Medium evolution of a static quark-antiquark pair in the large $N_{c}$ limit

We study the transitions between the different color states of a static quark-antiquark pair, singlet and octet, in a thermal medium. This is done non-perturbatively exploiting the infinite mass limit of QCD. This study is interesting because it can be used for future developments within the framework of Effective Field Theories (EFTs) and because it can be combined with other techniques, like lattice QCD or AdS/CFT, to gain non-perturbative information about the evolution of quarkonium in a medium. We also study the obtained expressions in the large $N_{c}$ limit. This allows us to learn lessons that are useful to simplify phenomenological models of quarkonium in a plasma.


I. INTRODUCTION
Heavy quarkonium suppression was proposed in [1] as a probe of the formation of a quark-gluon plasma. Since then, it has been intensively studied, both theoretically and experimentally. Quarkonium suppression has been observed at LHC and RHIC (see [2] for a recent review). However, the original goal of using quarkonium to obtain relevant information about the medium is not yet completely fulfilled.
On the theoretical side, three main mechanisms have been identified as relevant to understand the suppression pattern of quarkonium. Screening, thermally induced decay into octets and regeneration. Screening is the original mechanism studied in [1]. In a deconfined medium, static chromoelectric fields are screened at distances of the order or bigger than the Debye radius. This implies that, if the typical size of a particular bound state is much bigger than the Debye radius, then the bound state can not exist in such a medium. Another mechanism that can produce the dissociation of quarkonium is the existence of a thermally induced decay width. This process started to gain attention after the computation in [3].
The mechanism responsible for this imaginary part is the medium induced transition from color singlet to color octet [4]. Since then, many works have studied this mechanism and have identified it as important in a wide range of temperature regimes [4][5][6][7]. Finally, recombination [8,9] is a process that can counterbalance suppression in which two initially uncorrelated heavy quarks recombine inside the medium to form a new bound state. This * miguelangel.escobedo@usc.es effect might be relevant in charmonium since it is abundantly produced in heavy-ion collisions at LHC. Indeed, some models need the inclusion of this effect in order to reproduce experimental data [10,11].
A theoretical framework in which all these effects can be included in a consistent way is that of open quantum systems [12][13][14][15][16][17][18][19][20][21][22][23]. The heavy-quark pair is treated as an open system interacting with an environment formed by light quarks and gluons. The reduced density matrix of heavy quarks evolves following a master equation, which in the Markovian limit takes the form of a GKSL equation [24,25]. Examples of phenomenological applications of this approach which compare with experimental data can be found in [15,16,26].
At the moment, most of the derivations of the master equation have been done using perturbative arguments [13,19,[21][22][23] or assuming that all the thermal scales are much smaller than 1 r [15,16] (being r the typical radius). In this article, we would like to contribute towards a non-perturbative understanding by studying the problem in the static limit. This limit has already been used in the past to clarify some of the features of the framework of open quantum systems [16,22]. We would like to note also that, in the T = 0 case, using the EFT potential non-relativistic QCD (pNRQCD) [27][28][29] it is possible to use the potential computed in the static limit as an ingredient for computations in which the heavy quark mass is keep finite. In this paper we will argue that this is also true at finite temperature, although we will not prove it rigorously. We leave that for future developments within the EFT framework.
Note that our derivation in the static limit is valid for any relation between the particles' typical energy in the medium and the separation between the heavy quarks. Neither any relation between the time scales present in the problem is assumed. It is only in section III, in order to connect with heavy quarkonium physics, that this kind of assumptions is needed.
On top of this, we will also use the large N c limit in order to gain insight into the non-perturbative expressions that we obtained. This will allow us to gain some useful phenomenological insights, mainly the following: • Heavy quarks in an octet state evolve in time as uncorrelated particles.
• The rate of decay of an octet into a singlet is suppressed by a power of 1 • The survival probability of a singlet can be described with a modification of the effective action.
The outline of the article is as follows. In section II we discuss the conditional probabilities to find a static quark-antiquark pair in a given color state at time t provided that we know the color state at some other time t 0 . In section III, we argue how these results are relevant to the physics of heavy quarkonium. In section IV, we discuss what can be deduced from the use of the Fierz identity and the large N c limit. An alternative graphical representation that is useful to study the classical limit is given in section V. Next, in section VI, we discuss whether previous computations in the literature agree on the large N c limit with what we discussed on section IV. Finally, we give our conclusions in section VII.

II. CONDITIONAL PROBABILITIES IN THE STATIC LIMIT
As a starting point, we consider non-relativistic QCD (NRQCD) [30,31] in the limit in which the heavy quarks have an infinite mass where L QCD is the QCD Lagrangian including as degrees of freedom just gluons and light quarks, ψ is the field that annihilates an infinitely heavy quark and χ is the field that creates an infinitely heavy antiquark. With these fields, we try to create a static quarkantiquark pair that transforms as a color singlet. Taking guidance from the interpolating fields discussed in [29], we use where φ(y, Note, however, that the field S is just a combination of NRQCD operators and not the singlet field of pNRQCD. The later can be related under certain conditions and approximations with the field defined in eq. (2) by the matching procedure. The normalization is chosen such that if we have a state (in the Heisenberg picture) where |Ω is a normalized state of the theory without heavy quarks, then M|M = 1 always that d 3 R d 3 r|Ψ(R, r)| 2 = 1 is fulfilled. We can proceed similarly in the case of an Now the normalization is chosen such that if we have a state where |Ω A is a state of the theory without heavy quarks chosen such that |M ′ fulfills the Gauss's law and with the property The probability of finding the static pair in a singlet configuration is Equivalently, for the octet we have Using the Fierz identity, we can prove that The right-hand side operator leaves unchanged states which have one heavy quark and one heavy antiquark. Using this, we can prove that, in a suitably normalized state, it is fulfilled that We can generalize the results slightly by introducing the reduced density matrix projected respectively to the singlet or the octet state where W AB is a Wilson line in the adjoint representation Now let us look at eq. (1), we can introduce the field redefinitions where . Using this we can rewrite The tilde fields do not interact with the rest of the particles and have a trivial equation of motion. We can use this to conclude that Applying this formula to the singlet and octet field, we get and A word of caution regarding the time ordering is needed. We are going to use the Schwinger-Keldysh contour in the following derivation. In this framework, the correct time ordering is implemented according to the branch of the contour to which the field belongs. The order of the operators in the formulas illustrates how the color indices are multiplied. Now we can use the Fierz identity to derive the following relation Using this, we get that and where and These equations are more intuitively written in the graphical notation introduced in the appendix A. In fig. 1, we show the equation that corresponds to the singlet while in fig. 2, we do the same for the octet. Now let us introduce a state which contains a static quark pair in a singlet state and an arbitrary configuration of light quarks and gluons (such that the Gauss's law is fulfilled) where ρ l represents the state of light quarks and gluons. The meaning of D s is a generalization of the wave function we used in eq. (4) In this last equation, we have introduced a new notation that we are going to use now on. P s (R, r, t|ρ) is the conditional probability to find a singlet at a given time and position provided that the system is in a state ρ. Now on we name ρ s (without the tilde) a state such that D o (t 0 ) = 0 and ρ o a state such that D s (t 0 ) = 0. In other words, we consider states for which we know that at a given time, the heavy quarks are in a singlet (octet) state.
We can make a similar construction for the octet Then Note that in eq. (28) we have introduced a new matrix calledρ l . This is a state of light quarks and gluons such thatρ o fulfills Gauss's law. An approximation that is usually implicit when doing perturbative computations of quarkonium in a medium isρ l = ρ l . Imagine that we start with a generic ρ l , and we constructρ l by adding to ρ l an arbitrary gluonic configuration such thatρ o fulfills Gauss's law. Then, we can imagine that if the environment is big enough Tr(Aρ l ) ∼ Tr(Aρ l ) for any operator A. However, this might be problematic for non-perturbative computations. Therefore, we mark in our formulas the difference between ρ l andρ l . In fact, the observation thatρ l = ρ l in general is related with the fact that it is difficult to compute the octet potential in lattice QCD [33,34]. The octet free energy in lattice QCD is studied in [35].
To compute the conditional probability to find a singlet state at time t > t 0 provided that we know that at time t 0 the pair is also in a singlet state, we apply eq. (20) to get We find it convenient to represent this equation graphically in the way shown in fig  Analogously, we apply eq. (21) to get which can be represented graphically as done fig. 4.
Equivalently, we get the following results for the case in which we know that the pair is in an octet state at t 0 . The conditional probability of having a singlet at time t is which we represent graphically in fig. 5.
For the probability to find an octet we have 6. Graphical representation of the transition of an octet at time t 0 to an octet at time t.
which we represent graphically in fig. 6.
We focused on the study of conditional probabilities in this section. It is straight-forward to generalize these results to the study of D s,o (R, r, R ′ , r ′ ; t) given that we know that the system was in a singlet (or octet) state at time t 0 < t. However, this would have made the notation more cumbersome. Nevertheless, we do present results for r = r ′ in section V and in appendix B.

III. CONNECTION WITH THE PHYSICS OF HEAVY QUARKONIUM
In the rest of the sections of this paper, we focus on studying the static limit (m → ∞).
However, in this section, we argue that our results are also useful to describe bound states of heavy quarks with finite mass. The connection between the static potential and heavy quarkonium at T = 0 can be rigorously proven using pNRQCD. Non-relativistic EFTs exploits the separation between the scales relevant for quarkonium physics. They are the mass of heavy quarkonium m, the inverse of the typical radius 1 r (which scales like mv, with v ≪ 1) and the binding energy E (of order mv 2 ).
pNRQCD is the EFT that is obtained after integrating out the scales m and mv. The Lagrangian of pNRQCD can be ordered as an infinite sum of operators multiplied by Wilson coefficients with different powers of m and r. However, given a desired precision, there exists a power counting that ensures that only a finite number of operators needs to be taken into account. The heavy quarkonium potential is a Wilson coefficient of pNRQCD which, according to the power counting, needs to be taken into account at leading order.
The heavy quarkonium potential of pNRQCD is indeed equal to the static potential. This can be shown by noting that the matching to obtain the Wilson coefficients can be done in the limit m → ∞. Once this is done, and since every pNRQCD operator has a specific power of m, the matched potential can be used in any computation with a finite value of m.
More details on this procedure can be found in [29].
We will try to give here a more physical and intuitive idea of why this is possible. Static quarkonium is characterized by the fact that the positions of the heavy quark and the antiquark do not change in time. In the case of real quarkonium, the position of the heavy quarks will look almost static if we are interested in what happens at time scales much smaller than 1 mv 2 . The heavy quarkonium potential is obtained after integrating out the scale 1 r . Two types of particles are important in this regime: • Soft particles, with momentum and energy scaling like mv.
• Potential particles, with momentum of order mv and energy of order mv 2 .
From the point of view of soft particles, the heavy quarks are essentially frozen. Regarding potential particles, in most gauges, they are related to instantaneous interactions. The reason is that we can approximate their propagator with the limit of zero energy. The conclusion is that, for the physics that is encoded in the Wilson coefficients, the heavy quark and the antiquark do look like static particles. Now let us look at how we could generalize this result to the case 1 r ∼ T , where T is the temperature. Note that the case 1 r ≫ T was already studied in [15,16]. A rigorous way to do this would be to define a version of pNRQCD valid in this situation. However, since we know that there is an imaginary part of the potential, the effects of the scale 1 r ∼ T can not be encoded in an effective action because of the existence of dissipative effects. We would need a different type of EFT in which we consider operators that mix fields in different branches of the Schwinger-Keldysh contour (examples of this type of EFTs are found when studying hydrodynamics, see [36] and references therein).
This version of pNRQCD is still unknown, and its construction is out of the scope of this work. However, we can still use the intuitive and physical arguments related to the time scales. In order for a non-relativistic bound state to exist, we need that the time scale at which the effective interaction generated by the physics at the scale 1 r ∼ T becomes important is much larger than 1 r . In this case, in order to compute the effective interaction originating from the scale 1 r ∼ T and by the same arguments that are valid in the T = 0 case, to approximate real quarkonium by a static pair of heavy quarks is valid.
Finally, let us warn again that we made an abuse of notation naming S (singlet) and O A (octet) for the fields previously defined. More rigorously, they are interpolating fields of real pNRQCD singlets and octets (see [29]). The connection between these two kinds of fields is something that needs to be worked out. However, if the intuition from the T = 0 case is valid, we expect them to coincide at the leading order. The arbitrariness on the way we chose the space-like Wilson lines that make the fields S and O A transform in the desired way is related to this issue.

IV. USE OF THE FIERZ IDENTITY AND THE LARGE N c LIMIT
First, we discuss fig. 3 in the large N c limit while considering that g 2 N c ∼ 1 [37]. It scales like 1 in terms of N c . However, any diagram in which a gluon connects the two Wilson loops will be suppressed by a factor 1 These results suggest that, in this approximation, the survival probability of a singlet can be encoded in an effective Hamiltonian. T r(W SS (R, r; t, t 0 )ρ l ) is the Wilson loop from which the static potential is obtained in lattice QCD computations [38]. Now we look at fig. 9. We can see that in this case, both terms are of the same order but suppressed in the large N c limit, which means that the transition from an octet to a  fig. 11. The physical consequence is that, at leading order in this expansion, we can consider in this process the quark and the antiquark as uncorrelated particles.
The reader might have noted that several simplifications based on the multiplication of Wilson lines (for example that U † (t, t 0 , r)U(t, t 0 , r) = 1) were not used. This is because, although we are studying the evolution of ρ x (R, r, R, r, t), we want our discussion to illustrate what would happen in the more general r = r ′ case. Nevertheless, we discuss this which is just an illustration that probability is conserved. Analogously, looking at fig. 10, we find that This result is a combination of the conservation of probability with the fact that the transition from octets to singlets is suppressed in the large N c limit.
In summary, in the strict R = R ′ and r = r ′ limit, using the Fierz identity, we get the following results. Let us define Then we get These equations show that the octet is much more stable than the singlet. If we start from a singlet, at a later time we may find a singlet with zero probability (if it happens that S = 1). Instead, if we start with an octet, the probability to find an octet later is always equal or bigger than N 2 We can rewrite these results as transitions from D x (t 0 ) to D x (t) if we assume that ρ l ∼ρ l (now we do not write the coordinates explicitly but they are assumed to be the same as before).
Note that it is fulfilled that D s (t) + D o (t) = D s (t 0 ) + D o (t 0 ). If we assume that D s (t 0 ) and D o (t 0 ) are of the same size, then These results suggest the following physical interpretation. If D s and D o are of the same size, then the evolution of the singlet is independent of the number of octets and can be modeled with an effective Hamiltonian. Moreover, the transition of octet to singlet can be ignored. This evolution naturally leads to a state in which D s is much smaller than D o .
When D s is N 2 c times smaller than D o we reach a different regime. In this case, it is the octet the one that evolves independently of the singlet. However, the decay of the octet into singlet is an important contribution to D s (t). These qualitative results are not far from the toy model discussed in [22].

V. AN ALTERNATIVE GRAPHICAL REPRESENTATION AND THE CLASSI-CAL LIMIT
In this section, we are going to show that the Wilson loops that appear in our study have some similarities with other Wilson loops that appear in the study of the Color Glass Condensate (see [40] for a review), the dipole and the quadrupole [41]. To see this, we are going to introduce an alternative graphical representation which follows these rules: • Fields on the two branches of the Schwinger-Keldysh contour are represented in different colors. Blue for fields on the upper branch of the contour and garnet for the lower one. The classical limit is then recovered graphically by ignoring the color difference.
By classical, we mean that gauge fields have the same value in the two branches of the contour.
• We restrict ourselves to the case R = R ′ = 0 but, differently to the previous sections, we allow r and r ′ to be different. Since the three coordinates we are taking into account are not necessarily in the same plane, points in which different colors meet represent points in which the Wilson line has a cusp.
• Time always flows from left (t 0 ) to right (t). The position of the particles is given from the top to the bottom in the following order; first the quark at R + r 2 , next the quark at R ′ + r ′ 2 , next the antiquark at R ′ − r ′ 2 and finally at R − r 2 . This ordering seems a little bit weird, but it is the one that allows a direct connection with the quadrupole discussed in [41].
We can identify four building blocks from which we can obtain all the transitions that we discussed previously. Two of them correspond in the classical limit to two dipoles, while the other two correspond to quadrupoles. They are represented in fig. 12. To simplify the discussion, we have given them names which are not meant to be understood literally.
In the classical limit, a quark in the lower branch of the Schwinger-Keldysh contour is analogous to an antiquark in the upper branch, and vice versa. Then, we can either form two singlets by combining each quark with an antiquark or by combining a quark on the upper branch with a quark in the lower branch (and the same for the antiquarks). Singlet dipoles would correspond to the transition of two singlets made of a quark-antiquark pair, while quark dipoles would correspond to the transition of two singlets made of a quark in the upper branch and another in the lower branch, and an antiquark in the upper branch and another in the lower branch. The quadrupole and the inverse quadrupole represent transitions between these two ways of forming singlet states.
The conditional probabilities discussed in section II can be written in terms of these building blocks. A more compact way to write the results is to use the building blocks to describe the transitions from D x (t 0 ) to D x (t).
Using that the building blocks are defined such that they are of order 1 in the large N c counting, we can directly recover some of the results of the previous section.
An interesting observation is that the singlet to octet (octet to singlet) transition is zero if the expectation value of the quadrupole (inverse quadrupole) is equal to the expectation value of the square of two singlet dipoles. From the point of view of perturbation theory, this indicates that we have to take into account diagrams in which particles traverse the cut (in other words, connect fields on different branches of the Schwinger-Keldysh contour).
One lesson that we can take from the results in this section is the following. The operators defined in section II can be written in the classical limit in terms of gauge invariant dipoles and quadrupoles. In this way, the issues that forced us to use two different density matrices for light quarks and gluons (ρ l andρ l ) disappear in the classical limit. In this section, we check if the physical consequences that our static and large N c computation suggest are compatible with state-of-the-art perturbative derivations of the evolution equation. In particular, we will compare with [21,22] and [15,16] although these computations do not rely on the same assumptions. Before starting with the comparison, let us remark that, in our derivation, the only small parameter is 1 Nc since we take the exact limit M → ∞. In more realistic settings, there are other small parameters that might appear to spoil the naive power counting based on the large N c limit. For example, in the non-static case, we have to consider both bound and unbound states. Then the results will depend on the relation between the size of the bound states and the volume of the medium.
To start, let us first compare with the results in [15], which are valid in the limit 1 r ≫ T ≫ E where r is the typical radius and E is the binding energy. In that work, it was found that the evolution could be written as a GKSL equation In this equation, the reduced density matrix can be divided into two pieces, one for singlet and another for octet states The Hamiltonian is given by where the first term is the vacuum Hamiltonian and the parameter γ is defined as For a discussion on extractions of γ from lattice QCD, see [42,43]. In the last equation, E = ΩEΩ † with E the chromoelectric field and Ω(t, r) = U(t, −∞; r). Note that regarding the N c counting γ is of order one since g 2 ∼ O 1 Nc . There are two sets of collapse operators and where κ is the heavy quark diffusion coefficient [44,45]. Recent lattice QCD determinations can be found in [46][47][48]. Note that in this case also κ is of order one in the large N c counting. Now that we have reproduce the evolution equations of [15] we are ready to compare it with the static limit results that we obtained in section IV.
• The survival probability of the singlet is not suppressed by the large N c limit and can be encoded in an effective non-hermitic Hamiltonian h s + r 2 γ 2 −i r 2 κ 2 . This is compatible with eq. (34).
• The singlet to octet transition is also of order one in the large N c counting, as expected from our previous results. However, the octet to singlet transition goes like 1 . This can be easily seen by computing the only term in the GKSL equation which mixes singlets and octets • Finally, let us discuss the octet survival probability. Like that of the singlet, it is not suppressed by large N c effects. This agrees with our results in section IV. Regarding the observation that octet evolves like a pair of uncorrelated particles (see fig. 11 and its discussion in the text of section IV), it is challenging to check whether this is fulfilled or not in the pNRQCD results since, in this EFT, the degrees of freedom are directly singlets and octets.
We would have arrived at the same conclusions if we compared with the more general case presented in [16].
Now, let us compare with the results of [21]. We summarise here the main assumptions of that work.
• The interaction Hamiltonian that describes the coupling between the heavy quarks and the medium is H I = −g d 3 rA A 0 (r)n A (r), where n A is the color current associated with the heavy particles.
• The evolution of the density matrix is computed in the one gluon exchange approximation.
• The response of the plasma to the perturbation caused by the heavy quarks is fast compared to the characteristic time scales of the heavy quark motion Under these assumptions, the following evolution is found In these equations, and The meaning of V 12 , V 1 ′ 2 ′ , W (0), W c , W − , W a and W b can be found in [21]. These functions scale like 1 Nc in the large N c counting because they are proportional to g 2 . The following remarks are important: • The origin of V 12 , V 1 ′ 2 ′ , W (0) and W c are gluons that do not connect different branches of the Schwinger-Keldysh contour.
• W a and W (0) represent the effect of gluons that do not connect the quark with the antiquark.
Using these remarks, we comment on the compatibility of the results of [21] with the static limit large N c conclusions of this manuscript.
• The singlet to singlet transition is not suppressed by large N c effects and gets no contributions from terms connecting the two branches of the Schwinger-Keldysh contour.
• The singlet to octet transition is not suppressed while the octet to singlet is of order • The octet to octet transition is not suppressed by large N c effects. However, we can see that the only terms that contribute to L oo at leading order in N c are those who do not correlate the heavy quark and the antiquark.
With this, we finished the comparison with [21], we note that we would have arrived at the same conclusions if we would have compared with [22].

VII. CONCLUSIONS
In this paper, we have discussed how to define the conditional probabilities to evolve from a static pair in an octet or singlet state at a time t 0 into an octet or singlet state at time t. The results are written in terms of Wilson loops with the right transformation properties and they might be useful as a starting point for future investigations using non-perturbative techniques, as for example, lattice QCD or AdS/CFT correspondence. We also argued that the study of the color transitions in the static limit is useful for the study of real quarkonium if we are in a regime in which the correlation in time between thermal fields is shorter than the inverse of the binding energy. We leave the study of a more rigorous connection to future developments in the EFT framework.
We have also discussed the large N c limit. From which we obtain the following insights: • The survival probability of a singlet can be modeled with an effective Hamiltonian since diagrams that connect different branches of the Schwinger-Keldysh contour are suppressed.
• The effect of octet to singlet transitions in the evolution of the octet density is suppressed. The evolution of the octet can be approximated by the uncorrelated evolution of a quark and an antiquark. This is an approximation that has been used, for example, in [19,22] and which we see that is justified in the large N c limit.
• If we consider that the densities of singlets and octets are of the same size, the octet to singlet transition can be ignored. Then, the problem of solving the evolution of the singlet simplifies to evolution with an effective Hamiltonian. This justifies the computation of R AA solving a Schrödinger equation with an imaginary potential (see [49] for a recent example) in the large N c limit as long as the population of octet is not much bigger than that of singlets. Octets evolve as an uncorrelated pair of heavy particles that is sourced by the decay of singlets into octets. The situation changes when the population of octets is much bigger than that of singlets. Then singlet to octet transition can be ignored, but octet to singlet has to be taken into account to compute the number of singlets accurately.
One has to be careful when extrapolating these results from the static limit to the case of real quarkonium. In that case, one has to consider that the spectrum of the singlet contains bound states (and is discrete) while that of the octet does not (and is continuous). This could lead to additional suppressions that go like the inverse of the medium volume [22]. If medium corrections are smaller than the binding energy, these effects suppress further octet to singlet transitions. However, this does not need to be so in other cases.
We have also studied what happens when we assume that gauge fields behave classically.
Then we can write the transitions in terms of building blocks which are analogous to the dipoles and quadrupoles discussed in [41] with the difference that, in our case, the Wilson lines are along time-like and space-like directions while in their case, they have light-like and space-like directions. The classical approximation could be useful to construct fully gauge invariant operators that capture part of the physics of the case in which we have an octet at the initial time.
In this study, we have assumed a plasma formed by light quarks and gluons and only a pair of heavy particles. It would be interesting to study what happens when the number of heavy particles is increased. Notably, this is important to confirm if the large N c limit justifies the molecular chaos approximation, as our result suggests. Another possible future direction is to rigorously prove the connection between the static limit results and real quarkonium using non-relativistic EFTs.

Appendix A: Birdtrack notation
In this appendix, we introduce a diagrammatic representation suitable for the computations performed in this manuscript. It is essentially the birdtrack notation of [50] (see [51] for a previous use in the field of heavy-ion physics) with the only difference that our lines represent Wilson lines in the fundamental representation instead of Kronecker deltas.
• Quark fields have a color subscript while antiquark fields have a color superscript. In FIG. 14. Fierz identity in our graphical notation.
this notation, a line represents a Wilson line with an arrow going from its superscript to the subscript.
• The upper part of the diagram represent the position R + r 2 while the lower part represents R − r 2 . Whenever a diagram involves fields acting at different times, this will be indicated below the diagram. Therefore, vertical Wilson lines connect points at the same time and different positions while horizontal lines do the opposite.
• A Wilson line connecting fields with the same time and coordinate are equivalent to a Kronecker delta.
• A Kronecker delta relating indices in the adjoint representation is represented with dashed lines.
In fig. 13, we show the group generator T A and the definition of the singlet and octet field with this notation. In fig. 14 the Fierz identity is represented with this graphical notation.
Note that, for example, using the definitions of the singlet and octet field from fig. 13 and the Fierz identity, it is straightforward to derive eq. (9).