the generating functional in the Q-instanton background (D9) are indentical, From this, we read-off that the effective coupling is (G6) power Q. From the integrands on both sides of Eq. (G5) we infer that the tensor ω obeys the identity, where we made the color indices (a, b, c) and spinor indices (α, β) explicit now. Regarding the color structure of ω i , we see that they are required to carry the global color orientation U i , such that we can define the tensor ω via Furthermore, from the explicit form of the spinor ϕ R (C4) follows for the sum over color indices, where P R is the right-handed projection operator defined in (A4). This implies ω a αω a β = P αβ R .
With this, the integration over the gauge group orientation in the effective action (G1) can be carried out explicitly.
Since we have the same integral for different topological charge indices i, we can do the integration for fixed i following [4]. The final result is then given by taking this result to the power Q. We therefore explicitly evaluate Thus, for the gauge group SU (N c ) we have to carry-out the group integration where for U i is an element of SU (N c )/I Nc , with the stability group of the instanton I Nc , given by all SU (N c )transformations that leave the instanton configuration unchanged. For N c = 2 this is just the identity. dU i is the corresponding Haar measure. Hence, this integration is quite complicated for arbitrary N f and N c . For the present purposes, we restrict ourselves to color-singlet interactions only and use N f = 2 as an example. In general, this group integration will yield color-singlet and non-singlet terms in (G11). For two flavors, the color singlet part is extracted from where (non-singlet) refers to terms that lead to color-non-singlet effective interactions in (G1). c Nc is a N c -dependent constant. Since the gauge group orientation integral is performed on both sides of (G5), this factor cancels out. Of course, if we were interested also in the color-non-singlet channels, there are relative factors that do not cancel. Keeping this in mind, we find Now we can apply the identity for ω in (G10) to arrive at, Plugging this into (G1), we find for the pre-exponential factor with the coupling κ Q,LO given in (G6). We note that even though we explicitly used N f = 2, the color-singlet channel is given by a flavor-determinant for any N f , since the general structure of the gauge group integration (G13) is where σ(i) are permutations of i = 1, . . . , N f , see e.g. [49]. The present result therefore holds for any number of flavors.
This is not a proper effective action, since V (Q)+ eff,LO in not in the exponent. However, so far we considered the generating functional in the background of a single dilute Q-instanton. For a single dilute Q-anti-instanton, one simply has to replace the right-handed projection operator with the left-handed one, P R → P L , in (G16) to get V (Q)− eff,LO . We now assume that the field configurations of topological charge Q are described by a dilute gas of dilute Q-instantons and anti-instantons, i.e. a double-dilute limit for the topological sector of QCD. This generalizes the dilute instanton gas in [4] to arbitrary topological charge. The complete Q-instanton contribution to the functional integral is then given by a simple statistical ensemble, ν + and ν − are the numbers of instantons and anti-instantons. Hence, the resulting anomalous contribution to the effective action is Thus, to leading order in the double-dilute limit, the effective interaction induced by instantons of topological charge Q is a 2N f Q-quark interaction of the form However this interaction is non-local at leading order. To get a local interaction, we need to go beyond leading order. This is done explicitly next for the special case of topological charge Q = 2.
But first, we comment on the dilute gas of dilute instantons. In is conceivable that instantons of any topological charge contribute to the functional integral. Of course, in the semi-classical regime the contributions with higher topological charge are exponentially suppressed due to the factor exp(−8π 2 Q/g 2 ) in the instanton density. This picture is therefore not in conflict with lattice results on the topological charge at large temperature [33][34][35]. Still, these contributions can be present and the resulting anomalous contribution to the effective action of a dilute gas of dilute instantons and anti-instantons of all topological charges is While the the effective interactions are certainly small in the dilute instanton gas, they might have relevant phenomenological implications at lower energies.

The local interaction for Q = 2
We now repeat the same analysis for Q = 2, taking into account the results of App. E and F. As opposed to the leading-order analysis, this results in a local contribution to the effective action. To this end, we make the ansatz Following the arguments above, V (2)+ eff gives rise to the correlation function Again, we choose the coupling κ 2 and the tensors ω such that this correlation function is identical to the generating functional for Q = 2 in (F9), This expression holds for any N f . The overlap integral I 1 is given by (E7) and I N f for N f ≥ 2 by (F8). From this we infer that the effective coupling is given by The determination of ω and the integration over the gauge group are identical to our discussion above. The main difference here is that all propagators connect to the same point z, i.e. the correlation function is local. We therefore find for the color-singlet channel of the effective generating functional, With this, a dilute gas of dilute instantons and anti-instantons of topological charge Q = 2 gives rise to the local contribution to the effective action: with κ 2 given in (G24) and K 2,N f = 2 N f /(2N f )!. Owing to the overlap between the constituent-instantons, this interaction is local.

Appendix H: A low-energy model
Here, we discuss the details of the two-flavor linear sigma model (LSM) defined by the effective action, where the classical and anomalous (quantum) contributions to the effective Lagrangian are defined in (11). However, for the sake of generality, we include one more anomalous quartic term which is generated by 1-instantons, but has been neglected in the main text for reasons that become clear below. With this, the anomalous part of the effective Lagrangian is We setλ 3 = 0 in the main text. The the meson field is given by The equations of motion, where φ = (σ, a 0 , η, π), yield the vacuum expectation valueφ = (σ, 0, 0, 0) with Hence, for m 2 − χ 1 > 0σ is imaginary and the physical VEV isφ = 0. For m 2 − χ 1 < 0 G qu is spontaneously broken down to SU V (2) × Z A 2 . For vanishing anomalous terms, i.e. ∆L qu = 0, also U A (1) would be spontaneously broken, resulting in four Goldstone bosons π and η. In the presence of the anomalous terms U A (1) is broken explicitly and the spontaneous breaking of only SU A (2) results in the pions as the only Goldstone bosons. Due to isospin symmetry, there are only four distinct masses, In the symmetric phase,σ = 0, only the quadratic terms contribute to the masses directly and the Q = 1 term χ 1 induces a splitting of the chiral pairs (σ, π) and (η, a 0 ). The higher order couplings can only influence these masses via loop corrections in the symmetric phase. Inserting the VEV from Eq. (H5) yields for the masses in the broken phase The pion is the only Goldstone boson in the general case. With this we can explore the influence of the anomalous terms in the symmetric and the broken regime. To fix the masses in the vacuum, we use the following observables: f π =σ 0 = 93 MeV , m σ,0 = 400 MeV , m η,0 = 820 MeV , m a0,0 = 980 MeV .
The η mass is taken from [31]. For the other masses, we chose values compatible with [46]. Note that we identify σ with f 0 (500). Within the mean-field approximation and in absence of effects from topological charge Q > 1 andλ 3 = 0, all parameter, including χ 1 , are fixed by the vacuum masses. This then also fixes the amount of axial symmetry breaking above the chiral phase transition, as χ 1 is the only anomalous contribution to the masses in the symmetric phase. With nonvanishing χ 2 the vacuum mass spectrum can be fixed for different values for χ 2 and we can explore the influence of interactions induced by higher topological charge on the mass spectrum.
For a given χ 1 , the value of m 2 determines whether the symmetry is broken or not. So variations in m 2 can be related to variations in the temperature. To analyze how the mass spectrum changes as the symmetry is restored, we therefore assume that m 2 plays the role of temperature in the mean-field analysis. In order to explore the influence of the higher-order anomalous interactions, we fix the masses in the vacuum according to Eq. (H8) and study the mass spectrum as a function of m 2 for different values of the 2-instanton term χ 2 . We then find the following relations between the model parameters and the physical parameters, Since we have five model parameters, but use only four parameters to fix them, we choose χ 2 andλ 3 to be the free parameters for now. We not that in absence of the 2-instanton term, χ 2 = 0, andλ 3 = 0, the 1-instanton is fixed by the η mass, χ 1 = m 2 η,0 /2. Conversely, if U (1) A -breaking is only due to 2-instanton effects and χ 1 =λ 3 = 0, one finds χ 2 = m 2 η,0 /(2f 2 π ). The system has two characteristic scales in m 2 . The vacuum scale m 2 vac is the scale in m 2 where the masses in the broken phase in Eq. (H7) assume their vacuum values (H8). It can be read-off from the first equation in (H9), Furthermore, there is the critical scale m 2 crit of chiral symmetry breaking. It is defined as the value of m 2 where the VEVσ (H5) vanishes, Hence, for different values of χ 2 the characteristic scales of the system change. For a meaningful comparison of the masses for different χ 2 , we therefore define the reduced temperature and rewrite the mass in terms of t, t = 0 is the vacuum and t = 1 is where the phase transition occurs. The VEV of the σ as a function of t then becomes With this parametrization, the masses take very simple forms, We see that m σ (t) and m π (t) are independent of χ 2 . σ is the critical mode that becomes massless at the phase transition. The mass splitting between the chiral pairs (σ, π) and (η, a 0 ) in the symmetric phase is induced by χ 1 (t > 1) = m 2 η,0 /2. This mass splitting vanishes in the limit t → ∞. For χ 2 = 0 the η mass is independent of t in the broken phase. An interesting observation is that for χ 2 > 0, m η is a strictly decreasing function of t in the broken phase and strictly increasing in the symmetric phase. Hence, it has a minimum at the chiral phase transition. This behavior can therefore be attributed to corrections related to topological charge two.
Furthermore, in terms of the reduced temperature, the masses are independent ofλ 3 , so the 2-instanton-induced coupling χ 2 is the only relevant anomalous quartic interaction here. An analysis of the vacuum stability of the effective potential implies thatλ 3 ≤ m 2 σ,0 /4f 2 π ≈ 4.62. The choiceλ 3 = 0 we made in the main text is therefore innocuous. Most importantly, Fig. 1 is exactly the same for any value ofλ 3 . Our estimated values for the couplings χ 1 and χ 2 change, however. We find that the largerλ 3 < 0 is, the smaller χ 1 and χ 2 become, but it has to become very large to have a significant effect.
By bosonizing the multi-quark interactions generated by Q-instantons, the fermionic couplings κ Q can be related to the anomalous mesonic couplings χ Q . For two flavors, κ 1 is a four-quark coupling and can readily be bosonized by means of a Hubbard-Stratonovich transformation. The 2-instanton term κ 2 is an 8-quark interaction for two flavors, so more elaborate path integral bosonization techniques are necessary in general [22]. Here, we adopt a simplistic bosonization scheme motivated by low-energy models where mesons are coupled to quarks through Yukawa interactions, i.e. quark-meson models. On the equations of motion, the mesons are typically proportional to quark-bilinears and we make the simple ansatz based on (H3), Φ = 1 2M 2 (ψψ +ψγ 5 ψ) + (ψ τ ψ +ψγ 5 τ ψ) τ . (H16) M is a fundamental parameter of our effective theory with the dimension of a mass. By using the identity the instanton-induced quark determinant can then be rewritten as, and similarly for the anti-instanton term The 1-instanton induced effective interaction then becomes, and the 2-instanton induced effective interaction becomes, We therefore identify By plugging this into the expressions for the mesons masses above, the dependence on χ 1 and χ 2 is replaced by a dependence only on M , provided that we know the fermionic couplings κ 1 and κ 2 . This reduced the number of independent parameters to four. Given the four input parameters (H8), the effective action (H1) is uniquely determined at the mean-field level.  (14), versus ρΛ M S , for two massless quarks and three different temperatures at µ = 0.

Appendix I: The instanton density
The instanton density in the vacuum is given by (14). It depends on the constant and the running coupling constant g 2 (ρΛ M S ) at two loop order, is the renormalization mass scale of QCD in the modified minimal subtraction scheme. This expression is valid for small x, where log(x −2 ) is positive. By asymptotic freedom, the coupling g 2 (ρΛ M S ) is small at small ρ, so instantons are suppressed by the exponential of the classical action, 8π 2 /g 2 . Of necessity in a semi-classical computation, the exponential from the classical action dominates over the prefactor, ∼ g −12 , which arises from the Jacobian for the collective coordinates of the instanton [4]. Conversely, when ρ increases, so does the coupling g 2 (ρΛ M S ). The instanton density increases, but eventually decreases, suppressed by the prefactor from the Jacobian. The instanton density n 1 (ρΛ M S ) is illustrated in Fig. (3); as seen there, there is a natural maximum when ρ ∼ 0.50Λ M S in the vacuum.
For a single instanton, at a temperature T and quark chemical potential µ, we approximate the change to the instanton density as n 1 (ρ, T, µ) = exp − 2π 2 g 2 ρ 2 m 2 D − 12A(πρT ) 1 + where is the Debye mass at leading order, and A(x) = −1/12 log(1 + x 2 /3) + .0129(1 + 0.159/x 3/2 ) −8 [11,12]. The dominant term, ∼ ρ 2 m 2 D , is straightforward to understand. The topological charge is proportional to tr( E · B), where E and B are the color electric and magnetic fields. In any plasma, electrically charged particles screen static electric fields over distances ∼ 1/m D . Since instantons must carry color electric fields, just Debye screening is sufficient to suppress the instanton density. Needless to say, this argument only applies in a plasma where there is Debye screning, and not at low temperature.
For a single instanton at T = 0 and µ = 0, to one loop order the instanton density can be computed analytically either with puerile brute force [11] or cleverly [12]. The computation at µ = 0 is, unexpectedly, rather more difficult [14,20]. At nonzero µ, then, we only include the leading contribution of quarks to the Debye mass. Numerical computations at T = 0, though, show that for the instanton density, the difference between the complete result and that with just the leading term from the Debye mass is small, at most a few percent for all ρ and T . We comment that the instanton density to one loop order can be computed at µ = 0 numerically using the Gelfand-Yaglom method [6], as has been done for the computation of the one loop determinant in an instanton field for quarks of nonzero mass [26].
Using the elementary ansatzes of Eqs. (14), (15) and (I3), we can calculate numerically how the density changes with temperature and chemical potential. Consider first T = 0 and µ = 0. As illustrated in the left plot of Fig. (4), as the Debye mass increases the instanton density decreases smoothly. To have some definite measure, we define the temperature as that where the integrated instanton density is 1/10 th its value at zero temperature as T I . For three colors and two massless flavors, T 2f l I ≈ 0.71Λ M S ; for three massless flavors, T 3f l I ≈ 0.74Λ M S . Using Λ M S ≈ 332 MeV [46], for two flavors T 2f l I ≈ 236 MeV, and T 3f l I ≈ 246 MeV for three. We stress that these numerical values are, at best, merely suggestive. Under our naive ansatz for a dilute instanton gas, the instanton density is very sensitive to the choice of Λ M S ; after all, merely on dimensional grounds the instanton density is ∼ (Λ M S ) 4 . As discussed above, a dilute instanton gas is only applicable when fractional dyons can be ignored, for T > T χ .
At nonzero temperature, to date the results from lattice QCD find that above temperatures 300 − 400 MeV, the fall off with temperature is a power law, whose value follows from the classical action for a single instanton and the running of the coupling g 2 with temperature. The overall prefactor measured in lattice QCD is approximately ten times larger than the one loop result, but at high temperature perhaps this is ameliorated by the complete computation at two loop order [18]. It is still an open question as to whether topologically non-trivial fluctuations become dilute below [32,35] or above [33] the appropriate transition temperature. This is presumably due to a combination of effects from fractional dyons and instantons with integral topological charge, either as a liquid or a gas. For our purposes, which is frankly phenomenological, the moral which we draw is that a dilute instanton gas is not a preposterous assumption.
Consider next the case of zero temperature and nonzero quark chemical potential. As for temperature, the density of instantons are smoothly suppressed as µ increases. The integrated density of instantons, shown in the right plot of Fig. 4, is 1/10 th that in vacuum when µ 2f l I ≈ 2.44Λ M S for two flavors, and µ 3f l I ≈ 2.22Λ M S for three flavors. These correspond to µ 2f l I ≈ 810 MeV for two flavors, and µ 3f l I ≈ 737 MeV for three. Taking T χ ≈ 156 MeV [47], this is approximately ∼ 1.5 πT χ , While even the instanton density at one loop order is incomplete at µ = 0, we note that these are extremely high values of the quark chemical potential. they are almost into the perturbative regime, for µ > 1 GeV [50].
This gross disparity has a simple origin, and thus may persist a more careful analysis. In a thermal bath, or the Fermi sea of cold, dense quarks, instantons are suppressed primarily because of Debye screening. As can be seen from the expression for the Debye mass in Eq. (I4), the natural scale for the chemical potential is µ ≈ πT . Indeed, as the Euclidean energy of any fermion field is an odd multiple of πT , this balance between µ and πT is true of the propagator at tree level.
The weak dependence upon the quark chemical potential can also be understood in the limit of large N c . As N c → ∞ the coupling g 2 ∼ 1/N c , so that if the number of quark flavors N f is held fixed as N c → ∞, any effects of quarks are suppressed by ∼ 1/N c . In the plane of T and µ, large N c then generates a "quarkyonic" regime [51]. Our naive estimate for a dilute instanton gas is simply another illustration of this.
At present, numerical simulations of lattice QCD with classical computers can only provide can only provide results at nonzero temperature and µ ≤ T . Simulations of cold, dense quark matter may be possible with quantum computers, but will not be available for some time. This illustrates the virtue of using an effective model, such as a dilute gas of instantons.