Instanton-induced effects in interquark forces, light-front wave functions and formfactors

Exclusive processes are traditionally described by perturbative hard blocks and “distribution amplitudes" (DAs), matrix elements of operators of various chiral structure and twist. One paper (with I.Zahed) calculate instanton contribution to hard blocks, which is found comparable to perturbative one in few-GeV2 Q2 region of interest. Another paper aims at comprehensive wave functions of mesons, baryons and pentaquarks. The last ones are also included as 5-quark component of the baryons. The calculation, using ’t Hooft operator, gives x-dependence and magnitude of the antiquark PDF. It explains long standing issue of strong flavor asymmetry of antiquark sea. The third paper (also with I.Zahed) is semi-review on the instanton-sphaleron processes in QCD and electroweak theories, with emphasis on their possible experimental observation via double diffractive events at LHC and RHIC. Insert your english abstract here.


Brief history
The physics of the nonperturbative vacuum of strong interaction started even before QCD. Nambu and Jona-Lasinio (NJL) [1], inspired by BCS theory of superconductivity, have qualitatively explained that strong enough attraction of quarks can break S U(N f ) a chiral symmetry spontaneously and, among many other e↵ects, create near-massless pions.
Instantons, the basis for semiclassical theory of QCD vacuum and hadrons, has been discovered in 1970's [2], and soon t'Hooft has found their fermionic zero modes and formulated his famous e↵ective Lagrangian [3]. Not only it solved the famous "U A (1) problem" -by making the ⌘ 0 non-Goldstone and heavy -but it also produces a strong attraction in the σ and ⇡ channels. In the framework of the so called instanton liquid model (ILM) [4], it provided a microscopic (QCD-based) basis for chiral symmetry breaking, chiral perturbation theory and the pion properties. Its two parameters The subject of this talk is several other uses of instantons discussed in recent works. We will consider quark pair production and their role in isospin asymmetry of the nucleon "sea", inclusion ofĪI molecules in mesonic formfactors and forces between quarks in hadrons.

The topological landscape and sphaleron production processes
There was significant development in understanding the "topological landscape" and relation between instantons and sphaleron production processes, see [6]. Recently interest to it was renewed, due to possible sphaleron production at RHIC/LHC [7]. Let us focus on two main variables, for all static ( 3-dimensional and purely magnetic) configurations of the lowest energy, consistent with the walue of those coordinates. One of the coordinates is the topological Chern-Simons number, and the other is the size squared of the field If those are kept constant, adding Lagrange multiplies to the action one can find the energy and Chern-Simons number in parametric form where  = 0 corresponds to top of the barrier, known as "sphaleron". Production of sphaleron-path states can be described semiclassically using instantonantiinstanton "streamline" [8? , 9], and their explosion in Minkowski space-time by [10]. The new discussion is about estimates of how one can produce QCD sphalerons as topologicallycharged clusters in double-di↵ractive events, with a cluster of few GeV mass at the center. Its decay modes into 3 mesons were calculated using t'Hooft Lagrangian.

Mesons and baryon light-front wave functions and isospin asymmetry of the nucleon "sea"
High energy processes, like famous electron-nucleon deep inelastic scattering, produced rich "parton phenomenology", in form of parton distribution functions (PDFs), transverse distribution functions (TDFs) etc. Those are certain matrix elements of light front wave functions (LFWFs), which, however, did not obtained much attention. Furthermore, understanding of light front Hamiltonians is only in its initial stage.
In [11] I have calculated LFWFs for several mesons and baryons, including 5-quark component of the latter related to the isospin asymmetry puzzle. Canonical process of quark pair production via gluons is "flavor blind", and yet the antiquark sea of the nucleon is very asymmetric, as shown in Fig.2 (right). As noticed in [12], the t'Hooft Lagrangian is strongly flavor asymmetric, say d quark can produceūu pair (Fig.2 left) but notdd. If it would be the only process, thed/ū ratio would be 2, as there are two u quarks and only one d in the nucleon: not far from the data at certain x.
Calculating admixture of 5-q states to the nucleon LFWF, I was able to quantify this e↵ect and get the magnitude of the asymmetry and shape of first-generation antiquarks in agreement with the data.

THE 5-QUARK SECTOR OF THE BARYONS
relating it to the observed parameters of the exclusive processes (formfactors etc) and the valence quark PDFs, and the proceed with calculation of the other sectors of the physical state.
x 1 x 1 The Hamitonian matrix element corresponding to the diagram shown in Fig9 we calculated between the nucleon and each of the pentaquark wave functions, defined above, by the following 2+4 dimensional integral over variables in 3q and 5q sectors, related by certain delta functions The meaning of the delta functions is clear from the diagram, they are of course expressed via proper integration variables and numerically approximated by narrow Gaussians. After these matrix elements are calculated, the 5-quark "tail" wave function is calculated via standard perturbation theory expression The typical value of the overlap integral itself for different pentaquark state is ∼ 10 −3 , and using for effective plotted in Fig.5, as those include large perturbative c tributions, from gluon-induced quark pair product g →ūu,dd, dominant at very small x. However, th processes are basically flavor and chirality-independe while the observed flavor and spin asymmetries of the indicate that there must also exist some nonperturbat mechanism of its formation. For a general recent rev see [18]. As originally emphasized by Dorokhov and Koche [23], The 't Hooft topology-induced 4-quark interact leads to processes relating it to the observed parameters of the exclusive processes (formfactors etc) and the valence quark PDFs, and the proceed with calculation of the other sectors of the physical state.
The only diagram in which 4-quark interaction connects the 3 and 5 quark sectors, generating theū sea.
The Hamitonian matrix element corresponding to the diagram shown in Fig9 we calculated between the nucleon and each of the pentaquark wave functions, defined above, by the following 2+4 dimensional integral over variables in 3q and 5q sectors, related by certain delta functions The meaning of the delta functions is clear from the diagram, they are of course expressed via proper integration variables and numerically approximated by narrow Gaussians. After these matrix elements are calculated, the 5-quark "tail" wave function is calculated via standard perturbation theory expression The typical value of the overlap integral itself for different pentaquark state is ∼ 10 −3 , and using for effective couplingḠ the same value as we defined for G from the nucleon, namely ∼ 17GeV 2 , one finds that admixture of several pentaquarks to the nucleon is at the level of a percent. The resulting 5-quark "tails" of the nucleon and Delta baryons calculated in this way are given in the Appendix. The normalized distribution of the 5-th body, namelyū(x), over its momentum fraction is shown in Fig.10. One can see a peak at xū ∼ 0.05, which looks a generic phenomenon. The oscillations at large xū reflect strong correlations in the wave function between quarks, as well as perhaps indicate the insufficiently large functional basis used. This part of the distribution is perhaps numerically unreliable.

VII. PERTURBATIVE AND TOPOLOGY-INDUCED ANTIQUARK SEA
The results of our calculation cannot be directly compared to the sea quark and antiquark PDFs, already FIG. 10: The distribution over thatū in its momen tion, for the Nucleon and Delta 5-quark "tails" dashed, respectively). plotted in Fig.5, as those include large perturba tributions, from gluon-induced quark pair p g →ūu,dd, dominant at very small x. Howev processes are basically flavor and chirality-ind while the observed flavor and spin asymmetries indicate that there must also exist some nonper mechanism of its formation. For a general rece see [18].
As originally emphasized by Dorokhov and [23], The 't Hooft topology-induced 4-quark in leads to processes which are forbidden by Pauli principle applie modes. Since there are two u quarks and only the proton, one expects this mechanism to prod mored thanū.
The available experimental data, for the di of the sea antiquarks distributionsd −ū (from shown in Fig.11. In this difference the symmet production should be cancelled out, and there sensitive only to a non-perturbative contributio Few comments: (i) First of all, the sign of t ence is indeed as predicted by the topological in there are more anti-d than anti-u quarks; (ii) Second, since 2-1=1, this representation of directly give us the nonperturbative antiquark tion per valence quark, e.g. that ofū. This mea be directly compared to the distribution we c from the 5-quark tail of the nucleon and Delta Fig.10. (iii) The overall shape is qualitative similar, alth calculation has a peak at xū ∼ 0.05 while the ex which creates strong flavor asymmetry of the s up to \bar d/\bar u =2 relating it to the observed parameters of the exclusive processes (formfactors etc) and the valence quark PDFs, and the proceed with calculation of the other sectors of the physical state.
The only diagram in which 4-quark interaction connects the 3 and 5 quark sectors, generating theū sea.
The Hamitonian matrix element corresponding to the diagram shown in Fig9 we calculated between the nucleon and each of the pentaquark wave functions, defined above, by the following 2+4 dimensional integral over variables in 3q and 5q sectors, related by certain delta functions N |H|5q, i =Ḡ dsdtJ(s, t)ds dt du dw J(s , t , u , w ) (33) The meaning of the delta functions is clear from the diagram, they are of course expressed via proper integration variables and numerically approximated by narrow Gaussians. After these matrix elements are calculated, the 5-quark "tail" wave function is calculated via standard perturbation theory expression The typical value of the overlap integral itself for different pentaquark state is ∼ 10 −3 , and using for effective couplingḠ the same value as we defined for G from the nucleon, namely ∼ 17GeV 2 , one finds that admixture of several pentaquarks to the nucleon is at the level of a percent. The resulting 5-quark "tails" of the nucleon and Delta baryons calculated in this way are given in the Appendix. The normalized distribution of the 5-th body, namelyū(x), over its momentum fraction is shown in Fig.10. One can see a peak at xū ∼ 0.05, which looks a generic phenomenon. The oscillations at large xū reflect strong correlations in the wave function between quarks, as well as perhaps indicate the insufficiently large functional basis used. This part of the distribution is perhaps numerically unreliable. plotted in Fig.5, as those include la tributions, from gluon-induced q g →ūu,dd, dominant at very sma processes are basically flavor and while the observed flavor and spin a indicate that there must also exist mechanism of its formation. For a see [18]. As originally emphasized by Do [23], The 't Hooft topology-induce leads to processes which are forbidden by Pauli prin modes. Since there are two u qua the proton, one expects this mecha mored thanū.
The available experimental data of the sea antiquarks distributions shown in Fig.11. In this difference production should be cancelled ou sensitive only to a non-perturbativ Few comments: (i) First of all, ence is indeed as predicted by the to there are more anti-d than anti-u q (ii) Second, since 2-1=1, this repre mixing between N and All pentaquark calculated It is hard to plot function of 4 variables… focus.
Since this integral is a flavor non-singlet quantity where the contributions from gluon splitting cancel out in the di erence, the result is essentially scale independent. Therefore, there is no scale at which the sea quarks disappear by perturbative evolution, and this flavor di erence of the antiquark distributions must be a manifestation of non-perturbative aspects of quantum chromodynamics (QCD). Despite what one may hear, the proton is never just three valence quarks and glue.

A. Deep Inelastic Scattering
The relationships between the distributions of the quarks of various flavors and experimental data are covered by essentially all textbooks in high energy and nuclear physics. Here it will be quickly reviewed to define the notation and to point out the salient features of each technique. Figure 4 illustrates the kinematics for deep inelastic lepton scattering (DIS) with an incident lepton of four momentum p and outgoing momentum p 0 and a target of four momentum P. The momentum transfer through the virtual photon is q = p − p 0 and Q 2 = −q 2 > 0. If the scattering takes place from a very light constituent of mass m carrying a fraction x of the momentum of the target, the squared invariant mass of the quark after the collision is 2 2 2 2 2 Figure 2. The ratios ofd/ū measured by the NUSEA collabor tion [12] at a scale of 54 GeV 2 and NA-51 [11] at scales of 25-3 GeV 2 . The NUSEA analysis was based on a next-to-leading orde analysis assuming the other parton distributions were well described b 11 to the observed parameters of the exclusive ormfactors etc) and the valence quark PDFs, ceed with calculation of the other sectors of l state.

THE 5-QUARK SECTOR OF THE BARYONS
only diagram in which 4-quark interaction conand 5 quark sectors, generating theū sea.
itonian matrix element corresponding to the own in Fig9 we calculated between the nueach of the pentaquark wave functions, de-, by the following 2+4 dimensional integral les in 3q and 5q sectors, related by certain ions g of the delta functions is clear from the diaare of course expressed via proper integration d numerically approximated by narrow Gausr these matrix elements are calculated, the il" wave function is calculated via standard n theory expression l value of the overlap integral itself for differuark state is ∼ 10 −3 , and using for effective the same value as we defined for G from the mely ∼ 17GeV 2 , one finds that admixture entaquarks to the nucleon is at the level of The resulting 5-quark "tails" of the nucleon baryons calculated in this way are given in ix. The normalized distribution of the 5-th lyū(x), over its momentum fraction is shown ne can see a peak at xū ∼ 0.05, which looks a nomenon. The oscillations at large xū reflect elations in the wave function between quarks, erhaps indicate the insufficiently large funcused. This part of the distribution is perhaps unreliable.

VII. PERTURBATIVE AND LOGY-INDUCED ANTIQUARK SEA
lts of our calculation cannot be directly comhe sea quark and antiquark PDFs, already plotted in Fig.5, as those include large perturbative contributions, from gluon-induced quark pair production g →ūu,dd, dominant at very small x. However, these processes are basically flavor and chirality-independent, while the observed flavor and spin asymmetries of the sea indicate that there must also exist some nonperturbative mechanism of its formation. For a general recent review see [18]. As originally emphasized by Dorokhov and Kochelev [23], The 't Hooft topology-induced 4-quark interaction leads to processes which are forbidden by Pauli principle applied to zero modes. Since there are two u quarks and only one d in the proton, one expects this mechanism to produce twice mored thanū.
The available experimental data, for the dif f erence of the sea antiquarks distributionsd −ū (from [18]) is shown in Fig.11. In this difference the symmetric gluon production should be cancelled out, and therefore it is sensitive only to a non-perturbative contributions.
Few comments: (i) First of all, the sign of the difference is indeed as predicted by the topological interaction, there are more anti-d than anti-u quarks; (ii) Second, since 2-1=1, this representation of the data directly give us the nonperturbative antiquark production per valence quark, e.g. that ofū. This means it can be directly compared to the distribution we calculated from the 5-quark tail of the nucleon and Delta baryons, Fig.10. (iii) The overall shape is qualitative similar, although our calculation has a peak at xū ∼ 0.05 while the experimen-ian matrix element corresponding to the in Fig9 we calculated between the nuof the pentaquark wave functions, dethe following 2+4 dimensional integral n 3q and 5q sectors, related by certain the delta functions is clear from the diaf course expressed via proper integration merically approximated by narrow Gausese matrix elements are calculated, the ave function is calculated via standard eory expression e of the overlap integral itself for differstate is ∼ 10 −3 , and using for effective same value as we defined for G from the y ∼ 17GeV 2 , one finds that admixture quarks to the nucleon is at the level of resulting 5-quark "tails" of the nucleon ons calculated in this way are given in The normalized distribution of the 5-th (x), over its momentum fraction is shown an see a peak at xū ∼ 0.05, which looks a enon. The oscillations at large xū reflect ons in the wave function between quarks, ps indicate the insufficiently large func-. This part of the distribution is perhaps eliable.
. PERTURBATIVE AND Y-INDUCED ANTIQUARK SEA f our calculation cannot be directly comea quark and antiquark PDFs, already FIG. 10: The distribution over thatū in its momentum fraction, for the Nucleon and Delta 5-quark "tails" (solid and dashed, respectively). plotted in Fig.5, as those include large perturbative contributions, from gluon-induced quark pair production g →ūu,dd, dominant at very small x. However, these processes are basically flavor and chirality-independent, while the observed flavor and spin asymmetries of the sea indicate that there must also exist some nonperturbative mechanism of its formation. For a general recent review see [18]. As originally emphasized by Dorokhov and Kochelev [23], The 't Hooft topology-induced 4-quark interaction leads to processes which are forbidden by Pauli principle applied to zero modes. Since there are two u quarks and only one d in the proton, one expects this mechanism to produce twice mored thanū.
The available experimental data, for the dif f erence of the sea antiquarks distributionsd −ū (from [18]) is shown in Fig.11. In this difference the symmetric gluon production should be cancelled out, and therefore it is sensitive only to a non-perturbative contributions.
Few comments: (i) First of all, the sign of the difference is indeed as predicted by the topological interaction, there are more anti-d than anti-u quarks; (ii) Second, since 2-1=1, this representation of the data directly give us the nonperturbative antiquark production per valence quark, e.g. that ofū. This means it can be directly compared to the distribution we calculated from the 5-quark tail of the nucleon and Delta baryons, Fig.10. (iii) The overall shape is qualitative similar, although our calculation has a peak at xū ∼ 0.05 while the experimen-which creates strong flavor asymmetry of the sea up to \bar d/\bar u =2 G he only diagram in which 4-quark interaction con-3 and 5 quark sectors, generating theū sea.
mitonian matrix element corresponding to the shown in Fig9 we calculated between the nueach of the pentaquark wave functions, deve, by the following 2+4 dimensional integral ables in 3q and 5q sectors, related by certain ctions ing of the delta functions is clear from the diay are of course expressed via proper integration and numerically approximated by narrow Gausfter these matrix elements are calculated, the tail" wave function is calculated via standard ion theory expression al value of the overlap integral itself for differquark state is ∼ 10 −3 , and using for effective Ḡ the same value as we defined for G from the namely ∼ 17GeV 2 , one finds that admixture l pentaquarks to the nucleon is at the level of . The resulting 5-quark "tails" of the nucleon a baryons calculated in this way are given in ndix. The normalized distribution of the 5-th elyū(x), over its momentum fraction is shown One can see a peak at xū ∼ 0.05, which looks a henomenon. The oscillations at large xū reflect rrelations in the wave function between quarks, perhaps indicate the insufficiently large funcis used. This part of the distribution is perhaps lly unreliable.

VII. PERTURBATIVE AND OLOGY-INDUCED ANTIQUARK SEA
ults of our calculation cannot be directly comthe sea quark and antiquark PDFs, already FIG. 10: The distribution over thatū in its momentum fraction, for the Nucleon and Delta 5-quark "tails" (solid and dashed, respectively). plotted in Fig.5, as those include large perturbative contributions, from gluon-induced quark pair production g →ūu,dd, dominant at very small x. However, these processes are basically flavor and chirality-independent, while the observed flavor and spin asymmetries of the sea indicate that there must also exist some nonperturbative mechanism of its formation. For a general recent review see [18].
As originally emphasized by Dorokhov and Kochelev [23], The 't Hooft topology-induced 4-quark interaction leads to processes which are forbidden by Pauli principle applied to zero modes. Since there are two u quarks and only one d in the proton, one expects this mechanism to produce twice mored thanū.
The available experimental data, for the dif f erence of the sea antiquarks distributionsd −ū (from [18]) is shown in Fig.11. In this difference the symmetric gluon production should be cancelled out, and therefore it is sensitive only to a non-perturbative contributions.
Few comments: (i) First of all, the sign of the difference is indeed as predicted by the topological interaction, there are more anti-d than anti-u quarks; (ii) Second, since 2-1=1, this representation of the data directly give us the nonperturbative antiquark production per valence quark, e.g. that ofū. This means it can be directly compared to the distribution we calculated from the 5-quark tail of the nucleon and Delta baryons, Fig.10. (iii) The overall shape is qualitative similar, although our calculation has a peak at xū ∼ 0.05 while the experimen-mixing between N and All pentaquark calculated It is hard to plot function of 4 variables… 3 focus.
Since this integral is a flavor non-singlet quantity where the contributions from gluon splitting cancel out in the di erence, the result is essentially scale independent. Therefore, there is no scale at which the sea quarks disappear by perturbative evolution, and this flavor di erence of the antiquark distributions must be a manifestation of non-perturbative aspects of quantum chromodynamics (QCD). Despite what one may hear, the proton is never just three valence quarks and glue.

A. Deep Inelastic Scattering
The relationships between the distributions of the quarks of various flavors and experimental data are covered by essentially all textbooks in high energy and nuclear physics. Here it will be quickly reviewed to define the notation and to point out the salient features of each technique. Figure 4 illustrates the kinematics for deep inelastic lepton scattering (DIS) with an incident lepton of four momentum p and outgoing momentum p 0 and a target of four momentum P. The momentum transfer through the virtual photon is q = p − p 0 and Q 2 = −q 2 > 0. If the scattering takes place from a very light constituent of mass m carrying a fraction x of the momentum of the target, the squared invariant mass of the quark after the collision is (xP + q) 2 = x 2 P 2 − Q 2 + 2 x P · q ⇡ m 2 ⇡ 0 and the momentum fraction x = Q 2 /(2 P · q). Intuitively (at least to some) if one considers the target in a fast moving reference frame, the lifetime of each virtual state of the target is Lorentz dilated and the longitudinal extent of the target is Lorentz contracted so that a hard (large energy and momentum transfer) probe sees a collection of quarks that is frozen in time with the probability distribution f(x) for each flavor and interacts with the appropriate electro-weak cross section.
It can be proven for deep inelastic scattering that the cross section factorizes. (For details, see the discussion for example in Ref. [18]. ) where the sum is over all quark flavors and glue. The C i are hard scattering functions that are ultraviolet and infrared safe and calculable in perturbation theory. They are a function of quark flavor, the physical process (for example the nature of the vector boson being exchanged in DIS and the order of perturbation theory of the calculation), the renormalization scale µ 2 , the factorization scale µ 2 f and the strong coupling constant ↵s, but not the distribution of partons. The renormalization scheme eliminates the ultraviolet divergences of the hard scattering amplitude. The parton distributions for each flavor i, φ i/h , contain all the infrared sensitivity, are specific to the particular hadron, h, and depend on the factorization scale µf and the factorization scheme, but do not depend on the hard scattering process. If defined consistently, they are universal. Therefore, one can combine data from di erent kinds of exper- Figure 2. The ratios ofd/ū measured by the NUSEA collaboration [12] at a scale of 54 GeV 2 and NA-51 [11] at scales of 25-30 GeV 2 . The NUSEA analysis was based on a next-to-leading order analysis assuming the other parton distributions were well described by contemporaneous global fits( [13] [14]) and that nuclear corrections for deuterium are small. The curves are next-to-leading order global fits of CTEQ6, CTEQ10 [16] and CTEQ14 [17] in MS renormalization scheme, all at scales of 54 GeV 2 , to show how the parameterizations have changed over time, especially in the unmeasured region.
scale defines the separation of short-distance and long-distance e ects. By convention, one often sets the renormalization and factorization scales in deep inelastic scattering to Q 2 , but that is not necessary. For hadron-hadron reactions, one must integrate over the parton distributions of both the beam and target, but the separation of the hard scattering functions from the parton distributions remains. In such reactions, the choice of renormalization and factorization scales to be used is less obvious. In all cases, since the hard scattering functions depend on the order of perturbation theory, the scheme and scales, the parton distributions are not directly physical observables. It is not consistent to use parton distributions obtained from, for example, fits using next to leading order hard scattering functions in calculations done at a di erent order or to directly compare various parton distribution functions when they are not defined consistently.
Another important feature of QCD is that if the factorization and renormalization scales are taken as µ 2 = µ 2 f = Q 2 , then QCD allows one to calculate the parton distributions at higher Q 2 . This is the DGLAP QCD evolution of Dokshitzer [19], Gribov, Lipatov [20], Altarelli and Parisi [21].

"Dense instanton ensemble"
The original instanton liquid model focused on quark condensate and therefore relatively stand-alone instantons, whose zero modes gets collectivized. But instanton ensemble also includes close instanton-antiinstanton pairs, as it is also observed on the lattice. The stand-alone instantons are seen via "deep cooling" , during which the instanton-antiinstanton molecules get annihilated. In so far, the application of the "molecular component" of the semiclassical ensemble was made only in connection to phase transitions in hot/dense matter. Indeed, this component is the only one which survives at temperatures T > T c , where chiral symmetry is restored. Account for both components together started with [13]. The "molecular component" was also shown to be important at high baryonic densities, where it contributes to quark pairing and color superconductivity [14]. CloseĪI pairs has been qualitatively studied in recent work in [15] which studied their evolution during cooling, see Fig.3. Extrapolated to zero cooling (left side of the plots) one sees that while the instanton size fits previous expectations (1), the density seems to actually be significantly larger.   Tab. 2). The error bars for the instanton size show the width of the distribution rather than the standard deviation of the mean. For the density, the error bars incorporate an estimate of the systematic uncertainty associated to the distance filter discussed in the text.
when the density drops. We have checked that, if the Gradient flow is driven by the Iwasaki gauge action, the observed evolution does not differ from the one described above.

Momentum running, flow evolution and flavor effects
As we have discussed in Sec. 2, the running of two-and three-gluon Green's functions can be combined to yield a strong coupling definition, the estimate of which, computed from the gauge fields for a semiclassical multi-instanton ensemble leaves us with a very striking signature for the presence of instantons. In the following, we will use this coupling obtained from the lattice gauge fields to detect instantons by analyzing its running both at largeand small-momenta, as explained in Ref. [29], for all the set-up's of Tab. 1. We will then compare the obtained results with the ones stemming from instanton localization for the quenched case at = 4.2 derived in the previous section, and will also study their evolution with the Gradient flow time. Furthermore, we will take advantage of the other lattice -15 - Now, as Eq. (2.11) tells, the IR running of αMOM(k) is basically determined by the instanton density and, thus, a direct comparison of the results of the low-momenta fits shown in Fig. 6 makes possible the scrutiny of the evolution of this density with the number of flavors. Indeed, it suggests that the instanton density increases monotonously with the number of dynamical flavors. If we furthermore accept that the ratio δρ 2 /ρ 2 is the same for Nf = 0, Nf = 2 + 1 and Nf = 2 + 1 + 1, we obtain that the instanton density increases by a factor 1.3(1) from Nf = 0 to Nf = 2+1 and by a factor 1.5(1) from Nf = 0 to Nf = 2+1+1. These two ratios are nearly compatible within the errors and this suggests a very mild effect of the charm quark on the instanton density, certainly owing to its sizeable mass threshold. The role of the charm quark and the dependence on the quark masses has not been studied in detail yet but the fact that the instanton density grows with the presence of dynamical flavors has been previously reported, for instance, in Ref. [11]. In the previous reference it was interpreted as follows: it is well known that an isolated instanton produces a zero mode of the fermion determinant [11,48] and, as a consequence of this and the action of a given gauge field being weighted by [det(/ D + m)] NF , the existence of zero modes of the Dirac operator would be highly improbable in the chiral limit. Becoming more improbable the larger is the number of light fermion flavors. Thereupon, a large density of instantons can be understood as favoring the instanton superposition and thus suppressing the effect of isolated instantons. Or, in other words, it can be thought to enhance non-linear effects and pseudo-particle interactions destroying the zero-modes associated to the single instanton -20 - down at larger momentum transfer. This section also includes a subsection I C with a brief introduction to instanton e ects and related parameters.
Since the paper contains a lot of technical details, not so important for a first reading, we decided to collect all the results for the pion and rho meson in section II. The actual calculations start from the perturbative ones in section III A. Part of it is well known but it also has new contributions from meson's chirally non-diagonal contribution proportional to 2 /Q 2 , for the exact definition see (101). As discussed in subsection III E, these results can be generalized to a large set of e ective 4-quark scattering operators, as a substitute for one-gluon exchange. A simple warm-up calculation of this kind consists in taking the nton field are y only).

Scalar Form Factors of the Pseudoscalar Mesons
One can think of point-like scalar quantum, hitting one of the quarks with momentum transfer q µ to be the Higgs boson. If so, the corresponding couplings are Yukawa couplings λ q of the standard model: but, of course, it is unimportant for form factors. (For example, lattice groups use for convenience λ u = 1, λ d = 0.).
The amplitude of elastic scattering of a virtual scalar on a pion, with a perturbative one-gluon exchange between quarks, leads to the following scattering amplitude: Note that scalar amplitudes have negative overall sign, which does not matter as couplings λ q are arbitrary. This sign of course does not affect contribution to the form factor, the square bracket.
we calculated photon, scalar, graviton and for pion, rho and scalar a_0 (brother o they are mostly dominated by diagram c (N or just strong instanton fields, not zero

Hard and Semihard Exclus
and is therefore The function G V is avera Shuryak and Zahed (2020). T the other one coincides with The contribution of the z factor is zero This contribution vanishes a The summary plot of th flat distributions φ π (x) =φ circles) is the sum of both ch The instanton Born-style instanton diluteness paramet amplitude does not really c are V c , V d , and it is not show The instanton-induced V π c magnitude but has a differen the pion form factor for cor experimental data at the low parameters were specially tu diagram w 10.6 Hard and Semiha and is therefore The function GV Shuryak and Zahed ( the other one coincid The contribution o factor is zero

This contribution van
The summary plo flat distributions φπ ( circles) is the sum of The instanton Bo instanton diluteness p amplitude does not are Vc, Vd , and it is n The instanton-indu magnitude but has a d the pion form factor experimental data at parameters were spec diagram co one-gluon exchange instantons Figure 4. Left: two diagrams with quark propagators in the instanton fields. Right: vector formfactor of the pion Q 2 F ⇡ (Q 2 ). Open points correspond to some experimental data points, thin curve on the left is standard dipole fit. The dotted line on the top is the sum of perturbative and instanton contributions

Formfactors: including instantons in hard block
More recently, we have explored non-perturbative contributions to the "hard block" of mesonic form factors [16]. We calculated photon, scalar, graviton and dilaton FFs for pion, rho and scalar a 0 (brother of ⌘?). The field at the instanton center is rather strong, with comparable to Q 2 is the so called semi-hard region studied. One example, which was studied a lot experimentally, is electromagnetic pion formfactor, shown in Fig.4 right. It was found that they are mostly dominated by the diagram with three propagator made of non-zero Dirac modes (the left one on the left), not the one with Dirac zero modes. Therefore we included contributions of "dense instanton liquid" , taking the diluteness parameter  = ⇡ 2 n I+Ī ⇢ 4 to one, in plots like Fig. 4 right. As one can see, in this version the sum of perturbative and instanton-induced formfactor reproduce the data.
ged over the instanton size distribution, as explained in here is only a single integral over distributionφ, since the normalization and is just 1. ero mode ('t Hooft vertex) part to the vector pion form (e u + e d ) (10.35) fter the x integration is carried. e vector pion form factor is shown in Fig. 10.9, for π (x) = 1. The perturbative contributions V π a (closed iral structures of the pion density matrices. contributions to V π b is relatively close to V π a if the er κ = 1. To avoid misunderstanding, we note that V b onstitute a consistent account for instanton effects, as n in the summary plot. (squares) at κ = 1 is comparable to perturbative V π a in t dependence on Q 2 . Taken together (dots) they predict responding values of Q 2 , reasonably well joining the er end. We remind the reader that it is not a fit: no ned for this to happen.

ith 3 non-zero mode propagators
rd Exclusive Processes 261 is averaged over the instanton size distribution, as explained in 2020). There is only a single integral over distributionφ, since es with the normalization and is just 1. f the zero mode ('t Hooft vertex) part to the vector pion form ishes after the x integration is carried. t of the vector pion form factor is shown in Fig. 10.9, for x) =φπ (x) = 1. The perturbative contributions V π a (closed both chiral structures of the pion density matrices. rn-style contributions to V π b is relatively close to V π a if the arameter κ = 1. To avoid misunderstanding, we note that Vb really constitute a consistent account for instanton effects, as ot shown in the summary plot. ced V π c (squares) at κ = 1 is comparable to perturbative V π a in ifferent dependence on Q 2 . Taken together (dots) they predict for corresponding values of Q 2 , reasonably well joining the the lower end. We remind the reader that it is not a fit: no ially tuned for this to happen.

ntaining zero mode propagators (t'Hooft vertex)
The expressions for perturbative, non-zero-mode and zero mode parts are Next-twist distribution amplitudes ' P ⇡ , ' T ⇡ are associated with pseudoscalar γ 5 and tensor σ µ⌫ ones, the leading twist ' ⇡ is standard matrix element of the axial current γ µ γ 5 . For the plot all three are taken to be just constant, independent on x. (Therefore all contributions of the tensor DA, appearing as a derivative over x, becomes actually vanish.)

Instanton-induced inter-quark forces
Historically, hadronic spectroscopy got to solid foundation in 1970's, with discoveries of nonrelativistic quarkonia made of heavy c, b quarks. In the first approximation, those are well described by simple Cornell potential which correctly attributes the short-distance potential to perturbative gluon exchange, and its large distance O(r) contribution to the tension of a confining flux tube (the QCD string). One of the issues to be discussed in this paper is the nonperturbative origin of the inter-quark interaction at intermediate distances r ⇠ 0.2 − 0.5 fm.
Later developments [17,18] connected interquark potential to correlator of Wilson lines (central) Spin-dependent forces were related to such correlators with two magnetic fields (V S S , V tensor ), or magnetic and electric fields for spin-orbit one. To evaluate such nonlocal quantities one needs to use lattice simulation, or rely on certain model of the vacuum fields. In Fig.5 (left) we show "dense instanton model" to the central potential, compared to linear potential and its version from [19] including string quantum vibrations (resummed "Lusher terms"). One can see that instantons can comlpement the flux tube at intermediate distances.
One problem with electric flux tube model is that it does not provide magnetic fields, while instantons are self-dual and have B needed for spin forces. We calculated the instanton contributions to spin forces, see V S S comparison of lattice potential, perturbative and bservation is that these splittings grow along the formulae given above results are given in th One conclusion is basically in agreemen raw) is dramatically something important The phenomenolo monia to heavy-light  formulae given above, and then calculated their matrix elements at the 1P she results are given in the Table III. One conclusion is that "theory" (the sum of the last two raws) and the "latti basically in agreement. The other is that their disagreement with experiment (th raw) is dramatically growing for light quark systems. This implies that for light something important is missed, in the theory as it is developed up to this point.

D. Brief summary of the section
The phenomenological direction of this section has aimed at mesons, from heav monia to heavy-light and to light-light ones. Using data for spin splittings in 1S FIG. 4. Perturbative (upper plot) and instanton-induced (lower plot) spin-dependent potentials for charmonium. The black solid, blue dashed and red dash-dotted lines are for r 2 V SS , r 2 V SL , r 2 V T , in GeV 1 , versus r in n GeV 1 .

B. Lattice studies
Four decades later, after famous quarkonia discoveries and the formulation of perturbative quarkonium theory, static spin-spin potentials have been evaluated on the lattice, using correlators of Wilson lines with explicit field strengths. Koma and Koma found [19] that while V SS is indeed rather short range, it does not fit to vector+scalar exchange paradigm: a pseudoscalar glueball exchange has been proposed. Let us jump another  formulae given above, and then calculated their matrix elements at the 1P shell. The results are given in the Table III. One conclusion is that "theory" (the sum of the last two raws) and the "lattice" are basically in agreement. The other is that their disagreement with experiment (the upper raw) is dramatically growing for light quark systems. This implies that for light quarks something important is missed, in the theory as it is developed up to this point.

D. Brief summary of the section
The phenomenological direction of this section has aimed at mesons, from heavy charmonia to heavy-light and to light-light ones. Using data for spin splittings in 1S shell (2 14 distance squared r 2 V SS (GeV −1 ) versus ice measurements [20]. The dash-dotted ial (??) with δ = 0.6 GeV −1 . The blue ons with J = 1 and J = 0. The first ounded to 1 M eV ). The second gives ), the next two are those for (regulated) we also studied splittings of 1P states h,chi_0,chi_1,chi_2 and calculated matrix elements of VSS,VSL,VT also massless pion is due to zero modes (t' Hooft Lagrangian) correct mass of rho meson needs "molecular forces" to be included Figure 5. The central potential V c (left) and spin-spin r 2 V S S (r) (right) versus distance r, (GeV −1 ). The lattice result is one of the parameterization given in [20], the perturbative is Laplacian of (regulated) Coulomb term.
instanton-induced in Fig.5 (right). Note that the area below the perturbative and instantoninduced terms is comparable. The corresponding matrix elements (using Cornell wave functions with proper quark masses) is shown in the table. One can see that e.g. for charmonium the magntitude of spin-spin term is in agreement with lattice and the level splitting. We also considered L = 1 families of mesons, from heavy to light, and considered other spindependent potentials. We also discussedĪI molecules, which provide somewhat di↵erent potentials due to di↵erent pictures of their fields. e data for Vss and l, except in light-light mesons ntial forcc system multiplied by distance squared r 2 V SS (GeV −1 ) versus e is the exponential fit (19) to lattice measurements [20]. The dash-dotted d laplacian of the Coulomb potential (??) with δ = 0.6 GeV −1 . The blue stanton contribution.
" splittings of certain L = 0 mesons with J = 1 and J = 0. The first the experimental values (MeV) (rounded to 1 M eV ). The second gives lattice-based spin-spin potential (19), the next two are those for (regulated) -induced spin-spin forces. , and then calculated their matrix elements at the 1P shell. The e Table III. that "theory" (the sum of the last two raws) and the "lattice" are t. The other is that their disagreement with experiment (the upper growing for light quark systems. This implies that for light quarks is missed, in the theory as it is developed up to this point. f rho meson needs "molecular forces" to be included