Analytic continuation of 3-point functions of the conformal field theory

It is shown that the general 3-point function $<\Phi_{a} \Phi_{b} \Phi_{c}>$, with continuous values of charges $a, b, c$ of a statistical model operators, and the 3-point function of the Liouville model, could all be obtained by successive analytical continuations starting from the 3-point function of the minimal model.


Introduction.
Recent interest in the 3-point functions < Φ c Φ b Φ a > with continuous values of charges a, b, c, which do not satisfy the neutrality conditions of the Coulomb gas minimal models, is, principally, due to recently found realisations of these correlation functions in the context of statistical models, on the lattice : Potts model 3 spin correlation function [1], loop models [2].
En the other side, the interest in the Liouville model correlation function was always present, since 1981 [3].
The Liouville 3-point function was defined in [4,5] . The statistical model general 3-point function (of imaginary Liouville or Coulomb gas) was defined in [6]. See also the related work in [7].
In the present paper we rederive these results somewhat differently, by a sequence of analytical continuations, starting with the minimal model 3-point function [8][9][10].
The present study does not provide new results but gives some new methods and provides some unification, hopefully.
The objective of this Section is to show that one gets the general 3-point function Going back to our correlation functions, we shall continue the logarithm of the expression in (2.1), by using the integral representation of the logarithm of Γ-functions.
Let us define and similarly for g With these notations, the function in (2.1), which we shall note as C p n,m , takes the form: k (ρ) (2.30) and the function in (2.12), which we shall note as C (p ′ ,p) (n ′ ,n)(m ′ ,m) , takes the form: It is shown in the Appendix A that log g k (ρ), analytically continued in k, k → k − ρ ′ l, eq(2.25), is given by: For the analytic continuation of log g (α) k (ρ) we obtain, Appendix A: and similar expressions for log g (β) k−ρ ′ l (ρ) and log g Putting them together, by the eq.(2.30), we obtain To get the coefficient of log ρ, in its form above, we have used the neutrality condition on the parameters, eq.(2.22).
The factor ρ 2l(ρ ′ −1) is an another normalisation factor, being an exponent linear in l, produced this time in the process of analytic continuation . It could also be "symmetrized", so that l and k would appear en equal footing, by using the neutrality condition (2.21) en l and k. We shall do it later, because one extra factor of this type is still coming, will be obtained in the next Section.
For the time being we are dealing with correlation functions of Coulomb gas vertex operators, which is simpler.
They have their own nontrivial normalisation which will be specified later. 3 Analytic continuation to the general, unconstrained values of charges of the vertex operators in the 3point function.
To start, we shall reinterpret the results of the previous Section in the opposite direction. We shall consider that it has been shown that the general minimal model 3-point And we shall continue further, the function (C p n,m (ρ)) continued to the values of charges, of the operators in it, continuous values, a, b, c: We remind that in the process of analytic continuation in the Section 2, which results in the equality (3.1), k has been replaced with k − ρ ′ l, but also the charges α 1,m , α 1,n , α + 1,p has been replaced by α m ′ ,m , α n ′ ,n , α + p ′ ,p , comp. the comment preceding the eq.(2.25).
It is much easier to continue this way, in two steps, the general minimal model corre- instead of performing the continuation directly with a, b, c, being unconstrained.
To perform the second step in (3.3), we shall need the detailed expressions for k−ρ ′ l (ρ), obtained in the Appendix A. They are as follows: and similar expressions for log g (β) k−ρl (ρ) and log g (γ) k−p ′ l (p).
For (C p n,m (p)) continued we get the following expression: First we shall simplify the "polynomial" part in (3.7), the first part of it, the coefficient of e −t .
By using the neutrality conditions (2.21), on α, β, γ, we get (we remind that ρρ ′ = 1): To continue eventually to the general unconstrained values of charges, of the operators, we have to express everything in terms of these charges in particular the combination of the numbers of screenings k − ρ ′ l in (3.8). From (2.22): By eq.(2.16) where we have replaced α m ′ ,m , α n ′ ,n , α + p ′ ,p by a, b, c. For (3.9) we obtain For (3.8) we get: We remind that For the polynomical part of (3.7) we obtain: We shall simplify next the "exponential" part in (3.7), its second part. It has to be observed that, separately, the integrals of the polynomial part and of the exponential part in (3.7), they are divergent at t → 0. To manipulate them separately we should introduce the limit ǫ > 0, in the integrals, instead of 0, and assume that, finally, we shall take the limit ǫ → 0 when everything is put together, as it has been done already in the Appendix A.
For the first part we have done no transformations for the integration variable t, so we could keep it as it is in (3.14), although, more properly, we could have assumed that the lower limit of integration in (3.14) is ǫ.
But for the second part of (3.7) we do intend to transform the integration variable t, so that the explicit introduction of ǫ, for the lower limit of the integration, will be necessary at some point.
In fact, we shall start simplifying (or reorganising) the second part of (3.7) by transforming the variable t: We shall do all the transformations by ignoring, at first, the divergence at t = 0. But afterward we shall take specific care of the extra terms, the "anomaly" terms, being produced by this divergent limit.
With the change of the variable in (3.15), the second, exponental part of (3.7) takes form: We shall simplify next the notations and we shall express everything (k − ρ ′ l, α, β, γ) in terms of the charges a, b, c.
The Coulomb gas parameter √ ρ = α + , corresponds (is proportional) to the parameter b of the Liouville model, or to the parameter β in the imaginary Liouville [5,6]. As b and β are already in use, and the notations √ ρ = α + , √ ρ ′ = −α − would be slightly heavy, we shall use a single parameter, as in the Liouville model, but we shall note it h, so that Next: and, according to (3.9), or (2.22), We shall suppress also the tilde oft,t → t. Then the expression in (3.16) takes the form: Multiplying the numerator and the denominator in (3.21) by exp{ t 2h + ht 2 } we get: Next we obtain, by regrouping the terms: It can be presented as: For t → 0, the above integral takes the asymptotic form: This divergence, at t → 0, is compensated by the polynomial part (3.14), of the integral(3.7). But, saying it differently, the equality of the expressions under integrals in (3.14) and (3.25) implies that the polynomial part (3.14) could be distributed as in (3.25): so that the full integral (3.7), which is the sum of (3.14)=(3.26) and (3.24), takes the form: We find that everything is expressed in terms of the function Υ(x, h) [5,6] log so that log(C p n,m ) continued , which is the integral (3.7), takes the form: This is the function in [6], though not completely so.
First, the normalisation of operators used in [6] is different.
Second, we have putted the index "M" for Υ M , for "matter" (statistical model) which is slightly different from Υ(x, h) for Liouville, gravity, which has been introduced in [5].
The difference is in: In [6], the function which note Υ M (x, h) has been used in the form Υ(x + 1 h , h).
And third, there is one additional factor missing in (3.30), the way we derived it. We haven't calculated yet the anomaly term, which is produced because of our manipulations with the second, exponential part integral, which is divergent at t → 0 when taken separately. So far, in our derivation of (3.30), we have ignored this point. We shall take care of it now.
The second, exponential part in (3.7) should have been taken with the lower integration limit ǫ, instead of 0, before the change of the variable t in (3.15). We reproduce this integral somewhat symbolically: After the change of the variable in (3.15) we obtain The integral can be decomposed as follows: The first integral in above goes to join the first, polynomial part of (3.7) and gives finally the function in (3.30). But the second integral in (3.34) gives an additional term, which has been missed in our derivation of (3.30).
To calculate the second integral in (3.34) we could use all the transformations, for the expression under the integral, which has been done above. It could be taken in the form in (3.24), but with the limits of integration (ǫ, ǫ/ √ ρ ′ ) instead of (0, ∞). As t stays small, in the limits (ǫ, ǫ/ √ ρ ′ ), we can replace the expression under the integral by its limiting form, for t → 0, which has already been obtained in (3.25). In this way we get, for the second integral in (3.34), the following result: This is our anomaly. It has to be added to (3.29). With it, the formula in (3.30) takes the form: We remind that our objective was to continue the general minimal model 3-point with a, b, c unconstraint. By (3.1), the result of the first step of continuation, and (3.36), the result of the second step of continuation, we find so far: We have suppressed the factor Υ M (2α 0 ) in the denominator of (3.36), because Υ M (2α 0 ) = 1, as can be checked directly using the integral definition of the function Υ M (x) in (3.28).
Some specific values of Υ M (x) are listed in the Appendix B. With respect to the notations, we suppress sometimes the dependence of Υ M on h, which is implicite, by writing , as in (3.36) and in (3.39).
In eq.(3.39), the charges a, b, c in the r.h.s. are still having the discretized values of the degenerate charges: The remaining obstacle to fully continue to the continuous values of a, b, c, is the factor in (3.39). In particular, l is still given by (2.19).
The total ρ-factor in (3.39) could be symmetrized. It is easy to check, by using the Still, in the ρ-factor in (3.39), or in (3.42), there appear the numbers of screenings, k and l, in the form which could not be expressed fully by the charges a, b, c.
But it is clear, by the form of the ρ-factor in (3.42), that it could be removed by the renormalisation of the constants µ + and µ − in (2.9).
Specifically, if we give the following values for µ + , µ − : then the normalisation factor (2.20) will disappear, from the result for the integral (B.10) in [8], and the factor will appear, in front of the expression for < V + p ′ ,p (∞)V n ′ ,n (1)V m ′ ,m (0) > in (2.12). Then we perform the transformations of the Section 2 and of the present Section, to arrive to (3.39), but because of the extra factor (3.44) always present, unchanged during our transformations, the extra ρ factor in (3.39), in its form in (3.42), will be cancelled. We shall get, with the choice (3.43) for the values of the Coulomb gas constants µ + , µ − , the formula (3.39), without the ρ-factor. At this point we could finally continue a, b, c to the continuous values and we obtain, finally, the formula for 3-point function in the form with a, b, c taking general, continuous values.
We summarise that the formula (3.45) has all been obtained by the analytic continuation from general 3-point function for degenerate operators of the minimal model.
Though a specific normalisation of the Coulomb gas screening operators, or of the constants µ + , µ − , was required.
The formula (3.45) is that for the vertex operators with their nontrivial normalisation, N a for V a (z,z), etc., which will be specified in the next Section.
In the next Section we shall get a slightly different formula, compared to (3.45), for In the case of normalised operators We shall do it by using the original expression for the correlation functions, in terms of products of Γ functions, in its symmetrized form given below, and also by using the new expression, in terms of products of Υ functions, which should give the same values, when a, b, c are degenerate, eq.(4.1).
The fact that we have kept, in Sections 2 and 3, the charge α p ′ ,p always in its conjugate is not actually important for our derivations. We can relax now to the values of c in (4.1). The case of c = α + p ′ ,p will correspond, with the definition of c in (4.1), to α −p ′ ,−p , instead of α p ′ ,p . Which means that we shall allow for the indices to take also the negative values. We have kept, in the derivations of Section 2 and 3, one of the operators, in its conjugate form, V + p ′ ,p (z,z), in part for historical reasons, to make the transition from the original formulas of [9] smoother, not to become excessively general from the start, which is not needed.
We shall go back, in this Section, to the normalisation of µ + , µ − : which has been taken at the start, in the Section 2, with which the correlation function of vertex operators is of the form: when a, b, c take the degenerate values (4.1). We remind that and, by the neutrality condition for a, b, c in (4.1), The expression in (4.4) could additionally be symmetrized by transforming Γ's with −k, as follows: This gives: Sometimes the formula (4.8) is more convenient to make varions check.
We remind also that this same correlation function, expressed in terms of the function It is shown in the Appendix B, by using the formulas in (4.8) and (4.9), that we have the following particular results: 1.
I is the identity operator, I = 1. γ(ρ) = Γ(ρ)/Γ(1 − ρ). Z could be considered as the partition function of the Coulomb gas, because the function < III >=< 1 > is given by the Coulomb gas functional integral [7,8], without normalisation. So we note it as Z. The result (4.10) is obtained in the Appendix B, both with (4.8), by the analytic and also with (4.9).
Again, because of the relation (4.23), the norm of the conjugate operator is given by: The consistency with (4.18) -(4.21) could readily be verified.

Now, in general,
where (N a ) 2 is given by: The "naive" norms of vertex operators V a , V + a have been defined in [11], Section 9.1. They differ from the actual norms in (4.34) by the absence of the partition function Z, of the Coulomb gas, in the norm of V + a , N(V + a ) naive = 1/N a .
By the formula (4.9) we obtain: Here we assume that a = α n ′ ,n , and then, by (4.6) l = n ′ − 1, k = n − 1. Since , Appendix B, we obtain: Consistency could readily be checked (Appendix B) that: -consistent with (4.34). We could conclude that the formula (4.37), for the norm squared of the Coulomb gas vertex operator V a , is perfectly general. Though still limited to the degenerate values of a, a = α n ′ ,n , as we are still working with the formula (4.9) which contains the ρ factors, in which the numbers l, k are still present.
We shall now define the correlation function, properly normalised by 1 Z , and which is defined for the normalised operators (3.47). We find: We have assumed that a = α m ′ ,m , b = α n ′ ,n , c = α p ′ ,p . We obtain: We have used the formulas (4.6) for l and k. We observe that all the ρ-factors get cancelled and we find, finally, the normalised 3-point function for the normalised operators in the form: In this formula everything is expressed, analytically, in terms of charges a, b, c, so that we can continue the formula to the general, continuons values of charges.
The expression (4.41) is the formula (5.1) of [6], obtained there differently. Our point is that we have derived everything, by a series of analytic continuations, from the general 3-point function of the minimal model [9].

Analytic continuation of the function <<
towards the 3-point function of Liouville.

Analytic continuation of the function Υ M (x, h).
For convenience, we reproduce here the integral definition of log Υ M (x, h), eq.(3.28): We shall put wherex,h are considered to be real, positives, for the moment. (5.1.1) takes the form: As θ → π 2 , the poles approach the real axes, the integration line of t. For θ = π 2 the poles put themselves on the real axes and the integration line, over t, gets deformed accordingly, to avoid poles, Fig.2.
Next, the contour of integration could be deformed as shown in Fig.3.
Accordingly, the integration breaks into two part: the sum of integrals around poles, the first part, and the integral along the line l, Fig.3, the second part. We shall calculate them successively.  C n is a small closed contour around t n = 2πh · n, the pole due to the factor 1/ sinh( it 2h ). We get: The series in(5.1.7) is almost that for the log of ratio of two ϑ-functions: This formula is obtained in the Appendix C. Using (5.1.9), the result (5.1.7), for P oles, could be given as: The second part of (5.1.3) (with θ = π 2 ) is given by the integral along the line l, Fig.3: We shall break it into pieces as: In the above, we have introduced the lower limit of integration ǫ, with the limit ǫ → 0, because the integrals in (5.1.12), taking separately, are divergent at t → 0. Then we kept the integration contour as it is, t : ǫ → ∞, in the first integral. For this part, there are no poles and the integration line l, Fig.3, could be put back on the real axes. While for the integral of the second term in (5.1.11), the part which contains poles in the vicinity of l, Fig.3, we have turned the contour of integration, the line l, towards the imaginary axes, as is shown in Fig.4.
This contour then break in two, the small contour C ǫ around the origin, and the integral along the imaginary axes, for which we have changed the variable t = it, Fig.4.
This last integral is the third one in (5.1.12). We shall drop the tildes, oft, in third term of (5.1.12), in the following. In the second integral, over C ǫ in (5. where β 0 =h 2 + 1 2h (5.1.14) The first and the third integrals could be put together and the limit ǫ → 0 could be lifted. Altogether, we get Here Υ L (x,h) is the Υ function for the Liouville model, which was introduced in [5]: β 0 is given by(5.1.14). Now, putting together the two parts of (5.1.3) (with θ = π 2 ), the P oles, eq.(5.1.10), and I l , eq.(5.1.15), we obtain: We remind that The relation (5.1.18) is the formula (6.2) of [6], derived there by different methods.
One comment is in order, with respect to our derivation of the formula (5.1.18).
The ϑ functions in (5.1.18) are not defined for q = e iπh 2 withh real. We need to have In summary,h should be in the sector S, Fig.5, of its complex plane, for our analytic continuation had to make sense.
This implies that, at the start of our analytic continuation, h =h (eq.(5.1.2) for θ = 0) had to have a "small" imaginary part, positive. This implies in turn that our figures should slightly be deformed, as is indicated in Fig.6, Fig.7. Otherwise, the derivation stays as has been presented above, though in a somewhat (artificially) simplified context.
By the way, the fact that log Υ M (−ix, −ih), in (5.1.18), is well defined, by its integral, as was discussed above, is seen by the first figure in Fig.6: the poles stay away from the initial integration line, the real axes of t.

Analytic continuation of the function <<
We have suppressed the "tildes" ofx andh, which served us in the subsection 5.1 for the presentation purposes of the analytic continuation.    Replacing every Υ M in (4.41) by its analytically continued form, eq.(5.2.1), we obtain: We have suppressed the argument q in the functions ϑ 1 (ϑ 1 (πh2a, q) → ϑ 1 (πh2a), etc.) to compactify the expression a little bit.
Finally, the whole expression, in the r.h.s. of (5.2.3), preceding the part with Υ functions, could be reduced to: We obtain: Now we shall rewrite the equation (5.2.7) as follows: The idea to organise, in this way, the analytically continued expression for be continuous [13,14], to pass from the sum, over the intermediate states, to the integral, in the decomposition of the 4-point functions, could be organised by representing the initial sum as a sum over the residues, by adding an appropriate function which produce poles. And then the sum of the residues could be expressed by the appropriate integral.
The role of ϑ 1 functions in the denominator of (5.2.8), in the l.h.s., might be that of providing the necessary poles.
Saying it differently, the appearance of these ϑ 1 functions, the ones involving interactions (like ϑ 1 (πh(a+b+c−2β 0 ), q) etc.), in the analytic continuation from minimal models to Liouville, might be interpreted as a sign, or a proof, that in fact the intermediate states spectrum of the Liouville is going to be continuous.
We are, actually, starting talking of the possibility to obtain the 4 point functions of Liouville by the analytic continuation of the well defined 4 point functions of minimal models. If realised, the associativity, in particular, will not need to be proved, will be automatic.
The task should be more complicated than that of continuing the 3-point functions.
The possibility is to be attempted. For the moment we haven't yet much progressed in that direction.
The appearance of a product of "local" ϑ 1 functions, under the square root, in the l.h.s of (5. 6 Discussions. Historically, the first step of the analytical continuation, from the (1, n) operators 3point functions to the general (n ′ , n) minimal model operators 3-point functions, the continuation presented in the Section 2, it was realised long ago [15], as a by product of the curiosity, during my work on 3-point amplitudes of minimal models coupled to gravity [16]. At that time I have also defined the "naive" norms of vertex operators, V a (z,z) and V + a (z,z), the analyses described in [11], Section 9.1. This last curiosity was better justified, because in 3-point amplitudes of minimal models coupled to gravity, after cancellations, remain only products of norms of the operators. For instance, the question was raised in [6] with respect to apparent non-decoupling of some states from the outside of the minimal model (finite) Kac table of primary operators (the actual minimal model, not the generalised one). The problem that the 3point functions (or operator algebra constants) with particular operators from outside the Kac table, do not vanish, the way they are defined analytically by the direct calculation of the 3-point functions. This breaks the "fusion rules" of minimal models. So that some decouplings have to be added by hand.
We have seen the answer to this question in our work of [8 -10], where the operator algebra constants (3-point functions) have been derived from the structure of the 4-point functions of minimal models, which were the principal objects of [8 -10]. We have seen that in the sum over the intermediate states, in the case of actual minimal models, it happens that for a particular the primary operator, which is placed outside of the Kac table, its contribution to the sum over the intermediate states gets cancelled by the contribution of a descendent operator of the nearby channel of another primary operator, positioned inside the table. And in this way the "fusion rules" get restored, analytically, not by hand. One example of such "delicate decoupling", which could be seen only on the level of 4-point functions, is described in Section 9.2 of [11].
Above mentioned is just an example. But in general, we wish to stress again that, most likely, the proper, definite definition of 3-point functions should be provided by the 4-point ones. In particular, for the Liouville model. I am grateful to Marco Picco and Raoul Santachiara for numerous useful discussions.
A Formulas needed for the analytic continuation of C p n,m (ρ) in the Section 2.
In Section 2 we have defined the factors g k (ρ), G lk (ρ), g (n ′ ,n)(m ′ ,m) were given by the products in (2.30) and (2.31). To prove the formula (2.35), the first step of the analytic continuation, we have used the formulas (2.32), (2.33). We shall prove these relations now.
Taking log of (2.26) we get: Next we use the integral representation of Γ(x): One finds: In a similar way, by taking log of (2.27) we find: Next, replacing k by k − ρ ′ l in (A.5), we get: Our purpose is to compare log G lk(ρ) , (A.6), and log g k−ρ ′ l (ρ), the expression above.
For the polynomial part, the coefficient of e −t in (A.7), one finds: This agrees with the polynomial part in (A.6).
To give the exponential parts of (A.6) and (A.7) a similar appearance, we shall change the variable of integration t in the first integral of the exponential part in (A.6) as t → ρt, i.e. t = ρt, and shall we drop the tilde afterwards.
We ignore for the moment the fact that the integral, of the exponential part taken alone, is divergent at t → 0. We shall take care of the extra terms, the "anomaly" terms, which are due to this divergence, a little bit later.
After the change t → ρt in the first integral of the exponential part in (A.6), the whole exponential part of (A.6) takes the form: [e −ρt (1 − e −kρt+lt ) + e −t (1 − e kρt−lt )] + anomaly (A.9) Apart from the "anomaly", the expression above agrees with the exponential part of (A.7). The agreement for the polynomial parts of (A.6) and (A.7) has been established earlier, (A.8). So that we find the relation: log G lk (ρ) = log g k−ρ ′ l (ρ) + anomaly (A.10) Anomaly.
With respect to the divergence at t → 0, the integral in (A.6) for which we have changed the variable of integration, t → ρt, this integral should have been taken, more properly, in the form: After the change t → ρt, (t = ρt, and we drop the tilde afterwards) we obtain: The first integral in the expression above goes to joint the rest, the polynomial part and the second integral of the exponential part in (A.6), which also should have been treated properly with the limit ǫ → 0 for the lower limit of integration.
The second relation, eq.(2.33), is derived in the same way, by starting with the log's of products for g We shall show it with the relation (B.1). We get: } (A.14) As α 0 = h 2 − 1 2h , one gets: We consider -to compensate the extra terms in the polynomial part of (A.15). We have used the integral representation in (A.3).
For the sum of (A.15) and (A.16) we obtain: The first term in the integral above is already that of We shall transform, separately, the second and the third terms in (A.17).
By transforming t → t h in the integral of the second term in (A.17), we obtain: and we continue its transformation as follows: Now we shall transform, a little bit, the third term in the integral of (A.17): For the sum of (A.20) and (A.21), which is the sum of the second and the third terms in (A.17), we obtain: Together with he first term in (A.17) this gives Here "anomaly" stands for the term which we have disregarded so far, by ignoring the divergence, at t → 0, of the second integral in(A.17), while transforming t → t h .
Properly, we should have defined it as With t =t h , we obtain: We drop the tildes and we rewrite (A.25) as follows: The first integral goes to join our derivation of the relation (A.23), with the exception of the anomaly in it, while the second integral in (A.26) gives the anomaly.
We get: Finally, the relation (A.23) takes the form: This is the relation (B.1).

B
The function Υ M (x, h) and the list of some of its values. The list of results for particular 3-point functions which are used in the Section 4.
We shall remind here certain properties of the function Υ M (x) [5,6]. We shall list some particular values of this function. And we shall also derive several expressions for the 3-point functions which are used in the Section 4.
The function Υ M (x) ≡ Υ M (x, h), defined in (3.28), satisfies the following "quasiperiodicity" relations, with respect to translations by h and by 1/h: We have listed these properties in various forms, which are useful in actual calculations, though, evidently, (B.3) -(B.6) follow directly from (B.1), (B.2). The derivation of (B.1) is given in the Appendix A.
The obvious property, by (3.28), is that Next, one finds directly, from the integral form of log Υ M (x) in (3.28), that 1.
The integral in (3.28), which defines Υ M (x, h), is convergent for − 1 h < x < h. Outside, the function is defined by the analytic continuation: by the translations (B.1) -(B.6). For x → − 1 h , from above, and for x → h, from below, the integral in (3.28) is logarithmically divergent at t → ∞, which results, when evaluated, in the asymptotics in (B.8), (B.9).
3. In general, the zeros of Υ M (x), which we denote as x In fact, by (B.5) and (B.9), for x → h, one finds: 6. .
We shall derive now several results for the correlation functions, the results listed in the Section 4.
By the formula (4.8), if we put a = 0, b = 0, c = 0 directly, we shall get a problem, the expression will not be defined. But the integral, which gives (4.8), is defined, in fact, with a single condition, with l, k being integers. Separately, a, b, c do not have to be degenerate, to make the expression (4.8) valid. This allows to define the function < III >, with (4.8), as a limit a → 0, b → 0, c → 0, while keeping a + b + c = 0 to make the condition (B.22) satisfied, with l = k = 1.
We shall calculate it with (4.9), which is simpler.