Higher topological charge and the QCD vacuum

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 $\theta$. To explore this, topological susceptibilities and the production of axion dark matter are studied here. The former 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 $\theta$-dependence of QCD. The production of cold dark matter through the vacuum realignement 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. Within this approximation, the effects of multi-instantons are most pronounced in the absence of dynamical quarks, i.e. in quenched QCD. In this case, various qualitative features of topological susceptibilities at large temperature can be related to gauge field configurations with higher topological charge. Furthermore, multi-instanton effects can provide a topological mechanism to increase the amount of axion dark matter.


I. INTRODUCTION
The vacuum structure of interacting quantum field theories is inherently intriguing.In quantum chromodynamics (QCD) and other gauge theories, it is well known that this structure crucially depends on topological gauge field configurations [1][2][3][4][5].A vacuum state can be characterized by an integer, the topological charge, and topological gauge fields describe tunneling processes between topologically distinct realizations of the vacuum.The true vacuum is therefore a superposition of all possible topological vacuum configurations.It is characterized by a free parameter, the CP-violating topological phase θ , acting as a source for topological charge correlations.
The topological nature of the QCD vacuum has important phenomenological consequences.The axial anomaly in QCD is realized through topologically nontrivial fluctuations.They generate anomalous quarks correlations which explicitly break U(1) A and impact properties of hadrons [6].Most prominently, the large mass of the η meson is generated by topological effects [1-3, 7, 8].The fate of the axial anomaly at finite temperature also determines the order of the QCD phase transition in the limit of massless up and down quarks [9,10].Furthermore, a nonzero θ introduces CP-violating strong interactions.These would manifest in a nonvanishing electric dipole moment of the neutron, d n .Recent measurements give a stringent bound of |d n | < 1.8 × 10 −26 e cm [11], resulting in θ 10 −10 [12,13].This strongly suggests that there is no strong CP-violation.The question about the existence and nature of a physical mechanism to enforce θ = 0 remains to be answered.
One resolution of this strong-CP problem has been suggested by Peccei and Quinn (PQ) [14].It involves the introduction of a new complex scalar field, which has a global chiral U(1) PQ symmetry.The PQ symmetry is broken both explicitly, through the axial anomaly, and spontaneously.The resulting pseudo-Goldstone boson, the axion a(x) [15,16], couples to the gauge fields in exactly the same way as the θparameter, ∼ (a/ f a + θ ) tr F F , where F and F are the gluon field strength and its dual, and f a is the axion decay constant.This effectively promotes θ to a dynamical field, which has a physical value given by the minimum of its effective potential.Owing to the anomaly, the effective potential of axions is determined by topological gauge field configurations, and its minimum is at zero, thus resolving the strong CP-problem dynamically.Furthermore, the anomalous axion mass is very small and, in order to be consistent with observational bounds, axions couple very weakly to ordinary matter.This makes axions attractive dark matter candidates [17][18][19].Cold axions can be produced non-thermally through a field relaxation process known as vacuum realignement [20,21].The axion relaxes from a, possibly large, initial value to its vacuum expectation value at zero, thereby probing the global structure of the θ -vacuum.The production of cold dark matter through the vacuum realignment mechanism of axions therefore is particularly sensitive to the topological structure of the QCD vacuum.
While the nature of topological field configurations in the confined phase is still unsettled, at large temperatures where the gauge coupling is small, a semi-classical analysis is valid [22,23] and these field configurations can be described by instantons [24], see [25][26][27] for reviews.Finite temperature is crucial for a self-consistent treatment, since it effectively cuts off large-scale instantons.This is clearly shown by numerous first-principles studies of QCD on the lattice, both with dynamical quarks [28][29][30][31], and in the quenched limit [32][33][34][35].It has been demonstrated that the temperature dependence of the lowest topological susceptibility χ 2 , which measures the variance of the topological charge distribution, is in very good agreement with the corresponding prediction from a dilute gas of instantons for T 2.5T c , where T c is the pseudocritical temperature of the chiral phase transition.One can therefore assume that the topological structure of the QCD vacuum is described by instantons at large temperatures.This is of relevance also for axion physics, since, for example, the axion mass is proportional to the topological susceptibility, χ 2 = f 2 a m 2 a .Conventionally, only single-instantons, i.e. instantons with unit topological charge, are taken into account in the description of the topological structure of the QCD vacuum at large temperatures.The reason is that classically the action of an instanton with topological charge Q in a dilute gas is ∼ e −8π 2 |Q|/g 2 .Thus, in the limit of vanishing gauge coupling g, multi-instantons with Q > 1 are subject to a large exponential suppression.Furthermore, the moduli space of multi-instantons coincides with that of Q independent single-instantons, which are each characterized by a position z i , a size ρ i and an orientation in the gauge group U i , with i = 1, . . ., Q [36].Yet, multi-instantons are distinct selfdual gauge field configurations which, in general, cannot be constructed from single-instanton solutions [37,38].Since quantum corrections lead to an increasingly strong coupling towards lower energies, multi-instantons could give relevant corrections to the leading single-instanton contribution.Furthermore, it has been shown in [6] that there are 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 for single-instantons [1,2].Thus, one has to assume that the topological field configurations at large temperatures are instantons of arbitrary topological charge.
Given this motivation, the effects of multi-instantons on the vacuum structure of QCD are explored in this work.It is assumed that at sufficiently high temperature in the deconfined phase, the topological structure of QCD is described by a dilute gas of instantons of arbitrary topological charge.This leads to a modification of the known θ -dependence of the QCD vacuum.This modification is reflected in the distribution of topological charge, which is probed by topological susceptibilities.In addition, as outlined above, axion cosmology is an interesting application to showcase the effect of higher topological charge on the QCD vacuum.To this end, the production of cold axion dark matter via the vacuum realignment mechanism is investigated here as well.
A semi-classical description of multi-instantons requires knowledge of the partition function of QCD in their presence.To leading order in the saddle point approximation, the complete partition function is known for single-instantons at finite temperature [1,2,22,23].For multi-instantons, it is only known in certain limits, see [39,40] for reviews.The exact ADHM solutions for multi-instantons [37] can be expanded systematically if the size parameters ρ i are small against the separation |R i j | = |z i − z j | for i = j and i, j = 1, . . ., Q [38].To order ρ 3 /|R| 3 , the partition function of QCD in the background of a multi-instanton can be computed solely based on the knowledge of the single-instanton solution [6,41].This limit of small constituent-instantons (SCI) is used here.Due to the large suppression of the instanton density in the presence of dynamical quarks, it is expected that, at least in the large temperature regime, multi-instanton effects are most pronounced in quenched QCD, i.e. without dynamical quarks.Hence, quenched QCD provides a good laboratory to understand higher topological charge effects on the QCD vacuum.This work is organized as follows: The θ -dependence in a dilute gas of instantons is derived in Sec.II.The result is used to study topological susceptibilities in Sec.III.Their temperature dependence is evaluated numerically for quenched QCD in Sec.III A and discussed for QCD in Sec.III B. The focus there is on systematically investigating correction to the topological susceptibilities as instantons with increasing topological charge are included.The impact of multiinstantons on axion cosmology is studied in Sec.IV.First, the resulting axion effective potential and the production of axions via the vacuum realignment mechanism are discussed in Secs.IV A and IV B. In Sec.IV C axion production is studied numerically in the quenched approximation.Finally, a critical discussion of the approximations used here can be found in Sec.V, and a summary of the results in Sec.VI.Technical details, in particular regarding the instanton density, are given in the appendices.

II. θ -DEPENDENCE FROM A DILUTE GAS OF MULTI-INSTANTONS
The starting point is a modification of the old story of the θ -dependence of the QCD vacuum.In the context of the axial anomaly and the U(1) A problem in QCD, it has been realized that the QCD vacuum has more complicated structure which is related to the existence of topological gauge field configurations [3][4][5].Physical states can therefore be grouped into homotopy classes of field configurations with a given topological charge (or winding number) |n〉, where is the field strength tensor with D µ = ∂ µ + A µ the covariant derivative.Fµν = 1 2 ε µνρσ F ρσ is the corresponding dual field strength.Instantons of topological charge Q can be interpreted as tunneling from state |n〉 to state |n + Q〉, with the tunneling amplitude related to the exponential of the classical instanton action, e −S Q , and Due to such tunneling processes, |n〉 cannot describe the vacuum state in a unique way.Furthermore, states characterized by different winding numbers are related to each other by large gauge transformations.The true vacuum state, consistent with gauge symmetry, locality and cluster decomposition is the θ -vacuum, 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.
The vacuum amplitude between in-and out-states at t = −∞ and +∞, denoted by |θ 〉 − and + 〈θ | respectively, is then given by with ν = m − n.The part after the exponential in the second line of this equation describes the amplitude where inand out-state differ by a net-topological charge of ν.Using Eq. ( 1), the θ -dependent generating functional [θ ] ≡ + 〈θ |θ 〉 − can be represented by the Euclidean path integral where the multi-field Φ = (A µ , q, q, c, c) contains the gluon, quark and ghost fields and S[Φ] is the action of gauge-fixed Euclidean QCD.There can also be source terms, but they are suppressed here for the sake of brevity.
So far, no reference to the nature of the topological field configurations has been made.With the corresponding remarks in Sects.I and V in mind, assume that these configurations are given by instantons of arbitrary topological charge Q, which are called Q-instantons for simplicity.In addition, assume that the instantons are dilute, i.e. the spacetime distance between each Q-instanton is larger than their effective size.It has been demonstrated by numerous studies of QCD and Yang-Mills theory on the lattice that these assumptions are justified at temperatures above about 2.5T c [28][29][30][31][32][33][34][35].Note that, while the overall power of the topological susceptibility with respect to T measured on the lattice agrees very well with the predictions of the dilute instanton gas, the prefactor is off.This can likely be attributed to missing higher loop corrections, which can be substantial in the hot QCD medium, see, e.g., the discussion in [29].
What is new here is that the contributions of multiinstantons are also taken into account.Since their contributions to the path integral are proportional to e at weak coupling, single-instantons clearly dominate in the semi-classical regime.For a comprehensive discussion of the conventional dilute instanton gas, see [42].Yet, as discussed in Sec.I, there is no reason to assume that field configurations with higher topological charge are not present.Since the θ -dependence is directly linked to the topological structure of the vacuum, it is conceivable that it is affected by multi-instantons.Furthermore, it has been shown in [6] that there are effects that can be related uniquely to multi-instantons.Most importantly, corrections to the leading-order semi-classical results are largely unknown, and even less is known about the partition function in a multi-instanton background.This leaves a lot of room for exploratory studies.There are various possibilities to realize the vacuum amplitude in Eq. ( 4) for a given ν in a dilute gas of multiinstantons.If only single-instantons are taken into account, all configurations where n 1 instantons (Q > 0) and n1 antiinstantons (Q < 0) are distributed in spacetime in a dilute way are allowed, as long as ν = n 1 − n1 is fulfilled.This can be generalized straightforwardly for multi-instantons: One may 'sprinkle' spacetime with all possible combinations of The contribution of one Q-instanton A (Q) µ to the vacuum amplitude is given by the path integral in the background of where Φ is the fluctuating multi-field and Φ(Q) = (A (Q) µ , 0, 0, 0, 0) is the background field.The underlying assumption here is that the Q-instanton is the only topological field configuration.This is certainly guaranteed in a saddle point approximation about the background field, where only small fluctuations around the Q-instanton background are assumed.In the dilute limit, the θ -vacuum amplitude [θ ] is then completely described by a statistical ensemble of all possible combinations of Z Q 's.The resulting free energy density, F = − 1 ln , for a dilute gas of instantons of arbitrary topological charge in the the presence of a θ -term is: where = V /T is the Euclidean spacetime volume.The θ -vacuum entails a sum over all ν, and the Kronecker delta at the end of the second line enforces net winding ν for the sum over all possible combinations of (multi-) instantons.Since the sum over all possible multi-instanton configurations yields all possible net windings, one can drop both the sum over ν and the Kronecker delta.For reasons that become clear below, it is convenient to define the θ -dependent part of the free energy, Hence, while the θ -dependence generated only by singlesingle instantons is simply ∼ cos θ , the inclusion of multiinstantons reveals a much richer structure.This result can be viewed as an expansion of the free energy in terms of multi-instanton contributions.It is worth emphasizing that, since Q ∈ , the 2π-periodicity of the θ -vacuum is guaranteed also here.The corrections due to multi-instantons are overtones to the single-instanton contribution, which sets the fundamental frequency.
To evaluate this, knowledge of Z Q is required.For Q = 1, the partition function has been computed to leading oder in the saddle point approximation in vacuum in [1,2] and at finite temperature in [22,23].For Q > 1 it is only known in certain limits.One of these limits is the SCI limit mentioned in Sec.I.It exploits the fact that any Q-instanton is characterized by Q locations z i , sizes ρ i , and orientations in the gauge group U i ∈ SU(N c ) with i = 1, . . ., Q [36][37][38].The exact instanton solution can be expanded systematically in powers of ρ/|R| [38].To leading order in this expansion, if the sizes ρ i are small against the separation |R i j | = |z i − z j |, the Q-instanton can be interpreted as a superposition of Q constituent-instantons with unit topological charge.Remarkably, up to (ρ 3 /|R| 3 ), the path integral of both Yang-Mills theory and QCD factorizes into the contributions of the the individual constituent-instantons with Q = 1 [6,41], The combinatorial prefactor 1/Q! arises because the constituent-instantons can be treated as identical particles.See App.A for more details on Z 1 .In the leading-order SCI limit, the process at the bottom figure of Fig. 1 is approximated by the process shown in the top figure.However, its origin is a multi-instanton process, which according to Figure 2. The effective potential V (θ ) = ∆F SCI (θ ), defined in Eq. ( 9), for Z 1 = 0.1, 1, 3 and 6 (from dark to light red), normalized by its value at θ = π.While the potential retains its 2π-periodicity, strong anharmonicities arise with increasing Z 1 .
Eq. ( 7) gives a manifestly different contribution to the free energy as compared to corresponding single-instanton process.Furthermore, to (ρ 4 /|R| 4 ) and higher, the overlap between constituent-instantons has to be taken into account and Z Q cannot simply be related to Z 1 anymore [6].Thus, Eq. ( 8) does dot imply that multi-instantons are generally described by independent single-instantons, its simple form is merely a consequence of the choice to consider the SCI limit only up to (ρ 3 /|R| 3 ).In the leading-order SCI limit the sum over all topological charges in the free energy ( 7) can be performed analytically, Note that two different kinds of scale separations are relevant in the dilute multi-instanton gas in the SCI limit.First, the separation between multi-instantons has to be large against their effective sizes.This is what makes the gas dilute.Second, to ensure the SCI limit, the separation between the constituent-instantons within one multi-instanton has to be large against their sizes.For small Z 1 , one recovers the well-known result for This is expected since the effect of multi-instantons depends on the magnitude of the 'tunneling amplitude' Z Q .In the SCI limit, Q-instanton corrections are relevant for all Q with Q Z 1 .So for Z 1 1, the free energy is dominated by the conventional cos θ behavior, while for Z 1 1 sizable corrections to this behavior become relevant.This is illustrated in Fig. 2. Such corrections are referred to anharmonicities, since for small θ the free energy is modified with respect to the leading harmonic contribution ∼ θ 2 .As noted above, this might be misleading if one invokes an acoustic analogy, because multi-instanton contributions are overtones of the single-instanton contribution.as functions of temperature in a dilute gas of instantons of various topological charges to leading order in the limit of small constituent instantons in quenched QCD.

III. TOPOLOGICAL SUSCEPTIBILITIES
The free energy ∆F (θ ) generates moments of the topological charge distribution, the topological susceptibilities χ n , and all odd susceptibilities vanish, χ 2n+1 = 0.In the dilute instanton gas, the differences between the susceptibilities χ n are due to the the different powers of the topological charge, Q n .Hence, if only Q = 1 is taken into account, all susceptibilities are identical up to the sign, The topological susceptibilities in Eq. ( 11) resemble moments of the Poisson distribution in the SCI limit (8) and can be expressed as, where T n (z) are the Touchard polynomials defined in App.B. It is sometimes convenient to parametrize the θdependence of the free energy in terms of deviations from the second susceptibility χ 2 , Deviations from unity of the the expression in the square brackets hence are a measure for the anharmonicity of the θ -dependence of the free energy.The relations between the anharmonicity coefficients b 2n and the susceptibilities are In case only single-instantons are taken into account, it follows from Eq. ( 12) that the dilute instanton gas predicts constant values for these coefficients, For instantons of any topological charge in the SCI limit, however, one finds Thus, an explicit temperature dependence of the coefficients b n is generated by the effects of higher topological charge.
In the SCI limit it is possible to compute the free energy and all topological susceptibilities based on the knowledge of the partition function for the single-instanton, Z 1 .Here the results of [1,2,22,23,43,44] are used, with the instanton density n 1 (ρ, T ) discussed in App. A.
A. Quenched QCD Using these results, the topological susceptibilities including the effects of all topological charges can be computed from Eq. (13).For the dilute multi-instanton gas in the SCI limit, the size of the multi-instanton contributions to the θ -dependence depends on the size of Z 1 .As discussed in App.A, light quarks lead to a substantial suppression of the instanton density, so in order to study multi-instanton effects within the approximations used here, it is instructive to consider the quenched limit of QCD, i.e. using the instanton density of SU(3) Yang-Mills theory.
As shown in App.A, the renormalization group scale is set by the the instanton size relative to the renormalization scale parameter Λ. Thermal corrections to the instanton density only enter through the combination πT ρ [22,23].All scales are therefore measured relative to Λ here.Since the energy scale of thermal fluctuations is ∼ πT , the combination πT /Λ sets the relevant thermal scale.
In Fig. 3 the results for the first nonvanishing susceptibilities χ SCI 2 , −χ SCI as functions of temperature in a dilute gas of instantons of various topological charges to leading order in the limit of small constituent instantons in quenched QCD.
the effect of single-instantons, are compared to the results of dilute gases including also 2-and 3-instantons, as well as all possible Q-instantons in the SCI limit.In general, the effect of higher topological charge amplifies the temperature dependence of the susceptibilities.Within the range of temperatures considered here, the expansion of the dilute instanton gas in terms of the topological charge Q converges rapidly.This is expected since the instanton density itself is highly suppressed at large temperatures, such that higher powers become less relevant.are shown as functions of temperature in Fig. 4. As in Fig. 3, the conventional dilute single-instanton gas is compared to the result including any topological charge, as well as with the results of an expansion of the free energy up to Q = 2 and Q = 3.The temperature dependence of the anharmonicity coefficients is solely due to multi-instanton effects.Computations of b 2 on the lattice above T c show indications that, starting from the single-instanton value −1/12 at very high temperature, b 2 decreases slightly with decreasing temperature for T 2.5T c , before it starts rising towards small temperatures [32][33][34].As demonstrated here, multi-instanton corrections can explain this behavior qualitatively.However, more precise studies are required to corroborate this on the lattice.

B. QCD
As discussed after Eq. ( 10), the effects of gauge field configurations with higher topological charge in the dilute multiinstanton gas to leading order in the SCI limit depend on the size of the partition function in the presence of a singleinstanton, Z 1 .It is determined by the instanton density n 1 , which is significantly suppressed in the presence of light quarks, see App.A and in particular Fig. 8. Hence, within the approximations used here, the corrections to the leading single-instanton behavior are negligible in QCD.Explicit numerical calculations indeed show that, within the range of temperature considered here, Q-instanton corrections are on the level of 0.001% for Q = 2 and even smaller for Q ≥ 3. The present approximations are discussed critically in Sec.V.

IV. AXION COSMOLOGY
As outlined in Sec.I, the physics of axions is sensitive to the topological structure of the QCD vacuum.Higher topological charge effects on the cosmology of axions are explored here.

A. Axion effective potential
Having the θ -vacuum as the true vacuum with θ as a fundamental parameter of the theory begs the question which value is the physical one?Since θ = 0 implies C P-violation in QCD through F F , one can look corresponding processes in nature.As discussed in Sec.I, measurements of the neutron electric dipole moment put stringent lower bounds on θ , strongly suggesting that its physical value is zero.The question about the existence and nature of a physical mechanism to enforce this remains to be answered.
Among the possible resolutions of this problem, the PQ mechanism [14,45] is the one relevant for the present purposes.In this case, the standard model is augmented by an additional global chiral (axial) symmetry, U(1) PQ , and a complex scalar field (and possibly other fields which are irrelevant here) which is charged under U(1) PQ and couples to quarks.This symmetry is spontaneously broken at a scale f a , giving rise to a Goldstone boson, the axion a(x), related to the phase of the complex PQ field.The classical PQ symmetry entails a shift symmetry of a(x).Since U(1) PQ is a chiral symmetry, it is anomalous.The only non-derivative interactions of the axion are dictated by the chiral anomaly to be proportional to F F .The axion effective potential is therefore of the same form as the θ -term, and one can define an effective vacuum 'angle', Note that there is also a contribution from the finite quark masses to θ (x), but this is not relevant here since only θ (x) itself is of interest for the following discussion.For a more complete discussion, see, e.g., [46].The upshot is that the θ -angle is replaced by a dynamical field θ (x) (which will also be referred to as the axion for simplicity), so there is a physical value defined by the minimum of the axion effective potential.It follows from the discussion above that the axion effective potential V is identical to the free energy density ∆F .In the dilute multi-instanton gas Eqs. ( 7) and ( 9) yield: Hence, the superselection of the θ -parameter is avoided elegantly by effectively promoting it to a dynamical field.The effective potential in the SCI limit is shown for exemplary values for Z 1 in Fig. 2. Obviously, the vacuum expectation value is at 〈 θ 〉 = 0, which renders the QCD vacuum CPsymmetric.The topological susceptibilities computed in Sec.III can be interpreted directly as the axion mass and its higher order (non-derivative) self-interactions.

B. Vacuum realignement
The specifics of how the axion couples to the standard model besides the topological sector discussed above are model dependent.There is a class of 'invisible' axion models originating from [47][48][49][50], which are consistent with bounds from axion searches [51].These models require U(1) PQ to be spontaneously broken at a very high energy scale resulting in an axion decay constant of f a 10 9 GeV, rendering axions very light and their interactions faint.Furthermore, cold axions can be produced through a field-relaxation mechanism during the evolution of the universe, known as vacuum realignement [20,21].This makes axions viable candidates for dark matter.In the following, the possible implications of higher topological charge effects on the production of cold axions are discussed on a qualitative level.
For illustration, the simplest realization of the vacuum realignement mechanism is used.It is assumed that spontaneous PQ symmetry breaking occurs before inflation and that the reheat temperature is smaller than the temperature for PQ symmetry restoration.Note, however, that the qualitative statements made here are more general.If PQ symmetry is spontaneously broken in the very early universe, the resulting axion is essentially massless since the instanton effects that give rise to the axion mass, are negligible at T ∼ f a in this regime.Thus, the axion is strongly fluctuating around its vacuum expectation value within the range θ / f a ∈ [−π, π].In a sufficiently small patch in space right before inflation the axion field can be assumed to have a homogeneous value θ0 .Due to inflation, such a patch is blown up in size and it is possible to have a single homogeneous value θ0 for the axion within our causal horizon.Furthermore, inflation dilutes all relics from the PQ phase transition, such as topological defects, away.In the simplest realization of the vacuum realignment mechanism, one assumes that we live in one such domain.So it is assumed that the axion is homogeneous throughout the whole universe.Since fluctuations are redshifted away, it can be treated as a classical field.Thus, starting from the random initial value θ0 , called the misalignment angle, the axion evolves in time according to the classical equations of motion.For a more detailed discussion, see, e.g.[17,18].Within the standard model of cosmology, a homogeneous, isotropic, expanding universe with vanishing curvature is assumed.This is described by the Friedmann-Lemaitre-Robertson-Walker metric with the scale parameter a(t) (not to be confused with the axion).The Hubble parameter is The axion field described by the classical action has energy-momentum From this, one infers the time evolution of the homogeneous axion field according to its classical equation of motion, and the energy density of the axion, Strictly speaking, the time evolution in Eq. ( 25) is incomplete.There is an additional Friedmann equation determining the Hubble parameter, which depends on the energy density of the axion.For the timescales relevant here, the universe is to a good approximation radiation-dominated and the axions do not spoil that.Their energy density is negligible compared to the contributions of radiation to the Hubble parameter.
The time evolution of the axion field in Eq. ( 25) has a very simple heuristic interpretation if one assumes that anharmonicities in the axion effective potential are small.In this case one has ≈ m 2 a θ , cf.Eqs. ( 14) and ( 21).Naively (ignoring the explicit time dependence of H and m a ), Eq. ( 25) then has the form of a damped harmonic oscillator.Its qualitative behavior is determined by the damping ratio ζ = 3H/2m a .If the Hubble parameter dominates over the axion mass such that ζ > 1, the axion evolution is overdamped.Since the Hubble expansion is much smaller than the Compton wavelength of the axion at early times, the axion decays very slowly from the initial misalignment angle θ0 towards its vacuum expectation value.At later times, when a substantial axion mass is generated through instanton effects and the Hubble expansion slows down, the system enters the underdamped regime with ζ < 1.The axion then oscillates with decreasing amplitude around its vacuum expectation value θ = 0. Due to the strong time-dependence of H and m a this simple picture is merely suggestive.Eq. ( 25) is therefore solved numerically.However, the intuition from the damped oscillator helps to anticipate the possible effect that multiinstanton corrections have on the time evolution of the axion.Comparing the axion potential in Eq. ( 20) shown in Fig. 2 for Z 1 = 0.1 and Z 1 = 1, one sees that the anharmonicity indued by multi-instantons can lead to a flattening of the potential around the maxima.Thus, the 'frequency term' in the axion evolution equation (25), ∼ V ( θ / f a ), can become significantly smaller than the corresponding result for the effective potential induced by single-instantons.If the initial misalignment θ0 happens to be in this flattened region, the axion remains frozen for a longer time before it starts a damped oscillation around its vacuum expectation value for H 2  |V ( θ / f a )|.A similar scenario, but originating from a noncanonical kinetic term of the axions instead of topological field configurations, has been discussed in [52].Note that if Z 1 becomes large enough (e.g.Z 1 3 in Fig. 2), the anharmonic corrections in the SCI limit can induce a sign change in V ( θ / f a ) for intermediate values of θ .In this case it is possible to increase θ (t), before it starts oscillating.This possibility is commented on below.

C. Quenched-QCD axion
For the numerical solution of Eq. ( 25), it is assumed that the universe is radiation dominated.The Hubble parameter then is with the Planck mass m Pl ≈ 1.22 × 10 19 GeV.For the effective number of radiative degrees of freedom, g = 100 is chosen for simplicity, see, e.g., [53] for a more refined analysis.Converting temperature to time is done by using that radiation domination implies t(T ) = 1 2H(T ) .As for the topological susceptibilities, the axion effective potential is computed in the SCI limit in Eq. ( 20), with Z 1 discussed in App. A. As before, all scales are measured relative to Λ; the relevant time scale is Λ/πT .To highlight the effect of multi-instantons, an initial misalignment 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.Specifically, θ0 / f a = 3.14 and f a /Λ = 4 × 10 16 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. 5.The effect of the multi-instanton induced effective potential of Eq. ( 20) (solid lines) is compared to the single-instanton induced potential ∼ cos( θ / f a ) (dashed lines).For the present choice of parameters, one sees the qualitative behavior discussed above: the flattening of the potential due to the anharmonicities from multi-instanton corrections results in a longer period of overdamping, where the axion is essentially frozen at the misalignment angle.Once the Hubble expansion has slowed down sufficiently, the time evolution is determined by the curvature of the effective potential and the axion oscillates around its vacuum expectation value 〈 θ 〉 = 0 with decreasing amplitude.
Using this solution, the energy density of the axion can be computed from Eq. ( 26).The result, again for the multiand single-instanton induced potentials, is shown in Fig. 6.The energy density monotonously rises with time in the overdamped region, where it is almost exclusively due to the potential energy of the axion, and then slowly decreases in the oscillating regime at later times.In addition to the rate of decrease in the oscillating regime, the energy density today is crucially dependent on how long the axion is frozen in the overdamped regime.Since multi-instanton effects can prolong this phase and delay the start of oscillations, they can increase today's energy density of cold axions.In addition, owing to the steeper potential around the minimum, the axion field oscillates faster in the presence of multi-instanton corrections.Hence, the larger kinetic energy also leads to a larger energy density in the oscillating regime as compared to the single-instanton induced axion potential.In summary, higher topological charge effects can flatten the axion effective potential and therefore provide a topological mechanism to increase the amount of axion dark matter today.
Following the discussion after Eq. ( 10), in Sec.III B and App.A, the substantial suppression of the instanton density in the presence of light quarks leads to also a substantial suppression of multi-instanton effects for the present approximation to the topological structure of the vacuum.Hence, while the mechanism discussed above is still present in QCD, its effect could be negligible.
Note that the sign change of V seen in Fig. 2 for very large Z 1 could lead to an increase of θ (t) at intermediate times, additionally increasing also the energy density.However, Z 1 does never become this large here, so this possibility has not been further explored.

V. DISCUSSION
Before the results are summarized, a critical discussion of the assumptions and approximations made here is in order.
First, the actual size of the effects studied here obviously depends on the quantitative impact of field configurations of higher topological charge on the vacuum amplitude.A dilute gas and the (leading-order) SCI limit have been used.A dilute gas can only be valid at weak coupling for very high temperatures.Yet, multi-instanton effects are relevant if the classical suppression ∼ e −8π|Q|/g 2 becomes less strong.This is only possible away from the strict weak-coupling limit.Then, in turn, other effects, such as interactions between (multi-) instantons and (non-perturbative) quantum effects, also become increasingly relevant.Furthermore, it has been argued that the effects in QCD are small because light quarks suppress the instanton density.The dynamical generation of quark mass at lower temperatures might compensate this to some extent.To assess the relative importance of these different effects, a better understanding of the partition function in a Q-instanton background, Z Q , beyond leading order in the SCI limit is necessary.For this, the overlap between constituent-instantons has to be taken into account and Z Q cannot be approximated by single-instanton contributions [6].There is a priori no reason to assume that the the SCIlimit to leading order is sufficient to accurately describe Z Q , and is therefore the largest source of uncertainty here.
Second, it has been assumed that topological gauge field configurations at large temperatures, i.e. well within the deconfined phase, are described by instantons (or rather their finite-temperature cousins, sometimes called calorons [23]).It has been argued in [54] that the instanton picture is incompatible at large N c in the confined phase.However, neither is N c large, nor are quarks confined here.As mentioned already, the assumptions regarding the nature of topological field configurations made here are backed by numerous results for topological susceptibilities by first-principles lattice gauge theory methods.They all show that at large temperatures, T 2.5T c , the behavior of the susceptibilities with respect to T agrees with the predictions of a dilute instanton gas [28][29][30][31][32][33][34][35].Note that this is not contradicting the validity of a dilute multi-instanton gas, since, as shown here, multi-instanton corrections are small at large temperatures and might very well fit within the error bars of state-of-the-art lattice results.

VI. SUMMARY
It has been shown that gauge field configurations with higher topological charge modify the QCD vacuum.This is reflected in corrections to the conventional dependence on the CP-violating topological θ parameter.
At large temperatures well within the deconfined phase, the topological structure of QCD can be described by a dilute gas of instantons.Even though multi-instantons with topological charge Q > 1 are suppressed in the semi-classical weak-coupling limit, their contributions to the path integral are genuinely different from the single-instanton contribution.Hence, the picture of a dilute instanton gas has been generalized to include instantons of arbitrary topological charge.Note that this is very much in line with the findings in [6], where it has been shown that there are anomalous quark correlations which are only generated by multi-instantons.
In order to describe multi-instantons, the limit of small constituent-instantons has been used.There, the exact multiinstanton solutions are expanded to leading order in the limit where the sizes of the constituent-instantons are small against their separation.The QCD path integral in the background of a multi-instanton then factorizes, which facilitates analytical computations.At large temperature and with the approximations used, multi-instanton effects are most pronounced in the quenched limit of QCD, i.e. without dynamical quarks.In order to highlight these effects, mostly quenched QCD has been studied here.
There is a nice acoustics analogy regarding multiinstanton corrections to the θ -dependent free energy of QCD: they give rise to overtones to the fundamental frequency, which is set by single-instantons.In the dilute multiinstanton gas in the SCI limit, the θ -dependent free energy can be computed analytically, based on the known results for the single-instanton density.The resulting θ -dependence of QCD is reflected in the topological susceptibilities χ n .If only single-instanton effects are accounted for, all susceptibilities are proportional to the first non-vanishing susceptibility χ 2 .Higher topological charge contributions lift this 'degeneracy', and amplify the temperature dependence of these susceptibilities towards lower temperatures.This is most clearly seen in the anharmonicity coefficients b 2n ∼ χ 2n+2 /χ 2 , which are constant for single-instantons only.Multi-instantons give rise to a characteristic temperature dependence of the anharmonicity coefficients.This can be measured on the lattice.While existing results on the lowest coefficient b 2 in quenched QCD show indications of the behavior predicted here, more detailed studies are necessary at large temperatures.
An interesting application to showcase higher topological charge effects is axion cosmology.Since the axion effective potential is determined by the θ -dependence of QCD, it is also sensitive to the effects investigated here.Again, to accentuate these effects, a toy universe where axions are coupled to quenched QCD has been considered.The production of cold axion dark matter via the vacuum realignment mechanism has been studied as an example.Multi-instanton effects can flatten the axion effective potential around its maxima and, as a result, can delay the time where the evolution of the axion switches from the overdamped to the oscillating regime in an expanding, radiation-dominated universe.In addition, the axion oscillates faster.This leads to an overall increase in the energy density of axions at late times, as compared to the case where only single-instantons are taken into account.Hence, higher topological charge effects give rise to a mechanism that increases the amount of axion dark matter.
As discussed in the previous section, on the one hand, the effects studied here become very small if dynamical quarks are taken into account within the present approximations.On the other hand, multi-instanton effects can become relevant only in a regime where semi-classical and dilute approximations begin to break down, at least to leading order.Thus, a better understanding of the significance of higher topological charge effects requires a more detailed understanding of the impact of topological gauge field configurations on the vacuum amplitude of QCD.The leading-order SCI limit of a dilute multi-instanton gas used here can be viewed as a first step.Two major sources of uncertainty are the unknown vacuum amplitude in a multi-instanton background beyond the leading-order SCI limit, and neglected interactions between (multi-) instantons.

Acknowledgments -
The author is grateful to Rob Pisarski for valuable discussions and collaborations which inspired this work.

Appendix A: Instanton density
The instanton density used in Eq. ( 18), computed in [1,2,22,23,43,44], is in the modified subtraction scheme (MS): The renormalization scheme dependent constant d MS is with N c = 3 and, of course, N f = 0 for Yang-Mills.For the running coupling g(ρΛ) the two loop result both in the exponential and the pre-exponential is used [6], The in-medium corrections depend on the Debye mass at leading order, and the function is a numerical parametrization of the temperaturedependent part of the one-loop determinant in the instanton background [23,55].
The zero modes of quarks, which are generated in the presence of instantons, contribute to the instanton density through det 0 (M q ) in Eq. (A1).In the quenched limit, det 0 (M q ) = 1.In QCD it depends on the current quark mass matrix M q = diag(m u , m d , m s , m c , m b , m t ).Strictly speaking, the present analysis is only valid if M q can be viewed as a small perturbation of the Dirac operator such that the unperturbed quark-eigenmodes can be used and only the lowest eigenvalue is affected.For a more complete discussion, see [56,57].For the present purposes, a qualitative discussion is sufficient, though.Using the notation of Ref. [6], the quark zero modes related to a Q-instanton are denoted by ψ (Q) f i , where f = 1, . . ., N f is a flavor index and i = 1, . . ., Q.As the Q-instanton itself, each zero mode is characterized by Q positions, sizes and orientations in the gauge group.This is in line with an index theorem for (anti-) selfdual topological field configurations, which states that (anti-) instantons of topological charge Q are accompanied by N f Q (right-) lefthanded quark zero modes [58].It has been shown in [6] that to order ρ 3 /|R| 3 the quark zero modes ψ f i , each described by a single location, size and orientation.This property facilitates the factorization of the path integral in QCD.Hence, the quark zero mode determinant in Eq. (A1) is with respect to ψ (1) .det 0 (M q ) = det d 4 x f i ψ (1) † f i (x f i ) M f g q δ i j ψ (1) The zero modes are normalized to Since the quark mass matrix is diagonal in flavor, the quark zero mode determinant simply gives a factor det 0 (M q ) = where one can assume that all instanton sizes are the same in the SCI limit, i.e. ρ i = ρ for all i.Fig. 8 shows a comparison between Z 1 for QCD with and without dynamical quarks.In the former case, four quark flavors with m u = m d = 3 MeV, m s = 94 MeV and m c = 1.27GeV were chosen.The running of the masses and threshold effects in the running of the strong coupling have been neglected for simplicity.Z 1 is suppressed by about seven orders of magnitude if the four lightest quark flavors are taken into account.
This appendix is closed with a discussion on the spacetimevolume dependence of the partition function in the SCI limit, Eq. ( 8), and the corresponding free energy, Eq. ( 7).The path integral factorizes into independent contributions of 1-instantons in the SCI limit because the Q-instanton A (Q) (x) itself becomes a sum of well-separated 1-instantons A (1) (x − z i ), each sharply peaked around their location z i with i = 1, . . ., Q.Each 1-instanton (in singular gauge) fallsoff as ρ 2 i /|x −z i | 3 , so when these constituents are sufficiently widely separated and fluctuations around these configurations are small (as assumed in the SCI limit to leading order in the saddle point approximation), the path integral factorizes [6,39,40,59].The spacetime volume can then be decomposed into Q subvolumes i ⊂ , where each i contains the support of only one constituent-instanton A (1) (x − z i ), and Q i=1 i ⊂ .Hence, the partition function in the background of a singe Q-instanton in the SCI limit reads explicitly which demonstrates that the partition function behaves like that of Q independent subsystems.The i can in principle be computed directly from the partition function for a given A (Q) .Here i is estimated for convenience.To this end, note that one can assign an effective instanton size ρ(T ) to each constituent-instanton at any given temperature.It is given by the value of ρ at the maximum of the instanton density n 1 (ρ, T ), cf.Fig. 7. To ensure the validity of the SCI limit, the separation between constituent-instantons has to be large against their size.Hence, for a conservative estimate the largest effective instanton size is used for all temperatures.One can read-off from Fig. 7 that ρ(T ) is reduced with increasing temperature and define ρ ≡ ρ(T = 0).For the instanton density (A1) with the MS running strong coupling, one finds where Λ is the scale parameter of the renormalization scheme, see, e.g., Fig. 7.This is in agreement with estimates based on instanton liquids, e.g.[26].Since a Q-instanton behaves as ρ 2 i /|x − z i | 3 in the vicinity of z i , one can define the corresponding support by a 4-sphere of radius k ρ, with an arbitrary constant k 1.As long as the average instanton size is smaller than the separation of constituent-instantons, ρ |R|, this guarantees that the resulting i do not overlap.With this one finds The partition function for the Q-instanton in the SCI limit then becomes To make the final expression even simpler, k = 2(2/π 2 ) 1/4 ≈ 1.34 is chosen such that ¯ = Λ −4 .Since all scales are measured relative to Λ here, it is then sufficient to keep in mind that ¯ Λ 4 = 1.
It is worth emphasizing that the Q -behavior of Z Q is an artifact of our approximation.Beyond (ρ 3 /|R| 3 ) in the SCI limit, the overlap of constituent-instantons (and the corresponding quark zero modes) has to be taken into account.The factorization of the path integral, which has been exploited here, is then lost, but it is in principle possible integrate-out the separation of constituent-instantons.One is then left with one integration over the average position of the multi-instanton, which contributes as 1 to Z Q .This has been done explicitly for Q = 2 and (ρ 4 /|R| 4 ) in [6], where only the overlap between quark zero modes has to be taken into account.

5 5 m 6 z 4 e 4 0 m h 9 b 9 b 1 WFigure 1 .
Figure 1.Illustration of possible instanton configurations contributing to the vacuum amplitude + 〈2|1〉 − , which is part of the complete θ -vacuum amplitude in Eq. (3).The black sinosodial represents the vacuum, where each minimum corresponds to a different winding number ν. In-and out-states are marked by -and +, respectively.The blue lines show the vacuum transition due to instantons and the red lines the transitions due to anti-instantons.The size of the transition, i.e. the difference in winding from the starting point to the endpoint of an instanton, correspond to its topological charge.The upper figure illustrates a contribution where only single-instantons are considered.Exclusively configurations of such type are taken into account in the conventional dilute instanton gas.The lower figure shows a similar contribution, but including multi-instantons.The present analysis takes all possible configurations, including single-and multi-instantons, into account.

4 and χ SCI 6 areFigure 4 .
Figure 4. Anharmonicity coefficients b SCI 2 , b SCI 4 and b SCI 6 To study the effect of anharmonicities induced by multiinstantons, b SCI 2 , b SCI 4 and b SCI 6

Figure 5 .
Figure 5.Time evolution of the quenched-QCD axion.Multiinstanton effects delay the onset of the oscillating regime and increase the oscillation frequency.

Figure 6 .
Figure 6.Time evolution of the energy density of the quenched-QCD axion.Multi-instanton effects increase the energy density of axions at later times.