PHYSICAL REVIEW RESEARCH 2, 033359 (2020)

Higher topological charge and the QCD vacuum

Fabian Rennecke * Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA

(Received 14 April 2020; revised 2 July 2020; accepted 18 August 2020; published 3 September 2020)

It is shown that gauge field configurations with higher topological charge modify the structure of the QCD vacuum, which is reflected in its dependence on the CP-violating topological phase θ. To explore this, topological susceptibilities and the production of axion dark matter are studied here. The formers characterize the topological charge distribution and are therefore sensitive probes of the topological structure of QCD. The latter depends on the effective potential of axions, which is determined by the θ dependence of QCD. The production of cold dark matter through the vacuum realignment mechanism of axions can, therefore, be affected by higher topological charge effects. This is discussed qualitatively in the deconfined phase at high temperatures, where a description based on a dilute gas of instantons with arbitrary topological charge is valid. As a result, topological susceptibilities exhibit a characteristic temperature dependence due to anharmonic modifications of the θ dependence. Furthermore, multi-instanton effects give rise to a topological mechanism to increase the amount of axion dark matter.

DOI: 10.1103/PhysRevResearch.2.033359

I. INTRODUCTION strong CP violation. The question about the existence and nature of a physical mechanism to enforce θ = 0 remains to The vacuum structure of interacting quantum field theories be answered. is inherently intriguing. In quantum chromodynamics (QCD) One resolution of this strong-CP problem has been sug- and other gauge theories, it is well known that this structure gested by Peccei and Quinn (PQ) [14]. It involves the crucially depends on topological gauge field configurations introduction of a new complex scalar field, which has a global [1–5]. A vacuum state can be characterized by an integer, the chiral U(1) symmetry. The PQ symmetry is broken both topological charge, and topological gauge fields describe tun- PQ explicitly, through the axial anomaly, and spontaneously. The neling processes between topologically distinct realizations of resulting pseudo-Goldstone boson, the axion a(x)[15,16], the vacuum. The true vacuum is therefore a superposition of couples to the gauge fields in exactly the same way as the θ all possible topological vacuum configurations. It is character- parameter, ∼(a/ f + θ )trFF˜, where F and F˜ are the gluon ized by a free parameter, the CP-violating topological phase θ, a field strength and its dual and f is the axion decay constant. acting as a source for topological charge correlations. a This effectively promotes θ to a dynamical field, which has a The topological nature of the QCD vacuum has impor- physical value given by the minimum of its effective potential. tant phenomenological consequences. The axial anomaly in Owing to the anomaly, the effective potential of axions is QCD is realized through topologically nontrivial fluctuations. determined by topological gauge field configurations, and its They generate anomalous quarks correlations which explic- minimum is at zero, thus resolving the strong CP problem itly break U(1) and impact properties of hadrons [6]. Most A dynamically. Furthermore, the anomalous axion mass is very prominently, the large mass of the η meson is generated by small and, in order to be consistent with observational bounds, topological effects [1–3,7,8]. The fate of the axial anomaly at axions couple very weakly to ordinary matter. This makes finite temperature also determines the order of the QCD phase axions attractive dark matter candidates [17–19]. Cold ax- transition in the limit of massless up and down quarks [9,10]. ions can be produced nonthermally through a field relaxation Furthermore, a nonzero θ introduces CP-violating strong in- process known as vacuum realignment [20–23]. The axion teractions. These would manifest in a nonvanishing electric relaxes from a, possibly large, initial value to its vacuum dipole moment of the neutron, d . Recent measurements give n expectation value at zero, thereby probing the global structure a stringent bound of |d | < 1.8 × 10−26e cm [11], resulting n of the θ vacuum. The production of cold dark matter through in θ  10−10 [12,13]. This strongly suggests that there is no the vacuum realignment mechanism of axions therefore is particularly sensitive to the topological structure of the QCD vacuum. *[email protected] While the nature of topological field configurations in the confined phase is still unsettled, at large temperatures Published by the American Physical Society under the terms of the where the gauge coupling is small, a semiclassical analysis is Creative Commons Attribution 4.0 International license. Further valid [24,25] and these field configurations can be described distribution of this work must maintain attribution to the author(s) by instantons [26], see Refs. [27–29] for reviews. Finite and the published article’s title, journal citation, and DOI. temperature is crucial for a self-consistent treatment, since

2643-1564/2020/2(3)/033359(18) 033359-1 Published by the American Physical Society FABIAN RENNECKE PHYSICAL REVIEW RESEARCH 2, 033359 (2020) it effectively cuts off large-scale instantons. This is clearly 1,...,|Q| [42]. At leading order in this limit of small con- shown by numerous first-principles studies of QCD on the lat- stituent instantons (SCI), the partition function of QCD in the tice, both with dynamical quarks [30–35], and in the quenched background of a multi-instanton can be computed solely based limit [36–39]. It has been demonstrated that the temperature on the knowledge of the single-instanton solution. At next- dependence of the lowest topological susceptibility χ2, which to-leading order, correlations between constituent instantons measures the variance of the topological charge distribution, need to be taken into account [6,45,46]. The resulting genuine is in very good agreement with the corresponding prediction multi-instanton processes are studied in a qualitative manner from a dilute gas of instantons for T  2.5Tc, where Tc is the in the present work. Due to the large suppression of the instan- pseudocritical temperature of the chiral phase transition. One ton density in the presence of dynamical quarks, it is expected can therefore assume that the topological structure of the QCD that, at least in the large temperature regime, multi-instanton vacuum is described by instantons at large temperatures. This effects are most pronounced in quenched QCD, i.e., without is of relevance also for axion physics, since, for example, the dynamical quarks. Hence, quenched QCD provides a good axion mass is proportional to the topological susceptibility, laboratory to understand higher topological charge effects on χ = 2 2 2 fa ma. the QCD vacuum. Conventionally, only single-instantons, i.e., instantons with This work is organized as follows. The θ dependence in unit topological charge, are taken into account in the de- a dilute gas of multi-instantons is derived in Sec. II.This scription of the topological structure of the QCD vacuum requires knowledge of the multi-instanton contribution to the at large temperatures. The reason is that classically the ac- partition function, which is derived in detail in the SCI limit tion of an instanton with topological charge Q in a dilute in Sec. II A. The result is used to study topological suscepti- gas is ∼e−8π 2|Q|/g2 . Thus, in the limit of vanishing gauge bilities in Sec. III. Their temperature dependence is evaluated coupling g, multi-instantons with |Q| > 1 are subject to a numerically for quenched QCD in Sec. III A and discussed large exponential suppression. Furthermore, the moduli space for QCD in Sec. III B. The focus there is on systematically of multi-instantons coincides with that of |Q| independent investigating correction to the topological susceptibilities as single-instantons, which are each characterized by a position instantons with increasing topological charge are included. zi,asizeρi and an orientation in the gauge group Ui, with The impact of multi-instantons on axion cosmology is studied i = 1,...,|Q| [40]. Yet, multi-instantons are distinct self-dual in Sec. IV. First, the resulting axion effective potential and the gauge field configurations which, in general, cannot be inter- production of axions via the vacuum realignment mechanism preted as simple superpositions of single-instantons [41,42]. are discussed in Secs. IV A and IV B. In Sec. IV C, axion Since quantum corrections lead to an increasingly strong production is studied numerically in the quenched approxima- coupling towards lower energies, multi-instantons could give tion. Finally, a critical discussion of the approximations used relevant corrections to the leading single-instanton contribu- here can be found in Sec. V, and a summary of the results in tion. Furthermore, it has been shown in [6] that there are Sec. VI. Further technical details are given in the appendices. effects that are related uniquely to higher topological charge: higher order anomalous quark interactions are generated only by multi-instantons, which generalizes the classic analysis II. θ DEPENDENCE FROM A DILUTE GAS for single-instantons [1,2]. Thus one has to assume that the OF MULTI-INSTANTONS topological field configurations at large temperatures are in- The starting point is a modification of the old story of stantons of arbitrary topological charge. the θ dependence of the QCD vacuum. In the context of the Given this motivation, the effects of multi-instantons on axial anomaly and the U(1)A problem in QCD, it has been the vacuum structure of QCD are explored in this work. It realized that the QCD vacuum has more complicated structure is assumed that at sufficiently high temperature in the decon- which is related to the existence of topological gauge field fined phase, the topological structure of QCD is described configurations [3–5]. Physical states can therefore be grouped by a dilute gas of instantons of arbitrary topological charge. into homotopy classes of field configurations with a given This leads to a modification of the known θ dependence of the topological charge (or winding number) |n, QCD vacuum. It is reflected in the distribution of topological charge, which is probed by topological susceptibilities. In 1 4 addition, as outlined above, axion cosmology is an interesting n =− d x tr Fμν F˜μν ∈ Z, (1) application to showcase the effect of higher topological charge 16π 2 on the QCD vacuum. To this end, the production of cold axion dark matter via the vacuum realignment mechanism is where Fμν = [Dμ, Dν ] is the field strength tensor with Dμ = 1 μνρσ ∂μ + μ ˜μν = ρσ investigated here as well. A the covariant derivative. F 2 F is the A semiclassical description of multi-instantons requires corresponding dual field strength. Instantons of topological knowledge of the partition function of QCD in their pres- charge Q can be interpreted as tunneling from state |n to ence. At the next-to-leading order in the saddle point state |n + Q, with the tunneling amplitude related to the approximation, the complete partition function is known for exponential of the classical instanton action, e−8π 2|Q|/g2 .Due single-instantons at finite temperature [1,2,24,25]. For multi- to such tunneling processes, |n cannot describe the vacuum instantons, it is only known in certain limits, see Refs. [43,44] state in a unique way. Furthermore, states characterized by for reviews. The exact multi-instanton solutions [41] can be different winding numbers are related to each other by large expanded systematically if the size parameters ρi are small gauge transformations. The true vacuum state, consistent with against the separation |Rij|=|zi − z j| for i = j and i, j = gauge symmetry, locality and cluster decomposition is the θ

033359-2 HIGHER TOPOLOGICAL CHARGE AND THE QCD VACUUM PHYSICAL REVIEW RESEARCH 2, 033359 (2020) vacuum, +∞ |θ= e−inθ |n. (2) n=−∞ An important property of this state is that the value of θ cannot be changed by a gauge invariant operation. Hence, QCD falls into superselection sectors, where each θ labels a different - + theory. ν = −101 2 3 The vacuum amplitude between in and out states at t = −∞ and +∞, denoted by |θ− and +θ|, respectively, is then given by imθ −inθ +θ|θ− = e e +m|n− n m (3) iνθ = e +n + ν|n−, ν n - + with ν = m − n. The part after the exponential in the second ν = −101 2 3 line of this equation describes the amplitude where in- and out-state differ by a net-topological charge of ν.UsingEq.(1), the θ-dependent generating functional Z[θ] ≡ +θ|θ− can be FIG. 1. Illustration of possible instanton configurations con- represented by the Euclidean path integral tributing to the vacuum amplitude +2|1−, which is part of the complete θ-vacuum amplitude in Eq. (3). The black sinosodial rep- iθ 4 − S[ ] − d x tr Fμν F˜μν Z[θ] = D e 16π2 resents the vacuum, where each minimum corresponds to a different ν winding number ν. In- and out-states are marked by − and +, respec- (4) tively. The blue lines show the vacuum transition due to instantons 1 4 × δ ν + d x tr Fμν F˜μν , and the red lines the transitions due to anti-instantons. The size of the π 2 16 transition, i.e., the difference in winding from the starting point to the where the multifield = (Aμ, q, q¯, c, c¯) contains the gluon, endpoint of an instanton, correspond to its topological charge. The quark, and ghost fields and S[ ] is the action of gauge-fixed upper figure illustrates a contribution where only single-instantons Euclidean QCD. There can also be source terms, but they are are considered. Exclusively configurations of such type are taken into suppressed here for the sake of brevity. account in the conventional dilute instanton gas. The lower figure So far, no reference to the nature of the topological shows a similar contribution, but including multi-instantons. The field configurations has been made. With the corresponding present analysis takes all possible configurations, including single remarks in Secs. I and V in mind, assume that these con- and multi-instantons, into account. figurations are given by instantons. In addition, assume that the instantons are dilute, i.e., the spacetime distance between There are various possibilities to realize the vacuum ampli- each instanton is large against their effective size. It has been tude in Eq. (4) for a given ν in a dilute gas of multi-instantons. demonstrated by numerous studies of QCD and Yang-Mills If only single-instantons are taken into account, all configura- theory on the lattice that these assumptions are justified at tions where n instantons andn ¯ anti-instantons are distributed . 1 1 temperatures above about 2 5Tc [30–39]. Note that, while the in spacetime in a dilute way are allowed, as long as ν = overall power of the topological susceptibility with respect to n1 − n¯1 is fulfilled. This can be generalized straightforwardly T measured on the lattice agrees very well with the predictions for multi-instantons: One may “sprinkle” space-time with all of the dilute instanton gas, the prefactor is off. This can likely possible combinations of nQ Q-instantons, i.e., instantons with be attributed to missing higher loop corrections, which can be topological charge Q, andn ¯Q Q–anti-instantons, provided ν = substantial in the hot QCD medium, see, e.g., the discussion − Q Q(nQ n¯Q ) holds. This is illustrated in Fig. 1. If not oth- in Ref. [31]. erwise specified, Q > 0 is used from now on. The difference What is new here is that the contributions of multi- between positive and negative topological charge is made by instantons with arbitrary topological charge Q are also taken referring to instantons and anti-instantons, respectively. into account. Since their contributions to the path integral are (Q) The contribution of one multi-instanton Aμ to the vacuum − π 2|Q|/g2 proportional to e 8 at weak coupling, single-instantons amplitude is given by the path integral in the background of (Q) clearly dominate in the semiclassical regime. For a compre- Aμ , hensive discussion of the conventional dilute instanton gas, see Ref. [47]. Yet, as discussed above, there is no reason to as- (Q) iθQ = iθQ D −S[ + ¯ ], sume that field configurations with higher topological charge e ZQ e e (5) are not present. Since the θ dependence is directly linked to the topological structure of the vacuum, it is conceivable that where is the fluctuating multi-field and ¯ (Q) = (Q) it is affected by multi-instantons. Furthermore, it has been (Aμ , 0, 0, 0, 0) is the background field. The underlying shown in Ref. [6] that there are effects that can be related assumption here is that the Q-instanton is the only topological uniquely to multi-instantons. field configuration. This is certainly guaranteed in a saddle

033359-3 FABIAN RENNECKE PHYSICAL REVIEW RESEARCH 2, 033359 (2020) point approximation about the background field, where only then small fluctuations around the Q-instanton background are ∞ ∞ 1 1 + θ Z θ θ =− nQ n¯Q assumed. In the dilute limit, the -vacuum amplitude [ ] F ( ) ln ZQ V nQ!¯nQ! is then completely described by a statistical ensemble of all ν Q nQ=0 n¯Q=0 possible combinations of ZQ’s. (Q) θ − ν As mentioned above, a multi-instanton Aμ is described × i Q(nQ n¯Q ) δ e Q(n −n¯ ) by free parameters known as collective coordinates. They Q Q Q ρ can be interpreted as positions zi, sizes i and orientations 2 ∈ =− Z cos(Qθ ), (7) Ui SU(Nc )ofQ constituent instantons with unit topological V Q charge [40–42]. As discussed in more detail below, the parti- Q tion function ZQ can formally be written as an integral over where V = V/T is the Euclidean space-time volume. The θ these parameters. It has been shown in Refs. [6,45,46] that if vacuum entails a sum over all ν, and the Kronecker delta at | | the relative distances between the constituent instantons, Rij , the end of the second line enforces net winding ν for the sum ρ are small against their sizes, i, the path integral factorizes over all possible combinations of (multi-) instantons. Since into individual contributions of the constituent instantons. The the sum over all possible multi-instanton configurations yields underlying reason is that the interaction between instantons is all possible net windings, one can drop both the sum over ν very short-ranged. At this leading order in the SCI limit, the and the Kronecker delta. For reasons that become clear below, multi-instanton itself can be viewed as a simple superposition it is convenient to define the θ-dependent part of the free of single-instantons. Furthermore, large-scale instantons are energy, suppressed at finite temperature [24,25]. This is the founda- 2 tion for using a dilute gas of single-instantons to describe the F θ = F θ − F = Z − Qθ . ( ) ( ) (0) V Q[1 cos( )] (8) topological structure of QCD at large temperatures. Q Genuine multi-instanton processes can only contribute to θ Eq. (5) at large temperature if the constituent instantons are Hence, while the dependence generated only by single- single instantons is simply ∼cos θ, the inclusion of multi- close together. Since ZQ involves integrations over all pos- sible constituent instanton positions, there is a hierarchy of instantons reveals a much richer structure. This result can be instanton contributions. If all constituent instantons are far viewed as an expansion of the free energy in terms of multi- instanton contributions. It is worth emphasizing that, since apart, only single-instantons contribute; if two constituent ∈ π θ instantons are close together, genuine 2-instanton processes Q Z,the2 periodicity of the vacuum is guaranteed also contribute; and so forth until Q constituent instantons are close here. The corrections due to multi-instantons are overtones to together, which gives a genuine Q-instanton contribution. the single-instanton contribution, which sets the fundamental Since instanton interactions are short ranged, n-instantons frequency. with 1  n  Q, contribute to ZQ roughly proportional to a spacetime-volume factor VQ−n+1. One factor of volume V A. Multi-instanton contribution from small constituent always arises since only the relative positions of constituent instantons instantons matter. To evaluate the free energy explicitly, knowledge of ZQ is ZQ can therefore be decomposed into a part that is only required. As discussed above, only certain limits are known, (q) = ,..., − due to q-instanton processes ZQ with q 1 Q 1, and one of them being the SCI limit. It uses that the exact a genuine Q-instanton contribution ZQ, multi-instanton solution can be expanded systematically in powers of ρ/|R| [42]. To leading order in this expansion, the Q-instanton can be interpreted as a superposition of Q con- Q−1 stituent instantons with unit topological charge. Remarkably, = (q) + . to order ρ2/|R|2, the path integral of QCD factorizes into ZQ ZQ ZQ (6) q=1 the contributions of the individual constituent instantons with Q = 1[6,45,46], which yields the first term of Eq. (6), 1 Z (1) = ZQ. (9) (1) = = Q Q! 1 For ease of notation, Z1 0isusedforQ 1, such that Z1 = Z1. This forms the basis for a dilute gas of multi- The combinatorial prefactor 1/Q! arises because the con- instantons, which incorporates subleading corrections to the stituent instantons can be treated as identical particles. Z1 can conventional dilute instanton gas. The distinction between be expressed in terms of the single-instanton density n : 1 q-instanton and genuine Q-instanton processes in ZQ is neces- 4 sary to avoid double-counting in the total vacuum amplitude Z1 = d zdρ n1(ρ,T ) ≡ VZ¯1. (10) of the dilute gas. Since in the classical limit the contribution of Q single-instantons to the vacuum amplitude is exactly The single-instanton density only depends on the instanton the same as of a single Q-instanton, genuine multi-instanton size and the temperature. The integration over the instanton contributions are only due to quantum corrections. location z gives a factor of space-time volume V. n1 has been The dilute multi-instanton gas is a statistical ensemble of computed to next-to-leading order in the saddle point approxi- single- and multi-instanton processes ZQ. The resulting free mation in Refs. [1,2,24,25,48–50], see Appendix A for details. =−1 Z θ energy density, F V ln , in the presence of a term is Provided that the constituent instantons are sufficiently far

033359-4 HIGHER TOPOLOGICAL CHARGE AND THE QCD VACUUM PHYSICAL REVIEW RESEARCH 2, 033359 (2020) apart, any vacuum amplitude with net topological charge Q where N is the normalization of the collective coordinate mea- can be described by a superposition of single-instanton pro- sure. In the second line, the determinant over the zero modes (Q) cesses only. This is exemplified in the top figure of Fig. 1.The of quarks, det0 (J), has been separated from the instanton Q = 1 contribution in Eq. (7) accounts for all these processes. density for later convenience. It is assumed that J is only a For genuine multi-instanton processes, as shown in the small perturbation to the Dirac operator, such that it does not bottom figure of Fig. 1, correlations between constituent in- affect the nonzero modes of the quarks.1 stantons need to be taken into account. So while Eq. (9) To systematically compute ZQ[J] in the SCI limit, the holds in case the constituent instantons are far apart, distinct ADHM construction [41] is used. The following explicit con- multi-instanton contributions arise when they overlap to some struction is based on [42]forNc = 2. The straightforward extent. For the gauge fields themselves, these corrections are generalization to any number of colors is done in the end. The of order ρ4/|R|4 [6,42]. However, due to the existence of most general Q-instanton can be constructed algebraically quark zero modes in the presence of instantons, the first cor- from two matrices M and N that obey the simple constraints rection to the leading-order SCI-limit arises at order ρ3/|R|3 listed below. M is a (Q + 1) × Q matrix with quaternionic [6]. In Ref. [6] the correlation of constituent instantons has matrix elements, which can be represented as 2 × 2 matrices been computed explicitly for Q = 2. Here, this to generalized μ M = αμM . (14) to Q  2. ab ab μ α The generating functional in the background of a Q- Mab are real coefficients and μ are the basis quaternions, instanton is defined in Eq. (5). It is instructive to write this in α = 1 , − σ , terms of the QCD action in the chiral limit, i.e., with vanishing μ ( 2 i )μ (15) quark mass matrix Mq, and an explicit bilinear quark term, where σ are the Pauli matrices. For the construction of Q– anti-instantons, one simply has to replace αμ by (Q) Z [J] = D exp − S[ + ¯ ]| = Q Mq 0 α¯ μ = (12, iσ )μ. (16) M(x) is chosen to be linear in Euclidean space-time, x = 4 μ + d x ψ¯ (x) J(x) ψ(x) , (11) αμx , M(x) = B − Cx. (17) where the source J can be set equal to Mq to recover the original action. For the following computation to be valid B and C are constant (Q + 1) × Q matrices of rank Q. M is also in the chiral limit at large temperature, it is useful to required to obey the reality condition keep a more general source J, which may have nontrivial M†(x)M(x) = R(x), (18) flavor and spinor structure. To next-to-leading order in the × saddle point approximation, ZQ[J] can be written in term of with the real, invertible, quaternionic Q Q matrix R. Each the functional determinants of the fluctuating quark, gluon matrix element of a real quaternionic matrix is proportional to and ghost fields. However, the instanton collective coordinates α0 = 12. The quaternionic conjugate is defined as arise from translations, dilatations and global gauge rotations (M†)0 = M0 , (M†)i =−Mi . (19) which are symmetries of the system, but yield inequivalent ab ba ab ba instanton solutions. Thus, the collective coordinates corre- N(x) is a quaternionic (Q + 1) column vector that obeys the spond to zero mode directions of the gauge fields. Fluctuations two constraints around these directions cannot be assumed to be small and N†(x)M(x) = 0, need to be treated exactly. The functional integral over the (20) † zero modes is replaced by and integral over the collective N (x)N(x) = 12. coordinates, while the nonzero-modes can be computed in the Remarkably, Eqs. (18) and (20) determine M and N up to ordinary fashion [2,43,44]. 8Q free parameters, which corresponds exactly to the number In addition to the zero modes of gluons and ghost, due to of instanton collective coordinates for Nc = 2. The exact Q- the axial anomaly, the Dirac operator has Nf Q left-handed instanton is then simply given by (right-handed) quark zero modes in the presence of a Q- (Q) † instanton (Q–anti-instanton) [1,51–54], Aμ = N (x)∂μN(x). (21) (Q) (Q) To find M and N, one uses that Eqs. (18) and (20) allow γμ ∂μ + Aμ ψ = 0. (12) for the transformations M → SMT and N → SN, with an The quark zero modes are functions of the collective coordi- unitary (Q + 1) × (Q + 1) matrix S and an invertible, real nates. The resulting generating functional can be expressed in Q × Q matrix T, without changing the instanton solution. † terms of the multi-instanton density nQ, From Eqs. (17) and (18) follows that C C is a symmetric, real, † invertible matrix. T can therefore be chosen such that C C Q is congruent to 1 α . Furthermore, the first row of S can be 4 Q 0 ZQ[J] = N d zi dρi dUi nQ({zi,ρi,Ui}) (13) i=1 Q 1This is guaranteed if the source is identified with the mass matrix = 4 ρ { ,ρ, } (Q) , N d zi d i dUi n¯Q( zi i Ui ) det0 (J) of the light quarks. There are quantitative modifications for heavier i=1 quarks [74,75], but they are irrelevant here.

033359-5 FABIAN RENNECKE PHYSICAL REVIEW RESEARCH 2, 033359 (2020) chosen such that the first row of C vanishes, and the other ma- resulting matrix M is † trix elements can be defined as S + = C with i = 1....,Q ⎛ ⎞ (i 1)j ij q ··· q and j = 1....,Q + 1. After these transformations, B and C in 1 Q ⎜z1 − x ··· 0 ⎟ Eq. (17) assume the general form ⎜ ⎟ 2 M = ⎝ . . . ⎠ + O(ζ ). (28) . .. . v 0 ··· − B = , C = , (22) 0 zQ x b¯ 1Qα0 With this, Eq. (20) can be solved easily, = ,..., ⎛ ⎞ where v (q1 qQ ) is a row vector of arbitrary quater- u − ∗ nions qi and ⎜ x z1 · ⎟ − 2 q u 1 ⎜ (x z1 ) 1 ⎟ = √ ⎜ ⎟ + O ζ 2 , ¯ = δ + , N(x) ⎝ . ⎠ ( ) (29) bij ijzi bij (23) ξ0 . − x zQ ∗ · (x−z )2 qQ u is a Q × Q matrix with arbitrary quaternions zi and bij.The Q ¯ diagonal elements of b are chosen to be given by zi and bii = 0 where the normalization is determined by for all i = 1,...,Q. 12Q is the real, quaternionic Q × Q unit Q ρ2 matrix. It follows from Eq. (18) that the matrix b is symmetric, ξ = + i . = = − ∗ = 0(x) 1 (30) bij b ji, and from 2Im Rij Rij Rij 0 follows: (x − z )2 i=1 i ∗ ∗ 2(zi − z j ) bij − 2Re[(zi − zj) bij] u is an arbitrary unit quaternion. Different choices for u give gauge-equivalent instanton fields. The simplest choice is u = Q ∗ ∗ ∗ ∗ α , which corresponds to the singular gauge. Indeed, plugging = (q q − q q ) + (b b − b b ). (24) 0 j i i j kj ki ki kj this into Eq. (21) yields the Q-instanton to order ζ 2 in the SCI = k 1 limit, ∗ Here, denotes the quaternionic conjugate, which corre- Q − (Q) 1 μν † 2 (x zi )ν 4 sponds to the conjugate transpose of the matrix representation Aμ (x) = U σ¯ U ρ + O(ζ ). (31) ξ (x) i i i |x − z |4 in Eq. (14). Solving this equation is difficult in general. If the 0 i=1 i q are assumed to be real, then Eq. (24)issolvedbyb = 0. i ij The anti-self-dual matrixσ ¯ μν is defined as As shown in Ref. [42] (and below), this leads to ’t Hooft’s multi-instanton solution, where all constituent instantons have σ μν = 1 αμαν − αν αμ . ¯ 2 ( ¯ ¯ ) (32) the same orientation in the gauge group. More general, self-consistent solutions can be constructed For self-consistency of this solution, x cannot be too close to ∗ any of the constituent instanton locations zi. As long as by first noting that if bij is chosen such that Re[(zi − zj) bij] = 0, then Eq. (24) can be written as |x − zi|ρi, (33)

Q ζ 2 1 z − z the multi-instanton is guaranteed to be of order . Otherwise, b = i j q∗q − q∗q + (b∗ b − b∗ b ) . (25) ij 2 j i i j kj ki ki kj higher-order corrections need to be taken into account, since 2 |zi − z j| −1 k=1 for |x − zi|∼ρi,Eq.(31) can be of order ζ ; close to their center, constituent instantons cannot be assumed to be small. This equation can be solved systematically in the SCI limit. (Q) 4 Note that the next correction to Aμ is of order ζ ,soEq.(31) To this end, write the quaternion q in terms of its modulus i is accurate to order ζ 3. This observation is relevant for the and its phase, comparison to the quark zero modes below. q = ρ U , (26) Remarkably, Eq. (31) corresponds to ’t Hooft’s multi- i i i instanton solution, but with arbitrary gauge group orientations ∗ of the constituent instantons. Furthermore, since the single- where ρ = q q can readily be interpreted as the size of the i i i instanton in singular gauge is given by [26] ith constituent instanton and Ui ∈ SU(2) as its gauge group ρ ρ2 − orientation. In the SCI limit i is assumed to be small against (1) μν † (x z)ν | |=| − | ζ | | Aμ (x; z,ρ,U ) = Uσ¯ U , (34) Rij zi z j . Hence, one can replace qi by qi, with qi (x − z)2 (x − z)2 + ρ2 held fixed, and expand in powers of the small parameter ζ . ζ 3 Then, the first nonvanishing contribution to bij in the expan- the multi-instanton to order in the SCI limit can be viewed sion of Eq. (25)isoforderζ 2, as a superposition of single-instantons, − Q 1 zi z j ∗ ∗ 4 (Q) 1 (1) 4 b = (q q − q q ) + O(ζ ). (27) Aμ (x) = Aμ (x; zi,ρi,Ui ) + O(ζ ). (35) ij 2 |z − z |2 j i i j ξ (x) i j 0 i=1 ζ − 2 + ρ2 ≈ − 2 Solutions to Eq. (25)toanypowerin can be constructed From Eq. (33) follows (x zi ) i (x zi ) . Here and in from this solution by iteration. For small constituent instan- the following, this approximate identity is exploited whenever tons, the resulting series is guaranteed to be convergent. To it seems convenient. order ζ , one can therefore set bij = 0 even for arbitrary For Nc > 2, one can simply embed SU(2) into SU(Nc ) and quaternions qi. This is sufficient for the present purposes. The use the gauge group orientations Ui from this embedding [40].

033359-6 HIGHER TOPOLOGICAL CHARGE AND THE QCD VACUUM PHYSICAL REVIEW RESEARCH 2, 033359 (2020)

The corresponding collective coordinates are generated by all The zero mode in Eq. (38) is a left-handed Dirac spinor with ϕαc α global SU(Nc ) transformations that yield inequivalent instan- the right-handed constant spinor R , where is a spinor index ton solutions. Since the embedding of SU(2) into SU(Nc ) and c is a color index. For Q < 0 this is replaced by a left- αc † I ϕ ϕ ϕ / = has a stability group Nc , i.e., a subgroup of SU(Nc ) that handed spinor L . They are normalized to give R/L R L leaves the embedding unchanged, the collective coordinates 1N .TheQ = 1 zero modes are normalized to give c of the SU(Nc ) instanton correspond to the quotient group /I 4 ψ (1)† ,ρ, ψ (1) ,ρ, = . SU(Nc ) Nc . Hence, the group integration in Eq. (13) is over d x fi (x; zi i Ui ) fi (x; zi i Ui ) 1 (41) /I SU(Nc ) Nc , with dUi the corresponding Haar measure. Since, as proven above, the multi-instanton is a sim- ψ (Q) Note that the gauge group orientation of fi is that of the ith ple superposition of single-instantons in the SCI limit, it constituent instanton, Ui.Eq.(38) is a direct generalization of has been shown in Refs. [45,46] that the resulting gauge the result for Q = 2inRef.[6]. field contribution to the generating functional factorizes into The Q = 1 quark zero mode (39)isoforderζ , while the single-instanton contributions. Eq. (13) thus becomes overlap term (40)isoforderζ 2. Hence, Eq. (38) implies that Q genuine multi-instanton–induced quark zero modes appear SCI 1 4 (Q) at order ζ 3 in the SCI limit. At order ζ , one simply has Z [J] = d zi dρi dUi n¯1(ρi ) det (J). (36) Q Q! 0 ψ (Q) = ψ (1) + O ζ 3 i=1 fi fi ( ). In this case, also the quark zero mode determinant det0(J)inEq.(36) factorizes into independent SCI The superscript indicates that the SCI limit has been ap- Q = 1 contributions, which then leads to Eq. (9). This gives 3 plieduptoorderζ . precise meaning to what has been called the leading order in Next, the contribution of the quark zero mode determinant the SCI limit: it means that the small constituent instanton needs to be computed. This requires knowledge of the quark expansion is carried out up to order ζ 2. Consequently, no zero modes in the SCI limit. They are given by the solutions of genuine multi-instanton corrections appear at leading order. the Dirac equation, Eq. (12), in the background of the multi- The next-to-leading order is ζ 3. As shown here, while the instanton in Eq. (31). This has first been done in [6]forQ = 2. multi-instanton solution, Eq. (31), does not change at this Here, this is generalized to arbitrary Q. Using the results of order, the quark zero modes do. Hence, the only correction Refs. [53,54], the quark zero modes can be obtained directly to Eq. (36) at next-to-leading order in the SCI limit stems from the ADHM solution constructed above, from the quark zero mode determinant. This correction can ψ (Q) = ν (N†CR−1) · ϕ. (37) be computed from Eq. (38), again, as a direct generalization fi i of the computation in [6]. This is done next. This is a left-handed Weyl spinor. ϕ is a constant spinor and The quark zero mode determinant in Eq. (36)is ν a normalization constant. The flavor index, f = 1,...,Nf , = ,..., (Q) = 4 ψ (Q)† ψ (Q) , and the ‘topological charge’ index, i 1 Q, denote the det0 (J) det d x fi (x) J(x) gj (x) (42) Nf Q quark zero modes. Since the Dirac equation is diagonal × in flavor, one simply gets Nf copies of Q manifestly different where the determinant on the right hand side is of the QNf zero modes. Given the general form of the multi-instanton in QNf matrix spanned by the zero modes. It is sufficient to look ψ (Q) at the contribution of the diagonal elements, Eq. (21), it is a straightforward exercise to show that fi as defined above indeed solves Eq. (12). (Q) det (J)| With Eq. (28), R−1 can be computed from Eq. (18). C and 0 diag N are given in Eqs. (22) and (29). The resulting quark zero Nf Q = 4 ψ (Q)† ψ (Q) , modes for any Q are d x fi fi (x fi) J(x fi) fi (x fi) (43) f =1 i=1 ψ (Q) = ψ (1) ,ρ, fi (x) fi (x; zi i Ui ) the computation of the other contributions is completely − , ψ (1) ,ρ , , (38) equivalent. x are the locations of the sources. As discussed Xij(x zi ) fj (x; z j j Ui ) fi j=i above, the integration over all possible instanton locations gives rise to numerous terms which lead, schematically, to ∼ρn  where terms with n 4 have been dropped and Eq. (33) Eq. (6). For the dilute multi-instanton gas, one only needs to has been used to simplify the expression. This result is a sum identify the genuine Q-instanton contribution ZQ. With the of the single-instanton quark zero modes, zero mode solution in Eq. (38), it is straightforward to identify ρ the relevant contributions. The key observation is that every (1) 2 Ui i ψ (x; zi,ρi,Ui ) = zero mode has a finite overlap with each constituent instanton. fi π 2 3/2 (Q) Nc (x − z )2 + ρ2 ψ i i (39) This is reflected in the fact that each fi depends on all γ − constituent instanton locations at next-to-leading order in the × μ(x zi )μ ϕ | − | R SCI limit. This is not the case at leading order, where each x zi zero mode only depends on the location of one constituent SCI and a term which characterizes the overlap between the Q = 1 instanton. ZQ can therefore be extracted from the partition zero modes, function in Eq. (36) by considering the contributions to the quark zero mode determinant where all zero modes overlap ρ ρ |x − z | X (x, z ) = i j i . (40) at the same constituent instanton location. There are Q such ij i − 2 + ρ2 3/2 [(x zi ) i ] contributions. One of them, where all zero modes overlap at

033359-7 FABIAN RENNECKE PHYSICAL REVIEW RESEARCH 2, 033359 (2020) z1, reads explicitly arbitrarily. In particular, (47) is identical to Q Nf Q Nf Q α α ϕ†an n †anbn bncn ϕ ncn 4 dUi R Ui Ui R d x fi n n i=1 n=1 f =1 i=1 Nf Q Nf α α = ϕ†cn n ϕ ncn . (1)† (1) R R (48) × ψ (x f 1; z1,ρ1,U1) J(x f 1 ) ψ (x f 1; z1,ρ1,U1) f 1 f 1 n=1 f =1 Thus, assuming the source J is independent of color, the gauge Q group integration of the quark zero mode determinant in the × ψ (1)† ,ρ , ψ (1) ,ρ , fj (x fj; z1 1 Uj ) J(x fj) fj(x fj; z1 1 Uj ) SCI limit is trivial. The second and third lines of (44), as j=2 well as the other contributions both from different instanton Q locations and off-diagonal contributions to the determinant, × 2 , . X j1(x fj z j ) (44) therefore exactly correspond to the terms that arise from Q j=2 copies of the Q = 1 zero mode determinant. Defining the Q = 1 determinant as The other Q − 1 contributions are obtained by replacing z1 (1) = 4 ψ (1)† ,ρ, with one of the other constituent instanton locations. Note that det0,i (J) det d x fi fi (x fi; zi i Ui ) the terms in the second and third line of this expression only depend on one location z . The last term depends on all other 1 × J(x ) ψ (1)(x ; z ,ρ,U ) , (49) instanton locations and therefore quantifies the overlap. fi gi fi i i i The overlap term in (44) also depends on the source loca- where the determinant is only over flavor in this case, the ψ (Q) tions x fj. However, if all zero modes fj (x fj)arefaraway genuine Q-instanton contribution to the partition function in from z1, but closer other z j, then the overlap at these different the SCI limit, Eq. (36), is points would dominate. The resulting contribution is not part Q (q) 1 of ZQ, but of Z in Eq. (6). Only if all zero modes are SCI 4 (1) Q Z [J] = d zi dρi dUi n¯1(ρi ) det , (J) closer to one location z than to the others (while still being Q Q! 0 i i i=1 consistent with the SCI limit), i.e., Q 2Nf × X (zi, z j ) . (50) ρi |x fj− zi||x fj− z j|, (45) ji i=1 j=i the quark zero mode determinant generates a genuine Q- To next-to-leading order in the SCI limit, the multi-instanton instanton contribution. In this case, all zero modes for j = i contribution to the partition function factorizes into single- are instanton contributions, multiplied by a nonvanishing overlap term. (Q) (1) ψ || − || − | ≈− , ψ ,ρ, . J fj (x f j ) x fj zi x fj z j X ji(zi z j ) fj (x fj; zi i Uj ) To proceed, it is convenient to specify the source .The (46) canonical choice is the constant diagonal matrix fg = ρ δ fg, Jij i ijMq (51) 2 2 Hence, X (x fj, z j )in(44) can be replaced by X (z1, z j ), and j1 j1 with the quark mass matrix Mq. Due to the normalization of equivalently for all other contributions the quark zero modes, Eq. (41), the quark zero mode determi- Owing to the different orientations in the gauge group, the nant in Eq. (50) then becomes zero mode determinant has a nontrivial group structure. To Nf simplify this, one assumes that the source J does not depend det (1)(J) = det (1)(ρ M ) = ρ m , (52) on color. In the physical case, the source is identified with the 0,i 0,i i q i f f =1 quark mass matrix, J = ρMq, so this assumption is natural. One can then exploit that the partition function involves the where m f is the mass of the quark flavor f . Only the overlap integration over all possible orientations in the gauge group. term in Eq. (50) depends on the instanton locations now. More From the quark determinant, this is of the general form specifically, it depends on the relative locations Rij.Using the explicit form of X (40), the integration over the instanton > Q Nf Q locations is for Nf 1: a α a b b c α c ϕ† n n † n n n n ϕ n n , Q Q dUi R Ui Uj R (47) n n 2N = = 4 f , i 1 n 1 d zi X ji (zi z j ) i=1 i=1 j=i where the indices in and jn are drawn from 2Nf copies of Q the set {1,...,Q}. The group integrations then result in a ρ ρ |R | 2Nf = 4 4 i j ij δanbn δbncn d zi d Rij combination of products of , involving permutations [R2 + ρ2]3/2 i=1 j=i ij j of the indices and Nc-dependent coefficients [55,56]. Due to † the normalization of the spinors, ϕ ϕR = 1N , the integration Q R c − in expression (47) leads to fully contracted Kronecker deltas. = 4 ρ2Nf ρ4 2Nf , d zi cNf i j (53) The gauge group orientations can therefore be rearranged i=1 j=i

033359-8 HIGHER TOPOLOGICAL CHARGE AND THE QCD VACUUM PHYSICAL REVIEW RESEARCH 2, 033359 (2020) with the Nf -dependent constant π 2 + − = (Nf 1)! (2Nf 1)!. cNf (54) (3Nf − 3)! Defining the integrated overlap term as Q Q−1 2Nf 4−2Nf I , ({ρ }) = c ρ ρ , (55) Q Nf i Nf i j i=1 j=i the final result for the genuine Q-instanton contribution to the partition function at next-to-leading order in the SCI limit is Q SCI 1 4 Z = d z dρ n (ρ ) I , ({ρ }), (56) Q Q! i 1 i Q Nf i i=1 where the Q = 1 quark zero mode determinant has FIG. 2. The effective potential V (θ ) = F SCI (θ ), defined in ρ = been absorbed into the instanton density n1( i ) Eq. (59), for Z1 = 0.1, 1, 3, and 6 (from dark to light red), normalized ρ (1) ρ n¯1( i ) det0,i ( iMq ), which is discussed in Appendix A. by its value at θ = π. While the potential retains its 2π periodicity, It is worth emphasizing that Eq. (56)istheresultofa strong anharmonicities arise with increasing Z1. systematic expansion of the exact ADHM solution for the Q-instanton. There is only one integration over the instanton location left. It can be interpreted as the integration over the potentially large, but this is not relevant for the present analy- average location of the Q-instanton. The relative locations sis, where qualitative effects of multi-instantons are explored. have been integrated out to yield the overlap IQ,Nf .Thisisin Furthermore, while the magnitude of the contribution of the accordance with the general discussion that led to Eq. (6). overlap at order ζ 3 in the SCI limit is now known from the As reviewed in Appendix A, the instanton density n1(ρ) analysis above, it is unknown for higher orders. In particular depends nontrivially on ρ, so that the integrations over the the contributions from instantons themselves, not the quark ρi in Eq. (56) can only be done numerically for each Q and zero modes, is entirely unknown. Self-consistency of the SCI Nf . It is therefore not possible to express ZQ in closed, expansion at next-to-leading order also restricts the present analytical form here. For the present purposes, a simple ex- result to be valid only far away from the constituent instan- pression of ZQ is desirable in order to qualitatively study tons, see Eq. (33). Consistency in the full space-time region the θ-dependent free energy F (θ )(8). A simple estimate is requires higher orders in the SCI limit [42], where corrections facilitated by the fact that n1(ρ) has a pronounced peak at an to Eq. (36) become relevant. A quantitative analysis therefore instanton sizeρ ¯ , which can be interpreted as the effective size requires a more detailed computation of multi-instanton cor- a constituent instanton. Using this, the overlap term (55) can relations, which is beyond the scope of this work. be evaluated directly atρ ¯ for ZQ, and Eq. (56) becomes The free energy is obtained by plugging Eq. (57)into 4 (Q−1) Eq. (8). The advantage of using Eq. (57) is that the sum over (¯cN ρ¯ ) ZSCI ≈ V f Z¯ Q, (57) all topological charges can be carried out analytically. The Q − 1 (Q 1)! result is where Eq. (10) has been used. Since the instanton density θ SCI θ = 4 Z1 − θ Z1 cos , has a finite width,c ¯Nf can be quantitatively different from F ( ) 2 Z1 e cos(Z1 sin ) e (59) cNf . This result has an intuitive interpretation. The overlap term I , accounts for the correlation between Q constituent Q Nf with Z ≡ Z¯ /4. For small Z , one recovers the well-known instantons. It arises from the short-distance contribution of 1 1 1 result for Q = 1, the integration over their relative distances. The effective volume of a constituent instanton, in turn, is ∼ρ¯ 4. Hence, − SCI θ = ¯ − θ + O 2 . SCI ∼ ρ4(Q 1) ¯ Q F ( ) 2Z1(1 cos ) Z1 (60) ZQ ¯ Z1 reflects the effective geometric overlap of the constituent instantons that is necessary in order for them to be correlated. This also reflects the short-ranged nature of This is expected since the effect of multi-instantons depends instanton interactions. on the magnitude of the ‘tunneling amplitude’ ZQ.Inthe 4 The overlap contributionc ¯N ρ¯ is, in general, flavor- but SCI limit, Q-instanton corrections are relevant for all Q with f −   also temperature-dependent. Here, one final simplification is (Q 1) Z1.SoforZ1 1, the free energy is dominated by made by using that, as shown in Appendix A, the average the conventional cos θ behavior, while for Z1  1 sizable cor- instanton size is approximately determined by the renormal- rections to this behavior become relevant. This is illustrated ization scale parameter  viaρ ¯ ≈ 1/2. This motivates a in Fig. 2. Such corrections are referred to anharmonicities, simple approximation, since for small θ the free energy is modified with respect 2 − to the leading harmonic contribution ∼θ . As noted above, c¯ ρ¯ 4 ≈  4, (58) Nf this might be misleading if one invokes an acoustics analogy, which greatly simplifies the following analysis, as it then only because multi-instanton contributions are overtones of the requires knowledge of Z¯1. The error of such an estimate can be single-instanton contribution.

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III. TOPOLOGICAL SUSCEPTIBILITIES effects within the approximations used here, it is instructive to consider the quenched limit of QCD. In this case, the instanton The free energy F (θ ) generates moments of the topolog- density of SU(3) Yang-Mills theory is used. ical charge distribution, the topological susceptibilities χ , n As shown in Appendix A, the renormalization group scale ∂2n θ is set by the instanton size relative to the renormalization scale χ ≡ F ( ) 2n  ∂θ2n θ=0 parameter . Thermal corrections to the instanton density + (61) only enter through the combination πT ρ [24,25]. All scales 2(−1)n 1 = Q2n Z , are therefore measured relative to  here. Since the energy V Q Q scale of thermal fluctuations is ∼πT , the combination πT/ sets the relevant thermal scale. and all odd susceptibilities vanish, χ + = 0. In the dilute 2n 1 In Fig. 3, the results for the first nonvanishing suscepti- multi-instanton gas, the differences between the susceptibil- bilities χ SCI, −χ SCI and χ SCI are shown. The results for the ities χ are due to the different powers of the topological 2 4 6 n conventional dilute instanton gas, which only accounts for charge, Qn. Hence, if only Q = 1 is taken into account, all the effect of single-instantons, are compared to the results of susceptibilities are identical up to the sign, dilute gases including also 2- and 3-instantons, as well as all χ = − n+1 ¯ . possible Q-instantons in the SCI limit. In general, the effect 2n Q=1 2( 1) Z1 (62) of higher topological charge amplifies the temperature depen- In the SCI limit, the topological susceptibilities in Eq. (61) dence of the susceptibilities. This leads to the general trend resemble moments of the Poisson distribution and can be that the topological susceptibilities decreases faster with T be- expressed as fore they follow the behavior of the dilute single-instanton gas + χ SCI = − n 14 Z1 , at high temperatures. Such a behavior has been observed on 2n 2( 1) Z1 e T2n(Z1) (63) the lattice, see, e.g., Ref. [33]. Hence, multi-instanton effects where Tn(z) are polynomials which are related to the Touchard provide a microscopic explanation for this. polynomials defined in Appendix B. Within the range of temperatures considered here, the ex- It is sometimes convenient to parametrize the θ dependence pansion of the dilute instanton gas in terms of the topological of the free energy in terms of deviations from the second charge Q converges rapidly. This is expected since the instan- susceptibility χ2, ton density itself is highly suppressed at large temperatures, ∞ such that higher powers become less relevant. 1 2 2n F (θ ) = χ2θ 1 + b2n(T )θ . (64) To study the effect of anharmonicities induced by multi- 2 SCI SCI SCI n=1 instantons, b2 , b4 , and b6 are shown as functions of temperature in Fig. 4.AsinFig.3, the conventional di- Deviations from unity of the expression in the square brack- lute single-instanton gas is compared to the result including ets hence are a measure for the anharmonicity of the θ any topological charge, as well as with the results of an dependence of the free energy. The relations between the expansion of the free energy up to Q = 2 and 3. The temper- anharmonicity coefficients b and the susceptibilities are 2n ature dependence of the anharmonicity coefficients is solely χ 2 2n+2 due to multi-instanton effects. Computations of b2 on the b2n = . (65) (2n + 2)! χ2 lattice above Tc show indications that, starting from the single-instanton value −1/12 at very high temperature, b In case only single-instantons are taken into account, it fol- 2 decreases slightly with decreasing temperature for T  2.5T , lows from Eq. (62) that the dilute instanton gas predicts c before it starts rising towards small temperatures [35–38]. constant values for these coefficients, As demonstrated here, multi-instanton corrections can explain − n 2( 1) this behavior qualitatively. However, more precise studies are b2n = . (66) Q=1 (2n + 2)! required to corroborate this on the lattice. For instantons of any topological charge in the SCI limit, however, one finds B. QCD n 2(−1) T n+ (Z ) As discussed after Eq. (60), the effects of gauge field bSCI = 2 2 1 . 2n + (67) (2n 2)! T2(Z1) configurations with higher topological charge in the dilute multi-instanton gas to leading order in the SCI limit depend Thus an explicit temperature dependence of the coefficients on the size of the partition function in the presence of a bn is generated by the effects of higher topological charge. single-instanton, Z1. It is determined by the instanton density n1, which is significantly suppressed in the presence of light A. Quenched QCD quarks, see Appendix A and in particular Fig. 9. Hence, within Using these results, the topological susceptibilities includ- the approximations used here, the corrections to the leading ing the effects of all topological charges can be computed single-instanton behavior are negligible in QCD for most from Eq. (63). For the dilute multi-instanton gas in the SCI practical purposes. Explicit numerical calculations show that limit, the size of the multi-instanton contributions to the multi-instanton corrections only become relevant for suscepti- πT = . θ dependence depends on the size of Z1. As discussed in bilities of very high order. For example, at  1 5, the effect SCI Appendix A, light quarks lead to a substantial suppression of 2-instantons on b6 is about 0.002%, while it is about 32% SCI  of the instanton density, so in order to study multi-instanton for b20 . In either case, multi-instantons with Q 3 can be

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χ SCI −χ SCI χ SCI FIG. 3. Topological susceptibilities 2 (a), 4 (b), and 6 (c) as functions of temperature in a dilute gas of instantons of various topological charges to next-to-leading order in the limit of small constituent instantons in quenched QCD. neglected. The present approximations are discussed critically axion are dictated by the chiral anomaly to be proportional in Sec. V. to FF˜. The axion effective potential is therefore of the same form as the θ-term, and one can define an effective vacuum IV. AXION COSMOLOGY “angle,” θ¯ = θ + . As outlined in Sec. I, the physics of axions is sensitive to (x) fa a(x) (68) the topological structure of the QCD vacuum. Higher topolog- Note that there is also a contribution from the finite quark ical charge effects on the cosmology of axions are explored masses to θ¯ (x), but this is not relevant here since only θ¯ (x) here. itself is of interest for the following discussion. For a more complete discussion, see, e.g., Ref. [58]. A. Axion effective potential The upshot is that the θ angle is replaced by a dynamical θ¯ Having the θ vacuum as the true vacuum with θ as a funda- field (x) (which will also be referred to as the axion for sim- mental parameter of the theory begs the question which value plicity), so there is a physical value defined by the minimum is the physical one? Since θ = 0 implies CP-violation in QCD of the axion effective potential. It follows from the discussion through FF˜, one can look corresponding processes in nature. above that the axion effective potential V is identical to the

As discussed in Sec. I, measurements of the neutron electric free energy density F. In the dilute multi-instanton gas dipole moment put stringent lower bounds on θ, strongly Eqs. (8) and (59) yield suggesting that its physical value is zero. The question about 2 V θ/¯ f = Z − Qθ/¯ f the existence and nature of a physical mechanism to enforce ( a ) V Q[1 cos( a )] this remains to be answered. Q Among the possible resolutions of this problem, the PQ SCI θ/¯ ≈ 4 Z1 − θ/¯ Z1 cos( fa ) . mechanism [14,57] is the one relevant for the present pur- 2 Z1 e 2 cos[Z1 sin( fa )] e poses. In this case, the standard model is augmented by an (69) additional global chiral (axial) symmetry, U(1)PQ, and a com- plex scalar field (and possibly other fields which are irrelevant Hence, the superselection of the θ parameter is avoided el- here), which is charged under U(1)PQ and couples to quarks. egantly by effectively promoting it to a dynamical field. The This symmetry is spontaneously broken at a scale f ,giving effective potential in the SCI limit is shown for exemplary val- a rise to a Goldstone boson, the axion a(x), related to the phase ues for Z1 in Fig. 2. Obviously, the vacuum expectation value of the complex PQ field. The classical PQ symmetry entails is at θ¯=0, which renders the QCD vacuum CP-symmetric. ashiftsymmetryofa(x). Since U(1)PQ is a chiral symmetry, The topological susceptibilities computed in Sec. III can be it is anomalous. The only non-derivative interactions of the interpreted directly as the axion mass and its higher order

SCI SCI SCI FIG. 4. Anharmonicity coefficients b2 (a), b4 (b), and b6 (c) as functions of temperature in a dilute gas of instantons of various topological charges to next-to-leading order in the limit of small constituent instantons in quenched QCD.

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(nonderivative) self-interactions. Thus the temperature depen- field described by the classical action dence of axion self-interactions is modified by multi-instanton √ 4 1 μν effects. S [θ¯] = d x −g g ∂μθ∂¯ ν θ¯ − V (θ/¯ f ) , (72) a 2 a has energy-momentum B. Vacuum realignment 1 αβ The specifics of how the axion couples to the standard Tμν = ∂μθ∂¯ ν θ¯ − gμν g ∂αθ∂¯ β θ¯ − V (θ/¯ fa ) . (73) model besides the topological sector discussed above are 2 model dependent. There is a class of “invisible” axion models From this, one infers the time evolution of the homogeneous originating from [59–62], which are consistent with bounds axion field according to its classical equation of motion, from axion searches [63]. These models require U(1)PQ to be d2θ¯ dθ¯ dV (θ/¯ f ) spontaneously broken at a very high energy scale resulting in + 3H + a = 0, (74) 9 2 an axion decay constant of fa  10 GeV, rendering axions dt dt dθ¯ very light and their interactions faint. Furthermore, cold ax- and the energy density of the axion, ions can be produced through a field-relaxation mechanism during the evolution of the universe, known as vacuum re- 1 dθ¯ 2 ρ ≡ T = + V (θ/¯ f ). (75) alignment [20–23]. This makes axions viable candidates for a 00 2 dt a dark matter. In the following, the possible implications of higher topological charge effects on the production of cold Strictly speaking, the time evolution in Eq. (74) is incomplete. axions are discussed on a qualitative level. There is an additional Friedmann equation determining the For illustration, the simplest realization of the vacuum Hubble parameter, which depends on the energy density of realignment mechanism is used. It is assumed that sponta- the axion. For the timescales relevant here, the universe is to neous PQ symmetry breaking occurs before inflation and that a good approximation radiation-dominated and the axions do the reheat temperature is smaller than the temperature for not spoil that. Their energy density is negligible compared to PQ symmetry restoration. Note, however, that the qualitative the contributions of radiation to the Hubble parameter. statements made here are more general. If PQ symmetry is The time evolution of the axion field in Eq. (74) has a spontaneously broken in the very early universe, the resulting very simple heuristic interpretation if one assumes that an- harmonicities in the axion effective potential are small. In axion is essentially massless since the instanton effects that θ/¯ dV ( fa ) ≈ 2 θ¯ give rise to the axion mass, this case one has dθ¯ ma ,cf.,Eqs.(64) and (70). Naively (ignoring the explicit time dependence of H and m ), a 2 θ/¯ Eq. (74) then has the form of a damped harmonic oscillator. 2 d V ( fa ) −2 m = = f χ , (70) Its qualitative behavior is determined by the damping ratio a θ¯ 2 a 2 d θ¯=0 ζ = 3H/2ma. If the Hubble parameter dominates over the ax- ion mass such that ζ>1, the axion evolution is overdamped. ∼ are negligible at T fa in this regime. Thus, the axion Since the Hubble expansion is much smaller than the Comp- is strongly fluctuating around its vacuum expectation value ton wavelength of the axion at early times, the axion decays within the range θ/¯ fa ∈ [−π,π]. In a sufficiently small patch very slowly from the initial misalignment angle θ¯0 towards its in space right before inflation, the axion field can be assumed vacuum expectation value. At later times, when a substantial θ¯ to have a homogeneous value 0. Due to inflation, such a axion mass is generated through instanton effects and the patch is blown up in size and it is possible to have a single Hubble expansion slows down, the system enters the under- θ¯ homogeneous value 0 for the axion within our causal horizon. damped regime with ζ<1. The axion then oscillates with Furthermore, inflation dilutes all relics from the PQ phase decreasing amplitude around its vacuum expectation value transition, such as topological defects, away. In the simplest θ¯ = 0. realization of the vacuum realignment mechanism, one as- Due to the strong time dependence of H and ma this sim- sumes that we live in one such domain. So it is assumed ple picture is merely suggestive. Eq. (74) is therefore solved that the axion is homogeneous throughout the whole universe. numerically. However, the intuition from the damped oscilla- Since fluctuations are redshifted away, it can be treated as a tor helps to anticipate the possible effect that multi-instanton classical field. Thus, starting from the random initial value corrections have on the time evolution of the axion. From the θ¯ 0, called the misalignment angle, the axion evolves in time axion potential in Eq. (69) shown in Fig. 2 one sees that the according to the classical equations of motion. For a more anharmonicity indued by multi-instantons can lead to a flat- detailed discussion, see, e.g., Refs. [17,18]. tening of the potential around the maxima, or even turn them Within the standard model of cosmology, a homoge- into local minima. Thus, the ‘frequency term’ in the axion neous, isotropic, expanding universe with vanishing curvature evolution equation (74), ∼V (θ/¯ fa ), can become significantly is assumed. This is described by the Friedmann-Lemaitre- smaller than the corresponding result for the effective poten- Robertson-Walker metric tial induced by single-instantons. If the initial misalignment θ¯ happens to be in this flattened region, the axion remains 2 2 2 0 gμν = diag(1, −a (t ), −a (t ), −a (t )), (71) frozen for a longer time before it starts a damped oscillation 2 around its vacuum expectation value for H  |V (θ/¯ fa )|. with the scale parameter a(t ) (not to be confused with the Figure 5 shown an example for the multi-instanton induced = 1 da(t ) axion). The Hubble parameter is H(t ) a(t ) dt . The axion axion effective potential at three different times. A similar

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FIG. 6. Time evolution of the quenched-QCD axion. Multi- FIG. 5. Multi-instanton induced axion effective potential for instanton effects delay the onset of the oscillating regime and quenched QCD at three different times t¯ = /πT . increase the oscillation frequency. scenario, but originating from a noncanonical kinetic term of the axions instead of topological field configurations, has been anharmonicities from multi-instanton corrections results in discussed in Ref. [64]. Note that if Z1 becomes large enough, a longer period of overdamping, where the axion is essen- the anharmonic corrections in the SCI limit can induce a sign tially frozen at the misalignment angle. This can be read-off change in V (θ/¯ fa ) for intermediate values of θ¯. In this case, from Fig. 5, which shows that the multi-instanton induced it is possible to increase θ¯ (t ), before it starts oscillating. This axion effective potential flattens significantly in the over- possibility is commented on below. damped regime for t¯  0.54. In contrast, the single-instanton induced effective potential retains its cosine-shape throughout C. Quenched-QCD axion the whole time evolution. Once the Hubble expansion has slowed down sufficiently, the time evolution is determined by For the numerical solution of Eq. (74), it is assumed that the curvature of the effective potential and the axion oscillates the universe is radiation dominated. The Hubble parameter around its vacuum expectation value θ¯=0 with decreasing then is amplitude. π 3 4 2 8 T Using this solution, the energy density of the axion can H (T ) = g(T ) , (76) 90 m2 be computed from Eq. (75). The result, again for the multi- Pl and single-instanton induced potentials, is shown in Fig. 7. 19 with the Planck mass mPl ≈ 1.22 × 10 GeV. For the ef- The energy density monotonously rises with time in the over- fective number of radiative degrees of freedom, g = 100 is damped region, where it is almost exclusively due to the chosen for simplicity, see, e.g., Ref. [65] for a more refined potential energy of the axion, and then slowly decreases in analysis. Converting temperature to time is done by using the oscillating regime at later times. In addition to the rate of = 1 that radiation domination implies t(T ) 2H(T ) .Asforthe decrease in the oscillating regime, the energy density today topological susceptibilities, the axion effective potential is crucially depends on how long the axion is frozen in the computed in the SCI limit in Eq. (69), with Z1 discussed in Appendix A. As before, all scales are measured relative to ; the relevant time scale is  t¯ = . (77) πT To highlight the effect of multi-instantons, an initial misalign- ment angle close to the maximum of the potential, where the flattening of the potential is most pronounced, and a large axion decay constant, in order to push the onset of the underdamped regime to lower temperatures, are chosen. 16 Specifically, θ¯0/ fa = 3.14 and fa/ = 4 × 10 are used. Given that this is only a toy model, the value for the axion decay constant is not physical. The time evolution of the axion is shown in Fig. 6.The effect of the multi-instanton induced effective potential of Eq. (69) (solid lines) is compared to the single-instanton in- duced potential ∼cos(θ/¯ fa ) (dashed lines). For the present FIG. 7. Time evolution of the energy density of the quenched- choice of parameters, one sees the qualitative behavior dis- QCD axion. Multi-instanton effects increase the energy density of cussed above: the flattening of the potential due to the axions at later times.

033359-13 FABIAN RENNECKE PHYSICAL REVIEW RESEARCH 2, 033359 (2020) overdamped regime. Since multi-instanton effects can pro- is Nc large, nor are quarks confined here. As mentioned al- long this phase and delay the start of oscillations, they can ready, the assumptions regarding the nature of topological increase today’s energy density of cold axions. In addition, field configurations made here are backed by numerous results owing to the steeper potential around the minimum, the ax- for topological susceptibilities by first-principles lattice gauge ion field oscillates faster in the presence of multi-instanton theory methods. They all show that at large temperatures, corrections. Hence, the larger kinetic energy also leads to a T  2.5Tc, the behavior of the susceptibilities with respect larger energy density in the oscillating regime as compared to T agrees with the predictions of a dilute instanton gas to the single-instanton induced axion potential. In summary, [30–32,34,36–39]. Note that this is not contradicting the va- higher topological charge effects can flatten the axion effec- lidity of a dilute multi-instanton gas, since, as shown here, tive potential and therefore provide a topological mechanism multi-instanton corrections are small at large temperatures and to increase the amount of axion dark matter today. might very well fit within the error bars of state-of-the-art Following the discussion after Eq. (60), in Sec. III B and lattice results. Appendix A, the substantial suppression of the instanton den- sity in the presence of light quarks leads to also a substantial VI. SUMMARY suppression of multi-instanton effects for the present approx- imation to the topological structure of the vacuum. Hence, It has been shown that gauge field configurations with while the mechanism discussed above is still present in QCD, higher topological charge modify the QCD vacuum. This is its effect could be negligible. Only axion self-interactions for reflected in corrections to the conventional dependence on the very high order are modified substantially by multi-instantons CP-violating topological θ parameter. in QCD, but their overall magnitude is tiny. At large temperatures well within the deconfined phase, Note that the sign change of V seen in Fig. 2 for very large the topological structure of QCD can be described by a di- Z1 could lead to an increase of θ¯ (t ) at intermediate times, lute gas of instantons. Even though multi-instantons with additionally increasing also the energy density. However, Z1 topological charge Q > 1 are suppressed in the semiclassical does never become this large here in the relevant stage of weak-coupling limit, their contributions to the path integral the axion evolution, so this possibility has not been further are genuinely different from the single-instanton contribution. explored. Hence, the picture of a dilute instanton gas has been general- ized to include instantons of arbitrary topological charge. Note V. DISCUSSION that this is very much in line with the findings in [6], where it has been shown that there are anomalous quark correlations Before the results are summarized, a critical discussion of which are only generated by multi-instantons. the assumptions and approximations made here is in order. In addition to this conceptual result, a key technical result First, the actual size of the effects studied here obviously in the present work is the systematic derivation of the multi- depends on the quantitative impact of field configurations of instanton contribution to the partition function in the SCI limit higher topological charge on the vacuum amplitude. A dilute for arbitrary topological charges, which has led to Eq. (56). gas and the SCI limit have been used. A dilute gas can only This required the explicit expression for the quark zero modes be valid at weak coupling for very high temperatures. Yet, generated in the background of a multi-instanton given in multi-instanton effects are relevant if the classical suppression Eq. (38). The computational techniques presented here, which ∼e−8π|Q|/g2 becomes less strong and effective instanton sizes are based on the work in Refs. [6,42,45,46,53,54], pave the become larger. This is only possible away from the strict way to compute higher-order corrections to the partition func- weak-coupling limit at lower temperatures. Then, in turn, tion in a multi-instanton background in the SCI limit. other effects, such as interactions between (multi-) instantons There is a nice acoustics analogy regarding multi-instanton and (non-perturbative) quantum effects, also become increas- corrections to the θ-dependent free energy of QCD: they give ingly relevant. Furthermore, it has been argued that the effects rise to overtones to the fundamental frequency, which is set in QCD are small because light quarks suppress the instanton by single-instantons. In the dilute multi-instanton gas in the density. The dynamical generation of quark mass at lower SCI limit, the θ-dependent free energy can be computed ana- temperatures might compensate this to some extent. To as- lytically, based on the known results for the single-instanton sess the relative importance of these different effects, a better density. This leads to a modification of the conventional cos θ- understanding of the partition function in a Q-instanton back- behavior of the free energy. The resulting θ dependence of ground, ZQ, beyond the estimates based on the SCI limit used QCD is reflected in the topological susceptibilities χn. If only here, is necessary. For this, the overlap between constituent single-instanton effects are accounted for, all susceptibilities instantons has to be taken into account more accurately. There are proportional to the first nonvanishing susceptibility χ2. is apriorino reason to assume that the SCI limit to a low Higher topological charge contributions lift this “degeneracy” order is sufficient to accurately describe ZQ, and is therefore and amplify the temperature dependence of these susceptibil- the largest source of uncertainty here. ities towards lower temperatures. This is most clearly seen Second, it has been assumed that topological gauge field in the anharmonicity coefficients b2n ∼ χ2n+2/χ2, which are configurations at large temperatures, i.e., well within the de- constant for single-instantons only. Multi-instantons give rise confined phase, are described by instantons (or rather their to a characteristic temperature dependence of the anharmonic- finite-temperature cousins, sometimes called calorons [25]). ity coefficients. Strong indications for the behavior of the It has been argued in [66] that the instanton picture is incom- topological susceptibilities predicted here have been found on patible at large Nc in the confined phase. However, neither the lattice at temperatures above Tc. Hence, multi-instanton

033359-14 HIGHER TOPOLOGICAL CHARGE AND THE QCD VACUUM PHYSICAL REVIEW RESEARCH 2, 033359 (2020) effects can provide a microscopic explanation for various As a final remark, an interesting observation is that multi- qualitative features of topological susceptibilities observed on instanton effects lead to the possibility of metastable states the lattice in the deconfined phase. at θ =±π,cf.,Figs.2 and 5. This could have interesting An interesting application to showcase higher topological phenomenological implications due to the possibility of spon- charge effects is axion cosmology. Since the axion effec- taneous CP violation [72,73]. tive potential is determined by the θ dependence of QCD, it is also sensitive to the effects investigated here. Again, ACKNOWLEDGMENTS to accentuate these effects, a toy universe where axions are coupled to quenched QCD has been considered. The results The author is grateful to Rob Pisarski for valuable dis- on the topological susceptibilities directly apply to axion self- cussions, comments and collaborations, which inspired this interactions. The production of cold axion dark matter via work. It is supported by the U.S. Department of Energy under the vacuum realignment mechanism has been studied as an contract DE-SC0012704. example. Multi-instanton effects can flatten the axion effec- tive potential around its maxima and, as a result, can delay APPENDIX A: INSTANTON DENSITY the time where the evolution of the axion switches from the overdamped to the oscillating regime in an expanding, The single-instanton density n1 has been computed in radiation-dominated universe. In addition, the axion oscillates Refs. [1,2,24,25,48,49]. In the modified subtraction scheme faster. This leads to an overall increase in the energy density (MS), it reads of axions at late times, as compared to the case where only d 8π 2 6 8π 2 single-instantons are taken into account. Hence, higher topo- n (ρ,T ) = MS exp − logical charge effects give rise to a mechanism that increases 1 ρ5 g2 g2 the amount of axion dark matter. 2π 2 (A1) As discussed in the previous section, on the one hand, the × exp − ρ2m2 (T ) − 14A(πT ρ) 2 D effects studied here become very small if dynamical quarks g (1) are taken into account within the present approximations. On × det (Mq ). the other hand, multi-instanton effects can become relevant 0 only in a regime where semiclassical and dilute approxima- (1) det0 (Mq ) is the functional determinant in the space of quark tions begin to break down, at least to leading order. Thus, zero modes and Mq is the quark mass matrix. The renormal- a better understanding of the significance of higher topolog- ization scheme dependent constant dMS is ical charge effects requires a more detailed understanding 5/6 of the impact of topological gauge field configurations on 2e − . + . d = e 1 511374Nc 0 2614436Nf , (A2) MS 2 the vacuum amplitude of QCD. This work is a first step in π (Nc − 1)!(Nc − 2)! this direction. Two major sources of uncertainty are the un- known higher-order corrections to the vacuum amplitude in g is the running strong coupling. The renormalization group a multi-instanton background in the SCI limit, and neglected scale is chosen to be determined by the instanton size relative interactions between (multi-) instantons and anti-instantons. to the scale parameter , g(ρ). The two loop running both Higher topological charge effects at lower temperatures, in in the exponential and the pre-exponential is used [6], particular close to and in the confined phase have not been dis- (4π )2 β ln ln(x−2 ) cussed here at all. As mentioned in the beginning, the reason is g2(x) = 1 − 1 , (A3) β −2 β2 −2 that the nature of topological field configurations is unsettled 0 ln(x ) 0 ln(x ) in this case. A possibility is to study topological effects in “de- β = − / β = 2/ − / − formed” versions of QCD. One example is Yang-Mills theory with 0 (11Nc 2Nf ) 3 and 1 34Nc 3 (13Nc 3 on a small circle, where higher topological charge effects can 1/Nc )Nf . The in-medium corrections depend on the Debye be addressed systematically [67]. At low temperatures, chiral mass at leading order, perturbation theory is valid and the θ dependence can be stud- N N N ied in a controlled manner. This is possible without explicit 2 ,μ = 2 c + f 2 + f μ2 , mD(T ) g T (A4) knowledge of the nature of topological gauge field configura- 3 6 2π 2 tions, since the θ angle can be rotated to the phase of the quark and the function mass matrix by an axial transformation. The θ dependence − is then encoded in correlation functions of pseudo-Goldstone 1 x2 0.159 8 A(x) =− ln 1 + + 0.0129 1 + , (A5) bosons in the chiral expansion. This way, contributions from 12 3 x3/2 all topological charge configurations are taken into account and the free energy also deviates substantially from the simple is a numerical parametrization of the temperature-dependent cos θ behavior [68–71]. In this case, it is in principle possible part of the one-loop determinant in the instanton background to extract the contributions from different topological charges [24,25,50]. Note that the instanton density has, at least at the by means of a Fourier transformation of the free energy with loop order considered here, a large renormalization scheme respect to θ [68]. However, connecting these results to the dependence, the MS scheme, is just the canonical choice. present results is impossible since the regions of validity The instanton density for the pure SU(3) gauge theory [i.e., of chiral perturbation theory and dilute instantons do not Nf = 0 and det0(Mq ) = 1], which is used in the quenched overlap. approximation, is shown in Fig. 8. First, one clearly sees the

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FIG. 9. Comparison between the single-instanton partition func- FIG. 8. Instanton density n1(ρ) of quenched QCD for various tion Z1 of QCD with and without dynamical quarks. temperatures. APPENDIX B: TOUCHARD POLYNOMIALS suppression of the instanton density due to thermal correc- The evaluation of Eq. (61) in the SCI limit involves a tions. Second, the instanton density is peaked about summation of the form ∞ zQ 1 T (z) = z−1e−z Qn . (B1) ρ¯ ≈ . (A6) n (Q − 1)! 2 Q=1 This justifies the use of an effective instanton size in Sec. II A. The polynomials Tn are directly related to the Touchard poly- The zero-mode determinant of quarks in the background of nomials Tn via a single-instanton is given by n n T (z) = T (z), (B2) n k k (1) = 4 ψ (1)† ρ δ fgψ (1) . k=0 det0 (Mq ) det d x fi fi (x fi) i ijMq gj (x fi) and the Touchard polynomials are defined as (A7) ∞ zq T (z) = e−z qk . (B3) The quark zero modes are defined in Eq. (39). Since the quark k q! mass matrix is diagonal in flavor, and one can assume that all q=0 ρ = ρ instanton sizes are the same in the SCI limit (i.e., i for They can be written as all i), the determinant simply gives a factor n n T (z) = zk, (B4) Nf n k k=0 det0(Mq ) = ρm f , (A8) f =1 with the Stirling numbers of the second kind k where m is the constituent mass of quark flavor f . Strictly n 1 k f = (−1)i (k − i)n. (B5) speaking, this is only valid if Mq can be viewed as a small k i k! = perturbation of the Dirac operator such that the unperturbed i 0 quark-eigenmodes can be used and only the lowest eigenvalue The first few polynomials Tn are given by is affected. For a more complete discussion, see Refs. [74,75]. For the present purposes, a qualitative discussion is T0(z) = 1, sufficient. = + , 4 T1(z) 1 z Fig. 9 shows a comparison between Z1 = Z¯1/ , defined 2 in Eq. (10), for QCD with and without dynamical quarks. In T2(z) = 1 + 3z + z , = = the former case, four quark flavors with mu md 3MeV, 2 3 T3(z) = 1 + 7z + 6z + z , (B6) ms = 94 MeV, and mc = 1.27 GeV were chosen. The running 2 3 4 of the masses and threshold effects in the running of the strong T4(z) = 1 + 15z + 25z + 10z + z , coupling have been neglected for simplicity. Z1 is suppressed = + + 2 + 3 + 4 + 5, by about seven orders of magnitude if the four lightest quark T5(z) 1 31z 90z 65z 15z z flavors are taken into account. 2 3 4 5 6 T6(z) = 1 + 63z + 301z + 350z + 140z + 21z + z .

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